A Search for Rare Decay D0 --> mu+mu - High Energy Physics UPRM


A Search for Rare Decay D0 --> mu+mu - High Energy Physics UPRM

A Search for the Rare Decay D£¥¤¦£¨§ ¢¡byHugo R. Hernández MoraThesis submitted in partial fulfillment of the requirements for the degree ofMASTER OF SCIENCEinPhysicsUNIVERSITY OF PUERTO RICOMAYAGÜEZ CAMPUS2001Approved by:Angel M. López Berríos, Ph.D.President, Graduate CommitteeDateHéctor Méndez, Ph.D.Member, Graduate CommitteeDateDorial Castellanos, Ph.D.Member, Graduate CommitteeDateWolfgang A. Rolke, Ph.D.Representative of Graduate StudiesDateDorial Castellanos, Ph.D.Chairperson of the DepartmentDateL. Antonio Estévez, Ph.D. DateDirector of Graduate Studies

AbstractThis thesis presents a search for the rare decay D© based on data collectedby the Fermilab fixed target photoproduction experiment called FOCUS (FNAL-E831).This decay is an example of a flavor-changing neutral current process which, accordingto the Standard Model, should occur at a very low branching ratio of at most .A rate higher than this would be an indication of new physics. Using a blind analysismethod which consists of the observation of events outside a defined signal area, theoptimum selection criteria for candidate events was determined. This method minimizesbias by basing the choice on the background observed outside the signal region and noton maximizing the signal under study. The ratio of background outside to backgroundinside the signal region was determined using an independent real data sample. The resultfor the branching ratio of the D© decay at 90% Confidence Level upper limit iswhich reduces the published value by 50%. No evidence was found for this raredecay."! ii

ResumenEste trabajo de tesis presenta la búsqueda del decaimiento raro D© basadoen datos colectados por el experimento de fotoproducción de blanco fijo de Fermilabllamado FOCUS (FNAL-E831).Este decaimiento es un ejemplo de un proceso decambio de sabor a través de corrientes neutrales (FCNC) el cual, de acuerdo con elModelo Estándar, debería ocurrir a una muy baja probabilidad a un de orden . Unaprobabilidad mayor a ésta podría ser una indicación de nueva física. Usando un métodode análisis a ciegas que consiste en la observación de eventos fuera de una región definidacomo área de señal, se determinó un criterio óptimo de selección en la búsqueda de eventoscandidatos. El uso de este método de análisis permitió minimizar el prejuicio presente enla elección de nuestro criterio cual estaba basado en observar eventos de ruido fuera delárea de señal con la idea de no maximizar la señal bajo estudio. La razón de ruido fuera delárea de señal al ruido dentro de ésta se determinó usando una muestra real independiente.El resultado para la razón de porcentaje para el decaimiento D© mostró un límitesuperior (con un nivel de confianza de 90%) de#$ "! , el cual reduce en un 50% elvalor publicado hasta ahora. No se encontró evidencia alguna para este decaimiento raro.iii

To God,for my country: C O L O MBIA,to my beautiful city B A R R ANQUILLA,to my wife Diane,to my parents Fanny and Anthony and,to my brothers Madelaine and Jorge Luis.Los quiero muchísimo!!!To the thousands of countrymen who have fallen whileI have been in the “Isla del Encanto” because of the absurdconflict that lives within the boundaries of my harmedcountry, specially to a person who was an example to me andmy professional guide, Professor Lisandro Vargas Zapata,cowardly murdered by gunmen on february 23, 2001 at hisdoorstep in Barranquilla.iv

AcknowledgmentsThere are so many people that in one way or another have contributed in mydevelopment as a human being and as a professional.To mention them all would beimpossible. It is very difficult to even mention some before others. The most importantthing is that all of those who believed in me, will always hold a key to my heart.First of all. I thank God, that entity who has given us life and with that, a great numberof questions that we try to answer day by day. I thank Him for giving me the opportunity ofbeing born in C O L O MBIA, a country that even now, when it is going through a crisis, hasa face unknown to many. The face of a country that always fought for progress, of a hardworking country that is proud of its roots, that cries for those who have fallen and missesthose who are missing. A country that screams for freedom, but above all, there is a cry ofhope that the armed conflict that harms us will end.%'&(*),+.-)0/12&3&547698.+.-:1),?@? !I don’t want to leave behind my city, B A R R ANQUILLA, my “Curramba la Bella”.The city that watched me grow up. On her streets I learned to live life to the extreme. Itwas there where my life is centered and the crib of my professional future. I have greatexpectations for my professional development and I wish to offer those to my loved city.v

I thank Fanny Mora, my mother, for making me into what I am today. Without herunconditional help I wouldn’t heve gotten to this point. To her above all, I dedicate thistriumph. To my brothers, Madelaine and Jorge Luis, for believing in me and for taking careof my mother while I was absent. To my father Anthony Valle, for always being there inthe good times as well as in the bad times.P U E R TO RICO is a country that I will always have with me. A master’s degree, amarriage and a family. All this I take from you “Isla del Encanto”. I thank my wife Dianafor her unconditional help. Thank you for sharing my good times and my sorrow.An immensurable thanks to the friendship, professional and economic support thatDr. Angel M. López gave me. Thank you for everything you did during the time I workedfor you, and for always believing in me and trusting me. To professors Will E. Johns andHéctor Méndez for helping me out without asking and also to Dr. WeiJun Xiong. Thanksto Dr. Wolfgang Rolke for your cooperation in the statistical analysis for this work. Yourcontribution was of great importance for me. Overall, to all of those who work with me atthe HEP lab at Mayagüez. Those from the past: Alejandro Mirles, Carlos Rivera, MauricioSuárez, Salvador Carrillo, Fabiola Vázquez and Kennie Cruz; and those from the present:José A. Quiñones, Eduardo Luiggi, Alexis París, Luis A. Ninco and Carlos R. Pérez. Thankyou guys!!!To the Physics Department at Mayaguez Campus and its director when I became apart of the University, Dr. Dorial Castellanos (January, 1998), who gave me chance to makeit happen. I that all of those at the FOCUS Collaboration for giving me the opportunityvi

to work on their experiment. To all those people who work at EPSCoR but most of all toSandra Troche who was always there for me.To the great friendships I made in PR. To Juan Carlos Hernández, Carlos José andspecially to their parents José “Machito” Hernández and Juanita González. You were thefamily I needed throughtout these three years.Thanks to Mr. Joe Ortíz for helping me in my college studies, thank you for helpingme pursue the grounds of PR. I also thank Mr. Félix Rivera, and the Romero Valle familyas well as Valle Miranda family. Thanks for everything. I thank Mr. Rafael Sanjuan andhis family for being there when I needed it. A special thanks to Mrs. Martha Castaño, mayGod bless your noble soul.I thank my professors at the Universidad del Atlántico - Barranquilla: ArnaldoCohen and Oswaldo Dede, for making me look at science. To my colleagues and friendsArnulfo Barrios, Oscar Montesinos, Harkccynn Torregroza, Ericka Navarro and RafaelManga. I also thank my High School teachers from the Bienestar Social de la Policía- Barranquilla: Blas Tórres, Carlos Bray and Orlando Ortíz, specially Mr. ArmandoConsuegra (Barrabás).To all of these whom I have mentioned as well as those whom I have not, otherwiseI would never be able to finish...Mil y mil gracias!!!vii

AgradecimientosSon muchas las personas que de una u otra forma han contribuido en mi desarrollocomo ser humano y como profesional. Enumerarlas a todas resultaría imposible. Inclusivemencionar algunas primero que a otras resulta difícil para mí. Lo mas importante quequiero manifestar es que a todas estas personas que creyeron en mí por siempre las llevaréen mi corazón.Inicialmente quisiera darle las gracias a Dios, a ese ser que nos ha otorgado la viday con ella un sinnúmero de interrogantes que día a día tratamos de contestarnos. A Él leagradezco el haberme dado la oportunidad de nacer donde nací, mi patria C O L O MBIA, unpaís que aun cuando en estos momentos atraviesa por momentos de crisis posee una caraque muchos de nosotros desconocemos. La cara de un pueblo que por siempre ha luchadopor salir adelante, de un pueblo trabajador y orgulloso de sus raices, que llora a sus hijoscaidos y extraña a sus hijos ausentes. Un pueblo que lanza a todo pulmón un grito delibertad, pero sobre todo un grito de esperanza por la culminación de este conflicto armadoque nos acongoja. Queremos un país libre para todos!No quiero dejar de lado a mi B A R R ANQUILLA del alma, a mi Curramba la Bella.La ciudad que me vió crecer. En sus calles aprendí a vivir la vida en cada uno de losviii

extremos que ésta nos presenta. Y es allí donde se centra mi vida y mi futuro profesional.Resultan inmensas las expectativas de poder desarrollarme como profesional y ofrecer misservicios a la ciudad de mis amores.Le agradezco a mi madre, Fanny Mora, por hacer de mí lo que hoy en día soy. Sin elapoyo incondicional de parte de ella no hubiese sido posible la consecusión de cada uno demis éxitos. A ella más que a nadie dedico este triunfo que hoy día estoy consiguiendo. Amis hermanos, Madelaine y Jorge Luis, por creer en mí y por cuidar de mi madre durantemi ausencia. A mi padre, Anthony Valle, por estar allí siempre, tanto en las buenas comoen las malas.P U E R TO RICO es una tierra que llevaré conmigo por siempre. Una maestría, muchosamigos, un matrimonio y una familia. Todo esto me llevo de tus suelos “Isla del Encanto”.Agradezco enormemente la ayuda incondicional recibida de parte de Diana, mi esposa.Gracias por compartir a mi lado mis alegrías y soportar además las tristezas y los malosratos.Un agradecimiento más que enorme a la amistad, el apoyo profesional y económicoque me brindó el Dr. Angel M. López. Gracias por todo lo que hizo por mí durante elperiodo en el que estuve trabajando a su lado, por creer siempre en mí y por brindarme suconfianza. A los profesores Will E. Johns y Héctor Méndez por la ayuda que me brindaronsin nunca recibir una negativa de su parte. Al Dr. WeiJun Xiong por su colaboración y suapoyo a mi trabajo. Gracias por la ayuda brindada por parte del Dr. Wolfgang Rolke. Sucolaboración en el análisis estadístico para este trabajo de tesis fue de gran importancia.En general a todos los compañeros del laboratorio de HEP en Mayagüez. Los del pasado:ix

Alejandro Mirles, Carlos Rivera, Mauricio Suárez, Salvador Carrillo, Fabiola Vázquez yKennie Cruz; los del presente: José A. Quiñones, Eduardo Luiggi, Alexis París, Luis A.Ninco y Carlos R. Pérez. Muchachos, gracias!!!Al Departamento de Física del RUM y su director en el periodo (Enero de 1998)en que inicié estudios graduados en UPR. A ellos les agradezco el haberme brindado laoportunidad para que esto fuese una realidad. Agradecimientos muy especiales a todala gente de FOCUS Collaboration por darme la oportunidad de trabajar en este granexperimento. A todas las personas que trabajan en EPSCoR en especial a Sandra Trochepor su incondicional ayuda y por su constante colaboración.A las grandes amistades construidas durante mi permanencia en PR. A Juan CarlosHernández, Carlos José y en especial a sus padres José “Machito” Hernández y JuanitaGonzález. Ustedes fueron el soporte familiar que tanta ayuda me hizo en estos últimos tresaños tiempo que duró mi maestría.Gracias especiales a Joe Ortíz por apoyarme incondicionalmente en mis estudiosuniversitarios, gracias por tu patrocinio en mi empresa hacia PR. Igualmente le agradezcoa Félix Rivera, a la Familia Romero Valle al igual que a la Familia Valle Miranda. Graciaspor su patrocinio. Agradezco a Rafael Sanjuan y familia por siempre estar cuando losnecesitábamos. Gracias. A Martha Castaño, un millón de bendiciones para usted mi belladama.Agradezco a mis profesores de la Universidad del Atlántico - Barranquilla: ArnaldoCohen y Oswaldo Dede, por inculcarme los deseos de hacer ciencia. A mis colegas yx

amigos de universidad, Arnulfo Barrios, Oscar Montesinos, Harkccynn Torregroza, ErickaNavarro y Rafael Manga. Igualmente agradezco a todos mis profesores de Bachillerato enel Bienestar Social de la Policía - Barranquilla: Blas Tórres, Carlos Bray y Orlando Ortíz,especialmente a Armando Consuegra (Barrabás).A todas las personas que he mencionado al igual que a las muchas que dejé demencionar pues nunca terminaría este documento...Mil y mil gracias!!!xi

ContentsList of TablesxviiiList of FiguresxixList of Abbreviationsxxi1 Introduction 11.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Fundamental Particles . . . . . . . . . . . . . . . . . . . . . . . . 3Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Long and Short Distance Effects . . . . . . . . . . . . . . . . . . . 9Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10xii

1.2.1 Conservation of Angular Momentum and Helicity Suppression . . . 111.2.2 Conservation of the electrical charge . . . . . . . . . . . . . . . . . 121.2.3 Lepton Number Conservation . . . . . . . . . . . . . . . . . . . . 131.2.4 Baryon Number Conservation . . . . . . . . . . . . . . . . . . . . 141.2.5 Flavor Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 151.3 Cabibbo Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16a) Cabibbo-Favored . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19b) Cabibbo-Suppressed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20c) Doubly-Cabibbo-Suppressed . . . . . . . . . . . . . . . . . . . . . . . . 211.4 GIM Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5 Rare and Forbidden Decays . . . . . . . . . . . . . . . . . . . . . . . . . . 24a) Lepton Family Number Violation . . . . . . . . . . . . . . . . . . . . . 24b) Lepton Number Violating . . . . . . . . . . . . . . . . . . . . . . . . . 24c) Flavour-Changing Neutral-Currents . . . . . . . . . . . . . . . . . . . . 241.6 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.7 Confidence Intervals and Confidence Levels . . . . . . . . . . . . . . . . . 292 Previous Works 302.1 FCNC Decay Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30xiii

2.2 Previous Searches for Rare Charm Decays . . . . . . . . . . . . . . . . . . 322.2.1 The Hadroproduction WA92 Experiment . . . . . . . . . . . . . . 332.2.2 The Hadroproduction E791 Experiment . . . . . . . . . . . . . . . 342.2.3 The Photoproduction E687 Experiment . . . . . . . . . . . . . . . 362.3 FOCUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3.2 The FOCUS Photon Beam . . . . . . . . . . . . . . . . . . . . . . 392.3.3 Inside the FOCUS Experiment . . . . . . . . . . . . . . . . . . . . 403 Objectives 444 Procedure 464.1 Data Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Reconstruction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3 Data Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.1 Selection Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 50a) Vertexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501.ACB

4. Vertex z Positions . . . . . . . . . . . . . . . . . . . . . . . . 525. Vertex Isolations . . . . . . . . . . . . . . . . . . . . . . . . 53b) Particle Identification . . . . . . . . . . . . . . . . . . . . . . . 531. Muon Identification . . . . . . . . . . . . . . . . . . . . . . . 53i. Kaon Consistency for Muons . . . . . . . . . . . . . . . . . 54ii. Inner Muons . . . . . . . . . . . . . . . . . . . . . . . . . 55A) Number of Missed Muon Planes . . . . . . . . . . . . . 56B) Inner Muon Confidence Level . . . . . . . . . . . . . . 56iii. Outer Muons . . . . . . . . . . . . . . . . . . . . . . . . . 56A) Outer Muon Confidence Level . . . . . . . . . . . . . . 562. Hadron Identification . . . . . . . . . . . . . . . . . . . . . . 56i. Kaonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 57ii. Pion Consistency . . . . . . . . . . . . . . . . . . . . . . . 573. Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 584. Dimuonic Invariant Mass . . . . . . . . . . . . . . . . . . . . 584.3.2 FOCUS Data Selection Stages . . . . . . . . . . . . . . . . . . . . 59a) Pass1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59b) Skim1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59c) Skim2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59xv

5 Results and Conclusions 815.1 MisID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Background Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.3 Skimming Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.4 Cut Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.5 Confidence Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Bibliography 92xvii

List of Tables2.1 Chronological branching ratios for weak neutral-current decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1 Description of Superstreams . . . . . . . . . . . . . . . . . . . . . . . . . 61D©5 theGIHKJL4.2 Description of Superstream 1 . . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Sensitivity number as per Rolke and López . . . . . . . . . . . . . . . . . 694.4 An example of Rolke and López 90% Confidence Limits . . . . . . . . . . 704.5 Dimuonic Invariant Mass Cuts for Sidebands and Signal Regions . . . . . . 794.6 Cut values for each variable considered in the optimization process . . . . . 805.1 Best Cut Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2 Results of Cut Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 88xviii

List of Figures1.1 The Standard Model for Elementary Particles . . . . . . . . . . . . . . . . 51.2 Transformations permmited by the Standard Model . . . . . . . . . . . . . 81.3 Feynman diagram a weak process mediated by a virtual boson exchange. . . 101.4 A charged current interaction . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Feynman diagram for a weak transition . . . . . . . . . . . . . . . . . . . 171.6 Representation for the CKM matrix elements . . . . . . . . . . . . . . . . 201.7 Feynman diagram for a Cabibbo-Favored process . . . . . . . . . . . . . . 201.8 Feynman diagram for a Cabibbo-Suppressed process . . . . . . . . . . . . 211.9 Feynman diagram for a Doubly-Cabibbo-Suppressed process . . . . . . . . 211.10 Weak interactions in neutral-current processes . . . . . . . . . . . . . . . . 222.1 Feynman diagrams for the decay D© . . . . . . . . . . . . . . . . 312.2 Proton synchrotron and beamlines at Fermilab . . . . . . . . . . . . . . . . 382.3 The E831 Photon Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39xix

2.4 The FOCUS Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Trajectories through the FOCUS spectrometer . . . . . . . . . . . . . . . . 494.2 Production and decay vertices for the D© process . . . . . . . . . 514.3 Analysis overview of FOCUS . . . . . . . . . . . . . . . . . . . . . . . . 604.4 Monte Carlo generated process schematization for the decay D© . 664.5 D© Invariant Mass distributions . . . . . . . . . . . . . . . . . . 714.6 Dimuonic Invariant Mass comparison of Monte Carlo data fromand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75. D© E E D©4.7 Dimuonic invariant mass of D© KE generated by Monte Carlo . . . . 764.8 Background comparison between real and Monte Carlo data . . . . . . . . 775.1 MisID as a function of momentum . . . . . . . . . . . . . . . . . . . . . . 825.2 Typical Invariant Mass Distribution Used in Calculating the BackgroundRatio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3 Invariant Mass Distributions for Skim3 Cuts . . . . . . . . . . . . . . . . . 845.4 Invariant Mass Distributions for Skim4 Cuts . . . . . . . . . . . . . . . . . 855.5 Mass distributions for the best cut combination. . . . . . . . . . . . . . . . 875.6 Invariant Mass Distribution for Background Ratio using the best cutcombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.7 Event candidates for the D© process . . . . . . . . . . . . . . . . 90xx

List of AbbreviationsK/ML/Nerror,DBeO:Beryllium Oxide.BR:Branching Ratio.CERN:Organisation Européenne pour la Recherche Nucléaire.CITADL: Čerenkov Identification Through A Digital Likelihood.CKM Matrix: Cabibbo-Kobayashi-Maskawa Quark-Mixing Matrix.CP Violation: Charge-Parity Violation.CL:Confidence Level.CLP:Primary Vertex CL.CLS:Secondary Vertex CL.E687: Fermilab Experiment 687.E831: Fermilab Experiment 831.E791: Fermilab Experiment 791.Fermilab: Fermi National Accelerator Laboratory.FNAL-E831: Fermilab Experiment 831.FCNC:Flavor-Changing Neutral-Current.FOCUS: Fotoproduction Of Charm with an Upgraded Spectrometer.GIM Mechanism: Glashow-Iliopoulos-Maiani Mechanism.GeV:Giga Electron Volt.IMU:Inner Muon System.IMUCL: Inner Muon CL.ISOP:Primary Vertex Isolation.ISOS:Secondary Vertex Isolation.:Kaon Consistency to separate kaons from muons.LFNV:Lepton Family Number Violation.LNV:Lepton Number Violation.:Separation between the production and decay vertices divided bythe , on that measurement.MeV:Mega Electron Volt.MisID:Percent of events misidentified within any specific analysis.MISSPL: Number of Missed Muon Planes.M1: Magnet Number 1.xxi

M2: Magnet Number 2.: Lorentz Dimuonic Invariant Mass.OQPRPOMU:OMUCL:PAW:Pass1:PWC:Skim1:Skim2:Skim3:Skim4:SSD:ZVPRIM:ZVSEC:)::S:W(T*UTVXWZYNC[7\NC[^] :Outer Muon System.Outer Muon CL.Physics Analysis Workstation.Basic event reconstruction stage.Proportional Wire Chambers.First data reconstruction selection stage.Second data reconstruction selection stage.Third data reconstruction selection stage.Fourth data reconstruction selection stage.Silicon Microvertex Detectors.Primary Vertex z Position.Secondary Vertex z position.Kaonicity.Pion Consisitency.Difference between the Primary Vertex z position and the z targetedge divided by the uncertainty.Difference between the Secondary Vertex z position and the ztarget edge divided by the uncertainty.xxii

_ © F ). The main goal of the work reported in this thesis was the measurement ofChapter 1IntroductionThis thesis is a research work in a topic of high energy physics. It is also a formativework and could be used as a source of reference for students and people interested inknowing more about the physics related with the microscopic world around us.The Standard Model of electroweak interactions is used to understand the decaysof heavy quarks which are known to us.A significant observation of charm Flavor-Changing Neutral-Current decays, predicted to occur rarely (in the Standard Model theneutral-current interactions do not change flavor), would imply new physics beyond theStandard Model.Rare decay modes are probes of particle states or mass scales which cannot beaccessed directly.One of the most interesting processes to study is the D© .decay (all references to this decay also implies the corresponding charge-conjugate state1

1.1 The Standard Model 2`the branching ratio for this decay and a comparison to the actual 90% Confidence Levelupper limita0b "! .This chapter will display some basic and interesting concepts used for this study.First, an introduction to the Standard Model of elementary particles is done, next some ofthe conservation laws are discussed. The Cabibbo Theory and the GIM Mechanism willalso be discussed in this chapter. Basic information about Rare and Forbidden decays willbe shown defining three important related processes: Lepton Family Number Violation,Lepton Number Violation and Flavor-Changing Neutral-Currents. A short representationof Special Relativity will be discussed and finally some useful concepts about ConfidenceIntervals will be defined.1.1 The Standard ModelAt the beginning the2of century, physicists thought that all matter was composedof atoms as the fundamental building block. As the century entered adolescence, it wasdiscovered that atoms are constructed from electrons and a small positive charged nucleus.

particle 2 is in the first. Therefore, under interchange the wave-functionmIp§ q1.1 The Standard Model 3theory has been developed called the Standard Model [1–3]. The model describes all matterand forces in the universe (except gravity) and it is a package of two theories: the Unified`Electroweak [4, p.95] and Quantum Chromodynamics [4, p.85].1.1.1 Fundamental ParticlesThe Standard Model divides the elementary particles into two groupswhich are differentiated according to the behavior of multiparticle systems.calledfg62h369+hkl6=+h and/&i1

families of matter. These are divided into quarks with fractional electric chargesn(+o) and leptons with integer charge ) orn&no charge (neutrinos). At the samen n& (§ n n& ‚ and§‚1.1 The Standard Model 4`interchange must go to minus itself(1.2)rs J § mIpq rsFor non relativistic speeds, the total wave-function of the pair can be expressed as aq mIp§product of functions depending on spatial coordinates and spin orientation:(1.3)JuxpyhwzF),4Z& s:{ pyhwzkl+ swhere the spatialnmIpq, will describe the orbital motion of one particle about the other. Itrswvpart,xcan thatx happen is symmetric or antisymmetric under interchange and the function{spinmay be symmetric (spins parallel) or antisymmetric (spins antiparallel) under interchange.Bosons. They are: photon, gluon, W| and Z© . These particles have an integral spin(0, 1, 2,...) and are known as the intermediary particles because they are responsible forcarrying the force interactions. The photon is massless with spin 1 and is the intermediaryof the electromagnetic interaction. There is one kind of photon and two photons do notdirectly interact with one another. The gluon is massless with spin 1 and is the intermediaryof the strong interaction. There are eight types of gluons, each different from one anotheronly by a quantum number called color. The gluons interact with one another. The) and ), all of spin 1, are the intermediaries of the weakW|91.2~€Z (m} Z© 80.4~€Z (m}interaction. The W| and Z© do not interact with one another.Fermions. They have a half-integral spin ,...) and constitute the fundamental,‚o o (

1.1 The Standard Model 5`time, quarks are subdivided into six different particles: up (m} 1 to 5ƒ„7 ), charm1.15 to ) (m} and top 169 of¤to ) with electric charges (m} anddown 3 to ), strange 75 to ) and bottom179~€Z n& n4.0 to )o 1.35~…74.4~…7 (m} 170ƒ„Z ‚(m} (m} 9ƒ„7 . The six types of quarks are sometimes called flavors. Leptons are subdividedwith§into three different electrically charged particles,n& n0.511ƒ„7 the ), (m} electron,& the‚(m} ), and (m} 1.777~…7 the ), with their respective muon, 105.7ƒ†Z neutraltau,‡(mŠ 3 ) andˆ9Œ (mŠ ). Each particle,18.2ƒ„Z neutrinos,ˆ=‰ (mŠ eV),ˆ9‹quark or lepton, has its respective antiparticle with the same mass but with0.19ƒ„7opposite electriccharge. Figure 1.1 shows the fundamental particles of matter arranged into families by theStandard Model.Figure 1.1: The Standard Model for Elementary Particles.

The Standard Model 6Each of the three columns corresponds to a “family”. The relationships among the`1.1particles in a family are the same as those in the other two families. In a way we have threecopies of the same structure. In each of the families there is a lepton and a quark doublet.The charge of the members of the doublets differ by one& unit.The main differencebetween families are the masses of their members. The first family members are low masswhile the masses of the third family are much higher. This large difference in mass is oneof the principal open mysteries in elementary particle physics.1.1.2 InteractionsOf the known four forces, two, gravity and electromagnetism, have a long range andare common in every day life. The remaining two forces, the strong and weak nuclearforces, have such a small range that it is frequently compared to the size of an atomicnucleus.Due to its low strength, gravity is still not a complete part of the StandardModel. Another interaction that has an infinite range is the electromagnetic interactionwhose effects have been measured throughout immense scales.The strong interactionholds together the nucleus working against the electrostatic repulsion that occurs betweenprotons. Lastly, the weak interaction is known to be responsible for a large variety ofradioactive decays. Also, it is responsible for the interactions that occur between neutrinosand matter.Besides electric charge, quarks and gluons have another type of charge. It is calledthe color charge. The effects of the strong nuclear force or color force are only felt byquarks. It is a very strong force that binds quarks together forming particles. There are

1.1 The Standard Model 7three “colors” for quarks which are usually called “red”, “blue” and “yellow”. If you puttogether a red quark with an antired antiquark, the product is colorless. If you put together`a red, a blue and a yellow quark, the product is also colorless.Due to the nature of the color interaction, combinations with a net color charge arehighly unstable. If one starts with a system with no net color charge, it will never separateinto color charged pieces. The reason for this is believed to be that, due to the fact thatgluons have a color charge, there is an “anti-screening” effect whereby the effective forcebetween two with distance. This increased potential energy is convertedto quark-antiquark pairs in such a way that the separated pieces are each colorless. Thisis believed to be the reason no system with a net color charge has ever been directlyquarkskl+4g1

ŽuŽd’ ‘ “”“ “–• — —˜— —– “’‘ŽcŽs’ ‘ “”“ “–• — —˜— —– “’‘ŽtŽbŽeŽ›š‰Ž7žŽ›šŒ1.1 The Standard Model 8to the quarks. Here also the weak interaction can turn one flavor into another. However,it is not true to say that the weak force cannot act across families. It can, but with a much`reduced effect.QuarksLeptonsŽgœ(a)(b)Žš‹Figure 1.2: Transformations permmited by the Standard Model. (a) The effect of theweak force on the quarks. Transformations such as b u are also possible. (b)The effect of the weak force on the leptons. Transformations within families arerepresented by vertical arrows and across families are represented by horizontal anddiagonal arrows.ŸFor quarks, the electric charge changing transformations within families and acrossfamilies are dominant (the vertical transformations are more probable and the diagonaltransformations are less probable) as seen in Figure 1.2.This does not mean thattransformations due to neutral currents cannot occur. Quark transitions such as c u,in which the charge is the same, could occur through a neutral current . What happensis that transitions such as this one have a small probability of occurrence compared to theelectric charge changing transitions. In Section 1.3, this will be explained in more detail.©As mentioned before, the weak interactions of quarks as well as leptons are mediatedby the W , W and Z© . In the Standard Model leptons cannot have cross transformationsdirectly. Nevertheless, the Standard Model permits processes such as&3&

1.1 The Standard Model 9particles that have an electric charge including the W feel the electromagnetic force but thegravity force does not have an observable effect in particle physics.`Short and Long Distance Effects. Quantum mechanically interactions are dueto the exchange of virtual bosons.The fundamental color interaction between quarksis mediated by virtual gluons.Effects due to such fundamental processes involving afinite number of quark-gluon vertices are called short distance effects.The effectivestrength of this fundamental interaction is relatively small and Feynman diagrams (Seenext section.) and perturbation theory can be used to calculate them. However, the gluons(unlike the photon and the intermediaries of the weak interaction) carry color charge. Theconsequences of this color charge are profound. Gluon-gluon interactions are possible.These leads to an ”antiscreening” effect where the effective color interaction betweenquarks actually increases with distance instead of decreasing as in the electromagnetic andweak interactions. The result is that strong interactions between hadrons decrease withenergy since at higher energy the component quarks interact at shorter distances. At lowerenergies and longer distances the effective interaction is larger and it is not possible tocalculate such ”long distance effects” using perturbation theory.In hadron decay the fundamental decay mechanism may well be a weak interactionbetween the constituent quarks but as the decay products separate long distance effectsdue to the color interaction will play an important role.Some characteristics of thedecay will be determined by its fundamental weak nature but other characteristics willbe strongly affected by the long distance effects.For example, parity violation in adecay is an indication of its fundamental weak nature. However, the precise values oflifetimes and branching ratios can not be calculated with pure perturbation theory. Our

%1.2 Conservation Laws 10current best understanding of such decays uses our knowledge of the weak interactionin perturbation theory calculations together with phenomenological models for the long`distance effects of the color interaction. In the future it is hoped that lattice gauge theorynumerical calculations of the color interaction will achieve sufficient accuracy as to makephenomenological models unnecessary. These do not use perturbation theory to calculatecolor interaction effects.Feynman Diagrams. A Feynman diagram is a representation of the way particlesinteract with one another by the exchange of bosons. They are extremely useful in makingperturbation theory calculations because each representation is associated with formal rulesfor assigning vertex couplings, propagator terms, etc.eeeeFigure 1.3: Feynman diagram for a weak process mediated by a virtual bosonexchange.1.2 Conservation LawsAmong the most significant aids to understanding what takes place in the universe arethe conservation laws, given that they identify those types of interactions that cannot occur.

(1.4)N £¢ nN n¤n¢ n J ¡1.2 Conservation Laws 11It is a trustworthy rule of thumb in physics that whatever thing is not definitely prohibitedwill occur.`In classical mechanics, the invariants are the conservation of the electric charge,the conservation of linear momentum, the conservation of energy and mass, and theconservation of the angular momentum. In the quantum world, there are other invariantsas well that are preserved in some types of interactions but not others. For example, theisotopic spin is preserved in the strong interactions but not in interactions concerning theelectroweak force.1.2.1 Conservation of Angular Momentum and Helicity SuppressionThe helicity of a particle is defined as its spin projection along its direction of motion.A particle with spin vectorN and momentum p has helicityspin For there are only two possible helicity states,tI . If the helicity of a particleis positive, it is called right-handed, and left-handed if the helicity is negative. Theocorresponding states are denoted as¥§¦ and¥©¨ respectively.Due to the nature of the weak interaction, lepton states from weak decays arepredominantly left-handed.This helicity predominance is stronger the lower the massof the lepton. In the case of the neutrino, it is total. Only left-handed ˆ have beenobserved. For other leptons, the term helicity suppression is used. This means that right-

Conservation Laws 12`1.2handed leptons and left-handed antileptons are produced with much lower probability. Thehas spin zero so the two muons in must have opposite spin to conserveangular momentum. But they also have opposite momentum so they have helicity.D© D©theh3),j¥&However one is an antilepton. Thus, the weak decay D© is helicity suppressed.1.2.2 Conservation of the electrical chargeWhichever reaction, the entire charge of all the particles before the reaction has gotto be identical to the whole charge of all the particles following the reaction. For instance,in a beta decay, a neutron decays to form a proton and an electron plus an antineutrino.This begins with a zero charge and finishes one part negative and one part positive charge.Figure 1.4 shows the contribution of charge for each fermion involved in the reaction.+ z ¤ & ¤ ˆ–‰ (1.5)with charges: 0 +1n& n -1n& n0penWFigure 1.4: A charged current interaction. The neutron has a quark content of udd andˆ‰its charge . The proton quark content is uud andits charge . The electron as charge andthešas charge.isª5«9¬i­,®¯

Conservation Laws 13Conservation of electrical charge prevents the color force from materializing quarks`1.2such as u and d from energy (the total charge is not zero). However, a uc materializationwould conserve charge. Yet, such materialization is not seen in color force reactions. Theproperties of the color force dictate that the quark and antiquarks involved must be of thesame flavor. It is not possible to materialize a u and a c by the color force but you canmaterialize a u and a u.Flavor-changing processes can not occur via the color force or the electromagneticforce, only via the weak force. Since the color and electromagnetic forces satisfy moreconservation laws than the weak force, many particles can only decay weakly. This factallows us to study the weak force by studying those decays.1.2.3 Lepton Number ConservationA lepton number is a tag used to indicate which particles are leptons and which onesare not. Every lepton has a lepton number of , which is split into three different “values”:. On the other hand, every antilepton has of§a lepton number . Otherparticles will have a lepton number of are called lepton family numbers.andA§Œ ,A§‹ Aº‰ andA§Œ ,A§‹ .A»‰Both lepton number and lepton family number are always conserved according to theStandard Model. To understand the difference between them, consider the processD© ¤ & (1.6)where the lepton numbers and lepton family numbers are:

Conservation Laws 14`1.2A §Aº‰ § A§‹: 0 1 1: 0 0 1: 0 1 0For this reaction we can see that the lepton number is conserved but not the leptonfamily number. It is due to the fact that for this process in the initial state we do nothave any lepton but in the final state we have two leptons of different families. The decayis a Lepton Family Number Violation process which is not permitted to occurwithin the Standard Model.&3¼ | D©A process permitted by the Standard Model is(1.7)& ¤ ˆ9‰where the lepton numbers are¤ ˆ9‹conserved.A A»‰ Aº‹ : 1 1 1 1: 0 1 0 1: 1 0 1 0For the process above the lepton family number as well as the lepton number are1.2.4 Baryon Number ConservationAnalogous to leptons, the baryon number is a tag used to indicate which particles arebaryons and which ones are not. All baryons have a baryon number of . Each antibaryon

1.2 Conservation Laws 15`has a baryon number of§ . Every quark has a baryon number of9B2½ . Antiquarks havea baryon of§number and, therefore, each meson has a baryon number of . Everylepton has a baryon number of . In all reactions, the total baryon number of the particlesprior to the reaction is the same as the total baryon number following the reaction. For9B2½example,where the baryon numbers are+ z ¤ & ¤ ˆ–‰ (1.8)B: Lepton and baryon number conservation are ad hoc assumptions in the StandardModel in contrast to other conservation laws such as charge conservation which are relatedto the nature of the interactions. This brings up the possibility that lepton and baryonnumber conservation may be violated in some (as yet unobserved) processes.1.2.5 Flavor ConservationIn particle physics another frequently used term is flavor. Quarks have differentflavors such as: up, down, strange, etc. There are six flavors of quarks. The flavor numberfor each quark in the order of its mass is the following: ¾ J ,_ J § , H J ,, . ÀÁJÂ andÃÄJ § The strong and electromagnetic interactions conserve the quark flavor numbers while§ J ¿the weak interaction does not. Flavor change occurs mostly through the charged weak

However, these states are +63-1.3 Cabibbo Theory 16interactions since the flavor-changing neutral-current weak interaction is suppressed by theGIM mechanism. This is explained in detail in Section 1.4.`1.3 Cabibbo TheoryTransitions between quarks of different flavors occur only due to the weakinteraction. For the electromagnetic and color interactions all families behave the same.The only differences between them are their masses.When we talk about quarks ofdifferent flavors, we are referring to mass eigenstates.eigenstates of the total Hamiltonian (which includes the weak interaction) because, if theywere, flavor would be conserved in weak decays.The weak decay process can be described as follows. , a quark is createdin a mass (flavor) eigenstate via an electromagnetic or color interaction. This state canbe written as a linear superposition of weak eigenstates each of which has a well-definedAt-ÅJand different lifetime. It is said that the quark is in a “mixed” state. As time evolves, theweak eigenstate mix changes because of the different lifetime evolutions. At a time- later ,the state has acquired components of other mass eigenstates (other flavors) thus explainingflavor transitions.Only quarks with the same charge can mix. Due to the explicit form of the weakinteraction, it is only necessary to consider mixing of the§ 9B2½ quarks (this inducestransitions involving the¤ B2½ quarks also). The mixing is treated mathematically via theunitary Cabibbo-Kobayashi-Maskawa (CKM) [5, 6] matrix whose elements are

ÇÇÇÇÆÈdÉsÉÊËËËËJ ÌÇÇÇÇÆÈÍÎ7ÏÐÍÎZÑÒÍÓÎ7ÔÍFÕÑ ÍÕ@Ô ÍFÕ@ÏÍ cÏ Í cÑ Í cÔÊËËËËÌÇÇÇÇÆÈCabibbo Theory 17d`1.3sÊËËËË(1.9)ÌbIn the lenguage of Feynman diagrams the charged weak transition occurs at a vertexbÉsuch as that shown in Figure 1.5 where q is a¤ B2½ quark and qÉ is a§ 9B2½ quark.q%'qÉFigure 1.5: Feynman diagram for a weak transition.Mixing is treated mathematically by taking the coupling at that vertex to beproportional toÖÍ×Ø×lÙwhereÖ is the universal weak-coupling strength which is the samefor all three families. Notice the reason for the names of elementsÍ×Ø×Ùthe CKM matrixsince they are associated between¤with the and§coupling quarks. For example, 9B2½ B

ÇÇÇÇÆÈoH ‚ ¿ oH ‚ ¿ ‚& ÝäØåçæ§ ¿ Ho‚ § H o¿ o‚¿ ‚&ÝäØåçæ H oH o‚ § ¿ o¿ o‚¿ ‚&ÝäØåçæ ¿ o‚H ‚¿ o¿ oH § H oH o‚¿ ‚&ÝäØåçæ § H o¿ o‚ § ¿ oH o‚¿ ‚&ÝäØåçæ H o‚H ‚o‚ÇÇÇÇÆȧ ë oB ë í€ë ‚ pî § klï s§ § ë oB íðë o ë‚ p: § î § kyï s § í€ë o íðëþ9ÿ¡ whereø^ü is the Cabibbo ¢£ ¢ angle rad). ý¥þ=ÿ (ø^ü 1ñóò„ôöõ¤÷5ø7ùúò†ôöõû÷5ø^üIýCabibbo Theory 18`1.3ÊËËËË(1.10)ÌÍ JThose matrix elements with a simple form can be directly measured in a decayprocess and are found in the first row and third column. H ‚is known to deviate fromunity only in the sixth decimal place. Therefore,ÍÎ7ÏJâH o3ÛÍÎZÑ J ¿ o3ÛÍÕ@Ô J ¿ o‚andto an excellent approximation. phaseÜ The lies in the range ,with non-zero values breaking CP invariance for the weak interactions.E Š ‚ Ü ·é ‚ o‚ JèH cÔ ÍAn alternative approximate parametrization is given by Wolfenstein [11] using thethat¿fact are all small. The size of the Cabibbo angle 1 is setand then the other elements are expressed in terms of leading powers up to the thirdo*ê o ¿ asë„ì ‚ ¿ ê o‚ ¿ofëorder:ÊËËËË(1.11)ÌÍ JÛî andï as real numbers that are of order unity.havingíTransitions between quarks for the two family case are described by thetransformation

ÇÆ(1.13)ÍÓÎ7Ïͧ¦ cÏͧ¦ ÍÕÏ7ͧ¦ ¤ Í cÔ J Õ@Ô ¤ Î7Ô¦(1.14)nÍÕ@Ï7ÍFÕ@Ôn ¤ Í cÏ nÍÕ@Ï7ÍFÕ@Ôn JK Î7Ô ÍCabibbo Theory 19ÇÆdÉsÉ ÈÊËJ Ì4762h5Ú¥¤ Ú¥¤§Ú¥¤hikl+476

W¦¨ , , and © o ¨ cÏ Î7Ô¦Cabibbo Theory 20`1.3AαV*ubV tdγβCV s 12*cbBFigure 1.6: Representation for the CKM Matrix elements. An unitary triangle in thecomplex plane is formed by the CKM matrix ¨ elements .Õ@Ôthan the off-diagonal elements. Thus transformations within families are much more likely.Transitions proportional to4762hoÚ¥¤ are known as Cabibbo-Favored.Dcuu ¤dsuK §Figure 1.7: Feynman diagram for a Cabibbo-Favored process. A charm quark istransformed into a strange quark. The other transformation of a into u and dquarks in the second vertex is also Cabibbo-Favored.WIf we consider the process in which a D© is decaying into K E shown in Figure 1.7,a charm quark of the D© particle is transformed into a strange quark. This is Cabibbo-Favored. In the second vertex the W is decaying into a ud. This is also a Cabibbo-Favoredprocess because u and d are quarks of the same family.

W §Cabibbo Theory 21b) Cabibbo-Suppressed. Transitions proportional to hikl+ oÚ¥¤ are known as`1.3Cabibbo-Suppressed. These are transformations between the first and second families.Dcuu ¤ddu §Figure 1.8: Feynman diagram for a Cabibbo-Suppressed process. In this process acharm quark is changed to a down quark. The W decay vertex is Cabibbo-Favoredjust as in D© Ÿ K (Figure 1.7).In Figure 1.8 a charm quark turns into a down quark in a D© process decaying into a. The second -W vertex is a Cabibbo-Favored process.=8.)R1c) Doubly-Cabibbo-Suppressed. Processes in whichEtwo -W vertices E =8.),1are of the Cabibbo-Suppresed type are known as Doubly-Cabibbo-Suppressed. The decaymode D© KE is shown as an example of these processes (Figure 1.9).uK ¤WsDcuduFigure 1.9: Feynman diagram for a Doubly-Cabibbo-Suppressed process.For thisprocess both vertices, the W- and the -W, are Cabibbo-Suppressedprocesses.

1.4 GIM Mechanism 22`1.4 GIM MechanismThe process D© is a flavor changing neutral current process which issuppressed in the Standard Model due to the GIM mechanism, proposed in a famouspaper [12] by Glashow-Iliopoulus-Maiani in which they explain the suppression of flavorchangingneutral-currents. They suggested that, instead of a triplet of quarks (u, d, s), therewere two doublets of quarks (u, d) and (c, s). This suggestion required a new quark respectto the Cabibbo model, the charm quark. This prediction was made before the experimentaldiscovery of this quark.u_uc_cZ 0+(a)Z 0+(b)(d cos θc+ s sin θc )_ _(d cos θ c + s sin θ c )(s cos θc− d sin_ _(s cos θ c − d sinθ ) cθ c )Figure 1.10: Weak interactions in neutral-current processes. (a) Feynman diagrams fora neutral current in the three-quark model. (b) Including the charm quark, extra termsare added.If we have a © as carrier of a weak interaction as shown in Figure 1.10 we have that

ÊËÛ ÌÊËÛ Ì¦¥ J uu¤p dd4Z6

1.5 Rare and Forbidden Decays 24where the strangeness-changing part vanishes due to the effect of the charm quark. To firstorder, there are no strangeness-changing neutral-currents and similarly no charm-changing`neutral-currents. This cancellation of flavor-changing neutral-current processes occurs onlyin the first order Feynman diagrams (tree level). Flavor-changing neutral-currents can occurin the Standard Model through higher order terms (loop diagrams) which are suppresseddue to the smallness of the weak interaction coupling.1.5 Rare and Forbidden DecaysRare and forbidden decays are divided into three different groups: Lepton FamilyNumber Violating (LFNV), Lepton Number Violating (LNV) and Flavor Changing NeutralCurrent (FCNC).a) Lepton Family Number Violation. Processes of this type are forbiddenbecause they do not conserve the lepton family number. For instance, processes such asthe leptons belong to different families, are examples of this mode. These decays would | ¼ &Ï('Ñ*)+ | & ¼ and_&Ï('Ñ*), where + +.&3 isE or , and_& D©andsuggest the existence of heavy neutral leptons with non-negligible couplings to& and .b) Lepton Number Violating. Since there is no explanation in the Standard Modelfor the lepton number conservation, we can expect a violation at some presently unknownenergy range. Processes such as_&Ï('Ñ*) + .-/- in which leptons come from the samefamily and are of same charge are examples of LNV modes.c) Flavor-Changing Neutral-Current. In the Standard Model the neutral-current

5J32.j¥4oÛ (1.20)and (1.21)1.6 Special Relativity 25`interactions change of the flavor is at a very low level. Lower limits for charm-changingneutral-current are expected.Such decays are expected only in second-order in theelectroweak coupling in the Standard Model. Processes in the_system refer to the decays? and_+ - &Ï0'Ñ1), etc. Test for charm FCNC are limited to hadron decaysinto lepton pairs. - ? D©1.6 Special RelativityWe will go through a brief review of special relativity to establish notation andwhatever points are necessary for this work. If we have a particle that is moving anywhereclose to light speed, the relations of its energy and momentum will be different from thoseat a lower speed:andz q 4(1.19) Û J32j qwhere

55 q z4so ¤ pj¥4osoÛ { J zF45 (1.22) Û 2†J j¥4o p J oThis an invariant under Lorentz transformations for any pair of vectors 8Consider the situation where a primed system moves with the constant velocity4q1.6 Special Relativity 26`is the velocity of light in vacuum,j is the mass of the particle at the rest. From eq’s.(1.19), (1.20) and (1.21):4Note that if 4 J Û 5Jãj¥4o. Special units are used in particle physics such thatand 7 . Energy, momentum and mass are then all in units , and time andlength will be in units .4óJ of~…7 J of~…7Let us consider the displacement, a four-vector which is generalized by specialrelativity as the difference between two “events”, where both the spatial displacement andtime of occurrence are considered.8Û(9 (1.23)s JÄp@í © Ûí Ûí oÛí ‚s J p@í ©Û q í sÛ Û; p4g-Ûr í¹JThe invariant dot product concept is generalized as well8 8(1.24)ÃÄJ í ©Ã © § í à § í oà o § í ‚Ã;:µJ í ©Ã © § q í q à íandwhich transform like “event” vectors. Energy and three-momentum form such a vector.Ã8 íThis four-momentum vector is8 5 Ûz.< Ûz.= Ûz?>s(1.25)p í¹J

5É J@2©p5 § { z< s(1.28)Conservation of energy and momentum is used in order to reconstruct Aofrom the1.6 Special Relativity 27`with respect to the unprimed4system. is in the of¤direction and we will use{forr qthe fractional velocity of the coordinate system. Consider a system momentumqwith and zenergy 5 . The Lorentz transformation equations are:zÓÉ= J z.= Û zÓÉ> J z?>¥ (1.26)z É< J@2©pz.< § { 5 s (1.27)Under a Lorentz transformation, the scalar product of two four-vectors is invariant.Momentum and energy form a four-vector just like position and time. The?kX/&-:kyj¥& (‡ )of an unstable moving particle is calculated to be equal to 2F‡©where‡©is the particle’slifetime as measured in its rest frame.The Lorentz invariant mass of any system is its energy in a frame where it is at rest.In our analysis we are trying to identify decay processes of a short-lived particle into twoopposite charged particles. The decaying particle travels a negligible distance making itimpossible to determine the identity of the particle based on its trajectory. Therefore, animportant quantity in our work is the Lorentz invariant mass which is used to determine iftwo particles (whose momenta,qare known) are the daughters from the decay of aandqparticle of A mass into products of .z z oo andjmassj

Notice that AA5 q z ¤ q z oiÛ z?BÄJ q8 85 5then,¤ z osoAo o JÄp ¤ oso § p q z JÄp o ocan be calculated massesj from the assumed and the measuredin any A frame. is called the invariant ) mass of the two-particle o ,j oo ,j o o (j zA1.6 Special Relativity 28`candidate decay products:We can define the Lorentz invariant quantityBÂJ5 ¤ 5 o (1.29)where 5 ÝJ6jáÝo ¤ z–Ýo.z ¤Qqz oso(1.30)system. If these two particles did indeed have the massesjandj o and if they did indeedmomentaqz ,qcome from the decay of a particle A , then Ao will be equal to A within experimentalerrors. The calculation of invariant mass is a major tool in identifying decays.If we consider the D© process,j J j o J já‹¥} DCFEEMeV [4, p.23], thus J q z–‹ å ,q z o J q z–‹HG z andq5 5j o ‹9å ¤ z o ‹9åÛ oG ‹ J¹j oG ‹ ¤ z oG (1.31) ‹ J ‹=å oand the Lorentz invariant mass for the D© will be given by.IKJJ j o ‹ ¤ pez o ‹9å ¤ j o ‹ spz oG (1.32) ‹ ¤ j o ‹ s § G4762h5Ú‹7‹L zÓ‹9åzÓ‹ ‹7‹ owherez–‹=åÛzÓ‹ G andÚ‹Z‹ are measured quantities.

NOPRQTS;U1VWYXZVJ H0A1.7 Confidence Intervals and Confidence Levels 29`1.7 Confidence Intervals and Confidence LevelsSeveral concepts in statistics will be useful in our work. Confidence limits [13] referto those values that define an interval range of confidence (lower and upper boundaries)for an unknown parameter. If independent samples are taken repeatedly from the samepopulation and a confidence interval calculated for each sample, then a certain percentageof the intervals will include the real value of the parameter.A different concept is that of Confidence Level. This statistic is used in hypothesistesting where one calculates M a from the deviations of the observed data from thehypothesis. If the hypothesis is correct, high values of Moare unlikely. The confidenceolevel is defined as(1.33)where QTS[U\VDWis the ]_^ probability density which depends on the number of observations, ` .The confidence level (CL) is a random variable. If the hypothesis is correct, it has a uniformprobability density and all values of CL are equally likely. This may seem to make it notvery useful. However, if the hypothesis is not correct, high values of ] ^ (correspondingto low values of CL) will be likely. One can statistically differentiate the cases where thehypothesis is fulfilled by requiring a minimum CL value.

Chapter 2Previous Works2.1 FCNC Decay TheoryThe fundamental work concerning FCNC decays is the famous paper by Glashow,Iliopoulos & Maiani in which they propose the existence of the charm quark and discusshow it can explain the suppression of FCNC decays [12] through what is now called theGIM mechanism. This has been discussed in Section 1.4.With respect Dacb d$efdhg to which is a charm FCNC decay, the GIM mechanismleads to suppression of the tree-level diagrams such as Figure 2.1.a. In a 1997 paper [14],Pakvasa presented the latest calculation for this process. He found that the short distanceeffects are dominated by internal s-quark loop diagrams which are suppressed by thesmallness of the s quark mass and by helicity considerations (Section 1.2.1).30

jaW gqkja2.1 FCNC Decay Theory 31id$edlecucud gd g(a) Suppressed Diagram of First Order(b) Penguin Diagramcd eWeX?monm0pu(c) Box Diagramd gFigure 2.1: Feynman diagrams for the decay Dasrut e t g . (b) and (c) are second orderdiagrams with quark masses d, s, and b within the loops that give a rate proportionalto vxwy , being v y the mass of the strange quark. (a) represents a suppressed diagram offirst order.Pakvasa calculated the short distance effects to be of the order of z¥{ g?|\} . However,the long distance effects are large bringing Pakvasa’s calculation of the Standard Modelbranching ratio to zH{ g?|\~ for Da b d e d g .The long distance effects are due to intermediate states such as a , € a , € a ,ï ,ïZ orl and € € a [14].

ŠŠz‘i2.2 Previous Searches for Rare Charm Decays 322.2 Previous Searches for Rare Charm DecaysMany experiments have reported limits for the branching ratio of the rare decayDa b‚d e d g as is shown in Table 2.1 published by Particle Data Group in 2000 [4, p.566].Table 2.1: Chronological branching ratios for the ƒ…„‡†ˆz weak neutral-currentdecay Dacb d$efd‰g .Value Experiment Comment Year(at CL 90%) Namex‘ ŠŒ‹Ž˜…‘z‘ Šœ›x‘ Šœ›Š3F›ž‘ŠŒ¤Ž˜…‘Šœ›F›ž‘z‘ Š3{‘ ŠŒ¤Žz zE791 500 2000E789 nucleus 800 2000Beatrice Cu, W 350 1997E771 Si, 800 1996CLEO (4S) 1996Beatrice Cu, W 350 1995E653 emulsion 600 1995events 1994ARGUS 10 1988E789SPEC W 225 1986EMC Deep inelastic N 1985Š3F›ž‘In this section we will discuss the most recent experiments. We will also discussthe FOCUS predecessor experiment, E687, which searched for other FCNC charm decaysbut not Da bthe CERN 1 WA92 BEATRICE experiment. Then we present Fermilab’s 2 E791 and finallyE687.d e d g . The first experiment is the one that has obtained the lowest limit,1 CERN -Organisation Européenne pour la Recherche Nucléaire2 Fermilab -Fermi National Accelerator Laboratory

zi2.2 Previous Searches for Rare Charm Decays 332.2.1 The Hadroproduction WA92 ExperimentDuring the 1992-93 data-taking period of the BEATRICE Collaboration, thehadroproduction WA92 experiment [15] at CERN Super Proton Synchrotron was carriedout. In the interactions of 350 “•”—–‡ e particles in a W target during 1992 (W92 runs),charmed particles were produced as well as in the Cu target of 1992 and 1993 (Cu92 andCu93 runs respectively). The apparatus was made up of a 2 mm thick target followed bya beam hodoscope and by an in-target counter (IT), a high-resolution silicon-microstripdetector (SMD), a large magnetic spectrometer and a muon hodoscope.Candidate events Da b‚d e d gwere required to be in a mass range betweenEvents that came from target interactions as well as those in IT counters were accepted.F¨{«¥ŒzF¬D “•”—– . Dimuonic events were required and were selected by a dimuon trigger.Dimuonic events associated to a simple secondary vertex were differentiated from eventswhere both muons came from different secondary vertices. The Da b K g _e decayprocess was the normalizing mode which was detected in runs of Cu(W) applying the sameselection criteria for the dimuonic events except the ones for lepton identification.Monte Carlo simulation generators used Phytia 5.4 and Jetset 7.3 [16–19] tounderstand the hard processes and quark fragmentation.Fluka [20, 21] was used todetermine the characteristics of all other interaction products. Monte Carlo simulationsgenerated events that contained a pair of charmed particles.Dimuonic events comingfrom the Monte Carlo simulated Da b‚d e d g decays were treated in the same way asthe experimental data permitting the determination of the efficiencies and the acceptanceratios.

˜¥˜‘2.2 Previous Searches for Rare Charm Decays 34iNo Da­bd$efdhg candidate was found [15]. This result led to an upper limit on theerrors on the whole analysis procedure did not significantly alter this value.branching fraction B(Da b‚d e d g ) of ›. zz¥{ gŽ’ at 90% confidence level. Systematic2.2.2 The Hadroproduction E791 ExperimentEvents collected in the E791 experiment [22] were produced by a 500 “ª”¢–®$e beamin five target foils.This was a hadroproduction experiment in which track and vertexreconstruction were provided by 23 silicon microstrip planes and 45 wire chamber planes,plus two magnets.Muon identification was obtained from two planes of scintillationcounters. The experiment also included electromagnetic and hadronic calorimeters andtwo multi-cell Čerenkov counters that provided ‰¯€separation in the momentum range{[“•”—– .In the analysis stage, to separate “good” events from background noise, a separation, whereof the production vertex from the decay vertex was required by more z±°?²thanis the calculated longitudinal resolution. Also the secondary vertex was required to be°?²separated from the closest material in the target foils by more than², where °´³ ²is the‹§°´³separation uncertainty.E791 used as a Monte Carlo simulation generator Pythia/Jetset [23,24] and modeledthe effects of resolution, geometry, magnetic fields, multiple scattering, interactions in thedetector material, detector efficiencies and the analysis cuts. As normalizing mode in thestudy of the rare decays Da b d e d g , Da b Ÿ e Ÿ g and Da b‚d$µlŸ¥ , the Cabbibo-Favoredmode Da bK g e was used.

E791 used a p¢·©1¸X¹aWŠz2.2 Previous Searches for Rare Charm Decays 35Before the branching fractions were calculated for each of the channels studied,ianalysis technique. All the events within a ƒž¹»º range around theDa mass were “masked” in order to avoid bias in the cut selection due to the presence orabsence of a possible signal. The cut selection was based on the study of events generatedby Monte Carlo for signal events and real data for background events.Backgroundevents were chosen and studied in mass windows ƒ¼¹½º before and after the signal area.calculated using the method of Feldman & Cousins [25] to account for background,U1¿F¨¾ŠF¬The signal area was chosen as z{ . The 90% upper limit wasand then corrected for systematic errors by the method of Cousins & Highland [26].The upper limit was determined from the number of candidate events and the expectednumber of background events within the signal region. Only after the cuts were optimizedwas the signal area unmasked and then the number of events within the window was`«À*Á y †£Ầº¥Ã ÄŪẦÆÇà y\ÈYÉ Å§`«ÊËTÁ , where Ầº¥Ã Ä corresponds to the number of “good” candidatesof Da b d e d g , `«ÆÇà y1ÈÌÉ corresponds to hadronic decays with pions misidentified asmuons, and `«ÊËTÁ corresponds to combinatoric background arising primarily from falsevertices and partially reconstructed charm decays.No evidence for the rare decay Da b‚d e d g was found [22]. The 90% confidencelevel branching fraction limit wasstatistical errors from the fit in the normalization sample; statistical errors on the number of‹Í‘z¥{ gŽ’ . Systematic errors in this analysis includedMonte Carlo events for both the normalizing mode and the process studied; uncertaintiesin the calculation of MisID background; and uncertainties in the relative efficiency for eachmode.

··i2.2 Previous Searches for Rare Charm Decays 362.2.3 The Photoproduction E687 ExperimentProduction and decays of charm particles from high energy collisions with ahigh intensity photon beam and a beryllium target were studied by the Fermilab E687experiment [27] using a multiparticle spectrometer for particle identification and vertexingof charged hadrons and leptons.From the Tevatron beam, a photon beam frombremsstrahlung of secondary (Î*ϱÐ[†electrons “•”—– 350 ) hit a beryllium target. Chargedparticles were traced by four silicon microstrip detector stations. Those detectors werevery efficient for separating primary and secondary vertices. Charged particle momentumwas determined from deflections in two analysis magnets of opposite polarity with fivestations of multiwire proportional chambers. In order to identify electrons, pions, kaonsand protons, three multicell threshold Čerenkov counters were used.There were twoelectromagnetic calorimeters. Particles which passed the apertures of both magnets weredetected by the inner calorimeter which covered the forward solid angle. Those particlesthat only passed the first magnet were detected by the outer calorimeter covering the outerannular region. Two sections of thick steel muon filters divided four proportional tubeplanes which were used to identify muons in the forward solid angle [28].making three-track combinations of an event with the right particle identification. TheThe selection of candidates for De b Ñ?µe and De b € g e g started bycombination of the resulting three tracks were fit to a common secondary vertex.In order to detect muons with a momentum higher than › “ª”¢– , planes of scintillatormaterial in addition to four stations of proportional tubes, altogether known as the Muon

anching fractions in the U¬W‘i2.2 Previous Searches for Rare Charm Decays 37System, were used by the E687 Collaboration. A muon Confidence Level was calculatedwhen classifying candidate muon trajectories. At least 1% was required for the CL.In order to identify electrons, information of tracks associated with electromagneticshowers in either of the two electromagnetic calorimeters was used if they were consistentwith the electron hypothesis in the Čerenkov counters.The sensitivity to the measured branching ratios was optimized for each decay modestudied in E687. This sensitivity depended on the efficiency relative to the normalizingmode. This was calculated using a Pythia [29] Monte Carlo simulation generator along witha detailed spectrometer simulation, the amount of normalizing mode events (determinedfrom a fit to the invariant mass plot), and the amount of background in the signal region(determined using sidebands before and after the defined signal region).Monte Carlo signal shapes and background shapes derived from data were used tocalculate the 90% CL upper limit on the number of signal events in the corresponding massplot. Misidentification of hadrons was the primary background in the E687 analysis. Threetrack combinations with no lepton identification were used to determine the shape of thebackground. The probability of misidentifying a hadron as a lepton was introduced as aweight on each lepton candidate.No evidence for the fourteen exclusive modes of rare and forbidden decays wasobserved in this experiment [27].However, 90% CL upper limits on their absolutez¥{ŽgŽ~ range were determined.¥ {

i2.3 FOCUS 382.3 FOCUS2.3.1 IntroductionSince hadrons with quarks from the ÒHÓ and ÔÕÓ family exist only rarely in nature,we need a way to produce them copiously to study them effectively.The method ofphotoproduction in the experiment E831 [30] that took place in the Fermi NationalAccelerator Laboratory (Fermilab) near Chicago, Illinois was used to create particleswith the charm quark.The E831 experiment, or FOCUS (“Fotoproduction Of Charmwith an Upgraded Spectrometer”), was an upgrade version of the E687 experiment(Section 2.2.3) and was a heavy–flavor (produces particles withÒHÓ family, or higher,quarks) photoproduction experiment located at the Wide Band Photon Area.Figure 2.2: Proton synchrotron and beamlines at Fermilab. The FOCUS experiment islocated at the Wide Band Photon Area in the East Proton Area.

photons with a high energy (about 300 “•”—– ). At Fermilab the initial beam is a 800 “ª”¢–i2.3 FOCUS 392.3.2 The FOCUS Photon BeamTo photoproduce particles which contain the charm quark it is necessary to generateTevatron proton beam [31, p.10].Figure 2.3: The E831 Photon Beam (FOCUS Collaboration Figure).As shown in Figure 2.3, a photon beam was produced after three different stages.A neutral beam produced from “•”—– 800 protons incident on a liquid deuterium targetincluded a variety of neutral particles along with high energy photons from a decays.Charged secondaries and uninteracted protons were taken out of the beam and absorbedby the beryllium target dump (not shown). All the neutral secondary trajectories produced

¥i2.3 FOCUS 40were transmited within an aperture of 1.0 mr horizontally an 0.7 mr vertically with respectto the incident proton beam trajectories. To absorb a smaller portion of these secondaryphotons a Deuterium target material was used.The photons that were left produced a Ÿ e Ÿ g pair due to electromagnetic interactionsin passing through a 60% radiation length lead convertor. The pairs were separated fromthe neutrals using momentum selecting dipoles and then put together focusing them ontoanother target This momentum selection was necessary to purify the Ÿ g and Ÿ e beams byreducing their charged hadronic background considerably. A neutral dump was used toremove hadronic neutral particles.Finally, photons that came from bremsstrahlung were produced from the focusedŸ e Ÿ g beam which was passed through a 20% radiation length lead radiator. Recoiledelectrons were taken out of the photon beam and absorbed by an electron dump.2.3.3 Inside the FOCUS ExperimentFOCUS utilized a forward large aperture fixed-target multi-particle spectrometer,to be explained in Figure 2.4, to measure the interactions of high energy photons ona Ö±Ÿ¥× segmented target. This spectrometer had excellent particle identification andvertexing for charged hadrons and leptons.The experiment collected data during the zfixed target run and isinvestigating several topics in charm physics including high precision studies of charm¬¬˜¬¤semileptonic decays, Quantum Chromodynamics studies using double charm events,

2.3 FOCUS 41ia measurement of the absolute branching fraction for the Da meson, a systematicinvestigation of charm baryons and their lifetimes, and searches for Da mixing, CPviolation, rare and forbidden decays, and fully leptonic decays of the ¿e . ë¡ò¡ë í ß*àÂÞ ë íðÜß æ Ú¡Ü ë ù æè*ÜìóYÛ*Ú öë¡ò ë í ßÕà "$#&%!"'#&% ! ÛKèÝé çÜìÛ1í1Ú ë ßîà êDëö è*ÜìóÌÛ1Ú æö Ü ë óûóûÛðÜ ä ßûÙûàûÚ¡Û Ü æüý*þÝÿ¡ £¢¥¤¦ ðÿ¨§ ©¡ã—âï§âäDâ åæÛðÜ ØÙ1ÚÛ1í\ÚFÜìßîéè1óîàîÛ1Ú ë í ñ—òè ò ßÌÜ ë éÛ*ÚFÛ1Ü äÛÝÜÛðà1ô*ß*õ äßYÙYàYÚ ÛûÜ äà*à1ÛðÜ Û1í1ÚFÜ ßîéè1óîàîÛ\Ú ë í ñ—òè ò ßîÜ ë éÛ1ÚFÛ*Ü äèûø Ü ß à ÷è ò ßÌÜ ë éÛ1ÚFÛ*Ü äÞ/Ù1ß1àßÌøÌß æ í*ßÌùÌÛ ÷Þ_è*óÌàÌÛ*ÚØÙ1Ú ÛÝÜlÞ/Ù1ß1àã¢âïúâäâáDâãâäDâ åæÛÝèKé çÜìÛ1í1Ú ë ßîà êëÜ ë ó1ó1ÛðÜ ößÌøÌß æ í*ßîùÌÛ ÷Ü ë ó*ó1ÛÝÜ ößîøÌß æ í1ßÌùÌÛ ÷ã¢âïúâäâÞfÙ*ß*à ë ò Ú¡ÛðÜÜ è£ ÝÚÙYÛ æ öÛÝèKé çè ò ßÌÜ ë éÛ1ÚFÛ*Ü äÛðÜFÛðà*ô*ß*õ äßÌÙÌàYÚÛ*Ü æ äÞ_è*óÌàÌÛ*ÚFigure 2.4: The FOCUS Spectrometer (FOCUS Collaboration Figure).In the FOCUS spectrometer a BeO target was used to produce primary and secondaryparticles coming from interactions between the Tevatron photon beam and protons withinthe target (¢ Å p b ( Å p). The target was segmented in four separate parts in order tomaximize the percentage of charm decays which occured outside the target material.Two trigger counters and two silicon tracking systems were in the target region asshown in the Figure 2.4. Charged particles which emerged from the target were trackedby two systems of silicon microvertex detectors (SSD). The first system was mingled with

i2.3 FOCUS 42the experimental target while the second was just downstream of the target and consistedof twelve planes of microstrips arranged in three views. Altogether these two detectorsallowed to differentiate primary and secondary vertices with high resolution. Two magnets(M1 and M2) of opposite polarity with five stations of multiwire proportional chambers(PWC’s) were used to measure the momentum of charged particles using the deflection ofthe particles when passing through the magnets. There were three PWCs before M1 andtwo between M1 and M2.Three threshold multicell Čerenkov counters were used to identify electrons, pions,kaons and protons [31, p.16]. Two types of calorimeter were used to determine the particleenergy. The first calorimeters were two electromagetic calorimeters. The inner calorimeter,a lead glass block array, covered the central solid angle and detected particles which passedthrough the apertures of both magnets. The outer calorimeter covered the outer angularannulus described by particles that pass through the first magnet but not the second. Thesecond calorimeter type was a hadronic calorimeter consisting of iron and scintillating tileswhich was used primarily in the experiment trigger but was also used to reconstruct neutralhadrons.To identify muons, FOCUS had a detailed detecting system. Muons were identifiedin either a fine grained scintilator hodoscope with an iron filter (covering the inner region)or in an outer system that used resistive plate chambers and the iron yoke of the secondmagnet as a filter. The muon system was constituted by two sub-systems: the Inner MuonSystem which was composed of 9 planes of scintillator material, in which three were usedas triggers of data acquisition, and three iron filters [32, 33], and the Outer Muon System

i2.3 FOCUS 43composed of resistive plate chambers which detected muons with an angle greater than 125milliradians.In order to analyze the interactions in the spectrometer, the analog signals from eachof the detectors had to be digitalized and recorded. This was the task of the data acquisitionsystem. The FOCUS data acquisition system had to deal with input data in a number ofdifferent data formats, merge all this data into one stream, and put the output on 8 mmmagnetic tapes. Information from the different detectors was read out and then these datawere put on a data acquisition system bus.Data for each event was taken off the data adquisition system bus and placedinto a temporary memory system; data could be written to and read from this systemsimultaneously. Later on, the data was buffered on disk and written to tape. During therun, FOCUS collected more than z¥{*) events and reconstructed more than zH{ ’ .The FOCUS Collaboration has published four papers studying different topics incharm physics [34–39].

Chapter 3ObjectivesAs a departure point for this thesis, the following objectives were considered:1. The determination of the probability of misidentifying a meson as a muon (MisID)in FOCUS as a function of momentum.2. The development of software for the search for the rare decay Da b d e d g usingblind analysis methods.3. The preselection (from the large FOCUS data set) of a small and flexible set of eventcandidates for Da b d e d g as well as for the Da b K g e normalizing mode.4. The determination of the relative efficiency for the detection of Da b d e d g andDacb K g e in FOCUS using Monte Carlo simulation to generate and analyzeappropriate samples of the decays.44

3 Objectives 455. The determination of the background level in the Dacb‚d$é d‰g analysis by using aniindependent data set and the misid measurements.6. The determination of the 90% confidence region for the branching ratio of the decayDacb‚d$é dhg . In using the high statistics FOCUS data set and in optimizing theanalysis cuts, the objective is to improve existing measurements and obtain tighter(more precise) confidence limits.

Chapter 4ProcedureThis chapter will describe how the search for the rare decay Da bd e d g was carriedout. Initially, will be discussed the FOCUS data format and the reconstruction process.Then will be defined each one of the variables used in this study. Next, the various stagesof data reduction will be discussed. The use of a normalization mode will be explained.Basically the method used in this work is known as blind analysis which consists in theoptimization of the selection criteria by minimizing the background in the data sidebands.The use of Monte Carlo simulation to determine detection efficiencies as well as to studybackgrounds will be discussed. The methodology leads to the determination of confidenceintervals for the value of the Da­bdlé dhg branching ratio.46

i4.1 Data Format 474.1 Data FormatWe will begin by discussing the form of the data produced in the experiment. For this,it is advisable to be familiar with the concept of an “event”. This refers to one instance of aphoton interacting in the target producing various particles which travel through some or allof the detectors. Some of the particles produced decay in or near the target into secondarydecay products which are the ones actually detected. The experimental ideal is to use theinformation from the detectors in order to measure the characteristics of the decay. To thisend, it would be easier if the set of detector signals +¸·-,came from the decay of interest andits associated production partners. In practice, this is never perfectly achieved. The “raw”data consists of sets of signals which approach the ideal of coming from one and onlyone physical event. Each of these sets is called a “raw data event”. Typically it containsinformation about the physical event but also detector noise and information about particletrajectories unrelated to the physical event. Given the experimental difficulty, many rawdata events contain no interesting physical events.Each raw data event is an independent unit of data which is “reconstructed”separately.Reconstruction is the process whereby the physical characteristics of theunderlying particle trajectories are extracted from the raw detector data. The raw data eventis either discarded if it is not at all interesting or converted to a “reconstructed data event”which contains the values of the physical variables such as the number of trajectories thatwere registered by the PWC chambers, the momentum of the particles, etc. In FOCUS, thereconstruction process is called Pass1 (see Section 4.3.2). All data (raw and reconstructed)is stored on magnetic tape.

4.2 Reconstruction Process 48Reconstructed output tapes from Pass1 contain all kinds of physical events, instancesiof a myriad of interesting processes all in the same tape. There are thousands of tapes.The next step is to group events of the same kind with the goal of having all theevents of the process under study being contained in a small data file. The most efficientway to do this is to do it in steps called “skims”. At each skim level, stricter event selectioncriteria are applied and the data is separated into a larger number of groups each of whichcontains a smaller amount of data.4.2 Reconstruction ProcessTo have a more precise idea of how the data selection process is done in FOCUS,it is necessary to be familiar with the trajectory reconstruction process. A great deal ofthe information deposited in the FOCUS detectors are presented as hits which we haveto interrelate to be able to detemine which hits in different detectors correspond to oneparticular trajectory.Between the target and the first magnet (M1) we find the Silicon Microstrip Detector(SSD) which has four stations. After M1 the tracking is done by five multiwire proportionalchambers (PWCs). Each one of these, called P0, P1, P2, P3 and P4, has four planes.Outside of the magnets the trajectories will be straight line segments which will bereconstructed in the SSDs and PWCs. The trajectories will bend in the magnetic fields sothat the straight line segments will be different in the three regions: (1) before M1, (2)

4.2 Reconstruction Process 49ibetween M1 and M2 and (3) after M2. These will be reconstructed by (1) SSD, (2) P0, P1,P2 and, (3) P3 and P4.The process of SSD track reconstruction consists of finding trajectories with the SSDdetector. The steps are grouping hit strips into clusters of hits, finding each one of the threemeasurement directions to project the clusters, and then, combining these into trajectories.Then PWC trajectories are found. There must be hits in at least three chambers. Those withhits in all five PWCs are referred to as “tracks”. The ones with hits in only the first threechambers are referred to as “stubs”. The SSD and the PWC trajectories must be associatedwith one another to be able to get the necessary information for the reconstruction of thecharm decay. To identify a “link” the SSD and the PWC trajectories must be extrapolatedto the center of M1 and must have consistent slopes and intercepts.MagnetsOuter MuonR.P.C. ’sMuon HodoscopePhotonBeamµ +4BeamDirectionTarget32Muon Filters−µµ +P.W.C. s1−µFigure 4.1: Trajectories through the FOCUS spectrometer. A view of some trajectorieswithin the FOCUS spectrometer. The first and second trajectories are detected by theOuter Muon System. Trayectories two, three and four are detected using the InnerMuon System.

4.3 Data Selection 50Figure 4.1 shows some trajectories passing through the spectrometer. Trajectory 1icorresponds to a three-chamber track while the third and fourth trajectories (3 and 4 in thefigure) correspond to five-chamber tracks. The second trajectory (2) has hits in all the PWCchambers except in P3. This trajectory can still be reconstructed in the region after M2 bycombining the information from P0, P1, P2 and P4.4.3 Data Selection4.3.1 Selection VariablesIn this section we will discuss the different variables used for this work. They will bedivided into two types: vertexing and particle identification.a) Vertexing. Using the FOCUS microstrip detectors we can take advantage of thefact that charmed particles travel short distances within the spectrometer before they decaywhile non-charmed particles tend to either decay immediately at the production point or topass through the whole spectrometer without decaying. This means that for charmed decayswe can reconstruct two distinct vertices in the event: the primary or production vertex andthe secondary or decay vertex. A diagram of a Da b‚d$é dhg decay and its vertices is shownin Figure 4.2. By reconstructing the two vertices, we obtain several quantities on whichwe can do “cuts” to reduce non-charmed backgrounds. A cut refers to an event selectioncriteria which is implemented by requiring a certain range of values for a particular variable.A vertexing algorithm [40] is used to reconstruct a Da bd e d g decay candidate. It

4.3 Data Selection 51γPrimaryVertexπ00DπΣ +−µ − D +SecondaryVertexµµ−+iFigure 4.2: Production and decay vertices for Da r t e t g the process. Figure showsthe schematization of the production and decay vertices for Dasrut e t g the process.starts with two distinct trajectories that meet the requirements to be called d e d g a pair.The vertexing algorithm calculates a confidence level (CLS) for the hypothesis that thesetrajectories come from a common point and finds that point for the cases which have a„.0/21 z43 .Let us say that a useful vertex is found. The Da putative trajectory can be calculatedby adding the muon momentum vectors. Then the algorithm tries to find the primary vertexusing other trajectories in the event in combination with Da the candidate. The algorithmbegins by taking all of the two-trajectory combinations (one of which must be Da the andthe other must not be a daughter muon) and fits them to a vertex hypothesis again requiringa minimum CL of 1% [41]. If the trajectories are accepted, the algorithm tries to put moreand more trajectories in that vertex until the confidence level falls below 1%. The resultingprimary has a maximum number of daughters with a CL (CLP) greater than 1% [41].

°4.3 Data Selection 52Using information coming from the vertexing algorithm we can define the followingivariables to be used in the analysis for the rare decay Da b d e d g :1. L/5 . Short lived hadronic background is removed by requiring a minimumseparation between the primary and secondary vertices. The detachment cut will require a.s¯ minimum . where is the distance between the two vertices and is the error on thatmeasurement. Figure 4.2 illustrates this situation in which the errors on vertex positions°are shown as ellipses.2. Multiplicity of the Primary Vertex. This refers to the number of trajectoriesproduced in the interaction that occurs within the target.Higher multiplicity primaryvertices have less background.3. Vertex Confidence Levels. Each vertex is formed by a reconstruction algorithmthat has an associated confidence level. Due to measurement errors the trajectories willnot all intersect at the same point. However we can define error ellipses for each pairof trajectories (Figure 4.2). We then calculate the probability that, within their errors, allcandidate tracks are consistent with intersecting. Confidence levels above 1% are requiredfor minimal cuts on these values. Sometimes a larger cut is used to reject backgrounds.CLP stands for the primary vertex confidence level and CLS for the secondary.4. Primary and Secondary Vertex z Position. The FOCUS targets are thin flatwafers in the xy plane. Thus the primary vertex z position (ZVPRIM) becomes the criticalvariable in identifying good events where the primary vertex occurs within the target andbackground when it occurs outside the target. For certain long lived charmed particles, alarge percentage of the decays (secondary vertex) will occur outside the target segments

the target material and for °76 1£{ the vertex is outside. For the primary vertex we use °7698i4.3 Data Selection 53where there will be less background due to interactions of secondary particles with thetarget material. The secondary vertex z position (ZVSEC) can be used to maximize thesignal to noise ratio.Due to experimental resolution, these variables are not used directly but instead areused in a ratio with the experimental uncertainty. For this purpose we will define thevariable which is the difference between the z vertex position and the z target edgedivided by the uncertainty. Due to the four-part target segmentation we need to check if the°76vertex is occurring between the segments or within these. For °76xŠ{ the vertex is insideand for the secondary °76 P .5. Vertex Isolation. The primary vertex isolation, ISOP, gives a large confidencelevel when one of the trajectories in the secondary vertex is also consistent with being inthe primary. Therefore, the lower the ISOP value, there will be an unlikely chance thatany charm candidate trajectories will be in fact associated with the primary vertex. Thesecondary vertex isolation, ISOS, is the confidence level for a trajectory that is not in theprimary vertex to be in the secondary vertex together with the presumed decay daughters.This variable comes in handy for rejecting background from higher multiplicity decays byrequiring low values of ISOS.b) Particle Identification. This analysis concentrates on identifying two types ofparticles: muons and hadrons.1. Muon Identification. Muon identification is possible because muons are theonly charged particles which can penetrate large amounts of material. The method used

i4.3 Data Selection 54to detect muons is to place charged particle detectors behind a large amount of shieldingmaterial (typically steel). There are a pair of different kinds of detection systems for muonsin FOCUS. These are: the Inner Muon detector and the Outer Muon detector. The first oneuses common scintillator detector elements and the second uses resistive plate chambers todetect passing muons.For the inner region the system for muon detection uses three muon scintillatinghodoscope stations: MH1, MH2 and MH3. The first two stations register hits in x and ywhereas the third station registers hits in a uv plane which is oriented ¦the xy plane.{: with respect toThe algorithm of muon identification [33] was developed at the University of PuertoRico, Mayagüez Campus, with the purpose of calculating for each trajectory a confidencelevel to decide the degree of reliability in the classification of that trajectory as a real muon.The algorithm is based on the following considerations: the multiple Coulomb scatteringand the geometry of the scintillators that make up each one of the planes of the muonidentification system. It is described in detail by Ramirez [33] and Wiss [42, 43].There are two ways a muon identification algorithm can fail. One is to fail to identifya muon as a muon. This characteristic is measured as the algorithm’s efficiency. The otheris to identify a non-muon as a muon. That is measured by the misidentification. These twocharacteristics are correlated. A good algorithm achieves a balance between them.In the muon identification we are considering the following variables:i. Kaon Consistency for Muons. We can use the kaon consistency variable (see

i4.3 Data Selection 55hadron identification) to separate kaons from muons also, since pions and muons behavealmost identically in the Čerenkov system. The name given for this variable €¡¯d is .ii. Inner Muons. For the Inner Muon System (IMU) the algorithm begins byverifying if the candidate trajectories to be muons leave signal in the five PWC chambers.The trajectories are projected into the IMU planes and associated hits are identified bylooking within a circle around the projection .In the IMU system there are six MH planes.There must be four hits in theseplanes. We can proceed to calculate ] ^ which gives us the trajectory compatibility withthe MH hits. There are two other things that are taken into consideration when we do thiscalculation. These are the multiple Coulomb scattering that occurs inside the steel filtersand the granularity of the scintillator paddles.Lastly, an isolation variable is calculated which corresponds to the largest CL for anyother trajectory to have caused the same MH hits.Because there are steel filters in the IMU that can stop muons with low momentum,the number of required planes is reduced to two from the original number of four forcandidates with a momentum less than “•”—– 10 . This will increase the efficiency at lowmomentum.The efficiency for the muon identification algorithm is over 98% for momentumabove 10 “•”—– while misidentification is of the order of 0.5%. In-flight b d q4;decayscandidate to the misidentification rate. For this decay the pion and muon direction of flightare very closely matched.

›4.3 Data Selection 56Information obtained from the IMU is contained in the following variables:iA) Number of Missed Muon Planes. The number of missed muon planescorresponds to the number of MH planes without signal. In this analysis this variableis called MISSPL.B) Inner Muon Confidence Level. The muon identification algorithm calculatesa ] ^ value for each trajectory that passes a minimal requirement in the number of MHplanes with signals. Trajectories that do not correspond to muons tend to have higher ]_^because they have different behavior compared with muons [32, p.11]. The inner muon CL(IMUCL) is calculated from the ] ^ .iii. Outer Muons. Muons are identified in the Outer Muon System (OMU) almost thesame way as in the IMU. There is an internal magnetic field in the iron shield (M2) whichcan cause some problems when it comes to OMU identification because it deflects muons.Muons must be traced through the magnetic field in the M2 steel.Multiple Coulombscattering smearing must be applied to the traced position.range of muons in the M2.The OMU reconstruction algorithm is efficient for muons down to Ï“•”—– , theA) Outer Muon Confidence Level. In the same way as is calculated the IMUCL, theOMUCL is determined coming from the measurement of the ]_^ value for each trajectorythat passed a minimal requirement within the OMU.2. Hadron Identification. To identify hadrons such as µ , € µ , p¯ p, FOCUSdeveloped an algorithm called CITADL (Čerenkov Identification Through A Digital

k©U¥ƒk K WCB UkUW¥kUk4.3 Data Selection 57iLikelihood) [44] which calculates the probabilities for several hypothesis about the identityof a trajectory.The identities considered are electrons, pions, kaons and protons (andtheir antiparticles). The d$µ hypothesis is not considered separately from the $µ sincethe momentum range over which the two hypotheses can be separated is limited. In theČerenkov system the behavior of these two particles, muons and pions, is similar due totheir close mass values. For each of the four hypothesis, CITADL calculates the cone dueto the Čerenkov effect and the average number of photons emitted. Poisson statistics areused to calculate the probability that the observed size of the signals in the Čerenkov cellsis consistent with the identity hypothesis.From the probability < , the algorithm calculates the negative log-likelihoodEDGFA@WCBi. Kaonicity. It is defined as the difference between theÀûÁ y for the kaon and pionhypothesis.K W(4.1)=?>A@=?>A@Statistically, kaonicity is larger for real kaons that for real pions.ii. Pion Consistency. A variable called pion consistency or 7H =?I is used to identifypions.This variable is related to the probability that the pion hypothesis is better in

kŸnJÑ,ššJBn©kŸnJšz¥{{[“•”—–šz¥{{[“•”—–Ñ,Jn©L¥z¤k“ª”¢–š L “ª”¢–šUMMW^^4.3 Data Selection 58icomparison to other kinds of hypothesis. 7H =?I is defined as(4.2)=?>A@7H =?I=?>A@U\pš7+Ñ?ŸnWwhere thecorresponds to the minimal value of the likelihood fornWU1pall the hypothesis considered. Typically we use a cut 7H =?I 1œ{ of for pion identification.=?>A@š$+Ñ?ŸOther variables used in the analysis are:3. Momentum. To measure the momentum of a particle, FOCUS determined thedeflection angles in each one of the magnetic fields produced by M1 and M2. For threechambertrajectories the momentum is measured by M1 and for five-chamber trajectoriesthe momentum is measured by M2.The momentum resolution for trajectories measured in M1 is approximately° K(4.3)›…‘† { {z­Åand the resolution for trajectories measured in M2 is(4.4)° K›…‘† { {zz­ÅAt high momentum, the resolution is limited by the position resolution of the PWCsystem; at low momentum it is dominated by multiple Coulomb scattering.4. Dimuonic Invariant Mass. The mass of primary particles is calculated from the

i4.3 Data Selection 59information obtained from the particles into which these decay (Section 1.6). In our casethis is the two muon trajectories into which Da the decays.4.3.2 FOCUS Data Selection StagesThere are 6.5 billion photon interaction events on about 6000 8 mm magnetic tapeswhich constitute the raw data set of FOCUS. Obviously, this is way too much data foran individual researcher to handle. Therefore, the analysis was divided into three steps.Figure 4.3 shows a summary of the FOCUS analysis process. The reason for doing theselection in stages is to avoid having to handle the totality of the data again in the case thatsome problem is later found in the selection process. In a multistage procedure, one onlyneed begin at the stage previous to the problem.a) Pass1. This is the basic event reconstruction stage including track reconstructionand particle identification. It lasted from January, 1998 to October, 1998. The outputconsisted of “reconstructed events” made up of the values for the physical variables of theevent. It was put onto another set of 6000 8 mm magnetic tapes.b) Skim1. Here, the reconstructed data was divided into six “Superstreams”. Thiswas done in order to make the size of the data easier to handle. Approximately one halfof the events from Pass1 met the requirements for Skim1 and depending on the physicaltopics that were considered they were put into different Superstreams (see Table 4.1).Each Superstream consisted of 200–500 8 mm magnetic tapes. Skim1 was carried out atthe Universities of Colorado and Vanderbilt from October, 1998 to February, 1999.

4.3 Data Selection 60iReconstruction OverviewDAQMassStorage3% Raw Data3% Recon Data15% of Charm10 days of Skim1Pass1RawDataExpressAnalysisShift Monitor3% ReconCharm YieldMonitorMainline Tape PathExpressline Network PathSkim1ReconDataFNALDiskSkimmedDataSkim2Skim2InstitutionsFigure 4.3: Analysis overview of FOCUS. The solid lines show the path of the datawritten to 8 mm magnetic tapes. The dotted lines illustrate the distribution of largeamounts of data via the Internet, which was used to help collaborators get data quicklyfor studies and preliminary analyses (E. Vaandering’s Thesis Figure).c) Skim2. The six Skim1 Superstreams were divided into many other Substreams.Those events that did not pass the more specific cuts were not included but not all of theskims had additional cuts.

4.3 Data Selection 61iTable 4.1: Descriptions of Superstreams. There are about 30 different datasubsets grouped into six Superstreams based on physics topics and the typesof information present.KaySuper Physics Skim2Stream Topics Institution1 Semileptonic Puerto Rico2 Topological vertexing and Illinois3 Calibration CBPF, Brazil4 Baryons Fermilab5 Diffractive (light quark states) California, Davis6 Hadronic meson decays California, DavisTable 4.1 shows the five institutions involved in Skim2. There were 5–12 Substreamswritten from each of the Superstreams. Calibration data samples and many topics in physicswere brought together by these Substreams. Similar computing models between the Skim1and the Skim2 were used but they varied by institutions. Skim2 began in January, 1999 andfinished by June, 1999.Table 4.2 refers to the output of the Superstream 1 reconstructed and analyzed by theHEP Group of the University of Puerto Rico at Mayagüez Campus during January to Juneof 1999. More than 500 tapes of semileptonic events were skimmed and divided into fivedifferents subsamples.d) Skim3. It is in this stage where we really apply strong selection criteria whichlead us to obtain a reduced output yet still leaves us with a certain amount of flexibilityin the detailed analysis and optimization. It is very difficult to have to work with dozensof tapes in the detailed analysis. Skim3 reduced this number to one tape. Just the sameas in earlier stages, the idea with Skim3 was to obtain a more compact sample that wouldcontain the great majority of a certain kind of event.

‘°4.3 Data Selection 62iTable 4.2: Descriptions of Superstream 1. This Superstream was developed bythe University of Puerto Rico at Mayagüez Campus. There are five differentskims grouped according to physics topics.Sub Physics OutputStream Topics Tapes1 Semimuonic 262 Dileptonic/PPbar 453 Semielectronic with mesons 374 Semielectronic with baryons 275 Normalization 58The data in Substream 2 of Superstream 1 was used as the starting point in Skim3because it provided the highest efficiency for the detection Da b d e d g of . The mainidea of Skim3 was to require the existence of two linked muon candidate trajectorieswith opposite charge coming from a parent that was identified with a cut in the dimuonicinvariant mass around the mass of the Da (between 1.7 and 2.1 “ª”¢– ).In addition tothis, certain vertex identification conditions were required consisting of „N.POQ1 z43 andz43 . Furthermore, the production vertex was required to be separated from thea reduction in the data size from 45 tapes of 4.5 Gbytes each to one 3 Gbyte tape. The 3decay vertex by at least three times the separation error (.s¯„.0/R11 ). With this we achievedGbytes were easily kept on disk for greater convenience.At the end of Skim3, we continued to reduce the data by a process which we callSkim4. The selection criteria required for Skim4 were the following:These cuts were based on the experience acquired in previous FOCUS data analysis;these were known to have high efficiency.The output from Skim4 was a matrix of S¸ as an event file where m is the number

Jgp J\·+bpSKXXXXŠV4.4 Normalizing Mode 63iŠ3b Vertex isolation cuts Š 1% b Vertices z position, ¥§z¥{Ô Ã ËXW°76T8VU@‹b Daughter momentum 1 7 “•”—– b We required a MISSPL Yb IMUCL 1 z43 b OMUCL 1£z43of variables with information to be stored and n is the number of pair trajectories acceptedas muon candidates. The total data size did not exceed 500 Mbytes. This matrix is calledan ntuple, which is analysed using software developed by CERN called PAW (PhysicsAnalysis Workstation) [45] which lets us easily manipulate the information stored. It useshistograms, vectors and other familiar objects to provide the statistical or mathematicalanalysis interactively. An ntuple allows us to try different analysis criteria in a short timeinterval.4.4 Normalizing ModeThe object of this study is to determine at what frequency Da b d e d g do occur innature. This means that we will indirectly measure how many times this process occurswith respect to a large sample that contains many different processes.The branching ratio (BR) is defined as the fraction of decays to a particular channel.Mathematically(4.5)`dc$SŸeZ+*f Da bd e d g š7Z*+c'^—ŸÖ[Z]\D¸_^Ñ.©*¸'Ǹab\©¡+§†¸_c$SŸeZh+*f Dalš7Z*+c'^—Ÿ

gJ\·+UpUW†WX†UUX†UUpWWtArs;u; tU¿UnWK g eWXWX4.4 Normalizing Mode 64iIf we consider a well known process where the frequency at which it occurs is knownwith some certainty then we can use that process to measure the total number of Da ’sproduced.(4.6)¸_c$SŸeZh+*f Da š7Z*+c'^—Ÿìc'SŸeZj+]fa blkšmZ]+cn^¢ŸÖ[aa bokU1¿The process a b k is called the normalizing mode. In FOCUS a process coming¿from Da as primary particle is Da­bK g _e . It is known as a Golden Mode due to itslarge BR value. In the analysis for the search of the Da b d e d g process we can considerb K g e as the normalizing mode. Then the BR of the decay Da b‚d e d g can beDacalculated as:` ;q;(4.7)XŽWš7Z*+Da bÖpaÖ[aDa bd e d gš7Z]+XŽWǸrsThe number of produced particles is calculated as the number of observed particlesdivided by the (t efficiency ) of detection calculated by Monte Carlo simulations. Then eq.(4.7) turns intoK g eK g e(4.8)ÖpaDacb‚d e d gÖpaDacb`«ÀûÁ yDa b‚d e d gwhere the ratio tArs¯]t ;q;corresponds to the relative efficiency to be calculated making`«À*Á yDa bparallel Monte Carlo simulations for the two processes, Da­b d$efdhg and Dacb K g _e .BR(Da b K g e ) has been previously measured by other experiments as ¨ ¦§{ {z¥{ g ^[4, p.36].¬$‘The data events for the normalizing mode were selected using the same selection

i4.5 Monte Carlo Simulation 65criteria (except for particle identification) used to select the dimuonic data events. Thenormalizing mode data was obtained from Substream 5 of the Superstream 1.ThisSubstream contained hadronic decays selected with the same basic conditions as the otherSuperstream 1 events with the specific purpose of serving as a normalization sample.4.5 Monte Carlo SimulationBecause of the complexity of High Energy Physics experiments and the impossibilityof calculating every effect analytically, simulation methods are commonly used. Computershave pseudo-random number generators which are used to simulate approximations ofexpected results coming from the theoretical predictions. We call this simulation methodMonte Carlo. In FOCUS, generated Monte Carlo events were used as a measurement toolfor the efficiencies and also as an informative tool about the form of the background withinour dimuonic sample.Two mathematicians, the American J. Neyman and the Hungarian S. Ulam gavebirth to this method in Monaco in 1949 as an alternative solution to probabilistic aswell as deterministic problems.Using a computer, the Monte Carlo method performsstatistical sampling experiments by simulating random quantities, thereby providing uswith approximate calculations in High Energy Physics problems [46].FOCUS used the Pythia 6.127 [47] generator as a Monte Carlo simulator to modelcharm production by photon interactions with the target material. Pythia generated a ccpair where it was possible to specify the type of charm ( ¿a , ¿e , v e w , etc.) generated and

4.5 Monte Carlo Simulation 66ia specific decay path. Typically this was done for one of the charm quarks while the otherwas left free to hadronize and decay according to the known cross sections and branchingratios. To simulate all the processes within the FOCUS spectrometer, an algorithm calledROGUE was developed [48].Dxzy {}|0{~ Events. To calculate dimuonic efficiencies it was necessary togenerate Monte Carlo data which were required to pass through the same selectioncriteria that were applied to real data, in other words, the requirements set by Pass1,Skim1, Skim2, Skim3 and Skim4.800,000 events were generated for the processD 0µ+z€‚„ƒ … D† ƒo‡‰ˆ_‡¥Šn‹‰€c represented in Figure 4.4.γµ−Figure 4.4: Monte Carlo generated process schematization for the decay D†CŒR ˆ Š .c−The number of events provided us with enough sampling statistics to make accuratedeterminations of the detection efficiencies.DŽz K ‘j’ Events. 800,000 D† ƒK Š7“ ˆ events were generated. The sameselection criteria were applied as with the normalizing data.The decay channel D† ƒK Š “ ˆ is very important in this work because this is the

4.6 Analysis Methodology 67”normalizing mode used in BR(D†ƒo‡ ˆ ‡ Šthe ) measurement. In addition to that, thesame decay channel is present as possible background in our dimuonic sample.DŽ ‘ ’ ‘ and DŽ K ’ K Events. The processes D† ƒ “ ˆ “ ŠandD† ƒ Kˆ K Š were also considered in the background study. For the first process, 1.5million events were simulated and 4.3 million events for the second.4.6 Analysis Methodology4.6.1 Blind AnalysisOur Blind analysis excludes all the events within a defined symmetric area aroundthe value of D† the mass in the process of cut selection. This area is called the signal area.In other words, cut optimization is based on reducing background and not on maximizingthe signal under study. The point of this is not to introduce bias due to the possible apparentabsence or presence of a potential signal after having applied a specific group of selectioncuts. Once the optimization stage ended we opened the “hidden” area of the signal andcalculated the confidence interval.4.6.2 Rolke and López LimitsThe statistical method proposed by Rolke and López [49] was used to calculate thesensitivities and the confidence intervals.They suggest a technique to place limits on

”4.6 Analysis Methodology 68signals that are small when background noise is present. This technique is based on acombination of a two dimensional confidence region and a large sample approximation tothe likelihood-ratio test statistic. Automatically it quotes upper limits for small signals andalso two-sided confidence intervals for large samples.The technique of Feldman and Cousins [25] does not consider the amount ofuncertainty in the background.The new Rolke and López technique deals with thebackground uncertainty as a statistical error. The technique performs very well. It hasgood power and has correct coverage. This method can be used for two situations: whensidebands data give an estimated background rate and when it comes from Monte Carlo. Itcan also be used if there is a second background source in the signal region.To understand the results of this technique we will need the following notation: x willrepresent the observed events in our signal region, y will be the events in the backgroundregion • and will be the ratio of the sidebands background to the signal background. Theexpected background rate is –˜—Q›š¦• and the Sensitivity Number, œN … D† ƒo‡‰ˆ_‡Šn‹, isdefined as the average 90% upper limit for an ensemble of experiments having an averageof y events in the sidebands and no true signal. Table 4.3 gives the sensitivity numbers fordifferent values of the confidence • level, and y. During the various stages of this analysiscomputer routines provided by Rolke and López were used to calculate sensitivity numbersand confidence limits. The value • of for each set of cuts was determined as explained inSection 4.7. The sensitivity numbers were used to select the optimum set of cuts. Then theconfidence limits were computed from the values of x, y • and for that set of cuts. Examplesof confidence limits are given in Table 4.4

4.6 Analysis Methodology 69”Table 4.3: Sensitivity number as per Rolke and López. The number ofbackground events in the sidebands is y. The estimated background rate isC.L.C.L.–—ž›š¦• . Ÿ ¡ŸŸ¡•¢—¤£ •¢—¦¥ •¢—§£ •¢—¦¥0 2.21 2.21 4.52 4.521 3.31 2.82 6.22 5.35y2 3.97 3.17 7.12 6.143 4.62 3.49 8.17 6.454 5.29 4.03 8.99 7.125 5.90 4.23 9.77 7.436 6.37 4.60 10.52 8.007 6.86 4.79 11.21 8.308 7.23 5.14 11.85 8.869 7.64 5.32 12.51 9.0210 8.01 5.62 13.04 9.5211 8.40 5.78 13.55 9.7612 8.71 6.08 14.15 10.1413 9.09 6.19 14.63 10.3714 9.39 6.45 15.14 10.7615 9.71 6.59 15.64 10.964.6.3 Determination of Sensitivity and Confidence LimitsWe define the experimental sensitivity as the average upper limit for the branchingratio that would be obtained by an ensemble of experiments with the expected backgroundand no true signal. Therefore, the experimental sensitivity is calculated in the same way asthe branching ratio except that the sensitivity number is used for the number of observedsignal events. For each particular set of • cuts, is determined as explained in Section 4.7.3.is the predicted level of background, b. Having this in mind, we can determine the›š¦•

signal region and determined the corresponding confidence limits for the D† ƒ4.6 Analysis Methodology 70”Table 4.4: An example of Rolke and López 90% Confidence Limits. y is thenumber of events observed in the background region and and x is the numberof events observed in the signal region. A value of •„—ž¥ has been assumed.yx 0 1 2 3 40 0,2.21 0,2.27 0,1.58 0,1.28 0,0.91 0,3.65 0,3.22 0,3.44 0,3.06 0,2.872 0.42,5.3 0,4.87 0,4.43 0,4 0,4.413 0.97,6.81 0.08,6.37 0,5.93 0,5.49 0,5.054 1.54,8.25 0.74,7.8 0.03,7.36 0,6.91 0,6.475 2.16,9.63 1.42,9.18 0.74,8.73 0.07,8.28 0,7.836 2.82,10.98 2.12,10.53 1.45,10.08 0.8,9.62 0.17,9.177 3.5,12.3 2.83,11.84 2.18,11.39 1.54,10.94 0.91,10.488 4.2,13.59 3.55,13.14 2.91,12.68 2.29,12.23 1.67,11.779 4.92,14.87 4.29,14.42 3.66,13.96 3.04,13.5 2.43,13.0510 5.66,16.14 5.03,15.68 4.42,15.22 3.81,14.77 3.21,14.31experimental sensitivity as¨ … D† ƒo‡ ˆ ‡ Š ‹ —… D† ƒo‡‰ˆ_‡Š$‹œN… D† ƒK Š “ ˆ ‹®­°¯±œN©¡ª¬«… D† ƒK Š “ ˆ ‹ (4.9)­²q²Q³[´œN where, D† ƒ ‡µˆ_‡Šn‹is the sensitivity number (Section 4.6.2) which is determinedusing the Rolke and López routines from the number of events observed in the sidebands,…y, and the background ratio, • .In simple terms, the sensitivity is mainly dependent on the ratio œ[š of . We wishto minimize this ratio which means having low background with high efficiency.­²q²Once the optimum set of cuts had been determined, we proceeded to open the hidden‡‰ˆ'‡Š

equal to ½¾—”4.6 Analysis Methodology 71branching ratio by using eq. (4.9) and substituting the appropiate Rolke and López limitsin place œ[ of .4.6.4 Sidebands Background and Signal AreaIn this analysis, the signal area as well as the sidebands were defined from theinvariant mass distribution of Monte Carlo generated data (Figure 4.5a). The ability of theFOCUS Monte Carlo to predict signal width has been verified with other copious charmsignals.σ =·0.016 GeV·σ ¸|Figure 4.5: D† ƒ ‡ ˆ ‡ ŠInvariant Mass distributions. (a) distribution forgenerated Monte Carlo data with a gaussian fit. The gaussian width is equal to¹Vº¼»0.016. (b) Representation of the sidebands and the signal area for the FOCUS produceddata showing the signal and sideband areas. Skim3 cuts have been applied and thesignal area has been masked.The width of the invariant mass distribution for D† ƒ‡‰ˆn‡Ševents is approximately›¿À¿ÀÈ£ÂÁÄÃbÅuÆ . The signal area was taken as ÇN¥*½ around a D† mass of £ÁÉ

sideband was defined as £Í£¥Í£4.7 Background Study 72”ÃbÅuÆ[4, p.36]. Each one of the sidebands had a width equal to the signal area width butthe sidebands were not located immediately after the signal area. Between each sidebandand the signal area, a buffer area was left with a width also equal to ¥*½ . In this way the left¿ËÊ]ÌʢͦοÀÈ]¿ÏŸŸÐÍÑοÀŸŸ]Ì£ and right sideband £as shown in Figure 4.5b.²q²²q²4.7 Background Study4.7.1 IntroductionThe most important aspect of the analysis is the choice of a set of selection criteriawhich optimize the probability of observing the signal if it is present.Generally thischoice will not be one which eliminates all background because such cuts usually reducethe efficiency of detection of the signal to unacceptably small values.It is thereforeimportant to be able to predict the background level accurately in order to measure thesignificance of the candidate events. Due to the uncertainties in simulating the background,our analysis methodology relies mainly on the real data (in the sidebands) to predict thebackground in the signal. However, this methodology requires a determination of the ratioof sideband to signal • background, . This, in turn, requires a determination of the shape ofthe background. As is explained in this section, this shape is also obtained from real data.Fortunately, since they are not strong-interacting, muons are not produced copiouslyand it is even rarer that a pair of muons is produced in either a production or decay vertex.In addition, events where the pair of muons come from a non-D† two-body decay are

ƒÊ4.7 Background Study 73”eliminated by the invariant mass cut while those where the muon pair is formed at theproduction vertex (e.g. those from the Bethe-Heitler process) are eliminated by the ÒÓš*½cut. In fact, the main source of background in our final sample are events where one or twomesons have been misidentified as muons. Therefore the understanding of the backgroundstarted with a study of this misidentification using real data. These results were used tovalidate the capability of the Monte Carlo to simulate this effect. The Monte Carlo was thenused to study some known decay processes which could contribute to the background if thedecay products were misidentified as muons. Finally, the background ratio was determinedfrom an independent real data sample of events with no muons in the final state by assumingthat the background consisted of such events except that both decay products had beenmisidentified.4.7.2 MisID StudyThe goal of this study was to determine the misidentification probability as a functionof momentum. Samples of real and simulated K†«“ ˆ “ Šdecays were used for thispurpose.The real data was obtained from 140 raw data tapes. K†« decays were identified withthe following requirements: vertex conditions for a pair of opposite charged particles, thetwo pion trajectories must have left a signal in the five PWC chambers, and the dipionic›¿ÏÔ ÊÊÐͦÎmass in a K†« range around the mass (), being the K†« mass equal toÍÑ Õ¿›¿ÖÔ ŸÈ[4, p.32]. Furthermore, decay candidates had to have linked trajectories betweenÃ×ÅqÆthe silicon detectors and the PWCs. Events which pass these requirements are called±u±ÉÕ£

ƒ”4.7 Background Study 74K†« Type-9 [32, p.27]. By determining how many of these trajectories were identified asmuons one could determine the misidentification probability as a function of momentum.by artificially lowering the K†« lifetime thus forcing the K†« to decay near the target andA sample of simulated Type-9 K†«“ ˆ “ Šdecays was generated with Monte Carloobtaining a large sample of pions coming from the target. This sample was analyzed in thesame way as the real Type-9 K†« data obtaining consistent results between the two samples(See Section 5.1).4.7.3 Background Ratio StudyIn order to have a rough idea of the nature of the background, Monte Carlo was usedto generate two-body mesonic D† decays which were expected to contribute significantlyto the background. The first decay simulated D† ƒ “ ˆ “ Šwas . Figure 4.6 shows thecomparison of the dimuonic invariant mass distributions D† ƒ ‡ ˆ ‡ Šfor D† ƒ “ ˆ “ Šand .The D† ƒthe pion and muon mass. Since that difference is small, the shift in the peak is small“ ˆ “ Špeak is slightly shifted from the D† mass due to the difference betweenand the invariant mass cut will not help us to eliminate this background. Fortunately, theD† ƒ “ ˆ “ Šdecay is Cabibbo-Suppressed and the FOCUS muon system is very good atrejecting pions (low misidentification probabilities).The other background decay generated was D† ƒK Š “ ˆ . Figure 4.7 shows thedimuonic invariant mass distribution for D† ƒK Š7“ ˆ events. Due to the large massdifference between the kaon and the muon, the peak is much more shifted but it doesfall within the left sideband. D† ƒK Š “ ˆ is a Cabibbo-Favored decay. Fortunately, it lies

4.7 Background Study 75”Figure 4.6: Dimuonic Invariant Mass comparison of Monte Carlo data fromD†ÓŒÙØ ˆ Ø Šand D†ÓŒÚ ˆ Š .outside our signal area and can be almost entirely eliminated by the invariant mass cut.This distribution also shows an additional continuous background inside the signal region.This background is from the other charm particle in the event. (Charm is always producedin pairs). It is made up of semimuonic events where the neutrino is not observed or frompartially-reconstructed hadronic decays. It is very difficult to simulate all such backgroundcomponents and to know their relative abundance. This is why we do not rely on simulationto obtain the background sideband to signal ratio.As an alternative to simulation data, a sample of real data from Superstream 2 (also

4.7 Background Study 76”Figure 4.7: Dimuonic invariant mass of D† ŒK Š Ø ˆgenerated by Monte Carlo.called Global Vertex) was used to determine the background ratio. This superstream wasselected with a minimum of requirements and contains all backgrounds to our signal. Inparticular, no particle identification was used in making SS2. Our procedure consisted insubmitting this sample to the same skimming process as the real data except that no muonrequirement was used. It was only necessary to work Û with 10% of the total SS2 data inorder to obtain sufficient statistical power for our purposes.As an example of the dimuonic invariant mass distributions obtained with thisprocedure, Figure 4.8 shows the results when one applies tight cuts to the SS2 sample.

”4.7 Background Study 77The D† ƒK Š$“ ˆ is the dominant feature in the distribution but a D† ƒ “ ˆ “ Špeak isalso evident as well as other background. The D† ƒ“ ˆ “ Špeak is much smaller than theD† ƒK Š “ ˆ peak because it is Cabibbo-Suppressed. The peak from D† ƒ Kˆ K Š liescompletely outside any region of interest. As the cuts are varied, the ratio of the backgroundin the sidebands to that in the signal (• ) varies but the number of events in the histograms issufficient to allow a precise determination of this ratio which is calculated using weightedhistograms as explained below.D† Œ Š Ø ˆ D† ŒRØ ˆ Ø Š D† Œ Kˆ ŠFigure 4.8: Background comparison between real and Monte Carlo data. Distributionfrom SS2 data with tight cuts but no muon identification. These events constitute themain source of background when the decay products are misidentified as muons. Thereare evident peaks from the processes K , and Kalong with other background.

4.8 Optimization of the Selection Criteria 78In order to take into consideration the probability of the decay products being”misidentified as muons, the background ratio was determined from weighted histograms.Each entry in the dimuonic invariant mass histograms of the SS2 data (such as Figure 4.8)was weighted with the product of the momentum-dependent MisID probabilities for eachof the two decay products. Events with two muons were eliminated from the sample butevents with one muon were allowed in order to include semimuonic background. Theweight was calculated as follows:Ü —žÝ Þ¥ß}Ý4à (4.10)where Ýi—á£for muons, and Ýi—for non-muons. Then, we have that¿ÀÎãâ?äeå&æÎãâ?äeå&æΞâêäeåÕæ— çèÜÎãâ?äeå&æ— çèÜ£éߥ if neither decay product is identified as a muon.if one of the two decay products is identified as a muon.Results from this procedure are presented in Chapter 5. A value for the backgroundratio (• ) was obtained for each of the sets of cuts considered.4.8 Optimization of the Selection CriteriaThe optimization procedure of our selection criteria was the longest and mosttiresome phase of this effort. It established the final criteria for the selection of muoncandidate events. The aspect that was constantly kept in mind was to eliminate a largenumber of background events maintaining high efficiency values.

4.8 Optimization of the Selection Criteria 79For this procedure we started with the application of fixed (constant) cuts which”correspond to the cuts applied in the selection process including Skim4.Due tocorrelations, the optimization of the cuts required a procedure which considered a largenumber of cut sets. This implied a selection of the appropiate cut ranges for each one ofthese variables. Once each cut range had been determined we proceeded to optimize ourselection criteria with the idea of obtaining the best set of cuts to be applied in our searchof dimuonic events.4.8.1 Fixed Cuts and Variable Cut RangesIn addition to the cuts applied in the earlier stages, in this analysis the Skim4 cutswere used as fixed or constant requirements. Table 4.5 presents the fixed cuts applied to theinvariant mass variable which define the sideband and signal regions.Table 4.5: Dimuonic Invariant Mass Cuts for Sidebands and Signal Regions.VariableNameSidebandsSignal AreaCutAppliedòôïQë$ìöõ7÷ë (left)ë7ì-ímî_í‚ïñðóòòôïQë$ìöø$ø$î (right)ë7ìöø$ù'øôïñðóòë7ìöõ$î'îôïñðóò òôïQë$ìöõ$øníThe range of the variable cuts including the variables used for hadron identificationwere chosen on the basis of the experience with previous FOCUS analysis. These arepresented in Table 4.6.

ISOS ïþëÿ÷ ¡ ü&ë&÷¢üÕëÿ÷£ ë7üü í ë7ü îµü û4.8 Optimization of the Selection Criteria 80”Table 4.6: Cut values for each variable considered in the optimization process.Variable NameVertex IDCut AppliedCLSISOP ïþëÿ÷ ¡ ü&ë&÷¢üÕëÿ÷£Ò0š¦½úñûµü ínü*øúý÷‰ìö÷ë7ü¦÷ìG÷'ûµü4÷ì¨ë÷‰ü*ùK/ ‡ ¨ïñûµü]õ½¥¤§¦Muon IDMISSPLIMUCLMeson IDïþë7ü]ùµü*îúý÷‰ìö÷ë7ü¦÷ìG÷'îµü4÷ìG÷'û© … “K ‹“¥4.8.2 Optimization Based on SensitivityAll possible cut sets that could be formed from the cut values in Table 4.6 wereconsidered. The total number of cut sets was 26244. The sensitivity was calculated foreach cut set as presented in eq.(4.9). The optimum cut set was defined as the one with thelowest sensitivity.

ççÍ¡Chapter 5Results and Conclusions5.1 MisIDThe measurement of the percentage of pions that were misidentified as muonswas carried out using a large sample K†« ofevents. Results from this study are presented in Figure 5.1 for which the following muonidentification cuts were applied:ƒ “ ˆ “ Šdecays using Type-9 K†« dataInner Muon CL¦£.MISSPL¥ .We can see that for low momentum the probability of pion misidentification is greaterthan for high momentum for real data as well as for Monte Carlo simulated data. This is81

5.2 Background Ratio 82”Figure 5.1: MisID as a function of momentum.due to the fact that several effects that contribute to pion MisID, such as pion decays andmultiple Coulomb scattering, become larger at low momentum. The results from data andMonte Carlo are consistent within errors.5.2 Background RatioFigure 5.2 shows a typical weighted invariant mass distribution for background ratioof a specific set of cuts selection using Global Vertex sample. The background ratio • was

”5.3 Skimming Results 83calculated by integrating this histogram numerically in the sidebands as well as the signalregion.Figure 5.2: Typical Weighted Invariant Mass Distribution Used in Calculating theBackground Ratio.5.3 Skimming ResultsInvariant mass distributions with Skim3 selection criteria are shown in Figure 5.3 fordimuonic as well as normalizing samples.

5.3 Skimming Results 84”Figure 5.3: Invariant Mass Distributions for Skim3 Cuts. (a) Dimuon real data sample.(b) D†PŒÚ ˆ Š Monte Carlo sample. (c) Normalization real data sample. (d)Normalization Monte Carlo sample. The signal region has been masked for the dimuonreal data. The real data samples have large background levels with these loose cuts.Following the blind analysis methodology, only sideband events are presented for thedimuon real data. For the real data distributions we can see a higher level of background(Figure 5.3.a and Figure 5.3.c).Figure 5.4 shows the same distributions displayed in

”5.3 Skimming Results 85Figure 5.3 with the difference that these distributions correspond to events selected usingSkim4 criteria.Figure 5.4: Invariant Mass Distributions for Skim4 Cuts. A considerable reduction inthe level of background is achieved with the application of these stronger cuts.Although the background in the dimuon real data (Figure 5.4.a) has been reducedconsiderably with the Skim4 cuts, it is necessary to reduce it even further to obtain the

”5.4 Cut Optimization Results 86best measurement. The determination of the optimum cut set to achieve this reduction isdiscussed in the next Section.5.4 Cut Optimization ResultsUsing the cut ranges shown in Table 4.6, the values • of that had been previouslydetermined and the Rolke and López routines, the sensitivity was determined for eachof the 26244 different combinations of cuts. The lowest sensitivity was obtained for thefollowing cut set.Table 5.1: Best Cut Combination.7.01%$ #" !1%$ !1%% ' &1)(+*3,$ !." -1%& /-1%0& 0 #-8.0$ 11.02576 1 8 9 4 36;:= 9-5.0

5.4 Cut Optimization Results 87”Figure 5.5: Mass distributions for the best cut combination.For these cuts the dimuon mass distributions for the sidebands data and theD† ƒo‡ ˆ ‡ ŠMonte Carlo are presented in Figure 5.5.a and Figure 5.5.b. There are fiveevents in the sidebands. The efficiency for detecting D† ƒo‡ ˆ ‡ Šis 2.8%.The distributions for the normalizing mode are Figure 5.5.c and Figure 5.5.d. Forthe real data the yield corresponds to the number of events in the Gaussian curve. For the

¥£¨ … D† ƒ ‡ ˆ ‡ Š ‹ —…£Ì¥ÉÁ¥¥¿…5.4 Cut Optimization Results 88”Monte Carlo generated data, the efficiency­A¯±is calculated as the number of events in theGaussian curve divided by the total number of generated events.Our reconstruction process followed by the analysis algorithm let us observe 135432D† ƒK Š7“ ˆ events with a detection efficiency of 3.9%. Table 5.2 summarizes the resultsfor the best cut combination. For the backgound ratio study the results using the finalselection criteria are shown in Figure 5.6.Table 5.2: Results of Cut Optimization.5.725WV 8 @BADCFEHGJILKJM!N!O0.9PQASRTUKT[N aTUGJNbADc @BA^C_EHGJIKJM!N!Ò "XIYZO!T[C\R]YZOdYZN!e]TfRThgiThRkj l`M!m noYZIXp[l`qar2.96D ÷ts 2vuw2 Yyx CzTUYZN!C\j 5?{ 82.8D ÷ s K 6 u|Yyx C}TUY~N!Cj 5¡{ 8 3.9dYZN!e]TfRThgiThRkj 5}€‚Zƒ d„ 81.1¿ÏŸUsing the Rolke and López algorithm, the sensitivity number is determined to beÁ . Then the experimental sensitivity is calculated as¿ÀŸÔ Ì£ Š ࣠Šˆ‡(5.1)Ì›¿ÏŸ…£4É¿ÏÈ…ÌÕ¿ÀÈȆ…£ Š à—§££ Š àlimit ofThis result is encouraging since it is much lower than the existing 90% confidenceÔm¿measurements.£ Šˆ‡. This means that our measurement will be more precise than previous

5.5 Confidence Limits 89”Figure 5.6: Invariant Mass Distribution for Background Ratio using the best cutcombination.5.5 Confidence LimitsOnce our selection criteria were optimized, we proceeded to open the masked area.Applying the best cut combination to dimuonic events within our signal area we have twoevents in that region. Thus, we have five events in sidebands area (i—in the signal area (‰ —¦¥ ).É ) and two events¿ÀÊ›¿ÏŸUsing the Rolke and López routines, for • —§Éand, the 90% confidence level lower limit is 0 and the 90% confidence level upper limit¥ , the background rate (–j—¦mš4• ) is

… D† ƒo‡ ˆ ‡ Š ‹ —£ÌÉÉ¥¥¿£…5.5 Confidence Limits 90”Figure 5.7: Event candidates D†ÓŒÚ for the process. Figure display the only twoevents within the signal area which survives the optimum selection criteria in the studyof the rare decay ˆ.Š Š ˆ D†ÓŒÚis 5.34. The 90% upper confidence limit for the branching ratio is calculated as¿ÏÌ]ÔÌÕ¿ÀŸ…ÔÌ£ Š ࣠Šˆ‡(5.2)—¦¥¿ÀÈ…Ì›¿ÏÈÈ…£ Š ࣠Š à³p´

¿£…5.6 Conclusions 91No definitive evidence of the rare decay D† ƒo‡ ˆ ‡ Šwas found in this analysis.”However, our work allows us to set a lower 90% confidence level upper limit for this decay.5.6 ConclusionsMonte Carlo simulation of the level of meson misidentification in FOCUSwas confirmed by experimental measurements using real data.The misidentificationmeasurements were used as the basis of a method to determine the ratio of backgroundin the invariant mass sidebands versus background in the signal region. This method usedreal data and avoided systematic uncertainties in the use of simulation.It was possible to reduce the background in this analysis significantly using wellknownselection variables based on the characteristics of the signal being sought. Theblind analysis methodology provided an excellent way of optimizing the event selectioncriteria with a minimum of bias.The main conclusion obtained from this work was the reduction of the upper limit inthe branching ratio at a half of the value previously published by PDG. No evidence for therare decay D† ƒ ‡ ˆ ‡ Šwas found. The 90% CL branching ratio limit was ¥£ Šˆ‡.

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