FIRST OBSERVATION of the Ηb MESON and STUDY of THE

Home , Patricia Burchat

FIRST OBSERVATION OF THE ηb MESON AND STUDY OF THE

DECAYS Υ (3S) γηb AND Υ (2S) γηb → →

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Christopher West May 2010

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/nv861zm8623

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Rafe Schindler, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Patricia Burchat

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Jonathan Dorfan

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Michael Peskin

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

This dissertation presents the results of an analysis of data samples consisting of 122 million Υ (3S) decays and 90 million Υ (2S) decays collected with the BABAR detector operating at the PEP-II asymmetric-energy storage rings at SLAC National Accelera- tor Laboratory. The η meson is observed for the first time in the decay Υ (3S) γη b → b and its existence is confirmed in the decay Υ (2S) γη . The η mass extracted → b b from the Υ (3S) γη sample is 9388.9+3.1(stat) 2.7(syst) MeV/c2, correspond- → b −2.3 ing to a hyperfine mass splitting m m = 71.4+2.3(stat) 2.7(syst) MeV/c2. Υ (1S) − ηb −3.1 The branching fraction for the decay Υ (3S) γη is determined to be [4.8 → b 0.5(stat) 0.6(syst)] 10−4. An analysis of the decay Υ (2S) γη confirms the × → b

observation of the ηb meson and provides an additional measurement of the ηb mass: 9394.2+4.8(stat) 2.0(syst) MeV/c2. The branching fraction for the decay Υ (2S) γη −4.9 → b is determined to be [3.9 1.1(stat)+1.1(syst)] 10−4. As the measurements in the −0.9 × two samples are consistent, the masses are averaged to provide a combined value of m = 9390.9 3.2 MeV/c2. After canceling common systematic errors, the following ηb branching fraction ratio is determined: [Υ (2S) γη (1S)]/ [Υ (3S) γη (1S)] = B → b B → b 0.82 0.24(stat)+0.20(syst). −0.19

iv Acknowledgments

A Ph.D. dissertation is never completed without the help of many other people, and mine is no exception. First of all, I would like to thank my adviser, Rafe Schindler, for his guidance throughout the years and giving me the freedom to discover my

interests in physics at BABAR. A great deal of my knowledge of physics analysis comes directly from those with whom I have worked: Philippe Grenier, Peter Kim, Peter Lewis, Silke Nelson and Veronique Ziegler. In particular, I would like to thank

Peter Kim for spurring my interest in searches for undiscovered states, and the ηb in particular. My analysis of radiative decays beneﬁts from the knowledge I gained in electromagnetic calorimetry from Martin Kocian and Bill Wisniewski. The support of my Group E colleagues Walt Innes, Peter Lewis, Selina Li, Martin Perl and Stephen Sun made the task of completing my dissertation a pleasant experience. Finally, I would like to thank my parents, Bill and Claudia West, for always believing in me

and encouraging me to pursue my dreams.

v Contents

Abstract iv

Acknowledgments v

1 Introduction 1

1.1 The Standard Model ...... 2

1.2 Quantum Chromodynamics ...... 3

1.2.1 Historical Overview ...... 3

1.2.2 Asymptotic Freedom ...... 4

1.3 Theoretical Approaches to QCD ...... 5

1.3.1 Quark Potential Models ...... 5

1.3.2 Lattice QCD ...... 6

1.3.3 Non-relativistic quantum chromodynamics ...... 10

1.3.4 Potential non-relativistic quantum chromodynamics ...... 12

1.4 Bottomonium ...... 13

1.5 Electromagnetic transitions ...... 13

1.6 The ground state of bottomonium, the ηb meson ...... 15

1.7 Conclusion ...... 22

vi 2 The BABAR Detector 24

2.1 Physics Motivation ...... 24

2.1.1 CP violation ...... 24

2.1.2 Bottomonium Physics ...... 26

2.2 PEP-II ...... 26

2.3 The BABAR Detector ...... 26

2.4 Silicon Vertex Tracker ...... 27

2.5 Drift Chamber ...... 30

2.6 Detector of Internally Reﬂected Light ...... 34

2.7 Electromagnetic Calorimeter ...... 36

2.7.1 Design ...... 37

2.7.2 Calibration and Performance ...... 38

2.8 Superconducting Coil ...... 41

2.9 Instrumented Flux Return ...... 42

2.10 Trigger ...... 43

2.10.1 Level 1 Trigger ...... 44

2.10.2 Level 3 Trigger ...... 45

2.11 Datasets ...... 45

3 Analysis Overview 47

3.1 Introduction ...... 47

3.2 Backgrounds ...... 48

3.2.1 Non-peaking ...... 48

3.2.2 Υ (nS) γχ (mS), χ γΥ (1S) ...... 49 → b b → 3.2.3 e+e− γΥ (1S) ...... 49 → vii 3.3 Selection Criteria ...... 50

4 Study of the Decay Υ (3S) γη 52 → b 4.1 Selection Criteria ...... 52

4.1.1 Data set ...... 52

4.1.2 Event selection ...... 53

4.1.3 π0 veto optimization ...... 55

4.1.4 Investigation of a possible η veto ...... 62

4.1.5 N-1 cut plots ...... 64

4.2 Background to the Eγ spectrum ...... 66

4.2.1 Introduction ...... 66

4.2.2 Non-peaking Background ...... 67

4.2.3 Peaking Background from χbJ (2P ) → γΥ (1S) ...... 69

+ − 4.2.4 Peaking Background from e e → γISRΥ (1S) ...... 73

4.3 Fitting procedure ...... 81

4.3.1 Fit and Unblinding strategies ...... 81

4.3.2 Minimization Details ...... 82

4.4 Toy Studies ...... 84

4.4.1 Conclusions ...... 86

4.5 Fit on the 2.5 fb−1 Optimization Sample ...... 93

4.6 Fit to the inclusive photon spectrum ...... 93

4.7 Systematic Uncertainties ...... 96

5 Study of the Decay Υ (2S) γη 98 → b 5.1 Selection ...... 98

viii 5.2 Background Modeling ...... 102

5.2.1 Introduction ...... 102

5.2.2 Non-peaking Background ...... 102

5.2.3 e+e− γ Υ (1S) ...... 103 → ISR 5.2.4 Υ (2S) γχ (1P ), χ (1P ) γΥ (1S) ...... 114 → bJ bJ → 5.2.5 Υ (2S) Υ (1S)(η, π0) ...... 117 → 5.2.6 Υ (2S) Υ (1S)π0π0 ...... 119 → 5.3 Control Sample Studies ...... 122

5.3.1 Selection ...... 122

5.3.2 Monte Carlo ...... 123

5.4 Fit Procedure for Υ (2S) γη Sample ...... 129 → b 5.4.1 Introduction ...... 129

5.4.2 Fixed and ﬂoating parameters ...... 130

5.4.3 Fit to test sample, ISR yield ﬁxed ...... 131

5.4.4 Fit to test sample, ISR yield ﬂoating ...... 134

5.4.5 Fit to full sample with signal region blinded, ISR yield ﬁxed . 139

5.4.6 Fit to full sample with signal region blinded, ISR yield ﬂoating 139

5.5 Toy studies ...... 146

5.5.1 Introduction ...... 146

5.5.2 Toys with ISR yield ﬁxed to incorrect yield ...... 147

5.5.3 Conclusion ...... 148

5.6 Fit results ...... 149

5.7 Systematic Errors ...... 155

5.7.1 Sources of Systematic Errors ...... 155

ix 5.7.2 Signiﬁcance of Signal (Including Systematic Errors) ...... 156 5.7.3 Additional Fit Variations ...... 157

5.7.4 Branching Fraction Uncertainties ...... 160 5.8 Combination with Υ (3S) result ...... 164 5.8.1 Ratio of branching fractions ...... 164 5.8.2 New Υ (3S) γη branching fraction ...... 167 → b

5.8.3 Mass of the ηb ...... 168

6 Summary and Outlook 169

Appendices

A Use of EMC Timing to Improve π0 Veto 171

B Spurious Feature at 680 MeV 176 B.1 Description of problem ...... 176 B.2 Other investigations of spike ...... 177

Bibliography 183

x List of Tables

1.1 Predictions for the hyperﬁne splitting m m from lattice QCD Υ (1S) − ηb and perturbative QCD calculations...... 17

1.2 Predictions for the two-photon partial decay width, taken from [41],

and the total width determined by scaling the two-photon decay width using Eq. 1.12...... 20

1.3 Limits (95% conﬁdence level) on ηb two photon partial width times branching fraction from LEP...... 23

4.1 On-resonance datasets used in the selection optimization ...... 53

4.2 Selection eﬃciencies () for truth-matched signal MC and on-peak data

in the energy range 0.85 < Eγ < 0.95, in percent. The reconstruction eﬃciency on data is normalized to 100%...... 56

4.3 Comparison of single cut eﬃciencies and S/√B from ﬁtted χb yields and truth-matched signal MC. The background contribution is found

by integrating the background function from E 2σ to E + 2σ. 59 g,χb1 − g,χb2 xi 0 4.4 Signal to background study for the π veto. nχb and ∆nχb are the

ﬁtted χb signal yield and its error. B is the background determined by integrating the background function from E 2σ to E + 2σ. g,χb1 − g,χb2

S/√B of the χb peaks is derived from the ﬁtted χb yield and the

background underneath the peaks. The background in the ηb search region, is the integral of the background distribution in the search

region, deﬁned by 0.85 < pγ < 0.95...... 60

4.5 Signal to background study for π0 veto with additional 3σ timing cut.

The results of the timing from the simulation are not quoted, as the simulation matches the data poorly...... 61

4.6 Fitted background parameters from a ﬁt of the full data sample in the

side region 0.5 < Eγ < 0.6 GeV and 0.960 < Eγ < 1.2 GeV...... 69

4.7 Summary of χbJ (2P ) transitions. In the last column we give the number of expected events, corrected for eﬃciency, for 30 fb−1 of on resonance data...... 69

4.8 Fitted parameters for the χbJ (2P ) peaks in the blinded full dataset spectrum. A, N and σ are the transition point, tail parameter, and Gaussian width, respectively, of the Crystal Ball function. The oﬀset is deﬁned by E E ...... 71 γ,expected − γ,measured

4.9 Production cross section for e+e− γ Υ (1S) at √s = 10.3252 GeV → ISR (σ ), production cross section for e+e− γ Υ (1S) at √s = Υ (3S) → ISR

10.55 GeV (σΥ (4S)), and their ratio for various orders in perturbation theory. The assumed di-electron width of the Υ (1S) is 1.340 MeV. . . 75

xii 4.10 Number of e+e− γ Υ (1S) events from the Υ (3S) oﬀ-resonance → ISR and Υ (4S) oﬀ-resonance samples, and extrapolation to the Υ (3S) on-

resonance sample (25.598 fb−1). The errors are statistical only. . . . . 76

4.11 Fitted peak values (µ) and width (σ) of ηb peak position value distri-

bution in GeV for various combinations of ηb yield (in thousands) and photon peak positions...... 85

4.12 Mean and width of signal signiﬁcance distributions for various combi-

nations of ηb yield (in thousands) and photon peak positions. . . . . 86

4.13 Signal yields and width (σ) of signal yield distribution for various com-

binations of ηb yield (in thousands) and photon peak positions. . . . . 87

4.14 Mean and width of the ﬁtted ηb peak positions for diﬀerent yields of ISR events ...... 88

4.15 Mean and width of the fitted ηb signal significance for different yields of ISR events ...... 88

4.16 Mean and width of the ﬁtted ηb signal yield for diﬀerent yields of ISR events ...... 89

4.17 Mean and width of the ﬁtted ηb peak position in GeV for diﬀerent

generated and ﬁtted ηb widths. Generated peak position is 910 MeV and generated yield is 20k events...... 90

4.18 Mean and width of the fitted ηb signal significance in GeV for different

generated and ﬁtted ηb widths. Generated peak position is 910 MeV and generated yield is 20k events...... 91

xiii 4.19 Mean and width of the ﬁtted ηb signal yield in GeV for diﬀerent gen-

erated and ﬁtted ηb widths. Generated peak position is 910 MeV and generated yield is 20k events...... 92

4.20 Results of the ﬁt to the Eγ distribution assuming the nominal ηb width

of Γ(ηb) = 10 MeV...... 93

4.21 Results of the ﬁt to the Eγ distribution with diﬀerent assumed values

of Γ(ηb)...... 94

4.22 Systematic uncertainties for the measurement of the ηb yield and peak position (mass)...... 97

4.23 Systematic uncertainties for the measurement of the ηb branching fraction...... 97

5.1 Improvement of S/√B as the track and lateral moment selections are imposed. The overall normalization is arbitrary...... 99

5.2 Cut eﬃciencies for truth-matched signal MC determined by counting

events in the energy range 0.3 < Eγ < 0.7 GeV...... 100

5.3 Systematic errors on extrapolation of oﬀ-Υ (4S) ISR rate to on-Υ (2S) data...... 106

5.4 Selection eﬃciencies for truth-matched e+e− γΥ (1S) signal MC de- → termined by counting events, at Υ (4S) oﬀ-peak and Υ (2S) on-resonance

energies. The photons are counted in the energy ranges 0.4 < Eγ <

0.6 GeV and 0.85 < Eγ < 1.10 GeV for the Υ (2S) on-peak and Υ (4S) oﬀ-resonance energies, respectively...... 108

xiv 5.5 Number of e+e− γ Υ (1S) events from the Υ (2S) oﬀ-resonance → ISR and Υ (4S) oﬀ-resonance samples, and extrapolation to the Υ (2S) on-

resonance sample (13.4 fb−1), excluding the test sample. The errors are statistical only. We calculate that at the peak of the Υ (2S) resonance the cross section for radiative return to the Υ (1S) is 41.3 pb. The eﬃciencies for the Υ (2S) and Υ (3S) oﬀ-peak are determined from the

corresponding on-peak MC dataset...... 113

5.6 Branching fractions for the decays Υ (2S) γχ (1P ), χ (1P ) → bJ bJ → γΥ (1S), and the CM energies of the photon in the ﬁrst and second decay. The values are taken from the PDG...... 114

5.7 Theoretical branching fractions for the decays Υ (2S) γχ (1P ), χ (1P ) → bJ bJ → γΥ (1S), and the CM energies of the photon in the ﬁrst and second decay from [49]. In deriving the branching fractions from the radiative width, we use a full width of the Υ (2S) Γ = 31.98 2.63 keV. . . . . 115

5.8 Width of χcJ , J = 0, 1, 2 states from the PDG [79]. The widths of the

χbJ are expected to be smaller than those of the χcJ . This provides

the justiﬁcation for neglecting the χb width in the ﬁt...... 115

5.9 Doppler broadening due to momentum of χbJ relative to the CM frame. The value listed is the half-width of the Doppler broadening PDF. . . 117

xv 5.10 Result of simultaneous ﬁt to χbJ , J = 0, 1, 2 MC. The peak position, Doppler width and signal yield are labeled by their total angular mo-

mentum, J. The remaining parameters are shared between the three peaks. They are A, the transition point between the Gaussian and power tail components of the Crystal Ball function, in units of σ from the mean of the peak, and N, the power law parameter. Note that

there are 96,000 events in the J = 1 MC sample compared to 145,000 events in each of the J = 0, 2 data samples which must be taken into account in computing the eﬃciency...... 118

5.11 Cut eﬃciencies determined by counting events in χbJ (1P ) MC. . . . . 119

5.12 Parameters resulting from ﬁts to χbJ MC...... 124

5.13 γsoft spectrum fitted parameters, final fit...... 128

5.14 Parameters from ﬁt to test sample with the ISR yield ﬁxed. The sec-

tions of the table are yields, the ηb peak position, background param-

eters, and χb lineshape parameters. The ISR yield is ﬁxed in the ﬁt. . 134

5.15 Parameters from ﬁt to optimization sample with the ISR yield uncon-

strained. The sections of the table are yields, the ηb peak position,

background parameters, and χb lineshape parameters...... 138

5.16 Parameters from ﬁt to full sample with signal yield blinded. The sec-

tions of the table are yields, the ηb peak position, background param-

eters, and χb lineshape parameters. The ISR yield is ﬁxed to 17324, a value determined from an earlier study of the ISR yield...... 141

xvi 5.17 Fit parameters from ﬁt to full sample with signal yield blinded and

ﬂoating ISR yield. The sections of the table are yields, the ηb peak

position, background parameters, and χb lineshape parameters. . . . . 145

5.18 Signal yields, yield errors and pulls from toy studies with ηb width ﬁxed at 5 MeV. We show both the mean (µ) and width (σ) from a Gaussian ﬁt to the pull distribution...... 147

5.19 Signal peaks, peak errors and pulls from toy studies with ηb width ﬁxed at 5 MeV. We show both the mean (µ) and width (σ) from a Gaussian

ﬁt to the pull distributions...... 147

5.20 Signal yields, yield errors and pulls from toy studies with ηb width ﬁxed at 10 MeV. We show both the mean (µ) and width (σ) from a Gaussian ﬁt to the pull distribution...... 147

5.21 Signal peaks, peak errors and pulls from toy studies with ηb width ﬁxed at 10 MeV. We show both the mean (µ) and width (σ) from a Gaussian

ﬁt to the pull distributions...... 148

5.22 Fit parameters from ﬁt to full sample with ﬂoating ISR yield. The

sections of the table are yields, the ηb peak position, background pa-

rameters, and χb lineshape parameters...... 150

2 5.23 Signal yields and χ values for ﬁts with alternate ηb widths...... 152

5.24 Systematic uncertainties for the measurement of the ηb yield and peak position (mass). Errors with no sign given are taken to be symmetric. 157

5.25 Yield and signal signiﬁcance (statistical only) with several ﬁt variations.159

5.26 Comparison of selection eﬃciencies in χbJ (2P ) data versus χb0(2P ) MC.160

5.27 Systematic uncertainties on the selection eﬃciency, in percent. . . . . 163

xvii 5.28 Systematic uncertainties for the measurement of the ηb branching fraction, in percent...... 163

5.29 Yield of ηb signal in Υ (2S) and Υ (3S) analyses, the ratio of the two yields, and the deviation of the ratio from that of the 10 MeV ﬁt, as a

function of the ηb width assumed in the ﬁt. Υ (2S) and Υ (3S) are used as abbreviations for the Υ (2S) and Υ (3S) yields, respectively. . . . . 164

5.30 Systematic uncertainties (in %) on the selection eﬃciency in the Υ (3S) → γη and Υ (2S) γη analyses, in percent. Systematics which are as- b → b sumed to cancel in the branching fraction ratio are in italics...... 166 5.31 Systematic uncertainties on the branching fraction ratio, in percent. . 166

5.32 Systematic uncertainties on the new Υ (3S) γη branching fraction, → b in percent...... 167

xviii List of Figures

1.1 Cornell model potential for σ = 0.18 GeV2...... 7

1.2 Spectrum of b¯b levels from a recent lattice prediction [24]. Closed

and open symbols are from coarse (lattice spacing a 0.12 fm) and ≈ ﬁne (a 0.09 fm) lattices, respectively. Squares and triangles de- ≈ note unquenched and quenched results, respectively. Lines represent experiment...... 8

1.3 Spectrum of b¯b levels and possible transitions between levels. Figure from [36]...... 14

1.4 Spectrum of b¯b levels from a recent lattice prediction [24]. Closed and open symbols are from coarse (lattice spacing a 0.12 fm) and ≈ ﬁne (a 0.09 fm) lattices, respectively. Squares and triangles de- ≈ note unquenched and quenched results, respectively. Lines represent experiment...... 18

1.5 Hyperﬁne splitting M M as a function of renormalization scale Υ (1S) − ηb µ in leading order (dotted line), next to leading order (dashed line), leading log (dot-dashed line), and next-to-leading log (solid line) approximations. For the next-to-leading log result, the yellow band cor-

responds to a variation α (M ) = 0.118 0.003. Figure from [19]. . . 19 s Z xix 1.6 Theoretical predictions for Υ (3S) γη branching fraction and CLEO → b upper limit...... 22

2.1 BABAR detector longitudinal section...... 28

2.2 Silicon vertex tracker longitudinal section...... 29

2.3 Drift chamber longitudinal section. Measurements are in mm. . . . . 31

2.4 Drift chamber wire placement for ﬁrst four superlayers. Stereo angles

are in mrad...... 32

2.5 Single cell resolution as a function of distance from sense wire. . . . . 33

2.6 Speciﬁc energy loss versus momentum. Lines show predictions for various mass hypotheses from the Bethe-Bloch equation...... 34

2.7 Diagram of DIRC. Distances are not to scale...... 35

2.8 Cherenkov angle and timing resolution of the DIRC...... 36

2.9 Invariant mass plot of candidate D0 Kπ events before and after → applying loose kaon ID...... 37

2.10 Longitudinal section of the EMC. Lateral dimensions are shown in mm and angles in degrees...... 38

2.11 Fractional energy resolution σE/E versus energy as derived from various sources, compared to Monte Carlo expectations...... 40

2.12 EMC angular resolution as a function of photon energy...... 41

2.13 Geometry of barrel and endcap regions of the IFR...... 42

2.14 Pion rejection versus muon eﬃciency for track momentum 0.5 < p <

2.0 GeV/c (left) and 2 < p < 4 GeV/c (right)...... 43

xx 4.1 Photon CM energies for truth-matched signal photons. Photons inside the calorimeter ﬁducial region (top left), photons inside the barrel but

outside the ﬁducial region (top right). The top plots shown using the

same scale (bottom left). Photon energies versus cos(θγ,LAB) (bottom right). The blue (red) points show photons accepted (rejected) by the ﬁducial requirement. The gradual worsening of the resolution as one

moves away from θLAB = 0 is due to the increased photon path length through the DIRC...... 55

4.2 Cluster lateral moment (left), number of charged tracks in event (right), cosine of photon momentum direction with the thrust axis of the rest

of the event (bottom), for truth-matched signal (red) and data (blue). The distributions are shown prior to the application for any selection

criteria except a requirement of 0.85 < Eγ < 0.95. The arrows show the values of the selection criteria...... 57

4.3 S/√B derived from truth-matched signal MC and on-peak data. The

on-peak data is evaluated in a region 0.85 < Eγ < 0.95...... 58

0 4.4 S/√B of χb peaks versus π second photon energy cut. With (without) timing cut shown in blue (red). The overall scale is arbitrary. Note the suppressed zero...... 61

4.5 Eﬃciency of η veto on data and ηb signal MC, and change in S/√B obtained by applying the veto. No π0 veto is applied. Note the sup-

pressed zero...... 62

xxi 4.6 Eﬃciency of η veto on data and Monte Carlo, and change in S/√B obtained by applying the veto. A π0 veto is applied. Note that the

vertical scale is diﬀerent from that of Fig. 4.5...... 63

4.7 Normalized distribution of selection variables when all other cuts are applied in signal MC (blue) and data (red). The variables are, from

left to right, the number of tracks, the cluster lateral moment, cos θγ , cos θ and m ...... 65 | T | γγ 4.8 Full dataset blinded spectrum, used to determine the continuum background parametrization. This histogram shows the ﬁt while the data

is shown by the markers...... 68

4.9 Fit to the χbJ (2P ) peaks in the blinded full dataset subtracted spectrum. The medium ﬁgure is a zoom and the bottom shows the residuals...... 72

4.10 ISR photon c.m. energy distribution in e+e− γ Υ (1S) signal MC → ISR (data points). The top ﬁgure shows the truth-associated spectrum, the bottom ﬁgure the total spectrum. The superimposed signal lineshape corresponds to the Crystal Ball function (red). The background (green

line) is described using a exponential function. The total ﬁt function is represented by the blue curve...... 77

4.11 The photon c.m. energy distribution in Υ (3S) oﬀ-resonance data (data points). The superimposed ISR-signal lineshape corresponds to the Crystal Ball function (red). The background (green line) is described by a function of the form given in Eq. 4.2. The total ﬁt function is

represented by the blue curve...... 78

xxii 4.12 Shown in the top two plots is the photon c.m. energy distribution in e+e− γ Υ (1S) signal MC, generated at the Υ (4S) oﬀ-resonance → ISR energy. The photon c.m. energy distribution in the Υ (4S) oﬀ-resonance data sample is shown on the bottom. The ISR-signal lineshapes are shown in red(top) and black(bottom). The background (dashed-blue on top, green line on bottom) is described using an exponential func-

tion (top) or a function of the form given in Eq. 4.2 (bottom). The total ﬁt functions are represented by the blue curves...... 79

4.13 Inclusive photon spectrum in the below-Υ (4S) data, after background subtraction. The ﬁtted curve shown is the Crystal Ball Function which describes the data points very well...... 80

4.14 The inclusive photon c.m. energy distribution in the optimization sam-

ple (2.5 fb−1) of Υ (3S) on-resonance data. The top plot show the distribution together with the ﬁt results and the bottom plot shows the non-peaking background subtracted plot...... 94

4.15 (a) Inclusive photon spectrum in the region 0.50 < Eγ < 1.1 GeV. (b) Background subtracted photon spectrum in the signal region, showing

χbJ (2P ) peaks (red), ISR Υ (1S) (green), signal (blue) and the sum of the contributions (purple). (c) Signal peak after all backgrounds are subtracted...... 95

5.1 Number of truth-matched signal photons (top left) and background photons (top right) passing all cuts as the cuts on cos θ and E are | T | γ2 varied. S/√B (bottom) is computed using these two quantities. The

overall normalization is arbitrary...... 101

xxiii 5.2 Spectrum plotted with logarithmic y-axis...... 103

5.3 Fit of photon energy spectrum in ISR Υ (1S) MC at Υ (2S) energy using truth-matched (top) and all signal candidate photons (bottom). The blue curve is the Crystal Ball function used as the signal PDF and the red line is the polynomial background...... 104

5.4 Fit of MC photon energy spectrum in ISR Υ (1S) signal region in MC generated at Υ (4S) oﬀ-resonance energy. The signal PDF is a Crystal Ball function and the background PDF is an exponential...... 106

5.5 Fit of photon energy spectrum for ISR Υ (1S) production in off-resonance Υ (4S) data before (top) and after (bottom) subtracting the smooth background. The green (blue) curve in the top plot is the background (total) fit function. The normalized residuals from the fit are shown in the middle plot...... 109

5.6 Fit of photon energy spectrum of ISR Υ (1S) production in oﬀ-resonance Υ (2S) data before (top) and after (bottom) subtracting the smooth background. The green (blue) curve in the top plot is the background

(total) ﬁt function. The normalized residuals from the ﬁt are shown in the middle plot...... 110

5.7 Fit of MC photon energy spectrum in ISR Υ (1S) signal region in Υ (3S) oﬀ-peak MC. The signal PDF is a Crystal Ball function and the back-

ground PDF is an exponential...... 111

xxiv 5.8 Fit of photon energy spectrum for ISR Υ (1S) production in oﬀ-resonance Υ (3S) data before (top) and after (bottom) subtracting the smooth

background. The green (blue) curve in the top plot is the background (total) ﬁt function. The normalized residuals from the ﬁt are shown in the middle plot...... 112

5.9 Ratio of histograms of photon energy spectrum of χb2 decays before/after

a 25 cm bump distance cut (black). The red curve is the χb2 photon spectrum. Plot courtesy of Steve Sekula...... 116

5.10 Fit of zero-width χb0 (top), χb1 (middle), and χb2 (bottom) MC using truth-matched photons...... 120

5.11 Spectrum of truth-matched photons from the decays Υ (2S) ηΥ (1S)(top) → and Υ (2S) Υ (1S)π0π0 (bottom). Note that the x-axis is diﬀerent →

from that of the ﬁnal ﬁt, which is over the region 270 < Eγ < 800 MeV. 121

5.12 Photon spectrum before cuts for double radiative decay of Υ (2S) candidates. γ window (red): 60 200 MeV, γ window (blue): soft − hard 380 470 MeV...... 123 −

5.13 Monte Carlo χb0 peak with ﬁt superimposed. The curves displayed on the plot are the exponential background (short dotted light blue line), the signal Gaussian (dot-dash green line), the signal Crystal Ball function (red dashed line) and their sum (solid blue line). χ2 per degree of freedom = 193/91 ...... 124

5.14 Monte Carlo χb1 peak with ﬁt and normalized residual plot. The curves are as in Fig. 5.13. χ2 per degree of freedom = 94/81 ...... 125

xxv 5.15 Monte Carlo χb2 peak with ﬁt and normalized residuals plot. The curves are as in Fig. 5.13. χ2 per degree of freedom = 42/61 . . . . . 126

5.16 Fit to soft transitions in data with normalized residuals plot. The exponential background function is shown as a short dotted light blue line, each Gaussian is a dot-dash green line, the red dashed lines are the Crystal Ball functions and the solid blue line is the total ﬁt. χ2/ndof =

123/131...... 128

5.17 Fit to the test sample (top) and residuals from the fit (bottom). The ISR yield is fixed in the fit...... 132

5.18 Fit to the test sample after subtracting the non-peaking background (top) and the corresponding zoomed version (bottom). The χ2/ndof is equal to 93.8/94 = 0.98, corresponding to a ﬁt probability of 49%.

The histograms are χb (cyan), ISR (red), and ηb (blue). The ISR yield is ﬁxed in the ﬁt...... 133

5.19 Fit to the test sample (top) and normalized residuals from the fit (bottom). The ISR yield is floating in the fit...... 135

5.20 Fit to the test sample after subtracting the non-peaking background

(top) and the corresponding zoomed version (bottom). The χ2/ndof is equal to 93.7/94 = 0.98, corresponding to a ﬁt probability of 49%.

The histograms are χb (cyan), ISR (red), and ηb (blue)...... 137

5.21 Fit to the full data sample (top) and residuals from the ﬁt (bottom).

The ISR yield is ﬁxed in the ﬁt...... 140

xxvi 5.22 Fit to full data sample after background subtraction (top) and the corresponding zoomed plot (bottom). The cyan curve is the sum of

the χb peaks, and the red curve is the ISR peak. The histograms are

χb (cyan) and ISR (red). The ISR yield is ﬁxed in the ﬁt...... 142

5.23 Fit to the full data sample (top) and residuals from the ﬁt (bottom). 143

5.24 Fit to full data sample after background subtraction (top) and the

corresponding zoomed plot (bottom). The ηb signal region is excluded

in the residual plot. The cyan curve is the sum of the χb peaks, and

the red curve is the ISR PDF. The histograms are χb (cyan) and ISR (red). The ISR yield is ﬂoating in the ﬁt...... 144

5.25 Fit to the full data sample (top) and residuals from the ﬁt (bottom). 151

5.26 Fit to full data sample after smooth background subtraction (top), the corresponding zoomed plot (middle) and after all backgrounds are

subtracted (bottom). The cyan curve is the sum of the χb peaks, and

the red curve is the ISR PDF. The histograms are χb (cyan), ISR (red),

ηb (blue), and their sum (black). The ISR yield is ﬂoating in the ﬁt. . 153

5.27 Same as ﬁg. 5.26 but with the ηb PDF removed...... 154

5.28 Fit to test sample before π0 veto. The cyan curve is the sum of the

χb peaks, and the red curve is the ISR PDF. The histograms are χb

(cyan), ISR (red), and ηb (blue). The ISR yield is ﬂoating in the ﬁt. . 162

A.1 Deviation from average timing before cluster timing correction. The 48 ns shift from the long shaping time pre-ampliﬁers has already been subtracted. The vertical patterns correspond to diﬀerent crystal man-

ufacturers...... 173

xxvii A.2 Deviation from mean time after cluster timing correction...... 174 A.3 π0 peaks for signal candidate in data when combined with another

photon in the event. All π0 (blue), π0 candidates which are rejected by a 2.5σ timing cut (red), and the subtracted distribution (green). . 174 A.4 Candidate π0 which are rejected by timing cuts by cuts on timing signiﬁcance, σ ...... 175

B.1 Photon energy spectrum near 1 GeV with (red) and without (blue) the crystal edge correction turned on (left). Photon energy spectrum in CM frame for photons satisfying E 1GeV < 0.01GeV (right). 178 | γ,LAB − | B.2 Background subtracted energy spectrum after (left) and before (right)

the edge correction is turned on. The spectra are shown with non- peaking background subtracted (top) and all background subtracted (bottom)...... 178 B.3 Photon energy spectrum of clusters with lateral moment equal to zero,

and passing all analysis cuts...... 179 B.4 Fit to full data sample after background subtraction. The cyan curve

is the sum of the χb peaks, and the red curve is the ISR PDF. The

histograms are χb (cyan), ISR (red), ηb (blue), the extra peak (green) and their sum (black). The ISR yield is ﬂoating in the ﬁt. The top

plot displays the fit where the width of the spurious peak is fixed to detector resolution and the bottom displays the fit in which the width is allowed to float...... 181 B.5 Smooth background subtracted fit of energy spectrum with no primary

vertex correction...... 182

xxviii Chapter 1

Introduction

A fundamental test of the properties of Quantum Chromodynamics (QCD) is provided by measurements of the properties of bound states of quark-antiquark pairs.

Calculations in QCD are extremely complicated but in certain limits the equations of QCD simplify. In the limit of high momentum transfers, the coupling between the quark and antiquark is weak, a limit in which calculations based on perturbation theory are tractable. The nonrelativistic limit also provides additional simpliﬁcations because in this limit relativistic degrees of freedom are removed. States consisting of

a bottom quark and antiquark pair provide a system in which both of these limits apply, and are therefore particularly interesting. While many states consisting of b¯b pairs were discovered decades ago, the ground state of the b¯b system, referred to as

the ηb meson, has remained elusive. This dissertation presents an analysis of data

from the BABAR experiment that provides the ﬁrst observation of the ηb meson.

1 2 CHAPTER 1. INTRODUCTION

1.1 The Standard Model

Electrodynamics describes the interactions of electrically charged particles. A theory for properly treating electrodynamics within a quantum framework, quantum electrodynamics (QED), arose in the 1920s from the study of the the electromagnetic field in quantum mechanics. As well as subsuming classical electrodynamic phenomena, QED predicts lepton magnetic moments and small deviations from classical behavior in atoms. In 1925, Born, Heisenberg and Jordan took the first step by expanding the electromagnetic field in terms of a set of independent harmonic oscillators [1]. Dirac, in 1927, extended this work with a fully quantum mechanically treatment of the electromagnetic field [2].

The leading-order cross sections for many processes (i.e.; pair annihilation[3], bremsstrahlung [4], Bhabha scattering [5]) were successfully calculated in the 1930s. However, the calculation of higher-order corrections were fraught with technical prob-

lems related to divergences. In the late 1940s, Feynman, Schwinger, and Tomonaga discovered how to absorb these divergences into “renormalized” masses, charges and ﬁeld strength operators, a process known as renormalization [6, 7, 8].

Eventually it became understood that the criterion that the inﬁnities of a theory can be absorbed in this way is useful in classifying theories. Indeed, renormalizability is a central feature of the theories of the weak and strong interactions. In the

1960s, Glashow, Weinberg and Salam proposed the SU(2) U(1) theory of the weak ⊗ interactions [9], in which the renormalizability of the theory plays a crucial role. The strong interaction, described further below, is a renormalizable gauge theory based on the SU(3)c gauge group. These theories form collectively what is now referred to as the Standard Model of particle physics. 1.2. QUANTUM CHROMODYNAMICS 3

1.2 Quantum Chromodynamics

1.2.1 Historical Overview

In November 1974, a narrow resonance with a large mass ( 3095 GeV/c2) was discov- ∼ ered simultaneously at Brookhaven and SLAC [10]. The discovery of the J/ψ, as it is

now known, was puzzling as the J/ψ is exceptionally narrow for a meson of that mass. The SLAC measurements determined that the J/ψ decays primarily to hadrons and therefore decays via the strong interaction. Because there is a large amount of phase space for the decay, the J/ψ should have a short lifetime and decay very rapidly via

the strong interaction, leading to a large total width for the J/ψ resonance. The solution to the puzzle lay in the fact that the J/ψ was the ﬁrst example of a bound state of c and c¯ quarks to be discovered. Bound cc¯ states are referred to generically as charmonium, after the corresponding e+e− bound state, positronium. The ψ0, the

ﬁrst excited state of the bound cc¯ system, was discovered a couple weeks later. A few years later, in 1977, the corresponding bound state of b and ¯b quarks, the Υ , was discovered [16]; these states are known collectively as bottomonium.

The long lifetime of the J/ψ led to the realization that Quantum Chromodynamics (QCD) provided the correct description of strong interaction phenomena. QCD is a theory of spin-1/2 particles, called quarks, which carry a charge (called “color” charge)

under the non-Abelian group SU(3)c. Interactions between particles carrying color charges are mediated by spin-1 particles called gluons. The Lagrangian describing QCD [17] is N f 1 = q¯ (iγµD m )q G Gµν L k µ − k k − 4 µν Xk=1 where D = (∂ ig taAa ) is the gauge covariant derivative, Aa is the gluon ﬁeld, µ µ − s µ µ 4 CHAPTER 1. INTRODUCTION

a a Gµν is the gluon ﬁeld strength, mk are the quark masses, t are the generators of the

fundamental representation of SU(3) and the qk are the quark ﬁelds. This theory, combined with the SU(2) U(1) electroweak gauge theory, represents the Standard W ⊗ Y Model of particle physics, which describes all terrestrial particle physics phenomena.

The QCD coupling constant (gs) is smaller in processes involving high momentum transfer, a property known as asymptotic freedom [11, 12], described further below. This property implies that at high energy, the coupling constant is small enough that it is possible to use perturbation theory to describe QCD. As a result of the discovery of asymptotic freedom, for the ﬁrst time, it was possible to quantitatively

describe strong interaction phenomena [14, 15]. For simplicity, we will often refer

2 to the QCD analog of the QED ﬁne structure constant, αs = gs /(4π) as the QCD coupling constant.

1.2.2 Asymptotic Freedom

Calculations of Feynman diagrams in QCD are plagued with divergences resulting

from integrations over the momenta of all virtual particles. In order to create a mathematically well-defined theory, these integrals in QCD must be cut off at some energy µ, in a process known as regularization. Imposing the requirement that the final results do not depend upon the choice of µ leads to the requirement that the strength of the QCD coupling changes (or “runs”) with energy, determined by the

beta-function, β(αs).

The beta-function can be calculated in perturbation theory and is given by

dα α α 2 µ s β(α ) = 2α β s + β s + , (1.1) dµ ≡ s − s 0 4π 1 4π · · · 1.3. THEORETICAL APPROACHES TO QCD 5

where

11 4 β = C T n 0 3 A − 3 F f 34 20 β = C2 C T n 4C T n , . . . , 1 3 A − 3 A F f − F F f

and C = N = 3, C = (N 2 1)/(2N ) = 4/3 and T = 1/2. A c F c − c F

Because of the number of colors (Nc) and quark ﬂavors (nf ) observed in nature, the β function of QCD is negative, leading to a weakening of the QCD coupling at high

energies (or small spatial separations). Conversely, at low energies (or large spatial separations), the QCD coupling becomes large, and the potential energy between charges diverges, causing color charges (the QCD analog of electric charges) to be permanently conﬁned. This explains why free quarks have never been observed,

although they interact as nearly free particles in high energy deep inelastic scattering experiments.

The unexpected long life of the J/ψ can be explained by the asymptotic freedom of QCD. If this is the case, the gluons must carry the four-momentum of the decaying

particle, a large momentum, leading to a small value of αs. The small value of the strong coupling constant in turn suppresses the decay rate.

1.3 Theoretical Approaches to QCD

1.3.1 Quark Potential Models

The equations of QCD are exceptionally diﬃcult to solve. Due to the large value of

αs at low energy, perturbation theory, the primary tool for solving most problems in 6 CHAPTER 1. INTRODUCTION

quantum ﬁeld theory, is no longer applicable. As a result, for many problems in QCD, one is forced to use phenomenological models of the potential energy between a quark and an antiquark, known as potential models, to make predictions involving strong interaction phenomena. Originally, quark potential models were the only tool for understanding QCD phenomena, but they are slowly being replaced by calculations rigorously derived from QCD.

In potential models, one constructs an quark-antiquark potential with enough free parameters to ﬁt the spectrum observed by experiment. At short distances, due to asymptotic freedom, the strong coupling constant is small and so single gluon exchange dominates. This gives rise to a Coulombic form for the potential, analogous to that arising in QED from single photon exchange. At longer distances, color charges are conﬁned, which leads to the requirement that the potential energy between colored particles must diverge at long distances. A popular form of the potential, the so-called “Cornell potential”, is [18] 4 α (r) V (r) = s + σr −3 r where σ 0.18 GeV2 and gives qualitative agreement with the experimental spec- ≈ trum. A plot of this potential is given in Fig. 1.1. To this potential, spin-dependent terms describing spin-spin, spin-orbit and tensor interactions, similar to those describing the hydrogen atom, are added. Recently, progress has been made in deriving the quark-antiquark potential from perturbative QCD [20].

1.3.2 Lattice QCD

For calculations involving processes at low energies the strong coupling constant becomes of order one, rendering perturbative expansions in αs divergent. An alternate, 1.3. THEORETICAL APPROACHES TO QCD 7

2500 2000 1500

V(r) (MeV) 1000 500 0 -500 -1000 -1500 -2000 -2500 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r (fm)

Figure 1.1: Cornell model potential for σ = 0.18 GeV2. non-perturbative description of QCD, pioneered by Wilson, describes gauge ﬁelds in terms of closed loops on a discrete space-time lattice [21]. This description of QCD has the advantage of regularizing QCD in a way that manifestly preserves gauge invariance.

In lattice QCD, observables are calculated by numerically evaluating path integrals in Euclidean space-time by taking expectation values over a large number of gauge field configurations. Expectation values of an operator (φ) are calculated using the O equation dφ [φ] exp ( S[φ]) < >= O − . (1.2) O dφ exp ( S[φ]) R − where S[φ] is the action describing theR φ field and the denominator is a normal-

ization factor. This integral is calculated numerically by averaging over gauge ﬁeld conﬁgurations, using a procedure which ensures the proper exp ( S[φ]) weighting. − Statistical errors on these integrals can, in principle, be made arbitrarily small. 8 CHAPTER 1. INTRODUCTION

However, lattice QCD is extremely computationally intensive, limiting the precision with which it is practical to carry out these computations.

Until recently, most lattice simulations ignored light quark vacuum polarization (the so-called “quenched” approximation), which is computationally expensive to simulate. This is an uncontrolled approximation with errors which are diﬃcult to estimate. However, recent results, which include light quark vacuum polarization, have

succeeded in providing very precise results which are in agreement with experiment for a wide range of observables, including the heavy quarkonium spectrum [24] shown in Fig. 1.2, calculated by the UKQCD and HPQCD collaborations.

10.6

10.4 3S Mass (GeV) 2P 10.2 1D

10 2S 1P 9.8

Experiment 9.6 Quenched Unquenched 1S 9.4

Figure 1.2: Spectrum of b¯b levels from a recent lattice prediction [24]. Closed and open symbols are from coarse (lattice spacing a 0.12 fm) and ﬁne (a 0.09 fm) ≈ ≈ lattices, respectively. Squares and triangles denote unquenched and quenched results, respectively. Lines represent experiment.

Lattice calculations face several sources of systematic error, in addition to the errors discussed above. Calculations on any ﬁnite lattice can only approximate continuum QCD, and an extrapolation of results obtained at ﬁnite lattice spacing to zero 1.3. THEORETICAL APPROACHES TO QCD 9

lattice spacing is always necessary. This is very computationally intensive as, for a given volume, the computational requirements naively scale as the fourth power of

the inverse lattice spacing, and in practice the scaling is often worse. To reduce the systematic error due to discretization, smaller lattice spacings can be used in combination with a smaller simulation volume but, of course, the volume must fully contain the hadron of interest, necessitating a trade-oﬀ.

An additional source of uncertainty is due to the fact that integrals involving light quark vacuum polarization becomes noisier as the light quark masses are de- creased, requiring higher simulation statistics to achieve an identical statistical error

on the path integral. Simulations are performed with unphysically large light quark masses, and an extrapolation to physical light quark masses is necessary, incurring an additional systematic error.

In lattice simulations, full Lorentz symmetry is broken down to discrete symmetries, often creating artifacts which should disappear in the continuum limit. Lattice simulations require an understanding of how these lattice artifacts can arise. As the discretization of the action is not unique, these errors can be studied by altering the discretization.

In B meson physics, lattice techniques are often used to predict observables that are relevant in searches for physics beyond the Standard Model. The bottomonium spectrum gives a large number of experimentally well-known quantities which should be straightforward to predict using lattice techniques. Therefore, aside from the intrinsic interest in the prediction of the bottomonium spectrum, the agreement of the lattice QCD predictions with experiment constitutes an important validation of

lattice techniques. 10 CHAPTER 1. INTRODUCTION

1.3.3 Non-relativistic quantum chromodynamics

An important tool in the analysis of quantum field theories involving multiple energy scales is the effective field theory. An effective field theory (EFT) is obtained by integrating out degrees of freedom above a given energy scale, resulting in a simpler, but equivalent, theory in which calculations are more manageable.

Non-relativistic QED (NRQED) was the ﬁrst EFT for treating nonrelativistic bound states [30]. NRQED provides an ideal framework for the description of positronium (the bound state of an electron and positron) and muonium (a neutral eµ state) and greatly facilitates the calculation of higher order corrections. Some recent successes are the calculation to (α∈ ln α) corrections to the hyperﬁne splitting in O positronium (for a precision of 2.0 ppm) and the most precise value of mµ/me, which provides the best determination of the mass of the muon [32].

The successes of NRQED led to the development of its analogue in QCD, nonrelativistic QCD [31]. Nonrelativistic quantum chromodynamics (NRQCD) is the

eﬀective ﬁeld theory resulting from integrating out modes of energy and momentum describing heavy quark-antiquark pairs from QCD Green’s functions. Heavy quarkonium are characterized by three separate scales, often referred to as the hard, soft, and ultrasoft scales. The hard scale is associated with the mass m of the heavy quark.

The soft scale is determined by the relative momenta of the quark and anti-quark, mv. Finally, the ultrasoft scale is the kinetic energy E = mv2 of the heavy quark and anti-quark. All of these scales are assumed to be greater than the characteristic scale

of QCD, ΛQCD.

In the non-relativistic limit v c heavy quark-antiquark production is suppressed, 1.3. THEORETICAL APPROACHES TO QCD 11

decoupling the heavy quark and antiquark, and allowing a projection onto states containing exactly one heavy quark and antiquark. It is therefore suﬃcient to use Pauli spinors to describe the heavy quark and antiquark separately. NRQCD has obtained considerable successes, such as the quarkonium production rate at the Tevatron [34] and removing the infrared divergences in the calculation of the decay rates of P -wave quarkonium [33].

The NRQCD Lagrangian is an expansion of the form

O (µ) L = c (α (m), µ) n , (1.3) k s mk Xk

where the coefficients ck, called Wilson coefficients or matching coefficients, are determined by equating the Green’s functions of NRQCD and full QCD at the matching scale, µ. For example, the part of the NRQCD Lagrangian that is bilinear in the quark fields is, to order (m v4), O b

L † 1 2 1 4 cF = ψ iD0 + D + 3 D + σ gB 2mQ 8mQ 2mQ ·

cD cS + 2 (D gE gE E) + i 2 σ (D gE gE D) ψ. (1.4) 8mQ · − · 8mQ · × − × !

The third through sixth terms are the QCD analog of terms familiar from the

analysis of the hydrogen atom in QED: the relativistic correction, the hyperﬁne term, the Darwin term and the spin-orbit coupling, respectively.

A lattice simulation must contain the entire hadron within the simulation volume but the lattice spacing must remain ﬁne enough to resolve spatial details and the

decay in time of correlation functions, necessitating a large number of lattice points. 12 CHAPTER 1. INTRODUCTION

The largest momentum that can be simulated on a discrete lattice is π/L = π/(aN), whereas the smallest simulated momentum is π/a. As a result, the reduced range of

scales in NRQCD makes it ideal for lattice simulation.

1.3.4 Potential non-relativistic quantum chromodynamics

NRQCD contains additional degrees of freedom of order mv, as well as mv2. Potential non-relativistic quantum chromodynamics (pNRQCD) is derived from NRQCD by integrating out these degrees of freedom of order mv [35]. In this theory the matching

coeﬃcients between NRQCD and pNRQCD take the form of quark-antiquark poten- tials.

In this EFT the theory is simplified to an extent that, in some situations, analytic calculations of the spectrum are possible in a manner identical to perturbation theory with a Schrödinger-like potential. The potential used in potential model approaches to QCD arises naturally as a Wilson coefficient in the pNRQCD approach.

The pNRQCD Lagrangian has the schematic form

O (µ) L = c (α (m), µ) rnV (r, µ) n (1.5) k s mk Xk where V (r, µ) are matching coeﬃcients, r is the quark-antiquark separation, and µ is the renormalization scale. As an example of the sort of terms which arise in the expansion, the hyperﬁne splitting in the bottomonium ground state is given by

(2) 4CF π 2 σ1 + σ2 Hspin = DS2,s 2 S , S = (1.6) 3mb 2

(2) where σ1 and σ2 are the spin operators of the quark and anti-quark, and DS2,s is the 1.4. BOTTOMONIUM 13

expectation value of the operator

2 2 (2) 4πCF DS ,s (3) V 2 (r) = δ (r). (1.7) S ,s 3

1.4 Bottomonium

At short distances QCD is dominated by one-gluon exchange leading to a spectrum which is similar to the Coulomb spectrum. There are signiﬁcant corrections to the

Coulombic potential due to non-perturbative physics. The current picture of the bottomonium spectrum below BB¯ threshold is shown on Fig. 1.3, where the numerous electric dipole transitions are omitted. Symmetries under charge conjugation and parity impose selection rules on the allowed transitions.

1.5 Electromagnetic transitions

Electromagnetic transitions between bottomonium states are usually treated by potential models in a Schr¨odinger formalism, analogous to the treatment of transitions in atoms. Though transition rates can, in principle, be computed with lattice methods, only one group has attempted calculations of these rates, and only for transitions between charmonium states [37].

The lowest order electromagnetic transitions are the electric dipole (E1) and magnetic dipole (M1) transitions. Electric dipole transitions in quarkonium have ∆` = 1 and ∆s = 0 whereas the magnetic dipole transitions have ∆` = 0 and ∆s = 1. This results in a selection rule that, for dipole transitions, the parity of the state changes in a electric dipole transition but not in a magnetic dipole transition. 14 CHAPTER 1. INTRODUCTION

Figure 1.3: Spectrum of b¯b levels and possible transitions between levels. Figure from [36].

Electric dipole transitions can be calculated in a potential model using

4 Γ = e2 αC (2J + 1)E3 n L r n L 2 (1.8) E1 3 Q if f γ |h f f | | i ii|

where Cif is a statistical factor, equal to 1/9 for transitions between S and P states,

eQ is the charge of the quark, Jf is the total angular momentum of the ﬁnal state,

Eγ is the photon energy, r is the interquark separation, and the ni,f , Li,f are the quantum numbers of the initial and ﬁnal states.

Magnetic dipole transitions can be calculated in a potential model using

4 Γ = e2 αC (2J + 1)E3 n L r n L 2 (1.9) M1 3 Q if f γ |h f f | | i ii| 1.6. THE GROUND STATE OF BOTTOMONIUM, THE ηB MESON 15

where Cif is a statistical factor, equal to 1 for ∆` = 0 transitions.

The calculation of magnetic dipole transition rates in potential models is complicated due to several factors. In potential models the Lorentz structure of the confining potential (vector or scalar) is an arbitrary choice. Also, there is a dependence upon the anomalous magnetic moment of the quark. Finally, relativistic corrections contain an explicit dependence on the quark masses, which in potential models are simply fit parameters rather than rigorously defined quantities.

Magnetic dipole transitions can also be calculated in a model-independent fashion in pNRQCD [38]. However, the calculations performed to date are only applicable

2 to the weak-coupling regime (mv & ΛQCD), a condition which is not satisﬁed by the excited heavy quarkonium states.

All of these approaches assume that the photon energy is small enough that a

multipole expansion can be used to describe the electromagnetic ﬁeld. The multipole expansion converges for kr 1, a condition which may not be satisﬁed by the excited bottomonium states. Recently, another approach has been proposed in which avoids the multipole expansion by factorizing the transition amplitude into a hard scattering

amplitude and non-perturbative wavefunction [39], analogous to the factorization of the pion form factor at large momentum transfer [40].

1.6 The ground state of bottomonium, the ηb me-

son

The Υ (1S) meson, the ﬁrst bb bound state to be discovered, was found over 30 years ago yet until this study its pseudoscalar partner, the ηb meson, had still not been 16 CHAPTER 1. INTRODUCTION

discovered. The ηb meson, the ground state of bottomonium, is interesting because its dynamics are expected to be largely perturbative, due to its small spatial extent.

A measurement of the mass and width of the ηb meson would help to test lattice NRQCD, pNRQCD, and quark potential models in a regime where all three should be valid.

The expectations for the mass splitting between the ηb and Υ (1S) mesons vary from 36 to 100 MeV. A recent lattice calculation within the NRQCD framework by the HPQCD and UKQCD collaborations [24], including vacuum polarization for u, d, ands quarks, gives a prediction of 61(4)(12)(6) MeV/c2. The results are shown in Fig. 1.4. The errors are statistical/fitting and discretization errors; radiative corrections and relativistic corrections, respectively. In this calculation, the NRQCD matching coefficients are taken at lowest order in the expansion (tree level), and only operators up to order (α5m ) are included. A similar unquenched lattice O s b NRQCD calculation, done with a different fermion and gauge discretization (domain wall fermions and the Iwasaki gauge action [26, 27] at a single lattice spacing of 0.11 fm gives a result of 52.5 1.5 MeV/c2, where the errors are statistical only, ≈ and no continuum extrapolation is performed [28]. This hyperfine splitting is consistent with that obtained by the HPQCD and UKQCD collaborations on their 0.12 fm lattice. Another calculation of the hyperfine splitting was performed by the Fermilab

Lattice and MILC Collaborations [29] using the Fermilab action predicts a splitting of 54.0 12.4+1.2, where the second error results from the conversion from lattice −0.0 units to physical units.

The hyperﬁne splitting may also be calculated within perturbative QCD. Fig-

ure 1.5 shows the variation of the hyperﬁne splitting versus the renormalization 1.6. THE GROUND STATE OF BOTTOMONIUM, THE ηB MESON 17

scale at various levels of accuracy in the perturbative expansion. This calculation, performed in the pNRQCD framework, results in a hyperﬁne splitting of 41 +8 11(th)−9(αs) MeV [19]. A recent calculation of the hyperﬁne splitting using the static quark potential (the quark-antiquark potential in the limit m ) from perturba- b → ∞ tive QCD to (α5m ) gives m m gives 44 11 MeV [22]. Theoretical calculations O s b Υ − ηb which are rigorously based on QCD (not potential models) are summarized in Ta- ble 1.1.

Table 1.1: Predictions for the hyperﬁne splitting mΥ (1S) mηb from lattice QCD and perturbative QCD calculations. −

Calculation Hyperﬁne Splitting ( MeV/c2) Reference +8 pNRQCD 41 11(th)−9(αs) [19] Pertubative QCD 44 11 [22] Lattice NRQCD 61 4 12 6 [24] Lattice NRQCD 52.5 1.5 [28] Lattice NRQCD 54.0 12 .4+1.2 [29] −0.0

Non-perturbative corrections arise at order (m (Λ3 /m3)) m α6 5 MeV [23]. O b QCD b ≈ b s ≈ The leading non-perturbative corrections are dominated by the vacuum expectation value of gluon pairs, the gluon condensate,

1 G2 = 0 α G Gµν 0 (1.10) h i 4h | s µν | i which shifts the hyperﬁne splitting by

2 301066767 π αsG LO ∆Ehf = h 4 6 iEhf (1.11) 92480000 mb αs

LO 4 4 where Ehf = CF αs(µ)mb/3 is the hyperﬁne splitting at leading order in αs. 18 CHAPTER 1. INTRODUCTION

Figure 1.4: Spectrum of b¯b levels from a recent lattice prediction [24]. Closed and open symbols are from coarse (lattice spacing a 0.12 fm) and ﬁne (a 0.09 fm) lattices, respectively. Squares and triangles denote≈unquenched and quenched≈ results, respectively. Lines represent experiment.

The ηb meson is expected to decay almost exclusively to two gluons with a full width of around 10 MeV. The width of the ηb meson is expected to be smaller than that of the ηc due to a smaller value of the strong coupling constant at the b quark mass scale relative to the c quark mass scale (α (m ) 0.2 versus α (m ) 0.3). s b ≈ s c ≈

Predictions for the full width of the ηb are uncommon in the literature; rather, the di-photon width, the calculation of which is not beset with complications due to QCD radiative corrections, is usually computed. Assuming Γ Γ(η gg) and neglecting ≈ b → QCD radiative corrections, then the two-photon and two-gluon processes are equiva-

lent up to a vertex factor. The full width can then be calculated by multiplying the 1.6. THE GROUND STATE OF BOTTOMONIUM, THE ηB MESON 19

50 45

(MeV) 40 hfs

E 35 30 25 20 15 10 5 0 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 ν (GeV)

Figure 1.5: Hyperﬁne splitting M M as a function of renormalization scale µ Υ (1S) − ηb in leading order (dotted line), next to leading order (dashed line), leading log (dot- dashed line), and next-to-leading log (solid line) approximations. For the next-to- leading log result, the yellow band corresponds to a variation α (M ) = 0.118 0.003. s Z Figure from [19]. two-photon partial width by

2 α (m )2 α (m ) 9 s b 1 + 7.8 s b 2.4 104. (1.12) e4α2 π ≈ × b where α and αs are the electromagnetic and strong coupling constants, respectively, and e is the b-quark mass in units of e [45]. The factor 2/9 is a color factor and b | | the QCD radiative correction is the quantity in brackets. Table 1.2, summarized from [41], shows predicted widths for the ηb.

It has been suggested by Sanchis-Lozano that mixing with a light Higgs could modify the predictions [42] for both the hyperﬁne splitting and the width of the ηb.

Until this study, the ηb meson had not been observed. The large amount of phase 20 CHAPTER 1. INTRODUCTION

Table 1.2: Predictions for the two-photon partial decay width, taken from [41], and the total width determined by scaling the two-photon decay width using Eq. 1.12.

Author Γ(η γγ) (keV) Γ (MeV) b → tot Kim [46] 0.384 0.047 9.2 1.1 Munz¨ [47] 0.22 0.04 5.3 1.0 Chao [48] 0.46 11.0 Ebert [49] 0.35 8.4 Fabiano [50] 0.47 0.10 11.3 2.4 Gupta [51] 0.46 11.0 +0.019 Penin [52] 0.659 0.089(th.)−0.018(δαs) 0.015(exp.) 15.8 2.2 Laverty [53] 0.39, 0.42 9.4, 10.1

space available to ηb decays leads to extensive fragmentation of the gg decay, leading to few preferred exclusive ﬁnal states. The only experimental information about the decay of bottomonium states to exclusive hadronic ﬁnal states is provided by

the CLEO experiment, which has measured exclusive decays of χbJ (nP ), J = 0, 2, n = 1, 2 states [43]. No single exclusive hadronic decay mode has a branching fraction

−3 of greater than 10 . The χbJ (nP ), J = 0, 2 states decay via two gluons (as the ηb)

so their decays should be qualitatively similar to those of the ηb.

A promising way to produce and detect the ηb meson is via the photon produced

in magnetic dipole transitions from the Υ (3S) to the ηb. The rate can be calculated somewhat crudely in potential models. In the nonrelativistic approximation and including ﬁnite size corrections, the rate is given by

2 4 eb 2 3 Γ(Υ (3S) ηbγ) = α 2 I k (1.13) → 3 mb

where α is the ﬁne structure constant, e is the b quark charge in units of e , k is the b | |

photon momentum, and mb is the b quark mass. The overlap integral I is deﬁned by 1.6. THE GROUND STATE OF BOTTOMONIUM, THE ηB MESON 21

I = f j (kr/2) i (1.14) h | 0 | i where j0(x) is the spherical Bessel function of zeroth order. The spherical Bessel function takes into account the correction for the ﬁnite size of the bottomonium states [59], neglected in Eq. 1.9.

The transitions Υ (n0S) γη (nS) are conventionally separated into two types → b 0 0 based on the radial quantum numbers of the Υ (n S) and the ηb(nS): “allowed” (n = n) and “hindered” n0 = n. Since kr << 1, expanding j (kr/2) = 1 + k2r2/24 + 6 0 · · · in Eq. 1.14, and using the orthogonality of the wavefunctions, we see that the rate of hindered transitions is given entirely by higher-order and relativistic corrections. Therefore the predicted rate is subject to a substantial theoretical uncertainty. The expected branching fraction for the transition Υ (3S) γη ranges from 0.5 10−6 → b × to a few times 10−3 with most predictions around 10−3. Godfrey and Rosner give a recent review [60] of branching fractions and hyperﬁne splittings.

Figure 1.6, reproduced from the paper describing the ηb search by the CLEO experiment, shows the from the CLEO experiment and several predictions for the branching fraction Υ (3S) γη . → b

ALEPH [54], L3 [55], and DELPHI [56] have searched for the ηb meson in two- photon collisions, and provided 95% conﬁdence level upper limits on the product

Γ (η ) BR(η ) for various assumed decay modes [57]. The results are summa- γγ b × b rized in Table 1.3, which is reproduced from the report by the Heavy Quarkonium Working Group [57]. The best limits are from the ALEPH experiment. Assum-

ing an ηb two-photon decay width of 660 MeV, the ALEPH limits correspond to (η 4 charged) < 0.07 and (η 6 charged) < 0.20 at the 95% conﬁdence level. B b → B b → 22 CHAPTER 1. INTRODUCTION

: 2 Hyperfine Splitting MΥ(1S) - Mηb(1S) (MeV/c ) 30 50 70 90 110 130 150 3 | | | | | | | | | | | | | , Zambetakis,Byers 83 -3 , 2 Godfrey-Isgur 85 B

, Godfrey-Isgur 85 A

Branching Ratio in units of 10 90% CL UL CLEO-III

, Lahde,Nyfalt,Riska 99 A

0 880 900 920 940 960 980 1000 Eγ (MeV)

Figure 1.6: Theoretical predictions for Υ (3S) γηb branching fraction and CLEO upper limit. →

At hadron colliders, it is very diﬃcult to trigger on ηb decays into hadrons; instead,

one must trigger on rare decays of the ηb. In an unpublished analysis, CDF also has searched for the η , in the decay mode η J/ψJ/ψ. Three candidates are observed b b → compared to an expectation of 3.8 events [58].

1.7 Conclusion

Measurements of the properties of the ηb meson play an important role in the study

of nonrelativistic bound states in QCD. In this dissertation we observe the ηb meson 1.7. CONCLUSION 23

Table 1.3: Limits (95% conﬁdence level) on ηb two photon partial width times branching fraction from LEP.

Experiment ﬁnal state Γ B (keV) Reference γγ × ALEPH 4 charged < 0.048 [54] 6 charged < 0.132 [54] L3 K+K−π0 < 2.83 [55] 4 charged < 0.21 [55] 4 charged π0 < 0.50 [55] 6 charged < 0.33 [55] 6 charged π0 < 5.50 [55] π+π−η0 < 3.00 [55] DELPHI 4 charged < 0.093 [56] 6 charged < 0.270 [56] 8 charged < 0.780 [56]

for the first time, and measure relevant properties of the ηb. These properties include measurements of hindered magnetic dipole transition rates to the ηb, and the ηb mass, from which the hyperfine splitting ∆m = m m is obtained. From the Υ (1S) − ηb hyperfine splitting and an analytic calculation of the splitting [19], we obtain the strong coupling constant, αs, in a unique way. Chapter 2

The BABAR Detector

2.1 Physics Motivation

2.1.1 CP violation

A curious fact about our universe is that it contains matter. The conventional measure of the baryon content of the universe is the ratio of the number density of baryons to that of photons. This baryon-to-photon density ratio, as measured by the WMAP cosmic microwave background experiment, is η = n /n = (6.225 0.170) 10−10 [61]. b γ × The predominance of matter can be explained by out-of-equilibrium CP -violating reactions which violate baryon number in the early universe, the so-called Sakharov

mechanism [62]. The Standard Model contains all of the necessary ingredients to qualitatively explain the existence of matter in the universe; however, the quantitative prediction of the baryon-to-photon ratio in the Standard Model is much smaller than observed [63]. This leads to the suggestion that perhaps there are additional sources

of CP -violation in addition to those found within the Standard Model.

24 2.1. PHYSICS MOTIVATION 25

One of the primary motivations for the BABAR experiment is the study of CP - violation, through the time dependence of neutral B meson decays, in order to determine if there are sources CP -violation beyond those found in the Standard Model. The decay of the Υ (4S) resonance produces an entangled B0 B¯0 state. The B0 and − B¯0 mesons can oscillate into each other via box diagrams involving the W boson and u, c, t quarks. Before one of the mesons decays, there must be exactly one meson and one anti-meson; once either meson decays, this coherence is lost, and the other meson can oscillate freely before decaying. The measurement of the time dependence of the resulting particle-antiparticle asymmetries can be used to measure CP -violation parameters.

To achieve this goal it is necessary to determine the separation between the decay vertices of the two mesons, a task that is complicated by the B mesons being produced almost at rest (p 330 MeV) in the Υ (4S) frame. Since the lifetime of the B0 B ≈ −12 0 meson is τ 0 = (1.530 0.009) 10 seconds and the B meson is produced nearly B × at rest in the frame of the Υ (4S), the B0 meson travels only d = γvτ 30µm before ≈ decaying. With current technology this separation is too small to be measured with high enough accuracy. By constructing a collider in which the CM frame moves at relativistic speeds relative to the lab frame, as is done at the PEP-II-BABAR B-factory, the separation between the two decay vertices is improved.

For studies of CP -violation, it is critical to have a excellent vertex detector for very precise measurements of the position of the two decay vertices. However, it is also necessary to have general purpose detector with high reconstruction eﬃciency of neutral clusters and charged tracks, making the BABAR detector an excellent detector for bottomonium physics, as well as CP -violation physics. 26 CHAPTER 2. THE BABAR DETECTOR

2.1.2 Bottomonium Physics

Though the Υ (4S) resonance is ideal for studying B-meson decays, the transition

rates to other bottomonium resonances are overwhelmed by the rate of Υ (4S) decays to B B¯ pairs. The lower Υ resonances are better suited for studying bottomonium − transitions because they are below threshold for B B¯ production. −

2.2 PEP-II

The PEP-II e+e− storage rings were originally designed to operate at a center of

mass energy √s = 10.58 GeV, the peak of the Υ (4S) resonance. To support the primary purpose of the BABAR experiment, the center of mass is boosted relative to the laboratory frame by βγ = 0.56 by using an electron beam energy of 9.0 GeV and a positron beam energy of 3.1 GeV. This increases the average separation of the decay

vertices of B mesons resulting from Υ (4S) decays to roughly < βγcτ 0 > 260µm. B ≈ For data taking at the peak of lower resonances, the energy of positron beam is unchanged while the energy of the electron beam is reduced to 8.61 GeV and 8.07 GeV for the Υ (3S) and Υ (2S) resonances, respectively.

The direction of the electron beam and positron beam are referred to as the “forward direction” and “backward direction”, respectively.

2.3 The BABAR Detector

The BABAR detector, described in detail in [64] and pictured in Fig. 2.1, is composed

of ﬁve principal active subdetectors: 2.4. SILICON VERTEX TRACKER 27

Silicon Vertex Tracker (SVT) - provides dE/dx measurements and precise mea- • surements of track vertex parameters and is the only source of tracking for low

momentum tracks

Drift Chamber (DCH) - in combination with the SVT, measures momenta of • tracks and provides dE/dx measurements for particle ID

Calorimeter (EMC) - measures energies of electron and photon showers; also • used for electron and neutral hadron ID

Detector of Internally Reﬂected Light (DIRC) - provides particle identiﬁcation, • particularly π/K/p separation

Instrumented Flux Return (IFR) - provides muon and K0 identiﬁcation • L

2.4 Silicon Vertex Tracker

Trajectories of charged particles very close to the interaction point are measured using the Silicon Vertex Tracker (SVT). The SVT, shown in Fig. 2.2, consists of ﬁve layers of double-sided silicon strip detector in which the strips in the two sides are arranged perpendicular to each other, to provide z-position information. The close proximity of the silicon vertex tracker to the interaction point allows for an improved determination of the primary vertex of a physics event, relative to the precision determined in the absence of the silicon vertex tracker. While precision of the silicon vertex tracker is imperative in the the studies of time-dependent CP violation, it is less important in the study of bottomonium physics. 28 CHAPTER 2. THE BABAR DETECTOR

Detector C L Instrumented Flux Return (IFR)) 0 Scale 4m I.P. Barrel Superconducting BABAR Coordinate System Coil y 1015 1749 x Electromagnetic Cryogenic 1149 4050 1149 Calorimeter (EMC) z Chimney 370 Drift Chamber (DCH) Cherenkov Detector Silicon Vertex (DIRC) Tracker (SVT)

IFR Magnetic Shield 1225 Endcap for DIRC Forward 3045 End Plug Bucking Coil 1375 Support Tube 810 e– e+

Q4 Q2

Q1 3500 B1

Floor 3-2001 8583A50

Figure 2.1: BABAR detector longitudinal section.

The SVT complements the measurements of the drift chamber, a description of which appears in the following section. The drift chamber cannot adequately measure low momentum tracks (below about 120 MeV/c) and for these tracks the SVT must fully determine track parameters. The high precision measurements in the SVT are especially important for tracks with high momentum, as these have a small sagitta in the drift chamber. These high precision measurements are also critical for deriving an accurate Cherenkov angle in the DIRC.

The inner three layers of the SVT are necessary for accurately measuring decay vertices and are placed as close to the beam pipe as possible to reduce errors due 2.4. SILICON VERTEX TRACKER 29

580 mm Space Frame

Bkwd. support cone 520 mrad Fwd. support350 mrad cone e- Front end e + electronics

Beam Pipe

Figure 2.2: Silicon vertex tracker longitudinal section. to multiple scattering. The outer two layers are used for extrapolating SVT tracks to the drift chamber. These two layers are arranged in an arch shaped geometry to maximize the crossing angle of particles near the edge of the SVT acceptance.

It is critical to maintain a minimum of material in the SVT. Otherwise, multiple scattering degrades momentum resolution for charged tracks. Also, photon conver- sions in the material degrade photon measurements in two ways: the resulting e+e− pairs lose significant energy by bremsstrahlung before reaching the calorimeter or the leptons may be lost completely due to bending in the magnetic field, leading to reduced efficiency. As a result of its effect on photon efficiency, material further from the calorimeter is particularly harmful. At 90◦ the SVT is less than 0.04 radiation lengths thick.

The required resolution (about 80 µm) along the z-axis is determined by the requirement that the vertices of the B meson daughters of the Υ (4S) resonance are separately resolved, a capability necessary for time-dependent CP -violation studies. 30 CHAPTER 2. THE BABAR DETECTOR

Additionally, decay vertices of light charm mesons must be resolved, leading to a requirement of a resolution of 100µm in the x y plane. ≈ − Measurements of position and angle of tracks are dominated by the SVT measurements. The resolution of the helix parameters can be derived using cosmic rays that pass through the IP. An incident cosmic ray is reconstructed as two separate tracks and the diﬀerence between the reconstructed track parameters of the two tracks gives the resolution parameters. They are

σd0 = 23µm σφ0 = 0.43 mrad (2.1) σ = 29µm σ = 0.53 10−3 z0 tan λ × where d0, φ0, and z0 are the distance of closest approach, φ0 and z0 are the φ and z of the track at the distance of closest approach, and λ is the dip angle (the angle between the track momentum and the z-axis).

2.5 Drift Chamber

The principal device for measuring the trajectories of charged particles is a 40-layer drift chamber (DCH). The DCH also provides measurements of ionization loss dE/dx for particle ID, which are especially important for particle ID of low momentum tracks and tracks outside the acceptance of the DIRC. The DCH is also the only source of tracking for charged particles outside the acceptance of the SVT. This includes charged pions from KS decays, which may occur outside the SVT acceptance, due to the long K lifetime of 0.895 10−10s. In the central angular region of the detector this S × measurement complements the PID capabilities of the DIRC. In the extreme forward 2.5. DRIFT CHAMBER 31

and backward regions, where the DIRC is not present, and in the gaps between the DIRC bars in the barrel, the DCH is the only source of PID information.

The DCH is shown in Fig. 2.3. As charged particles traverse material they lose

630 1015 1749 68

Elec– tronics 809

485 27.4 1358 Be 17.2 236 e– 464 IP e+ 469

1-2001 8583A13

Figure 2.3: Drift chamber longitudinal section. Measurements are in mm.

energy by ionization, atomic excitation, and radiation. Hits are formed when charged particles ionize the molecules of the drift chamber gas and these electrons drift toward the anode. The drift chamber uses the drift velocity of electrons in a gas resulting from these ionizations to calculate the position of interactions of charged particles that propagate through the DCH. Hexagonal cells are used to obtain approximately circularly symmetric isochrones.

A diagram of the wire placement for the ﬁrst four superlayers is shown in Fig. 2.4. Longitudinal position information is obtained by placing 24 of the 40 layers at a small stereo angle (50-76 mrad) with respect to the z-axis. The overall resolution is roughly 140 µm, as shown in Fig. 2.5. For each cell, the relation between the time and

distance of closest approach to the wire is calibrated using e+e− and µ+µ− events. 32 CHAPTER 2. THE BABAR DETECTOR

To avoid bias in this calibration, the cell of interest is excluded from the calculation of the distance of closest approach.

16 0 15 0 14 0 13 0

12 -57 11 -55 10 -54 9 -52

8 50 7 48 6 47 5 45

4 0 3 0 2 0 1 0 Layer Stereo

4 cm Sense Field Guard Clearing

1-2001 8583A14

Figure 2.4: Drift chamber wire placement for ﬁrst four superlayers. Stereo angles are in mrad.

Information regarding dE/dx is obtained from the measured charge in each cell, corrected for several factors including global corrections for gas pressure and temper- ature, individual cell geometry, space charge buildup, non-linearities in energy loss at large dip angles, and variation of charge collection eﬃciency as a function of entrance angle. The dominant eﬀects on the resolution come from global corrections. The overall dE/dx resolution is observed to be 7.5%, close to the expected performance of 7%. This provides excellent K p separation for momenta less than 1 GeV/c2, as − shown in Fig 2.6. 2.5. DRIFT CHAMBER 33

0.4

0.3

0.2 Resolution (mm) 0.1

0 ±10 ±5 0 5 10 1-2001 8583A19 Distance from Wire (mm)

Figure 2.5: Single cell resolution as a function of distance from sense wire.

Hits are reconstructed to form tracks in 3-space using a Kalman ﬁlter approach [65] which includes dE/dx measurements, the distribution of material in the detector and magnetic ﬁeld inhomogeneities.

The transverse momentum resolution, σpt , is dominated by the measurements from the DCH. It it derived from cosmic ray events, as discussed in Section 2.4, and is p σ /p = (0.13 0.01)% t + (0.45 0.03)%. pt t × 1GeV where the ﬁrst term is related to the measurement accuracy, and the second is due to multiple Coulomb scattering. Multiple scattering is seen to dominate momentum resolution for all but the highest momentum tracks. 34 CHAPTER 2. THE BABAR DETECTOR

104 d p

K dE/dx

103π

e µ

10±1 1 10 1-2001 8583A20 Momentum (GeV/c)

Figure 2.6: Speciﬁc energy loss versus momentum. Lines show predictions for various mass hypotheses from the Bethe-Bloch equation.

2.6 Detector of Internally Reﬂected Light

In many analyses accurate particle identiﬁcation is necessary both for background reduction and the decision of the appropriate mass hypotheses to be used in vertexing. The Detector of Internally Reﬂected Light (DIRC) is shown schematically in Fig 2.7.

The study of CP violation requires the ability to tag the ﬂavor of one of the B mesons using kaons. The kaons produced in B decays have high momentum, generally above 1 GeV/c. When particles travel faster than the speed of light in a material, a

cone of light, referred to as a Cherenkov cone, is produced at an angle θC with respect

to the ﬂight direction, where cos θC = 1/(nβ), n is the refractive index of the medium and β = v/c. The particle ID system used in BABAR is a new type of Cherenkov detector in which the Cherenkov ring is imaged by propagating the Cherenkov light created in a quartz bar with a 1 2 cm2 cross section through the bar by internal × reﬂection, and out to a photon detector. The radiator material used in the DIRC is 2.6. DETECTOR OF INTERNALLY REFLECTED LIGHT 35

PMT + Base 10,752 PMT's

Standoff Purified Water Light Catcher Box

17.25 mm Thickness (35.00 mm Width) Bar Box

Track PMT Surface Trajectory Wedge Mirror

Bar Window

4.9 m 1.17 m 4 x 1.225m Bars { glued end-to-end { 8-2000 8524A6

Figure 2.7: Diagram of DIRC. Distances are not to scale.

composed of bars of synthetic fused silica. The DIRC bars are arranged cylindrically around the drift chamber in a 12-fold symmetry.

The DIRC provides π/K separation greater than 4σ for momenta less than 3 GeV/c. Fig 2.8 shows the Cherenkov angle and timing resolution of the DIRC. Below 700 MeV/c particle ID is based primarily on the dE/dx measurements of the DCH and SVT. The particle ID is demonstrated using D0 Kπ events in Fig. 2.9. →

Additionally, the DIRC can be used to tag photons which shower before reaching the calorimeter. 36 CHAPTER 2. THE BABAR DETECTOR

80000 (a) 60000 40000 20000

entries per mrad 0 -100 -50 0 50 100 ∆ θ C,γ (mrad)

80000 (b) 60000 40000 20000

entries per 0.2ns 0 -5 0 5

∆ tγ (ns)

Figure 2.8: Cherenkov angle and timing resolution of the DIRC.

2.7 Electromagnetic Calorimeter

The electromagnetic calorimeter (EMC) is designed to detect and measure the energies of photons and electrons of energies 20 MeV to 9 GeV with high precision and efficiency over a large angular region. The high light yield of CsI(Tl) crystals and high light collection efficiency allow for excellent photon energy resolution. The fine granularity of the detector allows accurate cluster position measurements which are crucial for π0 mass resolution at high π0 momentum, where it is dominated by angular

resolution. 2.7. ELECTROMAGNETIC CALORIMETER 37

Without DIRC x 10 2 1500 2

1000

entries per 5 MeV/c With DIRC 500

0 1.75 1.8 1.85 1.9 1.95 Kπ mass (GeV/c2)

Figure 2.9: Invariant mass plot of candidate D0 Kπ events before and after → applying loose kaon ID.

2.7.1 Design

The structure of the EMC is divided into two regions: a central angular region referred to as the barrel and an forward endcap. A backward endcap is not included in the design as most of the decay products from BB¯ decays are boosted in the forward direction. The EMC consists of 6580 trapezoidal CsI(Tl) crystals, 5760 of which are in the barrel. The length of the crystals varies from 16 to 17.5 radiation lengths

(X0,CsI = 1.85cm), where the the longer crystals are in the forward direction, where the maximum energy of a particle is higher. The front face dimensions are roughly

4.8 4.8cm2 and the longitudinal axis of the crystals point away from the beam spot × 38 CHAPTER 2. THE BABAR DETECTOR

to reduce leakage. A diagram of the EMC is provided in Fig. 2.10.

2359

1555 2295 External Support

1375 1127 1801 26.8˚ 920

38.2˚ 558 15.8˚ 22.7˚

Interaction Point 1-2001 1979 8572A03

Figure 2.10: Longitudinal section of the EMC. Lateral dimensions are shown in mm and angles in degrees.

The EMC electronics are designed to contribute negligibly to the overall energy resolution. This is achieved by using crystals with high light yield, highly efficient and uniform light collection and low-noise amplification. Crystals are wrapped in diffuse reflective material (TYVEK) and read out by a pair of 2 1 cm2 silicon PIN × diodes (Hamamatsu S2744-08).

2.7.2 Calibration and Performance

Frequent calibration of the EMC is necessary to reduce the contribution of the energy calibration to the energy resolution. Calibration of the electronics gain and removal of non-linearities is done by precision charge injection into the preampliﬁer input. Pedestal oﬀsets are measured using random triggers in the absence of beam.

The calibration of signal crystal energies is done using two diﬀerent techniques, 2.7. ELECTROMAGNETIC CALORIMETER 39

depending on the energy scale of interest. At low energies, crystals are calibrated using a radioactive source. A ﬂourine-containing liquid is activated by a beam of neutrons

and then piped along the front faces of the crystal. The decay 19F + n 16 N + α, → 16N 16 O∗ + β, 16O∗ 16 O + γ occurs with a half-life of 7 seconds, leaving no → → residual radioactivity. At high energies, the single crystal calibration is performed with Bhabha events. The Bhabha calibration constants are determined by minimizing

the sum 2 k k ci E (θ, φ) χ2 = i i − dep (2.2) σk k P ! X k where the sum runs over the single crystal energies i of all clusters k, the cluster

k energy resolution is σi, and the deposited cluster energy Edep(θ, φ) is determined from MC. At intermediate energies the single crystal calibration constant is determined from a logarithmic interpolation between the constants determined by the radioactive source and Bhabha calibrations.

After the single crystal energy scale as been determined it is necessary to correct these energies of clusters for shower leakage. This correction is determined using π0 events and µµγ events at low energy and high energy, respectively. At this stage the calorimeter energy scale is correctly determined. A ﬁnal empirical correction for photon energies as a function of proximity to crystal edges, is applied. This correction is applied for photon energies greater than Eγ,LAB > 1 GeV, where calorimeter position resolution is suﬃcient to resolve sub-crystal features.

The energy resolution of the EMC, derived from photons from π0 decays at low energy, and µµγ events at high energy is

σ (2.30 0.03 0.3)% E = (1.35 0.08 0.2)%, (2.3) E 4 E(GeV) ⊕ p 40 CHAPTER 2. THE BABAR DETECTOR

where E and σE are the energy and RMS error of a photon, measured in GeV. This parameterization is derived from data such as is shown in Fig. 2.11.

π0 →γγ Bhabhas 0.06 χc → J/ψ γ MonteCarlo

E 0.04

E σ

0.02

0.02 10-1 1.0 10.0 3-2001 8583A41 Photon Energy (GeV)

Figure 2.11: Fractional energy resolution σE/E versus energy as derived from various sources, compared to Monte Carlo expectations.

The angular resolution of the BABAR calorimeter is determined largely by the

Moli`ere radius of CsI(Tl), which is the radius in which 90% of an electromagnetic shower is contained, and the transverse granularity of the calorimeter. The angular resolution is determined from symmetric π0 and η decays to be

4.16 0.04 σθ = σφ = 0.00 0.04 mrad. (2.4) E(GeV) ⊕ ! p A parameterization of the angular resolution as a function of photon energy is shown in Fig. 2.12. 2.8. SUPERCONDUCTING COIL 41

π0 → γγ 12 MonteCarlo ) d

a 8 r m (

θ σ 4

0 0 1 2 3 3-2001 8583A42 Photon Energy (GeV)

Figure 2.12: EMC angular resolution as a function of photon energy.

2.8 Superconducting Coil

Radially just outside of the EMC is the superconducting coil. To measure the momenta of charged particles, the silicon vertex tracker and the drift chamber are contained within a 1.5 T longitudinal magnetic ﬁeld from a superconducting solenoid.

This magnetic field is provided using an operating current of 4596 A. The radial field does not exceed B = 0.15 T within the volume of the DCH. The photomultipliers | r| in the DIRC are sensitive to stray magnetic fields, requiring an additional magnet (the bucking coil) to offset these magnetic fields in the DIRC stand-off box, where

the DIRC photomultipliers are mounted. 42 CHAPTER 2. THE BABAR DETECTOR

2.9 Instrumented Flux Return

The instrumented ﬂux return (IFR) is used to identify muons and neutral hadrons

0 (primarily KL and neutrons). A magnetic flux return is situated immediately outside the cryostat of the superconducting coil to prevent any radial component of the magnetic field in the tracking volume. This flux return is instrumented to provide muon identification. The detection of muons is important in bottomonium physics to measure decays Υ (nS) µ+µ−. As muons are minimum ionizing particles, they are → able to penetrate large amounts of material, including the EMC. The steel and brass of the flux return is used as a hadron absorber.

The IFR (shown in Fig. 2.13) consists of a barrel and two endcap sections. The barrel is composed of six sections, arranged in a hexagonal pattern, covering the central region of the detector. The forward regions is composed to four doors, each covering half of a forward region.

Barrel 3200 FW 920 3200

19 Layers

18 Layers BW 1940 1250

End Doors 4-2001 8583A3

Figure 2.13: Geometry of barrel and endcap regions of the IFR. 2.10. TRIGGER 43

Near the beginning of the experiment, resistive plate chambers were used in the both the central and forward regions but due to a gradual degradation of eﬃciency, the barrel resistive plate chambers were replaced with limited streamer tubes in 2004. Figure 2.14, which shows pion rejection as a function of muon eﬃciency, demonstrates the performance of the limited streamer tubes in comparison with the failing resistive plate chambers.

Figure 2.14: Pion rejection versus muon eﬃciency for track momentum 0.5 < p < 2.0 GeV/c (left) and 2 < p < 4 GeV/c (right).

2.10 Trigger

In order to measure inclusive rates accurately, it is important to have a highly ef- ﬁcient trigger. This is achieved by combining information from calorimeter-based and track-based triggers. The complementarity of the two sets of triggers allows for

high eﬃciencies and accurate measurements of these eﬃciencies. The total trigger 44 CHAPTER 2. THE BABAR DETECTOR

eﬃciency is greater than 99% for BB¯ events and greater than 95% for continuum events.

Trigger decisions are made in two stages: Level 1, a hardware trigger, and Level 3, a software trigger based on partial reconstruction of physics events.

2.10.1 Level 1 Trigger

The Level 1 trigger used in physics data taking consists of drift chamber triggers (DCT and DCZ) and electromagnetic triggers (EMC). An additional IFR trigger is used mainly for diagnostic purposes.

The Level 1 drift chamber trigger is composed of seven diﬀerent trigger selections, or primitives, based on the pT of the tracks, their z position and the number of DCH superlayers crossed.

For triggering purposes, the calorimeter is divided into a 40 7 matrix with 40 × strips in the φ direction and 7 in the θ direction. The calorimeter triggers differs from that of the drift chamber in that its output is untriggered. Every 269 ns, all crystal energies above 20 MeV are summed and sent the electromagnetic trigger (EMT) along with the arrival time. EMT primitives are defined on the basis of the reconstructed cluster energy, with special triggers defined specifically to selected 1-prong Bhabha events for the EMC Bhabha calibration.

Composite objects are constructed from these primitives to require track-cluster matching or back-to-back objects. 2.11. DATASETS 45

2.10.2 Level 3 Trigger

The Level 3 drift chamber trigger ﬁnds tracks and performs track parameter ﬁts for

2 tracks with pT greater than 250 MeV/c . The Level 3 selections require events to

have events with at least one tight (high pT ) track or two loose tracks. The high pT track is required to have p > 600 MeV/c and satisfy the vertex constraint dIP < 1.0 T | 0 | cm and zIP z < 7.0 cm. The loose selection requires p > 250 MeV/c and a | 0 − IP| T vertex constraint of dIP < 1.5 cm and zIP z < 10.0 cm. | 0 | | 0 − IP| In the Level 3 electromagnetic trigger clusters are formed from crystals with an energy above 20 MeV and a cluster energy and time are computed. Clusters with an energy above 100 MeV that also have an event time within a 1.3µs window of the

event time are retained. Two calorimeter cluster selections are made, one to select high energy clusters and another to select events with high cluster multiplicity. The

former requires at least two clusters with ECM >350 MeV and the latter requires at least four clusters. Both selections require an event mass greater than 1.5 GeV.

For high multiplicity hadronic events, the Level 3 trigger is highly eﬃcient. For

simulated events from ηb decay, the trigger is better than 99.9% eﬃcient.

2.11 Datasets

The Υ (2S) data sample used in this analysis consists of 98.6 0.9 million Υ (2S) events, corresponding to an integrated luminosity of 14.4 fb−1, with an additional 1.5 fb−1 recorded 30 MeV below the Υ (2S) resonance for background studies. The Υ (3S) data sample used in this analysis consists of 121.8 1.2 million Υ (3S) events, corresponding to an integrated luminosity of 28.1 fb−1, with an additional 1.7 fb−1 46 CHAPTER 2. THE BABAR DETECTOR

recorded 30 MeV below the Υ (3S) resonance. The largest BABAR data sample, not used in this analysis, was taken at the peak of the Υ (4S) resonance and consists of 426 fb−1, or 467 106 bb events. In addition, × a sample of 44 fb−1 was collected 40 MeV below the Υ (4S) resonance. Chapter 3

Analysis Overview

3.1 Introduction

Very little is known about the ηb meson experimentally and theoretically. The hyper- ﬁne mass splitting between the Υ (1S) and the η , ∆m = m m is expected to b Υ (1S) − ηb be between 36 and 100 MeV, with the more reliable estimates in the range 40-60 MeV, and its width is expected to be in the range 5-15 MeV. However, its exclusive decay modes are completely unknown. While the ηb meson is expected to decay via two gluons (the lowest order QCD process allowed given the quantum numbers of the

ηb), the fragmentation and hadronization of the two gluons into exclusive ﬁnal states is not known. Due to the large amount of phase space available, the decay modes are expected to be of high multiplicity, with no single dominant decay mode. This

hinders searches for the ηb meson in its exclusive decays.

A more promising strategy consists searching for the ηb meson inclusively in radiative Υ (nS) decays. The lowest-order decay mode of the Υ (nS) states is the three

gluon decay, as the two gluon decay is forbidden by C-parity. Moreover, the decay

47 48 CHAPTER 3. ANALYSIS OVERVIEW

is OZI-suppressed. As a result, the Υ (nS) states have small hadronic decay widths, giving rise to large radiative branching fractions.

In this analysis we observe the ηb meson in the hindered magnetic dipole transitions Υ (nS) γη , n = 2, 3. In this transition the η meson appears as a peak in the → b b center-of-mass (CM) photon energy distribution at

s m2 E = − ηb γ 2√s where √s is the e+e− CM energy. Throughout this document we will refer to photon

2 energies in the CM frame, unless otherwise speciﬁed. With an ηb mass of 9.390 GeV/c the peak should appear near 910 (620) MeV in the transitions from the Υ (3S) (Υ (2S)). The calorimeter energy resolution is 25 (18) MeV at those energies, making us insen- sitive to the ηb natural width, unless it is very broad.

3.2 Backgrounds

In this section we discuss the background to this analysis, which will help motivate the selection criteria discussed in the following section.

3.2.1 Non-peaking

The largest source of real background photons to this analysis does not create peaks in the photon energy spectrum. This background, which is due to continuum (e+e− → qq¯, q = u, d, s, c) events, bottomonium decays and generic ISR production, produces a featureless, nearly exponential spectrum. 3.2. BACKGROUNDS 49

3.2.2 Υ (nS) γχb(mS), χb γΥ (1S) → →

The dominant source of real background photons that peaks near the signal region is due to the double radiative transitions Υ (nS) γχ (mS), χ γΥ (1S). Here m = 1 → b b → and m = 2 in the Υ (2S) and Υ (3S) analyses, respectively. While this background is separated by more than 100 MeV from the signal region, its yield is roughly 100 times the expected signal yield, making its parameterization important.

+ 3.2.3 e e− γΥ (1S) →

The most crucial background is due to the monochromatic photon from the process

e+e− γ Υ (1S), in which γ is a photon from initial state radiation (ISR) oﬀ of → ISR ISR one of the initial state leptons. Due to the small Υ (1S) η hyperﬁne splitting, this − b peak partially overlaps the signal peak, making an accurate knowledge of the ISR yield and peak position crucial. The cross section for ISR production of a narrow vector meson is given by

2 12π Γee σV (s) = W (s, xv) mV s ·

where Γee is the di-electron width of the vector meson, mV is the mass of the vector

meson, s is the center of mass energy, xv = 2Eγ /√s, and W (s, xv) is the probability

function of photon emission. W (s, xv) is often referred to as the “radiator function”. We use the radiator functions from ref. [66] to compute the cross section.

To evaluate the expected yield, the yield from the Υ (4S) oﬀ-resonance data sample is scaled by cross-section, luminosity, and eﬃciency. For example, to obtain the

expected ISR yield at the Υ (2S) resonance, NexpISR,on−Υ (2S), using the ﬁtted number 50 CHAPTER 3. ANALYSIS OVERVIEW

of ISR Υ (1S) events in the oﬀ-Υ (4S) dataset, NISR,off−Υ (4S), we use the formula

L σ N = on−Υ (2S) on−Υ (2S) on−Υ (2S) N . exp,Υ (2S) L σ ISR,off−Υ (4S) off−Υ (4S) off−Υ (4S) off−Υ (4S) The MC is generated using the BABAR VECTORISR generator, which is in turn based on the AfkQED package. The AfkQED package implements up to (α2) leading- O logarithmic corrections to the QED radiator function [67, 68].

3.3 Selection Criteria

The selection criteria can be separated into four categories: hadronic selection, photon quality, photon isolation and π0 veto.

As the two-gluon decays of the ηb are expected to have high track multiplicity, much larger than that expected for events created by events from quantum electrodynamic processes, in which the production of extra tracks is suppressed by powers of the ﬁne structure constant, α 1/137. As a result the purity of the event se- ≈ lection can be improved by requiring more than three charged tracks in an event. Two prong events are rejected using information on the event topology in the form of

Fox-Wolfram moments [69]. The ratio of the second to zeroth Fox-Wolfram moment is required to be less than 0.98.

The photon quality is improved by requiring the cluster lateral moment [70] to be

less than 0.55. An isolated photon will have a small, well-deﬁned cluster shape; the lateral moment cut removes photons which overlap with other photons in the event, as well as mis-identiﬁed charged tracks.

To remove photons from continuum (e+e− , qq¯, q = u, d, s, c) events the photon → 3.3. SELECTION CRITERIA 51

signal candidate is required to be isolated from the thrust axis [71] of the event. The thrust axis is the direction along which the longitudinal momentum of the event is maximized. Photons with cos θ > (0.7, 0.8) are rejected, where θ is the angle | T | T between the photon direction and the thrust axis of the rest of the event, in the CM frame. The selection on cos θ has the added advantage of rejecting photons | T | that are aligned with the thrust axis of the rest of the event, photons which are very

likely to suﬀer from degraded resolution due to overlaps with charged tracks or other photons. Chapter 4

Study of the Decay Υ (3S) γη → b

4.1 Selection Criteria

The selection criteria have been optimized using a sample of 2.5 fb−1 of on-resonance

data and ηb signal Monte Carlo (MC). This sample is excluded from the ﬁnal dataset. After optimizing S/√B using these samples, the validity the optimization is veriﬁed

using the photon peaks from the transitions χ (2P ) γΥ (1S), J = 1, 2. bJ →

4.1.1 Data set

The selection optimization uses the on-resonance datasets listed in Table 4.1 and ηb signal MC. The two on-peak datasets are taken from runs near the beginning and near the end of Υ (3S) running. The η signal MC simulates the decay Υ (3S) γη b → b 2 where the ηb decays to gg and uses the correct 1 + cos θCM angular distribution of

the signal photon. GEANT4 [72] is used for the detector simulation and the ηb is

decayed using JETSET [73] 7.4. In the optimization, the ηb is assumed to have a

52 4.1. SELECTION CRITERIA 53

mass of 9400 MeV and zero decay width.

Table 4.1: On-resonance datasets used in the selection optimization

Run range Integrated luminosity ( fb−1) 77924-78019 1.201 79501-79590 1.282 Total 2.483

4.1.2 Event selection

We select photon candidates from a standard BABAR list, consisting of single EMC bumps1which are not matched with any track, have a minimum lab energy of 0.030

GeV and have a lateral moment less than 0.8. To improve the purity of the photon selection, we require a cluster lateral moment of less than 0.55. In the optimization we use truth-matched photons. Truth-matching is a process in which the measured quantities of a reconstructed candidate are compared with those of the hits generated

in Monte Carlo. Requiring consistency between the two allows the selection of a pure sample of signal candidates.

The selection criteria are chosen to give the best S/√B using truth-matched signal Monte Carlo and on-peak data. For determining S/√B, we require the photon center-of-mass energy to lie within the range 0.85 < Eγ < 0.95 GeV. Distributions of the selection variables prior to the application of any selection criteria are shown

1Bumps are a way of dealing with overlapping energy distributions in the calorimeter. Bumps are in one-to-one correspondence with local maxima of energy deposition in a cluster. A cluster with a single local maximum of energy deposition is simply a bump. For clusters with multiple local maxima, the cluster is split into as many regions as there are local maxima. Weights are assigned to each crystal based on the distance to each local maximum in order to determine what fraction of its energy should be assigned to each local maximum. 54 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B in Fig. 4.2.

Hadronic events are selected by requiring the ratio of the second to zeroth Fox- Wolfram moments be less than 0.98 and requiring 4 charged tracks in the event. ≥

To reject continuum background, we require cos θ < 0.7, where θ is the angle | T | T in the center of mass frame between the photon momentum and the thrust axis of the rest of the event. The optimization of this variable is discussed below.

To reject backgrounds from π0 decays, the largest background to this analysis, we reject photon candidates in which the candidate photon combines with another photon in the event to form an invariant mass within 15 MeV of the nominal π0 mass. The second photon is chosen with the same primary selection criteria as the candidate photon. However, the photon lab energy is required to be greater than 50 MeV, a criteria that is optimized as discussed below.

We require the signal photon to lie within the barrel of the calorimeter with a selection of 0.762 < cos(θ ) < 0.890. The exclusion of the calorimeter endcap is − γ,LAB primarily to reduce the yield of peaking background from e+e− γ Υ (1S) events, → ISR where the ISR photon is detected. In the forward direction, the precise value of the selection criteria is chosen so that the photon does not fall into the crack between the endcap and barrel; in the backward direction, it is chosen so that the electromagnetic shower is completely contained by the calorimeter. In Fig. 4.1 we show how the values were determined, after a preliminary selection of cos(θγ,LAB) < 0.892 is made to remove the calorimeter endcap from consideration.

Selection eﬃciencies determined using truth-matched signal MC and on-peak data are listed in Table 4.2. 4.1. SELECTION CRITERIA 55

5000 22 20 4000 18 16 3000 14 12

Events/( 0.005 GeV ) Events/( 0.005 GeV ) 10 2000 8 6 1000 4 2 0 0 0.5 0.6 0.7 0.8 0.9 1 1.1 0.5 0.6 0.7 0.8 0.9 1 1.1 Eγ (GeV) Eγ (GeV)

0.08 Outside fiducial region 1.2 Inside fiducial region

0.07 (GeV)

γ 1.1 0.06 E 1 0.05 0.9

Normalized events 0.04 0.8 0.03 0.7 0.02 0.6 0.01 0.5 0 0.5 0.6 0.7 0.8 0.9 1 1.1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Eγ (GeV) cos(θγ)

Figure 4.1: Photon CM energies for truth-matched signal photons. Photons inside the calorimeter fiducial region (top left), photons inside the barrel but outside the fiducial region (top right). The top plots shown using the same scale (bottom left). Photon energies versus cos(θγ,LAB) (bottom right). The blue (red) points show photons accepted (rejected) by the fiducial requirement. The gradual worsening of the resolution as one moves away from θLAB = 0 is due to the increased photon path length through the DIRC.

4.1.3 π0 veto optimization

The largest background to this analysis is due to photons from π0 decays. We reduce this background by removing candidates which form a π0 with another photon in the event. The result of the optimization gives the following mass window:

m(γγ) > m (π0) 15 MeV (4.1) PDG 56 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

Table 4.2: Selection eﬃciencies () for truth-matched signal MC and on-peak data in the energy range 0.85 < Eγ < 0.95, in percent. The reconstruction eﬃciency on data is normalized to 100%.

Selection (MC) Total (MC) (Data) Total (Data) Reconstruction 0.629 0.629 1.000 1.000 Preliminary hadronic selection 0.977 0.615 0.787 0.787 Cluster lateral moment <0.55 0.995 0.612 0.967 0.762 Number of tracks 4 0.991 0.606 0.864 0.658 ≥ 0.762 < cos(θ ) < 0.890 0.901 0.546 0.788 0.518 − γ,LAB cos θ < 0.7 0.69 0.377 0.261 0.135 | T | π0 π0 < 15 MeV, | − PDG| 0.899 0.339 0.505 0.068 Eγ,2 > 15 MeV

though the expected signiﬁcance does not change substantially with a modiﬁed mass selection.

Unfortunately, this veto removes a large amount of signal. We improve the purity

of the veto by imposing a selection on the lab energy of the photon which combines

0 with the signal candidate to form a π , which we call Eγ2. Since the choice of selection criterion on the photon energy may, in principle, aﬀect the optimal position of the cos(θ ) selection, we perform a 2-d optimization of E and cos(θ ) . Figure 4.3 | T | γ2 | T | shows that the optimal cut is cos θ < 0.7 and E = 40 50 MeV. | T | γ2 −

Table 4.4 presents S/√B for values of the selection on Eγ2.

Choosing a photon energy cut which is too low decreases S/√B more than choosing a cut value which is too high. We conservatively use a selection of 50 MeV.

Since the π0 veto is the source of one of the larger signal selection ineﬃciencies, we investigate the use of selection criteria on other variables that may help improve

the purity of the π0 veto and thus improve the signal eﬃciency. Photon timing 4.1. SELECTION CRITERIA 57

0.05 0.16 0.14 0.04 0.12 0.03 0.1

Normalized events Normalized events 0.08 0.02 0.06 0.04 0.01 0.02

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 Cluster lateral moment Number of tracks

0.1

0.08

0.06 Normalized events

0.04

0.02

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 θ |cos( T)|

Figure 4.2: Cluster lateral moment (left), number of charged tracks in event (right), cosine of photon momentum direction with the thrust axis of the rest of the event (bottom), for truth-matched signal (red) and data (blue). The distributions are shown prior to the application for any selection criteria except a requirement of 0.85 < Eγ < 0.95. The arrows show the values of the selection criteria.

information from the calorimeter can be used to guarantee that the two photons used to form the candidate π0 arrive at the calorimeter at the same time, thereby rejecting beam background photons. The gain in S/√B obtained by using timing information is not large enough to justify its use. See Appendix A for details.

Since the properties of the decays of the ηb, in particular the photon spectrum and multiplicity, may not be correct in MC, the optimization is checked using photons from the nearby transitions χ (2P ) γΥ (1S). Table 4.3 shows that the χ (2P ) bJ → bJ →

Υ (1S) transitions are a reasonable model for ηb decays and thus can be used as a 58 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

120 B θ |cos( T)|<0.55

S/ 118 θ |cos( T)|<0.60 |cos(θ )|<0.65 116 T θ |cos( T)|<0.70 θ 114 |cos( T)|<0.75 θ |cos( T)|<0.80 θ 112 |cos( T)|<0.85 110 108 106 104 102 100 20 40 60 80 100 120 140 160 π0 Second photon energy [MeV]

Figure 4.3: S/√B derived from truth-matched signal MC and on-peak data. The on-peak data is evaluated in a region 0.85 < Eγ < 0.95.

check of the optimization.

In particular, the photon spectrum and multiplicity may not be correct in MC.

The cut on the lab energy of the second photon of the π0 is varied from 30 MeV to

150 MeV in 10 MeV steps. For each energy, we ﬁt the χb signal peak and extract the ﬁtted χb yield and the error on the χb yield, ∆χb. The following probability distribution function (PDF) is used to model the background shape:

2 A e−α(Eγ −µ)−β(Eγ −µ) where µ = 0.8 GeV is the center of the ﬁt window. Parameterizing the function about 4.1. SELECTION CRITERIA 59

Table 4.3: Comparison of single cut eﬃciencies and S/√B from ﬁtted χb yields and truth-matched signal MC. The background contribution is found by integrating the background function from E 2σ to E + 2σ. g,χb1 − g,χb2

Cut S/√B Eﬀ. (from χb peak) Eﬀ. (signal MC) No cut 101.5 - 0.629 Hadronic selection 109.8 0.973 0.977 4 charged tracks 107.2 0.903 0.995 ≥ Cluster lateral moment<0.55 113.2 0.997 0.991 0.762 < cos(θ ) < 0.890 109.6 0.928 0.901 − γ,LAB cos(θ ) < 0.7 135.2 0.672 0.690 | T | π0 π0 < 15 MeV, E > 15 MeV 164.7 0.849 0.899 | − PDG| γ,2

the center of the fit window rather than the origin reduces parameter correlations, making the fit more stable. An extra fit is made at 35 MeV where the significance is changing rapidly with energy. The background, B, is evaluated in a 2σ window around the mean of the peak. In Tables 4.4 and 4.5 we show the results, as well as the number of events in the ηb search region in data and MC. Figure 4.4 shows S/√B versus the cut on the energy of the second photon of the π0 candidate. 60 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B of the B 95. . error. in √ 0 its < S/ (MC) γ . p σ 2 and < kground + 2 54703 55845 56809 58373 60329 60960 61458 59518 61878 region 85 b . bac ,χ 0 yield g sig y E b b The η to signal σ b 2 χ eaks. − defined p 1 b (Data) ,χ the g fitted E region, the 182176 187692 193226 204338 224082 238481 241018 214594 254783 region h from sig are b searc b η χ underneath n the ∆ B function in √ and 143.4 144.9 145.6 146.5 145.5 145.8 146.1 146.7 146.4 S/ b kground χ n bac kground B the eto. bac 257424 266676 275806 294189 326153 339985 353494 310777 365883 v distribution 0 the b π and χ n the ∆ / yield 98.2 99.4 99.9 99.2 99.1 99.6 99.8 kground b 101.4 100.3 χ b for n χ bac tegrating in b χ the y n study b fitted 853 866 879 916 920 953 967 984 996 of ∆ the b χ tegral n kground from 83765 86059 87829 90925 93280 94398 96357 98205 99968 in determined bac ed the to is deriv energy is γ kground 30 35 40 50 60 70 80 90 100 Signal region, bac eaks h p 4.4: b the Second χ searc is able b T η B the 4.1. SELECTION CRITERIA 61

Table 4.5: Signal to background study for π0 veto with additional 3σ timing cut. The results of the timing from the simulation are not quoted, as the simulation matches the data poorly.

√ Second γ lab energy nχb ∆nχb nχb /∆nχb B S/ B ηb sig region (Data) 30 84788 859 98.7 261159 144.1 184355 35 86776 871 99.6 269605 145.3 189419 40 88431 877 100.8 278178 146.0 194624 50 91364 916 99.7 295692 146.8 205283 60 93574 928 100.8 311800 146.9 215211 70 94745 950 99.7 326864 145.9 224525 80 96502 964 100.1 340482 145.9 232935 90 98297 980 100.3 353954 146.1 241303 100 100008 997 100.3 366267 146.4 248944

150 B

S/ 149 148 147 146 145 144 143 142 141 140 20 40 60 80 100 120 140 160 π0 Second photon energy [MeV]

0 Figure 4.4: S/√B of χb peaks versus π second photon energy cut. With (without) timing cut shown in blue (red). The overall scale is arbitrary. Note the suppressed zero. 62 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

4.1.4 Investigation of a possible η veto

A secondary source of background photons comes from the decays of η mesons. Candidate η mesons are reconstructed in the η γγ decay mode by selecting pairs → of photons with an invariant mass 0.470 < mγγ < 0.620 GeV and a lab momentum

0.2 < pγγ < 10.0 GeV. A fit to the mγγ distribution is used to determine the width of the signal peak, σ. An event is vetoed if the signal photon combines with another photon in the event to form an η candidate within 2.5σ of the fitted peak of the η mass distribution. The requirement on the lab energy of the second photon is then varied to determine the optimal selection criterion. Figure 4.5 shows the efficiency on signal MC and on-resonance data, and S/√B for the η veto, versus the lab energy of the second photon. As S/√B is lower after applying the η veto, this simple η veto is not useful.

1.5 B S/ B S/ 1.4 Normalized signal efficiency 1.3 Normalized background efficiency 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 100 200 300 400 500 Second photon energy (MeV)

Figure 4.5: Eﬃciency of η veto on data and ηb signal MC, and change in S/√B obtained by applying the veto. No π0 veto is applied. Note the suppressed zero.

We ﬁnd that it is necessary to add a π0 veto to the η veto, as the η veto does 4.1. SELECTION CRITERIA 63

not improve S/√B without a π0 veto, due to the large number of π0 which create combinatorial background under the η peak. The second photon must not combine

to form a π0 within 15 MeV of the nominal π0 mass with any other photon in the event. Figure 4.6 shows the eﬃciency on signal MC and on-peak data, and S/√B for the η veto, versus the lab energy of the second photon, after the π0 veto is applied. The veto does improve S/√B somewhat but rejects very few η. The η veto is not

used in this analysis as its inclusion improves S/√B very little; imperfections in the simulation of the background photon spectrum could negate the eﬀects of the veto and induce substantial systematic errors in the eﬃciency.

1.05 B S/ B S/ 1.04 Normalized signal efficiency 1.03 Normalized background efficiency 1.02 1.01 1 0.99 0.98 0.97 0.96 0.95 100 200 300 400 500 Second photon energy (MeV)

Figure 4.6: Eﬃciency of η veto on data and Monte Carlo, and change in S/√B obtained by applying the veto. A π0 veto is applied. Note that the vertical scale is diﬀerent from that of Fig. 4.5. 64 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

4.1.5 N-1 cut plots

In optimizing selection criteria, one wishes to find variables with different distributions for signal and background. If two selection variables that are good discriminating variables initially are highly correlated, it is possible that once the optimal selection is imposed on one variable, the discrimination power of the other may be significantly reduced. A visual means of demonstrating that this is not the case may be found by applying all the selection criteria but one and observing that the discriminating power of the unused selection criteria persists. Figures 4.7 shows the selection variables when all but the given selection is applied. 4.1. SELECTION CRITERIA 65

0.05 0.16

0.14 0.04 0.12 0.1 0.03 Normalized events Normalized events 0.08 0.02 0.06 0.04 0.01 0.02 0 0 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of tracks Cluster lateral moment

0.04 0.05 0.045 0.035 0.04 0.03 0.035 0.025 0.03

Normalized events 0.02 Normalized events 0.025 0.015 0.02

0.01 0.015 0.01 0.005 0.005 0 -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 θ θ cos( γ) |cos( T)|

0.022 0.02 0.018 0.016 0.014 0.012 Normalized events 0.01 0.008 0.006 0.004 0.002 0 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 π0 mass (GeV/c2)

Figure 4.7: Normalized distribution of selection variables when all other cuts are applied in signal MC (blue) and data (red). The variables are, from left to right, the number of tracks, the cluster lateral moment, cos θ , cos θ and m . γ | T | γγ 66 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

4.2 Background to the Eγ spectrum

4.2.1 Introduction

There are two main types of backgrounds that contribute to the Eγ spectrum:

1. Non-peaking:

continuum and nonpeaking ISR background • Υ (3S) decays to bottomonium • Υ (1S) decays •

These three components have a smooth, non-peaking shape under the signal peak.

2. Peaking, next to signal region (850 to 950 MeV):

photons from χ (2P ) γΥ (1S) • bJ → photons from e+e− γ Υ (1S) • → ISR

The ﬁrst component will peak in the Eγ spectrum around 770 MeV. It is more than 100 MeV from the expected signal, but we anticipate one to two orders of magnitude more of these events than the signal.

The second component will give a peak in the Eγ spectrum at 855 MeV, and

will therefore overlap with the signal, especially if the ηb mass is near the low end of theoretical predictions.

It is extremely important to model both the line-shapes and the yields of these components correctly. 4.2. BACKGROUND TO THE Eγ SPECTRUM 67

4.2.2 Non-peaking Background

There are two components to the background: photons coming from qq¯ and generic ISR production, and photons coming from Υ (3S) decays to other bottomonium states. About one-third of photons from Υ (3S) decays result from Υ (1S) decays after transitions from the Υ (3S) to Υ (1S). We therefore considered classifying the background

from the Υ (1S) as a separate component.

There are two solutions to model the background components:

1. model each component with a separate PDF

2. model all components with the same PDF

The ﬁrst option requires that each component is precisely understood and modeled. We have investigated the ﬁrst option but decided that it was not a viable option. The qq¯ and generic ISR background could in principle be taken from on-resonance

data. We are recorded about 3 fb−1 of data just below the Υ (3S) resonance, which is about 10% of the on-resonance sample. Given the small amount of off-resonance data compared to on resonance data, a direct subtraction would not work, leading to large errors. One could get the line-shape from off-resonance and use it in the on resonance fit with the normalization floating. This also would have lead to large errors. For the

Υ (1S) background, we have investigated the possibility of building a sample of Υ (1S) decays by tagging the di-pion transitions Υ (3S) π+π−Υ (1S) and using sideband → subtraction to determine the photon spectrum. As the branching fraction for the di-pion transition is (4.48 0.21) 10−2, the statistics of the background-subtracted × spectrum was not high enough to be useful 68 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

We therefore decided to model the continuum background as a single PDF. We use the full data sample, but exclude the peaking regions (χb, γISRΥ (1S) and signal peaks). We tried several functions to ﬁt the resulting (“blinded”) spectrum and retained the following:

2 A C + e−αEγ −βEγ (4.2) The blinded spectrum of the full dataset with all selection cuts applied and the

ﬁt are shown on Fig. 4.8. The parameters of the ﬁtted function can be found in Table 4.6.

×103

500

400 Entries/5 MeV 300

200

100

0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 E(γ) (GeV)

Figure 4.8: Full dataset blinded spectrum, used to determine the continuum background parametrization. This histogram shows the ﬁt while the data is shown by the markers.

In the final fit, we will to float the continuum background parameters, with the starting value determined from the fit to the sideband region. 4.2. BACKGROUND TO THE Eγ SPECTRUM 69

Table 4.6: Fitted background parameters from a ﬁt of the full data sample in the side region 0.5 < Eγ < 0.6 GeV and 0.960 < Eγ < 1.2 GeV.

Parameter value C ( 4.000 0.046) 10−3 − × α 6.4028 0.0039 10−3 β 2.0011 0.0038× −

4.2.3 Peaking Background from χbJ (2P ) → γΥ (1S)

Introduction

The electromagnetic transitions χ (2P ) γΥ (1S) give three monochromatic lines bJ → with energies of 743, 764, and 777 MeV. Depending on the mass of the ηb, these three may be close to the signal. It is therefore important to properly model the line shapes of the three transitions. These transitions can also be used to determine the absolute photon scale and to validate our estimate of our signal reconstruction eﬃciency (for results, see Table 4.3). Table 4.7 summarizes the various χb(2P ) transitions.

Table 4.7: Summary of χbJ (2P ) transitions. In the last column we give the number of expected events, corrected for eﬃciency, for 30 fb−1 of on resonance data.

6 −1 Transition BR Eγ MeV N(10 ) for 30 fb Υ (3S) γχ (2P ) 0.131 86 → b2 Υ (3S) γχ (2P ) 0.126 99 → b1 Υ (3S) γχb0(2P ) 0.059 122 χ (2P→) γΥ (1S) 0.071 777 b2 → χb1(2P ) γΥ (1S) 0.085 764 χ (2P ) → γΥ (1S) 0.009 743 b0 → Υ (3S) γχb2(2P ), χb2(2P ) γΥ (1S) 0.0093 1.116 Υ (3S) → γχ (2P ), χ (2P ) → γΥ (1S) 0.0107 1.284 → b1 b1 → Υ (3S) γχ (2P ), χ (2P ) γΥ (1S) 0.0005 0.06 → b0 b0 → 70 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

We use the full dataset spectrum, with the e+e− γ Υ (1S) and signal regions → ISR (600 to 950 MeV) excluded.

Fit to the χbJ (2P ) peaks with blinded full dataset spectrum

We take the full dataset spectrum, blinded in the e+e− γ Υ (1S) and signal → ISR regions, and ﬁx the non-peaking background PDF as described above in 4.2.2. Each

of the χbJ (2P ) is ﬁtted with a Crystal Ball (CB) function, a function which is a Gaussian with a power law tail on one side. The Crystal Ball function is deﬁned as

2 (E−µ) E−µ exp ( 2 ) if > A f(E, A, N, µ, σ) = Norm − 2σ σ − ·  E−µ −N E−µ  C D if A  · − σ σ ≤ −  where N N N C = and D = A A A − | | | | | | Throughout this document, A, N, µ, and σ will be referred to as the “transi-

tion point”, “tail parameter”, “peak”, and “sigma”. In the parameterization of the

χbJ (2P ) peaks, we allow the widths to vary separately but the transition points and tail parameters are shared between the peaks. The relative ratio between the χbJ (2P ) is fixed from the value from the Particle Data Group [79]. The peak positions are allowed to float by a common amount with respect to the PDG values. We therefore measure an energy scale offset between the PDG values and the measured values. The fitted parameters are shown on Table 4.8. The result of the fit and the corresponding residuals can be seen in Fig. 4.9.

For the final fit, we fix all χbJ (2P ) peak parameters from this blinded fit. 4.2. BACKGROUND TO THE Eγ SPECTRUM 71

Table 4.8: Fitted parameters for the χbJ (2P ) peaks in the blinded full dataset spectrum. A, N and σ are the transition point, tail parameter, and Gaussian width, respectively, of the Crystal Ball function. The oﬀset is deﬁned by E E γ,expected − γ,measured

Parameter χb0(2P ) χb1(2P ) χb2(2P ) Crystal Ball transition point 0.977 0.005 same same Crystal Ball tail parameter 135 2 same same Crystal Ball σ parameter (MeV) 13.1 0.3 24.4 0.1 19.0 0.1 Energy scale oﬀset 0.0038 0.0001 same same 72 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

60000

50000

Entries/5 MeV 40000

30000

20000

10000

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Eγ (GeV)

25000

20000

15000 Entries/5 MeV

10000

5000

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Eγ (GeV)

-1 normalized residuals -2

-3 0.5 0.6 0.7 0.8 0.9 1 1.1 Eγ (GeV)

Figure 4.9: Fit to the χbJ (2P ) peaks in the blinded full dataset subtracted spectrum. The medium ﬁgure is a zoom and the bottom shows the residuals. 4.2. BACKGROUND TO THE Eγ SPECTRUM 73

+ − 4.2.4 Peaking Background from e e → γISRΥ (1S)

In this section, we investigate the presence of ISR Υ (1S) background. The photon center-of-mass (c.m.) energy for Υ (1S) production to ISR from the Υ (3S) is

q = E∗ 1 m2 /s = 856.4 MeV/c, beam − R 0

∗ 2 where Ebeam = 5.1776 GeV, mR = m(Υ (1S) = 9.4603 GeV/c and √s0 is the nominal c.m. energy. Therefore ISR Υ (1S) will constitute peaking background to the expected

ηb signal. It is crucial to precisely determine the ISR peak line-shape and yield.

In order to determine these two quantities, several options and channels have been investigated:

1. e+e− γ Υ (1S), Υ (1S) µ+µ−. → ISR →

2. ISR Signal MC.

3. Υ (3S) oﬀ-resonance Data.

4. Υ (4S) oﬀ-resonance Data.

ISR Signal MC

The ISR photon c.m. energy distribution in e+e− γ Υ (1S) signal MC is shown → ISR in Fig. 4.10 in which the Υ (1S) is assumed to decay generically. Truth-association is required for the top distribution of Fig. 4.10 which is ﬁtted with a CB function to describe the signal and an exponential as background function. Then, in order

to determine the efficiency, the non-truth-matched Monte Carlo with all selection 74 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B cuts applied (bottom plot of Fig. 4.10) is then fitted with a CB function where all parameters are fixed from the truth-matched fit.

We obtain a signal eﬃciency of = (5.78 0.09)%.

Υ (3S) Oﬀ-Resonance Data

We have used a sample of 2.4 fb−1 of Υ (3S) oﬀ-resonance data. At this CM energy, the production cross section of the e+e− γ Υ (1S) process is σ(e+e− γ Υ (1S)) → ISR → ISR ' 25.4 pb.

The ISR peak lineshape obtained from off-resonance data is fitted with a CB function with parameters fixed to those obtained from Fig. 4.10. A function of the form given in Eq. 4.2 is used to model the background. We observe 2773 473 events.

Υ (4S) Oﬀ-Resonance Data

We have used a sample of 43.9 fb−1 of Υ (3S) off-resonance data. At this c.m. energy, the production cross-section of the e+e− γ Υ (1S) process is σ(e+e− → ISR → γ Υ (1S)) 19.8 pb. The top distributions in Fig. 4.12 show the ISR photon c.m. ISR ' energy distribution from signal MC (generated at the Υ (4S) off-resonance energy). The background is fitted with an exponential. The bottom plots of Fig. 4.12 show the ISR photon c.m. energy distribution from the Υ (4S) off-resonance data sample. The background line-shape is fitted with a function of the form given in Eq. 4.2.

We determine the signal efficiency on MC to be = 6.16 0.12%. We measure 35759 1576 ISR events in the Υ (4S) off-resonance sample. The background- subtracted fit is shown in Fig. 4.13. 4.2. BACKGROUND TO THE Eγ SPECTRUM 75

+ − Table 4.9: Production cross section for e e γISRΥ (1S) at √s = 10.3252 GeV + − → (σΥ (3S)), production cross section for e e γISRΥ (1S) at √s = 10.55 GeV (σΥ (4S)), and their ratio for various orders in perturbation→ theory. The assumed di-electron width of the Υ (1S) is 1.340 MeV.

Calculation σΥ (3S) (pb) σΥ (4S) (pb) Ratio Asymmetric collider correction Benayoun, et. al., 2nd order 25.4 19.8 1.283 Yes Benayoun, et. al., 1st order 28.46 21.62 1.316 No Benayoun, et. al., 2nd order 26.12 20.21 1.292 No

Discussion

The cross section for ISR production of a narrow vector meson is given by

2 12π Γee σV (s) = W (s, xv) mV s ·

where Γee is the di-electron width of the vector meson, mV is the mass of the vector

meson, s is the center of mass energy, xv = 2Eγ /√s, and W (s, xv) is the probability

function of photon emission. W (s, xv) is often referred to as the “radiator function”.

To study the systematic error on the ratio of the ISR cross sections in off-resonance Υ (3S) and off-resonance Υ (4S) data, we compare the ratios predicted by different theoretical calculations, and the same calculation done to different orders in α. The calculation by Benayoun, et. al., is performed to order α2 and includes corrections

for soft multi-photon emission and hard collinear bremsstrahlung in the leading logarithmic approximation. The results are summarized in Table 4.9.

We extrapolate the number of e+e− γ Υ (1S) events from the measured yield → ISR with Υ (3S) oﬀ-resonance and Υ (4S) oﬀ-resonance samples, to the Υ (3S) on-resonance 76 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

sample (Medium: 25.598 fb−1). We assume that the ISR cross-sections are the same at the Υ (3S) oﬀ-resonance and on-resonance energies. The numbers are summarized in Tab 4.10.

Dataset Lumi Cross-Section Reconstruction Yield Extrapolation to [ fb−1 ] [pb] Eﬃciency Υ (3S) on-resonance Υ (3S) 2.415 25.4 5.78 0.09 2773 473 29393 5014 Υ (4S) 43.9 19.8 6.16 0.12 35759 1576 25153 1677

Table 4.10: Number of e+e− γ Υ (1S) events from the Υ (3S) oﬀ-resonance and → ISR Υ (4S) oﬀ-resonance samples, and extrapolation to the Υ (3S) on-resonance sample (25.598 fb−1). The errors are statistical only.

The extrapolated numbers from the two samples are in good agreement, with much better statistical errors from the Υ (4S) oﬀ-resonance sample. The systematic error on the extrapolation is taken to be 5% and results from the statistics available for the determination of the eﬃciency ratio (2%) and the variation of the ratio of the theoretical cross sections (3%). The largest contribution stems from the luminosity

error on the Υ (3S)-on-resonance dataset. In the final fit, the e+e− γ Υ (1S) → ISR yield will be fixed to the extrapolated value derived from the Υ (4S) off-resonance sample. The line-shape parameters will be taken from MC, which has worked well for the off-resonance samples. 4.2. BACKGROUND TO THE Eγ SPECTRUM 77

A RooPlot of "E_gamma" )

2 2400 A1 = 0.62 +/- 0.02 2200 AExp = -2.93 +/- 0.2 2000 Mean1 = 0.8367 +/- 0.0003 GeV/c2 1800 N1 = 3.4 +/- 0.3 1600 Sigma1 = 0.0204 +/- 0.0002 GeV/c2 1400 nExp = 1603 +/- 101 Events / ( 0.01 GeV/c 1200 nSig1 = 17148 +/- 160 1000 800 600 400 200 0 0.5 0.6 0.7 0.8 0.9 1 1.1 E_gamma (GeV/c2)

4 3 2 1 0 -1 -2 -3 Residuals -4 0.5 0.6 0.7 0.8 0.9 1 1.1 A RooPlot of "E_gamma"

AExpMC = -4.295 +/- 0.05 700 MeanISR = 0.8379 +/- 0.0004 nBg = 24481 +/- 180 600 nISR = 8594 +/- 129 500 Events / ( 0.005 GeV ) 400

300

200

100

0 0.5 0.6 0.7 0.8 0.9 1 1.1 E_gamma (GeV)

5 4 3 2 1 0 -1 -2 -3 Residuals -4 -50.5 0.6 0.7 0.8 0.9 1 1.1

+ − Figure 4.10: ISR photon c.m. energy distribution in e e γISRΥ (1S) signal MC (data points). The top figure shows the truth-associated spectrum,→ the bottom figure the total spectrum. The superimposed signal lineshape corresponds to the Crystal Ball function (red). The background (green line) is described using a exponential function. The total fit function is represented by the blue curve. 78 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

A RooPlot of "E_gamma" AExpMC = -4.295 +/- 0.05 20000 MeanISR = 0.8379 +/- 0.0004 700 C = 0.0003 +/- 0.0005 nBg = 24481 +/- 180 18000 MeanISR = 0.824 +/- 0.006 nISR = 8594 +/- 129 16000600 alpha = 6.4 +/- 0.2 beta = -1.34 +/- 0.2 14000 500 nBg = 820235 +/- 1020 12000 Events / ( 0.005 GeV ) nISR = 2773 +/- 473 400 10000

8000300

6000 200 4000 100 2000 0 0.5 0.6 0.7 0.8 0.9 1 1.1 E_gamma (GeV)

5 4 3 2 1 0 -1 -2 -3 Residuals -4 -50.5 0.6 0.7 0.8 0.9 1 1.1

Figure 4.11: The photon c.m. energy distribution in Υ (3S) oﬀ-resonance data (data points). The superimposed ISR-signal lineshape corresponds to the Crystal Ball function (red). The background (green line) is described by a function of the form given in Eq. 4.2. The total ﬁt function is represented by the blue curve. 4.2. BACKGROUND TO THE Eγ SPECTRUM 79

A RooPlot of "E_gamma" A1 = 0.58 +/- 0.06 ) 2 AExp = -4.114 +/- 0.08 250 N1 = 5 +/- 3 OffSet1 = 0.0124 +/- 0.0010 GeV/c2 Sigma1 = 0.0209 +/- 0.0008 GeV/c2 200 nExp = 7176 +/- 130 nSig1 = 3555 +/- 115 150 Events / ( 0.005 GeV/c

100

0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 E_gamma (GeV/c2)

4 3 2 1 0 -1 -2 Residuals -3 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 A RooPlot of "E_gamma" 3 ×10 AExpMC = -5.04 180 300 C = 0.0080 +/-MeanISR 0.0007 = 1.0 160 MeanISR = 1.0259nBg = +/-155000.0 0.0007 alpha = 2.7 +/-nISR 0.2 = 9000.0 140250 beta = 1.07 +/- 0.10 120 nBg = 3636546 +/- 2440 200 Events / ( 0.01 GeV )

Events / ( 0.005 GeV ) 100 nISR = 36825 +/- 1541 150 80

10060 40 50 20

0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 E_gamma (GeV)

5 4 3 2 1 0 -1 -2 -3 Residuals -4 -50.70.8 0.850.8 0.9 0.90.95 1 1.05 1.1 1.1 1.21.15 1.31.2

Figure 4.12: Shown in the top two plots is the photon c.m. energy distribution in + − e e γISRΥ (1S) signal MC, generated at the Υ (4S) off-resonance energy. The photon→c.m. energy distribution in the Υ (4S) off-resonance data sample is shown on the bottom. The ISR-signal lineshapes are shown in red(top) and black(bottom). The background (dashed-blue on top, green line on bottom) is described using an exponential function (top) or a function of the form given in Eq. 4.2 (bottom). The total fit functions are represented by the blue curves. 80 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

4000

3000 Entries/10 MeV 2000

1000

-1000 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 Eγ (GeV)

Figure 4.13: Inclusive photon spectrum in the below-Υ (4S) data, after background subtraction. The ﬁtted curve shown is the Crystal Ball Function which describes the data points very well. 4.3. FITTING PROCEDURE 81

4.3 Fitting procedure

In this section, we describe the ﬁtting procedure. We then test the ﬁtting procedure

under various background modeling options using “toy” MC. As a test of the ﬁt on

−1 real data, we ﬁt the 2.5 fb sample with all components, including an ηb signal.

4.3.1 Fit and Unblinding strategies

We extract the ηb yield using a binned likelihood ﬁt to the photon energy spectrum

in the range 0.5 < Eγ < 1.5 GeV. The low end of the ﬁt range is chosen to avoid the photons from the transitions Υ (3S) γχ (1P ) (the highest photon energy is from → bJ

the transition to the χb0(1P ) with an energy of 484 MeV).

We ﬁt each of the individual components in the global ﬁt as follows:

1. Continuum Background: the initial values for the PDF are taken from the full sample with the signal region blinded, as described in the previous section, but all parameters are allowed to ﬂoat.

2. χb: see previous section. All parameters except the yield are fixed, including the 3.8 MeV offset observed in the peak position. The offset is fixed to the same

value for each of the three χb peaks.

3. e+e− γ Υ (1S): the line-shape is taken from truth-matched MC. The yield → ISR is fixed to 25153 events and the peak position is fixed to 852.6 MeV, which is 3.8 MeV below the expected value. The shift is taken from the offset observed

in the peak position of the χbJ (2P ). 82 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

4. ηb signal: the line shape in taken from signal MC, with the yield and mean

value ﬂoating. The ηb width is ﬁxed to 10 MeV, near the center of the range of

theoretical predictions. As the ηb width is smaller than the detector resolution (25 MeV at 910 MeV), it is necessary to ﬁx the width to avoid ﬁt bias.

We ﬁt the ηb widths at 5, 10, 15, and 20 MeV, and quote the signal yield and signiﬁcance at each.

4.3.2 Minimization Details

Due to the fact that the non-peaking background level is very large, the value of the likelihood that we are minimizing in the fit also gets very large, of the order of -4 108. × To accurately find the minimum, the minimization program we use, MINUIT [77] needs to be sensitive to likelihood variations of the order of 10−9. However the default numerical integration setting in the RooFit fitting package is 10−7. Therefore in order to ensure that MINUIT correctly handles large likelihood values, we change the numerical integration settings so that the background PDF is evaluated to a precision of at least 10−9.

In the fit the MINUIT routines SEEK and SIMPLEX are invoked prior to the MINUIT routines MIGRAD and MINOS. This helps MINUIT to find the minimum faster and more accurately by providing a more reliable starting point for the fit. In the fit, we fix the non-peaking background parameters from the MIGRAD minimization and then run MINOS. Fixing these parameters after the MIGRAD minimization step ensure proper MINOS convergence so that the asymmetric errors are correctly obtained and that the covariance matrix is accurate. We verified that there is no bias by fixing the non-peaking parameters obtained from the MIGRAD minimization by 4.3. FITTING PROCEDURE 83

running MIGRAD and then MINOS without ﬁxing these parameters and found they do not change between MIGRAD and MINOS. 84 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

4.4 Toy Studies

We test the ﬁt procedure with “toy MC”. A “toy” is a dataset simulated with the PDFs and expected yields of all of the components in the ﬁt. The toy dataset is then

analyzed using the same fitting procedure that we plan to use in the fit to the full dataset. This procedure is repeated many times, and the distribution of the results is a means of studying bias in the analysis or fit procedure.

Each toy is generated according to the non-peaking background, χb0,1,2, ISR and

ηb PDF’s with a number of events for each distribution obtained to the expected number for the full-statistics sample.

The non-peaking background component is obtained from a generation of a set of 1000 distributions that are then used in the various toys. This is done in order to

avoid the regeneration of a large sample for every toy. The peaking components are generated according to the appropriate PDF with Poisson statistics.

The PDF’s used in the generation and in the ﬁt are the same. The signal PDF

consists of a convolution of a Breit-Wigner function with a Crystal Ball function. In

the ﬁt procedure the signal and χb yields and the signal mean are free parameters. The ISR yield and peak position is ﬁxed to the number expected by interpolation using Υ (3S) and Υ (4S) data (see section 4.2.4).

We produce 15 sets of 500 toys, with ηb peak position at 890, 895, 900, and 920

MeV, for a number of generated ηb events of 15000, 20000, 30000, respectively. We repeat the 20000 η event generation with a change in ISR yield of 1σ of the expected b number (see section 4.2.4). Tables 4.11 - 4.13 summarize the data presented in the

ﬁgures. 4.4. TOY STUDIES 85

We have approximately a 100% and 75% convergence rate with the MINUIT routines HESSE and MINOS, respectively.

A summary of the results of the toy fits follows below. We quote the mean and widths both obtained as mean and RMS from the histogram and as result of a likelihood fit with a Gaussian function. Tables 4.11-4.13 show the means and widths for the ηb peak position, signal yield and signal significance for toys with varied input yields and peak positions.

Table 4.11: Fitted peak values (µ) and width (σ) of ηb peak position value distribution in GeV for various combinations of ηb yield (in thousands) and photon peak positions. Yield[k]- µ RMS µ σ Mean Fit Peak[MeV] (histo) (fit) (fit) Error (fit) 15-890 0.890948 0.00409575 0.890947 0.00405628 0.00474713 20-890 0.89022 0.00275328 0.890256 0.00277593 0.00353754 30-890 0.890152 0.00170825 0.890156 0.00177796 0.00230443 15-895 0.896115 0.00327379 0.896051 0.00345211 0.00461712 20-895 0.895162 0.00265388 0.895142 0.00273411 0.0034851 30-895 0.895197 0.00175234 0.89516 0.0018837 0.00229684 15-900 0.900342 0.00362309 0.900299 0.00372508 0.00462206 20-900 0.900244 0.00259133 0.900242 0.00270207 0.00340684 30-900 0.900212 0.00169717 0.900195 0.00175836 0.00224902 15-910 0.910412 0.00345494 0.910376 0.00351703 0.00465174 20-910 0.910271 0.00263357 0.910255 0.00269485 0.00321852 30-910 0.91021 0.00171415 0.910205 0.00179092 0.00222286 15-920 0.92043 0.00355327 0.920442 0.0035783 0.0041995 20-920 0.920356 0.00238203 0.920354 0.00248524 0.00300483 30-920 0.920186 0.00162972 0.920234 0.00170445 0.00203371

Tables 4.14-4.16 show the means and widths for the ηb peak position, signal yield and signal signiﬁcance for toys with the ISR yields assumed in the ﬁt varied by 1σ.

For a third set of ﬁts, the ηb signal was generated and ﬁtted with a set of different values. Twenty thousand ηb signal events have been generated at 910 MeV. 86 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

Table 4.12: Mean and width of signal significance distributions for various combinations of ηb yield (in thousands) and photon peak positions. Yield[k]-Peak[MeV] Mean RMS µ (fit) σ (fit) 15-890 7.83605 1.02498 7.85965 1.0361 15-895 8.06881 0.969492 8.05292 0.972913 15-900 8.09415 0.92315 8.11207 0.935228 15-910 8.28476 0.994144 8.29852 1.00985 15-920 8.49536 1.09491 8.50442 1.10434 20-890 10.5732 0.802315 10.5708 0.808114 20-895 10.6853 0.869206 10.6946 0.884052 20-900 10.7343 0.93441 10.7261 0.955538 20-910 11.004 0.96755 11.006 0.970474 20-920 11.3467 0.916625 11.3486 0.921708 30-890 15.7379 0.97357 15.7354 0.986204 30-895 15.8712 1.02005 15.8723 1.03284 30-900 16.0679 1.01872 16.0621 1.02618 30-910 16.4415 1.03653 16.4413 1.04187 30-920 16.8559 1.03916 16.8524 1.05346

Tables 4.17-4.19 summarize the results.

4.4.1 Conclusions

As a result of the toy studies we determine that it is necessary to ﬁx the yield of the ISR background in order to avoid generating a spurious bias in the signal yield.

However, if the ISR yield is fixed in the fit, we observe no statistically significant fit bias. 4.4. TOY STUDIES 87

Table 4.13: Signal yields and width (σ) of signal yield distribution for various combinations of ηb yield (in thousands) and photon peak positions. Yield[k]- Mean RMS µ σ Mean Fit Peak[MeV] (histo) (fit) (fit) Error (fit) 15-890 14525.4 1964.03 14521.6 1972.7 1696.79 15-895 14800.7 1830.6 14804.3 1833.17 2081.69 15-900 14670.6 1703.28 14669.9 1697.72 2081.68 15-910 14631.7 1813.89 14634.1 1814.08 1871.53 15-920 14660.9 1901.45 14664.7 1900.09 1659.06 20-890 19730.5 1614.54 19730.7 1615.56 1984.26 20-895 19693.8 1680.32 19698.2 1684.94 1755 20-900 19581 1776.63 19578.1 1772.18 1818.11 20-910 19633.3 1782.41 19632.1 1779.46 1723.79 20-920 19709 1641.62 19707.5 1642.64 1618.35 30-890 29261.6 1771.28 29260.8 1769.28 1787.55 30-895 29222.7 1827.2 29226.3 1827.23 1768.07 30-900 29290.7 1797.22 29292.5 1794.08 1772.93 30-910 29405.5 1849.37 29399.6 1849.96 1676.37 30-920 29384.3 1797.98 29385.9 1799.82 1602.62 88 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

Table 4.14: Mean and width of the ﬁtted ηb peak positions for diﬀerent yields of ISR events

Yield[k]- ISR Yield µ RMS µ σ Mean Fit Peak[MeV] variation (histo) (fit) (fit) Error (fit) 20-895 -1σ 0.895214 0.00267746 0.895222 0.00275349 0.003451 def 0.895162 0.00265388 0.895142 0.00273411 0.0034851 +1σ 0.895236 0.00270055 0.895258 0.00275113 0.00352584 20-910 -1σ 0.910372 0.00253579 0.910368 0.0025499 0.0032419 def 0.920356 0.00238203 0.920354 0.00248524 0.00300483 +1σ 0.910428 0.00254886 0.910432 0.00260224 0.00323748 30-895 -1σ 0.895222 0.0017668 0.89526 0.00186308 0.00232902 def 0.895197 0.00175234 0.89516 0.0018837 0.00229684 +1σ 0.895193 0.00174673 0.895192 0.00185354 0.0022719 30-910 -1σ 0.91017 0.00171884 0.910134 0.00177383 0.00223415 def 0.91021 0.00171415 0.910205 0.00179092 0.00222286 +1 σ 0.910234 0.00174274 0.910237 0.00183471 0.00221993

Table 4.15: Mean and width of the fitted ηb signal significance for different yields of ISR events Yield[k]- ISR Yield µ RMS µ σ Peak[MeV] variation (histo) (fit) 20-895 -1σ 10.6773 0.919477 10.6838 0.922084 def 10.6853 0.869206 10.6946 0.884052 +1σ 10.6765 0.869455 10.6761 0.899883 20-910 -1σ 11.0093 0.975387 11.0117 0.985086 def 11.3467 0.916625 11.3486 0.921708 +1σ 11.0347 0.959051 11.0355 0.969631 30-895 -1σ 15.8987 1.00701 15.8918 1.01388 def 15.8712 1.02005 15.8723 1.03284 +1σ 15.8979 0.980098 15.8903 0.991624 30-910 -1σ 16.437 1.05575 16.4426 1.07596 def 16.4415 1.03653 16.4413 1.04187 +1σ 16.4465 1.04964 16.4483 1.05808 4.4. TOY STUDIES 89

Table 4.16: Mean and width of the fitted ηb signal yield for different yields of ISR events Yield[k]- ISR Yield µ RMS µ σ Mean Fit Peak[MeV] variation (histo) (fit) (fit) Error (fit) 20-895 -1σ 19670.2 1782.32 19669.1 1784.41 2104.64 def 19693.8 1680.32 19692.5 1681.75 1910.49 +1σ 19655.4 1667.71 19655.3 1671.06 1753.58 20-910 -1σ 19628 1807.38 19630.3 1806.99 1731.77 def 19709 1641.62 19707.2 1641.19 1630.47 +1σ 19658.6 1750.3 19657.4 1750.62 1695.41 30-895 -1σ 29254.6 1805.31 29254.1 1806.44 11624.6 def 29222.7 1827.2 29224.1 1828.5 12846.7 +1σ 29253.7 1729.84 29251.9 1730.86 1754.31 30-910 -1σ 29385.3 1866.55 29386.4 1865.2 1682.98 def 29405.5 1849.37 29405 1849.48 1688.15 +1σ 29356.7 1827.04 29358.1 1826.34 1683.94 90 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

Table 4.17: Mean and width of the fitted ηb peak position in GeV for different generated and fitted ηb widths. Generated peak position is 910 MeV and generated yield is 20k events. Width µ RMS µ σ Mean fit Generation/Fit (histo) (histo) (fit) (fit) Error (fit) Gen10-Fit5 0.910636 0.00243977 0.910784 0.00257991 0.00329063 Gen10-Fit10 0.910474 0.00252153 0.910462 0.00249457 0.00333059 Gen10-Fit20 0.909989 0.00255098 0.909937 0.00265931 0.00348627 Gen10-Fit30 0.909799 0.00253835 0.909701 0.00261397 0.00356103 Gen10-Fit40 0.909357 0.00269662 0.909282 0.00285952 0.00383623 Gen20-Fit5 0.910817 0.00328504 0.9109 0.00329494 0.00376129 Gen20-Fit10 0.91074 0.00338701 0.910723 0.00355834 0.00392965 Gen20-Fit20 0.910269 0.00304693 0.910302 0.00302705 0.00396037 Gen20-Fit30 0.910097 0.00319229 0.910158 0.00332871 0.00233492 Gen20-Fit40 0.909946 0.00348103 0.909774 0.00350768 0.00435495 Gen30-Fit5 0.911283 0.00404579 0.91137 0.00401984 0.00475909 Gen30-Fit10 0.911021 0.00426459 0.911082 0.0042754 0.00292865 Gen30-Fit20 0.910739 0.0033483 0.910815 0.00351227 0.00500995 Gen30-Fit30 0.910085 0.00391827 0.910071 0.00392727 0.00473963 Gen30-Fit40 0.910266 0.00433069 0.910222 0.00446246 0.00519436 4.4. TOY STUDIES 91

Table 4.18: Mean and width of the fitted ηb signal significance in GeV for different generated and fitted ηb widths. Generated peak position is 910 MeV and generated yield is 20k events. Width µ RMS µ σ Generation/Fit (histo) (histo) (fit) (fit) Gen10-Fit5 10.8975 1.0154 10.8919 1.00848 Gen10-Fit10 10.834 0.946236 10.8397 0.96988 Gen10-Fit20 10.9074 1.00702 10.8513 1.03539 Gen10-Fit30 10.8503 0.981161 10.8344 1.00777 Gen10-Fit40 10.5989 1.04747 10.609 1.05276 Gen20-Fit5 9.63371 1.11098 9.61667 1.11754 Gen20-Fit10 9.67862 1.01355 9.69615 1.01574 Gen20-Fit20 9.68583 0.953382 9.69444 0.943277 Gen20-Fit30 9.55482 1.01932 9.57456 1.03233 Gen20-Fit40 9.59009 0.870811 9.59677 0.891667 Gen30-Fit5 8.3953 1.03435 8.38889 1.03824 Gen30-Fit10 8.43543 1.09945 8.42347 1.09055 Gen30-Fit20 8.67627 1.00299 8.65741 1.00496 Gen30-Fit30 8.68735 0.99233 8.67857 0.965607 Gen30-Fit40 8.70747 0.86723 8.72222 0.878849 92 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

Table 4.19: Mean and width of the fitted ηb signal yield in GeV for different generated and fitted ηb widths. Generated peak position is 910 MeV and generated yield is 20k events. Width µ RMS µ σ Error µ Generation/Fit (histo) (histo) (fit) (fit) (fit) Gen10-Fit5 18079.5 1730 18083.8 1726.94 1479.06 Gen10-Fit10 19290.5 1765.44 19294.9 1770.46 4464.88 Gen10-Fit20 22084.3 1987.08 22085.4 1986.47 1897.02 Gen10-Fit30 24515.8 2211.15 24517.5 2207.98 1611.16 Gen10-Fit40 26543.1 2563.31 26544.9 2565.5 2232.38 Gen20-Fit5 15850.4 1879.12 15841.8 1894.16 1610.84 Gen20-Fit10 17156.6 1849.34 17155.7 1847.52 6202.75 Gen20-Fit20 19490.7 1956.97 19486.5 1955.01 1882.65 Gen20-Fit30 21641.5 2200.02 21644.7 2196.93 2042.02 Gen20-Fit40 24014.4 2172.39 24011.3 2173.77 2222.54 Gen30-Fit5 13780.3 1732.35 13667.1 1853.58 1592.09 Gen30-Fit10 14911.6 2021.61 14860.8 2091.38 1733.36 Gen30-Fit20 17390.3 2081.06 17389.2 2084.1 9140.98 Gen30-Fit30 19662.9 2301.82 19653.4 2300.29 2018.5 Gen30-Fit40 21728.2 2306.84 21731.5 2301.06 2189.8 4.5. FIT ON THE 2.5 FB−1 OPTIMIZATION SAMPLE 93

1 4.5 Fit on the 2.5 fb− Optimization Sample

We perform a fit to the test sample, with all components as described in the previous section, as a test of the procedure on real data. The results are shown in Fig. 4.14. The fit to the optimization sample is used as a check of the fit procedure before unblinding the data. Since the fit quality is excellent and the residuals show no strange features, we decided to unblind the full data sample.

4.6 Fit to the inclusive photon spectrum

We perform our fit to the unblinded, inclusive photon spectrum in 25.6 fb−1 of Υ (3S) data as described in section 4.3.1. The fit for an assumed ηb width of 10 MeV is shown in Fig. 4.15 and the corresponding fit parameters are shown in Table 4.20.

The numerical results for the ﬁts with the 4 diﬀerent ηb widths are summarized in Table 4.21.

Fit Parameter Value Signal yield 19152 2010 Nonpeaking background yield (2.38936 0.00060) 107 × 5 χbJ (2P ) yield (8.2184 0.022) 10 +2.1 × ηb peak position 917.3−2.8MeV Background parameter C ( 3.918 0.030) 10−3 − × Background parameter α 6.3987 0.0088 Background parameter β 1.990 0.0066 −

Table 4.20: Results of the ﬁt to the Eγ distribution assuming the nominal ηb width of Γ(ηb) = 10 MeV. 94 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

Γ(ηb) Nsignal Signiﬁcance (MeV) (ηb) (σ) 5 18050 1742 10.24 10 19152 2010 10.10 15 20348 2119 10.06 20 21314 2052 10.01

Table 4.21: Results of the ﬁt to the Eγ distribution with diﬀerent assumed values of Γ(ηb).

104 C = -0.00102 +/- 0.0002 OffSet = 0.012 +/- 0.008 GeV alpha = 6.09 +/- 0.03 103 beta = -1.467 +/- 0.01 nBg = 2317325 +/- 1550 nChiB = 82121 +/- 831 102 nEtaB = 2914 +/- 872 Events / ( 0.005 GeV )

1 0.5 0.6 0.7 0.8 0.9 1 1.1 Eγ (GeV)

5 4 3 2 01 -1 Residuals -2-3 -4-5 0.5 0.6 0.7 0.8 0.9 1 1.1

12000

10000

8000 Entries/ 10 MeV 6000

4000

2000

0.5 0.6 0.7 0.8 0.9 1 1.1 Eγ (GeV)

Figure 4.14: The inclusive photon c.m. energy distribution in the optimization sample (2.5 fb−1) of Υ (3S) on-resonance data. The top plot show the distribution together with the ﬁt results and the bottom plot shows the non-peaking background subtracted plot. 4.6. FIT TO THE INCLUSIVE PHOTON SPECTRUM 95

×103

500

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100

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8000

6000

4000

2000 Entries/ (0.020 GeV)

-2000

0.5 0.6 0.7 0.8 0.9 1 1.1 Eγ (GeV)

Figure 4.15: (a) Inclusive photon spectrum in the region 0.50 < Eγ < 1.1 GeV. (b) Background subtracted photon spectrum in the signal region, showing χbJ (2P ) peaks (red), ISR Υ (1S) (green), signal (blue) and the sum of the contributions (purple). (c) Signal peak after all backgrounds are subtracted. 96 CHAPTER 4. STUDY OF THE DECAY Υ (3S) γη → B

4.7 Systematic Uncertainties

For the systematic error on the signal yield (thus branching fraction) and peak position (thus ηb mass), we consider the following sources:

1. ηb width: our nominal result is obtained with an assumed width of 10 MeV. We repeat the ﬁt for widths of 5, 15 and 20 MeV and take the largest variation as the uncertainty.

2. The assumed ISR yield is varied by 1σ, with the error being the sum of the statistics and systematic error of our extrapolation (see 4.2.4).

3. To study the effect of the PDF parametrization of the different fit components, the following parameters are varied by 1σ:

ISR: N, A and σ of the Crystal Ball function • signal: vary N, A and σ of the Crystal Ball function • χ :vary N, A and σ of the Crystal Ball function as well as the • B χb0,χb1,χb2 common energy oﬀset

The resulting uncertainties are summarized in Table 4.22. For the measurement of the branching ratio, we have an additional source of uncertainty resulting from the luminosity uncertainty and the signal selection eﬃciency.

The latter is determined by comparing the selection efficiencies on the nearly χB peak in data and MC. We find these efficiencies to be 39% and 35% respectively, corresponding to a 12% systematic uncertainly. The determination of the χB selection efficiency also suffers from the uncertainty on the χB branching fraction in the PDG. 4.7. SYSTEMATIC UNCERTAINTIES 97

Table 4.22: Systematic uncertainties for the measurement of the ηb yield and peak position (mass).

Source ∆ N [Evts] ∆ peak pos. [MeV] ηb width, 5 MeV 1417 0.3 ηb width, 15 MeV 1545 0.2 ηb width, 20 MeV 2783 0.5 N , 1σ 108 0.5 ISR ISR PDF, N 1 0.1 ISR PDF, A 18 0.1 ISR PDF, σ 45 0.0 signal PDF, N 48 0.0 signal PDF, A 92 0.1 signal PDF, σ 5 0.0 χB PDF, N 6 0.0 χB PDF, A 135 0.1 χB PDF, σχB2 185 0.0 χB PDF, σχB1 378 0.1 χB PDF, σχB0 8 0.0 χB PDF, oﬀset 22 0.1 Total 2822 0.75

Table 4.23: Systematic uncertainties for the measurement of the ηb branching fraction.

Source ∆ N [Evts] ﬁt variations 2822 luminosity 1095 selection eﬃciency 3966 Total 4909 σ BF 1.310˙ −5 Chapter 5

Study of the Decay Υ (2S) γη → b

5.1 Selection

If the Υ (2S) decays to γηb, then the photon from this decay is 614 MeV based on the ηb mass determined from the Υ (3S). This is somewhat softer than the 920 MeV photon from the decay Υ (3S) γη . Also, the e+e− Υ (nS) production cross section is → b → larger for n = 2 than n = 3, leading to a smaller fraction of photons from continuum decays in the Υ (2S) analysis. On account of these diﬀerences, small diﬀerences are required in the analysis. The selection criteria are similar to those from the analysis of the decay Υ (3S) γη , with reoptimized cos θ and π0 cuts. → b | T | An additional change with respect to the Υ (3S) analysis is the removal of the crystal edge correction. This is done to avoid a feature in the energy spectrum, described in appendix B.

The selection optimization in this analysis borrows heavily from that of the search for the decay Υ (3S) γη . Hadronic events are selected by requiring 4 charged → b ≥ tracks. Photons are selected from single EMC bumps that are not matched with any

98 5.1. SELECTION 99

track, have a minimum lab energy of 0.030 GeV and have a cluster lateral moment less than 0.55. These selections are identical to those used in the Υ (3S) analysis.

Table 5.1 shows the improvement in S/√B derived from MC after each of these cuts.

Table 5.1: Improvement of S/√B as the track and lateral moment selections are imposed. The overall normalization is arbitrary.

Cut S/√B No cut 38.5 Hadronic selection 42.4 Cluster lateral moment<0.55 43.6 cos θ < 0.8 48.5 | T | m m 0 < 15 MeV, E > 40 MeV 55.8 | γγ − π ,peak| γ2

The largest background to this analysis is due to π0 decays. To select π0 candidates a requirement of m m 0 < 15 MeV is made, where m 0 = 133.3 MeV is | γγ − π ,peak| π ,peak 0 the peak of the reconstructed π mass distribution. The mγγ spectrum does not peak at the PDG mass as photon energy is calibrated to peak at the correct value

rather than the π0 mass. The value 15 MeV, which is about 2.5 times the π0 mass resolution, is the optimized value of the π0 mass window from the Υ (3S) analysis.

The optimization of the cos θ and π0 cuts is done to maximize S/√B in the | T |

photon energy range of the signal, from 0.5 < Eγ < 0.7 GeV. The signal is taken from Monte Carlo simulation of signal events and B is taken from on-peak Υ (2S) data. As generic bottomonium decays may not be well simulated in MC, we use on-peak Υ (2S) data to evaluate the background component. As the expected Υ (2S) γη → b signal is less than 1% of the background, the possible presence of the signal does not bias the optimization. 100 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

We use a 2-D optimization in cos θ and E , the energy of the non-signal can- | T | γ2 didate photon in a π0 candidate. Figure 5.1 shows the result of the 2-D optimization.

The best S/√B comes from a requirement of cos θ < 0.8 and E > 30 or 40 MeV. | T | γ2

We conservatively use a requirement of Eγ2 > 40 MeV in this analysis. Figure 5.1 shows the number of signal MC and on-peak data events selected for various cos θ | T |

and Eγ2 cuts. The ratio of the Υ (nS) production cross section to the continuum cross section is higher on the Υ (2S) resonance than on the Υ (3S). Thus, the fraction of background which comes from continuum events is smaller on the Υ (2S). Since the cos θ cut | T | is intended primarily to reject photons from continuum events, this cut need not be

as strict for the Υ (2S) analysis. The total signal efficiency determined from truth-matched signal MC is 35.7%, consistent with the value derived from fitting the Eγ spectrum, as described in the next section, 35.8 0.2. Table 5.2 gives the efficiencies for each cut.

Table 5.2: Cut eﬃciencies for truth-matched signal MC determined by counting events in the energy range 0.3 < Eγ < 0.7 GeV.

Cut Relative Eﬀ. (MC) Total Eﬀ. (MC) Pass L3 trigger + truth matched 0.732 0.732 Preliminary hadronic selection 0.977 0.715 >3 charged tracks 0.994 0.712 Cluster lateral moment <0.55 0.979 0.696 0.762 < cos(θγ,LAB) < 0.890 0.910 0.632 − cos θ < 0.8 0.794 0.502 | T | m m 0 < 15 MeV, E > 40 MeV 0.711 0.357 | γγ − π ,peak| γ2 5.1. SELECTION 101

×103

70000S 2000 B

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35000 600 30000 20 40 60 80 100 120 140 160 20 40 60 80 100 120 140 160 Eγ 2 [MeV] Eγ 2

60 B θ |cos( T)|<0.55

S/ θ 58 |cos( T)|<0.60 θ |cos( T)|<0.65 θ |cos( T)|<0.70 56 θ |cos( T)|<0.75 θ |cos( T)|<0.80 θ 54 |cos( T)|<0.85

46 20 40 60 80 100 120 140 160 Eγ 2 [MeV]

Figure 5.1: Number of truth-matched signal photons (top left) and background photons (top right) passing all cuts as the cuts on cos θT and Eγ2 are varied. S/√B (bottom) is computed using these two quantities.| The |overall normalization is arbitrary. 102 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

5.2 Background Modeling

5.2.1 Introduction

There are three principal sources of background to this analysis:

Non-peaking background due to udsc, generic ISR production and non-peaking • bottomonium decays

Peaking background due to e+e− γ Υ (1S) • → ISR

Peaking background due to Υ (2S) γχ (1P ), χ (1P ) γΥ (1S) • → bJ bJ →

In addition, we investigated the presence of two other types of backgrounds that we have subsequently found to be less important:

Υ (2S) Υ (1S)(η, π0) • →

Υ (2S) Υ (1S)π0π0 • →

5.2.2 Non-peaking Background

The non-peaking background is parameterized using the following function:

2 3 4 A e−c1x−c2x −c3x −c4x where x = E µ, where µ is the center of the ﬁt region. The choice of this function γ − is motivated by the fact that the non-peaking background is slowly varying when plotted with a logarithmic y-axis, as shown in Fig. 5.2. 5.2. BACKGROUND MODELING 103

106 Entries / ( 0.005 GeV )

105

0.3 0.4 0.5 0.6 0.7 0.8 Eγ (GeV)

Figure 5.2: Spectrum plotted with logarithmic y-axis.

This function diﬀers from the smooth background function used in the Υ (3S) →

γηb analysis. Additional terms in the exponential are necessary because the background shape changes more rapidly at lower energy. With a few exceptions, the bottomonium photon spectrum is a featureless smooth function in the ﬁt region. We will now examine the few exceptions.

+ 5.2.3 e e− γISRΥ (1S) → The ISR photon in the process e+e− γ Υ (1S) peaks at an energy of 547.2 MeV, → ISR which is near the expected signal at 614 MeV. Since the ISR photon peaks near the signal, it is very important to model it correctly. The parameters are ﬁxed from the ﬁt to signal MC shown in Fig. 5.3. 104 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

600

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Figure 5.3: Fit of photon energy spectrum in ISR Υ (1S) MC at Υ (2S) energy using truth-matched (top) and all signal candidate photons (bottom). The blue curve is the Crystal Ball function used as the signal PDF and the red line is the polynomial background.

To evaluate the expected yield, we use the procedure described in the Υ (3S) γη → b analysis. The yield from the Υ (4S) off-resonance data sample is scaled by cross- section, luminosity, and efficiency to find the expected ISR yield at the Υ (2S). For

example, to obtain the expected ISR yield at the Υ (2S) resonance, Nexp.ISR,on−Υ (2S),

using the fitted number of ISR Υ (1S) events in the off-Υ (4S) dataset, NISR,off−Υ (4S), we use the formula

L σ N = on−Υ (2S) on−Υ (2S) on−Υ (2S) N exp,Υ (2S) L σ ISR,off−Υ (4S) off−Υ (4S) off−Υ (4S) off−Υ (4S) 5.2. BACKGROUND MODELING 105

where Υ (nS) is the ISR eﬃciency from MC, σ(on,oﬀ)−Υ (nS) are the ISR cross sections. We use the radiator functions from Ref. [66] to compute the cross section. At the peak of the Υ (2S) resonance the cross section for radiative return to the Υ (1S) is 41.3 pb.

The ISR efficiency, 6.58 0.20%, and PDF are derived from the fit to MC in Fig. 5.4. Figure 5.5 shows the fit to the off-resonance Υ (4S) data sample that is used in deriving the expected ISR yield. The yield is 41799 1865, resulting in an extrapolated yield of 16721 746 1204 . The systematic errors are due to the statistical error in the fits to MC to determine the efficiency, the error on the luminosity ratio, and the error on efficiency ratio. As the luminosity of the Υ (2S) datasets was not final at the time of this analysis, the ratio of the Υ (4S) off-resonance and Υ (2S) on-resonance datasets (excluding the optimization sample) is taken from the estimate provided by Chris Hearty [74], L /L = 0.2946 0.0031. on−Υ (2S) off−Υ (4S)

Table 5.4 shows that the difference in efficiency at off-Υ (4S) and on-Υ (2S) energies is dominated by the π0 veto. At off-Υ (4S) energies we determine the π0 veto efficiencies by fitting the off-Υ (4S) data before and after applying the π0 veto. Fitting the data without the π0 veto gives 49068 2609 events, compared to 41799 1865 after the veto. Adding the fit errors in quadrature gives an efficiency of (85.2 4.3)%, compared to an expected value of 84.8% from MC. Taking the difference between the values in quadrature with the statistical error gives an error of 4.3% or 1793 events. There is not enough statistics at Υ (2S) energies to make a similar comparison. So, the efficiency for the π0 cut at on-Υ (2S) energies is determined by comparing the π0

0 veto eﬃciency on the nearby χb(1P ) peak before and after applying the π veto. This is determined using the test sample, which is easier to ﬁt than the full sample. We 106 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

find 137369 2044 events before the π0 veto compared to 78683 2515 after, for an efficiency ratio of 0.571 0.019. The full difference with respect to the expected value is added in quadrature with the statistical error is 4.1%, which we take as the error.

Table 5.3: Systematic errors on extrapolation of oﬀ-Υ (4S) ISR rate to on-Υ (2S) data.

Source Systematic uncertainty (%) MC statistics 3.9 Luminosity ratio 1.1 π0 cut efficiency (off-Υ (4S)) 4.3 π0 cut efficiency (on-Υ (2S)) 4.1 Total 7.2

350

300

250

200

Events / ( 0.005 GeV ) 150

100

0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Eγ (GeV)

Figure 5.4: Fit of MC photon energy spectrum in ISR Υ (1S) signal region in MC generated at Υ (4S) oﬀ-resonance energy. The signal PDF is a Crystal Ball function and the background PDF is an exponential. 5.2. BACKGROUND MODELING 107

We cross-check the estimate of the expected ISR yield using Υ (2S) and Υ (3S) off- resonance data. First, consider the Υ (2S) off-resonance data. In the fit shown in Fig. 5.6 we find a yield of 1921 822, with a significance of only 1.9σ. The peak of the ISR PDF is allowed to float in the fit, and the fitted peak is at 514 7 MeV, consistent with the expected peak at 519 MeV. The yield after scaling to the luminosity of the full on-resonance data sample is 16448 7038, consistent with the extrapolation using Υ (4S) off-resonance data.

The Υ (3S) oﬀ-resonance data can be used for an estimate of the expected ISR yield.

The ISR efficiency, 6.16 0.12%, and PDF are derived from the fit to MC in Fig. 5.7. The yield from the fit to the Υ (3S) off-resonance data in Fig. 5.8 is 3028 527, resulting in an extrapolated yield of 19380 3373.

The results of the three estimates of the expected ISR yield are summarized in Ta- ble 5.5. Using the result from the Υ (4S) off-resonance sample, we expect 1423 63 and 16721 746 1204 events in the Υ (2S) test sample and in the final fit, respectively. 108 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

Table 5.4: Selection efficiencies for truth-matched e+e− γΥ (1S) signal MC → determined by counting events, at Υ (4S) off-peak and Υ (2S) on-resonance energies. The photons are counted in the energy ranges 0.4 < Eγ < 0.6 GeV and 0.85 < Eγ < 1.10 GeV for the Υ (2S) on-peak and Υ (4S) off-resonance energies, respectively.

Υ (2S) Υ (4S) Selection Relative Total Relative Total Eff. (MC) Eff. (MC) Eff. (MC) Eff. (MC) Trigger + truth-match 0.112 0.112 0.125 0.125 Preliminary hadronic selection 0.927 0.104 0.918 0.114 >3 charged tracks 0.994 0.103 0.996 0.114 Cluster lateral moment <0.55 0.974 0.100 0.992 0.113 0.762 < cos(θ ) < 0.890 0.867 0.087 0.865 0.098 − γ,LAB cos θT < 0.8 0.803 0.070 0.803 0.079 E| > 40| MeV, γ2 0.683 0.048 0.848 0.067 m m 0 < 15 MeV | γγ − π | 5.2. BACKGROUND MODELING 109

×103 1.2 500 1 400 0.8 300 0.6 Events / ( 0.01 GeV ) Events / ( 0.005 GeV ) 200 0.4

1000.2

00.70.8 0.750.85 0.8 0.90.85 0.950.9 0.951 11.05 1.051.1 1.1 1.151.15 1.2 Eγ (GeV)

5 4 3 2 01 -1 Residuals -2-3 -4-5 0.70.8 0.85 0.8 0.9 0.9 0.95 1 1.05 1.1 1.1 1.2 1.15 1.31.2 hard Entries 5020407 5000 Mean 0.9565 RMS 0.1109 4000 Underflow 0 Overflow 0 Entries/5 MeV 3000 Integral 3.878e+04

2000

1000

-1000

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 E_γ (GeV)

Figure 5.5: Fit of photon energy spectrum for ISR Υ (1S) production in off-resonance Υ (4S) data before (top) and after (bottom) subtracting the smooth background. The green (blue) curve in the top plot is the background (total) fit function. The normalized residuals from the fit are shown in the middle plot. 110 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

1.2 90000

800001 70000 600000.8 50000 0.6

Events / ( 0.01 GeV ) 40000 300000.4

Events / ( 0.00416667 GeV ) 20000 0.2 10000

00.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Eγ (GeV)

1 5 0.9 4 0.8 3 0.7 2 0.6 1 0.5 0

Residuals -1 0.4 Residuals -2 0.3 -3 0.2 -4-5 0.1 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1000 800 600 400 200 0

Entries/( 0.010 GeV ) -200 -400 -600 -800 -1000 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Eγ (GeV)

Figure 5.6: Fit of photon energy spectrum of ISR Υ (1S) production in off-resonance Υ (2S) data before (top) and after (bottom) subtracting the smooth background. The green (blue) curve in the top plot is the background (total) fit function. The normalized residuals from the fit are shown in the middle plot. 5.2. BACKGROUND MODELING 111

700

600

500

400

300

200 Events / ( 0.00428571 GeV ) 100

0 0.5 0.6 0.7 0.8 0.9 1 1.1 Eγ (GeV)

Figure 5.7: Fit of MC photon energy spectrum in ISR Υ (1S) signal region in Υ (3S) oﬀ-peak MC. The signal PDF is a Crystal Ball function and the background PDF is an exponential. 112 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

A RooPlot of "E_gamma" CAExpMC = -0.00065 = -5.04 +/- 0.0008 1400 MeanISR = 0.8260.8 +/- 0.005 45000 alphanBg = = 155000.0 5.8 +/- 0.2 400001200 betanISR == -1.32 35000.0 +/- 0.2 35000 nBg = 1045899 +/- 1151 1000 nISR = 3028 +/- 527 30000 Events / ( 0.01 GeV ) 800 25000

20000600

15000 400 10000 200 5000

0 0.5 0.6 0.7 0.8 0.9 1 1.1 E_gamma (GeV)

5 4 3 2 1 0 -1 -2 -3 Residuals -4 -50.5 0.6 0.7 0.8 0.9 1 1.1 hard Entries 1048790 600 Mean 0.6918 RMS 0.1544 400 Underflow 0 Overflow 0 Entries/5 MeV 200 Integral 3034

-200

-400

-600 0.5 0.6 0.7 0.8 0.9 1 1.1 E_γ (GeV)

Figure 5.8: Fit of photon energy spectrum for ISR Υ (1S) production in off-resonance Υ (3S) data before (top) and after (bottom) subtracting the smooth background. The green (blue) curve in the top plot is the background (total) fit function. The normalized residuals from the fit are shown in the middle plot. 5.2. BACKGROUND MODELING 113 is ) and MC S (1 eak Υ statistical the on-p samples, are to eak 746 to 7038 3373 on-p errors onding ) olation return S 16721 e The (2 16445 19380 off-resonance Υ corresp ) Extrap S (4 the radiativ Υ sample. 1865 822 527 for and from test Yield the 1921 3028 section 41799 cross determined [%] off-resonance 20 11 12 . . . ) excluding 0 0 0 the are S ), 1 (2 − Υ eak 58 37 16 . . . fb 6 4 6 Efficiency the Reconstruction off-p resonance ) (13.4 ) S from S (3 ts (2 Υ Υ en [pb] sample 20.2 43.7 25.4 ev and ) the ) S of S Cross-Section (1 (2 Υ ] R Υ eak S 1 p I − γ the 43.9 fb 1.479 2.415 Lumi on-resonance [ the → ) for − S at e (2 + e Υ that of ) ) ) the er S S S b to efficiencies (2 (3 (4 Off-resonance Υ sample Υ Υ Num calculate The e . olation 5.5: W pb . able T dataset. only 41.3 extrap 114 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

5.2.4 Υ (2S) γχbJ (1P ), χbJ (1P ) γΥ (1S) → → Photons from the second decay in the process Υ (2S) γχ (1P ), χ (1P ) γΥ (1S), J = → bJ bJ → 0, 1, 2 peak just below the signal. The photons from the J = 0, 1, 2 states have energies of 391, 423, and 442 MeV, respectively. Table 5.6 lists the branching fraction for decays of the Υ (2S) to these states from the PDG and Table 5.7 lists the expected branching fractions from a representative potential model calculation [49]. As the branching fractions are poorly determined, we cannot ﬁx the relative rate of the J = 1 and J = 2 states from the PDG, as was done in the Υ (3S) analysis. Instead, we plan to ﬁx the rates from a control sample of γγµµ events. This procedure is described in Section 5.3.

Table 5.6: Branching fractions for the decays Υ (2S) γχ (1P ), χ (1P ) γΥ (1S), → bJ bJ → and the CM energies of the photon in the ﬁrst and second decay. The values are taken from the PDG.

J E1[ MeV ] E2 [ MeV ] First transition BF Second transition BF Product BF 0 162 391 (3.8 0.4) 10−2 < 6 10−2 (90% CL) - × × 1 130 423 (6.9 0.4) 10−2 (35 8) 10−2 (2.4 0.6) 10−2 2 110 442 (7.15 0.35)× 10−2 (22 4) × 10−2 (1.6 0.3) × 10−2 × × ×

The width of the χbJ states are not measured but are expected to be much smaller than the detector resolution (< 1 MeV for the J = 1, 2 states, and a few MeV for the

J = 0 state) [49]. The widths of the χbJ states are expected to be smaller than that of the corresponding χ states due to the smaller α at the b quark mass scale. The cJ S − width of the χcJ states, taken from the PDG, are listed in Table 5.8.

There is a Doppler broadening of the second transition peak (the χbJ decay) due to the motion of the χbJ relative to the CM frame. As the Doppler eﬀect is 5.2. BACKGROUND MODELING 115

Table 5.7: Theoretical branching fractions for the decays Υ (2S) → γχbJ (1P ), χbJ (1P ) γΥ (1S), and the CM energies of the photon in the ﬁrst and second decay from →[49]. In deriving the branching fractions from the radiative width, we use a full width of the Υ (2S) Γ = 31.98 2.63 keV.

J E1[ MeV ] E2 [ MeV ] First transition BF Second transition BF Product BF 0 162 391 5.07 10−2 4.4 10−2 0.22 10−2 1 130 423 7.77 × 10−2 39.1× 10−2 3.04 × 10−2 × × × 2 110 442 7.77 10−2 27.0 10−2 2.10 10−2 × × ×

Table 5.8: Width of χcJ , J = 0, 1, 2 states from the PDG [79]. The widths of the χbJ are expected to be smaller than those of the χcJ . This provides the justiﬁcation for neglecting the χb width in the ﬁt.

State Width [ MeV ]

χc0 10.2 0.70 χ 0.89 0.05 c1 χ 2.03 0.12 c2

non-negligible compared to the EMC energy resolution, we use rectangular functions convoluted with a Crystal Ball function to parameterize the PDF. The half-width of

the rectangular PDFs are listed in Table 5.9. In addition to the low side tail to the photon energy distribution due to electromagnetic shower leakage, there may be a high-side tail due to the overlap of photons from the rest of the event with the signal photon. Figure 5.9 shows a comparison of

1 χb2 signal MC with and without a bump distance cut of 25 cm. The plot demon-

strates that the high-side tail of the χb distribution is largely due to overlap with other photon candidates. We do not use the bump distance cut in the analysis, as many of the overlapping candidates are removed by the cluster lateral moment cut.

1Bumps are a way of dealing with overlapping energy distributions in the calorimeter. Bumps 116 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

A small Gaussian component is added as a means of parameterizing this high-side tail of the photon energy distribution, which is otherwise not modeled by the Crystal

Ball function. Since the ηb decays are not known, it is impossible to know the charged track and neutral cluster topology in ηb decays, and we will treat the high-side χb tail as a systematic eﬀect.

1.5

1.4

1.3

1.2 before/after bump isolation cut γ 1.1

Ratio of E 1 0.3 0.35 0.4 0.45 0.5 0.55 0.6 CM Eγ

Figure 5.9: Ratio of histograms of photon energy spectrum of χb2 decays before/after a 25 cm bump distance cut (black). The red curve is the χb2 photon spectrum. Plot courtesy of Steve Sekula.

As the χbJ Crystal Ball parameters will be fixed to be equal in the data fit, we perform a simultaneous fit to all three χb MC samples. Figure 5.10 shows fits to are in one-to-one correspondence with local maxima of energy deposition in a cluster. A cluster with a single local maximum of energy deposition is simply a bump. For clusters with multiple local maxima, the cluster is split into as many regions as there are local maxima. Weights are assigned to each crystal based on the distance to each local maximum in order to determine what fraction of its energy should be assigned to each local maximum. 5.2. BACKGROUND MODELING 117

Table 5.9: Doppler broadening due to momentum of χbJ relative to the CM frame. The value listed is the half-width of the Doppler broadening PDF.

State Broadening [ MeV ]

χb0 6.5 χb1 5.5 χb2 4.9

χbJ MC and Table 5.10 summarizes the resultant fitted PDF parameters. The fitted signal yields correspond to efficiencies of 0.257 0.001, 0.267 0.002, and 0.278 0.001 for J equal to 0, 1, and 2, respectively. Table 5.11 shows the efficiency of each cut on the “hard” transition of the different χbJ (1P ) peaks.

5.2.5 Υ (2S) Υ (1S)(η, π0) → In the decays Υ (2S) Υ (1S)(η, π0), the pseudoscalar is produced nearly at rest in the → Υ (2S) frame. As a result, the photons in the decay η γγ are nearly monochromatic, → with a Doppler broadening due to the momentum of the (π0, η) in the Υ (2S) frame. Though the peaks are not near the signal peak, they may need to be taken into account in order to obtain a reasonable background parameterization.

In the η transition, the η is produced with a momentum of p = 127 MeV in the Υ (2S) frame. The branching fraction of the decay Υ (2S) Υ (1S)η was measured → recently by CLEO to be (2.1+0.7 0.5) 10−4 [75]. The eﬃciency for this mode after −0.6 × all cuts is 0.106 0.034, leading to an expectation of 2000 800 events in 13.4 fb−1. Figure 5.11 shows the photon energy distribution from truth-matched photons, where

the photon is the daughter of the η and the η is the daughter of the Υ (2S).

The π0 decay mode is not yet measured, with a 90% CL upper limit of 3.7 10−4 × 118 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

Table 5.10: Result of simultaneous ﬁt to χbJ , J = 0, 1, 2 MC. The peak position, Doppler width and signal yield are labeled by their total angular momentum, J. The remaining parameters are shared between the three peaks. They are A, the transition point between the Gaussian and power tail components of the Crystal Ball function, in units of σ from the mean of the peak, and N, the power law parameter. Note that there are 96,000 events in the J = 1 MC sample compared to 145,000 events in each of the J = 0, 2 data samples which must be taken into account in computing the eﬃciency.

Fit parameter Fitted value Signal yield (J = 0) 37216 194 Signal yield (J = 1) 25647 162 Signal yield (J = 2) 40382 202 Peak position (J = 0) 0.3895 0.0001 GeV Peak position (J = 1) 0.4207 0.0001 GeV Peak position (J = 2) 0.4396 0.0001 GeV Crystal Ball A 0.586 0.007 Crystal Ball N 60 12 Crystal Ball σ (J = 0) 0.01004 0.00009 GeV Crystal Ball σ (J = 1) 0.0106 0.0001 GeV Crystal Ball σ (J = 2) 0.01086 0.00009 GeV Doppler half-width (J = 0) 0.0065 GeV Doppler half-width (J = 1) 0.0055 GeV Doppler half-width (J = 2) 0.0049 GeV Gaussian fraction 0.038 0.002 Wide Gaussian width 0.091 0.007 GeV

set by the CLEO experiment [75]. However, the π0 has a momentum of 531 MeV in

the Υ (2S) frame, considerably increasing the Doppler broadening of the peak. The eﬃciency for this mode after all cuts is 0.107 0.032 leading to 3500 events in the full data sample, assuming a branching fraction equal to the CLEO upper limit.

These backgrounds are negligible due to the huge non-peaking background; we

mention them here only for completeness. 5.2. BACKGROUND MODELING 119

Efficiency Efficiency Efficiency Cut (J = 0) (J = 1) (J = 2) Reconstruction + truth-match 0.772 0.764 0.776 Preliminary hadronic selection 0.927 0.927 0.927 > 3 charged tracks 0.993 0.993 0.993 Cluster lateral moment < 0.55 0.961 0.963 0.963 In calorimeter barrel 0.929 0.923 0.929 cos θ < 0.8 0.787 0.789 0.787 | T | E > 40 MeV, m m 0 < 15 MeV 0.523 0.551 0.567 γ2 | γγ − π |

Table 5.11: Cut eﬃciencies determined by counting events in χbJ (1P ) MC.

5.2.6 Υ (2S) Υ (1S)π0π0 → Photons from the decay Υ (2S) Υ (1S)π0π0 are not present in the signal region but →

constitute a signiﬁcant source of background in the region just below the χb peaks. The branching fraction for this decay is 9.0 0.8%, and the detection eﬃciency is 2.6 0.2%, leading to approximately 2.3 105 photons in the signal region E > × γ,CM 0.27 GeV (approximately 5.5% of the total signal yield). The MC for this sample is generated using amplitudes determined by CLEO in

their analysis of Υ (2S) ππΥ (1S) decays [76]. The distribution of truth-matched → signal photons is shown in Fig. 5.11 Since this background has a smooth, nonpeaking shape, this background component can be absorbed into the generic non-peaking background PDF. 120 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

900 800 700 600 500 400 Events / ( 0.001 GeV ) 300 200 100 0 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 Eγ (GeV)

600

500

400

300 Events / ( 0.001 GeV ) 200

100

0 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 Eγ (GeV)

900 800 700 600 500 400 Events / ( 0.001 GeV ) 300 200 100 0 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 Eγ (GeV)

Figure 5.10: Fit of zero-width χb0 (top), χb1 (middle), and χb2 (bottom) MC using truth-matched photons. 5.2. BACKGROUND MODELING 121

200 180 160 140

Entries/4 MeV 120 100 80 60 40 20 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Eγ (GeV)

45 40 35 30 Entries/4 MeV 25 20 15 10 5 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Eγ (GeV)

Figure 5.11: Spectrum of truth-matched photons from the decays Υ (2S) ηΥ (1S)(top) and Υ (2S) Υ (1S)π0π0 (bottom). Note that the x-axis is differen→t → from that of the final fit, which is over the region 270 < Eγ < 800 MeV. 122 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

5.3 Control Sample Studies

In contrast with the Υ (3S) γη analysis, the branching fractions to and from the χ → b bJ

states (here the χbJ (1P ) rather than the χbJ (2P ) transitions) are not well-measured. Hence, we need to measure their relative rates to fix in the fit to the inclusive photon spectrum. To determine the relative yields of the χbJ (1P ) peaks, we look at a control sample of Υ (2S) γχ (1P ), χ (1P ) γΥ (1S) events. Due to better absolute → bJ bJ → photon energy resolution at lower photon energies, the three overlapping peaks corresponding to the second transition in the decay (the “soft transitions”) are easier to separate than the first transition (the “hard transitions”). Therefore, we fit only the

soft transitions.

5.3.1 Selection

We select Υ (1S) candidates through their decay to µ+µ− and a vertex ﬁt is performed, including an Υ (1S) mass constraint. Pairs of photons are added to these candidates to form an Υ (3S) candidate.

Due to the loose selection criteria used, multiple Υ (2S) candidates are found in many events. To eliminate multiple candidates, we deﬁne candidate photons accord-

ing to Figure 5.12, with γsoft required to be between 60 and 200 MeV, and γhard between 380 and 470 MeV. We require exactly one photon in each energy region. To reduce photons from beam background, photons falling in the calorimeter endcap are rejected by the requirement cos(θ) < 0.89. 5.3. CONTROL SAMPLE STUDIES 123

45000

40000

35000

30000

25000

20000

15000

10000

5000

0.050 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Photon Energy (GeV)

Figure 5.12: Photon spectrum before cuts for double radiative decay of Υ (2S) candidates. γ window (red): 60 200 MeV, γ window (blue): 380 470 MeV. soft − hard − 5.3.2 Monte Carlo

We generated samples of 87,000, 110,000 and 110,000 events of Υ (2S) γχ , χ → bJ bJ → γΥ (1S), Υ (1S) µµ, for J=0,1,2, respectively. The signal peaks are fit with a sum of → a Crystal Ball and a Gaussian function, and the background is fit with an exponential function. These fits are shown in Figures 5.13, 5.14, and 5.15 and fit parameters are

given in Table 5.12. Using the signal yields from the fits above along with the number of Monte Carlo events analyzed for each peak, we find the Monte Carlo efficiencies = 0.286 0.003, χb0 = 0.246 0.003 and = 0.243 0.003. χb1 χb2 124 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

1400103

1200

2 100010

800 10

Events / ( 0.001 GeV ) 600

400 1 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 200 E (GeV)

4 02 0.090 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 -2

Residuals -4 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19

Figure 5.13: Monte Carlo χb0 peak with ﬁt superimposed. The curves displayed on the plot are the exponential background (short dotted light blue line), the signal Gaussian (dot-dash green line), the signal Crystal Ball function (red dashed line) and their sum (solid blue line). χ2 per degree of freedom = 193/91

Table 5.12: Parameters resulting from ﬁts to χbJ MC.

Parameter χb0 χb1 χb2 Crystal Ball A parameter 0.33 0.04 0.41 0.03 0.30 0.03 Crystal Ball tail parameter 10 5 3.3 0.5 6 2 Exponential parameter 7.2 2 7.9 3 12.3 2 − − − Gaussian fraction 0.28 0.03 0.30 0.02 0.39 0.02 Background yield 1748 227 1840 316 2862 290 Signal yield 24922 272 27100 353 26678 329 Energy scale oﬀset (MeV) 0.87 0.09 0.14 0.08 0.48 0.07 Crystal Ball σ parameter (MeV) 4.0 0.1 3.2 0.1 2.59 0.10 Gaussian σ parameter (MeV) 7.3 0.4 6.8 0.2 5.7 0.2 5.3. CONTROL SAMPLE STUDIES 125

103 1600 1400 102 1200 1000 80010 Events / ( 0.001 GeV ) 600

4001 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 200 E (GeV)

4 02 0.060 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 -2

Residuals -4 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15

Figure 5.14: Monte Carlo χb1 peak with ﬁt and normalized residual plot. The curves are as in Fig. 5.13. χ2 per degree of freedom = 94/81 126 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

2000 103 1800 1600 1400102 1200 1000 10 Events / ( 0.001 GeV ) 800 600

4000.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 E (GeV) 200 4 02 0.060 0.07 0.08 0.09 0.1 0.11 0.12 0.13 -2

Residuals -4 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13

Figure 5.15: Monte Carlo χb2 peak with ﬁt and normalized residuals plot. The curves are as in Fig. 5.13. χ2 per degree of freedom = 42/61

Data

The parameters from the fits to signal MC are used to specify the PDF properties for the peaks in the data spectrum. An energy scale offset is free in the fit to account for the limited precision of the EMC energy calibration and the uncertainty in the

overall χb mass scale; the peaks share a common energy scale oﬀset. To account for discrepancies in the modeling of calorimeter energy resolution, we allow some of the

resolution parameters to vary in the ﬁt. The χb0 peak is not signiﬁcant enough to 5.3. CONTROL SAMPLE STUDIES 127

allow the ﬁtter to determine the widths of the ﬁtting functions, but for the other two peaks the Gaussian σ parameter of the Crystal Ball and Gaussian PDFs are allowed

to float. Also left free in the fits are the exponential parameter for the background function and all of the yields. This fit is shown in Fig.5.16, with fitted parameters in Table 5.13. Given the PDG values for the branching fractions for the Υ (2S) γχ (1P ), χ → bJ bJ → γΥ (1S), Υ (1S) µµ processes, the Monte Carlo efficiencies predict less than 1600 → events for χ and 14600 3460 and 9390 1780 events for χ (1P ) and χ (1P ) b0 b1 b2 respectively in the data sample. The results are consistent with these expectations. The quantities of interest for the fit to the inclusive spectrum, the efficiency-corrected

relative yields for the three χbJ peaks, are as follows:

N / / N / = (8 2) 10−3, sig,χb0 χb0(1P ) sig,χb1 χb1(1P ) × N / / N / = 0.64 0.02. (5.1) sig,χb2 χb2(1P ) sig,χb1 χb1(1P ) 128 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

900900 alpha = -19.03 +/- 0.4 frac = 0.69 +/- 0.03 800 800 nBkg = 11324 +/- 199 700 700 nSig0 = 126 +/- 42 600 nSig1 = 12923 +/- 278 nSig2 = 8144 +/- 231 600500 offset = 0.0003 +/- 0.0001 GeV 500400 sigmaG1 = 0.0070 +/- 0.0002 GeV sigmaG2 = 0.0072 +/- 0.0003 GeV 300 400300 Events / ( 0.001 GeV ) Events / ( 0.001 GeV ) 200 300 100

0 2000.06 0.08 0.1 0.120.12 0.14 0.160.16 0.180.18 0.20.2 100 EE (GeV)(GeV)

900 4800 700 02600 0.06500 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0400 300 -2200 100 Residuals -40 0.060.06 0.080.08 0.10.1 0.120.12 0.140.14 0.160.16 0.180.18 0.20.2

Figure 5.16: Fit to soft transitions in data with normalized residuals plot. The exponential background function is shown as a short dotted light blue line, each Gaussian is a dot-dash green line, the red dashed lines are the Crystal Ball functions and the solid blue line is the total ﬁt. χ2/ndof = 123/131.

Table 5.13: γsoft spectrum fitted parameters, final fit.

Parameter Value Background exponential parameter 19.03 0.4 Gaussian fraction −0.69 0.03 Background yield 11324 199 χ (1P ) signal yield 126 42 b0 χb1(1P ) signal yield 12923 278 χ (1P ) signal yield 8144 231 b2 Energy scale oﬀset 0.3 0.1 MeV χ (1P ) Gaussian resolution 7.0 0.2 MeV b1 χ (1P ) Gaussian resolution 7.2 0.3 MeV b2 5.4. FIT PROCEDURE FOR Υ (2S) γη SAMPLE 129 → B

5.4 Fit Procedure for Υ (2S) γη Sample → b

5.4.1 Introduction

2 We extract the ηb yield using a binned χ ﬁt to the photon energy spectrum in the

range 0.27 < Eγ < 0.80 GeV. The low end of the ﬁt range is chosen to avoid the rapid change of the background shape below 250 MeV, and the photons from the soft transitions at 110-166 MeV.

A χ2 fit is used rather than a likelihood fit as round-off error prevents the likelihood fit from converging. The fitting software, MINUIT, terminates when it believes round- off error limits further improvement. The condition it uses is

EDM < √machinef

where EDM is the estimated distance to the minimum, 6 10−17 and machine ≈ × f is the function to be minimized. Since the likelihood is of order 6 108, the × minimization terminates once the EDM is less than four. This is insuﬃcient to ensure good convergence.

We have tested the fitting procedure on the test sample and the full sample prior to unblinding. The reader should be aware that there is a small difference between the fits in this section and the final fit in Section 5.6. After unblinding the data sample,

we observed a feature near 680 MeV in the subtracted spectrum which appeared unphysical (see Appendix B for a full description) . We determined its origin and after and removed its source, causing a slight change in the ﬁnal results. However, the ﬁts in this section are left as is to accurately document the studies that were done

prior to unblinding. 130 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

5.4.2 Fixed and ﬂoating parameters

Background parameters •

– Yield, ﬂoating

– c1, c2, c3, and c4, all ﬂoating

χ parameters • b

– Yield, ﬂoating

– χb1 and χb2 Crystal Ball σ parameters, ﬂoating

– χb0 Crystal Ball parameter ﬁxed to σ parameter of χb2. Will vary to study systematics

– A, the Crystal Ball transition point, common to all three peaks, ﬂoating

– N, the Crystal Ball power law parameter, ﬁxed from MC.

– An energy scale oﬀset, common to all peaks, ﬂoating

– Ratio of χbJ yields ﬁxed from γγµµ control sample

ISR parameters •

– Yield ﬂoating

– All lineshape parameters ﬁxed from MC.

– Mean ﬁxed to expected value based on PDG values of Υ (2S) and Υ (1S)

masses, 547.2 MeV, minus an energy scale oﬀset shared with χbJ (1P ) peak.

η parameters • b

– Yield ﬂoating 5.4. FIT PROCEDURE FOR Υ (2S) γη SAMPLE 131 → B

– All Crystal Ball parameters ﬁxed from MC

– Width ﬁxed to average of theoretical values, at 10 MeV. Will vary to study systematics.

– ηb signal peak allowed to vary in the range 580 < Eγ,CM < 650 MeV.

5.4.3 Fit to test sample, ISR yield ﬁxed

As a test of the fitting procedure described in Section 5.4.2 and the background shape, we fit the test sample used in the optimization. Figure 5.17 shows the fit to the test sample and Fig. 5.18 shows the corresponding non-peaking background-subtracted spectrum. The fit parameters are summarized in Table 5.14. 132 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

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Figure 5.17: Fit to the test sample (top) and residuals from the fit (bottom). The ISR yield is fixed in the fit. 5.4. FIT PROCEDURE FOR Υ (2S) γη SAMPLE 133 → B

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Figure 5.18: Fit to the test sample after subtracting the non-peaking background (top) and the corresponding zoomed version (bottom). The χ2/ndof is equal to 93.8/94 = 0.98, corresponding to a fit probability of 49%. The histograms are χb (cyan), ISR (red), and ηb (blue). The ISR yield is fixed in the fit. 134 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

Table 5.14: Parameters from fit to test sample with the ISR yield fixed. The sections of the table are yields, the ηb peak position, background parameters, and χb lineshape parameters. The ISR yield is fixed in the fit.

Fit parameter Value Non-peaking background yield 324422 3000 χ yield 78473 2333 b ISR yield 1423 η yield 2045 875 b η mean 0.616 0.007 GeV b c background parameter 2.98 0.08 1 c2 background parameter 2.6 0.6 c background parameter 2.9 1.6 3 − c background parameter 0.2 1.5 4 χ energy scale oﬀset 1.4 0.8 MeV b − χb CB transition parameter 0.65 0.07 χ CB σ parameter 0.0122 0.0007 GeV b1 χ CB σ parameter 0.0131 0.0008 GeV b2

5.4.4 Fit to test sample, ISR yield ﬂoating

As a test of the ﬁtting procedure and the background shape, we ﬁt the test sample

used in the optimization. If the background shape in the region of the ηb is good, the fitted ISR yield will be consistent with expectations. Figure 5.19 shows the fit to the test sample and Fig. 5.20 shows the corresponding non-peaking background- subtracted spectrum. The fit parameters are summarized in Table 5.15.

The η signal yield in the test sample is 2143 958, with a statistical signiﬁcance b of 2.2σ. The branching fraction derived from this yield is (including only statistical 5.4. FIT PROCEDURE FOR Υ (2S) γη SAMPLE 135 → B

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Figure 5.19: Fit to the test sample (top) and normalized residuals from the fit (bottom). The ISR yield is floating in the fit.

errors) 2143 958 B = N /(N ) = = (8.6 3.8) 10−4, sig Υ (2S) 6.95 106 0.358 × × · 136 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B where we have used the number of Υ (2S) events, 6.95 0.07 million Υ (2S), in the optimization sample. 5.4. FIT PROCEDURE FOR Υ (2S) γη SAMPLE 137 → B

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Figure 5.20: Fit to the test sample after subtracting the non-peaking background (top) and the corresponding zoomed version (bottom). The χ2/ndof is equal to 93.7/94 = 0.98, corresponding to a ﬁt probability of 49%. The histograms are χb (cyan), ISR (red), and ηb (blue). 138 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

Table 5.15: Parameters from ﬁt to optimization sample with the ISR yield uncon- strained. The sections of the table are yields, the ηb peak position, background parameters, and χb lineshape parameters.

Fit parameter Value Non-peaking background yield 3243630 3830 χ yield 78683 2516 b ISR yield 1701 1116 η yield 2143 958 b η peak 0.616 0.007 GeV b c background parameter 2.98 0.09 1 c2 background parameter 4.7 0.7 c background parameter 3.1 2.0 3 − c background parameter 0.4 1.7 4 χ energy scale oﬀset 1.4 0.8 MeV b − χb CB transition parameter 0.65 0.07 χ CB σ parameter 0.0122 0.0008 GeV b1 χ CB σ parameter 0.0132 0.0009 GeV b2

We have several indications of excellent ﬁt quality: the χ2/ndof of the ﬁt, the

2 offset of the χb peak, and the ISR yield. The χ /ndof is equal to 93.7/103 = 0.94, corresponding to a fit probability of 52%. The offset of the χb peaks is about an MeV. This is well within the accuracy of the EMC calibrations and comparable to the error due from the PDG values of the χb masses, 0.5 MeV. If there were serious problems with the fit of the χb peaks, the offset could be significantly larger. Finally, the ISR yield of 1701 1116 is consistent with the expected ISR yield of 1423 63. 5.4. FIT PROCEDURE FOR Υ (2S) γη SAMPLE 139 → B

5.4.5 Fit to full sample with signal region blinded, ISR yield

ﬁxed

Next, we fit the full sample, with the ISR yield fixed. The signal yield is blinded using the RooUnblindPrecision class of RooFit. We blind the histograms of the data and fitted background before performing the background subtraction. Figure 5.21 shows the result of the blinded fit and the corresponding residuals. The signal is much too

small to be seen before background subtraction so it is not necessary to blind the fit before background subtraction. The background subtracted plots are shown in fig. 5.22. The χ2 is 116.5 for 94 degrees of freedom, corresponding to a fit probability of 0.058.

5.4.6 Fit to full sample with signal region blinded, ISR yield

ﬂoating

This is the nominal ﬁt. To test the quality of the background parameterization in the

ηb region, we perform another fit to the full sample with the ISR yield floating. If the fitted ISR yield is consistent with expectations, we can conclude that the background parameterization is good in the ηb region. Figure 5.23 shows the result of the blinded fit and the corresponding residuals. As earlier, in Section 5.4.5, the signal is much too small to be seen before background subtraction so it is not necessary to blind the fit before background subtraction. The background subtracted plots are shown in 140 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

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Figure 5.21: Fit to the full data sample (top) and residuals from the fit (bottom). The ISR yield is fixed in the fit. 5.4. FIT PROCEDURE FOR Υ (2S) γη SAMPLE 141 → B

Table 5.16: Parameters from ﬁt to full sample with signal yield blinded. The sections of the table are yields, the ηb peak position, background parameters, and χb lineshape parameters. The ISR yield is ﬁxed to 17324, a value determined from an earlier study of the ISR yield.

Fit parameter Value Non-peaking background yield 40554500 10600 χb yield 945941 8106 ISR yield 17324 ηb yield blind ηb peak blind c1 background parameter 2.52 0.03 c background parameter 3.7 0.2 2 c3 background parameter 4.8 0.5 c background parameter −1.5 0.5 4 χb energy scale oﬀset 1.6 0.2 MeV χ CB transition point − 0.71 0.02 b χb1 CB σ parameter 0.0126 0.0002 GeV χ CB σ parameter 0.0138 0.0002 GeV b2

Fig. 5.24. The χ2 is 116.2 for 93 degrees of freedom, corresponding to a ﬁt probability of 0.052. 142 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

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Figure 5.22: Fit to full data sample after background subtraction (top) and the corresponding zoomed plot (bottom). The cyan curve is the sum of the χb peaks, and the red curve is the ISR peak. The histograms are χb (cyan) and ISR (red). The ISR yield is ﬁxed in the ﬁt. 5.4. FIT PROCEDURE FOR Υ (2S) γη SAMPLE 143 → B

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Figure 5.23: Fit to the full data sample (top) and residuals from the ﬁt (bottom). 144 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

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Figure 5.24: Fit to full data sample after background subtraction (top) and the corresponding zoomed plot (bottom). The ηb signal region is excluded in the residual plot. The cyan curve is the sum of the χb peaks, and the red curve is the ISR PDF. The histograms are χb (cyan) and ISR (red). The ISR yield is ﬂoating in the ﬁt. 5.4. FIT PROCEDURE FOR Υ (2S) γη SAMPLE 145 → B

Table 5.17: Fit parameters from ﬁt to full sample with signal yield blinded and ﬂoating ISR yield. The sections of the table are yields, the ηb peak position, background parameters, and χb lineshape parameters.

Fit parameter Value Non-peaking background yield 40558500 13429 χ yield 944609 8530 b ISR yield 15260 4106 ηb yield blind ηb mean blind c background parameter 2.53 0.03 1 c2 background parameter 3.7 0.2 c background parameter 4.6 0.7 3 − c background parameter 1.4 0.6 4 χ energy scale oﬀset 1.6 0.2 MeV b − χb CB transition point 0.71 0.02 χ CB σ parameter 0.0126 0.0002 GeV b1 χ CB σ parameter 0.0138 0.0003 GeV b2 146 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

5.5 Toy studies

5.5.1 Introduction

We generate toys with N = 1.04 106, the yield scaled from the test sample, and χb ×

NISR = 21000. The background PDF is taken from the fit to the full data sample. The background PDF is generated by first generating a histogram according to the expected background distribution, and then Poisson-fluctuating each bin. Generating the necessary 80 million events according to the PDF would take a prohibitive amount of time. Other PDFs are sampled using the toy generator in RooFit.

We generate toys with all combinations of Nηb = 10000, 20000, 30000 and widths of 5 and 10 MeV.

Tables 5.18 and 5.19 summarize the results for the toys generated with an ηb width of 5 MeV and Tables 5.20 and 5.21 summarize the results for the toys generated with

an ηb width of 10 MeV. The results are summarized in terms of a quantity called the “pull”, defined as N N Pull = fit − gen (5.2) σfit

where Nﬁt is the number of ﬁtted ηb events, Ngen is the number of generated events

and σfit is the error on the number of fitted ηb events. In the absence of fit bias, the pull has a Gaussian distribution with unit variance.

The fits behave reasonably. The convergence rate after MINOS is approximately 100%, compared to about 80% in the Υ (3S) fit, the difference being attributed to the improved numerical stability due to using a χ2 fit rather than a likelihood fit. We conclude from this study that there is no statistically significant bias in the signal yield or peak position for an assumed ηb width of 10 MeV. 5.5. TOY STUDIES 147

Table 5.18: Signal yields, yield errors and pulls from toy studies with ηb width ﬁxed at 5 MeV. We show both the mean (µ) and width (σ) from a Gaussian ﬁt to the pull distribution.

Yield (generated) Fitted yield Fitted error Pull (µ) Pull (σ) 10000 10720 2845 0.265 0.070 0.990 0.070 20000 20440 2838 0.172 0.099 0.989 0.070 30000 29820 2841 0.060 0.130 0.907 0.090 −

Table 5.19: Signal peaks, peak errors and pulls from toy studies with ηb width ﬁxed at 5 MeV. We show both the mean (µ) and width (σ) from a Gaussian ﬁt to the pull distributions.

Yield Fitted peak Fitted error Pull Pull (generated) (MeV) (MeV) (µ) (σ) 10000 616.1 5.27 0.002 0.100 0.973 0.070 20000 615.9 4.14 −0.026 0.093 0.934 0.066 30000 615.9 1.78 −0.070 0.130 0.897 0.090 −

Table 5.20: Signal yields, yield errors and pulls from toy studies with ηb width ﬁxed at 10 MeV. We show both the mean (µ) and width (σ) from a Gaussian ﬁt to the pull distribution.

Yield (generated) Fitted yield Fitted error Pull (µ) Pull (σ) 10000 9946 3237 0.002 0.117 0.817 0.083 − 20000 19660 3229 0.064 0.123 0.868 0.087 30000 29610 3255 −0.072 0.123 0.867 0.087 −

5.5.2 Toys with ISR yield ﬁxed to incorrect yield

To estimate the expected systematic error due to possibly fixing the ISR yield to an incorrect value, we generate toys with 17,000 ISR events, and fit the toys with fixed

ISR yields of 14,000 and 20,000. These toys are performed with an ηb yield of 10,000

events and an ηb width of 10 MeV. We generate 500 toy experiments in each case. 148 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

Table 5.21: Signal peaks, peak errors and pulls from toy studies with ηb width ﬁxed at 10 MeV. We show both the mean (µ) and width (σ) from a Gaussian ﬁt to the pull distributions.

Yield Fitted peak Fitted error Pull Pull (generated) (MeV) (MeV) (µ) (σ) 10000 616.4 6.82 0.013 0.136 0.952 0.096 20000 616.0 3.27 0.026 0.121 0.856 0.086 30000 616.0 2.48 −0.022 0.132 0.932 0.093

The pulls on the yield are 0.121 0.045 and 0.502 0.046 for the toys with the − ISR yield fixed to 14,000 and 20,000, respectively. The pulls on signal peak position are 0.067 0.051 and 0.089 0.051 for the toys with the ISR yield fixed to 14,000 − and 21,000, respectively. We conclude that fixing the ISR yield to an incorrect value

would generate a statistically signiﬁcant bias in the ηb yield.

5.5.3 Conclusion

The fit strategy is validated using the guidance of the toy studies described above. In particular, a fit bias would be generated by allowing the ISR yield to vary in the fit. There is no significant fit bias observed in our nominal fit. 5.6. FIT RESULTS 149

5.6 Fit results

This section describes the nominal ﬁt to the Υ (2S) data sample. This ﬁt uses the PDF

A exp c (E µ) + c (E µ)2 + c (E µ)3 + c (E µ)4 1 γ − 2 γ − 3 γ − 4 γ − where A, c , i = 1 4 are determined in the ﬁt and µ = 535 MeV is the center of the i − ﬁt region.

Figure 5.25 shows the fit and the residuals from the fit, and figure 5.26 shows the smooth background subtracted distributions. Table 5.22 lists the fit parameters from the nominal fit. The χ2/ndof of this fit is 115.1/93, corresponding to a fit probability of 6.0%.

2 Table 5.23 lists the signal yields and χ values for ﬁts with diﬀerent assumed ηb widths.

+3541+2362 The fitted ηb signal yield is 12806−3473−3094 events, where the systematic error is described in the following section. This yield corresponds to a branching fraction of (3.9+1.1+0.7) 10−4. The statistical significance of the peak is estimated as −1.1−0.9 × χ2(no signal) χ2(fixed mass), where χ2(fixed mass) is the χ2 of the fit with the − 2 2 ηpb signal included and χ (no signal) is the χ of the fit with the ηb PDF removed. The statistical significance estimated in this way is 3.7 standard deviations. The fit with the ηb peak excluded is shown in fig. 5.27.

After correcting the ηb peak position with the energy scale offset from the χb, we find a peak position of 609.3+4.6 1.9 MeV, including the systematic errors from −4.5 the next section. This corresponds to an η mass of 9394.2+4.8 2.0 MeV/c2 and b −4.9 150 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B a Υ (1S) η hyperfine splitting of 66.1+4.9 2.0 MeV/c2. This hyperfine splitting − b −4.8 is in excellent agreement with the value obtained from the Υ (3S) γη analysis, → b 71.4+2.3 2.7 MeV/c2. −3.1

Table 5.22: Fit parameters from ﬁt to full sample with ﬂoating ISR yield. The sections of the table are yields, the ηb peak position, background parameters, and χb lineshape parameters.

Fit parameter Value Non-peaking background yield 40545600 13200 +8442 χb yield 947177−8329 +4153 ISR yield 16762−4038 +3541 ηb yield 12806−3473 +4.6 ηb mean 607.9−4.5 MeV c1 background parameter 2.52 0.01 c background parameter 3.82 0.03 2 c3 background parameter 4.94 0.03 c background parameter −1.65 0.11 4 χb energy scale oﬀset 1.4 0.2 MeV χ CB transition point − 0.71 0.02 b χb1 CB σ parameter 12.6 0.2 MeV χ CB σ parameter 13.9 0.3 MeV b2 5.6. FIT RESULTS 151

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Figure 5.25: Fit to the full data sample (top) and residuals from the ﬁt (bottom). 152 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

2 Table 5.23: Signal yields and χ values for ﬁts with alternate ηb widths.

2 ηb width (MeV) Signal yield χ +2875 0 10987−2826 113.6 +3090 5 11630−3072 114.2 +3541 10 12806−3473 115.1 +3894 15 13728−3858 116.0 +4229 20 14541−4247 116.8 +4335 25 15308−4656 117.6 +4695 30 16052−5104 118.4 5.6. FIT RESULTS 153

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Figure 5.26: Fit to full data sample after smooth background subtraction (top), the corresponding zoomed plot (middle) and after all backgrounds are subtracted (bottom). The cyan curve is the sum of the χb peaks, and the red curve is the ISR PDF. The histograms are χb (cyan), ISR (red), ηb (blue), and their sum (black). The ISR yield is ﬂoating in the ﬁt. 154 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

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Figure 5.27: Same as ﬁg. 5.26 but with the ηb PDF removed. 5.7. SYSTEMATIC ERRORS 155

5.7 Systematic Errors

5.7.1 Sources of Systematic Errors

We evaluate the systematic errors in largely the same way as in the Υ (3S) analysis, the principal diﬀerence being the treatment of the signal eﬃciency.

For the systematic error on the signal yield (thus branching fraction) and peak

position (thus ηb mass), we consider the following sources:

1. ηb width: the nominal result is obtained with an assumed width of 10 MeV. We repeat the ﬁt for widths of 5, 15 and 20 MeV and use the variations to estimate the uncertainty.

2. To study the effect of the PDF parametrization of the different fit components, we vary the following parameters by 1σ:

ISR: vary N, A and σ of the Crystal Ball function • signal: vary N, A and σ of the Crystal Ball function. • χ : vary N of the Crystal Ball function, the ratio of the χ components, • b b

and ﬁx the χb0 σ to that of the χb1 rather than the χb2. An additional ﬁt

incorporating a high-side tail to the χbJ (1P ) is described below.

3. Smooth background: ﬁts are performed using an alternative smooth background

shape incorporating a 3rd-order polynomial in the exponential (i.e. c4 = 0) and a PDF with a smooth transition between two power laws with diﬀerent

exponents:

−Γ1 1/α −(Γ2−Γ1)α k (Eγ /E0) 1 + (Eγ /E0) (5.3) 156 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

4. bin width: the bin width is varied to 1, 2, 10 and 15 MeV.

5. An additional high side tail in the χbJ (1P ) peak may be produced by the coin-

cidental overlap of photons from χbJ (1P ) decays with particles from the rest of the event or beam debris. The systematic error due to this presence of this tail is

evaluated by taking the difference between the nominal fit and the same fit but with the high side tail included. We model this tail as a 90 MeV wide Gaussian

centered about the χbJ peaks. The parameters are taken from Table 5.10. The Gaussian width as well as the ratio of the Crystal Ball to Gaussian component

of the PDF are ﬁxed in the ﬁt.

For the estimation of the systematic uncertainty on the ηb peak position, we have all of the sources listed above plus the systematic error on the ηb peak position due to the imperfect knowledge of the photon energy scale. This uncertainty is estimated as half of the energy scale oﬀset (0.7 MeV) added in quadrature with the statistical error on the oﬀset (0.2 MeV) for a total of 0.7 MeV.

The resulting uncertainties are summarized in Table 5.24.

5.7.2 Signiﬁcance of Signal (Including Systematic Errors)

To evaluate the significance of the signal with systematics included we make all of the fit variations in the pessimistic direction of significance and refit the data. This procedure gives a significance, including systematic errors, of 3.0 standard deviations. The decrease in the systematic error is dominated by the background shape variation; the signal significance after making this variation alone is 3.0σ. 5.7. SYSTEMATIC ERRORS 157

Table 5.24: Systematic uncertainties for the measurement of the ηb yield and peak position (mass). Errors with no sign given are taken to be symmetric.

Source ∆ N [Evts] ∆ peak pos. [MeV] ηb width, 5 MeV -1176 -0.1 ηb width, 15 MeV 922 +0.2 ηb width, 20 MeV 1735 +0.3 ISR PDF, N 164 0.01 ISR PDF, A 216 0.07 ISR PDF, σCB 65 0.08 signal PDF, N 44 0.03 signal PDF, A 48 0.07 signal PDF, σCB 44 0.05 χbJ PDF, N 48 0.04 χb1/χb2 ratio 195 0.17 χb0/χb1 ratio 148 0.04 χb0 σCB +134 +0.02 Extra χbJ tail +1527 +0.03 exp P ol(3) PDF -2717 +1.3 Broken power law PDF +2562 +0.2 0 +651 (π ,η) transitions −71 0.2 +268 Bin width variation −696 1.1 +3545 Total, ﬁt variations −3066 1.8 Energy scale oﬀset 0 0.7 χb masses 0 0.4 +3545 Total −3066 1.9

5.7.3 Additional Fit Variations

Several additional ﬁt variations were performed as a cross-check. The ﬁt region was extended from 270 MeV to 250 MeV on the low end of the spectrum and from

800 MeV to 850 MeV on the high end, the bin width was varied between from 158 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

1 MeV and 15 MeV, and the cos(θ ) cut was varied between 0.7 and 0.95. Fi- | T | nally, fits were performed with the small Υ (2S) (π0, η)Υ (1S) contributions sub- → tracted before fitting. In these fits we assume (Υ (2S) ηΥ (1S)) = 2.1 10−4 B → × and (Υ (2S) π0Υ (1S)) = 3.7 10−4. The measured η yield changes by less than B → × b 0.2σ when these transitions are included. The branching fraction for the η and π0 transitions are the measured value and the 90% CL upper limit from CLEO.

The results are summarized in Table 5.25. The statistical significance is determined for each of the fits by (χ2 χ2) where χ2 is the chi-square statistic of the − 0 2 nominal fit and χ0 is the chi-squarep statistic of an identical fit with the ηb PDF removed. 5.7. SYSTEMATIC ERRORS 159 ) σ ( 3.67 3.62 3.50 3.74 3.74 3.56 3.22 3.84 3.52 3.03 3.52 3.84 3.82 3.40 4.41 3.64 3.46 3.13 Significance ariations. yield v fit eral nominal 71 696 793 771 sev 2717 2318 1854 2443 +3 +99 +80 − +237 +268 − − +651 +572 − +1990 +1860 − − +2562 − − from with only) Deviation 3482 3537 3166 3483 3314 3405 3158 3356 3482 3479 3486 3266 3476 3482 2982 3708 3857 4162 +3357 − +3572 − +3526 − +3502 − +3535 − +3537 − +3516 − +2981 − +3791 − +3851 − +4163 − (statistical Yield 10488 14796 12013 14666 10089 12809 13043 13074 12110 12707 12889 13457 13378 10952 15368 12735 10363 12035 significance MeV MeV γ γ MeV MeV MeV MeV γ 250 850 → signal PDF γ 15 10 2 1 0 γ to to π γ , to to to to → and ) 0 → S π PDF , η MeV MeV (1 ) , MeV MeV MeV MeV kground PDF ) Υ S Yield ) 5 5 5 5 S η 270 800 (1 bac , 7 75 85 9 95 (1 . . . . . Υ 0 0 0 0 0 0 0 Υ π to kground η π ( 5.25: 5.3 from from from from < < < < < from from kground | | | | | bac → → → ) ) ) ) ) eq. ) ) ) T T T T T able term bac to θ θ θ θ θ S S S T t width width width width (2 (2 (2 region region in from cos( cos( cos( cos( cos( Υ Υ Υ 0 | | | | | bin bin bin bin term fit fit = 5 x constan 4 ariation PDF c v Fit Extend Extend Change Change Change Change Use Set Add Add Require Require Require Require Require Subtract Subtract Subtract 160 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

5.7.4 Branching Fraction Uncertainties

For the measurement of the branching ratio, we have an additional source of uncertainty resulting from the signal selection eﬃciency. For the absolute photon detection eﬃciency, we use the BABAR standard 1.8%.

The hadronic selection eﬃciency systematic is determined using hadronic Υ (1S)

decays provided by the χbJ (2P ) peak in Υ (3S) data. The cross-check is done using Υ (3S) data rather than Υ (2S) data as the Υ (3S) data has better S/B and the background is easier to fit. The hadronic selection efficiency on data is the product of the efficiencies for the preliminary hadronic selection and the track number requirement

(from Table 5.26), and is (0.973 0.013) (0.903 0.012) = 0.879 0.017. The corre- × sponding eﬃciency from χ (2P ) signal MC is 0.928 0.993 = 0.921. The fractional bJ × diﬀerence added in quadrature with the statistical error gives a systematic error of 4.9%.

Table 5.26: Comparison of selection eﬃciencies in χbJ (2P ) data versus χb0(2P ) MC.

Selection Data Eﬃciency MC eﬃciency (J = 0) Reconstructed + truth-match - 0.780 Preliminary hadronic selection 0.973 0.013 0.928 > 3 charged tracks 0.903 0.012 0.993 Cluster lateral moment < 0.55 0.997 0.013 0.972 In calorimeter barrel 0.928 0.012 0.926 cos θ < 0.7 0.672 0.008 0.674 | T | E > 50 MeV, m m 0 < 15 MeV 0.849 0.008 0.840 γ2 | γγ − π ,peak|

To evaluate the cluster lateral moment selection eﬃciency systematic, we compare π0 yields in on-peak data and generic Υ (2S) MC before and after applying the cluster

lateral moment selection to the signal candidate. A sample of π0 events where one 5.7. SYSTEMATIC ERRORS 161

photon of the π0 candidate falls into the signal region is used. The signal candidate lies in the energy range 600 MeV to 700 MeV. The cluster lateral moment selection

eﬃciency in on-peak data is 94.6 0.2% compared to 95.0 0.2% in generic Υ (2S) MC. Adding the diﬀerence between the two numbers in quadrature with the statistical errors gives a systematic error of 0.5%.

For the thrust selection, we take the difference between the observed efficiency and the efficiency for a flat distribution, 0.6%, as the systematic.

0 The eﬃciency for the π selection is determined by comparing the χbJ (1P ) yield

0 on the nearby χbJ (1P ) peak before and after applying the π veto. This is determined using the test sample, which is easier to ﬁt than the full sample. We assume the same

0 χbJ (1P ) lineshape as in the fit to the data after the π veto, where the background is much smaller, making the χ (1P ) lineshape easier to determine. We find 137369 bJ 0 2044 χbJ (1P ) events before the π veto (the background-subtracted fit result may be seen in fig. 5.28) compared to 78683 2515 after, for an efficiency ratio of 0.571 0 0.019. The full difference between this and the χbJ (1P ) π selection efficiency in MC,

¯χb = 0.557 (see Table 5.11, weighted by relative rate), added in quadrature with the statistical error is 4.1%, which we take as the error. The errors are summarized in Table 5.27.

In the ﬁt of the energy spectrum with the π0 veto removed, the absolute ISR and

η yields are somewhat larger than expected. The ﬁtted yields are 6822 1699 and b 5466 1519, compared to 2083 178 and 1470 309, respectively. The expected ISR yield is the nominal expected ISR yield divided by the eﬃciency of the π0 veto on the ISR peak, and the expected ηb yield is computed similarly using the derived branching fraction. These yields are each about 2.5σ above the expected value but 162 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

given that the χ2 of the fit is essentially perfect (χ2/ndof = 104.7/96, corresponding to a fit probability of 25.5%) we consider this to be an unlikely statistical fluctuation

rather than a deﬁciency in the ﬁt. This view is supported by the upward statistical

ﬂuctuation in the ηb yield already seen in the nominal test sample ﬁt.

14000 12000 10000 8000 6000 Events / ( 0.005 GeV ) 4000 2000 0 -2000 0.3 0.4 0.5 0.6 0.7 0.8 Eγ, CM (GeV)

0 Figure 5.28: Fit to test sample before π veto. The cyan curve is the sum of the χb peaks, and the red curve is the ISR PDF. The histograms are χb (cyan), ISR (red), and ηb (blue). The ISR yield is ﬂoating in the ﬁt.

As a cross-check on the total efficiency systematic we can use the χb efficiency in data and MC. The sum the branching fractions (Υ (2S) γχ ) (χ J B → bJ B bJ → γΥ (1S)) from the PDG is (4.0 0.7)%. Since there areP91.6 million Υ (2S) events and 945000 8000 χ events in the full sample, the χ efficiency in data is (25.8 4.5)%. b b This is consistent with the MC efficiency of 27.1%. 5.7. SYSTEMATIC ERRORS 163

Table 5.27: Systematic uncertainties on the selection eﬃciency, in percent.

Source Uncertainty (in %) Photon detection 1.8 Hadronic selection 4.9 Photon selection 0.5 Thrust selection 0.6 π0 veto 4.1 Total 6.7

Table 5.28: Systematic uncertainties for the measurement of the ηb branching fraction, in percent.

Source Uncertainty (in %) +28 Fit variations −24 Υ (2S) counting 0.93 ηb MC eﬃciency 0.5 Selection eﬃciency 6.7 +29 Total −25 164 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

5.8 Combination with Υ (3S) result

In this section the results of the present analysis are combined with the results of the Υ (3S) analysis. Some of the systematic errors are common to the Υ (2S) and Υ (3S) analyses and partially cancel in the combination.

5.8.1 Ratio of branching fractions

The ηb width systematic partially cancels in the ratio of the Υ (2S) and Υ (3S) branching fractions. Table 5.29 lists the signal yields for diﬀerent assumed ηb widths, and demonstrates this cancellation. In the ratio of branching fractions the systematic

error due to the unknown ηb width is reduced from roughly 10% to 5%.

Table 5.29: Yield of ηb signal in Υ (2S) and Υ (3S) analyses, the ratio of the two yields, and the deviation of the ratio from that of the 10 MeV ﬁt, as a function of the ηb width assumed in the ﬁt. Υ (2S) and Υ (3S) are used as abbreviations for the Υ (2S) and Υ (3S) yields, respectively.

ηb width Υ (2S) Υ (3S) Υ (2S)/ Υ (3S) Deviation from nominal ratio (%) 5 11630 18050 0.644 -3.7 10 12806 19152 0.669 0.0 15 13728 20348 0.674 0.7 20 14541 21314 0.682 1.9

In order to obtain cancellation of uncertainties in the selection eﬃciencies in the

Υ (2S) and Υ (3S) analyses, it is necessary to treat the systematic errors identically in the two analyses. Table 5.26 compares the cut eﬃciencies in data and MC on

the χbJ (2P ) peak. The difference in cut efficiencies in MC among the three χbJ (2P ) peaks are negligible. The efficiency for the π0 cut in the Υ (3S) analysis is determined

0 by comparing the yield on the nearby χbJ (2P ) peak before and after applying the π 5.8. COMBINATION WITH Υ (3S) RESULT 165

veto. This is determined using the test sample, which is easier to fit than the full sample. We find an efficiency of 0.849 0.008, compared to a MC efficiency of 0.840. The full difference added in quadrature with the statistical error is 1.5%, which we take as the error.

Table 5.30 lists the error in the selection efficiencies for the Υ (3S) analysis, as well as the Υ (2S) numbers from Table 5.27. The photon detection efficiency and hadronic selection efficiency systematics in the Υ (3S) analysis are identical to those used in the Υ (2S) analysis. The photon selection efficiency in the Υ (2S) analysis is assumed to be equal to that of the Υ (3S) analysis. This is a conservative assumption as the LAT distribution is narrower for higher energy photons, resulting in a LAT cut inefficiency a factor of two smaller in the Υ (3S) analysis. The cos θ systematic is estimated in | T | the same way as in the Υ (3S) analysis.

Table 5.31 lists the uncertainties in the branching fraction ratio. The photon detection and hadronic selection eﬃciency is assumed to cancel in the branching fraction ratio. This gives systematic errors of 4.2% and 1.9% for the selection eﬃciencies of the

Υ (2S) and Υ (3S) analyses, respectively. Including systematic errors, the ratio of the two branching fractions is (Υ (2S) γη )/ (Υ (3S) γη ) = 0.82 0.24+0.20. By B → b B → b −0.19 properly cancelling the common systematic errors between the two analyses the systematic error is reduced; without taking into account this cancellation the systematic

+0.25 errors would be −0.22. 166 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

Table 5.30: Systematic uncertainties (in %) on the selection eﬃciency in the Υ (3S) γη and Υ (2S) γη analyses, in percent. Systematics which are assumed to cancel→ b → b in the branching fraction ratio are in italics.

Source Υ (3S) Υ (2S) Photon detection 1.8 1.8 Hadronic selection 4.9 4.9 Photon selection 0.5 0.5 Thrust cut 1.0 0.6 π0 veto 1.5 4.1 Total (excluding canceling systematics) 1.9 4.2 Total (including canceling systematics) 5.5 6.7

Table 5.31: Systematic uncertainties on the branching fraction ratio, in percent.

Source Uncertainty (in %) Υ (2S) selection eﬃciency 4.2 +24 Υ (2S) yield −22 Υ (3S) selection eﬃciency 1.9 Υ (3S) yield 2.5 +1.9 ηb width −3.7 +25 Total −23 5.8. COMBINATION WITH Υ (3S) RESULT 167

5.8.2 New Υ (3S) γηb branching fraction →

In the analysis of the decay Υ (3S) γη [78] the systematic error on the signal → b eﬃciency was determined by comparing the yield of χbJ (2P ) in data to the number of expected events, which is calculated from the known branching fractions, the

number of Υ (3S) events, and MC reconstruction eﬃciency of χbJ (2P ). The total uncertainty (22%) is obtained by adding the full diﬀerence (13%) in quadrature with

the uncertainties in the χbJ (2P ) branching fractions (18%). As a result of the large uncertainties in the χ (2P ) branching fractions, the Υ (3S) γη branching fraction bJ → b carried a substantial systematic error: (Υ (3S) γη ) = (4.8 0.5 1.2) 10−4. B → b × Using the error estimates from the previous section, we may provide a branching

fraction with a substantially reduced systematic error. The systematic error due to

ﬁt variations (2.5%) and the ηb width (11%) are the same as in the previous Υ (3S) analysis. The systematic error on the eﬃciency is, from Table 5.30 (now including the photon detection and hadronic selection systematic errors), 5.5%, giving a total

uncertainty of 12%. The new systematic errors are summarized in Table 5.32. This gives a ﬁnal branching fraction of (Υ (3S) γη ) = (4.8 0.5 0.6) 10−4. B → b ×

Table 5.32: Systematic uncertainties on the new Υ (3S) γηb branching fraction, in percent. →

Source Uncertainty (in %) Υ (3S) selection eﬃciency 5.5 Υ (3S) yield 2.5 ηb width 11 Total 13 168 CHAPTER 5. STUDY OF THE DECAY Υ (2S) γη → B

5.8.3 Mass of the ηb

The mass measurements in the Υ (2S) and Υ (3S) analyses (m = 9394.2+4.8 ηb −4.9 1.7 MeV/c2 and m = 9388.9+3.1 2.7 MeV/c2, respectively) are combined using ηb −2.3 the procedure used by the PDG [79]. Sections 5.2.1. and 5.2.2. of the Introduc-

tion section of the PDG describe the averaging procedure. The value obtained is 9390.8 3.2 MeV/c2. None of the systematic errors cancel in the combination. The only systematic error which could cancel is due to the energy scale oﬀset. If the energy scale was taken from a single isolated peak, this systematic would likely cancel. But since the energy scale

is in each case taken from three overlapping peaks, an uncertain portion of the energy scale oﬀset may actually be due to a deﬁciency in the parameterization of the peaks. Also, an assumption would need to be made regarding the energy dependence of the shift. So, we prefer to treat the systematic errors as though they are uncorrelated. Chapter 6

Summary and Outlook

The ηb has been unambigously observed for the ﬁrst time, and its mass measured. The radiative transition rates for the processes Υ (3S) γη and Υ (2S) γη have → b → b been determined, and are consistent with the rates expected for hindered magnetic dipole transitions to the ηb.

The measured hyperﬁne mass splitting between the Υ (1S) and the ηb is consistent with a recent unquenched lattice calculation (61 14 MeV, [24]). Intriguingly, the experimental ηb mass is several standard deviations above the predictions based on perturbative QCD (44 11 MeV, [22]) and non-relativistic QCD (39 11+9(α ) MeV), −8 s a discrepancy which is not yet understood.

Experimentally, future ηb analyses may attempt to measure the ηb width, possibly utilizing the improved energy resolution (but lower statistics) provided by converted photons.

Theoretical progress is necessary to fully utilize the information provided by these measurements. The lattice calculation of the ηb mass may be improved by including higher orders in the NRQCD Lagrangian (the calculation of [24] is only valid to

169 170 CHAPTER 6. SUMMARY AND OUTLOOK

(m v4)), and calculation of the matching coeﬃcients c . In pNRQCD, calculation O b i of higher order corrections (the current best is NLL order) and a better understanding of non-perturbative corrections would enable a better test of the pNRQCD framework. Additionally, bottomonium radiative transition rates could be calculated using lattice QCD, providing a model-independent test of the radiative transition rate. Appendix A

Use of EMC Timing to Improve π0 Veto

Timing information is used only crudely in BABARVery˙ loose selections are made on timing variables to ensure that all photons from a given event are selected for inclusion in an event. In this appendix we explore improvements in π0 selection using cluster timing variables.

In BABAR reconstruction code single crystal hits, referred to as digis, contain an energy and a time. The energy is determined from integrating the pedestal-subtracted waveform. The time is determined from the energy-weighted time of the samples which constitute the waveform.

We form an average cluster time by an energy-weighted time of

tcluster = iti/Ecluster i X where i indexes the crystals in the cluster and i and ti are the energies and times

171 172 APPENDIX A. USE OF EMC TIMING TO IMPROVE π0 VETO

of the digis which compose the cluster. The energy-weighting is done to emphasize the times of the digis in crystals with higher energy hits, whose times are determined from waveforms with better signal to noise.

A cluster time error is deﬁned similarly:

t2 σ2 = i i t2 tcluster E − cluster i cluster X The two photons from a real π0 will arrive at the calorimeter at the same time, whereas the times of photons from machine background will be uncorrelated with the time of the physics event. We deﬁne a timing “signiﬁcance” by

t t S = 1 − 2 2 2 σ1 + σ2 p where the ti and σi are the times and time errors of the clusters which from a candidate π0.

The timing information is uncalibrated; the time of the peak varies due to the different pre-amplifier shaping time constants and different CsI rise and decay times. Figure A.1 shows the deviation of the average photon time from the photon time

averaged across the calorimeter. Each bin is one crystal.

After subtracting this correction, we tested the correction on a subsample of the data to make sure the correction is stable. The deviation from the corrected mean time is shown in A.2. Figure A.3 shows the reduction of the background underneath the π0 mass peaks for a cut of 2.5σ in timing signiﬁcance. Figure A.4 shows the π0

candidates which are rejected by various cuts on timing signiﬁcance.

The diﬀerence between the arrival time of two photons, normalized by the error on 173

t-tave [ns] 30 index φ 100 20

80 10

60 0 Cluster centroid

40 -10

20 -20

0 -30 10 20 30 40 50 Cluster centroid θ index

Figure A.1: Deviation from average timing before cluster timing correction. The 48 ns shift from the long shaping time pre-ampliﬁers has already been subtracted. The vertical patterns correspond to diﬀerent crystal manufacturers.

that diﬀerence, is a useful variable for rejecting photons that arrive at the calorimeter

at different times. This makes a timing significance selection very useful for rejecting beam-related backgrounds, which are uncorrelated with the time of the physics event. While timing significance is not used in this analysis, it could be useful for analyses using soft photons. In particular, studies of the photons from the second transition in Υ (nS) γχ ((n 1)P ) γγΥ (1S), which have energies near 100 MeV, could → bJ − → use this selection to significantly reduce background. While significantly beyond the scope of this study, an investigation of the use of timing significance at the level of photon reconstruction could be beneficial. 174 APPENDIX A. USE OF EMC TIMING TO IMPROVE π0 VETO

t-tave [ns] 30 index φ 100 20

80 10

60 0 Cluster centroid

40 -10

20 -20

0 -30 10 20 30 40 50 Cluster centroid θ index

Figure A.2: Deviation from mean time after cluster timing correction.

7000

6000 Events/MeV

5000

4000

3000

2000

1000

0 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 π0 mass [GeV]

Figure A.3: π0 peaks for signal candidate in data when combined with another photon in the event. All π0 (blue), π0 candidates which are rejected by a 2.5σ timing cut (red), and the subtracted distribution (green). 175

10000 π0 mass with 1.5σ cut π0 mass with 2σ cut π0 mass with 2.5σ cut 8000 π0 mass with 3σ cut Events/MeV

6000

4000

2000

0 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 π0 mass [GeV]

Figure A.4: Candidate π0 which are rejected by timing cuts by cuts on timing significance, σ Appendix B

Spurious Feature at 680 MeV

B.1 Description of problem

In an early version of this analysis we observed an additional feature in the photon spectrum near 680 MeV, shown in Fig. B.2. Following investigation of its properties, we have attributed the spurious feature to the turn-on of the correction for energy loss in the gaps between crystals.

To understand the origin of the spike it is useful to review the photon energy calibration procedure. The energy scale for individual crystals is calibrated using a combination of a photons from a radioactive source and Bhabha events. Once the crystal energy scale is established, an overall correction for shower leakage and other

energy losses is applied. This correction to the photon energy scale is determined using µµγ events and π0 decays as a parameterization in photon energy and θ. At this stage the photon energy scale is, on average, correctly determined. However, to improve energy resolution for high energy clusters, cluster energies are corrected as a

function of their proximity to crystal edges. Since a single response function averaged

176 B.2. OTHER INVESTIGATIONS OF SPIKE 177

over the width of a crystal has been used, energies of photons in the central area are overcorrected, and those near the edge are undercorrected. The “edge correction”

raises the energy of photons with cluster centroids near the crystal boundaries and lowers the energy of those near the center of crystals.

For photon energies below 1 GeV in the lab frame the cluster position resolution

is not suﬃcient to accurately determine position on sub-crystal scales. As a result, an attempt at using a correction would lead to a smearing of the energy rather than an improvement in energy resolution. To avoid this, the correction is turned oﬀ.

Unfortunately, this edge correction leads to a spike in the photon energy spectrum near 1 GeV. Photons with energies above 1 GeV can be corrected to energies below 1 GeV whereas photons with energies below 1 GeV are not corrected. This gives rise to an excess below 1 GeV and a deﬁcit above 1 GeV in the lab frame, shown in

Fig. B.1. The spike will be smeared out considerably by the boost to the CM frame but a similar spike will be seen near the energy at which the edge correction turns on in the CM frame. As seen in Fig. B.1, which shows the photon energy spectrum in CM frame for photons satisfying E 1GeV < 0.01GeV, the minimum CM | γ,LAB − | energy at which the edge correction is applied is near 680 MeV.

Figure B.2 shows the background-subtracted energy spectrum with and without the crystal edge correction turned on.

B.2 Other investigations of spike

Before the source of the spike was determined many studies were performed to determined to show that the spike is not due to a fault in the electronics. We discuss

these studies below. 178 APPENDIX B. SPURIOUS FEATURE AT 680 MEV

1800 30000 1600 25000 1400 1200 20000 1000 15000 800 Entries / ( 0.005 GeV ) Entries / ( 0.005 GeV ) 10000 600 400 5000 200 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Eγ, LAB (GeV) Eγ, CM (GeV)

Figure B.1: Photon energy spectrum near 1 GeV with (red) and without (blue) the crystal edge correction turned on (left). Photon energy spectrum in CM frame for photons satisfying E 1GeV < 0.01GeV (right). | γ,LAB − |

10000 10000

8000 8000

6000 6000

4000 4000 Events / ( 0.005 GeV )

2000 Entries / ( 0.005 GeV ) 2000

0 0

-2000 -2000 0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.4 0.5 0.6 0.7 0.8 Eγ,CM (GeV) Eγ (GeV)

6000 6000 5000 5000 4000 4000 3000 3000 2000 2000 1000 1000 0 0 Entries / ( 0.010 GeV ) Entries / ( 0.010 GeV ) -1000 -1000 -2000 -2000 -3000 -3000 -4000 -4000 0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.4 0.5 0.6 0.7 0.8 Eγ (GeV) Eγ (GeV)

Figure B.2: Background subtracted energy spectrum after (left) and before (right) the edge correction is turned on. The spectra are shown with non-peaking background subtracted (top) and all background subtracted (bottom). B.2. OTHER INVESTIGATIONS OF SPIKE 179

We have veriﬁed that the photon candidates which constitute this feature are not localized to a particular time or location in the detector. The photon spectrum for clusters with a low crystal multiplicity does not show a feature, as would be expected if the spike is due to malfunctioning electronics. Fig. B.3 shows the energy spectrum of clusters with lateral moment equal to zero, which includes all clusters with one or two crystals. We have histogrammed the photon spectrum, separated according to

the crystal in which the cluster centroid is located (5760 crystals total) and using the same binning as the analysis. None of these histograms contains more than 86 entries in the region 650 < E < 750 MeV.

450

400

350

300

Events/ (0.005 GeV) 250

200

150

100

0.5 0.55 0.6 0.65 0.7 0.75 0.8 Eγ (GeV)

Figure B.3: Photon energy spectrum of clusters with lateral moment equal to zero, and passing all analysis cuts.

To determine the significance of the feature, we fit the data with an extra PDF for a monochromatic photon near 680 MeV. This fit, after background subtraction,

is displayed in Fig. B.4. To determine the signiﬁcance of the feature we ﬁt the data with an additional CB function describing a monochromatic photon near 680 MeV. The probability of observing such an excess, estimated using a χ2 distribution with one degree of freedom, is 0.0015. Since the excess is three bins wide and there are

106 bins, we assume that there are 106/3 35 independent locations in which the ≈ 180 APPENDIX B. SPURIOUS FEATURE AT 680 MEV

excess could occur. Including this trials factor of 35, the probability of observing an excess of this signiﬁcance or larger in any location in the spectrum is 0.05.

However, if the width of the feature is allowed to ﬂoat in the ﬁt, we obtain a value of 3 MeV, much less than the detector resolution at 680 MeV. Using a χ2 test we determine that the hypothesis that the width of the feature is inconsistent with the detector resolution at the level of 3.2σ.

We have performed other checks to eliminate other possibilities of software-related errors. We investigate the possibility that an improperly chosen position hypothesis for the photon could cause a spike in the photon energy spectrum, though there is no reason to believe that this is the case. Rather than using the primary vertex position we calculate the photon energy simply using the origin in the BABAR coordinate system and perform a ﬁt identical to the nominal ﬁt. The result is shown in Fig. B.5. B.2. OTHER INVESTIGATIONS OF SPIKE 181

10000

8000

6000

4000

Events / ( 0.005 GeV ) 2000

-2000 0.3 0.4 0.5 0.6 0.7 0.8 Eγ,CM (GeV)

10000

8000

6000

4000

Events / ( 0.005 GeV ) 2000

-2000 0.3 0.4 0.5 0.6 0.7 0.8 Eγ,CM (GeV)

Figure B.4: Fit to full data sample after background subtraction. The cyan curve is the sum of the χb peaks, and the red curve is the ISR PDF. The histograms are χb (cyan), ISR (red), ηb (blue), the extra peak (green) and their sum (black). The ISR yield is floating in the fit. The top plot displays the fit where the width of the spurious peak is fixed to detector resolution and the bottom displays the fit in which the width is allowed to float. 182 APPENDIX B. SPURIOUS FEATURE AT 680 MEV

10000

8000

6000

4000

Entries / ( 0.005 GeV ) 2000

-2000 0.3 0.4 0.5 0.6 0.7 0.8 Eγ,CM (GeV)

Figure B.5: Smooth background subtracted ﬁt of energy spectrum with no primary vertex correction. Bibliography

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