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Measurement of the decay K+-->[pi]+vv Bergbusch, Paul C. 2000

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M E A S U R E M E N T O F T H E D E C A Y K+ ir+vil by Paul C. Bergbusch M . S c , University of Br i t i sh Columbia, 1995 B.Sc. (Honors), Simon Fraser University, 1993 A thesis submitted in partial fulfillment of the requirements for the degree of D O C T O R OF P H I L O S O P H Y in the Faculty of Graduate Studies D E P A R T M E N T OF PHYSICS A N D A S T R O N O M Y We accept this thesis as conforming to the required standard. D . A . Bryman (supervisor), Dept. of Physics and Astronomy, U B C ' j . McKenna , Dept. of Physics and Astronomy, U B C D . F . Measday, Dept. of Physics and Astronomy, U B C A . Zhitnitsky, Dept. of Physics and Astronomy, U B C T H E UNIVERSITY OF BRITISH CO L U MBIA October 2000 © Paul C. Bergbusch, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada 0 c + \ 12-, 2oo0 Date • ) DE-6 (2/88) Abstract Experiment 787 at Brookhaven National Laboratory, is a sensitive search for the decay of a positively-charged kaon into a positively-charged pion, a neutrino, and an anti-neutrino: K+ —> TT+UP. This extremely rare decay is expected to occur in the Standard Model wi th a branching ratio of (0.8 ± 0.3) x 1 0 " 1 0 . It also serves as a hunting ground for new physics. Da ta collected between 1995 and 1997 contains one event consistent wi th K+ —> 7r + ^P, wi th background estimated to contribute 0.08 ± 0.02 events. Because this event survives the tightest data-selection cuts which further reduce the estimated background to 0.006 ± 0.002 events, it is likely due to K+ —• ir+vv. 3.24 x 10 1 2 kaons were collected wi th an acceptance for K+ of (0.21 ± 0.02)%, resulting in a K+ - * n+uP branching ratio of 1.5±i;a x 10~ 1 0 . The weak coupling of top to down quarks, parameterized by the Cabibbo-Kobayashi-Maskawa matrix element Vtd, is calculated to lie in the range 0.002 < |Vtd| < 0.04. The upper l imit on the K+ —• n+f branching ratio, for massless / and for no events observed, is 1.1 x 1 0 - 1 0 at the 90% confidence level. i i Table of Contents Abstract ii Table of Contents iii List of Figures vi List of Tables ix Acknowledgements xiii Preface xv 1 Introduction 1 2 Theory 7 2.1 Weak Interactions 7 2.2 K+ -> TT+UP 12 •2.3 Vtd 16 2.4 New Physics 17 3 Experiment 20 3.1 K a o n Product ion and Transport 21 3.2 Detector 24 3.2.1 Beamline Detectors 24 3.2.2 Target 28 3.2.3 Drift Chamber 29 3.2.4 Range Stack 32 i i i 3.2.5 Photon Veto 34 3.2.6 Monte Carlo Simulation ' 38 3.3 Da ta Acquisi t ion 40 4 Analysis 47 4.1 Backgrounds 47 4.1.1 Kn2 and Decays 52 4.1.1.1 1 2 C Giant Dipole Resonance Background 54 4.1.2 Beam Backgrounds and K a o n Charge Exchange 63 4.2 Analysis Strategy and Techniques 66 4.3 Da ta Analysis and Cuts 71 4.3.1 Function Cuts 77 4.4 Background Measurement Structure 88 4.4.1 K*2 Decay 92 4.4.2 Decay 94 4.4.3 Beam Backgrounds 98 4.4.4 K a o n Charge Exchange 105 4.5 Background Measurement Results 110 4.6 Outside-the-Box Tests 113 4.7 Signal Evaluation Cri ter ia 123 4.8 Search for Signal 125 5 Acceptance and Sensitivity 132 5.1 Counting Kaons 133 5.2 i \^ 2 -based Acceptance 134 5.3 K 7 r 2 -based Acceptance 135 5.4 7r s c a t-based Kinematic Acceptance 137 5-5 7r s c a t-based T D Acceptance 139 5.6 ^x-veto Accidental Loss . . 146 5.7 T • 2 Efficiency 146 iv 5.8 UMC-based Acceptances 151 5.9 K a o n Stopping Fraction, f„ 160 5.9.1 Measurement of fs 161 5.9.2 Measurement of the Branching Rat io 162 5.10 Summary 171 6 Final Results 175 6.1 K+ —> TT+VV Branching Ratio 175 6.2 K+ -» ir+f Upper L i m i t 176 6.3 Vtd 177 7 Conclusions 179 Bibliography 187 A B N L E787 Collaboration 192 B Personal Contributions 193 C Detailed Cut Descriptions 195 C l P A S S 1 195 C.2 PASS2 200 C.3 P A S S 3 202 C.3.1 Kinemat ic Pathology Cuts 202 C.3.2 Beam Pathology Cuts 208 C.3.3 Photon Veto Function 213 C.3.4 T D Function Cuts 214 C.3.5 Kn2 and K^2 Kinematic Function Cuts 223 C.3.6 Beam Function Cuts 227 C.4 Da ta Quali ty Cuts 232 D Glossary 235 v List of Figures 1.1 The fundamental particles and forces 3 2.1 Unitar i ty triangle in the complex (p, rf) plane 12 2.2 Short-distance Feynman diagrams for K+ —• TT+VV 13 2.3 A long-distance Feynman diagram for K+ —> ix+vv 15 2.4 Relationship between K+ —> K+VV and \Vtd\ in the {p,rj) plane 18 3.1 Accelerator facilities at B N L 22 3.2 Low energy separated beamline III at B N L 23 3.3 Side view of the E787 detector 25 3.4 Side view of the beam Cerenkov detector, and downstream views of the B W C 1 and B W C 2 detectors 27 3.5 Downstream view of the target 30 3.6 The "ultra-thin chamber" ( U T C ) 31 3.7 The R S shown in cross section. 4> is the angle in the (x,y) plane about the center of the detector 33 3.8 Pulses found in the upstream and downstream T D ' s in R S layers 11-14 for a pion track stopping in layer 12 35 3.9 The "barrel veto" ( B V ) and E C shown in cross section 36 4.1 R vs. P, and kinematic categorization of events passing the K^iX) trigger. . 49 4.2 R vs. P for the events in figure 4.1 which pass the full -K+ui>{\) trigger. . . . 50 4.3 Momentum phase space of the charged track from K+ decays 51 4.4 vs. E for range-tail events 58 4.5 tfj, vs. Efj, for range-tail events 59 v i 4.6 Momentum distribution of muon-band events 61 4.7 vs. Efis(diS) for muon-band events 62 4.8 Time spectra of single counter hits in the R S 64 4.9 A background estimate resulting from a bifurcated analysis 68 4.10 Outside-the-box correlation study 69 4.11 Event display of a successfully reconstructed event . 74 4.12 Close-up' of the target hits for the event shown in figure 4.11 75 4.13 Close-up of the RS hits for the event shown in figure 4.11 76 4.14 The P V function 82 4.15 The T D function 84 4.16 The K„2 kinematic function 86 4.17 The jfT 2^ kinematic function 87 4.18 The double-beam function 90 4.19 The C E X function 91 4.20 Kn2 background estimation structure 93 4.21 K^a background estimation structure 96 4.22 Single-beam kaon-entering background estimation structure 99 4.23 Single-beam pion-entering background estimation structure 101 4.24 Double-beam kaon-entering background estimation structure 102 4.25 Double-beam pion-entering background estimation structure 104 4.26 Target and IC displays of the " K I C " event 106 4.27 Target- and IC displays of the " T G G E O " event 107 4.28 Target display of the " T G D E D X " event I l l 4.29 R vs. E for events in the 1995-7 K+ —> n+uu data set passing all other cuts in the analysis 127 4.30 R vs. E for UMC-generated K+ -»• n+uu events 128 4.31 Display of the candidate K+ —> n+vv event 129 4.32 Close-up of the target and R S hits for the candidate K+ —> TT+VP event. . . . 130 vi i 5.1 T • 2 efficiency as a function of run number for K^2 and decays, and the Kfja/K^ ratio of efficiencies. 149 5.2 Number of T • 2-inefficient events for Kw2 (top) and (bottom) decays as a function of the azimuthal angle c/> of the track at the T counter (in ir radians). 150 7.1 Constraints from astrophysics and K+ —• n+f on the masses of the axion and familon 183 7.2 Experimental progress of the search for K+ —> ir+vD 185 v i i i List of Tables 3.1 Total 7 r + , 1 2 C interaction cross section, divided into 7 channels . 39 4.1 3-pulse signature in the T D of the stopping counter, for signal (K+ —• n+uP) and background (K+ —> ^ + / v M ) 54 4.2 Tools used to suppress background processes for K+ —> ir+v& 55 4.3 PASS1 cuts 73 4.4 P A S S 2 cuts 73 4.5 Definitions of the PASS2 output background data streams 77 4.6 P A S S 3 kinematic pathology cuts 78 4.7 P A S S 3 beam pathology cuts 79 4.8 P A S S 3 data quality cuts 80 4.9 P A S S 3 T D cuts 83 4.10 P A S S 3 kinematic function cuts 85 4.11 P A S S 3 beam function cuts 89 4.12 Numbers of events B* and C, CDi from the normalization and rejection branches, respectively, of figures 4.20 and 4.21, and the resulting Kn2 and normal-izations, rejections, and scaled background levels 112 4.13 Numbers of events J5j and C,CDf from the normalization and rejection branches, respectively, of figures 4.22 and 4.23, and the resulting single-beam normalizations, rejections, and scaled background levels 114 4.14 Numbers of events Bj and C, CD\ from the normalization and rejection branches, respectively, of figures 4.24 and 4.25, and the resulting double-beam normal-izations, rejections, and scaled background levels 115 4.15 Number of background events expected in the n+uP(l) signal region 116 ix 4.16 Predicted and observed numbers of Kw2 and K^2 background events 118 4.17 Predicted and observed numbers of beam background events 119 4.18 • Complete listing of cuts applied in the analysis, divided into pathology cut classes 120 4.19 Predicted and observed numbers of al l types of background events in the 2/3 1996-7 data sample, obtained by loosening al l functions at the same time. . . 121 4.20 Number of background events expected in the golden region 124 4.21 Number of background events expected in the tight golden region 125 5.1 KBUVC counting in the 1995-7 analysis. The variation in Ksiive wi th each year is primarily due to variation in the data collection time allotted to E787 each year 133 5.2 iC^-based acceptances of cuts 136 5.3 S E T U P cuts used in the Z ^ - b a s e d acceptance measurement 137 5.4 K 7 r2-based acceptances of cuts 138 5.5 S E T U P cuts used in the K^-based acceptance measurement 138 5.6 7r s c a t-based acceptances of kinematic cuts 140 5.7 S E T U P cuts used in the 7r s c a f-based kinematic acceptance measurement. . . 141 5.8 7r s c a t-based acceptances of T D cuts 142 5.9 S E T U P cuts used in the 7r s c a t-based T D acceptance measurement 143 5.10 Pion-nuclear absorption and decay in flight (NIDIF) contamination in the 1995 K+ -y ir+vD U M C data 144 5.11 Accidental-induced acceptance of the trigger/i-veto 147 5.12 Non-gap T • 2 efficiencies, e£ 9 2 , for K^2 and decays and kaon decays into 7r+uu and 7r/ 152 5.13 UMC-based K+ —• ir+vv trigger acceptance 153 5.14 UMC-based K+ —>• 7 r + / trigger acceptance 154 5.15 UMC-based n+uu fiducial acceptance 155 5.16 UMC-based irf fiducial acceptance 155 x 5.17 Measured raw values of R, E, and P for the and peaks, and linear scaling factors 157 5.18 Scaling factors which are applied to R, E, and P values from UMC-generated data 158 5.19 Smearing factors which are applied to scaled values of R, E, and P from UMC-generated data 158 5.20 Means and cr's of Gaussian fits to kinematic variables from real Kn2 data, and from U M C K^2 data after scaling and smearing 159 5.21 Means and cr's of Gaussian fits to kinematic variables from real data, and from U M C K^2 data after scaling and smearing 159 5.22 Summary of UMC-based acceptances for K+ —> -K+VV events. The quoted uncertainties are purely statistical 160 5.23 Summary of UMC-based acceptances for K+ —> n+f events. The quoted uncertainties are purely statistical 160 5.24 Cuts applied to ^ 2 ( 1 ) monitor data in order to measure the K^2 branching ratio 163 5.25 iiT^-based acceptance of cuts applied in the K^2 branching ratio measurement. 164 5.26 UMC-based acceptances of cuts applied in the branching ratio measurement. 165 5.27 Values of {K^/^K^ f ° r e a c n r u n period 165 5.28 Cuts applied to K^2{\) monitor data in order to measure the Kn2 branching ratio 167 5.29 Monitor-based acceptances of cuts applied in the Kn2 branching ratio mea-surement 169 5.30 UMC-based acceptances of cuts applied in the Kn2 branching ratio measurement. 170 5.31 Values of (KeBuve)K„2 for each run period 171 5.32 K + —> TT+VP single-event sensitivity of the 1995-7 analysis 172 5.33 K+ —> 7 r + / single-event sensitivity of the 1995-7 analysis 173 x i 6.1 Calculat ion of the probability that a K+ —> ir+f event, for massless / , has the R, E, and P values of the candidate event 177 C l Definitions of the categories of P V hits 215 C.2 P V function cut parameters and performance at the NpV < 1.0 cut position. 216 C.3 P V function 217 C.4 T D function 224 C.5 Kn2 kinematic function 226 C.6 K^2 kinematic function 228 C.7 Double-beam function "inside the box" 232 C.8 C E X function "inside the box" 233 x i i Acknowledgements Back in 1995, when I decided I wanted to continue wi th graduate schooling and enroll as a P h . D . candidate, I wasn't sure what I was getting into. M y undergraduate degree at Simon Fraser University started in biology and chemistry, and my intention was to eventually apply to medical school. However, chemistry appealed to me more than biology, and soon I realized that I enjoyed physics the most, so I ended up wi th an undergraduate degree in chemical physics. In my last year of undergrad, I took some nuclear science courses which I found very interesting, especially wi th respect to the study of nuclear reactions that are believed to take place in stars and in the early universe, which are responsible for the materials that now make up our bodies and planet. T R I U M F was just down the road, and I figured a graduate degree was probably a good idea, so I completed a M.Sc . degree at the University of Br i t i sh Columbia in experimental nuclear physics wi th Mike Hasinoff. Mike fostered my interest in subatomic physics, and when I started looking for P h . D . opportunities, Peter Gumplinger informed me of a particle physics experiment in New York which sounded interesting, in that it seemed to deal directly wi th the smallest fundamental particles that make up the matter around us. I thought the transition from nuclear to particle physics would be as natural and as smooth as my earlier transitions from biology to chemistry, atomic physics, and eventually to nuclear physics. However, I felt somewhat lost in the international, collaborative world of particle accelerators and high-energy physics, and I had to adjust to my new environment, much like other people in particle physics had to be accommodating and adjust to me. For this reason, I would like to thank Renee Poutissou, Toshio Numao, and John MacDonald for their considerable help. I 'd like to thank A k i r a Konaka, George Redlinger, Joe Mildenberger, and Norihito Muramatsu who played a large role in my P h . D . analysis and the completion of this dissertation. I 'd like to thank the many other bright and dedicated E787 scientists x i i i and technicians in Vancouver, New York, Princeton, and Japan who were instrumental in the acquisition and analysis of the high-quality data on which this thesis is based. I 'd like to thank Doug Bryman, my P h . D . supervisor, for his guidance and insight in helping me complete a P h . D . degree, which, as I've come to learn, is no small feat. Over the past 5 years these people have helped me grow a lot, both as a scientist and as a person. I 'd also like to thank the many good friends I've met over the years, whether at school or elsewhere, who have supported me and made my various pursuits, including graduate school, fulfilling experiences. Last, but most, I 'd like to thank my family - John, Dawn, Dave, Aaron, Mark , and Al ison - for al l they have meant to me over the years. They are an integral part of who I am, and this work is as much theirs as it is mine. x iv Preface This preface is designed wi th two purposes in mind: to give the reader some help in reading the thesis, and to outline some very basic reasons as to why one would want to pursue the topic of this thesis in the first place. Chapter 1 is an introduction to particles and forces, which may, in part, be accessible to non-scientific readers as well as to scientific readers. Non-scientific readers wi l l also hope-fully find this preface, the abstract, and parts of chapter 7 interesting. The remainder of the thesis is quite technical. However, the thesis in its entirety is writ ten as clearly and conceptually as possible so that particle physicists, as well as other scientists, can follow and fully understand it. Much detail is included, but hopefully not so much as to disturb textual flow. A glossary is included for aid in understanding jargon and acronyms used in the text. The following may not make much sense unti l much of the thesis is read, but here it is for the record: analysis "variables" and "cuts" are introduced in chapter 4, and are consistently denoted using italicized and C A P I T A L I Z E D text, respectively. Numbers of "events" which remain after reduction of the data are consistently denoted using the quantities B, C, CD, and M. "Scaled" numbers of events which are used as parameters in "function cuts" are con-sistently denoted using the quantity N. Numbers of events in the "normalization branches" of "bifurcated analyses" are denoted as B (B*) when optional "second bifurcations" are (are not) employed in the analyses. Now for the interesting question: why does one feel motivated to study particles and forces? A general answer, which can be used to justify scientific inquiry as a whole, is: scientific study inevitably leads to discoveries of new "phenomena", which require the devel-opment of new ideas in order to better understand how everything in the universe interacts and evolves. The scientific process by which this knowledge is obtained can be described xv very clearly in language and mathematics, and can be repeated many times in order to test the validity of the knowledge that is obtained. In this sense, one can have a lot of confidence in scientific ideas, which can then be used to solve human problems as well as to provide direction for human growth. Scientific understanding of how the world works can even bring satisfaction and peace of mind, to some people. Most people crave some kind of knowledge, be it scientific, religious, or otherwise, in order to understand and peacefully coexist wi th their world. A knowledge of why things are the way they are aids in human survival and plays a role in social and poli t ical stability. Whi le scientific ideas are aimed at alleviating struggle and improving quality of life through, for example, advances in medicine and engineering, religious and moral ideas are typically very personal and are aimed more at an acceptance of life's daily grind. Unlike scientific ideas, religious ideas usually cannot be tested, and are accepted more out of faith and intuition. If one is looking for an absolute "truth" or "meaning of life" however, then both science and religion, in my opinion, have thus far failed. No single religious idea has ever united the human species wi th a common view of existence. A n d while science is adept at finding patterns in nature which give insight into the structure of the world, new questions about the patterns, and new patterns themselves, always seem to emerge, which suggests that scientific inquiry is a never-ending process which can never result in "full understanding." Perhaps, then, there is no absolute truth, or grand concept, or single principle which mani-fests everything. A n d / o r perhaps conventional science and religion are intrinsically l imited in scope. Western science and religion seem to strive, somewhat, for "control" of the natural workings of the world, so that various "phenomena" can be predicted preceding, during, and after a human's life. Human beings can then exist wi th the least perceived amount of uncertainty and unhappiness. Scientists try to understand and predict a pattern in nature by isolating a system, "reducing" it down to its component parts, and using these parts to model reality. For example, particle physicists bui ld particle accelerators in order to isolate systems of particles and forces which emulate, to some degree, conditions in the early universe. They hope to be x v i able to construct patterns observed in the universe out of a small set of particles and forces which they observe in the laboratory. Since the advent of quantum mechanics, most scientists now realize that the very act of observing a system changes it, but the scientific approach is st i l l largely based on "objectivity," which aims to remove the observer from the observed. That is, while the laboratory approach of particle physics may help explain how a star evolves, it wi l l not be able to explain how human beings interact wi th and "experience" the rest of the universe, because this element has been removed from the system. The universe is not like this however: we are in this universe, so the system studied in the lab is necessarily artificial. For this reason, increased knowledge and insight about particles and forces might be more readily obtained by observing high-energy processes which are naturally occurring elsewhere in the universe, such as in other star systems and galaxies, because these systems have not been artificially constructed. St i l l , the act of observing anything causes us to see what we have unconsciously decided we want to see, and the "reality" of the situation has to somehow incorporate all of the observed, the observer, and the act of observation. One would hope to avoid subject/object, conscious/unconscious, and mind /body distinctions. In fact, " A science that attends to such relationships rather than to so-called discrete entities would be a science of what has been called 'participant observation,' and it is this type of holistic thinking which might hold the key to future human evolution" [1]. There is a proposed theory in physics which has been labelled the "theory of everything" ( T O E ) , which aims for a mathematical unification of quantum mechanics and the theory of general relativity. This theory attempts to unite the world of the very, very small (subatomic) wi th the world of the very, very big (cosmic). These worlds are definitely related, as is everything in the universe, because everything we can conceive of is part of the same universe. So the idea that these two worlds can both be described by the same mathematical language seems to make sense. However, the name of this theory may be an exaggeration. It's difficult to picture a theorist sitting in a room with a pen in his/her hand, wri t ing down some formula, and then saying "Wow - now we know everything." If we truly knew everything, then we would know why we are conscious entities who exist in time. In fact, every thought, feeling, action, observation, and "phenomenon" is a process in time. The big mystery then, it seems xv i i to me, which is difficult to address, is "what is time?" I spend a lot of time thinking about this question, because it seems to be important to my own peace of mind. I believe that some sort of tangible answer can be found, but not completely v ia conventional scientific inquiry. This is because the question is more than an inquiry; it probes the very act of inquiry itself. Furthermore, a complete and meaningful answer to this or any question, in my opinion, should be an answer that is clear and intu-itive to al l . For example, Charles Darwin's theory of "survival of the fittest" establishes an intimate and intuitive connection between the progression of time and the evolution of ter-restrial "life". Almost anyone can look at a giraffe and understand immediately that its long neck helps it to eat leaves at the tops of trees, and that this "adaptation" gives the giraffe a "survival advantage" which ensures its longevity wi thin time. A more general connection between the progression of time and the features of the universe could be extended to in-corporate not only the survival of terrestrial species, but also the structure and dynamics of galaxies, the stability of subatomic particles, and the development of human consciousness. I believe an understanding of time can at least be approached using scientific methods. In particular, studying fundamental particles and forces in the lab brings one into direct contact wi th the magic and mystery of the universe, and gives one a powerful set of mental tools for t rying to answer fundamental questions. Scientific tools on their own, at least for me, however, are not sufficient: pure intuition, feeling, and expression are equally valuable. I value personal interpretations of the world as much as I value scientific descriptions. Nev-ertheless, science is a dynamic field and it wi l l no doubt continue to grow and evolve in positive ways. The respect I have for fundamental science and the satisfaction that it brings me provided the motivation for 12 years of scientific training and work, which includes the work described in this thesis. xv i i i Chapter 1 Introduction A U matter, energy, and every process in the observable 15-billion-year-old universe can be "reduced" to a small set of fundamental constituents. There are two classes of material particles (quarks and leptons), three families or generations of material particles (from light to heavy), and four forces (gravitational, electromagnetic, strong, and weak). The forces and material particles are similar in the sense that a force between two particles is "mediated" v ia exchange of another particle. Two of the forces (gravity and electromagnetism) are "experienced" on large scales. For example, gravity keeps the earth in orbit around the sun, and keeps the atmosphere attached to the earth. Electromagnetism is responsible for the macroscopic properties of gases, liquids, and solids, including the friction between air and metal that enables flight. Conversely, the strong and weak forces are only effective at the subatomic level. For example, the strong force is responsible for the stability of the atomic nuclei that make up the earth and all l iving things. The weak force is responsible for the decay of the second and th i rd generations of quarks and leptons into the lighter first generation particles. A n y heavy quarks and leptons present in the early universe have long since decayed v ia the weak force, such that the vast majority of al l matter in. the universe, from the farthest, most ancient galaxies that can currently be seen, to nearby stars, planets, and people, is made out of the first generation particles. Particles can be described in terms of their electromagnetic charge, spin, and mass. There are six quarks and six leptons, both of which are grouped into "left-handed doublets" 1 Chapter 1. Introduction and "right-handed singlets". The quark doublets consist of a charge +2e/3 quark and a charge — e/3 quark, and the lepton doublets consist of a charge 0 lepton (neutrino) and a charge —e lepton, where e is the electromagnetic charge of the proton. The quarks and leptons are fermions (half-integral spin, e.g., spin = 1/2), whereas the force particles are bosons (integral spin, e.g., the electromagnetic force particle, the photon, has spin = 1). Also associated wi th every particle is an anti-particle wi th opposite electromagnetic charge but identical spin and mass. A l l quarks, leptons, and force particles, except the photon, gluon (strong force particle), and maybe the neutrinos and graviton (gravity force particle), have mass. The masses are believed to arise from "spontaneous symmetry breaking" which occurs due to the existence of a massive particle known as the Higgs boson (see, for example, the review in ref. [2]). A l l of these particles, except the graviton and Higgs boson, have been directly observed. The quarks, leptons, and force particles are shown in figure 1.1. Quarks interact v ia all four forces, whereas leptons only interact v ia three forces (all forces except the strong force). The forces have associated fields, and the mediating parti-cles are the "quanta" of these fields. The electromagnetic force is mediated by the photon, and is described by a quantum field theory known as "quantum electrodynamics" ( Q E D ) . The weak force is mediated by one of three particles, W+, W~, Z°, depending on the electro-magnetic charge of the particles involved in the weak process. The electromagnetic and weak forces are mathematically united in a single description known as Weinberg-Salam-Glashow S U ( 2 ) x U ( l ) electroweak theory [3, 4, 5]. This model has been very successful, most notably in its prediction and subsequent discovery of the W and Z bosons [6]. The strong force is mediated by gluons, and is described by a quantum field theory known as "quantum chromo-dynamics" ( Q C D ) . M u c h theoretical effort is focussed on "Grand Unified Theory" ( G U T ) which aims to extend the electroweak unification scheme to include strong interactions. The gravitational force is also believed to be mediated by a hypothetical particle called the gravi-ton, but a quantum field theory of gravity has not (yet) been realized. Gravi ty is described by the theory of general relativity, and much theoretical effort is focussed on the so-called "Theory of Everything" ( T O E ) which aims to incorporate gravity into G U T . The unification of the forces into a G U T or a T O E is desirable because it is believed 2 Chapter 1. Introduction Elementary Part icles CO o Z5 U C t up c h a r m top d s b down s t range b o t t o m GO c o Q_ a) V e V r e neut r ino fi neut r ino r neut r ino T e lec t ron m u o n tau g g luon 7 pho ton w,z W,Z bosons G grav i ton (0 CD O O CD O o Figure 1.1: The fundamental particles and forces. 3 Chapter 1. Introduction that at one time the universe was very hot and dense and essentially consisted of a single force/particle. The different forces and particles that are seen today "froze out" during the expansion and cooling of the universe caused by the "B ig Bang" , which occurred roughly 15 bil l ion years ago and gave rise to the still-expanding, still-cooling universe of today. For example, at a few millionths of a second after the B i g Bang, it is believed that quarks and gluons existed together in a hot plasma (near 2 t r i l l ion °C, evidence of which has recently been found [7]). After this point and up unti l the present day, the temperature of the expanding, cooling universe has been low enough such that the "charge" structure of the strong force is apparent. The strong force has three charges (named red, green, and blue), whereas the electromagnetic force has only a single charge (positive, and its opposite negative). Furthermore, the long-distance, low-temperature behaviour of the strong force is such that particles can only exist in chargeless states (e.g., red + anti-red, or all "colours" together: red + green + blue). That is, quarks are not directly detected as individual quarks (at least not wi th current technology operating below the quark-gluon plasma temperature). They are detected as quark + anti-quark pairs (mesons), or as three quarks bo'und together (baryons). .Mesons and baryons are collectively called "hadrons". Also, the strong, electromagnetic, and weak forces are described by "renormalizable" quantum field theories [8], which predict that the strengths of the forces are functions of energy. The strong coupling decreases wi th energy, whereas the electromagnetic and weak couplings increase wi th energy. Just as the electromagnetic and weak forces have the same intrinsic coupling at "electroweak" energy scales (> 80 G e V ) , one expects the strong, elec-tromagnetic, and weak forces to have the same intrinsic coupling at " G U T " energy scales (> 10 1 6 G e V - see, for example, ref. [9]). The matching of the strong, electromagnetic, and weak couplings can be made exact by invoking "supersymmetry" ( S U S Y ) , which predicts that each fundamental spin 1/2 fermion has a companion spin 0 boson, and each fundamental spin 1 or spin 2 boson has a companion spin 1/2 or spin 3/2 fermion. S U S Y particles have not (yet) been observed, perhaps due to their large masses predicted as a consequence of supersymmetry "breaking". Formulation of a successful G U T has also not yet been achieved. Formulation of a T O E requires compatibility of the two fundamental theories in physics: 4 Chapter 1. Introduction general relativity and quantum mechanics. A t very small distance scales, quantum-mechanical energy fluctuations ("quantum foam") are incompatible wi th the smooth, continuous space-time of general relativity. "String theory" [9] attempts to resolve this problem, and has credibility as a T O E because it naturally incorporates S U S Y , and accounts for the weak-ness of gravity relative to the other forces v ia introduction of "extra dimensions". These extra dimensions are not observed beyond the usual 3+1 spatial dimensions plus time due to "compactification". Other hypothetical particles (in addition to the Higgs boson and S U S Y particles) are "axions" and "familons", which are predicted to arise due to the spontaneous breaking of certain symmetries in nature. These particles may have non-zero mass, in which case they may be part of the "dark matter" of the universe. Most of the matter in the universe is believed to be dark matter, because the amount of visible matter in the universe is not sufficient to account for the observed rotation curves of galaxies and the dynamics of galaxy clusters and superclusters (as predicted by classical mechanics). Whi le searches for hypothetical particles and developement of a G U T and a T O E con-tinue, the S U ( 3 ) x S U ( 2 ) x U ( l ) "Standard Model" (SM) of particles and interactions is com-posed of three generations of quarks and leptons, together wi th electroweak theory and Q C D . The S M accounts for al l of the observed particles and describes a wide variety of processes wi th great precision. However, some basic questions sti l l remain. For example, why are there three generations of quarks and leptons? Is there a pattern to the specific masses of the fun-damental particles? A n d why are quarks and leptons different? One can probe the S M and look for answers to the above questions in two ways: (1) observe interactions in an energy regime that has not yet been explored (i.e., at very high energies), and (2) observe very rare, low-energy electroweak processes, because the S M allows for precise predictions of low-energy electroweak processes, and non-SM contributions to a rare process could be relatively large. The search for the decay K+ —> -K+VV utilizes the latter of these two approaches. In chapter 2, the S M theory of electroweak interactions, and its prediction for K+ —> 7r + ^P, are presented. The experimental method for measuring K+ —> n+uu is discussed in chapter 3. The analysis of data taken between 1995 and 1997 is described in chapters 4 and 5, and the 5 Chapter 1. Introduction results given in chapter 6. Conclusions are drawn in chapter 7. 6 Chapter 2 Theory This chapter continues the theoretical discussion of particle physics from the previous chapter, wi th emphasis on weak interactions and the decay K+ —> -K+VV. 2.1 Weak Interactions A s mentioned in the previous chapter, quarks interact v i a al l four forces, whereas leptons only interact v ia three forces (all forces except the strong force). Pu t another way, the strong force operates only on the quarks, not on the leptons, because only the quarks carry strong "charge". Each quark carries one of the three strong charges (red, green, or blue). The electromagnetic force operates on both quarks and leptons, but not on the lepton neutrinos because these do not carry electromagnetic charge. The pair of quarks or leptons in a given generation differs by exactly one unit of electromagnetic charge. The two quarks in each generation carry +2/3 and —1/3 units of electromagnetic charge, respectively, and the two leptons in each generation carry fj and —1 units of electromagnetic charge, respectively. The weak force operates on all quarks and leptons (as does the gravitational force). That is, all quarks and leptons carry weak "charge". Furthermore, the weak force mediators themselves (W+ ,W~, Z°) can carry one unit of electromagnetic charge, unlike the strong, electromagnetic, and gravitational mediators which do not carry electromagnetic charge (g, 7, and G). The weak force can transform a quark or lepton in a given generation into the other (lighter) quark or lepton from the same generation, while st i l l conserving 7 Chapter 2. Theory electromagnetic charge. The strong, electromagnetic, and gravitational forces cannot do this, and in fact are observed to also conserve particle "flavour". That is, a specific quark or lepton is stable and remains that specific quark or lepton regardless of whether or not it undergoes a strong, electromagnetic, or gravitational interaction. One might expect, however, that the weak force, similar to the other forces, at least conserves generation. This too is not the case. B o t h kaon decay (which involves decay of second-generation quark into a first-generation quark, s —> u) and neutron decay (which involves decay of a first-generation quark into the other, lighter first-generation quark, d —> u) are observed to occur, albeit at different rates. Recent evidence for neutrino oscillations and masses [10] suggests that the weak force also does not conserve generation in the lepton sector. However, inter-generational weak processes which involve decay of a second-generation quark into the the first-generation quark wi th the same electromagnetic charge (e.g., K\ —> fJ-+H~, and K° — K° mixing, both of which involve s —> d decay) are observed to occur at extremely low rates. That is, "neutral" weak interactions seem to conserve generation, whereas "charged" weak interactions do not. One can account for this behaviour, as well as maintain a general quark-generation-conserving structure for the weak force, by "rotating" the physical, massive quarks which undergo strong, electromagnetic, and gravitational in-teractions, into an analogous set of quarks which undergo weak interactions. This "change of basis" was originally performed v ia a unitary rotation matr ix of rank 2, which "mixes" the quarks in terms of a single parameter, the Cabibbo angle $c [11], or as the sine of this angle, A = sin#c = 0.22. A t this time, only three species of quarks were known (u, d, s) and the rank 2 rotation matrix required a fourth quark (c) which was subsequently discovered. The rotation of the physical, massive quark "eigenstates" into the weak quark eigenstates is referred to as the " G I M " (Glashow, Iliopoulos, Maiani) mechanism [12], and results in a cancellation of decay amplitudes such that "flavour-changing neutral currents" ( F C N C ) , i.e., weak processes which involve a quark changing into a different quark wi th the same electromagnetic charge (e.g., K\ —> p+p~), are not allowed. Later, the four-quark rotation matrix was generalized to the six-quark case. This rank 3 rotation matrix, called the " C K M " (Cabibbo, Kobayashi, Maskawa) matr ix [13], naturally 8 Chapter 2. Theory incorporates CP violation which had previously been discovered in neutral kaon decays [14]. Furthermore, the fifth and sixth quarks, b and t, which are the third-generation quarks, have since been found experimentally. Conventionally, one performs the C K M mixing by rotating the charge —e/3 quark mass eigenstates (d,s,b) relative to the charge +2e/3 quark mass eigenstates (u, c, t). That is, the weak eigenstates d', s', b' are related to the mass eigenstates d, s, b v ia the unitary C K M matrix: (d'\ lVud Vus Vub\ fd\ = Vcd Vcs Vcb \vtd vts vtb) The C K M matrix has four independent parameters which, in the leading order of the Wolfen-stein parameterization [15], are A,X,p,rj: (Vud Vus Vub\ ( l - A 2 / 2 A A\\p-iri)\ s' \b'J s (2.1) Vcd Vcs Vcb V Vtd Vts Vt,,) A l - A 2 / 2 \AX3{1- p-irj) -AX2 AX2 1 (2.2) J In this parameterization, the C K M matrix elements are writ ten in powers of A = s in^c = 0.22. The parameter rj describes CP violation in the S M in that non-zero values of rj break CP invariance for weak interactions. Current experimental ranges of the magnitudes of the C K M matrix elements are [16] /\Vud\ \VUS\ \Vub\\ ([0.9742,0.9757] [0.219,0.226] [0.002,0.005] \ \Wu\ \Vts\ ivy \Vt»\ J (2.3) [0.219,0.225] [0.9734,0.9749] [0.037,0.043] V [0.004,0.014] [0.035,0.043] [0.9990,0.9993]/ Knowledge of the C K M matrix elements comes from measurement of first-order and second-order weak processes. Those determined "directly" using first-order processes include [16]: • | Kid | — determined from superallowed nuclear (5 decay and from decay of the neutron, both of which involve d —> u decay. • IK, determined from K+ —> 7T°e+ue decay and from A decays, which involve s —> u decay. • \Vcd\ — determined from "inverse charm decay", i.e., neutrino or anti-neutrino produc-tion of c quarks off of valence d quarks. Chapter 2. Theory • \Vcs\ — determined using semi-leptonic D —> K decay and hadronic W decays. • | K b | — determined using semi-leptonic B —> D decays. • | K b I — determined from inclusive semi-leptonic decay of B mesons v i a b —> uWi, by measuring the lepton energy spectrum above the endpoint of b —* clvi decay. The result is interpreted in terms of i K b / V y , and is strongly model dependent. \^ub\ can also be extracted from exclusive semi-leptonic B —> ir and B —> p decay, but again there is significant theoretical model dependence of the result. • \Vtb\ — from measurement of the fraction of t quarks that decay semi-leptonically into b quarks as opposed to s or d quarks, one measures | V r t , | 2 / ( | V y 2 + \ Vts\2 + |K&| 2 ) -Further information on C K M matrix elements (particularly those involving coupling to t quarks) can be determined "indirectly" from flavour-changing second-order weak processes which involve an internal "loop" (e.g., K+ —> TV+VV, as described in section 2.2 and shown in figure 2.2). One predicts values for C K M matrix elements by assuming that the dominant contribution to the process comes from the t-quark loop and not from non-SM physics. Con-versely, the agreement of predicted and measured quantities can be used to-put constraints on new physics. Quantities measured using one-loop processes include: • \Vtt -Vtdl— from B°d - B°d mixing. • |Kd|/114s | — an upper l imit on this quantity comes from the lower l imit of the B°s, BQS mass difference as determined from a l imit on B°s — B°s mixing, and the ratio of hadronic matrix elements for B°s — BQS mixing and Bd — B® mixing as calculated using lattice Q C D . • |Vt s |/ |Vd>| — from observation of b —> S7 decays. • IVtd| — assuming three generations of quarks, \Vts\ ~ \Vd,\, so \Vtd\ can be extracted from the |Vtd | / |Vt s | ratio above measured using B physics. A s shown in section 2.3, I Vtd| can also be extracted from K+ —> n+uu decay, which is theoretically precise, but difficult to observe due to the rarity of this K decay channel (see section 2.2). 10 Chapter 2. Theory The C K M matr ix elements (in particular, Vcs and Vtd) can also be constrained using the assumption of unitarity of the quark mixing matrix (independent of the "rank" of this matrix, which is the number of quark generations). In fact, al l of the above direct and indirect information on the C K M matrix elements can be summarized in terms of the "unitarity triangle". A matrix V is unitary if VW = 1, which implies that if the complex conjugate of elements in one row (column) of a unitary matrix are multiplied by the corresponding elements in a different row (column), the products sum to zero. Appl ied to the first and third columns of the rank 3 C K M matrix in Eq . (2.2), this means that v:bvud + v;bvcd + vt*bvtd ~ v:b + xv;b + vtd = o (2.4) where the approximation Vud ~ Vtb — 1 has been made. This triangle in the complex (p, 77) plane is shown in figure 2.1. Because 77 parameterizes CP violation in the S M , the altitude or area of this triangle gives the S M contribution to CP violation. The location of the 77 7^  0 vertex is currently constrained by measurement of \Vub\, B mixing, and the C P -violating parameter e from neutral kaon decays. The altitude of the triangle may eventually be determined, given Vts, from the CP-violating process K\ —> ifivv, which offers a high-precision measurement of the imaginary part of Vt*s • Vtd. Also , in neutral B decays to CP eigenstates, e.g., Bd(Bd) —> 7T7r and Bd(Bd) —» ipKs, there is a direct relationship between CP-v io l a t i ng asymmetries and sin 20, where <p is one of the angles of the unitarity triangle, denoted by a, [3,7 in figure 2.1. Twice the area of this triangle, or any of 5 other unitarity triangles, is given by the Jarlskog invariant [17], whose non-zero value is a necessary and sufficient condition for C P violation with three generations. Measurement of the unitarity triangle is therefore a test of the S M , in that non-unitarity of the 3 x 3 C K M matr ix could imply more than three generations of quarks. Furthermore, measurement of 77 constrains the S M contribution to CP violation. The motivation for the search for K+ —> 7r+i/P stems from the desire to measure the small, imprecisely-determined C K M matrix element \Vtd\. Another goal is to search for non-SM physics. 11 Chapter 2. Theory Figure 2.1: Uni tar i ty triangle in the complex (p,rj) plane. 2 . 2 K+ —• <K+VV> K + —> -K+VV is a second-order F C N C which is highly suppressed in the S M . The Feynman diagrams for K+ —»• 7r+^P are shown in figure 2.2, and consist of a ' W - b o x " diagram and two "zT-penguin" diagrams. The weak amplitude for this process goes as i=u,c,t y '" i where the V^- are C K M matrix elements, the 7 M are Dirac matrices, is the momentum transfer, and the m; are quark masses. A s stated in the previous section, if the complex conju-gate of elements in one row (column) of a unitary matrix are multiplied by the corresponding elements in a different row (column), the products sum to zero. Appl ied to the first and second columns of the C K M matrix in E q . (2.2), this means that V*sVud + V*sVcd + V*sVtd — 0. So if the quark masses, m*, are equal, the assumed unitarity of the C K M matrix causes M. to vanish. However, the breaking of flavour symmetry, which results in the different quark masses, means that M. is non-zero and that K+ —• ir+vv can proceed at a very small rate. 12 Chapter 2. Theory * s. K _+ u. 7T d-W ) ±u ,c , t W e,p,r Figure 2.2: Short-distance Feynman diagrams for K+ —> n+vv. The greatest violator of quark mass equality is the top quark, which means that of the up, charm, and top-quark contributions to K+ —> n+uu, the top-quark contribution is the greatest, and the decay is sensitive to the weak coupling of top to down quarks, given by the C K M matrix element Vtd-The up-quark contribution to K+ —> ir+v9 is negligible, and the effective Hamiltonian for K+ —• •K+VV is given by [18] ^ / / = % 9 - " a E [V*sVcdXlNL + Vt:VtdX(xt)}(sd)v.A(P^)v.A (2.6) V 2 27rsm Uw l = e ^ T where Gp is the Fermi weak coupling constant, a is the electromagnetic fine structure con-stant, and 9w is the Weinberg or weak mixing angle. The dependence on the charged lepton masses, resulting from the VF-box diagram in figure 2.2, is negligible for the top-quark but not the charm-quark contribution to K+ —> n+vv, because the mass of the top quark is much greater than that of al l three charged leptons, whereas the mass of the charm quark is similar to that of the r lepton. 13 Chapter 2. Theory Q C D corrections to the quark loops are treated perturbatively and are contained in the G I M functions X(as, Xi,\nxi), where as is the strong coupling constant and xt is the square of the relative mass of quark i and the W boson, Xi = mf/M^. The logarithmic terms, hiXi, enter into X(as, x^ Inxi) at order a™ as l n m X j , where m — 0 , 1 , 2 , . . . , n + 1 [19]. For the top-quark loop, xt is of order 1 (2.17), so \n.xt is small (0.774). Furthermore, the strong coupling as{jj) is small (<C 1) at energy scales p = 0(mt). Therefore, X(a3, xt, l n x t ) need only be calculated to leading order in as, but al l orders in xt must be kept. The G I M function for the top-quark loop is calculated in the "leading-order logarithmic approximation" ( L L A ) [19] and denoted simply as X(xt), which is given by [18] X(xt) = V x • X0(xt) (2.7) xt\ 2 + xt Zxt - 6 1 where rjx = 0.994 summarizes Q C D corrections. Conversely, for the charm-quark loop, xc is small (0.0155), so | l n ( x c ) | is large (4.16). Furthermore, Q C D perturbation theory is less accurate for the smaller energy scales p = 0(mc) at which as(p) is larger. Therefore, X(as, xc, l n x c ) need only be calculated to leading order in xc, but non-leading orders in as must be kept. The G I M function for the charm-quark loop comes from a renormalization group calculation in the "next-to-leading-order logarithmic approximation" ( N L L A ) [18, 19], and is denoted simply as XlNL. Theoretical uncertainty in this term comes from the choice of renormalization scale and the charm quark mass, and is about 5%. Long-distance contributions to K+ —+ ir+vP, where a light meson or lepton is exchanged (e.g., see figure 2.3), are three orders of magnitude smaller than the short-distance diagrams of figure 2.2 which have heavy quarks in intermediate states [20, 21, 22]. Furthermore, the matrix element of the weak hadronic current, < TT+\J^\K+ >, is related v ia weak isospin to the well-measured first-order weak decay K+ —• n°e+ue, which has a branching ratio of (4.82 ± 0.06)% [16]. K+ —> 7r+vP is therefore theoretically "clean" and the branching ratio is given by [18] 14 Chapter 2. Theory-Figure 2.3: A long-distance Feynman diagram for K+ —> TT+UU. r ' 3 A t v , \\ , / y L / x c D / v v ^ A t ( ^ Po{X) + —^r-X\Xt) K + — rK+ A 5 "K~"J ' V A A = 4.11 x 10 - i i (2.9) (2.10) 2-7T2 s in 4 9w where A is the Wolfenstein A from E q . (2.2), A; = V*sVid, and A c is real to an accuracy of better than 1 0 - 3 . Ss\t is proportional to the Jarlskog invariant [17]. rK+ = 0.901 is the isospin-breaking correction for K+ —• 7r°e+ue relative to K+ —> ir+vv. Po(X) = 0.42 ± 0.06, defined for the charm sector, is a function of Q C D renormalization scale and mc. The S M prediction for the K+ —>• n+v9 branching ratio is B{K+ -* TT+VV) = (0.82 ± 0.32) x 10 -10 (2.11) where the uncertainty is dominated by the measurement uncertainty in the C K M matrix elements \Vcb\ and I K ^ / V ^ I which are used to constrain Xt. The total theoretical uncertainty in the K+ —> ir+vv branching ratio, resulting from the choice of Q C D renormalization scale in the charm sector, charm quark mass, weak isospin-breaking and two-loop electroweak corrections, is about 7%. Because the predicted rate is so low and lies in a narrow range, K+ —> n+uu serves as a sensitive probe of the quantum structure of S M flavour dynamics. 15 Chapter 2. Theory 2 . 3 Vta The measured value of the K+ —> TT+V& branching ratio can be used to extract the mag-nitude of the C K M matrix element Vtd- Rewrit ing E q . (2.9) using an "improved" Wolfenstein parameterization gives [18] 1 B{K+ - TT+UP) = K+AAX2(xt)- (af})' + (p 0 - p) a l ^2 Po is defined as Po = 1 + p, r] and a are defined by Po(X) A2x(Xty x2\ 77 = 7 7 1 _ T , a = 1 _ M • 2 -(2.12) (2.13) (2.14) (2.15) (2-16) E q . (2.12) defines an ellipse in the {p,rj) plane centered at (po,0). In the leading order of the Wolfenstein parameterization, <7->l, 77->77, p->p (2.17) and E q . (2.12) becomes B{K+ - TT+i/P) = K+A*Xz(xt) [rj2 + (po - p) which defines a circle in the (p, 77) plane centered at (po, 0) wi th radius ro given by 2 _ 1 B(K+ -> •n+vV) r ° ~ A 4 X 2 ( x t ) ' From E q . (2.2), the Wolfenstein parameterization of the C K M matr ix defines Vtd = A\*{\-p-iri). So (2.18) (2.19) (2.20) \Vtd\ = AX3Rt (2.21) 16 Chapter 2. Theory where, in the leading-order Wolfenstein parameterization, ^ 2 = ( l - p ) 2 + r/2 (2.22) which defines a circle in the (p, 77) plane centered at (1,0). Because Eqs. (2.18) and (2.19) define a circle in the (p, 77) plane wi th radius r 0 centered at (po,0), and E q . (2.22) defines a circle in the (p, 77) plane wi th radius Rt centered at (1,0), Vtd m a y be extracted by building a triangle wi th sides of length po — 1, ro and Rt. A s shown in figure 2.4, a th i rd circle from B physics provides a unique th i rd vertex to the triangle: this circle has radius Rb = \Vu*/Vcb\ and is centered at (0,0). No one side of the triangle can be longer than the sum of the two other sides, nor smaller than the absolute value of the difference between those sides, so These limits on Rt define the limits on \Vtd\ according to E q . (2.21). |p 0 — 1| and r 0 come from a combination of theoretical prediction and experimental measurement of quantities in Eqs. (2.13) and (2.19), respectively. Calculations of p 0 , r 0 , Rt, and \Vtd\ are described in section 6.3. Observation of a rate for K+ —* ir+vv which is different from the S M prediction, or a 7r+ momentum spectrum different from that for massless neutrinos and pure V — A couplings, could indicate physics beyond the S M . Possible extensions to the S M include: • K+ —> ir+X°, where X° is a Nambu-Goldstone boson, e.g., an axion, familon, or majoron [16, 20, 23], which arises when a global continuous symmetry is spontaneously broken in the vacuum. The boson's coupling to S M particles, or, equivalently, its mass, is suppressed by (i.e., is inversely proportional to) the energy scale of the symmetry breaking. • intermediate-state supersymmetric particles [20]. | | p o - 1| -r0\ <Rt< ||po - 1| + r 0 (2.23) 2.4 New Physics 17 Chapter 2. Theory V A Figure 2.4: Relationship between K+ —> ir+vv and \Vtd\ in the (p,rj) plane. • a fourth generation of quarks, which could interfere constructively or destructively wi th the top and charm contributions, or a fourth generation of leptons which would increase the number of channels available for K+ 7r+uu. • an enhanced effective sdZ vertex [24], which is currently constrained by measurement of direct CP violation, e'/e [16]. • right-handed neutrinos (associated wi th the majoron), which would alter the n+ mo-mentum spectrum. • lepton flavour violation K+ —> it+vivii. • a "fifth" force [20]. • K+ —• ir+XX, where X is any neutral weakly-interacting particle. Note that the K —* n decay system is well-suited to the search for massless Goldstone bosons (K+ —* TT+X°). The kaon and the pion are the lightest S M pseudoscalar (spin 0) particles so, by conservation of angular momentum, the other particle in a two-body K —> ir decay 1.8 Chapter 2. Theory-must also have spin 0. This means that, in the search for Goldstone bosons (which are scalar or pseudoscalar), there is no interference in the final state from massless S M scalar or non-scalar particles (e.g., the photon). 19 Chapter 3 Experiment As shown in the previous chapter, K+ —> n+ui> is theoretically clean and sensitive to top physics. The challenge then is to make K+ —»• K+VV experimentally clean, such that it can be measured and utilized as a sensitive probe of the S M . Because neutrinos are weakly-interacting neutral particles, the experimental signature for K+ —> ir+vv is simply K+ —> 7 r + , wi th nothing else observed. Potential backgrounds are the decays K+ —> 7r + 7T° (referred to as Kn2 decay) and K+ —> (referred to as K^2 decay), which have branch-ing ratios of 0.2116 ± 0.0014 and 0.6351 ± 0.0018, respectively [16]. These two-body decays, monochromatic in the kaon rest frame, are suppressed kinematically by l imit ing the search for K+ —> -K+VV to the region between the Kn2 and K^2 kinematic peaks, called the " 7 r + ^ ( l ) " region, or the region just below the Kv2 kinematic peak, called the u 7r + z/P(2)" region (see figure 4.3 of the next chapter). Kn2 decays are further suppressed v ia photon detection of 7 1 - 0 - • 77) a n d decays are further suppressed v ia ir+/fi+ particle identification (PID) . Other potential backgrounds include scattering of beam pions and kaon charge-exchange ( C E X ) : K+n —> K°p followed by —>• 7r+l~ui, where / _ is a muon or an electron. K+/TT+ P I D and t iming in the beam, and P I D of the kaon decay product can be used to suppress these backgrounds. Because the branching ratio for K+ —> -K+VV is on the order of 1 0 - 1 0 , suppression of each type of background must be at least 10 1 0 for a signal/background ratio > 1. These considerations motivate beamline and detector design. More details on the back-grounds to K+ —> 7 r + i /P and the methods used to suppress and estimate their contamination are given in chapter 4. 20 Chapter 3. Experiment The experiment was carried out at Brookhaven National Laboratory ( B N L ) , Upton, Long Island, New York, under the auspices of experiment 787 (E787). A total of 3.2 tr i l l ion kaons were collected in the detector over 3 separate run periods in 1995, 1996, and 1997. 3.1 Kaon Production and Transport The major accelerator facilities at B N L are shown in figure 3.1. Product ion of kaons begins wi th a volume of hydrogen gas. A n electrical arc is applied to the hydrogen in order to produce negatively-charged hydrogen ions (H~ ions), which are accelerated to 200 M e V in a linear accelerator ( L I N A C ) . The electrons are removed from the H - ions by thin carbon foils in order to produce H + ions (protons), which are accelerated to a momentum of 24 G e V / c in a pair of synchrotrons: a "booster" and the "alternating gradient synchrotron" ( A G S ) . 60 tr i l l ion protons (60 Tp) are extracted in 1.6-second "spills" once every 3.6 seconds from the A G S via resonant extraction [25, 26]. This beam, called "slow extracted beam" (SEB) , is split in a "switchyard" into 4 beams v ia 3 electrostatic wire splitters. The beams are then transported to 4 target stations: A , B , C and D . The C target is composed of platinum which extends 6 cm in the beam direction. Typica l A G S running conditions have 15 T p per 1.6-second spill at 24 G e V / c incident on the C target, which directly produce positively-charged kaons and pions. "Low energy separated beamline III" ( L E S B I I I , shown in figure 3.2) [27] collects and focusses the kaons produced at the C target for use by E787. The raw content of the beam emerging from the C target contains about 500 pions and 500 protons for every kaon, which are momentum-selected by a dipole magnet (C2D1 in figure 3.2). Kaons, pions, and protons of equal momenta have different velocities, so they arrive at the first electrostatic separator (C2BS-1 in figure 3.2) at different times. The separator applies an alternating voltage across the beam with amplitude, frequency, and phase such that kaons pass through the separator undeflected, but pions and protons are swept out of the beam. A second separator (C2BS-2 in figure 3.2) sweeps pions and muons out of the beam which arise from kaon and pion decay in the region between the two separators. The resulting beam is again momentum-selected by a second dipole magnet (C2D2 in figure 3.2). L E S B I I I also 21 Chapter 3. Experiment to RHIC LINAC Booster Figure 3.1: Accelerator facilities at B N L . contains a number of focussing quadrupole, sextupole, and octupole magnets and collimating slits, and has a total length of 19.6 m from the C target to the E787 target. The angular acceptance of L E S B I I I is 12 msr and the momentum acceptance is 4.5% F W H M . L E S B I I I provides the world's best kaon beam, wi th a flux of about 5 x 10 5 K+ per T p incident on the C target, and K+ : 7 r + and K+ : / J + ratios of about 4:1. Proton contamination is negligible due to the large deflection of protons by the separators. For comparison, the previous best kaon beam (provided by L E S B I ) had a K+ : 7 r + ratio of about 2:5. A t typical A G S running conditions of 15 T p per spill incident on the C target, about 7 x 10 6 K+ emerge from L E S B I I I per spill . These kaons were selected at 790 M e V / c in 1995, 730 M e V / c in 1996, 710 M e V / c in the first half of 1997 (1997a), and 670 M e V / c in the latter half of 1997 (1997b). The kaon beam momentum was successively lowered in order to improve the fraction of kaons which stop inside the E787 detector, which is mounted at the end of L E S B I I I inside a solenoidal magnet (C2D3 in figure 3.2). The kaons are slowed in degrader materials before entering the E787 target, so after kaon decay in flight, disappearance interactions, and scattering out of the beam, about 1.3 x 10 6 K+ enter the E787 target per spill . 22 Chapter 3. Experiment Figure 3.2: Low energy separated beamline III at B N L . 23 Chapter 3. Experiment 3.2 Detector A global view of the E787 detector is shown in figure 3.3. The basic components of the detector are: • beamline detectors for individual detection of beam kaons; • a scintillating fiber target for reconstruction of the kaon decay vertex and tracking of charged decay product(s); • a drift chamber for track-finding and measurement of momentum; • a range stack for energy and range measurements and P I D of the track; and • a system of photon detectors. In the discussion below, "upstream" / " downstream" refers to the direction opposite/along the kaon beam momentum. The coordinate system is defined such that z lies along the beam axis, wi th z increasing downstream, x is in the horizontal direction, increasing to the right, and y is in the vertical direction, increasing upwards, as viewed from downstream of the detector. The point (0,0,0) is defined as the geometric center of the drift chamber. 3.2.1 Beamline Detectors The function of the various detectors in the beam is to identify single kaons entering the target. The kaon beam passes through (in order) a Cerenkov counter, a "hole" counter, 2 wire chambers, a degrader and a hodoscope before entering the target. The Cerenkov counter, shown in figure 3.4, is located about 2 m upstream of the target center and independently detects beam kaons and pions based on the scheme of F i t ch and Motley [28]. The Cerenkov light is directed into a ring of 14 photo-multiplier tubes ( P M T ' s ) at large radius for (slow) kaons, or a ring of 14 P M T ' s at smaller radius for (faster) pions of the same momentum. The P M T ' s are read out by time-to-digital converters ( T D C ' s ) , and are multiplexed into an analog-to-digital converter ( A D C ) and a 500 M H z transient digitizer 24 Chapter 3. Experiment E787 DETECTOR BARREL fJ-VETD • Pb-GLASS COUNTER -BARREL ^ - VETO PHOTOTUBES RANGE STACK -RANGE STACK PHOTOTUBES I-CDUNTER OLLAR COUNTER Bed DEGRADER • END CAP ft-VETO - T A R u " END CAP $-VETO • DRIFT CHAMBER -RANGE STACK CHAMBERS Figure 3.3: Side view of the E787 detector. 25 Chapter 3. Experiment based on flash A D C ' s (hereafter referred to as a T D ) . The T D ' s sample voltage in 2 ns intervals [29] to provide information on the time development of a pulse. A kaon or pion is identified by a specified minimum number of P M T ' s that register pulses wi thin a specified time window. The hole counter is located at the upstream face of the first beam wire chamber, and is composed of two L-shaped plastic scintillators which jo in to form a rectangle. The rect-angular hole defines the acceptance region in the (x, y) plane for kaons in the beam. Each scintillator is instrumented wi th a P M T which is read out by an A D C and a T D C . The beam wire chambers are used for precision measurements of the (x,y) coordinate and time of beam particles so that multiple coincident beam particles can be detected. The first beam wire chamber ( B W C l ) , shown in figure 3.4, is located about 1.7 m upstream of the target center, and has 3 planes ( X , 7 1 , 7 2 ) of 12-/xm-diameter gold-plated tungsten anode wires strung 1.27 m m apart. The X-p lane has 144 wires strung vertically, and the 7 1 - and 72-planes each have 120 wires strung at ± 4 5 ° to the vertical. Each pair of 2 adjacent wires is read out by a single T D C . The second beam wire chamber ( B W C 2 ) , shown in figure 3.4, is located about 90 cm downstream from B W C 1 , and has 3 planes (X, (71,(72) of 12.7-/jm-diameter gold-plated tungsten anode wires strung 0.79 m m apart. The X-, 7 1 - , and 72-planes each have 96 wires strung vertically and at ± 1 2 0 ° to the vertical, respectively. The middle 72 wires of each plane are read out by T D C ' s in groups of 3 adjacent wires, while the outer 24 wires on each end are read out in groups of 6 adjacent wires. The "fast" gas in the chambers is a 4:1 mixture of CF 4 : isobutane, and the anode wires are held at approximately 3 k V (gain = 10 4). The degrader slows kaons down such that they stop in the target, and is composed of a cylinder of roughly 35 cm of BeO followed by 10 cm of P b O (called "lead glass"). The exact length of BeO is varied depending on the momentum of the kaon beam. The high density of BeO combined wi th its low atomic number provide good slowing power while minimizing the effects of multiple Coulomb scattering. The lead glass is active and was primarily designed to detect kaon-decay photons which travel back upstream from the target. However, it is also used to detect beam pions via Cerenkov light. The downstream face of the 11.2-cm-26 Chapter 3. Experiment Winston cones K PMT 7T PMT Beam n PMT K PMT Radiator Conical Mirror Parabolic Mirror downstream view of BWC1 downstream view of BWC2 -150 -100 100 150 x (mm) -150 -100 -50 100 150 x (mm) Figure 3.4: Top: Side view of the beam Cerenkov detector. Bot tom: Downstream views of the B W C 1 and B W C 2 detectors. Mult iplexed wires are shown as a single wire. The "active area" of B W C 1 is the area defined by the hole counter. 27 Chapter 3. Experiment diameter lead glass cylinder is located about 12 cm upstream of the target center. Light is collected from the sides of the cylinder by a 1.0-cm-thick lucite sleeve coupled wi th silicone gel. The lucite sleeve is glued on the upstream side to a lead glass sleeve, which is glued to 16 azimuthally-segmented trapezoidal lead glass pieces. Each of these pieces is instrumented wi th a fine-mesh P M T which can operate in the 1 T magnetic field immersing the detector (see section 3.2.3). The P M T ' s are located in a ring surrounding the BeO degrader and are read out by T D C ' s . Two P M T ' s were defective during the 1996 data-taking period, resulting in a decrease in beam pion detection power. The B4 hodoscope provides an {x,y) coordinate for particles which pass through the degrader, and is placed up against the upstream face of the target (10 cm upstream of the target center) for matching wi th the (x, y) coordinate of the kaon detected in the target. It consists of 2 0.25-inch-thick planes (U, V) of 8 plastic scintillating fingers each. Starting from the central pair of fingers and moving outwards in pairs, the dimensions of the fingers are 120 m m x 10.0 mm, 120 m m x 12.5 mm, 110 m m x 17.5 mm, and 80 m m x 20.0 mm, respectively. The long dimension of the fingers in the U, V planes is mounted at ± 5 5 ° to the vertical, respectively. The fingers are instrumented wi th P M T ' s which are read out by A D C ' s , T D C ' s , and T D ' s . More information on the Cerenkov counter, B W C 1 , B W C 2 , the degrader and the B4 hodoscope can be found elsewhere [30]. 3.2.2 Target The target is used to locate the kaon decay vertex and to track charged kaon decay products. It is also sensitive to photons. The target is located immediately downstream of the B4 hodoscope and consists of 413 5.0-mm-square, 28 3.5-mm-square, 72 2.0-mm-square, and 104 1.0-mm-square 3.1-m-long plastic scintillating fibers, packed axially to form a circular target roughly 12 cm in diameter (see figure 3.5). The 3.5-mm, 2.0-mm and 1.0-mm fibers are inserted in the gaps near the outer edge of the target. The 5.0-mm fibers are each connected to P M T ' s , whereas the 3.5-mm, 2.0-mm and 1.0-mm fibers are connected in groups to 8, 28 Chapter 3. Experiment 4, and 4 P M T ' s , respectively. The P M T ' s are read out by A D C ' s , T D C ' s , and 500 M H z transient digitizers based on G a A s charge-coupled devices (hereafter referred to as C C D ' s ) . The C C D ' s , similar to the T D ' s , sample voltage in 2 ns intervals [31]. Kaons travel along the fibers, leaving up to 80 M e V per fiber and typically exciting < 5 fibers. Fiducially-accepted pions from kaon decay travel perpendicular to the fibers, leaving about 1 M e V per fiber and exciting up to 25 fibers, depending on the location of the kaon decay vertex in the target. The fiducial region of the target is defined by two layers of 6 plastic scintillator counters arranged cylindrically around the target (see figure 3.5). The inner scintillators, called the I-counters (IC's), define an acceptable ^-region by tagging charged decay products after a kaon stops in the target and before decay products enter the drift chamber. The IC's are 6.4 m m thick at an inner radius of 6.0 cm and extend 24 cm downstream from the upstream face of the target. They are instrumented wi th P M T ' s which are read out by A D C ' s , T D C ' s , and T D ' s . The outer scintillators, called the V-counters (VC ' s ) , overlap the downstream edge of the IC's by 6 mm, and serve to detect particles which are downstream of the fiducial region of the target. The V C ' s are 5 m m thick and 1.96 m long, and are staggered wi th respect to the IC's . They are instrumented with P M T ' s which are read out by A D C ' s and T D C ' s . 3.2.3 Drift Chamber The "ultra-thin chamber" ( U T C ) , and the E787 detector as a whole, are immersed in a 1 T axial magnetic field, where the magnetic field lines point downstream. Due to the Lorentz force, a charged particle has a curved trajectory in a magnetic field, where the radius of curvature is proportional to the component of the particle's momentum which is perpendicular to the magnetic field. For a uniform field in the z direction then, the (x, y) component of a particle's momentum is given by p ( M e V / c ) ~ 3 • \q\ • /J(Tesla) • r(cm) (3.1) where |g| is the magnitude of the particle's charge in units of the proton charge e. The U T C ' s primary functions are therefore to provide momentum measurements of charged tracks with 29 Chapter 3. Experiment downstream view of target - 8 - 6 - 4 -2 Figure 3.5: Downstream view of the target. The 104 1.0-mm-square fibers in the target are not drawn. a resolution on the order of 1%, and to provide good tracking between the target and range stack. The U T C , shown in figure 3.6, is located just outside the IC at an inner radius of 7.85 cm and an outer radius of 43.31 cm. It consists of 12 layers of drift cells, grouped into 3 superlayers: 4 layers x 48 cells in the inner superlayer, 4 layers x 96 cells in the middle superlayer, and 4 layers x 144 cells in the outer superlayer. The cells have dimensions of between 11 and 19 mm, and are composed of 9 wires strung axially: a single 20-pm-diameter gold-plated tungsten anode sense wire surrounded by 8 100-^m-diameter gold-plated aluminum cathode wires arranged in a "square". Cathode wires are shared at the boundaries between cells. Cells in each layer are staggered by one-half cell with respect to neighbouring layers in order to resolve the left-right ambiguity. The gas in the superlayers is a 49.6%:49.6%:0.8% mixture of argon:ethane:ethanol (vdHft = 5 cm/ps). The cathode wires are grounded and the anode wires are held at 2 k V (gain = 8 x 10 4). Each anode wire is instrumented with an A D C and a T D C . The drift times to the anode wires provide (x, y) 30 Chapter 3. Experiment E787 Central Tracking Drift Chamber -20/tim W anode wires Cathode foil strips Cathode foils 25/tim Upilex with 2000A Cu strips Outer carbon fiber tube diameter 862mm Anode pre amp Cathode strip preamp Active length 508mm Super layer # 1 (192 Anodes) Superlayer # 2 (384 Anodes) Superlayer # 3 (576 Anodes) Figure 3.6: The "ultra-thin chamber" ( U T C ) . coordinates for tracks. A t the inner and outer radii of each superlayer are a helical array of cathode strips at a pitch angle of 45°. The 7-mm-wide strips are 1200 A copper coated wi th 300 A nickel, are separated by 1 mm, and are mounted on 25-yum-thick Kap ton foil. There are 48, 72, 108, 144, 180, and 216 strips on the 6 foils, from inner to outer, respectively. The centroid of induced charge on a cluster of strips provides a z coordinate with a resolution of about 1 mm. Each cathode strip is instrumented wi th an A D C and a T D C . Between the 3 superlayers are 2 inactive regions of nitrogen gas. Differential pressure in the 5 gas volumes supports the cathode foils (excluding the innermost and outermost foils, which are held in place by support tubes). The active length of the U T C is 50.8 cm for a solid angle acceptance of about 27T sr as seen by the target. The total mass in the measurement 31 Chapter 3. Experiment region (excluding the inner and outer support tubes and innermost and outermost foils) amounts to 2 x 1 0 - 3 radiation lengths. More information on the U T C can be found elsewhere [32]. 3.2.4 Range Stack The primary functions of the range stack (RS) are energy and range measurements of charged tracks, and PID. The RS, shown in figure 3.7, is located immediately outside the U T C at an inner radius of 45.1 cm and an outer radius of 89.6 cm. It is composed of 21 layers of plastic scintillator, each azimuthally segmented into 24 sectors. The innermost layer, called the T layer, is 6.35 mm thick and 52.0 cm long, and is used in the trigger to define the approximately 27r-sr solid angle acceptance of the range stack (RS), which roughly coincides with the solid angle acceptance of the U T C . The outer 20 layers are 19.05 mm thick and 1.82 m long, and are used for energy, range, and decay-sequence measurements of charged tracks, as well as detection of photons. Each end of each RS counter is instrumented with a P M T which is read out by an A D C . Groups of 4 P M T signals from the same end (upstream or downstream) of RS counters from 4 adjacent sectors in the same layer are multiplexed together and read out by a T D . The 4 multiplexed P M T signals are also input separately into the T D boards and read out as "flags" so that the pulse information in a T D channel can be assigned to 1 or more of the 4 multiplexed counters. The times of T D pulses on each of the upstream and downstream ends are found with respect to an electronically-generated reference pulse, called the "fiducial" pulse. The time of a hit in a RS counter is found from the average of the upstream and downstream T D times. The z location of a hit is found from the difference of the upstream and downstream T D times. The detector is designed such that pion tracks from K+ —• -K+VU stop in the RS, so that the 7T —> u. —> e decay sequence can be observed in the T D corresponding to the counter where the charged pion track came to rest (the "stopping counter"). A n example of T D pulses in and around the stopping counter for a pion track is shown in figure 3.8. The pion pulse can be anywhere between 1 and 30 MeV in the stopping counter, the muon from pion 32 Chapter 3. Experiment downstream view of RS - 1 0 0 - 8 0 - 6 0 - 4 0 - 2 0 0 20 40 60 80 100 x (cm) Figure 3.7: The R S shown in cross section. <p is the angle in the (x, y) plane about the center of the detector. 33 Chapter 3. Experiment decay at rest is 4.1 M e V confined to the stopping counter, and the positron from muon decay is up to 52 M e V and can be spread over several R S counters (up to 10 M e V per RS counter). These energies define the required dynamic range of the T D ' s . P ion pulses are typically 30 — 40 ns in width, so the T D sampling rate of 500 M H z (once every 2 ns) allows reconstruction of the pulse shapes so that double-pulse TT — » \i decays can be detected for pion decay times earlier than 10 ns (the n+ mean lifetime is 26 ns). The p+ mean lifetime is 2.2 ps, so a memory depth of between 5 and 7 ps is used to detect Michel positrons, which results in a large T D data volume. The readout time of the T D data is the l imit ing readout time in data acquisition (see section 3.3). More information on the R S and T D ' s can be found elsewhere [29, 33]. Located after layer 10 and layer 14 are range stack straw chambers (RSSC's) . They provide (x, y) and z coordinates of the track in the R S . The inner R S S C ' s consist of 2 layers of 24 straws per sector, and the outer R S S C ' s consist of 2 layers of 28 straws per sector, for a total of 2496 straws. Each straw is 3.4 m m in radius, and has a 38-pm-thick Kap ton skin, coated on the inside wi th C u / N i cathode, enclosing a gas volume of 25%:74.5%:0.5% argon:isobutane:water and a 50-pm-diameter gold-coated tungsten anode wire. The anode wires are held at about 3 k V , and the straw chambers are operated in "limited streamer" mode where ionizing particles cause a spark between the anode and cathode. Pairs of straws in the same layer and separated azimuthally by half a sector are connected by a "jumper card" at one end, and read out by T D C ' s at the other end. The time difference between the early pulse and the late pulse from the two connected straws gives the z position of the hit. More information on the R S S C ' s can be found elsewhere [34]. 3.2.5 Photon Veto A roughly 47r-sr array of photon detectors is used to suppress Kn2 decays and any other radiative processes. The primary photon veto consists of a barrel, upstream and downstream endcaps (EC's ) , an upstream and downstream collar (CO) , and a downstream microcollar ( C M ) . The barrel and endcap are shown in figure 3.9. 34 Chapter 3. Experiment 250 -50 Layer 14 Upstream + . e • 100 150 Tim (ra) 200 300 250 200 1150 o a 100 50 0 Layer 14 Downstream 100 BO 200 250 300 Time (IM) 250 J I I 1_ Layer 13 Upstream + e -SO 0 50 100 150 Tims (ra) 200 250 300 250 200 1150 0 100 so 0 _l I I I I 1— Layer 13 Downstream -i 1 1 1 1 * i 100 150Time (ra) 200 250 300 250 200 5 BOH Layer 12 Upstream + » + • v \ - — + e A -50 0 50 CO CO 200 Time (ra) 300 250 200 1150 Q no 50 0 J 1_ Layer 12 • • Downstream • 50 0 n 50 100 150 200 250 300 Time (ra) 250 200 150 • mo 50 0 _1 1_ Layer 11 Downstream -50 0 50 W W 200 250 300 Time (ns) Figure 3.8: Pulses found in the upstream and downstream T D ' s in R S layers 11-14 for a pion track stopping in layer 12. 35 Chapter 3. Experiment downstream view of EC and BV x (cm) Figure 3.9: The "barrel veto" ( B V ) and E C shown in cross section. 36 Chapter 3. Experiment The barrel is 1.90 m long and accounts for roughly 2/3 of the 4/7T-sr photon coverage. It surrounds the range stack at a minimum radius of 94.5 cm and a maximum radius of 145.3 cm, and consists of 24 azimuthal sectors each containing 8 modules. The 8 modules in each sector are split into 4 radially and 2 azimuthally, for a total azimuthal segmentation of 48. The azimuthal boundaries of each sector are t i l ted so that the inert inter-sector gaps do not project back to any part of the target. Each radial module, from inner to outer, consists of 16, 18, 20, and 21 layers of 1-mm-thick lead plus 5-mm-thick scintillator, respectively, for a total of 14.3 radiation lengths in the barrel. Each end of each module is instrumented wi th a P M T , which is read out by an A D C and a T D C . The fraction of total photon energy seen by the active regions (scintillator) of the barrel is about 28%. More information on the barrel can be found elsewhere [29]. The upstream and downstream endcaps are composed of 4 rings of undoped C s l crystals, where each crystal is pentagonal in cross section. The rings extend from an inner radius of 10 cm to an outer radius of 43 cm (similar to the U T C ) wi th a length of 25 cm (13.5 radiation lengths), and are mounted at the upstream and downstream ends of the U T C . The upstream E C has 13, 14, 21, and 27 crystals in the 4 rings, from inner to outer, and the downstream E C has 11, 13, 19, and 25 crystals, for a total of 143 crystals. The radial segmentation is staggered from ring to ring to minimize photon escape through the radial cracks. Fine-mesh P M T ' s are attached directly to the crystals (i.e., inside the magnetic field of the detector) for efficient light collection. The P M T ' s are read out by A D C ' s , T D C ' s , and C C D ' s . More information on the E C ' s and the fine-mesh P M T ' s can be found elsewhere [35, 36]. The collars and microcollar are used to detect photons which travel at small angles relative to the beamline and therefore miss the barrel and E C . The upstream and downstream collars are composed of 25 layers of 5-mm-thick plastic scintillator alternating wi th 24 layers of 1-mm-thick lead, stacked axially. They are located about 34 cm "behind" the back edge of the corresponding C s l E C crystals (with respect to the detector origin) and are 15 cm in length. They surround the beam line, extending from an inner radius of 10 cm to an outer radius of 25 cm, and are azimuthally segmented into 12 sectors, each instrumented wi th a P M T read out by an A D C and a T D C . The microcollar is composed of 8 layers 37 Chapter 3. Experiment of plastic scintillating fibers alternating wi th 7 layers of lead foil, stacked radially around the beamline and located just downstream of the downstream collar. Starting at an inner radius of 15.6 cm and extending to an outer radius of 20.0 cm (including air gaps between successive layers), the 8 layers of fibers contain 30, 31, 32, 33, 34, 35, 36, and 37 fibers each 2 m m in diameter, and the intervening 7 layers of lead are 0.41, 0.61, 0.61, 0.61, 0.61, 0.61, and 0.41 m m thick. The microcollar fibers lie parallel to the beamline, and are grouped into 4 azimuthal quadrants, each instrumented wi th a P M T which is read out by an A D C and a T D C . 3.2.6 Monte Carlo Simulation The detector, and various physical processes occurring wi th in the detector, are modelled by a Monte Carlo simulation called " U M C " [37]. U M C includes all detector elements except the beam counters (i.e., it includes everything except that which is upstream of the target), and generates all types of data except for T D and C C D data. Propagation of kaon decay products begins at the position of stopped kaons in the target. The kaon stopping distribution is found using real K^2 decays (from K^iX) monitor data: see section 3.3), where the muon track is wi thin ± 1 0 ° of the vertical. Corrections to the stopping distribution are applied for geometrical differences between the real detector and that simulated by U M C , and for ^ 2 ( 1 ) trigger bias. The kaon decay point in x, y, z, along wi th other information about the decay, are written to a "beam file". Beam files are made for the 4 different kaon beam momenta (see section 3.1): 790 M e V / c (1995), 730 M e V / c (1996), 710 M e V / c (1997a), and 670 M e V / c (1997b). Mul t ip le Coulomb scattering of charged muons and pions off various nuclei in the detector is calculated according to the theory of Moliere [16], wi th corrections for the spin of the scattered particle and the form factor of the scattering nucleus [38]. Hadronic interactions of positively-charged pions in plastic scintillator are calculated using a combination of data and phenomenological modelling [39]. The well-measured 7 r + , 1 2 C total cross section is divided into 7 channels, shown in table 3.1: elastic, "pseudo-elastic", 38 Chapter 3. Experiment channel process a(125 M e V ) elastic 7T+ 1 2 C -> 7T+ 1 2 C 214 mb pseudo-elastic 7T+ 1 2 C -> 7T+ 1 2 C * ,11 mb quasi-elastic 7T+ 1 2 C -> 7T+ p n B A 7T+ 1 2 C -> 7T+ n n C A 7T+ 1 2 C TT° N X 52 mb 31 mb 16 mb absorption 7T+ 1 2 C -> X (no 7T) 194 mb spallation 7T+ 1 2 C - » 7T+ X 120 mb Table 3.1: Total 7 r + , 1 2 C interaction cross section, divided into 7 channels, for 7 r + kinetic energy of 125 M e V . The meaning of A in U B A and n C A is that the product nucleus can be in either the ground or an excited state. * denotes an excited state, N denotes either p or n , and X denotes any nuclear or particle product or combination of products. Exci ted states in 1 2 C are typically the 4.4 and 9.6 M e V states, but levels up to 20 M e V have been observed. 3 quasi-elastic, absorption, and "spallation". The cross sections for each channel increase roughly linearly wi th ir+ kinetic energy, starting from 0 mb at about 0 M e V and peaking near 165 M e V , above which they decrease roughly linearly up to 200 M e V . The cross sections at 125 M e V (i.e., near the midpoint of the ir+vi>(l) signal region) are shown in table 3.1. Individual measurements of the elastic cross section at specific ir+ kinetic energies be-tween 60 and 205 M e V are about 10% uncertain; a fit is made to 5 data points in this range. Da ta on the pseudo-elastic cross section above 100 M e V indicates that it is at most a few percent of the elastic cross section, so it is arbitrarily assumed to be 5% of the elastic cross section over the entire energy range. The quasi-elastic cross sections are found using the well-measured inclusive 7 r + , 1 2 C and 7 r ~ , 1 2 C processes: n 1 2 C —•»• n n(p) 1 1 C A ( 1 1 B A ) , where A means that the product nucleus can be in either the ground or an excited state. These 2 processes can each be approximated by a linear combination of 3 "elementary" processes: (1) 7 r ± n —• ir^n, (2) n(p) —> 7 r ° j 9 ( ^ ) , and (3) 7 r ± p —> ^p. The well-measured 7 r ± , 1 2 C cross sections then give 2 independent ratios of the 3 elementary cross sections. These 2 ratios, combined wi th the well-measured 7r + p —»• 7 r + p cross section, are used to get approx-imate values of the 3 quasi-elastic cross sections. These underestimate the available data, but individual measurements of the quasi-elastic cross sections at energies between 60 and 205 M e V have large uncertainties (up to 50%). Individual measurements of the absorption 39 Chapter 3. Experiment cross section at energies between 60 and 205 M e V are about 10% uncertain; a fit is made to 5 data points in this range. Individual measurements of the total cross section at energies between 50 and 205 M e V are about 1% uncertain; a fit is made to 9 data points in this range. The spallation cross section is then the difference between the total cross section and the other 6 channels. Photon and electron interactions are calculated using the E G S electromagnetic shower simulation package [40]. The accuracy of U M C in modelling the detector is verified by comparing various geo-metric and kinematic variables for Kn2 and decays as calculated using UMC-generated and real data [41]. The accuracy of U M C in modelling 7 r + , 1 2 C interactions is verified by measuring the Kn2 branching ratio (see section 5.9.2). 3.3 Data Acquisition A s mentioned in section 3.1, roughly 1.3 x 10 6 kaons enter the E787 target per spill . A single kaon entering the target and decaying into the fiducial region of the detector defines an "event". A D C , T D C , C C D , and T D data from a K+ —• n+vv event typically totals between 70 and 90 kbytes. To accommodate data transfer speeds of about 14 — 17 Mbytes/s between digitizing hardware and the main computer, the number of events per spill must be reduced to about 100. This is done by a "trigger" which is designed to minimize the number of background events (e.g., Kn2 and K^2 decays) while maximizing the number of potential K+ —> 7T + J / P events. The trigger is composed of a fast level 0 trigger, and slower level 1.1 and level 1.2 triggers. The level 0 trigger must make decisions faster than the rate of kaons entering the target (i.e., sub-/xs), so it is composed entirely of logic pulses from fast detectors. The level 0 trigger for K+ — > TT+VV rejects events by a factor of about 550 — 750, and introduces 38 ns of deadtime for every coincident hit in the first and second layers of the R S (i.e., per T • 2 signal - see below). The level 1.1 and 1.2 triggers involve arithmetic processing of A D C and T D data, and operate on the lower-rate events that pass the level 0 trigger. The level 1.1 trigger has 40 Chapter 3. Experiment a rejection factor of about 13 after level 0, and introduces about 10 — 20 fxs of deadtime per level 0 trigger. The level 1.2 trigger has a rejection factor of about 2 after level 1.1, and introduces a deadtime of about 100 ps per level 1.1 trigger. The K+ —> 7r+uu trigger therefore has a rejection of about 14000 — 19000, and reduces the 1.3 x 10 6 kaons entering the detector to about 100 events per spill . The level 0 trigger is based on two t iming strobes which are used to issue gate signals to various detector subsystems. The "beam strobe" is defined by the time of a kaon detected by the Cerenkov counter (the CK signal) or the B4 hodoscope, whichever is later, and the "detector strobe" is defined by the time of coincident hits in the first and second layers of the R S (the T • 2 signal). The detector strobe is formed from a logical O R of T • 2 signals from each R S sector, so T • 2 cable lengths are tr immed to keep sector-to-sector t iming differences to < 1 ns. K a o n decay triggers use both strobes, but are initiated by the detector strobe. That is, signals which form the level 0 trigger are latched onto a bus which is updated with every new T • 2 signal. U p to 16 different level 0 triggers can be implemented based on the information on the trigger bus. For example, as mentioned at the beginning of chapter 3, K+ —> 7r+vi> events of interest are in the ir+vi>(l) and TT+VD(2) regions. The K+ —> ir+vu trigger is therefore defined as a logical O R of two different triggers, named ir+vi>(l) and rK+uu{2). The level 0 and level l . n components of these triggers are given by n+uu{l) = KB • IC • D C • T • 2 • (6 c t + 7ct) • (19 c t + 20 c t + 21 c i ) (3.2) • B V + E C • (LOr r l ( l ) • U S + L0r r l (2 ) • DS) • H E X • L l . l • L1.2 T T + ^ ( 2 ) = KB • IC • D C • T • 2 • 3 c i • 4 c t • 5 d • 6 c t • (13 c i + ... + 18 c 4) (3.3) •(19 + 20 + 21) • B V + E C • L0r r2 ( l ) • H E X • L l . l • L1.2 where • KB- a coincidence between CK (the definition of CK changed over 1995-7, varying between 5 and 9 kaon P M T ' s hit out of a total of 14), a hit in either plane of the B4 hodoscope, an energy sum of fiber hits in the target (at least 10 — 20 M e V summed), and a signal identifying the spill from the A G S . In 1995 (1996-7), the B4 signal had to 41 Chapter 3. Experiment fall wi thin about ± 2 0 ns ( ± 6 ns) of C# , and the target signal had to fall wi thin about ± 2 0 ns ( ± 3 0 ns) of CK- The KB signal identifies kaons entering the target, and has a 50 ns width as input into the trigger transmitter. This is to ensure that Kg is active when a T • 2 hit, arising from a kaon decay product, updates the trigger bus. The mean kaon lifetime is TK+ = 12.386 ns, so 1 - e - 5 0 / 1 2 - 3 8 6 = 98.2% of al l kaon decays are potentially accepted by the trigger. • IC: a hit in any I-counter within about ± 5 ns of the detector strobe. This selects kaon decay products leaving the target and entering the fiducial region of the detector. • D C : delayed coincidence. A n IC hit must be present some minimum time after C^-, typically set between 1.4 and 1.8 ns. This selects kaons which stop in the target and decay at rest. • T • 2: a hit in a layer 2 counter in the same sector and in coincidence wi th (within ± 2 0 ns of) a hit in the T layer. This selects charged tracks which have entered the R S . The T • 2 time is defined as the average of the times from each end of the layer 2 counter. In 1996 and subsequent years, the IC signal was added to T • 2 in order to reduce the rate of detector strobes arising from kaon interactions and production of secondaries in the degrader, which subsequently hit the first and second layers of the R S . Reducing the rate of detector strobes reduces the deadtime associated wi th the trigger. This is also the reason for making the T counters so thin: to reduce the probability of photon conversion in the T counters, so that detector strobes are more likely to arise only from charged tracks. T h i n T counters coupled to thin light guides also reduce the probability of photon conversion in the (inactive) light guides, so that photons associated wi th a kaon decay are less likely to be "hidden". • 6ct + let- a "charged-track" (ct) hit in layer 6 or layer 7 of the R S within ± 2 0 ns of the detector strobe, ct refers to the T • 2 sector or wi thin 2 sectors clockwise of the T • 2 sector (i.e., in the direction of curvature of the positively-charged track, as viewed from downstream of the detector). The ct label only applies to R S counters which 42 Chapter 3. Experiment are sector-adjacent to at least one other ct hit earlier in the R S track. This trigger condition rejects low-range tracks (e.g., K+ —> 3ir). • 19ct + 20ct + 21 c t : "online //-veto". There can be no ct hits in layers 19, 20, or 21 within ± 2 0 ns of the detector strobe. This rejects high-range tracks, such as p+ tracks from K^2 decays. • B V + E C : "online photon veto". Photon energy in the barrel and in the E C must be below a threshold value. This rejects decays. The summed energy of hits in the barrel within ± 1 0 ns of the detector strobe must be below 5.0 M e V visible energy (corresponding to a 17 M e V photon, due to "invisible" energy loss in the lead of the barrel). The energy of hits within ± 3 ns of the detector strobe in each E C crystal must be less than 20 M e V . (Due to the proximity of the E C to the beam, the E C time windows must be small and energy thresholds high in order to maintain high trigger acceptance). • L O r r l ( l ) • US + L0r r l (2 ) • DS: "refined range" for tracks, which rejects (a) low-range tracks that fail to reach R S layer 11, and (b) high-range tracks that pass the p-veto, because the track propagates at a small angle wi th respect to the beam axis and doesn't reach R S layer 19. L O r r l ( l ) and L0r r l (2 ) are refined range "masks" applied independently to tracks tagged as being in the upstream (US) half or downstream (DS) half of the R S , respectively. The range of a track is found from the range in the target (from the number of target fibers hit), plus the range outside of the target (from the polar angle and termination point of the track). The polar angle is estimated from the z of track hits (from the end-to-end time difference) in R S layers 11, 12, and 13. The termination of the track in the R S is found by the online "stopping counter finder" (SCF) , which defines the stopping counter as the outermost ct hit in the R S within ± 2 0 ns of the detector strobe. • H E X : "hextant cut". The 24 sectors of the range stack are grouped into 6 hextants of 4 sectors each (i.e., the same grouping as the R S T D multiplexing). Only 1 hextant 43 Chapter 3. Experiment can have hits wi thin ± 1 0 ns of the detector strobe, or 2 hextants if they are adjacent. This rejects multiple track events, and decays where the photon converts away from the track in the R S . • L l . l : level 1.1 trigger (all requirements above are part of the level 0 trigger). The L l . l trigger looks at the pulse height (PH) and pulse area (PA) of the pulse(s) in the T D of the stopping counter, and rejects events if the signature for 7r —> fj, decay is missing. For example, if there appears to be only a single pulse at track time (defined as a range of T D bins where the T D value is always > 0), then the P H / P A ratio of this "non-detached" pulse must be smaller than a certain value which is set by comparing P H / P A ratios for single and non-detached double pulses. The P H / P A ratio is smaller for double pulses than for single pulses because the P A increases whereas the P H remains the same when a second (muon) pulse is present on the ta i l of the first (pion) pulse. The L l . l requirement therefore rejects muon tracks in the R S . Events are also rejected if the P H of the non-detached pulse exceeds a maximum value. For events wi th "detached" pulses, where the first and second pulses are well separated in time, the first pulse is required to satisfy a minimum P A and maximum P H , and the second pulse must satisfy a minimum and maximum P A and occur before a maximum time relative to the first pulse. The "cut parameters" ( P H and P A requirements) were found using pion tracks from K^i decays in 1995, but from beam pion scattering events in subsequent years, because beam pions, unlike Kn2 pions, populate al l layers in the R S wi th good statistics. The net effect of this change was that the L l . l requirement was loosened in 1996. It was tightened in 1997 by also requiring a minimum P H / P A ratio for non-detached pulses. • L1.2: level 1.2 trigger. This trigger was implemented roughly midway through the 1997 run, and rejects events wi th more than 1 M e V A D C energy in any R S counter in the same hextant and layer as the stopping counter (excluding the stopping counter itself), and/or the counter in the layer above and in the same sector as the stopping counter. The A D C gate is about 100 ns wide starts about 10 ns before T • 2 time. The 44 Chapter 3. Experiment L1.2 trigger therefore rejects decays which pass L l . l due to "accidental" energy in and around the stopping counter providing the second pulse of the double-pulse n —> p decay signature. Also, a refinement to the H E X requirement is made by requiring events wi th two adjacent hit hextants to have different T • 2 and stopping hextants. L1.2 also removes events wi th invalid S C F assignments. In addition to collecting K+ —> ir+v9 triggers, various "monitor" triggers are collected for use in data quality assessment, detector calibrations, and acceptance and background measurements. For example, four triggers used in the acceptance calculations of chapter 5 are the Kn2(l), KV2(2), K^2(l), and n s c a t triggers: • Kn2(l) = KB • T • 2 • (6 c t + 7CT) • ( 19 d + 20 c t + 21 c t ) • KN2(2) = K B • IC • D C • T • 2 • (6 c t + 7CT) • (19 c t + 20 c , + 21 c t ) • H E X • L l . l • L1.2 . K^(l) = KB • T • 2 • (6 c t + 7CT) • ( 1 9 d + 20 c t + 21CT) • 7TSCAT = 7TB • D C • IC • T • 2 • (6 c t + 7 c t ) • (20 + 21) • B V + E C • H E X where TTB is the same as KB, but wi th the CK signal replaced by the (X signal for a beam pion detected by the Cerenkov counter. The monitor triggers are prescaled to reduce online deadtime and to create suitably-sized data samples. The 7 r + i / p ( l ) , 7T + I/P(2), and prescaled monitor triggers together form the "standard mix" trigger which is used for E787 data acquisition. A D C , T D C , C C D , and T D data is accumulated by up to 19 S L A C Scanner Processors (SSP's) [42] during the spill . A trigger SSP notifies the secondary SSP's when an event of interest has occurred, at which time the secondary SSP's independently read their data into memory. This occurs during the spill , and is the largest source of online deadtime (about 2 ms per event). The total online deadtime is therefore roughly 1.3 x 10 6 • [0.038 ps + 15 //s/650 + 100 ^s/(650 • 13)] + 100 • 2000 ps = 0.29 seconds, which is about 18% of the 1.6-second spill length. In the period between spills (typically about 2 seconds), a master SSP collects the data from each secondary SSP, assembles events, 45 Chapter 3. Experiment and sends the events to a Silicon Graphics Irix computer for online processing [43] and long-term storage. Part of the online processing involves analysis of events in order to quickly identify failures in the detector, electronics, and/or data acquisition systems. This analysis is referred to as the "quality of data" (QOD) assessment. The master SSP also collects data from the trigger SSP (e.g., monitor prescale factors) and the "scalers", which count various signals such as CK, KB, and T • 2 during the spill . Events are written to one of several 8-mm data-quality tapes, which are written in parallel in order to reduce the I / O time per event. Roughly 10300 tapes [33, 44, 45] of K+ —> ir+vi> data (about 30 Tbytes) were written during the 1995, 1996, and 1997 run periods combined. 46 C h a p t e r 4 A n a l y s i s The K+ —• n+uu data-taking hardware and online trigger were discussed in the pre-vious chapter. In this chapter, the characteristics of background events which satisfy the trigger, and the offline techniques and tools used to suppress the background are presented. Measurements of background and signal for the 1995-7 data set are described. The 1995 data has previously been analyzed twice [46, 47], giving rise to the observation of a single K+ —> TT+VV event and the following published results [46]: • B(K+ T T + I / P ) = 4.2±H x I O " 1 0 • B{K+ -> 7T+X°) < 3.0 x I O " 1 0 at 90% confidence level • 0.006 < \Vtd\ < 0.06 A s wi l l be shown in chapter 5, the current analysis and the full 1995-7 data set have a K+ —> ir+vv single-event sensitivity which is a factor of 2.8 greater than the published 1995 analysis and the 1995 data set alone. 4 . 1 Backgrounds Plots of range (in scintillator) vs. momentum for charged tracks which pass the K^i^) and n+vv(l) triggers are shown in figures 4.1 and 4.2, respectively. From the definitions given in section 3.3, the / ^ ( l ) trigger is a loose version of the ir+vu(l) trigger (for example, 47 Chapter 4. Analysis the Kn2(l) trigger does not include the online photon veto: B V + E C ) . These figures show potential backgrounds for K+ —• 7r + z/P, which are kinematically categorized as peak, range tai l , K N 2 peak, K V 2 range tai l , muon band, and pion band. Note that after online application of the •n+vvil) trigger (and before offline application of data-selection requirements), the major background is muon background. Events in the K ^ 2 peak are due to n+ tracks which have the expected values of range, energy, and momentum for a kaon decaying at rest into a ir+ and a n° (R = 30 cm, E — 108 M e V , P = 205 M e V / c ) . Events in the K ^ 2 peak are due to p+ tracks which have the expected values of range, energy, and momentum for a kaon decaying at rest into a fx+ and &uIJi{R = 54 cm, E = 152 M e V , P — 236 M e V / c ) . Events in the K T 2 and range tails have smaller values of range (and also energy) than the peak values due to elastic (inelastic) scattering in the R S . The scattering occurs after the track has passed through the U T C because the momenta of events in the range ta i l are the same as those in the peak. Muon-band events can arise from K+ —> p+ufl/y (referred to as radiative K ^ 2 , or decay), K+ —> TT°P+UIX (referred to as decay), K ^ 2 decay in flight, and/or decay wi th inelastic scattering in the target, such that range, energy, and momentum are smaller than the K ^ 2 peak values. Pion-band events are mainly due to pions in the beam which scatter into the detector (called 7rscat background). Pion-band events wi th range, energy, and momentum smaller than the K N 2 peak values can also arise from inelastic scattering of K V 2 pions in the target. The backgrounds for K+ —> TT+VV can be loosely grouped into two categories: K+-decay and non-/C + -decay backgrounds. The major i f + - d e c a y backgrounds are shown in figure 4.3. The present K+ —> ir+vv analysis focusses on the region between the K N 2 and K ^ 2 kinematic peaks which, as mentioned at the beginning of chapter 3, is referred to as the 7r + ^P( l ) region. Because the experimental signature for K+ —> -K+VV is a single ir+ track wi th nothing else observed, and because most kaon decays involve multiple charged tracks and/or photons and have small branching ratios and small fractions of final state phase space in the n+uu(l) region, the only decays which contribute significantly as background in the ir+vv(l) region are the K N 2 and K ^ 2 decays. The major non -K + -decay backgrounds come from pions in the beam which scatter into the detector, and from kaon charge exchange. 48 Chapter 4. Analysis E o 60 50 40 30 20 H 10 K^ 2 peok-muon bond pion bond ^ 2 range toil peak I—range ^ tail _i i i i_ 120 140 160 180 200 220 240 260 280 P (MeV/c) Figure 4.1: Range in scintillator (R) vs. momentum (P), and kinematic categorization of events passing the / ( ^ ( l ) trigger. The 7 r + ^ P ( l ) signal region is shown as a box. The production mechanisms for the listed categories of events are described in section 4.1 of the text. The peak momentum is reconstructed higher than the accepted value of 236 M e V / c [16] because a "pion hypothesis" has been used to calculate the momentum loss in the target (see section C l ) . 49 Chapter 4. Analysis £ OL. 60 50 40 h 30 20 10 * : i i I i i i J I I I I I I l i_ _L_L _l I I L_ 120 140 160 180 200 220 240 260 280 P (MeV/c) Figure 4.2: Range in scintillator (R) vs. momentum (P) for the events in figure 4.1 which pass the full 7r+vv(l) trigger. 50 Chapter 4. Analysis 0 5 0 100 150 2 0 0 2 5 0 5 0 0 Momentum (MeV/c) Figure 4.3: Momentum phase space of the charged track from K+ decays. The number in brackets next to the final state of each kaon decay channel is the branching ratio of that decay. The upper and lower shaded regions are the momentum components of the it+vv(l) and -K+VV(2) signal regions, respectively. The present analysis is concerned with the ir+vi?(l) region, which is defined as the signal region between the K^i (K+ —» 7r+7r°) and {K+ —* u+v^) kinematic peaks. 51 Chapter 4. Analysis 4.1.1 K^2 and Decays Because the K^2 and K^2 decays are two-body decays, they are monochromatic in the K+ rest frame. Therefore, they are suppressed kinematically by (1) imposing a "delayed coincidence," i.e., requiring that the kaon decay product be detected later than the kaon, such that the kaon decays from rest, and (2) cutting out the Kn2 and K^2 kinematic peaks. This results in a large loss in "acceptance" (i.e., a loss of potential K+ —> events): the kaon mean lifetime is 12 ns and K+ —> iv+vu decays are only accepted (after online + offline data-selection requirements) for kaons which decay at least about 5 ns after being detected. Furthermore, only the range, energy, and momentum phase space between the K^2 and kinematic peaks (i.e., the n+uu(l) region) is retained for K+ —> ir+vD. This region is shown as the high-momentum shaded region in figure 4.3, and only accounts for about 16% of the total K+ —> 7 r + i / i / phase space. Addi t ional kinematic suppression comes from requirements on the energy loss rate (dE/dx) and range-momentum correlation of charged tracks, which serve to separate pion from muon tracks and therefore to suppress K+ —> u.+ decays. K^2 decays are additionally suppressed v ia high-efficiency photon detection in the barrel, E C , R S , target, C O , C M , IC, and V C . decays are further suppressed by requiring observation of the 3-pulse n —> p —> e decay signature in the R S , given that muon tracks have a much different 2-pulse p —> e decay signature. K+ —-> 7T+7r° can imitate K+ —> ir+vu if ( la) the K^2 kaon decays in flight, boosting the charged pion into the n-+vv(l) region, or ( lb) the range (R), energy (E) and momentum (P) are "upshifted" into the 7 r + ^ P ( l ) region due to Gaussian resolution or non-Gaussian tails, and (2) the 25 — 225 M e V photons from 7r° —• 7 7 are missed. For ( la) to happen, the event has to fool the delayed coincidence. This can happen if the decay-in-flight kaon is missed in the beamline, and an earlier kaon, whose decay products are missed, is misidentified as the kaon responsible for the decay. The second kaon can overlap the first kaon and decay in the target, or decay earlier in the beamline (e.g., in the degrader) wi th the decay-pion overlapping the first kaon in the target and scattering into the detector. Contributions from ( lb) should be rare, unless the high-side non-Gaussian tails of R, E, and P are correlated. 52 Chapter 4. Analysis (2) is also rare as photon detection provides a measured 7r° suppression of about 1.7 x 10 6 (at an acceptance of 80%). K+ —> p+v^ can imitate K+ —> 7r+i/£/ if ( la) the kaon decays in flight, reverse-boosting the muon into the ir+uu(l) region, or ( lb) the muon inelastically scatters in the target, causing a "downshift" in R, E, and P, or ( lc) the muon inelastically scatters in the R S , causing a downshift in R and E, plus an independent downshift in P due to Gaussian resolution or a non-Gaussian tai l , and (2) the muon is misidentified as a pion. For (2) to happen, the dE/dx of the muon in the target and R S must be similar to that of a pion, and there must be a correlated or simultaneous downshift in R and P. Furthermore, the 3-pulse 7r e decay sequence must be faked in the R S . For a real 7r — > ji —» e decay, the first pulse, arising from termination of the pion track in the stopping counter, is expected to be between about 1 and 30 M e V ; the second pulse is 4.1 M e V , confined to the stopping counter, and is due to the muon from pion decay; and the third pulse is due to the 0 — 52 M e V electron shower from muon decay, typically spread over several R S counters. A muon track can fake the 3-pulse 7r —> p —> e signature if, in addition to the 2-pulse fx —• e decay, there is a th i rd pulse due to an accidental hit. Various combinations of these pulses are possible: the accidental hit may be responsible for the first (n) or second (p) pulse in the T D data of the stopping counter, or the th i rd (e) pulse in the T D data of the stopping and neighbouring counters. These " T D backgrounds" are referred to as pion-time accidental, muon-time accidental, and electron-time accidental (or early muon decay) background, respectively. The 3-pulse signature can also be faked if only 2 pulses are present (p —> e), but the first pulse is very large or very small or irregularly shaped, such that it fits better to a double pulse than to a single pulse. This type of T D background is referred to as "tail-fluctuation" background. Finally, muons from decay can inelastically scatter in the R S , exciting the giant dipole resonance ( G D R ) in 1 2 C which can de-excite by emitting a neutron. This neutron can travel slowly and unnoticed to the stopping counter, where it can provide the second pulse in the 3-pulse decay sequence (the first and thi rd pulses coming from n —> e decay). This type of T D background is referred to as G D R background. The different T D backgrounds are summarized in table 4.1. The G D R background is discussed 53 Chapter 4. Analysis 1st pulse 2nd pulse 3rd pulse signal: K+ — • TT+UU 7T A* e background: early muon decay e from p decay accidental muon-time accidental A4 accidental e from p decay pion-time accidental accidental e from p decay tail-fluctuation 2-pulse fit to a (j, pulse e from // decay G D R I* n emission from GDR-exc i t ed 1 2 C e from p decay Table 4.1: 3-pulse signature in the T D of the stopping counter, for signal (K+ —> 7r+vv) and background (if"1" —> in more detail in section 4.1.1.1. Radiative decay, K+ —» /u+ufi/y, is also a potential background because a photon wi th as little energy as 20 M e V can result in a muon wi th energy and momentum in the ir+vv(l) signal region (i.e., in the muon band at P = 215 M e V / c in figure 4.1). K+ —> p^v^ can therefore imitate K+ —• 7 r + z ^ if (1) the muon is downshifted in range due to Gaussian resolution or a non-Gaussian tai l , (2) the photon is missed, and (3) the muon is misidentified as a pion (as described for decays above). A range ta i l associated wi th events in the muon band may be significant (similar to the range ta i l associated wi th events in the momentum peak), and low energy photons are detected less efficiently than high-energy photons, so although the branching ratio for decay is small (less than 0.01 of the branching ratio for decay), radiative may be a significant source of background. The tools used to suppress Kn2, and other '-decay backgrounds are shown in table 4.2. The "software cuts" which attack i f + - d e c a y and non- i f + -decay backgrounds are outlined in section 4.3. 4.1.1.1 1 2 C Giant Dipole Resonance Background Giant Dipole Resonance ( G D R ) background is a particular case of background where both the 7r —> \JL —t e decay sequence in the R S is faked, and the muon track is kinematically shifted into the -K+VV(1) region. This correlation between T D and kinematic quantities can 54 Chapter 4. Analysis Background B R kinematics P V P I D mult, tracks CK,TT D C K+ - 0.64 V V K+ 7T+7T0 0.21 V V 0.08 V K + 3TT 0.07 V V V 5 x 10" 3 V V V i f + —> 7r + 77 1 x 10~ 6 V beam 7 r + V V V V V Table 4.2: Tools used to suppress background processes for K+ —> 7 r + i / P . " B R " (branching ratio) is the probability for a kaon to decay to a specific final state, "kinematics" refers to kinematic selection (R,E,P, dE/dx, R vs. P), " P V " (photon veto) refers to detection of photons, " P I D " (particle identification) refers to detection of the 7r —> fj, —> e decay sequence in the stopping counter, "mult, tracks" refers to detection of multiple charged tracks, " C ^ ] 7 r " refers to identification of kaons in the beam Cerenkov detector, and " D C " (delayed coincidence) refers to the online and offline requirements that the kaon's decay products be detected at least about 5 ns after the kaon is detected, such that the kaon decays from rest. lead to poor rejection for G D R background. Experimental evidence for the G D R background mechanism in the E787 K+ —> •n+uu data is, at present, suggestive but not conclusive. G D R background can arise as follows: the 152-MeV muon from K^2 decay inelastically scatters in the plastic target or R S , exciting the giant dipole resonance in 1 2 C via a vir tual photon. The muon thereby loses the energy of the resonance (23.2 M e V , V = 6 M e V [48]), which can place it in or near the ir+vv(l) kinematic signal region. The excited 1 2 C can de-excite by emitting a neutron, which can travel slowly and unnoticed to the stopping counter where it leaves a second pulse some time after the first track pulse, wi th energy similar to that of the muon from pion decay. If the electron from decay of the track muon is detected later than the neutron hit, then the 3-pulse pion decay sequence in the stopping counter can appear to be satisfied. More specifically, the 1 2 C photonuclear cross section wi th single neutron emission (pho-toneutron cross section) is significant at the G D R (about 7 mb [49]), and is significant for larger energies up to at least 37 M e V (between 1 and 4 mb for n , np, and In emission com-bined [48, 49]). Convolution of the photoneutron cross section near the G D R wi th the vir tual 55 Chapter 4. Analysis photon distribution emitted by muons gives the muon inelastic cross section associated with single neutron emission from 1 2 C . This cross section is about 0.02 mb, and has modest energy dependence for muons wi th energy between 50 and 300 M e V [41]. The inelastic scattering can occur early in the muon track (e.g., at low layers in the RS) , such that both range and energy are downshifted, placing the muon in the range ta i l . The muon energy loss can also occur in the target, in which case range, energy, and momentum are all downshifted, placing the muon in the muon band. Single neutrons are emitted from GDR-exc i ted 1 2 C with between 1 and 23 M e V , peaking near 4 M e V [41]. A 10 M e V neutron has an interac-tion length of about 12 cm in plastic scintillator, and travels slowly (0.14c = 4.3 cm/ns). In low-energy (n,p) scattering, the energy transfer distribution is flat, so half the protons get more than half of the neutron energy. A 10 M e V neutron can therefore give rise to a proton with between 6 and 10 M e V , which can leave between 2.5 and 5 M e V of visible energy (due to saturation of ionizing energy in the scintillator [50]) in the stopping counter a small time after the first track pulse. The mean pion lifetime is small (26 ns), and pion decay at rest gives rise to a muon wi th 4.1 M e V (3.0 M e V visible in the scintillator), so, wi th the track muon and its decay-electron providing the first and thi rd pulses in the stopping counter, the G D R mechanism can give rise to a 3-pulse signature similar to TT —• /i —> e decay. The evidence first found for G D R background is shown in figure 4.4. In the top row of figure 4.4, the time of the second pulse in the stopping counter relative to the first pulse, t^, is plotted vs. the kinetic energy of the muon, E, for higher-range (left) and lower-range (right) range-tail events. Two vertical bands are seen in each plot. The right band corresponds to the full energy muon (152 M e V ) ; the left band is about 22 M e V lower at 130 M e V . The right band is flat in for both plots (i.e., independent of range), but the left band is concentrated at small £M. Furthermore, the left band is wider (extending to lower energies) and is more heavily populated in the plot of lower-range events. range-tail events in the low-energy band likely arise from muon inelastic scattering in the R S , and pulses at small i M (close to track time) are more likely to be related to the track than those at large t^. Therefore, the range-tail events in the low-energy band which have small t/j, are suspected to be G D R background events. In contrast, the events in the full-energy 56 Chapter 4. Analysis band likely arise from muons which have undergone an elastic scatter, and the fact that the full-energy band is flat in suggests that the second pulse in the stopping counter is coming from accidental hits. The rejection of the "cuts" used to suppress T D background (see section 4.3) is about a factor of 10 smaller for range-tail events wi th small than it is for full-range peak events wi th large t^. This factor of 10 difference may be the difference in rejection between G D R and muon-time accidental background, which suggests that G D R background is the l imit ing background. The bottom 4 plots of figure 4.4 are projections of the top 2 plots onto the energy axis, for large (middle row) and small (bottom row) t^. The lower-range plots (right) have a much higher concentration of events in the low-energy band than the higher-range plots (left), especially at small t^. Note in the lower-range plots (right) that the population of events in the low-energy band shifts to lower energies for smaller t^. This can also be seen in the top right plot of figure 4.4 in that the suspected G D R events (i.e., the events clustered at small t/j, in the low-energy band) seem to have a correlation between and energy. A s the muon energy gets smaller (i.e., as the energy that the muon loses due to inelastic scattering gets larger), seems to get smaller. In figure 4.5, is plotted vs. the second-pulse energy in the stopping counter, E^, for the same range-tail events from the top right plot of figure 4.4. A correlation between t^ and E^ may be present, because the suspected G D R events (i.e., the events at small tM) seem to move to smaller t^ wi th increasing E^. Bo th the tfj, correlation wi th E in figure 4.4 and the t^ correlation wi th in figure 4.5 can be understood in terms of the G D R mechanism: the more energy that the muon loses to excite 1 2 C , the larger the energy and velocity that the de-excitation neutron can have, leading to shorter travel times to the stopping counter (smaller values of t^) and larger energy deposits in the stopping counter (larger values of E^). However, the experimental evidence for G D R background shown in figures 4.4 and 4.5 is merely suggestive. If G D R background is significant, then there should also be G D R background arising from muons scattering in the target, which would result in a momentum shift from the peak value of 236 M e V / c to < 211 M e V / c (for a loss of > 22 M e V ) , placing the muons in the muon band. A s shown in figure 4.6, the momentum distribution of 57 Chapter 4. Analysis to c 120 100 80 60 40 20 0 K m 2 R a n g e T a i l R = [42,45] cm — L ^ • 1 1 1 , , , * •,'V i i i 80 100 120 140 160 180 120 100 80 60 40 20 0 R < 42 cm _ — ,'t ' \ i „ * * . •* * / . * y • . * : - / %: • *• " : ' •. • - : "*. * \ i i i i i i i i i i i 80 100 120 140 160 180 t„ > 20 ns 20 h 0 , , , 80 100 120 140 160 180 20 10 0 ! i 11 n/7, i i i i t -i i i i 80 100 120 140 160 180 t„ < 20 ns 10 0 20 10 0 , . i 80 100 120 140 160 180 80 100 120 140 160 180 E(MeV) E(MeV) Figure 4.4: Top: Second-pulse time relative to first-pulse time in the stopping counter, £M, vs. p+ track energy for range-tail events at R = [42,45] cm (left) and .ft < 42 cm (right), after n o n - G D R T D backgrounds have been largely removed. The events clustered at < 20 ns and E < 130 M e V are suspected to be due to G D R background. Middle: Energy projection of the top plots for t^ > 20 ns. Bot tom: Energy projection of the top plots for t^ < 20 ns. 58 Chapter 4. Analysis 120 co c 100 80 60 40 20 K m 2 R a n g e T a i l 0 .* * * • ' J I I I I I I u I I • 7 8 EM (MeV) Figure 4.5: Second-pulse time relative to first-pulse time in the stopping counter, t^, vs. second-pulse energy, E^, for range-tail events, after n o n - G D R T D backgrounds have been largely removed. The triangular region at = [1, 5] M e V and £M < 10 ns is devoid of events due to application of T D tail-fluctuation "cuts" (see section 4.3) and the difficulty in finding a low-energy second pulse buried under the large-energy first pulse at small t^. 59 Chapter 4. Analysis muon-band events does in fact peak near 211 M e V / c , though the decrease in the number of events for P > 211 M e V / c may be partly due to the fact that the data sample used to make this plot contains only events which have R < 45 cm (see figures 4.1 and 4.2 for the range-momentum correlation of muon-band events). Also, the momentum distribution of muon-band events, while peaking near 211 M e V / c , is very broad. However, the distribution may be broad because the G D R in 1 2 C is broad. From figure 4.4, it appears that inelastic scattering of muons can result in energy shifts down to 105 M e V . If this loss occurs in the target, then the K^2 peak momentum can shift down to 182 M e V / c . O n the other hand, if muon-band events are mainly due to radiative decay, decay, and/or K^2 decay in flight, then the muon-band events themselves should have missing energy in the RS > 22 M e V if muon inelastic scattering is occurring. In figure 4.7, t^ is plotted vs. -E#s(diff) for the same muon-band events from figure 4.6, where £"#,5(diff) is the expected muon energy in the R S based on the momentum measured in the U T C , minus the measured muon energy in the R S . A band at 0 M e V is seen, but no band at ERs(diS) > 22 M e V is seen, which is where radiative K^, K^, and decay-in-flight events wi th an inelastic scatter in the RS should show up. The events seen at negative values of ER$(diff) are due to radiative or decay where a photon leaves energy on the charged track. Independent of whether the muon band arises from K^2 muons scattering inelastically in the target or from radiative K^2 decay, decay, and/or K^2 decay in flight: if there is a large component of G D R background present in the muon band, then the muon-band events should be concentrated at small i M , and the T D rejection of muon-band events at small should be similar to the T D rejection of range-tail events at small t^. The muon-band events may not be concentrated at as small as that for range-tail events however, because the neutron from muon-band G D R background (muon scattering in the target) has more distance to travel to the stopping counter than the neutron from range-tail G D R background (muon scattering in the RS) . Comparing the t^ distribution of events in the low-energy band of the top right plot in figure 4.4 wi th the t^ distribution in figure 4.7, the muon-band distribution of i M is much flatter than the low-energy range-tail distribution. However, the T D rejection in the muon band at small i ^ , similar to that in the K^2 range ta i l at small t^, 60 Chapter 4. Analysis M u o n B a n d 30 25 20 h 15 10 0 L _ l _ l l l I I I I I I I I I I l _ 180 190 200 210 220 230 P(MeV/c) Figure 4.6: Momentum distribution of muon-band events, after n o n - G D R T D backgrounds have been largely removed. Events are required to have R < 45 cm, E < 150 M e V , and P < 228 M e V / c . 61 Chapter 4. Analysis M u o n B a n d 120 100 80 60 40 20 ---• ..' . . . . --• ' • -• . ' I I I I I I I I I I I I I I I I * . . . .« • I- :'}••"*] • i i i i I i i i i i i i i i i i i - 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 30 40 50 E R S(diff) (MeV) Figure 4.7: Second-pulse time relative to first-pulse time in the stopping counter, tM, vs. ERS (diff) for muon-band events after n o n - G D R T D backgrounds have been largely removed. ERs(difJ) is the expected R S track energy (using momentum measured in the U T C ) minus the measured R S track energy. The events at negative values of ERS (diff) are due to radiative or K^z decay where a photon leaves energy on the charged track. 62 Chapter 4. Analysis is about a factor of 10 worse than that in the peak, which perhaps suggests that both the muon band and range ta i l have some component of T D background which is difficult to reject that is not present in the peak, e.g., G D R background. Also, events have been examined for neutron activity in R S counters other than the stopping counter, where the neutrons may arise due to de-excitation of GDR-exci ted 1 2 C . There is clear evidence of excess near-track-time activity off the track in low-range, low-energy range-tail events, shown as a peak near t = 0 ns in the rightmost plot in the middle row of figure 4.8. However, the expected difference in T D rejection between peak and range-tail events, based on this activity, is only about 4.9 [41]. However, any neutrons leaving energy in the same hextant and layer of a track counter were ignored in this study (because their T D times, due to multiplexing, tend to be shifted to 0 ns), and this is precisely where the neutron rate could be highest. Finally, a prediction of the relative number of peak and range-tail events, assuming that the range ta i l arises from the G D R mechanism, has been attempted by convoluting the vir tual photon distribution emitted by muons wi th the photoneutron cross section for 1 2 C [41]. The predicted number of range-tail events relative to peak events is about 10~ 5 , which is consistent wi th what is observed. So, based on al l of the above evidence, it is indeed possible that a large component of the l imit ing background from the range-tail and muon-band is due to G D R background. However, more data which is enhanced in this particular background mechanism is needed, and Monte Carlo modelling of the process needs to be attempted before the G D R background can be unambiguously identified. 4.1.2 Beam Backgrounds and Kaon Charge Exchange Beam background is grouped into 4 categories: "single beam" kaon- and pion-entering, and "double-beam" kaon- and pion-entering. Single-beam kaon-entering events are kaon decay-in-flight events, which are suppressed by requiring a delayed coincidence of the kaon and its decay product. Single-beam pion-entering events are events where a beam pion scatters into the detector. These are suppressed by requiring a delayed coincidence of the 63 Chapter 4. Analysis x> c o CD 6 uon 4 L 2 0 0 50 time (ns) 0 50 time (ns) 15 10 r ri If] J I I I I I L 0 50 time (ns) Figure 4.8: T ime spectrum of single counter hits in the R S (all energies), for range-tail events wi th R = [42,45] cm (top row), R < 42 cm (middle row), and muon-band events (bottom row), wi th track energy E > 140 M e V (left column), E = [125,140] M e V (middle column), and E < 125 M e V (right column) M e V . The deficiency of events around t = 0 ns is due to the online and offline photon vetos. Hits in the track hextant have been ignored. A n excess of near-track-time hits (near t = 0 ns) is observed for low-range range-tail events wi th E < 125 M e V (rightmost plot in the middle row), which is possibly a result of neutrons emitted from GDR-exc i ted 1 2 C . 64 Chapter 4. Analysis beam particle and the track particle, and by high-efficiency detection of beam pions in the Cerenkov detector (99.84% [30]). Single-beam background can imitate K+ —> TT+VV if the beam and/or track t iming is poorly reconstructed such that the delayed coincidence requirements are satisfied. Double-beam events are the same as single-beam events, except that an earlier kaon is present such that the delayed coincidence is fooled. These events are suppressed by looking for coincident activity in the beamline detectors and R S . Double-beam background can imitate K+ —> ir+vi> if the decay-in-flight kaon or scattering pion is missed in the beam, and the earlier kaon's decay products are missed. K a o n charge-exchange background arises from kaon charge-exchange, K+n —> K°p (e.g., in the target), followed by K\ — > n+l~Pi. The cross section for kaon charge exchange in carbon is estimated to increase rapidly from 0 to 11 mb between K+ energies of about 20 and 60 M e V , then increase more slowly to 15 mb at a K+ energy of about 100 M e V (see, for example, ref. [51]). The K° in the final state is a 1:1 linear combination of the short-lived Kg ( r = 0.08934 ns) and the long-lived K°L ( r = 51.7 ns). Neutral kaons are not slowed in the target, so potential background arising from prompt Ks decay is effectively removed by requiring a delayed coincidence of the kaon and its decay product. However, K°L particles are long-lived, and can decay into the fiducial volume of the detector if they travel slowly in the target. Also, the phase space of K® — • > -K+1~V\ is such that the final state 7 r + can be in the ir+vv(l) signal region if the final state l~ is of low energy, or the K\ decays in flight. Detection of both of the final-state charged tracks, and the kinematics (dE/dx and R vs. P) and P I D (via detection of ir —> p —> e in the stopping counter) of the l~ track can be used to suppress this process. C E X background can therefore imitate K+ — > -K+VV if ( la) K+n —» K°p occurs after the K+ has traversed the beamline detectors, or ( lb) K+n —> K°p occurs earlier in the beamline, and the K+ signal comes from a different kaon whose decay products are missed; (2) the K\ travels slowly such it decays, satisfying the delayed coincidence, into the fiducial volume of the detector; and (3a) the l~ from K\ —> ir+l~&i is missed, or (3b) the n+ is missed, and the l~ is misidentified as a pion as described in section 4.1.1. 65 Chapter 4. Analysis The tools used to suppress beam and C E X background are shown in table 4.2. The "software cuts" which attack K + - d e c a y and non-iC + -decay backgrounds are outlined in sec-t ion 4.3. 4.2 Analysis Strategy and Techniques The number of K+ —> -K+VV events expected in the combined 1995-7 data is on the order of 1 event, based on the number of collected kaons and the acceptance of data-selection requirements (see chapter 5), and the predicted K+ —> K+VV branching ratio (see section 2.2). Therefore, the goal in the offline analysis is to suppress the backgrounds in the ir+vv(l) signal region to an expected value of <C 1 background event, such that any events observed in this region can be unambiguously assigned to signal. This large suppression of backgrounds (by at least 10 1 0 ) makes estimation of the background in the signal region difficult, because any measurement involving low statistics is subject to large statistical fluctuations. Furthermore, the sequential development of data-selection requirements (i.e., "cuts") using smaller and smaller numbers of events can result in "bias", because it is difficult to ascertain whether or not a small number of pathological events is drawn from a larger population on which the cuts are supposed to be effective. For the same reason, the definition of a signal event based on examination of a single event is biased because it is always possible to invent some cut which wi l l reject a single event. To avoid bias, this analysis is a "blind" analysis. That is, background sources are identified a priori, and a signal region is defined (the Tr+vi?(l) region, often referred to as the "box") where the signal/background ratio is expected to be highest. Cuts to suppress background are developed using events which lie outside the box. Events in the box are not counted or examined unti l the cuts and the background estimates are final. To enhance the statistical power of the analysis, background measurements are made v ia "bifurcated" analyses. Each background is addressed by at least two uncorrelated cuts or groups of cuts, which can be independently "inverted" to create high-statistics background samples from the data. That is, background data samples can be created by selecting events 66 Chapter 4. Analysis which fail a specific cut. Wherever possible, background is extracted from the real data (as opposed to modelling the background wi th Monte Carlo simulations) so that al l possible background event pathologies are taken into account. The performance of other, uncorrelated cuts can then be tested on these data samples. A pictorial representation of a bifurcated analysis is shown in figure 4.9. The validity of this method relies on the assumption that the bifurcated cuts are uncorrelated. This assumption can be tested by loosening the bifurcated cuts simultaneously, re-measuring the background levels at these looser cut positions, and (after masking out the box) observing the numbers of events in these "outside-the-box" regions. If the number of events observed is greater than that predicted close to the box, a correlation between cuts may be present, which invalidates the bifurcated background estimate for the box. The outside-the-box correlation study is shown pictorially in figure 4.10. To detect any bias in the cuts, background measurements are performed on independent data samples. The data is partitioned into 1/3 and 2/3 samples, and cuts are designed and the background level measured using the 1/3 data samples. The background level is then re-measured using the independent 2/3 data sample. If the cuts are unbiased, the 1/3 and 2/3 data samples should give the same result (within statistical uncertainty). If the 2/3 mea-surement gives an anomalously high result which exceeds a predetermined acceptable level, then a predetermined contingency cut or cut-tightening is applied to reduce the background to the acceptable level. If this level is not met, then the 1/3 study and 2/3 test is repeated, although there is no longer a truly independent data sample available for testing the cuts. In either case, new cuts may be designed and applied purely as "safety cuts" (defined in section 4.3) based on the results from the 2/3 test. In the present analysis, cuts are init ial ly designed and the background level first measured using the 1/3 1995 data (the 1995 data has previously been examined [46, 47]). After testing on the 2/3 1995 data, the cuts are recalibrated for 1996-7 data using the 1/3 1996-7 data with high statistics (at least 100 events remaining after application of each cut, to minimize bias). New cuts (if necessary) are developed and the background level is measured using the 1/3 1996-7 data, followed by background measurement on the 2/3 1996-7 data. The 1996 67 Chapter 4. Analysis signal region B D A C invert cut1 B+D events B D A C invert cut2 C+D events 3 B D -A C cut2 B D A C apply cut2 B events B D A C apply cut1 R = (C+D ) /C if cut1,cut2 uncorrected, A / B = C / D A = B C / D bg = B / ( R - 1 ) = B C / D Figure 4.9: A background estimate resulting from a bifurcated analysis. Top: If the amount of background is linear in some cut parameters c u t l and cut2, and these parameters are uncorrelated, then the number of background events in region A of the cut l ,cut2 parameter space relative to that in B is equal to that in C relative to D . Middle: Count events that fail c u t l and pass cut2 to get the "normalization" B . Bottom: Select events that fail cut2, and measure the "rejection" of c u t l v ia R = ( C + D ) / C , where C and D are the numbers of events in regions C and D , respectively. Region A is never examined in this procedure. The background estimated to be present in region A is given by B / ( R - 1 ) = B C / D . 68 Chapter 4. Analysis bg = B C / D signal region o B D A C A cut2 predict T bg = &C/D' - B C / D mask out box and observe outside-the-box reg\on o o CO O CD / A ' C log scale —> Figure 4.10: Outside-the-box correlation study. The validity of the bifurcated background measurement procedure (see figure 4.9) relies on the assumption that c u t l and cut2 are uncorrelated. To test this, the background measurement can be made using larger and larger A regions, formed by loosening c u t l and cut2 by the same amount (in the above figure, by a factor of 10 for a total loosening of 10 x 10 = 100). The predicted number of events in the looser box can be compared to the observed number (after masking out the final box, which is only 1% of the loose box in this example) which should agree if no correlations are present. 69 Chapter 4. Analysis and 1997 data sets are combined into a single data set, because together they have roughly the same statistics as the 1995 data set. Finally, the bifurcated cuts are designed to have additional rejection beyond that which defines the background level in the box, so that a cut-tightening (box-shrinking) procedure can be used to evaluate candidate K+ —> ir+vv events. The specific procedure by which cuts are tightened is defined before examining any (small number of) events which lie in the box, in order to avoid bias. This cut tightening, and the cut loosening for the outside-the-box correlation test mentioned above, is performed using functions of the bifurcated cuts associated wi th each background type. The functions are defined in terms of a parameter N > 0 which is linear in background [52]. For example, a specific level of background can be achieved by requiring events to have N < 1. The requirement N < 2 then corresponds to a cut which lets in a factor of 2 more background events, and the requirement N < 0.33 corresponds to a cut which reduces the number of background events by a factor of 3. Effective cuts are those which reject background while keeping signal, so signal events are not linear in N, but rather concentrated at small values of N. Values of N are therefore a measure of the likelihood of an event to be signal: an event wi th a small N value is more likely to be signal than an event with a large N value. Equivalent functions can be defined in terms of a parameter A > 0 which is linear in signal. For example, if the requirement A < 1 corresponds to a cut which has a certain acceptance for signal, then the requirement A < 0.5 corresponds to a cut which reduces the number of signal events by a factor of 2. Again , effective cuts are those which reject background while keeping signal, so background events are not linear in A, but rather concentrated at large values of A. Values of A are therefore a measure of the likelihood of an event to be background: an event with a large A value is more likely to be background than an event wi th a small A value. N and A values are scaled such that the requirements N < 1 and A < 1 roughly correspond to the cut positions which give the background level and acceptance, respectively, of a previous analysis of 1995 data [46]. Because there is roughly a factor of 3 more data in the combined 1995-7 data set, events in the current analysis wi l l be required to have N < 0.33 in order to maintain the same background level as the 1995 analysis. Kaon-rate-independent cuts (e.g., 70 Chapter 4. Analysis kinematic cuts on the total range, energy, and momentum of a charged track) typically have less acceptance loss per gain in rejection than rate-dependent cuts (e.g., cuts on activity in the photon detectors, or the R S counter in which in the charged track comes to rest), so for a given background which is addressed by both rate-independent and rate-dependent cuts, the current analysis wi l l typically proceed wi th the rate-independent function cut set near N < 0.33, while the rate-dependent function cut remains set near N < 1. Once the cuts and signal evaluation functions have been defined, and achieve a back-ground level in the signal region of <C 1 event, the box is "opened". That is, all the cuts are applied and the surviving events (which, by definition, are signal events) are counted and examined. 4.3 Data Analysis and Cuts Mechanically, the analysis is performed by analyzing the data on tape wi th a computer program which consists of a number of software routines linked together. The routines unpack the data on an event-by-event basis, and calculate quantities associated wi th each event. Specific requirements on the values of these quantities are known as "cuts" because they serve to cut or remove background events from the K+ —> n+u9 data. The cuts can be loosely grouped into "reconstruction cuts", "pathology cuts", "function cuts", and "safety cuts". The reconstruction cuts require that a charged track be recon-structed in the target, U T C , and R S , such that the event is worthy of further analysis. The pathology cuts are typically applied at the ini t ia l stages of a bifurcated analysis in order to remove events which can contaminate the bifurcated data samples wi th correlations. A function cut is inverted to define one of these data samples, on which the performance of an-other, uncorrelated function cut is evaluated in order to make a background estimate. Safety cuts, like a l l cuts, are used to suppress background, but they are not used i n the background estimates because their performance is difficult to evaluate. This is usually because some event pathology was recognized only after background estimates were made using both the 1/3 and 2/3 data samples, so there is no independent data sample on which to test the cut 71 Chapter 4. Analysis designed to address this pathology. The reconstruction, pathology, and safety cuts are all "binary" cuts in that events either "pass" or "fail" these cuts. The function cuts are designed on a sliding scale, so that they can be loosened to perform the outside-the-box correlation test (see sections 4.2 and 4.6), and tightened to estimate the "likelihood" of an event to be a signal or background event (see sections 4.2 and 4.7). To reduce the 30 Tbytes of raw data into a volume suitable for t imely development of cuts and background estimation, the analysis is performed in three "passes". In the first pass (PASS1), basic event reconstruction is performed, some general, loose cuts are applied to reject the most obvious of the K^i-, and beam backgrounds, and the data is compacted such that the volume of raw data is reduced by a factor of about 10 while maintaining high acceptance for K+ —> -K+VV events. This creates a manageable number of output data tapes to be analyzed during the second pass (PASS2). The P A S S l cuts are listed and briefly described in table 4.3. A n event display of a successfully reconstructed event is shown in figures 4.11, 4.12 and 4.13. This event is viewed from downstream of the detector, such that a positively charged track curves clockwise in the magnetic field of the detector. These figures are also used for reference in the detailed description of al l cuts found in Appendix C. The PASS2 cuts are loose versions of some pathology and function cuts which are de-veloped further at the thi rd pass (PASS3). The PASS2 cuts are listed and briefly described in table 4.4. They are used in various combinations to define 3 background data streams, referred to as S K I M 1 , S K I M 2 , and S K I M 3 , shown in table 4.5. The S K I M 1 , S K I M 2 , and S K I M 3 data streams are used at PASS3 for study of K^, K^, and beam background, re-spectively. Note that each of these streams also serves as a "signal" stream in that no cuts are inverted (at this stage) to define a background data sample. The P A S S 3 analysis involves calculation of many quantities for each S K I M 1 , S K I M 2 , and S K I M 3 event, which are stored (grouped by event) in "ntuples". The ntuples are created, filled, and read by the Physics Analysis Workstation (PAW) program [53]. W i t h P A W one can loop through the events in the ntuples fairly quickly, filling histograms for the development of cuts, and/or applying cuts to measure background. To speed up the analysis even further, "summary" ntuples which contain only the information pertinent to making 72 Chapter 4. Analysis cut pass condition T R B I T the online trigger is satisfied R D . T R K a track is (crudely) reconstructed in the R S S T L A Y the online stopping counter and hextant agree wi th those found offline R S H E X there are no hits in the stopping layer and hextant which are not part of the track T R K T I M an average track time is found in the R S I N T I M E the non-track hits in the R S at track time sum to less than 10 M e V (e.g., no photons from decay are found in the RS) F I T P I the 2-pulse TT —• \x decay signature is found in the stopping counter U T C / R A N G E / T A R G E T a track is reconstructed in the U T C which lines up wi th the track in the R S , and the energy and range of the track in the R S are accurately calculated P D C the track particle does not arise from a high-momentum beam particle (> 280 M e V / c in the U T C ) L A Y 14 the track does not exit the R S and enter the R S support structure Table 4.3: PASS1 cuts. cut pass condition P V C U T no above-threshold photon energy is detected in the barrel, E C , or R S T G P V C U T no above-threshold photon energy is detected in the target T G P V T R no above-threshold photon energy is detected in the target T G R E C O N a kaon cluster and a pion track are reconstructed in the target T G C U T the kaon times in the target and B4 hodoscope are consistent, and the pion times in the target, IC, and R S are consistent wi thin measurement uncertainties; the energy deposit in the IC is consistent wi th that of a charged pion track P S C U T a pion is not present in the beamline at the same time as the track in the R S , based on B4 and Cn information, which means that the R S track is not likely to have arisen from a beam pion R S H E X 2 the track does not cross sectors in the stopping layer T D C U T there are no hits within ± 1 sector of the stopping counter which occur at the same time as the second pulse in the stopping counter Table 4.4: PASS2 cuts. 73 Chapter 4. Analysis E o 90 80 70 60 50 40 30 20 10 RUN 77Q71 Fvpnt 106116 -10 RSSC UTC hits J I L_J I 1_1 l_ ± RS hits hit target hits J I I I I I I I I I L J I I I I I I I I I I L 1 -40 -30 -20 -10 0 10 20 30 40 50 60 cm Figure 4.11: Event display of a successfully reconstructed event. 74 Chapter 4. Analysis RUN 22971 Event 106116 E o UTC-extrapolated track cm Figure 4.12: Close-up of the target hits for the event shown in figure 4.11. The top and bottom numbers in each fiber are the time (ns) and energy (MeV) of each hit, respectively. The kaon fibers have hits close to t = 0 ns, whereas the pion fibers have hits about 38 ns later, ensuring that the kaon decayed from rest, to is the angle in the (x, y) plane along the circular U T C track. 75 Chapter 4. Analysis Figure 4.13: Close-up of the R S hits for the event shown in figure 4.11. The left and right numbers in each counter are the time (ns) and energy (MeV) of each hit, respectively. 76 Chapter 4. Analysis S K I M 1 S K I M 2 S K I M 3 (beam) T G R E C O N T G R E C O N T G R E C O N T G C U T T G C U T T G C U T R S H E X 2 R S H E X 2 R S H E X 2 T G P V C U T T G P V C U T T G P V T R — P V C U T P V C U T T D C U T — T D C U T P S C U T P S C U T — Table 4.5: Definitions of the PASS2 output background data streams. the background estimates are created from the "full-record" ntuples. The P A S S 3 cuts are grouped into kinematic and beam pathology cuts, and P V , T D , kinematic, and beam function cuts. The kinematic and beam pathology cuts are listed and briefly described in tables 4.6 and 4.7, respectively. The function cuts are described in section 4.3.1. Some data quality cuts are also applied at PASS3 which are listed in table 4.8. Note, from sections C.3.1, C.3.2, and C.3.4, that some P A S S 3 cuts are modified from the 1995 versions for application to 1996-7 data ( C H I R F ) , and that some PASS3 cuts are applied only to the 1996-7 data ( P R O B Z , C H I R F _ N H Z , M A S S , O P S V E T O X K B , T D D F A 1 , T D E C O N , and T D V E L ) , mainly because of an increase in K^2 background observed in the 1996-7 data set (see sections 5.10 and C.3.4). Addi t ional technical information on the P A S S 3 cuts is available elsewhere [41]. 4.3.1 Function Cuts A s shown in figure 4.9 and stated in sections 4.2 and 4.3, a bifurcated background estimate involves "inversion" of a "function cut" in order to define a background data sam-ple, on which the performance of another, uncorrelated function cut is evaluated. The two, uncorrelated function cuts associated wi th a particular background type are called the "normalization" and "rejection" function cuts, respectively. A s stated in section 4.2, each function cut is designed on a sliding scale in terms of a parameter N, which is the scaled number of events remaining in a background data sample after the cut has been applied. Each function cut may also be defined in terms of a parameter A, which is the scaled accep-77 Chapter 4. Analysis cut pass condition U T C Q U A L the U T C track fit in the (x, y) plane is of high quality P R O B Z the U T C track fit in the (r, z) plane is of high quality Z U T O U T the track does not exit the side of the U T C L A Y V 4 the track stops in one of R S layers 11 through 18 inclusive C O S 3 D the track propagates within ± 3 0 ° of the vertical Z F R F the track does not exit or stop near the side of the R S L A Y E R 1 4 the track'does not stop or curl over in the outer R S S C C H I R F the track particle does not scatter in the R S as inferred from RS (x, y) and z and U T C z track hit coordinates C H I R F _ N H Z the track particle does not scatter in the R S as inferred from R S S C and U T C z information R S D E D X the energy deposit of the track particle in each R S counter is consistent wi th that of a pion T G D E D X the energy deposit of the track particle in the target is consistent with that of a pion P I G A P there are no gaps in the pion track in the target T G L I K E the target track fit is of high quality T G B 4 the B4 hit position and target kaon fiber hit positions are consistent wi th propagation of a kaon E I C K I N the measured IC energy is not greater than the energy loss expected from a charged pion track M A S S the mass of the particle which created the charged track is < 153 M e V / c 2 , such that the charged track does not arise from low-energy (high mass) muons which have inelastically scattered in the R S Table 4.6: PASS3 kinematic pathology cuts. 78 Chapter 4. Analysis cut pass condition TGCCDPF significant energy from the track particle is not hidden in target kaon fibers, as inferred from CCD information EPITG accidental energy does not overlap any target pion fibers E P I M A X K kaon energy does not spill into target pion fibers PHIVTX no back-to-back tracks are found in the target PHIVTX2 no back-to-back tracks are found in the target OPSVETO no back-to-back energy from the ir+ and nu from K^2 is detected in the target O P S V E T O i K B no back-to-back energy from the charged pion and the charged lepton from K\ decay is detected in the target TGEDGE significant energy from the track particle is not hidden in target edge fibers TGQUALT a kaon cluster and a pion track are reconstructed in the target TGER the energy and range of the track in the target are , consistent with a pion (not a muon) TARGF no gap between the kaon cluster and the pion track DTGTTP the track in the target is well-matched to the track in the UTC RTDIF the uncertainty in the range of the track in the target is small DRP the track particle does not scatter in the target TIMCON the kaon times in the target and B4 hodoscope are consistent, and the pion times in the target and RS are consistent within measurement uncertainties TIC the times of the track in the IC and RS are consistent within measurement uncertainties TGCCD kaon fibers are assigned correctly (from CCD information) EIC the energy deposit in the IC is consistent with that of a charged pion track KIC a second beam particle does not enter an IC and overlap the first beam particle which stops in the same IC TGGEO a second beam particle does not enter an IC and overlap either the first beam particle which stops in the same IC, or the decay product of the first beam particle which enters the same IC B4EKZ the energy in the B4 hodoscope, UTC-extrapolated z in the target, and energy in the target are all consistent with that of a kaon stopping in the target B4EKZJC if there are no pion fibers in the target, then the B4EKZ cut requirement is made more stringent TGZFOOL the kaon must decay from inside the target BHTRS the track in the RS does not arise from a beam particle, based on the fact that there is no activity in the beam hole counter at track time Table 4.7: PASS3 beam pathology cuts. 79 Chapter 4. Analysis cut pass condition B A D _ R U N all hardware involved in data-taking was operational B A D _ S T C the T D pulse-area-to-MeV calibration is valid for the stopping counter Table 4.8: PASS3 data quality cuts. tance of the applied cut. A function cut can be loosened to define large N and A values, and tightened to define small N and A values (see sections C.3.3, C.3.4, C.3.5 and C.3.6). The N and A values are scaled such that the requirements N < 1 and A < 1 roughly correspond to the cut position(s) which gives the background level and acceptance, respectively, of a previous analysis of 1995 data [46]. For K^i background, the K^2 kinematic function cut and the P V function cut (both described below) are the (uncorrelated) normalization and rejection function cuts, respectively. For K^2 background, the K^2 kinematic function cut and the T D function cut (both described below) are the (uncorrelated) normalization and rejection function cuts, respectively. Backgrounds need to be suppressed by an additional factor of 3 over the previous analysis, because there is roughly a factor of 3 more data in the combined 1995-7 data set. A s stated in section 4.2, the rate-independent function cuts (e.g., the Kn2 and K^2 kinematic function cuts) are therefore tightened by a factor of 3, whereas the rate-dependent function cuts (e.g., the P V and T D function cuts) stay at the same level of background suppression as the previous analysis, because the rate-dependent cuts have more loss in acceptance per gain in rejection than the rate-independent cuts. That is, events in the current analysis wi l l be required roughly to have K%2 and kinematic function values N < 0.33, and P V and T D function values N < 1.0. Beam background and C E X functions are constructed somewhat differently, due to overlap between the 4 differ-ent types of beam background (single-beam kaon- and pion-entering, and double-beam kaon-and pion-entering), and the fact that the C E X background estimate is based on Monte Carlo data (see section 4.4.4). Finally, note that if a function has A > 1.0 at the TV = 1.0 cut point, this corresponds to an "improvement" in the analysis, in that the current analysis has more acceptance than the previous analysis at the same level of background suppression. The P V function is shown in figure 4.14 and tabulated in table C.3 . It is plotted as Npy 80 Chapter 4. Analysis vs. Apv, where Npy is a measure of background level and Apy is a measure of acceptance. Each point on the function corresponds to a set of time windows (around track time) and energy thresholds (minimum energy) in the barrel, E C , R S , target, IC , V C , C O , and C M for detection of individual photon hits (see section C.3.3). Events wi th Npy < 1.0 are defined to pass the P V function cut, whereas events wi th Npy > 1.0 fail. The cut can be made looser or tighter by varying the value of Npy at which events are rejected (recall from section 4.2 that background is linear in iV) . Events can also be assigned a likelihood of having photon energy based on the value of Npy. The collection of PASS3 cuts identifying 7r e decays in the T D data of the stopping counter, referred to as the PASS3 T D cuts, are al l used to construct the T D function. The P A S S 3 T D cuts are divided into "fixed" and "variable" cuts, where the fixed cuts are applied at one specific level of rejection to define a discrete range of function values NTD > 10, and the variable cuts can be loosened or tightened in order to define a continuous range of function values NTD < 10. Each T D cut is designed to attack one of the five types of T D background listed in table 4.1. The T D cuts and the backgrounds they attack are summarized in table 4.9. The T D function at NTD < 10 is shown in figure 4.15 and tabulated in table C.4. Each point on the function for NTD < 10 corresponds to a different value of the "pion likelihood", as defined by a combination of the T D L I K 2 , T D L I K 3 , and T D D F A 2 cuts (see section C.3.4). Events wi th NTD < 1.003 are defined to pass the T D function cut, whereas events wi th NTD > 1-003 fail. The P A S S 3 Kn2 and kinematic function cuts are listed and briefly described in table 4.10. The Kn2 kinematic function is shown in figure 4.16 and tabulated in table C.5, where each point on the function corresponds to a specific lower l imit on range, energy, and momentum values. Function values NkiN,KN2 decrease (become more signal-like) as range, energy, and momentum values increase away from the Kn2 peak. Events wi th NkiN,K„2 ^ 0.3358 are defined to pass the kinematic function cut (i.e., pass the B O X and B O X ' cuts), whereas events wi th NkiN,K^2 > 0.3358 fail. The kinematic function is shown in figure 4.17 and tabulated in table C.6, where each point on the function corresponds to a specific upper l imit on values of momentum and x ( i ? P ) , where x(RP) is the measured 81 Chapter 4. Analysis N v s . A f o r t h e P V f u n c t i o n 10 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A, PV Figure 4.14: The P V function drawn as Npy vs. Apy. Events wi th Npv < 1.0 (indicated by the dashed line) pass the P V function cut. 82 er 4. Analysis cut background addressed pass condition fixed cuts: T D T C O N 7r-time accidental the average R S track time is consistent wi th track time in the stopping counter T D D F A 1 tail-fluctuation the 7r —• p double-pulse fit in the stopping counter is of good quality E V 5 early p decay no accidental activity in the R S at e time E L V E T O /i-time accidental no accidental activity in R S or barrel at p time T D F O O L //-time accidental no accidental activity along the R S track at p time T D E C O N p-time accidental the T D and A D C stopping-counter n energies are consistent T D V E L G D R the p time vs. energy, i.e., "velocity" is not indicative of a G D R de-excitation neutron variable cuts: T D L I K 2 p-time accidental the p pulse is unlikely to have arisen from an accidental T D L I K 3 early p decay the e pulse is unlikely to have arisen from an accidental T D D F A 2 tail-fluctuation the 7r —• p double-pulse fit in the stopping counter is of good quality Table 4.9: P A S S 3 T D cuts. 83 Chapter 4. Analysis Figure 4.15: The T D function drawn as NTD y s- ATD- Events wi th NTD < 1-003 (indicated by the dashed line) pass the T D function cut. 84 Chapter 4. Analysis cut pass condition R N G M O M the range-momentum correlation of the event places it outside the muon band (i.e., in the pion band) B O X range, energy, and momentum of the track fall between the Kn2 and K^2 peaks (i.e., the R B O X , E B O X , and P B O X cuts combined) R B O X range of the track falls between the and peaks E B O X energy of the track falls between the Kn2 and peaks P B O X momentum of the track falls between the Kn2 and peaks B O X ' range, energy, and momentum of the track, corrected for the polar angle, are larger than those of Kn2 peak events (i.e., the R B O X ' , E B O X ' , and P B O X ' cuts combined) R B O X ' range of the track, corrected for the polar angle, is larger than that of K^i peak events E B O X ' energy of the track, corrected for the polar angle, is larger than that of peak events P B O X ' range of the track, corrected for the polar angle, is larger than that of Kn2 peak events Table 4.10: PASS3 kinematic function cuts. minus expected range in the R S , divided by the range resolution (see the description of the R N G M O M cut in section C.3.5). Function values AT f c i l l ) ^ 2 decrease (become more signal-like) as momentum values decrease away from the peak and x{RP) values decrease away from the muon band towards the pion band. Events wi th N^K^ ^ 0.2681 are defined to pass the kinematic function cut (i.e., pass the B O X and R N G M O M cuts), whereas events wi th NkiN,K^2 > 0.2681 fail. The P A S S 3 beam function cuts are listed and briefly described in table 4.11. For the single- and double-beam kaon- and pion-entering normalization and rejection functions, as well as the C E X function described below, the N = 1.0, A = 1.0 point is defined to give the standard level of background suppression in the current analysis (i.e., Af and A are not scaled to the previous 1995 analysis like they are for the P V , T D , Kn2 and kinematic functions). So events are defined to have N < 1.0 if they satisfy the requirements of the beam and C E X function cuts described in section C.3.6). The looser points (Af > 1.0) on the single-beam kaon- and pion-entering normalization functions are found by loosening the requirements of the C K T R S , C K T A I L and C P I T R S , C P I T A I L cuts, respectively; the looser points on 85 Chapter 4. Analysis N v s , A f o r t h e K n 2 k i n e m a t i c f u n c t i o n CM c 10" 10 10 10 -2 0.8 0.9 At; kin,K7T2 Figure 4.16: The i f ^ kinematic function drawn as NkiN,KW2 v s - •^•kin,K^2- The "discontinuity" near NkiN,K^2 = 1 is due to the fact that Nkin^n2 values from 285.1 down to 0 are defined by tightening the B O X ' cut, which is a cut on R, E, and P corrected for polar angle of the track. Near NkiN,K„2 = 1-0 the B O X cut is also applied, in order to enforce hard cuts on R, E, and P independent of polar angle (see section C.3.5). Events wi th N^K^ < 0.3358 (indicated by the dashed line) pass the Kn2 kinematic function cut. 86 Chapter 4. Analysis N v s , A f o r t h e K m 2 k i n e m a t i c f u n c t i o n c 10 10 -1 10 h 0.4 0 .5 0.6 0 .7 0 .8 0.9 1 A k i n , M2 Figure 4.17: The kinematic function drawn as N^K^ v s - -^fcm.K^ • Events with Nkin,K„.2 ^ 0.2681 (indicated by the dashed line) pass the kinematic function cut. 87 Chapter 4. Analysis the single-beam kaon- and pion-entering rejection functions are both found by loosening the requirements of the D E L C cut; the looser points on the double-beam kaon- and pion-entering normalization functions are found by loosening the requirements of the B 4 T R S , B 4 T D and B 4 T R S , B 4 T D , P B N R S cuts, respectively; and the looser points on the double-beam kaon- and pion-entering rejection functions are found by loosening the requirements of the B W T R S , C K T R S , C K T A I L and B W T R S , C P I T R S , C P I T A I L cuts, respectively (see section C.3.6). There is clearly some overlap in the cuts which are loosened to define the N > 1.0 range of function values, which makes definition of independent N values for each beam function difficult. The functions are loosened for the purposes of the outside-the-box correlation study (see sections 4.2 and 4.6), in which case all 4 types of beam background are re-measured and totalled each time a single beam function is loosened. The single-beam background is very small (see section 4.5), so it is not necessary to define tight single-beam function values in the range N < 1.0. The double-beam kaon- and pion-entering functions are combined into one function in order to define tight double-beam function values in the range NBMI < 1-0. This function is shown in figure 4.18 and tabulated in table C.7, where each point on the function corresponds to a specific lower l imit on the minimum absolute time of all B W C l and B W C 2 hits (excluding those associated wi th the event kaon) relative to track time in the R S (see the description of the B W H R S cut in section C.3.6). Function values decrease (become more signal-like) as the lower l imit is raised. One C E X function is also defined at tight function values in the range NQEX < 1-0, shown in figure 4.19 and tabulated in table C.8, where each point on the function corresponds to a specific lower l imit on the kaon likelihood and decay time (see the descriptions of the B 4 E K Z and D E L C cuts in sections C.3.2 and C.3.6, respectively). Function values decrease (become more signal-like) as the lower l imit is raised. 4.4 Background Measurement Structure A s shown in figure 4.9, a bifurcated background estimate is given by B/(R— 1), where B is a "normalization" (the number of events which remain after application of al l cuts except 88 er 4. Analysis cut pass condition D E L C the kaon decays from rest B W T R S the track in the R S does not arise from a beam particle, based on the fact that there is no activity in the beam wire chambers at track time B W H R S the track in the R S does not arise from a beam particle, based on the fact that there is no activity in the beam wire chambers at track time C K T R S the track in the R S does not arise from a kaon decaying in flight, based on the fact that there is no activity in the kaon Cerenkov counter at track time C K T A I L the track in the R S does not arise from a kaon decaying in flight, based on the fact that there is no activity in the kaon Cerenkov counter at track time C P I T R S the track in the RS does not arise from a beam pion, based on the fact that there is no activity in the pion Cerenkov counter at track time C P I T A I L the track in the R S does not arise from a beam pion, based on the fact that there is no activity in the pion Cerenkov counter at track time P B N R S the track in the R S does not arise from a beam pion, based on the fact that there is no activity in the active degrader at track time B 4 D E D X the energy deposit the the B4 hodoscope is consistent wi th a kaon (not a pion) B 4 T R S the track in the R S does not arise from a beam particle, based on the fact that there is .no activity in the B 4 hodoscope at track time B 4 T D the track in the R S does not arise from a beam particle, based on the fact that there is no activity in the B4 hodoscope at track time (from multiple-pulse T D information) Table 4.11: PASS3 beam function cuts. 89 Chapter 4. Analysis N v s . A f o r t h e B M 2 f u n c t i o n 0.9 'BM2 Figure 4.18: The double-beam function at NBMI < 1-0, drawn as NBM2 V S . A B A T 2 - Events with NBM2 < 1-0 (indicated by the dashed line) pass the double-beam function cut. 90 Chapter 4. Analysis N v s . A f o r t h e C E X f u n c t i o n 0.9 A, CEX Figure 4.19: The C E X function at NCEX < 1-0, drawn as NCEX vs. ACEX-NCEX < 1-0 (indicated by the dashed line) pass the C E X function cut. Events with 91 Chapter 4. Analysis one, which is inverted) and R is a "rejection" (the measured rejection of the inverted cut). The sequence of cuts applied to get the normalization is referred to as the "normalization branch", and the sequence of cuts applied to measure the rejection is referred to as the "rejection branch" of a bifurcated analysis. It should be clear from figure 4.9 that the choice of which of the two bifurcated cuts is inverted to get the normalization and which is inverted to measure the rejection is arbitrary. The structures of the Kn2, beam, and C E X background estimates are given below. 4.4.1 K^2 Decay The structure of the PV-kinemat ic bifurcation used to estimate the Kn2 background is shown in figure 4.20. The normalization is found by inverting the P V function cut and ap-plying the Kn2 kinematic function cut, and the measured rejection is that of the P V function cut. The validity of the bifurcated Kn2 background measurement relies on the assumption that the P V and kinematic function cuts are uncorrelated. One potential mechanism of correlation is a photon from the 7r° "hiding" on the charged ir+ track. This would simultane-ously make photons harder to find and upshift the energy of the n+ track into the ir+vv(\) signal region. However, these correlated events are removed by applying the R S D E D X cuts (part of the kinematic pathology cuts described in section C.3.1) before the bifurcation is made (see figure 4.20), so that tracks, with energy loss in the R S inconsistent wi th that of a single pion are completely removed from the analysis. In fact, the Kn2 background esti-mation structure shown in figure 4.20 is designed so that, for events which survive to the bottoms of the normalization and rejection branches, any correlations between the detection of photons and kinematic upshifting of the ir+ should be small. Note that in the normalization branch a second bifurcation is employed in order to enhance statistics. The P V function cut is inverted, all other cuts in the analysis are applied except for the low-side E B O X and E B O X ' cuts, and the number of events remaining in the S K I M 1 data sample, B%?2, is counted. The rejection of the low-side E B O X and E B O X ' cuts is determined separately, by measuring the energy lineshape for events which pass loose 92 Chapter 4. Analysis SKIM1 all beam pathology and function cuts, all kinematic pathology cuts K | i 2 function cut (high-side RBOX,EBOX,PBOX), TD function cut, E | C < 4 MeV PV function cut 27 < R < 35 cm, 94<E<114MeV, 195<P<215MeV/c R,P KJI2 function cut (low-side RBOX.PBOX, RBOX'.PBOX') loose R,P Kn2 function cut (rdev>2.0, pdev>1.0) B Ein Kn2 CD PV PV function cut E Krc2 function cut (low-side EBOX.EBOX') B RP KK2 B Eout Kn2 PV Normalization Branch Rejection Branch Figure 4.20: Kn2 background estimation structure. The BlKv2 and C, CDpy quantities refer to numbers of events remaining at various stages in the Kn2 normalization and rejection branches, respectively, and correspond to the numbers of events in regions B , C , and C + D in figure 4.9, which are used to estimate the Kn2 background level (see section 4.5). The T D function cut does not include T D E C O N or T D V E L (these cuts were developed late in the analysis). Note that some of the beam and kinematic pathology cuts and T D cuts are only applied to the 1996-7 data (see sections C.3.1, C.3.2 and C.3.4). 93 Chapter 4. Analysis R B O X ' and E B O X ' cuts, namely, rdev > 2.0 and pdev > 1.0 (see section C.3.5). These cuts on R and P must be tight enough (i.e., near the box) such that correlations between energy and the other two kinematic quantities (mainly range) do not invalidate the second bifurcation. The Kn2 kinematic normalization is then given by B^P2 divided by the rejection minus 1 of the E B O X and E B O X ' cuts: B%™/B§°J? - 1. The P V rejection is found using K^i peak events in the S K I M 1 data sample which fail the K„2 kinematic function cut. The P V rejection is found during optimization of the P V function as described in section C.3.3. 4.4.2 Decay The structure of the TD-kinematic bifurcation used to estimate the background is shown in figure 4.21. Similar to the bifurcation, the normalization is found by invert-ing the T D function cut and applying the kinematic function cut, and the measured rejection is that of the T D function cut. The validity of the bifurcated background measurement relies on the assumption that the T D and kinematic function cuts are uncor-related. One potential source of correlation is the G D R background mechanism described in section 4.1.1.1, which simultaneously creates a second pulse in the stopping counter consis-tent wi th TT —> [i decay and downshifts the range and energy of the u,+ track into the TT+UU(1) signal region. This correlation mechanism was found late in the analysis v ia the outside-the-box correlation study (see section 4.6), and appears to be significant only in the 1996-7 data, possibly due to the use of narrower T D pulse shapes for the TT —> u. double-pulse fitting in the stopping counter (see section C.3.4). These correlated events are difficult to remove from the background estimation structure without a substantial loss of statistics, so they are suppressed to a negligible level by applying the M A S S cut as a safety cut (see section C.3.1) and the T D V E L cut as part of the T D function cut (see section C.3.4). Another potential source of correlation is p+ tracks which exit the side of the R S , such that range and energy of the u.+ track are downshifted into the TT+UU(1) signal region and, at the same time, an accidental "hidden" in the support structure of the R S leaves an isolated second pulse in 94 Chapter 4. Analysis the apparent stopping counter. These correlated events are located at large z in the R S and/or have polar angles far from 90° (see Appendix D ) , so they are removed from the K^2 background estimation structure by applying the C O S 3 D , Z F R F , and L A Y E R 1 4 cuts (see section C.3.1). Moreover, the background estimation structure shown in figure 4.21 is designed so that, for events which survive to the bottoms of the normalization and rejection branches, any correlations between the detection of the second pulse in the stopping counter and kinematic downshifting of the fi+ should be small. The number of events at the bottom of the normalization branch in figure 4.21, B*K , is the number of background events which fail the T D function cut but pass all other cuts in the analysis (except the M A S S cut, which is a safety cut for background). Most of these events have R and E values much smaller than, but P values very close to, the peak values. That is, these events appear to be range-tail events which have slightly small values of momentum. The small momentum values may be due to Gaussian resolution and/or non-Gaussian mismeasurement. background in the box is expected to arise from both the range ta i l and the muon band, so to estimate the contribution from the muon band, the P V function cut is removed from the normalization branch and the number- of surviving muon-band events, B^°^, is counted. The enhancement of the muon band due to non-application of the P V is measured using muon-band events with R, E, and P in the box, but failing the R N G M O M cut, and is found to be a factor of 2.3. The estimated number of muon band events at the bottom of the normalization branch is then B^band = B^°F//2.3. The muon-band contribution was found to be small in the 1995 data, and no evidence of enhanced muon-band background was found in the 1996-7 data, so it was not explicitly measured for the 1996-7 data. The 1995 normalization is therefore given by BK^ — B*KFI2 + B^and, whereas the 1996-7 normalization is simply B*K^. For the 1/3 1995 normalization, statistics for range-tail events were enhanced by using a second bifurcation. This enhancement is achieved by not applying the high-side P B O X cut in the normalization branch of figure 4.21. Instead, K^i^) monitor events, which are composed almost entirely of peak events, are used to find the rejection of the high-side P B O X cut (similar to the measurement of the kinematic function - see section C.3.5). 95 Chapter 4. Analysis SKIM2 PV function cut, all beam pathology and function cuts Kn2 function cut (low-side RBOX,EBOX,PBOX, RBOX',EBOX',PBOX') all kinematic pathology cuts TD function cut Kji2 function cut (high-side RBOX,EBOX,PBOX) B K\i2 low-side RBOX.EBOX COS3D,ZFRF,LAYER14, TDTCON.TDECON [high-side PBOX .and. R < 42 cm] (range tail) .or. [high-side PBOX .and. R < 42 cm .and. RNGMOM] (muon band) CD TD TD function cut TD Normalization Branch Rejection Branch Figure 4.21: background estimation structure. The B*K and C,CDTD quantities refer to numbers of events remaining at various stages in the normalization and rejection branches, respectively, and correspond to the numbers of events in regions B , C , and C + D in figure 4.9, which are used to estimate the background level (see section 4.5). For the 1/3 1995 background estimate, a second bifurcation is employed in the normalization branch. For the 1/3 and 2/3 1995 and 1/3 1996-7 background estimates, R < 45 cm is used in the definition of range-tail and muon-band events; the P V function cut is not applied to the rejection branch; and the beam cuts do not include C K T R S or C K T A I L , and involve a loose D E L C . More details can be found in section 4.4.2 of the text. The rejection of the T D function cut (as measured in the rejection branch) does not include the T D T C O N and T D E C O N cuts: these are applied as pathology cuts, higher up in the rejection branch, because events failing these cuts can correlate T D and kinematic cuts which are bifurcated here to estimate the background. Note that some of the beam and kinematic pathology cuts and T D cuts are only applied to the 1996-7 data (see sections C.3.1, C.3.2 and C.3.4). 96 Chapter 4. Analysis This second bifurcation gives a valid normalization because the momentum lineshape of range-tail events is not correlated with range (see figures 4.1 and 4.2) nor energy. The range-tail normalization for the 1 /3 1995 data is then given by the number of events at the bottom of the normalization branch in figure 4.21 without the high-side P B O X cut applied, divided by the rejection of the high-side P B O X cut as found using ^ 2 ( 1 ) monitor data. The rejection of the T D cuts is found by inverting the high-side P B O X cut and the R N G M O M cut on the S K I M 2 data sample. The low-side R B O X and E B O X cuts are applied to remove K^z events while maintaining statistics for muon-band events. The rejection measurements on 1/3 1995, 2/3 1995 and 1/3 1996-7 data were performed wi th al l beam cuts applied except C K T R S and C K T A I L , and using a loose version of the D E L C cut (tn — tx > 1 ns - see section C.3.6) in order to enhance statistics. range-tail and muon-band events were defined to have R < 45 cm. After the ini t ia l measurement of the T D rejection on the 1/3 1996-7 data sample, it was found that the T D rejection in the momentum peak is a strong function of range, varying from over 2000 for R > 45 cm to about 800 for 34 < R < 40 cm (possibly due in part to the G D R background mechanism - see section 4.1.1.1). Also, after the T D rejection measurement on the 2/3 1996-7 data, the muon-band rejection was found to be a strong function of P V (roughly a factor of 2 worse rejection for events wi th E and R in the box if the P V function cut is applied). The T D rejection in the box is most accurately given by the rejection of range-tail events close to the box, and the rejection of muon-band events (radiative decays, decays, decay in flight, and/or decay with inelastic scattering in the target) after the full P V has been applied, so range-tail and muon-band events were re-defined to have R < 42 cm, and the P V function cut as well as the complete set of beam cuts were applied. For the reduced number of events which satisfy these criteria, the T D rejection for muon-band events is slightly worse than that for Kya range-tail events [41], so the T D rejection in 1996-7 is estimated as the rejection of muon-band events wi th energy and range in the box, using muon-band events from both the 1/3 and 2/3 1996-7 S K I M 2 data samples. 97 Chapter 4. Analysis 4.4.3 Beam Backgrounds The structure of the bifurcation used to estimate the single-beam kaon-entering back-ground is shown in figure 4.22. Many cuts are applied to select a pure sample of kaons decaying in flight. The normalization is found by inverting the D E L C cut and applying the C K T R S , C K T A I L cuts, and the measured rejection is that of the D E L C cut. However, the rejection of the D E L C cut is measured using beam pions which scatter into the detector (se-lected by inverting the B 4 D E D X cut), not using kaons which decay in flight. This is because a sample of beam pion scatters is easy to define without using time information, and the time structure in various detector elements of a beam pion scatter should be similar to that of a kaon decaying in flight. The normalization and rejection branches therefore use completely different data samples and are believed to be uncorrelated. However, a second bifurcation between the C K T R S , C K T A I L and T D , P V function cuts is present in the normalization branch in order to enhance statistics, and to specifically select kaons in the beam which de-cay in flight. Therefore, the validity of the bifurcated single-beam kaon-entering background measurement relies on the assumption that the C K T R S , C K T A I L cuts are uncorrelated with the T D , P V function cuts with respect to single-beam kaon-entering background. This is expected to be the case because the C K T R S , C K T A I L cuts use beam-particle information from the Cerenkov counter, whereas the T D , P V function cuts mainly use decay-particle in-formation from the R S , barrel, and E C . Note that the rejection of the D E L C cut is measured 4 ways (in an attempt to enhance statistics): wi th and without the T D and kinematic cuts applied, each of these conditions wi th and without the B O X cut applied. The rejection of the D E L C cut is taken as the most conservative (minimum) value of the 4 measurements. The structure of the bifurcation used to estimate the single-beam pion-entering back-ground is shown in figure 4.23. Many cuts are applied to select a pure sample of beam pions scattering into the detector. The normalization is found by inverting the D E L C cut and applying the C P I T R S , C P I T A I L cuts, and the measured rejection is that of the D E L C cut. The rejection of the D E L C cut is measured using beam pions selected not by inverting the C P I T R S , C P I T A I L cuts, but rather by inverting the B 4 D E D X cut (similar to the single-98 Chapter 4. Analysis SKIM3 I RSDEDX,RNGMOM,BOX PV function cut PV function cut only applied in target applied everywhere except target 1 UTCQUAL,PROBZ,ZUTOUT, LAYV4,COS3D,ZFRF,LAYER14, CHIRF,CHIRF_NHZ,EICKIN, TGDEDX,PIGAP,TGLIKE,TGB4 BWTRS,B4TRS,B4TD, TGCCDPF,EPITG,EPIMAXK,PHIVTX,PHIVTX2,OPSVETO,TGEDGE, TGQUALT,TGER,TARGF,DTGTTP,RTDIF,DRP, TIMCON,TIC,TGCCD, EIC,KIC,TGGEO, TGZFOOL.BHTRS B4EKZ.B4EKZ IC DELC CPITRS,CPITAIL,PBNRS,B4DEDX CKTRS.or.CKTAIL PV and TD function cuts PVTD PV.or.TD function cut BCKin BM1K CKTRS,CKTAIL B BM1K B CKoul BM1K Normalization Branch B4DEDX(EB4<1.3 MeV) TD function cut, UTCQUAL,PROBZ,ZUTOUT, LAYV4,COS3D,ZFRF,LAYER14, CHIRF,CHIRF_NHZ,EICKIN CD (1) DELC DELC ,(1) 'DELC CD (2) DELC ,(2) 'DELC Rejection Branch Figure 4.22: Single-beam kaon-entering background estimation structure. The B B M l K and C , CDkDELC quantities refer to numbers of events remaining at various stages in the single-beam kaon-entering normalization and rejection branches, respectively, and correspond to the numbers of events in regions B , C, and C + D in figure 4.9, which are used to estimate the single-beam kaon-entering background level (see section 4.5). The T D function cut does not include T D E C O N or T D V E L (these cuts were developed late in the analysis). Note that some of the beam and kinematic pathology cuts and T D cuts are only applied to the 1996-7 data (see sections C.3.1, C.3.2 and C.3.4). 99 Chapter 4. Analysis beam kaon-entering rejection branch), so the normalization and rejection branches would appear to be uncorrelated. However, a second bifurcation between the C P I T R S , C P I T A I L and P B N R S , B 4 D E D X cuts is present in the normalization branch in order to enhance statistics, and to specifically select pions in the beam which scatter into the detector. There-fore, the validity of the bifurcated single-beam pion-entering background measurement relies somewhat on the assumption that the C P I T R S , C P I T A I L cuts are uncorrelated wi th the D E L C cut, as well as on the assumption that the C P I T R S , C P I T A I L cuts are uncorrelated wi th the P B N R S , B 4 D E D X cuts, with respect to single-beam pion-entering background. This is expected to be the case because the C P I T R S , C P I T A I L cuts use information from the Cerenkov counter, whereas the D E L C , P B N R S , and B 4 D E D X cuts use information from the target, lead glass, and B4 counters, respectively, al l of which are located downstream of the BeO degrader far from the Cerenkov counter. Note that the rejection of the D E L C cut is measured 4 ways, similar to the measurement for single-beam kaon-entering background. The structure of the bifurcation used to estimate the double-beam kaon-entering back-ground is shown in figure 4.24. Many cuts are applied to select a pure sample of events which have a kaon detected in the beam not only at the kaon detection time, but also at kaon de-cay time. The normalization is found by inverting the C K T R S , C K T A I L , B W T R S cuts and applying the B 4 T R S , B 4 T D cuts, and the measured rejection is that of the C K T R S , C K T A I L , B W T R S cuts. The validity of the bifurcated double-beam kaon-entering back-ground measurement relies on the assumption that the C K T R S , C K T A I L , B W T R S cuts are uncorrelated wi th the B 4 T R S , B 4 T D cuts wi th respect to double-beam kaon-entering background. This is expected to be the case because the C K T R S , C K T A I L , B W T R S cuts use information from the Cerenkov and beam wire counters, whereas the B 4 T R S , B 4 T D cuts use information from the B4 counter, which is located downstream of the BeO degrader far from the Cerenkov and beam wire counters. The structure of the bifurcation used to estimate the double-beam pion-entering back-ground is shown in figure 4.25. Many cuts are applied to select a pure sample of events which have a kaon detected in the beam, followed by a pion detected in the beam at kaon decay time. Similar to the double-beam kaon-entering background estimation structure, the normaliza-100 Chapter 4. Analysis SKIM3 RSDEDX,RNGMOM,BOX PV function cut applied only in target UTCQUAL,PROBZ,ZUTOUT, LAYV4,COS3D,ZFRF,LAYER14, CHIRF,CHIRF_NHZ,EICKIN, TGDEDX,PIGAP,TGLIKE,TGB4 PV function cut applied everywhere except target BWTRS,B4TRS,B4TD, TGCCDPF,EPITG,EPIMAXK,PHIVTX,PHIVTX2,OPSVETO,TGEDGE, TGQUALT,TGER,TARGF,DTGTTP,RTDIF,DRP, TIMCONJICJGCCD, EIC,KIC,TGGEO, TGZFOOL.BHTRS B4EKZ,B4EKZJC DELC B4DEDX(EB4<1.3MeV) CKTRS,CKTAIL,PV and TD function cuts TD function cut, UTCQUAL,PROBZ,ZUTOUT, LAYV4,COS3D,ZFRF,LAYER14, CHIRF,CHIRF_NHZ,EICKIN CPITRS.or.CPITAIL PBNRS,B4DEDX , PBB4 PBNRS.or.B4DEDX gCPlin BM1P CPITRS.CPITAIL CD (D DELC CD (2) DELC DELC B BM1P B CPIoul BM1P I C (1) 'DELC j(2) 'DELC Normalization Branch Rejection Branch Figure 4.23: Single-beam pion-entering background estimation structure. The BlBM1P and C, CDkDELC quantities refer to numbers of events remaining at various stages in the single-beam pion-entering normalization and rejection branches, respectively, and correspond to the numbers of events in regions B , C, and C + D in figure 4.9, which are used to estimate the single-beam pion-entering background level (see section 4.5). The T D function cut does not include T D E C O N or T D V E L (these cuts were developed late in the analysis). Note that some of the beam and kinematic pathology cuts and T D cuts are only applied to the 1996-7 data (see sections C.3.1, C.3.2 and C.3.4). 101 Chapter 4. Analysis SKIM3 I RSDEDX.BOX, PV function cut applied everywhere except target PV function cut applied only in target RNGMOM, TD function cut B4DEDX.DELC, TGQUALTJGER, TIMCON, TGZFOOL.BHTRS EPITG, DTGTTP,RTDIF,DRP, TICJGCCD, EIC.KIC.TGGEO CPITRS,CPITAIL,PBNRS CKTRS.or.CKTAIL.or.BWTRS TD function cut, UTCQUAL,PROBZ,ZUTOUT, LAYV4,COS3D,ZFRF,LAYER14, CHIRF,CHIRF_NHZ,EICKIN B4TRS.or.B4TD CPITRS.CPITAIL 1.5 < E M < 5.0 MeV at track time, kaon decay z > -7.0 cm CD TGCCDPF,EPIMAXK,PHIVTX, PHIVTX2,OPSVETO,TGEDGE, TARGF, B4EKZ,B4EKZ_IC, TGDEDX,PIGAP,TGLIKE,TGB4 B4in BM2K B4TRS.B4TD CK BWTRS,CKTRS,CKTAIL CK B TG BM2K B B4out BM2K Normalization Branch Rejection Branch Figure 4.24: Double-beam kaon-entering background estimation structure. The BZBM2K and C, CDCK quantities refer to numbers of events remaining at various stages in the double-beam kaon-entering normalization and rejection branches, respectively, and correspond to the numbers of events in regions B , C , and C + D in figure 4.9, which are used to estimate the double-beam kaon-entering background level (see section 4.5). The second bifurcation between target and B4 cuts in the normalization branch, and the requirement that the U T C -extrapolated kaon-decay z be more downstream than —7.0 cm in the rejection branch, are only used for the 1996-7 background estimates. More details can be found in section 4.4.3 of the text. The T D function cut does not include T D E C O N or T D V E L (these cuts were developed late in the analysis). Note that some of the beam and kinematic pathology cuts and T D cuts are only applied to the 1996-7 data (see sections C.3.1, C.3.2 and C.3.4). 102 Chapter 4. Analysis t ion is found by inverting the C P I T R S , C P I T A I L , B W T R S cuts and applying the B 4 T R S , B 4 T D , P B N R S cuts, and the measured rejection is that of the C P I T R S , C P I T A I L , B W T R S cuts. The validity of the bifurcated double-beam pion-entering background measurement re-lies on the assumption that the C P I T R S , C P I T A I L , B W T R S cuts are uncorrelated with the B 4 T R S , B 4 T D , P B N R S cuts wi th respect to double-beam pion-entering background. This is expected to be the case because the C P I T R S , C P I T A I L , B W T R S cuts use information from the Cerenkov and beam wire counters, whereas the B 4 T R S , B 4 T D , P B N R S cuts use information from the B4 and lead glass counters, which are located downstream of the BeO degrader far from the Cerenkov and beam wire counters. A second bifurcation between target cuts and B4 t iming cuts is used in the double-beam kaon- and pion-entering normalization branches in order to enhance statistics. This bifurcation is valid only if the K I C and T G G E O cuts (see section C.3.2) are applied earlier in the normalization branches in order to remove events which potentially correlate target and B4 cuts. For example, a " K I C " event is shown in figure 4.26, where a beam particle stops in an IC at t = 1.6 ns, leaving a large energy pulse (27 M e V ) which, after discrimination, can be up to 80 ns wide. This ini t ial beam particle therefore "masks out" the IC T D C hit of a second beam particle which enters the target 49 ns later through the same IC and propagates or decays in flight out the other side of the target. A " T G G E O " event is shown in figure 4.27. IC T D and target C C D information indicate that the first kaon stops at the target edge leaving a large energy pulse in a target fiber (39 M e V ) , then decays at 19 ns wi th the decay product leaving a large energy pulse in an IC (23 M e V ) . T D C hits in the kaon stopping fiber and IC are masked out for up to 80 ns, so that a second particle which enters the target at 46 ns near the decay point of the first kaon is not detected. These types of events tend to pass both target and B4 cuts, because only one particle plus apparent decay product can be seen in the target, and the second beam particle enters the target near the target edge, thereby missing the B4 hodoscope. Events in the double-beam kaon-and pion-entering normalization branches which fail the T G G E O cut were discovered after the 1/3 and 2/3 1995 background estimates had already been made, so the T G G E O cut is applied as a pathology cut in the beam background estimate only for 1996-7 data. It is 103 Chapter 4. Analysis SKIM3 RSDEDX.BOX, PV function cut applied everywhere except target PV function cut RNGMOM, applied only in target TD function cut B4DEDX.DELC, TGQUALTJGER, TIMCON, TGZFOOL.BHTRS EPITG, DTGTTP.RTDIF.DRP, TICJGCCD, EIC,KIC,TGGEO CKTRS.CKTAIL B4TRS.or.B4TD CKTRS.CKTAIL CPITRS.or.CPITAIL.or.BWTRS TD function cut, UTCQUAL,PROBZ,ZUTOUT, LAYV4,COS3D,ZFRF,LAYER14, CHIRF,CHIRF_NHZ,EICKIN kaon decay z > -7.0 cm CD TGCCDPF,EPIMAXK,PHIVTX, PHIVTX2,OPSVETO,TGEDGE, TARGF, B4EKZ.B4EKZJC, TGDEDX,PIGAP,TGLIKE,TGB4 , B4in BM2P B4TRS,B4TD,PBNRS CPI BWTRS,CPITRS,CPITAIL CPI BT° BM2P Normalization Branch Rejection Branch Figure 4.25: Double-beam pion-entering background estimation structure. The BlBM2P and C, C Dc PI quantities refer to numbers of events remaining at various stages in the double-beam pion-entering normalization and rejection branches, respectively, and correspond to the numbers of events in regions B , C , and C + D in figure 4.9, which are used to estimate the double-beam pion-entering background level (see section 4.5). The second bifurcation between target and B 4 cuts in the normalization branch, and the requirement that the U T C -extrapolated kaon-decay z be more downstream than —7.0 cm in the rejection branch, are only used for the 1996-7 background estimates. More details can be found in section 4.4.3 of the text. The T D function cut does not include T D E C O N or T D V E L (these cuts were developed late in the analysis). Note that some of the beam and kinematic pathology cuts and T D cuts are only applied to the 1996-7 data (see sections C.3.1, C.3.2 and C.3.4). g B4out BM2P 104 Chapter 4. Analysis applied as a safety cut to the 1995 data. Therefore, the second bifurcation is skipped in 1995: the target and B4 cuts are applied sequentially, and the number of surviving events counted to get the double-beam normalization directly. Note however that the T G G E O cut is ultimately applied to the 1995 data, so the actual beam background level in 1995 may be lower than estimated, i.e., closer to the second-bifurcated result than to the non-second-bifurcated result. Furthermore, wi th only K I C applied, correlations between target and B4 cuts appear to be small [41]. Nevertheless, the more conservative (no second bifurcation) result is used for the 1995 beam background estimate. The double-beam kaon-entering rejection of the C K T R S , C K T A I L , B W T R S cuts and pion-entering rejection of the C P I T R S , C P I T A I L , B W T R S cuts are measured in 2 ways: wi th the B O X cut, and wi th only the low-side B O X cut applied in an attempt to enhance statistics. The more conservative (minimum) value of rejection is used. The events which remain at the bot tom of the rejection branch appear to arise from B4 "splashes", where the kaon decays near the upstream edge of the target and leaves photon or delta-ray energy in the B4 counter, thereby causing the event to fail the B 4 T R S or B 4 T D cut but pass the B W T R S , C K T R S / C P I T R S , C K T A I L / C P I T A I L cuts. Because this contamination was only discovered after the 1/3 and 2/3 1995 background estimates had already been made, these events are partially removed only from the 1996-7 background estimation structure, while maintaining good statistics, by requiring that the UTC-extrapola ted z position of kaon decay in the target be more downstream than —7.0 cm. Roughly 1/3 of the events which otherwise survive the rejection branch are removed by this requirement. 4.4.4 Kaon Charge Exchange The kaon charge exchange background estimate does not involve a bifurcation like the Kiv2, K^, and beam background estimates above, because a sample of K°L —» -K+I'VI decays is difficult to extract from the data. Instead, Monte Carlo data is generated wi th arbitrary statistics, to which all cuts are applied in order to estimate the remaining C E X background level. 105 Chapter 4. Analysis Figure 4.26: Target and IC displays of the " K I C " event. The top and bottom numbers in each target fiber, and the left and right numbers in each IC, are the time (ns) and energy (MeV) of each hit, respectively. 106 Chapter 4. Analysis Figure 4.27: Target and IC displays of the " T G G E O " event. The top and bottom numbers in each target fiber, and the left and right numbers in each IC , are the time (ns) and energy (MeV) of each hit, respectively. 107 Chapter 4. Analysis Ks —> 7 r + 7r~ decay, wi th its two-body final state, has a clearly identifiable signature (unlike K\ —> K+1~DI decay), so the K\ production rate, momentum spectrum, decay vertex distribution, and pattern of target kaon fiber times and energies is taken from the measured Ks quantities, found using data collected wi th a Kg trigger [54]. This trigger is similar to the KW2(2) trigger shown in section 3.3, but wi th the D C requirement inverted, two T • 2 coincidences required, and the online P V applied. Background from K°L is expected only to be significant for "slow" K £ ' s (P < 100 M e V / c ) , and the production rate for these Kl's is approximated by MKo(P < lOOMeV/c) R k 1 = eKo • A P V • B(K°g - 7T+7T-) • KBlive/PS where MKo(P < lOOMeV/c) is the number of reconstructed K s —> 7 r + 7r _ decays passing the K + —y TT+V9 trigger wi th P < 100 M e V / c ; e^ -o is the K s —> 7r + 7T~ reconstruction efficiency (online + offline cuts) for Kg's produced wi th P < 100 M e V / c (0.082); APV is the acceptance of the online P V due to accidentals (0.75); B(KS —> 7 r + 7 r ~ ) is the K°s -* 7 r + 7r~ branching ratio (0.686); Ksuve is KB from section 3.3, corrected by the online trigger deadtime; and PS is the online K°s trigger prescale. This production rate is measured to be RKo = 4.51 x 1 0 - 5 [54]. The above quantities measured using Ks data are input into a Monte Carlo simulation which generates K \ —» 7 r + ^ ~ ^ ( ^ 3 ) and K \ ir+e~ve {K®3) decays. To increase the speed of U M C event generation, (1) events are discarded where the K\ leaves the target before decaying; (2) only semi-leptonic KQL decays wi th a n+ in the final state are accepted; (3) the final state ir+ must have P > 190 M e V / c , and the final state yT or e~ must have P > 100 M e V / c or P > 30 M e V / c , respectively. The equivalent U M C K B H v e is then the number of K\ events generated, multiplied by the rejection of conditions (1), (2) and (3), divided by the production rate RKo. Only the final cuts that are meaningful for U M C data (i.e., a l l cuts except for T D cuts and some beam cuts - see section 3.2.6) are applied to the U M C C E X data [41], and the number of remaining events normalized to KBUve of the K+ —> n+vv data. The C E X background for the 1995 data is therefore calculated as follows: 108 Chapter 4. Analysis W1995) = (^gax + . (4-2) Ko3 KBlive(1995) x 0.412^ ^ M p a s s x K ( U M r Ko, ) x KBHve(UMC,K°3) > AtrigxA^ / 1.53 x 10 1 2 x 0.588 ~ ^ X 7.8821 x IO*4 + 1.53 x 10 1 2 x 0.412N 0.00256 5 x ——- x 1 fiSQ9 y 1 014 / umc uv 1.6592 x I O 1 4 / 0.1197 x 0.1886 = 0.0045 events where Mpafs and Mpafs are the numbers of UMC-generated K°3 and K°3 events, respectively, that pass al l applied cuts; KBuVE(UMC, K®3) and KBuVE(UMC, K°3) are the "equivalent" K-BUve quantities for UMC-generated K®3 and K®3 events, respectively, as described above; KBHVE(1995) is K B L I V E from the 1995 K + -> n+uu data; 0.588 and 0.412 represent the relative branching ratios for K\ —> 7r + e~P e and K\ —> 7r +/z~PM; Atrig is the UMC-measured K+ —• ir+vv trigger acceptance; A^lv is the combined acceptance of the cuts which are applied to the C E X U M C data, as measured using K+ —* ir+vv U M C data; and A^a is the total acceptance for the K + —• TT+UU analysis. The factor Af^a/(Atrig x A^v) is simply a correction for K+ —> n+uu acceptance loss which is not modelled by U M C ( T D cuts, P V accidental loss, beam cuts, etc.). The values of Af^a, Atrig, and in the above equation for &<?OEX(1995) are taken from a previous analysis [46]. The C E X background level for 1996-7 data is calculated by normalizing the 1995 result to the 1996-7 KBuve: 1 7 x 10 1 2 &0CEX(1996-7) = 0.0045 x — — — = 0.0051 events. (4.3) J..0 X J_U A n event which fails only the T G D E D X cut was found after examination of the entire 1996-7 data sample (as part of the outside-the-box tests - see section 4.6). The target display of this event is shown in figure 4.28. Because the pion track appears to originate away from the kaon stopping position, the event could be interpreted as a K+ entering the target and undergoing C E X near t = 0 ns, followed by invisible propagation of K°L and subsequent K\ —• TT+1~PI decay near t = 6 ns. The existence of this event could 109 Chapter 4. Analysis indicate a loophole in the C E X background estimate, because the C E X background estimate is <S 1 event in the box even if the T G D E D X cut is turned off. To check whether or not a serious loophole exists, several cuts effective for C E X background are removed or loosened (to enhance statistics), and the number of events remaining in U M C and real data are compared. No serious discrepancies are found [41]. Nevertheless, the O P S V E T C L L K B safety cut (see section C.3.2) is added to suppress the C E X background further. 4.5 Background Measurement Results The KW2 and background estimates, and the numbers of events Bj and C, CDi from figures 4.20 and 4.21, are shown in table 4.12 for each of the 1/3 and 2/3 1995 and 1996-7 data sets. A s shown in figure 4.9, the K^2 and background levels are given by bgKii2 = BKIVJ(RPV - 1) and bgKll2 = BK^/{RTD - 1), respectively, where the KN2 normalization is given by BKir2 = B^J(BKT2/BK^) ( s e e section 4.4.1 and figure 4.20) and the Kyu normalization is given by BK)i2 = BK^(+B?bva°fdf/2.3) for 1996-7 (1995) data. BK(l2 for the 1/3 1995 data set also involves a second bifurcation of kinematic quantities (cuts on P are bifurcated wi th cuts on R and E) in order to enhance statistics from the range ta i l (see section 4.4.2 and figure 4.21). The rejection of the P V function cut was estimated with high statistics using the 1/3 1995 data, and confirmed on the 2/3 1995 and 1/3 1996-7 data samples. It is listed as Rpv = 50.1 ± 1.1 in table 4.12. The rejection of the T D function cut is given by RTD = CDTD/CTD (see section 4.4.2 and figure 4.21). If any Bj or C, CD\ quantity is 0, it is taken to be 1 when calculating the background level. The KN2 and K^2 background levels, listed as bg^2 and bgx^ in table 4.12, for the 1/3 and 2/3 data sets are scaled up by 3 and 3/2 (10/3 and 10/7 in 1995), respectively, to account for the fractions of the full data sets that they represent. The K^2 background level for the 2/3 1996-7 data set is not scaled up, however, because the KN2 normalization, BK„2, for the 2/3 1996-7 data set is actually that from the 1/3 and 2/3 1996-7 data sets combined. The K^2 normalization for the 2/3 1996-7 data set fluctuated down relative to that for 1/3 1996-7 (although the results are wi th in statistical uncertainty), and the 1/3 1996-7 measurement was performed 110 Chapter 4. Analysis Figure 4.28: Target display of the " T G D E D X " event. The top and bottom numbers in each target fiber, and the left and right numbers in the IC, are the time (ns) and energy (MeV) of each hit, respectively. The kaon decay product appears to originate two fibers to the left of the kaon decay fiber. Ill Chapter 4. Analysis 1995 1996-7 using 1/3 using 2/3 using 1/3 using 2/3 8 23 9 17 (full) oEin 443 1031 517 1374 (full) r>Eout BK„? 9 24 12 24 (full) BK„2 0.16 ± 0 . 0 8 0.54 ± 0 . 1 6 0.21 ± 0 . 0 9 0.30 ± 0 . 1 1 (full) Rpy 50.1 ± 1 . 1 bole*. 0.0110 ± 0 . 0 0 5 3 0.0156 ± 0 . 0 0 4 5 0.0128 ± 0 . 0 0 5 6 0.0060 ± 0.0022 JD* rjPVoff uband 231 34903/234 0 4 2 6 8 BKU2 2.0 ± 0 . 5 4.9 ± 2 . 1 6.0 ± 2 . 4 8.0 ± 2 . 8 CDTD 71130 136180 79173 8113 (full) CTD 86 146 95 14 (full) RTD 827 ± 89 933 ± 77 833 ± 85 580 ± 155 (full) bQK„2 0.0080 ± 0.0020 0.0075 ± 0.0033 0.0216 ± 0 . 0 0 9 1 0.0207 ± 0.0092 Table 4.12: Numbers of events and C, CD[ from the normalization and rejection branches, respectively, of figures 4.20 and 4.21 for each of the 1/3 and 2/3 1995 and 1996-7 data sets, and the resulting Kn2 and K^2 normalizations BK^K^ rejections Rpv,TD, and scaled back-ground levels bgx^^^ a s defined in the text of section 4.5. Note that the K^2 normalization, BK^I a n d the T D rejection, RTD, quoted for the 2/3 1996-7 data set are actually those from the "full" 1996-7 data set (1/3 ± 2/3). A l l quoted uncertainties are purely statistical. in a completely unbiased way (any new cuts introduced for 1996-7 that are part of the Kn2 normalization were designed by studying background), so a less-statistically-uncertain and more conservative result for the 1996-7 Kv2 normalization is obtained using the full 1996-7 data set. The rejection of the T D function cut, RTD in table 4.12, for the 2/3 1996-7 data set is also calculated using the 1/3 and 2/3 1996-7 data sets combined (see section 4.4.2). The single-beam background estimates, and the numbers of events B* and C, CD\ from figures 4.22 and 4.23, are shown in table 4.13 for each of the 1/3 and 2/3 1995 and 1996-7 data sets. A s shown in figure 4.9, the single-beam kaon- and pion-entering background levels are given by bgBMXK = BBMIK/(RDELC - 1) and bgBM\p = BBM1P/(RDELC ~ 1), respectively, where the single-beam kaon- and pion-entering normalizations are given by BBMIK = BB1IIK[{BBMIK/BBMTK) - 1] (see section 4.4.3 and figure 4.22) and B B M L P = BBMIP[(BBMIP/BBMIP) ~ !] ( s e e section 4.4.3 and figure 4.23). The rejection of the D E L C cut, RDELC, is taken from the minimum of four different CDDELC/CDELC values (see sec-112 Chapter 4. Analysis t ion 4.4.3 and figures 4.22 and 4.23). The double-beam background estimates, and the numbers of events Bj and C , CD\ from figures 4.24 and 4.25, are shown in table 4.14 for each of the 1/3 and 2/3 1995 and 1996-7 data sets. A s shown in figure 4.9, the double-beam kaon- and pion-entering back-ground levels are given by bgBM2K = BBM2K/(RCK ~ 1) and bgBM2P = BBM2P/(RCpi ~ 1), respectively, where the double-beam kaon- and pion-entering normalizations are given by BTG BTG BBM2K = pBAiJ^Aout and BBM2P = RBiinBff$4out (i.e., using the second bifurcation -BM2K ' BM2K BM2P' BM2P see section 4.4.3 and figures 4.24 and 4.25). If the second bifurcation is not used, then the double-beam kaon- and pion-entering normalizations are given by BBM2K = B*BM2K and BBM2P = B B M 2 P , where B*BM2K and B*BM2P are the numbers of events left at the bottom of the normalization branches in figures 4.24 and 4.25 after all cuts in these branches have been applied sequentially. The double-beam rejections, RCK and RCPI, are taken from the min-ima of two different (CD/C)CK values and two different (CD/C)CPI values (see section 4.4.3 and figures 4.24 and 4.25). The C E X background estimates for the 1995 and 1996-7 data sets are given in sec-t ion 4.4.4. The background levels at the final cut positions are summarized in table 4.15. The total number of background events expected in the TX+VV(\) signal region for the complete 1995-7 data set is 0.080 ± 0.020, based on the measurements using the 2/3 data sets in 1995 and 1996-7. This estimate does not include the second-bifurcation result for the 1995 double-beam background estimate. If the second bifurcation is used for al l years, then the expected number of background events for 1995-7 is 0.067 ± 0.014. 4.6 Outside-the-Box Tests After making background estimates for a particular data sample (e.g., 2/3 1995), and before moving on to analysis of the next data sample (e.g., 1/3 1996-7), two "outside-the-box" tests are performed in order to gain confidence that al l potential backgrounds have been measured and that the background estimates (see section 4.5) are accurate: 113 Chapter 4. Analysis 1995 1996-7 using 1/3 using 2/3 using 1/3 using 2/3 T3FVTU DBM\K r>CKin DBM\K 11 32 25 87 5+3 36+11 22+8 78+18 jDCKout nBMlK 6 28 1+2 7+1 BBMIK 2.54 ± 1.21 11.86 ± 3 . 0 1 0.81 + 0.59 2.64+1.03 DELC 2093 5079 2055 4062 DELC 0 0 0 0 rn{2) DELC 4127 10191 4498 9130 DELC 0 1 0 0 CD$ELC ( B O X off) 3389 8328 3377 6650 C{DILC ( B O X off) 0 0 0 1 CD™ELC ( B O X off) 6788 16849 7461 15105 C%RLC ( B O X off) 0 3 1 2 D ( 1 ) ^DELC 2093 ± 2092 5079 ± 5078 2055 ± 2054 4062 ± 4061 D ( 2 ) ^•DELC 4127 ± 4 1 2 6 10191 ± 1 0 1 9 0 4498 ± 4497 9130 ± 9 1 2 9 R $ E L C ( B O X off) 3389 ± 3388 8328 ± 8327 3377 ± 3376 6650 ± 6649 R ^ R L C ( B O X off) 6788 ± 6787 5616 ± 3 2 4 2 7461 ± 7460 7553 ± 5340 bgBMiK 0.0040 ± 0.0045 0.0033 ± 0.0034 0.0012 ± 0 . 0 0 1 5 0.0010 ±0 .0010 JDPBBA nBMlP jDCPIin DBM\P 9 76 33 201 34+3 29+29 80+2 47+77 JDCPIout DBM\P 5 8 1+1 1+2 BBMIP 0.63 ± 0 . 3 5 1.37 ± 0 . 5 3 1.32 + 1.22 1.79 + 1.73 bgBMip 0.0010 ±0 .0012 0.0004 ± 0.0004 0.0019 + 0.0026 0.0007 ± 0.0009 Table 4.13: Numbers of events Bj and C, CD\ from the normalization and rejection branches, respectively, of figures 4.22 and 4.23 for each of the 1/3 and 2/3 1995 and 1996-7 data sets, and the resulting single-beam normalizations BBMIK,BMIP, rejections RDELC, and scaled background levels bgsMiK,BMiP a s defined in the text of section 4.5. A l l quoted uncertain-ties are purely statistical. A s shown in the table, the single-beam kaon- and pion-entering normalizations are calculated separately for the 1996 and 1997 data sets, and then added. The rejection of the D E L C cut, RDELC, is taken from the minimum of four different values: RDELC a n ( f RDELC w i t n a n d without the B O X cut applied (see section 4.4.3 and figures 4.22 and 4.23). 114 Chapter 4. Analysis 1995 1996-7 Background using 1/3 using 2/3 using 1/3 using 2/3 r>'l 'U aBM2K lDBiin BM2K r>Biout °BM2K B%M2K ( n o 2 n d b i f 0 0 11 2 26 0 25 1 36 3 0 2 1 0 0 BBM2K 0.273 ± 0.304 0.154 ±0.151 0.040 ± 0.056 0.0278 ± 0.039 BBM2K (n° 2nd bif.) 1.0 ± 1.0 1.0 ± 1.0 — — CDCK 276 810 237 461 CcK CDCK (low-side BOX) CCK (low-side BOX) 2 577 4 7 1409 14 0 381 0 2 762 7 RCK 138 ± 97 116 ± 4 4 237 ± 236 2 3 1 ± 1 6 3 RCK (low-side BOX) 144 ± 72 101 ± 27 381 ± 380 109 ± 41 bgBM2K 0.0066 ± 0.0088 0.0022 ± 0.0022 0.0005 ± 0.0009 0.0004 ± 0.0006 bgBM2K (no 2nd bif.) 0.0243 ± 0.0298 0.0143 ± 0.0148 — — r,ra °BM2P QB4m JDBM2P BRMOP (no 2nd bif.) 2 67 2 163 4 141 5 283 2 0 1 0 0 5 BBM2P 0.060 ± 0.059 0.012 ±0.015 0.028 ± 0.032 0.088 ± 0.056 BBM2P (no 2nd bif.) 1.0 ± 1.0 1.0 ± 1.0 — — CDCPI 3812 9148 5108 9921 CCPI CDCPI (low-side BOX) CCPI (low-side BOX) 4 6248 7 4 15149 9 0 8780 3 6 17425 10 RCPI 953 ± 476 2287 ± 1143 5108 ± 5107 1654 ± 675 R C p i (low-side BOX) 893 ± 337 1683 ± 561 2927 ± 1689 1743 ± 551 bgBM2P 0.00022 ± 0.00024 0.00001 ± 0.00001 0.00003 ± 0.00004 0.00008 ± 0.00006 bgBM2P ( n o 2nd bif.) 0.00374 ± 0.00400 0.00085 ± 0.00090 — — Table 4.14: Numbers of events B* and C, CDi from the normalization and rejection branches, respectively, of figures 4.24 and 4.25 for each of the 1/3 and 2/3 1995 and 1996-7 data sets, and the resulting double-beam normalizations BBM2K,BM2P, rejections RCK,CPI, and scaled background levels bgBM2K,BM2P as defined in the text of section 4.5. A l l quoted uncertainties are purely statistical. The double-beam normalizations, BBM2K and BBM2P, and the resulting double-beam background levels, bgsM2K and bgsM2P, are calculated with and without the second bifurcation. The double-beam rejections, RCK and RCPI, are taken from the minima of two different values: wi th the B O X cut applied, and wi th only the low-side B O X cut applied (see section 4.4.3 and figures 4.24 and 4.25). 115 Chapter 4. Analysis bkgd 19 using 1/3 95 using 2/3 199 using 1/3 6-7 using 2/3 1995-7 B M l B M 2 C E X 0.0110 ± 0 . 0 0 5 3 0.0080 ± 0.0020 0.0051 ±0 .0054 0.0281 ± 0.0300 (0.0069 ± 0.0088) 0.0045 ± 0.0045 0.0156 ±0 .0045 0.0075 ± 0.0033 0.0037 ±0 .0038 0.0152 ± 0 . 0 1 4 9 (0.0022 ±0 .0022) 0.0045 ± 0.0045 0.0128 ±0 .0056 0.0216 ± 0 . 0 0 9 1 0.0031 ± 0.0037 0.0005 ± 0.0009 0.0051 ± 0 . 0 0 5 1 0.0060 ± 0.0022 0.0207 ± 0 . 0 0 9 2 0.0016 ± 0 . 0 0 1 8 0.0005 ± 0.0006 0.0051 ± 0.0051 0.0216 ±0 .0050 0.0282 ±0 .0098 0.0054 ±0 .0042 0.0157 ±0 .0149 (0.0027 ±0 .0023) 0.0096 ± 0.0068 total 0.0567 ±0 .0314 (0.0355 ±0 .0126) 0.0465 ± 0 . 0 1 6 9 (0.0335 ± 0.0084) 0.0431 ±0 .0124 0.0340 ± 0.0109 0.0804 ± 0 . 0 2 0 1 (0.0675 ±0 .0138) Table 4.15: Number of background events expected in the TT + ^P(1) signal region for the 1995 and 1996-7 data using estimates based on the 1/3 and 2/3 data samples. A l l quoted uncertainties are purely statistical. " B M l " and " B M 2 " refer to the sum of the kaon- and pion-entering single- and double-beam backgrounds, respectively. The 1995 B M 2 background estimates using the second bifurcation are shown in parentheses. The total background for 1995-7 is calculated using the 2/3 estimates. • a correlation study, performed by loosening the bifurcated branches of a background estimate and comparing predicted and observed numbers of events; • a search for events which fail only 1 of 16 possible "cut classes". The outside-the-box correlation study is described in section 4.2. Backgrounds are esti-mated using the bifurcation technique shown in figures 4.20 through 4.25 and the calculations described in section 4.5, but wi th the two function cuts associated wi th the normalization and rejection branches of each background estimate loosened at the same time. The number of events in each outside-the-box region, as defined by the looser cuts, is then counted and compared wi th the prediction. Results for various loosenings of the K^, K^ii single-beam and double-beam background normalization and rejection function cuts, as a function of pathology cuts applied (grouped into classes), are shown in tables 4.16 and 4.17 for the 2/3 1996-7 data sample. Background predictions and observations are made wi th each pathol-ogy cut class (defined in table 4.18) turned off, so that any discrepancies between predicted and observed numbers of events may be more readily assigned to a particular correlation mechanism. For example, in table 4.16, the discrepancy at the 10 x 10 outside-the-box cut position wi th the M A S S cut turned off (5 background events observed when 0.63 events were predicted) stimulated further study of correlations between T D and kinematic 116 Chapter 4. Analysis cuts, which led to the G D R background hypothesis (see section 4.1.1.1) and development of the T D V E L and M A S S cuts (see section C.3.4). Otherwise, predictions of the outside-the-box correlation study are consistent wi th observations in all cases, except perhaps at the loosest single-beam kaon-entering cut positions in table 4.17, where slightly more events are observed than predicted. However, this discrepancy is occurring very far from the final cut positions (possibly due to some breakdown in the background estimation structure for very loose cuts), so it is not worrisome. Background predictions and observations, as a function of pathology cuts applied, from a global loosening of al l function cuts at the same time on the 2/3 1996-7 data sample, are shown in table 4.19. Predictions are somewhat higher than observations, possibly due to correlations among background types. These correlations are partially taken into account by re-calibrating the functions every time a cut class is turned off and/or functions associated with other background types are loosened. For example, in figure 4.21, the Z U T O U T , C O S 3 D , Z F R F , L A Y E R 1 4 kinematic pathology cuts are turned off, and the K^2 kinematic, P V , and beam function cuts are all loosened when estimating the component of the background for the global loosening wi th the "kinz" cut class turned off (i.e., the second row in table 4.19). This is done so that additional background introduced by loosening the beam function cuts, above and beyond that introduced by loosening the T D and K^2 kinematic function cuts, is accounted for. However, it 's possible that loosening the beam function cuts can introduce the same additional K^2 background that loosening the T D and K^2 kinematic function cuts does (e.g., K^2 decay in flight events). This additional background is then predicted twice (in both the K^2 and beam background estimates), whereas it is observed in the outside-the-box region only once, making predictions higher than observations. In any case, observations are not larger than predictions, which gives confidence in the background estimates. Note that when a function is loosened, the inverted function (in the other branch of the bifurcation) is kept at the standard cut position, so that the tagging efficiency of the inverted cut for background events remains high. This tagging efficiency must be high so that events which correlate the bifurcated cuts (e.g., are more likely to pass a cut if they pass its 117 Chapter 4. Analysis cuts Kn2: P V x kin. K,2: T D x kin. turned off 10 x 10 50 x 50 50 x 300 10 x 10 159 x 134 1000 x 134 none predict: 1.04 10 47 0.42 13.7 119 observe: 1 10 48 2 12 111 mass predict: 1.04 10 48 0.63 34.9 295 observe: 1 10 49 5 37 272 kinz predict: 1.25 11 59 2.10 65.7 536 observe: 1 11 61 4 74 511 kqual predict: 1.46 14 56 1.43 46.8 388 observe: 2 14 58 3 63 357 kintg predict: 1.67 15 61 0.55 16.5 143 observe: 1 15 63 2 16 134 kdedx predict: 1.04 11 52 1.66 127.9 1026 observe: . 1 11 54 4 131 953 epitg predict: 1.24 12 59 0.50 16.5 143 observe: 1 12 61 2 13 132 tgtr predict: 1.05 11 49 0.41 13.8 119 observe: 1 11 50 2 12 111 timcon predict: 1.05 11 48 0.43 14.2 123 observe: 1 11 49 2 12 115 ic predict: 1.46 12 51 0.44 15.7 133 observe: 2 12 57 . 2 14 124 b4ekz predict: 1.04 10 47 0.46 14.1 121 observe: 1 10 48 2 12 112 bhtrs predict: 1.04 10 47 0.42 13.8 119 observe: 1 10 48 2 12 111 Table 4.16: Predicted and observed numbers of and background events in the 2/3 1996-7 data sample, obtained by loosening the P V and K^2 kinematic function cuts, and the T D and kinematic function cuts, respectively, by the factors shown. Background predictions and observations are made with various pathology cuts, grouped into classes (see table 4.18), turned off. The actual loosening factors given in the head of the table are only approximate, because the events introduced by turning off pathology cut classes may not be distributed evenly in the function values TV. The different functions are defined for different ranges in N (see sections C.3.3, C.3.4, and C.3.5), which is why the maximal loosening for each function cut is different in the head of the table. Note that predicted and observed numbers of events agree almost exactly at the maximal loosenings for the P V (factor of 50) and T D (factor of 1000) function cuts. This is because these loosest cut positions correspond to turning the cuts off, which is essentially equivalent to inverting the cuts, and it is on the inverted-cut data samples that the kinematic functions are calibrated. 118 Chapter 4. Analysis c u t s s i n g l e - b e a m k a o n - e n t e r i n g s i n g l e - b e a m p i o n - e n t e r i n g t u r n e d of f l oose 1 l oose2 l o o s e 3 l o o s e l l o o s e 2 l o o s e 3 n o n e p r e d i c t : 0 . 057 0 .94 8.9 0 .029 0.31 1.9 o b s e r v e : 0 0 14 0 0 2 k i n z p r e d i c t : 0 .073 1.15 10 .8 0 . 0 4 2 0 .42 2 .5 o b s e r v e : 0 1 16 0 0 2 k q u a l p r e d i c t : 0 .058 0.91 9.3 0 .029 0 .29 1.8 o b s e r v e : 0 0 14 0 0 2 k i n t g p r e d i c t : 0 . 0 7 0 1.13 11 .3 0 .038 0.41 2 .5 o b s e r v e : 0 0 16 0 0 2 k d e d x p r e d i c t : 0 .062 1.04 9.5 0 .038 0 .42 2 .6 o b s e r v e : 0 0 14 0 0 2 e p i t g p r e d i c t : 0 .145 1.57 13 .4 0 . 0 7 9 0 .58 3.2 o b s e r v e : 0 0 20 0 0 4 t g t r p r e d i c t : 0 . 0 7 3 1.05 9.6 0 .036 0 .32 1.9 o b s e r v e : 0 0 15 0 0 2 t i m c o n p r e d i c t : 0 .128 1.65 16 .7 0 . 0 6 2 0 .44 2 .5 o b s e r v e : 0 0 22 0 0 3 i c p r e d i c t : 0 . 077 1.00 9.0 0 . 0 3 5 0 .29 1.7 o b s e r v e : 0 0 14 0 0 2 b 4 e k z p r e d i c t : 0 .095 1.70 17 .2 0 . 0 5 0 0 .53 3.2 o b s e r v e : 0 1 18 0 0 4 b h t r s p r e d i c t : 0 . 067 1.07 9.8 0 .037 0.41 2 .5 o b s e r v e : 0 0 14 0 0 2 c u t s d o u b l e - b e a m k a o n - e n t e r i n g d o u b l e - b e a m p i o n - e n t e r i n g t u r n e d off l oose 1 l oose2 l o o s e 3 l o o s e l l o o s e 2 l o o s e 3 n o n e p r e d i c t : 0 . 0 0 3 7 0 .036 0.31 0 . 0 2 0 0 .16 1.2 o b s e r v e : 0 0 0 0 0 1 k i n z p r e d i c t : 0 . 0046 0 .036 0 .30 0 . 0 1 9 0 .16 1.2 o b s e r v e : 0 0 0 0 0 1 k q u a l p r e d i c t : 0 . 0 0 3 3 0 .031 0 .30 0 . 0 2 0 0 .16 1.2 o b s e r v e : 0 0 0 0 0 1 k i n t g p r e d i c t : 0 .0044 0 .037 0 .32 0 .026 0 .24 1.8 o b s e r v e : 0' 0 0 0 0 3 k d e d x p r e d i c t : 0 . 0039 0 .037 0.31 0 .024 0 .19 1.4 o b s e r v e : 0 0 0 0 0 2 e p i t g p r e d i c t : 0 . 0039 0 .035 0.31 0 .047 0 .53 4 .3 o b s e r v e : 0 0 0 0 0 4 t g t r p r e d i c t : 0 . 0036 0 .033 0 .30 0 . 0 2 3 0 .19 1.4 o b s e r v e : 0 0 0 0 0 1 t i m c o n p r e d i c t : 0 . 0 0 5 2 0 .039 0 .35 0 . 0 3 0 0 .28 2 .2 o b s e r v e : 0 0 0 0 0 1 i c p r e d i c t : 0 . 0069 0 .042 0.31 0 . 0 3 0 0 .30 2 .3 o b s e r v e : 0 0 0 0 1 2 b 4 e k z p r e d i c t : 0 .0064 0 .045 0 .36 0 .068 0 .78 6.3 o b s e r v e : 0 0 0 0 0 6 b h t r s p r e d i c t : 0 .0044 0 .036 0 .30 0 . 0 2 2 0 .17 1.2 o b s e r v e : 0 0 0 0 0 1 Table 4.17: Predicted and observed numbers of beam background events in the 2/3 1996-7 data sample, obtained by loosening the single-beam kaon-entering ( D E L C , C K T R S , C K -T A I L ) and pion-entering ( D E L C , C P I T R S , C P I T A I L ) function cuts, and the double-beam kaon-entering ( B W T R S , C K T R S , C K T A I L , B 4 T R S , B 4 T D ) and pion-entering ( B W T R S , C P I T R S , C P I T A I L , B 4 T R S , B 4 T D , P B N R S ) function cuts. Background predictions and observations are made with various pathology cuts, grouped into classes (see table 4.18), turned off. Because the single-beam functions involve cuts which are also used by the double-beam functions, it is difficult to define the individual single- and double-beam values at the outside-the-box cut positions, which is why they are labelled as "loosel, loose2, loose3" in the table. The functions are loosened such that loosel , loose2, and loose3 correspond to increases in the predicted amount of beam background by about 1, 2, and 3 orders of magnitude, respectively. 119 Chapter 4. Analysis cut class cut members 1. pv P V function 2. kp2 KT,2 kinematic function (low-side R B O X , E B O X , P B O X , and R B O X ' , E B O X ' , P B O X ' ) 3. t d T D function ( T D T C O N , T D D F A l , E V 5 , E L V E T O , T D F O O L , T D V E L , T D L I K 2 , T D L I K 3 , T D D F A 2 ) 4. km2 kinematic function (high-side R B O X . E B O X . P B O X , and R N G M O M ) 5. bmfn D E L C , B W T R S , C K T R S , C K T A I L , C P I T R S , C P I T A I L , P B N R S , B 4 D E D X , B 4 T R S , B 4 T D 6. mass M A S S 7. kinz Z U T O U T , C O S 3 D , Z F R F , L A Y E R 1 4 8. kqual U T C Q U A L , P R O B Z , C H I R F , C H I R F _ N H Z 9. kintg T G D E D X , P I G A P , T G L I K E , T G B 4 10. kdedx C H I M A X _ R S D E D X , C L _ R S D E D X , R S L I K E , T D E C O N 11. epitg T G C C D P F , E P I T G , E P I M A X K , P H I V T X , P H I V T X 2 , O P S V E T O , O P S V E T O _ L K B , T G E D G E 12. tgtr T G Q U A L T , T G E R , T A R G F , D T G T T P , R T D I F , D R P 13. t imcon T I M C O N , T I C , T G C C D 14. ic E I C K I N , E I C , K I C , T G G E O 15. b4ekz B 4 E K Z , B 4 E K Z J C , T G Z F O O L 16. bhtrs B H T R S Table 4.18: Complete listing of cuts applied in the analysis, divided into pathology cut classes. Classes 6—16 are turned off one by one in the outside-the-box correlation study. Classes 1 — 16 are used in the single-class-failure study. Note that some cuts are applied only to the 1996-7 data (see sections C.3.1, C.3.2 and C.3.4). bifurcated partner cut) are not eliminated by inverted cuts from the bifurcated background estimation structure. The outside-the-box correlation tests are very effective at detecting correlations between bifurcated cuts. For example, early in the analysis when designing the Kn2 kinematic function on the 1/3 1995 data, a correlation was discovered between kinematic quantities which are part of the second bifurcation in the Kn2 background estimation structure (cuts on E are bifurcated wi th cuts on R and P). This correlation was discovered v ia observation of roughly a factor of 3 more events than predicted at the K^2 outside-the-box cut positions (e.g., 20 events observed at the 50 x 50 cut position when 7.1 were expected). This was found to be mainly due to (1) a problem with the IC energy calculation, which affects al l of E, R, and P , and (2) a problem in the position of target fibers relative to the U T C (the "target rotation bug"), which resulted in a non-Gaussian high P ta i l to the K^2 peak. Possible problems wi th the IC were also suggested by more events observed than predicted (9 vs. 4) at the "loose3" double-beam kaon-entering outside-the-box cut position, when the cuts in the "ic" pathology class were turned off. The Kn2 correlations were removed by introducing the E I C K I N cut, and by using tighter cuts on R and P when measuring the rejection of the low-side E B O X cut. Possible correlations in the double-beam kaon-entering background 120 Chapter 4. Analysis cuts turned off 1« 4 ^osenii 25 ig facto 100 r 2500 none predict: observe: 0.276 0 2.59 2 9.74 7 62.9 38 kinz predict: observe: 0.548 0 5.04 3 20.78 14 110.7 69 kqual predict: observe: 0.349 0 3.75 2 17.07 12 97.7 52 kintg predict: observe: 1.004 0 4.76 3 16.08 10 93.8 62 kdedx predict: observe: 0.592 0 4.83 3 20.65 14 131.2 62 epitg predict: observe: 0.899 0 7.40 9 19.79 19 139.3 112 tgtr predict: observe: 0.612 0 4.67 3 14.20 8 60.8 41 t imcon predict: observe: 0.470 0 4.31 2 14.36 8 77.3 55 ic predict: observe: 0.771 0 6.02 2 20.77 10 127.0 49 b4ekz predict: observe: 0.810 0 6.51 3 30.80 18 331.5 201 bhtrs predict: observe: 0.784 0 6.32 2 23.05 7 206.5 38 Table 4.19: Predicted and observed numbers of al l types of background events in the 2/3 1996-7 data sample, obtained by loosening all functions at the same time. Background predictions and observations are made with various pathology cuts, grouped into classes (see table 4.18), turned off. The contributions from each of the K„2, and beam backgrounds are increased over the final background levels in the 7 r + z ^ ( l ) signal region by the approximate factors shown in the head of the table. 121 Chapter 4. Analysis estimate were removed by tightening the K I C cut, and by introducing the B 4 E K Z J C and T G G E O cuts. Late in the analysis, a potential correlation was found between T D and kinematic cuts in the 2/3 1996-7 data, v ia 5 events observed when 0.73 were expected at the 10 x 10 outside-the-box cut position. This apparent correlation was eventually attributed to G D R background [41], and the T D V E L and M A S S cuts were introduced to attack it. The Kn2 and K^2 outside-the-box correlation results shown in table 4.16 are those after the T D V E L and M A S S cuts were added, whereas the beam and global results in tables 4.17 and 4.19 do not include these cuts. The single-event failure study consists of a search for events which fail only one of the cut classes listed in table 4.18. Events which fail a single cut may be "close to the box", and may indicate an event pathology which has not been properly accounted for in the background estimates. For each of the 1/3 and 2/3 1995 and 1996-7 data sets, there are hundreds of events which fail just the K^i kinematic function cut. However, the vast majority of these events fail badly, and are well outside the range of meaningful function values (i.e., they are far from the box). The small number (between 1 and 3 for each data set) which have meaningful function values each fail at least two of the low-side R, E, and P B O X cuts. Furthermore, the number observed is consistent wi th the number expected from the Kn2 background estimate in the box and assumption of a linear increase in background wi th Nkin,K^2- For each data set there are also typically a few events which fail just the T D function cut, a few events which fail just the kinematic function cut, and up to roughly 10 events which fail just the beam function cuts. The T D failures always fail at least 2 of the fixed T D cuts, and therefore have maximal T D function values and are located far from the box. The failures either fail badly (are well outside the range of meaningful function values, far from the box), or are consistent with the number expected from the background estimate in the box and assumption of a linear increase in background wi th Nkin,K^2 • The beam function cut failures always fail multiple cuts which come from at least 2 different branches of either the single-beam or double-beam background estimation structures. 122 Chapter 4. Analysis The only potentially troubling event, which might indicate a problem in the background estimates, is in the 2/3 1996-7 data. This event fails only the "kintg" pathology class, and in fact fails only the T G D E D X cut. This event is discussed in section 4.4.4 and is believed to be a C E X background event. Further study of the C E X background did not indicate any loopholes in the C E X background estimate [41]. The O P S V E T O - L K B safety cut introduced to deal wi th this event was designed late in the analysis, and is not part of the beam and global outside-the-box correlation results shown in tables 4.17 and 4.19. 4.7 Signal Evaluation Criteria Prior to "opening the box" and examining any candidate events in the 7 r + ^ P ( l ) signal region, a "golden region" is defined where the background level is about a factor of 10 smaller than that estimated in section 4.5, such that candidate events can be assigned a likelihood to be signal based on whether or not they fall in this region. Given that the and Kn2 backgrounds are large compared to the beam backgrounds (see table 4.15), only the P V , T D , and Kn2 and K^2 kinematic function cuts are tightened by a factor of vTO, so that the Kn2 and backgrounds are each reduced by a factor of \ / l 0 • \ / l 0 = 10. The acceptance loss of the tighter P V cut reduces the background further. Likewise the acceptance loss of the tighter T D cut reduces the Kn2 background further. The acceptance loss of the tighter P V , T D , and Kn2 and K^2 kinematic function cuts combined reduces the single-beam, double-beam, and C E X backgrounds. The background rejection and signal acceptance of the golden region relative to the signal region are estimated as follows. The final P V cut requires Npy < 1.0, so the cut which is a factor of vTO tighter requires Npv < l / \ / T 0 - From table 4.14, this corre-sponds to the requirement Npy < 0.3093 which has 1.0000/0.3177 = 3.15 rejection and 0.6065/1.000 = 0.606 acceptance relative to the final cut position. The final T D cut re-quires NTD < 1-003. From table C.4, the cut which is a factor of \ / l 0 tighter requires NTD < 0.308, which has 1.003/0.308 = 3.26 relative rejection and 0.808/1.171 = 0.690 relative acceptance. The final Kn2 kinematic cut requires NkiN,KN2 ^ 0.3358. From ta-123 Chapter 4. Analysis bkgd cuts extra rej. no. of events relative acc. NPV < 0.3177, W ^ , ^ < 0.1175 13.06 0.00166 ± 0.00038 0.564 N T D < 0.308, N K I N T K F I 2 < 0.0828 17.43 0.00162 ± 0.00056 0.640 B M 1 — 2.771 0.00193 ± 0 . 0 0 1 5 2 1.0 B M 2 — 2.771 0.00097 ± 0.00083 1.0 C E X — 2.771 0.00346 ± 0.00246 1.0 total 7.00 0.00964 ± 0.00308 0.36 Table 4.20: Number of background events expected in the golden region for the 1995-7 data. The "no. of events" for each background type is obtained by dividing the final background levels in table 4.15 by the "extra rejection". The extra rejection comes from tighter cuts which have additional background suppression and acceptance loss. More details can be found in section 4.7 of the text. ble C.5, the cut which is a factor of ^/T0 tighter requires N^K^ ^ 0.1175, which has 0.3358/0.1175 = 2.86 relative rejection and 0.8433/0.9069 = 0.930 relative acceptance. The final Kpi kinematic cut requires N^K^ < 0.2681. From table C.6, the cut which is a factor of VlO tighter requires N K I N , K ) I 2 < 0.0828, which has 0.2681/0.0828 = 3.24 relative rejection and 0.9500/1.0237 = 0.928 relative acceptance. In the golden region, the background is therefore reduced by a factor of 3.15 • 2.86/0.690 = 13.06 wi th 0.606 • 0.930 = 0.564 acceptance. The background is reduced by a factor of 3.26-3.24/0.606 = 17.43 wi th 0.690-0.928 = 0.640 acceptance. The single-beam, double-beam, and C E X backgrounds are reduced by a factor of 1/(0.606-0.690-0.930-0.928) = 2.771 wi th no acceptance loss. A s shown in table 4.20, a total of 0 .00964±0.00308 background events is expected in the golden region for the complete 1995-7 data set (note that the second bifurcation result for the 1995 double-beam background estimate is used here, in order to define and achieve high rejections inside the box). This is a factor of 7.00 less than the background in the signal region (0.0675 ± 0.0138), at 0.564 • 0.640 = 36% of the signal-region acceptance. A s shown in table 4.20, the C E X background dominates in the golden region. In a previous analysis [46], the golden region was defined as having a factor of 10 less back-ground than the signal region. This can be done here (although it was done after signal event examination) by tightening the C E X function by a factor of 4. The final C E X cut 124 Chapter 4. Analysis bkgd cuts extra rej. no. of events relative acc. NPV < 0 . 3 1 7 7 , ^ , ^ < 0.1175 14.50 0.00149 ± 0.00035 0.564 NTD < 0.308, NKIN^2 < 0.0828 19.36 0.00146 ± 0.00050 0.640 B M 1 3.078 0.00174 ± 0 . 0 0 1 3 7 1.0 B M 2 — 3.078 0.00087 ± 0 . 0 0 0 7 5 1.0 C E X NCEX < 0.26 10.66 0.00090 ± 0.00064 0.9004 total 10.4 0.00646 ± 0.00180 0.33 Table 4.21: Number of background events expected in the tight golden region for the 1995-7 data. The "no. of events" for each background type is obtained by dividing the final background levels in table 4.15 by the "extra rejection". The extra rejection comes from tighter cuts which have additional background suppression and acceptance loss. More details can be found in section 4.7 of the text. requires NCEX < 1-0. From table C.8, the cut which is a factor of about 4 tighter requires NCEX < 0.26, which has 1.0/0.26 = 3.85 relative rejection and 0.9004 relative acceptance. Therefore, the C E X background i n table 4.20 is reduced by an additional factor of 3.85, and the Kn2, K/i2, single-beam and double-beam backgrounds are reduced by an additional factor of 1/0.9004 = 1.111. A s shown in table 4.21, the contributions from each background type in this tighter golden region are roughly equal, combining for a total of 0.00646 ± 0 . 0 0 1 8 0 back-ground events in the golden region for the complete 1995-7 data set. This is a factor of 10.4 less than the background in the signal region (0 .0675±0 .0138) , at 0.564-0.640-0.9004 = 33% of the signal region acceptance. 4.8 Search for Signal From section 4.5, backgrounds are estimated to contribute <C 1 event in the 7r + i /p( l) signal region, namely, 0.08 ± 0 . 0 2 events. The outside-the-box tests from section 4.6 revealed no flaws in the background estimation procedure, and signal evaluation criteria have been defined in section 4.7. According to the analysis strategy described in section 4.2, the box can now be opened with the understanding that any events observed in the box are defined to be signal. Figure 4.29 shows range in scintillator (R) vs. energy (E) for events passing al l other cuts 125 Chapter 4. Analysis in the complete 1995-7 data set. The rectangular box defined by the R B O X and E B O X cuts therefore indicates the ir+vD(l) signal region, where backgrounds are expected to contribute 0.08 ± 0.02 events. One event survives in the box, namely, the same event found in the published 1995 analysis [46]. The events clustered at E = 108 M e V are from decays where no photons are detected, the number of which is consistent wi th that expected from measurements of the photon detection inefficiency. The location of the candidate event in R vs. E wi th respect to the Monte Carlo ( U M C ) distribution of K+ —> TX+VV events is shown in figure 4.30. A display of the candidate event (event 42251 from run 23271) is shown in figure 4.31. The target, IC , and R S times and energies of this event are shown in figure 4.32. The event passes the tightest cuts defined by the P V , kinematic, kinematic, single-beam, double-beam, and C E X functions, which define Npv = 0, NkintK„2 = 0> ^kin,K^2 = 0> NBMI = 0, NBM2 = 0, and NCEX = 0. The T D function value for this event is NTD = 0.1343. Therefore, the event lies in the tight golden region (see tables 4.20 and 4.21). The probability of the event being due to one of the known backgrounds is therefore very small (about 0.6%), so the event is likely due to the decay K+ —> ir+vv. The kaon decay time of the candidate K+ —> TT+vi> event, defined as the R S track time minus the time of the kaon in the target, tRs - tu, is 24.0 ns (the mean kaon lifetime is 12.386 ns). This large delayed coincidence of the kaon and its decay particle guarantees that the kaon decayed from rest. The pion decay time in the stopping counter, t^, is 26.9 ns (the mean pion lifetime = 26.033 ns) and the decay-muon has an energy of 3.20 M e V (the expected scintillator-saturated value is 3 . 0 4 ± 0 . 4 5 M e V ) . The decay-muon has a decay time, te, of 3252.7 ns (the mean muon lifetime is 2197.03 ns) and the decay-electron has an energy of 50.66 M e V (the Michel spectrum peaks at the endpoint of 53 M e V ) . The kinematic quantities associated wi th the candidate event are R — 34.75 cm, E = 117.73 M e V and P = 218.17 M e V / c . The uncertainties in these quantities come from a linear interpolation of the R, E, and P resolutions for and decays at rest. From 126 Chapter 4. Analysis 46 i i i i i i i i i i i i i i i i i i i i i i i i | i i i i 44 — 42 — 40 - --38 — — 36 — — 34 — • • — 32 30 — •• • — 28 i i i i 1 i i i i 1 i i i i 1 i i i i 1 i i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 90 100 110 120 130 140 150 E (MeV) Figure 4.29: Range in scintillator (R) vs. energy (E) for events in the 1995-7 K+ —• n+uu data set passing all other cuts in the analysis. The it+vv{l) signal region is indicated by a box, and contains a single candidate event. 127 Chapter 4. Analysis g 4 6 42 40 38 36 34 32 30 28 "i—1—1 r 1 1 r i 1 1 r i 1 1 r i 1 r • • • • • • 0 • • . . . • • • • • • • • • • o ••••• • a a • n•P••••••P•• • » a • • • • • • • • • -- • = • • • • • • • • • • • • • • • • • • • • a « • • • a a • • • • • • • • • • • « • • • • • • • • • • Q • - • • O • • • • • • o • • • o D a D D a a - - - • • • • I !• , • • 3 • D| ) • • • • • 1 P • n • • O J I I L J I I L J I I L J I I I J I I L J L 90 100 110 120 130 140 150 E (MeV) Figure 4.30: Distr ibut ion of range in scintillator (R) vs. energy (E) for UMC-generated K+ —> TX+UD events (small open squares). Similar to figure 4.29, the ir+vv(l) signal region is indicated by a box, and contains the candidate event from the 1995-7 data, shown as the filled black circle. 128 Chapter 4. Analysis i ' •1 • i • • • • i • 1 1 11 • 1 1 • i • 1 1 11 • •1 • i • • • • i Figure 4.31: Display of the candidate K+ —> TT+VV event. Left: The event is viewed from downstream of the detector, and the pion track is shown to have several hits i n the target, an I C hit, anode wire hits i n a l l 3 superlayers of the U T C , and hits in layers 1 — 14 of the R S and the inner R S S C . The extraneous U T C hits in the second and third superlayers, located just below the 3 o'clock position, if fitted to a "track" and propagated back to a radius of 0, indicate that this "track" was produced about 16 cm upstream of the upstream edge of the target, somewhere in the degrader. The fact that the drift circles do not meet indicate that the hits in the second superlayer occurred about 30 ns before the track (i.e., near the time of the kaon). The extraneous U T C hits located just left of the track in the outer superlayer are at about the same z as the extraneous outer superlayer hits near 3 o'clock, and again the drift circles do not meet. The extraneous U T C hit in the second superlayer near 8 o'clock has no A D C value and there is no induced charge on the cathode strips, so it is likely due to electronic noise. Upper right: T D data in the stopping counter is indicated by crosses, which shows the double-pulse TT —• \x decay signature. Three fits of the T D data to the expected pulse shape are shown: for the first pulse, the second pulse, and the pulses combined. Lower right: The target region is blown up in order to see the kaon fibers (hatched) and pion fibers (open) more clearly. The C C D data from one of the kaon fibers (not the kaon decay fiber) is indicated by crosses, along wi th a fit of the data to the expected pulse shape. 129 Chapter 4. Analysis RUN 23271 Event 42251 - 3 - 2 -1 Figure 4.32: Top: Close-up of the target hits for the candidate K+ —> n+uu event. The top and bottom numbers in each target fiber, and the left and right numbers in the IC, are the time (ns) and energy (MeV) of each hit, respectively. The kaon fibers have hits close to t = 0 ns, whereas the pion fibers have hits about 24 ns later, ensuring that the kaon decayed from rest. The kaon entered the target near the location of the B4 hit, and left energy in 4 fibers before decaying. Bot tom: Close-up of the R S hits for the candidate K+ —> ir+vD event. The left and right numbers in each counter are the time (ns) and energy (MeV) of each hit, respectively. 130 Chapter 4. Analysis tables 5.20 and 5.21 of section 5.8 for the 1995 data, the resolutions are given by o(R) = 0.992 + (R - 30.349) • (2.217 0.992) = 0.05182 • R - 0.581 (4.4) (53.988 - 30.349) a{E) = 3.295 + {E - 108.391) • (4.336 3.295) = 0.02258 E + 0.848 (4.5) <T(P) = 0.992 + (154.494 - 108.391) ( P - 205.177) • (2.928 2.497) = 0.01417 P- 0.410 (4.6) (235.593 - 205.177) Plugging the event kinematics into Eqs. (4.4), (4.5) and (4.6) gives • R = 34.75 ± 1.22 cm • E = 117.73 ± 3 . 5 1 M e V • P = 218.17 ± 2 . 6 8 M e V / c More information about the candidate event can be found elsewhere [55]. The calculations of the K+ —> Tr+vi/ and K+ —• ir+f branching ratios and \ Vt(i\ are given in chapter 6, based on observation of this single event and the K+ —> n+uP and K+ —» n+f single-event sensitivities calculated in chapter 5. 131 Chapter 5 Acceptance and Sensitivity A s shown in the previous chapter, one candidate K+ —> ir+vv event was observed in the 1995-7 data. To calculate the branching ratio for K + — » it+vv, the total number of kaons collected must be counted. Also, the "acceptance" of the online and offline cuts must be calculated, which represents the fraction of potential K+ —> -K+VV events which survive the online and offline cuts. The "single-event sensitivity" of the analysis is the number of kaons multiplied by the acceptance. The K + —> n+uu branching ratio is then given by the number of observed K + —* TT+UP events divided by the single-event sensitivity for K + —> n+uu events. Because a large sample of K + —• n+uu events is not available, the acceptances of most cuts are measured using K^O-), ^ 2 ( 2 ) , and irscat monitor data (see section 3.3). The acceptances of some cuts must be measured using UMC-generated K + —> ix+vi> data because some cuts have specific acceptances for K+ —* ir+vv events which can not be estimated using monitor events. The total acceptance is divided into the K ^ - b a s e d acceptance, the K^-based acceptance, the 7r s c a t-based kinematic acceptance, the 7r s c a t-based T D acceptance, the ^-veto acceptance due to accidentals, the T • 2 efficiency, the UMC-based trigger, fiducial, and N I D I F (nuclear interactions and decay in flight) acceptances, and the stopping fraction fs. The complete measurement of the acceptance and sensitivity is tested by measuring the KW2 branching ratio and comparing with the accepted value. The K^-, K-K2~, nScat-, and UMC-based acceptance measurements are carefully con-structed so that specific types of acceptance loss by every cut are neither omitted nor double-132 Chapter 5. Acceptance and Sensitivity 1995 1996 1997a 1997b 1.53159 1.16161 0.378020 0.218325 add back bad runs: — +0.00324 — — multiple scalers: — -0 .035 — — missing T D fiducial pulses, target H V trips: — -0 .36% - 1 . 9 1 % -1 .17% bad stopping counters: -0 .3473% -0.0468% - 0 % -0.0120% 1.52627 1.12525 0.370800 0.215745 Table 5.1: KBuVe counting in the 1995-7 analysis. The variation in KBuve wi th each year is primarily due to variation in the data collection time allotted to E787 each year. counted. Kaons are counted and acceptances are calculated separately for the 1995, 1996, 1997a, and 1997b run periods due to changes in hardware and run conditions (e.g., kaon beam momentum - see section 3.1). 5.1 Counting Kaons A signal called KBuve is used to count the number kaons entering the detector. It is constructed from the KB signal (see section 3.3) and a "computer ready" signal. The "computer ready" signal indicates that online data processing and potential readout have been completed, such that al l detector systems are ready to accept data. The number of Ksiive signals represents the number of kaons that entered the target when the digitizing hardware was available for collection of signals arising from the kaon and its decay products. The KBuve scaler values (see section 3.3) are read and written to tape at the end of each spill . The values on tape are read, summed, and writ ten to loghTes during analysis. The logfiles from the PASS2 analysis, wi th some bad runs removed (via the B A D _ R U N cut - see section C.4), are used to get K*Blive in table 5.1. A s shown in table 5.1, some runs in 1996 that were supposed to be removed by the B A D _ R U N cut were mistakenly included in the P A S S 3 analysis, so their KBuve is added back in . Also, at the beginning of the 1996 run, scalers were counted in the same way as in 1995, when scaler values were distributed amongst many tapes. The resulting 8-fold multiple 133 Chapter 5. Acceptance and Sensitivity counting of Ksuve m early 1996 is removed. The B A D _ R U N cut also removes individual events if T D fiducial pulses are missing, and individual spills if target high voltage trips are detected (see section C.4). Note that high voltage trips are, in general, difficult to detect because signals from the high voltage power supplies, which indicate whether or not the high voltage was applied during the spill , are not collected as part of the data. Finally, the B A D _ S T C cut removes individual events from groups of runs where the T D area-to-MeV calibration fails in the stopping counter (see section C.4). These corrections, and the final K-BUvei are shown in table 5.1. 5.2 .K^-based Acceptance K+ — > iC'Vp events are, in general, single track kaon-decay events wi th no other activity in the detector. Therefore, they are similar to K+ — > it+vD events wi th respect to track reconstruction in the target, U T C , and RS; t iming and hit patterns in the beamline detectors and target; and photon activity. K^2 decays from the K^i^) monitor data (see section 3.3) are therefore used to measure the acceptances of • reconstruction cuts; • beam pathology and function cuts, except those involving track energy and scattering; • kinematic pathology cuts, except those involving track energy, scattering, and fiducial region; • al l online and offline P V cuts. The structure and results of the i ^ - b a s e d acceptance study are shown in table 5.2, which lists the number of surviving K^O-) monitor events from the 1995, 1996, 1997a, and 1997b data sets after each cut is applied, and the corresponding acceptance of each cut. A l l setup cuts in table 5.2 are defined in table 5.3. Acceptances are measured using "summary" ntuples (see section 4.3), except for the acceptance loss of the L A Y E R 1 4 cut due to accidental hits in the outer R S S C , which requires information included only in the 134 Chapter 5. Acceptance and Sensitivity full-record ntuples. After the 1995 P A S S l analysis, some of the P A S S l cuts were modified ( S T L A Y , I N T I M E , F I T P I , U T C / R A N G E ) , so these cuts changed for subsequent passes of the 1995 data. K^X) monitor events which pass al l other cuts in the i f ^ -based acceptance study are re-run through the 1995 P A S S l code in order to get the additional acceptance loss from the 1995 P A S S l S T L A Y , I N T I M E , and U T C / R A N G E cuts. Acceptances are grouped into the quantities ARE, A r e c o n , Arest, Apy, AaLc{4, and AK^, which are the R S reconstruction, U T C and target reconstruction, beamline and target pattern, P V , L A Y E R 1 4 accidental, and 1995 P A S S l acceptances, respectively. A l l of these acceptances are combined into the quantity AK^2 , which is defined as the K^-based acceptance. Note that the I C B I T (online I C requirement) acceptance is 1.0 for a l l years, because I C B I T is used as a setup cut in the i f^ -based acceptance measurement. This is done to make the definition of AK^2 equivalent for al l years, because the online IC bit was added to the K^X) trigger in 1996-7 (as part of T • 2 - see section 3.3). The fiducial and efficiency losses of I C B I T are absorbed into the UMC-based trigger acceptance (section 5.8) and the kaon stopping fraction, fs (section 5.9). Discussion of the value of AK^ is found in section 5.9. 5.3 Acceptance K+ —• 7r+7r° events are similar to K+ —> -K+VV events in that both involve a roughly minimum-ionizing pion track in the target which arises from kaon decay. Therefore, K%2 decays from the K„2(2) monitor data (see section 3.3) are used to measure the acceptances of the beam and kinematic pathology cuts which involve track energy and scattering in the target. The structure and results of the K^i-hased acceptance study are shown in table 5.4, which lists the number of surviving K^i^) monitor events from the 1995, 1996, 1997a, and 1997b data sets after each cut is applied, and the corresponding acceptance of each cut. A l l setup cuts in table 5.4 are defined in table 5.5. Note that the setup cuts in table 5.5 include the requirement of at least 200 M e V of photon energy in the barrel, E C , and R S combined, such that there is no photon energy in the target (i.e., similar to a K+ —> -K+VV event in the target, because Kv2 events have a total of 225 M e V photon energy). Note also that the 135 Chapter 5. Acceptance and Sensitivity cut 1995 (acc.) 1996 (acc.) 1997a (acc.) 1997b (acc.) S E T U P R D 45635 119105 39495 23990 RD.TRK 45635 (1.000) 119105 (1.000) 39495 (1.000) 23990 (1.000) TRKTIM 45635 (1.000) 118849 (0.998) 39492 (1.000) 23990 (1.000) ARD 1.0000 ± 0.0000 0.9979 ± 0.0001 0.9999 t 0.0000 1.0000 ± 0.0000 SETUP r e c o n 24749 66786 21386 13805 UTC/RANGE 24749 (1.000) 66786 (1.000) 21386 (1.000) 13805 (1.000) UTCQUAL 24245 (0.980) 64517 (0.966) 20417 (0.955) 13329 (0.966) PROBZ 24245 (1.000) 63935 (0.991) 20228 (0.991) 13202 (0.990) TARGET 24102 (0.994) 63640 (0.995) 20132 (0.995) 13159 (0.997) Arecon 0.9739 ± 0.0010 0.9529 ± 0.0008 0.9414 i 0.0016 0.9532 ± 0.0018 SETUPreat 36718 96912 30379 20119 ICBIT 36718 (1.000) 96912 (1.000) 30379 (1.000) 20119 (1.000) TIC 36397 (0.991) 96437 (0.995) 30247 (0.996) 20049 (0.997) TIMCON 35983 (0.989) 95748 (0.993) 30088 (0.995) 19954 (0.995) TGCCD 32338 (0.899) 89541 (0.935) 27635 (0.918) 18673 (0.936) DCBIT 29564 (0.914) 79835 (0.892) 25529 (0.924) 16859 (0.903) DELC 26607 (0.900) 71834 (0.900) 22578 (0.884) 14868 (0.882) CKTRS 26443 (0.994) 71218 (0.991) 22404 (0.992) 14757 (0.993) CKTAIL 25786 (0.975) 69329 (0.973) 21578 (0.963) 14286 (0.968) B4DEDX 25535 (0.990) 68424 (0.987) 21384 (0.991) 14159 (0.991) CPITRS 25484 (0.998) 68339 (0.999) 21353 (0.999) 14132 (0.998) CPITAIL 25469 (0.999) 68323 (1.000) 21348 (1.000) 14127 (1.000) PSCUT 25226 (0.990) 67920 (0.994) 21221 (0.994) 14058 (0.995) TARGF 24406 (0.967) 65836 (0.969) 20544 (0.968) 13617 (0.969) DTGTTP 24403 (1 000) 65830 (1.000) 20544 (1.000) 13617 (1.000) RTDIF 24157 (0.990) 65142 (0.990) 20286 (0.987) 13493 (0.991) TGQUALT 24157 (1.000) 65142 (1.000) 20286 (1.000) 13493 (1.000) PIGAP 23869 (0.988) 64483 (0.990) 20063 (0.989) 13393 (0.993) TGB4 22613 (0.947) 61087 (0.947) 19005 (0.947) 12752 (0.952) KIC 22405 (0.991) 60484 (0.990) 18854 (0.992) 12659 (0.993) TGGEO 22361 (0.998) 60362 (0.998) 18823 (0.998) 12644 (0.999) EIC 22137 (0.990) 59902 (0.992) . 18606 (0.988) 12557 (0.993) EICKIN 21586 (0.975) 58795 (0.982) 18237 (0.980) 12326 (0.982) B4EKZ 21374 (0.990) 58158 (0.989) 18043 (0.989) 12201 (0.990) B4EKZ.IC 21308 (0.997) 57935 (0.996) 17954 (0.995) 12160 (0.997) TGZFOOL 21308 (1.000) 57935 (1.000) 17954 (1.000) 12160 (1.000) TGCUT 21307 (1.000) 57934 (1.000) 17954 (1.000) 12160 (1.000) BWTRS 20711 (0.972) 56856 (0.981) 17615 (0.981) 11956 (0.983) BHTRS 20638 (0.996) 56508 (0.994) 17527 (0.995) 11906 (0.996) B4TRS 20508 (0.994) 56313 (0.997) 17444 (0.995) 11847 (0.995) B4TD 20383 (0.994) 56011 (0.995) 17359 (0.995) 11787 (0.995) PBNRS 19716 (0.967) 55401 (0.989) 16578 (0.955) 11312 (0.960) Arest 0.5370 ± 0.0026 0.5717 ± 0.0016 0.5457 ± 0.0029 0.5623 ± 0.0035 SETUPpv 10504 29224 8701 5905 HEX 9992 (0.951) 27794 (0.951) 8225 (0.945) 5619 (0.952) online PV (BV+EC) 9721 (0.973) 27177 (0.978) 8018 (0.975) 5523 (0.983) STLAY 9584 (0.986) 26897 (0.990) 7921 (0.988) 5471 (0.991) RSHEX 9357 (0.976) 26317 (0.978) 7748 (0.978) 5373 (0.982) INTIME 9331 (0.997) 26262 (0.998) 7732 (0.998) 5364 (0.998) PVCUT 9136 (0.979) 25738 (0.980) 7571 (0.979) 5273 (0.983) TGPVCUT 9111 (0.997) 25685 (0.998) 7555 (0.998) 5266 (0.999) TGPVTR 9111 (1.000) 25685 (1.000) 7555 (1.000) 5266 (1.000) PASS3 PV 8284 (0.909) 23284 (0.907) 6816 (0.902) 4870 (0.925) Apv 0.7887 t 0.0040 0.7967 ± 0.0024 0.7834 t 0.0044 0.8247 ± 0.0049 full-record 24816 23282 6782 4831 LAYER14acc 24644 23129 6740 4805 0.9931 t 0.0005 0.9934 ± 0.0005 0.9938 t 0.0010 0.9946 ± 0.0011 SETUPS1 Kfi2 PASSl STLAY PASSl INTIME PASSl UTC/RANGE 11250 11239 11216 11165 0.9924 ± 0.0008 0.4065 ± 0.0029 0.4303 ± 0.0018 0.3999 ± 0.0032 0.4396 ± 0.0039 Table 5.2: iC^-based acceptances of cuts. The quoted uncertainties are purely statistical. Table entries are described in section 5.2 of the text. The I C B I T and D C B I T cuts are the online IC and D C requirements (see section 3.3). The P R O B Z cut is not applied to the 1995 data (see section C.3.1). The various S E T U P ' S are defined in table 5.3. 136 Chapter 5. Acceptance and Sensitivity S E T U P component cuts S E T U P f l o B A D J R U N , B A D - S T C , K^(l) trigger, I C B I T , tIC - t C k > 5 ns, B 4 D E D X , U T C , T A R G E T S E T U P ^ e c o n B A D _ R U N , B A D . S T C , K„2(l) trigger, I C B I T , tIC - tCk > 5 ns, B 4 D E D X , C P I T R S , C P I T A I L , C K T R S , C K T A I L , B W T R S , B H T R S , ARD cuts, \tIC - tRS\ < 5 ns, P V ( n o B V ) , 120 < ERS < 150 M e V S E T U P r e s t B A D . R U N , B A D J 3 T C , K^l) trigger, I C B I T , ARD cuts, A r e c o n cuts, P V ( n o B V ) , 120 < ERS < 150 M e V , K M 2 P B O X , C O S 3 D S E T U P p y B A D _ R U N , B A D _ S T C , K^l) trigger, I C B I T , ARD cuts, Arecon cuts, Arest cuts, 120 < ERs < 150 M e V , K M 2 P B O X , C O S 3 D , stopping layer < 21 S E T U P S I C B I T , ARD cuts, A r e c o n cuts, Arest cuts, Apy cuts, 120 < ERS < 150 M e V , K M 2 P B O X , C O S 3 D , stopping layer < 21 Table 5.3: S E T U P cuts used in the K^-based acceptance measurement shown in table 5.2. ^ 2 ( 1 ) trigger is an offline reproduction of the ^ 2 ( 1 ) trigger (see section 3.3). tic, tck, tRs, and ERs are defined in Appendix D . P V ( n o B V ) is the P V function cut wi th the requirements in the barrel disabled. K M 2 P B O X is a 2a cut on the momentum. E V 5 and T D L I K 3 cuts are not part of the setup cuts because they require information on the electron from n —> /J, —»• e decay in the stopping counter, and Kn2(2) monitor data only includes T D data in the smaller time range consistent wi th ix —> p decay. Acceptances are grouped into the quantities A, p s and Atgkin, which are the target opposite-side and kinematic acceptances, respectively. These acceptances are combined into the quantity AKV2 , which is defined as the Kn2-based acceptance. 5.4 7T s c a£-based Kinematic Acceptance Events arising from beam pions scattering into the detector are similar to K+ —> ir+vv events in that both involve pion tracks in the R S wi th range, energy, and momentum roughly uniformly distributed throughout the 7r + ^P( l ) signal region. Therefore, "clean" beam pion scattering events (i.e., wi th no non-track activity in the RS) from the 7r s c a t monitor data (see section 3.3) are used to measure the acceptances of kinematic pathology and function cuts 137 Chapter 5. Acceptance and Sensitivity cut 1995 (acc.) 1996 (acc.) 1997a (acc.) 1997b (acc.) S E T U P o p s 15869 49593 12397 8103 O P S V E T O 15271 (0.962) 47658 (0.961) 11903 (0.960) 7791 (0.961) O P S V E T O J L K B 15271 (1.000) 47274 (0.992) 11793 (0.991) 7735 (0.993) A 0.9623 ± 0 . 0 0 1 5 0.9532 ± 0.0009 0.9513 ± 0 . 0 0 1 9 0.9546 ± 0.0023 SETXJPtgkin 14558 45273 11297 7441 T G D E D X 14347 (0.986) 44541 (0.984) 11080 (0.981) 7324 (0.984) T G E R 14345 (1.000) 44533 (1.000) 11078 (1.000) 7321 (1.000) T G L I K E 14090 (0.982) 43597 (0.979) 10860 (0.980) 7191 (0.982) E P I T G 14088 (1.000) 43587 (1.000) 10858 (1.000) 7190 (1.000) E P I M A X K 13844 (0.983) 42865 (0.983) 10686 (0.984) 7091 (0.986) T G E D G E 13831 (0.999) 42771 (0.998) 10659 (0.997) 7069 (0.997) T G C C D P F 13594 (0.983) 42427 (0.992) 10562 (0.991) 7011 (0.992) P H I V T X 13388 (0.985) 41832 (0.986) 10409 (0.986) 6918 (0.987) P H I V T X 2 13305 (0.994) 41614 (0.995) 10365 (0.996) 6879 (0.994) D R P 13230 (0.994) 41374 (0.994) 10303 (0.994) 6832 (0.993) Atgkin 0.9088 ± 0.0024 0.9139 ± 0 . 0 0 1 3 0.9120 ± 0 . 0 0 2 7 0.9182 ± 0 . 0 0 3 2 AK„2 0.8745 ± 0.0027 0.8711 ± 0 . 0 0 1 5 0.8676 ± 0.0031 0.8765 ± 0.0037 Table 5.4: K^-based acceptances of cuts. The quoted uncertainties are purely statistical. Table entries are described in section 5.3 of the text. The O P S V E T C L L K B cut is not applied to the 1995 data (see section C.3.2). The various S E T U P ' S are defined in table 5.5. Kn2 S E T U P component cuts S E T U P o p s B A D J I U N , B A D _ S T C , K^2(2) trigger, all reconstruction cuts, all beam function cuts, all beam and kinematic pathology cuts (except those whose acceptances are being measured in table 5.4), photon energy in b a r r e l ± E C + R D > 200 M e V , all T D cuts except T D E C O N , T D V E L , E V 5 , and T D L I K 3 , 2<7 cut on the K^2 peak R, E, and P SETUPtpfcin S E T U P o p s , Aops cuts, T G P V C U T Table 5.5: S E T U P cuts used in the K^-based acceptance measurement shown in table 5.4. K„2(2) trigger is an offline reproduction of the Kn2(2) trigger. 138 Chapter 5. Acceptance and Sensitivity which involve the momentum-energy correlation, range-momentum correlation, range-energy correlation, and scattering of pions in the R S . The structure and results of the 7r s c a t-based kinematic acceptance study are shown in table 5.6, which lists the number of surviving 7rscat monitor events from the 1995, 1996, 1997a, and 1997b data sets after each cut is applied, and the corresponding acceptance of each cut. A l l setup cuts in table 5.6 are defined in table 5.7. Because TTscat events do not arise from kaon decay in the target, the reconstruction of the track in the target is poor, leading to uncertainties in R, E, and P which define the 7r + i /P( l ) signal region (i.e., the B O X cut). The B O X cut is part of the setup cuts, and the piscat-based kinematic acceptance is a function of how tightly the B O X cut encompasses the pion band, so there is a systematic uncertainty in the acceptance which is estimated by varying the B O X cut. The appropriate variations in R, E, and P are found by comparing TTscat and pion mass resolutions. These are found to be 10.4 and 8.6 M e V / c 2 respectively, where mass is defined by ( P 2 — E2)/2E. So the fractional uncertainty in nscat target track reconstruction is roughly ^/(10.4) 2 - (8.6) 2/140 = 4.2%. The contributions of P and E to the mass resolution are assumed to be roughly equal, so their uncertainties are 4.2%/-\/2 = 3.0%. R scales approximately linearly wi th E, so the uncertainty in R is also expected to be 3.0%. Expanding and shrinking the B O X cut at the high and low sides by 3.0% define "large" and "small" B O X cuts which, when used in the setup, define lower and upper limits to the nscat-based kinematic acceptance, A1^^ and As™^. The systematic uncertainty is given by ± 0 . 5 • (Al££ - A%™!l). The final value for the 7r s c a t-based kinematic acceptance, A{££, comes from the measurement where the standard B O X cut is used in the setup. Note that the acceptance of the M A S S cut is evaluated only using the final B O X cut in the setup, because the pion mass resolution for K+ —• ix+vv events is similar to that for events, not nscat events. 5.5 7 r s c a £-based TD Acceptance Events arising from beam pions scattering into the detector are also similar to K+ —• ir+vv events in that both involve pion tracks which have stopping layers roughly uniformly 139 Chapter 5. Acceptance and Sensitivity cut 1995 (acc.) 1996 (acc.) 1997a (acc.) 1997b (acc.) large B O X , S E T U P ^ M A S S R N G M O M C H I M A X C L _ R S D E D X R S L I K E C H I R F ( x y ) C H I R F ( z ) C H I R F _ N H Z 424 746 (1.000) 710 (0.952) 705 (0.993) 688 (0.976) 635 (0.923) 619 (0.975) 611 (0.987) 611 (1.000) 1800 2807 (0.961) 2715 (0.967) 2691 (0.991) 2622 (0.974) 2423 (0.924) 2303 (0.950) 2234 (0.970) 2224 (0.996) 466 738 (0.944) 719 (0.974) 709 (0.986) 681 (0.961) 638 (0.937) 607 (0.951) 582 (0.959) 577 (0.991) 438 684 (0.961) 661 (0.966) 655 (0.991) 636 (0.971) 608 (0.956) 570 (0.938) 558 (0.979) 553 (0.991) 0.8190 ± 0 . 0 1 6 8 0.7615 ± 0 . 0 0 9 6 0.7382 ± 0.0197 0.7771 ± 0 . 0 1 8 6 final B O X , S E T U P . s c a t M A S S R N G M O M C H I M A X C L _ R S D E D X R S L I K E C H I R F ( x y ) C H I R F ( z ) C H I R F _ N H Z 424 424 (1.000) 406 (0.958) 403 (0.993) 396 (0.983) 375 (0.947) 371 (0.989) 365 (0.984) 365 (1.000) 1800 1730 (0.961) 1685 (0.974) 1671 (0.992) 1637 (0.980) 1545 (0.944) 1474 (0.954) 1429 (0.969) 1424 (0.997) 466 440 (0.944) 429 (0.975) 424 (0.988) 414 (0.976) 389 (0.940). 374 (0.961) 358 (0.957) 355 (0.992) 438 421 (0.961) 407 (0.967) 404 (0.993) 395 (0.978) 383 (0.970) 362 (0.945) 359 (0.992) 356 (0.992) J^final ^scat 0.8608 ± 0.0168 0.7911 ± 0 . 0 0 9 6 0.7618 ± 0 . 0 1 9 7 0.8128 ± 0 . 0 1 8 6 small B O X , S E T U P , . . . , M A S S R N G M O M C H I M A X C L _ R S D E D X R S L I K E C H I R F ( x y ) C H I R F ( z ) C H I R F _ N H Z 424 102 (1.000) 99 (0.971) 99 (1.000) 98 (0.990) 96 (0.980) 94 (0.979) 93 (0.989) 93 (1.000) 1800 378 (0.961) 369 (0.976) 365 (0.989) 358 (0.981) 346 (0.966) 337 (0.974) 328 (0.973) 325 (0.991) 466 99 (0.944) 97 (0.980) 97 (1.000) 96 (0.990) 94 (0.979) 89 (0.947) 86 (0.966) 85 (0.988) 438 70 (0.961) 69 (0.986) 69 (1.000) 67 (0.971) 67 (1.000) 64 (0.955) 64 (1.000) 62 (0.969) J^small TTscat. 0.9118 ± 0 . 0 1 6 8 0.8264 ± 0.0096 0.8107 ± 0 . 0 1 9 7 0.8513 ± 0 . 0 1 8 6 A 0.8608 ± 0 . 0 1 6 8 s t a * ± 0 . 0 4 6 4 ^ s t 0.7911 ± 0 . 0 0 9 6 s ' a t ± 0 . 0 3 2 4 ^ s t 0.7618 ± 0 . 0 1 9 7 s i a t ± 0 . 0 3 6 2 ^ s * 0.8128 ± 0 . 0 1 8 6 s t a 4 ± 0 . 0 3 7 1 ^ s t Table 5.6: 7r s c a i -based acceptances of kinematic cuts. Table entries are described in sec-tion 5.4 of the text. The M A S S and C H I R F _ N H Z cuts are not applied to the 1995 data (see section C.3.1). The various S E T U P ' S are defined in table 5.7. 140 Chapter 5. Acceptance and Sensitivity 7r s c a t S E T U P component cuts S E T U P W B A D . R U N , B A D _ S T C , al l PASS1 cuts, EBi < 1.3 M e V , \U - tRs\ < 5 ns, I C B I T , \tIC - tRS\ < 5 ns, T A R G F , D T G T T P , R T D IF, T G Q U A L T , T G Z F O O L , B H T R S , C K T R S , C K T A I L , P V in the R S only, al l T D cuts, C O S 3 D , L A Y V 4 final B O X 33 < R < 40 cm, 115 < E < 135 M e V , 211 < P < 229 M e V / c small B O X 34.0 < R < 38.8 cm, 118.4 < E < 131.0 M e V , 217.3 < P < 222.2 M e V / c large B O X 32.0 < R < 41.2 cm, 111.6 < E < 139.0 M e V , 204.7 < P < 235.8 M e V / c Table 5.7: S E T U P cuts used in the 7r s c a t-based kinematic acceptance measurement shown in table 5.6. E B i , tn, tRs, and tic are defined in Appendix D . distributed between layers 11 and 18, which is the range of stopping layers required by the L A Y V 4 cut. Therefore, "clean" beam pion scattering events (i.e., wi th no non-track activity in the RS) from the 7r s m t monitor data (see section 3.3) are used to measure the acceptances of online and offline T D cuts, which require observation of the n —> \i —> e decay signature in and around the stopping counter. The structure and results of the 7r s c a t-based T D acceptance study are shown in table 5.8, which lists the number of surviving irscat monitor events from the 1995, 1996, 1997a, and 1997b data sets after each cut is applied, and the corresponding acceptance of each cut. A l l setup cuts in table 5.8 are defined in table 5.9. After the 1995 PASS1 analysis, new T D pulse shape calibration files for 1995 were created, so the F I T P I cut (which involves 7r —> fi double-pulse fitting in the stopping counter) changed for subsequent passes of the 1995 data. 7r s c a i monitor events passing all other cuts in the 7r s c a 4-based T D acceptance study are re-run through the 1995 PASS1 code in order to get the additional acceptance loss from the 1995 PASS1 F I T P I cut. A concern in the T D acceptance measurement is TD-kinemat ic correlation. If the kine-matic setup cuts are correlated wi th T D cuts, then the application of these setup cuts wi l l likely result in a T D acceptance which is too high. For example, the R S D E D X cuts (CHI-M A X and C L _ R S D E D X ) may be correlated wi th the E L V E T O cut v ia muon-time accidentals along the track affecting the track dE/dx, thereby causing events to fail the R S D E D X cut as well as the E L V E T O cut. However, this correlation is expected to be small, because the dE/dx values from R S counter energies along the track come from T D P H energies instead 141 Chapter 5. Acceptance and Sensitivity cut 1995 (acc.) 1996 (acc.) 1997a (acc.) 1997b (acc.) S E T U P ^ 1153 4002 1124 1027 FITPI(bad data) 1135 (0.984) 3902 (0.975) 1100 (0.979) 996 (0.970) FITPI(counting) 800 (0.705) 2806 (0.719) 786 (0.715) 731 (0.734) L l l . and .L12 598 (0.748) 2522 (0.899) 664 (0.845) 555 (0.759) T D C U T 568 (0.950) 2432 (0.964) 643 (0.968) 547 (0.986) R S H E X 2 558 (0.982) 2388 (0.982) 643 (1.000) 547 (1.000) T D T C O N 554 (0.993) 2375 (0.995) 642 (0.998) 544 (0.995) T D D F A 1 554 (1.000) 2207 (0.929) 588 (0.916) 503 (0.925) E V 5 426 (0.769) 1695 (0.768) 449 (0.764) 437 (0.869) E L V E T O 393 (0.923) 1626 (0.959) 436 (0.971) 418 (0.957) T D F O O L 393 (1.000) 1624 (0.999) 435 (0.998) 418 (1.000) T D E C O N 393 (1.000) 1522 (0.937) 410 (0.943) 402 (0.962) T D V E L 393 (1.000) 1434 (0.942) 386 (0.941) 374 (0.930) T D L I K 2 359 (0.913) 1360 (0.948) 364 (0.943) 346 (0.925) T D L I K 3 356 (0.992) 1345 (0.989) 360 (0.989) 336 (0.971) T D D F A 2 336 (0.944) 1291 (0.960) 344 (0.956) 321 (0.955) A ( 1 ) TD 0.2914 ±0 .0134 0.3226 ± 0.0074 0.3060 ± 0 . 0 1 3 7 0.3126 ±0 .0145 S E T U P ^ 1055 3481 967 901 FITPI(bad data) 1040 (0.986) 3400 (0.977) 947 (0.979) 878 (0.974) FITPI(counting) 734 (0.706) 2468 (0.726) 679 (0.717) 649 (0.739) A count AFITPI 0.706 ± 0 . 0 1 4 0.726 ± 0.008 0.717 ± 0 . 0 1 5 0.739 ± 0 . 0 1 5 A area AFITPI 0.710 ± 0 . 0 2 8 0.719 ± 0 . 0 2 2 0.730 ± 0.030 L l l . and .L12 553 (0.753) 2224 (0.901) 575 (0.847) 497 (0.766) T D C U T 525 (0.949) 2147 (0.965) 560 (0.974) 492 (0.990) R S H E X 2 517 (0.985) 2110 (0.983) 560 (1.000) 492 (1.000) T D T C O N 515 (0.996) 2098 (0.994) 559 (0.998) 490 (0.996) T D D F A 1 515 (1.000) 1955 (0.932) 512 (0.916) 454 (0.927) E V 5 401 (0.779) 1504 (0.769) 393 (0.768) 396 (0.872) E L V E T O 370 (0.923) 1443 (0.959) 382 (0.972) 377 (0.952) T D F O O L 370 (1.000) 1442 (0.999) 381 (0.997) 377 (1.000) T D E C O N 370 (1.000) 1349 (0.936) 359 (0.942) 363 (0.963) T D V E L 370 (1.000) 1269 (0.941) 339 (0.944) 337 (0.928) T D L I K 2 339 (0.916) 1205 (0.950) 321 (0.947) 311 (0.923) T D L I K 3 336 (0.991) 1194 (0.991) 317 (0.988) 304 (0.977) T D D F A 2 317 (0.943) 1146 (0.960) 302 (0.953) 291 (0.957) 4(2) TD 0.3005 ± 0 . 0 1 4 1 0.3292 ± 0.0080 0.3123 ± 0 . 0 1 4 9 0.3230 ±0 .0156 Alow JiTD 0.2914 ±0 .0134 0.3226 ± 0.0074 0.3060 ± 0 . 0 1 3 7 0.3126 ±0 .0145 thigh 0.3047 ± 0 . 0 1 4 3 0.3338 ± 0 . 0 0 8 1 0.3167 ± 0 . 0 1 5 1 0.3275 ±0 .0158 S E T U P S 2464 — — — PASS1 F I T P I 2422 — — — s±TD 0.9830 ±0 .0026 — ATD C .2930 C .3282 0. 3114 0. 3201 ± 0 0096 s t a t ± 0 0 0 5 5 3 t a t ±0 .0102 s ' a * ± 0 . 0 1 0 7 s t a t ± 0 . 0065 s ? / s t ± 0 . QQSQsyst ± 0 . 0 0 5 4 s y s t ± 0 . 0 0 7 5 s ^ ' Table 5.8: 7r s c a t -based acceptances of T D cuts. Table entries are described in section 5.5 of the text. The T D D F A 1 , T D E C O N , and T D V E L cuts are not applied to the 1995 data (see section C.3.4). L1.2 became active in 1997b. The various S E T U P ' S are defined in table 5.9. 142 Chapter 5. Acceptance and Sensitivity iiscat S E T U P component cuts S E T U P S S E T U P T s c a t from table 5.6, wi th B O X , L A Y V 4 , R N G M O M , Z F R F , Z U T O U T , L A Y E R 1 4 , and E I C K I N , but without the T D cuts S E T U P ^ S E T U P S , C L _ R S D E D X , C H I M A X , C H I R F , C H I R F _ N H Z S E T U P S E B 4 < 1.5 M e V , C P I T R S , B W T R S , all P A S S l cuts, P V C U T , T G P V C U T , all A%£hi9h cuts, C O S 3 D , L A Y V 4 , and large B O X (as defined in table 5.6) Table 5.9: S E T U P cuts used in the 7r s c a t -based T D acceptance measurement shown in ta-ble 5.8. EBA is defined in Appendix D . of A D C energies when the A D C / T D P H energy ratio is found to be bigger than a critical value, thereby indicating the presence of an accidental (see U T C / R A N G E / T A R G E T in sec-t ion C l ) . A correlation may also exist between the C H I R F setup cut and T D pulse-fitting cuts (e.g., T D D F A 1 , T D D F A 2 , T D L I K 2 ) . A s described in section C.3.1, C H I R F is a x2 cut which involves the pion energy in the stopping counter, calculated as total pion plus decay-muon A D C energy, minus the muon energy found from a double-pulse fit to the T D data. This stopping counter pion energy could be affected by a poor n —• p double-pulse fit, which can happen when the pion decays early and the muon pulse is buried underneath the ta i l of the pion pulse. This usually makes the pion energy too small which, for example, can cause the event to preferentially fail both the C H I R F and T D D F A l cuts. To test these correlations, the T D acceptance is measured without and with the R S D -E D X and C H I R F cuts in the setup, giving rise to the values of A^D and A^D, respectively, in table 5.8. The R S D E D X , C H I R F correlations are shown to be < 3%, which is smaller than the statistical uncertainties in the T D acceptance measurements. In fact, the slightly lower acceptances seen when the R S D E D X and C H I R F cuts are removed from the setup may not be due to correlations, but rather due to the presence of muons and/or pion absorption and decay in flight (DIF) introduced into the data sample by turning these cuts off. This 143 Chapter 5. Acceptance and Sensitivity # events fail U F A T E N I D I F contamination no setup 23566 2482 10.5% S E T U P 1 5451 78 1.4% S E T U P 2 5408 71 1.3% S E T U P 3 5439 71 1.3% Table 5.10: Pion-nuclear absorption and decay in flight (NIDIF) contamination in the 1995 K+ —> it+vv U M C data. The total number of K+ — > ir+vp events in the U M C data sample, the number that fail the U F A T E cut (i.e., the number of events where the pion undergoes NI or D I F ) , and the resulting N I D I F contamination are listed for 4 different groups of setup cuts applied to the K+ — > it+uu data, "no setup" refers to no setup cuts applied. S E T U P 1 includes reconstruction, P V , and fiducial cuts, plus a two-sided R N G M O M cut to isolate the pion band, the B O X and C H I R F cuts. S E T U P 2 is S E T U P 1 plus the R S D E D X cut. S E T U P 3 is S E T U P 2 wi th the two-sided R N G M O M cut replaced by the standard R N G M O M cut (see section C.3.5). A N I D I F contamination of roughly 1.4% is indicated for K+ —• ir+vP events, even after kinematic cuts such as R S D E D X and C H I R F are applied. contamination is estimated to be as high as 5% [41]. However, it 's not clear whether the 3% change in T D acceptance comes from correlations or contaminations, so a lower l imit to the T D acceptance is assigned, Aj^ = A^\), based on the measurement without R S D E D X and C H I R F in the setup. Furthermore, a study of UMC-generated pions from K+ — » • n+uP shows that about 1.4% of these pions which pass al l kinematic setup cuts (including R S D E D X and C H I R F ) are absorbed or D I F (see table 5.10). Therefore, the nscat data sample used in the T D acceptance measurement may have a 1.4% contamination of pion absorption and D I F , even after the R S D E D X and C H I R F cuts are applied. The acceptance loss from these types of events is measured elsewhere (see section 5.8), so this loss should not be counted here. A n upper l imit to the T D acceptance is therefore assigned based on the measurement wi th R S -D E D X and C H I R F in the setup, plus 1.4%: AT%h = 1.014 • A%. The systematic uncertainty in the T D acceptance due to contaminations/correlations is given by ± 0 . 5 • (A^D1 ~ A^f))-The final value for the 7r s c a t-based T D acceptance, ATD, is given by the average of the high and low values, as shown in table 5.8. The F I T P I cut is the primary cut which requires a good 7r —> \i double-pulse fit in the stopping counter. In table 5.8, the acceptance of the F I T P I cut is measured before that of the other T D cuts by simply counting the number of 7r s c a t events that pass F I T P I . This 144 Chapter 5. Acceptance and Sensitivity method for measuring the acceptance of the F I T P I cut is called the "counting" method. It assumes that the irscat data sample has been appropriately selected such that none of the F I T P I acceptance loss arises from pion absorption or D I F events (which are accounted for elsewhere - see section 5.8). The only acceptance losses are assumed to arise from early 7r —> n decay (such that the muon pulse is buried underneath the pion pulse), accidentals in the stopping counter (which can cause confusion in the assignment of the pion, muon, and accidental pulses), early \i —> e decay (which again can sometimes cause confusion in the assignment of pion, muon, and electron pulses), and muon escape from the stopping counter. Another method for measuring the acceptance of the F I T P I cut, called the "area" method, attempts to account for each type of event that fails the F I T P I cut separately, such that any potential losses from pion absorption or D I F are explicitly excluded from the acceptance calculation. The area-method acceptance of the F I T P I cut is calculated according to MP i-FITPI = €/j,escape-MA + AMA Mp is the number of events in the n s c a t data sample that pass the F I T P I cut. MA is the integrated number of events under a pion-lifetime fit to these FITPI-passed events, and accounts for the F I T P I acceptance loss that arises from early TT —> \x decays where the muon pulse can't be seen. AMA is the "hand-scan correction", which is the number of events wi th good TT —> pi decay signatures which fail the F I T P I cut due to the presence of accidental and/or electron pulses, as described above. This number is found from a visual inspection of the T D pulses for events which fail F I T P I . escape = 0.982 [56] is a correction for missed muon pulses because the muon escapes the stopping counter. The area-method acceptance of the F I T P I cut is calculated in detail elsewhere [41], and the results are shown in table 5.5. The area-method acceptance agrees wi th the counting-method acceptance for each of the 1995, 1996, 1997a, and 1997b data sets, so pion absorption and D I F are not likely to significantly contaminate the 7 r s c a t data sample used to calculate the T D acceptance (perhaps due to the R S D E D X , C H I R F , and B O X cuts which are part of S E T U P ^ in table 5.8. These cuts select pion tracks which have the expected counter-by-counter energy loss and trajectory for charged pions whose total energy deposit in the R S is consistent wi th the momentum 145 Chapter 5. Acceptance and Sensitivity measured in the U T C ) . The agreement of the area-method and counting-method results further suggests that al l types of acceptance losses of the F I T P I cut have been accounted for. -For ease of calculation then, the counting-method results are used in the total T D acceptance calculation shown in table 5.8. 5.6 /x-veto Accidental Loss The TT+VV(\) trigger involves a muon veto which requires that no hit be present at track time in layer 19, 20, or 21 of the R S in a ct sector, defined by 19 c t + 20 c t + 21 c t , where a ct sector is defined as the T • 2 sector or wi thin 2 sectors clockwise of the T • 2 sector as viewed from downstream of the detector (see section 3.3). Candidate K+ —> ix+vv events can therefore be lost if an accidental gives rise to one of these hits. To estimate this acceptance loss, K^2 decays from the K^iX) monitor data (see section 3.3) are examined for events which have a hit in layer 19, 20, or 21 in a 10 ns pre-track-time window in any of 12 adjacent sectors opposite the stopping sector. Because this hit occurs before track time and is on the opposite of the detector, it is assumed to be unrelated to the decay and to arise from an accidental. The total number of events in the data sample is denoted by Mtot, and the number of events which satisfy the above definition of an accidental hit is denoted Macc. Because the trigger is sensitive to accidentals in a ± 2 0 ns window around the detector strobe, and because ct in 19 c t defines 3 sectors, the accidental muon-veto acceptance, Ajfc, is given by A^c = (Mtot - Macc • ^  • 130SeCtf°rS ) /Mtot = (Mtot - Macc)/Mtot. (5.2) ^ V 10 ns 12 sectors/ Values of Mtot, Mtot — Macc, and A^cc are shown in table 5.11. 5.7 T • 2 Efficiency The 7r + i /P( l ) trigger requires coincident hits in layers 1 and 2 and a hit in layer 6 or 7 of the R S , defined by T • 2 and 6 c t + 7ct (see section 3.3). Candidate K+ —> TT+VV events can therefore be lost if one of these counters is "inefficient", that is, if the scintillation light 146 Chapter 5. Acceptance and Sensitivity Mtot Mtot - Macc A acc AH 1995 106237 105432 0.9924 ± 0.0003 1996 93773 93119 0.9930 ± 0.0003 1997a 29822 29599 0.9925 ± 0.0005 1997b 19517 19400 0.9940 ± 0.0006 Table 5.11: Accidental-induced acceptance of the trigger ^x-veto, given by A?fc = (Mtot — Macc)/Mtot- Mtot and Macc are defined in section 5.6 of the text. The quoted uncertainties are purely statistical. induced by the charged pion track liberates 0 photoelectrons at the R S P M T photocathode, when a mean value of n photoelectrons is expected. The probability of seeing n photoelec-trons when n photoelectrons are expected is given by the Poisson distribution: n™ -P ( n ; n ) = — e~n (5.3) n\ So the probability of seeing 0 photoelectrons is P ( n = 0;n) = e-" (5.4) Layers 2, 6, and 7 respond to charged tracks with about 12 photoelectrons per M e V [57]. These counters are each about 2 cm thick, so a minimum ionizing particle deposits about 4 M e V per counter, giving rise to about 50 photoelectrons. The T (layer 1) counters, however, only give rise to about 3 photoelectrons per M e V [57], and are only about 0.6 cm thick. A minimum ionizing particle therefore deposits about 1.2 M e V , giving rise to about 4 photoelectrons. The probability of seeing 0 photoelectrons in layers 2, 6, and 7 is very small (e~ 5 0 ) , but it is not insignificant for the T counters ( e - 4 = 2%). Therefore, the T • 2 efficiency needs to be calculated in order to find the corresponding acceptance loss for K+ —> TV+UU events. The T • 2 efficiencies for K+ —> K+VV and K+ —> 7 r + / events are found by interpolating the T • 2 efficiencies measured for and KN2 decays. Moni tor data wi th the sole trigger requirement of KB (see section 3.3) is separated into K^ and KN2 decays according to the cuts in tables 5.24 and 5.28 of sections 5.9.1 and 5.9.2), respectively. The number of events with a valid T • 2, divided by the total number of events in each of the KN2 and K^2 data 147 Chapter 5. Acceptance and Sensitivity samples, gives the T • 2 efficiencies for both K^2 and K^2 decays. The K^2 and K^2 T • 2 efficiencies, shown as a function of run number in figure 5.1, are both about 93% in 1995, but the efficiency is as low as 83% and is about 5% lower than the Kn2 efficiency in 1996-7. The T • 2 inefficiency is mainly due to T-counter inefficiency, often when tracks are near the air gap between sectors as shown in figure 5.2. The top and bottom plots in figure 5.2 show the number of Kn2 and events (1995-7 data combined) wi th a missing T • 2 as a function of azimuthal angle </> of the track at the T counter. The azimuthal angle <fi is shown in figure 3.7 and defined in Appendix D . The T • 2 efficiencies vary from year to year (see figure 5.1), but figure 5.2 can be used to estimate the T • 2 inefficiency due to gaps between sectors, which is 4.1 ± 0.2% [41]. Summing over the 1995-7 data, the average Kn2 and K^2 T • 2 inefficiencies from figure 5.1 are about 8.5% and 11.7%, respectively. Excluding the gap inefficiency, the T • 2 inefficiency for Kn2 decays is therefore about 4.4%, and that for decays is about 7.6%. The K^2 and T • 2 inefficiencies are both somewhat higher than the 2% inefficiency estimated above, perhaps due to poorer R S counter quality than expected. They also do not agree with each other, perhaps due to differing energy deposits in the T counters for 109 M e V K^2 pions and 152 M e V muons. The run dependence of the T - 2 efficiency, particularly that for K^2 decays in 1997 (see figure 5.1) is not understood, though it may be related to the online R S energy calibration and adjustment of RS P M T high voltages. The T • 2 efficiencies for pion tracks arising from kaon decay into n+vu and nf are found as follows. According to E q . (5.4), the probability of seeing 0 photoelectrons from the T counter is given by P(0) = e _ s = 1 - e r . 2 - 0.041 (5.5) where CT-2 is the total T • 2 efficiency, and 0.041 is the gap inefficiency discussed above. The expected number of photoelectrons, n, is given by n = e q ~ - e - d ' L ° (5.6) where eq is the photocathode quantum efficiency; E is the energy deposited in the T counter; Eex is the scintillator molecular excitation energy; d is the distance between the photocathode 148 Chapter 5. Acceptance and Sensitivity T.2 efficiency as a function of run number : [ I! ft 1 i&JP!*. I - , , i i i i , , , , , , , , , i i i 1 i i i 22000 24000 26000 28000 30000 32000 34000 36000 : T . T i w , , , , , , , , , , , , i , , , 22000 24000 26000 28000 30000 32000 34000 36000 \ 1.1 -:- ft I1 : i . i i i i i i 1 ! 1 , , , , , , i , , , i , , , i 1 i i i I I I I I I i I I I I I I I I I I I I I U I I I I 22000 24000 26000 28000 30000 32000 34000 36000 run number Figure 5.1: T • 2 efficiency as a function of run number for KN2 and decays, and the K^/K-KZ ratio of efficiencies. The three clusters of data, from left to right, correspond to 1995, 1996, and 1997 data. 149 Chapter 5. Acceptance and Sensitivity cp at the T counter for T .2 misses 8 0 7 0 60 5 0 4 0 3 0 2 0 10 Ir 0 i f t _i i i i i i i i i_ _l I I I I I I ! L Entries i i r u _J I I L_ 1977 UU J I L 0 0.2 0 .4 0.6 0.8 1 1.2 1.4 1.6 1.8 al ^ 6 0 0 h -5 0 0 4 0 0 3.00 2 0 0 r 100 U 0 IDA Entries _i i i i i i i i i i i i i i_ L 2 1 2 5 7 _1 I I I I I L 0 0.2 0.4 0.6 0 .8 1.2 1.4 1.6 1.8 2 (pT (in TV radians) Figure 5.2: Number of T • 2-inefficient events for (top) and (bottom) decays as a function of the azimuthal angle 0 of the track at the T counter (in n radians). 150 Chapter 5. Acceptance and Sensitivity and the energy deposit; and L0 is the photon attenuation length in the scintillator. If d is fixed at some value (say, in the middle of the counter of length L such that d = L/2), then n = kE (5.7) Combining Eqs. (5.5) and (5.7) gives e T . 2 = l - 0.041 - e~kE ' (5.8) From E q . (5.7), k is the number of photoelectrons per M e V . €T-2{Kv2, K^2) is measured using monitor data (as shown in figure 5.1), whereas E for Kv2 and K^2 decays and kaon decays into TT+UU and 7r/ is taken from U M C data. er-2 and E for both K^2 and decays are plugged into E q . (5.8) to get 2 independent values of k. This is done by varying k and multiplying by E unt i l the mean of the 1 — 0.041 — e~kE distribution gives the measured average value of er-2 for each year. The final value of k is the average value of k calculated using Kn2 and decays. The E787 detector as simulated by U M C has realistic gaps between T counters, so T • 2 inefficiency due to gaps is accounted for elsewhere (Atrig, calculated in section 5.8). Therefore, only the non-gap T • 2 efficiency is calculated here, e ^ , which, in the case of kaon decay into -K+UI> and 7 r / , is found from the mean of the 1 — e~~kE distribution, where k is measured as described above, and E comes from U M C data. The systematic uncertainty in e^.2('K+vi>,T:f) is half the difference of the maximum and minimum values of €^92(7r+ui',Trf) as found using the Kn2 and values of k. The non-gap T • 2 efficiencies for Kn2 and Kpi decays are found by dividing out the gap efficiency (95.9 ± 0.2%) from the total T • 2 efficiencies measured using monitor data. Results are shown in table 5.12. 5.8 UMC-based Acceptances The acceptances of several cuts used in the analysis depend on the kinematics of pion tracks, and so cannot be estimated using Kn2, K^, or beam pion data. Instead, U M C -generated K+ —> ix+vD and K+ —> n+f data is used to account for 151 Chapter 5. Acceptance and Sensitivity eT92{K^) kK„2 4^2(^2) k K u 2 k 1995 0.9721 ± 0.0033 2.26 0.9645 ± 0.0022 2.61 2.44 0.976 ± 0 . 0 0 5 0.975 ± 0.005 1996 0.9420 ± 0 . 0 0 3 9 1.76 0.8849 ± 0.0024 1.66 1.71 0.930 ± 0 . 0 0 6 0.928 ± 0.006 1997a 0.9326 ± 0.0064 1.66 0.8967 ± 0 . 0 0 3 1 1.74 1.70 0.929 ± 0.004 0.927 ± 0 . 0 0 4 1997b 0.9437 ± 0 . 0 0 7 2 1.75 0.8821 ± 0 . 0 0 3 5 1.64 1.70 0.928 ± 0.008 0.926 ± 0 . 0 0 8 Table 5.12: Non-gap T • 2 efficiencies, for Kn2 and decays and kaon decays into 7T+ui> and 7 r / . The uncertainties in e^.^K^, K^2) are both statistical (counting events wi th missing T • 2) and systematic (from estimation of the gap inefficiency); the uncertainties in eT92(7r+ui>,iYf) are primarily systematic, k is the average of k^2 and kx^, which are the number of photoelectrons per M e V in the T counters, as found from Kn2 and decays, respectively. • the acceptance of the level 0 trigger (see section 3.3), in the absence of accidentals; • losses due to pion absorption and decay in flight, and pion decay in the gaps between R S counters, because the 7r —• fx —> e decay sequence is not observed in the stopping counter; • losses due to pion tracks which wrap back around on themselves, such that the trigger assigns the wrong stopping counter; • the acceptance of cuts sensitive to the length of pion tracks: L A Y Y 4 , C O S 3 D , L A Y E R 1 4 , Z F R F , and Z U T O U T ; • the acceptance of cuts on the kinematic phase space: B O X and B O X ' . As described in section 3.2.6, the generation of U M C data involves propagation of kaon decay products into the detector starting from a "real" distribution of kaon stops in the target, as stored in "beam files". K+ —* ix+vv and K+ —> n+f events are generated with pion-nuclear interactions and pion decay in flight (NIDIF) turned on and off, and acceptances of cuts are measured in both cases. The ratio of the NIDIF-on /NIDIF-o f f U M C acceptances then gives the acceptance loss due to pion-nuclear interactions and pion decay in flight. The 7 T + i/^( l ) level 0 trigger acceptance, Atrig, is shown for UMC-generated K+ —> ir+vv and K+ —> n+f events wi th the 1996 kaon stopping distribution in tables 5.13 and 5.14, respectively, which list the number of surviving events after each trigger condition is applied, 152 Chapter 5. Acceptance and Sensitivity N I D I F on # events (acc.) N I D I F off # events (acc.) K T T • 2 6 r f + 7ct p-veto B V E C H E X Refined Range 200000 77991 (0.3900) 54617 (0.7003) 54287 (0.9940) 54199 (0.9984) 54166 (0.9994) 54099 (0.9988) 23683 (0.4378) 100000 41051 (0.4105) 33600 (0.8185) 33592 (0.9998) 33592 (1.0000) 33592 (1.0000) 33592 (1.0000) 17991 (0.5356) Atrig (1T+V9) 0.1184 ± 0 . 0 0 0 7 0.1799 ± 0 . 0 0 1 2 Table 5.13: UMC-based K+ —> TT+VV trigger acceptance for 1996 data. The quoted uncer-tainties are purely statistical. Table entries are described in section 5.8 of the text. K T is the number of K+ —• n+uu events generated by U M C . The various trigger conditions are defined in section 3.3, including "//-veto" which is defined as (19 c t + 20 c t + 21 c t ) , and "refined range" which is defined as L O r r l ( l ) • US + L0r r l (2 ) • D S . and the corresponding acceptance of each condition, for pion N I D I F turned on and off. The acceptance of the online D C requirement is not part of Atrig because it is included in the K^-based acceptance measurement (see table 5.2). Note that Atrig for K+ —> n+uu is only about half of that for K+ —» n+f. Pions from K+ —> 7 r + w can have momenta between P — 0 and P = 227 M e V / c (see figure 4.3), whereas pions from the two-body decay K+ —> n+f always have P = 227 M e V / c (for massless / ) . Pions from K+ —> ix+vv are therefore less likely to penetrate layers 1 and 2, layers 6 or 7, and layer 11 of the R S , resulting in lower T • 2, 6 c t + 7ct, and refined range acceptances, respectively. The other, non-trigger UMC-based acceptances are shown for UMC-generated K+ —> TT+VP and K+ —> n+f events with the 1996 kaon stopping distribution in tables 5.15 and 5.16, respectively, which list the number of surviving events after each cut is applied, and the corresponding acceptance of each cut, for pion N I D I F turned on and off. The combined acceptance of the cuts in each of tables 5.15 and 5.16 is called the "fiducial acceptance", Aumc. U F A T E , U S T M E D , and U S T O P . H E X are logical bits which are false if: the pion is absorbed or decays in flight; the pion decays in the gap between R S counters; or the pion track wraps back on itself such that the trigger assigns the wrong stopping counter, 153 Chapter 5. Acceptance and Sensitivity N I D I F on # events (acc.) N I D I F off # events (acc.) K T T - 2 6 ct + 7ct li-veto B V E C H E X Refined Range 99998 43877 (0.4388) 36348 (0.8284) 35613 (0.9798) 35491 (0.9966) 35447 (0.9988) 35338 (0.9969) 24919 (0.7052) 50000 22708 (0.4542) 22708 (1.0000) 22570 (0.9939) 22570 (1.0000) 22570 (1.0000) 22570 (1.0000) 20171 (0.8937) 0.2492 ± 0.0014 0.4034 ± 0.0022 Table 5.14: UMC-based K+ —> ix+f trigger acceptance for 1996 data. The quoted uncertain-ties are purely statistical. Table entries are described in section 5.8 of the text. K T is the number of K+ —> n+f events generated by U M C . The various trigger conditions are defined in section 3.3, including "/i-veto" which is defined as (19 c t + 20 c t + 21 c t ) , and "refined range" which is defined as L O r r l ( l ) • US + L0r r l (2 ) • DS . respectively. The acceptance of the L A Y V 4 cut is omitted because it is included in Atrig: the refined range and p-veto requirements ensure that a pion stops in one of R S layers 11 through 18 inclusive. The acceptance of the Z F R F cut (see section C.3.1) is measured after adding 1.0 cm to the UTC-extrapolated z positions of the UMC-generated track in R S counters, because the real U T C and that simulated by U M C are offset by 1.0 cm. The P F B O X cut listed in table 5.16 is the "nf B O X " cut, defined as a 2<r cut around the pion peak resulting from kaon decay into a pion and a massless familon: 35.5 < R < 40.0 cm, 120 < E < 135 M e V and 221 < P < 229 M e V / c . The acceptance of the B O X ' cut is omitted from table 5.16 because events which pass the P F B O X cut automatically pass the B O X ' cut (see section C.3.5). Note that Aumc for K+ —> -K+VV is only about 2/3 of that for K+ —> 7r +/. This again is due to the different pion momenta spectra for K+ —> -K+VV and K+ —> 7r + / , resulting in different acceptances of the B O X cut. Before measuring the acceptances of the B O X , B O X ' , and P F B O X cuts, R, E, and P values from UMC-generated events must be scaled and smeared to match the real data. First , values of R, E, and P from the real data are scaled so that the Kn2 and K^2 peak values are the same as the accepted values [16]. The accepted values for Kn2 decay are 154 Chapter 5. Acceptance and Sensitivity N I D I F on N I D I F off cut # events (acc.) # events (acc.) setup 23636 17977 U F A T E 21157 (0.8951) 17977 (1.0000) U S T M E D 21001 (0.9926) 17871 (0.9941) U S T O P - H E X 20608 (0.9813) 17809 (0.9965) C O S 3 D 19931 (0.9671) 17250 (0.9686) L A Y E R 14 19862 (0.9965) 17241 (0.9995) Z F R F 18579 (0.9354) 16054 (0.9312) Z U T O U T 18578 (0.9999) 16054 (1.0000) B O X 6914 (0.3722) 6790 (0.4229) B O X ' 6680 (0.9662) 6532 (0.9620) [ B O X ' checkl] [6617] [(0.9570)] [6453] [(0.9504)] [ B O X ' check2] [6714] [(0.9711)] [6563] [(0.9666)] Aumc(-K+vi>) 0.2826 ± 0.0029 0.3634 ± 0.0036 Table 5.15: UMC-based ir+vv fiducial acceptance for 1996 data. The quoted uncertainties are purely statistical. Table entries are described in section 5.8 of the text. The "# events" after "setup" is the number of UMC-generated K+ — > n+vv events remaining at the bottom of table 5.13 after some reconstruction and beam pathology cuts are additionally applied. cut N I D I F on N I D I F off # events (acc.) # events (acc.) setup 24861 20161 U F A T E 21906 (0.8811) 20161 (1.0000) U S T M E D 21701 (0.9906) 19954 (0.9897) U S T O P _ H E X 20949 (0.9653) 19902 (0.9974) C O S 3 D 20059 (0.9575) 18994 (0.9544) L A Y E R 14 19923 (0.9932) 18961 (0.9983) Z F R F 18141 (0.9106) 17252 (0.9099) Z U T O U T 18138 (0.9998) 17249 (0.9998) B O X 11873 (0.6546) 12285 (0.7122) P F B O X 10442 (0.8795) 11352 (0.9241) A\umc(nf) 0.4200 ± 0.0031 0.5631 ± 0 . 0 0 3 5 Table 5.16: UMC-based nf fiducial acceptance for 1996 data. The quoted uncertainties are purely statistical. Table entries are described in section 5.8 of the text. The "# events" after "setup" is the number of UMC-generated K+ —• ir + f events remaining at the bottom of table 5.14 after some reconstruction and beam pathology cuts are additionally applied. 155 Chapter 5. Acceptance and Sensitivity R = 30.37 cm, E = 108.55 M e V and P = 205.14 M e V / c . The accepted values for decay are R = 54.34 cm, E = 152.49 M e V and P = 235.53 M e V / c . The scaling is performed via X = mx • Xraw + bx (5.9) where Xraw is the peak value of R, E, or P for Kn2 or decay, measured using S K I M 1 (see table 4.5) or K^X) monitor data (see section 3.3), respectively; and mx and bx are fitted linear scaling constants applied to Xraw such that X is the accepted value. The peak values of R, E, and P measured for K^i and decays (i.e., the values of Xraw) and the scaling factors mx and bx are shown for each of the 1995, 1996, and 1997 data sets in table 5.17. Note in table 5.17 that the peak value of E for decays is anomalously low in 1995. This is because the K^iX) monitor data was mistakenly analyzed wi th a "pion hypothesis", where the track is assumed to be that of a pion for the purposes of R, E, and P calculations. Roughly 3 M e V of muon energy in the stopping counter (arising from TT —> p decay) was mistakenly removed, though this should have no effect on the UMC-based acceptance measurement (see below). The scaling constants in table 5.17 are also applied to the raw values of R, E, and P from UMC-generated data. The scaled U M C values must be scaled again and smeared so that the peak positions and resolutions of UMC-generated and decays match those of the real data. The U M C scaling is performed v ia Xsc = m*-X + t£ (5.10) where X is given by E q . (5.9) in terms of the scaling factors mx and bx and the raw peak value Xraw for UMC-generated K„2 or decay; and mfc and b*c are fitted linear scaling constants applied to X such that the doubly-scaled value from UMC-generated data, Xsc, agrees wi th the scaled value from real data, X. The smearing of the U M C variables is performed v ia X™ = Xsc + r-aXsc(Xsc) (5.11) = Xsc + r • (mf m • Xsc + bfj 156 Chapter 5. Acceptance and Sensitivity 1995 1996 1997 Range: Kn2 peak (cm) 30.49 30.506 30.539 peak (cm) 54.58 54.506 54.549 mR 0.9949 0.9988 0.9983 bR 0.0409 -0.0979 -0.1181 Energy: Kn2 peak (MeV) 105.73 105.80 105.88 peak (MeV) 150.35 152.46 152.78 mE 0.9845 0.9415 0.9367 bE 4.4550 8.9402 9.3750 Momentum: Kn2 peak ( M e V / c ) 205.55 205.49 206.02 peak ( M e V / c ) 235.85 235.67 236.38 mp 1.0030 1.0070 1.0010 bp -1.0205 -1.7799 -1.0836 Table 5.17: Measured raw values of R, E, and P for the Kn2 and K^2 peaks, and linear scaling factors applied according to E q . (5.9) such that the scaled values of R, E, and P are the accepted values. where X%™ is a Gaussian distribution of R, E, or P wi th mean Xsc and resolution oxsc the same as the mean and resolution of the X distribution in E q . (5.9) from real Kn2 or data; and are fitted linear smearing constants applied to the U M C Xsc values such that the kinematic resolutions of UMC-generated and real data agree; and r is a Gaussian random variable wi th unit variance. The U M C scaling and smearing constants are re-calculated for each of the 1995, 1996, 1997a, and 1997b kaon stopping distributions and are summarized in tables 5.18 and 5.19. Real data typically has worse (larger) kinematic resolutions than UMC-generated data, due to the limited ability of U M C to simulate real processes in the real detector. However, the R resolution from U M C is slightly larger than that from the real data, so no smearing on R is performed. The matching of the R, E, and P peak positions and resolutions for real and UMC-generated Kn2 and K^2 decays is summarized in tables 5.20 and 5.21 for each of the 1995, 1996, 1997a, and 1997b kaon stopping distributions. Note in table 5.21 that the peak value of E for K^2 decays is anomalously high in 1995. This is because the 1995 E scaling factors in. table 5.17 were calculated using K^l) monitor data 157 Chapter 5. Acceptance and Sensitivity year Kc • R + m £ • E + 6£ 1995 1996 1997a 1997b 1.007 -R- 0.1378 1.008 -R- 0.1275 1.013 • R - 0.3192 1.010 • R - 0.2224 1.059 3.9199 1.015 - £ + 0.6787 1.012-£ + 1.1894 1.012- E + 1.0319 1.009 • P - 0.6436 1.008 • P - 0.2431 1.009 • P - 0.6247 1.012 P- 1.3390 Table 5.18: Scaling factors given in Eq. (5.10) which are applied to R, E, and P values from UMC-generated data, such that the R, E, and P values from UMC-generated data agree with those from real data. year ™%m • Rsc + b?m mfm-Esc + bfm "bsm rsc ' usm 1995 1996 1997a 1997b No Smearing No Smearing No Smearing No Smearing 0.02793 • Esc - 0.5525 0.02520 • Esc - 0.2827 0.02805 • Esc - 0.6036 0.02746 • Esc - 0.5099 0.00681 • Psc - 0.2493 0.01804 • Psc - 2.8055 0.01182 -Psc- 1.5674 0.01521 • Psc - 2.2377 Table 5.19: Smearing factors given in Eq. (5.11) which are applied to scaled values of R, E, and P from UMC-generated data, such that the resolutions in R, E, and P from U M C -generated data agree with those from real data. which had an anomalously low value of E (see discussion above). This should not affect the UMC-based acceptance measurement, because the final location of the E peak for decays is the same for UMC-generated and real data. The R, E, and P peaks from real K^i and data are also measured to be modest functions of the polar angle of the charged track, increasing by almost 1% between cos# = 0 and | cos#| =0.5 [58], where the polar (dip) angle is defined in Appendix D . The E and P dip-angle dependences are not modelled by U M C , so the dip-angle dependences of the E B O X ' and P B O X ' cuts (see section C.3.5) are disabled when measuring the total acceptance of the B O X ' cut in table 5.15. The systematic uncertainty in the acceptance of the B O X ' cut arising from the dip-angle dependences of the R, E, and P cut positions is estimated by fully enabling ("BOX' checkl" in table 5.15) and fully disabling ("BOX' check2") the dip-angle-dependences of the R, E, and P cut positions. The variation in acceptance is small, and therefore this systematic uncertainty is ignored. The UMC-based acceptances for the 1995, 1996, 1997a, and 1997b kaon stopping distri-158 Chapter 5. Acceptance and Sensitivity year R E P 1995 data: mean a 30.349 ± 0.002 0.992 ± 0.002 108.391 ± 0.007 3.295 ± 0.008 205.177 ± 0 . 0 0 5 2.497 ± 0.006 U M C : mean a 30.331 ± 0 . 0 1 7 1.019 ± 0 . 0 1 4 108.354 ± 0.059 3.388 ± 0.046 205.145 ± 0 . 0 2 7 2.532 ± 0.034 1996 data: mean a 30.344 ± 0 . 0 0 2 0.982 ± 0.003 108.385 ± 0.007 3.201 ± 0 . 0 0 9 205.183 ± 0 . 0 0 5 2.508 ± 0.007 U M C : mean a 30.330 ± 0 . 0 1 1 1.057 ± 0 . 0 1 4 108.294 ± 0.055 3.235 ± 0 . 0 4 1 205.242 ± 0.025 2.504 ± 0.033 1997a data: mean a 30.327 ± 0 . 0 0 4 0.998 ± 0.005 108.450 ± 0.019 3.209 ± 0 . 0 1 6 205.177 ± 0 . 0 0 9 2.450 ± 0 . 0 1 3 U M C : mean a 30.308 ± 0 . 0 1 9 1.034 ± 0 . 0 1 4 108.452 ± 0.056 3.259 ± 0 . 0 4 3 205.114 ± 0 . 0 1 9 2.461 ± 0 . 0 3 6 1997b data: mean a 30.329 ± 0.005 0.994 ± 0.006 108.486 ± 0.024 3.249 ± 0.020 205.142 ± 0 . 0 1 0 2.430 ± 0 . 0 1 7 U M C : mean a 30.318 ± 0 . 0 1 9 1.047 ± 0 . 0 1 5 108.420 ± 0.058 3.370 ± 0.045 205.135 ± 0 . 0 2 3 2.401 ± 0.034 Table 5.20: Means and a's of Gaussian fits to kinematic variables from real K^2 data, and from U M C K^2 data after scaling and smearing. year R E P 1995 data: mean a 53.988 ± 0.006 2.217 ± 0 . 0 0 6 154.494 ± 0 . 0 1 1 4.336 ± 0 . 0 1 3 235.593 ± 0.008 2.928 ± 0 . 0 0 7 U M C : mean a 53.929 ± 0 . 0 1 8 2.282 ± 0.020 154.389 ± 0.046 4.371 ± 0.040 235.633 ± 0 . 0 1 7 2.967 ± 0.020 1996 data: mean a 54.072 ± 0.006 2.215 ± 0 . 0 0 7 152.532 ± 0 . 0 1 1 4.108 ± 0 . 0 1 4 235.676 ± 0.006 2.984 ± 0 . 0 0 8 U M C : mean a 54.004 ± 0 . 0 1 8 2.303 ± 0 . 0 2 1 152.462 ± 0.032 4.210 ± 0 . 0 3 9 235.768 ± 0 . 0 1 0 3.023 ± 0 . 0 2 3 1997a data: mean a 54.083 ± 0 . 0 1 1 2.225 ± 0 . 0 1 3 152.532 ± 0.020 4.160 ± 0 . 0 2 5 235.679 ± 0 . 0 1 6 2.914 ± 0 . 0 1 4 U M C : mean a 54.014 ± 0 . 0 1 9 2.332 ± 0 . 0 2 1 152.499 ± 0.031 4.245 ± 0.040 235.629 ± 0 . 0 1 4 2.950 ± 0 . 0 2 2 1997b data: mean a 54.063 ± 0 . 0 1 4 2.209 ± 0 . 0 1 6 152.475 ± 0.025 4.218 ± 0 . 0 3 2 235.632 ± 0.020 2.932 ± 0 . 0 1 8 U M C : mean a 54.020 ± 0 . 0 1 8 2.306 ± 0.020 152.281 ± 0 . 0 3 3 4.185 ± 0 . 0 3 9 235.605 ± 0 . 0 1 5 2.983 ± 0.020 Table 5.21: Means and u's of Gaussian fits to kinematic variables from real Ku2 data, and from U M C K^2 data after scaling and smearing. 159 Chapter 5. Acceptance and Sensitivity year •Atrig A Anidif 1995 1996 1997a 1997b 0.1799 ± 0 . 0 0 1 2 0.1799 ± 0 . 0 0 1 2 0.1780 ± 0 . 0 0 1 2 0.1811 ± 0 . 0 0 1 2 0.3650 ± 0.0036 0.3634 ± 0.0036 0.3752 ± 0.0036 0.3657 ± 0 . 0 0 3 6 0.5144 ± 0 . 0 0 8 6 0.5118 ± 0 . 0 0 8 6 0.5069 ± 0.0085 0.5126 ± 0.0086 Table 5.22: Summary of UMC-based acceptances for K+ —> ir+vv events. The quoted uncertainties are purely statistical. year Atrig A Anidif 1995 1996 1997a 1997b 0.4021 ± 0.0022 0.4034 ± 0.0022 0.4018 ± 0.0022 0.4115 ± 0 . 0 0 2 2 0.5698 ± 0.0035 0.5631 ± 0 . 0 0 3 5 0.5764 ± 0.0035 0.5762 ± 0.0034 0.4674 ± 0.0058 0.4608 ± 0.0057 0.4584 ± 0.0057 0.4598 ± 0.0055 Table 5.23: Summary of UMC-based acceptances for K+ —> ir+f events. The quoted uncer-tainties are purely statistical. butions are summarized in table 5.22 for K+ —> n+vv events, and in table 5.23 for K+ —> 7T+/ events. The trigger and fiducial acceptances, Atrig and A u m c , are taken from the NIDIF-off values; the N I D I F acceptance, Anidif, is then taken from the NIDIF-on /NIDIF-o f f ratio of A • • A •f^-trig ^ -umc-5.9 Kaon Stopping Fraction, fs The acceptances of most of the online and offline cuts used in the K+ —» n+uu analysis are found directly through analysis of monitor and UMC-generated data, as described in the previous sections of this chapter. The remaining acceptance losses (e.g. fraction of kaons that stop in the target, and various detector inefficiencies) are lumped into a "fudge factor" called fs. This factor fs is found by normalizing the measured branching ratio to the accepted value [16]. The value of fs, and the K+ —* ir+vD acceptance measurement as a whole, are then tested by measuring the branching ratio and comparing with the accepted value [16]. A subset of the same cuts from the K+ —>• ix+vD analysis, and the same acceptance measurement structure, are used to measure the and K^i branching ratios. 160 Chapter 5. Acceptance and Sensitivity Because many cuts are common to the Ku2 and K^2 analyses, the acceptances of these cuts cancel out in the Kn2 branching ratio calculation, assuming that these cuts are insensitive to the kaon decay particle (i.e., pion or muon). If an anomalous or unstable value of the Kn2 branching ratio is measured, this could indicate problems in the K+ —> ix+vv acceptance structure, and/or to errors in the assumptions that various acceptances/efficiencies can be measured independent of the type of kaon decay particle, and/or to problems wi th U M C (e.g., simulation of pion-nuclear interactions). A l l appropriate K+ —> ix+vv cuts are applied when measuring the K^2 and Kn2 branching ratios, in order to test the K+ —• n+vv acceptance measurement as thoroughly as possible. 5.9.1 Measurement of fs The branching ratio is measured by analyzing ^ 2 ( 1 ) monitor data (see section 3.3). The cuts applied in the analysis are shown in table 5.24, which lists the number of ^ 2 ( 1 ) monitor events, M ^ 2 , from each of the 1995, 1996, 1997a, and 1997b data sets which remain after each cut is applied. The P V ( n o B V ) cut is the P V function cut, with the requirements in the barrel disabled because many K^2 decays have tracks which enter the barrel. The R T O T 4 0 cut requires R > 40 cm, and is implemented to remove K^2 and radiative K^2 decays not removed by P V ( n o B V ) . The acceptances of the cuts are measured using ^ 2 ( 1 ) monitor data and UMC-generated data, in much the same way way as the K^- and UMC-based acceptance measurements of sections 5.2 and 5.8. Table 5.25 lists the number of surviving ^ 2 ( 1 ) monitor events from the 1995, 1996, 1997a, and 1997b data sets after each cut used in the Ku2 branching ratio measurement is applied, and the corresponding acceptance of each cut. A l l setup cuts in table 5.25 are defined in table 5.3. Acceptances in table 5.25 are grouped into the quantities ARD, A r e c o n , Arest, Apv, and Abad(Km2), which are the R S reconstruction, U T C and target reconstruction, beamline and target pattern, P V , and bad data acceptances, respectively, for the cuts applied in the Ku2 branching ratio measurement. A l l of these acceptances are combined into the quantity AK^, which is analogous to the AK^ quantity from table 5.2. Table 5.26 lists the number of 161 Chapter 5. Acceptance and Sensitivity surviving UMC-generated K+ -> p+u^ events for the 1995, 1996, 1997a, and 1997b kaon stopping distributions after each trigger condition or cut used in the K^2 branching ratio measurement is applied, and the corresponding acceptance of each cut. Acceptances in table 5.26 are grouped into the quantities A^rfg, AK^2recon, and A^^kini which are the ^ 2 ( 1 ) trigger, UMC-based reconstruction, and kinematic acceptances, respectively, for the cuts applied in the branching ratio measurement. These acceptances are combined into the quantity A^™. Ax™trecon is not used in the calculation of the K^2 branching ratio, but it is measured here for later use in the K^2 branching ratio calculation. The measured K^2 branching ratio, B{K+ —> JJ^V^), is normalized to the accepted value of 0.6351 ± 0.0018 [16] in order to get the kaon "stopping fraction", fs, given by = B(K+ -> = MK^ 1 _ 0.6351 e - {K,2) • {KB{L)K^2 • AK^2 • Alfg • ATJMn 0-6351 where £ 7 ^ 2 ( ^ 2 ) is the "non-gap" T • 2 efficiency for K^2 decays (see section 5.7), and (K-BiLeJK^ is ^BUve (see section 5.1) corrected for online and offline K^il) monitor prescales [41] and shown in table 5.27. Using the numbers in tables 5.12, 5.24, 5.25, 5.26 and 5.27 and E q . (5.12), the values of fs for each run period are 1995: fs = 0.681 ± 0.007 s t a* ± 0 .014^ s t 1996: fa = 0.726 ± 0 .006 s t a t ± 0 .015^* 1997a: fs = 0.700 ± 0 .008 s t a t ± 0.015 s^* 1997b: f8 = 0.760 ± 0 .009 s t a t ± 0.016 S ! / S t where the systematic uncertainty i n fs comes from the uncertainty i n the T-2 gap inefficiency: 4 . 1 ± 0 . 2 % (see section 5.7). The average value of fs, weighted by Ae-fs-Ksuve (see section 5.10 and table 5.32), is 0.704 ± 0.004 s 4 a t ± 0 .009 s y s t . 5.9.2 Measurement of the K n 2 Branching Ratio The Kn2 branching ratio is measured by analyzing K^l) monitor data (see section 3.3). The cuts applied in the analysis are shown in table 5.28, which lists the number of K^l) monitor events, MK^2, from each of the 1995, 1996, 1997a, and 1997b data sets which 162 Chapter 5. Acceptance and Sensitivity cut 1995 1996 1997a 1997b 88989 223363 74999 43727 B A D _ R U N 87795 210984 69478 41473 B A D _ S T C 87772 210981 69478 41473 Kn2(l) trigger 87772 210981 69478 41473 R D . T R K 87771 210924 69478 41473 T R K T I M 87771 209516 69467 41467 U T C / R A N G E 82261 201981 66275 39979 U T C Q U A L 78878 191885 62013 38120 P R O B Z 78878 190072 61357 37761 T A R G E T 76702 187146 60439 37284 C O S 3 D 74377 182165 58711 36478 B 4 D E D X 73254 178520 57910 36110 C P I T R S 72621 177574 57599 35924 C P I T A I L 72521 177475 57555 35903 I C B I T 67934 176602 57469 35877 T I C 66699 173979 56695 35488 T I M C O N 65588 172079 56052 35127 T G C C D 57947 158028 50354 32517 D C B I T 51956 137028 45360 28665 D E L C 46258 122262 39662 24998 C K T R S 45822 120490 39172 24692 C K T A I L 44288 116286 37292 23716 B W T R S 41575 111591 35694 22910 B H T R S 41401 110864 35493 22810 T A R G F 39877 107039 34248 22010 D T G T T P 39867 107010 34246 22008 R T D I F 39434 105751 33775 21785 T G Q U A L T 39434 105751 33775 21785 P I G A P 38829 104382 33288 21538 T G B 4 36803 98986 31536 20498 K I C 36466 98005 31277 20342 T G G E O 36359 97739 31197 20308 B 4 E K Z 35847 96526 30800 20098 B 4 E K Z J C 35623 95881 30602 19959 T G Z F O O L 35623 95881 30602 19959 P V ( n o B V ) 30482 81954 25854 17426 R T O T 4 0 30470 81926 25840 17417 MKll2 30470 81926 25840 17417 Table 5.24: Cuts applied to K^2(l) monitor data in order to measure the Ka2 branching ratio. Table entries are described in section 5.9.1 of the text. The P R O B Z cut is not applied to the 1995 data, as is the case in the K+ —> it+vv analysis (see section C.3.1). 163 Chapter 5. Acceptance and Sensitivity cut 1995 (acc.) 1996 (acc.) 1997a (acc.) 1997b (acc.) S E T U P f i D 45635 119105 39495 23990 R D . T R K 45635 (1.000) 119105 (1.000) 39495 (1.000) 23990 (1.000) T R K T I M 45635 (1.000) 118849 (0.998) 39492 (1.000) 23990 (1.000) ARD 1.0000 ± 0.0000 0.9979 ± 0.0001 0.9999 ± 0.0000 1.0000 ±0 .0000 S E T U P r e c O T l 24749 66786 21386 13805 U T C / R A N G E 24749 (1.000) 66786 (1.000) 21386 (1.000) 13805 (1.000) U T C Q U A L 24245 (0.980) 64517 (0.966) 20417 (0.955) 13329 (0.966) P R O B Z 24245 (1.000) 63935 (0.991) 20228 (0.991) 13202 (0.990) T A R G E T 24102 (0.994) 63640 (0.995) 20132 (0.995) 13159 (0.997) A •Brecon 0.9739 ±0 .0010 0.9529 ± 0.0008 0.9414 ± 0 . 0 0 1 6 0.9532 ±0 .0018-S E T U P r e s t 36718 96912 30379 20119 I C B I T 36718 (1.000) 96912 (1.000) 30379 (1.000) 20119 (1.000) T I C 36397 (0.991) 96437 (0.995) 30247 (0.996) 20049 (0.997) T I M C O N 35983 (0.989) 95748 (0.993) 30088 (0.995) 19954 (0.995) T G C C D 32338 (0.899) 89541 (0.935) 27635 (0.918) 18673 (0.936) D C B I T 29564 (0.914) 79835 (0.892) 25529 (0.924) 16859 (0.903) D E L C 26607 (0.900) 71834 (0.900) 22578 (0.884) 14868 (0.882) C K T R S 26443 (0.994) 71218 (0.991) 22404 (0.992) 14757 (0.993) C K T A I L 25786 (0.975) 69329 (0.973) 21578 (0.963) 14286 (0.968) B 4 D E D X 25535 (0.990) 68424 (0.987) 21384 (0.991) 14159 (0.991) C P I T R S 25484 (0.998) 68339 (0.999) 21353 (0.999) 14132 (0.998) C P I T A I L 25469 (0.999) 68323 (1.000) 21348 (1.000) 14127 (1.000) T A R G F 24642 (0.968) 66223 (0.969) 20668 (0.968) 13685 (0.969) D T G T T P 24639 (1.000) 66217 (1.000) 20668 (1.000) 13685 (1.000) R T D I F 24392 (0.990) 65526 (0.990) 20410 (0.988) 13561 (0.991) T G Q U A L T 24392 (1.000) 65526 (1.000) 20410 (1.000) 13561 (1.000) P I G A P 24103 (0.988) 64866 (0.990) 20184 (0.989) 13460 (0.993) T G B 4 22801 (0.946) 61397 (0.947) 19098 (0.946) 12803 (0.951) K I C 22593 (0.991) 60787 (0.990) 18945 (0.992) 12708 (0.993) T G G E O 22549 (0.998) 60665 (0.998) 18914 (0.998) 12692 (0.999) B 4 E K Z 22314 (0.990) 60001 (0.989) 18708 (0.989) 12562 (0.990) B 4 E K Z J C 22244 (0.997) 59765 (0.996) 18616 (0.995) 12516 (0.996) T G Z F O O L 22244 (1.000) 59765 (1.000) 18616 (1.000) 12516 (1.000) B W T R S 21619 (0.972) 58623 (0.981) 18253 (0.981) 12307 (0.983) B H T R S 21541 (0.996) 58261 (0.994) 18160 (0.995) 12254 (0.996) 0.5867 ±0 .0026 0.6012 ±0 .0016 0.5978 ± 0.0028 0.6091 ± 0.0034 S E T U P p y 11553 30866 9627 6426 P V ( n o B V ) 10328 (0.894) 27535 (0.892)' 8481 (0.881) 5862 (0.912) APV 0.8940 ± 0 . 0 0 2 9 0.8921 ± 0 . 0 0 1 8 0.8810 ± 0 . 0 0 3 3 0.9122 ±0 .0035 S E T U P 6 a ( i 79122 221585 74826 43653 B A D - R U N 78066 (0.987) 209267 (0.944) 69315 (0.926) 41405 (0.949) B A D . S T C 78052 (1.000) 209264 (1.000) 69315 (1.000) 41405 (1.000) A T m 2 ( 1 ) trigger 78052 (1.000) 209264 (1.000) 69315 (1.000) 41405 (1.000) Abad{K u2) 0.9865 ± 0.0004 0.9444 ± 0.0005 0.9263 ± 0 . 0 0 1 0 0.9485 ± 0 . 0 0 1 1 AKU2 0.5039 ± 0.0028 0.4816 ± 0 . 0 0 1 7 0.4592 ± 0.0029 0.5023 ±0 .0036 Table 5.25: JC^-based acceptance of cuts applied in the branching ratio measurement. The quoted uncertainties are purely statistical. Table entries are described in section 5.9.1 of the text. The P R O B Z cut is not applied to the 1995 data, as is the case in the K+ —» n+vi? analysis (see section C.3.1). The various S E T U P ' S are defined in table 5.3. S E T U P ^ is simply the online IC requirement ( ICBIT) . 164 Chapter 5. Acceptance and Sensitivity cut 1995 (acc.) 1.996 (acc.) 1997a (acc.) 1997b (acc.) K T T - 2 19 c t + 20ct + 21ct 50000 22629 (0.452) 18777 (0.830) 50000 22632 (0.453) 18737 (0.828) 49999 22602 (0.452) 18674 (0.826) 49999 22806 (0.456) 18967 (0.832) 4 K " 2 0.3755 ± 0.0022 0.3747 ± 0.0022 0.3735 ± 0.0022 0.3793 ± 0.0022 U T C / R A N G E U T C Q U A L P R O B Z T A R G E T 18764 (0.999) 18761 (1.000) 18761 (1.000) 18703 (0.997) 18726 (0.999) 18717 (1.000) 18684 (0.998) 18638 (0.998) 18657 (0.999) 18649 (1.000) 18621 (0.998) 18576 (0.998) 18955 (0.999) 18949 (1.000) 18915 (0.998) 18887 (0.999) Aumc K^jTecon 0.9961 ± 0.0005 0.9947 ± 0.0005 0.9948 ± 0.0005 0.9958 ± 0.0005 C O S 3 D R T O T 4 0 18405 (0.984) 18403 (1.000) 18306 (0.982) 18305 (1.000) 18234 (0.982) 18231 (1.000) 18556 (0.982) 18555 (1.000) Aumc 0.9840 ± 0.0009 0.9821 ± 0 . 0 0 1 0 0.9814 ± 0 . 0 0 1 0 0.9824 ± 0 . 0 0 1 0 Aumc AKu2 0.3681 ± 0 . 0 0 2 2 0.3660 ± 0.0022 0.3646 ± 0.0022 0.3711 ± 0 . 0 0 2 2 Table 5.26: UMC-based acceptances of cuts applied in the branching ratio measurement. The quoted uncertainties are purely statistical. Table entries are described in section 5.9.1 of the text. K T is the number of K+ —> L i + u l l events generated by U M C . The various trigger conditions are defined in section 3.3. year 1995 392509 1996 1132804 1997a 385261 1997b 218617 Table 5.27: Values of {KB\{ve)Kii2 for each run period. 165 Chapter 5. Acceptance and Sensitivity remain after each cut is applied. The K P 2 B 0 X cut is a 3cr cut on the range, energy and momentum. The K P 2 S T O P cut requires the stopping layer to be between layers 8 and 15 inclusive. Table 5.29 lists the number of surviving K^i^) monitor events from the 1995, 1996, 1997a, and 1997b data sets after each cut used in the branching ratio measurement is applied, and the corresponding acceptance of each cut, grouped into the quantities ARD, Arecon, A r e s t , and Abad, which are the R S reconstruction, U T C and target reconstruction, beamline and target pattern, and bad data acceptances, respectively, for the cuts applied in the Kn2 branching ratio measurement. A l l setup cuts in table 5.29 are similar to those in table 5.25 which are defined in table 5.3, but wi th the following modifications: all S E T U P ' S include the requirement of at least 200 M e V of photon energy in the barrel and E C combined, such that there is no photon energy in the R S which can adversely affect track reconstruction (KW2 decays have a total of 225 M e V of photon energy); 75 < ERS < 105 M e V is required instead of 120 < ERS < 150 M e V in order to isolate the K„2 energy peak; the P V ( n o B V ) cut is not applied; and the K M 2 P B O X cut is replaced by the K P 2 B O X , K P 2 S T O P , and F I T P I cuts. S E T U P S is simply the online IC requirement ( I C B I T ) . ARD, Arecon, and Arest in table 5.29 are essentially the same quantities as those measured using K^iX) monitor data, shown in table 5.25). However, the measurements using K^l) monitor data are expected to give lower acceptances than the same measurements which use K^X) monitor data, because the ^ 2 ( 1 ) data sample after setup cuts are applied has a larger component of kaon-rate-dependent background, which can affect the performance of reconstruction and pathology cuts. The larger component of rate-dependent background arises from the fact that the . ^ 2 ( 1 ) trigger selects tracks which are not required to penetrate far into the R S , and are therefore closer to the beam axis where activity is highest. ARD, Arecon, and A r e s t are therefore most accurately measured using 7^ 2 (1) monitor data for both the and KV2 branching ratio measurements, and wi l l consequently "cancel out" when using fs to calculate the Kn2 branching ratio (see below). They are measured using K^l) monitor data and listed in table 5.29 mainly for the sake of investigating the assumption that these acceptances are independent of the type of kaon decay particle. Conversely, A^^K^) in 166 Chapter 5. Acceptance and Sensitivity cut 1995 1996 1997a 1997b A L L 88970 104092 34246 19386 B A D _ R U N 87677 102375 33438 19304 B A D . S T C 87536 102326 33438 19302 Kn2{l) trigger 87536 102326 33438 19302 R D _ T R K 87527 102266 33437 19302 T R K T I M 87527 101500 33411 19294 U T C / R A N G E 72258 91831 30147 17652 U T C Q U A L 67318 85934 27712 16575 P R O B Z 67318 85045 27378 16409 T A R G E T 62313 81725 26370 15886 C O S 3 D 55954 74470 23854 14730 B 4 D E D X 54907 72586 23408 14502 C P I T R S 54344 72111 23249 14412 C P I T A I L 54252 72070 23228 14404 I C B I T 51393 71806 23198 14386 T I C 50247 70499 22790 14161 T I M C O N 49371 69564 22501 14009 T G C C D 42531 62233 19700 12701 D C B I T 36678 51733 16956 10790 D E L C 32564 46129 14760 9450 C K T R S 32310 45548 14587 9357 C K T A I L 31246 43973 13892 9022 B W T R S 29482 42211 13313 8735 B H T R S 29377 41952 13254 8703 T A R G F 27855 39825 12623 8301 D T G T T P 27854 39820 12622 8300 R T D I F 27557 39278 12457 8189 T G Q U A L T 27557 39278 12457 8189 P I G A P 26998 38560 12203 8041 T G B 4 25213 36062 11408 7585 K I C 24538 35162 11184 7452 T G G E O 24343 34874 11086 7409 B 4 E K Z 23959 34413 10935 7316 B 4 E K Z J C 23857 34231 10875 7279 T G Z F O O L 23857 34231 10875 7279 F I T P I 8067 11534 3617 2522 K P 2 B O X 7294 10528 3282 2268 K P 2 S T O P 7272 10501 3272 2262 MK„2 7272 10501 3272 2262 Table 5.28: Cuts applied to Kn2(l) monitor data in order to measure the Kn2 branching ratio. Table entries are described in section 5.9.2 of the text. The P R O B Z cut is not applied to the 1995 data, as is the case in the K+ —> ix+vv analysis (see section C.3.1). 167 Chapter 5. Acceptance and Sensitivity table 5.29 is different from A^^K^) in table 5.25, and must be explicitly included in the Kn2 branching ratio measurement, because the B A D _ R U N and B A D _ S T C cuts can be dependent on stopping layer. AFITPI in table 5.29 is the acceptance of the F I T P I cut measured using irscat monitor data, similar to the measurement of the acceptance of the F I T P I cut from section 5.5, except that the nscat monitor events here are selected to fall into the K+ —> 7r+7r° kinematic region instead of the K+ —> ir+vD kinematic region. That is, the 7Tscat monitor events must pass the K P 2 B O X and K P 2 S T O P cuts instead of the B O X and L A Y V 4 cuts (see tables 5.8 and 5.9). Ajfc in table 5.29 is the /j-veto accidental acceptance measured using ^ 2 ( 1 ) monitor data as described in section 5.6 and shown also in table 5.11. ARD, Arecon, Arest, Abad{Kn2), AFITPI, and Aj?c are combined into the quantity Axn2 for use in the calculation of the Kv2 branching ratio. Table 5.30 lists the number of surviving UMC-generated K+ —> 7r+7r° events for the 1995, 1996, 1997a, and 1997b kaon stopping distributions after each trigger condition or cut used in the Kn2 branching ratio measurement is applied, and the corresponding acceptance of each cut. Acceptances in table 5.30 are grouped into the quantities A^g, A ^ ^ r e c o m a n d Au£^kin, which are the K^iX) trigger, UMC-based reconstruction, and kinematic accep-tances, respectively, for the cuts applied in the Kn2 branching ratio measurement. These acceptances are combined into the quantity A^™. The ratio of the and Kn2 UMC-based reconstruction acceptances, Au^recm/Au£^recon from tables 5.26 and 5.30, is used to account for a potential difference in and Kn2 event reconstruction due to the presence of photons in Kn2 decays. The structure of the UMC-based acceptance measurement in table 5.30 is identical to that used in the K+ —> ix+vv analysis (compare table 5.30 wi th tables 5.13 and 5.15), specifically wi th respect to measurement of the acceptances of the K^2{\) and ir+vv(l) triggers; the K P 2 S T O P and L A Y V 4 cuts; and the K P 2 B O X , B O X , and B O X ' cuts. The measurement of the K^2 branching ratio is therefore a test of the accuracy of U M C in simulating 7 r + , 1 2 C interactions. Furthermore, similar to measurement of the acceptances of the B O X and B O X ' cuts (see section 5.8), the acceptance of the K P 2 B O X cut is measured after scaling and smearing R, E, and P values from UMC-generated Kn2 events according 168 Chapter 5. Acceptance and Sensitivity c u t 1995 (acc . ) 1996 (acc . ) 1 9 9 7 a ( acc . ) 1 9 9 7 b (acc . ) S E T U P H D 10368 13829 4343 2 7 1 9 R D . T R K 10368 (1.000) 13829 (1.000) 4343 (1.000) 2 7 1 9 (1.000) T R K T I M 10368 (1.000) 13829 (1.000) 4343 (1.000) 2 7 1 9 (1.000) ARD 1.0000 ± 0 . 0 0 0 0 1.0000 ± 0 . 0 0 0 0 1.0000 ± 0 . 0 0 0 0 1.0000 ± 0 . 0 0 0 0 SETUPrecon 6375 8676 2 7 0 6 1809 U T C / R A N G E 6375 (1.000) 8676 (1.000) 2706 (1.000) 1809 (1.000) U T C Q U A L 6207 (0.974) 8387 (0.967) 2553 (0.943) 1725 (0.954) P R O B Z 6207 (1.000) 8305 (0.990) 2525 (0.989) 1707 (0.990) T A R G E T 6110 (0.984) 8195 (0.987) 2487 (0.985) 1688 (0.989) Arccon 0.9584 ± 0.0025 0.9446 ± 0.0025 0.9191 ± 0 . 0 0 5 2 0.9331 ± 0 . 0 0 5 9 S E T U P ™ , * 5 9 3 6 7917 2414 1581 I C B I T 5936 (1.000) 7 9 1 7 (1.000) 2414 (1.000) 1581 (1.000) T I C 5857 (0.987) 7849 (0.991) 2394 (0.992) 1570 (0.993) T I M C O N 5784 (0.988) 7765 (0.989) 2372 (0.991) 1559 (0.993) T G C C D 5152 (0.891) 7184 (0.925) 2149 (0.906) 1454 (0.933) D C B I T 4652 (0.903) 6391 (0.890) 1972 (0.918) 1309 (0.900) D E L C 4107 (0.883) 5700 (0.892) 1716 (0.870) 1182 (0.903) C K T R S 4070 (0.991) 5635 (0.989) 1698 (0.990) 1174 (0.993) C K T A I L 3924 (0.964) 5407 (0.960) 1608 (0.947) 1132 (0.964) B 4 D E D X 3898 (0.993) 5330 (0.986) 1594 (0.991) 1121 (0.990) C P I T R S 3866 (0.992) 5309 (0.996) 1586 (0.995) 1116 (0.996) C P I T A I L 3858 (0.998) 5307 (1 .000) 1585 (0.999) 1115 (0.999) T A R G F 3679 (0.954) 5079 (0.957) 1528 (0.964) 1073 (0.962) D T G T T P 3 6 7 9 (1.000) 5079 (1.000) 1528 (1.000) 1073 (1.000) R T D I F 3643 (0.990) 5023 (0.989) 1509 (0.988) 1064 (0.992) T G Q U A L T 3643 (1.000) 5023 (1.000) 1509 (1.000) 1064 (1 .000) P I G A P 3594 (0.987) 4961 (0.988) 1491 (0.988) 1052 (0.989) T G B 4 3395 (0.945) 4675 (0.942) 1404 (0.942) 1003 (0.953) K I C 3365 (0.991) 4622 (0.989) 1392 (0.991) 996 (0.993) T G G E O 3341 (0.993) 4592 (0.994) 1378 (0.990) 986 (0.990) B 4 E K Z 3272 (0.979) 4533 (0.987) 1355 (0.983) 974 (0.988) B 4 E K Z J C 3255 (0.995) 4512 (0.995) 1349 (0.996) 968 (0.994) T G Z F O O L 3255 (1 .000) 4512 (1.000) 1349 (1.000) 968 (1.000) B W T R S 3060 (0.940) 4320 (0.957) 1276 (0.946) 939 (0.970) B H T R S 3052 (0.997) 4295 (0.994) 1269 (0.995) 934 (0.995) •Arest 0.5142 ± 0.0065 0.5425 ± 0.0056 0.5257 ± 0.0102 0.5908 ± 0.0124 S E T U P 6 A D 19665 27161 8557 5005 B A D - R U N 19456 (0.989) 26742 (0.985) 8378 (0.979) 4987 (0.996) B A D . S T C 19430 (0.999) 26739 (1 .000) 8378 (1 .000) 4986 (1.000) Kn2(l) t r i g g e r 19430 (1.000) 26739 (1 .000) 8378 (1 .000) 4986 (1.000) Abad 0.9880 ± 0.0008 0.9845 ± 0.0008 0.9791 ± 0 .0015 0.9962 ± 0.0009 AFITPI .7117 ± .0120 .7063 ± .0069 .7111 ± .0134 .7171 ± .0142 0.9924 ± 0.0003 0.9930 ± 0.0003 0.9925 ± 0.0005 0.9940 ± 0.0006 0.3458 ± 0.0074 0.3557 ± 0.0052 0.3358 ± 0.0093 0.3932 ± 0 . 0 1 1 6 Table 5.29: Monitor-based acceptances of cuts applied in the K^i branching ratio mea-surement. The quoted uncertainties are purely statistical. Table entries are described in section 5.9.2 of the text. The P R O B Z cut is not applied to the 1995 data, as is the case in the K+ —> -K+VV analysis (see section C.3.1). 169 Chapter 5. Acceptance and Sensitivity cut 1995 (acc.) 1996 (acc.) 1997a (acc.) 1997b (acc.) K T T - 2 6ct + 7ct //-veto U F A T E U S T M E D U S T O P J H E X 19999 8988 (0.449) 7530 (0.838) 7197 (0.956) 6063 (0.842) 5920 (0.976) 5368 (0.907) 19999 9042 (0.452) 7597 (0.840) 7302 (0.961) 6156 (0.843) 6017 (0.977) 5480 (0.911) 19999 8867 (0.443) 7372 (0.831) 7079 (0.960) 5980 (0.845) 5872 (0.982) 5299 (0.902) 19996 9018 (0.451) 7542 (0.836) 7232 (0.959) 6097 (0.843) 5979 (0.981) 5422 (0.907) 0.2684 ± 0 . 0 0 3 1 0.2740 ± 0.0032 0.2650 ± 0 . 0 0 3 1 0.2712 ± 0 . 0 0 3 1 U T C / R A N G E U T C Q U A L P R O B Z T A R G E T 5355 (0.998) 5351 (0.999) 5351 (1.000) 5295 (0.990) 5448 (0.994) 5442 (0.999) 5422 (0.996) 5361 (0.989) 5268 (0.994) 5266 (1.000) 5249 (0.997) 5205 (0.992) 5396 (0.995) 5393 (0.999) 5378 (0.997) 5324 (0.990) Aumc K„2^econ 0.9864 ± 0 . 0 0 1 6 0.9783 ± 0.0020 0.9823 ± 0 . 0 0 1 8 0.9819 ± 0 . 0 0 1 8 K P 2 S T O P C O S 3 D K P 2 B O X 5083 (0.960) 4864 (0.957) 4517 (0.929) 5152 (0.961) 4926 (0.956) 4567 (0.927) 4998 (0.960) 4763 (0.953) 4420 (0.928) 5128 (0.963) 4939 (0.963) 4544 (0.920) Aumc 0.8531 ± 0.0049 0.8519 ± 0.0049 0.8492 ± 0.0050 0.8535 ± 0.0048 Aumc 0.2259 ± 0.0029 0.2284 ± 0 . 0 0 3 0 0.2211 ± 0 . 0 0 2 9 0.2273 ± 0.0029 Table 5.30: UMC-based acceptances of cuts applied in the K^2 branching ratio measurement. The quoted uncertainties are purely statistical. Table entries are described in section 5.9.2 of the text. U F A T E , U S T M E D , and U S T O P . H E X are defined in section 5.8. K T is the number of K+ —> 7r+7r° events generated by U M C . The various trigger conditions are defined in section 3.3, including "p-veto" which is defined as (19 c t + 20ct + 21ct). to tables 5.18 and 5.19 to match the real data. The branching ratio is given by B(K+ - T T + T T 0 ) = • j, ^ (5.13) enT%(Kn2) • (KBge)K^ • AFITPI • Afc • AK„2 • ATJ,kin • f. where £ ^ 2 ( ^ - 2 ) is the "non-gap" T • 2 efficiency for Kn2 decays (see section 5.7), and (^Biive)K„2 1 S analogous to (KB{{ve)Kp2 ^ r o m section 5.9.1 and shown in table 5.31. Using E q . (5.12), E q . (5.13) becomes B(K+^ TT+TT0) - 0 6351 M K ' 2 6 T ^ K ^ (KBL)K,2 AKlia-A%£ i B(K - 0.6351 ^ ^ { j ^ AK„2 • ATJ A F I T P I • A** ^ O f i ^ i ET92(K^) (KBL)K^ Apy Abad(K^) MKii2 e^2(Kn2) (KeJle)K„2 ATJ A F I T P I . A ^ Abad(K„2y The uncertainties in these quantities from tables 5.12, 5.24, 5.25, 5.26, 5.27, 5.28, 5.29, 5.30 and 5.31 are purely statistical, so, from Eq . (5.14), the values of the Kv2 branching ratio for 170 Chapter 5. Acceptance and Sensitivity year 1995 557517 1996 807044 1997a 273726 1997b 155559 Table 5.31: Values of (KBi{ve)K„2 f ° r e a c n r u n period. each run period are 1995: B(K+ -- + T T + T T 0 ) = 0.218 ± 0 .006 s t a t 1996: B(K+ -- * T T + T T ° ) = 0.210 ± 0.004 s t a* 1997a: B(K+ -- + T T + T T 0 ) = 0.212 ± 0.007 s ' a ' 1997b: B(K+ -f T T + T T 0 ) = 0.216 ± 0.007 s t a* The average value of the K n 2 branching ratio, weighted by Ae- fs- Ksuve (see section 5.10 and table 5.32), is 0.214 ± 0 .003 5 i a t . This agrees wi th the accepted value of 0.2116 ± 0.0014 [16], which implies that the value of / s , the U M C simulation of T T + , 1 2 C interactions, and the K + —• ix+vv acceptance measurement as a whole are accurate. Further comparisons of 7 r + , 1 2 C interactions in U M C wi th real data [41] indicate that U M C simulates these interactions to an accuracy of about ± 2 % . This is a measure of the systematic uncertainty in the measured branching ratio, given therefore by 0.214 ± 0.003 s t o* ± 0.004 S 2 / S i , which, as mentioned above, agrees wi th the accepted value. 5 . 10 Summary The acceptances for K + —> -K+UU and K + — > T T + / are combined wi th the number of kaons entering the target to get the single-event sensitivities shown in tables 5.32 and 5.33. In these tables, monitor- and U M C - b a s e d acceptances are combined into the quantity A, detector efficiencies are combined into the quantity e, the kaon stopping fraction is given by fs, and the number of kaons is given by Ksiive, such that the total acceptance is defined by Ae • fs and the sensitivity is given by Ae • fs • Kpuve-171 Chapter 5. Acceptance and Sensitivity 1995 1996 1997a 1997b Atrig 0.1799 ±0.0012 0.1799 ±0.0012 0.1780 ±0.0012 0.1811 ±0.0012 j^acc 0.99242 ± 0.00027 0.99303 ± 0.00027 0.99252 ± 0.00050 0.99401 ± 0.00055 Aumc 0.3650 ± 0.0036 0.3634 ± 0.0036 0.3752 ± 0.0036 0.3657 ± 0.0036 A-nidif 0.5144 ± 0.0086 0.5118 ±0.0086 0.5069 ± 0.0085 0.5126 ± 0.0086 AK„2 0.4065 ± 0.0029 0.4303 ± 0.0018 0.3999 ± 0.0032 0.4396 ± 0.0039 AK„2 0.8745 ± 0.0027 0.8711 ±0.0015 0.8676 ± 0.0031 0.8765 ± 0.0037 •A-xscat 0.8608 0.7911 0.7618 0.8128 ±0.0168"'°' ±0 .0096 s ' a t ±0.0197*'°' ±0 .0186 s ' a ' ±0.0464"""' ±0 .0324 s « s t ±0.0362"""' ±0 .0371 s " s ' ATD 0.2930 0.3282 0.3114 0.3201 ±0.0096"'°' ±0.0055 s'"' ±0.0102"'°' ±0.0107 s '° ' ±0.0065"""' ±0 .0056 s " s t ±0.0054"""' ±0.0075"" s' T-2 0.976 0.930 0.929 0.928 ±0.005 s " s * ±0.006*!"" ±0.004"""' ±0.008"""' fs 0.6808 0.7261 0.6997 0.7596 ±0.0068"'°' ± 0 . 0 0 5 7 s t a t 0.0077s'"' 0.0095s'0' ±0.0142 s" s* ±0.0151"""' 0.0146s""' 0.0158s"st Ae 0.00293 0.00301 0.00257 0.00314 ±0.00013"'°' ±0.00009"'°' ± 0 . 0 0 0 1 2 s t a ' ±0 .00015 s ' a t ±0.00017 s " s ' ±0.00013"""' ±0 .00013 s " s t ±0.00016"""' Ae-f. 0.00200 0.00218 0.00180 0.00238 ±0.00009"'°' ±0.00007"'°' ± 0 . 0 0 0 0 9 s t a ' ±0.00011 s '°' ±0.00012"" s t ±0.00011"" s' ±0 .00010 s " s t ±0.00013"""' 1.52627 1.12525 0.37080 0.21574 sensitivity (10a) 3.0482 2.4569 0.6666 0.5145 ±0.1375 s '°' ±0.0751"'°' ±0.0324 s '° ' ±0.0248"'°' ±0.1894 s« s* ±0.1214 s""' ±0.0366"""' ±0.0288"""' total sensitivity (10a) 6.686 ± 0.162s tat ± o.230 s y s t total 1/sensitivity (IO - 1 0 ) 1.496 ± 0.036"'°' ± 0.051""s* Table 5.32: K+ —> ir+vv single-event sensitivity of the 1995-7 analysis. 172 Chapter 5. Acceptance and Sensitivity 1995 1996 1997a 1997b Atrig 0.4021 ± 0.0022 0.4034 ± 0.0022 0.4018 ± 0.0022 0.4115 ± 0 . 0 0 2 2 ^ o c c 0.99242 ± 0.00027 0.99303 ± 0.00027 0.99252 ± 0.00050 0.99401 ± 0.00055 AUmc 0.5698 ± 0.0035 0.5631 ± 0.0035 0.5764 ± 0.0035 0.5762 ± 0.0034 •Anidif 0.4674 ± 0.0058 0.4608 ± 0.0057 0.4584 ± 0.0057 0.4598 ± 0.0055 AK»2 0.4065 ± 0.0029 0.4303 ± 0.0018 0.3999 ± 0.0032 ' 0.4396 ± 0.0039 AK*2 0.8745 ± 0.0027 0.8711 ± 0 . 0 0 1 5 0.8676 ± 0.0031 0.8765 ± 0.0037 A-nscat 0.8608 0.7911 0.7618 0.8128 ± 0 . 0 1 6 8 " ' ° ' ± 0 . 0 0 9 6 " ' ° ' ± 0 . 0 1 9 7 " ' ° ' ± 0 . 0 1 8 6 " ' ° ' ± 0 . 0 4 6 4 " " " ' ± 0 . 0 3 2 4 " " " ' ± 0 . 0 3 6 2 " " " ' ± 0 . 0 3 7 1 " " " ' ATD 0.2930 0.3282 0.3114 0.3201 ± 0 . 0 0 9 6 " ' ° ' ± 0 . 0 0 5 5 " ' ° ' ± 0 . 0 1 0 2 " ' ° ' ± 0 . 0 1 0 7 " ' ° ' ± 0 . 0 0 6 5 " " " ' ± 0 . 0 0 5 6 " " " ' ± 0 . 0 0 5 4 " " " ' ± 0 . 0 0 7 5 " " " ' rig T-2 0.975 0.928 0.927 0.926 ± 0 . 0 0 5 " " " ' ± 0 . 0 0 6 " " " ' ± 0 . 0 0 4 " " " ' ± 0 . 0 0 8 " " " ' fs 0.6808 0.7261 0.6997 0.7596 ± 0 . 0 0 6 8 " ' ° ' ± 0 . 0 0 5 7 " ' ° ' 0 . 0 0 7 7 " ' ° ' 0 . 0 0 9 5 " ' ° * ± 0 . 0 1 4 2 " " " ' ± 0 . 0 1 5 1 " " " ' 0.0146"""' 0.0158"""' Ae 0.00929 0.00939 0.00804 0.01006 ± 0 . 0 0 0 3 9 " ' ° ' ± 0 . 0 0 0 2 4 " ' ° ' ± 0 . 0 0 0 3 6 " ' ° ' ± 0 . 0 0 0 4 4 " ' ° ' ± 0 . 0 0 0 5 4 " " " ' ± 0 . 0 0 0 4 2 " " " ' ± 0 . 0 0 0 4 1 " " " ' ± 0 . 0 0 0 5 2 " " " * Ae-fs 0.00633 0.00682 0.00563 0.00764 ± 0 . 0 0 0 2 7 " ' " ' ± 0 . 0 0 0 1 8 " ' ° ' ± 0 . 0 0 0 2 6 " ' ° ' ± 0 . 0 0 0 3 5 " ' ° ' ± 0 . 0 0 0 3 9 " " " ' ± 0 . 0 0 0 3 4 " " " ' ± 0 . 0 0 0 3 1 " " " ' ± 0 . 0 0 0 4 3 " " " ' KBlive(W12) 1.52627 1.12525 0.37080 0.21574 sensitivity (10 9 ) 9.6544 7.6694 2.0859 1.6488 ± 0 . 4 1 3 7 " ' ° ' ± 0 . 2 0 7 5 " ' ° ' ± 0 . 0 9 7 0 " ' ° ' ± 0 . 0 7 5 6 " ' ° ' ± 0 . 5 9 9 7 " " " ' ± 0 . 3 7 9 1 " " " ' ± 0 . 1 1 4 5 " " " ' ± 0 . 0 9 2 4 " " " ' total sensitivity (10H) 21.059 ± 0 . 4 7 9 s t a t j_ o 725s y s £ total 1/sensitivity ( I O - 1 0 ) 0.475 ± 0 . 0 1 1 " ' ° ' ± 0.016"""' Table 5.33: K+ —> ir+f single-event sensitivity of the 1995-7 analysis. 173 Chapter 5. Acceptance and Sensitivity Note, in table 5.32, that the acceptance of the T D cuts, ATD, increases in 1996-7 relative to 1995. This is mostly due to an increased level 1.1 trigger acceptance (see section 3.3), which is offset in large part by losses from additional cuts designed to attack the increase in tail-fluctuation and G D R background (the M A S S cut in A n s c a t , and the T D D F A 1 , T D E C O N , and T D V E L cuts in ATD, which are only applied to the 1996-7 data set - see sections C.3.1 and C.3.4). The increase in these backgrounds is thought to be mainly due to the use of narrower T D pulse shapes for double-pulse fitting by F I T P I (see section C.3.4), which is estimated to increase the F I T P I acceptance by about 6%. The drop in ejf2 in 1996-7 relative to 1995 is thought to be due to deteriorating R S scintillator or light guide quality. fs and Ac • fs (i.e., sensitivity per KBUVC which represents the general quality of the data taking and analysis) increase from year to year, except for a drop in 1997a. The increase in fs is likely due to a gradual lowering of kaon beam momentum between 1995 and 1997 (see section 3.1). The drop in Ae • fs in 1997a seems to come mainly from a low value of fs and from acceptance losses of cuts such as U T C Q U A L , T G C C D , P B N R S , the P V cuts, M A S S , R S L I K E , C H I R F ( z ) , L l . l , and T D D F A 1 . Also, the target H V problem (part of the B A D _ R U N cut) was at its worst in 1997a. Hardware problems and/or higher kaon rate/poorer beam quality may be responsible for the drop in performance in 1997a. The total K+ — > K+VV single-event sensitivity for the combined 1995-7 analysis is a factor of 4.2/1.5 = 2.8 greater than the published 1995 sensitivity of (4.2±§;£) x I O - 1 0 [46]. 174 Chapter 6 Fina l Results In this chapter, the number of candidate K+ —> n+uu events from chapter 4 is divided by the K+ —> n+uu single-event sensitivity from chapter 5 to calculate the K+ —y n+vD branching ratio. Similarly, the number of K+ —y n+f events is inferred here and divided by the K+ —y 7 r + / single-event sensitivity from chapter 5 to calculate the K+ —y ir+f branching ratio. Finally, the K+ —y TV+UU branching ratio is used to calculate a value of the C K M matrix element \Vtd\ (see section 2.1). 6.1 K + — • 7r+v9 Branching Ratio A s shown in section 4.8, one candidate K+ —y ir+isp decay was observed in the 1995-7 data set. The K+ — > TT+U9 branching ratio is therefore 1.0 divided by the K+ —y ir+vv single-event sensitivity from table 5.32, which gives 1.50 x 10~ 1 0 . Uncertainties in measured quantities wi th Gaussian distributions are typically quoted as ± lc r , which corresponds to a 68% confidence interval. For a Poisson process where 1.0 events are observed, the lower bound of the 68% confidence interval is given by the mean, n , of the Poisson distribution for which the probability of observing 1 or more events is (1 — 0.68)/2 = 0.16. From E q . (5.3), this lower bound is found by solving P(n > l ; n ) = 1 - P(n = 0;n) = 1 - e~" = 0.16 (6.1) 175 Chapter 6. Final Results which gives n = 0.17435. The upper bound of the 68% confidence interval is given by the mean, n, of the Poisson distribution for which the probability of observing 1 or fewer events is 0.16. Solving P{n < l ; n ) = P(n = 0; n) + P{n = 1; n) = e _ " + n e - " = 0.16 (6.2) gives n = 3.2885. The 68% confidence interval therefore spans the range [0.17435,3.2885] events [59]. Div id ing this by the K+ —> TT+VV single-event sensitivity from table 5.32 gives a K+ —> it+vv branching ratio range of [0.261,4.92] x 1 0 - 1 0 , or B(K+ n+uu) = 1.50t?;g x 1 0 " 1 0 (6.3) 6.2 K+ 7 T + / U p p e r Limit The signal region for kaon decays into a pion and a massless familon, K+ —• n+f, defined as the "rr/" region, is the same as the 7r + i /p ( l ) signal region wi th the additional requirement of the P F B O X cut. The P F B O X cut is a 2a cut around the pion peak resulting from K+ —> T T + / decay (see section 5.8), and was defined before looking in the irf region. That is, the search for K+ —> T T + / , similar to the search for K+ —* -n+vv, was performed blind. The candidate K+ —> n+uu event (see section 4.8) does not fall into the TT/ region. Results of fits to the R, E, and P peaks of UMC-generated K+ —> n+f decays, where values of R, E, and P from U M C have been scaled and smeared by the quantities in ta-bles 5.18 and 5.19 to match real data, are shown in table 6.1. The R, E, and P values of the candidate event given in section 4.8 are also listed in table 6.1, along wi th their distances from the K+ —> T T + / peak values. The candidate event is located 2.65<r, 2.64cr, and 3.21cr lower than the K+ —>• n+f peak in R, E, and P, respectively. To find the probability that a K+ —> 7 r + / event has these values of R, E, and P, presumably due to a, pion-nuclear interaction and/or Gaussian resolution, the probability that a event is located this far or farther from the K^i peak is measured. Results of fits to the R, E, and P peaks of decays are shown in table 6.1, along with the number of events MK^2 used to perform the fits, which is the number of decays from the K^X) monitor data (see section 3.3) which 176 Chapter 6. Final Results Range Energy Momentum K+ —> 7 r + / , mean ±cr Candidate event, mean ±cr Distance to the irf peak Kn2 mean ±<r 37.93 ± 1.201 34.75 ± 1.22 2.65o-30.43 ± 0 . 8 8 1 1 127.6 ± 3.734 117.73 ± 3 . 5 1 2.64a 108.7 ± 2 . 8 0 8 227.0 ± 2 . 7 5 5 218.17 ± 2 . 6 8 3.21(7 205.1 ± 2 . 2 5 8 M £ 9 at R,E,P< [2.65,2.64,3.21]a 6398 16 Probabil i ty that a jftT+ —> 7 r + / event has candidate values of R, E, P 0.250 ± 0.062% Table 6.1: Calculat ion of the probability that a K+ —> n+f event, for massless / , has the R, E, and P values of the candidate event. Table entries are described in section 6.2 of the text. pass all cuts except the P V function cut for detector subsystems outside of the target, the B O X cut, and the L A Y V 4 cut (see section C.3). Also shown in table 6.1 is the number of Kn2 decays located as far or farther from the Kn2 peak as the candidate event is from the K+ —• n+f peak, which, when divided by M]/ , gives the probability that a K+ —* 7 r + / event has the R, E, and P values of the candidate event. A s shown in table 6.1, this proba-bili ty is only 0.3%, which suggests that the candidate event is not due to K+ —> 7 r + / decay. Based on no K+ —> n+f decays observed, the upper l imit on the K+ —> n+f branching ratio at the 90% confidence level is given by the mean, n, of the Poisson distribution for which the probability of observing 0 events is 1 — 0.90 = 0.10. From E q . (5.3), this upper l imit is found by solving P(n = 0;n) = e - " = 0.10 (6.4) which gives n = 2.3026. Div id ing this by the K+ —> n+f single-event sensitivity from table 5.33 gives an upper l imit to the K+ —> ir+f branching ratio at the 90% confidence level of 1.09 x I O " 1 0 . 6.3 Vtd A s stated in section 2.3, the measured value of the K+ —> ir+vu branching ratio can be used to extract the magnitude of the C K M matrix element Vtd- From E q . (2.18), the 177 Chapter 6. Final Results K+ —> ir+vv branching ratio defines a circle in the (p, 77) plane (see figure 2.4). This circle has radius ro given by E q . (2.19) which is obtained from 1. the K+ —> ix+vv branching ratio range quoted in section 6.1 ([0.26,4.92] x 10~ 1 0 ) ; 2. A = Vcb/X2 from E q . (2.2), wi th A = 0.22 and Vcb = 0.040 ± 0.003 [18]; 3. X(xt) as defined in Eqs. (2.7) and (2.8), wi th rjx = 0.994 and xt = mf/M^, where mt = mt(mt) = (166 ± 5) G e V / c 2 [18] and Mw = 80.41 M e V / c 2 [16]; and 4. K+ = 4.11 x I O " 1 1 [18]. The circle is centered at (po,0), where p0 is given by E q . (2.13) and is obtained using tabulated values of Po(X) [18] and A and X(xt) as given above. p0 and r 0 define limits on Rt according to E q . (2.23), where Rt is the length of one of the sides of the unitarity triangle and is related to Vtd v ia E q . (2.21). Looping over the full ranges of al l variables, the limits on Vtd from the measured K+ —> -K+UU branching ratio are 0.0024 < \Vtd\ < 0.038 (6.5) These limits on Vtd are less restrictive than those which follow from assumption of unitarity of the quark mixing matrix, which are given in E q . (2.3): 0.004 < \Vtd\ < 0.014. 178 Chapter 7 Conclusions The decay K+ —> ir+vu involves a flavour-changing neutral weak current, and therefore cannot occur at first order ("tree level") in the Standard Mode l (SM) . However, flavour symmetry is "broken" as evidenced by the widely different quark masses, which allows K+ —> -K+UU to proceed at second order ("one-loop level"). K+ —> it+vD is therefore particularly sensitive to weak interactions of the highest mass quark (in the S M , this is the top quark), and is highly suppressed in the S M with a predicted branching ratio of (0.8 ± 0.3) x 1 0 - 1 0 . The uncertainty i n the branching ratio is dominated by measurement uncertainties in the Cabibbo-Kobayashi-Maskawa ( C K M ) quark mixing matr ix elements \Vcb\ and I K ^ / V ^ . The theoretical uncertainty is only 7%. Because the predicted branching ratio is so low and lies in a narrow range, measurement of K+ —> ix+vv offers a high-precision test of the 2-class, 3-generation, 4-force structure of the S M . It can also be used to hunt for non-SM physics. Positive identification of K+ — > n+vv requires suppression of potential backgrounds by at least a factor of 10 1 0 . The experimental signature for K+ — > IT+VV is K+ —» ir+ wi th nothing else observed. Known background processes which can mimic this signature include the first-order weak decays K+ —• 7r+7r° (Kw2) and K+ — > (J^v^ (K^), scattering of pions in the beam, and kaon charge exchange. Kn2 and decays are two-body decays, which means that the charged track from these decays is monochromatic in the kaon rest frame. These decays are therefore suppressed by requiring kaons to stop before they decay, and restricting the search for K+ — > n+uP to the kinematic window between the and peaks. Kn2 decays are further suppressed by high-efficiency photon detection. K^2 decays 179 Chapter 7. Conclusions are further suppressed by /J,+/TY+ particle separation v ia the different energy loss rate and range-momentum correlation of muons and pions i n detector materials. Also , the detector is constructed such that the TT+ track from K+ —> it+vD is confined within, so decays are additionally suppressed by requiring identification of the rr —• n —> e decay sequence in the detector. Backgrounds arising from pion scattering in the beam and kaon charge exchange are suppressed by K+/n+ particle identification in the beam, and by requiring non-coincident beamline and decay-volume detector activity. Experiment 787 (E787) at Brookhaven National Laboratory ( B N L ) has adopted a "blind" approach to the search for K+ —> TX+VV, in an attempt to minimize analysis "bias" which can arise due to small statistics. A small number of K+ —> n+vv events is expected in the 1995-7 data, based on the number of collected kaons, the acceptance of data-selection requirements, and the predicted K+ —> n+vv branching ratio. Therefore, a signal region, or "box", is defined where the signal/background ratio is expected to be the highest, and small numbers of events in the box are not used to define the properties of background or signal events. Instead, potential backgrounds are identified a priori, and techniques are devised to measure the background level in the box using events which lie outside the box. Events in the box are not examined unti l the background estimates are final. The goal in the analysis is to reduce the expected background to <§C 1 event in the box. Then, if events are observed in the box, they wi l l have a high probability to be signal, i n part because they have not previously been examined at any point in the analysis. To reduce the effects of small-statistics bias on the background measurements, back-grounds are estimated using "bifurcated" analyses. That is, each background is addressed by two uncorrelated cuts or sets of cuts. Each of the two cuts is inverted to provide a high-statistics, outside-the-box background data sample, on which the performance of the other cut is measured. Correlations between the cuts are verified to be small by comparing predicted and observed numbers of background events in regions outside the box, defined by loosening both cuts simultaneously. The degree to which bias has entered the analysis is measured by dividing the data into 1/3 and 2/3 samples. Cuts were designed using the 1/3 data sample, and then verified to perform as expected on the independent 2/3 data sample. 180 Chapter 7. Conclusions The total number of background events expected in the box for data collected between 1995 and 1997 is 0.08 ± 0.02 events. Opening the box revealed one event inside. This event passed the most stringent cut-tightening criteria, which were designed before examining the event, and further reduce the estimated background in the box to 0.006 ± 0.002 events. Therefore, the observed event is likely a K+ — • TT+VV signal event. The probability that a kaon decaying into a pion and a massless familon gives rise to the measured kinematic quantities of the candidate event is 0.3%. Assuming then that the event is due to K+ —> ir+vv, the K+ —> TT+VV branching ratio is 1.5l?;2 x 1 0 - 1 0 . Based on no K+ —> TT+f events observed, the upper l imit on the K+ —> ir+f branching ratio at the 90% confidence level is 1.1 x 1 0 - 1 1 (for massless / ) . The K+ — • TT+VV branching ratio is sensitive to the weak coupling of top to down quarks which, as parameterized by the C K M quark mixing matrix element Vtd, is calculated to lie in the range 0.002 < \Vtd\ < 0.04. The measured branching ratio for K+ —> TT+VV, agrees wi th the S M prediction, (0 .8±0 .3) x 10~ 1 0 , although the statistical uncertainty associated wi th observation of one event makes it difficult to draw any conclusions about the presence or absence of non-SM contributions to the decay. One calculation [18] suggests that if the K+ —> TT+VV branching ratio is found to be > 1.67 x 1 0 - 1 0 , then this would be compelling evidence for new physics. Aga in however, the statistical uncertainty of the current measurement does not allow any definitive statements to be made about contributions from non-SM physics. Nevertheless, the K — > TT decay system is unique in that it involves decay of the two lightest S M pseudoscalar (spin 0) particles. B y conservation of angular momentum, the other particle in a two-body K —• ir decay must also have spin = 0. Therefore, there are no two-body K —• n decays which have a massless S M scalar or non-scalar particle (e.g., the photon) in the final state. The breaking of a global continuous symmetry in nature at a high energy scale gives rise to a massless Goldstone boson (scalar or pseudoscalar), so the K —y TT decay system is a clean system for searching for Goldstone bosons associated with the s —> d flavour/family change (e.g., familons, or axions coupled to this flavour/family change). The upper l imit on the K+ —> TT+f width is related to a l imit on the energy scale 181 Chapter 7. Conclusions F of the symmetry breaking v ia [60] r (^ -7r / ) = I ^ ^ f ^ 3 | F 1 ( 0 ) | 2 ' (7.1) where j3 = 1 — m^/mK, gv is the weak vector coupling constant, and Fi(q2) is the K+ —> 7r + weak vector form factor. According to the conserved vector current hypothesis, gy = 1, and in the l imit of exact flavour SU(3) symmetry, F i (0) = 1 [60]. The branching ratio B is related to the width T and mean lifetime r v ia T = Bh/r, so 1 ml rKfl_rnty ( ? 2) s d \ 16TT B(K+ -> 7T+/) h \ m\ where F^d is the energy scale of the symmetry breaking which gives rise to a Goldstone boson that couples vectorially to the s —> d flavour/family change. Using the K+ —> ir+f upper l imit of 1.09 x 1 0 - 1 1 , E q . (7.2) gives F s ^ > 1.8 x 10 1 2 G e V . The energy scale of the symmetry breaking is inversely proportional to the Goldstone boson's coupling to S M particles, or, equivalently, to the Goldstone boson's mass v ia [16] m = 0.62 x 10~ 3 eV • (10 1 0 G e V / F ) (7.3) So the mass of the familon or axion which couples vectorially to the s —> d flavour/family change is restricted to be m < 3.5 x 1 0 - 6 eV. The allowed window of masses is found to be (1.0 < m < 3.5) x 1 0 - 6 eV, using astrophysical constraints on the axion mass [16] shown in figure 7.1. The experimental progress of the search for K+ —> n+uu is shown in figure 7.2. Da ta from 1998-9 is currently being analyzed, and has a single-event sensitivity similar to the combined 1995-7 data set. W i t h E949, the successor experiment to E787 at B N L , it is hoped to eventually achieve a K+ —•> n+vv single-event sensitivity of (8 — 15) x 1 0 - 1 2 v ia improved data-taking conditions in 2001-3. This sensitivity should allow for observation of about 10 K+ —> ir+vv events, which wi l l reduce the statistical uncertainty in the branching ratio measurement, and allow for a more precise determination of \Vtd\- Specifically, the duty factor of the A G S spill wi l l be increased from about 41% to 64%; the number of protons per spill on the kaon production target wi l l be increased from about 15 T p / s p i l l to 65 T p / s p i l l ; 182 Chapter 7. Conclusions mA (eV) I I I axion limits familon limits Figure 7.1: Constraints from astrophysics [16] and K+ —>• n+f on the masses of the axion and familon. The axion mass limits come from comparisons of predicted and observed energy losses from globular cluster stars and supernova 1987a, and from considerations of axion production i n the early universe. The "step" in the mass constraint from dark matter considerations indicates the theoretical uncertainty in this constraint. The familon is a "flavour-coupled axion", so both astrophysics and K+ —> 7 r + / can be used to set an allowed mass region for the familon. 183 Chapter 7. Conclusions the kaon beam momentum wi l l be kept low (at about 700 M e V / c , as in 1997) in order to maintain a high kaon stopping fraction in the target; the data acquisition wi l l be improved via reduced trigger and readout deadtimes and online accidental losses; the "barrel veto liner" wi l l replace the outer 3 layers of the R S to provide improved photon detection; new scintillator for layers 1 — 5 of the R S wi l l be installed, which is expected to increase the T • 2 efficiency; a new B4 hodoscope wi th greater segmentation wi l l be installed in order to improve the spatial resolution of beam particles; another active degrader wi th photon and charged particle detection abilities may replace part of the BeO degrader; and running periods wi l l be extended to > 25 weeks/year, symbiotic wi th the operation of the A G S , in 2000 and afterwards, as an injector for the "relativistic heavy-ion collider" ( R H I C ) . The bl ind analysis technique used by E787 to search for K+ —> n+uu is useful when a small number of K+ — • *K+VV events is expected, such that it is desirable to define a signal region which is never probed for the purposes of signal or background event definition, and where backgrounds are estimated to contribute <C 1 event. Because E949 expects to collect on the order of 10 K+ —> IT+VV events in a comparably-sized signal region, it may no longer be desirable to define such a signal region. Instead, the background in the signal region could be directly measured, and subtracted from the number of events observed in this region. Or, the signal region could be defined on a sliding scale, and the K+ —» ir+vH branching ratio could be found as a function of background level (which should be constant, if the background is well understood). Furthermore, bifurcated analyses become more difficult as the expected signal increases, because this signal wi l l significantly contaminate a background data sample when a single cut is inverted to define the data sample. Therefore, new background estimation techniques may be needed for analysis of E949 data. In order to measure the K+ —> TT+UU branching ratio and \Vtd\ to high precision ( ± 1 0 % ) , it is necessary to observe on the order of 100 K+ —> ir+vu events. To collect this data in a reasonable amount of time, kaon rates would have to be much higher than those used in E787/E949. However, the E787/E949 detector is l imited wi th respect to the kaon rate that it can handle, due to the need for detection of photons in order to suppress decays, and identification of the TT —> \x —> e decay sequence in order to suppress decays. Bo th 184 Chapter 7. Conclusions -6 g 1 0 o cr cn -lie"7 O c o m I* „ - 8 2*10 K+—>7vW Experimental Progress t ^ -9 10 - 1 0 10 -11 10 - 1 2 10 . Asano F 7 P 7 F Q d Q — ^ -T • 90% CL • Branch • Est, Sir • v 1995 result . upper L t m u ing Ratio igle-Event S< < jnsitivity Th > < is work > Stanc t terd Model R ange Existii E787 data . . . , , , , , , , , Rc K+ >ughly 10 events , , , 1980 1985 1990 1995 2000 Data Acquisition Year Figure 7.2: Experimental progress of the search for K+ —• TT+VV. The upper l imit on the K+ —> 7r+z/z/ branching ratio at the 90% confidence level is shown as triangles [61, 62], the measured branching ratio for the 1995 analysis [46] and this work [63] are shown as circles, and the estimated K+ —> TT+VV single-event sensitivities for the complete E787 data and the E949 experiment are shown as squares, each as a function of year in which data is acquired. The current theoretical prediction for the K+ —> TT+VV branching ratio [18] is shown as the hatched region. The point labelled "Asano" comes from an experiment which pre-dates E787 at B N L [61]. 185 Chapter 7. Conclusions of these requirements can cause large losses in acceptance when a high rate of accidentals is present. A new detector, wi th better rate capabilities and the abili ty to collect close to 100 K+ —> n+uP events in 2 years wi th a background level of about 10 events has been proposed, which would collect kaon decays in flight rather than requiring the kaons to stop before decaying. Because the momentum of a kaon decaying in flight is not known to the near-perfect precision of a stopped kaon, the in-flight technique of the proposed " C K M " experiment at Fermilab [64] may perhaps suffer from worse kinematic resolution of and K^2 decays, resulting in worse kinematic suppression of these decays, than the stopped-kaon technique of E787/E949. However, the muon and photon vetos of the C K M detector are better than E787/E949, which allows for the collection of kaon decays at a high rate. Beam-related backgrounds in the C K M experiment are suppressed v ia use of a high purity kaon beam (> 70%). Backgrounds arising from scattering of kaons, muons, and pions in detector materials can potentially be understood much better in the C K M experiment because the amount of material that these particles traverse can be varied. Furthermore, the C K M detector, similar to the E787/E949 detector, allows for redundant measurements of the kinematics of charged pion tracks, which in turn allows for high background suppression and reliable background estimation. However, while the in-flight technique of the C K M experiment has the potential for making high-precision measurements of the K+ — » n+uP branching ratio and \ Vtd\ in a short amount of time, it is not well-tested whereas the stopped-kaon technique of E787/E949 is. Therefore, the in-flight technique may face unanticipated challenges, particularly in the realm of background suppression. Experiment 787 at Brookhaven National Laboratory has been successful in observing one K+ —> n+uP decay in a sample of 6.7 bil l ion kaon decays. The positive identification of this extremely rare signal is made possible by excellent understanding of the E787 detector, and by excellent understanding and suppression of potential background processes for K+ —> "K+VV. 186 B i b l i o g r a p h y [1] M . Berman, The Reenchantment of the World, F i f th printing, Cornell University Press, Ithaca, New York, 1994 (pg. 147). [2] M . J . Veltman, Sci. Amer. 255, 76 (1986). [3] S. Glashow, Nuc l . Phys. 22, 579 (1961). [4] S. Weinberg, Phys. Rev. Lett . 19, 1264 (1967). [5] A . Salam, Elementary Particle Physics (Nobel Symp. No. 8), Almqvis t and Wilse l l , Stockholm, 367 (1968). [6] G . Arnison et al. (UA1 collaboration), Phys. Lett . 122B, 103 (1983). [7] cern.web.cern.ch/CERN/Announcements/2000/NewStateMatter [8] F . M a n d l and G . Shaw, Quantum Field Theory, John Wi ley and Sons, Toronto, 1993. [9] B . Greene, The Elegant Universe, W . W . Norton & Company, N . Y . , 1999. [10] Y . Fukuda et al, Phys. Rev. Lett. 81, 1562 (1998). [11] N . Cabibbo, Phys. Rev. Lett . 10, 531 (1963). [12] S.L. Glashow, J . Iliopoulos, and L . Maian i , Phys. Rev. D 2, 1285 (1970). [13] M . Kobayashi and T . Maskawa, Prog. Theor. Phys. 49, 652 (1973). [14] J . H . Christenson et al, Phys. Rev. Lett. 13, 138 (1964). [15] L . Wolfenstein, Phys. Rev. Lett . 51, 1945 (1983). 187 Bibliography [16] C . Caso et al. (Particle Da ta Group), "The Review of Particle Physics", European Physical Journal C3, 1 (1998). [17] C . Jarlskog, Phys , Rev. Lett. 55, 1039 (1985). [18] G . Buchalla and A . J . Buras, Nucl . Phys. B548, 293 (1999). [19] G . Buchalla and A . J . Buras, Nucl . Phys. B412, 106 (1994). [20] J.S. Hagelin and L . S . Littenberg, Prog. Part. Nuc l . Phys. D23, 1 (1989). [21] D . Rein and L . M . Sehgal, Phys. Rev. D39, 3325 (1989). [22] M . L u and M . B . Wise, Phys. Lett . B337, 133 (1994). [23] F . Wilczek, Phys. Rev. Lett . 49, 1549 (1982). [24] L . Silvestrini, e-print hep-ph/9906202. [25] www. ags.bnl.gov/~kbrown/seb/sebsetup. h tml [26] www.ags.bnl.gov/~kbrown/seb/overview.html [27] J . Doornbos et al, Nucl . Instr. Meth . A 4 4 4 , 546 (2000). [28] V . F i t ch and R . Motley, Phys. Rev. 101, 496 (1956). [29] M . S . A t i y a et al, Nucl . Instr. Meth . A321, 129 (1992). [30] P. Ki tch ing and T . Nakano, "The New Beam Instrumentation", E787 Technical Note No. 237 (1992). Unpublished. [31] D . A . Bryman et al, Nucl . Instr. Meth . A396, 394 (1997). [32] E . W . Blackmore et al, Nuc l . Instr. Meth . A404, 295 (1998). [33] T . K . Komatsubara, "Brief Summary of the Status of Upgraded-E787 in 1995", E787 Technical Note No. 305 (1995). Unpublished. 188 Bibliography [34] R . A . McPherson, "Chasing the Rare Decay K+ — > -K+VV" , Princeton University, P h . D . Thesis, November 1995. [35] I . -H. Chiang et al, I E E E Trans. Nucl . Sci. 42, 394 (1995). [36] T . K . Komatsubara et al, Nucl . Instr. Meth . A 4 0 4 , 315 (1998). [37] The version of U M C used in this analysis is version 6. [38] P. Meyers, " A Modified Version of the U M C Mul t ip le Scattering Routine M S C A T 1 " , E787 Technical Note No. 77 (1985). Unpublished. [39] A . J . Stevens, "Nuclear Interactions in C H revisited", E787 Technical Note No. 140 (1987). Unpublished. [40] W . R . Nelson et al, "The E G S 4 Code System", S L A C 265, S L A C (1985). [41] P . C . Bergbusch et al, " r r + i /p( l ) Analysis of the 1995-7 Da ta Set", E787 Technical Note No. 365 (2000). Unpublished. [42] H . Brafman et al, I E E E Trans. Nucl . Sci. , 32, 1 (1985). [43] M . Burke et al, I E E E Trans. Nucl . Sci. , 41 , 131 (1994). [44] S. Ket te l l and T . Komatsubara, "Some Analysis of the 1996 Da ta Set", E787 Technical Note No. 337 (1997). Unpublished. [45] S. Ket te l l , "Summary of 1997 Pas s l " , E787 Technical Note No. 338 (1997). Unpub-lished. [46] S. Adler et al, Phys. Rev. Lett. 79, 2204 (1997). [47] J .R . Stone, "Capturing the Rare Decay K+ —> n+ui>", Princeton University, P h . D . Thesis, November 1998. [48] F . Ajzenberg-Selove, Nucl . Phys. A 5 0 6 , 1 (1990). [49] S.C. Ful tz et al, Phys. Rev. 143, 790 (1966). 189 Bibliography [50] J . B . Birks , Proc. Phys. Soc. A 6 4 , 874 (1951). [51] H-S Ng , "Low Energy (K+,12C) Charge Exchange Cross Section Measurements, U n i -versity of Alber ta , M.Sc . Thesis, 2000. [52] A . Konaka et al, "Tr+vv Analysis Memo No. 4", E787 1995 Analysis Memo-4 (1997). Unpublished. [53] O. Couet, " P A W - Physics Analysis Workstation", C E R N Program Library entry Q 1 2 1 , C E R N (1993). [54] A . Konaka et al, " A note on the charge exchange background estimate", E787 1995 Analysis Memo-Cex (1997). Unpublished. [55] A . Konaka et al, " A note for the analysis checking committee (2): Results of a detailed examination", E787 1995 Analysis Memo-6 (1997). Unpublished. [56] A . S . Turcot, "The Search for the Decay K+ -» n+vu", University of Vic tor ia , P h . D . Thesis, 1994. [57] M . Diwan, "E787 Energy Resolution Studies Part 1", E787 Technical Note No. 362 (1998). Unpublished. [58] A . Konaka et al, "Background and acceptance studies", E787 1995 Analysis Memo-3 (1996). Unpublished. [59] S. Ferneyhough, "Some General Results on Poisson Statistics", E787 Technical Note No. 326 (1997). Unpublished. [60] J . Feng et al, Phys. Rev. D 5 7 , 5875 (1998). [61] Y . Asano et al, Phys. Lett . B 1 0 7 , 159 (1981). [62] S. Adler et al, Phys. Rev. Lett . 76, 1421 (1996). [63] S. Adler et al, Phys. Rev. Lett . 84, 3768 (2000). 190 Bibliography [64] R. Coleman et al, " A Proposal for a Precision Measurement of the Decay K+ --K+UU and Other Rare K+ Decay Processes at Fermilab Using the M a i n Injector' www.fnal.gov/projects/ckm/documentation/public/proposal/ckm_proposal.ps.gz [65] A . Desjardins, "Study of Range Stack d E / d x Calibrations and Likel ihood Analysis ' E787 Technical Note No. 363 (1998). Unpublished. [66] S. Wi lks , Mathematical Statistics, J . Wi ley and Sons, 1962. 191 Appendix A B N L E787 Collaboration S. Adler , M . S . At iya , I -H. Chiang, M . V . Diwan, J.S. Frank, J .S. Haggerty, V . Jain, S .H. Ket te l l , T . F . K y c i a , K . K . L i , L . S . Littenberg, C . Ng , R . C . Strand, C . W i t z i g Brookhaven National Laboratory M . Kazumori , T . K . Komatsubara, M . K u r i k i , N . Muramatsu, A . Otomo, S. Sugimoto K E K , Tanashi-branch T. Inagaki, S. Kabe, M . Kobayashi, Y . Kuno , T . Sato, T . Shinkawa, Y . Yoshimura K E K , Tsukuba Y . K i sh i , T . Nakano, T . Sasaki Osaka University M . Ardebi l i , A . O . Bazarko, M . R . Convery, M . M . Ito, D . R . Marlow, R . A . McPherson, P . D . Meyers, F . C . Shoemaker, A . J . S . Smith, J .R . Stone Princeton University M . A o k i , E . W . Blackmore, P . C . Bergbusch, D . A . Bryman, A . Konaka, J . A . Macdonald, J . Mildenberger, T . Numao, P. Padley, J . - M . Poutissou, R . Poutissou, G .Redl inger T R I U M F P. Ki tch ing and R . Soluk University of Alberta 192 Appendix B Personal Contributions Due to the highly collaborative nature of particle physics experiments, it has become customary in P h . D . theses to include some information on the major contributions of the author to the experiment. The duties/contributions of the present author to the E787 experiment in the years 1995-2000 include: • hardware — tuning and maintenance of the full C C D system, including target, E C , barrel, and beam detector C C D ' s — maintenance of the U T C and B W C gas systems — maintenance of the U T C post-amps, A D C ' s , and T D C ' s — maintenance of the B W C post-amps — tests of the demultiplexed B W C l — feasibility study of implementing T D C ' s on the R S • data acquisition — maintenance of the SSP's — maintenance of the computer system used in data acquisition, and some computer system administration — acquisition and storage of monitor data 193 Appendix B. Personal Contributions • calibrations - al l T D time calibrations: R S , R S flags, E C , IC, B4, Cerenkov, and strobes - T D pulse height —> M e V calibration — T D pulse shape calibrations: R S , B4 • processing — check of al l calibrations required at P A S S l - development of P A S S l source code and run scripts - verification of the quality of P A S S l data reduction - tape-loading at T R I U M F , and coordination of parallel P A S S l processing using machines at Princeton, B N L , and in Japan • analysis — elucidation and suppression of T D backgrounds — outside-the-box studies —> G D R background correlation mechanism — all aspects of the acceptance and sensitivity measurements, except U M C • calculations - Kn2, K+ —+ TT+VV, and K+ —• n+f branching ratios/upper limits -— allowed mass range of the familon 194 Appendix C Detailed Cut Descriptions .1 P A S S 1 The PASS1 cuts, in order of application, are described below. • T R B I T : This is an offline reproduction of the online K+ —> n+ui> trigger requirement (see section 3.3). • R D _ T R K : This track reconstruction cut requires that a track be found in the R S . This is the first reconstruction cut that is applied because accidental hit rates in the detector are lowest farthest from the beamline. Only one positively-charged track is found, beginning at the T • 2 hit closest in time to the detector strobe (i.e., closest in time to the T • 2 hit found by the trigger), and extending at least to layer 6. Gaps between R S layers are allowed at sector crossings occurring up to layer 12. The t iming of hits comes from the T D ' s , and each hit must have a min imum energy (0.1 M e V for the T counters; 0.5 M e V for layers 2 - 2 1 ) . • S T L A Y : For the track which satisfies the R D _ T R K cut requirements, the stopping layer and hextant as found by the offline track reconstruction must be the same as those found by the trigger. The stopping layer found by the trigger is the outermost layer wi th a "charged-track" {ci) hit (see section 3.3), so this cut rejects highly-curved tracks which cross hextants in the stopping layer. For example, muon tracks from 195 Appendix C. Detailed Cut Descriptions decays can cross hextants in layer 21, due to high track curvature or re-entrance of the track muon or decay-electron into the R S from the barrel. This cut also rejects KW2 decays which have non-contiguous energy deposition in the R S due to a charged track at low layers and a photon splash at higher layers in the ct sectors. Events with undefined T D times in the stopping counter are also rejected. • R S H E X : For the track that satisfies the R D _ T R K cut requirements, if there is a hit in the stopping hextant and layer with more than 1 M e V A D C energy which is not part of the track fit, then the event is rejected. This cut rejects decays which may have a false n —• JJL decay signature. This can happen because R S counters in the same hextant and layer are multiplexed together into the same T D (see section 3.2.4), so if the T D flags fail for some reason, the accidental (non-track) hit in the stopping layer could be assigned to a second, later hit the stopping counter, thereby faking the ix —> p decay signature. This cut also rejects decays where a photon converts near the charged pion track. • T R K T I M : For hits on the R S track that satisfies the R D _ T R K cut requirements, T D time information must be available such that an average track time, tp_s, can be found. Otherwise the event is rejected. • INTIME: This cut rejects events which have hits in the R S which occur at track time, but are located outside of the track, and sum to more than 10 M e V . This cut primarily rejects decays which have photon energy in the R S . • FITPI: This is the lowest-level muon/pion particle identification cut in the R S , which requires that a crude 7r — > p double-pulse signature be present in the stopping counter. Loose conditions are imposed on the x2 °f the double-pulse fit; the muon energy recorded on each end of the stopping counter, El^ and E2U, as well as the geometric mean muon energy, E^ = ^El^ • i?2 M ) ; the pion decay time; and the muon z relative to the pion z, where z comes from end-to-end t iming, i.e., tl — t2, where tl and t2 are the times of a pulse recorded on the upstream and downstream ends of a R S counter, 196 Appendix C. Detailed Cut Descriptions respectively. This cut rejects decays. • U T C / R A N G E / T A R G E T : This track reconstruction cut requires that a track be found in the U T C which links up wi th the track in the R S . Track reconstruction in the target is also attempted in order to refine the track fit in the U T C , but the specific target track reconstruction requirements of the T A R G E T cut are disabled at P A S S 1 . The (x, y) projection of tracks in the U T C is reconstructed using the times of anode-wire hits which, for a constant electron drift time in the gas, are fit to circular isochrones (see figure 4.11). The polar angles of tracks in the U T C and the z component of the particles' momenta are found from linear fits to the (r, z) coordinates of induced charge on the helical cathode strips. More information on the U T C track-fitting can be found elsewhere [32]. The U T C track that points to the T • 2 sector of the R S track is found, followed by reconstruction of the track in the target, using extrapolation of this U T C track back into the target. Track hits in the target ("pion fibers") are found in a 2-cm-wide "swath" centered on the U T C extrapolation, and kaon hits in the target ("kaon fibers") are found in clusters which touch the swath, based on a likelihood function which uses the fiber energy, time, and distance from the UTC-extrapola ted track to assign pion fibers (see figure 4.12). The energy, time, and distance likelihoods are assigned based on an empirical calibration. Photon hits in the target ("gamma fibers") are defined as hits which are not assigned to the track, but lie wi thin ± 6 ns of the average time of the pion fibers (t^), and are closer to t^ than they are to the average time of the kaon fibers (.#). The kaon decay vertex is defined as the center of the kaon fiber on the swath which is farthest from the target-entry kaon fiber. The (x, y) coordinate of target entry is given by the (x, y) coordinate of the kaon in the B4 hodoscope. If a track is successfully reconstructed in the target, the U T C fit is re-done using the additional (x, y) coordinates of the target pion fibers. If this fit fails, the original fit (without the pion fibers) is re-done. Detailed range and energy measurements of the track in the R S are performed. The 197 Appendix C. Detailed Cut Descriptions A D C energies of R S track counters are compared to T D pulse heights: if the A D C energy/TD P H ratio is anomalously large, this indicates the presence of an accidental hit in the A D C gate, because maximum pulse height (as opposed to total pulse area) is fairly insensitive to the presence of multiple (low energy) pulses. In this case, for all track counters excluding the stopping counter, the P H energy is used in place of the A D C energy. In the stopping counter, a second pulse from the pion-decay muon is expected, so the A D C energy, minus the muon energy as inferred from the n —* p double-pulse fit required by the F I T P I cut, is always used. These energies are summed, and corrections are made for energy losses in dead material (RS scintillator wrapping and the RSSC ' s ) , as well as saturation of ionizing energy in the plastic scintillator, to get the total energy deposited by the track in the R S . The range of the track in the R S is found by fitting the (x, y) coordinates of the track counters, sector crossings, and R S S C hits, and the stopping-counter energy, to a Monte Carlo track (see figure 4.13). The Monte Carlo track is composed of numerous circular segments, found by propagating the track in steps of 3 M e V energy loss through the R S , for given values of track momentum and the angle in the (x, y) plane between the U T C and R S tracks, at the R S entry point. The R S is treated as a solid block of scintillator, and the momentum and relative angle in (x, y) into the R S are treated as free parameters. The Monte Carlo track which gives the best fit to the R S data for given ini t ia l momentum and relative angle in (x, y) is used to get the range of the track in the R S up to the stopping counter. The range in the stopping counter is estimated from the measured energy deposit in this counter. In the above analysis sequence, if no tracks are reconstructed in the U T C (due to failure in finding an isochrone, or an ambiguity in the z fitting), or no match is found between the U T C and R S tracks, or the range calculation in the R S fails for some reason, then the event is rejected. • PDC: Events where the track momentum measured in the U T C is greater than 280 M e V / c are rejected. These events can arise from beam pions scattering into the de-198 Appendix C. Detailed Cut Descriptions tector, and are very unlikely to arise from K+ —> ir+vv (for which the maximum pion momentum is 227 M e V / c ) . • LAY14: Tracks which stop in layer 14 and are pointed upstream in the polar angle 120° < 9 < 180° (cos 0 < —0.5) are rejected. This is because the support structure for the R S lies in this region and could "hide" track energy. It could also hide accidental activity, which may result in an isolated second pulse in the stopping counter causing a fake n —• fj, decay signature. A t P A S S 1 , the total range, energy, and momentum of a track are also calculated. The range in the target is defined as the distance along the UTC-f i t t ed track, extrapolated from the point of closest approach to the kaon decay vertex, to the edge of the target. Similarly, the range in the IC is defined as the pathlength of the U T C track extrapolation through the IC. The total range is then the R S range (as described above), plus the target and IC ranges, plus the equivalent range in plastic scintillator for the track in the U T C (i.e., conversion of the U T C inner wall , cathode foils, gas volume, and outer wall into an equivalent interaction length of plastic scintillator). The total energy of the track comes from the energy in the RS (as described above), the measured A D C energy in the target, the energy in the IC (as described below), plus a correction for energy loss in the dead material of the U T C (inner wall, foils, gas volume, and outer wall). The A D C and T D energies in the IC come from the IC which lies along the U T C track extrapolation. If a second IC close in azimuth to the UTC-extrapolated track ( -0 .4 < A 0 < 0.1 radians) has a hit wi th less than 3 M e V A D C energy within ± 3 ns of track time, then the track likely hit two adjacent IC's and the A D C energy from the second IC is added to the A D C and T D energies of the first IC. Also, if a V C close in azimuth to the UTC-extrapolated track ( | A 0 | < 0.55 radians) has a hit wi th less than 2 M e V A D C energy within ± 3 ns of track time, and the kaon stopped deep in the target (at z > 10 cm, where the center of the U T C is at z — 0 cm) giving rise to a U T C track wi th polar angle near 90° (-0.1 < cos (9 < 0.3), then the track likely hit a V C and the A D C energy from this V C is added to the A D C and T D energies of the track IC. The resulting, corrected A D C energy for the IC is used unless (1) the expected energy in 199 Appendix C. Detailed Cut Descriptions the IC, from the UTC-extrapolated range in the IC and empirical tables of pion dE/dx in scintillator, differs from the corrected A D C energy by more than 1 M e V , but differs from the corrected T D energy by less than 1 M e V , or (2) tRS differs from the IC T D C time by more than 5 ns, but differs from the IC T D time by less than 5 ns. In either case the corrected T D energy for the IC is used. The total momentum comes from the momentum measured in the U T C , converted to range in scintillator for a particle of this momentum, using empirical tables of pion or muon range in scintillator as a function of total energy. This range is added to the range in the target, IC , inner U T C wall and half the U T C gas volume as calculated from an extrapolation of the U T C track through these materials. This total range is then converted back to total momentum using the same empirical tables of pion or muon range in scintillator as a function of total energy. C .2 P A S S 2 • P V C U T : This "photon veto" cut primarily rejects decays. Events are rejected that have more than 2 M e V (visible) in the barrel in a ± 2 ns window, or more than 3.5 M e V in the E C or R S (excluding the track energy) in a ± 1 ns window around track time. • T G P V C U T : This "target photon veto" cut primarily rejects Kw2 decays. Events are rejected that have more than 5 M e V (total) in target gamma fibers in a ± 1 ns window around track time. • T G P V T R : This cut is a loose version of T G P V C U T for study of beam background pathologies in the target. Events are rejected that fail T G P V C U T only if they (1) satisfy the n+uu(2) trigger, or (2) have tn — tx < 2 ns but don't satisfy the ix+vv(2) trigger. • T G R E C O N : This cut is identical to the P A S S l T A R G E T cut, and requires that a track be reconstructed in the target. The T G R E C O N cut rejects events that have no 200 Appendix C. Detailed Cut Descriptions kaon cluster in the target due to failure in finding the kaon hits, or have more than 5 kaon-like clusters, or have more than 150 fibers in one kaon cluster, or have no kaon clusters connected to the swath. Events are also rejected that have no pion track along the swath if the kaon cluster is away from the target edge. • T G C U T : This "target" cut rejects events which are poorly reconstructed in the target or IC . Events are rejected that fail any one of the T G R E C O N , T I M C O N , E I C , or T I C cuts (see section C.3.2 for a description of the latter 3 cuts). • P S C U T : This "pion scattering" cut primarily rejects events arising from scattering of beam pions. Events are rejected that fail a loose version of any one of the B 4 D E D X , B 4 T R S , or C P I T R S cuts (see section C.3.6 for a description of the standard versions of these cuts). That is, events are rejected that have small B 4 energy deposition (less than 1.2 M e V , which is indicative of a beam pion), B4 activity at track time (within ± 1 . 5 ns of tRs — 0.5 ns), or Cn activity at track time (within ± 1 . 0 ns of tRs, wi th at least 5 pion tubes firing). • R S H E X 2 : Events are rejected if the track crosses sectors in the stopping layer, where the sectors are in the same hextant and each counter has a hit wi th more than 1 M e V A D C energy. Similar to the R S H E X cut (see section C l ) , this cut rejects decays which may have a false rr —• p decay signature, because the track hit in the non-stopping counter gets assigned to a second, later hit in stopping counter, thereby faking the TX —> LI decay signature and simultaneously decreasing the track energy. • T D C U T : This " T D " cut is a loose version of the E L V E T O cut (see section C.3.4) and rejects decays that pass the F I T P I cut (see section C l ) because an accidental hit provided the second pulse in the stopping counter, thereby faking the TJ —> LL decay signature. This accidental may be present as a splash of energy in several neighbouring R S counters, so these events are rejected if there is at least one hit w i t h i n . ± 2 . 5 ns of muon time, in a R S counter within ± 1 sector of the stopping counter, which has good T D data from each end of the counter. 201 Appendix C. Detailed Cut Descriptions C.3 PASS3 C.3.1 Kinematic Pathology Cuts For each kinematic pathology cut, the cut position is set by examining the distribution of the cut variable in a sample of events which passes al l other kinematic pathology and function cuts. For kinematic pathology cuts attacking decays, S K I M 1 events are used which fail the photon veto function cut, and pass the T D function cut and the beam pathology and function cuts (see subsequent sections for descriptions of these cuts). The E B O X cut on the Kw2 (low) side is not applied in order to enhance statistics, which implici t ly assumes that the l imit ing Kn2 background events have a correlated mismeasurement of range and momentum, and an uncorrelated upward fluctuation in energy, placing them in the PNN(1) signal region. For kinematic pathology cuts attacking Ka2 background, S K I M 2 events are used which fail the T D function cut, and pass the P V function cut and the beam pathology and function cuts. The P B O X cut on the (high) side is not applied in order to enhance statistics, which implici t ly assumes that the l imit ing Ku2 background events have a correlated downshift in range and energy, and an uncorrelated mismeasurement of momentum, placing them in the ir+vv(\) signal region. The kinematic pathology cuts are described in more detail below. • U T C Q U A L : A U T C track-fitting likelihood is formed from the number of (x,y) coordinates (anode wire hits) and number of U T C layers used in the fit, as well as the number of unused anode wire hits which are wi thin 1.5 cm of the U T C track. For 1995 data the likelihood also includes the number of z coordinates (cathode strip clusters) used in the fit. For 1996-7 data, the U T C z coordinates are not input into the U T C Q U A L likelihood; instead, they are used in the P R O B Z cut (see below). Events are rejected for which the U T C Q U A L likelihood value is < 10~ 5 . • P R O B Z : Events are rejected which have a x2 probability < 10~ 6 for the fit of U T C z hits to a straight track. This cut was designed in the late stages of an earlier 1995 202 Appendix C. Detailed Cut Descriptions analysis [47], and subsequent investigation revealed that it may have been tuned on a small number of events, leading to a biased cut position. However, the (normalized) K^2 background in the 1/3 1996-7 data sample is found to be larger than that in the 2/3 1995 data by almost a factor of 2, presumably due to the use of narrower T D pulse shapes in 1996-7 for the TJ —>• ui double-pulse fit in the stopping counter, which increases the probability of a a successful double-pulse fit (see section C.3.4). Furthermore, the P R O B Z cut's performance was confirmed on the untested 2/3 1996-7 data, so any bias in the cut is likely to be small. Therefore, the cut is applied to 1996-7 data to help reduce background, but to be conservative wi th respect to potential bias, this cut is not applied to the 1995 data. • ZUTOUT: Events where the UTC-extrapolated track has \z\ > 25 cm at the outer radius of the U T C (such that the track exits the side of the U T C ) are rejected, because these tracks can lose energy in the dead material of the U T C electronics. • L A Y V 4 : The track must stop in one of R S layers 11 through 18 inclusive. Events wi th lower stopping layers have a high contamination of Kn2 decays; events wi th higher stopping layers have a high contamination of decays. • COS3D: Events where the track is pointed downstream or upstream in the polar angle 0° < 9 < 60° or 120° < 9 < 180° (| cos<9| < 0.5) are rejected. This is because tracks at these angles are likely to enter the dead material of the R S support structure and therefore have "hidden" energy. Accidentals can also enter from the support structure and leave an isolated pulse in the exit counter, thereby faking the double-pulse TJ —> \x decay signature in the stopping counter. decays which fail the C O S 3 D cut are therefore more likely to be downshifted in range and energy towards the TT+UU(1) signal region, and more likely to pass the T D cuts. That is, K^2 decays which fail the C O S 3 D cut can correlate T D and kinematic cuts, so these decays must be removed from a data sample before bifurcating the T D and kinematic cuts to measure the background (see sections 4.2 and 4.4.2). 203 Appendix C. Detailed Cut Descriptions • ZFRF: Events are rejected at large z in the R S , because the track may have entered the dead material of the R S support structure, which is a potential mechanism for correlation between T D and kinematic cuts in the background estimate (similar to the C 0 S 3 D cut above). Specifically, if the track in the R S has \z\ > 35 cm (for stopping layer = 11 or 12), \z\ > 40 cm (stopping layer = 13), \z\ > 30 cm (stopping layer = 14), or \z\ > 50 cm (stopping layer = 15, 16, 17 or 18), where z comes from extrapolation of the U T C track to the stopping layer, then the event is rejected. • LAYER14: Events are rejected which appear to stop in layer 14 of the R S , but also have a hit in the outer R S S C in the stopping sector or one sector clockwise. This cut removes events which potentially lose a large amount of energy in the R S S C (this energy is not included in the track energy). It also removes events which potentially curl around and re-enter layer 14, causing a fake double-pulse rr —> p decay signature in the stopping counter. These events can correlate T D and kinematic cuts, so they must be removed when making the background estimate (similar to the C O S 3 D cut above). • CHIRF: This cut removes events which have "kinked" R S tracks due to scattering in one or both of the (re, y) and (r, z) planes. The (x, y) part of the cut removes events where the Monte Carlo track fit in the R S (see U T C / R A N G E / T A R G E T in section C l ) gives a x2 probability < 0.01 (1995 data) or < 0.05 (1996-7 data). The cut was tightened in 1996-7 due to the increase in K^2 background in 1996-7 mentioned in the P R O B Z cut description above. The 1995 cut position was found using peak events (see figure 4.1) from S K I M 2 , but it was later found that muon band events have a worse ta i l in C H I R F (x, y) probabilities, possibly due to the fact that muon band events need undergo less violent scatters than peak events in order to get into the TT+VD(1) signal region. Therefore, the 1996-7 cut position was found using muon band events, which are statistically enhanced i n the S K I M 2 data sample by loosening the P V function cut. The (r, z) part of the cut comes from the residuals between the z position of the track calculated from end-to-end t iming in each counter 204 Appendix C. Detailed Cut Descriptions of the R S (excluding the T counter, stopping counter, and counters in a layer where more than one sector is hit) and the z position as extrapolated from the U T C track. These residuals are divided by the end-to-end t iming resolution for each R S counter and summed in order to form a x2 quantity. For 1995 data, the end-to-end timing resolution of a R S counter, cr, is parameterized by u = a + eb+cE, where a, b, c are constants, and E is the geometric mean T D P H energy of the track in that counter. Events wi th a x2 probability < 0.01 are rejected. For 1996-7 data, a was found to be more appropriately defined as a — a + b • ecE for the same a, b, c from 1995, which resulted in a re-scaling of probabilities, such that to maintain the same acceptance as the 1995 version, the cut in 1996-7 rejected events wi th probabilities < 0.0001. Furthermore, the relationship between the z from the UTC-extrapola ted track and the z from R S end-to-end t iming was found to be non-linear at large z in 1995, so the residuals in z were based on a polynomial fit. However, further investigation showed that the relationship is actually linear, and that the non-linear behaviour was due to selection of events, based on z and polar angle, in the data sample used to calibrate the z residuals. Because a polynomial fit was used to find the x2 f ° r 1995 data, the C H I R F z cut is looser for 1996-7 data than it is for 1995 data. • CHIRF_NHZ: This cut involves calculation of another ^-fitting probability, this time from the residuals between the z of the UTC-extrapola ted track and the z from the cluster in each of the inner and outer R S S C layers which is closest in (x, y) to the U T C -extrapolated track. These residuals are divided by the z resolution for each R S S C and summed in order to form a x2 quantity. The z resolution of a R S S C is an empirical function of R S S C layer and the mean time and time difference of hits on the straws. Events are rejected that have < 5 z clusters from the U T C , and have either a R S 2 - f i t t ing x2 probability < 1 0 ~ 3 3 (see the C H I R F cut above) or a R S S C z-fitting x2 probability < 10~ 7 . A version of this cut was originally designed in the late stages of an earlier 1995 analysis [47] and, like the P R O B Z cut, was believed to be more biased than later observed. Therefore, similar to the P R O B Z cut, this cut is applied to 1996-7 205 Appendix C. Detailed Cut Descriptions data to help reduce KU2 background, but to be conservative wi th respect to potential bias, this cut is not applied to the 1995 data. • RSDEDX: The differences between the expected and measured energy deposits in each R S counter, assuming the particle is a pion, are divided by the measured energy resolution in each layer and summed to form a x2 quantity. The expected energy in each counter is found starting from the stopping counter and moving backwards, using the energy that the pion has entering the counter, the range in the counter calculated by the Monte Carlo propagation of the fitted track (see U T C / R A N G E / T A R G E T in section C l ) , and empirical tables of dE/dx for pions in scintillator. The energy res-olution in each R S layer is an empirical function of layer and stopping layer. Events are rejected which have x2 > 5 from any one counter (called the C H I M A X cut), or a combined x2 probability < 0.02 (the C L J I S D E D X cut). The logs of the differences between expected and measured energy are also combined into a likelihood, and events are rejected which have a likelihood value < I O " 9 (the R S L I K E cut [65]). The C H I -M A X cut is effective at removing KN2 decays which have photon energy on the track. KN2 decays which fail the C H I M A X cut are therefore more likely to be upshifted in energy towards the IT+VV{1) signal region, and more likely to pass the P V cuts. That is, K^2 decays which fail the C H I M A X cut can correlate P V and kinematic cuts, so these decays must be removed from a data sample before bifurcating the P V and kinematic cuts to measure the K^2 background (see sections 4.2 and 4.4.1). The C L _ R S D E D X and R S L I K E cuts are effective at rejecting muon tracks from K^2 decays. • TGDEDX: This cut rejects events wi th dE/dx in the target inconsistent wi th that of a pion. Events are rejected which have (1) RTG > 12 cm, or (2) ETG > 28 M e V , or (3) 9.5 • ETG > 28 • RTG, or (4) 10 • ETG < 21.5 • {RTG ~ 2), where RTG is the UTC-extrapola ted range from the kaon decay vertex to the edge of the target, and ETG is the total energy of the pion fibers. This cut is effective at removing events with lepton tracks in the target (e.g., K^2 and C E X background). • P IGAP: Events are rejected if the track in the target has a gap between neighbouring 206 Appendix C. Detailed Cut Descriptions pion fibers greater than 1.5 cm. This cut removes events which may have abnormally large range because the track particle travels in the air gap between rows of target fibers. • TGLIKE: Events are rejected if the combined likelihood value of all pion fibers in the target, based on fiber time, energy, and distance from the UTC-extrapola ted track (see U T C / R A N G E / T A R G E T in section C l ) is < 1CT 3- 2 . Events are also cut if the combined likelihood value using only the distances from the UTC-extrapolated track is < 10- 2 - 3 . • T G B 4 : This cut examines the matching of the (x,y) coordinates of the kaon hit in the B 4 hodoscope and the kaon cluster in the target. Events are rejected if (1) the distance between the B4 (x, y) coordinate and the nearest kaon fiber is > 2.0 cm, or (2) the distance between the B4 (x, y) coordinate and the t ip of the kaon cluster farthest from the kaon decay vertex is > 2.0 cm, or (3) the distance between the kaon decay vertex and the nearest kaon cluster t ip is > 0.7 cm, or (4) the distance from the kaon decay vertex to the closest pion fiber is > 1.5 cm. This last condition removes events wi th ambiguous, incorrect target entrance and kaon decay fiber assignments, which arise when the B4 hit position is located near the middle of the kaon cluster in the target. K a o n cluster "tips", where the (x, y) coordinate of the B4 hit is at the t ip corresponding to the point of target entry, and the kaon decay vertex is at the other tip, can be seen in figure 4.12. • EICKIN: The expected energy deposit of a pion in the IC is calculated using the energy of the pion (from U T C momentum), range in the IC (from a U T C extrapola-tion), and empirical tables of dE/dx for pions in scintillator. Events wi th measured IC energy greater than the expected energy by more than 1.5 M e V are rejected. This cut removes Kn2 decays wi th photon activity in the IC , thereby causing a correlated upshift in track energy and momentum towards the 7r + ^ i / ( l ) signal region. These quan-tities are bifurcated and therefore assumed to be uncorrelated in the Kn2 background estimate (see sections 4.2 and 4.4.1). 207 Appendix C. Detailed Cut Descriptions • MASS: This cut attacks KU2 decays which can give rise to G D R background due to inelastic scattering of muons in the R S (see section 4.1.1.1). When analyzing K+ —> •K+UU data, the momentum of the muon from K^i decay is reconstructed at 238 M e V / c (instead of the accepted value of 236 M e V / c [16]), because the track is assumed to be that of a pion. That is, the total momentum of a track is found using the U T C -extrapolated range of the track in the target, IC, inner U T C wall and half the U T C gas volume, converted to momentum assuming that the track particle is a pion (see section C l ) , so the momentum loss in these materials is overestimated, which, when added to the momentum measured in the U T C , causes the total momentum to be overestimated. The kinetic energy of a muon is 152 M e V , which is downshifted to < 130 M e V if the muon excites the G D R in carbon. Therefore, potential G D R background events are rejected if M = ( ( P 2 - E2)/(2E) > (238 2 - 130 2 ) / (2-130) = 153 M e V / c 2 i.e., ( ( P 2 - E2)/(2E) > 1 5 3 M e V / c 2 The M A S S cut is applied as a safety cut only to the 1996-7 data for the reasons outlined in section C.3.4. C.3.2 Beam Pathology Cuts • T G C C D P F : In the C C D data of any kaon target fiber, if a second pulse is found whose time relative to the first pulse is within ± 5 ns of tn — tx and whose energy is more than 5 M e V , then reject the event. This cut removes events which may have a significant fraction of the track energy hidden in kaon fibers, because the A D C energy of pion fibers is defined as the track energy in the target. • EPITG: Events which have more than 5 M e V A D C energy in any target pion fiber are rejected. These events likely have K^i photon or accidental energy overlapping the track energy in the target. 208 Appendix C. Detailed Cut Descriptions • E P I M A X K : The circular U T C track defines an azimuthal angle LO in the (x, y) plane about the center of the U T C track, as shown in figure 4.12 and described in A p -pendix D . The kaon fiber wi th the maximum value of LO, uoKiax, is likely to be the kaon decay fiber. This cut requires that pion fibers close to this kaon fiber, specifically, with uo in the range toKiax to uoKulx + 0.06 radians, each have < 3 M e V energy. Otherwise, it is likely that there is kaon energy overlapping the pion (track) energy, and the event is rejected. • P H I V T X : The value of u^™ (smallest LO for a kaon fiber) minus uo™m (smallest LO for a pion fiber, which wi l l be negative for any pion fibers on the non-track side of the kaon decay vertex) must be < 0.0028. Otherwise the event is rejected. This cut rejects events which have back-to-back tracks in the target (e.g., Kn2 decays wi th photon conversion in the target, or K°L —• TT+1~VI events). • PHIVTX2: The value of LOD^CAY (uo of the kaon decay fiber) minus uo™in must be < 0. Otherwise the event is rejected. Lod^cay is usually close to 0, so, similar to the P H I V T X cut, this cut rejects events wi th pion fibers on the non-track side of the kaon decay vertex. This cut was designed only after backgrounds had been measured using both the 1/3 and 2/3 1995 data samples, so it is used in the beam background estimate only for 1996-7 data. It is applied purely as a safety cut for 1995 data. • OPSVETO: Target fibers on the non-track side of the kaon decay fiber, which have hits wi thin ± 4 ns of the target pion time, must have a combined energy E^G < 2 M e V . Otherwise the event is rejected. This cut rejects K^2 events, similar to the P H I V T X and P H I V T X 2 cuts. • OPSVETO_LKB: Events are rejected if E^G > 0.5 M e V and Ikbeam < 200 (see the B 4 E K Z cut below). This cut was designed only after backgrounds had been measured using al l of the 1/3 and 2/3 1995 and 1996-7 data samples, and addresses a suspected C E X background event found in the 2/3 1996-7 data which only fails the T G D E D X cut (see sections 4.4.4 and 4.6). This cut is therefore not applied to the 1995 data, and 209 Appendix C. Detailed Cut Descriptions is applied as a safety cut to the 1996-7 data. The requirement Ikbeam < 200 includes most of the beam background, and E^G > 0.5 M e V has as low an energy threshold as possible to generalize the cut. • T G E D G E : Events are rejected which have more than 4 M e V energy in any target edge fiber wi thin ± 5 ns of track time. Due to multiplexing, the edge fibers are not used in the target reconstruction or the track energy calculation, which is valid as long as the edge fiber energy is small. • T G Q U A L T : This cut is identical to the T G R E C O N cut (see section C.2). • T G E R : Events where the pion has small energy but substantial range in the target, ETG < 1 M e V when RTG > 2 cm, are rejected. These target tracks may be muon tracks, or may have energy hidden in target dead material. • T A R G F : Events wi th more than 0.6 cm between the kaon cluster and nearest pion fiber in the target are rejected, because the pion track may not arise from decay of this kaon. • D T G T T P : This cut requires consistency between two slightly different extrapolations of the U T C track to the target edge. • RTDIF: The path length of the pion in the target is calculated twice, using a U T C extrapolation from the two tips of the kaon cluster to the target edge. Half the differ-ence in these two path lengths represents the maximum error in target range due to kaon decay vertex assignment. If this error is > 1.5 cm, the event is rejected. • DRP: The maximum distance spanned by the pion fibers perpendicular to the tangent to the U T C track at u = 0 can be no more than 30% of the total UTC-extrapolated range in the target. Otherwise the event is rejected. This cut rejects events with kinked tracks in the target due to pion scattering, which can lead to correlated effects in track range, energy and momentum. 210 Appendix C. Detailed Cut Descriptions • T IMCON: This cut requires consistency between the time of the kaon as detected by the target, tx, and the time of the kaon as detected by the B 4 hodoscope, tBAstrobe (must be wi thin ± 4 ns of each other). This cut also requires consistency between the time of the pion as detected by the target, tn, and the time of the pion as detected by the R S , tRs (must be wi thin ± 5 ns of each other). • TIC: The IC T D C time or the IC T D time must be wi th in ± 5 ns of tRS. Otherwise the event is rejected, because the IC energy may not be due to the track, leading to correlated mismeasurements of track energy and momentum. • T G C C D : This cut rejects events wi th misassigned kaon fibers, by comparing C C D times of the kaon fibers, txcco, to the R S track time and the B 4 strobe time. Specifi-cally, for each kaon fiber, 2 • txccD — tp.s — tBAstrobe < 0 ns is required, and txccD must be wi thin ± 3 . 5 ns of tBAstrobe- Otherwise the event is rejected. • EIC: Similar to the E I C K I N cut in section C.3.1, the measured A D C energy in the track IC must be wi thin ± 5 M e V of .the expected IC energy, or the measured T D en-ergy must be wi thin ± 4 M e V of the expected energy. Otherwise the event is rejected, because the IC energy may not be due to the track, leading to correlated mismeasure-ments of track energy and momentum. • KIC: This cut addresses events where the kaon cluster is near an IC, and this IC has a T D C hit at kaon time. These events can arise from a combination of 2 beam particles, where the first particle enters the target and stops in an IC , wi th no decay products detected, and the second particle enters the target from this same IC and decays in flight or exits the target through a different IC . Because the first particle stops in the IC, it leaves a large energy pulse in the IC which, after discrimination, can be up to 80 ns wide. The IC hit from the second beam particle is therefore "masked out" for up to 80 ns after the IC hit from the first beam particle. These types of events tend to correlate target and B4 cuts, because the second beam particle enters the target through a dead IC, such that it doesn't go through the B4 hodoscope, and 211 Appendix C. Detailed Cut Descriptions is reconstructed as the decay product of the first beam particle in the target. Target and B4 cuts are bifurcated and therefore assumed to be uncorrelated in the double-beam background estimate (see sections 4.2 and 4.4.3), so these events are found and removed if there are IC T D C hits within ± 3 ns of tx when the kaon fibers are within 3.7 cm of the IC and are located at the edge of the target (at radii > 5.2 cm). If the IC is on the track, then the event is removed only if the IC hit is closer to kaon time than track time, i.e., tjc < {tx + *BS)/2 ns. • t g g e o : Double-beam background events are rejected which have the following sig-nature: (1) both particles enter the target near the target edge or from an IC, and (2) the first particle or its decay product leaves a large energy pulse in the IC, such that the IC T D C hit of the second target-entering particle is "masked out". This cut removes events which can correlate target and B4 cuts, similar to the K I C cut above. The events addressed specifically by T G G E O were recognized after background mea-surements had already been made on both the 1/3 and 2/3 1995 data samples, so this cut is used in the beam background estimate only for 1996-7 data. It is applied purely as a safety cut for 1995 data. Events are rejected if pion fibers which lie within 1.0 cm of the minimum-a> pion fiber are located at the target edge (at radii > 4.5 cm), and the nearest IC to these pion fibers is not related to the RS track and has a hit with E > 3 M e V energy at time tic < tRs + 5 ns. The IC condition is also applied to the next-nearest IC if it is within 1.5 cm of the pion fibers (i.e., the pion fibers are near an IC sector boundary). Events are also rejected if kaon fibers which lie within 1.0 cm of the kaon decay fiber are located at the target edge (at radii > 4.5 cm), and the nearest IC to these kaon fibers is (1) not related to the RS track and has a hit with either (a) tic within ± 3 ns of tx, or (b) E > 3 M e V at tic < tRs + 5 ns; or (2) related to the RS track and has a hit within ± 3 ns of tx, and is closer to kaon time than track time, i.e., tic < {tx + tfis)/2 ns (this is the same condition as the K I C cut). Conditions (1) and (2) are also applied to the next-nearest IC if it is with in 1.5 cm of the kaon fibers (i.e., the kaon fibers are near an IC sector boundary). 212 Appendix C. Detailed Cut Descriptions • B 4 E K Z : A kaon likelihood is formed from the energy deposited in the B4 hodoscope, the UTC-extrapolated kaon stopping z in the target, and the kaon energy in the target. The likelihood measure Ikbeam is calculated using empirical tables of the correlation between these 3 quantities for kaons. Events are rejected if Ikbeam < 2. • B 4 E K Z J C : If there are no pion fibers in the target, then the event is rejected if Ikbeam < 15. These events may have kaon energy in the IC, leading to correlated mismeasurements of track energy and momentum. • T G Z F O O L : This cut ensures that the kaon enters the target. Events are rejected if the kaon z stopping position (from a U T C extrapolation) is farther upstream than — 15 cm. These events can have poor B4 and target information, and therefore poten-tially correlate B4 and target cuts which are bifurcated and therefore assumed to be uncorrelated in the double beam background estimate (see sections 4.2 and 4.4.3). • B H T R S : Events are rejected which have activity at the beam hole counter within ± 5 ns of track time. This cut rejects double-beam background, where the track-time beam particle may be located in the beam halo, missing the wire chambers, B4 hodoscope, and target. C.3.3 Photon Veto Function The P V function cut is optimized for rejection at different values of acceptance using, as far as possible, "pure" data samples of Kn2 and decays. The K^i data sample is the 1/3 1995 SKIM1 data, with beam and kinematic cuts applied to isolate events in the kinematic peak. The data sample is the 1995 K^X) monitor data (see section 3.3), with beam and kinematic cuts applied to isolate the kinematic peak, and a layer 21 veto to remove events which exit the RS and leave energy in the barrel. Because the SKIM1 data already has online, PASS1, and PASS2 P V cuts applied (online B V + E C , H E X ; PASS1 INTIME; and PASS2 RSHEX2, T G P V C U T ) , these same cuts are also applied to the K^l) monitor data. 213 Appendix C. Detailed Cut Descriptions The optimization consists of an automated search of the parameter space of time windows (around track time) and energy thresholds (minimum energy) for detection of hits, which are grouped into the 17 categories listed in table C l . Values of rejection for Kn2 decays are found simultaneously wi th values of acceptance for Ka2 decays using the data samples described above, and the time windows and energy thresholds are varied according to an optimization algorithm [58] which selects those that give the maximum rejection at a specified value of acceptance. Figure 4.14 shows the P V background level, Npy, as a function of the acceptance, Apy. Events with Npy < 1.0, Apy < 1.0 are defined to pass the P V function cut. This point was chosen to give a total photon veto acceptance of 0.80, including losses from all levels of P V . Measurements using Ku2(l) monitor data give a total acceptance of 0.90 from B V + E C , H E X , I N T I M E , P V C U T and T G P V C U T (see table 5.2). The APV < 1.0 position therefore corresponds to a P V function cut acceptance of 0.89. The 1995 parameters for this cut point are shown in table C.2, which give a P V function cut rejection and acceptance of 50.1 ± 1.1 and 0.893 ± 0.005, respectively. The P V function is therefore defined for the range Npy = [0,50] and is tabulated in table C.3. The P V rejection was confirmed wi th the 2/3 1995 S K I M 1 data, and the function was re-optimized for application to 1996-7 data using the 1/3 1996-7 data set, giving slightly more rejection than 1995 at the same acceptance loss point. C.3.4 TD Function Cuts The T D background (described in table 4.1) increased significantly in 1996-7 relative to 1995, mainly due to a z-dependent correction to the calibrated T D pulse shapes which are used for ix —> fi double-pulse fitting to T D data. The correction narrowed the shapes, and therefore increased the probability of a successful double-pulse fit at small pion decay times, t^. The acceptance of the F I T P I cut increased by about 6%, but the amount of tail-fluctuation background (resulting from double-pulse fits to a single large or abnormally-shaped pulse) also increased significantly. The narrower shapes may additionally be re-sponsible for the suspected increase in G D R background seen in 1996-7, because the G D R 214 Appendix C. Detailed Cut Descriptions detector cat. time and energy information barrel B V double-ended T D C , double-ended A D C B V 1 double-ended T D C , single-ended A D C B V 2 double-ended T D C , no A D C B V 3 single-ended T D C , double-ended A D C B V S single-ended T D C , single-ended A D C endcaps E C C C D , excluding upstream inner ring E C 1 C C D , inner ring of upstream E C only range stack R D double-ended T D R D 1 double-ended T D time, single-ended energy, not in track hextant R D 2 double-ended T D time, no A D C energy R D 3 single-ended T D time, double-ended energy, not in track hextant R D S single-ended T D time, single-ended energy, not in track hextant target T G C C D I-counters IC T D C and A D C V-counters V C T D C and A D C collars C O T D C and A D C micro collar C M T D C and A D C Table C l : Definitions of the categories of hits whose time windows and energy thresholds are optimized to get the best P V function cut rejection for a given value of acceptance. A "double-ended" quantity means that the quantity is recorded at both the upstream and downstream ends of a R S or barrel counter. In the R S and barrel, the upstream and down-stream times of a pulse must not differ by more than 14 ns in order for the pulse to be defined as having double-ended time information. In the R S , energy comes from the geometric mean of the upstream and downstream T D PH ' s . If one end has T D P H = 0 (the T D time on that end can st i l l be valid), then the energy comes from the T D P H on the other end. If both ends have T D P H = 0, then the A D C energy on each end is used. Therefore, single-ended R S energy is defined as single-ended A D C energy, and double-ended R S energy is defined as double-ended T D P H energy, single-ended T D P H energy, or double-ended A D C energy. "No A D C energy" means that there may have been single- or double-ended T D P H energy, but there was no A D C energy. 215 Appendix C. Detailed Cut Descriptions cat. time window (ns) E • J-'min (MeV) ( M - ) s (M„)f Rej. Acc . B V 2.25 ± 5 . 2 5 0.20 10977 1914 5.7 35334 34420 0.974 B V 1 8.00 ± 8 . 0 0 4.00 1923 1914 1.0 34433 34420 1.000 B V 2 3.50 ± 0 . 5 0 — 1918 1914 1.0 34428 34420 1.000 B V 3 -3 .50 ± 1.00 5.20 1919 1914 1.0 34426 34420 1.000 B V S -6 .50 ± 2 . 5 0 3.60 1914 1914 1.0 34420 34420 1.000 E C 0.25 ± 2 . 5 0 2.40 8615 1914 4.5 35089 34420 0.981 E C 1 0.25 ± 2 . 2 5 1.60 2545 1914 1.3 34756 34420 0.990 R D 1.25 ± 3 . 2 5 0.80 4403 1914 2.3 35666 34420 0.965 R D 1 -1 .50 ± 0 . 5 0 4.40 1920 1914 1.0 34436 34420 1.000 R D 2 -2 .00 ± 0.00 8.20 1914 1914 1.0 34420 34420 1.000 R D 3 -3 .00 ± 0.50 6.80 1918 1914 1.0 34427 34420 1.000 R D S -2 .00 ± 0 . 5 0 4.40 1914 1914 1.0 34421 34420 1.000 T G -0 .75 ± 2 . 7 5 3.40 2077 1914 1.1 34605 34420 0.995 IC -1 .00 ± 2 . 5 0 1.20 2563 1914 1.3 34577 34420 0.995 V C -2 .50 ± 3 . 5 0 1.20 2065 1914 1.1 34511 34420 0.997 C O 1.00 ± 2 . 0 0 1.00 2121 1914 1.1 34515 34420 0.997 C M 1.00 ± 2 . 0 0 1.00 1925 1914 1.0 34422 34420 1.000 Table C.2: P V function cut parameters and performance at the Npy < 1.0 cut position. Each cut category is defined in table C l . T ime windows (around track time) and energy thresholds (minimum energy) for detection of hits in each category are listed, along with the rejection and acceptance of each cut category for the time and energy parameters listed. Rejections are given by (M^) s / (Mn) f, where (Mn)f is the "final" number of K^2 decays which remain after application of the cut category to a "starting" number of Kn2 decays, (M„)s. Acceptances are given by ( M M ) S / ( M / J ) / , where ( M M ) / is the "final" number of Kp2 decays which remain after application of the cut category to a "starting" number of K^2 decays, (M^)s. The listed-time windows and energy thresholds result in a total P V function cut rejection and acceptance of 50.1 ± 1.1 and 0.893 ± 0.005, respectively. 216 Appendix C. Detailed Cut Descriptions NPV APV NPV APV NPV APV NPV APV 4 9 . 8 9 2 4 1 1 1 9 5 3 7 9 1 0 1 . 0 9 4 9 1 . 1 1 3 4 1 . 0 1 3 7 0 . 5 6 2 7 0 . 8 5 6 8 4 7 . 6 3 2 7 1 1 1 9 5 3 7 7 6 4 1 . 0 9 4 7 1 . 1 0 7 1 1 . 0 1 3 0 0 . 5 6 2 2 0 . 8 5 6 5 4 5 . 9 3 3 1 1 1 1 9 5 3 7 0 8 5 1 . 0 9 4 1 1 . 0 8 2 5 1 . 0 1 0 3 0 . 5 5 0 2 0 . 8 4 8 2 4 1 . 6 0 4 0 1 1 1 9 3 3 6 9 7 0 1 . 0 9 4 0 1 . 0 8 1 0 1 . 0 1 0 1 0 . 5 4 6 5 0 . 8 4 5 6 4 1 . 6 0 2 9 1 1 1 9 3 3 6 8 2 9 1 . 0 9 3 8 1 . 0 7 8 9 1 . 0 0 9 9 0 . 5 4 6 0 0 . 8 4 5 2 3 3 . 3 1 2 9 1 1 1 8 8 3 6 6 9 3 1 . 0 9 3 7 1 . 0 5 4 9 1 . 0 0 7 2 0 . 4 6 4 5 0 . 7 7 2 5 2 8 . 5 2 4 5 1 1 1 8 4 3 6 6 4 6 1 . 0 9 3 6 1 . 0 4 7 5 1 . 0 0 6 4 0 . 4 6 2 9 0 . 7 7 1 3 2 8 . 2 7 3 2 1 1 1 8 4 3 6 6 3 5 1 . 0 9 3 5 1 . 0 2 8 7 1 . 0 0 4 0 0 . 4 6 2 4 0 . 7 7 1 1 2 7 . 9 6 5 0 1 1 1 8 4 3 6 6 0 9 1 . 0 9 3 5 1 . 0 2 4 6 1 . 0 0 3 5 0 . 4 6 0 8 0 . 7 7 0 2 2 0 . 9 6 3 4 1 1 1 7 6 3 6 3 2 7 1 . 0 9 3 1 1 . 0 0 4 7 1 . 0 0 0 7 0 . 4 5 9 8 0 . 7 6 9 4 1 7 . 0 4 4 9 1 1 1 6 9 3 1 9 3 8 1 . 0 8 8 1 1 . 0 0 3 1 1 . 0 0 0 5 0 . 4 3 4 7 0 . 7 4 7 2 1 6 . 2 8 9 4 1 1 1 6 7 3 1 3 5 3 1 . 0 8 7 4 1 . 0 0 0 0 1 . 0 0 0 0 0 . 4 3 2 6 0 . 7 4 5 3 1 4 . 7 8 5 8 1 1 1 6 2 3 1 2 9 6 1 . 0 8 7 3 0 . 9 7 4 4 0 . 9 9 6 0 0 . 4 2 8 4 0 . 7 4 1 7 1 4 . 4 2 2 2 1 1 1 6 1 3 1 2 3 3 1 . 0 8 7 2 0 . 9 5 0 4 0 . 9 9 2 2 0 . 4 2 6 9 0 . 7 4 0 3 1 4 . 0 9 9 3 1 1 1 6 0 2 9 1 5 9 1 . 0 8 4 1 0 . 9 4 0 4 0 . 9 9 0 5 0 . 4 1 5 4 0 . 7 2 9 0 1 3 . 8 6 5 2 1 1 1 5 8 2 8 9 2 9 1 . 0 8 3 8 0 . 9 2 4 2 0 . 9 8 7 7 0 . 4 1 4 8 0 . 7 2 8 5 1 3 . 6 9 2 8 1 1 1 5 7 2 7 7 6 4 1 . 0 8 1 7 0 . 8 8 6 1 0 . 9 8 0 0 0 . 4 1 1 7 0 . 7 2 5 3 1 2 . 2 8 6 8 1 1 1 4 9 2 7 5 8 1 1 . 0 8 1 3 0 . 8 7 2 0 0 . 9 7 7 3 0 . 4 0 0 2 0 . 7 1 2 6 1 1 . 8 4 1 2 1 1 1 4 6 2 7 5 2 4 1 . 0 8 1 2 0 . 8 7 1 0 0 . 9 7 7 0 0 . 3 9 6 6 0 . 7 0 8 9 1 1 . 6 3 6 9 1 1 1 4 5 2 7 4 8 7 1 . 0 8 1 1 0 . 8 6 6 2 0 . 9 7 6 0 0 . 3 9 5 0 0 . 7 0 7 1 1 0 . 9 6 7 6 1 1 1 4 0 2 7 1 9 4 1 . 0 8 0 4 0 . 8 6 4 2 0 . 9 7 5 6 0 . 3 5 9 5 0 . 6 6 2 1 1 0 . 9 3 0 5 1 1 1 4 0 2 6 9 0 2 1 . 0 7 9 8 0 . 8 4 1 7 0 . 9 6 9 9 0 . 3 5 8 9 0 . 6 6 1 4 1 0 . 5 9 8 2 1 1 1 3 7 2 6 1 6 0 1 . 0 7 8 3 0 . 8 4 0 6 0 . 9 6 9 6 0 . 3 5 4 8 0 . 6 5 5 6 1 0 . 5 1 7 8 1 1 1 3 6 2 5 2 1 9 1 . 0 7 6 3 0 . 8 2 5 0 0 . 9 6 5 6 0 . 3 5 1 6 0 . 6 5 3 5 1 0 . 2 8 0 6 1 1 1 3 4 2 5 1 6 7 1 . 0 7 6 2 0 . 8 2 2 4 0 . 9 6 4 9 0 . 3 4 5 9 0 . 6 4 7 1 9 . 6 5 9 4 1 1 1 2 7 2 3 7 9 8 1 . 0 7 3 3 0 . 8 1 9 2 0 . 9 6 4 1 0 . 3 4 5 4 0 . 6 4 6 4 9 . 4 9 7 9 1 1 1 2 5 2 3 5 9 5 1 . 0 7 2 8 0 . 8 1 7 7 0 . 9 6 3 6 0 . 3 4 4 3 0 . 6 4 4 9 9 . 2 8 0 6 1 1 1 2 3 2 3 5 5 3 1 . 0 7 2 7 0 . 8 1 3 5 0 . 9 6 2 4 0 . 3 4 3 8 0 . 6 4 4 1 9 . 1 6 4 1 1 1 1 2 1 2 2 7 3 8 1 . 0 7 0 5 0 . 7 9 8 3 0 . 9 5 7 9 0 . 3 3 0 2 0 . 6 2 2 7 9 . 0 1 2 0 1 1 1 1 9 2 0 7 9 9 1 . 0 6 4 0 0 . 7 9 4 1 0 . 9 5 6 6 0 . 3 2 8 1 . 0 . 6 1 9 6 8 . 3 1 0 9 1 1 1 1 0 1 9 1 1 7 1 . 0 5 8 9 0 . 7 8 8 4 0 . 9 5 4 7 0 . 3 2 1 8 0 . 6 1 0 8 7 . 2 3 4 1 1 1 0 9 4 1 9 0 4 9 1 . 0 5 8 7 0 . 7 8 2 7 0 . 9 5 2 9 0 . 3 1 9 7 0 . 6 0 8 9 7 . 1 8 3 4 1 1 0 9 3 1 8 2 4 5 1 . 0 5 5 8 0 . 7 8 0 6 0 . 9 5 2 2 0 . 3 1 7 7 0 . 6 0 6 5 7 . 1 6 8 2 1 1 0 9 3 1 8 0 8 3 1 . 0 5 5 1 0 . 7 6 8 0 0 . 9 4 7 9 0 . 3 0 9 3 0 . 5 9 2 5 7 . 1 1 3 4 1 1 0 9 2 1 7 9 0 0 1 . 0 5 4 4 0 . 7 6 3 3 0 . 9 4 6 4 0 . 2 9 6 8 0 . 5 7 4 9 6 . 9 8 1 7 1 1 0 8 9 1 5 8 7 8 1 . 0 4 6 2 0 . 7 6 0 2 0 . 9 4 5 3 0 . 2 8 2 1 0 . 5 5 8 0 6 . 9 3 1 0 1 1 0 8 7 1 5 4 3 9 1 . 0 4 4 0 0 . 7 3 7 2 0 . 9 3 6 5 0 . 2 8 0 6 0 . 5 5 5 5 6 . 7 7 5 9 1 1 0 8 3 1 5 2 0 4 1 . 0 4 2 7 0 . 7 1 4 7 0 . 9 2 8 5 0 . 2 7 9 0 0 . 5 5 2 9 6 . 5 4 7 5 1 1 0 7 8 1 5 0 4 2 1 . 0 4 1 9 0 . 6 9 7 0 0 . 9 2 1 4 0 . 2 7 2 7 0 . 5 4 1 6 6 . 4 8 2 2 1 1 0 7 6 1 5 0 2 6 1 . 0 4 1 8 0 . 6 9 5 9 0 . 9 2 1 0 0 . 2 6 2 3 0 . 5 2 6 1 6 . 2 8 4 2 1 1 0 7 0 1 4 8 3 8 1 . 0 4 0 7 0 . 6 9 5 4 0 . 9 2 0 8 0 . 2 6 0 7 0 . 5 2 3 6 5 . 6 1 4 9 1 1 0 5 0 1 4 2 2 2 1 . 0 3 7 3 0 . 6 8 9 7 0 . 9 1 8 3 0 . 2 6 0 2 0 . 5 2 2 7 5 . 4 0 9 1 1 1 0 4 3 1 3 9 7 6 1 . 0 3 5 8 0 . 6 6 0 4 0 . 9 0 5 0 0 . 2 5 8 1 0 . 5 1 8 9 5 . 3 6 5 2 1 1 0 4 1 1 3 7 7 2 1 . 0 3 4 5 0 . 6 5 9 9 0 . 9 0 4 8 0 . 1 9 4 4 0 . 3 9 3 9 4 . 8 0 1 5 1 1 0 1 9 1 3 3 9 1 1 . 0 3 2 1 0 . 6 5 3 1 0 . 9 0 1 8 0 . 1 8 2 3 0 . 3 7 1 4 4 . 7 8 6 3 1 1 0 1 7 1 3 1 1 9 1 . 0 3 0 4 0 . 6 2 0 2 0 . 8 8 6 8 0 . 1 7 9 7 0 . 3 6 7 1 4 . 7 6 8 5 1 1 0 1 7 1 2 9 3 1 1 . 0 2 9 1 0 . 6 1 9 6 0 . 8 8 6 5 0 . 1 7 1 4 0 . 3 5 1 5 4 . 7 5 0 8 1 1 0 1 6 1 2 8 2 7 1 . 0 2 8 2 0 . 6 1 8 6 0 . 8 8 6 1 0 . 1 3 9 0 0 . 2 8 5 8 4 . 7 3 3 5 1 1 0 1 5 1 2 6 4 9 1 . 0 2 6 9 0 . 6 1 7 6 0 . 8 8 6 1 0 . 1 3 7 9 0 . 2 8 5 2 4 . 6 9 8 0 1 1 0 1 3 1 2 3 8 2 1 . 0 2 4 8 0 . 6 1 1 8 0 . 8 8 3 6 0 . 1 3 5 3 0 . 2 8 3 1 4 . 5 6 1 7 1 1 0 0 6 1 2 2 6 8 1 . 0 2 3 9 0 . 5 8 7 3 0 . 8 7 1 3 0 . 1 3 3 8 0 . 2 8 1 6 4 . 5 5 5 9 1 1 0 0 6 1 2 1 5 8 1 . 0 2 2 9 0 . 5 8 4 1 0 . 8 6 9 7 0 . 1 3 2 2 0 . 2 7 9 1 4 . 4 8 8 5 1 1 0 0 2 1 2 0 6 4 1 . 0 2 2 1 0 . 5 8 1 5 0 . 8 6 8 3 0 . 1 2 5 9 0 . 2 6 6 2 4 . 4 2 8 4 1 0 9 9 9 1 1 5 0 5 1 . 0 1 7 3 0 . 5 8 1 0 0 . 8 6 8 0 0 . 1 2 4 3 0 . 2 6 4 7 4 . 2 4 5 6 1 0 9 8 7 1 1 3 5 8 1 . 0 1 6 0 0 . 5 7 6 3 0 . 8 6 5 3 0 . 1 2 1 2 0 . 2 5 9 1 4 . 2 4 2 9 1 0 9 8 7 1 1 3 3 8 1 . 0 1 5 8 0 . 5 7 3 7 0 . 8 6 3 8 0 . 1 1 7 6 0 . 2 5 1 2 4 . 1 6 3 0 1 0 9 8 1 1 1 2 9 6 1 . 0 1 5 3 0 . 5 6 8 4 0 . 8 6 0 5 0 . 1 1 6 0 0 . 2 4 9 1 4 . 0 0 8 4 1 0 9 6 8 1 1 2 8 5 1 . 0 1 5 2 0 . 5 6 4 3 0 . 8 5 7 8 Table C.3: P V function. Npy is a measure of background level, and Apy is a measure of acceptance, for different time windows (around track time) and energy thresholds (minimum energy) for detection of hits in each category listed in table C l . Events with Npy < 1.0 pass the P V function cut. This function is drawn in figure 4.14. 217 Appendix C. Detailed Cut Descriptions background, similar to tail-fluctuation background, is concentrated at small t^. The G D R background was "discovered" only after backgrounds had been measured using all of the 1/3 and 2/3 1995 and 1996-7 data sets, and the T D E C O N , T D V E L and M A S S cuts were designed to reduce it to a negligible level. Because the G D R background was found after the background measurements on the 2/3 1996-7 data sample had been performed, the M A S S cut is applied only as a safety cut to the 1996-7 data. However, to get a potentially more accurate measurement of the T D rejection, the T D E C O N cut is applied as a pathology cut, and the rejection of the T D V E L cut is used in the calculation of total T D rejection for 1996-7 data. These cuts were designed wi th large statistics using the 1/3 1996-7 data sample and were found to work as expected when applied to the 2/3 1996-7 data, so any bias in these cuts should be minimal . The P A S S 3 T D cuts are described in more detail below. • TDTCON: If an accidental hit is present in the stopping counter prior to track time, it's possible for this accidental to be assigned to pion time, and the track hit to be assigned to muon time, such that the TT —> p double-pulse decay signature is satisfied. This cut attacks pion-time accidental background by comparing the stopping-counter pion time to tRs- If the stopping-counter pion time is earlier (or later) than tRs by more than 1.5 ns, the event is rejected. Events wi th pulse misassignment in the stopping counter can correlate T D and kinematic cuts, because these events can have both mismeasured track energy, and a fake TT —> n decay signature. T D and kinematic cuts are bifurcated and therefore assumed to be uncorrelated in the background estimate (see sections 4.2 and 4.4.2), so the T D T C O N cut is applied as a pathology cut when measuring the T D rejection (see figure 4.21). • TDDFA1: This cut is applied only to the 1996-7 data, in order to remove the extra tail-fluctuation background introduced by the use of narrower pulse shapes for double-pulse fitting to T D data. It is comprised of the weighted sum of three stopping-counter variables: (1) the log of: the product of single-pulse fit x 2 ' s from the upstream and downstream T D data, divided by that from double-pulse fits; (2) the log of the product of single-pulse fit x2's from the upstream and downstream T D data; and (3) second-218 Appendix C. Detailed Cut Descriptions pulse time relative to first-pulse time, _M. The weights (0.37522, 0.092414, and 0.07189, respectively) are found from a discriminant function analysis [66] of 1636 pion tracks and 1636 muon tracks in the R S . The pion events are taken from the 7 T s c a t monitor data (see section 3.3), and the muon events are those that remain in the 1/3 1996-7 S K I M 2 data after application of al l beam and Kn2 kinematic cuts, the C O S 3 D , Z F R F , and L A Y E R 14 cuts (see section C.3.1), the E V 5 , E L V E T O , T D F O O L , T D L I K 2 , and T D L I K 3 cuts (see below), and the requirement t^ < 10 ns. Events are rejected which have T D D F A 1 values < 4.2. • E V 5 : This cut attacks early muon decay background by looking for accidental activity in the R S at electron time. Electron hits are found within ± 1 sector and ± 2 layers of the stopping counter, and valid electrons must: (1) have a hit in the stopping sector wi thin ± 1 layers of the stopping layer; (2) have at least one hit wi th at least 1 M e V recorded on both of the upstream and downstream ends; (3) occur at least 20 ns after muon time; and (4) have z wi thin ± 2 . 7 ns of the pion, where z comes from the end-to-end time difference of the electron hit in the counter closest to the stopping counter. Furthermore, the electron can have no more than 56 M e V summed over all hits in the R S at electron time. Otherwise the event is rejected. • ELVETO: This cut attacks muon-time accidental background by looking for accidental activity in the R S and barrel at muon time. The time windows, energy thresholds, and search region are optimized for rejection and acceptance, similar to the P V optimization described in section C.3.3. • TDFOOL: This cut attacks muon-time accidental background by looking for acci-dental activity along the track at muon time. A double-pulse fit to the T D data is attempted in the two counters previous to the stopping counter, and if either counter returns a single-pulse fit x2 divided by that from a double-pulse fit > 4 on each of the upstream and downstream ends, wi th the second pulse occurring within ± 5 ns of muon time in the stopping counter and having at least 2.2 M e V energy, then the event is rejected. 219 Appendix C. Detailed Cut Descriptions • T D E C O N : This cut requires consistency between T D and A D C pion energies in the stopping counter. The A D C energy is the total gated energy in the stopping counter, minus the fraction of muon energy (found in the T D ) lying wi th in the A D C gate (which is about 80 ns wide, starting just before pion time). Kinemat ic cuts typically use the A D C pion energy in the stopping counter, whereas T D cuts use the T D pion energy. Therefore, events failing T D E C O N can correlate T D and kinematic cuts if, for example, the calculated fraction of muon energy inside the A D C gate is a function of muon energy. decays with a muon-like accidental pulse in the stopping counter wi l l pass the T D cuts, and may also preferentially pass the kinematic cuts because the calculated fraction of "muon" energy inside the A D C gate, for this muon energy, moves the track energy towards the TT+V&(1) signal region. T D and kinematic cuts are bifurcated and therefore assumed to be uncorrelated in the background estimate (see sections 4.2 and 4.4.2), so the T D E C O N cut is applied as a pathology cut when measuring the T D rejection (see figure 4.21). This cut was designed based on evidence of large TD-kinematic correlations in the 1996-7 data (e.g., arising from G D R background - see section 4.6), so this cut is only applied to the 1996-7 data. Specifically, events are rejected which have epitc > (37.5 • elastjcor/'40. — 0.5), where epitc and elastjcor are the geometric means of the T D and A D C pion energies, respectively, from the upstream and downstream ends of the stopping counter. This cut was designed using the 1/3 1996-7 data, and found not to be particularly effective when applied to the 2/3 1996-7 data, so T D - A D C pion energy inconsistency in the stopping counter is unlikely to be a major source of TD-kinematic correlation. • T D V E L : This cut is designed to attack G D R background v ia the time-energy relation ("velocity") of emitted neutrons, shown figure 4.5. Specifically, events are rejected which have t^ < (—2.5 • + 17.5). This cut was designed based on evidence of G D R background in the 1996-7 data, so this cut is only applied to the 1996-7 data. • TDLIK2: This cut attacks muon-time accidental background (as well as tail-fluctuation background) by forming a likelihood out of quantities specific to the muon from TT —> /i 220 Appendix C. Detailed Cut Descriptions decay. This likelihood is constructed from a sum of the logs of probabilities that the muon has given values of energy, z relative to the pion (where z comes from the end-to-end time difference and from the ratio of energies on each end), and time rel-ative to the pion. The probabilities are taken from signal/background distributions, where the background distributions are measured using the 1368 events in the 1/3 1995 S K I M 2 data which remain after application of al l beam cuts, except the C K T R S and C K T A I L cuts, cuts specific to 1996-7, and using a loose version of the D E L C cut; U - t K > l ns; the Kn2 (low) side of the R B O X and E B O X cuts; the C O S 3 D , Z F R F , and L A Y E R 1 4 cuts (see section C.3.1); and a subset of the fixed T D cuts ( E V 5 , E L -V E T O , and T D F O O L , described above). The C K T R S and C K T A I L cuts are turned off, and the D E L C cut is loosened, in order to enhance statistics. Meaningful values of this likelihood range from —70 (background-like) to 0 (signal-like). The T D function cut pass condition, NTD < 1.003, corresponds to events wi th T D L I K 2 likelihood values > -2 .128. • TDLIK3: This cut attacks early muon decay background by forming a likelihood out of quantities specific to the electron from rr —> fi —• e decay. This likelihood is constructed from a sum of the logs of probabilities that the electron has given values of energy, z relative to the pion (where z comes from the end-to-end time difference in the electron counter closest to the stopping counter), and time relative to the muon. The muon energy and muon time relative to the pion are also part of this likelihood. The probabilities are taken from signal/background distributions, which are measured in the same way as the T D L I K 2 distributions. Meaningful values of this likelihood range from —40 (background-like) to 0 (signal-like). The T D function cut pass condition, NTD < 1-003, corresponds to events wi th T D L I K 3 likelihood values > -2.440. • TDDFA2: This cut attacks tail-fluctuation background which remains after applica-t ion of al l previous T D cuts. It is comprised of the weighted sum of five stopping-counter variables: (1) the log of: the product of single-pulse fit x2's from the upstream and downstream T D data, divided by that from double-pulse fits; (2) muon energy; 221 Appendix C. Detailed Cut Descriptions (3) muon z relative to pion z from end-to-end time differences; (4) the log of the prod-uct of single-pulse fit x2's from the upstream and downstream T D data; and (5) the minimum of the muon energy from the upstream and downstream ends. The weights (0.53288, -0.57242, -0.17664, 0.42072, and 0.42446, respectively) are found from a discriminant function analysis [66] of 52 pion tracks and 52 muon tracks in the R S . The pion events are taken from the 7Tscat monitor data (see section 3.3), and the muon events are those that remain in the 1/3 1995 S K I M 2 data wi th tM < 10 ns which re-main after application of loose T D L I K 2 and T D L I K 3 cuts ( T D L I K 2 > - 4 , T D L I K 3 > —4) to the T D L I K 2 background sample mentioned above. Meaningful values of T D D F A 2 range from —1 (background-like) to 4 (signal-like). The T D function cut pass condition, NTD 5_ 1-003, corresponds to events wi th T D D F A 2 values > 2.605. T D background in the n+vD(l) signal region likely comes from some combination of range-tail and muon-band events, because range-tail events already have R and E in the box, and muon-band events can have R, E, and P all in the box (see figure 4.1). Conversely, peak events have R, E, and P well away from the box. Therefore, only range-tail and muon-band events are used to define the T D function. Starting wi th 381 range-tail and 489 band events from the 1/3 1995 S K I M 2 data which pass the E V 5 , E L V E T O , and T D F O O L cuts (i.e., from the T D L I K 2 background sample mentioned above), and a larger sample of ftscat events from S K I M 3 data, the parameter space of possible values of T D L I K 2 , T D L I K 3 , and T D D F A 2 is scanned to find the optimal value of acceptance for the S K I M 3 irscat events, for different values of rejection of the S K I M 2 muon events [41]. The T D rejection is required to be the same in the range ta i l and band at each point on the function, so that T D rejection is uncorrelated (to some degree) wi th muon track kinematics. Figure 4.15 shows the T D background level, NTD, a s a function of the acceptance, ATD, for these events. Events with NTD f_ 1-003 are defined to pass the T D function cut. This point was chosen to give a total T D function cut rejection for S K I M 2 range-tail and muon-band events of 1000 (using all of the "fixed" and "variable" T D cuts as defined in table 4.9). The T D function is therefore defined for the range NTD = [0,1000]. Events which fail the fixed T D cuts have this maximal T D function value; events which pass the fixed T D function cuts have the smaller NTD values 222 Appendix C. Detailed Cut Descriptions tabulated in table C.4, depending on their values of T D L I K 2 , T D L I K 3 , and T D D F A 2 . After confirming the T D rejection wi th the 2/3 1995 S K I M 2 data, the function was re-optimized for application to 1995-7 data using the statistics of the full 1995 S K I M 2 data sample. C.3.5 Kn2 and Kinematic Function Cuts The Kn2 and Ku2 kinematic functions involve loosening/tightening the following cuts. • R N G M O M : The expected R S range, RRS, and its resolution, <T(RRS), are calibrated as a function of momentum in the U T C for muons in the muon band. A X ' ^ e quantity is formed using the measured R S range, and events are rejected that have x(RP) = (RRS - RRXS)IC(RRS) > 2.0. • BOX: This is a cut on the 3 major kinematic variables of a track (R,E,P), which selects the 7r + ^P(l) kinematic region between the K„2 and peaks. Events pass which satisfy the following 3 conditions: 1. RBOX: 33 < R < 40 cm. 2. EBOX: 115 < E < 135 M e V . 3. PBOX: 211 < P < 229 M e V / c . The R B O X , E B O X , and P B O X cuts are direct cuts on total range, energy and momentum of a track. Addi t iona l cuts on the variables (X — Xpeak)/ax, where X is R, E, or P , and where the peak positions Xpeak and resolutions ax are calibrated as a function of polar angle [41], are imposed on the (low) side of the box: • BOX': Events pass which satisfy the following 3 conditions: 1. RBOX' : rdev > 2.80, where rdev = {R- Rpeakj/oR 2. EBOX': edev > 2.453, where edev = (E - Epeak)/aE 3. PBOX' : pdev > 2.625, where pdev = ( P - Ppeak)/<Jp 223 Appendix C. Detailed Cut Descriptions NTD ATD - TDLIK2 TDLIK3 TDDFA2 NTD A T D TDLIK2 TDLIK3 TDDFA2 9.796 1 29630 -59.150 -35.848 -1 .726 1 191 1.21127 -2 128 -2 624 2 235 9.242 1 29599 -37.758 -35.848 -1.726 1 139 1.20094 -2 128 -2 440 2 235 8.772 1 29562 -37.758 -17.312 -1.726 1 034 1.17840 -2 128 -2 440 2 550 8.333 1 29524 -36.414 -17.312 -1.726 1 003 1.17126 -2 128 -2 440 2 605 8.265 1 29518 -36.414 -16.040 -1.726 0 987 1.16756 -2 086 -2 440 2 605 8.135 1 29505 -35.868 -16.040 -1.726 0 899 1.14728 -1 918 -2 440 2 605 8.009 1 29493 -35.868 -13.992 -1.726 0 883 1.14665 -1 918 -2 432 2 605 7.048 1 29399 -15.624 -13.992 0 827 0 852 1.14039 -1 876 -2 432 2 605 6.897 1 29392 -15.624 -13.992 1 051 0 836 1.13632 -1 876 -2 352 2 605 6.818 1 29386 -15.624 -13.992 1 171 0 820 1.13218 -1 848 -2 352 2 605 6.771 1 29380 -15.246 -13.992 1 171 0 799 1.12636 -1 820 -2 352 2 605 6.097 1 29280 -11.844 -11.688 1 171 0 773 1.11847 -1 820 -2 352 2 662 6.008 1 29274 -11.844 -9.544 1 171 0 731 1.10213 -1 736 -2 352 2 662 5.925 1 29267 -11.844 -8.568 1 171 0 711 1.09205 -1 694 -2 352 2 662 5.397 1 29198 -9.156 -8.416 1 171 0 705 1.09036 -1 694 -2 328 2 662 5.282 1 29180 -8.652 -8.416 1 171 0 637 1.05072 -1 694 -2 000 2 662 4.974 1 29079 -7.742 -7.808 1 171 0 627 1.04702 -1 694 -2 000 2 691 4.781 1 29023 -7.742 -7.808 1 566 0 564 1.01327 -1 554 -2 000 2 691 4.504 1 28904 -6.916 -7.224 1 566 0 554 1.00983 -1 540 -2 000 2 691 4.028 1 28679 -5.474 -5.928 1 566 0 543 1.00551 -1 526 -2 000 2 691 3.934 1 28666 -5.320 -5.880 1 566 0 538 1.00300 -1 526 -1 984 2 691 3.887 1 28654 -5.320 -5.000 1 566 0 522 0.99324 -1 526 -1 984 2 760 3.866 1 28647 -5.320 -4.840 1 566 0 486 0.96731 -1 526 -1 864 2 760 3.830 1 28635 -5.208 -4.840 1 566 0 465 0.95122 -1 470 -1 864 2 760 3.610 1 28553 -4.746 -4.840 1 566 0 444 0.93613 -1 470 -1 808 2 760 3.600 1 28547 -4.746 -4.696 1 566 0 408 0.89568 -1 470 -1 808 2 958 3.339 1 28359 -4.298 -4.592 1 566 0 392 0.89367 -1 470 -1 800 2 958 3.302 1 28328 -4.228 -4.592 1 566 0 371 0.87395 -1 470 -1 800 3 038 2.973 1 27971 -3.654 -4.568 1 566 0 340 0.85228 -1 470 -1 800 3 112 2.931 1 27959 -3.626 -4.568 1 566 0 334 0.84790 ' -1 470 -1 800 3 126 2.910 1 27946 -3.626 -4.568 1 621 0 308 0.80820 -1 470 -1 680 3 126 2.894 1 27933 -3.626 -4.304 1 621 0 287 0.78353 -1 470 -1 624 3 126 2.847 1 27883 -3.626 -3.848 1 621 0 266 0.76381 -1 470 -1 584 3 126 2.816 1 27852 -3.584 -3.848 1 621 0 256 0.75848 -1 470 -1 576 3 126 2.800 1 27833 -3.584 -3.848 1 656 0 235 0.75022 -1 470 -1 560 3 126 2.795 1 27827 -3.584 -3.832 1 656 0 214 0.71684 -1 302 -1 560 3 126 2.591 1 27558 -3.332 -3.832 1 656 0 209 0.71296 -1 288 -1 560 3 126 2.414 1 27345 -3.136 -3.832 1 656 0 199 0.70407 -1 260 -1 560 3 126 2.403 1 27339 -3.136 -3.696 1 656 0 183 0.67658 -1 240 -1 520 3 136 2.351 1 27270 -3.108 -3.696 1 656 0 172 0.65554 -1 230 -1 490 3 146 2.330 1 27238 -3.094 -3.696 1 656 0 157 0.61384 -1 160 -1 460 3 156 2.126' 1 26900 -2.926 -3.696 1 656 0 141 0.58422 -1 100 -1 450 3 166 2.085 1 26781 -2.926 -3.696 1 848 0 131 0.54665 -1 090 -1 390 3 176 2.074 1 26750 -2.912 -3.696 1 848 0 120 0.52304 -1 080 -1 360 3 186 2.048 1 26662 -2.912 -3.696 1 922 0 110 0.50776 -1 070 -1 340 3 196 1.959 1 26337 -2.786 -3.696 1 922 0 094 0.44665 -0 980 -1 300 3 216 1.766 1 25761 -2.618 -3.696 1 922 0 068 0.40307 -0 900 -1 290 3 226 1.651 1 25141 -2.618 -3.696 2 195 0 063 0.38704 -0 880 -1 280 3 236 1.641 1 25072 -2.618 -3.232 2 195 0 057 0.34890 -0 870 -1 270 3 356 1.620 1 24922 -2.618 -3.232 2 216 0 052 0.33231 -0 860 -1 260 3 386 1.609 1 24846 -2.618 -3.096 2 216 0 047 0.31641 -0 850 -1 250 3 416 1.452 1 23744 -2.380 -3.096 2 216 0 042 0.29656 -0 840 -1 240 3 456 1.437 1 23682 -2.366 -3.096 2 216 0 037 0.28240 -0 830 -1 230 3 476 1.348 1 23037 -2.268 -3.096 2 216 0 031 0.26756 -0 810 -1 220 3 486 1.332 1 22912 -2.268 -2.920 2 216 0 026 0.25667 -0 800 -1 210 3 496 1.322 1 22786 -2.268 -2.920 2 235 0 021 0.24609 -0 790 -1 200 3 506 1.259 1 22016 -2.170 -2.920 2 235 0 016 0.15485 -0 620 -1 190 3 516 1.212 1 21428 -2.170 -2.624 2 235 0 010 0.14402 -0 610 -1 180 3 536 1.202 1 21309 -2.156 -2.624 2 235 0 005 0.12999 -0 590 -1 170 3 546 Table C.4: T D function. NTD is measure of K^2 background level, and ATD is a measure of acceptance, for different lower limits on the allowed values of T D L I K 2 , T D L I K 3 , and T D D F A 2 (see section C.3.4). Events wi th NTD < 1-003 pass the T D function cut. This function is drawn in figure 4.15. 224 Appendix C. Detailed Cut Descriptions The low sides of the R B O X , E B O X , and P B O X cuts correspond to cuts at 2.4, 2.0, and 2.2 in rdev, edev, and pdev, respectively. The KN2 kinematic function is based on simultaneous, uniform loosening/tightening of the R B O X ' , E B O X ' and P B O X ' cuts. It is measured using KN2 decays from the 1/3 1995 S K I M 1 data which fail the P V function cut, pass all beam cuts, pass the T D function cut, and pass the high-side B O X cut (to remove K^2 decays). For the function at "loose" cut positions (Nkin,K^2 > 1-0) > e v e n t s are first required to fail the low-side R B O X , E B O X , and P B O X cuts. The R B O X ' , E B O X ' , and P B O X ' cuts are then loosened, and the number of events counted at each step. For the function at "tight" cut positions {Nkin,K^2 — 1-0), events are first required to pass the low-side R B O X , E B O X , and P B O X cuts. The R B O X ' , E B O X ' , and P B O X ' cuts are then tightened to give the same rejection for al l 3 kinematic quantities at each step, assuming each quantity has a Gaussian distribution. Note, from section 4.4.1 and figure 4.20, that R and P are bifurcated wi th E in order to get the KN2 normalization. This "second bifurcation" is valid as long as R and P are not correlated wi th E for K^2 background. From table C.5, "# events" is the number of events in the S K I M 1 data sample described above which pass the listed cuts on R and P, " N ( E B O X ) " is "# events" divided by the rejection of the cut on E (i.e., the KN2 normalization for the 1/3 1995 data, as described in section 4.4.1), and "N(seen)" is the observed number of events which simultaneously pass the cuts on R, E, and P. The agreement of N ( E B O X ) wi th N(seen) verifies the absence of a correlation of R and P wi th E for KN2 background. Figure 4.16 shows the KK2 kinematic background level, Nkin,Kn2, a s a function of the acceptance, Akin,K„2- Events with Nktn,K„2 < 0.3358 are defined to pass the KN2 kinematic function cut. The function is defined for the range Nkin,K^2 = [0,285.1] and is tabulated in table C.5. Note that at the N k i n ^ 2 = 0.3358 position in table C.5 that # events = 7 and N ( E B O X ) = 0.127. In table 4.12 for the 1/3 1995 data however, B^p = # events = 8 and BK^ = N ( E B O X ) = 0.16 for the 1/3 1995 data. This is because some cuts/calibrations were changed slightly between measurement of the KN2 kinematic function, and the measurement of KN2 background in the 1/3 1995 data. The kinematic function is based on the R N G M O M cut and the high side of the 225 Appendix C. Detailed Cut Descriptions tt ^peafc O-R tt ttpeak (JF. r' ^peak Op Akin,Kn2 # events N ( E B O X ) N(seen) 1.70 1.157 1.435 285.1200 1.2845 468 104.3 108 1.75 1.220 1.491 221.7600 1.2673 391 80.7 84 1.80 1.283 1.547 190.0800 1.2508 328 63.4 72 1.85 1.345 1.602 142.5600 1.2319 272 49.1 54 1.90 1.407 1.658 102.9600 1.2162 223 37.4 39 1.95 1.468 1.713 81.8400 1.1995 186 28.2 31 2.00 1.529 1.767 66.0000 1.1831 163 22.1 25 2.05 1.589 1.822 55.4400 1.1637 145 17.3 21 2.10 1.649 1.876 42.2400 1.1427 122 13.1 16 2.15 1.708 1.931 36.9600 1.1251 97 9.4 14 2.20 1.767 1.985 31.6800 1.1086 84 7.2 12 2.25 1.826 2.039 21.1200 1.0910 68 5.0. 8 2.30 1.884 2.093 10.5600 1.0745 50 2.9 4 2.35 1.942 2.146 7.9200 1.0606 41 1.8 3 2.40 2.000 2.200 1.0996 1.0323 14 0.417 1 2.45 2.057 2.253 1.0512 1.0218 14 0.398 1 2.50 2.115 2.307 0.9306 1.0089 13 0.353 0 2.55 2.171 2.360 0.8868 0.9932 13 0.336 0 2.60 2.228 2.413 0.7772 0.9755 12 0.294 0 2.65 2.285 2.466 0.6745 0.9601 11 0.256 0 2.70 2.341 2.519 0.5792 0.9421 10 0.219 0 2.75 2.397 2.572 0.4364 0.9255 8 0.165 0 2.80 2.453 2.624 0.3358 0.9069 7 0.127 0 2.85 2.508 2.677 0.2893 0.8930 7 0.110 0 2.90 2.564 2.729 0.2418 0.8775 7 0.092 0 2.95 2.618 2.782 0.1708 0.8610 6 0.065 0 3.00 2.673 2.834 0.1175 0.8433 5 0.045 0 3.05 2.728 2.887 0.0742 0.8233 4 0.028 0 3.10 2.783 2.939 0.0541 0.8058 4 0.021 0 3.15 2.838 2.991 0.0436 0.7906 4 0.017 0 Table C.5: Kn2 kinematic function. Nkin,K„2 is a measure of K^2 background level, and Akin,^ is a measure of acceptance, for different lower l imits on the allowed values of (X — Xpeak)/ax (see section C.3.5). Events wi th Nkin,Kn2 ^ 0.3358 pass the Kn2 kine-matic function cut. This function is drawn in figure 4.16. events", " N ( E B O X ) " , and "N(seen)" are defined in section C.3.5 of the text. 226 Appendix C. Detailed Cut Descriptions P B O X cut. It is measured using decays from the 1/3 1995 S K I M 2 data which fail the T D function cut, pass al l beam cuts, pass the P V function cut, and pass the low-side B O X cut (to remove Kn2 decays). For the function at "loose" cut positions (N^K^ > 0.2681), events are first required to pass the high-side R B O X and E B O X cuts. The R N G M O M cut and the high side of the P B O X cut are then loosened, and the number of events counted at each step. The two cuts are loosened such that an equal number of range-tail and muon-band events are added at each step. For the function at "tight" cut positions {Nkin,K^2 ^ 0.2681), there are too few events to measure a function. However, there are indications (see section 4.4.2) that the l imit ing background is dominated by range-tail events, i.e., decays where the muon interacts in the R S , such that range and energy are downshifted, wi th an uncorrelated momentum mismeasurement placing the muon in the box. The function inside the box can then be measured by tightening the P B O X cut on a large-statistics sample of K^2 peak events available from Ka2(l) monitor data (see section 3.3), and counting the number of events which remain at each step. The R N G M O M cut is also tightened to define the Kn2 kinematic function inside the box, but due to lack of a high-statistics sample of muon-band events, the rejection of the R N G M O M cut cannot be measured into the box with any reliability. Therefore, the muon band is assumed to be Gaussian in the quantity x(RP) — (PRS — RRS)/G{RRS) (see above), and the R N G M O M cut is tightened to get the same rejection per step for muon-band events as the tightening of the P B O X cut gives for Ku2 range-tail events. Figure 4.17 shows the kinematic background level, Nkin,Ky.2i a s a function of the acceptance, Akin,K^2- FJvents wi th N ^ K ^ ^ 0.2681 are defined to pass the K^2 kinematic function cut. The function is defined for the range Nkin,K^2 = [0,45.15] and is tabulated in table C.6. C .3 .6 Beam Function Cuts Beam functions are measured by loosening/tightening the following cuts: • D E L C : Events are rejected where the kaon decay product is detected soon after the kaon is detected, such that the kaon may have decayed in flight. This cut rejects 227 Appendix C. Detailed Cut Descriptions x(RP) P # events ^kin,Ktj.2 •Akin,Kfj,2 7.790 238.00 261 45.1530 1.0757 7.530 237.80 243 • 42.0390 1.0757 7.450 237.60 234 40.4820 1.0757 7.020 237.30 214 37.0220 1.0757 6.970 237.10 206 35.6380 1.0757 6.900 237.00 198 34.2540 1.0757 6.830 236.90 186 32.1780 1.0757 6.590 236.50 173 29.9290 1.0757 6.450 236.20 157 27.1610 1.0757 6.310 236.00 145 25.0850 1.0756 6.240 235.70 133 23.0090 1.0756 6.170 235.50 123 21.2790 1.0756 6.100 235.10 112 19.3760 1.0756 5.990 234.80 102 17.6460 1.0754 5.880 234.60 90 15.5700 1.0751 5.770 234.50 79 13.6670 1.0749 5.720 234.20 71 12.2830 1.0748 5.650 234.00 65 11.2450 1.0746 5.420 233.60 52 8.9960 1.0740 5.260 233.30 42 7.2660 1.0738 5.170 233.10 38 6.5740 1.0731 5.090 232.80 34 5.8820 1.0727 4.760 232.40 25 4.3250 1.0719 4.690 232.30 21 3.6330 1.0715 4.560 232.10 19 3.2870 1.0712 4.150 231.80 15 2.5950 1.0701 4.060 231.60 13 2.2490 1.0675 3.770 231.20 11 1.9030 1.0643 3.740 230.50 9 1.5570 1.0587 2.900 229.80 5 0.8650 1.0490 2.290 229.30 3 0.5190 1.0351 2.000 229.00 234 0.2681 1.0237 1.852 228.00 142 0.1293 0.9912 1.732 227.00 91 0.0828 0.9500 1.635 226.00 69 0.0628 0.8924 1.534 225.00 47 0.0428 0.8301 1.435 224.00 37 0.0337 0.7541 1.393 223.00 26 0.0237 0.6767 1.320 222.00 22 0.0201 0.5918 1.259 221.00 14 0.0127 0.5093 1.204 220.00 11 0.0100 0.4317 1.121 219.00 9 0.0082 0.3526 Table C.6: kinematic function. A 7 ^ ^ 2 *s a measure of background level, and Akin,Kn2 i s a measure of acceptance, for different upper l imits on the allowed values of x(RP) and P (see section C.3.5). Events wi th N^K^ ^ 0.2681 pass the kinematic function cut. This function is drawn in figure 4.17. "# events" is the number of S K I M 2 (N^K^ > 0.2681) or K^il) monitor ( A ^ m , ^ — 0.2681) events which are used to define the NkiN,K^2 values, which is why # events is discontinuous at N^K^ ~ 0.2681 (more details can be found in section C.3.5 of the text). 228 Appendix C. Detailed Cut Descriptions single-beam background. Events are rejected unless they satisfy al l of the following conditions: 1. U — tx > 2 ns 2. U -tK > 5 ns if \tK - tB4strobe\ > 1.5 ns 3. tn — tx > 5 ns if \tn — tus\ > 1.5 ns 4. t.^  — tx > 5 ns if there are no pion fibers in the target (in which case tn comes from the IC) 5. — tx > 4 ns if the kaon energy in the target is small (< 50 M e V ) 6. tn — tx > 3 ns if there are less than 4 pion fibers in the target 7. tn — tx > 3 ns if Ikbeam < 200 (this is primarily intended to attack C E X back-ground) 8. t„ — tx > 4 ns when any kaon fiber has time farther than 2.5 ns from the average kaon time 9. tn — tx > 4 ns when any pion fiber has time farther than 3.5 ns from the average pion time. • BWTRS: Events are rejected that have hits in B W C 1 or B W C 2 wi th in ± 5 ns of tRS. This cut rejects double-beam background, where a second beam particle (detected by B W C 1 or B W C 2 ) scatters into the R S to produce the apparent decay product of the event kaon. • BWHRS: Events are rejected where the minimum absolute time of all hits in B W C l and B W C 2 (excluding those due to the event kaon) relative to R S track time is < 0 ns (that is, this cut does nothing). However, this time requirement is tightened to define the double-beam function (see below). This cut rejects double-beam background, where a second beam particle (detected by B W C l or B W C 2 ) scatters into the R S to produce the apparent decay product of the event kaon. 229 Appendix C. Detailed Cut Descriptions • C K T R S : Events are rejected that have a CK hit wi thin ± 2 ns of tRs, where at least 5 kaon tubes fired. This cut rejects single-beam background, where the event kaon decays in flight, and double-beam background, where a second beam kaon decays in flight to produce the track detected in the R S . • C K T A I L : This cut is similar to C K T R S , but it uses the trailing-edge T D C time to infer the time of a CK pulse. CK pulses are typically 20 ns wide, so hits occurring up to 20 ns after a previous hit may not have a leading-edge T D C time. This cut therefore attacks double-beam background, where a second beam kaon decays in flight to produce the track detected in the R S , and enters the Cerenkov detector less than 20 ns after the event kaon. Events are rejected depending on the value of tn — tK-1- t„ — tK < 15 ns: the event is rejected if there are CK hits wi thin ± 2 ns of tRs with at least 5 kaon tubes firing 2. 15 < tn — tK < 25 ns: the event is rejected if there are CK hits within +5 or —2 ns of tRs wi th at least 1 kaon tube firing 3. tw — tK > 25 ns: the event is rejected if there are CK hits wi thin ± 2 ns of tRs with at least 5 kaon tubes firing. • CPITRS: Events are rejected that have a (X hit wi th in ± 2 ns of £RS , where at least 5 pion tubes fired. This cut rejects double-beam background, where a beam pion scatters into the R S to produce the apparent decay product of the event kaon. • CPITAIL: This cut is similar to C P I T R S , but it uses the trailing-edge T D C time to infer the time of a (X pulse. (X pulses are typically 20 ns wide, so hits occurring up to 20 ns after a previous hit may not have a leading-edge T D C time. This cut therefore attacks double-beam background, where a beam pion scatters into the R S to produce the apparent decay product of the event kaon, and enters the the Cerenkov detector less than 20 ns after a previous (X hit (presumably due to the event kaon). Events are rejected if there are (X hits wi thin ± 2 ns of tRs wi th at least 5 pion tubes firing. 230 Appendix C. Detailed Cut Descriptions • PBNRS: Events are rejected that have 2 or more lead glass phototube hits within ± 5 ns of tns- This cut rejects double-beam background, where a beam pion (detected by the lead glass) scatters into the R S to produce the apparent decay product of the event kaon. • B 4 D E D X : Events are rejected that have energy in the B 4 hodoscope < 1.5 M e V . This cut rejects single- and double-beam background arising from pions in the beam, because pions have smaller dE/dx than kaons of the same momentum. • B 4 T R S : Events are rejected that have hits in the B4 counter wi thin ± 3 ns of tRs-This cut rejects double-beam background, where a second beam particle is responsible for the apparent decay product of the event kaon. • B 4 T D : Events are rejected which have second pulses in the T D data of the B4 ho-doscope within ± 4 ns of tRs, where the pulse fitting to T D data returns a single-pulse fit x2 divided by that of a double-pulse fit > 2.5, and the energy of the second pulse is more than 20 M e V . This cut removes double-beam background, where a second beam particle is responsible for the apparent decay product of the event kaon. Individual single- and double-beam normalization and rejection functions are not mea-sured directly due to overlap between the different beam background types. Rather, the beam function cuts which are used in each of the normalization and rejection branches of the bifurcated background estimation structures (see figures 4.22, 4.23, 4.24 and 4.25 of section 4.4.3) are loosened for the purpose of the outside-the-box correlation study (see sec-tions 4.2 and 4.6, and the beam background re-measured at each step. When loosening the beam function cuts associated wi th the single-beam kaon- and pion-entering rejection branches, the rejections are measured with enhanced statistics, obtained by using a loose B O X cut in the setup (see figures 4.22 and 4.23). When loosening the beam function cuts associated wi th the double-beam kaon- and pion-entering rejection branches, the rejections are measured wi th enhanced statistics, obtained by applying only the low side of the B O X cut in the setup (see figures 4.24 and 4.25). 231 Appendix C. Detailed Cut Descriptions M O U T 1V1BM2K mBM2P pass condition (low-side box) (low-side box) NBM2 ABM2 | B W H R S | > 0.0 ns 14 9 1.00 1.0000 B W H R S > 1.0 ns 13 9 0.93 0.9892 B W H R S > 2.0 ns 13 8 0.93 0.9766 B W H R S > 3.0 ns 12 8 0.86 0.9615 B W H R S > 4.0 ns 11 8 0.79 0.9457 B W H R S > 5.0 ns 7 7 0.52 0.9279 Table C.7: Double-beam function "inside the box". NBM2 is a measure of double-beam background level, and ABM2 is a measure of acceptance, for different lower limits on the minimum absolute time of all hits in B W C 1 and B W C 2 (excluding those due to the event kaon) relative to R S track time (see the description of the B W H R S cut in section C.3.6 of the text). This function is drawn in figure 4.18. MBUM2K (low-side box) and M j g j ^ 2 P (low-side box) are the numbers of surviving events at the bottom of the rejection branches in figures 4.24 and 4.25, respectively, after the B W H R S cut in the table is applied. Figure 4.18 shows the total double-beam background level, NBM2, as a function of the acceptance, ABM2, for tight cut positions (NBM2 < 1-0). The function is measured using the 2/3 1995 S K I M 3 data, by tightening the B W H R S cut and counting the events which survive at the bottom of the rejection branches in figures 4.24 and 4.25. It is defined for the range NBM2 = [0.52,1.0] and is tabulated in table C.7. Figure 4.19 shows the C E X background level, NCEX, as a function of the acceptance, ACEX, for tight cut positions (NCEX < 1-0). The function is measured using UMC-generated C E X data, by tightening the B 4 E K Z and D E L C cuts and counting the number of events which survive after al l other (UMC-appropriate) cuts have been applied (see section 4.4.4). It is defined for the range NCEX = [0.26,1.0] and is tabulated in table C.8. C . 4 Data Quality Cuts • BAD_RUN: Dur ing data-taking, a log is kept of various hardware failures. Runs that are deemed un-analyzable due to hardware problems are omitted from the analysis. Some hardware problems are only found offline: they are removed by imposing various conditions on the data. For example, in 1997 there was a period when the high voltages 232 Appendix C. Detailed Cut Descriptions pass condition # events NCEX ACEX Ikbeam > 2 23 1.00 1.0000 Ikbeam > 3 21 0.91 0.9934 Ikbeam > 4 20 0.87 0.9876 Ikbeam > 5 18 0.78 0.9811 Ikbeam > 6 14 0.61 0.9763 Ikbeam > 7 14 0.61 0.9706 Ikbeam > 8 14 0.61 0.9653 Ikbeam > 9 13 0.57 0.9612 Ikbeam > 10 13 0.57 0.9573 (U ~ tK)/iDC > 1-0 13 0.57 0.9573 (U-tK)/iDC > 1-1 10 0.43 0.9373 (** - *x)/i£>c > 1.2 8 0.35 0.9214 (*«• - tK)/iDC > 1-3 6 0.26 0.9004 Table C.8: C E X function "inside the box". NCEX is a measure of C E X background level, and ACEX is a measure of acceptance, for different lower l imits on the kaon likelihood (Ikbeam -see the description of the B 4 E K Z cut in section C.3.2) and decay time (see the description of the D E L C cut in section C.3.6 of the text). This function is drawn in figure 4.19. inc is an index which refers to al l subconditions of the D E L C cut, which are tightened simultaneously. That is, (tn — tK)/iDC > 1-0 refers to the standard D E L C cut, and {t^ — t^/iDC > 1-1 means that the veto time window in each subcondition of the D E L C cut is widened by 10%, e.g., from tw — tx > 2 ns to tw — tx > 2.2 ns. "# events" is the number of events remaining in the UMC-generated C E X data after al l UMC-appropr ia te cuts and the cuts in the table are applied. 233 Appendix C. Detailed Cut Descriptions to the target P M T ' s were failing. In this case, events are rejected on a spill-by-spill basis by requiring that the number of KB hits relative to CK hits in a spill be > 0.21 (see section 3.3). Other failures found offline include T D failures: if the "fiducial" pulse (a reference pulse for T D timing) is missing from any T D channel, then the event is rejected. • BADJSTC: There are periods of runs for which the T D pulse-area-to-MeV calibration in a particular R S counter fails. If the stopping counter for an event is one of these counters, then the event is rejected. These types of events can preferentially fool both the T D and kinematic cuts, because something is wrong wi th the pulses in the T D data. T D and kinematic cuts are bifurcated and therefore assumed to be uncorrelated in the background estimate (see sections 4.2 and 4.4.2), so the B A D J S T C cut is applied as a pathology cut when measuring backgrounds. 234 Appendix D Glossary acceptance — a measure of a data-selection cut's abili ty to retain desired events (signal) while removing unwanted events (background) from a data sample. The ac-ceptance of a cut is the fraction of signal events which survive the cut. accidental — describes an energy deposit in some detector element which arises from a real particle which is not associated wi th the decay products of the event kaon. A D C — analog-to-digital converter: hardware used to digitize the energy of a pulse. A G S — Alternat ing Gradient Synchrotron: the primary proton accelerator at B N L . azimuthal angle, <f> — the angle about the center of the detector, perpendicular to the beam axis, which is the angle of cylindrical symmetry of the detector. That is, 0 is the angle in the (x, y) plane that a radial line, originating at the center of the detector, makes wi th respect to the radial line extending horizontally from the center of the detector to the right, as viewed from downstream (see figure 3.7). 4> ranges from 0° to 360°. A positive increment in <fi corresponds to a counter-clockwise displacement (as viewed from downstream). azimuthal angle, UJ — the angle in the (x, y) plane that a radial line, originating at the center of the circular U T C track, makes wi th respect to the radial line extending from the center of the track to the point on the track closest to the kaon decay vertex 235 Appendix D. Glossary (see figure 4.12). to ranges from 0° to 360°. A positive increment in u corresponds to a displacement in the direction of propagation of the track. • B 4 — a hodoscope consisting of 2 planes of 8 fingers of plastic scintillator, placed against the upstream face of the target. • baryon — a particle composed of 3 quarks. • beam strobe — a signal used by the trigger which is defined by the time of the beam particle as detected in the Cerenkov counter or B4 hodoscope, whichever is later. • bias — non-reproducible behaviour of an analysis on independent data samples, due to analysis design using a small number of events which may or may not represent a larger population of events. • bifurcate — prepare a data sample by inverting a cut, and use this data sample to evaluate the performance of another uncorrelated cut. • blind analysis — an analysis of data where the true result is hidden from the person analyzing the data, such that the data itself does not influence the analysis. This approach is useful when only a a small number of signal events are expected from the data, because it avoids the problem of small-statistics bias. In the E787 analysis of K+ —• 7r+uu data, a signal region is defined where the signal/background ratio is expected to be highest, and background and signal characteristics are not defined by examining events in this region. Instead, background contamination is estimated using events which lie outside of the signal region, and is suppressed such that the estimated background in the signal region is <C 1 event. Events in the signal region therefore have a high probability to be signal, and are not counted or examined unti l the background estimates are final. • B M 1 — single-beam background • B M 2 — double-beam background 236 Appendix D. Glossary • BNL — Brookhaven National Laboratory, New York, U S A • boson — a particle wi th integral spin (e.g., spin = 1). • box — a region in a multi-dimensional parameter space where the signal/background ratio is expected to be highest (also known as the "signal region"). • BOX — a cut on the minimum and maximum range, energy, and momentum of a charged track. • BR — branching ratio • B V — barrel veto • B W C l , BWPC1, MWPC1 — upstream beam wire chamber • BWC2, BWPC2, MWPC2 — downstream beam wire chamber • Ctf — a signal indicating detection of a beam kaon by the Cerenkov detector • Cw — a signal indicating detection of a beam pion by the Cerenkov detector • C C D — charge-coupled device. In the case of E787, the C C D ' s are 500 M H z transient digitizers based on a G a A s charge-coupled devices which sample and digitize voltage in 2 ns intervals. • Cerenkov detector — detects charged particles whose speed in a medium is greater than the speed of light, such that a moving dipole is induced in the material, the radiation from which constructively interferes in a cone around the particle's path. The angle of the cone depends on the particle's speed in the material. • C E R N — European Laboratory for Particle Physics, Geneva, Switzerland • C E X — background arising from kaon charge exchange • x2 — a measure of how well a model fits to data, usually defined by 237 Appendix D. Glossary where (xi,yi) are data points wi th standard deviation cr*, and y(xi) is the model being fit. If the CTJ are correct, then a good fit is indicated when the x2 P e r degree of freedom, X 2 /d .o . f . , is equal to 1, where d.o.f. = n — m wi th n the number of data points and m the number of model parameters. • C K M matrix — Cabibbo-Kobayashi-Maskawa quark mixing matrix • C M — microcollar detector • C O — collar detector • CP — quantum-mechanical symmetry operation defined by the combination of charge conjugation (C) and parity inversion (P) • ct — charged track: used to denote a R S counter which is in the T • 2 track sector or within 2 sectors clockwise (as viewed from downstream) of the T • 2 track sector. • cut — hardware (online) or software (offline) requirement that an event must satisfy certain criteria. Hardware cuts make up a "trigger" which is used to acquire data, and software cuts are part of a computer program which is used to analyze the data after acquisition. Cuts are used to separate background events from signal events. • D C — "delayed coincidence" trigger requirement • delayed coincidence — requirement that the kaon decay products be detected later than the kaon, such that the kaon decays from rest. • dE/dx — energy loss of a particle per unit distance travelled in a material. • detector strobe — a signal used by the trigger which is defined by the time of coincident hits in the T and 2 counters on the track. • DIF — decay in flight: characterizes a particle which decays while in motion. • dip angle — see polar angle. 238 Appendix D. Glossary • discriminant function analysis — an analysis of various parameters associated wi th two different data sets wi th the intention of forming a linear combination of the parameters which is maximally different for the two data sets. • discriminator — logic hardware which takes an analog pulse as input, and outputs a constant voltage signal of variable duration when the input pulse is above an adjustable threshold voltage. • downstream — in the direction of the kaon beam momentum • E — total energy deposited by the track particle in the detector. • Efj, — energy of the second pulse in the stopping counter (e.g., for pion tracks, this is usually the energy of the muon from pion decay). • ERS — energy deposited by the track particle in the R S . • ETc — energy deposited by the track particle in the target. • EJPQ — total energy at kaon decay time in target fibers located on the non-track side of the kaon decay fiber. • El — energy of a pulse as recorded on the upstream end of a R S or barrel counter. • E2 — energy of a pulse as recorded on the downstream end of a R S or barrel counter. • E787 — Experiment 787 at B N L : search for K+ -> TX+VV • E926 — Experiment 926 at B N L : search for K°L -> -K°VV • E949 — Experiment 949 at B N L : successor experiment to E787 • E C — C s l endcap detector. Also used to refer to the E C trigger requirement. • electromagnetic shower detector — detects electrons and high-energy photons. High-energy photons (gamma rays) are detected v ia their conversion into e + , e~ pairs, which has highest efficiency in the field of atoms of high Z (atomic number). This 239 Appendix D. Glossary is followed by emission of bremsstrahlung photons by the electrons, which in turn convert into more e + , e~ pairs unti l all the energy of the original photon is dissipated in a "shower" of electrons. The scintillation light created by the electrons can be converted into an electrical pulse by a P M T . • event — a single kaon entering the target and decaying into the fiducial region of the detector. • f8 — stopping fraction: a "fudge" factor used to account for the fraction of kaons which stop in the target, and any other cut acceptances/efficiencies that are not calculated explicitly. • F C N C — flavour-changing neutral current • Fermilab — see F N A L • fermion — a particle wi th half-integral spin (e.g., spin = 1/2). • fiducial pulse — an electronically-generated pulse which is used for reference when finding the times of pulses in T D data. • flags — see T D flags. • F N A L — Fermi National Accelerator Laboratory, Chicago, U S A • function cut — an offline cut which is inverted to define a background data sample, on which the performance of another, uncorrelated function cut is evaluated in order to make a bifurcated background estimate. A function cut is designed on a sliding scale in terms of a parameter N (scaled number of background events) or A (acceptance), so that it can be loosened to perform the outside-the-box correlation test (see sections 4.2 and 4.6), and tightened to estimate the "likelihood" of an event to be a signal or background event (see sections 4.2 and 4.7). • F W H M — full width of a peak at the half-maximum value 240 Appendix D. Glossary • gamma fiber — target fiber with energy at kaon decay time, but located away from the pion track. • G D R background — giant dipole resonance background, whereby positively-charged muons excite the giant dipole resonance in 1 2 C , which de-excites by emitting a single neutron. The neutron travels slowly and leaves an isolated second pulse in the counter where the muon comes to rest, faking the double-pulse TT —> /J, decay signature in the stopping counter. The stopping counter information is therefore indicative of a pion track instead of a muon track. • geometric mean energy — y/El x E2, where El and E2 are upstream and down-stream energies, respectively. • G I M mechanism — Glashow-Iliopoulos-Maiani mechanism, whereby flavour-changing neutral weak currents are forbidden. • G U T — grand unified theory: theory which unites the electromagnetic, weak, and strong forces into a single mathematical description. • hadron — any particle which is a bound state of quarks (see meson and baryon). • H E X — trigger requirement for summed hextant energies in the R S . • hextant — a group of 4 adjacent R S sectors, which are multiplexed by layer into the T D ' s . • hodoscope — a combination of several detector elements arranged in space such that particle tracks can be identified. Hodoscopes are usually constructed from scintillation counters which have short-duration output pulses which can be used for triggering purposes. • IC — I-counter. Also used to refer to the IC trigger requirement. • KB — a coincidence signal between the CK, summed B4 , and summed target signals indicating the presence of a beam kaon. 241 Appendix D. Glossary • Ksuve — KB signal in coincidence wi th the "computer ready" signal. • — K+ —> n+v^ decay • peak — kinematic region characterized by range, energy, and momentum values consistent wi th the p+ from decay (R = 54 cm, E = 152 M e V , P = 236 M e V / c ) . • range tail — kinematic region characterized by the expected peak value of momentum, but range (and energy) values smaller than the peak values due to elastic (inelastic) scattering of muons in the detector. • Kfj,3 — K+ --> 7 r V + t V decay • ir+e~ue decay • KU — Kl- • 7T+/J,~UU decay • - K+ —> p + i ' f i j decay, also called radiative decay • Kn2 K+ -•> 7 T + 7 r ° decay • Kn2 peak — kinematic region characterized by range, energy, and momentum values consistent wi th the TT + from Kn2 decay (R = 30 cm, E = 108 M e V , P = 205 M e V / c ) . • K n 2 range tail — kinematic region characterized by the expected peak value of momentum, but range (and energy) values smaller than the peak values due to elastic (inelastic) scattering of pions in the detector. • kaon fiber — target fiber wi th pulse time and energy and location consistent with the event kaon. • K E K — Koh-Enerugi i Kasokuki Kenkyu K i k o u (High Energy Accelerator Research Organization), Japan • layer — 1 of 21 radial segments of the R S 242 Appendix D. Glossary • LESBIII — low-energy separated beamline III: the kaon beamline at B N L which provides a high intensity and relatively high purity kaon beam for use by E787. • L I N A C — linear accelerator: a machine which accelerates charged particles through alternating voltages in a straight path. A L I N A C is used at the first stage of particle acceleration at B N L , to accelerate H ~ ions. • Ikbeam — a kaon likelihood quantity formed from the energy deposited in the B4 hodoscope, the UTC-extrapolated kaon stopping z in the target, and the kaon energy in the target. • L L A — Leading-Order Logarithmic Approximat ion • meson — a particle composed of a quark, anti-quark pair. • minimum ionizing particle — a moderately relativistic charged particle which loses energy in a medium primarily through ionization at a rate of about 2 M e V • c m 2 / g , almost independent of the medium. For particles travelling faster than the atomic electrons (i.e., faster than about c|g|/137, where c is the speed of light and \q\ is magnitude of the particle's charge in units of the proton charge e), the mean rate of energy loss in a medium \dE/dx\ is given by the Bethe-Bloch equation [16], and init ial ly falls as 1//32, where /? is the particle's velocity in units of c. The mean rate of energy loss reaches a broad minimum at 7 ~ 3.2, where 7 = (1 — / 3 2 ) - 0 5 . The energy loss rate increases slowly for 7 > 4 so, in practical cases, most relativistic charged particles have energy loss rates close to the minimum, i.e., 2 M e V • c m 2 / g , almost independent of the medium. • Monte Carlo — see U M C . • muon band — kinematic region in range-momentum space wi th correlated values of range and momentum consistent with a track, and values of range and momentum which are smaller than the K^2 peak values. Events in the muon band can arise from 243 Appendix D. Glossary K+ -»• n+u^ (radiative K^), K+ —> n°p+u^ (K^), decay in flight, and/or Ka2 decay wi th inelastic scattering in the target. • N I D I F — pion "nuclear interactions and decay in flight", which can be turned on or off when generating Monte Carlo data (i.e., when propagating pions through the detector). • N L L A — Next-to-Leading-Order Logarithmic Approximat ion • n o r m a l i z a t i o n b r a n c h — one of two branches in a bifurcated background estimate. The events in this branch are required to fail a cut whose rejection is measured in the "rejection" branch. A l l other cuts in the analysis are applied to these events, and the number that remain is referred to as the background "normalization". This normalization, divided by the rejection calculated in the rejection branch, gives an estimated number of background events. • n t u p l e — a computer data file containing quantities associated wi th each event. N tu -ples for the E787 analysis are produced and read by P A W . • offl ine — any system (e.g., software cuts) which operates on data once it has been collected and stored on disk/tape. • o n l i n e — any system (e.g., digitizing hardware, trigger) that operates on the data as it is being collected. • P — total momentum of a track. • P A — pulse area • P A S S l — in i t ia l processing of the raw data, wi th the intention of reducing the total data volume by a factor of about 10 while maintaining high acceptance for K+ —> n+vv. • P A S S 2 — separation of the P A S S l output data into background data samples suitable for studying background processes. 244 , Appendix D. Glossary • P A S S 3 — high-level analysis of the P A S S 2 output data samples, and storage of various quantities in ntuples for subsequent study using P A W . • pathology cut — an offline cut that is typically applied at the ini t ia l stages of a bifurcated analysis in order to remove events which can contaminate the bifurcated data samples wi th correlations. • P A W — Physics Analysis Workstation: data analysis and plott ing program developed at C E R N . • P H — pulse height • PID — particle identification • PID cuts — see T D cuts. • pion band — kinematic region in range-momentum space wi th correlated values of range and momentum consistent with- a TT+ track. Events in the pion band can arise from beam pions which scatter into the detector, Kn2 decay in flight, and/or Kn2 decay wi th inelastic scattering in the target. • pion fiber — target fiber wi th pulse time and energy and location consistent wi th a charged kaon decay product which is subsequently tracked in the U T C . • Kscat background — potential background for K+ —> ir+vi> due to pions in the beam which scatter into the detector, giving rise to a pion track wi th similar kinematics as K+ —> IT+VI> decay. • P M T — photo-multiplier tube: consists of a photocathode and a system of "dyn-odes". Scintil lation or Cerenkov light (produced by charged particles in a medium) is collected at the photocathode, where the light kicks out electrons v ia the photoelectric effect. The electrons are accelerated through a sequence of voltages applied to cathodes (dynodes), each acceleration knocking out more electrons at each cathode surface such that the resulting "avalanche" of electrons results in a detectable electrical pulse. 245 Appendix D. Glossary • 7r+vu(l) — refers to the K+ —> -K+UU signal region kinematically located between the monochromatic and peaks. • n+ui>(2) — refers to the K+ —> n+uu signal region kinematically located just below the KW2 peak. • polar angle, 0 — the angle in the (r, z) plane between the beam axis and the U T C track. 0 ranges from 0° (track propagates directly downstream from the kaon decay vertex) to 180° (track propagates directly upstream from the kaon decay vertex). • P V — photon veto • Q C D — quantum chromodynamics: quantum field theory which describes the strong force. • Q E D — quantum electrodynamics: quantum field theory which describes the electro-magnetic force. • quantum field theory — see Q C D and Q E D . • R — total range of a track. • RTG — range of a track in the target (as found from an extrapolation of the U T C track from the kaon decay vertex to the outer edge of the target). • reconstruction cut — an offline data-reduction cut that requires an event to have a charged track reconstructed in the target, U T C ,