c 2006 by Paras P. Naik. All rights reserved.I must thank the many teachers from Addison schools who...

MEASUREMENT OF THE RELATIVE AMPLITUDE AND STRONG PHASEBETWEEN D

0 → K∗+K− AND D0 → K∗+K−

VIA DALITZ PLOT ANALYSIS OF D0 → K+K−π0 DECAYS

BY

PARAS P. NAIK

B.S., University of Illinois at Urbana-Champaign, 2000M.S., University of Illinois at Urbana-Champaign, 2001

DISSERTATION

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2006

Urbana, Illinois

Abstract

I present physics concepts, which are useful to understand our analyses, and describe

the CLEO III and CLEO-c experiments at the Cornell Electron Storage Ring. I also

present motivations for a Dalitz plot analysis of the Cabibbo-suppressed charmed

meson decay mode D0 → K+K−π0 at CLEO. The analysis uses 9.0 fb−1 of data

collected at√

s ≈ 10.58 GeV with the CLEO III detector. We find the strong

phase difference δD ≡ arg(

A(D0→K∗+K−)

A(D0→K∗+K−)

)= 332◦ ± 8◦ ± 11◦ and relative amplitude

rD ≡∣∣∣A(D

0→K∗+K−)A(D0→K∗+K−)

∣∣∣ = 0.52 ± 0.05 ± 0.04. This measurement indicates significant

destructive interference between D0 → K+(K−π0)K∗− and D0 → K−(K+π0)K∗+

in the D0 → K+K−π0 Dalitz plot region where these two modes overlap. The fit

includes the K∗± and φ resonances and a non-resonant amplitude, and the measured

fit fractions for each resonance (with statistical uncertainty only) are (46.1 ± 3.1)%

for the K∗+, (12.3 ± 2.2)% for the K∗−, (14.9 ± 1.6)% for the φ, and (36.0 ± 3.7)%

for the non-resonant contribution. We find δD = 313◦ ± 9◦ (stat.) and an amplitude

ratio of rD = 0.52 ± 0.05 (stat.) from a second fit which substitutes scalar κ±

(mass 878 MeV/c2, width 499 MeV/c2) amplitudes for the non-resonant amplitude.

The measured fit fractions for each resonance (with statistical uncertainty only) are

(48.1 ± 4.5)% for the K∗+, (12.9 ± 2.6)% for the K∗−, (16.1 ± 1.9)% for the φ,

(12.6 ± 5.8)% for the κ+, and (11.1 ± 4.7)% for the κ−. We also investigate the

D0 → K+K−π0 Dalitz plot in 281 pb−1 of data collected at√

s ≈ 3.77 GeV with the

CLEO-c detector. We find results which are consistent with the CLEO III analysis.

I conclude by summarizing our results and present a brief appendix detailing the

K-matrix formalism.

iii

In memory of my grandfather,

Dr. Y.G. Naik (Physics)

Principal, Gujarat College

Dean, Faculty of Science

Gujarat University

Ahmedabad, Gujarat, India.

1906-1976

He would have loved to read this.

iv

Acknowledgments

I appreciate this opportunity to thank those people in my life who have brought out

the best in me, and thus directly helped me to pursue and complete the highest degree

awarded in Physics.

First, and foremost, I must thank my thesis advisor, Mats Selen. Mats has always

been available to discuss physics and assist me in my research, and he has always been

supportive of my personal and professional goals. Hopefully someday I will have the

opportunity to teach and guide someone towards success as well has he has.

I must thank my parents, Pradeep and Aruna, and my brother, Samar. I am

fortunate to have had the kind of support and friendship that they have provided me

over the years. I also thank my extended family and family friends who have always

treated me like their own son or brother.

I must thank Heather Gorman, without whom I wonder where I would have found

the motivation to complete my studies and move on to new challenges. Her patience,

affection, and consideration have helped me to achieve a lot in a short amount of

time. I am excited and pleased that the next phase of my life will begin in Ithaca,

where we first met.

I must thank the University of Illinois and the Department of Physics here, where I

have continued my education for the last ten years. I must also thank the University of

Illinois for awarding me a University Fellowship for the Fall 2005 semester, providing

me with the assistance to remain a research assistant when I returned from Ithaca.

I must thank the many collaborators I had while working on particle physics

research as part of the CLEO experiment. I must first thank the “original” (room) 487

v

crew of Scott Davis, Chris Sedlack, and Jeremy Williams for their helpful assistance

and endless supply of comedic one-liners. I must thank Norm Lowrey for helping

me acclimate myself to living and working in Ithaca. I must also thank Eric White,

who was pivotal in helping me assemble an additional chapter investigating the most

recent CLEO data.

I must thank Charles Plager for his help and support in understanding the impor-

tant physics and the relevant tools for studying Dalitz plots, and Tim Bergfeld whose

work has guided Illinois CLEO Dalitz plot analyses. I also must thank Bob Eisen-

stein, Jim Wiss, Inga Karliner, Doris Kim, and Topher Cawlfield for their assistance

with my research.

I must thank the entire Charm/Dalitz group for their help with various theoretical

and technical issues, including Steve Dytman who helped me to study the K-matrix

formalism. I would particularly like to thank Jim Napolitano, Jon Rosner, Mikhail

Dubrovin, and Hanna Mahlke-Kruger for all of their comments and assistance in

publishing a paper based on my research.

I must especially thank David Asner for his continuing assistance and involvement

with my research, and for offering me a job as a postdoctoral fellow with Carleton

University, so that I may continue to pursue scientific research in the field of elemen-

tary particle physics. I am looking forward to the opportunity.

I must thank the support staff for the Illinois High Energy Physics Group for

helping me arrange for travel and expenses related to research. I also must thank

the computing staffs at both Illinois and Cornell for providing me with the tools and

help necessary to complete computing projects.

I must thank my committee members, Kevin Pitts (chair), John Stack, and Brian

Fields, who have happily made their time and schedules available to remain apprised

of my research and encourage me to succeed.

I must thank the many teachers from Addison schools who have helped me along

vi

the way, especially: Tom Bookler, for helping me explore a world of curiosity from

elementary school through junior high; Walter Raczynski and especially Mark Ailes,

for piquing my interest in physics; and David Porter, for being a wonderful Calculus

teacher and writing a letter of recommendation that I am sure led to my first position

as a teaching assistant (TA).

I must thank the Department of Physics for giving me the opportunity to teach

as an undergraduate and as a graduate student. I appreciate the support of Gary

Gladding and Tim Stelzer during my years of involvement with the Physics Education

Research Group. I thank Johnetta Wilde and Cindy Hubert for their assistance and

conversations while I was a teaching assistant. I also thank all of my fellow TAs and

professors whom I worked with while teaching and as Mentor TA.

I must thank the friends and colleagues who have helped me and/or kept me

grounded throughout the years, including: Matthew Eisenbraun, Eric Rasmussen,

Melissa Headley, Nikhil Trivedi, Tolgar Alpagut, Lambros Tsonis, Eugene and Adele

Torigoe, Jim and Chimei Schneider, Andrew and Emily Meyertholen, Adam and

Laura Rasmussen, Joe Zimmerman, Amy Rund, John Phillips, Curtis Jastremsky,

Melissa Cole, Clinton Cronin, Matt Chasse, Pete and Adrienne Zweber, “Big” Mike

Bell, Ryan Small, Mike Watkins, Brian O’Shea, Evan Graves, Paco Jain, Kurt and

Jennifer Haberkamp, Steven Poulakos, Bob Drost, Brian Lang, Justin Hietala, Steve

Hugo, Jerry Codner and Patty McNally, Jocelyn and Nick Voss, Tim and Lynde Klein,

Dom Ricci, Selina Li, Mike Weinberger, Gregg and Jana Thayer, Tina Majethia, Matt

and Katie Shepherd, Christine Crane, Batbold Sanghi, Lisa Gay, Joscelyn Fisher, and

Ahren Sadoff.

I also thank anyone whom I have forgotten to mention and you, the reader of this

dissertation.

This material is based upon work supported by the Department of Energy under

Grant DE FG02 91ER40677.

vii

Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

1 Relevant Physics Concepts . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Quark Decay and the CKM Matrix . . . . . . . . . . . . . . . . . . . 21.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Violation of Discrete Symmetries . . . . . . . . . . . . . . . . . . . . 71.5 Unitarity and the CKM Angles . . . . . . . . . . . . . . . . . . . . . 81.6 Measuring γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Four-Momentum and Invariant Mass . . . . . . . . . . . . . . . . . . 131.8 Resonances and Interference . . . . . . . . . . . . . . . . . . . . . . . 141.9 The Dalitz Plot Formalism . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Model for Three-body Decays . . . . . . . . . . . . . . . . . . . . . . 172.1 Breit-Wigner Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Decay Amplitudes for a Particular Spin . . . . . . . . . . . . . . . . . 202.3 Amplitudes on the Dalitz Plot . . . . . . . . . . . . . . . . . . . . . . 22

3 CLEO, CESR, and LEPP . . . . . . . . . . . . . . . . . . . . . . . . 243.1 LEPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 CESR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 The CLEO III Detector . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Silicon Vertex Detector . . . . . . . . . . . . . . . . . . . . . . 283.3.2 Drift Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.3 dE/dx Particle Identification . . . . . . . . . . . . . . . . . . 313.3.4 RICH Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.5 Crystal Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . 373.3.6 Solenoid Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.7 Muon Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.8 Trigger and Data Acquisition Systems . . . . . . . . . . . . . 38

3.4 The CLEO-c Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 40

viii

3.4.1 The ZD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5 Reconstructing Events . . . . . . . . . . . . . . . . . . . . . . . . . . 433.6 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 D0 → K+K−π0 in CLEO III . . . . . . . . . . . . . . . . . . . . . . . 454.1 Current Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Data Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Event Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4 The Dalitz Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.5 The Matrix Element . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.6 Method of Maximum Likelihood . . . . . . . . . . . . . . . . . . . . . 574.7 Efficiency and Background . . . . . . . . . . . . . . . . . . . . . . . . 60

4.7.1 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.8 Overview of Fitting Technique . . . . . . . . . . . . . . . . . . . . . . 644.9 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.10 Fit Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.10.1 Nominal Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.10.2 κ Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.10.3 Results of Nominal Fit and κ Fit . . . . . . . . . . . . . . . . 744.10.4 Other Fits to the Entire Dalitz Plot . . . . . . . . . . . . . . . 854.10.5 Fits to Partial Regions of the Dalitz Plot . . . . . . . . . . . . 994.10.6 Floating mκ, Γκ Fit to the Entire Dalitz Plot . . . . . . . . . . 113

4.11 Systematic Errors for rD and δD in our Nominal Fit . . . . . . . . . . 1164.12 Branching Ratio Cross-check . . . . . . . . . . . . . . . . . . . . . . . 1224.13 U -spin Symmetry Check . . . . . . . . . . . . . . . . . . . . . . . . . 1234.14 The Effect of our Uncertainties on γ . . . . . . . . . . . . . . . . . . 1244.15 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5 D0 → K+K−π0 in CLEO-c . . . . . . . . . . . . . . . . . . . . . . . . 1275.1 Effects of Quantum Correlations . . . . . . . . . . . . . . . . . . . . . 1275.2 Data Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.3 Event Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.4 CLEO-c Dalitz Plot Fit . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

A The K-matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . 138A.1 The K operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138A.2 The K-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A.3 K-matrix Examples and Argand Plots . . . . . . . . . . . . . . . . . 141

A.3.1 One Pole, One Decay Channel . . . . . . . . . . . . . . . . . . 141A.3.2 Two Poles, One Decay Channel . . . . . . . . . . . . . . . . . 142

A.4 Application to Dalitz Plot Analysis . . . . . . . . . . . . . . . . . . . 145A.4.1 Lorentz-Invariant T -Matrix . . . . . . . . . . . . . . . . . . . 145

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A.4.2 The �P -Vector Formalism . . . . . . . . . . . . . . . . . . . . . 146A.4.3 Practical Application . . . . . . . . . . . . . . . . . . . . . . . 147

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Author’s Biography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

x

List of Tables

2.1 Normalized Blatt-Weisskopf Barrier Factors (FP (mab) = 1 when mab =mr) for the particle P decay vertex. zP is defined as follows: zP (mab) =r2P p2

P (mab). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 Branching ratios of intermediate D0 modes which may decay to thethree-body mode K+K−π0. . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 The masses and widths of resonances r considered in this analysis[1, 45–48]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Best-fit parameters for the efficiency shape. . . . . . . . . . . . . . . . 614.4 Best-fit parameters for the background shape. . . . . . . . . . . . . . 644.5 Summary of fits. See Table 4.2 for more information about each reso-

nance, and see the following figures for details. . . . . . . . . . . . . . 714.6 Amplitude, phase, and fit fraction results for our nominal fit (Fit 1)

with statistical errors. . . . . . . . . . . . . . . . . . . . . . . . . . . 764.7 Amplitude, phase, and fit fraction results for Fit 3 (with κ± instead of

a non-resonant contribution) with statistical errors. . . . . . . . . . . 854.8 Summary of systematic checks and systematic errors. “<” means a

small change. * RICH selection criteria are used when RICH info isavailable for kaons with momentum pK > 500 MeV. ** dE/dx selec-tion criteria are used when pK < 500 MeV or when RICH info is notavailable for kaons with momentum pK > 500 MeV. . . . . . . . . . 118

4.9 Details about systematic checks. * RICH selection criteria are usedwhen RICH info is available for kaons with momentum pK > 500 MeV.** dE/dx selection criteria are used when pK < 500 MeV or whenRICH info is not available for kaons with momentum pK > 500 MeV. 120

4.10 Best-fit parameters for the background shape when the backgroundparameters are allowed to float. . . . . . . . . . . . . . . . . . . . . . 121

4.11 Best-fit parameters for the efficiency shape when the efficiency param-eters are allowed to float. . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.12 Comparison of estimated branching ratios from this analysis to pub-lished branching ratios [32–34,48] based on our nominal Dalitz plot fit.We accounted for the fact that K∗± decays to K±π0 only one-third ofthe time, and φ decays to K+K− only about one-half of the time. . . 123

5.1 Amplitude, phase, and fit fraction results for Fit 1-c with statisticalerrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

xi

5.2 Amplitude, phase, and fit fraction results for Fit 3-c with statisticalerrors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3 Comparison of predicted to measured ratios of fit fractions when com-paring CLEO-c data (correlated D0s) to CLEO III data (uncorrelatedD0s). The errors on the predictions are propagated from the totalerrors on rK∗K and δK∗K . The errors on the ratios from data are sta-tistical errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

A.1 Partial widths used to calculate �F for interfering K∗ resonances. . . . 150

xii

List of Figures

1.1 The quarks and leptons of the Standard Model. Each row consists ofparticles with the same electric charge and the columns are groupedinto three generations each. . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 One example of a D0 → K+K−π0 decay. . . . . . . . . . . . . . . . . 41.3 The electromagnetic interaction e+e− → qq. . . . . . . . . . . . . . . 51.4 The Unitarity Triangle showing the common definitions of the three

weak phases α (also referred to as φ2), β (φ1), and γ (φ3), as well asthe relevant CKM matrix elements. . . . . . . . . . . . . . . . . . . . 9

2.1 D0 → abc through an ab resonance. The initial and final statesboth have no angular momentum, so the intermediate spin states aresummed over. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Since the D0 is a spinless particle, a vector (spin 1) decay particlesuch as the φ must be in a |1, 0〉 angular momentum state in the D0

rest frame (when we quantize along the π0 direction). The daughtersof the φ therefore decay preferentially parallel to its momentum (theamplitude has a Y 1

0 ∝ cos(θ) dependence in the D0 rest frame). . . . 222.3 |A|2 for the vector K−π0 resonance K∗−. When the π0 comes out

opposite to the K∗− direction, it is almost at rest and corresponds toθ = 0◦. The K− is almost at rest when θ = 180◦. When the K−

and π0 come out perpendicular to the K∗− (θ = 90◦), they have equalmomentum. We can see the preference for a decay to occur parallel tothe momentum of the K∗− as a cos2(θ) dependence on the Dalitz plot(the square of the cos(θ) amplitude dependence mentioned in Figure2.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 A diagram of CESR. . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 The CLEO III Detector. . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 End view and quarter-section view of the CLEO III Silicon Detector. 283.4 Charge collection and multiplication in the drift chamber. . . . . . . . 293.5 Using tilted stereo wires to obtain z information in the drift chamber.

Tilted wires are projected as arrows coming out of the page in the r-φprojection; position along a tilted wire indicates the z of the track helixnear that wire. The closest wires to the track (in three dimensions)are highlighted (in green). . . . . . . . . . . . . . . . . . . . . . . . . 30

xiii

3.6 CLEO III dE/dx vs. momentum showing the π, K and p bands.Leptons are suppressed where possible. The dE/dx scale is chosen tobe 1 for electrons and positrons from e+e− → e+e− events. . . . . . . 32

3.7 A section of the RICH detector. . . . . . . . . . . . . . . . . . . . . . 343.8 How true kaons (simulated) would look at various momenta (binned)

in terms of the range of possible values for Δχ2dE/dx (Equation 3.2)

for dE/dx information and the range of possible values for Δχ2RICH

(Equation 3.5) for RICH information, as defined in the text. Values ofless then zero indicate that the kaon hypothesis is more likely. . . . . 36

3.9 How the trigger tiles read shower information from the calorimeter. . 393.10 Cross section of the CLEO-c detector in the r-z plane. . . . . . . . . . 413.11 An isometric view of the CLEO-c inner wire chamber. . . . . . . . . . 42

4.1 Shown are examples of (a) a non-resonant decay, (b) a neutral inter-mediate state (φ), (c) a positively charged intermediate state (K∗+),and (d) a negatively charged intermediate state (K∗−). . . . . . . . . 46

4.2 Histograms of (−2 lnLK)−(−2 lnLπ) from RICH information, numberof photons nγ used to determine the RICH information, and σK , thenumber of standard deviations away from the expected dE/dx for atrue kaon, for the positive (1st row) and negative (2nd row) kaons afterall other selection criteria have been applied. . . . . . . . . . . . . . . 49

4.3 Distribution of (a) mK+K−π0 for |ΔM | < 1 MeV/c2 and (b) ΔM for1.84 < mK+K−π0 < 1.89 GeV/c2 after passing all other selection cri-teria discussed in the text. The solid curves (red) show the results offits to the mK+K−π0 and ΔM distributions, respectively. The back-ground level in each plot is shown by a dashed horizontal line (black).The vertical lines in (a) and the left-most set of vertical lines in (b)denote the signal region. The right-most set of vertical lines in figure(b) denote the ΔM sideband used for estimation of the backgroundshape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 (a) The Dalitz plot distribution for D0 → K+K−π0 candidates. (b)-(d) Projections onto the m2

K+π0 , m2K−π0 , and m2

K+K− axes of the resultsof Fit 1 (discussed in Section 4.10) showing both the fit (curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 53

xiv

4.5 An example of the effect of a non-resonant (NR) contribution interfer-ing with two K∗ resonances. The K∗s in this example have the sameamplitude but a relative phase with each other such that the two reso-nances have maximal destructive interference with each other, and theamplitude aNR is set to be 5aK∗ . When the relative phase of the NRcontribution to the K∗+ phase, δNR, is 0◦ or 180◦ there is not a notice-able effect on the K∗ lobes. However, a phase δNR = 90◦ makes theK∗− enhanced (depleted) at low (high) K+π0 invariant mass squared,and the K∗+ enhanced (depleted) at high (low) K−π0 invariant masssquared. A phase δNR = 270◦ makes the K∗+ enhanced (depleted)at low (high) K−π0 invariant mass squared, and the K∗− enhanced(depleted) at high (low) K+π0 invariant mass squared. . . . . . . . . 54

4.6 Dalitz plot projections properly rotated to show how they are obtainedfrom the Dalitz plot. The thicker straight lines correspond to reso-nances which decay to the two particles in the respective invariantmass-squared projection, and the thinner straight lines correspond toreflections which do not decay to the two particles in the respectiveinvariant mass-squared projection. . . . . . . . . . . . . . . . . . . . . 56

4.7 (a) Scatter plot of Monte Carlo events used to study the efficiencyacross the Dalitz plot and (b-d) projections of these events along withthe result of a fit using the efficiency function. . . . . . . . . . . . . . 62

4.8 (a) Scatter plot of ΔM versus mK+K−π0 for event candidates satisfyingall other requirements. The black boxed area (the bottom box) is theSignal Region which contains our signal candidates, and the red boxedarea (the top box) is the Background Region which contains eventswhich we use to estimate the shape of our background. (b) a Dalitzplot of the events from the Background Region of plot (a). . . . . . . 63

4.9 The projected background events from the ΔM sideband and the resultof fitting the background Dalitz plot to a 2-D cubic polynomial plusnon-interfering K∗ and φ contributions (line). . . . . . . . . . . . . . 65

4.10 Interfering K∗+ and K∗− resonances on the Dalitz plot. The waywe have defined our amplitudes results in a phase convention wheremaximal destructive interference between the K∗s occurs at δD = 0◦.Maximal constructive interference occurs when δD = 180◦. Note thatthe effect of the interference is most noticeable in the region where theK∗ resonances overlap. . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.11 Dalitz plot projections for Fit 1 showing both the fit (red curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 72



4.14 The signal and background in the Dalitz plot projections of Fit 1. . . 77

xv

4.15 Signal portion of Fit 1 and the component resonance amplitudes squared(m2

K+π0 projection). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.16 Signal portion of Fit 1 and the component resonance amplitudes squared

(m2K+K− projection). . . . . . . . . . . . . . . . . . . . . . . . . . . . 79


K−π0 projection). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.18 The signal and background in the Dalitz plot projections of Fit 3. . . 814.19 Signal portion of Fit 3 and the component resonance amplitudes squared

(m2K+π0 projection). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


K+K− projection). . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.21 Signal portion of Fit 3 and the component resonance amplitudes squared

(m2K−π0 projection). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.22 Fit 1 projections (blue, dashed line) overlayed on top of Fit 3 projec-tions (green, solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.23 Dalitz plot projections for Fit 3a showing both the fit (red curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 87










4.33 Fit Regions (a) K∗ bands (b) No φ 1.05 (c) No φ 1.15 (d) No φ 1.26(e) “L” area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.34 Dalitz plot projections for Fit 1x showing both the fit (red curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 102



xvi

4.37 (a) The absolute value of the phase distribution of our nominal fit overthe entire Dalitz plot subtracted from the phase distribution of thenominal fit with the φ simply removed (aφ is set to be 0). This showsthe effect the φ resonance has on the overall phase at different pointsin phase space on the Dalitz plot. Removing only a region where theφ has its peak is not enough to remove all regions of the Dalitz plotwhose phase is noticeably affected by the φ resonance. (b) The sameplot as (a) except that areas outside the K∗ bands are excluded. Notethat the phase distribution is not affected by the φ in the vast majorityof the K∗ bands region, so it is definitely appropriate to exclude the φfrom any K∗ bands fit. . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.38 Dalitz plot projections for Fit 1n showing both the fit (red curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.39 Dalitz plot projections for Fit 1np showing both the fit (red curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.40 Dalitz plot projections for Fit 1na showing both the fit (red curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.41 Dalitz plot projections for Fit 1nap showing both the fit (red curve)and the binned data sample. . . . . . . . . . . . . . . . . . . . . . . . 110

4.42 Dalitz plot projections for Fit 1nb showing both the fit (red curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.43 Dalitz plot projections for Fit 1nbp showing both the fit (red curve)and the binned data sample. . . . . . . . . . . . . . . . . . . . . . . . 112

4.44 Dalitz plot projections for Fit 1s showing both the fit (red curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.45 Dalitz plot projections for Fit 3f showing both the fit (red curve) andthe binned data sample. . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.1 Plots of mbc and ΔE for all events passing the selection criteria inSection 5.2, with the signal region within the dashed (brown) lines ofboth figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.2 Dalitz plot of K+K−π0 candidates. . . . . . . . . . . . . . . . . . . . 1335.3 Dalitz plot projections for Fit 1-c. . . . . . . . . . . . . . . . . . . . . 1345.4 Dalitz plot projections for Fit 3-c. . . . . . . . . . . . . . . . . . . . . 135

A.1 (left) An Argand Plot of the K∗(892) T -matrix for all values of m2Kπ.

(right) A plot of |T |2 vs. m2Kπ. . . . . . . . . . . . . . . . . . . . . . . 142

A.2 (left) An Argand Plot of the K∗(1410) T -matrix for all values of m2Kπ.

(right) A plot of |T |2 vs. m2Kπ. . . . . . . . . . . . . . . . . . . . . . . 143

A.3 (left) An Argand Plot of T = TK∗(892) + TK∗(1410) for all values ofm2

Kπ < 5. (right) A plot of |TK∗(892) + TK∗(1410)|2 vs. m2Kπ. . . . . . . 143

A.4 (left) An Argand Plot of T calculated from KK∗(892) + KK∗(1410) for allvalues of m2

Kπ < 5. (right) A plot of |T |2 vs. m2Kπ. . . . . . . . . . . . 144

xvii

A.5 A diagram of the process where a resonance is produced from a D0

decay. The resonance may decay to a number of decay channels, ifpossible. The amplitude for the decay channel with particles a and bcan be found in the ab component of the final state vector �F . . . . . . 146

A.6 A plot of the magnitude of the �F -vector amplitude to the Kπ decaychannel squared vs. Kπ invariant mass squared for destructively in-terfering K∗ resonances. . . . . . . . . . . . . . . . . . . . . . . . . . 150

A.7 An example of a fit using the P -vector formalism. . . . . . . . . . . . 152

xviii

List of Abbreviations

APS American Physical Society.

BR Branching Ratio.

C Charge Conjugation.

CESR Cornell Electron Storage Ring.

CHESS Cornell High Energy Synchrotron Source.

CKM Cabibbo-Kobayashi-Maskawa.

CP Charge-Parity.

DAQ Data Acquisition System.

DP Dalitz Plot.

FF Fit Fraction.

LEPP Laboratory for Elementary-Particle Physics.

LINAC Linear Accelerator.

MC Monte Carlo.

NR Non-resonant.

P Parity Inversion.

Ph.D. Doctor of Philosophy.

RF Radio Frequency.

RICH Ring Imaging Cherenkov.

S.L. Significance Level.

Stat. Statistical Error.

Std. Dev. Standard Deviations.

T Time Reversal.

Vs. Versus.

xix

List of Symbols

G Giga. 109

M Mega. 106

k Kilo. 103

c Centi. 10−2

m Milli. 10−3

μ Micro. 10−6

n Nano. 10−9

p Pico. 10−12

f Femto. 10−15

B Bytes. (A unit representing data)

b Barns. (A unit representing area, 1 b = 10−28 m2)

m Meters. (A unit representing length)

eV Electron Volts. (A unit representing energy, 1 eV ≈ 1.6 × 10−19 Joules)

Hz Hertz. (A unit representing frequency, 1 Hz = 1 s−1)

s Seconds. (A unit representing time)

◦ Degrees. (A unit representing angle)

rad Radians. (A unit representing angle, 2π rad = 360◦)

O Order

→ Decays To

�x Vector x

x Anti-particle of x

X Operator X

X Lorentz Invariant Version of X

xx

1 Relevant Physics Concepts

1.1 The Standard Model

The Standard Model of particle physics, which summarizes our current understanding

of matter, includes six quarks and six leptons grouped into three generations (see

Figure 1.1) [1]. All of the quarks and leptons are spin 12

fermions. Each of the

particles in Figure 1.1 has an antiparticle which has the same mass and spin but a

change in sign of additive quantum numbers, most importantly electric charge (d has

charge − e3, d has charge + e

3; −e is the electric charge of the electron). The quarks and

leptons, along with their antiparticles, are assumed to be the fundamental particles

of matter in the theory.

There are three forces that affect the interaction of particles: strong, electroweak,

and gravity. The effects of gravity are neglected in the Standard Model because of

its relative weakness in comparison to the other forces [2]. The electroweak force at

current experimental energies can be thought of in terms of separate electromagnetic

and weak interactions. Quarks interact via the electromagnetic, weak, and strong

chargecharge

+2e3− e3

(ud

) (cs

) (tb

)6 Quarks, 3 Generations

chargecharge

0−e

(νe

e

) (νμ

μ

) (ντ

τ

)6 Leptons, 3 Generations

Figure 1.1: The quarks and leptons of the Standard Model. Each row consists ofparticles with the same electric charge and the columns are grouped into three gen-erations each.

1

forces; charged leptons only interact through the electromagnetic and weak forces, and

neutral leptons only interact via the weak force [1]. The electromagnetic interaction

is mediated by a massless, spin-1 photon. The weak interaction is mediated by the

massive, spin-1, charged W± and neutral Z bosons. The strong force is mediated by

massless, spin-1 gluons.

Isolated leptons can readily be found in nature, but quarks have only been ob-

served in bound states of two types: mesons, which consist of a quark bound to an

antiquark (qq); and baryons, which consist of three quarks (qqq) or three antiquarks

(qqq). The mesons and baryons are collectively known as hadrons due to their ability

to interact via the strong force [2].

1.2 Quark Decay and the CKM Matrix

The weak interaction allows one type of quark to change into another type of quark,

also known as changing flavor, by emitting or absorbing a charged W boson. Gener-

ally, a quark will decay within its generation unless forbidden by energy conservation.

Less frequently, a particle will decay into another generation. This ability to decay

into another generation is due to a difference in the weak and mass eigenstates for the

quarks. A charge − e3

quark’s weak eigenstates (|d′〉, |s′〉, |b′〉) are related to its mass

eigenstates (|d〉, |s〉, |b〉) by the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing

matrix, VCKM in Equation 1.1 [1]. We have the freedom to define the charge +2e3

quarks’ weak eigenstates to be the same as their mass eigenstates (Equation 1.2), so

that the VCKM contains all of the information about the quark mixing.

⎛⎜⎜⎜⎜⎝

d′

s′

b′

⎞⎟⎟⎟⎟⎠ = VCKM

⎛⎜⎜⎜⎜⎝

d

s

b

⎞⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎝

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

⎞⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

d

s

b

⎞⎟⎟⎟⎟⎠ (1.1)

2

⎛⎜⎜⎜⎜⎝

u′

c′

t′

⎞⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎜⎝

u

c

t

⎞⎟⎟⎟⎟⎠ (1.2)

The coupling strength used in determining a quark’s interaction probability am-

plitude is defined by gVij where i and j are types (or “flavors”) of quarks and g is the

universal weak coupling constant [3]. An antiquark’s interaction probability ampli-

tude is the complex conjugate, gV ∗ij . The diagonal elements of the CKM matrix are

close to 1 since quarks prefer to decay to their own generation if allowed; however,

all of the off-diagonal elements have magnitudes that are non-zero within 90% con-

fidence limits. Conservation of probability requires the CKM matrix to be unitary,

assuming there are only 3 generations of quarks. The current global fit which uses

all available measurements of Vij and the unitarity constraint [1] gives the following

for the magnitudes of the CKM elements:

VCKM =

⎛⎜⎜⎜⎜⎝

0.97383+0.00024−0.00023 0.2272+0.0010

−0.0010 (3.96+0.09−0.09) × 10−3

0.2271+0.0010−0.0010 0.97296+0.00024

−0.00024 (42.21+0.10−0.80) × 10−3

(8.14+0.32−0.64) × 10−3 (41.61+0.12

−0.78) × 10−3 0.999100+0.000034−0.000004

⎞⎟⎟⎟⎟⎠ . (1.3)

It should be noted that elements of the CKM matrix are allowed to be complex

numbers.

Two examples of how the weak interaction occurs can be seen in a decay we are

interested in, D0(cu) → K+(us)K−(us)π0(uu−dd√2

). The particular Feynman diagram

shown in Figure 1.2 is just one of many that could be drawn for the D0 → K+K−π0

decay. The first weak decay, c → W+s, allows the c quark to change into an s quark.

The second weak decay occurs when the W+ boson decays into a u and s quark.

Decays that involve quarks within the same generation, such as c → W+s are known

3

c

u_

s

u_

u_

D0

W+

K-

K+

π0

u

u

s_

gg

Figure 1.2: One example of a D0 → K+K−π0 decay.

as Cabibbo-favored decays since the diagonal CKM matrix elements are relatively

large, while decays that involve quarks of different generations, such as W+ → us,

are known as Cabibbo-suppressed. It turns out that in all possible Feynman diagrams

for the D0 → K+K−π0 decay, there must be at least one Cabibbo-suppressed decay,

and thus the decay itself is referred to as Cabibbo-suppressed also.

Mesons may also decay by way of the strong force. Often a meson may be in

a highly energetic state, such as when individual quarks of the meson are traveling

quickly enough to separate. It is energetically preferred to create new quark-antiquark

pairs in order to lower the energy of the original meson by forming two separate

particles [4]. An example of this type of decay in Figure 1.2 occurs after the W+ → us

vertex, when a uu pair is created to hadronize the separating u and s quarks.

The strong force is itself responsible for the existence of hadrons. Quarks have a

property called “color,” which is conserved in strong interactions (this is an analog

to charge which is conserved in electromagnetic interactions). Color is a necessary

4

e+

e-

q

q_

γ∗

Figure 1.3: The electromagnetic interaction e+e− → qq.

quality of quarks, because, without it, protons (uud) would not be able to have two

u quarks in the same state due to the Pauli exclusion principle for fermions. Quarks

may have three types of colors, antiquarks have three types of anticolors, and gluons

carry color-anticolor combinations. Bound states of quarks (hadrons) must have no

net color. This is the case when quarks of all three colors (or all three anticolors)

form a bound state (baryons) or when a quark has one color and an antiquark has

the anticolor (mesons).

A relevant lepton electromagnetic interaction is that of an electron (e−) − positron

(e+) annihilation into a virtual photon, which then produces a quark-antiquark pair

(see Figure 1.3). This interaction is utilized in some particle accelerators (such as

the Cornell Electron Storage Ring) to create qq pairs which separate, hadronize, and

decay to create mesons and baryons that particle physicists are interested in.

1.3 Symmetries

In physics, there are certain fundamental symmetries which lead to conservation (in-

variance) of physical quantities. Some of these are continuous symmetries, while

others are non-continuous, or discrete. An example of a continuous symmetry is time

translation, where a physical system at time t is compared to an identical physical

system at time t + Δt, where Δt is an non-zero interval of time. If we are stand-

5

ing at sea level and throw a ball up vertically (neglecting air resistance) at time t,

Newton’s Second Law, �F = m�r, would correctly predict the ball’s acceleration under

the influence of gravity. If we throw the ball at time t + Δt under the same ini-

tial conditions, we still expect Newton’s Second Law to provide the same result for

the acceleration. Newton’s Second Law is unchanged under a time translation; the

acceleration is preserved under this symmetry.

An example of a discrete symmetry is time reversal (T ), where t is replaced with

−t. This replacement would reverse velocities, momenta, and angular momenta, but

would not reverse displacements or accelerations. Newton’s Second Law still holds

under this transformation. Imagine the motion of the ball in the previous example

thrown upward at time t. If we reversed time by applying the T symmetry, we would

see that the reverse process would appear to be the same, the process being symmetric

with respect to the moment at which the ball is at it’s maximum height.

Another discrete symmetry is parity inversion (P ), which is the spatial inversion

of position with respect to the origin. We can define a parity operator, P , which

performs this operation:

P�r = P (x, y, z) = (−x,−y,−z) = −�r. (1.4)

Velocities, accelerations, and momenta will all be reversed under parity inversion, but

angular momenta will not. Angular momentum is a vector product of a displacement

and a momentum, and is preserved under a parity inversion:

P �L = P (�r × �p) = −�r ×−�p = �r × �p = �L. (1.5)

Quantities such as �r which change sign under parity transformation have a parity

eigenvalue, P = −1. Quantities such as �L which do not change sign under parity

6

transformation have a parity eigenvalue, P = 1, and are called Pseudovectors. Note

that applying the parity operator twice is the same as applying the identity operator,

P 2 = I. Hadrons are eigenstates of the parity operator [5]. The convention is that

quarks have positive parity (+1), and anti-quarks have negative parity (−1). The

parity of a composite system of quarks is the product of the parities of each quark

multiplied by a factor of (−1)L, where L is the orbital angular momentum of the

hadron. Also, the photon is a vector (spin 1) particle and has an intrinsic parity

P = −1.

One more example of a discrete symmetry is charge conjugation symmetry (C).

The charge conjugation operator, C, reverses all internal quantum numbers, including

electric charge, thus turning a particle into its anti-particle. For example, C|K+〉 =

|K−〉. Most hadrons are not eigenstates of C, but those that are (|π0〉, for example)

have charge conjugation quantum number C = (−1)L+S, where S is the spin of the

hadron. Like P , applying C twice to a particle returns the same particle, thus C2 = I.

1.4 Violation of Discrete Symmetries

C and P symmetries hold under strong decays, but are violated in weak decays [1].

Charge-Parity (CP ) symmetry is the combination of C symmetry and P symmetry.

CP symmetry was once thought to hold true for weak decays, but a violation of this

symmetry has been observed in neutral kaons and the neutral B-meson system [1].

The term “CP violation” refers to violation of the combined CP symmetry. CP

violation has not yet been observed in decays of D-mesons. In the case that CP

violation is negligible, the rates of conjugate decays are equal (for example, we would

expect as many D0 mesons to decay to K∗−K+ as we would expect D0(cu) mesons

to decay to K∗+K−). This is an acceptable assumption for this analysis, because CP

violation is predicted to be negligibly small for D decays in the Standard Model [1].

7

One of the main motivations for a better understanding of, and the continued

search for, CP violation is to explain why matter is preferred over anti-matter in

our universe. While CP violation is not sufficient, it is necessary to explain this

dominance of matter [6]. There are few tests of processes in which CP symmetry

is violated. When there is a violation, such as for neutral kaons, it is consistent

with the Standard Model. However, the Standard Model violation is not a large

enough asymmetry to explain a matter-dominated universe. Looking deeper into CP

violation and its parameters may lead to observations of physics beyond the Standard

Model which would explain the matter-antimatter asymmetry.

So far, no experimental evidence for CPT violation, the violation of the combined

C, P , and T symmetries, has been observed [1].

1.5 Unitarity and the CKM Angles

We have the freedom to parameterize the CKM matrix in terms of a single phys-

ical phase because it is possible to redefine the quark mass eigenstates to absorb

other phases. An explicit parametrization of this type is known as the Wolfenstein

parametrization [1]:

VCKM =

⎛⎜⎜⎜⎜⎝

1 − λ2

2λ Aλ3(ρ − iη)

−λ 1 − λ2

2Aλ2

Aλ3(1 − ρ − iη) −Aλ2 1

⎞⎟⎟⎟⎟⎠ + O(λ4), (1.6)

where higher order terms in λ are ignored because λ is small and real. A, ρ, and η

are real numbers of order unity.

We expect that the inner product of any two of the columns, or any two of the

rows, of the CKM matrix is zero because a unitary matrix is orthogonal. We may

apply this unitarity constraint to the first and third columns of the CKM matrix [1]

8

γ

α

β

*

Vud

Vub

* *Vtd

Vtb

Vcd

Vcb

Figure 1.4: The Unitarity Triangle showing the common definitions of the three weakphases α (also referred to as φ2), β (φ1), and γ (φ3), as well as the relevant CKMmatrix elements.

which yields:

VudV∗ub + VcdV

∗cb + VtdV

∗tb = 0. (1.7)

Equation 1.7 is interesting because Vub and Vtd are both matrix elements that are

complex in the O(λ2) Wolfenstein parametrization. A unitarity triangle (Figure 1.4)

can represent Equation 1.7 in the complex plane [1].

The size of the area, JCP

2≈ λ6A2η

2, of the triangle in Figure 1.4 determines the

amount of charge-parity symmetry (CP ) violation in weak quark decays and in the

current global fit is measured to be JCP

2= (1.54+0.08

−0.09) × 10−5 [1]. There are five

additional triangles that can be made which enable us to overconstrain the area if we

have information about the angles and sides of the triangles. The CKM angles α (also

referred to as φ2), β (φ1), and γ (φ3) of Figure 1.4 can, in principle, be measured by

CP asymmetries in B-meson decays, while the sides can also be measured by decays

involving the associated quarks.

9

1.6 Measuring γ

γ is a challenging CKM angle to determine experimentally. Of course, we can con-

strain γ with the knowledge of α and β assuming that the unitarity triangle is closed,

but a direct measurement would provide a test of unitarity itself and thus would be

sensitive to physics beyond the Standard Model.

There are a few promising ways in which γ may be measured directly from B-

meson decays. Some of these methods include measuring decays whose amplitudes

have only a relative CKM phase γ and a relative strong phase δ, measuring B →ππ, Kπ, or KK decays, partial reconstruction of B → D(∗)±π∓ decays, and measuring

Bs decays such as Bs → ρ0K0S and Bs → D0φ [7].

A method of measuring γ via B±(ub, ub) → DK± decays where the neutral D-

meson decays to CP eigenstates was proposed by Gronau and Wyler [8]. Atwood,

Dunietz, and Soni show that choosing final states which are not CP eigenstates

can lead to large direct CP violation which can also give significant information

about γ [9]. Grossman, Ligeti, and Soffer [10] have proposed a method for a direct

measurement of γ by studying B± → DK±, where the neutral D meson (D0/D0)

decays to the Cabibbo-suppressed final states K∗+K− or K∗−K+. The final states

both have sizable rates in D decay and are not CP eigenstates.

In this formalism (where we ignore terms of O(λ4) and higher order terms in λ),

we define:

A∓B ≡ A(B∓ → D0K∓) (1.8)

A∓B ≡ A(B∓ → D0K∓) (1.9)

A∓B

A∓B

= rBei(δB∓γ), (1.10)

10

and

A∓D ≡ A(D0 → K∗±K∓) (1.11)

A∓D ≡ A(D

0 → K∗±K∓) (1.12)

A∓D

A∓D

= rDeiδD , (1.13)

where A is the decay amplitude for the respective decay, δB is the relative strong

phase between A+B and A+

B, the CKM angle γ is the weak phase between A+B and

A+B, δD is the relative strong phase between A−

D and A−D, and rB and rD are real

and positive. There is no weak phase between A−D and A−

D since D0 → K∗+K− and

D0 → K∗+K− both have amplitudes proportional to V ∗csVus(= VcsV

∗us, since Vus and

Vcs are real up to O(λ3)).

Four interesting B-decay amplitudes can then be defined as follows:

A[B− → K−(K∗+K−)D] = |ABAD|(1 + rBrDei(δB+δD−γ)) (1.14)

A[B− → K−(K∗−K+)D] = |ABAD|eiδD(rD + rBei(δB−δD−γ)) (1.15)

A[B+ → K+(K∗−K+)D] = |ABAD|(1 + rBrDei(δB+δD+γ)) (1.16)

A[B+ → K+(K∗+K−)D] = |ABAD|eiδD(rD + rBei(δB−δD+γ)), (1.17)

where we define AB ≡ A−B and AD ≡ A−

D. The CKM angle γ can then be determined

with the knowledge of all of the other, measurable quantities in Equations 1.14, 1.15,

1.16, and 1.17.

An important ingredient in this proposition is the knowledge of the relative com-

plex amplitude rDeiδD between AD and AD, which, in the absence of CP viola-

tion in D decays, is the same as the relative complex amplitude between A+D =

11

A(D0 → K∗−K+) and AD = A(D0 → K∗+K−). The primary objective of the anal-

ysis described in this dissertation is to measure rD and δD via the three-body decay

D0 → K+K−π0. We expect the amplitude for K∗+ → K+π0 to be the same as the

amplitude for K∗− → K−π0. In this case, rDeiδD can be determined by the interfer-

ence between the D0 → K+(K−π0)K∗− and D0 → (K+π0)K∗+K− decay modes, as

discussed in Chapter 4.

Similar analyses can also be done with B− → K−(ρ±π∓)D decays [10], B → DK∗0

decays [7], and B− → K−Dmulti−body (where the D-meson decays to a multi-body final

state which both the D0 and D0

have large probability to decay to, such as K0Sπ+π−)

[11]. The latter method has already been used to make a direct measurement of

γ, with the Belle Collaboration measuring γ = (68+14−15 ± 13 ± 11)◦ and the BaBar

Collaboration measuring γ = (67 ± 28 ± 13 ± 11)◦, where the last uncertainty is

due to the D decay modeling [1]. A more recent result by Belle using multiple

B+ → D(∗)K(∗)+ decay modes is γ = (53+15−18 ± 3 ± 9)◦ [12].

Determining the length of sides of the unitarity triangle can also be a helpful way

to constrain γ, if our assumptions are correct and the unitarity triangle is closed. The

sides of the unitarity triangle corresponding to VudV∗ub and VcdV

∗cb can be measured

in b quark decays to u or c quarks, respectively, such as in b → u(c)lν, where l is

a charged lepton and ν is the corresponding neutrino [1]. The side of the unitarity

triangle corresponding to VtdV∗tb can be measured via measurements of B0-B

0and

B0s -B

0

s oscillations [1].

As of October 2006, the CKM angle γ is inferred to be 59.0◦+9.2◦−3.7◦ from various

experimental and theoretical constraints [13].

12

1.7 Four-Momentum and Invariant Mass

In relativistic mechanics, some fundamental information about a particle is defined

by that particle’s four-momentum, pμ. Four-momentum is similar to non-relativistic

momentum, p, except that the energy of the particle is also one of the elements:

pμ = (E

c, px, py, pz), (1.18)

where E2 = (mc2)2 +(pc)2. The Minkowski inner product of pμ with itself has a very

simple result:

pμpμ =

(E

cpx py pz

)⎛⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

⎞⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎝

Ec

px

py

pz

⎞⎟⎟⎟⎟⎟⎟⎟⎠

=E2

c2− p2 = m2c2. (1.19)

Thus, if we can know the four-momentum of a particle, we will know the particle’s

mass. Since momentum is conserved, we can use these principles to find the masses

of particles which we cannot observe. For example, consider the decay φ → K+K−.

Since momentum is conserved, pμφ = pμ

K+ + pμK− . We may not be able to see the

φ with a detector because it decays too quickly, but suppose we have found two

oppositely charged kaons and we are curious about their origin. We can determine

(pK+ + pK−)μ(pK+ + pK−)μ = m2K+K−c2 and define mK+K− as the invariant mass

of the two kaon system. If these kaons truly came from a φ, we must find that

mK+K− = mφ, since pφμpφμ ≡ m2

φc2.

13

1.8 Resonances and Interference

The decay D0 → K+K−π0 may occur directly, with one particle (the D0) decaying

into three particles without any intermediate states. However, the D0 may also decay

into two particles, followed by a subsequent two-body decay of one of those particles.

The three possibilities are: D0 → K+R−1 , R−

1 → K−π0; D0 → K−R+2 , R+

2 → K+π0;

and D0 → π0R03, R0

3 → K+K−. Ri represents a short-lived intermediate particle

which is called a resonance.

Quantum mechanical interference occurs when a physical process has multiple in-

distinguishable paths to completion. If there could be two or more different processes

for D0 → K+K−π0, and we can not distinguish which process participated in the

decay, we expect to observe interference between the processes.

1.9 The Dalitz Plot Formalism

The D0, K+, K−, and π0 mesons are all spin zero particles. In studying the decay

D0 → K+K−π0, it is important to consider how many degrees of freedom are required

to completely describe the kinematics. In the spin zero D-meson rest frame, there are

three decay particles each with four degrees of freedom (their four-momenta) for a

total of 12 unknowns. There is one constraint for each daughter particle mass (known

by experiment) and four additional constraints from the conservation of momentum

and energy in the decay. Finally, the three degrees of freedom describing the spatial

orientation of the decay are irrelevant since we have a spin zero D-meson. Thus, only

two independent degrees of freedom remain from the original twelve.

Since four-momentum is conserved during our decay, we know that:

pμD0 = pμ

K+ + pμK− + pμ

π0 . (1.20)

14

The Minkowski inner products of each side of Equation 1.20 with themselves should

also be equal. Such an operation, along with some rearranging, results in an equation

in terms of four constants and three invariant masses:

m2D0 + m2

K+ + m2K− + m2

π0 = m2K−π0 + m2

K+π0 + m2K+K− , (1.21)

where we define the invariant mass mab for particles a and b as:

mab =

√(pa + pb)μ(pa + pb)μ

c2. (1.22)

It is possible to form three possible two-particle invariant mass-squared variables

in a three-body decay. In our case, these are the mass-squared variables m2K+π0 ,

m2K−π0 , and m2

K+K− shown in Equation 1.21. These three invariant-mass squared

variables can be measured for each D0 → K+K−π0 candidate. Since the left side of

Equation 1.21 is a constant, only two of the invariant mass-squared variables on the

right side of the equation are independent. Hence all unique information about an

event is contained in any one of the three possible pairs of these variables.

A scatter plot of two of the mass-squared variables for a series of events is referred

to as a Dalitz plot [14] and can give us insight into the internal structure of the decay.

For example, suppose D0 → K+R−1 and then R−

1 → K−π0. Then we would see events

along the m2K−π0 axis of the Dalitz plot with the mass of the R−

1 resonance squared,

m2R−

1

. Meanwhile, the profile of the band along the orthogonal variable tells us about

its spin (this will be discussed further in Chapter 2, Section 2.3). A Dalitz plot

contains a wealth of information about the substructure of three-body decays (as well

as being a beautiful example of easily observable quantum-mechanical interference).

When several bands are present, their relative size and their interference in regions

of overlap tell us about the relative complex amplitudes of the resonances. Thus,

15

looking at the Dalitz plot gives us an idea of what resonances the decay may have

gone through and how they interfere.

The kinematically-allowed range of invariant mass-squared values define all of the

possible states (or “phase space”) of the system. Choosing a pair of invariant mass-

squared variables (rather than invariant mass, for example) as our degrees of freedom

is advantageous, as the decay fraction for any point in the phase space defined by

the two chosen invariant mass-squared variables is proportional to the square of the

matrix element [15], “M ” ≡ 〈K+K−π0|M|D0〉, describing the decay amplitude:

dΓ =|M|2

256π3m3D

dm2abdm2

bc, (1.23)

where a, b, and c could be any cyclic permutation of {K+, K−, and π0}. Note that in

the absence of intermediate resonances (which means a featureless matrix element),

scatter plots of any two invariant mass-squared variables will yield uniform distribu-

tions. It is also very advantageous that the relativistic invariance of the invariant

masses means that the Lorentz frame in which they are evaluated is irrelevant.

16

2 Model for Three-body Decays

The general amplitude for a D0 decay to a particle c and a resonance r, where r then

decays to particles a and b, and a, b, and c are pseudo-scalars (spin 0), is [15]:

AD0→(ab)rc =∑

λ

〈ab|rλ〉 Tr(mab) 〈crλ|D0〉

= Zl BD0

L BrL Tr(mab), (2.1)

where the sum is over the helicity states λ of r, L is the orbital angular momentum

between r and c, l is the orbital angular momentum between a and b (the spin of r),

Zl describes the angular distribution of the final state particles, BD0

L and BrL are the

barrier factors for the production of rc and of ab, and Tr is the dynamical function

describing the resonance r. There are multiple choices for the modeling of Zl, BL, and

Tr, which vary by experiment and the required level of detail necessary to describe a

particular three-body decay. Usually the resonances are modeled with a Breit-Wigner

formalism as will be discussed in this chapter. However, some recent analyses have

been done using a K-matrix formalism [16] with the P -vector approximation [17]. The

K-matrix formalism and reasons for considering it are discussed briefly in Appendix

A.

2.1 Breit-Wigner Formalism

The amplitude for a D0 decay to a particle c and a resonance r, where r then decays to

particles a and b, may be derived from the Breit-Wigner propagator and the Feynman

17

D0

c a

b

ab resonanceF

DF

rε

λ ελ

ελ

ε

λ

∗

λ

Σ

Figure 2.1: D0 → abc through an ab resonance. The initial and final states both haveno angular momentum, so the intermediate spin states are summed over.

rules, and is defined as [14]:

AD0→(ab)rc = (pD0 − pc)μ(Breit-Wigner propagator)μν(pa − pb)ν

= (pd − pc)μ

∑λ

εμ∗λ εν

λ

m2r − m2

ab − imrΓ(mab)(pa − pb)ν . (2.2)

This decay is represented in Figure 2.1. The amplitude in Equation 2.2 only

applies to point particles. In order to account for the finite size of our particles, we

add decay form factors at each vertex, FD and Fr:

AD0→(ab)rc = FD(mab)(pd − pc)μ

∑λ

εμ∗λ εν

λ

m2r − m2

ab − imrΓ(mab)(pa − pb)νFr(mab). (2.3)

These form factors are unknown, but in practice they are set to the Blatt-Weisskopf

penetration factors [18] as shown in Table 2.1, where rD0 or rr is the “radial parame-

ter” of the decaying meson (D0 or r, respectively). We choose to have rD0 = 5 hcGeV

for

the D0 and rr = 1.5 hcGeV

for the intermediate resonance. These values are consistent

with previous Dalitz plot analyses [4, 14, 19], and small changes in rD0 and rr were

shown in those analyses to have a very small effect on the fit results.

18

Spin FP (mab)

0 1

1

√1 + zP (mr)1 + zP (mab)

2

√9 + 3zP (mr) + z2

P (mr)9 + 3zP (mab) + z2

P (mab)

Table 2.1: Normalized Blatt-Weisskopf Barrier Factors (FP (mab) = 1 when mab = mr)for the particle P decay vertex. zP is defined as follows: zP (mab) = r2

P p2P (mab).

Γ(mab) in the denominator of Equation 2.3 is the mass-dependent width [20] and

depends on the spin of the resonance:

Γ(J)(mab) = Γrmr

mab

(pr(mab)

pr(mr)

)2J+1

F (J)r

2. (2.4)

FD, Fr, and Γ(mab) are dependent on the breakup momentum of the related

particle D or r at invariant mass mab. It helps to recall the general expression for the

breakup momentum, pP , which is the magnitude of the momentum of either daughter

(masses m1 and m2) from the two-body decay of a particle P (mass m12) in the P

rest frame:

pP =

√(m2

12 − (m1 + m2)2)(m212 − (m1 − m2)2)

4m212

. (2.5)

We can compare Equations 2.1 and 2.3 to see how each of the parts of Equation

19

2.1 is modeled:

Zl = (pd − pc)μ

∑λ

εμ∗λ εν

λ(pa − pb)ν ,

BD0

L = F(J)D (mab),

BrL = F (J)

r (mab),

and Tr(mab) =1

m2r − m2

ab − imrΓ(mab). (2.6)

2.2 Decay Amplitudes for a Particular Spin

Equation 2.3 may be evaluated for resonances having different spins. For a scalar

(spin 0) resonance the result is:

Ascalar =1

m2r − m2

ab − imrΓ(0)(m2ab)

. (2.7)

For a vector (spin 1) resonance, the spin sum in the numerator of Equation 2.3 is

evaluated to [21]:

−gμν +pμ

abpνab

mab

. (2.8)

With this information, we can evaluate Equation 2.3 for a vector resonance as:

Avector =F

(1)D (m2

ab) · F (1)r (m2

ab)

m2r − m2

ab − imrΓ(1)(m2ab)

·[m2

ca − m2bc +

(m2D − m2

c)(m2b − m2

a)

m2ab

]. (2.9)

The spin 2 resonance amplitude is also used in our analysis. A partial derivation

20

is in reference [14]. The result of this derivation is the following amplitude:

Atensor =F

(2)D (m2

ab) · F (2)r (m2

ab)

m2r − m2

ab − imrΓ(2)(m2

ab)·⎡

⎢⎣ −13(m2

ab − 2m2D − 2m2

c +(m2

D−m2c)2

m2ab

)(m2ab − 2m2

a − 2m2b +

(m2a−m2

b)2

m2ab

)

+(m2bc − m2

ca +(m2

D−m2c)(m2

a−m2b)

m2ab

)2

⎤⎥⎦ . (2.10)

Higher spin intermediate states were not necessary for our analysis. The non-

resonant amplitude is simply a constant (it doesn’t change at different points on the

Dalitz plot).

In our analysis, the parametrization of resonances which decay r → bc or r →ca (in m2

bc and m2ca invariant mass squared variables, respectively) are obtained

from Equations 2.7, 2.9, and 2.10 by simple cyclic permutations of the {a, b, c} =

{K+, K−, π0} particles and {m2ab,m

2bc,m

2ca} = {m2

K+K− ,m2K−π0 ,m2

π0K+} variables.

This is an important specification to make. Note that swapping a and b in Equa-

tion 2.9 will result in the amplitude being multiplied by −1 (which is equivalent to

a phase shift of 180◦). Thus, we resolve any ambiguities by specifying that we use

cyclic permutations when defining amplitudes. However, the order of the particles

that we choose to cyclically permute is also important. For example, if we choose

permutations of {K−, K+, π0} instead of {K+, K−, π0} in our definitions of the vec-

tor resonances, then each vector resonance will have a 180◦ phase shift. This would

affect the interference between vector resonances and those contributions which do

not have a phase shift based on particle order (such as the non-resonant contribution).

When comparing Dalitz plot fits from different experiments, it is crucial to account

for differences in phase conventions.

21

D0

K-

J = 1J = 0 J = 0

θπ

0

K+

φ

Figure 2.2: Since the D0 is a spinless particle, a vector (spin 1) decay particle suchas the φ must be in a |1, 0〉 angular momentum state in the D0 rest frame (when wequantize along the π0 direction). The daughters of the φ therefore decay preferentiallyparallel to its momentum (the amplitude has a Y 1

0 ∝ cos(θ) dependence in the D0

rest frame).

2.3 Amplitudes on the Dalitz Plot

Although not obvious from looking at Equation 2.9, this amplitude is proportional

to cos(θ) in the D0 rest frame, where θ is the angle between the momentum vector

of the resonance and the momentum vector of one of the daughters as shown in

Figure 2.2. However, this should not be surprising because the D0 has no spin and

the resonance has spin 1, therefore the angular dependence in the decay amplitude

should be represented by the Y 01 spherical harmonic, which is proportional to cos(θ).

This causes an interesting feature in the Dalitz plot variables orthogonal to the one

in which the resonance decays. Because of the cos(θ) dependence of the amplitude,

we will find less events near the center of the resonant band in the orthogonal Dalitz

plot variable and more events near the edges, as seen in Figure 2.3. Similarly, we

expect scalar (spin 0) resonances to have no preferred angular direction (Y 00 ), and

tensor (spin 2) resonances to have an angular dependence in the decay amplitude

represented by Y 02 ∝ 3 cos2(θ) − 1.

22

0.5

1.5

1.0

1.0

0.5

1.5

mK -

π 02 (GeV 2/c 4) m

K+ π

02 (G

eV2 /c

4 )

θ = 180°

θ = 90°

θ = 0°

mK*

2

Figure 2.3: |A|2 for the vector K−π0 resonance K∗−. When the π0 comes out oppositeto the K∗− direction, it is almost at rest and corresponds to θ = 0◦. The K− is almostat rest when θ = 180◦. When the K− and π0 come out perpendicular to the K∗−

(θ = 90◦), they have equal momentum. We can see the preference for a decay tooccur parallel to the momentum of the K∗− as a cos2(θ) dependence on the Dalitzplot (the square of the cos(θ) amplitude dependence mentioned in Figure 2.2).

23

3 CLEO, CESR, and LEPP

We obtain the data for our analysis using the Cornell Electron Storage Ring (CESR)

and CLEO detector at the Laboratory for Elementary-Particle Physics (LEPP).

LEPP is located on the Cornell University campus in Ithaca, NY. There is one inter-

action region at the south end of CESR which houses the CLEO detector. The CLEO

detector first took data in 1979, and has been upgraded on numerous occasions. For

our analyses, we have used data collected by the CLEO III [22–24] and CLEO-c [25]

versions of the detector.

3.1 LEPP

The main mission of LEPP is to study nature’s fundamental particles and the laws

that govern them. LEPP is also responsible for the development of technology behind

accelerators which are a major tool in learning about particle physics. The CLEO

experiment and CESR are integral to fulfilling LEPP’s charge. In addition to its

role in particle physics, LEPP is also responsible for providing x-rays to scientists via

the Cornell High Energy Synchrotron Source (CHESS), which harnesses synchrotron

radiation from the particles which are stored in CESR. This intense radiation, in the

form of x-rays, is used by surface physicists, medical biologists, and others to study

the microscopic structure of many materials.

24

e-

Transfer Linee+

Transfer Line

LINAC

Converter

CLEO

Synchrotron

CESR

Gun

e+

e-

Figure 3.1: A diagram of CESR.

3.2 CESR

CESR (Figure 3.1) is a symmetric e+e− collider capable of running at center-of-mass

energies between approximately 3 and 11 GeV. While “CESR” can be taken to refer

specifically to the storage ring, it can also refer to the entire apparatus used to create

and accelerate positrons and electrons. The entire apparatus consists mainly of a

linear accelerator (LINAC), synchrotron, and the storage ring. The storage ring

stores both electrons and positrons and has a circumference of 768 m. It is located

underneath the Robison Alumni Fields, the Robert J. Kane Sports Complex, the

Friedman Wrestling Center, and the Kite Hill parking lot.

Electrons are produced from a heated filament in an electron gun, and the elec-

trons are collected in a “prebuncher” which compresses electrons into packets. The

electron packets are accelerated in the LINAC using varying electric fields generated

by radio frequency (RF) cavities. The electrons have an energy of approximately 300

MeV at the end of the 30 m LINAC. About half-way (15 m) down the LINAC, the

electron beam may collide with a movable tungsten target. Collisions with the tung-

25

sten target create showers of many particles which include positrons. The positrons

are separated from the other particles (mostly electrons, x-ray photons, and protons)

and accelerated in the remainder of the LINAC to an energy of about 150 MeV.

Bunches of electrons and positrons from the LINAC are injected in opposite di-

rections into the synchrotron. The synchrotron is a few meters smaller in radius than

the storage ring and is located in the same tunnel. In the synchrotron, the particles

are accelerated by four 3-meter long linear accelerators, and contained by a series

of dipole bending magnets. The magnetic fields are increased as the energy of the

particles is increased in order to continuously contain the particles. After reaching

their final energy, the particles are transferred to the storage ring.

In the storage ring, electrons and positrons are guided by dipole bending magnets

and are focused by a series of quadrupole and sextupole magnets. The beams lose

energy by synchrotron radiation which occurs as charged particles move in a curved

path, so superconducting RF cavities are used to maintain the beam energy, thus

keeping the particles in their orbits. The electrons and positrons are stored in the

same vacuum beam pipe, which means that care must be taken to avoid unwanted col-

lisions. Electrostatic separators are used to bend the paths of electrons and positrons

into trajectories which only intersect in the interaction region. This configuration is

known as a “pretzel”-shaped orbit, although in reality the deviations of the electrons

and positrons are minimal. For low-energy running (center-of-mass energies near 4

GeV), “wiggler” magnets induce synchrotron radiation, but this has a minimal effect

on the beam trajectory.

The electrons and positrons are not continually placed into the storage ring, but

are rather located in “bunches”, which are grouped into “trains”. CESR can be

configured to store up to 9 trains with a maximum of 5 bunches each. The bunches

in the trains are separated by 14 ns, and the trains themselves are separated by 284 ns.

When an electron and a positron collide, they either annihilate, producing a virtual

26

Solenoid Coil BarrelCalorimeter

Drift Chamber

Silicon Vertex Detector / Beampipe

EndcapCalorimeter

IronPolepiece

Barrel MuonChambers

MagnetIron

Rare EarthQuadrupole

SCQuadrupoles

SC QuadrupolePylon

Ring Imaging Cherenkov Detector

Figure 3.2: The CLEO III Detector.

photon which decays into a pair of fermions, or scatter. The possible decay fermions

depend on the beam energy, but are always produced in a matter-antimatter pair.

It is also possible that the initial electron and positron radiate two photons, which

subsequently collide. For our analyses, we are mainly interested in D0’s and D0’s

produced from cc pairs generated in continuum at or near the Υ(4S) center-of-mass

energy (CLEO III) or generated at the ψ(3770) center-of-mass energy (CLEO-c).

3.3 The CLEO III Detector

The CLEO III detector shown in Figure 3.2 consists of (from the beam line outwards)

a four-layer silicon strip vertex detector, a 47-layer wire drift chamber using a helium-

propane gas mixture (60:40), a ring imaging Cherenkov (RICH) detector [26], an

electromagnetic crystal calorimeter (made of cesium iodide doped with thallium), a

superconducting solenoid maintaining a 1.5 T magnetic field parallel to the beam line,

27

Figure 3.3: End view and quarter-section view of the CLEO III Silicon Detector.

and a muon chamber system [22, 25]. For the particles in which we are interested,

charged mesons and π0 mesons, the most interesting parts of the detector are the

tracking system, the RICH, and the calorimeter.

3.3.1 Silicon Vertex Detector

The tracking system, consisting of the silicon vertex detector and the drift chamber,

covers 93% of the 4π solid angle around the interaction region. The silicon vertex

detector (Figure 3.3) provides accurate track position measurements close to the

interaction point in perpendicular distance from beam line (r), azimuthal angle (φ),

and parallel distance along beam line (z) coordinates. It consists of four 300 μm thick

detection layers which circle around the beam line at 2.5, 3.76, 7.0, and 10.1 cm. Each

detection layer is built of 5.0 cm × 2.5 cm silicon detectors. Each individual detector

has silicon strips every 50 μm along r − φ (on the obverse side) and 100 μm along

z (on the reverse side). Each layer consists of a proportionally increasing number

28

circle of closestapproach

14 mm

electronsdrift and multiply

sense wirefield wire

char

ged

part

icle

traj

ecto

ry

electron

Figure 3.4: Charge collection and multiplication in the drift chamber.

of detectors along φ and z (starting with 7 along φ and 3 along z in the 2.5 cm

layer); a total of 447 silicon detectors are used to make the four layers. Precision

information about the direction of the pion in D∗ → Dπ decays allows for a D∗ − D

mass difference resolution as good as 0.19 MeV/c2 [25,27].

3.3.2 Drift Chamber

The drift chamber covers a region in r of 13.2 cm to 82.0 cm from the beamline. The

inner wall of the drift chamber is 2 mm thick expanded acrylic with 20 μm aluminum

skins. The outer radial wall is made up of two layers of 0.8 mm thick aluminum

cylindrical shells. The drift chamber is made up of 9796 drift cells and is filled with a

60% helium, 40% propane gas mixture. At the center of each cell is a 20 μm diameter

gold-plated tungsten sense wire, and the sense wire is surrounded by eight 110 μm

diameter gold-plated aluminum field wires. The cell forms a nearly square shape 14

mm across. A 2100 V potential difference is applied between the sense wire and the

field wires, creating an electric field.

29

back FRONT

a track pointing toward the FRONT of the detectorwire projection

z

stereo anglesexaggerated

stereo wire

s tilted counter−clockwise

stereo wires tilte

d clockwise

axial wires (untilted)

FRONTview

Figure 3.5: Using tilted stereo wires to obtain z information in the drift chamber.Tilted wires are projected as arrows coming out of the page in the r-φ projection;position along a tilted wire indicates the z of the track helix near that wire. Theclosest wires to the track (in three dimensions) are highlighted (in green).

A charged particle will ionize the gas mixture when it passes through a cell (see

Figure 3.4), and the ionized electrons are then attracted to the sense wire. The

electric field near the sense wire is very strong and causes the ionized electrons to

ionize more atoms. This creates a avalanche of electrons on the sense wire. A distance

of closest approach to the sense wire can be calculated based on the transit time of

the electron pulse from the wire, and the timing of collisions in the detector. The

wire which carried the pulse is noted, and is used in concert with information from

other wires to help determine the trajectory of the charged particle.

The drift chamber consists of 16 layers of axial sense wires (parallel to the beam-

line) and 31 sense wire layers which alternate in small stereo angles to provide some

information about the position in z of the tracks. The wires in a stereo layer are

skewed in the φ direction so that the φ coordinate at one end of the wire is offset by

a small angle from the other end (see Figure 3.5). The stereo angle varies from 21 to

30

28 mrad with respect to the beam axis and alternate in each subsequent layer.

The outer radial wall is lined with 1 cm wide cathode pads. The cathode pads

provide a longitudinal position measurement at the outer radius. The cathodes, when

used with the stereo layers, improve the spatial measurement in the z direction. The

combined z resolution is on the order of a couple millimeters.

Overall, the drift chamber has an average position resolution of 100 μm and

momentum resolution of 40 MeV/c at p = 5.3 GeV/c. [25, 27].

3.3.3 dE/dx Particle Identification

The energy loss per unit length, dE/dx, and momentum of a particle are correlated

for different charged particles which may leave a track in the detector. The nature

of this energy loss is the ionization or atomic excitation of moderately relativistic

particles traveling through matter. At the energies of the CLEO experiment, dE/dx

is a function of only the particle’s velocity, as determined by Bethe and Bloch [1].

Together with a measurement of the particle’s momentum, the mass of the particle

can be determined from the velocity and momentum as follows:

m = p

√1

v2− 1

c2. (3.1)

Figure 3.6 shows how different charged hadrons have different correlations between

their momentum and energy loss. Notice that in this energy range dE/dx ∝ 1v2 , and

thus the curve has a relationship to the inverse square of the momentum as well. We

can use our knowledge of these correlations to perform particle identification with the

dE/dx information.

If one had a particular track’s momentum and the dE/dx information it is possible

to determine how many standard deviations away from a particular particle type the

track’s dE/dx information lies. It is clear from Figure 3.6 that the effectiveness of

31

p (GeV/c)

dE/

dx

0

1

2

3

4

5

0 0.25 0.5 0.75 1 1.25 1.5

π

K

p

Figure 3.6: CLEO III dE/dx vs. momentum showing the π, K and p bands. Leptonsare suppressed where possible. The dE/dx scale is chosen to be 1 for electrons andpositrons from e+e− → e+e− events.

32

using dE/dx information to determine type of particle is dependent on momentum.

Areas of overlap in Figure 3.6 are regions where the ability to distinguish particle

type is lost. Figure 3.6 shows that distinguishing kaons and pions with dE/dx starts

to become effective at momenta below approximately 700 MeV/c. If the number of

standard deviations, σ, for a given particle type is 3 or less (in the lower momentum

region where we can distinguish kaons and pions), a positive identification may be

made.

Typically dE/dx information can be used to distinguish two types of particles

(the π and the K for example) by looking at the χ2 difference Δχ2dE/dx:

Δχ2dE/dx = σ2

K − σ2π, (3.2)

where particles with Δχ2dE/dx < 0 are more likely to be kaons than pions.

3.3.4 RICH Detector

The RICH detector (Figure 3.7) is located between the drift chamber and the calorime-

ter and covers 83% of 4π. Charged particles passing through 10 mm thick lithium

fluoride crystals create a ring of Cherenkov photons that expands by traveling through

a nitrogen expansion gap which is 16 cm in length. For the region of the RICH de-

tector which surrounds the interaction region, the interface of the radiating lithium

fluoride crystals and the nitrogen is shaped as a sawtooth rather than flat in order

to prevent total internal reflection of Cherenkov light from incident tracks normal

to the boundary. Further from the interaction area, total internal reflection is not a

problem, and flat crystals are used. After traveling through the expansion gap, the

Cherenkov photons are detected via conversion into photo-electrons by interaction

with a methane-triethylamine mixture. These photo-electrons are multiplied by a

multi-wire proportional chamber mechanism. The position of the photon conversion

33

Methane - TEA

CaF2

Windows

N2 Expansion

Gap

K / π

LiF Radiator

Drif

t Cha

mb

er

θc

Figure 3.7: A section of the RICH detector.

34

is determined by induced charge on cathode pads at the end of the wire chamber [25].

The RICH is an important tool in distinguishing charged kaons (K±) from charged

pions (π±), because the mass of a charged K± or π± can be inferred from the opening

angle of the ring of photons and the particle’s momentum. Cherenkov radiation is

produced by the charged particles in the lithium fluoride crystal at an angle θc to the

trajectory of the track. The angle depends on the velocity of the particle, v, and the

index of refraction, nLiF = 1.5, of the lithium fluoride as follows:

cos θc =c

vnLiF

. (3.3)

Of course, it follows that the minimum velocity that a particle must have in order

to radiate in the detector is approximately 23c. The (rest) mass m of the particle can

then be determined from θc and the measured magnitude of momentum p, since the

relativistic momentum of the particle is given by:

p =mv√1 − v2

c2

. (3.4)

Information from the RICH is used to determine a likelihood for a particular

particle hypothesis. A likelihood L for each particle type is calculated from the

number of photons which are within 5 standard deviations of the expected ring size

for that particle type. An effective χ2 for that particular particle type is -2 ln(L).

Thus, we can separate particles in a familiar way by using:

Δχ2RICH = −2 ln(LK) − (−2 ln(Lπ)), (3.5)

where particles with Δχ2RICH < 0 are more likely to be kaons than pions.

It is important to note that tracks with a higher number of associated RICH

photons will have a higher quality of likelihood determination. This is because the

35

0 1 2 3 4Momentum (GeV)

-400

-200

0

Δχ

2 dE/

dx

0 1 2 3 4Momentum (GeV)

-400

-200

0

(b)

(a)

Δχ

2 RIC

H

Figure 3.8: How true kaons (simulated) would look at various momenta (binned) interms of the range of possible values for Δχ2

dE/dx (Equation 3.2) for dE/dx information

and the range of possible values for Δχ2RICH (Equation 3.5) for RICH information,

as defined in the text. Values of less then zero indicate that the kaon hypothesis ismore likely.

number of photons, Nγ, is determined within a 3 standard deviation band about

the expected radius, even though the likelihood is determined within a 5 standard

deviation band about the expected radius. So it is possible to have likelihoods which

are non-zero, even when Nγ is zero! Typically, choosing tracks with Nγ ≥ 3 is

necessary for reliable particle identification. Typical selection requirements for the

RICH are able to identify 92% of kaons with a pion fake rate of 8%1 [25].

The particle ID capabilities of dE/dx and the RICH fortunately compliment each

other. In the case of K/π separation, RICH information is not good below a mo-

mentum of about 700 MeV/c because the kaons do not radiate in the RICH at lower

1This is for a sample of tracks from D0 → K−π+ decays (produced via D∗+ → D0π+ decays)with momenta in between 0.7 and 2.7 GeV/c.

36

momenta. The dE/dx separation is good below about 700 MeV/c, but is not very

useful above 700 MeV/c. Figure 3.8 shows the ability of each particle identification

system to identify kaons at various momenta.

RICH and dE/dx information may be combined into a single overall chi-squared

difference:

Δχ2 = −2 ln(LK) − (−2 ln(Lπ)) + σ2K − σ2

π, (3.6)

where particles with Δχ2 < 0 are more likely to be kaons.

3.3.5 Crystal Calorimeter

The 7800-crystal thallium-doped cesium iodide electromagnetic calorimeter is used

mainly to help find photons and identify electrons, and covers 93% of the 4π solid

angle. A π0 decays to two photons about 98.8% of the time [1], so when looking for

a π0 we aim to reconstruct it from these two photons. These photons interact with

the calorimeter creating showers of charged particles and additional photons. The

scintillation light from these showers is detected by silicon photodiodes located on the

back of each crystal, and the signal is used to determine the energy of the photon.

The face of each crystal is a 5 by 5 cm square, and the length of each crystal is 30 cm.

The crystals are oriented to point approximately towards the interaction region. The

calorimeter has a mass resolution for π0 → γγ of approximately 6 MeV/c2 depending

on photon energies and locations [25]. The calorimeter is optimized to capture all

of the energy of electrons and photons which interact with it, as the length of each

crystal is approximately 16 radiation lengths.

37

3.3.6 Solenoid Coil

The solenoid coil is a large superconducting magnet that is cooled by liquid helium in

order to produce a strong, uniform magnetic field of 1.5 T. This field causes charged

particles to make curved paths in the detector. The degree and the direction of a

particle’s curvature can be used to help identify the particle. The coil encompasses

all of the detector elements except for the muon detectors.

3.3.7 Muon Detectors

The muon detector system consists of three “superlayers” which contain many plastic

tubes surrounding anode wires. When a charged particle passes through the muon

detector, an electrical signal is generated in a similar manner to those signals gener-

ated in the drift chamber. There are layers of iron which serve to stop most particles

which would otherwise escape the detector. Muons are able to penetrate the iron,

and the depth to which the muon travels helps to identify it. About 85% of the

solid angle is covered by the muon detectors. This analysis does not use any of the

information provided by the muon detector.

3.3.8 Trigger and Data Acquisition Systems

Events are recorded by the Data Acquisition System (DAQ). All events cannot be

recorded because they happen at a rate which is much too fast to record. Also, many

events are relatively uninteresting, and thus not worth recording. A relatively large

amount of time is required to reconstruct the event and write the event information

from the detector to disk, so only events that contain interesting physics are recorded.

The triggering system designed and built by the University of Illinois performs

pattern recognition algorithms which determine which events are good candidates for

recording. After information from the detector becomes available, the trigger chooses

38

1 crystal

Figure 3.9: How the trigger tiles read shower information from the calorimeter.

to store an event based on multiple selection criteria. The trigger uses information

from tracks in the drift chamber and showers in the calorimeter to make its decisions

without fully reconstructing the event. The tracking trigger is further broken down

into axial and stereo components.

The axial layers have 1696 wires and information from each wire is read out

individually. There are 8100 wires in the stereo layers, so in order to get information

more quickly, the wires are read out in groups of 16 wires in 4× 4 wire groups called

“blocks”. Each block is read out to the trigger as a single piece of information, which

represents a wire hit or lack thereof. The information from both the stereo and axial

trigger is correlated into a proto-track, which is fed into the Level 1 (global) trigger.

Also fed into the trigger is information from the calorimeter. The trigger basically

looks for how much energy is deposited in the calorimeter, and where. However, a

low energy shower that is split across multiple crystals may not have enough energy

39

to surpass the threshold for what would constitute a hit in the calorimeter. In order

to overcome this difficulty, the trigger reads out the energy from the calorimeter in

groups of 64 crystals called “tiles”. There is overlap in the tiles, such that a signal

in one crystal will appear in four different tiles, as seen in Figure 3.9. At least one

tile will contain the majority of a shower. A signal in a single crystal will show up in

four tiles, and this is accounted for by the calorimeter trigger.

The Level 1 trigger looks at these tracks and showers and determine if the event is

interesting. The Level 1 Trigger looks for sets of tracks and showers which correspond

to predefined categories. These categories, or “trigger lines”, each look for basic event

properties to determine if the event should be recorded.

If a particular trigger line is satisfied, the name of this line is recorded so that

events can later be sorted into groups, making it easy to access general event categories

of interest for analysis.

Should the event pass the trigger, all of the event information from the detector

is recorded to the DAQ. The DAQ takes about 30 μs to record information about an

event, and can do so at a rate of 500 Hz. A typical data size for a single event is

25 kB and the data transfer bandwidth is 6 MB/s. The average detector deadtime

(time when the detector cannot collect any information) is 3%.

3.4 The CLEO-c Detector

The CLEO-c Detector (Figure 3.10) is basically identical to the CLEO-III detector,

except that the magnetic field is lowered from 1.5T to 1.0T and the silicon vertex

detector is replaced with a small inner wire chamber called the ZD (Figure 3.11).

The purpose of these changes is to help in the detection of particles with significantly

lower momentum, due to a decrease in running energy for the experiment from the

(CLEO-III) energy range of the Υ resonance family to the (CLEO-c) energy range of

40

CsIElectromagnetic

Calorimeter

(End Cap)

CsIElectromagnetic

Calorimeter

Drift ChamberOuter Endplate

Drift Chamber EndplateSmall Radius Section

Drift ChamberInner Skin

cos θ = 0.93

Interaction Point

CESR

ZD Inner DriftChamber

Ring Imaging CherenkovDetector

Figure 3.10: Cross section of the CLEO-c detector in the r-z plane.

41

Figure 3.11: An isometric view of the CLEO-c inner wire chamber.

the J/ψ resonance family.

3.4.1 The ZD

The ZD is a six-layer inner drift chamber which is located between the beam pipe

and the main (CLEO-III) drift chamber. It consists of 300 drift cells and is filled

with a 60% helium, 40% propane gas mixture. At the center of each cell is a sense

wire, and the sense wire is surrounded by eight field wires. The wires are made of the

same materials as those in the main drift chamber. The cell forms a nearly square

shape 10 mm across. A 1900 V potential difference is applied between the sense wire

and the field wires. All of the layers in the ZD are arranged at stereo angles, with

the innermost layer being at a stereo angle of 4.4◦ with respect to the beam axis and

the outermost layer being at a stereo angle of 5.8◦ with respect to the beam axis.

The process of charged particle detection is identical to the process of detection in

the main drift chamber.

The ZD covers the region which is between 4.1 and 11.7 cm from the beamline.

42

The inner wall is made of aluminum which is 1 mm thick, and the outer radial wall

is made of mylar which is 127 μm thick. The resolution in the position of charged

particles along the z direction is 680 μm. The ZD has a momentum resolution, σp

p, of

about 0.4% for charged particles at normal incidence. The ZD is the only source of

z information for a charged particle with transverse momentum less than 67 MeV/c.

3.5 Reconstructing Events

After an event is read out, the next step is to process the data into a form useful

for analysis. This analysis process is performed by an off-line computer code called

“Pass2”, which performs the full reconstruction and fitting of the tracks, and the clus-

tering of calorimeter showers (there is also a code called “Pass1” which is performed

on the data as it is taken, but this is mostly to ensure the quality of the data).

The first step is to determine the set of calibration constants for the events, which

help convert the raw information from the detector into meaningful physical qualities,

and which can also help remove noise from the detector. After this step, higher level

reconstruction occurs. This includes building the tracks and showers, and matching

them. Quantities which are used for identifying tracks, such as dE/dx and RICH

information, are also calculated.

Pass2 also reconstructs short lived particles such as the π0 which decay into two

photons that are only detected in the calorimeter. Showers from the calorimeter are

combined to determine if the showers had the right energy to be a π0. A list of these

particles is created for data analysis.

43

3.6 Monte Carlo Simulation

In order to prevent bias due to selection criteria, examine sources of background, and

determine detector efficiency, particle experimentalists need the ability to create a

simulated data sample, which we call “Monte Carlo” (MC).

Creation of Monte Carlo is done in multiple stages. The first stage, using the

simulation program QQ [28], simulates the e+ - e− collision and primary decays of

the particles at the particular beam energy and initial conditions of the beam. The

Monte Carlo uses a decay file that gives the probability of each possible decay of a

particle, and one of the decay chains is randomly selected based on these probabilities.

The output of QQ is the four-momentum of each daughter particle. The next

simulation program, GEANT [29], takes the QQ particles and runs them through

a simulation of the detector. All of the interactions of the particles in the detector

are simulated, including bremsstrahlung radiation and interactions with material.

GEANT also accounts for resolution effects, detector efficiencies, and noise. GEANT

can be set to use the same calibration constants as any particular set of recorded

events, so that different detector settings may be used to generate MC samples in

order to represent all sets of events.

All of the simulated detector responses are put into a file which looks very much

like the one that is stored to the DAQ for real events, except that it also contains

information from QQ. This means that for simulated data, we can have the informa-

tion about which particles decayed and what they decayed into. With real data we

can only make hypotheses. The simulation can then be run through Pass2, just like

real events, and the MC can be analyzed in the same way as the data.

44

4 D0 → K+K−π0 in CLEO III

The main goal of the analysis which we describe here1 is to measure the strong phase

difference δD and relative amplitude rD between D0 → K∗−K+ and D0 → K∗+K−

decays, which is required for the extraction of γ via the method of Grossman, Ligeti,

and Soffer [10] as discussed in Section 1.6. We are further motivated by a paper by

Rosner and Suprun [30] that points out the sensitivity to δD using D0 → K+K−π0

produced in e+e− → ψ(3770) → D0D0

(CLEO-c), though the analysis presented here

relies on D0 mesons from D∗+ meson decays in e+e− continuum (non-BB) production

at√

s ≈ 10.58 GeV (CLEO-III).

4.1 Current Knowledge

This is the first analysis of the resonant substructures of D0 → K+K−π0 and their

interference. The only published measurement of the branching ratio (decay proba-

bility) was done by the CLEO collaboration in 1996 [31]. The current world average

for the branching ratio (BR) is BR(D0 → K+K−π0) = (0.13 ± 0.04%) [1]. Some

other published branching ratios of D0 decays which decay to our three-body mode

are shown in Table 4.1. For examples of possible Feynman diagrams for this decay,

see Figure 4.1.

1Some of the material in this chapter is our work previously published by the American PhysicalSociety (APS) in their journal, Physical Review D. The journal reference is: C. Cawlfield et al.(CLEO collaboration), Phys. Rev. D 74, 031108(R) (2006). Copyright to the article was transferredto APS. However, the transfer of copyright agreement states: “The author(s) shall have the followingrights:... The right, after publication by APS, to use all or part of the Article without revision ormodification, including the APS-formatted version, in personal compilations or other publicationsof the author’s own works...”

45

(a)

c

u_

s

u

u

u

u

s

_

_

_

D0

W+

K-

π0

K+

Nonresonant D0 K+ K-π0

(b)

c

u_

s

s

u

u_

u

s_

s_

D0

W+

K+

π0

φ

K-

D0 π0

K+ K-

(c)

c

u_

s

u_

u_

D0

W+

K-

K+

π0

u

s_ K*+

D0 K*+ K-

K+ π0

u

u

s_

(d)

c

u_

u_

ss

u

u

s

u_

u_

_

D0

W+

K-K*-

K+

π0

D0 K*- K +

K-π0

Color suppressed φ

g

gg

g

gg

g

g

u_

Figure 4.1: Shown are examples of (a) a non-resonant decay, (b) a neutral intermedi-ate state (φ), (c) a positively charged intermediate state (K∗+), and (d) a negativelycharged intermediate state (K∗−).

Decay Mode BR ReferenceD0 → K∗+K− (0.37 ± 0.08)% [1,32,33]D0 → K∗−K+ (0.20 ± 0.11)% [1,32]

D0 → φπ0 (0.076 ± 0.005)% [34]

Table 4.1: Branching ratios of intermediate D0 modes which may decay to the three-body mode K+K−π0.

46

Three-body decays of D mesons are expected to be dominated by resonant two-

body decays [35–39] and the well established Dalitz plot analysis technique [15] can

be used to explore their relative amplitudes and phases. The CLEO collaboration

has published Dalitz plot analyses for several three-body D0 decays over the past few

years [4,14,40–44] and the work described here closely follows the methods developed

in these previous analyses. The two most similar analyses in terms of technique are

CLEO analyses of D0 → K−π+π0 [14] and D0 → π−π+π0 [4, 40] decays.

4.2 Data Sample

This analysis uses an integrated luminosity2 of L = 9.0 fb−1 of e+e− collisions at

√s ≈ 10.58 GeV provided by the Cornell Electron Storage Ring (CESR). The data

were collected with the CLEO III detector described in Chapter 3.

To suppress backgrounds and to tag the flavor D0(D0), we require our D candidate

to come from a D∗+(cd) or D∗−(cd). The D0 mesons are reconstructed in the decay

sequence D∗+ → π+s D0, where the sign of the slow pion π+

s (π−s ) tags the flavor of the

D0(D0) at the time of its production. We call the charged pion ‘slow’ due to its small

expected momentum based on the relatively small mass difference between D∗+ and

D0.

The detected charged particle tracks must reconstruct to within 5 cm of the in-

teraction point along the beam pipe and within 5 mm perpendicular to the beam

pipe (the typical beam spot is 300 μm in the horizontal dimension, 100 μm in the

vertical dimension, and 10 mm in the longitudinal dimension). The cosine of the

angle between a track and the nominal beam axis must be between −0.9 and 0.9 in

2Instantaneous luminosity Linst. is a measure of the collision rate per unit area at the interactionpoint. The rate of a process with cross section σ is dN

dt = Linst.σ, where N is the number of timesthe process occurs. We may integrate the instantaneous luminosity over a period of time to get theintegrated luminosity L =

∫ Linst.dt. This way we may obtain N for a process by simply multiplyingthe integrated luminosity by the cross section for the process: N = Lσ.

47

order to assure that the particle is in the fiducial volume of the detector. The πs can-

didates are required to have momenta 150 ≤ pπs ≤ 500 MeV/c to remove particles

which are too slow to be properly detected and to remove pions which have too much

momentum to be good slow pion candidates. Kaon candidates are required to have

momenta 200 ≤ pK ≤ 5000 MeV/c.

Candidate kaon tracks that have momenta greater than or equal to 500 MeV/c

are selected based on information from the Ring Imaging Cherenkov (RICH) detector

if at least four photons associated with the track are detected. The pattern of the

Cherenkov photon hits in the RICH detector is fit to both a kaon and a pion hypoth-

esis, each with its own likelihood LK and Lπ. We require (−2 lnLK)− (−2 lnLπ) < 0

for a kaon candidate to be accepted. Candidate kaon tracks without RICH informa-

tion or with momentum below 500 MeV/c are required to have specific energy loss

in the drift chamber within 2.5 standard deviations of that expected for a true kaon

(|σK | < 2.5). Figure 4.2 shows plots of (−2 lnLK) − (−2 lnLπ), number of photons

nγ, and σK for the positive and negative kaons after all other selection criteria have

been applied.

The π0 candidates are reconstructed from all pairs of electromagnetic showers

that are not associated with charged tracks. To reduce the number of fake π0s from

random shower combinations, we require that each shower have an energy greater than

100 MeV and be in the barrel region of the detector. The two-photon invariant mass

is required to be within 2.5 standard deviations of the known π0 mass. To improve

the resolution on the π0 three-momentum, the γγ invariant mass is constrained to

the known π0 mass.

48

-250 -200 -150 -100 -50 0 500

100

200

300

400

500

[(- 2 ln LK) - (-2 ln L

π)]

K+

RICH

-250 -200 -150 -100 -50 0 500

100

200

300

400

[(- 2 ln LK) - (-2 ln L

π)]

K-

RICH

dE/dx

-20 0 20 400

100

200

300

400

500

600

dE/dx

σK-

σK+

-20 0 20 400

100

200

300

400

500

600

0 10 20 30 400

200

400

600RICH

[nγ]

K+

0 10 20 30 400

200

400

600RICH

[nγ]

K-

Figure 4.2: Histograms of (−2 lnLK)−(−2 lnLπ) from RICH information, number ofphotons nγ used to determine the RICH information, and σK , the number of standarddeviations away from the expected dE/dx for a true kaon, for the positive (1st row)and negative (2nd row) kaons after all other selection criteria have been applied.

49

Figure 4.3: Distribution of (a) mK+K−π0 for |ΔM | < 1 MeV/c2 and (b) ΔM for1.84 < mK+K−π0 < 1.89 GeV/c2 after passing all other selection criteria discussedin the text. The solid curves (red) show the results of fits to the mK+K−π0 and ΔMdistributions, respectively. The background level in each plot is shown by a dashedhorizontal line (black). The vertical lines in (a) and the left-most set of vertical linesin (b) denote the signal region. The right-most set of vertical lines in figure (b) denotethe ΔM sideband used for estimation of the background shape.

50

4.3 Event Candidates

We reconstruct the decay chain D∗+ → π+s D0, D0 → K+K−π0 with the requirement

that the D∗+ momentum be at least as large as one-half of its maximum allowed value

(pD∗max =

√E2

beam

c2− M2

D∗+c2 = 4.88 GeV /c). This suppresses large combinatoric

backgrounds and also removes D∗s from B-meson decays (D∗-mesons which come

from B-mesons at rest cannot have a momentum of more than about 2.26 GeV/c).

The D0 candidate invariant mass mK+K−π0 and invariant D∗+ − D0 mass difference

ΔM ≡ mK+K−π0π+s− mK+K−π0 − (mD∗+ − mD0) are calculated for each candidate,

where mD∗+ and mD0 are the world average values of 2010.0±0.4 MeV and 1864.5±0.4

MeV, respectively, taken from Reference [1]. The distributions of mK+K−π0 and ΔM

are shown in Figure 4.3. We fit each of the distributions to the sum of two bifurcated

Gaussians plus a background shape which is constant (for mK+K−π0) or parabolic

(for ΔM). A sum of two bifurcated Gaussians is defined as the sum of two Gaussian

(Normal) distributions with different widths on either side of the mean (μ). The

mean is the same for both Gaussians. The form used for each Gaussian is:

A

2σA

σA+σB

σA

√2π

e−x−μ

2σ2A for x < μ

A

2σB

σA+σB

σB

√2π

e−x−μ

2σ2B for x >= μ, (4.1)

where A, σA, σB, and μ are parameters in the fit for each Gaussian, x is the variable

representing the distribution we are interested in (mK+K−π0 or ΔM), and the ratio of

both of the widths for the two Gaussians is same (σB2

σB1= σA2

σA1). This shape accounts

for the fact that the signal is slightly broader at lower mass due to uncertainty in

reconstructing the π0 momentum. The parabolic background is a simple fit to a 2nd

order function A(x − a)2 + B(x − a) + C, where a is the x offset, and A, B, and C

are the constants which are fitted.

51

The purpose of these fits is to determine the signal fraction, which is the proportion

of signal events (the events we believe to be true D0s) to background events in the

“signal region” where we expect the vast majority of our signal to be. We select a

signal region defined by 1.84 < mK+K−π0 < 1.89 GeV/c2 and |ΔM | < 1 MeV/c2,

which contains 735 D0 → K+K−π0 candidates. We determine the signal fraction by

integrating the signal fit over the signal region, and dividing by the integral of the

combined signal and background fit over the signal region. We do this separately for

the mK+K−π0 and ΔM plots, finding signal fractions of (79.1±5.3)% and (84.6±6.3)%,

respectively. The average signal fraction is then (81.8 ± 6.3 ± 2.8)%, where the

systematic error is half of the difference between the signal fraction from the fits to

mK+K−π0 and ΔM .

4.4 The Dalitz Plot

We expect CP violation in D decay to be negligible and assume the amplitudes for

D0 → K∗−K+ and D0 → K∗+K− are equal to the amplitudes for charge-conjugated

modes D0 → K∗+K− and D

0 → K∗−K+, respectively. This allows us to double

our statistics in a single Dalitz plot by combining flavor-tagged D0 → K+K−π0 and

D0 → K+K−π0 candidates and choosing the m2

K−π0 variable for one to be the m2K+π0

variable for the other (and vice versa). The inclusion of charge conjugate modes is

implied throughout this chapter.

Figure 4.4(a) shows the Dalitz plot distribution (a scatter plot of m2K+π0 vs.

m2K−π0) for the D0 → K+K−π0 candidates satisfying the requirements described

in Sections 4.2 and 4.3. Charge conjugation is implied, so D0 → K+K−π0 candidates

are on this plot as well, but they are plotted as m2K−π0 vs. m2

K+π0 and superimposed

with the D0 candidates. The density of events is clearly not uniform across the Dalitz

plot, indicating that there is physics which dictates the probability density function.

52

Figure 4.4: (a) The Dalitz plot distribution for D0 → K+K−π0 candidates. (b)-(d)Projections onto the m2

K+π0 , m2K−π0 , and m2

K+K− axes of the results of Fit 1 (discussedin Section 4.10) showing both the fit (curve) and the binned data sample.

53

0.5

1

1.5 0.5

1

1.50

500

1000

0.5

1

1.5

0.5

1

1.5 0.5

1

1.50

500

1000

0.5

1

1.5

0.5

1

1.5 0.5

1

1.50

500

1000

0.5

1

1.5

0.5

1

1.5 0.5

1

1.50

500

1000

0.5

1

1.5

δNR

= 0° δNR

= 90°

δNR

= 180° δNR

= 270°

mK -

π 02 (GeV 2/c 4) m

K+ π

02 (G

eV2 /c

4 )

mK*

2m K*

2 mK -

π 02 (GeV 2/c 4) m

K+ π

02 (G

eV2 /c

4 )

mK*

2

m K*2

mK -

π 02 (GeV 2/c 4) m

K+ π

02 (G

eV2 /c

4 )

mK*

2

m K*2 m

K -π 0

2 (GeV 2/c 4) mK

+ π0

2 (GeV

2 /c4 )

mK*

2

m K*2

Figure 4.5: An example of the effect of a non-resonant (NR) contribution interferingwith two K∗ resonances. The K∗s in this example have the same amplitude but arelative phase with each other such that the two resonances have maximal destructiveinterference with each other, and the amplitude aNR is set to be 5aK∗ . When therelative phase of the NR contribution to the K∗+ phase, δNR, is 0◦ or 180◦ thereis not a noticeable effect on the K∗ lobes. However, a phase δNR = 90◦ makes theK∗− enhanced (depleted) at low (high) K+π0 invariant mass squared, and the K∗+

enhanced (depleted) at high (low) K−π0 invariant mass squared. A phase δNR = 270◦

makes the K∗+ enhanced (depleted) at low (high) K−π0 invariant mass squared, andthe K∗− enhanced (depleted) at high (low) K+π0 invariant mass squared.

54

The enhanced bands perpendicular to the m2K−π0 and m2

K+π0 axes at an invariant

mass-squared of m2Kπ ≈ 0.8 GeV2/c4 correspond to K∗(892)− and K∗(892)+ reso-

nances, respectively. The φ(1020) can be seen as a diagonal band along the upper

right edge of the plot. Recall the reason why the φ resonance appears this way on

the m2K+π0 vs. m2

K−π0 plot: Due to energy and momentum conservation, m2K+K−

is a function of m2K+π0 and m2

K−π0 . The vector (spin 1) nature of these resonances

is evident from the depleted region in the middle of each band, which comes from

the cos(θ) dependence discussed in Section 2.3. The nearly missing bottom lobe of

the K∗(892)− band and the enhanced left lobe of the K∗(892)+ band hint that these

resonances are interfering with opposite phases with an S-wave amplitude (such as

the non-resonant contribution) under these resonances. An example of how the non-

resonant contribution can affect the K∗ lobes is shown in Figure 4.5.

The major resonances can be more easily be seen in the projections of the Dalitz

plots (Figure 4.4(b)-(d)), as peaks in the two-particle invariant masses. There are

multiple peaks in the projection plots, but some are simply reflections of other reso-

nances. To visualize both peaks that are resonances in a certain projection and peaks

that are reflections, we can arrange the projections as in Figure 4.6.

4.5 The Matrix Element

In most Dalitz plot analyses, choosing the form of the matrix element M is the most

difficult part of the whole analysis procedure. The isobar model, a coherent sum of

Breit-Wigner amplitudes, is the simplest approach, but is known to only be reliable

in cases where the resonances being modeled are fairly narrow, far from threshold,

and well separated. This should be a reasonable approximation for the dominant

K∗(892) and φ contributions in this analysis, but perhaps not as reasonable for other

resonant components. For the purposes of this analysis the isobar model is sufficient,

55

K*+

K*-

φ

Figure 4.6: Dalitz plot projections properly rotated to show how they are obtainedfrom the Dalitz plot. The thicker straight lines correspond to resonances which decayto the two particles in the respective invariant mass-squared projection, and thethinner straight lines correspond to reflections which do not decay to the two particlesin the respective invariant mass-squared projection.

56

Resonance r mr (GeV/c2) Γr (GeV/c2) SpinK∗(892)± 0.8917 0.0508 1φ(1020) 1.0190 0.0043 1

non-resonant uniform uniforma0(980) 0.9910 0.0690 0f0(1370) 1.3500 0.2650 0

K∗0(1430)± 1.4120 0.2940 0

K∗2(1430)± 1.4260 0.0985 2f0(1500) 1.5070 0.1090 0f ′

2(1525) 1.5250 0.0730 2κ± 0.8780 0.4990 0

Table 4.2: The masses and widths of resonances r considered in this analysis [1,45–48].

but other models do exist and have been tested in this analysis (see Appendix A).

We express the complex amplitude for a D0 decay to the jth quasi-two-body state

as ajeiδjB(k)

j , where aj is real and positive and B(k)j is the Breit-Wigner amplitude for

resonance j with spin k of Equation 2.3 described in Chapter 2. The amplitudes B(k)j

have no particular normalization.

We consider thirteen resonant components (see Table 4.2) as well as a uniform

non-resonant contribution described by the simple complex amplitude aNReiδNR . The

matrix element M is most generally written as a sum over the individual resonant

amplitudes and the non-resonant contribution:

M =∑

j

ajeiδjB(k)

j + aNReiδNR . (4.2)

4.6 Method of Maximum Likelihood

Once we have a set of D0 → K+K−π0 candidates, we want to match these data with

a given theoretical model which constitutes M. The most useful general method

of choosing the free parameters in a given model of M is the method of maximum

likelihood [1].

57

The basic idea of the method of maximum likelihood is that the best model is

consistent with the observation. We begin by assuming that our event distribution

came from the probability distribution function, L, whose form is what we expect from

theory, and L has parameters whose values are unknown. The method of maximum

likelihood forms a likelihood which is the product over all of our events of the value

of L for each event. By finding an optimal set of parameters, we seek to maximize

this likelihood. This effectively determines the parameters for which the data is most

likely.

We can explain why the method of maximum likelihood works. If the observed

events do not match where our model expects the events to be, L at those events

will be very low, and L at each of the events will multiply to a small number. If

the observed events are where our model expects them to be, then for each point

the probability distribution function will be high, and L at each of the events will

multiply to a large number. In order to find the best model based on the physics

which we believe dictates the probability distribution function, we adjust the free

parameters until we get the highest value of∏

events

L.

For our analysis, we can form a probability distribution function which has as

its parameters the amplitudes and phases of the various resonances and non-resonant

contribution described in Section 4.5, and the signal fraction f as described in Section

4.3:

L = fE(y, z)|M|2∫

E(y, z)|M|2 dDP + (1 − f )B(y , z )∫

B(y , z ) dDP , (4.3)

where M is the matrix element “model” for the decay, E is a polynomial representing

the efficiency shape, and B is a polynomial representing the background shape. E

and B are described in Section 4.7. M, E, and B are all functions of the two Dalitz

plot variables chosen for fitting. (1 − f) is the fraction of event candidates that are

58

believed to be background. In the probability distribution function L, the efficiency-

corrected matrix element squared and the background shape are each normalized over

the Dalitz plot “area” (DP) defined by the kinematic limits of the D0 decay to the two

Dalitz plot variables. These normalizations ensure that L represents what we believe

to be probability distribution function; the integral of the probability distribution

function over DP is 1, as expected.

The likelihood,∏

events

L, tells us how likely it is that our data on the Dalitz plot fit

into a model with a particular set of parameters. We use a MINUIT [49] based fitter

to minimize the negative logarithmic likelihood function L:

L =∑

events

−2 ln L, (4.4)

which is effectively the same as maximizing∏

events

L. The efficiency and background

shapes are determined as described in Section 4.7. We allow our signal fraction f to

float in the fits, unless otherwise noted.

Note that one can multiply all of the pieces of the matrix element by a complex

constant (i.e. an amplitude and phase) and the probability distribution function

would be exactly the same. Thus, we can only fit for the shape of |M|2, not its

absolute value.

Since Dalitz plot analyses are only sensitive to relative phases and amplitudes, we

may arbitrarily define the amplitude and phase for one of the two-body decay modes.

The mode with the largest rate, K∗+K−, is assigned an amplitude aK∗+K− = 1

and phase δK∗+K− = 0◦. Measuring the remaining amplitudes and relative phases

of the resonances, which will be free parameters in our fit, thus gives us a detailed

understanding of the substructure of this decay.

At this point, we need to choose the invariant-mass squared variables we will use

to parameterize our Dalitz plot. We choose to fit the Dalitz plot in (m2K+π0 , m2

K+K−)

59

rather than (m2K−π0 , m2

K+π0) or (m2K+K− , m2

K−π0). m2K+K− is chosen because the

momentum resolution of tracks is better than the momentum resolution of π0s. m2K+π0

is chosen because the resonance with the largest rate, K∗+, decays to K+π0.

4.7 Efficiency and Background

4.7.1 Efficiency

The kinematically-allowed range of invariant mass-squared values define all of the

possible states (or “phase space”) of the system. The efficiency of our ability to

reconstruct a D0 → K+K−π0 event is not constant over the entire phase space.

Since a fit to the Dalitz plot will yield only relative values for amplitudes, only the

shape of the efficiency across the Dalitz plot is important, not its absolute value. We

study efficiency by generating a large Monte Carlo sample of D0 → K+K−π0 signal

events, which are equally likely to decay at any point in the allowed phase space, and

running these events through the same analysis code that is used on data. The shape

of the remaining event distribution is a direct measure of the efficiency.

A 2-D cubic polynomial is used to fit the shape of the efficiency. We only need to

fit to two of the three possible invariant mass combinations x ≡ m2K−π0 , y ≡ m2

K+π0 ,

and z ≡ m2K+K− , and choose to fit to y and z. The form of the efficiency function

E(y, z) is given by:

E(y, z) = E0 + Eyy + Ezz

+ Eyyy2 + Eyzyz + Ezzz

2

+ Eyyyy3 + Eyyzy

2z + Eyzzyz2 + Ezzzz3. (4.5)

Table 4.3 shows the best-fit parameters for the efficiency shape. Figure 4.7 shows

the distribution of events used to determine the efficiency across the Dalitz plot as

60

E0 Ey Ez Eyy Eyz

1 (fixed) -3598 ± 1027 977.4 ± 488.4 3459 ± 1053 2686 ± 995.3

Ezz Eyyy Eyyz Eyzz Ezzz

-244.7 ± 298.7 -755.3 ± 283.2 -1184 ± 372.0 -423.5 ± 217.2 -18.85 ± 65.54

Table 4.3: Best-fit parameters for the efficiency shape.

well as the result of our fit to these events.

4.7.2 Background

Figure 4.3 shows that background D0 → K+K−π0 candidates are clearly present even

after all selection requirements are imposed. Since there is no way to sort individual

events into “signal” and “background” categories, we must try to understand the

fraction of these events and their collective shape on the Dalitz plot since this will

allow us to take backgrounds into account when extracting physics information from

our Dalitz plot fit.

Our approach is to select events in a ΔM sideband region (the “Background

Region”), which are known to be almost entirely background, and use them to model

the shape of background events in the signal region. Figure 4.8(a) is a scatter plot

of the D∗+ − D0 mass difference versus D0 mass for D0 → K+K−π0 candidates

satisfying all other selection requirements. The lower rectangular region represents

the “Signal Region” and hence the events that form the data sample we use for

our Dalitz plot analysis. In other words, each point inside the lower rectangle of

Figure 4.8(a) corresponds to a point on the Dalitz plot shown in Figure 4.4(a). The

top rectangular region in Figure 4.8(a) contains very little signal and is used to model

the shape of the background across the Dalitz plot and Figure 4.8(b) is a Dalitz plot of

the events in this region. The background events are fitted to extract the background

shape parameters. The signal and background regions have the same extent along

the D0 mass axis, and the extent of the background sideband along the ΔM axis is

61

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )

0.7

1.1

1.5

1.9

0.7

1.1

1.5

1.9

mK+π0

2 (GeV

2/c

4)

mK-π

02 (GeV

2 /c4 )(a

)

(d)

(b)

(c)

Events per 0.03 GeV2/c

4

Fig

ure

4.7:

(a)

Sca

tter

plo

tof

Mon

teC

arlo

even

tsuse

dto

study

the

effici

ency

acro

ssth

eD

alit

zplo

tan

d(b

-d)

pro

ject

ions

ofth

ese

even

tsal

ong

wit

hth

ere

sult

ofa

fit

usi

ng

the

effici

ency

funct

ion.

62

0.7 1.1 1.5 1.9

0.7

1.1

1.5

1.9

1.7 1.8 1.9 2.0-10

-5

0

5

10

15

mK-

π02 (GeV2/c4)

mK

+π

02 (GeV

2 /c4 )

mK+K-

π0 (GeV/c2)

ΔM

(MeV

/c2 )

(a) (b)

mD0

Figure 4.8: (a) Scatter plot of ΔM versus mK+K−π0 for event candidates satisfyingall other requirements. The black boxed area (the bottom box) is the Signal Regionwhich contains our signal candidates, and the red boxed area (the top box) is theBackground Region which contains events which we use to estimate the shape of ourbackground. (b) a Dalitz plot of the events from the Background Region of plot (a).

defined to be 0.003 GeV /c2 < ΔM < 0.010 GeV /c2.

Note that this technique assumes that the background events in both the ΔM

signal and ΔM sideband regions have the same distribution across the Dalitz plot.

This is a good assumption since ΔM depends predominantly on our measurement of

the slow pion from the D∗ decay and only weakly on the D0 decay itself.

In order to model the background, we use a 2-D cubic polynomial plus terms

to represent particles which we expect to be in the background. The form of the

63

background function B(y, z) is given below:

B(y, z) = B0 + Byy + Bzz

+ Byyy2 + Byzyz + Bzzz

2

+ Byyyy3 + Byyzy

2z + Byzzyz2 + Bzzzz3

+ BK∗+ |A(K∗+)|2

+ Bφ|A(φ)|2

+ BK∗−|A(K∗−)|2. (4.6)

The last three terms in B represent K∗+, K∗−, and φ particles which are present in

the background but are not from D0 → K+K−π0 decays. Note that these amplitudes

do not interfere.

B0 By Bz Byy Byz

1 (fixed) 20.58 ± 64.32 11.80 ± 35.89 92.65 ± 60.44 -89.03 ± 55.36

Bzz Byyy Byyz Byzz Bzzz

9.596 ± 25.08 -43.19 ± 23.56 18.84 ± 17.76 8.572 ± 11.94 -1.482 ± 4.688

BK∗+ Bφ BK∗−

.02742 ± .02200 .03254 ± .02597 .01933 ± .01939

Table 4.4: Best-fit parameters for the background shape.

Table 4.4 shows the best-fit parameters for the background shape. Figure 4.9

shows the projected background events from the ΔM sideband and the result of

fitting the background Dalitz plot with the background function.

4.8 Overview of Fitting Technique

In the case of our D0 → K+K−π0 analysis we can guess what the main quasi two-body

contributions are just by examining the Dalitz plot in Figure 4.4(a). We can see clear

enhancements that correspond to D0 → K+K∗−, D0 → K−K∗+ and D0 → φπ0. Any

64

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.9:

The

pro

ject

edbac

kgr

ound

even

tsfr

omth

eΔ

Msi

deb

and

and

the

resu

ltof

fitt

ing

the

bac

kgr

ound

Dal

itz

plo

tto

a2-

Dcu

bic

pol

ynom

ialplu

snon

-inte

rfer

ing

K∗

and

φco

ntr

ibuti

ons

(lin

e).

65

matrix element we hypothesize for the D0 → K+K−π0 decay must include these three

resonances. However, there may be additional broad resonances present which are

not clearly visible in Figure 4.4(a). It is precisely this, the inclusion and parametriza-

tion of broad components that are difficult to distinguish from non-resonant decays,

efficiency variations, and/or backgrounds, that is the main challenge in most Dalitz

analyses.

The main goal of this analysis is to measure the strong phase difference δD and

relative amplitude rD. In terms of the amplitudes and phases in the matrix element,

δD and rD are defined by the following:

rDeiδD =aK∗−K+

aK∗+K−ei(δK∗−K+−δK∗+K− ), (4.7)

where rD in Equation 4.7 is defined as real and positive. The strong phase difference

is equivalent to the overall phase difference for these decays due to our assumption

that CP violation in D decays is negligible. δD is the difference in phase between the

D0 → K+K∗− and D0 → K−K∗+ components on this Dalitz plot, and can therefore

be extracted directly from a fit to these data.

Measuring δD and rD is in principle very simple, however in practice it is compli-

cated by a couple of things. One issue is that the width of the φ is comparable to

our resolution. thus in order to model the φ properly we considered the possibility of

convoluting this resonance with our invariant mass squared resolution in our model.

However, we believe that the width of the φ is close enough to our resolution so that

we may use the documented width of the φ in our analysis and choose to evaluate the

systematic uncertainty on rD and δD as a result of this choice. Also, the uncertainty

in modeling possible broad resonant contributions, especially resonant contributions

in regions of the Dalitz plot which affect δD and rD, may affect our results. We will

approach this problem by adding various additional resonant components which seem

66

0.5

1

1.5 0.5

1

1.50

500

1000

0.5

1

1.5

δD

= 0° δD

= 180°

mK -

π 02 (GeV 2/c 4) m

K+ π

02 (G

eV2 /c

4 )

mK*

2

m K*2 m

K -π 0

2 (GeV 2/c 4) mK

+ π0

2 (GeV

2 /c4 )

mK*

2

m K*2

0.5

1

1.5 0.5

1

1.50

500

1000

0.5

1

1.5

Figure 4.10: Interfering K∗+ and K∗− resonances on the Dalitz plot. The way wehave defined our amplitudes results in a phase convention where maximal destructiveinterference between the K∗s occurs at δD = 0◦. Maximal constructive interferenceoccurs when δD = 180◦. Note that the effect of the interference is most noticeable inthe region where the K∗ resonances overlap.

reasonable and investigating their effect on rD and δD.

It is important to note that our sign convention for our amplitudes implies that

δD ≡ δK∗−K+ − δK∗+K− = 0◦ indicates maximal destructive interference between the

K∗ amplitudes (see Figure 4.10). This is due to our choice of defining amplitudes

which decay to different resonant two-body modes by simple cyclic permutations, as

discussed in Section 2.2.

The unbinned maximum likelihood fitter used was originally developed by Tim

Bergfeld for his study of D0 → K−π+π0 [14] and has subsequently been modified

by many others doing CLEO Dalitz plot analyses. Recent modifications include the

capability to fit only parts of the Dalitz plot (see Subsection 4.10.5) and the ability

to implement the K-matrix formalism (see Appendix A).

The fitter excludes events which are not within the kinematic limits of the D0 →K+K−π0 decay. 24 of the 735 events on the Dalitz plot fall outside these limits in

the (m2K+π0 , m2

K+K−) plane and are excluded, leaving 709 events which are fitted. An

event may be excluded because it is background (the tracks and showers reconstruct to

a particle with the D0 mass, but it is not a D0 → K+K−π0 decay because conservation

67

of momentum would not allow it) or because it is signal and the candidate falls just

outside the kinematic boundary due to limits in how well we can measure the decay.

The efficiency and background functions, E and B, respectively, are those de-

scribed in Section 4.7, the signal fraction f (see Equation 4.4) is allowed to float in

the fit, and the fits were done in the (m2K+π0 , m2

K+K−) plane, unless otherwise stated.

In our fitter, for resonances that decay to the third two-particle combination, K− and

π0, m2K−π0 is calculated from the two mass-squared variables that are used in the fit.

In order to get some idea of how much of each type of resonant three-body decay

is in each particular fit, the fit fraction (FF ) for the resonant component j of a given

fit may be calculated as follows:

FFj =

∫ ∣∣ajeiθjBj

∣∣2dDP∫ |M|2dDP , (4.8)

where we may substitute “1” for “Bj” when calculating the fit fraction for the non-

resonant contribution.

We don’t get the exact fraction of how much of each resonance is in the fit,

since the amplitude squared in the numerator of Equation 4.8 only tells us what we

would expect if that amplitude was not interfering with other resonant amplitudes.

Therefore, the sum over all the fit fractions j, known as the total fit fraction, does

not necessarily have to add up to 100%. A total fit fraction of greater than 100%

implies overall destructive interference (the denominator of Equation 4.8 is small),

while a total fit fraction of less than 100% implies overall constructive interference

(the denominator of Equation 4.8 is large).

68

4.9 Goodness of Fit

Judging the quality of a Dalitz plot fit can be difficult. We may make some qualitative

judgements by simply looking at our fits, although the best way to do this isn’t

obvious. There are two quantitative methods that we use. The first is a very natural

extension of the fitting method. Since we are minimizing the summed log likelihood

(see Equation 4.4), we can simply look at the negative logarithmic likelihood function

(L) and see how it differs between fits. L does not in itself give us a very intuitive

indication of the quality of the fit. However, fits which have a significantly higher

value of L for the same number of events are considered to be inferior. That said,

it is important to be wary of fits which have a significantly lower value of L because

our model has too many parameters.

Another measure of goodness-of-fit is the Significance Level (S.L.) which is calcu-

lated for each fit which used the entire Dalitz plot:

S. L. =1

2

(1 − erf

(L − < L >

σ<L>

√2

)), (4.9)

where L is the negative logarithmic likelihood function, < L > is the expected mean

for the negative logarithmic likelihood function assuming the events are truly dis-

tributed according to the likelihood function which gives the best fit, and σ<L> is

the standard deviation associated with < L >. The procedure is detailed in Refer-

ences [14, 50]. It is important to note that the S.L. only gives a measurement of the

goodness of fit assuming the fit function correctly describes the distribution.

There are other considerations which can be very useful in determining the via-

bility of a fit. If there are too many resonances in a fit, it is likely that the fit fraction

of some of those resonances will be insignificant. If a particular fit has insignificant

resonances, it is highly likely that the resonance content of the fit does not represent

what is in our data (at least to the precision which we can measure). Also, if the sig-

69

nal fraction that the fit prefers differs substantially from the measured signal fraction

(as measured in Section 4.3), it is likely that the fit is not accurate.

4.10 Fit Results

The strategy we used when fitting the data was to start with the minimum sensible

number of Dalitz plot components (in our case: K∗+, K∗−, φ, and a “non-resonant”

contribution, all of which are allowed to interfere with each other) and then observe

the effects of excluding the non-resonant part and of including additional resonances

to the fit. Table 4.2 gives a list of the resonances that were tried. The first three rows

represent the components of the “nominal” fit and the rest are various additional

resonances.

The fit projections for each fit are shown in Figures 4.11 through 4.32, 4.34 through

4.36, and 4.38 through 4.45, and a descriptive summary of all fits is given in Table 4.5.

The first thirteen fits and the last fit were done using data across the entire Dalitz

plot, and the other fits were partial fits that excluded specific regions of the Dalitz

plot (reasons for which are discussed in Subsection 4.10.5).

4.10.1 Nominal Fit

Figure 4.11 (Fit 1) shows the results of the simplest possible reasonable fit to the

data, the nominal fit. In this fit the matrix element M is composed of the three

obvious resonances seen in Figure 4.4(a) (K∗+, K∗−, and φ) plus a uniform non-

resonant (NR) component. Notice that these provide a very good fit to the data. A

measurement of δD = 332◦ (or equivalently, −28◦) indicates destructive interference

between the K∗s.

Three-body decays of D mesons are expected to be dominated by resonant two-

body decays [35–39], so it is worthwhile to see what the fit looks like without having

70

Fit

Res

onan

ces

use

dR

egio

nL

S.L

.f

(%)

r Dδ D

φK

∗±N

Rκ±

K∗± 0

a0

f 0K

∗± 2f 2

1x

xx

Who

leD

P−9

7.4

18.6

%85

.9±

5.6

0.52

±0.

0533

2◦±

8◦

2x

xW

hole

DP

−27.

015

.4%

56.8

±3.

40.

60±

0.06

171◦

±28

◦

3x

xx

Who

leD

P−1

10.9

17.2

%80

.1±

4.7

0.52

±0.

0531

3◦±

9◦

3ax

xx

xW

hole

DP

−117

.919

.9%

86.2

±5.

50.

50±

0.06

318◦

±9◦

4x

xx

xW

hole

DP

−122

.322

.8%

86.3

±5.

40.

43±

0.07

327◦

±12

◦

5x

xx

xW

hole

DP

−112

.627

.5%

84.9

±5.

30.

46±

0.07

314◦

±12

◦

6x

xx

Who

leD

P−8

0.6

28.0

%78

.9±

5.4

0.35

±0.

0514

◦±

16◦

7x

xx

1370

Who

leD

P−6

7.6

17.5

%65

.4±

4.4

0.53

±0.

0624

2◦±

19◦

8x

xx

x1370

Who

leD

P−1

03.9

26.0

%82

.2±

5.2

0.35

±0.

088◦

±27

◦

9x

xx

x1370

xW

hole

DP

−113

.429

.1%

82.5

±4.

40.

30±

0.06

45◦±

34◦

10x

xx

xx

1370

xW

hole

DP

−146

.636

.0%

86.3

±5.

70.

44±

0.09

353◦

±18

◦

11x

xx

1500

Who

leD

P−1

01.4

17.7

%86

.1±

5.9

0.55

±0.

0532

4◦±

10◦

12x

xx

xW

hole

DP

−98.

421

.5%

84.7

±6.

10.

53±

0.05

332◦

±8◦

1xx

xK

∗ba

nds

825.

5N

/A87

.7±

5.6

0.50

±0.

0533

3◦±

14◦

2xx

K∗

band

s88

5.1

N/A

61.3

±6.

30.

68±

0.07

290◦

±27

◦

3xx

xK

∗ba

nds

829.

1N

/A84

.6±

6.1

0.57

±0.

0630

4◦±

17◦

1nx

xN

oφ

1.05

331.

2N

/A80

.5±

6.2

0.52

±0.

0533

7◦±

9◦

1np

xx

xN

oφ

1.05

321.

6N

/A84

.1±

6.3

0.50

±0.

0533

4◦±

9◦

1na

xx

No

φ1.

1522

0.6

N/A

84.9

±5.

90.

50±

0.05

343◦

±9◦

1nap

xx

xN

oφ

1.15

215.

8N

/A84

.9±

6.0

0.48

±0.

0533

8◦±

11◦

1nb

xx

No

φ1.

2612

3.5

N/A

83.2

±6.

10.

50±

0.05

338◦

±10

◦

1nbp

xx

xN

oφ

1.26

122.

1N

/A83

.1±

6.4

0.50

±0.

0633

1◦±

12◦

1sx

x“L

”ar

ea33

1.2

N/A

82.1

±6.

20.

50±

0.05

334◦

±10

◦

3fx

xx

Who

leD

P−1

18.3

16.2

%79

.7±

5.3

0.53

±0.

0530

6◦±

11◦

Tab

le4.

5:Sum

mar

yof

fits

.See

Tab

le4.

2fo

rm

ore

info

rmat

ion

abou

tea

chre

sonan

ce,an

dse

eth

efo

llow

ing

figu

res

for

det

ails

.

71

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.11

:D

alit

zplo

tpro

ject

ions

for

Fit

1sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

72

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.12

:D

alit

zplo

tpro

ject

ions

for

Fit

2sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

73

a non-resonant contribution. Figure 4.12 (Fit 2) shows what happens when we simply

remove the non-resonant (NR) component from Fit 1. While the fit may look visually

acceptable, the fit wants the signal fraction, f , to be significantly smaller (≈ 57%)

than the value measured in Section 4.3 (≈ 82%). L, the negative logarithmic likeli-

hood function, is about 70 higher than for the nominal fit.

4.10.2 κ Fit

Removing the non-resonant contribution had a significant negative effect on the nom-

inal fit, hence we expect there to be a large, broad S-wave component in the region

of low Kπ mass in order to create the type of interference observed on our Dalitz

plot. Having a non-resonant contribution certainly provides this S-wave component,

but it is also possible to have a large low Kπ mass S-wave if there is a broad scalar

resonance which decays to Kπ. While such a resonance has not been published in

the Review of Particle Physics [1], there is some evidence for a low-mass broad scalar

resonance which decays to Kπ called the κ [45]. Since we expect three-body decays of

D mesons to be dominated by resonant two-body decays, the κ is worth considering

in our fit as an alternative to the non-resonant contribution.

Figure 4.13 (Fit 3) shows what happens when we replace the non-resonant (NR)

component with a κ+ and a κ−. The fit remains good and rD is unaffected, but the

phase difference δD changes from the nominal fit, although the change is not drastic.

The signal fraction from the fit is once again consistent with our measured signal

fraction.

4.10.3 Results of Nominal Fit and κ Fit

Table 4.6 contains the results for our nominal fit with statistical errors, including

calculated fit fractions. The signal fraction from the fit is (86 ± 6)%. Since the K∗+

74

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.13

:D

alit

zplo

tpro

ject

ions

for

Fit

3sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

75

Resonance Amplitude Phase Fit FractionK∗+ 1.00 (fixed) 0◦ (fixed) (46.1 ± 3.1)%K∗− 0.52 ± 0.05 332◦ ± 8◦ (12.3 ± 2.2)%φ 0.64 ± 0.04 326◦ ± 9◦ (14.9 ± 1.6)%

Non-Resonant 5.62 ± 0.45 220◦ ± 5◦ (36.0 ± 3.7)%Total Fit Fraction: (109.3 ± 5.5)%

Table 4.6: Amplitude, phase, and fit fraction results for our nominal fit (Fit 1) withstatistical errors.

amplitude and phase were fixed, we require an independent method to determine the

statistical errors on the fit fractions. We use the full covariance matrix from the fitter

and final parameter values to generate 5000 sample parameter sets. For each set, the

fit fractions are calculated and recorded in histograms. These histograms are fit with

a Gaussian to extract their width, which is used as the statistical error on the fit

fraction.

The signal and background in the Dalitz plot projections of the nominal fit (Fit

1) is shown in Figure 4.14. A graphical version of how much of each resonance is in

the total signal for Fit 1 is shown in the projections in Figures 4.15, 4.16, and 4.17.

Note that the amounts of each individual resonance are those from the numerator

of the fit fraction in Equation 4.8, and so the sum of the areas of each resonance in

Figures 4.15, 4.16, and 4.17 do not add up to the area of the signal. The total fit

fraction of more than 100% for Fit 1 implies overall destructive interference.

Fit 3, in which we replaced the non-resonant amplitude with a κ+ and a κ−

amplitude, is also a reasonable match to the data. While we are not certain with

what seriousness the κ resonances should be considered, we present results for the fit

with the κ for the purpose of review.

Table 4.7 contains the results for Fit 3 with statistical errors, including calculated

fit fractions. We again generate sample parameter sets to determine the statistical

errors on the fit fractions. The signal fraction from the fit is (80 ± 5)%.

76

01

23

0122436

48

60

72

01

23

020

40

60

80

10

0

12

0

01

23

0612182430

36


4

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )

Tota

lSi

gn

alB

ackg

rou

nd

Fig

ure

4.14

:T

he

sign

alan

dbac

kgr

ound

inth

eD

alit

zplo

tpro

ject

ions

ofFit

1.

77

0 1 2 3

Even

ts p

er 0

.03

GeV

2 /c4

mK+

π02 (GeV2/c4)

0

12

24

36

48

60

72

Signal|�

φ|2

|�K*+|

2

|�K*-

|2

|�NR

|2

Figure 4.15: Signal portion of Fit 1 and the component resonance amplitudes squared(m2

K+π0 projection).

78

0 1 2 30

20

40

60

80

100

120

Even

ts p

er 0

.03

GeV

2 /c4

mK+K-

2 (GeV2/c4)

Signal|�

φ|2

|�K*+|

2

|�K*-

|2

|�NR

|2


K+K− projection).

79

0 1 2 30

6

12

18

24

30

36

Even

ts p

er 0

.03

GeV

2 /c4

mK-

π02 (GeV2/c4)

Signal|�

φ|2

|�K*+|

2

|�K*-

|2

|�NR

|2


K−π0 projection).

80

01

23

0122436

48

60

72

01

23

020

40

60

80

10

0

12

0

01

23

0612182430

36


4

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )

Tota

lSi

gn

alB

ackg

rou

nd

Fig

ure

4.18

:T

he

sign

alan

dbac

kgr

ound

inth

eD

alit

zplo

tpro

ject

ions

ofFit

3.

81

0 1 2 30

12

24

36

48

60

72

Even

ts p

er 0

.03

GeV

2 /c4

mK+

π02 (GeV2/c4)

Signal|�

φ|2

|�K*+|

2

|�K*-

|2

|�κ

+|2

|�κ-|

2


K+π0 projection).

82

0 1 2 30

20

40

60

80

100

120

Even

ts p

er 0

.03

GeV

2 /c4

mK+K-

2 (GeV2/c4)

Signal|�

φ|2

|�K*+|

2

|�K*-

|2

|�κ

+|2

|�κ-|

2


K+K− projection).

83

0 1 2 30

7.5

15

22.5

30

Even

ts p

er 0

.03

GeV

2 /c4

mK-

π02 (GeV2/c4)

Signal|�

φ|2

|�K*+|

2

|�K*-

|2

|�κ

+|2

|�κ-|

2


K−π0 projection).

84

Resonance Amplitude Phase Fit FractionK∗+ 1.00 (fixed) 0◦ (fixed) (48.1 ± 4.5)%K∗− 0.52 ± 0.05 313◦ ± 9◦ (12.9 ± 2.6)%φ 0.65 ± 0.05 334◦ ± 12◦ (16.1 ± 1.9)%κ+ 1.78 ± 0.43 109◦ ± 17◦ (12.6 ± 5.8)%κ− 1.60 ± 0.29 128◦ ± 17◦ (11.1 ± 4.7)%

Total Fit Fraction: (99.0 ± 9.3)%

Table 4.7: Amplitude, phase, and fit fraction results for Fit 3 (with κ± instead of anon-resonant contribution) with statistical errors.

The signal and background in the Dalitz plot projections of Fit 3 is shown in

Figure 4.18. A graphical version of how much of each resonance is in the total signal

for Fit 3 is shown in the projections in Figures 4.19, 4.20, and 4.21. Note that the

amounts of each individual resonance are those from the numerator of the fit fraction

in Equation 4.8, and so the sum of the areas of each resonance in Figures 4.19, 4.20,

and 4.21 do not add up to the area of the signal. For Fit 3 the total fit fraction is not

statistically conclusive as overall destructive or constructive interference. However,

a phase of 313◦ ± 9◦ (alternatively, −47◦ ± 9◦) does imply destructive interference

between the K∗ resonances.

To compare the fits, we can overlay them as shown in Figure 4.22. They look very

similar but do have very subtle differences.

4.10.4 Other Fits to the Entire Dalitz Plot

Figure 4.23 (Fit 3a) shows what happens when we add the non-resonant (NR) compo-

nent to the set of resonances in Fit 3. The fit remains good and the phase difference

δD and relative amplitude rD change slightly from Fit 3. The κ+ does not have a

significant fit fraction, though (FFκ+ = (1.6 ± 1.8)%).


component with four rather broad resonances (κ± and K∗0(1430)±). The fit improves

85

mK

+K-2 (G

eV2 /c

4 )m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fit

1Fi

t 3

Fig

ure

4.22

:Fit

1pro

ject

ions

(blu

e,das

hed

line)

over

laye

don

top

ofFit

3pro

ject

ions

(gre

en,so

lid

line)

.

86

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.23

:D

alit

zplo

tpro

ject

ions

for

Fit

3ash

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

87

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.24

:D

alit

zplo

tpro

ject

ions

for

Fit

4sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

88

slightly and the phase difference δD remains fairly unchanged from Fit 1, while rD

decreases. The K∗0 ’s do not seem to have significant fit fractions, though (FFK∗+

0=

(8.9 ± 7.6)%, FFK∗−0

= (0.9 ± 2.2)%).

Figure 4.25 (Fit 5) shows what happens when we include both a non-resonant

(NR) component and the K∗0(1430)±. In this case we clearly have redundancy in the

basis set since the fit fractions come out completely nonsensical: the NR contribution

is (FFNR = (127±51)%) and the K∗0 ’s are barely significant (FFK∗+

0= (15.0±13.2)%,

FFK∗−0

= (19.0 ± 11.6)%). Indeed, the total fit fraction for all resonances is ≈ 224%

which implies destructive interference on an unrealistic scale.


component with K∗0(1430)±. In this case the K∗

0(1430)− resonance does not have a

significant fit fraction (FFK∗−0

= (1.3±2.0)%). This is not hard to understand: We are

attempting to replace a broad non-resonant background with high-mass resonances

whose widths are not big enough to have much influence on the lower mass regions

of the Dalitz plot. Since this basis set is insufficient to cover the Dalitz plot, the fit

in this case is pulled in an unnatural direction.

The same argument holds for Figure 4.27 (Fit 7). The replacement of the NR

component with a0(980) (FFa0 = (2.9± 1.6)%) and f0(1370) is not enough to satisfy

the need for a broad low mass feature that a simple non-resonant component can

satisfy very well. Also, the signal fraction for this fit is much lower than the measured

signal fraction.

Figures 4.28 and 4.29 (Fits 8 and 9) show the effects of including all of the ad-

ditional resonances tried in Fits 6 and 7 (i.e. K∗0(1430), a0 and f0), with the extra

addition of the K∗2(1430) in Fit 9. The summed minus log likelihood improves, but

there are many non-significant components.

Figure 4.30 (Fit 10) shows the result of using the same resonances used in Fit 9

with the addition of the κ resonances. In this case we clearly have redundancy in the

89

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.25

:D

alit

zplo

tpro

ject

ions

for

Fit

5sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

90

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.26

:D

alit

zplo

tpro

ject

ions

for

Fit

6sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

91

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.27

:D

alit

zplo

tpro

ject

ions

for

Fit

7sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

92

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.28

:D

alit

zplo

tpro

ject

ions

for

Fit

8sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

93

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.29

:D

alit

zplo

tpro

ject

ions

for

Fit

9sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

94

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.30

:D

alit

zplo

tpro

ject

ions

for

Fit

10sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

95

basis set due to a number of insignificant resonances.

Figure 4.31 (Fit 11) shows a fit that uses the resonances in Fit 1 along with the

f0(1500). This fit did not converge in the fitter at first with the default starting

values (aj = 1 and δj = 0 for all resonances j). When I changed the starting values

for amplitudes and phases of the fit to those in Fit 1, the fitter did converge. rD is

consistent with Fit 1 and δD is lower, but the f0(1500) is not significant in this fit

(FFf0(1500) = (1.0 ± 1.1)%).

Figure 4.32 (Fit 12) is a fit that uses the resonances in Fit 1 along with the

f ′2(1525). δD and rD remain fairly unchanged from Fit 1, but the fit fraction for the

f ′2(1525) is not significant (FFf ′

2(1525) = (0.3 ± 0.6)%).

It is useful to summarize what we have learned from Fits 1-12. First, the Dalitz

plot is well fitted by a simple combination of K∗+, K∗−, φ, and a uniform non-resonant

contribution (Fit 1). The non-resonant contribution to Fit 1 is significant; its fit frac-

tion is large (≈ 36%) and removing it makes the fit significantly less accurate (Fit 2).

The resonance that provides the best progress in replacing the non-resonant contri-

bution is the scalar κ; however, the existence of the κ is unconfirmed by Reference [1].

The addition of other sub-modes can improve the appearance of the fit but are not

well motivated since these tend to have masses and widths which do not make them

ideally suited for providing the broad low-mass feature that the simple non-resonant

component can provide. When too many resonances which decay to the same par-

ticles (for example, many resonances which decay to K+K−) are used the fitter has

problems, as evidenced by fit fractions which indicate destructive interference at lev-

els which do not make much sense or multiple resonances which may be insignificant

in the fit.

96

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.31

:D

alit

zplo

tpro

ject

ions

for

Fit

11sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

97

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.32

:D

alit

zplo

tpro

ject

ions

for

Fit

12sh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

98

4.10.5 Fits to Partial Regions of the Dalitz Plot

When examining the mass-squared projections for Fits 1-12 we see a small but con-

spicuous discrepancy between data and fit at around 1 GeV2 /c4 in both Kπ pro-

jections (there is a small deficit in the data compared to the fit). In the course of

understanding this feature (also known as “the dip”) we investigated a variety of

different matrix element models M. Several of these involved various K-matrix ap-

proaches (see Appendix A), none of which succeeded in modeling the data better

than the approach using Breit-Wigner resonances we have described so far in this

note. Fortunately, in the course of these studies we developed the ability to perform

partial fits over the Dalitz plot, motivated by the desire to understand exactly where

on the Dalitz plot the model discrepancy was the worst.

It is evident that most of the information relevant to extracting the relative am-

plitude rD and phase difference δD will be contained in the “+” shaped region that

is defined by the two K∗ bands on the Dalitz plot (i.e. the vertical and horizontal

bands of Figure 4.4(a)). In fact, the idea for determining δD described by Rosner

and Suprun [30] involves a simple counting experiment which basically compares the

yield of events in a region of overlap between K∗+ and K∗− bands to the yield in

the non-overlapping parts of the two bands. Thus, most of the information about δD

should be contained within the overlapping K∗ bands.

We should be able to study any systematic effect of our model’s inability to

describe areas of the Dalitz plot which are not in the K∗ signal region by separately

fitting the full Dalitz plot and the much smaller K∗ signal region with the same model

and looking for any changes to the extracted values of rD and δD. We also used other

choices for regions of the Dalitz plot to see if the fits for rD and δD change much.

The results of these studies are shown in Figures 4.34-4.36, 4.38-4.44, and Table 4.5.

See Figure 4.33 for a visual definition of each type of reduced fit region.

99

(a) (b)

(c) (d)

(e)

Black points (triangles) are includedCyan points (squares) are excluded

0.7 1.1 1.5 1.9

0.7

1.1

1.5

1.9

0.7 1.1 1.5 1.9

0.7

1.1

1.5

1.9

0.70.7 1.11.1 1.51.5 1.91.9

0.70.7

1.11.1

1.51.5

1.91.9

0.70.7 1.11.1 1.51.5 1.91.9

0.70.7

1.11.1

1.51.5

1.91.9

mK-

π02 (GeV2/c4)

mK

+π

02 (GeV

2 /c4 )

mK-

π02 (GeV2/c4)

mK

+π

02 (GeV

2 /c4 )

mK-

π02 (GeV2/c4)

mK

+π

02 (GeV

2 /c4 )

mK-

π02 (GeV2/c4)

mK

+π

02 (GeV

2 /c4 )

0.70.7 1.11.1 1.51.5 1.91.9

0.70.7

1.11.1

1.51.5

1.91.9

mK-

π02 (GeV2/c4)

mK

+π

02 (GeV

2 /c4 )

Figure 4.33: Fit Regions (a) K∗ bands (b) No φ 1.05 (c) No φ 1.15 (d) No φ 1.26 (e)“L” area.

100

To fit partial areas of the Dalitz plot, we use the same efficiency and background

functions which we determined from the whole Dalitz plot. However, only events

which are in the fitting region are used in the negative logarithmic likelihood function

L, and the efficiency and background are both set to zero outside the fitting region

(in order to get the correct normalization over the Dalitz plot).

Fits to the K∗ Bands

In our ‘K∗ bands’ fits we use K∗ bands which are defined as being bounded by

(mK∗ − ΓK∗)2 and (mK∗ + ΓK∗)2 along each of the m2Kπ axes (see Figure 4.33(a)).

These bands are twice as big in mKπ as the bands referred to in Reference [30]. The

K∗ bands region contains 334 D0 → K+K−π0 candidates.

Figure 4.34 (Fit 1x) shows the result of a partial fit to just the K∗ bands using

the nominal simple model that includes just K∗+, K∗−, plus a uniform non-resonant

component. We immediately notice that the strong phase difference and the K∗

amplitude ratio are about the same as in the nominal fit (Fit 1), the fit fraction of

the non-resonant component is greatly reduced (by almost a factor of two, although

this is not strange given the reduced fitting area), and there are no-longer any “dips”

around 1 GeV2 along the m2Kπ axes.

This seems to be a very good fit. Furthermore, the fact that δD and rD are

unchanged from their nominal values shows that our result should be rather insensitive

to the details of the modeling of the components which are outside the K∗ bands.

Figure 4.35 (Fit 2x) shows the effect of using the K∗s but ignoring the non-

resonant component. The quality of the fit is clearly worse than the nominal “K∗-

band” fit shown in Fit 1x — the summed negative log likelihood, L, increases by 60

and the fitted signal fraction is considerably lower than that for Fit 1x. This gives us

further confidence that a broad S-wave feature is present.

Figure 4.36 (Fit 3x) shows the effect of replacing the non-resonant contribution

101

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.34

:D

alit

zplo

tpro

ject

ions

for

Fit

1xsh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

102

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.35

:D

alit

zplo

tpro

ject

ions

for

Fit

2xsh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

103

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.36

:D

alit

zplo

tpro

ject

ions

for

Fit

3xsh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

104

0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.4

0.6

0.8

1

1.2

1.4

1.6

1.8

±30o

0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.4

0.6

0.8

1

1.2

1.4

1.6

1.8(a) (b)

0

mK-

π02 (GeV2/c4)

mK

+π

02 (GeV

2 /c4 )

mK-

π02 (GeV2/c4)

Figure 4.37: (a) The absolute value of the phase distribution of our nominal fit overthe entire Dalitz plot subtracted from the phase distribution of the nominal fit withthe φ simply removed (aφ is set to be 0). This shows the effect the φ resonance hason the overall phase at different points in phase space on the Dalitz plot. Removingonly a region where the φ has its peak is not enough to remove all regions of theDalitz plot whose phase is noticeably affected by the φ resonance. (b) The same plotas (a) except that areas outside the K∗ bands are excluded. Note that the phasedistribution is not affected by the φ in the vast majority of the K∗ bands region, soit is definitely appropriate to exclude the φ from any K∗ bands fit.

to the nominal fit with the κ. δD is much lower than in Fit 1x, but with a large

statistical uncertainty, while rD is higher. However, rD and δD are consistent with

Fit 3. This fit seems good for the most part, although the κs are less than 2 standard

deviations significant (FFκ+ = (7.6 ± 4.4)%, FFκ− = (10.1 ± 5.6)%).

Fits to Regions which Exclude the φ

An alternative set of fits we tried were fits that removed the φ resonance. The φ is

a narrow but significant part of the Dalitz Plot and does have an effect. We do not

believe these fits to be as useful as the K∗ bands fits, because simply removing the

φ from our fitting region is not enough to remove its effects on the Dalitz plot (see

Figure 4.37). However, by simply removing the φ we do not remove as many of our

105

statistics as in the K∗ bands fit.

Figure 4.38 (Fit 1n) shows the effect of excluding from the fit only the peak of the

φ resonance (Figure 4.33(b)). Specifically, we exclude m2K+K− between 1.02 GeV2 and

1.05 GeV2. The remaining region contains 613 D0 → K+K−π0 candidates. We again

use our simplest model consisting of just K∗+, K∗−, plus a uniform non-resonant

component, and we again obtain a good fit with the same value of rD and a fairly

unchanged, but slightly higher, value of δD. We believe that the φ still affects the

region of this plot, so leaving it out of the fit is likely causing the shift in δD.

Figure 4.39 (Fit 1np) uses the same fit region as Fit 1n. We add the φ back in to

go along with K∗+, K∗−, plus a uniform non-resonant component. We obtain a good

fit and δD is more in line with Fit 1 than Fit 1n was, although rD decreases slightly.

The fit fraction for the φ is not very significant, as expected.

Figure 4.40 (Fit 1na) shows the effect of excluding m2K+K− less than 1.15 GeV2

from the fit (Figure 4.33(c)). We fit with the same resonances as in Fit 1n. This

region contains 573 D0 → K+K−π0 candidates. δD increases compared to Fit 1,

perhaps for the same reason as in Fit 1n, and rD decreases slightly when compared

to Fit 1.

Figure 4.41 (Fit 1nap), shows a fit which is the same as Fit 1na but with the φ

added. This fit did not converge in the fitter at first with the default starting values.

When I changed the starting values for amplitudes and phases of the fit to those in

Fit 1, the fitter did converge. The fit fraction for the φ is not significant, as expected.

δD is now a few degrees less than in Fit 1na, and rD decreases further.

Figure 4.42 (Fit 1nb) shows the effect of excluding m2K+K− less than 1.26 GeV2

from the fit (Figure 4.33(d)). We fit with the same resonances as in Fit 1n. This

region contains 552 D0 → K+K−π0 candidates. rD is slightly lower than Fit 1, and

δD is still greater than the δD in Fit 1, but still consistent with Fit 1.

Figure 4.43 (Fit 1nbp) uses the same fit region as Fit 1nb. We add the φ back in

106

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.38

:D

alit

zplo

tpro

ject

ions

for

Fit

1nsh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

107

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.39

:D

alit

zplo

tpro

ject

ions

for

Fit

1np

show

ing

bot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

108

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.40

:D

alit

zplo

tpro

ject

ions

for

Fit

1na

show

ing

bot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

109

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.41

:D

alit

zplo

tpro

ject

ions

for

Fit

1nap

show

ing

bot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

110

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.42

:D

alit

zplo

tpro

ject

ions

for

Fit

1nb

show

ing

bot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

111

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.43

:D

alit

zplo

tpro

ject

ions

for

Fit

1nbp

show

ing

bot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

112

to go along with K∗+, K∗−, plus a uniform non-resonant component. rD is the same

as in Fit 1nb, but δD is more consistent with what is found in Fit 1. However, the φ

fit fraction is not significant, as expected.

Fits to the “L” Region

Figure 4.44 (Fit 1s) excludes events for which both m2K+π0 and m2

K−π0 are greater

than 1.10 GeV2 (Figure 4.33(e)). This is an “L” shaped region in m2K+π0 and m2

K−π0

which contains 549 D0 → K+K−π0 candidates. This region is a compromise between

the K∗ bands fit and the no φ fit regions. We again use just K∗+, K∗−, and a uniform

non-resonant component. We again obtain a good fit. It is interesting that these fit

constraints remove the φ also, but δD and rD are fairly unchanged compared to Fit

1.

4.10.6 Floating mκ, Γκ Fit to the Entire Dalitz Plot

Finally, we do a fit which allows the mass and width of the κ to float, since the mass

and width of the κ are not well determined by other experiments.

Figure 4.45 (Fit 3f) shows a fit to the entire Dalitz plot with the resonance set:

{K∗+, K∗−, κ+, κ−, and φ} where the κ mass and width are allowed to float. The fit

to the data is good. The fit prefers mκ = 855 ± 15 MeV and Γκ = 251 ± 48 MeV.

The relative phase δD between K∗− and K∗+ falls to 306◦, and rD remains consistent

with Fit 1.

113

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.44

:D

alit

zplo

tpro

ject

ions

for

Fit

1ssh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

114

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

4.45

:D

alit

zplo

tpro

ject

ions

for

Fit

3fsh

owin

gbot

hth

efit

(red

curv

e)an

dth

ebin

ned

dat

asa

mple

.

115

4.11 Systematic Errors for rD and δD in our

Nominal Fit

We consider systematic errors from experimental sources and from the decay model

separately. Our general procedure is to change some aspect of the analysis and

interpret the change in the values of the amplitude ratio rD and phase difference

δD as an estimate of the associated systematic uncertainty. Contributions to the

experimental systematic uncertainties arise from our models of the background, the

efficiency, the signal fraction, and the event selection. The model systematic error

arises from uncertainty in the choice of resonances used to fit the Dalitz plot.

Part of our systematic error is the variation of δD and rD based on different models

used for our matrix element. We also expect that changing the fitting region will be

part of our systematic error. Also, fits that have resonance choices which lead to very

unreasonable fit fractions and amplitudes should also be excluded. This leaves the

set of fits {1, 1n, 1np, 1na, 1nap, 1nb, 1nbp, 1x, 1s, 3, 3f, 4, 5, 11, 12}. Within these

fits, we believe that fits that are in “no φ” regions which did not include the φ may

still have lingering interference effects from the φ which affect the K∗s. So we will

define fits {1, 1x, 1np, and 1s} within the category of changing the fitting region on

the Dalitz plot, and we characterize fits {1, 3, 3f, 4, 5, 11, 12, 1np, 1nap, 1nbp} to

be within the category of changing our model.

We can also do a fit to (m2K−π0 , m2

K+π0) rather than (m2K+π0 , m2

K+K−). In the

(m2K−π0 , m2

K+π0) fit, the third Dalitz plot variable, m2K+K− , is calculated from m2

K−π0

and m2K+π0 . Because we do not enforce an energy-momentum constraint on all three

invariant-mass squared variables measured, the distribution for calculated m2K+K− is

not the same as that for measured m2K+K− . Unfortunately, this difference we found is

large enough such that we believe the φ resolution is much worse in calculated m2K+K−

[Γ ≈ 30 MeV] than measured m2K+K− [Γ ≈ 5 MeV]. However, we repeat the Dalitz fit

116

for the new choice of Dalitz plot variables as a means to understand the systematics

of our choice of invariant mass-squared variables. This fit used a background and

efficiency parametrization based on a best fit to m2K−π0 and m2

K+π0 . We fixed the

signal fraction to 81.8% in the (m2K−π0 , m2

K+π0) fit and the (m2K+π0 , m2

K+K−) fit for

the systematic comparison. We did not expect the φ to be fit very well, and its

amplitude and phase are inconsistent between the two fits.

Table 4.8 lists our systematic checks and systematic uncertainties. Systematic

Check 1 represents our modeling error, and the other systematic errors represent our

experimental error. We take the square root of the sample variance of the amplitudes

and phases from the nominal result (Fit 1) compared to the results in the series of

variation fits used for systematics checks as a measure of the systematic uncertainty

[51]. We find systematic errors of ±2.9◦(experimental) ± 10.6◦(model) for δD and

±0.016(experimental) ± 0.038(model) for rD. Adding systematic errors in this way

results in a model systematic error for δD that is less than the difference in δD when

comparing Fit 1 to Fit 3. We add the experimental and model systematic uncertainty

in quadrature to obtain the total systematic uncertainty shown in Table 4.8.

Our systematic error is dominated by the model dependence, and the largest

deviations from the nominal fit were observed in the series of fits where we replaced the

non-resonant contribution with the κ±. If fits including a κ± resonance are removed

from consideration, then the systematic errors on δD and rD decrease from ±11◦

and ±0.04 to ±8◦ and ±0.03, respectively, and the remaining systematic uncertainty

is dominated by fits including the K0(1430)±. The largest experimental systematic

uncertainties are ±8◦ for δD when allowing the background parameters to float, and

±0.05 for rD when allowing the efficiency parameters to float, as described below.

Table 4.9 provides details about our systematic checks. “Lo Acceptance” refers to

tighter selection criteria (less events pass), whereas “Hi Acceptance” refers to looser

selection criteria (more events pass). Generally the signal fraction is allowed to float

117

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and

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.“<

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fois

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sw

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>50

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/dx

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mp K

>50

0M

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118

in the fitter as it was for the nominal fit, unless otherwise stated.

Systematic checks 1-3 are described above.

In systematic checks 4a and 4b, we could only tighten our selection criteria due

to pre-existing limits placed to make the data accessible.

Systematic checks 5, 7, 8, 9, and 10 are simple variation checks where we increase

and decrease our selection criteria and see how it affects our results.

Systematic check 6 is a cut on a subtraction of how likely the particle is to be a

kaon vs. how likely it is to be a pion. Since 0 is the point where a kaon hypothesis is as

likely as a pion hypothesis, it was hard to determine an amount of change for a test of

systematics. Instead, our guideline was to choose variations which did not change the

number of events accepted drastically from the 709 accepted in the nominal fit. 680

events were accepted for the “Lo Acceptance” RICH systematic fit and 783 for the

“Hi Acceptance” RICH systematic fit. The number of events accepted decreased by

4.1% for tighter selection criteria and increased by 10.4% for looser selection criteria.

Systematic check 11 is a check to see how much a change in the signal fraction

affects the fitter. The signal fraction determined from the ΔM and mK+K−π0 fits is

81.8% ± 6.3%(stat.) ± 2.8%(syst.), while the signal fraction from the nominal Dalitz

fit (Fit 1) is 86 ± 6%. The central values differ by about 0.7 standard deviations.

In this check, we simply varied the signal fraction from the original signal fraction to

different values which could be reasonable given the statistical error.

Systematic check 12 adds a term to the negative logarithmic likelihood function

L which allows the signal fraction to float but penalizes L depending on how far the

signal fraction floats from the original signal fraction f0 = 81.8% ± 6.3%(stat.). This

term looks like:

Penalty Term = (f − f0

σf

)2. (4.10)

119

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120

For the purpose of systematic checks 11 and 12, we ignore the comparatively small

(± 2.8%) systematic error on the signal fraction.

Systematic check 13 adds a term to the function L which allows the background

parameters to float up to one standard deviation away from their original values, but

penalizes L depending on how far the background parameters float from the original

parameters in Table 4.4. This term looks like:

Background Covariance Term = Σij(Bi − Bio)Vij(Bj − Bjo), (4.11)

where Bi is one of the background parameters and Bio is the original value of that

parameter, and where V is the background covariance matrix which relates variations

in the parameter Bj to variations in Bi. The fit requires that the resonant background

parameters can only vary by 1 standard deviation.

Table 4.10 shows the best-fit parameters for the background shape when the back-

ground parameters are allowed to float, with a background covariance matrix penalty

term.

B0 By Bz Byy Byz

1 (fixed) 20.58 ± 0.1355 11.96 ± 0.2474 92.66 ± 0.2343 -88.91 ± 0.2687Bzz Byyy Byyz Byzz Bzzz

10.22 ± 0.4594 -43.16 ± 0.4716 18.95 ± 0.4086 9.040 ± 0.5994 0.4483 ± 1.260BK∗+ BK∗− Bφ

0.04942 ± 5.208x10−5 .006606 ± .001932 0.03932 ± 3.311x10−7

Table 4.10: Best-fit parameters for the background shape when the background pa-rameters are allowed to float.

Systematic check 14 is to change the fit by allowing the efficiency parameters to

all float within 1 standard deviation of their original values. We did not use the

efficiency covariance matrix to further constrain this fit.

Table 4.11 shows the best-fit parameters for the efficiency shape when the effi-

121

ciency parameters are allowed to float.

E0 Ey Ez Eyy Eyz

1 (fixed) -4459 ± 149 1458 ± 92 3261 ± 57 2562 ± 188

Ezz Eyyy Eyyz Eyzz Ezzz

-46.54 ± 36.23 -597.9 ± 31.9 -830.5 ± 70.0 -640.2 ± 261.8 -81.78 ± 13.12

Table 4.11: Best-fit parameters for the efficiency shape when the efficiency parametersare allowed to float.

4.12 Branching Ratio Cross-check

As a cross-check of our data, we estimate the branching ratio of D0 → K+K−π0 from

our data and compare it to the published value. Branching ratio measurements are

not the focus of this analysis, so systematic errors have not been investigated. Based

on our mK+K−π0 fit, we have a total of 627 ± 30 signal events. We measured our

average efficiency3 for finding the decay chain D∗+ → π+slowD0; D0 → K+K−π0 from

the signal Monte Carlo (discussed in Section 4.7) to be ε = (5.83 ± 0.09)%. From this,

we calculated how many D0 → K+K−π0 we expected in our data sample (our yield

divided by ε). We may estimate the total number of D0s expected from continuum

D∗+s in our data sample, based on the integral cross-section for continuum D∗+

production near√

s = 10.6 GeV [52]: σ(e+e− → D∗+X) = (583 ± 8 ± 33 ± 14) pb,

where the fourth error stems from external branching fraction uncertainties. By

dividing our yield by our efficiency to determine the number of D0 → K+K−π0

decays in our sample, taking that result and dividing by the total number of D0s we

expect from continuum D∗+s, and knowing BR(D∗+ → π+s D0) = (67.7 ± 0.5)% [1],

we estimate BR(D0 → K+K−π0) = (0.30±0.02)%, which is significantly higher than

3The average efficiency ε is simply the number of MC events we are able to reconstruct (4077.3± 64.43) as D∗+ → π+

slowD0; D0 → K+K−π0, using the same computer code we use to reconstructdata events, divided by the total number of generated signal (D∗+ → π+

slowD0; D0 → K+K−π0)MC events (69922). This is the same sample used to create the MC Dalitz Plot in Section 4.7.

122

the previous measurement of BR(D0 → K+K−π0) = (0.13 ± 0.04%) [1, 31].

Combining our fit fractions with known values for BR(K∗± → K±π0) and BR(φ →K+K−) [1], we also estimate branching ratios of the resonant decay modes. We find

BR(D0 → K∗+K−) = (0.38 ± 0.04)%, BR(D0 → K∗−K+) = (0.10 ± 0.02)%, and

BR(D0 → φπ0) = (0.084 ± 0.012)%. These branching ratios are consistent with

published measurements [1, 32–34] and are listed in Table 4.12.

A recent4 measurement of BR(D0 → K+K−π0) = (0.334±0.004±0.006±0.012)%

by the BaBar Collaboration [53] is consistent with our estimate from CLEO III data.

Decay Mode Published BR Estimated BRD0 → K∗+K− (0.37 ± 0.08)% (0.38 ± 0.04)%D0 → K∗−K+ (0.20 ± 0.11)% (0.10 ± 0.02)%

D0 → φπ0 (0.076 ± 0.005)% (0.084 ± 0.012)%D0 → K+K−π0 non-resonant No Measurement (≈ 0.09)%

Table 4.12: Comparison of estimated branching ratios from this analysis to publishedbranching ratios [32–34, 48] based on our nominal Dalitz plot fit. We accounted forthe fact that K∗± decays to K±π0 only one-third of the time, and φ decays to K+K−

only about one-half of the time.

4.13 U-spin Symmetry Check

The U -spin symmetry interchanging s and d quarks predicts the following for D0

decays to a pseudoscalar meson and a vector meson [54]:

A(D0 → π+ρ−) = −A(D0 → K+K∗−) (4.12)

and

A(D0 → π−ρ+) = −A(D0 → K−K∗+), (4.13)

4This measurement was published very recently (on 15 November 2006) so, due to time con-straints, this chapter has not been updated to include this result as “current knowledge”. However,the journal reference is cited in the References.

123

where A is the respective dimensionless invariant amplitude for each decay. Dividing

Equation 4.12 by Equation 4.13, and assuming that A(K∗+ → K+π0) = A(K∗− →K−π0), gives:

A(D0 → π+ρ−)

A(D0 → π−ρ+)=

aK∗−K+

aK∗+K−ei(δK∗−K+−δK∗+K− ). (4.14)

Assuming a phase convention such that a phase difference of 0◦ indicates maximal

destructive interference between ρ− and ρ+, and assuming A(ρ+ → π+π0) = A(ρ− →π−π0), we can use the recently published results of a Dalitz plot analysis of D0 →π+π−π0 [40] to evaluate the left hand side of Equation 4.14:

(0.65 ± 0.03 ± 0.02)ei(356◦±3◦±2◦)

which may be compared to the right hand side of Equation 4.14 which comes from

this analysis:

(0.52 ± 0.05 ± 0.04)ei(332◦±8◦±11◦).

We do not expect these results to necessarily be consistent, since U -spin symmetry

does not always make correct predictions [54]. However, any symmetry-breaking

effects are important to identify.

4.14 The Effect of our Uncertainties on γ

It is interesting to look at how the uncertainties in our measurements of rD and δD

affect the measurement of γ. For convenience we repeat the equations (discussed in

124

Section 1.6) which may be used to determine γ from δD and rD:

A[B− → K−(K∗+K−)D] = |ABAD|(1 + rBrDei(δB+δD−γ)) (4.15)

A[B− → K−(K∗−K+)D] = |ABAD|eiδD(rD + rBei(δB−δD−γ)) (4.16)

A[B+ → K+(K∗−K+)D] = |ABAD|(1 + rBrDei(δB+δD+γ)) (4.17)

A[B+ → K+(K∗+K−)D] = |ABAD|eiδD(rD + rBei(δB−δD+γ)), (4.18)

If we calculated γ via Equations 4.15 or 4.17, the size of the uncertainty in γ due

to uncertainty in δD is simply equal to the size of the uncertainty in δD, and does not

depend on rD. However if we use Equations 4.16 or 4.18, the uncertainty in γ due to

the uncertainty in rD or δD is more difficult to know. We may define:

A′Eq. 4.16 =

A[B− → K−(K∗−K+)D]

|ABAD| (4.19)

A′Eq. 4.18 =

A[B+ → K+(K∗+K−)D]

|ABAD| . (4.20)

Then we can determine the real parts of the partial derivatives of γ to get an idea

of how the uncertainty in γ is affected by the uncertainty in δD or rD when using

Equations 4.16 or 4.18:

Re

(∂γ

∂δD

)Eq. 4.16

= Re

[A′

Eq. 4.16

A′Eq. 4.16 − rDeiδD

]− 1 (4.21)

Re

(∂γ

∂rD

)Eq. 4.16

= Im

[1

A′Eq. 4.16 e−iδD − rD

](4.22)

Re

(∂γ

∂δD

)Eq. 4.18

= Re

[A′

Eq. 4.18

rDeiδD − A′Eq. 4.18

]+ 1 (4.23)

Re

(∂γ

∂rD

)Eq. 4.18

= Im

[1

rD − A′Eq. 4.18 e−iδD

], (4.24)

where Im[x] is the imaginary part of x and Re[x] is the real part of x. The uncertainty

125

in γ in these cases also depends on the values of δD and rD themselves. Imaginary

parts of the partial derivatives do not affect γ, these represent effects which rescale

the amplitudes but do not affect the phase.

The value of γ from direct measurements (not including the recent measurement

by Belle in Reference [12]) is γ = (63+15−12)

◦ [1]. We expect that the effect on the

uncertainty of γ due to the uncertainties of δD and rD from this analysis is comparable

to the current errors from direct measurements of γ.

4.15 Conclusion

In conclusion, we have examined the resonant substructure of the decay D0 →K+K−π0 using the Dalitz plot analysis technique. We observe resonant K∗+K−,

K∗−K+, and φπ0 contributions. We also observe a significant S-wave modeled as a

κ±K∓ or a non-resonant contribution. Other models of the S-wave (including the K-

matrix formalism discussed in Appendix A) were not used but should be considered

and tested in future, higher-statistics analyses. We determine δD = 332◦ ± 8◦ ± 11◦

and rD = 0.52 ± 0.05 ± 0.04. The measurements of δD and rD seem robust against

variations in signal selection as well as background and efficiency determination.

126

5 D0 → K+K−π0 in CLEO-c

The final phase of the CLEO experiment began June 2003 and is called CLEO-c.

CLEO-c uses mostly the same detector as CLEO III, but replaces the silicon vertex

detector with a small drift chamber [25]. CLEO-c is expected to record about 1500

resonant decays of the type useful for determining the strong phase δD, an amount

that will allow a determination of cos(δD) with an uncertainty of ±0.27 or better [30].

A preliminary look at D0 → K+K−π0 in CLEO-c is presented here.

5.1 Effects of Quantum Correlations

To tag the flavor of the D0(D0), we have to take advantage of the fact that at

√s ≈ 3.77 GeV a D0 can only be produced by the process e+e− → D0D

0. If we

can cleanly tag the other side of the decay to a mode which primarily decays from a

D0, we will be able to say with confidence that the side we are interested in (which

decays to K+K−π0) is a D0, and vice versa. However, the results of our fit may

be modified by the effects of quantum correlations of the C = −1 initially produced

state1 e+e− → D0D0. According to Reference [55], the rates, R, for a decay which

has one side decaying to a flavor-tag mode f and the other side decaying to a non-CP

eigenstate f ′ or its conjugate mode f′are:

Rf ′,f = BR(D0 → f ′)BR(D0 → f)(1 + r2f ′r2

f − rf ′rfv−f ′,f ) (5.1)

Rf′,f

=BR(D0 → f

′)BR(D0 → f)(r2

f ′ + r2f − rf ′rfv

+f ′,f )

r2f ′

, (5.2)

1C = Charge Conjugation eigenvalue

127

where f is the conjugate of f . The rate for a decay which has one side decaying to a

flavor-tag mode and the other side decaying to a CP = +1 eigenstate S+ is:

RS+,f = BR(D0 → S+)BR(D0 → f)(1 + r2f + rfzf ). (5.3)

In both cases:

rj =

∣∣∣∣〈j|D0〉〈j|D0〉

∣∣∣∣ (5.4)

δj = − arg

(〈j|D0〉〈j|D0〉

)= −(δstrong + δweak + 180◦) (5.5)

v±f ′,f = 2 cos δf ′ cos δf ± 2 sin δf ′ sin δf (5.6)

zf = 2 cos δf , (5.7)

and CP is assumed to be conserved for D decays. The conjugate states j and j are

defined such that rj ≤ 1.

For the resonant modes on our Dalitz plot, we would like to know how their

rates are affected by quantum correlations of the original D0D0

state. We have

two conjugate non-CP eigenstate modes D0 → K∗+K− and D0 → K∗−K+, and a

CP = +1 mode D0 → φπ0. In order to calculate rK∗K and δK∗K we simply use

the information from our CLEO III analysis where the D0s were uncorrelated (see

Chapter 4)2. We use:

rK∗K =

∣∣∣∣〈K∗−K+|D0〉〈K∗+K−|D0〉

∣∣∣∣ = 0.52 ± 0.05 ± 0.04, (5.8)

2Our phase convention for δD is consistent with the phase convention of δstrong in Reference [55].

128

and

δK∗K = − arg

(〈K∗−K+|D0〉〈K∗+K−|D0〉

)= −(δstrong + δweak + 180◦)K∗K

= −(332◦ + 0◦ + 180◦) ± 8◦ ± 11◦

= −512◦ ± 8◦ ± 11◦

= 208◦ ± 8◦ ± 11◦. (5.9)

For this calculation we will assume our tag mode is f = K−π+. rKπ is known for this

mode to be [55]:

rKπ =

∣∣∣∣〈K+π−|D0〉〈K−π+|D0〉

∣∣∣∣ = 0.0612 ± 0.0015, (5.10)

and we assume

δKπ = − arg

(〈K+π−|D0〉〈K−π+|D0〉

)= −(δstrong + δweak + 180◦)Kπ

= −(0◦ + 180◦ + 180◦)

= −360◦

= 0◦. (5.11)

We may then calculate the ratios of expected rates for correlated D0s in terms of

the uncorrelated rates (the branching ratios, BR) from Equations 5.1 through 5.11

and find:

RK∗+K−,Kπ

RK∗−K+,Kπ

= (0.87 ± 0.03) · BR(D0 → K∗+K−)

BR(D0 → K∗−K+), (5.12)

RK∗+K−,Kπ

Rφπ0,Kπ

= (0.94 ± 0.06) · BR(D0 → K∗+K−)

BR(D0 → φπ0), (5.13)

129

and

RK∗−K+,Kπ

Rφπ0,Kπ

= (1.08 ± 0.07) · BR(D0 → K∗−K+)

BR(D0 → φπ0). (5.14)

where the errors are propagated from the total errors on rK∗K and δK∗K . We do not

include the error on rKπ since the error is relatively small, and we assume δKπ to be

identically zero [55].

These results may vary if another tag mode f is used, although this calculation

is representative of how the other tag modes (K−π+π0 and K−π+π+π−) will affect

the results.

5.2 Data Sample

This analysis uses an integrated luminosity of 281 pb−1 of e+e− collisions at√

s ≈3.77 GeV provided by the Cornell Electron Storage Ring (CESR). The data were

collected with the CLEO-c detector [25]. Charge conjugation is implied as in Chapter

4. Flavor tag modes used in this analysis include D0 → K−π+, D0 → K−π+π0, and

D0 → K−π+π+π−.

The analysis uses a standard set of selection criteria for the charged kaons and

neutral pions. The detected charged particle tracks must reconstruct to within 5 cm of

the expected interaction point along the beam pipe and within 5 mm perpendicular

to the beam pipe. Kaon candidates are required to have momenta 50 ≤ pK ≤2000 MeV/c. The cosine of the angle θ between a track and the nominal beam axis

must be between −0.93 and 0.93 in order to assure that the particle is in the fiducial

volume of the detector.

Candidate kaon tracks that have momenta greater than or equal to 700 MeV/c are

selected based on combined information from the Ring Imaging Cherenkov (RICH)

130

detector and dE/dx if at least three photons associated with the track are detected.

The combined likelihood used in this case is that defined by Equation 3.6 and the

selection requirement is Δχ2 ≤ 0. Candidate kaon tracks without RICH information

or with momentum below 700 MeV/c are required to have specific energy loss in the

drift chamber within 3 standard deviations of that expected for a true kaon (|σK | < 3).

We still require Δχ2 ≤ 0, except that in the case where only dE/dx information is

used, we ignore the RICH terms in Equation 3.6. The RICH is also not used for

| cos θ| > 0.8, to avoid an inefficiency near the edge of the RICH for kaons.

The π0 candidates are reconstructed from all pairs of electromagnetic showers that

are not associated with charged tracks. The two-photon invariant mass is required to

be within 3 standard deviations of the known π0 mass. To improve the resolution on

the π0 three-momentum, the γγ invariant mass is constrained to the known π0 mass.

5.3 Event Candidates

The D0 candidate beam-constrained mass mbc ≡ 1c2

√E2

beam − p2c2 and ΔE ≡ E −Ebeam are calculated for each candidate, where Ebeam =

√s

2is the electron (or positron)

beam energy, and p and E are the reconstructed momentum and energy, respectively,

of the decay particles. The distributions of mbc and ΔE are shown in Figure 5.1.

We select a signal region defined by 1.86 < mbc < 1.87 GeV/c2 and |ΔE| <

30 MeV/c2, which contains 211 D0 → K+K−π0 candidates. There is very little

background thanks to the clean D-meson decays at√

s ≈ 3.77 GeV.

5.4 CLEO-c Dalitz Plot Fit

Figure 5.2 shows the Dalitz plot distribution (a scatter plot of m2K+π0 vs. m2

K−π0)

for the D0 → K+K−π0 candidates in our signal region satisfying the requirements

131

h4Entries 246

Mean 1.865

RMS 0.003834

1.85 1.86 1.87 1.880

4

8

12

16

20

24Entries 246

Mean 1.865

RMS 0.003834

0π

-K+Kbcm (GeV/c2)

Entries 246

Mean -0.003417

RMS 0.02654

0

5

10

15

20

25 Entries 246

Mean -0.003417

RMS 0.02654

0π

-K+KE (MeV)Δ

Even

ts p

er 4

00 k

eV/c

2

Even

ts p

er 2

MeV

40 80-40-80 0

Signal Region

Figure 5.1: Plots of mbc and ΔE for all events passing the selection criteria in Section5.2, with the signal region within the dashed (brown) lines of both figures.

described in Sections 5.2 and 5.3.

Fitting the Dalitz plot is done by the procedure outlined in Chapter 4, except

that we assume that our efficiency and background functions are uniform (E(y, z) =

1, B(y, z) = 1) for our preliminary look at this data. Figure 5.3 shows the results

of the simplest possible reasonable fit to the data, Fit 1-c. In this fit the matrix

element M is composed of the three nominal resonances (K∗+, K∗−, and φ) plus a

uniform non-resonant (NR) component. The amplitudes, phases, and fit fractions

are shown in Table 5.1. The signal fraction was allowed to float in the fit, and is

f1−c = (92 ± 10)%. rD and δD are consistent with our nominal fit (Fit 1) result

from Chapter 4, but keep in mind that the amplitudes and phases of Fit 1-c are not

adjusted for effects of quantum correlations.

Figure 5.4 shows the results of Fit 3-c, which has a matrix element M composed

of K∗+, K∗−, κ+, κ−, and φ resonances. The κ has the mass and width shown in

Table 4.2. The amplitudes and phases are shown in Table 5.2. The signal fraction

132

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

mK-

π02 (GeV2/c4)

mK

+π

02 (GeV

2 /c4 )

Figure 5.2: Dalitz plot of K+K−π0 candidates.

Resonance Amplitude Phase Fit FractionK∗+ 1.00 (fixed) 0◦ (fixed) (50.2 ± 5.6)%K∗− 0.52 ± 0.07 333◦ ± 14◦ (13.8 ± 3.6)%φ 0.63 ± 0.07 311◦ ± 18◦ (16.0 ± 3.4)%

Non-Resonant 4.87 ± 0.93 245◦ ± 9◦ (29.2 ± 7.5)%Total Fit Fraction: (109.2 ± 10.6)%

Table 5.1: Amplitude, phase, and fit fraction results for Fit 1-c with statistical errors.

133

11.

52

2.5

305101520253035

0.4

0.6

0.8

11.

21.

41.

61.

82

051015202530

0.4

0.6

0.8

11.

21.

41.

61.

82

05

1015202530354045 Events per 0.0583 GeV2/c

4

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4


4

Sig

nal

Bac

kgro

un

dTo

tal

Fig

ure

5.3:

Dal

itz

plo

tpro

ject

ions

for

Fit

1-c.

134


4

mK

+K-2 (G

eV2 /c

4 ) m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4


4

11.

52

2.5

305101520253035

0.4

0.6

0.8

11.

21.

41.

61.

82

051015202530

0.4

0.6

0.8

11.

21.

41.

61.

82

05

1015202530354045Si

gn

alB

ackg

rou

nd

Tota

l

Fig

ure

5.4:

Dal

itz

plo

tpro

ject

ions

for

Fit

3-c.

135

was allowed to float in the fit, and is f3−c = (91±7)%. rD and δD are consistent with

our κ fit (Fit 3) result from Chapter 4, although rD is higher in Fit 3-c. Keep in mind

that the amplitudes and phases of Fit 3-c are not adjusted for effects of quantum

correlations.

Resonance Amplitude Phase Fit FractionK∗+ 1.00 (fixed) 0◦ (fixed) (39.3 ± 4.1)%K∗− 0.62 ± 0.10 311◦ ± 16◦ (15.4 ± 3.9)%φ 0.68 ± 0.08 332◦ ± 21◦ (17.1 ± 3.7)%κ+ 2.91 ± 0.58 194◦ ± 14◦ (28.7 ± 6.9)%κ− 1.34 ± 0.49 217◦ ± 25◦ ( 6.2 ± 2.8)%

Total Fit Fraction: (106.7 ± 10.1)%

Table 5.2: Amplitude, phase, and fit fraction results for Fit 3-c with statistical errors.

For the purpose of checking if quantum correlations have any effect, we calculate

ratios of fit fractions from CLEO-c and CLEO III data and compare them to what

is predicted from Equations 5.12 through 5.14, as shown in Table 5.3.

Ratio (X) Predicted Xcorrelated

Xuncorrelated

Xcorrelated(Fit 1-c)Xuncorrelated(Fit 1)

Xcorrelated(Fit 3-c)Xuncorrelated(Fit 3)

FF (K∗+)FF (K∗−)

0.87 ± 0.03 0.97 ± 0.33 0.68 ± 0.24FF (K∗+)

FF (φ)0.94 ± 0.06 1.01 ± 0.28 0.77 ± 0.22

FF (K∗−)FF (φ)

1.08 ± 0.07 1.04 ± 0.41 1.12 ± 0.46

Table 5.3: Comparison of predicted to measured ratios of fit fractions when comparingCLEO-c data (correlated D0s) to CLEO III data (uncorrelated D0s). The errors onthe predictions are propagated from the total errors on rK∗K and δK∗K . The errorson the ratios from data are statistical errors.

136

6 Conclusions

In an analysis using 9.0 fb−1 of data collected at√

s ≈ 10.58 GeV with the CLEO III

detector, we find the strong phase difference δD ≡ arg(

A(D0→K∗+K−)

A(D0→K∗+K−)

)= δK∗−K+ −

δK∗+K− = 332◦ ± 8◦ ± 11◦ and relative amplitude rD ≡∣∣∣A(D

0→K∗+K−)A(D0→K∗+K−)

∣∣∣ =aK∗−K+

aK∗+K−=

0.52 ± 0.05 ± 0.04. This measurement indicates significant destructive interference

between D0 → K+(K−π0)K∗− and D0 → K−(K+π0)K∗+ in the Dalitz plot region

where these two modes overlap.

The fit includes the K∗± and φ resonances and a non-resonant amplitude. The

measured fit fractions for each resonance (with statistical uncertainty only) are (46.1

± 3.1)% for the K*+, (12.3 ± 2.2)% for the K*−, (14.9 ± 1.6)% for the φ, and (36.0

± 3.7)% for the non-resonant contribution.

A second fit including the scalar κ± (mass 878 MeV/c2, width 499 MeV/c2) am-

plitudes instead of a non-resonant amplitude results in a strong phase difference of

δD = 313◦ ± 9◦ (stat.) and an amplitude ratio of rD = 0.52 ± 0.05 (stat.) The

measured fit fractions for each resonance (with statistical uncertainty only) are (48.1

± 4.5)% for the K*+, (12.9 ± 2.6)% for the K*−, (16.1 ± 1.9)% for the φ, (12.6 ±5.8)% for the κ+, and (11.1 ± 4.7)% for the κ−.

We also investigate the D0 → K+K−π0 Dalitz plot from 281 pb−1 of data collected

at√

s ≈ 3.77 GeV with the CLEO-c detector. We find that the Dalitz plot exhibits

the same behavior as the Dalitz plot seen in the CLEO III analysis, which is confirmed

by a fit to the data.

137

A The K-matrix Formalism

In two-body scattering of the type ab → cd, the cross section in the partial wave J

is [16]:

σJfi =

(4π

q2i

)(2J + 1)|T J

fi(s)|2, (A.1)

where ‘i’ and ‘f ’ stand for the initial and final states, s is the square of the center-of-

mass energy, qi is the breakup momentum (see Equation 2.5 for the definition) in the

initial system, and T Jfi is the transition amplitude from the initial state to the final

state. In the following sections, it is important to note that we suppress references to

the total angular momentum J . Therefore, a separate K-matrix is required for each

partial wave.

A.1 The K operator

The scattering operator, S, is the operator which gives the amplitude, Sfi, that an

initial state |i〉 will be found in the final state |f〉 [16]:

Sfi = 〈f |S|i〉. (A.2)

We may separate S into a part where the initial and final states do not interact (the

identity operator, I) and a part that defines the possible transitions, which we define

138

to be the transition operator T :

S = I + 2iT . (A.3)

The factors of 2 and i (here i refers to the imaginary number√−1) have been intro-

duced for convenience. Since probability must be conserved, the scattering operator

S must be unitary:

SS† = S†S = I . (A.4)

We can multiply each side of Equation A.3 by its conjugate transpose (on the left,

then separately on the right) and we find:

T − T † = 2iT †T = 2iT T † (A.5)

We may write this in terms of inverse operators with a small amount of algebraic

manipulation, and then transform it further:

(T †)−1 − T−1 = 2iI

(T−1 + iI)† = T−1 + iI (A.6)

This allows us to introduce a new hermitian operator, K defined as:

K−1 = T−1 + iI (A.7)

Then we may eliminate the inverse operators in Equation A.7 through multiplying

by K and T from left and right and vice versa:

T = K + iT K = K + iKT (A.8)

139

Notice that T and K commute.

Both T and S may now be expressed only in terms of K:

T = K(I − iK)−1

= (I − iK)−1K, (A.9)

and S = (I + iK)(I − iK)−1

= (I − iK)−1(I + iK) (A.10)

A.2 The K-matrix

Resonances appear as a sum of poles in the K-matrix, which contains the informa-

tion regarding how the K operator relates initial to final states. When resonances

dominate the scattering amplitudes we write the K-matrix elements as [16]:

Kij =∑

R

gRi(m)gRj(m)

m2R − m2

, (A.11)

where

g2Ri(m) = mRΓRi(m). (A.12)

The sum is over the number of resonances each with mass mR which decay to two-

body channels i and j, and ΓRi(m) is the partial width for the scattering of the initial

state through the resonance R. The partial widths are given by the expression:

ΓRi(m) =ΓRi(mR)

ρi(mR)[F

(J)Ri (m)]2ρi(m), (A.13)

140

where

ρi(m) =2pi

m, (A.14)

F(J)Ri is the normalized Blatt-Weisskopf barrier factor of Table 2.1 for the decay of

R (spin J) to decay channel i, and pi is the breakup momentum for the decay of

resonance R into decay channel i. It should be noted that ΓRi(mR) is precisely the

measured decay width of resonance R to channel i.

A.3 K-matrix Examples and Argand Plots

A.3.1 One Pole, One Decay Channel

For only one resonance which decays to a kaon and a π0 at pole mass mR, K is simply

written as:

KR→Kπ=mRΓR(mKπ)

m2R − m2

Kπ

(A.15)

We may determine the T -matrix which contains the information about the transition

from initial to final states from Equation A.9 and we find:

TR→Kπ =mRΓR(mKπ)

m2R − m2

Kπ − imRΓR(mKπ)(A.16)

As with the Breit-Wigner formalism, we expect the rate at a particular value of m2Kπ

to be proportional to the amplitude squared, |T |2. Also, we can plot the imaginary

part of T (Im[T ]) vs. the real part of T (Re[T ]) for all values of m2Kπ in order to

confirm that unitarity is conserved. We call this type of plot an Argand plot.

The Argand plot for the decay of the K∗(892) resonance to Kπ is shown in

Figure A.1. Note that the Argand plot is a circle of unit diameter with its center at

141

-0.4 -0.2 0.2 0.4

0.2

0.4

0.6

0.8

1 2 3 4 5

0.2

0.4

0.6

0.8

1

1.2

mKπ

2

|T|2

Re[T]

Im[T]

mKπ

2 = mK*(892)

2

Re[T] = 0

1

Figure A.1: (left) An Argand Plot of the K∗(892) T -matrix for all values of m2Kπ.

(right) A plot of |T |2 vs. m2Kπ.

{Re[T ] = 0, Im[T ] = 12}. This circle is called the unitarity circle, and the physically

allowed values of T should remain at or within this circle, in order to preserve the

unitarity of S.

The Argand plot for the decay of the K∗(1410) resonance to Kπ is shown in

Figure A.2. Even though the resonance has a different mass and width, the Argand

plot looks identical to the Argand plot for a lone K∗(892) resonance.

A.3.2 Two Poles, One Decay Channel

Naively, we could add the amplitudes (or T -matrices, in this case) for the K∗(1410)

and K∗(892) in order to determine the combined amplitude for the decays of the two

Kπ resonances, as we do in the Breit-Wigner formalism. We can see the Argand

plot for the sum of T -matrices in Figure A.3. It is very clear that unitarity is no

longer conserved, as the path of the Argand plot is not the circular shape we expect.

The Argand plot does go near {Re[T ] = 0, Im[T ] = 1}, but not at the peaks of

142

-0.4 -0.2 0.2 0.4

0.2

0.4

0.6

0.8

mKπ

2

Re[T]

Im[T]

mKπ

2 = mK*(1410)

2

Re[T] = 0

1 2 3 4 5

0.2

0.4

0.6

0.8

1

1.2

|T|2

1

Figure A.2: (left) An Argand Plot of the K∗(1410) T -matrix for all values of m2Kπ.

(right) A plot of |T |2 vs. m2Kπ.

mKπ

2

Re[T]

Im[T]Re[T] = 0 (not exactly at the peaks...)

-0.4 -0.2 0.2 0.4 0.6

0.2

0.4

0.6

0.8

1 2 3 4 5

0.2

0.4

0.6

0.8

1

1.2

T ≠ 0 (!)

1

|T|2

Figure A.3: (left) An Argand Plot of T = TK∗(892)+TK∗(1410) for all values of m2Kπ < 5.

(right) A plot of |TK∗(892) + TK∗(1410)|2 vs. m2Kπ.

143

mKπ

2

Re[T]

Im[T] Re[T] = 0

-0.4 -0.2 0.2 0.4

0.2

0.4

0.6

0.8

1

1 2 3 4 5

0.2

0.4

0.6

0.8

1

1.2

T = 0

|T|2

Figure A.4: (left) An Argand Plot of T calculated from KK∗(892) + KK∗(1410) for allvalues of m2

Kπ < 5. (right) A plot of |T |2 vs. m2Kπ.

the resonances. Also the Argand plot noticeably does not pass through {Re[T ] =

0, Im[T ] = 0} between the two resonances, as would be expected if we stayed on

the unitarity circle. The K∗(892) and K∗(1410) are separated in m2Kπ enough where

|T |2 is not affected too much, but for resonances which are close to each other the

preservation of unitarity decreases further.

We can preserve unitarity by adding the pole terms in the K-matrix and then

finding the associated T -matrix, instead of adding the T -matrices for the individual

resonances. For two resonances at different pole masses, both decaying to Kπ we can

define the K-matrix as the combination of the K-matrices for the scattering through

resonance R and resonance Q:

KKπ = KR→Kπ + KQ→Kπ, (A.17)

where the form of KQ→Kπ is the same as Equation A.15 (Qs replacing Rs).

144

The Argand plot of T when derived from the addition of K∗(892) and K∗(1410)

K-matrices is shown in Figure A.3. Unitarity is now preserved, and |T |2 becomes

zero between the two resonances.

A.4 Application to Dalitz Plot Analysis

A.4.1 Lorentz-Invariant T -Matrix

The T -matrix as we have defined it so far is not Lorentz invariant. The Lorentz

invariant T -matrix, which we call T , is given by [16]:

T = (ρ)−12 T (ρ)−

12 , (A.18)

where the “phase-space matrix” is a diagonal matrix which has:

ρii =2pi

m, (A.19)

where pi is the breakup momentum of channel i. All of the other elements of ρ are

zero.

The Lorentz invariant K-matrix, K, is given by:

K = (ρ)−12 K(ρ)−

12 , (A.20)

and then:

T = (I − iKρ)−1K, (A.21)

145

D0 P (1 - iK)-1a

b

c

possibledecay

channels

Final State

rescatteringProduction

Figure A.5: A diagram of the process where a resonance is produced from a D0 decay.The resonance may decay to a number of decay channels, if possible. The amplitudefor the decay channel with particles a and b can be found in the ab component of thefinal state vector �F .

A.4.2 The �P -Vector Formalism

Strictly, the K-matrix formalism applies only to s-channel resonances observed in the

two-body scattering of the type ab → cd. However, if we assume that the two-body

system in the final state is an isolated one and that the two particles do not interact

with the rest of the final state, we may generalize the K-matrix formalism to the

production of resonances.

We may define the production vector �P [17] as

Pi =∑

R

β0RgRi(m)

m2R − m2

, (A.22)

where we require the poles of the �P -vector to be the same as the pole of the K-matrix.

�P is a n-dimensional vector where n is the number of decay channels. The factors β0R

are complex amplitudes with units of energy which carry the production information

of the resonance R.

The production vector, �P , may be thought of as the virtual two-particle states

which the resonances decay to (see Figure A.5). These virtual states scatter to the

146

final state which is described by the final state vector, �F :

�F = (I − iK)−1 �P = TK−1 �P . (A.23)

A.4.3 Practical Application

In its Lorentz invariant form, the production vector is written as:

�P = ρ− 12 �P , (A.24)

and the final state vector is written as:

�F = ρ− 12 �F , (A.25)

so the invariant final state vector can be written simply as:

�F = T K−1 �P. (A.26)

One Pole, One Channel

From Equations A.15, A.22, and A.24 we can determine the �P -vector for a single

resonance decaying to the Kπ channel:

PKπ =β0

R

√mRΓR→Kπ(mKπ)

(m2R − m2

Kπ)√

ρKπ

. (A.27)

We can rescale β0R as follows:

βR =β0

R√mR

ΓR→Kπ(mR)ρKπ(mR)

(A.28)

147

and then use Equations A.26 and A.28 to get the �F -vector:

FKπ =βRmR

ΓR→Kπ(mR)ρKπ(mR)

m2R − m2

Kπ − imRΓR(mKπ)× F

(J)R→Kπ(m) (A.29)

Excluding the barrier factor, Equation A.29 is the K-matrix justification for the Breit-

Wigner model we define in Chapter 2, since the amplitude can always have arbitrary

normalization in a Dalitz plot analysis, so long as the probability distribution function

for the fit is properly normalized.

Two Poles, Two Channels

One of the motivations for thinking about the K-matrix formalism was the small dip

at m2Kπ ≈ 1 GeV2 /c4 in our data as seen in the m2

Kπ projections of Figure 4.11. Also

we wondered if the dip affected our measurements of rD and δD. Our concerns were

mostly resolved by changing the fitting axes from (m2K−π0 ,m2

K+π0) to (m2K+π0 ,m2

K+K−)

and by evaluating the systematic errors due to fitting different regions of the Dalitz

plot where the dip was not a factor. However, at the time we wanted to see how using

the K-matrix formalism with the �P -vector approximation could help us fit our data.

We believed that it was possible that the dip near m2Kπ ≈ 1 GeV2 /c4 may have

had to do with the interference of two J = 1 resonances in the Kπ decay channel. We

were certain that one of those resonances was the K∗(892) that we could clearly see,

and we thought the interfering resonance would be the next higher K∗ resonance, the

K∗(1410). We also thought that the preferred decay of the K∗(1410) to the K∗(892)π

channel might have the effect of causing a dip in the amplitude at K∗(892)π threshold

which is just above 1 GeV2 /c4.

To construct the �P -vector which represents the K∗(892) and K∗(1410) decays to

Kπ and K∗π, we should note that we may separate the �P -vector into two parts which

148

each represent the two resonances:

Pi = PK∗(892)i + PK∗(1410)i, (A.30)

where we define:

PRi =β0

R

√mRΓR→i(m)

(m2R − m2)

. (A.31)

for resonance R and decay channel i.

We choose the first �P -vector element to represent the Kπ decay channel, and

the second �P -vector element represents the K∗(892)π decay channel. The K∗(892)

resonance cannot decay to the K∗(892)π channel, so there is only one non-zero element

in the K∗(892) �P -vector:

�PK∗(892) =

⎛⎜⎝ β0

K∗(892)

√mK∗(892)ΓK∗(892)→Kπ(m)

0

⎞⎟⎠ × 1

(m2K∗(892) − m2)

. (A.32)

The K∗(1410) resonance can decay the both channels, so we can write the corre-

sponding �P -vector as:

�PK∗(1410) =

⎛⎜⎝ β0

K∗(1410)

√mK∗(1410)ΓK∗(1410)→Kπ(m)

β0K∗(1410)

√mK∗(1410)ΓK∗(1410)→K∗π(m)

⎞⎟⎠ × 1

(m2K∗(1410) − m2)

.(A.33)

The amplitude for these J = 1 resonances decaying to the Kπ channel is then

simply the first component of the corresponding �F -vector which comes from the total

�P -vector (�PK∗(892)+ �PK∗(1410)). The resulting amplitude is different for varying choices

of partial widths and βs. An example of the type of amplitude with a dip near 1

GeV is shown in Figure A.6. In the case of maximum destructive interference, the

relative complex phase between β0K∗(892) and β0

K∗(1410) is equal to zero, and the �F -

149

0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1

mKπ

2 (GeV2/c4)

|FK

π|2

Figure A.6: A plot of the magnitude of the �F -vector amplitude to the Kπ decaychannel squared vs. Kπ invariant mass squared for destructively interfering K∗ res-onances.

vector is exactly zero in between the resonances. The partial widths used are shown

in Table A.1. The magnitudes of the βs were arbitrarily set to make |FKπ|2 ≈ 1 at

the resonance peaks. In an actual Dalitz plot analysis, of course, the βs would be

free parameters in the fit.

We attempted a fit (to the Dalitz plot shown in Section 4.4) which used the

above P -vector formalism for the Kπ resonances, and also had a Breit-Wigner φ

Resonance (R) Decay Channel (i) Partial Width (ΓRi(mR))K∗(892) Kπ 52.7 MeV /c2

K∗(1410) Kπ 15.2 MeV /c2

K∗(1410) K∗π 190.9 MeV /c2

Table A.1: Partial widths used to calculate �F for interfering K∗ resonances.

150

resonance and a non-resonant contribution. The only differences in the P -vector

formalism we used for the fit are that we chose ΓK∗(1410)→Kπ = 15.0 MeV /c2 and

ΓK∗(1410)→K∗π = 200 MeV /c2. We fitted to the Dalitz plot variables (m2K−π0 ,m2

K+π0)

in this example, and the projections of the fit are shown in Figure A.7.

The dip could also be the result of interference between S-wave resonances. In

a future high-statistics Dalitz plot analysis of D0 → K+K−π0 it will be worth con-

sidering that the dip could possibly be an interference in the S-wave for the κ and

K∗0(1430) resonances, which could be fit with a K-matrix parametrization [56]. This

may possibly involve a large coupling of the K∗0(1430) to Kη′ (mη′ = 957.78 ± 0.14

MeV, Γη′ = 0.203 ± 0.016 MeV [1]). Another interpretation of the dip from Refer-

ence [56] is a possible preference for resonances which would decay to Kπ to decay

to Kη instead (mη = 547.51 ± 0.18 MeV, Γη = 1.30 ± 0.07 keV [1]). If this is the

case, there would be an enhancement at Kη threshold in the D0 → K+K−η Dalitz

plot. However, this interpretation is considered to be less likely since the coupling of

a J = 0, P = +1 resonance1 to Kη is expected to be suppressed [56].

1J = Total angular momentum, P = Parity eigenvalue

151

mK

+K-2 (G

eV2 /c

4 )m

K-π

02 (GeV

2 /c4 )

mK

+π

02 (GeV

2 /c4 )


4

Fig

ure

A.7

:A

nex

ample

ofa

fit

usi

ng

the

P-v

ecto

rfo

rmal

ism

.

152

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155

Author’s Biography

Paras Naik was born in Elmhurst, Illinois on 4 May 1978 to Pradeep and Aruna Naik

of Wood Dale, Illinois. After almost two years his younger brother, Samar, joined the

family. When he was three years old he moved to Addison, Illinois, which became his

hometown. During his formative years, he attended Lincoln Elementary School and

Indian Trail Junior High School. He then proceeded to Addison Trail High School

and graduated at the top of his class. He entered the University of Illinois at Urbana-

Champaign in the fall of 1996. He received his Bachelor’s Degree in Engineering

Physics with Honors in January 2000. He continued at the University of Illinois and

received his Master’s Degree in Physics in May 2001 while being active in the Physics

Education Research Group. He pursued his Ph.D. in Physics by participating in the

High Energy Physics Group. During his time with the group he worked on the CLEO

experiment in Ithaca, NY, where he resided between August 2003 and July 2005.

Upon his return he successfully published a research paper entitled “Measurement of

Interfering K∗+K− and K∗−K+ Amplitudes in the Decay D0→K+K−π0”2 with the

assistance of the CLEO Collaboration. He will continue particle physics research as

a Postdoctoral Fellow with Carleton University.

2C. Cawlfield et al. (CLEO collaboration), Phys. Rev. D 74, 031108(R) (2006).

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