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The measurement of the rare kaon decay k-plus to pi-plus, neutrino and anti-neutrino Ives, Joss 2008

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THE MEASUREMENT OF THE RARE DECAY K-PLUS TO PI-PLUS, NEUTRINO, AND ANTI-NEUTRINO by JOSS IVES B.Sc., The University of Saskatchewan, 2000 M.Sc., The University of Saskatchewan, 2003  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  December 2008 c Joss Ives, 2008  Abstract Brookhaven National Laboratory experiment E949 was designed to search for the rare K meson decay K + → π + ν ν¯, a decay sensitive to physics beyond the Standard Model. While previous data analyses dealt with the high π + momentum region accessible for this reaction, this thesis concentrates on the lower range between 140 and 199 MeV/c. Analysis of this low π + momentum region was performed to search for additional evidence of the process K + → π + ν ν¯. A blind analysis technique was used to avoid bias when developing the selection criteria used to suppress the competing background processes. The blind analysis technique was based on identifying background sources a priori and only examining the signal region once all selection criteria and background estimates had been finalized. The background estimates were performed using a technique known as a “bifurcation method”, which relied on using two uncorrelated selection criteria to suppress each background source. The analysis of an exposure of 1.71×1012 K + decays resulted in an ob+0.320 servation of three events with an estimated background of 0.927 ± 0.168(stat.) −0.237 (sys.)  events and a single event sensitivity of (4.28 ± 0.43) × 10−10 . Using a likelihood method, the three candidate events observed here were combined with the previous E787 and E949 −10 results, yielding a branching ratio of B(K + → π + ν ν¯) = 1.73+1.15 at the 68% −1.05 × 10  confidence level. This branching ratio is consistent with the prediction of the Standard Model, (0.85 ± 0.07) × 10−10 .  ii  Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 1 Introduction . . . . . . . . . . . . . . . 1.1 Theory and Motivation . . . . . . . . 1.2 CP -Violation and Weak Interactions 1.3 K + → π + ν ν¯ in the Standard Model . 1.4 Physics Beyond the Standard Model 1.5 History of K + → π + ν ν¯ Experiments 1.6 My Role in the Analysis . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . 1 . 2 . 2 . 8 . 12 . 14 . 15  2 Experimental Method . 2.1 Kaon Beam . . . . . . 2.2 Beam Instrumentation 2.3 Target . . . . . . . . . 2.4 Drift Chamber . . . . 2.5 Range-Stack . . . . . . 2.6 Photon Veto . . . . . . 2.7 Trigger . . . . . . . . . 2.8 Summary of 2002 Data  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collection  3 Analysis Overview . . . . . . . . 3.1 Overview of Backgrounds . . . . 3.2 Analysis Strategy and Methods 3.3 Event Reconstruction . . . . . . 3.4 Selection Criteria . . . . . . . . 3.5 Monte Carlo Simulation . . . .  . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  17 18 20 29 31 32 35 42 48  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  50 52 54 60 69 76  iii  4 Backgrounds . . . . . . . . . . . . . . . 4.1 Kπ2 Target-Scatter Background . . . 4.2 Kπ2 Range-Stack-Scatter Background 4.3 Kπ2γ Background . . . . . . . . . . . 4.4 Muon Background . . . . . . . . . . 4.5 Ke4 Background . . . . . . . . . . . . 4.6 Single-Beam Background . . . . . . . 4.7 Double-Beam Background . . . . . . 4.8 Charge Exchange Background . . . . 4.9 Other Backgrounds . . . . . . . . . . 4.10 Background Summary . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  78 79 94 98 103 107 114 119 127 134 136  5 Validity Checks . . . . . . . . . . . . . . . . 5.1 Outside-the-Box Studies . . . . . . . . . . 5.2 Single-Cut and Double-Cut Failure Studies 5.3 Background Cross-Contamination Studies  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  138 138 144 149  6 Signal Acceptance and Sensitivity . . . . . . . . . . . . 6.1 Acceptance Factors from Kµ2 Monitor Trigger Events . 6.2 Acceptance Factors from πscatter Monitor Trigger Events 6.3 Acceptance Factors from Kπ2 Monitor Trigger Events . 6.4 Acceptance Factors Using Monte Carlo . . . . . . . . . 6.5 Acceptance Summary . . . . . . . . . . . . . . . . . . . 6.6 Correction to T•2 Trigger Acceptance . . . . . . . . . . 6.7 K + Stopping Fraction . . . . . . . . . . . . . . . . . . 6.8 Measurement of the Kπ2 Branching Ratio . . . . . . . . 6.9 Single-Event Sensitivity . . . . . . . . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  165 166 170 180 182 183 183 186 186 189  7 Results . . . . . . . . . . . . . . . . . . . . . . 7.1 Cell Definitions . . . . . . . . . . . . . . . . 7.2 Examination of the Signal Region . . . . . . 7.3 Calculation of K + → π + ν ν¯ Branching Ratio  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  190 190 194 211  8 Conclusion  . . . . . . . . . . .  . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 A E787 to E949 Upgrades . . . A.1 Beam Upgrades . . . . . . . A.2 Detector Upgrades . . . . . A.3 Trigger and DAQ Upgrades  . . . .  . . . .  . . . .  . . . .  iv  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  227 227 227 228  B Data Acquisition, Storage, and B.1 Data Acquisition . . . . . . . B.2 PASS0 . . . . . . . . . . . . . B.3 PASS1 and PASS2 Processing B.4 PASS3 Analysis . . . . . . . .  Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . .  C Target Pulse Data Analysis . . . . . C.1 Creation of Standardized Pulses . . C.2 Overview of the Fitter . . . . . . . C.3 Optimization of the Error Input . . C.4 Hold and Release Double-Pulse Fit  . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  229 229 231 231 231  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  232 233 234 235 238  D Detailed List of Cuts . . . . . . . . . . D.1 PASS1 Cuts . . . . . . . . . . . . . . D.2 PASS2 Cuts . . . . . . . . . . . . . . D.3 Kinematic Cuts . . . . . . . . . . . . D.4 Phase Space Cuts . . . . . . . . . . . D.5 Beam Cuts . . . . . . . . . . . . . . D.6 Delayed Coincidence Cuts . . . . . . D.7 Target Quality Cuts . . . . . . . . . D.8 π + → µ+ → e+ Decay-Sequence Cuts D.9 Photon Veto Cut . . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . .  241 241 242 244 246 247 251 252 258 260  E Target Pulse Cuts . . . . . . . . . . . . . . E.1 Description of CCDPUL . . . . . . . . . E.2 Description of CCDBADFIT . . . . . . . E.3 Description of CCDBADTIM . . . . . . E.4 CCDPUL Optimization . . . . . . . . . . E.5 CCDBADTIM Parameters . . . . . . . . E.6 Updating the De-multiplexing Algorithm  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  266 266 270 271 274 277 280  Ratio . . . . . . . . . . . . . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  284 284 285 286  F Acceptance Factors for the K + → π + π 0 Branching F.1 Acceptance Factors Using Monte Carlo . . . . . . . . F.2 Acceptance Factors from Kπ2 Monitor Trigger Events F.3 Calculation of Total Acceptance . . . . . . . . . . . .  v  List of Tables 4.1 Definition of the classes of events used to measure the PV rejection in the kinematic signal region for Kπ2 target-scatter backgrounds. . . . . . . . . 4.2 Kπ2 target-scatter loose photon veto rejections . . . . . . . . . . . . . . . 4.3 Rejection of the tight photon veto for class 12 with various combinations of loose and tight versions of the setup cuts . . . . . . . . . . . . . . . . 4.4 Summary of the loose Kπ2 target-scatter background evaluation . . . . . 4.5 Summary of the tight Kπ2 target-scatter background evaluation . . . . . 4.6 Summary of the Kπ2 Range-Stack-scatter background evaluation . . . . . 4.7 Summary of values used to determine A(Kπ2 ) and A(Kπ2γ ) . . . . . . . . 4.8 Summary of Kπ2γ background evaluation . . . . . . . . . . . . . . . . . . 4.9 Summary of the muon background evaluation . . . . . . . . . . . . . . . 4.10 Rejection of RTGPV·OPSVETO for loose rejection branch . . . . . . . . . . . 4.11 Rejection of RTGPV·OPSVETO for tight rejection branch . . . . . . . . . . . 4.12 Ke4 background summary . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Summary of the single-beam background evaluation . . . . . . . . . . . . 4.14 Summary of the double-beam KK -background evaluation . . . . . . . . . 4.15 Summary of the double-beam KP -background evaluation . . . . . . . . . 4.16 Summary of the total expected double-beam background . . . . . . . . . 4.17 Summary of the charge-exchange background evaluation . . . . . . . . . 4.18 Methods of suppression of various K + decays . . . . . . . . . . . . . . . 4.19 Total expected background in the loose signal region . . . . . . . . . . . 4.20 Total expected background . . . . . . . . . . . . . . . . . . . . . . . . . .  . 85 . 86 . . . . . . . . . . . . . . . . . .  88 91 92 96 101 102 106 112 112 113 117 124 125 125 131 135 136 137  Summary of PV1 Outside-the-Box Study . . . . . . . . . . . . . . . . . . . Summary of PV2 Outside-the-Box Study . . . . . . . . . . . . . . . . . . . Summary of EPI outside-the-box study . . . . . . . . . . . . . . . . . . . . Single-Cut Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/3 Double-cut failures (detailed) . . . . . . . . . . . . . . . . . . . . . . . 2/3 Double-cut failures (detailed) . . . . . . . . . . . . . . . . . . . . . . . Pion acceptance of muon bifurcation cuts . . . . . . . . . . . . . . . . . . . Correcting for muon contamination in target-scatter photon veto rejection Acceptance and rejection of double-beam bifurcation cuts . . . . . . . . . . Correcting target-scatter normalization branch for double-beam contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Correcting target-scatter rejection branch for KK contamination . . . . . .  140 141 143 146 147 148 151 155 157  5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10  vi  158 160  5.12 Correcting target-scatter rejection branch for KP contamination . . . . . . 160 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24  Setup cuts for the Kµ2 -based acceptance measurements . . . . Range-stack-reconstruction acceptance . . . . . . . . . . . . . Target and UTC reconstruction acceptance . . . . . . . . . . . Target and Beam acceptance . . . . . . . . . . . . . . . . . . . Online and offline photon veto acceptance . . . . . . . . . . . Kµ2 -based acceptance summary . . . . . . . . . . . . . . . . . Setup cuts for the πscatter -based acceptance measurements . . . BAD STC acceptance . . . . . . . . . . . . . . . . . . . . . . The “small” and “large” kinematic signal regions . . . . . . . Range-Stack-kinematic acceptance . . . . . . . . . . . . . . . . Range-Stack-kinematic acceptance in the small kinematic box Range-Stack-kinematic acceptance in the large kinematic box . π + → µ+ → e+ identification acceptance (TD1) . . . . . . . . π + → µ+ → e+ identification acceptance (TD2) . . . . . . . . πscatter -based acceptance summary . . . . . . . . . . . . . . . . Setup cuts for the Kπ2 -based acceptance measurements . . . . UTC acceptance . . . . . . . . . . . . . . . . . . . . . . . . . OPSVETO acceptance . . . . . . . . . . . . . . . . . . . . . . Target kinematic acceptance . . . . . . . . . . . . . . . . . . . Kπ2 -based acceptance summary . . . . . . . . . . . . . . . . . Monte-Carlo-based acceptance . . . . . . . . . . . . . . . . . . Total acceptance . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Kπ2 branching ratio measurements . . . . . . . . The single event sensitivity summary . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . .  167 167 168 169 171 172 172 173 175 176 176 177 178 179 179 180 181 181 182 183 184 184 187 189  7.1 7.2 7.3 7.4  Acceptance and background summary of each cell . . . . . . . . . Acceptance losses and additional rejection for each background . . Summary of the π + kinematics and S/B for candidate events . . . Kπ2 Momentum, energy and range for runs containing candidates  . . . .  . . . .  . . . .  . . . .  . . . .  191 192 195 195  B.1 Digitizing electronics for E949 . . . . . . . . . . . . . . . . . . . . . . . . . 230 C.1 Components of acceptance for various target fitter fixes . . . . . . . . . . . 239 D.1 D.2 D.3 D.4 D.5  Time and energy thresholds for ELVETO . . . . . . . . . . . . . . . . Photon veto cut parameters . . . . . . . . . . . . . . . . . . . . . . . Photon veto cut parameters when both-ends requirement was not met Very loose PV parameters . . . . . . . . . . . . . . . . . . . . . . . . Very loose PV parameters when both-ends requirement was not met .  . . . . .  . . . . .  . . . . .  259 263 264 264 265  E.1 Parameter optimization for CCDPUL, CCDBADFIT and EPIONK . . . . 278 F.1 Monte-Carlo-based components of acceptance for the Kπ2 branching ratio calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 vii  F.2 F.3 F.4 F.5 F.6  Setup cuts for the Kπ2 branching ratio calculation . . . . . . Ard for the Kπ2 branching ratio calculation . . . . . . . . . . Arecon for the Kπ2 branching ratio calculation . . . . . . . . . Abm−tg for the Kπ2 branching ratio calculation . . . . . . . . Acceptance summary for the Kπ2 branching ratio calculation  viii  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  286 286 287 287 288  List of Figures The decay KL0 → µ+ µ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Unitarity triangles in the ρ − η plane . . . . . . . . . . . . . . . . . . . . . 7 Unitarity triangle as determined by B and K decays . . . . . . . . . . . . 7 First-order weak flavor-changing processes . . . . . . . . . . . . . . . . . . 8 Second-order weak processes that contribute to the K + → π + ν ν¯ branching ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 LHT predictions for the relationship between B(K + → π + ν ν¯) and B(KL0 → π 0 ν ν¯) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1 1.2 1.3 1.4 1.5  2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18  Schematic views of the upper half of the E949 detector . . . . . . . . Schematic view of the AGS complex . . . . . . . . . . . . . . . . . . . Low-energy separated beam line III at BNL . . . . . . . . . . . . . . ˇ Schematic of Cerenkov counter . . . . . . . . . . . . . . . . . . . . . . Schematic of the Upstream Photon Veto . . . . . . . . . . . . . . . . Cross-sectional views of the BWPCs . . . . . . . . . . . . . . . . . . Schematic of the Active Degrader . . . . . . . . . . . . . . . . . . . . Downstream and schematic cross-sectional views of the B4 Hodoscope End and side views of the target . . . . . . . . . . . . . . . . . . . . . Schematic of the UTC . . . . . . . . . . . . . . . . . . . . . . . . . . End view of the Range-Stack . . . . . . . . . . . . . . . . . . . . . . . End view of the Range-Stack Straw Chambers . . . . . . . . . . . . . Radiation length versus polar angle for photon detectors . . . . . . . End view of the Barrel Veto and Barrel Veto Liner . . . . . . . . . . End and side views of the upstream End Cap . . . . . . . . . . . . . Collar and Microcollar . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic of the Downstream Photon Veto . . . . . . . . . . . . . . . Number of E787/E949 kaon decays as a function of data-taking days  . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  3.1 Momentum spectrum of the Standard Model K + → π + ν ν¯ process . . . . 3.2 The range versus momentum for events passing πν ν¯(1) or πν ν¯(2) triggers 3.3 A schematic explanation of the bifurcation method. Figure reproduced from [37]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Schematic representation of the outside-the-box study . . . . . . . . . . . 3.5 Reconstruction of an event in x-y view. . . . . . . . . . . . . . . . . . . . 3.6 Double-pulse fit of the CCD information in the Kaon decay vertex fiber . ix  . . . . . . . . . . . . . . . . . .  18 19 20 22 24 25 27 28 30 31 33 35 36 38 39 40 41 49  . 51 . 53 . . . .  57 58 61 66  3.7 Double and Triple-pulse TD fits in the RS stopping counter 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17  . . . . . . . . 67  Schematics of Kπ2 target-scatter and regular Kπ2 . . . . . . . . . . . . Flowchart of Kπ2 target-scatter bifurcations . . . . . . . . . . . . . . . Momentum distributions of Kπ2 target-scatter bifurcations . . . . . . . Flowchart of Kπ2 Range-Stack-scatter bifurcations . . . . . . . . . . . . Kinetic energy distribution of the π + from Kπ2γ events in Monte Carlo Flowchart of muon bifurcation branches . . . . . . . . . . . . . . . . . . Typical Ke4 event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e+ and π − kinetic energy vs. π + momentum for Ke4 events . . . . . . . Flowchart of Ke4 normalization branch . . . . . . . . . . . . . . . . . . Observable absorption energy of π − stopped in the Range Stack . . . . A schematic of the single-beam background processes . . . . . . . . . . Flowchart of Single-Beam Normalization Branches . . . . . . . . . . . . A Schematic of the double-beam background processes . . . . . . . . . Flowchart of Double-Beam Normalization Branches . . . . . . . . . . . Flowchart of Double-Beam Rejection Branches . . . . . . . . . . . . . . A Schematic of the problematic charge-exchange background processes Flowcharts of charge-exchange bifurcation branches . . . . . . . . . . .  . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . .  80 83 90 97 100 104 107 108 109 111 114 118 119 122 126 127 129  5.1 Flowchart of muon contamination in target-scatter rejection branch . . . . 154 6.1 Distributions of the reconstructed π + mass . . . . . . . . . . . . . . . . . . 175 6.2 Flowchart of cuts applied to measure NKπ2 . . . . . . . . . . . . . . . . . . 187 6.3 Kπ2 branching fraction versus rate . . . . . . . . . . . . . . . . . . . . . . . 188 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 7.17  Energy vs. range of candidate events passing all other cuts . . . . . . . . Event parameter displays for Candidate A . . . . . . . . . . . . . . . . . Event parameter displays for Candidate B . . . . . . . . . . . . . . . . . Event parameter displays for Candidate C . . . . . . . . . . . . . . . . . Quantities related to timing consistency in reconstruction cuts . . . . . . Quantities related to reconstruction cuts in the target . . . . . . . . . . . Quantities related to UTC and Range Stack reconstruction . . . . . . . . Quantities related to target kaon reconstruction . . . . . . . . . . . . . . Quantities related to single beam detection and photon veto . . . . . . . Quantities related to Range Stack kinematics . . . . . . . . . . . . . . . Quantities related to pion particle identification from TD variables . . . . Quantities related to pion particle identification from kinematic variables More quantities related to single beam detection and photon veto . . . . Signal-like probability for the three candidates . . . . . . . . . . . . . . . Xobs and CLs for the three E949-PNN2 candidates . . . . . . . . . . . . Xobs and CLs for the three E949-PNN2 candidates . . . . . . . . . . . . Comparison of Standard Model, previous PNN1 only, and final E787/E949 branching ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  . . . . . . . . . . . . . . . .  196 197 198 199 201 202 203 204 206 207 208 209 210 212 215 218  . 219  C.1 Target CCD fitter low-count uncertainty fix . . . . . . . . . . . . . . . . . 237 C.2 Effect of first bin uncertainty fix on target fitter . . . . . . . . . . . . . . . 240 D.1 Illustration of the photon veto optimization process . . . . . . . . . . . . . 262 E.1 E.2 E.3 E.4 E.5  An event that failed the CCDBADTIM cut . . . . . . . . . . . . . . . Setup cuts for CCDBADFIT, CCDPUL, EPIONK acceptance sample . Setup cuts for CCDBADFIT, CCDPUL, EPIONK rejection sample . . CCDBADTIM distribution for t1 - tk . . . . . . . . . . . . . . . . . . . Newly Rejected CCDPUL event due to new de-multiplexing algorithm .  xi  . . . . .  . . . . .  273 276 276 279 283  Glossary A  Acceptance;  AD  Active Degrader;  ADC  Analog-to-Digital Converter;  AGS  Alternating Gradient Synchrotron;  B4  B4 Hodoscope;  BEAMCUTS  The set of beam cuts consisting of B4DEDX, B4CCD, B4TRS, BWTRS, CKTRS, CKTAIL, CPITRS, CPITAIL, UPVTRS, RVTRS, B4ETCON, TGGEO, TGQUALT, TIMCON and TGTCON. See Appendix D for descriptions of the individual cuts;  BeO  BeO Beryllium-oxide;  bg  Background-level expressed as a number of events expected in the signal region;  BNL  Brookhaven National Laboratory;  BV  Barrel Veto;  BVL  Barrel Veto Liner;  BWPC  Beam Wire proportional chambers;  BWPC1  Upstream beam wire proportional chamber;  BWPC2  Downstream beam wire proportional chamber;  CO  Collar;  CCD  500 MHz transient digitizers based on a gallium arsenide chargedcoupled device;  CEX  Charged Exchange;  CK  ˇ Kaon Cerenkov Counter; xii  Cπ  ˇ Pion Cerenkov Counter;  CKM  Cabibbo-Kobayashi-Maskawa;  CP  Charge-Parity;  CsI  Cesium Iodide;  ct  Charged track;  CUT1  The set of bifurcation cuts that are inverted to create the normalization branch;  CUT2  The set of bifurcation cuts that are inverted to create the rejection branch upon which to measure the rejection of CUT1;  DAQ  DAQ Data Acquisition;  DIF  Decay-in-Flight;  DPV  Downstream Photon Veto;  EC  End Cap;  EC1  Upstream End Cap;  EC2  Downstream End Cap;  etot  Kinetic energy of the charged track;  FCNC  Flavor-Changing Neutral Current;  FERA  Fast Encoding and Readout ADC;  IC  I-counter;  Kµ2  K + → µ+ νµ ;  Kµ2γ  K + → µ+ νγ;  Kµ3  K + → µ+ π 0 ν;  Ke4  K + → π + π − e+ ν;  Kπ2  K + → π+π0;  Kπ2γ  K + → π + π 0 γ;  KBlive  The number of K + s that entered the target while the detector was live (available to register a trigger);  xiii  KINCUTS  The set of kinematic cuts consisting of LAYER14, COS3D, LAYV4, ZFRF, ZUTOUT, UTCQUAL, PRRF, RSDEDX and RNGMOM. See Appendix D for descriptions of the individual cuts;  LED  Light-Emitting diode;  LESBIII  Low-Energy Separated Beamline III;  µCO  Microcollar;  PAW  Physics Analysis Workstation;  PMT  Photo-Multiplier Tube;  PNN1  The signal region above the Kπ2 momentum peak;  PNN2  The signal region below the Kπ2 momentum peak;  ptot  Momentum of the charged track;  PV  Photon Veto;  R  Rejection;  RSSC  RSSC Range-Stack Straw Chamber;  RSSCAT  The name of the group of cuts consisting of the range stack track quality cuts RSDEDX and PRRF. See Appendix D for descriptions of the individual cuts;  rtot  Range in plastic scintillator of the charged track from the kaon decay point to the stopping counter in the Range-Stack;  RV  RV Ring Veto;  SCF  Online Range-Stack stopping-counter-finder  SES  Single Event Sensitivity;  SM  Standard Model;  T•2  Coincidence between first two layers of the Range-Stack in the same sector;  TD  Transient Digitizers;  TDC  TDC Time-to-Digital Converter;  TDCUTS  Either the loose set of TD cuts TDLOOSE or the tight set of TD cuts TDTIGHT depending on the context; xiv  TDLOOSE  The set of TD cuts consisting of IPIFLG, ELVETO, TDFOOL and the loose version of TDVARNN. See Appendix D for descriptions of the individual cuts;  TDTIGHT  The set of TD cuts consisting of IPIFLG, ELVETO, TDFOOL, EV5 and the tight version of TDVARNN. See Appendix D for descriptions of the individual cuts;  TG  Target;  TGCUTS  The set of target quality cuts consisting of B4EKZ, TGZFOOL, EPITG, EPIMAXK, TARGF, DTGTTP, RTDIF, DRP, TGKTIM, EIC, TIC, TGEDGE, TGDEDX, TGENR, TGER, PIGAP, TGB4, KIC, PHIVTX, OPSVETO, TGLIKE, TIMKF, NPITG, ALLKFIT, TPICS, EPIONK, CHI567, CHI5MAX, VERRNG, ANGLI, CCD31FIB, CCDBADFIT, CCDBADTIM, CCDPUL. See Appendix D for descriptions of the individual cuts;  θ  Polar angle of charged track determined by UTC;  RS  Range-Stack;  tk  Average time of the kaon fiber hits in the target;  tpi  Average time of the pion fiber hits in the target;  trs  Time of the charged track in the Range Stack;  UMC  E949 Monte Carlo simulation;  UPV  Upstream Photon Veto;  UTC  Ultra-Thin Drift Chamber;  VC  V-counter;  WLS  Wavelength-Shifting;  X0  Radiation Lengths.  xv  Chapter 1 Introduction The Standard Model (SM) describes the interaction of elementary particles through the electroweak and strong forces, but does not include a quantum description of gravity. The SM leaves many questions unanswered, such as: what caused the dominance of matter over anti-matter in our Universe (known as the baryon asymmetry), how do neutrinos acquire their mass, and why are there three copies of each lepton and quark? In the search for new physics beyond the Standard Model, two approaches can be taken. The first approach is to search directly for heavy particles by producing them at increasingly powerful accelerators or by observation of very-high-energy cosmic rays. The second approach is to measure to very high precision, quantities predicted by the SM in an attempt to find discrepancies between the measured and predicted quantities. One example of this second approach is the measurement of the branching ratio of the rare decay K + → π + ν ν¯. This chapter discusses the branching ratio of K + → π + ν ν¯ as predicted by the SM and by other new physics models. This chapter also contains a brief history of measurements of the K + → π + ν ν¯ branching ratio.  1  1.1  Theory and Motivation  The Sakharov conditions [90] are regarded as the necessary conditions to explain the baryon asymmetry in our universe. Any theory for baryon asymmetry must satisfy the three Sakharov conditions: (1) baryon number violating reactions must exist, (2) C and CP -violation must exist, and (3) there must be a deviation from equilibrium. The CP -transformation combines the two transformations of charge conjugation (C ) and parity (P ). Under the C -transformation, each particle is converted to its own antiparticle via a change in sign of all the internal quantum numbers (charge, strangeness, baryon number, etc...), and under the P -transformation, space is inverted (r → −r). Under the combined CP -transformation, a left-handed electron e− L comes out a righthanded positron e+ R . Presently, the sources of CP -violation in the SM only account for a small portion of the CP -violation needed to produce the observed baryon asymmetry. Particle physics experiments are constantly searching for new sources of CP -violation, and rare decays such as K + → π + ν ν¯ are sensitive to new CP -violating physics. The first observation of a CP -violating decay occurred at Brookhaven National Laboratory (BNL) in 1964, in an experiment involving the neutral K meson sector [36]. Thus far, violation of CP -symmetry has been observed only in the B and K meson sectors [1, 17], from rare decay amplitudes and from the mixing of the neutral mesons with their anti-particles.  1.2  CP-Violation and Weak Interactions  In the SM, the six flavors of quarks are classified into three generations or families: up (u) and down (d), charm (c) and strange (s), and top (t) and bottom (b). For each of these families, the first listed quark has an electric charge of +2/3 and the second has an electric charge of −1/3. Only through weak interactions can these quarks transform from  2  one to another. The charged-current processes, mediated by the charged intermediate W + or W − bosons, allow a quark to transform into another quark having a different charge (such as d → u + W − ). The neutral-current process, mediated by the neutral Z 0 boson, cannot directly transform a quark into another quark of different flavor but same charge, i.e. s → d. This type of transformation, known as a flavor-changing neutral current (FCNC) process, is forbidden at tree level and suppressed at the one-loop level due to the Glashow-Iliopoulos-Maiani (GIM) mechanism [47]. To understand the origin of the GIM mechanism, we can revisit a time when physicists believed only three quarks existed: u, d and s. Cabibbo [29] introduced an angle (θC ) to show how the strength of the weak transformations d → u and s → u as mediated by the W − differed only by factors of cos θC and sin θC , respectively. This theory failed to explain why the experimental limits on the rate of FCNC processes such as KL0 → µ+ µ− and K + → π + ν ν¯ were so much lower than the calculated rates. In 1970, the GIM mechanism was proposed to solve this problem by introducing a fourth quark, now called the charm quark. In this theory, there exist the weak eigenstates d and s which participate in the weak interactions instead of the regular d and s mass eigenstates. The relationships between the weak and mass eigenstates is written as            d   cos θC sin θC   d   =  . s − sin θC cos θC s  (1.1)  At tree level, the FCNC processes are forbidden as shown by the following expression  3  µ  d W  −  u, c  KL0  νµ  W− s  µ  Figure 1.1: The decay KL0 → µ+ µ− .  for the neutral current, J 0 , expressed without the γ-matrices: J0 = u ¯u + c¯c + d¯ d + s¯ s , ¯ + s¯s) cos2 θC + (dd ¯ + s¯s) sin2 θC =u ¯u + c¯c + (dd  (1.2)  ¯ + s¯d − ds ¯ − s¯d) sin θC cos θC , + (ds ¯ + s¯s. =u ¯u + c¯c + dd ¯ and s¯d cancel each other due to the positive and negative Here the FCNC terms, such as ds sin θC terms in the rotations of the weak eigenstates. The suppression of FCNC processes at the one-loop level can be understood by looking at the KL0 → µ+ µ− decay (Figure 1.1). In the four-quark model, the virtual quark can be either a u or a c. The contributions due to the two different virtual quarks cancel out, since the decay amplitude for the u quark is proportional to sin θC cos θC and that for the c quark is proportional to − sin θC cos θC . If the masses of the u and c quarks were identical the decay would be strictly forbidden, but due to the difference in their masses it proceeds at a very suppressed rate. The weak eigenstate model can be extended to include all six quarks where the relationship between the weak eigenstates d , s , b and mass eigenstates d, s, b is given by  4  the Cabibbo-Kobayashi-Maskawa (CKM) matrix [66],        d   Vud Vus Vub     s = V    cd Vcs Vcb    b Vtd Vts Vtb       d     s .     b  (1.3)  The SM does not predict the CKM matrix elements, so their magnitudes and phases must be determined experimentally. The constraints of unitarity connect different matrix elements such that the CKM matrix can be expressed using as few as 4 different parameters. One standard method of parameterizing the matrix involves the real angles θ12 , θ23 , θ13 and the CP -violating angle δ:   −iδ  c12 c13 s12 c13 s13 e   −iδ V = c12 c23 − s12 s23 s13 e−iδ s23 c13  −s12 c23 − c12 s23 s13 e  s12 c23 − c12 s23 s13 e−iδ −c12 c23 − s12 s23 s13 e−iδ c23 c13      ,    (1.4)  where cij = cos θij and sij = sin θij , with the i and j representing 1,2,3, the quark generation labels. The angle θ12 is the same Cabibbo angle θC seen earlier in the fourquark scheme. An approximation proposed by Wolfenstein [98] sets a parameter λ equal to s12 , the sine of the Cabibbo angle. The other parameters are real numbers that are of order unity: A, ρ and η. The parameter η describes CP -violation in the SM and a nonzero value of this parameter breaks CP -invariance for weak interactions. The following representation of the matrix expresses the elements in terms of powers of λ = sin θc ≈ 0.22:  V      =     1−  λ2 2  −λ  3  λ 1−  λ2 2  Aλ3 (1 − ρ − iη) −Aλ2  5    Aλ (ρ − iη)    + O(λ4 ). Aλ2   1  (1.5)  Unitarity (V † V = 1) of the CKM matrix implies six unitarity conditions, all of which can be expressed as unitarity triangles which are geometric representations of the unitarity conditions in the complex plane, where the areas of each of the unitarity triangles are equal to half of the Jarlskog invariant [59], JCP = s212 s23 s13 c12 c23 c13 sin δ,  (1.6)  with the definitions of sij and cij being the same as defined earlier for Equation (1.4). The unitarity condition of interest for the decay K + → π + ν ν¯ is Vud Vub ∗ + Vcd Vcb ∗ + Vtd Vtb ∗ = 0.  (1.7)  If the Wolfenstein parameterization from Equation (1.5) is used with the approximations Vud  Vtb∗  1 and Vcd  −λ, Equation (1.7) can be rewritten as ∗ Vub − λVcb∗ + Vtd = 0.  (1.8)  The triangle shown in Figure 1.2 results from this equation if all sides of the unitarity triangle are normalized such that the baseline has a length equal to 1. In this normalized unitarity triangle, the apex is given by two Wolfenstein parameters, ρ¯ = ρ(1 − η¯ = η(1 −  λ2 ) 2  λ2 ) 2  and  [27].  Thus far, K and B mesons are the only mesons that have shown evidence of CP violation in their decay processes. As can be seen in Figure 1.3, comparisons of ρ¯ and η¯ from independent measurements in the K and B sectors can be used to either verify the Standard Model description of CP -violation as explained by the CKM phase, or to signal new physics.  6  – –  (ρ,η) |V  td |  /A  λ3  |V  ub  /V  cb |  /λ  α  γ  β  (0,0)  (1,0)  Figure 1.2: A unitarity triangle in the ρ − η plane. The length of the bottom side is normalized to unity and the lengths of the other two sides can be expressed by the CKM matrix elements |Vtd |/Aλ3 and |Vub /Vcb |/λ. Reprinted figure with permission from S. Adler et al. (E949 Collaboration), Phys. Rev. D 77 052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003. Copyright 2008 by the American Physical Society.  – –  (ρ,η)  ∆MBd/∆MBs  α  –  –  KL→π0νν B0d→ψKs γ (0,0)  K+→π+νν β (1,0)  (ρ0,0)  Figure 1.3: The unitarity triangle as determined by B and K decays. The angle β measured from the CP-violating asymmetry (ACP ) in the decay Bd0 → J/ψKs0 can be compared with that from the ratio B(KL0 → π 0 ν ν¯)/B(K + → π + ν ν¯). The magnitude of Vtd extracted from the ratio of mass differences (∆MBs /∆MBd ) from Bs −Bd mixing can be compared with that extracted from B(K + → π + ν ν¯). Reprinted figure with permission from S. Adler et al. (E949 Collaboration), Phys. Rev. D 77 052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003. Copyright 2008 by the American Physical Society.  7  u  u  –  –  s  u  W+  +  e  νe  Figure 1.4: The first-order weak charged-current K + → π 0 e+ νe decay (left), which is allowed in the Standard Model. The first-order neutral-current K + → π + ν ν¯ diagram (right) shows a process which is not allowed in the Standard Model. Figures reproduced from [72].  1.3  K + → π +ν ν¯ in the Standard Model  As discussed in Section 1.2, the decay K + → π + ν ν¯ is a FCNC process, prohibited in the SM at tree level (see Figure 1.4), but allowed at the one-loop level. The leading order diagrams describing this decay are shown in Figure 1.5. ¯ as seen in the diagrams of The weak amplitude for the process s¯ → (¯ u, c¯, t¯) → d, Figure 1.5, is represented as  M∼  Vis∗ Vid i=u,c,t  γ µ qµ + m i , q 2 − m2i  (1.9)  where the index i = u, c, t denotes the quark flavor, Vij represents the CKM matrix element, γ µ the Dirac matrices, qµ the momentum transfer, and mi the quark masses. Unitarity of the CKM matrix would cause M to vanish if the quarks all had equal masses, however, the variation of quark masses due to the breaking of flavor symmetry allows this decay to proceed at a very small rate. Due to its very large mass relative to the up and charm quarks, the top quark provides the dominant contribution to the K + → π + ν ν¯ branching ratio through the coupling of top to down quarks, the CKM matrix element Vtd . The following calculation of the branching ratio K + → π + ν ν¯ follows the method from  8  u  u  – ––  u,c,t  –  s  W  –  d  + +  + +  e ,µ ,τ  W  –  ν u  W  –  s  +  – ––  u  –  –  Z  u,c,t  W  u,c,t  –  ν  u  – ––  s  d  0  ν  u – ––  u,c,t  –  –  d  +  W 0  Z  ν  + –  ν  ν  Figure 1.5: The second-order weak processes that contribute to the K + → π + ν ν¯ branching ratio are a “Box” diagram (upper) and two “Z-penguin” diagrams (lower). Reprinted figure with permission from S. Adler et al. (E949 Collaboration), Phys. Rev. D 77 052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003. Copyright 2008 by the American Physical Society.  9  [28] and [25]. The low-energy effective Hamiltonian in the SM can be written as GF α SM Hef f = √ 2 2π sin2 ΘW  Vcs∗ Vcd X l (xc ) + Vts∗ Vtd X(xt ) (¯ sd)V −A (ν¯l νl )V −A ,  (1.10)  l=e,µ,τ  where the index l = e, µ, τ denotes lepton flavor, ΘW is the electroweak mixing angle, α is the fine structure constant, and GF is the Fermi coupling. The functions X l (xc ) and 2 X(xt ), where xj ≡ m2j /MW , summarize the contributions from the charm and top quarks  respectively and include QCD corrections at next-to-next-to-leading order (NNL0). The uncertainty in the function X(xt ) is dominated by the experimental uncertainty on the top quark mass in the minimal subtraction scheme, a renormalization scheme frequently used in quantum chromodynamics. The top quark mass in the minimal subtraction scheme is mt (mt ) = (163.01 ± 1.43) GeV [48], giving a value for X(xt ) of X(xt ) = 1.443 ± 0.017.  (1.11)  The perturbative charm contribution is described in terms of the parameter  Pc (X) ≡  1 λ4  2 e 1 X (xc ) + X τ (xc ) 3 3  = 0.372 ± 0.015,  (1.12)  where λ = 0.2255 was used [25]. Using the above definitions and relationships, the K + → π + ν ν¯ branching ratio can be written as +  +  B(K → π ν ν¯) = κ+ ·  Imλt X(xt ) λ5  2  +  Reλt Reλc (Pc (X) + δPc,u ) + 5 X(xt ) λ λ  2  , (1.13)  where λi ≡ Vis∗ Vid are the CKM matrix elements, and δPc,u = 0.04 ± 0.02 represents the  10  long-distance contributions as calculated in [53]. The term  κ+ = (5.173 ± 0.025) × 10  −11  λ 0.225  8  ,  (1.14)  calculated in [78], summarizes the remaining factors that follow from Equation (1.10), in particular the relevant hadronic mixing elements that can be extracted from leading-order semi-leptonic decays of K + , KL and KS mesons. Equation (1.13) describes in the ρ¯ − η¯ plane an ellipse with small eccentricity, namely (σ η¯)2 + (¯ ρ − ρ¯0 )2 =  σB(K + → π + ν ν¯) , κ ¯ + |Vcb |4 X 2 (xt )  (1.15)  where ρ¯0 ≡ 1 + σ≡  λ4 Pc (X) , |Vcb |2 X(xt )  λ2 1− 2  (1.16)  −2  ,  (1.17)  and κ+ ≡  κ+ = (7.87 ± 0.04) × 10−6 . 8 λ  (1.18)  Using (1.13), the SM branching ratio of K + → π + ν ν¯ is predicted to be B(K + → π + ν ν¯) = (0.85 ± 0.07) × 10−10 .  (1.19)  A precise measurement of B(K + → π + ν ν¯) is regarded to be one of the cleanest ways to extract |Vtd |. This is due to the following factors: 1. The long-distance contributions to the branching ratio are small [51] and under control. The most recent calculation [53] gives an enhancement to the branching ratio of (6 ± 3)%; 2. The uncertainty from the hadronic matrix element has been reduced to less than 11  1% by recent theoretical and experimental developments [78]; and 3. Recent improvements in the calculation of the charm quark contribution [25] have reduced the theoretical uncertainties to ∼ 5%.  1.4  Physics Beyond the Standard Model  Since the theoretical uncertainty in the SM prediction of B(K + → π + ν ν¯) is small, a precise measurement can serve as a stringent test of the SM and provide an effective probe for new physics beyond the SM. Predictions of B(K + → π + ν ν¯) have been made for many models beyond the SM and a precise measurement can constrain or reject these models. Since the current experimental limit for the complementary KL0 → π 0 ν ν¯ decay is B(KL0 → π 0 ν ν¯) < 6.7 × 10−8 at 90% confidence level [9] and the SM prediction is B(KL0 → π 0 ν ν¯) = (2.49 ± 0.39) × 10−11 [78], many new physics models have the freedom to predict much larger enhancements to B(KL0 → π 0 ν ν¯) than to B(K + → π + ν ν¯). Some new physics models are discussed briefly below. In the Littlest Higgs with T-Parity (LHT) model [34], new massive particles are introduced. These new particles include massive partners to the SM particles, but unlike Supersymmetric models [84], the massive partners have the same spin-statistics as their SM partners. Under the new T-parity symmetry, most of these new massive partners are odd and the SM particles even, allowing for the lightest particle with odd T-parity to be stable. The massive quark partners, known as “mirror quarks”, have flavor-violating interactions which can impact FCNC processes such as K + → π + ν ν¯. Figure 1.6 shows that the predicted relationship between B(K + → π + ν ν¯) and B(KL0 → π 0 ν ν¯) lies along two branches. The first branch is parallel to the upper bound on the model-independent Grossman-Nir limit, B(KL0 → π 0 ν ν¯)/B(K + → π + ν ν¯) < 4.4 [50]. In this branch, B(KL0 → π 0 ν ν¯) can be as high as 5 × 10−10 while B(K + → π + ν ν¯) stays within the bounds of the measured −10 branching ratio of B(K + → π + ν ν¯) = 1.47+1.30 [11]. The second branch predicts −0.89 × 10  12  Br KL Π0 ΝΝ -10  5·10  -10  4·10  -10  3·10  -10  2·10  -10  1·10  -10  1·10  -10  2·10  -10  3·10  -10  4·10  -10  Br K  Π ΝΝ  5·10  Figure 1.6: Littlest Higgs with T-Parity (LHT) model predictions for the relationship between B(K + → π + ν ν¯) and B(KL0 → π 0 ν ν¯). The shaded area represents the 1σ-range for the previous −10 [11]. The black dotted line shows the E949 measurement of B(K + → π + ν ν¯) = 1.47+1.30 −0.89 × 10 model independent Grossman-Nir limit, B(K L0 → π 0 ν ν¯)/B(K + → π + ν ν¯) < 4.4 [50]. The blue data points represent the most general LHT model predictions, and the red, green and gold data points represent further constrained scenarios as discussed in [23]. Figure reproduced from [23].  that B(KL0 → π 0 ν ν¯) remains close to its SM prediction, but B(K + → π + ν ν¯) can vary from 1 × 10−10 to 5 × 10−10 . In the minimal 3-3-1 model, the SM SU (2)L gauge group is extended to SU (3)L resulting in an additional neutral gauge boson, the Z , which is able to transmit FCNCs at tree-level. The resulting enhancement to B(K + → π + ν ν¯) above the SM prediction ranges up to approximately 65% when the Z mass is small (1 TeV) [88]. It may also be possible that the “nothing” in the observation of K + → π + + “nothing” is actually the observation of K + → π + X 0 , where X 0 is a weakly interacting particle such as a familon, axion or majoron [51]. These particles are Nambu-Goldstone bosons (massless) or pseudo-Nambu-Goldstone bosons (small mass) that arise from spontaneously broken symmetries such as global family symmetries (familon), the Peccei-Quinn symmetry (axion), or baryon-lepton invariance (majoron). The limit set on the branching ratio 13  of K + → π + X 0 by previous E949 analyses is B(K + → π + X 0 ) < 0.73 × 10−10 at 90% confidence level [11].  1.5  History of K + → π +ν ν¯ Experiments  The first published limit on B(K + → π + ν ν¯) came in 1969 from a heavy liquid bubble chamber experiment at the Argonne Zero Gradient Synchrotron. Their initial 90% confidence level upper limit of 10−4 [31] was improved four years later to 5.7 × 10−5 [74] after final analysis of the experiment. A counter/spark-chamber experiment at the Berkeley Bevatron, sensitive only to the most energetic π + s, improved the limit to 1.4 × 10−6 in 1971. They improved their limit to 5.6 × 10−7 [30] by combining their previous data with data acquired after reconfiguring their detector to be sensitive to π + s in the 60 to 105 MeV energy range. In 1981, the limit was improved to 1.4×10−7 [13] by an experiment at the KEK Proton Synchrotron using a setup similar to that used by the Berkeley Bevatron experiment. The E787 experiment, later superseded by the E949 experiment, was initiated at Brookhaven National Laboratory in the early 1980s. From data collected between 1988 and 1991, the E787 experiment achieved an upper limit on the branching ratio at 90% confidence limit of 2.4 × 10−9 [2] and 1.7 × 10−8 [16], for the PNN1 and PNN2 regions, respectively. The PNN1 (PNN2) region is that where the π + momentum is greater (less) than that from K + → π + π 0 as shown in Figure 3.1. The detector and beam line were upgraded between 1992 and 1994 [67], and data collection resumed in 1995 and continued through 1998. In these data, two events were observed in the PNN1 region yielding a branching ratio measurement of B(K + → π + ν ν¯) = −10 1.57+1.75 [3, 4, 5]. In the PNN2 region, one event, consistent with background, −0.82 × 10  was observed and the limit placed on the branching ratio by measurements in this region was improved to 2.24 × 10−9 [6, 7]. 14  Between 1999 and 2001, the detector and beam line were again upgraded [18], as were the trigger and data acquisition [100]. The E787 experiment was rechristened E949, and despite approval to run for a total of 60 weeks, was funded for only 12 weeks of running, which occurred in 2002. In these data, a third event in the PNN1 region was observed and the combined branching ratio from all three events observed in the PNN1 region was −10 measured to be B(K + → π + ν ν¯) = 1.47+1.30 [11]. This thesis describes the search −0.89 × 10  for K + → π + ν ν¯ in the PNN2 region using the 2002 data.  1.6  My Role in the Analysis  After the demise of BNL experiment E926 “KOPIO” in 2005, I joined the E949 collaboration at a time when their analysis of the data in the PNN2 region had just begun. The results presented in this thesis were reproduced from E949 collaboration internal documents [55] and [56], for both of which I was also one of the authors. These internal documents were cited throughout this thesis to attribute figures and results of studies to others within the collaboration. This section briefly summarizes the work for which I was responsible. In Chapter 4 “Backgrounds”, I was responsible for all the Kπ2 -scatter background results (Sections 4.1 and 4.2) and the normalization result for Kπ2γ (Section 4.3). I also independently verified the results presented for the single-beam (Section 4.6), doublebeam (Section 4.7), and muon (Section 4.4) backgrounds, as well as the results of the normalization/data branch studies for the Ke4 (Section 4.5) and charge exchange (Section 4.8) backgrounds. In Chapter 5 “Validity Checks”, I was responsible for all the validation studies other than the single-cut and double-cut failure studies. In Chapter 7 “Results”, I was responsible for the “signal probability analysis” study. I verified the calculation for the probability of observed candidates being due to background 15  only and also verified, using independently developed code, many of the likelihood analysis results presented in [56]. Appendices C and E describe work for which I was completely responsible. Appendix C describes the improvements I made to the analysis of target pulse data known as the “target CCD fitter”. Appendix E describes my work on the class of cuts known as the target pulse cuts. This work included improvements to the cuts CCDPUL and CCDBADTIM and development of the cut CCDBADTIM. These target pulse cuts were critical in the suppression of the large K + → π + π 0 target-scatter background and work on these cuts accounted for a large fraction of my contributions to this analysis.  16  Chapter 2 Experimental Method In Brookhaven National Laboratory (BNL) experiment E949, the low-momentum (710 MeV/c) K + beam was stopped in an active scintillator target. Identification of the K + → π + ν ν¯ decay involved the observation of a π + in the absence of other coincident activity. The charged decay particle was identified as a π + by the measured momentum, energy and range, and by the observation of the π + → µ+ → e+ decay sequence. The detector featured full 4π-steradian photon veto coverage and the entire E949 spectrometer was situated in a 1-Tesla solenoidal magnetic field along the beam direction. The detector is shown in Figure 2.1. E949 superseded BNL experiment E787, with upgrades taking place from 1999-2001. A summary of these upgrades can be found in Appendix A. This chapter briefly describes the various detector systems and triggers used in the E949 experiment. The coordinate system used when describing the detector is as follows: • The z -direction was positive along the direction of travel of the beam (downstream); • The x - and y-directions were the horizontal and vertical directions with respect to the z -direction. The y-direction pointed vertically up. Details of the data acquisition, storage and processing can be found in Appendix B.  17  Figure 2.1: Schematic side (a) and end (b) views of the upper half of the E949 detector. Reprinted figure with permission from S. Adler et al. (E949 Collaboration), Phys. Rev. D 77 052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003. Copyright 2008 by the American Physical Society.  2.1  Kaon Beam  The K + beam was produced by a high-intensity 21.5 GeV/c proton beam from the Alternating Gradient Synchrotron (AGS) at Brookhaven National Laboratory (BNL). A schematic view of the AGS accelerator complex is shown in Figure 2.2. During a 2.2 s spill, approximately 6.5 × 1013 protons arrived at the K + production target, with spills occurring every 5.4 s. The K + production target was made of 6 cm thick platinum located on a water cooled base, with the maximum target temperature being measured at about 700◦ C. The Low Energy Separated Beam [40] (LESB III), shown in Figure 2.3, collected and transported K + s emitted at 0◦ , along with the 500 π + s and 500 protons at the production target for every K + . The beam was momentum-selected by the first dipole magnet (D1 in Figure 2.3). Two electromagnetostatic separators swept away protons and π + s from the beam resulting in a K + to π + ratio of 3:1 and negligible proton contamination when the beam arrived at the target. The angular acceptance of LESB III was 12 msr and the momentum acceptance was 4.5% FWHM, with the mean beam momentum arriving at 18  Figure 2.2: Schematic view of the AGS complex, which consisted of a 200 MeV LINAC, a booster and a synchrotron. The secondary beam lines were located in the area marked “experimental area”. Figure reproduced from [37].  19  Figure 2.3: Low-energy separated beam line III at BNL. The E949 solenoid magnet is also shown at the end of LESB III. Figure reproduced from [37].  the target being 710 MeV/c. The typical conditions during data collection were 3.5 × 106 K + s entered the E949 ˇ target per spill and the instantaneous rates in the Cerenkov Counter were 6.3 MHz for K + s and 1.5 - 2.5 MHz for π + s.  2.2  Beam Instrumentation  The purposes of the beam instrumentation were K + identification, K + momentum degradation, and detection of additional particles coincident with the K + and its decay particles. In the order encountered by the K + beam, these detectors were: • The B0 counter, a large scintillator counter which counted all charged particles in the beam; ˇ • The Cerenkov counter which identified particles as K + or π + ; 20  • The Upstream Photon Veto (UPV), a 3.1 radiation length plastic scintillator with a large beam hole; • Two beam wire chambers (BWPC) which monitored the beam profile and identified multiple incoming particles; • Passive (BeO) and active (copper and scintillator sandwich) degraders that slowed the kaons such that they came to rest in the target; • The Beam Hodoscope (B4) which detected the position of incoming beam particles just before they entered the target; • The Ring Veto (RV), an annular plastic scintillator at the same position along the beam direction as the B4 Hodoscope, but at a larger radius. ˇ The Cerenkov counters, BWPCs, degraders and B4 hodoscope are shown in Figure 2.1. The B0 counter did not enter into the analysis, but is mentioned here for completeness.  2.2.1  ˇ Cerenkov Counter  ˇ The Cerenkov counter identified beam particles as kaons or pions and was located just ˇ downstream of the B0 counter. Figure 2.4 shows a side-view of the Cerenkov counter. The acrylic radiator was 2.5 cm thick and had a refractive index n of 1.49. Based on ˇ this refractive index the threshold for Cerenkov radiation in the radiator was  βCˇ =  1 = 0.671, n  (2.1)  and the threshold for total internal reflection (TIR) was  βTIR =  n2  1 = 0.905. −1  21  (2.2)  ˇ Figure 2.4: Schematic of the side-view of the Cerenkov counter. The incoming K + and resulting + ˇ ˇ Cerenkov light are shown in blue. The incoming π and resulting Cerenkov light are shown in red. Figure reproduced from [37].  ˇ At a momentum of 710 MeV/c, βK + = 0.82 and βπ+ = 0.98. The Cerenkov light from the kaons was transmitted at the downstream surface of the radiator and reflected by a parabolic mirror to the outer ring of 14 EMI [43] 9964KB photomultiplier tubes (PMTs), ˇ ˇ collectively referred to as the K + Cerenkov counter (CK ). The Cerenkov light from the pions experienced total internal reflection at the downstream surface of the radiator and was reflected to the inner ring of 14 EMI9964KB PMTs, collectively referred to as the ˇ π + Cerenkov counter (Cπ ). The signals from the PMTs were split with 90% being sent to time-to-digital converters (TDCs) via LRS3412 discriminators and the remaining 10% being sent to ×10 amplifiers whose output was sent to 500 MHz transient digitizers based on a gallium arsenide charge-coupled device (CCD) [26]. Gallium arsenide was used because it is a suitable material for construction of high speed devices due to its high electron mobility. The pulse-height information in every 2-ns interval was recorded by the CCDs to reproduce the time development of the pulses and to detect two particles  22  ˇ close in time to each other. At the trigger level, a minimum threshold of 5 Cerenkov discriminator channels firing was used for kaons for the KB trigger and for pions for the πB trigger (see Section 2.7 for trigger definitions). The CCD information was used offline for discriminating between multiple incoming beam particles of the same type when the TDCs were unable to distinguish between pile-up signals.  2.2.2  Upstream Photon Veto  ˇ Downstream of the Cerenkov counter was the upstream photon veto (UPV). Figure 2.5 shows this 3.1-radiation-length detector made of 12 square layers, 28.4 cm in height and width, of Bicron [89] BC404 plastic scintillator and lead or copper sandwich. The scintillator layers were 2 mm thick. The five upstream-most layers of lead were 2 mm thick and the six remaining lead layers were 1 mm thick. The downstream-most layer was 2.2 mm thick copper instead of lead and formed part of the box holding the layers together. The upstream end of the box was 3.175-mm thick aluminum. There was a 175 mm wide by 40 mm high slot through the detector to allow the beam to pass through. The detector was divided horizontally into two modules of six scintillator segments each. Each scintillator segment was read out by twenty-one 1-mm diameter double-clad wavelength shifting (WLS) fibers coupled to one of the two Hamamatsu R1924 PMTs [52] used to read out the entire detector. The PMT signals were sent to ADCs, TDCs and CCDs. This detector was designed to detect photons traveling in the upstream direction, but it was found that the beam rate was too large and this overwhelmed the detector. Instead, the UPV was used to veto beam particles coincident with the time of the K + decay.  2.2.3  Beam Wire Chambers  The two beam wire proportional chambers (BWPCs) monitored the beam profile and identified multiple incoming particles. The first chamber (BWPC1) was located down-  23  Figure 2.5: Schematic of the Upstream Photon Veto. The figure on the left shows the detector looking from the beam view. The figure in the upper-right shows a cross-section and the figure on the lower-right an enlarged cross-section. The K + beam traveled through the horizontal slot in the center of the detector. Figure reproduced from [72].  24  V-plane  Y (cm)  Y (cm)  Downstream view of BWPC1  150  U-plane  100  Downstream view of BWPC2  100  V-plane  80  U-plane  60 40  50  20 0  0  -20 -50  -40 -60  -100  -150 -150  -100  -50  X-plane  -80  X-plane 0  50  100  -100 -100  150 X (cm)  -80  -60  -40  -20  0  20  40  60  80 100 X (cm)  Figure 2.6: Cross-sectional views of the beam wire proportional chambers: the upstream BWPC1 (left) and the downstream BWPC2 (right). Figure reproduced from [37].  stream of the UPV and the second chamber (BWPC2) was located 1.0 m downstream from the first. Both chambers were filled with a recirculated mixture of CF4 (80%) and Isobutane (20%). Cross-sectional views of the BWPCs are shown in Figure 2.6. BWPC1 contained three planes of 12-µm-diameter gold-plated tungsten anode wires with 1.27 mm wire spacing. The x-plane consisted of 144 anodes running along the vertical. The 120 wires in each of the u- and v-planes were oriented at ±45◦ to the vertical, respectively. These wires were multiplexed by 2 in the readout channels. The total active area of the chamber was 17.8 cm in the horizontal by 5.08 cm in the vertical. The cathode foils were 25-µm thick aluminized mylar coated with carbon and the anodecathode distance was 3.18 mm. BWPC2 contained three planes of 12-µm-diameter gold-plated tungsten anode wires with 0.8 mm wire spacing and 120 anodes per plane. The directions of the anodes were vertical in the x-plane, and ±60◦ to the vertical in the u- and v-planes, respectively. Among the 120 anode wires, the central 72 were multiplexed by 3 and the remaining wires were multiplexed by 6 to the readout channels, for a total of 32 readout channels 25  per plane. The cathode foils were 8-µm thick single-sided aluminized mylar coated with carbon and the anode-cathode distance was 1.6 mm.  2.2.4  Degraders  Downstream of BWPC1 were the degraders whose purpose was to slow the kaons such that they came to rest in the target before decaying. The lengths of the degraders were chosen such that mean stopping position in the z-direction for a K + in the target was halfway along the of the 24-cm long I-Counter which was used to tag charged decay products before they entered the drift chamber. The upstream degrader was inactive and consisted of 11.11 cm of beryllium oxide (BeO) followed by 0.48 cm of lucite. BeO was chosen due to its high density (3.0 g/cm3 ) and low atomic number. The low atomic number minimizes multiple scattering and thus minimized the spread from the beam axis of the kaons as they came to rest in the target. The reconstruction of the kaon decay in the target was considered less reliable when the decay occurred close to the radial edge of the target. The downstream degrader was called the Active Degrader (AD) and consisted of 40 layers of 2-mm thick disks of Bicron BC404 scintillator alternated with 2.2-mm thick disks of copper. The diameters of these disks were 13.9 cm and 13.6 cm, respectively. The AD was split into 12 azimuthal segments, with each segment being read out by 14 Bicron BCF99-29-AA-MC wavelength shifting (WLS) fibers sent to a single Hamamatsu R1924 PMT. The output from each PMT was sent to a TDC and a CCD. The PMT signals were multiplexed in groups of four and sent to ADCs. A schematic view of the Active Degrader is shown in Figure 2.7. Measurements from the AD allowed identification of beam particles and detection of activity coincident with K + decays.  26  Figure 2.7: Schematic of the Active Degrader. The removed wedge represents one of the sectors. Figure reproduced from [72].  2.2.5  Beam Hodoscope  Downstream of the degraders was the Beam Hodoscope (B4) whose purpose was to detect the entrance position of beam particles into the target and to identify beam particles as K + or π + by energy loss. As shown in Figure 2.8, the two 11.8-cm diameter planes of the B4 Hodoscope were oriented at angles of ±33.5◦ , labeled u and v, respectively, with respect to the horizontal. Each plane consisted of 16 “z-shaped” Bicron BC404 scintillator fingers with a 7.2-mm pitch. The cross-section of each “z-shaped” finger, was 6.35-mm thick in the middle and 3.175-mm at the edges as shown in Figure 2.8. This shape was chosen to reduce the inactive regions and improve the spacial resolution. Three Bicron BCF990290AA-MC WLS fibers were embedded in each finger and connected to a single Hamamatsu H3165-10 PMT read out by TDC, ADC and CCD.  27  Hit the middle  6.35mm  Hit the edges  Beam  3.65mm  7.30mm  Figure 2.8: The upper figure shows the B4 Hodoscope from the beam view. The segments of the fingers are shown as solid lines for the upstream segments and dashed lines for the downstream segments. The outer ring shows the WLS fibers used to read out the scintillator fingers. The lower figure shows a schematic cross-section of one of the B4 layers with the approximate positions of the three WLS fibers (small circles) in each segment. Figure reproduced from [37].  28  2.2.6  Ring Veto  The Ring Veto (RV) was located at the same position along the beam direction as the B4 Hodoscope, but at a larger radius. The RV was designed to veto particles passing through the perimeter of the B4 Hodoscope. This annular detector was composed of two 180 ◦ arcs of 3.27-mm thick Bicron scintillator with an inner diameter that varied between 11.85 cm and 12.0 cm, and an outer diameter of 14.55 cm. Each of the two elements was read out by a Hamamatsu H3165-10 PMT whose signal was sent to ADC, TDC and CCD.  2.3  Target  The target (shown in Figure 2.9) was a 12-cm diameter cylinder, 3.1 m in length, which was made of 413 5-mm square Bicron BCF10 scintillating fibers. These fibers ran in the direction parallel to the beam and had 0.09-mm thick inactive cladding. Smaller fibers, known as edge fibers, were used to fill the gaps near the outer edge of the target. Each of the 5-mm fibers was connected to a Hamamatsu R1635-02 PMT. The edge fibers were multiplexed into groups of 12 and each group was read out by a single PMT. The PMT signals were sent to ADCs, TDCs and CCDs. Typical energy deposits in an individual target fiber were on the order of tens of MeV for K + s traveling along the fiber, and only ∼1 MeV for π + s passing perpendicularly through a fiber, where bench tests [46] measured the photoelectron production from these fibers to be 30 photoelectrons per MeV. Pattern recognition was used to assign fibers with deposited energy to the K + path (kaon fibers) or the π + path (pion fibers) from the K + decay (see Section 3.3.4). To cover this large dynamic range, two separate CCD channels were used to read-out the target fibers. Each fiber was read out individually by a “high-gain” CCD channel. The sum of 4-6 randomly positioned fibers was read out by a “low-gain” CCD channel. A double-pulse fit was performed on the CCD information for both the “low-gain” and “high-gain” CCD channels to identify the fiber in which the 29  25 cm  + K Beam  V−Counter  y  Target  z I−Counter scintillator  I−Counter light guide  Figure 2.9: End (left) and side (right) views of the target. The CCD pulse-shape information for typical K + fibers is shown in the two lower plots “a” and “b”, and for typical π + fibers in the two upper plots “c” and “d”. The target is surrounded the plastic scintillator I-Counter (IC) and V-Counter. Figure reproduced from [37].  decay occurred and to decouple the energies deposited by the K + and the π + in that fiber. The target CCD fitter is described in more detail in Appendix C. The fiducial region of the target was defined by two layers of six plastic scintillator counters surrounding the target as shown in Figure 2.9. The inner scintillators (I-counters or ICs) surrounded the target and tagged charged decay products before they entered the drift chamber. The IC was 6.4 mm thick with an inner radius of 6.0 cm and extended 24 cm from the upstream face of the target. The outer scintillators (V-Counters or VCs) overlapped the downstream edge of the IC by 6 mm and detected particles that decayed downstream of the fiducial region of the target. The VC was 5 mm thick and 1.96 m long. To prevent gaps, the VC elements were rotated by 30◦ with respect to the IC elements. Each IC and VC element was read out by an EMI 9954KB PMT whose signal was sent to ADC, TDC and a 500 MHz transient digitizer (TD) based on a flash ADC [14].  30  Figure 2.10: Schematic of the Ultra-Thin Chamber. Figure modified from [37].  2.4  Drift Chamber  The drift chamber, called the Ultra Thin Chamber (UTC), was located just outside of the I-counter, with an inner radius of 7.85 cm, an outer radius of 43.31 cm and length of 51 cm. The UTC consisted of three superlayers (see Figure 2.10), each of which had four layers of axially strung 20-µm anode wires made of gold-coated tungsten. Each of the anode wires was surrounded by eight 100-µm grounded gold-coated aluminum cathode wires to form a cell, with adjacent anodes sharing cathode wires. The cells in adjacent layers were staggered by one half of a cell width to resolve the left-right ambiguity. The superlayers were filled with a 49.8%:49.8%:0.4% mixture of argon, ethane and ethanol. The cathode wires were grounded and the anode wires maintained at 2 kV, resulting in a drift velocity of 5 cm/µs and a gain of 8 × 104 . Each anode wire was instrumented with an ADC and a TDC. The drift time to the anode wires was used to determine the x-y positions of the charged track. Each superlayer was sandwiched on either side by a plane of cathode strips, which 31  were aligned helically at a pitch angle of 45◦ . The cathode strips had a width of 7 mm, a separation of 1 mm and were made of 1200 ˚ A thick copper coated with 300 ˚ A thick nickel, and were mounted on a 25 µm thick Kapton foil. Each cathode strip was instrumented with an ADC and a TDC. The cathode strips were used to determine the track position along the z -direction by means of weighted mean of the induced signal on the anode wires. The two inactive regions between superlayers were filled with nitrogen gas. Differential pressures of approximately 2 mbar in the five gas volumes supported the cathode foils. The total mass of the UTC was only 2 × 10−3 radiation lengths, excluding the inner and outer support tubes with the attached foils. The UTC position resolutions (RMS) were approximately 175 µm in the x - and ydirections and 1 mm in the z -direction. The curvature of the charged particle track in the UTC due to the magnetic field was used to determine the momentum. The momentum resolution (∆P/P ) for the two-body Kπ2 decay was 1.1% [22].  2.5  Range-Stack  The Range-Stack (RS) consisted of both scintillator counters and position-sensitive embedded straw chambers, and was located just outside of the UTC with an inner radius of 45.08 cm and an outer radius of 84.67 cm. The RS provided energy and range measurements, information for the observation of the π + → µ+ → e+ decay sequence and measurement of photon activity.  2.5.1  Range-Stack Scintillator  The RS consisted of 19 layers of Bicron BC408 plastic scintillator, azimuthally segmented into 24 sectors as shown in Figure 2.11. The innermost layer, known as the T-counter, was 0.635 cm thick and 52 cm long. The T-counter defined the fiducial volume for the charged decay products and was thinner than the remaining RS layers to suppress the rate 32  Y (cm)  Downstream view of Range Stack  100  Gap  80 60 40 20  T-counter  0  RSSC layer 1  -20  RSSC layer 2  -40 -60 -80 -100 -100  -80  -60  -40  -20  0  20  40  60  80 100 X (cm)  Figure 2.11: End view of the Range-Stack. The two superlayers of Range-Stack Straw Chambers are shown after RS layers 10 and 14. Figure reproduced from [37].  from photon conversions. Each of the T-counter scintillators was read out by seventeen 1-mm-diameter Bicron BCF-92 multiclad WLS fibers coupled to a Hamamatsu R1398 PMT at both the upstream and downstream ends. Layers 2-18 were 1.905 cm thick and 1.82 m long and each scintillator was coupled through light guides to EMI 9954KB PMTs at each end. Layer 19 was installed as part of the E949 upgrade and was used mainly to veto long range charged particles such as muons. This layer was 1.0 cm thick and had the same length and read out method as layers 2-18. Signals from each PMT were sent to ADCs, TDCs and analog fan-in modules. The TDCs recorded up to 16 hits in a 10.5 µs time window. A time window of this length was  33  required to reliably capture information from the µ+ decay, where the µ+ has a lifetime of 2.2 µs [99]. Each group of four sectors was grouped into a hextant and the signals from the four PMTs in a given layer on a given end of a hextant were summed via the fan-in modules and sent to a TD. The TDs recorded the charge in 2 ns intervals (500 MHz sampling) with a resolution of 8 bits. This provided sufficient pulse-shape information to separate pulses as close as 5 ns apart. The time window used for the TDs was 8 µs until run 47959, 4 µs until run 48227, and 2.5 µs for the remaining runs. To determine the z -position of the interaction in a counter, it was possible to use the end-to-end time differences or the end-to-end ADC information differences.  2.5.2  Range-Stack Straw Chambers  Since the Range-Stack produced only very rough position information, two superlayers of Range-Stack Straw Chambers (RSSCs) were embedded after the 10th and 14th layers of the Range-Stack as shown in Figure 2.11. The position of a hit chamber provided x-y position information and the end-to-end time differences provided the z -position information. Each superlayer consisted of two staggered layers of 3.4 mm radius straw chambers which operated in self-quenching streamer mode at 3450 V. The inner superlayer is shown in Figure 2.11. The gas mixture in these chambers was 67% argon and 33% isobutane with a trace of water. The 50-µm-diameter gold-coated tungsten anode wires of the straw chambers ran parallel to the beam direction. The layers of the inner superlayer had 24 straws per sector with a length of 97.8 cm. The layers of the outer superlayer had 28 straws per sector with a length of 113.0 cm. The z -position resolution was 1.5 cm [73] and more details on the RSSCs can be found in [77].  34  1.2 cm  50 micron Carbon Fiber  Anode Wires (50 micron W with Au coated)  Straws : Kapton foil (38 micron) with Cu/Ni cathode  Figure 2.12: End view of the inner superlayer of Range-Stack Straw Chambers. The Straw Chambers ran parallel to the beam direction. Figure reproduced from [72].  2.6  Photon Veto  The E949 detector consisted of the hermetic system of photon detectors shown in Figure 2.1. The photon veto system consisted of almost every detector system in the E949 detector. Detectors whose sole purpose was the detection of photon activity were the Barrel Veto (BV), the Barrel Veto Liner (BVL), the upstream and downstream End Caps (ECs), the upstream and downstream Collars (CO), the downstream Microcollar (µCO) and the Downstream Photon Veto (DPV). Detectors that were part of the photon veto system, but also served other purposes were the AD, target, IC, VC and RS. For a given event, the regions of the target, IC and RS traversed by the charged track were excluded from the photon veto. The thickness of the photon veto in radiation lengths as a function of polar angle is shown in Figure 2.13.  2.6.1  Barrel Veto  The Barrel Veto (BV) was located in the outermost barrel region with an inner radius of 94.5 cm, an outer radius of 145.3 cm, and a length of 1.9 m. The BV covered 2/3 of the 4π sr solid angle photon veto coverage and had a thickness of 14.3 radiation lengths. The BV consisted of 48 azimuthal sectors and four radials layers as shown in Figure 2.14. From innermost to outermost, the radial layers consisted of 16, 18, 20 and 21 alternating layers of 1-mm thick lead and 5-mm thick Bicron BC408 plastic scintillator. The azimuthal boundaries between sectors were tilted so that it was not possible for a photon originating from the target to travel along a gap between sectors. Approximately 30% of the energy 35  Thickness of PV  Upstream  25  25  20  20  15  15  10  10  5  5  0  -1  -0.5  0  0.5  0  1  X0 vs costh  -1  -0.99  -0.98  -0.97  -0.96  X0 vs costh  -0.95  Downstream 25  20  15  10  5  0 0.95  0.96  0.97  0.98  0.99  X0 vs costh  1  Figure 2.13: Radiation length “X0” versus the cosine of the polar angle “costh” for the photon detector system. The thickness is shown for all angles (top-left), angles closest to the upstream end (top-right), and angles closest to the downstream end (bottom-right). Figure reproduced from [44], courtesy of of Brookhaven National Laboratory.  36  deposited in the BV was deposited in the scintillator [15]. The upstream and downstream ends of each module were read out by EMI 9821KB PMTs and the signals were sent to ADCs and TDCs.  2.6.2  Barrel Veto Liner  As part of the E949 upgrades (Appendix A), the 2.29 radiation length Barrel Veto Liner (BVL) was installed to replace layers 20 and 21 of the E787 RS for the purpose of improved photon detection. As shown in Figure 2.14, the BVL consisted of 48 2.2-m long azimuthal sectors. Each sector consisted of 12 layers of 1-mm thick lead and 5-mm thick Bicron BC408 plastic scintillator. The BVL was expected to increase the photon veto rejection of Kπ2 decays by a factor of two. The upstream and downstream ends of each module were read out by EMI 9821KB PMTs and the signals sent to ADCs and TDCs. Each group of eight adjacent sectors was grouped to form a hextant. Each end of a hextant was read out by a TD. Approximately 30% of the energy deposited in the BVL was deposited in the scintillator.  2.6.3  End Caps  The two End Cap (EC) photon detectors accounted for roughly one-third of the 4π sr photon veto coverage [35]. The upstream EC (EC1) is shown in Figure 2.15 and consisted of 75 undoped Cesium Iodide (CsI) crystals grouped into four rings. The downstream EC (EC2) consisted of 68 crystals, also grouped into 4 rings. The crystals were 25 cm in length, which was equivalent to 13.5 radiation lengths. The PMTs were coupled directly to the crystals via a Sylgard [41] cookie formed over the PMT and a ultraviolet transmitting optical filter that passed only the fast component of the CsI scintillation light and blocked the slow component. The fast component had a decay time of approximately 30 ns at 305 nm and the slow component had a decay time of 680 ns, contributing approximately  37  y(cm)  150  100  50  0  -50  -100  -150 -150  -100  -50  0  50  100  150 x(cm)  Figure 2.14: End view of the Barrel Veto (four outermost layers) and Barrel Veto Liner (innermost layer). The sets of numbers show the numbering of radial layer and sector. Figure reproduced from [72].  38  Figure 2.15: End view (left) and schematic of the End Cap assembly (right). Figure reproduced from [37].  20% of the total light output. Fine-mesh PMTs [68] were used due to their ability to maintain high gains in strong magnetic fields. Hamamatsu R5545 2” PMTs were used for the inner ring crystals and Hamamatsu R5545 3” PMTs for the crystals in the three outer rings. The PMT signals were sent to ADCs, CCDs, and constant fraction discriminators (CFDs). The output from the CFDs was used as part of the online photon veto and was also sent to TDCs. Due to being quite near the beam line, the ECs were exposed to a high count-rate environment and as a result were prone to accidental hits occurring earlier than a given photon hit thereby reducing the efficiency of the photon veto. Offline pulse-finding was performed on the CCD information to separate double pulses close in time and to provide timing information.  2.6.4  Collar  The upstream and downstream Collar (CO) detectors were located beyond the ECs (see Figure 2.1) and covered the small angle region surrounding the beam line. The downstream CO consisted of twenty-five 2-mm thick lead sheets alternating with 5-mm thick 39  y(cm)  y(cm) 20  20  10  10  0  0  -10  -10  -20  -20  -20  -10  0  10  20  -20  x(cm)  -10  0  10  20  x(cm)  Figure 2.16: Upstream Collar (left), and downstream Collar and Microcollar (right). Element numbers are listed. Figure reproduced from [72].  Bicron BC404 scintillator sheets, providing a thickness along the beamline of about 9 radiation lengths. The upstream CO was of similar construction, but only half the thickness of the downstream one. The COs were each segmented into 12 azimuthal sectors. Each layer of the downstream CO was read out by 16 Bicron BCF99-29AA WLS multiclad fibers glued into grooves in the scintillator layer [82]. All fibers in a given sector were read out together. The fibers were polished and aluminized on one end to provide a reflective surface and the other end was coupled to a 1.17-m long lucite light guide. Each light guide was coupled to an EMI 9954KB PMT in a low magnetic field outside of the magnet. Each layer of the upstream CO was read out out by a lucite light guide coupled to the same type of PMTs outside of the magnet. Signals from the PMTs for each of the COs were sent to ADCs and TDCs.  2.6.5  Microcollar  The Microcollar surrounded the beamline directly downstream of the downstream CO. It had an inner diameter of 15.6 cm, an outer diameter of 20.0cm and a length of 53.0 cm. 40  . Reflector PMT  PMT  700  PMT  200  PMT  700  250 10.1 beam axis  1  1.5  Lead 2  .  .  .  Scintillator  25  26  300  Figure 2.17: Schematic of the Downstream Photon Veto. All dimensions are in mm. Figure reproduced from [72].  It consisted of eight azimuthal sectors containing eight layers of 2 mm scintillating fibers separated by seven layers of 60 µm Pb. The fibers in each group of two adjacent sectors were read out by an EMI9954 PMT whose signal was sent to an ADC and TDC.  2.6.6  Downstream Photon Veto  The Downstream Photon Veto (DPV) was located at the far downstream end of the detector, after the target PMTs. The DPV, shown in Figure 2.17, was a 700 mm by 700 mm square that intersected the beam path. It consisted of 26 sheets of 10-mm thick scintillator alternating with 1.5-mm thick lead, for a total thickness of 7.3 radiation lengths. Each scintillator layer was read out simultaneously by four PMTs whose signals were sent to TDCs and ADCs.  41  2.7  Trigger  The E949 πν ν¯ signal triggers selected K + → π + ν ν¯ events from the large number of K + decays and scattered beam particles, with requirements on the range of the π + track, the observation of a π + → µ+ decay signature in the RS, the absence of other activity at the time of the observed π + track and the presence of a K + at an appropriately earlier time. The trigger was composed of a fast level-0 trigger and a slower level-1 trigger. The level-0 trigger made its decision entirely from logic pulses from various detectors, had a rejection of 103 and introduced 38 ns seconds of dead time for every coincident hit in the first two layers of the RS (known as the T•2 trigger) [100]. Rejection is the ratio of the number of events encountered by that trigger to the number of events accepted by that trigger. The dead time is the amount of time that the detector is unable to trigger on subsequent events due to being busy processing information from a previously triggered event. The level-1 trigger, composed of the level-1.1 and level-1.2 triggers, operated on the events passing the level-0 trigger and involved partial processing of ADC and TD data. The level-1.1 trigger had a rejection factor of 12 after level-0, and introduced 10 to 20 µs of dead time per level-0 trigger. The level-1.2 trigger had a rejection factor of 2 after level-1.1 and introduced a dead time of up to 100 µs per level-1.1 trigger. The total rejection of these triggers reduced the 2.6 × 106 K + decays per spill to about 100 events.  2.7.1  Signal Triggers  Two signal triggers were used: πν ν¯(1) and πν ν¯(2) for the kinematic signal regions PNN1 and PNN2, respectively. The PNN1 analysis used only events passing the πν ν¯(1) trigger as signal data while the PNN2 analysis used events passing either the πν ν¯(1) or πν ν¯(2)  42  triggers. These triggers had the following requirements:  πν ν¯(1) ≡ KB · DC · IC · (T • 2) · (6ct + 7ct ) · 19ct · (BV + BVL + EC) · L0rr1 · zfrf · HEX · L1.1 · L1.2,  (2.3)  πν ν¯(2)1 ≡ KB · DC · IC · (T • 2) · 3ct · 4ct · 5ct · 6ct · (13ct + · · · + 18ct ) · 19ct · (BV + BVL + EC) · L0rr2 · HEX · L1.1 · L1.2,  (2.4)  πν ν¯(2)2 ≡ KB · DC · IC · (T • 2) · 3ct · 4ct · 5ct · 6ct · (13ct + · · · + 18ct ) · 19ct · (BV + BVL + EC) · L0rr2 · HEX · L1.1 · L1.2 · (PS16 + Cπ ),  (2.5)  which are explained in more detail below. In the trigger conditions described below, the designation “ct” refers to the Range-Stack sectors associated with the T•2 hit plus the next two clockwise sectors, where clockwise was the direction that a positive particle moved in the magnetic field. ˇ KB - The K + beam condition required a coincident hit in the kaon Cerenkov Counter (tCK ), the B4 Hodoscope, the AGS spill gate, and an analog sum of the energy of the hit fibers in the target of at least 20 MeV. This requirement ensured that the beam particle entering the target was a kaon and served as a beam strobe (BS) for the trigger. The AGS spill gate indicated the time in which the K + s produced during the 2.2-s long AGS proton spill should have been arriving at the target. T•2 - A coincidence hit was required between the first two layers (T-Counter and layer 2) of the RS in the same sector. This condition required that a charged track from from the kaon decay entered the Range-Stack. This T•2 time typically served as the detector strobe (DS) that gated all of the ADCs not associated with the beam system or target and provided a common stop for many of the TDCs. 43  IC - At least one of the hits in the I-Counter was required to be coincident with the T•2 time. This condition ensured that a charged track from the kaon decay entered the fiducial region of the detector. DC - The online delayed coincidence condition (DC), was that the IC time (tIC ) was ˇ required to be at least 2 ns later than the time in the Kaon Cerenkov Counter (tCK ). This delayed coincidence requirement ensured that beam particle was a kaon that had decayed at rest rather than a beam pion that scattered into the active region of the detector or a kaon that decayed in flight. zfrf - An additional requirement defining the active region of the detector was imposed for the πν ν¯(1) trigger. This required the charged track to stop in the active region of the RS scintillators and not the second superlayer of the RSSCs (after RS layer 14). This condition did not need to be applied to the πν ν¯(2) trigger since the range conditions of the πν ν¯(2) trigger did not allow the pions to reach RS layer 13. It also removed long-range tracks with hits at large z -positions. 6 ✂✁ + 7 ✂✁ - For the πν ν¯(1) trigger, the charged track was required to reach RS layer 6 or 7. This condition suppressed the K + → π + π − π + and K + → π + π 0 π 0 backgrounds. 3 ✂✁ · 4 ✂✁ · 5 ✄✁ · 6 ✂✁ - For the πν ν¯(2) trigger, the charged track (ct) was required to reach layer 6 and have hits in all previous layers. This condition suppressed backgrounds from 3 or more body decays such as K + → π + π − π + and K + → π + π 0 π 0 . 19 ✂✁ - For the πν ν¯(1) trigger, the Kµ2 background was suppressed by requiring that the charged track did not reach the 19th layer of the RS. (13 ✂✁ + · · · + 18 ✂✁ ) · 19 ✄✁ - For the πν ν¯(2) trigger, the charged track was not allowed to reach Range-Stack layers 13 through 19. This condition suppressed the Kµ2 background and other long-range charged tracks beyond the PNN2 kinematic region.  44  L0rr1, L0rr2 - These conditions were a refined calculation of the charged track range, which included the number of target fibers hit and the deepest layer of penetration of the track as corrected for polar angle by a measurement of the z -position of the track from flash TDCs on RS layers 3 and 11-13. The deepest later of penetration was determined by the online stopping-counter finder (SCF). The event was rejected if this calculated range was too long to be consistent with a charge track due to a π + in the kinematic region of interest (L0rr1 for PNN1 and L0rr2 for PNN2). For L0rr2, the only information used was the number of target fibers hit. These conditions suppressed the Kµ2 background and other long-range charged tracks beyond the kinematic region of interest. BV + BVL + EC - Online photon veto in the BV, BVL and EC rejected events if there was activity coincident with the T•2 time. For each of the BV and BVL counters, digital mean-timers [100] were used to determine the mean time of the discriminated PMT signals from either end of the counter. HEX - Only one RS hextant was allowed to have hits coincident with T•2, or two hextants if they were adjacent. This condition rejected events with multiple tracks and events with photon activity in the RS. L1.1 - The level-1.1 trigger required a signature of π + → µ+ decay in the online stopping counter. The height (PH) and the area (PA) of the pulse(s) as recorded by the TDs in the stopping counter were compared and the event was rejected if the ratio PH/PA was too large and there was no evidence of a detached pulse. This condition removed events without the π + → µ+ decay signature as the ratio was smaller for double pulse activity than for single pulse. This decision was made by a control board with Application Specific Integrated Circuits (ASICs) on each TD board. L1.2 - The level-1.2 trigger used data digitized by the ADCs and consisted of three con-  45  ditions. The first condition was the “L1.1 afterburner”, which rejected events with accidental activity near the stopping counter. This rejected muon events that passed L1.1 due to accidental activity near the stopping counter and faked the double-pulse π + → µ+ decay signature in the stopping counter. The second condition was the “HEX afterburner”, which rejected events where one of the two adjacent hextant hits was not due to the charged track. This rejected events that passed the HEX trigger condition due to accidental activity. The third condition rejected events where the SCF assignment was not meaningful, such as finding a stopping layer beyond layer 19. PS16+C☎  ˇ - An online pion Cerenkov (Cπ ) veto was imposed during run 49151, when the  beam separator broke down and the rate of beam pions was found to be too high. Approximately 39.4% of the data were taken before this condition became part of the trigger, changing the πν ν¯(2) trigger from πν ν¯(2)1 to πν ν¯(2)2 . In addition to events passing the Cπ veto, 1/16th of the events were passed regardless of the Cπ veto to allow measurement of the Cπ veto performance.  2.7.2  Monitor Triggers  In addition to the πν ν¯ signal triggers, there were monitor triggers for calibration and normalization, as well as triggers for other physics modes. Only the monitor triggers used throughout the rest of this analysis are described here. A complete list of E949 triggers is available from [63]. The monitor triggers were prescaled to reduce the impact on the total dead time and the monitor trigger data were taken simultaneously with πν ν¯ signal data to ensure the experimental conditions were the same for all analyzed data.  46  K✆  2  Monitor Trigger  Since the final state of this decay contains no photons or additional charged particles, it was used for many of the acceptance measurements. This sample was used to determine the fraction of beam kaons coming to rest in the target by normalizing the measured Kµ2 branching ratio to the world average. This sample was also used to optimize the rangemomentum consistency cut and the target quality cuts dealing with energy deposits. The Kµ2 trigger condition was defined as  Kµ2 ≡ KB · (T • 2) · (6ct + 7ct ) · (17ct + 18ct + 19ct ). K☎  2  (2.6)  Monitor Trigger  The final state of this decay contained two photons from the π 0 decay and a mono-energetic π + . The π + from this sample was used to check the measurement of the charged track momentum, energy and range, as well as to study particle identification in the RangeStack. The photons were used to study the photon veto. The Kπ2 trigger condition was defined as  Kπ2 ≡ KB · (T • 2) · (6ct + 7ct ) · (19ct ).  (2.7)  πscatter Monitor Trigger This sample contained beam pions that scattered into the fiducial volume of the RS. They were identified as beam pions by the πB trigger, which was the same as the KB ˇ trigger, except that a coincident hit was required in the Pion Cerenkov Counter instead ˇ of the Kaon Cerenkov Counter. Due to these pions not being mono-energetic, they were kinematically similar to πν ν¯ in the RS. This sample was used to calibrate the ionization energy loss of the pions and for studying the acceptance of many selection criteria applied  47  to the RS. The πscatter trigger condition was defined as  πscatter ≡ πB · DC · (T • 2) · IC · (6ct + 7ct ) · BV + BVL + EC · HEX.  2.8  (2.8)  Summary of 2002 Data Collection  The physics run for the E949 experiment took place over 12 weeks from March to June of 2002. The data collected during this period corresponded to a total of 1.71 × 10 12 K + s that entered the target (KBlive ). This total represents the total recorded by the scalers that counted the number of KB triggers for each spill for only “good runs”. Various runs were discarded due to technical flaws known at the time of data collection or discovered during the analysis. Runs discarded in this way resulted in a lower overall KBlive , but did not introduce bias into the final measurement since monitor and πν ν¯ signal trigger data were discarded for all discarded runs. Figure 2.18 shows the accumulated KBlive for the various running periods of E787 and E949. The proton intensity at the AGS for E949 was twice that of E787 and the total K + exposure for E949 was about 30% of that for E787 due to the significantly shorter running period.  48  Figure 2.18: Number of accumulated K + s that entered the target for E787 and E949 as a function of data-taking days. Figure reproduced from [37].  49  Chapter 3 Analysis Overview This chapter discusses the key analysis methods and steps used to estimate the background level, measure the acceptance, and finally obtain the branching ratio for K + → π + ν ν¯. The blind analysis technique [6, 7, 8, 11] was designed to avoid bias when creating the selection criteria (“cuts”) used to suppress the backgrounds that could mimic the K + → π + ν ν¯ signal. This technique was based on identifying the competing background processes a priori. Knowing these competing background processes, two kinematic phase space regions having a high signal-to-background ratio were identified. The first region, known as PNN1, was above the K + → π + π 0 (Kπ2 ) peak and covered the momentum region for the charged π + from 211 MeV/c to 229 MeV/c. The second region, known as PNN2, was below the Kπ2 peak and covered the momentum region from 140 MeV/c to 199 MeV/c. These two regions are shown in Figure 3.1 as part of the total momentum spectrum of the π + from the K + → π + ν ν¯ decay. This analysis was concerned with the PNN2 signal region and unless otherwise specified, all mentions of “the signal region” refer specifically to the PNN2 signal region. To estimate the background level in the signal region for each type of background, a bifurcation method [6, 7, 8, 11] was used. This method relied on two uncorrelated sets of cuts suppressing each type of background. Multiple validity checks were performed to 50  Figure 3.1: The momentum spectrum of the π + for the Standard Model K + → π + ν ν¯ process showing the PNN1 and PNN2 signal regions. Also shown are the charged product momentum spectra of the seven highest branching ratio K + decay modes along with their branching ratios in parentheses. The processes are not shown to scale. Figure modified from [37].  51  look for correlations between sets of bifurcation cuts, search for flaws in the analysis and check for cross-contamination in the background evaluations. The components of acceptance were measured using monitor trigger data where possible, and Monte Carlo simulation data when appropriate samples could not be made using monitor trigger data. To validate the acceptance measurement, the Kπ2 branching ratio was measured. Based on the acceptance measurement and the total K + exposure, the single event sensitivity was determined, where the single event sensitivity was the branching ratio that would have corresponded to one candidate in the absence of background. Once the background measurements were finalized and the acceptance measurements completed, the signal region was divided into nine cells whose relative signal-to-background varied by approximately a factor of four across the cells. The signal region was examined and all observed events in the signal region were considered signal candidates. Based on a likelihood analysis incorporating the acceptance, background and observed signal events in each cell, the branching ratio from this analysis was determined. Using the likelihood analysis, these results were combined with results from the previous E787 and E949 analyses [11, 6, 7] to obtain a combined E787/E949 branching ratio.  3.1  Overview of Backgrounds  Figure 3.2 shows the distribution of the range in plastic scintillator (rtot) versus momentum (ptot) of the outgoing charged particle for events that passed the πν ν¯(1) or πν ν¯(2) triggers. Events from competing background processes could mimic K + → π + ν ν¯ and become background if they fell into the kinematic signal region PNN2BOX and passed all analysis cuts. These competing background processes could be misidentified as K + → π + ν ν¯ decays when combinations of the following failure modes occurred: (1) photons and other particles escaped detection, (2) the pion was reconstructed incorrectly due to charged-particle scattering, (3) the overlapping tracks from different particles in52  Figure 3.2: The range in plastic scintillator (rtot) versus momentum (ptot) distribution of the outgoing charged particle for events passing the πν ν¯(1) or πν ν¯(2) triggers. The momentum was determined assuming pion mass for all particles. The kinematic signal region is labeled “PNN2BOX”. Figure reproduced from [72].  terfered with kinematic measurements or particle-identification cuts, or (4) beam particles emulated stopped-K + decays.  3.1.1  Stopped-K +-Decay Backgrounds  The dominant background in this analysis was from Kπ2 events where the photons escaped detection and the π + scattered in the target or Range-Stack such that the event fell into the kinematic signal region. The radiative Kπ2 decay (Kπ2γ ) became a background when all three photons escaped detection. The K + → µ+ νγ (Kµ2γ ) and K + → µ+ π 0 ν (Kµ3 ) processes were the primary source of muon-based backgrounds when the muon was misidentified as a pion and the photons escaped detection. The K + → π + π − e+ ν (Ke4 ) decay became most problematic when the π − and e+ had low energies and escaped 53  detection.  3.1.2  Beam-Based Backgrounds  The beam-based backgrounds came from beam-pions scattering into the fiducial region of the detector (πscatter ) or beam kaons decaying in flight. The single-beam background originated from πscatter and kaon decay-in-flight events that passed the delayed coincidence requirement between the incoming beam particle and the outgoing charged decay product due to incorrectly measured beam or Range-Stack timing. For the πscatter events, the beam pion also had to have been misidentified as a kaon. The double-beam background came from events where the decay products from an initial kaon were missed, and a second beam particle was subsequently missed by the beam-line detectors. This second particle could have been a πscatter or kaon decay-inflight. This type of event satisfied the delayed coincidence requirement between incoming beam kaon (the initial kaon) and outgoing decay product (the second beam particle) times. The charge-exchange (CEX) background came from the charge exchange interaction K + n → K 0 p in the target. The most problematic decays that result from this charge exchange process were KL0 → π + e− νe and KL0 → π + µ− νµ , both of which have π + kinematics that overlap the kinematic signal region.  3.2 3.2.1  Analysis Strategy and Methods Blind Analysis Method  A blind analysis method was developed to search for K + → π + ν ν¯ signal candidates amongst the many competing background processes. This method was based on (1) iden-  54  tifying the background sources a priori, and (2) keeping the signal region hidden until the background and acceptance analyses were completed. The signal region was defined by the application of all analysis cuts, thus, to avoid examining the signal region, two methods were used to create signal-like samples and samples rich in a specific type of background. The first was to use samples created from monitor triggers (see Section 2.7.2 for monitor trigger definitions). The second was to use events passing the PNN1 or PNN2 triggers and invert a cut that distinguished signal from background. By inverting a cut that suppressed a specific background, a sample of that background was created and the effectiveness of other cuts in suppressing that background could be measured. Depending on the circumstance, both of these methods could be used to create signal-like or background-like samples for use in developing cuts.  3.2.2  Bifurcation Method to Evaluate Backgrounds Using Data  Estimation of background levels in this analysis was performed using a bifurcation method in which each background was heavily suppressed by at least two sets of cuts designed to be generally uncorrelated. Studies such as the outside-the-box and the single-cut and double-cut failure studies, detailed in Chapter 5, were used to establish the level of correlation between these cuts. For each source of background, a sample of that background was created by applying setup cuts that suppressed other background sources, leaving a sample made almost entirely of the desired background. Using this sample, the bifurcation method was used to estimate the number of events of that background that would remain in the signal region after all analysis cuts have been applied. Figure 3.3 illustrates the bifurcation method using the parameter space of the uncorrelated sets of bifurcation cuts, CUT1 and CUT2. The region “A” represents the signal region and contains A events from a sample which has passed both CUT1 and CUT2. Region “D” contains the D events which have failed  55  both CUT1 and CUT2. The total number of events in the sample is A + B + C + D. If the rejection of one of the cuts does not depend on the rejection of the other cut (they are uncorrelated), the ratios of the events in the four regions will be A/B = C/D. Rearranging this algebraic expression allows the number of background events in the signal region A to be estimated without being directly measured: A = BC/D. In practice, the E949 bifurcation analysis was done through two branches: a normalization branch to measure B, and a rejection branch to determine the ratio R = (C + D)/C, where R was known as the rejection of CUT1. In this analysis, the convention was that the number of events in the region “B” was referred to as the normalization and assigned the notation N . Given these definitions, the number of background events in the signal region was estimated by bg =  N , R−1  (3.1)  which is algebraically equivalent to A = BC/D.  3.2.3  Two Data Sets  After reconstructing events passing the πν ν¯(1) or πν ν¯(2) triggers and processing them to remove unreliable data, the data were divided into “1/3” and “2/3” data sets selected uniformly from the entire data set. The analysis cuts were developed and optimized using the 1/3 data set. Once the final cut positions were set, the 2/3 data set was used to evaluate the final background levels, removing potential bias introduced by optimizing the cuts on the 1/3 data set.  3.2.4  Validity Checks  This section briefly describes the validity checks performed to verify the reliability of the analysis strategy. The validity checks discussed are the single-cut and double-cut failure studies, the outside-the-box studies, and the contamination studies used to determine the 56  Figure 3.3: A schematic explanation of the bifurcation method. Figure reproduced from [37].  57  Figure 3.4: A schematic representation of the outside-the-box study. Figure reproduced from [37].  degree of contamination in a background estimate due to other background processes. Outside-the-Box Studies The “outside-the-box” studies tested the assumption that the cuts used in the bifurcation method were uncorrelated. This was done by simultaneously loosening the two bifurcation cuts CUT1 and CUT2 so that the four regions A, B, C and D became A , B , C and D as shown in Figure 3.4. The background study was performed in the same way as the standard background evaluations using the bifurcation method, except that the number of events in region A was directly measured with the signal region masked out. This resulted in a region known as the “outside-the-box” region. The measured number of events in 58  the “outside-the-box” region was compared to the value B C /D − BC/D. Correlations between the bifurcation cuts were present if the measured and predicted “outside-thebox” background levels did not agree. Significant correlations between bifurcation cuts would generally result in the background level being underestimated. The details of these studies are discussed in Section 5.1. Single-Cut and Double-Cut Failure Studies The single-cut and double-cut failure studies involved cataloging and examining events which failed only one or two sets of correlated cuts. Since the method used to estimate backgrounds depended on each background being suppressed by two uncorrelated sets of cuts, examining the events failing only a single set of correlated cuts provided a clear way to discover flaws in the analysis. Potential flaws included cuts not operating as designed to reject the appropriate background, new background processes not accounted for in the existing background estimates, and loop-holes by which a background event not accounted for in the background estimates could become a signal candidate. Analysis flaws discovered by examination of the single-cut and double-cut failure events were generally fixed by the modification of existing cuts or by the introduction of new safety cuts. Safety cuts are high acceptance cuts that target the specific analysis flaw. These studies were performed on both the 1/3 and 2/3 data sets, with consistency of the single-cut and double-cut failure rates between the two data sets providing a check for the potential bias introduced by optimizing the analysis cuts using the 1/3 data set. The details of these studies are discussed in Section 5.2. Cross-Contamination in Background Estimates The background contamination studies were designed to determine if contributions from any of the background processes were being double-counted by being included in more than one of the background estimates, causing the total background to be overestimated. 59  The details of these studies are discussed in Section 5.3.  3.3  Event Reconstruction  Reconstruction of each event was performed under the assumption that it was a K + → π + ν ν¯ event with only a single π + track in the detector. Background events with multiple charged decay tracks or multiple beam particles were greatly suppressed by this assumption as they typically failed one or more reconstruction steps. The reconstruction took place in a series steps, listed here and described in more detail in the rest of the section. These steps were as follows: 1. The initial time of the beam particle was determined from the beam instrumentation. 2. Clustering was performed in the Range-Stack providing track time, a guide for tracking in the UTC, and identification of the stopping counter. 3. The track was located in the UTC with based on the clustering information from the Range Stack. 4. In the target, K + and π + fibers were identified and clustered, and the fiber in which the K + decay occurred was identified. 5. Second iterations of both the UTC and target fitting were performed. The energy loss and range were found in the target. 6. Track fitting in the IC was used to find the energy loss and range in the IC. 7. Track fitting was performed in the RS. The energy loss and range in the RS were found. 8. The total range, energy and momentum were determined. The x-y view of the results of a typical event reconstruction is shown in Figure 3.5. 60  Y (cm)  10  ✝✟✞✡✠☞☛✍✌✏✎ ✑✓✒ ✎✕✔✗✖✘✌✙✠☞✚  0  ✣✤✌✙✥✙✦✍✧✗✚★✎☞✠☞✩✓✥✏✎✂✌✙✪ ✣✮✭✯✎✂✠☞✞✍✥✂✬  -20  ✑✓✒ ✎  ✣✮✭✓✭✗✢  ✣✤✌✙✥✙✦✍✧✓✚★✎✂✠★✩✗✥✏✎☞✌✙✪ ✛✜✝✜✢✫✎✂✠✂✞✡✥✄✬  -30  -40  ✑✗✒ ✎✂✚  ✛✜✝✜✢  -10  ✰ ✰  ✭✴✎☞✦✲✶✓✶ ✒ ✧✓☛ ✥✙✦✍✩✗✧✲✎✂✌✙✠  ✵ ✣✮✭✱✎☞✠✂✞✡✥✄✬✱✥✙✦✲✩✓✧✳✎☞✌✙✠☞✚ ✭✴✌✙✥✏✎✂✦✍✠✤✥✏✠☞✦✍✚☞✚ ✒ ✧✗☛  -50  Y (cm)  -60  -60  -50  -40  -30  -20  -10  0  10 X (cm)  4  2  0  -2  -4  -6  -8  -4  -2  0  2  4  6  8 X (cm)  Figure 3.5: Reconstruction of an event in x-y view. The top figure shows reconstruction in the target, UTC and Range-Stack. The bottom shows an enlargement of reconstruction in the target. Figure reproduced from [37].  61  3.3.1  Beam Time Measurements  Beam times were found from TDC and CCD information for each of the CK (tCK ), Cπ (tCπ ), BWPC (tbw ), and B4 (tbm ) detectors. Average TDC times from the clusters were used to determine these times with the CCD measurements being used to discriminate between beam particles close in time. In the CK and Cπ counters, coincident TDC hits were clustered and the times averaged to find the time of the incoming particle. The same process was used for CCD hits. In the BWPCs, hits from beam particles were reconstructed when two or three of the three planes had coincident TDC hits. The time of the incoming particles was taken from an average of the TDC hits. In the B4 beam hodoscope, counters from the two layers were clustered based on coincident TDC or CCD hits. Energy-weighted averages were used to determine the time and x-y position of clusters. The measured energy deposits were used to differentiate between and identify beam kaons and beam pions.  3.3.2  Clustering in the Range Stack  Based on the coincidence with the online T•2 time (tds ), the offline T•2 sector was identified from TDC hits in the first two layers of the RS. Moving outward and clockwise from this T•2 sector, coincident hits within 10 ns of the T•2 time and with more than 0.5 MeV were included in the cluster. A good cluster involved at least six consecutive layers from inside to outside. The track time trs was determined from the average of all hit times in the good cluster. The outermost and most clockwise counter in the cluster was identified as the offline stopping counter.  62  3.3.3  Initial Track Finding in the UTC  Two separate fits were used to reconstruct the UTC track. The curvature of the fit in the x-y plane was used to determine the charged particle’s transverse momentum. The fit in the r-z plane was used to determine the polar angle of the charged particle and the resulting slope was used to convert from the measured transverse momentum to the total momentum. For the fit in the x-y plane, clusters in the hit wires were identified. A circle fit was performed using wire positions and drift distances with left-right ambiguity resolution. For the fit in the r-z plane, the cathode foils from the clusters were used. A straightline fit was performed, where the z position of the cluster was determined using a ratio method suggested in Reference [64]. This ratio method involved using the three cathode strips with the largest energy as measured by the ADCs and locating the centroid using an energy-weighted mean. When more than one track pointed at the offline T•2 sector, the good UTC track was defined as the one closest to the first crossing point (from inward to outward) between RS sectors, or closest to the clockwise edge of the offline stopping counter otherwise.  3.3.4  Initial Target Reconstruction  The first step of the target reconstruction was to classify the hit target fibers into K + , π + , and γ fibers using their timing and energy information. Note that at the point of this initial classification, each fiber could fall into multiple categories. A fiber was classified as • K + if it had an energy greater than 4 MeV and was coincident with the beam strobe tbs ; • Or π + if it had between 0.1 and 10.0 MeV of energy and was coincident with track time in the Range-Stack trs ;  63  • Or γ if it did not fall into the other two categories and had an energy above 0.1 MeV. The fibers were clustered into K + and π + paths based on geometry. The π + fibers had to lie along a strip (typically 1 cm in width) along the UTC track extrapolated into the target. The designation of each fiber as a π + or γ fiber could be be switched based on likelihood values calculated from energy, time and distance from the extrapolated UTC track. The event failed target reconstruction if there was more than one K + or more than one π + track, or if these two tracks were not geometrically adjacent in the x-y plane. The decay vertex was identified in the K + fiber closest to the extrapolated UTC track and furthest away from the x-y position of the B4 hit. The position of the track exit from the target was identified. Fibers were reclassified from π + to opposite-side-π + fibers if they were located on the opposite side of the decay vertex as the track exit position. These hits were typically due to tracks from decays with multiple charged decay products. The average times of the K + and π + hits were defined as tK and tpi , respectively. The sums of the K + and π + energies were defined as EK and Epi , respectively. This reconstruction process was performed as many as three times per event. For the first pass, the B4 position information was not used. After the first pass of target reconstruction, the B4 reconstruction was performed again with the hit closest to the average time of the kaon fiber hits being designated as that due to the K + beam particle. A second pass of the target reconstruction was then performed using this B4 information. Another round of B4 reconstruction followed by a third pass of the target reconstruction was performed if the solution found from the second pass was not satisfactory. Further details on the target reconstruction can be found in [71].  64  3.3.5  Second Iteration of UTC Reconstruction  A second iteration of the UTC reconstruction was performed using the information from the first iteration of the target fitter as constraints.  3.3.6  Second Iteration Target Reconstruction  Then a second iteration of the target reconstruction was performed. In addition to the steps described for the first iteration of the target reconstruction, double-pulse fits were performed to search for π + or other energy hidden in the K + fibers using the CCD information from each of the K + fibers. The target pulse data fitter is discussed in more detail in Appendix C. In the case where the difference between tpi and tK was greater than 15 ns, any energy hidden in the second pulse of the vertex fiber was subtracted from EK and assigned to Epi . The 15 ns threshold was the minimum time difference for which it was possible for the fitter to reliably identify the K + and π + pulses in a fiber. Figure 3.6 shows a double-pulse fit with hidden π + energy. The π + energy loss Etg and range rtg were found in the target based on the target reconstruction.  3.3.7  Track Fitting in the IC  A good track in the IC was one with hits in only one or two adjacent sectors. In the case of a sector crossing (two adjacent hits), the energy EIC was the sum of the two energies and the time tIC was the energy weighted sum of the times from the two sectors. The range RIC was computed as the length of the extrapolated UTC track from the inner to the outer IC radius.  3.3.8  Reconstruction in the Range-Stack  The first step in RS reconstruction was to search for the π + → µ+ → e+ decay signature by fitting the TD information in the stopping counter using single-, double- and triple65  160 140 120 100 80 60 40 20 05 0 -5 0  20  40  60  80  100  120  100  120  K fiber 445 Raw High Gain 35 30 25 20 15 10 5 05 0 -5 0  20  40  60 80 K fiber 445 Raw Low Gain  Figure 3.6: Double-pulse fit of the CCD information in the Kaon decay vertex for for the highgain (top) and low-gain (bottom) CCDs. The energy of the second pulse was significantly higher than the expected 1 MeV energy from a pion headed directly toward the fiducial region of the detector and may have indicated a scatter.  66  5 100  0 -5  Residual  Residual  5 0 -5  250  80  200 60  150  40  100  20  0  20  40  60  80  End 1 (RAW)  100  120  5 0 -5  225 200  0  140 Residual  0  0  20  40  60  80  End 1 (RAW)  100  120  140 5 0 -5  100  175  Residual  50  80  150 125  60  100 40  75 50  20  25 0  0  20  40  60  80  End 2 (RAW)  100  120  0  140  0  20  40  60  80  End 2 (RAW)  100  120  140  Figure 3.7: Double-pulse fit (left) to the TD pulse information in the RS stopping counter for the upstream (top) and downstream (bottom) readouts. Triple-pulse fit (right) to the TD pulse information in the RS stopping counter for the upstream (top) and downstream (bottom) readouts.  pulse assumptions. The fits were performed by varying the leading edges, heights and pedestals of template pulses, prepared for each individual counter using πscatter monitor trigger data. Examples of double- and triple-pulse fits are shown in Figure 3.7 and further details on the TD fitter can be found in [57]. In addition to observation of the π + → µ+ decay signature, these fits allowed the energy deposited by the muon Eµ in the stopping counter to be determined. The RS track fit in the x-y plane was performed using (1) the entrance point provided by the UTC track extrapolation, (2) the sector crossing points, (3) the RSSC hit positions, and (4) the expected range predicted by energy losses in the RS layers. The χ2 of the fitted track was minimized by changing the incident momentum and angle of entrance to the RS, with the shape of the track taking into account effects of energy loss in each counter as given by the Bethe-Bloch equation for a 1 T magnetic field. Due to this correlation between energy and range these quantities could not be treated as uncorrelated in the bifurcation analysis. Many of these quantities are labeled in Figure 3.5 and discussed in 67  more detail below. Sector crossing points allowed for precise measurement of track positions in the x-y plane. Events were rejected if they had more than two sector crossings. Hits in the RSSCs provided another precise position measurement in the x-y plane. Adjacent hits in Straw Chambers were formed into clusters and the track cluster was identified as that closest to the UTC track extrapolation. For each layer, the x-y position of the hit was taken from the average of the track hits and the time of the hit from the earliest hit. The energy loss in each RS counter was determined using ADC information. A correction was applied to remove the additional energy recorded by the ADC from accidental activity by using the TD pulse height information. The total energy loss of the track in the RS (Ers ) was the sum of the energy losses in all the RS counters corrected for the the energy deposited by the µ+ and e+ in the stopping counter. The polar angle in the RS was determined by a straight-line fit in the r-z plane. The z -positions in the RS counters and the Straw Chambers were determined from end-to-end time (and charge for the RS counters) differences, with average resolutions of 4-5 cm and 1.5 cm, respectively. The z -position of the extrapolated UTC track was also used in the fit. The range in the RS (Rrs ) was calculated from the path length of the fitted track in the x-y plane, corrected for the polar angle.  3.3.9  Kinematic Measurements of the Track  The total range of the charged track (rtot) was the sum of Rrs , RIC and Rtg . The total energy (etot) was the sum of Ers , EIC and Etg . The total momentum (ptot) was obtained from the UTC with corrections from the range in the target and IC. Due to these corrections, there was some correlation between momentum and range and they could not be treated as uncorrelated in the bifurcation analysis. All three kinematic quantities were  68  corrected for tiny contributions due to inactive material in the UTC.  3.4  Selection Criteria  The cuts used to define the signal region and suppress backgrounds were optimized by maximizing their ability to suppress backgrounds while maximizing signal acceptance. The analysis was performed in three stages, called passes. The PASS1 cuts removed events which could not be reconstructed and obvious background events. The PASS2 cuts were a set of cuts which could be grouped into various combinations to enhance specific backgrounds. The PASS3 cuts were the final analysis cuts that were optimized to minimize the acceptance loss of signal events while maximizing their ability to reject the targeted background. The PASS3 cuts were grouped into categories of kinematic cuts, phase space cuts, delayed coincidence cuts, beam cuts, target quality cuts, π + → µ+ → e+ decay-sequence cuts, and photon veto cuts.  3.4.1  Loose and Tight Cuts  The primary (loose) signal region was defined by the application of all of the analysis cuts described in this section. Four sets of analysis cuts were tightened and various combinations of these sets of cuts were used to divide the primary signal region into nine cells having varying levels of signal-to-background. The sets of cuts that were tightened were (1) the kinematic phase space cuts “PNN2BOX”, (2) the delayed coincidence cuts “DELCO”, (3) the π + → µ+ → e+ decay-sequence cuts “TDCUTS ”, and (4) the photon veto “PVCUT”. These cells were used in the likelihood method to determine the K + → π + ν ν¯ branching ratio from the observed candidates (Section 7.1). The cell having the highest signal-to-background was defined by the application of the tight versions of each of the four sets of cuts and is referred to as the “tight signal region” throughout the rest of this thesis. 69  3.4.2  PASS1 Cuts  The most basic analysis cuts were the PASS1 cuts. These loose cuts removed obvious Kπ2 , Kµ2 and beam background events, as well as events which could not be reconstructed. Details of the PASS1 cuts are discussed further in Appendix D.1.  3.4.3  PASS2 Cuts  The PASS2 cuts consisted of looser versions of the final analysis cuts. The PASS2 cuts were applied in certain combinations to enhance specific backgrounds, reducing processing time when developing cuts and evaluating background levels. Details of the PASS2 cuts are discussed further in Appendix D.2.  3.4.4  PASS3 Cuts  The PASS3 cuts were the final analysis cuts, grouped into categories of kinematic cuts, phase space cuts, delayed coincidence cuts, beam cuts, target quality cuts, π + → µ+ → e+ decay-sequence cuts, and photon veto cuts.  3.4.5  Kinematic Cuts  The kinematic cuts, collectively referred to as KINCUTS, were grouped into the following categories: Fiducial cuts - These cuts ensured that the charged decay product passed only through and came to rest in the fiducial region of the detector, where the fiducial region was the active region of the detector in which events were accepted for further analysis. These cuts further tightened the conditions from the πν ν¯(1) and πν ν¯(2) triggers placed on the parts of the detector through which the charged decay product could travel or come to rest.  70  Track Reconstruction Cuts - These cuts required good track reconstruction in the UTC, and good matching between RS track reconstruction and UTC track extrapolation. dE/dx Cuts in Range-Stack - These cuts removed events with energy deposits in the RS inconsistent with those of a π + . These cuts removed events where a photon or extra track deposited energy that was measured by the ADCs. These cuts also removed events that scattered in the Range-Stack such as Kπ2 Range-Stack scatter. Range-Momentum Consistency Cut - Since muons typically had a longer range than pions for equivalent momenta, this cut suppressed muon backgrounds by requiring that the range of the charged track was consistent with that for pions. Details of the kinematic cuts are discussed further in Appendix D.3.  3.4.6  Phase Space Cuts  To suppress Kπ2 events, the upper limits on the phase space cut (PNN2BOX) were chosen to be below the Kπ2 monochromatic peak (Figure 3.2) in momentum, energy and range. The upper bounds of this cut were located 2.5 × σptot below the Kπ2 momentum peak, 2.5 × σetot below the energy peak and 2.75 × σrtot below the range peak. The standard deviations on the measured kinematic quantities (σptot , σetot , σrtot ) were measured using reconstructed Kπ2 events. The lower bounds of this cut were chosen to suppress or kinematically exclude the troublesome many-body decays that could most easily mimic the K + → π + ν ν¯ decay. The acceptance conditions for the loose version of the phase space cut (BOXLOOSE) were: 140.0 MeV/c ≤  ptot  ≤ 199.0 MeV/c;  60.0 MeV ≤  etot  ≤ 100.5 MeV;  12.0 cm ≤  rtot  ≤ 28.0 cm.  71  The phase space cut was one of the sets of cuts that was further tightened to define the tight signal region. The lower bounds of the tight phase space cut (BOXTIGHT) were raised to suppress the Ke4 background, whose π + momentum peaked at around 160 MeV/c as shown in Figure 4.8. The lower momentum bound of 165 MeV/c was chosen to maximize Ke4 suppression while minimizing the loss of signal acceptance. The upper momentum bound was lowered from 199 MeV/c to 197 MeV/c to increase Kπ2 suppression without having a significant impact on signal acceptance. The acceptance conditions for the tight version of the phase space cut (BOXTIGHT) were: 165.0 MeV/c ≤  ptot  ≤ 197.0 MeV/c;  72.0 MeV ≤  etot  ≤ 100.0 MeV;  17.0 cm ≤  rtot  ≤ 28.0 cm.  The optimization of BOXTIGHT is described in more detail in [55].  3.4.7  Beam Cuts  The beam cuts, collectively referred to as BEAMCUTS, were designed to suppress singlebeam and double-beam backgrounds. Single-beam backgrounds were suppressed by a particle identification cut that ensured that the beam particle was a kaon based on its energy deposition in B4. Double-beam backgrounds were suppressed by searching for and removing events with extra beam particles in the beam instrumentation at track time (trs ). The beam cuts included cuts that (1) enforced timing consistency between CCD and TDC information in B4, between B4 and the average target K + fiber hit times, and between the track time in the Range-Stack and the average target π + fiber hit times, and (2) removed double-beam events with specific signatures known to cause the target reconstruction to find incorrect solutions. Details of the beam cuts are discussed further in Appendix D.5. 72  3.4.8  Delayed Coincidence Cuts  The delayed coincidence cuts suppressed single-beam backgrounds by enforcing a minimum time between the average time of the kaon fiber hits (tk ) and the average time of the pion fiber hits (tpi ). The delayed coincidence cuts were one of the sets of cuts that was further tightened to define the tight signal region. Details of the delayed coincidence cuts are discussed further in Appendix D.6.  3.4.9  Target Quality Cuts  The target quality cuts, collectively referred to as TGCUTS, were designed to select signal-like events based on quantities measured in the target. These cuts targeted specific backgrounds and/or event signatures. These cuts rejected events that fell into the following categories: • Events where the energy deposited by a π + in a target fiber was larger than that expected for a π + traveling through the target along the extrapolated UTC track. This energy is a signature of π + scattering in the target, or could have been evidence of a photon in a π + fiber, or evidence of a photon or π + hidden in a K + fiber. These cuts suppressed the Kπ2 target-scatter background, any backgrounds involving photons in the final state and events where a K + fiber was misidentified as a π + fiber. • Events with poor agreement between the target π + track fitter and the target π + fibers as determined using χ2 -like quantities. These cuts suppressed events that scattered in the target. • Events where the beam particle did not behave as a K + . A likelihood function was constructed using the z -position of the kaon decay in the target as determined by UTC track extrapolation, the expected z -position of the kaon decay as determined 73  by the total energy deposited in target kaon fibers, and the energy deposited by the kaon in the B4 Hodoscope. These conditions suppressed the beam pion scatter single-beam background. • Events where the π + cluster did not appear to originate from one of the K + cluster target fibers. This cut suppressed CEX and double-beam backgrounds. • Events where there was a large uncertainty in the target pion path length from the target reconstruction. This suppressed double-beam and Kπ2 target scatter events. • Events that did not meet consistency requirements for times or energies between the K + or π + tracks and those in other detector systems. • Events with charged track dE/dx in the target inconsistent with that of a pion. This suppressed backgrounds where the charged track was due to a lepton, such as muon-based backgrounds. • Events that did not meet position consistency requirements between the K + decay vertex, the beam particle in B4, and the target K + and π + clusters. • Events with target π + fibers on both sides of the K + decay vertex as shown in Figure 4.7. This suppressed double-beam events and decays with multiple charged products, such as Ke4 . • Events that showed evidence of hidden energy as determined by likelihood functions comparing target π + fiber positions, times and energies to the extrapolated UTC track. • Events that did not meet consistency requirements between the target K + fiber times and their distances to the vertex in the x-y plane or the z -positions of their hits as determined by the deposited energy.  74  • Events where the target pulse data fitter, described in Appendix E, failed to find acceptable solutions to both the single-pulse and double-pulse assumptions in any of the K + fibers having energy above a certain threshold. • Events where there was coincident activity between the IC and a nearby K + fiber. This event signature was often an indication of double beam background. Details of the target quality cuts are discussed further in Appendix D.7.  3.4.10  π + → µ+ → e+ Decay-Sequence Cuts  The π + → µ+ → e+ decay-sequence cuts, collectively referred to as TDCUTS, were designed to reject events in which a µ+ caused the charged track. These cuts searched in and around the Range-Stack stopping counter for activity consistent with the observation of the π + → µ+ → e+ decay sequence. The signature for this decay sequence, based on the pulse-shape information from the TDs, was 1. Three pulses were found in the stopping counter, corresponding to the π + , µ+ and e+ of the π + → µ+ νµ and µ+ → e+ ν¯µ νe decays. 2. The 4.1 MeV of kinetic energy from the µ+ was observed as about a 3 MeV pulse due to saturation [19]. Since the path length of a µ+ coming from the decay of a π + at rest is approximately 1.4 mm in the RS scintillator counters, most of these muons did not exit the stopping counter. 3. Most of the positrons from the µ+ → e+ ν¯µ νe decays exited the stopping counter due to having up to 53 MeV of kinetic energy; thus the positron should be observed to deposit energy in and around the stopping counter. Events were rejected if the above signatures were not observed or if it was determined that the observed signatures were faked by accidental activity. 75  The π + → µ+ → e+ decay-sequence cuts were one of the sets of cuts that was further tightened to define the tight signal region. Details of these cuts are discussed further in Appendix D.8.  3.4.11  Photon Veto Cut  The photon veto cut (PVCUT) rejected events with activity above threshold in any of the photon veto subsystems coincident with Range-Stack track time trs . These subsystems were the BV, BVL, RS, EC, target, IC, VC, CO, µCO, AD and DPV. The photon veto cut was one of the sets of cuts that was further tightened to define the tight signal region. The photon veto cut was also loosened below the nominal level for some background and acceptance studies. The photon veto cut is discussed further in Appendix D.9.  3.5  Monte Carlo Simulation  Detector responses were modeled by a Monte Carlo simulation package, referred to internally as “UMC”. The simulation modeled all detector systems other than the beam instrumentation upstream of the target. The simulation samples were generated with the same information and in the same format as data except that pulse-shape information from CCDs and TDs was not generated. The accuracy and performance of the simulation were verified by comparisons of kinematic variables between simulation and data for Kπ2 and Kµ2 decays [33].  3.5.1  Simulation of K + Propagation  A “beam file” was created from analysis of the K + stopping distribution using Kµ2 monitor trigger data. This file contained the B4 hit position; the K + stopping position and target  76  fiber element number; and the number of fiber hits, their time, energy and fiber element number for K + and accidental hits. The target patterns were the same in the simulation and the experimental data. Using this information, a simulated K + decay started from the stopping fiber at the given K + stopping position.  3.5.2  Simulation of K + Decay Products  A number of K + decays were simulated for various studies: K + → π + ν ν¯ signal, Kµ2 , Kπ2 , Ke4 , Kπ2γ , and a number of the CEX processes discussed in Section 4.8. Photon and electron interactions and their energy deposits were calculated using routines from the electromagnetic-shower simulation package EGS4 [86]. For other charged particles, the energy deposits were calculated by summing the energy losses of each ionization along the steps taken by the particles. Using the Bethe-Bloch formula, the total average energy deposited along each step was calculated. The multiple Coulomb scatterings of the charged particles with various nuclei in the detector were calculated according to the theory of Moliere [32], with corrections for the spin of the scattered particle and the form factor of the nucleus [79]. The hadronic interactions of positively charged pions in the plastic scintillators were calculated using a combination of data and phenomenological models [95]. The π − absorption was modeled based on experimental data of stopped π − in the Range Stack [97].  3.5.3  Simulation of Trigger  A comparison between simulated Kµ2 triggers and data was used to derive a weight function to correct for the trigger bias introduced by generating the beam file used in the simulation.  77  Chapter 4 Backgrounds To obtain an unbiased estimate of the total background level, the data were split into two sets and the signal region was not examined until after the background evaluation studies were finalized. The “1/3 data set” was used to optimize the cuts and the resulting background levels. The final background levels were measured on the “2/3 data set”. The process of creating the “1/3” and “2/3” data sets was described in Section 3.2.3. This chapter describes the background evaluation procedure and resulting background levels for each of the background categories described in Section 3.1. Throughout this chapter, results from the “1/3” and “2/3” data sets have been scaled by the appropriate factor to represent the background level for the entire data set. In cases where the background sample size was reduced to zero events due to high suppression of the given background, the calculation assumed one event remained. Throughout this chapter, the backgrounds were estimated for both the loose and tight signal regions (Section 3.4.1). The estimates in the tight signal region were used as a validity check against the scaling method used to divide the signal region into nine cells as detailed in Section 7.1.  78  4.1  Kπ2 Target-Scatter Background  The decay K + → π + π 0 (Kπ2 ) was responsible for the dominant background in this analysis. For this decay to become a background, the photons from the π 0 decay have to be missed and the π + has to lose some undetected energy, perhaps via nuclear interactions, such that it falls into the kinematic signal region (PNN2BOX). This scattering can happen in the target (Kπ2 target-scatter) or in the Range Stack (Kπ2 Range-Stackscatter), but the target-scatter component dominates this background. The evaluation of the Range-Stack-scatter background is discussed in Section 4.2. The topology of the most problematic type of Kπ2 target-scatter was that of a π + initially traveling along the kaon fibers and scattering into the active region of the detector. There were two reasons why this problematic type of target-scatter was difficult to reject. The first reason was that some energy deposited in the target by the scattering π + was hidden in a kaon fiber. The second reason was a consequence of the π + and π 0 emerging back-to-back. A π + initially traveling along the kaon fibers meant that the π 0 was also traveling parallel to the beam direction and the resulting photons from its decay were directed at the upstream or downstream ends of the detector. As can be seen in Figure 2.13, these small angles are where the photon veto is the weakest. Figure 4.1 contrasts this type of event with a regular Kπ2 event. The Kπ2 target-scatters were classified into two non-exclusive categories. The first category, known as “z -scatters”, occurred when the π + traveled in the beam direction, scattered in a kaon fiber and into the active region of the detector. The troublesome decay described above falls into this category. The second category, known as “xy-scatters”, occurred when the π + scattered outside of the kaon fibers and the scatter was visible in the xy-plane. Section 4.1.3 describes how samples with varying degrees of “z -scatters” and “xy-scatters” were created to measure the photon veto rejection for these events in the rejection branch.  79  Figure 4.1: Schematic of the regular K π2 (left) and the Kπ2 target-scatter (right) processes. For a regular Kπ2 event (left), the back-to-back correlation between the directions of the π + and π 0 mean that a π + directed at the active region of the detector results in the photons from the π 0 being directed toward a region of the detector where the photon veto is strongest such as the Barrel Veto. For the most troublesome type of K π2 target-scatter event (right), a π + initially traveling along the kaon fibers and scattering toward the active region of the detector results in the π 0 also traveling parallel to the beam direction. The photons from the π 0 decay are then directed toward the upstream or downstream ends of the detector, where the photon veto is weakest.  80  The most effective methods of suppressing this background were the detection of the photons from the π 0 decay using the photon veto PVCUT and the detection of the elastic or inelastic scatter of the π + in the target using the target quality cuts TGCUTS, many of which suppress events with additional energy in the target thus suppressing π + scatters. The bifurcation cuts used for the Kπ2 target-scatter background evaluation were the photon veto PVCUT (CUT1) and the target quality cuts TGCUTS (CUT2).  4.1.1  Potential Sources of Contamination  Since the photon veto was effective at suppressing all background process involving photons, a sample of Kπ2 target-scatter events created by inverting the photon veto was expected to have contamination from other background processes containing photons. It was also expected that other background processes could result in measurable and nonnegligible amounts of contamination in both the normalization and rejection branches for the Kπ2 target-scatter background. The Kπ2 target-scatter normalization branch was found to contain Kπ2 Range-Stackscatter events and this contamination was exploited to estimate the Kπ2 Range-Stack scatter background (see Section 4.2). The Kπ2 target-scatter normalization branch was corrected for this Range-Stack-scatter contamination as is discussed in Section 4.1.2. It was determined that the Kπ2 target-scatter normalization branch was likely to contain a large amount of contamination due to Kπ2γ events due to the extra photon and decay-pion kinematics that fell mostly in the kinematic signal region. This contamination study is detailed in Section 5.3.4 and the resulting correction to the Kπ2 target-scatter background is discussed in Section 4.1.5. Studies to measure the contamination due to muon and double-beam (Sections 5.3.2 and 5.3.3, respectively) events were performed and found negligible contamination in both the Kπ2 target-scatter normalization and rejection branches. Note that in the Kπ2 target-  81  scatter background studies, the setup cuts included the cuts designed to heavily suppress the muon and double-beam backgrounds thus the contamination due to the background processes was expected to be small. Based on the bifurcation cuts used for the Ke4 background normalization branch (Section 4.5.1), it can be argued that the number of Ke4 events contaminating the Kπ2 targetscatter normalization branch must have been less than the number of events that were in the Ke4 normalization branch, and thus the Ke4 contamination was negligible. The Ke4 background normalization branch was created by inverting the combination of OPSVETO and TGPV and applying the remaining analysis cuts. For the 1/3 data set, the events remaining at the end of the Ke4 normalization branch, before the application of CCDPUL, were visually examined and were found to be consistent with Ke4 events. For the 1/3 (2/3) data set, there were 4 (7) events remaining at the end of this normalization branch. The Kπ2 target-scatter normalization branch can be characterized as being selected by the application of OPSVETO, the inversion of the photon veto, and the application of all other analysis cuts. A sample of events chosen by OPSVETO and inverted TGPV is a subset of the Kπ2 target-scatter normalization branch. Since OPSVETO rejects Ke4 events, the number of Ke4 events found in the sample chosen by OPSVETO and inverted TGPV will be less than those chosen by inverting the combination of OPSVETO and TGPV and thus the Ke4 contamination in the Kπ2 target-scatter normalization branch was negligible.  4.1.2  Kπ2 Target-Scatter Normalization Branch  The Kπ2 target-scatter normalization branch was created by inverting the loose photon veto PV60 (CUT1) and applying all the other analysis cuts. The “60” in PV60 refers to the fact that the parameters for the loose photon veto were optimized to accept approximately 60% of the signal-like events. The loose photon veto was inverted in the  82  Kπ2 Target-Scatter Bifurcations VALID TRIG, P2PSCUT, P2TGCUT, P2TGPVCUT, BEAMCUTS, DELCO ∗ , TDCUTS ∗ , KINCUTS, PNN2BOX∗ Normalization Branch  Rejection Branch  PV60  CCD31FIB, CCDBADTIM, CCDBADFIT, CCDPUL · EPIONK · B4EKZ N1  PVCUT∗  TGCUTS  N2  Ntgscat  RPVCUT = N1 /N2  Figure 4.2: Flowchart showing the K π2 target-scatter bifurcations. Cuts in italics refer to named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used for the background evaluation in the loose box and the tight version for the background evaluation in the tight box. The target quality cuts in the rejection branch are those for class 12.  normalization branches for both the loose and tight background evaluations because inverting the tight photon veto would have resulted in examining a portion of the loose signal region. Figure 4.2 shows a schematic representation of the normalization branch for the loose and tight Kπ2 target-scatter background evaluations. The number of events remaining at the end of this normalization branch (Ntgscat ) was corrected for the contamination due to nrsscat Kπ2 Range-Stack-scatter events, the calculation of which is detailed in Section Section 4.2.1,  ntgscat = Ntgscat − nrsscat .  (4.1)  The resulting number of Kπ2 target-scatter normalization events are shown in Tables 4.4 and 4.5.  83  4.1.3  Kπ2 Target-Scatter Rejection Branch  Monte Carlo simulations [58] of Kπ2 decays have shown that for events where the π + enters the active region of the UTC and Range-Stack, the distribution of π 0 decay photons is much more uniform for events where the π + had scattered in the target than for events where the π + scatter had not occurred. In events where the π + scatter did not occur, the directions of the photons from the π 0 decay were strongly correlated to the direction of the π + . When the π + scattered in the target, this correlation was obscured resulting in this more uniform distribution of photons from the π + scatter events. Due to the variation with polar angle of the total thickness of the photon veto (see Figure 2.13), the photon veto performance was not expected to be uniform in polar angle. As a result, the performance of the photon veto was expected to be different for the Kπ2 target-scatter events than for the Kπ2 -peak events. To measure the rejection of the photon veto for target-scatter events, 12 classes (classes 2-13 in Table 4.1) of Kπ2 target-scatter events were created by applying and inverting various combinations of the target quality cuts, creating samples with varying mixtures of “xy-scatter” and “z -scatter” events. Table 4.1 shows the cuts used to make the 12 classes and Figure 4.2 shows the rejection branch for class 12. Table 4.2 for each of the 12 classes for both the loose photon veto (PV60) and the tight photon veto (PV30). The directional correlation between the π + and the photons from the π 0 was obscured by using the given combinations of cuts to create classes of target-scatter samples, making classes 2-13 suitable for measuring the photon veto rejection of target-scatter events. Class 1 was not suited to measure the photon veto rejection of the target-scatter events since it consisted of events where the directional correlation between the π + and the photons from the π 0 was still present due the application of all TGCUTS and the use of the kinematic region KP2BOX instead of the kinematic signal region PNN2BOX. Class 12, considered to be the richest in z -scatters, was chosen to measure the photon  84  CLASS  Applied and Inverted TGCUTS  1  all TGCUTS, KP2BOX  2 3  CCDPUL · EPIONK CCDPUL · EPIONK, all others  4  CCDPUL, EPIONK, TGZFOOL, EIC, OPSVETO, others  5 6  CCDPUL · EPIONK · CHI567 · VERRNG CCDPUL · EPIONK · CHI567 · VERRNG, all others  7 8  CHI567 · VERRNG CHI567 · VERRNG, all others  9 CCDPUL · EPIONK · CHI567 · VERRNG, KIC, PIGAP, TARGF, TPICS  10 11  B4EKZ B4EKZ, all others  12 13  CCDPUL · EPIONK · B4EKZ CCDPUL · EPIONK · B4EKZ, all others  Table 4.1: Definition of the classes of events used to measure the PV rejection of K π2 targetscatter events in the kinematic signal region. The notation “all others” refers to the remaining TGCUTS not applied in that class also being applied or inverted. All Classes that had either CCDPUL applied or CCDPUL inverted had the three associated safety cuts (CCDBADFIT, CCDBADTIM and CCD31FIB) applied. Class 1 was used to measure the PV rejection of K π2 Range-Stack-scatter events and not K π2 target-scatter events. Since class 1 used events from the Kπ2 -peak instead of the kinematic signal region, all of the TGCUTS could be applied without examining the signal region.  85  Before CLASS  PV60  PV30  Photon Veto  After  Rejection  After  Rejection  1  1/3 2/3  61410 122581  36 106  1706±284 1156±112  13 44  4724±1310 2786±420  2  1/3 2/3  24396 49032  9 21  2711±903 2335±509  3 8  8132±4695 6129±2167  3  1/3 2/3  2776 5495  3 2  925±534 2748±1942  1 1  2776±2776 5495±5495  4  1/3 2/3  4159 8092  3 1  1386±800 8092±8092  0 1  4159±4159 8092±8092  5  1/3 2/3  29899 59871  12 22  2492±719 2721±580  4 8  7475±3737 7484±2646  6  1/3 2/3  4170 8452  3 3  1390±802 2817±1626  1 1  4170±4170 8452±8452  7  1/3 2/3  24574 49636  6 18  4096±1672 2758±650  1 7  24574±24574 7091±2680  8  1/3 2/3  353 644  0 0  353±353 644±644  0 0  353±353 644±644  9  1/3 2/3  23736 47463  10 19  2374±750 2498±573  3 7  7912±4568 6780±2563  10  1/3 2/3  11037 22037  4 10  2759±1379 2204±697  1 2  11037±11037 11019±7791  11  1/3 2/3  45 64  0 0  45±45 64±64  0 0  45±45 64±64  12  1/3 2/3  26317 52621  10 22  2632±832 2392±510  4 8  6579±3289 6578±2325  13  1/3 2/3  3319 6503  3 2  1106±639 3252±2299  1 1  3319±3319 6503±6503  Table 4.2: The rejection of loose photon veto (PV60) and tight photon veto (PV30) rejection for the Kπ2 target-scatter rejection branch. Class 1 was not considered as it used the KP2BOX kinematic box. For each class, the rejection “Rejection” of the photon veto was determined for each of PV30 and PV60. To determine the rejection, the number of events before the photon veto was applied “Before Photon Veto” was divided by the number of events remaining after the photon veto was applied “After”. If zero events remained after the photon veto was applied, the calculation of the rejection assumed that one event remained. Only the classes having sufficient statistics for a meaningful measurement were considered.  86  veto rejection and the other classes having adequate statistics were used to determine the systematic uncertainty in this rejection. The inverted cuts (CCDPUL, EPIONK and B4EKZ) in class 12 rejected events with secondary energy deposits in target kaon fibers consistent with a scattered π + . CCDPUL rejected events where a second pulse having more than 1.25 MeV of energy was found in a kaon fiber. EPIONK rejected events where a fiber classified as both a kaon and pion fiber had more than 1.25 MeV of pion energy. B4EKZ rejected events where the z -position of the decay vertex found by the UTC did not agree with the expected z -position as determined by the total energy deposited in the kaon fibers. Due to the loss of statistics in the rejection branch for the background evaluation in the tight signal region, the rejection of the tight (PV30) photon veto was measured using a rejection branch that used the loose versions of PNN2BOX, TDCUTS and DELCO. In doing this it was assumed that the rejections of the tight photon veto for each of the classes was the same when applying the loose and tight versions of PNN2BOX, TDCUTS and DELCO. The “30” in PV30 refers to the fact that the parameters for the tight photon veto were optimized to accept approximately 30% of the signal-like events. Table 4.3 shows that the rejection of the tight photon veto did not change within statistical uncertainty when applying the tight versions of these cuts as compared to the loose versions of these cuts. When determining the systematic uncertainty of the tight photon veto rejection, the measurements using the tight kinematic box BOXTIGHT were treated as an additional class (called BOXTIGHT) and ended up being used to determine the lower bounds on the systematic uncertainty as shown in Table 4.5. Tables 4.4 and 4.5 summarize the photon veto rejections and other values used in the background evaluation.  87  Setup Cuts  Events Before  Events After  PV30 Rejection  1/3 Data Set All Loose BOXTIGHT DELCO6 TDTIGHT All Tight  26317 19741 22780 19624 12725  4 4 2 2 1  6579±3289 4935±2467 11390±8054 9812±6938 12725±12725  2/3 Data Set All Loose BOXTIGHT DELCO6 TDTIGHT All Tight  52621 39481 45574 39287 25471  8 7 6 6 5  6578±2325 5640±2132 7596±3101 6548±2673 5094±2278  Table 4.3: Rejection of the tight photon veto cut PV30 for class 12 with various combinations of loose and tight versions of the setup cuts for the 1/3 sample and 2/3 sample. The columns “Events Before” and “Events After” show the number of events remaining before and after the application of PV30, respectively. The column “PV30 Rejection” is the rejection of the tight photon veto PV30, determined by taking the quotient of numbers found in the “Events Before” and “Events After” columns.  88  4.1.4  Kπ2 Target-Scatter Background Evaluation  Figure 4.3 shows the momentum distributions for the events remaining at various stages of performing the background estimate using the bifurcation method. The loose Kπ2 target-scatter background was evaluated using the expression loose bgtgscat  nloose tgscat , = RPV60 − 1  (4.2)  where nloose tgscat was the number of events at the end of the loose normalization branch corrected for Kπ2 Range-Stack-scatter contamination as shown in Equation (4.1), and RPV60 was the rejection of the loose photon veto as measured on the loose rejection branch. For the background evaluation in the tight signal region, the loose photon veto PV60 was inverted to avoid examining the signal region and the results of the entire background evaluation were scaled by the ratio of the loose and tight photon veto rejections. The K π2 target-scatter background in the tight signal region was evaluated using the expression tight bgtgscat =  ntight RPV60 tgscat × , RPV60 − 1 RPV30  (4.3)  where ntight tgscat was the number of events at the end of the tight normalization branch corrected for Kπ2 Range-Stack-scatter contamination, RPV60 was the rejection of the loose photon veto, and RPV30 was the rejection of the tight photon veto. Tables 4.4 and 4.5 show the summary of all values used to determine the Kπ2 targetscatter background levels in the loose and tight signal regions, respectively. These background evaluations were scaled to the full data set for both the 1/3 and 2/3 data samples.  89  5  10  4  10  3  10  2  Events/(2.5MeV/c)  Events/(2.5MeV/c)  10  B+D  B  10  10  5  10  4  10  3  10  2  C+D  10  C  1  1 140  160  180  200  220  240  140  Momentum (MeV/c)  160  180  200  220  240  Momentum (MeV/c)  Figure 4.3: The bifurcation cuts used for the loose K π2 target-scatter background evaluation were the photon veto PV60 (CUT1) and the target quality cuts TGCUTS (CUT2). A schematic representation of the phase space of these two cuts after setup cuts to remove other backgrounds have been applied is shown on top with the arrows representing the cuts becoming increasingly loose. In the left plot, the momentum (ptot) distribution of the events remaining in the loose normalization branch after the inversion of the photon veto PV60 (black, regions B+D) and and after the application of all TGCUTS (blue, region B). In the right plot, the momentum (ptot) distribution of the events remaining in the loose rejection branch after class 12 has been chosen (black, regions C+D) and after the photon veto PV60 has been applied (red, region C). The estimate of the background events in the signal region “A” is given by “BC/D”.  90  1/3  2/3  Normalization (Uncorrected) loose Ntgscat  528  1131  Normalization (Corrected for Kπ2 Range-Stack-scatter) nloose tgscat  515.5 ± 23.1 +1.2 −1.1  1107.7 ± 33.8+2.9 −2.8  Rejection RPV60 (CLASS12)  2632±832  2392±510  RPV60 (max.)  4096±1672 (CLASS7)  2758±650 (CLASS7)  RPV60 (min.)  2374±750 (CLASS9)  2204±697 (CLASS10)  2632 ± 832+1464 −258  2392 ± 510+366 −188  0.588 ± 0.188 +0.065 −0.211  0.695 ± 0.150+0.061 −0.094  RPV60  Background (Before correction for Kπ2γ ) loose bgtgscat (uncorrected)  Kπ2γ Background loose bgK π2γ  0.0514 ± 0.0086 +0.0042 −0.0038  0.0757 ± 0.0073 +0.0062 −0.0056  Background loose bgtgscat  0.537 ± 0.188 +0.069 −0.215  0.619 ± 0.150 +0.067 −0.100  Table 4.4: The summary of the loose K π2 target-scatter background evaluation. For the photon loose , the first uncertainty is statistical and veto rejections RPV60 and background evaluations bgtgscat the second uncertainty systematic. The maximum and minimum photon veto rejections are labeled to show which class was used to determine the systematic uncertainties in R PV60 and loose . bgtgscat  91  1/3  2/3  Normalization (Uncorrected) tight Ntgscat  265  512  Normalization (Corrected for Kπ2 Range-Stack-scatter) ntight tgscat  259.1 ± 16.4 +0.6 −0.7  499.7 ± 22.8 +1.1 −1.3  Rejection (Tight Photon Veto) RPV30 (CLASS12)  6579±3289  6578±3289  RPV30 (max.)  8132±4695 (CLASS2)  7484±2646 (CLASS5)  RPV30 (min.)  4935±2467 (BOXTIGHT)  5640±2132 (BOXTIGHT)  6579 ± 3289+1553 −1644  6578 ± 2325+906 −938  RPV30  Rejection (Loose Photon Veto) RPV60  2632 ± 832+1464 −258  2392 ± 510+366 −188  0.118 ± 0.059 +0.075 −0.023  0.114 ± 0.041+0.019 −0.014  Background (Before correction for Kπ2γ ) tight bgtgscat (uncorrected)  Kπ2γ Background tight bgK π2γ  0.0122 ± 0.0038 +0.0010 −0.0010  0.0188 ± 0.0034 +0.0016 −0.0014  Background tight bgtgscat  0.106 ± 0.059 +0.076 −0.024  0.095 ± 0.041 +0.020 −0.016  Table 4.5: The summary of the tight K π2 target-scatter background evaluation. For the photon loose , the first uncertainty is statistical and the veto rejections and background evaluations bg tgscat second uncertainty systematic. The maximum and minimum photon veto rejections are labeled to show which class was used to determine the systematic uncertainties in R PV30 , where the class called “BOXTIGHT” was described in Section 4.1.3. The rejection for the loose photon veto PV60 was taken from Table 4.4  92  4.1.5  Correction to Background for Kπ2γ Contamination  The contamination due to Kπ2γ events was only present in the normalization branch since any Kπ2γ contamination in the rejection branch was heavily suppressed by selecting events which scattered in the target. The Kπ2γ contamination in the normalization branch was enhanced by inverting the photon veto due to the presence of the additional photon in the Kπ2γ events. An upper limit of 30.0 ± 7.5% was found (Section 5.3.4) for the Kπ2γ contamination in the normalization branch using Kπ2 monitor trigger events under the assumption that all monitor trigger events found in the kinematic signal region were Kπ2γ . Based on this upper limit and the fact that the size of the Kπ2γ background estimate was typically about 10% of the Kπ2 target-scatter background estimate or more, it was reasonable to conclude that the Kπ2γ background was being entirely double-counted between its own estimate and the Kπ2 target-scatter background estimate. To correct for this double-counting, the Kπ2γ background was subtracted from the Kπ2 target-scatter background as shown in Tables 4.4 and 4.5.  93  4.2  Kπ2 Range-Stack-Scatter Background  As was described in Section 4.1, there were two types of Kπ2 scatter backgrounds: targetscatter and Range-Stack-scatter. For Kπ2 scatters in the Range-Stack to become a background the momentum of the π + had to have been mis-measured in addition to the target-scatter conditions of the missing the energy lost in the scatter and the missing of the two photons from the π 0 decay. For this reason, the Kπ2 Range-Stack-scatter background was expected to be much smaller than the Kπ2 target-scatter background.  4.2.1  Range-Stack-Scatter Normalization Branch  The number of events Ntgscat left at the end of the target-scatter normalization branch actually consisted of ntgscat target-scatter with contamination due to nrsscat Range-Stackscatter events: Ntgscat = ntgscat + nrsscat .  (4.4)  As will be described in this section, the number of Range-Stack-scatter events nrsscat was determined by studying the target-scatter and the Range-Stack-scatter normalization branches. The most effective cuts against the Kπ2 Range-Stack-scatter background were the Range-Stack track quality cuts RSDEDX and PRRF, the PNN2BOX momentum cut (PNN2PBOX), and the photon veto. The cuts RSDEDX and PRRF were collectively referred to as RSCT. The Range-Stack normalization branch consisted of the same cuts as were used for the target-scatter normalization with RSCT inverted (see Figure 4.4). This sample was heavily contaminated with target-scatter events due to the inefficiency of the RSCT cuts. The acceptance ARSCT of RSCT for Kπ2 target-scatter events falling into the kinematic signal region was expected to be the same as for the signal. Thus the acceptance of these cuts was taken from the Range-Stack kinematic acceptance measurements found in 94  Section 6.2.2, with systematic uncertainties determined in the same way as described in that section. The rejection RRSCT of RSCT was measured using events in the kinematic region known as the Kπ2 range-tail. This region consisted of the Kπ2 -peak momentum region (KP2PBOX), but having a range and energy consistent with the signal region (PNN2REBOX). These were Kπ2 -peak events that did not scatter in the target and thus their momentum was still in the Kπ2 -peak region, but they scattered in the Range-Stack and their range and energy measurements fell into the range and energy of the signal region. Figure 4.4 shows the samples used to measure the rejection of the RSCT cuts and Table 4.6 summarizes the results. With the performance of the RSCT cuts measured, it was possible to use equation 4.4 and the following equation for the number of events Nrsscat remaining at the end of the Kπ2 Range-Stack-scatter normalization to determine the quantity nrsscat : 1 − ARSCT × ntgscat + (RRSCT − 1) × nrsscat = Nrsscat . ARSCT  (4.5)  The nrsscat values which resulted are shown in Table 4.6.  4.2.2  Range-Stack-Scatter Rejection Branch  The photon veto rejection of the Range-Stack-scatter events was assumed to be the same as for the Kπ2 -peak events since the distribution of photons coming from the π 0 for π + particles entering the fiducial region of the detector will be the same for the Kπ2 -peak and the Kπ2 Range-Stack-scatter events. The Kπ2 -peak rejection branch from Figure 4.4 shows the sample used to measure this photon veto rejection RPVCUT−KP2 and Table 4.6 summarizes the results.  95  Loose  Tight  1/3 ARSCT N4 N5 RRSCT Ntgscat Nrsscat nrsscat N2 N3 RPVCUT−KP2  2/3  1/3  2/3  Acceptance of RSCT 0.888 ± 0.001 ± 0.012 0.894 ± 0.002 +0.010 −0.012 642 80 8.03±0.84  Rejection of RSCT 1355 307 192 44 7.06±0.47 6.98±0.97  739 118 6.26±0.53  528 153 12.5 ± 2.4+1.2 −1.2  Normalization Numbers 1131 265 281 66 +2.9 +0.6 23.3 ± 3.5−3.0 5.9 ± 1.7−0.7  512 124 12.3 ± 2.6+1.3 −1.5  61410 36 1706±184  Photon Veto Rejection (Kπ2 peak) 122581 41103 106 10 1156±112 4110±1300  82387 31 2658±477  Background bgrsscat (1/3)  0.0220  bgrsscat (2/3)  0.0303  ± 0.0056 +0.0021 −0.0021 ± 0.0054 +0.0038 −0.0039  0.0043 ± 0.0019 +0.0004 −0.0005  0.0069 ± 0.0019 +0.0007 −0.0009  Table 4.6: The summary of the Kπ2 Range-Stack-scatter background evaluation. For values having two sets of uncertainties, the first is statistical and the second systematic. See Figure 4.4 for definitions of N2 - N5 .  4.2.3  Range-Stack-Scatter Background Evaluation  The final background level due from Kπ2 Range-Stack-scatter events was given by  bgrsscat =  nrsscat RPVCUT−KP2 − 1  ,  (4.6)  where the 1/3 and 2/3 data sets were normalized to the full data set. A summary of the Kπ2 Range-Stack-scatter background evaluation is found in Table 4.6.  96  Kπ2 Range-Stack Scatter Bifurcations VALID TRIG, P2PSCUT, P2TGCUT, P2TGPVCUT, BEAMCUTS, DELCO ∗ , TDCUTS ∗ , TGCUTS Rejection Branch  Normalization Branch  PNN2BOX KP2BOX  KP2PBOX, PNN2REBOX  LAYER14, COS3D, LAYV4, ZFRF, ZUTOUT, UTCQUAL, RNGMOM  LAYER14, COS3D, LAYV4, ZFRF, ZUTOUT, UTCQUAL, RNGMOM  RSDEDX .OR. PRRF  LAYER14, COS3D, LAYV4, ZFRF, ZUTOUT, UTCQUAL, RNGMOM  N1  RSDEDX, PRRF  PV60  N2  PVCUT  Nrsscat  N4  RSDEDX, PRRF  N5  RRSCT = N4 /N5  N3  RPVCUT−KP2 = N2 /N3 Figure 4.4: Flowchart showing the K π2 Range-Stack-scatter bifurcations. Cuts in italics refer to named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used for the background evaluation in the loose box and the tight version for the background evaluation in the tight box. The photon veto rejection R PVCUT−KP2 is the photon veto rejection for events in the Kπ2 peak.  97  4.3  Kπ2γ Background  The background from the decay K + → π + π 0 γ (Kπ2γ ) was expected to be small relative to the K + → π + π 0 (Kπ2 ) background due to (1) the smaller branching ratio and (2) the additional photon against which to veto. The γ emitted in this decay can be produced by either direct emission (DE) or inner bremsstrahlung (IB). The inner bremsstrahlung process dominates these two γ emission processes with a branching ratio of 2.75 ± 0.15 × 10−4 for Tπ+ in the range of 55 MeV to 90 MeV [99]. The direct emission process has a branching ratio of 4.3 ± 0.7 × 10−6 for the same Tπ+ range [99]. Due to the difficulty of isolating a Kπ2γ sample in data, evaluation of this background was performed using a combination of data and Monte Carlo simulation. The simulation was used to predict the number of Kπ2γ events that would be found in the signal region given a certain number of Kπ2 events found in the Kπ2 -peak kinematic box KP2BOX using data. This was done by determining the relative acceptance of the π + from Kπ2 and Kπ2γ events, and determining the extra rejection from the photon veto due to the additional photon.  4.3.1  Relative Rate of π + Acceptance for Kπ2 and Kπ2γ Events  The quantity κ represents the relative rate of acceptance of the π + for Kπ2 and Kπ2γ events given (1) the acceptance A of the events generated in Monte Carlo for passing the trigger conditions and offline cuts, and (2) the branching ratio B of the process. The following expression is used for κ,  κ=  B (Kπ2 ) × A(Kπ2 ) . B (Kπ2γ ) × A(Kπ2γ )  (4.7)  To determine the acceptances A, about 2 × 105 Kπ2 and 5 × 105 Kπ2γ Monte Carlo 98  events were generated. The events were required to pass either the PNN1 or PNN2 trigger conditions in simulation without the online photon veto, L1.1, L1.2 or L0rr2 triggers. The events were then required to pass all available offline cuts including the appropriate kinematic region cuts, but excluding the photon veto. The kinematic boxes used were the Kπ2 -peak kinematic box KP2BOX for the Kπ2 events and the PNN2 kinematic box PNN2BOX for the Kπ2γ events. The numbers of events used to determine the acceptance values are shown in Table 4.7. Also needed to determine κ were the branching ratios of Kπ2 and Kπ2γ . The branching ratio of the direct emission Kπ2γ process was ignored since it is known to be two orders of magnitude smaller than the inner bremsstrahlung Kπ2γ process. The branching ratio for the inner bremsstrahlung Kπ2γ process was only given over the π + energy range of 55 - 90 MeV. To extrapolate this branching ratio to the full energy spectrum available to the decay π + , the Monte Carlo simulation was used to determine this energy spectrum available to the π + as shown in Figure 4.5. Using this spectrum, the effective branching ratio B(Kπ2γ ) for Tπ+ between 0 and 106 MeV was calculated, 106  dN B(Kπ2γ ) =  0 90  dN  × (2.75 ± 0.15) × 10−4 = (1.11 ± 0.06) × 10−3 .  (4.8)  55  The branching ratio for Kπ2 was 0.2092 ± 0.0012 [99]. The resulting values for κ in the loose and tight signal regions are shown in Table 4.7. It was found that κ was fairly insensitive to the applied cuts and was determined mainly from the relative branching ratios.  4.3.2  Additional Photon Veto Rejection  The performance of the photon veto in rejecting events based on the photons from the π0 decay was expected to be the same for both Kπ2 and Kπ2γ events. However due to 99  Figure 4.5: Kinetic energy distribution of the π + from Kπ2γ events in Monte Carlo. Figure reproduced from [55].  the third decay photon, Kπ2γ events had additional photon veto rejection above that for Kπ2 events. A single photon rejection function was created by convolving the spatial and energy distribution of the third photon (from the simulation) with the detector detection inefficiency for a single photon (from data [83]). The additional photon veto rejection factor (Rγ ) was determined by applying this single photon rejection function to the 11,305 Kπ2γ events remaining after the loose offline cuts and 7,409 Kπ2γ events remaining after the tight offline cuts, respectively. The resulting rejection factors, taken from [55], were Rγloose = 5.04 ± 0.10.  (4.9)  Rγtight = 5.11 ± 0.11  (4.10)  and  4.3.3  Kπ2γ Background Evaluation  To evaluate the background level bg from the Kπ2γ process, the number of Kπ2 events (NKπ2 −peak ) found in the Kπ2 -peak kinematic box KP2BOX was used to predict the num100  Number of Events N1 : N2 : N3 : N4 :  Total events produced Passed PNN1 or PNN2 trigger Passed loose offline cuts Passed tight offline cuts  Aloose = N3 /N1 Atight = N4 /N1  NKπ2  NKπ2γ  199986 30625 9776 7608  499973 64217 11035 7409  0.0489±0.0005 0.0221±0.0002 0.0380±0.0004 0.0148±0.0002  κloose κtight  417 ± 24 483 ± 28  Table 4.7: Summary of Monte Carlo simulation events produced and remaining after various cuts have been applied as were used to determine A(K π2 ) and A(Kπ2γ ). The cuts tightened for the tight offline cuts measurement N4 were DELCO for both types of events, and the PNN2BOX for Kπ2γ events. Table reproduced from [55].  ber of Kπ2γ events that would have been found in the signal region. This prediction was done by using the simulation to determine κ, the relative rate of π + acceptance for Kπ2 and Kπ2γ events, and by using a combination of simulation and photon veto data to determine Rγ , the additional rejection due to the third photon from the Kπ2γ events. The following expression was used to evaluate the background level due to the Kπ2γ process:  bg =  NKπ2 −peak , κ · Rγ  (4.11)  where NKπ2 −peak was the value N3 from Figure 4.4 and Table 4.6, and the other terms were determined previously in this section. This background was normalized to the full data set for the 1/3 and 2/3 data sets. Table 4.8 summarizes the results of the Kπ2γ background evaluation.  101  Loose  Tight Normalization  NKπ2 −peak (1/3) NKπ2 −peak (2/3)  36 106  10 31  Relative Rate of π + Acceptance κ  417±24  483±28  Additional PV Rejection Rγ  5.04±0.10  5.11±0.11 Background  bg(1/3) bg(2/3)  +0.0042 0.0514 ± 0.0086 −0.0038  +0.0062 0.0757 ± 0.0073 −0.0056  +0.0010 0.0121 ± 0.0038 −0.0010  +0.0016 0.0188 ± 0.0034 −0.0014  Table 4.8: Summary of the Kπ2γ background evaluation. The relative rate of π + acceptance “κ” for Kπ2 and Kπ2γ events and the additional photon veto rejection “R γ ” are defined in the text. The first uncertainty in the background bg is statistical (from N Kπ2 −peak ) and the second is systematic (from κ and Rγ ). The background bg was normalized to the full data set for the 1/3 and 2/3 data sets. The values for κ and R γ were taken from [55].  102  4.4  Muon Background  The decay K + → µ+ νµ (Kµ2 ) has a very high branching ratio, but due to the µ+ momentum of 236 MeV/c, it would need to lose a large amount of undetected energy to simulate K + → π + ν ν¯ in the PNN2 kinematic region, and therefore it was highly suppressed. The muon background came primarily from the K + → µ+ νγ (Kµ2γ ) and K + → µ+ π 0 ν (Kµ3 ) decays, each of which have a µ+ momentum spectrum that extends into the kinematic signal region. For these decays to be backgrounds, the muon had to be misidentified as a pion and the photon(s) must have escaped detection. The main cuts used to suppress muon backgrounds are the particle identification cuts in the Range-Stack and the photon veto. The bifurcation cuts used for the muon background evaluation were the loose π + → µ+ → e+ decay sequence cuts TDLOOSE (CUT1) and the range-momentum consistency cut RNGMOM (CUT2). The RNGMOM cut required that the measured range of the charged decay particle was consistent with that predicted for a π + given the measured momentum.  4.4.1  Muon Normalization Branch  The muon normalization branch was created by inverting the loose π + → µ+ → e+ decay sequence cuts TDLOOSE (CUT1) and applying all other analysis cuts. The loose version of the π + → µ+ → e+ decay sequence cuts was inverted in the normalization branches for both the loose and tight background evaluations since inverting the tight version TDTIGHT would have resulted in examining a portion of the loose signal region. Inverting TDLOOSE resulted in a sample rich in K + decays with muons since these cuts were designed to strongly suppress events having muon-like signatures in detector subsystems outside the target. Figure 4.6 shows a schematic representation of the normalization branch used for the loose and tight muon background evaluations.  103  4.4.2  Muon Rejection Branch  To create the sample upon which the rejection of TDLOOSE (CUT1) was measured, RNGMOM (CUT2) was inverted. Setup cuts were applied to this sample to remove Kπ2 decays and beam backgrounds. Figure 4.6 shows a schematic representation of the rejection branch used for the loose and tight muon background evaluations. Muon Bifurcations VALID TRIG, P2TGCUT, P2PSCUT, BEAMCUTS, DELCO ∗ , P2TGPVCUT  Normalization Branch  Rejection Branch  T DLOOSE  RNGMOM  PNN2BOX∗ , KINCUTS, TGCUTS, PVCUT∗  PNN2BOX∗ , KINCUTS (excluding RNGMOM), TGCUTS (excluding CHI567 & CHI5MAX), PVPNN1 N1  Nmuon  TDCUTS ∗  N2  RT DCU T S = N1 /N2  Figure 4.6: Flowchart showing the muon bifurcation branches. Cuts in italics refer to named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used for the background evaluation in the loose box and the tight version for the background evaluation in the tight box.  4.4.3  Muon Background Evaluation  loose The loose muon background bgmuon was evaluated using the expression  loose bgmuon =  loose Nmuon  RT DLOOSE − 1 104  ,  (4.12)  loose where Nmuon was the number of events at the end of the loose muon normalization branch  and RT DLOOSE was the rejection of TDLOOSE. This background was scaled to the full data set from the 1/3 or 2/3 data samples. For the background evaluation in the tight signal region, TDLOOSE was inverted in order to not examine the signal region and the results of the entire background evaluation scaled by the ratio of the rejections of TDLOOSE and TDTIGHT. The tight muon tight background bgmuon was evaluated using the expression  tight bgmuon  =  tight Nbeam  RT DLOOSE − 1  ×  RT DLOOSE , RT DT IGHT  (4.13)  tight where Nmuon was the number of events at the end of the tight muon normalization branch,  and RT DLOOSE and RT DT IGHT were the rejection of TDLOOSE and TDTIGHT, respectively. This background estimate was scaled to the full data set for the 1/3 and 2/3 data samples. Table 4.9 summarizes the muon background evaluations.  105  Muon Background Summary 1/3  2/3 Normalization  loose Nmuon  0  1  tight Nmuon  0  1  Rejection (TDLOOSE ) N1 N2 RT DLOOSE  10328 84 123.0 ± 13.4  20488 154 133.0 ± 10.7  9277 17 546 ± 132  18411 45 409 ± 61  Rejection (TDTIGHT )  N1 N2 RT DT IGHT  Background loose bgmuon tight bgmuon  0.0246 ± 0.0246 0.0114 ± 0.0114  0.0055 ± 0.0055 0.0037 ± 0.0037  Table 4.9: Summary of the muon background evaluation. See Figure 4.6 for definitions of N 1 and N2 for RT DLOOSE and RT DT IGHT .  106  4.5  Ke4 Background  The K + → π + π − e+ ν (Ke4 ) decay has a branching ratio of approximately 4 × 10−5 and could be a serious background due to the lack of photons to veto and a substantial fraction of the decay π + phase-space falling in the kinematic signal region. The target event display of a typical Ke4 event is shown in Figure 4.7.  6 4 2 0 -2 -4 -6 -6  -4  -2  0  2  4  6  Figure 4.7: The target event display of a typical K e4 event. Red squares indicate the K + fibers, blue squares the π + fibers, and green the opposite-side-π + and γ fibers. The two green tracks to the right of the decay vertex (blue circle) show the tracks of the π − and e+ . The top number shown in each fiber is the time of the hit and the bottom number is the energy deposited.  This background was most problematic when the low-energy π − and e+ did not leave the target and escaped detection by depositing all their energy in kaon fibers or insensitive 107  materials. For a combined total kinetic energy (T2 ) of the π − and e+ below 100 MeV, the π + momentum peaked at around 160 MeV/c (see Figure 4.8), which was in the loose kinematic signal region. As is described in Appendix D.4, the tight kinematic signal region was chosen to kinematically exclude most of the Ke4 background while minimizing  Ptot (MeV)  the acceptance loss of signal events. ENTRIES  240  6595  220 200 180 160 140 120 100  0  20  40  60  80  100  T2=Tpi- + Te+ (MeV) Figure 4.8: Total kinetic energy (T 2 ) of the π − and the e+ versus π + momentum (Ptot ) for Monte Carlo Ke4 events that pass the trigger conditions. Figure reproduced from [55].  The most effective cuts in finding the additional energy deposited in the target from the low-energy π − and e+ were the target sub-system of the photon veto (TGPV), the opposite side pion-fiber veto (OPSVETO) and the pion-energy cut from target-CCD pulse fitting (CCDPUL). After inverting the combination of OPSVETO and TGPV to create the normalization branch, there remained no set of cuts with which to create a Ke4 -rich data sample to measure the rejection of OPSVETO and TGPV so Monte Carlo simulation was used.  4.5.1  Normalization Branch Using Data  The Ke4 normalization branch used data and was created by inverting the combination of the TGPV and OPSVETO cuts (CUT1), such that an event failing either of these cuts 108  remained in the normalization branch. All other analysis cuts other than TGGEO were applied. TGGEO was highly correlated with the inverted cuts and was thus excluded from the normalization branch. Figure 4.9 shows all the cuts applied in the normalization branch. Ke4 Normalization Branch VALID TRIG, P2TGCUT, P2PSCUT, KINCUTS, PNN2BOX∗ , BEAMCUTS (excluding TGGEO), DELCO∗ , TDCUTS ∗ , PVCUT∗ (excluding TGPV)  TGPV · OPSVETO  TGCUTS (excluding OPSVETO)  NKe4 Figure 4.9: Flowchart showing the K e4 normalization branch. Cuts in italics refer to named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used for the background evaluation in the loose box and the tight version for the background evaluation in the tight box. The loose version of TGPV was inverted for both the loose and tight normalization branches. Table 4.12 shows the number of events left at the end of this normalization branch.  4.5.2  Rejection Branch Using Monte Carlo  Monte Carlo simulation was used to determine the total rejection of the combination of the TGPV and OPSVETO cuts for Ke4 events. A total of approximately 2 × 108 Ke4 events with T2 < 50 MeV were produced, where T2 was the combined kinetic energy of the π − and the e+ from the Ke4 decay. This constraint on T2 was used to enhance the statistical power of the simulation by concentrating on the most troublesome phase space of the Ke4 decay. Before measuring the rejection of the combination of the TGPV and OPSVETO cuts, these events were required to pass all possible cuts in simulation. The Monte Carlo simulation used two different measures of energy deposited in target 109  fibers to reproduce the effects of OPSVETO, TGPV and CCDPUL. The first quantity Txtg was the total energy deposited in all fibers not classified as kaon or pion fibers by the reconstruction algorithm. This represented the energy available to the TGPV and OPSVETO cuts. The second quantity Ehide was the total energy deposited in kaon fibers by the π − and e+ . This represented the energy available to the CCDPUL cut. The main source of uncertainty in the simulation for this study was from the energy of absorbed π − particles. The π − absorption was modeled based on experimental data of stopped π − in the Range Stack [97]. The absorption of the π − by carbon dominated and the excited nucleus de-excited by gamma emission or particle evaporation. The resulting gammas and neutrons were likely to escape the target and were not simulated in the Monte Carlo simulation. Since the charged particles resulting from particle evaporation had a short range, they were treated in Monte Carlo as having deposited all their energy in the same fiber as the original π − absorption. Figure 4.10 shows the additional absorption energy of the π − which was determined experimentally by taking the difference between the measured absorption energy in the Range Stack and the π − energy as determined from its momentum. Due to the resolution of the two measurements used to determine the total π − absorption energy, it was expected that a small number of events would have a negative energy. This distribution was sampled in Monte Carlo to determine the total π − absorption energy with energies below zero being treated as zero. Since the correlation between the energy deposited by particles in the simulation and in the data was not precisely known, the rejection thresholds for the quantities Txtg and Ehide were independently varied over a range of energies as shown in Tables 4.10 and 4.11. The rejection of the combined TGPV and OPSVETO cuts (represented by the quantity Txtg ) was determined based on events above the Ehide threshold. The central value for this rejection was used for the background estimation while the variations based on Txtg and Ehide provided the systematic uncertainty. The Ke4 background summary table (Table 4.12) shows the values for the rejection as used in the background estimate. 110  Probability/5 MeV  0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0  -50  0  50  100  150  Energy of stopped pi- (MeV)  Figure 4.10: Observable absorption energy of π − stopped in the Range Stack as used by the Monte Carlo simulation. This is the additional energy observed from a stopped π − beyond that expected from the energy of the π − as determined by momentum. Figure reproduced from [37].  4.5.3  Ke4 Background Evaluation  The Ke4 background was evaluated using the expression  bgKe4 =  NKe4 RTGPV·OPSVETO − 1  ,  (4.14)  where NKe4 was the number of events at the end of the Ke4 normalization branch and RTGPV·OPSVETO was the rejection of TGPV and OPSVETO as measured using Monte Carlo. The results are summarized in Table 4.12.  111  Txtg < 0.6 MeV Ehide Ehide Ehide Ehide  < 1.6 MeV < 2.5 MeV < 4.0 MeV < 10.0 MeV  Txtg < 1.2 MeV  Txtg < 1.8 MeV  2250/66 = 34 2250/86 = 26 2250/98 = 23 6769/100 = 68 6769/129 = 52 6769/149 = 45 34992/202 = 173 34992/288 = 122 34992/335 = 104 97100/627 = 155 97100/888 = 109 97100/1105 = 88  Table 4.10: Rejection of RTGPV·OPSVETO as a function of Ehide for the loose rejection branch. In a given cell, the first number shows the number of events remaining after the E hide condition was applied, and the second number shows the number of events remaining after the T xtg condition was applied. The Txtg condition represented the cuts TGPV and OPSVETO within the simulation. Table reproduced from [55].  Txtg < 0.6 MeV Ehide Ehide Ehide Ehide  Txtg < 1.2 MeV  Txtg < 1.8 MeV  < 1.6 389/18 = 22 389/20 = 19 389/22 = 18 < 2.5 2282/23 = 99 2282/26 = 88 2282/31 = 74 < 4.0 15105/43 = 351 15105/53 = 285 15105/65 = 232 < 10.0 37174/160 = 232 37174/206 = 180 37174/269 = 138  Table 4.11: Rejection of RTGPV·OPSVETO as a function of Ehide for the tight rejection branch. In a given cell, the first number shows the number of events remaining after the E hide condition was applied, and the second number shows the number of events remaining after the T xtg condition was applied. The Txtg condition represented the cuts TGPV and OPSVETO within the simulation. Table reproduced from [55].  112  Loose 1/3  Tight  2/3  1/3  2/3  Normalization NKe4  4  6  1  0  Rejection RTGPV·OPSVETO  52+121 −29  88+263 −70 Background  bgKe4 (1/3) bgKe4 (2/3)  0.235 ± 0.118+0.310 −0.166  0.176 ± 0.072+0.233 −0.124  0.034 ± 0.034+0.142 −0.026  0.017 ± 0.017+0.071 −0.013  Table 4.12: The values used for the K e4 background evaluation are shown with the results from the 1/3 and 2/3 data samples normalized to the full data set. The first uncertainty of bg Ke4 is statistical from NKe4 and the second uncertainty is the systematic uncertainty determined from the range of values found for RTGPV·OPSVETO . The values for RTGPV·OPSVETO were taken from [55].  113  Upstream Photon Veto  π BWPC1  I Counter  BWPC2  K Decay in flight Cerenkov Counter  BeO Degrader  Active Degrader  B4 Counter  Target  π  π Scattering  Figure 4.11: A schematic of the single-beam background processes. The top figure shows a kaon decay-in-flight and the bottom a beam pion scattering into the fiducial region of the detector. Figure reproduced from [37].  4.6  Single-Beam Background  The beam backgrounds were a collection of backgrounds due to beam pions scattering into the fiducial region of the detector or beam kaons decaying in flight. For events having only a single beam particle, the delayed coincidence cuts (DELC3 for the loose box and DELC6 for the tight box) removed all properly reconstructed events of this type. DELC3 (DELC6) required that the kaon decayed at least 3 ns (6 ns) after entering the target. Both the scattered pion and the kaon decay-in-flight failed this minimum time difference condition between outgoing particles from the target and incoming particles from the beam. These types of events could imitate K + → π + ν ν¯ if the event was poorly reconstructed such that the beam and/or RS timing wasn’t measured correctly and the event passed the delayed coincidence requirements. This type of background was referred to as single-beam background and schematic representations of these two single-beam background processes are shown in Figure 4.11. A kaon decay-in-flight looked like a regular kaon-stop to the beam-line detectors and the decay products looked like any other kaon stop decay to rest of the detector. As a result the cuts designed to suppress stopped-kaon background were also effective against 114  decay-in-flight kaon events. The kinematic box provided additional suppression due to the Lorentz boost of the π + resulting from the K + decaying in flight. ˇ The scattered beam pions were heavily suppressed by the high efficiency Cerenkov and B4 counters that could distinguish between K + and π + in the beam. Scattered beam pions that were not suppressed by the beam-line counters and delayed coincidence cuts could simulate K + → π + ν ν¯ when scattering into the fiducial region of the detector if the kinematics of the π + were the same as a signal π + since there were no photons against which to veto.  4.6.1  Single-Beam Normalization Branch  For the single-beam background evaluation, the normalization branch bifurcation cut (CUT1) was the delayed coincidence cut DELC3. To avoid examining the signal region for the background evaluation in the tight signal region, only the loose cut DELC3 was inverted and the results of the background evaluation scaled to take into account the additional rejection of the tight version (DELC6) of the cut. Figure 4.12 shows the normalization branch and Table 4.13 the resulting event counts.  4.6.2  Single-Beam Rejection Branch  To create the sample used to measure the rejection of DELC3, a loose version of the B4DEDX cut was inverted. Inverting this cut demanded that the beam particle was a pion by requiring that the energy b4abm atc in the B4 at beam time was less than 1.0 MeV. Figure 4.12 shows the rejection branch and Table 4.13 the resulting event counts. It was assumed that the rejections of the delayed coincidence cuts DELC3 and DELC6 were the same for scattered beam pions as for decay-in-flight kaons. To preserve statistics in the single-beam background study, the very loose photon veto PV90 was applied in the rejection branch. The Active Degrader and target photon veto subsystems were excluded  115  from this photon veto as indicated by the notation “noTG, noAD”.  4.6.3  Single-Beam Background Evaluation  loose The loose single-beam background bg1bm was written as  loose bg1bm =  loose N1bm , RDELC3 − 1  (4.15)  loose where N1bm was the number of events at the end of the loose single-beam normalization  branch and RDELC3 was the rejection of DELC3. These values are summarized in Table 4.13. This background was scaled to the full data set for the 1/3 and 2/3 data samples. The expression used to evaluate the tight single-beam background was tight bg1bm =  tight RDELC3 N1bm × , RDELC3 − 1 RDELC6  (4.16)  tight where N1bm was the number of events at the end of the tight single-beam normalization  branch, RDELC3 and RDELC6 were the rejections of DELC3 and DELC6, respectively as measured on the loose rejection branch. The scaling by the ratios of the rejections of DELC3 and DELC6 was required because DELC3 was inverted in the tight single-beam normalization branch in order to not look in the loose signal region and this scaling results in a normalization value that would have been measured had DELC6 been inverted. This expression simplifies for the 1/3 data set since the rejection of DELC3 was used for both DELC3 and DELC6 due to statistical limitations. Table 4.13 summarizes the results of the single-beam background evaluation.  116  1/3  2/3 Normalization  loose N1bm  1  0  tight N1bm  1  0  Rejection (DELC3) N1 N2 RDELC3  6483 0 6483 ± 6483  12850 2 6425 ± 4543  3913 0 3913 ± 3913  7780 1 7780 ± 7780  Rejection (DELC6)  N1 N2 RDELC6  Background loose bg1bm tight bg1bm  0.00046 ± 0.00046 0.00023 ± 0.00023 0.00046 ± 0.00046 0.00019 ± 0.00019  Table 4.13: Summary of the single-beam background evaluation. Note that for the 1/3 data set, the rejection of DELC6 was lower than DELC3 due to statistical limitations. For this reason, tight the rejection of DELC3 from the 1/3 data set was used when determining bg 1bm . See Figure 4.12 for definitions of N1 and N2 for RDELC3 and RDELC6 . Table reproduced from [56].  117  Single-Beam Bifurcations VALID TRIG, P2TGCUT, B4CCD, B4TRS, BWTRS, UPVTRS, RVTRS, B4ETCON, TGGEO, TGQUALT, TIMCON, TGTCON, PNN2BOX∗ , TDCUTS ∗ Normalization Branch  Rejection Branch  DELC3  b4abm atc < 1.0  B4DEDX, CKTRS, CKTAIL, CPITRS, CPITAIL, PVCUT∗ , TGCUTS  EPITG, TGER, TARGF, TIC, DTGTTP, RTDIF, EPIMAXK, DRP, PHIVTX, EIC, OPSVETO, KIC, TGEDGE, TGZFOOL, TGDEDX, TGLIKE, TGB4, PIGAP, PV90(noTG,noAD) N1  N1bm DELCO∗  N2  RDELCO∗ = N1 /N2  Figure 4.12: Flowchart showing the single-beam bifurcation branches. Cuts in italics refer to named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used for the background evaluation in the loose box and the tight version for the background evaluation in the tight box.  118  π K  Decay in flight  K  Stop  π π  Scattering Stop  K  Figure 4.13: A schematic of the double-beam background processes. The top figure shows the KK class (kaon-kaon) and the bottom figure the KP class (kaon-pion). Figure reproduced from [37].  4.7  Double-Beam Background  Section 4.6 discussed how a single beam pion or decay-in-flight kaon could become singlebeam background. The other way that a scattered beam pion or decay-in-flight kaon could become background was if it was missed by the beam-line detectors and an earlier beam kaon satisfied the kaon requirements in the beam-line detectors. For this type of event to become background, the decay products of the early kaon had to be missed and the scattered beam pion or decay-in-flight kaon had to be missed by the beam-line detectors. This type of background was referred to as double-beam background and was subdivided into two classes. The first class, KK, was an early kaon followed by a decay-in-flight kaon. The second class, KP, was an early kaon followed by a scattered beam pion. Schematic representations of these two double-beam background processes are shown in Figure 4.13. ˇ The discrimination between pions and kaons by the Cerenkov counters allowed the two classes of double-beam background, KK and KP, to be evaluated separately.  4.7.1  Double-Beam Normalization Branch  The bifurcation cut (CUT1KK ) for the KK double-beam normalization branch, used to select events with an early kaon followed by a decay-in-flight kaon, was the collection of  119  cuts CKTRS, CKTAIL and BWTRS. For the KP double-beam normalization branch, the bifurcation cut (CUT1KP ) used to select events with an early kaon followed by a scattered beam pion, was the collection of cuts CPITRS, CPITAIL and BWTRS. The Active Degrader (ADPV) and target photon veto (TGPV) subsystems were excluded from the initial application of the photon veto in these normalization branches as indicated by the notation “noTG, noAD”. Both of these subsystems were applied as part of the secondary bifurcations of these normalization branches. Due to low statistics, each of the normalization branches were bifurcated into two branches containing uncorrelated sets of cuts. The notation is shown for the KK class, but this can be replaced with KP for the early kaon followed by a scattered beam pion class of double-beam background. The first branch consisted of B4TRS, B4CCD, TGGEO, B4DEDX, the target photon veto TGPV, and the target-quality cuts TGCUTS excluding EPITG, DTGTTP, RTDIF, DRP, EIC, TIC, TGER and KIC. The number of events remaining after these cuts were applied was called nKK . The second branch measured the rejection rKK of the active degrader ADPV. The final normalization NKK was then defined as NKK =  nKK . rKK  (4.17)  Figure 4.14 shows the schematic representation of the double-beam normalization branches and the results for the measurements on this normalization branch are shown in Tables 4.14 and 4.15.  4.7.2  Double-Beam Rejection Branch  For both the KK and KP classes of double-beam background, the bifurcation cut used to create the rejection sample (CUT2) was the pair of cuts B4TRS and B4CCD. It was assumed that the B4 counters used for CUT2 were uncorrelated with the beam wire ˇ chambers and Cerenkov counters used for CUT1 due to the multiple scattering that 120  occurred in the degraders between the two sets of systems. Inverting CUT2 selected events that had activity in the B4 counters at trs , the time of the charged track in the RangeStack. The sample of events created by inverting CUT2 was contaminated by decays from stopped kaons where one of the decay products was directed upstream and detected by the B4 counters. To correctly measure the rejections of CUT1 KK and CUT1KP , a sample of pure double-beam events was desired so KPIGAP was applied after CUT2 was inverted to remove the contamination due to stopped kaon decays in the rejection sample. The cut KPIGAP required that kaon and pion clusters in the target were spatially disconnected, thus applying this cut selected events that were geometrically inconsistent with the decay of a single stopped K + . The rejection of CUT1KK and CUT1KP for this subset of spatially disconnected double-beam events was expected to be consistent with the rejection of these cuts for all double-beam events since there was no significant correlation of these cuts with the proximity of the double-beam particles. ˇ To select the KK class of events in the rejection branch, the Cerenkov pion-veto cuts CPITRS and CPITAIL were applied. This sample of KK events was further purified by requiring that the energy deposited in the B4 by the second beam particle at RangeStack time b4ars was consistent with the energy deposited by a kaon (between 1.1 and 5.0 MeV). ˇ To select KP events in the rejection branch, the Cerenkov kaon-veto cuts CKTRS and CKTAIL were applied. This sample of KP events was further purified by requiring that b4ars was consistent with the energy deposited by a pion (less than 1.1 MeV). After each of the KK and KP rejection branch samples were purified, the rejections of CUT1KK and CUT1KP were measured. Figure 4.15 shows the schematic representation of the double-beam rejection branches and the results for the measurements on this rejection branch are shown in Tables 4.14 and 4.15.  121  Double-Beam Normalization Branches VALID TRIG, P2TGCUT, PNN2BOX∗ , KINCUTS, DELCO∗ , TGQUALT, TIMCON, UPVTRS, RVTRS, TGTCON, B4ETCON  PVCUT∗ (noTG,noAD), TDCUTS ∗ , EPITG, DTGTTP, RTDIF, DRP, EIC, TIC, TGER, KIC  KK Branch  KP Branch (early runs only)  CPITRS, CPITAIL  CKTRS, CKTAIL  CKTRS · CKTAIL · BWTRS  CPITRS · CPITAIL · BWTRS  N1 B4TRS, B4CCD, TGGEO, B4DEDX, TGPV∗ , TGCUTS (excluding EPITG, DTGTTP, RTDIF, DRP, EIC, TIC, TGER, KIC)  ADPV  N3 B4TRS, B4CCD, TGGEO, B4DEDX, TGPV∗ , TGCUTS (excluding EPITG, DTGTTP, RTDIF, DRP, EIC, TIC, TGER, KIC)  N2  nKK  ADPV  N4  nKP  rKK = N1 /N2  rKP = N3 /N4  Figure 4.14: Flowchart showing the double-beam normalization branches Cuts in italics refer to named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used for the background evaluation in the loose box and the tight version for the background evaluation in the tight box.  4.7.3  Double-Beam Background Evaluation  The KP background evaluation had an additional complication due to the addition of ˇ the Cerenkov pion-veto to the PNN2 trigger part of the way through data collection (see Section 2.7.1). Due to the lack of statistics for KP -type events after this trigger change, the KP background was evaluated using events before the trigger change and scaled to represent the entire run. This scaling was done using the inverse of the fraction of the  122  before total KBlive before the trigger change fKB , live  before fscale = 1/fKB live  = 1/0.394  (4.18)  = 2.54, before where the value for fKB was taken from [72]. To verify the method of applying the live  scaling factor to a subset of the run for the KP double-beam background evaluation, two additional measurements were performed to show that these backgrounds were consistent before and after the trigger change, as detailed in [72]. The first was to use only the PNN1 trigger to evaluate the KP double-beam background before and after the PNN2 trigger change. The second was to evaluate the KK double-beam before and after the PNN2 trigger change. When scaling each of these measurements to the full run, each of these measurements found the backgrounds to be consistent before and after the PNN2 trigger change. Figure 4.15 shows the schematic representation of the double-beam bifurcations and Table 4.15 the results for the measurements on this rejection branch. The KK double-beam background bgKK was written as  bgKK =  NKK , RKK − 1  (4.19)  where NKK was the number of events left at the end of the KK class double-beam normalization branch and RKK was the rejection of CUT1KK (BWTRS, CKTRS and CKTAIL). The KP double-beam background bgKP was written as  bgKP = fscale ×  NKP , RKP − 1  (4.20)  where NKP was the number of events left at the end of the KP class double-beam normalization branch, RKP was the rejection of CUT1KP (BWTRS, CPITRS and CPITAIL), 123  Loose 1/3  Tight 2/3  1/3  2/3  Normalization nKK  0  0  0  0  N1 N2 rKK  1363 182 7.49±0.52  2699 325 8.30±0.43  212 22 9.64±1.94  462 48 9.62±1.32  NKK  0.134±0.134  0.120±0.120  0.104±0.104  0.104±0.104  Rejection N5 N6 RKK  790 7 113±43  1576 4 394±197  148 1 148±148  269 1 269±269  Background 1/3  2/3  loose bgKK  0.00359±0.00359  0.00046±0.00046  tight bgKK  0.00212±0.00212  0.00058±0.00058  Table 4.14: The summary of the double-beam KK -background evaluation. See Figures 4.14 and 4.15 for definitions of the normalization and rejection quantities, respectively. Table reproduced from [56].  and fscale was the scaling factor applied so that the background evaluation as measured before the trigger change represented the entire run. These expressions describe both the loose and tight background evaluations for both double-beam classes.  124  Loose 1/3  Tight 2/3  1/3  2/3  Normalization nKP  0  0  0  0  N3 N4 rKP  2289 221 10.36±0.66  4435 464 9.56±0.42  494 36 13.72±2.20  939 92 10.21±1.01  NKP  0.097±0.097  0.105±0.105  0.073±0.073  0.098±0.098  Rejection N7 N8 RKP  1179 2 590±417  2467 4 617±308  261 0 261±261  541 0 541±541  Background 1/3  2/3  loose bgKP  0.00126±0.00126  0.00065±0.00065  tight bgKP  0.00095±0.00095  0.00069±0.00069  Table 4.15: The summary of the double-beam KP-background evaluation. See Figures 4.14 and 4.15 for definitions of the normalization and rejection quantities, respectively. Table reproduced from [56].  Loose  Tight  1/3  2/3  1/3  2/3  bgKK  0.00359±0.00359  0.00046±0.00046  0.00212±0.00212  0.00058±0.00058  bgKP  0.00126±0.00126  0.00065±0.00065  0.00095±0.00095  0.00069±0.00069  bg2bm (total)  0.00485±0.00380  0.00111±0.00080  0.00307±0.00232  0.00127±0.00090  Table 4.16: The summary of the total expected double-beam background.  125  Double-Beam Rejection Branches VALID TRIG, P2TGCUT, PNN2BOX∗ , DELCO∗ , TDCUTS ∗ , RSDEDX, RNGMOM  TGQUALT, TIMCON, B4DEDX, UPVTRS, RVTRS, B4ETCON, PVCUT ∗ (noTG,noAD), CHI5MAX, CHI567, TGER, TGZFOOL  B4TRS · B4CCD, KPIGAP  KK Branch  KP Branch (early runs only)  CPITRS, CPITAIL  CKTRS, CKTAIL  1.1 < b4ars < 5.0  b4ars < 1.1  N5  N7  CKTRS, CKTAIL, BWTRS  CPITRS, CPITAIL, BWTRS  N6  N8  RKK = N5 /N6  RKP = N7 /N8  Figure 4.15: Flowchart showing the double-beam rejection branches. Cuts in italics refer to named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used for the background evaluation in the loose box and the tight version for the background evaluation in the tight box.  126  π Charge Exchange  K  KL  l  ν  Figure 4.16: A schematic of the problematic semileptonic charge-exchange background processes KL0 → π + e− νe and KL0 → π + µ− νµ . Figure reproduced from [37].  4.8  Charge Exchange Background  The charge-exchange (CEX) background came from the charge exchange interaction K + n → K 0 p. The K 0 could be either the shorter lifetime KS0 state (0.1 ns) or the longer ¯ 0 , it is also possible for lifetime KL0 state (51 ns). Through oscillation of the K 0 to K hyperon (a strange baryon) production to become another charge exchange background source. This mode is discussed in more detail in Section 4.8.4. The main KS0 decay channel that could simulate K + → π + ν ν¯ was KS0 → π + π − . Since it is such a prompt decay, it was very heavily suppressed by the delayed coincidence cut DELCO and any contributions to background from these KS0 decays were determined as part of the single-beam background estimation (see Section 4.6). The more problematic KL0 decay modes were the semileptonic decays KL0 → π + e− νe and KL0 → π + µ− νµ with branching ratios of 40.56% and 27.05%, respectively [99], where these branching ratios are for the sum of the possible charge states for each decay mode. The kinematics of the decay π + from these decay modes fell inside the kinematic signal region used for this analysis. A schematic representation of these semileptonic decays is shown in Figure 4.16. The two main methods of suppressing the problematic KL0 decay modes were (1) detection of the excess energy deposited in the target from the negatively charged lepton, and (2) identification of gaps between the incoming kaon and outgoing pion tracks in the target. Due to the longer life (51 ns) and non-interacting nature of the KL0 , there was a gap between where the kaon track ended and where the outgoing (daughter) pion track 127  began.  4.8.1  Method Used to Evaluate CEX Background  To effectively use the bifurcation method on data, two uncorrelated sets of inverted cuts need to be able to isolate the background in question. The only cut that was capable of isolating this background was TARGF, which identified gaps greater than one fiber width (0.6 cm) between the kaon and pion fiber clusters. Any cuts that detected the excess energy deposited in the target suppressed many different backgrounds and could not have been inverted to isolate only the CEX background. The background evaluation for CEX varied slightly from the bifurcation method used for most of the other backgrounds (see Section 3.2.2), but it also relied on geometric arguments to determine the number of background events without examining the signal region. A schematic representation of this background evaluation is shown in Figure 4.17. The normalization branch was created by applying KPIGAP (a tighter version of TARGF) using data with NKPIGAP events left at the end of the branch. Using Monte Carlo, the normalization branch was reproduced using the cuts available to Monte Carlo with MKPIGAP events remaining at the end of the branch. A similar branch was created in Monte Carlo using the TARGF cut instead of KPIGAP, with MTARGF events remaining after TARGF has been applied. Measurement of the number of events NTARGF that would remain in the data study if TARGF were applied instead of KPIGAP was forbidden since it would have involved examining the signal region. Instead geometric arguments were used to show that the following ratios were equal for the number of events remaining after each of TARGF or KPIGAP were applied in the data and Monte Carlo studies: NTARGF MTARGF = . NKPIGAP MKPIGAP  (4.21)  To retain statistics in measuring NKPIGAP , two groups of cuts were excluded from this 128  Data  Monte Carlo PNN2BOX∗ ,  VALID TRIG, P2TGCUT, KINCUTS, BEAMCUTS, DELCO2, TDCUTS ∗ , PVCUT∗ (noTG)  Monte Carlo Setup Cuts  TIC, EIC, TGZFOOL, EPITG, EPIMAXK, EPIONK, TIMKF, KIC, NPITG, TGER, DTGTTP, RTDIF, DRP, TGKTIM, TGEDGE, TGDEDX, TGENR, PIGAP, TGLIKE, TGB4, PHIVTX, TPICS  TARGF  NTARGF  Monte Carlo Target Quality Cuts  KPIGAP  TARGF  MTARGF  NKPIGAP  KPIGAP  MKPIGAP  DELCO∗ , B4EKZ, TGPV, OPSVETO, CCDPUL  DELCO∗ , B4EKZ, TGPV, OPSVETO, CCDPUL NEXCL  MEXCL  CHI567, CHI5MAX, VERRNG, ANGLI, TGFITALLK, CCDBADTIM, CCDBADFIT, CCD31FIB  bgCEX  Figure 4.17: Flowchart showing the charge-exchange data and Monte Carlo branches. Cuts in italics refer to named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used for the background evaluation in the loose box and the tight version for the background evaluation in the tight box. The dashed boxes indicate a branch inaccessible to data due to being in the signal region. Instead the measurements were performed on an equivalent branch using Monte Carlo.  normalization branch. The first group consisted of cuts where CEX events were expected to have rejection above the acceptance loss of signal-like events: DELCO, B4EKZ, TGPV, OPSVETO and CCDPUL. The rejection of these cuts was measured on the MTARGF events remaining after TARGF was applied in the Monte Carlo study:  REXCL =  MTARGF . MEXCL  (4.22)  The second group consisted of cuts where the CEX events were expected to behave like 129  signal events and thus the measured acceptance losses of these cuts (Section 6.1.3) were used. These cuts were CHI567, CHI5MAX, VERRNG, ANGLI, TGFITALLK, CCDBADTIM, CCDBADFIT and CCD31FIB, and their combined acceptance will be referred to as Aloss . The background level bgCEX in the signal region was determined by correcting the calculated number of events NTARGF for the performance of the cuts excluded from the normalization branch, bgCEX =  Aloss × NTARGF . REXCL  (4.23)  Combining the expressions from Equations (4.21), (4.22) and (4.23) the expression for the background can be written as  bgCEX = NKPIGAP ×  4.8.2  MEXCL × Aloss . MKPIGAP  (4.24)  The CEX Monte Carlo Study  The branches of the CEX Monte Carlo study were evaluated using approximately 3.42 × 108 KL0 → π + µ− ν¯µ events in Monte Carlo. These events were generated with production points, momentum distributions, and corresponding B4 and kaon fiber information from KS0 → π + π − data [92]. Figure 4.17 shows the cuts applied to measure the quantities MKPIGAP and MEXCL . The measurement of the quantity MEXCL involved simulating the performance of the cuts TGPV, OPSVETO and CCDPUL. The same method as described for the Ke4 background (Section 4.5.2) was used to determine the systematic uncertainty associated with the possible mismatch between the target-fiber energy scale in Monte Carlo and data. The Monte Carlo simulation energy Txtg was the total energy deposited in photon veto and opposite-side pion fibers and was varied over the range of 0.6 to 1.8 MeV. This energy was used to simulate the performance of TGPV and OPSVETO. The Monte Carlo simulation energy Ehide was the total energy deposited in the kaon fibers from the KL0 decay 130  Loose  Tight Data  NKPIGAP (1/3)  3  1  NKPIGAP (2/3)  0  0 Monte Carlo  MEXCL  50+33 −10  6+6 −2  MKPIGAP  4136  3332 Acceptance Loss  Aloss  0.687 ± 0.001  0.687 ± 0.001  Background  bg(1/3) bg(2/3)  0.076 ±  0.013 ±  0.044+0.058 −0.015 0.013+0.010 −0.003  0.0038 ± 0.0038+0.0038 −0.0013  0.0019 ± 0.0019+0.0019 −0.0006  Table 4.17: The summary of the charge-exchange background evaluation. Values for M EXCL and MKPIGAP taken from [55]. See Figure 4.17 for definitions of the measured quantities N KPIGAP , MEXCL and MKPIGAP .  products and was varied over the range of 1.5 to 5.0 MeV. The values of “Txtg = 1.2 MeV” and “Ehide = 2.5 MeV” were used to determine the mean number of events MEXCL and the variations due to the ranges of Txtg and Ehide were used to determine the systematic uncertainty. Table 4.17 shows a summary of results of these measurements made on the KPIGAP and TARGF branches using Monte Carlo.  4.8.3  CEX Background Evaluation  Equation (4.24) shows the expression used to evaluate the CEX background level using a combination of data and Monte Carlo. Table 4.17 summarizes the results of this background evaluation, where the results for the 1/3 and 2/3 data sets are scaled to represent the entire data set.  131  4.8.4  Hyperon Background  This section discusses the method by which hyperon production can become a background process and presents arguments as to why the existing charge exchange background estimate takes hyperon production into account. The production of hyperons is a multi-stage process. First, a K 0 is produced via the ¯ 0 [49], which charge exchange interaction K + n → K 0 p. Next, the K 0 oscillates to K ¯ 0 can then interact with a nucleon (N ), producing a contains a strange quark. The K hyperon (Y ) and pion: ¯ 0 + N → Y + π. K  (4.25)  There are a number of processes by which a π + having a momentum which overlaps the PNN2 region can be produced: K 0 + p → π 0 + Σ+ → π 0 + (p+ π 0 or nπ + ), ... → π + + Σ0 → π + + (Λ0 γ) → π + + ((p+ π − or nπ 0 )γ), ... → π + + Λ → π + + (p+ π − or nπ 0 ), K0  0  +n→π +Σ  0  0  0  +  (4.26) + −  0  → π + (Λ γ) → π + ((p π or nπ )γ),  ... → π − + Σ+ → π − + (p+ π 0 or nπ + ), ... → π + + Σ− → π + + (nπ − ), where the Σ and Λ particles are hyperons. The cross-sections for hyperon production are not well determined. At a KL0 momentum of 168 MeV/c, the total cross-section is 481 mb/(carbon nucleus) and 70 mb/H [91]. This agrees well with a total hyperon production cross-section of 90 mb per proton for K − at 160 MeV/c [75], where the K − and KL0 cross-sections are expected to have only small differences due to isospin differences and electromagnetic effects. According to [75], the KL0 cross-sections should approximately double in the lower momentum region of 0 to 100  132  MeV/c. Based on this information and an assumption of an entirely polystyrene target, ¯ 0 in the target was estimated to be between 37.3 and 75.6 cm, the mean free path of the K ¯ 0 travels 6 cm in which corresponds to a survival probability of 73% to 85% after the K the target [56]. As a result the KPIGAP cut, which selected events with a gap between the incoming kaon and outgoing pion tracks, should have been effective at selecting both the semi-leptonic KL0 and hyperon production events. Thus the charge exchange normalization branch should have consisted of both these types of events. Similar to the K 0 ¯ 0 results in potential background processes case, only the long-lived component of the K due to the delayed coincidence cut DELCO removing the short-lived component. The rejection of TGPV, CCDPUL and OPSVETO was estimated using the semileptonic KL0 decays in simulation. For the hyperon production events, particles such as π − , Σ and Λ were also produced. Due to their relatively short lifetimes, it is expected that they would deposit their energy in the target and thus be rejected by TGPV, CCDPUL and OPSVETO. Thus the rejection of TGPV, CCDPUL and OPSVETO for charge exchange events, determined in Section 4.8.2, should underestimate the rejection power of these cuts on hyperon events. Based on the above arguments and the large systematic error associated with the rejection of TGPV, CCDPUL and OPSVETO, it was decided that the estimation of the charge exchange background presented in this section also takes hyperon production into account.  133  4.9  Other Backgrounds  The previous sections in this chapter discussed the backgrounds which were identified as being significant sources of background for this analysis. Table 4.18 shows a list of K + decays which could have contributed to backgrounds in the PNN2 region. This list contains both decays which were considered in background studies earlier in the chapter and decays which were considered to have negligible contributions to the background due to the following reasons: • Many decay modes were excluded due to the charged products being kinematically excluded from the signal region by having kinematic quantities lower than the ranges covered by the kinematic signal region PNN2BOX. Products from K + decays-inflight could have been boosted into the kinematic signal region, but these types of events were accounted for in the single- and double-beam studies. • Decays having three or more photons and no π + had small branching ratios. The combination of photon veto and π + → µ+ → e+ particle identification suppressed these backgrounds to a level where they were not considered significant sources of background. • Decay modes having only an e+ and no π + or µ+ were highly suppressed by the π + → µ+ → e+ decay sequence conditions and dE/dx measurements. • The remaining decay modes were heavily suppressed by having a combination of multiple charged products, photons against which to veto, π + → µ+ → e+ particle identification where no decay π + was present, and very small branching ratios.  134  Background  Branching Ratio  K + → µ+ ν K + → π+π0 K + → π+π+π−  0.6344 0.2092 0.0559  2  0.0498 0.0332 0.01757  2 2 4  0.0062 0.000275 0.000269  1 3 3  0.000104 < 0.0001 < 0.00006  1 3  K + → π 0 e+ ν K + → π 0 µ+ ν K + → π+π0π0 K + → µ+ νγ K + → π+π0γ K + → π 0 e+ νγ  K + → π+π+π−γ K + → π + 3γ K + → e+ νν ν¯ K + → π + π − e+ ν K + → π 0 µ+ νγ K + → π 0 π 0 e+ ν  K + → e+ ν K + → e+ νγ K + → π + π − µ+ ν  0.0000409 0.000024 0.000022  0.0000076 < 0.000006 < 0.000005  K + → µ+ νµ+ µ− K + → e+ νe+ e− K + → e+ νµ+ µ−  < 0.00000041 0.000000025 0.000000017  Extra TG Energy  Fails π + ID  Number of Photons  3 4  0.0000155 0.0000152 0.000014  K + → π+π0π0γ K + → µ+ νν ν¯ K + → π 0 π 0 e+ νγ  K + → π 0 π 0 π 0 e+ ν K + → π + γγ K + → µ+ νe+ e−  Kinematically Excluded  1 5 4  < 0.0000035 0.00000110 0.000000071  6 2  Table 4.18: Various K + decays with their branching ratios [99]. A in the column “Kinematically Excluded” indicates that the process is kinematically excluded from the PNN2 signal region. A in the column “Extra TG Energy” indicates that the decay is suppressed by cuts that reject events with additional energy in the target. A in the column “Fails π + ID” that there is no π + in the final state and thus is suppressed by the π + → µ+ → e+ decay-sequence cuts. The right-most column indicates the number of photons in the final state.  135  Loose Kπ2 TG-scatter Kπ2 RS-scatter Kπ2γ Muon Ke4  1/3  2/3  0.537 ±0.188  0.0220 ±0.0056  0.0514 ±0.0086  0.0246 ±0.0246 0.235 ±0.118  +0.069 −0.215 +0.0021 −0.0021 +0.0042 −0.0038  +0.310 −0.166  0.619 ±0.150  0.0303 ±0.0054  0.0757 ±0.0073  0.0114 ±0.0114 0.176 ±0.072  Single-Beam  0.00046±0.00046  0.00023±0.00023  Double-Beam  0.00485±0.00380  0.00111±0.00080  CEX  0.076 ±0.044  Total  0.951 ±0.228  +0.058 −0.015 +0.443 −0.402  0.013 ±0.013  0.927 ±0.168  +0.067 −0.100 +0.0038 −0.0039 +0.0062 −0.0056  +0.233 −0.124  +0.010 −0.003 +0.320 −0.237  Table 4.19: The summary of the total expected background in the loose signal region from the various components.  4.10  Background Summary  The evaluation of the final background levels came from the 2/3 sample with the results from the 1/3 sample providing a check against large biases introduced when tuning the cuts. The total background level in the loose signal region was estimated to be 0.927 ± +0.320 +0.095 0.168 −0.237 and in the tight signal region 0.144 ± 0.045 −0.032 , where the first uncertainty  is statistical and the second systematic. Recall that the signal region was divided into nine cells (see Section 7.1) and the cell designed to have the highest signal-to-background was designated the tight signal region. Tables 4.19 and 4.20 summarize the background levels for both data sets in the loose and tight signal regions, respectively. Results from both data sets were scaled to represent the entire data set. The individual and total background levels were statistically consistent between the 1/3 and 2/3 data sets.  136  Tight Kπ2 TG-scatter Kπ2 RS-scatter Kπ2γ Muon Ke4  1/3  2/3  0.106 ±0.059  +0.076 −0.024  0.0121 ±0.0038  +0.0010 −0.0010  0.034 ±0.034  +0.142 −0.026  0.0043 ±0.0019  0.0055 ±0.0055  Single-Beam  0.00046±0.00046  Double-Beam  0.00212±0.00212  CEX Total  0.0038 ±0.0038  0.177 ±0.069  +0.0004 −0.0005  0.095 ±0.041  +0.020 −0.016  0.0188 ±0.0034  +0.0016 −0.0014  0.017 ±0.017  +0.071 −0.013  0.0069 ±0.0019  0.0037 ±0.0037  +0.0007 −0.0009  0.00019±0.00019 0.00058±0.00058 +0.0038 −0.0013 +0.223 −0.053  0.0019 ±0.0019  0.144 ±0.045  +0.0019 −0.0006 +0.095 −0.032  Table 4.20: The summary of the total expected background in the tight signal region from the various components.  137  Chapter 5 Validity Checks The validity studies detailed in this chapter were used to verify the reliability of the analysis strategy as motivated in Section 3.2.4.  5.1  Outside-the-Box Studies  The general method used to evaluate the number of events in the outside-the-box region is described here, with the details of each individual outside-the-box study described in its own respective section. These outside-the-box studies tested the assumption that the bifurcation cuts, used to estimate a given background, were uncorrelated. The outside-the-box studies involved loosening either set of bifurcation cuts used for the Kπ2 target-scatter background evaluation: the photon veto or the “EPI” cuts collection of cuts: CCDBADFIT, CCDBADTIM, CCDPUL and EPIONK. With the cuts loosened, the Kπ2 target-scatter, Range-Stack-scatter and Kπ2γ backgrounds were reevaluated. The remaining backgrounds were scaled using appropriate acceptance loss factors from the loosened and nominal levels of the cuts. The estimate of the total background level in the outside-the-box region was taken as the difference between the total background level evaluated with the loosened cuts and  138  the total background level in the signal region. The 2/3 data set was used for all outsidethe-box background evaluations. The number of events in the outside the box region was measured directly using the full data set and compared to the outside-the-box estimate. Three outside-the-box studies were performed, examining the following regions: • PV1 - Between the loose photon veto PV60 and the very loose photon veto PV90, where the “60” or “90” refers to the fact that the parameters for the photon veto were optimized to accept approximately 60% or 90% of the signal-like events; • PV2 - Between the very loose photon veto PV90 and the even looser PNN1-level photon veto PVPNN1; • EPI - Between the pion energy thresholds of 1.25 MeV and 2.5 MeV for the cuts CCDBADFIT, CCDBADTIM, CCDPUL and EPIONK. Note that for all of these studies, the Kπ2γ background contributed only to the uncertainty and not the central value of the total outside-the-box background estimates due to the way the correction to the Kπ2 target-scatter background due to Kπ2γ contamination was applied (see Section 4.1.5).  5.1.1  PV1 Region  For many of the backgrounds, loosening the photon veto increased the background by the ratio of the acceptance of PV90 to PV60. These backgrounds were Ke4 , CEX, muon and beam and the scaling factor was given by 0.8855 A(PV90) = = 1.428. A(PV60) 0.6199  (5.1)  For the remaining backgrounds, the new background level due to loosening of the photon veto was evaluated using PV90 instead of PV60. Table 5.1 shows the backgrounds due to PV60, PV90 and the resulting PV1 outside-the-box background. 139  Background Kπ2 -tgscat Kπ2 -rsscat Kπ2γ Ke4 CEX Muon 1bm 2bm-KK 2bm-KP Total  PV60  PV90  OTB (PV1)  +0.061 0.695 ± 0.150 −0.094  +1.13 9.58 ± 0.63 −1.00  +1.23 8.89 ± 0.64 −1.06  +0.004 0.030 ± 0.005 −0.004  +0.018 0.143 ± 0.022 −0.018  +0.022 0.113 ± 0.023 −0.022  +0.233 0.176 ± 0.072 −0.124  +0.333 0.251 ± 0.103 −0.177  +0.100 0.075 ± 0.031 −0.053  +0.006 0.076 ± 0.007 −0.006  +0.010 0.013 ± 0.013 −0.003  0.0114 ± 0.0114  0.00023 ± 0.00023  +0.029 0.357 ± 0.016 −0.026  +0.014 0.019 ± 0.019 −0.004  0.0163 ± 0.0163  0.00033 ± 0.00033  +0.023 0.281 ± 0.018 −0.020  +0.004 0.006 ± 0.006 −0.001  0.0049 ± 0.0049  0.00010 ± 0.00010  0.00046 ± 0.00046  0.00065 ± 0.00065  0.00020 ± 0.00020  +0.31 0.93 ± 0.17 −0.23  +1.53 10.02 ± 0.64 −1.23  +1.38 9.09 ± 0.65 −1.15  0.00065 ± 0.00065  0.00093 ± 0.00093  0.00028 ± 0.00028  Table 5.1: Summary of the PV1 Outside-the-Box Study. Scaling by a factor of A(PV90)/A(PV60) = 0.8855/0.6199 = 1.428 was used for the K e4 , CEX, muon and beam backgrounds. The remaining backgrounds were re-evaluated using PV90. For values having two sets of uncertainties, the first is statistical and the second systematic. The central value for Kπ2γ was treated as zero as the contribution due to this background was included in the K π2 target-scatter value.  The total number of background events predicted in the PV1 outside-the-box study +1.38 was Npred = 9.09 ± 0.65(stat.) −1.15 (sys.). When the number of events in this region  was measured directly, 3 events were found. Treating the central value of 9.09 events as the mean of a Poisson distribution gave a probability of 0.02 of observing 3 or less events. When this probability was re-evaluated at the upper and lower bounds of the total uncertainty of Npred , the probability range was [0.01, 0.05]. To help determine if this lower than expected number of events was a statistical anomaly or an indication of anti-correlation between the photon veto and EPI cuts, this outside-the-box study was repeated looking at the region between the PV90 and the PNN1-level photon veto PVPNN1.  140  5.1.2  PV2 Region  This study examined the outside-the-box region between PV90 and PVPNN1. Scaling was used to determine the background level in the expanded box for Ke4 , CEX, muon and beam. The scaling factor was 0.9248 0.8855 A(PVPNN1) A(PV90) − = − = 0.064. A(PV60) A(PV60) 0.6199 0.6199  (5.2)  As with the PV1 outside-the-box study, the Kπ2 -scatter and Kπ2γ backgrounds were re-evaluated for the expanded regions. Table 5.2 shows the backgrounds due to PV90, PVPNN1 and the resulting PV2 outside-the-box background. Background Kπ2 -tgscat Kπ2 -rsscat Kπ2γ Ke4 CEX Muon 1bm 2bm-KK 2bm-KP Total  PV90  PVPNN1  OTB (PV2)  +1.133 9.584 ± 0.626 −1.000  +12.09 41.63 ± 1.74 −6.65  +13.09 32.04 ± 1.85 −7.78  +0.018 0.143 ± 0.022 −0.018  +0.054 0.449 ± 0.067 −0.056  +0.073 0.305 ± 0.070 −0.074  +0.333 0.251 ± 0.103 −0.177  +0.348 0.266 ± 0.107 −0.185  +0.015 0.011 ± 0.005 −0.008  +0.029 0.357 ± 0.016 −0.026  +0.014 0.019 ± 0.019 −0.004  0.0163 ± 0.0163  0.00033 ± 0.00033  +0.090 1.091 ± 0.028 −0.079  +0.061 0.734 ± 0.018 −0.053  +0.015 0.019 ± 0.019 −0.005  +0.0006 0.0008 ± 0.0008 −0.0002  0.00034 ± 0.00034  0.00001 ± 0.00001  0.0170 ± 0.0170  0.0007 ± 0.0007  0.00065 ± 0.00065  0.00068 ± 0.00068  0.00003 ± 0.00003  +1.53 10.02 ± 0.64 −1.23  +12.60 42.38 ± 1.75 −6.98  +13.24 32.36 ± 1.85 −7.92  0.00093 ± 0.00093  0.00097 ± 0.00097  0.00004 ± 0.00004  Table 5.2: Summary of the PV2 Outside-the-Box Study. Scaling was used for the K e4 , CEX, muon and beam backgrounds. The remaining backgrounds were re-evaluated in both the PVPNN1 and PV90 regions. For values having two sets of uncertainties, the first is statistical and the second systematic. The central value for K π2γ was treated as zero since the contribution due to this background was included in the K π2 target-scatter value.  The total number of background events predicted in the PV2 outside-the-box study +13.24 was Npred = 32.36 ± 1.85(stat.) −7.92 (sys.). When the number of events in this region was  141  measured directly, 34 events were found. This number of observed events agrees with the predicted number within statistical uncertainty. Assuming a Poisson distribution of mean √ 32.36 can be approximated by a Gaussian of mean 32.36 and σ = 32.36, the probability of observing 34 or fewer events was 0.61. When this probability was re-evaluated at the upper and lower bounds of the total uncertainty of Npred , the probability range was [0.05, 0.98].  5.1.3  EPI Region  For the purpose of this study, the cuts CCDBADFIT, CCDBADTIM, CCDPUL and EPIONK will be called “EPI” cuts. The pion energy threshold for these cuts was loosened from 1.25 MeV (“EPI=1.25”) to 2.5 MeV (“EPI=2.5”) for this outside-the-box study. Scaling by a factor of A(EPI = 2.5) 0.6862 = = 1.4995 A(EPI = 1.25) 0.4576  (5.3)  was used to determine the Ke4 , CEX, muon and beam backgrounds in the expanded box. The normalization branch for Kπ2γ was re-evaluated and the Kπ2 -scatter backgrounds completely re-evaluated to determine the background levels in the expanded box. Table 5.3 shows the backgrounds due to EPI=1.25, EPI=2.5 and the resulting EPI outside-the-box background. The total number of background events predicted by the EPI outside-the-box study was Npred = 0.79 ± 0.35(stat.) +0.30 −0.37 (sys.). When the number of events in this region was measured directly, 0 events were found. Treating the central value of 0.79 events as the mean of a Poisson distribution gave a probability of 0.45 of observing 0 events. When this probability was re-evaluated at the upper and lower bounds of the total uncertainty of Npred , the probability range was [0.29, 0.62].  142  Background Kπ2 -tgscat Kπ2 -rsscat Kπ2γ Ke4 CEX Muon 1bm 2bm-KK 2bm-KP Total  EPI=1.25  EPI=2.5  OTB (EPI)  +0.061 0.695 ± 0.150 −0.094  +0.066 1.361 ± 0.314 −0.229  +0.160 0.666 ± 0.348 −0.290  +0.006 0.076 ± 0.007 −0.006  +0.010 0.121 ± 0.009 −0.009  +0.004 0.045 ± 0.012 −0.003  +0.004 0.030 ± 0.005 −0.004  +0.233 0.176 ± 0.072 −0.124  +0.010 0.013 ± 0.013 −0.003  0.0114 ± 0.0114  0.00023 ± 0.00023  +0.008 0.057 ± 0.008 −0.008  +0.349 0.264 ± 0.108 −0.186  +0.015 0.019 ± 0.019 −0.005  0.0171 ± 0.0171  0.00034 ± 0.00034  +0.011 0.026 ± 0.009 −0.011  +0.116 0.088 ± 0.036 −0.062  +0.005 0.006 ± 0.006 −0.002  0.0057 ± 0.0057  0.00011 ± 0.00011  0.00046 ± 0.00046  0.00069 ± 0.00069  0.00023 ± 0.00023  +0.31 0.93 ± 0.17 −0.23  +0.45 1.72 ± 0.33 −0.44  +0.30 0.79 ± 0.35 −0.37  0.00065 ± 0.00065  0.00097 ± 0.00097  0.00032 ± 0.00032  Table 5.3: Summary of the EPI outside-the-box study. Scaling by a factor of A(EPI-2.5)/A(EPI1.25) = 1.499 was used for the Ke4 , CEX, muon and beam backgrounds. The remaining backgrounds were re-evaluated using the loosened CCDPUL, CCDBADFIT, CCDBADTIM and EPIONK cuts. For values having two sets of uncertainties, the first is statistical and the second systematic. The central value for K π2γ was treated as zero as the contribution due to this background was included in the Kπ2 target-scatter value.  5.1.4  Consistency of the Outside-the-Box Studies  The purpose of the outside-the-box was to look for correlations between the bifurcation cuts used for the largest background, Kπ2 target-scatter. This section discusses the combined probabilities of the observed number of events from each study given the predictions. Given the probabilities of the observed number of events from two outside-the-box studies p1 and p2 , the combined probability of the two observations is given by [39]  p12 = p1 p2 (1 − lnp1 p2 ) .  (5.4)  Combining the probabilities of the observations from the three outside-the-box studies (0.02, 0.61 and 0.45) gave a combined probability of 0.14. When this combined probability was re-evaluated at the upper and lower bounds of the constituent probabilities, the 143  probability range was [0.01, 0.40]. These probabilities indicated that the observed and predicted number of events in the outside-the-box studies were generally consistent, giving confidence in the background estimates and their associated systematic uncertainties.  5.2  Single-Cut and Double-Cut Failure Studies  For the single-cut and double-cut failure studies, the cuts were were divided into 13 groups based on correlations. The cuts contained in each group are listed below: 1. BOX - This group of cuts, known as PNN2BOX, defined the kinematic signal region with cuts on the π + momentum (ptot), energy (etot) and range in plastic scintillator (rtot). 2. PV - This group was a single cut, the standard loose photon veto (PV60) including neither the target photon veto (TGPV) nor the active degrader (ADPV). 3. ADPV - This group was a single cut, the active degrader (ADPV). 4. DELC3 - This group was a single cut, the loose delayed coincidence (DELC3). 5. B4EKZ - This group was a single cut, the beam likelihood cut (B4EKZ). 6. TGZ - This group was a single cut, the target fiducial region cut (TGZFOOL) which demanded that the K+ -decay vertex was within the target fiducial volume. 7. ETG - This group of cuts removed events that had extra energy in the target: TGPV, OPSVETO. 8. EKAON - This group of cuts removed events that had large pion energy in kaon fibers: CCDPUL, CCDBADFIT, CCDBADTIM, CCD31FIB, EPIONK, TIMKF. 9. ICGEO - This group of cuts removed events that had beam-particle activity in the I-Counters: TGGEO, KIC 144  10. TD - This group of cuts was used for particle identification in the Range-Stack counters: PIFLG, ELVETO, TDFOOL, TDNN, RSHEX, RSHEX2. 11. KIN - This group of cuts was a collection of cuts based on kinematic constraints in various detector sub-systems: COS3D, ZFRF, ZUTOUT, UTCQUAL, TICCON, EICCON, RNGMOM, PRRF, RSDEDX, LAYER14. 12. BEAM - This group of cuts removed single- and double-beam events: BWTRS, CKTRS, CKTAIL, CPITRS, CPITAIL, B4DEDX, B4TRS, B4CCD, TIMCON, UPVTRS, RVTRS. 13. OTHER - This group of cuts was a large group of cuts that didn’t fit into any of the first 12 groups: TGQUALT, NPITG, EPITG, EPIMAXK, TGER, TARGF, DTGTTP, RTDIF, TGKTIM, TGEDGE, TGDEDX, TGENR, PIGAP, TGLIKE, TGB4, PHIVTX, CHI567, CHI5MAX, ALLKFIT, TPICS, TGTCON, B4ETCON, DRP.  5.2.1  Single-Cut Failures  The single-cut failure study consisted of examining the signal data for events failing only one of the 13 groups of cuts detailed above. Examination of these events provided a clear way to discover flaws in the analysis. Events were classified as “true” single-cut failures if they were classified as single-cut failures and failed only a single cut within that group. The single-cut study was performed twice. The initial study was performed only on the 1/3 signal data. All events, other than those that failed multiple photon veto cuts, were visually inspected. Two events exposing potential analysis flaws were found in this initial study, prompting the creation of the CCDBADTIM and early BVL safety cuts as discussed in Section 5.2.3. A third event was found in this initial study, exposing an error in the way energy de-multiplexing was being handled in the low-gain target fiber  145  Group  1/3  2/3  BOX PV ADPV DELC3 B4EKZ TGZ ETG EKAON ICGEO TD KIN BEAM OTHER  41 221 0 0 0 0 1 3 1 0 3 0 3  (0) 114 (0) (22) 494 (38) (0) 2 (2) (0) 0 (0) (0) 0 (0) (0) 0 (0) (0) 3 (0) (2) 3 (3) (1) 0 (0) (0) 1 (1) (2) 1 (0) (0) 0 (0) (1) 1 (1)  Total  273 (28) 619 (45)  Table 5.4: The number of single-cut failures listed by group. For each data set, the first number is the number of events which fail only cuts the given group. The numbers in parenthesis are the “true” single-cut failures, events which fail only one of the individual cuts within the given group. The numbers shown for the “true” PVCUT group do not include the pass 2 photon veto P2PVCUT since this cut was composed of multiple subsystems: EC, RD and BV. When P2PVCUT was included in the “true” single-cut failures, there was only 1 event in this group for the 1/3 data set and 6 for the 2/3 data set. Table was reproduced from [56].  CCDs. Appendix E.6 details the way in which this error was fixed and the subsequent improvements that were made to the de-multiplexing algorithm. The second study was performed on both the 1/3 and 2/3 signal data sets after the final cut positions were set and the 2/3 data set was used to evaluate the final background levels. The results of this study are found in Table 5.4 and show that the rate of single-cut failures was consistent between the 1/3 and 2/3 data sets. The single-cut failures, other than those failing multiple photon veto cuts, were visually inspected. It was concluded from the visual inspection of the “true” photon veto failure events, that these events were Kπ2 decays where one or two very energetic photons were converted and contained within a single photon detector.  146  Group 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.  1.  2.  3.  4. 5. 6.  7.  8.  9. 10.  11.  BOX 47518 61 2 3 1 176 79 11 353 934 PV 47518 233 39 17 8 93 1328 30 48 179 ADPV 61 233 - 1 3 9 2 2 DELC3 2 39 1 1 B4EKZ 3 17 1 TGZ 1 8 ETG 176 93 3 36 3 1 EKAON 79 1328 9 36 1 ICGEO 11 30 3 TD 353 48 2 121 KIN 934 179 2 1 1 1 1 121 BEAM 3 37 OTHER 26 808 2 2 19 5 1 1 1  Total  12.  13.  3 26 37 808 2 2  1  19 5 1 1 1 1 -  49167 50338 313 45 21 9 331 1458 45 525 1241 41 866  Table 5.5: The number of double-cut failures listed by group for the 1/3 sample. All blank entries represent zero events failing that combination of groups. Table was reproduced from [56].  5.2.2  Double-Cut Failures  The double-cut failure study consisted of examining the signal data for events failing cuts in exactly two of the 13 groups of cuts detailed above. The results for the 1/3 data set are shown in Table 5.5 and for the 2/3 data set in Table 5.6. As with the single-cut failures, it was found that the rate of double-cut failures was consistent between the 1/3 and 2/3 data sets.  5.2.3  Safety Cuts  Two events exposing potential analysis flaws were found when examining events from the initial single-cut failure study on the 1/3 data set. These events prompted the creation of two safety cuts: CCDBADTIM and the early BVL. These cuts had minimal acceptance loss and were devised to target the signatures of the two events. The first event was an ADPV single-cut failure. From reconstruction, the event was 147  Group 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.  1.  2.  3.  4.  5. 6.  7.  8.  9.  BOX 94557 141 9 6 2 352 187 23 PV 94557 533 96 32 4 182 2557 82 ADPV 141 533 6 DELC3 9 96 1 1 B4EKZ 6 32 1 1 TGZ 2 4 ETG 352 182 56 3 EKAON 187 2557 6 1 1 56 2 ICGEO 23 82 3 2 TD 731 113 2 3 KIN 1885 372 8 2 1 4 BEAM 16 60 2 1 2 OTHER 95 1575 7 8 30 25 2  Total  10.  11.  12.  13.  731 1885 16 95 113 372 60 1575 8 7 2 8 2 2 3  1 4  1 2  30 25 2  268  268 1 4  1 2  4 2 -  98004 100163 695 117 42 6 627 2844 112 1117 2545 84 1748  Table 5.6: The number of double-cut failures listed by group for the 2/3 sample. All blank entries represent zero events failing that combination of groups. Table was reproduced from [56].  identified as a Kπ2 target-scatter with one of the photons being detected in the Active Degrader. Comparisons between the raw target-CCD data and the solutions found by the target CCD fitter revealed a large second pulse (∼ 8 MeV) in one of the kaon fibers that was not found by the fitter. Subsequent studies found common incorrect target CCD fitter solutions that could be found when checking for consistency between the pulse times found using the target CCD fitter and those expected from target reconstruction. The cut CCDBADTIM was created to remove these types of events and is discussed in more detail in Appendix E. The second event was a PV single-cut failure in the BVL. Examination of this event and data from previous studies [92] revealed that two photons from a single π 0 decay converting in the same BVL element caused the timing of these photon hits to be mismeasured. The “early BVL” cut was added to the photon veto, removing events having a mean time in the BVL between -5.0 and -2.0 ns, a time difference between the hits on each end of less than 4.0 ns, and energy of the hit of greater than 10.0 MeV. 148  5.3  Background Cross-Contamination Studies  As discussed in 4.1.1, a number of contamination studies were performed on the Kπ2 target-scatter background. This section details data-driven studies that found negligible contamination due to muon and double-beam events in the Kπ2 target-scatter background estimate. Also found in this section are the details of a study used to determine the upperlimit of Kπ2γ contamination in the Kπ2 target-scatter normalization branch, supporting the method used to correct for Kπ2γ contamination in the Kπ2 target-scatter background discussed in Section 4.1.5. Details of the contamination due to Kπ2 Range-Stack-scatter in the Kπ2 target-scatter normalization branch were previously discussed in Sections 4.1.2 and 4.2.1 and are not discussed further in this section. The subject of contamination due to Ke4 in the Kπ2 target-scatter normalization branch, which was argued to be negligible in Section 4.1.2, is also not discussed further in this section.  5.3.1  Effects of Contamination Background Estimates  The method used to quantify the muon and double-beam contamination in the Kπ2 targetscatter background is discussed in this section and the calculations detailed in the two subsequent sections. For the Kπ2 target-scatter background, estimated using the bifurcation method, the effects of a generic contamination were as follows. The number of events remaining at the end of the normalization branch Ntgscat was inflated due to contamination events in addition the target-scatter events. The effect on the rejection of CUT1 (RPV60 ) varied depending on whether or not the rejection of the contamination events was higher or lower than that of the target-scatter events. If the rejection of the contamination events was higher (lower) than that of the target-scatter events, the rejection was overestimated (underestimated). The typical effect of these mis-measurements of the normalization and rejection was that the Kπ2 target-scatter background was overestimated, but a sufficiently 149  overestimated rejection could have caused the background to be underestimated.  5.3.2  Muon Contamination  The bifurcation cuts used to evaluate the muon background were used to estimate the contamination of muon events in the Kπ2 target-scatter normalization and rejection branches and thus the effect of this contamination on the background evaluation. These cuts were the loose TD cuts TDLOOSE (CUT1) and the range-momentum consistency cut RNGMOM (CUT2). Acceptance and Rejection of the Muon Bifurcation Cuts To perform a data-driven estimate of the muon contamination in the Kπ2 target-scatter normalization and rejection branches, the acceptance of these cuts on Kπ2 target-scatter events (Aπ ) and the rejection of these cuts on muon events (Rµ ) had to be evaluated. The rejection of RNGMOM (RRNGMOM ) for muon events was measured on the muon background normalization branch (Figure 4.6) after setup cuts and the inversion of TDLOOSE. The rejection of TDLOOSE (RTDCUT02 ) for muon events was measured in the muon background rejection branch by inverting RNGMOM (Table 4.9). The combined rejection of these cuts Rµ was Rµ = RRNGMOM × RTDLOOSE , = (28.29 ± 1.06) × (133.0 ± 10.7),  (5.5)  = 3764 ± 333. The acceptance of these cuts (Aπ ) was measured using a modified version of the rejection branch used in the Kπ2 target-scatter background estimate. The modifications were that TDLOOSE and RNGMOM were removed from the setup cuts and the kinematic box was changed from PNN2BOX box to the Kπ2 -peak kinematic box KP2BOX. 150  After the setup cuts were applied, the 12 classes (2-13) described in Table 4.1 were applied and the performance of RNGMOM and TDLOOSE was measured before and after application of the photon veto PV60 as shown in Table 5.7. Ignoring the classes with very low statistics, the measured acceptance of these muon bifurcation cuts were equal before and after the application of the photon veto cut within statistical uncertainty. The acceptance Aπ was taken as the average of the highest and lowest acceptance values measured before the application of PV60 with the difference between these extreme values setting the bounds for the uncertainty:  (5.6)  Aπ = 0.809 ± 0.030.  CLASS 2 3 4 5 6 7 8 9 10 11 12 13  BEFORE PV60 293455/363196 = 120164/148540 = 120568/149124 = 364667/451267 = 175663/217075 = 179337/222017 = 27644/34214 = 343309/424682 = 59863/74471 = 5883/7383 = 316971/392405 = 132619/164163 =  0.808 0.809 0.809 0.808 0.809 0.808 0.808 0.808 0.804 0.797 0.808 0.808  AFTER PV60 ± ± ± ± ± ± ± ± ± ± ± ±  0.001 0.001 0.001 0.001 0.001 0.001 0.002 0.001 0.002 0.005 0.001 0.001  280/343 115/140 109/138 346/428 159/199 166/214 16/25 331/412 58/72 5/5 303/368 127/152  = = = = = = = = = = = =  0.816 0.821 0.790 0.808 0.799 0.776 0.640 0.803 0.806 1.000 0.823 0.836  ± ± ± ± ± ± ± ± ± ± ± ±  0.021 0.032 0.035 0.019 0.028 0.029 0.096 0.020 0.047 0.000 0.020 0.030  Table 5.7: The acceptance of RNGMOM×TDCUT02 measured on K π2 -peak events before and after the application of the photon veto cut PV60 for each of the 12 classes (2-13) from the K π2 target-scatter rejection branch.  Muon Contamination in the Normalization Branch To determine the amount of muon contamination in the Kπ2 target-scatter normalization branch, the number of events N left at the end of the normalization branch was treated 151  as being made up of either muon (Nµ ) or pion (Nπ ) events,  N = N π + Nµ .  (5.7)  With the performance known for the muon bifurcation cuts (RNGMOM and TDLOOSE ) with respect to pions (Aπ ) and muons (Rµ ), these cuts were moved to the bottom of the Kπ2 target-scatter normalization branch. The number of events remaining before RNGMOM and TDLOOSE were applied was denoted M. These M events were also made up of pion or muon events as given by  M=  Nπ + N µ Rµ . Aπ  (5.8)  The amount of muon contamination left at the end of the normalization branch was represented by the quantity f , Nµ , N −1 Aπ M N . = Aπ Rµ − 1  f=  (5.9)  Taking the acceptance (Aπ = 0.809±0.030), rejection (Rµ = 3764±333), and measured values N = 1131 and M = 12980, the value of f was determined,  f = (2.72 ± 0.26) × 10−3 .  (5.10)  After being corrected for muon contamination, the number of events at the end of the Kπ2 target-scatter normalization branch was given by N = N (1 − f ), = 1128 ± 34. 152  (5.11)  This was corrected for Range-Stack contamination nrs , which gave the final corrected normalization: ntgscat = N − nrs = (1128 ± 34) − (23.3 ± 3.5)  (5.12)  = 1105 ± 34. Muon Contamination in the Rejection Branch The method used to determine the amount of muon contamination in the rejection branch was very similar to that for the normalization branch except the amount of contamination had to be measured before and after the Kπ2 target-scatter bifurcation cut PV60. The number of events left at the end of the Kπ2 target-scatter rejection branch (Figure 4.2) was denoted N2 where only the results for CLASS12 were used for this part of the study. The number of events before the photon veto was applied was denoted N1 and the photon veto rejection RPV given by  RPV =  N1 N2  (5.13)  It was possible to examine the amount of muon contamination both before and after the photon veto was applied by treating N1 and N2 as being made up of muon and pion events as with the normalization branch method: N1 = Nπ1 + Nµ1 ,  (5.14)  N2 = Nπ2 + Nµ2 . Again, the performances of the muon bifurcation cuts (RNGMOM and TDLOOSE ) with respect to pions (Aπ ) and muons (Rµ ) were used to solve for the fraction of the events which were contamination muons.  153  VALID TRIG, P2PSCUT, P2TGCUT, P2TGPVCUT, BEAMCUTS, DELC3, BOXLOOSE, KINCUTS (excluding RNGMOM), CLASS12 After PV60  Before PV60  PV60  M1 RNGMOM, TDLOOSE  M2  RNGMOM, TDLOOSE  N1  N2  Figure 5.1: Flowchart showing the muon contamination study in the K π2 target-scatter rejection branch. Cuts in italics refer to named groups of cuts.  Figure 5.1 shows a flowchart with the cuts applied and the quantities measured for this part of the contamination study. To determine the muon contamination remaining before the photon veto was applied, the number of events remaining before the application of PV60 was measured with (N1 ) and without (M1 ) the muon bifurcation cuts applied. The same was done to determine the muon contamination present after the application of PV60 by measuring the events remaining at the end of the rejection branch with (N2 ) and without (M2 ) the muon bifurcation cuts applied. The following equations show the breakdown of pion and muon events in the quantities M1 and M2 : Nπ1 + Nµ1 Rµ , Aπ Nπ2 M2 = + Nµ2 Rµ . Aπ M1 =  (5.15)  The amounts of muon contamination before and after PV60 was applied were represented  154  by the quantities f1 and f2 , respectively,  f1 = f2 =  1 Aπ M −1 N1  Aπ Rµ − 1 2 Aπ M −1 N2 Aπ Rµ − 1  , (5.16) .  Using the measured values summarized in Table 5.8, The corrected photon veto rejection was given by RPV60 =  N1 (1 − f1 ) N2 (1 − f2 )  (5.17)  = 2666 ± 843  Quantity  Before PV60  After PV60  Muon bifurcation cuts not applied  M1 = 94424  M2 = 652  Muon bifurcation cuts applied  N1 = 52621  N2 = 22  f1 = (1.76 ± 0.16) × 10−4  f2 = (7.63 ± 1.80) × 10−3  f -value Corrected value RPV60 = N1 /N2  N1 = N1 (1 − f1 ) = 52612 ± 229  2410 ± 518  N2 = N2 (1 − f2 ) = 21.8 ± 4.7  Table 5.8: Table showing the values used to arrive at the K π2 target-scatter photon veto rejection after the effects of muon contamination were removed.  155  Background Estimate Corrected for Muon Contamination The numbers from the contamination study in the normalization and rejection branches were used to estimate the background with the effects of muon contamination removed, ntgscat 3 × , 2 RPV − 1 1105 ± 34 3 , = × 2 (2410 ± 518) − 1  bgtgscat =  (5.18)  = 0.688 ± 0.150. Since the central value of the nominal Kπ2 target-scatter background evaluation (0.695 ± +0.061 0.150 −0.094 ) and this value agree to approximately 1%, the muon contamination in the Kπ2  target-scatter background was considered to be negligible. Note that the values used for this Kπ2 target-scatter background comparison were both values without the corrections for Kπ2γ contamination applied.  5.3.3  Double-Beam Contamination  Due to a lack of acceptance and rejection information for the rejection branch bifurcation cuts (CUT2) for double-beam background, only the normalization branch bifurcation cuts (CUT1) were used in this study. The rejection of CKTRS, CKTAIL and BWTRS for KK -type events will be denoted RKK and the rejection of CPITRS, CPITAIL and BWTRS for KP -type events will be denoted RKP . These rejections were taken from the double-beam rejection branches: Table 4.14 for RKK and Table 4.15 for RKP . The combined acceptance of these groups of cuts for pion events, denoted AKK and AKP , were taken from the beam acceptance measurement (Table 6.4). The sample used for that measurement used Kµ2 monitor trigger data with cuts applied to ensure the event looks like a single K + decay with no photons. These values are summarized in Table 5.9.  156  Cuts Acceptance Rejection  KK Branch  KP Branch  CKTRS·CKTAIL·BWTRS  CPITRS·CPITAIL·BWTRS  AKK = 0.8973 ± 0.0002  AKP = 0.9159 ± 0.0002  RKK = 394 ± 197  RKP = 617 ± 308  Table 5.9: Acceptance of pion events and rejection of double-beam events for the double-beam bifurcation cuts.  Double-Beam Contamination in the Normalization Branch The method for determining the double-beam contamination in the Kπ2 target-scatter normalization branch was the same as that described for muon contamination, but with a different set of cuts for each the KK and KP double-beam contaminations. Since the contamination due to each of these backgrounds was expected to be very small, the KK contamination was ignored for the KP contamination study and the KP contamination ignored for the KK contamination study. The following discussion lays out the equations used to determine the amount of KK double-beam contamination, but the same equations all apply for the KP double-beam contamination with the KP notation replacing the KK notation. To determine the amount of KK contamination in the normalization branch, the number of events N left at the end of the normalization branch was treated as being made up of Nπ pion events and NKK KK double-beam events. Written in equation form, this looks like:  N = Nπ + NKK .  (5.19)  With the performance known for the KK double-beam rejection branch bifurcation cuts (CKTRS, CKTAIL and BWTRS) with respect to Kπ2 target-scatter events (AKK ) and KK double-beam events (RKK ), these cuts were moved to the bottom of the Kπ2 target-scatter normalization branch. The number of events remaining before these cuts were applied was denoted M. These M events were made up of pion and KK double-beam 157  events as given by  M=  Nπ + RKK NKK . AKK  (5.20)  The amount of KK contamination left at the end of the normalization branch was represented by the quantity f , given by the expression AKK Nn − 1 f= . AKK RKK − 1  (5.21)  Table 5.10 shows the values used to determine the fractional contamination due to KK and KP double-beam events in the Kπ2 target-scatter normalization branch. Note that the final corrected normalization ntgscat was corrected for both the given double-beam contamination and the range-stack scatter component nrs . KK Branch  KP Branch  M  1186  1160  N  1131  1131  f -value ntgscat  fKK = −0.00017 ± 0.00009 fKP = −0.00011 ± 0.00005  TG-Scatter Normalization corrected for f -value and nrs ntgscat = 1108 ± 34 = N (1 − f ) − nrs ntgscat = 1108 ± 23  Table 5.10: Correcting for double-beam contamination in the K π2 target-scatter normalization branch.  Double-Beam Contamination in the Rejection Branch The method used to determine the double-beam contamination in the Kπ2 target-scatter rejection branch was also similar to that described for muon contamination with the KK or KP double-beam bifurcation cuts replacing the muon bifurcation cuts. As with the double-beam contamination in the normalization branch study, contamination due to one type of double-beam process (KK or KP ) was ignored when studying the other. 158  The following discussion lays out the equations used to determine the amount of KK double-beam contamination, but the same equations all apply for the KP double-beam contamination with the KP notation replacing the KK notation. As with the muon contamination in the rejection branch, the amount of contamination was measured before and after the photon veto was applied. The amount of KK double-beam contamination, both before and after the photon veto was applied, was examined by treating the N1 events before the application of PV60 and the N2 events after the application of PV60 as being made up of double-beam and pion events as was done for the normalization branch: N1 = Nπ1 + NKK1 ,  (5.22)  N2 = Nπ2 + NKK2 . These KK double-beam bifurcation cuts were applied immediately before the end of the branch (after the photon veto) which gave M2 events before the KK double-beam bifurcation cuts and N2 after. The same was done by applying these KK double-beam bifurcation cuts immediately before the photon veto which gave M1 events before the KK double-beam bifurcation cuts and N1 after. The breakdown of pion and doublebeam events in the quantities M1 and M2 was as follows: Nπ1 + NKK1 RKK , AKK Nπ2 M2 = + NKK2 RKK . AKK M1 =  (5.23)  The amount of KK double-beam contamination before and after the photon veto was applied was represented by the quantities fM and fN , respectively, using the same definition as introduced for muons (Section 5.3.2). Tables 5.11 and 5.12 show the values used to arrive at values for the PV60 rejection after being corrected for each of the doublebeam processes. 159  Quantity  Before PV60  After PV60  Double-beam bifurcation cuts not applied  M1 = 55873  M2 = 25  Double-beam bifurcation cuts applied  N1 = 52621  N2 = 22  f1 = (−13.3 ± 6.7) × 10−5  f2 = (5.48 ± 2.16) × 10−5  f -value Corrected value RPV60 (KK) = N1 /N2  N1 = N1 (1 − f1 ) = 52628 ± 230  2392 ± 510  N2 = N2 (1 − f2 ) = 22.0 ± 4.7  Table 5.11: The KK Double-Beam Contamination in the K π2 target-scatter rejection branch. This table shows the values used to arrive at the photon veto rejection after the effects of KK double-beam contamination were removed.  Quantity  Before PV60  After PV60  Double-beam bifurcation cuts not applied  M1 = 54563  M2 = 25  Double-beam bifurcation cuts applied  N1 = 52621  N2 = 22  f1 = (−8.90 ± 4.46) × 10−5  f2 = (−7.18 ± 14.11) × 10−5  f -value Corrected value RPV60 (KP ) = N1 /N2  N1 = N1 (1 − f1 ) = 52626 ± 229  2392 ± 510  N2 = N2 (1 − f2 ) = 22.0 ± 4.7  Table 5.12: KP Double-Beam Contamination in the K π2 target-scatter rejection branch. This table shows the values used to arrive at the photon veto rejection after the effects of KP doublebeam contamination were removed.  160  Background Estimates Corrected for Double-Beam Contamination The numbers from the double-beam contamination studies in the normalization and rejection branches were used to estimate the backgrounds with the effects of each of the studied double-beam processes removed: ntgscat 3 × , 2 RPV60 − 1 1108 ± 34 3 , = × 2 (2392 ± 510) − 1  bgKK =  (5.24)  = 0.695 ± 0.150; and ntgscat 3 × , 2 RPV60 − 1 1108 ± 34 3 , = × 2 (2392 ± 510) − 1  bgKP =  (5.25)  = 0.695 ± 0.150. Since the central values of each of these corrected backgrounds agreed to better than 1% with the nominal value of 0.695 ± 0.150, the contamination due to both types of double-beam contamination was considered negligible. Note that the values used for this Kπ2 target-scatter background comparison were both values without the corrections for Kπ2γ contamination applied. Double-Beam Contamination Follow-Up Study A follow-up study was performed to test the assumption that the KP contamination could be ignored for the KK contamination study and vice versa. In this study, Equations (5.19)  161  and (5.20) were replaced with a set of 3 equations, N = Nπ + NKK + NKP , Nπ NKP , + NKK RKK + AKK AKK Nπ NKK = + NKP RKP . + AKP AKP  MKK = MKP  (5.26)  For these equations MKK and MKP were the M -values from Table 5.10 for the KK and KP branches, respectively. The definitions of the combined acceptances AKK and AKP were the same as AKK and AKP except the acceptance of BWTRS was replaced with the inverse of the rejection for BWTRS for that specific background, from the doublebeam rejection branch. To determine the amount of KK and KP contamination, the values NKK and NKP were determined from the above set of 3 equations and the f -values determined as NKK , N NKP . = N  fKK = fKP  (5.27)  The results from this follow-up study were consistent with the original double-beam contamination studies, showing it was reasonable to assume that at the given levels of contamination, the KP contamination could be ignored for the KK contamination study and vice versa.  5.3.4  Upper Limit of Kπ2γ Contamination  This study was performed to estimate the upper limit of Kπ2γ contamination in the Kπ2 target-scatter normalization branch using Kπ2 monitor trigger data. This study involved two kinematic regions, the Kπ2 -peak region KP2BOX and the kinematic signal region PNN2BOX. The L1.1 and L1.2 triggers and all analysis cuts other than the photon  162  veto PVCUT were applied to the sample, leaving npeak events in the KP2BOX and ntail events in the PNN2BOX. After applying all analysis cuts, these samples should have been dominated by events where a beam K + decayed into a π + and any number of photons since the photon veto was not applied. Due to the high branching ratio and pion identification cuts, the npeak events should have been made up entirely of Kπ2 events. The ntail events should have been a mixture of Kπ2 -scatter and Kπ2γ events. For the purpose of an upper limit study, it was assumed that the Kπ2γ contamination in the Kπ2 monitor trigger data was large enough that ntail was made up entirely of Kπ2γ events. The fraction of Kπ2γ events in the Kπ2 target-scatter normalization branch was expressed as g=  Ng , Ns + N g  (5.28)  where Ng was the number of Kπ2γ events in the Kπ2 target-scatter normalization branch and Ns was the number of Kπ2 -scatter events in the Kπ2 target-scatter normalization branch. The value for Ns + Ng was Ntgscat = 1131, taken from Table 4.4. The relative rate of ntail to npeak from this study was given by ntail , npeak 16 , f= 5778 f=  (5.29)  f = 0.00277 ± 0.00069. Next, the Kπ2 target-scatter normalization branch (Figure 4.2) was reproduced using the kinematic region KP2BOX instead of PNN2BOX, leaving Np = 122473 events at the end of that normalization branch. Based on the relative rate f determined above, the upper limit on the number of Kπ2γ events (Ng ) in the Kπ2 target-scatter normalization branch was Ng = f × N p .  163  (5.30)  Finally, Equation (5.28) was rewritten as Ng , Ns + N g f × Np = , Ntgscat (0.00277 ± 0.00069) × 122473 , = 1131  g=  (5.31)  = 0.300 ± 0.075, which was the upper limit on Kπ2γ contamination in the Kπ2 target-scatter normalization branch. Section 4.1.5 details how this upper limit contributed to the process of correcting the Kπ2 target-scatter background for Kπ2γ contamination.  164  Chapter 6 Signal Acceptance and Sensitivity To accurately measure the branching ratio of K + → π + ν ν¯, the components of acceptance of the many cuts applied to the signal data needed to be carefully measured. As with the rest of this analysis, experimental data were used instead of Monte Carlo simulation wherever possible. The Kπ2 , Kµ2 and πscatter monitor trigger data were used to measure the components of acceptance of most of the cuts, with the Monte Carlo simulation being used to measure primarily those cuts involving decay phase space and trigger efficiencies. Since the monitor trigger data was collected in parallel with the signal data, effects due to running conditions such as rate effects were automatically taken into account. To measure the acceptance “acc” of a cut, the cut is applied to a sample of N events, leaving n events that survive the cut. Thus the acceptance is defined as  acc =  n . N  (6.1)  The acceptance is the probability of success of a cut and is thus described by a binomial distribution. The uncertainty on the acceptance measurement is then given by the  165  standard deviation on a measurement of N trials,  σacc = =  6.1  1 N  N · acc · (1 − acc), acc · (1 − acc) . N  (6.2)  Acceptance Factors from Kµ2 Monitor Trigger Events  With a single charged track, and no photons or other activity in the detector, Kµ2 and K + → π + ν ν¯ events were topologically very similar with respect to event reconstruction and hit patterns in many subsystems. The Kµ2 monitor trigger events were used to measure the acceptance of event reconstruction cuts in the Range-Stack (ARS ), event reconstruction cuts in the target and UTC (Arecon ), and the acceptance of the photon veto (APV ). These monitor trigger events were also used to measure the beam cuts and the target-quality cuts that did not involve pion energies (Abeam ). Table 6.1 shows the setup cuts applied to create the samples used for each of these categories of cuts.  6.1.1  Range-Stack Tracking Acceptance  To measure the acceptance ARS (Table 6.2) of the Range-Stack tracking, the setup cuts SetupRS were used to create a sample consisting of good tracks arriving at the RangeStack. This was done by ensuring successful reconstruction in the target (TRIGGER, ICBIT) and UTC (UTC, UTC QUAL), both of which are independent of Range-Stack reconstruction. To ensure it was a stopped K + decay, B4DEDX made sure that the beam particle had an energy consistent with a K + , and a modified delayed-coincidence condition (tIC − tCK > 5ns) ensured that the kaon did not decay in flight. The modified ˇ delayed coincidence condition, using the kaon time in the Cerenkov counters and the hit 166  Kµ2 Setup Categories  Measured Quantities  SetupRS  ARS  TRIGGER, ICBIT, tIC − tCK > 5ns, B4DEDX, UTC, UTC QUAL  Setuprecon  Arecon  TRIGGER, ICBIT, tIC − tCK > 5ns, B4DEDX, CPITRS, CPITAIL, CKTRS, CKTAIL, BWTRS, RDTRK, TRKTIM, |tIC − tRS | < 5ns, PVCUT(noBV,noBVL)  Setupbeam  Abeam  TRIGGER, ICBIT, RDTRK, TRKTIM, RDUTM, KM2PBOX, COS3D  SetupPV  APV  Setupbeam , Abeam cuts, stopping layer < 19  Setup Cuts  Table 6.1: The setup cuts applied for the K µ2 -based acceptance measurements. The notation “Abeam cuts” means that all the cuts whose acceptance were measured to determine the quantity Abeam were applied as setup cuts. KM2PBOX selected events in the momentum range 226 MeV/c < ptot < 246 MeV/c. The photon veto “PVCUT(noBV,noBVL)” was applied with the Barrel-Veto and Barrel-Veto Liner subsystems excluded. Table reproduced from [56].  time in the IC, was used because the standard delayed coincidence cut DELC3 indirectly required a reconstructed track in the Range-Stack.  6.1.2  Target and UTC Reconstruction Efficiency  To measure the reconstruction efficiency Arecon (Table 6.3) of the target and UTC, the setup cuts Setuprecon were applied to create a sample which required a K + decay (B4DEDX, tIC − tCK > 5ns) without beam contamination (CPITRS, CPITAIL, CKTRS, CKTAIL, Loose  Cut Events SetupRS RD TRK TRKTIM ARS  Tight  Acceptance  2967140 2967140 1.0000 ± 0.0000 2966943 0.9999 ± 0.0000 0.9999 ± 0.0000  Events  Acceptance  2967140 2967140 1.0000 ± 0.0000 2966943 0.9999 ± 0.0000 0.9999 ± 0.0000  Table 6.2: The acceptance of the range-stack-reconstruction cuts using K µ2 monitor trigger events. Table reproduced from [56].  167  Loose  Cut Events Setuprecon RDUTM TARGET Arecon  Tight  Acceptance  1542443 1541571 0.9994 ± 0.0000 1541571 1.0000 ± 0.0000 0.9994 ± 0.0000  Events  Acceptance  759060 758792 0.9996 ± 0.0000 758792 1.0000 ± 0.0000 0.9996 ± 0.0000  Table 6.3: The acceptance of the target and UTC reconstruction cuts using K µ2 monitor trigger events. Table reproduced from [56].  BWTRS). The sample also required that a good track crossed through the UTC to the Range-Stack (|tIC −tRS | < 5ns, RD TRK, TRKTIM), and required no photons in order to avoid possible contamination from Kµ2γ . To avoid self-vetoing from events where the µ+ passed through the Range-Stack and into the Barrel-Veto and Barrel-Veto Liner, these subsystems were excluded from the photon veto applied to this sample as indicated by the notation “PVCUT(noBV,noBVL)”.  6.1.3  Beam and Target-Quality Acceptance  To measure the acceptance Abeam (Table 6.4) of the beam and selected target-quality cuts, the setup cuts Setupbeam were applied to select K + decays with a single charged track, no photons, no beam contamination and no other activity in the detector. Successfully reconstructed Kµ2 events were chosen by restricting the momentum to the Kµ2 peak (KM2PBOX), ensuring the charged track entered the active region of the detector (COS3D), and ensuring the quality of the track (RD TRK, TRKTIM, RDUTM). The cuts measured by this sample were ordered to obtain the most meaningful acceptance for each individual cut. For example, most of the target-quality cuts requires a successful target reconstruction, so TGQUALT was applied before any of the other cuts.  168  Loose  Cut Events Setupbeam TGCUT TGQUALT NPITG TIMCON TGTCON B4ETCON DCBIT DELCO PSCUT B4DEDX BWTRS CPITRS CPITAIL CKTRS CKTAIL B4TRS B4CCD UPVTRS RVTRS TGGEO B4EKZ TGZFOOL TARGF DTGTTP RTDIF TGKTIM EICCON TICCON PIGAP TGB4 PHIVTX CCDPUL EPIONK CCDBADTIM CCD31FIB TIMKF VERRNG ANGLI ALLKFIT TPICS KIC Abeam  3824854 3741291 3610937 3610937 3605667 3566647 3531329 3110649 2665661 2528550 2514694 2308180 2304287 2303213 2288540 2251649 2193877 2164219 2128633 2126603 2041316 1861055 1838070 1778937 1778930 1761888 1744527 1697720 1697716 1682926 1588984 1541372 694731 691595 684667 684658 628179 585499 585135 577756 577003 576825  Tight  Acceptance 0.9782 ± 0.0001 0.9652 ± 0.0001 1.0000 ± 0.0000 0.9985 ± 0.0000 0.9892 ± 0.0001 0.9901 ± 0.0001 0.8809 ± 0.0002 0.8569 ± 0.0002 0.9486 ± 0.0001 0.9945 ± 0.0001 0.9179 ± 0.0002 0.9983 ± 0.0000 0.9995 ± 0.0000 0.9936 ± 0.0001 0.9839 ± 0.0001 0.9743 ± 0.0001 0.9865 ± 0.0001 0.9836 ± 0.0001 0.9990 ± 0.0000 0.9599 ± 0.0001 0.9117 ± 0.0002 0.9876 ± 0.0001 0.9678 ± 0.0001 1.0000 ± 0.0000 0.9904 ± 0.0001 0.9901 ± 0.0001 0.9732 ± 0.0001 1.0000 ± 0.0000 0.9913 ± 0.0001 0.9442 ± 0.0002 0.9700 ± 0.0001 0.4507 ± 0.0004 0.9955 ± 0.0001 0.9900 ± 0.0001 1.0000 ± 0.0000 0.9175 ± 0.0003 0.9321 ± 0.0003 0.9994 ± 0.0000 0.9874 ± 0.0002 0.9987 ± 0.0001 0.9997 ± 0.0000  0.1508 ± 0.0002  Events 3824854 3741291 3610937 3610937 3605667 3566647 3531329 3110649 2191189 2074552 2063095 1892260 1889137 1888271 1878953 1867769 1818255 1798933 1770430 1768838 1696457 1544226 1525163 1475963 1475956 1461737 1456412 1417410 1417407 1405081 1327709 1283527 633868 630732 624344 624335 572339 533386 533050 526144 525413 525245  Acceptance 0.9782 ± 0.0001 0.9652 ± 0.0001 1.0000 ± 0.0000 0.9985 ± 0.0000 0.9892 ± 0.0001 0.9901 ± 0.0001 0.8809 ± 0.0002 0.7044 ± 0.0002 0.9468 ± 0.0001 0.9945 ± 0.0001 0.9172 ± 0.0002 0.9983 ± 0.0000 0.9995 ± 0.0000 0.9951 ± 0.0001 0.9940 ± 0.0001 0.9735 ± 0.0001 0.9894 ± 0.0001 0.9842 ± 0.0001 0.9991 ± 0.0000 0.9591 ± 0.0001 0.9103 ± 0.0002 0.9877 ± 0.0001 0.9677 ± 0.0001 1.0000 ± 0.0000 0.9904 ± 0.0001 0.9964 ± 0.0001 0.9732 ± 0.0001 1.0000 ± 0.0000 0.9913 ± 0.0001 0.9449 ± 0.0002 0.9667 ± 0.0001 0.4938 ± 0.0004 0.9951 ± 0.0001 0.9899 ± 0.0001 1.0000 ± 0.0000 0.9167 ± 0.0004 0.9319 ± 0.0003 0.9994 ± 0.0000 0.9870 ± 0.0002 0.9986 ± 0.0001 0.9997 ± 0.0000  0.1373 ± 0.0002  Table 6.4: The acceptance of the target and beam cuts using K µ2 monitor trigger events. Table reproduced from [56].  169  6.1.4  Photon Veto Acceptance  To measure the acceptance APV (Table 6.5) of the photon veto cuts, the setup cuts SetupPV were applied to create a sample of successfully reconstructed Kµ2 events without additional beam particles at decay time. To measure the acceptance of the photon veto, the same conditions as used for measuring Abeam were needed, but an even cleaner sample was created by also applying all the cuts measured for Abeam . Additionally, the requirement that the muon stopped prior to the 19th Range-Stack layer (stopping layer < 19) was imposed so that muons penetrating into the BVL and BV were not used for the measurement. Since the Kµ2 monitor trigger had no online PV requirement, it was possible to measure the acceptance of online and offline photon veto cuts with this sample.  6.1.5  Kµ2 -Based Acceptance Summary  The total acceptance AKµ2 was the product of the four categories of components of acceptance measured using Kµ2 monitor trigger events:  AKµ2 = ARS × Arecon × Abeam × APV .  (6.3)  Table 6.6 summarizes the results.  6.2  Acceptance Factors from πscatter Monitor Trigger Events  The πscatter monitor trigger selected beam pions that scattered into the active region of the detector. These events were similar to K + → π + ν ν¯ in that the πscatter events covered the entire phase space of the kinematic signal region PNN2BOX and they provided a single π + in the Range-Stack with a continuous stopping layer distribution. The πscatter  170  Loose  Cut Events SetupPV LHEX HEXAFTER PVONLINE LAY20or21 STLAY RSHEX PVCUT TGPVCUT TGPVTR  62556 58388 56244 53832 53413 52910 50992 49039 48558 48558  TGPV ICPV VCPV COPV MCPV ECinner ECouter EC 2nd RSPV BVPV BVLPV ADPV EARLYBV DPV EARLYBV L PV60  47044 46996 46966 46707 46702 43191 37652 37390 34680 32182 31668 30132 30106 30103 30103 -  APVCUT APVALL  Tight  Acceptance 0.9334 ± 0.0010 0.9633 ± 0.0008 0.9571 ± 0.0009 0.9922 ± 0.0004 0.9906 ± 0.0004 0.9637 ± 0.0008 0.9617 ± 0.0009 0.9902 ± 0.0005 1.0000 ± 0.0000  0.9688 ± 0.0008 0.9990 ± 0.0002 0.9994 ± 0.0001 0.9945 ± 0.0003 0.9999 ± 0.0001 0.9248 ± 0.0012 0.8718 ± 0.0016 0.9930 ± 0.0004 0.9275 ± 0.0013 0.9280 ± 0.0014 0.9840 ± 0.0007 0.9515 ± 0.0012 0.9991 ± 0.0002 0.9999 ± 0.0001 1.0000 ± 0.0000 -  0.6199 ± 0.0022 0.4812 ± 0.0020  Events 56294 52530 50621 48449 48069 47609 45855 44092 43661 43661 40121 40007 39933 39778 39768 31655 25258 23395 16681 15318 15108 14439 14433 14432 14432 14120  Acceptance 0.9331 ± 0.0011 0.9637 ± 0.0008 0.9571 ± 0.0009 0.9922 ± 0.0004 0.9904 ± 0.0004 0.9632 ± 0.0009 0.9616 ± 0.0009 0.9902 ± 0.0005 1.0000 ± 0.0000  0.9189 ± 0.0013 0.9972 ± 0.0003 0.9981 ± 0.0002 0.9961 ± 0.0003 0.9997 ± 0.0001 0.7960 ± 0.0020 0.7979 ± 0.0023 0.9262 ± 0.0016 0.7130 ± 0.0030 0.9183 ± 0.0021 0.9863 ± 0.0009 0.9557 ± 0.0017 0.9996 ± 0.0002 0.9999 ± 0.0001 1.0000 ± 0.0000 0.9784 ± 0.0021  0.3234 ± 0.0022 0.2508 ± 0.0018  Table 6.5: The online and offline acceptance of the photon veto cuts using K µ2 monitor trigger events. The acceptance APVCUT refers specifically to the acceptance of the cut PV60 (loose) or PV30 (tight). The component of acceptance A PVALL is the product of all components of acceptance measured in this table. Table reproduced from [56].  171  ARS Arecon Abeam APVALL AKµ2  Loose  Tight  0.9999 ± 0.0000 0.9994 ± 0.0000 0.1508 ± 0.0002 0.4812 ± 0.0020  0.9999 ± 0.0000 0.9996 ± 0.0000 0.1373 ± 0.0002 0.2508 ± 0.0018  0.0725 ± 0.0003 0.0344 ± 0.0003  Table 6.6: The Kµ2 -based acceptance summary Table reproduced from [56].  πscatter Setup Categories  Measured Quantities  SetupBAD STC  ABADSTC  RD TRK, TRKTIM, STLAY, UTC, RDUTM, PDC, ICBIT, b4abm2 < 1.3MeV, |tpi − tRS | < 5ns, |tIC − tRS | < 5ns, TARGF, DTGTTP, RTDIF, TGQUALT, TGZFOOL, CKTRS, CKTAIL, PVCUT(onlyRS), COS3D, LAYV4, BOXLOOSE  SetupRS−kin  ARS−kin , Asmall RS−kin , large ARS−kin  SetupBAD STC , BAD STC, TDCUTS  Setupµ→π→e  ATD1 , ATD2  SetupBAD STC , BAD STC, RNGMOM, ZFRF, ZUTOUT, LAYER14, UTCQUAL, EIC  Setup Cuts  Table 6.7: The setup cuts applied for the π scatter -based acceptance measurements. The quantity b4abm2 is the energy deposited in the B4 near beam time. Table reproduced from [56].  monitor trigger events were used to measure the acceptance of Range-Stack stopping counter reliability (ABADSTC ), kinematic and quality cuts in the UTC and Range-Stack (ARS−kin ), and particle identification in the Range-Stack (Aπ→µ→e ). Table 6.7 shows the setup cuts applied for each of these categories of cuts.  6.2.1  Range-Stack Stopping Counter Reliability  The setup cuts SetupBADSTC were applied to create the sample used to measure the acceptance ABADSTC (Table 6.8) of BAD STC, a cut that removed events when the TD in the Range-Stack stopping counter was not working properly. This sample required that  172  Loose  Cut Events SetupBAD STC BAD STC ABADSTC  Tight  Acceptance  Events  Acceptance  153716 153474 0.9984 ± 0.0001  74214 74093 0.9984 ± 0.0001  0.9984 ± 0.0001  0.9984 ± 0.0001  Table 6.8: The acceptance of BAD STC using π scatter monitor trigger events. Table reproduced from [56].  the events had a single beam π + that scattered in the target and entered the Range-Stack. For this sample, beam kaons were removed (b4abm2 < 1.3MeV, CKTRS, CKTAIL), the tracks in the target and Range-Stack were required to be created from the same particle (|tpi − tRS | < 5ns, |tIC − tRS | < 5ns), the loose kinematic signal region was chosen (BOXLOOSE), and a well reconstructed track was required (the remaining cuts). Additionally, the Range-Stack subsystems of the photon veto “PVCUT(onlyRS)” were applied to remove coincident activity in the Range-Stack.  6.2.2  Range-Stack-Kinematic Acceptance  To measure the acceptance ARS−kin (Table 6.10) of the kinematic and quality cuts in the UTC and Range-Stack, the setup cuts SetupRS−kin were applied to create a sample of events that had a single beam π + that scattered in the target and decayed at rest in the Range-Stack. In addition to the setup cuts SetupBADSTC , the π + → µ+ → e+ decaysequence cuts TDCUTS were applied to remove events where the π + decayed in flight in the Range-Stack. Good target reconstruction required good classification of the kaon and pion fibers, but since the incoming particle was a beam pion and not a kaon, the clustering based on times and energies was not as reliable as for a stopped-kaon decay. This poor target reconstruction led to less well-measured momentum (ptot), range (rtot) and energy (etot) of the outgoing particle. To determine the systematic uncertainty associated with 173  these larger uncertainties in momentum, range and energy, the kinematic signal region cuts (BOXLOOSE) were all loosened and tightened by one standard deviation to create small and large kinematic boxes, respectively. The relative resolutions of the reconstructed π + mass,  mπ =  ptot2 − etot2 , 2 · etot  (6.4)  from πscatter and Kπ2 events were used to determine the size of the shifts to the kinematic signal region. The distributions of mπ for these events are shown in Figure 6.1, where σπscatter was 13.8 MeV and σKπ2 was 8.4 MeV. Taking the mean reconstructed mass from the Kπ2 events as 139.4 MeV, the relative uncertainty in the reconstructed mass of these two types events was determined, √ 13.82 − 8.42 = 7.8%. 139.4  (6.5)  The contributions to the resolution of the reconstructed mass from the momentum and √ energy were roughly the same, so the uncertainties in each were taken as 7.8%/ 2 = 5.5%. Additionally, the range scaled with energy so the uncertainty in rtot was also taken as 5.5%. The boundaries of the nominal loose kinematic signal region (BOXLOOSE) were loosened and tightened by 5.5% to create BOXSMALL and BOXLARGE as shown in Table 6.9. These loosened and tightened kinematic signal regions were applied in SetupRS−kin large to measure Asmall RS−kin (Table 6.11) and ARS−kin (Table 6.12), respectively. The variations large between the components of acceptance Asmall RS−kin and ARS−kin were used to determine the  systematic uncertainty for the measurement ARS−kin . The final values for ARS−kin are shown in Table 6.15.  174  Figure 6.1: Distributions of the reconstructed π + mass from πscatter (top) and Kπ2 events (bottom). Figure reproduced from [56].  BOXLOOSE BOXSMALL BOXLARGE  Momentum ptot (MeV/c)  Energy etot (MeV)  Range rtot (cm)  140 − 199 147.7 − 188.1 132.3 − 209.9  60.0 − 100.5 63.3 − 95.0 56.7 − 106.7  12 − 28 12.7 − 26.5 11.3 − 29.5  Table 6.9: The “small” and “large” versions of the loose kinematic signal regions as created by loosening and tightening the nominal PNN2BOX by one standard deviation (5.5%). Table reproduced from [56].  175  Loose  Cut Events SetupRS−kin UTCQUAL RNGMOM RSDEDXMAX RSDEDXCL RSLIKE PRRF1 PRRFZ ARS−kin  88719 84373 82845 80449 76828 76828 76196 73596  Tight  Acceptance 0.9510 ± 0.0007 0.9819 ± 0.0005 0.9711 ± 0.0006 0.9550 ± 0.0007 1.0000 ± 0.0000 0.9918 ± 0.0003 0.9659 ± 0.0007  0.8295 ± 0.0013  Events 32932 31672 31161 30355 29048 29048 28841 27862  Acceptance 0.9617 ± 0.0011 0.9839 ± 0.0007 0.9741 ± 0.0009 0.9569 ± 0.0012 1.0000 ± 0.0000 0.9929 ± 0.0005 0.9661 ± 0.0011  0.8461 ± 0.0020  Table 6.10: The acceptance of the Range-Stack-kinematic cuts using π scatter monitor trigger events. The “Tight” label indicates the tightening of TDCUTS and PVCUT(onlyRS). Table reproduced from [56].  Loose  Cut Events Setupsmall RS−kin UTCQUAL RNGMOM RSDEDXMAX RSDEDXCL RSLIKE PRRF1 PRRFZ LAYER14 Asmall RS−kin  63400 60350 59251 57778 55375 55375 55017 53324 53324  Tight  Acceptance 0.9519 ± 0.0009 0.9818 ± 0.0005 0.9751 ± 0.0006 0.9584 ± 0.0008 1.0000 ± 0.0000 0.9935 ± 0.0003 0.9692 ± 0.0007 1.0000 ± 0.0000  0.8411 ± 0.0015  Events 29195 27906 27396 26746 25685 25685 25548 24778 24778  Acceptance 0.9558 ± 0.0012 0.9817 ± 0.0008 0.9763 ± 0.0009 0.9603 ± 0.0012 1.0000 ± 0.0000 0.9947 ± 0.0005 0.9699 ± 0.0011 1.0000 ± 0.0000  0.8487 ± 0.0021  Table 6.11: The acceptance of the Range-Stack-kinematic cuts in the small version of the loose kinematic box BOXSMALL using πscatter monitor trigger events. The “Tight” label indicates the tightening of TDCUTS and PVCUT(onlyRS). Table reproduced from [56].  176  Loose  Cut Events Setuplarge RS−kin UTCQUAL RNGMOM RSDEDXMAX RSDEDXCL RSLIKE PRRF1 PRRFZ LAYER14 Alarge RS−kin  110317 104830 102909 99517 94726 94726 93737 90176 90176  Tight  Acceptance 0.9503 ± 0.0007 0.9817 ± 0.0004 0.9670 ± 0.0006 0.9519 ± 0.0007 1.0000 ± 0.0000 0.9896 ± 0.0003 0.9620 ± 0.0006 1.0000 ± 0.0000  Events 51078 48730 47846 46347 44201 44201 43806 42205 42205  0.8174 ± 0.0012  Acceptance 0.9540 ± 0.0009 0.9819 ± 0.0006 0.9687 ± 0.0008 0.9537 ± 0.0010 1.0000 ± 0.0000 0.9911 ± 0.0005 0.9635 ± 0.0009 1.0000 ± 0.0000  0.8263 ± 0.0017  Table 6.12: The acceptance of the Range-Stack-kinematic cuts in the large version of the loose kinematic box BOXLARGE using πscatter monitor trigger events. The “Tight” label indicates the tightening of TDCUTS and PVCUT(onlyRS). Table reproduced from [56].  6.2.3  π + → µ+ → e+ Identification Acceptance  To measure the acceptance Aπ→µ→e (Tables 6.13 and 6.14) of the Range-Stack particle identification cuts, the setup cuts Setupπ→µ→e were applied to create a sample of events that had a single beam π + that scattered in the target and entered the Range-Stack. In addition to the setup cuts SetupBADSTC , the Range-Stack kinematic cuts were applied to ensure that the track in the Range-Stack was due to a π + . Since the πscatter monitor trigger did not include the online L1.1 and L1.2 triggers, the components of acceptance of these online cuts were also measured with this sample. Some small correlations existed between some of the cuts measured in ARS−kin and Aπ→µ→e . A µ+ accidental along the Range-Stack track could have been rejected by both the dE/dx condition of RSDEDX and by the µ+ → e+ decay requirement of EV5. The other correlation was between the dependence of the Range-Stack stopping-counter energy in PRRF and the TD-pulse fitting information used by TDNN. To examine the effects of these correlations, the cuts measured for Aπ→µ→e were measured without (ATD1 , Table 6.13) and with (ATD2 , Table 6.14) RSDEDX and PRRF applied as setup cuts. The 177  Loose  Cut Events Setupπ→µ→e PIFLG RSHEX2 LEV1.1 LEV1.2 TDCUT ELVETO TDFOOL TDNN EV5 ATD1  126239 104055 102123 82659 69374 65186 62425 62208 58607 58607  Tight  Acceptance 0.8243 ± 0.0011 0.9814 ± 0.0004 0.8094 ± 0.0012 0.8393 ± 0.0013 0.9396 ± 0.0009 0.9576 ± 0.0008 0.9965 ± 0.0002 0.9421 ± 0.0009 1.0000 ± 0.0000  Events 64210 53280 52271 42382 38160 35907 34453 34343 29016 24264  0.4643 ± 0.0014  Acceptance 0.8298 ± 0.0015 0.9811 ± 0.0006 0.8108 ± 0.0017 0.9004 ± 0.0015 0.9410 ± 0.0012 0.9595 ± 0.0010 0.9968 ± 0.0003 0.8449 ± 0.0020 0.8362 ± 0.0022  0.3779 ± 0.0019  Table 6.13: The acceptance of the π + → µ+ → e+ cuts using πscatter monitor trigger events. Table reproduced from [56].  measurement for ATD2 included additional acceptance loss due to π + absorption and π + decay-in-flight for which a 1.4% correction factor was estimated using Monte Carlo [55] and applied to ATD2 . The value for Aπ→µ→e was taken as the average of ATD1 and ATD2 , with the systematic uncertainty taken from the difference.  6.2.4  πscatter -Based Acceptance Summary  The total acceptance Aπscatter was the product of the three categories of components of acceptance measured using πscatter monitor trigger events:  Aπscatter = ABADSTC × ARS−kin × Aπ→µ→e . Table 6.15 summarizes the results.  178  (6.6)  Loose  Cut Events Setupπ→µ→e RSDEDX, PRRF PIFLG RSHEX2 L1.1 L1.2 TDCUT ELVETO TDFOOL TDNN EV5 ATD2 (uncorrected) ATD2  107124 107124 90161 88616 72545 61913 58288 55833 55655 52472 52472  Tight  Acceptance  Events  0.8417 ± 0.0011 0.9829 ± 0.0004 0.8186 ± 0.0013 0.8534 ± 0.0013 0.9415 ± 0.0009 0.9579 ± 0.0008 0.9968 ± 0.0002 0.9428 ± 0.0010 1.0000 ± 0.0000  0.4898 ± 0.0015  0.4967 ± 0.0015  55113 55113 46466 45640 37347 34125 32155 30859 30774 26060 21820  Acceptance  0.8431 ± 0.0016 0.9822 ± 0.0006 0.8183 ± 0.0018 0.9137 ± 0.0015 0.9423 ± 0.0013 0.9597 ± 0.0011 0.9972 ± 0.0003 0.8468 ± 0.0021 0.8373 ± 0.0023  0.3959 ± 0.0021  0.4015 ± 0.0021  Table 6.14: The acceptance of the π + → µ+ → e+ cuts using πscatter monitor trigger events. The acceptance ATD2 (uncorrected) was the acceptance before the correction factor of 1.014 was applied. This correction factor corrected for π + decay-in-flight and π + absorption in the stopping counter. Table reproduced from [56].  Loose  Tight  ABADSTC ARS−kin Aπ→µ→e  0.9984±0.0001 0.8295±0.0013±0.012 0.4805±0.0015±0.016  0.9984±0.0001 0.8461±0.0020+0.003 −0.020 0.3897±0.0021±0.012  Aπscat  0.3980±0.0014±0.014  0.3292±0.002  +0.010 −0.013  Table 6.15: The πscatter -based acceptance summary Table reproduced from [56].  179  Kπ2 Setup Categories  Measured Quantities  Setuputc  Autc  TRIGGER, RD TRK, TRKTIM, STLAY, BAD STC  Setupopsveto  Aopsveto  Setuputc , UTC, RDUTM, PDC, BEAMCUTS, DELCO, KINCUTS, TGCUTS (excluding ATG−kin and OPSVETO), TDCUTS, KP2BOX  SetupTG−kin  ATG−kin  Setupopsveto , OPSVETO, TGPVCUT  Setup Cuts  Table 6.16: The setup cuts applied for the K π2 -based acceptance measurements. The notation “Aopsveto cuts” means that all the cuts whose acceptance were measured to determine the quantity Aopsveto were applied as setup cuts. Table reproduced from [56].  6.3  Acceptance Factors from Kπ2 Monitor Trigger Events  The Kπ2 monitor trigger events were similar to K + → π + ν ν¯ in that they both had a single outgoing π + coming from the incoming K + , thus Kπ2 monitor trigger events were used to measure the acceptance of cuts requiring good decay-vertex determination in the target. The Kπ2 monitor trigger events were used to measure the acceptance of the PASS1 UTC cut (AUTC ), OPSVETO (Aopsveto ), and the target kinematic cuts (ATG−kin ). Table 6.16 shows the setup cuts applied for each of these categories of cuts.  6.3.1  UTC Acceptance  To measure the acceptance Autc (Table 6.17) of the PASS1 UTC cut, the setup cuts Setuputc were applied to create a sample with valid reconstruction of events in the target and Range-Stack.  180  Cut Setuputc UTC AUTC  Events  Acceptance  1502895 1417906 0.9435 ± 0.0002 0.9435 ± 0.0002  Table 6.17: The acceptance of UTC using K π2 monitor trigger events. Table reproduced from [56].  Cut Setupopsveto OPSVETO Aopsveto  Events  Acceptance  64024 62370 0.9742 ± 0.0006 0.9742 ± 0.0006  Table 6.18: The acceptance of OPSVETO using K π2 monitor trigger events. Table reproduced from [56].  6.3.2  OPSVETO Acceptance  To measure the acceptance Aopsveto (Table 6.18) of OPSVETO, the setup cuts Setupopsveto were applied to create a sample of events with valid reconstruction in the target, UTC and Range-Stack, along with the requirement of no secondary beam particles. These setup cuts consisted of all analysis cuts other than the photon veto PVCUT, OPSVETO and the target quality cuts TGCUTS measured in ATG−kin . The group of cuts BEAMCUTS was applied to remove secondary beam particles. The kinematic box cut KP2BOX and the rest of the listed cuts were applied to ensure good Kπ2 decays.  6.3.3  Target Kinematic Acceptance  The cuts measured in the acceptance ATG−Kin (Table 6.19) were target-kinematic cuts that required valid reconstruction in the target, UTC and Range-Stack along with the requirements of no secondary beam particles and the decay product to be a π + . The setup cuts SetupTG−kin were applied to create a sample similar to that used for Aopsveto , with the additional application of OPSVETO and the online target photon veto TGPVCUT 181  Loose  Cut Events SetupTG−kin TGDEDX TGER TGENR TGLIKE1 TGLIKE2 EPITG EPIMAXK TGEDGE DRP CHI567 CHI5MAX ATG−kin  61687 61017 61000 58984 57931 57005 51086 51086 50802 50716 44324 44323  Tight  Acceptance 0.9891 ± 0.0004 0.9997 ± 0.0001 0.9670 ± 0.0007 0.9821 ± 0.0006 0.9840 ± 0.0005 0.8962 ± 0.0013 1.0000 ± 0.0000 0.9944 ± 0.0003 0.9983 ± 0.0002 0.8740 ± 0.0015 1.0000 ± 0.0000  Events 37295 36883 36873 35594 34946 34381 30874 30874 30715 30658 26823 26822  0.7185 ± 0.0018  Acceptance 0.9890 ± 0.0005 0.9997 ± 0.0001 0.9653 ± 0.0010 0.9818 ± 0.0007 0.9838 ± 0.0007 0.8980 ± 0.0016 1.0000 ± 0.0000 0.9949 ± 0.0004 0.9981 ± 0.0003 0.8749 ± 0.0019 1.0000 ± 0.0000  0.7192 ± 0.0023  Table 6.19: The acceptance of the target kinematic cuts using K π2 monitor trigger events. Table reproduced from [56].  to remove additional activity in the target.  6.3.4  Kπ2 -Based Acceptance Summary  The total acceptance AKπ2 was the product of the three categories of components of acceptance measured using Kπ2 monitor trigger events:  AKπ2 = Autc × Aopsveto × ATG−kin .  (6.7)  Table 6.20 summarizes the results.  6.4  Acceptance Factors Using Monte Carlo  The acceptance losses of the online trigger (Atrigger ), and the phase space and solid angle cuts (Abox ) were measured using approximately 105 K + → π + ν ν¯ Monte Carlo events. 182  Loose  Tight  Autc Aopsveto ATG−kin  0.9435 ± 0.0002 0.9435 ± 0.0002 0.9742 ± 0.0006 0.9735 ± 0.0008 0.7185 ± 0.00181 0.7192 ± 0.0023  AKπ2  0.6604 ± 0.0018  0.6606 ± 0.0023  Table 6.20: The Kπ2 -based acceptance summary Table reproduced from [56].  The acceptance losses due to pion decay-in-flight and pion nuclear interactions were also measured due to the inclusion of these processes in Monte Carlo. The cut UFATE required that the pion stopped without decay or interaction. The cut USTMED required that the pion stopped in a scintillator counter in the Range-Stack. The cut USTOP HEX required that the offline reconstructed stopping counter agreed with the true stopping counter. These three cuts used information taken directly from the Monte Carlo event and not the subsequent reconstruction. The cut SETUP was a requirement that the reconstructed momentum ptot was less than 300 MeV.  6.5  Acceptance Summary  The total acceptance Atotal of signal events due to all analysis cuts was the product of the components of acceptance from each of the monitor trigger data and Monte Carlo:  Atotal = AKµ2 × Aπscatter × AKπ2 × AMC .  (6.8)  These values are summarized in Table 6.22.  6.6  Correction to T•2 Trigger Acceptance  A correction factor was determined to account for the fact that the simulated T•2 trigger in Monte Carlo did not include acceptance losses due to the geometric ( 183  geom )  and counter  Cut T•2 3ct · 4ct · 5ct · 6ct Trigger (πν ν¯(1) or πν ν¯(2)) Atrigger  99999 39227 27575 26288 0.2629 ± 0.0014  SETUP UFATE USTMED USTOP HEX COS3D LAYER14 ZFRF ZUTOUT  25793 22688 22517 21743 20870 20838 20175 20148  BOXLOOSE  9552  Aloose box  0.3703 ± 0.0030  Aloose MC  0.0974 ± 0.0009  BOXTIGHT  7758  Atight box  0.3008 ± 0.0029  Atight MC  0.0791 ± 0.0009  Table 6.21: The acceptance of the online trigger, phase space and solid angle cuts using Monte Carlo. Table reproduced from [56].  Loose AKµ2 Aπscatter AKπ2 AMC Atotal  Tight  0.0725 ± 0.0003 0.3980 ± 0.0014 ± 0.014 0.6604 ± 0.0018 0.0974 ± 0.0009  (1.857 ± 0.021 ± 0.065) × 10−3  0.0344 ± 0.0003 0.3292 ± 0.0020+0.010 −0.013 0.6606 ± 0.0023 0.0791 ± 0.0009  −3 (0.592 ± 0.009+0.018 −0.024 ) × 10  Table 6.22: The total acceptance of all online and offline cuts. Table reproduced from [56].  184  (  counter )  inefficiencies of the T-Counters. The T•2 trigger was a coincidence between the  first two layers of the RS (T-Counter and layer 2) in the same sector. The geometric inefficiency was due to tracks passing through the azimuthal gap between two adjacent T-Counters. The counter inefficiency was due to events that did not produce a large enough signal to pass our threshold. The counter inefficiency was expressed as  counter  = e−kE ,  (6.9)  where k was the number of photoelectrons produced per MeV and E was the mean energy deposited in the T-counters. Both inefficiencies were originally determined in [33] using Kπ2 and Kµ2 events from KB monitor trigger data (see Section 2.7 for trigger definitions). The KB monitor trigger data were used since the Kπ2 and Kµ2 triggers contain the T•2 trigger as one of their conditions. Reconstructed events pointing at the T-Counters were checked against their T•2 online trigger state to determine the inefficiencies. The values corresponding to the Kπ2 monitor trigger data were used for the T•2 trigger inefficiencies in this analysis due to their similarity to K + → π + ν ν¯ PNN2 pions. The geometric inefficiency was found to be  geom  = 0.0286 ± 0.0027.  For the counter inefficiency, k values of 1.74 and 1.62 photoelectrons per MeV were found. The first value was for all sectors apart from two that had hardware problems for which the second value applied. Monte Carlo K + → π + ν ν¯ events in the PNN2 kinematic signal region were used to determine the mean energy deposited in the T-counters as a function of momentum, giving a counter inefficiency of  counter  = 0.0210 ± 0.0027 [37].  Subtracting the two inefficiencies from unity gave a T•2 trigger efficiency of  T•2  = 0.9505 ± 0.0012stat. ± 0.0143sys. ,  185  (6.10)  where the 1.5% systematic uncertainty accounted for the variation observed when further constraining the z-position requirement of the reconstructed track passing through the T-counters as detailed in [33].  6.7  K + Stopping Fraction  The K + stopping fraction quantified the number of beam kaons that decayed in the target relative to the number of kaons that satisfied the KB trigger requirement. These ˇ kaons decayed after the Cerenkov counter with the daughter satisfying the B4 and target requirements of the KB trigger or they deposited some energy in the target and exited without decaying. This fraction was obtained as part of the E949 PNN1 analysis [33] by normalizing the total kaon exposure to the Kµ2 branching ratio using Kµ2 monitor trigger events. The stopping fraction was found to be  fs = 0.7740 ± 0.0011.  6.8  (6.11)  Measurement of the Kπ2 Branching Ratio  The measurement of the Kπ2 branching ratio served as a consistency check of the acceptance measurements detailed earlier in this chapter. The Kπ2 monitor trigger data were used for this measurement. The online prescale factor for these monitor trigger data was 163840 for runs earlier than and including run 48045. For runs after 48045, the prescale factor was 131072. The total number of stopped Kπ2 events in the monitor trigger data (NKπ2 ) was measured by applying the cuts shown in Figure 6.2. Acceptance measurements were performed for all the triggers and cuts applied to make the NKπ2 measurement. Details of the acceptance measurements used to determine the total acceptance AKπ2 are found in Appendix F.  186  TRIGGER, BAD STC, RD TRK, TRKTIM, RDUTM, ICBIT, DCBIT, COS3D, UTCQUAL, BEAMCUTS (excluding B4CCD, B4TRS)  DELC3, IPIGAP, PV60(noBV,noBVL), KP2BOX, KP2STOP  TIC, TARGF, DTGTTP, RTDIF, TGB4, KIC, B4EKZ, TGZFOOL  NKπ2 Figure 6.2: The cuts applied to measure N Kπ2 .  NKπ2 Prescale Factor KB live AKπ2 B (Kπ2 )  All Runs  Prescale 163840  Prescale 131072  144989 131926 1.792 × 1012 0.04833 ± 0.00047  2973 163840 0.052 × 1012 0.04921 ± 0.00084  141926 131072 1.740 × 1012 0.04830 ± 0.00047  0.2213 ± 0.0022  0.1905 ± 0.0051  0.2216 ± 0.0016  Table 6.23: The summary of the Kπ2 branching ratio measurements for all runs and for the two different prescale factors. Details for the constituent components of acceptance making up the total acceptance AKπ2 are given in Appendix F. Table reproduced from [55].  The Kπ2 branching ratio was measured using B K + → π+π0 =  PRESCALE · NKπ2 , KBlive · AKπ2  (6.12)  where PRESCALE was the prescale factor and KBlive was the total number of K + decays. Table 6.23 summarizes the results of this calculation for all runs (weighted average) and for the two different prescale factors. The average measured branching ratio of 0.2213±0.0022 overestimates the world average value of 0.2092 ± 0.0012 [99] by approximately 5.8%. It was also observed that the range of variation in the average measured branching ratio was approximately 4.4% as a function of rate as shown in Figure 6.3. 187  BR(Kπ2)  0.23  All runs prescale 163840  0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15  0  0.5  1  1.5  2  2.5  3 Rate  Figure 6.3: The Kπ2 branching fraction versus rate for all runs (large points) and for runs with prescaler 163840 (small points). Rate is measured in 10 6 KBlive per second during the spill. The arrow shows the average rate. Figure modified from [55].  188  Atotal T •2  fs KBlive SES  Loose  Tight  Source  (1.857 ± 0.021 ± 0.065) × 10−3  −3 0.592 ± 0.009 +0.018 −0.024 × 10  Table 6.22  0.9505 ± 0.0012 ± 0.0143  Section 6.6  0.7740 ± 0.0011  Ref. [92]  1.7096 × 1012  (4.28 ± 0.43) × 10−10  Section 2.8  (13.13 ± 1.31) × 10−10  Table 6.24: The summary of the values used to determine the single event sensitivity. For values with two sets of uncertainties, the first uncertainty is statistical and the second systematic. The total uncertainties of SES reflect the 10% total uncertainty applied to the acceptance due to consistency issues when measuring B K + → π + π 0 . Table reproduced from [55].  A relative systematic uncertainty of 10% was assigned to the total signal acceptance to account for the concerns discussed above and the inconsistencies in the measured branching ratios for the two different prescale factors.  6.9  Single-Event Sensitivity  The single event sensitivity (SES ) was the branching ratio that would have corresponded to one candidate in the absence of background. It was determined by  SES −1 = Atotal ×  T•2  × fs × KBlive ,  where Atotal was the total acceptance of all online and offline cuts,  (6.13)  T•2  was the T•2 trigger  efficiency, fs was the fraction of beam kaons stopping in the target, and KBlive was the total number of K + decays in the detector for the full data set. Table 6.24 summarizes SES for the loose and tight signal regions. The relative uncertainty associated with the single event sensitivity was conservatively set to 10% based on the uncertainties in the total acceptance and the discrepancy between the measured Kπ2 branching fraction and the world average (Section 6.8).  189  Chapter 7 Results 7.1  Cell Definitions  The signal region was divided into nine cells using the loose and tight versions of the kinematic phase space cuts (PNN2BOX), the delayed coincidence cuts (DELC), the π + → µ+ → e+ decay-sequence cuts (TDCUTS ), and the photon veto (PVCUT). Each of these cuts was used to divide the signal region into two, with the first region being defined by the application of the tight version of the cut and the second region being the rest of the signal region outside of the region defined by the tight version of the cut: KIN1 ≡ BOXTIGHT,  KIN2 ≡ BOXTIGHT · BOXLOOSE,  DC1 ≡ DELC6,  DC2 ≡ DELC6 · DELC3,  TD1 ≡ TDTIGHT,  TD2 ≡ TDTIGHT · TDLOOSE,  PV1 ≡ PV30,  PV2 ≡ PV30 · PV60.  Based on these regions, the signal region was first divided into two major regions defined by KIN1 and KIN2 . The major region defined by KIN1 was sub-divided into 8 cells based on all possible permutations of the regions defined by the other three cuts. The cell defined by the application of the tight version of each of the four cuts (cell 1) was the tight signal region and had the highest signal-to-background ratio of all the cells. 190  Cell Number 1 2∗ 3 4  ∗  5 6 7 8 9∗  Cuts  Acc.  Total Background  S/B  KIN1 · TD1 · DC1 · PV1  +0.047 0.314 0.152 ± 0.027 −0.035  0.84  KIN1 · TD1 · DC2 · PV1  0.031 0.019 ±  0.66  KIN1 · TD1 · DC1 · PV2  KIN1 · TD1 · DC2 · PV2  KIN1 · TD2 · DC1 · PV1  KIN1 · TD2 · DC1 · PV2  KIN1 · TD2 · DC2 · PV1  KIN1 · TD2 · DC2 · PV2 KIN2  0.287 0.243 ±  +0.054 0.044 −0.047 +0.007 0.005 −0.004  0.47  0.028 0.027 ±  +0.008 0.006 −0.005 +0.011 0.007 −0.008  0.42  0.066 0.059 ±  +0.012 0.011 −0.011  0.45  +0.002 0.007 0.005 ± 0.001 −0.001  0.57  +0.177 0.188 0.379 ± 0.074 −0.120  0.20  0.073 0.038 ±  +0.002 0.006 0.007 ± 0.001 −0.001  0.78  0.35  Table 7.1: Relative acceptance and background summary of each cell. The components of acceptance (“Acc.”) were normalized such that the entire signal region was normalized to have an acceptance equal to one. The signal-to-background ratios (“S/B”) were calculated assuming B (K + → π + ν ν¯) = 1.73 × 10−10 . The cells denoted with an ‘∗ ’ were the cells within which the candidate events were observed. Table modified from [56].  The ninth cell was the major region defined by KIN2 , and it had the lowest signal-tobackground ratio. These cell definitions are shown in Table 7.1.  7.1.1  Acceptance of Each Cell  For each of the four cuts used to define the cells, the additional acceptance loss when tightening a cut was defined as the ratio of the acceptance of the tight version of the cut to the acceptance of the loose version of the cut as determined in Chapter 6. These acceptance losses are summarized in Table 7.2. The relative acceptance of each cell was then calculated by the product of the appropriate acceptance losses for the cuts used to define the cell. For example, cell 2 was defined by the tight versions of each of PNN2BOX, TDCUTS and DELCO, and by the  191  Background  PNN2BOX  TDCUTS  DELC  PVCUT  0.911  0.522  Acceptance Loss 0.812  0.812  Rejection Above Acceptance Loss Kπ2 -tgscat Kπ2 -rsscat Kπ2γ Ke4 CEX Muon Beam  1.63 1.63 1.20 2.70 -  3.08 -  6.7 1.0  2.75 2.75 2.75 -  Table 7.2: Summary of the additional rejection above the acceptance loss for each background when tightening the kinematic phase space cuts (PNN2BOX), the delayed coincidence cuts (DELC), the π + → µ+ → e+ decay-sequence cuts (TDCUTS ), and the photon veto (PVCUT). Blank entries indicate that there was no rejection above the acceptance loss indicated in the “Acc. Loss” row. Table modified from [56].  region between the loose and tight versions of PVCUT, so the acceptance of that cell was A(cell 2) = A(KIN1 ) × A(TD1 ) × A(DC1 ) × A(PV2 ), = 0.812 × 0.812 × 0.911 × (1 − 0.522),  (7.1)  = 0.287. Using these relative components of acceptance, the entire signal region had a relative acceptance of one. The relative components of acceptance of each of the nine cells are summarized in Table 7.1.  7.1.2  Background Levels in Each Cell  In Chapter 4, the bifurcation method was used to estimate the backgrounds in the two signal regions known as the “loose” and the “tight” signal regions. In the context of the discussion of dividing the signal region into cells, the “loose” signal region was the  192  entire signal region which was divided into the nine cells. The “tight” signal region was equivalent to cell number 1, which was defined by the application of the tightened versions of each the cuts used to define the cells. What follows is a discussion of the scaling method used to estimate the background levels in each of the nine cells. The scaling method was verified by comparing the scaled result in cell number 1 to that found using the bifurcation method to directly estimate the background level in the “tight” signal region from Chapter 4. Using the scaling method, the total background level in each cell was calculated by estimating the contribution of each type of background to the total in that cell. For each of the tightened cuts, additional rejection above the acceptance loss was gained for some of the backgrounds, as summarized in Table 7.2. When evaluating the background level in each of the cells, the single- and double-beam backgrounds were treated as a single background called “Beam”. This simplification was possible due to the tiny contribution to the total background from the “Beam” background. Tightening the kinematic phase space cuts from BOXLOOSE to BOXTIGHT resulted in rejection above the acceptance loss for the Kπ2 -scatter and Ke4 backgrounds. The lower bounds of BOXTIGHT were chosen to heavily suppress Ke4 and this additional rejection was determined using the Monte Carlo simulation. The additional rejection for the Kπ2 target-scatter and Kπ2 Range-Stack-scatter backgrounds was determined using the loose and tight Kπ2 target-scatter normalization branches (Section 4.1.2) while taking into account the additional acceptance loss of the DELCO and TDCUTS between the two branches. It was assumed that the rejection of the photon veto for Kπ2 -scatter events was not correlated with PNN2BOX as discussed in Section 4.1.3. For the remaining backgrounds, the additional acceptance loss of BOXTIGHT above that of BOXLOOSE was used. Tightening the π + → µ+ → e+ decay-sequence cuts from TDLOOSE to TDTIGHT 193  resulted in rejection above the acceptance loss only for the muon background. The additional rejection on the muon background was taken from Table 4.9. For the remaining backgrounds, the additional acceptance loss of TDLOOSE above that of TDCUTS was used. Tightening the delayed coincidence cut from DELC3 to DELC6 resulted in rejection above the acceptance loss for the charge exchange background, measured using the Monte Carlo simulation. This cut should also have heavily suppressed the single-beam background, but the statistical limitations on the background estimate of this very tiny background resulted in no significant rejection above the acceptance loss being measured when tightening DELCO. Since the contributions of the single-beam background were very small, it was treated in the same way as the remaining backgrounds, where the change in background level was accounted for by the additional acceptance loss of DELC6 above that of DELC3. Tightening the photon veto from PV60 to PV30 resulted in rejection above the acceptance loss for the Kπ2 -scatter and Kπ2γ backgrounds. This additional rejection was determined from Table 4.5. For the remaining backgrounds, the additional acceptance loss of PV30 above that of PV60 was used. Using the scaling method the total background in cell number 1, which was equivalent +0.047 to the “tight” signal region, was 0.152 ± 0.027 −0.035 . Using the bifurcation method to  directly estimate the background in the “tight” signal region, the total background was +0.0952 0.1441 ± 0.0448 −0.0319 . These two values were consistent.  7.2  Examination of the Signal Region  Examination of the signal region revealed three candidate events. The kinematics of these events, along with the four previous E787/E949 K + → π + ν ν¯ candidates, are shown in Figure 7.1 and summarized in Table 7.3. Displays of various event parameters for the 194  ptot  etot  rtot  (MeV/c)  (MeV)  (cm)  Candidate A Candidate B Candidate C  161.4 188.4 191.3  76.1 95.6 98.0  95A (E787-PNN1) 96B (E787-PNN2) 98C (E787-PNN1) 02A (E949-PNN1)  218.2 180.7 213.8 227.3  117.7 86.3 117.1 128.9  S/B  Ref.  17.3 24.2 26.1  0.20 0.47 0.42  Cell 9 Cell 2 Cell 4  34.7 22.1 33.9 39.2  59 0.17 8 1.1  [11] [6] [11] [7]  Table 7.3: Summary of the π + kinematics and signal-to-background ratio (S/B) for all K + → π + ν ν¯ candidates. The signal-to-background ratios were calculated assuming B (K + → π + ν ν¯) = 1.73 × 10−10 . Table reproduced from [56].  Candidate A Candidate B Candidate C All E949 Data  ptot  etot  rtot  (MeV/c)  (MeV)  (cm)  204.3 ± 2.2 205.2 ± 2.4 205.3 ± 2.3 204.9 ± 2.3  110.3 ± 3.2 108.1 ± 3.0 108.8 ± 3.3 108.8 ± 3.0  30.6 ± 1.5 30.3 ± 0.9 30.4 ± 0.8 30.3 ± 0.9  Table 7.4: Summary of momentum (ptot), energy (etot) and range (rtot) measurements for Kπ2 events in the runs containing candidates. Table reproduced from [56].  three candidates are found in Figures 7.2, 7.3, and 7.4. The three candidates were found in cells 2, 4 and 9 as indicated by the ‘∗ ’ adjacent to these cell numbers in Table 7.1.  7.2.1  Consistency of Kπ2 Kinematics  For each of the runs containing an E949 PNN2 candidate, the momentum, energy and range of the π + from the Kπ2 events were compared to those found for the entire data set as shown in Table 7.4. For each of the runs these quantities were found to be consistent with those found for the entire data set.  195  Figure 7.1: Energy (etot) vs. range (rtot) of all candidate events passing all other cuts. The squares represent the events selected by this analysis. The circles and upward-pointing triangles represent the events selected by the E787 and E949 PNN1 analyses, respectively. The downwardpointing triangles represent the events selected by the E787 PNN2 analysis. The solid (dashed) lines represent the limits of the PNN1 and PNN2 signal regions for the E949 (E787) analyses. No kinematic cuts were applied to the simulated K + → π + ν ν¯ events (light gray). Despite the smaller signal region displayed in this figure, the PNN1 analyses were 4.2 times more sensitive than the PNN2 analyses. The events shown near Energy = 108 MeV were K π2 events that survived the photon veto. These events were predominantly events from the PNN1 analyses due to the higher sensitivity and less stringent photon veto cuts. The light gray points are simulated K + → π + ν ν¯ events that would be accepted by the πν ν¯(1) or πν ν¯(2) triggers.  196  100 6  80 60  4  40  2  20 0  0  -20  -2  -40  -4  -60 -80 -60  -40  -20  0  20  40  60  80  100  -6  -4  -2  0  2  4  5  500  40  30  300  25  200  140 120 100 80 60 40 20 05 0 -5 0  0  -5  35  400  100 05 0 -5 0  Residual  600  6  20 15 10 5  20  40  60  80  Decay Vertex 78 Raw High  100  120  0  0  20  40  60  End 1 Up (RAW)  80  100  120  Residual  -100 -100 -80  -6  5  40  0 -5  35 30 25 20 15 10 5 0  20  40  60  80  Decay Vertex 78 Raw Low  100  120  0  20  40  60  End 2 Down (RAW)  80  100  120  Figure 7.2: Some event parameter displays for Candidate A. Top-left is the end-view showing the UTC track fit where the circles along the track show the hits in the UTC. The layers hit in the RS and RSSC are shown outside the UTC. Additional hits shown in green show hits that were out of time with the event, in this case they were at least 40 ns before the K + entered the target. Top-right shows a magnification of the target where the red squares are the K + fibers and the blue ones are the π + fibers. Bottom-left shows the target CCD data for the vertex fiber where the upper plot is the “high-gain” CCD and the lower plot is the “low-gain”. The x-axis shows time in ns and the y-axis shows the pulse-height. The purple dashed line shows the result of the single-pulse fit for each of the CCD channels. Bottom-right shows the TD data in the π + stopping counter. The x-axis shows time in ns and the y-axis shows the pulse-height. The solid red line shows the result of the fit for the single-pulse assumption and the solid black line shows the result of the fit for the double-pulse assumption. Figures reproduced from [56].  197  100 6  80 60  4  40  2  20 0  0  -20  -2  -40  -4  -60 -80  120 100 80 60 40 20 05 0 -5 0  -40  -20  0  20  40  60  80  100  -6  -4  -2  0  2  4  6  Residual  450 400 350 300 250 200 150 100 50 05 0 -5 0  -60  5 0  225  -5  200 175 150 125 100 75 50 25  20  40  60  80  Decay Vertex 110 Raw High  100  120  0  0  20  40  60  80  End 1 Up (RAW)  100  120  Residual  -100 -100 -80  -6  5  0 -5  140 120 100 80 60 40 20 0  20  40  60  80  Decay Vertex 110 Raw Low  100  120  0  20  40  60  80  End 2 Down (RAW)  100  120  Figure 7.3: Some event parameter displays for Candidate B. Top-left is the end-view showing the UTC track fit where the circles along the track show the hits in the UTC. The layers hit in the RS and RSSC are shown outside the UTC. Additional hits shown in green show hits that were out of time with the event, in this case they were at least 50 ns before or 40 ns after the K + entered the target. Top-right shows a magnification of the target where the red squares are the K + fibers, the blue ones are the π + fibers and the green ones are the γ fiber. The energies in each of these γ fibers were significantly below the target photon veto threshold. Bottom-left shows the target CCD data for the vertex fiber where the upper plot is the “high-gain” CCD and the lower plot is the “low-gain”. The x-axis shows time in ns and the y-axis shows the pulse-height. The purple dashed line shows the result of the single-pulse fit for each of the CCD channels. Bottom-right shows the TD data in the π + stopping counter. The x-axis shows time in ns and the y-axis shows the pulse-height. The solid red line shows the result of the fit for the single-pulse assumption and the solid black line shows the result of the fit for the double-pulse assumption. Figures reproduced from [56].  198  100 6  80 60  4  40  2  20 0  0  -20  -2  -40  -4  -60 -80  90 80 70 60 50 40 30 20 10 05 0 -5 0  -40  -20  0  20  40  60  80  100  -6  -4  -2  0  2  4  6  Residual  400 350 300 250 200 150 100 50 05 0 -5 0  -60  5  180  0  160  -5  140 120 100 80 60 40 20  20  40  60  80  Decay Vertex 131 Raw High  100  120  0  0  20  40  60  End 1 Up (RAW)  80  100  120  Residual  -100 -100 -80  -6  5  0 -5  200 175 150 125 100 75 50 25 0  20  40  60  80  Decay Vertex 131 Raw Low  100  120  0  20  40  60  End 2 Down (RAW)  80  100  120  Figure 7.4: Some event parameter displays for Candidate C. Top-left is the end-view showing the UTC track fit where the circles along the track show the hits in the UTC. The layers hit in the RS and RSSC are shown outside the UTC. Additional hits shown in green show hits that were out of time with the event, in this case they were at least 50 ns before or 35 ns after the K + entered the target. Top-right shows a magnification of the target where the red squares are the K + fibers and the blue ones are the π + fibers. Bottom-left shows the target CCD data for the vertex fiber where the upper plot is the “high-gain” CCD and the lower plot is the “low-gain”. The x-axis shows time in ns and the y-axis shows the pulse-height. The purple dashed line shows the result of the single-pulse fit for each of the CCD channels. Bottom-right shows the TD data in the π + stopping counter. The x-axis shows time in ns and the y-axis shows the pulse-height. The solid red line shows the result of the fit for the single-pulse assumption and the solid black line shows the result of the fit for the double-pulse assumption. Figures reproduced from [56].  199  7.2.2  Signal Probability Analysis  To verify that the properties of each of the candidates were consistent with signal, quantities classified into four categories were checked: reconstruction quality, kaon identification quality, pion identification quality and single particle quality. For each cut, distributions of the important quantities used to discriminate signal from background were generated for signal-like events and the properties of each of the candidate events compared to these distributions. For each cut, the same sample that was used to measure the acceptance of the cut in Chapter 6 was also used to generate the reference distributions due to signallike events. The expectation was that an event which was consistent with a signal would have a fairly flat distribution when the probabilities due to each quantity were compared together. Figures 7.5 through 7.13 show the comparisons between the quantities from the candidate events and the reference distributions. For many of the cuts that target only a single specific background, a reference distribution for background-like events is also included. Reconstruction quality To make sure the candidates were of good reconstruction quality the quantities checked fell into the categories of timing consistency (Figure 7.5), target reconstruction (Figure 7.6), and UTC reconstruction (Figure 7.7). Kaon identification quality To make sure the beam particle in the candidate events was a kaon the quantities checked ˇ involved the B4 counters, Cerenkov hits at beam time and some target quantities (Figure 7.8).  200  Figure 7.5: Quantities related to timing consistency in reconstruction cuts. The abscissa label contains the quantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. For the reference signal-like distribution (black), the ordinate axis shows the number of events per abscissa bin. The measured quantity for each candidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green and candidate C in blue. Note that for some quantities a candidate has multiple entries and that when multiple quantities fell in the same abscissa bin, information from only one candidate was displayed. Sample for the signal-like distribution was from Kµ2 monitor trigger data.  201  Figure 7.6: Quantities related to reconstruction cuts in the target. The abscissa label contains the quantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. For the reference signal-like distribution (black), the ordinate axis shows the number of events per abscissa bin. The measured quantity for each candidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green and candidate C in blue. Note that for some quantities a candidate has multiple entries and that when multiple quantities fell in the same abscissa bin, information from only one candidate was displayed. Sample for the signal-like distribution was from Kµ2 monitor trigger data.  202  Figure 7.7: Quantities related to UTC and Range Stack reconstruction. The abscissa label contains the quantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. The reference signal-like (black) and background-like (purple) distributions were each normalized to an area equal to one, thus the ordinate axis shows the normalized number of events per abscissa bin. The measured quantity for each candidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green and candidate C in blue. Note that for some quantities a candidate has multiple entries and that when multiple quantities fell in the same abscissa bin, information from only one candidate was displayed. The sample for the reference signal-like distribution was from π scatter monitor trigger data in PNN2BOX and the sample for the reference background-like distribution was from K π2 monitor trigger data in KP2BOX.  203  Figure 7.8: Quantities related to target kaon reconstruction. The abscissa label contains the quantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. For the reference signal-like distribution (black), the ordinate axis shows the number of events per abscissa bin. The measured quantity for each candidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green and candidate C in blue. Note that for some quantities a candidate has multiple entries and that when multiple quantities fell in the same abscissa bin, information from only one candidate was displayed. Sample for the signal-like distribution was from Kµ2 monitor trigger data.  204  Pion identification quality To make sure the charged track in the candidate events was due to a pion the quantities checked fell into the categories of energy deposited in the IC counters (Figure 7.9), RangeStack kinematics (Figure 7.10), Range-Stack TD variables (Figure 7.11), and kinematics from Monte Carlo π + ν ν¯ events (Figure 7.12). Single particle quality To make sure there was only a single decay particle in the candidate events the quantities checked were related to beam detectors at track time in the Range-Stack (Figures 7.9 and 7.13). Signal-like Probability Based on the reference distributions, the signal-like probability was computed for each candidate for each of the quantities as summarized in Figure 7.14. These probabilities were determined using a cumulative integral for each of the reference distributions and were formed such that low probability corresponded to more background-like and high probability corresponded to more signal-like quantities. For distributions where one side was signal-like and the other side background-like, the probability distribution was built going from low probability on the background-like side to high probability on the signal-like side. For quantities where extreme values were background-like and median values were signal-like, the probability distribution was constructed to be very low for extreme values and high for median values. For quantities where extreme values were signal-like and median values were background-like, the probability distribution was constructed to be very high for extreme values and low for median values. Figure 7.14 shows the probabilities of each component of reconstruction quality, single  205  Figure 7.9: Quantities related to pion identification, single beam detection and photon veto. The top-left plot is related to pion identification and the rest of the plots to single beam detection and photon veto. The abscissa label contains the quantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. For the reference signal-like distribution (black), the ordinate axis shows the number of events per abscissa bin. The measured quantity for each candidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green and candidate C in blue. Note that for some quantities a candidate has multiple entries and that when multiple quantities fell in the same abscissa bin, information from only one candidate was displayed. Sample for the signal-like distribution was from K µ2 monitor trigger data. Overflow channels correspond to no hits or hits that are very far out of time with the quantity being checked.  206  Figure 7.10: Quantities related to pion identification by Range Stack kinematics. The abscissa label contains the quantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. The reference signal-like (black) and background-like (purple) distributions were each normalized to an area equal to one, thus the ordinate axis shows the normalized number of events per abscissa bin. The measured quantity for each candidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green and candidate C in blue. Note that for some quantities a candidate has multiple entries and that when multiple quantities fell in the same abscissa bin, information from only one candidate was displayed. The sample for the reference signal-like distribution was from π scatter monitor trigger data in PNN2BOX and the sample for the reference background-like distribution was from K π2 monitor trigger data in KP2BOX.  207  Figure 7.11: Quantities related to pion particle identification from TD variables. The abscissa label contains the quantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. The reference signal-like (black) and background-like (purple) distributions were each normalized to an area equal to one, thus the ordinate axis shows the normalized number of events per abscissa bin. The measured quantity for each candidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green and candidate C in blue. Note that for some quantities a candidate has multiple entries and that when multiple quantities fell in the same abscissa bin, information from only one candidate was displayed. The sample for the reference signal-like distribution was from π scatter monitor trigger data in PNN2BOX and the sample for the reference background-like distribution was from K µ2 monitor trigger data since these cuts were designed to suppress muon-based backgrounds.  208  Figure 7.12: Quantities related to pion particle identification from kinematic variables. The abscissa label contains the quantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. For the reference signal-like distribution (black), the ordinate axis shows the number of events per abscissa bin. The measured quantity for each candidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green and candidate C in blue. Note that for some quantities a candidate has multiple entries and that when multiple quantities fell in the same abscissa bin, information from only one candidate was displayed. Sample for the signal-like distribution was from UMC K + → π + ν ν¯ events passing πν ν¯(1) or πν ν¯(2) triggers.  209  Figure 7.13: More quantities related to single beam detection and photon veto. The abscissa label contains the quantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. For the reference signal-like distribution (black), the ordinate axis shows the number of events per abscissa bin. The measured quantity for each candidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green and candidate C in blue. Note that for some quantities a candidate has multiple entries and that when multiple quantities fell in the same abscissa bin, information from only one candidate was displayed. Sample for the signal-like distribution was from K µ2 monitor trigger data. Overflow channels correspond to no hits or hits that are very far out of time with the quantity being checked.  210  beam K + requirements, decay π + requirements and single decay product requirements. These probabilities, other than the “single particle quality”, showed fairly flat distributions for each candidate, which was consistent with the expected signal distributions. The quantities from the “single particle quality” were typically associated with particle vetoes where probabilities being peaked at 1 was expected due to the binary nature of the quantities used.  7.2.3  Background and Signal Fluctuation Probabilities  Based on the signal acceptance and the estimated background in each cell, the probability that the three observed candidates were due to background only was 3.7% and the probability that the three observed candidates were due to a combination of background processes and SM predicted signal was 5.7%. The probability that all seven E949 and E787 K + → π + ν ν¯ candidates (Table 7.3) were due to background was 0.1%.  7.3  Calculation of K + → π +ν ν¯ Branching Ratio  This section describes the likelihood analysis used to measure the branching ratio B(K + → π + ν ν¯) based on the method described in [60].  7.3.1  Branching Ratio Using Maximum Likelihood  Assuming the signal and background processes obeyed Poisson statistics, the probability of observing exactly di events in the ith cell having an estimated background level of bi and expected signal level of si was P (di |si + bi ) = e−(bi +si )  211  (bi + si )di . di !  (7.2)  Figure 7.14: Signal-like probabilities (abscissa) for the three candidates. The ordinate axis shows the number of events per abscissa bin. The probabilities were calculated for each candidate for each of the entries appearing in Figures 7.5 through 7.13 and classified into the four categories defined in the text: reconstruction quality (“Reconstruction”), single beam K + requirements (“Kaon ID”), decay π + requirements (“Pion ID”) and single decay product requirements (“Single Particle ID”). Candidates are colour coded: candidate A in red (column 1), candidate B in green (column 2), and candidate C in blue (column 3).  212  For a background only assumption, the probability was  P (di |bi ) = e−bi  b di . di !  (7.3)  For m cells, a combination of di events in each of the i cells was given the index α. Given the assumption of signal and background, the probability of observing the exact combination α was m  Pα (s + b) =  P (di |si + bi )  i=1 m  =  e  −(bi +si ) (bi  i=1  + s i ) di . di !  (7.4)  The probability of the same combination given the assumption of background only was m  Pα (b) = i=1 m  =  P (di |bi ) e  di −bi bi  i=1  di !  (7.5)  .  A likelihood ratio Xα , defined as m  Xα ≡  Xi , i=1 m  e  −(bi +si ) (bi  ≡  + s i ) di di !  bdi i i=1 −b i e di ! m si = e−si (1 + )di , bi i=1  ,  (7.6)  was used to compare the probability of the signal and background assumption to the background-only assumption for a combination α. The likelihood ratio for the distribution of the 3 candidates in the 9 cells of this analysis 213  was denoted Xobs . To determine B(K + → π + ν ν¯), B was varied to maximize Xobs given the following relationship between B and the expected signal in each cell si : si = A i ×  B , SES  (7.7)  where SES was the single event sensitivity in the loose signal region from Table 6.24 and Ai was the fraction of the total acceptance of the ith cell from Table 7.1. The value of B that maximized Xobs was taken as the branching ratio. Thus far in the discussion of the likelihood analysis, uncertainties in the signal and background levels have not been discussed. Based on [60], the uncertainties on the background (σbi ) and signal (σsi ) levels were taken into account using a gaussian convolution, and the likelihood ratio was replaced with ✷✸ − m  ∞ 0  ∞ 0  e  X=  (s − si )2 (b − bi )2 + 2σs2i 2σb2i ✸✷  i=1  − ∞ 0  ∞ 0  e  ✹✺  2πσs2i σb2i ✺✹ (s − si )2 (b − bi )2 + 2σs2i 2σb2i 2πσs2i σb2i  Xi ds db .  (7.8)  ds db  The resulting curve for Xobs as a function of B is shown in the top plot of Figure 7.15. The value of B that maximized Xobs , 7.89 × 10−10 , was the branching ratio based on the candidates observed in this analysis. The next section discusses the method used to determine the uncertainty of this branching ratio.  214  Likelihood Ratio (Xobs)  3.5 3 2.5 2 1.5 1 0.5  Confidence Level (CLs)  0  0  2.5  5  7.5  10  12.5  15  17.5  20  22.5  Branching Ratio (10  25  -10  )  1 0.8 0.6  68% CL Interval  0.4 0.2 0  0  2.5  5  7.5  10  12.5  15  17.5  20  22.5  25  Branching Ratio (10-10)  Figure 7.15: Likelihood ratio Xobs (top) and confidence level CLs (bottom) for the three candidates from the E949-PNN2 analysis. The value of B (7.89 × 10 −10 ) that maximized Xobs is indicated with a vertical line (top).  215  7.3.2  Branching Ratio Uncertainty Using Confidence Level Interval  The sum of all probabilities Ps+b which had a likelihood ratio Xα less than or equal to Xobs gives the confidence level for the signal plus background assumption, m  CLs+b ≡ Ps+b (Xα < Xobs ) =  e−(bi +si ) Xα <Xobs  i=1  (bi + si )di di !  .  (7.9)  The confidence level for the background-only assumption was m  CLb ≡ Pb (Xα < Xobs ) =  e−bi Xα <Xobs  i=1  bdi i di !  .  (7.10)  Based on these two confidence levels, the confidence level for the signal was defined as  CLs ≡  CLs+b . CLb  (7.11)  Again the branching ratio B was treated as a parameter and varied, with the resulting curve for the quantity CLs shown in the bottom plot of Figure 7.15. The 68% confidence level interval (equivalent to ±σ) was taken as the interval corresponding to the branching ratios at CLs values equal to 0.84 and 0.16. The branching ratio determined by maximizing Xobs was taken as the central value and the resulting branching ratio for the three observed candidates in this analysis was +9.26 B(K + → π + ν ν¯) = 7.89−5.10 × 10−10 .  7.3.3  (7.12)  Combined E787/E949 Branching Ratio  For the combined E787 and E949 result (all candidates shown in Table 7.3), the di , bi and si information was determined for each cell of each analysis and information was treated 216  as one large data set. The resulting Xobs and CLs plots are shown in Figure 7.16, which gives a final branching ratio due to all K + → π + ν ν¯ candidates of −10 B(K + → π + ν ν¯) = 1.73+1.15 . −1.05 × 10  (7.13)  The CLs plot was also used to get the 90% confidence level limit of B(K + → π + ν ν¯) < 3.35×10−10 . This result was consistent with the standard model prediction of B SM (K + → π + ν ν¯) = (0.85 ± 0.07) × 10−10 [25].  217  Likelihood Ratio (Xobs)  160 140 120 100 80 60 40 20 0  Confidence Level (CLs)  New Previous  0  1  2  3  4  5  Branching Ratio (10  -10  6  )  1 0.8 0.6  New Previous  68% CL Interval  0.4 0.2 0  0  1  2  3  4  5  6  Branching Ratio (10-10)  Figure 7.16: Likelihood ratio Xobs (top) and confidence level CLs (bottom) for the new and previously published [11] combined results for all E787 and E949 analyses. analysis. The values of B that maximized Xobs for the respective results are indicated with vertical lines (top).  218  Figure 7.17: Comparison of the Standard Model (0.85 ± 0.07 × 10 −10 ), previous PNN1 only −10 [11]), and final E787/E949 (1.73 +1.15 × 10−10 ) branching ratios. (1.47+1.30 −0.89 × 10 −1.05  219  Chapter 8 Conclusion BNL experiment E949 was an upgrade of the E787 experiment, designed to measure the rare K + → π + ν ν¯ decay. The search took place in the π + momentum region of 140 to 199 MeV/c for this analysis. A blind analysis technique was used to analyze the 1.71 × 10 12 stopped K + decays collected in the E949 detector. This technique was based on keeping the signal region hidden until all selection criteria for signal had been finalized, estimates of all background levels completed, and the acceptance of the signal region determined. The background levels were estimated from data using a bifurcation method, where two uncorrelated cuts with significant rejection of a specific background were used to estimate the background level in the signal region using measurements outside of the signal region. The signal region was divided into nine cells, whose relative signal-to-background levels varied by a factor of 4. The estimated background was 0.927 ± 0.168(stat.) +0.320 −0.237 (sys.) events. Based on acceptance measurements, the single event sensitivity was (4.28 ± 0.43) × 10−10 . Examination of the signal region yielded 3 candidate events. Based on these 3 candidates a likelihood analysis was used to determine the branching ratio, B(K + → π + ν ν¯) = −10 (7.89+9.26 at 68% confidence level. Based on the signal acceptance and the −5.10 ) × 10  estimated background in each cell, the probability that these three events were due to 220  background only was 3.7% and the probability that the three observed candidates were due to a combination of background processes and SM predicted signal was 5.7%. Using the likelihood analysis, all E787 and E949 results were combined to give a −10 branching ratio of 1.73+1.15 at 68% confidence level. This branching ratio is −1.05 × 10  consistent with the Standard Model prediction of (0.85 ± 0.07) × 10−10 . The probability that all observed E787 and E949 candidates were due to background only was 0.1%. E949 has made a significant step in confirming the SM’s detailed predictions involving second-order weak interactions, and it has advanced the state-of-the art in the experimental technique of measuring small effects. However, given the level of statistical uncertainty associated with this B(K + → π + ν ν¯) measurement, it was not possible to make any definitive conclusions regarding physics beyond the SM. It was very unfortunate that an executive decision by the Department of Energy (DOE) resulted in the E949 experiment being terminated after only 20% of the approved beam time as the potential for strong hints of physics beyond the SM were within reach. Future measurements of B(K + → π + ν ν¯) are being planned. The NA62 experiment, currently in preparation at CERN, aims to collect 65 K + → π + ν ν¯ events over two years of data-taking using a decay-in-flight experiment [10]. A letter of intent for a decay-at-rest experiment similar to E949 was presented at J-PARC [69]. 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[96] Struck Innovative Systeme, Harksheider Str. 102A, 22399 Hamburg, Germany. [97] C. Witzig, “Pi- Absorption in the Range Stack”, E787 Technical Note 278 (1994), Unpublished. [98] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983). [99] W.-M. Yao et al. (Particle Data Group), Journal of Physics G 33, 1 (2003). [100] T. Yoshika et al. (E949 Collaboration), IEEE Trans. Nucl. Sci. 51, 199 (2004).  226  Appendix A E787 to E949 Upgrades The E949 experiment was a successor to experiment E787 at BNL [15], whose data taking was completed in 1998. The experiment was quite similar to E787, with the following upgrades to the beam and detector [18], and to the trigger and data acquisition [100].  A.1  Beam Upgrades  The proton intensity at the production target was increased by a factor of two, resulting in about twice the beam rate as E787. The mean K + momentum was 710 MeV/c throughout E949 data collection. For E787 the mean K + momentum ranged from 670 to 790 MeV/c [62] over the entire life of the experiment.  A.2  Detector Upgrades  The detector upgrades were as follows: • The two outer-most layers of the Range-Stack were replaced by the Barrel Veto Liner (BVL), adding 2.3 radiation lengths to the photon veto. The BVL was a lead/scintillator sandwich sampling calorimeter like the Barrel Veto.  227  • The Active Degrader (AD), Upstream Photon Veto (UPV), and Downstream Photon Veto (DPV) were installed to add photon veto coverage at small polar angle (θ). • One third of the RS scintillation counters were replaced to increase the light output. • A gain monitor system was installed in the RS, operated by blue LED flashers. This allowed changes in the gains of the PMTs to be tracked on a spill by spill basis and even within a spill, improving energy calibration. • The UTC read-out electronics were upgraded to operate in the higher rate environment. • The RSSC read-out electronics were upgraded to improve resolution of z -position measurements.  A.3  Trigger and DAQ Upgrades  The trigger and DAQ upgrades were as follows: • The level 0 trigger board was upgraded [100]. • Digital mean-timer modules were installed to improve the performance of the online photon rejection [100].  228  Appendix B Data Acquisition, Storage, and Processing The E949 experiment collected its physics data for 12 weeks from March through June of 2002.  B.1  Data Acquisition  Detector analog and discriminated signals were digitized by ADC, TDC, TD and CCD systems. The ADCs and TDCs were commercial units, and the TD and CCD waveform digitizers (WFD) were custom built. Data from a triggered event was digitized and stored in a buffer module or local crate controller. At the end of the spill, all buffered data from the spill were transferred to the host computer. Table B.1 shows a summary of the digitizing electronics that were used. For the Fastbus systems, SLAC Scanner Processor (SSP) modules [24] served as crate controllers as well as performing the tasks of reading out, reformatting and buffering the data from the front end after each trigger were accepted. The CAMAC ADCs were readout through the FERA bus by a Fastbus Struck [96] 370 QDP DSP module. The  229  Type  Model  Standard  Resolution  Subsystems  ADC  LRS 4300B LRS 1881  CAMAC Fastbus  10 bits 13 bits  RS,BV,BVL,EC,beam target, UTC  TDC  LRS 3377 LRS 1879 LRS 1876  CAMAC Fastbus Fastbus  0.5 ns 2 ns 1 ns  RS, BVL UTC,BV,target EC,RSSC,beam  TD  Fastbus  500 MHz sampling 8 bits, 10 µs depth  RS,BVL,IC  CCD  Fastbus  500 MHz sampling 8 bits, 256 ns depth  beam, target, EC  WFD  Table B.1: Digitizing electronics for E949. The ADCs and TDCs were commercial units (LeCroy [70]), and TD and CCD waveform digitizers (WFD) were custom built. Reprinted table with permission from S. Adler et al. (E949 Collaboration), Phys. Rev. D 77 052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003. Copyright 2008 by the American Physical Society.  CAMAC TDCs were readout by custom-built DYC3 modules [42] which pushed the data into VME memory boards. The readout time per event, as determined by the slowest crate, was typically 850 µs. At the end of each spill, the data from the Fastbus buffer memories were readout via the cable segment (12-15 MB/s) by Struck 340 SFI modules, each controlled by a Motorola [85] VME 2604 single-board computer (SBC) running VxWorks. The VME memory boards were readout by a separate SBC. Data were transferred from the SBCs to the host computer (SGI [93] Origin 200, designated bnlku9) via 9 MB/s per link ethernet through a simple network switch. Event fragments from the readout segments were combined by event builder processes running on the host computer. The πν ν¯ triggers were written to two DLT-7000 drives at 5 MB/s per drive; a third DLT drive was used to log monitor triggers. Under typical running conditions, 300 events were written per spill with a typical event size of 80 kB. This was well within the maximum possible throughput of the system  230  of about 50 MB per spill. The DAQ dead time was due entirely to the speed of the event-by-event readout of the front-end electronics at the crate level. The total dead time introduced by the trigger and DAQ was typically 26%. A slow control system, based on the MIDAS [80] framework, ran independently of the main DAQ system, and was used to monitor a variety of experiment conditions, including crate voltages and temperatures.  B.2  PASS0  After the 12 week physics data run had completed, PASS0 processing was performed. For each monitor trigger, the data were staged, sorted and written to dedicated DLT tapes.  B.3  PASS1 and PASS2 Processing  PASS1 and PASS2 processing were performed in sequence on the various monitor and signal trigger data by first staging raw data from DLT tapes to hard-drive disks. Next, the data were unpacked and event reconstruction was performed using a cluster of 25 Linux-based computers. Fortran routines and shell scripts were used to decode the raw digitized information written to tape, run event processing and create PAW [87] ntuples containing event information.  B.4  PASS3 Analysis  Subsequent analysis of the PAW ntuples employed PAW comis functions and kumac scripts to apply the desired cuts to the event ntuples. The output from this stage typically took the form of histograms of the desired quantities. The ntuples and cut functions were available in public areas such that data analysis could be performed by multiple analyzers in parallel. 231  Appendix C Target Pulse Data Analysis The implementation of the target pulse data analysis, known as the “target CCD fitter”, for the analysis of experimental data from the E787 experiment is described in [38] with improvements to the fitter described in [21]. Further improvements were made to the fitter for the E949 analysis as described in [45]. This appendix describes the target CCD fitter and details the modifications and improvements made to the fitter for this analysis, summarized as follows: • The uncertainties associated with the first bin and with bins containing less than ten counts were increased to de-weight these bins in the fit. • Bins with zero counts were included in the minimization for the fit if they occurred within the pulse. They were previously excluded. • Hold and release fitting was implemented for the double-pulse fits.  The four-  parameter double-pulse fit was changed from a one-stage fit to a two-stage fit. In the first stage, the time for the first pulse was fixed at the time found from the single-pulse fit while the other three parameters were allowed to wander. In the second stage, the values from these three parameters were used as initial guesses and all four parameters were allowed to wander. 232  • The maximum number of target fibers that were fit was increased to 31 from 15. • The previous restriction that the two pulses in a double-pulse fit had to be within ±10 ns of each other was lifted due to a number of observed events with undesirable double-pulse fit solutions. Additional small bug fixes found in the previous fitter code were also made by the author. Throughout this appendix, the nomenclature “fiber channel” will be used when discussing each the high-gain and low-gain CCD channels for each target fiber. For example, a statement beginning with “For each fiber channel” will describe a situation in which each of the high-gain and each of the low-gain CCD channels for each target fiber was examined. For single-pulse fits, the single pulse was often referred to as the “kaon pulse”. For double-pulse fits, the first pulse was often referred to as the “kaon pulse” and the second as the “pion pulse”.  C.1  Creation of Standardized Pulses  For each fiber channel, a collection of standardized K + pulses was created using Kµ2 monitor trigger data and the following criteria for fibers identified as K + fibers: • The energy of the K + fiber EK was greater than 4 MeV; • The pulse did not saturate the 8-bit dynamic range of the CCD; • The time of the K + fiber was within ±3 ns of time zero; • The event passed the target reconstruction criteria. These pulse shapes were normalized to unit area, aligned with a timing granularity of 0.5 ns by interpolation, and these average “standardized” shapes stored. The effects due to saturated pulses and outliers in each time bin were removed by determining the average 233  from the peak of a Gaussian fit. Based on these standardized pulses, the reliability of each of the fiber channels was determined with a typical reliability of 0.99.  C.2  Overview of the Fitter  For each fiber, the fitting procedure was performed on each of the low-gain and the highgain CCD information independently. Fits were only attempted on fiber channels that passed the following criteria: • The reliability for that fiber channel had to be greater than 0.8 for that run. • The energy had to be greater than the threshold energy for 90% efficiency for a CCD hit for that fiber channel. Typical thresholds were 2 MeV for low-gain and 0.5 MeV for high-gain CCD channels. • The time for that target kaon fiber had to be earlier than track time trs . • The edge finder had to find a good leading edge. Additionally, a maximum of 31 low-gain and 31 high-gain CCD channel fits were stored. The cut ALLKFIT removed events where a fit was performed on neither the high-gain nor the low-gain CCD channels of a given target fiber if the energy in that fiber was above 3 MeV. A single-pulse fit was attempted for each fiber channel passing the above criteria. The single-pulse fit used two parameters, the kaon-pulse amplitude and the time. Pedestal subtraction had already occurred previous to the target CCD fitter so the pedestal was not included in the fit. If the probability of the single-pulse fit was less than 25%, a double-pulse fit was performed. The double-pulse fit used a second copy of the standardized kaon pulse and added it to the first pulse. The double-pulse fit used four parameters, the amplitudes and times for each of the kaon and pion pulses. 234  These fits minimized χ2 by varying the above-mentioned parameters using the MINUIT function minimization library [81]. The uncertainty input for each bin of height N of the pulse used experimentally determined uncertainty parameters:  High-gain = 0.74 + 0.69 × Low-gain = 1.21 + 0.35 ×  √ N,  (C.1)  √ N,  (C.2)  where the first term was a constant related to instrumental noise and the second term scaled with the square-root of the number of photo-electrons in the pulse. This general form for the uncertainties was retained in the modifications made by the author, but two modifications were made as detailed in the following section. For these fits, time bins which have saturated the 8-bit dynamic range of the CCDs were not included in the fit and the fit was performed using only the unsaturated bins.  C.3  Optimization of the Error Input  A sample of Kµ2 monitor trigger data was used for optimizing the uncertainty inputs for the target fitter. For the sample, the following cuts were applied as setup cuts: TGQUALT, DELC, NPITG, TARGET, TGCUT, UTC, RD TRK, TRKTIM, RDUTM. This left approximately 50,000 events to which CCDBADFIT, CCDPUL and EPIONK were applied. The resulting components of acceptance of these cuts are shown in the row labeled “No fixes” of Table C.1 before the fixes detailed below.  C.3.1  Low-Count Error Fix  Two sub-samples of the Kµ2 monitor trigger data sample described above were created by choosing only pulses that had above a 25% probability for the single-pulse fit. One sub-sample was created for low-gain CCD fits meeting this criteria and another for the 235  high-gain CCD fits meeting this criteria. One at a time, each time bin was removed from the single-pulse fit and the difference between the pulse-height for that bin as predicted by the fitter and the actual number of counts “predicted−actual ” was tabulated as a function of the actual number of counts in that bin. The bin numbering in this study was such that bin number 1 was the first bin found to have a non-zero pulse-height. For each time bin, the shape given by the Equation (C.1) or (C.2) was plotted against the above distribution, with a typical comparison plot being shown in Figure C.1. It was found that for time bins having counts (N ) below 10, the uncertainty functions (Equations C.1 and C.2) were underestimating the uncertainty. Plots similar to C.1 were visually examined for each time bin and it was found that time bins having counts less than 10 were consistently underestimated, with the turn-up in the distribution occurring typically in the first 5 to 10 time bins. This was fixed by applying the uncertainty corresponding to N = 10 counts for all channels having 10 or less counts.  C.3.2  First Bin Error Fix  It was found that the fit was very sensitive to the pulse-height of the first bin. A very low number of counts (such as 1 or 2) in the first bin tended to give a very large contribution to the chi-squared of the fit due to the reference pulse predicting a larger number of counts. This sensitivity to very low pulse-heights in the first bin was found to be present even with the low-count uncertainty fix applied. As shown in Figure C.2, a reasonable looking fit can have a large chi-squared contribution due to the first bin, resulting in a fit probability of zero and thus causing it to fail the fit. The contribution due to this first-bin sensitivity was reduced by doubling the uncertainty associated with the first bin. The improved components of acceptance due to this fix are shown in the row labeled “First bin and low-count uncertainty fixes” in Table C.1. The results of the single- and double-pulse fits were examined for many (40-50) events where the “low-count error”  236  Figure C.1: This plot shows the results of leaving the second bin out of the fitter and comparing the actual counts in that bin to those predicted by the fit. The x -axis shows the number of counts in that second bin. The y-axis shows the difference predicted−actual counts for only positive values√of this difference. The line shows the uncertainty for each bin from the equation 0.74 + 0.69 × counts.  237  and “first bin uncertainty” fixes resulted in improvements of the fit probability from zero to some non-zero value. Examination of these events showed that these fixes improved the fitter’s ability to identify single-pulse events while creating a negligible number of false-positives (none were observed in the events examined).  C.3.3  Intermediate Zero-Count Bins Included in the Fit  The previous fitter did not include intermediate bins having zero counts in the fit. This was changed so that the fitter first identified the first and last bins of the pulse. Then up to the first 30 bins of this pulse were fit with bins having zero counts being included. Bins which were identified as saturated were excluded from the fit. The resulting components of acceptance due to this fix relative to the two previous fixes detailed in this section are shown in the row labeled “First bin and low-count uncertainty fixes + 0-count bins included” in Table C.1. This fix resulted in a slight drop in the combined acceptance of CCDBADFIT, CCDPUL and EPIONK due to an increase in single-pulse fit probabilities below 25% resulting in more double-pulse fits being performed. A fiber channel can only cause an event to fail CCDBADFIT or CCDPUL if a double-pulse fit was performed.  C.4  Hold and Release Double-Pulse Fit  The fitter was modified so that the double-pulse fit became a two-stage process. For the first stage of the double-pulse fit process, the kaon-pulse time was fixed at the time returned from the single-pulse fit. The three other parameters (kaon-pulse amplitude, pion-pulse time and pion-pulse time) were allowed to wander in the fit. For the second stage of the double-pulse fit, the values returned from the first stage of the double-pulse fit were used as the initial guesses. The sample used to optimize this modification to the fitter was Kµ2 monitor trigger 238  Fix  CCDBADFIT  CCDPUL  ALL  No fixes  0.797  0.454  0.362  First bin and low-count uncertainty fixes  0.876  0.518  0.453  First bin and low-count uncertainty fixes + 0-count bins included  0.881  0.504  0.443  Table C.1: Components of acceptance for various target fitter fixes. The “ALL” column shows the combined acceptance of the set of cuts consisting of CCDBADFIT, CCDPUL and EPIONK applied sequentially. The acceptance of the EPIONK cut was 0.999 for all 3 situations so it was not given a column in the table. The row “First bin and low-count uncertainty fixes” includes both the first bin uncertainty fix (doubling the uncertainty for this bin) and the low-count uncertainty fix (assigning an uncertainty equal to that for 10 counts for all bins having less than 10 counts). The row “First bin and low-count uncertainty fixes + 0-count bins included” includes the above fixes in addition to including intermediate zero count bins in the fit.  data with the following cuts applied: TARGET, TGCUT, UTC, RD TRK, TRKTIM, RDUTM, TGQUALT, DELC and NPITG. After these cuts were applied, 7021 events remained. With the two-stage fit used instead of the previous one-stage fit, the total acceptance of CCDBADFIT and CCDPUL went from 0.402 to 0.451.  239  Figure C.2: The high-gain single-pulse probability of 0 when the uncertainty associated with the first bin was treated as usual. This was due to the first bin having only two counts versus where the shape of the reference pulse predicted a larger number of counts. Had the first bin had a pedestal subtraction that left it with zero counts, the next bin would have been used for the fit and the resulting fit would have been a non-zero probability. To reduce the effect of this sensitivity to the first bin, the uncertainties assigned to the first bin were always doubled. When the fit was performed with this increased first bin uncertainty, the single-fit probability for the high-gain was 0.069 instead of the zero probability shown in this figure.  240  Appendix D Detailed List of Cuts D.1  PASS1 Cuts  These cuts were applied to all the background studies and thus were excluded from the lists of setup cuts for the individual bifurcation branches.  D.1.1  Event Reconstruction Quality Cuts  TRBIT - The event passed the πν ν¯(1) or πν ν¯(2) triggers. BAD RUN - Removed runs where problems occurred in the beam, detector or electronics. RD TRK - The charged track was reconstructed in the Range-Stack. TRKTIM - An average track time was found in the Range-Stack. STLAY - There was agreement in the Range-Stack stopping counter as identified by the offline track finding routine and the online Stopping Counter Finder (SCF). BAD STC - Runs were identified during calibration in which a set of TD channels had unreliable signals. For these runs, events were removed if the Range-Stack stopping 241  counter contained these unreliable TD channels. UTC - The charged track was reconstructed in the UTC. RDUTM - The extrapolation of the reconstructed UTC track intersected the triggered T•2 sector. TARGET - The event was reconstructed in the target. PDC - The momentum of the charged track as measured by the UTC was less than 280 MeV/c. This removed high-momentum beam particles and failures in UTC pattern recognition.  D.1.2  Muon Background Rejection Cuts  FITPI - The double-pulse π + → µ+ signature was observed in the fitted TD pulses from the Range-Stack stopping counter. RSHEX - Event was rejected if there were additional hits in (1) the counter after the Range-Stack stopping counter, or (2) the hextants that did not have track hits. RSHEX2 - Event was rejected if the charged track in the Range-Stack crossed from one sector to another in the stopping layer.  D.2  PASS2 Cuts  The PASS2 cuts consisted of looser versions of the PASS3 cuts. The PASS2 cuts were applied in certain combinations to enhance specific backgrounds, reducing processing time when developing cuts and evaluating background levels. P2PVCUT - A very loose photon veto was applied in the Barrel Veto (±2.0 ns, 1.5 MeV), End Cap (±1.5 ns, 3.5 MeV), and Range-Stack (±1.5 ns, 3.0 MeV). Events 242  were rejected if total coincident energy was above the listed threshold for the listed time window about the Range-Stack track time trs . P2TGCUT - Events failed this target reconstruction cut if they did not meet all of the following conditions: • The kaon decay vertex identified by target reconstruction was found inside the target. • The average time of the kaon fiber hits (tk ) and the B4 Hodoscope hit time (tb4 ) were within 4.0 ns. • The average time of the pion fiber hits (tpi ) and the Range-Stack track time (trs ) were within 5.0 ns. • The I-Counter time (tIC ) and the Range-Stack track time (trs ) were within 5.0 ns. This condition involved a small time offset: |tIC − trs + 0.3ns| ≤ 5.0ns. • The energy deposited in the I-Counter was within 4 MeV of that expected for a minimum ionizing pion track. P2TDCUT - Event was rejected if accidental activity in the Range-Stack coincident with the second pulse in the Range-Stack stopping counter exceeded a threshold. P2PSCUT - Beam pion events were removed by requiring that (1) the energy deposited in the B4 Hodoscope was consistent with a kaon (> 1.0 MeV), and (2) there was no beam particle coincident with the T•2-trigger time. These beam-particle times ˇ were determined by the times of hits in the B4 Hodoscope and the pion Cerenkov counters. P2TGPVCUT - Event was rejected if the total coincident activity in the target photon fibers at track time was above 5 MeV. The coincidence condition with respect to Range-Stack track time was |tγ − trs + 0.8 ns| ≤ 1.0 ns. 243  DELCO2 - Required that the beam kaon decayed at least 2.0 ns after entering the target. The times used for this condition were the average time of the kaon fiber hits (tk ) and the average time of the pion fiber hits (tpi ).  D.3  Kinematic Cuts  The kinematic cuts, collectively referred to as KINCUTS, were designed to remove events having kinematics inconsistent with signal.  D.3.1  Fiducial Cuts  LAYER14 - Event was rejected if it came to rest in the second Range-Stack Straw Chamber layer. This condition was accomplished by rejecting stopping layer 14 events having a prompt RSSC hit in the same sector or one sector clockwise of the stopping counter. COS3D - Event was rejected if the cosine of the track polar angle was greater than 0.5. This removed events likely to interact with the dead material of the Range-Stack support structure. LAYV4 - Event was rejected if the charged track did not come to rest in between RangeStack stopping layers 6 to 18 inclusive. ZFRF - The charged track was required to come to rest inside of the Range-Stack fiducial region based on the stopping z -position. This condition was applied for stopping layers 11-12 (|z ≤ 35 cm|), 13 (|z ≤ 40 cm|), 14 (|z ≤ 30 cm|), and 15-18 (|z ≤ 50 cm|). ZUTOUT - The charged track was required to pass through the UTC outer layer within the active region (|z ≤ 25 cm|). 244  D.3.2  Track Reconstruction Cuts  UTCQUAL - Good track reconstruction was required in the UTC based on a likelihood condition in the x-y plane. The quantities considered in this likelihood condition were the number of hits used in the UTC track fit, the number of hit UTC layers, and the number of unused hits within 1.5 cm of the fitted track. PRRF - Event was rejected if the charged-track scattered in the Range-Stack, as determined by matching between Range-Stack track reconstruction and UTC track extrapolation. The following conditions were examined: • Event was rejected if it failed a χ2 probability condition for the Range-Stack track reconstruction in the x-y plane. The Range-Stack track fitting used sector crossing positions, RSSC hit positions and the energy deposit in the stopping counter. • Event was rejected if the track reached the first layer of RSSCs and the matching between the UTC track extrapolation and the RSSC hit positions was poor. • Event was rejected if the matching in the r-z plane was poor between the UTC track extrapolation and the track hit positions in the Range-Stack counters as determined by end-to-end timing.  D.3.3  dE/dx Cuts in Range-Stack  RSDEDX - The following conditions based on energy deposits in the Range-Stack were examined: • Event was rejected if the energy deposited in any Range-Stack track counter was inconsistent with that expected for a pion. The deviation between mea-  245  sured Eimeas and expected Eiexp energies for the ith track counter was given by: logEimeas − logEiexp χi = , σi  (D.1)  where the event was rejected if |χi | ≥ 4. • Event was rejected if the confidence level calculated from the probability of χ2 from Equation (D.1) was less than 0.04. • Event was rejected based on a likelihood condition constructed from the energy deposits in the track counters. • Event was rejected based on additional constraints on the fit times and energies determined by the fitpi4 routine [57].  D.3.4  Range-Momentum Consistency Cut  RNGMOM - Event was rejected if the measured range rtot and the expected range based on momentum rexp were not consistent under the assumption of a charged track due to a pion. This cut targeted muon backgrounds due to the different regions of the range-momentum phase-space occupied by pions and muons. Event was rejected if the deviation χr−p was greater than 2.2, where  χr−p =  D.4  rtot − rexp . σrtot  (D.2)  Phase Space Cuts  The kinematic constraints of the PNN2 phase space (PNN2BOX) were split into loose (BOXLOOSE) and tight (BOXTIGHT) versions. For each cut, the outgoing charged track was required to pass the listed kinematic conditions on momentum (ptot), energy (etot) and range in plastic scintillator (rtot). 246  BOXLOOSE - The acceptance conditions for the loose kinematic box were: 140.0 MeV/c ≤  ptot  ≤ 199.0 MeV/c;  60.0 MeV ≤  etot  ≤ 100.5 MeV;  12.0 cm ≤  rtot  ≤ 28.0 cm.  BOXTIGHT - The acceptance conditions for the tight kinematic box, created to heavily suppress the Ke4 background, were: 165.0 MeV/c ≤  ptot  ≤ 197.0 MeV/c;  72.0 MeV ≤  etot  ≤ 100.0 MeV;  17.0 cm ≤  rtot  ≤ 28.0 cm.  There were two additional sets of kinematic constraints that did not define the signal region, but were used for various studies in this analysis: KP2BOX to choose events in the Kπ2 peak, and KM2BOX to choose events in the Kµ2 peak. Figure 3.2 shows where these other kinematic boxes were found with respect to the kinematic signal region PNN2BOX in range versus momentum phase space. KP2BOX - The acceptance conditions for the Kπ2 -peak kinematic region were: 199.0 MeV/c ≤  ptot  ≤ 215.0 MeV/c;  100.5 MeV ≤  etot  ≤ 115.0 MeV;  28.0 cm ≤  rtot  ≤ 35.0 cm.  KM2BOX - The acceptance conditions for the Kµ2 -peak kinematic region were:  D.5  226.0 MeV/c ≤  ptot  37.0 cm ≤  rtot.  ≤ 246.0 MeV/c;  Beam Cuts  The beam cuts, collectively referred to as BEAMCUTS, were designed to remove events with beam-pions scattering into the fiducial region of the detector and double-beam events. 247  D.5.1  Particle Identification  B4DEDX - Beam pion events were removed by requiring that the energy deposited in the B4 Hodoscope was consistent with a kaon (> 1.1 MeV).  D.5.2  Double-Beam Cuts  B4CCD - Double-beam events were removed based on fits in the B4 Hodoscope CCDs fitted with a double-pulse assumption. A double-pulse signature was considered present when the ratio of χ2 for the double-pulse fit versus the single-pulse fit was greater than 2.5 for second-pulse amplitudes above a certain threshold. Event was rejected when the difference between the average time from hit modules having having double-pulse signatures and trs was greater than 3.5 ns. B4TRS - Event was rejected if there was activity in the B4 Hodoscope at track time in the Range-Stack trs . This activity was considered present if (1) the average TDC time from B4 hit modules was within 2.5 ns of trs , or (2) the average CCD time from B4 hit modules was within 1.5 ns of trs when the B4 energy sum was above 0.7 MeV. BWTRS - Event was rejected if there was activity in the Beam-Wire Chamber at track time in the Range-Stack trs . This activity was considered present if there was a hit cluster in any of the Beam-Wire Chambers within 4.5 ns of trs . ˇ CKTRS - Event was rejected if the average TDC or CCD time of the Cerenkov kaon counter hits was within 2.0 ns of trs . CKTAIL - Event was rejected if the average time of the trailing edge of pulses in the ˇ Cerenkov kaon counters minus the average TDC width was within twindow of trs . The time window varied based on the difference between the average time of the kaon fiber hits tk and the average time of the pion fiber hits tpi : 248  • twindow = 3 ns when tpi − tk < 15.0 ns; • twindow = 3.5 ns when 15.0 ns ≤ tpi − tk < 25.0 ns; • twindow = 3 ns when 15.0 ns ≤ tpi − tk . ˇ CPITRS - Event was rejected if the average TDC or CCD time of the Cerenkov pion counter hits was within 2.0 ns of trs . CPITAIL - Event was rejected if the average time of the trailing edge of pulses in the ˇ Cerenkov pion counters minus the average TDC width was within twindow of trs . The time window varied based on the difference between the average time of the kaon fiber hits tk and the average time of the pion fiber hits tpi : • twindow = 3 ns when tpi − tk < 15.0 ns; • twindow = 3.5 ns when 15.0 ns ≤ tpi − tk < 25.0 ns; • twindow = 3 ns when 15.0 ns ≤ tpi − tk . UPVTRS - Event was rejected if there was a hit in the Upstream Photon Veto (UPV) coincident with trs . Timing information from both the CCDs (tCCD ) and TDCs (tTDC ) was used. Event was rejected if either condition was met: • −3.5 ns < tCCD − trs < 2.4 ns; • −3.75 ns < tTDC − trs < 2.5 ns. RVTRS - Event was rejected if there was a hit in the Ring Veto (RV) within 4 ns of trs . Timing information from both the CCDs and TDCs was used.  D.5.3  Beam Pathology Cuts  Pathology cuts were cuts designed to remove specific types of abnormal events that were not properly suppressed by the regular cuts designed to suppress the various backgrounds. 249  These undesirable pathologies could have been due to behaviors such as inconsistencies between quantities measured in different detectors or known abnormal behavior in a reconstruction algorithm. B4ETCON - Event was rejected if the timing or energy information in the B4 Hodoscope was inconsistent between the CCDs and the TDCs. Event was rejected if the times from these two systems were not within 2.0 ns of each other or if the energies from these two systems were not within 1.5 MeV. TGGEO - Event was rejected by this cut if it had certain topological signatures associated with with double-beam events known to fool target reconstruction. The three types of signatures for rejected events were: • Both beam-particles entered the target from the target edge or I-Counter; • The first beam-particle or its charged decay product deposited a large amount of energy in the I-Counter; • The charged decay product of the first beam-particle was not detected due to decaying downstream or decaying late. The second beam-particle scattered such that it missed some of the beam counters, but by multiple scattering enters the target and intersects with target fibers hit by the first beam-particle. TGQUALT - The target reconstruction algorithm successfully reconstructed the event with at least one of the target fibers being classified as a pion fiber. TIMCON - Event was rejected if it failed to meet either of the following timing consistency checks: • The average time of the kaon fiber hits (tk ) and the B4 Hodoscope hit time (tbm ) were within 3.0 ns;  250  • The average time of the pion fiber hits (tpi ) was consistent with track time in the Range-Stack (trs ), −4.75 ns ≤ tpi − trs ≤ 3.75 ns. TGTCON - Event was rejected if the time difference between an individual target kaon fiber and the average time of the kaon fiber hits tk was greater than an energy dependent value varying between 2.05 and 3.68 ns.  D.6  Delayed Coincidence Cuts  The delayed coincidence cuts, collectively referred to as DELCO, rejected the same type of events as the beam cuts. The delayed coincidence cuts were split into loose (DELC3) and tight (DELC6) versions. Each version of the cut ensured that the incoming K + had come to rest before decaying by requiring a minimum time difference delco between the average time of the kaon fiber hits (tk ) and the average time of the pion fiber hits (tpi ):  tpi − tk ≥ delco.  (D.3)  DELC3 - Event was required to meet the minimum delayed coincidence requirement of delco = 3 ns. Additional constraints were placed on the delayed coincidence requirement when the timing consistency between detector systems was degraded. The cut threshold was the maximum of the following conditions, with a minimum value of 3 ns: • delco = 5 ns when the difference between the average time of the kaon fiber hits (tk ) and the B4 Hodoscope hit time (tb4strob) was greater than 1.0 s. • delco = 6 ns when the difference between the average time of the pion fiber hits (tpi ) and track time in the Range-Stack (trs ). was greater than 1.5 s. • delco = 5 ns when tpi was determined using I-Counter hit time (tIC ) instead 251  of the average time of the pion fiber hits (tpi ). • delco = 4 ns when the energy deposit in target kaon fibers is less than or equal to 50 MeV. • delco = 4 ns if the time of any of the individual target kaon fibers differed from the average time of the kaon fiber hits (tk ) by more than 2.0 ns. • delco = 4 ns if the time of any of the individual target pion fibers differed from the average time of the pion fiber hits (tpi ) by more than 3.5 ns. DELC6 - Event was required to meet the tight delayed coincidence requirement of delco = 6 ns. Additional constraints like those in DELCO3 were not applied.  D.7  Target Quality Cuts  The target quality cuts, collectively referred to as TGCUTS, were designed to select events having good event signatures in the target. Note that the cut KPIGAP was not included in the group TGCUTS, nor was it applied as an analysis cut used to define the signal region. B4EKZ - Event was rejected if the beam particle identified as a kaon by event reconstruction did not behave like a kaon. A likelihood function was created using the z -position of the kaon decay in the target as determined by UTC track extrapolation, the expected z -position of the kaon decay as determined by the total energy deposited in target kaon fibers, and the energy deposited by the kaon in the B4 Hodoscope. This cut suppressed the beam pion scatter single-beam background. A tighter rejection condition was placed on this likelihood function if tpi was determined using I-Counter timing information instead of the average hit times in the target pion fibers.  252  TGZFOOL - Event was rejected if the z -position of the kaon decay vertex was too close to the upstream end of the target or was not in the fiducial region of the target at all. This z -position was determined by UTC track extrapolation. EPITG - Event was rejected if an individual target pion fiber had an energy greater than 3.0 MeV. This rejected potential pion target-scatters since the nominal energy deposited in a pion fiber was 1.2 MeV. TARGF - Event was rejected if the minimum distance between the target kaon and pion fibers was greater than one fiber (0.6 cm). This cut suppressed CEX and double-beam backgrounds. DTGTTP - Event was rejected if the charged-product tracks in the target and UTC were not well-matched at the target edge. RTDIF - Event was rejected if the uncertainty in the calculation of the pion path length in the target was greater than 1.5 cm. DRP - Event was rejected if there was a kink in the target pion track as determined by a large spread between the minimum and maximum radius of the UTC reconstructed track. TGKTIM - Event was rejected if the time of an individual target kaon fiber hit (tik ) was more than 3.5 ns later than the B4 Hodoscope hit time (tB4 ). Event was also rejected if 2tik − tB4 − trs > −1.0 ns, where trs is track time in the Range-Stack. EIC - Consistency was required between the energy measured in the I-Counter (EIC ) and the expected energy (Eexp ) based on path length in the I-Counter, −5.0 MeV ≤ EIC − Eexp < 1.75 MeV. TIC - Event was rejected if the I-Counter hit time tIC and track time in the Range-Stack trs were not within 5.0 ns of each other. 253  TGEDGE - Event was rejected if there was more than 4.0 MeV in a multiplexed target edge-fiber PMT within 5.0 ns of trs . TGDEDX - Events was rejected if the charged track dE/dx in the target was inconsistent with that of a pion. This cut used a likelihood function based on the following quantities of the charged track: momentum (ptot), range in the target , energy deposited in the target and expected range in the target based on the measured energy deposited in the target and the momentum ptot. TGENR - Event was rejected if the total energy of the hit pion fibers in the target was not in the range of 1 to 28 MeV. PIGAP - Event was rejected if a gap greater than 1.5 cm was found between target pion fibers. This cut was tightened to reject gaps greater than 1.0 cm when the z -position of the track in the I-Counter was less than -7.0 cm and the cosine of the polar angle was negative (charged-track pointed upstream). TGB4 - Consistency was required between the positions of the target kaon decay vertex, the kaon and pion clusters, and the beam particle in the B4 Hodoscope. The following conditions were examined: • Event was rejected if the distance in the xy-plane between the hit position in the B4 Hodoscope and the nearest target kaon cluster tip was greater than 1.8 cm. The kaon cluster tips were the two kaon fibers furthest apart from each other. • Event was rejected if the distance in the xy-plane between the kaon decay vertex and the nearest kaon cluster tip was greater than 0.7 cm. • Event was rejected if the distance in the xy-plane between the kaon decay vertex and the nearest pion fiber was greater than 1.5 cm.  254  PHIVTX - Event was rejected if it had back-to-back charged decay-product tracks in the target. OPSVETO - Event was rejected if the total energy in target opposite-side pion fibers was more than 1.0 MeV within 4.0 ns of the average pion fiber hit time tpi . This energy threshold was reduced to 0.5 MeV if the likelihood function from B4EKZ was less than 200. Opposite-side pion fibers were pion fiber hits found on the opposite side of the kaon cluster relative to the main pion cluster (see Section 3.3). TGLIKE - Event was rejected if a target pion fiber showed evidence of hidden energy as determined by two target pion-fiber likelihood functions. The first likelihood function was constructed for each pion fiber from the distance to the extrapolated UTC track, the time and the energy of the pion fiber hit. The rejection condition for the first likelihood function was based on the average likelihood for all pion fibers. A second likelihood function was constructed using only the distance between the pion fiber hit positions and the extrapolated UTC track and had a tighter rejection threshold than the first likelihood function. TIMKF - Event was rejected if the times of the target kaon fiber hits were not consistent with a kaon approaching the kaon decay vertex. This consistency was checked by tabulating the times of the kaon fiber hits against the distance to the decay vertex in the x-y plane and against the range of the kaon as determined by deposited energy. NPITG - Event was rejected if no target fibers were identified as pion fibers. ALLKFIT - Event was rejected if any of the target kaon fibers having more than 3.0 MeV of energy were not successfully fit by the target CCD fitter. A fiber was considered successfully fit if the probability of the fit for the single-pulse or double-pulse assumptions was greater than 0.01. See Appendix C for more detailed discussion of the target CCD fitter. 255  TPICS - Event was rejected if the standard deviation of hit times for the target pion fibers was greater than 4.0 ns. EPIONK - Event was rejected if a target fiber classified as both a kaon and a pion fiber had more than 1.25 MeV assigned to the pion pulse. The target reconstruction was able to find pion hits in kaon fibers when the average hit time for pion fibers tpi was at least 15 ns greater than the average hit time for kaon fibers tk . CHI567 - Event was rejected if the probability of a sum of three χ2 -like quantities was below a certain threshold. These χ2 -like quantities had contributions from target pion fibers and the target π + track fitter [72] as follows: • χ25 - Contributions for hit pion fibers which were part of the reconstructed pion track were based on observed versus expected energy. • χ26 - Contributions for fibers with no energy, but lying along the projected pion track were based on the minimum distance between the projected track and the corners of the fiber. This forced the fitted track to go between the fibers. • χ27 - Contributions for hit pion fibers which were not part of the reconstructed pion track were based on their distance from the track. CHI5MAX - Event was rejected if the contribution to χ25 (see CHI567) due to any single fiber was greater than 10. VERRNG - Event was rejected if the fitted track in the target did not intersect the vertex fiber as identified by target reconstruction. ANGLI - Event was rejected if the range of the charged track in the target was less than 2.0 cm, and the angle between the track from target reconstruction and the UTC extrapolated track was greater than 0.01 radian.  256  KIC - Event was rejected if a hit in the I-Counter was coincident with target kaon time and kaon fibers were found near this counter. CCDBADFIT - Event was rejected if the target CCD fitter was unable to make a successful fit for a kaon fiber having more than 1.25 MeV of energy. The fit was considered unsuccessful if the probability of the fit for both the single-pulse and double-pulse assumptions was zero. This cut is discussed in more detail in Appendix E. CCDBADTIM - Event was rejected if the target CCD fitter found a known incorrect solution for a kaon fiber having more than 1.25 MeV of energy. These known incorrect solutions were as follows: • The first-pulse time from the double-pulse fit was less than -9.98 ns. • The second-pulse time from the double-pulse fit was less than -4.99 ns. • The first-pulse time from the double-pulse fit t1 was inconsistent with the average target kaon fiber hit time tk . The rejection conditions were t1 − tk < −6 ns or t1 − tk > 7 ns. • The time from the single-pulse fit was less than -9.98 ns when the probability of the single-pulse fit was greater than 0.25 and thus the double-pulse fit was not performed. • The time from the single-pulse fit t0 was inconsistent with tk when the probability of the single-pulse fit was greater than 0.25. The rejection conditions were t0 − tk < −6 ns or t0 − tk > 7 ns. This cut is discussed in more detail in Appendix E. CCD31FIB - Event was rejected if the 31st fiber fit by the target CCD fitter had a fit probability for the single-pulse assumption of less than 0.25. When this occurred, 257  the results of the subsequent fit for the double-pulse assumption were not stored correctly. CCDPUL - Event was rejected if the second-pulse found in a kaon fiber by the target CCD fitter had more than 1.25 MeV of energy and was coincident with target pion time tpi . This coincidence condition was quite loose: −7.5 ns < t2 − tpi < 10.0 ns. This cut is discussed in more detail in Appendix E. KPIGAP - This was not one of the analysis cuts used to define the signal region. This cut was used to identify events where the pion track did not emerge directly from the target kaon fibers. This cut, used in the charge exchange background evaluation (see Section 4.8), was designed to provide a cleaner sample of this type of event than that produced by inverting TARGF. The target reconstruction algorithm identified potential pion fibers having large energy deposits as photon fibers. Allowing these photon fibers (within 3.0 ns of Range-Stack track time trs ) to fill the gap created a cleaner sample of events having true gaps between the kaon and pion fibers.  D.8  π + → µ+ → e+ Decay-Sequence Cuts  The π + → µ+ → e+ decay-sequence cuts, collectively referred to as TDCUTS, were designed to reject events with a µ+ as the primary charged particle from the kaon decay. This set of cuts was split into loose (TDLOOSE ) and tight (TDTIGHT ) versions. The loose set TDLOOSE consisted of P2TDCUT, IPIFLG, ELVETO, TDFOOL and a loose version of TDNN. The tight set TDTIGHT consisted of P2TDCUT, IPIFLG, ELVETO, TDFOOL, EV5 and a tight version of TDNN. IPIFLG - Event was rejected if the π + → µ+ decay sequence in the Range-Stack stopping counter was emulated by an accidental hit in conjunction with the muon track. The event was rejected if the time of the first pulse, obtained from TD double-pulse 258  fitting, was not within 2.5 ns of Range-Stack track time trs . ELVETO - Event was rejected if there was accidental activity coincident with the time of the second pulse in the Range-Stack stopping counter. The coincident energy was searched for in the Range-Stack, Barrel Veto, Barrel Veto Liner and End Caps, and the event was rejected if the total coincident energy in a given subsystem was above the threshold energy shown in Table D.1. Time Window (ns)  Category  Both-ends hit category Range-Stack (RS) ±3.00 RS (TD) ±0.25 Barrel Veto (BV) ±1.25 Barrel Veto Liner (BVL) ±2.75  Energy Threshold (MeV) 0.20 5.20 0.20 0.20  Single-end hit category RS single energy, both times ±7.00 RS both energy, single time ±4.50 RS single energy, single time ±8.75 RS no energy, both times ±5.00 RS (TD) single energy, single time ±3.00 BV single energy, both times ±3.00 BV both energy, single time ±0.25 BV single energy, single time ±3.00 BV no energy, both times ±5.75 BVL both energy, single time ±0.75 BVL single energy, single time ±5.00 BVL no energy, both times ±5.50  0.20 9.40 6.60 3.20 1.60 0.40 1.80 0.10 1.00 -  End Cap (EC)  22.00  Other category  ±0.25  Table D.1: The energy threshold and time window relative to muon time in the Range-Stack for each category of accidental hits for the ELVETO cut. Table reproduced from [55].  TDFOOL - Event was rejected if accidental activity was causing the second-pulse in the Range-Stack stopping counter. The second-pulse was considered to have come from 259  accidental activity when there was double-pulse activity present in one of the two layers preceding the stopping counter having second-pulse timing consistent with the second-pulse found in the stopping counter. TDNN - Based on a neural-net function, events having tail fluctuations in the TD pulse or K + decays involving µ+ particles were rejected. The five variables used to construct the neural-net function were the TD single-pulse fit χ2 , the ratio of χ2 for single-pulse fit to double-pulse fit, the time and energy of the fitted second pulse, and the difference between both ends for the second pulse time. The cut position was tighter for TDTIGHT than for TDLOOSE (see Section 6.2.3). EV5 - Event was rejected if the signature of the µ+ → e+ decay was not present in and around the Range-Stack stopping counter. The signature of this decay was a cluster of hits on only one side of the stopping counter, consistent with the time of the third pulse in the stopping counter. This cut was included in the named group of cuts T DT IGHT , but not in T DLOOSE.  D.9  Photon Veto Cut  The event was rejected if there was activity in any of the photon veto subsystems coincident with Range-Stack track time trs . The photon veto (PVCUT) used to define the signal region was split into loose (PV60) and tight (PV30) versions. The photon veto was loosened even further (PV90) for use in the outside-the-box studies. Table D.2 shows the photon veto parameters for each subsystem with the ability to detect photons, where activity above the energy threshold and falling within the time window (defined relative to trs ) caused the event to fail the photon veto cut. Table D.3 shows the parameters used for the BV, BVL and RS when the both-end requirements for time and energy were not met. In addition to the parameters listed in the tables, the PASS2 photon veto P2PVCUT  260  was always applied as part of this photon veto cut. The optimization of the photon veto was based on a process of optimizing the parameters for each of the photon veto subsystems to maximize background rejection while maintaining a reasonable signal acceptance. The PV optimization for E949 had previously been performed for the PNN1 analysis [83], but had to be re-optimized due to the PNN2 analysis including TD hits in the BVL [37]. The acceptance and rejection samples used for this optimization were the Kµ2 monitor trigger data and Kπ2 -peak events from signal data respectively. The optimization process started from an initial set of cut parameters for each subsystem (time offsets, time window and energy thresholds). The cut parameters were changed in small steps, one subsystem at a time, and the acceptance and rejection remeasured. It was considered to be an improved set of parameters if both the acceptance and rejection were increased or if one was increased without the other being lowered. The optimization process continued until no further gains in rejection could be attained without losing acceptance. Nine different goal levels of acceptance were chosen and the entire optimization process was repeated for each of these goal levels of acceptance to produce the profile curve of maximum achievable rejections as functions of acceptance as shown in Figure D.1. The details of this optimization can be found in [37]. The Active Degrader was not used in the E949 PNN1 analysis, but was used in this PNN2 analysis since it was essential in suppressing the most problematic type of Kπ2 target-scatter events as described in 4.1. The optimization of the AD, which required great care be taken not to veto on beam activity at beam time, is detailed in [37]. An additional set of parameters (“early BV”) was added to the PV to remove a class of events observed in the photon veto optimization acceptance sample. These events had large hits in the BV in the time range of 5 to 35 ns earlier than trs . It was found that these hits, due to accidentals, blinded the BV TDCs so that they could not register the hits from the decay photons. The ADC gates for the BV were approximately 50 ns wide so the energy from both the decay photon and the accidental was measured and these 261  Figure D.1: Illustration of the optimization process used to determine the photon veto parameters. The “Profile curve” defines the maximum achievable rejection as a function of acceptance, along which it is no longer possible to further increase the acceptance or rejection without decreasing the other. Reprinted figure with permission from S. Adler et al. (E949 Collaboration), Phys. Rev. D 77 052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003. Copyright 2008 by the American Physical Society.  262  PV60 (Loose) Category  Timing (ns) Offset Window  BV 2.25 early BV -20.70 BVL 3.15 RS 0.05 EC 1.80 EC inner-ring 0.99 EC 2nd pulse -1.60 TG -0.25 IC 1.25 VC -2.40 CO 2.90 µCO -1.60 AD 3.00 DPV 2.50 early BVL -3.50  7.95 15.0 7.55 4.30 6.15 4.64 4.07 2.40 3.25 4.15 2.95 3.90 5.00 7.50 1.50  PV30 (Tight)  Energy (MeV) 0.20 30.00 0.30 0.30 0.40 0.20 10.60 2.00 5.00 6.80 0.60 3.00 0.60 0.00 10.00  Timing (ns) Offset Window 1.35 -22.5 3.15 2.25 1.75 -2.45 -1.51 -2.15 3.20 -0.20 2.15 -0.60 3.00 2.50 -3.50  8.85 15.0 7.55 5.55 7.75 11.55 4.19 4.40 6.10 7.25 2.95 3.90 5.00 7.50 1.50  Energy (MeV) 0.70 30.0 0.30 0.60 0.20 0.20 1.70 1.40 5.00 6.00 1.60 0.60 0.60 0.00 10.00  Table D.2: The parameters for the loose (PV60) and tight (PV30) photon vetoes. The parameters shown for BV, BVL and RS are for when the detector met the requirement that both ends obtained measurements of time and energy. Table reproduced from [56].  blinded BV modules recorded unusually large energies. These events were rejected by adding the “early BV” parameters with a threshold energy of 30 MeV.  263  Category  PV60 (Loose) hit-ends Energy Time  BV BV BV BVL RS RS RS  both single single both both single single  single both single single single both single  Timing (ns) Offset Window 3.05 4.80 -8.10 -5.65 -2.85 6.60 -6.80  15.95 1.50 8.50 11.80 0.70 1.35 1.22  PV30 (Tight)  Energy (MeV) 1.00 1.40 1.60 8.19 5.20 0.00 3.40  Timing (ns) Offset Window 0.55 4.00 -8.30 -5.65 0.01 3.70 -11.54  13.05 3.10 6.90 11.80 5.36 6.10 4.53  Energy (MeV) 0.40 0.60 1.00 8.19 0.20 0.00 0.60  Table D.3: The parameters for the loose (PV60) and tight (PV30) photon vetoes when the requirement was not met that both ends obtained measurements of time and energy. Table reproduced from [56].  PV90 (Very Loose) Category  Timing (ns) Offset Window  BV early BV BVL RS EC EC inner-ring EC 2nd pulse TG IC VC CO µCO AD DPV early BVL  -0.15 -19.15 0.35 -0.85 0.15 -0.35 -2.75 0.25 -0.50 -0.25 2.60 -1.50 3.00 2.50 -3.50  4.00 15.0 1.75 1.45 1.80 2.30 0.32 1.50 2.75 1.50 1.23 2.50 5.00 7.50 1.50  Energy (MeV) 0.50 30.00 0.40 0.20 2.20 1.20 18.80 5.20 13.00 3.80 1.80 3.60 0.60 0.00 10.00  Table D.4: The parameters for the very loose (PV90) photon veto. The parameters shown for BV, BVL and RS are for when the detector met the requirement that both ends obtained measurements of time and energy. Table reproduced from [56].  264  Category  BV BV BV BVL RS RS RS  PV90 (Very Loose) hit-ends Energy Time  Timing (ns) Offset Window  both single single both both single single  0.55 4.00 -8.30 -5.65 0.01 3.70 -11.54  single both single single single both single  13.05 3.10 6.90 11.80 5.36 6.10 4.53  Energy (MeV) 0.40 0.60 1.00 8.19 0.20 0.00 0.60  Table D.5: The parameters for the very loose (PV90) photon veto when the requirement was not met that both ends obtained measurements of time and energy. Table reproduced from [56].  265  Appendix E Target Pulse Cuts The original CCDPUL cut, as it was used in the analysis of the PNN2 kinematic region for the E787 experiment, performed the function of both the CCDBADFIT and CCDPUL cuts described in this appendix. It was split into two for this analysis since CCDBADFIT was meant as a safety cut against events where the target CCD fitter was unable to find valid solutions. The CCDPUL cut was inverted for many of the classes used in the Kπ2 target-scatter rejection branch (Section 4.1.3) and a much cleaner sample of target-scatter events was created by applying CCDBADFIT and inverting CCDPUL than by inverting both of them. The CCDBADTIM cut was a safety cut created as a result of an event observed in the single-cut failure study (Section 5.2) that enforced timing consistency between the times found using the target CCD fitter and those expected from target reconstruction. This cut had a very similar overall structure to CCDBADFIT and CCDPUL.  E.1  Description of CCDPUL  The CCDPUL cut removed events that had kaon fibers with second-pulse energies above 1.25 MeV and coincident with the average hit time of the target pion fibers tpi . The  266  CCDPUL cut had three stages: • In the first stage, the low-gain and high-gain CCDs of each kaon fiber was examined to determine if a double-pulse fit had been successfully performed. • In the second stage, an algorithm determined if the information from the high-gain CCD, the low-gain CCD or a combination of the two should be used for each fiber that had successful double-pulse fits. • In the third stage, each fiber from the second stage was examined for second pulse energies above a 1.25 MeV and rejected the event if the second pulse was coincident with the global pion time tpi .  E.1.1  CCDPUL First Stage  For each kaon fiber, the results of the fits from the low-gain and high-gain CCDs were examined independently. The high-gain CCD data were not passed onto the second stage if any of the following conditions were met: • The single-pulse fit probability was greater than 0.25 as the double-pulse fit was only performed if the single fit probability was less than or equal to 0.25. • The number of bins in the second pulse having a non-zero amplitude was less than 3. • The double-pulse fit probability was equal to zero. The same conditions as above were applied to the low-gain CCDs with one additional condition, if the fiber was found to be multiplexed with at least one other fiber having activity within ±5 ns of tk or tpi then the target low-gain CCD de-multiplexing algorithm would determine the first- and second-pulse energies of the fiber being examined. This de-multiplexing algorithm is detailed in Section E.6. 267  E.1.2  CCDPUL Second Stage  During the second stage of CCDPUL, an algorithm decided for each kaon fiber which CCD information was to be used in the third CCDPUL stage: the low-gain, the high-gain, or a weighted average of the two. If only the high-gain or or only the low-gain CCD data for a given fiber was passed on from the first CCDPUL stage, that is what was used. If the first stage passed on both the high-gain and the low-gain CCD information for a given fiber, the decision of how to use this information in the third CCDPUL stage was based on the following conditions, which were checked in sequence: • An average of the high-gain and low-gain CCD information was used if the fiber energy as determined by ADC was between 10 MeV and 30 MeV and the fractional uncertainty in the second-pulse amplitude for both the low-gain and high-gain CCD channels was greater than 0.05. A typical pulse started to saturate at around 25 MeV so the ADC energy condition included high-gain CCD information that showed small amounts of saturation. The second-pulse energy and the relative time between the first and second pulses were the quantities calculated via the weighted average. For the remaining quantities (such as fit probabilities), the information from the high-gain channel was used. The following equations show how the weighted average was determined for a quantity x :  whi =  dA2hi ; dA2hi + dA2lo  wlo =  dA2lo ; dA2hi + dA2lo  x = xhi whi + xlo wlo ; where dAlo and dAhi were the uncertainties on the fitted second-pulse amplitudes of the low-gain and high-gain double pulse fits, respectively.  268  • The high-gain CCD information was used if it had not saturated. • The low-gain CCD information was used if the high-gain CCD had saturated and the ADC energy in the fiber was greater than 40 MeV. The low-gain target CCDs typically started to saturate at around 40 MeV. • The most appropriate CCD channel was chosen based on the expected quality of the double-pulse fit given the fiber energy Ek (from ADC) and the time difference between the average time of the pion fiber hits tpi and the average time of the kaon fiber hits tk , The high-gain CCD information was used if either of these sets of conditions were met: (15 < Ek < 25) and (tpi − tk > 12); (25 < Ek < 40) and (tpi − tk > 20). The low-gain CCD information was used if either of these sets of conditions were met: (15 < Ek < 25) and (tpi − tk < 12); (25 < Ek < 40) and (tpi − tk < 20). This information was then passed onto the third stage.  E.1.3  CCDPUL Third Stage  The information for all fibers that were passed onto the third stage were checked for second-pulse energy above 1.25 MeV when the timing of the second pulse was consistent with the average pion fiber hit time tpi . The event failed if any fiber in stage three met both of the following conditions: 269  • The second-pulse energy was above 1.25 MeV. • The quantity deltat fell between -7.5 and 10 ns inclusive. The quantity deltat was a measure of consistency between tpi and tk , and the first (t1 ) and second (t2 ) pulse times from the double-pulse fit for that kaon fiber,  deltat = (t2 − t1 ) − (tpi − tk ) .  E.2  (E.1)  Description of CCDBADFIT  The CCDBADFIT safety cut removed events where the target CCD fitter was unable to successfully fit a kaon fiber. This occurred when the probabilities for both the single- and double-pulse fits were equal to zero. As with CCDPUL, this cut also had three stages.  E.2.1  CCDBADFIT First Stage  For each kaon fiber, the results of the fits from the low-gain and high-gain CCDs were examined independently. The high-gain CCD data were not passed onto the second stage if any of the following conditions were met: • The single-pulse fit probability was greater than 0.25 as the double-pulse fit was only performed if the single fit probability was less than or equal to 0.25. • The number of bins in the second pulse having a non-zero amplitude was less than 3 when the double-pulse fit probability was greater than zero. • The double-pulse fit probability was equal to zero and the single-fit probability was greater than zero. The same conditions as above were applied to the low-gain CCD for each kaon fiber.  270  E.2.2  CCDBADFIT Second Stage  During the second stage of CCDBADFIT, a similar decision making process to that used in CCDPUL was used to determine which CCD information was to be passed onto the third stage for each kaon fiber: the low-gain or the high-gain. The difference in CCDBADFIT was that a weighted average of the low-gain and high-gain was not used, but the rest of the decision making sequence was the same.  E.2.3  CCDBADFIT Third Stage  All kaon fibers passed onto the third stage were checked for probabilities equal to zero for both the single- and double-pulse fits. If this condition was met and the fiber energy as determined by the ADC was greater than 1.25 MeV, the event failed the CCDBADFIT cut.  E.3  Description of CCDBADTIM  The CCDBADTIM cut was a safety cut designed to reject events if the target CCD fitter found a known incorrect solution for a kaon fiber having more than 1.25 MeV of energy. The known incorrect solutions fell into two main categories 1. The fitter fell into a local minimum where it attempted to fit the tail of the reference pulse to the actual pulse. The signature for this was a time of -9.9939 ns for the single-pulse fit (t0 ) or for the first pulse of the double-pulse fit (t1 ). Times lower than -9.9939 ns were stored in the ntuple as -9.9939 ns due to range limits placed on these variables when stored to the ntuple. A time of -4.9939 represented the same situation for the second pulse of the double-pulse fit. 2. Due to a large second pulse in the fiber, the fitter fit the second pulse as the primary pulse as shown in Figure E.1. The signature of this failure mode was either t0 or 271  t1 was inconsistent with the average time of the kaon fiber hits (tk ). CCDBADTIM had three similar stages to those found in CCDPUL and CCDBADFIT. An additional fourth stage existed in CCDBADTIM whose purpose was to reject events with known incorrect solutions for the single-pulse fit times. The first stage of CCDBADTIM was identical to that for CCDPUL (Section E.1.1). The second stage of CCDBADTIM was identical to that for CCDBADFIT (Section E.2.2). The third and fourth stages of CCDBADTIM are described below.  E.3.1  CCDBADTIM Third Stage  In the third stage of CCDBADTIM each kaon fiber passed on from the second stage was checked to see if the fiber energy as determined by the ADC was greater than 1.25 MeV. If the energy was above threshold and any of the following conditions were met, the event failed the cut. • The time of the first-pulse in the double-pulse fit t1 was less than -9.98 ns. • The time of the second-pulse in the double-pulse fit t2 was less than -4.99 ns. • The time t1 was not consistent with the average time of the kaon fiber hits tk . The flagging conditions were t1 − tk < −6ns or t1 − tk > 7ns. This choice of cutting parameters is discussed in Section E.5.  E.3.2  CCDBADTIM Fourth Stage  In the fourth stage of CCDBADTIM, the times found in the single-pulse fit were examined for both the high-gain and the low-gain CCDs for each kaon fiber. This process was completely independent of the first three stages of the cut and an event failed CCDBADTIM if it failed due to the conditions laid out in the first three stages or if it failed due to the conditions checked in the fourth stage. The fourth part of the cut was designed 272  70 60 50 40 30 20 10 05 0 -5 0  20  40  12.5 10 7.5 5 2.5 0 -2.5 -5 -7.5 -105 0 -5 0  20  40  60  80  100  120  60  80  100  120  Decay Vertex 438 Raw High  Decay Vertex 438 Raw Low  Figure E.1: An event that failed the CCDBADTIM cut due to the fitter trying to fit the energetic second pulse as the kaon pulse. The single-fit probabilities for both the high-gain (top) and lowgain CCDs were above 0.5 as shown in the row “Prob” and column “Single”. The times of the single-pulse fits (row “Time”) show that the second pulse in each of the high-gain and low-gain CCDs were fit as the single-pulse since the fit times agreed with t pi (“tpi 39.892”) and not tk (“tk 2.776”).  273  to reject events where a known incorrect solution was found for the single-pulse fit when (1) a double-pulse fit was not performed or (2) a good solution to the double-pulse fit was not found. For each fiber, each of the two CCD channels were checked to see if they had a singlepulse fit probability above 0.25 and that the energy from ADC was above 1.25 MeV. If so and either of the following conditions were met, the examined CCD channel for that kaon fiber was flagged for possible rejection by CCDBADTIM. • The single-pulse fit time t0 was less than -9.98 ns. • The single-pulse time (t0 ) was not consistent with tk . The failing conditions were t1 − tk < −6ns or t1 − tk > 7ns. This choice of cutting parameters is discussed in Section E.5. A fiber having a CCD channel that has been flagged for possible rejection by CCDBADTIM caused the event to fail CCDBADTIM if • Both CCD channels were flagged for possible rejection by CCDBADTIM; • One CCD channel was flagged for possible rejection by CCDBADTIM and the other CCD channel had a double-pulse fit probability of zero; • One CCD channel was flagged for possible rejection by CCDBADTIM and information from the other CCD channel was missing.  E.4  CCDPUL Optimization  Based on the previous PNN2 analysis of data from the E787 experiment [20], the secondpulse energy threshold above which would cause an event to be rejected by CCDPUL was initially set to 1.5 MeV. The coincidence window for the deltat coincidence condition (Equation E.1) between the second-pulse time of the double-pulse fit t2 and the average 274  time of the pion fibers hits tpi was initially set to ±10ns. Due to correlations between the cuts CCDBADFIT, CCDPUL and EPIONK, the optimization of these cuts was performed together. The acceptance sample used for the optimization of CCDBADFIT, CCDPUL and EPIONK was the Kµ2 monitor trigger data with very similar setup cuts as were used for measuring the components of acceptance of Abeam (Section 6.1.3). Figure E.2 shows the setup cuts applied to measure the acceptance of CCDBADFIT, CCDPUL and EPIONK with 136263 events remaining upon which to measure the acceptance. These cuts selected signal-like events for the target reconstruction by selecting K + decays with a single charged track, no photons, no beam contamination and no other activity in the detector. To minimize the µ+ scatters in the target, the events were further constrained to be in the Kµ2 -peak by restricting the momentum ptot to be between 229 and 245 MeV/c (PTOT 229 245) and restricting the energy deposited in the Range-Stack (E rs ) to be between 120 and 150 MeV (ERS 120 150). The BV and BVL subsystems were excluded from the photon veto to gain a larger acceptance for Kµ2 events. The rejection sample used for the optimization of CCDBADFIT, CCDPUL and EPIONK was very similar to the Kπ2 target-scatter normalization branch (Section 4.1.2). This sample used πν ν¯(1) and πν ν¯(2) triggers from the 1/3 data set with an inverted photon veto, where the target was excluded from the photon veto. The setup cuts are shown in Figure E.3. The optimization was performed by varying the second-pulse energy threshold (E2 ), and the lower and upper bounds (tlo and thi ) of the timing coincidence quantity deltat (Equation E.1). Recall that an event failed CCDPUL if any of the kaon fibers had a second-pulse energy above E2 and had second-pulse timing coincident with the outgoing charged decay product as indicated by deltat falling within the bounds of tlo and thi . These three parameters (E2 , tlo and thi ) were varied and the optimal values were chosen based on maximizing the product of acceptance and rejection from their respective 275  P2PSCUT, P2TGCUT, RDUTM, TARGET, RD TRK, TRKTIM, UTC, PTOT 229 245, ERS 120 150  TGQUALT, COS3D, ZUTOUT, B4DEDX, BWTRS, B4TRS, CKTRS, CPITRS, TIMCON, DELC3, PV60(noBV,noBVL)  B4EKZ, TGZFOOL, EPIMAXK, TGER, TARGF, DTGTTP, RTDIF, DRP, TGKTIM, EIC, TIC, TGEDGE, TGENR, PIGAP, TGLIKE, TGB4, PHIVTX, NPITG, TIMKF, VERRNG, ANGLI, KIC, EPITG, ALLKFIT N1 CCDBADFIT, CCDPUL, EPIONK  N2  Figure E.2: Setup cuts for the CCDBADFIT, CCDPUL, EPIONK acceptance sample using K µ2 monitor trigger data. The acceptance was the ratio of the number of events N 2 /N1 , where N1 was 136263 events.  P2PSCUT, P2TGCUT, P2TGPVCUT, UTC, RD TRK, TRKTIM, RDUTM, STLAY BEAMCUTS, DELC3, KINCUTS, PNN2BOX∗  PV60(noTarget)  TGCUTS (excluding CCDBADFIT, CCDPUL, EPIONK, CCD31FIB, CCDBADTIM)  N1 CCDBADFIT, CCDPUL, EPIONK  N2  Figure E.3: Setup cuts for the CCDBADFIT, CCDPUL, EPIONK rejection sample using πν ν¯(1) and πν ν¯(2) triggers from the 1/3 data set. The rejection was the ratio of the number of events N1 /N2 , where N1 was 3692 events.  276  samples. For similar values of Acceptance × Rejection, it was preferred to choose the parameters that corresponded to the largest acceptance. Note that the energy parameter E2 was also used in the cuts CCDBADFIT and EPIONK. Table E.1 shows the components of acceptance and rejection when a single parameter was varied and the other two were held fixed. Although the optimization allowed all the parameters to vary at the same time, the table presents only this subset of the parameter space for the sake of readability. Based on this optimization, the final values for the parameters were as follows: • E2 = 1.25MeV; • tlo = −7.5ns; • thi = 10.0ns.  E.5  CCDBADTIM Parameters  The section discusses the choice of time parameters for the CCDBADTIM cut:  t1 − tk < −6, t1 − tk > 7  (E.2)  t0 − tk < −6, t0 − tk > 7.  (E.3)  and  as originally described in Section E.3. A sample was created using Kµ2 monitor trigger data with all the cuts shown in Figure E.2 applied, including CCDBADFIT, CCDBADTIM and EPIONK. The time difference t1 − tk was plotted for each kaon fiber passed onto the third stage of CCDBADTIM (Section E.3.1) that also passed the conditions t1 > −9.98 ns and t2 > −4.99 ns in the third stage. Figure E.4 shows the distribution of these events. The target CCD 277  tlo = −10ns, thi = 10ns E2 (MeV)  Acceptance  Rejection  Acceptance × Rejection  0.75 0.875 1.0 1.125 1.25 1.375 1.5 1.75 2.0  0.299 0.324 0.353 0.384 0.415 0.447 0.479 0.538 0.591  9.764 9.079 8.625 7.916 7.289 6.561 6.097 5.287 4.585  2.917 2.944 3.041 3.037 3.025 2.935 2.923 2.847 2.708  E2 = −10ns, thi = 10ns tlo (ns)  Acceptance  Rejection  Acceptance × Rejection  -6 -7 -7.5 -8 -9 -10 -12 -15 -18  0.475 0.453 0.445 0.437 0.425 0.415 0.401 0.388 0.382  6.142 6.592 6.787 6.877 7.126 7.289 7.486 7.638 7.709  2.918 2.988 3.017 3.005 3.025 3.025 3.001 2.966 2.946  E2 = −10ns, tlo = −10ns thi (ns)  Acceptance  Rejection  Acceptance × Rejection  6 7 7.5 8 9 10 12 15 18  0.434 0.428 0.425 0.423 0.419 0.415 0.409 0.402 0.397  6.809 6.946 7.005 7.077 7.163 7.289 7.514 7.782 7.977  2.952 2.971 2.979 2.993 2.998 3.025 3.070 3.128 3.169  Table E.1: Acceptance and rejection results for the combined group of cuts CCDPUL, CCDBADFIT and EPIONK. The three parameters E 2 , tlo and tlo were varied to maximize the product of acceptance and rejection. Each of the three sections of the table show one of the three parameters being varied while the other two were held constant.  278  Figure E.4: Setting the allowed time window for consistency between t 1 and tk in the CCDBADTIM cut. The plot shows t1 − tk for the sample of events described in the text of Section E.5. The bounds for the cut were set to t 1 − tk < −6 ns and t1 − tk > 7 ns as indicated by the red lines on the plot.  information for many events falling outside these bounds were visually examined and were found to be consistent with the second target CCD fitter failure mode described in this section. A similar study was performed for the time difference t0 − tk and the same bounds as chosen for t1 − tk were found to be appropriate. The final acceptance of the safety cut CCDBADTIM was measured to be 0.99 as detailed in Section 6.1.3.  279  E.6  Updating the De-multiplexing Algorithm  An event in the single-cut failure study for the 1/3 sample data sample (Section 5.2.1) revealed a mistake in the way low-gain target fiber CCD information was being demultiplexed for use in the CCDPUL cut. The low-gain target fiber CCDs were multiplexed in groups of five fibers that were spatially found far apart from each other within the target. When more than one fiber in a multiplexed group had a hit at the same time, the amplitude information for that group had to be de-multiplexed using ADC information from each of the fibers to determine if second-pulse activity in a studied fiber was due to activity in that fiber of in another fiber within the same group.  E.6.1  Previous De-multiplexing Algorithm  In the previous de-multiplexing algorithm the second-pulse energy E2 found by the doublepulse fitter was corrected for the ADC energies from the other fibers in the group for energy within ±5 ns of tpi . This energy came from fibers assigned as pion, opposite-side pion or gamma fibers (see Section 3.3.4 for a discussion of target fiber assignment). The sum of the coincident energy from the other fibers was called the multiplexed pion energy and this energy was subtracted from the E2 value found by the fitter to give a corrected second-pulse energy E2 (corr.). The total ADC energy for a fiber was split between the two pulses in a double-pulse fit to give an energy E1 to the first pulse and an energy E2 to the second pulse. This energy was split up based on the relative amplitudes of the two pulses as found by the double-pulse fitter. The flaw in the just described de-multiplexing algorithm was that if any multiplexed pion energy was found, the sum of E1 and E2 (corr.) was not equal to the original ADC energy of the fiber. Since the target fiber ADCs were not multiplexed, the corrected energies of the two pulses should always have added up to the total ADC energy for that fiber. 280  E.6.2  New De-multiplexing Algorithm  The algorithm was modified so that the ADC energies near tk and near tpi from the other fibers in a group were taken into account when dividing the ADC energy in the studied fiber between the first and second pulse. The steps in the new de-multiplexing algorithm were as follows: 1. The multiplexed pion energy was determined using the same method described in Section E.6.1. 2. ADC energies within ±5 ns of tk from the other kaon fibers within the group were summed to create the multiplexed kaon energy. 3. The pion and kaon multiplexed energies were added to the ADC energy of the studied fiber to get a total energy as seen by the low-gain CCD channel of that fiber. 4. This total CCD energy was split between the first and second pulses according the ratio of the fitted amplitudes to give E1 and E2 , respectively. 5. The kaon multiplexed energy was subtracted from E1 to give the corrected energy E1 (corr.) and the pion multiplexed energy was subtracted from E2 to give the corrected energy E2 (corr.). 6. If either of the resulting corrected energies was below 0.001 MeV, it was assigned an energy of 0.001 MeV and the other pulse was assigned the remaining ADC energy ensuring the total energy from the two pulses was equal to the ADC energy of the fiber. This new de-multiplexing algorithm resulted in slightly less than 1% additional events failing the CCDPUL cut in both the acceptance and rejection samples used for the optimization of CCDPUL (Section E.4). Figure E.5 shows an event that passed CCDPUL 281  when the previous de-multiplexing algorithm was used, but failed the cut after the new algorithm was implemented. These events typically had a corrected E2 that was slightly below the 1.25 MeV threshold when the old algorithm was used and slightly above the 1.25 MeV threshold when the new algorithm was used.  282  180 160 140 120 100 80 60 40 20 05 0 -5 0  20  40  45 40 35 30 25 20 15 10 5 05 0 -5 0  20  40  60  80  100  120  60  80  100  120  Kaon fiber 278 Raw High  Kaon fiber 278 Raw Low  Old Algorithm  New Algorithm  Multiplexed kaon and pion energies (MeV) Ek (mux) Eπ (mux)  N/A 1.58  0. 1.58  Fitted pulse energies from ADC energy (MeV) Total Energy E1 E2  12.28 9.53 2.75  12.28 + 0.00 + 1.58 = 13.86 10.76 3.10  Corrected energies (MeV) E1 (corr.) E2 (corr.)  9.53 2.75 − 1.58 = 1.17  10.76 3.10 − 1.58 = 1.52  Figure E.5: Newly Rejected CCDPUL event due to new de-multiplexing algorithm. Using the old algorithm, the event passed CCDPUL with 1.17 MeV in the second pulse (E 2 (corr.)). Using the new algorithm, the event passed CCDPUL with E 2 (corr.) = 1.17 MeV. The multiplexed pion energy Eπ (mux) came from a pion fiber in the same group that had a time of 5.98 ns and energy of 1.58 MeV.  283  Appendix F Acceptance Factors for the K + → π +π 0 Branching Ratio This appendix details the acceptance measurements used to find the total acceptance (Aπ2 ) used in the measurement of the Kπ2 branching ratio.  F.1  Acceptance Factors Using Monte Carlo  kin The trigger (AMC trig ) and kinematic (Atrig ) components of acceptance were measured using  Kπ2 events from the Monte Carlo simulation. The acceptance losses due to pion decayin-flight and pion nuclear interactions were also measured due to the inclusion of these processes in Monte Carlo. The cut KP2STOP required that the stopping layer be between layers 8 and 15 inclusive. The cut UFATE required that the pion stopped without decay or interaction. The cut USTMED required that the pion stopped in the Range-Stack scintillator. The cut USTOP HEX required that the offline reconstructed stopping counter agreed with the true stopping counter. These four cuts used information taken directly from the Monte Carlo event and not the subsequent reconstruction. The cut SETUP was a requirement that the reconstructed momentum ptot was less than 300 MeV. The  284  Cut SETUP T•2 6ct + 7ct 19ct UFATE USTMED USTOP HEX AMC trig  Events  Acceptance  99993 44891 37605 36986 31222 30518 27426  0.4489 0.8377 0.9835 0.8442 0.9775 0.8987  0.2743 ± 0.0014  RDUTM UTCQUAL TARGET+TGQUALT  27426 26910 25659  1.0000 0.9812 0.9535  KP2STOP COS3D KP2BOX  24639 23671 20213  0.9603 0.9607 0.8539  AMC kin  0.7878 ± 0.0026  Table F.1: Monte-Carlo-based components of acceptance for the K π2 branching ratio calculation. The cuts “RDUTM”, “UTCQUAL” and “TARGET+TGQUALT” were used as setup cuts for the acceptance AMC kin . Table reproduced from [55].  results for these measurements are found in Table F.1.  F.2  Acceptance Factors from Kπ2 Monitor Trigger Events  The setup cuts applied to measure each of Ard , Arecon and Abm−tg are shown in Table F.2. For each of the measurements, three additional setup cuts were applied as part of the TRIGGER cut: (1) RNGMOM to remove Kµ2 events, (2) the offline stopping layer had to be found in the Range Stack, and (3) the momentum was required to be in the range of 0MeV/c < ptot < 300MeV/c. The second and third cuts were used to ensure good reconstruction. The results of these acceptance measurements are found in Tables 285  Kµ2 Setup Categories  Measured Quantities  Setuprd  Ard  TRIGGER, ICBIT, tIC − tCK > 5ns, B4DEDX, UTC, TARGET  Setuprecon  Arecon  TRIGGER, ICBIT, tIC − tCK > 5ns, B4DEDX, CPITRS, CPITAIL, CKTRS, CKTAIL, BWTRS, RVUPV, Ard cuts  Setupbm−tg  Abm−tg  TRIGGER, ICBIT, Ard cuts, Arecon cuts, KP2BOX, KP2STOP, IPIFLG, COS3D  Setup Cuts  Table F.2: The setup cuts applied for the components of acceptance measured for the K π2 branching ratio calculation. The notation “A rd cuts” means that all the cuts whose acceptance were measured to determine the quantity A rd were applied as setup cuts. Table reproduced from [55].  Cut  Events  Acceptance  Setuprd RD TRK TRKTIM  490579 490579 490579  1.0000 1.0000  Ard  1.0000 ± 0.0000  Table F.3: The acceptance Ard for the Kπ2 branching ratio calculation. Table reproduced from [55].  F.3, F.4 and F.5.  F.3  Calculation of Total Acceptance  The total acceptance, Aπ2 , used in the measurement of the Kπ2 branching ratio was the product of the components of acceptance determined in this appendix thus far in addition to four more acceptance factors previously determined in the PNN1 analysis [33]: MC acc AKπ2 = AMC trig × Akin × Ard × Arecon × Abm−tg × Aµ × Aipiflg × fs ×  286  Kπ2 T•2  (F.1)  Cut  Events  Acceptance  Setuprecon RDUTM UTCQUAL TARGET+TGQUALT  449621 449621 407402 386491  1.0000 0.9061 0.9487  Arecon  0.8596 ± 0.0005  Table F.4: The acceptance Arecon for the Kπ2 branching ratio calculation. Table reproduced from [55].  Cut  Events  Acceptance  Setupbm−tg TIC TIMCON TGTCON DCBIT DELC CKTRS CKTAIL B4DEDX CPITRS CPITAIL TARGF DTGTTP RTDIF TGQUALT PIGAP TGB4 KIC TGGEO B4EKZ B4ETCON TGZFOOL BWTRS RVUPV  336407 336173 334866 331041 293685 257163 250305 243233 239389 237015 236833 226972 226965 224789 224789 222488 207545 204785 169847 157338 156673 154703 148013 145339  0.9993 0.9961 0.9886 0.8872 0.8756 0.9733 0.9718 0.9842 0.9901 0.9992 0.9584 1.0000 0.9904 1.0000 0.9898 0.9328 0.9867 0.8294 0.9264 0.9958 0.9874 0.9568 0.9819  Abm−tg  0.4320 ± 0.0009  Table F.5: The acceptance Abm−tg for the Kπ2 branching ratio calculation. Table reproduced from [55].  287  AMC trig AMC kin Ard Arecon Abm−tg Aacc µ Aipiflg fs  Value  Source  0.2743 ± 0.0014  Table F.1  0.7878 ± 0.0026  1.0000 ± 0.0000  0.8596 ± 0.0005  Table F.1 Table F.3 Table F.4  0.4320 ± 0.0009  Table F.5  0.8350 ± 0.8350  [33]  0.9931 ± 0.0002  0.7740 ± 0.0011  Kπ2 T•2  0.9383 ± 0.0027  AKπ2  0.04833 ± 0.00047  [33] [33] [33]  Table F.6: Summary of the components of acceptance used to find A Kπ2 for the Kπ2 branching ratio calculation.  The conditions for these additional acceptance factors were the same for PNN2 as they were for PNN1. These acceptance factors were: • Aacc - the acceptance loss due to the 19ct requirement in the Kπ2 monitor trigger ✆ (Section 2.7.2); • Aipiflg - Acceptance of the IPIFLG cut as measured using πscatter monitor trigger data; • f ✻ - K + stopping fraction (discussed previously in Section 6.7); •  2  ✼✾✿ ✽ 2 T  - Correction due to T•2 trigger inefficiencies for Kπ2 events. This correction  for K + → ν ν¯ events in the PNN2 phase space was discussed in Section 6.6. The resulting acceptance AKπ2 was 0.04833 ± 0.00047, where the values used in this acceptance calculation can be found in Table F.6.  288  

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