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P.R.B. Sauli

Department of Physics

McGill University, Montreal

July 1991

A thesis submitted 1.0 the Faculty of Graduate Studies and ResearcJ. ln partial fulfillment of the requirements for a

Masters degree in Physics

@P.R.B. Saull19~l

Abstract

Data taken with the ARGUS detector at DESY, Hamburg, specifically BB pairs pro

duced from e+ e- collisions at the energy of the T( 4S) resonance, are used to investigate

the dccay channel B- -. pot-v. Observation of a signal would be conclusive evidence

that the CI" 1\1 nldtl'ix element Vub is non-zero, a necessary condition for the validity of

the KobaYlt.shi-Maskawa cxplanation for CP violation. The recoillllass technique is em

plo)ed to try to isolate signal events. Monte Carlo data are used to lTlodel the signal

background, which is ,10minated by reasOl.ably weIl undcrstood b -. c decays. Using the

rnodel of Wirbel, Stech, and Bauer, a model-dependp[l/J upper limit of 1.6 x 10-2 is placed

Oll the value of 1\I;,bl at 90% confidence. The results, however, suggest that further study

of the b -. c background is warranted. In particular, B meson tra:1Sitions to states with

highel' mass thall the DO may contribute significantly.

Abstrait

Des données collectées a l'aide du detecteur ARGUS situé à DESY, Hambourg, spéci

fiquement des mésons BB produits lors de collisions e+e- à l'énergie de resonance T4S,

sont utilil>ées pour mettre en evidence la transition B- ~ poe-v. L'observation d'un

signal suffirait à prouver que l'élément CKM Vub est non-nul, une conditIOn nécessaire à

la \'alidité de l'explication de la violation CP proposée par Kobayashi et Maskawa. Une

technique de masse de recul e~t employée pour isoler le signal recherché. Des donnés

Monte Carlo modèlcnt la forme du bruit de fond qui est dominé par les transitions b -. c

bien conoues. En utilisant le modèle de Wirbel, Stech, et Bauer, une limit supérieure de

1.6 x 10-2 a été calculée pour la valeur de lVubl avec une confiance de 90%. Cependant,

les résultats indiquent que des études plus approfondies des transitions b -t c doivent

êtr(' effectuées, En particulier, les transitit'ns des mésons B à des états ayants une masse

supérieure au D- peuvent contribuer de façon significative.

Il

Acknowledgments

1 thank Dr. David B. MacFarlane for providing me with a tremendous opportunity

to learn (and an ambitious project to boot) and for being more thct.n patiel't with me on

numerous occasions. It has been a pleasure and an honour to have bren his first, and

perhaps most difficult to graduate, Masters student. 1 also acknowledge his keen insight

and useful comments.

Several other people have played important roles in the development of my thesis. A

few things need to be said. To Paul Mercure, who made an often unsympathetic computer

system a little less formidable, you can have your $*&!%!*@ disk space back now; to Ken

McLean, who always seemed to derive sorne kind of sick pleasure out of debugging my

programs, l'lI never try to match you drink for drink again; to George Tsipolitis, who

had an annoying habit of looking over my shoulder ail the time, but who more tha.n made

up for it when he lost to me in squash, 1 hope you enjoy those last commentsj and to

Dave Gilkinson who is forever roaming the physics building, quatsch!. A special thanks

also goes to each of Kevin, Edwin, Catherine, Katerina, Themis, aU those who dared

to play soecer witb me, and, of course, Wendy, for making the time Pass a little more

enjoyably than it should have. Finally, 1 would like to thank my parents and sisters for

their encouragement and kind support over the years, and for the oceasionjJl handout.

And to aU those who kept bugging me about my thesis, now that it's been completed,

VOU HAD BETTER READ THE DAMN THING!

1 regret nothing, not even the awfully long time it bas taken me to produce an awfully

dull result. The time has passed enjoyably and l've learned a great deal. 1 fear 1 could

have done much more; but as 1 may have already set a new standard in the longevity of

the Masters student, 1 bid thee aU farewell, and suggest we reconvene at the nearest pub.

Contents

Abstract

A cknowledgements

Table of Contents

List oC Figures

List of Tables

Personal Contributions

Foreword

1 The Standard Model

1.1 Introduction .....

1.2 Qnn and the Strong Force ..

1.3 Multiplet Structure ..

1.4 Resonances . . . . .

1.5 Development of the Standard Model .

1.6 The Weak Lagrangian ........ .

1. 7 The CKM Matrix and CP Violation ..

1.8 e+e- Collisions .... . . . . . . ...

2 B Meson Physics

2.1 The Upsilon Resonances

2.2 Semileptonic B Meson Physics .

iii

ii

iii

vii

xi

xii

xiv

1

1

4

6

7

8

9

12

. . . . . 14

16

16

20

..

CONTENTS

2.3 Free Quark Decay . . . . . . . . . . . . . . . . .

2.4 Models of Exclusive Semileptonic B Decay ..

2.4.1 Introduction................

2.4.2 The Model of Wirbel, Stech and Bauer

2.4.3 The Model of Korner and Schuler . . .

2.4.4 The Model of Isgur, Scora, Grinstein, a.nd Wise

2.4.5 Lepton Spectra, Phase Space, and Decay Rates

3 The Detector

3.1 Introduction .......... .

3.2 The Accelerator and Bearn .. .

3.3 Location of the Detector a..ld Harnessing of the Bearn .

3.4 The Mi:.gnetic Field .

3.5 The Vertex Chamber

3.6 The Main Drift Charnber ....

3.7 The Time-of-Flight System (TOF) ..

3.8 The Electromagnetic Calorimeter

3.9 The Muon Chambers

3.10 The A RG US Trigger

3.10.1 The Fast Pretrigger .. .

3.10.2 The Slow Trigger .... .

3.11 On-line Da.ta Acquisition ... .

3.12 Data Analysis .......... .

3.13 Luminosity Measurement ... .

3.14 Charged Particle Identification. .

3.14.1 Likelihoods for General Particle Identification

3.14.2 Likelihoods for Lepton Identification ., ....

4 Analysis

4.1 Introduction......

4.2 Experimental Data . . . . . . . .

4.3 Monte Carlo Data. . . .

IV

22

27

27

31

32

33

35

38

38

40

41

41

42

43

46

48

51

51

52

53

53

55

.17

58

58

59

61

61

61

64

CONTEN'J'S

4.3.1 modelling b·-+ c BB Decays

4.3.2 modelling 0 -+ U Events ..

4.4 Multihadron Event Selection . . . . .

4.5 Fast Lepton Selection ........ .

4.5.1 Lepton Momentum Cuts . . .

4.6 Continuum Suppression

4.6.1 Thrust Axis Cut

4.6.2 Momentum Cuts

4.7 Signal Enhancement ... . . . . . •

4.7.1 The Recoil Mass Technique . . . . .

4.8 modelling the Background

4.9 The Xl Function .......... .

4.9.1 Minimization of the X2 Function ..

4.10 Observations and Interpretation of Results

4.11 Systematic Error Analysis ......... .

4.12 Upper Limits on BR(B -+ p°tv) and IVubl 4.13 Recommendations and Conclusions

Appendi)(

A The Development of the Standard Model

A.1 Experimental Constraints

A.2 Theoretical Constraints ..

A.3 Gauge Theories. . . . . . . ....

A.4 Spontaneous Symmetry 8reaking and The Higgs Mecbanism

B The Least Squares Principle

B.l General . . . . . . . . . . .

B.2 The Xl Function Used in Cbapter 4

B.3 Calculation of Error Bars for Fits

C MINUIT

v

64

66

68

72

72

73

73

77

81

82

94

97

98

· .. 109

· .. 112

· .. 113

· . 114

116

· .116

· .. 111

· .118

· .. 120

122

· . 122

· .125

· .126

128

CONTENTS vi

D A Proposed Method for Increasing Sensitivity 130

• Bibliography 137

..

..

List of Figures

0.1 The elusive charmless semileptonic decay mode, B- -+ pOL-v. . ..•... xiv

1.1 Multiplet structure exhibited by the lowest lying spin-O and spin-l mesons. 7

2.1 Muon pair mass distribution as measured at FNAL [24]. 17

2.2 The dominant T(IS) and T(4S) decay channels. . . . . . 18

2.3 Cross-section Cor e+e- -+ hadrons showing the T resonances [24]. 19

2.4 Lepton spectrum from pure free qu~.rk decay moJel. . . . . . . . . 23

2.5 Leading-order bremsstrahlung (a) and gluon exchange (b) in free quark

decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24

2.6 Invariant mass distribution of the hadronic system produced by the Altarelli

model for different values of the Fermi parameter. (Hagiwartha, Martin,

and Wade, Nud. Phys. 8327 (1989) 570). ..... . . . . . . . . . . . .. 26

2.7 Feynman diagram for quark decay showing how pole dominance can dictate

Corm Cactor dependences. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29

2.8 Lepton momentum distributions for B -+ ptii events as generated by

MOPEK for the three Corm factor models. . . . . . . . . . . . . . . . . .. 35

2.9 Meson momeutum distributions for B -+ pt;; events as generated by MOPEK

Cor the three Corm Cactor models. .. . . . . . . . . . . . . . . . . . . . .. 36

2.10 Phase space available Cor B -+ pw transitions. The phase space is actually

somewhat larger owing to the width of the resonance. . . . . . . . . . 37

3.1 The ARGUS detector. Section shown is parallel to the beam axis. 1:

muon chambers; 2: shower counters; 3: TOF counters; 4: dljft chamber;

5: vertex chamber; 6: iron yùkej 7: solenoid coils; 8: compensation coils;

9: mini-,8-quadrupoles. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39

vii

LIST OF FIGURES

3.2 The DESY synchrotron and DORIS II storage ring. . . . . . .

3.3 Specifie energy loss for a multihadron sam pIe of 10 000 events.

viii

40

45

3.4 Mass squared versus momentum as derived from the ARGUS TOF system. 41

4.1 Centre-of-mass energies for available data. . . . . . . . . . . . . . . . . .. 63

4.2 Comparison of the B+ -+ DO, D·°tv lepton spectra for the three Corm factor

models discussed in the text. The D and D- contributions are present in

the ratios predicted by the different models. ............... .. 65

4.3 Pion and non-pion (hatched) contributions to the photon spectrum as gen

erated by MOPEK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 70

4.4 Normalized distributions of eharged and total multiplieity for BB decays

as generated by MOPEK-MINIMC. 71

4.5 QG""" for T(4S) data. . . . . . . . . 74

4.6 Invariant mass plot for e+e- candidates when one candidate is a sclcdcd

fast lepton. ................................... 75

4.7 Angle between thrust axis and electrons (Pt> 1 CeV / c ) for continuum data. 76

4.8 Lepton momentum spectrum from T( 4S) data. . . . 78

4.9 Levton momentum spectrum from continuum data. 78

4.10 Lepton momentum spectrum from Monte Carlo BB data. 79

4.11 Lepton momentum spectrum from Monte Carlo b -+ U decays. 79

4.12 Lepton spectra for Continuum and 1"(4S) decays befOTe application of the

thrust cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80

4.13 Invariant mass distribution for 1!'+1f- combinations, as selccted from T(4S)

decays to Cast leptons, having momentum greater than 800GeV je. . . . .. 83

4.14 Momentum spectrum for pO candidates from MOPEK-SIMARG BIJ de-

eays (solid histogram) and continuum (error bars) as comparcd to theory

(arbitrary normalization). . . . . . . . . . . . . . . . . . . . . . . . . . .. 84

4.15 Monte Carlo recoil mass signal generated according to the ISCW modcl for

a single decaying B on the T( 4S). The tai) on the electron distribution Îs

due to bremsstrahlung in the detector. .................. .. 85

LIST OF FIGURES

4.16 Comparison of the signal width for motionless and boosted B mesons,

demonstrating the dominance of the unknown B momentum on the signal

IX

width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86

4.17 Monte Carlo recoil mass plot Cor full BB events containing one B -+ porv. 87

4.18 Recoil mass spectrum Crom T(48) data. . . . 87

4.19 Recoil mass spectrum Crom continuum data.

4.20 Recoil mass spectrum Crom Monte Carlo B B data ..

4.21 Direct T( 48) recoil mass spectrum. . . . . . . . . .

4.22 Monte Carlo recoil mass spectrum for RB decays showing contributions

88

88

89

from real and fake pO candidates. ..... . . . . . . . . . . . . 90

4.23 Angular distribution oC pO-lepton pairs(arbitrary normalization). 91

4.24 Recoil mass spectrum from T(48) data aCter application of pO - f angle cut. 93

4.25 Recoil mass spectrum from continuum data after application of pO -l angle

eut.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93

4.26 Recoil mass spectrum Crom Monte Carlo BB data after application of pO_f

angle cu t. ....................... . . . . . . . . . . . .. 95

4.27 Recoil mass spectrum Crom signal data aCter application oC pO - f angle cut. 95

4.28 Fit oC the momentum spectrum for electrons with Pt > 1.6GeV le. The

signal is modelled using the ISGW model. .............. 103

4.29 Fit of recoil mass spectrum for electrons with Pt > 1.6GeV le. The signal

is modelled using the ISG W model. . . . . . . . . . . . . . . . . . . . . . . 103

4.30 Fit of the momentum spectrum Cor muons with Pt > 1.6GeV le. The signal

is modelled using the ISGW model. ...................... 104

4.31 Fit oC recoil mass spectrum for muons with Pt > 1.6GeV le. The signal is

modelled using the ISGW mode!. . . . . . . . . . . . . . . . . . . . 104

4.32 Fit of the momentum spectrum for electrons with Pt > 1.8GeV le. The

signal is modelled using the WSB mode!. ................... 105

4.33 Fit oC recoil mass spectrum for electrons with Pt > 1.8GeV 1 e. The signal

is modelled using tbe WSB model. ...................... 105

4.34 Fit of the momentum spectrum for muons with Pt > 1.8GeV le. The signal

is modelled using the WSB model. ...................... 106

i

'!..*

LIST OF FIGURES x

4.35 Fit of reeoil mass spectrum for muons with Pt > 1.8GcV je. The signal is

modelled using the WSB model. . . . . . . . . . . . . . . . . . . . . . . . . 106

4.36 Fit of the momentum speetrum for electrons with Pt > 2.ÙGeV le. The

signal is modelled using the KS model. . . . . . . . . . . . . . . . . . . . . 107

4.37 Fit of reeoil mass spectrum for electrons with Pt > 2.0GeV le. The signal

is modelled using the KS model. . . . . . . . . . . . . . . . . . . . . . . . . 107

4.38 Fit of the momentum speetrum for muons with Pt > 2.0GeV le. The signal

is modelled using the KS model. . . . . . . . . . . . , . . . . . . . . . . . . 108

4.39 Fit of reeoil mass spectrum for muons with Pt > 2.0GeV je. The signal is

modelled using the KS model. . . . . . . . . . . . . . . . . . . . . . . . 108

0.1 Plot of eos 0 0 for single B signal events (MOPEK-MINIMC) .... · . 131

0.2 Plot of cos 0 0 for real signal events (WSB model). · . 133

0.3 Plot of cos 0 0 for Monte Carlo B B events. · . 134

0.4 Plot of cos 0 0 for T ( 4S) events. .. · . 134

0.5 Plot of cos 0 0 for continuum events. · . 135

0.6 Result of fit for electruD data. .. · . 135

0.7 Result of fit for muon data. ... · . 136

List of Tables

1.1 Fundamental partides of the standard model. . . . . . . . . . . . . . . .. 2

2.1 Upper limits on sorne charmless hadronic decay modes of B rnesons as

measured at ARGUS [44]. ...• . . . . . . . . . . . . . . . . . . . • . .. 21

2.2 Theoretical and experimental branching ratios (or B --+ p°til. The numbers

in parenthcses are computed (rom the theoretical rates using the values

provided in the texte . . . . . . • . . . . . . . . . . . . . . • . . . . . . .. 36

4.1 Statistics for ARGUS experimental data.

4.2 Statistics for the multiha.dron dataset. .

4.3 Continuum scaling factors as derived from the numbers of leptons beyond

the kinematic limit for B deca.y: compare to the luminosity ratio of r =

2.30 ± .05 ............. .

4.4 List of cuts made on the data. . .

4.5 Results from simultaneous fits of the electron momentum and recoil mass

62

71

81

92

spectra. El'rors are parabolic. . ........................ 100

4.6 Results from simultaneous fits of the muon momentum and recoil mass

spectra. Errors are parabolic. . ........................ 101

4.7 Upper limits on the CKM matrix element Vub from fitting the recoil mass

and lepton rnomenturn spectra. . . . . . . . . . . . . . . . . . . . . . . . . 113

0.1 Results frorn fit of cos ao distributions. Model used for signal is that of

Wirbel, Stech, and Bauer. Errors are parabolic ................ 133

xi

LIST OF TABLES XII

Personal Contributions

My contribution to the ARGUS experiment bas been smalt. The entire analysilJ was

earried out in the Department of Physics at McGill University, Montreal, and there was

no opportunity to participate in the goings-on at DESY, Ha.mburg. As a Masters st'j

dent, 1 was not required to fulfil any duties as a member of the ARGUS collaboration.

Nevertheless, 1 have had the opportunity to make myseIr useful in a Cew ways.

My major responsibility was to keep the ARGUS experimental dataset at McGill up

to date. This involved the translation of the minidst EXMUHA tapes {or expcrirncnts

6 and 7 from IBM ta VAX format, allowing the tapes to be read with a VAX 1l/785

and several other VAX stations loeated at MeGiIl. The updated RUNFILE for thcsc ne\\'

experiments had also ta be translated. 1 was responsible for installing and modiCying SOIl1('

ARGUS software, most notably MOPEK, MINIMC, and LUND. ln particular, 1 Cound an

important bug in a heavily used LUND subroutine which had been modified to gencrat(~

inclusive B decays (ACCMM model). The problem turned out ta be a consequence oC

the physies involved, and 1 spent many months trying to rewrite the routine so that

the program would generate reasonably I4physical" events. This turned out to be a rather

diffieult task, howeverj a proper approach to the problem would have required a significant

modification of the ACCMM model itself. As a consequence, the modifications 1 proposed

were only partly satisfactory and not incorporated into the ARGUS software for gcncral

use.

The topie for my thesis and the method used {or the analysis were both the brainstorm

of my thesis adviser, Dr. David B. MacFarlane. 1 managed to contribute here and thcrc,

but after two and a haU years of work, it is difficult to isolate the exact role 1 played in the

development of the analysis. 1 made my own data selection, and gencrated ail Monte Carlo

data, with the exception of the SIMARG dataset which was sent ovcr frorn Germany. The

entire analysis was carried out by me, and any computer programs beyond the standard

ARGUS software were writt.en by me. The interpretation o{ tbe rcsults, the conclusions,

and the suggestions made are aIl my own, but obviously 1 have been iufiuenced by many

observations made by Dr. MacFarlane. In addition, 1 bad sorne influence on the final

LIST OF TABLES xiii

form of the X2 function used for the fits, although 1 cannot take any credit for its general

form. 1 consider my only truly original contribution to be the discovery of a potentially

useful method for improving sensitivity to the signal transition, described in Appendix D.

To the best of my knowledge, this method has not been suggested heretofore.

..

LIST OF TABLES xiv

Figure 0.1: The elusive charmless semileptonic decay mode, B- -+ pOl-v.

Foreword

This thesis deals with one particular rare semileptonic charmless B dccay channel

(Fig. 0.1), the transition B- -+ pOl-v 1. Only transitions to electrons and muons are

considered. Tau decays in the ARGUS detedor always involve the tau neutrino and this

severely complicates the analysis. Electrons and muons on the other hand arc identified

with high efliciency. The pO decays into two charged pions (~ 100% branching ratio)

almost instantly alter it is produced. These remain undecayed in the detcctor, and are

used to reconstruct the pO. The neutrino goes undetected but its presence can be inferred

Crom the Cour-momentum oC the other decay products and the B meson.

The B- -+ pot-v transition is extremely important to study because it can only

proceed if the Cabbibo-Kobayashi-Maskawa matrix element Vvi> :s non-zero. This clement

quantiCys the strength of the coupling between the b and the u quark. A nOil-1.cro valu~

is a necessary condition for the validity of the Kobayashi-Maskawa explanation for CP

violation [1,2]. When the author first embarked on his search for the dccay mode in 1988,

1 Unless stated otherwise, for aU decay schemes referred to in this thes;s, the charge conjugate bansition is implied as weil.

1 ...

LIST OF TABLES xv

there was no evidence that V ... wu non-zero. Since then, both CLEO [3] and ARGUS [4]

have measured a reasonably strong signal in the B meson lepton momentum spectrum

in the region kinematically Corbidden to charmed B decays, and ARGUS has managed

to reconstruct fully some of the decays responsible for this signal [5]. Extraction of V",

from this signal is severely model-dependtat, however, and a signal in an exclusive decay

channel such as B- -t (pO,w)t-ji is preferable. Unfortunately, these transitions are rare

and are plagued by enormous backgrounds. For example, the large width of the pO and

heavy pion production inherent in aIl high energy interactions make it difficult to isolate

the resonance. In addition, the directions along which B mesons are produced cannot be

resolved by the ARGUS detector. This incomplete knowledge of their moment a translates

into a severe broadening of the signal width which is currently unavoidable. The present

experimental upper limit for the B- --+ pot-Ti branching ratio is about 0.1% [6], but all

thcoretical predictions set the rate at at least a Cactor of three lower.

Several previous attempts have been made to observe the signal at ARGUS, to no

avail 2. In the present analysis, a new method is used that exploits the fact that the

domi na,nt b ..... c backgrounds cao be reasonably well modelled with Monte Carlo data.

2See for example Ref. [6J.

l

Chapter 1

The Standard Mode}

1.1 Introduction

Over the p~t several decades, physicists have sought to account for an tbe properties of

subatomic particles and properly de~cribe the interactions amongst thcm. An extensive

proliferation of particles, first observed in the 50's and 60's, has been shown to display

many symmetries and interrelations that indicate a simpler underlying structure. Severa)

models have been applied over the years with varying degrees of success in the attcmpt ta

describe what is observed. Eventually, the now widely-accepted standard model emergt'd

as a reliable, testable, and highly predictive theory. As experimental physics maves inta

higher-and-higher energy regions, new opportunities arise to test the standard mode!. In

particular, the discovery of the T resonances has enabled not only further testing of this

model, but, for the first time, the determination oC a few of its unfixed p .... rametcrs ln

fact, the simple existence of the T resonances was support for the standard model in its

own right. To date, all experimental studies of these resonances confirm the validity of

the model. In light of this, the theory presented here will be restricted to that of the

standa.rd model.

Tbe four forces observed in nature are gravit y, electromagnetism, the strong force and

the weak force (TaMe 1.1). At the energies a.vailable to experimental physicists today,

the gravitational force is much too insignificant to play an importd.ut role in particle

interactions; therefore, the standard model seeks to explain only the interactions due to

the last three forces. The incorporation of gravit y and tht! unification of the four forces

at much higher energies remains a theoretical objective and is left to the rcalm of grand

1

CHAPTER 1. THE STANDARD MODEL

Fermions Il Mass (Ge\'/c"l) 1 Charge 1 Quark Il Lepton 1 Charge 1 ~1ass (GeV/è)

First Genera.tion 0.311 0.008" -1/3 d e -1 0.0005 0.3 0.004 2/3 u Ile 0 < 1.8 X 10-8

Second Generation 0.5 0.15 -1/3 ~ p -1 0.106 1.5 1.2 2/3 c Il,, 0 < 2.5 X 10-4

Thlrd Generation 5 4.7 -1/3 b T -1 1.784

> 46 > 46 2/3 t liT 0 < 3.5 X 10-2

Il (Current masses) Il (Constituent Masses)

L Bosons Force Mediator Charge Mass (GeVIc?) Spin

Gravitational G 0 0 2 Electromagnetic "Y 0 < 3 X 10-36 1

Weak W± ±1 81 1 Weak zo 0 91 1

Strong 91· .. 98 0 0 1

[ o > 24 Ion

Table 1.1: Funda.mental particles of the standard mode\.

2

CHAPTER 1. THE STANDARD MODEL 3

unified theories (GUT's). Tbe electromagnetic and weak forces have already becn united

in the eledroweak foree and therefore the standard model is more correctly considercd a

mathematical description of the electroweak and strong forces.

There are severa} partic1es involved in the standard model. Leptons are particles that

experience the weak (oree but not the strong. They include the electroD, muon, and tau

and their associated neutrinos. The first three have both mass and charge and th('refore

rnay participate in electromagnetic interactions. The neutrinos have neither mass 1 nor

chu&e and hence intera.ct only weakly with other particlesj they a.re therefore etTcctivcly

impossible to detect in practical collider detectors. Although the tau neutrino has not becll

directly ohserved, its existence can be irferred from the large missing momentuni obscrved

in tau decays. Also in this model are six massive quarks, the fundamental building

blocks of aU mesons and baryons. They experience the strong, weak, and electromagndi<

interactions and have Cractional charges: 2/3 for the u, r. and t quarks and - 1/3 for the

d, sand b quarks. In order to overcome a difficulty in the theory, quarks have hecn

endowed with an additional quantum number called colour which is rcsponsible for the

strong interaction. Although the top quark has not yel beeu sccn. thcrc is ('vidcnC{' for

its existence Crom the absence ot' Bavour-changing neutraJ currents and the ohsf'rvatioll of

BB mixing at ARGUS (7) and CLEO [SI. The standard model requîrcs that the top quark

exist, and it is thererore included here as one of the fundamental particles of nature.

Ail the above particles are spin 1/2 fermions, and must thercfore respect the Pauli

exclusion principle which states that no two identical fermions can OCCupY thc same space

time point. It is thereCore natural to associate them with matter [9]. Indecd, the basic

building blocks of matter, electrons, protons, and neutrons, are ail Cermions. Bosons,

on the other band, nccd not satisfy the exclusion principle, and create fields with which

the fermions interact. Interactions between 'particles th us occur through the exchange

of virtual (off mass shell) spin l gauge bosons which arise in the tbcory quite naturally

from tbe principle of gauge invariance (Appcndix A). The massless, uncharged photon is

the mediator oC the electromagnetic Corce whiIe the very heavy chargcd W's and neutral

Z mediate the weak interaction. Eight massJess gluons carrying colour charge but not

'The possibility that neutrinos are endowed with a tin y mass bas not been ruled out lIowever. the standard model in it.c present Corm requires that this mass be zero. See Appendix A.

CHAPTER 1. THE STANDARD MODEL

gravitational force occurs through the exchange of musless spin 2 gravitons.

Tbe photon is the only gauge boson that is easily observable in nature. This is a

consequence of the differing strengths of the four forces, and the uncertainty principle.

Ali particles that couple to a given gauge boson couple with a universal strengtb given by

a dimensionless coupling constant. The strength of the force is therefore dependent upon

this coupling constant. Gravit y, the weakest force, bas a. very smaJl coupling constant

and a single graviton bu a low probability of interacting witb matter and being detected.

AJthough the unified electroweak force (the next strongest force) has one coupling COD

stant, it manifests itself as two forces that appear very difl'erent in strengtb. This is a

result of the heavy mass of the W's and Z relative to the masses of the particles to which

they couple. The gauge bosons must therefore be produced in a virtual state and can

only exist for a very short time as dictated by the uncertainty principle. As a result, the

weak force must be of very short range in contrast to the electromagnetic force which is of

infinite range due to the massless photon. Massless gluons could conceivably travel long

distances but they carry the colour charge; the principle of confinement, to be discussed

in the next section, only permits colourless objects to exist in nature and so gluons are

never directly seen. Thus, the photon, which cao travellong distances and interact witb

matter through the fairly strong electromagnetic force, is the only boson readily visible

at natura! energies.

In addition to the above fields, the standard model requires the existence of the neutra'

Higgs scalar field in order to account for the masses of the particles (Appendix A). As

yet, there is no experimental evidt!Dce for the Higgs field.

1.2 QCD and the Strong Force

The strong force is responsible {or binding quarks into hadronic states of mesons and

baryons. The valence quark configuration of the mesons is of the form qq while that of the

baryons is qqq. (The situation is, of course, much more messy than this because the valence

quark configuration is embedded in what can aptly be described as a "virtual" soup of

tluctuating qq and gluonic states, as allowed by the uncertainty principle.) These appear

to he the only quark comhinations permitted hy nature. This h:><" heen accounted for


througb colour tbeory, or quantum chromodynamics (QCD), by postulating the existence

oC a new quantum number cal1ed colour [10]. According t~ this hypothesis, each quark

carries one of three possible colours, red, green or blue, white the mediators of tht" strong

force, the gluons, also carry colour and 50 interact with e&ch other in a complt"x manner.

The three quark colour fields form a. colour triplet; the strong interactions are invariant

under SU(3) transformations of these fields. Colour theory stipulates that only colour

singlets are a1lowed in nature, i.e., only colourless states are possible. Meson states are

thus Cormed Crom a quark and antiquark of the same colour, while baryon statf's are

formed Crom three quarks eacb of a different colour.

Because an isolated quark is a member of a colour triplet it cannot exist alonc, and

indeed Cree quarks bave never been observed. An attempts to isolate quarks from tht>ir

hadronic states result in the extraction of quark-antiquark pairs from the vacuum in order

to form colour singlets. This is a direct consequence of the nature oC the strong coup1ing

('()nstant, Os. Unlike the QED coupling constant, Os grows as the interquark distall("f'

widens, increasing tbe Corce between quarks and confining them within the badronic state.

If the interaction energy is sufficient, it becomes possible to produce rea.l quark-antiqua.rk

pairs from the vacuum allowing the more energetically favourable process oC fragmentation

to take place. The original quarks then combine with the new qq pairs to form colourless

hadronic states.

Although QCD has been a very successful theory, it has not demonstrated the widf'spread

applicability of QED. This is because the small electromagnetic coupling allows one to do

perturbation theory and calculate as a.ccurately as necessary, while the strong interactl~n

coupling constant is much larger and exhibits a dramatic increase in strength at large dis

tances. In this limit, lacking the aid of perturbation theory, there is no satisfactory way to

go about calculating. (There has been sorne recent success with lattice theories whcrchy

the space-time continuum is broken down into a discrete Cour-dirncnsionallattice of Lnite

size in order to calculate integrals using Monte Carlo sampling rncthods. These attempts

are still in the early stages of development, however, and it may be sorne time berore thcy

prove to be useful [9).) At the other limit of very sman distance, the coupling constant

goes to zero and the quarks, although confined within hadronic states, are effectively free

(asymptotic freedom). It is only in this region tbat perturbat.ion theory can be applied

. , ..


with any reliability. Therefore, QCD is currently a theory of limited applicability .

1.3 Multiplet Structure

Mesons and baryons can be grouped very naturally into singkts, octets and decuplets,

representations of the symmetry group SU(3). It was this SU(3) structure that even

tually led Gell-Mann and Zweig to postulate the existence of a quark triplet as a more

Cundamental set of particles [Il}. The three quarks u, d, and $ combine to form all the

known low mass hadrons. The SU(3) symmetry is then just the invariance of the strong

interaction under unitary transformations of these three quark fields. (Note that this

global symmetry has no connection with the local SU(3) colour symmetry responsible for

the strong interaction.) The symmetry is not perfect, however, owing to the mass differ

ences between the u, d, and s quarks. The current masses given in Table 1.1 are hardly

degenerat«:', however, and it would appear that the SU(3) and even the SU(2) isospin

symmetries should be badly broken. This is in fact not the case: isospin symmetry holds

at the 1 - 2% level while SU(3) is good at the 15 - 20% level. The real reason that these

symmetries hold so weil is that aU the quark mas&es are small compared to the hadronic

quark interaction energies which are at least of the order of Ac ~ 0.2 - O.3GeV [12]. In

this energy range, the masses of the u and d quarks are clearly negligible leading to an

almost exact isospin symmetry white the larger $ quark mass yields a more approximate

SU(3) symmetry. The masses of the c, b, and t quarks are clearly much too big for the

larger SU(4) etc. symmetries to be exhibited.

The SU(3) symmetry can be displayed by arranging the hadrons according to their

isospin projection 13 and strong hypercharge Y = S + B quantum numbers (Fig. 1.1).

These two correspond to the diagonal generators of SU(3). Each horizontal row is a

multiplet of the isospin subgroup, while the vertical scale represents different degrees of

strangeness since baryon number is constant for a given representation. The masses in a

given isospin multiplet are ail very similar indicating the near degeneracy of the u and d

quarks. Each location in a multiplet represents a different possible particle state, specified

by the quantum numbers Y and 13. The total isospin 1 is also a good quantum number,

but this is not displayed in the diagram. For ex ample, although the w and pO appear to

1


y=s y=s

+

o

~------~----~---------/3 -1 _1 0 ! 1 l 2

SPIN 0 NONET

+

o

K-+ 892

~-------~----~--------h -1 _1 0 1 l 2 2

SPIN 1 NONET

Figure 1.1: Multiplet structure exbibited by the lowest lying spin-O and spin-l mesons.

occupy the same position, tbe former has total isospin 1=0 while the latter has 1= 1. This

distinguishes them from eacb other.

1.4 Resonances

Resonances are very short-lived particles witb definite values of electric charge, baryon

number, mass, spin, isospin, parity, lifetime, etc. They have aIl the properties of cle

mentary particles except that they have very short lifetimes due to their ~trong decay

channels. As a result, the mass of a resonance is not an absolute quantity but rather has

assodated a weIl defined widtb. This is a consequence of the uncertainty principle: how

weIl the energy E of astate can be doefined is directly dependcnt upon the amollnt of time

!lT available to measure it,

" AE ~ !lT'

In the rest frame of the resonance, !lE = !lM and the time avaiJable is just the lifetime

of the particle, 80

r

CHAPTER J. THE STANDARD MODEL 8

For example, the pO resonance decays very quickly into two pions

with a total resonance width of 150MeV. This means that the pO has a lifetime of about

4.4 x 10-24 seconds, typical of strongly decaying particles. Pions on the other hand have

comparatively negligible widths. The weakly decaying charged pion with a lifetime of

2.6 x 1O-8s has effectively no width while the faster electromagnetic decay of the neutral

pion (0.8 x 1O-16S) produces a width of about 7eV.

Because a resonance decays so quickly, it bas a path lengtb that is' much too small

to be detected. At present, the only way to determine the presence of a resonance is to

observe the decay productsj these must have an invariant mass consistent with the mother

partic1e. For example, a pO meson with Ep = IGeV has a typical pathlength given by

where Tlllb and To are respectively the lifetimes of the pO in the lab and rest frames. As

this cannot be rewlved by the ARGUS detedor, po particles are inferred by plotting the

invariant mass of all pairs of oppositely charged pions. The po then shows up as peak

of width f = 150M~V centered at a nominal mass of Mp = 770MeV (Fig. 4.13). There

is a large background component due to uncorrelated pions; this makes the po resonance

difficult to isolate. The shape of this curve can be approximated by a Breit-Wigner:

(f/2)2

1.5 Development of the Standard Model

The history of the standard model displays a fruitful interplay between theory and exper

iment. Present day discoveries in the particle physics world have carried on in this spirit

and all indications are that future discoveries will follow in the same vein. An attempt is

made in Appendix A to provide a Bavour of the developmental process, as weIl as to shed

some light on a few of the subtleties of the standard model.

..

......


1.6 The Weak Lagrangian

As the mé in emphasis in this work is on weak decays oC the B meson, only the charged

current weak Lagrangian of the quark and leptonic sectors will be given in this section.

The quarks and leptons can be arranged into three familie3 of left-handed SU(2) doublets

a.nd right-handed singlets 1. The doublets L are given by

( :~ ) L (;~) L (;~) L (~) L (:) L (!) L •

This is a. reflection of the maximal parity violat:on of weak decays observed in nature: the

left- a.nd right-handed components of a spinor field do not experience the weak interaction

with equal strengths. Because right-handed spinors do not participate in charged-currcnt

weak interaction::; at aIl, the asymmetry is termed "maximal".

The leptonic sector of the Lagrangian can be written in terms of these doublets as

= -i 0( ilL "fIA eL W: + eL,'" /IL W;) + /J, T terms

= -i 9 rn(iï,~(l - ,5)eW: + ë,~(l - ,5)lIlVjA-) + Il, T terms 2v2

where Il, e, and W:t are respectively the neutrino, electron, and charged intermediate

boson fields. The 4 x 4 Dirac matrices ," and ,5 satisfy the relations

p, Il = 0, 1,2, 3. (1.1 )

The adjoint spinner, tb is defined as t/J = tbt,o.

In the quark sector, the situation is complicated by the fact that the weak interaction

quark eigenstates q: are not the same as the quark mass eigenstates q" but are rclatcd

by unitary transformations. This mixing allows charged-current (but not neutral-current)

transitions between different families to occur as parametrized by the Cabbibo-Kobayashi

Maskawa (CKM) matrix, Va,. This is discussed in the next section. (If such "cross family"

currents were permitted in the leptonic sector, they would manifcst themselvcs in lepton

IThere is no theoretical reason wby more ramilies could Dot exist, but recent measuremenf.8 of the width of the Z [13] indicate that nI = 3, if one assumes that additional families would bave hght neutrmOfl:


nurnber violating decays such as p- -. e-II."e' They are forbidden in the standard model

due to the assumption that m., = O.) The quark sector Lagrangian is given by

L .9\';j (- "(1 5)d W+ + -d 11(1 5) W-) 9-,.A: = -'2V2 Ua"Y - 'Y j Il J"Y -..., Uj "

= -i 9 m(Vt6dü 'Y"(1 - ...,S)dW: + V".,d...,l1(l - ...,S)uW;) 2v2

+V.,.ü'Y"(l- 'YS) .. W: + V".i"Y"'(l- ...,S)uW;) + ... )

where the u, and dJ respectively represent surns over up and down type quarks.

Each of the lepton and quark weak Lagrangians have a romrnon V-A structure,

(1.2)

AU known weak processes have been round to respect this forrn of interaction. The first

and second terms terms transform respectively as vector and axial vector under Lorentz

transformations. The V-A Corm ensures tbat only leCt·handed (right-handed) particles

(antiparticles) participate in weak interactions. This Collows frorn tbe definition of right

and left·handed particles,

so that with the help of 1.1 and the identity (1 - "Y5 )(1 - ...,S)/2 = 1 - "Ys, the weak

Lagrallgian can be rewritten as

.p...,II(l- ...,5)",

= .p...,II(l- ...,S)u.:p",

= .p(l + 'Y5h ll (1;'Y').p

= ",t(l - "Ysh°'Y""'L

= [(1 - 'Y5)"'P"Y°'Y""'L

= 2";jJ L "Y"'" L·

In the relativistic limit, mIE -+ 0, the belicity projection operators assume the form

E± = (1~"'§) [14]; therefore, left-banded particles bave pure negative helicity. This can

be exploited very etl'ectively in charmless semileptonic b ~ U decays (see Section 4.7.1),

where the lepton and li quark are relativistic and must therefore have negative helicities.

.'


The spin-l W must be emitted with negative helicity as well in order to preserve the

spin of the b quark. The decaying W produces a lepton with negative helicity and an

antineutrino with positive helicity. The only way to respect the helicity of the W is for

the electron to be emitted roughly in the direction of the W, i.e., opposite to that of the

u quark. Therefore, the lepton and u quark tend to be produced back-to-back.

The two particle seetors can be combined in a convenient (orm in order to eonnect

with the current-current effective Fermi theory:

L = L(J"W+ + Jl'tW-) ec V2 " l'

where the charged eurrent J" is given by

1 1 JI' = lIe"Y"2(1- "Y5 )e + ü"Y"2(1 - "Y 5 )d' + ... (lA)

and d'etc. provide the mixing oC the quark mass states (see next section),

This gives the coupling of charged currents to the vector bosons. The decay of a W

produces another current, and so the Lagrangian for complete weak decays is given by

L = 92

J" (g"" - q"q,,; M'lv) J"'t, 2 q'l -l'rlw

where the expression within parentheses is the boson propagator. At low encrgics, the

mornentum transfer q2 is much sm aller than Ml.- and 50 the propagator becomes simply

g,,"/Mlv and yields the well-known phenomenological Fermi interaction,

4GF L = - v'2 JP J"f, (1.5)

with g2 4GF

2Mlv = J2' The above effective Lagrangian is quite sa.tisCactory for practical B physics. To see

this, consider the channel through which the W can achieve the greatest virtual mass,

B ... 1ftii. The W will acquire the largest mass when a stationary pion is produced at

rest in the frame of the B, and 50

q!caz _ (Ms - M,y '" 0 004 Ma. - Ma, _. . Alter the matrix element is squared, this is a correction oC the order of 1 % and clearly

negligible.


1.7 The CKM Matrix and CP Violation

The weak interaction is special in that, unlike the electromagnetic and strong forces, it

is not invariant under parity (P) or charge conjugation (C). Parity replaces the spatial

coordinates i by -Xi invariance under P is essentially the statement that "Ieft" and "right"

ca.nnot be defined in an absolute sense. Charge conjugation changes a particle into its own

antiparticle, additive quantum numbers changing sign in the process. Charge conjugation

invariance implies that a world composed of antiparticles would not be distinguishable

(rom our own. For exa.mple, the spectrallines of an antia.tom would be identical to those of

an atom [15]. Time reversai (T), a third operation, corresponds to an inversion of time2•

Conservation of C and P had been experimentally observed for both electromagnetic

and strong interactions and was thougbt to hold (or weak interactions as weil unti11957

when the first conclusive evidence (or parity violation was found [17]. Parity is violated

maximally because only le(t-banded neutrinos exist in nature.

For a while, the combined operat\on o( C and P (CP) appeared to rema,in a good

symmetrYi but in 1964, an examination o( kaon decays showed that CP in turn was

violated [18]. The efl'ect was small, however, (o( the order of 0.1%), and to date CP

violation bas not been observed in any other system, although, if one believes in the KM

mecbanism discussed below, B decays are expected to be an excellent area in which to

searcb [19].

Unfortunately, the origin of CP violation is still not understood. Two classes of models

exist to try to account (or it, superweak and milliwea.k. The superweak model [20] pos

tulates the existence of a new CP violation interaction that has a coupling sm aller than

second-order weak interactions. This interaction can only be observed in kaon decays

because it is the ooly system sensitive to sucb negligible forces. This is rather unsatis

(ying because the theory is not testable. Milliweak models, on tbe other band, assume

that a small fraction (10-3 ) of the weak interaction is CP-violating and should there

fore show up in other systems. One member of milliweak models is the very attractive

Kobayashi- Maskawa mechanism.

ln 1973, Kobayashi and Maslcawa [21] suggested that a small imaginary component

, According to the CPT theorem, ail interactions are invariant under the combined operations of C, p. and T (16).


in the quark mixing matrix could account for CP violation3 • The weak quark eigenstates

q' are related to the physical quark mass states q through a 3 x 3 matrix, the Cabbibo

Kobayashi-Maskawa matrix (CKM matrix). The only theoretical constraint on the quark

mixing matrix is that it be unitary. In general, a 3 x 3 unitary ma.trix can be parametrized

by three real paramet~rs and one imaginary phase. Using the parametrization of Chau

and Keung [22), one can write

with

v=( ( 1.6)

Here,s" = sinIJ" and Ca, = cos 8". The four para.meters are thercfore labeled IJ12 ,013,023,

and 1513• In general, 613 :f. 0 results in CP violation.

An important consequence of unitarity is that the complex inner product of any two

columns or rows must vanish. For CP violation to take place, it is c1ear that in general

the terms of each of these six inner products must be complex. One could plot them tip to

tail in the complex plane and form a triangle which has area J = lS12S13S23C12C~3C23S613'

regardless of which of the six inner products one considers [23]. If CP is conserved, the

terms become real, and the triangle collapses. It turns out that if the KM mechanism is

responsible for CP violation, ail CP violating effects are proportion al to J [l, 2]. It is

easy to see that if any element of the CKM matrix 1.6 is zero, then J likewise is zero,

implying that the K-M mechanism cannot a.ccount (or CP violation.

It is imperative, therefore, to determine whether any of the clements in tbe CKM

matrix elements are zero. Currently, tbe ouly ones in doubt are those characterizing the

couplings to b and to t. Assuming the absence of Havour cbanging neutral currents, the

observation of BB mixing at both ARGUS [7] and CLEO [8] imply that V,d is non-zero,

and observation of B.B. mixing would indicate a non-zero value of Va.. Of particula.r

31n fad, K.)bayashi and Maskawa actually postulated the existence of the third generation of quarks and leptons. Within the framework of the then existing four quark mode), there Wall no way of lDcorporating the CP violating phase inlo the 2 x 2 Cabbibo Matrix. A six quark mode) requiring a. 3 x 3 unitary matrÏJc does this naturally.

-..


importance to this work are the non-zero values CLEO [3] and ARGUS [4] have recently

ascribed to Vu6 from the investigation of the endpoint of the lepton momentum spedrum

Crom semileptonic B decays. Observation of the B -+ pOlïï channel would provide an

independent measure oi this matrix element. It now seems likely that ail elements of the

CKM matrix are non-zero. However, the actual values are a1so significant. The CKM

matrix elements, although parameters in the standard model, are heavily related to the

quark mass matrices and could provide sorne insight into the origin of the mass hierarchy

oC the standard model [24].

1.8 e+ e- Collisions

Electron-positron collisions provide a véry clean method for high energy particle produc

tion. In contrast to proton-proton and proton-antiproton collisions, where the parton

momenta are not known, the energy of the interaction can be fixed quite accurately. In

addition, electrons being point-like particles, the number of possible types of interactions

is Iimited and 1.11 background processes MC Cairly weil understood from QED. The proton,

along with three valence quarks, contains a sel. of virtual gluons and qq pairs making it diC

ficult to isolate the desired processes [25]. (In fact, gluon-gluon interactions can dominate

pp collisions.) U nCortunately, e+ e- collisions are rather difficult to achieve at high energy.

This is because the most practical way oC accelerating electrons is using a synchrotron.

The eledron energy suffers severely from synchrotron radiation, the radiation emitted

tangential to the path of the electron as it is bent in the magnets. At very high centre-of

mass energies, an enormous amount of power is required to main tain the beams, and it

is no longer cost-effective to build such machines. Proton beams suffer from synchrotron

radiation as weil, but this effect is down bya factor of (~)4 = 10-13 [26] and the effect is

negligible in comparison. One way of avoiding energy losses due to synchrotron radiation

is to use a linear accelerator. Although particles still radiate during acceleration, the

effect is negligible [27]. However, linear accelerators have to be improved in other areas -'

beCore they can become effective (SLe). LEP at CERN is the e+e- synchrotron with the

highest beam energy to date, 60GeV, although an upgrade to 100GeV is scheduled. The

Russians plan to achieve 500GeV lot, Serpukhov by 1996 [22J. In contrast, the maximum

l


proton beam energy attained 10 far is about 1TeV at both HERA and Fermilab, and the

SSC is expected to achieve a 20TeV beam energy.

There are two types of ~)rocesses that CUl occur in e+e- collisions: annihilation and

scattering. Those interactions which do not involve annihilation are Bhabha events (where

a single virtual photon is exchanged between electron and positron) and 1'"'( events (whcre

virtual photons emitted {rom the beam partides collide and produce mesons). Anni

hilation of the original beam particles produces fermion-antifermion pairs of leptons or

quarks.

There are two possible annihilation channels for electron-positron collisions. Thf!Se

are to a virtual photon (JP = 1-) or a virtual Z (JP = 1- or }+) [251. The Z being

very massive, at the energy of the T resonances this channel is down by the factor M}

in the propagator and cannot corn pete with the photon channel. The cross section for

annihilation into energetic fermiûn-antifermion pairs ({J ~ 1), includiog QCD correction~

to second order in o., is given by 122]

(1.7)

For quark production, this expression applics only at energics weIl away (rom rcsonancc

regions_ A summation over ail quark pairs not including br, and tt yields an approximatc

expression for the continuum hadrJnic cross section at the T(4S) energy_ Using a value

Os = 0.2 [14], and correctly summing over quark colours yields

Comparing this to the resonance component of the visible cross section at the T( 4S) energy

at DORIS, (0.87 ± 0.06)nb (Chapter 4), it is c.lear that less than 25% of multihadron data

accumulated at the T(4S) results {rom BB deca.ys.

Chapter 2

B Meson Physics

2.1 The Upsilon Resonances

ln 1973, Kobayashi and Maskawa [21] proposed their mechanism for CP violation. This

involved the hypothesis of a 3rd generation of quarks and leptons for which there was

then no experimental evidence. Glashow, Iliopoulos, and Maiani had already suggested

the existence of a fourth quark to account for the non-observation of flavour-cha.nging

neutral currents [28], and its discovery in 1974 [29] by way of the J/W made the theory

nicely symmetric aflcl elegant. There was no need for a 3rd generation of particles.

ln 1975, however, evidence for a new heavy lepton, the tau, began to accumulate, also

implying the existence of a 3rd generation of quarks. Experimental verification of this

came with the discovery of the r resonance in 1977 [30] , quickly interpreted as a bound

state of a new quark, bottom, and its antiquark. Although its counterpart, the top quark,

has not yet been directly observed, there is little doubt that it will be discovered. Indeed,

the phenomenon of BB mixing observed first at ARGUS is strong evidence for the top

quark.

The T resonance was first seen at Fermilab in 1977 [30] through the protess

(2.1)

at 400GeV. The mass distribution of the 1'+1'- pair (Fig. 2.1) indicated an eilhancement

at roughly 9.5GeV, with a shape suggesting two or more narrow resonances close together

but not resolvable with the detector used. These were labeled the T(IS) and T(2S).

The large electromagnetic channel was a. tell-tale sign that sorne new particle was

responsible for the en han cement. Rad the resonances been due to Any of the four quarks

16

..

CHAPTER 2. B MESON PHYSICS

, " .. '.

.. :;

.. ·n .. 10 E

"

1

\ _, fi.

1 • . ~ ~.

T ~t

• 10 12 Moss G.V

Figure 2.1: Muon pair mass distribution as measured at FNAL [24].

17

already known to exist, any electromagnetic channel would have been swamped by the

much faster strong interaction and not seen. Clearly, the strong decay channel was being

inhibited somehow. The resonances were interpreted therefore as bound states of a new

quark and its antiquark, bb, that did not bave enough energy to extract a light qUt.Lrk

pair out of the quark sea. (The electromagnetic channel bas only to compete with the

significantly suppressed three-gluon channel which is down by a factor of Q~ [Fig. 2.2].

Three-gluon emission is the only way to create a colour singlet.) This interpretation was

bOfne out by new experiments done on e+e- machines.

The proton-target collisions were not very efficient sources of of T resonances. Due to

the parton motion in protons, the production of the resonance is spread out over a wide

range of pp centre of mass energies. Furthermore, the composite nature of protons gives

rise ta many background processes. Because the T resonances decayed electromagneti

cally, it was clear that they could be produced very effectively in e+e- machines from the

decayof the virtual photon. This was done both in 1978 at DORIS [311 and in 1980 at

CESR [32]. By varying the centre-of-mass energy of the colliding beams in the region of

~ J

lo..

CHAPTER 2. B MESON PHYSICS 18

e-g

b

T(IS) g Hadrons

;;

e+ T(IS)

e-:} B-,BD

T(4S)

Figure 2.2: The dominant T(IS) and T( 4S) decay channels.

the T resonances and counting the rate of hadronic events occurring, it wu possible to

resolve c1early three sharp peaks located at 9.46, 10.02 and IO.38GeV (Figure 2.3). This

third resonance wu named the T(3S). The narrowness of the peaks reinforced the notion

that the strong decay was suppressed. Furthermore, the distance between resonances was

similar to that exhibited by the J/'ifI resonances, strongly supporting the b quark bound

state hypothesis. By measuring the width of the electromagnetic decay T -+ e+ e- , it was

possible to ascertain the charge of the constituents (33). The -1/3 result was compatible

with the new quark hypothesis.

In 1980, the T(4S) was discovered at 10.58GeV with a width of about 25MeV [34].

This large width suggested that the b;; pairs were being produced with enough energy

to extract uu or dd pairs from the vacuum, and so decay strongly into two B mesons.

Several observations confirmed this hypothesis. First, the B mesons could only decay

weakly and tbere should thererore have beeo a strong B -+ X Cil channel producing an

excess of leptons at the T(4S) eoergy. This was in fact obsen·('(l :1;1. AIso, the lepton

...

'.'


• 17 1. i. 1· .' .1

1 2 ----- 1 :> ,

1.l1li

IUSI

?CISI

1 A\'J ._. ___ .... !LI 1 T"------

1.50 '0.00 '040

CINTER OF MASS ENE"GY ICitV)

T (45.

'050 '0.10

Figure 2.3: Cross-section for e+e- -. hadrons showing the T resonances [24].

19

spectra matched what one would expect from heavy meson decay and was oot consistent

with charm decay alone. Finally, it wu possible to reconstruct exclusive B decays in a

few channels and measure the B mass [36, 37, 38). This wa..'1 round to be greater than half

the T(3S) mass but less than half the T( 4S) mass, demonstrating that it was energetically

possible to produce BB pairs at the T(4S) but not the T(3S).

Until recently, the 1(48) was thought to decay solely ioto BB pairs. No other form

of decay had ever been round. For example, one possible way to coofirm the existence

of exotic decays would be to observe a signal in the momentum region p > Pkom /2. No

evidence of a signal had been observed [39]. Furthermore, the possibility that an excitcd

Bis produced has been ruled out from energy considerations.

Recently, however, J/t production in a momentum region kinematicalJy forbidden for

B decays has been observed at a rate of 0.22% [40, 41]. This can ooly be attributable

to the process T(48) --. J/~X and is the first indication of non BB production hy the

T(45). A few models exist which try to account for this. 10 particular, it is conceivable

that the T(4S) contains a smalt admixture of light quark or gluon matter Along with the

.r .

; ..


bb pair [42]. Presumably, tbe b6 pair would be in a colom octet configuration and upon

annihilation would produce detectable photons at energies of 4-5GeV. A searcb for tbis

pboton signal is presently underway at ARGUS.

2.2 Semileptonic B Meson Physics

Tbe 1977 discovery and subsequent study of tbe T resonances bas opened up a whole new

arel witbin whicb one cao test tbe standard model. In particular, the weak decay of B

mesons produced by the T(45) resonance promises to be an important tool for examining

CP violation. According to the standard model, B mesons must be capable of decaying

directly to light charmless mesons tbrough tbe weak interaction, i.e., the b quark should

be able to decay into a u quark. As outlined in Cbapter l, the CKM matrix element Vub

must be non-zero. If this is not tbe case, an explanation for CP violation would not be

possible witbin the framework of tbe standard model.

There are esseotially two difFereot approaches to examining the b -+ U contribution

to B decay. One can either study ail the data inclusively and look for cbaracteristics

ooly attributable to the charmless channel, or one can seareh for an exclusive channel

that requires the b -+ U transition. Inclusive studies of B meson decays ofFer the added

advaotage of greater statistics but quickly run into problems if one wisbes to extract an

actual number for Vub. Investigation of the average kaon yield per B decay (dift'erent for

b -+ C and b -. u decays) is one example of an inclusive study but one which has not

provided very meaningful numbers [43].

A mucb more powerful method bas been tbe investigation of the leptonic momentum

spectrum from B decays. Tbe effectiveness of the inclusive lepton momentum spectrum

method arises Crom the large difference in mass between charrncd and up quarks. Be

cause of tbis mass ditrerence, leptons from charmless B decays cao populate a region,

Pt ?,2.3GeV le, tbat cannot he reacbed by cbarmed B decays. A signal in tbis region

aCter continuum background subtraction would indicate the existence of the b -+ U tran

sition. In Cact, both CLEO [3] and ARGUS [4] have recently round a significaot signal in

this region and set tbe value of IVubIVc61 at 0.11 ± 0.01 [4]. Unfortunately, this value is

highly model-dependent and must be accepted with sorne reservation .

1


Charmless Hadronic B decays Decay Mode 8ranching Ratio (90% CL)

11'+11'0 < 2.4 x 10-4 7r+I1'+7r- < 4.5 x 10-4

l'7r+ < 1.5 x 10-4 ,...+7r+I1'-7r- < 6.7 x 10-4

0°11'+ 1 < 9.0 x 10-4 Wlr+ < 4.0 x 10-4 '111'+ < 7.0 x 10-4 p+p- < 2.2 x 10-3

Table 2.1: Upper limits on some cbarmless hadronic decay modes of B mesons as measurcd at ARGUS [44].

Exclusive B meson decays, while more difficult to isolate due to statistics, provide

a much hetter means of extracting Vub from the data because they are not a.c; strongly

model-dependent. Furthermore, there is less ambiguity involved in the interprctation of

a signal. In light of these considerations, a signal in an exclusive channel would Dot only

he desirable but preferable.

B meson decay occurs by way of the weak interaction both semileptonically and

hadronically. Although several charmless hadronic B decay channels are predicted by

the standard model, investigation of the semileptonic channels is preCerable Crom both a

theoretical and experimental viewpoint. While it is true that Cully hadronic channels do

not involve the elusive neutrino and can be completely reconstructed, hadronic currents

are not well understood. In general, hadronic decays Învolve at least two su ch currents

and these cannot be treated independently because of the non-perturbative nature of

QCD. As indicated in Chapter l, the coupling constant is large and promotes multiple

gluon exchange within and between currents. Nevertbeless, these channels have becn ex

amined in great detail [44], and will continue to be studied in the future (Table 2.1) _ A

signal would indicate a nonzero matrix element, but it would be difficult to interpret the

result in terms of an actual value for Kb unless further progrcss is made towards under

standing the hadronization process. In contrast, semileptonic dccaj , are much easier to

understand because they involve a single hadronic current, and a leptonic current which

is Cully understood. Under very mild assumptions (Section 2.4), the two currents ca.n be

t , .


treated in an independent manner, making calculations much easier and more reliable.

Furthermore, the ARGUS detector has excellent lepton identification capabilities. The

missing neutrino is easily taken into account by the method discussed in Chapter 4.

It may seem that the semileptonic channel is without complications. The number of

different approaches to semileptonic B decay documented in the past few years is a tell

tale sign that this is not the case. The hadronic sector remains to be understood. While

the weak decay of an isolated b quark is perfectly calculable, there is stiJl no consensus

as to how the remaining u quark and spectator quark arrange themselves into a bound

state. There are essentially two dift'erent approaches to this problem and these will be

discussed here. It is b"th convenient and enlightening to compare these models to that

of free quark decay and so such a model will be described first.

2.3 Free Quark Decay

The decay of an isolated b quark is dictated by the weak interaction coupling described

earHer. Specifically, the relevant low-energy form of this interaction is given by the Fermi

coupling as

(2.2)

where Ut is the Dirac spinor for either the u or c quark. A straightforward calcuJation of

the differential rate yields [45]

dfo G}mf 2(ZM - Z)2[ Tx = 9671'3 % (1- %)3 (1 - z)(3 - 2z) + (1 - zM)(3 - x)]

( 2E, m f 2

x = -, ZM = 1 - (-) ), m6 m6

(2.3)

whiJe the total rate is given by

(2.4)

where

(2.5)

This Cree quark lepton spectrum is shown in Figure 2.4 Cor b -+ U and b -+ C decays with

PB = 0, mu = 0.15, me: = 1.56 and m6 = 5.12.


tr/dx 1.0

(orbi har)' uni te)

0.8

0.6

0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0

x=2Ee/m~

Figure 2.4: Lepton spectrum from pure {ree quark dccay mode\.

23

Because the b quark is bound with a spectator antiquark in a B meson, it become~

necessary to include bound-state effects and gluon exchange mechanisms in order to model

more correctly the actual decay of the B decay. Several models attempt to do this. A

commonly-used example is the model due to Altarelli et al. (ACCMM model) [45] which

ascribes to the spectator quark a Fermi motion PF with a gaussian distribution. The

spectator quark is assumed to be on mass shell during the disintcgration of the b quark

which must therefore have a virtual mass given by

(2.6)

in order to respect energy-mornentum conservation. The Cree quark process is then QCD

corrected by including soft gluon effeds due to real gluon hremsstrahlung and virtual

gluon loop contributions (Fig. 2.5), the former tending to decrease the total rate expccted

{rom a straight {ree quark decay scenario, and the latter tending to increase it [46]. The

end result is an o"erall decrease in the total rate along with a depletion or the lepton

end point as follows:

~

{


b ~ '= . c 6 7

'" c

(a)

b 6' c 6 Z. c ~

6 7 ~

c

(b)

Figure 2.5: Leading-order bremsstrahlung (a) and gluon exchange (b) in (ree quark deeay.

dI' dfo [ 20. - ( )] {20. 2 ] - = - 1 - -G Z,E exp --ln (1- z) dz dz 3,.. 3,..

(2.7)

where G(x, E) arises (rom leading-order QCD corrections, while the exponential attempts

to take into aceount a eutoff effect due to soft gluon contributions at the endpoint. The

final spedrum is then obtained by Colding this eorreded spedrum with the initial motion

of the b quark, and boosting the whole system in aceordanee with PB at the energy of the

T(45) (PB::::: 350MeV le).

Althougl 'lis model is able to reproduce the lepton spectrum, it has a. major disad

vantage that is typical of ail inclusive models: there are too many free parameters which

cannot be reliably estimated [47, 49]. The average Fermi momentum PF, for example, is

not specified by any theoretical model and is used as a fitting parameter. AIso, while the

b quark mass is in theory specified by energy momentum conservation, it is dependent

upon the values oC both the Fermi motion parameter and the spectator mass whieh must

be estimated. In addition, the mass m, of the quark into which the b decays is not exactly

CHAPTER 2. B MLSON PHYSICS 25

known. This severely a.ffects phase space calculations, indicating that binding efl'ects are

of great importance in the decay process. (Note that the decay rate varies as the firth

power of 1n,. A reasonable variation of m, (rom 4.8 to 5.2GeV Ic' results in a 50% increase

in the total rate. The eft'ect is even more dramatic when m, is allowed to vary witbin

acceptable limits as well [49].) The models for exclusive semileptonic decay circurnvent

this problem to a large extent by working with measurable meson masses as opposed to

constituent quark masses.

A much more disturbing aspect of inclusive models is that final-state meson production

is ignored. No attempt is made to ensure tbat the final qd system ("q" here represents

the c or u quark into which the b decays. "d" represents the spectator quark) has an

invariant mass consistent with the formation of a real meson. While it is clearly possible

that the quarks could fragment into an n-partic1e "on mass" system, it has been shown

experimentally that such multihadron final states are not prominent in semileptonic B

decay. Approximately 80% of semileptonic B decays occur through the channels 8 -+ Dtii

and B -+ D*w [41]. Because both the D and D* have very small widths, it is dear thal

inclusive models describe a very unphysical process. A dramatic demonstration of this

can be seen in Fig. 2.6 where the invariant mass of the qd system is plotted for differcnt

values of < PF >. The location of the D and D- are also shown. It is clear tha.t the

model is consistent with the dominance of D and D* production in B decay provided

one assumes negligible Fermi motion of the quarks. In order to fit the experimental data,

however, values of < PF > around 300 to 450MeV Ic are required, inconsistent with D

and D* dominance. Additional strong effeds could be included to soCten up the recoil

mass spectrum but these would alter the lepton spectrum [50]. This inability to account

for physical meson masses is by far the most unsatisfying aspect of inclusive modcls. (An

attempt was made by the author using a Monte Carlo program to try to force the final cil

state to respect the available meson masses without altering the lepton spectrum shape.

This involved bringing the ci system on shell by altering the moment a. of the neutrino and

ël quark without violating energy-momentum conservation. Inadequate rc:,ul b forced the

author to abandon the Altarelli model in favour of the more physical Corm factor models.)

Evidently, there are sorne significant difficulties inherent in the inclusive mode} ap

proach. Exclusive ~l1odels attempt to surmount these difficulties by calculating each decay


K. Il.,..,.,. ,,111. l '·IIIft. litt .... ,

':'> ~

':'",

~5! .. 250 • >" -r"

v t: v

.' ......... .:.".-..:. ...... 0

. --1 • 22 24 26

0 o· H. Ui.V,

Figure 2.6: Invariant mass distribution of the hadronic system produced by the Altarelli model for different values of the Fermi parameter. (Hagiwartha, Martin, and Wade, Nud. Phys. B327 (1989) 570).


channel explicitly. A model Cor semileptonic B decay is then construded by summing over

all final states. The remainder of this Chapter deals with the theory of exclusive models.

2.4 Models of Exclusive Semileptonic B Decay

2.4.1 Introduction

The exclusive models work at the meson level as opposed to the quark level. In this

way, it is possible to avoid working with unknown or at least unreliable constituent quark

masses. In what follows, the process described will be that oC b -+ C transitions, but it

should be clear that the same applies Cor b -+ U decays with the appropriate substitutions.

Using Eq. 1.5, the amplitude Cor the decay process is

A =< Xtiï! H 1 B>

with H = ~Jt"J:

(2.8)

(2.9)

where Jt~ and Jr are given by the leptonic and hadronic current terms in Eq. 1.4. The

amplitude can be decomposed into a simple product of hadronic and leptonic matrix

elements if one assumes [51] that 1) the weak decay time scale is much smaller than the

hadronic timescale (because Mw is large) and so proceeds uncomplicated by strong effects

and 2) that all final~state interactions, sort gluon emissions, and bound-state effccts can

be absorbed into the initial and final-state wave Cunctions. Thus, the transition amplitude

becomes

(2.10)

The starting point for all the exclusive models entails writing down the Most general

covariant Corm for the matrix element,

H~ =< X(PX,&x) 1 V" - A~ 1 B(PB) >. (2.11)

For spinless Mesons X, the axial vector contribution vanishes and, in the notation of

ReC.[47), one is leCt with 1

< x (px ) 1 V" 1 B(PB) >= f+(PB + Px)" + f-(PB - px)" (2.12)

1 A diff'erent deeomposition of the Corm factors ia uaed in [52, 53]. There, the rnatrix elernent ill broken down into terrns orthogonal and parallel to the mornentum transfer 9 in order to easily pick out the longitudinal and transverse Corm factors [54).

. t

{


white for spin 1 states X· with polarization € (f' Px = 0)

(2.13)

and

The form factors f-J:, 0-J:, / and 9 are Lorentz scalars and can thereCore depend only upon

(PB + px )2, (PB - Px )2, pL, and p~. The latter two scalars are simply the constants

Mi and Ml and can be disregarded. The second term is the square of the momentum

transfer to X, q2, while the first term can be expressed in terms oC the three others:

(2.15)

Therefore, the form factors cao sim ply ~e considered as functions of q2. The moment a

PB and Px are arranged into symmetric and antisymmt~tric combinations to simplify

calculations and to take advantage of the natur:J q2 variable. The form factors contain

ail the information about non-perturbative QCD in the decay process and modify phase

space accordingly.

The problem is thus reduced to the determination of the (orm factors f-J:, a-J:, f and g.

Essentially, this involves employing quark model wave functions to calculate the matrix

element 2.11, and comparing the Iesult to the general covariant forms given by 2.12- 2.14.

It would then be 3. simple ma.tter to read oft' the form factors. Unfortunately, proper form

factor calculations require a good understanding oC non-perturbative QCD and therein

lies the root of ail difficulties involved in the modelling process. Typically, the way around

this is to a priori ascribe to the form factors sorne q2 dependence based on reasonable

assumptions. The majority of theorists advocate a functional form based on dispersion

relations [47], that of nearest-pole dominance:

(2.16)

where mp is the mass of the nearest resonance with the appropriate quantum numbers [48].

Figure 2.7 illustrates the rationale behind this form of q2 dependence. A modification of

the external lioe directions for the b and c quarks shows how the production of the W

l


1

b

ii • ii

Figure 2.7: Feynman diagram for quark decay showing how pole dominance can dictate form factor dependences.

can be viewed as the annihilation of a lié pair. The doser the invaria.nt mass of the pair

to the mass of sorne ph/sical state, the more probable the tran~ition. The form of 2.16

correctly describes the expected fall·ofP of the decay rate with decreasing q2, and cnsures

asymptotic q2 -+ 00 current conservation [55]; however, there is little justification for

assuming that the neaJ'est pole contribution dominates. The B; states, for example, are

densely spaced compared to the B mass and there is no really good reason for choosing

one over another. In addition, the exact positions of sorne poles are not prccisely known.

Although approximations suffice in most cases, this is of particula.r con cern in the case

of b -+ U transitions, where the poles lie very close to Jq'!n.az causing the form factors

t.o blow up in this region [49, 55]. The resulting over-sensitivity to pole positions is a

major reason for theorists' inability to reliably predict rates for charmless B decay. One

"solution" would be to allow mp to become a free parameter [50], but this is aesthctically

unpleasing as it takes away from the predictive power of the model.

The overall normalizations of the form factors are determined by calculating 2.11 for

a particular value of q2, usually q2 = 0 or q2 = q!gz. These correspond respectively to

minimum and maximum four-momentum transfer, and to maximum and minimum recoil,

2The sm aller the 92 value, the faster the c quark ia ejected, and the less likely the resultins cd pair "tIl form ioto a D or Ir meson due to the dramatie spatial wave function mismatch.


i.e three-momentum transfer. In the Cormer case, the virtual W is produced with zero

mass 3 leaving the meson and W the most energy possible to recoil against each other,

with momcntum Pmo~ calculated as follows:

.. .. .. ~ .. (MB, 0) = (Ex, P)+(Ew, -P) = (Ex, P)+(P, -P) (because Mw = 0)

P = Ms-Ex = MB-(P2+M} )1/2

P_ Mi - M1 - 2MB .

(2.17)

The leptoml produced from the subsequent W decay must have zero momentum in the

W rest frame by energy conservation. Consequently, in the rest frame of the decaying B,

the leptons are produced collinearly with a common momentum given by lPmA%. In the

latter case, q2 = q!.u' the W is produced with the larges!. mass possible; this is achit'ved

if all the available energy, Ms - Mx, is converted into ma.4iS, leaving t!J~ W and X at

rest. The leptons are therefore produced back to back with a common momentum of

iVq'!noz' (This, however, is not the maximum po!sible momentum for the lepton. The

lepton endpoint is actually given by 2.17. It is clear that that same expression is valid

for any collinear configuration of massless leptons; the special case of Pli = 0 provides the

electron or muon with the maximum possible momentum.)

Each of the above approaches has been adopted in a variety of different models. The

Most promising are summarized below. It should be noted that none of these models

accounts for multi-particle final states. It has been argued [56] Lhat, particularly in the

case of decays to charmless Mesons, the fast light quark is much too energetic in general

to be bound into a single Meson state. The endpoint region is dominated by low q2

events and can be expected to be strongly populated by many-body final-state decays.

For example, two pions in a relative S wave, a very prominent resonance in 1r1r scattering,

could conceivably contribute to the lepton endpoint as much as the p resonance. Another

'In fad, the W cannot have Jero mass if it ia ta decay into massive leptons. Most model., however, neglect the masses of the these leptons, a pradice which is certainly valid for decay. ta electrons and muons al this energy, but obviously not (or decay. to tau •.


thing to note is that all models below, except the first, neglect the mass of the lepton

and cannot describe tau decays. In the limit of zero mass, leptons (antileptons) in weak

decays are always negatively (positively) polarized and the form factors f- and 0+ do Dol

contribute [57]. This i5 not the case in tau decays where the lepton is not light-like and

positive polarizations are allowed. The first model beJow provides expressions for ail Corm

factors and can be used to describe decays to tau. It is therefore used for generation of

Monte Carlo B B events at ARGUS. The others explicitly assume massless leptons and

calculate only those matrix elements relevant to such decays.

2.4.2 The Mode} of Wirbel, Stech and Bauer

The WSB model [52] is the most detailed to date. The matching process described above

is done at maximum recoil, q1 = O. The form factors are assumed to have the Corm of 2.16

with the overall normalization determined in what is termed the infinite momentum frame

(IMF). The decay i5 viewed in a frame boosted in the direction of the leptons such that

the leptons are at rest. For massless leptons this means that one has to boost to the frame

of a light-like particle, which is impossible; therefore, limits are used in the calculation.

The large boost implies that the massive mesons must approach infinite momentum in

this limit (IMF)". One assumes that the hadronic wavefunctions of the B and X take on

limiting forms which remain the same for 3011 decays at 91 = O. This greatly 8implifies

matrix element calculations. The wave function calculations are more reliable in this high

momentum limit because uncertain parton masses play a much lesser role in comparison

to their momenta.

The initial and final meson states are constructed as relativistic bound states of the

valence quarks in terms of creation and annihilation operatorc;:

1 K, m, J, J. >= v'2,(27r)3/'l E f cPk1<Pk'lc53(K - k l - k2)~(J, J" m, Sb "2; kt, k2) 10 > ••• 2

(2.18)

where K is the magnitude and direction of the infini te boost, k, are tbe momenta of

the boosted valence quarks, z is the fraction of longitudinal momentum carried by the

4For deeays to the massive tau, tbe minimum q: ia Dot zero and it is physieally possible ta boost to the rest Crame oC the W. Tbe IMF calculation is approximate in tbia case ainee the booet ia no lonser "infini te" .

(


spectator quark, and m is the mass of the meson in question. The internai wave functions,

~, used in the expression are cbosen to be relativistic scalar harmonie oscillator solutions.

Substituti(,'D into Eq.2.11, aCter expressing the quark current J~ ill terms of creation and

annihilation operators, yields expressions for the form factor normalizations in terms of

an overlap factor between initial and final state mesons. For example, the form factor f+

Aboye becomes

1 ( 2) = ftPkTfJdztPx(kT,z)tPs(kT,Z) + q 1 2/ 2 , -q ms-

(2.19)

where the numerator is a just a simple constant.

This model suffers from all the problcms inherent in the assumption of nearest-pole

dominance ol the form factors. These have already been outlined above. In addition, the

overall normalization of the form factors involves a calculation of wave function overlaps

in a configuration where the spectator quark has very low rnomentum compared to the

ejected c quark. Expecting that the harmonie oscillator wavefunctions will accurately

describe the D or D- in this very unlikely configuration is probably overly optimistic.

This is especially important in the case of 6 -. u transitions where the u and spectator

quark have markedly asymmetric shares of the cbarmless meson momentum [49].

2.4.3 The Model of Korner and Scbuler

Karner and Scbuler alSO calculate in the IMF [53]. However, rather than relying on some

question able model in order to describe hadronic states, they exploit the fundamental

principle of duality in order to match Cree quark decay helicity amplitudes to those of

hadronic dccays. The spectator quark is assumed to remain spin-inert, i.e, not to exve

rience any spin flip in the decay process. There should then be a strong correspondence

between the helicity amplitudes of actual meson decays and those of free quark decay. A

q2 = 0 match is performed between the perfectly calculable free quark helicity amplitudes

and those from the helicity sta.te form factor decomposition of Eq. 2.11 (see foot note at

the bottom of page 27). Matching at maximum 1 Px 1 ensures that one is far enough away

from the region where current-meson intermediate states could strongly influence the he

licity structure. A proportionality factor is included in the overall uormalization !n order

to take into account incomplete wavefunction overlap of the initial- and final-state mesons.

The model provides no way of calculating these factors and they are obtained from the

&

l


work of WSB. The form factors are again assumed ta have a nearest-pole dominance q'

dependence, but whereas 2.16 has monopole behaviour, the forms used by the KS model

can also have di pole behaviour. This is consistent with QCD power-counting rules [58}.

AIso, a.lthough there are several possible (bë)· poles ta choose from, one common mas! Îs

employed for ail form factors, in contrast to the WSB model which sets different values

for each factor.

One criticism of this model is that a.lthough explicit quark wave functions need not

be employed for the shapes of the helicity amplitudes, the proportionality factor included

to take into account incomplete overlaps does inevitably require either sorne intelligent

guesswork or the application of quark model wave functions. The overalJ rates are there

fore subject to sorne uncertainty. Another point is that the assumption of spcctator

spin-inertness may not be valid. The IMF boost is performed in the direction directly

opposite ta that of the emitted c quark. Fermi motion will provide the spectator quark

with sorne momentum transverse to this direction and thus subject it to Wigner rotations

as it is attracted to the jetting c quark [49]. Whateve~ its original spin state, the spcctator

acquires a small probability of having the opposite spin when viewed in the IMF frame.

2.4.4 The Model of Isgur, Scora, Grinstein, and Wise

The modelof Isgur, Scora, Grinstein, and Wise (ISGW) uses well-studied non-relativistic

quark model wave functions to calculate hadronic matrix elements at q' :::::: q!Clz [59]. This

is the configuration where the c quark simply replaces the b upon dec'1Y. The internai

m(',4ion of the cU state is then comparable to typical meson bound-stale Cermi motions

and it is natural to employ the non-relativistic quark model to estimate wave Cunctions.

The approach of ISGW is somewhat different from those of WSB and KS in that an

attempt is made to calculate not only the overall normalization but a1~o the Cunctional q'

dependence of the Corm factors. They do this by calculating matrix elements Cor values

of Ipx 1 near as weIl as at minimum recoil. Bound state wave functions are then built up

using the moc.lc meson method. Valence quarks are assigned momenta depending upon

the value of Ipx 1. A match is then performed in the usual manner exccpt tbat Ipx 1 tenns

are kept in order to provide tbe the q' dependence using 2. '.5. Form factors are found to


bave the form :.4 ( [1, 4):.4] /(Px)' ~ 1 - 6r • + m' Px

where r, is the size of the hadron, m a quark mass, and 4 a constant. For b -t c decays this

expression can be shown to be equivalent to 2.16 over most. of the q' range [50,60]. Tbe

diff'erence for b -t u decays is somewhat more marked and account.s for the widely varying

rate predictions among the models. The ISGW model can make use of the wealth of

information available from quark model calculations and predict rl'.tes for decay to higher

energy statt"s. It is found that these contribute significantly to the total rate and can be

greater than decays to the low Iying p and 7r states. However, the sum of aIl channels

cannot account for the total inclusive rate predicted by the ACCMM model.

A drawback of this method of calculation is that the matching procedure is performed

in the region or phase space weil aw~y from the lepton momentum end point and one must

extrapolate into the q2 = 0 regioll in order to get the endpoint form [53]. In the case

of b -+ C decays, no difficultï arises; however, for b -+ U decays, where the q2 variable

ranges over a much larger intervaI, small uncertainties in meson radii can lead to dramatic

uncertainties in endpoint rates. Mainly for this reason, a factor of two uncertainty is

ascribed to the rate in this region [49]. This bas strong consequences for an experimental

evaluation of VQ using the endpoint spectrum. Another observation is that if one is

going to include Pi- terms in the quark model calculations, one should ideally work with

relativistic quark model wave fundions, where such terms are important.

Recently, two of the authors of this model have discovered a. new, model-independent

method for calculating form factors when the masses of the original and final quarks are

very heavy compared to flQCD [61,621. They exploit certain flavour and spin symmet.ries

which arise in this limit, and show that ail form factors are determined by a single universal

function of q2. A consequence of this symmetry is that a model-independent value for Vc6

can be obtained veryeasily from experirnent. Unfortunately, the form fador /- needs to

be determined (for q!caz ) and this can ooly be extracted from B -t Drv, somewhat of

a challenge. This newfound symmetry is a1so applicable to decays to p, although the u

quark is comparable in mass to AQCD (~ 0.2 - O.3GeV). This is because the form factors

for this decay can be directly related to the Cabbibo suppressed D -t P form factors. A

detaited study of D -+ ptu decays is thereCore warranted [63].

1


.. ,. 0.060

0.01&8

0.036

0.021&

0.012

0.0 0.0

• ISGW

• WSB

• KS

1.0

." . .~. . .. ;& # -

J-..~ .. , ..-~ . --.. . .-~ .-.. ~ .: -.

•• • • .. ... •

2.0 Lepton Momentum (GeV/c)

35

3.0

Figure 2.8: Lepton momentum distributions for B -+ ptii events as generated by MOPEK for the three form factor models.

2.4.5 Lepton Spectra, Phase Space, and Decay Rates

The B -+ ptii momentum spedra for the three form factor models discussed above are

plotted in figures 2.8 and 2.9 respectively. The available pbase space for this decay,

assuming the p to have zero width, is shown in Figure 2.10. Using TB = l.I5ps [64],

1 Vub/Vch 1 = 0.11 [41], and IVchl = 0.046 (41), the branching ratios for the transition can

be calculated from the decay widths predicted by the models. These are providcd in

Table 2.2 along \Vith the upper limits from Ref. [6}.

( j.


dI'/dp 0.030

0.02'1

0.018

0.012

0.006

0.0 0.0

e e e#

e.&

~ / fi> ,

~ .-t .. . -••• e ••

e •• e.

1.0

e lSGW

• WSB • KS

2.0 ,,-Momentum (GeV/c)

36

3.0

Figure 2.9: Meson momentum distributions for B ~ plii events as generated by MOPEK for the three form factor models.

BR(B -+ p0l:ii) from Theory and Experiment

Il " ISGW 1 WSB 1 KS

" Theory (Xl0- U TSIVu6Fl ) 4.2 13.1 16.5 (0.94 x 10-4) (2.9 X 10-4 ) (3.7 X 10-4 )

Experiment < 8.4 x 10-4 < 10.4 X 10-4 < 9.0 X 10-4

Table 2.2: Theoretica.l and experimental branching ratios for B ~ p°tu. The numbers in parentheses are computed from the theoretical rates using the values provided in the text.

...


20.0

15.0

10.0

5.0

Maximum Recoil 0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Electron Momentum (GeV/e)

Figure 2.10: Phase space available {or B -+ ptii transitions. The phase space is actually somewhat larger owing to the width of the resonance.

," 1

Chapter 3

The Detector

3.1 Introduction

The ARGUS detector [65] is a universal, cylindrical, magnetic detector designed to study

the T resonances produced in higb energy e+e- annihilation collisions. Its most powerful

attribute is its excellent ability to identify charged particles, absolutely essential for the

extraction of important pbysics from large combinatorial backgrounds. As well, it provides

good photon energy resolution and high precision momentum measurement over a nearly

411" solid angle. Located in the DORIS II storage ring at the DESY laboratory in Hamburg,

Germany, it has been taking data successfully aince 1982.

The detector comprises the following components (Fig. 3.1):

Vertex Chamber. Installed in 1984, the vertex chamber bas significantlyadded to the

vertex locating power of the detector. Information from it can be combined with

the main drift chamber to improve track reconstruction.

Main Drift Chamber. The main drift chamber is the heart of the detector. It isolates

the curved paths of individual charged partic1es in the magnetic field, and measures

their momenta. As weIl it supplies specifie ionization values necessary for particle

identification.

Time-of-Flight Counters. The time-of-8ight counters measure the flight times of par

ticles originating at the interaction point. Combined with momentum and track

information this provides an estimate of the rest mass of the particle. In addition,

the counters aid in on-line luminosity monitoring.

38

CHAPTER 3. THE DETECTOR 39

ARGUS ..

Figure 3.1: The ARGUS detector. Section shown is parallel to the beam axis. 1: muon chambersj 2: shower countersj 3: TOF countersj 4: drift chamber; 5: vertex cbamber; 6: iroD yoke; 7: solenoid coils; 8: compensation coils; 9: mini-,8-quadrupolcs.

CHAPTER 3. THE DETECTOR

DORIS II

C1\VSTAL SALL !xpe;men&

Figure 3.2: The DESY 3ynchrotron and DORIS II storage ring.

40

Electromagnetic Calorimeter. The electromagnetic calorimeter measures electron and

photon energies. In addition, shower energies and shapes are used to distinguish

between hadrons/muons and electrons.

Muon Chambers. The muon chambers serve as Bags for penetrating charged particles.

Each of these components will be discussed in more detail in the following sections.

3.2 The Accelerator and Beam

ARGUS studies the T resonances through collisions of electron and positron beams sup

plied by the DORIS II storage ring (Fig. 3.2). The beams are produced by separate small

energy linear accelerators (LINAC). Because positrons are not as plentifully produced as

electrons, these bave to be gradually accumulated in a small st'1ragc ring (PIA). Bunches

of 2.7 x 1010 particles are then introduced into the main synchrotron DESY, accelerated

to an energy compatible with that of the DORIS II, about 9.0GeV to IlGeV, and subse

quently injected. The storage ring operates in single bunch mode at a frequency of 500Hz


and is capable of storing currents of 30 to SOmA. Bunches collide once every microsecond

with beam dimensions of 480pm and 85pm ;n the z and y directions respect i vely. An

average luminosity of 3.3 x 1031cm-2s-1 is achieved (22).

3.3 Location of the Detector and Harnessing of the Bearn

The ARGUS detector is located between two of the horizont.al focussing quadrupolcs of

the DORIS Il storage ring. DORIS II was constructed from the old double storage ring,

DORIS 1. As a consequence of this original design, the beam has to be diverted downwards

with the use of vertical bending magnets, and interactions take place 20cm below the plane

of the arcs. Thus, the detector components are spared the high intensity synchrotron

radiation from the main horizontal bending magnets. The detector is further shi('lded

from the low energy radiation from these vertical bends through the use of prott'ctive

absorbers called "scrapers" and a thin layer of lead paint applied to the inside of the

beam pipe.

Vertical strong focussing quadrupoles, termed "mini-p", are used to maximize the

luminosity. To achieve the desired rate, it was round necessary to install the quadrupoles

in a position where they protrude into the detector. Compensation coils surround them

to prevent the main field from leaking in and degrading the luminosity.

3.4 The Magnetic Field

A O.755T magnetic field strf!Dgth is deveJoped using thirteen 3m diameter coils which draw

a total current of 4520A. This field value was chosen as a suitablc compromise bctween

good high momentum track resolution and good low momentum particle detection. A

higher field creates larger track curvatures for fast particles allowing them to be more

accuratel:;' reconstruded. However, too high a field causes low momentum particles to

curl up too tightly to be deteded in the vertex and dflft chambers. The chosera field

strength off ers good wide range momentum resolution and allows detedion of particles

with trausverse moment a as 10was 30Mev/c. To ensure that the magnetic field does not

deftect the beam, additional compensating coils have been placcd just inside the minj·beta


quadrupoles along part of the beam patb.

In order to reconstruct accurately tbe momentum of any particle, it is imperative

that tbe magnetic field strength be weil known througbout the detedor, especially in

the main drift chamber where most of the information for momentum determination is

obtained. Through the use of several Hall probes, the field was carefully mapped prior to

the installation of the detedor components in 1982 and found to be symmetric in the ~

direction within the measurement uncertainty (0.1 %) The field values were then c\)rrected

for an estimated misalignment of the probes and adjusted to satisfy Maxwell's equations.

The field is presently modelled to within 0.2% witb a polynomial parametrization in each

of the drift chamber layers. The uncertainty accounts (or the balance oC tbe systematic

error on momentum measurement in the drift chamber.

3.5 The Vertex Chamber

Instal!~d in 1985 around the beam pipe and between the compensation coils, the vertex

chamler [66) is a bigh resolution drift chamber witb a solid angle coverage of 95%. It is lm

long and has inner and outer radii of 5cm and 14cm respectively. Strung in the axial di

rection, 594 gold-plated tungsten-rhenium sense wires (20l'm) and 1412 copper-beryllium

field wires (127I'm) combine to form small close-packed hexagonal cells. In contrast to the

main drift chamber, this configuration allows transverse coordinate measurement only. To

inhibit multiple scattering, the inner and outer cylindrical walls are constructed of a thin

carbon fibre epoxy composite. The whole chamber is filled to a pressure of 1.5 bar with

C02 doped with a 0.3% water vapour component to prevent wire degradation.

The spatial resolution as a fundion of drift distance is determined directly from the

standard deviation o( the distribution of residuals, a residual being the difference between

the measured drift distance and the value obtained from the track fitting procedure. Res

olution deterioration at small and lorge drilt distances can be accounted for by small ion

ization statistics and non-circular isochrones respectively. The chamber has significantly

increased the efficiency o( locating K~ and A 0 vertices decaying clo~;e to the interaction

point. As weil, any track information (rom it can be added to that of the drift chamber

to improve momentum resolution.


In 1990, the ARGUS vertex chamber wu replaced with a. superior micro-vertex drift

chamber (1' VDC) [67] which takes advantage of recent advances in drift charnber design.

The chamber gas is CO, with an admixture of 20% propane. Unfortunately, a short

in sorne drift wires occurred, and a.bout ten percent of the chamber volume has become

ineffective. To prevent further degradation, it was deemed prudent to reduce the operating

voltage, and therefore the pressure, by about 25%. Under these conditions, the pVDC

has a melon single track resolution of about 35pm with a vertex resolution of about 60pm

to 80pm. These are preliminary results (rom an ongoing analysis and are expected to

improve. It is hoped that increasing the propane component to 30% will enable a return

to the originally specified gas pressure without haviog to increase the operating voltage.

The vertex chamber is expeeted to provide good resolution for D flight paths. As a

by-product, a cleaner data pool for b --. u transitions may be aehieved [681 .

3.6 The Main Drift Chamber

The drift chamber [69] serves as the central track finder for the ARGUS detector. Its

excellent spatial resolution and highly efficient traek finding eapabilitics wmbine to pro

duce aeeurate event reconstruction. It 1.150 supplies good energy 1055 measurerncnts for

particle identification, and plays a central role in the operation of the slow trigger.

Loeated inside the calorimeter, the cham ber extends 2m in the axial direction and

has inner and outer diameters of 30cm and 172cm respectively. A total of 5940 sense

wires and 24,588 field wires are strung between 30mm aluminum end plates to form

a homogeneous small-cell pattern, essential for quick and efficient track reconstruction.

This pattern consists of 36 concentrie lay~rs of drift cells, each eell having a cross section

of 18 x 18.8mm2 , nearly optimal for dE/dx resolution. The number of layers was chosen as

a suitable compromise between two competing factors: to improve track recognition a.nd

sharpen dE/dx resolution a large number of layers is preferable, while a sm aller number

helps to minimize multiple scattering and avoid excessive amounts of readout wires and

electronics.

To determine the axial position of track hits, the sense wires possess successive stereo

angles of 0°, +0°,0°, -0°, etc. with 0 increasing as ..;r from 40 mrad to 80 mrad. These

J


values were chosen in order to maintain almost circular drift time isochrones for most z

values.

The chamber originally employed a mixture of 97% propane and 3% methylal as the

amplifying gas. The large radiation length and slow diffusion rate of propane provide

good momentum resolution, while its nurow Landau distribution optimizes the dE/dx

resolutioD for low momentum particles, essential for the study of the T resonances. An

additional 0.2% of water vapour had to be added to this mixture to prevent the buildup

of deposits on the field wires. The deposits encouraged charge buildup which eventually

produced discharges resulting in large, un&Cceptable wire currents. This problem, known

as the Malter effed, was cured by the water additive.

The sense wires are supplied with 2930V via IMO resistors with field wires kept at

ground. This arrangememt leads to a total gas amplification of 10· and creates ground

wire surface fields of 25 kV lem.

For track reconstruction purposes, it is important to be able to relate drift times

to drift distances. This so called drift time - space relation (TSR) is obtained from

a sam pie of Bhabha scattering events using an iterati"e procedure. An approximate

TSR is used to reconstruct the tracks of these events. The distances of closest approach

are then compared to the TDC values and the fundional relation smoothed to produce

a better TSR approximation. Repeated application of this process produces the final

parametrization.

The momentum resolution of the drift chamber is directly dependent upon its spatial

resolution, which can be determined by comparison of measured drift distances with fitted

track-wire distances from the reconstruction process. Results indicate that the resolution

sufrers from low ionization statistics at small distances and excessive deviations from

circular isochrones at large distances. The momentum resolution at high momentum

predominantly refleds uncertainties in track measurements and is given by

a('PI')/PT = .OO9PT[GeV le},

while for low momentum, multiple scattering dominates leading to a resolution of

, i


dE/d. 25.0

(keV/cm)

20.0

15.0

10.0

5.0

0.0

......

0.05 0.10

.: P .: . . ~.

~ .' '." :- . . " . .......

0.50 1.00

P (GeV/e)

Figure 3.3: Specifie energy 10ss for a multihadron sample of 10 000 events.

45

The absolute momentum seale is determined and monitored by analysing Ks -. "'+1r

deeays. The resulting systematic error on the mass scale is less than 0.2%.

The distribution of charge deposited on each wire hit by a traek is a measure of the

ionization power of a particle and is used to determine its specifie energy loss or dE/dx.

This energy 10ss is described by a Landau distribution.

Each dE/dx value needs to be corrected for two effects. First, the amount of charge

collected on a wire is strongly dependent upon the angle the track makes with the wire.

Second, due to timing, the tails oC avalanches having large drift distances escape charge

measurement. Once these etrects have been taken into account, the dEfdx values are

8.veraged, leaving out the largest 30% and smallest 10% to ensure an approximate Gaussian

distribution. The specifie energy loss for a sam pie of multihadron events is shown in

Figure 3.3.

The driCt cha.mber dEfdx measurements are an important part of the particle identi

fication process, discussed further in Section 3.14.1.


3.7 The Time-of-Flight System (TOF)

The time-of-flight system [70] serves a three fold purpose. Its most important role is

to measure charged particle flight times which, when cou pied with reconstructed track

lengths, lead to partide velocities. These are combined with the drift chamber momenta

to identify charged particles by their rest masses. 5econdary, but still important, uses

are to &id in on-line luminosity monitoring and to contribute to the makeup of the fast

energy trigger, both of which are discussed in later sections.

The TOF system forms a cylindrical shell covering 92% of the full solid angle and

surrounds the drift chamber, flush with the inside face of the calorimeter. The barrel of

the shell comprises 64 scintillation counters while the endcaps each have 48. The signal

from each barrel counter is led to a phototube at each end via light guides, while the

end cap counters are viewed by a single phototube each. Because these phototubes require

a field (ree region to operate, they are located between the coils and the flux return yoke,

and are protected from stray fields with lem thick soft iron. The light guide cross section

must be constant and eq'lal to the size of the phototube cathode. Therefore, the size of

the counters wu limited by the largest size hole that could safely be made in the yoke

without disturbing the flux return. In order to ensure homogeneous performance of the

detector, each phototube is supplied with its own high voltage source. These are adjusted

to match all the gains.

To measure the t,ime-o(-Bight of a particle, a TDC counter is started for each scintillator

when a coincidenCf. occurs bet ween the bunch crossing signal and the (ast trigger (to be

discussed below). The signal output from a counter that has been hit is split so that 80%

of it is used to stop its TDC counter aCter a cable delay correction and a descriminator

test. The other 20% is used to measure the charge on the photomultipler tube in order

to correct the TDC value (or a charge-dependent discriminator crossing time.

The TOF system was designed with the T(45) in mind. A TOF resolution of 300ps

would adequately discriminate between pions and kaons up to 600MeV le and kaons and

protons up to 900MeV Ic. Because most secondaries in B decay have momentum less than

IGeV le, this seemed to be a good target to aim (or. A resolution of 230ps for hadrons

is achieved. Monte Carlo studies, however, indicate that the lowest possible resolution is

r .


1.5

(GeV~

1.0

0.5

0.0

-0.5 0.1

:.

0.5 1.0

, .. . . .

P (GeV/e)

47

Figure 3.4: Mass squared versus momentum as derived from the ARGUS TOF system.

170ps [71 J. Uncertainty in tbe buncb crossing time, time instability of the phototubes, and

a pulse height correction parametrization which is invalid for high and low pulse values

a11 account for the discrepancy between these last two values.

From the corrected TOF values and the momentum values obtained from reeonstructed

tracks in the drift chamber, it is a straight-forward problem to obtain mass estimates (or

different charged tracks. With this method alone, a three-standard-deviation separation

of electrons from pions up to 230Me VIe, pions {rom kaons up to 700Mev 1 c, and kaons

from protons up to 1200MeV le can be made. ln practice, however, not ail charged

tracks have mass estimates. In multihadron events, about 80% of cltarged tracks with PT

greater than 120Mev le are associa.ted with use(ul TOF informa.tion. Tracks with lower

PT curl up before reaching the TOF counters. ln addition, multiple hits on a eounter

create ambiguity. Figure 3.4 shows Mf OF versus momentum for a sample of (ast lepton

multihadron events.

(


3.8 The Electromagnetic Calorimeter

The ARGUS electromagnetic shower counters [72J perform the important task of deter

mining the energies of electrons and photons as weil as the directions of these photons.

ln addition, they provide a method for diseriminating between hadrons or muons and

electrons. Other roles in the ARGUS trigger system and in luminosity monitoring will be

discussed in later sections.

The electromagnetic shower counter! form a eylindrical section located outside the

TOF counters and inside the magnet coils. Installation of the calorimeter inside the

solenoids dramatieally cuts down on the amount of material between the eounters and

the interaction point (as low as 0.16 radiél.tion lengths in the barrel region) thus providing

excellent low-momentum photon detection efficieney.

Twenty rings of 64 counters each in the barrel combined with 5 rings of varying number

of counters in each of the end caps break up the calorimeter into 1760 different segments.

Each counter is made of alternating tayers of lead and sCÎntillator and has an overaliiength

equivalent to 12.5 radiation lengths. Two different shapes are used to allow compact fitting

into the detector. A wavelength shifter is eonnected to each counter to collect the signal

and convey it to a smalt phototube via an adiabatïe light guide. This assembly is then

combined with another of the same shape to form a module, the wavelength sbifters being

sandwiched in the middle but separated by aluminized mylar to prevent cross talk. A

quartz light fibre is connected to each counter to allow constant on-li ne monitoring of the

phototubes.

The shower counters serve as an important component of the fast trigger (to be dis

cussed below). It becomes necessary therefore to treat variations in phototube gains and

module light collection efficiencies in such a way as to equalize trigger thresholds. This

is achieved in the same manner as in the TOF system, each phototube being supplied by

its own separate variable high voltage supply.

The absolute energy calibration of the shower counters is a fairly involved procedure.

An electron, positron or photon that is absorbed by the counters produces a pulse height in

the photomultiplier output proportional to its energy. The determination of tms constant

of proportionality is complicated by several factors. In any particular eounter the amount

..


of energy converted into ligbt and detected by the scintillators will only be a fradion of

the actual energy deposited (roughly 30% on average), and is dependent on tbe loca.tion

of the impact point due to spatial variations in light collection efficiencies and sigmù

transport. In addition, the energy of a parti de will be deposited over several counters,

the amount detected by ea.ch varying according to the angle and location of impact.

This is an obviou! consequence of the cylindric&.l geornetry of the ca.lorimeter (particle

production is isotropie) and the interference of its support structure and other dead space

contributors. Furtherrnore, losses along the scintillators and waveguide paths need to

to be taken into account. Finally, the photomultipliers are subject to time-dependent

gain variations which must be continually monitored in order to correctly interprct pulse

heights.

The above complex interplay of factors demands that an iterative procedure be uscd

to determine a set of constants that accurately interpret shower counter signal pulse

heights. The electrons and positrons of a samplc ur 5 x lOs Bhabha events (selccted in a

manner that minimizes the number of radiative Dhd.bha events) are tracked through the

drift chamber in order to determine their positions, directions and momenta whcn they

hit the calorimeter, although the moment a are known quite accurately already (rom the

beam energy. With the aid of a Monte Carlo analysis, a set of three equations relating

pulse heights, detector geometry, detedor efficiencies, and time-dependent ga.in variations

is solved iteratively to transform known input and output information into information

about the effect of the calorimeter on the output signal.

The energy resolution of high energy electrons and positrons is determincd from the

Bhabba energy spectrum, while that of high energy photons is found similarly with the

channel e+e- -+ "'t'Y. Due to the excellent energy resolution of convertcd photons, the

mass distribution of ".0 and ,,0 mesons with one converted photon directly renects the

resolution for low energy photons. The resolutions are parametrizcd as follows:

0'( E) = C(O 0722 0.0652)

E ~ \U. + E[GeV] (barrel) (3.1 )

( 2 0.0762

) 0.075 + E[GeV) (endcaps) (3.2)

the constant factor being attributable to the support structure (Monte Carlo result).

CHAPTER 3. THE DETECTOI

White the directions of electrons and positrons in the ARGUS detedor are easily de

rived from drift chamber information, photon directions cao only be inferred from shower

information. The method used involves weighting the positions of the counters hit. with

their pulse heights and determining the centre of gravit y of the shower for both azimuthal

and polar angles. The relation between these ealculated values and the true values is

deterrnined through Monte Carlo simulation. The angular resolution is determined by

applying this procedure to a sample of electrons and positrons from 8habha scattering

events where the directions of impact are known from drift cham ber information. Re

sults indicate that a resolution of 13 mrad (equivalent to an average of 1.5 cm in spatial

resolutioo) exists in the barrel region whereas in the endcaps the resolution achieved is

beUer by about 30% due to the smaller eounter granularity. The resolution degrades with

decreasing photon energy and is obviously physically limited by counter granularity.

The shower counters provide important information for partide identification, espe

cialty the separation of electrons from hadrons. First of all, electrons tend to deposit

ail or most of their energy in the counters. However, hadronic showers will often ex tend

past the shower counters and into the flux return, with the result that only a fraction of

the hadronic energy is deposited. Consequently, there is little or no correlation between

shower energy and momentum for hadrons, while for electrons there is a strong correlation.

Requiring the energy to be within a few standard deviations of the measured momentum

excludes a large fraction of hadrons, white maintaining a high efficiency for electrons.

Second, hadrons tend to spread their energy over a larger area than electrons. A tateraI

cut has been devised to take advantage of this facto Essentially it involves weighting the

location of the bit counters with their deposited energy to find the "('..en~er" of the shower

and then dctermining the variance of this distribution alter leaving out the two counters

with largest energy deposit: E _ .ç . .(~ - r)2E.

/Q' - L.J < >2 i=l r

(0 < J'Q' < 1)

(3.3)

(3.4)

This eut only works at higher energies because hadrons do not shower below 600Mev.

Muons can also be separated from hadrons and electrons because oC their high penetration.

This manifests itself in a smaIler number of counters being hit.


3.9 The Muon Chambers

The muon chambers [73] serve as fla.gs for penetrating charged particles. Tbe system of218

chambers i5 arranged in tbree layen, two outside the detector and one just inside the yoke.

In t • .,ta.l, they cover 93% of the total solid angle. There is a large a.mount of absorption

material between the chambers and tbe interaction point (3.3 and 5.1 radiation lengths

for inner and outer cbambers respectivelY)j this inhibits hadronic punch.through and

minimizes misidentification. PracticaJ application of the chambers is restricted to muons

of momentum greater than 700 and 1l00MeV le respectively. (This does not hinder the

study of the charmless B cbannel considered in this paper because the bulk of leptons are

predicted by tbeory to have momentum greater than IGeV le.) With the help of off·line

cosmic data analysis, the average efficiency {or each layer has been dctermined to be 98%.

Each chamber consists of a side-by-side arrangement of eight proportional tubes of

cross section al area 56 x 56mm1 and lengths varying between lm and 4m. These are filled

with a nonHammable argon-proplUle mixture [92:8] and supplied by a voltage which is

sufficiently high (2.35 kV) 50 as not to limit tbe chamber efficiency. Bearn tests indicatc

that the largest loss of efficiency occurs at the boundary walls of the tubes.

3.10 The ARGUS Trigger

Bunch crossings at DORIS II oceur at the rate of 106 per second. In practice, however,

many events are of little interest to the experimenter and can be rejccted. The decision

to do so must be made within Il's; otherwise, an event occurring at the next crossing

will be lost. Limited computer speeds rnake it impossible to carry out a complete evcnt

analysis, and an efficient method must be employed to screen each cvent quickly. This is

the role of the ARGUS trigger system.

Events which are accepted for off-Hne analysis neecl to pass a two-stage trigger test.

The first-Ievel trigger (fast pretrigger) bas to complete the clecision making process in

one microsecond, and is therefore restrictcd to information {rom the TOF and showcr

counters. This is available within 200n8 of the collision. The conditions required (or

the fast pretrigger to accept an event are discusscd below. If an event {ails to meet

these criteria, aIl counters are c1eared and the trigger returncd to a "trigger ready" state.


Otherwise, the signaIs are read out for analysis by the second level trigger (slow trigger).

The fast. pretrigger then lies in a donnant state until the event is completely transferred

to the on·Hne computer or is vetoed by the second level trigger.

The ARGUS fast pretrigger has a mean trigger rate of 100Hz. Allowing for 1 % dead

time, this gives the slow trigger lOOI'S to make the final decision on an event. Using data

from the main drift ch'amher and the TOF counters, a fast two-dimensional track search

is made ("Little Track Finder"). To carry out the procedure quickly, a hardwire de vice

with software support is used. This is discussed lurther helow. The second-Ievel trigger

reduces the trigger rate by an order of magnitude.

3.10.1 The Fast Pretrigger

The fast pretrigger makes use of TOF and shower counter information by combiniog

couoter signais into independent trigger units which each Act as a single output. These

outputs have to satisfy certain conditions in order for the event to be considered further.

Four different subtriggers are employed to do this, each having its own individual threshold

requirements.

The total energy trigger (ETOT) requires that the linear sum of shower counter ener

gies in either hemisphere of the calorimeter he greater than 700MeV. It is used to detect

events with balanced energy deposit, such as e+e- -+ "'t'Y and Bhabha scattering. It is also

used as an on-line correction for backgrounds and a monitor for the running conditions

of the storage ring.

The high energy shower trigger (HESH) segments each hemisphere of the calorimeter

into 8 groups of 10 adjacent rows of barrel counters with two rows of overlap between

thern (endcaps are not used). Designed to detect particles carrying a large amount of

the total energy, it triggers only if the summed energy of any HESH group exceeds about

IGeV.

The charged particle pretrigger (CP PT) consists of 16 trigger units in each hemisphere

of the detector. r.ach unit contains four TOF counters overlapped by six rows of shower

counters with adjacent units sharing two rows ol shower counters. The trigger searches

for events with one or more charged track candidates in each hemisphere by requiring the

coincidence of TOF and shower signals as weil as a minimum ('If ~n, r,.v energy deposit


in at least one unit per hemisphere.

The coïncidence matrix trigger (CMT) combines CPPT groups at the sa.me azimuthal

angle in each hemisphere into single CMT units. At least one pair oC tracks opposite in tP is

required. Typically, for a given CMT group signal, at least one of tbe seven CMT groups

opposite in t/J must also register a signal. This topology is characteristic oC two-photoll

events, for which the trigger was designed.

These four subtriggers, along with an additional cosmic ray trigger (Cor off·line testing

of detector components) and O.lHz pulser (to estimate random noise), are intcrprcte(f by

the pretrigger collector. Provided the fast subtrigger ~);gnals are in coincidence with the

bunch crossing signal and another event is not being currently processed (indicatcd bya

"trigger ready" signal discussed below), the pretrigger collector will pcrform an OR on

aU pretrigger signaIs and start further event processing.

3.10.2 The Slow Trigger

Tbe slow trigger or Little Track Finder (LTF) uses driCt chamber and TOF information to

find tracks in the r- q, plane. To minimise deadtime and still maintain triggcr flexibility, a

software supported hardwired device is used to analyse each event as it is acccptcd by the

pretrigger. The chamber and TOF bits are read into wire input boards (WIll). The LTF

tben accesses Crom memory a set of up to 2000 "masks" wbicb contain the hit sequences

of possible track roads. The masks are used in conjunction with the WIBs to search for

acceptable bit patterns, also stored in memory, within these masks. If the requircd numbcr

of tracks is found (dependent on pretrigger), the event is acccpted and the inCormation

obtained by the LTF Ced to the on-Hne computer; otherwise, the electronics arc rcset

and the "trigger ready" state is returned. Tbe procedure has a track finding efficicn<-y of

nearly 97%, and the maximum deadtime incurred is only 0.2%.

3.11 On-line Data Acquisition

The ouput Crom the various ARGUS detedor components is collccted and digitizcd bya

CAMAC crate system. A system crate controller enables control over the many CAMAC

modules according to commands received by the on-line computer.


Data from the CAMAC sytem is read out in an extremely efficient manner by a fast

microprocessor called a CAMAC booster (CAB). Each event can be processed into its

final format in less than 1.2 ms. An additional 1.8 ms is required to synchronize the

booster wita the on-line computer and to transfer all event information to it. Without

the booster, the event read-in time would be as high as 40 ms, severely increasing dead

time. Tbe CAB, acting in conjunction with the LTF which limits the event output rate

to about 5Hz, keeps dead time to a minimum, typically less than 5%.

The on-Hne computer, a DEC PDP 11/45, receives data from the CAB on an event

by-event basis and stores it in a buffer. In addition, it is responsible for running the entire

data acquisition software, manipulating data Bow according to parameters modifiable by

the shift operator, and controlling on-line calibration procedures of the various detector

components. Events are dumped to a VAX 11/780 in a direct memory access manner

which permits bigh speed transfer of data so as not to impede the data flow rate. The

buffering of data in the POP allows for synchronization between it and the VAX. The

VAX performs sorne changes in tbe record format of the events, as well as sorne monitoring

(sec bclow) , before dumping data to yet another computer, an IBM. The data are written

to a disk data set which is periodically dumped to tape. In the process, the final event

format is created and further monitoring carried out. Two raw data tapes, about 80,000

cvcnts, are thcn concatenated to form permanent EXDATA tapes. These are used for

off-Hne analysis (next section).

Continuous on-Hne monitoring of the machine, of the detector, and of the data acqui

sition process is absolutely essential in or der to optimize performance and mainta.in the

quaHty of data collection. This is an important role of the VAX 11/780 computer. As

events pass through the VAX buffer, they are unpacked and used to create histograms

of sorne of the more important variables. These are made ava.ilable directly to the shift

operator, allowing irnmediate recognition of any system faHures. Th, monitoring process

includes in part: hit frequencies and pulse heights for aIl detector <.Omponents, as we))

as drift times for OC wires in order to detect ADC or TDC fa.ilures; frequency distribu

tion of masks found by the LTF in order to uncover slow trigger fa.i1ures; rates for all

the triggers so as to get an idea of the storage ring background conditions. There is, in

addition, a statie monitoring system which watches for unacceptably large variations in

..


detedor temperatures, gas pressures, magnet current, high voltage, and eledronic current

supplies .

3.12 Data Analysis

The EXOATA tapes are used for off-line analysis by the ARGUS reconstruction program.

This involves several stages of data analysis which will be hriefty outlined below. A more

detailed description can be found in Ref. [65}.

A major step of the program is pattern recognition in the drift chamber. The (ollowing

five parameters are used to parametrize the helical paths made by charged particles in

the rnagnetic field: K, the curvature of the helix; do, the distance of closest approach to

the longitudinal (symmetry) axis of the detector (z - axis); ifJo, the azimuthal anglf' of thf'

track tangent at closest approachj Zo, the z-coordinate of closcst approach to the origin.

and cot 9, the cotangent of the angle the track makes with the z - axis. The origin lie~

on the z - axis at the centre of the detector.

• Two-Dimensional Pattern Recognition: This first step of the reconstruction

program uses hits among paraxial wires in the drift chamber to recognizc two

dimensional track projections; rough estimates for K, do, and fjJ are obtaincd. No

actual fitting is done at this stage. Rather, triplets of wire hits are combined and

exhaustively searched for circular consistency and similar K values.

• Three-Dimensional Pattern Recognition: This step includes the information

from stereo wires as weil. When completed, approximate values for ail track parant

eters are available, and all hits are either associated with tracks or thrown out as

background.

• Track Reconstruction: A rigorous least squares fit ')f the ideal track, as specÏ

fied by the track parameters, is made to the drift distance data. Spatial variations

of the magnetic field as well as energy los ses due to ionization are taken into ac

count. The fit procedure yields the optimum track parameter values along with

their uncertainties.


• Vertex Chamber Reconstruction: Each reeonstructed track is extrapolated

back into the vertex ebamber, and a new fit is made witb the additional information

from VDC hits along ils path.

• Vertex Reconstruction: The main interaction vertex is determined by fitting as

Many possible reconstructed tracks to a cam mon point of origin. Tracks that fit

poorly ta this main vertex are excIuded. A seareb for secondary vertices (K~ decay,

AO deeay, photon conversion) is then carried out using these rejected tracks. A final

attempt is made to a,uociate tracks from the main vertex with any of these new

vertices.

• TOF Reconstruction The reconstruction of particle times-of-Bight is earried out.

Traeks from the main drift cham ber are assodated with hits recorded in the TOF

counters, and assigned velodties.

• Shower Counter Reconstruction The energies and positions of particles inter·

acting with the calorimeter are determined. The shapes of the reconstructed showers

are used for partic1e identification.

• Muon Chamber Analysis Each hit in the muon chambers is affiliated with a

charged track. This information is then used for part.icle identification.

EXDATA events analysed by the ARGUS program, and which either 1) have at least

two tracks pointing to the interaction region or 2) are consistent with e+e- -+ "YI, e+ e- -+

"fIl' or Bhabha events, are written to EXPDST tapes. Tl,i~ first criterion is an attempt to

reduce bearn-wall and beam-gas evcnts. Bearn-wall events arise from beam particles which

suffer small momentum changes upstream due to bremsstrahlung or coulomb scattering

by residual molecules in the vacuum. These particles become poorly focussed and can

strike the bearn-wall. Beam·gas events are a result of imperfect vacuum conditions. The

EXDATA tapes contain Many unwanted background events due to these two types of

interactions. As these events generally originate far from the collision point for the heams,

one cao significantly reduce the number of unwanted events by demanding that at least

sorne tracks be traceable to the interaction region.

•

~.


Bhabha events are maintained in this selection 50 as to provide a me ans for calculating

the luminosity as discussed in Section 3.13 below. These events, however, are excluded

from the next and final stage o( data. selection relevant to this analysis, the crea.tion of

EXMUHA or multihadron tapes. Events selected in this stage must have at least three

tracks pointing to the main vertex or an energy deposit of at least a 1.7GeV in the shower

couoters. The data are theo re(ormatted ioto a more accessible "minidst" format which

excludes a substantial portion of the original raw data. The minid~t form is convenient

(or physics analysis.

Tbe analysis presenled in this thesis was performed using the high level ARGUS

Kinematical Analysis Language (KAL) [74], although another melhod directly using the

ARGUS reconstruction program ean also be employed. KAL greatly simplifies the analysis

process, however, by hiding a number of data manipulation and physics routines (written

in FORTRAN) behind an extremely simple yel versatile set of commands. The program

is linked to a high-performance graphies package (usually GEP) 175).

3.13 Luminosity Measurement

The luminosity is determined from the pure QED, and hence weIl understood, Bhabha

scaUering process e+e- - e+e-. By counting the rate of these events occurring within a

given solid angle, the differentiaJ luminosity can be ascertained to he

dNBhœbhœ L = dt /UBlaœbhœ.

On-Iine, as weil as off-line, luminosity measurements are carried out in order to get sorne

idea of the rate of data intake and quality of machine performance.

The on-lioe monitoring process employs the endcaps of the calorimeter, as weil as the

TOF counters directly in (ront of them. Each endcap is segmented ioto 8 scctors of 22

shower counters and 3 TOF counters each. Bhabha events are j".l"tcd by requiring an

energy deposit of at least a IGev into each of two diagonally OppU::.ltc sectors, along with

a coincidence of TOF and shower counter signaIs for each sector. C"ing the Bhabha cross

section for the endcap solid angle,

lOOGeV2

(TBladho = 58nh X El ' Ctrl


the number of Bhabha events can be eonverted into an on-Hne luminosity measurement.

Tbe off-line luminosity measurement involves fully reeonstructed Bhabha events from

the barrel alone. Tracks must have momentum greater than IGeV le, shower energy

greater than O.6GeV, and form an opening angle of not less than 165 degrees. Using a

Monte Carlo simulation, the total radiatively corrected Bhabha cross section for these

cuts is determined to be [76}

100Gey3 tTSA06/lt" = 11.38nb x E3 .

cm

correct to within about ±1.8%.

3.14 Charged Particle Identification

3.14.1 Likelihoods for General Particle Identification

Charged particles are identified by reconstructing particle masses through the use of

time-of-flight and dE/dx measurements combined with momentum information from the

drift charnber. Depending on the particle momentum, unambiguolls identification of a

particle may not be possible. In practice, a likelibood function is developed enabling the

experimenter to make probability cuts for allowed mass hypotheses. A track is considered

to be a pion, for exarnple, if the relative likelihood for the pion hypothesis is greater than

sorne value, usually of the order of 1 % to 5%.

The likelihood fULction is developed from X3 values deterrnined for each particle from

dE/dx and TOF measurements. From specifie energy loss measurements, one computes

for cach hypothesis,

X~(dE/dx) = (dE/~x - dE./~x:")l tT dE/dI. + tT'A

(i = e,I',1r,K,p) (3.5)

where dE/dx!h is the specifie energy loss value expected from theory. The uncertainty in

the theoretical values derives from the uncertainty in the momentum. From time-of-flight

measurements a similar expression is constructed:

X?(TOF) = (1/~ - 1/{J~h)2 D'TOF + D'CA

(i = e,,",, 1r, K,p). (3.6)

Here {J is the particle velocity determined from TOF measurements and D',,, once aga.in

arises from an impr:.rfect knowledge oC the particle momentum. As the two sets oC X2

..


values are derived (rom completely independent measurements, one can combine them to

form a. single set of X2 values for euh hypothesis,

x~ = X~(dE/dx) + X~(TOF). (3.7)

The likelihood function for each hypothesis is then given by

(3.8)

and used to calculate normalized likelihood ratios:

(3.9)

Here, the weights w' are used to take into account a priori relative particle abundanccs.

About 80% o( ail measured charged particles can be unambiguously identifierl witt. 'l'OF

and dE/dx information.

For the purposes of this analysis, the weight of each hypothesis, exccpt the pion

hypothesis, is set to one. The pion hypothesis is chosen to have a weight of five to reOect

the {act that pions are on average produced five times as orten as any other mcasurable

particle. A charged track is accepted for a given hypothesis if the likelihood ratio for that

hypothesis exceeds 1 %.

3.14.2 Likelihoods for Lepton Identification

Section 3.14.1 describes how charged tra.cks are assigned possible mass hypotheses usillg

TOF and dE/dx measurements. For lepton identification, information beyond TOF and

dE/dx values is available and can be incorporated into the likelihood function to provide

a better rejection of background [65J.

The interaction of electrons with the shower counters is quite different from hadrons

and muons. As has already been discussed in Chapter 3, therc exists a strong correlation

between electron momentum and shower energy deposit, while no such corrdation cxists

for hadrons. AIso, the lateral spread of showers is much more compact for c1ectrons titan

it is for hadrons. Muons appear as minimum ionizing particles in the counters and can

be clearly separated from electrons. Based on these observations, a Iikelihood function

r

, ,


particular to el,ectrons wu developed. From measured parameter values, the probability

p~ that a track belongs to a particular hypothesis k is determined for each method i

capable of distinguishing electrons, and a normalized likelihood function is computed as

we• II ,:

~e= __ ~ ___ ,=_T_O~~~4E~/~ü~,~sc ____ _

E w' II pt (3.10)

lr=e,,,,ft.K,, i=TOF.4Elü.SC

where the weights wt reftect parti cie abundances. This likelihood function, termed

LH EW ES, is an extremely powerful tool for distinguishing electrons. The efficiency

for identifying electrons is weil over 90% for momenta greater than 500MeV le and the

misidentification rate better than half a percent. For this analysis, all tracks having

LH El-V ES greater than 50% are considered electrons.

Muons are identified with a similar likelihood function, but information from the

muon chambers is used as well. If a reconstructed track cannot be assigned to a hit in

the muon chambers, the muon hypothesis is immediately rejected. Electrons, which are

always absorbed in the detector bel ore reaching the muon chambers, cannot survive this

restriction and are thererore not considered as one of the hypotheses in this likelihood

function, which is given by

w" • II If. \" = _= __ i_=_TO_F:....;.,Ii_E.:../-==ü:::::,s~c...;.;.," __ _

" E w' II ~. (3.11)

'=l',.,K,p ,=TOF,IiE/ü,SC,I'

Once again, this is a v~ry powerful method for distinguishing muons. Due to hadronic

punchthrough and muons decaying in-ftight, the IDÏsidentification rate is of the order of

2% for pions and kaons 165]. The likelihood function is tenned LH MZAI, with a eut of

LH M Z Al > 50% being employed to select muons for this analysis.

..

Chapter 4

Analysis

4.1 Introduction

This chapter details the methods used to try to extract signal events from a strong back

ground produced by charmed B decays and continuum events at the encrgy of the T( 4S).

Data taken from ARGUS runs on the T( 4S) energy are used as a source of B mesoll in

formation. The continuum background is weil described by data taken in the contin""m

at energies slightly below the T( 4S), white reliable Monte Carlo data are used to model

the dominant b --+ C decay channel. A fit of the T( 4S) data is then made and an upper

limit placed on the branching ratio for the B --+ pot;; transition.

4.2 Experimental Data

The data used were recorded by the ARGUS detector at DESY during five separate

running periods between 1982 and 1988. Poorly or incompletely recorded runs are not

included as part of the data pool, and only continuum data taken al energies just below

the 1(4S) are deemed acceptable for analysis for reasons outlined below. Data rccorded

during sc ans of the T(4S) are also excluded because the centre-of-mass energies differ (rom

that of the peak cross section. CODsequenty, these data contain B rnesons with different

boosts. This is rather difficult to mode}, and the amount of scan data does not warrant

the effort involved. The totalluminosity for this restricted data sample is 167.1pb-1 and

70.7pb-1 for the T(4S) and continuum respectively.

Although continuum events below the T(4S) do not produce B mesons and therefore

cannot exhibit a true signal, it is necessary to apply to this data ail cuts used on the

61

f ..

CHAPTER 4. J.NALYSIS 62

ARGUS Experimental Data T( 4S) Energy 10.58 GeV

Continuum Energy 10.43 GeV - 10.55 GeV T(4S) Luminosity 167.1 pb-'

Continuum Luminosity 70.7 pb-1

Continuum Luminosity (scaled) 72.7 pb- ' Continuum scaling factor 2.30

Table 4.1: Statistics for ARGUS experimenta.l data.

T(4S) data. This is because the T(4S) resonance sits on a large continuum background

and this background cannot be completely distinguished from pure BB events. The data

collected from just below the T( 4S) energy provide a pure continuum data set identical

in every way to the continuum data at the T( 4S) except in energy. This is ensured by

considering only continuum data at energies very close to the T( 4S) so that no differences

in decay schemes are exhibited. (For example, were the continuum energy used very much

lower than the T( 4S) energy, one might expect a significantly sm aller number of particles

to be produced. As this analysis requires cuts on the pion multiplicity, such an effect is

unacceptable. )

The small difference in energy ma.nifests i~self in two ways. First, the cross-section for

continuum events decreases as the inverse of the square of the centre-of-mass energy, ~

(see Eq. 1.7). To account correctly for the amount of continuum data at the T(4S), one

nceds therefore to scale the continuum luminosity by the ratio of the cross sections at the

two energies, S4S Re

Le( corrected) = Le x - x D • Sc -.... 5

(4.1)

An additional factor has been included ta take into account radiative corrections to the

cross section. An obvious generalization of equation 4.1 is used for the ARGUS continuum

data poo] which contains runs at differcnt energies. From the values listed in Table 4.1,

the continuum scaling factor is computed to be r = 2.30. (An error of 270 is ascribed to

this ratio to take into account small uncertainties in the luminosities and in the energy

scaling procedure.) This number is emv10yed to produce continuum-subtracted T( 4S)

distributions.

CHAPTER 4. ANALYSIS 63

Number of Runs 5000.0 • ,

. 4000.0 -

i 3000.0 - 1

~ 2000.0 t-

,. -1 1000.0

1 t- -

0.0 & l, J La. l 9.0 9.5 10.0 10.5 11.0

ECM (GeV)

Figure 4.1: Centre-of-mass energies for available data.

Second, particle energies are slightly lower in the continuum and one must scale the

lepton energies, for example, by the ratio of the beam energies in order to mode} correclly

the continuum under the T( 4S). This is an approximation and can be implementcd

reliably only for small perturbations in the bea.m energy.

Figure 4.1 shows the centre-of-mass energies of the available data. The continuum

data used are restricted to energies greater than 10.43GeV and less than 1O .. 55GeV.

As will be discussed in more detail in Section 4.6.2, an alternative method exists

for determining the scaling factor from the data. Although subjcct to fewei possible

systematic errors, the approach is statistics limited.

After counting the number of events in the two data samples, the information in

Table 4.1 can be used to determine the total number of BB pair:, iu the T(4S) data pool.

This is calculated in a later section, alter cuts are applied in order to exc1ude beam-gas


and bearn-wall events. 5uch eveDts may not be present in quantities proportional to

the !uminosity because they depeDd on storage ri Dg operating conditions, and thererore

CaDrot be properly accounted for using the luminosity ratio.

4.3 Monte Carlo Data

4.3.1 Modelling b -+ c BB Decays

Event Generation

As has been discussed in Chapter 2, the probability that the T( 45) resonance decays to

anything but BH can be considered negligiblej therefore, once the continuum component,

obtained in the manDer described above, is subtracted, the resulting T( 45) distributions

(termed "direct") re8ect the physics of decaying BB pairs. To describe these decays, a

Monte Carlo program, MOPEK (MOnte carlo Program for Event Kinematics), has been

developed at ARGUS. An interplay between theoretical and experimental results provides

branching ratios and differential rates for the B meson dccay modes which are reCerenced

by the generator. Two-, three-, and Cour-body decays respecting these decay probabil

ities are then simulated, yielding a model which does a very good job of reproducing

experimental data [77].

Of particular importance to this analysis are the lepton momentum distributions aris

ing Crom semileptonic B and D decays. Severa! models are availablej for this work, the

model of Wirbel, Stech, and Bauert is employed. For b -+ C decays, the lepton momen

tum spectra for these diffel'cnt models closely resemble each other (Fig. 4.2) and are here

considered model-independent.

For the purposes oC this analysis, a Monte Carlo da.ta pool of 2 x 105 T(4S) events

are used. The centre-of-mass energy is 10.580GeV with a bearn energy spread of 7MeV.

Charged BB pairs are assumed to be produced with a slightly higher probability than

neutral BR pairs according to the ratio 55: 45. The mass difference between charged and

neutral B mesons is small (Ms+ - Mso = 2.0 ± 1.1 ± 0.3MeV /c2 [22]) and one expects

the branching ratios to be about equal. However, different theoretical predictions of the

l,\s noted in Cbapter 2, tbis model bas an advantage in tbat it ac:c:ounts (or semileptonic: decays to tau.

CHAPTER 4. ANALYSIS

d /dp 0.08

0.06

0.04

0.02

0.0 0.0 0.5

65

1.0 1.5 2.0 2.5 3.0

Lepton Momentum (GeV/c)

Figure 4.2: Comparison of the B+ -+ DO, D-°tv lepton spectra for the three form factor models discussed in the text. The D and D- contributions are present in the ratios predicted by the different models.


branching ratios suggest that charge effects may increase the fraction of charged BB pairs

ta as rnuch as 60% [83]. (For exarnple, the work of Ref. [78] indicates that, as a result of

radiative decays, charged pairs are favoured in the ratio 55:45. There is sorne dispute over

this result however.) The generated events also account for the eff'ects of BB mixingj the

mixing parameter is set to :t = 0.7 to reflect ARGUS rneasurernents [7]. The relathely

large error associated with the recent ARGUS measurement of the semileptonic b -+ u

rate (Eq. 4.2) requires that these events be generated separately (Section 4.3.2) in order

ta allow this cornponent to be varied independently of the b -+ C background.

Detector Simulation

In arder to make use of this Monte Carlo data, there must be sorne way of incorpo

rating the effects of the detector. Events generated by MOPEK are Ced through the

ARGUS detector simulator program SIMARG [79] which uses Monte Carlo methods to

model detector efficiencies, acceptances, and resolutions. The program is built upon the

GEANT [801 package which supplies a general framework for rnodelling ,geometry and

tracking. MOPEK supplies the initial-state direction ('nd energy of each particle and

SIMARG models the interactions that take place with the detedor along each trajectory.

After adjusting for trigger effects (TRIGGR), the output information frorn SIMARG can

be considered equivalent to the output (rom the actual ARGUS detector. This is then

analysed with the sarne ARGUS reconstruction program that is used on the real data

(Chapter 3).

4.3.2 Modelling b --+ U Events

Event Generation

Thcory predicts that the B -+ pofV channel is of the order of 5 to 10% of ail b -+ u

semileptonic transitions [46, 59] and recent ARGUS results place the rate of b -+ utv

events at only a small fraction of the dominant b -+ cf1ï rate [5,41, 81]:

f(b -+ uliï) f(b -+ diï) = (1.5 ± 0.5)%. (WSB model) ( 4.2)

Clearly, the probability that both mesons in a BB pair decay to charmless semiIeptonic

states is negligible. To model b -+ U transitions, therefore, one B is always decayed via

...


the b --+ C transition, with the other undergoing the much less likely b -+ U tr&l1sition.

Two types of b -+ u data sets are requil"ed for this analysis, one exhibiting ail possible

b -+ util transitions and the other containing the B .... ptJfïj channel alone. The latter is

generated usîng MOPEK but the {or mer requîres the use of a reliable free quark dccay

model. The generator for these decays is the Lund JETSET program (Version 6.2).

For the pure B -+ ptJliï signal, the three models outlined in Chaptcr 2 are used. Only

charged B mesons were generated because these are responsible for the signal. Each of

the three data sets has the B+ decaying via any possible (semileptonic or hadronic) dccay

channel, and the B- undergoing signal decays according to one of the WSB, KS, or ISGW

models. The difference between the lepton spectra in B+ and B- decays is quite negligible

for this analysis. For each model, a data set of 2 x 104 events was gcncrated.

The second type of data set required is a model of the inclusive leptoll spedrulll for

aIl possible b -+ utii transitions. The (orm factor models described in Chapter 2 rakulittf'

ex-:lusive differential decay rates to only a. few of the lowest lying meson state.' and, u nk:-.:-.

one restricts oneself to the endpoint region, cannot provide a complete description of ttw

inclusive mornentum spectrum. Although there is sorne doubt as to th<, validit.y of tl\('

ACCMM free quark decay mode} oear the endpoint, it is expected to do a rea.'lonahle

job at }ower lepton energies [49, 56). At any rate, this analysis requires a modcl for the

inclusive lepton spectrum over a. mueh larger momentum intcrval than the endpoint and

the ACCMM model is the best available.

Because the model describes decays at the quark level, it cannot be implemenlcd within

the MOPEK framework, and a different generator is used, the Lund JETSI-:T progra.m

(LUND). LUND generates e+e- annihilation events and decays jets according to the wdl

known LUND fra.gmentation scheme. The entire decay chain is done at the quark leve!.

The generator is best suited for bigh energy jet fragmentation but is applicable al the levci

required for tbe ACCMM model. A B meson pair, whose kincmatics are consistent with

the 1'(4S), is \oaded ioto the generator and allowed to decay according to the ACCMM

mode!. As will be recalled from Chapter 2, this involves ascribing a Fermi mOmf'utllHl

to the spectator quark which fixes the virtual mass of the b quark in the B rneson. The

b quark is deeayed to a u or c with LUND then dressing the final u or c quark and the

spectator quark into hadronic states. Although the ACCMM modcl does not provide a


prescription Cor Corming these final states, the leptonie seetor oC the inclusive data set is

generated exactly aecording to the ACCMM model. Allowing LUND to take care oC the

badronic sector is expected to he acceptable beeause the generator ha., heen "tuned up"

at ARGUS to properly reftect experimental and theoretieal analyses of different B decay

schemes. A total of 5 x 104 events was generated in the same manner as the excluJive

B -+ p°tv data. The mass oC the u quark and the Cermi momentum value, the two

paramcters in the model, were set ta O.325GeV Je2 and O.30GeV Je respect i vely.

Detector Simulation

ln practice one ean only ._.ode} a detector as weil as one understands it. The ARGUS

deleclor, however, has been extensively studied and its performance constantly moni

tored. As a rcsult the simulated interactions between a particle and the detector can

get extremcly detailed, and unavoidably time-consuming. For most prelirninary analysis

purposes, it is more useful to employa much faster although less precise program, MIN

IMC. Whilc the MINIMC program must necessarily be less accurate than SIMARG, it is

fairly reliable for most purposes. The major difference betwecn the two simulators is that

MINIMe is a parametrized mode} that does not record hits on drift chamber wires, and

thereforc has no track reconstruction. Both the b -+ U FQD events and the B -+ p°tv

signal events were run through MINIMe. As will he clear later, using SIMARG to mode}

the B -+ p°tfï signal would have been unnecessarily detailed in Iight of the final signal

parameter errors.

4.4 Multihadron Event Selection

The EXMUH A tapes are the source of data used in this analysis. As indicated in Chapter

3, these are events which pass sorne crude multiplicity cuts made in an atternpt to exclude

bcam-wall. beam-gas, and Bhabha events. Harder multipHcity cuts are made (or this

analysis ta cut out unwanted radiative Bhabha events, e+e- -+ p.+ P.-1 events, and "'11

interaction events. Although in theory radiative decays only have two charged particles

in the final statè, photon conversions in the detector can ea:.ily increase this number to

four and contaminate the EXMUHA sam pIe.


Requiring that events satisfy

NçAarged ~ 4 (4.3)

and

N tot = Ncho.rged + .5N-y(.lGeV < E.., < 1.2GeV) + N-y(E.., > 1.2GeV) ~ 6 (4.4)

yields a multihadron data set 2. It should be noted that eledrons from photon conversions

in the detector are Rot included in the value of NcAo.rged. Any pair of oppositely charged

secondary vertex tracks consistent with the electron hypothesis and having an invariant

mass less than 100MeV /c2 is considered to be a converted photon and includcd in the

neutral multiplicity COUot. The second term in the total multiplicity expression takcs

into account the fact that high energy photons arise predominantly from lH'utral pion

decay. The factor of .5 ensures that one is correctly counting them. Figure 4.3 show~

the neutral pion contribution to the photon spectrum due to T( 4S) decays a.<; gent'rat('d

by MOPEK. The third term in Eq. 4.4 reflects the inability of the ARG US detedor to

resolve photons from high energy pion showers in the calorimetcr becausc the opt'ning

angle of the two photons is small. Reconstructed converted photons with energy grcat<,r

than 1.2GeV cODtribute to the second term, however, because the electron and positron

can be differentiated from the shower through tracking.

The multiplicity cuts only marginally decrease the acceptance for 1(45) evcllts. This

is because B mesons from the T( 4S) undergo a cascading decay sequence that ncccssarily

yields a large number of particles in the final state. The average charged multiplicity

for T( 4S) decays is 10.99 ± 0.06 ± 0.2 while that for photons is 10.00 ± 0 .. 53 ± 0.50 [83].

The normalized T (4S) charged and total multiplicity spectra a.s gellerated using MO P EK

MINIMe are shown in Figure 4.4. The acceptanc\.o for T( 4S) decays ha.<; bccn compute(J

to be (97.3 ± .5)%. using the Monte Carlo MOPEK-SIMARG data set.

There are good reasons for wanting to reduce the contribution from QED background

sources. Although continuum subtraction in principle yields the propcr T (4S) distribu

tions, pro cesses tbat do not proceed througb the e+e- annihilation channel are not present

2The multihadron selection cuts used in this work parallel those used in t II!. theses of T. Ruf and J .C. Gabriel [6, 82].


~/6p pet Event 1.5

1.0

0.5

0.0 0.0 1.0 2.0 3.0

Photon Momentum (GeV/e)

Figure 4.3: Pion and non-pion (hatched) contributions to the photon spectrum as generated by MO P EK.

in amollnts proportiona.l to the lurninosity. The rate for bearn-wall interaction events is

severely afTected by the quality of beam focussing, which is machine-dependent. The

beam-gas event rate is likewise time-dependent, suffering frorn the quality of the vacuum

in the storage ring. Two-photon interactions can aIso be expected to be inftuenced by the

focussing of the beam. In addition, the cross section for two-photon events does not scale

with energy in the same manner as annihilation events, and tbe process for correcting

the continuum luminosity for centre-of-mass energy cannot take this into account. It is

imperative, therefore, to reduce the contamina.tion from these events as much as possible.

Finally, a.lthough the remaining low multiplicity annihilation channels mentioned above

will be properly subtracted, they are of no interest for this analysis. Excluding them

increases the statistical power of the data sam pIe.

The numbers of events in the multihadron data sets for continuum and T(4S) data

are given in Table 4.2. Using the continuum scaling Cactor of r = 2.30 ± 0.05, one can

CHAPTER 4. ANALYSIS

, of Ewnta 0.20

0.15

0.10

0.05

0.0 0.0 5.0 10.0

--. Totol 1

11

15.0 20.0

Multiplicily

71

25.0

Figure 4.4: Normalized distributions of charged and total multiplicity (or BB decays as generated by MOPEK-MINIMC.

Multihadron Dataset T( 4S) Luminosity 167.1 pb- I

Continuum Luminosity 70.7 pb- I

Continuum Luminosity (scaled) 72.7 pb- I

Continuum scaling factor (2.30 ± 0.05)% Number of events at T(4S) resonance 610,618

Number of events in the continuum 204,203 Acceptance for T(4S) decays 0.973 ± 0.00.)

Number of charged B Mesons 159,000 ± 11,000 Calculated cross section (0.87 ± 0.06) nb

Table 4.2: Statistics for the multihadron dataset.

DHAPTER 4. ANALYSIS 72

compute the total number of charged B mesons in the data sample:

NB* = 2·0.55· (N4s - Ne' r)/"BB = 159000 ± 11000 (4.5)

"ere it is assumed that the T( 4S) decays 55% of the time into charged B mesons (no

uncertainty is ascribed to this value). The uncertainty in the number of charged B mesons

is dominated by the uneertainty in the scaling ratio.

Using the number of BB pairs, the cross section for e+e- -+ T( 4S) events is determined

to be NT(4S) (

f7Y(4S) = L = 0.87 ± 0.06)nb T(4S)

consistent with other measurements [82]. This value should be interpreted as the visible

cross section in the DORIS II storage ring and not an estimate of the rf'.,onance cross

section. The T( 4S) has a natural width of (23.8 ± 2.2)MeV [22]; the 7MeV energy spread

of the storage ring contributes significantly to the visible width of the resonance.

4.5 Fast Lepton Selection

4.5.1 Lepton Moment'lm Cuts

The first step towards extracting a signal from the data is to require that there be at

least one fast lepton per event. A preliminary momentum cut of p/ep'on ~ l.OGeV le is

mad,'! on ail lepton candidates. It is clear from Figure 2.8 that cutting at this momentum

preserves essentially ail of the signal; the cut is required to reduce the contribution from

secondary decays. (Stronger momentum cuts are made later in order to exploit the ract

that the signal lepton momentum spectrum is much harder than that due to the dominant

B -+ D·l:iï and B -+ Dtii transitions.) A eut on the maximum value of the lepton

momentum (J"ep,on $ 3.0GeV Ic when one takes into account detector resolutions and the

motion of the B mesons) should evidently be appliedj for the moment, this cut is left out

for reasons to be made clear in Section 4.6.2.

Angular cuts are a1so made on the lepton candidates. Due to a high muon misiden

tification rate in the detedor endcaps, muons making an a.ngle less than about 45°

(Icos 61 > 0.7) with respect to the beam axis of the detector are excl uded. There is

also an angle cut made on electron candidates, but for a different reason. An attempt is


made to suppress electrons resulting Crom photon-photon interaction events. Two pho

ton events arise {rom the collision of virtual photons emitted from both the electron and

positron beam particles as they pass close to each other, resulting in low energy hadrons

being deposited in the detector. No annihilation of the beam particles takes place and

they usually scat ter undetected at smalt angles down the bearn ripe. Occasionally, how

ever, a high-Q2 "rr event will result in one of the initial beam particles being scaltered

into the detector. Such an electron will necessarily have an energy lower than the beam

energy and could CaU into the signal filvL1entum range oC P,ep,o,. $ 3.0GeV le. To elim

inate this background, one takes advantage oC the fa.ct that the beam particle scaltcred

down the bearn-pipe will,show up as a large missing momentum in the evcnt (provided

few energetic particles go undetected or are lost in the trad:ing routine). The direction

of the missing momentum should correlate with the direction of the beam partic1e that

has opposite charge to tbat oC the scattered electron (i for positrons, -z (or e1cctrons).

A plot o( ... . Pml.s . Z

QCmau = Qt'i.'h 'Jo( 1 ... 1 pmu~

(4.6)

therefore displays an enhancement duc to 'Y, events near +1 (Fig. 4.5). Accepting events

\Vith QCmw ::; 0.9 cuts out a good number of unwanted electrons (rom this source.

Finally, electrons from converted photons are significantly reduccd by excluding a.ny

electron candidate which, when combined with any other oppositely charged track con

sistent with the electron bypothesis (likelihood of 1 % or greater) has an invariant ma.'iS of

less than lOOMeV le2• Figure 4.6 shows the converled photon signal.

4.6 Continuum Suppression

Ail cuts described thus far have done little to distinguish B B decays frorn continuum

eventsj it is clear from Figure 2.3 that a substantial number of continuum evcnts occur

at the 1(4S) energy, and that sorne method must be employed to suppress them.

4.6.1 Thrust Axis eut

There are different ways of suppressing the continuum with respect to T( 4S) decays but

these all exploit the same basic chara.cteristic of continuum events. Being substantially

'" J

CHAPTER 4. ANALYSIS

• of El.chou 1500.0

1000.0

500.0

0.0 -1.0 -0.5 0.0 0.5

Figure 4.5: QCmi .. for T( 48) data.

74

1.0

QCmiss

CHAPTER 4. ANALYSIS

, of Elec\ ·on Pairs 2S00.0

2000.0

Photon 1500.0 Con .... r.lon.

1000.0

500.0

0.0 0.0 1.0

75

2.0 3.0 4.0 e-+e- Invariant Mass (GeV/c ')

Figure 4.6: Invariant mass plot for e+e- candidates when one candidate is a selectcd fast lepton.

greater in energy than the t/J resonances, ail qq pairs produced in the cOlltinuum at the

T( 4S) energy tend to hadronize into particles having momenta. a.long the original directions

of the decaying quarks. This is the well-known jet-like st.ructure of qq events away from

resonance regions (see, for example, Ref. [25]). At the energy of the i(4S), 10 .58G<,V,

the emission of hard gluons (e+ e- -+ qqg) is small and two-jet evcnls dominatl' 3. 0,14'

therefore attempts to determine an axis along which the original qq pair is assumcd to

be produced. Several possible event shape variables have been devcloped [851. One sueh

variable, th rus t, involves the determination of an event thrust axis, dcfincd as the unit

vector fi. which maximizes the following expression:

T _ {El Ii. . ni} - m~x ~ 1"'1

n l.J1 PI (4.7)

where the Pa are the measured tracks for a. given event. (Tbe actual value of T is not

of interest for this analysis - it essentially re8ects to what degree the dctected particles

cluster around the thrust axis, loosely analogous ta variance.) The hadronizatian proccss

3Three-jet events become observable at about 30GeY wltb a probability of QS/7r [84].

,

\ ..

CHAPTER 4. ANALYSIS

1 of Electrons 2500.0

2000.0

1500.0 + +

1000.0 + + +

+ 500.0

0.0 -1.0

Ac:cept

-+ R •• onone. ++ +

+++.. ~~ .... ++~ ................. +~ ... -

ontlnuu ..

-0.5 0.0 0.5 ElectTon-Thrust Axis Angle

+

+ + +

1.0

76

Figure 4.7: Angle between thrust axis and electrons (Pt> IGeV le ) for continuum data.

produces very energetic hadrons. Consequently, leptons from semileptonic decays of these

continuum particles suffer large boosts along the jet orientation and are expected to be

slrongly correlaled in direction with the thrust axis. In contrast, B mesons a.re produced

almost at rest at the ï( 4S) and decay isotropically. Although a thrust axis can be

defined for this case according to the above equation, it is clear that this axis is quite

meaningless and that no correlation is expected between it and the direction of fast

leptons. The angular correlation, cos (J, between electrons and the thrust axis is shown

in Figure 4.7. A cut of Icos81 ~ 0.7 is used to enhance BR events while suppressing

continuum contributions. The cut suppresses the continuum by a factor of about 16 for

events with leptons havi!lg momentum above IGeV le while harder lepton cuts of Pt > 1.6

to Pt > 2.0 suppress the continuum by a factor of 10. About 60% of signal and non-signal

B B events pass this eut.


4.6.2 Momentum Cuts

The other eut used to suppress continuum events exploits the fact that, assuming 100%

decay to BB, particles produced by the T(45) resonanee are kinematically restricted to

momenta less than 2.82GeV le. (This result is arrived at in a straightforward manner by

considering the extreme case whereby a B meson decays to two massless particles that

are boosted according to the initial motion of the B. In reality, sueh a decay channd docs

not exist but it serves to provide an ultimate upper limit. More correctly, one should

consider the B -+ 7r1:ii deeay channel which yields effectively the sa.me number anyway.)

T~is limit is due to the intermediate production of heavy B mesons. Continuum events

are not constrained by the production of B mesons and can conœivably yield particl('s

with momenta up to halC the centre-oC-mass energy (e.g., ~+e- -+ qq -+ 11'11', although

such an event would obviously not pass the multiplicity cuts). Any event containing a

charged (leptons excluded) or neutral particle with momentum greater than 3GeV je or

3.5GeV jc respectively is rejected as a continuum event. These values are greater th an

the kinematic limit in order to take into account detector rcsolutiolls.

The lepton spectra resulting from the application of ail cuts describcd 50 far art' shown

in Figures 4.8 to 4.11 for tbe various data pools. Note that the distribution Crom tll('

b -+ U data contains a sm.ü] admixture of b -+ C leptons. These are produced by thf' 8

which does not undergo b -+ u transitions.

The reason why leptons are ignored in the kinematic Jirnit cut is to allow an additional

method to be employed for determioing the continuum scaling Cactor, r. Because leptons

in the region glcater than 3GeV le can only he ascribp.d to continuum evcnts, the ratio

of the numher of leptons in the T(4S) data sample to the continuum data sample (aft.cr

scaling the mûmenta of leptons in the continuum data sam pie by the ccntn~-oC-mass

energies) directly provides a new estimate oC r. In principle, this mcthod is much better

than scaling by the luminosity ra.tios. To justify the luminosity ratio technique, one must

demonstrate that the lepton identification efficiency and hadron misidcntification rates

are constant with time; otherwise, the total luminosities gathered in the continuum and

on the T(4S) may Dot reBect the total number of leptons identified for analysis. One

could circumvent having to worry about this by alternating frequently betwecn running

" i

...

CHAPTER 4. ANALYSIS

, of Leptons 1200.0 ---1000.0 +t

+ ++++ +++ Electrons 800.0 + Muons

~+ 600.0

400.0

200.0

0.0 1.0 2.0 3.0 4.0

Lepton Nomentum (GeV/c)

Figure 4.8: Lepton momentum spectrum from ï(4S) data.

1 of Leptons 200.0

150.0

100.0

50.0

0.0 1.0 2.0

+++ Electrons

Muons

J.O 4.0 Lepton Nomentum (GeV/c)

Figure 4.9: Lepton momentum spectrum from continuum data.

78

CHAPTER 4. ANALYSIS

, of Leptons 1500.0

1000.0

500.0

0.0 1.0

+++ Electrons

Muon.

2.0 3.0 4.0

lepton Momentum (OeV/c)

Figure 4.10: Lepton momentum spectrum from Monte Carlo BB data.

, of leptons 800.0

600.0

400.0

200.0

0.0 1.0 2.0

+++ Electrons

Muons

J.O 4.0 lq>ton Momentum (GeV/c)

Figure 4.11: Lepton momentum spectrum from Monte Carlo b --t u decays.

79

CHAPTER 4. ANALYSIS

1 of Electrons 600.0

400.0

200.0

0.0 2.0

+ Resononce

+ +

2.5

80

' ...... .....,.,...,....., Ille .... lit .... lnW'fIIIII ."".,... ....... IIdInt ...... fIIIh ~ .... fectar.

3.0 3.5 4.0 Electron Momentum Distribution (GeV/c)

Figure 4.12: Lepton spectra (or Continuum and T( 4S) decays beforc application of the thrust eut.

on the T( 4S) and off [86], thereby ensuring cancellation of long range drift effects in

the detector. ARGUS obtains data over long continuous runs at one energy, however,

and an investigation of the mean lepton efficiency per unit luminosity indicatcs a non

negligible time variation. Although there is sorne hope that this effect averages out over

time, a more valid method for determining the continuum scaling factor is preferred. The

procedure adopted here is to compare the numbers of lertons from continuum and T( 45)

data in the region 2.8 < Pt < 4.0GeV le, kinematically forbidden to BB decays. 4 The one

disadvantage of the method is that it suffers from low statistics. Figure 4.12 shows the

lepton spectra for continuum and T( 45) decays after ail cuts exccpt the thrust eut have

been applied. The continuum spectra have been scaJed by the centre-of-rnass encrgies.

Table 4.3 shows the numbers of electrons and muons along with the scaling factors derivcd

from them. Within statistical errors, the values agree with the corrected lurninosity ratio

4 Although imperCect detector resolutions enable lE'ptoDs Crom 6 -+ u decays to populate the reglon 2.8 < Pl < 3.0GeV Ic as weil, tbis contribution is expected to be negligible in comparison tn that of the continuum. In considering this larger interval, one can dramatically iocrease the static;ti,." "r the sam pie because the continuum lepton spectrum ralls off exponentially with energy.


Continuum Scaling Factors e 1 po

T(4S) 604

1 839

Continuum 268 369 r 2.25 ± 0.23 1 2.27 ± 0.20

Table 4.3: Continuum scaling factors as derived from the numbers of leptons beyond the kinematic limit for B decay: compare to the luminosity ratio of r = 2.30 ± .05.

of 2.30, but have a much larger uncertainty.

4.7 Signal Enhancement

Evidence for b -+ U decays was found in 1989 at ARGUS by examining the lepton mo

mentum spectrum in the region accessible only to charmless B decays [4]. Subtracting the

continuum component yielded a smali but statistically significant excess of leptons that

could only be attributed to these events. The region examined was from 2.3 to 2.6GeV je,

which is essentially populated by charmless semileptonic decays to the 'Ir ,P,W,77 and 11'

mesons. None of these channels dominates and the pO channel is expected to account

for only a small fraction of an already very small number of leptons. The only 'way to

determine the pO contribution to this signal would be to analyse individual events one

by one and try to reconstruct the entire decay scheme. This has in fact been done [5],

but the particular B -+ p°tii channel has not been seen. Even if such events were re

constructed, there are not enough statistics available in this limited region to make any

kind of meaningful statement about the branching ratio. For example, using the model

of Korner and Schuler, which predicts the largest rate for this channel, one expects only

about 20 leptons in the region 2.3 - 2.6GeV jc (!VubjVcb 1 = 0.11 ± 0.01) before any cuts

are applied. It is clear, then, that an accu rate determination of the branching ratio relies

on increased statistics. One way to a.chieve this is to search for a signal over a larger mû

mentum interval, provided sorne way of suppressing the b -+ C background can be found.

The recoil mass technique has been chosen for this purpose.

...


4.7.1 The Recoil Mass Technique

The B -+ p°tv c~annel is a very good channel to seare!: for evidence for b --+ U decays

because it involves three detectable charged particles in the final state. Fast leptons are

easily identified with the ARGUS detedor. The pO decays almost instantly to two charged

pions (~ 100% branching ratio) wbose four-momenta are constrained by the mass of the

po. The neutrino presents an easily remedied problem. Its presence goes undeteded but

can be inCerred from the kinemé'.ties oC the other particles. This is possible becausc, Cor

a signal event, the po and lepton result from the decay of a B meson almost at rest,

thus providing a eonstraint for determining e neutrino four-momentum. The pO, l, and

v moment a must be consistent with that of 1.', B. The presence of the neutrino can

therefore be inferred through a calculation of the missing mass as follows:

PB = Pp + Pt + pv

"'-+ pv = PB - pp - Pt

2 - 2 (E E E )2 ( ... ... ... )2 ""-+ mil = PlI = B - p - t - PB - pp - Pt (4.8)

This is termed the recoil mass method as one is trying to determine the mass of the object

recùiling against the pO - l system. AlI quantities in Equation 4.8 are known with the

exception of the B momentum. The magnitude can be derived from

1- 1 1 J;"2 Ml PB = VL.·B - B = (4.9)

but tht> direction remains unknown because the ftight paths of B mesons are too short to

be resolved by the ARGUS detector. However, the magnitude is small compared to that

of the pO and lepton moment a and one can approximate the recoil mass by neglccting the

motion of the B altogether:

(4.10)

(Note that the magnitude of the B meson momentum is not used at aIl here. This is

ra.ther disconcerting as a potentially useful bit of information is being discarded. The

author recognizes thi~ shortcoming and suggest& a way of avoiding it in Appendix O.)


, of Pion Pairs 4000.0 , , •

+ + +++

3000.0 ~ + +. ~ +

+ +

~

2000.0 + +

+ c.n.ddn ,.., ". ++ +

+ ... 1000.0 - ... -

+-4-... -- ---- --..

0.0 1 1

0.0 0.5 1.0 1.5 2.0

ft+1\'- Invariant Mass (CeV/c2)

Figure 4.13: Invariant mass distribution for 11'+11'- combinations, as selected from T( 4S) decays to fast leptons, having momentum greater than 800GeV le.

The objective, therefore, is to select ail possible 1r+1I'-l combinations (the fast lepton

is not considered a pion even if it satisfies the pion hypothesis) in a given event, where

the 11'+1r- mass is consistent with the pO, and calculate the recoil mass according to

Equation 4.10. A signal should then appear as an enhancement at zero recoil mass

reflecting the presence of the massless neutrino. The signal necessarily lies on a very

large background due to the large number of 1r1l' combinat ions possible in a given event.

Charged pions are present wtth an average multiplicity of about eight, and assnm' '~

an even number of positively and negatively charged pions, one expects an average of

4 x 4 = 16 11'+11'- pairs per event. Cutting on the invariant mass of the 11'11' system reduces

the number of possible pO candidates but not as effectively as one would hope. The pO,

with a mass and width of 770GeV /c2 and 150GeV /c2 respectively, is a very broad object

and, in order not to drastically reduc.:. the signal efficiency, a modest cut of 620GeV /c2

< Mn < 920GeV /c2 is applied. A quick glance at Figure 4.13 shows that a. subf-+,antial

number of fake pO candidates survive this eut.

One ean of course also eut on the momentum of the 1r1r system. Figure 4.14 gives


~/dp

0.06

0.05 • ISGW

• WSB 0.04 a KS

0.03 .. •

t • 0.02 • • • • 0.01

0.0 0.0 1.0 2.0 3.0 4.0

p' Momentum (GeV/c)

Figure 4.14: Momentum spectrum for pO candidates from MOPEK-SIMARG BD decays (solid histoglo.l.m) and continuum (error bars) as compared to theory (arbitrary norma\ization).

the Monte Carlo moment.um spectrum for pO candidates from BB and continuum clecdys

as compared to the expected distribution !rom the signal. A cut of Pp > 800MeV je is

applied. A maximum momentum constraint of Pp < 3GeV Je , refleeting the kincmatic

limit for pO production in the signal channel, is also placed on the po. Even after thef,c cuts,

many po candidates exist. (For the recoil mass plots, the lepton mornenta arc reslrictcd

to less than 3GeV Je as weIl. It is only for the lepton momentum distributions thal this

constraint is relaxed.)

The signal recoil mass spectrum, as generated by MOPEK and run through MINIMe,

is plotted in figure 4.15. The pure signal, that resulting from cuts made on the decays

of single B mesons, appea.rs as a rather broad peak centered at zero recoil. Detector

resolutions contribute to the width, !:>ut the major factor is the neglected motion of tlH'

B meson. This is demonstrated in Figure 4.16 where signal evcnts from motion\ess 11

mesons are compared to the true signal. Unfortunately, signal events always come along

with another indistinguishable B, ~.nd so in reality these events produce a recoil mass

CHAPTER 4. ANALYSIS

, of p-lepton Pair' ~.O ~~r-~~~~~~~~~~~~-r-r-r~~-,

300.0

200.0

100.0

0.0 -10.0 -5.0

- Huon.

0.0 5.0

Recoil Moss2for Signol

10.0

[(GeV/C 1)']

85

Figure 4.15: Monte Carlo recoil mass signal gener!.\ted accordillg to the ISGW model for a single decaying B on the T( 4S). The tail on the electron distribution is due to bremsstrahlung in the detf!ctor.

CHAPTER 4. ANALYSIS

Number of Events

1000.0 f ' ,

800.0

600.0

400.0

200.0

0.0 -10.0 -5.0

BIs at rest

S's in motion

86

•

0.0 5.0 10.0

Recoil Moss2 fOf Signol [(GeV/c')']

Figure 4.16: Comparison of the signal width for motionless and boosted B mcsons, dClllonstrating the dominance of the unknown B momentum on the signal width.

spectrum that looks like that of Figure 4.17. The signal is diluted by a large back:~:-:)lJnd

due to fake pO candidates. Another eut is introduced later to enhance the pure signal over

this background.

The recoil mass 3pectra for non-signal Monte Carlo BB events as weIl as for the expcr

imentall'(4S) and continuum data are shown in Figures 4.18 to 4.20. The T(4S) "direct"

recoil mass spectrum, obtained from subtracting tbe continuum component (scaling

factor from the luminosity ratio = 2.30) from the T( 4S) data, is shown in Figure 4.21.

'î'he overall shape of the BB spectrum is fairly easily understood. The negative recoil

mass contribution is a result of inter-mixing of the B decay products. If a 1r1d combination

cornes from the same B, it physically has to recoil against one or severa! particlps which

neC'essarily have an invariant mass greater than zero. Ali such entries must have positive

recoil mass values (neglecting detector resolution effects). Th;s is not the case when

the combination is split between the two B mes ons because the particJe mnmcnta no

longer neecl to be consistent with the decay of a single B (the energy of the system can he

greater than 5 278GeV or the momentum greater than the kinematic limit of 2.82GeV je).

.. 1 1


'of p-lepton Pairs ~.O ~~~~~~~~r-~r-~~~~r-~~~~

600.0 +++ Electron.

- Huon.

400.0

200.0

+ 0.0

-10.0 -5.0 0.0 5.0 10.0

Recoil Moss2 for Signal [(~v/c et]

Figure 4.17: Monte Carlo recoil mass plot for full BB events containing one B -+ p°tii.

1 of p-tepton Poirs 800.0

480.0

360.0

240.0

120.0

0.0 -10.0

+++ Elechone

- Huon.

-5.0 0.0 5.0 10.0

Recoll Ho .. 2 ~G.V/ce 1]

Figure 4.18: Recoil mass spectrum from ï{-lS) data.

CHAPTER 4. ANALYSIS

f of p-lepton Pairs 80.0

48.0

36.0

24.0

12.0

0.0 -10.0

+++ Electron.

- Huon.

-5.0 0.0 5.0 10.0

Figure 4.19: Recoil mass spectrum from continuum data.

, of p-Lepton Pa;rs 900.0

720.0

640.0

360.0

100.0

0.0 -10.0

+ .... Elechon.

- Huone

, ft

-5.0

1 t • • t

+ +

0.0 5.0 10.0

R.coll Ha88 2 ~GeV/c'f]

Figure 4.20: Recoil mass spectrum from Monte Carlo B B data.

88

CHAPTER" ANALYSIS

, of p-i.epton Pairs 800.0

480.0

380.0

240.0

laJ.O

0.0 -10.0

.... Electron.

- Muone

-5.0 0.0 5.0 10.0

Recoll H0882 ~GeV/c'r]

Figure 4.21: Direct T(4S) reeoil mass speclrum.

Therefore, su ch mixed cornbinations tan populate the entire spectrum range.

89

Figure 4.7.1 shows the breakdown of the recoil spectrum as generated by MOPEK.

Tho~~ 7r7rf combinations from a. single B eonstitute only a small fraction of the total

spectrum. The peaking of this contribution around 3 to 4 GeV2 jc4 reBects the masses of

the D and D- rncsons which aecount for almost all semileptonic B deeays (Eq. 4.11). The

combinations arising from inter-mixing produce a less loealized reeoil distribution, as ex

peeted; however, it is not entirely obvious wby the shape should still exhibit a peak in the

3 to 4 Gey2 je· range. Essentially, this is due to the rather soft fake pO momentum spee

trum whieh iooks mu ch like that expected from the non-mixed comhinations. The recoil

mass values on average still reflect the dominant D-D- semileptonie deeay modes. The

Q\'craU shape of the reeoil mass spectrum can be surprisingly weIl modelled by generating

isotropie pO and lepton moment a with uniform probability on t.he intervals 0.7 -1.2GeV je

and 1 - 2.4Gt" V je respeetively, and eomp11ting the reeoil mass. This suggests that the

shape is prcdorninantly a reBection of the functional form of the reeoil mass, with the

physics of B deeays entering only insofar as it dictates the energy regimes of the lepton


1 of p-t.epton Pairs 800.0

400.0

200.0

0.0 -10.0 -5.0 0.0 5.0 10.0

Recoll "oee 2 ~GeV/c·r]

1: Contribution from combinations having a true reconstructed pO along with a lepton from the same B meson.

2: Contribution from combinations having a real pO along with a lepton from the 0IJpOslte

B meson.

3: Contribution frorn combinations having a fake po (with the daughter pions coming from tbe 5ame B meson) along with a lepton frorn the sarne B meson. This component combined with the first provides the expected contribution from a single decaying B meson.

4: Contribution from combinations having a fake pO (with the daughter pions coming from the same B me!.. ..... ll) along with a lepton from the OpposIte B rne:,on.

5: Contribution frorn combinations having a fake pO with the daughter pions coming from different B mesons.

Figure 4.22: Monte Carlo recoil mass spectrum for BB decays showing contributions from real and fake pO candidates.

CHAPTER 4. ANALYSIS

, of p-lepton Pairs 800.0

600.0

400.0

200.0

0.0 -1.0 -0.5 0.0 0.5 1.0

p-lepton Angle

Figure 4.23: Angular distribution of pO-lepton pairs(arbitrary normalization).

91

and pO. This is borne out by the continuum data, which have nothing to do with B decay,

but neverthelc3s display the same basic recoil mass shape as BB data.

Another eut can be made in order to exploit the V - A nature of weak decays. The

V - A structure is responsible for the faet that only left-handed quarks and leptons (right

hi!.nded antiparticles) participate in weak decays. Consequently, the lepton and u quark

in b -+ U decays tend to be produced back-to-baek (Chapter 1). Therefore, assuming that

the spectator gets dragged along in the process, a eut on the p - lepton angle can be

cmployed very cffectively. Figure 4.23 gives the p -lepton angular distribution for BB

decays, continuum decays, and signal deeays. To maximize the signal-to-baekground ratio,

a cut of cos Opl < -0.5 is required. The recoil mass spectra, Figures 4.24 to 4.27, change

shape slightly. The most marked change occurs with the signal spectrum which is more

localized about zero recoil, making it much casier to distinguish from the background.

A list of all cuts applied to the data is given in Table 4.4.

An examinatioll of the direct 1(4S) recoil mass plots shows wh:.t a formidable task it

is to extract a sign".i. In the region of zero recoil mass, there is no indication whatsoevcr


CUTS

Event Selection:

N tot = Ncharged + O.5N"'(.lGeV < E"f < 1.2GeV) + N..,(E"'( > 1.2GeV) 2: 6

Kinematical Cuts:

Pcharged < 3.0CeV le

Pneutral < 3.5CeV le

O.SGeV je < Pp < 3.0CeV le

Lepton Requirements:

LHEWESjLHMZAI > 50%

1.0GeV je < Pl < 3.0CeV je

exclude endeaps for l': I~·.il < 0.7

excludt e from "Y"Y events: Qm, .. x Pmu" . Z < 0.9

exclude e from converted photons: mee' > O.lGeV je'J

Topology Cuts:

V - A eut: eos(Opl) < -0.5

thrust eut: IPi· Til < 0.7

Table 4.4: List of cuts made on the data.

1 ..

CHAPTER 4. ANALYSIS


200.0

150.0

100.0

50.0

0.0 -10.0

93

... Electron.

- Huon.

-5.0 0.0 5.0 10.0

A.co 1 l Ha .. 2 ~G.V le' 'l

Figure 4.24: Reeoil mass spectrum 'from T( 4S) data after application of pO - l angle eut.

, of p-lepton Pairs -40.0

30.0

20.0

10.0

0.0 -10.0

+.+ E leetron.

- Huon.

-5.0 0.0 5.0 10.0

A.coll Ha811 2 ~G.V/c·t]

Figure 4.25: Recoil mass spectrum from continuum data after application of po -1 angle eut.


of an enhancement that could be attributed to the B -+ p°tu transition. What is more

disturbing is that the background upon which the signal should appear is not ftat, and

there is no theoretical prediction available for its shape. Any signal present must be

distinguishable (rom this background; without knowing the shape of the latter, this is Dot

possible.

It should be noted that the simplest way of getting an upper limit for the signal rate

is to assume that the errors represent the largest possible signal rate, i.e., that the signal

is so small that the recoil mass distribution essentially is the background and that the

signal must be less than the statistical ftuctuations of this background. U nfortunatcly,

the value obtained with this rnethod is not very restrictive. It is much more advtilltageous

to find sorne way of reliably modelling the background.

4.8 Modelling the Background

Before detailing the procedure actually used in this analysis to model the signal back

ground, one other possible method is outlined briefly. This uses the spectrum from

wrong-charge combinat ions to constrain a second-order polynomial fit of the direct T( 4S)

recoil mass distribution in the signal region [6]. Instead of reconstructing pO rnesons using

1r+1r- pairs, wrong sign 7r+1r+ pairs are used. The resulting distribution, aCter correcting

for the expected difference in the number of wrong- and right-sign combinat ions per event

(derived {rom theory), looks much like the right-charge distribution, and is assumed to

model the signal background. The resulting upper limit is comparable to that of other

methods which use the lepton spectrum endpoint alone to search for a signal. However,

there are three main reasons {or rejecting this way of modelling the background. The

primary reason is that there is no justification for modelling the background with a poly

nomial of second order. A second criticism questions the validity of constructing the

wrong-charge recoil spectrum from data that also contains the signal. Finally, simply

using wrong-charge combinations to model pO candidates ignores the rather important

contributions due to resonances. As a result of these weaknesses, the method remains

subject to controversy.

The method adopted here {or modelling the signal background is to use B B Monte

--------------------------- -----

CHAPTER 4. ANALYSIS

, of p--lepton Pairs 400.0

300.0

200.0

100.0

0.0 -10.0

- Huon.

-5.0 0.0 5.0 10.0

R.co 1 l Ha •• 2 ~G.V/c· 1]

95

Figure 4.26: Recoil mass spectrum fron. Monte Culo BB data aCter application of pO - t angle eut.

, of p-tepton Pairs 400.0

300.0

200.0

100.0

0.0 -10.0

ft +++ Electruna

- Muon.

-5.0 O~ ~o 1~0

fteco 1 l Ha •• 2 ~GeV/c' 1]

Figure 4.27: Recoil mass spcctrum Crom signal data after application of pO -l angle eut.


Carlo data as generated by MOPEK and SIMARG. This is expected to be a much morc>

reliable method because the background is comprised tota.lly of b -... c events which

are fairly well understood. Furthermore, employment of the fast lepton cut means the

resulting data set comprises essentia.lly semileptonic B and D decays, and thesc are v~ry

well understood. As noted in Section 4.3, the MOPEK-MMARG combination models B

decays with very good agreement with experiment. The one dra.wback that could present

problems is thl\t the model used to generate the B decays, that of Wirbel, Stech, and

Bauer, accounts for only two charmed semileptonic B decay modes. Experiment, however,

indicates that these channels, decays to D and De, cannot account for the observed total

semileptonic rate:

BR(BO -+ De-t+ïï) = (5.4 ± 0.9 ± 1.3)% [41]

BR(BO -+ D- t+ïï} = (1.7 ± 0.6 ± 0.4)% [87]

BR(B -+ Xlv) = (10.3 ± 0.7 ± 0.4)% [82]. (4.11)

Semileptomc decays to higher mass charmed states or multi·partic1estates are expected to

make up the rest of the tota.l rate, but these are not provided for in the Monte Carlo. To

circumvent this problem, h",rder cuts on the lepton momentum spectrum are employcd,

exploiting the fact that higher mass states result in lower momentum cutoffs. At higher

momenta, the dominant contribution to the lepton spectrum is thus due mainly to the

D and D- channels. These cuts still maintain a high efficiency for the signal as the

momentum spectrum for the B ..... p°tiï channel is expected to be quite hard (Figure 2.8).

The procedure, then, is to compare the B B recoil spectrum to the direct T (4S) spec

trum. The process is a bit more involved than this because one can also make use of the

fad that the lepton spectra must be properly modelled by ti.ae Monte Ca.rlo da.ta as wcll.

Furthermore, the direct T( 4S) spectra depend heavily on the continuum scaling factor

which is not precisely known and cao be varied within its range of uncertainty to produce

slightly different direct spectra. The end result is that a simultaneous fit of the recoil and

lepton momentum spectra is deemed to be the best way of extracting an upper limit for

the signal from the data. A X2 function is constructed to quantify the goodness of fit.

CliAPTER 4. ANALYSIS 97

4.9 The X2 Function

A simultaneous fit of the lepton momentum spectra and the recoil mass spectra needs to

incorporate the following features. ~·'irst, the continuum scaling factors for electrons and

muons as calculated from the momentum spectrum above the kinematic limit for B decays

have large uncertainties associated with them due to low statistics. The factor arrived at

from the corrected luminosity ratio has sm aller uncertainty if one ignores a possible time

variation of efficiencies. Second, the lepton momentum spectrum from the BB Monte

Carlo data dot.9 not include the inclusive b -+ u signal recently Cound at ARGUS. The

b -+ ut'ii transition is expected to proceed at (3.2 ± 0.7)% the rate oC the equivalent b -+ c

channel (ACCMM model, see below), and likewise has large errors associated with it.

The pararneters multiplying the continuum spectra and inclusive lepton spectra can be

eithec fixed or allowed to vary freely in light of these considerations. It was decided that

constraining terms would be introduced into the standard X2 Cunction used Cor the fit to

t,he data to take into account the uncertainties in measurement 5. The scaling parameters

for the BB Monte Carlo spectra (.\8B) and the signal recoil mass spectra (>.SlG) are both

allowed to vary Creely. The contribution of the signal to the lepton momentum spectrum

is expected to be small compared to the inclusive b -+ U contribution. Therefore, the

signal contribution is neglected in the fit to the momentum spectrum6• III iJ1C recoil mass

spectrum, the contribution from (non-signal) inclusive charmless B decays is drastically

reduced in comparison to the signal by the pO mass and V-A cuts and can be ignored.

Moreover, the shape resembles that due to the Monte Carlo BB data for b -+ C transitions,

and Its contribution can be expected to be absorbed by the BB scaling parameter.

The form of the X2 function is therefore as follows:

2 _ L [T(4S), - .\8sBBi - >'cCONT, - '\b_u BTOU,]2

X - 1 148, + .\~BBBi + .\bCONT, p.

L [1(48), - .\8sBB, - ),.cCONT. - .\sIGSIGir

+ m~"C T4S. + ),.~BBBi + .\bCONT,

5The constraining terms are entirely consistent with the X2 rormalism (Appendix 4.12). 6Ir one believes in the theoretical ACCMM model, one can consider the signal lepton spectrum to be

taken into account at sorne level anyhow, althougb its rate is futed witb respect to the inclusive 6 -+ u rate.

10

-- -- --------------------------CHAPTER4. ANALYSŒ 98

+ (~C - r)2 + (ÀlI_U - À::!'U) 2

a(r) 0-(>'11-.. ) ( 4.12)

The first term is responsible for fitting the lepton momentum sector and the second (or

the recoil mass sector. The third term constrains the continuum multiplier '\C to the valut'

of r, white the final term constra.ins the inclusive b ..... u lepton spectrum to its measured

strength, '\~u. The summations run over the bins of the histograms being fitted. The

T(4S), continuum, and BB histograms are represented by the obvious symbols. BTOU.

gives the entry of tbe ie" bin of the inclusive lepton moment.um spectrum for b -+ u

transitions, while SIG. represents the entry of the if" bin of the recoil mass spectrum fOf

the B -+ pot;; signal. The expected value ,\:~ .. and uncertainty q of Àb ..... are calculated •

below. The denominators of the first two terms are arrived at in a straightforward mannef,

outlined in Appendix B. The rationate behind the form of the X2 function is also outlincd

in that section.

4.9.1 Minimization of the X2 Fonction

Minimization of equation 4.12 is done with a widely available and popular computer

package called MINUIT (Appendix C). The minimization process provides an estimate

for the parameters ÀB8, '\C, À,,_ .. , and '\SIG. The values of Àc and Àb_u are put in

by hand ab initio and 50 the estimates of these parameters are biased and should not

be interpreted as new and independent measurements. The valut' Jf ÀB8 is entirely a

reflection of the number of Monte Carlo events generated to model the background, and

is, therefore, arbitrary. However, the parameter ÀSlG, being Cree of constraint, has the

potential to provide an improved upper limit for the signal branching ratio.

The Monte Carlo recoil mass signal and b --+ u spectra are scaled 50 that the pa·

rameter values in the fit represent the actual nurnber of events observed. The proper

normalizations are obtained in the manner described in the following.

Normalization of the the Signal Contribution

The signal contribution is generated with one B decaying according to the signal channel

(50% electrons, 50% muons) and the other undergoing general B decay. Dividing the

recoil mass distributions by the number of events generated for each lepton (1 x 104) gives

CHAPTER4. ANALYSffl 99

the expected distribution for a single event. The final value for ~SIG then represents the

total number oC signal events observed.

Normalization of the b -+ u Spectrum Contribution

The determination of the normalization for this spectrum is a little more involved but

nonetheJess straightforward. It requires the ARGUS result from [5] in arder to ealcuJate

the expeetf:d branehing ratio for charmless semileptonie decays,

BR(B -+ Xul+/I, 2.3 < Pt < 2.6GeV le) = (5.4 ±'0.9 ± 0.8)% ) BR(B ~ XI+II,2.0 < Pt < 2.3GeV le) (4.13

This has heen arrived at tbrough a detailed study of the lepton endpoint spectrurn and

is modcl-independent. Detector efficiencies and the motion of the B mesons have been

taken into aceount. Assuming that the contribution Crom b -+ C deeays is negligible iü

the higher m')mp.i1tum region, and from b ~ u decays is negligibJe in the Jower region, it

is possible to split the ratio as follows:

BRu(2.3 < Pt < 2.6GeV /e) BRc BRc BRu BRc(2.0 < Pt < 2.3GeV/c) = (5.4 ± 1.2)%· BRu (4.14)

whcre the subscripts represent semileptonic decay to a particular quark. Using the AC

CMM model, one can simply integrate over the appropriate range of the spectrum to

determine

BRu fll-u BRc = (5.4 ± 1.2)%· III-c = (5.4 ± 1.2)% • (.049/.083) = (3.2 ± 0.7)% ( 4.15)

Using the ARGUS result for the inclusive semileptonic B branching ratio, this leads to

BR(b -+ ulV) = (3.2 ± 0.7)%. (10.3 ± 0.7 ± 0.2)% = (3.3 ± 1.0) x 10-3• ( 4.16)

However, the result is quite model-dependent, with an estimated uneertainty of about

50%. There are a total of 145000 ± 10000 1'(48) decays in the experimental data set, of

which 2 x (478 ± 145) can be expected to undergo eharmless semileptonic B decay to a

particular lepton (electron or muon). The factor of two arises from the fact that either

B in the resonance ean decay through this channel. The number of T( 4S) charmless

semilcptonic decays for a given lepton is therefore set ~b-u = 956 ± 300. As the Monte


Fit Results for ELECTRONS 1 ISGW 1 WSB 1 KS

Continuum Scaling Factor from Lepton nurnber Ratio Àc 2.36 ± 0.20 2.35 ± 0.20 2.35 ± 0.20

Pt> 1.6GeV À,,_u 546 ± 302 538 ± 303 535 ± 303 ÀSIG -254 ± 116 -234 ± 127 -287 ± 126

X2/ D.O .F. 1.33 1.32 1.32

U.L. @90%C.L 84 91 89

.\( 2.31 ± 0.20 2.29 ± 0.20 2.30 ± 0.20 Pt> 1.8GeV Àb-u 723 ± 373 718 ± 313 710 ± 313

.\SIG -282 ± 112 -330 ± 130 -327 ± 123 X2/D.O.F. 0.88 0.87 0.86

U .L. @90%C.L 67 78 69 ---Àc 2.27 ± 020 2.27 ± 0.20 2.27 ± 0.20

Pt> 2.0GeV À,,_u 713 ± 332 710 ± 332 704 ± 332

ÀSIG -98 ± 107 -125 ± 131 -120 ± 118 X2/D.O.F. 0.84 0.83 0.83

V.L. @90%C.L 121 148 129 --.

Continuum Scahng Factor Crom Lummoslty RatIO Àc 2.31 ± 0.05 2.30 ± 0.05 2.30 ± 0.05

Pt> 1.6GeV Àb_u 558 ± 298 546 ± 299 516 ± 299 ÀSIG -247 ± 112 -279 ± 12·1 -280 ± 123

X2/ D.O .F . 1.33 1.32 1.32

U.L. @90%C.L 83 88 87

Àc 2.30 ± 0.05 2.30 ± 0.05 2.30 ± 0.05 Pt> 1.8GeV Àb_u 724 ± 309 716 ± 309

1

710 ± 309 ÀSIG -281 ± 109 -331 ± 12f,

1

--327 ± 120

X2/D.O.F. 0.88 0.87 0.86 U .L. @90%C.L 63 74 62

Àc 2.30± 0.05 2.30 ± 0.05 2.30 ± 0.05 Pt> 2.0GeV Àb-u 705 ± 328 701 ± 328 696 ± 328

ÀSIG -101 ± 105 -129 ± 129 -123 ± 116

X2/D.O.F. 0.84 0.83 0.83 U .L. @90%C.L 120 146 128

Table 4.5: Results from simultaneous fits of the electron rnorncntum and recoil mass spectra. Errors are parabolic.

l' i


Fit Results (or MUONS 1 ISGW 1 WSB 1 KS

Continuum Scaling Factor from Lepton number Ratio .\c 2.43 ± 0.18 2.42 ± 0.18 2.42 ± 0.18

Pt> 1.6GeV .\~u 968 ± 325 974 ± 326 977 ± 326 .\f$IG 24 ± 148 45 ± 172 51 ± 158

X2/ D.O.F. 1.2 1.2 1.2

V.L. @90%C.L 257 309 291

.\c 2.40 ± 0.18 2.38 ± 0.18 2.39 ± 0.18 Pt> 1.8GeV .\,,-u 1048 ± 332 1060 ± 333 1061 ± 333

.\SIG 91 ± 141 140 ± 167 131 ± 153 X2

/ D.O.F. 1.3 1.29 1.29 U.L. @90%C.L 293 373 344

. .\c 2.42 ± 0.18 2.42 ± 0.18 2.42 ± 0.18

Pt> 2.0GeV .\~u 1274 ± 348 1289 ± 348 1291 ± 348 .\SIG 125 ± 137 200 ± 180 172 ± 155

X~/D.O.F. 1.42 1.41 1.41 V.L. @90%C.L 314 443 380

Continuum Scaling Factor from Luminosity Ratio .\(" 2.31 ± 0.05 2.3 ± 0.05 2.31 ± 0.05

Pt> 1.6GeV '\b-u 997 ± 321 1003 ± 321 1005 ± 321 >'SIG 36 ± 145 58 ± 169 63 ± 155

X:l/D.O.F. 1.21 1.21 1.21 U.L. @90%C.L 261 314 296

.\c 2.31 ± 0.05 2.31 ± 0.05 , 2.3Ï ± 0.05 Pt> 1.8GeV )~u 1073 ±328 1083 ± 328 ' 1084 ± 328

>'SIG 99 ± 139 149 ± 105 140 ± 151 X:l/D.O.F. 1.31 1.30 1.30-

U.L. @90%C.L 297 377 348

.\c 2.31 ± 0.05 2.31 ± 0.05 2.31 ± 0.05 Pt> 2.0GeV .\~u 1305 ± 344 1319 ± 344 1320 ± 344

.\SIG 132±135 208 ± 178 179 ± 152 X2/D.O.F. 1.43 1.42 1.42

U.L. @90%C.L 316 446 383

Table 4.6: Results from simultaneous fits of the muon momcntum and recoil mass spectra. Errors are parabolic.

..


Carlo dataset contains both leptons in the ratio 50:50, the proper normalization for the

histograms is to divide by 25000, one halt of the total number of events generated .

Shown in Tables 4.5 and 4.6 are the results of the minimization of 4.12. Examples

of the fits are shown in Figures 4.28 to 4.39. The error bars {or the fit points have

had the parameter errors incorporated ioto them aecording to the expression derivcd in

Appendix B by the author. The recoil mass and lepton spcctra are fit simulta.ncously for

aH cases, with independent fits made for each of the electron and muon data. pools. The

lepton spectrum is fit for three different lower limits of Pt: 1.6, 1.8, and 2.0GeV le. The

upper limit is fixed at 3GeV le. Increasing this up~,er limit has little effect on the best

fit values. For a. given lower momenLum limit, the corresponding rccoil mass spectrum is

used, i.e, only those pOL combinations arising from P, greater than this lowcr Iimit. The

range (or the recoil mass spectrum is kept at ·-1O.OGeV2 Je" ta lO.OGey2 Je" for ail fits.

Two sets of rand 0'( r) values, as derived from the eorrected luminosity ratio and lepton

eounting in the region above the kinematie limit for B deeay, h~ve bccn included. Each

of the three form factor models from Chapter 3 has in tllrn becn uscd to modcl the signal.

CHAPTER 4. ANALYSIS

, of leptons 1200.0 ~~~~~T-~~~~~~~~ __ ~-r~~~-,

1000.0

800.0

600.0

'100.0

200.0

0.0 1.0 2.0

- T(4S)

t FIT

3.0

P>1.6 GeV/c ISGW

Il.0

lepton t.Comentum (GeV/c)

5.0

103

Figure 4.28: Fit of the momentum spectrum for electrons with Pt > 1.6GeV le. The signal is modellcd using the ISGW model.


200.0

160.0

120.0

80.0

110.0

0.0 -10.0 -5.0

- T(4S)

+ FIT

P>1.6 GeV/c ISGW

0.0 5.0 10.0

A.coll "0882 ~GeV/c'f]

Figure 4.29: Fit of recoil mass spec.trum for eledrons with Pt > 1.6GeV je. The signal is mode lIed using the ISGW model.

..

."

CHAPTER 4. ANALYSIS

1 of leptons 900.0

150.0

600.0

"50.0

300.0

150.0

0.0 1.0 2.0

T(45)

FIT

3.0

P> 1.6 CeV/c ISGW

lepton Momentum (GeV/c)

104

5.0

Figure 4.30: Fit of the momentum spectrum for muons with Pt > L6GeV le. The sigllal is modelled using the ISGW mode}

, of p-Lepton Pairs 180.0

150.0

120.0

90.0

60.0

30.0

0.0 -10.0 -5.0

T(45)

t FIT

P>1.6 GeV/c

ISGW

0.0 5.0 10.0

Recoll Haut ~GeV/c·r]

Figure 4.31: Fit of reeoil mass spectrum for muons with Pt > 1.6GeV le. The signal is modelled using the ISGW model.

CHAPTER 4. ANALYSIS

, of Leptons 1200.0

1000.0

800.0

60a.0

'wa.o

200.0

a.o 1.0 2.0

- T(45)

t FIT

3.0

P>1.8 GeV/c WSB

Il.a Lepton Momentum (GeV/c)

105

5.0

Figure 4.32: Fit of the momentum spectrum for eleclrons with Pt > 1.8GeV je. The signal is modellcd using the WSB mode!.

/1 of p-lepton Pairs 150.0 •• i

125.0

100.0

75.0

50.0

25.0

0.0 -10.0 -5.0

T(45)

t FIT

P>1.8 GeV/c WSB

0.0 5.0 10.0

Reco 1 l MOlU 2 ~GeVlc' f]

Figure 4.33: Fit of reeoil mass spectrum for electrons with Pl > 1.8GeV je. The signal is modelled using the \VSB moc~d.

.'

CHAPTER 4. ANALYSIS

, of Leptons

900.0

150.0

600.0

1150.0

300.0

150.0

0.0 1.0 2.0

- T(45)

+ fIT

3.0

P>1.8 GeV/c WS8

Il.0


106

5.0

Figure 4.34: Fit of the momentum spectrum for muons with Pt > 1.8GcV je. The signal is modelled using the WSB model.

1 of p-Lepton Pairs 120.0

100.0

80.0

60.0

ijO.O

20.0

0.0 -10.0 -5.0

- T(45)

+ FIT

P>1.8 GeV/c WSB

0.0 5.0 10.0 Recoll Hasa 2 ~GeV/c'f]

Figure 4.35: Fit of reeoil mass spectrum for muons with Pt > 1.8GeV je. The signal is modelled using the WSB model.

CHAPTER 4. ANALYSIS

, of leptons 1200.0 ~~~~~~~-r-r-r-T~ __ ~~-'~~~~~

1000.0

800.0

600.0

1100.0

200.0

0.0 1.0 2.0

- T(4S)

t FIT

3.0

P>2.0 GeV/c KS

Il.0


5.0

107

Figure 4.36: Fit of the momentum spectrum for electrons with Pt > 2.0GeV je. The signal is modelled using the KS mode!.

, of p-Lepton Pairs 60.0

50.0

110.0

30.0

20.0

10.0

0.0 -10.0 -5.0

- T(4S)

+ FIT

P>2.0 GeV/c KS

0.0 5.0 10.0

R.coll "au 2 ~G.v/c·f]

Figure 4.37: Fit of reeoil mass speetrum for electrons with Pt > 2.0GeV je. The signal is modelled using the KS model.

CHAPTER 4. ANALYSIS

, of Leptons 900.0

750.0

600.0

1150.0

300.0

150.0

0.0 1. a 2.0

T(45)

FIT

3.0

P>2.0 a.V/c KS

Il.0


108

5.0

Figure 4.38: Fit of the momentum spectrum for muons with Pt > 2.0GeV Je. The signal is modelled using the KS model.

, of p~epton Pairs 60.0

50.0

1&0.0

30.0

20.0

10.0

0.0 -10.0 -5.0

- T(45)

+ FIT

P>2.0 GeV/c KS

0.0 5.0 10.0

A.co 1 l Ha •• 2 ~G.V le'l]

Figure 4.39: Fit of reeoil mass spectrum for muons witb Pt > 2.0GeV Je. The signal is modelled using the KS model.

f'


4.10 Observations and Interpretation of Results

Not surprisingly, the '\6-u and "\C results agree within error with the expected result. This

is a consequence of the constraining terms in the fit function. However, the muon and

electron results for ,\6 ..... u barely overlap, the former predicting qui te a high contribution,

the latter a fairly low one. The continuum values also display this same character; in fact,

without the constraining term, the muon data yield an absurdly high value for '\C, of the

order of 2.6 to 2.8.

In ail fits, the WSB model predicts the highest upper limit for the number of signal

events, the KS model the next highest. This is a direct re8ection of the form of the signal

lepton spectrum predicted by the different models. The WSB model has the softest

spectrum; consequently, it has the lowest acceptance (all other differences between the

models being relatively equaF) and requires more events to account for the signal.

The values for '\SIG are rather disturbing. The electron data predict a negative value

in general whereas the muon data predict a value more consistent with zero but positive.

Theelectron value only overlaps zero for the lower momentum cutoff of Pt > 2.0GeV/c. As

alluded to in Section 4.8, this could be a direct reftection of the c.ontribution due to higher

mass states in b -+ C transitions, unaccounted for by the BB Monte Carlo. The muon data

result also steadily increases as the lower momentum cutofl' is increased. In fact, the values

for ,\6 ..... u display this same trend, suggesting that the b -+ u components are needed less

and less to compensate for the higher mass states, enabling their parameters to reflect the

true b -+ U contributions more accurately. Indeed, one would expect that the distributions

from higher mass states could be modelled to first order, albeit roughly, by subtracting

a small b -+ u component from the BB distributions. This is consistent with what is

observed. It is probably prudent, therefore, to assign the results for Pt > 2.0GeV le much more weight, especially as they predict more conservative lower limits for "\SIG.

Unfortunately, this is a very hard eut, and the histograms begin to look rather sparsely

filled, the continuum contributions particularly !:o. The assumption of Gaussian statistics

is not as strong, and the upper limits become questionable. ft might be wiser to use the

maximum likelihood method for fitting this data; the contents of any bin can be treated

7The rates predicted by the models are very dlfl'erent, but these do Dot enter the problem until ODe wisbes to establish an upper limit (or Vu •.


as a Poisson variable, which has a weil defined probability distribution regardless of the

number of enlries. Alternatively, one could use bigger binning, the drawback being that

for distributions with fine detail, information is lost. Ultimately, the best solution is to

acquire more statistics, espedally from the continuum.

The X2/nOF values are low for the electron fits and rather high for the muon fils.

These values should be interp~eted with sorne ca.ution, however, because the non-linear

nature of the problem means that the fit fUDction cannot be considered as having a

a. strictly X2 distribution. Nevertheless. the difference betwecn t.he electron and muon

X2/DOF is statistically significant. One would not expect such a large difference unless

there were sorne basic differences between the electron and muon spectra which have not

been modelled properly. Tilere are a few possibilities. One of the main problems is that

of bremsstrahlung, both in the detector and in the initial state. Auother, perhaps less

important, problem is an imperfect knowledge of the muon misidentification rate. These

two effects will be discussed below. A third thing to note is thal, because of thcir mass

difference, electrons and muons do not share exactly the same production channels, and

those that are different may not be properly modelled. This effect is small, although

perhaps Dot negligible, for data with a high value for the momentum cu toIT. In that case,

the only channels to worry about would be electrons and muons resulting from J /1/J and

T decay. However, these contributions are well understood. More significant would be

the contribution from sorne o{ the less-understood hadronic decay channels. Fast pions

misidentified as muons would not be properly accounted {or.

For a given energy, the rate at which a particle will undergo bremsstrahlung is inversely

proportional to the square of its mass, and increases with energy [27}. The rate is therefore

quite different for electrons and muons. At ARGUS energies, muon bremsstrahlung is

negligible, but electron bremsstrahlung is quite an importan~ effect. The rac:!iation cross

section for a particle traversing a medium is very well understoodj the eITect is included

ta high accuracy in SIMARG, which can there{ore be expected to mode} properly the

interaction of electrons with the detector. There is no correction for bremsstrahlung in

MINIMe, however, but the effect has been inc1uded at the MOPEK level in a reasonable

manner. The corrections are certainly much more important for charmed BB dccays

which dominate the T(4S) distributions. The b -+ U contributions to the spcctra arc small


enough and sufficiently difl'erent from the continuum and BB contributions 50 that the

corrections have o!lly a minor efl'ect on the measured number of signal events. Much more

important, however, is the efFect of radiative decays, those resulting from bremsstrahlung

at the decay vertex. This proc~s is not so well understood and there is no consensus as

to the exact form or rate of the cross Section [82]. The contribution is expected to be

small, however, compared to bremsstrahlung in the detector, but not negligible8 . Vertex

corrections are Rot included in MOPEK and have been ignored in this analysis. While

not crucial for the b -+ U Monte Carlo, as argued above for the case of bremsstrahlung

in the detector, it is oC major importance for BB Monte Carlo data, which is responsible

for modelling the bulk of the background. A small correction to this data could easily

swamp the much sm aller b -+ U contributions. For example, the experimental non b -+ u

B B momentum spectrum should be slightly softer than that predicted by the uncorrected

Mont.e Carlo. It al 50 has its endpoint much more depleted because the bremsstrahlung

cross section increases with energy for relativistic particles. The difference between this

true spectrum and the one from the uncorrected data should therefore resemble a harder

momentum spectrum. A smaller b -+ u compone nt would be required for the fit, and this

is what is observed: the '\b-u estimate from the electron data is mu ch smaller than that

for the muon data. This suggests that one cannot ignore the effect of radiative decays.

It would appear, therefore, that one should abandon the electron data results and

use the results from the muon data. UnCortunately, the fits to these data are ra.ther

poor. This is partly the result of an incomplete knowledge of the muon misidentification

rate. Bec.t\use pions are very close in mass to muons, tbey are often misidentified as

such. Unfortunately, a good number of "identified" muons in the continuum are, in fad,

pions, and unless the misidentificatio& rates are accurate and constant, the continuum

data Olay not be properly subtracted from the T( 4S) data. The situation is much worse

in MINIMe, wherc the muon misidentification rate is crudely modelled. This would imply

that the b -+ U distributions are not accurate. However, this should not be so large an

effect as to yield such poor fits, and a better explanation is desirable. Perhaps the reason

is simply that higher n,ass states have been neglected, and that, for the electron data,

the additional neglect of vertex corrections somehow compensates to yield improved fits.

'See, for example, Ref. [78J.


Further hypothesis along these lines would be difficult to justify, however, and 50 tht"

author passes on to an analysis of thE' systematic errors.

4.11 Systematic Error Analysis

In light of the rather unsatisfa.ctory results, a detailed anal!'sis of the systematic errors

is unwarranted. Only the systematic errors arising (rom the uncertainties in the contin

uum scaling factor, the form factor rnodels, and the inclusive lepton spectrum rnodel are

investigated.

As rnentioned above, there is sorne doubt as to the reliability o! using the luminosity

ratio to get the continuum sr.aling factor. The systematic error inlroduccd by this un

certainty is round by changing the value of r used in the fits by 2% and refitting. The

best fit values for >"SIG change by at most 2%, with the upper limits changing by less

than 1%. Alternatively, one could simply use the larger errors resulting Crom the lepton

number ratio; this is equivalent to including the systematic error in the statistical crror.

Using the larger errors yields about a 2% change in the upper limit. A '2% unccrtainty is

therefore ascribed to the upper limit on the signal parameter.

The models for B --+ potii decay all predict the same basic shape for the recoil mass

signal. This is because the signal shape is dictated for the most part by the motion of

the B mesons and the detectol" resolutions. Only the normalizations vary, due to diffcrent

acceptances. The differing values of >"SIG are a reflection of the acceptances alone, a.nd the

shape oC the spcctrum has negligible eff'ect. ThereCore, thtre is no nccd to do a detailed

a.nalysis of how the shape of the signal spectrum affects the results. The systematic error

arising from the uncertainty in the tr..msition Corm factors can be extracted directly Crom

the variation of the observed number of signal events from model to mode!.

This is not the case, however, for the b -+ U inclusive lepton spectrum. The ACCMM

model is the only model used to generate this contribution, and there arc many reasons

why it may not be a.ccura.te. Not only the total rate but also the shape oC the inclusive

b -+ u component should be considered as uncertain. Indeed, the ACCMM modcl :tself

does not predict a definite shape but parametrizes tbe possible Corms in terms of mt., mu,

and PF. Therefore, to get a rough idea of how the shape of the charmless inclusive lepton

~----------_ ..


Upper Limits on BR(B -+ p°tii) and lVu,,1 ISGW WSB KS

# of Signal Events @90% C.L. 350 490 420 BR(B -+ p°tv)@90%C.L. < 2.35 x 10 ·3 < 3.30 X 10-3 < 2.82 X 10-3

rS/G (lVu,,11 x 10-11 ) 4.2 13.1 16.5 I\-~I CQ90% C.L. < 2.37 x 10 .~ < 1.6 x 10 .~ < 1.3 x 10-':

Table 4.7: Upper limits on the CKM matrix element Vu" from fitting the recoil mass and lepton momentum spectra.

spectrum effects >'SIG, a reasonable alteration of the b --+ u spectrum used in the fits is

performed; the data is then refit using these slightly harder and softer spectra. Thert'

is about a 7% change in the best fit values, while the upper limits change byabout 3%.

An independent estimate9 of the systematic erTor due to the rate is made by changing

the value of >':~". The value used in this analysis was derived using information from

the endpoint, specifically the ratio h::u..J - used in Equation 4.15; models tend to disagree .-c by a factor of about two in this region. Varying >':~u by 50% introduces a 15% change

in -'6 ... u and a 6% change in the :lpper limit, implying that the uncertainty in the overall

normalization of the inclusive lepton spectrum is the dominant systematic error.

From the above considerations, the upper limits for ).SIG are increased by 10%, a

conservative estimate of the effect of systematic errors. The new upper limits are given

in Table 4.7 using the values of r and u(r) derived from the lepton number ratio. For the

reasons discussed in Section 4.10, only the results from the data pool containing muons

with momentum greater than 2.0GeV le are used.

4.12 Upper Limits on BR(B --+ pO tu) and IVubl Using the upper limits on the number of signal events observed (Table 4.7), along with

the number of B mesons ealeulated in Section 4.4, an upper limit on BR(B -+ p°tv)

for each model can be calculated. Upper limits on /Vubl can also be extracted usmg

8There is certainly no reason to believe that tbe systematic errors arising from the rate and the shape are independent of each other. In fact, it is clear from the procedure used to derive the total number of 6 -+ u events tbat they are directly dependent UpOD each other. Bowever, only a rough idea of the systematic error is sought, and this cao be aehieved without resorting to a detaHed analysis.

,. -1


TB = 1.15 ± .l.tps [64], and the absolu te rates predicted by each of the models. The

results are displayed in Table 4.7. The upper limits on the branching ratios are round

to be of the order of three tirnes larger than those predicted by Ref. (6}. Comparison

with the value extracted from an investigation of the lepton endpoint, IV"b/Vc61 x IYcbl = (.5 ± .05) x 10-2 [41], would suggest that, given the present amount of data, the method

used in this analysis is insensitive to any signal that may be present.

4.13 Recommendations and Conclusions

It is clear {rom the above observations that the method used in this analysis is not suffi·

ciently sensitive to observe a signal, and is plagued with important problems. Although

the errors are large and several effects occur simultaneously, a (ew trends can be pickcd

out that might indicate the path to take towards improving the results. In principle,

the method is powerful, but is only as good as the Monte Carlo used to model the data.

Both event generation and detector simulation need to be improved. In particular, highcr

mass states need to be studied and included in the Monte Carlo. As already not.'d,

the observed inclusive semileptonic rate is not completely accounted for by the exclusive

B -+ Dtv and B -+ D-W rates. lt is likely that higher mass states conlribule to the tolal

rate, and although thtir contributions would be smaller, it is nol unreasonabl":! to expcct

that these contributions would significantly affect the signal values. Vertex corrections to

semileptonic decays also need to be better understood. This effect is small as welJ 1 but

the results of this analysis suggest it cannot be ignored. As for detector simulation, the

muon misidentification rate needs to be determined more accurateJy, and it may be wise

to generate all Monte Carlo data using SIMARG.

Once all the systematic errors are reduced, there will still remain the slatistical errors.

Subtraction of the continuum introduces large errors that swamp the signal. If the sen·

sitivity could be increased, it rnight be possible to observe the signal without having to

increase the statistics. However, ail attempts, whether by the author or others, to improve

the sensitivity have failed. The limiting factors are the width of the pO and the motion

of the B mesons. To date, the only way to "identify" po mesons is to cut on the ,..+~-

(

CHAPTER4. ANALYSŒ 115

mass. Unfortunately, there is an enonnous background associated with this cut10• One

cannot reduce the width of the pO, obviously, but if a good theoretical model is developed

that caon reliably pred.ict the polarization of the pO in the transition, one might be able

to exploit the p7r angle in the po rest frame. This may improve the ability to distinguish

po mesons and increase the sensitivity for the transition. The sensitivity would also be

improved by sharpening the width of the recoil mass signal. This requires knowledge of

the directions in which the B mesons are produced, impossible ta determine using the

vertex charnher. The new J'Vertex detector, which has a vertex resolution of no hetter

tban 60pm, willlikely not be able to pick out the Bight paths (23pm) either. D meson

vertices will be resolved, however, and this may help to get a rough idea of the B direc

tion, or perhaps "tag" the charge of the B. Although the directions of the B mesons are

not easily determined, the magnitude of their momenta is known from the centre-of-mass

energy. This important bit of information is discarded {or want of a means of exploiting

it. It would certainly be adva.ntageous to include it somehow, and one possible way of

doing this is suggested in Appendix D.

In conclusion, using the WSB model, an upper limit of 3.3 x 10-3 has been placed on

BR(B -+ p°tii) at 90% confidence; this corresponds ta a value of Vub less than 1.6 x 10-2•

Tbe results oC the analysis indicate that a better understanding of the b -+ C ba.ckground

is necessary to improve these limita.

IOOne solution might be to examine the channel B - w°tv. The ",0 has a mass comparable to the pO

but a width of only 8.5MeV. Bowever, the dominant deeay mode is to two charged pions and one neutral pion, a ratber cballenging object to reconstruct. Futbermore, poor photon energy resolutioD would widen the invariant mass peak.

•

'J'

Appendix A

The Development of the Standard Model

A.1 Experimental Constraints

There are several experimental observations that need to be incorporated ioto the standard

model. A few of the more important on es are outlined here. It should be cmphasized now

that these observations are solely experimental and there is as yet no theoretical basis for

them. Indeed, the major role of grand unified theories is to illuminate the mcchanisrns

responsible for these mysteries.

The most trivial observation that needs to be Încorporated is the mass hierarchy of the

particles. Except for their masses, the leptons and quarks combine to form three identical

families of particles (Table 1.1). Because the quarks are confined within hadrons, sorne

difficulty arises in trying to determine their masses. Two different sets of values are

given. The current masses are those which have been determined from current algcbra

and represent the masses of quarks that are observed when a hadron is probcd by a

weak or electromagnetic current. The constituent masses include in addition the average

kinetic energy of the quarks in the hadron (of the order Ac) and represent roughly the

fraction of energy carried by the quarks when confined in hadronic states. The mass of

the t quark is still unknown although it has been constrained by both experiment and

theory. Although neutrinos are commonly accepted to be massless, there is no theoretical

reason why this should be 50. The standard model as it stands, however, requires that

ml' = 0 [88]. This is essentially a consequence of the non-observation of right handed

neutrinos which are necessary to provide a mass term in the Lagrangian. Although it is

116

APPENDIX A. THE DEVELOPMENT OF THE STANDARD MODEL 117

possible within the framework of the standard model to give mass to the neutrino without

requiring the existence of right-handed neutrinos, one does so at the expense of lepton

number conservation, which is clearly unacceptable [12, 88]. Sorne grand unified theories

do provide acceptable mechanisms for providing the neutrino with mass [12], but these

are outside the scope of this work. (The possibility that neutrinos have mass has dramatic

consequences for cosmological theories of the uni verse. A sufficiently large neutrino mass

cou Id provide the universe with enough mass to eventually collapse upon itself [89].)

The coupling constants also need to be determined from experiment. Just as the weak

and eledromagnetic coupling constants have been related to each other, it is hoped that

the strong and gravitational coupling constanta may sorne day be included and aIl forces

related under sorne grand unification scheme.

The observed universal conservation of lepton number, baryon number, and electric

charge ail help to neglect otherwise theoretically possible terms in the Lagrangian. The

conservation of parity and charge conjugation quantum numbers in electromagnetic and

weak interactions likewise constu . .ins the form of these interactions. The absence of right

handed neutrinos in natuce implies that parity is violated rnaximally in weak interactions

and this needs to be incorporated into the theory as weIl. The discovery of CP violation

in neutral kaon decays (Section 1.7) suggested the existence of a third generation and this

was subsequently confirmed. Many other experimental observations have had important

consequences for the standard model, but these will not be mentioned here.

A.2 Theoretical Constraints

The most obvious mathematical constraint that must be satisfied is that of Lorentz invari

ance. Cou pied with translational invariance, this yields the larger constraint of Poincaré

invariance. Also, aIl forms of interaction must be local so that the physics can be de

scribed simply by understanding what is occurring at a given point in space time. This

basically Îs a statement that processes at other points in space-time are irrelevant [90]. In

addition. the principle of causality ensures that processes occurring simultaneously but

at different points in space cannot influence each other (a consequence of the postulate

that nothing travels faster than light).

..

1

APPENDIX A. THE DEVELOPMENT OF THE STANDARD ~IODEL 118

A.3 Gauge Theories

Each of the three forces of the standard model bas been developed theoretically by as

suming local gauge invariance. The importance and meaning of this symmetry will briefly

be expounded here.

It is possible to reproduce the electromagnetic Lagrangian sim ply by demanding th"t

the theory be invariant under local transformations of the phase of the field descri bing a

charged particlej i.e. the Lagrangian and hence the physics remain unchangcd undcr a

transformation that changes the phase of the field by different indepcndent amoullls al

every point in space time [14]. Invariance can only be satisfied by postulating the exist(!nce

of a gauge boson field, the photon, which couples to the charged particle in a way that

counteracts the effects of local phase changes. 1 Such gauge invariance as it is cé.\llcd was

long thought to be just a mathematical device that greatly simplificd the managcbility

of Ma...xweU's equations. One cou Id choose a gauge (coulomb, Lorentz) in order to solve

the equations in a particular case. However, work done by Weyl as carly as the 1920'5

and later revived and emphasized for its importance has shown thal this invariance in

fact accounts for and indeed forces the existence of the photon, the elcctromagnclic gauge

boson [91].

It has been possible to exploit this powerful constraint of gauge invariance in the elec

troweak and colour theories and produce similar gauge bosons. In the electrowcak thcory,

allleCt-handed quarks and leptons are arranged into d.>ublets. A global SU(2) transfor

mation acting on these doublets and transforming its members into each other can be

shown to leave the Lagrangian invariant. This is termed weak isospin symmetry. Another

group of U(l) transformations, responsible for hypercharge, also leaves the Lagrangian

invariant. In order for the Lagrang5an t.o be further invariant under local transformations

of the combined SU(2) x U(l) symmetry group, it is necessary to incorporate the spin

1 charged W's and neutral Z. The Lagrangian is then said to exhibit SU(2)/e/l X U(l)y

gauge invariance. Colour theory exhibits SU(3) gauge invariance. In this case, the La

grangian is invariant under all local SU(3) transformations acting on triplets of colour

fields. The gauge bosons required to permit this are the 8 gluons. The standard model is

l Note that su eh a local variation in a parameter differs from the globalsymmetries of Lorentz invariance (same transformation is made for all points in spaee-time) or isotopie spin invariance.


considered an SU(3)c x SU(2), x U(l)y gauge-invariant theory.

Why should local gauge invariance he a symmetry of the Lagrangian? The application

of gauge invariance secms ad hoc at best. However, a simple argument shows that local

invariance is in fact a much more natural s.fmmetry than global invariance. A Lagrangian

that is only invariant under global changes of a field phase, for example, would suggest tbat

there is sorne mcchani~m that allows each phase at two differeIlt points in space to "know"

instantly what the other is doing. This violates causality and makes global invariance

undesirable. Hy introdudng the freedom for independent phase changes, different points

in space are no longer connected, and such a mechanism is no longer required. Another

very important reason for assuming gauge invariance to be a good symmetry is that it

can be shown that ail gauge theories that are spontaneously broken (see next section)

are renorrnalizable [92]. While there is no fundamentaJ principle demanding the use

of renormalizable theories, there are a few important factors that strongly encourage

it. First, a non-renormalizable theory that describes what is experimentally ohserved

indkatcs that there is new physics at distance sca.les smaller than can he reached at

available energies [93]. (For example, the non-renormalizability of the point coupling

Fermi interaction is a tell tale sign that at higher energies the interaction is in fact not a

point coupling at all but involves the exchange of an Întermediate vectnr boson.) Because

the standard model postulates that leptons, quarks, and gauge bosons are fundamental

particles, a non-renormalizable theory is unacceptable. Of course, it rnay be that these

supposedly fundamental particles are in fact composite particlesj this possihility is left to

the realm of composite model grand unified theories, however, and is outside the bounds

of the standard model. Second, renormalizable theories offer the exciting possihility of

being able to relate the masses of the different quarks and leptons, a major objective of all

grand unified theories [88]. To date, the origin of the quark and lepton rnass hierarchy bas

remained a complete mystery. Finally, renormalizable theories are calcu!a.ble_ It would

make little sense to employa theory that cannot handle the infinities inherent in allloop

diagrams. By scaling (renormalizing) the c01Jpling constants and fields, these divergences

can be made to disappear, leaving a more ma.nageable theory_

A more disturbing question, still unanswered, is why does nature decide that sorne

symmetries are gauged while others are not? For example, wby are the symmetries that


are responsible for lepton number and baryon number conservation not gauged? This is

yet another mystery that will hopefully find a solution amongst the different grand unified

theories.

A.4 Spontaneous Symmetry Breaking and The Higgs Mechanism

Requiring 8U(2) x U(I) gauge invariance is not enough to reproduce the eledroweak

Lagrangian. This is because there is no mechanism to give the gauge bosons and fermions

mass. The Dirac mass term can be written

which is not SU(2) invariant because tPL is a member of an SU(2) doublet while tPn is an

SU(2) singlet. Inserting thesc mass terms ",--ould therefore destroy the renormalizahility

of the theory, so critically dependent upon gauge symmetry. In order to generatc gaugf'

boson and fermion masses, it becomes nec~ssary to introduce e!eI11entary sca.lar lIiggs

fields. These couple invariantly to the gauge bosons and fermions and so mu~t he' in th(>

Corm of a complex SU(2) doublet. More important, they can interact with each other in a

manner that produces a potential with a minimum that has a non-zero expcdation value,

i.e., the iowest energy state is not the one with no particles. The particle quanta in the

theory then correspond to quantum fluctuations about this non-zero expectation value. Oy

re-expressing the Lagrangian in terms of a Higgs field that fluctl \tes about this non-zcro

expectation, it becomes possible to generate the necessary mass terms. This is known as

spontaneous symmetry breaking; although the Lagra.ngian still remains invariant undcr

local transformations, the symmetry of the states is broken and thc rCl>ulting particlc

spectrum gives little indication of an underlying symmetry. The SU(2), x U(l)y local

ga.uge invariance is sa.id to be hidden.

Although the Higgs mechanism provides for the ma..<;ses of the fermions and incorpo

rates nicely into the standard electroweak model, a few rather disturbing featurcs arisc.

For example, in its present form, the Higgs mechanism is uoable to reduce the total oum

ber of independent parameters in the model; pa.rticle masses a.re effectively replac:cd by

an equal number of seemingly arbitrary coupling constants (the couplings bctwccn the

------------------------------- ---

-i

APPENDIX A. TlIE DEVELOPMENT OF THE STANDARD MODEL 121

Higgs field and fermions). This is a "a rather awkward way to 'explain' the existence

of masses and their magnitudes" [94], as there does not seem to be a universal coupling

involved. Also, the mass of the Higgs is not accounted for by the theory, further adding

to the Hst of parameters in the model. The fact that the Higgs bas yet to be observed

Curther cripples an already doubtful mechanism. Still, it remains the simplest and most

promising construct available for generating mass in the standard model.

Appendix B

The Least Squares Principle1

B.l General

A set of N independent measurements y.(%i) is made, one at each of the observational

points XI ••• ZN. Let J(OlJ02 .•• lhix) he sorne parameter·dependent model that attempts to

predict the true value '1. associated with each x •. The least squares (LS) prillciple asserts

that the best values of the unknown parameters, !l = ~, are those that minimizc t.he

expression N

X 2 = Ew.(y. - J.)2 (B.1) i=1

where Wi is the weight given to the i lh observation and re8ects the accuracy of the mea

surement. If one equates w. to the precision of the measurements, ~ the least squares ", principle becomes

X 2 ~ (Yi - J.)2 .. = L...J 2 = mInImum.

i=l (1.

(B.2)

It is natural to make this connection between the weight and the precision: on average,

a given Yi cannot he expected to lie any doser to the true value '1. ~ fi(n.. = ~) than its

uncertainty in measurement2, i.e. u 2(y. - J.) ~ u? Equation B.2 assumes that f is precisely known once the hesl fit values of the O. are

found. If, however, the fi have some uncertainty associated with them, as is the case, for

example, when 1 is constructed from a set of histograms, tbis should be reflccted in the

weight of (y. - li)2. Following the logic used to justify the cboicc of w. used in Eq. B.2,

lTbe formulation of tbis section follow8 clœely that of referenee [95], Chapter 10. 'lIf J eorrectly modela the observables, tbis would imply that for large N, X'l /N will be roughly equaJ

to one. A lignifieaotly higher value tban tbis would lugest an inadequate model has been used, while a mueh lower value would indicate tbat the uncertainties on the measurements have been overestimated.

122

"

"

APPENDIX B. THE LEAST SQUARES PRINCIPLE 123

and assuming that the Yi measurements are independent of those that yield the form of

/, one is led to the assignment

(8.3)

and the LS principle becomes

X2 __ ~ b/i -Ii )2 . . ~ 2 = mlDlmum. i=l (1. + (l'l(fi) (8.4)

The value of U 2(/i) is dictated by the dependence / has on the measurements.

When the observations Yi are the contents of a histogram, the variances (I~ cao he

deduced in a straight forward manner. Each bin has sorne true value 'Ii a.c;sociated with

it; thereCore, the recorded number of entries must have a Poisson distribution with Mean

and variance TT.. Although in general the 'Ii are not known, for a large number of entries

one can saCely make the approximation '1. ~ y,. Furthermore, for large '1. ('11 ~ 10), the

Poisson distribution closely approximates a Gaussian distribution. The observables can

then be considered as distributed normally with variance y,.

The LS principle makes no statement about how an observable y. should be distributed

statistically. If, however, the Yi are distributed normally about their true values 'Ii, / is

lineal'ly dependen~ upon its parameters, and the weights are independent of these param

eters, then the values of X'l resulting from severa} experiments will have a X'l distribution

with v = N degrees of freedom. This allows one ta make a more quantitative statement

about Any fit. Assuming the model to be correct, the probahility that a worse value for

X2 exists can be calculated as

CL = 1~ fez; v)dz X.la

(B.5)

where feZ; Il) is the X2 distribution for JI degrees of freedom.

In general, the Taylor's expansion of X2 about its minimum is given by

2 2 1 ~ ( 82 X

2 ) .. .. X (~) = Xmin + 2 ~ 8080; . (0, - D.)(O] - D,) + , ..

• , 'J !=! (B.6)

The terms linear in O. are, hy definition, zero. Any contribution from terms cubic or higher

in 0 is an indication of the nonlinearity of the problem. If / is linear in the parameters and

the weights are constants, these terms do Dot existe If, in addition, the observables are

t


distributed multinormally about their real values, the coefficients of the quadratic terms

are directly related to the covariance matrix (error matrix) of the parameters,

.. .... 1 (J'l X' [ J-I

V;,(t) = E[(II; - 8;)(11, - 8,)) = 2" (all.8O,) t=. (B.i)

thus providing a direct estimate of the parameter errors and correlations. The error

ascribed to the i th parameter is just the square root of "'ii. For the ideal conditions

listed above, these errors are Gaussian and correspond to one standard deviatioDs on thr

parameters. In the more general case of a non-quadratic X2, however, the covariance

matrix is still a.ssumed to be related to the coefficients of the quadratic terms. The

presence of higher order terms is ignored and it is Dot possible to make a general statemenl

about the distribution al propel ties of the parameters. A necessary consequence of this

is that the probability that the true value of any parameter lies within its errors can

no longer he considered to he 68%. The errors are termed parabolic. In any case, for

prohlerns that are not too non-linear, these parabolic errors are approximately Gaussian.

For the general non-linear problem, to place an upper limit on a parameter, 0" wbose

fitted value, Ô. is consistent with zero, it is conventional to compute a likelihood function,

L(Oi), for aIl values of (Ji on the range [Bi - c~8., Ô. + c~B.I. Here fj.Ô. is the parabolic

error calculated from the covariance matrix while c is any real number that defines tbe

interval within which one expects to find /Ji more than, for example, 99% of the time3.

The minimization procedure is carried out with aU parameters free to vary except 0,; this

parameter is systematically stepped along the interval defincd above. For each val ue of

X!un(fJi ), exp(-x!.m/2) is plotted versus B., and the final distribution normalizcd to unit

area. The resuIting function is termed the likelihood function and represents the best

estimate of the distrihutional properties of the parameter 0,. The upper lirnit, Oun" on

this parameter can then be round from the equation

19~pp l'·PP -00 L(O,)dO. = -00 exp( -X~n(O.)/2)dO, = CL (B.8)

where CL is sorne confidence limit, usually 90 or 95%.

3It is not possible to determine tbis region in advance without knowing the distributional propertics of 8~; however, assuming rougbly Gaussian errOf8, a typical first choice would be c = 4.


B.2 The X2 Function Used in Chapter 4

The least squares principle has been used to construct the function of Eq. 4.12:

x2 = E [T(4S)j - >'ssBBi - ~cCONTi - >',,_uBTOU.]'

.Elep6in T4Si + >'~BBBj + XbCONTi

(B.9)

The parameters (j are just >'BBI >'c, >. .... ,., and >'SIGI and the observation points are

the bins of the histograms. (The author invites the reader to agonize over an elegant

way of describing t~e observation points of the last two terms.) The first thing to note

is that this fit Cunction employs not one but four4 Cunctions to fit the data and thereCore

differs Crom the single function fit described above. It is obvious, however, that having

four functions instead of one only complicates the formulation of the problem, and does

not reduce the applicability of the least squares p!'inciple.

Most of the bins of the histograms used in the fits have a large number of entries;

therfore, the contents of aDy given bin can be considered as normally distributed with

mean ahd variance given by the Dumber of entries. The weights can easily be calculated

in accordance with Eq. B.3. For example, the weights for the first summation in Eq. B.9

are givcn by

(8.10)

The ,\ .... " term has been droppe<! because it is negligible in comparison with the other

terms. Including it would only further the non-linear nature of the problem. The >'SIG

term is dropped from the weights in the second summation for the same reason.

4The last two terms in Eq. B.9 have functions/ = '\C and J = '\.-u. with observations (r ± C1(r» and P::!'u ± C1('\._u)) respectively.


B.3 Calculation of Error Bars for Fits

The crror bars for the recoil mass and lepton momentulll spectrum fits arc cakulat.ed

using the best fit values ft and the covariance matrix V(~) produced by rvlINUIT. The

derivation is straightforward but tedious, and will only be skctchcd hert'5. The t. lIeol <'1 jea 1

model uscd for the fits is the linear sum of tluee histograllls, X, }', and Z, lll<' [th hill of

the sum given by

(B. Il)

XI' y;, and Zi are variables independent of each other and of the paratllet('ts, O •. TIlt'

parameters cannot be considered as simple constants; rather, they arc correlatt'd variahles

as dictated by the error matrix V(ft). Given the following propcrtics of Ill1tOrrf'lat(·d i\lId

corrclated variables,

E(pq) = E(p)E( q)

E(pq) = E(p)E(q) + Vpq

l'(p + q) = E(p) + E(q)

(T2(p) = E(p2) _ E2(p)

( uncorl'.)

( llncorr.)

( corr.)

(general)

(gcneral),

the errOIS on the fit values can be derived as follows:

-[E(OIX.) + E((}2Y.) + E(03Z.W

= (T2(01.\'.) + 0'2(02}~) + 0'2(03Z,)

+2E(X.)E(}~){E(OI02) - E(OdE(02)}

+2E(XI )E(Z.){E(OI03) - E(Od E(03)}

+2E(Y;)E(Z.){E(0203} - E(02)E(03)}

= (T2(01.\,) + 0'2(02~) -1- (T2(03Z,) + 2X.Y. Vl2 + 2X .Z. V13 + 2Y.Z, V2:l

5This is the author's own derivation.

APPENDIX B. THE LEAST SQUARES PR/NO/PLE 127

22 22 22 -,-- ----T = 0lUX, + 02UY, + 03(1Z, + (X" Y" Z,)V(n.)(X" Y" Z,)

~ O~U}, + O~(1? + O~Ol + (X" Va, Z,)V(~)(X" Va, Z,)T

Appendix C

MINUIT!

The minimization of equation 4.12 is done entirely in the framework of the MINUIT

package for multiparameter function minimization and shape analysis. The package offers

several methods of function minimization, and is capable of performing a proper error

analysis for both lincar and non-linear functional dependences on the parameters. A

routine to calculate confidence interva.ls is also availahle, but this feature is only applicable

to linear or nearly linear models. These routines were not cmployed for this work. lnstead,

a program that calls MINUIT routines is used to calculate the likelihood function (sec

Appendix B) for the signal parameterj this is then integratcd to provide an upper lirnit

at any desired confidence level.

The routine used to minimize the X2 function is called MIGRAD. A byproduct of

minimization using MIGRAD is the covariance matrix that provides parabolic crrors for

the parameters. The minimization proceeds very quickly using a technique from a c1ass of

methods termed variable metric [97]. This particular variable metric method belongs to

R. Fletcher who first introduced it in 1970. There is no good rcason to go into the details,

however, and only a basic outline of the general variable metric method is given here.

There are three steps involved. First, initial parameter values 0 are given. If the gradient

V(O) is not provided as weB, it is calculated. An estimate oC the covariance matrix v., is

made but nced not he accu rate. Second, using these values, a new minimum is searched

for in the direction of v.} . V(O)J so that the new values of '0 becomc 0: = O. - ov.J . V(O) •.

The new gradient is calculated as well for "1. The last step is to repeat the procedure aCter

a. suit able updating of the error matrix is exccuted. It is at this step that one finds the

lInformation OD MINUIT ean be round in Rer. [96]

128

APPENDIX C. MINUIT 129

variations among the variable metric techniques aince the error matrix cao be estimated

through a variety o( different schemes. Once & certain criterion (or minimization has been

met, the process cornes to a halt complete with error matrix.

..

l-

Appendix D

A Proposed Method for Increasing Sensitivity

The author has discovered a new method for potentially increasing sensitivity to the

signal transition. As indicated in Section 4.7.1, B mesons are produced back-to-back

at the 1(45) with a smaU, calculable momentum of about PB =350MeV. However, the

direction along which the mesons are produced is not known. Therfore, to employ the

recoil mass technique, their motion must be neglectedj as a consequence, a potentially

useful bit of data, PB, is ignored. It is possible to exploit this information, however, if

pOL combinations that falI into the recoil mass signal region are examincd for consistency

with the decay of B mesons having moment a of 350MeV.

Without neglecting the motion of the B mesan, the recoil mass value for a pOL pair is

given by Eq. 4.8:

MREC2' = (EB - Ep - Et)' - (PB - Pp - Pt)'

= MREC2- Il PB 11 2 +2PB . (pp + Pt)

= MREC2- Il PB 11 2 +2 Il PB Il . Il (pp + Pt) Il . cos a (D.I)

where MREC2 is the usual recoil mass value when the motion of the B is neglected. The

only unknown is a, the angle between the B meson and the sum of the pO aud lepton

momenta. Because the width of the recoil mass signal is due predominantly to the motion

of the B mef;ons, for any real signal event it is usually possible to change the B direction in

momentum space until it yields a recoil mass value of MREC2' = O. This is only physically

possible if 1 cos 0 0 I~ l, where 0 0 is the value of a necessary to satisy MREC2' = 0 in

130

f

APPENDIX D. A PROPOSED METHOD FOR INCREASING SENSITIVITY 131

, of p-Iepton pairs 600.0 ~ __ ~~~'-~~~'--r~----~-r~~'-~-' • 1

1&00.0

200.0 '"

0.0 -1&.0

+ + +

+

-+--t-

-+-.-.-. 1

-2.0 0.0

-

--'

2.0 1&.0 Cos cr.

Figure D.1: Plot of cos 0 0 for single B signal events (MOPEK-MINIMC).

Eq. D.1:

(D.2)

Signal events would al ways meet this requirement if not for imperfect detector resolutions

(Fig. D.1). On the other hand, pOl combinations from the signal region of the T(4S),

continuum, and B B MREC2' data need not satisfy this condition because they are not

physically constrained by the momentum of a decaying B meson.

The procedure then is to plot cos 0 0 for all recoil mass values (MREC2') which fall into

the region -1. 5Ge y'l / c" to 1. SGe y2 / c4 . For the calculation of M REC2', i t is assumed

that PB is always directed along the z-axis1• It is advantageous to eut on MREC2' rather

than MREC2 due to a s. ,all increase in sensitivity to the signal. Furthermore, this

technique attempts to exploit the effects of the B momentum on the signal; it is therfore

more natural to employa recoil mass expression that incorporates PB. To calculate the

continuum contribution, t!le magnitude of PB is set to 350MeY and the lepton and pO

energies scaled by E .. s / Ec. The cos 0 0 distributions for the difl'erent data pools are shown

lClearly, any direction can be chosen for PB.


in Figures 0.2 to D.5. Only leptons with momentum greater than 1.6GeV are used. To

demonstrate the ability of the technique to distinguish the signal, a simultaneous fit such

as that advocated in Chapter 4 is not employedj rather, a single fit of the T( 4S) cos 0 0

distribution is made with the BB, continuum, and signal data. The continuum scaling

factor constraining term used in 4.12 is also incorporated into the fit function. The scalillg

factor values and errors are taken from the lepton number ratio beyond the endpoint as

calculated in Section 4.5.

The results of the fit are shown in Table D.I and Figures 0.6 and D.7. The eledron

and muon upper limits contradict each other. Sorne atternpt is made to account for this

in Chapter 4. The errors on the signal parameter values are larger than those for the

simultaneous fits of the recoil mass and lepton spectra described in Chapter 4. There

are a few reasons for this. First, only a small fractIon of the recoit mass spectrum is

used to produce the (OS 0 0 plots and as a consequence, the statistics are poor. Second,

the shape of the signal distribution is not as easily distinguishable from the continuum

and BB distributions as it is for the recoil mass distributions. Finally, the results arc

not for simultaneous fits. L,cludi~g the lepton momentum and recoil mass spectra in the

fits would introduce further constraints on the parameters and lower the errors. Besides

increasing the size of the data pool, this would be the best approach to improving the

errors. Unfortunatel:y, time constraints prevented the author from carrying out such an

analysis in the present thesis.

To the best of the author's knowledge, this technique for isolating signal events has

not. heretofore been suggested. Unfortunately, the power of the method is not very weil

demonstrated by this analysis because of the absense of a signal. The technique would

be more useful in the analysis of charmed decays like B - D-lri where significant recoil

mass signaIs are observed.

., \

APPENDIX D. A PROPOSED METHOD FOR INCREASING SENSITNITY 133

, of p-Ieplon pairs 0.030

0.025 ~

0.020

0.015 ~

0.010

0.005

0.0 -11.0

- e

+ '"

.

-2.0

1

1 -1

1 1 -

--

1 0.0 2.0 Il.O

Cos a.

Figure D.2: Plot of cos lko for real signal events (WSB model).

[!eSülts of fits of the cos lko distributions 1

/1 -'c 1 >'SIG Il X2/ D•O.F Il

Electrons

112.32 ± 0.251 -728 ± 615 " 1.24 Il Muons

112.28 ± 0.251 394 ± 674 Il 2.47 Il

Table D.1: Results from fit of cos lko distributions. Model used for signal is that of Wirbel, Stech, and Bauer. Errors are parabolic .

r i' ~'


, of p-Iepton pairs 300.0

250.0

200.0

150.0 l-

100.0

50.0 f0-

0.0 -4.0

- e

+ P-

r

T

I 1 1

I 1

-2.0

•

1

0.0

.-1

1

2.0 Cos CIo

Figure 0.3: Plot of cos 0 0 for Monte Carlo BB events.


100.0

80.0

60.0

40.0

20.0

0.0 -14.0

-t

1 ,

e

P-

1

1

l 1 J 1

-2.0 0.0 2.0 Cos CI.

Figure 0.4: Plot of cos 0 0 for T(4S) events.

.

4.0

14.0

l


, of p-Iep ton pairs 15.0

12.5 -

10.0 -7.S

5.0 -

2.5

0.0 -11.0

- e

t II-

1

1 1

-2.0

1

0.0 2.0 Cos CI.

Figure D.5: Plot of cos Qo for continuum events.


125.0 ~

100.0

75.0

50.0 ...

25.0

0.0 -11.0

-+

,

1(45)

FIT 1

1 1

_1_

-2.0

1

1

1

0.0

T

~

1

2.0 Cos CI.

Figure D.6: Result of fit for electron data.

·

·

·

·

Il.O

~

·

·

·

Il.O

t. f, l'


, of p-Ieplon pairs 180.0

150.0

120.0

90.0 ~

60.0 ~

30.0 ~

0.0 -'i.o

-+

r r

T(45)

FIT

1 1

-2.0

+ 1

0.0

1 1

1 2.0 Cos CI.

Figure D.7: Result of fit for muon data.

.

.

Il.0

r ,

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