Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
[if-you-see-this-DO-NOT-CLICK]
ubc_spring_2009_ives_joss.pdf [ 13.48MB ]
Metadata
JSON: 1.0066858.json
JSON-LD: 1.0066858+ld.json
RDF/XML (Pretty): 1.0066858.xml
RDF/JSON: 1.0066858+rdf.json
Turtle: 1.0066858+rdf-turtle.txt
N-Triples: 1.0066858+rdf-ntriples.txt
Original Record: 1.0066858 +original-record.json
Full Text
1.0066858.txt
Citation
1.0066858.ris

Full Text

THE MEASUREMENT OF THE RARE DECAY K-PLUS TOPI-PLUS, NEUTRINO, AND ANTI-NEUTRINObyJOSS IVESB.Sc., The University of Saskatchewan, 2000M.Sc., The University of Saskatchewan, 2003A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2008c Joss Ives, 2008AbstractBrookhaven National Laboratory experiment E949 was designed to search for the rareK meson decay K+ !  +  , a decay sensitive to physics beyond the Standard Model.While previous data analyses dealt with the high  + momentum region accessible for thisreaction, this thesis concentrates on the lower range between 140 and 199 MeV/c. Anal-ysis of this low  + momentum region was performed to search for additional evidence ofthe process K+ !  +  . A blind analysis technique was used to avoid bias when devel-oping the selection criteria used to suppress the competing background processes. Theblind analysis technique was based on identifying background sources a priori and onlyexamining the signal region once all selection criteria and background estimates had been nalized. The background estimates were performed using a technique known as a \bifur-cation method", which relied on using two uncorrelated selection criteria to suppress eachbackground source. The analysis of an exposure of 1:71 1012 K+ decays resulted in an ob-servation of three events with an estimated background of 0:927 0:168(stat:) +0:320 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 E949results, yielding a branching ratio of B(K+ !  +  ) =  1:73+1:15 1:05  10 10 at the 68%con dence level. This branching ratio is consistent with the prediction of the StandardModel, (0:85 0:07) 10 10.iiContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Theory and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 CP-Violation and Weak Interactions . . . . . . . . . . . . . . . . . . . . . 21.3 K+ !  +  in the Standard Model . . . . . . . . . . . . . . . . . . . . . . 81.4 Physics Beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . 121.5 History of K+ !  +  Experiments . . . . . . . . . . . . . . . . . . . . . 141.6 My Role in the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Experimental Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Kaon Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Beam Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Range-Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6 Photon Veto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.7 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.8 Summary of 2002 Data Collection . . . . . . . . . . . . . . . . . . . . . . . 483 Analysis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 Overview of Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 Analysis Strategy and Methods . . . . . . . . . . . . . . . . . . . . . . . . 543.3 Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.5 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76iii4 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1 K 2 Target-Scatter Background . . . . . . . . . . . . . . . . . . . . . . . . 794.2 K 2 Range-Stack-Scatter Background . . . . . . . . . . . . . . . . . . . . . 944.3 K 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.4 Muon Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.5 Ke4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.6 Single-Beam Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.7 Double-Beam Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.8 Charge Exchange Background . . . . . . . . . . . . . . . . . . . . . . . . . 1274.9 Other Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.10 Background Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365 Validity Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.1 Outside-the-Box Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.2 Single-Cut and Double-Cut Failure Studies . . . . . . . . . . . . . . . . . . 1445.3 Background Cross-Contamination Studies . . . . . . . . . . . . . . . . . . 1496 Signal Acceptance and Sensitivity . . . . . . . . . . . . . . . . . . . . . . . 1656.1 Acceptance Factors from K 2 Monitor Trigger Events . . . . . . . . . . . . 1666.2 Acceptance Factors from  scatter Monitor Trigger Events . . . . . . . . . . . 1706.3 Acceptance Factors from K 2 Monitor Trigger Events . . . . . . . . . . . . 1806.4 Acceptance Factors Using Monte Carlo . . . . . . . . . . . . . . . . . . . . 1826.5 Acceptance Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.6 Correction to T 2 Trigger Acceptance . . . . . . . . . . . . . . . . . . . . . 1836.7 K+ Stopping Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1866.8 Measurement of the K 2 Branching Ratio . . . . . . . . . . . . . . . . . . . 1866.9 Single-Event Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907.1 Cell De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907.2 Examination of the Signal Region . . . . . . . . . . . . . . . . . . . . . . . 1947.3 Calculation of K+ !  +  Branching Ratio . . . . . . . . . . . . . . . . . 2118 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222A E787 to E949 Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227A.1 Beam Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227A.2 Detector Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227A.3 Trigger and DAQ Upgrades . . . . . . . . . . . . . . . . . . . . . . . . . . 228ivB Data Acquisition, Storage, and Processing . . . . . . . . . . . . . . . . . 229B.1 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229B.2 PASS0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231B.3 PASS1 and PASS2 Processing . . . . . . . . . . . . . . . . . . . . . . . . . 231B.4 PASS3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231C Target Pulse Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 232C.1 Creation of Standardized Pulses . . . . . . . . . . . . . . . . . . . . . . . . 233C.2 Overview of the Fitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234C.3 Optimization of the Error Input . . . . . . . . . . . . . . . . . . . . . . . . 235C.4 Hold and Release Double-Pulse Fit . . . . . . . . . . . . . . . . . . . . . . 238D Detailed List of Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241D.1 PASS1 Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241D.2 PASS2 Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242D.3 Kinematic Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244D.4 Phase Space Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246D.5 Beam Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247D.6 Delayed Coincidence Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . 251D.7 Target Quality Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252D.8  + !  + ! e+ Decay-Sequence Cuts . . . . . . . . . . . . . . . . . . . . . 258D.9 Photon Veto Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260E Target Pulse Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266E.1 Description of CCDPUL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266E.2 Description of CCDBADFIT . . . . . . . . . . . . . . . . . . . . . . . . . . 270E.3 Description of CCDBADTIM . . . . . . . . . . . . . . . . . . . . . . . . . 271E.4 CCDPUL Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274E.5 CCDBADTIM Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 277E.6 Updating the De-multiplexing Algorithm . . . . . . . . . . . . . . . . . . . 280F Acceptance Factors for the K+ !  + 0 Branching Ratio . . . . . . . 284F.1 Acceptance Factors Using Monte Carlo . . . . . . . . . . . . . . . . . . . . 284F.2 Acceptance Factors from K 2 Monitor Trigger Events . . . . . . . . . . . . 285F.3 Calculation of Total Acceptance . . . . . . . . . . . . . . . . . . . . . . . . 286vList of Tables4.1 De nition of the classes of events used to measure the PV rejection in thekinematic signal region for K 2 target-scatter backgrounds. . . . . . . . . . 854.2 K 2 target-scatter loose photon veto rejections . . . . . . . . . . . . . . . . 864.3 Rejection of the tight photon veto for class 12 with various combinationsof loose and tight versions of the setup cuts . . . . . . . . . . . . . . . . . 884.4 Summary of the loose K 2 target-scatter background evaluation . . . . . . 914.5 Summary of the tight K 2 target-scatter background evaluation . . . . . . 924.6 Summary of the K 2 Range-Stack-scatter background evaluation . . . . . . 964.7 Summary of values used to determine A(K 2) and A(K 2 ) . . . . . . . . . 1014.8 Summary of K 2 background evaluation . . . . . . . . . . . . . . . . . . . 1024.9 Summary of the muon background evaluation . . . . . . . . . . . . . . . . 1064.10 Rejection of RTGPV OPSVETO for loose rejection branch . . . . . . . . . . . . 1124.11 Rejection of RTGPV OPSVETO for tight rejection branch . . . . . . . . . . . . 1124.12 Ke4 background summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.13 Summary of the single-beam background evaluation . . . . . . . . . . . . . 1174.14 Summary of the double-beam KK-background evaluation . . . . . . . . . . 1244.15 Summary of the double-beam KP-background evaluation . . . . . . . . . . 1254.16 Summary of the total expected double-beam background . . . . . . . . . . 1254.17 Summary of the charge-exchange background evaluation . . . . . . . . . . 1314.18 Methods of suppression of various K+ decays . . . . . . . . . . . . . . . . 1354.19 Total expected background in the loose signal region . . . . . . . . . . . . 1364.20 Total expected background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.1 Summary of PV1 Outside-the-Box Study . . . . . . . . . . . . . . . . . . . 1405.2 Summary of PV2 Outside-the-Box Study . . . . . . . . . . . . . . . . . . . 1415.3 Summary of EPI outside-the-box study . . . . . . . . . . . . . . . . . . . . 1435.4 Single-Cut Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.5 1/3 Double-cut failures (detailed) . . . . . . . . . . . . . . . . . . . . . . . 1475.6 2/3 Double-cut failures (detailed) . . . . . . . . . . . . . . . . . . . . . . . 1485.7 Pion acceptance of muon bifurcation cuts . . . . . . . . . . . . . . . . . . . 1515.8 Correcting for muon contamination in target-scatter photon veto rejection 1555.9 Acceptance and rejection of double-beam bifurcation cuts . . . . . . . . . . 1575.10 Correcting target-scatter normalization branch for double-beam contami-nation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1585.11 Correcting target-scatter rejection branch for KK contamination . . . . . . 160vi5.12 Correcting target-scatter rejection branch for KP contamination . . . . . . 1606.1 Setup cuts for the K 2-based acceptance measurements . . . . . . . . . . . 1676.2 Range-stack-reconstruction acceptance . . . . . . . . . . . . . . . . . . . . 1676.3 Target and UTC reconstruction acceptance . . . . . . . . . . . . . . . . . . 1686.4 Target and Beam acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . 1696.5 Online and o ine photon veto acceptance . . . . . . . . . . . . . . . . . . 1716.6 K 2-based acceptance summary . . . . . . . . . . . . . . . . . . . . . . . . 1726.7 Setup cuts for the  scatter-based acceptance measurements . . . . . . . . . . 1726.8 BAD STC acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736.9 The \small" and \large" kinematic signal regions . . . . . . . . . . . . . . 1756.10 Range-Stack-kinematic acceptance . . . . . . . . . . . . . . . . . . . . . . . 1766.11 Range-Stack-kinematic acceptance in the small kinematic box . . . . . . . 1766.12 Range-Stack-kinematic acceptance in the large kinematic box . . . . . . . . 1776.13  + !  + ! e+ identi cation acceptance (TD1) . . . . . . . . . . . . . . . 1786.14  + !  + ! e+ identi cation acceptance (TD2) . . . . . . . . . . . . . . . 1796.15  scatter-based acceptance summary . . . . . . . . . . . . . . . . . . . . . . . 1796.16 Setup cuts for the K 2-based acceptance measurements . . . . . . . . . . . 1806.17 UTC acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.18 OPSVETO acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.19 Target kinematic acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.20 K 2-based acceptance summary . . . . . . . . . . . . . . . . . . . . . . . . 1836.21 Monte-Carlo-based acceptance . . . . . . . . . . . . . . . . . . . . . . . . . 1846.22 Total acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1846.23 Summary of K 2 branching ratio measurements . . . . . . . . . . . . . . . 1876.24 The single event sensitivity summary . . . . . . . . . . . . . . . . . . . . . 1897.1 Acceptance and background summary of each cell . . . . . . . . . . . . . . 1917.2 Acceptance losses and additional rejection for each background . . . . . . . 1927.3 Summary of the  + kinematics and S/B for candidate events . . . . . . . . 1957.4 K 2 Momentum, energy and range for runs containing candidates . . . . . 195B.1 Digitizing electronics for E949 . . . . . . . . . . . . . . . . . . . . . . . . . 230C.1 Components of acceptance for various target  tter  xes . . . . . . . . . . . 239D.1 Time and energy thresholds for ELVETO . . . . . . . . . . . . . . . . . . . 259D.2 Photon veto cut parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 263D.3 Photon veto cut parameters when both-ends requirement was not met . . . 264D.4 Very loose PV parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 264D.5 Very loose PV parameters when both-ends requirement was not met . . . . 265E.1 Parameter optimization for CCDPUL, CCDBADFIT and EPIONK . . . . 278F.1 Monte-Carlo-based components of acceptance for the K 2 branching ratiocalculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285viiF.2 Setup cuts for the K 2 branching ratio calculation . . . . . . . . . . . . . . 286F.3 Ard for the K 2 branching ratio calculation . . . . . . . . . . . . . . . . . . 286F.4 Arecon for the K 2 branching ratio calculation . . . . . . . . . . . . . . . . . 287F.5 Abm tg for the K 2 branching ratio calculation . . . . . . . . . . . . . . . . 287F.6 Acceptance summary for the K 2 branching ratio calculation . . . . . . . . 288viiiList of Figures1.1 The decay K0L !  +  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Unitarity triangles in the    plane . . . . . . . . . . . . . . . . . . . . . 71.3 Unitarity triangle as determined by B and K decays . . . . . . . . . . . . 71.4 First-order weak  avor-changing processes . . . . . . . . . . . . . . . . . . 81.5 Second-order weak processes that contribute to the K+ !  +  branchingratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 LHT predictions for the relationship between B(K+ !  +  ) and B(K0L ! 0  ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Schematic views of the upper half of the E949 detector . . . . . . . . . . . 182.2 Schematic view of the AGS complex . . . . . . . . . . . . . . . . . . . . . . 192.3 Low-energy separated beam line III at BNL . . . . . . . . . . . . . . . . . 202.4 Schematic of  Cerenkov counter . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Schematic of the Upstream Photon Veto . . . . . . . . . . . . . . . . . . . 242.6 Cross-sectional views of the BWPCs . . . . . . . . . . . . . . . . . . . . . 252.7 Schematic of the Active Degrader . . . . . . . . . . . . . . . . . . . . . . . 272.8 Downstream and schematic cross-sectional views of the B4 Hodoscope . . . 282.9 End and side views of the target . . . . . . . . . . . . . . . . . . . . . . . . 302.10 Schematic of the UTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.11 End view of the Range-Stack . . . . . . . . . . . . . . . . . . . . . . . . . . 332.12 End view of the Range-Stack Straw Chambers . . . . . . . . . . . . . . . . 352.13 Radiation length versus polar angle for photon detectors . . . . . . . . . . 362.14 End view of the Barrel Veto and Barrel Veto Liner . . . . . . . . . . . . . 382.15 End and side views of the upstream End Cap . . . . . . . . . . . . . . . . 392.16 Collar and Microcollar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.17 Schematic of the Downstream Photon Veto . . . . . . . . . . . . . . . . . . 412.18 Number of E787/E949 kaon decays as a function of data-taking days . . . 493.1 Momentum spectrum of the Standard Model K+ !  +  process . . . . . 513.2 The range versus momentum for events passing    (1) or    (2) triggers . 533.3 A schematic explanation of the bifurcation method. Figure reproducedfrom [37]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4 Schematic representation of the outside-the-box study . . . . . . . . . . . . 583.5 Reconstruction of an event in x-y view. . . . . . . . . . . . . . . . . . . . . 613.6 Double-pulse  t of the CCD information in the Kaon decay vertex  ber . . 66ix3.7 Double and Triple-pulse TD  ts in the RS stopping counter . . . . . . . . 674.1 Schematics of K 2 target-scatter and regular K 2 . . . . . . . . . . . . . . 804.2 Flowchart of K 2 target-scatter bifurcations . . . . . . . . . . . . . . . . . 834.3 Momentum distributions of K 2 target-scatter bifurcations . . . . . . . . . 904.4 Flowchart of K 2 Range-Stack-scatter bifurcations . . . . . . . . . . . . . . 974.5 Kinetic energy distribution of the  + from K 2 events in Monte Carlo . . 1004.6 Flowchart of muon bifurcation branches . . . . . . . . . . . . . . . . . . . . 1044.7 Typical Ke4 event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.8 e+ and   kinetic energy vs.  + momentum for Ke4 events . . . . . . . . . 1084.9 Flowchart of Ke4 normalization branch . . . . . . . . . . . . . . . . . . . . 1094.10 Observable absorption energy of   stopped in the Range Stack . . . . . . 1114.11 A schematic of the single-beam background processes . . . . . . . . . . . . 1144.12 Flowchart of Single-Beam Normalization Branches . . . . . . . . . . . . . . 1184.13 A Schematic of the double-beam background processes . . . . . . . . . . . 1194.14 Flowchart of Double-Beam Normalization Branches . . . . . . . . . . . . . 1224.15 Flowchart of Double-Beam Rejection Branches . . . . . . . . . . . . . . . . 1264.16 A Schematic of the problematic charge-exchange background processes . . 1274.17 Flowcharts of charge-exchange bifurcation branches . . . . . . . . . . . . . 1295.1 Flowchart of muon contamination in target-scatter rejection branch . . . . 1546.1 Distributions of the reconstructed  + mass . . . . . . . . . . . . . . . . . . 1756.2 Flowchart of cuts applied to measure NK 2 . . . . . . . . . . . . . . . . . . 1876.3 K 2 branching fraction versus rate . . . . . . . . . . . . . . . . . . . . . . . 1887.1 Energy vs. range of candidate events passing all other cuts . . . . . . . . . 1967.2 Event parameter displays for Candidate A . . . . . . . . . . . . . . . . . . 1977.3 Event parameter displays for Candidate B . . . . . . . . . . . . . . . . . . 1987.4 Event parameter displays for Candidate C . . . . . . . . . . . . . . . . . . 1997.5 Quantities related to timing consistency in reconstruction cuts . . . . . . . 2017.6 Quantities related to reconstruction cuts in the target . . . . . . . . . . . . 2027.7 Quantities related to UTC and Range Stack reconstruction . . . . . . . . . 2037.8 Quantities related to target kaon reconstruction . . . . . . . . . . . . . . . 2047.9 Quantities related to single beam detection and photon veto . . . . . . . . 2067.10 Quantities related to Range Stack kinematics . . . . . . . . . . . . . . . . 2077.11 Quantities related to pion particle identi cation from TD variables . . . . . 2087.12 Quantities related to pion particle identi cation from kinematic variables . 2097.13 More quantities related to single beam detection and photon veto . . . . . 2107.14 Signal-like probability for the three candidates . . . . . . . . . . . . . . . . 2127.15 Xobs and CLs for the three E949-PNN2 candidates . . . . . . . . . . . . . 2157.16 Xobs and CLs for the three E949-PNN2 candidates . . . . . . . . . . . . . 2187.17 Comparison of Standard Model, previous PNN1 only, and  nal E787/E949branching ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219xC.1 Target CCD  tter low-count uncertainty  x . . . . . . . . . . . . . . . . . 237C.2 E ect of  rst bin uncertainty  x on target  tter . . . . . . . . . . . . . . . 240D.1 Illustration of the photon veto optimization process . . . . . . . . . . . . . 262E.1 An event that failed the CCDBADTIM cut . . . . . . . . . . . . . . . . . 273E.2 Setup cuts for CCDBADFIT, CCDPUL, EPIONK acceptance sample . . . 276E.3 Setup cuts for CCDBADFIT, CCDPUL, EPIONK rejection sample . . . . 276E.4 CCDBADTIM distribution for t1 - tk . . . . . . . . . . . . . . . . . . . . . 279E.5 Newly Rejected CCDPUL event due to new de-multiplexing algorithm . . . 283xiGlossaryA 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 TGT-CON. See Appendix D for descriptions of the individual cuts;BeO BeO Beryllium-oxide;bg Background-level expressed as a number of events expected inthe 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 charged-coupled device;CEX Charged Exchange;CK Kaon  Cerenkov Counter;xiiC 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 nor-malization branch;CUT2 The set of bifurcation cuts that are inverted to create the re-jection 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;K 2 K+ !  +  ;K 2 K+ !  +  ;K 3 K+ !  + 0 ;Ke4 K+ !  +  e+ ;K 2 K+ !  + 0;K 2 K+ !  + 0 ;KBlive The number of K+s that entered the target while the detectorwas live (available to register a trigger);xiiiKINCUTS The set of kinematic cuts consisting of LAYER14, COS3D,LAYV4, ZFRF, ZUTOUT, UTCQUAL, PRRF, RSDEDX andRNGMOM. See Appendix D for descriptions of the individualcuts;LED Light-Emitting diode;LESBIII Low-Energy Separated Beamline III; CO Microcollar;PAW Physics Analysis Workstation;PMT Photo-Multiplier Tube;PNN1 The signal region above the K 2 momentum peak;PNN2 The signal region below the K 2 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 stacktrack quality cuts RSDEDX and PRRF. See Appendix D fordescriptions of the individual cuts;rtot Range in plastic scintillator of the charged track from the kaondecay point to the stopping counter in the Range-Stack;RV RV Ring Veto;SCF Online Range-Stack stopping-counter- nderSES Single Event Sensitivity;SM Standard Model;T 2 Coincidence between  rst two layers of the Range-Stack in thesame sector;TD Transient Digitizers;TDC TDC Time-to-Digital Converter;TDCUTS Either the loose set of TD cuts TDLOOSE or the tight set ofTD cuts TDTIGHT depending on the context;xivTDLOOSE The set of TD cuts consisting of IPIFLG, ELVETO, TDFOOLand the loose version of TDVARNN. See Appendix D for de-scriptions 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 fordescriptions of the individual cuts;TG Target;TGCUTS The set of target quality cuts consisting of B4EKZ, TGZFOOL,EPITG, EPIMAXK, TARGF, DTGTTP, RTDIF, DRP, TGK-TIM, EIC, TIC, TGEDGE, TGDEDX, TGENR, TGER, PI-GAP, 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 individualcuts; Polar angle of charged track determined by UTC;RS Range-Stack;tk Average time of the kaon  ber hits in the target;tpi Average time of the pion  ber 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.xvChapter 1IntroductionThe Standard Model (SM) describes the interaction of elementary particles through theelectroweak and strong forces, but does not include a quantum description of gravity. TheSM leaves many questions unanswered, such as: what caused the dominance of matterover anti-matter in our Universe (known as the baryon asymmetry), how do neutrinosacquire 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 betaken. The  rst approach is to search directly for heavy particles by producing them atincreasingly powerful accelerators or by observation of very-high-energy cosmic rays. Thesecond approach is to measure to very high precision, quantities predicted by the SM inan attempt to  nd discrepancies between the measured and predicted quantities. Oneexample of this second approach is the measurement of the branching ratio of the raredecay K+ !  +  .This chapter discusses the branching ratio of K+ !  +  as predicted by the SM andby other new physics models. This chapter also contains a brief history of measurementsof the K+ !  +  branching ratio.11.1 Theory and MotivationThe Sakharov conditions [90] are regarded as the necessary conditions to explain thebaryon asymmetry in our universe. Any theory for baryon asymmetry must satisfy thethree Sakharov conditions: (1) baryon number violating reactions must exist, (2) C andCP-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 anti-particle 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 right-handed positron e+R. Presently, the sources of CP-violation in the SM only account fora 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+ !  +  are sensitive to new CP-violating physics.The  rst observation of a CP-violating decay occurred at Brookhaven National Labo-ratory (BNL) in 1964, in an experiment involving the neutral K meson sector [36]. Thusfar, 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 theiranti-particles.1.2 CP-Violation and Weak InteractionsIn the SM, the six  avors of quarks are classi ed into three generations or families: up(u) and down (d), charm (c) and strange (s), and top (t) and bottom (b). For each ofthese families, the  rst listed quark has an electric charge of +2=3 and the second has anelectric charge of  1=3. Only through weak interactions can these quarks transform from2one to another. The charged-current processes, mediated by the charged intermediateW+ or W bosons, allow a quark to transform into another quark having a di erentcharge (such as d ! u + W ). The neutral-current process, mediated by the neutralZ0 boson, cannot directly transform a quark into another quark of di erent  avor butsame charge, i.e. s ! d. This type of transformation, known as a  avor-changing neutralcurrent (FCNC) process, is forbidden at tree level and suppressed at the one-loop leveldue to the Glashow-Iliopoulos-Maiani (GIM) mechanism [47].To understand the origin of the GIM mechanism, we can revisit a time when physicistsbelieved only three quarks existed: u, d and s. Cabibbo [29] introduced an angle ( C) toshow how the strength of the weak transformations d ! u and s ! u as mediated by theW di ered only by factors of cos  C and sin  C, respectively. This theory failed to explainwhy the experimental limits on the rate of FCNC processes such as K0L !  +  andK+ !  +  were so much lower than the calculated rates. In 1970, the GIM mechanismwas proposed to solve this problem by introducing a fourth quark, now called the charmquark. In this theory, there exist the weak eigenstates d0 and s0 which participate inthe weak interactions instead of the regular d and s mass eigenstates. The relationshipsbetween the weak and mass eigenstates is written as0B@ d0s01CA =0B@ cos  C sin  C sin  C cos  C1CA0B@ ds1CA: (1.1)At tree level, the FCNC processes are forbidden as shown by the following expression3K0Lsu; cdW W     Figure 1.1: The decay K0L ! +  .for the neutral current, J0, expressed without the  -matrices:J0 =  uu +  cc +  d0d0 +  s0s0;=  uu +  cc + (  dd +  ss) cos2  C + (  dd +  ss) sin2  C+ (  ds +  sd  ds  sd) sin C cos  C;=  uu +  cc +  dd +  ss:(1.2)Here the FCNC terms, such as  ds and  sd cancel each other due to the positive and negativesin  C terms in the rotations of the weak eigenstates.The suppression of FCNC processes at the one-loop level can be understood by lookingat the K0L !  +  decay (Figure 1.1). In the four-quark model, the virtual quark canbe either a u or a c. The contributions due to the two di erent virtual quarks cancel out,since the decay amplitude for the u quark is proportional to sin  C cos  C and that forthe c quark is proportional to  sin  C cos  C. If the masses of the u and c quarks wereidentical the decay would be strictly forbidden, but due to the di erence in their massesit proceeds at a very suppressed rate.The weak eigenstate model can be extended to include all six quarks where the rela-tionship between the weak eigenstates d0, s0, b0 and mass eigenstates d, s, b is given by4the Cabibbo-Kobayashi-Maskawa (CKM) matrix [66],0BBBB@d0s0b01CCCCA=0BBBB@Vud Vus VubVcd Vcs VcbVtd Vts Vtb1CCCCA0BBBB@dsb1CCCCA: (1.3)The SM does not predict the CKM matrix elements, so their magnitudes and phases mustbe determined experimentally.The constraints of unitarity connect di erent matrix elements such that the CKMmatrix can be expressed using as few as 4 di erent parameters. One standard method ofparameterizing the matrix involves the real angles  12,  23,  13 and the CP-violating angle :V =0BBBB@c12c13 s12c13 s13e i  s12c23  c12s23s13e i c12c23  s12s23s13e i s23c13s12c23  c12s23s13e i  c12c23  s12s23s13e i c23c131CCCCA; (1.4)where cij = cos  ij and sij = sin  ij, with the i and j representing 1,2,3, the quarkgeneration labels. The angle  12 is the same Cabibbo angle  C seen earlier in the four-quark scheme. An approximation proposed by Wolfenstein [98] sets a parameter  equalto s12, the sine of the Cabibbo angle. The other parameters are real numbers that are oforder unity: A,  and  . The parameter  describes CP-violation in the SM and a non-zero value of this parameter breaks CP-invariance for weak interactions. The followingrepresentation of the matrix expresses the elements in terms of powers of  = sin  c  0:22:V =0BBBB@1  22  A 3(  i )  1  22 A 2A 3(1   i )  A 2 11CCCCA+O( 4): (1.5)5Unitarity (V yV = 1) of the CKM matrix implies six unitarity conditions, all of which canbe expressed as unitarity triangles which are geometric representations of the unitarityconditions in the complex plane, where the areas of each of the unitarity triangles areequal to half of the Jarlskog invariant [59],JCP = s212s23s13c12c23c13 sin  ; (1.6)with the de nitions of sij and cij being the same as de ned earlier for Equation (1.4).The unitarity condition of interest for the decay K+ !  +  isVudVub + VcdVcb + VtdVtb = 0: (1.7)If the Wolfenstein parameterization from Equation (1.5) is used with the approximationsVud ’ V  tb ’ 1 and Vcd ’  , Equation (1.7) can be rewritten asV  ub   V  cb + Vtd = 0: (1.8)The triangle shown in Figure 1.2 results from this equation if all sides of the unitaritytriangle are normalized such that the baseline has a length equal to 1. In this normalizedunitarity triangle, the apex is given by two Wolfenstein parameters,  =  (1   22 ) and =  (1  22 ) [27].Thus far, K and B mesons are the only mesons that have shown evidence of CPviolation in their decay processes. As can be seen in Figure 1.3, comparisons of  and  from independent measurements in the K and B sectors can be used to either verify theStandard Model description of CP-violation as explained by the CKM phase, or to signalnew physics.6(0,0) (1,0)(r–,h– )abg|Vtd | / Al 3|Vub / Vcb| / lFigure 1.2: A unitarity triangle in the    plane. The length of the bottom side is normalizedto unity and the lengths of the other two sides can be expressed by the CKM matrix elementsjVtdj=A 3 and jVub=Vcbj= . Reprinted  gure with permission from S. Adler et al. (E949 Col-laboration), Phys. Rev. D 77 052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003.Copyright 2008 by the American Physical Society.(0,0) (1,0) (r0,0)(r–,h– )abgKL→p0nn–K+→p+nn–DMBd/DMBsB0d→yKsFigure 1.3: The unitarity triangle as determined byB andK decays. The angle  measured fromthe CP-violating asymmetry (ACP ) in the decay B0d !J= K0s can be compared with that fromthe ratio B(K0L !  0  )=B(K+ !  +  ). The magnitude of Vtd extracted from the ratio ofmass di erences ( MBs= MBd) from Bs Bd mixing can be compared with that extracted fromB(K+ !  +  ). Reprinted  gure with permission from S. Adler et al. (E949 Collaboration),Phys. Rev. D 77 052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003. Copyright2008 by the American Physical Society.7us–uu–e+neW+Figure 1.4: The  rst-order weak charged-current K+ !  0e+ e decay (left), which is allowedin the Standard Model. The  rst-order neutral-current K+ !  +  diagram (right) shows aprocess which is not allowed in the Standard Model. Figures reproduced from [72].1.3 K+ !  +  in the Standard ModelAs discussed in Section 1.2, the decay K+ !  +  is a FCNC process, prohibited in theSM at tree level (see Figure 1.4), but allowed at the one-loop level. The leading orderdiagrams describing this decay are shown in Figure 1.5.The weak amplitude for the process  s ! ( u;  c;  t) !  d, as seen in the diagrams ofFigure 1.5, is represented asM Xi=u;c;tV  isVid   q + miq2  m2i ; (1.9)where the index i = u; c; t denotes the quark  avor, Vij represents the CKM matrixelement,   the Dirac matrices, q 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  avor symmetry allows thisdecay to proceed at a very small rate. Due to its very large mass relative to the up andcharm quarks, the top quark provides the dominant contribution to the K+ !  +  branching ratio through the coupling of top to down quarks, the CKM matrix elementVtd.The following calculation of the branching ratio K+ !  +  follows the method from8us–ud–u–,c–,t–n–ne+,m+,t+W+ W–us–u–,c–,t–ud–n–nu–,c–,t–Z0W+us–ud–n–nu–,c–,t–Z0W+ W+Figure 1.5: The second-order weak processes that contribute to the K+ !  +  branchingratio are a \Box" diagram (upper) and two \Z-penguin" diagrams (lower). Reprinted  gurewith 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 So-ciety.9[28] and [25]. The low-energy e ective Hamiltonian in the SM can be written asHSMeff = GFp2  2 sin2  WXl=e; ;  V  csVcdXl(xc) + V  tsVtdX(xt) ( sd)V  A( l l)V  A; (1.10)where the index l = e;  ;  denotes lepton  avor,  W is the electroweak mixing angle,  is the  ne structure constant, and GF is the Fermi coupling. The functions Xl(xc) andX(xt), where xj  m2j=M2W , summarize the contributions from the charm and top quarksrespectively and include QCD corrections at next-to-next-to-leading order (NNL0).The uncertainty in the function X(xt) is dominated by the experimental uncertaintyon the top quark mass in the minimal subtraction scheme, a renormalization schemefrequently used in quantum chromodynamics. The top quark mass in the minimal sub-traction scheme is mt(mt) = (163:01 1:43) GeV [48], giving a value for X(xt) ofX(xt) = 1:443 0:017: (1.11)The perturbative charm contribution is described in terms of the parameterPc(X)  1 4 23Xe(xc) + 13X (xc) = 0:372 0:015; (1.12)where  = 0:2255 was used [25].Using the above de nitions and relationships, the K+ !  +  branching ratio can bewritten asB(K+ !  +  ) =  +  " Im t 5 X(xt) 2+ Re c (Pc(X) +  Pc;u) +Re t 5 X(xt) 2#;(1.13)where  i  V  isVid are the CKM matrix elements, and  Pc;u = 0:04 0:02 represents the10long-distance contributions as calculated in [53]. The term + = (5:173 0:025) 10 11  0:225 8; (1.14)calculated in [78], summarizes the remaining factors that follow from Equation (1.10), inparticular the relevant hadronic mixing elements that can be extracted from leading-ordersemi-leptonic decays of K+, KL and KS mesons.Equation (1.13) describes in the    plane an ellipse with small eccentricity, namely(  )2 + (   0)2 =  B(K+ !  +  ) +jVcbj4X2(xt) ; (1.15)where 0  1 +  4Pc(X)jVcbj2X(xt); (1.16)   1  22  2; (1.17)and +   + 8 = (7:87 0:04) 10 6: (1.18)Using (1.13), the SM branching ratio of K+ !  +  is predicted to beB(K+ !  +  ) = (0:85 0:07) 10 10: (1.19)A precise measurement of B(K+ !  +  ) is regarded to be one of the cleanest waysto extract jVtdj. This is due to the following factors:1. The long-distance contributions to the branching ratio are small [51] and undercontrol. The most recent calculation [53] gives an enhancement to the branchingratio of (6 3)%;2. The uncertainty from the hadronic matrix element has been reduced to less than111% by recent theoretical and experimental developments [78]; and3. Recent improvements in the calculation of the charm quark contribution [25] havereduced the theoretical uncertainties to  5%.1.4 Physics Beyond the Standard ModelSince the theoretical uncertainty in the SM prediction of B(K+ !  +  ) is small, aprecise measurement can serve as a stringent test of the SM and provide an e ectiveprobe for new physics beyond the SM. Predictions of B(K+ !  +  ) have been madefor many models beyond the SM and a precise measurement can constrain or reject thesemodels. Since the current experimental limit for the complementary K0L !  0  decayis B(K0L !  0  ) < 6:7  10 8 at 90% con dence level [9] and the SM prediction isB(K0L !  0  ) = (2:49 0:39) 10 11 [78], many new physics models have the freedomto predict much larger enhancements to B(K0L !  0  ) than to B(K+ !  +  ). Somenew physics models are discussed brie y below.In the Littlest Higgs with T-Parity (LHT) model [34], new massive particles are intro-duced. These new particles include massive partners to the SM particles, but unlike Su-persymmetric models [84], the massive partners have the same spin-statistics as their SMpartners. Under the new T-parity symmetry, most of these new massive partners are oddand 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  avor-violating interactionswhich can impact FCNC processes such as K+ !  +  . Figure 1.6 shows that the pre-dicted relationship between B(K+ !  +  ) and B(K0L !  0  ) lies along two branches.The  rst branch is parallel to the upper bound on the model-independent Grossman-Nirlimit, B(K0L !  0  )=B(K+ !  +  ) < 4:4 [50]. In this branch, B(K0L !  0  ) canbe as high as 5  10 10 while B(K+ !  +  ) stays within the bounds of the measuredbranching ratio of B(K+ !  +  ) = 1:47+1:30 0:89  10 10 [11]. The second branch predicts121·10-10 2·10-10 3·10-10 4·10-10 5·10-10BrHK+ fip+ nn L1·10-102·10-103·10-104·10-105·10-10BrHKLfip0nn LFigure 1.6: Littlest Higgs with T-Parity (LHT) model predictions for the relationship betweenB(K+ ! +  ) and B(K0L ! 0  ). The shaded area represents the 1 -range for the previousE949 measurement of B(K+ ! +  ) = 1:47+1:30 0:89 10 10 [11]. The black dotted line shows themodel independent Grossman-Nir limit, B(K0L !  0  )=B(K+ !  +  ) < 4:4 [50]. The bluedata points represent the most general LHT model predictions, and the red, green and gold datapoints represent further constrained scenarios as discussed in [23]. Figure reproduced from [23].that B(K0L !  0  ) remains close to its SM prediction, but B(K+ !  +  ) can varyfrom 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)Lresulting in an additional neutral gauge boson, the Z0, which is able to transmit FCNCsat tree-level. The resulting enhancement to B(K+ !  +  ) above the SM predictionranges up to approximately 65% when the Z0 mass is small (1 TeV) [88].It may also be possible that the \nothing" in the observation of K+ !  ++ \noth-ing" is actually the observation of K+ !  +X0, where X0 is a weakly interacting particlesuch as a familon, axion or majoron [51]. These particles are Nambu-Goldstone bosons(massless) or pseudo-Nambu-Goldstone bosons (small mass) that arise from spontaneouslybroken symmetries such as global family symmetries (familon), the Peccei-Quinn symme-try (axion), or baryon-lepton invariance (majoron). The limit set on the branching ratio13of K+ !  +X0 by previous E949 analyses is B(K+ !  +X0) < 0:73  10 10 at 90%con dence level [11].1.5 History of K+ !  +  ExperimentsThe  rst published limit on B(K+ !  +  ) came in 1969 from a heavy liquid bubblechamber experiment at the Argonne Zero Gradient Synchrotron. Their initial 90% con -dence level upper limit of 10 4 [31] was improved four years later to 5:7 10 5 [74] after nal analysis of the experiment.A counter/spark-chamber experiment at the Berkeley Bevatron, sensitive only to themost energetic  +s, improved the limit to 1:4 10 6 in 1971. They improved their limitto 5:6 10 7 [30] by combining their previous data with data acquired after recon guringtheir detector to be sensitive to  +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 ProtonSynchrotron using a setup similar to that used by the Berkeley Bevatron experiment.The E787 experiment, later superseded by the E949 experiment, was initiated atBrookhaven National Laboratory in the early 1980s. From data collected between 1988and 1991, the E787 experiment achieved an upper limit on the branching ratio at 90%con dence 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  + momentum is greater (less)than that from K+ !  + 0 as shown in Figure 3.1.The detector and beam line were upgraded between 1992 and 1994 [67], and datacollection resumed in 1995 and continued through 1998. In these data, two events wereobserved in the PNN1 region yielding a branching ratio measurement ofB(K+ !  +  ) =1:57+1:75 0:82  10 10 [3, 4, 5]. In the PNN2 region, one event, consistent with background,was observed and the limit placed on the branching ratio by measurements in this regionwas improved to 2:24 10 9 [6, 7].14Between 1999 and 2001, the detector and beam line were again upgraded [18], as werethe trigger and data acquisition [100]. The E787 experiment was rechristened E949, anddespite 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 observedand the combined branching ratio from all three events observed in the PNN1 region wasmeasured to be B(K+ !  +  ) = 1:47+1:30 0:89  10 10 [11]. This thesis describes the searchfor K+ !  +  in the PNN2 region using the 2002 data.1.6 My Role in the AnalysisAfter the demise of BNL experiment E926 \KOPIO" in 2005, I joined the E949 collabo-ration 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 internaldocuments [55] and [56], for both of which I was also one of the authors. These internaldocuments were cited throughout this thesis to attribute  gures and results of studies toothers within the collaboration. This section brie y summarizes the work for which I wasresponsible.In Chapter 4 \Backgrounds", I was responsible for all the K 2-scatter backgroundresults (Sections 4.1 and 4.2) and the normalization result for K 2 (Section 4.3). I alsoindependently veri ed the results presented for the single-beam (Section 4.6), double-beam (Section 4.7), and muon (Section 4.4) backgrounds, as well as the results of thenormalization/data branch studies for the Ke4 (Section 4.5) and charge exchange (Section4.8) backgrounds.In Chapter 5 \Validity Checks", I was responsible for all the validation studies otherthan the single-cut and double-cut failure studies.In Chapter 7 \Results", I was responsible for the \signal probability analysis" study. Iveri ed the calculation for the probability of observed candidates being due to background15only and also veri ed, using independently developed code, many of the likelihood analysisresults presented in [56].Appendices C and E describe work for which I was completely responsible. AppendixC describes the improvements I made to the analysis of target pulse data known asthe \target CCD  tter". Appendix E describes my work on the class of cuts knownas the target pulse cuts. This work included improvements to the cuts CCDPUL andCCDBADTIM and development of the cut CCDBADTIM. These target pulse cuts werecritical in the suppression of the large K+ !  + 0 target-scatter background and workon these cuts accounted for a large fraction of my contributions to this analysis.16Chapter 2Experimental MethodIn Brookhaven National Laboratory (BNL) experiment E949, the low-momentum (710MeV/c) K+ beam was stopped in an active scintillator target. Identi cation of theK+ !  +  decay involved the observation of a  + in the absence of other coincidentactivity. The charged decay particle was identi ed as a  + by the measured momentum,energy and range, and by the observation of the  + !  + ! e+ decay sequence. Thedetector featured full 4 -steradian photon veto coverage and the entire E949 spectrometerwas situated in a 1-Tesla solenoidal magnetic  eld along the beam direction. The detectoris shown in Figure 2.1. E949 superseded BNL experiment E787, with upgrades takingplace from 1999-2001. A summary of these upgrades can be found in Appendix A.This chapter brie y describes the various detector systems and triggers used in theE949 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 tothe z-direction. The y-direction pointed vertically up.Details of the data acquisition, storage and processing can be found in Appendix B.17Figure 2.1: Schematic side (a) and end (b) views of the upper half of the E949 detector.Reprinted  gure with permission from S. Adler et al. (E949 Collaboration), Phys. Rev. D 77052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003. Copyright 2008 by the Ameri-can Physical Society.2.1 Kaon BeamThe K+ beam was produced by a high-intensity 21.5 GeV/c proton beam from the Al-ternating Gradient Synchrotron (AGS) at Brookhaven National Laboratory (BNL). Aschematic view of the AGS accelerator complex is shown in Figure 2.2. During a 2.2 sspill, approximately 6:5  1013 protons arrived at the K+ production target, with spillsoccurring every 5.4 s. The K+ production target was made of 6 cm thick platinum locatedon a water cooled base, with the maximum target temperature being measured at about700 C.The Low Energy Separated Beam [40] (LESB III), shown in Figure 2.3, collected andtransported K+s emitted at 0 , along with the 500  +s and 500 protons at the productiontarget for every K+. The beam was momentum-selected by the  rst dipole magnet (D1in Figure 2.3). Two electromagnetostatic separators swept away protons and  +s fromthe beam resulting in a K+ to  + ratio of 3:1 and negligible proton contamination whenthe beam arrived at the target. The angular acceptance of LESB III was 12 msr and themomentum acceptance was 4.5% FWHM, with the mean beam momentum arriving at18Figure 2.2: Schematic view of the AGS complex, which consisted of a 200 MeV LINAC, a boosterand a synchrotron. The secondary beam lines were located in the area marked \experimentalarea". Figure reproduced from [37].19Figure 2.3: Low-energy separated beam line III at BNL. The E949 solenoid magnet is also shownat 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 E949target per spill and the instantaneous rates in the  Cerenkov Counter were 6.3 MHz forK+s and 1.5 - 2.5 MHz for  +s.2.2 Beam InstrumentationThe purposes of the beam instrumentation were K+ identi cation, K+ momentum degra-dation, and detection of additional particles coincident with the K+ and its decay parti-cles. In the order encountered by the K+ beam, these detectors were: The B0 counter, a large scintillator counter which counted all charged particles inthe beam; The  Cerenkov counter which identi ed particles as K+ or  +;20 The Upstream Photon Veto (UPV), a 3.1 radiation length plastic scintillator witha large beam hole; Two beam wire chambers (BWPC) which monitored the beam pro le and identi edmultiple incoming particles; Passive (BeO) and active (copper and scintillator sandwich) degraders that slowedthe kaons such that they came to rest in the target; The Beam Hodoscope (B4) which detected the position of incoming beam particlesjust before they entered the target; The Ring Veto (RV), an annular plastic scintillator at the same position along thebeam direction as the B4 Hodoscope, but at a larger radius.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  Cerenkov CounterThe  Cerenkov counter identi ed beam particles as kaons or pions and was located justdownstream 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 onthis refractive index the threshold for  Cerenkov radiation in the radiator was  C = 1n = 0:671; (2.1)and the threshold for total internal re ection (TIR) was TIR =r 1n2  1 = 0:905: (2.2)21Figure 2.4: Schematic of the side-view of the  Cerenkov counter. The incoming K+ and resulting Cerenkov light are shown in blue. The incoming  + and resulting  Cerenkov light are shown inred. Figure reproduced from [37].At a momentum of 710 MeV/c,  K+ = 0:82 and   + = 0:98. The  Cerenkov light fromthe kaons was transmitted at the downstream surface of the radiator and re ected by aparabolic mirror to the outer ring of 14 EMI [43] 9964KB photomultiplier tubes (PMTs),collectively referred to as the K+  Cerenkov counter (CK). The  Cerenkov light from thepions experienced total internal re ection at the downstream surface of the radiator andwas re ected to the inner ring of 14 EMI9964KB PMTs, collectively referred to as the +  Cerenkov counter (C ). The signals from the PMTs were split with 90% being sentto time-to-digital converters (TDCs) via LRS3412 discriminators and the remaining 10%being sent to  10 ampli ers whose output was sent to 500 MHz transient digitizers basedon a gallium arsenide charge-coupled device (CCD) [26]. Gallium arsenide was usedbecause it is a suitable material for construction of high speed devices due to its highelectron mobility. The pulse-height information in every 2-ns interval was recorded bythe CCDs to reproduce the time development of the pulses and to detect two particles22close in time to each other. At the trigger level, a minimum threshold of 5  Cerenkovdiscriminator channels  ring was used for kaons for the KB trigger and for pions for the B trigger (see Section 2.7 for trigger de nitions). The CCD information was used o inefor discriminating between multiple incoming beam particles of the same type when theTDCs were unable to distinguish between pile-up signals.2.2.2 Upstream Photon VetoDownstream of the  Cerenkov counter was the upstream photon veto (UPV). Figure 2.5shows this 3.1-radiation-length detector made of 12 square layers, 28.4 cm in height andwidth, of Bicron [89] BC404 plastic scintillator and lead or copper sandwich. The scin-tillator layers were 2 mm thick. The  ve upstream-most layers of lead were 2 mm thickand the six remaining lead layers were 1 mm thick. The downstream-most layer was 2.2mm 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 wideby 40 mm high slot through the detector to allow the beam to pass through. The de-tector was divided horizontally into two modules of six scintillator segments each. Eachscintillator segment was read out by twenty-one 1-mm diameter double-clad wavelengthshifting (WLS)  bers coupled to one of the two Hamamatsu R1924 PMTs [52] used toread 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 itwas 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 ChambersThe two beam wire proportional chambers (BWPCs) monitored the beam pro le andidenti ed multiple incoming particles. The  rst chamber (BWPC1) was located down-23Figure 2.5: Schematic of the Upstream Photon Veto. The  gure on the left shows the detectorlooking from the beam view. The  gure in the upper-right shows a cross-section and the  gureon the lower-right an enlarged cross-section. The K+ beam traveled through the horizontal slotin the center of the detector. Figure reproduced from [72].24Downstream view of BWPC1X (cm)Y (cm)V-plane U-planeX-plane-150-100-50050100150-150 -100 -50 0 50 100 150Downstream view of BWPC2X (cm)Y (cm) V-plane U-planeX-plane-100-80-60-40-20020406080100-100 -80 -60 -40 -20 0 20 40 60 80 100Figure 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 downstreamfrom the  rst. Both chambers were  lled with a recirculated mixture of CF4 (80%) andIsobutane (20%). Cross-sectional views of the BWPCs are shown in Figure 2.6.BWPC1 contained three planes of 12- m-diameter gold-plated tungsten anode wireswith 1.27 mm wire spacing. The x-plane consisted of 144 anodes running along thevertical. The 120 wires in each of the u- and v-planes were oriented at  45 to thevertical, respectively. These wires were multiplexed by 2 in the readout channels. Thetotal active area of the chamber was 17.8 cm in the horizontal by 5.08 cm in the vertical.The cathode foils were 25- m thick aluminized mylar coated with carbon and the anode-cathode distance was 3.18 mm.BWPC2 contained three planes of 12- m-diameter gold-plated tungsten anode wireswith 0.8 mm wire spacing and 120 anodes per plane. The directions of the anodes werevertical 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 remainingwires were multiplexed by 6 to the readout channels, for a total of 32 readout channels25per plane. The cathode foils were 8- m thick single-sided aluminized mylar coated withcarbon and the anode-cathode distance was 1.6 mm.2.2.4 DegradersDownstream of BWPC1 were the degraders whose purpose was to slow the kaons suchthat they came to rest in the target before decaying. The lengths of the degraders werechosen such that mean stopping position in the z-direction for a K+ in the target washalfway along the of the 24-cm long I-Counter which was used to tag charged decayproducts 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 thusminimized 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 thedecay occurred close to the radial edge of the target.The downstream degrader was called the Active Degrader (AD) and consisted of 40layers of 2-mm thick disks of Bicron BC404 scintillator alternated with 2.2-mm thick disksof copper. The diameters of these disks were 13.9 cm and 13.6 cm, respectively. The ADwas split into 12 azimuthal segments, with each segment being read out by 14 BicronBCF99-29-AA-MC wavelength shifting (WLS)  bers sent to a single Hamamatsu R1924PMT. The output from each PMT was sent to a TDC and a CCD. The PMT signalswere multiplexed in groups of four and sent to ADCs. A schematic view of the ActiveDegrader is shown in Figure 2.7. Measurements from the AD allowed identi cation ofbeam particles and detection of activity coincident with K+ decays.26Figure 2.7: Schematic of the Active Degrader. The removed wedge represents one of the sectors.Figure reproduced from [72].2.2.5 Beam HodoscopeDownstream of the degraders was the Beam Hodoscope (B4) whose purpose was to detectthe entrance position of beam particles into the target and to identify beam particles asK+ or  + by energy loss. As shown in Figure 2.8, the two 11.8-cm diameter planes ofthe B4 Hodoscope were oriented at angles of  33.5 , labeled u and v, respectively, withrespect to the horizontal. Each plane consisted of 16 \z-shaped" Bicron BC404 scintillator ngers with a 7.2-mm pitch. The cross-section of each \z-shaped"  nger, was 6.35-mmthick in the middle and 3.175-mm at the edges as shown in Figure 2.8. This shape waschosen to reduce the inactive regions and improve the spacial resolution. Three BicronBCF990290AA-MC WLS  bers were embedded in each  nger and connected to a singleHamamatsu H3165-10 PMT read out by TDC, ADC and CCD.27Hit the edges6.35mm7.30mmBeam 3.65mmHit the middleFigure 2.8: The upper  gure shows the B4 Hodoscope from the beam view. The segmentsof the  ngers are shown as solid lines for the upstream segments and dashed lines for thedownstream segments. The outer ring shows the WLS  bers used to read out the scintillator ngers. The lower  gure shows a schematic cross-section of one of the B4 layers with theapproximate positions of the three WLS  bers (small circles) in each segment. Figure reproducedfrom [37].282.2.6 Ring VetoThe Ring Veto (RV) was located at the same position along the beam direction as the B4Hodoscope, but at a larger radius. The RV was designed to veto particles passing throughthe perimeter of the B4 Hodoscope. This annular detector was composed of two 180 arcsof 3.27-mm thick Bicron scintillator with an inner diameter that varied between 11.85 cmand 12.0 cm, and an outer diameter of 14.55 cm. Each of the two elements was read outby a Hamamatsu H3165-10 PMT whose signal was sent to ADC, TDC and CCD.2.3 TargetThe target (shown in Figure 2.9) was a 12-cm diameter cylinder, 3.1 m in length, whichwas made of 413 5-mm square Bicron BCF10 scintillating  bers. These  bers ran in thedirection parallel to the beam and had 0.09-mm thick inactive cladding. Smaller  bers,known as edge  bers, were used to  ll the gaps near the outer edge of the target. Eachof the 5-mm  bers was connected to a Hamamatsu R1635-02 PMT. The edge  bers weremultiplexed into groups of 12 and each group was read out by a single PMT. The PMTsignals were sent to ADCs, TDCs and CCDs.Typical energy deposits in an individual target  ber were on the order of tens of MeVfor K+s traveling along the  ber, and only  1 MeV for  +s passing perpendicularlythrough a  ber, where bench tests [46] measured the photoelectron production from these bers to be 30 photoelectrons per MeV. Pattern recognition was used to assign  berswith deposited energy to the K+ path (kaon  bers) or the  + path (pion  bers) fromthe K+ decay (see Section 3.3.4). To cover this large dynamic range, two separate CCDchannels were used to read-out the target  bers. Each  ber was read out individually bya \high-gain" CCD channel. The sum of 4-6 randomly positioned  bers was read out bya \low-gain" CCD channel. A double-pulse  t was performed on the CCD informationfor both the \low-gain" and \high-gain" CCD channels to identify the  ber in which the29+zy25 cm V−CounterTargetK   Beam light guideI−CounterscintillatorI−CounterFigure 2.9: End (left) and side (right) views of the target. The CCD pulse-shape informationfor typical K+  bers is shown in the two lower plots \a" and \b", and for typical  +  bers inthe 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  + in that ber. The target CCD  tter is described in more detail in Appendix C.The  ducial region of the target was de ned by two layers of six plastic scintillatorcounters surrounding the target as shown in Figure 2.9. The inner scintillators (I-countersor ICs) surrounded the target and tagged charged decay products before they entered thedrift chamber. The IC was 6.4 mm thick with an inner radius of 6.0 cm and extended24 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 decayeddownstream of the  ducial 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 sentto ADC, TDC and a 500 MHz transient digitizer (TD) based on a  ash ADC [14].30Figure 2.10: Schematic of the Ultra-Thin Chamber. Figure modi ed from [37].2.4 Drift ChamberThe drift chamber, called the Ultra Thin Chamber (UTC), was located just outside ofthe I-counter, with an inner radius of 7.85 cm, an outer radius of 43.31 cm and length of51 cm. The UTC consisted of three superlayers (see Figure 2.10), each of which had fourlayers of axially strung 20- m anode wires made of gold-coated tungsten. Each of theanode wires was surrounded by eight 100- m grounded gold-coated aluminum cathodewires to form a cell, with adjacent anodes sharing cathode wires. The cells in adjacentlayers were staggered by one half of a cell width to resolve the left-right ambiguity. Thesuperlayers were  lled 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 adrift velocity of 5 cm/ s and a gain of 8 104. Each anode wire was instrumented withan ADC and a TDC. The drift time to the anode wires was used to determine the x-ypositions of the charged track.Each superlayer was sandwiched on either side by a plane of cathode strips, which31were aligned helically at a pitch angle of 45 . The cathode strips had a width of 7 mm, aseparation of 1 mm and were made of 1200  A thick copper coated with 300  A thick nickel,and were mounted on a 25  m thick Kapton foil. Each cathode strip was instrumentedwith an ADC and a TDC. The cathode strips were used to determine the track positionalong the z-direction by means of weighted mean of the induced signal on the anode wires.The two inactive regions between superlayers were  lled with nitrogen gas. Di erentialpressures of approximately 2 mbar in the  ve gas volumes supported the cathode foils.The total mass of the UTC was only 2 10 3 radiation lengths, excluding the inner andouter support tubes with the attached foils.The UTC position resolutions (RMS) were approximately 175  m in the x- and y-directions and 1 mm in the z-direction. The curvature of the charged particle track in theUTC due to the magnetic  eld was used to determine the momentum. The momentumresolution ( P=P) for the two-body K 2 decay was 1.1% [22].2.5 Range-StackThe Range-Stack (RS) consisted of both scintillator counters and position-sensitive em-bedded straw chambers, and was located just outside of the UTC with an inner radiusof 45.08 cm and an outer radius of 84.67 cm. The RS provided energy and range mea-surements, information for the observation of the  + !  + ! e+ decay sequence andmeasurement of photon activity.2.5.1 Range-Stack ScintillatorThe RS consisted of 19 layers of Bicron BC408 plastic scintillator, azimuthally segmentedinto 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 de ned the  ducial volume for thecharged decay products and was thinner than the remaining RS layers to suppress the rate32Downstream view of Range StackX (cm)Y (cm)T-counterRSSC layer 1RSSC layer 2Gap-100-80-60-40-20020406080100-100 -80 -60 -40 -20 0 20 40 60 80 100Figure 2.11: End view of the Range-Stack. The two superlayers of Range-Stack Straw Chambersare shown after RS layers 10 and 14. Figure reproduced from [37].from photon conversions. Each of the T-counter scintillators was read out by seventeen1-mm-diameter Bicron BCF-92 multiclad WLS  bers coupled to a Hamamatsu R1398PMT at both the upstream and downstream ends. Layers 2-18 were 1.905 cm thick and1.82 m long and each scintillator was coupled through light guides to EMI 9954KB PMTsat each end. Layer 19 was installed as part of the E949 upgrade and was used mainly toveto long range charged particles such as muons. This layer was 1.0 cm thick and hadthe same length and read out method as layers 2-18.Signals from each PMT were sent to ADCs, TDCs and analog fan-in modules. TheTDCs recorded up to 16 hits in a 10.5  s time window. A time window of this length was33required to reliably capture information from the  + decay, where the  + has a lifetimeof 2.2  s [99]. Each group of four sectors was grouped into a hextant and the signalsfrom the four PMTs in a given layer on a given end of a hextant were summed via thefan-in modules and sent to a TD. The TDs recorded the charge in 2 ns intervals (500 MHzsampling) with a resolution of 8 bits. This provided su cient pulse-shape informationto separate pulses as close as 5 ns apart. The time window used for the TDs was 8  suntil run 47959, 4  s until run 48227, and 2.5  s for the remaining runs. To determinethe z-position of the interaction in a counter, it was possible to use the end-to-end timedi erences or the end-to-end ADC information di erences.2.5.2 Range-Stack Straw ChambersSince the Range-Stack produced only very rough position information, two superlayersof Range-Stack Straw Chambers (RSSCs) were embedded after the 10th and 14th layersof the Range-Stack as shown in Figure 2.11. The position of a hit chamber provided x-yposition information and the end-to-end time di erences provided the z-position informa-tion. Each superlayer consisted of two staggered layers of 3.4 mm radius straw chamberswhich operated in self-quenching streamer mode at 3450 V. The inner superlayer is shownin Figure 2.11. The gas mixture in these chambers was 67% argon and 33% isobutanewith a trace of water. The 50- m-diameter gold-coated tungsten anode wires of the strawchambers ran parallel to the beam direction. The layers of the inner superlayer had 24straws per sector with a length of 97.8 cm. The layers of the outer superlayer had 28straws 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].34Anode Wires(50 micron W with Au coated)Straws : Kapton foil (38 micron)with Cu/Ni cathode1.2 cm50 micron Carbon FiberFigure 2.12: End view of the inner superlayer of Range-Stack Straw Chambers. The StrawChambers ran parallel to the beam direction. Figure reproduced from [72].2.6 Photon VetoThe E949 detector consisted of the hermetic system of photon detectors shown in Figure2.1. The photon veto system consisted of almost every detector system in the E949detector. Detectors whose sole purpose was the detection of photon activity were theBarrel Veto (BV), the Barrel Veto Liner (BVL), the upstream and downstream End Caps(ECs), the upstream and downstream Collars (CO), the downstream Microcollar ( CO)and the Downstream Photon Veto (DPV). Detectors that were part of the photon vetosystem, but also served other purposes were the AD, target, IC, VC and RS. For a givenevent, the regions of the target, IC and RS traversed by the charged track were excludedfrom the photon veto. The thickness of the photon veto in radiation lengths as a functionof polar angle is shown in Figure 2.13.2.6.1 Barrel VetoThe Barrel Veto (BV) was located in the outermost barrel region with an inner radiusof 94.5 cm, an outer radius of 145.3 cm, and a length of 1.9 m. The BV covered 2/3 ofthe 4 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 alternatinglayers of 1-mm thick lead and 5-mm thick Bicron BC408 plastic scintillator. The azimuthalboundaries between sectors were tilted so that it was not possible for a photon originatingfrom the target to travel along a gap between sectors. Approximately 30% of the energy35Thickness of PV0510152025-1 -0.5 0 0.5 1X0 vs costhUpstream0510152025-1 -0.99 -0.98 -0.97 -0.96 -0.95X0 vs costhDownstream05101520250.95 0.96 0.97 0.98 0.99 1X0 vs costhFigure 2.13: Radiation length \X0" versus the cosine of the polar angle \costh" for the photondetector system. The thickness is shown for all angles (top-left), angles closest to the upstreamend (top-right), and angles closest to the downstream end (bottom-right). Figure reproducedfrom [44], courtesy of of Brookhaven National Laboratory.36deposited in the BV was deposited in the scintillator [15]. The upstream and downstreamends of each module were read out by EMI 9821KB PMTs and the signals were sent toADCs and TDCs.2.6.2 Barrel Veto LinerAs 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 improvedphoton detection. As shown in Figure 2.14, the BVL consisted of 48 2.2-m long azimuthalsectors. Each sector consisted of 12 layers of 1-mm thick lead and 5-mm thick BicronBC408 plastic scintillator. The BVL was expected to increase the photon veto rejectionof K 2 decays by a factor of two. The upstream and downstream ends of each modulewere read out by EMI 9821KB PMTs and the signals sent to ADCs and TDCs. Eachgroup of eight adjacent sectors was grouped to form a hextant. Each end of a hextant wasread out by a TD. Approximately 30% of the energy deposited in the BVL was depositedin the scintillator.2.6.3 End CapsThe two End Cap (EC) photon detectors accounted for roughly one-third of the 4 srphoton veto coverage [35]. The upstream EC (EC1) is shown in Figure 2.15 and consistedof 75 undoped Cesium Iodide (CsI) crystals grouped into four rings. The downstreamEC (EC2) consisted of 68 crystals, also grouped into 4 rings. The crystals were 25 cm inlength, which was equivalent to 13.5 radiation lengths. The PMTs were coupled directly tothe crystals via a Sylgard [41] cookie formed over the PMT and a ultraviolet transmittingoptical  lter that passed only the fast component of the CsI scintillation light and blockedthe slow component. The fast component had a decay time of approximately 30 ns at305 nm and the slow component had a decay time of 680 ns, contributing approximately37x(cm)y(cm)-150-100-50050100150-150 -100 -50 0 50 100 150Figure 2.14: End view of the Barrel Veto (four outermost layers) and Barrel Veto Liner (in-nermost layer). The sets of numbers show the numbering of radial layer and sector. Figurereproduced from [72].38Figure 2.15: End view (left) and schematic of the End Cap assembly (right). Figure reproducedfrom [37].20% of the total light output. Fine-mesh PMTs [68] were used due to their ability tomaintain high gains in strong magnetic  elds. Hamamatsu R5545 2" PMTs were used forthe inner ring crystals and Hamamatsu R5545 3" PMTs for the crystals in the three outerrings. 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 wasalso sent to TDCs.Due to being quite near the beam line, the ECs were exposed to a high count-rateenvironment and as a result were prone to accidental hits occurring earlier than a givenphoton hit thereby reducing the e ciency of the photon veto. O ine pulse- nding wasperformed on the CCD information to separate double pulses close in time and to providetiming information.2.6.4 CollarThe upstream and downstream Collar (CO) detectors were located beyond the ECs (seeFigure 2.1) and covered the small angle region surrounding the beam line. The down-stream CO consisted of twenty- ve 2-mm thick lead sheets alternating with 5-mm thick39x(cm)y(cm)-20-1001020-20 -10 0 10 20 x(cm)y(cm)-20-1001020-20 -10 0 10 20Figure 2.16: Upstream Collar (left), and downstream Collar and Microcollar (right). Elementnumbers are listed. Figure reproduced from [72].Bicron BC404 scintillator sheets, providing a thickness along the beamline of about 9 ra-diation lengths. The upstream CO was of similar construction, but only half the thicknessof the downstream one. The COs were each segmented into 12 azimuthal sectors. Eachlayer of the downstream CO was read out by 16 Bicron BCF99-29AA WLS multiclad bers glued into grooves in the scintillator layer [82]. All  bers in a given sector were readout together. The  bers were polished and aluminized on one end to provide a re ectivesurface and the other end was coupled to a 1.17-m long lucite light guide. Each light guidewas coupled to an EMI 9954KB PMT in a low magnetic  eld outside of the magnet. Eachlayer of the upstream CO was read out out by a lucite light guide coupled to the sametype of PMTs outside of the magnet. Signals from the PMTs for each of the COs weresent to ADCs and TDCs.2.6.5 MicrocollarThe Microcollar surrounded the beamline directly downstream of the downstream CO. Ithad an inner diameter of 15.6 cm, an outer diameter of 20.0cm and a length of 53.0 cm.401 210.1PMT PMTPMTPMT200 250Reflector.    .    .700700LeadScintillatoraxisbeam.3001.525 26Figure 2.17: Schematic of the Downstream Photon Veto. All dimensions are in mm. Figurereproduced from [72].It consisted of eight azimuthal sectors containing eight layers of 2 mm scintillating  bersseparated by seven layers of 60  m Pb. The  bers in each group of two adjacent sectorswere read out by an EMI9954 PMT whose signal was sent to an ADC and TDC.2.6.6 Downstream Photon VetoThe Downstream Photon Veto (DPV) was located at the far downstream end of thedetector, after the target PMTs. The DPV, shown in Figure 2.17, was a 700 mm by700 mm square that intersected the beam path. It consisted of 26 sheets of 10-mmthick scintillator alternating with 1.5-mm thick lead, for a total thickness of 7.3 radiationlengths. Each scintillator layer was read out simultaneously by four PMTs whose signalswere sent to TDCs and ADCs.412.7 TriggerThe E949    signal triggers selected K+ !  +  events from the large number of K+decays and scattered beam particles, with requirements on the range of the  + track, theobservation of a  + !  + decay signature in the RS, the absence of other activity at thetime of the observed  + 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. Thelevel-0 trigger made its decision entirely from logic pulses from various detectors, had arejection of 103 and introduced 38 ns seconds of dead time for every coincident hit in the rst two layers of the RS (known as the T 2 trigger) [100]. Rejection is the ratio of thenumber of events encountered by that trigger to the number of events accepted by thattrigger. The dead time is the amount of time that the detector is unable to trigger onsubsequent events due to being busy processing information from a previously triggeredevent. The level-1 trigger, composed of the level-1.1 and level-1.2 triggers, operated onthe events passing the level-0 trigger and involved partial processing of ADC and TDdata. The level-1.1 trigger had a rejection factor of 12 after level-0, and introduced 10to 20  s of dead time per level-0 trigger. The level-1.2 trigger had a rejection factor of 2after level-1.1 and introduced a dead time of up to 100  s per level-1.1 trigger. The totalrejection of these triggers reduced the 2:6 106 K+ decays per spill to about 100 events.2.7.1 Signal TriggersTwo signal triggers were used:    (1) and    (2) for the kinematic signal regions PNN1and PNN2, respectively. The PNN1 analysis used only events passing the    (1) triggeras signal data while the PNN2 analysis used events passing either the    (1) or    (2)42triggers. These triggers had the following requirements:   (1)  KB DC IC (T 2) (6ct + 7ct) 19ct  (BV + BVL + EC) L0rr1 zfrf  HEX L1:1 L1:2; (2.3)   (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)   (2)2  KB DC IC (T 2) 3ct  4ct  5ct  6ct  (13ct +   + 18ct) 19ct  (BV + BVL + EC) L0rr2 HEX L1:1 L1:2 (PS16 + C ); (2.5)which are explained in more detail below. In the trigger conditions described below, thedesignation \ct" refers to the Range-Stack sectors associated with the T 2 hit plus thenext two clockwise sectors, where clockwise was the direction that a positive particlemoved in the magnetic  eld.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 thehit  bers in the target of at least 20 MeV. This requirement ensured that the beamparticle entering the target was a kaon and served as a beam strobe (BS) for thetrigger. The AGS spill gate indicated the time in which the K+s produced duringthe 2.2-s long AGS proton spill should have been arriving at the target.T 2 - A coincidence hit was required between the  rst two layers (T-Counter and layer2) of the RS in the same sector. This condition required that a charged track fromfrom the kaon decay entered the Range-Stack. This T 2 time typically served asthe detector strobe (DS) that gated all of the ADCs not associated with the beamsystem or target and provided a common stop for many of the TDCs.43IC - At least one of the hits in the I-Counter was required to be coincident with the T 2time. This condition ensured that a charged track from the kaon decay entered the ducial region of the detector.DC - The online delayed coincidence condition (DC), was that the IC time (tIC) wasrequired 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 thathad decayed at rest rather than a beam pion that scattered into the active regionof the detector or a kaon that decayed in  ight.zfrf - An additional requirement de ning the active region of the detector was imposedfor the    (1) trigger. This required the charged track to stop in the active regionof the RS scintillators and not the second superlayer of the RSSCs (after RS layer14). This condition did not need to be applied to the    (2) trigger since the rangeconditions of the    (2) trigger did not allow the pions to reach RS layer 13. Italso removed long-range tracks with hits at large z-positions.6a0a2a1 + 7a0a2a1 - For the    (1) trigger, the charged track was required to reach RS layer 6 or7. This condition suppressed the K+ !  +   + and K+ !  + 0 0 backgrounds.3a0a2a1  4a0a2a1  5a0a3a1  6a0a2a1 - For the    (2) trigger, the charged track (ct) was required to reachlayer 6 and have hits in all previous layers. This condition suppressed backgroundsfrom 3 or more body decays such as K+ !  +   + and K+ !  + 0 0.19a0a2a1 - For the    (1) trigger, the K 2 background was suppressed by requiring that thecharged track did not reach the 19th layer of the RS.(13a0a2a1 +    + 18a0a2a1 )  19a0a3a1 - For the    (2) trigger, the charged track was not allowedto reach Range-Stack layers 13 through 19. This condition suppressed the K 2background and other long-range charged tracks beyond the PNN2 kinematic region.44L0rr1, L0rr2 - These conditions were a re ned calculation of the charged track range,which included the number of target  bers hit and the deepest layer of penetration ofthe track as corrected for polar angle by a measurement of the z-position of the trackfrom  ash TDCs on RS layers 3 and 11-13. The deepest later of penetration wasdetermined by the online stopping-counter  nder (SCF). The event was rejected ifthis calculated range was too long to be consistent with a charge track due to a  + inthe kinematic region of interest (L0rr1 for PNN1 and L0rr2 for PNN2). For L0rr2,the only information used was the number of target  bers hit. These conditionssuppressed the K 2 background and other long-range charged tracks beyond thekinematic region of interest.BV + BVL + EC - Online photon veto in the BV, BVL and EC rejected events if therewas 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 discriminatedPMT signals from either end of the counter.HEX - Only one RS hextant was allowed to have hits coincident with T 2, or twohextants if they were adjacent. This condition rejected events with multiple tracksand events with photon activity in the RS.L1.1 - The level-1.1 trigger required a signature of  + !  + decay in the online stoppingcounter. The height (PH) and the area (PA) of the pulse(s) as recorded by theTDs in the stopping counter were compared and the event was rejected if the ratioPH/PA was too large and there was no evidence of a detached pulse. This conditionremoved events without the  + !  + decay signature as the ratio was smaller fordouble pulse activity than for single pulse. This decision was made by a controlboard with Application Speci c 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-45ditions. The  rst condition was the \L1.1 afterburner", which rejected events withaccidental activity near the stopping counter. This rejected muon events that passedL1.1 due to accidental activity near the stopping counter and faked the double-pulse + !  + decay signature in the stopping counter. The second condition was the\HEX afterburner", which rejected events where one of the two adjacent hextanthits was not due to the charged track. This rejected events that passed the HEXtrigger condition due to accidental activity. The third condition rejected eventswhere the SCF assignment was not meaningful, such as  nding a stopping layerbeyond layer 19.PS16+Ca4 - An online pion  Cerenkov (C ) veto was imposed during run 49151, when thebeam 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 ofthe trigger, changing the    (2) trigger from    (2)1 to    (2)2. In addition toevents passing the C veto, 1/16th of the events were passed regardless of the C veto to allow measurement of the C veto performance.2.7.2 Monitor TriggersIn addition to the    signal triggers, there were monitor triggers for calibration andnormalization, as well as triggers for other physics modes. Only the monitor triggers usedthroughout the rest of this analysis are described here. A complete list of E949 triggersis available from [63]. The monitor triggers were prescaled to reduce the impact on thetotal dead time and the monitor trigger data were taken simultaneously with    signaldata to ensure the experimental conditions were the same for all analyzed data.46Ka5 2 Monitor TriggerSince the  nal state of this decay contains no photons or additional charged particles, itwas used for many of the acceptance measurements. This sample was used to determinethe fraction of beam kaons coming to rest in the target by normalizing the measured K 2branching ratio to the world average. This sample was also used to optimize the range-momentum consistency cut and the target quality cuts dealing with energy deposits. TheK 2 trigger condition was de ned asK 2  KB (T 2) (6ct + 7ct) (17ct + 18ct + 19ct): (2.6)Ka4 2 Monitor TriggerThe  nal state of this decay contained two photons from the  0 decay and a mono-energetic +. The  + from this sample was used to check the measurement of the charged trackmomentum, energy and range, as well as to study particle identi cation in the Range-Stack. The photons were used to study the photon veto. The K 2 trigger condition wasde ned asK 2  KB (T 2) (6ct + 7ct) (19ct): (2.7) scatter Monitor TriggerThis sample contained beam pions that scattered into the  ducial volume of the RS.They were identi ed as beam pions by the  B trigger, which was the same as the KBtrigger, except that a coincident hit was required in the Pion  Cerenkov Counter insteadof the Kaon  Cerenkov Counter. Due to these pions not being mono-energetic, they werekinematically similar to    in the RS. This sample was used to calibrate the ionizationenergy loss of the pions and for studying the acceptance of many selection criteria applied47to the RS. The  scatter trigger condition was de ned as scatter   B DC (T 2) IC (6ct + 7ct) BV + BVL + EC HEX: (2.8)2.8 Summary of 2002 Data CollectionThe physics run for the E949 experiment took place over 12 weeks from March to Juneof 2002. The data collected during this period corresponded to a total of 1:71 1012 K+sthat entered the target (KBlive). This total represents the total recorded by the scalersthat counted the number of KB triggers for each spill for only \good runs". Various runswere discarded due to technical  aws known at the time of data collection or discoveredduring the analysis. Runs discarded in this way resulted in a lower overall KBlive, but didnot introduce bias into the  nal measurement since monitor and    signal trigger datawere discarded for all discarded runs. Figure 2.18 shows the accumulated KBlive for thevarious running periods of E787 and E949. The proton intensity at the AGS for E949 wastwice that of E787 and the total K+ exposure for E949 was about 30% of that for E787due to the signi cantly shorter running period.48Figure 2.18: Number of accumulated K+s that entered the target for E787 and E949 as afunction of data-taking days. Figure reproduced from [37].49Chapter 3Analysis OverviewThis chapter discusses the key analysis methods and steps used to estimate the backgroundlevel, measure the acceptance, and  nally obtain the branching ratio for K+ !  +  .The blind analysis technique [6, 7, 8, 11] was designed to avoid bias when creatingthe selection criteria (\cuts") used to suppress the backgrounds that could mimic theK+ !  +  signal. This technique was based on identifying the competing backgroundprocesses a priori. Knowing these competing background processes, two kinematic phasespace regions having a high signal-to-background ratio were identi ed. The  rst region,known as PNN1, was above the K+ !  + 0 (K 2) peak and covered the momentumregion for the charged  + from 211 MeV/c to 229 MeV/c. The second region, known asPNN2, was below the K 2 peak and covered the momentum region from 140 MeV/c to199 MeV/c. These two regions are shown in Figure 3.1 as part of the total momentumspectrum of the  + from the K+ !  +  decay. This analysis was concerned with thePNN2 signal region and unless otherwise speci ed, all mentions of \the signal region"refer speci cally to the PNN2 signal region.To estimate the background level in the signal region for each type of background, abifurcation method [6, 7, 8, 11] was used. This method relied on two uncorrelated setsof cuts suppressing each type of background. Multiple validity checks were performed to50Figure 3.1: The momentum spectrum of the  + for the Standard Model K+ !  +  processshowing the PNN1 and PNN2 signal regions. Also shown are the charged product momentumspectra of the seven highest branching ratio K+ decay modes along with their branching ratiosin parentheses. The processes are not shown to scale. Figure modi ed from [37].51look for correlations between sets of bifurcation cuts, search for  aws in the analysis andcheck for cross-contamination in the background evaluations.The components of acceptance were measured using monitor trigger data where possi-ble, and Monte Carlo simulation data when appropriate samples could not be made usingmonitor trigger data. To validate the acceptance measurement, the K 2 branching ratiowas measured. Based on the acceptance measurement and the total K+ exposure, the sin-gle event sensitivity was determined, where the single event sensitivity was the branchingratio that would have corresponded to one candidate in the absence of background.Once the background measurements were  nalized and the acceptance measurementscompleted, the signal region was divided into nine cells whose relative signal-to-backgroundvaried by approximately a factor of four across the cells. The signal region was examinedand all observed events in the signal region were considered signal candidates. Basedon a likelihood analysis incorporating the acceptance, background and observed signalevents in each cell, the branching ratio from this analysis was determined. Using thelikelihood analysis, these results were combined with results from the previous E787 andE949 analyses [11, 6, 7] to obtain a combined E787/E949 branching ratio.3.1 Overview of BackgroundsFigure 3.2 shows the distribution of the range in plastic scintillator (rtot) versus mo-mentum (ptot) of the outgoing charged particle for events that passed the    (1) or   (2) triggers. Events from competing background processes could mimic K+ !  +  and become background if they fell into the kinematic signal region PNN2BOX andpassed all analysis cuts. These competing background processes could be misidenti edas K+ !  +  decays when combinations of the following failure modes occurred: (1)photons and other particles escaped detection, (2) the pion was reconstructed incorrectlydue to charged-particle scattering, (3) the overlapping tracks from di erent particles in-52Figure 3.2: The range in plastic scintillator (rtot) versus momentum (ptot) distribution ofthe outgoing charged particle for events passing the    (1) or    (2) triggers. The momentumwas determined assuming pion mass for all particles. The kinematic signal region is labeled\PNN2BOX". Figure reproduced from [72].terfered with kinematic measurements or particle-identi cation cuts, or (4) beam particlesemulated stopped-K+ decays.3.1.1 Stopped-K+-Decay BackgroundsThe dominant background in this analysis was from K 2 events where the photons escapeddetection and the  + scattered in the target or Range-Stack such that the event fellinto the kinematic signal region. The radiative K 2 decay (K 2 ) became a backgroundwhen all three photons escaped detection. The K+ !  +  (K 2 ) and K+ !  + 0 (K 3) processes were the primary source of muon-based backgrounds when the muonwas misidenti ed as a pion and the photons escaped detection. The K+ !  +  e+ (Ke4) decay became most problematic when the   and e+ had low energies and escaped53detection.3.1.2 Beam-Based BackgroundsThe beam-based backgrounds came from beam-pions scattering into the  ducial region ofthe detector ( scatter) or beam kaons decaying in  ight.The single-beam background originated from  scatter and kaon decay-in- ight eventsthat passed the delayed coincidence requirement between the incoming beam particle andthe outgoing charged decay product due to incorrectly measured beam or Range-Stacktiming. For the  scatter events, the beam pion also had to have been misidenti ed as akaon.The double-beam background came from events where the decay products from aninitial kaon were missed, and a second beam particle was subsequently missed by thebeam-line detectors. This second particle could have been a  scatter or kaon decay-in- ight. This type of event satis ed the delayed coincidence requirement between incomingbeam kaon (the initial kaon) and outgoing decay product (the second beam particle)times.The charge-exchange (CEX) background came from the charge exchange interactionK+n ! K0p in the target. The most problematic decays that result from this charge ex-change process were K0L !  +e  e and K0L !  +    , both of which have  + kinematicsthat overlap the kinematic signal region.3.2 Analysis Strategy and Methods3.2.1 Blind Analysis MethodA blind analysis method was developed to search for K+ !  +  signal candidatesamongst the many competing background processes. This method was based on (1) iden-54tifying the background sources a priori, and (2) keeping the signal region hidden untilthe background and acceptance analyses were completed. The signal region was de nedby the application of all analysis cuts, thus, to avoid examining the signal region, twomethods were used to create signal-like samples and samples rich in a speci c type ofbackground. The  rst was to use samples created from monitor triggers (see Section2.7.2 for monitor trigger de nitions). The second was to use events passing the PNN1 orPNN2 triggers and invert a cut that distinguished signal from background. By invertinga cut that suppressed a speci c background, a sample of that background was createdand the e ectiveness of other cuts in suppressing that background could be measured.Depending on the circumstance, both of these methods could be used to create signal-likeor background-like samples for use in developing cuts.3.2.2 Bifurcation Method to Evaluate Backgrounds Using DataEstimation of background levels in this analysis was performed using a bifurcation methodin which each background was heavily suppressed by at least two sets of cuts designedto be generally uncorrelated. Studies such as the outside-the-box and the single-cutand double-cut failure studies, detailed in Chapter 5, were used to establish the level ofcorrelation between these cuts.For each source of background, a sample of that background was created by applyingsetup cuts that suppressed other background sources, leaving a sample made almost en-tirely of the desired background. Using this sample, the bifurcation method was used toestimate the number of events of that background that would remain in the signal regionafter all analysis cuts have been applied. Figure 3.3 illustrates the bifurcation methodusing 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 whichhas passed both CUT1 and CUT2. Region \D" contains the D events which have failed55both CUT1 and CUT2. The total number of events in the sample is A + B + C + D. Ifthe rejection of one of the cuts does not depend on the rejection of the other cut (theyare uncorrelated), the ratios of the events in the four regions will be A=B = C=D. Re-arranging this algebraic expression allows the number of background events in the signalregion A to be estimated without being directly measured: A = BC=D.In practice, the E949 bifurcation analysis was done through two branches: a normaliza-tion 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 thatthe number of events in the region \B" was referred to as the normalization and assignedthe notation N. Given these de nitions, the number of background events in the signalregion was estimated bybg = NR  1; (3.1)which is algebraically equivalent to A = BC=D.3.2.3 Two Data SetsAfter reconstructing events passing the    (1) or    (2) triggers and processing themto remove unreliable data, the data were divided into \1/3" and \2/3" data sets selecteduniformly from the entire data set. The analysis cuts were developed and optimized usingthe 1/3 data set. Once the  nal cut positions were set, the 2/3 data set was used toevaluate the  nal background levels, removing potential bias introduced by optimizingthe cuts on the 1/3 data set.3.2.4 Validity ChecksThis section brie y describes the validity checks performed to verify the reliability of theanalysis strategy. The validity checks discussed are the single-cut and double-cut failurestudies, the outside-the-box studies, and the contamination studies used to determine the56Figure 3.3: A schematic explanation of the bifurcation method. Figure reproduced from [37].57Figure 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 StudiesThe \outside-the-box" studies tested the assumption that the cuts used in the bifurcationmethod were uncorrelated. This was done by simultaneously loosening the two bifurcationcuts CUT1 and CUT2 so that the four regions A, B, C and D became A0, B0, C0 andD0 as shown in Figure 3.4. The background study was performed in the same way as thestandard background evaluations using the bifurcation method, except that the number ofevents in region A0 was directly measured with the signal region masked out. This resultedin a region known as the \outside-the-box" region. The measured number of events in58the \outside-the-box" region was compared to the value B0C0=D0  BC=D. Correlationsbetween the bifurcation cuts were present if the measured and predicted \outside-the-box" background levels did not agree. Signi cant correlations between bifurcation cutswould generally result in the background level being underestimated. The details of thesestudies are discussed in Section 5.1.Single-Cut and Double-Cut Failure StudiesThe single-cut and double-cut failure studies involved cataloging and examining eventswhich failed only one or two sets of correlated cuts. Since the method used to estimatebackgrounds depended on each background being suppressed by two uncorrelated sets ofcuts, examining the events failing only a single set of correlated cuts provided a clearway to discover  aws in the analysis. Potential  aws included cuts not operating asdesigned to reject the appropriate background, new background processes not accountedfor in the existing background estimates, and loop-holes by which a background event notaccounted for in the background estimates could become a signal candidate. Analysis  awsdiscovered by examination of the single-cut and double-cut failure events were generally xed by the modi cation of existing cuts or by the introduction of new safety cuts. Safetycuts are high acceptance cuts that target the speci c analysis  aw. These studies wereperformed on both the 1/3 and 2/3 data sets, with consistency of the single-cut anddouble-cut failure rates between the two data sets providing a check for the potential biasintroduced by optimizing the analysis cuts using the 1/3 data set. The details of thesestudies are discussed in Section 5.2.Cross-Contamination in Background EstimatesThe background contamination studies were designed to determine if contributions fromany of the background processes were being double-counted by being included in morethan one of the background estimates, causing the total background to be overestimated.59The details of these studies are discussed in Section 5.3.3.3 Event ReconstructionReconstruction of each event was performed under the assumption that it was a K+ ! +  event with only a single  + track in the detector. Background events with multiplecharged decay tracks or multiple beam particles were greatly suppressed by this assump-tion as they typically failed one or more reconstruction steps. The reconstruction tookplace 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 fortracking in the UTC, and identi cation of the stopping counter.3. The track was located in the UTC with based on the clustering information fromthe Range Stack.4. In the target, K+ and  +  bers were identi ed and clustered, and the  ber in whichthe K+ decay occurred was identi ed.5. Second iterations of both the UTC and target  tting were performed. The energyloss and range were found in the target.6. Track  tting in the IC was used to  nd the energy loss and range in the IC.7. Track  tting was performed in the RS. The energy loss and range in the RS werefound.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 3.5.60X (cm)Y (cm)-60-50-40-30-20-10010-60 -50 -40 -30 -20 -10 0 10a6a8a7a10a9a12a11a14a13a16a15a17a19a18 a15a21a20a23a22a24a13a25a9a12a26a27a28a6a28a29 a17a23a18 a15a2a26a30a31a13a25a32a25a33a14a34a19a26a35a15a2a9a35a36a23a32a16a15a12a13a25a37a27a28a6a28a29a38a15a2a9a2a7a10a32a3a39a30a31a13a25a32a25a33a14a34a23a26a35a15a12a9a12a36a19a32a16a15a2a13a25a37a30a41a40a42a15a2a9a12a7a14a32a2a39a43a30a41a40a44a15a12a9a2a7a10a32a3a39a44a32a25a33a45a36a19a34a46a15a12a13a25a9a12a26a40a47a13a25a32a16a15a2a33a14a9a31a32a16a9a12a33a14a26a12a26 a18 a34a23a11a48a30a41a40a19a40a23a29 a17a19a18 a15a40a47a15a12a33a45a49a19a49 a18 a34a19a11a32a25a33a14a36a23a34a45a15a2a13a25a9a43X (cm)Y (cm)-8-6-4-2024-4 -2 0 2 4 6 8Figure 3.5: Reconstruction of an event in x-y view. The top  gure shows reconstruction inthe target, UTC and Range-Stack. The bottom shows an enlargement of reconstruction in thetarget. Figure reproduced from [37].613.3.1 Beam Time MeasurementsBeam times were found from TDC and CCD information for each of the CK (tCK), C (tC ), BWPC (tbw), and B4 (tbm) detectors. Average TDC times from the clusters wereused to determine these times with the CCD measurements being used to discriminatebetween beam particles close in time.In the CK and C counters, coincident TDC hits were clustered and the times averagedto  nd 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 thethree planes had coincident TDC hits. The time of the incoming particles was taken froman average of the TDC hits.In the B4 beam hodoscope, counters from the two layers were clustered based oncoincident TDC or CCD hits. Energy-weighted averages were used to determine the timeand x-y position of clusters. The measured energy deposits were used to di erentiatebetween and identify beam kaons and beam pions.3.3.2 Clustering in the Range StackBased on the coincidence with the online T 2 time (tds), the o ine T 2 sector wasidenti ed from TDC hits in the  rst two layers of the RS. Moving outward and clockwisefrom this T 2 sector, coincident hits within 10 ns of the T 2 time and with more than0.5 MeV were included in the cluster. A good cluster involved at least six consecutivelayers from inside to outside. The track time trs was determined from the average of allhit times in the good cluster. The outermost and most clockwise counter in the clusterwas identi ed as the o ine stopping counter.623.3.3 Initial Track Finding in the UTCTwo separate  ts were used to reconstruct the UTC track. The curvature of the  t inthe x-y plane was used to determine the charged particle’s transverse momentum. The t in the r-z plane was used to determine the polar angle of the charged particle and theresulting slope was used to convert from the measured transverse momentum to the totalmomentum.For the  t in the x-y plane, clusters in the hit wires were identi ed. A circle  t wasperformed using wire positions and drift distances with left-right ambiguity resolution.For the  t in the r-z plane, the cathode foils from the clusters were used. A straight-line  t was performed, where the z position of the cluster was determined using a ratiomethod suggested in Reference [64]. This ratio method involved using the three cathodestrips with the largest energy as measured by the ADCs and locating the centroid usingan energy-weighted mean.When more than one track pointed at the o ine T 2 sector, the good UTC track wasde ned as the one closest to the  rst crossing point (from inward to outward) betweenRS sectors, or closest to the clockwise edge of the o ine stopping counter otherwise.3.3.4 Initial Target ReconstructionThe  rst step of the target reconstruction was to classify the hit target  bers into K+, +, and   bers using their timing and energy information. Note that at the point of thisinitial classi cation, each  ber could fall into multiple categories. A  ber was classi ed as K+ if it had an energy greater than 4 MeV and was coincident with the beam strobetbs; Or  + if it had between 0.1 and 10.0 MeV of energy and was coincident with tracktime in the Range-Stack trs;63 Or  if it did not fall into the other two categories and had an energy above 0.1MeV.The  bers were clustered into K+ and  + paths based on geometry. The  +  bershad to lie along a strip (typically 1 cm in width) along the UTC track extrapolated intothe target.The designation of each  ber as a  + or   ber could be be switched based on likelihoodvalues calculated from energy, time and distance from the extrapolated UTC track. Theevent failed target reconstruction if there was more than one K+ or more than one  +track, or if these two tracks were not geometrically adjacent in the x-y plane.The decay vertex was identi ed in the K+  ber closest to the extrapolated UTC trackand furthest away from the x-y position of the B4 hit. The position of the track exitfrom the target was identi ed. Fibers were reclassi ed from  + to opposite-side- +  bersif 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  + hits were de ned as tK and tpi, respectively. Thesums of the K+ and  + energies were de ned as EK and Epi, respectively.This reconstruction process was performed as many as three times per event. Forthe  rst pass, the B4 position information was not used. After the  rst pass of targetreconstruction, the B4 reconstruction was performed again with the hit closest to theaverage time of the kaon  ber 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 reconstructionwas performed if the solution found from the second pass was not satisfactory.Further details on the target reconstruction can be found in [71].643.3.5 Second Iteration of UTC ReconstructionA second iteration of the UTC reconstruction was performed using the information fromthe  rst iteration of the target  tter as constraints.3.3.6 Second Iteration Target ReconstructionThen a second iteration of the target reconstruction was performed. In addition to thesteps described for the  rst iteration of the target reconstruction, double-pulse  ts wereperformed to search for  + or other energy hidden in the K+  bers using the CCDinformation from each of the K+  bers. The target pulse data  tter is discussed in moredetail in Appendix C. In the case where the di erence between tpi and tK was greaterthan 15 ns, any energy hidden in the second pulse of the vertex  ber was subtracted fromEK and assigned to Epi. The 15 ns threshold was the minimum time di erence for whichit was possible for the  tter to reliably identify the K+ and  + pulses in a  ber. Figure3.6 shows a double-pulse  t with hidden  + energy. The  + energy loss Etg and range rtgwere found in the target based on the target reconstruction.3.3.7 Track Fitting in the ICA good track in the IC was one with hits in only one or two adjacent sectors. In the caseof a sector crossing (two adjacent hits), the energy EIC was the sum of the two energiesand the time tIC was the energy weighted sum of the times from the two sectors. Therange RIC was computed as the length of the extrapolated UTC track from the inner tothe outer IC radius.3.3.8 Reconstruction in the Range-StackThe  rst step in RS reconstruction was to search for the  + !  + ! e+ decay signatureby  tting the TD information in the stopping counter using single-, double- and triple-65K fiber 445 Raw High GainK fiber 445 Raw Low GainK fiber 445 Resid Sngl HGK fiber 445 Resid Sngl LG0204060801001201401600 20 40 60 80 100 120051015202530350 20 40 60 80 100 120-5050 20 40 60 80 100 120-5050 20 40 60 80 100 120Figure 3.6: Double-pulse  t of the CCD information in the Kaon decay vertex for for the high-gain (top) and low-gain (bottom) CCDs. The energy of the second pulse was signi cantly higherthan the expected 1 MeV energy from a pion headed directly toward the  ducial region of thedetector and may have indicated a scatter.660501001502002500 20 40 60 80 100 120 14002550751001251501752002250 20 40 60 80 100 120 140-5050 20 40 60 80 100 120 140End 1 (RAW)End 2 (RAW)End 1 2P (Raw-Fit)/sResidualEnd 2 2P (Raw-Fit)/sResidual-5050 20 40 60 80 100 120 1400204060801000 20 40 60 80 100 120 1400204060801000 20 40 60 80 100 120 140-5050 20 40 60 80 100 120 140End 1 (RAW)End 2 (RAW)End 1 2P (Raw-Fit)/sResidualEnd 2 2P (Raw-Fit)/sResidual-5050 20 40 60 80 100 120 140Figure 3.7: Double-pulse  t (left) to the TD pulse information in the RS stopping counterfor the upstream (top) and downstream (bottom) readouts. Triple-pulse  t (right) to the TDpulse information in the RS stopping counter for the upstream (top) and downstream (bottom)readouts.pulse assumptions. The  ts were performed by varying the leading edges, heights andpedestals of template pulses, prepared for each individual counter using  scatter monitortrigger data. Examples of double- and triple-pulse  ts are shown in Figure 3.7 and furtherdetails on the TD  tter can be found in [57]. In addition to observation of the  + !  +decay signature, these  ts allowed the energy deposited by the muon E in the stoppingcounter to be determined.The RS track  t in the x-y plane was performed using (1) the entrance point providedby 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  2 of the  ttedtrack was minimized by changing the incident momentum and angle of entrance to theRS, with the shape of the track taking into account e ects of energy loss in each counteras given by the Bethe-Bloch equation for a 1 T magnetic  eld. Due to this correlationbetween energy and range these quantities could not be treated as uncorrelated in thebifurcation analysis. Many of these quantities are labeled in Figure 3.5 and discussed in67more detail below.Sector crossing points allowed for precise measurement of track positions in the x-yplane. 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 wasidenti ed as that closest to the UTC track extrapolation. For each layer, the x-y positionof the hit was taken from the average of the track hits and the time of the hit from theearliest hit.The energy loss in each RS counter was determined using ADC information. A correc-tion was applied to remove the additional energy recorded by the ADC from accidentalactivity by using the TD pulse height information. The total energy loss of the track inthe RS (Ers) was the sum of the energy losses in all the RS counters corrected for the theenergy deposited by the  + and e+ in the stopping counter.The polar angle in the RS was determined by a straight-line  t in the r-z plane. Thez-positions in the RS counters and the Straw Chambers were determined from end-to-endtime (and charge for the RS counters) di erences, with average resolutions of 4-5 cm and1.5 cm, respectively. The z-position of the extrapolated UTC track was also used in the t. The range in the RS (Rrs) was calculated from the path length of the  tted track inthe x-y plane, corrected for the polar angle.3.3.9 Kinematic Measurements of the TrackThe total range of the charged track (rtot) was the sum of Rrs, RIC and Rtg. The totalenergy (etot) was the sum of Ers, EIC and Etg. The total momentum (ptot) was obtainedfrom the UTC with corrections from the range in the target and IC. Due to these correc-tions, there was some correlation between momentum and range and they could not betreated as uncorrelated in the bifurcation analysis. All three kinematic quantities were68corrected for tiny contributions due to inactive material in the UTC.3.4 Selection CriteriaThe cuts used to de ne the signal region and suppress backgrounds were optimized bymaximizing their ability to suppress backgrounds while maximizing signal acceptance.The analysis was performed in three stages, called passes. The PASS1 cuts removedevents which could not be reconstructed and obvious background events. The PASS2cuts were a set of cuts which could be grouped into various combinations to enhancespeci c backgrounds. The PASS3 cuts were the  nal analysis cuts that were optimized tominimize the acceptance loss of signal events while maximizing their ability to reject thetargeted background. The PASS3 cuts were grouped into categories of kinematic cuts,phase space cuts, delayed coincidence cuts, beam cuts, target quality cuts,  + !  + ! e+decay-sequence cuts, and photon veto cuts.3.4.1 Loose and Tight CutsThe primary (loose) signal region was de ned by the application of all of the analysiscuts described in this section. Four sets of analysis cuts were tightened and variouscombinations of these sets of cuts were used to divide the primary signal region into ninecells having varying levels of signal-to-background. The sets of cuts that were tightenedwere (1) the kinematic phase space cuts \PNN2BOX", (2) the delayed coincidence cuts\DELCO", (3) the  + !  + ! e+ decay-sequence cuts \TDCUTS", and (4) the photonveto \PVCUT". These cells were used in the likelihood method to determine the K+ ! +  branching ratio from the observed candidates (Section 7.1). The cell having thehighest signal-to-background was de ned by the application of the tight versions of eachof the four sets of cuts and is referred to as the \tight signal region" throughout the restof this thesis.693.4.2 PASS1 CutsThe most basic analysis cuts were the PASS1 cuts. These loose cuts removed obvious K 2,K 2 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 CutsThe PASS2 cuts consisted of looser versions of the  nal analysis cuts. The PASS2 cutswere applied in certain combinations to enhance speci c backgrounds, reducing processingtime when developing cuts and evaluating background levels. Details of the PASS2 cutsare discussed further in Appendix D.2.3.4.4 PASS3 CutsThe PASS3 cuts were the  nal analysis cuts, grouped into categories of kinematic cuts,phase space cuts, delayed coincidence cuts, beam cuts, target quality cuts,  + !  + ! e+decay-sequence cuts, and photon veto cuts.3.4.5 Kinematic CutsThe kinematic cuts, collectively referred to as KINCUTS, were grouped into the followingcategories:Fiducial cuts - These cuts ensured that the charged decay product passed only throughand came to rest in the  ducial region of the detector, where the  ducial region wasthe active region of the detector in which events were accepted for further analysis.These cuts further tightened the conditions from the    (1) and    (2) triggersplaced on the parts of the detector through which the charged decay product couldtravel or come to rest.70Track Reconstruction Cuts - These cuts required good track reconstruction in theUTC, and good matching between RS track reconstruction and UTC track extrap-olation.dE/dx Cuts in Range-Stack - These cuts removed events with energy deposits in theRS inconsistent with those of a  +. These cuts removed events where a photonor extra track deposited energy that was measured by the ADCs. These cuts alsoremoved events that scattered in the Range-Stack such as K 2 Range-Stack scatter.Range-Momentum Consistency Cut - Since muons typically had a longer range thanpions for equivalent momenta, this cut suppressed muon backgrounds by requiringthat 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 CutsTo suppress K 2 events, the upper limits on the phase space cut (PNN2BOX) were chosento be below the K 2 monochromatic peak (Figure 3.2) in momentum, energy and range.The upper bounds of this cut were located 2:5   ptot below the K 2 momentum peak,2:5   etot below the energy peak and 2:75  rtot below the range peak. The standarddeviations on the measured kinematic quantities ( ptot,  etot,  rtot) were measured usingreconstructed K 2 events. The lower bounds of this cut were chosen to suppress orkinematically exclude the troublesome many-body decays that could most easily mimicthe K+ !  +  decay. The acceptance conditions for the loose version of the phase spacecut (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.71The phase space cut was one of the sets of cuts that was further tightened to de nethe tight signal region. The lower bounds of the tight phase space cut (BOXTIGHT)were raised to suppress the Ke4 background, whose  + momentum peaked at around 160MeV/c as shown in Figure 4.8. The lower momentum bound of 165 MeV/c was chosento maximize Ke4 suppression while minimizing the loss of signal acceptance. The uppermomentum bound was lowered from 199 MeV/c to 197 MeV/c to increase K 2 suppressionwithout having a signi cant impact on signal acceptance. The acceptance conditions forthe 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 CutsThe beam cuts, collectively referred to as BEAMCUTS, were designed to suppress single-beam and double-beam backgrounds. Single-beam backgrounds were suppressed by aparticle identi cation cut that ensured that the beam particle was a kaon based on itsenergy deposition in B4. Double-beam backgrounds were suppressed by searching for andremoving 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 andTDC information in B4, between B4 and the average target K+  ber hit times, andbetween the track time in the Range-Stack and the average target  +  ber hit times,and (2) removed double-beam events with speci c signatures known to cause the targetreconstruction to  nd incorrect solutions. Details of the beam cuts are discussed furtherin Appendix D.5.723.4.8 Delayed Coincidence CutsThe delayed coincidence cuts suppressed single-beam backgrounds by enforcing a mini-mum time between the average time of the kaon  ber hits (tk) and the average time ofthe pion  ber hits (tpi). The delayed coincidence cuts were one of the sets of cuts thatwas further tightened to de ne the tight signal region. Details of the delayed coincidencecuts are discussed further in Appendix D.6.3.4.9 Target Quality CutsThe target quality cuts, collectively referred to as TGCUTS, were designed to selectsignal-like events based on quantities measured in the target. These cuts targeted spe-ci c backgrounds and/or event signatures. These cuts rejected events that fell into thefollowing categories: Events where the energy deposited by a  + in a target  ber was larger than thatexpected for a  + traveling through the target along the extrapolated UTC track.This energy is a signature of  + scattering in the target, or could have been ev-idence of a photon in a  +  ber, or evidence of a photon or  + hidden in a K+ ber. These cuts suppressed the K 2 target-scatter background, any backgroundsinvolving photons in the  nal state and events where a K+  ber was misidenti edas a  +  ber. Events with poor agreement between the target  + track  tter and the target  + bers as determined using  2-like quantities. These cuts suppressed events thatscattered in the target. Events where the beam particle did not behave as a K+. A likelihood function wasconstructed using the z-position of the kaon decay in the target as determined byUTC track extrapolation, the expected z-position of the kaon decay as determined73by the total energy deposited in target kaon  bers, and the energy deposited bythe kaon in the B4 Hodoscope. These conditions suppressed the beam pion scattersingle-beam background. Events where the  + cluster did not appear to originate from one of the K+ clustertarget  bers. This cut suppressed CEX and double-beam backgrounds. Events where there was a large uncertainty in the target pion path length from thetarget reconstruction. This suppressed double-beam and K 2 target scatter events. Events that did not meet consistency requirements for times or energies betweenthe K+ or  + 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 asmuon-based backgrounds. Events that did not meet position consistency requirements between the K+ decayvertex, the beam particle in B4, and the target K+ and  + clusters. Events with target  +  bers on both sides of the K+ decay vertex as shown inFigure 4.7. This suppressed double-beam events and decays with multiple chargedproducts, such as Ke4. Events that showed evidence of hidden energy as determined by likelihood functionscomparing target  +  ber positions, times and energies to the extrapolated UTCtrack. Events that did not meet consistency requirements between the target K+  bertimes and their distances to the vertex in the x-y plane or the z-positions of theirhits as determined by the deposited energy.74 Events where the target pulse data  tter, described in Appendix E, failed to  ndacceptable solutions to both the single-pulse and double-pulse assumptions in anyof the K+  bers having energy above a certain threshold. Events where there was coincident activity between the IC and a nearby K+  ber.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  + !  + ! e+ Decay-Sequence CutsThe  + !  + ! e+ decay-sequence cuts, collectively referred to as TDCUTS, weredesigned to reject events in which a  + caused the charged track. These cuts searched inand around the Range-Stack stopping counter for activity consistent with the observationof the  + !  + ! e+ decay sequence. The signature for this decay sequence, based onthe pulse-shape information from the TDs, was1. Three pulses were found in the stopping counter, corresponding to the  +,  + ande+ of the  + !  +  and  + ! e+   e decays.2. The 4.1 MeV of kinetic energy from the  + was observed as about a 3 MeV pulsedue to saturation [19]. Since the path length of a  + coming from the decay of a + at rest is approximately 1.4 mm in the RS scintillator counters, most of thesemuons did not exit the stopping counter.3. Most of the positrons from the  + ! e+   e decays exited the stopping counter dueto having up to 53 MeV of kinetic energy; thus the positron should be observed todeposit energy in and around the stopping counter.Events were rejected if the above signatures were not observed or if it was determinedthat the observed signatures were faked by accidental activity.75The  + !  + ! e+ decay-sequence cuts were one of the sets of cuts that was furthertightened to de ne the tight signal region. Details of these cuts are discussed further inAppendix D.8.3.4.11 Photon Veto CutThe photon veto cut (PVCUT) rejected events with activity above threshold in any of thephoton veto subsystems coincident with Range-Stack track time trs. These subsystemswere the BV, BVL, RS, EC, target, IC, VC, CO,  CO, AD and DPV.The photon veto cut was one of the sets of cuts that was further tightened to de nethe tight signal region. The photon veto cut was also loosened below the nominal levelfor some background and acceptance studies. The photon veto cut is discussed further inAppendix D.9.3.5 Monte Carlo SimulationDetector responses were modeled by a Monte Carlo simulation package, referred to in-ternally as \UMC". The simulation modeled all detector systems other than the beaminstrumentation upstream of the target. The simulation samples were generated with thesame information and in the same format as data except that pulse-shape informationfrom CCDs and TDs was not generated. The accuracy and performance of the simulationwere veri ed by comparisons of kinematic variables between simulation and data for K 2and K 2 decays [33].3.5.1 Simulation of K+ PropagationA \beam  le" was created from analysis of the K+ stopping distribution using K 2 monitortrigger data. This  le contained the B4 hit position; the K+ stopping position and target76 ber element number; and the number of  ber hits, their time, energy and  ber elementnumber for K+ and accidental hits. The target patterns were the same in the simulationand the experimental data. Using this information, a simulated K+ decay started fromthe stopping  ber at the given K+ stopping position.3.5.2 Simulation of K+ Decay ProductsA number of K+ decays were simulated for various studies: K+ !  +  signal, K 2,K 2, Ke4, K 2 , and a number of the CEX processes discussed in Section 4.8. Photonand electron interactions and their energy deposits were calculated using routines fromthe electromagnetic-shower simulation package EGS4 [86]. For other charged particles,the energy deposits were calculated by summing the energy losses of each ionizationalong the steps taken by the particles. Using the Bethe-Bloch formula, the total averageenergy deposited along each step was calculated. The multiple Coulomb scatterings ofthe charged particles with various nuclei in the detector were calculated according to thetheory of Moliere [32], with corrections for the spin of the scattered particle and the formfactor of the nucleus [79]. The hadronic interactions of positively charged pions in theplastic scintillators were calculated using a combination of data and phenomenologicalmodels [95]. The   absorption was modeled based on experimental data of stopped   in the Range Stack [97].3.5.3 Simulation of TriggerA comparison between simulated K 2 triggers and data was used to derive a weightfunction to correct for the trigger bias introduced by generating the beam  le used in thesimulation.77Chapter 4BackgroundsTo obtain an unbiased estimate of the total background level, the data were split intotwo sets and the signal region was not examined until after the background evaluationstudies were  nalized. The \1/3 data set" was used to optimize the cuts and the resultingbackground levels. The  nal background levels were measured on the \2/3 data set". Theprocess of creating the \1/3" and \2/3" data sets was described in Section 3.2.3. Thischapter describes the background evaluation procedure and resulting background levelsfor 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 factorto represent the background level for the entire data set. In cases where the backgroundsample 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 tightsignal regions (Section 3.4.1). The estimates in the tight signal region were used as avalidity check against the scaling method used to divide the signal region into nine cellsas detailed in Section 7.1.784.1 K 2 Target-Scatter BackgroundThe decay K+ !  + 0 (K 2) was responsible for the dominant background in this anal-ysis. For this decay to become a background, the photons from the  0 decay have tobe missed and the  + has to lose some undetected energy, perhaps via nuclear interac-tions, such that it falls into the kinematic signal region (PNN2BOX). This scatteringcan happen in the target (K 2 target-scatter) or in the Range Stack (K 2 Range-Stack-scatter), but the target-scatter component dominates this background. The evaluation ofthe Range-Stack-scatter background is discussed in Section 4.2.The topology of the most problematic type of K 2 target-scatter was that of a  +initially traveling along the kaon  bers and scattering into the active region of the detec-tor. There were two reasons why this problematic type of target-scatter was di cult toreject. The  rst reason was that some energy deposited in the target by the scattering + was hidden in a kaon  ber. The second reason was a consequence of the  + and 0 emerging back-to-back. A  + initially traveling along the kaon  bers meant that the 0 was also traveling parallel to the beam direction and the resulting photons from itsdecay were directed at the upstream or downstream ends of the detector. As can be seenin Figure 2.13, these small angles are where the photon veto is the weakest. Figure 4.1contrasts this type of event with a regular K 2 event.The K 2 target-scatters were classi ed into two non-exclusive categories. The  rstcategory, known as \z-scatters", occurred when the  + traveled in the beam direction,scattered in a kaon  ber and into the active region of the detector. The troublesome decaydescribed above falls into this category. The second category, known as \xy-scatters",occurred when the  + scattered outside of the kaon  bers and the scatter was visible inthe 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 inthe rejection branch.79Figure 4.1: Schematic of the regular K 2 (left) and the K 2 target-scatter (right) processes. Fora regular K 2 event (left), the back-to-back correlation between the directions of the  + and  0mean that a  + directed at the active region of the detector results in the photons from the  0being directed toward a region of the detector where the photon veto is strongest such as theBarrel Veto. For the most troublesome type of K 2 target-scatter event (right), a  + initiallytraveling along the kaon  bers and scattering toward the active region of the detector results inthe  0 also traveling parallel to the beam direction. The photons from the  0 decay are thendirected toward the upstream or downstream ends of the detector, where the photon veto isweakest.80The most e ective methods of suppressing this background were the detection of thephotons from the  0 decay using the photon veto PVCUT and the detection of the elasticor inelastic scatter of the  + in the target using the target quality cuts TGCUTS, many ofwhich suppress events with additional energy in the target thus suppressing  + scatters.The bifurcation cuts used for the K 2 target-scatter background evaluation were thephoton veto PVCUT (CUT1) and the target quality cuts TGCUTS (CUT2).4.1.1 Potential Sources of ContaminationSince the photon veto was e ective at suppressing all background process involving pho-tons, a sample of K 2 target-scatter events created by inverting the photon veto wasexpected to have contamination from other background processes containing photons. Itwas also expected that other background processes could result in measurable and non-negligible amounts of contamination in both the normalization and rejection branches forthe K 2 target-scatter background.The K 2 target-scatter normalization branch was found to contain K 2 Range-Stack-scatter events and this contamination was exploited to estimate the K 2 Range-Stackscatter background (see Section 4.2). The K 2 target-scatter normalization branch wascorrected for this Range-Stack-scatter contamination as is discussed in Section 4.1.2.It was determined that the K 2 target-scatter normalization branch was likely tocontain a large amount of contamination due to K 2 events due to the extra photon anddecay-pion kinematics that fell mostly in the kinematic signal region. This contaminationstudy is detailed in Section 5.3.4 and the resulting correction to the K 2 target-scatterbackground is discussed in Section 4.1.5.Studies to measure the contamination due to muon and double-beam (Sections 5.3.2and 5.3.3, respectively) events were performed and found negligible contamination in boththe K 2 target-scatter normalization and rejection branches. Note that in the K 2 target-81scatter background studies, the setup cuts included the cuts designed to heavily suppressthe muon and double-beam backgrounds thus the contamination due to the backgroundprocesses was expected to be small.Based on the bifurcation cuts used for the Ke4 background normalization branch (Sec-tion 4.5.1), it can be argued that the number of Ke4 events contaminating the K 2 target-scatter normalization branch must have been less than the number of events that were inthe Ke4 normalization branch, and thus the Ke4 contamination was negligible. The Ke4background normalization branch was created by inverting the combination of OPSVETOand TGPV and applying the remaining analysis cuts. For the 1/3 data set, the events re-maining 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 K 2 target-scatter normalization branch can be characterized as being selected bythe application of OPSVETO, the inversion of the photon veto, and the application ofall other analysis cuts. A sample of events chosen by OPSVETO and inverted TGPV isa subset of the K 2 target-scatter normalization branch. Since OPSVETO rejects Ke4events, the number of Ke4 events found in the sample chosen by OPSVETO and invertedTGPV will be less than those chosen by inverting the combination of OPSVETO andTGPV and thus the Ke4 contamination in the K 2 target-scatter normalization branchwas negligible.4.1.2 K 2 Target-Scatter Normalization BranchThe K 2 target-scatter normalization branch was created by inverting the loose photonveto PV60 (CUT1) and applying all the other analysis cuts. The \60" in PV60 refersto the fact that the parameters for the loose photon veto were optimized to accept ap-proximately 60% of the signal-like events. The loose photon veto was inverted in the82VALID TRIG, P2PSCUT, P2TGCUT, P2TGPVCUT, BEAMCUTS, DELCO , TDCUTS ,KINCUTS, PNN2BOX PV60TGCUTSNormalization BranchCCD31FIB, CCDBADTIM, CCDBADFIT,CCDPUL  EPIONK  B4EKZPVCUT N1N2Rejection BranchNtgscat RPVCUT = N1=N2K 2 Target-Scatter BifurcationsFigure 4.2: Flowchart showing the K 2 target-scatter bifurcations. Cuts in italics refer to namedgroups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used forthe background evaluation in the loose box and the tight version for the background evaluationin 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 in-verting the tight photon veto would have resulted in examining a portion of the loosesignal region. Figure 4.2 shows a schematic representation of the normalization branchfor the loose and tight K 2 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 K 2 Range-Stack-scatter events, thecalculation of which is detailed in Section Section 4.2.1,ntgscat = Ntgscat  nrsscat: (4.1)The resulting number of K 2 target-scatter normalization events are shown in Tables 4.4and 4.5.834.1.3 K 2 Target-Scatter Rejection BranchMonte Carlo simulations [58] of K 2 decays have shown that for events where the  + entersthe active region of the UTC and Range-Stack, the distribution of  0 decay photons ismuch more uniform for events where the  + had scattered in the target than for eventswhere the  + scatter had not occurred. In events where the  + scatter did not occur,the directions of the photons from the  0 decay were strongly correlated to the directionof the  +. When the  + scattered in the target, this correlation was obscured resultingin this more uniform distribution of photons from the  + scatter events. Due to thevariation with polar angle of the total thickness of the photon veto (see Figure 2.13), thephoton veto performance was not expected to be uniform in polar angle. As a result, theperformance of the photon veto was expected to be di erent for the K 2 target-scatterevents than for the K 2-peak events.To measure the rejection of the photon veto for target-scatter events, 12 classes (classes2-13 in Table 4.1) of K 2 target-scatter events were created by applying and invertingvarious combinations of the target quality cuts, creating samples with varying mixturesof \xy-scatter" and \z-scatter" events. Table 4.1 shows the cuts used to make the 12classes and Figure 4.2 shows the rejection branch for class 12. Table 4.2 for each of the12 classes for both the loose photon veto (PV60) and the tight photon veto (PV30). Thedirectional correlation between the  + and the photons from the  0 was obscured by usingthe given combinations of cuts to create classes of target-scatter samples, making classes2-13 suitable for measuring the photon veto rejection of target-scatter events. Class 1was not suited to measure the photon veto rejection of the target-scatter events since itconsisted of events where the directional correlation between the  + and the photons fromthe  0 was still present due the application of all TGCUTS and the use of the kinematicregion KP2BOX instead of the kinematic signal region PNN2BOX.Class 12, considered to be the richest in z-scatters, was chosen to measure the photon84CLASS Applied and Inverted TGCUTS1 all TGCUTS, KP2BOX2 CCDPUL EPIONK3 CCDPUL EPIONK, all others4 CCDPUL, EPIONK, TGZFOOL, EIC, OPSVETO, others5 CCDPUL EPIONK CHI567 VERRNG6 CCDPUL EPIONK CHI567 VERRNG, all others7 CHI567 VERRNG8 CHI567 VERRNG, all others9 CCDPUL EPIONK CHI567 VERRNG, KIC, PIGAP, TARGF, TPICS10 B4EKZ11 B4EKZ, all others12 CCDPUL EPIONK B4EKZ13 CCDPUL EPIONK B4EKZ, all othersTable 4.1: De nition of the classes of events used to measure the PV rejection of K 2 target-scatter events in the kinematic signal region. The notation \all others" refers to the remainingTGCUTS not applied in that class also being applied or inverted. All Classes that had eitherCCDPUL 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 2Range-Stack-scatter events and not K 2 target-scatter events. Since class 1 used events from theK 2-peak instead of the kinematic signal region, all of the TGCUTS could be applied withoutexamining the signal region.85Before PV60 PV30CLASS Photon Veto After Rejection After Rejection1 1/3 61410 36 1706 284 13 4724 13102/3 122581 106 1156 112 44 2786 4202 1/3 24396 9 2711 903 3 8132 46952/3 49032 21 2335 509 8 6129 21673 1/3 2776 3 925 534 1 2776 27762/3 5495 2 2748 1942 1 5495 54954 1/3 4159 3 1386 800 0 4159 41592/3 8092 1 8092 8092 1 8092 80925 1/3 29899 12 2492 719 4 7475 37372/3 59871 22 2721 580 8 7484 26466 1/3 4170 3 1390 802 1 4170 41702/3 8452 3 2817 1626 1 8452 84527 1/3 24574 6 4096 1672 1 24574 245742/3 49636 18 2758 650 7 7091 26808 1/3 353 0 353 353 0 353 3532/3 644 0 644 644 0 644 6449 1/3 23736 10 2374 750 3 7912 45682/3 47463 19 2498 573 7 6780 256310 1/3 11037 4 2759 1379 1 11037 110372/3 22037 10 2204 697 2 11019 779111 1/3 45 0 45 45 0 45 452/3 64 0 64 64 0 64 6412 1/3 26317 10 2632 832 4 6579 32892/3 52621 22 2392 510 8 6578 232513 1/3 3319 3 1106 639 1 3319 33192/3 6503 2 3252 2299 1 6503 6503Table 4.2: The rejection of loose photon veto (PV60) and tight photon veto (PV30) rejectionfor the K 2 target-scatter rejection branch. Class 1 was not considered as it used the KP2BOXkinematic box. For each class, the rejection \Rejection" of the photon veto was determined foreach of PV30 and PV60. To determine the rejection, the number of events before the photonveto was applied \Before Photon Veto" was divided by the number of events remaining after thephoton veto was applied \After". If zero events remained after the photon veto was applied, thecalculation of the rejection assumed that one event remained. Only the classes having su cientstatistics for a meaningful measurement were considered.86veto rejection and the other classes having adequate statistics were used to determinethe systematic uncertainty in this rejection. The inverted cuts (CCDPUL, EPIONK andB4EKZ) in class 12 rejected events with secondary energy deposits in target kaon  bersconsistent with a scattered  +. CCDPUL rejected events where a second pulse havingmore than 1.25 MeV of energy was found in a kaon  ber. EPIONK rejected events wherea  ber classi ed as both a kaon and pion  ber had more than 1.25 MeV of pion energy.B4EKZ rejected events where the z-position of the decay vertex found by the UTC didnot agree with the expected z-position as determined by the total energy deposited in thekaon  bers.Due to the loss of statistics in the rejection branch for the background evaluation inthe tight signal region, the rejection of the tight (PV30) photon veto was measured usinga rejection branch that used the loose versions of PNN2BOX, TDCUTS and DELCO. Indoing this it was assumed that the rejections of the tight photon veto for each of the classeswas the same when applying the loose and tight versions of PNN2BOX, TDCUTS andDELCO. The \30" in PV30 refers to the fact that the parameters for the tight photon vetowere optimized to accept approximately 30% of the signal-like events. Table 4.3 showsthat the rejection of the tight photon veto did not change within statistical uncertaintywhen applying the tight versions of these cuts as compared to the loose versions of thesecuts. When determining the systematic uncertainty of the tight photon veto rejection, themeasurements using the tight kinematic box BOXTIGHT were treated as an additionalclass (called BOXTIGHT) and ended up being used to determine the lower bounds onthe systematic uncertainty as shown in Table 4.5.Tables 4.4 and 4.5 summarize the photon veto rejections and other values used in thebackground evaluation.87Setup Events Events PV30Cuts Before After Rejection1/3 Data SetAll Loose 26317 4 6579 3289BOXTIGHT 19741 4 4935 2467DELCO6 22780 2 11390 8054TDTIGHT 19624 2 9812 6938All Tight 12725 1 12725 127252/3 Data SetAll Loose 52621 8 6578 2325BOXTIGHT 39481 7 5640 2132DELCO6 45574 6 7596 3101TDTIGHT 39287 6 6548 2673All Tight 25471 5 5094 2278Table 4.3: Rejection of the tight photon veto cut PV30 for class 12 with various combinationsof 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 theapplication of PV30, respectively. The column \PV30 Rejection" is the rejection of the tightphoton veto PV30, determined by taking the quotient of numbers found in the \Events Before"and \Events After" columns.884.1.4 K 2 Target-Scatter Background EvaluationFigure 4.3 shows the momentum distributions for the events remaining at various stagesof performing the background estimate using the bifurcation method.The loose K 2 target-scatter background was evaluated using the expressionbgloosetgscat = nloosetgscatRPV60  1; (4.2)where nloosetgscat was the number of events at the end of the loose normalization branchcorrected for K 2 Range-Stack-scatter contamination as shown in Equation (4.1), andRPV60 was the rejection of the loose photon veto as measured on the loose rejectionbranch.For the background evaluation in the tight signal region, the loose photon veto PV60was inverted to avoid examining the signal region and the results of the entire backgroundevaluation were scaled by the ratio of the loose and tight photon veto rejections. The K 2target-scatter background in the tight signal region was evaluated using the expressionbgtighttgscat = ntighttgscatRPV60  1  RPV60RPV30 ; (4.3)where ntighttgscat was the number of events at the end of the tight normalization branchcorrected for K 2 Range-Stack-scatter contamination, RPV60 was the rejection of the loosephoton 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 K 2 target-scatter background levels in the loose and tight signal regions, respectively. These back-ground evaluations were scaled to the full data set for both the 1/3 and 2/3 data samples.8911010 210 310 410 5140 160 180 200 220 240Momentum (MeV/c)Events/(2.5MeV/c) B+DB11010 210 310 410 5140 160 180 200 220 240Momentum (MeV/c)Events/(2.5MeV/c)C+DCFigure 4.3: The bifurcation cuts used for the loose K 2 target-scatter background evaluationwere the photon veto PV60 (CUT1) and the target quality cuts TGCUTS (CUT2). A schematicrepresentation of the phase space of these two cuts after setup cuts to remove other backgroundshave been applied is shown on top with the arrows representing the cuts becoming increasinglyloose. In the left plot, the momentum (ptot) distribution of the events remaining in the loosenormalization branch after the inversion of the photon veto PV60 (black, regions B+D) and andafter 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). Theestimate of the background events in the signal region \A" is given by \BC/D".901/3 2/3Normalization (Uncorrected)Nloosetgscat 528 1131Normalization(Corrected for K 2 Range-Stack-scatter)nloosetgscat 515:5 23:1 +1:2 1:1 1107:7 33:8+2:9 2:8RejectionRPV60(CLASS12) 2632 832 2392 510RPV60(max.) 4096 1672 2758 650(CLASS7) (CLASS7)RPV60(min.) 2374 750 2204 697(CLASS9) (CLASS10)RPV60 2632 832+1464 258 2392 510+366 188Background (Before correction for K 2 )bgloosetgscat(uncorrected) 0:588 0:188 +0:065 0:211 0:695 0:150+0:061 0:094K 2 BackgroundbglooseK 2 0:0514 0:0086 +0:0042 0:0038 0:0757 0:0073 +0:0062 0:0056Backgroundbgloosetgscat 0:537 0:188 +0:069 0:215 0:619 0:150 +0:067 0:100Table 4.4: The summary of the loose K 2 target-scatter background evaluation. For the photonveto rejections RPV60 and background evaluations bgloosetgscat, the  rst uncertainty is statistical andthe second uncertainty systematic. The maximum and minimum photon veto rejections arelabeled to show which class was used to determine the systematic uncertainties in RPV60 andbgloosetgscat.911/3 2/3Normalization (Uncorrected)Ntighttgscat 265 512Normalization(Corrected for K 2 Range-Stack-scatter)ntighttgscat 259:1 16:4 +0:6 0:7 499:7 22:8 +1:1 1:3Rejection (Tight Photon Veto)RPV30(CLASS12) 6579 3289 6578 3289RPV30(max.) 8132 4695 7484 2646(CLASS2) (CLASS5)RPV30(min.) 4935 2467 5640 2132(BOXTIGHT) (BOXTIGHT)RPV30 6579 3289+1553 1644 6578 2325+906 938Rejection (Loose Photon Veto)RPV60 2632 832+1464 258 2392 510+366 188Background (Before correction for K 2 )bgtighttgscat(uncorrected) 0:118 0:059 +0:075 0:023 0:114 0:041+0:019 0:014K 2 BackgroundbgtightK 2 0:0122 0:0038 +0:0010 0:0010 0:0188 0:0034 +0:0016 0:0014Backgroundbgtighttgscat 0:106 0:059 +0:076 0:024 0:095 0:041 +0:020 0:016Table 4.5: The summary of the tight K 2 target-scatter background evaluation. For the photonveto rejections and background evaluations bgloosetgscat, the  rst uncertainty is statistical and thesecond uncertainty systematic. The maximum and minimum photon veto rejections are labeledto show which class was used to determine the systematic uncertainties in RPV30, where theclass called \BOXTIGHT" was described in Section 4.1.3. The rejection for the loose photonveto PV60 was taken from Table 4.4924.1.5 Correction to Background for K 2 ContaminationThe contamination due to K 2 events was only present in the normalization branch sinceany K 2 contamination in the rejection branch was heavily suppressed by selecting eventswhich scattered in the target. The K 2 contamination in the normalization branch wasenhanced by inverting the photon veto due to the presence of the additional photon inthe K 2 events. An upper limit of 30:0  7:5% was found (Section 5.3.4) for the K 2 contamination in the normalization branch using K 2 monitor trigger events under theassumption that all monitor trigger events found in the kinematic signal region wereK 2 . Based on this upper limit and the fact that the size of the K 2 backgroundestimate was typically about 10% of the K 2 target-scatter background estimate or more,it was reasonable to conclude that the K 2 background was being entirely double-countedbetween its own estimate and the K 2 target-scatter background estimate. To correct forthis double-counting, the K 2 background was subtracted from the K 2 target-scatterbackground as shown in Tables 4.4 and 4.5.934.2 K 2 Range-Stack-Scatter BackgroundAs was described in Section 4.1, there were two types of K 2 scatter backgrounds: target-scatter and Range-Stack-scatter. For K 2 scatters in the Range-Stack to become a back-ground the momentum of the  + had to have been mis-measured in addition to thetarget-scatter conditions of the missing the energy lost in the scatter and the missingof the two photons from the  0 decay. For this reason, the K 2 Range-Stack-scatterbackground was expected to be much smaller than the K 2 target-scatter background.4.2.1 Range-Stack-Scatter Normalization BranchThe number of events Ntgscat left at the end of the target-scatter normalization branchactually consisted of ntgscat target-scatter with contamination due to nrsscat Range-Stack-scatter events:Ntgscat = ntgscat + nrsscat: (4.4)As will be described in this section, the number of Range-Stack-scatter events nrsscatwas determined by studying the target-scatter and the Range-Stack-scatter normalizationbranches.The most e ective cuts against the K 2 Range-Stack-scatter background were theRange-Stack track quality cuts RSDEDX and PRRF, the PNN2BOX momentum cut(PNN2PBOX), and the photon veto. The cuts RSDEDX and PRRF were collectivelyreferred to as RSCT. The Range-Stack normalization branch consisted of the same cutsas 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 ine ciencyof the RSCT cuts.The acceptance ARSCT of RSCT for K 2 target-scatter events falling into the kinematicsignal region was expected to be the same as for the signal. Thus the acceptance ofthese cuts was taken from the Range-Stack kinematic acceptance measurements found in94Section 6.2.2, with systematic uncertainties determined in the same way as described inthat section.The rejection RRSCT of RSCT was measured using events in the kinematic regionknown as the K 2 range-tail. This region consisted of the K 2-peak momentum re-gion (KP2PBOX), but having a range and energy consistent with the signal region(PNN2REBOX). These were K 2-peak events that did not scatter in the target and thustheir momentum was still in the K 2-peak region, but they scattered in the Range-Stackand their range and energy measurements fell into the range and energy of the signalregion. Figure 4.4 shows the samples used to measure the rejection of the RSCT cuts andTable 4.6 summarizes the results.With the performance of the RSCT cuts measured, it was possible to use equation 4.4and the following equation for the number of events Nrsscat remaining at the end of theK 2 Range-Stack-scatter normalization to determine the quantity nrsscat:1 ARSCTARSCT  ntgscat + (RRSCT  1) nrsscat = Nrsscat: (4.5)The nrsscat values which resulted are shown in Table 4.6.4.2.2 Range-Stack-Scatter Rejection BranchThe photon veto rejection of the Range-Stack-scatter events was assumed to be the sameas for the K 2-peak events since the distribution of photons coming from the  0 for  +particles entering the  ducial region of the detector will be the same for the K 2-peakand the K 2 Range-Stack-scatter events. The K 2-peak rejection branch from Figure 4.4shows the sample used to measure this photon veto rejection RPVCUT KP2 and Table 4.6summarizes the results.95Loose Tight1/3 2/3 1/3 2/3Acceptance of RSCTARSCT 0:888  0:001  0:012 0:894  0:002 +0:010 0:012Rejection of RSCTN4 642 1355 307 739N5 80 192 44 118RRSCT 8.03 0.84 7.06 0.47 6.98 0.97 6.26 0.53Normalization NumbersNtgscat 528 1131 265 512Nrsscat 153 281 66 124nrsscat 12:5  2:4+1:2 1:2 23:3  3:5+2:9 3:0 5:9 1:7+0:6 0:7 12:3  2:6+1:3 1:5Photon Veto Rejection (K 2 peak)N2 61410 122581 41103 82387N3 36 106 10 31RPVCUT KP2 1706 184 1156 112 4110 1300 2658 477Backgroundbgrsscat(1/3) 0:0220  0:0056 +0:0021 0:0021 0:0043  0:0019 +0:0004 0:0005bgrsscat(2/3) 0:0303  0:0054 +0:0038 0:0039 0:0069  0:0019 +0:0007 0:0009Table 4.6: The summary of the K 2 Range-Stack-scatter background evaluation. For valueshaving two sets of uncertainties, the  rst is statistical and the second systematic. See Figure4.4 for de nitions of N2 - N5.4.2.3 Range-Stack-Scatter Background EvaluationThe  nal background level due from K 2 Range-Stack-scatter events was given bybgrsscat = nrsscatRPVCUT KP2  1; (4.6)where the 1/3 and 2/3 data sets were normalized to the full data set. A summary of theK 2 Range-Stack-scatter background evaluation is found in Table 4.6.96PNN2BOXRSDEDX .OR. PRRFLAYER14, COS3D, LAYV4,ZFRF, ZUTOUT,UTCQUAL, RNGMOMPV60Normalization BranchKP2BOXLAYER14, COS3D, LAYV4,ZFRF, ZUTOUT,UTCQUAL, RNGMOMRSDEDX, PRRFPVCUTN2N3N1KP2PBOX, PNN2REBOXLAYER14, COS3D, LAYV4,ZFRF, ZUTOUT,UTCQUAL, RNGMOMRSDEDX, PRRFN4N5Rejection BranchVALID TRIG, P2PSCUT, P2TGCUT, P2TGPVCUT, BEAMCUTS, DELCO , TDCUTS ,TGCUTSNrsscatRPVCUT KP2 = N2=N3RRSCT = N4=N5K 2 Range-Stack Scatter BifurcationsFigure 4.4: Flowchart showing the K 2 Range-Stack-scatter bifurcations. Cuts in italics referto named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut wasused for the background evaluation in the loose box and the tight version for the backgroundevaluation in the tight box. The photon veto rejection RPVCUT KP2 is the photon veto rejectionfor events in the K 2 peak.974.3 K 2 BackgroundThe background from the decay K+ !  + 0 (K 2 ) was expected to be small relativeto the K+ !  + 0 (K 2) background due to (1) the smaller branching ratio and (2) theadditional photon against which to veto.The  emitted in this decay can be produced by either direct emission (DE) or innerbremsstrahlung (IB). The inner bremsstrahlung process dominates these two  emissionprocesses with a branching ratio of 2:75 0:15 10 4 for T + in the range of 55 MeV to90 MeV [99]. The direct emission process has a branching ratio of 4:3 0:7 10 6 for thesame T + range [99].Due to the di culty of isolating a K 2 sample in data, evaluation of this backgroundwas performed using a combination of data and Monte Carlo simulation. The simulationwas used to predict the number of K 2 events that would be found in the signal regiongiven a certain number of K 2 events found in the K 2-peak kinematic box KP2BOXusing data. This was done by determining the relative acceptance of the  + from K 2and K 2 events, and determining the extra rejection from the photon veto due to theadditional photon.4.3.1 Relative Rate of  + Acceptance for K 2 and K 2 EventsThe quantity  represents the relative rate of acceptance of the  + for K 2 and K 2 events given (1) the acceptance A of the events generated in Monte Carlo for passing thetrigger conditions and o ine cuts, and (2) the branching ratio B of the process. Thefollowing expression is used for  , = B(K 2) A(K 2)B(K 2 ) A(K 2 ): (4.7)To determine the acceptances A, about 2  105 K 2 and 5  105 K 2 Monte Carlo98events were generated. The events were required to pass either the PNN1 or PNN2 triggerconditions in simulation without the online photon veto, L1.1, L1.2 or L0rr2 triggers.The events were then required to pass all available o ine cuts including the appropriatekinematic region cuts, but excluding the photon veto. The kinematic boxes used werethe K 2-peak kinematic box KP2BOX for the K 2 events and the PNN2 kinematic boxPNN2BOX for the K 2 events. The numbers of events used to determine the acceptancevalues are shown in Table 4.7.Also needed to determine  were the branching ratios of K 2 and K 2 . The branchingratio of the direct emission K 2 process was ignored since it is known to be two ordersof magnitude smaller than the inner bremsstrahlung K 2 process. The branching ratiofor the inner bremsstrahlung K 2 process was only given over the  + energy range of 55- 90 MeV. To extrapolate this branching ratio to the full energy spectrum available tothe decay  +, the Monte Carlo simulation was used to determine this energy spectrumavailable to the  + as shown in Figure 4.5. Using this spectrum, the e ective branchingratio B(K 2 ) for T + between 0 and 106 MeV was calculated,B(K 2 ) =Z 1060dNZ 9055dN (2:75 0:15) 10 4 = (1:11 0:06) 10 3: (4.8)The branching ratio for K 2 was 0:2092 0:0012 [99].The resulting values for  in the loose and tight signal regions are shown in Table 4.7.It was found that  was fairly insensitive to the applied cuts and was determined mainlyfrom the relative branching ratios.4.3.2 Additional Photon Veto RejectionThe performance of the photon veto in rejecting events based on the photons from the 0 decay was expected to be the same for both K 2 and K 2 events. However due to99Figure 4.5: Kinetic energy distribution of the  + from K 2 events in Monte Carlo. Figurereproduced from [55].the third decay photon, K 2 events had additional photon veto rejection above that forK 2 events. A single photon rejection function was created by convolving the spatial andenergy distribution of the third photon (from the simulation) with the detector detectionine ciency for a single photon (from data [83]). The additional photon veto rejectionfactor (R ) was determined by applying this single photon rejection function to the 11,305K 2 events remaining after the loose o ine cuts and 7,409 K 2 events remaining afterthe tight o ine cuts, respectively. The resulting rejection factors, taken from [55], wereRloose = 5:04 0:10: (4.9)andRtight = 5:11 0:11 (4.10)4.3.3 K 2 Background EvaluationTo evaluate the background level bg from the K 2 process, the number of K 2 events(NK 2 peak) found in the K 2-peak kinematic box KP2BOX was used to predict the num-100Number of Events NK 2 NK 2 N1: Total events produced 199986 499973N2: Passed PNN1 or PNN2 trigger 30625 64217N3: Passed loose o ine cuts 9776 11035N4: Passed tight o ine cuts 7608 7409Aloose = N3=N1 0.0489 0.0005 0.0221 0.0002Atight = N4=N1 0.0380 0.0004 0.0148 0.0002 loose 417 24 tight 483 28Table 4.7: Summary of Monte Carlo simulation events produced and remaining after variouscuts have been applied as were used to determine A(K 2) and A(K 2 ). The cuts tightened forthe tight o ine cuts measurement N4 were DELCO for both types of events, and the PNN2BOXfor K 2 events. Table reproduced from [55].ber of K 2 events that would have been found in the signal region. This prediction wasdone by using the simulation to determine  , the relative rate of  + acceptance for K 2and K 2 events, and by using a combination of simulation and photon veto data to de-termine R , the additional rejection due to the third photon from the K 2 events. Thefollowing expression was used to evaluate the background level due to the K 2 process:bg = NK 2 peak  R ; (4.11)where NK 2 peak was the value N3 from Figure 4.4 and Table 4.6, and the other termswere determined previously in this section. This background was normalized to the fulldata set for the 1/3 and 2/3 data sets. Table 4.8 summarizes the results of the K 2 background evaluation.101Loose TightNormalizationNK 2 peak(1/3) 36 10NK 2 peak(2/3) 106 31Relative Rate of  + Acceptance 417 24 483 28Additional PV RejectionR 5.04 0.10 5.11 0.11Backgroundbg(1/3) 0:0514 0:0086 +0:0042 0:0038 0:0121 0:0038 +0:0010 0:0010bg(2/3) 0:0757 0:0073 +0:0062 0:0056 0:0188 0:0034 +0:0016 0:0014Table 4.8: Summary of the K 2 background evaluation. The relative rate of  + acceptance\ " for K 2 and K 2 events and the additional photon veto rejection \R " are de ned in thetext. The  rst uncertainty in the background bg is statistical (from NK 2 peak) and the secondis systematic (from  and R ). The background bg was normalized to the full data set for the1/3 and 2/3 data sets. The values for  and R were taken from [55].1024.4 Muon BackgroundThe decay K+ !  +  (K 2) has a very high branching ratio, but due to the  + momen-tum of 236 MeV/c, it would need to lose a large amount of undetected energy to simulateK+ !  +  in the PNN2 kinematic region, and therefore it was highly suppressed. Themuon background came primarily from the K+ !  +  (K 2 ) and K+ !  + 0 (K 3)decays, each of which have a  + momentum spectrum that extends into the kinematicsignal region. For these decays to be backgrounds, the muon had to be misidenti ed as apion and the photon(s) must have escaped detection.The main cuts used to suppress muon backgrounds are the particle identi cationcuts in the Range-Stack and the photon veto. The bifurcation cuts used for the muonbackground evaluation were the loose  + !  + ! e+ decay sequence cuts TDLOOSE(CUT1) and the range-momentum consistency cut RNGMOM (CUT2). The RNGMOMcut required that the measured range of the charged decay particle was consistent withthat predicted for a  + given the measured momentum.4.4.1 Muon Normalization BranchThe muon normalization branch was created by inverting the loose  + !  + ! e+ de-cay sequence cuts TDLOOSE (CUT1) and applying all other analysis cuts. The looseversion of the  + !  + ! e+ decay sequence cuts was inverted in the normalizationbranches for both the loose and tight background evaluations since inverting the tightversion 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 cutswere designed to strongly suppress events having muon-like signatures in detector subsys-tems outside the target. Figure 4.6 shows a schematic representation of the normalizationbranch used for the loose and tight muon background evaluations.1034.4.2 Muon Rejection BranchTo create the sample upon which the rejection of TDLOOSE (CUT1) was measured,RNGMOM (CUT2) was inverted. Setup cuts were applied to this sample to removeK 2 decays and beam backgrounds. Figure 4.6 shows a schematic representation of therejection branch used for the loose and tight muon background evaluations.VALID TRIG, P2TGCUT, P2PSCUT, BEAMCUTS, DELCO , P2TGPVCUTTDLOOSEPNN2BOX , KINCUTS, TGCUTS,PVCUT Normalization BranchRNGMOMPNN2BOX , KINCUTS (excludingRNGMOM), TGCUTS (excluding CHI567 &CHI5MAX), PVPNN1TDCUTS N1N2Rejection BranchNmuonRTDCUTS = N1=N2Muon BifurcationsFigure 4.6: Flowchart showing the muon bifurcation branches. Cuts in italics refer to namedgroups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used forthe background evaluation in the loose box and the tight version for the background evaluationin the tight box.4.4.3 Muon Background EvaluationThe loose muon background bgloosemuon was evaluated using the expressionbgloosemuon = NloosemuonRTDLOOSE  1; (4.12)104where Nloosemuon was the number of events at the end of the loose muon normalization branchand RTDLOOSE was the rejection of TDLOOSE. This background was scaled to the fulldata set from the 1/3 or 2/3 data samples.For the background evaluation in the tight signal region, TDLOOSE was inverted inorder to not examine the signal region and the results of the entire background evalua-tion scaled by the ratio of the rejections of TDLOOSE and TDTIGHT. The tight muonbackground bgtightmuon was evaluated using the expressionbgtightmuon = NtightbeamRTDLOOSE  1  RTDLOOSERTDTIGHT ; (4.13)where Ntightmuon was the number of events at the end of the tight muon normalization branch,and RTDLOOSE and RTDTIGHT were the rejection of TDLOOSE and TDTIGHT, respec-tively. This background estimate was scaled to the full data set for the 1/3 and 2/3 datasamples.Table 4.9 summarizes the muon background evaluations.105Muon Background Summary1/3 2/3NormalizationNloosemuon 0 1Ntightmuon 0 1Rejection (TDLOOSE)N1 10328 20488N2 84 154RTDLOOSE 123:0 13:4 133:0 10:7Rejection (TDTIGHT)N1 9277 18411N2 17 45RTDTIGHT 546 132 409 61Backgroundbgloosemuon 0:0246 0:0246 0:0114 0:0114bgtightmuon 0:0055 0:0055 0:0037 0:0037Table 4.9: Summary of the muon background evaluation. See Figure 4.6 for de nitions of N1and N2 for RTDLOOSE and RTDTIGHT .1064.5 Ke4 BackgroundThe K+ !  +  e+ (Ke4) decay has a branching ratio of approximately 4  10 5 andcould be a serious background due to the lack of photons to veto and a substantial fractionof the decay  + phase-space falling in the kinematic signal region. The target event displayof a typical Ke4 event is shown in Figure 4.7.-6-4-20246-6 -4 -2 0 2 4 6Figure 4.7: The target event display of a typical Ke4 event. Red squares indicate the K+  bers,blue squares the  +  bers, and green the opposite-side- + and   bers. The two green tracksto the right of the decay vertex (blue circle) show the tracks of the   and e+. The top numbershown in each  ber is the time of the hit and the bottom number is the energy deposited.This background was most problematic when the low-energy   and e+ did not leavethe target and escaped detection by depositing all their energy in kaon  bers or insensitive107materials. For a combined total kinetic energy (T2) of the   and e+ below 100 MeV,the  + momentum peaked at around 160 MeV/c (see Figure 4.8), which was in the loosekinematic signal region. As is described in Appendix D.4, the tight kinematic signalregion was chosen to kinematically exclude most of the Ke4 background while minimizingthe acceptance loss of signal events.1001201401601802002202400 20 40 60 80 100ENTRIES            6595T2=Tpi- + Te+ (MeV)Ptot (MeV)Figure 4.8: Total kinetic energy (T2) of the   and the e+ versus  + momentum (Ptot) for MonteCarlo Ke4 events that pass the trigger conditions. Figure reproduced from [55].The most e ective cuts in  nding the additional energy deposited in the target fromthe low-energy   and e+ were the target sub-system of the photon veto (TGPV), theopposite side pion- ber veto (OPSVETO) and the pion-energy cut from target-CCD pulse tting (CCDPUL). After inverting the combination of OPSVETO and TGPV to createthe normalization branch, there remained no set of cuts with which to create a Ke4-richdata sample to measure the rejection of OPSVETO and TGPV so Monte Carlo simulationwas used.4.5.1 Normalization Branch Using DataThe Ke4 normalization branch used data and was created by inverting the combinationof the TGPV and OPSVETO cuts (CUT1), such that an event failing either of these cuts108remained in the normalization branch. All other analysis cuts other than TGGEO wereapplied. TGGEO was highly correlated with the inverted cuts and was thus excludedfrom the normalization branch. Figure 4.9 shows all the cuts applied in the normalizationbranch.VALID TRIG, P2TGCUT, P2PSCUT, KINCUTS, PNN2BOX ,BEAMCUTS (excluding TGGEO), DELCO , TDCUTS ,PVCUT (excluding TGPV)TGPV  OPSVETOTGCUTS (excluding OPSVETO)NKe4Ke4 Normalization BranchFigure 4.9: Flowchart showing the Ke4 normalization branch. Cuts in italics refer to namedgroups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut was used forthe background evaluation in the loose box and the tight version for the background evaluation inthe tight box. The loose version of TGPV was inverted for both the loose and tight normalizationbranches. Table 4.12 shows the number of events left at the end of this normalization branch.4.5.2 Rejection Branch Using Monte CarloMonte Carlo simulation was used to determine the total rejection of the combination ofthe TGPV and OPSVETO cuts for Ke4 events. A total of approximately 2  108 Ke4events with T2 < 50 MeV were produced, where T2 was the combined kinetic energy ofthe   and the e+ from the Ke4 decay. This constraint on T2 was used to enhance thestatistical power of the simulation by concentrating on the most troublesome phase spaceof the Ke4 decay. Before measuring the rejection of the combination of the TGPV andOPSVETO cuts, these events were required to pass all possible cuts in simulation.The Monte Carlo simulation used two di erent measures of energy deposited in target109 bers to reproduce the e ects of OPSVETO, TGPV and CCDPUL. The  rst quantityTxtg was the total energy deposited in all  bers not classi ed as kaon or pion  bers bythe reconstruction algorithm. This represented the energy available to the TGPV andOPSVETO cuts. The second quantity Ehide was the total energy deposited in kaon  bersby the   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 energyof absorbed   particles. The   absorption was modeled based on experimental data ofstopped   in the Range Stack [97]. The absorption of the   by carbon dominated andthe excited nucleus de-excited by gamma emission or particle evaporation. The resultinggammas and neutrons were likely to escape the target and were not simulated in theMonte Carlo simulation. Since the charged particles resulting from particle evaporationhad a short range, they were treated in Monte Carlo as having deposited all their energy inthe same  ber as the original   absorption. Figure 4.10 shows the additional absorptionenergy of the   which was determined experimentally by taking the di erence betweenthe measured absorption energy in the Range Stack and the   energy as determinedfrom its momentum. Due to the resolution of the two measurements used to determinethe total   absorption energy, it was expected that a small number of events would havea negative energy. This distribution was sampled in Monte Carlo to determine the total  absorption energy with energies below zero being treated as zero.Since the correlation between the energy deposited by particles in the simulation andin the data was not precisely known, the rejection thresholds for the quantities Txtg andEhide 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 quantityTxtg) was determined based on events above the Ehide threshold. The central value for thisrejection was used for the background estimation while the variations based on Txtg andEhide provided the systematic uncertainty. The Ke4 background summary table (Table4.12) shows the values for the rejection as used in the background estimate.110Energy of stopped pi- (MeV)Probability/5 MeV00.020.040.060.080.10.120.140.160.18-50 0 50 100 150Figure 4.10: Observable absorption energy of   stopped in the Range Stack as used by theMonte Carlo simulation. This is the additional energy observed from a stopped   beyond thatexpected from the energy of the   as determined by momentum. Figure reproduced from [37].4.5.3 Ke4 Background EvaluationThe Ke4 background was evaluated using the expressionbgKe4 = NKe4RTGPV OPSVETO  1; (4.14)where NKe4 was the number of events at the end of the Ke4 normalization branch andRTGPV OPSVETO was the rejection of TGPV and OPSVETO as measured using MonteCarlo. The results are summarized in Table 4.12.111Txtg < 0:6 MeV Txtg < 1:2 MeV Txtg < 1:8 MeVEhide < 1:6 MeV 2250/66 = 34 2250/86 = 26 2250/98 = 23Ehide < 2:5 MeV 6769/100 = 68 6769/129 = 52 6769/149 = 45Ehide < 4:0 MeV 34992/202 = 173 34992/288 = 122 34992/335 = 104Ehide < 10:0 MeV 97100/627 = 155 97100/888 = 109 97100/1105 = 88Table 4.10: Rejection of RTGPV OPSVETO as a function of Ehide for the loose rejection branch.In a given cell, the  rst number shows the number of events remaining after the Ehide conditionwas applied, and the second number shows the number of events remaining after the Txtg con-dition was applied. The Txtg condition represented the cuts TGPV and OPSVETO within thesimulation. Table reproduced from [55].Txtg < 0:6 MeV Txtg < 1:2 MeV Txtg < 1:8 MeVEhide < 1:6 389/18 = 22 389/20 = 19 389/22 = 18Ehide < 2:5 2282/23 = 99 2282/26 = 88 2282/31 = 74Ehide < 4:0 15105/43 = 351 15105/53 = 285 15105/65 = 232Ehide < 10:0 37174/160 = 232 37174/206 = 180 37174/269 = 138Table 4.11: Rejection of RTGPV OPSVETO as a function of Ehide for the tight rejection branch.In a given cell, the  rst number shows the number of events remaining after the Ehide conditionwas applied, and the second number shows the number of events remaining after the Txtg con-dition was applied. The Txtg condition represented the cuts TGPV and OPSVETO within thesimulation. Table reproduced from [55].112Loose Tight1/3 2/3 1/3 2/3NormalizationNKe4 4 6 1 0RejectionRTGPV OPSVETO 52+121 29 88+263 70BackgroundbgKe4(1/3) 0:235 0:118+0:310 0:166 0:034 0:034+0:142 0:026bgKe4(2/3) 0:176 0:072+0:233 0:124 0:017 0:017+0:071 0:013Table 4.12: The values used for the Ke4 background evaluation are shown with the results fromthe 1/3 and 2/3 data samples normalized to the full data set. The  rst uncertainty of bgKe4 isstatistical from NKe4 and the second uncertainty is the systematic uncertainty determined fromthe range of values found for RTGPV OPSVETO. The values for RTGPV OPSVETO were taken from[55].113CerenkovCounterUpstreamPhotonVetoBeODegraderActiveDegraderB4CounterKDecay in flightpBWPC2BWPC1TargetI CounterScatteringppFigure 4.11: A schematic of the single-beam background processes. The top  gure shows a kaondecay-in- ight and the bottom a beam pion scattering into the  ducial region of the detector.Figure reproduced from [37].4.6 Single-Beam BackgroundThe beam backgrounds were a collection of backgrounds due to beam pions scatteringinto the  ducial region of the detector or beam kaons decaying in  ight.For events having only a single beam particle, the delayed coincidence cuts (DELC3for the loose box and DELC6 for the tight box) removed all properly reconstructed eventsof this type. DELC3 (DELC6) required that the kaon decayed at least 3 ns (6 ns) af-ter entering the target. Both the scattered pion and the kaon decay-in- ight failed thisminimum time di erence condition between outgoing particles from the target and in-coming particles from the beam. These types of events could imitate K+ !  +  if theevent was poorly reconstructed such that the beam and/or RS timing wasn’t measuredcorrectly and the event passed the delayed coincidence requirements. This type of back-ground was referred to as single-beam background and schematic representations of thesetwo single-beam background processes are shown in Figure 4.11.A kaon decay-in- ight looked like a regular kaon-stop to the beam-line detectors andthe decay products looked like any other kaon stop decay to rest of the detector. As aresult the cuts designed to suppress stopped-kaon background were also e ective against114decay-in- ight kaon events. The kinematic box provided additional suppression due tothe Lorentz boost of the  + resulting from the K+ decaying in  ight.The scattered beam pions were heavily suppressed by the high e ciency  Cerenkovand B4 counters that could distinguish between K+ and  + in the beam. Scattered beampions that were not suppressed by the beam-line counters and delayed coincidence cutscould simulate K+ !  +  when scattering into the  ducial region of the detector if thekinematics of the  + were the same as a signal  + since there were no photons againstwhich to veto.4.6.1 Single-Beam Normalization BranchFor the single-beam background evaluation, the normalization branch bifurcation cut(CUT1) was the delayed coincidence cut DELC3. To avoid examining the signal regionfor the background evaluation in the tight signal region, only the loose cut DELC3 wasinverted and the results of the background evaluation scaled to take into account theadditional rejection of the tight version (DELC6) of the cut. Figure 4.12 shows thenormalization branch and Table 4.13 the resulting event counts.4.6.2 Single-Beam Rejection BranchTo create the sample used to measure the rejection of DELC3, a loose version of theB4DEDX cut was inverted. Inverting this cut demanded that the beam particle was apion by requiring that the energy b4abm atc in the B4 at beam time was less than 1.0MeV. 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 DELC6were the same for scattered beam pions as for decay-in- ight kaons. To preserve statisticsin the single-beam background study, the very loose photon veto PV90 was applied in therejection branch. The Active Degrader and target photon veto subsystems were excluded115from this photon veto as indicated by the notation \noTG, noAD".4.6.3 Single-Beam Background EvaluationThe loose single-beam background bgloose1bm was written asbgloose1bm = Nloose1bmRDELC3  1; (4.15)where Nloose1bm was the number of events at the end of the loose single-beam normalizationbranch and RDELC3 was the rejection of DELC3. These values are summarized in Table4.13. 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 wasbgtight1bm = Ntight1bmRDELC3  1  RDELC3RDELC6 ; (4.16)where Ntight1bm was the number of events at the end of the tight single-beam normalizationbranch, RDELC3 and RDELC6 were the rejections of DELC3 and DELC6, respectively asmeasured on the loose rejection branch. The scaling by the ratios of the rejections ofDELC3 and DELC6 was required because DELC3 was inverted in the tight single-beamnormalization branch in order to not look in the loose signal region and this scaling resultsin a normalization value that would have been measured had DELC6 been inverted. Thisexpression simpli es for the 1/3 data set since the rejection of DELC3 was used for bothDELC3 and DELC6 due to statistical limitations.Table 4.13 summarizes the results of the single-beam background evaluation.1161/3 2/3NormalizationNloose1bm 1 0Ntight1bm 1 0Rejection (DELC3)N1 6483 12850N2 0 2RDELC3 6483 6483 6425 4543Rejection (DELC6)N1 3913 7780N2 0 1RDELC6 3913 3913 7780 7780Backgroundbgloose1bm 0:00046 0:00046 0:00023 0:00023bgtight1bm 0:00046 0:00046 0:00019 0:00019Table 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,the rejection of DELC3 from the 1/3 data set was used when determining bgtight1bm . See Figure4.12 for de nitions of N1 and N2 for RDELC3 and RDELC6. Table reproduced from [56].117VALID TRIG, P2TGCUT, B4CCD, B4TRS, BWTRS, UPVTRS, RVTRS, B4ETCON, TGGEO,TGQUALT, TIMCON, TGTCON, PNN2BOX , TDCUTS DELC3B4DEDX, CKTRS, CKTAIL, CPITRS,CPITAIL, PVCUT , TGCUTSNormalization Branchb4abm atc < 1.0EPITG, TGER, TARGF, TIC, DTGTTP,RTDIF, EPIMAXK, DRP, PHIVTX, EIC,OPSVETO, KIC, TGEDGE, TGZFOOL,TGDEDX, TGLIKE, TGB4, PIGAP,PV90(noTG,noAD)DELCO N1N2Rejection BranchN1bmRDELCO = N1=N2Single-Beam BifurcationsFigure 4.12: Flowchart showing the single-beam bifurcation branches. Cuts in italics refer tonamed groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut wasused for the background evaluation in the loose box and the tight version for the backgroundevaluation in the tight box.118KKpStopDecay in flightKScatteringStopppFigure 4.13: A schematic of the double-beam background processes. The top  gure shows theKK class (kaon-kaon) and the bottom  gure the KP class (kaon-pion). Figure reproduced from[37].4.7 Double-Beam BackgroundSection 4.6 discussed how a single beam pion or decay-in- ight kaon could become single-beam background. The other way that a scattered beam pion or decay-in- ight kaon couldbecome background was if it was missed by the beam-line detectors and an earlier beamkaon satis ed the kaon requirements in the beam-line detectors. For this type of eventto become background, the decay products of the early kaon had to be missed and thescattered beam pion or decay-in- ight kaon had to be missed by the beam-line detectors.This type of background was referred to as double-beam background and was subdividedinto two classes. The  rst class, KK, was an early kaon followed by a decay-in- ight kaon.The second class, KP, was an early kaon followed by a scattered beam pion. Schematicrepresentations of these two double-beam background processes are shown in Figure 4.13.The discrimination between pions and kaons by the  Cerenkov counters allowed the twoclasses of double-beam background, KK and KP, to be evaluated separately.4.7.1 Double-Beam Normalization BranchThe bifurcation cut (CUT1KK) for the KK double-beam normalization branch, used toselect events with an early kaon followed by a decay-in- ight kaon, was the collection of119cuts 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 ascattered beam pion, was the collection of cuts CPITRS, CPITAIL and BWTRS. TheActive Degrader (ADPV) and target photon veto (TGPV) subsystems were excludedfrom the initial application of the photon veto in these normalization branches as indicatedby the notation \noTG, noAD". Both of these subsystems were applied as part of thesecondary bifurcations of these normalization branches.Due to low statistics, each of the normalization branches were bifurcated into twobranches containing uncorrelated sets of cuts. The notation is shown for the KK class, butthis can be replaced with KP for the early kaon followed by a scattered beam pion classof double-beam background. The  rst branch consisted of B4TRS, B4CCD, TGGEO,B4DEDX, the target photon veto TGPV, and the target-quality cuts TGCUTS excludingEPITG, DTGTTP, RTDIF, DRP, EIC, TIC, TGER and KIC. The number of eventsremaining after these cuts were applied was called nKK. The second branch measuredthe rejection rKK of the active degrader ADPV. The  nal normalization NKK was thende ned asNKK = nKKrKK: (4.17)Figure 4.14 shows the schematic representation of the double-beam normalizationbranches and the results for the measurements on this normalization branch are shown inTables 4.14 and 4.15.4.7.2 Double-Beam Rejection BranchFor both the KK and KP classes of double-beam background, the bifurcation cut usedto create the rejection sample (CUT2) was the pair of cuts B4TRS and B4CCD. It wasassumed that the B4 counters used for CUT2 were uncorrelated with the beam wirechambers and  Cerenkov counters used for CUT1 due to the multiple scattering that120occurred in the degraders between the two sets of systems. Inverting CUT2 selected eventsthat had activity in the B4 counters at trs, the time of the charged track in the Range-Stack. The sample of events created by inverting CUT2 was contaminated by decays fromstopped kaons where one of the decay products was directed upstream and detected bythe B4 counters. To correctly measure the rejections of CUT1KK and CUT1KP, a sampleof pure double-beam events was desired so KPIGAP was applied after CUT2 was invertedto remove the contamination due to stopped kaon decays in the rejection sample. The cutKPIGAP required that kaon and pion clusters in the target were spatially disconnected,thus applying this cut selected events that were geometrically inconsistent with the decayof a single stopped K+. The rejection of CUT1KK and CUT1KP for this subset of spatiallydisconnected double-beam events was expected to be consistent with the rejection of thesecuts for all double-beam events since there was no signi cant correlation of these cuts withthe proximity of the double-beam particles.To select the KK class of events in the rejection branch, the  Cerenkov pion-veto cutsCPITRS and CPITAIL were applied. This sample of KK events was further puri ed byrequiring that the energy deposited in the B4 by the second beam particle at Range-Stack time b4ars was consistent with the energy deposited by a kaon (between 1.1 and5.0 MeV).To select KP events in the rejection branch, the  Cerenkov kaon-veto cuts CKTRS andCKTAIL were applied. This sample of KP events was further puri ed by requiring thatb4ars 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 puri ed, the rejections ofCUT1KK and CUT1KP were measured. Figure 4.15 shows the schematic representation ofthe double-beam rejection branches and the results for the measurements on this rejectionbranch are shown in Tables 4.14 and 4.15.121VALID TRIG, P2TGCUT, PNN2BOX , KINCUTS, DELCO , TGQUALT, TIMCON, UPVTRS,RVTRS, TGTCON, B4ETCONPVCUT (noTG,noAD), TDCUTS , EPITG, DTGTTP, RTDIF, DRP, EIC, TIC, TGER, KICCPITRS, CPITAILCKTRS  CKTAIL  BWTRSB4TRS, B4CCD, TGGEO,B4DEDX, TGPV ,TGCUTS (excluding EPITG,DTGTTP, RTDIF, DRP, EIC,TIC, TGER, KIC)ADPVN1N2KK BranchCKTRS, CKTAILCPITRS  CPITAIL  BWTRSB4TRS, B4CCD, TGGEO,B4DEDX, TGPV ,TGCUTS (excluding EPITG,DTGTTP, RTDIF, DRP, EIC,TIC, TGER, KIC)ADPVN3N4KP Branch (early runs only)nKK rKK = N1=N2 nKP rKP = N3=N4Double-Beam Normalization BranchesFigure 4.14: Flowchart showing the double-beam normalization branches Cuts in italics refer tonamed groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut wasused for the background evaluation in the loose box and the tight version for the backgroundevaluation in the tight box.4.7.3 Double-Beam Background EvaluationThe KP background evaluation had an additional complication due to the addition ofthe  Cerenkov pion-veto to the PNN2 trigger part of the way through data collection (seeSection 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 torepresent the entire run. This scaling was done using the inverse of the fraction of the122total KBlive before the trigger change f beforeKBlive ,fscale = 1=f beforeKBlive= 1=0:394= 2:54;(4.18)where the value for f beforeKBlive was taken from [72]. To verify the method of applying thescaling factor to a subset of the run for the KP double-beam background evaluation, twoadditional measurements were performed to show that these backgrounds were consistentbefore and after the trigger change, as detailed in [72]. The  rst was to use only the PNN1trigger to evaluate the KP double-beam background before and after the PNN2 triggerchange. The second was to evaluate the KK double-beam before and after the PNN2trigger change. When scaling each of these measurements to the full run, each of thesemeasurements found the backgrounds to be consistent before and after the PNN2 triggerchange. Figure 4.15 shows the schematic representation of the double-beam bifurcationsand Table 4.15 the results for the measurements on this rejection branch.The KK double-beam background bgKK was written asbgKK = NKKRKK  1; (4.19)where NKK was the number of events left at the end of the KK class double-beam normal-ization branch and RKK was the rejection of CUT1KK (BWTRS, CKTRS and CKTAIL).The KP double-beam background bgKP was written asbgKP = fscale  NKPRKP  1; (4.20)where NKP was the number of events left at the end of the KP class double-beam normal-ization branch, RKP was the rejection of CUT1KP (BWTRS, CPITRS and CPITAIL),123Loose Tight1/3 2/3 1/3 2/3NormalizationnKK 0 0 0 0N1 1363 2699 212 462N2 182 325 22 48rKK 7.49 0.52 8.30 0.43 9.64 1.94 9.62 1.32NKK 0.134 0.134 0.120 0.120 0.104 0.104 0.104 0.104RejectionN5 790 1576 148 269N6 7 4 1 1RKK 113 43 394 197 148 148 269 269Background1/3 2/3bglooseKK 0.00359 0.00359 0.00046 0.00046bgtightKK 0.00212 0.00212 0.00058 0.00058Table 4.14: The summary of the double-beam KK-background evaluation. See Figures 4.14 and4.15 for de nitions of the normalization and rejection quantities, respectively. Table reproducedfrom [56].and fscale was the scaling factor applied so that the background evaluation as measuredbefore the trigger change represented the entire run. These expressions describe both theloose and tight background evaluations for both double-beam classes.124Loose Tight1/3 2/3 1/3 2/3NormalizationnKP 0 0 0 0N3 2289 4435 494 939N4 221 464 36 92rKP 10.36 0.66 9.56 0.42 13.72 2.20 10.21 1.01NKP 0.097 0.097 0.105 0.105 0.073 0.073 0.098 0.098RejectionN7 1179 2467 261 541N8 2 4 0 0RKP 590 417 617 308 261 261 541 541Background1/3 2/3bglooseKP 0.00126 0.00126 0.00065 0.00065bgtightKP 0.00095 0.00095 0.00069 0.00069Table 4.15: The summary of the double-beam KP-background evaluation. See Figures 4.14 and4.15 for de nitions of the normalization and rejection quantities, respectively. Table reproducedfrom [56].Loose Tight1/3 2/3 1/3 2/3bgKK 0.00359 0.00359 0.00046 0.00046 0.00212 0.00212 0.00058 0.00058bgKP 0.00126 0.00126 0.00065 0.00065 0.00095 0.00095 0.00069 0.00069bg2bm(total) 0.00485 0.00380 0.00111 0.00080 0.00307 0.00232 0.00127 0.00090Table 4.16: The summary of the total expected double-beam background.125VALID TRIG, P2TGCUT, PNN2BOX , DELCO , TDCUTS , RSDEDX, RNGMOMTGQUALT, TIMCON, B4DEDX, UPVTRS, RVTRS, B4ETCON, PVCUT (noTG,noAD),CHI5MAX, CHI567, TGER, TGZFOOLB4TRS  B4CCD, KPIGAPCPITRS, CPITAIL1.1 < b4ars < 5.0CKTRS, CKTAIL, BWTRSN5N6KK BranchCKTRS, CKTAILb4ars < 1.1CPITRS, CPITAIL, BWTRSN7N8KP Branch (early runs only)RKK = N5=N6 RKP = N7=N8Double-Beam Rejection BranchesFigure 4.15: Flowchart showing the double-beam rejection branches. Cuts in italics refer tonamed groups of cuts. Cuts denoted with an ‘*’ indicate that the loose version of the cut wasused for the background evaluation in the loose box and the tight version for the backgroundevaluation in the tight box.126ChargeExchangeLKKnlpFigure 4.16: A schematic of the problematic semileptonic charge-exchange background processesK0L ! +e  e and K0L ! +    . Figure reproduced from [37].4.8 Charge Exchange BackgroundThe charge-exchange (CEX) background came from the charge exchange interactionK+n ! K0p. The K0 could be either the shorter lifetime K0S state (0.1 ns) or the longerlifetime K0L state (51 ns). Through oscillation of the K0 to  K0, it is also possible forhyperon (a strange baryon) production to become another charge exchange backgroundsource. This mode is discussed in more detail in Section 4.8.4.The main K0S decay channel that could simulate K+ !  +  was K0S !  +  . Sinceit is such a prompt decay, it was very heavily suppressed by the delayed coincidence cutDELCO and any contributions to background from these K0S decays were determined aspart of the single-beam background estimation (see Section 4.6).The more problematic K0L decay modes were the semileptonic decays K0L !  +e  eand K0L !  +    with branching ratios of 40.56% and 27.05%, respectively [99], wherethese branching ratios are for the sum of the possible charge states for each decay mode.The kinematics of the decay  + from these decay modes fell inside the kinematic signalregion used for this analysis. A schematic representation of these semileptonic decays isshown in Figure 4.16.The two main methods of suppressing the problematic K0L decay modes were (1)detection of the excess energy deposited in the target from the negatively charged lepton,and (2) identi cation of gaps between the incoming kaon and outgoing pion tracks in thetarget. Due to the longer life (51 ns) and non-interacting nature of the K0L, there was agap between where the kaon track ended and where the outgoing (daughter) pion track127began.4.8.1 Method Used to Evaluate CEX BackgroundTo e ectively use the bifurcation method on data, two uncorrelated sets of inverted cutsneed to be able to isolate the background in question. The only cut that was capable ofisolating this background was TARGF, which identi ed gaps greater than one  ber width(0.6 cm) between the kaon and pion  ber clusters. Any cuts that detected the excessenergy deposited in the target suppressed many di erent backgrounds and could not havebeen inverted to isolate only the CEX background.The background evaluation for CEX varied slightly from the bifurcation method usedfor most of the other backgrounds (see Section 3.2.2), but it also relied on geometricarguments to determine the number of background events without examining the signalregion. A schematic representation of this background evaluation is shown in Figure 4.17.The normalization branch was created by applying KPIGAP (a tighter version ofTARGF) using data with NKPIGAP events left at the end of the branch. Using MonteCarlo, the normalization branch was reproduced using the cuts available to Monte Carlowith MKPIGAP events remaining at the end of the branch. A similar branch was created inMonte Carlo using the TARGF cut instead of KPIGAP, with MTARGF events remainingafter TARGF has been applied. Measurement of the number of events NTARGF that wouldremain in the data study if TARGF were applied instead of KPIGAP was forbidden sinceit would have involved examining the signal region. Instead geometric arguments wereused to show that the following ratios were equal for the number of events remaining aftereach of TARGF or KPIGAP were applied in the data and Monte Carlo studies:NTARGFNKPIGAP =MTARGFMKPIGAP : (4.21)To retain statistics in measuring NKPIGAP, two groups of cuts were excluded from this128VALID TRIG, P2TGCUT, KINCUTS, PNN2BOX ,BEAMCUTS, DELCO2, TDCUTS ,PVCUT (noTG)TIC, EIC, TGZFOOL, EPITG, EPIMAXK,EPIONK, TIMKF, KIC, NPITG, TGER, DTGTTP,RTDIF, DRP, TGKTIM, TGEDGE, TGDEDX,TGENR, PIGAP, TGLIKE, TGB4, PHIVTX, TPICSTARGFDELCO , B4EKZ, TGPV,OPSVETO, CCDPULCHI567, CHI5MAX, VERRNG,ANGLI, TGFITALLK,CCDBADTIM, CCDBADFIT,CCD31FIBNEXCLNTARGFKPIGAPNKPIGAPMonte Carlo Setup CutsMonte Carlo Target Quality CutsTARGFDELCO , B4EKZ,TGPV, OPSVETO,CCDPULMEXCLMTARGFKPIGAPMKPIGAPbgCEXData Monte CarloFigure 4.17: Flowchart showing the charge-exchange data and Monte Carlo branches. Cuts initalics refer to named groups of cuts. Cuts denoted with an ‘*’ indicate that the loose versionof the cut was used for the background evaluation in the loose box and the tight version for thebackground evaluation in the tight box. The dashed boxes indicate a branch inaccessible to datadue to being in the signal region. Instead the measurements were performed on an equivalentbranch using Monte Carlo.normalization branch. The  rst group consisted of cuts where CEX events were expectedto 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 eventsremaining after TARGF was applied in the Monte Carlo study:REXCL = MTARGFMEXCL: (4.22)The second group consisted of cuts where the CEX events were expected to behave like129signal events and thus the measured acceptance losses of these cuts (Section 6.1.3) wereused. These cuts were CHI567, CHI5MAX, VERRNG, ANGLI, TGFITALLK, CCD-BADTIM, CCDBADFIT and CCD31FIB, and their combined acceptance will be referredto as Aloss. The background level bgCEX in the signal region was determined by correctingthe calculated number of events NTARGF for the performance of the cuts excluded fromthe normalization branch,bgCEX = AlossREXCL NTARGF: (4.23)Combining the expressions from Equations (4.21), (4.22) and (4.23) the expression for thebackground can be written asbgCEX = NKPIGAP  MEXCLMKPIGAP Aloss: (4.24)4.8.2 The CEX Monte Carlo StudyThe branches of the CEX Monte Carlo study were evaluated using approximately 3:42 108 K0L !  +    events in Monte Carlo. These events were generated with productionpoints, momentum distributions, and corresponding B4 and kaon  ber information fromK0S !  +  data [92]. Figure 4.17 shows the cuts applied to measure the quantitiesMKPIGAP and MEXCL.The measurement of the quantity MEXCL involved simulating the performance of thecuts TGPV, OPSVETO and CCDPUL. The same method as described for the Ke4 back-ground (Section 4.5.2) was used to determine the systematic uncertainty associated withthe possible mismatch between the target- ber energy scale in Monte Carlo and data.The Monte Carlo simulation energy Txtg was the total energy deposited in photon vetoand opposite-side pion  bers and was varied over the range of 0.6 to 1.8 MeV. This energywas used to simulate the performance of TGPV and OPSVETO. The Monte Carlo sim-ulation energy Ehide was the total energy deposited in the kaon  bers from the K0L decay130Loose TightDataNKPIGAP(1=3) 3 1NKPIGAP(2=3) 0 0Monte CarloMEXCL 50+33 10 6+6 2MKPIGAP 4136 3332Acceptance LossAloss 0:687 0:001 0:687 0:001Backgroundbg(1=3) 0:076 0:044+0:058 0:015 0:0038 0:0038+0:0038 0:0013bg(2=3) 0:013 0:013+0:010 0:003 0:0019 0:0019+0:0019 0:0006Table 4.17: The summary of the charge-exchange background evaluation. Values for MEXCL andMKPIGAP taken from [55]. See Figure 4.17 for de nitions of the measured quantities NKPIGAP,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 andthe variations due to the ranges of Txtg and Ehide were used to determine the systematicuncertainty. Table 4.17 shows a summary of results of these measurements made on theKPIGAP and TARGF branches using Monte Carlo.4.8.3 CEX Background EvaluationEquation (4.24) shows the expression used to evaluate the CEX background level usinga combination of data and Monte Carlo. Table 4.17 summarizes the results of this back-ground evaluation, where the results for the 1/3 and 2/3 data sets are scaled to representthe entire data set.1314.8.4 Hyperon BackgroundThis section discusses the method by which hyperon production can become a backgroundprocess and presents arguments as to why the existing charge exchange background esti-mate takes hyperon production into account.The production of hyperons is a multi-stage process. First, a K0 is produced via thecharge exchange interaction K+n ! K0p. Next, the K0 oscillates to  K0 [49], whichcontains a strange quark. The  K0 can then interact with a nucleon (N), producing ahyperon (Y ) and pion: K0 + N ! Y +  : (4.25)There are a number of processes by which a  + having a momentum which overlaps thePNN2 region can be produced:K0 + p !  0 +  + !  0 + (p+ 0 or n +);::: !  + +  0 !  + + ( 0 ) !  + + ((p+  or n 0) );::: !  + +  !  + + (p+  or n 0);K0 + n !  0 +  0 !  0 + ( 0 ) !  + + ((p+  or n 0) );::: !   +  + !   + (p+ 0 or n +);::: !  + +   !  + + (n  );(4.26)where the  and  particles are hyperons.The cross-sections for hyperon production are not well determined. At a K0L momen-tum 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 K0L cross-sections are expected to have only smalldi erences due to isospin di erences and electromagnetic e ects. According to [75], theK0L cross-sections should approximately double in the lower momentum region of 0 to 100132MeV/c. Based on this information and an assumption of an entirely polystyrene target,the mean free path of the  K0 in the target was estimated to be between 37.3 and 75.6 cm,which corresponds to a survival probability of 73% to 85% after the  K0 travels 6 cm inthe target [56]. As a result the KPIGAP cut, which selected events with a gap betweenthe incoming kaon and outgoing pion tracks, should have been e ective at selecting boththe semi-leptonic K0L and hyperon production events. Thus the charge exchange normal-ization branch should have consisted of both these types of events. Similar to the K0case, only the long-lived component of the  K0 results in potential background processesdue to the delayed coincidence cut DELCO removing the short-lived component.The rejection of TGPV, CCDPUL and OPSVETO was estimated using the semi-leptonic K0L decays in simulation. For the hyperon production events, particles such as   , and  were also produced. Due to their relatively short lifetimes, it is expected that theywould deposit their energy in the target and thus be rejected by TGPV, CCDPUL andOPSVETO. Thus the rejection of TGPV, CCDPUL and OPSVETO for charge exchangeevents, determined in Section 4.8.2, should underestimate the rejection power of thesecuts on hyperon events. Based on the above arguments and the large systematic errorassociated with the rejection of TGPV, CCDPUL and OPSVETO, it was decided thatthe estimation of the charge exchange background presented in this section also takeshyperon production into account.1334.9 Other BackgroundsThe previous sections in this chapter discussed the backgrounds which were identi edas being signi cant sources of background for this analysis. Table 4.18 shows a list ofK+ decays which could have contributed to backgrounds in the PNN2 region. This listcontains both decays which were considered in background studies earlier in the chapterand decays which were considered to have negligible contributions to the background dueto the following reasons: Many decay modes were excluded due to the charged products being kinematicallyexcluded from the signal region by having kinematic quantities lower than the rangescovered by the kinematic signal region PNN2BOX. Products from K+ decays-in- ight could have been boosted into the kinematic signal region, but these types ofevents were accounted for in the single- and double-beam studies. Decays having three or more photons and no  + had small branching ratios. Thecombination of photon veto and  + !  + ! e+ particle identi cation suppressedthese backgrounds to a level where they were not considered signi cant sources ofbackground. Decay modes having only an e+ and no  + or  + were highly suppressed by the + !  + ! e+ decay sequence conditions and dE/dx measurements. The remaining decay modes were heavily suppressed by having a combination ofmultiple charged products, photons against which to veto,  + !  + ! e+ particleidenti cation where no decay  + was present, and very small branching ratios.134Background Branching Kinematically Extra Fails NumberRatio Excluded TG Energy  + ID of PhotonsK+ !  + 0.6344 X XK+ !  + 0 0.2092 X 2K+ !  + +  0.0559 X XK+ !  0e+ 0.0498 X 2K+ !  0 + 0.0332 X 2K+ !  + 0 0 0.01757 X 4K+ !  +  0.0062 X 1K+ !  + 0 0.000275 3K+ !  0e+  0.000269 X 3K+ !  + +   0.000104 X X 1K+ !  +3 < 0:0001 3K+ ! e+   < 0:00006 XK+ !  +  e+ 0.0000409 XK+ !  0 +  0.000024 X 3K+ !  0 0e+ 0.000022 X 4K+ ! e+ 0.0000155 X XK+ ! e+  0.0000152 X 1K+ !  +   + 0.000014 XK+ !  + 0 0 0.0000076 X 5K+ !  +   < 0:000006 XK+ !  0 0e+  < 0:000005 X 4K+ !  0 0 0e+ < 0:0000035 X X 6K+ !  +  0.00000110 2K+ !  + e+e 0.000000071 X XK+ !  +  +  < 0:00000041 X XK+ ! e+ e+e 0.000000025 X XK+ ! e+  +  0.000000017 X XTable 4.18: Various K+ decays with their branching ratios [99]. A X in the column \Kine-matically Excluded" indicates that the process is kinematically excluded from the PNN2 signalregion. A X in the column \Extra TG Energy" indicates that the decay is suppressed by cutsthat reject events with additional energy in the target. A X in the column \Fails  + ID" thatthere is no  + in the  nal state and thus is suppressed by the  + !  + ! e+ decay-sequencecuts. The right-most column indicates the number of photons in the  nal state.135Loose 1/3 2/3K 2 TG-scatter 0.537  0.188 +0:069 0:215 0.619  0.150 +0:067 0:100K 2 RS-scatter 0.0220  0.0056 +0:0021 0:0021 0.0303  0.0054 +0:0038 0:0039K 2 0.0514  0.0086 +0:0042 0:0038 0.0757  0.0073 +0:0062 0:0056Muon 0.0246  0.0246 0.0114  0.0114Ke4 0.235  0.118 +0:310 0:166 0.176  0.072 +0:233 0:124Single-Beam 0.00046 0.00046 0.00023 0.00023Double-Beam 0.00485 0.00380 0.00111 0.00080CEX 0.076  0.044 +0:058 0:015 0.013  0.013 +0:010 0:003Total 0.951  0.228 +0:443 0:402 0.927  0.168 +0:320 0:237Table 4.19: The summary of the total expected background in the loose signal region from thevarious components.4.10 Background SummaryThe evaluation of the  nal background levels came from the 2/3 sample with the resultsfrom the 1/3 sample providing a check against large biases introduced when tuning thecuts. The total background level in the loose signal region was estimated to be 0:927 0:168 +0:320 0:237 and in the tight signal region 0:144 0:045 +0:095 0:032, where the  rst uncertaintyis statistical and the second systematic. Recall that the signal region was divided intonine cells (see Section 7.1) and the cell designed to have the highest signal-to-backgroundwas designated the tight signal region. Tables 4.19 and 4.20 summarize the backgroundlevels for both data sets in the loose and tight signal regions, respectively. Results fromboth data sets were scaled to represent the entire data set. The individual and totalbackground levels were statistically consistent between the 1/3 and 2/3 data sets.136Tight 1/3 2/3K 2 TG-scatter 0.106  0.059 +0:076 0:024 0.095  0.041 +0:020 0:016K 2 RS-scatter 0.0043  0.0019 +0:0004 0:0005 0.0069  0.0019 +0:0007 0:0009K 2 0.0121  0.0038 +0:0010 0:0010 0.0188  0.0034 +0:0016 0:0014Muon 0.0055  0.0055 0.0037  0.0037Ke4 0.034  0.034 +0:142 0:026 0.017  0.017 +0:071 0:013Single-Beam 0.00046 0.00046 0.00019 0.00019Double-Beam 0.00212 0.00212 0.00058 0.00058CEX 0.0038  0.0038 +0:0038 0:0013 0.0019  0.0019 +0:0019 0:0006Total 0.177  0.069 +0:223 0:053 0.144  0.045 +0:095 0:032Table 4.20: The summary of the total expected background in the tight signal region from thevarious components.137Chapter 5Validity ChecksThe validity studies detailed in this chapter were used to verify the reliability of theanalysis strategy as motivated in Section 3.2.4.5.1 Outside-the-Box StudiesThe general method used to evaluate the number of events in the outside-the-box regionis described here, with the details of each individual outside-the-box study described inits own respective section. These outside-the-box studies tested the assumption that thebifurcation cuts, used to estimate a given background, were uncorrelated.The outside-the-box studies involved loosening either set of bifurcation cuts used forthe K 2 target-scatter background evaluation: the photon veto or the \EPI" cuts col-lection of cuts: CCDBADFIT, CCDBADTIM, CCDPUL and EPIONK. With the cutsloosened, the K 2 target-scatter, Range-Stack-scatter and K 2 backgrounds were re-evaluated. The remaining backgrounds were scaled using appropriate acceptance lossfactors from the loosened and nominal levels of the cuts.The estimate of the total background level in the outside-the-box region was taken asthe di erence between the total background level evaluated with the loosened cuts and138the total background level in the signal region. The 2/3 data set was used for all outside-the-box background evaluations. The number of events in the outside the box region wasmeasured 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 vetowere 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-levelphoton veto PVPNN1; EPI - Between the pion energy thresholds of 1.25 MeV and 2.5 MeV for the cutsCCDBADFIT, CCDBADTIM, CCDPUL and EPIONK.Note that for all of these studies, the K 2 background contributed only to the uncer-tainty and not the central value of the total outside-the-box background estimates due tothe way the correction to the K 2 target-scatter background due to K 2 contaminationwas applied (see Section 4.1.5).5.1.1 PV1 RegionFor many of the backgrounds, loosening the photon veto increased the background by theratio of the acceptance of PV90 to PV60. These backgrounds were Ke4, CEX, muon andbeam and the scaling factor was given byA(PV90)A(PV60) =0:88550:6199 = 1:428: (5.1)For the remaining backgrounds, the new background level due to loosening of the photonveto was evaluated using PV90 instead of PV60. Table 5.1 shows the backgrounds due toPV60, PV90 and the resulting PV1 outside-the-box background.139Background PV60 PV90 OTB (PV1)K 2-tgscat 0:695 0:150 +0:061 0:094 9:58 0:63 +1:13 1:00 8:89 0:64 +1:23 1:06K 2-rsscat 0:030 0:005 +0:004 0:004 0:143 0:022 +0:018 0:018 0:113 0:023 +0:022 0:022K 2 0:076 0:007 +0:006 0:006 0:357 0:016 +0:029 0:026 0:281 0:018 +0:023 0:020Ke4 0:176 0:072 +0:233 0:124 0:251 0:103 +0:333 0:177 0:075 0:031 +0:100 0:053CEX 0:013 0:013 +0:010 0:003 0:019 0:019 +0:014 0:004 0:006 0:006 +0:004 0:001Muon 0:0114 0:0114 0:0163 0:0163 0:0049 0:00491bm 0:00023 0:00023 0:00033 0:00033 0:00010 0:000102bm-KK 0:00046 0:00046 0:00065 0:00065 0:00020 0:000202bm-KP 0:00065 0:00065 0:00093 0:00093 0:00028 0:00028Total 0:93 0:17 +0:31 0:23 10:02 0:64 +1:53 1:23 9:09 0:65 +1:38 1:15Table 5.1: Summary of the PV1 Outside-the-Box Study. Scaling by a factor ofA(PV90)/A(PV60) = 0.8855/0.6199 = 1.428 was used for the Ke4, CEX, muon and beambackgrounds. The remaining backgrounds were re-evaluated using PV90. For values having twosets of uncertainties, the  rst is statistical and the second systematic. The central value forK 2 was treated as zero as the contribution due to this background was included in the K 2target-scatter value.The total number of background events predicted in the PV1 outside-the-box studywas Npred = 9:09  0:65(stat:) +1:38 1:15(sys:). When the number of events in this regionwas measured directly, 3 events were found. Treating the central value of 9.09 eventsas the mean of a Poisson distribution gave a probability of 0.02 of observing 3 or lessevents. When this probability was re-evaluated at the upper and lower bounds of thetotal uncertainty of Npred, the probability range was [0:01; 0:05].To help determine if this lower than expected number of events was a statisticalanomaly or an indication of anti-correlation between the photon veto and EPI cuts, thisoutside-the-box study was repeated looking at the region between the PV90 and thePNN1-level photon veto PVPNN1.1405.1.2 PV2 RegionThis study examined the outside-the-box region between PV90 and PVPNN1. Scalingwas used to determine the background level in the expanded box for Ke4, CEX, muonand beam. The scaling factor wasA(PVPNN1)A(PV60)  A(PV90)A(PV60) =0:92480:6199  0:88550:6199 = 0:064: (5.2)As with the PV1 outside-the-box study, the K 2-scatter and K 2 backgrounds werere-evaluated for the expanded regions. Table 5.2 shows the backgrounds due to PV90,PVPNN1 and the resulting PV2 outside-the-box background.Background PV90 PVPNN1 OTB (PV2)K 2-tgscat 9:584 0:626 +1:133 1:000 41:63 1:74 +12:09 6:65 32:04 1:85 +13:09 7:78K 2-rsscat 0:143 0:022 +0:018 0:018 0:449 0:067 +0:054 0:056 0:305 0:070 +0:073 0:074K 2 0:357 0:016 +0:029 0:026 1:091 0:028 +0:090 0:079 0:734 0:018 +0:061 0:053Ke4 0:251 0:103 +0:333 0:177 0:266 0:107 +0:348 0:185 0:011 0:005 +0:015 0:008CEX 0:019 0:019 +0:014 0:004 0:019 0:019 +0:015 0:005 0:0008 0:0008 +0:0006 0:0002Muon 0:0163 0:0163 0:0170 0:0170 0:0007 0:00071bm 0:00033 0:00033 0:00034 0:00034 0:00001 0:000012bm-KK 0:00065 0:00065 0:00068 0:00068 0:00003 0:000032bm-KP 0:00093 0:00093 0:00097 0:00097 0:00004 0:00004Total 10:02 0:64 +1:53 1:23 42:38 1:75 +12:60 6:98 32:36 1:85 +13:24 7:92Table 5.2: Summary of the PV2 Outside-the-Box Study. Scaling was used for the Ke4, CEX,muon and beam backgrounds. The remaining backgrounds were re-evaluated in both thePVPNN1 and PV90 regions. For values having two sets of uncertainties, the  rst is statisticaland the second systematic. The central value for K 2 was treated as zero since the contributiondue to this background was included in the K 2 target-scatter value.The total number of background events predicted in the PV2 outside-the-box studywas Npred = 32:36 1:85(stat:) +13:24 7:92 (sys:). When the number of events in this region was141measured directly, 34 events were found. This number of observed events agrees with thepredicted number within statistical uncertainty. Assuming a Poisson distribution of mean32.36 can be approximated by a Gaussian of mean 32.36 and  = p32:36, the probabilityof observing 34 or fewer events was 0.61. When this probability was re-evaluated atthe upper and lower bounds of the total uncertainty of Npred, the probability range was[0:05; 0:98].5.1.3 EPI RegionFor the purpose of this study, the cuts CCDBADFIT, CCDBADTIM, CCDPUL andEPIONK will be called \EPI" cuts. The pion energy threshold for these cuts was loosenedfrom 1.25 MeV (\EPI=1.25") to 2.5 MeV (\EPI=2.5") for this outside-the-box study.Scaling by a factor ofA(EPI = 2:5)A(EPI = 1:25) =0:68620:4576 = 1:4995 (5.3)was used to determine the Ke4, CEX, muon and beam backgrounds in the expanded box.The normalization branch for K 2 was re-evaluated and the K 2-scatter backgroundscompletely 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 EPIoutside-the-box background.The total number of background events predicted by the EPI outside-the-box studywas Npred = 0:79 0:35(stat:) +0:30 0:37(sys:). When the number of events in this region wasmeasured directly, 0 events were found. Treating the central value of 0.79 events as themean of a Poisson distribution gave a probability of 0.45 of observing 0 events. Whenthis probability was re-evaluated at the upper and lower bounds of the total uncertaintyof Npred, the probability range was [0:29; 0:62].142Background EPI=1.25 EPI=2.5 OTB (EPI)K 2-tgscat 0:695 0:150 +0:061 0:094 1:361 0:314 +0:066 0:229 0:666 0:348 +0:160 0:290K 2-rsscat 0:030 0:005 +0:004 0:004 0:057 0:008 +0:008 0:008 0:026 0:009 +0:011 0:011K 2 0:076 0:007 +0:006 0:006 0:121 0:009 +0:010 0:009 0:045 0:012 +0:004 0:003Ke4 0:176 0:072 +0:233 0:124 0:264 0:108 +0:349 0:186 0:088 0:036 +0:116 0:062CEX 0:013 0:013 +0:010 0:003 0:019 0:019 +0:015 0:005 0:006 0:006 +0:005 0:002Muon 0:0114 0:0114 0:0171 0:0171 0:0057 0:00571bm 0:00023 0:00023 0:00034 0:00034 0:00011 0:000112bm-KK 0:00046 0:00046 0:00069 0:00069 0:00023 0:000232bm-KP 0:00065 0:00065 0:00097 0:00097 0:00032 0:00032Total 0:93 0:17 +0:31 0:23 1:72 0:33 +0:45 0:44 0:79 0:35 +0:30 0:37Table 5.3: Summary of the EPI outside-the-box study. Scaling by a factor of A(EPI-2.5)/A(EPI-1.25) = 1.499 was used for the Ke4, CEX, muon and beam backgrounds. The remaining back-grounds were re-evaluated using the loosened CCDPUL, CCDBADFIT, CCDBADTIM andEPIONK cuts. For values having two sets of uncertainties, the  rst is statistical and the sec-ond systematic. The central value for K 2 was treated as zero as the contribution due to thisbackground was included in the K 2 target-scatter value.5.1.4 Consistency of the Outside-the-Box StudiesThe purpose of the outside-the-box was to look for correlations between the bifurcationcuts used for the largest background, K 2 target-scatter. This section discusses the com-bined 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-boxstudies p1 and p2, the combined probability of the two observations is given by [39]p12 = p1p2 (1 lnp1p2): (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 probabilitywas re-evaluated at the upper and lower bounds of the constituent probabilities, the143probability range was [0:01; 0:40]. These probabilities indicated that the observed andpredicted number of events in the outside-the-box studies were generally consistent, givingcon dence in the background estimates and their associated systematic uncertainties.5.2 Single-Cut and Double-Cut Failure StudiesFor the single-cut and double-cut failure studies, the cuts were were divided into 13 groupsbased on correlations. The cuts contained in each group are listed below:1. BOX - This group of cuts, known as PNN2BOX, de ned the kinematic signal regionwith cuts on the  + momentum (ptot), energy (etot) and range in plastic scintillator(rtot).2. PV - This group was a single cut, the standard loose photon veto (PV60) includingneither 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  ducial region cut (TGZFOOL) whichdemanded that the K+-decay vertex was within the target  ducial 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 bers: CCDPUL, CCDBADFIT, CCDBADTIM, CCD31FIB, EPIONK, TIMKF.9. ICGEO - This group of cuts removed events that had beam-particle activity in theI-Counters: TGGEO, KIC14410. TD - This group of cuts was used for particle identi cation in the Range-Stackcounters: PIFLG, ELVETO, TDFOOL, TDNN, RSHEX, RSHEX2.11. KIN - This group of cuts was a collection of cuts based on kinematic constraintsin 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, UP-VTRS, RVTRS.13. OTHER - This group of cuts was a large group of cuts that didn’t  t into anyof the  rst 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 FailuresThe single-cut failure study consisted of examining the signal data for events failing onlyone of the 13 groups of cuts detailed above. Examination of these events provided a clearway to discover  aws in the analysis. Events were classi ed as \true" single-cut failuresif they were classi ed 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 onthe 1/3 signal data. All events, other than those that failed multiple photon veto cuts,were visually inspected. Two events exposing potential analysis  aws were found in thisinitial study, prompting the creation of the CCDBADTIM and early BVL safety cutsas discussed in Section 5.2.3. A third event was found in this initial study, exposing anerror in the way energy de-multiplexing was being handled in the low-gain target  ber145Group 1/3 2/3BOX 41 (0) 114 (0)PV 221 (22) 494 (38)ADPV 0 (0) 2 (2)DELC3 0 (0) 0 (0)B4EKZ 0 (0) 0 (0)TGZ 0 (0) 0 (0)ETG 1 (0) 3 (0)EKAON 3 (2) 3 (3)ICGEO 1 (1) 0 (0)TD 0 (0) 1 (1)KIN 3 (2) 1 (0)BEAM 0 (0) 0 (0)OTHER 3 (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  rst numberis the number of events which fail only cuts the given group. The numbers in parenthesis arethe \true" single-cut failures, events which fail only one of the individual cuts within the givengroup. The numbers shown for the \true" PVCUT group do not include the pass 2 photonveto P2PVCUT since this cut was composed of multiple subsystems: EC, RD and BV. WhenP2PVCUT was included in the \true" single-cut failures, there was only 1 event in this groupfor 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  xed and the subsequentimprovements 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 nal cut positions were set and the 2/3 data set was used to evaluate the  nal backgroundlevels. The results of this study are found in Table 5.4 and show that the rate of single-cutfailures was consistent between the 1/3 and 2/3 data sets. The single-cut failures, otherthan those failing multiple photon veto cuts, were visually inspected. It was concludedfrom the visual inspection of the \true" photon veto failure events, that these events wereK 2 decays where one or two very energetic photons were converted and contained withina single photon detector.146Group 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.1. BOX - 47518 61 2 3 1 176 79 11 353 934 3 262. PV 47518 - 233 39 17 8 93 1328 30 48 179 37 8083. ADPV 61 233 - 1 3 9 2 2 24. DELC3 2 39 1 - 1 25. B4EKZ 3 17 - 16. TGZ 1 8 -7. ETG 176 93 3 - 36 3 1 198. EKAON 79 1328 9 36 - 1 59. ICGEO 11 30 3 - 110. TD 353 48 2 - 121 111. KIN 934 179 2 1 1 1 1 121 - 112. BEAM 3 37 - 113. OTHER 26 808 2 2 19 5 1 1 1 1 -Total 49167 50338 313 45 21 9 331 1458 45 525 1241 41 866Table 5.5: The number of double-cut failures listed by group for the 1/3 sample. All blankentries represent zero events failing that combination of groups. Table was reproduced from[56].5.2.2 Double-Cut FailuresThe double-cut failure study consisted of examining the signal data for events failing cutsin exactly two of the 13 groups of cuts detailed above. The results for the 1/3 data set areshown 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 2/3data sets.5.2.3 Safety CutsTwo events exposing potential analysis  aws were found when examining events from theinitial single-cut failure study on the 1/3 data set. These events prompted the creation oftwo safety cuts: CCDBADTIM and the early BVL. These cuts had minimal acceptanceloss and were devised to target the signatures of the two events.The  rst event was an ADPV single-cut failure. From reconstruction, the event was147Group 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.1. BOX - 94557 141 9 6 2 352 187 23 731 1885 16 952. PV 94557 - 533 96 32 4 182 2557 82 113 372 60 15753. ADPV 141 533 - 6 8 74. DELC3 9 96 - 1 1 2 85. B4EKZ 6 32 1 - 1 26. TGZ 2 4 -7. ETG 352 182 - 56 3 2 1 1 308. EKAON 187 2557 6 1 1 56 - 2 3 4 2 259. ICGEO 23 82 3 2 - 210. TD 731 113 2 3 - 26811. KIN 1885 372 8 2 1 4 268 - 1 412. BEAM 16 60 2 1 2 1 - 213. OTHER 95 1575 7 8 30 25 2 4 2 -Total 98004 100163 695 117 42 6 627 2844 112 1117 2545 84 1748Table 5.6: The number of double-cut failures listed by group for the 2/3 sample. All blankentries represent zero events failing that combination of groups. Table was reproduced from[56].identi ed as a K 2 target-scatter with one of the photons being detected in the ActiveDegrader. Comparisons between the raw target-CCD data and the solutions found bythe target CCD  tter revealed a large second pulse ( 8 MeV) in one of the kaon  bersthat was not found by the  tter. Subsequent studies found common incorrect target CCD tter solutions that could be found when checking for consistency between the pulse timesfound using the target CCD  tter and those expected from target reconstruction. Thecut CCDBADTIM was created to remove these types of events and is discussed in moredetail in Appendix E.The second event was a PV single-cut failure in the BVL. Examination of this eventand data from previous studies [92] revealed that two photons from a single  0 decayconverting in the same BVL element caused the timing of these photon hits to be mis-measured. The \early BVL" cut was added to the photon veto, removing events havinga mean time in the BVL between -5.0 and -2.0 ns, a time di erence between the hits oneach end of less than 4.0 ns, and energy of the hit of greater than 10.0 MeV.1485.3 Background Cross-Contamination StudiesAs discussed in 4.1.1, a number of contamination studies were performed on the K 2target-scatter background. This section details data-driven studies that found negligiblecontamination due to muon and double-beam events in the K 2 target-scatter backgroundestimate. Also found in this section are the details of a study used to determine the upper-limit of K 2 contamination in the K 2 target-scatter normalization branch, supportingthe method used to correct for K 2 contamination in the K 2 target-scatter backgrounddiscussed in Section 4.1.5. Details of the contamination due to K 2 Range-Stack-scatterin the K 2 target-scatter normalization branch were previously discussed in Sections 4.1.2and 4.2.1 and are not discussed further in this section. The subject of contamination dueto Ke4 in the K 2 target-scatter normalization branch, which was argued to be negligiblein Section 4.1.2, is also not discussed further in this section.5.3.1 E ects of Contamination Background EstimatesThe method used to quantify the muon and double-beam contamination in the K 2 target-scatter background is discussed in this section and the calculations detailed in the twosubsequent sections.For the K 2 target-scatter background, estimated using the bifurcation method, thee ects of a generic contamination were as follows. The number of events remaining atthe end of the normalization branch Ntgscat was in ated due to contamination events inaddition the target-scatter events. The e ect on the rejection of CUT1 (RPV60) varieddepending on whether or not the rejection of the contamination events was higher orlower than that of the target-scatter events. If the rejection of the contamination eventswas higher (lower) than that of the target-scatter events, the rejection was overestimated(underestimated). The typical e ect of these mis-measurements of the normalization andrejection was that the K 2 target-scatter background was overestimated, but a su ciently149overestimated rejection could have caused the background to be underestimated.5.3.2 Muon ContaminationThe bifurcation cuts used to evaluate the muon background were used to estimate the con-tamination of muon events in the K 2 target-scatter normalization and rejection branchesand thus the e ect of this contamination on the background evaluation. These cuts werethe loose TD cuts TDLOOSE (CUT1) and the range-momentum consistency cut RNG-MOM (CUT2).Acceptance and Rejection of the Muon Bifurcation CutsTo perform a data-driven estimate of the muon contamination in the K 2 target-scatternormalization and rejection branches, the acceptance of these cuts on K 2 target-scatterevents (A ) and the rejection of these cuts on muon events (R ) had to be evaluated.The rejection of RNGMOM (RRNGMOM) for muon events was measured on the muonbackground normalization branch (Figure 4.6) after setup cuts and the inversion of TD-LOOSE. The rejection of TDLOOSE (RTDCUT02) for muon events was measured in themuon background rejection branch by inverting RNGMOM (Table 4.9). The combinedrejection of these cuts R wasR = RRNGMOM  RTDLOOSE;= (28:29 1:06) (133:0 10:7);= 3764 333:(5.5)The acceptance of these cuts (A ) was measured using a modi ed version of therejection branch used in the K 2 target-scatter background estimate. The modi cationswere that TDLOOSE and RNGMOM were removed from the setup cuts and the kinematicbox was changed from PNN2BOX box to the K 2-peak kinematic box KP2BOX.150After the setup cuts were applied, the 12 classes (2-13) described in Table 4.1 wereapplied and the performance of RNGMOM and TDLOOSE was measured before andafter application of the photon veto PV60 as shown in Table 5.7. Ignoring the classeswith very low statistics, the measured acceptance of these muon bifurcation cuts wereequal before and after the application of the photon veto cut within statistical uncertainty.The acceptance A was taken as the average of the highest and lowest acceptance valuesmeasured before the application of PV60 with the di erence between these extreme valuessetting the bounds for the uncertainty:A = 0:809 0:030: (5.6)CLASS BEFORE PV60 AFTER PV602 293455/363196 = 0.808  0.001 280/343 = 0.816  0.0213 120164/148540 = 0.809  0.001 115/140 = 0.821  0.0324 120568/149124 = 0.809  0.001 109/138 = 0.790  0.0355 364667/451267 = 0.808  0.001 346/428 = 0.808  0.0196 175663/217075 = 0.809  0.001 159/199 = 0.799  0.0287 179337/222017 = 0.808  0.001 166/214 = 0.776  0.0298 27644/34214 = 0.808  0.002 16/25 = 0.640  0.0969 343309/424682 = 0.808  0.001 331/412 = 0.803  0.02010 59863/74471 = 0.804  0.002 58/72 = 0.806  0.04711 5883/7383 = 0.797  0.005 5/5 = 1.000  0.00012 316971/392405 = 0.808  0.001 303/368 = 0.823  0.02013 132619/164163 = 0.808  0.001 127/152 = 0.836  0.030Table 5.7: The acceptance of RNGMOM TDCUT02 measured on K 2-peak events before andafter the application of the photon veto cut PV60 for each of the 12 classes (2-13) from the K 2target-scatter rejection branch.Muon Contamination in the Normalization BranchTo determine the amount of muon contamination in the K 2 target-scatter normalizationbranch, the number of events N left at the end of the normalization branch was treated151as being made up of either muon (N ) or pion (N ) events,N = N + N : (5.7)With the performance known for the muon bifurcation cuts (RNGMOM and TD-LOOSE) with respect to pions (A ) and muons (R ), these cuts were moved to thebottom of the K 2 target-scatter normalization branch. The number of events remainingbefore RNGMOM and TDLOOSE were applied was denoted M. These M events werealso made up of pion or muon events as given byM = N A + N R : (5.8)The amount of muon contamination left at the end of the normalization branch wasrepresented by the quantity f,f = N N ;= A MN  1A R  1:(5.9)Taking the acceptance (A = 0:809 0:030), rejection (R = 3764 333), and measuredvalues 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 theK 2 target-scatter normalization branch was given byN0 = N(1 f);= 1128 34:(5.11)152This was corrected for Range-Stack contamination nrs, which gave the  nal correctednormalization:n0tgscat = N0  nrs= (1128 34) (23:3 3:5)= 1105 34:(5.12)Muon Contamination in the Rejection BranchThe method used to determine the amount of muon contamination in the rejection branchwas very similar to that for the normalization branch except the amount of contaminationhad to be measured before and after the K 2 target-scatter bifurcation cut PV60.The number of events left at the end of the K 2 target-scatter rejection branch (Fig-ure 4.2) was denoted N2 where only the results for CLASS12 were used for this part ofthe study. The number of events before the photon veto was applied was denoted N1 andthe photon veto rejection RPV given byRPV = N1N2(5.13)It was possible to examine the amount of muon contamination both before and afterthe photon veto was applied by treating N1 and N2 as being made up of muon and pionevents as with the normalization branch method:N1 = N 1 + N 1;N2 = N 2 + N 2:(5.14)Again, the performances of the muon bifurcation cuts (RNGMOM and TDLOOSE)with respect to pions (A ) and muons (R ) were used to solve for the fraction of theevents which were contamination muons.153VALID TRIG, P2PSCUT, P2TGCUT, P2TGPVCUT, BEAMCUTS, DELC3, BOXLOOSE,KINCUTS(excluding RNGMOM), CLASS12RNGMOM, TDLOOSEBefore PV60M1N1PV60RNGMOM, TDLOOSEM2N2After PV60Figure 5.1: Flowchart showing the muon contamination study in the K 2 target-scatter rejectionbranch. Cuts in italics refer to named groups of cuts.Figure 5.1 shows a  owchart with the cuts applied and the quantities measured forthis part of the contamination study. To determine the muon contamination remainingbefore the photon veto was applied, the number of events remaining before the applicationof 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 applicationof 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 quantitiesM1 and M2:M1 = N 1A + N 1R ;M2 = N 2A + N 2R :(5.15)The amounts of muon contamination before and after PV60 was applied were represented154by the quantities f1 and f2, respectively,f1 = A M1N1  1A R  1;f2 = A M2N2  1A R  1:(5.16)Using the measured values summarized in Table 5.8, The corrected photon veto rejec-tion was given byR0PV60 = N1(1 f1)N2(1 f2)= 2666 843(5.17)Quantity Before PV60 After PV60Muon bifurcation M1 = 94424 M2 = 652cuts not appliedMuon bifurcation N1 = 52621 N2 = 22cuts appliedf-value f1 = (1:76 0:16) 10 4 f2 = (7:63 1:80) 10 3Corrected value N01 = N1(1 f1) N02 = N2(1 f2)= 52612 229 = 21:8 4:7R0PV60 = N01=N02 2410 518Table 5.8: Table showing the values used to arrive at theK 2 target-scatter photon veto rejectionafter the e ects of muon contamination were removed.155Background Estimate Corrected for Muon ContaminationThe numbers from the contamination study in the normalization and rejection brancheswere used to estimate the background with the e ects of muon contamination removed,bg0tgscat = 32  n0tgscatR0PV  1;= 32  1105 34(2410 518) 1;= 0:688 0:150:(5.18)Since the central value of the nominal K 2 target-scatter background evaluation (0:695 0:150 +0:061 0:094) and this value agree to approximately 1%, the muon contamination in the K 2target-scatter background was considered to be negligible. Note that the values used forthis K 2 target-scatter background comparison were both values without the correctionsfor K 2 contamination applied.5.3.3 Double-Beam ContaminationDue to a lack of acceptance and rejection information for the rejection branch bifurcationcuts (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 denotedRKK and the rejection of CPITRS, CPITAIL and BWTRS for KP-type events will bedenoted 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 groupsof cuts for pion events, denoted AKK and AKP, were taken from the beam acceptancemeasurement (Table 6.4). The sample used for that measurement used K 2 monitortrigger data with cuts applied to ensure the event looks like a single K+ decay with nophotons. These values are summarized in Table 5.9.156KK Branch KP BranchCuts CKTRS CKTAIL BWTRS CPITRS CPITAIL BWTRSAcceptance AKK = 0:8973 0:0002 AKP = 0:9159 0:0002Rejection RKK = 394 197 RKP = 617 308Table 5.9: Acceptance of pion events and rejection of double-beam events for the double-beambifurcation cuts.Double-Beam Contamination in the Normalization BranchThe method for determining the double-beam contamination in the K 2 target-scatternormalization branch was the same as that described for muon contamination, but witha di erent set of cuts for each the KK and KP double-beam contaminations. Since thecontamination due to each of these backgrounds was expected to be very small, the KKcontamination was ignored for the KP contamination study and the KP contaminationignored for the KK contamination study.The following discussion lays out the equations used to determine the amount of KKdouble-beam contamination, but the same equations all apply for the KP double-beamcontamination with the KP notation replacing the KK notation. To determine theamount of KK contamination in the normalization branch, the number of events N leftat the end of the normalization branch was treated as being made up of N pion eventsand NKK KK double-beam events. Written in equation form, this looks like:N = N + NKK: (5.19)With the performance known for the KK double-beam rejection branch bifurcationcuts (CKTRS, CKTAIL and BWTRS) with respect to K 2 target-scatter events (AKK)and KK double-beam events (RKK), these cuts were moved to the bottom of the K 2target-scatter normalization branch. The number of events remaining before these cutswere applied was denoted M. These M events were made up of pion and KK double-beam157events as given byM = N AKK+ RKKNKK: (5.20)The amount of KK contamination left at the end of the normalization branch was rep-resented by the quantity f, given by the expressionf = AKKnN  1AKKRKK  1: (5.21)Table 5.10 shows the values used to determine the fractional contamination due to KKand KP double-beam events in the K 2 target-scatter normalization branch. Note thatthe  nal corrected normalization n0tgscat was corrected for both the given double-beamcontamination and the range-stack scatter component nrs.KK Branch KP BranchM 1186 1160N 1131 1131f-value fKK =  0:00017 0:00009 fKP =  0:00011 0:00005TG-Scatter Normalization corrected for f-value and nrsn0tgscat = N(1 f) nrs n0tgscat = 1108 23 n0tgscat = 1108 34Table 5.10: Correcting for double-beam contamination in the K 2 target-scatter normalizationbranch.Double-Beam Contamination in the Rejection BranchThe method used to determine the double-beam contamination in the K 2 target-scatterrejection branch was also similar to that described for muon contamination with the KKor KP double-beam bifurcation cuts replacing the muon bifurcation cuts. As with thedouble-beam contamination in the normalization branch study, contamination due to onetype of double-beam process (KK or KP) was ignored when studying the other.158The following discussion lays out the equations used to determine the amount of KKdouble-beam contamination, but the same equations all apply for the KP double-beamcontamination with the KP notation replacing the KK notation. As with the muoncontamination in the rejection branch, the amount of contamination was measured beforeand after the photon veto was applied.The amount of KK double-beam contamination, both before and after the photonveto was applied, was examined by treating the N1 events before the application of PV60and the N2 events after the application of PV60 as being made up of double-beam andpion events as was done for the normalization branch:N1 = N 1 + NKK1;N2 = N 2 + NKK2:(5.22)These KK double-beam bifurcation cuts were applied immediately before the end ofthe branch (after the photon veto) which gave M2 events before the KK double-beambifurcation cuts and N2 after. The same was done by applying these KK double-beambifurcation cuts immediately before the photon veto which gave M1 events before theKK double-beam bifurcation cuts and N1 after. The breakdown of pion and double-beam events in the quantities M1 and M2 was as follows:M1 = N 1AKK+ NKK1RKK;M2 = N 2AKK+ NKK2RKK:(5.23)The amount of KK double-beam contamination before and after the photon vetowas applied was represented by the quantities fM and fN, respectively, using the samede nition as introduced for muons (Section 5.3.2). Tables 5.11 and 5.12 show the valuesused to arrive at values for the PV60 rejection after being corrected for each of the double-beam processes.159Quantity Before PV60 After PV60Double-beam bifurcation M1 = 55873 M2 = 25cuts not appliedDouble-beam bifurcation N1 = 52621 N2 = 22cuts appliedf-value f1 = ( 13:3 6:7) 10 5 f2 = (5:48 2:16) 10 5Corrected value N01 = N1(1 f1) N02 = N2(1 f2)= 52628 230 = 22:0 4:7R0PV60(KK) = N01=N02 2392 510Table 5.11: The KK Double-Beam Contamination in the K 2 target-scatter rejection branch.This table shows the values used to arrive at the photon veto rejection after the e ects of KKdouble-beam contamination were removed.Quantity Before PV60 After PV60Double-beam bifurcation M1 = 54563 M2 = 25cuts not appliedDouble-beam bifurcation N1 = 52621 N2 = 22cuts appliedf-value f1 = ( 8:90 4:46) 10 5 f2 = ( 7:18 14:11) 10 5Corrected value N01 = N1(1 f1) N02 = N2(1 f2)= 52626 229 = 22:0 4:7R0PV60(KP) = N01=N02 2392 510Table 5.12: KP Double-Beam Contamination in the K 2 target-scatter rejection branch. Thistable shows the values used to arrive at the photon veto rejection after the e ects of KP double-beam contamination were removed.160Background Estimates Corrected for Double-Beam ContaminationThe numbers from the double-beam contamination studies in the normalization and re-jection branches were used to estimate the backgrounds with the e ects of each of thestudied double-beam processes removed:bg0KK = 32  n0tgscatR0PV60  1;= 32  1108 34(2392 510) 1;= 0:695 0:150;(5.24)andbg0KP = 32  n0tgscatR0PV60  1;= 32  1108 34(2392 510) 1;= 0:695 0:150:(5.25)Since the central values of each of these corrected backgrounds agreed to better than1% with the nominal value of 0:695  0:150, the contamination due to both types ofdouble-beam contamination was considered negligible. Note that the values used for thisK 2 target-scatter background comparison were both values without the corrections forK 2 contamination applied.Double-Beam Contamination Follow-Up StudyA follow-up study was performed to test the assumption that the KP contamination couldbe ignored for the KK contamination study and vice versa. In this study, Equations (5.19)161and (5.20) were replaced with a set of 3 equations,N = N + NKK + NKP;MKK = N AKK+ NKKRKK + NKPA0KK;MKP = N AKP+ NKKA0KP+ NKPRKP:(5.26)For these equations MKK and MKP were the M-values from Table 5.10 for the KK andKP branches, respectively. The de nitions of the combined acceptances A0KK and A0KPwere the same as AKK and AKP except the acceptance of BWTRS was replaced withthe inverse of the rejection for BWTRS for that speci c background, from the double-beam rejection branch. To determine the amount of KK and KP contamination, thevalues NKK and NKP were determined from the above set of 3 equations and the f-valuesdetermined asfKK = NKKN ;fKP = NKPN :(5.27)The results from this follow-up study were consistent with the original double-beamcontamination studies, showing it was reasonable to assume that at the given levels ofcontamination, the KP contamination could be ignored for the KK contamination studyand vice versa.5.3.4 Upper Limit of K 2 ContaminationThis study was performed to estimate the upper limit of K 2 contamination in theK 2 target-scatter normalization branch using K 2 monitor trigger data. This studyinvolved two kinematic regions, the K 2-peak region KP2BOX and the kinematic signalregion PNN2BOX. The L1.1 and L1.2 triggers and all analysis cuts other than the photon162veto PVCUT were applied to the sample, leaving npeak events in the KP2BOX and ntailevents in the PNN2BOX. After applying all analysis cuts, these samples should have beendominated by events where a beam K+ decayed into a  + and any number of photons sincethe photon veto was not applied. Due to the high branching ratio and pion identi cationcuts, the npeak events should have been made up entirely of K 2 events. The ntail eventsshould have been a mixture of K 2-scatter and K 2 events. For the purpose of an upperlimit study, it was assumed that the K 2 contamination in the K 2 monitor trigger datawas large enough that ntail was made up entirely of K 2 events.The fraction of K 2 events in the K 2 target-scatter normalization branch was ex-pressed asg = NgNs + Ng; (5.28)where Ng was the number of K 2 events in the K 2 target-scatter normalization branchand Ns was the number of K 2-scatter events in the K 2 target-scatter normalizationbranch. 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 byf = ntailnpeak;f = 165778;f = 0:00277 0:00069:(5.29)Next, the K 2 target-scatter normalization branch (Figure 4.2) was reproduced usingthe kinematic region KP2BOX instead of PNN2BOX, leaving Np = 122473 events at theend of that normalization branch. Based on the relative rate f determined above, theupper limit on the number of K 2 events (Ng) in the K 2 target-scatter normalizationbranch wasNg = f  Np: (5.30)163Finally, Equation (5.28) was rewritten asg = NgNs + Ng;= f  NpNtgscat;= (0:00277 0:00069) 1224731131 ;= 0:300 0:075;(5.31)which was the upper limit on K 2 contamination in the K 2 target-scatter normalizationbranch. Section 4.1.5 details how this upper limit contributed to the process of correctingthe K 2 target-scatter background for K 2 contamination.164Chapter 6Signal Acceptance and SensitivityTo accurately measure the branching ratio of K+ !  +  , the components of acceptanceof the many cuts applied to the signal data needed to be carefully measured. As withthe rest of this analysis, experimental data were used instead of Monte Carlo simulationwherever possible. The K 2, K 2 and  scatter monitor trigger data were used to measurethe components of acceptance of most of the cuts, with the Monte Carlo simulation beingused to measure primarily those cuts involving decay phase space and trigger e ciencies.Since the monitor trigger data was collected in parallel with the signal data, e ects dueto running conditions such as rate e ects 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 de ned asacc = nN : (6.1)The acceptance is the probability of success of a cut and is thus described by a bino-mial distribution. The uncertainty on the acceptance measurement is then given by the165standard deviation on a measurement of N trials, acc = 1NpN  acc (1 acc);=racc (1 acc)N : (6.2)6.1 Acceptance Factors from K 2 Monitor TriggerEventsWith a single charged track, and no photons or other activity in the detector, K 2 andK+ !  +  events were topologically very similar with respect to event reconstructionand hit patterns in many subsystems. The K 2 monitor trigger events were used tomeasure the acceptance of event reconstruction cuts in the Range-Stack (ARS), eventreconstruction cuts in the target and UTC (Arecon), and the acceptance of the photonveto (APV). These monitor trigger events were also used to measure the beam cuts andthe target-quality cuts that did not involve pion energies (Abeam). Table 6.1 shows thesetup cuts applied to create the samples used for each of these categories of cuts.6.1.1 Range-Stack Tracking AcceptanceTo measure the acceptance ARS (Table 6.2) of the Range-Stack tracking, the setup cutsSetupRS were used to create a sample consisting of good tracks arriving at the Range-Stack. This was done by ensuring successful reconstruction in the target (TRIGGER,ICBIT) and UTC (UTC, UTC QUAL), both of which are independent of Range-Stackreconstruction. To ensure it was a stopped K+ decay, B4DEDX made sure that thebeam particle had an energy consistent with a K+, and a modi ed delayed-coincidencecondition (tIC  tCK > 5ns) ensured that the kaon did not decay in  ight. The modi eddelayed coincidence condition, using the kaon time in the  Cerenkov counters and the hit166K 2 Setup Measured Setup CutsCategories QuantitiesSetupRS ARS TRIGGER, ICBIT, tIC  tCK > 5ns,B4DEDX, UTC, UTC QUALSetuprecon Arecon TRIGGER, ICBIT, tIC  tCK > 5ns,B4DEDX, CPITRS, CPITAIL, CKTRS, CKTAIL,BWTRS, RDTRK, TRKTIM, jtIC  tRSj < 5ns,PVCUT(noBV,noBVL)Setupbeam Abeam TRIGGER, ICBIT, RDTRK, TRKTIM,RDUTM, KM2PBOX, COS3DSetupPV APV Setupbeam, Abeam cuts, stopping layer < 19Table 6.1: The setup cuts applied for the K 2-based acceptance measurements. The notation\Abeam cuts" means that all the cuts whose acceptance were measured to determine the quantityAbeam were applied as setup cuts. KM2PBOX selected events in the momentum range 226MeV/c < ptot < 246 MeV/c. The photon veto \PVCUT(noBV,noBVL)" was applied with theBarrel-Veto and Barrel-Veto Liner subsystems excluded. Table reproduced from [56].time in the IC, was used because the standard delayed coincidence cut DELC3 indirectlyrequired a reconstructed track in the Range-Stack.6.1.2 Target and UTC Reconstruction E ciencyTo measure the reconstruction e ciency Arecon (Table 6.3) of the target and UTC, thesetup cuts Setuprecon were applied to create a sample which required a K+ decay (B4DEDX,tIC  tCK > 5ns) without beam contamination (CPITRS, CPITAIL, CKTRS, CKTAIL,Cut Loose TightEvents Acceptance Events AcceptanceSetupRS 2967140 2967140RD TRK 2967140 1:0000 0:0000 2967140 1:0000 0:0000TRKTIM 2966943 0:9999 0:0000 2966943 0:9999 0:0000ARS 0:9999 0:0000 0:9999 0:0000Table 6.2: The acceptance of the range-stack-reconstruction cuts using K 2 monitor triggerevents. Table reproduced from [56].167Cut Loose TightEvents Acceptance Events AcceptanceSetuprecon 1542443 759060RDUTM 1541571 0:9994 0:0000 758792 0:9996 0:0000TARGET 1541571 1:0000 0:0000 758792 1:0000 0:0000Arecon 0:9994 0:0000 0:9996 0:0000Table 6.3: The acceptance of the target and UTC reconstruction cuts using K 2 monitor triggerevents. Table reproduced from [56].BWTRS). The sample also required that a good track crossed through the UTC to theRange-Stack (jtIC tRSj < 5ns, RD TRK, TRKTIM), and required no photons in order toavoid possible contamination from K 2 . To avoid self-vetoing from events where the  +passed through the Range-Stack and into the Barrel-Veto and Barrel-Veto Liner, thesesubsystems were excluded from the photon veto applied to this sample as indicated bythe notation \PVCUT(noBV,noBVL)".6.1.3 Beam and Target-Quality AcceptanceTo measure the acceptance Abeam (Table 6.4) of the beam and selected target-qualitycuts, the setup cuts Setupbeam were applied to select K+ decays with a single chargedtrack, no photons, no beam contamination and no other activity in the detector. Suc-cessfully reconstructed K 2 events were chosen by restricting the momentum to the K 2-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). Thecuts measured by this sample were ordered to obtain the most meaningful acceptance foreach individual cut. For example, most of the target-quality cuts requires a successfultarget reconstruction, so TGQUALT was applied before any of the other cuts.168Cut Loose TightEvents Acceptance Events AcceptanceSetupbeam 3824854 3824854TGCUT 3741291 0:9782 0:0001 3741291 0:9782 0:0001TGQUALT 3610937 0:9652 0:0001 3610937 0:9652 0:0001NPITG 3610937 1:0000 0:0000 3610937 1:0000 0:0000TIMCON 3605667 0:9985 0:0000 3605667 0:9985 0:0000TGTCON 3566647 0:9892 0:0001 3566647 0:9892 0:0001B4ETCON 3531329 0:9901 0:0001 3531329 0:9901 0:0001DCBIT 3110649 0:8809 0:0002 3110649 0:8809 0:0002DELCO 2665661 0:8569 0:0002 2191189 0:7044 0:0002PSCUT 2528550 0:9486 0:0001 2074552 0:9468 0:0001B4DEDX 2514694 0:9945 0:0001 2063095 0:9945 0:0001BWTRS 2308180 0:9179 0:0002 1892260 0:9172 0:0002CPITRS 2304287 0:9983 0:0000 1889137 0:9983 0:0000CPITAIL 2303213 0:9995 0:0000 1888271 0:9995 0:0000CKTRS 2288540 0:9936 0:0001 1878953 0:9951 0:0001CKTAIL 2251649 0:9839 0:0001 1867769 0:9940 0:0001B4TRS 2193877 0:9743 0:0001 1818255 0:9735 0:0001B4CCD 2164219 0:9865 0:0001 1798933 0:9894 0:0001UPVTRS 2128633 0:9836 0:0001 1770430 0:9842 0:0001RVTRS 2126603 0:9990 0:0000 1768838 0:9991 0:0000TGGEO 2041316 0:9599 0:0001 1696457 0:9591 0:0001B4EKZ 1861055 0:9117 0:0002 1544226 0:9103 0:0002TGZFOOL 1838070 0:9876 0:0001 1525163 0:9877 0:0001TARGF 1778937 0:9678 0:0001 1475963 0:9677 0:0001DTGTTP 1778930 1:0000 0:0000 1475956 1:0000 0:0000RTDIF 1761888 0:9904 0:0001 1461737 0:9904 0:0001TGKTIM 1744527 0:9901 0:0001 1456412 0:9964 0:0001EICCON 1697720 0:9732 0:0001 1417410 0:9732 0:0001TICCON 1697716 1:0000 0:0000 1417407 1:0000 0:0000PIGAP 1682926 0:9913 0:0001 1405081 0:9913 0:0001TGB4 1588984 0:9442 0:0002 1327709 0:9449 0:0002PHIVTX 1541372 0:9700 0:0001 1283527 0:9667 0:0001CCDPUL 694731 0:4507 0:0004 633868 0:4938 0:0004EPIONK 691595 0:9955 0:0001 630732 0:9951 0:0001CCDBADTIM 684667 0:9900 0:0001 624344 0:9899 0:0001CCD31FIB 684658 1:0000 0:0000 624335 1:0000 0:0000TIMKF 628179 0:9175 0:0003 572339 0:9167 0:0004VERRNG 585499 0:9321 0:0003 533386 0:9319 0:0003ANGLI 585135 0:9994 0:0000 533050 0:9994 0:0000ALLKFIT 577756 0:9874 0:0002 526144 0:9870 0:0002TPICS 577003 0:9987 0:0001 525413 0:9986 0:0001KIC 576825 0:9997 0:0000 525245 0:9997 0:0000Abeam 0:1508 0:0002 0:1373 0:0002Table 6.4: The acceptance of the target and beam cuts using K 2 monitor trigger events. Tablereproduced from [56].1696.1.4 Photon Veto AcceptanceTo measure the acceptance APV (Table 6.5) of the photon veto cuts, the setup cutsSetupPV were applied to create a sample of successfully reconstructed K 2 events with-out additional beam particles at decay time. To measure the acceptance of the photonveto, the same conditions as used for measuring Abeam were needed, but an even cleanersample was created by also applying all the cuts measured for Abeam. Additionally, therequirement 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 forthe measurement. Since the K 2 monitor trigger had no online PV requirement, it waspossible to measure the acceptance of online and o ine photon veto cuts with this sample.6.1.5 K 2-Based Acceptance SummaryThe total acceptance AK 2 was the product of the four categories of components of ac-ceptance measured using K 2 monitor trigger events:AK 2 = ARS  Arecon  Abeam  APV: (6.3)Table 6.6 summarizes the results.6.2 Acceptance Factors from  scatter Monitor TriggerEventsThe  scatter monitor trigger selected beam pions that scattered into the active region ofthe detector. These events were similar to K+ !  +  in that the  scatter events coveredthe entire phase space of the kinematic signal region PNN2BOX and they provided asingle  + in the Range-Stack with a continuous stopping layer distribution. The  scatter170Cut Loose TightEvents Acceptance Events AcceptanceSetupPV 62556 56294LHEX 58388 0:9334 0:0010 52530 0:9331 0:0011HEXAFTER 56244 0:9633 0:0008 50621 0:9637 0:0008PVONLINE 53832 0:9571 0:0009 48449 0:9571 0:0009LAY20or21 53413 0:9922 0:0004 48069 0:9922 0:0004STLAY 52910 0:9906 0:0004 47609 0:9904 0:0004RSHEX 50992 0:9637 0:0008 45855 0:9632 0:0009PVCUT 49039 0:9617 0:0009 44092 0:9616 0:0009TGPVCUT 48558 0:9902 0:0005 43661 0:9902 0:0005TGPVTR 48558 1:0000 0:0000 43661 1:0000 0:0000TGPV 47044 0:9688 0:0008 40121 0:9189 0:0013ICPV 46996 0:9990 0:0002 40007 0:9972 0:0003VCPV 46966 0:9994 0:0001 39933 0:9981 0:0002COPV 46707 0:9945 0:0003 39778 0:9961 0:0003MCPV 46702 0:9999 0:0001 39768 0:9997 0:0001ECinner 43191 0:9248 0:0012 31655 0:7960 0:0020ECouter 37652 0:8718 0:0016 25258 0:7979 0:0023EC 2nd 37390 0:9930 0:0004 23395 0:9262 0:0016RSPV 34680 0:9275 0:0013 16681 0:7130 0:0030BVPV 32182 0:9280 0:0014 15318 0:9183 0:0021BVLPV 31668 0:9840 0:0007 15108 0:9863 0:0009ADPV 30132 0:9515 0:0012 14439 0:9557 0:0017EARLYBV 30106 0:9991 0:0002 14433 0:9996 0:0002DPV 30103 0:9999 0:0001 14432 0:9999 0:0001EARLYBV L 30103 1:0000 0:0000 14432 1:0000 0:0000PV60 - - 14120 0:9784 0:0021APVCUT 0:6199 0:0022 0:3234 0:0022APVALL 0:4812 0:0020 0:2508 0:0018Table 6.5: The online and o ine acceptance of the photon veto cuts using K 2 monitor triggerevents. The acceptance APVCUT refers speci cally to the acceptance of the cut PV60 (loose)or PV30 (tight). The component of acceptance APVALL is the product of all components ofacceptance measured in this table. Table reproduced from [56].171Loose TightARS 0:9999 0:0000 0:9999 0:0000Arecon 0:9994 0:0000 0:9996 0:0000Abeam 0:1508 0:0002 0:1373 0:0002APVALL 0:4812 0:0020 0:2508 0:0018AK 2 0:0725 0:0003 0:0344 0:0003Table 6.6: The K 2-based acceptance summary Table reproduced from [56]. scatter Setup Measured Setup CutsCategories QuantitiesSetupBAD STC ABADSTC RD TRK, TRKTIM, STLAY, UTC, RDUTM,PDC, ICBIT, b4abm2 < 1:3MeV,jtpi  tRSj < 5ns, jtIC  tRSj < 5ns, TARGF,DTGTTP, RTDIF, TGQUALT, TGZFOOL,CKTRS, CKTAIL, PVCUT(onlyRS),COS3D, LAYV4, BOXLOOSESetupRS kin ARS kin, AsmallRS kin, SetupBAD STC, BAD STC, TDCUTSAlargeRS kinSetup ! !e ATD1, ATD2 SetupBAD STC, BAD STC, RNGMOM, ZFRF,ZUTOUT, LAYER14, UTCQUAL, EICTable 6.7: The setup cuts applied for the  scatter-based acceptance measurements. The quantityb4abm2 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 stoppingcounter reliability (ABADSTC), kinematic and quality cuts in the UTC and Range-Stack(ARS kin), and particle identi cation in the Range-Stack (A ! !e). Table 6.7 shows thesetup cuts applied for each of these categories of cuts.6.2.1 Range-Stack Stopping Counter ReliabilityThe setup cuts SetupBADSTC were applied to create the sample used to measure theacceptance ABADSTC (Table 6.8) of BAD STC, a cut that removed events when the TD inthe Range-Stack stopping counter was not working properly. This sample required that172Cut Loose TightEvents Acceptance Events AcceptanceSetupBAD STC 153716 74214BAD STC 153474 0:9984 0:0001 74093 0:9984 0:0001ABADSTC 0:9984 0:0001 0:9984 0:0001Table 6.8: The acceptance of BAD STC using  scatter monitor trigger events. Table reproducedfrom [56].the events had a single beam  + 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 sameparticle (jtpi  tRSj < 5ns, jtIC  tRSj < 5ns), the loose kinematic signal region waschosen (BOXLOOSE), and a well reconstructed track was required (the remaining cuts).Additionally, the Range-Stack subsystems of the photon veto \PVCUT(onlyRS)" wereapplied to remove coincident activity in the Range-Stack.6.2.2 Range-Stack-Kinematic AcceptanceTo measure the acceptance ARS kin (Table 6.10) of the kinematic and quality cuts in theUTC and Range-Stack, the setup cuts SetupRS kin were applied to create a sample ofevents that had a single beam  + that scattered in the target and decayed at rest in theRange-Stack. In addition to the setup cuts SetupBADSTC, the  + !  + ! e+ decay-sequence cuts TDCUTS were applied to remove events where the  + decayed in  ight inthe Range-Stack.Good target reconstruction required good classi cation of the kaon and pion  bers,but since the incoming particle was a beam pion and not a kaon, the clustering basedon times and energies was not as reliable as for a stopped-kaon decay. This poor targetreconstruction led to less well-measured momentum (ptot), range (rtot) and energy(etot) of the outgoing particle. To determine the systematic uncertainty associated with173these larger uncertainties in momentum, range and energy, the kinematic signal regioncuts (BOXLOOSE) were all loosened and tightened by one standard deviation to createsmall and large kinematic boxes, respectively.The relative resolutions of the reconstructed  + mass,m = ptot2  etot22 etot ; (6.4)from  scatter and K 2 events were used to determine the size of the shifts to the kinematicsignal region. The distributions of m for these events are shown in Figure 6.1, where  scatter was 13.8 MeV and  K 2 was 8.4 MeV. Taking the mean reconstructed mass fromthe K 2 events as 139.4 MeV, the relative uncertainty in the reconstructed mass of thesetwo types events was determined,p13:82  8:42139:4 = 7:8%: (6.5)The contributions to the resolution of the reconstructed mass from the momentum andenergy were roughly the same, so the uncertainties in each were taken as 7:8%=p2 = 5:5%.Additionally, the range scaled with energy so the uncertainty in rtot was also taken as5.5%. The boundaries of the nominal loose kinematic signal region (BOXLOOSE) wereloosened and tightened by 5.5% to create BOXSMALL and BOXLARGE as shown in Ta-ble 6.9. These loosened and tightened kinematic signal regions were applied in SetupRS kinto measure AsmallRS kin (Table 6.11) and AlargeRS kin (Table 6.12), respectively. The variationsbetween the components of acceptance AsmallRS kin and AlargeRS kin were used to determine thesystematic uncertainty for the measurement ARS kin. The  nal values for ARS kin areshown in Table 6.15.174Figure 6.1: Distributions of the reconstructed  + mass from  scatter (top) and K 2 events(bottom). Figure reproduced from [56].Momentum Energy Rangeptot (MeV/c) etot (MeV) rtot (cm)BOXLOOSE 140 199 60:0 100:5 12 28BOXSMALL 147:7 188:1 63:3 95:0 12:7 26:5BOXLARGE 132:3 209:9 56:7 106:7 11:3 29:5Table 6.9: The \small" and \large" versions of the loose kinematic signal regions as createdby loosening and tightening the nominal PNN2BOX by one standard deviation (5.5%). Tablereproduced from [56].175Cut Loose TightEvents Acceptance Events AcceptanceSetupRS kin 88719 32932UTCQUAL 84373 0:9510 0:0007 31672 0:9617 0:0011RNGMOM 82845 0:9819 0:0005 31161 0:9839 0:0007RSDEDXMAX 80449 0:9711 0:0006 30355 0:9741 0:0009RSDEDXCL 76828 0:9550 0:0007 29048 0:9569 0:0012RSLIKE 76828 1:0000 0:0000 29048 1:0000 0:0000PRRF1 76196 0:9918 0:0003 28841 0:9929 0:0005PRRFZ 73596 0:9659 0:0007 27862 0:9661 0:0011ARS kin 0:8295 0:0013 0:8461 0:0020Table 6.10: The acceptance of the Range-Stack-kinematic cuts using  scatter monitor triggerevents. The \Tight" label indicates the tightening of TDCUTS and PVCUT(onlyRS). Tablereproduced from [56].Cut Loose TightEvents Acceptance Events AcceptanceSetupsmallRS kin 63400 29195UTCQUAL 60350 0:9519 0:0009 27906 0:9558 0:0012RNGMOM 59251 0:9818 0:0005 27396 0:9817 0:0008RSDEDXMAX 57778 0:9751 0:0006 26746 0:9763 0:0009RSDEDXCL 55375 0:9584 0:0008 25685 0:9603 0:0012RSLIKE 55375 1:0000 0:0000 25685 1:0000 0:0000PRRF1 55017 0:9935 0:0003 25548 0:9947 0:0005PRRFZ 53324 0:9692 0:0007 24778 0:9699 0:0011LAYER14 53324 1:0000 0:0000 24778 1:0000 0:0000AsmallRS kin 0:8411 0:0015 0:8487 0:0021Table 6.11: The acceptance of the Range-Stack-kinematic cuts in the small version of the loosekinematic box BOXSMALL using  scatter monitor trigger events. The \Tight" label indicatesthe tightening of TDCUTS and PVCUT(onlyRS). Table reproduced from [56].176Cut Loose TightEvents Acceptance Events AcceptanceSetuplargeRS kin 110317 51078UTCQUAL 104830 0:9503 0:0007 48730 0:9540 0:0009RNGMOM 102909 0:9817 0:0004 47846 0:9819 0:0006RSDEDXMAX 99517 0:9670 0:0006 46347 0:9687 0:0008RSDEDXCL 94726 0:9519 0:0007 44201 0:9537 0:0010RSLIKE 94726 1:0000 0:0000 44201 1:0000 0:0000PRRF1 93737 0:9896 0:0003 43806 0:9911 0:0005PRRFZ 90176 0:9620 0:0006 42205 0:9635 0:0009LAYER14 90176 1:0000 0:0000 42205 1:0000 0:0000AlargeRS kin 0:8174 0:0012 0:8263 0:0017Table 6.12: The acceptance of the Range-Stack-kinematic cuts in the large version of the loosekinematic box BOXLARGE using  scatter monitor trigger events. The \Tight" label indicatesthe tightening of TDCUTS and PVCUT(onlyRS). Table reproduced from [56].6.2.3  + !  + ! e+ Identi cation AcceptanceTo measure the acceptance A ! !e (Tables 6.13 and 6.14) of the Range-Stack particleidenti cation cuts, the setup cuts Setup ! !e were applied to create a sample of eventsthat had a single beam  + that scattered in the target and entered the Range-Stack. Inaddition to the setup cuts SetupBADSTC, the Range-Stack kinematic cuts were appliedto ensure that the track in the Range-Stack was due to a  +. Since the  scatter monitortrigger did not include the online L1.1 and L1.2 triggers, the components of acceptanceof these online cuts were also measured with this sample.Some small correlations existed between some of the cuts measured in ARS kin andA ! !e. A  + accidental along the Range-Stack track could have been rejected by boththe dE=dx condition of RSDEDX and by the  + ! e+ decay requirement of EV5. Theother correlation was between the dependence of the Range-Stack stopping-counter energyin PRRF and the TD-pulse  tting information used by TDNN. To examine the e ects ofthese correlations, the cuts measured for A ! !e were measured without (ATD1, Table6.13) and with (ATD2, Table 6.14) RSDEDX and PRRF applied as setup cuts. The177Cut Loose TightEvents Acceptance Events AcceptanceSetup ! !e 126239 64210PIFLG 104055 0:8243 0:0011 53280 0:8298 0:0015RSHEX2 102123 0:9814 0:0004 52271 0:9811 0:0006LEV1.1 82659 0:8094 0:0012 42382 0:8108 0:0017LEV1.2 69374 0:8393 0:0013 38160 0:9004 0:0015TDCUT 65186 0:9396 0:0009 35907 0:9410 0:0012ELVETO 62425 0:9576 0:0008 34453 0:9595 0:0010TDFOOL 62208 0:9965 0:0002 34343 0:9968 0:0003TDNN 58607 0:9421 0:0009 29016 0:8449 0:0020EV5 58607 1:0000 0:0000 24264 0:8362 0:0022ATD1 0:4643 0:0014 0:3779 0:0019Table 6.13: The acceptance of the  + !  + ! e+ cuts using  scatter monitor trigger events.Table reproduced from [56].measurement for ATD2 included additional acceptance loss due to  + absorption and  +decay-in- ight for which a 1.4% correction factor was estimated using Monte Carlo [55]and applied to ATD2. The value for A ! !e was taken as the average of ATD1 and ATD2,with the systematic uncertainty taken from the di erence.6.2.4  scatter-Based Acceptance SummaryThe total acceptance A scatter was the product of the three categories of components ofacceptance measured using  scatter monitor trigger events:A scatter = ABADSTC  ARS kin  A ! !e: (6.6)Table 6.15 summarizes the results.178Cut Loose TightEvents Acceptance Events AcceptanceSetup ! !e 107124 55113RSDEDX, PRRF 107124 55113PIFLG 90161 0:8417 0:0011 46466 0:8431 0:0016RSHEX2 88616 0:9829 0:0004 45640 0:9822 0:0006L1.1 72545 0:8186 0:0013 37347 0:8183 0:0018L1.2 61913 0:8534 0:0013 34125 0:9137 0:0015TDCUT 58288 0:9415 0:0009 32155 0:9423 0:0013ELVETO 55833 0:9579 0:0008 30859 0:9597 0:0011TDFOOL 55655 0:9968 0:0002 30774 0:9972 0:0003TDNN 52472 0:9428 0:0010 26060 0:8468 0:0021EV5 52472 1:0000 0:0000 21820 0:8373 0:0023ATD2(uncorrected) 0:4898 0:0015 0:3959 0:0021ATD2 0:4967 0:0015 0:4015 0:0021Table 6.14: The acceptance of the  + !  + ! e+ cuts using  scatter monitor trigger events.The acceptance ATD2(uncorrected) was the acceptance before the correction factor of 1.014was applied. This correction factor corrected for  + decay-in- ight and  + absorption in thestopping counter. Table reproduced from [56].Loose TightABADSTC 0.9984 0.0001 0.9984 0.0001ARS kin 0.8295 0.0013 0:012 0.8461 0.0020+0:003 0:020A ! !e 0:4805 0:0015 0:016 0:3897 0:0021 0:012A scat 0.3980 0.0014 0.014 0.3292 0.002 +0:010 0:013Table 6.15: The  scatter-based acceptance summary Table reproduced from [56].179K 2 Setup Measured Setup CutsCategories QuantitiesSetuputc Autc TRIGGER, RD TRK, TRKTIM,STLAY, BAD STCSetupopsveto Aopsveto Setuputc, UTC, RDUTM, PDC,BEAMCUTS, DELCO, KINCUTS,TGCUTS (excluding ATG kin and OPSVETO),TDCUTS, KP2BOXSetupTG kin ATG kin Setupopsveto, OPSVETO, TGPVCUTTable 6.16: The setup cuts applied for the K 2-based acceptance measurements. The nota-tion \Aopsveto cuts" means that all the cuts whose acceptance were measured to determine thequantity Aopsveto were applied as setup cuts. Table reproduced from [56].6.3 Acceptance Factors from K 2 Monitor TriggerEventsThe K 2 monitor trigger events were similar to K+ !  +  in that they both had a singleoutgoing  + coming from the incoming K+, thus K 2 monitor trigger events were used tomeasure the acceptance of cuts requiring good decay-vertex determination in the target.The K 2 monitor trigger events were used to measure the acceptance of the PASS1 UTCcut (AUTC), OPSVETO (Aopsveto), and the target kinematic cuts (ATG kin). Table 6.16shows the setup cuts applied for each of these categories of cuts.6.3.1 UTC AcceptanceTo measure the acceptance Autc (Table 6.17) of the PASS1 UTC cut, the setup cutsSetuputc were applied to create a sample with valid reconstruction of events in the targetand Range-Stack.180Cut Events AcceptanceSetuputc 1502895UTC 1417906 0:9435 0:0002AUTC 0:9435 0:0002Table 6.17: The acceptance of UTC using K 2 monitor trigger events. Table reproduced from[56].Cut Events AcceptanceSetupopsveto 64024OPSVETO 62370 0:9742 0:0006Aopsveto 0:9742 0:0006Table 6.18: The acceptance of OPSVETO using K 2 monitor trigger events. Table reproducedfrom [56].6.3.2 OPSVETO AcceptanceTo measure the acceptance Aopsveto (Table 6.18) of OPSVETO, the setup cuts Setupopsvetowere applied to create a sample of events with valid reconstruction in the target, UTC andRange-Stack, along with the requirement of no secondary beam particles. These setupcuts consisted of all analysis cuts other than the photon veto PVCUT, OPSVETO andthe target quality cuts TGCUTS measured in ATG kin. The group of cuts BEAMCUTSwas applied to remove secondary beam particles. The kinematic box cut KP2BOX andthe rest of the listed cuts were applied to ensure good K 2 decays.6.3.3 Target Kinematic AcceptanceThe cuts measured in the acceptance ATG Kin (Table 6.19) were target-kinematic cutsthat required valid reconstruction in the target, UTC and Range-Stack along with therequirements of no secondary beam particles and the decay product to be a  +. The setupcuts SetupTG kin were applied to create a sample similar to that used for Aopsveto, withthe additional application of OPSVETO and the online target photon veto TGPVCUT181Cut Loose TightEvents Acceptance Events AcceptanceSetupTG kin 61687 37295TGDEDX 61017 0:9891 0:0004 36883 0:9890 0:0005TGER 61000 0:9997 0:0001 36873 0:9997 0:0001TGENR 58984 0:9670 0:0007 35594 0:9653 0:0010TGLIKE1 57931 0:9821 0:0006 34946 0:9818 0:0007TGLIKE2 57005 0:9840 0:0005 34381 0:9838 0:0007EPITG 51086 0:8962 0:0013 30874 0:8980 0:0016EPIMAXK 51086 1:0000 0:0000 30874 1:0000 0:0000TGEDGE 50802 0:9944 0:0003 30715 0:9949 0:0004DRP 50716 0:9983 0:0002 30658 0:9981 0:0003CHI567 44324 0:8740 0:0015 26823 0:8749 0:0019CHI5MAX 44323 1:0000 0:0000 26822 1:0000 0:0000ATG kin 0:7185 0:0018 0:7192 0:0023Table 6.19: The acceptance of the target kinematic cuts using K 2 monitor trigger events. Tablereproduced from [56].to remove additional activity in the target.6.3.4 K 2 -Based Acceptance SummaryThe total acceptance AK 2 was the product of the three categories of components ofacceptance measured using K 2 monitor trigger events:AK 2 = Autc  Aopsveto  ATG kin: (6.7)Table 6.20 summarizes the results.6.4 Acceptance Factors Using Monte CarloThe acceptance losses of the online trigger (Atrigger), and the phase space and solid anglecuts (Abox) were measured using approximately 105 K+ !  +  Monte Carlo events.182Loose TightAutc 0:9435 0:0002 0:9435 0:0002Aopsveto 0:9742 0:0006 0:9735 0:0008ATG kin 0:7185 0:00181 0:7192 0:0023AK 2 0:6604 0:0018 0:6606 0:0023Table 6.20: The K 2-based acceptance summary Table reproduced from [56].The acceptance losses due to pion decay-in- ight and pion nuclear interactions were alsomeasured due to the inclusion of these processes in Monte Carlo. The cut UFATE requiredthat the pion stopped without decay or interaction. The cut USTMED required that thepion stopped in a scintillator counter in the Range-Stack. The cut USTOP HEX requiredthat the o ine reconstructed stopping counter agreed with the true stopping counter.These three cuts used information taken directly from the Monte Carlo event and not thesubsequent reconstruction. The cut SETUP was a requirement that the reconstructedmomentum ptot was less than 300 MeV.6.5 Acceptance SummaryThe total acceptance Atotal of signal events due to all analysis cuts was the product of thecomponents of acceptance from each of the monitor trigger data and Monte Carlo:Atotal = AK 2  A scatter  AK 2  AMC: (6.8)These values are summarized in Table 6.22.6.6 Correction to T 2 Trigger AcceptanceA correction factor was determined to account for the fact that the simulated T 2 triggerin Monte Carlo did not include acceptance losses due to the geometric ( geom) and counter183Cut99999T 2 392273ct  4ct  5ct  6ct 27575Trigger (   (1) or    (2)) 26288Atrigger 0:2629 0:0014SETUP 25793UFATE 22688USTMED 22517USTOP HEX 21743COS3D 20870LAYER14 20838ZFRF 20175ZUTOUT 20148BOXLOOSE 9552Aloosebox 0:3703 0:0030AlooseMC 0:0974 0:0009BOXTIGHT 7758Atightbox 0:3008 0:0029AtightMC 0:0791 0:0009Table 6.21: The acceptance of the online trigger, phase space and solid angle cuts using MonteCarlo. Table reproduced from [56].Loose TightAK 2 0:0725 0:0003 0:0344 0:0003A scatter 0:3980 0:0014 0:014 0:3292 0:0020+0:010 0:013AK 2 0:6604 0:0018 0:6606 0:0023AMC 0:0974 0:0009 0:0791 0:0009Atotal (1:857 0:021 0:065) 10 3 (0:592 0:009+0:018 0:024) 10 3Table 6.22: The total acceptance of all online and o ine cuts. Table reproduced from [56].184( counter) ine ciencies of the T-Counters. The T 2 trigger was a coincidence between the rst two layers of the RS (T-Counter and layer 2) in the same sector. The geometricine ciency was due to tracks passing through the azimuthal gap between two adjacentT-Counters. The counter ine ciency was due to events that did not produce a largeenough signal to pass our threshold. The counter ine ciency was expressed as counter = e kE; (6.9)where k was the number of photoelectrons produced per MeV and E was the mean energydeposited in the T-counters.Both ine ciencies were originally determined in [33] using K 2 and K 2 events fromKB monitor trigger data (see Section 2.7 for trigger de nitions). The KB monitor triggerdata were used since the K 2 and K 2 triggers contain the T 2 trigger as one of theirconditions. Reconstructed events pointing at the T-Counters were checked against theirT 2 online trigger state to determine the ine ciencies. The values corresponding to theK 2 monitor trigger data were used for the T 2 trigger ine ciencies in this analysis dueto their similarity to K+ !  +  PNN2 pions. The geometric ine ciency was found tobe  geom = 0:0286 0:0027.For the counter ine ciency, k values of 1.74 and 1.62 photoelectrons per MeV werefound. The  rst value was for all sectors apart from two that had hardware problems forwhich the second value applied. Monte Carlo K+ !  +  events in the PNN2 kinematicsignal region were used to determine the mean energy deposited in the T-counters as afunction of momentum, giving a counter ine ciency of  counter = 0:0210 0:0027 [37].Subtracting the two ine ciencies from unity gave a T 2 trigger e ciency of T 2 = 0:9505 0:0012stat:  0:0143sys:; (6.10)185where the 1.5% systematic uncertainty accounted for the variation observed when furtherconstraining the z-position requirement of the reconstructed track passing through theT-counters as detailed in [33].6.7 K+ Stopping FractionThe K+ stopping fraction quanti ed the number of beam kaons that decayed in thetarget relative to the number of kaons that satis ed the KB trigger requirement. Thesekaons decayed after the  Cerenkov counter with the daughter satisfying the B4 and targetrequirements of the KB trigger or they deposited some energy in the target and exitedwithout decaying. This fraction was obtained as part of the E949 PNN1 analysis [33] bynormalizing the total kaon exposure to the K 2 branching ratio using K 2 monitor triggerevents. The stopping fraction was found to befs = 0:7740 0:0011: (6.11)6.8 Measurement of the K 2 Branching RatioThe measurement of the K 2 branching ratio served as a consistency check of the accep-tance measurements detailed earlier in this chapter. The K 2 monitor trigger data wereused for this measurement. The online prescale factor for these monitor trigger data was163840 for runs earlier than and including run 48045. For runs after 48045, the prescalefactor was 131072. The total number of stopped K 2 events in the monitor trigger data(NK 2) was measured by applying the cuts shown in Figure 6.2. Acceptance measurementswere performed for all the triggers and cuts applied to make the NK 2 measurement. De-tails of the acceptance measurements used to determine the total acceptance AK 2 arefound in Appendix F.186TRIGGER, BAD STC, RD TRK, TRKTIM, RDUTM, ICBIT, DCBIT,COS3D, UTCQUAL, BEAMCUTS (excluding B4CCD, B4TRS)DELC3, IPIGAP, PV60(noBV,noBVL), KP2BOX, KP2STOPTIC, TARGF, DTGTTP, RTDIF, TGB4, KIC, B4EKZ, TGZFOOLNK 2Figure 6.2: The cuts applied to measure NK 2.All Runs Prescale 163840 Prescale 131072NK 2 144989 2973 141926Prescale Factor 131926 163840 131072KBlive 1:792  1012 0:052  1012 1:740  1012AK 2 0:04833  0:00047 0:04921  0:00084 0:04830  0:00047B(K 2) 0:2213  0:0022 0:1905  0:0051 0:2216  0:0016Table 6.23: The summary of the K 2 branching ratio measurements for all runs and for the twodi erent prescale factors. Details for the constituent components of acceptance making up thetotal acceptance AK 2 are given in Appendix F. Table reproduced from [55].The K 2 branching ratio was measured usingB K+ !  + 0 = PRESCALE  NK 2KBlive  AK 2; (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 forthe two di erent prescale factors. The average measured branching ratio of 0:2213 0:0022overestimates 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 branchingratio was approximately 4.4% as a function of rate as shown in Figure 6.3.187    All runsprescale 163840RateBR(Kp2)0.150.160.170.180.190.20.210.220.230 0.5 1 1.5 2 2.5 3Figure 6.3: The K 2 branching fraction versus rate for all runs (large points) and for runs withprescaler 163840 (small points). Rate is measured in 106 KBlive per second during the spill. Thearrow shows the average rate. Figure modi ed from [55].188Loose Tight SourceAtotal (1:857  0:021  0:065)  10 3  0:592  0:009 +0:018 0:024  10 3 Table 6.22 T 2 0:9505  0:0012  0:0143 Section 6.6fs 0:7740  0:0011 Ref. [92]KBlive 1:7096  1012 Section 2.8SES (4:28  0:43)  10 10 (13:13  1:31)  10 10Table 6.24: The summary of the values used to determine the single event sensitivity. For valueswith two sets of uncertainties, the  rst uncertainty is statistical and the second systematic. Thetotal uncertainties of SES re ect the 10% total uncertainty applied to the acceptance due toconsistency issues when measuring B K+ ! + 0 . Table reproduced from [55].A relative systematic uncertainty of 10% was assigned to the total signal acceptanceto account for the concerns discussed above and the inconsistencies in the measuredbranching ratios for the two di erent prescale factors.6.9 Single-Event SensitivityThe single event sensitivity (SES) was the branching ratio that would have correspondedto one candidate in the absence of background. It was determined bySES 1 = Atotal   T 2  fs  KBlive; (6.13)where Atotal was the total acceptance of all online and o ine cuts,  T 2 was the T 2 triggere ciency, fs was the fraction of beam kaons stopping in the target, and KBlive was thetotal number of K+ decays in the detector for the full data set. Table 6.24 summarizesSES for the loose and tight signal regions.The relative uncertainty associated with the single event sensitivity was conservativelyset to 10% based on the uncertainties in the total acceptance and the discrepancy betweenthe measured K 2 branching fraction and the world average (Section 6.8).189Chapter 7Results7.1 Cell De nitionsThe signal region was divided into nine cells using the loose and tight versions of thekinematic phase space cuts (PNN2BOX), the delayed coincidence cuts (DELC), the  + ! + ! e+ decay-sequence cuts (TDCUTS), and the photon veto (PVCUT). Each of thesecuts was used to divide the signal region into two, with the  rst region being de ned bythe application of the tight version of the cut and the second region being the rest of thesignal region outside of the region de ned 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  rst divided into two major regionsde ned by KIN1 and KIN2. The major region de ned by KIN1 was sub-divided into 8cells based on all possible permutations of the regions de ned by the other three cuts.The cell de ned 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.190Cell Cuts Acc. Total S/BNumber Background1 KIN1  TD1  DC1  PV1 0.314 0:152 0:027 +0:047 0:035 0.842 KIN1  TD1  DC1  PV2 0.287 0:243 0:044 +0:054 0:047 0.473 KIN1  TD1  DC2  PV1 0.031 0:019 0:005 +0:007 0:004 0.664 KIN1  TD1  DC2  PV2 0.028 0:027 0:006 +0:008 0:005 0.425 KIN1  TD2  DC1  PV1 0.073 0:038 0:007 +0:011 0:008 0.786 KIN1  TD2  DC1  PV2 0.066 0:059 0:011 +0:012 0:011 0.457 KIN1  TD2  DC2  PV1 0.007 0:005 0:001 +0:002 0:001 0.578 KIN1  TD2  DC2  PV2 0.006 0:007 0:001 +0:002 0:001 0.359 KIN2 0.188 0:379 0:074 +0:177 0:120 0.20Table 7.1: Relative acceptance and background summary of each cell. The components ofacceptance (\Acc.") were normalized such that the entire signal region was normalized to havean acceptance equal to one. The signal-to-background ratios (\S/B") were calculated assumingB(K+ ! +  ) = 1:73  10 10. The cells denoted with an ‘ ’ were the cells within which thecandidate events were observed. Table modi ed from [56].The ninth cell was the major region de ned by KIN2, and it had the lowest signal-to-background ratio. These cell de nitions are shown in Table 7.1.7.1.1 Acceptance of Each CellFor each of the four cuts used to de ne the cells, the additional acceptance loss whentightening a cut was de ned as the ratio of the acceptance of the tight version of thecut to the acceptance of the loose version of the cut as determined in Chapter 6. Theseacceptance losses are summarized in Table 7.2.The relative acceptance of each cell was then calculated by the product of the ap-propriate acceptance losses for the cuts used to de ne the cell. For example, cell 2 wasde ned by the tight versions of each of PNN2BOX, TDCUTS and DELCO, and by the191Background PNN2BOX TDCUTS DELC PVCUTAcceptance Loss0.812 0.812 0.911 0.522Rejection Above Acceptance LossK 2-tgscat 1.63 - - 2.75K 2-rsscat 1.63 - - 2.75K 2 1.20 - - 2.75Ke4 2.70 - - -CEX - - 6.7 -Muon - 3.08 - -Beam - - 1.0 -Table 7.2: Summary of the additional rejection above the acceptance loss for each backgroundwhen tightening the kinematic phase space cuts (PNN2BOX), the delayed coincidence cuts(DELC), the  + ! + !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 modi ed from [56].region between the loose and tight versions of PVCUT, so the acceptance of that cell wasA(cell 2) = A(KIN1) A(TD1) A(DC1) A(PV2);= 0:812 0:812 0:911 (1 0:522);= 0:287:(7.1)Using these relative components of acceptance, the entire signal region had a relativeacceptance of one. The relative components of acceptance of each of the nine cells aresummarized in Table 7.1.7.1.2 Background Levels in Each CellIn Chapter 4, the bifurcation method was used to estimate the backgrounds in the twosignal regions known as the \loose" and the \tight" signal regions. In the context ofthe discussion of dividing the signal region into cells, the \loose" signal region was the192entire signal region which was divided into the nine cells. The \tight" signal region wasequivalent to cell number 1, which was de ned by the application of the tightened versionsof each the cuts used to de ne the cells. What follows is a discussion of the scaling methodused to estimate the background levels in each of the nine cells. The scaling method wasveri ed by comparing the scaled result in cell number 1 to that found using the bifurcationmethod to directly estimate the background level in the \tight" signal region from Chapter4.Using the scaling method, the total background level in each cell was calculated byestimating the contribution of each type of background to the total in that cell. For eachof the tightened cuts, additional rejection above the acceptance loss was gained for someof the backgrounds, as summarized in Table 7.2.When evaluating the background level in each of the cells, the single- and double-beambackgrounds were treated as a single background called \Beam". This simpli cation waspossible due to the tiny contribution to the total background from the \Beam" back-ground.Tightening the kinematic phase space cuts from BOXLOOSE to BOXTIGHT resultedin rejection above the acceptance loss for the K 2-scatter and Ke4 backgrounds. The lowerbounds of BOXTIGHT were chosen to heavily suppress Ke4 and this additional rejectionwas determined using the Monte Carlo simulation. The additional rejection for the K 2target-scatter and K 2 Range-Stack-scatter backgrounds was determined using the looseand tight K 2 target-scatter normalization branches (Section 4.1.2) while taking intoaccount the additional acceptance loss of the DELCO and TDCUTS between the twobranches. It was assumed that the rejection of the photon veto for K 2-scatter eventswas not correlated with PNN2BOX as discussed in Section 4.1.3. For the remainingbackgrounds, the additional acceptance loss of BOXTIGHT above that of BOXLOOSEwas used.Tightening the  + !  + ! e+ decay-sequence cuts from TDLOOSE to TDTIGHT193resulted in rejection above the acceptance loss only for the muon background. The ad-ditional rejection on the muon background was taken from Table 4.9. For the remainingbackgrounds, the additional acceptance loss of TDLOOSE above that of TDCUTS wasused.Tightening the delayed coincidence cut from DELC3 to DELC6 resulted in rejec-tion above the acceptance loss for the charge exchange background, measured using theMonte Carlo simulation. This cut should also have heavily suppressed the single-beambackground, but the statistical limitations on the background estimate of this very tinybackground resulted in no signi cant rejection above the acceptance loss being measuredwhen tightening DELCO. Since the contributions of the single-beam background were verysmall, it was treated in the same way as the remaining backgrounds, where the changein background level was accounted for by the additional acceptance loss of DELC6 abovethat of DELC3.Tightening the photon veto from PV60 to PV30 resulted in rejection above the ac-ceptance loss for the K 2-scatter and K 2 backgrounds. This additional rejection wasdetermined from Table 4.5. For the remaining backgrounds, the additional acceptanceloss of PV30 above that of PV60 was used.Using the scaling method the total background in cell number 1, which was equivalentto the \tight" signal region, was 0:152  0:027 +0:047 0:035. Using the bifurcation method todirectly estimate the background in the \tight" signal region, the total background was0:1441 0:0448 +0:0952 0:0319. These two values were consistent.7.2 Examination of the Signal RegionExamination of the signal region revealed three candidate events. The kinematics of theseevents, along with the four previous E787/E949 K+ !  +  candidates, are shown inFigure 7.1 and summarized in Table 7.3. Displays of various event parameters for the194ptot etot rtot S/B Ref.(MeV/c) (MeV) (cm)Candidate A 161.4 76.1 17.3 0.20 Cell 9Candidate B 188.4 95.6 24.2 0.47 Cell 2Candidate C 191.3 98.0 26.1 0.42 Cell 495A (E787-PNN1) 218.2 117.7 34.7 59 [11]96B (E787-PNN2) 180.7 86.3 22.1 0.17 [6]98C (E787-PNN1) 213.8 117.1 33.9 8 [11]02A (E949-PNN1) 227.3 128.9 39.2 1.1 [7]Table 7.3: Summary of the  + kinematics and signal-to-background ratio (S/B) for all K+ ! +  candidates. The signal-to-background ratios were calculated assuming B(K+ ! +  ) =1:73  10 10. Table reproduced from [56].ptot etot rtot(MeV/c) (MeV) (cm)Candidate A 204:3 2:2 110:3 3:2 30:6 1:5Candidate B 205:2 2:4 108:1 3:0 30:3 0:9Candidate C 205:3 2:3 108:8 3:3 30:4 0:8All E949 Data 204:9 2:3 108:8 3:0 30:3 0:9Table 7.4: Summary of momentum (ptot), energy (etot) and range (rtot) measurements forK 2 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 foundin cells 2, 4 and 9 as indicated by the ‘ ’ adjacent to these cell numbers in Table 7.1.7.2.1 Consistency of K 2 KinematicsFor each of the runs containing an E949 PNN2 candidate, the momentum, energy andrange of the  + from the K 2 events were compared to those found for the entire data setas shown in Table 7.4. For each of the runs these quantities were found to be consistentwith those found for the entire data set.195Figure 7.1: Energy (etot) vs. range (rtot) of all candidate events passing all other cuts. Thesquares represent the events selected by this analysis. The circles and upward-pointing trianglesrepresent the events selected by the E787 and E949 PNN1 analyses, respectively. The downward-pointing 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+ !  +  events (light gray). Despite thesmaller signal region displayed in this  gure, the PNN1 analyses were 4.2 times more sensitivethan the PNN2 analyses. The events shown near Energy = 108 MeV were K 2 events thatsurvived the photon veto. These events were predominantly events from the PNN1 analyses dueto the higher sensitivity and less stringent photon veto cuts. The light gray points are simulatedK+ ! +  events that would be accepted by the    (1) or    (2) triggers.196-100-80-60-40-20020406080100-100 -80 -60 -40 -20 0 20 40 60 80 100-6-4-20246-6 -4 -2 0 2 4 6Decay Vertex  78 Raw HighDecay Vertex  78 Raw LowDecay Vertex  78 Resid SnDecay Vertex  78 Resid Sn01002003004005006000 20 40 60 80 100 1200204060801001201400 20 40 60 80 100 120-5050 20 40 60 80 100 120-5050 20 40 60 80 100 120End 1 Up (RAW)End 2 Down (RAW)End 1 2P (Raw-Fit)/sResidualEnd 2 2P (Raw-Fit)/sResidual05101520253035400 20 40 60 80 100 12005101520253035400 20 40 60 80 100 120-5050 20 40 60 80 100 120-5050 20 40 60 80 100 120Figure 7.2: Some event parameter displays for Candidate A. Top-left is the end-view showingthe UTC track  t where the circles along the track show the hits in the UTC. The layers hit inthe RS and RSSC are shown outside the UTC. Additional hits shown in green show hits thatwere out of time with the event, in this case they were at least 40 ns before the K+ entered thetarget. Top-right shows a magni cation of the target where the red squares are the K+  bersand the blue ones are the  +  bers. Bottom-left shows the target CCD data for the vertex  berwhere the upper plot is the \high-gain" CCD and the lower plot is the \low-gain". The x-axisshows time in ns and the y-axis shows the pulse-height. The purple dashed line shows the resultof the single-pulse  t for each of the CCD channels. Bottom-right shows the TD data in the  +stopping counter. The x-axis shows time in ns and the y-axis shows the pulse-height. The solidred line shows the result of the  t for the single-pulse assumption and the solid black line showsthe result of the  t for the double-pulse assumption. Figures reproduced from [56].197-100-80-60-40-20020406080100-100 -80 -60 -40 -20 0 20 40 60 80 100-6-4-20246-6 -4 -2 0 2 4 6Decay Vertex 110 Raw HighDecay Vertex 110 Raw LowDecay Vertex 110 Resid SnDecay Vertex 110 Resid Sn0501001502002503003504004500 20 40 60 80 100 1200204060801001200 20 40 60 80 100 120-5050 20 40 60 80 100 120-5050 20 40 60 80 100 120End 1 Up (RAW)End 2 Down (RAW)End 1 2P (Raw-Fit)/sResidualEnd 2 2P (Raw-Fit)/sResidual02550751001251501752002250 20 40 60 80 100 1200204060801001201400 20 40 60 80 100 120-5050 20 40 60 80 100 120-5050 20 40 60 80 100 120Figure 7.3: Some event parameter displays for Candidate B. Top-left is the end-view showingthe UTC track  t where the circles along the track show the hits in the UTC. The layers hit inthe RS and RSSC are shown outside the UTC. Additional hits shown in green show hits thatwere out of time with the event, in this case they were at least 50 ns before or 40 ns after theK+ entered the target. Top-right shows a magni cation of the target where the red squares arethe K+  bers, the blue ones are the  +  bers and the green ones are the   ber. The energiesin each of these   bers were signi cantly below the target photon veto threshold. Bottom-leftshows the target CCD data for the vertex  ber where the upper plot is the \high-gain" CCDand the lower plot is the \low-gain". The x-axis shows time in ns and the y-axis shows thepulse-height. The purple dashed line shows the result of the single-pulse  t for each of the CCDchannels. Bottom-right shows the TD data in the  + stopping counter. The x-axis shows timein ns and the y-axis shows the pulse-height. The solid red line shows the result of the  t for thesingle-pulse assumption and the solid black line shows the result of the  t for the double-pulseassumption. Figures reproduced from [56].198-100-80-60-40-20020406080100-100 -80 -60 -40 -20 0 20 40 60 80 100-6-4-20246-6 -4 -2 0 2 4 6Decay Vertex 131 Raw HighDecay Vertex 131 Raw LowDecay Vertex 131 Resid SnDecay Vertex 131 Resid Sn0501001502002503003504000 20 40 60 80 100 12001020304050607080900 20 40 60 80 100 120-5050 20 40 60 80 100 120-5050 20 40 60 80 100 120End 1 Up (RAW)End 2 Down (RAW)End 1 2P (Raw-Fit)/sResidualEnd 2 2P (Raw-Fit)/sResidual0204060801001201401601800 20 40 60 80 100 12002550751001251501752000 20 40 60 80 100 120-5050 20 40 60 80 100 120-5050 20 40 60 80 100 120Figure 7.4: Some event parameter displays for Candidate C. Top-left is the end-view showingthe UTC track  t where the circles along the track show the hits in the UTC. The layers hit inthe RS and RSSC are shown outside the UTC. Additional hits shown in green show hits thatwere 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 magni cation of the target where the red squares are theK+  bers and the blue ones are the  +  bers. Bottom-left shows the target CCD data for thevertex  ber 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 showsthe result of the single-pulse  t for each of the CCD channels. Bottom-right shows the TD datain the  + 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  t for the single-pulse assumption and the solid blackline shows the result of the  t for the double-pulse assumption. Figures reproduced from [56].1997.2.2 Signal Probability AnalysisTo verify that the properties of each of the candidates were consistent with signal, quanti-ties classi ed into four categories were checked: reconstruction quality, kaon identi cationquality, pion identi cation quality and single particle quality. For each cut, distributionsof the important quantities used to discriminate signal from background were generatedfor signal-like events and the properties of each of the candidate events compared to thesedistributions. For each cut, the same sample that was used to measure the acceptance ofthe cut in Chapter 6 was also used to generate the reference distributions due to signal-like events. The expectation was that an event which was consistent with a signal wouldhave a fairly  at distribution when the probabilities due to each quantity were comparedtogether. Figures 7.5 through 7.13 show the comparisons between the quantities from thecandidate events and the reference distributions. For many of the cuts that target onlya single speci c background, a reference distribution for background-like events is alsoincluded.Reconstruction qualityTo make sure the candidates were of good reconstruction quality the quantities checkedfell into the categories of timing consistency (Figure 7.5), target reconstruction (Figure7.6), and UTC reconstruction (Figure 7.7).Kaon identi cation qualityTo make sure the beam particle in the candidate events was a kaon the quantities checkedinvolved the B4 counters,  Cerenkov hits at beam time and some target quantities (Figure7.8).200Figure 7.5: Quantities related to timing consistency in reconstruction cuts. The abscissa labelcontains the quantity plotted in lowercase and the cut from which that quantity is plotted inuppercase. For the reference signal-like distribution (black), the ordinate axis shows the numberof events per abscissa bin. The measured quantity for each candidate is shown as a colour-codedvertical line: candidate A in red, candidate B in green and candidate C in blue. Note that forsome quantities a candidate has multiple entries and that when multiple quantities fell in thesame abscissa bin, information from only one candidate was displayed. Sample for the signal-likedistribution was from K 2 monitor trigger data.201Figure 7.6: Quantities related to reconstruction cuts in the target. The abscissa label containsthe 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 eventsper abscissa bin. The measured quantity for each candidate is shown as a colour-coded verticalline: candidate A in red, candidate B in green and candidate C in blue. Note that for somequantities a candidate has multiple entries and that when multiple quantities fell in the sameabscissa bin, information from only one candidate was displayed. Sample for the signal-likedistribution was from K 2 monitor trigger data.202Figure 7.7: Quantities related to UTC and Range Stack reconstruction. The abscissa labelcontains the quantity plotted in lowercase and the cut from which that quantity is plotted inuppercase. The reference signal-like (black) and background-like (purple) distributions wereeach normalized to an area equal to one, thus the ordinate axis shows the normalized number ofevents per abscissa bin. The measured quantity for each candidate is shown as a colour-codedvertical line: candidate A in red, candidate B in green and candidate C in blue. Note thatfor some quantities a candidate has multiple entries and that when multiple quantities fell inthe same abscissa bin, information from only one candidate was displayed. The sample forthe reference signal-like distribution was from  scatter monitor trigger data in PNN2BOX andthe sample for the reference background-like distribution was from K 2 monitor trigger data inKP2BOX.203Figure 7.8: Quantities related to target kaon reconstruction. The abscissa label contains thequantity plotted in lowercase and the cut from which that quantity is plotted in uppercase. Forthe reference signal-like distribution (black), the ordinate axis shows the number of events perabscissa 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 quantitiesa 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 wasfrom K 2 monitor trigger data.204Pion identi cation qualityTo make sure the charged track in the candidate events was due to a pion the quantitieschecked fell into the categories of energy deposited in the IC counters (Figure 7.9), Range-Stack kinematics (Figure 7.10), Range-Stack TD variables (Figure 7.11), and kinematicsfrom Monte Carlo  +  events (Figure 7.12).Single particle qualityTo make sure there was only a single decay particle in the candidate events the quantitieschecked were related to beam detectors at track time in the Range-Stack (Figures 7.9 and7.13).Signal-like ProbabilityBased on the reference distributions, the signal-like probability was computed for eachcandidate for each of the quantities as summarized in Figure 7.14. These probabilitieswere determined using a cumulative integral for each of the reference distributions andwere formed such that low probability corresponded to more background-like and highprobability 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-likeside to high probability on the signal-like side. For quantities where extreme values werebackground-like and median values were signal-like, the probability distribution was con-structed to be very low for extreme values and high for median values. For quantitieswhere extreme values were signal-like and median values were background-like, the prob-ability distribution was constructed to be very high for extreme values and low for medianvalues.Figure 7.14 shows the probabilities of each component of reconstruction quality, single205Figure 7.9: Quantities related to pion identi cation, single beam detection and photon veto. Thetop-left plot is related to pion identi cation and the rest of the plots to single beam detectionand photon veto. The abscissa label contains the quantity plotted in lowercase and the cut fromwhich 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 eachcandidate is shown as a colour-coded vertical line: candidate A in red, candidate B in green andcandidate C in blue. Note that for some quantities a candidate has multiple entries and thatwhen multiple quantities fell in the same abscissa bin, information from only one candidate wasdisplayed. Sample for the signal-like distribution was from K 2 monitor trigger data. Over owchannels correspond to no hits or hits that are very far out of time with the quantity beingchecked.206Figure 7.10: Quantities related to pion identi cation by Range Stack kinematics. The abscissalabel contains the quantity plotted in lowercase and the cut from which that quantity is plottedin uppercase. The reference signal-like (black) and background-like (purple) distributions wereeach normalized to an area equal to one, thus the ordinate axis shows the normalized number ofevents per abscissa bin. The measured quantity for each candidate is shown as a colour-codedvertical line: candidate A in red, candidate B in green and candidate C in blue. Note thatfor some quantities a candidate has multiple entries and that when multiple quantities fell inthe same abscissa bin, information from only one candidate was displayed. The sample forthe reference signal-like distribution was from  scatter monitor trigger data in PNN2BOX andthe sample for the reference background-like distribution was from K 2 monitor trigger data inKP2BOX.207Figure 7.11: Quantities related to pion particle identi cation from TD variables. The abscissalabel contains the quantity plotted in lowercase and the cut from which that quantity is plottedin uppercase. The reference signal-like (black) and background-like (purple) distributions wereeach normalized to an area equal to one, thus the ordinate axis shows the normalized number ofevents per abscissa bin. The measured quantity for each candidate is shown as a colour-codedvertical line: candidate A in red, candidate B in green and candidate C in blue. Note thatfor some quantities a candidate has multiple entries and that when multiple quantities fell inthe same abscissa bin, information from only one candidate was displayed. The sample for thereference signal-like distribution was from  scatter monitor trigger data in PNN2BOX and thesample for the reference background-like distribution was from K 2 monitor trigger data sincethese cuts were designed to suppress muon-based backgrounds.208Figure 7.12: Quantities related to pion particle identi cation from kinematic variables. Theabscissa label contains the quantity plotted in lowercase and the cut from which that quantity isplotted in uppercase. For the reference signal-like distribution (black), the ordinate axis showsthe number of events per abscissa bin. The measured quantity for each candidate is shown asa 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 quantitiesfell in the same abscissa bin, information from only one candidate was displayed. Sample for thesignal-like distribution was from UMC K+ ! +  events passing    (1) or    (2) triggers.209Figure 7.13: More quantities related to single beam detection and photon veto. The abscissalabel contains the quantity plotted in lowercase and the cut from which that quantity is plottedin uppercase. For the reference signal-like distribution (black), the ordinate axis shows thenumber of events per abscissa bin. The measured quantity for each candidate is shown as acolour-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 quantitiesfell in the same abscissa bin, information from only one candidate was displayed. Sample forthe signal-like distribution was from K 2 monitor trigger data. Over ow channels correspondto no hits or hits that are very far out of time with the quantity being checked.210beam K+ requirements, decay  + requirements and single decay product requirements.These probabilities, other than the \single particle quality", showed fairly  at distribu-tions for each candidate, which was consistent with the expected signal distributions.The quantities from the \single particle quality" were typically associated with particlevetoes where probabilities being peaked at 1 was expected due to the binary nature ofthe quantities used.7.2.3 Background and Signal Fluctuation ProbabilitiesBased on the signal acceptance and the estimated background in each cell, the probabil-ity that the three observed candidates were due to background only was 3.7% and theprobability that the three observed candidates were due to a combination of backgroundprocesses and SM predicted signal was 5.7%. The probability that all seven E949 andE787 K+ !  +  candidates (Table 7.3) were due to background was 0.1%.7.3 Calculation of K+ !  +  Branching RatioThis section describes the likelihood analysis used to measure the branching ratioB(K+ ! +  ) based on the method described in [60].7.3.1 Branching Ratio Using Maximum LikelihoodAssuming the signal and background processes obeyed Poisson statistics, the probabilityof observing exactly di events in the ith cell having an estimated background level of biand expected signal level of si wasP(dijsi + bi) = e (bi+si) (bi + si)didi! : (7.2)211Figure 7.14: Signal-like probabilities (abscissa) for the three candidates. The ordinate axis showsthe number of events per abscissa bin. The probabilities were calculated for each candidate foreach of the entries appearing in Figures 7.5 through 7.13 and classi ed into the four categoriesde ned in the text: reconstruction quality (\Reconstruction"), single beam K+ requirements(\Kaon ID"), decay + requirements (\Pion ID") and single decay product requirements (\SingleParticle ID"). Candidates are colour coded: candidate A in red (column 1), candidate B in green(column 2), and candidate C in blue (column 3).212For a background only assumption, the probability wasP(dijbi) = e bi bdidi!: (7.3)For m cells, a combination of di events in each of the i cells was given the index  .Given the assumption of signal and background, the probability of observing the exactcombination  wasP (s + b) =mYi=1P(dijsi + bi)=mYi=1e (bi+si) (bi + si)didi! :(7.4)The probability of the same combination given the assumption of background only wasP (b) =mYi=1P(dijbi)=mYi=1e bi bdiidi!:(7.5)A likelihood ratio X , de ned asX  mYi=1Xi; mYi=1e (bi+si) (bi + si)didi!e bi bdiidi!;=mYi=1e si(1 + sibi)di;(7.6)was used to compare the probability of the signal and background assumption to thebackground-only assumption for a combination  .The likelihood ratio for the distribution of the 3 candidates in the 9 cells of this analysis213was denoted Xobs. To determine B(K+ !  +  ), B was varied to maximize Xobs giventhe following relationship between B and the expected signal in each cell si:si = Ai  BSES ; (7.7)where SES was the single event sensitivity in the loose signal region from Table 6.24 andAi was the fraction of the total acceptance of the ith cell from Table 7.1. The value of Bthat maximized Xobs was taken as the branching ratio.Thus far in the discussion of the likelihood analysis, uncertainties in the signal andbackground levels have not been discussed. Based on [60], the uncertainties on the back-ground ( bi) and signal ( si) levels were taken into account using a gaussian convolution,and the likelihood ratio was replaced withX =mYi=1R10R10e a50a51 (s0  si)22 2si +(b0  bi)22 2bia52a532  2si 2bi Xi ds0db0R10R10e a50a51 (s0  si)22 2si +(b0  bi)22 2bia52a532  2si 2bi ds0db0: (7.8)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 onthe candidates observed in this analysis. The next section discusses the method used todetermine the uncertainty of this branching ratio.21400.511.522.533.50 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25Branching Ratio (10-10)Likelihood Ratio (Xobs)00.20.40.60.810 2.5 5 7.5 10 12.5 15 17.5 20 22.5 2568% CL IntervalBranching Ratio (10-10)Confidence Level (CLs)Figure 7.15: Likelihood ratio Xobs (top) and con dence level CLs (bottom) for the three can-didates from the E949-PNN2 analysis. The value of B (7:89  10 10) that maximized Xobs isindicated with a vertical line (top).2157.3.2 Branching Ratio Uncertainty Using Con dence Level In-tervalThe sum of all probabilities Ps+b which had a likelihood ratio X less than or equal toXobs gives the con dence level for the signal plus background assumption,CLs+b  Ps+b(X < Xobs) =XX <Xobs mYi=1e (bi+si) (bi + si)didi!!: (7.9)The con dence level for the background-only assumption wasCLb  Pb(X < Xobs) =XX <Xobs mYi=1e bi bdiidi!!: (7.10)Based on these two con dence levels, the con dence level for the signal was de ned asCLs  CLs+bCLb: (7.11)Again the branching ratio B was treated as a parameter and varied, with the resultingcurve for the quantity CLs shown in the bottom plot of Figure 7.15. The 68% con dencelevel interval (equivalent to   ) was taken as the interval corresponding to the branchingratios at CLs values equal to 0.84 and 0.16. The branching ratio determined by maxi-mizing Xobs was taken as the central value and the resulting branching ratio for the threeobserved candidates in this analysis wasB(K+ !  +  ) =  7:89+9:26 5:10  10 10: (7.12)7.3.3 Combined E787/E949 Branching RatioFor the combined E787 and E949 result (all candidates shown in Table 7.3), the di, bi andsi information was determined for each cell of each analysis and information was treated216as one large data set. The resulting Xobs and CLs plots are shown in Figure 7.16, whichgives a  nal branching ratio due to all K+ !  +  candidates ofB(K+ !  +  ) =  1:73+1:15 1:05  10 10: (7.13)The CLs plot was also used to get the 90% con dence level limit of B(K+ !  +  ) <3:35 10 10. This result was consistent with the standard model prediction of BSM(K+ ! +  ) = (0:85 0:07) 10 10 [25].2170204060801001201401600 1 2 3 4 5 6NewPreviousBranching Ratio (10-10)Likelihood Ratio (Xobs)00.20.40.60.810 1 2 3 4 5 6NewPrevious68% CL IntervalBranching Ratio (10-10)Confidence Level (CLs)Figure 7.16: Likelihood ratio Xobs (top) and con dence level CLs (bottom) for the new andpreviously published [11] combined results for all E787 and E949 analyses. analysis. The valuesof B that maximized Xobs for the respective results are indicated with vertical lines (top).218Figure 7.17: Comparison of the Standard Model (0:85  0:07  10 10), previous PNN1 only(1:47+1:30 0:89  10 10 [11]), and  nal E787/E949 (1:73+1:15 1:05  10 10) branching ratios.219Chapter 8ConclusionBNL experiment E949 was an upgrade of the E787 experiment, designed to measure therare K+ !  +  decay. The search took place in the  + momentum region of 140 to 199MeV/c for this analysis. A blind analysis technique was used to analyze the 1:71 1012stopped K+ decays collected in the E949 detector. This technique was based on keepingthe signal region hidden until all selection criteria for signal had been  nalized, estimatesof all background levels completed, and the acceptance of the signal region determined.The background levels were estimated from data using a bifurcation method, where twouncorrelated cuts with signi cant rejection of a speci c background were used to estimatethe 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 levelsvaried by a factor of 4.The estimated background was 0:927  0:168(stat:) +0:320 0:237(sys:) events. Based on ac-ceptance 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 candi-dates a likelihood analysis was used to determine the branching ratio, B(K+ !  +  ) =(7:89+9:26 5:10)  10 10 at 68% con dence level. Based on the signal acceptance and theestimated background in each cell, the probability that these three events were due to220background only was 3.7% and the probability that the three observed candidates weredue 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 abranching ratio of  1:73+1:15 1:05  10 10 at 68% con dence level. This branching ratio isconsistent with the Standard Model prediction of (0:85 0:07) 10 10. The probabilitythat all observed E787 and E949 candidates were due to background only was 0.1%.E949 has made a signi cant step in con rming the SM’s detailed predictions involvingsecond-order weak interactions, and it has advanced the state-of-the art in the experimen-tal technique of measuring small e ects. However, given the level of statistical uncertaintyassociated with this B(K+ !  +  ) measurement, it was not possible to make any de ni-tive conclusions regarding physics beyond the SM. It was very unfortunate that an execu-tive decision by the Department of Energy (DOE) resulted in the E949 experiment beingterminated after only 20% of the approved beam time as the potential for strong hints ofphysics beyond the SM were within reach. Future measurements of B(K+ !  +  ) arebeing planned. The NA62 experiment, currently in preparation at CERN, aims to collect65 K+ !  +  events over two years of data-taking using a decay-in- ight experiment[10]. A letter of intent for a decay-at-rest experiment similar to E949 was presented atJ-PARC [69]. These experiments should be able to make de nitive conclusions regardingthe presence or absence of new physics contributions to this decay if the branching ratiois comparable to the central value found by E949.221Bibliography[1] K. Abe et al. (Belle Collaboration), Phys. Rev. Lett. 87 091802 (2001).[2] S. Adler et al. (E787 Collaboration), Phys. Rev. Lett. 76 1421 (1996).[3] S. Adler et al. (E787 Collaboration), Phys. Rev. Lett. 79 2204 (1996).[4] S. Adler et al. (E787 Collaboration), Phys. Rev. Lett. 84 3768 (2000).[5] S. Adler et al. (E787 Collaboration), Phys. Rev. Lett. 88 041803 (2002).[6] S. Adler et al. (E787 Collaboration), Phys. Lett. B 537 211 (2002).[7] S. Adler et al. (E787 Collaboration), Phys. Rev. D 70 037102 (2004).[8] S. Adler et al. (E949 Collaboration), Phys. Rev. D 77 052003 (2008).[9] J.K. Ahn et al. (E391a Collaboration), Phys. Rev. Lett. 100, 201802 (2008).[10] F. Ambrosino et al. (NA62 Collaboration), \Proposal to Measure the Rare DecayK+ !  +  at the CERN SPS", (2005). http://na48.web.cern.ch/NA48/NA48-3/Documents/P326.pdf[11] V.V. Anisimovsky et al. (E949 Collaboration), Phys. Rev. Lett. 93 031801 (2004).[12] A.V. Artamonov et al. (E949 Collaboration), (2008), arXiv:0808.2459v1 [hep-ex].[13] Y. Asano et al., Phys. Lett. B 107, 159 (1981).[14] M. Atiya et al., Nucl. Instrum. Methods Phys. Res., Sect. A 279, 180 (1989).[15] M. Atiya et al. (E787 Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A321, 129 (1992).[16] M. Atiya et al. (E787 Collaboration), Phys. Rev. D 48, R1 (1993).[17] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 87 091801 (2001).[18] B. Basseleck et al. (E949 Collaboration), BNL Report No. BNL-67247, TRIUMFReport No. TRI-PP-00-06 (1999). http://www.phy.bnl.gov/e949/[19] J. B. Birks, Proc. Phys. Soc. A64, 874 (1951).222[20] B. Bhuyan, \Search for the rare decay K+ !  +  ", Ph.D. thesis (2003).[21] B. Bhuyan, \Analysis of 1997 data in the pnn2 region", E787 Technical Note 391(2003), Unpublished.[22] E.W. Blackmore et al., Nucl. Instr. Meth. A404, 295 (1998).[23] M. Blanke et al., J. High Energy Phys. 0701, 066 (2007).[24] H. Brafman et al., IEEE Trans. Nucl. Sci. 32, 336 (1985).[25] J. Brod & M. Gorbahn (2008), arXiv:0805.4119 [hep-ph].[26] D.A. Bryman et al., Nucl. Instrum. Methods Phys. Res., Sect. A 396, 394 (1997).[27] A.J. Buras, M.E. Lautenbacher, and G. Ostermaier, Phys. Rev. D 50, 3433 (1994).[28] A.J. Buras, F. Schwab, and S. Uhlig (2007), arXiv:hep-ph/0405132v3.[29] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963).[30] G.D. Cable et al., Phys. Rev. D 8. 3807 (1973).[31] U. Camenrini et al., Phys. Rev. Lett. 23. 326, (1969).[32] C. Caso et al., European Physical Journal C3, 1 (1998).[33] S. Chen et al., \2002    (1) Data Analysis", E949 Technical Notes K-034 and K-038 (2003), Unpublished.[34] H. C. Cheng and I. Low, JHEP 0309, 051 (2003).[35] I.H. Chiang et al., IEEE Trans. Nucl. Sci. 42, 394 (1995).[36] J.H. Christienson et al., Phys. Rev. Lett. 13 138 (1964).[37] I. Christidi, \Search for the rare decay K+ !  +  with p + < 199 MeV/c", Ph.D.thesis (2006).[38] M. Diwan, J. Frank and V. Jain, \An algorithm to  t pulses in target CCD data",E787 Technical Note 374 (1999), Unpublished.[39] W.T. Eadie, D. Drijard, F.E. James, M. Roos and B. Sadoulet, Statistical Methodsin Experimental Physics, North-Holland (1971), pp.292-283.[40] J. Doornbos et al., Nucl. Instr. Meth. A444, 546 (2000).[41] Dow Corning Corporation, South Saginaw Road, Midland, Michigan 48686, UnitedStates.[42] The reference manual can be found at http://ppd.fnal.gov/elec/dyc3.223[43] Electron Tubes Inc., 100 Forge Way, Unit F, Rockaway, New Jersey 07866, UnitedStates.[44] The original  gure can be found at http://www.phy.bnl.gov/e949/detector/photon veto/.[45] J.S. Frank, \Study of the target CCD pulse  tting analysis", E949 Technical NoteK-045 (2005), Unpublished.[46] Results from bench tests of the target can be found athttp://www.phy.bnl.gov/e949/detector/tracking/tt/general/general.html.[47] S.L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D1 1585 (1970).[48] Personal communication with Martin Gorbahn.[49] D. Gri ths, \Introduction to Elementary Particles", John Wiley & Sons Inc. (1987).[50] Y. Grossman and Y. Nir, Phys. Lett. B398, 163 (1997).[51] J.S. Hagelin and L.S. Littenberg, Prog. Part. Nucl. Phys 23, 1 (1989).[52] Hamamatsu Photonics K.K., 312-5, Shimokanzo, Iwata City, Shizuoka Pref., 438-0193, Japan.[53] G. Isidori, F. Mescia, and C. Smith, Nucl. Phys. B718, 319 (2005).[54] G. Isidori et al., J. High Energy Phys. 0608, 064 (2006).[55] J. Ives et al., \Analysis of the 1/3 E949 pnn2 data", E949 Technical Note K-073 (6December 2007), Unpublished. Note that this work is cited to attribute to a co-authorthe reproduced results or  gure.[56] J. Ives et al., \Analysis of the 2/3 E949 pnn2 data", E949 Technical Note K-074 (24April 2008), Unpublished. Note that this work is cited to attribute to a co-authorthe reproduced results or  gure.[57] D. Ja e, \FITPI: Triple-pulse  tting and other modi cations", E949 Technical NoteK-029 (2002), Unpublished.[58] V. Jain, \Simulation of elastic scatters of  + in the target from Kp2 decays", E787Technical Note 375 (1999), Unpublished.[59] C. Jarlskog and R.Stora, Phys. Lett. B208, 268 (1988).[60] T. Junk, Nucl. Instr. Meth. Phys. Res. A434, 435 (1999).[61] K. Mizouchi, \Experimental Search for the Decay  0 !   ", Ph.D. thesis (2006).[62] M. Aoki et al., \An experiment to measure the branching ratio B(K+ !  +  ",E949 Technical Note K-001 (1998), Unpublished.224[63] S. Kettel, T. Yoshioka, \Trigger de nitions - (2002)", E949 Technical Note K-030(2002), Unpublished.[64] N. Khovansky et al., Nucl. Instr. Meth. A351, 317 (1994).[65] J.H. Klems, R.H. Hiledebrand, and R. Steining, Phys. Rev. D 4, 66 (1971).[66] M. Kobayashi & T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).[67] T. Komatsubara, \Brief Summary of The Status of Upgraded-E787 in 1995", E787Technical Note 305 (1995), Unpublished.[68] T.K. Komatsubara et al. (E787 Collaboration), Nucl. Instr. Meth. A404, 315 (1998).[69] T.K. Komatsubara, T. Nakano and T. Nomura, \Letter of Intent for Study of theRare Decay K+ !  +  with Stopped Kaon Beam at J-PARC" (2002). http://www-ps.kek.jp/jhf-np/LOIlist/pdf/L04.pdf[70] LeCroy Corporation, 700 Chestnut Ridge Road, Chestnut Ridge, NY 10977-6499,United States.[71] B. Lewis, \A new Target Reconstruction and Target Scatter Algorithm", E949 Tech-nical Note K-063 (2006), Unpublished.[72] B. Lewis, \The Quest for the Rare Decay K+ !  +  ", Ph.D. thesis (2008).[73] X. Li et al., \Improvement to the Range Stack Straw Chamber Electronics", E949Technical Note K-023 (2002), Unpublished.[74] D. Ljung and D. Cline, Phys. Rev. D 8, 1307 (1973).[75] A.D. Martin, Nucl. Phys. B179, 33 (1981).[76] W.J. Marciano and Z. Pars, Phys. Rev. D 53, R1 (1996).[77] R.A. McPherson, "Chasing the Rare Decay K+ !  +  ", Ph.D. Thesis (1995).[78] F. Mescia and C. Smith, Phys. Rev. D 76 034017 (2007)[79] P. Meyers, \A modi ed Version of the UMC Multiple Scattering Routine MSCAT1",E787 Technical Note 77 (1985), Unpublished.[80] The reference manual can be found at http://midas.triumf.ca.[81] The reference manual can be found atwwwinfo.cern.ch/asdoc/minuit/minmain.html.[82] O. Mineev et al. (E949 Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A494, 362 (2002).[83] K. Mizouchi, \Experimental Search for the Decay  0 !   ", Ph.D. thesis (2006).225[84] T. Morii, C.S. Lim, and S.M. Mukherjee, \The Physics of the Standard Model andBeyond", World Scienti c Co. Pte. Ltd. (2004).[85] Motorola Inc., 1303 E. Algonquin Road, Schaumburg, Illinois 60196, United States.[86] W.R. Nelson et al., "The EGS4 Code System", SLAC Report No. 265 (1985).[87] The reference manual can be found at wwwinfo.cern.ch/asd/paw/reference manual/index.html.[88] C. Promberger, S. Schatt, and F. Schwab, Phys. Rev. D 75, 115007 (2007).[89] Compagnie de Saint-Gobain, Les Miroirs, 18, avenue d’Alsace, 92400 Courbevoie,France.[90] A.D. Sakharov, JETP Lett 5 (1967) 24.[91] G.A. Sayer et al., Physical Review 169, 1045 (1968).[92] T. Sekiguchi, \Measurement of the K+ !  +  Branching Ratio", Ph.D. thesis(2004).[93] SGI, 1140 E. Arques Ave., Sunnyvale, CA 94085, United States.[94] C. Smith (2007), arXiv:0710.2883v2 [hep-ph].[95] A.J. Stevens, \Nuclear Interactions in CH revisited", E787 Technical Note 140(1987), Unpublished.[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).226Appendix AE787 to E949 UpgradesThe E949 experiment was a successor to experiment E787 at BNL [15], whose data takingwas completed in 1998. The experiment was quite similar to E787, with the followingupgrades to the beam and detector [18], and to the trigger and data acquisition [100].A.1 Beam UpgradesThe proton intensity at the production target was increased by a factor of two, resulting inabout twice the beam rate as E787. The mean K+ momentum was 710 MeV/c throughoutE949 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 UpgradesThe detector upgrades were as follows: The two outer-most layers of the Range-Stack were replaced by the Barrel VetoLiner (BVL), adding 2.3 radiation lengths to the photon veto. The BVL was alead/scintillator sandwich sampling calorimeter like the Barrel Veto.227 The Active Degrader (AD), Upstream Photon Veto (UPV), and Downstream PhotonVeto (DPV) were installed to add photon veto coverage at small polar angle ( ). 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  ashers. Thisallowed changes in the gains of the PMTs to be tracked on a spill by spill basis andeven within a spill, improving energy calibration. The UTC read-out electronics were upgraded to operate in the higher rate environ-ment. The RSSC read-out electronics were upgraded to improve resolution of z-positionmeasurements.A.3 Trigger and DAQ UpgradesThe 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 onlinephoton rejection [100].228Appendix BData Acquisition, Storage, andProcessingThe E949 experiment collected its physics data for 12 weeks from March through June of2002.B.1 Data AcquisitionDetector analog and discriminated signals were digitized by ADC, TDC, TD and CCDsystems. The ADCs and TDCs were commercial units, and the TD and CCD waveformdigitizers (WFD) were custom built. Data from a triggered event was digitized and storedin a bu er module or local crate controller. At the end of the spill, all bu ered data fromthe spill were transferred to the host computer. Table B.1 shows a summary of thedigitizing electronics that were used.For the Fastbus systems, SLAC Scanner Processor (SSP) modules [24] served as cratecontrollers as well as performing the tasks of reading out, reformatting and bu eringthe data from the front end after each trigger were accepted. The CAMAC ADCs werereadout through the FERA bus by a Fastbus Struck [96] 370 QDP DSP module. The229Type Model Standard Resolution SubsystemsADC LRS 4300B CAMAC 10 bits RS,BV,BVL,EC,beamLRS 1881 Fastbus 13 bits target, UTCTDC LRS 3377 CAMAC 0.5 ns RS, BVLLRS 1879 Fastbus 2 ns UTC,BV,targetLRS 1876 Fastbus 1 ns EC,RSSC,beamWFD TD Fastbus 500 MHz sampling RS,BVL,IC8 bits, 10  s depthCCD Fastbus 500 MHz sampling beam, target, EC8 bits, 256 ns depthTable 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 withpermission 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 So-ciety.CAMAC TDCs were readout by custom-built DYC3 modules [42] which pushed the datainto VME memory boards. The readout time per event, as determined by the slowestcrate, was typically 850  s.At the end of each spill, the data from the Fastbus bu er memories were readoutvia the cable segment (12-15 MB/s) by Struck 340 SFI modules, each controlled by aMotorola [85] VME 2604 single-board computer (SBC) running VxWorks. The VMEmemory boards were readout by a separate SBC. Data were transferred from the SBCsto the host computer (SGI [93] Origin 200, designated bnlku9) via 9 MB/s per linkethernet through a simple network switch. Event fragments from the readout segmentswere combined by event builder processes running on the host computer. The    triggerswere written to two DLT-7000 drives at 5 MB/s per drive; a third DLT drive was used tolog monitor triggers.Under typical running conditions, 300 events were written per spill with a typicalevent size of 80 kB. This was well within the maximum possible throughput of the system230of about 50 MB per spill. The DAQ dead time was due entirely to the speed of theevent-by-event readout of the front-end electronics at the crate level. The total dead timeintroduced by the trigger and DAQ was typically 26%.A slow control system, based on the MIDAS [80] framework, ran independently of themain DAQ system, and was used to monitor a variety of experiment conditions, includingcrate voltages and temperatures.B.2 PASS0After the 12 week physics data run had completed, PASS0 processing was performed. Foreach monitor trigger, the data were staged, sorted and written to dedicated DLT tapes.B.3 PASS1 and PASS2 ProcessingPASS1 and PASS2 processing were performed in sequence on the various monitor andsignal trigger data by  rst staging raw data from DLT tapes to hard-drive disks. Next,the data were unpacked and event reconstruction was performed using a cluster of 25Linux-based computers. Fortran routines and shell scripts were used to decode the rawdigitized information written to tape, run event processing and create PAW [87] ntuplescontaining event information.B.4 PASS3 AnalysisSubsequent analysis of the PAW ntuples employed PAW comis functions and kumac scriptsto apply the desired cuts to the event ntuples. The output from this stage typically tookthe form of histograms of the desired quantities. The ntuples and cut functions wereavailable in public areas such that data analysis could be performed by multiple analyzersin parallel.231Appendix CTarget Pulse Data AnalysisThe implementation of the target pulse data analysis, known as the \target CCD  tter",for the analysis of experimental data from the E787 experiment is described in [38] withimprovements to the  tter described in [21]. Further improvements were made to the tter for the E949 analysis as described in [45]. This appendix describes the target CCD tter and details the modi cations and improvements made to the  tter for this analysis,summarized as follows: The uncertainties associated with the  rst bin and with bins containing less thanten counts were increased to de-weight these bins in the  t. Bins with zero counts were included in the minimization for the  t if they occurredwithin the pulse. They were previously excluded. Hold and release  tting was implemented for the double-pulse  ts. The four-parameter double-pulse  t was changed from a one-stage  t to a two-stage  t. Inthe  rst stage, the time for the  rst pulse was  xed at the time found from thesingle-pulse  t while the other three parameters were allowed to wander. In thesecond stage, the values from these three parameters were used as initial guessesand all four parameters were allowed to wander.232 The maximum number of target  bers that were  t was increased to 31 from 15. The previous restriction that the two pulses in a double-pulse  t had to be within 10 ns of each other was lifted due to a number of observed events with undesirabledouble-pulse  t solutions.Additional small bug  xes found in the previous  tter code were also made by the author.Throughout this appendix, the nomenclature \ ber channel" will be used when discussingeach the high-gain and low-gain CCD channels for each target  ber. For example, astatement beginning with \For each  ber channel" will describe a situation in which eachof the high-gain and each of the low-gain CCD channels for each target  ber was examined.For single-pulse  ts, the single pulse was often referred to as the \kaon pulse". Fordouble-pulse  ts, the  rst pulse was often referred to as the \kaon pulse" and the secondas the \pion pulse".C.1 Creation of Standardized PulsesFor each  ber channel, a collection of standardized K+ pulses was created using K 2monitor trigger data and the following criteria for  bers identi ed as K+  bers: The energy of the K+  ber EK was greater than 4 MeV; The pulse did not saturate the 8-bit dynamic range of the CCD; The time of the K+  ber 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 0.5ns by interpolation, and these average \standardized" shapes stored. The e ects due tosaturated pulses and outliers in each time bin were removed by determining the average233from the peak of a Gaussian  t. Based on these standardized pulses, the reliability ofeach of the  ber channels was determined with a typical reliability of 0.99.C.2 Overview of the FitterFor each  ber, the  tting procedure was performed on each of the low-gain and the high-gain CCD information independently. Fits were only attempted on  ber channels thatpassed the following criteria: The reliability for that  ber channel had to be greater than 0.8 for that run. The energy had to be greater than the threshold energy for 90% e ciency for aCCD hit for that  ber channel. Typical thresholds were 2 MeV for low-gain and 0.5MeV for high-gain CCD channels. The time for that target kaon  ber had to be earlier than track time trs. The edge  nder had to  nd a good leading edge.Additionally, a maximum of 31 low-gain and 31 high-gain CCD channel  ts were stored.The cut ALLKFIT removed events where a  t was performed on neither the high-gainnor the low-gain CCD channels of a given target  ber if the energy in that  ber was above3 MeV.A single-pulse  t was attempted for each  ber channel passing the above criteria. Thesingle-pulse  t used two parameters, the kaon-pulse amplitude and the time. Pedestalsubtraction had already occurred previous to the target CCD  tter so the pedestal wasnot included in the  t.If the probability of the single-pulse  t was less than 25%, a double-pulse  t wasperformed. The double-pulse  t used a second copy of the standardized kaon pulse andadded it to the  rst pulse. The double-pulse  t used four parameters, the amplitudes andtimes for each of the kaon and pion pulses.234These  ts minimized  2 by varying the above-mentioned parameters using the MI-NUIT function minimization library [81]. The uncertainty input for each bin of height Nof the pulse used experimentally determined uncertainty parameters:High-gain = 0:74 + 0:69 pN; (C.1)Low-gain = 1:21 + 0:35 pN; (C.2)where the  rst term was a constant related to instrumental noise and the second termscaled with the square-root of the number of photo-electrons in the pulse. This generalform for the uncertainties was retained in the modi cations made by the author, but twomodi cations were made as detailed in the following section. For these  ts, time binswhich have saturated the 8-bit dynamic range of the CCDs were not included in the  tand the  t was performed using only the unsaturated bins.C.3 Optimization of the Error InputA sample of K 2 monitor trigger data was used for optimizing the uncertainty inputsfor the target  tter. 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 EPIONKwere applied. The resulting components of acceptance of these cuts are shown in the rowlabeled \No  xes" of Table C.1 before the  xes detailed below.C.3.1 Low-Count Error FixTwo sub-samples of the K 2 monitor trigger data sample described above were createdby choosing only pulses that had above a 25% probability for the single-pulse  t. Onesub-sample was created for low-gain CCD  ts meeting this criteria and another for the235high-gain CCD  ts meeting this criteria. One at a time, each time bin was removed fromthe single-pulse  t and the di erence between the pulse-height for that bin as predicted bythe  tter and the actual number of counts \predicted actual" was tabulated as a functionof the actual number of counts in that bin. The bin numbering in this study was such thatbin number 1 was the  rst 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 timebins having counts (N) below 10, the uncertainty functions (Equations C.1 and C.2) wereunderestimating the uncertainty. Plots similar to C.1 were visually examined for eachtime bin and it was found that time bins having counts less than 10 were consistentlyunderestimated, with the turn-up in the distribution occurring typically in the  rst 5 to10 time bins. This was  xed by applying the uncertainty corresponding to N = 10 countsfor all channels having 10 or less counts.C.3.2 First Bin Error FixIt was found that the  t was very sensitive to the pulse-height of the  rst bin. A very lownumber of counts (such as 1 or 2) in the  rst bin tended to give a very large contributionto the chi-squared of the  t due to the reference pulse predicting a larger number ofcounts. This sensitivity to very low pulse-heights in the  rst bin was found to be presenteven with the low-count uncertainty  x applied. As shown in Figure C.2, a reasonablelooking  t can have a large chi-squared contribution due to the  rst bin, resulting in a  tprobability of zero and thus causing it to fail the  t. The contribution due to this  rst-binsensitivity was reduced by doubling the uncertainty associated with the  rst bin.The improved components of acceptance due to this  x are shown in the row labeled\First bin and low-count uncertainty  xes" in Table C.1. The results of the single- anddouble-pulse  ts were examined for many (40-50) events where the \low-count error"236Figure C.1: This plot shows the results of leaving the second bin out of the  tter and comparingthe actual counts in that bin to those predicted by the  t. The x-axis shows the number ofcounts in that second bin. The y-axis shows the di erence predicted actual counts for onlypositive values of this di erence. The line shows the uncertainty for each bin from the equation0:74 + 0:69  pcounts.237and \ rst bin uncertainty"  xes resulted in improvements of the  t probability from zeroto some non-zero value. Examination of these events showed that these  xes improvedthe  tter’s ability to identify single-pulse events while creating a negligible number offalse-positives (none were observed in the events examined).C.3.3 Intermediate Zero-Count Bins Included in the FitThe previous  tter did not include intermediate bins having zero counts in the  t. Thiswas changed so that the  tter  rst identi ed the  rst and last bins of the pulse. Thenup to the  rst 30 bins of this pulse were  t with bins having zero counts being included.Bins which were identi ed as saturated were excluded from the  t.The resulting components of acceptance due to this  x relative to the two previous xes detailed in this section are shown in the row labeled \First bin and low-count un-certainty  xes + 0-count bins included" in Table C.1. This  x resulted in a slight dropin the combined acceptance of CCDBADFIT, CCDPUL and EPIONK due to an increasein single-pulse  t probabilities below 25% resulting in more double-pulse  ts being per-formed. A  ber channel can only cause an event to fail CCDBADFIT or CCDPUL if adouble-pulse  t was performed.C.4 Hold and Release Double-Pulse FitThe  tter was modi ed so that the double-pulse  t became a two-stage process. Forthe  rst stage of the double-pulse  t process, the kaon-pulse time was  xed at the timereturned from the single-pulse  t. The three other parameters (kaon-pulse amplitude,pion-pulse time and pion-pulse time) were allowed to wander in the  t. For the secondstage of the double-pulse  t, the values returned from the  rst stage of the double-pulse t were used as the initial guesses.The sample used to optimize this modi cation to the  tter was K 2 monitor trigger238Fix CCDBADFIT CCDPUL ALLNo  xes 0.797 0.454 0.362First bin and 0.876 0.518 0.453low-count uncertainty  xesFirst bin andlow-count uncertainty  xes 0.881 0.504 0.443+ 0-count bins includedTable C.1: Components of acceptance for various target  tter  xes. The \ALL" column showsthe combined acceptance of the set of cuts consisting of CCDBADFIT, CCDPUL and EPIONKapplied sequentially. The acceptance of the EPIONK cut was 0.999 for all 3 situations so it wasnot given a column in the table. The row \First bin and low-count uncertainty  xes" includesboth the  rst bin uncertainty  x (doubling the uncertainty for this bin) and the low-countuncertainty  x (assigning an uncertainty equal to that for 10 counts for all bins having lessthan 10 counts). The row \First bin and low-count uncertainty  xes + 0-count bins included"includes the above  xes in addition to including intermediate zero count bins in the  t.data with the following cuts applied: TARGET, TGCUT, UTC, RD TRK, TRKTIM,RDUTM, TGQUALT, DELC and NPITG. After these cuts were applied, 7021 eventsremained. With the two-stage  t used instead of the previous one-stage  t, the totalacceptance of CCDBADFIT and CCDPUL went from 0.402 to 0.451.239Figure C.2: The high-gain single-pulse probability of 0 when the uncertainty associated withthe  rst bin was treated as usual. This was due to the  rst bin having only two counts versuswhere the shape of the reference pulse predicted a larger number of counts. Had the  rst binhad a pedestal subtraction that left it with zero counts, the next bin would have been used forthe  t and the resulting  t would have been a non-zero probability. To reduce the e ect of thissensitivity to the  rst bin, the uncertainties assigned to the  rst bin were always doubled. Whenthe  t was performed with this increased  rst bin uncertainty, the single- t probability for thehigh-gain was 0.069 instead of the zero probability shown in this  gure.240Appendix DDetailed List of CutsD.1 PASS1 CutsThese cuts were applied to all the background studies and thus were excluded from thelists of setup cuts for the individual bifurcation branches.D.1.1 Event Reconstruction Quality CutsTRBIT - The event passed the    (1) or    (2) triggers.BAD RUN - Removed runs where problems occurred in the beam, detector or electron-ics.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 identi ed by theo ine track  nding routine and the online Stopping Counter Finder (SCF).BAD STC - Runs were identi ed during calibration in which a set of TD channels hadunreliable signals. For these runs, events were removed if the Range-Stack stopping241counter contained these unreliable TD channels.UTC - The charged track was reconstructed in the UTC.RDUTM - The extrapolation of the reconstructed UTC track intersected the triggeredT 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 280MeV/c. This removed high-momentum beam particles and failures in UTC patternrecognition.D.1.2 Muon Background Rejection CutsFITPI - The double-pulse  + !  + signature was observed in the  tted TD pulses fromthe Range-Stack stopping counter.RSHEX - Event was rejected if there were additional hits in (1) the counter after theRange-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 onesector to another in the stopping layer.D.2 PASS2 CutsThe PASS2 cuts consisted of looser versions of the PASS3 cuts. The PASS2 cuts wereapplied in certain combinations to enhance speci c backgrounds, reducing processing timewhen developing cuts and evaluating background levels.P2PVCUT - A very loose photon veto was applied in the Barrel Veto ( 2.0 ns, 1.5MeV), End Cap ( 1.5 ns, 3.5 MeV), and Range-Stack ( 1.5 ns, 3.0 MeV). Events242were rejected if total coincident energy was above the listed threshold for the listedtime window about the Range-Stack track time trs.P2TGCUT - Events failed this target reconstruction cut if they did not meet all of thefollowing conditions: The kaon decay vertex identi ed by target reconstruction was found inside thetarget. The average time of the kaon  ber hits (tk) and the B4 Hodoscope hit time(tb4) were within 4.0 ns. The average time of the pion  ber 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 5.0ns. This condition involved a small time o set: jtIC  trs + 0:3nsj 5:0ns. The energy deposited in the I-Counter was within 4 MeV of that expected fora minimum ionizing pion track.P2TDCUT - Event was rejected if accidental activity in the Range-Stack coincidentwith the second pulse in the Range-Stack stopping counter exceeded a threshold.P2PSCUT - Beam pion events were removed by requiring that (1) the energy depositedin the B4 Hodoscope was consistent with a kaon (> 1:0 MeV), and (2) there wasno beam particle coincident with the T 2-trigger time. These beam-particle timeswere determined by the times of hits in the B4 Hodoscope and the pion  Cerenkovcounters.P2TGPVCUT - Event was rejected if the total coincident activity in the target photon bers at track time was above 5 MeV. The coincidence condition with respect toRange-Stack track time was jt  trs + 0:8 nsj 1:0 ns.243DELCO2 - 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  ber hits (tk)and the average time of the pion  ber hits (tpi).D.3 Kinematic CutsThe kinematic cuts, collectively referred to as KINCUTS, were designed to remove eventshaving kinematics inconsistent with signal.D.3.1 Fiducial CutsLAYER14 - Event was rejected if it came to rest in the second Range-Stack StrawChamber layer. This condition was accomplished by rejecting stopping layer 14events having a prompt RSSC hit in the same sector or one sector clockwise of thestopping 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-Stacksupport structure.LAYV4 - Event was rejected if the charged track did not come to rest in between Range-Stack stopping layers 6 to 18 inclusive.ZFRF - The charged track was required to come to rest inside of the Range-Stack  ducialregion based on the stopping z-position. This condition was applied for stoppinglayers 11-12 (jz  35 cmj), 13 (jz  40 cmj), 14 (jz  30 cmj), and 15-18 (jz  50 cmj).ZUTOUT - The charged track was required to pass through the UTC outer layer withinthe active region (jz  25 cmj).244D.3.2 Track Reconstruction CutsUTCQUAL - Good track reconstruction was required in the UTC based on a likelihoodcondition in the x-y plane. The quantities considered in this likelihood conditionwere the number of hits used in the UTC track  t, the number of hit UTC layers,and the number of unused hits within 1.5 cm of the  tted track.PRRF - Event was rejected if the charged-track scattered in the Range-Stack, as de-termined by matching between Range-Stack track reconstruction and UTC trackextrapolation. The following conditions were examined: Event was rejected if it failed a  2 probability condition for the Range-Stacktrack reconstruction in the x-y plane. The Range-Stack track  tting used sectorcrossing positions, RSSC hit positions and the energy deposit in the stoppingcounter. Event was rejected if the track reached the  rst layer of RSSCs and the match-ing between the UTC track extrapolation and the RSSC hit positions waspoor. Event was rejected if the matching in the r-z plane was poor between the UTCtrack extrapolation and the track hit positions in the Range-Stack counters asdetermined by end-to-end timing.D.3.3 dE/dx Cuts in Range-StackRSDEDX - The following conditions based on energy deposits in the Range-Stack wereexamined: Event was rejected if the energy deposited in any Range-Stack track counterwas inconsistent with that expected for a pion. The deviation between mea-245sured Eimeas and expected Eiexp energies for the ith track counter was given by: i = logEimeas  logEiexp i ; (D.1)where the event was rejected if j ij 4. Event was rejected if the con dence level calculated from the probability of  2from Equation (D.1) was less than 0.04. Event was rejected based on a likelihood condition constructed from the energydeposits in the track counters. Event was rejected based on additional constraints on the  t times and energiesdetermined by the fitpi4 routine [57].D.3.4 Range-Momentum Consistency CutRNGMOM - Event was rejected if the measured range rtot and the expected rangebased on momentum rexp were not consistent under the assumption of a chargedtrack due to a pion. This cut targeted muon backgrounds due to the di erentregions of the range-momentum phase-space occupied by pions and muons. Eventwas rejected if the deviation  r p was greater than 2.2, where r p = rtot rexp rtot: (D.2)D.4 Phase Space CutsThe kinematic constraints of the PNN2 phase space (PNN2BOX) were split into loose(BOXLOOSE) and tight (BOXTIGHT) versions. For each cut, the outgoing chargedtrack was required to pass the listed kinematic conditions on momentum (ptot), energy(etot) and range in plastic scintillator (rtot).246BOXLOOSE - 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 heavilysuppress 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 de ne the signalregion, but were used for various studies in this analysis: KP2BOX to choose events in theK 2 peak, and KM2BOX to choose events in the K 2 peak. Figure 3.2 shows where theseother kinematic boxes were found with respect to the kinematic signal region PNN2BOXin range versus momentum phase space.KP2BOX - The acceptance conditions for the K 2-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 K 2-peak kinematic region were:226.0 MeV/c  ptot  246.0 MeV/c;37.0 cm  rtot.D.5 Beam CutsThe beam cuts, collectively referred to as BEAMCUTS, were designed to remove eventswith beam-pions scattering into the  ducial region of the detector and double-beam events.247D.5.1 Particle Identi cationB4DEDX - Beam pion events were removed by requiring that the energy deposited inthe B4 Hodoscope was consistent with a kaon (> 1:1 MeV).D.5.2 Double-Beam CutsB4CCD - Double-beam events were removed based on  ts in the B4 Hodoscope CCDs tted with a double-pulse assumption. A double-pulse signature was consideredpresent when the ratio of  2 for the double-pulse  t versus the single-pulse  t wasgreater than 2.5 for second-pulse amplitudes above a certain threshold. Event wasrejected when the di erence between the average time from hit modules havinghaving 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 inthe Range-Stack trs. This activity was considered present if (1) the average TDCtime from B4 hit modules was within 2.5 ns of trs, or (2) the average CCD timefrom B4 hit modules was within 1.5 ns of trs when the B4 energy sum was above0.7 MeV.BWTRS - Event was rejected if there was activity in the Beam-Wire Chamber at tracktime in the Range-Stack trs. This activity was considered present if there was a hitcluster in any of the Beam-Wire Chambers within 4.5 ns of trs.CKTRS - Event was rejected if the average TDC or CCD time of the  Cerenkov kaoncounter hits was within 2.0 ns of trs.CKTAIL - Event was rejected if the average time of the trailing edge of pulses in the Cerenkov kaon counters minus the average TDC width was within twindow of trs.The time window varied based on the di erence between the average time of thekaon  ber hits tk and the average time of the pion  ber 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.CPITRS - Event was rejected if the average TDC or CCD time of the  Cerenkov pioncounter hits was within 2.0 ns of trs.CPITAIL - Event was rejected if the average time of the trailing edge of pulses in the Cerenkov pion counters minus the average TDC width was within twindow of trs.The time window varied based on the di erence between the average time of thekaon  ber hits tk and the average time of the pion  ber 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 CutsPathology cuts were cuts designed to remove speci c types of abnormal events that werenot properly suppressed by the regular cuts designed to suppress the various backgrounds.249These undesirable pathologies could have been due to behaviors such as inconsistenciesbetween quantities measured in di erent detectors or known abnormal behavior in areconstruction algorithm.B4ETCON - Event was rejected if the timing or energy information in the B4 Hodoscopewas inconsistent between the CCDs and the TDCs. Event was rejected if the timesfrom these two systems were not within 2.0 ns of each other or if the energies fromthese two systems were not within 1.5 MeV.TGGEO - Event was rejected by this cut if it had certain topological signatures associ-ated with with double-beam events known to fool target reconstruction. The threetypes of signatures for rejected events were: Both beam-particles entered the target from the target edge or I-Counter; The  rst beam-particle or its charged decay product deposited a large amountof energy in the I-Counter; The charged decay product of the  rst beam-particle was not detected dueto decaying downstream or decaying late. The second beam-particle scatteredsuch that it missed some of the beam counters, but by multiple scattering entersthe target and intersects with target  bers hit by the  rst beam-particle.TGQUALT - The target reconstruction algorithm successfully reconstructed the eventwith at least one of the target  bers being classi ed as a pion  ber.TIMCON - Event was rejected if it failed to meet either of the following timing consis-tency checks: The average time of the kaon  ber hits (tk) and the B4 Hodoscope hit time(tbm) were within 3.0 ns;250 The average time of the pion  ber hits (tpi) was consistent with track time inthe Range-Stack (trs),  4:75 ns  tpi  trs  3:75 ns.TGTCON - Event was rejected if the time di erence between an individual target kaon ber and the average time of the kaon  ber hits tk was greater than an energydependent value varying between 2.05 and 3.68 ns.D.6 Delayed Coincidence CutsThe delayed coincidence cuts, collectively referred to as DELCO, rejected the same typeof 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+ hadcome to rest before decaying by requiring a minimum time di erence delco between theaverage time of the kaon  ber hits (tk) and the average time of the pion  ber hits (tpi):tpi  tk  delco: (D.3)DELC3 - Event was required to meet the minimum delayed coincidence requirementof delco = 3 ns. Additional constraints were placed on the delayed coincidencerequirement when the timing consistency between detector systems was degraded.The cut threshold was the maximum of the following conditions, with a minimumvalue of 3 ns: delco = 5 ns when the di erence between the average time of the kaon  berhits (tk) and the B4 Hodoscope hit time (tb4strob) was greater than 1.0 s. delco = 6 ns when the di erence between the average time of the pion  berhits (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) instead251of the average time of the pion  ber hits (tpi). delco = 4 ns when the energy deposit in target kaon  bers is less than or equalto 50 MeV. delco = 4 ns if the time of any of the individual target kaon  bers di eredfrom the average time of the kaon  ber hits (tk) by more than 2.0 ns. delco = 4 ns if the time of any of the individual target pion  bers di eredfrom the average time of the pion  ber hits (tpi) by more than 3.5 ns.DELC6 - Event was required to meet the tight delayed coincidence requirement ofdelco = 6 ns. Additional constraints like those in DELCO3 were not applied.D.7 Target Quality CutsThe target quality cuts, collectively referred to as TGCUTS, were designed to select eventshaving good event signatures in the target. Note that the cut KPIGAP was not includedin the group TGCUTS, nor was it applied as an analysis cut used to de ne the signalregion.B4EKZ - Event was rejected if the beam particle identi ed as a kaon by event recon-struction did not behave like a kaon. A likelihood function was created using thez-position of the kaon decay in the target as determined by UTC track extrapola-tion, the expected z-position of the kaon decay as determined by the total energydeposited in target kaon  bers, and the energy deposited by the kaon in the B4Hodoscope. This cut suppressed the beam pion scatter single-beam background. Atighter rejection condition was placed on this likelihood function if tpi was deter-mined using I-Counter timing information instead of the average hit times in thetarget pion  bers.252TGZFOOL - Event was rejected if the z-position of the kaon decay vertex was too closeto the upstream end of the target or was not in the  ducial region of the target atall. This z-position was determined by UTC track extrapolation.EPITG - Event was rejected if an individual target pion  ber had an energy greaterthan 3.0 MeV. This rejected potential pion target-scatters since the nominal energydeposited in a pion  ber was 1.2 MeV.TARGF - Event was rejected if the minimum distance between the target kaon andpion  bers was greater than one  ber (0.6 cm). This cut suppressed CEX anddouble-beam backgrounds.DTGTTP - Event was rejected if the charged-product tracks in the target and UTCwere not well-matched at the target edge.RTDIF - Event was rejected if the uncertainty in the calculation of the pion path lengthin 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 alarge spread between the minimum and maximum radius of the UTC reconstructedtrack.TGKTIM - Event was rejected if the time of an individual target kaon  ber hit (tik)was more than 3.5 ns later than the B4 Hodoscope hit time (tB4). Event was alsorejected 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) andthe 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-Stacktrs were not within 5.0 ns of each other.253TGEDGE - Event was rejected if there was more than 4.0 MeV in a multiplexed targetedge- ber PMT within 5.0 ns of trs.TGDEDX - Events was rejected if the charged track dE=dx in the target was inconsis-tent with that of a pion. This cut used a likelihood function based on the followingquantities of the charged track: momentum (ptot), range in the target , energydeposited in the target and expected range in the target based on the measuredenergy deposited in the target and the momentum ptot.TGENR - Event was rejected if the total energy of the hit pion  bers in the target wasnot in the range of 1 to 28 MeV.PIGAP - Event was rejected if a gap greater than 1.5 cm was found between targetpion  bers. This cut was tightened to reject gaps greater than 1.0 cm when thez-position of the track in the I-Counter was less than -7.0 cm and the cosine of thepolar 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. Thefollowing conditions were examined: Event was rejected if the distance in the xy-plane between the hit position inthe B4 Hodoscope and the nearest target kaon cluster tip was greater than 1.8cm. The kaon cluster tips were the two kaon  bers furthest apart from eachother. Event was rejected if the distance in the xy-plane between the kaon decayvertex 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 decayvertex and the nearest pion  ber was greater than 1.5 cm.254PHIVTX - Event was rejected if it had back-to-back charged decay-product tracks inthe target.OPSVETO - Event was rejected if the total energy in target opposite-side pion  berswas more than 1.0 MeV within 4.0 ns of the average pion  ber hit time tpi. Thisenergy threshold was reduced to 0.5 MeV if the likelihood function from B4EKZ wasless than 200. Opposite-side pion  bers were pion  ber hits found on the oppositeside of the kaon cluster relative to the main pion cluster (see Section 3.3).TGLIKE - Event was rejected if a target pion  ber showed evidence of hidden energyas determined by two target pion- ber likelihood functions. The  rst likelihoodfunction was constructed for each pion  ber from the distance to the extrapolatedUTC track, the time and the energy of the pion  ber hit. The rejection condition forthe  rst likelihood function was based on the average likelihood for all pion  bers.A second likelihood function was constructed using only the distance between thepion  ber hit positions and the extrapolated UTC track and had a tighter rejectionthreshold than the  rst likelihood function.TIMKF - Event was rejected if the times of the target kaon  ber hits were not consistentwith a kaon approaching the kaon decay vertex. This consistency was checked bytabulating the times of the kaon  ber hits against the distance to the decay vertex inthe x-y plane and against the range of the kaon as determined by deposited energy.NPITG - Event was rejected if no target  bers were identi ed as pion  bers.ALLKFIT - Event was rejected if any of the target kaon  bers having more than 3.0MeV of energy were not successfully  t by the target CCD  tter. A  ber was con-sidered successfully  t if the probability of the  t for the single-pulse or double-pulseassumptions was greater than 0.01. See Appendix C for more detailed discussion ofthe target CCD  tter.255TPICS - Event was rejected if the standard deviation of hit times for the target pion bers was greater than 4.0 ns.EPIONK - Event was rejected if a target  ber classi ed as both a kaon and a pion  berhad more than 1.25 MeV assigned to the pion pulse. The target reconstruction wasable to  nd pion hits in kaon  bers when the average hit time for pion  bers tpiwas at least 15 ns greater than the average hit time for kaon  bers tk.CHI567 - Event was rejected if the probability of a sum of three  2-like quantities wasbelow a certain threshold. These  2-like quantities had contributions from targetpion  bers and the target  + track  tter [72] as follows:  25 - Contributions for hit pion  bers which were part of the reconstructed piontrack were based on observed versus expected energy.  26 - Contributions for  bers with no energy, but lying along the projected piontrack were based on the minimum distance between the projected track andthe corners of the  ber. This forced the  tted track to go between the  bers.  27 - Contributions for hit pion  bers which were not part of the reconstructedpion track were based on their distance from the track.CHI5MAX - Event was rejected if the contribution to  25 (see CHI567) due to any single ber was greater than 10.VERRNG - Event was rejected if the  tted track in the target did not intersect thevertex  ber as identi ed by target reconstruction.ANGLI - Event was rejected if the range of the charged track in the target was less than2.0 cm, and the angle between the track from target reconstruction and the UTCextrapolated track was greater than 0.01 radian.256KIC - Event was rejected if a hit in the I-Counter was coincident with target kaon timeand kaon  bers were found near this counter.CCDBADFIT - Event was rejected if the target CCD  tter was unable to make asuccessful  t for a kaon  ber having more than 1.25 MeV of energy. The  t wasconsidered unsuccessful if the probability of the  t for both the single-pulse anddouble-pulse assumptions was zero. This cut is discussed in more detail in Ap-pendix E.CCDBADTIM - Event was rejected if the target CCD  tter found a known incorrectsolution for a kaon  ber having more than 1.25 MeV of energy. These knownincorrect solutions were as follows: The  rst-pulse time from the double-pulse  t was less than -9.98 ns. The second-pulse time from the double-pulse  t was less than -4.99 ns. The  rst-pulse time from the double-pulse  t t1 was inconsistent with theaverage target kaon  ber hit time tk. The rejection conditions were t1  tk < 6 ns or t1  tk > 7 ns. The time from the single-pulse  t was less than -9.98 ns when the probabilityof the single-pulse  t was greater than 0.25 and thus the double-pulse  t wasnot performed. The time from the single-pulse  t t0 was inconsistent with tk when the prob-ability of the single-pulse  t was greater than 0.25. The rejection conditionswere 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  ber  t by the target CCD  tter had a  tprobability for the single-pulse assumption of less than 0.25. When this occurred,257the results of the subsequent  t for the double-pulse assumption were not storedcorrectly.CCDPUL - Event was rejected if the second-pulse found in a kaon  ber by the targetCCD  tter had more than 1.25 MeV of energy and was coincident with target piontime 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 de ne the signal region. Thiscut was used to identify events where the pion track did not emerge directly fromthe target kaon  bers. 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 thanthat produced by inverting TARGF. The target reconstruction algorithm identi edpotential pion  bers having large energy deposits as photon  bers. Allowing thesephoton  bers (within 3.0 ns of Range-Stack track time trs) to  ll the gap created acleaner sample of events having true gaps between the kaon and pion  bers.D.8  + !  + ! e+ Decay-Sequence CutsThe  + !  + ! e+ decay-sequence cuts, collectively referred to as TDCUTS, weredesigned to reject events with a  + as the primary charged particle from the kaon decay.This set of cuts was split into loose (TDLOOSE) and tight (TDTIGHT) versions. Theloose set TDLOOSE consisted of P2TDCUT, IPIFLG, ELVETO, TDFOOL and a looseversion 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  + !  + decay sequence in the Range-Stack stoppingcounter was emulated by an accidental hit in conjunction with the muon track. Theevent was rejected if the time of the  rst pulse, obtained from TD double-pulse258 tting, was not within 2.5 ns of Range-Stack track time trs.ELVETO - Event was rejected if there was accidental activity coincident with the timeof the second pulse in the Range-Stack stopping counter. The coincident energywas 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 wasabove the threshold energy shown in Table D.1.Time EnergyCategory Window Threshold(ns) (MeV)Both-ends hit categoryRange-Stack (RS)  3.00 0.20RS (TD)  0.25 5.20Barrel Veto (BV)  1.25 0.20Barrel Veto Liner (BVL)  2.75 0.20Single-end hit categoryRS single energy, both times  7.00 0.20RS both energy, single time  4.50 9.40RS single energy, single time  8.75 6.60RS no energy, both times  5.00 -RS (TD) single energy, single time  3.00 3.20BV single energy, both times  3.00 1.60BV both energy, single time  0.25 0.40BV single energy, single time  3.00 1.80BV no energy, both times  5.75 -BVL both energy, single time  0.75 0.10BVL single energy, single time  5.00 1.00BVL no energy, both times  5.50 -Other categoryEnd Cap (EC)  0.25 22.00Table D.1: The energy threshold and time window relative to muon time in the Range-Stackfor 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 theRange-Stack stopping counter. The second-pulse was considered to have come from259accidental activity when there was double-pulse activity present in one of the twolayers preceding the stopping counter having second-pulse timing consistent withthe second-pulse found in the stopping counter.TDNN - Based on a neural-net function, events having tail  uctuations in the TD pulseor K+ decays involving  + particles were rejected. The  ve variables used to con-struct the neural-net function were the TD single-pulse  t  2, the ratio of  2 forsingle-pulse  t to double-pulse  t, the time and energy of the  tted second pulse,and the di erence between both ends for the second pulse time. The cut positionwas tighter for TDTIGHT than for TDLOOSE (see Section 6.2.3).EV5 - Event was rejected if the signature of the  + ! e+ decay was not present in andaround the Range-Stack stopping counter. The signature of this decay was a clusterof hits on only one side of the stopping counter, consistent with the time of thethird pulse in the stopping counter. This cut was included in the named group ofcuts TDTIGHT, but not in TDLOOSE.D.9 Photon Veto CutThe event was rejected if there was activity in any of the photon veto subsystems coin-cident with Range-Stack track time trs. The photon veto (PVCUT) used to de ne thesignal region was split into loose (PV60) and tight (PV30) versions. The photon vetowas loosened even further (PV90) for use in the outside-the-box studies. Table D.2 showsthe photon veto parameters for each subsystem with the ability to detect photons, whereactivity above the energy threshold and falling within the time window (de ned relativeto trs) caused the event to fail the photon veto cut. Table D.3 shows the parameters usedfor the BV, BVL and RS when the both-end requirements for time and energy were notmet. In addition to the parameters listed in the tables, the PASS2 photon veto P2PVCUT260was always applied as part of this photon veto cut.The optimization of the photon veto was based on a process of optimizing the pa-rameters for each of the photon veto subsystems to maximize background rejection whilemaintaining a reasonable signal acceptance. The PV optimization for E949 had previ-ously been performed for the PNN1 analysis [83], but had to be re-optimized due to thePNN2 analysis including TD hits in the BVL [37]. The acceptance and rejection samplesused for this optimization were the K 2 monitor trigger data and K 2-peak events fromsignal data respectively. The optimization process started from an initial set of cut pa-rameters for each subsystem (time o sets, time window and energy thresholds). The cutparameters were changed in small steps, one subsystem at a time, and the acceptance andrejection remeasured. It was considered to be an improved set of parameters if both theacceptance and rejection were increased or if one was increased without the other beinglowered. The optimization process continued until no further gains in rejection could beattained without losing acceptance. Nine di erent goal levels of acceptance were chosenand the entire optimization process was repeated for each of these goal levels of acceptanceto produce the pro le curve of maximum achievable rejections as functions of acceptanceas 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 thisPNN2 analysis since it was essential in suppressing the most problematic type of K 2target-scatter events as described in 4.1. The optimization of the AD, which requiredgreat 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 classof events observed in the photon veto optimization acceptance sample. These events hadlarge hits in the BV in the time range of 5 to 35 ns earlier than trs. It was found thatthese hits, due to accidentals, blinded the BV TDCs so that they could not register thehits from the decay photons. The ADC gates for the BV were approximately 50 ns wideso the energy from both the decay photon and the accidental was measured and these261Figure D.1: Illustration of the optimization process used to determine the photon veto parame-ters. The \Pro le curve" de nes the maximum achievable rejection as a function of acceptance,along which it is no longer possible to further increase the acceptance or rejection without de-creasing the other. Reprinted  gure with permission from S. Adler et al. (E949 Collaboration),Phys. Rev. D 77 052003 (2008), http://link.aps.org/abstract/PRD/v77/e052003. Copyright2008 by the American Physical Society.262PV60 (Loose) PV30 (Tight)Category Timing (ns) Energy Timing (ns) EnergyO set Window (MeV) O set Window (MeV)BV 2.25 7.95 0.20 1.35 8.85 0.70early BV -20.70 15.0 30.00 -22.5 15.0 30.0BVL 3.15 7.55 0.30 3.15 7.55 0.30RS 0.05 4.30 0.30 2.25 5.55 0.60EC 1.80 6.15 0.40 1.75 7.75 0.20EC inner-ring 0.99 4.64 0.20 -2.45 11.55 0.20EC 2nd pulse -1.60 4.07 10.60 -1.51 4.19 1.70TG -0.25 2.40 2.00 -2.15 4.40 1.40IC 1.25 3.25 5.00 3.20 6.10 5.00VC -2.40 4.15 6.80 -0.20 7.25 6.00CO 2.90 2.95 0.60 2.15 2.95 1.60 CO -1.60 3.90 3.00 -0.60 3.90 0.60AD 3.00 5.00 0.60 3.00 5.00 0.60DPV 2.50 7.50 0.00 2.50 7.50 0.00early BVL -3.50 1.50 10.00 -3.50 1.50 10.00Table D.2: The parameters for the loose (PV60) and tight (PV30) photon vetoes. The parame-ters shown for BV, BVL and RS are for when the detector met the requirement that both endsobtained measurements of time and energy. Table reproduced from [56].blinded BV modules recorded unusually large energies. These events were rejected byadding the \early BV" parameters with a threshold energy of 30 MeV.263Category PV60 (Loose) PV30 (Tight)hit-ends Timing (ns) Energy Timing (ns) EnergyEnergy Time O set Window (MeV) O set Window (MeV)BV both single 3.05 15:95 1.00 0.55 13:05 0.40BV single both 4.80 1:50 1.40 4.00 3:10 0.60BV single single -8.10 8:50 1.60 -8.30 6:90 1.00BVL both single -5.65 11:80 8.19 -5.65 11:80 8.19RS both single -2.85 0:70 5.20 0.01 5:36 0.20RS single both 6.60 1:35 0.00 3.70 6:10 0.00RS single single -6.80 1:22 3.40 -11.54 4:53 0.60Table D.3: The parameters for the loose (PV60) and tight (PV30) photon vetoes when therequirement was not met that both ends obtained measurements of time and energy. Tablereproduced from [56].PV90 (Very Loose)Category Timing (ns) EnergyO set Window (MeV)BV -0.15 4.00 0.50early BV -19.15 15.0 30.00BVL 0.35 1.75 0.40RS -0.85 1.45 0.20EC 0.15 1.80 2.20EC inner-ring -0.35 2.30 1.20EC 2nd pulse -2.75 0.32 18.80TG 0.25 1.50 5.20IC -0.50 2.75 13.00VC -0.25 1.50 3.80CO 2.60 1.23 1.80 CO -1.50 2.50 3.60AD 3.00 5.00 0.60DPV 2.50 7.50 0.00early BVL -3.50 1.50 10.00Table D.4: The parameters for the very loose (PV90) photon veto. The parameters shownfor BV, BVL and RS are for when the detector met the requirement that both ends obtainedmeasurements of time and energy. Table reproduced from [56].264Category PV90 (Very Loose)hit-ends Timing (ns) EnergyEnergy Time O set Window (MeV)BV both single 0.55 13.05 0.40BV single both 4.00 3.10 0.60BV single single -8.30 6.90 1.00BVL both single -5.65 11.80 8.19RS both single 0.01 5.36 0.20RS single both 3.70 6.10 0.00RS single single -11.54 4.53 0.60Table D.5: The parameters for the very loose (PV90) photon veto when the requirement wasnot met that both ends obtained measurements of time and energy. Table reproduced from [56].265Appendix ETarget Pulse CutsThe original CCDPUL cut, as it was used in the analysis of the PNN2 kinematic regionfor the E787 experiment, performed the function of both the CCDBADFIT and CCDPULcuts described in this appendix. It was split into two for this analysis since CCDBADFITwas meant as a safety cut against events where the target CCD  tter was unable to  ndvalid solutions. The CCDPUL cut was inverted for many of the classes used in the K 2target-scatter rejection branch (Section 4.1.3) and a much cleaner sample of target-scatterevents was created by applying CCDBADFIT and inverting CCDPUL than by invertingboth of them.The CCDBADTIM cut was a safety cut created as a result of an event observed in thesingle-cut failure study (Section 5.2) that enforced timing consistency between the timesfound using the target CCD  tter and those expected from target reconstruction. Thiscut had a very similar overall structure to CCDBADFIT and CCDPUL.E.1 Description of CCDPULThe CCDPUL cut removed events that had kaon  bers with second-pulse energies above1.25 MeV and coincident with the average hit time of the target pion  bers tpi. The266CCDPUL cut had three stages: In the  rst stage, the low-gain and high-gain CCDs of each kaon  ber was examinedto determine if a double-pulse  t had been successfully performed. In the second stage, an algorithm determined if the information from the high-gainCCD, the low-gain CCD or a combination of the two should be used for each  berthat had successful double-pulse  ts. In the third stage, each  ber from the second stage was examined for second pulseenergies above a 1.25 MeV and rejected the event if the second pulse was coincidentwith the global pion time tpi.E.1.1 CCDPUL First StageFor each kaon  ber, the results of the  ts from the low-gain and high-gain CCDs wereexamined independently. The high-gain CCD data were not passed onto the second stageif any of the following conditions were met: The single-pulse  t probability was greater than 0.25 as the double-pulse  t wasonly performed if the single  t probability was less than or equal to 0.25. The number of bins in the second pulse having a non-zero amplitude was less than3. The double-pulse  t probability was equal to zero.The same conditions as above were applied to the low-gain CCDs with one additionalcondition, if the  ber was found to be multiplexed with at least one other  ber havingactivity within  5 ns of tk or tpi then the target low-gain CCD de-multiplexing algorithmwould determine the  rst- and second-pulse energies of the  ber being examined. Thisde-multiplexing algorithm is detailed in Section E.6.267E.1.2 CCDPUL Second StageDuring the second stage of CCDPUL, an algorithm decided for each kaon  ber which CCDinformation was to be used in the third CCDPUL stage: the low-gain, the high-gain, or aweighted average of the two. If only the high-gain or or only the low-gain CCD data fora given  ber was passed on from the  rst CCDPUL stage, that is what was used. If the rst stage passed on both the high-gain and the low-gain CCD information for a given ber, the decision of how to use this information in the third CCDPUL stage was basedon the following conditions, which were checked in sequence: An average of the high-gain and low-gain CCD information was used if the  berenergy as determined by ADC was between 10 MeV and 30 MeV and the fractionaluncertainty in the second-pulse amplitude for both the low-gain and high-gain CCDchannels was greater than 0.05. A typical pulse started to saturate at around 25MeV so the ADC energy condition included high-gain CCD information that showedsmall amounts of saturation. The second-pulse energy and the relative time betweenthe  rst and second pulses were the quantities calculated via the weighted average.For the remaining quantities (such as  t probabilities), the information from thehigh-gain channel was used. The following equations show how the weighted averagewas determined for a quantity x:whi = dA2hidA2hi + dA2lo ;wlo = dA2lodA2hi + dA2lo;x = xhiwhi + xlowlo;where dAlo and dAhi were the uncertainties on the  tted second-pulse amplitudesof the low-gain and high-gain double pulse  ts, 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 andthe ADC energy in the  ber was greater than 40 MeV. The low-gain target CCDstypically started to saturate at around 40 MeV. The most appropriate CCD channel was chosen based on the expected quality ofthe double-pulse  t given the  ber energy Ek (from ADC) and the time di erencebetween the average time of the pion  ber hits tpi and the average time of the kaon ber hits tk, The high-gain CCD information was used if either of these sets ofconditions 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 weremet:(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 StageThe information for all  bers that were passed onto the third stage were checked forsecond-pulse energy above 1.25 MeV when the timing of the second pulse was consistentwith the average pion  ber hit time tpi. The event failed if any  ber in stage three metboth 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 deltatwas a measure of consistency between tpi and tk, and the  rst (t1) and second (t2)pulse times from the double-pulse  t for that kaon  ber,deltat = (t2  t1) (tpi  tk) : (E.1)E.2 Description of CCDBADFITThe CCDBADFIT safety cut removed events where the target CCD  tter was unable tosuccessfully  t a kaon  ber. This occurred when the probabilities for both the single- anddouble-pulse  ts were equal to zero. As with CCDPUL, this cut also had three stages.E.2.1 CCDBADFIT First StageFor each kaon  ber, the results of the  ts from the low-gain and high-gain CCDs wereexamined independently. The high-gain CCD data were not passed onto the second stageif any of the following conditions were met: The single-pulse  t probability was greater than 0.25 as the double-pulse  t wasonly performed if the single  t probability was less than or equal to 0.25. The number of bins in the second pulse having a non-zero amplitude was less than3 when the double-pulse  t probability was greater than zero. The double-pulse  t probability was equal to zero and the single- t probability wasgreater than zero.The same conditions as above were applied to the low-gain CCD for each kaon  ber.270E.2.2 CCDBADFIT Second StageDuring the second stage of CCDBADFIT, a similar decision making process to that used inCCDPUL was used to determine which CCD information was to be passed onto the thirdstage for each kaon  ber: the low-gain or the high-gain. The di erence in CCDBADFITwas that a weighted average of the low-gain and high-gain was not used, but the rest ofthe decision making sequence was the same.E.2.3 CCDBADFIT Third StageAll kaon  bers passed onto the third stage were checked for probabilities equal to zero forboth the single- and double-pulse  ts. If this condition was met and the  ber energy asdetermined by the ADC was greater than 1.25 MeV, the event failed the CCDBADFITcut.E.3 Description of CCDBADTIMThe CCDBADTIM cut was a safety cut designed to reject events if the target CCD  tterfound a known incorrect solution for a kaon  ber having more than 1.25 MeV of energy.The known incorrect solutions fell into two main categories1. The  tter fell into a local minimum where it attempted to  t the tail of the referencepulse to the actual pulse. The signature for this was a time of -9.9939 ns for thesingle-pulse  t (t0) or for the  rst pulse of the double-pulse  t (t1). Times lowerthan -9.9939 ns were stored in the ntuple as -9.9939 ns due to range limits placed onthese variables when stored to the ntuple. A time of -4.9939 represented the samesituation for the second pulse of the double-pulse  t.2. Due to a large second pulse in the  ber, the  tter  t the second pulse as the primarypulse as shown in Figure E.1. The signature of this failure mode was either t0 or271t1 was inconsistent with the average time of the kaon  ber 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 eventswith known incorrect solutions for the single-pulse  t times.The  rst 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 StageIn the third stage of CCDBADTIM each kaon  ber passed on from the second stage waschecked to see if the  ber 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 eventfailed the cut. The time of the  rst-pulse in the double-pulse  t t1 was less than -9.98 ns. The time of the second-pulse in the double-pulse  t t2 was less than -4.99 ns. The time t1 was not consistent with the average time of the kaon  ber hits tk. The agging conditions were t1  tk <  6ns or t1  tk > 7ns. This choice of cuttingparameters is discussed in Section E.5.E.3.2 CCDBADTIM Fourth StageIn the fourth stage of CCDBADTIM, the times found in the single-pulse  t were exam-ined for both the high-gain and the low-gain CCDs for each kaon  ber. This process wascompletely independent of the  rst three stages of the cut and an event failed CCDBAD-TIM if it failed due to the conditions laid out in the  rst three stages or if it failed dueto the conditions checked in the fourth stage. The fourth part of the cut was designed272Decay Vertex 438 Raw HighDecay Vertex 438 Raw LowDecay Vertex 438 Resid SnDecay Vertex 438 Resid Sn0102030405060700 20 40 60 80 100 120-10-7.5-5-2.502.557.51012.50 20 40 60 80 100 120-5050 20 40 60 80 100 120-5050 20 40 60 80 100 120Figure E.1: An event that failed the CCDBADTIM cut due to the  tter trying to  t the energeticsecond pulse as the kaon pulse. The single- t probabilities for both the high-gain (top) and low-gain CCDs were above 0.5 as shown in the row \Prob" and column \Single". The times of thesingle-pulse  ts (row \Time") show that the second pulse in each of the high-gain and low-gainCCDs were  t as the single-pulse since the  t times agreed with tpi (\tpi 39.892") and not tk(\tk 2.776").273to reject events where a known incorrect solution was found for the single-pulse  t when(1) a double-pulse  t was not performed or (2) a good solution to the double-pulse  t wasnot found.For each  ber, each of the two CCD channels were checked to see if they had a single-pulse  t probability above 0.25 and that the energy from ADC was above 1.25 MeV. Ifso and either of the following conditions were met, the examined CCD channel for thatkaon  ber was  agged for possible rejection by CCDBADTIM. The single-pulse  t time t0 was less than -9.98 ns. The single-pulse time (t0) was not consistent with tk. The failing conditions weret1  tk <  6ns or t1  tk > 7ns. This choice of cutting parameters is discussed inSection E.5.A  ber having a CCD channel that has been  agged for possible rejection by CCDBAD-TIM caused the event to fail CCDBADTIM if Both CCD channels were  agged for possible rejection by CCDBADTIM; One CCD channel was  agged for possible rejection by CCDBADTIM and the otherCCD channel had a double-pulse  t probability of zero; One CCD channel was  agged for possible rejection by CCDBADTIM and informa-tion from the other CCD channel was missing.E.4 CCDPUL OptimizationBased on the previous PNN2 analysis of data from the E787 experiment [20], the second-pulse energy threshold above which would cause an event to be rejected by CCDPUL wasinitially 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  t t2 and the average274time of the pion  bers hits tpi was initially set to  10ns. Due to correlations between thecuts CCDBADFIT, CCDPUL and EPIONK, the optimization of these cuts was performedtogether.The acceptance sample used for the optimization of CCDBADFIT, CCDPUL andEPIONK was the K 2 monitor trigger data with very similar setup cuts as were usedfor measuring the components of acceptance of Abeam (Section 6.1.3). Figure E.2 showsthe setup cuts applied to measure the acceptance of CCDBADFIT, CCDPUL and EPI-ONK with 136263 events remaining upon which to measure the acceptance. These cutsselected signal-like events for the target reconstruction by selecting K+ decays with asingle charged track, no photons, no beam contamination and no other activity in thedetector. To minimize the  + scatters in the target, the events were further constrainedto be in the K 2-peak by restricting the momentum ptot to be between 229 and 245MeV/c (PTOT 229 245) and restricting the energy deposited in the Range-Stack (Ers)to be between 120 and 150 MeV (ERS 120 150). The BV and BVL subsystems wereexcluded from the photon veto to gain a larger acceptance for K 2 events.The rejection sample used for the optimization of CCDBADFIT, CCDPUL and EPI-ONK was very similar to the K 2 target-scatter normalization branch (Section 4.1.2).This sample used    (1) and    (2) triggers from the 1/3 data set with an invertedphoton veto, where the target was excluded from the photon veto. The setup cuts areshown 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  bers had asecond-pulse energy above E2 and had second-pulse timing coincident with the outgoingcharged 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 cho-sen based on maximizing the product of acceptance and rejection from their respective275P2PSCUT, P2TGCUT, RDUTM, TARGET, RD TRK, TRKTIM, UTC, PTOT 229 245,ERS 120 150TGQUALT, 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, ALLKFITCCDBADFIT, CCDPUL, EPIONKN1N2Figure E.2: Setup cuts for the CCDBADFIT, CCDPUL, EPIONK acceptance sample using K 2monitor trigger data. The acceptance was the ratio of the number of events N2=N1, where N1was 136263 events.P2PSCUT, P2TGCUT, P2TGPVCUT, UTC, RD TRK, TRKTIM, RDUTM, STLAYBEAMCUTS, DELC3, KINCUTS, PNN2BOX PV60(noTarget)TGCUTS(excluding CCDBADFIT, CCDPUL, EPIONK, CCD31FIB, CCDBADTIM)CCDBADFIT, CCDPUL, EPIONKN1N2Figure E.3: Setup cuts for the CCDBADFIT, CCDPUL, EPIONK rejection sample using   (1)and    (2) triggers from the 1/3 data set. The rejection was the ratio of the number of eventsN1=N2, where N1 was 3692 events.276samples. For similar values of Acceptance  Rejection, it was preferred to choose theparameters that corresponded to the largest acceptance. Note that the energy parameterE2 was also used in the cuts CCDBADFIT and EPIONK.Table E.1 shows the components of acceptance and rejection when a single parameterwas varied and the other two were held  xed. Although the optimization allowed all theparameters to vary at the same time, the table presents only this subset of the parameterspace for the sake of readability. Based on this optimization, the  nal values for theparameters were as follows: E2 = 1:25MeV; tlo =  7:5ns; thi = 10:0ns.E.5 CCDBADTIM ParametersThe section discusses the choice of time parameters for the CCDBADTIM cut:t1  tk <  6; t1  tk > 7 (E.2)andt0  tk <  6; t0  tk > 7: (E.3)as originally described in Section E.3.A sample was created using K 2 monitor trigger data with all the cuts shown in FigureE.2 applied, including CCDBADFIT, CCDBADTIM and EPIONK. The time di erencet1  tk was plotted for each kaon  ber 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 inthe third stage. Figure E.4 shows the distribution of these events. The target CCD277tlo =  10ns, thi = 10nsE2 (MeV) Acceptance Rejection Acceptance  Rejection0.75 0.299 9.764 2.9170.875 0.324 9.079 2.9441.0 0.353 8.625 3.0411.125 0.384 7.916 3.0371.25 0.415 7.289 3.0251.375 0.447 6.561 2.9351.5 0.479 6.097 2.9231.75 0.538 5.287 2.8472.0 0.591 4.585 2.708E2 =  10ns, thi = 10nstlo (ns) Acceptance Rejection Acceptance  Rejection-6 0.475 6.142 2.918-7 0.453 6.592 2.988-7.5 0.445 6.787 3.017-8 0.437 6.877 3.005-9 0.425 7.126 3.025-10 0.415 7.289 3.025-12 0.401 7.486 3.001-15 0.388 7.638 2.966-18 0.382 7.709 2.946E2 =  10ns, tlo =  10nsthi (ns) Acceptance Rejection Acceptance  Rejection6 0.434 6.809 2.9527 0.428 6.946 2.9717.5 0.425 7.005 2.9798 0.423 7.077 2.9939 0.419 7.163 2.99810 0.415 7.289 3.02512 0.409 7.514 3.07015 0.402 7.782 3.12818 0.397 7.977 3.169Table E.1: Acceptance and rejection results for the combined group of cuts CCDPUL, CCD-BADFIT and EPIONK. The three parameters E2, tlo and tlo were varied to maximize theproduct of acceptance and rejection. Each of the three sections of the table show one of thethree parameters being varied while the other two were held constant.278Figure E.4: Setting the allowed time window for consistency between t1 and tk in the CCD-BADTIM cut. The plot shows t1  tk for the sample of events described in the text of SectionE.5. The bounds for the cut were set to t1  tk < 6ns and t1  tk > 7ns as indicated by thered lines on the plot.information for many events falling outside these bounds were visually examined andwere found to be consistent with the second target CCD  tter failure mode described inthis section.A similar study was performed for the time di erence t0 tk and the same bounds aschosen for t1  tk were found to be appropriate.The  nal acceptance of the safety cut CCDBADTIM was measured to be 0.99 asdetailed in Section 6.1.3.279E.6 Updating the De-multiplexing AlgorithmAn 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  ber CCD information was being de-multiplexed for use in the CCDPUL cut. The low-gain target  ber CCDs were multi-plexed in groups of  ve  bers that were spatially found far apart from each other withinthe target. When more than one  ber in a multiplexed group had a hit at the same time,the amplitude information for that group had to be de-multiplexed using ADC informa-tion from each of the  bers to determine if second-pulse activity in a studied  ber wasdue to activity in that  ber of in another  ber within the same group.E.6.1 Previous De-multiplexing AlgorithmIn the previous de-multiplexing algorithm the second-pulse energy E2 found by the double-pulse  tter was corrected for the ADC energies from the other  bers in the group for energywithin  5 ns of tpi. This energy came from  bers assigned as pion, opposite-side pionor gamma  bers (see Section 3.3.4 for a discussion of target  ber assignment). The sumof the coincident energy from the other  bers was called the multiplexed pion energyand this energy was subtracted from the E2 value found by the  tter to give a correctedsecond-pulse energy E2(corr.).The total ADC energy for a  ber was split between the two pulses in a double-pulse t to give an energy E1 to the  rst pulse and an energy E2 to the second pulse. Thisenergy was split up based on the relative amplitudes of the two pulses as found by thedouble-pulse  tter. The  aw in the just described de-multiplexing algorithm was that ifany multiplexed pion energy was found, the sum of E1 and E2(corr.) was not equal tothe original ADC energy of the  ber. Since the target  ber ADCs were not multiplexed,the corrected energies of the two pulses should always have added up to the total ADCenergy for that  ber.280E.6.2 New De-multiplexing AlgorithmThe algorithm was modi ed so that the ADC energies near tk and near tpi from the other bers in a group were taken into account when dividing the ADC energy in the studied ber between the  rst and second pulse. The steps in the new de-multiplexing algorithmwere as follows:1. The multiplexed pion energy was determined using the same method described inSection E.6.1.2. ADC energies within  5 ns of tk from the other kaon  bers within the group weresummed to create the multiplexed kaon energy.3. The pion and kaon multiplexed energies were added to the ADC energy of thestudied  ber to get a total energy as seen by the low-gain CCD channel of that ber.4. This total CCD energy was split between the  rst and second pulses according theratio of the  tted amplitudes to give E1 and E2, respectively.5. The kaon multiplexed energy was subtracted from E1 to give the corrected energyE1(corr.) and the pion multiplexed energy was subtracted from E2 to give the cor-rected energy E2(corr.).6. If either of the resulting corrected energies was below 0.001 MeV, it was assigned anenergy of 0.001 MeV and the other pulse was assigned the remaining ADC energyensuring the total energy from the two pulses was equal to the ADC energy of the ber.This new de-multiplexing algorithm resulted in slightly less than 1% additional eventsfailing the CCDPUL cut in both the acceptance and rejection samples used for the op-timization of CCDPUL (Section E.4). Figure E.5 shows an event that passed CCDPUL281when the previous de-multiplexing algorithm was used, but failed the cut after the newalgorithm was implemented. These events typically had a corrected E2 that was slightlybelow the 1.25 MeV threshold when the old algorithm was used and slightly above the1.25 MeV threshold when the new algorithm was used.282Kaon fiber   278 Raw HighKaon fiber   278 Raw LowKaon fiber   278 Resid SnKaon fiber   278 Resid Sn0204060801001201401601800 20 40 60 80 100 1200510152025303540450 20 40 60 80 100 120-5050 20 40 60 80 100 120-5050 20 40 60 80 100 120Old Algorithm New AlgorithmMultiplexed kaon and pion energies (MeV)Ek(mux) N/A 0.E (mux) 1.58 1.58Fitted pulse energies from ADC energy (MeV)Total Energy 12.28 12:28 + 0:00 + 1:58 = 13:86E1 9.53 10.76E2 2.75 3.10Corrected energies (MeV)E1(corr.) 9.53 10.76E2(corr.) 2:75  1:58 = 1:17 3:10 1:58 = 1:52Figure E.5: Newly Rejected CCDPUL event due to new de-multiplexing algorithm. Using theold algorithm, the event passed CCDPUL with 1.17 MeV in the second pulse (E2(corr.)). Usingthe new algorithm, the event passed CCDPUL with E2(corr.) = 1.17 MeV. The multiplexed pionenergy E (mux) came from a pion  ber in the same group that had a time of 5.98 ns and energyof 1.58 MeV.283Appendix FAcceptance Factors for theK+ !  + 0 Branching RatioThis appendix details the acceptance measurements used to  nd the total acceptance(A 2) used in the measurement of the K 2 branching ratio.F.1 Acceptance Factors Using Monte CarloThe trigger (AMCtrig) and kinematic (Akintrig) components of acceptance were measured usingK 2 events from the Monte Carlo simulation. The acceptance losses due to pion decay-in- ight and pion nuclear interactions were also measured due to the inclusion of theseprocesses in Monte Carlo. The cut KP2STOP required that the stopping layer be betweenlayers 8 and 15 inclusive. The cut UFATE required that the pion stopped without decayor interaction. The cut USTMED required that the pion stopped in the Range-Stack scin-tillator. The cut USTOP HEX required that the o ine reconstructed stopping counteragreed with the true stopping counter. These four cuts used information taken directlyfrom the Monte Carlo event and not the subsequent reconstruction. The cut SETUPwas a requirement that the reconstructed momentum ptot was less than 300 MeV. The284Cut Events AcceptanceSETUP 99993T 2 44891 0.44896ct + 7ct 37605 0.837719ct 36986 0.9835UFATE 31222 0.8442USTMED 30518 0.9775USTOP HEX 27426 0.8987AMCtrig 0.2743  0.0014RDUTM 27426 1.0000UTCQUAL 26910 0.9812TARGET+TGQUALT 25659 0.9535KP2STOP 24639 0.9603COS3D 23671 0.9607KP2BOX 20213 0.8539AMCkin 0.7878  0.0026Table F.1: Monte-Carlo-based components of acceptance for theK 2 branching ratio calculation.The cuts \RDUTM", \UTCQUAL" and \TARGET+TGQUALT" were used as setup cuts forthe acceptance AMCkin . Table reproduced from [55].results for these measurements are found in Table F.1.F.2 Acceptance Factors from K 2 Monitor TriggerEventsThe setup cuts applied to measure each of Ard, Arecon and Abm tg are shown in TableF.2. For each of the measurements, three additional setup cuts were applied as part ofthe TRIGGER cut: (1) RNGMOM to remove K 2 events, (2) the o ine stopping layerhad to be found in the Range Stack, and (3) the momentum was required to be in therange of 0MeV=c < ptot < 300MeV=c. The second and third cuts were used to ensuregood reconstruction. The results of these acceptance measurements are found in Tables285K 2 Setup Measured Setup CutsCategories QuantitiesSetuprd Ard TRIGGER, ICBIT, tIC  tCK > 5ns,B4DEDX, UTC, TARGETSetuprecon Arecon TRIGGER, ICBIT, tIC  tCK > 5ns,B4DEDX, CPITRS, CPITAIL, CKTRS, CKTAIL,BWTRS, RVUPV, Ard cutsSetupbm tg Abm tg TRIGGER, ICBIT, Ard cuts, Arecon cuts,KP2BOX, KP2STOP, IPIFLG, COS3DTable F.2: The setup cuts applied for the components of acceptance measured for the K 2branching ratio calculation. The notation \Ard cuts" means that all the cuts whose acceptancewere measured to determine the quantity Ard were applied as setup cuts. Table reproduced from[55].Cut Events AcceptanceSetuprd 490579RD TRK 490579 1.0000TRKTIM 490579 1.0000Ard 1:0000 0:0000Table F.3: The acceptance Ard for the K 2 branching ratio calculation. Table reproduced from[55].F.3, F.4 and F.5.F.3 Calculation of Total AcceptanceThe total acceptance, A 2, used in the measurement of the K 2 branching ratio was theproduct of the components of acceptance determined in this appendix thus far in additionto four more acceptance factors previously determined in the PNN1 analysis [33]:AK 2 = AMCtrig  AMCkin  Ard  Arecon  Abm tg  Aacc  Aipi g  fs   K 2T 2 (F.1)286Cut Events AcceptanceSetuprecon 449621RDUTM 449621 1.0000UTCQUAL 407402 0.9061TARGET+TGQUALT 386491 0.9487Arecon 0:8596 0:0005Table F.4: The acceptance Arecon for the K 2 branching ratio calculation. Table reproducedfrom [55].Cut Events AcceptanceSetupbm tg 336407TIC 336173 0.9993TIMCON 334866 0.9961TGTCON 331041 0.9886DCBIT 293685 0.8872DELC 257163 0.8756CKTRS 250305 0.9733CKTAIL 243233 0.9718B4DEDX 239389 0.9842CPITRS 237015 0.9901CPITAIL 236833 0.9992TARGF 226972 0.9584DTGTTP 226965 1.0000RTDIF 224789 0.9904TGQUALT 224789 1.0000PIGAP 222488 0.9898TGB4 207545 0.9328KIC 204785 0.9867TGGEO 169847 0.8294B4EKZ 157338 0.9264B4ETCON 156673 0.9958TGZFOOL 154703 0.9874BWTRS 148013 0.9568RVUPV 145339 0.9819Abm tg 0:4320 0:0009Table F.5: The acceptance Abm tg for the K 2 branching ratio calculation. Table reproducedfrom [55].287Value SourceAMCtrig 0.2743  0.0014 Table F.1AMCkin 0.7878  0.0026 Table F.1Ard 1.0000  0.0000 Table F.3Arecon 0.8596  0.0005 Table F.4Abm tg 0.4320  0.0009 Table F.5Aacc 0.9931  0.0002 [33]Aipi g 0.8350  0.8350 [33]fs 0.7740  0.0011 [33] K 2T 2 0.9383  0.0027 [33]AK 2 0.04833  0.00047Table F.6: Summary of the components of acceptance used to  nd AK 2 for the K 2 branchingratio calculation.The conditions for these additional acceptance factors were the same for PNN2 as theywere for PNN1. These acceptance factors were: Aacca5- the acceptance loss due to the 19ct requirement in the K 2 monitor trigger(Section 2.7.2); Aipi g - Acceptance of the IPIFLG cut as measured using  scatter monitor triggerdata; fa54 - K+ stopping fraction (discussed previously in Section 6.7);  a55a57a562Ta58 2 - Correction due to T 2 trigger ine ciencies for K 2 events. This correctionfor K+ !   events in the PNN2 phase space was discussed in Section 6.6.The resulting acceptance AK 2 was 0:04833  0:00047, where the values used in thisacceptance calculation can be found in Table F.6.288

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 20 0
China 19 28
United Kingdom 9 0
Netherlands 2 0
City Views Downloads
Beijing 8 0
Unknown 8 1
Washington 6 0
Dongguan 4 0
Ashburn 4 0
Berkeley 3 0
Shenzhen 3 28
Liverpool 3 0
Ürümqi 2 0
Hangzhou 2 0
Wilmington 2 0
Mountain View 2 0
Amsterdam 2 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0066858/manifest

Comment

Related Items