Search for Radiative Decays of D Mesons at the Babar

Home , Radiative process

SEARCH FOR RADIATIVE DECAYS OF D0 MESONS AT THE BABAR DETECTOR

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

Joseph J. Regensburger, M.S.

*****

The Ohio State University

2008

Dissertation Committee: Approved by

Klaus Honscheid, Adviser Richard Kass Adviser Richard Furnstahl Graduate Program in Thomas Humanic Physics Roberto G. Rojas-Teran ABSTRACT

I detail my work searching for the radiative decays of D0 mesons, e.g. D0 → φγ, within 381.7fb−1 of e+e− data collected by the BABAR detector at the PEP-II asymmetric-energy e+e− collider at SLAC from 1999-2006. Such decays are not well described under perturbative techniques typically used to estimate the frequency of decays under the Standard Model of Particle Physics (SM), and as such are valuable laboratories to investigate quantum chromodyanmics calculations (QCD).

I speciﬁcally examine the Cabibbo-suppressed (CS) D0 φγ decay as well as → ∗0 search for the yet unobserved Cabibbo-favored (CF) decay, D0 K γ. I measure → the branching fractions of each mode relative to the decay D0 K−π+ and ﬁnd →

(D0 φγ)=(2.73 0.30 0.26) 10−5 B → ± ± × and

∗0 (D0 K γ)=(3.22 0.20 0.27) 10−4 B → ± ± × These results are preliminary and currently under review by the BABAR collaboration ahead of publication. In these expressions the ﬁrst error is due to statistical sources and the second is due to systematic sources.

ii ACKNOWLEDGMENTS

This work would not have been possible without the expertise and advice of many people both in and out of the Physics community. I would like to thank my wife, Tammi, for her love, encouragement, and patience. This work could not have been accomplished without her emotional and intellectual support. She has been beyond generous with her time, love, and advice. I also thank my family for all of their support over these many years.

I would like to thank the entire Ohio State BABAR group. Foremost I would like to thank my advisor, Dr. Klaus Honscheid, for his wisdom and generosity throughout my time at Ohio State, and Dr. Amir Rahimi for his advice and friendship over the course of this analysis. This work would have never succeeded without their input.

I also extend my gratitude to Dr. Dirk Hufnagel for establishing the OSU BABAR

Monte Carlo Farm and later for his time and eﬀort training me to assume his role as the site coordinator for the Farm. On that same note, I would like to thank the

OSU Physics Computer Group, in particular Tim Randles and J.D. Wear for their help in maintaining the OSU BABAR computer system.

I am grateful for the extraordinary contributions of the PEP-II team in achieving the excellent luminosity and machine conditions that have made this work possible, as well as the BABAR computing group, and the BABAR Charm Analysis Working

Group for their input during the process.

iii I would also like to thank all of the professors and teachers who in the past have helped my development as a scientist. These include Mrs. Pam Mathis at

McNicholas High School, Drs. Haowen Xi and Barry Cobb at Bowling Green State

University, each of whom gave me my ﬁrst exposure to academic research, and Dr.

Monroe Rabin at the University of Massachusetts at Amherst.

Finally I would like to thank all of my past and present colleagues. At Ohio State

Don Burdette, James Morris, and Iulian Hetel have been a constant source of good company, good humor, and good collaboration. At the University of Massachusetts

Don Blair, Deniz Kaya, Ozgur Yavuzcetin, Andrew Varnon, and James Heﬂin have long been great friends.

This work is supported by the US Department of Energy.

iv VITA

February 26, 1975 ...... Born - Cincinnati, Ohio

August 1993-May 1997 ...... Bowling Green State University, Bowling Green, OH

Spetember 1999-May 2002 . . .University of Massachusetts at Amherst, Amherst, MA June 2002 - ...... The Ohio State University, Columbus, OH

FIELD OF STUDY

Major Field: Physics

Studies in Experimental Particle Physics: Professor Klaus Honscheid

v TABLE OF CONTENTS

Page

Abstract...... ii

Acknowledgments...... iii

Vita...... v

ListofTables...... xi

ListofFigures ...... xiv

Chapters:

1. StandardModel...... 1

1.1 IntroductiontoParticlePhysics ...... 1

1.2 FundamentalParticles ...... 3

1.3 FundamentalForces...... 3

1.3.1 Electromagneticforce...... 6

1.3.2 Strongforce...... 7

1.3.3 Weakforce ...... 9

1.3.4 Measuring Branching Fractions ...... 11

vi 2. Search for Radiative Decays ...... 15

2.1 PreviousExperimentalStudies...... 16

2.2 RadiativePenguins ...... 16

2.2.1 Long Distance Contributions ...... 20

2.3 ExperimentalChallenges ...... 24

2.3.1 HelicityAngle...... 28

2.3.2 Other Discriminating Observables ...... 30

3. The BABAR DetectorandPEP-IIStorageRing ...... 33

3.1 PEP-IIStorageRing ...... 35

3.1.1 CenterofMassEnergy ...... 36

3.1.2 LuminosityMeasurement...... 36

3.2 The BABAR Detector ...... 40

3.2.1 Silicon Vertex Tracker ...... 42

3.2.2 TheDriftChamber ...... 44

3.2.3 Directed Internally Reﬂected Cherenkov Light (DIRC) Detector 44

3.2.4 Electomagnetic Calorimeter ...... 47

3.2.5 InstrumentedFluxReturn ...... 51

3.3 ParticleIdentiﬁcation...... 53

3.4 Monte Carlo Simulation of the BABAR Detector ...... 56

3.4.1 MCSimulation ...... 58

3.4.2 Distributed Computing at BABAR ...... 59

4. Experimental Details and Datasets ...... 61

4.1 EventSelection ...... 61

vii 4.2 MonteCarloandDataSampleSizes...... 63

4.3 AdditionalEventSelection ...... 67

5. Experimental Search for Radiative D0 Decays ...... 80

5.1 D0 φγ Analysis ...... 80 → 5.1.1 D0 φγ SignalShape ...... 82 → 5.1.2 D0 φπ0 Background ...... 86 → 5.1.3 D0 φη Background...... 88 → 5.1.4 Remaining Background ...... 92

5.1.5 SummaryofSignalShapes ...... 95

5.1.6 FittingMethodValidation ...... 95

∗0 5.2 D0 K γ Analysis ...... 101 → ∗0 5.2.1 D0 K γ SignalShape...... 103 → 5.2.2 D0 K−π+π0 BackgroundShape ...... 105 → ∗0 5.2.3 D0 K η BackgroundShape ...... 106 → 5.2.4 Combinatoric Background Shape ...... 110

5.2.5 Additional Backgrounds ...... 112

5.2.6 ValidatingofFittingProcedure ...... 113

∗0 5.2.7 Fitting Results While Varying (D0 K γ) ...... 115 B →

6. SignalShapeValidation ...... 117

6.1 SignalShapeControlSamples ...... 117

6.1.1 D0 K0γ Dataset...... 119 → s 6.1.2 D0 K0π0 Dataset ...... 121 → s ∗0 6.1.3 D0 K γ Helicity Sideband Dataset ...... 123 →

viii 6.2 Obtaining Correction Between Data and MC ...... 124

6.3 Applying Corrections when Fitting Data ...... 129

6.3.1 Summary of Signal Shape Correction ...... 130

7. Results...... 133

7.1 D0 φγ Results ...... 135 → ∗0 7.2 D0 K γ Results...... 137 → 7.3 ValidatingResults...... 138

8. Systematics ...... 143

8.1 FittingSystematics ...... 144

8.1.1 Fixed PDF Systematics ...... 144

8.1.2 Control Sample Correction Systematics ...... 149

8.1.3 D0 Vη SignalShape...... 151 → 8.1.4 Combinatoric Signal Shape ...... 152

8.2 Tracking Reconstruction and Vertexing ...... 153

8.3 ParticleIdentiﬁcation...... 154

8.3.1 D0 φγ PIDSystematic ...... 156 → ∗0 8.3.2 D0 K γ PIDSystematic ...... 156 → 8.3.3 D0 K−π+ ReferenceMode ...... 157 → 8.3.4 PIDSystematicSummary ...... 157

8.4 PhotonSystematics...... 163

8.5 π0 Veto ...... 164

8.6 CutVariation ...... 166

8.7 SystematicSummary ...... 168

ix 9. Conclusion...... 169

Appendices:

A. D0 K−π+ ReferenceMode ...... 174 → A.1 EventSelection ...... 174

A.2 FittingProcedure...... 176

A.3 SystematicUncertainties ...... 176

B. ProbabilityDistributionFunctions ...... 179

B.1 CrystalBallLineshape(CB) ...... 179

B.2 Gaussian...... 180

B.3 Chebychev Polynomials ...... 180

B.4 ∆M BackgroundShape ...... 180

Bibliography ...... 182

x LIST OF TABLES

Table Page

1.1 Important properties of the fundamental Standard Model matter par-

1 ticles [1]. All of the listed particles have spin 2 . The electric charge is written in terms of the electric charge of an electron e. The upper

limits displayed are at the 90% conﬁdence level (C.L.)...... 4

1.2 SummaryofForceCarriers...... 6

1.3 Inventory of Mesons Important in Radiative Analyses ...... 13

2.1 The current experimental status and theoretical predictions for the

branching fraction of weak radiative charm decays. Statistical and

systematic uncertainties are quoted for D0 φγ. The upper limits → for the remaining three modes are quoted at 90% Conﬁdence Limit

(CL)...... 17

2.2 Short distance contributions to radiative processes ...... 19

2.3 Summary of Theoretical Amplitude Predictions ...... 24

3.1 Crosssectionsforthedominantmodes ...... 39

3.2 LikelihoodPIDSelection ...... 57

4.1 MonteCarloSampleSize...... 66

xi 4.2 DataSampleSize ...... 66

4.3 Analysis D0 Vγ Cuts ...... 77 → 4.4 D0 φγ CutEfficiencies...... 78 → ∗0 4.5 D0 K γ CutEfficiencies ...... 79 → 0 5.1 Correlation coefficients between D mass and cos(θH )...... 82

5.2 Final Fit Parameters for D0 φγ PDF ...... 85 → 5.3 Final Fit Parameters for D0 φπ0 PDF...... 88 → 5.4 Final Fit Parameters for D0 φη PDF...... 91 → 5.5 Final Fit Parameters for Combinatoric BG PDF ...... 93

5.6 D0 φγ FitProcedureSummary...... 96 → ∗0 5.7 Final Fit Parameters for D0 K γ signalshape ...... 105 → 5.8 Final Fit Parameters for D0 K−π+π0 signal shape ...... 106 → ∗0 5.9 Final Fit Parameters for D0 K η signalshape ...... 108 → 5.10 Final Fit Parameters for combinatoric background ...... 110

∗0 5.11 D0 K γ FitProcedureSummary ...... 113 → 6.1 D0 K0π0 ControlSampleFitResults ...... 122 → s 6.2 D0 K0π0 ControlSampleFitResults ...... 123 → s 6.3 D0 K−π+π0 ControlSampleFitResults ...... 124 → 6.4 CorrectionParameters ...... 128

6.5 Run Dependency of Correction Ratios ...... 129

6.6 Correction Factors using two Control Samples ...... 129

8.1 Yields with independent Control Samples correction applied . . . . . 152

8.2 The change in yields as the D0 Vη signal shape is allowed to ﬂoat. 152 →

xii 8.3 The change in yields as the combinatoric BG signal shape is allowed

toﬂoat...... 153

8.4 SummaryofthePIDsystematicerrors ...... 158

8.5 Single Photon Systematic Corrections ...... 164

8.6 Summary of Selection Based Systematics ...... 167

8.7 Summaryofallsystematicerrors ...... 168

A.1 SampleofunskimmedDataandMC ...... 178

xiii LIST OF FIGURES

Figure Page

1.1 Examples of Electromagnetic Processes ...... 7

1.2 ExampleofQCDProcess...... 8

1.3 ExamplesofWeakProcesses...... 14

2.1 ExampleofRadiativePenguins ...... 18

2.2 Long distance contributions to radiative D0 processes ...... 21

2.3 D0 K−π+ InvariantMass ...... 26 → 2.4 φ helicityangledeﬁnition...... 30

2.5 Signal and Background Distributions ...... 32

3.1 PEP-IIschematicrepresentation...... 35

3.2 Total recorded BABAR luminosity ...... 38

3.3 BABAR schematicrepresentation ...... 42

3.4 BABAR SVTschematicrepresention ...... 43

3.5 BABAR DCHSchematicView...... 45

3.6 Schematic and example of DIRC internal reﬂection ...... 48

3.7 PIDSeparationPlots...... 49

3.8 EMCSchematic...... 49

xiv 3.9 EMCEnergyResponse...... 51

3.10 High Voltage Power Supply and GUI ...... 54

4.1 π0 Veto ...... 68

4.2 M(D∗+) M(D0) Cut Selection – D0 φγ Analysis ...... 70 − → ∗0 4.3 M(D∗+) M(D0) Cut Selection – D0 K γ Analysis ...... 71 − → 0 ∗0 4.4 cos(θH ) Cut Selection – D K γ Analysis...... 74 → 4.5 CutOptimization...... 76

5.1 Fits to D0 φγ ...... 84 → 5.2 D0 φπ0 Projections ...... 87 → 5.3 Fits to D0 φη ...... 90 → 5.4 Two Dimensional Combinatoric Background ...... 94

5.5 CombinedSignalFits...... 98

5.6 Varying (D0 φγ)...... 100 B → ∗0 5.7 D0 K γ SignalEvents ...... 104 → 5.8 D0 K−π+π0 BackgroundShape...... 107 → ∗0 5.9 D0 K η SignalEvents ...... 109 → ∗0 5.10 Combinatoric Background to D0 K γ ...... 111 → 5.11 CombinedSignalShapeFit ...... 114

∗0 5.12 Varying (D0 K γ)...... 116 B → 6.1 Comparing Control Samples and Background ...... 120

6.2 FitsofMCControlSamples ...... 126

6.3 FitsofDataControlSamples ...... 127

0 0 0 6.4 Fit to D K γ Data Allowing for D Ksγ Signal ...... 131 → s → 7.1 D0 φγ FittoData...... 136 →

xv ∗0 7.2 D0 K γ FittoData ...... 138 → 7.3 (D0 φγ)measuredforeachrun ...... 140 B → ∗0 7.4 (D0 K γ)measuredforeachrun...... 141 B → 0 ∗0 ∗0 7.5 N(D K γ) as a function of K cos(θH ) ...... 142 → ∗0 8.1 Correlations between D0 K γ parameters...... 147 → 8.2 N(D0 Vγ) while varying fit parameters on data ...... 149 → 8.3 N(D0 Vγ) while varying correction ratios on data ...... 151 → 8.4 D0 φγ PIDefficiencyplots ...... 159 → ∗0 8.5 D0 K γ Kaon PID efficiency plots ...... 160 → ∗0 8.6 D0 K γ PionPIDefficiencyplots ...... 161 → ∗0 8.7 D0 K γ Kaon PID efficiency plots ...... 162 → 8.8 π0 VetoEfficiency...... 165

8.9 N(D0 Vγ) while varying cut values ...... 167 → A.1 D0 K−π+ Fit ...... 176 →

xvi CHAPTER 1

Standard Model

1.1 Introduction to Particle Physics

Elementary Particle Physics seeks to understand what the fundamental building blocks of all nature are, what comprises what we observe, and what forces drive our universe. Initially the concept that matter had a fundamental building block, the atom, was a philosophical question, pondered by great minds from ancient India to Greece. These early particle physicists used reason and observation to attempt to understand what is fundamental to all matter. Observation of the visible world suﬀered from being constrained to the macroscopic, and therefore lacked the ability to probe beyond what was seen by the human eye.

By looking past what can be observed by the unaided eye greater understanding of what is common to all of matter was achieved. The microscope enabled mankind to probe into the ﬁner divisions of matter, such as cells. Finding commonalities between organisms in the biological world began to suggest that the search for what is fundamental was reasonable.

In 1897, J. J. Thomson utilized the cathode ray tube to determine the charge to mass ratio of electrons, by measuring the deﬂection of cathode rays in a magnetic

1 ﬁeld. With this apparatus he was able to ﬁnd evidence that the electric charge came in discrete quanta, further suggesting that the idea of “the fundamental” was again reasonable.

This was followed a decade later by Earnest Rutherford’s famous Gold Foil experiment. Rutherford was able to determine the existence of the atomic nucleus by directing helium ions onto a thin sheet of gold foil and observing the backscatter of incident particles. After another decade Rutherford discovered the proton, one of the constituent particles of the nucleus.

In 1930 James Chadwick discovered evidence of a neutral particle with a mass slightly larger than the proton, now known as the neutron. With the discovery of electrons, protons, and neutrons the picture of the atom was seemingly complete.

This was contradicted in 1936 by the observation of a particle more massive than electron but with equal charge. This particle is known today as the muon (µ) and oﬀered the ﬁrst glance of generations of matter beyond the particles seen within the atom. In 1964, Gell-Mann and Zweig suggested that neutrons and protons were not fundamental, rather the idea that these nucleons were comprised of “quarks”.

Over the past half of a century the community of Particle Physicists has developed the “Standard Model (SM) of Particle Physics”. This model includes twelve unique and fundamental particles and four force carriers. The Standard Model has successfully described a vast cross section of physical phenomena, including the electromagentic force, nuclear decay, and the binding of positively charged protons within a nucleus. Further, all experimental tests to date have validated SM. Gravity is the lone force not included within SM.

Within the next few years the ﬁnal particle predicted under the SM, called the

2 Higgs boson, may be observed. With its observation, one of the most complete, well tested, and veriﬁed theoretical models of our physical world will be ﬁnalized. While the studies may help add detail to a small and somewhat forgotten aspect of this vast and comprehensive picture.

1.2 Fundamental Particles

The Standard Model includes twelve unique and elementary particles and their anti-matter companions. These twelve can be divided into two classes, quarks and leptons. Quarks are bound together by a force known as the strong force to form protons and neutrons. Leptons include electrons, muons and taus, as well as their nearly massless partner , neutrinos.

Leptons and quarks each are classiﬁed into three diﬀerent grouping known as generations. The vast majority of the world we live in is comprised of the lightest of these generation. For quarks this generation contains the up quark (u) and the down quark (d). For leptons the lightest generation is made up of the electron (e−) and the electron neutrino (νe). Protons and neutrons are constructed from u and d quarks, protons from the uud combination and neutrons from udd combination.

The other two generations are comprised of more massive and shorter lived quarks and leptons. Table 1.1 shows the observed quarks and leptons.

1.3 Fundamental Forces

Interactions between the matter particles shown in Table 1.1 are governed by four fundamental forces: electromagnetic, weak, strong, and gravitational. The gravitational force, though the most ubiquitous in the macroscopic, is also the weakest.

3 Lepton Charge (e) Mass (MeV/c2) electron (e) -1 0.511 −6 electron neutrino (νe) 0 < 3 10 × muon (µ) -1 105.658

muon neutrino (νµ) 0 < 0.19 tau (τ) -1 1776.90 0.20 ± tau neutrino (ντ ) 0 < 18.2 Quark Charge (e) Mass (GeV/c2) up (u) +2/3 0.001 – 0.003 down (d) -1/3 0.003 – 0.007 charm (c) +2/3 1.25 0.09 ± strange (s) -1/3 0.095 0.025 ± top (t) +2/3 172.5 2.7 ± bottom (b) -1/3 4.20 0.07 ±

Table 1.1: Important properties of the fundamental Standard Model matter par- 1 ticles [1]. All of the listed particles have spin 2 . The electric charge is written in terms of the electric charge of an electron e. The upper limits displayed are at the 90% conﬁdence level (C.L.).

Gravity is nearly 10−37 times weaker than the electromagnetic force, the other force we are familiar with in the macroscopic world. The weak force is responsible for nuclear β decay, in which a neutron (n) decays into a proton (p), electron (e−) and electron anti-neutrino (¯νe, note theν ¯ is meant to denote an anti-particle rather than a particle). The strong force is what binds together quarks to form neutrons and protons.

These forces act between quarks and leptons by exchanging particles known as force carriers. The force carriers are classiﬁed as gauge bosons, meaning they have integer spin, whereas quarks and leptons are classiﬁed as fermions, meaning they

4 have spin 1/2. Force carriers communicate forces between particle. For example the electromagnetic force between two charged particles is carried from one charged object to another by a virtual photon (denoted by the symbol, γ). By emitting and absorbing photons two charged objects are either attracted or repelled. Other examples of force carriers are the electrically charged bosons (W +, W −) or the neutral boson Z0, all of which carry the weak force, and the gluon which carries the strong force.

Leptons and quarks are diﬀerentiated by how they couple to these force carriers.

Leptons are able to couple to either photons or weak force carriers, but leptons do not experience the strong interaction and therefore do not couple to gluons.

Quarks interact with all the force carriers, and therefore are the only fermions to experience the strong force. The strong force is just that, STRONG, approximately

100 times greater than the electromagnetic force. However, it is only felt at very short distance, on the order of the size of an atomic nucleus, about 10−15m.

The Standard Model does not describe gravity. At this point gravity can be theoretically described as being carried by the graviton, a boson carrying spin 2.

The graviton has not been observed, but is predicted to be massless and act at inﬁnite distances. While all massive objects experience gravitational forces, it is a very small eﬀect when compared to any of the three other forces, and can be safely ignored for our purposes.

Table 1.2 summarizes some of the important characteristics of the force carriers described under the Standard Model as well as the graviton.

5 Carrier Charge (e) Spin Mass ( GeV/c2) Coupling Constant Force γ 0 1 0 1/137 Electromagnetic W ± -1 1 80.403 0.029 10−6 Weak ± Z0 0 1 91.1876 0.021 10−6 Weak ± g 0 1 0 100 Strong graviton 0 2 0 10−39 Gravitational

Table 1.2: A summary of the force carriers included in the Standard Model and the graviton. The coupling constant is deceptive, as it is not truly constant. The strength of these interactions is dependent on the energy scale at which the interaction takes place.

1.3.1 Electromagnetic force

The electromagnetic is the force felt by electrically charged objects. Other than gravity it is the most familiar force in everyday life. Within Quantum Electro-

Dynamics (QED) theory the electromagnetic force is propagated by the photon (γ).

The photon is massless and electrically neutral. Photons can couple to any charged particle, these include quarks, charged leptons, and W ± bosons. Since the photon is massless the force can act over inﬁnite distance, but its strength falls oﬀ as 1/r2, where r is the distance between charged objects.

In general, interactions between particles can be represented using “Feynman diagrams”. These are tools used to calculate the probabilities of processes occurring within the standard model. Two examples of these diagrams are shown in Figs.

1.1(a) and 1.1(b). Each of these reactions are the fundamental interactions of QED.

In each case an electrically charged particle (shown with a solid line) couples to a photon (shown with a wavy line). The coupling location is known as a vertex and carries with it the coupling constant, α, shown in Table 1.2. This coupling constant

6 l± q

γ γ

time

l± q (a) (b)

Figure 1.1: Examples of Electromagnetic Processes. (a) Shows a photon coupling to a charged lepton, l±. (b) Shows the photon coupling to a quark, q. In each case time is taken as propagating up the page. .

is a reﬂection of the strength of an interaction. In the electromagnetic case the coupling constant is on the order of 10−2 speciﬁcally, α = 1/137.

1.3.2 Strong force

The strong interaction is carried by gluons which do not couple to electric charge, as seen with photons in electromagnetic interactions. Rather gluons couple to particles carrying “color” charge. Quarks are the only fermions which carry color charge.

The color charge of a quark can take on three diﬀerent values: red (R), blue (B), and green (G). Gluons also carry color charge, but rather than assuming a single value like quarks, gluons carry both a color and an anti-color. Figure 1.2 shows one example of a (RG¯) gluon coupling to two up quarks, one carrying R color charge

7 u(G)

g(RG¯)

u(R)

Figure 1.2: Examples of Quantum Color-Dynamics (QCD) process. Shown here a u quark carrying R color charge couples to a gluon with RG¯ color charge and a u quark with G color charge. In this way the color charge is conserved in this interaction. .

and the second carrying G color charge.

Quarks are bound together with gluons to form “colorless” objects, which are objects with zero net color charge. This can occur in many ways, but the two most common are either by forming a color/anti-color pair (for example RR¯, GG¯, or

BB¯), or the combination of three quarks each carrying a diﬀerent color charge, and forming an RGB triplet. In this way color charge is not meant literally as visual color, but rather color is meant to evoke the idea of blending three independent colors together to form a colorless or “white” object.

When quarks are bound by gluons into colorless particles, the composite particle is known as a hadron. Hadrons are categorized into two classes: baryons and mesons. Baryons are objects with 1/2+ n spin, where n is a natural number. All well observed baryons appear to be composites of three quarks. As mentioned earlier this includes protons (uud) and neutrons (udd). Although it is theoretically possible

8 for ﬁve quarks to bind together into a colorless pentaquark, all observations of these states have been discredited.

In contrast mesons are hadrons with spin n, where n is a non-negative integer.

All well observed mesons are composites of a quark and an anti-quark. An example of this is the π+ meson, with quark content ud¯. The branch of the SM which explains how quarks are bound together by gluons is known as Quantum Chromodynamics

(QCD).

1.3.3 Weak force

The charged weak force carriers (W + or W −) can couple to either leptons or quarks. In order to conserve charge, a W − boson must couple with quarks and leptons of diﬀerent charge. This can occur in two ways. In leptonic events W −

− − couples a lepton, l , and a neutrino, νl. In hadronic events a W couples a quark, q, of charge +2/3 and a second quark, q′ of charge 1/3. Each of these examples is − shown in Figs. 1.3(a) and 1.3(b).

The weak interaction can be contrasted with the electromagnetic interaction in how it couples to different types, or flavors, of quarks. Where the electromagnetic interaction only coupled into two quarks of the same flavor, the weak interaction can couple quarks of different flavors. This allows for interactions such as, d W − + u. → Through coupling quarks of different flavors, the weak interact allows for flavor changing currents. This can be seen in β decays. In this process a neutron, with quark content udd, decays into a proton, with quark content uud, as well as an

− electron, e , and anti-neutrino,ν ¯e. The process is shown in Fig. 1.3(c).

9 CKM Matrix

The process shown in Fig. 1.3(c) shows a charged lepton coupling to a neutrino of the same family. The weak interaction also allows for coupling of quarks between families. Typically we can divide the quarks into three generations based on the mass of each quark. u c t (1.1) d s b The weak force allows for coupling not just between u and d, c and s, and t and b, but also between u and s, u and b, c and d, etc. In this way we can re-express the generations using Eq. 1.2. This expression shows u coupling to d′ which is a linear combination of d, s, and b. In this way the weak eigenstates can be represented as linear combinations of mass eigenstates. Similiar relationships can be drawn for s′ and ‘b′. This is done using the matrix shown in Eq. 1.3.

u c t (1.2) d′ s′ b′

′ d Vud Vus Vub d ′ s = Vcd Vcs Vcb s (1.3)  ′      b Vtd Vts Vtb b 2       In Eq. 1.3 Vij is the probability of quark, qi, weakly coupling with quark, qj. | | The matrix is called the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The CKM matrix is an arena which oﬀers Experimental Particle Physicist a wide number of tests of the standard model. Assuming the three quark model is complete the CKM matrix must be unitary, that is U †U = I, where U † is the complex transpose of U and I is the identity matrix. If this is the case then the quark q, with change 1/3 − will always transition into one of the three observed quarks with charge 2/3. The

10 current experimental limits on the CKM matrix are shown in Eq. 1.4 [1].

0.9739 0.9751 0.221 0.227 0.0029 0.0045 − − − V = 0.221 0.227 0.9730 0.9744 0.039 0.044 (1.4) CKM   0.0048− 0.014 0.037 − 0.043 0.9990 − 0.9992 − − −   1.3.4 Measuring Branching Fractions

While the SM provides a strong theoretical framework to understand Particle

Physics, experimentalists are primarily concerned with what can actually be measured. For example, we cannot directly measure the CKM matrix elements. Instead we are able to measure how frequently a particle decays into a particular ﬁnal state.

By knowing the frequency of a particular decay, constraints can be placed upon theoretical parameters such as the CKM matrix values. The frequency with which a particle X decays into a ﬁnal state Y + Z is known as a branching fraction, and is expressed in Eq. 1.5.

N(X Y Z) (X Y Z) = → (1.5) B → N(X)

In this expression, N(X Y Z) is the number of instances particle X decays into → ﬁnal state particles Y and Z, and N(X) represents the total number of X particles created.

While we can eﬀectively measure the number of X particles decaying into the

ﬁnal state Y Z, it is often impractical to measure N(X). Instead we measure the frequency of a particular decay in ratio to the frequnecy of a well measured decay.

This is known as a reference mode. In our case, we are measuring decays of D0 → ∗0 Vγ, where V can take the form of either a φ or K meson. We measure the

11 branching fraction of these modes in ratio to the well measured decay D0 K−π+, → using Eq. 1.6. As shown in Table 1.3 the branching fraction of D0 K−π+ is → measured with an uncertainty less than 2%.

0 0 0 − + (D Vγ) NMeasured(D Vγ) ǫReconstruction(D K π ) B 0 → − + = 0→ 0 → − + (1.6) (D K π ) ǫReconstruction(D Vγ) NMeasured(D K π ) B → → → 0 0 We are able to measure the number of reconstructed D Vγ events, NMeasured(D → → Vγ), and the number of reconstructed D0 K−π+ events, N(D0 K−π+), using → → 0 the BABAR dataset. The reconstruction eﬃciencies, ǫReconstruction(D Vγ) and → 0 − + ǫReconstruction(D K π ), are estimated using Monte Carlo (MC) simulations of → the BABAR detector.

Note to Reader

Throughout this analysis I will be taking charge conjugation as implicit. Charge conjugation refers to a particle’s anti-particle analog. For example when I refer to

∗0 studies of the decay D0 K γ, this is meant to include the analogous anti-particle → decay, D¯0 K∗0γ. → Over the course of this analysis we look only at mesons decaying to mesons and photons. Table 1.3 summarized the mesons important to our analysis.

12 Name Mass ( GeV/c2) Spin Quark Prominent Content Decay(s) B D∗+ 2.010 0.4 1 cd¯ D0π+ 0.677 0.005 ± ± D0 1.865 0.17 0 cu¯ K−π+π0 0.135 0.006 ± K−π+ 0.0382± 0.007 ∗0 ± K η 0.019 0.008 φπ0 (7.4 0.±5) 10−4 φη (1.4 ± 0.4) × 10−4 ± × φ 1.019 0.00002 1 ss¯ K−K+ 0.491 0.006 ∗0 ± ± K 0.892 0.00026 1 sd¯ K−π+ 2/3 ± K0 0.498 0.000022 0 sd¯ π−π+ 0.692 0.00005 s ± ± K− 0.494 0.000016 0 su¯ - - ± η 0.548 0.00018 0 uu¯ γγ 0.394 0.002 ± ± π+ 0.140 (3.5 10−7) 0 ud¯ - - ± × π0 0.135 (6 10−7) 0 uu¯ γγ 0.988 0.032 ± × ±

Table 1.3: A list of mesons which are of importance to this analysis. The quark content reﬂects the valence quarks. The prominent decay channels listed are those used extensively in this analysis and the branching fraction, , reﬂects the current world average for the meson to decay through this channelB [1]. K− and π+ have a small probablity of decaying within the detector volume, and are therefore considered stable for our purposes. Charge conjugation is implied throughout.

13 νl q′

W − W −

l− q

(a) (b)

ν u d u ¯e e−

W −

u d d (c)

Figure 1.3: Examples of Weak Processes. (a) Shows a W − boson coupling − to a charged lepton, l which in turn scatters intonu ¯ l. While this reaction deals with negatively charged leptons an identical process can be drawn for + + − l W ν¯l. (b) Shows a W boson coupling with a quark, q, which in turn→ scatters to a quark of diﬀerent ﬂavor, q′. In order to conserve charge q′ must have charge equal to q + 1. (c)Shows the weak process which causes β decay. .

14 CHAPTER 2

Search for Radiative Decays

The focus of this analysis is the study of radiative decays of D0 mesons, and in particular the decays shown in Eqs. 2.1 and 2.2

∗0 D0 K γ → ∗0 K K−π+ (2.1) → D0 φγ → φ K−K+ (2.2) →

Analogous radiative decays have been a topic of great interest in B meson physics for much of the past ﬁfteen years. The ﬁrst observation of the decay B0 K∗0γ → was made in 1993 and has been cited over 500 times since the original publication

[2].

Radiative decays of charmed mesons have not had the same level of attention, either experimentally or theoretically. In this document I describe the search for radiative decays of D0 mesons as conducted at the BABAR detector, located at the

Stanford Linear Accelerator Center (SLAC). The results will be compared to current theoretical predictions and previous experimental searches.

15 2.1 Previous Experimental Studies

There have been two documented experimental searches for radiative decays of

D0 mesons. In 1998, the CLEO II experiment, located at the Cornell Electron

Storage Ring (CESR), studied radiative decays of the type D0 Vγ where V was → ∗0 one of four vector mesons, (K , ρ0, ω, and φ). This search saw no evidence for radiative decays, and instead set upper limits for the branching fractions of each of these modes with a 90% conﬁdence level. The CLEO II experiment was limited by a dataset approximately 1.2% the size of the current BABAR dataset.

In 2003, the Belle experiment, located at High Energy Accelerator Research

Organisation (KEK) in Tsukuba, Japan, measured the branching fraction of D0 → φγ, serving as the ﬁrst observation of a radiative decay of D0. This measurement was made with a dataset approximately 20% the size of the current BABAR dataset.

Table 2.1 shows the current theoretical predictions, the experiment upper limits

∗0 for D0 K γ, D0 ρ0γ, and D0 ωγ, as well as the Belle measurement of → → → (D0 φγ). Using the larger BABAR dataset we hope to be able to observe and B → improve upon the precision of these previous studies.

2.2 Radiative Penguins

The greater interest in radiative decays of B mesons arises from the implications these decays have upon the CKM matrix elements shown in Eq. 1.4, speciﬁcally Vts

∗ and Vtd. These elements are small but accessible by measuring the decays B K γ → and B ργ respectively. → Radiative decays of B mesons proceed largely through so called radiative “penguin” diagrams, two examples of which are shown in Fig. 2.1. In these processes a

16 Mode Theoretical Predictions Experimental B ( 10−5)B Status( 10−5) [3, 4,× 5, 6, 7, 8, 9] × 0 +0.70 +0.15 D φγ 0.1 3.4 2.40−0.61(stat)−0.17(sys) [10] → ∗0 − D0 K γ 7.0 80. < 76. [11] → − D0 ρ0γ 0.1 6.3 < 24. [11] D0 → ωγ 0.1 − 0.9 < 24. [11] → −

Table 2.1: The current experimental status and theoretical predictions for the branching fraction of weak radiative charm decays. Statistical and systematic uncertainties are quoted for D0 φγ. The upper limits for the remaining three modes are quoted at 90% Conﬁdence→ Limit (CL)

ﬂavor changing neutral current arises through W − emission and rescattering with an internal quark. Prior to rescattering either the internal quark or W − couples with a photon. The latter is shown in both Figs. 2.1(a) and 2.1(b). In these decays the internal quark can take on three possible ﬂavors: t, c, or u in B meson decays, and b, s, or d in D meson decays.

Theoretical predictions for the branching fractions of these decays have been based upon the partial decay width, Γ. The partial decay width is related to the branching fraction using Eq. 2.3

Γ(X Y Z) = Γtot(X) (X XY ) (2.3) → B → 1 = τ (2.4) Γtot(X)

In Eqs. 2.3 and 2.4, Γtot(X) is the total decay width of a particle X and τ is the lifetime of X.

Theoretical calculations for the partial decay width of Q qγ, are made while → ignoring the spectator quark contributions (d¯ in Fig. 2.1(a) andu ¯ in Fig. 2.1(b)),

17 Vbj Vjs Vci Viu

b t, c, u s c b,s,d u 0 0 0 0 B K∗ D ρ

d¯ u¯ (a) (b)

Figure 2.1: (a) Feynman diagram of a radiative penguin decay of the B0 meson. In this ﬁgure Vbj and Vjs refer to the CKM matrix elements between a b or s quark and a j quark, where j is either t, c, or u. Contributions from all possible quarks must be considered by summing over the internal quarks, t, c, and u. (b) An example of the analogous decay of D0 mesons. In this case the virtual quark takes the form of a b, s, or d quark.

and are expressed in Eq 2.5. [5]

3 (0) 5 ∗ 2 Γ m V VqiF (mi) (2.5) Q→qγ ∝ Q| Qi | i=1 X x3 5x2 2x 3x2ln(x) 2x3 + 5x2 x 3x3ln(x) F (x)= − − + + − (2.6) ± 4(x 1)3 2(x 1)4 4(x 1)3 − 2(x 1)2 − − − − In Eq. 2.5, mQ refers to the mass of the initial heavy quark. In the case of B decays this is the b quark while Q = c in D decays. The sum shown in Eq. 2.5 is taken over the three internal quarks. VQi and Vqi are the CKM matrix elements between the initial state quark and internal quark and the ﬁnal state quark and internal quark, respectively. Finally, F (mi), is a function strictly of the internal quark mass, shown in Eq. 2.6. The sign of the ﬁrst term in Eq. 2.6 is equal to the charge of the internal quark. As the masses of internal quarks become large F (mi) is dominated by the

ﬁnal term in Eq. 2.6. This leads to contributions from the most massive ﬂavor of quark, t, dominating. This s shown in Table 2.2, F (mi) is greatest when x = M(t).

Because B penguin decays proceed dominantly through internal t quarks, they

18 ∗ Quark F (Mi) V VqiF (mi) | Qi | B u 2.27 10−9 1.29 10−12 - c 2.03 × 10−4 7.34× 10−6 - t 0×.39 1.53 × 10−2 - × b sγ - - 1.29 10−4 → × d 1.57 10−9 3.36 10−10 - s 2.92 × 10−7 6.26× 10−8 - b 3.31 × 10−4 3.17 × 10−8 - × × c uγ - - 1.39 10−17 → ×

Table 2.2: The amplitudes each internal quark line contribute to the overall decay rate of Q qγ is shown. The calculated branching fraction is appears in the ﬁnal → 0 column. This is found by multiplying the decay width ΓQ→qγ by the measured value (B D0lν). The values for all contributions are taken from Ref [5] B →

are of great interest in studying the ratio of CKM matrix elements, Vtd/Vts. The decays B (ρ,ω)γ and B K∗γ provide the ability to measure these rare quark → → transitions. The diagram for the former can be written exactly as Fig. 2.1(a) only exchanging the ﬁnal state s quark with a d and changing Vjs to Vjd. The ratio has been recently measured [12], and the result is shown in Eq. 2.7.

Vtd +0.021 | | = 0.200−0.020 0.015 (2.7) Vts ± | | Also shown in Table 2.2 is the very small branching fraction of charm meson penguin diagrams. Even with enhancements the largest predicted branching fractions using penquin diagrams are on the order of 10−8. This number places measuring radiative D decays an order of magnitude smaller than the sensitivity of any running experiment.

19 2.2.1 Long Distance Contributions

The approach shown in Section 2.2 calculates radiative decays using contributions strictly from penguin processes. While these are the dominant contributions for radiative B decays, radiative charm decays are predicted to proceed predomi- nantly through long distance effects. A process is labeled as a long distance effect if it is so dominated by strong force processes that is no longer calculable using preturbative QCD. As a result, the theoretical predictions of long distance effects often vary significant dependent upon the models used to estimate long distance contributions. Measurements of long distance effects provide constraints on nonpre- turbative QCD calculations.

There have been several attempts to estimate the long distance eﬀects contributing to radiative charm decays. As shown in Table 2.1, these predictions vary by as much as an order of magnitude. One consistent feature across of all of these predictions is that the rates for long range contributions are signiﬁcantly larger than the penguin diagram processes previously described. Based on long distance calculations radiative D decays can be measured using current high luminosity high energy physics experiments.

When considering the theoretical predictions, we can look at the two most comprehensive theoretical studies [5, 9]. As shown in Fig. 2.2 each of these theoretical studies focused on three long distance diagrams contributing to radiative D0 decays. Figs. 2.2(a) and 2.2(b) are examples of so called pole diagrams. In the case of Fig. 2.2(a) a D0 transitions to a virtual pseudoscalar meson, P , through a W + exchange between the D0 quark lines. A photon then couples to either of the P

∗0 meson’s quark lines. After this, occurrs P transitions into either a K or φ meson.

20 c s c s

¯ 0 ¯ 0 D0 P K∗ (φ) D0 V K∗ (φ) u¯ d(s) u¯ d(s) (a) (b)

c s 0 K¯ ∗ (φ)

d¯(¯s) D0 u

u¯ (c)

Figure 2.2: Long distance contributions to radiative D0 processes. The diagrams shown in (a) and (b) are pole diagrams. These are considered parity conserving processes. Diagram (a) shows a transition from a D0 to a virtual pseudoscalar meson, P through internal W + exchange. P then emits a photon and transitions ∗0 to either a K or φ meson. This is in contrast to (b) where photon emission proceeds W + exchange resulting in the transition from a D0 to a vector meson, V . It is important to note that both diagrams show the photon emitting from the upper most quark line, while in principle it can be emitted from either quark line. Diagram (c) shows the radiative process proceeding through Vector Meson Dominance (VMD). Where (a) and (b) are strictly considered parity conserving, (c) can have both a parity conserving and a parity violating component.

A similar process is shown in Fig. 2.2(b), but here the photon emission takes place prior to the W + exchange.

Another contributing process is shown in Fig. 2.2(c). The diagram shown here demonstrates the radiative decay which occurs through a quark-antiquark annihila- tion. This diagram proceeds through a process known a Vector Meson Dominance

(VMD). VMD was first considered in the context of photon scattering off a proton target. When this scattering occurs it is theoretically possible for a photon to produce a virtual quark-antiquark pair which scatters off one of the quarks within

21 the target proton. Using VMD models it is possible to relate the cross section of photon-proton scattering to the cross section π-proton scattering [13]. In the case of radiative decays the process is reversed, rather than an incident photon coupling to a pair of virtual quarks, the virtual quarks annihilate to produce a photon.

Using contributions from each of these diagrams the decay width of radiative processes can be expressed using Eq. 2.8.

3 0 q 2 2 Γ(D Vγ) = PC + PV (2.8) → 4π |A | |A | 2 m 0 m q = D 1 V (2.9) 2 − m 0 D ! N i PC = (2.10) A APC i=1 XM i PV = (2.11) A APV i=1 X Eq. 2.8 relates the decay width to q, the photon energy within the D0 rest frame, and the amplitudes of parity conserving (PC) and parity violating (PV) processes,

0 denoted as PC and PV , respectively. The photon energy in the D rest frame is A A shown in Eq. 2.9, where mV is the mass of the vector meson and mD0 is the nominal

0 mass of the D meson. PC and PV are each taken to be the linear sum of the A A parity conserving or parity violating amplitudes.

Figs. 2.2(a) and 2.2(b) represent parity conserving processes. The parity conservation of Fig. 2.2(c) depends on the state of the V V ′ system, where V is either

∗0 K or φ and V ′ is the virtual vector meson. If V V ′ is in a P-wave state, then parity is conserved; if V V ′ is in either an S or D-wave state, then parity is violated [5].

Refs. [5] and [9] considered contributions from the three diagrams through different models. Ref. [5] calculated the contributions from all diagrams using a

22 combination of experimental results and lattice QCD calculations. The authors use limited experimental data to model contributions from Fig. 2.2(c). Their estimata- tions were made by examining data from D MV processes and approximating → coupling of V γ. The experimental data was limited at the time of the publica- → tion, leading to large uncertainties. While the paper does not quote these directly, based upon the range of values used, the uncertainties may be on the order of 30%.

For labeling purposes we call these studies (PD-VMD).

Ref. [9] used a combination of Heavy Quark Eﬀective Theory and Chiral La- grangians (HQET-CL) to estimate contributions from the processes shown in Fig.

2.2. The approaches of Ref. [5] and [9] result in very similar estimations for the amplitude of each diagram. These are summarized in Table 2.3, with contributions from Fig. 2.2(a) labeled I , and contributions from Fig. 2.2(b) labeled II . APC APC Contributions from Fig. 2.2(c) can have both a parity conserving and a parity violating component, labeled III and I respectively. APC APV The main discrepancy between these two predictions is the amount of interference between the pole diagrams. Where Ref [5] suggests that the pole diagram contributions are nearly canceled, Ref. [9] make no such assertion. Overall our analysis should clarify several topics of interest:

∗0 Potentially observe a new long distance decay D0 K γ and conﬁrm a • → previous measurement of D0 φγ. →

Deliver a measurement of an additional long distance process, which may help • reduce uncertainties in other long distance calculations.

Perhaps clarify the amount of interference between pole diagrams. •

23 ∗0 D0 K γ D0 φγ → → PD-VMD HQET-CL PD-VMD HQET-CL I −8 −1 PC ( 10 GeV ) 5.6 6.4 0.7 1.8 AII × −8 −1 PC ( 10 GeV ) -5.9 6.2 -1.6 1.34 AIII × −8 −1 PC ( 10 GeV ) 3.8 - (0.6, 3.5) - AI (×10−8GeV −1) (5±.1, 6.8) 5.5 ±(0.9, 2.1) 1.8 APV × (D0 Vγ)[∗]( 10−5) (7, 12) 35. (0.1, 3.4) 1.9 B → ×

Table 2.3: A summary of theoretical amplitude predictions (amplitudes carrying units of 10−8GeV −1) based on two models, (HQET-CL), and VMD and Pole Di- agrams (VMD-PD), It should also be noted that the authors of Ref [9] make no assumptions about the relative sign of the amplitudes. [∗] The branching fraction is −12 calculated based on Γtot = 1.6 10 GeV . ×

2.3 Experimental Challenges

In the BABAR experiment we are only able to directly detect a limited set of particles. These include charged pions (π±), kaons (K±), protons (p±), electrons

(e±), and muons (µ±), as well as photons (γ). Further we are only able to measure a limited number of observables. These include charge, momentum (p), energy loss (dE/dx), Cherenkov angle (θc), and position over time. The challenge of any experiment is understanding how to translate these observables into a measurement with physical meaning.

If we are able to identify every charged track correctly as either a pion, kaon, proton, etc, we can correctly deﬁne the four vector of each candidate. The four vectors can be added together and the invariant mass of the sum can be found. If we sum together two candidates, the invariant mass of the combination is deﬁned

24 using Eq. 2.14.

µ pi = (Ei, p¯i) (2.12)

M(12)2 = (pµ + pµ) (p1 + p2 ) (2.13) 1 2 • µ µ M(12)2 = m2 + m2 + 2(E E p¯ p¯ ) (2.14) 1 2 1 2 − 1 • 2

2 By measuring momentap ¯1 andp ¯2, and correctly identifying each candidate, M(12) can be related back to a single source. Taking the decay D0 K−π+ as an example, → we add the four vectors of every kaon and pion in an event. After this M(K−π+)2 can be calculated for each K−π+ pair. The distribution of M(K−π+) across all candidates should have a peak, if K− and π+ arise from a common source, such as a D0. This can be seen in Fig. 2.3, which shows a strong peak in the M(K−π+) distribution near the nominal D0 mass.

The peak seen in the invarient M(K−π+) distribution near the nominal D0 mass can be interpreted as K−π+ being produced by a decaying D0. While this works well in principle, there are several types of events which result in fake signals near the D0 mass which confuse our measurements.

The simplest background to model is from random events. On average there are 11 charged tracks in each event at the BABAR detector. These are produced by many possible decays. Two random and unrelated tracks can combine resulting in an invariant mass near the nominal D0 mass. For the most part these events have a ﬂat or at least easily modeled background shape. In Fig. 2.3 the ﬂat background below the D0 mass results largely from these random combinations.

Another background arises from incorrectly identifying particles. For example in

25 ) 2 105 Entries / (2 MeV/c

104

103

1.7 1.75 1.8 1.85 1.9 1.95 2 M(K- π+) (GeV/c2)

Figure 2.3: The invariant mass combination of K− and π+ candidates.

D0 K−π+ kaons can be misidentified as pions and vice versa. The technical de- → tails of correct particle identification will be discussed in more detail in Section 3.3, but in general BABAR is able to correctly discriminate between kaons and pions more than 95% of the time. This still leaves a small percentage of misidentified particles.

This result in background peaking in the vicinity of the nominal D0 mass. In the event of a kaon being misidentiﬁed as a pion the resulting M(K−π+) distribution will be shifted to lower mass values, relative to the nominal D0 mass. Conversely misidentifying a pion as a kaon will cause a peak in the high end of the M(K−π+) mass distribution.

Another type of background, which is particularly problematic to our analysis, are decays with nearly the same ﬁnal state, but with an additional ﬁnal state particle. For example D0 K−π+π0 events in which only the K− and π+ are →

26 reconstructed cause the small peak occurring near 1.70 GeV/c2 in Fig. 2.3. Because

D0 K−π+π0 events peak far removed from the nominal D0 mass, they can be → easliy separated from real signal events. This is not the case for our analysis, where

D0 Vπ0 overlaps very strongly with D0 Vγ events. → → ∗0 In the context of measuring D0 K γ and D0 φγ, a signiﬁcant background → → arises from both π0 and η meson decays. As shown in Table 1.3 both π0 and η decay into pairs of photons. The π0’s which are produced in the decays: D0 φπ0 and → ∗0 D0 K π0, produce particularly prominent backgrounds to radiative D0 events. → If the detector misses one of these photons, a fake signal closely mimicking a true radiative decay results.

Reconstructing D0 φπ0 decays as D0 φγ events, will produce an invari- → → ant, M(φγ), which strongly peaks near the nominal D0 mass. This is shown in

Fig. 2.5(a). This can be contrasted with the invariant mass distribution of true

D0 φγ events, shown in Fig. 2.5(g). The background is less signiﬁcant from → D0 φη decays. As with D0 φπ0 events, one of the photons produced by the → → η is not detected. However, the M(φγ) distribution of D0 φη events peaks well → below the nominal D0 mass, as shown in Fig. 2.5(d). All distributions shown in

Fig. 2.5(d) are obtained by running MC simulations of these decays as they would occur within the BABAR detector.

The greatest complication from D0 K−π+π0 and D0 φπ0 decays is not → → that they give rise to a peak in the invarient mass just below the nominal D0 mass, but that they occur at rates far above the predicted rates for the signal modes:

∗0 D0 K γ and D0 φγ. Comparing the predicted branching fractions shown in → → Table 2.1 to the measured branching fractions for the backgrounds shown in Table

27 ∗0 1.3, it is clear that for every D0 K γ decay we would expect to see at least 176 → D0 K−π+π0 events. In order to make any practical measurement we must be → able to reduce these background events and also to accurately estimate the invariant mass distribution of the remaining D0 K−π+π0 events. →

2.3.1 Helicity Angle

Fortunately we can remove a large source of background from D0 K−π+π0 → ∗0 and D0 φπ0 by examining the angular distribution of K and φ decays products. → ∗0 For an object with non-zero spin, such as K , φ, or γ, the helicity is deﬁned with

Eq. 2.15. σ¯ p¯ λ = • (2.15) p | | Whereσ ¯ is the vector of spin components andp ¯ is the momentum vector. A spin

1 particle can assume three values for helicity, λ 1, 0, 1. A helicity value of +1 ∈ − occurs when spin is aligned parallel to the axis of motion, a value of 1 occurs when − spin is aligned antiparallel to the axis of motion, and value of 0 occurs when the spin is aligned prependicular to the axis of motion.

Photons present a special case. Since the photon is massless, there exists no frame in which it can be considered at rest. Therefore its spin must either be parallel or antiparallel to its axis of motion, meaning its helicity must be 1. Helicity is a ± ∗0 conserved quantity, meaning that if the photon has λ = 1, either K or φ must have helicity λ = 1. Conversely π0 and η have zero spin, meaning in each case − λ = 0. Since these particle are produced by the spinless D0 candidate the helicity

∗0 state of φ or K is ﬁxed to λ = 0.

28 The implication of this diﬀerence is made apparent when looking at the spherical harmonics. These functions are used to describe the angular distribution of spins of

∗0 a free particle, speciﬁcally K or φ, and are shown in Eq. 2.16 When λ = 1 this ± forces m = 1, and if λ = 0 then m must be also be 0. ±

Y 1 (θ,φ) P m(θ)e−imφ m ∝ 1 Y 1(θ,φ) cos(θ) 0 ∝ Y 1 (θ,φ) sin(θ)e±iφ (2.16) ±1 ∝

In this expression, θ is the polar coordinate, φ is the azimuthal coordinate, and m is the componet of spin along the axis of motion. The angular distribution is equal to Y l 2. From this we see that a vector meson, V , if produced by D0 Vγ must | m| → have m = 1 and if produced by decays such as D0 Vπ0 have m = 0. This ± → results in a V from D0 Vγ decays having a sin2(θ) helicity distribution, and a V → from D0 Vπ0 or D0 Vη decays having a cos2(θ) helicity distribution → → While V is very short lived, we can eﬀectively measure its angular distribution, by examining the distribution of its decay products. The helicity angle, θH can be deﬁned as the angle between the three momentum of one of V ’s decay products and the three momentum of the D0, both measured in V ’s rest frame. A graphical representation of this is seen in Fig. 2.4. By measuring helicity angle we can better separate signal from the π0 and η backgrounds.

∗0 As mentioned earlier the major background to D0 K γ is from D0 → → K−π+π0 decays. This ﬁnal state is reached through many possible intermediate

∗0 ∗0 states. About 14% of these events proceed through D0 K π0. These K candi- → 2 dates have an angular distribution of cos (θH ). The other intermediate states have

29 K+

θH

D0 X(γ, π0, orη)

K−

Figure 2.4: A model of the helicity angle for D0 φX events. If X is a scaler particle the distribution should be proportional to cos→2. If X is a photon the distribution will follow sin2.

a more complicated structure. We will discuss how these are with dealt in Section

5.2. The helicity distributions of D0 φγ, D0 φπ0 and D0 φη candidates are → → → shown in Figs. 2.5(h), 2.5(b), and 2.5(e).

2.3.2 Other Discriminating Observables

In order to reduce signiﬁcant background from random events, studies of D0 mesons often require that a candidate D0 be produced as a result of the decay,

D∗+ D0π+ . The mass diﬀerence, or Q value, between a D0 meson and a → slow D∗+ meson is very small, with a nominal value of 0.1455 GeV/c2. By detecting the

+ ∗+ slow pion, πs , produced by a decaying D , we can define the mass difference as as ∆M = M(Vγπ+) M(Vγ). Th advantage of this method is that the ∆M s − distribution is narrow and somewhat independent of the M(Vγ) distribution. This allows us to select a narrow region of ∆M and remove a significant number random background events.

30 The mass diﬀerence also helps reduce background from D0 Vπ0 and D0 Vη → → events. Figs. 2.5(i), 2.5(c), and 2.5(f) show the mass diﬀerence distributions for

D0 φγ events as well as the distribution from background processes, D0 φπ0 → → and D0 φη. → We further reduce the eﬀect of D0 Vπ0 events by attempting to reconstruct → the other photon produced by a π0 decay. By fully reconstructing the π0 decay, we can then veto these events from our dataset. In order to implement the veto we calculated the invariant M(γγ′) between the candidate photon in the decay,

D0 Vγ, and every other photon, γ′. All candidate photons near the nominal π0 → mass, that is where M(γγ′) falls in the range (0.115, 0.150) GeV/c2, are discarded.

The implementation and eﬀectiveness of the mass diﬀerence, helicity angle and

π0 veto will be discussed in greater detail later in this document.

31 ) ) 2 2 2200 3000 2000 5000 1800 2500 1600 4000 1400 2000 Entries / (0.05) Entries / (0.5 MeV/c

Entries / (0.010 GeV/c 1200 3000 1500 1000 2000 800 1000 600 400 500 1000 200 0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 0.14 0.145 0.15 0.155 0.16 0 2 *+ 0 2 D Mass (GeV/c ) Cos(θH) M(D ) - M(D ) (GeV/c ) (a) (b) (c) ) ) 2 2 1000 1000 1000

800 800 800

600 Entries / (0.5 MeV/c 600 Entries / (0.05) 600 Entries / (0.010 GeV/c

400 400 400

200 200 200

0 1.7 1.75 1.8 1.85 1.9 1.95 2 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 0.14 0.145 0.15 0.155 0.16 0 2 *+ 0 2 D Mass (GeV/c ) Cos(θH) M(D ) - M(D ) (GeV/c ) (d) (e) (f) ) )

2 14000 2 20000 18000 12000 3000 16000 10000 2500 14000 12000 8000 Entries / (0.05) Entries / (0.5 MeV/c

Entries / (0.010 GeV/c 2000 10000 6000 8000 4000 1500 6000 4000 2000 1000 2000 0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 0.14 0.145 0.15 0.155 0.16 0 2 *+ 0 2 D Mass (GeV/c ) Cos(θH) M(D ) - M(D ) (GeV/c ) (g) (h) (i)

0 ∗ 0 Figure 2.5: D , Cos(θH ) and M(D ) M(D ) distributions for signal MC events from φπ0(a, b, c) φη(d, e, f) and φγ(g,− h, i) signal MC, all analyzed as D0 φγ. → The region of overlap in the D0 mass distribution is greatest between φπ0 and φγ. Both helicity and mass diﬀerence cuts will help discriminate between φγ and each background. All distributions are taken with minimal cuts. After vetoing photons consistent with arising from π0 and cutting on the mass diﬀerence the background events decrease. Although the overall background decreases it also takes on a stronger peaking character.

32 CHAPTER 3

The BABAR Detector and PEP-II Storage Ring

Our study of radiative charm decays was conducted using data collected at the BABAR detector located at the Positron-Electron Project (PEP-II) asymmetric- energy e+e− storage ring and collider operating at the Stanford Linear Accelerator

Center (SLAC) in Menlo Park, CA.

The BABAR experiment derives its name from being designed as a B-meson factory, a facility which is constructed speciﬁcally to produce B and B¯ in equal quantities. The desire to produce this equilibrium arose out of the need to study Charge

Parity (CP) violation.

BABAR produces B mesons by operating at energies equal to the rest mass of the Υ (4S) resonance, √s = 10.58GeV , and slightly greater than the BB¯ threshold.

This is provides a dataset rich in B mesons with nearly 100% of Υ (4S) resonances decaying into either B+B− or B0B¯0. (The actual fraction of Υ (4S) resonances decays into non-BB¯ states is less than 4 10−2 [1].) × PEP-II was designed as a e+e− collider. Having fundamental particles such as e− in the inital state allows for clean signals with a relatively low multiplicity of charged particles created in each event. This can be compared against pp¯ experiments which see an order of magnetude greater event multiplicity, depending on

33 beam energy. Leptonic collisions, such as e+e−, directly produce a limited number of states, as shown in a few examples below:

e+e− qq¯ → e+e− l+l− → e+e− l+l−γ → (3.1)

In these expressions q represents a quark, l represents a lepton, and γ represents a photon. The average BABAR event contains approximately 11 charged particles.

The BABAR detector is not only a BB¯ factory, but also produces a large number of cc, τ +τ −, reactions. Some of the most interesting physics in recent years has arisen out of the charm sector. The BABAR experiment actually produces about

30% more direct charm quark events than B meson events. A direct charm quark event is deﬁned as the interaction e+e− cc¯. The number of charm events and a → small event multiplicity, oﬀer analysts an excellent opportunity to study rare charm physics, such as radiative decays which are the topic of this thesis.

In addition to B meson and charm studies, the BABAR dataset has been used to successfully measure mixing of D mesons, examine rare decays of τ leptons, investigate charmonium states, and study the possible existence of pentaquarks.

In the following section I will brieﬂy describe the PEP-II storage rings and the

BABAR detector. These sections should be viewed as a general overview of the great engineering achievements which comprise both the PEP-II facility and the BABAR detector. Each of these are discussed in great detail elsewhere. [14]

34 3.1 PEP-II Storage Ring

The PEP-II storage ring and e+e− collider is actually a pair of independent storage rings, one circulating electrons and the second circulating positrons in the opposite direction. The ﬁrst ring maintains an electron beam at energies of 9.0 GeV.

The second ring houses a beam of positrons, at energies of 3.1 GeV. In the Center of

Mass (CM) system, the energy of the two rings is 10.58 GeV, equivalent to the rest mass of Υ (4S). The facility was originally designed to operate at an instantaneous luminosity of 3 1033 cm−2s−1 but has surpassed the original design requirements × by recording instantaneous luminosities in excess of 1.2 1034 cm−2s−1. × A 3 km linear accelerator (LINAC) is used as an injector of high momentum electron and positron beams. The beams are accelerated along the LINAC until they reach their target energies, and injected into the two independent storage, each with a circumference of 2.2km. Fig. 3.1 shows a schematic of the storage ring and

LINAC at PEP-II.

Figure 3.1: A schematic representation of the PEP-II storage ring and collider. The red line is the path taken by positrons and and blue line is the path taken by electrons.

35 3.1.1 Center of Mass Energy

The energy diﬀerential between the electron and positron beams results in a

Lorentz boost of βγ = 0.56 to the CM system, with β and γ deﬁned in Eqs. 3.2 and

3.3, respectively.

v β = (3.2) c 1 γ = (3.3) 1 β2 − In Eq. 3.2 v is the velocity of the CM systemp in the lab frame and c is the speed of light. The asymmetric design induces a separation in produced B meson pairs. This allows for precise measurements of lifetime diﬀerence between B0 and B¯0 meson, essential for CP violation studies.

3.1.2 Luminosity Measurement

The original design requirements called for instantaneous luminosities of 3 × 1033 cm−2s−1. The luminosity is a measure of the number of collisions in a segment of time, and can be estimated using Eq. 3.4. [15]

I−I+ 1 (3.4) 2 2 2 L ∝ e fc 4πσy (σztan(θ/2)) + σz

+ In this expression fc is the beam crossingp frequency, I is the current of the positron

− beam, I is the current of the electron beam. σz and σy are the beam sizes in the longitudinal and horizontal directions. Finally, θ is the crossing angle of the two beams at the interaction point (IP). A four fold increase in luminosity was achieved by raising I+ from 0.75A to 1.90A and I− from 2.14A to 3.00A, while at the same time reducing the horizontal beam, σy, area by approximately 10%. [16]

A more convenient unit to measure luminosity is the “barn” (b). A barn is

36 approximately the cross sectional area of an atomic nucleus, 1 b = 10−24cm2. Trans- lating the instantaneous luminosity into units of pb−1s−1, we see an instantaneous luminosity of = 1.2 10−2pb−1s−1, or about 1fb−1/day. The integrated luminos- L × ity recorded over the life cycle of the BABAR detector can be seen in Fig. 3.2.

Every e+e− interaction can scatter into numerous final states, each with different probability. The likelihood of an interaction resulting in a particular final state, for example qq¯, is known as the cross section. Cross sections have units of area and is noted with the symbol σ. The units are a consequence of the classical mechanics conception, where two objects of differing areas collide. In this picture the cross section would be proportional to the overlap in area between the two objects.

In the case of e+e− qq¯ interactions the cross section is instead a measure of → the likelihood that an e+e− interaction will produce a qq¯ pair. When examining the entire BABAR dataset the total number of qq¯ pairs produced in an operational period can be calculated using Eq. 3.5

Nqq = σqq L (3.5) ¯ ¯ ×

In this expression Nqq¯ is the number of qq¯ pairs produced, σqq¯ is the cross section for a qq¯ pair to be produced by an e+e− interaction, and L is the total integrated luminosity. The cross sections for the dominant modes observed at the BABAR detector are shown in Table 3.1.

37 iue32 h uioiyrcre ythe by recorded luminosity The 3.2: Figure fbt h aarcre hl unn teege equivale energies at running sum while a the recorded is to data This PEP-II. the luminosity. by recorded both delivered total of luminosity the shows the line show red line The blue The time. unn utblwthe below just running n sntue nayo h tde icse nti document. this in discussed studies the of any in used not is and hw ngen eety aahsbe eodda energies at recorded been has the data to Recently, equivalent green. in shown

Integrated Luminosity [fb-1] 100 200 300 400 500 0 Υ 2000 (4 S BaBar RecordedLuminosity:504.26/fb PEP IIDeliveredLuminosity:524.41/fb as hw nlgtbu,addt eoddwhile recorded data and blue, light in shown mass, ) BaBar RecordedY(4s):432.72/fb BaBar RecordedY(3s):24.44/fb 2001 Off PeakLuminosity:47.10/fb Off Peak Recorded LuminosityY(3s) Recorded LuminosityY(4s) Recorded Luminosity Delivered Luminosity Υ BaBar (3

2002 Run 1-7 S etms.Ti aai hw npurple in shown is data This mass. rest ) Υ 2003 (4 S as nw so ekand peak oﬀ as known mass, ) 38

2004

2005 B A 2006 B AR eetrover detector As of2008/02/1700:00 2007

2008 nt e+e− σ(nb) → cc 1.30 uu 1.35 dd 0.35 ss 0.35 bb 1.05 τ +τ − 0.94 µ+µ− 1.16 e+e− 40

Table 3.1: The cross sections for the dominant modes seen at BABAR while operating at the Υ (4S) resonance.

39 3.2 The BABAR Detector

In order to successfully study a wide variety of topics in Particle Physics, the

BABAR detector needs to provide the following:

Maximal detector acceptance in the Center of Mass (CMS) of the colliding • beam. As mentioned earlier the CMS of the beams is boosted forwarded

relative to the laboratory frame. As a consequence the detector has been con-

structed asymmetrically, with many detector elements placed in the forward

direction relative to the e− beam.

Excellent vertex resolution, needed to study CP violation. To achieve this, • solid state detector elements are placed within a radius 32mm from the in-

teraction point. While our analysis does not study CP violation, we greatly

+ beneﬁt by this close proximity as it enables us to detect the slow pions, πslow, produced by the decay, D∗+ D0π+ . → slow

Charged particles tracking over a wide range of transverse momenta, 0.060 GeV/c < •

pt < 4 GeV/c.

Successful discrimination between e, µ, π, K, and p candidates across a wide • range of momenta. This is particularly important in our analysis, speciﬁcally

in separating kaons and pions.

Detect photons over a wide range of energies, (0.020 < Eγ < 5) GeV. This is • again a primary requirement when studying radiative D0 decays.

The ability to identify neutral hadrons. While this is not needed for our • studies, it has been useful to conﬁrm CP measurements.

40 To achieve these goals the BABAR detector was constructed out of ﬁve subdetector elements and a 1.5T superconducting magnetic coil. The ﬁve elements are:

The Silicon Vertex Tracker (SVT) provides precise tracking information on • charged particles. It also serves as the sole tracking device for low momentum

charged particles.

The Drift Chamber (DCH) is a helium gas ﬁlled tracking chamber. The DCH • measures the momentum of medium through high energy charged tracks. It

also measures the energy loss of particles as they travel through the chamber,

which aids in particle identiﬁcation (PID).

A Detector of Internally Reﬂected Cherenkov light (DIRC) allows greater PID • of charged hadrons.

A CsI Electromagnetic Calorimeter (EMC) is used to measure the energy of • incident photons and helps in electron identiﬁcation.

All of these elements are enclosed in the 1.5T magnetic coil, used to determine • the electric charge of candidates. The generated magnetic ﬁeld also induces

curvature to the trajectory of charged tracks, allowing for momentum mea-

surements.

An Instrumented Flux Return (IFR) is used for both µ and neutral hadron • identiﬁcation.

Two schematic views of the BABAR detector are shown in Fig. 3.3. In the following sections we will describe the detector elements in more detail as well as discuss PID and MC simulations needed to conduct analyses at BABAR .

41 Detector C L Instrumented Flux Return (IFR)) 0 Scale 4m I.P. Barrel Superconducting 0 Scale 4m BABAR Coordinate System Coil IFR Barrel y 1015 1749 BABAR Coordinate System x Electromagnetic y Cryogenic 1149 4050 1149 Calorimeter (EMC) z Cutaway Chimney 370 Drift Chamber Superconducting Section x (DCH) Cherenkov Coil z Detector Silicon Vertex (DIRC) Tracker (SVT) DIRC EMC DCH IFR Magnetic Shield 1225 Endcap for DIRC Forward 3045 SVT End Plug IFR Cylindrical RPCs Corner Bucking Coil Plates 1375 Support Tube 810 e– e+ Earthquake Tie-down Gap Filler Q4 Plates Q2

Q1 3500 3500 B1 Earthquake Isolator

Floor Floor 3-2001 3-2001 8583A50 8583A51 (a) (b)

Figure 3.3: A schematic representation of the BABAR detector, shown (a) in cross section and (b) along the beam line.

3.2.1 Silicon Vertex Tracker

The SVT is needed to provide precision tracking of charged particles near the interaction point of the detector. The SVT covers a complete 2π angle in the region perpendicular to the beam pipe, known as transverse direction, and while also subtending a polar angle between 20.1o and 150.2o measured relative to the beam pipe. The SVT is the only device capable of measuring low momentum charged tracks, (these are tracks with a transverse momentum, pt < 0.120 GeV/c). Tracking of low momentum particles is vital in reconstructing pions produced by the decay

D∗+ D0π+ . → slow The SVT is constructed out of ﬁve concentric layers of double sided silicon detectors. The inner side of each layer consists of silicon strips oriented along the beam pipe or z-axis. The outer side of each layer consists of silicon strips oriented along

42 Beam Pipe 27.8mm radius

Layer 5a Layer 5b 580 mm Space Frame Layer 4b Layer 4a Bkwd. support cone 520 mrad 350 mrad Layer 3 Fwd. support cone Layer 2 e- Front end e + Layer 1 electronics

Beam Pipe (a) (b)

Figure 3.4: A schematic representation of the SVT, shown as (a) view from the beam line and (b) shows the cross section. Shown in (b) is also the arch design of the outer two layers.

the longitudinal direction or φ axis.

The inner three layers are placed at radii of 32mm, 40mm, and 54mm, as measured radially from the interaction point. These layers are arranged in a barrel structure, with the layers placed at a constant distance from the beam line. The remaining two layers are placed at radii of 124mm and 144mm. Rather than being arranged along the barrel, the ﬁnal two layers are arranged in an arch. This provides greater solid angle coverage and creates a greater surface perpendicular to the motion of charged particles towards the edges of detector acceptance. The cross section and beam axis views of the SVT are shown in Fig. 3.4.

Overall the SVT is able to resolve z position measurements with a precision of

70µm and measurements within the x, y plane with a precision of less than 40µm.

The SVT is also able to measure the speciﬁc energy dE/dx of charged tracks. This provides a source of PID and allows for a 2σ separation between kaons and pions, for track momenta less than 0.500 GeV/c.

43 3.2.2 The Drift Chamber

The DCH serves as the principle position and momentum measurement device in the BABAR detector. The detector is constructed as a 40 layer wire drift chamber, ﬁlled with a gas mixture of 80% helium and 20% isobutane. The low density gas is used to reduce multiple scattering. The DCH covers the entire 2π azimuthal angle and covers a polar range of (17.2o, 152.6o) along the beam axis. A schematic representation is shown in Fig. 3.5.

The layers are grouped into ten superlayers, with each superlayer having the same wire orientation. Each superlayer is then placed in stereo at angles ranging between (45, 76)mrad. This allows for three dimensional tracking information. ± Fig. 3.5(a) shows the inner four superlayers.

The DCH is able to measure the position of charged particles with spatial resolutions below 30µm and angular resolutions of 0.43mrad. The resolution of a particle’s transverse momentum has been measured to be:

σp /pt = (0.13 0.01)%pt + (0.45 0.03)% (3.6) t ± ±

In this expression σpt is the absolute resolution of the transverse momentum, pt.

The DCH also measures energy loss of a charged particle, dE/dx. This enables for

PID for charged tracks with momenta less than 0.700 GeV/c.

3.2.3 Directed Internally Reﬂected Cherenkov Light (DIRC) Detector

The DIRC detector is a novel detector of Cherenkov radiation, vital to the PID needs of the BABAR experiment. Cherenkov light results when high momentum charged particles travel through a dense material. If the velocity of a particle, v, is

44 16 0 15 0 14 0 13 0

12 -57 11 -55 10 -54 9 -52 630 1015 1749 68

8 50 7 48 6 47 Elec– 5 45 tronics 809 4 0 3 0 485 27.4 1358 Be 17.2 236 2 0 – IP 1 0 e 464 e+ Layer Stereo 469 4 cm Sense Field Guard Clearing 1-2001 1-2001 8583A14 8583A13 (a) (b)

Figure 3.5: A schematic representation of the DCH. (a) shows the orientation of the ﬁrst four super layers of the DCH. (b) Shows a cross section of the DCH detector.

45 greater than the velocity of light within the propagating material, c/n (where n is the index of refraction of the propagating material), a shock wave of electromagnetic radiation is emitted. The shockwave is emitted in a cone with an angle θC :

1 cos(θ ) = (3.7) C nβ

In this expression β = v/c as equal to Eq. 3.2. In the operation of the DIRC this light is internally reﬂected inside fused silica bars with n = 1.473. Fig. 3.6(b) shows the light internally propagating down on of the fused silica crystals.

The DIRC is comprised of 144 silica bars arranged in a 12 sided polygon immediately outside the DCH. The Cherenkov light travels down the 4.9m bars to a dome

filled with purified water. The water has an index of refraction close to that of the silica crystals, reducing the distortion of the light cone and reflection back into the silica. The Cherenkov light is measured using 11000 Photo Multiplier Tubes (PMT) placed around the outside of the dome. The dome is placed away from the boosted

CM frame. Fig. 3.6(a) shows the schematic of the DIRC.

Using the momentum measurements from the DCH and SVT the momentum dependence of θC can be used to determine the mass of a charged track. The dependence of momentum on θC is shown in Fig. 3.7(c). This ﬁgure shows the contours for all the diﬀerent charged tracks measured by BABAR . As can be seen here, the separation between pions and kaons is large for momenta less than 3 GeV/c.

The PID delivered by the DIRC is invaluable for the BABAR experiment. Fig.

3.6(c) compares the invariant M(K−π+) distribution seen while searching for D0 → K−π+. The top plot shows the M(K−π+) without using the DIRC’s kaon PID

46 information, while the bottom plot shows the same using the DIRC’s kaon PID information. As is shown here a great deal of background can be removed with little eﬀect on the actual signal.

3.2.4 Electomagnetic Calorimeter

The EMC is the sole device in the BABAR detector used to detect photons. Con- structed out of 6580 thallium doped CsI crystals, a schematic of the EMC is shown in Fig. 3.8. Photon energies between 0.020GeV and 4.0GeV are measured using electromagnetic showers which are produced by photons incident upon the EMC’s scintillating material. The energy showers is measured using photodiodes. The resulting energy resolution is shown in Eq. 3.8, and spatial resolution is shown in Eq.

3.9.

σ 1% E = 1.2% (3.8) E 4√E ⊕ 3mrad σθ,φ = + 2mrad (3.9) √E

When a particle is incident on the EMC, it produces a shower which is not conﬁned to one crystal, and instead the energy can be distributed among many adjacent crystals. The energy deposited across a grouping of crystals is known as a cluster.

The energy response of inorganic crystals is a well studied phenomena. A great deal of this work was done during the Crystal Ball Experiment [17, 18]. The Crystal

Ball Experiment was designed to detect neutral particles using a NaI EMC. During their experiment the energy response of the NaI EMC was modeled using Eq. 3.10,

47 PMT + Base ~11,000 PMT's

Purified Water

17.25 mm Thickness Light (35.00 mm Width) Catcher Bar Box Track Trajectory Wedge PMT Surface Mirror

Bar Window Standoff Box 91 mm 10mm 4.90 m 1.17 m

4 x 1.225 m Synthetic Fused Silica Bars glued end-to-end (a) (b)

Without DIRC x 10 2 1500 2

1000

With DIRC entries per 5 MeV/c 500

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

Figure 3.6: (a) The schematic of the DIRC detector. (b) Photograph showing light internally reﬂected through one of the DIRC crystals. (c) Shows the eﬀect of using PID information from the DIRC on kaons involved in D0 K−π+ reconstruction. →

48 dE/dx vs momentum

4 50 10 850 p d d e p 40 BABAR µ

dE/dx (a.u) 800

30 K K

750

(mrad) π 20 C

π θ 10 3 π 10 µ 700 K e e

80% truncated mean (arbitrary units) µ 0 p 650 -1 0 1 2 3 4 5 -1 10 1 10 10 1 p (GeV) Track momentum (GeV/c) pLab (GeV/c) (a) (b) (c)

Figure 3.7: The separation of the types of charged candidates for each PID detector component. (a) Shows the separation due to the SVT. This provides good separation for tracks less than 0.500 GeV/c. (b) Shows the separation using dE/dx from the DCH. Kaons and pions are well distinguished for momenta less than 0.700 GeV/c. (c) Shows the diﬀerence in Cherenkov angle, θc, as a function of transverse momentum. Across all plots e denotes electrons, µ-muons, π-pions, K-kaons, p-protons, and d-deuterons.

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 3.8: A schematic representation of the EMC

49 known as the Crystal Ball (CB) lineshape.

2 − (x−µ) x−µ 2σ2 e , if σ > α CB(x; α,n,µ,σ)= x−µ x−µ − (3.10) ( A(B )−n, if α − σ σ ≤ − n 2 n |α| − 2 A = |α| e , B = n α |α| − | | CB is a Gaussian for x>µ σα and a power loss tail for x<µ σα. − − This model was developed to account for the incomplete energy deposition of photons in the EMC. Incomplete energy deposition is the results of atomic eﬀects and energy leakage. A consequence of this is an asymmetric energy response. This feature is shown in Fig. 3.9.

Fig. 3.9 shows the diﬀerence in measured and actual energy of photons produced

∗0 by MC simulations of the decay D0 K γ. The energy loss tail can be seen in the → asymmetry of the low end tail. The response shown in Fig. 3.9 can be well modeled with Eq. 3.10.

This energy response has great importance to our analysis. The energy of photons must be measured in order to reconstruct a D0 candidate. Therefore this asymmetry needs to be modeled properly in order to accurately measure M(Vγ).

The EMC is also used to identify electrons. Electrons are identified based on shower energy, lateral shower moments, and track momentum. A electron candidate leaves a track in the DCH. Therefore if a cluster in the EMC is matched to a track in the DCH and the clusteris is identified as an electron. However, if an EMC shower is unmatched to a track in the DCH the shower is identified as a photon.

50 900

800

700

600 Entries / (0.008)

500

400

300

200

100

0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Eγ(Measured) - E (Incident) 2 γ E (Measured) + E (Incident) γ γ

Figure 3.9: The energy response of the EMC, taken with MC simulation comparing the generated photon, Eγ(Incident) with the measured photon energy, Eγ(Measured).

3.2.5 Instrumented Flux Return

The IFR is not used in our analysis, but is included here both to complete the description the BABAR detector, and also to discuss some of my earlier work at Ohio

State. When I ﬁrst joined the BABAR collaboration, I was involved with upgrading the IFR from a Resistive Plate Chamber (RPC) system to a Limited Streamer Tube

(LST) system.

The IFR is used to detect muons and long-lived neutral hadrons. The current

LST system is placed just outside the 1.5T magnet. LST, or Iarocci Tubes, are made out of extruded plastic, coated with resistive graphite. Each LST is comprised of four chambers. A sensor wire is placed in the center of each of the four chambers.

The wire is held at high voltage, (5 6)kV , and the tube is ﬁlled with a mixture of −

51 C02, Ar, and isobutane. When a charged particle travels through the tube, the gas is ionized and the wire collects the ionization and a current is read.

As documented earlier [19], a sizable loss of eﬃciency was observed in the muon detection over the life time of the RPC. BABAR set out to upgrade the IFR from

RPC to LST technology [20], in order to reverse this degradation. A strenuous quality control (QC) procedure was developed to avoid future loss in performance.

This included two classes of tests. The ﬁrst test measured the detection eﬃciency as a function of applied voltage on each tube. The second checked for run away discharges within the gas.

In the ﬁrst test, the voltage applied to the tubes was ramped in steps of 100V from 4000V up to 6000V , and the signal induced by cosmic rays was counted along the ramping process. The counting eﬃciency is maximal at voltages close to 5000V .

The measured ﬂux of cosmic rays plateaued for voltages above 5000V . If the measured ﬂux remained relatively constant for voltages above 5000V the tube passed the signals rate test.

The second QC test measured a tube’s response in the presence of Cs137, a radioactive isotope which decays through β emission. The radioactive source was scanned over each LST tube and the current induced by the β source was measured as a high voltage was applied to the sensor wire. The signals for each chamber was measured as the source transversed the tube. If the current dissipated after the source was moved away from a given chamber, the tube passed the QC test. This

QC testing took place at both Ohio State and Princeton University.

The LST upgrade required a robust high voltage (HV) power supply system.

52 Overall the IFR detector contains 1164 LST tubes, each with 4 independent segments. Therefore the HV power supplies must be able to control 4656 independent

LST segments, simultaneously. Furthermore, the power supply must prevent run away discharges within the gas, as these discharges can damage the tubes. Ohio

State designed and built the High Voltage Power supplies used in the LST installation, shown in Fig. 3.10(a).

I contributed to this eﬀort by designing and implementing a graphical user interface (GUI) used to communicate with the Rabbit Semiconductor microcontroller used to control the HV Power Supply. The GUI was designed to help in QC testing by allowing an operator to control the voltage, read the induced current, and record and reset current trips. The GUI interfaced with a SQL database to help keep dili- gent records of this QC testing over time. The GUI also allowed for a experimental burn-in of the LST. Under the burn-in an operator could establish a QC procedure to ramp up and down the applied voltage over time as well as establish fault conditions to either continue or abandon a QC test. Fig. 3.10(b) shows the front end of the GUI. Overall the LST upgrade has been a very successful upgrade. In the past three years of operations there have been no failures in opterations.

3.3 Particle Identiﬁcation

Correct PID is of extreme importance in high energy physics studies. For PID, the Cherenkov angle shown in Eq. 3.7 is combined with the Bethe-Bloch energy loss parameterization:

dE Z 1 1 2m c2T δ(βγ) = Kz2 ln e max β2 (3.11) dx General A β2 2 I2 − − 2

53 (a)

(b)

Figure 3.10: (a) One of the power supplies used for the LST installation. (b) A screen shot of the graphical user interface used to communicate with a High Voltage Power supply.

54 In this expression Z is the atomic number of the absorber, ze is the charge of the incident particle, A is the atomic mass of the absorber, Tmax is the maximum kinetic energy which can be imparted to a single electron in a collision, I is the mean excitation energy, K is a constant. Eq. 3.11 is a complicated expression but when measured in the SVT and DCH, energy loss is modeled with Eqs. 3.12 and

3.13, respectively.

dE b2 b3 = b1β (βγ) (3.12) dx SVT

dE a1 ′ a4 = (a2 β ln(a3 +(βγ) ) (3.13) dx DCH β′ − − p a5 β′ = (3.14) E In these expressions bi and ai are constant parameters set by the SVT and DCH working groups [21]. These expressions are dependent only on the velocity of an incident particle. Using the momentum information from the DCH the contours shown in Figs. 3.7(a) and 3.7(b) are calculated for each charged particle measured at BABAR .

A particle hypothesis is assigned to each charged candidate. Based on this assumption the pull of dE/dx is deﬁned using Eq. 3.15.

dE dE pull = dx Measured − dx Bethe−Bloch (3.15) σ

With the error deﬁned as:

γ −b6 b4 5 σ = b5 + b (3.16) SVT dE 7 N dx SVT ! r SVT −a7 −a8 NDCH p a σ = a | | 1+ 9 (3.17) DCH 6 40 p p2 t t The pull distributions for both the SVT and DCH are obtained using pure samples of each type of measured particle. These are ﬁt with a Gaussian. The Gaussian

55 serves as the Probability Density Function (PDF) for the pull distribution, and consequently that a particular particle hypothesis is correct.

When a candidate of unknown type is detected, its pull for each possible particle type is measured. The pull is then related to the PDF obtained using control samples for each type of particle. Using the pull value of the PDF a likelihood value is determined for each particle hypothesis. The DIRC likelihood is quantiﬁed using a lookup table, rather than a Gaussian ﬁt. The probabilities from each of the three sub detectors are formed into a single likelihood value:

= DIRC DCH SVT (3.18) L L L L

A given candidate is classiﬁed as a kaon, pion, etc, based on the ratio of its likelihood as compared to the other candidates. As part of the BABAR experiment these are grouped into general classes of VeryLoose, Loose, Tight, VeryTight. The speciﬁc criteria are shown in Table 3.2

3.4 Monte Carlo Simulation of the BABAR Detector

When the BABAR experiment began, the collaboration adopted the medical practice of blinding data. This practice helps prevent biasing the results of an analysis.

With blinding in place there is a need to develop a study procedure that would replicate results and verify their authenticity without examining real data. This is accomplished using MC simulations of the BABAR detector. Using MC, analysts have the ability to develop analysis procedures, study backgrounds, understand reconstruction eﬃciencies, and validate their analyses without biasing the analyses using real data.

The process of generating BABAR MC events is distributed over twenty diﬀerent

56 List Kaons (KLH) Pions (piLH) NotPion (K)/( (K)+ (π)) > 0.20 - L L L VeryLoose (K)/( (K)+ (π)) > 0.50 (K)/( (K)+ (π)) < 0.98 L(K)/(L(K)+ L(p)) > 0.18 L (p)/(L(p)+ L(π)) < 0.98 L L L L L L Loose (K)/( (K)+ (π)) > 0.8176 (K)/( (K)+ (π)) < 0.82 L L L L L L (K)/( (K)+ (p)) > 0.18 (p)/( (p)+ (π)) < 0.98 L L L L L L Tight (K)/( (K)+ (π)) > 0.9 (K)/( (K)+ (π)) < 0.5 L(K)/(L(K)+ L(p)) > 0.2 L (p)/( L(p)+ L(π)) < 0.98 L L L L L L VeryTight (K)/( (K)+ (π)) > 0.9 (K)/( (K)+ (π)) < 0.2 L(K)/(L(K)+ L(p)) > 0.2 L (p)/(L(p)+ L(π)) < 0.5 L L L L L L

Table 3.2: A summary of the PID selections made for kaons and pions. In this expression (p) is the likelihood value for protons. Additional cuts are made on electron likelihoodL for VeryLoose lists for kaons and Loose lists for pions and muon likelihood for VeryTight lists.

sites through out the BABAR collaboration. The Ohio State University BABAR group operates one of these the facilities. The MC generated at Ohio State is merged with

MC generated across the collaboration and is used in nearly every physics analysis undertaken at BABAR.

The bank of computers at OSU, known as “the farm”, generates 1.5 million MC events a day. This production is generated using 56 computers with Linux Red Hat

Enterprise 3 installed, containing a total of 110 AMD CPUs. In order to insure proper operation and contribution to BABAR these computers and the data they generate must be monitored daily. For the past two years I have been the primary contact in charge of OSU MC farm, maintaining and monitoring the hardware, soft- ware, and database located at OSU.

57 3.4.1 MC Simulation

The MC simulation of the detector takes place over a four step process.

Generate a speciﬁc physics process. •

Simulate the passage of particles through the detector. •

Simulate detector response. •

Reconstruct physics events from the detector response. •

The generation of a speciﬁc physics process takes place by simulating the initial e+e− interaction. Seven particular e+e− events are simulated. These include e+e− → Υ (4S) B0B0, e+e− Υ (4S) B+B−, e+e− cc, e+e− (lightquarks, u, d, s) → → → → → e+e− µ+µ− +X, e+e− τ +τ −, and e+e− e+e−. The passage from free quarks → → → into a hadronic state is simulated. Once a hadronic state is formed, MC simulates its decay into ﬁnal state particles, measurable by the BABAR detector. For the majority of hadronic events, the JetSet event generator is used [22].

Once the initial particles and decay products are generated by JetSet, their passage through the detector as well as the detector’s response are modeled using GEANT4 [23]. The interaction of particles through the detector must include electromagnetic showers through the EMC, ionization loss in the SVT and DCH, multiple scattering in the SVT, etc. In order to accurately model the detector, the material composition of the detector must be known throughout the active area.

This includes not only detector elements, but also structural supports.

58 Detector response to events must also be simulated. In order to accurately simulate the detector, background processes must be added. These include beam interactions, residual signals from previous events, and other non-physics processes. The conditions of the detector over diﬀerent periods in time are simulated and recorded in a database. These condition and conﬁguration databases are currently stored in

ROOT ﬁles which are accessed when the simulations are produced.

The ﬁnal step of reconstruction can be done well after the MC simulation is run.

But in order to achieve greater eﬃciency of computer resources, a limited amount of the reconstruction is run immediately after simulating the detector response. This includes reconstructing a charged particle by matching hits in the SVT and DCH, reconstructing photons by the shower shape in the EMC, etc.

3.4.2 Distributed Computing at BABAR

A single Υ (4S) event can take 8 seconds to simulate. When one considers the amount of data needed to simulate the full BABAR dataset, it becomes clear that the

MC needs of collaboration will not be satisﬁed by only one farm. With this in mind the multi-site farm system was developed. In this system, known as “embarrassingly parallel” computing, a series of independent jobs are distributed across a wide net- work. Once complete the jobs are merged into a single dataset. With no interaction between jobs needed while running, processes can be easily made parallel.

The BABAR farm system is comprised of SLAC and 19 oﬀ site facilities. SLAC accounts for about 30% of the MC production used at BABAR. The OSU farm has

59 contributed 1.12 billion events to the BABAR MC dataset or about 10% of the oﬀsite production since 2005.

60 CHAPTER 4

Experimental Details and Datasets

This section describes the analysis strategy employed to study the radiative decays:

∗0 D0 K γ (4.1) → D0 φγ (4.2) → Furthermore, we discuss the MC datasets used to establish and test our analysis procedure, as well as the data used to measure the branching fraction of each mode.

As mentioned in Section 1.3.4, we measure the branching fraction, (D0 Vγ) B → relative to (D0 K−π+), as shown in Eq. 1.6. The use of a reference mode B → is needed to resolve uncertainties concerning the number of D∗+ mesons that are created at BABAR. This particular reference mode was chosen to reduce systematic uncertainties associated with Particle Identifaction (PID), as well as provide a measured mode with a small uncertainty in the decay rate, (D0 K−π+) = B → (3.82 0.07)% [1]. ± 4.1 Event Selection

The initial datasets for each mode were taken from the whole BABAR dataset using very broad selection criteria. More stringent criteria will be determined and

61 applied once the initial datasets are constructed.

∗0 The decays D0 K γ and D0 φγ are examined using the charged decay → → modes of both vector mesons, shown in Eqs. 4.3 and 4.4

∗0 K K−π+ (4.3) → φ K−K+ (4.4) →

We require that kaon candidates in each decay satisfy the NotPion PID requirements and that pions satisfy the VeryLoose PID requirements shown in Table 3.2. The

∗0 four vectors of K− and π+ are summed to form the four vector of K . This

∗0 combination must have an invariant mass in the vicinity of the nominal K mass,

(0.797 < M(K−π+) < 0.997) GeV/c2. Similarly the invariant mass of the K−K+ combination must fall within the range, (0.99 < M(K−K+) < 1.05) GeV/c2.

Photon candidates are taken from EMC clusters not paired with any charged tracks seen in the DCH. The energy deposition within the EMC cluster is required to be greater than 0.030 GeV. The distribution of a cluster among ECM crystals is described with the by a variable call the lateral moment. The lateral moment is a measure of the spread of a cluster over individual EMC crystals. The functional form is shown in Eq. 4.5.

2 E r2 Lat. Moment = 1 i=1 i i (4.5) − N 2 Pi=1 Eiri

In this expression Ei is energy in ith EMC crystalP (ordered from highest to lowest energy) and ri is the radial distance from the center of a cluster, and N is the number of crystals within a cluster. A lateral moment close to zero mean that nearly all of a shower’s energy being deposited into one or two crystals. A lateral moment that is close to one corresponds to a shower’s energy being evenly deposited over many

62 EMC crystals. Good photon candidates typically have a lateral moment greater than zero and less than 0.8. Only photons meeting this requirement are selected for this analysis.

Our D0 candidates are constructed by adding a candidate photon’s four vector

∗0 to that of either K or φ. The production vertex of the photon is set to the common vertex of either K−π+ or K−K+. Doing this ignores the very short lifetime

∗0 of K and φ. The invariant mass of a D0 candidate must fall within the range

(1.60 < M(Vγ) < 2.10) GeV/c2.

As mentioned in previous sections, a common technique when studying D0 mesons is requiring that a D0 candidate be produced by the decay D∗+ D0π+ . → slow The diﬀerence between the mass of the D∗+ candidate and the masses of its decay products: D0 and π+ is known as the Q-value. The Q-value for this decay is small, leading to the π+ being a low momentum track. We accept π+ candidates with momenta less than 0.450 GeV/c2 and that approach the beam spot within 1.5cm in

+ the radial direction and 10cm along the z axis. The four vector of the πs is added to the D0 candidate four vector. The diﬀerence in the invariant masses of the D∗+ and D0 candidate must be, M(Vγπ+) M(Vγ) < 0.190 GeV/c2. s −

4.2 Monte Carlo and Data Sample Sizes

We develop our analysis using MC simulations of the detector, which enables us to validate our analysis on simulated events and not to bias our results on real data.

In order to estimate our reconstruction eﬃciency, model our signal, and predict the prominent backgrounds, we simulate the following modes using the JetSet event

63 generator and a GEANT 4 based simulation of the detector:

D∗+ D0π+ ; D0 φγ; φ K−K+ (4.6) → slow → → ∗0 ∗0 D∗+ D0π+ ; D0 K γ; K K−π+ (4.7) → slow → → D∗+ D0π+ ; D0 φπ0; φ K−K+ (4.8) → slow → → D∗+ D0π+ ; D0 φη; φ K−K+ (4.9) → slow → → D∗+ D0π+ ; D0 K−π+ (4.10) → slow →

The MC events which explictly contain events simulating the decay chains shown in Eqs. 4.6 and 4.7, are analyzed and formed into a dataset as described in Section

4.1. Similarly, decays serving as the dominant background to D0 φγ, shown → in Eqs. 4.8 and 4.9, are also generated and analyzed the exact same manner as

D0 φγ events. The signal and background datasets are used to evaluate tech- → niques to reduce contributions from background without adversely effecting the signal efficiency. Once backgrounds are reduced, the remaining signal events are used to calculate the reconstruction efficiency of signal and to model the distribution of signal. The residual background datasets are used to model how the remaining background will complicate our measurement. An MC sample simulating the decay chain shown in Eq. 4.10 is also used to estimate the reconstruction efficiency of our reference mode.

We use the simulated e+e− qq¯ processes to estimate the background from → random events, evaluate other specific backgrounds, and optimize our selection criteria without biasing the results using data. These processes are defined as generic, because the process from q to final state particles is meant to be a cross section of ob- servational data recorded over the last several decades. Ideally generic MC simulates

64 the vast majority of the events recorded at the BABAR dectector. The speciﬁcally generic processes we examine include e+e− Υ (4S) (where Υ (4S) can decay into → charged or neutral B mesons), e+e− cc, and e+e− uds. These generic samples → → are analyzed using the selections detailed in Section 4.1. The generic sample of

∗0 e+e− cc contains a plentiful amount of D0 K−π+π0 and D0 K η so it is → → → not necessary to explicitly generate these decays individually. These event classes

∗0 serve as the dominant backgrounds to D0 K γ. We fed MC of e+e− l−l+ → → through our analysis and saw no meaningful contributions. The sample sizes of each

MC dataset are shown in Table 4.2, with the equivalent luminosity of the generic data sets estimated using the cross section.

After establishing our analysis using MC we ran over data recorded the BABAR detector from 1999-2006. Over this time data was collected in ﬁve distinct run periods. Shown in Table 4.2 are the integrated luminosities for each run used in this analysis.

65 Mode Number of Events (106) Cross Section (nb) ( fb−1) ∗+ 0 + 0 + − L D D πslow; D φγ; φ K K 0.268 - - ∗+ → 0 + 0 → 0 → + − D D πslow; D φπ ; φ K K 0.268 - - ∗+ → 0 + 0 → → + − D D πslow; D φη; φ K K 0.286 - - ∗+ → 0 + 0 → ∗0 → ∗0 − + D D πslow; D K γ; K K π 0.286 - - ∗+ → 0 + 0 → − + → D D πslow; D K π 0.618 - - e+e−→ cc → 768.4 1.3 591.1 → e+e− uu,dd,ss 796.0 2.09 380.9 → e+e− B0B0 557.7 0.53 1052.3 e+e− → B+B− 603.9 0.53 1139.4 →

66 Table 4.1: The MC sample size used in the analysis as well as the approximate luminosity of each sample.

Run Number On Peak Luminosity ( fb−1) Oﬀ Peak Luminosity ( fb−1) 1 20.4 2.6 2 61.1 6.9 3 32.3 2.5 4 100.3 10.1 5 131.0 14.5 Total 345.1 36.6

Table 4.2: The data recorded at BABAR while running at energies equivalent to the υ(4S) rest mass (On Peak) and at energies just below the υ(4S) rest mass (Oﬀ Peak). 4.3 Additional Event Selection

With our datasets in place, the background from D0 φπ0 and D0 K−π+π0 → → events is drastically reduced by vetoing all candidates with a photon that may arise from the decay π0 γγ. A photon that is produced by a π0 decay can be paired → with another photon in the event to form an invariant mass near the nominal π0 mass. We therefore veto any candidate photon that can be matched with any other recorded photon with the invariant mass of the combination so that falls within the range (0.105 < M(γγ) < 0.150) GeV/c2. The eﬀectiveness of the π0 veto was measured by examining the resulting M(Vγ) distributions of MC of the signal modes:

∗0 D0 φγ and D0 K γ, as well as the M(Vγ) distribution of the prominent π0 → → backgrounds. These are shown in Figs. 4.1 with the red lines denoting the distributions prior to imposing the veto and the blue lines showing the distributions after the veto.

Signiﬁcant contributions from background processes were removed, while maintaining a large fraction of signal, by selecting only events where the diﬀerence in quantity ∆M = M(Vγπ+) M(Vγ) is near the nominal value of M(D∗+) M(D0). s − − As mentioned earlier the Q-value of decay D∗+ D0π+ is very small. This dis- → slow tribution of ∆M exhibits a strong and narrow peak near 145.5 MeV/c2, for real

D∗+ D0π+ events. The ∆M distribution of real D∗+ D0π+ ; D0 Vγ → slow → slow → events can be ﬁt to a linear combination of a Gaussian distribution, modeling

D0 Vγ events, and a background distribution shown in Eq. 4.11 →

∆M−M π+ c2 c1 ∆M ∆M F (∆M; c1,c2,c3, Mπ+ )= 1 e + c3 1 (4.11) − M + M + − ! π π

67 ) ) 5000 2 7000 2 *0 D0 → K γ -- Before π0 Veto

0 0 *0 D → φ γ -- Before π Veto D0 → K γ -- After π0 Veto

6000 D0 → φ γ -- After π0 Veto 4000

5000

Entries /(5 MeV/c 3000 Entries / (5 MeV/c 4000

3000 2000

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0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2) D0 Mass (GeV/c2)

(a) (b) ) 2 0 0 0 1000 D → φ π -- Before π Veto

D0 → φ π0 -- After π0 Veto ) 2

800 18000

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12000 Entries / (5 MeV/c

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0 1.7 1.75 1.8 1.85 1.9 1.95 2 0 0 2 1.7 1.75 1.8 1.85 1.9 1.95 2 D Mass (GeV/c ) D0 Mass (GeV/c2)

Figure 4.1: Using MC data, the eﬀect of a π0 veto for the (a) signal D0 φγ MC ∗0 → events, (b) signal D0 K γ MC events, and background from (c) D0 φπ0 MC events and (d) D0 →K−π+π0. (a) and (c) show the distributions prior→ to the veto in red and after the→ veto in blue. In (b) and (d) the distributions prior to the veto are shown in blue and after in red.

68 We model signal MC of D0 φγ using a linear combination of a Gaussian dis- → tribution and the background expressed in Eq. 4.11. MC is then ﬁt using an unbinned extended maximum likelihood method which determines the mean and standard deviation of the Gaussian distribution. We then select only events in the range [µ 3σ, µ + 3σ], where µ is the mean of the Gaussian distribution and σ − is the distribution’s standard deviation. This corresponds to a selection region of

∗0 [0.1435, 0.1475] GeV/c2. Similarly we ﬁt the mass diﬀerence for D0 K γ MC → events and found the same selection region.

Nearly all signal is preserved by cutting between 3σ of the mean of the Gaussian ± distribution, but such a tight cut signiﬁcantly reduces contributions from additional background processes. This can be seen in Figs. 4.2 and 4.3. The distributions for

∗0 D0 φγ and D0 K γ MC events are shown in Figs. 4.2(a) and 4.3(a), respec- → → tively. The remaining plots show the distributions for MC events of (b) D0 Vπ0 → (c) D0 Vη, and (d) combinatoric background events. → Both the π0 veto and the ∆M selection greatly reduce contributions from background events while leaving a large percentage of true signal candidates. We can further improve our ability to distinguish D0 Vγ events from background by → tightening selections on the following:

The vector meson mass: •

∗0 – For D0 K γ this corresponds to M(K−π+) → – For D0 φγ this corresponds to M(K−K+). →

The photon momentum in the CM frame: PCMS(γ). •

∗+ ∗+ The D momentum in the CM frame: PCMS(D ). • 69 ) )

2 16000 2 4500

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) 500 ) 2 2 3500

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Figure 4.2: The mass diﬀerence distributions for the MC events of (a) signal D0 → φγ mode as well as the backgrounds (b) D0 φπ0 and (c) D0 φη and (d) the combinatoric cc background. Events between→ the vertical lines are→ selected.

70 ) ) 80000 16000 2 2 c c

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GeV 5000 ) 45000 2 c

4000 GeV 40000 35000 3000 30000 25000 2000 20000 Events / ( 0.0005 15000 Events / ( 0.0005 1000 10000 5000 0 0 0.14 0.145 0.15 0.155 0.16 0.14 0.145 0.15 0.155 0.16 *± 0 GeV *± M(D ) - M(D ) ( ) M(D ) - M(D0) (GeV) c2 c2 (c) (d)

∗0 Figure 4.3: The mass diﬀerence distributions for the (a) signal D0 K γ mode ∗0→ as well as the backgrounds (b) D0 K−π+π0 and (c) D0 K η and (d) the combinatoric background. Events between→ the vertical lines are→ selected.

71 ∗0 The helicity angle of either K or φ: cos(θH ). •

A φ meson mass distribution is very narrow, with a width of 4 MeV/c2. Most true φ mesons can be preserved by cutting on the invariant mass distribution, of (1.01 <

M(K−K+) < 1.03) GeV/c2.

All remaining selection criteria are determined by optimizing signal signiﬁcance as deﬁned by Eq.4.12.

N(D0 Vγ) σ = → (4.12) N(D0 Vγ)+ N(BG) → Where N(D0 Vγ) is thep number of signal events, N(D0 Vγ), weighted → → according to the assumed branching fractions: (D0 φγ) = 2.5 10−5 and B → × ∗0 (D0 K γ) = 2.1 10−5. N(BG) is the sum of generic processes, π0 back- B → × grounds, and η backgrounds. Each of these datasets are weighted according to the cross section of each mode, shown in Table 3.1.

The correlations between the observables listed above results in σ not being independent for each quantity. To account for this, rather than optimizing significance for each observable independently, we measure significance as a function of two observables. This means optimizing significance across three pairs of observables: both the high and low selection values of M(K−π+), the high and low selection values of

∗0 K helicity angle, the minimal γ and D∗+ momenta.

∗0 We use helicity differently in the D0 φγ and D0 K γ studies. The helicity → → distribution of both signal and background for D0 φγ is well understood and → predictable. As a result we will use this observable in the final fit. In the case of

∗0 ∗0 D0 K γ, though D0 K π0 has a helicity distribution proportional to cos2, → → D0 K−π+π0 can proceed through many other resonances. This leads to a far → 72 more complicated helicity distribution. This can be seen in Fig. 4.4(b), where the peak seen near 0.6 is largely due to D0 K−ρ+ events. An accurate model − → of the helicity distribution requires an accurate measure the multitude of different contributions to the final state D0 K−π+π0. Since we do not expect our MC to → fully model all intermediate states, rather than fitting the helicity distribution for

∗0 D0 K γ, we simply select events from a region in which signal modes are favored. → Such complications are not a problem in D0 φγ, where background from non-φ → D0 K−K+π0 events is reduced to zero by the very narrow M(φ) distribution. → The selection of the optimal helicity angle requirement is made by varying cuts on the high and low values of helicity between [0.3, 0.8] on the high end and [ 0.7, 0.2] − − on the low end. We vary each cut independently and measure significance as defined in Eq. 4.12 for each set of cuts. We find that significance is maximized for a cut on

∗0 helicity, ( 0.30 < cos(θH ) < 0.65), in the K mode. − Cuts on M(K−π+) are varied between [0.830, 0.860] GeV/c2 on the low side and

[0.930, 0.960] GeV/c2 on the high side. As each cut is varied independently, the sig- niﬁcance was measured. The selection region with the greatest observed signiﬁcance is found to be (0.848 < M(K−π+) < 0.951) GeV/c2. This region corresponds to a

∗0 ∗0 width slightly less than Γ(K ) on the low side and slightly greater than +Γ(K ) − ∗0 ∗0 on the high side, where Γ(K ) is the lifetime of the K . This slight asymmetry helps to reduce background from D0 K−π+π0 events which proceed through → ∗0 non-K channels, speciﬁcally D0 K−ρ+. The invariant M(K−π+) distribution → ∗0 of these events peaks at masses less than the nominal K mass.

By requiring a high momentum D∗+ candidate backgrounds from BB¯ events are greatly reduced. D∗+ candidates produced as a result of B D∗+X, where X is →

73 4000 1000

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0 ∗0 Figure 4.4: The cos(θH ) distributions for the (a) signal D K γ mode as well as ∗0 → the backgrounds (b) D0 K−π+π0 and (c) D0 K η and (d) the combinatoric → ∗0 → ∗+ ∗+ 0 background after cuts on M(K ), PCMS(γ), PCMS(D ), and (M(D ) M(D )) as well as a π0 veto have been applied. Events between the vertical lines ar−e selected.

74 another unreconstructed state, often carry low momentum. The energy which could be delivered to the D∗+ is shared with X leading to a lower momentum D∗+. By requiring higher momentum D∗+ candidates, we can nearly eliminate the background from B events. Medium to high momentum photons are produced by D0 Vγ → events. By requiring a minimum energy for our photon candidates, background from random events can be greatly reduced.

Cuts on both the D∗+ and photon momenta are varied independently between

[2.60, 2.80] GeV/c2 for D∗+ and [0.40, 0.80] GeV/c for photons. The signiﬁcance of

D0 φγ is measured for each pair of selection values, using Eq. 4.12. This value is → ∗+ maximal for PCMS(γ) > 0.540 GeV/c and PCMS(D ) > 2.62 GeV/c. These cuts are

∗0 applied to both the D0 φγ and D0 K γ analyses. The measured signiﬁcance → → as a function of cuts on (a) the low and high values of helicity, (b) the low and high values of M(K−π+), and (c) the D∗+ and γ CMS momenta are shown in Fig. 4.5.

The ﬁnal event selection criteria are detailed in Table 4.3. We measure the ef-

ficiency for the signal modes, the prominent backgrounds, and the generic qq¯ backgrounds based on the initial sample sizes shown in Table 4.1. The efficiencies are shown in Tables 4.3 and 4.5, the former shows the efficiency of D0 φγ datasets → ∗0 and the latter shown the efficiency of D0 K γ datasets. →

75 ) 2 0.8 0.96 13.5

) High 14.95 H θ 0.955 0.7 13 14.9 Cos(

12.5 0.95 14.85

0.6 ) Cut High (GeV/c + π

12 - 0.945 14.8

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11 0.4 14.7 0.935

10.5 14.65 0.3 0.93 10 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 0.83 0.835 0.84 0.845 0.85 0.855 0.86 - + 2 Cos(θH) Cut Low M(K π ) Cut Low (GeV/c )

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0.8 4.95 0.75 4.9

) (GeV/c) 0.7 γ ( 0.65 CMS 4.85 P 0.6

0.55 4.8 0.5

0.45 4.75 0.4 2.6 2.65 2.7 2.75 2.8 *+ PCMS(D ) (GeV/c) (c)

− + Figure 4.5: Significance as a function of (a) cos(θH ) cut, (b) M(K π ) cut, and (c) D∗+ and γ CMS momenta. Here red corresponds to regions of high significance, as compared to blue whihc represents regions of low significance.

76 ∗0 Candidate Track D0 φγ Selection D0 K γ Selection → → Kaons KLHTight pions - piLHVeryTight [∗] Photons PCMS > 0.54 (0.0 < Lateral Moment < 0.8) V [∗] (1.01 M(K−K+) 1.03) (0.848 < M(K−π+) < 0.951) ≤ ≤ cos(θH ) < 0.9 ( 0.30 < cos(θH ) < 0.65) +[∗] | | − πs (RXY < 15mm) (RZ < 100mm) (0.10 < PCMS < 0.45) D0[∗] (1.70 < M(D0) < 2.02) P (χ2,N) > 0.001 D∗+[∗] (0.1435 < M(D∗+) M(D0) < 0.1475) − PCMS > 2.62

Table 4.3: Analysis cuts made to elucidate the D0 Vγ signal. [ ] Please note that masses are represented in units of GeV/c2 and momenta→ are represented∗ in units of GeV/c.

77 Cut ǫ(D0 φγ) ǫ(D0 φπ0) ǫ(D0 φη) → → → Post Reconstruction Eﬃciency 25.9% 11.0% 1.5% π0 Veto 21.5% 3.1% 1.0% KLHTight 14.7% 2.1% 0.7% ∆M Cut 13.4% 1.7% 0.3% M(φ) 12.0% 1.5% 0.3% ∗+ PCMS(D ) 11.9% 1.5% 0.3% PCMS(γ) 11.3% 1.4% 0.3% cos(θH ) Cut 10.8% 1.0% 0.2% Cut ǫ(cc)( 10−5) ǫ(uds)( 10−5) ǫ(B+B−)( 10−6) ǫ(B0B0)( 10−6) × × × × Post Reconstruction 8.20 2.31 6.79 8.04 Eﬃciency π0 Veto 2.96 0.99 2.69 3.21 KLHTight 0.81 0.31 1.24 1.54 ∆M Cut 0.23 0.074 0.31 0.38 M(φ) 0.15 0.044 0.17 0.25 ∗+ PCMS(D ) 0.14 0.042 0.16 0.23 PCMS(γ) 0.11 0.032 0.15 0.22 cos(θH ) Cut 0.090 0.028 0.13 0.20

∗+ 0 + 0 Table 4.4: Cumulative cut eﬃciencies for the signal (D D πslow; D φγ), ∗+ 0 + 0 0 ∗+→ 0 + 0 → prominent backgrounds (D D πslow; D φπ & D D πslow; D φη), and the generic modes. Please→ note that the “Post→ Skim Reconstruc→ tion” Eﬃciency→ refers to the number of reconstructed events with the inital loose cuts applied when searching for D∗+ D0π+ ; D0 φγ. → slow →

78 ∗0 ∗0 Cut ǫ(D0 K γ; ǫ(D0 K−π+π0) ǫ(D0 K η) ∗0 → → → K K−π+) → Post Reconstruction Eﬃciency 21.7% 2.2% 1.21% ∆M Cut 19.0% 1.5% 0.67% π0 Veto 16.3% 0.4% 0.57% PID Cuts 13.2% 0.4% 0.46% ∗0 M(K ) 11.0% 0.2% 0.35% ∗+ PCMS(D ) 10.9% 0.2% 0.34% PCMS(γ) 10.3% 0.2% 0.32% Cos(θH ) Cut 6.4% 0.03% 0.070% Cut ǫ(cc)( 10−4) ǫ(uds)( 10−5) ǫ(B+B−)( 10−5) ǫ(B0B0)( 10−5) × × × × Post Reconstruction 3.00 8.26 3.50 4.80 Eﬃciency ∆M Cut 0.88 2.02 0.83 1.19 π0 Veto 0.35 0.86 0.33 0.49 PID Cuts 0.25 0.50 0.23 0.35 ∗0 M(K ) 0.15 0.31 0.13 0.20 ∗+ PCMS(D ) 0.14 0.29 0.12 0.18 PCMS(γ) 0.11 0.24 0.10 0.17 Cos(θH ) Cut 0.048 0.11 0.049 0.077

∗+ 0 + 0 ∗0 Table 4.5: Cumulative cut efficiencies for the signal (D D πslow; D K γ), ∗+ 0 + 0 − + 0 ∗+ 0 + → 0 ∗0 → (D D πslow; D K π π ), and (D D πslow; D K η. Also included→ are the efficiencies→ for the generic MC. The→ efficiencies of the→ generic samples are measured relative to Table 4.1.

79 CHAPTER 5

Experimental Search for Radiative D0 Decays

5.1 D0 φγ Analysis →

The branching ratio for this mode has previously been measured by BELLE to be, (D0 φγ)=(2.40+0.70(stat.)+0.15(sys.)) 10−5 [10]. This result was obtained B → −0.61 −0.17 × by fitting the invariant φγ mass distribution, M(φγ), using a binned maximum likelihood fit. Our study differs from this approach by fitting both M(φγ) and cos(θH ) using an unbinned two-dimensional maximum likelihood fit. By fitting both distributions simultaneously we are able to reconstruct a higher precentage of true D0 φγ candidates than was possible in the BELLE analysis. In addition our → study is based a larger overall dataset than was used by BELLE and also avoids the systematic uncertainties of a binned fitting procedure.

Our initial studies described in the previous section were limited strictly to MC generated events. In order to diﬀerentiate contributions from signal events, π0 and

η backgrounds and combinatoric events, MC was divided into four distinct classes.

D0 φγ events • →

D0 φπ0 events • →

D0 φη events • → 80 The remaining generic MC events. •

We modeled the invariant M(φγ) and cos(θH ) distributions of each class independently, using MC to obtain and fix the signal shapes. The signal shapes were added together in linear combination to form a composite Probablity Distribution Func- tion (PDF). Using an unbinned extended maximum likelihood fit method (EMLM), the composite PDF is fit to data with the normalizations allowed to float and taken as the yield of each class of event.

EMLM minimizes log( ), where the likelihood, , is deﬁned in Eq. 5.1. − L L − m n N m e j=1 j = nj j(M(φγ),cos(θH );κ ¯j (5.1) L PN! P i=1 j=1 ! Y X In this expression, m denotes the number of component PDFs within the composite PDF, while N denotes the number of events in our analysis. The likelihood is taken by summing over all component PDF’s, j(M(φγ),cos(θH );κ ¯j), and taking P the product across all N candidates in the dataset. Each PDF is deﬁned by k pa-

th rameters denoted byκ ¯j = (κ1,κ2, ...κk). Here j denotes the j component PDF.

These parameters,κ ¯j, are ﬁxed using MC for the four event classes.

When constructing a two dimensional PDF, we assume no correlation between

M(φγ) and cos(θH ) observables. This is verified by calculating the correlation coefficients between M(φγ) and cos(θH ) in each MC event class. In all cases the correlation coefficients are small, on the order of 10−2. This validates the assumption, that no correlation exists between the M(φγ) and cos(θH ). Table 5.1 shows the correlation between these two quantities in both signal and generic MC.

81 0 Dataset σD Mass, cos(θH ) D0 φγ 0.62 10−2 D0 → φπ0 −0.63 × 10−2 D0 → φη − 5.4 ×10−2 cc → −3.9 × 10−2 − × uds 2.2 10−2 B+B− −0.29× 10−2 − × B0B0 2.3 10−2 − ×

0 Table 5.1: The correlation coeﬃcients between D mass and cos(θH ).

5.1.1 D0 φγ Signal Shape → The invariant mass distribution, M(φγ), of the D0 φγ event class is fit to a → linear combination of a Crystal Ball (CB) line shape and a flat polynomial, where the flat polynomial is given by a zeroth order Chebychev polynomial, T0(x) = 1. As discussed in Section 3.2.4, the CB line shape is used to model energy loss associated with reconstructing photons in the EMC. The CB functional form is shown in Eq.

3.10 and restated below.

2 − (x−µ) x−µ 2σ2 e , if σ > α CB(x; α,n,µ,σ)= x−µ x−µ − (5.2) ( A(B )−n, if α − σ σ ≤ −

n 2 n − |α| A = e 2 α n| | B = α α − | | | |

In this expression the order, n, is fixed (n = 5), while all other terms are allowed to float in the fit. A flat polynomial is included in the M(φγ) distribution to include events on the ends of the M(φγ) window. The flat portion of the PDF is not

82 included in the ﬁnal ﬁtting PDF.

The cos(θH ) distribution is fit to a linear combination of a second order Cheby- chev polynomial, T2, detailed in Appendix B, and a flat background, represent by a zeroth order Chebychev polynomial, T . The total PDF used to fit the D0 φγ 0 → event class is formed by taking the product of the M(φγ) and cos(θH ) distributions with no correlation built in. The PDF is expressed in Eq. 5.3.

0 0 (M(φγ),cos(θH );κ ¯) = N(D φγ) PD →φγ → ×

(CB(M(φγ); α, n = 5,µ,σ) T (cos(θH ); m,q)) + × 2

N(Flat) (T (M(φγ)) T (cos(θH ))) (5.3) × 0 × 0

In this expression, CB is the CB line shape shown in Eq. 3.10, T2 is a sec-

ond order Chebychev polynomial with linear and quadratic coeﬃcients: mcos(θH ),

qcos(θH ), respectively. N(Flat) is the coefficient of the constant (flat) background, and T0(M(φγ)) and T0(cos(θH )) are the flat components of the M(φγ) and cos(θH ) distributions. After Eq. 5.3 is fit to generated D0 φγ MC, all parameters are → fixed. The normalization, N(D0 Vγ) is the event yield returned by the fitter. →

The two dimensional distribution of M(φγ) and cos(θH ) is shown in Fig. 5.1(a).

Figs. 5.1(b) and 5.1(c) show the dataset projected onto the M(φγ) and cos(θH ) axes as well as the fits results. Figs 5.1(b) and 5.1(c), the χ2/N values are shown as a rough quantification of the “goodness-of-fit”. Since we are using an unbinned

EMLM, this is not meant to reflect the true quality of the fit. The parameter values corresponding to the optimal fit are shown in Table 5.2.

The D0 φγ eﬃciency is deﬁned in Eq. 5.4. → 0 0 NSignal(D φγ) ǫ(D φγ)= 0→ (5.4) → NGenerated(D φγ) → 83 )

H 140 θ 0.8

Cos( 0.6 120

0.4 100

0.2 80 -0 60 -0.2

-0.4 40

-0.6 20 -0.8 0 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

(a)

) 700 2 1200 2 600 χ /N = 316.90/315 1000 500 Events / ( 0.05 ) 800 400

Events / ( 0.001 GeV/c 600 300

400 200

100 200 χ2/N = 28.87/33

0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 0 2 D Mass (GeV/c ) Cos(θH)

2 i 6 6 χ

± 4 4 2 2 2 i χ

0 0 ± -2 -2 -4 -4 -6 -6

D0 Mass (GeV/c2) -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 (b) (c)

Figure 5.1: (a) A two dimensional event surface for D0 φγ events extracted from → 0 signal MC. Fits to the (b) M(φγ) and (c) cos(θH ) distributions for D φγ events. The flat background is shown with a dotted line in each case. The→ residuals are plotted beneath either fit. The red line specifies a 2σ difference between the fitted shape and data.

84 Parameter Final Fit Value 0 NSignal(D φγ) 28901.1 173.6 0 → ± NFlat(D φγ) 646.4 43.2 → ± α(D0 φγ) 1.13 0.02 µ(D0 → φγ) 1.8606 ± 0.0002 σ(D0 → φγ) 0.0179 ± 0.0001 → 0 ± mcos(θH )(D φγ) 0.0037 0.096 q (D0 →φγ) 0.699± 0.011 cos(θH ) → − ±

Table 5.2: The final values for D0 φγ PDF obtained by fitting D0 φγ signal → → MC. NSignal and NFlat are the normalizations for both the signal and flat background components. The terms α, µ, and σ are parameters of the M(φγ) distribution and correspond to the parameters shown in Eq. 3.10. The terms mCos(θH ) and q (D0 φγ) are the linear and quadratic coefficients of the quadratic Cos(θH ) → polynomial used to fit the cos(θH ) distribution, represented as m and q in Eq. 5.3.

0 0 Using the value of NSignal(D φγ) shown in Table 5.2 and NGenerated(D φγ) → → shown in Table 4.1, the eﬃciency is found to be ǫ(D0 φγ) = (10.8 0.07)%. → ±

85 5.1.2 D0 φπ0 Background →

0 The M(φγ) and cos(θH ) distributions are estimated using MC simulating D → φπ0 events. These events are fed through our D0 φγ analysis. The resulting → two dimensional distribution of M(φγ) and cos(θH ) is ﬁt using a two dimensional

EMLM, described earlier.

In this case, the M(φγ) distribution is modeled using a linear combination of the CB line shape shown Eq. 3.10 and a ﬂat background. The order of the CB is set to n = 3 and the remaining parameters are allowed to ﬂoat freely. A CB lineshape is used to account for the asymetric energy resolution of the EMC, and also to account for the missing energy resulting from the imcomplete reconstruction of a second photon produced by π0. This results in a distribution with a much longer decay tail, as shown in Fig. 5.2(b).

The helicity distribution is modeled with a linear combination of a second order

Chebychev polynomial, T2(cos(θH ); m,q) (where m and q are the ﬁrst and second order coeﬃcients of the polynomial) and a zeroth order Chebychev polynomial,

T0(cos(θH ). The total PDF is shown in Eq. 5.5.

0 0 0 0 (M(φγ),cos(θH )) = N(D φπ ) PD →φπ → ×

(CB(M(φγ); α, n = 3,µ,σ) T (cos(θH ); m,q)) + × 2

N(Flat) (T (M(φγ)) T (cos(θH ))) (5.5) × 0 × 0

The resulting ﬁts are shown in Figs. 5.2(b) and 5.2(c), while Fig. 5.2(a) shows

0 0 the two dimensional distribution of M(φγ) and cos(θH ) obtained from D φπ → MC. The parameters obtained from ﬁtting D0 φπ0 MC are ﬁxed to the values → shown in Table 5.3.

86 ) H θ 0.8 14

Cos( 0.6 12 0.4 10 0.2 8 -0 6 -0.2 4 -0.4

-0.6 2

-0.8 0 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

(a) )

2 160 200 140 χ2/N = 64.27/59 180 χ2/N = 39.18/33 120 160 140 100 120 Events / ( 0.05 )

Events / ( 0.005 GeV/c 80 100 60 80 60 40 40 20 20 0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 D0 Mass (GeV/c2) Cos(θH) 2 i 2 i χ χ

± 6 6 4 ± 4 2 2 0 0 -2 -2 -4 -4 -6 -6 1.7 1.75 1.8 1.85 1.9 1.95D0 Mass 2 (GeV/c2) -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 (b) (c)

Figure 5.2: (a)The two dimensional distribution of M(φγ) mass vs Helicity for 0 0 D φπ events. Projections of both (b) the M(φγ) distribution and (c) cos(θH ) distribution→ for D0 φπ0 events. →

87 Parameter Final Fit Value 0 0 NSignal(D φπ ) 2493.1 51.6 0 → 0 ± NFlat(D φπ ) 96.7 16.7 → ± α(D0 φπ0) 0.353 0.032 µ(D0 → φπ0) 1.834 ±0.0018 σ(D0 → φπ0) (2.44 0.±013) 10−2 → 0 0 ± × −2 mCos(θH )(D φπ ) (0.31 2.48) 10 q (D0 →φπ0) 0.999± 0.006× Cos(θH ) → ±

Table 5.3: The final values for D0 φπ0 PDF obtained by fitting D0 φπ0 signal → → MC. NSignal and NFlat are the normalizations for both the signal and flat background components. The terms α, µ, and σ are Crystal Ball line shape parameters used to model the M(φγ) distribution. The terms m and q (D0 φπ0) are the Cos(θH ) Cos(θH ) → linear and quadratic coefficients of the quadratic polynomial used to fit the cos(θH ) distribution, and correspond to the terms m and q seen in Eq. 5.5.

5.1.3 D0 φη Background →

0 The impact that D φη decays have upon the M(φγ) and cos(θH ) distribu- → tions is assessed by generating MC which simulates D0 φη decays. These events → are fed through our D0 φγ analysis and the resulting two dimensional M(φγ) → and cos(θH ) distribution is ﬁt using a two dimensional EMLM.

The M(φγ) distribution of this event class is modeled using a linear combination of a Gaussian and a linear background, denoted by T . Similar to D0 φγ 1 → and D0 φπ0 these events could be modeled using a CB line shape; however, our → D0 mass window removes the trace of a radiative tail, seen within the two previous classes of events. Hence only a Gaussian is used. The helicity distribution is modeled with a linear combination of a second order Chebychev polynomial, T2(cos(θH ); m,q)

(where m and q are the ﬁrst and second order coeﬃcents of the polynomial) and a

88 ﬂat polynomial. The total PDF is shown in Eq. 5.6.

0 0 (M(φγ),cos(θH )) = N(D φη) PD →φη → ×

(Gs(M(φγ); µ, σ) T (cos(θH ); m,q)+ × 2 0 ′ N(Flat) (T (M(D ; m )) T (cos(θH ))) (5.6) × 1 × 0

In this expression µ and σ are the mean and standard deviation of the Gaussian

(Gs). The Chebychev polynomials are represented by T1 for the first order and T2 for the second order. The values m and q are the first and second order coefficients of a general Chebychev polynomial.

The resulting two dimensional distribution of M(φγ) and cos(θH ) is shown in

Fig. 5.3(a). Figs. 5.3(b) and 5.3(c) show the projections of the ﬁts on M(φγ) and cos(θH ), respectively. After the optimal ﬁt is found using MC, the parameters in

Eq. 5.6 are ﬁxed to the values shown in Table 5.4.

89 ) H θ 0.8 10

Cos( 0.6 8 0.4

0.2 6 -0 4 -0.2

-0.4 2 -0.6 0 -0.8 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

(a)

) 90 2

80 60 2 2 70 χ /N = 20.09/59 50 χ /N = 24.03/33

60 Events / ( 0.05 ) 40 50 Events / ( 0.005 GeV/c 40 30

30 20 20 10 10

0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 0 2 D Mass (GeV/c ) Cos(θH) 2 i 2 i χ χ

± 6 ± 6 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 0 2 Cos(θ ) 1.7 1.75 1.8 1.85 1.9 1.95D Mass 2 (GeV/c ) -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 H (b) (c)

Figure 5.3: (a) Shows the two dimensional distribution of D0 φη events, while → the projection of the ﬁts is shown in (b) M(φγ) and (c) cos(θH ) distributions. The ﬂat background is shown with a dotted line in each case.

90 Parameter Final Fit Value 0 NSignal(D φη) 537.7 24.1 0 → ± NFlat(D φη) 60.4 10.2 → ± µ(D0 φη) 1.71 0.003 σ(D0 → φη) (2.07 0±.17) 10−2 → 0 ± × mD0 Mass(D φη) 0.68 0.22 0 → − ± mCos(θH )(D φη) 0.012 0.054 q (D0 →φη) −0.996 ±0.016 Cos(θH ) → ±

Table 5.4: The final values for D0 φη PDF obtained by fitting D0 φη signal → → MC. NSignal and NFlat are the normalizations for both the signal and flat background components. The terms µ and σ are parameters of a Gaussian distribution used to model the M(φγ) distribution. An additional background to M(φγ) is modeled 0 with a linear polynomial with mD0 Mass(D φη) used to model the slope. This corresponds to m′ seen in Eq. 5.6 The terms m→ and q (D0 φη) are the Cos(θH ) Cos(θH ) → linear and quadratic coefficients of the quadratic polynomial used to fit the cos(θH ) distribution and are written as m and q in Eq. 5.6.

91 5.1.4 Remaining Background

The ﬁnal class of background within our study is comprised of the subset of generic cc events which do not contain any of the previously discussed event classes, as well as generic uds, B+B−, and B0B0 events. This dataset is labeled “combinatoric background”. These four generic modes are individually weighted to match the uds sample’s equivalent luminosity of 340fb−1. This was the maximum available luminosity for this mode when the analysis was developed. Each event is weighted by w(X)= (uds)/ (X), where (X) is the equivalent luminosity of each generic L L L sample as shown in Table 4.1.

Combinatoric background events are ﬁt using a two dimensional maximum likelihood method (MLM). Here MLM is used rather than EMLM because we are not

ﬁtting a linear combination of PDFs, but rather a single PDF. The M(φγ) distribution of these events is modeled with a second order Chebychev polynomial. The helicity distribution is also modeled with a second order Chebychev polynomial.

These are represented in Eq. 5.7.

BG(M(φγ),cos(θH )) = T (M(φγ); m ,q ) P 2 M(φγ) M(φγ) ×

T2(cos(θH ); mcos(θH ),qcos(θH ))

(5.7)

In this expression m and q are the ﬁrst and second order coeﬃcients of the polynomials.

Figs. 5.4(b) and 5.4(c) show the projections of the M(φγ) distribution and cos(θH ) distribution as well as the optimal ﬁt. The full two dimensional surface is

92 Parameter Final Fit Value −1 mD0 Mass(Comb BG) ( 5.32 0.59) 10 − ± × −1 qD0 Mass(Comb BG) (2.15 0.53) 10 ± × −2 mCos(θH )(Comb BG) (2.78 6.31) 10 q (Comb BG) (2.40 ± 0.53) × 10−1 Cos(θH ) ± ×

Table 5.5: The ﬁnal values for the combinatoric background PDF obtained by ﬁtting generic MC events. Both the M(φγ) and cos(θH ) distributions are modeled with quadratic polynomials. In each case m and q are the linear and quadratic terms of the polynomials.

shown in Fig. 5.4(a). Once the optimal ﬁt is reached, the parameters are ﬁxed to the values shown in Table 5.5.

∗0 We also look for possible reﬂections from D0 K γ events caused by incor- → rectly identifying a pion as a kaon. Reﬂections from these events are reduced by cutting tightly on the φ meson invariant mass and by applying a tight PID require-

∗0 ∗0 ment. Feeding down signal MC of D0 K γ; K K−π+ through our D0 φγ → → → analysis we measure the impact of these events. Using the highest predicted branch-

∗0 ing fraction of D0 K γ we see only 1.4 events surviving the ﬁnal cuts; therefore, → we can safely ignore contributions from these events.

We look for background due to D0 K−π+π0 events in which the true pion → is incorrectly identiﬁed as a kaon. Using D0 K−π+π0 events within our cc MC → sample, we observe 52.3 of these events surviving our ﬁnal analysis cuts. This background does not have any structure in either the M(φγ) or helicity distributions.

This allows us to integrate background from D0 K−π+π0 events into our general → combinatoric background shape. We further look for reﬂections due to decays such

93 ) 7 H θ 0.8 6 Cos( 0.6 5 0.4

0.2 4

-0 3

-0.2 2 -0.4 1 -0.6 0 -0.8 -1 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

(a) ) 2 35 2 35 χ /N = 23.33/34 2 30 χ /N = 53.30/62 30 Events / ( 0.05 ) 25 25

20 20 Events / ( 0.005 GeV/c

15 15

10 10

5 5

0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 0 2 D Mass (GeV/c ) Cos(θH) 2 i 2 i χ χ

± 6 ± 6 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 0 2 Cos(θ ) 1.7 1.75 1.8 1.85 1.9 1.95D Mass 2 (GeV/c ) -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 H (b) (c)

Figure 5.4: (a) The two dimensional histogram of the combinatoric background taken from generic MC. Fits to (b) M(φγ) distribution and (c) cos(θH ) distribution on combinatoric background events.

94 0 0 0 0 0 0 + 0 0 0 0 as: D φKs; Ks π π , D φπ π , Ds φπ π , D ρ γ, and Ds ρ γ. → → → → → → In all cases no signiﬁcant signal is seen.

5.1.5 Summary of Signal Shapes

Each of the four PDF’s are added in a linear combination to form a composite

PDF used to model data. With all parameters fixed only the normalizations of each mode are allowed to float in the final fit. In this way the yield of D0 φγ, → N(D0 φγ), can be taken as the total number of reconstructed signal events. → The full linear combination is shown in Eq. 5.8. Each of the component PDFs are summarized in Table 5.6

0 Total(M(φγ),cos(θH ); N¯) = N(D φγ) 0 (M(φγ),cos(θH )) + P → PD →φγ 0 0 N(D φπ ) 0 0 (M(φγ),cos(θH )) + → PD →φπ 0 N(D φη) 0 (M(φγ),cos(θH )) + → PD →φη

N(BG) BG(M(φγ),cos(θH )) P N¯ = (N(D0 φγ),N(D0 φπ0),N(D0 φη),N(BG)) → → → (5.8)

5.1.6 Fitting Method Validation

Even with the signal shape fixed, it is not clear that a fitting procedure will be able to fully discriminate between D0 φγ and D0 φπ0 events. The M(φγ) → → invariant mass distributions for these two classes of events have a large overlap. In order to understand how well our fitting procedure differentiates these events in data, we will apply the full fitting procedure on MC events, representing a realistic

95 Fit Component (M(φγ) (cos(θH )) P P D0 φγ Crystal Ball(n=5) Quadratic → D0 φπ0 Crystal Ball(n=3) + Flat Quadratic + Flat → D0 φη Gaussian + Linear Quadratic + Flat → Combinatoric Quadratic Quadratic

Table 5.6: Summary of each component used in the overall ﬁt.

estimate of data.

When forming the combined MC dataset, generic MC events are selected with a probability ρ = (uds)/ (Mode), where (Mode) is the equivalent luminosity of L L L the total generic sample of a given mode, as shown in Table 4.1. The luminosity of uds MC is used as a scale factor to ensure ρ 1. As generic cc contains D0 φπ0 ≤ → events, this event class is simple taken from generic MC. Speciﬁc contributions of

D0 φγ MC, and D0 φη MC are weighted according to the assumed branching → → fractions for each mode as shown in Eqs. 5.9 and 5.10.

(D0 φγ) = 2.5 10−5 (5.9) B → × (D0 φη) = 1.4 10−4 (5.10) B → ×

The equivalent luminosity of a speciﬁc MC sample is found using Eq. 5.11.

N(D0 φX) (D0 φX)= → L → σ(cc) R(D∗+) (D∗+ D0π+ ) (D0 φX) (φ K−K+) × × B → slow × B → × B → (5.11)

In Eq. 5.11, N(D0 φX) is the total number of generated D0 φX events; → → σ(cc) is the cross section for cc events; R(D∗+) is the percentage of D∗+ candidates appearing in a cc event; (D∗+ D0π+ ) is the world average for the branching B → slow fraction of D∗+ D0π+ ; (D0 φX) is the presumed branching fraction for a → slow B → 96 given signal mode; and (φ K−K+) is the branching fraction for φ K−K+. B → → In the case of D0 φγ this corresponds to 214 candidates occurring within the test → sample.

Once a combined MC dataset is generated it is ﬁt to the linear combination shown in Eq. 5.8. In this EMLM ﬁt, the normalizations of all PDF’s, N, are allowed to

float freely and are taken as the yields of each mode. The fit to the combined dataset is shown in Fig. 5.5. The measured yield of N(D0 φγ) = 207.8 23.1 compares → ± well with the actual input sample size of 214 candidates. The fit is conducted on the range ( 0.9 < cos(θH ) < 0.9. This range was choosen by fitting the combined − MC dataset with successively tighter cuts applied along the helicity angle. For each cut the significance was estimated with σ = N(D0 φγ)/δN(D0 φγ), → → where δN(D0 φγ) is the statistical uncertainty of N(D0 φγ). A cut of, → → cos(θH ) < 0.9, was found to have optimal significance. | | To further verify our fitting and measurement procedure we vary the number of

D0 φγ MC events which are combined with generic MC, in essence varying the → input branching fraction. In addition to varying the scale of D0 φγ MC events, → we also divide our MC samples from the other modes into ﬁve distinct sets. These are then mixed together to correspond to a total luminosity = (cc)/5. Generic L L the sample of generic uds events limits our overall sample size, in terms of the total equivalent luminosity shown in Table 4.1. Background from uds events is very small, and exlcuding these events does not eﬀect the result background distribution.

Therefore, rather than limiting this cross check by using uds we will simply not add contributions from uds events.

The branching fraction is measured for each sample and then compared to the

97 ) 2 60

50 χ2/N = 70.91/76

e+ e- → B+B- e+ e- → B0B0 40 e+ e- → u u, d d, s s e+ e- → c c D0 → φ η D0 → φ π0 D0 → φ γ Combinatoric Background Fit 0 0 Events / ( 0.004 GeV/c 30 Combinatoric Background + D → φ π Fit Combinatoric Background + D0 → φ π0 + D0 → φ η Fit Total Fit MC Data 20

0 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

4 2 2 i

χ 0

± -2 -4 -6 1.7 1.75 1.8 1.85 1.9 1.95 2 (a)

100 e+ e- → B+ B- e+ e- → B0 B0 e+ e- → u u, d d, s s e+ e- → c c D0 → φ η 2 D0 → φ π0 χ /N = 39.62/32 D0 → φ γ 80 Combinatoric Background Fit Combinatoric Background + D0 → φ π0 Fit Combinatoric Background + D0 → φ π0 + D0 → φ η Fit Total Fit Run 1-5 Data Events / ( 0.05 ) 60

-0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 Cos(θH) 2 i χ

± 6 4 2 0 -2 -4 -6 Cos(θ ) -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 H (b)

Figure 5.5: Fits to (a) M(φγ) invariant mass distribution and (b) cos(θH ) distribution on all events. The sum of all PDF’s are shown in solid blue lines, with the D0 φπ0 components in dashed blue and combinatoric contributions in dashed → red. Signal D0 φγ events are shown in the yellow histograms, signal D0 φπ0 events in the grey→ histograms, signal D0 φη events are shown in orange,→ and combinatoric background events in purple,→ green, and red.

98 input value. The measured branching fractions, as shown in Fig. 5.6, hold well to the input values. The red line in Fig. 5.6 correspond to the ideal solution, a line of slope 1 and intercept 0. Based on this result we observe no bias in our ﬁtting procedure.

The branching fraction for each dataset is estimated relative to the well studied decay D0 K−π+. The relative branching fraction can be measured using Eq. → 5.12.

(D0 φγ) N(D0 φγ) ǫ(D0 K−π+) 1 B → = → → (D0 K−π+) N(D0 K−π+) × ǫ(D0 φγ) × (φ K+K−) B → → → B → (5.12)

In this expression, N(D0 K−π+) is the number of reconstructed D0 K−π+ → → events and ǫ(D0 K−π+) is the reconstruction eﬃciency for D0 K−π+ decays. → →

99 ×10-6

) Measured B(Measured) = B(Input)

γ 0 Measured B(D → φ γ) φ 30 →

0 25 B(D 20

-6 0 ×10 0 5 10 15 20 25 30 35 40 B(D 0 → φ γ) Input

Figure 5.6: The result of our ﬁt procedure as the weight of D0 φγ MC dataset is → varied. The red line denotes (Nin = Nout).

100 0 5.2 D0 K∗ γ Analysis →

∗0 The CLEO II collaboration set the current upper limit of (D0 K γ) < B → 7.6 10−4, with 90% conﬁdence. This upper limit was obtained using a dataset × with a luminosity 1/30 of the current BABAR dataset. Considering the theoretical predictions placed on the branching fraction in the range of [0.7, 8.0] 10−4, a mea- × ∗0 surement of (D0 K γ) is well within the scope of the larger BABAR dataset. B → ∗0 When this mode is investigated through the charged decay mode of the K meson, K¯ ∗0 K−π+, a large background from D0 K−π+π0 events com- → → plicates any measurement. Unlike the measurement of D0 φγ, where a rel- → atively small background from D0 φπ0 decays is seen, the measurement of → ∗0 (D0 K γ) is greatly eﬀected by the large branching fraction of D0 K−π+π0, B → → (D0 K−π+π0) = (13.5 0.6)%. Such a large rate causes these events to dom- B → ± ∗0 inate the M(K γ) distribution. Background from these events is reduced by a

π0 veto, but the contributions from the surviving events still overwhelm the real

∗0 D0 K γ signal. → Fitting the helicity distribution is not as practical as it was in the D0 φγ anal- → ∗0 ysis. While the helicity angle for D0 K γ events will follow a sin2 distribution, → distribution of D0 K−π+π0 background events is more complicated. The helicity → distribution of these events is largely dependent upon the intermediate resonances contributing to D0 K−π+π0. These include: → D0 K−ρ+; ρ+ π+π0 • → →

D0 K∗−π+; K∗− K−π0 • → →

∗0 ∗0 D0 K π0; K K−π+ • → → 101 And at least 7 other intermediate states. • The ﬁrst three are the most plentiful intermediate states and are modeled within our MC sample. The remaining intermediate states are largely ignored in the MC generator, because of uncertainities in their amplitudes and phases. Each of these states can interfere with the others and produce a helicity distribution not easily modeled in MC. Additionally the relative size of each mode is uncertain, imposing greater uncertainity in the helicity distribution of MC events.

The result of all these factors is that the D0 K−π+π0 helicity distribution → within our MC sample cannot be trusted to faithfully represent the helicity dis-

∗0 tribution in data. Therefore we choose not to ﬁt both the M(K γ) and cos(θH ) distributions simultaneously. We will instead modify our approach and cut directly

∗0 on the helicity angle, ﬁtting only M(K γ), using an unbinned EMLM.

∗0 ∗0 In order to measure the number of D0 K γ events using the M(K γ) dis- → tribution, we must estimate the invariant distribution for four distinct classes of events:

∗0 1. D0 K γ events →

2. D0 K−π+π0 events →

∗0 3. D0 K η events →

4. The remaining combinatoric background events.

∗0 We use MC to simulate each of these events. Speciﬁcally D0 K γ MC is gen- → ∗0 erated to populate the ﬁrst class. D0 K−π+π0 and D0 K η datasets are → → created by segregating generic cc MC based upon the JetSet generator level information. The remaining set of generic MC events are placed into a combinatoric

102 background class.

∗0 ∗0 The invariant M(K γ) distributions, i(M(K γ)) where i is between 1 and 4, P are ﬁt independently for each of the four classes of MC events. In this case the number i corresponds to the list of classes discussed in the previous paragraph. Once

∗0 ﬁt, the shapes of i(M(K γ)) are ﬁxed and the invariant mass spectrum of real P data is modeled using the linear combination shown in Eq. 5.13

i=4 ∗0 ∗0 total(M(K γ); N¯)= Ni i(M(K γ)) (5.13) P P i=1 X In this expression N¯ is the vector of normalizations across the four classes of events,

Ni.

0 5.2.1 D0 K∗ γ Signal Shape → ∗0 ∗0 The M(K γ) distribution of D0 K γ events is modeled with a linear com- → bination of a CB line shape (shown in Eq. 3.10) and a zeroth order Chebychev

∗0 ∗0 polynomial, T (M(K γ)). Our MC dataset containing D0 K γ events is fit to 0 → this model using an unbinned EMLM. The order, n, of the CB is fixed to 5 while all other terms are allowed to float freely. Eq. 5.14 shows the PDF used to model

∗0 D0 K γ events. →

∗0 ∗0 ∗0 (M(K γ)) = N(CB) CB(M(K γ); α, n = 5µ, σ) PD0→K γ × ∗0 N(Flat) T (M(K γ)) (5.14) × 0

In this expression N(CB) and N(Flat) are the normalizations of the CB line shape and ﬂat background. The parameters α, µ, and σ correspond to those shown in Eq.

3.10. The optimal ﬁt results are shown in Table 5.7, and Fig. 5.7 shows the ﬁt to

∗0 the D0 K γ event class. → 103 ) 2 400 2 350 χ /N = 297.21/315

300

250

Events / ( 0.001 GeV/c 200

150

100

0 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

4 2 2 i

χ 0

± -2 -4 -6

1.7 1.75 1.8 1.85 1.9 1.95d0Mass 2 (GeV/c2)

∗0 ∗0 Figure 5.7: The M(K γ) distribution for the signal Monte Carlo D0 K γ events →

104 Parameter Final Fit Value 0 ∗0 NCB(D K γ) 18209 140 0→ ∗0 ± NFlat(D K γ) 671.7 45 →∗0 ± α(D0 K γ) 1.12 0.021 → ∗0 ± µ(D0 K γ) 1.8599 0.0002 → ∗0 ± σ(D0 K γ) 0.0198 0.0002 → ±

∗0 Table 5.7: The final parameters for the D0 K γ signal shape obtained by fitting ∗0 → signal MC D0 K γ events. The MC dataset is fit to a CB lineshape of order → 0 ∗0 0 ∗0 5 over a flat background. The terms NCB(D K γ) and NFlat(D K γ) are the normalizations of the CB and flat background→ respectively. The remaining→ terms, α, µ, σ are parameters corresponding to Equation 3.10.

∗0 The reconstruction eﬃciency of D0 K γ is calculated by taking the ratio of → ∗0 N(CB) in Table 5.7 to the number of generated D0 K γ MC events shown in → ∗0 Table 4.1. This is found to be, ǫ(D0 K γ) = (6.37 0.04)%. The signal shape → ± ∗0 of D0 K γ events is ﬁxed by setting the CB parameters to the values shown in → Table 5.7.

0 + 0 5.2.2 D K−π π Background Shape → The D0 K−π+π0 MC dataset is extracted from our sample of generic cc MC → events by examining the JetSet generator information. If an event contains the decay D∗+ D0π+ ; D0 K−π+π0, the event is ﬂagged and later placed in → slow → the D0 K−π+π0 MC dataset. A smaller sample of pure D0 K−π+π0 MC → → is used to verify this procedure and to calculate the reconstruction eﬃciency of

D0 K−π+π0. → The background from D0 K−π+π0 is modeled using a linear combination of →

105 a zeroth order Chebychev polynominal background and a CB line shape (shown in

Eq. 3.10) of order, n = 3. The PDF used to model these events is expressed in Eq.

5.15.

∗0 ∗0 0 − + 0 (M(K γ)) = N(CB) CB(M(K γ); α, n = 3,µ,σ) PD →K π π × ∗0 N(Flat) T (M(K γ)) (5.15) × 0

In this expression N(CB) and N(Flat) are the normalization of the CB line shape and ﬂat background. The parameters α, µ, and σ correspond to those shown in Eq.

3.10. The values of these parameters are obtained by an unbinned EMLM and are shown in Table 5.7. The optimal ﬁt can be seen in Fig. 5.7.

Parameter Final Fit Value 0 − + 0 NCB(D K π π ) 8042.7 108.9 0→ − + 0 ± NFlat(D K π π ) 2097.5 76.5 α(D0 K→−π+π0) 0.298 ±0.021 µ(D0 → K−π+π0) 1.8367 ± 0.0014 σ(D0 → K−π+π0) (2.28 0.±11) 10−2 → ± ×

Table 5.8: The final parameters for the D0 K−π+π0 signal shape obtained by fitting D0 K−π+π0 events within generic MC.→ The dataset is fit to a CB lineshape → 0 − + 0 0 of order 3 over a flat background. The terms NCB(D K π π ) and NFlat(D K−π+π0) are the normalizations of the CB and flat→ background. The remaining→ terms, α, µ, σ are parameters corresponding to Equation 3.10.

0 5.2.3 D0 K∗ η Background Shape → Applying the same method used to populate the D0 K−π+π0 MC dataset, → ∗0 a D0 K η MC dataset is extracted from our generic cc MC events tagged as → ∗0 containing the full decay: D∗+ D0π+ ; D0 K η. These events are modeled → slow → 106 ) 200 2 180 160 χ2/N = 131.11/155 140

120

100 Events / ( 0.002 GeV/c

0 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

4 2 2 i

χ 0

± -2 -4 -6 1.7 1.75 1.8 1.85 1.9 1.95 2

∗0 Figure 5.8: The M(K γ) distribution for D0 K−π+π0 signal MC events. →

107 with a linear combination of a Gaussian and a ﬁrst order Chebychev polynomial,

∗0 T1(M(K γ)), as shown in Eq 5.16.

∗0 ∗0 ∗0 (M(K γ)) = N(Gaussian) Gs(M(K γ); µ, σ) PD0→K η × ∗0 N(Linear) T (M(K γ); m) (5.16) × 1

∗0 In this expression G(M(K γ); µ, σ) is a Gaussian of mean, µ, and standard deviation σ and the ﬁrst order coeﬃcient of T1 is represented by m. N(Gaussian) and

N(Linear) are the normalizations of the Gaussian and polynomial.

∗0 The D0 K η MC dataset is fit using an unbinned EMLM. All terms are → allowed to float freely in the fit. The optimal fit corresponds to the values shown in Table 5.9, and the resulting fit is displayed in Fig. 5.9. After completing this procedure, the parameters are fixed to the values shown in Table 5.9.

Parameter Final Fit Value 0 ∗0 NGaussian(D K η) 1034.9 45.7 0 → ∗0 ± NLinear(D K η) 251.4 59.8 m → 0.729± 0.050 ∗0 − ± µ(D0 K η) 1.718 0.001 → ∗0 ± σ(D0 K η) 0.025 0.001 → ±

∗0 Table 5.9: The final parameters for the D0 K η signal shape obtained by fitting ∗0 → D0 K η events within generic MC. The dataset is fit to a Gaussian over a → 0 ∗0 0 ∗0 linear background. The terms NGaussian(D K η) and NLinear(D K η) are the normalizations of the Gaussian and linear→ background. The terms→µ and σ are parameters of the Gaussian, while m is the slope of the linear polynomial.

108 ) 2 140 120 χ2/N = 162.86/155 100

80 Events / ( 0.002 GeV/c 60

0 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

4 2 2 i

χ 0

± -2 -4 -6 1.7 1.75 1.8 1.85 1.9 1.95 2

∗0 ∗0 Figure 5.9: The M(K γ) distribution for D0 K η MC events →

109 5.2.4 Combinatoric Background Shape

∗0 ∗0 ∗0 With the D0 K γ, D0 K η, and D0 K−π+π0, invariant M(K γ) → → → mass distributions ﬁxed, the remaining combinatoric background shape is modeled

∗0 by a second order Chebychev polynomial, T2(M(K γ); m,q). The combinatoric background MC dataset is based on events from cc generic MC which contain neither

∗0 D0 K−π+π0 or D0 K η, as well as B0B0, B+B−, and uds generic MC. Each → → of the four generic modes are weighted based upon the cross sections shown in Table

4.1, and formed into a single dataset. The dataset luminosity is set relative to the uds cross section.

∗0 The Chebychev polynomial is respresented as, T2(M(K γ); m,q), where m and q are the first and second order coefficients. The MC dataset is fit using an unbinned

MLM. The resulting parameters are shown in Table 5.10 and the ﬁt can be seen in

Fig. 5.10. The combinatoric background shape is ﬁxed by setting m and q to the values shown in Table 5.10.

Parameter Final Fit Value q 0.155 0.025 m 0.645± 0.025 − ±

Table 5.10: The ﬁnal parameters for the combinatoric background signal shape ∗0 obtained by generic MC events not containing either D0 K−π+π0 or D0 K η. The dataset is ﬁt to a quadratic polynomial, where m is→ the slope of the polynomial→ and q is the quadratic term.

110 ) 2

60 χ2/N = 136.41/158 50

40 Events / ( 0.002 GeV/c 30

0 1.7 1.75 1.8 1.85 1.9 1.95 2 M(D0) (GeV/c2)

4 2 2 i

χ 0

± -2 -4 -6 1.7 1.75 1.8 1.85 1.9 1.95 2

∗0 Figure 5.10: The M(K γ) distribution for the generic MC events not ﬂagged as ∗0 either D0 K−π+π0 or D0 K η. → →

111 5.2.5 Additional Backgrounds

∗0 Additional backgrounds to D0 K γ can arise from either D0 φγ or D0 → → → ρ0γ. In the case of D0 φγ, a K+ can be misidentified as a π+, furthermore → in D0 ρ0γ, a π− prodcued by a decaying rho0 can be misidentified as a K−. → Since both D0 φγ and D0 ρ0γ are Cabibbo suppressed states, the effect from → → these reflections is mitigated by there expected small rates. These reflections are further reduced by strenuous PID requirements. To measure their impact we feed

∗0 down MC of D0 φγ and D0 ρ0γ through our D0 K γ analysis. Using → → → the world average for (D0 φγ) we expect only 1.1 of these events to reflect B → ∗0 into the M(K γ) mass window. Using the D0 ρ0γ MC we see only 0.02% of → ∗0 D0 ρ0γ events reflecting into the M(K γ) mass window. This is a slightly → lower rate is seen for D0 K−π+π0 events. If we use the CLEO II upper limit → of 2.4 10−4 we expect to see N(D0 ρ0γ) < 8.0 events. If we use the largest × → predicted theoretical branching fraction, (D0 ρ0γ) = 1.0 10−5, we expect to B → × see N(D0 ρ0γ) < 0.3 events. In either case background from these and other → ∗0 radiative charm processes is of little consequence to the D0 K γ analysis. Other → ∗0 possible sources of backgrounds, such as Ds K γ have no measureable reflection. → The PDF used to fit data is a linear combination of the four component PDFs discussed previously. In the fit to data the normalizations of each PDF are allowed to

ﬂoat freely, while the component PDFs themselves are ﬁxed. Table 5.11 summarizes each of the component PDFs and the datasets used to obtain the signal shapes.

112 ∗0 Fit Component (M(K γ) Data Set Used ∗0 P ∗0 D0 K γ Crystal Ball D0 K γ → (n = 5) Signal→ MC D0 K−π+π0 Crystal Ball D0 K−π+π0 → (n = 3)+ Flat events→ from cc MC ∗0 ∗0 D0 K η Gaussian+Linear D0 K η → → events from cc MC Combinatoric Polynomial Generic MC

Table 5.11: Summary of each component used in the ﬁnal ﬁtting procedure.

5.2.6 Validating of Fitting Procedure

In order to validate that we can are able to resolve the signal contributions over the three background processes, we simulate data using a realistic ensemble of MC events. This is comprised of MC events from all four classes, weighted according to the cross sections of generic MC, as shown in Table 4.1 and the presumed branching

∗0 fraction of signal, (D0 K γ) = 2.1 10−4. Events from both generic and signal B → × modes are selected with a probability, ρ = (uds)/ (Mode), where (Mode) is the L L L equivalent luminosity of either generic or signal MC.

MC data is ﬁt using an unbinned EMLM, with the normalization of the D0 → ∗0 ∗0 K γ component taken as the yield of D0 K γ events. We input a total sample → ∗0 ∗0 of 1459 D0 K γ events and measure N(D0 K γ) = 1517.4 93.0. Overall → → ± we see good agreement between our input sample and the measured yeild. The ﬁt to the MC ensemble is shown in Fig. 5.11. The histograms show the full weighted

MC sample.

113 ) 2 2 350 χ /N = 81.79/76

300 e+ e- → B+B- e+ e- → B0B0 e+ e- → u u, d d, s s e+ e- → c c 0 *0 D → K η D0 → K- π+ π0 250 0 *0 D → K γ Combinatoric Background Fit Combinatoric Background + D0 → φ π0 Fit Combinatoric Background + D0 → φ π0 + D0 → φ η Fit 200 Total Fit

Events / ( 0.004 GeV/c Run 1-5 Data

150

100

1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2) 2 i χ

± 6 4 2 0 -2 -4 -6 1.7 1.75 1.8 1.85 1.9 1.95D0 Mass 2 (GeV/c2)

∗0 Figure 5.11: The M(K γ) distribution for the combined MC dataset. The sum of all ∗0 component PDFs is shown in solid blue. The sum of D0 K η + D0 K−π+π0 + Combinatoric terms is shown with a solid black line. The→ sum of D0 →K−π+π0 + Combinatoric terms is shown with a dashed red line and the quadratic→ combinatoric term is shown with a dashed blue line. The histograms represent signal D0 ∗0 ∗0 → K γ events in yellow, D0 K η events in orange, D0 K−π+π0 in gray and combinatoric events in purple.→ →

114 0 5.2.7 Fitting Results While Varying (D0 K∗ γ) B → As a further test of the robustness of our ﬁtting procedure we vary the weight of our signal sample to correspond to branching fractions in the range, (D0 B → ∗0 K γ) = [0.0, 4.0] 10−4. As can be seen in Figure 5.12 the measured decay rates × agree well with the input branching fractions.

Using the branching fraction found when no signal is added, the upper limit

∗0 of (D0 K γ) < 2.74 10−5 @ 95% C.L. is found. This is well beneath the B → × lowest predicted branching fraction. Therefore, using the current BABAR dataset

∗0 a measurement of (D0 K γ) is well within reach. The branching fraction is B → measured relative to D0 K−π+ using to Eq. 5.17. In this expression we assume → ∗0 (K K−π+) = 2/3, based on conservation of isospin. B → ∗0 ∗0 (D0 K γ) N(D0 K γ) ǫ(D0 K−π+) 1 B 0 → − + = 0 → − + → ∗0 ∗0 (D K π ) N(D K π ) × ǫ(D0 K γ) ! × (K K−π+) B → → → B → (5.17)

115 5

Measured 4 -4 10 × ) γ

3 *0 K

→

0 2 B(D

0 0.5 1 1.5 2 2.5 3 3.5 4 *0 B(D 0 → K γ) × 10-4 Input

Figure 5.12: The result of our ﬁt procedure as the weight of each dataset is varied. The red line denotes (Nin = Nout).

116 CHAPTER 6

Signal Shape Validation

It is essential to our analysis that the background from D0 h+h′−π0 decays → is well understood, where h and h′ are charged hadrons. The M(Vγ) distributions of these types of events are shown in Figs. 2.5, 5.2(b), and 5.8. In each of these

ﬁgures, the invariant M(Vγ) distribution of D0 Vπ0 events peaks just beneath → the nominal D0 mass, and nearly coincident to the peak of D0 Vγ events. Given → the close overlap between the mass distributions from D0 Vπ0 events and the → expected mass distributions from D0 Vγ, any inconsistencies between the dis- → tributions predicted by MC and the distributions observed in data will distort any measurement of (D0 Vγ). B → 6.1 Signal Shape Control Samples

Events of the type D0 Pγ, where P is a pseudoscalar meson, are forbidden → under conservation of angular momentum. When looking for these events in data the majority of events seen would be from either D0 Pπ0 or D0 Pη decays. → → By observing the invariant M(Pγ) distribution in data we can validate how well

D0 Vπ0 events are modelled within MC. → We analyzed the BABAR dataset, searching for D0 K0γ events. We search for → s 117 this mode not only because Ks is a pseudoscalar meson, but also because it decays into two charged mesons π+π−, hence closely resembling the ﬁnal state of our signal modes. Since D0 K0γ events are forbidden under conservation of angular mo- → s mentum the majority of events seen in our signal region are due to D0 K0π0 and → s D0 K0η. → s A second set of data used to validate MC, was obtained by examining events

∗0 from the D0 K γ dataset, taken from the far sidebands of the helicity angle → distribution, deﬁned by ( cos(θH ) > 0.9). The helicity angle distribution of both | | ∗0 D0 K γ events and D0 K−π+π0 events can be seen in Figs. 4.4(a) and → → 4.4(b). The far sidebands of helicity are rich in D0 K−π+π0 events with little → ∗0 ∗0 contribution from the D0 K γ events. The M(K γ) distribution of these events → can be compared between MC and real data and used as a second control sample to validate the background from D0 Vπ0. → Since the invariant mass distribution of D0 Vγ events and D0 Vπ0 events → → overlap to such a large degree, we may not be able to perceive discrepancies in the invariant mass distribution of real D0 Vγ events. Therefore, the invariant mass → distribution of signal D0 Vγ events obtained using MC must also be validated → against an appropriate control sample in data. Unfortunately no clean samples of

D0 Vγ decays are available without outside pollution from D0 Vπ0 events. → → We do have clean samples of fully reconstructed D0 Pπ0 events, where π0 decays → via π0 γγ. Reconstructing π0 through this dominant decay channel allows us to → incorporate all of the inherent problems of properly modeling photons in MC. Specif- ically we analyze the BABAR dataset in search of decays of the type D0 K0π0. → s

118 Fig. 6.1 compares the invariant mass distributions for the π0 background processes and D0 Vγ processes to the invariant mass of our three control samples: →

D0 K0γ • → s

0 ∗0 D K γ with the selection, cos(θH ) > 0.9 • → | |

D0 K0π0 • → s

The accuracy of the invariant mass distributions predicted in MC was validated using these control samples. The observed descrepencies between MC and data are used to correct the signal shape distributions of D0 Vγ and D0 Vπ0 in MC. → → 6.1.1 D0 K0γ Dataset → s We consider D0 mesons, which are produced by a D∗+ decay, and in turn decay through the following channels: D0 φπ0, D0 K−π+π0, and D0 K0π0. → → → s 0 We analyze MC events from these decays as if they were D Xsγ, where Xs → is a strange meson. Figs. 6.1(a) and 6.1(b) shows D0 K0π0 events compared → s to D0 φπ0 and D0 K−π+π0 events. The distributions are taken from our → → D0 K0γ MC dataset, formed by applying the following selection criteria: → s

∗+ PCMS(D ) > 2.62 GeV/c •

PCMS(γ) > 0.54 GeV/c •

(0.1435 < M(D∗+) M(D0) < 0.1475) GeV/c2 • −

(0.490 < M(π+π−) < 0.505) GeV/c2 •

VeryTight PID on Ks pions. •

119 ) ) 2 2 0 0 500 - 140 D → φ π MC D0 → K π+ π0 0 0 0 D → Ks π MC 120 0 0 0 400 D → Ks π

100

300 Entries / (10 MeV/c Entries / (10 MeV/c 80

60 200

100

0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 1.7 1.75 1.8 1.85 1.9 1.95 2 M(D0) (GeV/c2) M(D0) (GeV/c2)

(a) (b) ) )

2 180 2 500 0 0 D → φ π MC D0 → K- π+ π0 160 D0 → K- π+ π0 (Helicity SB)

D0 → K- π+ π0 (Helicity SB) 140 400

120

300 Entries / (10 MeV/c 100 Entries / (10 MeV/c

80 200 60

40 100

0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 1.7 1.75 1.8 1.85 1.9 1.95 2 M(D0) (GeV/c2) M(D0) (GeV/c2)

2000 Entries / (5 MeV/c Entries / (10 MeV/c

1000 1500

1000

500

0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 1.7 1.75 1.8 1.85 1.9 1.95 2 M(D0) (GeV/c2) M(D0) (GeV/c2)

(e) (f)

Figure 6.1: MC of each of the three control samples are compared with MC background for D0 Vπ0 and D0 Vγ events. (a), and (c) compare D0 φπ0 MC 0 →0 0 0 → − + 0 → to the D Ks π and (D K π π )cos(θH )>0.9 MC samples respectively. (b) and (d) show→ the same comparison→ only with D0 K−π+π0 events. (e) and (f) → 0 0 0 compare distributions of the dataset from fully reconstructed D Ks π MC to ∗0 → D0 φγ and D0 K γ MC. → →

120 ∗0 The ﬁrst three cuts are identical to those used in both the D0 φγ and D0 K γ → → studies. We require that the two charged pions used to construct Ks candidates share a common vertex with a χ2 probability greater than 0.001.

The D0 K0γ MC dataset is constructed from generic cc, uds, B0B0, and → s B+B− MC. These generic events are classiﬁed into three distinct groups using JetSet generator information:

D0 K0π0 events • → s

D0 K0η events • → s

The remaining generic qq¯ events •

All three events classes are modeled using the same functional forms as were used

0 0 ∗0 0 in both D φγ and D K γ studies. The M(Ksγ) distribution of D → → → 0 0 Ks π MC events are modeled with a CB line shape of order n = 3. The M(Ksγ) distribution of D0 K0η MC candidates are are modeled by a Gaussian. The → s

M(Ksγ) mass distribution of the remaining combinatoric BG events are modeled by a second order Chebychev polynomial. Each event class is fit separately and for the fit to data, the signal shapes are fixed to the optimal fit values, summarized in

Table 6.1.

6.1.2 D0 K0π0 Dataset → s ∗0 We analyze D0 φγ and D0 K γ MC and compare the resulting M(Vγ) → → 0 0 0 0 mass distributions to the M(Ksπ ) distribution of fully reconstructed D K π → s MC. We apply the same selection criteria on K0 as was used for D0 K0γ. The s → s candidate π0 is taken from combinations of photons with invariant masses in the

121 PDF Parameter Value ′ −4 2 µ 0 0 0 (1.832 7.2 10 ) GeV/c D →Ks π 0 0 0 ± × 0 0 ′ −2 2 CBD →Ks γ(D Ks π ) σ 0 0 0 (3.03 0.056) 10 GeV/c D →Ks π → ′ ± × α 0 0 0 0.564 0.021 D →Ks π ± 2 0 0 0 0 µD →Ks η (1.730 0.0015) GeV/c 0 0 GsD →Ks γ(D Ks η) ± −2 2 0 0 → σD →Ks η (3.39 0.12) 10 GeV/c ± × −1 2 0 0 TD →Ks γ(Combo BG) m (3.71 0.22) 10 GeV/c q −(8.86 ±2.12) ×10−2( GeV/c2)2 ± ×

Table 6.1: The values obtained for the individual component PDF. 0 0 0 0 0 0 0 0 PDFD →Ks γ(D Ks π ) parameters are determined using D Ks π MC events, 0 → 0 → analyzed as D Ks γ. These values correspond to the CB line shape shown in Eq. → 0 0 0 0 0 0 3.10. PDFD →Ks γ(D Ks η) values are set using D Ks η MC events analyzed as D0 K0γ. PDF (Combo→ BG) is a second order Chebychev→ polynomial, m and → s q are the ﬁrst and second order components respectively.

range (0.110 < M(γγ) < 0.150) GeV/c2. A minimum requirement is placed on the

0 0 π momentum, PCMS(π ) > 0.54 GeV/c.

0 0 0 0 The M(Ksπ ) distribution of D K π events is modeled using a CB line → s shape of order n = 5. These events are obtained from generic cc MC, by examining generator level information and searching for decays of the type D∗+ → D0π+ ; D0 K0π0. After fit convergence, the values of µ, σ, and α are fixed slow → s to those shown in Table 6.2. The remaining MC dataset is comprised of generic cc events not containing a D0 K0π0 event, as well as generic uds, B0B0, and B+B− → s MC. These events are modeled with a first order Chebychev polynomial, the shape of which is fixed using generic MC. Table 6.2 summarizes the optimal fit parameters.

122 PDF Parameter Value −5 2 0 0 0 µD →Ks π (1.860 8.8 10 ) GeV/c 0 0 0 ± × −2 2 CB 0 0 0 (D K π ) σ 0 0 0 (2.36 0.008) 10 GeV/c D →Ks π → s D →Ks π ± × 0 0 0 αD →Ks π 1.14 0.007 ± −1 2 T 0 0 0 (Combo BG) m (6.26 0.11) 10 GeV/c D →Ks π − ± ×

Table 6.2: The values obtained for the individual component PDF. PDF (D0 0 0 0 0 0 → Ks π ) parameters are determined using D Ks π MC events. These values correspond to the CB line shape shown in Eq.→ 3.10. PDF (Combo BG) is a first order Chebychev polynomial, m is the first coefficient.

0 6.1.3 D0 K∗ γ Helicity Sideband Dataset → A clean dataset of D0 K−π+π0 events is obtained using the nearly exact cuts → ∗0 used on the D0 K γ dataset, with the exception of selecting events from the → sidebands of helicity, cos(θH ) > 0.9. The MC events are modeled in the same | | ∗0 manner as detailed in Section 5.2, the only exception being that D0 K γ MC is → not included and not modeled.

∗0 MC samples of D0 K−π+π0 and D0 K η are obtained by examining → → the generated MC. The D0 K−π+π0 dataset is comprised of generic cc MC → events that contain the decay chain, D∗+ D0π+ ; D0 K−π+π0. While the → slow → ∗0 D0 K η dataset is comprised of generic cc MC events that contain the decay, → ∗0 D∗+ D0π+ ; D0 K η. The remaining sample of cc MC events is mixed → slow → together with generic MC of uds, B+B−, and B0B0 in order to form a combinatoric sample.

∗0 The M(K γ) distribution of D0 K−π+π0 events is modeled with a CB of → ∗0 order n = 3. The distribution of D0 K η events is modeled with a Gaussian; →

123 PDF Parameter Value −4 2 µD0→K−π+π0 (1.833 8.7 10 ) GeV/c 0 − + 0 ± × −2 2 CB 0 ∗0 (D K π π ) σ 0 − + 0 (2.59 0.064) 10 GeV/c D →K γ → D →K π π ± × αD0→K−π+π0 0.421 0.018 ± 2 0 ∗0 µ 0 ∗0 (1.720 0.0011) GeV/c ∗0 D →K η GsD0→K γ(D K η) ± −2 2 ∗0 → σD0→K η (2.36 0.09) 10 GeV/c ± × −1 2 ∗0 TD0→K γ(Combo BG) m (7.25 0.44) 10 GeV/c q −(1.89 ±0.42) ×10−1( GeV/c2)2 ± ×

Table 6.3: The values obtained for the individual component PDF. 0 − + 0 0 − + 0 ∗0 PDFD0→K γ(D K π π ) parameters are determined using D K π π → ∗0 → MC events, analyzed as D0 K γ. These values correspond to the CB line shape → 0 ∗0 0 ∗0 ∗0 shown in Eq. 3.10. PDFD0→K γ(D K η) values are set using D K η MC ∗0 → → events analyzed as D0 K γ. PDF (Combo BG) is a second order Chebychev polynomial, m and q are→ the ﬁrst and second order components.

while the combinatoric BG is modeled with a second order Chebychev polynomial.

Each of these distributions are fit using the event classes mentioned previously. The resulting optimal fit parameters are fixed and shown in Table 6.3.

6.2 Obtaining Correction Between Data and MC

We quantify diﬀerences between data and MC by examining the change in the

CB line shape parameters within each. This is done by multiplying the three ﬁxed

CB parameters, (α, µ, σ), with a set of correction factors, (Rα,Rµ,Rσ), and redeﬁning the CB used in each control sample as shown in Eqs. 6.1, 6.2, and 6.3.

0 ′ ′ ′ CBD0→K0γ(M(D ); µ 0 0 0 , σ 0 0 0 , α 0 0 0 , n = 3) s D →Ks π D →Ks π D →Ks π → 0 ′ ′ ′ CB 0 0 (M(D ); R µ 0 0 0 ,R σ 0 0 0 ,R α 0 0 0 , n = 3) D →Ks γ µ D →Ks π σ D →Ks π α D →Ks π

(6.1)

124 0 CB 0 0 0 (M(D ); µ 0 0 0 , σ 0 0 0 , α 0 0 0 , n = 5) D →Ks π D →Ks π D →Ks π D →Ks π → 0 0 0 0 0 0 0 0 0 0 0 0 0 CBD →Ks π (M(D ); RµµD →Ks π ,RσσD →Ks π ,RααD →Ks π , n = 5)

(6.2)

0 CB 0 − + 0 (M(D ); µ 0 − + 0 , σ 0 − + 0 , α 0 − + 0 , n = 3) D →K π π D →K π π D →K π π D →K π π → 0 CBD0→K−π+π0 (M(D ); RµµD0→K−π+π0 ,RσσD0→K−π+π0 ,RααD0→K−π+π0 , n = 3)

(6.3)

′ ′ ′ 0 The values µ 0 0 0 , σ 0 0 0 , and α 0 0 0 in Eq. 6.1 are ﬁxed using D D →Ks π D →Ks π D →Ks π → K0π0 MC from the D0 K0γ control sample. These values are shown in Table s → s 0 0 0 6.1. Fully reconstructed D K π MC is used to determine µ 0 0 0 , σ 0 0 0 , → s D →Ks π D →Ks π

0 0 0 and αD →Ks π . The terms are ﬁxed to the values shown in Table 6.2. Finally

0 − + 0 µ 0 − + 0 , σ 0 − + 0 , and α 0 − + 0 are ﬁxed using D K π π MC, as D →K π π D →K π π D →K π π → shown in Table 6.3.

The remaining terms, (Rα,Rµ,Rσ), are used to allow the CB line shapes to

ﬂoat. These are constrained to be the same across all three control samples, in essence averaging the correction across all three.

An equal number of events in both MC and data are selected randomly across all control samples. The invariant mass distributions of these events are ﬁt with an

EMLM using the combined PDF’s for each control sample. Whn the ﬁt it performed the correction terms, RX , and the normalizations for each class are permitted to

float. This procedure is applied first using MC events and then using events from real data. The results of the fit to MC events are displayed in Fig. 6.2. The results of the data fit are shown in Fig. 6.3.

Once both data and MC are ﬁt, we calculate the ratio of the correction terms

125 ) )

2 2 250

0 B0 B MC B+ B- MC 0 B0 B MC UDS MC + - B B MC c c MC 200 UDS MC D*+ → D0 π+; D0 → K- π+ π0 c c MC *+ 0 + 0 *0 *+ 0 + 0 0 200 D → D π ; D → K η D → D π ; D → Ks π MC Run 1-5 MC *+ 0 + 0 D → D π ; D → Ks η MC Total PDF Fit - Run 1-5 MC P(D0 → K π+ π0) + Background Total PDF Fit Background 0 0 P(D → Ks π ) + Background Background 150 150 Events / ( 0.0025 GeV/c Events / ( 0.0025 GeV/c 100 100

50 50

0 0 1.7 1.75 1.8 1.85 1.9 1.95 2 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2) D0 Mass (GeV/c2)

(a) (b)

0 0

) B B MC - 2 B+ B MC UDS MC 350 c c MC *+ 0 + 0 0 D → D π ; D → Ks π MC Run 1-5 MC Total PDF Fit 300 Background PDF Fit

250

200 Events / ( 0.0025 GeV/c 150

100

0 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

(c)

Figure 6.2: MC of each of the three control samples are fit with a single set of 0 0 0 ∗0 correction factors. (a) Fits MC of D K γ, (b) fits MC of D K γ, cos(θH > → s → | | 0.9, and (c) fits MC of D0 K0π0. → s

126 200 ) )

2 2 250

0 180 B0 B MC B+ B- MC 0 B0 B MC UDS MC + - B B MC c c MC 160 UDS MC D*+ → D0 π+; D0 → K- π+ π0 c c MC *+ 0 + 0 *0 *+ 0 + 0 0 200 D → D π ; D → K η D → D π ; D → Ks π MC Run 1-5 Data *+ 0 + 0 D → D π ; D → Ks η MC Background - 140 Run 1-5 Data P(D0 → K π+ π0) + Background Total PDF Fit Total PDF 0 0 P(D → Ks π ) + Background Background 120 150

100 Events / ( 0.0025 GeV/c Events / ( 0.0025 GeV/c 80 100

40 50

0 1.7 1.75 1.8 1.85 1.9 1.95 2 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2) D0 Mass (GeV/c2)

(a) (b)

0 0

) B B MC 450 - +

2 B B MC UDS MC c c MC *+ 0 + 0 0 D → D π ; D → Ks π MC Data Runs 1-5 400 Total PDF Fit Background PDF Fit

350

300

250

200Events / ( 0.0025 GeV/c

150

100

1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

(c)

Figure 6.3: Data of each of the three control samples are fit with a single set of correc- 0 0 0 ∗0 tion factors. (a) Fits data of D Ks γ, (b) fits data of D K γ, cos(θH > 0.9, 0 0 0 → → | | and (c) fits data of D Ks π . In each plot data is shown with markers and MC is shown with histograms.→ One particular interesting feature of these plots is the subtle shift in the peak of D0. Also a small shift in the decaying tail of (b) between MC and data can be observed. Each of the small and difficult to precieve changes need to be reintegrated into our final fit.

127 RX (Data) Parameter (X) RX Data RX MC RX (MC) µ 1.0021 0.0002 0.999 0.0001 1.0022 0.00026 ± ± ± σ 0.951 0.011 0.991 0.011 0.960 0.016 ± ± ± α 0.977 0.029 1.097 0.037 0.890 0.040 ± ± ±

Table 6.4: Comparing ﬁt parameters found in both data and MC. The ratio, RX , is the ﬂoating correction parameter applied to either data or the combined MC dataset. The ratio between these two is used to tweak both the D0 Vπ0 and D0 Vγ signal shapes derived from MC. → →

found between each. These are defined as the correction to the CB signal shapes that were obtained using MC. The ratios are fixed to the values shown in the fourth column of Table 6.4 and are applied the D0 Vγ and D0 Vπ0 signal shapes → → when fitting data.

By allowing the correction terms to float, we are in effect allowing the signal shapes of D0 Vγ and D0 Vπ0 to float. By refitting MC as well as fitting → → data and comparing the ratio of the two corrections, the effect of allowing the signal shapes to float is absorbed in ratio.

These correction factors are a means to both validate MC and correct for subtle inaccuracies in MC models. By applying these we inheritally induce systematic uncertainities into the final results which are difficult to quantify. We will detail our systematic studies later in this document, but mention some details here to preface these discussions. For systematic studies it is important to observe any run dependence on the correction factors. To do this we divide both data and MC by run number. Combined MC and data are each fit for every run and the correction

128 Run Number Rµ(Data) Rσ(Data) Rα(Data) Rµ(MC) Rσ(MC) Rα(MC) 1 1.0021 0.0005 0.956 0.035 0.873 0.072 2 1.0042 ± 0.0006 0.880 ± 0.030 0.790 ± 0.075 3 1.0029 ± 0.0009 0.957 ± 0.057 0.774 ± 0.136 4 1.0008 ± 0.0006 1.014 ± 0.039 1.066 ± 0.101 ± ± ± 5 1.0041 0.0011 0.868 0.052 0.616 0.123 ± ± ±

Table 6.5: The correction ratios are found for each run. These will be used as a systematic correction for each run.

Datasets used Rµ(Data) Rσ(Data) Rα(Data) Rµ(MC) Rσ(MC) Rα(MC) 0 0 0 −4 D Ks π 1.0029 (3. 10 ) 0.937 0.019 0.878 0.048 0 → 0 ± × −4 ± ± D Ks γ 1.0014 (14.0 10 ) 1.015 0.067 1.107 0.295 → ∗0 ± × ± ± D0 K γ 1.0031 (9.0 10−4 0.901 0.047 0.791 0.095 → ± × ± ±

Table 6.6: The correction ratios found when ﬁtting the three control samples, independently.

factors measured. Table 6.5 shows the correction ratios across Runs 1-5.

For an additional systematic study we measure the correction factors for each control sample independently. These corrections are applied to D0 φγ and D0 → → ∗0 K γ signal shapes to measure the stablity of our results when applying diﬀerent correction ratios. The correction ratios for each control sample are shown in Table

6.6.

6.3 Applying Corrections when Fitting Data

The need for these corrections can be demonstrated using the D0 K0γ dataset → s from Runs 1-5. Rather than assuming that no real D0 K0γ events exist, we allow → s 129 for their possibility by adding an additional PDF. This PDF models the M(Ksγ)

∗0 distribution of forbidden D0 K0γ events, and is set equal to the M(K γ) CB → s ∗0 shape of D0 K γ MC events. → By assuming the MC model is completely accurate, no corrections are applied to either the D0 K0γ or D0 K0π0 signal shapes. Data is ﬁt using an unbinned → s → s EMLM. The normalization of D0 K0γ events is found to be: N(D0 K0γ) = → s → s 597.9 142.2. The result of not applying correction between MC and data is that ± a forbidden mode is observed with 4σ signiﬁcance.

The ﬁt is shown in Figure 6.4. With the correction applied to both the D0 → K0π0 and D0 K0γ signal shapes the ﬁtter returns a normalization of N(D0 s → s → K0γ)= 114..4 134.8. Which is consistent with seeing no D0 K0γ events. s − ± → s There is a small possiblity that this fake signal is due D0 ρ0(770)γ events → originating from the low end of the ρ0(770) mass region. We verify this is not the case

2 by ﬁtting candidates within the high mass Ks sideband, 0.525 GeV/c > M(Ks) >

0.512 GeV/c2. With no correction applied, no evidence for a signal is found, with the ﬁt returning a normalization N(D0 K0γ)= 37.9 42.8. → s − ± 6.3.1 Summary of Signal Shape Correction

The discrepancies between MC and data are quantiﬁed by comparing the CB line shape parameters in data and MC for the three control samples. In each case we divide MC of each control sample into distinct classes and ﬁt each class independently.

Once all MC classes, across all control samples are ﬁt, the signal shapes are ﬁxed.

The correction ratios shown in Table 6.4 are obtained as follows:

130 ) 2 Real Data 350 e+ e- → B+B- e+ e- → B0B0 e+ e- → u u, d d, s s e+ e- → c c D0 → V η 300 D0 → V π0 D0 → V γ Combinatoric Background Fit Combinatoric Background + D0 → V π0 Fit Total Fit 250 Run 1-5 Data

200 Events / ( 0.002 GeV/c

150

100

0 1.7 1.75 1.8 1.85 1.9 1.95 2 D0 Mass (GeV/c2)

4 2 2 i

χ 0

± -2 -4 -6 1.7 1.75 1.8 1.85 1.9 1.95 2

0 0 0 0 Figure 6.4: We have allowed for a D Ksγ signal on Data. The D Ks π signal shape is derived directly from MC.→ Not applying any correction to→ the MC signal shape results in a fake signal. This can be mitigated by transforming the MC 0 shape using D Ksγ as a control sample →

131 We augment the D0 Vγ and D0 Vπ0 signal shapes by multiplying the • → →

CB parameters by a single set of correction factors, RX , X µ, σ, α. ∈

MC and real data for each control sample are ﬁt by allowing RX to ﬂoat freely, •

for each. The set of correction ratios on MC are deﬁned as Rx(MC), while

the set of correction ratios in data are deﬁned as RX (Data).

The correction ratios are deﬁned as RX (Data)/RX (MC). (Shown in Table • 6.4)

The set of correction ratios are applied to the D0 Vπ0 and D0 Vγ signal • → → ∗0 shapes which are used to ﬁt D0 φγ and D0 K γ mass spectrum. → →

We use the independent correction ratios and the run dependent correction • ratios for systematic studies, to be detailed later.

132 CHAPTER 7

Results

The following is a summary of the measurement techniques used to extract

(D0 Vγ). We used data recorded by the BABAR detector during run peri- B → ods 1-5. The yield of D0 φγ events was found using a two dimensional unbinned →

Extended Maximum Likelihood Method to ﬁr the M(φγ) and cos(θH ) distributions to a a linear combination of the PDFs, which are summarized in Table 5.6, as well as below:

The M(φγ) and helicity angle distributions of D0 φγ events are modelled • → with a CB line shape of order n = 5 and a second Chebychev polynomial,

respectively.

The M(φγ) and helicity angle distributions of D0 φπ0 events are modelled • → with a CB line shape of order n = 3 and a second Chebychev polynomial,

respectively.

The M(φγ) and helicity angle distributions of D0 φη events are modelled • → with a Gaussian distribution and a second Chebychev polynomial, respectively.

The M(φγ) and helicity angle distributions of Combinatoric events are each • modelled a second Chebychev polynomial.

133 ∗0 The yield of D0 K γ events was found using an unbinned EMLM ﬁt over the → ∗0 invariant M(K γ) distribution, using the PDFs summarized in Table 5.11, as well as below:

∗0 ∗0 The M(K γ) and helicity angle distributions of D0 K γ events are mod- • → elled with a CB line shape of order n = 5 and a second Chebychev polynomial,

respectively.

∗0 The M(K γ) and helicity angle distributions of D0 K−π+π0 events are • → modelled with a CB line shape of order n = 3 and a second Chebychev poly-

nomial, respectively.

∗0 ∗0 The M(K γ) and helicity angle distributions of D0 K η events are mod- • → elled with a Gaussian distribution and a second Chebychev polynomial, re-

spectively.

∗0 The M(K γ) and helicity angle distributions of Combinatoric events are each • modelled a second Chebychev polynomial.

The PDF’s modeling D0 Vγ and D0 Vπ0 events were corrected using the → → technique described in Chapter 6 and the correction factors summarized in Table

∗0 6.4. The branching fractions for D0 φγ and D0 K γ were measured relative → → to D0 K−π+. The details of the D0 K−π+ measurement are discussed in → → ∗0 Appendix A. The reconstruction eﬃciencies for both D0 φγ and D0 K γ are → → obtained using MC simulating each mode.

134 7.1 D0 φγ Results →

The ﬁt of D0 φγ events in data found a total yield shown in Eq. 7.1 →

N(D0 φγ) = (242.6 24.8) (7.1) → ±

The data and the resulting fit fit are displayed in Fig. 7.1. The reconstruction efficiency, ǫ(D0 φγ) = (10.8 0.07)%, was determined using MC as described in → ± Section 5.1. We measured the relative branching fraction using Eq. 5.12 in Section

5.1. Since only the charged decay mode of φ was used to measure D0 φγ, the → number of D0 φγ events is scaled by the branching fraction, (φ K−K+) = → B → (49.2 0.6)% [1]. ± N(D0 K−π+) = (335.1 3.99) 103 events were measured from the reference → ± × mode at an eﬃciency of ǫ(D0 K−π+) = (5.25 0.17)%. This results in the → ± relative branching fraction shown in Eq. 7.2. Using the current world average of

(D0 K−π+)=(3.82 0.07)% [1], the absolute branching fraction is calculated B → ± in Eq. 7.3. (D0 φγ) B → = (7.15 0.78) 10−4 (7.2) (D0 K−π+) ± × B → (D0 φγ)=(2.73 0.30) 10−5 (7.3) B → ± × This result agrees with the previous BELLE measurement, within statistical ﬂuctu- ation, but with reduced statistical uncertainties [10]. Furthermore our result compares well with theoretical predictions.

135 Data )

2 100 Fit to Data

P(D0 → φ η) + P(D0 → φ π0) + P(BG) Fit

80 P(D0 → φ π0) + P(BG) Fit

P(BG) Fit

2 D0 → φ η MC 60 χ /N = 57.92/60 D0 → φ π0 MC

Events / ( 0.005 GeV/c Generic MC 40

0 1.7 1.75 1.8 1.85 1.9 1.95 2 M(φ γ) (GeV/c2)

6 4

2 i 2 χ 0

± -2 -4 -6 1.7 1.75 1.8 1.85 1.9 1.95 2 (a)

90 Data Fit to Data 80 P(D0 → φ η) + P(D0 → φ π0) + P(BG) Fit P(D0 → φ π0) + P(BG) Fit P(BG) Fit 70 0 D → φ η MC D0 → φ π0 MC

Events / ( 0.025 ) Generic MC 60 χ2/N = 59.87/68 50

-0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 Cos(θH)

4 2 2 i

χ 0

± -2 -4 -6 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 (b)

Figure 7.1: Fit of the (a) M(φγ) spectrum and (b) helicity for D0 φγ events in data recorded over run periods 1-5. →

136 0 7.2 D0 K∗ γ Results →

∗0 The M(K γ) distribution of data is modeled using a linear combination of the

PDFs summarized in Table 5.11. Using an unbinned EMLM, we measured a yield

∗0 of D0 K γ shown in Eq. 7.4 →

∗0 N(D0 K γ) = (2285.8 113.2) (7.4) → ±

∗0 Using MC we determine that D0 K γ events are reconstructed at an eﬃciency → ∗0 of ǫ(D0 K γ) = (6.37 0.04)%. This is compared to the reference mode, → ± D0 K−π+. The same D0 K−π+ dataset and selection criteria are used in → → the D0 φγ analysis, reiterating the previous measurement of N(D0 K−π+)= → → (335.1 3.99) 103 events measured at an eﬃciency of ǫ(D0 K−π+) = (5.25 ± × → ± 0.17)%.

∗0 Since only the charged decay mode, K K−π+, was used to reconstruct → ∗0 D0 K γ, the number of measured events must be scaled by 3/2, as shown in → Eq. 5.17. Using this equation, the relative branching fraction is shown in Equation

7.5. The absolute branching fraction is calculated using the current world average of (D0 K−π+)=(3.82 0.07)% [1] and shown in Equation 7.6. B → ± ∗0 (D0 K γ) B → = (8.43 0.51) 10−3 (7.5) (D0 K−π+) ± × B → ∗0 (D0 K γ)=(3.22 0.20) 10−4 (7.6) B → ± × The error here reflects statistical sources of uncertainty. Fig. 7.2 shows the fit to data. The branching fraction reflected in Eq. 7.6 rests within the range of theoretical predictions.

137 ) 400 2 Data χ2/N = 121.61/124 Fit to Data 350 *0 P(D0 → K η) + P(D0 → K- π+ π0) + P(BG) Fit

P(D0 → K- π+ π0) + P(BG) Fit 300 P(BG) Fit

*0 250 D0 → K η MC D0 → K- π+ π0 η MC

200 Generic MC Events / ( 0.0025 GeV/c

150

100

1.75 1.8 1.85 1.9 1.95 2 *0 M(K γ) (GeV/c2)

6 4

2 i 2 χ 0

± -2 -4 -6 1.7 1.75 1.8 1.85 1.9 1.95 2

∗0 Figure 7.2: Fit of the D0 Mass spectrum for D0 K γ events in data over the period of Run 1-5. →

7.3 Validating Results

One means of validating our measurement is to rerun the EMLM on data recorded during each run period independently. When this is done, the run dependent corrections, shown in Table 6.5, are applied on data from each run. This gives us ﬁve statistically independent results to compare against. The branching fractions for

D0 φγ show very clean consistency for all runs. This is shown in Fig. 7.3, where → all measurements are statistically consistent.

∗0 The run dependent measurements for D0 K γ are also statistically consis- → tent across all runs. We do note a positive ﬂuctuation in Run 3. While this upward measurement is high it still remains within statistical uncertainties of the nominal branching fraction. These results are shown in Fig. 7.4

138 The systematic uncertainty arising from run dependence is quantified using split sample statistics [24]. This is a technique to determine systematic uncertainty in statistically independent measurements. We start with the anzatz that χ2/(N 1) = 1, − where χ2 is the sum of the square difference between each measurement and the average measurement. This is shown in Eq. 7.7. If χ2/(N 1) > 1, when considering − only statistical fluctuations, then the remaining sources of uncertainty must be entirely systematic. N i χ2 = B − B (7.7) σ2 i=1 i X In this expression N represents the number of runs, i is the measured branching B fraction for a single run, the measured branching fraction across all runs, and σi B is the statistical uncertainty for i. The average statistical uncertainty is expressed B in Eq. 7.8. This is scaled by χ2/(N 1), in order to force χ2/(N 1) = 1. − − p 1 σ = (7.8) 5 2 i=1 1/σi q σ˜ = σ Pχ2/(N 1) (7.9) − p The scaled uncertainty is meant to take into account both statistical variations as well as systematic uncertainties in a given run. This leaves two cases,σ ˜ > σstat. and

2 2 σ˜ σstat.. In the case of the ﬁrst, the systematic uncertainty is σsys = σ˜ σ . ≤ − stat. In the second case all variations can be attributed only to statistical sources.p

Using this method on D0 φγ analysis, we see χ2/(N 1) = 2.1/4. As a → − result we do not assign any systematic uncertainty due to run dependence. This is

∗0 repeated for D0 K γ, with χ2/(N 1) = 3.2/4. Again this means we do not → − assign systematic uncertainty to run dependence.

∗0 We do not directly measure either φ or K , but instead only measure their decay

139 ×10-6 ) γ 40 φ

→

B(D Branching Fraction -- Run X

30 Branching Fraction -- All Runs

B ± δ B -- All Runs

15 Branching Fraction -- Run X

Branching Fraction -- All Runs

10 B ± δ B -- All Runs 1 1.5 2 2.5 3 3.5 4 4.5 5 Run Number

Figure 7.3: Measured value of (D0 φγ) for each run. B →

products. Because of this there is a question about whether or not we are observing

∗0 true φ and K . The φ meson is extremely narrow, meaning very little pollution from other states can reﬂect in our signal shape. Additionally real D0 φγ events → follow a sin2 distribution in helicity. This is included in our ﬁtting procedure. As a result there is little chance we are seeing anything other than φ mesons.

∗0 ∗0 The same features are not seen in the D0 K γ analysis. K is very broad → and the helicity is not included in the ﬁt, therefore alternative evidence is needed to

∗0 assure we are seeing true D0 K γ events. Evidence that we are observing true → ∗0 D0 K γ decays is seen by rerunning our ﬁt in diﬀerent regions of helicity. If we → ∗0 ∗0 are observing true D0 K γ decays, the yield of D0 K γ should hold well to → →

140 ×10-3 ) γ

K 0.6

→

0 0.55 Branching Fraction -- Run X B(D

0.5 Branching Fraction -- All Runs

B ± δ B -- All Runs 0.45

0.4

0.35

0.3

0.25

1 1.5 2 2.5 3 3.5 4 4.5 5 Run Number

∗0 Figure 7.4: Measured value of (D0 K γ) for each run. B →

a sin2 distribution. We see a very clear agreement, as is shown in Fig. 7.5. This

∗0 serves as strong evidence we are seeing true D0 K γ events. →

141 ) 0.5 γ

*0 K

→ *0 0.45 0 0 Ni(D → K γ) Measured *0 (D 0 NTotal(D → K γ)

Total 0.4 )/N γ

*0 *0 N(D0 K ) K i → γ 0.35 Predicted 0 *0 → NTotal(D → K γ) 0 (D i

N 0.3

0.25

0.2

0.15

0.1

0.05

0 -1 -0.5 0 0.5 1 Cos(θH) Region

∗0 Figure 7.5: Data is refit for different regions of K cos(θH ) and the normalization ∗0 of D0 K γ is extracted for each region. The normalization are compared to the → 2 theoretical expectation that the distribution hold to a sin (θH ), corrected for the ∗0 reconstruction efficiency based on D0 K γ MC. →

142 CHAPTER 8

Systematics

Our measurements are based on the number of signal events reconstructed by the

EMLM fitting procedure, and the reconstruction efficiency derived using MC. Each of these values are subject to systematic uncertainties. These uncertainties must be quantified in order to assure accurate errors onto our measurements. As a result we consider systematic errors arising from two sources: those effecting the number of measured events, and those effecting the derived reconstruction efficiency.

Uncertainties of the first type are mainly realized by the method used to model the signal and background PDFs. These uncertainties are quantified by examining the dependence of our yields as we vary the component PDFs and the corrections applied to 0 and 0 0 . PD →Vγ PD →V π The second class of systematic uncertainty arises when the signal efficiency predicted by MC is not fully in agreement with the actual reconstruction efficiency. The reconstruction efficiency is effected by tracking inaccuracies in MC, inconsistent PID efficiencies, π0 veto effectiveness, and variations in cut selection.

143 8.1 Fitting Systematics

We consider two distinct sources of systematic uncertainty resulting from our

fitting procedure. For the first, we quantify the uncertainties caused by fixing the signal shapes for D0 Vγ, D0 Vπ0, D0 Vη, and the combinatoric back- → → → ground prior to fitting real data. For the second source, we quantify the systematic uncertainty arising from applying the ratio corrections to 0 and 0 0 PD →Vγ PD →V π obtained using MC.

8.1.1 Fixed PDF Systematics

Each of the PDFs used in the ﬁts are ﬁxed using datasets containing only one

MC event class. By ﬁxing these PDFs, the statistical uncertainties of the parameters are ignored when ﬁtting data. We attempt to incorporate these uncertainties by

first assuming the fixed parameters are Gaussian distributed, with a mean equal to the optimal fit value and a standard deviation equal to the statistical error returned by the fit. For example as shown on Table 5.2, α(D0 φγ) = 1.13 0.02, we then → ± consider α to be generated by a Gaussian: G(α; µ = 1.13, σ = 0.02).

By considering every fit value being Gaussian distributed, we generate an ensemble of random numbers arising from these distributions. These random values are then fed back into the component PDFs and used to refit data. For each iteration we measure the yield of D0 Vγ, and when complete, measure the corresponding → spread of N(D0 Vγ). The percent variation, defined as σ/µ, is assigned as the → systematic uncertainty due to fixing the component PDFs, where σ is the standard deviation of the Gaussian and µ is the mean of the Gaussian.

This procedure may suﬃce in returning a reasonable systematic uncertainty, but

144 it ignores inherent correlations between fit parameters. This is particularly acute for the CB parameters. Eq. 3.10 expresses a CB line shape as a piecewise defined function with a switch over defined to be x = µ ασ. This switch induces large − correlations across µ, α, and σ. This correlation tends to increase the statistical uncertainties of individual parameters, meaning that when these correlations are ignored a greater variation is seen in the systematic uncertainty.

Uncorrelated random numbers can be made correlated using a matrix, M. The elements of M deﬁne the correlation coeﬃcients between parameters. This matrix can be decomposed using a Cholesky factorization, which separates M into the product of L and LT , where LT is the transpose of L and L is a lower triangular matrix.

If r is a row vector of uncorrelated random numbers, these can be correlated by

T multiplying r by L: rc = r L . × We use the correlation matrix return by RooFit to deﬁne M. For an example,

∗0 Eq. 8.1 shows the correlation matrix obtained from ﬁtting the D0 K γ event → class. ∗0 σD0→K γ NCB NFlat α µ σ NCB 1.00 0.22 0.041 0.012 0.065 N 0.22− 1.000− 0.13 0.038 0.20 M = Flat (8.1) F it α −0.041 0.13 1.000 − 0.60− 0.63 µ −0.012 0.038 0.60− 1.00 0.53 σ 0.065 − 0.20− 0.63 0.53− 1.000 − − This is decomposed into L and LT . We then generate N uncorrelated random numbers, each drawn from a Gaussian of mean, µ = 0, and standard deviation,

∗0 σ = 1. Here N is the number of free parameters, in the case of D0 K γ, N = 5. → The vector of uncorrelated random numbers is denoted byr ˆ. We correlate these by

T T multiplying by L :r ˆc =rL ˆ . Since each row of L has a norm of 1, L preserves the length ofr ˆ. The implication is that the components ofr ˆc though correlated are

145 still consistent with being drawn from a Gaussian with a mean of 0 and a standard deviation of 1. Such a distribution can be turned in a Gaussian of arbitrary width and mean using Eq. 8.2.

gµ,σ = σ g , + µ (8.2) × 0 1

In this expression, g0,1, is a number drawn from a Gaussian with a mean of zero and a standard deviation of one, while gµ,σ represents a random number drawn from a Gaussian with a mean, µ, and a standard deviation, σ. Using this procedure

∗0 a large number of correlated random numbers are generated for the D0 K γ → PDF parameters. The correlations between random values are shown in Eq. 8.3.

The correlations of the generated ensemble agree very well with the initial values returned by the ﬁtter (shown in Eq. 8.1).

∗0 σD0→K γ NCB NFlat α µ σ NCB 1.00 0.22 0.039 0.013 0.066 N 0.22− 1.000− 0.13 0.038 0.20 M = Flat (8.3) Generated α −0.039 0.13 1.000 − 0.60− 0.63 µ −0.013 0.038 0.60− 1.00 0.53 σ 0.066 − 0.20− 0.63 0.53− 1.000 − − To visualize the agreement between the generate ensemble and expected distribution, Fig. 8.1 shows the generated ensemble against the mean and correlation proﬁle of the input values.

We assume that there is no correlation between component PDFs, e.g. the parameters of 0 0 and 0 are uncorrelated. This is largely a safe as- PD →V π PD →Vγ sumption, considering each PDF is used to model independent event classes.

Once an ensemble of correlated random numbers is generated for each event class, these are formed into a single parameter set. We iterate through each set of correlated random numbers, substituting these for the optimal ﬁt parameters for

146 ) ) ) ) ) γ γ γ γ γ 180

18800 V 18800 V 18800 V 18800 V 18800 V → → → → →

0 0 0 0 0 160

18600(D 18600(D 18600(D 18600(D 18600(D 140

18400 18400 18400 18400 18400 120 Crystal~Ball Crystal~Ball Crystal~Ball Crystal~Ball Crystal~Ball N N N N N 100 18200 18200 18200 18200 18200 80 18000 18000 18000 18000 18000 60

17800 17800 17800 17800 17800 40

20 17600 17600 17600 17600 17600 0 17600 17800 18000 18200 18400 186000 18800 450 500 550 600 650 700 750 8000 850 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.180 1.2 1.22 1.859 1.8595 1.86 1.86050 1.861 0.01880.0190.01920.01940.01960.01980.020.02020.02040.02060 NCrystal~Ball(D → V γ) NFlat (D → V γ) α(D → V γ) µ(D → V γ) σ(D → V γ) ) ) ) ) ) γ γ γ γ γ 180 850 V 850 V 850 V 850 V 850 V → → → → →

0 0 0 0 0 160

800(D 800(D 800(D 800(D 800(D 140 Flat Flat Flat Flat Flat

750N 750N 750N 750N 750N 120 700 700 700 700 700 100 650 650 650 650 650 80

600 600 600 600 600 60

550 550 550 550 550 40

500 500 500 500 500 20

450 450 450 450 450 0 17600 17800 18000 18200 18400 18600 18800 450 500 550 600 650 700 750 800 850 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.859 1.8595 1.86 1.8605 1.861 0.01880.0190.01920.01940.01960.01980.020.02020.02040.0206 0 0 0 0 0 NCrystal~Ball(D → V γ) NFlat (D → V γ) α(D → V γ) µ(D → V γ) σ(D → V γ) ) ) ) ) )

1.22γ 1.22γ 1.22γ 1.22γ 1.22γ 220

V V V V V 200

→ 1.2 1.2→ 1.2→ → 1.2 1.2→

0 0 0 0 0 180

1.18(D 1.18(D 1.18(D 1.18(D 1.18(D α α α α α 160 1.16 1.16 1.16 1.16 1.16 140 1.14 1.14 1.14 1.14 1.14 120 1.12 1.12 1.12 1.12 1.12 100 1.1 1.1 1.1 1.1 1.1 80 1.08 1.08 1.08 1.08 1.08 60 1.06 1.06 1.06 1.06 1.06 40 1.04 1.04 1.04 1.04 1.04 20 1.02 1.02 1.02 1.02 1.02 0 17600 17800 18000 18200 18400 186000 18800 450 500 550 600 650 700 750 8000 850 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.180 1.2 1.22 1.859 1.8595 1.86 1.86050 1.861 0.01880.0190.01920.01940.01960.01980.020.02020.02040.02060 NCrystal~Ball(D → V γ) NFlat (D → V γ) α(D → V γ) µ(D → V γ) σ(D → V γ) ) ) ) ) )

1.861γ 1.861γ 1.861γ 1.861γ 1.861γ 200 V V V V V 180 → → → → →

0 0 0 0 0

(D (D (D (D (D 160

1.8605µ 1.8605µ 1.8605µ 1.8605µ 1.8605µ 140 120 1.86 1.86 1.86 1.86 1.86 100 80

1.8595 1.8595 1.8595 1.8595 1.8595 60 40

1.859 1.859 1.859 1.859 1.859 20 0 17600 17800 18000 18200 18400 186000 18800 450 500 550 600 650 700 750 8000 850 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.180 1.2 1.22 1.859 1.8595 1.86 1.86050 1.861 0.01880.0190.01920.01940.01960.01980.020.02020.02040.02060 NCrystal~Ball(D → V γ) NFlat (D → V γ) α(D → V γ) µ(D → V γ) σ(D → V γ) ) ) ) ) ) γ γ γ γ γ

0.0206 V 0.0206 V 0.0206 V 0.0206 V 0.0206 V 3500 → → → → →

0 0 0 0 0 0.0204 0.0204 0.0204 0.0204 0.0204 (D (D (D (D (D 3000 σ σ σ σ σ 0.0202 0.0202 0.0202 0.0202 0.0202 2500 0.02 0.02 0.02 0.02 0.02

0.0198 0.0198 0.0198 0.0198 0.0198 2000 0.0196 0.0196 0.0196 0.0196 0.0196 1500 0.0194 0.0194 0.0194 0.0194 0.0194 1000 0.0192 0.0192 0.0192 0.0192 0.0192

0.019 0.019 0.019 0.019 0.019 500

0.0188 0.0188 0.0188 0.0188 0.0188 0 17600 17800 18000 18200 18400 18600 18800 450 500 550 600 650 700 750 800 850 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.859 1.8595 1.86 1.8605 1.861 0.01880.0190.01920.01940.01960.01980.020.02020.02040.0206 0 0 0 0 0 NCrystal~Ball(D → V γ) NFlat (D → V γ) α(D → V γ) µ(D → V γ) σ(D → V γ)

Figure 8.1: The correlation between fit values are shown with the blue ellipses. These show the correlation profile between two independent fit parameters. The fit values are represented by the blue dots in the center of each plot. We generate a large (M = 25000) ensemble of correlated random numbers. The distribution of these events is shown in the surface plot. This empirically shows the agreement between the initial fit values and the generated ensemble.

147 each signal shape. We then measure the normalization, N(D0 Vγ), with each → new step of component parameters. The distributions of N(D0 Vγ) are shown in → Fig. 8.2. This can be easily fit with a Gaussian. The systematic uncertainty is taken as the percent standard deviation of N(D0 Vγ). We generate 1000 ensembles for → ∗0 D0 K γ and 2500 ensembles for D0 φγ. The systematic uncertainty is found → → ∗0 to be 4.35% for D0 φγ and 3.78% for D0 K γ. → → Through correlating parameters, we are able to significantly reduce the systematic uncertainties due to fixing the component PDFs. If we repeat this procedure with no correlation assumed between parameters, the uncertainties are 12.8% for

∗0 D0 K γ and 5.65% for D0 φγ. By correlating the random ﬁt values we are → → able to reduce systematic uncertainty by nearly a factor of 3.

In summary we ﬁnd the systematic uncertainties due to ﬁxing the component

PDFs using the following method:

Fit a distribution with n free parameters. •

Decompose the n n correlation matrix, M, into the matrix L, such that • × M = LT L. ×

Generate a set of n uncorrelated random numbers from a Gaussian with a •

mean of 0 and a standard deviation of 1. Label the set [r1, r2, ..., rn].

Form a vector using the generated random numbers,r ˆ =(r , r , ..., rn). • 1 2

T Correlate the random numbers by multiplyingr ˆc =r ˆ L . • ×

Substitute the correlated random numbers into the original distribution. •

148 700 220 Varying Fit Parameter with Correlation Varying Fit Parameters with Correlation Varying Fit Parameters without Correlation Varying Fit Parameters without Correlation Optimal Fit Value Optimal Fit Value 200 600 180

500 160

140 Events / ( 6.18769 ) Events / ( 56.6265 ) 400 120

100 300

200 60

40 100 20

0 150 200 250 300 350 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 0 *0 N(D → φ γ) N(D0 → K γ)

(a) (b)

∗0 Figure 8.2: The normalization of (a) D0 φγ and (b) D0 K γ while the → → parameters of all component PDFs were varied in a correlated manner.

Reﬁt data using the new set of random numbers and obtain the normalization • of D0 Vγ. →

Fit the resulting distribution of N(D0 Vγ) to a Gaussian G(N(D0 • → → Vγ); µ, σ)

Deﬁne the relative systematic uncertainty as σ/µ. •

8.1.2 Control Sample Correction Systematics

As described in Chapter 6 we model the M(Vγ) distributions for several event classes based on MC events. The yield, N(D0 Vγ), is sensitive to proper modeling → to the D0 Vγ and D0 Vπ0 signal shapes. As a result when we apply the signal → → shapes obtained using MC to data, we must impose corrections on the signal shapes.

These are obtained by averaging the corrections over three control samples:

149 D0 K0γ • → s

0 ∗0 D K γ with cos(θH ) > 0.9 • → | |

D0 K0π0 • → s

These corrections are subject to systematic uncertainties based on statistical variations when ﬁnding the average correction, as well as variations in the corrections between control samples.

We implement the same technique discussed in the Section 8.1 to generate an ensemble of correlated random numbers, consistent with the correction ratios shown in Table 6.4. These are once again drawn from Gaussian distributions with a mean equal to the optimal ﬁt value and a standard deviation equal to the statistical error on the ﬁt. We correlate these correction ratios based on the error matrix returned by RooFit. Using these distributions, a large ensemble of correlated random numbers are generated and used for the correction ratios. The resulting distributions of

∗0 D0 φγ and D0 K γ are shown in Fig. 8.3. Based on the percent standard → → deviation, we see a systematic uncertainty of 1.21% for D0 φγ and 3.87% for → ∗0 D0 K γ. → As an alternative study we simply use the corrections found for each control sample separately to determine the systematic uncertainty. Using the corrections for each control sample, as shown in Table 6.6, the yield of D0 Vγ is measured → and summarized on Table 8.1. The systematic uncertainty is assigned by ﬁnding the standard deviation of the yield when apply the three corrections separately. This

∗0 is found to be 2.97% for D0 φγ and 4.30% for D0 K γ. As this is a more → →

150 70 Fit to Correction Variation Correlated Varying of the Correction Factors

120 Correlated Varying of the Correction Factors 60 Fit to Correction Variation

Events / ( 2 ) 100 50 Events / ( 22.6 )

80 40

60 30

40 20

20 10

0 0 200 210 220 230 240 250 260 270 280 290 2000 2100 2200 2300 2400 2500 2600 0 *0 N(D → φ γ) N(D0 → K γ)

(a) (b)

∗0 Figure 8.3: The normalization of (a) D0 φγ and (b) D0 K γ while correction → → ratios are varied on the D0 Vγ and D0 Vπ0 signal shapes. → →

conservative estimate, this is taken as the systematic uncertainty of the correction between MC and data.

8.1.3 D0 V η Signal Shape → The signal shape of D0 Vη events does not overlap with the D0 Vγ signal → → shape to the same extent as 0 0 . The impact of properly modeling such events PD →V π has a lesser effect on the yield of N(D0 Vγ). A conservative approach to finding → the systematic is to allow the 0 signal shape to float freely when fitting data. PD →V η ∗0 When this is done very little change is seen in N(D0 K γ), as compared to → a noticeable change for N(D0 φγ). The systematic uncertainty in both case is → taken as the percent change in N(D0 Vγ). Table 8.2 summarizes the change in → N(D0 Vγ) when the D0 Vη signal shape is allowed to float. → →

151 ∗0 Control Sample Correction N(D0 φγ) N(D0 K γ) 0 0 0 → → D Ks π 235.0 24.4 2078.8 109.7 0 → 0 ± ± D Ks γ 250.5 26.2 2108.1 112.4 → ∗0 ± ± D0 K γ 235.7 23.9 2288.5 110.3 → ± ± σ 2.97% 4.30%

Table 8.1: The yield of N(D0 Vγ) with independent Control Samples correction applied. →

∗0 N(D0 φγ) N(D0 K γ) → → P (D0 Vη) Fixed 242.6 24.8 2285.8 113.2 P (D0 → Vη) Float 232.1 ± 24.2 2291.9 ± 137.9 → ± ± σ 4.52% 0.26%

Table 8.2: The change in yields as the D0 Vη signal shape is allowed to ﬂoat. →

8.1.4 Combinatoric Signal Shape

Similar to D0 Vη, the combinatoric background shape is obtained using MC. → A typical cross check used to validate the model of these events is examining events for an observable sideband in data. Unfortunately, background from D0 Vπ0 → events is seen across numerous observables, making it difficult to find a region in data which is not effected by D0 Vπ0 events. As an alternative, the systematic → uncertainty due to the combinatoric background is measured by allowing the background shape to float when fitting data. We used the percent change in N(D0 Vγ) → to assign the systematic uncertainty associated with the combinatoric background shape. The results are summarized in Table 8.3.

152 ∗0 N(D0 φγ) N(D0 K γ) → → P (BG) Fixed 242.6 24.8 2285.8 113.2 P (BG) Float 240.7 ± 27.0 2304.9 ± 127.2 ± ± σ 0.79% 0.84%

Table 8.3: The change in yields as the combinatoric BG signal shape is allowed to ﬂoat.

8.2 Tracking Reconstruction and Vertexing

Following the recipe from the BABAR tracking eﬃciency group [25] a ﬂat systematic uncertainty of 0.8% per track is assigned for higher momentum charged candidates. This compares to a 1.2% systamatic uncertainity for lower momentum

+ charged candidates, speciﬁcally πslow. If these uncertainties are added linearly, a to- ∗0 tal systematic uncertainty of 2.8% would be used for both D0 φγ and D0 K γ → → analyses. However the branching fraction for each mode is measured in ratio to

D0 K−π+. The systematic effect from charged tracks seen in both signal decays → is largely mimicked in D0 K−π+. When the branching fraction is measured in → ratio to D0 K−π+, these systematic uncertainties cancel out. → Completely allowing the systematic uncertainties to cancel in ratio ignores residual effects between kaon and pions. The residual uncertainties in the systematic corrections for kaons and pions are 0.5% and 0.3%, respectively [26] . Adding these residual effects in quadrature leads to a 0.6% tracking systematic for D0 φγ → ∗0 analysis. The tracking systematic uncertainty of (D0 K γ) when measured in B → reference to (D0 K−π+) cancels each other out. B → In both analyses the production vertex of our candidate photon is set using a

153 vertexing algorithm. Both analyses use a cut on vertex χ2 probability of 0.001.

Systematic eﬀects of such a vertex cut are well studied [27, 28]. These studies document systematic uncertainties of 1.0% or less. We simply apply a 1.0% systematic for vertexing in each analyses. Adding the tracking and vertexing systematics quadratically the total systematic is taken as 1.2% the D0 φγ analysis and 1.0% → ∗0 for the D0 K γ analysis. → 8.3 Particle Identiﬁcation

The systematic errors associated with PID are well studied by the PID Analysis

Working Group (AWG) at BABAR. The PID eﬃciencies on MC are typically higher than those seen in data. In order to bring MC closer to data a process known as

“PID Tweaking” was developed. In this procedure control samples of clean events are maintained in data and MC, and used to calibrate the selection eﬃciency between both. These include:

D∗+ D0π+ ; D0 K−π+, used to calibrate the kaon and pion PID. • → slow →

Λ pπ−, used to calibrate proton PID. • →

K0 π+π−, used to calibrate pion PID. • s →

e−e+ µ+µ−γ, used to calibrate µ PID. • →

e−e+ e+e−γ, used to calibrate e PID. • →

τ 3π, used to calibrate pion PID. • →

Using these control samples, PID performance tables include measurements of the

PID eﬃciencies, ǫ, in data and MC. PID Tweaking only successfully identiﬁes

154 tracks in MC with a probability ǫ(Data)/ǫ(MC) when ǫ(MC) > ǫ(Data). This has the eﬀect of reducing the overall PID eﬃciency. For example, a kaon carrying

ǫ(Data) = 70% and ǫ(MC) = 72% will be properly identiﬁed in MC with a probability of 97%.

Instituting PID tweaking greatly reduces PID systematic uncertainties. However the control samples used to populate the PID tables are topologically different from our decays. The systematic uncertainty for PID is measured as the residual difference in PID efficiency between our events and the control samples used to determine

PID tweaks.

PID success is highly dependent both on particle momentum and its location within the detector, the former is shown in Fig 3.7. The systematic uncertainty is assigned as the average diﬀerence in PID eﬃciency between MC of our signal modes and one of the PID control samples in data [29].

The PID control sample most topologically similar to our signal modes is D∗+ → D0π+ ; D0 K−π+. The PID AWG recommends the following cuts to render a slow → clean sample:

0.1445 GeV/c2 < M(D∗) M(D0) < 0.1465 GeV/c2. • −

1.845 GeV/c2 < D0 < 1.885 GeV/c2 •

1.99 GeV/c2 < M(D∗) < 2.03 GeV/c2 • The PID eﬃciency, ǫ, is deﬁned in equation 8.4.

N 0 (with P ID) ǫ(PID)= D →XY (8.4) ND0→XY (without P ID) This is measured in bins of particle momentum and polar angle, θ, and the systematic error is assigned using the following procedure:

155 Apply all cuts on signal MC, excluding PID selection of a single meson, h. •

Take momentum and polar angle of h as measured in the lab frame and break • each into bins of 0.1 GeV/c and 0.075 rad between [0., 3.5] GeV/c and [0,π] rad

respectively.

Count the number of mesons, h in each bin before and after the PID cut is • applied.

Measure eﬃciency, as deﬁned in equation 8.4, in each bin. •

Take the diﬀerence of ǫ between signal Monte Carlo and the D∗+ D0π+ ; D0 • → slow → K−π+ control sample in data at each bin.

Define the systematic uncertainty as the weighted average of the difference in • efficiency in each bin between signal MC and the PID control sample, weighting

the diﬀerence in each bin by the number of events in signal MC.

8.3.1 D0 φγ PID Systematic → The Tight PID selectors are applied to the daughter pion of the D∗+ D0π+ ; D0 → slow → K−π+ PID control sample. Figures 8.4 (a) and (b) show the efficiency of the Tight selector when applied to K+ and K− while (c) and (d) show the same PID list on the D∗ control sample in data. Taking the weighted average of the difference in efficiency gives to a systematic effect of 2.32% for K+ and 2.67% for K−.

0 8.3.2 D0 K∗ γ PID Systematic → Figures 8.5 (a) and (b) show the eﬃciency of the Tight selector when applied to

K+ and K− while (c) and (d) show the same PID list on the D∗ control sample in

156 data. Figures 8.6 (a) and (b) show the eﬃciency of the VeryTight selector applied to π+ and π− while (c) and (d) show the same PID list on the D∗ control sample in data. The average diﬀerence is found to be σ(K+) = 2.26%, σ(K−) = 2.35%,

σ(π+) = 0.81%, and σ(π−) = 0.84%.

0 + 8.3.3 D K−π Reference Mode → (D0 Vγ) is measured in reference to (D0 K−π+). However, the selec- B → B → tions of this reference mode are somewhat diﬀerent than the PID control samples.

For completeness these events are taken through the same procedure to ﬁnd PID systematic uncertainties. As expected the systematic error does not rise to the level seen in the ﬁrst two cases, σ(K+) = 1.53%, σ(K−) = 1.26%, σ(π+) = 0.63%, and

σ(π−) = 0.75%. Figure 8.7 shows the eﬃciency in terms of momentum and polar angle.

8.3.4 PID Systematic Summary

The systematic uncertainty for each mode are summarized in Table 8.4. The total systematic uncertainty is found by adding the uncertainty for kaons and pions in quadrature. The uncertainty for an individual particle is taken as the charged particle with the largest diﬀerence between signal and reference mode. For example,

∗0 the total systematic uncertainty for D0 K γ is found by taking the residual → between K− in signal and reference, then adding it in quadrature with the diﬀerence

∗0 between π+ in signal and reference. The total systematic uncertainty for D0 K γ → is 1.10%. In the case of D0 φγ the systematic of only K+ is oﬀset by the reference →

157 mode, giving a systematic of 2.88%.

∗0 (D0 K γ) σ B → = 1.10% (8.5) (D0 K−π+) B → ! (D0 φγ) σ B → = 2.88% (8.6) (D0 K−π+) B →

Mode σ(K−)(%) σ(K+)(%) σ(π+)(%) σ(π−)(%) D0 φγ 2.67 2.32 - - → ∗0 D0 K γ 2.35 2.26 0.81 0.84 → D0 K−π+ 1.26 1.53 0.63 0.75 →

Table 8.4: Summary of PID systematic errors for each mode

158 ) )

- 1

+ 1 3 3 (K (K 0.9 0.9 θ θ 2.5 0.8 2.5 0.8 0.7 0.7 2 2 0.6 0.6 1.5 0.5 1.5 0.5 0.4 0.4 1 0.3 1 0.3 0.2 0.2 0.5 0.5 0.1 0.1 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0 0.5 1 1.5 2 2.5 3 3.5 + - (a) P(K ) (b) P(K )

) 1 ) - 1 + 3 3 (K (K 0.9 0.9 θ θ 2.5 0.8 2.5 0.8 0.7 0.7 2 2 0.6 0.6

1.5 0.5 1.5 0.5 0.4 0.4 1 0.3 1 0.3 0.2 0.2 0.5 0.5 0.1 0.1 0 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3 3.5 + - (c) P(K ) (d) P(K )

Figure 8.4: (a) and (b) show the efficiency of KLHTight as applied to signal MC for D0 φγ. (a) show K+ and (b) showing K− efficiencies. (c) and (d) show the control→ sample efficiency with the same kaon PID cut.

159 1 1 3 3 0.9 0.9 2.5 0.8 2.5 0.8 0.7 0.7 ), KLHTight ), KLHTight -

+ 2 2 0.6 0.6 (K (K θ θ 1.5 0.5 1.5 0.5 0.4 0.4 1 0.3 1 0.3 0.2 0.2 0.5 0.5 0.1 0.1 0 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 + - (a) P(K ) (b) P(K ) 1 1 3 3 0.9 0.9 2.5 0.8 2.5 0.8 0.7 0.7 ), KLHTight ), KLHTight -

+ 2 2 0.6 0.6 (K (K θ θ 1.5 0.5 1.5 0.5 0.4 0.4 1 0.3 1 0.3 0.2 0.2 0.5 0.5 0.1 0.1 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 + - (c) P(K ) (d) P(K )

Figure 8.5: (a) and (b) show the efficiency of KLHTight as applied to signal MC ∗0 for D0 K γ. (a) showing K+ and (b) showing K− efficiencies. (c) and (d) show the control→ sample efficiency with the same kaon PID cut.

160 1 1 3 3 0.9 0.9 2.5 0.8 2.5 0.8 0.7 0.7 2 2 0.6 0.6

), piLHVeryTight 0.5 ), piLHVeryTight 0.5 1.5 - 1.5 + π π ( ( 0.4 0.4 θ θ 1 0.3 1 0.3 0.2 0.2 0.5 0.5 0.1 0.1 0 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 + - (a) P(π ) (b) P(π ) 1 1 3 3 0.9 0.9 2.5 0.8 2.5 0.8 0.7 0.7 2 2 0.6 0.6 0.5 ), piLHVeryTight ), piLHVeryTight 0.5 - 1.5 - 1.5 π π ( 0.4 ( 0.4 θ θ 1 0.3 1 0.3 0.2 0.2 0.5 0.5 0.1 0.1 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 - - (c) P(π ) (d) P(π )

Figure 8.6: (a) and (b) show the efficiency of piLHVeryTight as applied to signal ∗0 MC for D0 K γ. (a) showing π+ and (b) showing π− efficiencies. (c) and (d) show the control→ sample efficiency with the same kaon PID cut.

161 + 1 + 1

π 3 π 3

- 0.9 - 0.9 K K 0.8 0.8 → 2.5 → 2.5

0 0.7 0 0.7

); D 2 ); D 2 - + 0.6 0.6 (K (K θ θ 1.5 0.5 1.5 0.5 0.4 0.4 1 0.3 1 0.3 0.2 0.2 0.5 0.5 0.1 0.1 0 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 + - (a) P(K ) KLHTight (b) P(K ) KLHTight

+ 1 + 1

π 3 π 3

- 0.9 - 0.9 K K

→ 2.5 0.8 → 2.5 0.8

0 0 0.7 0.7

); D 2 ); D 2 - + 0.6 0.6 π π ( ( θ θ 1.5 0.5 1.5 0.5 0.4 0.4 1 0.3 1 0.3 0.2 0.2 0.5 0.5 0.1 0.1 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 + - (c) P(π ) piLHVeryTight (d) P(π ) piLHVeryTight

Figure 8.7: (a) and (b) show the eﬃciency of KLHTight as applied to signal MC for D0 K−π+. (a) showing K+ and (b) showing K− eﬃciencies. (c) and (d) show the same→ for pions in D0 K−π+ signal MC with a pion PID cut of piLHVeryTight. →

162 8.4 Photon Systematics

The systematic uncertainty for photon reconstruction is measured by the Neu- tral AWG at BABAR [30]. The decays of τ leptons into π0s are used to estimate the systematic uncertainties associated π0 reconstruction. This in turn is used to estimate the systematic uncertainties of photon reconstruction.

The π0 systematics are determined by examining the ratios shown in Eqs. 8.7 and 8.8. N(τ + π+ν)(data) r + + = → (8.7) τ →π ν N(τ + π+ν)(MC) → N(τ + π+ν)(data) r + + + + 0 = → (8.8) τ →ρ ν;ρ →π π N(τ + ρ+ν; ρ+ π+π0)(MC) → → The first expression is dependent on the reconstruction efficiency of charged particles, while the second expression is dependent both on the reconstruction efficiency of charged particles and the reconstruction efficiency of π0. Some fraction of π0 are known as Merged. These are typically high energy π0s which decay into two nearly collinear photons. These photons are constructed as a single cluster in the EMC.

Merged π0s typically have momentum above 2.5GeV , meaning that single clusters seen below this value are typically single photons. With this in mind the ratio in

Eq. 8.9 is calculated as a function of photon energy in data and MC:

0 Nγ(Unmerged π ) 0 (8.9) Nγ(Unmerged + Merged π ) Table 8.5 shows the diﬀerence of Eq. 8.9 as measured in data and MC. Taking the weighted average of both the relative eﬃciency and statistical error leads to a sys-

∗0 tematic eﬀect of σ(D0 φγ)=(1.07 0.23)% and σ(D0 K γ)=(1.00 0.22)%. → ± → ± A second approach is to take half of the systematic uncertainty of π0 reconstruction, 3.0%. When the uncertainty in (τ π+ν) and (τ ρ+ν), is accounted B → B → 163 for the resulting systematic uncertainty of photon reconstruction is 1.8%. This is a more conservative measure than the values previously quoted. Therefore, this is taken as the systematic uncertainty of our photon reconstruction.

E(γ)(GeV ) Rel. Eﬀ. Stat Err. 0.03 0.5 0.993 0.001 − 0.5 1.0 0.984 0.001 − 1.0 1.5 0.988 0.002 − 1.5 2.0 0.990 0.003 − 2.0 2.5 0.999 0.003 −

Table 8.5: Single photon relative eﬃciency as a function of photon energy

8.5 π0 Veto

To assess diﬀerences in the π0 veto eﬃciency between data and MC, we use the

0 D Ksγ dataset as a control sample. The veto is measured in data and MC of → this dataset as a function of Nγ, the number of photons in an event. The systematic uncertainty is taken as the average difference in efficiency, ǫ, between data and MC, weighted by the number of candidates. The efficiency is assigned to be the ratio of the number of candidates after applying our π0 veto, to the number of candidates prior to applying our π0 veto. Figure 8.8 shows the comparison between data and

MC. Taking the weighted average gives a systematic eﬀect of:

σ(π0 V eto) = 1.8% (8.10)

We also measure the eﬃciency as a function of photon energy and as a two

164 0.6 Veto)

0 Data π ( 0.5∈ Monte Carlo

0.4

0.3

0.2

0.1

0 5 10 15 20 25 Nγ

Figure 8.8: The eﬃciency of the π0 in both data and MC as a function of the number of photons in a given event.

165 dimensional function of photon energy and Nγ. In each case the systematic eﬀect is found to be less than or equal to 1.8%. Neither of these methods give a more conservative estimate, therefore we used the 1.8% as the systematic uncertainity.

8.6 Cut Variation

To account for systematic errors due to cut selection, we vary cuts on helic-

∗+ ity, M(V Meson), ∆M, PCMS(γ), and PCMS(D ) around the optimized selection

∗0 range. As the cuts are varied, the yields of both D0 φγ and D0 K γ are → → measured, and the variation of N(D0 Vγ)/ǫ(D0 Vγ) is taken as the system- → → atic uncertainty associated with cut selection. For each cut selection we remeasured both N(D0 Vγ) using data and ǫ(D0 Vγ) using the D0 Vγ MC sample. → → → In an attempt to remove bias in selecting individual cut values, we randomly choose new values for all selection parameters within a certain range of the optimal selection. For each new set of selections we measure both the number of reconstructed signal events and the reconstruction eﬃciency for each cut selection. The most conservative approach is to assume all selections are equally probable and uncorrelated. We therefore draw the randomized selection values from uniform distributions. Table 8.6 summarizes the range of each cut variable. We measure both the eﬃciency of each selection using MC and the number of reconstructed events

∗0 from data. The total number of events for both D0 K γ and D0 φγ is → → ﬁt with a Gaussian distribution. The percent deviation is taken as the systematic

∗0 uncertainty due to cut variation, resulting in a 4.48% effect in D0 K γ and a → 5.41% effect in D0 φγ. Figs. 8.9(a) and 8.9(b) show the distributions and the → fits for both modes.

166 ∗0 Cut D0 K γ Selection D0 φγ Selection → → cos(θH ) Low [ 0.39, 0.21] [ 0.95, 0.85] − − − − cos(θH ) High [0.56, 0.74] [0.85, 0.95] ∆M( GeV/c2) Low [0.142, 0.145] [0.142, 0.145] ∆M( GeV/c2) High [0.146, 0.149] [0.146, 0.149] PCMS(γ) ( GeV/c) [0.48, 0.60] [0.48, 0.60] ∗+ PCMS(D ) ( GeV/c) [2.60, 2.64] [2.60, 2.64] M(V Meson)( GeV/c2) Low [0.842, 0.852] [1.005, 1.015] M(V Meson)( GeV/c2) High [0.945, 0.957] [1.025, 1.035]

Table 8.6: Summary of the selection range used to determine selection systematic ∗0 uncertainties. Note that M(V Meson) refers to either M(φ) or M(K ).

24 35 22 20 30

Entries / (20) 18 Entries / (400) 25 16 14 20 12

15 10 8 10 6 4 5 2 0 0 32000 34000 36000 38000 40000 42000 2000 2200 2400 2600 2800 N(D0 → K γ)/∈ N(D0 → φ γ)/∈(D0 → φ γ)

(a) (b)

∗0 Figure 8.9: The normalization of (a) D0 K γ and (b) D0 φγ while the selection values for each cut are varied. → →

167 8.7 Systematic Summary

The total systematic uncertainty is taken as the quadratic sum of all systematic components. These are summarized in Table 8.7. The item Reference Mode Skim

Eﬀ. is detailed in Appendix A. The total systematic errors of 9.59% and 8.31%

∗0 for D0 φγ and D0 K γ respectively are found by adding all systematic → → contributions in quadrature.

∗0 Systematic σ(D0 φγ) (%) σ(D0 K γ) (%) → → Tracking and Vertexing 1.20 1.00 Particle ID 2.88 1.10 Photon Reconstruction 1.80 1.80 π0 Veto 1.80 1.80 Combinatoric BG Shape 0.79 0.84 D0 Vη Signal Shape 4.52 0.26 → Fixing Signal Shapes 3.78 4.35 Correcting D0 Vγ and D0 Vπ0 2.97 4.30 → → Run Dependence 0.00 0.00 Reference Mode Skim Eﬀ [*]. 1.50 1.50 Cut selection 5.41 4.48 Total Systematic Eﬀect 9.59 8.31

Table 8.7: Summary of all systematic errors for each mode. [*] Appendix A details the systematic contribution for inconsistencies in skim eﬃciency between data and MC. The total systematic uncertainty is found by adding all systematic estimates in quadrature.

168 CHAPTER 9

Conclusion

We measure the branching fraction of the decay, D0 φγ, as well as the → ∗0 previously unobserved decay, D0 K γ, relative to the well measured process, → D0 K−π+. The results for each decay mode are shown in Eq. 9.1 and 9.2. These → are translated into absolute branching fractions using the current world average for

(D0 K−π+). The absolute branching fractions are shown in Eqs. 9.3 and 9.4 B → (D0 φγ) B → = (7.15 0.78 0.69) 10−4 (9.1) (D0 K−π+) ± ± × B → ∗0 (D0 K γ) B → = (8.43 0.51 0.70) 10−3 (9.2) (D0 K−π+) ± ± × B → (D0 φγ) = (2.73 0.30 0.26) 10−5 (9.3) B → ± ± × ∗0 (D0 K γ) = (3.22 0.20 0.27) 10−4 (9.4) B → ± ± ×

The measurement of D0 φγ conﬁrms a previous experimental search [10], while → ∗0 the observation of D0 K γ represents the ﬁrst measurement of this mode. While → theoretical predictions vary greatly, both measurements present good agreement with the order of magnitude of these predictions. Particularly good agreement is seen for theoretical predictions based on HQET and CL. Using this hybrid approach

∗0 Ref. [9] predicts, (D0 K γ) = 3.5 10−4 and (D0 φγ) = 1.9 10−5. B → × B → ×

169 The most straightforward theoretical description of radiative charm decays is made using VMD. VMD processes describe a quark and an antiquark annihilating into a single photon, an example of which is shown in Fig. 2.2(c). Under this model the quark/anti-quark pair is considered a virtual vector meson. For both

∗0 D0 K γ and D0 φγ the virtual vector meson takes the form of ρ0. As a → → consequence the ﬁrst order Feynman diagram for VMD is nearly identical to the di-

∗0 agrams for D0 K ρ0 and D0 φρ0. In D0 Vρ0 decays the quark/anti-quark → → → pair hadronize into a ρ0, rather than annihilating. VMD calculations for radiative processes use measurements of these modes to predict the scale of radiative processes.

This approach was ﬁrst advanced in Ref. [5]. In this technique the rate of radiative decays is proportional to fT and αρ. fT is the fraction transversely polarized

0 0 0 0 ρ mesons produced by D Vρ , and α 0 is the probability of ρ γ [11]. A → ρ → reasonable check of VMD contributions is made by comparing the relative branching

∗0 ∗0 fractions of D0 φγ and D0 K γ against D0 φρ0 and D0 K ρ0. The → → → → ∗0 branching fractions for D0 φρ0 and D0 K ρ0 are shown in Eq. 9.5 [1]. → →

∗0 ∗0 (D0 K ρ0, K K−π+) = (1.00 0.22)% B → → ± (D0 φρ0, φ K−K+) = (6.7 0.6) 10−4 B → → ± × (9.5)

In principle, for VMD to proceed, the ρ0 must by transversely polarized. Experimen-

∗0 tal this has been found to be the case in nearly 100% of the observed D0 K ρ0 → decays. No such angular analysis has been studied for D0 φρ0 decays. For com- → parison purposes, we consider ρ0 to be 100% transversely polarized in D0 Vρ0. →

170 Our measurements show excellent agreement with the ratio of D0 Vρ0 states, as → shown in Eqs. 9.6 and 9.7.

0 − + (D φγ,φ K K ) −2 B → ∗0 →∗0 = (6.26 0.71 0.79) 10 (9.6) (D0 K γ, K K−π+) ± ± × B → →

0 0 − + (D φρ ,φ K K ) −2 B → ∗0 →∗0 = (6.7 1.6) 10 (9.7) (D0 K ρ0, K K−π+) ± × B → → If we assume all contributions are from VMD type processes and it is assumed that

0 0 all ρ are transversely polarized, αρ can be estimated with the ratio of (D Vγ) B → 0 0 and (D Vρ ). In both cases we see αρ 0.021. A very crude approximation B → ≈ can be made for the expected branching fraction using only VMD: (D0 Vγ) B → ≈ 0 0 αEM (D Vρ ), where αEM is the ﬁne structure constant: αEM = 1/137 [5]. B → Hence we are seeing nearly a factor of three greater (D0 Vγ) then predicted B → using only VMD. This suggests that we are seeing enhancements from processes other than VMD.

Theoretical prediction made in Refs [5] and [9] each incorporate pole diagrams into the overall calculations. As shown in Table 2.3, both models predict nearly all amplitudes within 20%. The greatest disagreement between the two models is how these pole diagrams interfere. Ref. [5] predicts that the pole diagrams will interfere such that any contribution almost entirely cancels. Ref [9] makes no predictions about these diagrams either canceling or enhancing each other.

Comparing our results in Eqs. 9.3 and 9.4 to those predicted in Table 2.3, our results show closer agreement to predictions made while assuming little or no interference between pole diagrams. It is important to consider that both sets of predictions relied upon measurements made a decade ago. Over the course of the last

171 decade experimental measurements have been made with greater precision. Our measurements may spark some renewed theoretical interest as well as interest in measuring other radiative charm decays.

Actual measurements of D0 ωγ and D0 ρ0γ may be just within the capa- → → bilities of existing e+e− detectors. Most predictions for these modes are on the order of 1 10−5. If the D0 φγ measurement is used as an analog to the D0 ρ0γ B ≈ × → → analysis, it may be possible to see up to 90 events within a dataset equivalent to that of BABAR. While it is concievable to measure this at a number of facilities, a signiﬁcant background from D0 π+π−π0 decays must be overcome. Considering → the width of ρ0, little background can be removed through a tight M(ρ0) selection.

Instead a tight helicity cut can be used to remove some D0 ρ0γ, but residual → contributions from D0 ρ±π∓ are more difficult to eliminate. → Measurement of D0 ωγ will in all likelihood be at the edge of the datasets → currently availabled at HEP facilities. Techniques will be needed to remove combinatoric contributions from 3 photons in the final state, two from π0 and one direct photon from D0. Additionally, the scales of the prominent backgrounds: D0 ωπ0 → and D0 ωη, have not been measured. As recently as 2006, an upper limit of → (D0 ωπ0) < 2.6 10−4 has been set [31]. A measurement of D0 ωη is ex- B → × → pected in the near future. If these modes are measured, D0 ωγ may be the most → reasonable candidate for the next examination of radiative D0 decays. Both D+ and Ds mesons are also expected to decay radiatively, largely through long distance processes similar to those described here. These may offer another strong channel

+ to investigate radiative charm processes. In particular, Ds ρ γ is a opportunity → to study color favored radiative charm decays.

172 In summary we have conﬁrmed a previous observation of D0 φγ and provided → ∗0 the ﬁrst measurement of D0 K γ. These results suggest that while VMD pro- → cesses play an important role in radiative decays, pole diagram contributions cannot be ignored when considering radiative charm physics.

173 APPENDIX A

0 + D K−π Reference Mode →

Due to uncertainties on the order of 10% in the overall number of D∗+’s produced at BABAR a measurement of the D0 Vγ absolute branching fraction is impractical. → Instead we use the a well studied decay, D∗+ D0π+ ; D0 K−π+, as a reference → slow → mode. We measure the rate of (D0 Vγ) relative to the observed number of B → events in the reference mode, D0 K−π+. The branching fraction of our signal → mode is calculated using Equation 1.6, shown in the Chapter 1

A.1 Event Selection

For the analysis of D0 Vγ events, the vast BABAR dataset was whittled down → using a process known as “skimming”. Events which met a broad set of selection criteria were ﬂagged and used for the D0 Vγ analysis. Events were skimmed that → satisﬁed the following set of selections:

A photon candidate with a minimum momentum of 250 MeV/c. •

A π+ candidate with a maximum momentum of 250 MeV/c. • s

A vector meson candidate from any one of the following criteria: •

174 – A φ candidate comprised of kaons with NotPion PID and an invariant

mass (0.99 < M(K−K+) < 1.05) GeV/c2

∗0 – A K candidate comprised of K− candidates with NotPion PID and

pions with VeryLoose PID. The invariant mass must be in the range

(0.792 < M(K−π+) < 0.992) GeV/c2.

– A ρ0 candidate comprised of oppositely charged pions with an invariant

mass in the range (0.30M(π+π−) < 1.30) GeV/c2.

– A ω candidate comprised of oppositely charged pions with Loose PID

and a π0 comprised of photons with an invariant mass (0.110 < M(γγ) <

0.150) MeV/c2. The invariant mass must be in the range (0.680 < M(π+π−π0) <

0.860) GeV/c2.

The invariant mass must be in the range: (1.60 < M(Vγ) < 2.10) GeV/c2. •

The mass diﬀerence must be in the range (M(Vγπ+) M(Vγ)) < 0.200 GeV/c2. • s −

These selections do not explicitly look for D0 K−π+ events. However the very → broad selection criteria a very large sample of D0 K−π+ candidate remain with → the skimmed dataset. Kaon candidates are required to pass Tight PID and pions were required to pass VeryTight PID. The invariant mass, M(K−π+), must be within the window, [1.80, 1.92] GeV/c2.A D∗+ tag is required to maintain consistency with D0 Vγ analyses. The diﬀerence in mass between a candidate D∗+ → and D0 must fall within the range [0.1435, 0.1475] GeV/c2. We also require that

∗+ PCMS(D ) > 2.62 GeV/c.

175 ) 2 ) 2 χ /N = 173.74/92 2

104 104 Monte Carlo χ2/N = 168.25/92

3 103 10

2 Events / ( 0.00125 GeV/c Events / ( 0.00125 GeV/c 10 102 1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.8 1.82 1.84 1.86 1.88 1.9 1.92 D0 Mass (GeV/c2) D0 Mass (GeV/c2) 2 i 2 i χ χ

6 6 ± 4 ± 4 2 2 0 0 -2 -2 -4 -4 (a) -61.8 1.82 1.84 1.86 1.88 1.9 1.92 (b) -61.8 1.82 1.84 1.86 1.88 1.9 1.92

Figure A.1: The D0 K−π+ signal is fit to a double Gaussian with the mean and standard deviation→ of each allowed to float freely. (a) Shows data from Runs 1-5 is plotted in with black markers while overlayed on MC events, shown with the histogram. (b) Shows the fit to MC.

A.2 Fitting Procedure

Signal D0 K−π+ events are ﬁt to a double Gaussian with the mean and → standard deviation of each Gaussian allowed to ﬂoat freely. Background events are

ﬁt to a linear polynomial. Figure A.1(a) shows the ﬁt to data from Runs 1-5, while

(b) shows the ﬁt to MC. The reconstruction eﬃciency is estimated using signal

MC of D∗+ D0π+ ; D0 K−π+ to be ǫ = (5.25 0.17)%. We measure → slow → ± N(D0 K−π+)=(3.35 0.040) 105 within our data sample. → ± × A.3 Systematic Uncertainties

The systematic uncertainties due to PID eﬃciencies have been estimated in

Section 8.3. However an additional systematic uncertainty is introduced when taking D0 K−π+ candidates that are within the skimmed dataset. We do → 176 not look for D0 K−π+ events explicitly within the skim, therefore systematic → discrepancies would arise if simulation does not match actual data in modeling

0 − + D K π + γRandom. To quantify any diﬀerences we run a subset of both data → and MC without using any skim. The dataset sizes are detailed in Table A.1. At the same time we ﬂag events which within the unskimmed dataset that pass our skim selection.

The D0 K−π+ signal is fit for both the entire dataset as well as the skimmed → 0 − + 0 − + events. We measure the ratio Npass(D K π )/Nall(D K π ) in data and → → 0 − + 0 − + MC, where Npass(D K π ) are the number of reconstructed D K π candi- → → 0 − + dates from the skimmed dataset. skim and Nall(D K π ) are the total number → of reconstructed D0 K−π+ candidates. We measure this ratio as 0.233 0.012 and → ± 0.235 0.007 in data and MC. The difference between these values is (0.2 1.4)%, ± ± where the second number is the statistical uncertainty in the difference. This is found by adding the percent uncertainties in quadrature. Adding the absolute difference and statistical uncertainty in quadrature, we find a total systematic effect of 1.5%.

177 Mode Number of Events (106) ( fb−1) L e+e− cc 51.9 40.0 → e+e− uu, dd,ss 86.5 41.4 → e+e− B0B0 24.1 45.5 → e+e− B+B− 30.3 57.3 → Run 1 On Peak 21.3 - Run 2 On Peak 31.7 - Run 3 On Peak 22.8 - Run 4 On Peak 61.8 - Run 5 On Peak 49.6 -

Table A.1: The MC and data sample sizes taken from unskimmed data. This subset is used to verify the skim eﬃciencies between data and MC.

178 APPENDIX B

Probability Distribution Functions

This appendix is provided as a reference for the PDF’s used in this analysis.

B.1 Crystal Ball Lineshape (CB)

The Crystal Ball (CB) lineshape is used to model processes which intrinsically have energy loss. Speciﬁcally in this analysis CB are used to model both incomplete photon energy deposition in the EMC and an unreconstrcuted photon produced by decaying π0. The function form has been shown previously (Eq. 3.10), but is reiterated here for completeness.

2 − (x−µ) x−µ 2σ2 e , if σ > α CB(x; α,n,µ,σ)= x−µ x−µ − (B.1) ( A(B )−n, if α − σ σ ≤ − n 2 n |α| − 2 A = |α| e , B = n α |α| − | | The CB is a piecewise continuous function through its ﬁrst derivative. The CB is a

Gaussian for values greater than the threshold (x>µ ασ), and a power loss tail − for values less than the threshold. The threshold couple the free terms: µ, α, and σ.

This couple leads to high correlation in these parameters when the function is ﬁt.

179 B.2 Gaussian

The Gaussian function, as shown in Eq. B.2, is one of the most ubiquitous PDF’s in science. In our analysis ∆M of signal events is modeled using a Gaussian.

2 1 − (x−µ) Gs(x; µσ) = e 2σ2 (B.2) r2πσ B.3 Chebychev Polynomials

Chebychev polynomials are solutions the diﬀerential equation shown in Eq. B.3.

d2y dy (1 x2) x + n2y = 0 (B.3) − dx2 − dx

In this expression n is an integer constant. The solutions, Fn, for n = 0, n = 1, and n = i are shown in Eq. B.4. Also shown in Eq. B.5 is the functional form of an nth order Chebychev polynomial used to ﬁt data in our analysis.

F0(x) = 1

F1(x) = x

Fi(x) = 2xTi (x) Ti (x) (B.4) −1 − −2 n

Tn(x) = 1+ ajFj(x) (B.5) j=1 X The Chebychev polynomials are orthogonal and therefore offer lower correlations between coefficients when used in fitting. This results in far more stable fit when

Chebychev polynomials are used over standard polynomials [32].

B.4 ∆M Background Shape

The distribution used to model ∆M = M(Vγπ+) M(Vγ), is shown in Eq. s −

4.11 and restated in Eq. B.6. In this expressions ci are free parameters and Mπ+ is

180 ﬁxed to the mass of π+. The pion mass serves as a cutoﬀ for the background shape, which is set to zero for ∆M < Mπ+ .

∆M−M π+ c2 c1 ∆M ∆M F (∆M; c1,c2,c3, Mπ+ )= 1 e + c3 1 (B.6) − M + M + − ! π π

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