ph.d. thesisepx.phys.tohoku.ac.jp/eeweb/paper/2011_dthesis_yhorii.pdfph.d. thesis first evidence of...

Ph.D. Thesis

First Evidence of the Suppressed B Meson Decay

B− → DK− Followed by D → K+π−

and Extraction of the CP -Violating Angle φ3

（B中間子の稀崩壊B− → DK−, D → K+π−

の初の証拠とCP 非保存角φ3の測定）

Department of Physics, Tohoku University

Horii Yasuyuki

December 2010

Abstract

We report the first evidence for the suppressed decay chain B− → DK− followedby D → K+π−, where D indicates a D0 or D0 state, with a significance of 4.1σ. Aninterference of the two decay paths, corresponding to the two intermediate D states,provides an important information on the CP -violating angle φ3. We measure the ratioRDK of the suppressed decay rate to the favored decay rate, obtained in the chainB− → DK−, D → K−π+, and the asymmetry ADK between the charge-conjugatedecays of the suppressed mode as

RDK = [1.63+0.44−0.41(stat)+0.07

−0.13(syst)] × 10−2,

ADK = −0.39+0.26−0.28(stat)+0.04

−0.03(syst).

The results are based on a data sample recorded with the Belle detector at the KEKBe+e− storage ring, corresponding to 772 × 106 BB pairs produced at the Υ(4S) reso-nance.

A method based on Bayesian statistics including the observables for the chainsB− → DK− followed by D → CP -eigenstates provides

φ3 = 89 +13

−16

(modulo 180), which is consistent with the expectation obtained from the Cabibbo-Kobayashi-Maskawa framework in the standard model. Our study strongly encouragesthe precise measurements of φ3 at the next-generation B factories, which will play animportant role for investigating new physics.

1

Acknowledgements

I would like to express my deep appreciation to my supervisor, Prof. H. Yamamoto,for his great guidance and supervision. His teachings are definitely essential for my un-derstanding of quantum field theory, phenomenology of the B mesons, and accelerator-based experiments.

I am heartily thankful to an analysis coordinator of the Belle experiment, Dr. K.Trabelsi, for his bright ideas and suggestions for the analyses, the presentations, andthe publications. This thesis would not have been possible without his supports.

I appreciate the members of Tohoku University. Dr. T. Sanuki, Dr. T. Nagamine,Dr. Y. Onuki, and Dr. Y. Takubo, for their valuable assistance and encouragement.Prof. A. Yamaguchi and Mr. N. Kikuchi, for their great introduction on particlephysics. Mr. K. Kousai, Ms. Y. Minekawa, Mr. H. Tabata, Mr. K. Ito, Mr. T.Kusano, Mr. R. Sasaki, Mr. K. Itagaki, Mr. D. Okamoto, Mr. Y. Sato, Mr. Z. Suzuki,Mr. T. Honda, Mr. H. Nakano, Mr. K. Negishi, Mr. T. Saito, Mr. D. Kamai, Ms. E.Kato, Mr. H. Katsurayama, Ms. M. Kikuchi, and Mr. Y. Ono, as very nice colleagues.

I am grateful to the members of the Belle collaboration. Prof. Y. Sakai, for his vitalguidance from the initial to the final level. Prof. T. E. Browder, Dr. Z. Dolezal, Dr. N.Muramatsu, Prof. G. Mohanty, Dr. A. Zupanc, Mr. M. Prim, Dr. M. Nakao, Dr. H.Ozaki, Dr. K. Sakai, Ms. M. Watanabe, Dr. N. Taniguchi, and Dr. M. Iwabuchi, fortheir great contributions to the analysis. All the other members, for keeping high qualityof the detector and the software framework. I also appreciate the KEKB acceleratorgroup, for the high luminosity provided by the excellent operation.

Last but not the least, I would like to express my gratitude to my parents, brother,and wife. The project would not have been completed without their moral support.

Contents

1 Introduction 11

2 KM Mechanism and CP -Violating Angle φ3 132.1 Interactions between Quarks and W Bosons . . . . . . . . . . . . . . . 132.2 KM Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Unitarity Triangle and CP -Violating Angles . . . . . . . . . . . . . . . 152.4 Methods for Measuring φ3 . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 GLW Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 ADS Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.3 Dalitz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Measurements of φ3 as of Summer 2010 . . . . . . . . . . . . . . . . . . 202.5.1 Result of Dalitz Method on B− → [KSπ+π−]DK− . . . . . . . . 202.5.2 Comparison of Constraints on CKM Parameters . . . . . . . . . 21

2.6 Motivation to Analyze B− → [K+π−]DK− . . . . . . . . . . . . . . . . 22

3 Experimental Apparatus and Analysis Tools 233.1 KEKB Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Belle Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Silicon Vertex Detector (SVD) . . . . . . . . . . . . . . . . . . . 273.2.2 Central Drift Chamber (CDC) . . . . . . . . . . . . . . . . . . . 293.2.3 Aerogel Cherenkov Counter (ACC) . . . . . . . . . . . . . . . . 293.2.4 Time-of-Flight Counter (TOF) . . . . . . . . . . . . . . . . . . 323.2.5 Electromagnetic Calorimeter (ECL) . . . . . . . . . . . . . . . . 343.2.6 KL and Muon Detector (KLM) . . . . . . . . . . . . . . . . . . 353.2.7 Trigger and Data Acquisition . . . . . . . . . . . . . . . . . . . 36

3.3 Analysis Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.1 K±/π± Identification . . . . . . . . . . . . . . . . . . . . . . . . 373.3.2 Continuum Suppression . . . . . . . . . . . . . . . . . . . . . . 38

4 Event Selection 414.1 Charged Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Reconstruction of D Meson . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Reconstruction of B Meson . . . . . . . . . . . . . . . . . . . . . . . . 42

3

4 CONTENTS

4.4 Rejection of Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . 434.4.1 B− → [K+K−]Dπ− . . . . . . . . . . . . . . . . . . . . . . . . . 434.4.2 B− → [K−π+]Dh− . . . . . . . . . . . . . . . . . . . . . . . . . 434.4.3 Continuum Background . . . . . . . . . . . . . . . . . . . . . . 45

5 Discrimination of Continuum Background 475.1 Likelihood Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Introduction of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3 NeuroBayes Neural Network . . . . . . . . . . . . . . . . . . . . . . . . 515.4 Neural Network Training . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.4.1 Contribution of Each Variable . . . . . . . . . . . . . . . . . . . 555.4.2 Check of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.5 Tests on Several Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 565.6 Test by Removing Three Variables . . . . . . . . . . . . . . . . . . . . 57

6 Signal Extraction 596.1 Probability Density Function (PDF) . . . . . . . . . . . . . . . . . . . 59

6.1.1 Functions for ∆E . . . . . . . . . . . . . . . . . . . . . . . . . . 596.1.2 Functions for NB . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Signal Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2.1 Fit to Favored Modes . . . . . . . . . . . . . . . . . . . . . . . . 656.2.2 Fit to Suppressed Modes . . . . . . . . . . . . . . . . . . . . . . 65

7 Estimation of Peaking Backgrounds 757.1 Peaking Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2 Estimation Using the M(Kπ) Sidebands . . . . . . . . . . . . . . . . . 767.3 Check Using MC Samples . . . . . . . . . . . . . . . . . . . . . . . . . 79

8 Observables Relative to Angle φ3 838.1 Detection Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.2 Ratio RDh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 838.3 Asymmetry ADh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 858.4 Comparison of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

9 Extraction of Angle φ3 89

10 Conclusion 93

A Details on Neural Network Training 95

B Demonstration of the Fit to MC Samples 109B.1 Fit to Favored Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109B.2 Fit to Suppressed Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 109

List of Figures

2.1 Unitarity Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Diagrams for the B− → D0K− and B− → D0K− Decays . . . . . . . . 162.3 Complex Triangles of Amplitudes in the GLW Method . . . . . . . . . 172.4 Complex Triangles of Amplitudes in the ADS Method . . . . . . . . . . 182.5 Result of the Dalitz-Plot Analysis for B− → D(∗)K−, D → KSπ+π− . . 202.6 Constraints on the Unitarity Triangle Provided by the CKMfitter Group 212.7 Diagrams for the Decay B− → DK−, D → K+π− . . . . . . . . . . . . 22

3.1 KEKB Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 History of the Integrated Luminosity . . . . . . . . . . . . . . . . . . . 253.3 Cross Section for e+e− → Hadrons around the Υ Resonances . . . . . . 253.4 Belle Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Configuration of SVD1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.6 Configuration of SVD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.7 Overview of the CDC Structure . . . . . . . . . . . . . . . . . . . . . . 303.8 The pt Resolution and the dE/dx Measurement at CDC . . . . . . . . 303.9 Arrangement of ACC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.10 Distributions of the Number of Photoelectrons Obtained from ACC . . 323.11 Dimension of a TOF/TSC Module . . . . . . . . . . . . . . . . . . . . 333.12 Mass Distribution Obtained from TOF Measurements . . . . . . . . . . 333.13 Overall Configuration of ECL . . . . . . . . . . . . . . . . . . . . . . . 343.14 Schematic Diagrams of KLM Counters . . . . . . . . . . . . . . . . . . 353.15 Schematic View of the Trigger System . . . . . . . . . . . . . . . . . . 363.16 Overview of the Data Acquisition System . . . . . . . . . . . . . . . . . 373.17 Scatter Plot of the Track Momentum and the Likelihood Ratio P (K/π) 383.18 Momentum Coverage of the Detectors Used for K±/π± Separation . . . 39

4.1 Distributions of P (K/π) and M(Kπ) for a Signal MC Sample . . . . . 424.2 Distributions of Mbc and ∆E for a Signal MC Sample . . . . . . . . . . 434.3 Distributions of M(KK) for the Veto of B− → Dπ−, D → K+K− . . . 444.4 Distributions of M(Kπ)exchanged for the Veto of B− → DK−, D → K−π+ 444.5 Distributions of M(Kπ)exchanged for the Veto of B− → Dπ−, D → K−π+ 454.6 Distributions of ∆M for the Veto of D− → D0π− . . . . . . . . . . . . 46

5

6 LIST OF FIGURES

5.1 Distributions Relative to KSFW . . . . . . . . . . . . . . . . . . . . . . 485.2 Distributions of | cos θB| . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Discrimination of Continuum Background by Likelihood Method . . . . 495.4 Distributions of the Inputs in the Neural Network . . . . . . . . . . . . 525.4 Distributions of the Inputs in the Neural Network (Continued) . . . . . 535.5 Discrimination of Continuum Background by Neural Network Method . 545.6 Check of the Stability of the Neural Network . . . . . . . . . . . . . . . 565.7 Comparison of the Neural Network Outputs for Several Samples . . . . 575.8 Neural Network without Three Inputs . . . . . . . . . . . . . . . . . . . 57

6.1 Two-Dimensional Distributions on ∆E and NB for MC Samples . . . . 606.2 Distributions of ∆E for Signal MC Samples . . . . . . . . . . . . . . . 616.3 Distributions of ∆E for MC Samples of the Dh Feed-Across . . . . . . 626.4 Distributions of ∆E for MC Samples of the BB Background . . . . . . 626.5 Distributions of ∆E for MC Samples of the qq Background . . . . . . . 636.6 Distributions of NB for Signal MC Samples . . . . . . . . . . . . . . . 646.7 Distributions of NB for MC Samples of the qq Background . . . . . . . 646.8 Result of the Fit to the Favored Modes . . . . . . . . . . . . . . . . . . 666.9 Projections of ∆E and NB for the Favored Dπ Sample . . . . . . . . . 676.10 Projections of ∆E and NB for the Favored DK Sample . . . . . . . . . 686.11 Result of the Fit to the Suppressed modes . . . . . . . . . . . . . . . . 706.12 Projections of ∆E and NB for the Suppressed Dπ Sample . . . . . . . 716.13 Projections of ∆E and NB for the Suppressed DK Sample . . . . . . . 72

7.1 Distribution of M(Kπ) for a MC Sample of B− → Dπ−, D → K+K− . 767.2 Distribution of M(Kπ) for a MC Sample of B− → DK−, D → K−π+ . 777.3 Distribution of M(Kπ) for a MC Sample of B− → K+K−π− . . . . . . 777.4 Projections of ∆E and NB for the M(Kπ) Sidebands of DK . . . . . . 787.5 Projections of ∆E and NB for the M(Kπ) Sidebands of Dπ . . . . . . 787.6 Projections of ∆E and NB for a MC Sample of B− → Dπ−, D → K+K− 807.7 Projections of ∆E and NB for a MC Sample of B− → DK−, D → K−π+ 807.8 Projections of ∆E and NB for a MC Sample of B− → Dπ−, D → K−π+ 81

8.1 Distribution of√

−2 ln (L/Lmax) for Estimating the Significance . . . . 85

9.1 Result of the φ3 Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

A.1 Distributions of Network Output and Purity Curve for MC Samples . . 96A.2 Correlation Matrix of the Input Variables . . . . . . . . . . . . . . . . . 97A.3 Distributions in the Training Relative to LR(KSFW) . . . . . . . . . . 98A.4 Distributions in the Training Relative to ∆z . . . . . . . . . . . . . . . 99A.5 Distributions in the Training Relative to cos θK

D . . . . . . . . . . . . . 100A.6 Distributions in the Training Relative to | cos θB| . . . . . . . . . . . . . 101A.7 Distributions in the Training Relative to rtag . . . . . . . . . . . . . . . 102

LIST OF FIGURES 7

A.8 Distributions in the Training Relative to | cos θT| . . . . . . . . . . . . . 103A.9 Distributions in the Training Relative to dDh . . . . . . . . . . . . . . . 104A.10 Distributions in the Training Relative to QBQK . . . . . . . . . . . . . 105A.11 Distributions in the Training Relative to cos θD

B . . . . . . . . . . . . . 106A.12 Distributions in the Training Relative to ∆Q . . . . . . . . . . . . . . . 107

B.1 Result of the Fit to the Favored Modes on MC . . . . . . . . . . . . . . 110B.2 Projections of ∆E and NB for the Favored Dπ Sample on MC . . . . . 111B.3 Projections of ∆E and NB for the Favored DK Sample on MC . . . . 112B.4 Result of the Fit to the Suppressed Modes on MC . . . . . . . . . . . . 114B.5 Projections of ∆E and NB for the Suppressed Dπ Sample on MC . . . 115B.6 Projections of ∆E and NB for the Suppressed DK Sample on MC . . . 116

List of Tables

5.1 Performance of the Discrimination of the Continuum Background . . . 545.2 Importance and Correlations of the Inputs in the Neural Network . . . 55

6.1 List of the Parameters in the Fit to the Favored Modes . . . . . . . . . 696.2 List of the Parameters in the Fit to the Suppressed Modes . . . . . . . 73

8.1 Summary of the Efficiency, the Yield, and the Observable RDh . . . . . 858.2 Summary of the Systematic Uncertainties for RDh and ADh . . . . . . 868.3 Comparison of the Results for RDh and ADh . . . . . . . . . . . . . . . 87

A.1 List of the Input Variables . . . . . . . . . . . . . . . . . . . . . . . . . 95

B.1 List of the Parameters in the Fit to the Favored Modes on MC . . . . . 113B.2 List of the Parameters in the Fit to the Suppressed Modes on MC . . . 117

9

Chapter 1

Introduction

For a long time, the science has been taken as an empirical and experimental activityinto the nature. On the other hand, the metaphysics is one of non-empirical inquiriesand known as “natural philosophy.” Recently, the development of philosophy has pro-vided the idea that the science is in fact a part of metaphysics. This concept is basedon the standpoint that the science intends to connect the scientific knowledge to thereality of the nature whereas this connection can not be confirmed even empirically orexperimentally.

However, the resulting knowledge of the science has actually revolutionized ourview of the world, and transformed our society. Even if we can not reach the truth,the development of science provides effective insights into the nature and fertilizes ourlives.

The particle physics is a primary means of inquiry into the basic working of thenature. In this field, every phenomena are ruled by universal laws and principles whosenatural realm is at scales of time and distance far removed from our direct experience. Inthe twentieth century, a standard model has been developed with various confirmationsof many of its aspects. In this framework, the matters are constructed from quarks andleptons, and the strong and electroweak interactions are mediated by gauge bosons.

One of the great constitutions of the standard model is the Cabibbo-Kobayashi-Maskawa (CKM) picture [1, 2], which was proposed by M. Kobayashi and T. Maskawain 1973 by extending the idea of the quark mixing by N. Cabibbo. This picture nat-urally explains the CP violation in K0 decays observed in 1964 [3], by introducingthe third generation of quarks. In 1977 and 1995, bottom and top quarks, known asthird-generation quarks, are observed, respectively [4, 5, 6]. A conclusive experimentalresult is the CP violation in B0 decays observed in 2001 [7, 8]. Up to now, plenty ofexperimental outputs agree with the CKM picture in high precisions.

Despite the great success, there are various reasons for the standard model notto be an ultimate theory. An example is the fact that there are several absences ofexplanations on such as dark matter, gravitation, etc. Nowadays, precise measurements

11

12 CHAPTER 1. INTRODUCTION

of the CKM parameters are considered to be ones of methods for revealing the extendedphysics called “new physics.” Any significant discrepancy between the expected andmeasured values would be a signature of new physics.

The parameter φ3 (also known as γ) is one of the worst-determined CKM parametersand further constraints are strongly demanded. Since the measurement of φ3 is obtainedtheoretically cleanly from the tree-dominated decays, it provides a unique basis forsearching for new physics.

In this thesis, we report the first evidence of the suppressed decay chain1 B− → DK−

followed by D → K+π−, where D indicates a D0 or D0 state. The two decay paths,corresponding to the two intermediate D states, interfere and provide important infor-mation on φ3. We use a data sample, that contains 772 × 106 BB pairs, recorded atthe Υ(4S) resonance with the Belle detector at the KEKB e+e− collider. For discrim-inating the large backgrounds from e+e− → qq (q = u, d, s, c) processes, we employ anew method based on a neural network technique.

In Chapter 2, we introduce the Kobayashi-Maskawa (KM) mechanism and the meth-ods for measuring φ3. In Chapter 3, we describe the experimental apparatus and anal-ysis tools. In Chapter 4, 5, and 6, we show the event selection, the discrimination ofthe qq backgrounds, and the signal extraction, respectively. In Chapter 7, we estimatethe contribution from the backgrounds remaining after the rejections. In Chapter 8, wemeasure the observables relative to φ3, and in Chapter 9, we extract φ3.

1 Charge-conjugate modes are implicitly included throughout the thesis unless otherwise stated.

Chapter 2

KM Mechanism and CP -ViolatingAngle φ3

2.1 Interactions between Quarks and W Bosons

In the standard model, the masses of quarks arise from the Yukawa interactions withthe Higgs fields. The Yukawa terms are described by left-handed quark doublet q′L andright-handed down-type (up-type) quark singlet d′

R (u′R) both in the weak-interaction

basis, as

LYukawa,quark = −ydij(q

′Liφ)d′

Rj − yuij(q

′Liϵφ

∗)u′Rj + h.c., (2.1)

where yd and yu are complex constants, i and j are the generation labels, φ is theHiggs doublet, and ϵ is the 2 × 2 antisymmetric tensor. The mass terms are producedby introducing a spontaneous symmetry breaking, where the Higgs field acquires avacuum expectation value ⟨φ⟩ = (0, v/

√2), as

Lmass,quark = −vyd

ij√2

(d′Lid

′Rj) −

vyuij√2

(u′Liu

′Rj) + h.c. (2.2)

The quark states can be transformed by unitary matrices so that the terms for i = jdrop out, leading the mass eigenstates.

The interactions of physical quarks and W bosons are then given by

Lint,qW = − g√2

[(ULγµV DL

)W+

µ +(DLγµV †UL

)W−

µ

], (2.3)

where g is a dimensionless constant, γµ is a Dirac matrix, U shows up-type quarks(u, c, t), D shows down-type quarks (d, s, b), W±

µ are the W -boson fields, and the V isa unitary matrix connecting the mass eigenstates and the weak-interaction states. Thematrix V is called as the CKM matrix [1, 2], and usually expressed as

V =

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

, (2.4)

13

14 CHAPTER 2. KM MECHANISM AND CP -VIOLATING ANGLE φ3

which parameterizes the mixing of quarks in the weak interactions with W bosons.

2.2 KM Mechanism

Under the CP transformation, the Lint,qW is invariant only when V = V ∗, since

(CP )Lint,qW (CP )† = − g√2

[(ULγµV ∗DL

)W+

µ +(DLγµ(V ∗)†UL

)W−

µ

], (2.5)

which is supported by (CP )W+µ (CP )† = −W−

µ and (CP )ψaγµψb(CP )† = −ψbγ

µψa

with ψa,b to be general fermion states. Therefore, the CP violation occurs when one ormore elements of V are complex numbers.

Assuming n generations of quarks, the n-by-n complex matrix V generally has 2n2

of degrees of freedom. Since V is unitary, n2 of them are removed by the unitarityconditions:

n∑j=1

VijV∗kj = δik (i, k = 1, 2, ..., n). (2.6)

Other sources for reducing the degrees of freedom are the phases of quarks, which aremeaningless in the quantum mechanics. Excluding the overall phase, which does notchange V , 2n − 1 degrees of freedom consequently disappear. Then, the degrees offreedom in V are

2n2 − n2 − (2n − 1) = (n − 1)2. (2.7)

Considering the case without complex elements, the unitary condition of V is equiv-alent to the orthogonal conditions:

n∑j=1

VijVkj = δik (i ≤ k). (2.8)

The maximum number of real elements in V is thus

n2 − (n + 1)n

2=

n(n − 1)

2. (2.9)

Therefore, the number of possible complex components in V is obtained from Eq. (2.7)by subtracting Eq. (2.9) as

(n − 1)2 − n(n − 1)

2=

(n − 1)(n − 2)

2. (2.10)

When n = 3, V has at least one complex element, which reveals the CP violation in theinteractions of quarks and W bosons. M. Kobayashi and T. Maskawa won the Nobelprize in physics 2008 for the discovery of the origin of the CP violation which predictsthe existence of at least three families of quarks in nature.

2.3. UNITARITY TRIANGLE AND CP -VIOLATING ANGLES 15

2.3 Unitarity Triangle and CP -Violating Angles

The CKM matrix V can be parameterized as an expansion on λ = sin θC ∼ 0.2 (θC isthe Cabibbo angle),

V =

1 − 12λ2 λ Aλ3(ρ − iη)

−λ 1 − 12λ2 Aλ2

Aλ3(1 − ρ − iη) −Aλ2 1

+ O(λ4), (2.11)

where A, η, and ρ are real parameters having magnitudes O(1) [9]. Of the unitaryconditions in Eq. (2.6), six vanishing combinations indicate the triangles in complexplanes. The most commonly used unitarity triangle (Figure 2.1) arises from

VudV∗ub + VcdV

∗cb + VtdV

∗tb = 0, (2.12)

for which the three terms have comparable magnitudes O(λ3).The angles of the unitarity triangle are defined as

φ1 ≡ arg

(VcdV

∗cb

−VtdV ∗tb

), (2.13)

φ2 ≡ arg

(VtdV

∗tb

−VudV ∗ub

), (2.14)

φ3 ≡ arg

(VudV

∗ub

−VcdV ∗cb

), (2.15)

where φ1, φ2, and φ3 are also known as β, α, and γ, respectively. Since these phasesare fundamental parameters of the standard model, the determinations are of primaryimportance, and any significant discrepancy between the expected and measured valueswould be a signature of new physics.

Figure 2.1: Unitarity triangle.


2.4 Methods for Measuring φ3

In the usual quark phase convention in Eq. (2.11), large complex phases appear onlyin Vub and Vtd. Therefore, the angle φ3 is expressed as

φ3 ∼ −arg(Vub). (2.16)

The measurement of φ3 is equivalent to the extraction of the phase of Vub relative tothe phases of other CKM matrix elements except for Vtd.

Several proposed methods for measuring φ3 exploit the interference in the decayB− → DK− (D = D0 or D0), which occurs when D0 and D0 decay to common finalstate [10, 11, 12, 13]. Figure 2.2 shows the diagrams for B− → DK−. The “tree”contributions enable theoretically clean determinations. The φ3 is extracted with themagnitude of ratio of amplitudes rB and the strong phase difference δB defined by

rB =

∣∣∣∣A(B− → D0K−)

A(B− → D0K−)

∣∣∣∣ , δB = δ(B− → D0K−) − δ(B− → D0K−). (2.17)

These are crucial parameters in the measurement of φ3. The value of rB is thought to bearound 0.1, by taking a product of the ratio of the CKM matrix elements |VubV

∗cs/VcbV

∗us|

and the color suppression factor.

W−

u

b

B−

u

u

c

sK−

D0

W−

u

bB−

u

c

u

s

D0

K−

Figure 2.2: Diagrams for the B− → D0K− and B− → D0K− decays, where theinformation of φ3 is included in the transition b → u.

The methods of measuring φ3 using the decay B− → DK− can be categorized bythe following D decays. In the sections below, we describe several techniques. Notethat the D0–D0 mixings can safely be neglected for current precision of φ3, while it ispossible to take them into account explicitly without a need of time-dependent analysisof the B decay [14].

2.4.1 GLW Method

The Gronau-London-Wyler (GLW) method to extract φ3 use the D decays to the CPeigenstates [11]. In this method, the branching ratios B(B− → D0K−), B(B− →D0K−), and B(B− → DCP+K−) are separately determined, where DCP+ is the CPeigenstate DCP+ = (D0 + D0)/

√2 and can be reconstructed from K+K−, π+π−, etc.

2.4. METHODS FOR MEASURING φ3 17

The phase difference between the amplitudes A(B− → D0K−) and A(B− → D0K−) isδB −φ3, while the one for B+ decays is δB +φ3. Thus, the phase 2φ3 could be extractedby measuring three sides of the triangles in the complex plane as shown in Figure 2.3.

Figure 2.3: Complex triangles of amplitudes in the GLW method.

Unfortunately, a problem arises from the small magnitude of A(B− → D0K−) ex-pected by small rB. Possible two ways of tagging D0 flavor, through semi-leptonicdecays and through hadronic decays, are both likely to be impractical. However, theavailability of using B(B− → DCP−K−), where DCP− = (D0 − D0)/

√2 can be re-

constructed from KSπ0, KSφ, etc., overcomes the problem (dashed lines in Figure 2.3).The additional constraints enable the numerical extraction of φ3 rather than geometricalmethod. Then, the observables

RCP± ≡ B(B− → DCP±K−) + B(B+ → DCP±K+)

B(B− → D0K−) + B(B+ → D0K+)

= 1 + r2B ± 2rB cos δB cos φ3, (2.18)

ACP± ≡ B(B− → DCP±K−) − B(B+ → DCP±K+)

B(B− → DCP±K−) + B(B+ → DCP±K+)

= ±2rB sin δB sin φ3/RCP±, (2.19)

are measured and taken as simultaneous equations for solving for φ3. Possible fourequations could constrain the three unknowns rB, δB and φ3.

Note that the effects of the CP violation to the observables are limited because ofsmall rB, and thus precise measurements of the observables are needed for obtainingeffective constraint on φ3. Also, the observables in Eqs. (2.18) and (2.19) are invariantunder the three transformations

(i) φ3 → δB, δB → φ3,(ii) φ3 → −φ3, δB → −δB,(iii) φ3 → φ3 + π, δB → δB + π,

and the φ3 and δB have eight solutions in the region 0–360.


2.4.2 ADS Method

The effects of the CP violation could be enhanced if the final state is chosen so that theinterfering amplitudes have comparable magnitudes [12]. In the Atwood-Dunietz-Soni(ADS) method, Cabibbo-favored D0 decay and doubly Cabibbo-suppressed D0 decayare chosen as the following D decays to B− → DK−. Denoting the final state of the Ddecay as f , the comparable magnitudes of A(B− → [f ]D0K−) and A(B− → [f ]D0K−)provide a relatively large interfering effect on A(B− → [f ]DK−) as shown in Figure 2.4.Therefore, significant information of φ3 is included in the branching ratio B(B− →[f ]DK−).

Figure 2.4: Complex triangles of amplitudes in the ADS method.

The usual observables are described as

RADS ≡ B(B− → [f ]DK−) + B(B+ → [f ]DK+)

B(B− → [f ]DK−) + B(B+ → [f ]DK+)

= r2B + r2

D + 2rBrD cos (δB + δD) cos φ3, (2.20)

AADS ≡ B(B− → [f ]DK−) − B(B+ → [f ]DK+)

B(B− → [f ]DK−) + B(B+ → [f ]DK+)

= 2rBrD sin (δB + δD) sin φ3/RADS, (2.21)

where rD = |A(D0 → f)/A(D0 → f)| and δD = δ(D0 → f)−δ(D0 → f). By measuringthe observables for two final states for which rD and δD are known, sufficient informationare obtained to solve for φ3. The most important final state f is K+π−, while otherpossibilities are K+ρ−, K+a−

1 , and K∗+π−. Important note is that the observables inthe GLW method, Eqs. (2.18) and (2.19), are possible additional inputs; the combinedapproach is sometimes called as the GLW+ADS method.

2.4. METHODS FOR MEASURING φ3 19

The value of δD for f = K+π− is measured with no fold ambiguity by a fit includingseveral experimental results relative to the D meson decays [15]. Therefore, the numberof solutions of φ3 in 0–360 when we include the observables for f = K+π− could bereduced to two on

φ3 → φ3 + π, δB → δB + π, δD → δD.

2.4.3 Dalitz Method

The above methods can be generalized to the cases for three-body D decays [13].The CP -violating effects appear in the interfering amplitudes in the Dalitz plane.Denoting the particles coming from the D decay d1, d2, and d3, the amplitude ofB− → [d1d2d3]DK− is proportional to

M− = f(s12, s13) + rBei(δB−φ3)f(s12, s13), (2.22)

where sij are defined as Dalitz plot variables sij ≡ (pi+pj)2 with the momenta p1, p2, and

p3 for d1, d2, and d3, respectively, and f(s12, s13) and f(s12, s13) are the amplitudes of thedecays D0 → d1d2d3 and D0 → d1d2d3, respectively. The value of φ3 is extracted by a fiton s12 and s13 or on the Dalitz plot. The method is sensitive to φ3 in parts of the Dalitzplot including the resonances. The most important D decay is the Cabibbo-favoredD → KSπ+π−, which has D → KSρ0, D → K∗±(892)π∓, and higher resonances. Thedecay D → KSK+K− is also an effective mode.

The magnitudes of f(s12, s13) and f(s12, s13) can be determined from the Dalitzplot of the flavor specific D decays, where the flavor is tagged by the charge of thesoft pion in D∗− → D0π−. However, the phases are not measurable without furthermodel-dependent assumptions. A possibility for solving the problem is to assume theBreit-Wigner functions for the resonances, and take a sum with non-resonant term.The phases are obtained simultaneously with the magnitudes by fitting the Dalitz plotof the D decays. However, the model assumption causes nontrivial uncertainty on φ3.

A model-independent approach is to obtain the phases of D-decay amplitudes si-multaneously with φ3, rB, and δB by the fits on binned regions of the Dalitz plot.Enough constraints to solve for φ3, rB, δB, and D-decay parameters should be obtainedby taking sufficiently large number of binned regions. Also, larger number of bins ispreferred to avoid that the interference terms average out by integrating in the regions.Since smaller number of bins is primarily favored to obtain more statistics for each fit,the number of bins should be determined in the trade-off of the above situations.

Another possibility of model-independent method is to obtain the phases of D-decay amplitudes independently at charm factories by using the ψ(3770) resonance.The particle ψ(3770) decays into a pair of D mesons. By measuring the decay rate ofthis coherent state, the phases in the interfering amplitudes can be extracted.


2.5 Measurements of φ3 as of Summer 2010

2.5.1 Result of Dalitz Method on B− → [KSπ+π−]DK−

The measurement of φ3 as of summer 2010 is dominated by the Dalitz method on thedecay B− → [KSπ+π−]DK− [16, 17]. The most precise measurement on this modeis obtained by the Belle collaboration using a data sample that contains 657 × 106

BB pairs. The decay B− → D∗K− is also analyzed similarly by reconstructing theD∗ from Dπ0 or Dγ to improve the sensitivity, where the parameters r∗B and δ∗B areintroduced. The amplitudes f(s12, s13) and f(s12, s13) are obtained by a large sample ofD → KSπ+π− decays produced in continuum e+e− annihilation, where the isobar modelis assumed with Breit-Wigner functions for resonances. By obtaining the backgroundfractions in several kinematic variables, the fit on Dalitz plot is performed with theparameters x± = r± cos (δB ± φ3) and y± = r± sin (δB ± φ3), where the rB is separatedfor B± as r±. The results are shown in Figure 2.5 for B− → DK− and B− → D∗K−.The separations with respect to the charges of B± indicate an evidence of the CPviolation. From the results of the fits, the value of φ3 is measured to be

φ3 = 78.4 +10.8

−11.6(stat) ± 3.6(syst) ± 8.9(model), (2.23)

where rB = 0.161+0.040−0.038 ± 0.011+0.050

−0.010, r∗B = 0.196+0.073−0.072 ± 0.013+0.062

−0.012, δB = 137.4 +13.0

−15.7 ±4.0± 22.9, and δ∗B = 341.7 +18.6

−20.9 ± 3.2± 22.9. The model error is due to the uncer-tainty in determining f(s12, s13) and f(s12, s13). Model-independent measurements arestrongly demanded for further constraints on φ3.

-0.4

-0.2

0

0.2

0.4

-0.4 -0.2 0 0.2 0.4

B+→DK+

B–→DK–

x

y

-0.4

-0.2

0

0.2

0.4

-0.4 -0.2 0 0.2 0.4

B+→D*K+

B–→D*K–

B→[Dπ0]D*KB→[Dγ]D*K with ∆δ=1800

x

y

Figure 2.5: Results of the fits for B− → DK− (left) and B− → D∗K− (right) sampleson the Dalitz analyses for D → KSπ+π−, where the contours indicate 1, 2, and 3 (left)and 1 (right) standard-deviation regions.

2.5. MEASUREMENTS OF φ3 AS OF SUMMER 2010 21

2.5.2 Comparison of Constraints on CKM Parameters

The CKMfitter group provide a global analysis of determining the CKM parameters [18].The result as of summer 2010 is shown on ρ, η plane in Figure 2.6, where ρ + iη =−(VudV

∗ub)/(VcdV

∗cb); ρ = ρ(1 − λ2/2 ...) and η = η(1 − λ2/2 ...). All the measurements

are consistent with each other.The numerical results on direct measurements of the CKM angles are given as

φ1 = 21.15 +0.90

−0.88 , (2.24)

φ2 = 89.0 +4.4

−4.2 , (2.25)

φ3 = 71 +21

−25 . (2.26)

The error of φ3 is larger than the one in Eq. (2.23), an important reason of which isthe resulting smaller value of rB = 0.103+0.015

−0.024 enhancing generally the uncertainty onφ3. An indirect measurement of φ3, or an expectation of φ3 from other measurementsassuming the CKM picture, gives φ3 = 67.2 ± 3.9, which is consistent with the directmeasurement. High-precision direct measurements of φ3 are strongly demanded to testfurther the CKM picture. Any significant discrepancy between the measurements ofthe parameters would be a signature of new physics.

3φ

3φ

2φ

2φ

dm∆Kε

Kε

sm∆ & dm∆

ubV

1φsin 2

(excl. at CL > 0.95)

< 01

φsol. w/ cos 2

excluded at CL > 0.95

2φ

1φ

3φ

ρ-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

η

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5excluded area has CL > 0.95

ICHEP 10

CKMf i t t e r

Figure 2.6: Unitarity triangle and constraints on it from various measurements, as ofsummer of 2010, provided by the CKMfitter group.


2.6 Motivation to Analyze B− → [K+π−]DK−

As shown in the previous section, it is fundamentally important to improve the sen-sitivity for φ3 in a model-independent manner. A reliable possibility is to apply theGLW+ADS method, while another one is to apply the Dalitz method in a model-independent way by using the binned analysis or the ψ(3770) information. Of thepossible two ways, the subject in this thesis is the former.

The decay B− → [K+π−]DK− (Figure 2.7) plays a quite important role in a model-independent measurement of φ3 via the GLW+ADS method. The observables, denotedas RDK and ADK , are obtained from Eqs. (2.20) and (2.21) with f = K+π− as

RDK ≡ B(B− → [K+π−]DK−) + B(B+ → [K−π+]DK+)

B(B− → [K−π+]DK−) + B(B+ → [K+π−]DK+)

= r2B + r2

D + 2rBrD cos (δB + δD) cos φ3, (2.27)

ADK ≡ B(B− → [K+π−]DK−) − B(B+ → [K−π+]DK+)

B(B− → [K+π−]DK−) + B(B+ → [K−π+]DK+)

= 2rBrD sin (δB + δD) sin φ3/RDK . (2.28)

For the parameters rD and δD, experimental inputs can be used [15]. The comparablemagnitudes of rB ∼ 0.1 and rD = 0.058±0.001 enhance the relative effects of the termsincluding φ3 in both RDK and ADK . Previous studies of this decay mode have notfound a significant signal yield [19, 20]. It is very important to find an evidence of thesignal, and measure the observables as precisely as possible.

W−

W−

W−

W+

u

B−

K−

u

u

c

s

bD

0 K+

π−

u

s

d

u

B−

u

b

u

c

u

sK−

D0 π

−

K+

u

d

s

u

Figure 2.7: Diagrams for the decay B− → [K+π−]DK−. The color-suppressed B decayfollowed by the Cabibbo-favored D decay interferes with the color-favored B decayfollowed by the doubly Cabibbo-suppressed D decay. Comparable magnitudes of theamplitudes enhance the effects of CP violation.

Chapter 3

Experimental Apparatus andAnalysis Tools

The data set is collected with the Belle detector at the KEKB accelerator located at theHigh Energy Accelerator Research Organization (KEK) in Japan. In this chapter, wedescribe the experimental apparatus and analysis tools. All the developments and theconstructions shown in this chapter have been done by the collaborators. The authorhas only contributed to the detector operation and the management of the database.

3.1 KEKB Accelerator

Figure 3.1 shows a schematic view of the KEKB accelerator [21]. The electrons emittedfrom the thermionic gun are collected to make the bunches, parts of which are injectedinto a tungsten target causing the positron bunches. After accelerated, the electronand positron beams are transferred to the separated storage rings named High EnergyRing (HER) and Low Energy Ring (LER), respectively. The radio-frequency cavitiesaccelerate the particles, while the dipole (quadrupole) magnets bend (focus) them. Thebeams are collided at the interaction point (IP), making variety of interesting reactions.

Thanks to the continuous efforts by the KEKB accelerator group, many advanta-geous features are realized. A steady operation is performed with the current of morethan one ampere and the bunch size of O(100) µm for the horizontal and O(1) µmfor the vertical. To avoid parasitic collisions and reduce beam-related backgrounds,the beam lines have a crossing angle of 22 mrad. A continuous injection mode, toadd particles to the bunches without significant dead time at the detector, had beenintroduced in early 2004. A special superconducting radio-frequency cavities (“crab”cavities), which kick the beams in the horizontal plane, had been installed to makethe head-on collisions while retaining the crossing angle of beams (“crab” crossing) inFebruary 2007. By controlling the off-energy particles using skew sextupole magnets,

23

24 CHAPTER 3. EXPERIMENTAL APPARATUS AND ANALYSIS TOOLS

world’s highest luminosity1 of L = 2.11×1034 cm−2s−1 had been achieved in June 2009.The total integrated luminosity had reached 1000 fb−1, which is one of the primarytargets of the KEKB project, by finishing the data taking in June 2010. Figure 3.2shows the history of the integrated luminosity.

Most of the data are taken with the e+e− center-of-mass (CM) energy that cor-responds to the Υ resonances from e+e− → bb (Figure 3.3), while several parts aretaken with sideband energy for the studies of non-resonant components mainly frome+e− → qq (q = u, d, s, c). The integrated luminosity of 711 fb−1 is recorded at theΥ(4S) resonance with the energy 8.0 GeV and 3.5 GeV for HER and LER, respectively,where 772 × 106 BB pairs are produced by the decays of the Υ(4S) mesons.

Figure 3.1: KEKB accelerator. The green and red boxes show the radio-frequencycavities for supplying missing energy caused by the radiations.

1The luminosity is described as L = N+N−f/4πσ∗xσ∗

y , where N± is the number of particles e± perbunch, f is the frequency of the collision, and σ∗

x,y is the beam size at IP in x or y direction.

3.1. KEKB ACCELERATOR 25

Time [year]2000 2002 2004 2006 2008 2010

]-1

Inte

grat

ed L

umin

osity

[fb

0

200

400

600

800

1000

Figure 3.2: History of the integrated luminosity.

Figure 3.3: Cross section for e+e− → hadrons normalized by e+e− → µ+µ− around theCM energy to be 10 GeV, where the figure is taken from Ref. [22]. The Υ resonancesfrom e+e− → bb are placed on the continuum processes mainly from e+e− → qq (q =u, d, s, c).


3.2 Belle Detector

Belle detector [23] is a general-purpose detector consists of a barrel, forward, and back-ward components. Figure 3.4 shows the configuration of the Belle detector. A siliconvertex detector (SVD) and a central drift chamber (CDC) provide precision trackingand vertex measurements. The dE/dx measurement by CDC and the information froman array of aerogel-threshold Cherenkov counters (ACC) and a time-of-flight scintilla-tion counters (TOF) enable the identification of charged pions and kaons. An electro-magnetic calorimeter (ECL) composed of CsI(Tl) crystals detect the electromagneticparticles. The above detectors are located inside a superconducting solenoid coil thatmaintains 1.5 T magnetic field. The outermost subsystem is a KL and muon detector(KLM).

Two inner detector configurations were used. A 2.0 cm beam pipe and a 3-layersilicon vertex detector were used up to summer of 2003 for the first sample of 152 ×106BB pairs, while a 1.5 cm beam pipe, a 4-layer silicon detector, and a small-cell innerdrift chamber were used from summer of 2003 to record the remaining 620 × 106BBpairs. Almost all descriptions including figures in this section are based on Ref. [23].

Figure 3.4: Overview of the Belle detector.

3.2. BELLE DETECTOR 27

3.2.1 Silicon Vertex Detector (SVD)

Silicon detectors are based on the p-n junction diodes operated at reverse bias. Theionization currents caused by particles passing through the depleted region are detectedand measured. We use double-sided silicon strip detectors (DSSDs) manufactured byHamamatsu Photonics.2 One side, called n-side, of the bulk has highly doped stripsoriented perpendicular to the beam direction to measure the z coordinate, where z isdefined as the opposite of the positron beam direction. The other side, called p-side,has longitudinal strips to measure the φ coordinate, where φ is the azimuthal anglearound the z axis. The DSSDs reinforced by boron-nitride support ribs constitute aladder to cover large range of polar angle θ from z axis, and the ladders construct a layerto cover all angle of φ. Multiple layers enable the tracking and vertex measurements.The readout chain of DSSDs is based on CMOS-integrated circuit placed outside of thetracking volume. The design is constrained to reduce multiple Coulomb scattering.

Figure 3.5 shows the side and end views of the SVD, SVD1, used up to the summerof 2003. There are three layers, where 8, 10, and 14 ladders are comprised in the inner,middle, and outer layers, respectively. Covered angle range is 23 < θ < 139, whichcorresponds to 86% of the full solid angle. The readout electronics are based on theVA1 integrated circuits, which have excellent noise performance and reasonably goodradiation tolerance of 0.2–1 Mrad depending on the version. The impact parameterresolutions σrφ and σz, where r is the distance from z axis, are measured using cosmicrays. Obtained values are

σrφ =

√19.22 +

(54.0

pβ sin3/2 θ

)2

µm, σz =

√42.22 +

(44.3

pβ sin5/2 θ

)2

µm, (3.1)

where p and β are the momentum in GeV/c and the velocity divided by c of the particle,respectively.

A new vertex detector, SVD2, was installed in summer of 2003 [24]. Figure 3.6shows the configuration of SVD2. There are four layers, where 6, 12, 18, and 18 laddersare comprised in the first, second, third, and fourth layers, respectively. It has largercoverage of 17 < θ < 150, which corresponds to 92% of the full solid angle. Thereadout electronics are based on the VA1TA integrated circuits, which have excellentradiation tolerance of more than 20 Mrad. The impact parameter resolutions are

σrφ =

√21.92 +

(35.5

pβ sin3/2 θ

)2

µm, σz =

√27.82 +

(31.9

pβ sin5/2 θ

)2

µm. (3.2)

The performance is slightly better than the one of SVD1, mainly owing to the smallerradius of the first layer.

2http://www.hamamatsu.com/


Figure 3.5: Configuration of SVD1.

Figure 3.6: Configuration of SVD2.


3.2.2 Central Drift Chamber (CDC)

Gas detectors localize the ionization generated by charged particles. In a drift chamber,electrons released in the gas volume drift towards the anodes, producing avalanchesin the increasing field. The drift time of the electrons can be used to measure thelongitudinal position of a trajectory. Multiple wires enable the reconstruction of a track,the curvature of which provides the momentum measurement. Additional importantinformation is the energy deposit dE/dx usable for the particle identification.

The overview of the CDC structure up to the summer of 2003 is shown in Figure 3.7.The angular coverage of 17 < θ < 150 is provided by an asymmetric configuration inz direction. The chamber has 50 cylindrical layers of anode wires and 8400 drift cells.There are 32 wire layers parallel to the z axis (“axial wires”) for the r–φ measurementand 18 wire layers slanted off the z axis (“stereo wires”) for the z measurement. Themost inner part includes three cathode layers for higher precision of low-momentumparticles, which is beneficial for the trigger. In summer of 2003, the inner three layershave been replaced by two small-cell layers for making a space of SVD2, maintainingthe performance of the trigger. The sense wires are gold-plated tungsten with diametersof 30 µm to maximize the electric drift field, while the field wires are made of unplatedaluminum to reduce the material. A low-Z gas mixture of 50% helium and 50% ethaneis employed to minimize the multiple-Coulomb scattering, where the ethane componentpermits a good dE/dx resolution.

The spacial resolution for particles passing near the center of the drift space is about100 µm. The resolution of transverse momentum pT in GeV/c is measured using cosmicrays as

σpT

pT

(%) =√

(0.20pT )2 + (0.29/β)2, (3.3)

for which the pT -dependent distribution is shown in Figure 3.8 (a). No apparent sys-tematic effects due to the particle charge are observed. A truncated-mean method isemployed to estimate the most probable dE/dx, where the largest values are discardedto minimize occasional fluctuations of the Landau tail. The resolution of dE/dx is mea-sured to be 7.8% for the momentum range from 0.4 GeV/c to 0.6 GeV/c. Figure 3.8 (b)shows a scatter plot on dE/dx and momentum obtained using collision data, as well asthe expected mean for different particles. Clear separations between pions, kaons, andprotons are obtained for a momentum range up to around 1 GeV/c.

3.2.3 Aerogel Cherenkov Counter (ACC)

Cherenkov radiation is emitted by a charged particle when the velocity is greater thanthe speed of propagation of light in a material medium. Denoting the refractive indexof the matter n, and the velocity, mass, and momentum of the particle β, m, and p,respectively, the condition for producing the radiation is given as

n >1

β=

√1 +

(m

p

)2

. (3.4)


747.0

790.0 1589.6

880

702.2 1501.8

Central Drift Chamber

5

10

r

2204

294

83

Cathode part

Inner part

Main part

ForwardBackward

eeInteraction Point

17°150°

y

x100mm

y

x100mm

- +

Figure 3.7: Overview of the CDC structure. The lengths are in units of mm.

2 (

pT -

pT )

/(p

T +

pT )

] (%

)

0

0.4

0.8

1.2

1.6

2

0 1 2 3 4 5p ( GeV/c )

Fit

( 0.201 ± 0.003 ) % p ( 0.290 ± 0.006 ) %/

dow

n u

p d

ow

n u

pσ

[ √

T

T

+ β

(a) The pt resolution for cosmic rays. The fittedresult (solid curve) and the ideal expectation forβ = 1 particles (dotted curve) are shown.

(b) Plot on dE/dx and momenta of particles ob-tained from collision data. Separations betweenpions, kaons, protons, and electrons are seen.

Figure 3.8: The performances of CDC.


There is a momentum region where a particle emits Cherenkov light while anotherheavier particle does not. Thus, the Cherenkov detector can identify charged particleshaving different masses by choosing the index n for the interested momentum region.

The identifications of K± and π± are very important for studying the B mesondecays. The ACC is constructed to compensate the momentum range of 1.5–3.5 GeV/c,which is not covered by the dE/dx measurement by CDC and time-of-flight informationfrom TOF. We use aerogels with the refractive indices from 1.01 to 1.03 depending onthe polar-angle region. A highly hydrophobic aerogel was developed and produced tokeep the transparency. The fine-mesh photomultiplier tubes (FM PMTs) are attachedto the aerogels, since the ACC is operated in a high magnetic field. Figure 3.9 showsthe configuration of ACC. It consists of 960 counter modules segmented into 60 cells forthe barrel part and 228 modules arranged in 5 concentric layers for the end-cap part.All the counters are arranged in a semi-tower geometry, pointing to IP.

The performance of the ACC is checked by using the decay chain D∗− → D0π−

followed by D0 → K+π−. Since the mass difference between D∗− and D0 is small, themomentum of π− from the D∗− decay is characteristically low. Then, the K+ and π−

from the D0 decay can be identified by the charges of the tracks, safely neglecting thedoubly Cabibbo-suppressed decay D0 → K−π+. Figure 3.10 shows the distributions ofthe number of photoelectrons, where good K/π separation is obtained for each module.

B (1.5Tesla)

BELLE Aerogel Cherenkov Counter

3" FM-PMT2.5" FM-PMT

2" FM-PMT

17°

127 ° 34 °

Endcap ACC

885

R(BACC/inner)

1145

R(EACC/outer)

1622 (BACC)

1670 (EACC/inside)

1950 (EACC/outside)

854 (BACC)

1165

R(BACC/outer)

0.0m1.0m2.0m3.0m 2.5m 1.5m 0.5m

n=1.02860mod.

n=1.020240mod.

n=1.015240mod.

n=1.01360mod.

n=1.010360mod.

Barrel ACC

n=1.030228mod.

TOF/TSC

CDC

Figure 3.9: The arrangement of ACC.


00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40

Ent

ries

/pe/

trac

k

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40E

ntri

es/p

e/tr

ack

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40

Ent

ries

/pe/

trac

k

00.10.20.30.40.50.60.70.80.9

1

0 10 20 30 40

Ent

ries

/pe/

trac

k

Figure 3.10: Distributions of the number of photoelectrons for K± and π∓ in D∗∓

decays. Each plot corresponds to the different set of modules with a different refractiveindex.

3.2.4 Time-of-Flight Counter (TOF)

Scintillation counters utilize the ionization produced by charged particles to generateoptical photons. We employ plastic scintillators with FM PMTs for improving theparticle identification. The time of flight t of a particle is given by

t =l

cβ=

l

c

√1 +

(m

p

)2

, (3.5)

where l, β, p, and m are the path length, the velocity, the momentum, and the mass,respectively. Given the values of l and p, the measurement of t provides an identificationof m. Figure 3.11 shows the dimension of a module, which contains two trapezoidally-shaped TOF counters and one thin trigger scintillation counter (TSC). The latter isused for keeping the fast trigger rate below 70 kHz. There are 64 modules located at aradius 1.2 m from IP, covering the barrel part 34 < θ < 120.


By combining the measurements on the forward and backward FM PMTs, the timeresolution of 100 ps is achieved, which provides an effective identification of particlesof momenta below 1.2 GeV/c. Figure 3.12 shows the mass distribution obtained fromTOF measurements for the particles with momenta below 1.2 GeV/c. Clear peakscorresponding to pions, kaons, and protons are seen.

TSC 0.5 t x 12.0 W x 263.0 L

PMT 122.0

182. 5 190. 5

R= 117. 5

R= 122. 0R=120. 05

R= 117. 5R=117. 5

PMT PMT

- 72.5 - 80.5- 91.5

Light guide

TOF 4.0 t x 6.0 W x 255.0 L1. 0

PMTPMT

ForwardBackward

4. 0

282. 0 287. 0

I.P (Z=0)

1. 5

Figure 3.11: Dimension of a TOF/TSC Module

Figure 3.12: Mass distribution from TOF measurements for momenta below 1.2 GeV/c.The data points are in good agreement with a prediction for σTOF = 100 ps (histogram).


3.2.5 Electromagnetic Calorimeter (ECL)

When an electron or a photon is incident on a thick absorber, an electromagnetic cascadeis induced by pair production and bremsstrahlung. An electromagnetic calorimeterutilizes the generated shower for measuring the energy deposition and the position. Acomparison with the momentum measurement provides the identification of electrons.The overall configuration of the ECL is shown in Figure 3.13. The ECL is based on anarray of tower-shaped CsI(Tl) crystals that project to the vicinity of IP. The CsI(Tl)crystals have various nice features such as a large photon yield, weak hygroscopicity,and mechanical stability. The ECL consists of the barrel section of 3.0 m in length andthe annular end-caps at z = +2.0 and z = −1.0 m from IP, covering the angular range17 < θ < 150. The entire system contains 8736 counters and weighs 43 tons. Thereadout is based on an independent pair of silicon PIN photodiodes and charge-sensitivepreamplifiers attached at the end of each crystal.

The energy resolution is measured by a beam test to be

σE

E(%) =

√(0.066

E

)2

+

(0.814√

E

)2

+ 1.342 (E in GeV), (3.6)

where the value is affected by the electronic noise (1st term), the shower leakage fluctua-tion (2nd and 3rd terms), and the systematic effect such as the uncertainty of calibration(3rd term). The spacial resolution is approximately found to be 0.5 cm/

√E (E in GeV).

Figure 3.13: Overall configuration of ECL.


3.2.6 KL and Muon Detector (KLM)

The KLM system makes it possible to identify KL’s and muons by the alternating layersof 4.7 cm-thick iron plates and charged particle detectors. There are 14 iron layersand 15 (14) detector layers in the barrel (end-cap) region. The iron plates provide 3.9interaction length of material, in addition to 0.8 interaction length of ECL. The KL thatinteracts in the iron plates or ECL produces a shower of ionizing particles, the locationof which with respect to IP determines the flight direction of KL. The muons travelmuch farther with smaller deflections on average than strongly interacting hadrons,which enables the discrimination between muons and other charged hadrons.

The charged particle detectors are based on glass-electrode-resistive plate counters(RPCs). The schematic diagrams of the barrel and end-cap modules are shown inFigure 3.14. The counters have two parallel plate electrodes with high bulk resistivity(≥ 1010 Ωcm) separated by a gas-filled gap. An ionizing particle traversing the gapinduces a streamer in the gas that results in a local discharge. The discharge generatesa signal on external pickup strips, and the location and the time are recorded. Thenumber of KL clusters per event is in good agreement with the prediction. Typicalmuon identification efficiency is 90% with a fake rate around 2%.

Gas output

220 cm

+HV

-HV

Internal spacers

Conducting Ink

Gas input

(a) Barrel RPC. (b) End-cap RPC.

Figure 3.14: Schematic diagrams of the internal spacer arrangement for (a) barrel and(b) end-cap RPCs.


3.2.7 Trigger and Data Acquisition

The events of our interest are smaller in the cross section than the background pro-cesses including the beam backgrounds, and they have to be triggered by a restrictivecondition. Our trigger system consists of a hardware trigger and a software trigger.Figure 3.15 shows the schematic view of the hardware trigger system. The CDC andTSC are used to obtain trigger signals for charged particles. The ECL provides totalenergy deposit and cluster counting of crystal hits. The KLM gives additional informa-tion on muons. An extreme forward calorimeter (EFC), which is a radiation-hard BGO(Bismuth Germanate, Bi4Ge3O12) crystal calorimeter covering 6.4–11.5 and 163.3–171.2, is used for tagging two photon events as well as Bhabha events. The triggersignals of the sub-triggers are combined to characterize the event type in the centraltrigger system called the global decision logic. The software trigger is designed to be im-plemented in the online computer farm. The efficiency for multi-hadronic data samplesis more than 99.5%.

Figure 3.15: Schematic view of the trigger system.

3.3. ANALYSIS TOOLS 37

The data acquisition is based on a distributed-parallel system working at 500 Hzwith a deadtime fraction of less than 10%. Figure 3.16 shows the overview of thedata acquisition system. Information from the seven sub-detectors are combined into asingle event record by an event builder. The output of the event builder is transferredto an online computer farm, where an event filtering is applied after the fast eventreconstruction. Then, the data are sent to a mass storage system via optical fibers.

Front-endelec.

Q-to-Tconverter

MasterVME

MasterVME

MasterVME

Front-endelec.

Q-to-Tconverter

MasterVME

Front-endelec.

Q-to-Tconverter

MasterVME

MasterVME

Front-endelec.

Q-to-Tconverter

Eventbuilder

Onlinecomp.farm

Datastoragesystem

Tapelibrary

Sequencecontrol

Globaltriggerlogic

TDCLRS1877S

TDCLRS1877S

TDCLRS1877S

TDCLRS1877S

TDCLRS1877S

TDCLRS1877S

2km

SVD

CDC

ACC

TOF

KLM

ECL

VMEFlash ADC

Front-endelectronics

Subsystemtrigger logics

MasterVME

TDCLRS1877S

TRG

Belle Data Acquisition System

Front-endelec.

Q-to-Tconverter

EFC

Hitmultiplexer

Figure 3.16: Overview of the data acquisition system.

3.3 Analysis Tools

3.3.1 K±/π± Identification

The K±/π± identification is based on the dE/dx measurement by CDC, the Cherenkov-light yield in ACC, and the time-of-flight information from TOF. The likelihood func-


tions of K± (π±) for the three detector responses are combined as a product to obtainthe kaon (pion) likelihood LK (Lπ). The likelihood ratio P (K/π) is then calculated as

P (K/π) =LK

LK + Lπ

, (3.7)

from which we discriminate between K± and π±.The decay D∗− → D0π− followed by D0 → K+π− can be reconstructed without

particle identification requirements by utilizing the soft-pion charge as in Section 3.2.3.This feature is exploited to validate the method of the K±/π± identification. Figure 3.17shows a scatter plot of the track momentum and the likelihood ratio P (K/π) for K± andπ±. Clear separation in a momentum range up to around 4 GeV/c is obtained, whichis achieved by the different momentum coverages of the detectors shown in Figure 3.18.

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1

P(K/π)

Mom

entu

m (

GeV/c

)

Figure 3.17: A scatter plot of the track momentum and the likelihood ratio P (K/π) forK± (red circle) and π± (blue circle) obtained from the data of the decay chain D∗− →D0π− followed by D0 → K+π−. Strong concentration in the region of P (K/π) ∼ 1 (∼0) is observed for K± (π±) over a wide momentum region up to about 4 GeV/c.

3.3.2 Continuum Suppression

In many analyses of the B meson decays, large backgrounds arise from the continuumprocesses e+e− → qq (q = u, d, s, c). The cross section of the transition e+e− → qq issimilar to the one of e+e− → BB at the Υ(4S) resonance. Several methods to reducethe background events exploit the event topology, where a qq (BB) event has a jet-like

3.3. ANALYSIS TOOLS 39

0 1 2 3 4p (GeV/c)

dE/dx (CDC)

TOF (only Barrel)

Barrel ACC

Endcap ACC

∆ dE/dX ∼ 5 %

∆ T ∼ 100 ps (r = 125cm )

n = 1.010 ∼ 1.028

n = 1.030( only flavor tagging )

Figure 3.18: Momentum coverage of the detectors used for K±/π± separation.

(spherical) shape. The event topology is characterized on the basis of the Fox-Wolframmoments [25]. The l-th moment is defined by the l-th Legendre polynomial Pl as

Hl ≡∑i,j

|pi||pj|Pl(cos θij), (3.8)

where pi and pj are the momenta in the CM frame of the i-th and j-th particles,respectively, θij is the angle between pi and pj, and the sum is over the particles in thefinal state. Note that the overall constant is ignored here for simplicity.

SFW

The Fox-Wolfram moments can be divided as

Hl = Hssl + Hso

l + Hool , (3.9)

where Hssl is the moment on combinations of two particles “s” from the reconstructed

B, Hsol is the moment on combinations of a particle “s” from the reconstructed B and

a particle “o” not from the reconstructed B, and Hool is the moment on combinations

of two particles “o” not from the reconstructed B. We define a Fisher discriminant [26]named SFW (Super Fox-Wolfram) by the divided Fox-Wolfram moments as

SFW ≡∑l=2,4

αl

(Hso

l

Hso0

)+

4∑l=1

βl

(Hoo

l

Hoo0

), (3.10)

where the moments with negligible correlations to the four-momentum of the B can-didate are employed. The αl and βl are the Fisher coefficients, which are selected tomaximize the power of separation. The SFW is an effective discriminant for suppressingthe continuum background.


KSFW

In order to improve the discrimination, the SFW is modified to derive so-called KSFW(Kakuno Super Fox-Wolfram) [27, 28] by considering the charges of the particles, themissing mass of the event, and the normalization factors. The KSFW is defined as

KSFW ≡4∑

l=0

Rsol +

4∑l=0

Rool + γ

Nt∑n=1

|pt,n|, (3.11)

where the explanations of the terms are the following.

• The Rsol is obtained by dividing the particles “o” not from the reconstructed B

into the charged particles “c” and the neutral particles “n” and adding the missingmass “m” as

Rsol =

αcl H

′scl + αn

l H′snl + αm

l H′sml

Ee+e− − EB

, (3.12)

where Ee+e− and EB are the energy in the CM frame of e+e− and B, respec-tively, and αc

l , αnl , and αm

l are the Fisher coefficients for the charged, neutral, andmissing-mass categories, respectively. The H

′scl , H

′snl , and H

′sml are the modified

Fox-Wolfram moments with respect to the charges of particles. The normalizationEe+e− − EB is included to remove the small correlation of Hso

0 to the B energy.The terms for “n” and “m” are zero for l = 1, 3, and we totally have 11 terms forl = 0–4.

• The Rool is similar to the corresponding term in SFW, while the moment is modi-

fied with respect to the charges, noted as H′′ool , and the normalization is replaced

to remove the correlation to the B energy as

Rool =

βlH′′ool

(Ee+e− − EB)2. (3.13)

There are totally 5 terms corresponding to l = 0–4.

• The γ∑Nt

n=1 |pt,n| is the scalar sum of the transverse momenta pt of all particlesin the event multiplied by a Fisher coefficient γ. The n represents particle index,and the Nt is the number of particles in the event.

The KSFW has totally 17 (= 11+5+1) Fisher coefficients. Since the KSFW is stronglycorrelated with the missing mass, the values of the coefficients are optimized dependingon the divided missing-mass regions. The discriminants for the divided regions arecombined by the likelihood method.

The definition of KSFW was obtained primarily for the analysis of B0 → ππ, whileit is also very efficient for other modes and widely used. The distributions of KSFWwill be shown later for B− → DK−.

Chapter 4

Event Selection

We detect the suppressed B meson decay B− → [K+π−]DK− as well as the favored de-cay B− → [K−π+]DK−. The same selection criteria are used for both decays wheneverpossible in order to cancel systematic uncertainties. The decays B− → [K+π−]Dπ−

and B− → [K−π+]Dπ− are also analyzed as reference modes. The ratio RDπ and theasymmetry ADπ are introduced similarly to Eqs. (2.27) and (2.28). The Dπ− modehave an analogous event topology and is Cabibbo-enhanced relative to the DK− mode,while the effects of CP -violation are smaller.

A kaon or a pion is indicated by a “h” meson. Also, a particle that originatesdirectly from a B meson is referred to as a “prompt” particle.

4.1 Charged Tracks

All candidate tracks are required to have a point of closest approach to the beam linewithin ± 5 mm of IP in the direction perpendicular to the beam axis and ± 5 cm inthe direction parallel to the beam axis.

4.2 Reconstruction of D Meson

The D meson candidates are reconstructed from pairs of oppositely charged tracks.For each track, we apply a particle identification requirement based on the K/π likeli-hood ratio P (K/π) = LK/(LK + Lπ), where the distribution for a signal Monte Carlo(MC) sample is shown in Figure 4.1 (a). We use the requirements P (K/π) > 0.4 andP (K/π) < 0.7 for the kaon and pion candidates, respectively. The efficiency to identifya kaon or a pion is 90–95%, while the probability of a pion (kaon) being misidenti-fied as a kaon (pion) is around 15%. The invariant mass of the Kπ pair must bewithin ∼ 3σ of the nominal D mass [22]: 1.850 GeV/c2 < M(Kπ) < 1.880 GeV/c2.The M(Kπ) distribution for a signal MC sample is shown in Figure 4.1 (b). After theM(Kπ) requirement, tracks from the D candidate are refitted with their invariant massconstrained to the nominal D mass to improve the momentum determinations.

41

42 CHAPTER 4. EVENT SELECTION

)πP(K/0 0.2 0.4 0.6 0.8 1

Pro

bab

ility

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) P (K/π) for kaon (red) and pion (blue).

)2) (GeV/cπM(K

1.8 1.82 1.84 1.86 1.88 1.9 1.92P

rob

abili

ty0

0.05

0.1

0.15

0.2

(b) M(Kπ) distribution.

Figure 4.1: Distributions of P (K/π) and M(Kπ) for a signal MC sample.

4.3 Reconstruction of B Meson

The B meson candidates are reconstructed by combining a D candidate with a promptcharged hadron candidate. We apply particle identification requirements P (K/π) > 0.6and P (K/π) < 0.4 for B− → DK− and B− → Dπ−, respectively. The efficiencyto identify a kaon or a pion is 85–90%, while the probability of a pion (kaon) beingmisidentified as a kaon (pion) is around 10%. Tighter criteria are employed to suppressthe contamination between DK− and Dπ−. The signal is identified using the beam-energy-constrained mass (Mbc) and the energy difference (∆E) defined, in the CMframe, as Mbc =

√E2

beam − |pB|2 and ∆E = EB − Ebeam, where Ebeam is the beamenergy and pB and EB are the momentum and energy of the B meson candidate,respectively. The distributions for a signal MC sample are shown in Figure 4.2. Werequire 5.271 GeV/c2 < Mbc < 5.287 GeV/c2 (±3σ), and fit the ∆E distribution in−0.1 GeV < ∆E < 0.3 GeV to extract the signal yield.

In the rare cases where there are more than one candidate in an event (0.3% forB− → DK− and 0.8% for B− → Dπ−), we select the best candidate on the basis of aχ2 determined from the difference between the measured and nominal values of M(Kπ)and Mbc. The former is the value without the D-mass constraint on the tracks. Forthe calculation of χ2, the values 5.0 MeV/c2 and 2.7 MeV/c2 are used as the errors ofM(Kπ) and Mbc, respectively.

4.4. REJECTION OF BACKGROUNDS 43

)2 (GeV/cbcM5.25 5.26 5.27 5.28 5.29 5.3 5.31

Pro

bab

ility

0

0.05

0.1

0.15

0.2

(a) Mbc distribution.

E (GeV)∆-0.1 -0.05 0 0.05 0.1

Pro

babi

lity

0

0.05

0.1

0.15

0.2

(b) ∆E distribution.

Figure 4.2: Distributions of Mbc and ∆E for a signal MC sample.

4.4 Rejection of Backgrounds

4.4.1 B− → [K+K−]Dπ−

For B− → [K+π−]DK−, a possible background arises from the decay B− → [K+K−]Dπ−,which has the same final state and can peak in the signal window. To reject this back-ground, we veto the events that satisfy | M(KK) − 1.865 GeV/c2 |< 0.025 GeV/c2.Figure 4.3 shows the distributions of M(KK) for the background and signal MC sam-ples. Relative loss of signal efficiency by the requirement is 0.8%. The number of eventsremaining after the veto will be estimated later by using the M(Kπ) sideband sampleof data.

4.4.2 B− → [K−π+]Dh−

The favored decay B− → [K−π+]Dh− can also produce a peaking background if boththe kaon and the pion from the D decay are misidentified and the particle assign-ments are interchanged. We thus veto the events for which the invariant mass ofthe Kπ pair is inside the D mass window when the mass assignments are exchanged:|M(Kπ)exchanged − 1.865 GeV/c2| < 0.020 GeV/c2. Figures 4.4 and 4.5 show the dis-tributions of M(Kπ)exchanged for the background and signal MC samples. Relative lossof signal efficiency by the requirement is 11.3%. The number of events remaining afterthe veto will be estimated later by using the M(Kπ) sideband sample of data.


M(KK)1.8 1.82 1.84 1.86 1.88 1.9 1.92

Pro

bab

ility

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

M(KK)1.5 2 2.5 3 3.5 4 4.5 5

Pro

bab

ility

0

0.005

0.01

0.015

0.02

0.025

0.03

Figure 4.3: The distributions of M(KK) for MC samples of the background B− →[K+K−]Dπ− (left) and the signal B− → [K+π−]DK− (right). The vertical lines showthe veto region.

exchanged)πM(K

1.8 1.82 1.84 1.86 1.88 1.9 1.92

Pro

bab

ility

0

0.02

0.04

0.06

0.08

0.1

0.12

exchanged)πM(K

1.2 1.4 1.6 1.8 2 2.2 2.4

Pro

bab

ility

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Figure 4.4: The distributions of M(Kπ)exchanged for MC samples of the backgroundB− → [K−π+]DK− (left) and the signal B− → [K+π−]DK− (right). The solid verticallines show the veto region.

4.4. REJECTION OF BACKGROUNDS 45

exchanged)πM(K

1.8 1.82 1.84 1.86 1.88 1.9 1.92

Pro

bab

ility

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

exchanged)πM(K

1.2 1.4 1.6 1.8 2 2.2 2.4

Pro

bab

ility

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Figure 4.5: The distributions of M(Kπ)exchanged for MC samples of the backgroundB− → [K−π+]Dπ− (left) and the signal B− → [K+π−]Dπ− (right). The solid verticallines show the veto region.

4.4.3 Continuum Background

A large background arises from the continuum processes e+e− → qq (q = u, d, s, c). Inorder to remove D∗− → D0π− decays included in such a process, we employ a variable∆M defined as the mass difference between the D∗− and D candidates, where the D∗−

candidate is reconstructed from the D candidate used in the B reconstruction anda π− candidate not used in the B reconstruction. When there are multiple chargedtracks not used in the B reconstruction, we select the one for which the value of ∆Mis closest to 0.14 GeV/c2. Figure 4.6 shows the ∆M distributions for qq backgroundand signal, where a good agreement between MC and data samples is seen. We require∆M > 0.15 GeV/c2, which removes 20% of qq background with relative loss of signalefficiency of 3.9%. Note that the requirement on ∆M also removes the D∗− → D0π−

from BB events.After the ∆M requirement, the continuum background is discriminated by using

a neural network technique based on ten variables. For the output of neural network,named NB , we apply a loose requirement NB > −0.6, which retains 96% of the signaland removes 74% of the background according to MC samples. The variable NB isfitted with ∆E to extract the signal yield. The details about NB are described in thenext chapter.


veto)±M (for D*∆0.12 0.14 0.16 0.18 0.2 0.22 0.24

Pro

bab

ility

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Figure 4.6: The ∆M distributions for the qq background (blue) and the signal (red).The histograms are obtained from MC samples, while the dots with error bars areobtained from data samples. For data, we use the sideband sample indicated by0.15 GeV < ∆E < 0.30 GeV for the qq background and the B− → [K−π+]Dπ− samplein |∆E| < 0.05 GeV for the signal. The consistencies in different ∆E regions and indifferent signal samples are checked by MC samples. The solid vertical line shows thevalue for the veto.

Chapter 5

Discrimination of ContinuumBackground

A large background arises from the continuum processes e+e− → qq (q = u, d, s, c). Inthis chapter, we describe a new approach to discriminate between the signal and thecontinuum background using a neural network method.

5.1 Likelihood Method

In this section, we review the previous method [19] of the discrimination between thesignal and the continuum background. We employed two variables, KSFW and | cos θB|,where θB is the polar angle of the B candidate in the CM frame. Figures 5.1 and5.2 show the distributions of KSFW and | cos θB| for MC samples. The KSFW wasobtained for seven missing-mass regions. In the variable | cos θB|, the signal follows asin2 θB distribution, while the background is approximately uniform.

To combine the information from KSFW and | cos θB|, a likelihood ratio LR wasobtained by the likelihoods for KSFW (LKSFW) and | cos θB| (L| cos θB |) as

LR =Lsig

KSFWLsig| cos θB |

LsigKSFWLsig

| cos θB | + LbgKSFWLbg

| cos θB |

, (5.1)

where the superscript “sig” (“bg”) indicates the likelihood for signal (background).Figure 5.3 (a) shows the LR distributions for the signal and background MC samples.A plot on the rate of background rejection and the signal efficiency for each x in therequirement LR > x is shown in Figure 5.3 (b). The optimized requirement in theprevious analysis was LR > 0.9, which rejects 99% of background retaining 45% ofsignal.

47

48 CHAPTER 5. DISCRIMINATION OF CONTINUUM BACKGROUND( a ) ( b )( c )( e )( g )

( d )( f )

( h )I D o f m i s s i n g m a s s r e g i o n K S F W f o r r e g i o n 0

K S F W f o r r e g i o n 1K S F W f o r r e g i o n 3K S F W f o r r e g i o n 5

K S F W f o r r e g i o n 2K S F W f o r r e g i o n 4K S F W f o r r e g i o n 6

Figure 5.1: The distributions of (a) index of missing-mass region and (b)–(h) KSFWfor each missing-mass region. The regions are defined by the missing-mass square mm2

as mm2 < −0.5 GeV/c2 (region 0), −0.5 GeV/c2 < mm2 < 0.3 GeV/c2 (region 1),0.3 GeV/c2 < mm2 < 1.0 GeV/c2 (region 2), 1.0 GeV/c2 < mm2 < 2.0 GeV/c2 (re-gion 3), 2.0 GeV/c2 < mm2 < 3.5 GeV/c2 (region 4), 3.5 GeV/c2 < mm2 < 6.0 GeV/c2

(region 5), and mm2 > 6.0 GeV/c2 (region 6). The red and blue histograms correspondto the signal and background MC samples, respectively.

5.1. LIKELIHOOD METHOD 49

|Bθ|cos 0 0.2 0.4 0.6 0.8 1

Pro

bab

ility

0

0.005

0.01

0.015

0.02

0.025

Figure 5.2: The | cos θB| distributions for the signal (red) and background (blue) MCsamples.

LR0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

No

rmal

ized

nu

mb

er o

f ev

ents

0

0.01

0.02

0.03

0.04

0.05

0.06

(a) The LR distributions for the signal (red) andbackground (blue) MC samples.

Signal efficiency0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Bac

kgro

un

d r

ejec

tio

n

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Plot of the background rejection and the sig-nal efficiency for each requirement on LR.

Figure 5.3: Discrimination between signal and continuum background by the likelihoodmethod based on KSFW and | cos θB|.

50 CHAPTER 5. DISCRIMINATION OF CONTINUUM BACKGROUND

5.2 Introduction of Variables

To improve the discrimination between the signal and the continuum background, weintroduce ten variables listed in the following. For each variable, the distributions forMC and data samples are shown in Figure 5.4.

• LR(KSFW): a likelihood ratio of KSFW defined by LsigKSFW/(Lsig

KSFW + LbgKSFW).

The distribution is peaked at one for the signal and zero for the background.

• | cos θT|: absolute value of the cosine of the angle between the thrust axes of the Bcandidate and the rest of the event, where the thrust axis is oriented in such a waythat the sum of momentum projections has a maximum value. For the rest of theevent, we use all charged tracks, for which the pion mass is assigned, and photonswhich have the energies larger than 0.1 GeV. The distribution is approximatelyuniform for the signal and strongly peaked at one for the background.

• ∆z: vertex separation between the B candidate and the remaining tracks [29].The absolute value tends to be larger for the signal than the background, sincethe B meson has larger life time.

• cos θKD : cosine of the angle between the direction of the K from the D decay and

the opposite direction to the B in the rest frame of D. The distribution for thesignal is about flat because of the zero spins of the particles, while the one for thebackground is peaked mostly at one.

• rtag: expected B flavor dilution factor [30] that ranges from zero for no flavorinformation to unity for unambiguous flavor assignment. The signal events tendto have higher values than the background.

• | cos θB|: absolute value of the cosine of the polar angle of the B candidate inthe CM frame. The signal follows a sin2 θB distribution, while the background isapproximately uniform.

• ∆Q: difference of charge sums in the D hemisphere and the opposite hemisphereafter removing the B candidate, where we apply opposite signs for B− and B+

candidates. The value is around zero for the signal and slightly shifted for the ccbackground.

• QBQK : product of the charge of the B candidate and the charge sum of kaonsnot used in the B reconstruction. Since the B− (B+) meson tends to producethe K− (K+) meson, the value for the signal is likely to be lower than zero. Thevalue is zero for most background events.

• dDh: distance of closest approach between trajectories of the D and h candidates.The value is close to zero for the signal, while the value can be larger for the ccbackground for which the h candidate can be secondary from a decay of a particlewith longer life time.

5.3. NEUROBAYES NEURAL NETWORK 51

• cos θDB : cosine of the angle between the D direction and the opposite direction to

the Υ(4S) in the rest frame of B. The distribution is approximately flat for thesignal, while the one slopes slightly for the cc background.

5.3 NeuroBayes Neural Network

For combining the variables, we employ the NeuroBayes package [31], which is a highlysophisticated tool for multivariate analysis of correlated data on the basis of Bayesianstatistics. An automated preprocessing of the input variables is followed by a three-layered (input, hidden, and output layers) neural network. The complex relationshipsbetween the input variables are learnt by using a provided dataset such as simulateddata, and transformed into the output for analyzing the data of interest. The outputcan be utilized both for classification and shape reconstruction.

The preprocessing enables to find the optimal starting point for the subsequentnetwork training. All variables can be normalized and linearly decorrelated such thatthe covariance matrix of new set of input variables becomes the unit matrix. Binaryor discrete variables are automatically recognized and treated accordingly. A veryimportant option is to handle the variables with a default value or delta-function, whichcan e.g. occur if the vertex fitting is failed for obtaining ∆z. The preprocessing is veryrobust and efficient for powerful classification.

In the neural network, it is of vital importance to improve the generalization, whichis realized by an advanced regularization and pruning techniques. Employing regular-ization based on Bayesian statistics practically eliminates the risk of overtraining andenhances the generalization abilities. Insignificant network connections and even entirenodes are removed to ensure that the network learns only the physical features of data.The resulting network represents the minimal topology needed to correctly reproducethe characteristics of the data while being insensitive to statistical fluctuations.

5.4 Neural Network Training

The neural network is trained with MC samples for signal and continuum background.For the latter, we loosen the requirement on Mbc as 5.20 GeV/c2 < Mbc < 5.29 GeV/c2

to obtain larger number of events.1 We use 200,000 events for each sample after theevent-selection requirements. Figure 5.5 (a) shows the result of the training, where theoutput is denoted as NB . A plot on the rate of background rejection and the signalefficiency for each x in the requirement NB > x is shown in Figure 5.5 (b). In Table5.1, we numerically summarize the performance with a comparison to the likelihoodmethod. The output NB is fitted with ∆E to extract the signal yield. The expectedsignificance of the signal is increased by about a factor two compared to the previousanalysis. The details about the training are described in Appendix A.

1 The correlation factors to Mbc for all input variables are less than 5%.


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5.4. NEURAL NETWORK TRAINING 53

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Figure 5.5: Discrimination between the signal and the continuum background by theNeuroBayes method based on ten variables.

Background Signal efficiency (%) Rate forrejection (%) Likelihood Meth. NeuroBayes Meth. signal efficiency

99 42 60 1.4395 69 81 1.1790 79 88 1.1180 88 94 1.07

Table 5.1: Summary table of the performance of the discrimination between the signaland the continuum background. The error of the efficiency is about 1%. The Neu-roBayes method significantly improves the discrimination compared to the likelihoodmethod.

5.4. NEURAL NETWORK TRAINING 55

5.4.1 Contribution of Each Variable

As a check of the training, we obtain the powers of individual variables as well as thecorrelation factors as in Table 5.2, where the explanations of contents are the following.

• Only this: the significance for single variable. The quantity is obtained as thecorrelation of a variable to the output multiplied by

√n, where n is the sample

size. The computation does not take into account other variables.

• Without this: the significance loss when a variable is removed. This is the loss ofcorrelation multiplied by

√n when only one variable is removed from the input

set and the total correlation to the output is recomputed.

• Corr. to others: the correlation of a variable to all the others, computed with thecomplete correlation matrix.

The values of “Only this” are higher for the variables which have the distributionsclearly different between the signal and the background. The value of “Without this”is based on the “Only this”, becoming lower if the value of “Corr. to others” is higher.A large correlation exists for the pair of LR(KSFW) and | cos θT|, since both variablesutilize the event topology. Some correlations exist for several variables, all of whichare reasonable since common particles are used in the definitions. All the values inTable 5.2 make sense according to the distributions in Figure 5.4 and the definitions.

Variable Only this Without this Corr. to others

LR(KSFW) 409 136 0.80| cos θT| 355 63 0.76

∆z 229 110 0.29cos θK

D 189 76 0.27rtag 186 56 0.38

| cos θB| 150 67 0.19∆Q 143 39 0.28

QBQK 142 46 0.30dDh 141 60 0.21

cos θDB 87 40 0.14

Table 5.2: The powers of individual variables and the correlation factors.


5.4.2 Check of Stability

To check whether the overtraining occurs or not, we apply the trained network to theMC samples independent of the training. Both the signal and background MC samplescontain 100,000 events, where the same selection criteria as the training are applied.Figure 5.6 shows the distributions of resulting outputs as well as the ones from thetraining. The consistent results indicate that there is no significant overtraining.

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Figure 5.6: Check of the stability of the neural network. The NB distributions areshown for the training (histograms) and the test (dots with error bars) applied on theMC samples of the signal (red) and the background (blue). Linear (left) and logarithmic(right) scales are used for the vertical axes. The consistent results indicate that thereis no significant overtraining.

5.5 Tests on Several Samples

In this section, we apply the trained network to several different samples. Figure 5.7 (a)shows the results for different Mbc requirements on a background MC sample. We apply5.20 GeV/c2 < Mbc < 5.29 GeV/c2 and 5.271 GeV/c2 < Mbc < 5.287 GeV/c2, whichcorrespond to the selections in the training and in the analysis, respectively. Similardistributions are obtained, validity of which is supported by the small correlationsbetween the input variables and Mbc. The result provides a confidence to use the wideMbc region in the training.

Figure 5.7 (b) shows the results for the MC and data samples. For the latter, we usethe B− → [K−π+]Dπ− in |∆E| < 0.05 GeV for signal and the sideband ∆E > 0.15 GeVof the B− → [K+π−]DK− for background. Obtained NB distributions for MC and datasamples are in good agreement. The small discrepancy is not due to the difference ofthe signal modes or the ∆E regions but due to the difference in MC and data. We willovercome this difficulty by using the data sample to obtain the functions for the fit.

5.6. TEST BY REMOVING THREE VARIABLES 57

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Figure 5.7: Comparison of the neural network outputs for several samples.

5.6 Test by Removing Three Variables

The variables ∆z, cos θKD , and dDh have some discrepancies between MC and data

samples for the background (Figure 5.4). Figure 5.8 shows the results when the threevariables are removed from the input set. The MC and data samples are in betteragreement, which encourages us to be confident of the NeuroBayes method.

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D , and dDh are removed from the input set.

Chapter 6

Signal Extraction

The signal yield is extracted from the two-dimensional distribution of ∆E and NB usingthe extended unbinned maximum likelihood fit. The fit is applied simultaneously forB− and B+ samples using the parameters NB− +NB+ and (NB− −NB+)/(NB− +NB+),where NB± is the yield of B±, to obtain RDh and ADh directly. Also, the fit is appliedsimultaneously for DK and Dπ samples, where the Dπ (DK) “feed-across” in DK (Dπ)sample caused by the misidentification is parameterized by the efficiency difference inthe two samples. We describe the probability density function (PDF) in Section 6.1followed by the signal extraction in Section 6.2. Note that the fit on the data sampleis not applied before all the strategy is fixed.

6.1 Probability Density Function (PDF)

The components of the fit are divided into the Dh signal, the Dh feed-across, the back-ground from e+e− → BB (denoted as BB background), and the continuum backgroundfrom e+e− → qq (denoted as qq background). For each component, the correlation be-tween ∆E and NB is found to be small as in Figure 6.1. We thus obtain the PDF bytaking a product of individual PDFs for ∆E and NB . The functions are obtained fromdata samples as many as possible to reduce the systematic uncertainties. A demonstra-tion of the fit on MC samples is shown in Appendix B.

6.1.1 Functions for ∆E

The functions for ∆E are explained in the following.

• Dh signal: we use a sum of two Gaussians with a common mean. The parametersare floated in the fit to the favored modes, and fixed to the results in the fit tothe suppressed modes. Common parameters are used for the DK and Dπ signals,the validity of which is checked by MC samples as shown in Figure 6.2.

59

60 CHAPTER 6. SIGNAL EXTRACTION

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• Dh feed-across: we use a sum of an asymmetric Gaussian1 and a Gaussian with acommon mean. For h = π, the mean and the right-side width of the asymmetricGaussian are floated in the fit to the favored modes, and fixed to the results inthe fit to the suppressed modes. The remaining parameters are obtained from thefit to the favored Dπ signal in the data sample where the kaon mass is assignedto the prompt pion, as shown in Figure 6.3 (a). For h = K, the parameters areobtained from a MC sample as shown in Figure 6.3 (b), since the DK signal israther smaller than the Dπ signal and the data sample cannot be used.

• BB background: we use a free exponential function, the validity of which ischecked by MC samples as shown in Figure 6.4. In this component, there arecontributions from B− → D∗h−, B− → Dρ−, and so on, which populate thenegative ∆E region. An exponential function is suited to model their tails as wellas the combinatorial contribution located in whole region.

• qq background: we use a free linear function, the validity of which is checked byMC samples as shown in Figure 6.5. The validity of using a linear function is alsochecked by a data sample of the NB sideband: NB < 0.

1 The shift caused by the incorrect mass assignment makes the shape of the ∆E distributionasymmetric.

6.1. PROBABILITY DENSITY FUNCTION (PDF) 61

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Figure 6.2: The ∆E distributions for signal MC samples fitted with a sum of twoGaussians. The shape parameters are floated in the fit to the mode B− → [K−π+]Dπ−,and fixed to the results for the other modes.


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Figure 6.3: The ∆E distributions for the Dh feed-across samples. A fit is applied toobtain the values of the shape parameters.

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Figure 6.4: The ∆E distributions for MC samples of the BB background. Each samplehas two times larger size than the data sample. A fit is applied using free exponentialfunction.

6.1. PROBABILITY DENSITY FUNCTION (PDF) 63

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Figure 6.5: The ∆E distributions for MC samples of the qq background. Each samplehas two times larger size than the data sample. A fit is applied using free linear function.The smaller yield for B− → [K+π−]Dπ− is due to the two same-sign pions needed forthe reconstruction.

6.1.2 Functions for NB

The functions for NB are explained in the following. The histogram functions areemployed for all components.

• Dh signal: we obtain the shape for the favored signal from the data sample. Thehistogram in the region |∆E| < 0.01 GeV is used by subtracting the one for theregion 0.15 GeV < ∆E < 0.30 GeV for which we multiply the weight 0.02/0.15.The same shape is used for the suppressed signal, validity of which is checked byMC samples as shown in Figure 6.6.

• Dh feed-across: we obtain the shape for h = π by a similar way as the one for theDh signal, where the region |∆E − 0.05 GeV| < 0.01 GeV is used. We obtain theshape for h = K from a MC sample, since the DK signal is rather smaller thanthe Dπ signal and the data sample cannot be used.

• BB background: we obtain the shape from MC samples.

• qq background: we obtain the shape from the data sample of the sideband forwhich we require 5.20 GeV/c2 < Mbc < 5.26 GeV/c2 and 0.15 GeV < ∆E <0.30 GeV. Figure 6.7 shows the NB distributions for MC samples. There is smallcontribution from the BB background for the data sample (0.5%–1.0% dependingon the modes), which is subtracted according to a MC study.


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Figure 6.7: The NB distributions for MC samples of the qq background. Each samplehas two times larger size than the data sample. The fit is applied using a histogramfunction obtained from the sideband of the MC sample.

6.2. SIGNAL EXTRACTION 65

6.2 Signal Extraction

The signal yield is extracted with the PDF described in the previous section. The fitis applied separately for the favored modes and the suppressed modes.

6.2.1 Fit to Favored Modes

Figure 6.8 shows the result of the fit to the favored modes. In this fit, the ratio of theDK yields in the Dπ and DK samples is fixed to the value obtained from the decayD∗− → D0π− followed by D0 → K+π−, and the asymmetry for the qq background isfixed to zero. Figures 6.9 and 6.10 show the projections for several regions, where goodquality of the fit is obtained for each region. Table 6.1 shows the list of the parametersin the fit.

The yield for the Dπ (DK) signal is consistent with the expectation from the pre-vious analysis [19]; the increase of the number of events is consistent with the factor1.8 (2.6) which is obtained by multiplying the increases of the efficiency and the data.2

The asymmetry for each component is very close to zero, as expected. Note that theratio of the Dπ yields in the DK and Dπ samples is 1.70 ± 0.04 times larger than theone for the MC samples (Table B.1), which is consistent with the value 1.69 ± 0.10obtained from the decay D∗− → D0π− followed by D0 → K+π−.

6.2.2 Fit to Suppressed Modes

Figure 6.11 shows the result of the fit to the suppressed modes. In this fit, all shapeparameters for the Dh signal and the ratio of the Dπ yields in the DK and Dπ samplesare fixed to the results for the favored modes. Figures 6.12 and 6.13 show the projectionsfor several regions, where good quality of the fit is obtained for each region. Table 6.2shows the list of the parameters in the fit.

The yield and asymmetry for the Dπ signal are 165+19−18 and (−4±11)%, respectively,

which are consistent with the expected values 169 and −2%, respectively, obtainedfrom the previous analysis [19]. The yield for the DK signal is 56.0+15.1

−14.2, for whichthe statistical significance is 4.4σ. The significance including the systematic error isobtained in Section 8.2. The asymmetry for the DK signal is (−39+26

−28)%, which isconsistent with the value (−86+49

−50)% obtained by BaBar [20]. Assuming rB = 0.1,the yield (asymmetry) for the DK signal varies from 5 to 87 (varies in most of theregion from −100% to 100%) depending on the values of φ3 and the strong phases.Therefore, our measurement provides important information for measuring φ3. Notethat the asymmetry for the BB background in the DK sample, for which non-zerovalue can be generated by B− → D∗K−, is (30 ± 17)%.

2 The difference of the efficiency increase between the Dπ and DK analyses is mainly due to thedifferent requirements for rejecting the qq background in the previous analysis.


E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.005

x 0

.04

)

0200

400

600800

10001200

1400

1600

1800

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.005

x 0

.04

)

0200400600

8001000

12001400

16001800

2000

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

020406080

100120140160180200220

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

020406080

100120140160180200220

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.005

x 0

.04

)

0

0.01

0.02

0.03

0.04

0.05

0.06

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.005

x 0

.04

)

0

0.01

0.02

0.03

0.04

0.05

0.06

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(a) The histograms of the data samples (upper) and the fitted PDFs (lower).

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

500

1000

1500

2000

2500

3000

3500

4000

/ndf = 2.792χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

500

1000

1500

2000

2500

3000

3500

4000

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

500

1000

1500

2000

2500

3000

3500

4000

/ndf = 2.772χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

500

1000

1500

2000

2500

3000

3500

4000

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

100

200

300

400

500

600

/ndf = 1.142χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

100

200

300

400

500

600

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

100

200

300

400

500

600

/ndf = 0.762χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

100

200

300

400

500

600

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

2000

4000

6000

8000

10000

12000

/ndf = 0.962χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

2000

4000

6000

8000

10000

12000

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

2000

4000

6000

8000

10000

12000

/ndf = 0.942χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

2000

4000

6000

8000

10000

12000

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

200

400

600

800

1000

1200

1400

1600

1800

2000/ndf = 1.172χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

200

400

600

800

1000

1200

1400

1600

1800

2000

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

200

400

600

800

1000

1200

1400

1600

1800

2000/ndf = 0.642χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

200

400

600

800

1000

1200

1400

1600

1800

2000

(b) The ∆E distributions (upper) and the NB distributions (lower), both of which are obtained byprojecting all fitted regions. The fitted sample is shown with dots with error bars and the PDF isshown with the solid blue curve, for which the components are shown with thicker long-dashed red(DK), thinner long-dashed magenta (Dπ), dash-dotted green (BB background), and dashed blue (qqbackground).

Figure 6.8: The result of the fit to the favored modes. The distributions for Dπ−, Dπ+,DK−, and DK+ are shown from left to right.


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

50

100

150

200

250

300

/ndf = 0.782χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

50

100

150

200

250

300

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

50

100

150

200

250

300

350

400

450

/ndf = 0.732χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

50

100

150

200

250

300

350

400

450

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

500

1000

1500

2000

2500

3000

3500/ndf = 2.672χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

500

1000

1500

2000

2500

3000

3500

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

10

210

310

/ndf = 2.672χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

10

210

310

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

100

200

300

400

500

600/ndf = 1.352χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

100

200

300

400

500

600

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

2000

4000

6000

8000

10000

12000/ndf = 1.042χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

2000

4000

6000

8000

10000

12000

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

10

210

310

410

/ndf = 1.042χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

10

210

310

410

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60/ndf = 0.692χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

(a) The projections for the Dπ− mode.

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

50

100

150

200

250

300

/ndf = 1.092χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

50

100

150

200

250

300

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

50

100

150

200

250

300

350

400

450

/ndf = 0.932χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

50

100

150

200

250

300

350

400

450

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

500

1000

1500

2000

2500

3000

3500/ndf = 2.332χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

0

500

1000

1500

2000

2500

3000

3500

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

10

210

310

/ndf = 2.332χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 5 M

eV

10

210

310

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

100

200

300

400

500

600/ndf = 0.682χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

100

200

300

400

500

600

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

2000

4000

6000

8000

10000

12000/ndf = 1.022χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

2000

4000

6000

8000

10000

12000

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

10

210

310

410

/ndf = 1.022χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

10

210

310

410

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60/ndf = 1.022χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

(b) The projections for the Dπ+ mode.

Figure 6.9: The projections for the favored Dπ sample. The ∆E distributions forNB < 0, 0 < NB < 0.5, NB > 0.5, and NB > 0.5 (log-scaled) are shown fromleft to right (upper). The NB distributions for ∆E < −0.05 GeV, |∆E| < 0.04 GeV,|∆E| < 0.04 GeV (log-scaled), and ∆E > 0.15 GeV are shown from left to right (lower).


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70/ndf = 0.792χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

90/ndf = 0.682χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

90

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

50

100

150

200

250

300

350

400

450

/ndf = 1.232χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

50

100

150

200

250

300

350

400

450

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

50

100

150

200

250

300

/ndf = 1.072χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

50

100

150

200

250

300

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

160

180/ndf = 0.802χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

160

180

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

200

400

600

800

1000

/ndf = 0.792χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

200

400

600

800

1000

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

300

350

400

450

/ndf = 1.122χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

300

350

400

450

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35/ndf = 0.842χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

(a) The projections for the DK− mode.

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70/ndf = 0.922χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

90/ndf = 0.672χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

90

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

50

100

150

200

250

300

350

400

450

/ndf = 0.732χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

50

100

150

200

250

300

350

400

450

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

50

100

150

200

250

300

/ndf = 0.462χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

50

100

150

200

250

300

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

160

180/ndf = 0.502χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

160

180

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

200

400

600

800

1000

/ndf = 0.802χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

200

400

600

800

1000

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

300

350

400

450

/ndf = 0.822χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

300

350

400

450

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35/ndf = 1.122χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

(b) The projections for the DK+ mode.

Figure 6.10: The projections for the favored DK sample. The ∆E distributions forNB < 0, 0 < NB < 0.5, NB > 0.5, and NB > 0.9 are shown from left to right (upper).The NB distributions for ∆E < −0.05 GeV, |∆E| < 0.04 GeV, 0.05 GeV < ∆E <0.1 GeV, and ∆E > 0.15 GeV are shown from left to right (lower).


Component Type Parameter Value

General Yield 49164+245−244

Asymmetry (−0.97 ± 0.45)%Dπ Mean 0.0008 ± 0.0001

D. Gaussian σ1 0.0111 ± 0.0002(∆E) σ2/σ1 1.92+0.05

−0.04

Yield1/Yield (72.6+2.5−2.6)%

DK General Yield 3394+68−69

Asymmetry (2.3 ± 1.9)%General Yield in DK/Yield in Dπ (8.93 ± 0.16)%

Mean 0.0466 ± 0.0004Dπ in DK σ1,right 0.0223 ± 0.0004

D. Bif. Gaussian σ1,left/σ1,right 0.631 (fixed)(∆E) σ2/σ1,right 3.49 (fixed)

Yield1/Yield 95.8% (fixed)General Yield in Dπ/Yield in DK 11.2% (fixed)

Mean −0.0463 (fixed)DK in Dπ σ1,right 0.0139 (fixed)


Yield1/Yield 85.1% (fixed)General Yield 3525+131

−133

BB in Dπ Asymmetry (−0.4 ± 2.5)%Exponential (∆E) Coefficient −16.6+0.6

−0.7


BB in DK Asymmetry (0.1 ± 3.9)%Exponential (∆E) Coefficient −36.4+2.6

−2.7


qq in Dπ Asymmetry 0 (fixed)Linear (∆E) Slope −0.25 ± 0.03


qq in DK Asymmetry 0 (fixed)Linear (∆E) Slope −0.28 ± 0.04

Table 6.1: The list of parameters in the fit to the favored modes.


E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

2

4

6

8

10

12

14

16

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

2

4

6

8

10

12

14

16

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

1

2

3

4

5

6

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

1

2

3

4

5

6

7

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

0.002

0.004

0.006

0.008

0.01

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

0.002

0.004

0.006

0.008

0.01

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0.0005

0.001

0.0015

0.002

0.0025

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80/ndf = 0.562χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80/ndf = 0.552χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80/ndf = 0.702χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80/ndf = 0.832χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120/ndf = 0.912χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120/ndf = 1.142χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120/ndf = 1.182χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120/ndf = 0.932χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120


Figure 6.11: The result of the fit to the suppressed modes. The distributions for Dπ−,Dπ+, DK−, and DK+ are shown from left to right.


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30/ndf = 0.822χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

16

18

20/ndf = 0.792χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

16

18

20

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40/ndf = 0.572χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25/ndf = 0.532χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70/ndf = 0.682χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40/ndf = 0.542χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30/ndf = 0.642χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

16

18

20/ndf = 0.372χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

16

18

20

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40/ndf = 0.682χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25/ndf = 0.542χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70/ndf = 0.692χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40/ndf = 0.962χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40


Figure 6.12: The projections for the suppressed Dπ sample. The ∆E distributions forNB < 0, 0 < NB < 0.5, and NB > 0.5 are shown from left to right (upper). The NBdistributions for ∆E < −0.05 GeV, |∆E| < 0.04 GeV, and ∆E > 0.15 GeV are shownfrom left to right (lower).


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40/ndf = 0.922χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30/ndf = 0.612χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30/ndf = 0.812χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

/ndf = 0.342χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25/ndf = 0.232χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40/ndf = 0.412χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25/ndf = 0.592χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

50/ndf = 0.802χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

50


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40/ndf = 1.012χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30/ndf = 0.552χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30/ndf = 0.482χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

/ndf = 0.332χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25/ndf = 0.542χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40/ndf = 0.872χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25/ndf = 0.562χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

50/ndf = 0.722χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

50


Figure 6.13: The projections for the suppressed DK sample. The ∆E distributions forNB < 0, 0 < NB < 0.5, NB > 0.5, and NB > 0.9 are shown from left to right (upper).The NB distributions for ∆E < −0.05 GeV, |∆E| < 0.04 GeV, 0.05 GeV < ∆E <0.1 GeV, and ∆E > 0.15 GeV are shown from left to right (lower).



General Yield 165.0+19.1−18.1

Asymmetry (−4.3+11.0−10.9)%

Dπ Mean 0.0008 (fixed)D. Gaussian σ1 0.0111 (fixed)

(∆E) σ2/σ1 1.92 (fixed)Yield1/Yield 72.6% (fixed)

DK General Yield 56.0+15.1−14.2

Asymmetry (−38.6+25.5−27.8)%

General Yield in DK/Yield in Dπ 8.93% (fixed)Mean 0.0466 (fixed)

Dπ in DK σ1,right 0.0223 (fixed)D. Bif. Gaussian σ1,left/σ1,right 0.631 (fixed)

(∆E) σ2/σ1,right 3.49 (fixed)Yield1/Yield 95.8% (fixed)

General Yield in Dπ/Yield in DK 11.2% (fixed)Mean −0.0463 (fixed)

DK in Dπ σ1,right 0.0139 (fixed)D. Bif. Gaussian σ1,left/σ1,right 1.29 (fixed)

(∆E) σ2/σ1,right 4.77 (fixed)Yield1/Yield 85.1% (fixed)


BB in Dπ Asymmetry (−10.2+18.4−18.5)%

Exponential (∆E) Coefficient −10.2+2.8−4.5


BB in DK Asymmetry (29.7+17.4−17.1)%

Exponential (∆E) Coefficient −9.9+2.3−2.9



General Yield 3171 ± 61qq in DK Asymmetry 0 (fixed)

Linear (∆E) Slope −0.11 ± 0.03

Table 6.2: The list of parameters in the fit to the suppressed modes.

Chapter 7

Estimation of Peaking Backgrounds

In this chapter, we describe the backgrounds which peak inside the signal window.Since all the backgrounds have approximately flat M(Kπ) distributions, the yields of thebackgrounds are estimated by using M(Kπ) sidebands of data. We use 1.815 GeV/c2 <M(Kπ) < 1.845 GeV/c2 and 1.885 GeV/c2 < M(Kπ) < 2.005 (1.915) GeV/c2 for DK−

(Dπ−). The regions are taken to avoid the effects of the misidentification backgroundsfrom B− → [K+K−]Dh− and B− → [π+π−]Dπ−.

7.1 Peaking Backgrounds

In this section, we list the peaking backgrounds, where the M(Kπ) distribution is shownfor each component.

• The decay B− → [K+K−]Dπ− remaining after the veto on M(KK), described inSection 4.4, can have a peak under the signal B− → [K+π−]DK−. In Figure 7.1,the M(Kπ) distribution is shown, where one million MC events are analyzed with-out the M(Kπ) requirement. A constant function fits well, which indicates thevalidity of using the M(Kπ) sidebands to estimate the background contribution.

• The favored decay B− → [K−π+]Dh− remaining after the veto on M(Kπ)exchanged,described in Section 4.4, can have a peak under the signal B− → [K+π−]Dh−. InFigure 7.2, the M(Kπ) distribution for h = K (similar for h = π) is shown, whereone million MC events are analyzed without the M(Kπ) requirement. A constantfunction fits well, which indicates the validity of using the M(Kπ) sidebands toestimate the background contribution.

• The charmless decay B− → K+K−π− can also be a peaking background for thesignal B− → [K+π−]DK−.1 In Figure 7.3, the M(Kπ) distribution is shown,where one million MC events are analyzed without the M(Kπ) requirement. Forthe MC generation, we take no interference and use a sum of f0π

− (14.9%), K∗0K−

1 The decay rate of B− → K+π−π− is thought to be much smaller and the signal has not seen yet.

75

76 CHAPTER 7. ESTIMATION OF PEAKING BACKGROUNDS

(6.9%), K∗00 K− (24.5%), and non-resonant K+K−π− (53.7%). This model is

based on a theoretical calculation [32], which is consistent with the observationby BaBar [33]. A constant function fits well, which indicates the validity of usingthe M(Kπ) sidebands to estimate the background contribution.

)2) (GeV/cπM(K1.85 1.9 1.95 2

Eve

nts

/ 5

MeV

0

10

20

30

40

50

60

70

80/NDF = 1.272χ

)2) (GeV/cπM(K1.85 1.9 1.95 2

Eve

nts

/ 5

MeV

0

10

20

30

40

50

60

70

80

Figure 7.1: The M(Kπ) distribution fitted by a constant function for a MC sample ofthe background B− → [K+K−]Dπ− in the analysis of B− → [K+π−]DK−.

7.2 Estimation Using the M(Kπ) Sidebands

For estimating the contributions from the peaking backgrounds, we use the events inthe M(Kπ) sidebands of data. Figures 7.4 and 7.5 show the results of the fits, wherethe same method as the one for the signal extraction is applied. By taking into accountthat the sideband region is five (two) times larger than the signal region, we obtain anexpected background yield of −1.9+3.7

−3.5 (−3.2+7.0−6.4) events for DK− (Dπ−). Since the

yields are consistent with zero, we do not subtract these backgrounds from the signalyields but instead include the uncertainties in the systematic errors.

7.2. ESTIMATION USING THE M(Kπ) SIDEBANDS 77

)2) (GeV/cπM(K1.85 1.9 1.95 2

Eve

nts

/ 5

MeV

0

10

20

30

40

50

60/NDF = 0.852χ

)2) (GeV/cπM(K1.85 1.9 1.95 2

Eve

nts

/ 5

MeV

0

10

20

30

40

50

60

Figure 7.2: The M(Kπ) distribution fitted by a constant function for a MC sample ofthe background B− → [K−π+]DK− in the analysis of B− → [K+π−]DK−. It is quitesimilar for the background B− → [K−π+]Dπ− in the analysis of B− → [K+π−]Dπ−.

)2) (GeV/cπM(K1.85 1.9 1.95 2

Eve

nts

/ 5

MeV

0

100

200

300

400

500

600/NDF = 1.102χ

)2) (GeV/cπM(K1.85 1.9 1.95 2

Eve

nts

/ 5

MeV

0

100

200

300

400

500

600

Figure 7.3: The M(Kπ) distribution fitted by a constant function for a MC sample ofthe charmless background B− → K+K−π− in the analysis of B− → [K+π−]DK−.


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70

80

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70

80

Figure 7.4: ∆E (NB > 0.9) and NB (|∆E| < 0.03 GeV) distributions for the M(Kπ)sidebands of data in the analysis of B− → [K+π−]DK−. The fitted sample is shownwith dots with error bars and the PDF is shown with the solid blue curve, for whichthe components are shown with dashed red (signal), dashed green (BB background),and dashed blue (qq background).

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

Figure 7.5: ∆E (NB > 0.9) and NB (|∆E| < 0.03 GeV) distributions for the M(Kπ)sidebands of data in the analysis of B− → [K+π−]Dπ−. The fitted sample is shownwith dots with error bars and the PDF is shown with the solid blue curve, for which thecomponents are shown with dashed magenta (signal), dashed green (BB background),and dashed blue (qq background).

7.3. CHECK USING MC SAMPLES 79

7.3 Check Using MC Samples

For a check of our study based on the M(Kπ) sidebands of data, we estimate thebackground contributions using MC samples. We use the same MC samples as the onesin Section 7.1, and apply a fit for the events in the signal region of M(Kπ). The resultsare consistent with the estimation in Section 7.2.

• Figure 7.6 shows the result of the fit for the background B− → [K+K−]Dπ− inthe analysis of B− → [K+π−]DK−. Estimated number of events which contributeto the signal extraction is

NBB × BR× Eff. = 772 million × (1.92 × 10−5) × (35.4 ± 9.4)/1 million

= 0.5 ± 0.1, (7.1)

where NBB is the number of BB pairs in data, BR is the branching ratio forthe background, and Eff. is the efficiency in the analysis of B− → [K+π−]DK−

obtained by the fit. The errors on NBB and BR are negligible compared to theerror on Eff.

• Figures 7.7 and 7.8 show the results of the fits for the background B− → [K−π+]Dh−

in the analysis of B− → [K+π−]Dh−. Estimated number of events which con-tribute to the signal extraction is

NBB × BR× Eff. = 772 million × (1.44 × 10−5) × (28.9 ± 7.6)/1 million × 1.6

= 0.5 ± 0.1 for h = K and (7.2)

NBB × BR× Eff. = 772 million × (1.89 × 10−4) × (14.5 ± 6.6)/1 million × 1.6

= 3.4 ± 1.5 for h = π. (7.3)

Since the K/π misidentification probability is different between MC and datasamples, we multiply the calibration factor 1.6 ± 0.1 obtained from the decayD∗− → D0π− followed by D0 → K+π−, where the error 0.1 is safely neglected inthe calculation.

• For the charmless decay B− → K+K−π−, we count the events which satisfythe selection criteria and obtain 1790 over one million generated events. Theestimated contribution is calculated to be 6.9 ± 1.0 using the branching ratioobtained by BaBar [33]: (5.0± 0.7)× 10−6. Note that the value is conservativelytaken to be larger by the counting, since the ∆E distribution is slightly wider andthe number obtained by the fit should become smaller.2 Also, it is conservativeto take no interferences of the resonances in the MC generation. In any case, theresult is close to the value obtained in Section 7.2.

2 If we apply a fit, the number 5.7 ± 0.8 is obtained, while the quality of the fit is slightly low.


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Figure 7.6: ∆E (NB > 0.9) and NB (|∆E| < 0.03 GeV) distributions for a MC sampleof B− → [K+K−]Dπ− in the analysis of B− → [K+π−]DK−. The fitted sample is shownwith dots with error bars and the PDF is shown with the solid blue curve, for whichthe components are shown with dashed red (signal), dashed green (BB background),and dashed blue (qq background).

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Figure 7.7: ∆E (NB > 0.9) and NB (|∆E| < 0.03 GeV) distributions for a MC sampleof B− → [K−π+]DK− in the analysis of B− → [K+π−]DK−. The fitted sample is shownwith dots with error bars and the PDF is shown with the solid blue curve, for whichthe components are shown with dashed red (signal), dashed green (BB background),and dashed blue (qq background).

7.3. CHECK USING MC SAMPLES 81

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Figure 7.8: ∆E (NB > 0.9) and NB (|∆E| < 0.03 GeV) distributions for a MC sampleof B− → [K−π+]Dπ− in the analysis of B− → [K+π−]Dπ−. The fitted sample is shownwith dots with error bars and the PDF is shown with the solid blue curve, for which thecomponents are shown with dashed magenta (signal), dashed green (BB background),and dashed blue (qq background).

Chapter 8

Observables Relative to Angle φ3

In this chapter, we evaluate the ratio RDK and the asymmetry ADK defined in Eqs. (2.27)and (2.28), respectively. The observables for Dπ, RDπ and ADπ, are similarly obtained.

8.1 Detection Efficiency

The detection efficiencies are obtained from signal MC samples, where 100,000 eventsare generated for each mode. A calibration factor for the requirements of particleidentifications, 0.96 ± 0.01 (0.94 ± 0.01) for DK (Dπ), is multiplied, where the decayD∗− → D0π− followed by D0 → K+π− is used for the estimation. The resulting valuesare listed in Table 8.1.

8.2 Ratio RDh

From the detection efficiency (ϵ) and the signal yield (N), we evaluate the ratio RDh as

RDh =N[K+π−]Dh−/ϵ[K+π−]Dh−

N[K−π+]Dh−/ϵ[K−π+]Dh−, (8.1)

for which the charge-conjugate modes are implicitly included. We obtain

RDK = [1.63+0.44−0.41(stat)+0.07

−0.13(syst)] × 10−2, (8.2)

RDπ = [3.28+0.38−0.36(stat)+0.12

−0.18(syst)] × 10−3, (8.3)

where the values are summarized in Table 8.1.The systematic errors (Table 8.2) are subdivided as follows.

(i) ∆E PDFs: The uncertainties due to the PDFs of ∆E for the DK signal, the Dπsignal, and the Dπ feed-across are obtained by varying the shape parameters by±1σ. Those due to the DK feed-across are obtained by varying the width andthe mean by ±10%, which is the difference observed in the data and MC samplesfor the Dπ feed-across. The total uncertainty is the quadratic sum of them: +2.1

−1.8%(+1.3−1.2%) for RDK (RDπ).

83

84 CHAPTER 8. OBSERVABLES RELATIVE TO ANGLE φ3

(ii) NB PDFs: The uncertainties from the PDFs of NB for the DK and Dπ signals(the Dπ feed-across) are estimated by obtaining PDFs from 0.01 GeV < |∆E| <0.02 GeV (0.01 GeV < |∆E−0.05 GeV| < 0.02 GeV) instead of |∆E| < 0.01 GeV(|∆E − 0.05 GeV| < 0.01 GeV). Those due to the DK feed-across and the BBbackground are estimated by applying the PDF of the DK signal instead ofMC-based PDF. Those due to the qq background are estimated by using differentsideband regions in the space of Mbc and ∆E: 5.20 GeV/c2 < Mbc < 5.26 GeV/c2

with −0.1 GeV < ∆E < 0.15 GeV and Mbc > 5.26 GeV/c2 with 0.15 GeV <∆E < 0.3 GeV. The total uncertainty is the quadratic sum of them: +3.4

−3.0%(±3.1%) for RDK (RDπ).

(iii) Yields and asymmetries in the fit: The uncertainty due to the rate of the Dhyields in the DK and Dπ samples is obtained by varying the value by ±1σ.1 Theuncertainty due to the asymmetry of the qq background is obtained by varyingthe value by ±0.02 (±0.00) for DK (Dπ), where the error of the asymmetry forthe favored DK (Dπ) signal is used.2 The total uncertainty is the quadratic sumof them: +0.8

−0.9% (±0.1%) for RDK (RDπ).

(iv) Fit bias: Potential fit bias is checked by generating 10,000 pseudo experiments.We obtain almost standard-Gaussian pull distributions, and take the product ofthe mean of pull and the error of yield, providing −1.1% (−0.5%) for RDK (RDπ).

(v) Peaking backgrounds: The uncertainties of the yields of the backgrounds whichpeak under the signal were described in Section 7.2. The corresponding uncer-tainty in RDK (RDπ) is −6.6% (−4.2%).3

(vi) Efficiency: The uncertainties in detection efficiencies mainly arise from MC statis-tics and the uncertainties in the efficiencies of particle identifications. The esti-mated value is ±1.7% (±1.5%) for RDK (RDπ).

The total systematic error is the sum in quadrature of the above uncertainties.

The significances of RDK and RDπ are estimated using√−2 ln (L0/Lmax), where

Lmax is the maximum likelihood and L0 is the likelihood when the signal yield is con-strained to be zero. The distribution of the likelihood L is obtained by convolvingthe likelihood in the fit and an asymmetric Gaussian for taking the systematic errorinto account. Figure 8.1 shows the distributions of

√−2 ln (L/Lmax). The significance

for RDK (RDπ) is estimated to be 4.1σ (9.2σ), indicating the first evidence for thesuppressed DK signal.

1 The σ is the error in the estimation by the decay D∗− → D0π− followed by D0 → K+π− forh = K and the error in the fit to the favored modes for h = π.

2 Momentum distributions of the particles used for the B reconstruction are similar in the Dh signaland in the qq background.

3 The interference between the peaking background and the signal is negligible, since the “physical”width of the D meson is O(meV).

8.3. ASYMMETRY ADH 85

Table 8.1: Summary of the results. The two errors for the ratio RDh are statistical andsystematic, respectively.

Mode Efficiency (%) Yield RDh [Significance]B− → [K+π−]DK− 33.6 ± 0.4 56.0+15.1

−14.2 (1.63+0.44+0.07−0.41−0.13) × 10−2 [4.1σ]

B− → [K−π+]DK− 33.2 ± 0.4 3394+68−69

B− → [K+π−]Dπ− 36.5 ± 0.4 165.0+19.1−18.1 (3.28+0.38+0.12

−0.36−0.18) × 10−3 [9.2σ]B− → [K−π+]Dπ− 35.7 ± 0.4 49164+245

−244

]-2 10×R [ 0 0.5 1 1.5 2 2.5

)m

ax-

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Figure 8.1: The distributions of√−2 ln (L/Lmax) for DK (left) and Dπ (right). The

dashed curve shows the distribution for the likelihood in the fit, while the solid curveshows the distribution obtained by including the systematic error.

8.3 Asymmetry ADh

We obtain the asymmetry ADh in the suppressed decays as

ADK = −0.39+0.26−0.28(stat)+0.04

−0.03(syst), (8.4)

ADπ = −0.04 ± 0.11(stat)+0.02−0.01(syst). (8.5)

The systematic errors (Table 8.2) are subdivided as follows. The uncertainties relatedto the fit are obtained by the same ways as the ones for RDh. The uncertainty due tothe yield of the peaking background is obtained to be +0.03 (+0.00) for ADK (ADπ) byvarying the signal yield (denominator of the asymmetry). The uncertainty due to theasymmetry of the peaking background, where the value is 0.00± 0.10 for the charmlessdecay [33], is negligible. Possible bias due to the charge asymmetry of the detector

86 CHAPTER 8. OBSERVABLES RELATIVE TO ANGLE φ3

is estimated to be the error of the asymmetry for the favored signal, ±0.02 (±0.00)for ADK (ADπ). The total systematic error is the sum in quadrature of the aboveuncertainties.

Table 8.2: Summary of the systematic uncertainties for RDh and ADh. We use “· · ·”for the sources which do not contribute.

Source RDK RDπ ADK ADπ

∆E PDFs +2.1−1.8%

+1.3−1.2% ±0.01 ±0.00

NB PDFs +3.4−3.0% ±3.1% +0.02

−0.01 ±0.01Yields and asymmetries in fit +0.8

−0.9% ±0.1% ±0.01 ±0.00Fit bias −1.1% −0.5% −0.01 −0.00Peaking backgrounds −6.6% −4.2% +0.03 +0.00Efficiency ±1.7% ±1.5% · · · · · ·Detector asymmetry · · · · · · ±0.02 ±0.00Combined +4.4

−7.8%+3.7−5.6%

+0.04−0.03

+0.02−0.01

8.4 Comparison of Results

In this section, we perform a comparison of the results between different analyses.Table 8.3 shows the previous values by Belle [19] and BaBar [20] as well as the resultsin this analysis. All the results for RDπ and ADπ are consistent with the value (3.31±0.08) × 10−3 [15] for which the decay B− → D0π−, being small relative to the decayB− → D0π− in the standard model, is neglected. The value of RDK in this analysisis ∼ 1σ higher than the previous value by Belle. The reason can be statistical, sincemore than 60% events are independent from the ones in the previous analysis.4 Thevalues of ADK are consistent with each other. Note that the contents of the systematicuncertainties in the three analyses are quite similar, while the values for this analysisare reduced by statistical advantages and careful studies. Additional uncertainties areincluded in the results by BaBar since they fit Mbc, the value of which is not changedeven when a misidentification is applied.

4 The increase of the data sample is only 18% (657 × 106 BB pairs to 772 × 106 BB pairs), whilemany events are added by removing tight requirement for the continuum suppression.

8.4. COMPARISON OF RESULTS 87

Table 8.3: Comparison of the results for RDh and ADh.

This Analysis Previous Result by Belle Latest Result by BaBarRDK [×10−2] 1.63 +0.44 +0.07

−0.41 −0.13 0.78 +0.62 +0.20−0.57 −0.28 1.1 ± 0.6 ± 0.2

RDπ [×10−3] 3.28 +0.38 +0.12−0.36 −0.18 3.40 +0.55 +0.15

−0.53 −0.22 3.3 ± 0.6 ± 0.4ADK −0.39 +0.26 +0.04

−0.28 −0.03 −0.1 +0.8−1.0 ± 0.4 −0.86 ± 0.47 +0.12

−0.16

ADπ −0.04 ± 0.11 +0.02−0.01 −0.02 +0.15

−0.16 ± 0.04 0.03 ± 0.17 ± 0.04

Chapter 9

Extraction of Angle φ3

The measurement of φ3 is obtained by including the observables RCP± and ACP± andthe strong phase δD. The list of the inputs is the following:

RDK = 0.0163 ± 0.0045, (9.1)

ADK = −0.39 ± 0.28, (9.2)

RCP+ = 1.18 ± 0.08, (9.3)

ACP+ = 0.24 ± 0.06, (9.4)

RCP− = 1.09 ± 0.08, (9.5)

ACP− = −0.10 ± 0.07, (9.6)

δD = 338.0 ± 11.2, (9.7)

where the values for RCP±, ACP±, and δD are taken from Ref. [15]. For each error,we take the sum in quadrature of the statistical and systematic errors. We have sevenequations for four unknown parameters φ3, rB, δB, and δD.

The method for extracting φ3 is based on Bayesian statistics with Gaussian assump-tions for the observables. We minimize the χ2(φ3, rB, δB, δD) given by

χ2(φ3, rB, δB, δD) =∑

i

(Oimeasured −Oi(φ3, rB, δB, δD))2

σOi2

, (9.8)

where Oimeasured and σOi are the measured values and the errors in Eqs. (9.1)–(9.7),

Oi(φ3, rB, δB, δD) are the observables as the functions of φ3, rB, δB, and δD, and theindex i runs over the seven observables.

The result (Figure 9.1) is

φ3 = 89 +13

−16 , (9.9)

rB = 0.12 ± 0.02, (9.10)

δB = 34 +12

−10 , (9.11)

δD = 329 ± 10, (9.12)

89

90 CHAPTER 9. EXTRACTION OF ANGLE φ3

modulo 180 for φ3 and δB. The projections of χ2(φ3, rB, δB, δD) for φ3 and δB areasymmetric, and two standard deviation regions are

39 < φ3 < 115, (9.13)

14 < δB < 78. (9.14)

No significant deviations from other measurements (Section 2.5 or Refs. [16, 17, 18]),including an indirect measurement assuming the standard model, are found.1

1 The deviations for δB are slightly higher but less than 3σ, being estimated by using the likelihooddistributions of the results.

91

)° (3

φ0 20 40 60 80 100 120 140 160 180

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Figure 9.1: Result of the φ3 fit. The contours indicate 1 and 2 standard deviationregions. Of the two solutions for φ3 in 0–360, a solution in 0–180 is selected.

Chapter 10

Conclusion

In summary, we report study of the suppressed decay B− → [K+π−]DK− using 772×106

BB pairs collected with the Belle detector. A new method to discriminate the qqbackground combining ten variables with a neural network [31], a D∗− veto, a two-dimensional fit to extract the signal, as well as a 20% increase of the data sample enablea significant improvement on the measurement compared to the previous analysis [19].We obtain the first evidence of the signal with a significance of 4.1σ, resulting in themost precise measurements of the ratio RDK of the suppressed decay rate to the favoreddecay rate, obtained in the decay B− → [K−π+]DK−, and the asymmetry ADK betweenthe charge-conjugate decays of the suppressed mode as

RDK = [1.63+0.44−0.41(stat)+0.07

−0.13(syst)] × 10−2,

ADK = −0.39+0.26−0.28(stat)+0.04

−0.03(syst).

The decay B− → Dπ− is also analyzed as a reference, for which the decay rate isrelatively large whereas the CP -violating effect is small. We obtain

RDπ = [3.28+0.38−0.36(stat)+0.12

−0.18(syst)] × 10−3,

ADπ = −0.04 ± 0.11(stat)+0.02−0.01(syst),

which are consistent with the expectations of the standard model.The results for DK constitute important ingredients in a model-independent ex-

traction of φ3 based on a minimum χ2 method including the observables for the decaysB− → DCP±K− [15]. We obtain

φ3 = 89 +13

−16 ,

which is consistent with the expectation from the CKM picture in the standard model [18].Our study strongly encourages the precision measurements of φ3 at the next-generationB factories, which will play an important role for investigating new physics.

93

Appendix A

Details on Neural Network Training

This appendix presents the details about the neural network training. Table A.1 showsthe summary table for input variables. The rank of importance (“rank”), the identifi-cation number (“node”), the name (“name”), and the preprocessing flag (“preprocess”)are listed for all variables. The explanations on the preprocessing flags are the following.

• 12: transform to Gaussian with no delta-function

• 14: regularized spline fit with no delta-function

• 15: regularized monotonous spline fit with no delta-function

• 18: use map for unordered classes with no delta-function

• 34: regularized spline fit with delta-function

rank node name preprocess

1 2 LR(KSFW) 122 4 ∆z 343 7 cos θK

D 144 3 | cos θB| 155 5 rtag 186 6 | cos θT| 147 11 dDh 158 10 QBQK 189 8 cos θD

B 1410 9 ∆Q 14

Table A.1: List of the input variables.

In the following pages, we show the figures related to the training. The explanationsare included in the captions.

95

96 APPENDIX A. DETAILS ON NEURAL NETWORK TRAINING

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97

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Figure A.2: Correlation matrix of input variables listed in the order of node numbers.


Phi-T Teacher

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2.75e-10 0.0277 0.0338 0.0416 0.0482 0.0534 0.0597 0.0667 0.0746 0.0837 0.0936 0.1047388 0.1171148 0.1310492 0.146851 0.1640648 0.1830448 0.2042399 0.2275901 0.2533394 0.2815452 0.3123611 0.345751 0.3812787 0.4187142 0.4572417 0.4968576 0.5366098 0.5771414 0.6168376 0.6543156 0.6901087 0.7237328 0.7547134 0.7838663 0.809666 0.8333567 0.8544834 0.873159 0.8896185 0.9035593 0.9163926 0.927394 0.937153 0.9456847 0.9532662 0.9609323 0.9684365 0.975033 0.9814217 0.9982257

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Figure A.3: The equalized distributions for the input variable (1st), the signal purity(2nd), the distributions after the preprocessing (3rd), and the purity curve on theefficiency (4th) for LR(KSFW). In the 1st and 3rd figures, the red corresponds tothe signal while the black corresponds to the background. In the 4th figure, the redcorresponds to the result after the training using all variables and the black correspondsto the one for single variable.

99

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Figure A.4: The equalized distributions for the input variable (1st), the signal puritywith a spline fit (2nd), the distributions after the preprocessing (3rd), and the puritycurve on the efficiency (4th) for ∆z. In the 1st and 3rd figures, the red corresponds tothe signal while the black corresponds to the background. In the 4th figure, the redcorresponds to the result after the training using all variables and the black correspondsto the one for single variable. We use a delta-function for the cases when the the vertexfitting is failed.


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-1 -0.962 -0.927 -0.889 -0.85 -0.808 -0.768 -0.728 -0.687 -0.646 -0.602 -0.556 -0.506 -0.455 -0.4 -0.347 -0.292 -0.236 -0.181 -0.125 -0.067 -0.00873 0.0499 0.1075713 0.1634557 0.218463 0.2732971 0.3269012 0.379716 0.4304181 0.4783987 0.5225568 0.5648809 0.6053668 0.6429166 0.6802329 0.714852 0.7473704 0.7779744 0.8063201 0.8320078 0.8559407 0.8775756 0.8971701 0.9152564 0.932026 0.9473825 0.9615176 0.9749992 0.9877721 0.9999992

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Figure A.5: The equalized distributions for the input variable (1st), the signal puritywith a spline fit (2nd), the distributions after the preprocessing (3rd), and the puritycurve on the efficiency (4th) for cos θK

D . In the 1st and 3rd figures, the red correspondsto the signal while the black corresponds to the background. In the 4th figure, the redcorresponds to the result after the training using all variables and the black correspondsto the one for single variable.

101

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NeuroBayes

Input node 3 : abs(cosb)

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1.45e-06 0.016 0.0321 0.048 0.0641 0.0802 0.0964938 0.1124538 0.1288756 0.1450071 0.1611016 0.1775736 0.1938143 0.2104979 0.2266705 0.24291 0.2595239 0.2761568 0.2930995 0.3103176 0.3275277 0.3445148 0.3620794 0.3798557 0.3973091 0.4152758 0.433512 0.4514781 0.4697107 0.4883716 0.5071638 0.5261865 0.5456281 0.5654906 0.5856385 0.6059673 0.6264638 0.647095 0.668848 0.690815 0.7135924 0.7368053 0.7602122 0.7851865 0.8112947 0.8375867 0.8656838 0.8955527 0.9280084 0.9619358 0.9999925

bin #10 20 30 40 50 60 70 80 90 100

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Figure A.6: The equalized distributions for the input variable (1st), the signal puritywith a spline fit (2nd), the distributions after the preprocessing (3rd), and the puritycurve on the efficiency (4th) for | cos θB|. In the 1st and 3rd figures, the red correspondsto the signal while the black corresponds to the background. In the 4th figure, the redcorresponds to the result after the training using all variables and the black correspondsto the one for single variable.


Phi-T Teacher

NeuroBayes

Input node 5 : abs(qr)

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0 0 0 0 0 0 0 0.0476 0.0476 0.0476 0.0476 0.0476 0.0476 0.0476 0.0952381 0.0952381 0.0952381 0.0952381 0.1428571 0.1428571 0.1428571 0.1904762 0.1904762 0.2380952 0.2380952 0.2380952 0.2857143 0.3333333 0.3333333 0.3809524 0.4285714 0.4285714 0.4761905 0.5238096 0.5238096 0.5714286 0.5714286 0.5714286 0.6190476 0.6190476 0.6666667 0.7142857 0.7142857 0.7619048 0.7619048 0.8095238 0.8571429 0.9047619 0.952381 0.952381 1

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Figure A.7: The equalized distributions for the input variable (1st), the signal purity(2nd), the distributions after the preprocessing (3rd), and the purity curve on theefficiency (4th) for rtag. In the 1st and 3rd figures, the red corresponds to the signalwhile the black corresponds to the background. In the 4th figure, the red correspondsto the result after the training using all variables and the black corresponds to the onefor single variable.

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NeuroBayes

Input node 6 : abscost

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0 0.0366 0.0728 0.1093428 0.1457525 0.1813001 0.217179 0.2525885 0.2871957 0.3213383 0.3552231 0.388943 0.4217492 0.454019 0.4858158 0.5171553 0.5470883 0.5765514 0.6058583 0.6330591 0.6597762 0.6844749 0.707554 0.7297946 0.7506995 0.7698839 0.7884737 0.8049811 0.8212116 0.836199 0.8502011 0.8630762 0.8749147 0.8859401 0.8962022 0.9059699 0.9151261 0.9238764 0.9320868 0.9396653 0.9468443 0.9534637 0.9596761 0.9656221 0.9713357 0.9767764 0.9819016 0.9867579 0.9914272 0.9958457 0.9999998

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Figure A.8: The equalized distributions for the input variable (1st), the signal puritywith a spline fit (2nd), the distributions after the preprocessing (3rd), and the puritycurve on the efficiency (4th) for | cos θT|. In the 1st and 3rd figures, the red correspondsto the signal while the black corresponds to the background. In the 4th figure, the redcorresponds to the result after the training using all variables and the black correspondsto the one for single variable.


Phi-T Teacher

NeuroBayes

Input node 11 : dista_hd

PrePro:

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5.51e-08 0.000121 0.000243 0.000364 0.000487 0.00061 0.000734 0.000859 0.000984 0.00111 0.00124 0.00137 0.0015 0.00164 0.00178 0.00192 0.00206 0.00221 0.00237 0.00253 0.00269 0.00285 0.00302 0.0032 0.00339 0.00359 0.0038 0.00401 0.00424 0.00448 0.00474 0.00502 0.00532 0.00566 0.00602 0.00643 0.00689 0.00742 0.00803 0.00877 0.00968 0.0108 0.0123 0.0144 0.0175 0.0226 0.0313 0.0461 0.0744 0.1508375 14.89745

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Figure A.9: The equalized distributions for the input variable (1st), the signal puritywith a spline fit (2nd), the distributions after the preprocessing (3rd), and the puritycurve on the efficiency (4th) for dDh. In the 1st and 3rd figures, the red correspondsto the signal while the black corresponds to the background. In the 4th figure, the redcorresponds to the result after the training using all variables and the black correspondsto the one for single variable.

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NeuroBayes

Input node 10 : qbqk

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-8 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 12

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Figure A.10: The equalized distributions for the input variable (1st), the signal purity(2nd), the distributions after the preprocessing (3rd), and the purity curve on theefficiency (4th) for QBQK . In the 1st and 3rd figures, the red corresponds to the signalwhile the black corresponds to the background. In the 4th figure, the red correspondsto the result after the training using all variables and the black corresponds to the onefor single variable.


Phi-T Teacher

NeuroBayes

Input node 8 : cosdb

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-1 -0.965 -0.929 -0.892 -0.853 -0.814 -0.772 -0.731 -0.689 -0.645667 -0.6 -0.554 -0.507 -0.461 -0.414 -0.366 -0.316 -0.267 -0.218 -0.168 -0.119 -0.0706 -0.021 0.0275 0.0756 0.1221359 0.1685728 0.2136354 0.2585407 0.3030383 0.3461289 0.3875996 0.4296722 0.4692934 0.5089386 0.5473464 0.5843067 0.6203035 0.6551127 0.6889002 0.721117 0.7532679 0.7847027 0.8152196 0.8439682 0.8721639 0.8996782 0.9258696 0.9514158 0.9762384 0.9999955

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Figure A.11: The equalized distributions for the input variable (1st), the signal puritywith a spline fit (2nd), the distributions after the preprocessing (3rd), and the puritycurve on the efficiency (4th) for cos θD

B . In the 1st and 3rd figures, the red correspondsto the signal while the black corresponds to the background. In the 4th figure, the redcorresponds to the result after the training using all variables and the black correspondsto the one for single variable.

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Input node 9 : dq

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Figure A.12: The equalized distributions for the input variable (1st), the signal puritywith a spline fit (2nd), the distributions after the preprocessing (3rd), and the puritycurve on the efficiency (4th) for ∆Q. In the 1st and 3rd figures, the red correspondsto the signal while the black corresponds to the background. In the 4th figure, the redcorresponds to the result after the training using all variables and the black correspondsto the one for single variable.

Appendix B

Demonstration of the Fit to MCSamples

Using the method of configuring PDF described in Section 6.1, we demonstrate the fitto MC samples, in which all known decays are allowed. Each sample has two timeslarger size than the data sample.

B.1 Fit to Favored Modes

Figure B.1 shows the result of the fit to the favored modes. Figures B.2 and B.3 showthe projections for several regions, where good quality of the fit is obtained for eachregion. Table B.1 shows the list of the parameters in the fit. All resulting values areconsistent with the inputs.

B.2 Fit to Suppressed Modes

Figure B.4 shows the result of the fit to the suppressed modes. Figures B.5 and B.6show the projections for several regions, where good quality of the fit is obtained foreach region. Table B.2 shows the list of the parameters in the fit.

The yields for the Dπ and DK signals are obtained to be 428+26−25 and 81+16

−15, respec-tively, which are consistent with the included numbers 416 and 80, respectively. Theasymmetry of the DK signal is obtained to be (43± 19)%, which is consistent with theinput 38%. The asymmetries for the other components are consistent with the inputs 0.The fits are also applied for several other possible inputs. All the results are consistentwith the inputs.

109

110 APPENDIX B. DEMONSTRATION OF THE FIT TO MC SAMPLES

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

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Figure B.1: The result of the fit to the favored modes on MC. The distributions forDπ−, Dπ+, DK−, and DK+ are shown from left to right.

B.2. FIT TO SUPPRESSED MODES 111

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0

200

400

600

800

1000

1200

/ndf = 0.722χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

200

400

600

800

1000

1200

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5000

10000

15000

20000

25000

30000/ndf = 0.942χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5000

10000

15000

20000

25000

30000

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

210

310

410

/ndf = 0.942χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

210

310

410

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120/ndf = 0.502χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120


Figure B.2: The projections for the favored Dπ sample on MC. The ∆E distributionsfor NB < 0, 0 < NB < 0.5, NB > 0.5, and NB > 0.5 (log-scaled) are shown fromleft to right (upper). The NB distributions for ∆E < −0.05 GeV, |∆E| < 0.04 GeV,|∆E| < 0.04 GeV (log-scaled), and ∆E > 0.15 GeV are shown from left to right (lower).


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120/ndf = 0.622χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

160

180

/ndf = 0.772χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

160

180

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

200

400

600

800

1000

1200

/ndf = 0.712χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

200

400

600

800

1000

1200

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

100

200

300

400

500

600

700

800

900/ndf = 0.572χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

100

200

300

400

500

600

700

800

900

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

300/ndf = 0.702χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

300

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0200400600800

1000120014001600180020002200

/ndf = 0.892χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0200400600800

1000120014001600180020002200

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

100

200

300

400

500

600

700

/ndf = 0.932χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

100

200

300

400

500

600

700

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

50/ndf = 0.842χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

50


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120/ndf = 0.722χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

160

180

/ndf = 0.582χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

160

180

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

200

400

600

800

1000

1200

/ndf = 0.822χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

200

400

600

800

1000

1200

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

100

200

300

400

500

600

700

800

900/ndf = 0.642χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

100

200

300

400

500

600

700

800

900

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

300/ndf = 0.932χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

300

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0200400600800

1000120014001600180020002200

/ndf = 0.812χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0200400600800

1000120014001600180020002200

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

100

200

300

400

500

600

700

/ndf = 0.852χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

100

200

300

400

500

600

700

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

50/ndf = 1.232χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

50


Figure B.3: The projections for the favored DK sample on MC. The ∆E distributionsfor NB < 0, 0 < NB < 0.5, NB > 0.5, and NB > 0.9 are shown from left to right(upper). The NB distributions for ∆E < −0.05 GeV, |∆E| < 0.04 GeV, 0.05 GeV <∆E < 0.1 GeV, and ∆E > 0.15 GeV are shown from left to right (lower).



General Yield 104666 ± 349Asymmetry (−0.10 ± 0.31)%

Dπ Mean 0.0011 ± 0.0000D. Gaussian σ1 0.0100 ± 0.0001

(∆E) σ2/σ1 1.86 ± 0.05Yield1/Yield (80.8+1.9

−2.1)%DK General Yield 7715 ± 96

Asymmetry (−0.8 ± 1.2)%General Yield in DK/Yield in Dπ (5.24 ± 0.08)%

Mean 0.0516 ± 0.0003Dπ in DK σ1,right 0.0191 ± 0.0003






−177

BB in Dπ Asymmetry (−1.7 ± 1.6)%Exponential (∆E) Coefficient −18.9+0.5

−0.6


BB in DK Asymmetry (1.5 ± 3.2)%Exponential (∆E) Coefficient −34.2+2.0

−2.1





Table B.1: The list of parameters in the fit to the favored modes on MC.


E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

5

10

15

20

25

30

35

40

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

5

10

15

20

25

30

35

40

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

2

4

6

8

10

12

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

2

4

6

8

10

12

14

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0.0005

0.001

0.0015

0.002

0.0025

0.003

E (GeV)∆-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

NB-0.6

-0.4-0.20

0.20.4

0.60.81

Eve

nts

/ ( 0

.01

x 0.

04 )

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0.0018


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

160/ndf = 0.842χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

160

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

160/ndf = 1.042χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

160

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140/ndf = 1.092χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140/ndf = 1.112χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

20

40

60

80

100

120

140

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

160

180

200/ndf = 1.172χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

160

180

200

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

160

180

200/ndf = 1.432χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

160

180

200

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

/ndf = 1.302χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250

/ndf = 1.312χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

50

100

150

200

250


Figure B.4: The result of the fit to the suppressed modes on MC. The distributions forDπ−, Dπ+, DK−, and DK+ are shown from left to right.


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

/ndf = 0.832χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

/ndf = 0.972χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

90/ndf = 0.682χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

90

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45/ndf = 0.922χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

/ndf = 0.632χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

/ndf = 0.912χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

/ndf = 0.982χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

/ndf = 0.662χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

90/ndf = 0.652χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

80

90

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45/ndf = 0.672χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

45

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

/ndf = 1.102χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

120

140

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

/ndf = 0.962χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60


Figure B.5: The projections for the suppressed Dπ sample on MC. The ∆E distributionsfor NB < 0, 0 < NB < 0.5, and NB > 0.5 are shown from left to right (upper). TheNB distributions for ∆E < −0.05 GeV, |∆E| < 0.04 GeV, and ∆E > 0.15 GeV areshown from left to right (lower).


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70/ndf = 1.062χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40/ndf = 0.922χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

/ndf = 1.252χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

16

18

20/ndf = 0.662χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

16

18

20

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

/ndf = 0.802χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70/ndf = 0.812χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

/ndf = 0.992χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

/ndf = 0.982χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100


E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70/ndf = 1.052χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

10

20

30

40

50

60

70

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40/ndf = 0.662χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

/ndf = 0.992χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

5

10

15

20

25

30

35

40

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

16

18

20/ndf = 0.522χ

E (GeV)∆-0.1 0 0.1 0.2 0.3

Eve

nts

/ 10

MeV

0

2

4

6

8

10

12

14

16

18

20

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

/ndf = 0.402χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70/ndf = 1.022χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

10

20

30

40

50

60

70

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

/ndf = 0.902χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

5

10

15

20

25

30

35

40

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100

/ndf = 1.062χ

NB-0.5 0 0.5 1

Eve

nts

/ 0.0

4

0

20

40

60

80

100


Figure B.6: The projections for the suppressed DK sample on MC. The ∆E distri-butions for NB < 0, 0 < NB < 0.5, NB > 0.5, and NB > 0.9 are shown from leftto right (upper). The NB distributions for ∆E < −0.05 GeV, |∆E| < 0.04 GeV,0.05 GeV < ∆E < 0.1 GeV, and ∆E > 0.15 GeV are shown from left to right (lower).




Asymmetry (2.0+5.8−5.9)%

Dπ Mean 0.0011 (fixed)D. Gaussian σ1 0.0100 (fixed)

(∆E) σ2/σ1 1.86 (fixed)Yield1/Yield 80.8% (fixed)

DK General Yield 81.3+16.3−15.4

Asymmetry (43 ± 19)%General Yield in DK/Yield in Dπ 5.24% (fixed)

Mean 0.0516 (fixed)Dπ in DK σ1,right 0.0191 (fixed)






−24

BB in Dπ Asymmetry (12 ± 17)%Exponential (∆E) Coefficient −16.0+3.4

−4.5


BB in DK Asymmetry (−4 ± 16)%Exponential (∆E) Coefficient −3.2+1.5

−1.6





Table B.2: The list of parameters in the fit to the suppressed modes on MC.

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