A Search for the Rare Decay D - Recinto Universitario de...

A Searchfor theRareDecayD��

by

Hugo R. HernandezMora

Thesissubmittedin partialfulfillment of therequirementsfor thedegreeof

MASTER OFSCIENCE

in

Physics

UNIVERSITY OFPUERTO RICO

MAYAGUEZ CAMPUS

2001

Approvedby:

AngelM. LopezBerrıos,Ph.D. DatePresident,GraduateCommittee

HectorMendez,Ph.D. DateMember, GraduateCommittee

Dorial Castellanos,Ph.D. DateMember, GraduateCommittee

WolfgangA. Rolke,Ph.D. DateRepresentativeof GraduateStudies

Dorial Castellanos,Ph.D. DateChairpersonof theDepartment

L. Antonio Estevez,Ph.D. DateDirectorof GraduateStudies

Abstract

This thesispresentsa searchfor theraredecayD � �� basedon datacollected

by the Fermilabfixed target photoproductionexperimentcalled FOCUS(FNAL-E831).

This decayis an exampleof a flavor-changingneutralcurrentprocesswhich, according

to the StandardModel, should occur at a very low branchingratio of at most � � �� .A rate higher than this would be an indication of new physics. Using a blind analysis

methodwhich consistsof the observation of events outsidea definedsignal area, the

optimumselectioncriteria for candidateeventswasdetermined.This methodminimizes

biasby basingthe choiceon the backgroundobserved outsidethe signal region andnot

on maximizing the signal understudy. The ratio of backgroundoutsideto background

insidethesignalregion wasdeterminedusinganindependentrealdatasample.Theresult

for thebranchingratio of theD �� decayat 90%ConfidenceLevel upperlimit is�� "! whichreducesthepublishedvalueby 50%.No evidencewasfoundfor this rare

decay.

ii

Resumen

Estetrabajodetesispresentala busquedadel decaimientoraroD �� basado

en datoscolectadospor el experimentode fotoproduccion de blanco fijo de Fermilab

llamado FOCUS (FNAL-E831). Este decaimientoes un ejemplo de un procesode

cambio de sabora traves de corrientesneutrales(FCNC) el cual, de acuerdocon el

ModeloEstandar, deberıa ocurrir a unamuy bajaprobabilidada un ordende � � �� . Una

probabilidadmayora estapodrıa serunaindicacion de nueva fısica. Usandoun metodo

deanalisisa ciegasqueconsisteenla observacion deeventosfueradeunaregion definida

comoareadesenal,sedetermino uncriterio optimodeseleccionenla busquedadeeventos

candidatos.El usodeestemetododeanalisispermitio minimizarel prejuiciopresenteen

la eleccion de nuestrocriterio cual estababasadoen observar eventosde ruido fueradel

areadesenalconla ideadenomaximizarla senalbajoestudio.La razonderuidofueradel

areadesenal al ruido dentrodeestasedetermino usandounamuestrareal independiente.

El resultadoparala razon deporcentajeparael decaimientoD �� mostro un lımite

superior(con un nivel de confianzade 90%) de�� #�$� � �"! , el cual reduceen un 50% el

valorpublicadohastaahora.No seencontro evidenciaalgunaparaestedecaimientoraro.

iii

To God,

for mycountry:COLOMBIA ,

to mybeautifulcity BARRANQUILLA ,

to mywifeDiane,

to myparentsFanny andAnthony and,

to mybrothersMadelaine andJorge Luis.

Los quiero muchısimo!!!

To thethousandsof countrymenwhohavefallenwhile

I have beenin the “Isla del Encanto”becauseof theabsurd

conflict that lives within the boundariesof my harmed

country, speciallyto apersonwhowasanexampleto meand

my professionalguide, ProfessorLisandroVargasZapata,

cowardly murderedby gunmenon february23, 2001at his

doorstepin Barranquilla.

iv

Acknowledgments

There are so many people that in one way or anotherhave contributed in my

developmentas a humanbeing and as a professional. To mention them all would be

impossible. It is very difficult to even mentionsomebeforeothers. The most important

thing is thatall of thosewho believedin me,will alwaysholda key to my heart.

Firstof all. I thankGod,thatentitywhohasgivenuslife andwith that,agreatnumber

of questionsthatwetry to answerdayby day. I thankHim for giving metheopportunityof

beingborn in COLOMBIA , a countrythatevennow, whenit is goingthrougha crisis,has

a faceunknown to many. Thefaceof a countrythatalwaysfought for progress,of a hard

working countrythat is proudof its roots,thatcriesfor thosewho have fallenandmisses

thosewhoaremissing.A countrythatscreamsfor freedom,but aboveall, thereis acry of

hopethatthearmedconflict thatharmsuswill end. %'&�(*),+.-�)0/�12&3&547698.+.-:1<;0/�6=1>),?@? !I don’t want to leave behindmy city, BARRANQUILLA , my “Currambala Bella”.

Thecity thatwatchedmegrow up. On her streetsI learnedto live life to theextreme. It

wastherewheremy life is centeredandthe crib of my professionalfuture. I have great

expectationsfor my professionaldevelopmentandI wish to offer thoseto my lovedcity.

v

I thankFanny Mora, my mother, for makingme into what I amtoday. Without her

unconditionalhelp I wouldn’t heve gottento this point. To her above all, I dedicatethis

triumph.To my brothers,MadelaineandJorgeLuis, for believing in meandfor takingcare

of my motherwhile I wasabsent.To my fatherAnthony Valle, for alwaysbeingtherein

thegoodtimesaswell asin thebadtimes.

PUERTO RICO is a countrythat I will alwayshave with me. A master’s degree,a

marriageanda family. All this I take from you “Isla del Encanto”.I thankmy wife Diana

for herunconditionalhelp.Thankyou for sharingmy goodtimesandmy sorrow.

An immensurablethanksto the friendship,professionalandeconomicsupportthat

Dr. Angel M. Lopezgaveme. Thankyou for everythingyou did duringthetime I worked

for you, andfor alwaysbelieving in meandtrustingme. To professorsWill E. Johnsand

HectorMendezfor helpingmeout without askingandalsoto Dr. WeiJunXiong. Thanks

to Dr. WolfgangRolke for your cooperationin thestatisticalanalysisfor this work. Your

contribution wasof greatimportancefor me. Overall, to all of thosewho work with meat

theHEPlabatMayaguez.Thosefrom thepast:AlejandroMirles, CarlosRivera,Mauricio

Suarez,SalvadorCarrillo, FabiolaVazquezandKennieCruz;andthosefrom thepresent:

JoseA. Quinones,EduardoLuiggi, Alexis Parıs,Luis A. NincoandCarlosR.Perez.Thank

youguys!!!

To thePhysicsDepartmentat MayaguezCampusandits directorwhenI becamea

partof theUniversity, Dr. Dorial Castellanos(January, 1998),whogavemechanceto make

it happen.I thatall of thoseat theFOCUS Collaboration for giving metheopportunity

vi

to work on their experiment.To all thosepeoplewho work at EPSCoRbut mostof all to

SandraTrochewho wasalwaystherefor me.

To the greatfriendshipsI madein PR.To JuanCarlosHernandez,CarlosJose and

speciallyto their parentsJose “Machito” HernandezandJuanitaGonzalez. You werethe

family I neededthroughtoutthesethreeyears.

Thanksto Mr. JoeOrtız for helpingmein my collegestudies,thankyou for helping

mepursuethegroundsof PR.I alsothankMr. Felix Rivera,andtheRomeroValle family

aswell asValle Mirandafamily. Thanksfor everything. I thankMr. Rafael Sanjuanand

his family for beingtherewhenI neededit. A specialthanksto Mrs. MarthaCastano,may

Godblessyournoblesoul.

I thank my professorsat the Universidad del Atlantico - Barranquilla: Arnaldo

CohenandOswaldoDede,for makingme look at science.To my colleaguesandfriends

Arnulfo Barrios, OscarMontesinos,Harkccynn Torregroza,Ericka Navarro and Rafael

Manga. I also thank my High Schoolteachersfrom the BienestarSocial de la Policıa

- Barranquilla: Blas Torres, Carlos Bray and Orlando Ortız, specially Mr. Armando

Consuegra(Barrabas).

To all of thesewhomI have mentionedaswell asthosewhomI have not, otherwise

I wouldneverbeableto finish...

Mil y mil gracias!!!

vii

Agradecimientos

Sonmuchaslaspersonasquedeunau otra formahancontribuido enmi desarrollo

comoserhumanoy comoprofesional.Enumerarlasa todasresultarıa imposible.Inclusive

mencionaralgunasprimero que a otrasresultadifıcil paramı. Lo masimportanteque

quieromanifestaresquea todasestaspersonasquecreyeronenmı por siemprelasllevare

enmi corazon.

Inicialmentequisieradarlelasgraciasa Dios, a eseserquenoshaotorgadola vida

y con ella un sinnumerode interrogantesquedıa a dıa tratamosdecontestarnos.A El le

agradezcoel habermedadola oportunidaddenacerdondenacı, mi patriaCOLOMBIA , un

paıs queauncuandoenestosmomentosatraviesapor momentosdecrisisposeeunacara

quemuchosdenosotrosdesconocemos.La caradeun puebloquepor siemprehaluchado

por salir adelante,deun pueblotrabajadory orgullosodesusraices,quellora a sushijos

caidosy extrana a sushijos ausentes.Un puebloque lanzaa todo pulmon un grito de

libertad,perosobretodoungrito deesperanzapor la culminaciondeesteconflictoarmado

quenosacongoja.Queremosun paıs librepara todos!

No quierodejarde ladoa mi BARRANQUILLA del alma,a mi Currambala Bella.

La ciudadque me vio crecer. En suscallesaprendı a vivir la vida en cadauno de los

viii

extremosqueestanospresenta.Y esallı dondesecentrami vida y mi futuro profesional.

Resultaninmensaslasexpectativasdepoderdesarrollarmecomoprofesionaly ofrecermis

serviciosa la ciudaddemisamores.

Le agradezcoami madre,Fanny Mora,por hacerdemı lo quehoy endıasoy. Sinel

apoyo incondicionaldepartedeellanohubiesesidoposiblela consecusiondecadaunode

mis exitos. A ella masquea nadiededicoestetriunfo quehoy dıa estoy consiguiendo.A

mis hermanos,Madelainey JorgeLuis, por creerenmı y por cuidardemi madredurante

mi ausencia.A mi padre,Anthony Valle, por estarallı siempre,tantoenlasbuenascomo

enlasmalas.

PUERTO RICO esunatierraquellevareconmigoporsiempre.Unamaestrıa,muchos

amigos,unmatrimonioy unafamilia. Todoestomellevo detussuelos“Isla delEncanto”.

Agradezcoenormementela ayudaincondicionalrecibidade partede Diana, mi esposa.

Graciaspor compartira mi lado mis alegrıasy soportarademaslas tristezasy los malos

ratos.

Un agradecimientomasqueenormea la amistad,el apoyo profesionaly economico

queme brindo el Dr. Angel M. Lopez. Graciaspor todo lo quehizo por mı duranteel

periodoenel queestuve trabajandoa sulado,por creersiempreenmı y por brindarmesu

confianza.A losprofesoresWill E. Johnsy HectorMendezpor la ayudaquemebrindaron

sin nuncarecibir unanegativadesuparte.Al Dr. WeiJunXiong por sucolaboracion y su

apoyo a mi trabajo.Graciaspor la ayudabrindadapor partedel Dr. WolfgangRolke. Su

colaboracion en el analisis estadıstico paraestetrabajode tesisfue de granimportancia.

En generala todoslos companerosdel laboratoriodeHEPenMayaguez.Los del pasado:

ix

AlejandroMirles, CarlosRivera,Mauricio Suarez,SalvadorCarrillo, FabiolaVazquezy

KennieCruz; los del presente:Jose A. Quinones,EduardoLuiggi, Alexis Parıs, Luis A.

Nincoy CarlosR. Perez.Muchachos,gracias!!!

Al Departamentode Fısica del RUM y su directoren el periodo(Enerode 1998)

en que inicie estudiosgraduadosen UPR. A ellos les agradezcoel habermebrindadola

oportunidadparaque esto fueseuna realidad. Agradecimientosmuy especialesa toda

la gentede FOCUS Collaboration por darmela oportunidadde trabajaren estegran

experimento.A todaslaspersonasquetrabajanen EPSCoRen especiala SandraTroche

por suincondicionalayuday por suconstantecolaboracion.

A lasgrandesamistadesconstruidasdurantemi permanenciaenPR.A JuanCarlos

Hernandez,CarlosJose y en especiala suspadresJose “Machito” Hernandezy Juanita

Gonzalez.Ustedesfueronel soportefamiliarquetantaayudamehizoenestosultimostres

anostiempoqueduro mi maestrıa.

Graciasespecialesa JoeOrtız por apoyarmeincondicionalmenteen mis estudios

universitarios,graciaspor tu patrocinioenmi empresahaciaPR.Igualmentele agradezco

a Felix Rivera,a la FamiliaRomeroValle al igual quea la Familia Valle Miranda.Gracias

por su patrocinio. Agradezcoa Rafael Sanjuany familia por siempreestarcuandolos

necesitabamos.Gracias.A MarthaCastano, un mill on debendicionesparaustedmi bella

dama.

Agradezcoamisprofesoresdela UniversidaddelAtlantico- Barranquilla: Arnaldo

Coheny Oswaldo Dede,por inculcarmelos deseosde hacerciencia. A mis colegasy

x

amigosdeuniversidad,Arnulfo Barrios,OscarMontesinos,HarkccynnTorregroza,Ericka

Navarroy RafaelManga.Igualmenteagradezcoa todosmis profesoresdeBachilleratoen

el BienestarSocialdela Policıa - Barranquilla: BlasTorres,CarlosBrayy OrlandoOrtız,

especialmenteaArmandoConsuegra(Barrabas).

A todaslas personasque he mencionadoal igual que a las muchasque deje de

mencionarpuesnuncaterminarıaestedocumento...

Mil y mil gracias!!!

xi

Contents

List of Tables xviii

List of Figures xix

List of Abbreviations xxi

1 Intr oduction 1

1.1 TheStandardModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 FundamentalParticles . . . . . . . . . . . . . . . . . . . . . . . . 3

Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Long andShortDistanceEffects . . . . . . . . . . . . . . . . . . . 9

FeynmanDiagrams. . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 ConservationLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

xii

1.2.1 Conservationof AngularMomentumandHelicity Suppression. . . 11

1.2.2 Conservationof theelectricalcharge . . . . . . . . . . . . . . . . . 12

1.2.3 LeptonNumberConservation . . . . . . . . . . . . . . . . . . . . 13

1.2.4 BaryonNumberConservation . . . . . . . . . . . . . . . . . . . . 14

1.2.5 Flavor Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 CabibboTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

a)Cabibbo-Favored . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

b) Cabibbo-Suppressed. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

c) Doubly-Cabibbo-Suppressed. . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 GIM Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.5 RareandForbiddenDecays. . . . . . . . . . . . . . . . . . . . . . . . . . 24

a)LeptonFamily NumberViolation . . . . . . . . . . . . . . . . . . . . . 24

b) LeptonNumberViolating . . . . . . . . . . . . . . . . . . . . . . . . . 24

c) Flavour-ChangingNeutral-Currents. . . . . . . . . . . . . . . . . . . . 24

1.6 SpecialRelativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.7 ConfidenceIntervalsandConfidenceLevels . . . . . . . . . . . . . . . . . 29

2 Previous Works 30

2.1 FCNCDecayTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

xiii

2.2 PreviousSearchesfor RareCharmDecays. . . . . . . . . . . . . . . . . . 32

2.2.1 TheHadroproductionWA92 Experiment . . . . . . . . . . . . . . 33

2.2.2 TheHadroproductionE791Experiment . . . . . . . . . . . . . . . 34

2.2.3 ThePhotoproductionE687Experiment . . . . . . . . . . . . . . . 36

2.3 FOCUS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3.2 TheFOCUSPhotonBeam . . . . . . . . . . . . . . . . . . . . . . 39

2.3.3 InsidetheFOCUSExperiment. . . . . . . . . . . . . . . . . . . . 40

3 Objectives 44

4 Procedure 46

4.1 DataFormat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 ReconstructionProcess. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3 DataSelection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 SelectionVariables . . . . . . . . . . . . . . . . . . . . . . . . . . 50

a) Vertexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1. ACB<D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2. PrimaryVertex Multiplicity . . . . . . . . . . . . . . . . . . . 52

3. Vertex ConfidenceLevels . . . . . . . . . . . . . . . . . . . . 52

xiv

4. Vertex z Positions . . . . . . . . . . . . . . . . . . . . . . . . 52

5. Vertex Isolations . . . . . . . . . . . . . . . . . . . . . . . . 53

b) ParticleIdentification . . . . . . . . . . . . . . . . . . . . . . . 53

1. Muon Identification . . . . . . . . . . . . . . . . . . . . . . . 53

i. KaonConsistency for Muons . . . . . . . . . . . . . . . . . 54

ii. InnerMuons . . . . . . . . . . . . . . . . . . . . . . . . . 55

A) Numberof MissedMuonPlanes . . . . . . . . . . . . . 56

B) InnerMuonConfidenceLevel . . . . . . . . . . . . . . 56

iii. OuterMuons. . . . . . . . . . . . . . . . . . . . . . . . . 56

A) OuterMuonConfidenceLevel . . . . . . . . . . . . . . 56

2. HadronIdentification . . . . . . . . . . . . . . . . . . . . . . 56

i. Kaonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

ii. PionConsistency . . . . . . . . . . . . . . . . . . . . . . . 57

3. Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4. DimuonicInvariantMass . . . . . . . . . . . . . . . . . . . . 58

4.3.2 FOCUSDataSelectionStages. . . . . . . . . . . . . . . . . . . . 59

a) Pass1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

b) Skim1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

c) Skim2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xv

d) Skim3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 NormalizingMode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5 MonteCarloSimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

a) D �� Events . . . . . . . . . . . . . . . . . . . . . . . . 66

b) D K �FE Events. . . . . . . . . . . . . . . . . . . . . . . . 66

c) D E E � andD K K � Events . . . . . . . . . . . . . . 67

4.6 AnalysisMethodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6.1 Blind Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.6.2 RolkeandLopezLimits . . . . . . . . . . . . . . . . . . . . . . . 67

4.6.3 Determinationof Sensitivity andConfidenceLimits . . . . . . . . . 69

4.6.4 SidebandsBackgroundandSignalArea . . . . . . . . . . . . . . . 71

4.7 BackgroundStudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.7.2 MisID Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.7.3 BackgroundRatioStudy . . . . . . . . . . . . . . . . . . . . . . . 74

4.8 Optimizationof theSelectionCriteria . . . . . . . . . . . . . . . . . . . . 78

4.8.1 FixedCutsandVariableCut Ranges. . . . . . . . . . . . . . . . . 79

4.8.2 OptimizationBasedonSensitivity . . . . . . . . . . . . . . . . . . 80

xvi

5 Resultsand Conclusions 81

5.1 MisID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 BackgroundRatio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3 SkimmingResults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4 CutOptimizationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.5 ConfidenceLimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Bibliography 92

xvii

List of Tables

2.1 Chronologicalbranchingratiosfor the GIHKJL� weakneutral-currentdecayD 5�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1 Descriptionof Superstreams. . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Descriptionof Superstream1 . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Sensitivity numberasperRolkeandLopez . . . . . . . . . . . . . . . . . 69

4.4 An exampleof RolkeandLopez90%ConfidenceLimits . . . . . . . . . . 70

4.5 DimuonicInvariantMassCutsfor SidebandsandSignalRegions . . . . . . 79

4.6 Cutvaluesfor eachvariableconsideredin theoptimizationprocess. . . . . 80

5.1 BestCutCombination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Resultsof Cut Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 88

xviii

List of Figures

1.1 TheStandardModel for ElementaryParticles . . . . . . . . . . . . . . . . 5

1.2 Transformationspermmitedby theStandardModel . . . . . . . . . . . . . 8

1.3 Feynmandiagramaweakprocessmediatedby avirtual bosonexchange.. . 10

1.4 A chargedcurrentinteraction . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Feynmandiagramfor aweaktransition . . . . . . . . . . . . . . . . . . . 17

1.6 Representationfor theCKM matrixelements . . . . . . . . . . . . . . . . 20

1.7 Feynmandiagramfor aCabibbo-Favoredprocess . . . . . . . . . . . . . . 20

1.8 Feynmandiagramfor aCabibbo-Suppressedprocess . . . . . . . . . . . . 21

1.9 Feynmandiagramfor aDoubly-Cabibbo-Suppressedprocess. . . . . . . . 21

1.10 Weakinteractionsin neutral-currentprocesses. . . . . . . . . . . . . . . . 22

2.1 Feynmandiagramsfor thedecayD �� . . . . . . . . . . . . . . . . 31

2.2 ProtonsynchrotronandbeamlinesatFermilab . . . . . . . . . . . . . . . . 38

2.3 TheE831PhotonBeam. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

xix

2.4 TheFOCUSSpectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1 TrajectoriesthroughtheFOCUSspectrometer. . . . . . . . . . . . . . . . 49

4.2 Productionanddecayverticesfor theD �� process . . . . . . . . . 51

4.3 Analysisoverview of FOCUS . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4 MonteCarlogeneratedprocessschematizationfor thedecayD �� . 66

4.5 D �� InvariantMassdistributions . . . . . . . . . . . . . . . . . . 71

4.6 DimuonicInvariantMasscomparisonof MonteCarlodatafromD E E �andD �� .�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.7 Dimuonicinvariantmassof D K ��E generatedby MonteCarlo . . . . 76

4.8 BackgroundcomparisonbetweenrealandMonteCarlodata . . . . . . . . 77

5.1 MisID asa functionof momentum . . . . . . . . . . . . . . . . . . . . . . 82

5.2 Typical InvariantMassDistribution Usedin Calculatingthe BackgroundRatio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 InvariantMassDistributionsfor Skim3Cuts . . . . . . . . . . . . . . . . . 84

5.4 InvariantMassDistributionsfor Skim4Cuts . . . . . . . . . . . . . . . . . 85

5.5 Massdistributionsfor thebestcut combination. . . . . . . . . . . . . . . . 87

5.6 Invariant Mass Distribution for BackgroundRatio using the best cutcombination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.7 Eventcandidatesfor theD �� process . . . . . . . . . . . . . . . . 90

xx

List of Abbreviations

BeO: Beryllium Oxide.

BR: BranchingRatio.

CERN: OrganisationEuropeennepourla RechercheNucleaire.

CITADL: Cerenkov IdentificationThroughA Digital Likelihood.

CKM Matrix: Cabibbo-Kobayashi-MaskawaQuark-MixingMatrix.

CP Violation: Charge-Parity Violation.

CL: ConfidenceLevel.

CLP: PrimaryVertex CL.

CLS: SecondaryVertex CL.

E687: FermilabExperiment687.



Fermilab: FermiNationalAcceleratorLaboratory.

FNAL-E831: FermilabExperiment831.

FCNC: Flavor-ChangingNeutral-Current.

FOCUS: FotoproductionOf Charmwith anUpgradedSpectrometer.

GIM Mechanism: Glashow-Iliopoulos-MaianiMechanism.

GeV: GigaElectronVolt.

IMU: InnerMuonSystem.

IMUCL: InnerMuonCL.

ISOP: PrimaryVertex Isolation.

ISOS: SecondaryVertex Isolation.

K/ M : KaonConsistency to separatekaonsfrom muons.

LFNV: LeptonFamily NumberViolation.

LNV: LeptonNumberViolation.

L/ N : Separationbetweentheproductionanddecayverticesdividedby

theerror, D , on thatmeasurement.

MeV: MegaElectronVolt.

MisID: Percentof eventsmisidentifiedwithin any specificanalysis.

MISSPL: Numberof MissedMuonPlanes.

M1: MagnetNumber1.

xxi

M2: MagnetNumber2.OQPRP: LorentzDimuonicInvariantMass.

OMU: OuterMuonSystem.

OMUCL: OuterMuonCL.

PAW: PhysicsAnalysisWorkstation.

Pass1: Basiceventreconstructionstage.

PWC: ProportionalWire Chambers.

Skim1: Firstdatareconstructionselectionstage.

Skim2: Seconddatareconstructionselectionstage.

Skim3: Third datareconstructionselectionstage.

Skim4: Fourthdatareconstructionselectionstage.

SSD: SiliconMicrovertex Detectors.

ZVPRIM: PrimaryVertex z Position.

ZVSEC: SecondaryVertex zposition.SW( T*U ): Kaonicity.T�VXWZY : PionConsisitency.NC[7\ : Dif ferencebetweenthePrimaryVertex z positionandthez target

edgedividedby theuncertainty.NC[^] : Dif ferencebetweentheSecondaryVertex z position and the z

targetedgedividedby theuncertainty.

xxii

Chapter 1

Intr oduction

This thesisis a researchwork in a topicof highenergy physics.It is alsoa formative

work and could be usedas a sourceof referencefor studentsand peopleinterestedin

knowing moreaboutthephysicsrelatedwith themicroscopicworld aroundus.

The StandardModel of electroweak interactionsis usedto understandthe decays

of heavy quarkswhich are known to us. A significant observation of charm Flavor-

ChangingNeutral-Currentdecays,predictedto occur rarely (in the StandardModel the

neutral-currentinteractionsdo not changeflavor), would imply new physicsbeyond the

StandardModel.

Rare decaymodesare probesof particle statesor massscaleswhich cannotbe

accesseddirectly. One of the most interestingprocessesto study is the D �� .��decay(all referencesto this decayalso implies the correspondingcharge-conjugatestate_ ��F�� ). Themaingoalof thework reportedin this thesiswasthemeasurementof

1

`1.1 TheStandardModel 2

the branchingratio for this decayanda comparisonto the actual90% ConfidenceLevel

upperlimit a � �0�b� � �"! .This chapterwill display somebasicand interestingconceptsusedfor this study.

First, anintroductionto theStandardModel of elementaryparticlesis done,next someof

the conservation laws arediscussed.The CabibboTheoryandthe GIM Mechanismwill

alsobediscussedin this chapter. BasicinformationaboutRareandForbiddendecayswill

be shown defining threeimportantrelatedprocesses:LeptonFamily NumberViolation,

LeptonNumberViolation andFlavor-ChangingNeutral-Currents.A shortrepresentation

of SpecialRelativity will bediscussedandfinally someusefulconceptsaboutConfidence

Intervalswill bedefined.

1.1 The Standard Model

At thebeginningof the�2�<ced

century, physiciststhoughtthatall matterwascomposed

of atomsasthe fundamentalbuilding block. As the centuryenteredadolescence,it was

discoveredthatatomsareconstructedfrom electronsandasmallpositivechargednucleus.

This nucleuswassubsequentlyfound to be composedof protonswhich have a positive

charge, and neutrons(except the Hydrogennucleus),which are uncharged. However,

duringthelatterhalf of thecentury, physicistsdeterminedthatneitherprotonsnorneutrons

arefundamental.A betterdescriptionaboutthe matterin the universewasobtainedand

now weknow thatall matteris madefrom oneor moreof sixteenelementaryparticles.

To explain the natureof theseelementaryparticlesandthe forcesbetweenthem,a


theoryhasbeendevelopedcalledtheStandardModel[1–3]. Themodeldescribesall matter

andforcesin theuniverse(exceptgravity) andit is a packageof two theories:theUnified

Electroweak[4, p.95]andQuantumChromodynamics[4, p.85].

1.1.1 FundamentalParticles

TheStandardModel dividestheelementaryparticlesinto two groupscalled fg62h369+�hand /�&i1<j kl6=+�h whicharedifferentiatedaccordingto thebehavior of multiparticlesystems.

Bosonsobey theBose-Einsteinstatisticsandfermionsobey theFermi-Diracstatistics.The

Bose-Einsteinstatisticsmeansthat any numberof identicalbosonscanexist in the same

quantumstate.This is very essentialin shapingthestatisticalbehavior of a largenumber

of bosons.On theotherhand,in theFermi-Diracstatisticstwo fermionscannotcoexist in

thesamequantumstate.Whicheverstatisticsa particlefollows will determinethe m wave

function’s symmetryunderinterchangeof particles. If thesetwo particleshave identicaln m n o (wave-functionsquared),theprobabilityof particle1 beingin onespotandparticle2

at another, mustbeequalto theprobability that thefirst particleis in thesecondspotand

particle2 is in thefirst. Therefore,underinterchangethewave-function mIp � qr�s JKt0m#p qr�swhere

qr is therelativeposition.

For identical bosonsthe Bose-Einsteinsymmetryprinciple statesthat the wave-

functionunderinterchangemustgo to plusitself

mIp � qr�s JumIp qr�s (1.1)

and, for identical fermions the Pauli principle states that the wave-function under


interchangemustgo to minusitself

mIp � qr�s J � mIp qr�s (1.2)

For nonrelativistic speeds,thetotal wave-functionof thepair canbeexpressedasa

productof functionsdependingonspatialcoordinatesandspinorientation:

n mIp qr�swv Jux�pyhwzF),4Z& s:{ pyhwz�kl+ s (1.3)

wherethespatialpart, x , will describetheorbitalmotionof oneparticleabouttheother. It

canhappenthat x is symmetricor antisymmetricunderinterchangeandthespinfunction {maybesymmetric(spinsparallel)or antisymmetric(spinsantiparallel)underinterchange.

Bosons . They are:photon, gluon, W | andZ . Theseparticleshaveanintegralspin

(0, 1, 2,...) andareknown asthe intermediaryparticlesbecausethey areresponsiblefor

carryingtheforceinteractions.Thephotonis masslesswith spin1 andis theintermediary

of the electromagneticinteraction. Thereis onekind of photonandtwo photonsdo not

directly interactwith oneanother. Thegluonis masslesswith spin1 andis theintermediary

of thestronginteraction.Thereareeight typesof gluons,eachdifferentfrom oneanother

only by a quantumnumbercalledcolor. The gluonsinteractwith oneanother. The W |(m } 80.4 ~��Z� ) andZ (m } 91.2 ~��Z� ), all of spin1, aretheintermediariesof theweak

interaction.TheW | andZ donot interactwith oneanother.

Fermions . They have a half-integral spin ( �o , �o ,...) andconstitutethe fundamental

familiesof matter. Thesearedivided into quarkswith fractionalelectriccharges(+o� n & n

and� �� n & n ) andleptonswith integercharge(

� � n & n ) or no charge(neutrinos). At thesame


time, quarksare subdivided into six different particles: up (m } 1 to 5 ��7� ), charm

(m } 1.15to 1.35 ~��7� ) andtop (m } 169to 179 ~��Z� ) with electricchargesof� o� n & n and

down(m } 3 to 9 ��7� ), strange (m } 75 to 170 ��Z� ) andbottom(m } 4.0 to 4.4 ~��7� )

with� �� n & n . Thesix typesof quarksaresometimescalledflavors. Leptonsaresubdivided

into threedifferentelectricallycharged particles,the electron, & (m } 0.511 ��7� ), the

muon, � (m } 105.7 ��Z� ), andthe tau, � (m } 1.777 ~��7� ), with their respectiveneutral

neutrinos,�=� (m � 3 eV), �9� (m � 0.19 ��7� ) and �9� (m � 18.2 ��Z� ). Eachparticle,

quarkor lepton,hasits respectiveantiparticlewith thesamemassbut with oppositeelectric

charge. Figure1.1shows thefundamentalparticlesof matterarrangedinto familiesby the

StandardModel.

Figure1.1: TheStandardModel for ElementaryParticles.


Eachof the threecolumnscorrespondsto a “f amily”. The relationshipsamongthe

particlesin afamily arethesameasthosein theothertwo families.In awaywehavethree

copiesof thesamestructure.In eachof thefamiliesthereis a leptonanda quarkdoublet.

The charge of the membersof the doubletsdiffer by one & unit. The main difference

betweenfamiliesarethemassesof their members.Thefirst family membersarelow mass

while themassesof thethird family aremuchhigher. This largedifferencein massis one

of theprincipalopenmysteriesin elementaryparticlephysics.

1.1.2 Interactions

Of theknown four forces,two, gravityandelectromagnetism, have a long rangeand

are commonin every day life. The remainingtwo forces,the strong and weaknuclear

forces,have sucha small rangethat it is frequentlycomparedto the size of an atomic

nucleus. Due to its low strength,gravity is still not a completepart of the Standard

Model. Another interactionthat hasan infinite rangeis the electromagneticinteraction

whoseeffects have beenmeasuredthroughoutimmensescales. The strong interaction

holdstogetherthenucleusworking againsttheelectrostaticrepulsionthatoccursbetween

protons. Lastly, the weak interactionis known to be responsiblefor a large variety of

radioactivedecays.Also, it is responsiblefor theinteractionsthatoccurbetweenneutrinos

andmatter.

Besideselectriccharge,quarksandgluonshave anothertypeof charge. It is called

the color charge. The effectsof the strongnuclearforce or color force areonly felt by

quarks. It is a very strongforce that bindsquarkstogetherforming particles. Thereare


three“colors” for quarkswhich areusuallycalled“red”, “blue” and“yellow”. If you put

togethera redquarkwith anantiredantiquark,theproductis colorless.If you put together

a red,ablueandayellow quark,theproductis alsocolorless.

Due to thenatureof thecolor interaction,combinationswith a netcolor chargeare

highly unstable.If onestartswith a systemwith no netcolor charge,it will never separate

into color chargedpieces. The reasonfor this is believed to be that, dueto the fact that

gluonshave a color charge,thereis an“anti-screening”effect wherebytheeffective force

betweentwo quarkskl+�4g1<&3)�h3&9h with distance.This increasedpotentialenergy is converted

to quark-antiquarkpairsin sucha way that the separatedpiecesareeachcolorless.This

is believed to be the reasonno systemwith a net color charge has ever beendirectly

observed, including both quarksor gluons. However, throughthe useof deepinelastic

scatteringexperiments,the interactionsof gluonsor quarkswith incomingprojectileshas

beeninferred.

Only groupsof two quarks(mesonsformedof a quarkandan antiquark)or three

quarks(baryonsformedof threequarksor threeanti-quarks),havebeenseen.Collectively,

thesegroups are called hadrons. On the other hand, leptons can exist without the

companionshipof otherparticlessincethey have no color charge. Thenet forcebetween

color neutralsystems(thestrongforce)actually �,&34g12&3)"h3&9h with distance.

Theweakinteractionis felt by bothquarksandleptons.Thereis a deeprelationship

betweenthe weakinteractionandthe family structureof quarksandleptons. In the case

of theleptons,it is only possibleto changeleptonswithin thesamefamily into eachother.

Thus,theelectroncanbeturnedinto �9� but not into a muon.Theweakinteractioncanact

within the leptonfamilies,but not betweenthem. Therearecomplicationswhenit comes


to the quarks.Herealsotheweakinteractioncanturn oneflavor into another. However,

it is not trueto saythat theweakforcecannotactacrossfamilies. It can,but with a much

reducedeffect.

Quarks�u

�d

��

��

��

�c

�s

��

��

��

�t

�b

��

Leptons�e

��

�g�

��

�7�

��

(a) (b)

Figure1.2: Transformationspermmitedby theStandardModel. (a) Theeffect of theweak force on the quarks. Transformationssuchas b � u are also possible. (b)The effect of the weak force on the leptons. Transformationswithin families arerepresentedby vertical arrows andacrossfamiliesarerepresentedby horizontalanddiagonalarrows.

For quarks,theelectriccharge changingtransformationswithin familiesandacross

families are dominant(the vertical transformationsare more probableand the diagonal

transformationsare less probable)as seen in Figure 1.2. This does not mean that

transformationsdueto neutralcurrentscannotoccur. Quark transitionssuchasc u,

in which thecharge is thesame,couldoccurthrougha neutralcurrent . Whathappens

is that transitionssuchasthis onehave a smallprobabilityof occurrencecomparedto the

electricchargechangingtransitions.In Section1.3,this will beexplainedin moredetail.

As mentionedbefore,theweakinteractionsof quarksaswell asleptonsaremediated

by theW , W � andZ . In theStandardModel leptonscannothavecrosstransformations

directly. Nevertheless,theStandardModelpermitsprocessessuchas &3 �&<�� where

the & & � first convertsto a Z which thenturnsinto � � � . It is importantto notethatall


particlesthathaveanelectricchargeincludingtheW feel theelectromagneticforcebut the

gravity forcedoesnothaveanobservableeffect in particlephysics.

Shor t and Long Distance Effects . Quantummechanicallyinteractionsaredue

to the exchangeof virtual bosons. The fundamentalcolor interactionbetweenquarks

is mediatedby virtual gluons. Effects due to such fundamentalprocessesinvolving a

finite numberof quark-gluonverticesare called short distanceeffects. The effective

strengthof this fundamentalinteractionis relatively small and Feynmandiagrams(See

next section.)andperturbationtheorycanbeusedto calculatethem.However, thegluons

(unlike thephotonandtheintermediariesof theweakinteraction)carrycolor charge. The

consequencesof this color charge are profound. Gluon-gluoninteractionsare possible.

Theseleadsto an ”antiscreening”effect where the effective color interactionbetween

quarksactuallyincreaseswith distanceinsteadof decreasingasin theelectromagneticand

weak interactions. The result is that stronginteractionsbetweenhadronsdecreasewith

energy sinceat higherenergy thecomponentquarksinteractat shorterdistances.At lower

energies and longer distancesthe effective interactionis larger and it is not possibleto

calculatesuch”long distanceeffects”usingperturbationtheory.

In hadrondecaythe fundamentaldecaymechanismmaywell bea weakinteraction

betweenthe constituentquarksbut as the decayproductsseparatelong distanceeffects

due to the color interactionwill play an important role. Somecharacteristicsof the

decaywill be determinedby its fundamentalweak naturebut other characteristicswill

be strongly affected by the long distanceeffects. For example, parity violation in a

decayis an indication of its fundamentalweak nature. However, the precisevaluesof

lifetimes andbranchingratioscannot be calculatedwith pureperturbationtheory. Our

`1.2 ConservationLaws 10

currentbestunderstandingof suchdecaysusesour knowledgeof the weak interaction

in perturbationtheory calculationstogetherwith phenomenologicalmodelsfor the long

distanceeffectsof thecolor interaction.In the future it is hopedthat latticegaugetheory

numericalcalculationsof thecolor interactionwill achieve sufficient accuracy asto make

phenomenologicalmodelsunnecessary. Thesedo not useperturbationtheoryto calculate

color interactioneffects.

Feynman Diagrams . A Feynmandiagramis a representationof thewayparticles

interactwith oneanotherby theexchangeof bosons.They areextremelyusefulin making

perturbationtheorycalculationsbecauseeachrepresentationis associatedwith formalrules

for assigningvertex couplings,propagatorterms,etc.

%e

e

e

e

Figure 1.3: Feynman diagram for a weak processmediatedby a virtual bosonexchange.

1.2 Conservation Laws

Amongthemostsignificantaidsto understandingwhattakesplacein theuniverseare

theconservationlaws,giventhatthey identify thosetypesof interactionsthatcannotoccur.


It is a trustworthy rule of thumbin physicsthatwhatever thing is not definitelyprohibited

will occur.

In classicalmechanics,the invariantsare the conservation of the electric charge,

the conservation of linear momentum,the conservation of energy and mass,and the

conservationof the angularmomentum.In thequantumworld, thereareotherinvariants

aswell that arepreserved in sometypesof interactionsbut not others. For example,the

isotopicspin is preserved in thestronginteractionsbut not in interactionsconcerningthe

electroweakforce.

1.2.1 Conservation of Angular Momentum and Helicity Suppression

Thehelicityof aparticleis definedasits spinprojectionalongits directionof motion.

A particlewith spinvector N andmomentump hashelicity

¡ J N �£¢n N n¤n ¢ n (1.4)

For spin �o thereareonly two possiblehelicity states,tI� . If thehelicity of a particle

is positive, it is called right-handed,and left-handedif the helicity is negative. The

correspondingstatesaredenotedas ¥§¦ and ¥©¨ respectively.

Due to the natureof the weak interaction, lepton statesfrom weak decaysare

predominantlyleft-handed. This helicity predominanceis strongerthe lower the mass

of the lepton. In the caseof the neutrino, it is total. Only left-handed � have been

observed. For otherleptons,the termhelicity suppressionis used.This meansthat right-


handedleptonsandleft-handedantileptonsareproducedwith muchlowerprobability. The

D hasspin zeroso the two muonsin D �� musthave oppositespin to conserve

angularmomentum.But they alsohaveoppositemomentumsothey havethe h3),j�& helicity.

Howeveroneis anantilepton.Thus,theweakdecayD �� is helicity suppressed.

1.2.2 Conservation of the electrical charge

Whichever reaction,theentirechargeof all theparticlesbeforethereactionhasgot

to beidenticalto thewholechargeof all theparticlesfollowing thereaction.For instance,

in a betadecay, a neutrondecaysto form a protonandan electronplus an antineutrino.

This beginswith a zerochargeandfinishesonepartnegativeandonepartpositivecharge.

Figure1.4shows thecontributionof chargefor eachfermioninvolvedin thereaction.

+ z � & � �� (1.5)

with charges: 0 +1n & n -1

n & n 0

W n

p

��

e

Figure1.4: A chargedcurrentinteraction.Theneutronhasaquarkcontentof udd andits charge is ª5«9¬i,® ¯<®<°²±7¬i,® ¯2®<°²±7¬i,® ¯<®�³$´"® ¯<® . Theprotonquarkcontentis uud andits charge is ªµ«9¬i,® ¯<®¶ª·«9¬i,® ¯<®<°¸±7¬i,® ¯<®,³¹±=® ¯<® . Theelectronhas °µ±=® ¯<® aschargeandthe� � has ´"® ¯<® ascharge.


Conservationof electricalchargepreventsthecolor forcefrom materializingquarks

suchasu andd from energy (thetotal charge is not zero). However, a uc materialization

would conserve charge. Yet, suchmaterializationis not seenin color forcereactions.The

propertiesof thecolor forcedictatethat thequarkandantiquarksinvolvedmustbeof the

sameflavor. It is not possibleto materializea u anda c by the color force but you can

materializeau andau.

Flavor-changingprocessescannot occurvia the color force or the electromagnetic

force, only via the weak force. Sincethe color andelectromagneticforcessatisfymore

conservation laws thanthe weakforce,many particlescanonly decayweakly. This fact

allowsusto studytheweakforceby studyingthosedecays.

1.2.3 Lepton Number Conservation

A leptonnumberis a tagusedto indicatewhichparticlesareleptonsandwhichones

arenot. Every leptonhasa leptonnumberof � , which is split into threedifferent“values”:

Aº� , A§� and A§� . On the otherhand,every antileptonhasa leptonnumberof� � . Other

particleswill havea leptonnumberof�. A»� , A§� and A§� arecalledleptonfamily numbers.

Both leptonnumberandleptonfamily numberarealwaysconservedaccordingto the

StandardModel. To understandthedifferencebetweenthem,considertheprocess

D � � & � (1.6)

wheretheleptonnumbersandleptonfamily numbersare:


A : 0 �1 1Aº� : 0 0 1A§� : 0 �1 0

For this reactionwe canseethat the leptonnumberis conservedbut not the lepton

family number. It is due to the fact that for this processin the initial statewe do not

have any leptonbut in thefinal statewe have two leptonsof differentfamilies.Thedecay

D �� | &3¼ is aLeptonFamily NumberViolationprocesswhich is notpermittedto occur

within theStandardModel.

A processpermittedby theStandardModel is

�9� � & � � �9� (1.7)

wheretheleptonnumbersare

A : 1 1 1 1A»� : 0 1 0 1Aº� : 1 0 1 0

For the processabove the lepton family numberaswell as the leptonnumberare

conserved.

1.2.4 Baryon Number Conservation

Analogousto leptons,thebaryonnumberis a tagusedto indicatewhichparticlesare

baryonsandwhich onesarenot. All baryonshave a baryonnumberof � . Eachantibaryon


hasa baryonnumberof� � . Every quarkhasa baryonnumberof �9B2½ . Antiquarkshave

a baryonnumberof� �9B2½ and,therefore,eachmesonhasa baryonnumberof

�. Every

leptonhasa baryonnumberof�. In all reactions,thetotal baryonnumberof theparticles

prior to the reactionis the sameasthe total baryonnumberfollowing the reaction. For

example,

+ z � & � �� (1.8)

wherethebaryonnumbersare

B: � � � �Lepton and baryonnumberconservation are ad hoc assumptionsin the Standard

Model in contrastto otherconservationlawssuchaschargeconservationwhicharerelated

to the natureof the interactions. This brings up the possibility that lepton and baryon

numberconservationmaybeviolatedin some(asyetunobserved)processes.

1.2.5 Flavor Conservation

In particle physicsanotherfrequentlyusedterm is flavor. Quarkshave different

flavorssuchas:up,down, strange,etc. Therearesix flavorsof quarks.Theflavor number

for eachquark in the order of its massis the following: ¾ J � , _ J � � , H J � ,¿ J � � , ÀÁJÂ� and ÃÄJ � � .Thestrongandelectromagneticinteractionsconservethequarkflavor numberswhile

the weak interactiondoesnot. Flavor changeoccursmostly throughthe charged weak

`1.3 CabibboTheory 16

interactionssincetheflavor-changingneutral-currentweakinteractionis suppressedby the

GIM mechanism.This is explainedin detail in Section1.4.

1.3 Cabibbo Theory

Transitions between quarks of different flavors occur only due to the weak

interaction. For the electromagneticandcolor interactionsall familiesbehave the same.

The only differencesbetweenthem are their masses. When we talk about quarksof

different flavors, we are referring to masseigenstates.However, thesestatesare +�63-eigenstatesof thetotal Hamiltonian(which includestheweakinteraction)because,if they

were,flavor wouldbeconservedin weakdecays.

The weakdecayprocesscanbe describedasfollows. At -ÅJ � , a quarkis created

in a mass(flavor) eigenstatevia an electromagneticor color interaction. This statecan

bewritten asa linearsuperpositionof weakeigenstateseachof which hasa well-defined

anddifferentlifetime. It is saidthat thequarkis in a “mixed” state.As time evolves,the

weakeigenstatemix changesbecauseof thedifferentlifetime evolutions.At a latertime - ,thestatehasacquiredcomponentsof othermasseigenstates(otherflavors)thusexplaining

flavor transitions.

Only quarkswith the samecharge canmix. Due to the explicit form of the weak

interaction, it is only necessaryto considermixing of the� �9B2½ quarks(this induces

transitionsinvolving the� � B2½ quarksalso). Themixing is treatedmathematicallyvia the

unitaryCabibbo-Kobayashi-Maskawa(CKM) [5,6] matrixwhoseelementsare

`1.3 CabibboTheory 17ÆÇÇÇÇ

Èd ÉsÉb É

Ê¶ËËËËÌ J

ÆÇÇÇÇÈÍ�Î7ÏÐÍ�ÎZÑÒÍÓÎ7ÔÍFÕ@Ï ÍFÕ�Ñ Í�Õ@ÔÍ c Ï Í c Ñ Í c Ô

Ê¶ËËËËÌÆÇÇÇÇÈ

d

s

b

Ê¶ËËËËÌ (1.9)

In thelenguageof Feynmandiagramsthechargedweaktransitionoccursat a vertex

suchasthatshown in Figure1.5whereq is a� � B2½ quarkandq É is a

� �9B2½ quark.

%' q q É

Figure1.5: Feynmandiagramfor aweaktransition.

Mixing is treated mathematicallyby taking the coupling at that vertex to be

proportionalto Ö Í�×Ø×lÙ where Ö is the universalweak-couplingstrengthwhich is the same

for all threefamilies. Notice the reasonfor the namesof the CKM matrix elementsÍ�×Ø× Ù

sincethey areassociatedwith thecouplingbetween� � B<½ and

� �9B2½ quarks.For example,

amplitudesfor theprocessesb c andb u areproportionaltoÍ�Õ@Ô

andÍ�Î7Ô

, respectively.

TheCKM matrix is in generalcomplex but, dueto theunitarity condition,it canbe

shown thatthemostgeneralform containsonly four realparameterswhich canbechosen

to be threeangles( Ú � o9Û Ú o � Û Ú � � ) andonephaseÜ � � [7–10]. The anglesÚ¶ÝßÞ arerelatedto

the mixing betweenthe families k and à . Defining H�ÝßÞáJâ476<hÅÚ¶ÝßÞ and¿ ÝßÞ Jãh3ky+²Ú¶ÝßÞ this

parametrizationgives:


Í JÆÇÇÇÇÈ

H � o H � � ¿ � o H � � ¿ � � & � Ý�äØåçæ� ¿ � o H o � � H � o ¿ o � ¿ � � & Ý�äØåçæ H � o H o � � ¿ � o ¿ o � ¿ � � & Ý�äØåçæ ¿ o � H � �¿ � o ¿ o � � H � o H o � ¿ � � & Ý�äØåçæ � H � o ¿ o � � ¿ � o H o � ¿ � � & Ý�äØåçæ H o � H � �


Thosematrix elementswith a simple form can be directly measuredin a decay

processandare found in the first row and third column. H � � is known to deviate from

unity only in the sixth decimalplace. Therefore,Í�Î7Ï JâH � o3Û Í�ÎZÑ J ¿ � o3Û Í�Õ@Ô J ¿ o � andÍ c Ô JèH o � to anexcellentapproximation.The phaseÜ � � lies in the range

�·é Ü � � � � E ,

with non-zerovaluesbreakingCPinvariancefor theweakinteractions.

An alternative approximateparametrizationis given by Wolfenstein[11] usingthe

fact that¿ � o*ê ¿ o � ê ¿ � � areall small. Thesizeof theCabibboangle1 is setas ë�ì ¿ � o

andthentheotherelementsareexpressedin termsof leadingpowersof ë up to the third

order:

Í JÆÇÇÇÇÈ

� � ë o B � ë í�ë � p�î � klï s� ë � � ë o B � íðë oíðë � p:� � î � kyï s � í�ë o �


having í Û î and ï asrealnumbersthatareof orderunity.

Transitions between quarks for the two family case are described by the

transformation1 ñóò�ôöõ¤÷5ø7ù�ú�ò�ôöõû÷5ø^üIý þ9ÿ�� whereø^ü is theCabibboangle( ø^ü ý�þ=ÿ �� rad).

`1.3 CabibboTheory 19ÆÇ

È d ÉsÉÊ¶ËÌ J

ÆÇÈ 4762h5Ú�� hikl+ Ú�� hikl+ Ú�� 476<h�Ú��

Ê¶ËÌÆÇÈ d

s

Ê¶ËÌ (1.12)

In this approximation,the transitionc s andu d areproportionalto 4Z6<h o Ú��while the transitionsc d and s u are proportionalto h3ky+ o Ú�� . This is a good

approximationsince¿ � o Juh3ky+áÚ�� ê ¿ o � ê ¿ � � .

A relation betweenthe two smallestelementsof the CKM matrixÍÓÎ7Ô

andÍ c Ï is

obtainedby applyingtheortogonalityconditionto thefirst andthird columns:

ÍÓÎ7Ï¶Í��Î7Ô � Í�Õ�Ï7Í��Õ@Ô � Í c Ï¶Í��c Ô J � (1.13)

whereeachtermin thesumis of theorder ë � . In theparametrizationgivenabove,ÍFÕ@Ô Û ÍFÕ@Ï

andÍ c Ô arereal,andby using

ÍÓÎZÏ } Í c Ô }Ä� andÍ�Õ@Ï � � weobtain

Í �Î7Ôn Í�Õ@Ï7ÍFÕ@Ô n � Í c Ïn Í�Õ@Ï7ÍFÕ@Ô n JK� (1.14)

This equationis representedgeometricallyby an “unitarity” triangle [7] in the

complex plane. The lengthsof the two uppersidesareproportionalto themagnitudesof

theleastwell known elementsof theCKM matrix,ÍÓÎZÔ

andÍ c Ï asshown in theFigure1.6

TheStandardModelof CPviolationcanbetestedby measuringthesidesandangles

of theunitarity triangleto testwhetherthey really form a triangle.

a) Cabibbo-F avored . Thediagonalelementsof theCKM matrix aremuchlarger


A

C BVs12 cb

ubV Vtd*

*

γ β

α

Figure1.6: Representationfor the CKM Matrix elements.An unitary trianglein thecomplex planeis formedby theCKM matrixelements� �Î7Ô , � c Ï , and � o � �Õ@Ô .

thantheoff-diagonalelements.Thustransformationswithin familiesaremuchmorelikely.

Transitionsproportionalto 4762h o Ú�� areknown asCabibbo-Favored.

D� W

u

c

u

s

d

u �

K�

Figure 1.7: Feynman diagramfor a Cabibbo-Favored process. A charm quark istransformedinto a strangequark. The other transformationof a W into u and dquarksin thesecondvertex is alsoCabibbo-Favored.

If weconsidertheprocessin whichaD is decayinginto K � E shown in Figure1.7,

a charmquarkof the D particle is transformedinto a strangequark. This is Cabibbo-

Favored.In thesecondvertex theW is decayingintoaud. Thisis alsoaCabibbo-Favored

processbecauseu andd arequarksof thesamefamily.


b) Cabibbo-Suppressed . Transitionsproportional to hikl+ o Ú�� are known as

Cabibbo-Suppressed.Thesearetransformationsbetweenthefirst andsecondfamilies.

D� W

u

c

u

d

d

u �

�

Figure 1.8: Feynmandiagramfor a Cabibbo-Suppressedprocess. In this processacharmquarkis changedto a down quark. TheW decayvertex is Cabibbo-Favoredjustasin D � K �� (Figure1.7).

In Figure1.8acharmquarkturnsinto adown quarkin a D processdecayinginto a

E E � . Thesecond =8.)R1�� -W vertex is aCabibbo-Favoredprocess.

c) Doub ly-Cabibbo-Suppressed . Processesin which two =8.),1�� -W vertices

areof theCabibbo-Suppresedtypeareknown asDoubly-Cabibbo-Suppressed.Thedecay

modeD K �E � is shown asanexampleof theseprocesses(Figure1.9).

D� W

u

c

u

d

s

uK�

�

Figure 1.9: Feynmandiagramfor a Doubly-Cabibbo-Suppressed process. For thisprocessboth vertices, the W- �� and the �� -W, are Cabibbo-Suppressedprocesses.

`1.4 GIM Mechanism 22

1.4 GIM Mechanism

The processD �� is a flavor changingneutral current processwhich is

suppressedin the StandardModel due to the GIM mechanism,proposedin a famous

paper[12] by Glashow-Iliopoulus-Maianiin which they explain thesuppressionof flavor-

changingneutral-currents.They suggestedthat,insteadof atriplet of quarks(u, d, s), there

weretwo doubletsof quarks(u, d) and(c, s). Thissuggestionrequiredanew quarkrespect

to theCabibbomodel,thecharmquark.Thispredictionwasmadebeforetheexperimental

discoveryof this quark.

Z

Z 0

0 +

+

θ(d cos + s sinθc c )

(d cos + s sinθc )θc

_

_u

u

c

c

θ )c

θc )

θ

θc

c

(s cos − d sin

(s cos − d sin

_

__

_

(a)

(b)

Figure1.10: Weakinteractionsin neutral-currentprocesses.(a)Feynmandiagramsfora neutralcurrentin thethree-quarkmodel. (b) Includingthecharmquark,extra termsareadded.

If wehavea ascarrierof aweakinteractionasshown in Figure1.10wehave that

`1.4 GIM Mechanism 23

theeigenstatesof theHamiltonianfor theweakinteractionare:

ÆÇÈ u

d ÉÊ¶ËÌ Û

ÆÇÈ c

sÉÊ¶ËÌ Û

ÆÇÈ t

b ÉÊ¶ËÌ (1.15)

In thethree-quarkmodel,mixing thed ands quarksby theCabibboangle,oneobtain

d É�J d 4Z6<h5Ú�� s hiky+áÚ�� sÉ J s 4Z6<h�Ú�� d hiky+áÚ�� (1.16)

whered É andsÉ areanorthogonaltransformof d ands. Thentheneutral-currentcoupling

will beof theform

¥ Í ¥ � J uu� p dd 4Z6<h o Ú�� ss hiky+ o Ú�� s� �� !"$#�%

� p sd�

sd s h3ky+ Ú��4762h5Ú�� !"$#�% �(1.17)

Experimentally, very small rates for flavor-changing( G ¿ J � ) neutral-current

processesareobserved but eq.(1.17)saysthat G ¿ J � shouldbe significant. This was

theproblemwith thethree-quarkmodelwhich wassolvedby thecharmquarkprediction.

Includinga doubletcorrespondingto thecharmquarkwe obtainfor theweak-interaction

neutral-currentmatrixelement

¥ Í ¥ � J uu�

cc� p dd

�ss s 4Z6<h o Ú�� p ss

�dd s hikl+ o Ú�� !"$#�%

� p sd�

sd�

sd�

sd s hikl+áÚ��4Z6<h5Ú�� !"$#�% �(1.18)

`1.5 RareandForbiddenDecays 24

wherethestrangeness-changingpartvanishesdueto theeffectof thecharmquark.To first

order, therearenostrangeness-changingneutral-currentsandsimilarly nocharm-changing

neutral-currents.Thiscancellationof flavor-changingneutral-currentprocessesoccursonly

in thefirst orderFeynmandiagrams(treelevel). Flavor-changingneutral-currentscanoccur

in the StandardModel throughhigherorderterms(loop diagrams)which aresuppressed

dueto thesmallnessof theweakinteractioncoupling.

1.5 Rareand Forbidden Decays

Rareandforbiddendecaysaredivided into threedifferentgroups: LeptonFamily

NumberViolating(LFNV), LeptonNumberViolating(LNV) andFlavor ChangingNeutral

Current(FCNC).

a) Lepton Famil y Number Violation. Processesof this type are forbidden

becausethey do not conserve the leptonfamily number. For instance,processessuchas

D � | & ¼ and_ & Ï(' Ñ*) +� �� | & ¼ and

_ & Ï(' Ñ*) +��.�� &3 , where + is E or , and

the leptonsbelongto differentfamilies,areexamplesof this mode. Thesedecayswould

suggesttheexistenceof heavy neutralleptonswith non-negligible couplingsto & and � .

b) Lepton Number Violating. Sincethereis noexplanationin theStandardModel

for theleptonnumberconservation,we canexpecta violation at somepresentlyunknown

energy range.Processessuchas_ & Ï(' Ñ*) + �.-¶ /-¶ in which leptonscomefrom thesame

family andareof samechargeareexamplesof LNV modes.

c) Flavor-Changing Neutral-Current. In theStandardModel theneutral-current

`1.6 SpecialRelativity 25

interactionschangeof the flavor is at a very low level. Lower limits for charm-changing

neutral-currentare expected. Such decaysare expectedonly in second-orderin the

electroweakcouplingin theStandardModel. Processesin the_

systemreferto thedecays

D � ? ? � and_ & Ï0' Ñ1) + - - � , etc.Testfor charmFCNCarelimited to hadrondecays

into leptonpairs.

1.6 SpecialRelativity

We will go through a brief review of specialrelativity to establishnotation and

whateverpointsarenecessaryfor this work. If wehaveaparticlethatis moving anywhere

closeto light speed,therelationsof its energy andmomentumwill bedifferentfrom those

at a lowerspeed:

qz J32�j q4 Û (1.19)

and

5 J32.j�4 o Û (1.20)

where

2�J �6 � � { o and { J 44 Û (1.21)


4 is the velocity of light in vacuum,j is the massof the particleat the rest. From eq’s.

(1.19),(1.20)and(1.21):

5 o J p qz�4 s o � p�j�4 o s o Û { J zF45 Û 2�J 5j�4 o (1.22)

Note that if 4 J � Û 5 Jãj�4 o . Specialunits areusedin particlephysicssuchthat

4óJ � and 7 J � . Energy, momentumandmassarethenall in unitsof ~��7� , andtime and

lengthwill bein unitsof ~��7� �� .Let us considerthe displacement,a four-vector which is generalizedby special

relativity asthedifferencebetweentwo “events”,whereboththespatialdisplacementand

timeof occurrenceareconsidered.

8í¹J p�4g- Û r Û ; Û(9 s JÄp@í Û í � Û í o¶Û í � s J p@í Û qí s Û (1.23)

Theinvariantdot productconceptis generalizedaswell

8í � 8ÃÄJ í Ã � í � Ã � � í o Ã o � í � Ã;:µJ í Ã � qí � qÃ (1.24)

This an invariant underLorentz transformationsfor any pair of vectors8í and

8Ãwhich transformlike “event” vectors. Energy andthree-momentumform sucha vector.

This four-momentumvectoris

8í¹J p 5 Û z.< Û z.= Û z?> s (1.25)

Considerthe situationwherea primedsystemmoveswith the constantvelocityq4


with respectto the unprimedsystem.q4 is in the directionof

� r andwe will use { for

thefractionalvelocity of thecoordinatesystem.Considera systemwith momentumqz and

energy5

. TheLorentztransformationequationsare:

zÓÉ= J z.= Û zÓÉ > J z?>�� (1.26)

z É< J@2©p�z.< � { 5 s � (1.27)

5 É J@2©p 5 � { z�< s (1.28)

Undera Lorentztransformation,thescalarproductof two four-vectorsis invariant.

Momentumandenergy form a four-vectorjust like positionandtime. The ? kX/�&¶-:kyj�& ( � )of an unstablemoving particle is calculatedto be equalto 2F� where � is the particle’s

lifetime asmeasuredin its restframe.

TheLorentzinvariantmassof any systemis its energy in a framewhereit is at rest.

In our analysiswe aretrying to identify decayprocessesof a short-livedparticleinto two

oppositechargedparticles. The decayingparticle travels a negligible distancemaking it

impossibleto determinethe identity of the particlebasedon its trajectory. Therefore,an

importantquantityin our work is theLorentzinvariantmasswhich is usedto determineif

two particles(whosemomenta,qz � and

qz o areknown) arethedaughtersfrom thedecayof a

particleof massA into productsof massj � and j o .Conservationof energy andmomentumis usedin orderto reconstructA o from the


candidatedecayproducts:

qz?BÄJ qz � � qz oiÛ 5 BÂJ 5 � � 5 o � (1.29)

WecandefinetheLorentzinvariantquantity

A o� o JÄp 8z � � 8z o s o then, A o� o JÄp 5 � � 5 o s o � p qz � �Qqz o s o (1.30)

where5 Ý�J 6 jáÝ o � z�Ý o .

Noticethat A � o canbecalculatedfromtheassumedmassesj � , j o andthemeasured

momentaqz � , qz o in any frame. A � o is calledtheinvariant( j � , j o ) massof thetwo-particle

system.If thesetwo particlesdid indeedhavethemassesj � and j o andif they did indeed

comefrom the decayof a particle A , then A � o will be equalto A within experimental

errors.Thecalculationof invariantmassis amajortool in identifyingdecays.

If we considertheD �� process,j � J j o J já��} � �DC��FE�E MeV [4, p.23]

andqz � J qz�� å , qz o J qz��HG , thus

5 o�=å J j o�9å � z o�9å Û 5 o� G J¹j o� G � z o� G (1.31)

andtheLorentzinvariantmassfor theD will begivenby

A o�7� J � j o� � �.IKJ pez o�9å � j o� s p�z o� G � j o� s � zÓ�9å zÓ� G 4762h5Ú¶�7��L (1.32)

wherez��=å Û zÓ� G and Ú¶�Z� aremeasuredquantities.

`1.7 ConfidenceIntervalsandConfidenceLevels 29

1.7 ConfidenceInter valsand ConfidenceLevels

Severalconceptsin statisticswill beusefulin our work. Confidencelimits [13] refer

to thosevaluesthat definean interval rangeof confidence(lower andupperboundaries)

for an unknown parameter. If independentsamplesare taken repeatedlyfrom the same

populationanda confidenceinterval calculatedfor eachsample,thena certainpercentage

of theintervalswill includetherealvalueof theparameter.

A differentconceptis thatof ConfidenceLevel. This statisticis usedin hypothesis

testing where one calculatesa M o from the deviations of the observed data from the

hypothesis.If the hypothesisis correct,high valuesof M o areunlikely. The confidence

level is definedas

H0A J NO�PRQTS;U1V�WYXZV (1.33)

whereQTS[U\VDW is the ]_^ probabilitydensitywhichdependsonthenumberof observations, .

Theconfidencelevel (CL) is arandomvariable.If thehypothesisis correct,it hasauniform

probabilitydensityandall valuesof CL areequallylikely. This mayseemto make it not

very useful. However, if the hypothesisis not correct,high valuesof ] ^ (corresponding

to low valuesof CL) will be likely. Onecanstatisticallydifferentiatethecaseswherethe

hypothesisis fulfilled by requiringaminimumCL value.

Chapter 2

Previous Works

2.1 FCNC DecayTheory

The fundamentalwork concerningFCNC decaysis the famouspaperby Glashow,

Iliopoulos & Maiani in which they proposetheexistenceof thecharmquarkanddiscuss

how it canexplain the suppressionof FCNC decays[12] throughwhat is now calledthe

GIM mechanism.Thishasbeendiscussedin Section1.4.

With respectto D acb d$efdhg which is a charmFCNC decay, the GIM mechanism

leadsto suppressionof thetree-level diagramssuchasFigure2.1.a.In a 1997paper[14],

Pakvasapresentedthe latestcalculationfor this process.He found that theshortdistance

effects are dominatedby internal s-quark loop diagramswhich are suppressedby the

smallnessof thes quarkmassandby helicity considerations(Section1.2.1).

30

i2.1 FCNCDecayTheory 31

j ac u

d gd$e

(a)SuppressedDiagramof First Order k

j ac u

d gdle

(b) PenguinDiagram

X?mon�m0pW g

qW e

u

c

d g

d e

(c) Box Diagram

Figure2.1: Feynmandiagramsfor thedecayD asrut e t g . (b)and(c)aresecondorderdiagramswith quarkmassesd, s, andb within the loopsthatgive a rateproportionalto vxwy , being v y themassof thestrangequark.(a) representsasuppresseddiagramoffirst order.

Pakvasacalculatedthe shortdistanceeffectsto be of the orderof z�{ g?|\} . However,

the long distanceeffectsare large bringing Pakvasa’s calculationof the StandardModel

branchingratio to zH{ g?|\~ for D a b d e d g .

Thelong distanceeffectsaredueto intermediatestatessuchas � a , � a , � a , ï , ïZ� or

�l� and � � a [14].

i2.2 PreviousSearchesfor RareCharmDecays 32

2.2 PreviousSearchesfor RareCharm Decays

Many experimentshave reportedlimits for the branchingratio of the rare decay

D a b�d e d g asis shown in Table2.1publishedby ParticleDataGroupin 2000[4, p.566].

Table2.1: Chronologicalbranchingratiosfor the ��z weakneutral-currentdecayD acb d$efd�g .

Value Experiment Comment Year(atCL 90%) Name��x� z�{ g�� E791 � g 500 �� 2000� z �� z�{ g�~ E789 � nucleus800 �� 2000�� z � z�{ g�� Beatrice � g Cu,W 350 �� 1997��x� z�{Zg�� E771 � Si, 800 �� 1996�3��F�� z�{Zg�~ CLEO ��el��g¡ £¢ (4S) 1996��¤�� z�{ g�� Beatrice � g Cu,W 350 �� 1995��F�� z�{ g�~ E653 � g emulsion600 �� 1995�3�� z � z�{ g�~ E789 ¥ �.� z§¦ �.�F¨ events 1994��¤�� { � z�{ g�~ ARGUS � e � g 10 �� 1988� z � z � z�{ g�~ SPEC �?© g W 225 �ª�� 1986�3��F�� z�{Zg w EMC Deepinelasticdhg N 1985

In this sectionwe will discussthe most recentexperiments. We will alsodiscuss

theFOCUSpredecessorexperiment,E687,which searchedfor otherFCNCcharmdecays

but not D a b d e d g . The first experimentis the onethat hasobtainedthe lowest limit,

theCERN1 WA92 BEATRICE experiment.Thenwe presentFermilab’s2 E791andfinally

E687.1CERN-OrganisationEuropeennepourla RechercheNucleaire2Fermilab-FermiNationalAcceleratorLaboratory


2.2.1 The Hadroproduction WA92 Experiment

During the 1992-93 data-taking period of the BEATRICE Collaboration, the

hadroproductionWA92 experiment[15] at CERN SuperProtonSynchrotronwascarried

out. In the interactionsof 350 �� e particlesin a W target during 1992(W92 runs),

charmedparticleswereproducedaswell asin theCu targetof 1992and1993(Cu92and

Cu93runsrespectively). Theapparatuswasmadeup of a 2 mm thick target followedby

a beamhodoscopeandby an in-target counter(IT), a high-resolutionsilicon-microstrip

detector(SMD), a largemagneticspectrometerandamuonhodoscope.

Candidateevents D a b�d e d g were required to be in a mass range between

z �F¨ {«¥�z �F¬D� �� . Dimuoniceventswererequiredandwereselectedby a dimuontrigger.

Eventsthat camefrom target interactionsaswell asthosein IT counterswereaccepted.

Dimuoniceventsassociatedto a simplesecondaryvertex weredifferentiatedfrom events

where both muonscamefrom different secondaryvertices. The D a b K g �_e decay

processwasthenormalizingmodewhichwasdetectedin runsof Cu(W)applyingthesame

selectioncriteriafor thedimuoniceventsexcepttheonesfor leptonidentification.

Monte Carlo simulation generatorsused Phytia 5.4 and Jetset7.3 [16–19] to

understandthe hard processesand quark fragmentation. Fluka [20, 21] was used to

determinethe characteristicsof all other interactionproducts. Monte Carlo simulations

generatedevents that containeda pair of charmedparticles. Dimuonic eventscoming

from the Monte Carlo simulatedD a b�d e d g decayswere treatedin the sameway as

the experimentaldatapermittingthe determinationof the efficienciesandthe acceptance

ratios.


No D ab d$efdhg candidatewasfound [15]. This resultled to anupperlimit on the

branchingfraction B(D a b�d e d g ) of �.� z � z�{ g�� at 90% confidencelevel. Systematic

errorson thewholeanalysisproceduredid notsignificantlyalterthis value.

2.2.2 The Hadroproduction E791Experiment

Eventscollectedin theE791experiment[22] wereproducedby a500 �ª��®�$e beam

in five target foils. This was a hadroproductionexperimentin which track and vertex

reconstructionwereprovidedby 23 silicon microstripplanesand45 wire chamberplanes,

plus two magnets. Muon identification was obtainedfrom two planesof scintillation

counters. The experimentalso includedelectromagneticand hadroniccalorimetersand

two multi-cell Cerenkov countersthat provided ��¯�� separationin the momentumrange� ¥ � {[�� .

In theanalysisstage,to separate“good” eventsfrom backgroundnoise,a separation

of the productionvertex from the decayvertex wasrequiredby morethan z �±°?² , where°?² is thecalculatedlongitudinalresolution.Also thesecondaryvertex wasrequiredto be

separatedfrom theclosestmaterialin the target foils by morethan ��°´³² , where °´³² is the

separationuncertainty.

E791usedasaMonteCarlosimulationgeneratorPythia/Jetset[23,24] andmodeled

theeffectsof resolution,geometry, magneticfields,multiple scattering,interactionsin the

detectormaterial,detectorefficienciesandtheanalysiscuts. As normalizingmodein the

studyof theraredecaysD a b d e d g , D a b � e � g andD a b�d$µl��¶ , theCabbibo-Favored

modeD a b K g � e wasused.


Before the branchingfractionswere calculatedfor eachof the channelsstudied,

E791 useda p�· ©1¸ X analysistechnique. All the eventswithin a ��¹»º rangearoundthe

D a masswere“masked” in orderto avoid biasin thecut selectiondueto thepresenceor

absenceof a possiblesignal.Thecut selectionwasbasedon thestudyof eventsgenerated

by Monte Carlo for signal events and real data for backgroundevents. Background

eventswerechosenandstudiedin masswindows �¼¹½º beforeandafter the signalarea.

The signal areawas chosenas z �F¨��¾� ¹ U1¿ a W � z �F¬ { . The 90% upper limit was

calculatedusing the methodof Feldman& Cousins[25] to accountfor background,

and then correctedfor systematicerrors by the methodof Cousins& Highland [26].

The upperlimit wasdeterminedfrom the numberof candidateeventsand the expected

numberof backgroundeventswithin thesignalregion. Only afterthecutswereoptimized

was the signal areaunmasked and then the numberof eventswithin the window was

`«À*Á y �£`Âº�Ã�Ä�Åª`ÂÆÇÃ y\ÈYÉ Å�`«Ê�ËTÁ , whereÂº�Ã�Ä correspondsto thenumberof “good” candidates

of D a b d e d g , `«ÆÇÃ y1ÈÌÉ correspondsto hadronic decayswith pions misidentified as

muons,and `«Ê�ËTÁ correspondsto combinatoricbackgroundarisingprimarily from false

verticesandpartially reconstructedcharmdecays.

No evidencefor the raredecayD a b�d e d g wasfound [22]. The 90% confidence

level branchingfraction limit was ��Í� z�{ g�� . Systematicerrorsin this analysisincluded

statisticalerrorsfrom thefit in thenormalizationsample;statisticalerrorsonthenumberof

MonteCarloeventsfor both thenormalizingmodeandtheprocessstudied;uncertainties

in thecalculationof MisID background;anduncertaintiesin therelativeefficiency for each

mode.


2.2.3 The Photoproduction E687Experiment

Production and decaysof charm particles from high energy collisions with a

high intensity photonbeamand a beryllium target were studiedby the FermilabE687

experiment[27] usingamultiparticlespectrometerfor particleidentificationandvertexing

of charged hadrons and leptons. From the Tevatron beam, a photon beam from

bremsstrahlungof secondaryelectrons( Î*Ï±Ð[� 350 �� ) hit a beryllium target. Charged

particleswere tracedby four silicon microstrip detectorstations. Thosedetectorswere

very efficient for separatingprimaryandsecondaryvertices.Chargedparticlemomentum

was determinedfrom deflectionsin two analysismagnetsof oppositepolarity with five

stationsof multiwire proportionalchambers.In orderto identify electrons,pions,kaons

and protons, three multicell thresholdCerenkov counterswere used. There were two

electromagneticcalorimeters.Particleswhich passedtheaperturesof both magnetswere

detectedby the innercalorimeterwhich coveredthe forward solid angle. Thoseparticles

thatonly passedthefirst magnetweredetectedby theoutercalorimetercoveringtheouter

annularregion. Two sectionsof thick steelmuon filters divided four proportionaltube

planeswhich wereusedto identify muonsin theforwardsolid angle[28].

The selectionof candidatesfor D e b Ñ?µ · ¶ · e and D e b � g � e � g startedby

making three-trackcombinationsof an event with the right particle identification. The

combinationof theresultingthreetrackswerefit to acommonsecondaryvertex.

In orderto detectmuonswith amomentumhigherthan � �ª�� , planesof scintillator

materialin additionto four stationsof proportionaltubes,altogetherknown astheMuon


System,wereusedby theE687Collaboration.A muonConfidenceLevel wascalculated

whenclassifyingcandidatemuontrajectories.At least1% wasrequiredfor theCL.

In orderto identify electrons,informationof tracksassociatedwith electromagnetic

showersin eitherof thetwo electromagneticcalorimeterswasusedif they wereconsistent

with theelectronhypothesisin theCerenkov counters.

Thesensitivity to themeasuredbranchingratioswasoptimizedfor eachdecaymode

studiedin E687. This sensitivity dependedon the efficiency relative to the normalizing

mode.ThiswascalculatedusingaPythia[29] MonteCarlosimulationgeneratoralongwith

a detailedspectrometersimulation,the amountof normalizingmodeevents(determined

from a fit to the invariantmassplot), andthe amountof backgroundin the signalregion

(determinedusingsidebandsbeforeandafterthedefinedsignalregion).

Monte Carlo signalshapesandbackgroundshapesderived from datawereusedto

calculatethe90%CL upperlimit onthenumberof signaleventsin thecorrespondingmass

plot. Misidentificationof hadronswastheprimarybackgroundin theE687analysis.Three

track combinationswith no leptonidentificationwereusedto determinethe shapeof the

background.The probability of misidentifyinga hadronasa leptonwasintroducedasa

weighton eachleptoncandidate.

No evidencefor the fourteenexclusive modesof rare and forbiddendecayswas

observed in this experiment [27]. However, 90% CL upper limits on their absolute

branchingfractionsin the U ¬ ¥ � { W � z�{�g�~ rangeweredetermined.

i2.3 FOCUS 38

2.3 FOCUS

2.3.1 Intr oduction

Sincehadronswith quarksfrom the ��ÒHÓ and ��ÔÕÓ family exist only rarely in nature,

we needa way to producethem copiously to study them effectively. The methodof

photoproductionin the experiment E831 [30] that took place in the Fermi National

AcceleratorLaboratory (Fermilab) near Chicago, Illinois was used to createparticles

with the charmquark. The E831 experiment,or FOCUS(“FotoproductionOf Charm

with an Upgraded Spectrometer”), was an upgradeversion of the E687 experiment

(Section2.2.3) and was a heavy–flavor (producesparticleswith � ÒHÓ family, or higher,

quarks)photoproductionexperimentlocatedat theWide BandPhotonArea.

Figure2.2: ProtonsynchrotronandbeamlinesatFermilab. TheFOCUSexperimentislocatedat theWide BandPhotonAreain theEastProtonArea.

i2.3 FOCUS 39

2.3.2 The FOCUSPhotonBeam

To photoproduceparticleswhich containthecharmquarkit is necessaryto generate

photonswith a high energy (about300 �� ). At Fermilabthe initial beamis a 800 �ª��Tevatronprotonbeam[31, p.10].

Figure2.3: TheE831PhotonBeam(FOCUSCollaborationFigure).

As shown in Figure2.3, a photonbeamwasproducedafter threedifferentstages.

A neutralbeamproducedfrom 800 �� protonsincident on a liquid deuteriumtarget

includeda variety of neutralparticlesalong with high energy photonsfrom � a decays.

Chargedsecondariesanduninteractedprotonswere taken out of the beamandabsorbed

by theberyllium targetdump(not shown). All theneutralsecondarytrajectoriesproduced

i2.3 FOCUS 40

weretransmitedwithin anapertureof 1.0mr horizontallyan0.7mr verticallywith respect

to the incidentprotonbeamtrajectories.To absorba smallerportion of thesesecondary

photonsa Deuteriumtargetmaterialwasused.

Thephotonsthatwereleft produceda � e � g pair dueto electromagneticinteractions

in passingthrougha 60%radiationlengthleadconvertor. Thepairswereseparatedfrom

theneutralsusingmomentumselectingdipolesandthenput togetherfocusingthemonto

anothertargetThis momentumselectionwasnecessaryto purify the � g and � e beamsby

reducingtheir charged hadronicbackgroundconsiderably. A neutraldumpwasusedto

removehadronicneutralparticles.

Finally, photonsthat camefrom bremsstrahlungwere producedfrom the focused

� e � g beamwhich was passedthrougha 20% radiation length lead radiator. Recoiled

electronsweretakenoutof thephotonbeamandabsorbedby anelectrondump.

2.3.3 Inside the FOCUSExperiment

FOCUSutilized a forward large aperturefixed-target multi-particle spectrometer,

to be explained in Figure 2.4, to measurethe interactionsof high energy photonson

a segmented Ö±��× target. This spectrometerhad excellent particle identification and

vertexing for chargedhadronsandleptons.

The experiment collected data during the z ¬�¬�� ¥ ¬�¤ fixed target run and is

investigatingseveral topics in charmphysicsincluding high precisionstudiesof charm

semileptonicdecays,Quantum Chromodynamicsstudiesusing double charm events,

i2.3 FOCUS 41

a measurementof the absolutebranching fraction for the D a meson, a systematic

investigationof charm baryonsand their lifetimes, and searchesfor D a mixing, CP

violation, rareandforbiddendecays,andfully leptonicdecaysof the ¿ e .

Ø�Ù1Ú ÛÝÜlÞ/Ù1ß1àáDâ ã�â äDâ å æç ÛÝèKéê�ë ÜìÛ1í1Ú ë ßîà

ã�â ï§â äDâ å æ Ø�Ù1Ú ÛðÜñ�ò Û1í\ÚFÜìßîé�è1óîàîÛ1Ú ë íä è ò ßÌÜ ë é�Û*ÚFÛ1Ü

ä ÛðÜFÛðà*ô*ß*õä ßÌÙÌàYÚ�Û*Ü æö Ü ë ó*ó1ÛÝÜ÷ ßîøÌß æ í1ßÌùÌÛ ã�â ïúâ ä�â

ã�â ïúâ ä�âä ÛÝÜ�Ûðà1ô*ß*õä ßYÙYàYÚ�ÛûÜÞ_è*óÌàÌÛ*Ú

ü�ý*þÝÿ��ðÿ��

Þ_è*óÌàÌÛ*Ú

� ë�ò�ë í ß*àÂÞ ë íðÜ�ß æ Ú�Ü ë ù æö Ü ë óûóûÛðÜ ä ßûÙûàûÚ�Û Ü æö è*ÜìóÌÛ1Ú æ � à*à1ÛðÜñ�ò Û1í1ÚFÜ ßîé�è1óîàîÛ\Ú ë íä è ò ßîÜ ë é�Û1ÚFÛ*Ü÷ èûø Ü ß àä è ò ßÌÜ ë é�Û1ÚFÛ*Ü

ö Ü ë ó1ó1ÛðÜ÷ ßÌøÌß æ í*ßîùÌÛ

Þ/Ù1ß1à÷ ßÌøÌß æ í*ßÌùÌÛ

ÞfÙ*ß*à� ë ò Ú�ÛðÜç ÛÝèKéä è ò ßÌÜ ë é�Û1ÚFÛ*Ü�ÝÚ Ü è��ö Ù��YÛ æ

ö è*ÜìóYÛ*Ú� ë�ò ë í ßÕà �� ç ÛKèÝéêDë ÜìÛ1í1Ú ë ßîà! "$#&%!�"'#&%!�"'#&%

Figure2.4: TheFOCUSSpectrometer(FOCUSCollaborationFigure).

In theFOCUSspectrometeraBeOtargetwasusedto produceprimaryandsecondary

particlescomingfrom interactionsbetweentheTevatronphotonbeamandprotonswithin

the target ( ¢ Å p b ( Å p). The targetwassegmentedin four separatepartsin orderto

maximizethepercentageof charmdecayswhichoccuredoutsidethetargetmaterial.

Two trigger countersandtwo silicon trackingsystemswere in the target region as

shown in the Figure2.4. Chargedparticleswhich emergedfrom the target weretracked

by two systemsof silicon microvertex detectors(SSD).Thefirst systemwasmingledwith

i2.3 FOCUS 42

theexperimentaltargetwhile thesecondwasjust downstreamof the targetandconsisted

of twelve planesof microstripsarrangedin threeviews. Altogetherthesetwo detectors

allowedto differentiateprimaryandsecondaryverticeswith high resolution.Two magnets

(M1 andM2) of oppositepolarity with five stationsof multiwire proportionalchambers

(PWC’s) wereusedto measurethemomentumof chargedparticlesusingthedeflectionof

the particleswhenpassingthroughthe magnets.TherewerethreePWCsbeforeM1 and

two betweenM1 andM2.

Threethresholdmulticell Cerenkov counterswereusedto identify electrons,pions,

kaonsandprotons[31, p.16].Two typesof calorimeterwereusedto determinetheparticle

energy. Thefirst calorimetersweretwo electromageticcalorimeters.Theinnercalorimeter,

a leadglassblockarray, coveredthecentralsolidangleanddetectedparticleswhichpassed

throughthe aperturesof both magnets.The outercalorimetercoveredthe outerangular

annulusdescribedby particlesthatpassthroughthefirst magnetbut not thesecond.The

secondcalorimetertypewasahadroniccalorimeterconsistingof iron andscintillatingtiles

whichwasusedprimarily in theexperimenttriggerbut wasalsousedto reconstructneutral

hadrons.

To identify muons,FOCUShada detaileddetectingsystem.Muonswereidentified

in eithera fine grainedscintilatorhodoscopewith aniron filter (coveringtheinnerregion)

or in an outersystemthat usedresistive platechambersandthe iron yoke of the second

magnetasa filter. Themuonsystemwasconstitutedby two sub-systems:theInnerMuon

Systemwhich wascomposedof 9 planesof scintillatormaterial,in which threewereused

astriggersof dataacquisition,andthreeiron filters [32,33], andtheOuterMuon System

i2.3 FOCUS 43

composedof resistiveplatechamberswhichdetectedmuonswith ananglegreaterthan125

milliradians.

In orderto analyzetheinteractionsin thespectrometer, theanalogsignalsfrom each

of thedetectorshadto bedigitalizedandrecorded.Thiswasthetaskof thedataacquisition

system.The FOCUSdataacquisitionsystemhadto dealwith input datain a numberof

differentdataformats,merge all this datainto onestream,andput the outputon 8 mm

magnetictapes.Informationfrom thedifferentdetectorswasreadout andthenthesedata

wereput onadataacquisitionsystembus.

Data for each event was taken off the data adquisition systembus and placed

into a temporarymemorysystem;datacould be written to and read from this system

simultaneously. Later on, the datawasbufferedon disk andwritten to tape. During the

run,FOCUScollectedmorethan z�{*) eventsandreconstructedmorethan zH{ � .The FOCUSCollaborationhaspublishedfour papersstudyingdifferent topics in

charmphysics[34–39].

Chapter 3

Objectives

As adeparturepoint for this thesis,thefollowing objectiveswereconsidered:

1. Thedeterminationof theprobabilityof misidentifyinga mesonasa muon(MisID)

in FOCUSasa functionof momentum.

2. The developmentof software for the searchfor the raredecayD a b d e d g using

blind analysismethods.

3. Thepreselection(from thelargeFOCUSdataset)of asmallandflexible setof event

candidatesfor D a b d e d g aswell asfor theD a b K g � e normalizingmode.

4. The determinationof the relative efficiency for the detectionof D a b d e d g and

D acb K g � e in FOCUS using Monte Carlo simulation to generateand analyze

appropriatesamplesof thedecays.

44

i3 Objectives 45

5. Thedeterminationof thebackgroundlevel in theD acb�d$e´d�g analysisby usingan

independentdatasetandthemisidmeasurements.

6. Thedeterminationof the90%confidenceregion for thebranchingratioof thedecay

D acb�d$e´dhg . In using the high statisticsFOCUSdataset and in optimizing the

analysiscuts, the objective is to improve existing measurementsandobtaintighter

(moreprecise)confidencelimits.

Chapter 4

Procedure

Thischapterwill describehow thesearchfor theraredecayD a b d e d g wascarried

out. Initially, will be discussedthe FOCUSdataformat and the reconstructionprocess.

Thenwill bedefinedeachoneof thevariablesusedin this study. Next, thevariousstages

of datareductionwill be discussed.The useof a normalizationmodewill be explained.

Basically the methodusedin this work is known asblind analysiswhich consistsin the

optimizationof theselectioncriteriaby minimizing thebackgroundin thedatasidebands.

Theuseof MonteCarlosimulationto determinedetectionefficienciesaswell asto study

backgroundswill bediscussed.Themethodologyleadsto thedeterminationof confidence

intervalsfor thevalueof theD ab dle´dhg branchingratio.

46

i4.1 DataFormat 47

4.1 Data Format

Wewill begin bydiscussingtheformof thedataproducedin theexperiment.For this,

it is advisableto befamiliarwith theconceptof an“event”. This refersto oneinstanceof a

photoninteractingin thetargetproducingvariousparticleswhichtravel throughsomeor all

of thedetectors.Someof theparticlesproduceddecayin or nearthetarget into secondary

decayproductswhich aretheonesactuallydetected.Theexperimentalideal is to usethe

informationfrom thedetectorsin orderto measurethecharacteristicsof thedecay. To this

end,it wouldbeeasierif thesetof detectorsignalscame+�¸ ·-, from thedecayof interestand

its associatedproductionpartners.In practice,this is never perfectlyachieved. The“raw”

dataconsistsof setsof signalswhich approachthe ideal of coming from one and only

onephysicalevent. Eachof thesesetsis calleda “raw dataevent”. Typically it contains

informationaboutthephysicaleventbut alsodetectornoiseandinformationaboutparticle

trajectoriesunrelatedto the physicalevent. Given the experimentaldifficulty, many raw

dataeventscontainno interestingphysicalevents.

Each raw data event is an independentunit of data which is “reconstructed”

separately. Reconstructionis the processwhereby the physical characteristicsof the

underlyingparticletrajectoriesareextractedfrom theraw detectordata.Theraw dataevent

is eitherdiscardedif it is not at all interestingor convertedto a “reconstructeddataevent”

which containsthevaluesof thephysicalvariablessuchasthenumberof trajectoriesthat

wereregisteredby thePWCchambers,themomentumof theparticles,etc. In FOCUS,the

reconstructionprocessis calledPass1(seeSection4.3.2).All data(raw andreconstructed)

is storedonmagnetictape.

i4.2 ReconstructionProcess 48

Reconstructedoutputtapesfrom Pass1containall kindsof physicalevents,instances

of amyriadof interestingprocessesall in thesametape.Therearethousandsof tapes.

The next stepis to group eventsof the samekind with the goal of having all the

eventsof theprocessunderstudybeingcontainedin a small datafile. Themostefficient

wayto do this is to do it in stepscalled“skims”. At eachskimlevel, strictereventselection

criteriaareappliedandthedatais separatedinto a largernumberof groupseachof which

containsasmalleramountof data.

4.2 ReconstructionProcess

To have a morepreciseideaof how the dataselectionprocessis donein FOCUS,

it is necessaryto be familiar with the trajectoryreconstructionprocess.A greatdealof

the information depositedin the FOCUSdetectorsare presentedashits which we have

to interrelateto be able to deteminewhich hits in differentdetectorscorrespondto one

particulartrajectory.

Betweenthetargetandthefirst magnet(M1) wefind theSiliconMicrostripDetector

(SSD) whichhasfour stations.After M1 thetrackingis doneby fivemultiwire proportional

chambers(PWCs). Eachoneof these,calledP0,P1,P2,P3andP4,hasfour planes.

Outsideof themagnetsthe trajectorieswill bestraightline segmentswhich will be

reconstructedin theSSDsandPWCs.Thetrajectorieswill bendin themagneticfieldsso

that the straightline segmentswill be different in the threeregions: (1) beforeM1, (2)

i4.2 ReconstructionProcess 49

betweenM1 andM2 and(3) afterM2. Thesewill bereconstructedby (1) SSD,(2) P0,P1,

P2and,(3) P3andP4.

Theprocessof SSDtrackreconstructionconsistsof findingtrajectorieswith theSSD

detector. Thestepsaregroupinghit stripsinto clustersof hits,findingeachoneof thethree

measurementdirectionsto projecttheclusters,andthen,combiningtheseinto trajectories.

ThenPWCtrajectoriesarefound.Theremustbehits in at leastthreechambers.Thosewith

hits in all five PWCsarereferredto as“tracks”. Theoneswith hits in only thefirst three

chambersarereferredto as“stubs”. TheSSDandthePWCtrajectoriesmustbeassociated

with oneanotherto beableto get thenecessaryinformationfor the reconstructionof the

charmdecay. To identify a “link” theSSDandthePWCtrajectoriesmustbeextrapolated

to thecenterof M1 andmusthaveconsistentslopesandintercepts.

µ

µ

µ+

−

−

µ+

Direction

Magnets

Beam

Muon Filters

PhotonBeam

Muon HodoscopeR.P.C. ’sOuter Muon

P.W.C. s

Target

3

2

1

4

Figure4.1: TrajectoriesthroughtheFOCUSspectrometer. A view of sometrajectorieswithin theFOCUSspectrometer. Thefirst andsecondtrajectoriesaredetectedby theOuter Muon System. Trayectoriestwo, threeand four aredetectedusing the InnerMuonSystem.

i4.3 DataSelection 50

Figure4.1 shows sometrajectoriespassingthroughthe spectrometer. Trajectory1

correspondsto a three-chambertrackwhile thethird andfourth trajectories(3 and4 in the

figure)correspondto five-chambertracks.Thesecondtrajectory(2) hashits in all thePWC

chambersexceptin P3. This trajectorycanstill bereconstructedin theregion afterM2 by

combiningtheinformationfrom P0,P1,P2andP4.

4.3 Data Selection

4.3.1 SelectionVariables

In thissectionwewill discussthedifferentvariablesusedfor thiswork. They will be

dividedinto two types:vertexing andparticleidentification.

a) Vertexing . UsingtheFOCUSmicrostripdetectorswe cantake advantageof the

factthatcharmedparticlestravel shortdistanceswithin thespectrometerbeforethey decay

while non-charmedparticlestendto eitherdecayimmediatelyat theproductionpointor to

passthroughthewholespectrometerwithoutdecaying.Thismeansthatfor charmeddecays

we canreconstructtwo distinctverticesin theevent: theprimaryor productionvertex and

thesecondaryor decayvertex. A diagramof aD a b�d$e´dhg decayandits verticesis shown

in Figure4.2. By reconstructingthe two vertices,we obtainseveral quantitieson which

we cando “cuts” to reducenon-charmedbackgrounds.A cut refersto aneventselection

criteriawhichis implementedby requiringacertainrangeof valuesfor aparticularvariable.

A vertexing algorithm[40] is usedto reconstructa D a b d e d g decaycandidate.It


µ

µ

π

π

µ −

D+

0

+

−

γ

Σ +

0D

Primary

Vertex

Vertex

Secondary

−

Figure4.2: Productionanddecayverticesfor theD a r t e t g process.Figureshowstheschematizationof theproductionanddecayverticesfor theD asrut e t g process.

startswith two distinct trajectoriesthat meetthe requirementsto be calleda d e d g pair.

Thevertexing algorithmcalculatesa confidencelevel (CLS) for thehypothesisthat these

trajectoriescomefrom a commonpoint andfinds that point for the caseswhich have a

��.0/21 z43 .

Let ussaythatausefulvertex is found.TheputativeD a trajectorycanbecalculated

by addingthemuonmomentumvectors.Thenthealgorithmtriesto find theprimaryvertex

usingothertrajectoriesin theevent in combinationwith theD a candidate.Thealgorithm

beginsby takingall of thetwo-trajectorycombinations(oneof which mustbetheD a and

theothermustnotbeadaughtermuon)andfits themto avertex hypothesisagainrequiring

a minimumCL of 1% [41]. If thetrajectoriesareaccepted,thealgorithmtriesto put more

andmoretrajectoriesin thatvertex until theconfidencelevel fallsbelow 1%. Theresulting

primaryhasamaximumnumberof daughterswith aCL (CLP) greaterthan1% [41].


Usinginformationcomingfrom thevertexing algorithmwecandefinethefollowing

variablesto beusedin theanalysisfor theraredecayD a b d e d g :

1. L/ 5 . Short lived hadronicbackgroundis removed by requiring a minimum

separationbetweentheprimaryandsecondaryvertices.Thedetachmentcut will requirea

minimum .s¯ ° where . is thedistancebetweenthetwo verticesand ° is theerroron that

measurement.Figure4.2 illustratesthis situationin which the errorson vertex positions

areshown asellipses.

2. Multiplicity of the Primary Vertex. This refersto the numberof trajectories

producedin the interactionthat occurswithin the target. Higher multiplicity primary

verticeshave lessbackground.

3. Vertex ConfidenceLevels. Eachvertex is formedby a reconstructionalgorithm

that hasan associatedconfidencelevel. Due to measurementerrorsthe trajectorieswill

not all intersectat the samepoint. However we can defineerror ellipsesfor eachpair

of trajectories(Figure4.2). We thencalculatetheprobability that,within their errors,all

candidatetracksareconsistentwith intersecting.Confidencelevelsabove1% arerequired

for minimal cutson thesevalues. Sometimesa larger cut is usedto rejectbackgrounds.

CLP standsfor theprimaryvertex confidencelevel andCLSfor thesecondary.

4. Primary and SecondaryVertex z Position. The FOCUStargetsare thin flat

wafersin thexy plane.Thustheprimaryvertex z position(ZVPRIM) becomesthecritical

variablein identifying goodeventswheretheprimaryvertex occurswithin the targetand

backgroundwhenit occursoutsidethe target. For certainlong livedcharmedparticles,a

large percentageof the decays(secondaryvertex) will occuroutsidethe target segments


wheretherewill be lessbackgrounddue to interactionsof secondaryparticleswith the

target material. The secondaryvertex z position (ZVSEC) canbe usedto maximizethe

signalto noiseratio.

Dueto experimentalresolution,thesevariablesarenot useddirectly but insteadare

usedin a ratio with the experimentaluncertainty. For this purposewe will definethe

variable °76 which is the differencebetweenthe z vertex position and the z target edge

dividedby theuncertainty. Dueto thefour-parttargetsegmentationweneedto checkif the

vertex is occurringbetweenthesegmentsor within these.For °76x� { thevertex is inside

thetargetmaterialandfor °76 1£{ thevertex is outside.For theprimaryvertex we use °7698andfor thesecondary°76 P .

5. Vertex Isolation. The primary vertex isolation, ISOP, givesa large confidence

level whenoneof the trajectoriesin thesecondaryvertex is alsoconsistentwith beingin

the primary. Therefore,the lower the ISOPvalue, therewill be an unlikely chancethat

any charmcandidatetrajectorieswill be in fact associatedwith the primary vertex. The

secondaryvertex isolation,ISOS, is theconfidencelevel for a trajectorythat is not in the

primaryvertex to bein thesecondaryvertex togetherwith thepresumeddecaydaughters.

This variablecomesin handyfor rejectingbackgroundfrom highermultiplicity decaysby

requiringlow valuesof ISOS.

b) Partic le Identification . This analysisconcentrateson identifying two typesof

particles:muonsandhadrons.

1. Muon Identification . Muon identification is possiblebecausemuonsare the

only chargedparticleswhich canpenetratelarge amountsof material. The methodused


to detectmuonsis to placechargedparticledetectorsbehinda large amountof shielding

material(typically steel).Thereareapairof differentkindsof detectionsystemsfor muons

in FOCUS.Theseare:theInnerMuondetectorandtheOuterMuondetector. Thefirst one

usescommonscintillatordetectorelementsandthesecondusesresistiveplatechambersto

detectpassingmuons.

For the inner region the systemfor muon detectionusesthreemuon scintillating

hodoscopestations:MH1, MH2 andMH3. The first two stationsregisterhits in x andy

whereasthethird stationregistershits in auvplanewhich is oriented¦ � {�: with respectto

thexyplane.

Thealgorithmof muonidentification[33] wasdevelopedat theUniversityof Puerto

Rico, MayaguezCampus,with thepurposeof calculatingfor eachtrajectorya confidence

level to decidethedegreeof reliability in theclassificationof thattrajectoryasarealmuon.

Thealgorithmis basedon the following considerations:themultiple Coulombscattering

and the geometryof the scintillatorsthat make up eachone of the planesof the muon

identificationsystem.It is describedin detailby Ramirez[33] andWiss[42,43].

Therearetwo waysamuonidentificationalgorithmcanfail. Oneis to fail to identify

a muonasa muon.Thischaracteristicis measuredasthealgorithm’sefficiency. Theother

is to identify anon-muonasamuon.Thatis measuredby themisidentification.Thesetwo

characteristicsarecorrelated.A goodalgorithmachievesabalancebetweenthem.

In themuonidentificationweareconsideringthefollowing variables:

i. Kaon Consistencyfor Muons. We canusethe kaonconsistency variable(see


hadronidentification)to separatekaonsfrom muonsalso,sincepionsandmuonsbehave

almostidenticallyin theCerenkov system.Thenamegivenfor this variableis �¡¯�d .

ii. Inner Muons. For the Inner Muon System(IMU) the algorithm begins by

verifying if thecandidatetrajectoriesto bemuonsleave signalin thefive PWCchambers.

The trajectoriesare projectedinto the IMU planesand associatedhits are identified by

lookingwithin acirclearoundtheprojection.

In the IMU systemthere are six MH planes. There must be four hits in these

planes.We canproceedto calculate] ^ which givesus the trajectorycompatibility with

theMH hits. Therearetwo otherthingsthataretakeninto considerationwhenwe do this

calculation.Thesearethemultiple Coulombscatteringthatoccursinsidethe steelfilters

andthegranularityof thescintillatorpaddles.

Lastly, anisolationvariableis calculatedwhichcorrespondsto thelargestCL for any

othertrajectoryto havecausedthesameMH hits.

Becausetherearesteelfilters in theIMU thatcanstopmuonswith low momentum,

the numberof requiredplanesis reducedto two from the original numberof four for

candidateswith a momentumlessthan10 �� . This will increasethe efficiency at low

momentum.

The efficiency for the muon identificationalgorithm is over 98% for momentum

above 10 �� while misidentificationis of theorderof 0.5%. In-flight � b d q4; decays

candidateto themisidentificationrate.For thisdecaythepionandmuondirectionof flight

areverycloselymatched.


Informationobtainedfrom theIMU is containedin thefollowing variables:

A) Number of Missed Muon Planes. The number of missed muon planes

correspondsto the numberof MH planeswithout signal. In this analysisthis variable

is calledMISSPL.

B) Inner Muon ConfidenceLevel. The muon identificationalgorithmcalculates

a ] ^ value for eachtrajectorythat passesa minimal requirementin the numberof MH

planeswith signals.Trajectoriesthatdo not correspondto muonstendto have higher ]_^becausethey havedifferentbehavior comparedwith muons[32,p.11].TheinnermuonCL

(IMUCL) is calculatedfrom the ] ^ .iii. Outer Muons. Muonsareidentifiedin theOuterMuonSystem(OMU) almostthe

sameway asin theIMU. Thereis aninternalmagneticfield in theiron shield(M2) which

cancausesomeproblemswhenit comesto OMU identificationbecauseit deflectsmuons.

Muons must be tracedthroughthe magneticfield in the M2 steel. Multiple Coulomb

scatteringsmearingmustbeappliedto thetracedposition.

TheOMU reconstructionalgorithmis efficient for muonsdown to Ï � �� , the

rangeof muonsin theM2.

A) Outer Muon ConfidenceLevel. In thesamewayasis calculatedtheIMUCL, the

OMUCL is determinedcomingfrom themeasurementof the ]_^ valuefor eachtrajectory

thatpassedaminimal requirementwithin theOMU.

2. Hadron Identification . To identify hadronssuchas � µ , � µ , p ¯ p, FOCUS

developed an algorithm called CITADL (Cerenkov Identification Through A Digital


Likelihood)[44] whichcalculatestheprobabilitiesfor severalhypothesisabouttheidentity

of a trajectory. The identitiesconsideredare electrons,pions, kaonsand protons(and

their antiparticles). The d$µ hypothesisis not consideredseparatelyfrom the �$µ since

the momentumrangeover which the two hypothesescanbe separatedis limited. In the

Cerenkov systemthe behavior of thesetwo particles,muonsandpions,is similar dueto

their closemassvalues.For eachof thefour hypothesis,CITADL calculatestheconedue

to the Cerenkov effect andthe averagenumberof photonsemitted. Poissonstatisticsare

usedto calculatetheprobabilitythattheobservedsizeof thesignalsin theCerenkov cells

is consistentwith theidentity hypothesis.

From the probability < , the algorithm calculates the negative log-likelihood

k =?>A@ U © WCB ¥ �EDGF <cÃ for eachparticlehypothesis© .Variablesusedin hadronidentificationare:

i. Kaonicity . It is definedasthedifferencebetweenthe k ÀûÁ y for thekaonandpion

hypothesis.

� k U � K WCB k =?>A@ U � W ¥ k =?>A@ U K W (4.1)

Statistically, kaonicityis largerfor realkaonsthatfor realpions.

ii. Pion Consistency. A variablecalledpion consistency or �7H =?I is usedto identify

pions. This variable is relatedto the probability that the pion hypothesisis better in


comparisonto otherkindsof hypothesis.�7H =?I is definedas

�7H =?I B k =?>A@ U\p � n J Ñ , �7+ J Ñ?� n © n�W ¥ k =?>A@ U � W (4.2)

wherethe k =?>A@ U1p � n J Ñ , �$+ J Ñ?� n © n�W correspondsto the minimal valueof the likelihoodfor

all thehypothesisconsidered.Typically weuseacut of �7H =?I 1�{ for pion identification.

Othervariablesusedin theanalysisare:

3. Momentum. To measurethe momentumof a particle,FOCUSdeterminedthe

deflectionanglesin eachoneof the magneticfieldsproducedby M1 andM2. For three-

chambertrajectoriesthemomentumis measuredby M1 andfor five-chambertrajectories

themomentumis measuredby M2.

Themomentumresolutionfor trajectoriesmeasuredin M1 is approximately

° K� � { � { �� z�{�{[�� zÅ L z ¤ �ª�� M ^ (4.3)

andtheresolutionfor trajectoriesmeasuredin M2 is

° K� � { � {�z �� z�{�{[�� zÅ L �� ª�� M ^ (4.4)

At high momentum,theresolutionis limited by thepositionresolutionof thePWC

system;at low momentumit is dominatedby multipleCoulombscattering.

4. Dimuonic Invariant Mass. Themassof primaryparticlesis calculatedfrom the


informationobtainedfrom theparticlesinto which thesedecay(Section1.6). In our case

this is thetwo muontrajectoriesinto which theD a decays.

4.3.2 FOCUSData SelectionStages

Thereare6.5 billion photoninteractioneventson about60008 mm magnetictapes

which constitutethe raw dataset of FOCUS.Obviously, this is way too much datafor

an individual researcherto handle. Therefore,the analysiswasdivided into threesteps.

Figure4.3 shows a summaryof the FOCUSanalysisprocess.The reasonfor doing the

selectionin stagesis to avoid having to handlethetotality of thedataagainin thecasethat

someproblemis later found in theselectionprocess.In a multistageprocedure,oneonly

needbegin at thestagepreviousto theproblem.

a) Pass1 . This is thebasiceventreconstructionstageincludingtrackreconstruction

and particle identification. It lastedfrom January, 1998 to October, 1998. The output

consistedof “reconstructedevents”madeup of thevaluesfor thephysicalvariablesof the

event. It wasputontoanothersetof 60008 mmmagnetictapes.

b) Skim1 . Here,the reconstructeddatawasdivided into six “Superstreams”.This

wasdonein orderto make the sizeof the dataeasierto handle. Approximatelyonehalf

of the eventsfrom Pass1met the requirementsfor Skim1 anddependingon the physical

topics that were consideredthey were put into differentSuperstreams(seeTable4.1).

EachSuperstreamconsistedof 200–5008 mm magnetictapes.Skim1wascarriedout at

theUniversitiesof ColoradoandVanderbiltfrom October, 1998to February, 1999.


Skim2

Reconstruction Overview

Skimmed

3% Raw Data

3% Recon Data

15% of Charm

10 days of Skim1

StorageMass

DAQ

Pass1

Skim1

Skim2

FNAL

Disk

Mainline Tape Path

Expressline Network Path

Express

Analysis

MonitorCharm Yield

Shift Monitor

3% Recon

Institutions

DataRaw

DataRecon

Data

Figure4.3: Analysisoverview of FOCUS.The solid lines show the pathof the datawritten to 8 mm magnetictapes. The dottedlines illustrate the distribution of largeamountsof datavia theInternet,whichwasusedto helpcollaboratorsgetdataquicklyfor studiesandpreliminaryanalyses(E. Vaandering’s ThesisFigure).

c) Skim2 . Thesix Skim1Superstreamsweredividedinto many otherSubstreams.

Thoseeventsthatdid not passthe morespecificcutswerenot includedbut not all of the

skimshadadditionalcuts.


Table 4.1: Descriptionsof Superstreams.There are about 30 different datasubsetsgroupedinto six Superstreamsbasedon physicstopicsandthe typesof informationpresent.

Super Physics Skim2Stream Topics Institution

1 Semileptonic PuertoRico2 Topologicalvertexing andK a y Illinois3 Calibration CBPF, Brazil4 Baryons Fermilab5 Diffractive (light quarkstates) California,Davis6 Hadronicmesondecays California,Davis

Table4.1showsthefiveinstitutionsinvolvedin Skim2.Therewere5–12Substreams

writtenfromeachof theSuperstreams.Calibrationdatasamplesandmany topicsin physics

werebroughttogetherby theseSubstreams.SimilarcomputingmodelsbetweentheSkim1

andtheSkim2wereusedbut they variedby institutions.Skim2beganin January, 1999and

finishedby June,1999.

Table4.2refersto theoutputof theSuperstream1 reconstructedandanalyzedby the

HEPGroupof theUniversityof PuertoRico at MayaguezCampusduringJanuaryto June

of 1999. More than500tapesof semileptoniceventswereskimmedanddividedinto five

differentssubsamples.

d) Skim3 . It is in this stagewherewe really apply strongselectioncriteria which

leadus to obtaina reducedoutputyet still leavesus with a certainamountof flexibility

in the detailedanalysisandoptimization. It is very difficult to have to work with dozens

of tapesin the detailedanalysis.Skim3 reducedthis numberto onetape. Justthe same

asin earlierstages,the ideawith Skim3wasto obtaina morecompactsamplethatwould

containthegreatmajorityof acertainkind of event.


Table4.2: Descriptionsof Superstream1. This Superstreamwasdevelopedbythe University of PuertoRico at MayaguezCampus.Therearefive differentskimsgroupedaccordingto physicstopics.

Sub Physics OutputStream Topics Tapes

1 Semimuonic 262 Dileptonic/PPbar 453 Semielectronicwith mesons 374 Semielectronicwith baryons 275 Normalization 58

The datain Substream2 of Superstream1 wasusedasthe startingpoint in Skim3

becauseit provided the highestefficiency for the detectionof D a b d e d g . The main

idea of Skim3 was to require the existenceof two linked muon candidatetrajectories

with oppositechargecomingfrom a parentthatwasidentifiedwith a cut in thedimuonic

invariant massaroundthe massof the D a (between1.7 and 2.1 �ª�� ). In addition to

this, certainvertex identificationconditionswererequiredconsistingof �N.POQ1 z43 and

��.0/R1 z43 . Furthermore,the productionvertex wasrequiredto be separatedfrom the

decayvertex by at leastthreetimestheseparationerror( .s¯ ° 1 � ). With this we achieved

a reductionin thedatasizefrom 45 tapesof 4.5 Gbyteseachto one3 Gbytetape. The3

Gbyteswereeasilykepton disk for greaterconvenience.

At the endof Skim3, we continuedto reducethe databy a processwhich we call

Skim4.Theselectioncriteriarequiredfor Skim4werethefollowing:

Thesecutswerebasedon theexperienceacquiredin previousFOCUSdataanalysis;

thesewereknown to havehighefficiency.

Theoutputfrom Skim4wasamatrixof S � ¸ asaneventfile wherem is thenumber

i4.4 NormalizingMode 63

b Vertex isolationcuts � 1% b Verticeszposition, ¥�z�{ � V �3�b °76T8VU@� b S K Ô Ã�ËXW �b Daughtermomentum1 7 �� b WerequiredaMISSPL Y �b IMUCL 1 z43 b OMUCL 1£z43

of variableswith informationto bestoredandn is thenumberof pair trajectoriesaccepted

asmuoncandidates.Thetotal datasizedid not exceed500Mbytes. This matrix is called

an ntuple, which is analysedusing software developedby CERN called PAW (Physics

AnalysisWorkstation)[45] which letsuseasilymanipulatetheinformationstored.It uses

histograms,vectorsand other familiar objectsto provide the statisticalor mathematical

analysisinteractively. An ntupleallows usto try differentanalysiscriteria in a shorttime

interval.

4.4 Normalizing Mode

Theobjectof this studyis to determineat what frequency do D a b d e d g occurin

nature. This meansthat we will indirectly measurehow many timesthis processoccurs

with respectto a largesamplethatcontainsmany differentprocesses.

Thebranchingratio (BR) is definedasthefractionof decaysto aparticularchannel.

Mathematically

Ö[Z]\D¸_^�Ñ.©*¸'`Nab\ J ©�+�� `dc$S p �eZ�+*f D a b d e d g �7Z*+ X c'^�� Xg + J \ · ¸_c$S p �eZh+*f D al�7Z*+ X c'^�� X (4.5)

i4.4 NormalizingMode 64

If weconsiderawell known processwherethefrequency atwhich it occursis known

with somecertaintythen we can usethat processto measurethe total numberof D a ’sproduced.g + J \ · ¸_c$S p �eZh+*f D a �7Z*+ X c'^�� X � `ic'S p �eZj+]f ¿ a blk � n �mZ]+ X cn^�� XÖ[a U1¿ a bok W (4.6)

Theprocess¿ a b k is calledthenormalizingmode. In FOCUSa processcoming

from D a asprimary particle is D ab K g �_e . It is known asa GoldenMode due to its

largeBR value.In theanalysisfor thesearchof theprocessD a b d e d g we canconsider

D a b K g � e as the normalizingmode. Then the BR of the decayD a b�d e d g canbe

calculatedas:

Öpa U D a b d e d g W � ` ;q; U �7Z*+ X�W`Nr�s U �7Z]+ X�W Ö[a U D a b K g � e W (4.7)

Thenumberof producedparticlesis calculatedasthenumberof observedparticles

dividedby theefficiency ( t ) of detectioncalculatedby MonteCarlosimulations.Theneq.

(4.7) turnsinto

Öpa U D acb�d e d g W � `«ÀûÁ y U D a b�d e d g W`«À*Á y U D a b K g � e WtAr�st ;u; Öpa U D acb K g � e W (4.8)

where the ratio tAr�s�¯]t ;q; correspondsto the relative efficiency to be calculatedmaking

parallelMonte Carlo simulationsfor the two processes,D ab d$efdhg andD acb K g �_e .

BR(D a b K g � e ) hasbeenpreviouslymeasuredbyotherexperimentsas ��¨�� ¦�{ � { ¬$� z�{ g ^[4, p.36].

The dataeventsfor the normalizingmodewere selectedusing the sameselection

i4.5 MonteCarloSimulation 65

criteria (except for particle identification)usedto selectthe dimuonic dataevents. The

normalizing mode data was obtainedfrom Substream5 of the Superstream1. This

Substreamcontainedhadronicdecaysselectedwith thesamebasicconditionsastheother

Superstream1 eventswith thespecificpurposeof servingasa normalizationsample.

4.5 Monte Carlo Simulation

Becauseof thecomplexity of High Energy Physicsexperimentsandtheimpossibility

of calculatingeveryeffectanalytically, simulationmethodsarecommonlyused.Computers

have pseudo-randomnumbergeneratorswhich are usedto simulateapproximationsof

expectedresultscomingfrom the theoreticalpredictions.We call this simulationmethod

MonteCarlo. In FOCUS,generatedMonteCarloeventswereusedasa measurementtool

for theefficienciesandalsoasaninformativetool abouttheform of thebackgroundwithin

our dimuonicsample.

Two mathematicians,the AmericanJ. Neyman and the HungarianS. Ulam gave

birth to this method in Monaco in 1949 as an alternative solution to probabilistic as

well as deterministicproblems. Using a computer, the Monte Carlo methodperforms

statisticalsamplingexperimentsby simulating randomquantities,therebyproviding us

with approximatecalculationsin High Energy Physicsproblems[46].

FOCUSusedthePythia6.127[47] generatorasa Monte Carlo simulatorto model

charmproductionby photoninteractionswith the target material. Pythiagenerateda cc

pair whereit waspossibleto specifythetypeof charm( ¿ a , ¿ e , v ew , etc.) generatedand

i4.5 MonteCarloSimulation 66

a specificdecaypath.Typically this wasdonefor oneof thecharmquarkswhile theother

wasleft freeto hadronizeanddecayaccordingto theknown crosssectionsandbranching

ratios. To simulateall theprocesseswithin theFOCUSspectrometer, analgorithmcalled

ROGUEwasdeveloped[48].

D xzy {}|0{�~ Events . To calculatedimuonic efficiencies it was necessaryto

generateMonte Carlo data which were required to passthrough the sameselection

criteria that were applied to real data, in other words, the requirementsset by Pass1,

Skim1, Skim2, Skim3 and Skim4. 800,000 events were generatedfor the process�z�� D � �o��_��n�� c representedin Figure4.4.

µ

µ

c−

−

+D 0

γ

Figure4.4: MonteCarlogeneratedprocessschematizationfor thedecayD �C�R� � � � .

Thenumberof eventsprovideduswith enoughsamplingstatisticsto make accurate

determinationsof thedetectionefficiencies.

D �z� K ��j� Events . 800,000D � � K �7� � eventsweregenerated.The same

selectioncriteriawereappliedaswith thenormalizingdata.

The decaychannelD � � K � � � is very importantin this work becausethis is the

�4.6 AnalysisMethodology 67

normalizingmodeusedin the BR(D � �o� � � � ) measurement.In addition to that, the

samedecaychannelis presentaspossiblebackgroundin ourdimuonicsample.

D � � � � � � and D � � K � K � Events . The processesD � � � � � � and

D � � K � K � were also consideredin the backgroundstudy. For the first process,1.5

million eventsweresimulatedand4.3million eventsfor thesecond.

4.6 AnalysisMethodology

4.6.1 Blind Analysis

Our Blind analysisexcludesall the eventswithin a definedsymmetricareaaround

thevalueof theD � massin theprocessof cut selection.This areais calledthesignalarea.

In otherwords,cut optimizationis basedon reducingbackgroundandnot on maximizing

thesignalunderstudy. Thepointof this is notto introducebiasdueto thepossibleapparent

absenceor presenceof a potentialsignalafterhaving applieda specificgroupof selection

cuts. Oncethe optimizationstageendedwe openedthe “hidden” areaof the signaland

calculatedtheconfidenceinterval.

4.6.2 Rolkeand LopezLimits

Thestatisticalmethodproposedby Rolke andLopez[49] wasusedto calculatethe

sensitivities and the confidenceintervals. They suggesta techniqueto place limits on


signalsthat are small when backgroundnoiseis present. This techniqueis basedon a

combinationof a two dimensionalconfidenceregion anda largesampleapproximationto

thelikelihood-ratioteststatistic.Automaticallyit quotesupperlimits for smallsignalsand

alsotwo-sidedconfidenceintervalsfor largesamples.

The techniqueof Feldmanand Cousins [25] does not consider the amount of

uncertaintyin the background. The new Rolke and Lopez techniquedealswith the

backgrounduncertaintyasa statisticalerror. The techniqueperformsvery well. It has

goodpower andhascorrectcoverage.This methodcanbeusedfor two situations:when

sidebandsdatagiveanestimatedbackgroundrateandwhenit comesfrom MonteCarlo. It

canalsobeusedif thereis asecondbackgroundsourcein thesignalregion.

To understandtheresultsof thistechniquewewill needthefollowing notation:x will

representtheobservedeventsin our signalregion, y will be theeventsin thebackground

region and � will be theratio of thesidebandsbackgroundto thesignalbackground.The

expectedbackgroundrate is ��Q�� and the SensitivityNumber, �N� � D � �o��_��n� , is

definedastheaverage90%upperlimit for anensembleof experimentshaving anaverage

of y eventsin thesidebandsandno truesignal.Table4.3givesthesensitivity numbersfor

differentvaluesof theconfidencelevel, � andy. During thevariousstagesof this analysis

computerroutinesprovidedby RolkeandLopezwereusedto calculatesensitivity numbers

andconfidencelimits. Thevalueof � for eachsetof cutswasdeterminedasexplainedin

Section4.7.Thesensitivity numberswereusedto selecttheoptimumsetof cuts.Thenthe

confidencelimits werecomputedfrom thevaluesof x, y and � for thatsetof cuts.Examples

of confidencelimits aregivenin Table4.4


Table 4.3: Sensitivity number as per Rolke and Lopez. The number ofbackgroundeventsin the sidebandsis y. The estimatedbackgroundrate is�� . �� ¡

C.L.

��¡C.L.

y �¢�¤£ �¢�¦¥ �¢�§£ �¢�¦¥0 2.21 2.21 4.52 4.521 3.31 2.82 6.22 5.352 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 10.96

4.6.3 Determination of Sensitivity and ConfidenceLimits

We definethe experimentalsensitivity asthe averageupperlimit for the branching

ratio thatwouldbeobtainedby anensembleof experimentswith theexpectedbackground

andno truesignal.Therefore,theexperimentalsensitivity is calculatedin thesamewayas

thebranchingratio exceptthat thesensitivity numberis usedfor thenumberof observed

signalevents.For eachparticularsetof cuts, � is determinedasexplainedin Section4.7.3.�� is the predictedlevel of background,b. Having this in mind, we candeterminethe


Table4.4: An exampleof Rolke andLopez90% ConfidenceLimits. y is thenumberof eventsobserved in thebackgroundregion andandx is thenumberof eventsobservedin thesignalregion. A valueof ��¥ hasbeenassumed.

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 3.21,14.31

experimentalsensitivity as¨ � D � �o� � � � � � �N� � D � �o��_��$��N©�ª¬« � D � � K � � � �®°¯�±�²q²Q³[´ � D � � K � � � � (4.9)

where, �N� � D � � �µ�_��n� is the sensitivity number(Section4.6.2) which is determined

usingtheRolke andLopezroutinesfrom thenumberof eventsobservedin thesidebands,

y, andthebackgroundratio, � .

In simpleterms,thesensitivity is mainlydependenton theratioof �[�¶� �²q² . Wewish

to minimizethis ratiowhichmeanshaving low backgroundwith highefficiency.

Oncetheoptimumsetof cutshadbeendetermined,weproceededto openthehidden

signal region and determinedthe correspondingconfidencelimits for the D � � ��'��


branchingratio by usingeq. (4.9) andsubstitutingtheappropiateRolke andLopezlimits

in placeof �[� .

4.6.4 SidebandsBackground and Signal Ar ea

In this analysis,the signal areaas well as the sidebandswere definedfrom the

invariantmassdistributionof MonteCarlogenerateddata(Figure4.5a).Theability of the

FOCUSMonte Carlo to predictsignalwidth hasbeenverified with othercopiouscharm

signals.

0.016 GeVσ =·σ·|¸

Figure 4.5: D � � � � � � Invariant Mass distributions. (a) distribution forgeneratedMonteCarlodatawith a gaussianfit. Thegaussianwidth is equalto 0.016¹Vº¼»

. (b) Representationof thesidebandsandthesignalareafor theFOCUSproduceddatashowing the signalandsidebandareas. Skim3 cutshave beenappliedand thesignalareahasbeenmasked.

Thewidth of theinvariantmassdistributionfor D � � ��n�� eventsis approximately

equalto ½¾� �¿À £ÂÁÄÃbÅuÆ . The signalareawastaken as ÇN¥*½ arounda D � massof £ ¿ÀÈ Á�É

�4.7 BackgroundStudy 72ÃbÅuÆ [4, p.36]. Eachoneof thesidebandshada width equalto thesignalareawidth but

thesidebandswerenot locatedimmediatelyafter thesignalarea.Betweeneachsideband

andthesignalarea,abuffer areawasleft with awidth alsoequalto ¥*½ . In thisway theleft

sidebandwasdefinedas £ ¿ËÊ]Ì�Ê¢Í¦Î ²q² Í £ ¿ÀÈ] £ andright sideband£ ¿Ï� ¥ �ÐÍÑÎ ²q² Í £ ¿À��]Ìasshown in Figure4.5b.

4.7 Background Study

4.7.1 Intr oduction

Themostimportantaspectof theanalysisis thechoiceof a setof selectioncriteria

which optimize the probability of observingthe signal if it is present. Generallythis

choicewill not beonewhich eliminatesall backgroundbecausesuchcutsusuallyreduce

the efficiency of detectionof the signal to unacceptablysmall values. It is therefore

importantto be able to predict the backgroundlevel accuratelyin order to measurethe

significanceof thecandidateevents.Dueto theuncertaintiesin simulatingthebackground,

our analysismethodologyreliesmainly on the real data(in the sidebands)to predict the

backgroundin thesignal.However, this methodologyrequiresa determinationof theratio

of sidebandto signalbackground,� . This, in turn, requiresadeterminationof theshapeof

thebackground.As is explainedin this section,this shapeis alsoobtainedfrom realdata.

Fortunately, sincethey arenot strong-interacting,muonsarenotproducedcopiously

andit is evenrarerthata pair of muonsis producedin eithera productionor decayvertex.

In addition, eventswherethe pair of muonscomefrom a non-D � two-body decayare

�4.7 BackgroundStudy 73

eliminatedby the invariantmasscut while thosewherethe muon pair is formed at the

productionvertex (e.g. thosefrom the Bethe-Heitlerprocess)areeliminatedby the ÒÓ�*½cut. In fact,themainsourceof backgroundin ourfinal sampleareeventswhereoneor two

mesonshavebeenmisidentifiedasmuons.Thereforetheunderstandingof thebackground

startedwith a studyof this misidentificationusingreal data. Theseresultswereusedto

validatethecapabilityof theMonteCarloto simulatethiseffect. TheMonteCarlowasthen

usedto studysomeknown decayprocesseswhichcouldcontributeto thebackgroundif the

decayproductsweremisidentifiedasmuons.Finally, thebackgroundratiowasdetermined

fromanindependentrealdatasampleof eventswith nomuonsin thefinal stateby assuming

that the backgroundconsistedof sucheventsexcept that both decayproductshad been

misidentified.

4.7.2 MisID Study

Thegoalof thisstudywasto determinethemisidentificationprobabilityasafunction

of momentum. Samplesof real and simulatedK � « � � � � � decayswere usedfor this

purpose.

Therealdatawasobtainedfrom 140raw datatapes.K � « decayswereidentifiedwith

thefollowing requirements:vertex conditionsfor a pair of oppositechargedparticles,the

two pion trajectoriesmusthave left a signal in the five PWC chambers,andthe dipionic

massin a rangearoundtheK � « mass(

�¿ÏÔ Ê�ÊÐÍ¦Î ±u± ÍÑ Õ¿ ÉÕ£ Ê ), beingtheK � « massequalto �¿ÖÔ ��È Ã×ÅqÆ [4, p.32].Furthermore,decaycandidateshadto havelinkedtrajectoriesbetween

the silicon detectorsand the PWCs. Eventswhich passtheserequirementsare called


Type-9K � « [32, p.27]. By determininghow many of thesetrajectorieswereidentifiedas

muonsonecoulddeterminethemisidentificationprobabilityasa functionof momentum.

A sampleof simulatedType-9K � « � � � � � decayswasgeneratedwith MonteCarlo

by artificially lowering the K � « lifetime thusforcing the K � « to decaynearthe target and

obtaininga largesampleof pionscomingfrom thetarget.Thissamplewasanalyzedin the

samewayastherealType-9K � « dataobtainingconsistentresultsbetweenthetwo samples

(SeeSection5.1).

4.7.3 Background Ratio Study

In orderto havea roughideaof thenatureof thebackground,MonteCarlowasused

to generatetwo-bodymesonicD � decayswhich wereexpectedto contributesignificantly

to the background. The first decaysimulatedwas D � � � � � � . Figure 4.6 shows the

comparisonof thedimuonicinvariantmassdistributionsfor D � � � � � � andD � � � � � � .

TheD � � � � � � peakis slightly shiftedfrom theD � massdueto thedifferencebetween

the pion and muon mass. Sincethat differenceis small, the shift in the peakis small

andthe invariantmasscut will not helpus to eliminatethis background.Fortunately, the

D � � � � � � decayis Cabibbo-SuppressedandtheFOCUSmuonsystemis very goodat

rejectingpions(low misidentificationprobabilities).

The other backgrounddecaygeneratedwas D � � K � � � . Figure 4.7 shows the

dimuonic invariant massdistribution for D � � K �7� � events. Due to the large mass

differencebetweenthe kaon and the muon, the peak is much more shifted but it does

fall within theleft sideband.D � � K � � � is aCabibbo-Favoreddecay. Fortunately, it lies


Figure 4.6: Dimuonic Invariant Mass comparison of Monte Carlo data fromD �Ó�ÙØ � Ø � andD �Ó�Ú� � � � .

outsideour signalareaandcanbe almostentirely eliminatedby the invariantmasscut.

This distributionalsoshowsanadditionalcontinuousbackgroundinsidethesignalregion.

This backgroundis from theothercharmparticlein theevent. (Charmis alwaysproduced

in pairs). It is madeup of semimuoniceventswheretheneutrinois not observedor from

partially-reconstructedhadronicdecays.It is verydifficult to simulateall suchbackground

componentsandto know theirrelativeabundance.Thisis whywedonotrely onsimulation

to obtainthebackgroundsidebandto signalratio.

As analternative to simulationdata,a sampleof realdatafrom Superstream2 (also


Figure4.7: Dimuonicinvariantmassof D � � K � Ø � generatedby MonteCarlo.

calledGlobalVertex) wasusedto determinethebackgroundratio. This superstreamwas

selectedwith a minimum of requirementsandcontainsall backgroundsto our signal. In

particular, no particleidentificationwasusedin makingSS2. Our procedureconsistedin

submittingthis sampleto thesameskimmingprocessastherealdataexceptthatno muon

requirementwasused.It wasonly necessaryto work with Û 10%of the total SS2datain

orderto obtainsufficientstatisticalpower for our purposes.

As an example of the dimuonic invariant massdistributions obtainedwith this

procedure,Figure 4.8 shows the resultswhenone appliestight cuts to the SS2sample.


The D � � K �$� � is the dominantfeaturein the distribution but a D � � � � � � peakis

alsoevidentaswell asotherbackground.TheD � � � � � � peakis muchsmallerthanthe

D � � K � � � peakbecauseit is Cabibbo-Suppressed.The peakfrom D � � K � K � lies

completelyoutsideany regionof interest.As thecutsarevaried,theratioof thebackground

in thesidebandsto thatin thesignal( � ) variesbut thenumberof eventsin thehistogramsis

sufficient to allow a precisedeterminationof this ratio which is calculatedusingweighted

histogramsasexplainedbelow.

Figure4.8: BackgroundcomparisonbetweenrealandMonteCarlodata.Distributionfrom SS2datawith tight cutsbut no muonidentification.Theseeventsconstitutethemainsourceof backgroundwhenthedecayproductsaremisidentifiedasmuons.Thereareevidentpeaksfrom theprocessesD � � K � Ø � , D � �RØ � Ø � andD � � K � K �alongwith otherbackground.

�4.8 Optimizationof theSelectionCriteria 78

In order to take into considerationthe probability of the decay productsbeing

misidentifiedasmuons,the backgroundratio wasdeterminedfrom weightedhistograms.

Eachentryin thedimuonicinvariantmasshistogramsof theSS2data(suchasFigure4.8)

wasweightedwith theproductof themomentum-dependentMisID probabilitiesfor each

of the two decayproducts.Eventswith two muonswereeliminatedfrom the samplebut

eventswith one muon were allowed in order to include semimuonicbackground. The

weightwascalculatedasfollows: Ü ��Ý Þ�ß}Ý4à (4.10)

where Ýi�á£ ¿À for muons,and Ýi� Îãâ?äeå&æfor non-muons.Then,wehave that

çèÜ � Îãâ?äeå&æ £éß Î�âêäeåÕæ ¥ if neitherdecayproductis identifiedasamuon.çèÜ � Îãâ?äeå&æif oneof thetwo decayproductsis identifiedasamuon.

Resultsfrom this procedurearepresentedin Chapter5. A valuefor thebackground

ratio ( � ) wasobtainedfor eachof thesetsof cutsconsidered.

4.8 Optimization of the SelectionCriteria

The optimization procedureof our selectioncriteria was the longest and most

tiresomephaseof this effort. It establishedthe final criteria for the selectionof muon

candidateevents. The aspectthat wasconstantlykept in mind was to eliminatea large

numberof backgroundeventsmaintaininghighefficiency values.


For this procedurewe startedwith the applicationof fixed (constant)cuts which

correspondto the cuts applied in the selection processincluding Skim4. Due to

correlations,the optimizationof the cuts requireda procedurewhich considereda large

numberof cut sets.This implied a selectionof the appropiatecut rangesfor eachoneof

thesevariables.Onceeachcut rangehadbeendeterminedwe proceededto optimizeour

selectioncriteriawith the ideaof obtainingthebestsetof cutsto beappliedin our search

of dimuonicevents.

4.8.1 Fixed Cuts and Variable Cut Ranges

In additionto the cutsappliedin the earlierstages,in this analysisthe Skim4 cuts

wereusedasfixedor constantrequirements.Table4.5presentsthefixedcutsappliedto the

invariantmassvariablewhichdefinethesidebandandsignalregions.

Table4.5: DimuonicInvariantMassCutsfor SidebandsandSignalRegions.

Variable CutName Applied

Sidebands ë7ì-ímî_í�ïñðóò òôïQë$ìöõ7÷¶ë (left)ë7ìöø$ù'øôïñðóò òôïQë$ìöø$ø$î (right)SignalArea ë7ìöõ$î'îôïñðóò òôïQë$ìöõ$øní

Therangeof thevariablecutsincluding thevariablesusedfor hadronidentification

were chosenon the basisof the experiencewith previous FOCUSanalysis. Theseare

presentedin Table4.6.


Table4.6: Cut valuesfor eachvariableconsideredin theoptimizationprocess.

VariableName Cut Applied� Vertex IDÒ0��½ úñûµü ínü*øCLS úý÷�ìö÷¶ë7ü�÷¶ìG÷'ûµü4÷¶ì�ëISOP ïþëÿ÷ �� ü&ë&÷ � � üÕëÿ÷ � �ISOS ïþëÿ÷ �� ü&ë&÷ � � üÕëÿ÷ � �½�� ÷�ü*ù� Muon IDK/ � ïñûµü]õMISSPL ïþë7ü]ùµü*îIMUCL úý÷�ìö÷¶ë7ü�÷¶ìG÷'îµü4÷¶ìG÷'û� MesonID� � � K � � ë7ü� ¶ü í�� ë7ü � îµü � û

4.8.2 Optimization Basedon Sensitivity

All possiblecut setsthat could be formed from the cut valuesin Table 4.6 were

considered.The total numberof cut setswas26244. The sensitivity wascalculatedfor

eachcut setaspresentedin eq.(4.9).Theoptimumcut setwasdefinedastheonewith the

lowestsensitivity.

Chapter 5

Resultsand Conclusions

5.1 MisID

The measurementof the percentageof pions that were misidentified as muons

was carried out using a large sampleof K � « � � � � � decaysusing Type-9 K � « data

events.Resultsfrom this studyarepresentedin Figure5.1 for which thefollowing muon

identificationcutswereapplied:

ç InnerMuonCL �¦£ ¡ .ç MISSPL

Í ¥ .Wecanseethatfor low momentumtheprobabilityof pionmisidentificationis greater

thanfor high momentumfor realdataaswell asfor MonteCarlosimulateddata. This is

81

�5.2 BackgroundRatio 82

Figure5.1: MisID asa functionof momentum.

dueto the fact thatseveraleffectsthatcontribute to pion MisID, suchaspion decaysand

multiple Coulombscattering,becomelargerat low momentum.Theresultsfrom dataand

MonteCarloareconsistentwithin errors.

5.2 Background Ratio

Figure5.2showsa typicalweightedinvariantmassdistribution for backgroundratio

of a specificsetof cutsselectionusingGlobalVertex sample.Thebackgroundratio � was

�5.3 SkimmingResults 83

calculatedby integratingthis histogramnumericallyin thesidebandsaswell asthesignal

region.

Figure 5.2: Typical WeightedInvariant Mass Distribution Used in CalculatingtheBackgroundRatio.

5.3 Skimming Results

Invariantmassdistributionswith Skim3selectioncriteriaareshown in Figure5.3for

dimuonicaswell asnormalizingsamples.


Figure5.3: InvariantMassDistributionsfor Skim3Cuts.(a)Dimuonrealdatasample.(b) D �P�Ú� � � � Monte Carlo sample. (c) Normalization real data sample. (d)NormalizationMonteCarlosample.Thesignalregionhasbeenmaskedfor thedimuonrealdata.Therealdatasampleshave largebackgroundlevelswith theseloosecuts.

Following theblind analysismethodology, only sidebandeventsarepresentedfor the

dimuonrealdata. For the realdatadistributionswe canseea higherlevel of background

(Figure 5.3.a and Figure 5.3.c). Figure 5.4 shows the samedistributions displayedin


Figure5.3 with thedifferencethat thesedistributionscorrespondto eventsselectedusing

Skim4criteria.

Figure5.4: InvariantMassDistributionsfor Skim4Cuts. A considerablereductioninthelevel of backgroundis achievedwith theapplicationof thesestrongercuts.

Although the backgroundin the dimuonreal data(Figure5.4.a)hasbeenreduced

considerablywith the Skim4 cuts, it is necessaryto reduceit even further to obtain the

�5.4 CutOptimizationResults 86

bestmeasurement.Thedeterminationof theoptimumcut setto achieve this reductionis

discussedin thenext Section.

5.4 Cut Optimization Results

Using the cut rangesshown in Table4.6, the valuesof � that hadbeenpreviously

determinedand the Rolke and Lopezroutines,the sensitivity was determinedfor each

of the 26244differentcombinationsof cuts. The lowestsensitivity wasobtainedfor the

following cut set.

Table5.1: BestCut Combination.�� 7.0��! #" $1%��! %� $1%& �'� �1%�)(+* ,1- ��!�."�� $3�/- 0 & � �

1% #- 0 & � �1%1 ��2 $8.03 4 576 1 8 91.06;:=<?> 9-5.0


Figure5.5: Massdistributionsfor thebestcut combination.

For these cuts the dimuon mass distributions for the sidebandsdata and the

D � �o� � � � Monte Carlo arepresentedin Figure5.5.aandFigure5.5.b. Therearefive

eventsin thesidebands.Theefficiency for detectingD � �o� � � � is 2.8%.

The distributionsfor the normalizingmodeareFigure5.5.candFigure5.5.d. For

therealdatatheyield correspondsto thenumberof eventsin theGaussiancurve. For the


MonteCarlogenerateddata,theefficiency A¯�± is calculatedasthenumberof eventsin the

Gaussiancurvedividedby thetotal numberof generatedevents.

Our reconstructionprocessfollowedby theanalysisalgorithmlet usobserve135432

D � � K �7� � eventswith a detectionefficiency of 3.9%.Table5.2summarizestheresults

for the bestcut combination. For the backgoundratio study the resultsusing the final

selectioncriteriaareshown in Figure5.6.

Table5.2: Resultsof Cut Optimization.

@BADCFEHGJILKJM!N!O PQASR�TUK 5WV 85.72"XI�YZO!T[C\R]YZO @BA^C_EHGJI�KJM!N!O`T[N �aTUGJNbADc0.9�dYZN!e]TfR�ThgiThRkj l`M!m noYZIXp[l`qar2.96

D ÷ts 2vuw2 � Yyx CzTUYZN!C\j 5?{ 82.8

D ÷ s K� 6 u|Yyx C}TUY~N!C�j 5�{ 8

3.9�dYZN!e]TfR�ThgiThRkj 5}��Z� �d� 81.1

Using the Rolke andLopezalgorithm, the sensitivity numberis determinedto be¥ ¿Ï� Á . Thentheexperimentalsensitivity is calculatedas¨ � D � � � � � � � � ¥ ¿À� Á£ Ì É Ô Ì ¥Ì�¿Ï�� £ � à¥ ¿ÏÈ�� £ � à ÌÕ¿ÀÈ�È�� £ � à �§£ ¿ £4É � £ �� (5.1)

This result is encouragingsinceit is muchlower thanthe existing 90% confidence

limit of

Ôm¿ £ � £ �� . This meansthatour measurementwill bemoreprecisethanprevious

measurements.

�5.5 ConfidenceLimits 89

Figure 5.6: Invariant Mass Distribution for BackgroundRatio using the best cutcombination.

5.5 ConfidenceLimits

Onceour selectioncriteriawereoptimized,we proceededto openthemaskedarea.

Applying thebestcut combinationto dimuoniceventswithin our signalareawe have two

eventsin that region. Thus,we have five eventsin sidebandsarea( �i� É ) andtwo events

in thesignalarea( � �¦¥ ).UsingtheRolke andLopezroutines,for � �§É ¿ÀÊ ¥ , thebackgroundrate( �j�¦�m�4� ) is �¿Ï�

and,the90%confidencelevel lower limit is 0 andthe90%confidencelevel upperlimit

�5.5 ConfidenceLimits 90

Figure5.7: Eventcandidatesfor theD �Ó�Ú� � � � process.Figuredisplaytheonly twoeventswithin thesignalareawhichsurvivestheoptimumselectioncriteriain thestudyof theraredecayD �Ó�Ú� � � � .

is 5.34.The90%upperconfidencelimit for thebranchingratio is calculatedas

³p´ � D � �o� � � � � � É ¿ÏÌ]Ô£ Ì É Ô�Ì ¥ÌÕ¿À�� £ � à¥ ¿ÀÈ�� £ � à Ì�¿ÏÈ�È�� £ � à �¦¥ ¿ £ � £ �� (5.2)

�5.6 Conclusions 91

No definitive evidenceof the rare decayD � �o� � � � was found in this analysis.

However, ourwork allowsusto seta lower90%confidencelevel upperlimit for thisdecay.

5.6 Conclusions

Monte Carlo simulation of the level of meson misidentification in FOCUS

was confirmedby experimentalmeasurementsusing real data. The misidentification

measurementswereusedas the basisof a methodto determinethe ratio of background

in theinvariantmasssidebandsversusbackgroundin thesignalregion. This methodused

realdataandavoidedsystematicuncertaintiesin theuseof simulation.

It waspossibleto reducethe backgroundin this analysissignificantly usingwell-

known selectionvariablesbasedon the characteristicsof the signal being sought. The

blind analysismethodologyprovided an excellentway of optimizing the event selection

criteriawith aminimumof bias.

Themainconclusionobtainedfrom thiswork wasthereductionof theupperlimit in

thebranchingratioatahalf of thevaluepreviouslypublishedby PDG.No evidencefor the

raredecayD � � � � � � wasfound.The90%CL branchingratio limit was ¥ ¿ £ � £ �� .

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