Study of the B0 to Ds*X decays at Belle · 2013-09-08 · Study of the B0!D sXdecays at Belle A Thesis Submitted to the Tata Institute of Fundamental Research, Mumbai for the degree

Study of the B0 → D∗sX decays at Belle

A Thesis

Submitted to theTata Institute of Fundamental Research, Mumbai

for the degree of Doctor of Philosophyin Physics

by

Nikhil Jayant Joshi

School of Natural SciencesTata Institute of Fundamental Research

Mumbai

15th September 2010

to them . . .who pumped courage in my lungs,so that I could exhale fresh thoughts!who injected their identity in my veins,so that I could B +ve!!

. . . to aai and baba, my parents.

DECLARATION

This thesis is a presentation of my original research work.Wherever contributions of other are involved, every effort ismade to indicate this clearly, with due reference to the literat-ure, and acknowledgement of collaborative research and dis-cussions.

The work was done under the guidance of Professor Tariq Aziz,at the Tata Institute of Fundamental Research, Mumbai.

Nikhil Jayant Joshi

In my capacity as supervisor of the candidate’s thesis, I certifythat the above statements are true to the best of my knowledge.

Professor Tariq Aziz

Date :

Acknowledgments

THE time has come when I must uncover a secret, I have always kept to myself . . . andso, I lied . . . I lied, that it is my thesis. The deception doesn’t rest in the emphasis

that this is my work, but I will be certainly taking advantage of the mute contributionsand support from all of them, if I am to claim that it is only me, who is responsible forthis success-story. And, on the eve of summarizing my research contribution into a thesisdissertation, I wish to recall all those, without inscribing whose names my thesis willalways remain incomplete.

I take this opportunity to thank Prof. Tariq Aziz, who not only listened to my crazy-looking ideas, but also worked restlessly to shape them up into meaningful concepts. Itis only due to the complete but graduated freedom, offered by him, which allowed meto develop an independent approach as well as understanding in the field of high energyphysics, in general and in my research topic, in particular.

Most of the thesis work has been carried out in meticulous invigilation of Dr. KarimTrabelsi of KEK, Japan. His prudent help in defining my research projects and his confirm-ing approach in bringing them to concrete conclusions have been the key ingredients ofthis work. I am grateful to him for his guidance as well as constant encouragements incarrying out a good quality research.

Prof. Yoshihide Sakai is one name I can not have any less reverence for. . . for teach-ing me by example, what dedication toward science means. Though I could never graba chance to work directly under him, his constant presence in all the Belle meetings, en-lightening guidance and persistent efforts to improve the analyses have always left mespellbound.

I would like to thank Prof. Eunil Won, Dr. Kenkichi Miyabashi and Dr. Ichiro Adachi fortaking deep interest in refereeing our analysis and aiding us establish its correctness andauthenticity. I also wish to thank my colleagues in the Belle Collaboration for helping meintegrate smoothly within the scientific environment at KEK.

I wish to express my gratitude toward Prof. Amol Dighe, Dr. Soumen Dutta and Dr.Ashutosh Alok of Department of Theory, TIFR, and Dr. Naveen Gaur of Delhi University forspending their invaluable time explaining complex theoretical concepts to me and clearingmy doubts and queries, which substantiated the basic motivational framework behind thiswork. Special thanks to Prof. Dighe for proof-reading and correcting the chapter 1 herein.I will never forget the homely atmosphere at Naveen’s house, in which I was absorbedeasily by bhabhi-ji and Aniruddha just as another family member of theirs.

Special thanks to Anagha, Supriya, Ganesh, Lalit, Atul, Bhargav, and Harshal andGayatri. I endorsed myself less into a mirror than in their hearts. Thanking them would

v

ACKNOWLEDGMENTS

just be a formality to conceal my inabilities to truly reciprocate their importance in mylife. I also owe special thanks to Mayuri, who affirmed by negation my growing youngerin time.

I thank my friends at KEK: Seema didi, Tapas, Manmohan, Himansu, Puneet-ji, PuneetTyagi, Rajeev, Abhay, Karina, Kim, Pavel, Sunghyon, Hyuncheong Ha, Horii, sakai, Jimmy,Tanuja, Vishal and Vipin; and my friends at TIFR: Arun, Abinash, Seema, Suvadeep,Garima-didi, Devdutta, Anirban, Ajay, Gautam and Taniya, Chandrasheel and Prachi, Aratiand Satej, Kadir, Vandna, Samarth, Aditya, Vinod, Ashok, Bunty, Jeet and Shilphy, Ashwinand Neha, Anand, Swanand, Aparna, Venkatesh, Ravi, Chinmoy, Sayali, Lakshmi, and allthose who made my TIFR stay comfortable and enjoyable.

It wouldn’t be an exaggeration to say, that all my strength to create something mean-ingful has been epicentered at the firm supporting pivot - a more than homely atmosphere- offered by Maitreyee, Aditya, Bharati-kaku and Vivek-kaka, by Samruddhi, Jyoti-kaku andAnand-kaka, and also by Surendra-kaka and by many others, whom I call just as “kaka” or“kaku”.

I am indebted to the high energy physics secretariats, both at KEK and TIFR, espe-cially to Nakaa-san, Imai-san, Yamaguchi-san, and Minal-ji and Divekar-ji, for maintaininga friendly environment and helping me adopt fast to it. I thank Nishida-san at KEK andDeshpande-ji at TIFR for helping me with my computation system related issues.

I formally wish to thank my TIFR synopsis committee members: Dr. Shashi Dugad, Dr.Nilamani Mathur, Dr. Rudrojyoti Palit, Dr. Kajari Mazumdar, and Dr. Gagan Mohanty, fortheir careful scrutiny and comments on the work presented here.

Nothing is more precious than having a wonderful family at home. This work wouldhave been completely ripped off colours, had I not been received with their appreciatingand smiling faces, every time I needed them. Their approval for pursuing the field ofresearch in basic sciences, particularly in this “moneytonous” world, will always remainthe backbone behind my success. I thank my father, Mr. Jayant, my mother Mrs. Priyanka,and my sisters Nimisha and Prachi for being with me.

And ultimately, to those who always believed that they deserve, but could not find theirnames here: it is only because whatever technological advances we may experience, noscript has ever been developed which truly represents the language of heart . . . For yourpresence in this work can not be spelt, but only be felt. And like the gold sunk in the coreof the earth, deeper you are, higher is your virtue and tougher to surface your name.

- Nikhil

vi

CONTENTS

Contents

Acknowledgments v

Synopsis xi

Bibliography xxiii

0 Prologue 10.1 Baryonic Asymmetry in the Universe (BAU) . . . . . . . . . . . . . . . . . . 10.2 The Dawn of strangeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Theoretical framework and motivation 31.1 C, CP and CPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 CP Violation within SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Kobayashi-Maskawa (KM) Mechanism . . . . . . . . . . . . . . . . . 61.3 CP Violation in neutral B-meson system . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Oscillations: B0 − B0 mixing . . . . . . . . . . . . . . . . . . . . . . 91.3.2 Evolution: B0 − B0 decays . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Classification of CP violating sources . . . . . . . . . . . . . . . . . . . . . . 121.4.1 CP violation in decay . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.2 CP violation in mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.3 CP violation in interference . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 Time Evolution at the Υ(4S) . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.6 CKM vis-à-vis B-factories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.7 sin(2φ1 + φ3) from B0 → D∗∓π± decays . . . . . . . . . . . . . . . . . . . . 181.8 B0 → D∗sh decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.8.1 Sensitivity to RD∗π . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.8.2 Sensitivity to |Vub| . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.8.3 Previous Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Experimental Setup 292.1 Υ(4S): The BB warehouse . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 KEKB accelerator and Belle Detector . . . . . . . . . . . . . . . . . . . . . . 31

2.2.1 KEKB Asymmetric-energy Collider . . . . . . . . . . . . . . . . . . . 322.2.2 Vertexing: Silicon vertex detector (SVD) . . . . . . . . . . . . . . . . 342.2.3 Tracking: Central Drift Chamber (CDC) . . . . . . . . . . . . . . . . 362.2.4 Particle Identification (PID) . . . . . . . . . . . . . . . . . . . . . . . 37

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CONTENTS

2.2.5 Electromagnetic Calorimeter (ECL) . . . . . . . . . . . . . . . . . . . 432.3 Analysis Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.3.1 Data Summary Tables (DST) . . . . . . . . . . . . . . . . . . . . . . 452.3.2 Monte Carlo Simulations (MC) . . . . . . . . . . . . . . . . . . . . . 452.3.3 Data skims and Index files . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Analysis:Monte Carlo Studies 493.1 Signal Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1.1 D+s candidate selection . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.2 D∗+s Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.1.3 B0 → D∗+s π− and B0 → D∗−s K+ . . . . . . . . . . . . . . . . . . . . 56

3.2 Background Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.1 Dominant B decays . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.2 Rare B events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.3 Continuum events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 Signal Extraction Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.3.1 Mbc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.3.2 Mbc-∆E 2D fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.3.3 ∆E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Analysis:MC to data 834.1 Data Driven Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Data Sideband studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.1 ∆M sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2.2 Mbc sidebands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2.3 Off-resonance data studies . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3 Fitting Algorithm for signal Extraction . . . . . . . . . . . . . . . . . . . . . 884.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3.2 Number of fit-parameters . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4 Control studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.4.1 Correcting for PDF parameters: Fudge factors . . . . . . . . . . . . . 914.4.2 Correcting for signal Efficiency . . . . . . . . . . . . . . . . . . . . . 94

4.5 Bias in the fitting program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.1 Ensemble Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.2 Toy MC Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Results And Systematics 1015.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2.1 Branching fractions for D+s decays . . . . . . . . . . . . . . . . . . . 104

5.2.2 Branching fractions for background modes . . . . . . . . . . . . . . . 1075.2.3 Charged-track finding efficiency . . . . . . . . . . . . . . . . . . . . . 1075.2.4 Photon Detection Efficiency . . . . . . . . . . . . . . . . . . . . . . . 1085.2.5 PID efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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CONTENTS

5.2.6 K0S reconstruction efficiency . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.7 Rtotal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.2.8 NBB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.2.9 MC Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2.10 PDF Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.2.11 Fit Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.2.12 Other Negligible Sources . . . . . . . . . . . . . . . . . . . . . . . . . 1175.2.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.3 Results: revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.4 Signal Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6 Conclusion and discussion 1236.1 RD∗π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.2 φ3 estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.3 |Vub| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.4 B0 → D∗+s ρ−: an immediate prospective . . . . . . . . . . . . . . . . . . . . 128

6.4.1 Helicity formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.4.2 Signal Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.4.3 Background MC Studies . . . . . . . . . . . . . . . . . . . . . . . . . 132

A Standard Model of Particle Physics 135A.1 Schema of modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135A.2 Spectrum of SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136A.3 Success of SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

B Analysis Tools 141B.1 Blind Analysis Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141B.2 Fisher Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

B.2.1 Theory of resolving power . . . . . . . . . . . . . . . . . . . . . . . . 143B.2.2 Linear Discriminator in multivariate analysis . . . . . . . . . . . . . . 144

B.3 Kinematical constrain-fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Glossary 149

Bibliography 151

Index 157

List of Publications 159

ix

Synopsis

THE phenomenon of CP violation arises due to the presence of one or more complexphases in the Lagrangian, which can not be removed by simple re-phasing of the fields

representing the particle content of the field theory. Particularly within the framework ofthe standard model (SM), CP violation is achieved via the Kobayashi-Maskawa (KM)quark flavor mixing mechanism, according to which the weak interaction eigenstates arenot the same as the mass eigenstates. This results in addition of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix, which in general is complex, to the SM Lagrangian [1].In the three quark-generations scenario, CKM matrix introduces one single complex phasein the Lagrangian and is the only source of all the CP violation. In general, differentgenerations of quarks mix with differing strengths and are categorised “Cabibbo-favored”or “Cabibbo-suppressed” accordingly. The mixing matrix is necessarily unitary in nature,giving rise to relations between certain parameters of it, which can be visualised as closedtriangles in the corresponding parameter spaces. In particular, the one related to the B-meson system is referred as the unitarity triangle (UT), throughout this work. Measuringprecisely the angles of this triangle would thus serve as an imperative testing ground forthe validity of the KM picture of CP violation within the SM and is the main goal ofresearch conducted at the B-factories like at KEKB.

There are three possible ways in which CP violation can manifest:

• The amplitudes of a B decay to a state f and its CP conjugate process may differ.This is termed as CP violation in decay or direct CP violation. This is the onlysource of CP violation for a charged B meson,

• CP violation in mixing, which occurs when the mass eigenstates differ from theCP eigenstates and hence oscillate. This is possible only in case of the neutral Bmesons, since charge conservation does not allow mixing in the charged modes.

• Even with no CP violation in the mixing or in decay as depicted above, it is possiblethat CP violation manifests via interference terms between decays with mixing andwithout mixing. This is the CP violation in interference between mixing anddecay.

Measuring the time evolution of the decay rates of certain processes, where the decayproducts are accessible both to a B0 and to its CP conjugate state B0 can lead to CPviolating observable, which can then be connected to the parameters in UT. In particular,time-dependent CP analysis of the B0(B0) → D∗∓π± decays provides a theoreticallyclean approach for estimating RD∗π sin(2φ1 + φ3) [2], where φ1 and φ3 are the angles of

xi

SYNOPSIS

the UT and RD∗π is the ratio of the doubly Cabibbo-suppressed decay (DCSD) amplitude(Fig. 1(b)) to the Cabibbo-favored decay (CFD) amplitude (Fig. 1(a)). Given RD∗π andthe angle φ1, it is possible to extract φ3 from the this analysis, which is the least accuratelymeasured parameter of the UT. However, extracting the angle φ3 is not possible from thisstudy alone and requires an independent measurement of RD∗π. Unlike the B0 → D∗∓π±

system, B0 → D∗+s π−, which is dominantly a spectator process with a b → u transition(Fig. 1(c)), does not have contributions from the B0 decaying to the same final stateand can provide a clean experimental access to RD∗π. Under the SU(3) flavor symmetrybetween D∗ and D∗s , the RD∗π is given by

RD∗π = tan θC

(fD∗

fD∗s

)√B(B0 → D∗+s π−)

B(B0 → D∗−π+)(1)

where θC is the Cabibbo angle, fD∗ and fD∗s are the meson form factors, and the B’s standfor the corresponding branching fractions. The B0 → D∗+s π−, in addition, does not havea penguin pollution and hence can in principle provide a way to determine |Vub| [3].

u

d

c

dD∗−

π+

W+

d

bB0

c

d

u

dπ−

D∗+W+

d

bB0

c

s

u

dπ−

D∗+s

W+

d

bB0

(a) (b) (c)

W+

cd

du

D∗−

π+

b

d

B0 W+

ud

dc

π−

D∗+

b

d

B0 W+

cs

su

D∗−s

K+

b

d

B0

(d) (e) (f)

Figure 1: Feynman diagrams for (a) Cabibbo-favored decay B0 → D∗−π+, (b)doubly Cabibbo-suppressed decay B0 → D∗+π−, (c) SU(3) flavor symmetricB0 → D∗+s π−, (d) color suppressed W -exchange contribution to B0 → D∗−π+,

(e) to B0 → D∗+π− and (f) decay B0 → D∗−s K+.

In contrast to the B0 → D∗∓π± decays shown in Figs. 1(d) and (e), the B0 → D∗+s π−

decay lacks contribution from the W -exchange amplitude, as the quark-antiquark pairwith the same flavor, required for such a diagram, is absent in the final state. We assumethe W -exchange contribution in D∗∓π± to be negligible, while deriving equation (1). Thesize of the W -exchange diagram can be estimated from the B0 → D∗−s K+ decay, whichproceeds only through W -exchange (Fig. 1(f)). The B(B0 → D∗−s K+) was expected tobe enhanced due to rescattering effects [4]. However, a recent study based on similarprocesses confirms the absence of such enhancement [5].

It should be noted that the above study can give a clean estimate of the angle φ3,though a discrete ambiguity always exists, due to the inability to identify the strong phasesfrom the weak phases present in the processes. A time-dependent angular analysis of

xii

B0 → D∗∓ρ± is desirable, which would have the same weak phases as in B0 → D∗∓π±

but different strong phases. Being a B → V V process, where V is a vector, the variouspolarization modes has wider informational scope. The B0 → D∗+s ρ− process plays rolesimilar to the B0 → D∗+s π− in this case, though now the presence of the π0 on top of theunknown polarization configuration obscures the analysis.

This work mainly concentrates on the measurement of the branching fractions for thedecays B0 → D∗+s π− and B0 → D∗−s K+ with a data sample consisting of 657 × 106 BBpairs, collected with the Belle detector at the KEKB asymmetric-energy e+e− collider [10].We demonstrate, that the conventional methods of signal extraction, though simpler inapplicability, are not reliable in terms of accuracy of the measurement. On the contrary,the method employed in this work is known to separate the signal from a wider vari-ety of backgrounds and gives more realistic estimates for the signal branching fractions,though is less preferred due to the technical complexities involved in the reconstructionof a photon in the final state. We further extend the technique to include polarizationmeasurements in case of a more challenging B0 → D∗+s ρ− analysis.

While B0 → D+s π− and B0 → D−s K

+ decays have been observed previously byBelle [6] and BaBar [7], respectively, the observations of the modes B0 → D∗+s π− andB0 → D∗−s K+ have only recently been reported by BaBar with a smaller dataset [8, 9].Consequently, in the estimate of RD∗π = [1.81+0.16

−0.13(stat) ± 0.09(syst) ± 0.04(th)]% fromBaBar, the overall error is still dominated by the statistical uncertainties and can be im-proved with a larger data-set collected with the Belle detector.

The Belle detector is a large-solid-angle magnetic spectrometer that consists of a sil-icon vertex detector, a 50-layer central drift chamber (CDC), an array of aerogel thresholdCherenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation coun-ters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals located insidea superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-returnyoke located outside the solenoid is instrumented to detect K0

L mesons and to identifymuons. The detector is described in detail elsewhere [11]. Two different inner detectorconfigurations were used. For the first sample of 152 × 106 BB pairs, a 2.0 cm radiusbeam-pipe and a 3-layer silicon vertex detector were used; for the latter 505 × 106 BBpairs, a 1.5 cm radius beam-pipe with a 4-layer silicon vertex detector and a small-cellinner drift chamber were used [12].

The D∗+s in the signal is reconstructed in three D+s modes: φπ+ with φ → K+K−,

K∗(892)0K+ with K∗(892)0 → K−π+, and K0SK

+ with K0S → π+π−1. Charged tracks are

selected with requirements based on the impact parameter relative to the interaction point(IP). For charged particle identification (PID), the combined information from the specificionization (dE/dx) in the CDC and measurements from the TOF and ACC are used. Atlarge momenta (p > 2.5 GeV/c) only ACC measurement and dE/dx are used. We assignthe likelihood values LK (Lπ) for being a kaon (pion) to each charged track. The tracksare identified based on the ratio RK/π = LK/(LK + Lπ), which peaks at 1 for real kaonsand at 0 for real pions. The identification efficiency is 85% (92%) with a pion (kaon)fake-rate of 8% (15%).

φ (K∗(892)0) mesons are required to have an invariant mass within ±14 MeV/c2

1Inclusion of the charge-conjugate states is implicit throughout this work unless otherwise stated.

xiii

SYNOPSIS

(±75 MeV/c2) of the nominal φ (K∗(892)0) mass2. We reconstruct the K0S candidates

from the π+π− pairs, requiring the invariant mass to be within ±10 MeV/c2 (∼ ±3σ)of the nominal K0

S mass. The K0S candidate is further required to pass the momentum-

dependent selection criteria based on its vertex topology, the flight length in the r − φplane and the daughter π± momentum distribution [13]. The D+

s candidate selectionmass window is chosen to be ∼ ±3σ around the D+

s nominal mass, respectively in thethree modes: φπ+, K∗(892)0K+ and K0

SK+. To reduce the combinatorial background,

we tighten the PID cut for the kaon accompanying the K∗(892)0. The D+s candidate is

further constrained kinematically to have a mass equal to the nominal value.D∗+s mesons are reconstructed by combining the D+

s candidates with a photon. Thephotons are reconstructed as energy deposits in the ECL and are required to have energiesgreater than 60 MeV (100 MeV) in the barrel (endcap) region covering the polar angle,32 < θ < 128 (17 < θ < 32 (forward endcap) and 128 < θ < 150 (backwardendcap)). The D∗+s candidate is selected on the basis of its ∆M = MD+

s γ−MD+

sprofile,

where MD+s γ

and MD+s

are the invariant masses of the D+s γ system and the D+

s candidate,respectively. To reduce the combinatorial background due to low energy photons, werequire that cos θD∗+s > −0.6 (−0.7) in case of B0 → D∗+s π− (B0 → D∗−s K+), whereθD∗+s is defined as the angle between the flight direction of the photon and the directionopposite to the B0 flight in the D∗+s rest frame. We perform the mass-constrained fit tothe D∗+s candidate. This improves the momentum resolution by 25%.

The B0 candidates, reconstructed by combining a D∗+s candidate with an oppositelycharged pion/kaon track, are identified by the energy difference, ∆E =

∑iEi − Ebeam

and the beam-energy constrained mass, Mbc =√E2

beam − (∑

i ~pi)2, where Ebeam is the

beam energy in the Υ(4S) rest frame (CM frame) and ~pi and Ei are the momentum andenergy of the ith daughter of the B0 in the CM frame. We keep B0 candidates with ∆E tobe within ±0.2 GeV and Mbc between 5.2 GeV/c2 and 5.3 GeV/c2 for further analysis.

The dominant background is expected to come from the e+e− → qq (q = u, d, s and cquarks) continuum process. To suppress this background, we use the event topology inthe CM frame to distinguish the spherically symmetric BB events from the jet-like con-tinuum events. A likelihood function R = Lsig/(Lsig +Lbkg) is prepared by combining theFisher discriminant, based on a set of modified Fox-Wolfram moments [14, 15] and thecos θB, where θB is the polar angle of the B0 meson flight direction in the CM frame. Theangle θB follows a sin2 θB distribution for the BB events, while remaining flat in case ofcontinuum events. The selection criteria for R are determined by maximizing a figure-of-merit (FoM), S/

√S +B, where S and B are the number of signal and background events

determined from large Monte Carlo (MC) samples [16], with statistics corresponding toabout 100 (5) times data in case of signal (background) MC. The signal yield, S is ob-tained assuming the latest branching fraction measurements [17]. Figure 2 shows resultof the optimization studies, for the case of B0 → D∗+s π− reconstructed in the φπ mode.In case of B0 → D∗+s π− (B0 → D∗−s K+), we require R to be greater than 0.45 (0.45) forthe φπ mode, 0.50 (0.60) for the K∗(892)0K mode, and 0.40 (0.40) for the K0

SK mode.This requirement suppresses typically 80% of the continuum background, while retaining85% of the signal.

We observe about 15% of the events with more than one B0 candidate. For these2By nominal mass we mean the world-average value from ref. [17].

xiv

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Figure 2: Optimization study in case of B0 → D∗+s π− reconstructed in the φπmode. The red points correspond to the FoM curve obtained with MC backgroundin the signal ∆M region. The green (blue) curve is obtained with the FoM study

done with data (MC) ∆M sideband.

events we choose the candidate with the Mbc value closest to the nominal B0 mass. Thisprocedure selects the correct B0 candidate in about 92% of the cases. Only events withthe Mbc value between 5.27 GeV/c2 and 5.29 GeV/c2 are considered for further analysis,while the signal is extracted performing a fit to the ∆E distribution. We define the fitregion to be |∆E| < 0.2 GeV.

A MC sample of BB events is used to determine possible backgrounds that can enterthe ∆E fit region. In both signal modes (i.e. B0 → D∗sh, where h is a charged K or π),about 45% of the background comes from the decays involving a D+ → K−π+π+ or aD+ → K0

Sπ+ sub-decay, which form a fake D+

s , when a π+ from the D+ is misidentifiedas a K+. However, these decays do not show any peaking structure and are spread overthe entire fit region due to the addition of a random photon.

Some of the rare B decays involving a true D(∗)+s , on the other hand, show non-

negligible peaking structures in the ∆E fit region: the B0 → D+s π− (B0 → D−s K

+)events populate the region around 150 MeV due to the extra photon added, the B0 →D∗+s ρ− (B+ → D∗−s K+π+) events populate the region around −150 MeV, losing the π0

(π+), while the events B0 → D+s ρ− (B+ → D−s K

+π+) are spread around −50 MeV,

xv

SYNOPSIS

with the extra photon compensating the lost pion. These backgrounds are representedby probability distribution functions (PDF) with fixed yields. MC samples are used todetermine the PDF parameters as well as the efficiencies for the peaking modes. The yieldsare calculated assuming the most recent known values for their branching fractions [17,9]. Apart from the backgrounds discussed above, the two B0 signal modes cross-feed eachother. We use signal MCs to determine the PDFs for the cross-feeds. ∆M sidebands in thedata are used to verify the consistency of the MC background predictions with the data.

The signal PDF is defined as a Crystal Ball (CB) line-shape [18] with a broad Gaussianand is determined using signal MC samples generated in eachD+

s mode. The signal as wellas the peaking background PDFs are subsequently corrected for the inconsistencies in theMC predictions, representing the real data, using the B0 → D∗+s D− control sample. Thecombinatorial backgrounds in each mode are accounted for by adding linear functions.

We determine the branching fractions from an unbinned extended maximum likeli-hood fit on the ∆E variable simultaneously in the three D+

s decay modes of each signalmode. To account for the cross-feeds between the signal modes due to the misidenti-fication of the prompt track, the two modes are fitted simultaneously, with the B0 →D∗−s K+ signal yield in the proper candidate events determining the cross-feed amount ofthe same in the B0 → D∗+s π−fit region, and vice verse. The fit is performed with 14 freeparameters: the branching fractions of the signal modes (2), and the yields and slopes ofthe first-order polynomials representing the combinatorial background in each of the threeD+s modes (12). Figure 3 shows the simultaneous fit results performed on the three D+

s

modes in both signal modes.The major source of systematic uncertainty in the branching fraction measurement

of B0 → D∗+s π−(B0 → D∗−s K+) is the uncertainties in the branching fractions of theD+s decays, which amount to 5.9% (6.2%). We estimate the total uncertainty from other

sources to be 9.4% (8.8%).We obtain B(B0 → D∗+s π−) = (1.75± 0.34 (stat)± 0.17 (syst)± 0.10 (B))× 10−5 and

B(B0 → D∗−s K+) = (2.02±0.33 (stat)±0.18 (syst)±0.14 (B))×10−5 with a significance of6.1σ and 8.0σ, respectively, where the systematic uncertainties on the signal yield as well asthe statistical uncertainties are included into the significance evaluation. The significanceis defined as

√−2 ln(L0/Lmax), where Lmax (L0) are the likelihoods for the best fit with

the signal branching fraction under concern allowed to vary (fixed to zero). Figure 4shows the maximization curves of the likelihood with and without systematics included.

Using the observed value for the B0 → D∗+s π− branching fraction, the latest valuesfor B(B0 → D∗−π+) = (2.76 ± 0.13) × 10−3, tan θC = 0.2314 ± 0.0021 [17], and thetheoretical estimation of the fD+

s/fD+ = (1.164 ± 0.006 (stat) ± 0.020 (syst)) [19], we

obtain,

RD∗π = (1.58± 0.15(stat)± 0.10(syst)± 0.03(th))%,

where the first error is statistical, the second corresponds to the experimental system-atic uncertainty and the third accounts for the theoretical uncertainty in the fD+

s/fD+

estimation. We have assumed the equality between fDs/fD and the vector meson decayconstants ratio, fD∗s/fD∗ . The quenched QCD approximation [20] as well as the heavyquark effective theory predictions [21] point toward an uncertainty of about 1% due tothe above assumption, which is included in our estimation of RD∗π.

xvi

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SK mode) and the B0 → D∗−s K+ mode ((d)-(f)). Signal peaks areshown with solid lines while the solid-filled histograms represent the cross-feedcontributions from the other B0 signal mode. The long-dashed lines correspondto the contribution from the B → Dsπ (B → DsK) and the dot-dashed linesto that from the B0 → D

(∗)+s ρ− (B+ → D

(∗)−s K+π+) events. The dotted lines

correspond to the combinatorial background.

The value we obtain for RD∗π, though consistent with the theoretical expectation of2%, is slightly less than the previous estimation [9].

The observed value for the B0 → D∗−s K+ branching fraction is two orders of mag-nitude lower than the CFD B0 → D∗−π+ amplitude. This can be understood purely interms of the exchange amplitude and does not show any evidence for the enhancementdue to rescattering effects, which predicts the two amplitudes to be comparable. Also fromthe same comparison, the effect of the W -exchange processes in the B0 → D∗∓π±, and

xvii

SYNOPSIS

Figure 4: Likelihood maximization scans for B0 → D∗+s π−(B0 → D∗−s K+) fit.The blue (red) curve signifies maximization with (without) systematics added.

consequently in the RD∗π above, can be concluded to be negligible.In conclusion, we report the most precise measurement of the B0 → D∗+s π− and B0 →

D∗−s K+ decay branching fractions. This improves the precision with which the parameterRD∗π can be estimated, and thus the prospect of observing CP violating effects in theD∗±π∓ system. Figure 5 summarizes the effect of our study on the global fit performed inorder to constrain the value for | sin(2φ1 +φ3)|. We follow a procedure similar to that usedby the CKMfitter group and described in [22]. The latest HFAG summer 2009 averages on

)|3

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Figure 5: Constraints on | sin(2φ1 + φ3)| using RD∗π estimation from BaBar [9](left) and Belle (right).

B → D∗π are used for the constrain plots. The most-probable value for | sin(2φ1 + φ3)|appears to be 1.0+0.0

−0.2, which can be translated into a constrain on φ3 using sin(2φ1) as anadditional input. We obtain the most probable value for φ3 to be (77+20

−23).

|Vub| estimation

Extraction of the CKM element |Vub| using the nonleptonic decays, such asB0 → D∗+s π−(ρ−),is considered an alternative approach to the semi-leptonic decays [23]. Following the

xviii

treatment given in [3], we estimate

|Vub| = [2.98+0.27−0.29(stat)± 0.38(syst)]× 10−3 (2)

which is consistent with the earlier measurements from the semileptonic decays. We usedB(B0 → D∗+s D−) = (7.5 ± 1.6) × 10−3 and |Vcb| = 41.6 ± 0.6) × 10−3 from Particle datagroup summary. It should be noted, that the above calculations are performed with theassumption of general factorization being valid to both B0 → D∗+s π− and B0 → D∗+s D−

and is open for additional checks and validations.However, if found theoretically valid, the above estimation can be redone in order

to improve over the experimental systematics. The calculation demands ratio betweenthe branching fractions of B0 → D∗+s π− and B0 → D∗+s D−. The dominant part of thesystematic error comes from the D∗+s reconstruction, which being common to both, conbe cancelled if in stead of measuring individual branching fractions, the ratio is measuredperforming a simultaneous fit to the same data.

Study of B0 → D∗+s ρ−

The signal extraction techniques can easily be extend to include more challenging raredecays of B0. Assuming the SU(3) symmetry between D∗ and D∗s , one would expecta branching fraction of about as high as four times that of B0 → D∗+s π−. Although, thepoor reconstruction efficiency of the additional π0 in the former reduces the efficiency by afactor of 3.5 compared to the latter. The background on the contrary is about thrice higherthan in B0 → D∗+s π−. Opposed to the previous analysis, in which the signal modes areexpected to have a single polarization mode, B0 → D∗+s ρ− has three possible polarizationmodes, composition of which is planned to be measured in this analysis and we removecuts on the D∗+s helicity distributions, which in turn increases the background further. Asa consequence, if the signal is found to have branching ratio equal to the expected valueabove, the significance with which it can be observed in Belle would be about 1.5 timeslower than in B0 → D∗+s π− measurement. Hence, we

• include the full data-set taken till date, which adds another 100 fb−1 to the previ-ously used sample,

• measure theD∗+s and ρ− helicities simultaneously with the branching fraction: Over-all it will be a 3D fit performed simultaneously on three D+

s modes.

The background MC study as well as all validity checks on data sidebands have beencompleted. We chose the B0 → D∗−ρ+ decay as the control study, in order to check thecorrectness of the signal extraction procedure as well as obtaining the correction factorsfor any inconsistencies in MC predictions, representing the real data. We are working onthe preparation and toy MC studies to remove any procedural biases introduced in thefinal fitting code. Figure 6 shows comparison between the MC background and the data∆M sideband. The two samples are in very good agreement as far as the signal extractionvariables are concerned.

The final fit to the data is yet to be performed and intended to be a 3D unbinnedmaximum likelihood fit done simultaneously to three D+

s modes.

xix

http://pdglive.lbl.gov/listings1.brl?quickin=Y


SYNOPSIS

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Figure 6: Comparison between the MC predictions and data for ∆E (left), D∗+spolarization (middle) and ρ− polarization (right). The data points represent thedata sideband and the solid histograms indicate various components of the back-

ground predicted by the MC.

Research Values

The research content of this work, which give us confidence in presenting it as my thesistoward the doctoral degree, is many-fold. Not every aspect is of equal significance to insertinto the final thesis, though worth mentioning in brief

• Demonstration of reliability and applicability of a technique, which is otherwise con-sidered obscure as well as complicated;

• successful measurement of B0 → D∗+s π− and B0 → D∗−s K+ decay branching frac-tions and subsequently the most precise estimation of RD∗π, so far, which is beensummarized into a publication, ready to be submitted to PRD (RC);

• exploration of applicability of B0 → D∗+s π−in estimation of |Vub|;

• Extension of the method to include more obscure and hence challenging measure-ment of B0 → D∗+s ρ−;

• Overall following B decay modes were analysed:

xx

B0 → D∗+s π− ... branching fraction measurementsB0 → D∗−s K+

B0 → D∗+s ρ−

B0 → D∗+s D− ... control sample for B0 → D∗shB+ → D∗−s K+π+ ... control sample for B0 → D∗+s ρ−in Mbc

B0 → D∗−ρ+ ... control sample for B0 → D∗+s ρ−in ∆EB+ → χc1(J/ψγ)K+

... photon systematics studiesB− → D∗0π−

• debugging and re-tuning the Bell Analysis Framework (BASF); and

• Maintenance of DSSD at KEK.

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BIBLIOGRAPHY

Bibliography

[1] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963); M. Kobayashi and T. Maskawa, Prog.Theor. Phys. 49, 652 (1973).

[2] I. Dunietz and R. G. Sachs, Phys. Rev. D 37, 3186 (1988). Erratum: Phys. Rev. D 39,3515 (1989); I. Dunietz, Phys. Lett. B 427, 179 (1998); D. A. Suprun, C. -W. Chiangand J. L. Rosner, Phys. Rev. D 65, 054025 (2002).

[3] C. S. Kim et al., Phys. Rev. D 63, 094506 (2001).

[4] B. Blok, M. Gronau and J. L. Rosner, Phys. Rev. Lett. 78, 3999 (1997).

[5] M. Gronau and J. L. Rosner, Phys. Lett. B 666, 185 (2008).

[6] P. Krokovny et al. (Belle Collab.), Phys. Rev. Lett. 89, 231804 (2002).

[7] B. Aubert et al. (BaBar Collab.), Phys. Rev. Lett. 90, 181803 (2003).

[8] B. Aubert et al. (BaBar Collab.), Phys. Rev. Lett. 98, 081801 (2007).

[9] B. Aubert et al. (BaBar Collab.), Phys. Rev. D 78, 032005 (2008).

[10] S. Kurokawa and E. Kikutani, Nucl. Instrum. Meth. A 499, 1 (2003), and other papersincluded in this volume.

[11] A. Abashian et al. (Belle Collab.), Nucl. Instrum. Meth. A 479, 117 (2002).

[12] Z. Natkaniec et al. (Belle SVD2 Group), Nucl. Instrum. Meth. A 560, 1 (2006).

[13] K.-F. Chen et al. (Belle Collab.), Phys. Rev. D 72, 012004 (2005).

[14] The Fox-Wolfram moments were introduced in G. C. Fox and S. Wolfram, Phys. Rev.Lett. 41, 1581 (1978). The Fisher discriminant used by Belle, based on modifiedFox-Wolfram moments (SFW), is described in K. Abe et al. (Belle Collab.), Phys. Rev.Lett. 87, 101801 (2001) and K. Abe et al. (Belle Collab.), Phys. Lett. B 511, 151(2001).

[15] S. H. Lee et al. (Belle Collab.), Phys. Rev. Lett. 91, 261801 (2003).

[16] For MC event generation, EvtGen, described in D. J. Lange, Nucl. Instrum. Meth. A462, 152 (2001) is used, while the detector performance is simulated using GEANT,described in R. Brun et al., CERN-DD-78-2-Rev, CERN-DD-78-2, Jul 1978.

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BIBLIOGRAPHY

[17] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667, 1 (2008).

[18] J. E. Gaiser (Crystal Ball Collab.), Ph.D. Thesis, Stanford University, Appendix-F,SLAC-R-255 (1982).

[19] E. Follana et al., Phys. Rev. Lett. 100, 062002 (2008).

[20] D. Becirevic et al., hep-lat/0011075 and the references therein.

[21] M. Neubert, Phys. Rept. 245, 259 (1994).

[22] J. Charles et al., Eur. Phys. J. C 41, 1 (2005).

[23] D. Choudhury et al., Phys. Rev. D 45 217 (1992).

xxiv

0Prologue

IN 1928, while Paul M Dirac was accounting for the existence of negative energy states inthe spectrum of spin-1/2 particle field theory [1], he generated innumerable new ques-

tions not only on the scratch-book of a physicist but also in the mind of naive spectatorsof the scientific development. The toughest among them would be to explain a layperson,“what does an anti-particle mean?”. Not really its conceptual content, but it is the awfullyabstract imagination required to understand the idea of an anti-particle, which makes theexplanation incomprehensible. It is genuinely challenging to imagine something, whichhas the same status as the ubiquitous particle in theory, but has absolutely no trace ofexistence in experience. So, why do we need it, if it does not exist? The answer is, anti-particles have been observed in the laboratories and they do exist, but not easily. And thisindeed is the real query: why aren’t they as abundant as their theoretical counterparts, theparticles?

Or in summary, if for every particle, there exists an antiparticle, where has all theanti-part of the universe disappeared? Is the universe distributed into patches of matter-and antimatter- dominated clusters and we happen to live in just one of them, whichaccidentally happens to be matter dominated? If so, why? What separated them? And ifnot, then how does one account for this natural bias for particles?

0.1 Baryonic Asymmetry in the Universe (BAU)

All attempts to locate antimatter dominating patches in the observable universe, includingthe most recent WMAP observations [2], have been unsuccessful and support possibilityof a natural selection over accidental matter dominance. Using the latest WMAP data, thebaryon asymmetry observed in the universe can be quantified into a term, which is expec-

1

PROLOGUE

ted to remain constant during expansion of the early universe, including the inflationaryphase:

ηasym =nB − nB

s≈ 10−10 (1)

where nB, nB are the baryon and anti-baryon densities and s is the entropy density. Thepositive ηasym implies baryon dominance and one needs to develop theories and mechan-isms to generate such an imbalance within the existing knowledge.

The matter-antimatter imbalance observed in the universe can be realised only in twoways: (a) a universe evolving out of initial conditions tuned to support the baryon asym-metry observed today, or (b) a perfectly symmetric early universe with processes gener-ating the baryon asymmetry during evolution. A biased universe as an initial conditionseems an implausible scenario, since it demands an enormous fine tuning - an excess ofone baryon in 30 million baryon-anti-baryon pairs - to start with. Also, the inflationaryphase of the evolution washes out all the baryonic asymmetry present in the initial stages.As a result, the present asymmetry in the matter is believed to be generated dynamicallyduring the evolution.

Sakharov [3] proposed a viable way of dynamically generating a baryonically asym-metric world out of symmetric initial conditions. To generate the matter-antimatter im-balance, it is necessary and sufficient to have

1. Existence of (at least one) baryon number violating process(es),

2. C and CP violation, and

3. Momentary out-of-thermal-equilibrium phase during evolution

Except for the baryon number conserving processes, other two conditions have beenestablished experimentally to a convincing degree and we, in particular, will concentrateon the efforts put in to understand the phenomenon of CP violation, in this work.

0.2 The Dawn of strangeness

While one part of the world was struggling for ideas to account for the missing spectrumof the already known particle domain, the others were facing population explosion viadiscovering new particles and resonances. It was not only the number of newly observedparticles, but also their puzzling behaviour with respect to their decay processes, whichdeeply urged for improved physical understanding. The most notable among them wasthe discovery of the neutral kaons in 1947 and their decay properties, which later led tothe proposal of a new quark state, s, termed as strange, owing to the strangely long livedkaon state. Surprisingly, it is the same neutral kaon system, which put the assumption ofthe parity conservation in weak interactions to test [4, 5] and later for the first time, ledto the observation of the CP violation in 1964 [6].

In conclusion, the nature has chosen to disrespect the CP symmetry, as per Sakharov’sexpectations. And it remains to resolve the immediate quest for introducing the CP viol-ating behaviour into the standard wisdom, gained so far, while quantifying the size of theCP violation.

2

1Theoretical framework and

motivation

This chapter discusses the conceptual background required to de-velop the phenomenology of the CP violation in the framework ofthe standard model of particle physics. The general formalism, hencedeveloped is studied in the context of the neutral B-meson systems.In particular, the Kobayashi-Maskawa mechanism is explored, indepth. An attempt is made to build up the motivation for the studydeveloped in the latter chapters.

1.1 C, CP and CPT

WHILE developing the standard model of particle physics standard model (SM) andthe underlying field theory one conservatively demands invariance of the Lag-

rangian only under proper orthochronous Lorentz group, which is required by any relativ-istic field theory. On the other hand, the Minkowsky metric allows for a higher space-timesymmetry group and also includes two additional discrete symmetries, the parity and thetime reversal. As a consequence, it is natural to extend the symmetry group to the fullLorentz group and check the invariance properties, given some field theory, representingthe physical phenomenon at hand. Knowing the transformation properties of the funda-mental fields in the theory can then guide one in constructing the correct Lagrangian ofthe field theory. The basic ingredients of this extension is summarized here (table 1.1) inthe context of a fermion field ψ(t, ~x), which constitutes the matter-building blocks in SM.

The parity operation, P, reverses the sign of the spatial components of a Lorentz 4-

3

THEORETICAL FRAMEWORK AND MOTIVATIONS

vector: (t, ~x)→ (t,−~x) in particular, effect of which is

Pψ(t, ~x)P† = eiθpγ0ψ(t,−~x)hermitian−−−−−−−→

conjugationPψ†(t, ~x)P† = e−iθpψ†(t,−~x)γ0 (1.1)

where θp is an arbitrary phase. It can be shown, that the effect of P is equivalent to thereversal of the momentum, without affecting the spin.

The time reversal operation, T , which reverses the sign of the time co-ordinate, i.e(t, ~x)→ (−t, ~x) acts as

T ψ(t, ~x)T † = eiθtγ1γ3ψ(−t, ~x)hermitian−−−−−−−→

conjugationT ψ†(t, ~x)T † = e−iθtψ†(−t, ~x)γ3γ1 (1.2)

Or equivalently, the T operation reverses the momentum as well as spin directions.The charge conjugation C, transforms a particle field into its anti-particle, without

affecting the spin and momentum, or

Cψ(t, ~x)C† = (ψ(t, ~x)γ0γ2)T (1.3)

It is possible to construct states with definite parity eigenvalues, called intrinsic paritiesof the state. Similarly, the eigenstates of the charge conjugation operator, C can be attachedintrinsic C-parities. However, since C does not commute with other conserved charges,only those particles, which have all the charges equal to zero, can be assigned an intrinsicC-parity. In particular, for a aa bound system, where a is one of the elementary particles,in a well defined angular momentum state l, the operation of P picks up an extra signof (−1)l, owing to the transformation of the spherical harmonics representing the angularwave function. In case of C, an overall sign of (−1)l+s = (−1)×(−1)l×(−1)s+1, where thefirst factor corresponds to the fermionic field swapping, the second to the relative angularmomentum flip, similar to the case in P, and the third factor is due to the symmetry withrespect to the overall spin of the bound state. As a consequence, if such decay occurs viaCP -respecting process, it results in a C-parity coherent A− A pair, where A is either equalto a, in case of leptons or some composite system thereof, such as a meson.

Table 1.1: Effect of the parity (P), the charge conjugation (C) and the timereversal (T ) operations on a fermionic (anti-fermionic) field ψ(ψ) and the a P−P

bound state with definite orbital angular momentum, l.

Effect ofField

Parity, P Charge conjugation, C Time Reversal, Tψ(t, ~x) eiθpγ0ψ(t,−~x) (ψ(t, ~x)γ0γ2)T eiθtγ1γ3ψ(−t, ~x)

ψ(t, ~x) e−iθpψ(t,−~x)γ0 (ψ(t, ~x)γ0γ2)T e−iθtγ1γ3ψ(−t, ~x)(aa)sl (−1)l+1(aa)sl (−1)l+s(aa)sl -

At the advent of the parity violation, it was believed that not the parity alone but thecombination, CP is the symmetry of the nature, which was later observed to be violatedin the weak decays. Following the quantum field theoretic implications, it is now believedthat the CP symmetry must be violated whenever time-reversal symmetry is broken andvice verse, preserving CPT symmetry. Throughout this work, we assume the CPT to bea respectable symmetry of the nature.

4

1.2. CP VIOLATION WITHIN SM

1.2 CP Violation within SM

The so-far-well-understood sector of the particle physics appears to follow the gauge struc-ture of SU(3)C × SU(2)L × U(1)Y , where C, L and Y denote the color, the left-handedchirality and the electroweak hypercharge symmetry respectively, generally termed as theSM of particle physics. Any effort to introduce the CP violating effects into the currentscenario, must either be fit into the SM or must attempt to extend the SM in a systematicway, without disturbing the current picture significantly. The experimental observationsimply that the standard interaction of the gauge boson fields with other particle fields isCP invariant. This comes from the freedom of choice of the phases of gauge boson fields,which allows one to have real gauge couplings. As a consequence, the CP violation isalways introduced into the Higgs boson sector of the theory. There are three ways this canbe achieved [7]:

• A CP violating Higgs potential, in case of more than one Higgs fields,

• CP violating Yukawa interactions between the Higgs bosons and the fermions, or

• Spontaneously violatingCP via vacuum expectation values of various Higgs fields [8]

In case of the SM, with only one Higgs doublet field, the only possibility for introducingCP violation is in the complex coefficients of the Yukawa interaction. In order to explorethe avenues, of including CP violation mechanisms, present within SM, it is necessary tounderstand the current picture of the elementary particle physics. However, not to disturbthe flow of the development of the central theme, we defer it to appendix A.1, where avery short introduction to the SM is provided and we continue here, with the most popularas well as successful model of CP violation within SM, namely the Kobayashi-Maskawa orKM mechanism.

In order to explain the ratio between the amplitudes for a charged kaon decayingto leptonic final states to that of a charged pion decaying to the same state, which occursthrough a u→ s(d)+W+ transition in case of a kaon (pion) decay, Cabibbo introduced theidea of flavor mixing, in 1963 [9]. According to this concept, the actual weak eigenstates,coupled to the W± in such decays are superpositions of the d and s flavors, i.e.

(d′

s′

)=

(cos θC sin θC− sin θC cos θC

)(ds

)(1.4)

About an year later, in 1964 CP violation was first observed in the neutral kaon sys-tem [6]. Kobayashi and Maskawa extended Cabibbo’s idea of flavor mixing to accom-modate the phenomena of CP violation within the SM, by proposing possibility of thethird generation [10]. The quark flavor mixing mechanism introduced by Kobayashi andMaskawa was certainly a bold step, considering the fact that not even the charm quark,the heavier member of the second quark family, was hinted from the experimental obser-vations, so far.

5


1.2.1 Kobayashi-Maskawa (KM) Mechanism

KM proposed that the weak interaction (V −A) Lagrangian expressed in terms of the flavoreigenstates as,

L(flavor)weak =

g√2

(W+µ u′γµ(1− γ5)d′ +W−µ d

′γµ(1− γ5)u′)

(1.5)

where summation over all u-type and d-type quark fields is implicit, to take the form

L(mass)weak =

g√2

(W+µ uγ

µ(1− γ5)V d+W−µ dγµ(1− γ5)V †u

)(1.6)

when expressed in terms of the mass eigenstates1. The resulting mixing matrix V havingthe form, d′

s′

b′

=

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

dsb

(1.7)

where the unprimed (primed) quarks denote the mass (flavor) eigenstates, is expected tobe unitary. The complex phases entering through the unitarity matrix are the only sourcesof CP violation, in this picture. However, many complex phases from V can be removed,just by re-phasing the fields.

uα → eiψαuα dβ → eiψβdβ (1.8)

under which, the elements of V undergo,

Vαβ → ei(ψβ−ψα)Vαβ (1.9)

As an effect, only those quantities continue to remain physically relevant, which are re-phasing invariant. It is worth reviewing some of the properties of the CKM matrix.

CP violating parameters

A unitary matrix with ng generations has ng × ng independent parameters. The ng u-type and ng d-type quark fields can be re-phased arbitrarily, up to an overall phase, toabsorb (2ng − 1) parameters of V which are of no physical significance. The remaining(ng−1)2 phases are the capable candidates to be the CP violating phases. The orthogonalsubgroup of the unitary group in ng generations-space accounts for ngC2 = ng(ng − 1)/2rotation angles (sometimes referred as the Euler’s angles). In totality, the number of leftout phases, which have physical meaning is

Nphases = (ng − 1)2 − ng(ng − 1)

2=

1

2(ng − 1)(ng − 2) (1.10)

It is evident that to have a CP violating phase in the KM picture within SM, the minimumnumber of generations required is three, in which case Nphases = 1 and we will alwaysassume, ng = 3, unless otherwise stated explicitly.

1To generate the required CP violating effects, KM proposed existence of a third generation consisting ofthe beauty, b and top, t quark doublet. This condition will be demonstrated as necessary for the CP violation,in the next section.

6

1.2. CP VIOLATION WITHIN SM

Phase Invariants of CKM

The quark field re-phasing transforms the CKM matrix elements as given in equation (1.9).The most trivial phase-invariants are the magnitudes of these elements. The next simplestinvariants are the quartets,

Qαiβj = VαiVβjV∗αjV

∗βi (1.11)

Exploiting the unitarity of V , it is possible to show that in case of ng = 3, the imaginaryparts of all the quartets are equal up to a sign [11, 12, 13]. One defines, the Jarlskoginvariant |J | as,

J = =(Quscb) (1.12)

which is an alternate way of realising, that Nphases = 1 for ng = 3.

Parametrization of CKM matrix

In case of three generations, the CKM matrix has four re-phasing invariant - and hencephysically meaningful - parameters, one of them being responsible for all the CP viol-ation observed. To make these facts manifest in the formalism, the CKM matrix can beconveniently parametrized. The most standard parametrization, underscoring all featuresand caveats is due to Chau and Keung [14]:

V =

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e

iδ s23c13

s12s23 − c12c23s13eiδ −c12s23 − s12c23s13e

iδ c23c13

(1.13)

where cij ≡ cos θij and sij ≡ sin θij . On the contrary, the Wolfenstein parametriza-tion [15], being phenomenologically motivated, is the most suited and widely used para-metrization.

V =

1− λ2/2 λ Aλ3(ρ− iη)−λ 1− λ2/2 Aλ2

Aλ3(1− ρ− iη) −Aλ2 1

+O(λ4) (1.14)

To summarize the main features of the Wolfenstein parametrization,

• |λ| = |Vus| ≈ 0.22

• The more off-diagonal one goes, the λ dependence increases, and hence the strengthsdecrease. Flavors among different generations show mixing with different strengths.The relatively stronger ones are referred as “Cabibbo-favored”, while the weaker aretermed “Cabibbo-suppressed”.

• The only complex parameter, iη present in the parametrization, resides in the mostoff-diagonal entries, representing Vub and Vtd.

• Among the four parameters, λ and A are very precisely measured, while ρ and η arenot known very well. It is usual practice to represent the notion of CKM parameterson the ρ− η plane, as discussed later in figure 1.3.

7


Unitarity of V

The unitarity conditions on the six off-diagonal elements of V †V , can be written as,

∑αεui

Vαd1V∗αd2 = 0

VudV∗us + VcdV

∗cs + VtdV

∗ts = 0 . . . K system

(λ) (λ) (λ5)VudV

∗ub + VcdV

∗cb + VtdV

∗tb = 0 . . . Bd system

(λ3) (λ3) (λ3)VusV

∗ub + VcsV

∗cb + VtsV

∗tb = 0 . . . Bs system

(λ4) (λ2) (λ2)

(1.15)

∑βεdi

Vu1βV∗u2β = 0

VudV∗cd + VusV

∗cs + VubV

∗cb = 0 . . . D0 system

(λ) (λ) (λ4)VudV

∗td + VusV

∗ts + VubV

∗tb = 0

(λ3) (λ3) (λ3)VcdV

∗td + VcsV

∗ts + VcbV

∗tb = 0

(λ4) (λ2) (λ2)

(1.16)

where the approximate λ dependence for each term is mentioned below that term, whilealso denoting the meson system, where all the terms occurring in a particular identitycan be naturally accessed. Each of the relations in equations (1.15) and (1.16) can beimagined as three complex vectors adding up to a resultant of zero, arranged in a closedtriangle: the unitary triangle (UT). Out of these, the one corresponding to theBd(u)-systemwill be referred to as UT hereafter, with the angles of UT termed as,

φ1 = arg

(−VcdV

∗cb

VtdV∗tb

)= arg(−Qtbcd)

φ2 = arg

(− VtdV

∗tb

VudV∗ub

)= arg(−Qubtd) (1.17)

φ3 = arg

(−VudV

∗ub

VcdV∗cb

)= arg(−Qcbud)

Note, that merely by their construction, the angles satisfy the identity,∑

i φi = π mode 2πand are not linearly independent. The UT can be imagined in a phase-invariant way, asshown in figure 1.1, since its sides and the angles between them are parametrized interms of phase invariants. It is easy to show, that the area of the triangle turns out to be

A =1

2|VcdV ∗cb|h

=1

2|Qudcb| sinφ3 =

1

2=(Qudcb)

=|J |2

Figure 1.1: (Right) Unitary triangle and (left) the area of UT can be shown to behalf the magnitude of the Jarlskog’s invariant.

8

1.3. CP VIOLATION IN NEUTRAL B-MESON SYSTEM

equal to the Jarlskog invariant defined in equation (1.12).In conclusion,

• According to equations (1.15) and (1.16), it is possible to test the CKM picture invarious meson systems.

• In particular, Bd system is the most prolific among all: a naïve comparison of thesizes of the sides making the triangles in each case, in terms of their λ dependenceimplies that in most of the cases, one of the sides is at least two orders of magnitudesmaller than the other two, flattening the triangle considerably.

• The other system, which can offer an equally open unitary triangle, requires a t−quark bound state, which is impossible due to the t quark decay.

• Even though theBd system opens up the potential for observing CP violation effects,by restricting to the Bd system for experimental sensitivity, one does not lose thegenerality, for areas of the unitary triangles and hence the size of CP violation in allthe systems is equal to the Jarlskog invariant.

The avenues for observing CP violation in the B-meson system is discussed in brief.

1.3 CP Violation in neutral B-meson system

1.3.1 Oscillations: B0 − B0 mixing

The flavor eigenstates corresponding to the neutral B0 − B0 system are bound states of ab-quark and a d-antiquark and vice verse.

|B0〉 ≡ (bd) |B0〉 ≡ (bd)

With zero angular momentum of these bound state, under the CP transformation,they behave as

CP|B0〉 = eiξB |B0〉 CP|B0〉 = e−iξB |B0〉 (1.18)

It is possible to construct the CP eigenstates from these two flavor eigenstates.

|B±〉 =1√2

(|B0〉 ± eiξB |B0〉) (1.19)

The particle detectors, probing the higher states via their decay products, are sensitiveto the flavor eigenstates. But, the CP violating effects can be seen the most conveni-ently through the evolution of the CP eigenstates. While the flavor symmetry is onlyapproximate symmetry of the Hamiltonian, if the CP symmetry is not respected in theweak interactions and hence does not commute with the weak Hamiltonian, none of thetwo sets of eigenstates above coincide with the mass eigenstates. Consequently, the CPeigenstates above are (almost) never realized directly in the real experiments.

The light and heavy mass eigenstates can be written as,

|BL/H〉 =|B∓〉+ ε|B±〉√

1 + |ε|2

=1√

2(1 + |ε|2)

[(1 + ε)|B0〉 ± (1− ε)|B0〉

](1.20)

9


The significance of the ε is evident from the above equation: it specifies the deviation of themass eigenstates from the CP eigenstates and hence the extent of CP violation. However,the absolute value of ε and not the ε itself is the measure of the CP violation, because ofit’s phase-convention dependency. Equivalent to equation (1.20), it is customary to writethe mass eigenstates as,

BH = p|B0〉+ q|B0〉 BL = p|B0〉 − q|B0〉 (1.21)

with the normalization, |p|2 + |q|2 = 1. For the mass eigenstates, the time evolutionequation follows rather simple behaviour, though the 2 × 2 Hamiltonian H leading theevolution is not hermitian, responsible for the decay of the system. It can be decomposedinto two hermitian components,

H = M − i

2Γ (1.22)

withM † = M Γ† = Γ (1.23)

where M and (i/2)Γ represent the B0− B0 evolution via virtual and via real decay states,respectively. It is possible to express the CP violation observables p and q completely in

b

d

W W

d

bu, c, t

u, c, t

B0 B0

b

d

d

b

B0 B0

W

W

u, c, t u, c, t

Figure 1.2: Box diagram for B0 − B0 mixing in the SM.

terms of the components of H. The CPT invariance implies H11 = H22, where the indices1, 2 indicate the flavor eigenstates. The off-diagonal components of H, which account forthe mixing induced oscillations in the B0− B0 system shown in figure 1.2, are particularlyimportant for the CP violation phenomenon. The 〈B0|H|BH〉 − 〈B0|H|BL〉 gives,

q

p=

∆M − i2∆Γ

2H12=

∆M − i2∆Γ

2M12 − iΓ12(1.24)

where ∆M = MH −ML and ∆Γ = ΓH − ΓL. The diagonalization of H implies

q

p= ±

√M∗12 − i

2Γ∗12

M12 − i2Γ12

(1.25)

It should be noted, that an observable persists to be meaningful with respect to CPviolation, if it remains re-phasing invariant. The off-diagonal elements of M and Γ, whichare potential sources of generating CP violating effects in mixing are not re-phasing in-variant. If the meson fields are re-phased as, |B0〉 → eiα|B0〉 and |B0〉 → eiα|B0〉,

M12 → ei(α−α)M12 Γ12 → ei(α−α)Γ12 (1.26)

10

1.3. CP VIOLATION IN NEUTRAL B-MESON SYSTEM

However, the termM∗12Γ12 is re-phasing invariant and any deviation from the real-valuednessof it can imply CP violation in mixing. As a consequence, the quantity, $ = arg(M∗12Γ12)will carry any CP violating effect, if exists in the mixing.

In summary,

1. The CP violation observable ε and hence the p and q carry the complete informationabout the extent of CP violation, and CP invariance holds if and only if q/p = 1.

2. In particular, the magnitude of sin$ imply the amount of CP violation, present inmixing.

3. The ratio, q/p is fixed by the diagonalization of H.

4. The mass eigenstates are not orthogonal, if CP is violated :

〈BH |BL〉 = |p|2 − |q|2 (1.27)

1.3.2 Evolution: B0 − B0 decays

If a pure |B0〉 beam is prepared at time t = 0 in the laboratory, the probability of observinga |B0〉 can be obtained from

〈B0|B0(t)〉 = 〈B0|e−iHt|B0(0)〉

and from equation (1.21), for |B0〉 and |B0(0)〉,

|B0(t)〉 = f+(t)|B0〉+q

pf−(t)|B0〉

|B0(t)〉 =p

qf−(t)|B0〉+ f+(t)|B0〉 (1.28)

wheref±(t) =

1

2e−iM1te−

12

Γ1t[1± e−i∆Mte

12

∆Γt]

(1.29)

Similarly, for a B0 decaying to a state f with amplitudes A(f) = 〈f |H|B0〉 and A(f) =〈f |H|B0〉,

Γ(B0(t)→ f) ∝ e−Γ1t|A(f)|2K+(t) +K−(t)

∣∣∣∣qp∣∣∣∣2 |ρ(f)|2

+2Re

[L∗(t)

(q

p

)ρ(f)

] (1.30)

and a similar expression for Γ(B0(t)→ f), where

ρ(f) = A(f)/A(f)

K±(t) = 1 + e∆Γt ± 2e12

∆Γt cos ∆Mt

L∗(t) = 1− e∆Γt + 2ie12

∆Γt cos ∆Mt

Independent of the mixing betweenB0 and B0, CP violation can still be present in decays,through terms like, A(f) in equation (1.30). Similar to the case of mixing, it is possible

11


to construct re-phasing invariant quantities, which are capable of carrying CP violatingeffects. It can be shown that in addition to the B0(B0) re-phasing, if the states f and fare re-phased, only the quantities,∣∣∣∣qp

∣∣∣∣ , |A(f)|, |A(f)|, and λf =q

p

A(f)

A(f)(1.31)

are re-phasing invariant and can be potential candidates of carrying CP violation effects.Besides, one would expect equalities, such as

|A(f)| = |A(f)| |A(f)| = |A(f)| (1.32)

to hold iff CP is a good symmetry of the Hamiltonian, i.e. (CP)H(CP)† = H.

1.4 Classification of CP violating sources

Having summed up the time evolution of a B0 (and of a B0) into equations (1.28) and(1.30), it is possible to classify the various ways in which an observable CP violating effectcan be realized in a neutral meson system, and in particularly, in the B0 − B0 system.

1.4.1 CP violation in decay

This is also known as the Direct CP violation and exists when equation (1.32) does nothold. Also, due to the freedom of re-phasing, the discrepancy can arise only due to twocompeting complex phases and can not occur due to any one phase. In the present scen-ario, the discrepancy in the sizes of A(f) and A(f) can be attributed to the interferencebetween the two terms corresponding to two different diagrams contributing to A(f) aswell as A(f). It is consequent, that at least two diagrams contributing to the same processare required to observe this effect.

The phases occurring in the amplitudes can have two independent origins:

1. The phases present in the Lagrangian, or weak phasesthese phases come from any complex parameter present in the Lagrangian and hencedue to the hermitian conjugation appear with the opposite sign in the CP conjugateamplitude: A(f) and A(f) have weak phases with opposite signs.

2. The phases appearing due to long distance scatterings, or strong phasesthese phases can occur even when the Lagrangian is real and need not appear withopposite signs in the conjugated terms of the Lagrangian

The amplitudes can be written as,

A(f) =∑i

Aiei(δi+φi) A(f) = ei(ξf−ξB)

∑i

Aiei(δi−φi) (1.33)

Hence, for any final state f , ∣∣∣∣A(f)

A(f)

∣∣∣∣ =

∣∣∣∣∣∑

iAiei(δi−φi)∑

iAiei(δi+φi)

∣∣∣∣∣ (1.34)

12

1.4. CLASSIFICATION OF CP VIOLATING SOURCES

The weak phases φi’s differ from each other, only if CP is violated.

CP violation in decays:∣∣∣∣A(f)

A(f)

∣∣∣∣ 6= 1 (1.35)

The CP asymmetry in this case is usually expressed as,

aCP =1− |A/A|21 + |A/A|2 (1.36)

It is the only possibility of observing CP violation in charged B mesons, since there isno mixing.

1.4.2 CP violation in mixing

CP violation in mixing is also referred to as indirect CP violation and is a consequenceof the fact that the CP eigenstates differ from the mass eigenstates only if the CP is notconserved, which causes the evolution of the CP eigenstates in a mixed manner, discussedin section 1.3.1.

CP violation in mixing:∣∣∣∣qp∣∣∣∣ 6= 1, sin$ 6= 0 (1.37)

This type of CP violation has been fully exploited in the neutral kaon system, due to itsfar separated lifetime eigenvalues. The effect of indirect CP violation is expected to berather small (≈ O(10−2)) in the neutral B meson system. Calculating the CP observablesq/p or $ has turned out to be challenging due to the large hadronic uncertainties.

1.4.3 CP violation in interference

If both CP observables, |q/p| and the decay amplitudes, happen to match their expectedvalues under CP invariance, CP violation will not be present in the mixing as well as inthe decays. Even under such scenario, CP violation can reveal itself via the relative phasebetween these two quantities. As listed in equation (1.31), the quantity λf is a potentialCP violation observable, which depends, in addition to q/p and |A(f)|/|A(f)|, on therelative phase between the two. This CP violating effect is typically seen as, due to theinterference between amplitudes with and without mixing. If some CP eigenstate, f isavailable for both B0 and B0 to decay into, then due to the mixing, CP observable effectscan be extracted, from λf .

CP violation in interference: λf 6= 1 (1.38)

Starting with a pure B0 and a pure B0 at time t=0, and comparing the time evolutionproperties of their decay into the CP eigenstate f , the CP asymmetry can be observed as

afCP =Γ(B0(t)→ fCP )− Γ(B0(t)→ fCP )

Γ(B0(t)→ fCP ) + Γ(B0(t)→ fCP )(1.39)

It will be shown in section 1.5, that

afCP = CfCP cos(∆Mt)− SfCP sin(∆Mt) (1.40)

13


and, the CfCP encapsulates the effect of the CP violation in decays, while SfCP is non-trivial iff CP is violated in the interference between mixing and decay. Even in case of nodirect CP violation, afCP is non-zero, as expected.

In case of no direct CP violation, i.e. CfCP = 0, the asymmetry can be related to theparameters in the Lagrangian, in a theoretically clean way. For example, in the “golden”decay mode of B0 → J/ψK0

S , where the CP asymmetry is dominated by a single CPviolating phase, the CP violating parameters can be cleanly related to the parameters inthe electroweak Lagrangian. In cases with non-zero CfCP , it is not obvious to connect theCP asymmetries to the Lagrangian.

In general, CP violation either in mixing or in decay is sufficient for the CP violationin the interference, but none of them are necessary for the latter.

1.5 Time Evolution at the Υ(4S)

In case of the Υ(4S), which is a spin-1 bb bound state, decaying electromagnetically to aB−B pair, with both spin-0 mesons, the produced mesons fly with an angular momentumof l = 1. As discussed in section 1.1, the resulting system of the B − B pair has a definiteparity and C-parity of (−1)l+s+1 = 1 and (−1)l+s = −1, respectively. Consequently, thetwo mesons are locked in a C-parity singlet (C-odd) state and are said to be quantumentangled [16]. Such a system can be written as,

|(B0B0)C=−(0)〉 =1√2

[|B0~k〉|B0−~k〉 − |B

0~k〉|B0−~k〉]

(1.41)

As an effect of this coherence, no member of the pair can start oscillating randomly onproduction, but the pair, as a whole, oscillates in a coherent manner. As a consequence, notwo B0’s or two B0’s can exist at a time. The coherence persists until one of the memberdecays, after which the other can oscillate at random. Observing the time evolution ofsuch a system can lead to interesting CP violation effects.

Let us assume that at time t = 0, such a coherent pair is formed. The amplitude forone of the members (say with momentum ~k in the Υ(4S) frame) decaying to a state f attime t1 and the other (with momentum −~k) to state g at time t2 is given by

〈f t1; g t2|(B0B0C=−)〉 =

1√2

[〈f |B0

~k(t1)〉〈g|B0

−~k(t2)〉 − 〈f |B0~k(t1)〉〈g|B0

−~k(t2)〉]

(1.42)

Inserting the expressions for |B0(−)~k

(t1(2))〉 and |B0(−)~k

(t1(2))〉, from equation (1.28), we

have

〈f t1; g t2|(B0B0C=−)〉 =

1√2

[a(f+(t1)f−(t2)− f−(t1)f+(t2)

)(1.43)

+ b(f+(t1)f+(t2)− f−(t1)f−(t2)

) ]where

a =p

qA(f)A(g)− q

pA(f)A(g)

b = A(f)A(g)− A(f)A(g)

14

1.5. TIME EVOLUTION AT THE Υ(4S)

Hence the decay rate for this process is given by,

|〈f t1; g t2|(B0B0C=−)〉|2 = e−Γ(t1+t2)

( |a+ b|28

e−Γy∆t +|a− b|2

8eΓy∆t (1.44)

+|b|2 − |a|2

4cos(Γx∆t) +

=(ab∗)

2sin(Γx∆t)

)where x = ∆M

Γ , y = ∆ΓΓ and ∆t = (t1 − t2). The times t1 and t2 are measured in the

rest frame of the respective mesons. The Υ(4S) rest mass being only marginally above themass of the two B mesons, the mesons are produced almost at rest in the Υ(4S) frame. Itis practically insignificant, then if the times are measured in the meson rest frames or inΥ(4S) frame.

Since ∆Γ/Γ ≈ O(10−2), in practice, precision as well as accuracy of measuring theindividual decay times, t1 and t2 can be severely limited by the e+e− bunch lengths andthe detector resolution. The situation becomes even worse when the two mesons areformed at rest, as in case of a Υ(4S) decay. This issue remains one of the main guidelinesbehind the e+e− experimental design.

It is worth studying the time averaged decay rates, in the light of their observationalpotential for the CP violating signatures. It can be shown [17], that the complete timeaveraged decay rates follow,

|〈f ; g|(B0B0C=−)〉|2 =

1

4Γ2

[(|b|2 + |a|2)

1

1− y2+ (|b|2 − |a|2)

1− x2

(1 + x2)2

](1.45)

Two subtle points worth noting here are

1. The time averaged rates in equation (1.45) are functions of the ratios x = ∆M/Γrather than just of ∆M and hence both ∆M as well as Γ would decide the size ofthe CP violating effects in the neutral B meson sector.

2. The time dependence in equations (1.44) and (1.45) comes only through ∆t =(t1 − t2), which is a signature of the C-odd correlation.

To study the time evolution of a B, it is necessary that its flavour is identified in ad-vance. The flavour of a B meson can be identified or tagged by the flavour specific decays,which are accessible to a B0 and not to B0 and vice verse. One of the most adopted meth-ods is the flavor tagging, in which one of the B’s from the Υ(4S) decay is tagged using itsflavor specific decays, while studying the time evolution of the other. Due to the mixing, itis not always possible to guess the flavor of one B, just by tagging the flavor of the other,in general. However, in case of the Υ(4S) decays, the correlation between the two B’sconfirm that the B’s must be of opposite flavors, at the time of tagging.

In case of an experiment, where one of the B’s (say B0) has been tagged, using itsflavor specific decay (say, semi-leptonic SL), so that

A(g) = A(g) = 0 (1.46)

while the time evolution of the other B decaying to a CP eigenstate fCP is observed,

A(f) = A(f) A(f) = A(f) (1.47)

15


Equation (1.44) reduces to

|〈f tCP ; g ttag|(B0B0C=−)〉|2 = e−Γ(tCP+ttag)|A(g)|2|A(f)|2 × (1.48)( |λf + 1|2

8e−Γy∆t +

|λf − 1|28

eΓy∆t

+1− |λf |2

4cos(Γx∆t) +

=(λf )

2sin(Γx∆t)

)where λf = (q/p)(A(f)/A(f)), as defined in equation (1.31). Based on both experimentalobservations as well as theoretical arguments [18], one would expect no CP violation inmixing in B decays in SM: |q/p| = 1. Also, to a very good approximation, for the B-mesonsystem ∆Γ ∼ 0. Within these approximations, with x = ∆M/Γ

|〈f tCP ; g ttag|(B0B0C=−)〉|2 =

|A(g)A(f)|2e−Γ(tCP+ttag)

4× (1.49)(

1 + |λf |2 + (1− |λf |2) cos(Γx∆t) + 2=(λf ) sin(Γx∆t))

and with a similar expression for the tagged B0 decay, where now B0 → gtag and B0 →fCP .

Hence, the CP asymmetry in tagged decays into CP eigenstate f can be written as,

aCP (t) =Γ(B0(t)→ f)− Γ(B0 → f)

Γ(B0(t)→ f) + Γ(B0 → f)

=

(1− |λf |21 + |λf |2

)cos(∆M∆t) +

(2=λf

1 + |λf |2)

sin(∆M∆t) (1.50)

Under the assumption of no CP violation in mixing in B decays, |λf |, and hence the coef-ficient in front of the cosine term in equation (1.50), is sensitive only to the discrepanciesin A(f) and A(f) = A(f), i.e. to the CP violation in decay or direct CP violation. On theother hand, the coefficient of the sine term measures the amount of CP violation in theinterference between mixing and decays.

A few special cases may arise,

• A subclass of the above decays, where the decay of aB0 and B0 in to a CP eigenstatef is dominated by a single weak phase, λf turns out to be a pure phase. Also, thereis no direct CP violation, since |λf | = 1. As a result,

aCP (t) ∝ sin 2(φw − φq/p) sin(∆M∆t) (1.51)

and the time-dependent asymmetry directly measures the weak phase φw in theLagrangian.

• As a signature of the coherent system, the time dependence in aCP comes through∆t, which can take values in (−∞,∞) and the odd parity of sin(∆M∆t) can washout the sensitivity to the CP violation in interference, when time-integrated decaysare measured, in an experiment with poor vertexing abilities. This need not happenfor a incoherent system, such as in hadronic colliders.

16

1.6. CKM VIS-À-VIS B-FACTORIES

1.6 CKM vis-à-vis B-factories

The KM mechanism with three generations within SM, allows for only four phase in-variant, physically meaningful parameters, one of which is a pure complex phase, ac-commodating all the CP violating effects (see, section 1.2.1). In principle, these fourparameters could be fixed if three angles and any one side of the UT are measured pre-cisely. Though, as emphasized earlier, the three angles defined in equation (1.17) satisfythe identity

∑i φi = π mode 2π, merely by their construction and are not independent.

Hence, it would not be sufficient to measure the three angles. On the other hand, it is notonly the KM mechanism which is on trial, but also the various assumptions made, whiledeveloping the formalism: about the number of generations being three, about CPT be-ing invariant, or SM being sufficient explanation of the particle physics, etc; are equallyvulnerable. As a consequence, it is necessary to over-constrain the CKM matrix, usingcopious measurements, yielding overlapping information.

Since the arrival of the B-factories, numerous methods have been developed to extractinformation not only to observe the CP violation in B meson system, but also to extractobservables to quantify various CKM matrix parameters [19]. Figure 1.3 shows the globalfit performed by the CkmFitter group in winter 2009 [20], using the latest world averagesfor all the CKM parameters (left) with the current estimates of the three angles of theunitarity triangle published by the Particle data group (right).

3φ

2φ

2φ

dm∆

Kε

Kε

sm∆ & dm∆

SLubV

ν τubV

1φsin 2

(excl. at CL > 0.95)

< 01

φsol. w/ cos 2

2φ

1φ

3φ

ρ−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0

η

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

excl

uded

are

a ha

s C

L >

0.9

5

Moriond 09

CKMf i t t e r

AngleWorld Average

degree ()

φ1 21.1± 0.9

φ2 88+6−5

φ3 77+30−32

Figure 1.3: (left) Global fit to the CKM parameters by the CkmFitter group, shown inMoriond’09 and (right) the current world averages for the three angles of the unitaritytriangle, as per quoted by the Particle data group. Note, that the other value of φ1 =

68.9± 0.9 is disfavored in SM by decays such as B0 → J/ψK∗, B → D∗D∗K0S .

It can be concluded from the table in 1.3, that while relatively precise estimates of theangles φ1(∼ 4%) and φ2(∼ 10%) have been achieved, the angle φ3 is plagued with a hugeuncertainty of ∼ 45% and severely limits our knowledge about the KM mechanism withinSM, as far the current understanding goes. The two basic reasons behind why φ3 is poorlyknown are,

1. Various examples of B decays exist where the final state is a CP eigenstate. In suchcases, many hadronic uncertainties in the ratio |A(f)|/|A(f)| cancel out, cleaning offthe theoretical uncertainties. Not many such possibilities have been found in case ofφ3 determination.

17

http://www.ckmfitter.in2p3.fr





2. In many such modes, which satisfy f = fCP condition, the amplitudes are foundto be dominated by a single weak phase. Under such circumstances, it is relativelyeasy to relate the experimental observables directly to CKM parameters, as discussedpreviously.

Many approaches have been suggested which can potentially overcome the limitationsof dealing with non-CP eigenstates, which share a common central concept: The angle φ3

is the relative phase between b→ cus and b→ ucs processes and if the uc(cu) hadronizesinto a singleD0(D0), which subsequently decays into a CP eigenstate, then both processeslead to a common final state and can give rise to CP violating effects via interference [21].

1. Direct approaches:measurements which yield information directly about φ3

• Time dependent analysis of Bs → ρ0K0S

If only tree level diagrams contribute, this mode can measure sin 2φ3. However,it suffers a large penguin pollution, in practice. Theoretically, the angle 2φ3

is allowed to take values near π constraining the sin 2φ3 to lie around zero,making its extraction inconvenient experimentally. Also, this decay branchingfraction is expected to be extremely small O(10−7). Hence, theoretically as wellas experimentally, it does not appear to be a very promising mode.

• ADS [22], GLW [23, 24], GGSZ [25] methodsThese methods exploit the so called triangle relations between the suppressedmodes B → D0X, B → D0X and B → D0

fCPX. However, the GLW method

suffers from flattening of the triangle in the triangle relation, as a result ofCabibbo-suppression in two of its sides. ADS method surpasses the limitationsin GLW, by extending the analysis to include non-CP eigenstates. While, GGSZextended the idea from ADS to include three body decays of D, introducing thepower of Dalitz analysis [26] into the original GLW framework and has beenproved to be the most sensitive approach of extracting φ3.

2. Indirect approaches:measurement which require information on other CKM angles

• B0 → D(∗)∓π±(ρ±, a±1 )these modes measure sin(2φ1 +φ3) and require information on φ1 to extract thevalue of φ3. We discuss these measurements in detail, in the next section.

1.7 sin(2φ1 + φ3) from B0 → D∗∓π± decays

The decays B0 → D∗∓π± constitute the simplest example of the modes, which can offerindirect, but theoretically clean, access to the CKM phase φ3. The theoretical cleanlinesscomes from the fact, that these modes do not receive any penguin pollution, which wouldotherwise obscure the interpretation of the extracted phases. The final states f = D∗∓π±

are not CP eigenstates. However, the same decay state, say D∗−π+, is attainable to bothB0 as well as B0 and can reveal CP violating effects via interference between the twodecays. The amplitudes in the two cases, however correspond to Feynman diagrams with

18

1.7. SIN(2φ1 + φ3) FROM B0 → D∗∓π± DECAYS

different CKM elements and hence, differ considerably in strengths. Figure 1.4 showsthe Feynman diagrams for a B0 decaying to D∗−π+ (left) and to its conjugate final state(right). It can be seen that the former proceeds through a b → cud transition and is

u

d

c

d

D∗−

π+

W+

b

B0

V ∗ud

Vcbd

c

d

u

d

π−

D∗+

W+

b

B0

V ∗cd

Vubd

Figure 1.4: Dominant Feynman diagrams for the Cabibbo-favored decay (CFD)B0 → D∗−π+ (left) and the doubly Cabibbo-suppressed decay (DCSD) B0 →

D∗+π− (right).

said to be “Cabibbo-favored decay (CFD)”, whereas the latter involves a b→ ucd process,which suffers suppression from two off-diagonal CKM elements and will henceforth bereferred as the “doubly Cabibbo-suppressed decay (DCSD)”. As a reason, this methodrequires extracting the weak phase via comparing rates of contrasting strengths, unlikeother methods, where the two amplitudes are usually comparable.

We denote by f± = D∗∓π± (i.e. sign on pion corresponds to the subscript of f). Usingequations (1.49), the time dependent decay rates for the above four processes can bewritten as,

|〈f±|B0〉|2 = |Af± |2e−Γt

2×(

1 + |λf± |2 + (1− |λf± |2) cos(∆M∆t) + 2=(λf±) sin(∆M∆t))

(1.52)

|〈f±|B0〉 = |Af± |2e−Γt

2

∣∣∣∣pq∣∣∣∣2 ×(

1 + |λf± |2 − (1− |λf± |2) cos(∆M∆t)− 2=(λf±) sin(∆M∆t))

(1.53)

where |p/q|2 = 1 in the second equation above. It is easy to see, that

λf+ = 1λf+

= A(D∗−π+)A(D∗−π+)

λf− = 1λf−

= A(D∗+π−)A(D∗+π−)

=DCSD

CFD(1.54)

The smallness of CP violation in mixing in the B-meson system allows one to assert thatq/p is very close to 1 and can be assumed to be a pure phase. Indeed, it has been foundthat |M12 |Γ12|, and equation (1.25) reduces to

q

p≈√M∗12

M12=

M∗12

|M12|∼ exp(i argM∗12) (1.55)

which can be estimated from the box diagrams shown in figure 1.2 to be (within SM)

q

p= −ei(ξB+ξd+ξb)

V ∗tbVtdVtbV

∗td

= −ei(ξB+ξd+ξb)e−2iφ1 (1.56)

19


where we use the identity arg(z) = arg(1/z∗), while rearranging the CKM matrix elementsto relate to the CKM angle φ1. Using this relation, we have

λf+ = λD∗π =q

p

AD∗−π+

AD∗−π+

=(−ei(ξB+ξd−ξb)e−2iφ1

)(V ∗cdVubV ∗cbVud

)(〈D∗−π+|B0〉〈D∗−π+|B0〉

)(1.57)

The two amplitudes in the last term are difficult to connect by simple arguments and maycarry additional spurious weak phases and a strong phase δ. The re-phasing invariantnature of λf can be exploited to expect overall cancellation of the spurious weak phases.The CKM elements can be rearranged to show,

V ∗cdVubV ∗cbVud

= |V∗cdVubV ∗cbVud

| exp(i arg(VudV

∗ub

VcdV∗cb

)) = |V∗cdVubV ∗udVcb

| exp(−i(π + φ3)) (1.58)

Overall, λD∗π in equation (1.57) has the form

λD∗π = RD∗π exp[i(2φ1 + φ3 + δ)] (1.59)

Measuring the time-dependent decay rates of a tagged B0 or a tagged B0 and fitting thedistributions to the expressions in equations (1.52) and (1.53), one can extract informa-tion about RD∗π and sin(2φ1 + φ3 ± δ) from,

S±D∗π(∆t) = (1 +R2D∗π)± (1−R2

D∗π) cos(∆M∆t) (1.60)

A±D∗π(∆t) = −2RD∗π sin(2φ1 + φ3 ± δ) sin(∆M∆t) (1.61)

where the S±D∗π(∆t) and A±D∗π(∆t) are the symmetric and antisymmetric components ofthe time dependent decay rates distributions in a B0(B0) decay.

A more detailed theoretical analysis of the B0 → D∗∓π± decays can be found in [27],with the most recent experimental endeavours at Belle [28, 29] and BaBar [30]. We drawattention to the main features of this analysis:

• It is possible to make an educated guess about the size of RD∗π

RD∗π =

∣∣∣∣V ∗cdVubV ∗cbVud

∣∣∣∣ r = | − λ2CKM(ρ2 − iη)|r ∼ 0.02r (1.62)

where λCKM is one of the Wolfenstein parameters of CKM matrix, introduced inequation (1.14) and must not be confused with the λf , and r ∼ O(1) is the hadronicpart of the ratio between the decay constants. It is clear, that the sensitivity to theCP asymmetry is low.

• Using the fits to the symmetric part of the decay rate distributions, RD∗π can beestimated, in principle. However, the smallness of RD∗π critically limits the stat-istical sensitivity in this measurement. One of the main reasons behind this is thelarge B0−B0 mixing amplitude compared to DCSD, which introduces overwhelmingirreducible background in the latter.

20

1.8. B0 → D∗SH DECAY

• It has been proposed to use the decays of the chargedB meson, such asB+ → D∗+π0

to obtain the direct measurement of the ratio RD∗π [27]. However, this particulardecay has not yet been established experimentally and only an upper bound existsfor its branching fraction (< 3.6 × 10−6) [31]. Also, the comparison between thecharged mode decay and B0 → D∗−π+ relies strongly on the naïve factorization andis supposed to carry large hadronic uncertainties.

• The other strategy is to use the decay B0 → D∗+s π−, which can be associated withthe DCSD assuming the flavor SU(3) symmetry between the d-quark and the s-quark.This approach is discussed in detail, in the next section.

• It is evident from equation (1.59) that this method is plagued with a non-zero strongphase δ, which can not be distinguished from the weak phases, without making fur-ther assumptions. This introduces up to a four-fold ambiguity in the determination ofthe weak phase φ3, depending upon the situation. It is hence preferable to make anindependent measurement of sin(2φ1 + φ3) in decays like B0 → D∗ρ or B0 → D∗a1,where the strong phase may differ from δ in B0 → D∗π decays [27]. The B0 → D∗ρin particular, being a B → V V decay, is of great importance due to its informationcontent [32] and one is required to do a time dependent angular analysis to extractthe full information.

• On the contrary, this method can prove to be very effective in detecting deviationsfrom the SM: If the strong phase δ is not very close to 0 or π, the ambiguity betweensin(2φ1 + φ3) and cos δ can be resolved and the former can be determined with aprecision of 5% or better.

1.8 B0 → D∗sh decay

1.8.1 Sensitivity to RD∗π

The B0 → D∗+s π− is a b → ucs process, as shown in figure 1.5 and can be related tothe DCSD B0 → D∗+π− diagram in figure 1.4, if one assumes the flavor SU(3) sym-metry between the d-quark and the s-quark. The experimental difficulties discussed in

c

s

u

d

π−

D∗+s

W+

b

B0

V ∗cs

Vubd

Figure 1.5: Feynman diagram for the decay B0 → D∗+s π−.

the earlier section, especially the mixing-induced, overwhelming cross-feed to the DCSD,prevents one from accessing RD∗π from the time dependent studies of the B0 → D∗∓π±

decays alone and a supplementary method is highly sought which can provide this in-put externally. Unlike the B0 → D∗∓π±, there is no background contribution to the

21


B0 → D∗+s π− from the B0 decaying to the same final states, i.e. B0 9 D∗+s π−. Hence,under the assumption of SU(3) flavor symmetry, one would expect the DCSD amplitudeto be equal to the B0 → D∗+s π−amplitude, provided the Feynman diagrams for the twoprocesses are identical and as a consequence, RD∗π in equation (1.59) can be expressedas,

RD∗π = |λD∗π| =∣∣∣∣A(D∗−π+)

A(D∗−π+)

∣∣∣∣ SU(3)−−−−−−→symmetry

∣∣∣∣A(D∗+s π−)

A(D∗−π+)

∣∣∣∣ (1.63)

and can be calculated, if A(D∗+s π−) is known experimentally.However, this equality rests strongly on the validity of two assumptions made:

• SU(3) flavor symmetryThe difference between the two diagrams at the quark level manifests only throughthe CKM elements Vcd and Vcs and can be accounted for by adding the Cabibbofactor of θC (see, page 5). On the contrary, due to the quark confinement, the quarksalways hadronize to give a color singlet meson in the final state and an isolated quarkis never observed. As a result, the test for the validity of SU(3) symmetry must beextended to include the hadronization effects. The hadronic details of a meson, M isgenerally summed up into a quantity fM , termed as the meson decay constant. Thedesired quantities in case at hand, then would be the decay constants, fD∗ and fD∗s .Correcting for the SU(3) flavor symmetry breaking, one has,

RD∗π = tan θC

(fD∗

fD∗s

) ∣∣∣∣A(D∗+s π−)

A(D∗−π+)

∣∣∣∣ (1.64)

Various approaches exist to probe the fD∗(s)

:

– Quantum Chromodynamics (QCD):Resulting from the asymptotically free nature of the strong interactions, it isnot usually possible to study the same perturbatively, without invoking the ef-fective theoretic regime, under additional approximations. Under the heavyquark symmetry (HQS) [33], in which the heavy quarks2 are assumed infinitelymassive and the hadronic properties become independent of the flavor and spinof the heavy quark, the heavy quark is usually assumed to provide the centralstatic charges, while surrounded with the light quark-gluon sea. Any deviationfrom the HQS predictions, due to finite masses of heavy quarks, is calculatedperturbatively in the regime of the heavt quark effective theory (HQET), as aseries in 1/mQ.

– Lattice Monte Carlo techniques:The logic of HQS-HQET can be extended to non-perturbative, but computa-tional techniques. For computational convenience, the lattice QCD (LQCD)approach treats the heavy quark as providing static charges to the hadron and

2A heavy quark is the one with a mass very large compared to the intrinsic chiral symmetry breakingscale, ΛQCD ∼ 0.2 GeV. When such a heavy quark Q, bound inside a hadron, has a Compton wavelength of1/mQ, much smaller than the typical hadronic distance of 1 fm, the heavy-quark mass becomes insignificant,as long as the low energy properties of the hadron are concerned.In that case, the strong interaction of theheavy quark with other light quarks and gluons is described by an effective theory, which is invariant underthe change of flavor and spin of Q.

22


is identified with a lattice point on a discrete space-time grid, while the lightquarks introduce the dynamically interacting quark-sea field. The decay con-stant is then calculated on the lattice by Monte Carlo techniques. So far, thishas proved to be the best non-experimental method for fM determination.

– Experimental Measurements:The cleanest avenues of determining the decay constants experimentally is thesemi-leptonic or leptonic decays of a charged pseudoscalar, i.e. D+

(s) → l+νl.

Figure 1.6 shows the Feynman diagram for the purely leptonic decay of a D+(s).

The final state being purely leptonic, are completely free from the hadronic

W+

D∗+(s)

d(s)

c l+

νl

Vcd(Vcs)

Figure 1.6: Purely leptonic decay of a D∗+s .

uncertainty and the only hadronic quantity fD(s)can be estimated from the de-

cay rate, if the CKM matrix element is known with good precision. Thoughavailable for a pseudoscalar, this approach is not accessible in case of the cor-responding vector meson, since an excited state such as a vector meson readilydecays to its ground-state pseudoscalar by a radiative or pionic decay and notdirectly into a semileptonic decay associated.

– other:There exist a number of alternate, theoretically somewhat moderate approachesto surpass difficulties in the field theoretic formalisms. Non-relativistic quantummechanical approximations (NRQM) to the effective QCD Lagrangian is one ofsuch efforts.

Table 6.1 in chapter 6 summarizes the current known values for the D meson decayconstants from various approached discussed above.

• Identical Feynman ProcessesThe analogy between the DCSD and the B0 → D∗+s π− is limited to the diagramsshown in figures 1.4 and 1.5, since in addition to the aforementioned spectatordiagrams, the B0 → D∗∓π± decays also have a contribution from W -exchange amp-litudes, as shown in figure 1.7. The B0 → D∗+s π−on the contrary, lacks such a

W+

b

d

B0

c

d

du

π+

D∗−

W+

b

d

B0

u

d

dc

D∗+

π−

Figure 1.7: W -exchange contributions to the CFD (left) and to the DCSD (right).

contribution, since a quark-antiquark pair with the same flavor, required in the final

23


state, is absent. As a result, the strategy discussed above of utilizing SU(3) symmetrybetween the two processes to obtain RD∗π would fail, if the W -exchange contribu-tion in DCSD is sizable relative to the external-W diagram in 1.4. Due to the internalgluonic pop-up in the W -exchange process, the amplitude is expected to be smaller(∼ α2

s ≈ O(0.04)) than the process with external-W (∼ O(1)). However, it neednot be negligible and makes it necessary to check the relative strengths of the twodiagrams contributing to DCSD. It is easy to see, that the relative strengths of thetwo diagrams would be comparable in case of CFD and DCSD, i.e.

A(CFDW−exchange)

A(CFDdominant)=A(DCSDW−exchange)

A(DCSDdominant)(1.65)

The B0 → D∗−s K+ decay dominantly occurs via a W -exchange diagram, shown infigure 1.8 (left) and can be related to the W -exchange CFD using the SU(3) d − s

W+

b

d

B0

c

s

su

K+

D∗−s

u

d

c

d

D∗−s

π+

W+

b

B0

V ∗ud

Vcb

d

u

s

sK+

D∗−

Figure 1.8: Feynman diagram for B0 → D∗−s K+ decay via a W -exchange dia-gram (left) and via CFD final-state rescattering (right).

flavor symmetry. Since this is the only contribution in former, the size of CFD W -exchange amplitude can be assessed through measuring the B0 → D∗−s K+ decayamplitude and the above comparison can be extended to

A(CFDW−exchange)

A(CFDdominant)=A(B0 → D∗−s K+)

A(CFDdominant)=A(DCSDW−exchange)

A(DCSDdominant)(1.66)

Hence, whether the W -exchange contributions in B0 → D∗∓π± can be neglected ornot can be decided from if the B0 → D∗−s K+ amplitude is found to be negligiblerelative to CFD or not.

The only subtle issue in using B0 → D∗−s K+ to assess the size of the CFD W -exchange amplitude is that a B0 decaying to D∗−π+ via a b → c process, just bymere rearrangement of the internal quarks later, may turn into a D∗−s K+ outcome,enhancing the apparent amplitude of the B0 → D∗−s K+ process, as shown in fig-ure 1.8 (right). This effect is called final-state rescattering and is expected to en-hance the B0 → D∗−s K+ amplitude up to as large as the size of CFD [34]3. As aresult, the comparison between B0 → D∗−s K+ amplitude and CFD amplitude wouldsimultaneously yield information about the size of rescattering effects in the former.

3It should be noted, that on the basis of a recent phenomenological study [35], performed with the latestmeasurements on processes, expected to show rescattering enhancements, such effects in B0 → D∗−s K+ arepredicted to be negligible.

24


1.8.2 Sensitivity to |Vub|

It has been emphasized previously that the spectator process shown in figure 1.5 is theonly dominant Feynman diagram contributing to B0 → D∗+s π− decay and it does notreceive interference from a W -exchange amplitude, similar to the B0 → D∗∓π± system.It turns out that the B0 → D∗+s π−, also lacks any contribution from a Penguin process.This special feature of B0 → D∗+s π− decay is not a coincidence, but a consequence of thefact that a quark-antiquark pair qq of same flavor, required in a W -exchange or a Penguindiagram, is absent in the B0 → D∗+s π− final-state. As a result, the overall amplitude forthe B0 → D∗+s π− decay is a purely b → u transition with a d− spectator and is expectedto be a clean avenue for accessing |Vub|, one of the CKM elements.

The importance of the CKM element Vub is impossible to overestimate, as Vub is oneof the most off-diagonal elements of the CKM matrix, containing the only complex phasepresent in the 3-generation KM mechanism. The fate of the KM mechanism as a correcttheory for the CP violation is hence dictated effectively by values Vub takes. On a com-pletely different note, some anomalous behaviour, in terms of a discrepancy larger than3 standard deviations between the lattice QCD - in particular, those by HPQCD collab-oration [36] - estimates and the world-average of the experimental measurements, hasalready been observed for the D+

s meson decay constant fD+s

4, as seen in table 6.1. Thisfailure of lattice QCD in predicting a fD+

svalue, which is consistent with the experimental

observations, appears more alarming in the light of the fact, that the same procedure pre-dicts fD+ - the D+ meson decay constant - in quite close agreement with the experimentalobservations. This discrepancy has been considered more seriously for yet another reason,even though there have been incidences in the past of observing discrepancies as largeas 3σ, which later disappeared. With increasingly powerful computing techniques, thesystematic uncertainties in the lattice QCD calculations seem to play a role less signific-ant in accounting for any bias and in order to compensate for the discrepancy one hasto rely strongly on the uncertainties in the experimental results, which are dominated bythe statistical fluctuations, unlike situations in the past. Though a statistical fluctuationas large as 3σ in the experimental values is not improbable, fluctuating all experimentalmeasurements always in the higher side of the true value is considered highly unrealistic.This “fD+

spuzzle” [37] is now being attributed to the signatures of new physics and hence

it becomes crucial to measure the SM predictions of D+s meson properties more accurately,

which in turn demands fairly good knowledge about Vub.

The semi-leptonic decays of a B meson are considered to be the best avenues forobtaining precise measurements of the CKM elements, including Vub. As already beenpointed out, such decays provide the cleanliest approach, due to very little hadronic un-certainties present in the final-states. For the same reason, constant efforts have been putin to extract information from the semi-leptonic decays. Nonetheless, because only a singleamplitude contribute to the decay, nonleptonic processes such as B0 → D∗+s π− are con-sidered promising alternative resources for extracting CKM elements, in general and |Vub|,in particular, with precisions compatible with those in semi-leptonic measurements [38].

4With most recent CLEO-c results and excluding some of the earlier inconsistent measurements from theaverage, this discrepancy is reduced to about 2 standard deviations.

25


Based on the generalized factorization scheme, it can be shown [39], that

B(B0 → D∗+s π−)

B(B0 → D∗+s D−)∝∣∣∣∣VubVcb

∣∣∣∣2[FBπ1 (m2

D∗s)

FBD1 (m2D∗s

)

]2

(1.67)

where FBπ1 (m2D∗s

) and FBD1 (m2D∗s

) are the B → π and B → D transition form factorscalculated at the D∗+s invariant mass, respectively. The B0 → D∗+s D− is a b→ c Cabibbo-favored process, as shown in figure 1.9 (right). The parallel between this diagram and

c

s

c

d

D−

D∗+s

W+

b

B0

V ∗cs

Vcbd

W+

b

d

B0

s

c

cd

D−

D∗+s

u, c, t

Figure 1.9: Feynman diagrams for the B0 → D∗+s D− process. In addition to thedominant b→ c spectator process (left), B0 → D∗+s D− also receives contribution

from b→ s Penguin process (right).

that for the B0 → D∗+s π− process clearly illustrates why the ratio between the two canyield |Vub/Vcb|.

It is necessary however, to bear in mind the limitations of this method, which mayarise due to break-down of the generalized factorization mechanism assumed in derivingthe relation in equation (1.67) or if the contribution due to the Penguin process shown infigure 1.9 (right), which contributes only to B0 → D∗+s D− and not to the B0 → D∗+s π−

decay, is found to be large.

What is the Generalized Factorization scheme?QCD is a non-perturbative field theory describing color dynamics of the strong interactions.

Even though non-perturbative, the asymptotic freedom in QCD Lagrangian allows one to separatethe high energy physics from the low energy physics under certain conditions. Equivalently said,the long distance effects can be factored out from the short distance processes, systematically in theregime of effective field theory by using Wilson’s Operator Product Expansion or OPE techniques.

The factorization scheme can be visualized as follows: In a spectator process such as that forB0 → D(∗)−π+, one can assume that the D(∗)− meson almost completely takes over the totalenergy available in the B0 decay, while leaving very little space for the prompt pion. As a con-sequence, the quark-constituents of the pion do not have enough energy to receive long distanceinteractions, particularly from the colored constituents of the D(∗)− meson, and the hadronizationdynamics of the pion is very similar to that of a pion popping out of vacuum. This scheme ofseparating short and long distance dynamics into distinct parts is termed as the naïve factoriza-tion scheme [40, 41, 42] and the phenomenological extension of the naïve factorization scheme,to include process-dependent Penguin effects and non-factorisable contributions like W -exchangeamplitudes, is termed as the generalized factorization. Technically, the generalized factorization

26


implies a neat separation of the interaction matrix as

Γ(B0 → D(∗)−π+) ∝ VcbV ∗ud × a(D(∗)π)× 〈D(∗)−|cγµ(1− γ5)b|B0〉〈π0|dγµ(1− γ5)u|0〉 (1.68)

where a(D(∗)π) includes any non-factorisable effects or corrections from non-spectator diagrams,in general.

Intuitively, it is clear why applicability of the naïve factorization is usually restricted to caseswith a light-mass companion of the D(∗)+ meson, e.g. a pion above. Though less readily visualizedotherwise, validity of the factorization formalism must be understood in terms of the success ofWilson’s OPE method and hence can be generalized to spectator diagrams in B0 → D∗+s π− decayas well, even when diagrammatically the pion appears to come from the B0 emitting a D∗+s . Onthe contrary, application of the factorization going beyond the light-companion processes, such asin B0 → D∗+s D− decay, may be questioned and does not follow trivially from the factorizationscheme, naïvely developed for the B0 → D(∗)π decays. Beneke et al. has proved the validity of fac-torization for the B-decay amplitude, in case of light-meson companion decays, using perturbativeQCD formalism [43], though it is generally believed that the factorization is more fundamental toQCD, than just a remnants of perturbative regime. Tests for generalized factorization have beensuggested in case of heavy-meson D-companions [44], though no conclusive evidence has beenfound with the current experimental statistic.

1.8.3 Previous Measurements

Branching fraction for the B0 → D∗+s π− and B0 → D∗−s K+ decays have been measuredby BaBar collaboration, earlier using a data-set consisting of 230×106 BB events [45] andrecently with a larger data sample of 381× 106 BB events [46]. Table 1.2 summarizes theresults from these analyses.

Table 1.2: Branching fraction measurements prior to our studies. The first un-certainty in all the entries is statistical, while the second comes from systematicsand the third, whenever present, is due to the theoretical uncertainties in the cal-culations. The numbers in parenthesis show the significance for the observation

to be different from zero.

Sample size Branching fractions (×10−5)NBB (×106) B0 → D∗+s π− B0 → D∗−s K+ RD∗π(%)

230 2.8± 0.6± 0.5 (6σ) 2.0± 0.5± 0.4 (5σ) 1.87± 0.19± 0.19

381 2.6+0.5−0.4 ± 0.3 (6.8σ) 2.4± 0.4± 0.2 (7.4σ) 1.81+0.17

−0.14 ± 0.12± 0.10

It is evident from table 1.2, that the RD∗π estimate, based on all previous measure-ments, is dominated by the statistical uncertainty, which can be reduced by using a largerdata sample. Using a much bigger sample of 657 × 106 BB events, collected at the KEKB-factory using the Belle detector, it is expected to achieve an improved estimate for theRD∗π. A comprehensive study on time dependent CP analysis of the B0 → D∗∓π± sys-tem has also been carried out at Belle, once using the full reconstruction techniques [28]

27


and again, adopting partial reconstruction approach [29] to increase the statistics. Theavailability of all essential ingredients, required for the CKM angle φ3 extraction, make itstrongly demanding that the measurements of B0 → D∗+s π− and B0 → D∗−s K+ branchingfractions be carried out5.

5Throughout this work, inclusion of the charge-conjugated processes is implied unless otherwise stated.

28

2Experimental Setup

A brief introduction to the particle accelerator at KEK is given.The basic designing guidelines behind the Belle detector archi-tecture are derived, keeping their importance in the context ofthe analysis under study in mind. The basic particle identific-ation techniques within the Belle analysis framework are out-lined. quality monitoring techniques are explained.

2.1 Υ(4S): The BB warehouse

THE fundamental goal of the B-factory experiments1 is to explore the phenomenon ofCP violation in the B-meson system. The term B-factory has the literal meaning

of an experiment designed to produce B mesons populously. The potential of a B-mesonsystem, in revealing CP violating effects, has been underscored previously, in chapter 1. Inparticular, the openness of the CKM triangle, which allows for a wider experimental accessto the phenomenon of CP violation without loss of generality, is the key determinant inselecting this system. Among all possible methods of producing a B-meson population inthe laboratory, the Υ(4S) has the highest merits, for the following reasons:

• Υ(4S) is produced copiously in an e+e− collider and predominantly (> 96%) decays toa BB pair.Figure 2.1 shows the Feynman diagram for the process e+e− → qq and the corres-ponding production cross section as a function of the center of mass (CM) energy.

1Among the two major functional B-factories in the world: one at the Stanford in USA and the other atTsukuba in Japan, we would concentrate on the latter.

29

EXPERIMENTAL SETUP

The decay, being purely electromagnetic in nature the cross-section is proportional

Figure 2.1: Feynman diagram for the e+e− → qq production (left) and theproduction cross section against the center of mass energy,

√s.

γ∗e−

e+

q

q

to the electric-charge squared. In particular, at CM energies just above twice theb-quark rest mass, the cross-section shows a prominent enhancement, correspond-ing to the Υ(4S) threshold, which is a bb bound state. Table 2.1 summarizes theproduction cross sections of various final states at the Υ(4S) threshold energy.

Table 2.1: The production cross-sections at Υ(4S) threshold energy,√s = 10.58

GeV, within an experimental acceptance of 92%⊗ 4π .

q Cross-section (nb)σ(e+e− → qq)

b 1.05c 1.30s 0.35d 0.35u 1.39

τ− 0.94µ− 1.16e− ∼ 40

An e+e− → qq event, where q is one of the light quarks u, d, s, or a c− quark, whichconstitutes the continuous background under the Υ(4S) threshold, are collectivelycalled continuum events.

• Absence of higher Bq statesThe Υ(4S) rest mass, being just marginally above the BdBd and less than that re-quired for production of any higher states like Bs, produces only Bd or Bu states.

30

2.2. KEKB ACCELERATOR AND BELLE DETECTOR

Hence experimentally, the situation is not obscured by the presence of higher Bstates, which would happen in case of a Υ(5S) or in a pp collider.

• Quantum entanglement of the BB pairThe BB pair produced is quantum entangled in a C odd coherent state, which allowsfor flavor identification of one of the B’s, at the time of tagging the other. This is notpossible in case of a Υ(5S) decaying to a C even Bq − Bq state as well as in the ppcollider, where the mesons are uncorrelated.

As a result, all the B factories are prevalently tuned to work at the Υ(4S) threshold,though at times a continuous energy scan is conducted, in order to

1. establish the optimal experimental conditions to produce Υ(4S) the most efficiently,

2. understand and estimate the continuum background under the threshold,

3. study physics at other Υ(XS) resonances.

Table 2.2 shows distribution of the data collected at the KEK B-factory (KEKB) inJapan, since its commissioning in 1999 till Nov’09. At the time of finalizing this work,

Table 2.2: Data collected at various CM energies at KEK B-factory as of November2009.

√s fb−1 a

Υ(4S) 710.526Υ(1S) 5.712Υ(2S) 6.523Υ(3S) 2.950Υ(5S) 50.592

off-resonance 83.271b

athe integrated luminosity calculation hasan uncertainty of 1.4% coming from theBhabha generator.bonly Υ(4S) off-resonance data mentioned

here.

only 604.297 fb−1 of Υ(4S) data was available for analysis.

2.2 KEKB accelerator and Belle Detector3

In the context of CP violation studies in the B-meson system, the importance of measur-ing the B-meson lifetime and the time-dependent decay rates can not be overemphasized.It was developed in the previous chapter, that the size of the CP violating effects wouldseverely constrict the freedom of choice in the experimental designing at various levels.It not only demands for collecting sizable statistics, but also imposes severe constrainson the quality of the data collected and hence serves as a strong guiding principle in thedesigning of the experimental setup. In the time-dependent analyses, one has to measure

3This is intended to be a brief introduction to the experimental environment, mainly in the context ofthe current work. A detailed description of the KEKB accelerator and the Belle detector can be found in [47]and [48], respectively

31

EXPERIMENTAL SETUP

the distribution of the decay time of the B-meson, with respect to the tagging time. Theaccuracy as well as precision with which the individual decay time measurements can beperformed will decide the sensitivity to the CP violating effects in such measurements.In case of the B-mesons, where the lifetimes are nearly the same, i.e ∆Γ/Γ ∼ O(10−2),the e+e− bunch lenghts as well as the detector resolution would turn out to be an un-avoidable obstacle. One way around will be to restrict the CP violation studies to thetime-integrated decay rates. But, as discussed in section 1.5 the CP violation in interfer-ence is completely washed out in time-integrated decay rates of the C-odd systems, i.e.at Υ(4S) threshold. Also equation (1.45) suggests that the time-integrated decay rateswould be functions of x = ∆M/Γ and y = ∆Γ/Γ and one can not avoid the need forestimating the lifetimes, in principle. The fact that the Υ(4S) rest mass being marginallyabove that of the BB combination invariant mass, only about 300 MeV per B meson isavailable as the kinetic energy in the Υ(4S) rest frame. This worsens the situation, as faras decay time measurements are concerned.

As a result, an experiment must be designed to have high precision vertexing andtracking facilities, while implementing ways to reduce the relative errors in the decay timemeasurements.

2.2.1 KEKB Asymmetric-energy Collider

One ingenious plan toward reducing uncertainties in the decay time measurements isto work with an asymmetric e+e− collider, where the produced Υ(4S) will experience aboost in the lab frame and the B pairs would travel considerable distance before decay-ing and the distances measured with respect to the interaction point (IP) can be trans-lated into their decay times. More the boost one gives to the Υ(4S), larger the dis-tances and better the precision in measurements. One can not continue extrapolatingthis scenario, as the stationary-detector coverage would soon take over and one wouldlose significantly in the detector acceptance. As a consequence, an optimal configuration

Figure 2.2: KEKB collider

for the e+e− asymmetry is to be planned. KEKB inparticular, chose to collide a high energy (e− beam)ring (HER) of 8 GeV energy on a low energy (e+

beam) ring (LER) with 3.5 GeV energy. This corres-ponds to a boost of

β ∼ EHER − ELER

EHER + ELER=

4.5

11.5= 0.39 (2.1)

and the center-of-mass (CM) energy of

ECM =√s

= 2√EHER × ELER

= 10.58 GeV/c2 (2.2)

which corresponds to the Υ(4S) threshold energy,as desired. Figure 2.2 shows the design of the KEKBcollider and the resulting beam energy profile is shown in figure 2.3 with the specifica-tions (left) at the KEKB. The beam energy is estimated from reconstructing decays with

32


a Cabibbo-favored b → c transition, or B → D(∗)π decays for every 500 pb−1 data-set.With nominal reconstruction efficiencies, this statistic corresponds to about 100 eventsper sample. The shift in the mean of the Mbc distribution (for definition of Mbc see, sec-tion 3.1.3) is linearly related to the shift in Ebeam for these events and can be used toobtain average Ebeam value for these events (for details, see Belle Note 567 and referencestherein). The two beams collide at a finite crossing-angle of ±11 mrad in the interaction

beamE5.287 5.288 5.289 5.29

# o

f E

ven

ts

0

500

1000

1500

2000

0.0000073± = 5.2889472 µ

0.0000058± = 0.0003470 σ

beamE5.287 5.288 5.289 5.29

# o

f E

ven

ts

0

500

1000

1500

2000 Beam Energy

LER HER

Energy E (GeV) 3.5 8.0Energy spread σE (GeV) 7.1× 10−4 6.7× 10−4

CM energy a √s (GeV) 10.58

Boost factor βγ 0.42a µ in the figure corresponds to half the total CM energy, i.e

√s/2.

Figure 2.3: KEKB asymmetric e+e− collider design showing the interaction re-gion (IR) in the Tsukuba experimental Hall (left) and specifications (right).

region (IR) at the Tsukuba experimental hall. The data is collected with the help of theBelle detector, which is a 4π-coverage detector surrounding the IR. A cartoon of the cross-section of the Belle detector depicting various sub-detectors and their relative positions isshown in figure 2.4. We describe the major components of the Belle detector with theirsignificance to the current analysis. Various components of the detector are arranged ina cylindrically symmetric configuration and are described using their cylindrical coordin-ates, (r, φ, z) with the positive z-axis in the direction opposite to the positron beam.

Solenoid 1.5 T ; SuperconductingBeam pipea double-wall 0.5mm Be; He-cooledSVD 4 layers of 300 µm-silicon sensorsCDC 50 anode (18 stereo) and 3 cathode layersACC 960 + 228 aerogel cells, 1.01 ≤ n ≤ 1.03TOF 4cm scintillator, 128 segmentsECL 6624 + 1152(F) + 960(B) CsI (Tl) crystalsKLMb 14 layers RPC; 5cm FeEFC 32 (φ) × 5 (θ) BGO

a not shown in figure. b same as KL and muon Detector.

Figure 2.4: The cross-section of the Belle Detector (left) and the specificationsof its components (right).

33

http://belle.kek.jp/secured/belle_note/gn567/

EXPERIMENTAL SETUP

2.2.2 Vertexing: Silicon vertex detector (SVD)

Given the asymmetric conditions corresponding to the Lorentz boost of βγ = 0.42 at KEKB,and the B meson lifetime of ∼ 1.5× 10−12 s [49], the typical flight length of a B meson atKEKB is

∆z = ∆t× cβγ ∼ 189 µm (2.3)

This implies a vertexing resolution of ∼ 60µm or better would be required to achieve reli-able measurements of the CP observables. Placing the vertexing assembly in the vicinityof IR may require robustness against large beam backgrounds as well as high multipleCoulomb scattering effects. These requirements put stringent constraints on the design ofthe IR, in general and of the vertexing detector, in particular.

In Belle detector, vertexing with the desired precision is achieved by placing arraysof double-sided silicon strip detectors (DSSD) in cylindrical configuration surrounding theIR, covering polar region of 17 < θ < 150. The junction side with the p+ strips implantedon the n-bulk of the DSSD, helps in r − φ measurements, while the ohmic side with n+

readout strips is used for the z measurements4. Each DSSD has 1280 sense (960 readout)strips of p+ (n+) implantation with pitch-width equal to 25 µm (42 µm).

Two different configurations were used for collecting the data. Out of the total Υ(4S)data mentioned in table 2.2, the first 140.9fb−1 were collected using three concentric cyl-indrical layers of DSSDs, while the remaining data is collected with four layers of DSSDs.The vertexing assembly is collectively called the Silicon Vertex Detector (SVD) and thetwo configurations are denoted by SVD1 [50] and SVD2 [51] . Table 2.3 summarizes themajor differences between the two configurations.

Table 2.3: Comparison between SVD1 and SVD2 designs

SVD1 SVD2

Beam pipe radius (cm) 2.0 1.5SVD Layer radii (cm) 3 layers: 3.0, 4.5, 6.0 4 layers: 2.0, 4.35, 7.0, 8.8CDC inner region 3 Cathode layers 2 layers, smaller cell chamber

The detector performance and hence the reconstruction efficiencies differ in SVD1 andSVD2 data-sets, and care must be taken when using the full data-set, which is the case athand.

The SVD2 performance studies (see, Belle Note 715) show that the impact parameterresolution can be expressed as a function of the track momentum, p (GeV/c):

σz(µm) = (26.3± 0.4)⊕ (32.9± 0.8)/(pβ sin5/2 θ)

σr(µm) = (17.4± 0.3)⊕ (34.3± 0.7)/(pβ sin3/2 θ)

(2.4)

where, the first uncertainty is treated as the intrinsic resolution of the vertex detector andthe second is the effect of multiple Coulomb scattering. This implies a resolution of 60 µm

4role of the p+ and n+ implants in measuring r − φ and z coordinates is swapped in later upgradation,i.e. in SVD2 configuration. (ref: Belle Note 794)

34




or better for the track momentum of 500 MeV in both along the beam direction as wellas in the transverse plane. Figure 2.5 shows the impact parameter resolution along thebeam (left) and in the plane transverse (right) with respect to the track momentum. The

Figure 2.5: Impact-parameter resolution of SVD1.0 and SVD2.0 in the beamdirection (left) and in the direction transverse to beam (right) w.r.t. the trackpseudo-momentum pβ sin5/2 θ (pβ sin3/2 θ). The data points are experimental ob-servations taken with cosmic rays, where as the dashed curves are the fits per-

formed resulting in relations in equation (2.4). (ref: Belle Note 715)

method of vertexing is usually an involved process and receives inputs from many othersub-detectors and track reconstruction algorithms. The vertex determination of an eventusually requires extrapolating the constituent tracks reconstructed in the tracking systemand can be considerably affected by the uncertainties in the momenta of the tracks. Theuncertainty in a track momentum is statistical in nature and comes from the fact that theactual particle flight is sampled with a finite number of hits in the tracking system. Asa result, the uncertainty on the decay vertex reduces, when the number of tracks or thenumber of hits per track increase. Although a separate tracking device is employed forthis purpose, tracks with low momenta are not accessible to this system, mainly becauseof the physical position of the tracking system away from the IR. In order to identifythe particle type associated with a charged track, the whole detector assembly is placedinside a magnetic field, so that the information about the momentum of the track from itscurvature can be extracted. At KEKB a 1.5 T superconducting solenoidal magnet is usedfor this purpose. As an effect, many low momentum particles spiral back towards the IReven before reaching the tracking detector. The hits in the SVD is the only source to accesssuch low momentum tracks. The SVD hits are added to the tracking algorithm for thispurpose.

Figure 2.6 shows distribution of the signed impact parameters5, i.e. the distance of the

5The impact parameter is assigned the sign of the charge in order to improve efficiency and significanceof the tracking algorithm.

35


EXPERIMENTAL SETUP

closest approach to the IP, of all the tracks coming from the decay of a B-meson or oneits cascade decay products along the beam direction (left) and in the plane transverse toit (right). On the other hand, the tracks which do not emerge from a B decay are not

z(cm)δ-6 -4 -2 0 2 4 6

# of

trac

ks

0

10

20

30

40

50

60310× z∆

r(cm)δ-0.3 -0.2 -0.1 0 0.1 0.2 0.3

# of

trac

ks

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

610× r∆

Figure 2.6: Signed Impact Parameter distribution of the tracks at Belle, withrespect to the IP.

expected to show any such geometric affinity toward the interaction region. To reject suchbackground we require a track to have |δz| < 4 cm and |δr| < 0.2 cm.

2.2.3 Tracking: Central Drift Chamber (CDC)

In a uniform magnetic field, the most general trajectory of a charged particle is a helix,which can be parametrized by five parameters in general. The tracking assembly in theBelle detector is placed inside a 1.5 T superconducting magnet with ∆Bz/Bz ∼ 3%. Evenunder practical difficulties due to the energy losses, multiple scattering effects and the non-uniformity in the magnetic field, Belle employs the helix parametrization of the chargedparticle tracks, following that used in TOPAZ collaboration [52]. This parametrization,corrected for the energy loss and multiple scattering effects, works reasonably well, ex-cept for the extreme end-regions of the detector, where magnetic field non-uniformitiesdominate (Belle Note 148).

The charged particle tracking is provided by wire drift chamber: central drift chamber(CDC) [53, 54]. To comply with the geometry of the experiment, CDC is designed in anasymmetric fashion along the z axis, with a coverage of 17 < θ < 150. The chamberhas 50 cylindrical layers organized in 11 super-layers, each containing three to six axial orsmall-angle stereo layers.

Majority of the decay particles of a B meson have a momentum less than 1 GeV/c. Re-ducing the amount of multiple scattering is hence important for improving the momentumresolution and hence, a low-Z gas is used. Since low-Z gases have smaller photoelectriccross-sections than argon-based gases, they help in reducing the background from the syn-chrotron radiation. A gas mixture of 50% helium and 50% ethane has been selected forthe CDC filling gas. This mixture has a long radiation length of 640 m, and a drift velocitythat saturates at 4 cm/µs at a relatively low electric field. For a square-cell drift chamberdue to the large field non-uniform inherent to its geometry, it is important to have an early

36



saturating gas. Even with the low Z mixture, a good dE/dx resolution is obtained with thelarge ethane component. Figure 2.7 shows result of the pt resolution calibration studiesperformed with the cosmic rays data.

Figure 2.7: pt dependence of pt resolution, obtained with the cosmic ray data.The solid curve shows the fit result, while the dotted curve represents the ideal

expectations of (0.118%pt ⊕ 0.195%) for β = 1 particles.

The cosmic ray data study implies a pt (expressed in GeV/c) dependence of

σptpt

[%] = ((0.201± 0.003)pt ⊕ (0.290± 0.006)/β) (2.5)

Figure 2.8 shows a typical momentum distribution of the tracks observed in the Belledetector. The blue histogram corresponds to the particles resulted from a B decay andcan have momenta as high as 4 GeV/c. Such tracks will be termed as prompt tracks, inorder to distinguish them from the low momentum tracks, for example coming from a D+

s

decay represented by the red histogram. From the expression (2.5) it is evident that formost of the tracks coming from a D+

s decay the momentum resolution σpt/pt would notbe better than 0.4%, while for the prompt track the resolution is expected to be around0.8− 1.0% or more, during the experimental runs. The CDC in situ performance is studiedwith the e+e− → µ+µ− events for the tracks with momenta in the range 4 to 5.2 GeV/c,and with the mass resolution studies of the K0

S reconstructed from the π+π− pairs for thelow momentum ranges of 1 GeV/c or less. The pt resolution is found to be 1.64% in theformer case, whereas about 0.68% in the later.

2.2.4 Particle Identification (PID)

Protons (p), electrons (e±), muons (µ±), kaons (K±) and pions (π±) are the most quali-fied candidates to leave tracks in SVD and CDC. For efficient event reconstruction, it is ofextreme importance to effectively identify the type of particle correctly, while reducing the

37

EXPERIMENTAL SETUP

(GeV/c)lab

p0 0.5 1 1.5 2 2.5 3 3.5

# o

f tr

acks

0

0.01

0.02

0.03

0.04

0.05

0.06 Track Momentum

daughter+sD

prompt track

Normalized to unit area

Figure 2.8: The lab frame momentum distribution for the D+s daughter tracks

(red) and the B0 daughter or prompt charged tracks (blue).

misidentification probabilities. The former improves the signal reconstruction efficiency,whereas the latter gives better control over the background.

The ionisation loss dE/dx measured in the CDC can be used for identification. How-ever, the loss profiles for these particles have considerable overlaps over the momentumranges of interests. The asymmetric Landau-distributed dE/dx against particle momentacan introduce additional complications. Overall, the ionization loss in CDC alone is usuallynot sufficient for identification purposes.

Belle implements Bayesian approach for calculating the likelihoods Lk for a chargedtrack observed to be of a particular kind k, depending upon a set of observation Ai made.For K/π identification in particular, Ai constitute

• the ionization loss dE/dx measured in CDC,

• the time-of-flight measurement in TOF, and

• the photon yield in the ACC

And assuming these measurements to be statistically independent, the kaon (pion) likeli-hood function can be built as

LK(π) =∏i

PK(π)(Ai) = PK(π)(dE/dx)× PK(π)(TOF)× PK(π)(ACC) (2.6)

According to Neyman-Pearson lemma, the most powerful decision function, which can beconstructed out of these likelihood functions is the ratio [55]

RK/π =LK

LK + Lπ(2.7)

which by construction, peaks at one for a real kaon and at zero for a real pion. Thelikelihoods are calculated by combining the ionization loss dE/dx measured in CDC withthe information from the threshold cerenkov counter ACC, which is basically a binarydecision detector and the time-of flight measurements in the TOF, described below.

38


Ionization loss (dE/dx) in CDC

Figure 2.9 shows the mean ionization loss 〈dE/dx〉 for various charged tracks againstthe track momentum. The truncated-mean method is employed to calculate the expected

Figure 2.9: Mean ionization loss 〈dE/dx〉 vs. track momentum observed in testbeam data.

ionization loss for a track. Due to the occasional knock-out of an atomic electron by ahigh momentum ionizing particle, fluctuations can arise introducing the Landau tail in thedE/dx distribution. In the truncated-mean method, largest 20% of the measured dE/dxvalues are discarded, while taking the average. This minimizes risks involved due to theoccasional large stochastic fluctuations in the Landau tail.

The studies with minimum ionizing pions from the K0S show a 〈dE/dx〉 resolution of

7.8% for momenta between 0.4 and 0.6 GeV/c, while resolution for Bhabha and µ-pairevents is found to be about 6%.

Time Of Flight Detector (TOF)

Detection PrincipleThe time-of-flight detectors measures the time taken by a particle with momentum p totravel a distance L between the “start” and the “stop” counter. The situation is depictedin figure 2.10. If the momentum p is known from the tracking measurements, the particlecan be identified from its mass, obtained with the TOF measurement. Fast timing devicesare needed for measuring the masses precisely.

Specifications and PerformanceIn Belle the fast timing is achieved with trigger scintillation counters or TSCs. A timing

resolution of 100 ps is possible with TSCs, which can allow effective K/π identificationup to momentum of 1.2 GeV/c, for a flight length of 1.2 m. The minimum transversemomentum required to reach the TOF placed at a radius of 1.2 m from the IP is 280MeV/c. TOF covers a polar angle range of 30 to 120.

39

EXPERIMENTAL SETUP

Lm

p

t =L

v

= L

(1 +

m2

p2

)1/2

... with p = mβγ

For very large momentum p compared to theparticle masses m1 and m2,

t1 − t2 =L

2p2(m2

1 −m22)

Figure 2.10: The principle behind a time-of-flight measurement.

TOF measurement is excluded from the likelihoods for tracks with momenta higherthan 1.2 GeV/c.

Aerogel Cerenkov Chamber (ACC)

Detection PrincipleIf a charged particle travels through a dielectric medium of refractive index n with avelocity v exceeding the speed of light c/n in that medium, the electromagnetic wave-front, produced due to the polarization of the medium, traverses slower than the particleas shown in figure 2.11. The developed radiation cone has the characteristic angle of

θ

ctn

vt

cos θ =(ct/n)

vt

=1

βn

⇒ ∃ θ iff β ≥ 1

nor p ≥ m√

n2 − 1

Figure 2.11: The development of the Cerenkov radiation cone in the detectormedium.

θ = cos−1(1/βn). It is easy to see that a Cerenkov cone will develop only for the cases,where the track momentum satisfies the condition, p ≥ m/

√(n2 − 1). For n ∼ 1.02, this

corresponds to a threshold momentum of pthr ∼ 695 MeV/c for a pion (mπ ∼ 140 MeV/c2),while a kaon (mK ∼ 495 MeV/c2) gives Cerenkov yield only after pthr ∼ 2.5 GeV/c. Thisis the working principle behind the ACC, a threshold Cerenkov counter, with which thekaon-pion identification can be extended much beyond the momentum reach of dE/dxmeasurements by the CDC and the time-of-flight measurements in the TOF.

Specifications and PerformanceIn the Belle detector, silica aerogel Cerenkov technology is adopted. Silica aerogel can

be imagined as a mixture of air bubbles embedded in Silica glass giving rise to an effective

40


refractive index equal to the weighted-average of the two components. Hence, with silicaaerogel it is possible to tune n between 1.003 and 1.013, extending the K/π separationmomentum range up to 4 GeV/c. Since low momentum K/π separation up to 1.2 GeV/c isprovided by TOF system, Belle uses only a single refractive index aerogel layer. To matchthe aerogel performance with the momentum distribution angular profiles, n is variedbetween 1.010− 1.030 depending upon the polar angle of the module, while the refractiveindices are controlled with about 3% precision.

The refractive index n depends upon the wavelength of the radiation, giving rise todispersion of light, and the temperature. The temperature dependence is usually negli-gible.

Likelihoods and RK/πThe likelihood for dE/dx is parametrized as a Gaussian function,

PK(π)(dE/dx) =1

σdE/dxexp

−

[(dE/dx)obs − (dE/dx)K(π)]2

4σ2dE/dx

(2.8)

where (dE/dx)obs and (dE/dx)K(π) are respectively the observed and expected dE/dxmeasurements in CDC for particle hypothesis of either a kaon (K) or a pion (π). Theresolution of dE/dx in CDC is denoted by σdE/dx.

The likelihood for TOF measurements is given by

PK(π)(TOF) =1∏i

σiexp

−∑i

(tiobs − tiK(π))(Ei)−1(tiobs − tiK(π))

(2.9)

where the sum is over all TOF hits, and ti is the vector with two timing signals from thePMT at the two ends of the TOF counter, while Ei is the 2 × 2 error matrix and σi’s aretiming resolutions.

The ACC is expected to give zero photoelectric yield Npe for a kaon and a finite numberof photons for pions. However, emission of δ rays and other electronics as well as PMTbackgrounds add a non-negligible tail on the higher energy side of the distribution, whichdoes not allow Npe to be described by a simple function and complicates the choice ofparticle hypothesis selection threshold NK/π

thr . The likelihood for ACC is given by,

PK(π)(ACC) =

ε (Npe ≥ NK/π

thr )

1− ε (Npe ≤ NK/πthr )

(2.10)

where ε is the expected efficiency for a given particle hypothesis to be correct.The efficiency and performance of RK/π prepared by combining these likelihoods de-

pends on the momentum in general. Figure 2.12 shows the kaon identification efficiencyas a function of the particle momentum in the lab frame. The probability of a pion fakingas a kaon or pion fake rate is also shown. The corresponding plots are obtained fromMonte Carlo (MC) simulation studies of the D∗+ → D0π+ (D0 → K−π+) decays, whereidentifying the hadron is feasible without relying on the particle identification functions.

41

EXPERIMENTAL SETUP

Figure 2.12: The kaon identification efficiency (red) and the pion fake rate (blue)of the ratio RK/π as a function of the particle momentum in the lab frame, de-termined from the Monte Carlo studies of the D∗+ → D0π+ (D0 → K−π+)

decays.

It can be seen that corresponding to the momentum ranges in which a kaon or piontrack shown in figure 2.8 can lie, the kaon identification efficiency is better than 85% witha pion fake rate of 15%. Figure 2.13 (right) shows RK/π distribution for the tracks infigure 2.8 against the track momentum. The red (blue) points are for kaon (pion) tracks.The overall RK/π distribution is shown on the left.

πK/R0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(G

eV/c

)la

bp

0.5

1

1.5

2

2.5

3

3.5 πK/Track Momentum - R

πK/R0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

# o

f tr

acks

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

πtrue

true K

L(TOF)⊗ L(ACC) ⊗ = L(dE/dx) πK/R

Figure 2.13: Right: The RK/π distribution for the tracks shown in figure 2.8 as afunction of the track momentum and Left: Rk/π distribution for a true pion (blue)

and a true kaon (red).

42


2.2.5 Electromagnetic Calorimeter (ECL)

Apart from the charged particles, a big fraction of the final-state particles of a B decaycomes from photons and neutral pions; the latter subsequently decaying to a photon pair.Consequently, having an efficient detection mechanism for the photons is essential for aneffective reconstruction of any B decay involving a neutral pion or a photon in the final-state. On the other hand, many charged particles lose energy due to radiation, whichneeds to be accounted. While most of the photons coming from a π0 decay have energiesbelow 100 MeV, a thorough understanding of the photons in the low momentum regionalso helps controlling the beam background contamination. Use of electromagnetic Calor-imeter comprised of CsI(Tl) crystals is made to achieve superior photon detection. Thechoice of the material was based on its high density (4.51 g/cm3), short radiation length(1.86 cm), high light output, radiation hardness and cost-effective superiority over others.

Working Principle

The CsI lattice provides scaffold-like frame for light-matter interaction, where processessimilar in nature to fluorescence can take place, while the thallium impurities introduceintermediate energy states between the valence and the conduction bands, increasing theprobability of an electronic-transition to the excited states, which subsequently return tothe ground state via vibrational intermediate-states. The excited centers return to theground state without any significant delay, but radiating light of lower frequencies thanthe absorbed photon and can be collected using photo-multiplier tubes. This process isknown as scintillation.

Design and Performance

The complete calorimeter is comprised of a 3.0 m long barrel section with the inner radiusof 1.25 m and annular end-caps located at z = +2.0 m and z = −1.0 m from the IP.The entire system contains 8816 highly segmented arrays of the Cs(I) crystal, covering thepolar angle region of 17 < θ < 150, which corresponds to a total coverage of 91% the 4π.Each crystal is arranged to point towards the IP, with a tilt of 1.9 in both θ and φ directionsto avoid the photon escaping through the gap of crystals. The size of each CsI(Tl) crystalis optimized to have about 80% of the total energy deposited by the photon, generatedat the center of the crystal, contained in itself. The 30 cm length, providing interactionlength of about 16.2 X0 is chosen to avoid deterioration of the energy resolution at highenergies due to the fluctuations in the shower leakage, out of the rear of the counter.

The overall energy resolution of the ECL at Belle can be written as,

σEE

=0.0066%

E⊕ 1.53%

E1/4⊕ 1.18% (2.11)

where, the first term corresponds to the intrinsic noise, the second comes from the stochasticfluctuations in the showering process and the third is a result of the shower leakage in thecrystal.

43

EXPERIMENTAL SETUP

Cluster Identification

The cluster identification process starts withenlisting the energy deposits in each crystalcell. A generic cluster is identified over agroup comprised of one central crystal withits neighbours and next-neighbours, makingup an assembly of 5 × 5 crystal array, suchthat the energy deposited is maximum in thecentral crystal. If two neighbouring clustershave common crystals, contribution from the

0

100

200

300

400

500

54

32

1 12

34

5

common crystal to a cluster is weighed according to the relative strength of the clusterenergy, calculated by summing energy deposit-es of all cells in the assembly, except thecommon ones.

The quality of a cluster is decided on the basis of the fraction of the total 5× 5 clusterenergy concentrated in the 3×3 crystal array out of the total 5×5 array, termed as E9/E25.Figure 2.14 shows the E9/E25 distribution for a correctly reconstructed photon from a sig-nal (blue) and for a fake photon cluster (red). A cluster is identified as a good photon

25/E9E0.75 0.8 0.85 0.9 0.95 1

# o

f E

net

ries

0

0.02

0.04

0.06

0.08

0.1

0.12 25/E9E

γReconstructed

true

fake

Figure 2.14: The E9/E25, energy deposited in a 3× 3 crystal array over that in a5× 5 array. The blue(red) histogram corresponds to the distribution of a correctly

reconstructed (fake) photon cluster.

event requiring the E9/E25 to be greater than 0.85. The ECL clusters which are depositedin the proximity of any charged track extrapolated from the hits in the CDC are associatedwith the charged track as radiation photons. Figure 2.15 shows the momentum distribu-tion of final-state photons from the D∗+s decay. These photons have energies around 200MeV and equation (2.11) implies an energy resolution of about 2.5 %.

To reduce the low momentum beam background photons, clusters with energies lessthan 60 MeV are rejected. In the end-cap regions, i.e. regions closer to beam directions,this criterion is tightened to 100 MeV.

44

2.3. ANALYSIS FRAMEWORK

(GeV)labγE

0 0.1 0.2 0.3 0.4 0.5 0.6

# of

Ene

trie

s

0

0.02

0.04

0.06

0.08γLab-frame E

Figure 2.15: Lab energy distribution for a photon from a D∗+s decay.

2.3 Analysis Framework

2.3.1 Data Summary Tables (DST)

The raw data collected in the form of electronic pulses read-out from various sub-detectorassemblies at Belle is reprocessed to perform preliminary IP profiling by vertexing, track-finding and fitting, and particle identification. The raw data is then converted into phys-ically more meaningful quantities. Many redundancies in the information can be reducedby using certain relations and constrains defining physical boundaries. The post-processeddata of every collision event is compiled into various data summary tables (DST) using thedatabase tools based on postgress/MySQL. At Belle use of Panther tables [56] (or BelleNote 130) is made, which is compatible with the C/C++ data structure and the entriesare readily accessible from the analysis codes written individually by the user. Table 2.4summarizes various DST’s used in the current analysis, with a brief explanation about theinformation they contain. A more detailed information can be obtained from the Pantherusers’ manual.

2.3.2 Monte Carlo Simulations (MC)

In order to have an optimal signal selection versus background rejection procedure, oneneeds to understand the behavioral properties and differences between the signal and thebackground events in the data. It is usually not possible to separate these event in thedata itself. On the other hand, any data-driven procedure to optimize signal selection overbackground rejection is prone to introduce, what is generally termed as, experimenter’sbias. To avoid such limitations, the Monte Carlo simulation techniques are adopted togenerate data-like samples, provided enough input information about the physical pro-cesses involved in the decays and the detector performance is known to a good accuracy.Such samples are generated using mathematical models, representing the data and all

45



EXPERIMENTAL SETUP

Table 2.4: DST and their information contents used in the current analysis

Table Name Parameter Entriesa

mdst_charged Charged tracks charge, track momentum,vertex, no. of hits

mdst_gamma ECL clusters energy, position, matchingtrack in CDC, E9/E25, point-ers to ECL cluster

mdst_vee V- type decays (e.g. K0S) momentum, flight length, in-

formation and pointers to thedaughters

mdst_pi0 neutral pionsb momentum, mass constrain-fit χ2, pointers to daughterphotons

gen_hepEvt Event generator information vertex, momentum, decay re-lations

a and the errors on themb reconstructed with photon pairs in mdst_gamma

the information about the generation process is available, which can be used to tag andseparate the type of events.

Event Generation

To generate the events based on detector-independent physical processes involved in thedecay we use EvtGen [57], an event generator specially developed for decays related toB-meson, likely to be occurring at a B-factory. In case of existence of different models forthe same decay, the choice is made according to the need by the user and can be suppliedas an input to the decay file, where evolution of a process is specified.

Detector Performance

To include the detector performance effects, the EvtGen generated events are passedthrough a Geant based MC simulator gsim [58]. In order to achieve near-to-the-real-experimental conditions, during each experimental run, for collecting the real data, thedetector configuration and performance is monitored and saved into data-files, which canlater be fed to gsim to mimic run-dependent conditions while generating the MC samples.This reduces the rick of generating inconsistencies between the more idealistic MC samplesand the realistic data sample. Generating run-dependent MC samples allows one to modelthe real data with accuracies as high as 90%. The disagreement of about 10% between thereal data and the MC sample has to be taken care of by either correcting with the help ofsome efficient control sample or by adding this effect into overall systematic uncertainty,whenever applicable.

The statistical errors on various distributions studied by binning the samples usually

46

2.3. ANALYSIS FRAMEWORK

are Poissonian, i.e. proportional to the square-root of the bin content. As a consequence,in order to amplify significant features in a distribution, which otherwise would be hiddenunder the statistical fluctuations in the real data, large MC samples are used in this study.Table 2.5 summarizes various MC samples, with their statistical sizes relative to the realdata, used for this study.

Table 2.5: Type of MC samples, with their sizes relative to the real data, used inthis study

MC type size Description(times data)

signal 100 One of the B’s decaying to signal events,while the other decaying generically

BB 5 both B’s decaying generically; does notinclude rare decays observed recently

rare 50 rare B decays; generated with dominantB decays excluded in generation

continuum 5 e+e− → qq decays underlying the Υ(4S)resonance

2.3.3 Data skims and Index files

Through the first experimental run, Belle has collected huge amount of data till date.Though it is important to have large statistic in order to establish results with good sig-nificance, it is highly unlikely that every event collected will be of interest in a specificanalysis. Neither does all the events have equal probability of satisfying the selection re-quirements in an analysis: starting from the signal events which (are expected to) have thehighest probabilities to the background events which have no similarities with the signalevents and do not satisfy any of the selection criteria. In cases, where signal is expectedto occur rarely, most of the analysis time is spent on the events which do not carry anysignificance for the study at hand. At the time of optimization, it might be required tostudy the backgrounds iteratively and in such cases the computational time per iterationplays an important role.

It is possible to first analyse the whole data-set, with loose selection requirements6,to check whether an event has potential of satisfying the actual selection criteria or not.A separate index file can then be prepared which will contain result of this study. Sub-sequently running the analysis code on these index files can save computation time wastedin searching for signal in events unlikely to have cleared even the loser criteria, while pre-paring the index file. It is possible skim the original data size down to only 1% or evenless. For our study we skimmed the data and the background MC samples using skim-ming criteria developed for selecting any event with a D∗+s like candidate. The skimmingefficiency is found to be 4%.

6loose selection also allows for accepting a larger background, needed for sideband studies.

47

3Analysis:

Monte Carlo Studies

10 cm

BELLE This chapter discusses the details about the reconstruction of a Bcandidate from the B → D∗sX decay. This involves various steps,such as selecting a signal event, understanding and devising re-jection techniques for various background events and subsequentlyemploying optimization techniques to maximize the signal extrac-tion significance. Monte Carlo samples are used for optimization,with blinding the signal region in real data.

3.1 Signal Event Reconstruction

IF an initial-state particle i decays to two or more final-state daughters f1, f2, . . . andif the 4-momenta of the daughters fj are known, it is possible to reconstruct the 4-

momentum of i using the energy-momentum conservation laws. While it is always possibleto select all probable combinations of fj ’s, the invariant mass of the combination will fallin a region around the mass of i only if

• it is the true combination of the final-state, and

• the 4-momentum of each of fj ’s is correctly reconstructed

Irrespective of whatever immediate daughters a B-meson decays to, the detector issensitive to only a few final-state particles, which are usually stable and do not decay

49

ANALYSIS: MONTE CARLO STUDIES

further. Hence, one usually has to reconstruct the B candidate starting from these stablefinal-state particles detected in the experiment and move upwards on the decay cascade,step-by-step looking for the mother particle every time in the invariant mass distributionof the daughter particle combinations1. As a result, the overall efficiency with which aB candidate can be reconstructed depends upon how efficiently a mother particle can bereconstructed using the daughters at each step in the decay cascade. Keeping this in mind,a reconstruction strategy is always decided on the basis of ease and cleanliness in detectingthe final-states, so as to have

1. highest signal selection efficiency, as well as

2. less ambiguity in selecting the correct daughter combinations, reducing the amountof background from wrong combinations contaminating signal region.

To be able to draw meaningful inferences, the selected modes are expected to havehigh probability (branching fractions) of occurrence and hence highest possible statisticswithin the choices available.

In the present case, We reconstruct a D∗+s candidate in following decay channels ofD+s and combine it with appropriate X(∼ π±,K±, ρ±, D±, . . . ) to form a B event. The

Table 3.1: D∗+s reconstruction decay channels. The respective branching fractionsshown here are taken from those published by Particle data group.

BranchingDecay Mode

fraction (%)

D∗+s → D+s γ 94.2± 0.7

D+s → φπ+ 4.39± 0.34

φ→ K+K− 4.89± 0.05D+s → K∗(892)0K+

2.6± 0.4K∗(892)0 → K−π+

D+s → K0

SK+

1.49± 0.09K0S → π+π−

three D+s decay modes will henceforth be called the φπ mode, the K∗(892)0K mode and

the K0SK mode, respectively.

3.1.1 D+s candidate selection

• φπ mode:

A φ candidate is reconstructed combining oppositely charged kaon pairs. The relat-ively narrow mass width of the φ resonance and low underlying background allowsfor a nominal kaon identification requirement ofRK/π > 0.1. Figure 3.1 (left) shows

1this is when a particle is reconstructed fully, i.e. when all the daughters are used explicitly for recon-structing the mother; in case of partial reconstruction techniques, the missing part of the 4-momentum isusually looked at for.

50


3.1. SIGNAL EVENT RECONSTRUCTION

the invariant mass distribution for the kaon pairs. The resonant peak [59] is fittedwith a Breit-Wigner distribution, represented by the blue curves, while the red curveindicates the combinatorial background. The kaon combinations, for which the in-variant mass lies between ±14 MeV/c2, a width corresponding to ∼ 3Γ of the φresonance, is identified as a φ candidate.

1.000 1.010 1.020 1.030 1.040MK+K- (GeV/c2)

0

1000

2000

3000

4000

5000

# of

Eve

nts

MINUIT Likelihood Fit to Plot 16&1B0 → Ds*hFile: *dssth.hbk 3-JAN-2009 11:28Plot Area Total/Fit 29971. / 29971.Func Area Total/Fit 29970. / 29970.

Fit Status 2E.D.M. 6.436E-06

Likelihood = 324.7χ2= 293.7 for 40 - 5 d.o.f., C.L.= 0.00 %Errors Parabolic MinosFunction 1: Breit-WignerAREA 32368. ± 187.9 - 0.000 + 0.000MEAN 1.0195 ± 1.8878E-05 - 0.000 + 0.000WIDTH 4.67105E-03 ± 3.9298E-05 - 0.000 + 0.000Function 2: Chebyshev Polynomial of Order 1NORM 0.20999 ± 0.2790 - 0.000 + 0.000CHEB01 1.66750E+05 ± 2.2154E+05 - 0.000 + 0.000

1.935 1.945 1.955 1.965 1.975 1.985 1.995Mφπ (GeV/c2)

0

1000

2000

3000

4000

5000

6000

# of

Eve

nts



Likelihood = 55.5χ2= 53.8 for 40 - 7 d.o.f., C.L.= 1.3%Errors Parabolic MinosFunction 1: Two Gaussians (sigma)AREA 30754. ± 157.4 - 0.000 + 0.000MEAN 1.9684 ± 2.3896E-05 - 0.000 + 0.000SIGMA1 2.99844E-03 ± 6.1399E-05 - 0.000 + 0.000AR2/AREA 0.37252 ± 2.4975E-02 - 0.000 + 0.000DELM∗ 0.0000 ± 0.000 - 0.000 + 0.000SIG2/SIG1 2.0663 ± 3.1249E-02 - 0.000 + 0.000Function 2: Chebyshev Polynomial of Order 1NORM 27830. ± 964.5 - 0.000 + 0.000CHEB01 5.36314E-03 ± 4.1191E-02 - 0.000 + 0.000

Figure 3.1: MK+K− distribution (left) and, Mφπ distribution (right) for D+s →

φπ+ signal events. The blue curves represent the signal peak, whereas the combin-atorial backgrounds are represented by the red curves. The selection requirements

on respective plots are shown with vertical lines.

A φ candidate is further combined with a π+ to form a D+s candidate. Figure 3.1(b)

shows the signal D+s mass distribution. The D+

s candidate is selected with a ±13MeV/c2 (∼ 3σw, the weighted width of the double Gaussian) window around theD+s nominal mass.

• K∗(892)0K mode:

The K∗(892)0meson is reconstructed by combining a kaon with an oppositely chargedpion. Figure 3.2(a) shows the invariant mass distribution of the kaon-pion pair. TheK∗(892)0 resonance peak and the flat combinatorial background are fitted with aBreit-Wigner shape and a linear order Chebyshev polynomial, respectively. The com-bination is identified as a K∗(892)0 candidate, whenever the invariant mass lieswithin ±75 MeV/c2 (∼ 1.5Γ) region around the nominal K∗(892)0resonance mean.

A K∗(892)0 candidate is combined with a K+ to form a D+s candidate. Figure 3.2(b)

shows the invariant mass distribution for K∗(892)0K+ combinations. The invariantmass is required to be within ±15 MeV/c2 (∼ 3σw) region around the D+

s nominalmass.

51


0.75 0.85 0.95 1.05MKπ (GeV/c2)

0

500

1000

1500

# of

Eve

nts



Likelihood = 150.1χ2= 147.0 for 40 - 5 d.o.f., C.L.=0.113E-12%Errors Parabolic MinosFunction 1: Breit-WignerAREA 16200. ± 169.5 - 0.000 + 0.000MEAN 0.89349 ± 3.1400E-04 - 0.000 + 0.000WIDTH 5.30357E-02 ± 7.7142E-04 - 0.000 + 0.000Function 2: Chebyshev Polynomial of Order 1NORM -4361.0 ± 300.8 - 0.000 + 0.000CHEB01 -0.10659 ± 4.5793E-02 - 0.000 + 0.000

1.935 1.945 1.955 1.965 1.975 1.985 1.995MK*K (GeV/c2)

0

1000

2000

3000

# of

Eve

nts



Likelihood = 54.8χ2= 54.9 for 40 - 7 d.o.f., C.L.=0.970 %Errors Parabolic MinosFunction 1: Two Gaussians (sigma)AREA 18384. ± 145.0 - 0.000 + 0.000MEAN 1.9684 ± 3.1287E-05 - 0.000 + 0.000SIGMA1 3.01036E-03 ± 2.9047E-05 - 0.000 + 0.000AR2/AREA 0.37456 ± 1.3263E-02 - 0.000 + 0.000DELM∗ 0.0000 ± 0.000 - 0.000 + 0.000SIG2/SIG1 2.1906 ± 3.7447E-02 - 0.000 + 0.000Function 2: Chebyshev Polynomial of Order 1NORM 8057.7 ± 551.2 - 0.000 + 0.000CHEB01 -0.19646 ± 8.3336E-02 - 0.000 + 0.000

Figure 3.2: MK−π+ distribution (left) and, MK∗0K distribution (right) for SignalD+s → K∗(892)0 K+ events. The blue curve represent the signal peak, while

the combinatorial background is represented by the red curves. The vertical linesindicate the K∗(892)0candidate selection criterion.

• K0SK mode:

A K0S candidate is reconstructed from its vø-type decays to a pair of oppositely

charged pions. Since a K0S can not decay to a charged kaon, it is possible to identify

the daughter pions just on the basis of kinematics and without invoking the PID.For this reason, all the charged tracks found in an event are assigned a pion mass.Any kaon track is shifted by 49 MeV in energy due to the wrong mass assignment(see digression 2 on page 63) and does not enter into the K0

S mass region, nullify-ing the possibility of kaon faking. TheK0

S candidates hence reconstructed, which fallaround the K0

S nominal mass are further required to pass the momentum-dependentselection criteria based on

1. fl: the flight length of K0S in the transverse (r − φ) plane,

2. dr = min(|dr1|, |dr2|): where dr1(2) are the signed impact parameters of thedaughter tracks,

3. zdist: distance between the two tracks at the intersection (i.e. the K0S decay

vertex), and

4. φ: angle between K0S momentum and the vertex vector with respect to IP.

This selection requirement is optimized for three K0S momentum regions: p < 500

MeV/c, 500 MeV/c < p < 1 GeV/c, and p > 1 GeV/c (for details see Belle Note323). Figure 3.3(a) shows the invariant mass distribution for the K0

S which satisfy

52




the selection criterion. We further demand the K0S candidate to have an invariant

mass within ±10 MeV/c2 around the K0S nominal mass.

0.485 0.490 0.495 0.500 0.505 0.510MKs (GeV/c2)

0

500

1000

1500

2000

2500

# of

Eve

nts

MINUIT Likelihood Fit to Plot 16&3B0 → Ds*hFile: Generated internally 3-JAN-2009 10:31Plot Area Total/Fit 18201. / 18201.Func Area Total/Fit 18201. / 18201.


Likelihood = 43.3χ2= 43.9 for 40 - 7 d.o.f., C.L.= 9.7%Errors Parabolic MinosFunction 1: Two Gaussians (sigma)AREA 17639. ± 159.0 - 0.000 + 0.000MEAN 0.49774 ± 1.7254E-05 - 0.000 + 0.000SIGMA1 1.63543E-03 ± 4.6029E-05 - 0.000 + 0.000AR2/AREA 0.35144 ± 3.4413E-02 - 0.000 + 0.000DELM∗ 0.0000 ± 0.000 - 0.000 + 0.000SIG2/SIG1 2.1511 ± 6.4721E-02 - 0.000 + 0.000Function 2: Chebyshev Polynomial of Order 1NORM 22577. ± 3580. - 0.000 + 0.000CHEB01 0.10674 ± 9.9613E-02 - 0.000 + 0.000

1.935 1.945 1.955 1.965 1.975 1.985 1.995MKsK (GeV/c2)

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Likelihood = 35.4χ2= 35.5 for 40 - 7 d.o.f., C.L.= 35.3%Errors Parabolic MinosFunction 1: Two Gaussians (sigma)AREA 15192. ± 128.1 - 0.000 + 0.000MEAN 1.9687 ± 4.3210E-05 - 0.000 + 0.000SIGMA1 3.95475E-03 ± 1.0855E-04 - 0.000 + 0.000AR2/AREA 0.35990 ± 3.7601E-02 - 0.000 + 0.000DELM∗ 0.0000 ± 0.000 - 0.000 + 0.000SIG2/SIG1 1.9791 ± 4.7333E-02 - 0.000 + 0.000Function 2: Chebyshev Polynomial of Order 1NORM 2892.8 ± 574.8 - 0.000 + 0.000CHEB01 -7.39676E-03 ± 0.1749 - 0.000 + 0.000

Figure 3.3: Invariant mass distribution for the K0S candidates satisfying the selec-

tion criteria (left) and the D+s candidates reconstructed in the K0

SK mode (right).The signal peaks are shown in blue, while the red curves represent the combinat-

orial backgrounds. The vertical lines indicate the selection regions.

A K0S candidate is combined with a charged kaon to form a D+

s candidate. Fig-ure 3.3(b) shows the invariant mass distribution for the combination reconstructedfrom signal events. The D+

s mass peak is fitted with a double Gaussian. The flatbackground is represented by the linear order Chebyshev polynomial. A D+

s candid-ate is required to have an invariant mass within ±17 MeV/c2 (∼ 3σw) around theD+s nominal mass.

The invariant mass of a D+s meson is a very well defined physical quantity: a sharp

peak centered at 1968.49 MeV/c2 with a width of only 0.34 MeV/c2. In such cases, theeffect of the uncertainties introduced due to poor detector performance in measuring themomenta of their daughter tracks can be clearly seen in their invariant mass distributions.Taking advantage of the precise kinematical relations between mother and daughter 4-momenta, possible due to the precisely known mass of the mother, it is possible to performan overall fit to all the daughter track phase-space measurements, involved in the decay(see section B.3). For theD+

s candidates, which satisfy all the selection criteria, the daugh-ter tracks are fitted to accommodate the kinematical constrain due to the precise D+

s mass:we perform a mass-constrain-fit, using the Kalman-fitting techniques [60] (also discussedin Belle Note 193 and Belle Note 194). This helps improve the momentum resolution byabout 4%.

53




3.1.2 D∗+s Reconstruction

A D+s candidate is combined with a photon to form a D∗+s candidate. Alike D+

s , the massof a D∗+s meson is known to a very good precision: i.e. (2112.3±0.5 MeV/c2). This impliesa very precise definition of the mass difference,

∆M = MD∗+s−MD+

s= MD+

s γ−MD+

s(3.1)

The mass difference between a D+s meson and a D∗+s meson is around 143 MeV/c2, which

is reflected in the lab momentum distribution of the photons shown in figure 2.15. D∗+s 4-momentum reconstructed from the daughter momenta is plagued due to the uncertaintiesin the D+

s momentum as well as in the photon momentum. On the contrary, resolution of∆M defined in equation (3.1) is dominated by the photon momentum resolution only.

The energy available to a D∗+s daughter photon lies in the range where many of thephotons from a π0 decay are found . In addition, a lot of beam background photonspopulate the low momentum region. To reduce the low momentum photon background,we do not accept a D∗+s candidate if its helicity angle θD∗+s , defined as the angle betweenthe flight direction of the photon and the direction opposite to the B0 meson momentumin the D∗+s rest frame [61], is found to satisfy cos θD∗+s < −0.6 (−0.7) in case of B0 →D∗+s π−(B0 → D∗−s K+). This reduces about 43% of the background against only 10%signal loss (see digression 1)

Digression 1. In order to remove the background photons, three possible strategies are con-sidered:

1. Eγ:Requiring the photon energy to be largerthan some critical valueEc helps reducingthe background below Ec, though a con-siderable fraction of the signal photon isalso lost, if the requirement is not chosenoptimally. Considering the substantialoverlap observed between the signal (red)and background (blue) photon distribu-tions shown in figure 3.4, this scenariodoesn’t appear promising.

2. π0 veto:Rejecting photon combinations whichhave invariant masses around the nom-inal π0 mass reduces 45% of the π0 photonbackground against only 15% lose in thesignal events. Although, photons fromπ0 constitute only 50% of the total back-ground (histogram shown in cyan in fig-ure 3.5) and hence is not effective whenconsidered overall background photons.

3. D∗+s helicity cos θD∗+s

:D∗+s → D+

s γ being a V → PV decay,where V (P ) is a vector (pseudo-scalar)meson, the D∗+s helicity follows a (1 −cos2 θ) distribution [61]. On the otherhand, most of the background photonstend to have helicities around 180 withrespect to the high momentum D∗+s . Thiskinematical difference between a signalevent and a background event on the heli-city plot can be exploited to reduce thebackground. Figure 3.5 shows the helicitydistributions for the correctly reconstruc-ted signal events and the fake or back-ground events.

The helicity distribution, cos θD∗s

proves tobe the most effective in separating the sig-nal photons from the background. Figure 3.6shows the optimization curves for the threeD+s modes for each of B0 → D∗+s π− and

B0 → D∗−s K+. The optimization is per-

54


formed using signal MC against data ∆M side-bands. We demand cos θD∗

s> −0.6 for the

former, while cos θD∗s> −0.7 for the latter.

B0 → Ds* K

File: *sig.hbkID IDB Symb Date/Time Area Mean R.M.S.

16 510 1 070523/0605 7625. 0.2076 7.6137E-02

0.00 0.10 0.20 0.30 0.40 0.50Eγ (GeV)

0

100

200

300

No.

of E

vent

s/ 1

0 M

eV b

in

γ from Ds*

γ from π0

16 500 2 070523/0606 4028. 0.1426 7.2786E-02

Figure 3.4: Photon energy distri-bution for the correctly reconstruc-ted signal (red) and fake (blue)

events.

This requirement reduces about 43% of thebackground, retaining about 91% of the signal.

cos(θDs*hel)

File: *sig.hbkID IDB Symb Date/Time Area Mean R.M.S.

16 1 1 070528/1746 1.000 -1.4820E-02 0.4442

-1.0 -0.5 0.0 0.5 1.0cos(θDs*

hel)

0.000

0.010

0.020

0.030

0.040

0.050

γ from Ds*

γ from bkg (total)

γ from π0

16 3 2 070528/1754 1.0000 -0.4474 0.4690 16 2 2 070528/1754 0.6290 -0.4377 0.4737

Figure 3.5: D∗+s helicity distribu-tion for the correctly reconstructedsignal (red) and fake (blue) events.The cyan histogram shows contri-

bution from π0 photons.

Figure 3.6: cos θD∗+s optimization curves for the B0 → D∗+s π− mode (left) andthe B0 → D∗−s K+ mode(right). The red curve is for the φπ mode, the green for

the K∗(892)0K modeand the blue for the K0SK mode.

55


Figure 3.7 (a-c) shows the ∆M distribution for the events reconstructed in the threeD+s modes in B0 → D∗+s π−. A D∗+s candidate is selected requiring ∆M to be between 128

0.10 0.125 0.15 0.175 0.20∆m (GeV/c2)

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MINUIT Likelihood Fit to Plot 16&1dstk. m_diff axisFile: *dstpi_phi.hbk 16-JUL-2007 18:11Plot Area Total/Fit 19035. / 19035.Func Area Total/Fit 19035. / 19035.


Likelihood = 219.2χ2= 219.0 for 100 - 7 d.o.f., C.L.=0.367E-09%Errors Parabolic MinosFunction 1: Chebyshev Polynomial of Order 1NORM 44596. ± 1102. - 0.000 + 0.000CHEB01 -0.57030 ± 2.3454E-02 - 0.000 + 0.000Function 2: Two Gaussians (sigma)AREA 14575. ± 157.4 - 0.000 + 0.000MEAN 0.14516 ± 5.3899E-05 - 0.000 + 0.000SIGMA1 3.88712E-03 ± 1.2736E-04 - 0.000 + 0.000AR2/AREA 0.45371 ± 3.2368E-02 - 0.000 + 0.000DELM∗ 0.0000 ± 0.000 - 0.000 + 0.000SIG2/SIG1 2.2967 ± 6.1423E-02 - 0.000 + 0.000

φ modeabs(∆E) < 50 MeV

0.10 0.125 0.15 0.175 0.20∆m (GeV/c2)

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MINUIT Likelihood Fit to Plot 16&1dstk. m_diff axisFile: *dstpi_kstar.hbk 16-JUL-2007 18:20Plot Area Total/Fit 14023. / 14023.Func Area Total/Fit 14023. / 14023.


Likelihood = 193.2χ2= 192.5 for 100 - 7 d.o.f., C.L.=0.646E-06%Errors Parabolic MinosFunction 1: Chebyshev Polynomial of Order 1NORM 32466. ± 856.6 - 0.000 + 0.000CHEB01 -0.56213 ± 2.3977E-02 - 0.000 + 0.000Function 2: Two Gaussians (sigma)AREA 10776. ± 118.1 - 0.000 + 0.000MEAN 0.14502 ± 5.7801E-05 - 0.000 + 0.000SIGMA1 3.76938E-03 ± 1.3492E-04 - 0.000 + 0.000AR2/AREA 0.49703 ± 2.6073E-02 - 0.000 + 0.000DELM∗ 0.0000 ± 0.000 - 0.000 + 0.000SIG2/SIG1 2.3703 ± 6.0591E-02 - 0.000 + 0.000

K* mode

abs(∆E) < 50 MeV

0.10 0.125 0.15 0.175 0.20∆m (GeV/c2)

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MINUIT Likelihood Fit to Plot 16&1dstk. m_diff axisFile: *dstpi_kshort.hbk 16-JUL-2007 18:16Plot Area Total/Fit 11818. / 11818.Func Area Total/Fit 11818. / 11818.


Likelihood = 188.6χ2= 185.4 for 100 - 7 d.o.f., C.L.=0.416E-05%Errors Parabolic MinosFunction 1: Chebyshev Polynomial of Order 1NORM 25231. ± 807.8 - 0.000 + 0.000CHEB01 -0.63361 ± 2.9779E-02 - 0.000 + 0.000Function 2: Two Gaussians (sigma)AREA 9294.9 ± 115.2 - 0.000 + 0.000MEAN 0.14520 ± 6.5869E-05 - 0.000 + 0.000SIGMA1 3.75940E-03 ± 1.4986E-04 - 0.000 + 0.000AR2/AREA 0.46975 ± 3.7407E-02 - 0.000 + 0.000DELM∗ 0.0000 ± 0.000 - 0.000 + 0.000SIG2/SIG1 2.3683 ± 7.4772E-02 - 0.000 + 0.000

KS mode

abs(∆E) < 50 MeV

Figure 3.7: ∆M distribution for three D+s decay modes: the φπ mode (left), the

K∗(892)0K mode (middle), and the K0SK mode(right).

MeV/c2and 162 MeV/c2and is further applied mass-constrain-fit to the nominal D∗+s mass(see B.3). Mass-constraining the D∗+s candidate helps improve the momentum resolutionby 32% (figure B.4).

3.1.3 B0 → D∗+s π− and B0 → D∗−s K+

A B0 meson is reconstructed by combining a D∗+s candidate with an oppositely chargedpion or kaon. Following the conventional procedure, it is possible to apply selection re-quirements on the B0 meson 4-momentum or equivalently, its invariant mass. Use oftwo almost uncorrelated kinematical variables - the beam constrained mass Mbc and theenergy difference ∆E - is made, instead.

∆E and Mbc

The two-body decay of a Υ(4S) to a BB pair results in a B with a fixed energy E∗B andmomentum p∗B in the CM frame. In principle,

E∗B =n∑i=1

E∗i =mΥ(4S)

2= 5.29GeV |p∗B| = |

n∑i=1

~p∗i | = 340MeV/c, (3.2)

where, (E∗i , ~p∗i ) is the CM frame 4-momentum of the i-th particle. The variables E∗B and p∗B

can be used to identify a B0 candidate. However, use of an equivalent set of uncorrelatedkinematical variables (∆E, Mbc) is made, where

∆E = E∗B − Ebeam and Mbc =√E2

beam − p∗2B , (3.3)

The advantage of using these variables lies in the facts:

56


• Ebeam is known to a very good accuracyEbeam has a resolution of about 3 MeV (see figure 2.3). The B energy EB, on theother hand, is derived from the track momenta of the final state particles, whichhave resolutions of about 1-5 MeV/c each. As a result EB resolution usually rangesfrom 10 MeV to 40 MeV, depending upon the kinematical details of the decay beingreconstructed.

• ∆E is dominated by the track momentum resolutionThe energy of the track is an unknown unmeasured quantity determined from thetrack momentum after the mass-assignment. As a result the energy resolution ofeach track is dominated by the track momentum uncertainty. ∆E as defined above,has the dominant contribution from the track momenta and no effect of Ebeam meas-urements. As a result, ∆E is highly sensitive to the overall momentum of the B0

candidate.

• Shape of Mbc is dominated by Ebeam

The resolution of Mbc can be estimated from the errors on Ebeam and p∗i propagatingaccording to

σMbc=

√√√√(Ebeam

Mbc

)2

σ2Ebeam

+∑j

(∑i ~p∗i

Mbc

)2

(βγ)2σ2pj

It is easy to see that the second factor above is doubly suppressed due to the factthat in CM frame, B has momentum of only 340 MeV/c or (

∑i ~p∗i ) Mbc and

βγ = 0.42. The resolution of Mbc is governed mainly by the beam energy resolutionσEbeam

, since Ebeam ∼Mbc. (In a similar way, it can be shown that ∆Mbc ∝ ∆Ebeam,a relation which has been exploited to obtain Ebeam calibration as explained onpage 33.)

As mentioned earlier, another advantage of using ∆E and Mbc is that these vari-ables are almost uncorrelated: a fact manifest in the 2-dimensional (2D) plot shown infigure 3.8. All the events which have Mbc within 5.2 GeV/c2 and 5.3 GeV/c2 and ∆Ebetween ± 200 MeV are considered for further analysis and this region is referred to asthe fit-region. The signal events however populate a very tiny sub-region, termed as signal-region or signal box, around ∆E equal to 0 MeV and Mbc equal to the invariant mass ofthe B0 meson, i.e. ∼ 5.28 GeV/c2.

Throughout the optimization, or maximization of the significance for observing signalevents over the background, the signal region is blinded in the real data in order to reducethe possibility of introducing (experimenter’s) bias. This optimization technique is usuallycalled Blind Analysis Technique (see, section B.1).

Three distinct possible strategies for extracting signal can be devised:

1. using Mbc

because of the sensitivity only to Ebeam, the Mbc distribution can be parametrizedindependent of the kinematical details of the signal (as well as background). As areason, the probability density functions (PDF) are very well defined.This robustnessof Mbc has been exploited in all previous measurements [45, 46] and offers the mostefficient way to parametrize signal with the least possible uncertainties.

57


E∆

-0.2 -0.1 0 0.1 0.2

bcM

5.2

5.22

5.24

5.26

5.28

5.3 0

5000

10000

15000

0 10000 20000

E∆

bcM

Figure 3.8: ∆E against Mbc distribution for the signal events.

2. using ∆Ein contrast toMbc, the sensitivity of ∆E to the kinematical details of the signal makesit a powerful signal to background discriminant. However, the leakage effects andpoor detector performance in reconstructing neutral final-states obscure the signaland background PDF parametrization and hence use of ∆E for signal extraction hasbeen limited to the only charged-final-states decays.

3. performing a Mbc-∆E 2D fitsignal can be extracted performing a fit to the distributions on a 2D plot, combiningrobustness ofMbc parametrization and the power of ∆E as a signal over backgrounddiscriminant. A 2D fit also offers a higher (∼ 10%) statistical sensitivity over a 1D fitand hence is desirable in cases of studies involving rare decays.

It is evident that in deciding the best possible strategy for signal extraction, the role ofbackground events is crucial and hence a thorough understanding of the background pro-cesses, which can potentially contaminate the fit-region, is required.

Note, while studying the backgrounds in ∆E fit-region, we require the events to comefrom the Mbc signal-region and vice verse.

58

3.2. BACKGROUND ESTIMATION

3.2 Background Estimation

Background events in the fit-region can be classified according to the physical processesinvolved and subsequently according to the probability of occurrence of such events.

• BB backgroundA Υ(4S) decayed to a BB pair can enter into the fit-region depending upon thedegree of similarity of the event kinematics to that of the signal

– reflection backgrounda B meson decay involving a D+ → K∗(892)0π+ or a D+ → K0

Sπ+ process via

the pion misidentification mimics a D+s decay to a K∗(892)0K+ or a K0

SK+,

respectively. Due to the dominance of the Cabibbo-favored b→ c process, theseevents have the highest potential among B decays entering the fit-region. Also,the reflection background enters more readily in the K∗(892)0K mode com-pared to the K0

SK mode, since the D+ → K∗(892)0π+ or D+ → K−π+π+

decays contributing to former have effectively 10 times higher branching frac-tion than the D+ → K0

Sπ+ decay entering the latter.

– charmless backgrounda B meson decay (usually a many-body decay) to the same final-state as of asignal event

– rare backgrounda Cabibbo-suppressed B decay; usually involving a D+

s or a D∗+s meson in thedecay.

– cross-feed backgroundthe two signal processes, namely B0 → D∗+s π− and B0 → D∗−s K+, can feedacross the fit-region of each other due to the prompt track misidentification.

– combinatorial backgroundinvolving random or wrongly reconstructed combinations of the final-state particles.

• continuum backgroundthese decays are similar to the combinatorial background events discussed above,except that they do not come from a Υ(4S) decaying to a BB pair but from thee+e− → qq continuum events underlying the threshold.

3.2.1 Dominant B decays

The various types of BB background events listed above, show different behaviors on the∆M as well as ∆E plots depending upon the nature of the decay, thus allowing one toseparate them and devise ways of reducing their population in the fit region. Table 3.2summarizes the characteristic properties seen in the ∆M and ∆E distributions of thesedecays.

Among various BB background processes the most dominant contribution comes fromthe reflection background. These events corresponds to a B → D(∗)+X decay, involvinga Cabibbo-favored b → c transition and hence constitute the largest single componentof the total B-meson decays. As an effect, they have the highest potential to populate

59


Table 3.2: Various types of BB decays contributing to the background and theirgeneric behaviour in ∆M and ∆E distributions.

type ofon ∆M on ∆E

background

reflection remains flat peaks at slightly shifted values fromzero due to misidentification (seedigression 2 on page 63)

charmless remains flat can peak at zero if the decay has ex-actly same final-state particle con-tent as signal.

rare can peak whenever a D∗+s ispresent

peaks at values shifter from zerodepending upon the final-state con-tent of the decay

cross-feed peaks in the signal-region peaks at slightly shifted values fromzero due to misidentification (seedigression 2 on page 63)

combinatorial remains flat remains flat

the fit-region. Even though dominant, the background coming from a B decay to a D∗+ isstatistically suppressed, since only (1.6±0.4)% of the total population ofD∗+ mesons decayto a D+ with a photon emission. As a result, the photon present in an event correspondingto this class of the background is almost always picked up randomly and does not holdany kinematical relation to the D+ mimicking as a D+

s . Consequently these decays do notpeak in the ∆M distribution and show broad, smeared-out peaks in the ∆E distribution.Moreover, different B decays, such as B0 → D+π−, B0 → D+ρ− populate distinctlyseparated regions of ∆E fit-region, depending upon the energetics of the adjusted particle-content required to match the signal final-state: addition of a photon in B0 → D+π−, lossof the π0 in addition to a photon gained in B0 → D+ρ−, etc. As a result, no significantpeaking structures are observed in the overall ∆E distribution.

The charmless background is observed to be negligible. Due to their low probability ofoccurrence, most of the rare B decays are either not observed experimentally and only anupper bound is known or only an evidence of their existence is established so far. All thenecessary kinematical inputs related to the rare decays, such as their branching fractions,angular distributions or other phase-space details, can not be provided as precisely astheir dominant B decays counterpart in advance, but become available as new improvedmeasurements are carried out. Hence, these rare decays are not included in the BBMC samples, but are separately generated and studied. The discussion related to therare background studies is deferred to the next section 3.2.2. The two B signal modesthemselves being rare decays, the cross-feed effects of the two are discussed along with therare background studies.

Figure 3.9 summarizes the BB MC observations, with rare backgrounds excluded, inthe fit-regions of B0 → D∗+s π− (left) and B0 → D∗−s K+ (left). Five MC samples of sizeseach equal to that of real data are used to reduce the statistical uncertainties.

60


Ds*+π-: ∆E for background

-0.20 -0.10 0.00 0.10 0.20∆E (GeV)

0

5

10

15

# of

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nts

B+ → D(*) 0ρ+

B0 → D*-π+

B0 → D-π+

B0 → D*-ρ+

B0 → D-ρ+

B+ → Ds(*)-K+π+

rest bkg

Ds*-K+: ∆E for BBar background

-0.20 -0.10 0.00 0.10 0.20∆E (GeV)

0

1

2

3

4

# of

eve

nts

B+ → Ds*-K+π+

B+ → Ds-K+π+

rest background

Figure 3.9: BB background in the ∆E fit region of B0 → D∗+s π− (left) andB0 → D∗−s K+ (right). Five MC samples of size each equal to that of real data wasused for this study. The rare B decays are not included in these samples, but are

studied separately.

The background level in ∆E fit-region of B0 → D∗−s K+ is lower compared to that inB0 → D∗+s π− fit-region, because of

1. amplitudes for the corresponding reflection background decays, such asB → D(∗)K(∗),are Cabibbo-suppressed, which is responsible for effective reduction of the reflectionbackground in this case,

2. average multiplicity of charged kaons, i.e. average number of charged kaons perevent is only one as opposed to that of six-seven for a charged pion, reducing theprobability of occurrence of combinatorial background events,

3. shift in ∆E caused due to an addition or loss of a kaon is large enough to move theevents outside the fit-region.

The pion to kaon misidentification is the main cause behind a D+ mimicking as a D+s

in the reflection background. As a consequence, to reduce the reflection events in the fit-region, we tighten the PID requirement on the kaon accompanying the K∗(892)0 in D+

s

to RK/π > 0.8. No significant reflection background is observed in the K0SK mode due to

lower branching fraction for the D+ → K0Sπ

+ process and unlike the K∗(892)0K mode,we do not apply any tighter constrain on the accompanying charged kaon.

A further reduction in the reflection background can be achieved by applying aD+ massveto: rejecting the D+

s candidates in the K∗(892)0K mode and K0SK mode which have in-

variant masses around the nominal D+ mass after re-assigning the kaon (or misidentifiedpion) the correct pion mass. This procedure reduces about 82% of the reflection back-ground, with a signal loss of 19%. Due to its wide-spread profile, instead of prominentlypeaking behaviour, in the ∆E fit-region we do not apply a D+ veto if ∆E is chosen to be

61


the variable for signal extraction. A different strategy might be required in case if Mbc isto be used (see section 3.3.1).

In conclusion, the dominant BB decays do not show any significant peaking structure inthe ∆E fit-region of both B0 → D∗+s π− as well as B0 → D∗−s K+ decays.

3.2.2 Rare B events

As mentioned in the previous section, due to the limited knowledge available about thesedecays at the time of BB MC production, the usually Cabibbo-suppressed and hence rareB decays are separately generated as updates on the same become available. Anotheradvantage of generating rare events separately is the possibility of generating huge stat-istics of rare modes. While generating the rare MC samples, the dominant B decays canbe excluded, as the MC samples from them are already available. Since in MC generationthe total probability for a B decay must always be scaled to unity, exclusion of dominantdecays from the generation effectively amplifies the probabilities of occurrence (i.e. effect-ive branching fractions) for the rare modes. As a result, the rare MC samples used in thisanalysis have huge statistics of rare modes, equivalent to 50 times the size of real data.

Figure 3.10 shows the rare B decay events entering the ∆E fit-region of B0 → D∗+s π−

(left) and B0 → D∗−s K+ (right). Many B decays to a true D+s or a true D∗+s not only enter

the fit-region more readily, but also show prominent peaking structures. Due to its sens-

Ds*+π-: ∆E for rare background

-0.20 -0.10 0.00 0.10 0.20∆E (GeV)

0

4

8

12

16

# of

eve

nts

B0 → Ds*+π-

B0 → Ds+π-

B0 → Ds*-K+

B0 → Ds-K+

B0 → Ds*+ρ-

B0 → Ds+ρ-

B+ → Ds*+ρ0

B+ → Ds+ρ0

rest background

Figure 3.10: Rare B decays entering the ∆E fit-region of B0 → D∗+s π−(left)and B0 → D∗−s K+(right). A hatched histogram represents a D∗+s mode, while a

solid-filled histogram in the same colour represents corresponding D+s mode.

itivity to momentum resolution of the final-state particles, the peak positions in the ∆Edistribution for a particular B decay can be explained on the basis of whether some final-state particle is added or lost from the original decay in order to match the signal modefinal-state configuration. Moreover, the two signal modes being present in the rare MCsamples, their cross-feeding effect into each other due to the kaon-pion misidentificationof the prompt track can be seen in figure 3.10. The cross-feeding events are shifted in ∆E

62


by about 49 MeV, due to the kinematical difference built into due to the misidentification(see digression 2).

The strength of ∆E variable as a signal to background discriminator can be clearlyseen from figure 3.10. Despite many background modes, having kinematical similaritiesto the signal modes, tend to have significant peaking contributions in the fit-region, noneof them can be mistaken for signal because of the distinguishing properties of their PDFs,such a mean or spread of the distributions.

Digression 2. A track with a lab-frame momentum p corresponding to a particle hypothesis ihas a CM energy

E∗i = γ(Ei + βp)

= γp

√1 +m2i

p2+ β

= γp

(1 +

m2i

2p2+ β

). . . p mi

where, mi is the mass assigned to the track consistent with particle hypothesis i. As a result, a piontrack misidentified as a kaon would hence be shifted in energy by

E∗K − E∗π =γ

2p(m2

K −m2π) ∼ 0.122

p. . . p in GeV/c (3.4)

where, we substituted γ = 1.086 for KEKB, mK+ ∼ 494 MeV/c2 and mπ+ ∼ 139 MeV/c2 fromParticle data group. For the prompt tracks with lab-frame momenta around 2.5 GeV/c (figure 2.8),misidentifying a pion (kaon) track as a kaon (pion) shifts the track energy higher (lower) by about49 MeV, a shift responsible for the two signal modes cross-feeding the ∆E fit-region of each other.

In summary, the rare modes which show peaking behaviour can be listed according totheir most probable position of occurrence on the ∆E plot:

• In B0 → D∗+s π−

1. B0 → D+s π− : with an extra photon - shifted to positive ∆E values,

2. B0 → D∗−s K+ cross-feed: with K → π faking - shifted to negative ∆E values,

3. B0 → D+s ρ−: with a lost π0, but extra photon - shifted to negative ∆E values

but again moved back towards ∆E= 0

4. B0 → D∗+s ρ−: with a lost π0 - shifted to negative ∆E values

• In B0 → D∗−s K+

1. B0 → D−s K+ : with an extra photon - shifted to positive ∆E values;

2. B0 → D∗+s π− cross-feed : with a π → K faking - shifted to positive ∆E values;

Due to the peaking nature of these decays in the fit-region, it is important to estimatethe overall effect of them as precisely as possible, in order to be able to quantify the

63



amount of signal present in the data sample. For this reason, it is necessary to obtaininformation about their PDFs, efficiency with which they enter the fit-regions and theirfrequency of occurrence in data or respective branching fractions. With this information,it is then possible to fix yields or number of events of each of these background modeswith their correct distributions, up to the best known values.

PDFs

To determine the PDFs as accurately as possible,

1. Separate MC samples are generated for each rare B mode listed above, in each D+s

mode separately. Size of a sample is chosen to have at least 100 times rare eventsbeing studied compared to that expected in the real data;

2. Fits are made to the ∆E distributions, where a PDF is the nearest approximation tothe shape observed with the minimal set of parameters needed to be fixed. To beable to correct for any difference in parameter values between MC and the real dataat a later stage, attempt is made to restrict the choice of PDFs which offer parametersclose in meaning to mean (µ) and spread (σ) of a Gaussian distribution;

3. Because in all the peaking modes observed a D+s is always present, the PDF para-

meters are expected to have consistent values within the three D+s decay modes.

We observe no significant discrepancy in any of the peaking modes. As a result,one single PDF for each rare mode in all the three D+

s modes is prepared. This isachieved through merging the MC samples in three D+

s modes of a rare decay. Fig-ure 3.11 summarizes the PDFs fixed from the MC samples of the rare B decays underconcern.

Table 3.3 lists various PDFs chosen to fit the ∆E distributions shown in figure 3.11.

Table 3.3: PDFs chosen to fit the ∆E distributions of the rare modes listed above.

Rare B Mode reffBranching ratio

PDF(×10−5)

B0 → D∗+s π−

B0 → D+s π− Landau (2.9± 0.2)× 10−1 (2.5± 0.4± 0.2)

B0 → D∗−s K+ cross-feed CB plus Gaussian (1.6± 0.3)× 10−1 ‡ not fixedB0 → D+

s ρ− double Gaussian (6.8± 0.4)× 10−2 (1.1+0.9

−0.8 ± 0.3)B0 → D∗+s ρ− double Gaussian (4.7± 0.3)× 10−2 (4.4+1.3

−1.2 ± 0.8)

B0 → D∗−s K+

B0 → D−s K+ Landau (2.0± 0.3)× 10−1 (2.9± 0.4± 0.2)

B0 → D∗+s π− cross-feed CB plus Gaussian (4.2± 0.3)× 10−2 ‡ not fixedB0 → D−s K

+π+ † double Gaussian (4.3± 0.4)× 10−2 (20.2± 1.3± 3.8)B0 → D∗−s K+π+ † double Gaussian (4.0± 0.1)× 10−2 (16.7± 1.6± 3.5)

† from BB MC studies (see figure 3.9)‡ denotes ε(D∗+s π− CF)/ε(D∗+s π− signal), etc (see section 3.2.2).

64


-0.20 -0.10 0.00 0.10 0.20 0.30∆E (GeV)

0

250

500

750

1000

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(G

eV/c

2 )

MINUIT Likelihood Fit to Plot 16&1B0 → Ds

*+π-

File: *dsubpi-sb0.hbk 9-SEP-2008 22:52Plot Area Total/Fit 6207.0 / 6207.0Func Area Total/Fit 6206.7 / 6206.7


Likelihood = 97.8χ2= 92.6 for 40 - 5 d.o.f., C.L.=0.421E-04%Errors Parabolic MinosFunction 1: LandauAREA 3.68892E+05 ± 6081. - 0.000 + 0.000XLAN 0.13928 ± 4.4462E-04 - 0.000 + 0.000WIDTH 1.77291E-02 ± 2.0608E-04 - 0.000 + 0.000Function 2: Chebyshev Polynomial of Order 1NORM 1024.9 ± 74.93 - 0.000 + 0.000CHEB01 1.0087 ± 1.6434E-02 - 0.000 + 0.000

from B0 → Ds+π-

φ mode

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*+π-

File: *dsubrho-both-sb0.hbk 10-SEP-2008 00:10Plot Area Total/Fit 14467. / 14467.Func Area Total/Fit 14466. / 14466.


Likelihood = 154.8χ2= 151.1 for 40 - 5 d.o.f., C.L.=0.224E-13%Errors Parabolic MinosFunction 1: CB Line ShapeAREA 15652. ± 367.2 - 0.000 + 0.000MEAN -4.88882E-02 ± 1.3346E-03 - 0.000 + 0.000SIGMA 6.55897E-02 ± 1.6007E-03 - 0.000 + 0.000ALPHA∗ 0.60000 ± 0.000 - 0.000 + 0.000N∗ 3.0000 ± 0.000 - 0.000 + 0.000Function 2: Chebyshev Polynomial of Order 1NORM 3171.0 ± 343.5 - 0.000 + 0.000CHEB01 -1.0257 ± 4.4676E-06 - 0.000 + 0.000

from B → Ds+ρ(0/-)

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*+π-

File: *dsstrho-both-sb0.hbk 9-SEP-2008 23:37Plot Area Total/Fit 7541.0 / 7541.0Func Area Total/Fit 7531.3 / 7531.3


Likelihood = 136.3χ2= 133.9 for 40 - 5 d.o.f., C.L.=0.169E-10%Errors Parabolic MinosFunction 1: Two Gaussians (sigma)AREA 9948.1 ± 200.4 - 0.000 + 0.000MEAN -0.17660 ± 9.8834E-04 - 0.000 + 0.000SIGMA1 2.86429E-02 ± 7.1163E-04 - 0.000 + 0.000AR2/AREA 0.18542 ± 1.3785E-02 - 0.000 + 0.000DELM∗ 0.0000 ± 0.000 - 0.000 + 0.000SIG2/SIG1 3.2616 ± 8.6504E-02 - 0.000 + 0.000

from B → Ds*+ρ0/-

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File: *dsubk.hbk 9-SEP-2008 12:46Plot Area Total/Fit 7096.0 / 7096.0Func Area Total/Fit 7095.6 / 7095.6


Likelihood = 110.1χ2= 104.8 for 40 - 5 d.o.f., C.L.=0.688E-06%Errors Parabolic MinosFunction 1: LandauAREA 4.29187E+05 ± 5491. - 7582. + 0.000XLAN 0.12440 ± 8.0896E-05 - 4.8554E-04 + 4.8948E-04WIDTH 1.76746E-02 ± 3.2637E-05 - 2.7791E-04 + 2.8525E-04Function 2: Chebyshev Polynomial of Order 1NORM 672.98 ± 51.35 - 51.45 + 54.03CHEB01 1.0070 ± 1.7949E-02 - 2.3684E-02 + 0.000

from Ds-K+

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File: *temp/dsubkpi.hbk 9-SEP-2008 14:13Plot Area Total/Fit 5379.0 / 5379.0Func Area Total/Fit 5378.9 / 5378.9


Likelihood = 104.3χ2= 99.0 for 40 - 3 d.o.f., C.L.=0.148E-04%Errors Parabolic MinosFunction 1: CB Line ShapeAREA 13295. ± 333.1 - 0.000 + 0.000MEAN -0.16903 ± 5.0168E-03 - 0.000 + 0.000SIGMA 0.11159 ± 3.2394E-03 - 0.000 + 0.000ALPHA∗ 0.20000 ± 0.000 - 0.000 + 0.000N∗ 3.0000 ± 0.000 - 0.000 + 0.000

from B+ → Ds-K+π+

-0.20 -0.10 0.00 0.10 0.20∆E (GeV)

0

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*-K+

File: *temp/dsstKpi-sb0-best1.hbk 9-SEP-2008 17:52Plot Area Total/Fit 1891.0 / 1891.0Func Area Total/Fit 1889.3 / 1889.3


Likelihood = 26.0χ2= 25.7 for 40 - 5 d.o.f., C.L.= 87.5%Errors Parabolic MinosFunction 1: Two Gaussians (sigma)AREA 2936.9 ± 170.5 - 0.000 + 0.000MEAN -0.18612 ± 2.8995E-03 - 0.000 + 0.000SIGMA1 2.96659E-02 ± 2.0449E-03 - 0.000 + 0.000AR2/AREA 0.28257 ± 2.8155E-02 - 0.000 + 0.000DELM∗ 0.0000 ± 0.000 - 0.000 + 0.000SIG2/SIG1 3.7064 ± 0.2334 - 0.000 + 0.000

from B+ → DS*-K+π+

Figure 3.11: PDF models representing the rare modes ∆E distributions and thein the B0 → D∗+s π− mode (top row): B0 → D+

s π− (left), B0 → D+

s ρ− (middle)

and B0 → D∗+s ρ− (right) and in the B0 → D∗−s K+ mode (II row): B0 → D−s K+

(left), B+ → D−s K+π+ (middle) and B+ → D∗−s K+π+ (right).

Efficiencies

The same MC samples used for PDF determination are used to determine efficiencies.Because all the rare modes of interest have a D+

s in the decay, the differences within theefficiencies of these modes must come from the kinematical differences introduced due to

65


the non-D+s counterpart of the decay process. Also, within different D+

s decay modes,

reff =ε(rare mode)

ε(signal)

∣∣∣∣φπ

=ε(rare mode)

ε(signal)

∣∣∣∣K∗0K

=ε(rare mode)

ε(signal)

∣∣∣∣K0SK

(3.5)

is expected to hold, as the kinematical difference between two D+s modes are cancelled in

the ratio. No discrepancy is observed in any of the rare mode reff and within the three D+s

modes it is observed to be consistent. A common reff to three D+s modes is used for a rare

mode, which is calculated according to the weighted average. The advantage of using reff

instead of the absolute efficiencies is that any correction applied to the signal efficienciesin order to compensate for any difference between MC and real data will be propagatedautomatically to all the corresponding peaking backgrounds. Table 3.3 shows reff valuesobtained for various rare modes. Use of a common reff in three D+

s modes reduces thenumber of parameters being fixed and the sources of uncertainties hence introduced.

Since the B0 → D∗+s π− cross-feed is kinematically nearer to the B0 → D∗+s π− signalthan B0 → D∗−s K+ and vice verse, reff for the cross-feed is calculated within the samemode, even if the cross-feed of one signal mode appears as a background in the fit-regionof the other. Also, reff for the cross-feeds, calculated this way, must be same as the ratiobetween the prompt track fake-rate (i.e. misidentification efficiency) and the identificationefficiency. Table 3.4 compares the reff estimated from the MC samples with the valuesobtained using the PID efficiency studies used to obtain figure 2.12.

Table 3.4: Comparison between cross-feed reff calculated from MC samples andthat obtained in the PID efficiency studies.

reff

signal MC KID Tables

B0 → D∗+s π− 0.04± 0.01 0.0433± 0.0004B0 → D∗−s K+ 0.16± 0.02 0.1916± 0.0002

Branching fraction

The latest branching fractions quoted by Particle data group are used for fixing the yieldsof the rare modes, except the cross-feed contributions. Since, the branching fraction of thecross-feeding background in one signal mode is same as the quantity being measured in theother signal mode, a simultaneous fitting technique can be devised, which would adjustthe cross-feed yield in one mode simultaneously with branching fraction measurement inthe other and vice verse. This strategy is discussed at length a little later (in section 4.3).The branching fractions used for the rare modes under discussion are enlisted in table 3.3.

3.2.3 Continuum events

Continuum events, as opposed to generic BB events, do not come from a Υ(4S) decay andhence are not expected to show any characteristic behaviour on both ∆E as well as Mbc

66



plots. These events are uniformly distributed over the entire fit-region in ∆E and overmost part of the Mbc fit-region with a tapering end at a threshold value set by the beamenergy: a distribution usually parametrized empirically as an ARGUS function [62]. Inshort, these events do not show any harmful peaking structures in the fit-regions. However,the continuum events are expected to be the most dominant source of overall background,since

• B decays without a D+s have relatively small probabilities to satisfy all the signal

selection criteria as seen in 3.2.1,

• B decays with a D+s are usually Cabibbo-suppressed, whereas

• continuum events are rich in D+s ’s as well as pions and kaons, contributing to huge

combinatorial background.

Figure 3.12 shows the continuum background (blue) seen in the Mbc fit-region, as com-pared to the BB background (red). The continuum background observed is about 5 timeshigher in size compared to the BB background. As a result, a large amount of this back-

E (GeV)∆-0.2 -0.1 0 0.1 0.2

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E (GeV)∆-0.2 -0.1 0 0.1 0.2

# of

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20

30

40

50

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BBar + rare

cont

+K*-s D→0B

Figure 3.12: Continuum background entering theMbc fit-region ofB0 → D∗+s π−

(left) and B0 → D∗−s K+ (right) decays. The blue and red histograms signifycontributions from the continuum and BB events, respectively.

ground present in the fit-region under the signal peak can affect the significance withwhich signal events can be observed and proper measures are necessary to be taken inorder to reduce this background, thus improving upon the significance.

The generic BB events, coming from a Υ(4S) decay demonstrate a more sphericaltopology compared to the continuum events developing into two back-to-back jets. Inaddition, since a Υ(4S) is never produced, these events do not share any well-definedkinematical properties when taken to the CM frame and the hadronic jets emerging outof these events remain constricted within cones of very narrow opening angles along thejet axes, which coincide with the beam axis due to the boost. This topological differencein the behaviour can be exploited to discriminate between these two classes. The mosttrivial choice of basis for expressing sphericity is provided by the spherical harmonics

67


Y ml (Ω): a perfect spherical distribution is incorporated into the term with l = 0 or S-

wave mode and deviations from the spherically symmetric distributions can be understoodperturbatively with non-zero coefficients of higher order terms with l = 1, 2, . . . or P -,D- wave modes, . . . To distinguish the continuum 4-momentum geometry from that of theBB events, we use the multipole moments of the 4-momenta defined as,

Hl =

(4π

2l + 1

) +l∑m=−l

∣∣∣∣∣∑i

Y ml (Ωi)

|~pi|√s

∣∣∣∣∣2

=∑i,j

|~pi||~pj |sPl(cos θij) (3.6)

where, i, j runs over all final-state particles, Ωi is the solid angle defining the momentumof ith final-state, Pl is the l-th order Legendre polynomial and θij is the angle between ~piand ~pj .

The Hl’s constitute a set of rotationally invariant observables that can be used to dif-ferentiate the two background types. Use of these moments was first suggested by Fox andWolfram to distinguish two jet e+e− → qq events from more spherical three jet e+e− → qqgevents [63], where g signifies a gluon emission. We use a slightly modified version of theoriginal idea of Fox-Wolfram, where all the missing 4-momentum in an event is identifiedwith an imaginary particle while calculating the Hl’s [64]. This procedure has been foundto be more effective in comparison with original one.

Energy-momentum conservation requires H0 = 1 and H1 = 0. The number of Hl’srequired to differentiate effectively between two classes is case-dependent and in caseswhere continuum background is not too fierce first two moments are sufficient to sup-press the background to a tolerable level. On the other hand, inclusion of higher ordermoments become increasingly necessary with increasing background levels, as is the casein hand. First seven moments are used in this study. Though easy to calculate indi-vidually, the process of optimization for the background suppression becomes more andmore involved as number of variables increase. The most efficient scheme to deal withsuch multi-dimensional optimization processes was given by R. A. Fisher [65]. FollowingFisher’s procedure2, which is discussed briefly in section B.2, a single linear variable F ,termed as the Super Fox-Wolfram variable [66], is prepared as

F =∑l

αlHl (3.7)

where, the coefficients αl’s are to be chosen, such that F provides the maximum separationbetween a signal or BB event and a continuum event. These coefficients are determinedusing signal and continuum MC samples and subsequently, a likelihood LFisher is preparedusing the F .

Because the Hl’s are calculated in the CM frame, they do not carry information re-garding the flight directions of B produced and hence the polar angle distribution of theB offers an additional discriminating variable, which is uncorrelated with any of the Hl’sincluded in F . The spin-1 Υ(4S) decaying to two spin-0 B mesons is expected to followa cos2 θ angular distribution in the CM frame, as opposed to a nearly flat distribution forthe non-Υ(4S) continuum events. Figure 3.13 shows the B polar angle θB distribution forsignal (blue) and continuum (red) events.

2Fisher’s discrimination procedure works also in more complicated situations, such as multi-dimensionalvariables with non-zero correlations among them.

68


Bθcos-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

# of

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0.02

0.03

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continuum

B polar Angle Distibution

Figure 3.13: B flight polar angle distribution for the signal and continuumevents.

Combining the PDF for the polar angle distribution with F in equation (3.7), a likeli-hood

Ltotal =Psig

Psig + Pback

=Pcos θB (sig)× PFisher(sig)

Pcos θB (sig)× PFisher(sig) + Pcos θB (back)× PFisher(back)

=Pcos θB (sig)× LFisher(sig)

Pcos θB (sig)× LFisher(sig) + Pcos θB (back)× LFisher(back)(3.8)

and further a likelihood ratio Rtotal can be prepared, which peaks at one for a signal orBB event and at zero for a continuum event. Figure 3.14 shows the Rtotal distribution fora signal and continuum events.

Optimization

The huge combinatorial background due to the continuum events is mainly responsible forlowering the significance of signal observation. It is easy to check that a mild suppressioncriterion - or equivalently a loose selection requirement such asRtotal near zero - will leaveenormous background untouched in the fit-region hiding effectively some signal eventsbehind the large statistical fluctuations in the background, while a very tight suppressioncriterion - with Rtotal close to unity - will suppress all the background at an expense ofmassive reduction in signal efficiency. An optimal point can be reached for an intermediatevalue of Rtotal, which can be estimated by performing a figure-of-merit (FoM) study, i.e.studying the effect ofRtotal variation onNs/

√(Ns +Nb) 3, whereNs (Nb) are the number

of signal (background) events observed in the signal-region. For effective separation and

3Ns/√Ns +Nb is roughly the significance for observing Ns signal events, out of (Ns + Nb) events and

hence with a Poissonian error of√

(Ns +Nb).

69


for obtaining a more realistic optimal point, it is necessary to carry out the optimizationstudy when all the background, including BB and rare events, is present under the signalpeak. The FoM curve is obtained by scanning the Rtotal from zero to one in regular steps,and determining Ns and Nb from signal and background samples, respectively, by fittingthe distributions to pre-determined PDFs for the two and calculating the area under thefits inside the fit-region. We performed the FoM study on three statistically independentbackground samples:

• Background MC in ∆M signal-regionThe Ns and Nb are determined from signal and background (BB + rare + con-tinuum) MC samples, each scaled to real data size, respectively. In ideal cases withMC representing the real data exactly, this would be the most realistic procedure.Though because of limited a priori knowledge, MCs differ from real data and cannot be relied, unless checked for dangerous discrepancies.

• Real data SidebandsTheNb is calculated from real data ∆M sidebands (defined in section 4.2.1), insteadof background MC samples. This removes risks of using unrealistic MC samples forbackground determination. It can also serve as a check for the FoM curve obtainedfrom solely MC studies. The only difficulty in a faithful comparison is that the ∆Mdata sidebands are completely devoid of true D∗+s candidates, while the rare andcontinuum background MC samples have a non-negligible contribution coming fromdecays with D∗+s , as discussed previously.

• Background MC ∆M sideband regionThis sample provides an intermediate method of comparing the two curves obtainedabove.

Figure 3.15 shows the FoM curves obtained from the three samples for B0 → D∗+s π−

(top row) and B0 → D∗−s K+ (bottom row) signal-regions for the three D+s modes, from

left to right: φπ mode, K∗(892)0K mode and K0SK mode. No significant discrepancy is

totalR0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

# o

f tr

acks

0

0.05

0.1

0.15

0.2

signal

continuum

Continuum Discriminator

Figure 3.14: Likelihood ratio Rtotal distribution for the signal (blue) and con-tinuum (red) events.

70




500 0 -32 070729/1601 20.00 0.000 0.000

0.00 0.40 0.80LR (total)

0

2

4

6

S/s

qrt(

S+

B)

on MC SigBand

on MC SideBand

on data SideBand

B0 → Ds* πφ mode

510 0 -33 070729/1601 20.00 0.000 0.000 520 0 -34 070729/1602 20.00 0.000 0.000



500 0 -32 070729/1638 20.00 0.000 0.000

0.00 0.40 0.80LR (total)

0

1

2

3

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5

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qrt(

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B)

on MC SigBand

on MC SideBand

on data SideBand

B0 → Ds* πK* mode

510 0 -33 070729/1638 20.00 0.000 0.000 520 0 -34 070729/1639 20.00 0.000 0.000



500 0 -32 070729/1548 20.00 0.000 0.000

0.00 0.40 0.80LR (total)

0

1

2

3

4

5

6

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qrt(

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B)

on MC SigBand

on MC SideBand

on data SideBand

B0 → Ds* πKS mode

510 0 -33 070729/1549 20.00 0.000 0.000 520 0 -34 070729/1549 20.00 0.000 0.000



500 0 -32 070729/1708 20.00 0.000 0.000

0.00 0.40 0.80LR (total)

0

1

2

3

4

5

6

S/s

qrt(

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B)

on MC SigBand

on MC SideBand

on data SideBand

B0 → Ds* Kφ mode

510 0 -33 070729/1709 20.00 0.000 0.000 520 0 -34 070729/1709 20.00 0.000 0.000



500 0 -32 070729/1716 20.00 0.000 0.000

0.00 0.40 0.80LR (total)

0

1

2

3

4

5

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qrt(

S+

B)

on MC SigBand

on MC SideBand

on data SideBand

B0 → Ds* KKS mode

510 0 -33 070729/1716 20.00 0.000 0.000 50 0 -34 070729/1716 20.00 0.000 0.000



500 0 -32 070729/1655 20.00 0.000 0.000

0.00 0.40 0.80LR (total)

0

1

2

3

4

5

S/s

qrt(

S+

B)

on MC SigBand

on MC SideBand

on data SideBand

B0 → Ds* KK* mode

510 0 -33 070729/1655 20.00 0.000 0.000 520 0 -34 070729/1656 20.00 0.000 0.000

Figure 3.15: FoM for the combined likelihood, Rtotal for the three D+s modes:

(left) φπ mode, (middle) K∗0K mode, (right) K0SK mode. (Red: using MC signal

band, Blue: using MC sideband, Green: on data sideband)

observed and the three curves obtained are found to be consistent within the statisticalerrors, in all the six cases. As a result, the optimal point is fixed following the curve ob-tained using the background MC in ∆M signal-region (red curve). Table 3.5 summarizesthe optimal points on the FoM curves in the six cases. Figure 3.16 shows the continuum

Table 3.5: The optimal points on the FoM curves for the six cases.

φπ K∗0K K0SK

D∗+s π− 0.45 0.50 0.40D∗−s K+ 0.45 0.60 0.40

background contribution compared to the total after applying the suppression require-ments on Rtotal listed in table 3.5.

A further reduction in both generic BB as well as continuum background can beachieved by realising the fact that the helicity of a D+

s daughter vector meson, such asa φ or a K∗(892)0, decaying to two pseudo-scalars follows a (1 − cos2 θ) angular depend-ence, where helicity is defined as the angle between the momentum of the pseudo-scalarand the direction opposite to the momentum of D+

s in the vector frame of reference. Onthe contrary, a φ or K∗(892)0 meson which is not aD+

s daughter remains uniformly distrib-

71


B0 → Ds*+π-


16 5 1 081018/0659 613.1 -1.2268E-02 0.1163

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Aftere LR cut

16 3 1 081018/0718 137.6 -1.2863E-02 0.1162

Figure 3.16: Continuum background compared to the total after the suppressionrequirement on Rtotal has been applied in case of B0 → D∗+s π− (left) and B0 →

D∗−s K+ (right).

uted on the helicity plots. We require the | cos θφ/K∗(892)0 | > 0.3. This reduction criterionreduces about 23% of the total background in the φ (K∗(892)0) modes, while retains morethan 97% of the signal.

3.3 Signal Extraction Strategy

Equipped with all necessary ingredients required, the decision regarding signal extractionstrategy deferred in section 3.1.3 can now be took up once again.

3.3.1 Mbc

Discussed in previous section with relation to the ∆E fit-region, it is worth recapitulatinghow the signal, if found to be exactly in equal amounts as previous measurements, withall the background components behave on the Mbc fit-region. Figure 3.17 shows the sum-mary of all the events entering theMbc fit-region ofB0 → D∗+s π− (left) andB0 → D∗−s K+

(right) decays. All the distributions are scaled to the real data size in order to draw a morerealistic estimate. The signal branching fraction is assumed equal to the latest Particle datagroup values. As mentioned previously, the signal as well as background PDFs are well-behaved in the Mbc fit-region, due to the Ebeam dominance. In both B0 → D∗+s π− andB0 → D∗−s K+ fit-regions, the respective signal modes show prominent peaking structuresabove the ARGUS-like background distributions. The Mbc appears to be highly promisingvariable for signal extraction, supporting the strategy followed in all previous measure-ments, prima facie. However, though the robustness in parametrization provided by theMbc allows comfortable control over signal PDF, the same is responsible for ill-behavednature of some of the BB and rare decays. Since the shape of signal distribution in Mbc

is dominated by the Ebeam resolution and is insensitive to the exact kinematical details of

72



3.3. SIGNAL EXTRACTION STRATEGY

B0 → Ds* K


16 540 1 070725/1403 746.4 5.248 2.5986E-02

5.200 5.225 5.250 5.275 5.300Mbc (GeV/c

2)

0

20

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60

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c-2 b

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Bkg MC

BBar component

All modes together

16 530 1 070725/1403 617.1 5.242 2.4427E-02 16 520 1 070725/1353 194.9 5.248 2.4278E-02

B0 → Ds* K


16 540 1 070725/1325 303.6 5.250 2.6310E-02

5.200 5.225 5.250 5.275 5.300Mbc (GeV/c

2)

0

10

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Bkg MC

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All modes together

16 530 1 070725/1324 233.3 5.241 2.4120E-02 16 520 1 070725/1315 41.04 5.253 2.4794E-02

Figure 3.17: Expected signal and background distribution in the Mbc fit-regionof B0 → D∗+s π− (left) and B0 → D∗−s K+ (right) decays. The signal distributionis shown in red, while contribution from BB (continuum) events is represented

by the green (blue) histograms.

the signal events, as shown in section 3.1.3, many background events with kinematicalstructures similar to the signal, tend to peak in the signal-region. Note, that in ∆E, anybackground mode which shows a peaking behaviour and has final-state content differentfrom signal is always shifted away from the signal-region. In particular, due to dominantBB events no significant peaking structures were seen in the ∆E fit-region.

The role of reflection D+ events mimicking as a D+s event has been discussed in re-

lation to the ∆E fit-region. Due to their wide-spread smeared-out profile, they did notpose serious trouble then. The situation in respect to Mbc does not appear so promising,depicted in figure 3.18 (right). Since the B0 → D∗+s π− decay is not present in the BBMC, one would not expect any peaking structure around the nominal B mass - a signa-ture of the signal events. However, the green histogram shows a prominent peak in thesignal-region. The reflection background event enter the fit-region through a pion to kaonmisidentification and if this misidentified pion is re-assigned its correct mass, the originalD+ will be retrieved. Figure 3.18 (left) shows the invariant mass distribution for the re-flection background D+

s candidates after re-assigning the pion its correct mass. A peakaround the D+ mass is clearly seen.

To reduce the reflection background, potentially mimicking the signal in Mbc, we

• apply a D+-mass veto: reject events around the nominal D+ mass after correctingthe pion misidentification, and/or

• tighten PID requirement on the track to reduce misidentification probability.

Figure 3.18 (right) shows the effect of applying various combinations of D+-veto and PIDtightening on the BB events. It is evident that the rejection requirement is not 100%effective and the peaking behaviour of the background near the signal-region seems un-

73


/ ndf 2χ 26.61 / 35

Yield 2.86± 27.61

Mean 0.000± 1.868

Sigma 0.000540± 0.005651

Back1 21.8± 118.9

Back2 11.55± -62.34

)2 (GeV/c+DM1.84 1.86 1.88 1.9

# of

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40

/ ndf 2χ 26.61 / 35

Yield 2.86± 27.61

Mean 0.000± 1.868

Sigma 0.000540± 0.005651

Back1 21.8± 118.9

Back2 11.55± -62.34

/ ndf 2χ 26.61 / 35

Yield 2.86± 27.61

Mean 0.000± 1.868

Sigma 0.000540± 0.005651

Back1 21.8± 118.9

Back2 11.55± -62.34

+K0(892)*

K → +sD

+ D→ +π0(892)*

K

Distribution+DM

-π*+s D→0B

)2 (GeV/cbcM5.2 5.22 5.24 5.26 5.28 5.3

)2#

of E

vent

s/ (

2.5

MeV

/c

0

2

4

6

8 No cut

+D veto on M215 MeV/c

veto + KID > 0.827.5 MeV/c

MC: 4 StreamsBB

DistributionbcM

-π*+s D→0B

Figure 3.18: (left) The invariant mass distribution for the reflection D+ eventsmimicking a D+

s , after correcting pion-kaon misidentification. The blue-solidcurve shows the overall fit, while the red curve signifies theD+ mass peak. (right)

Effect of applying D+-veto on the BB background in Mbc fit-region.

avoidable. In general, it is possible to subtract the peaking background in the signal-region, provided one can determine the yield and PDF precisely. However, this method isnot very desirable,

• if the background information is solely determined from MC and not from the realdata, or

• if there exist yet unknown or known but unmeasured background modes.

In conclusion, Mbc offers a very robust PDF parametrization for signal as well as mostpart of the background events. Although, it tends to overestimate signal yield, due to someill-behaved background modes mimicking the signal.

It is worthwhile however, to confirm the completeness of our understanding about thesignal and all the background entering the fit-region, by reproducing results obtained inthe previous measurements. Since, the previous measurements [45, 46] have been carriedout using Mbc variable, we further push this study to quickly perform a consistency check,by

1. obtaining the PDFs for signal and background Mbc distributions,

2. correcting the PDFs determined from MC using a suitable control sample, and

3. perform a fit to the real data in order to obtain the branching fractions

Since all these steps will be discussed at length in latter sections in relation with the ∆Esignal extraction method, we summarise the main findings of our study using the Mbc

technique.Figure 3.19 shows the fit done to Mbc distribution for the signal events in φπ mode.

In case of multiple B candidates satisfying the signal selection criteria (see section 3.3.2),only the candidate with the least absolute value for ∆E is chosen. For further analysis, we

74


Figure 3.19: (left) Mbc distribution for B0 → D∗+s π− signal events in φπ mode.(right) Parameters for the PDF in Mbc for signal events.

consider only those events which lie inside the ∆E signal-region (roughly defined to be|∆E| < 35 MeV).

The signal peak is fitted with a Gaussian with an ARGUS function [62] representing thebackground underlying. The mean and spread for the signal PDF, namely the Gaussian,are found to be 5.2801 ± 0.001 GeV/c2and 2.74 ± 0.01 MeV/c2, respectively. The mostprobable value for Mbc, which also coincides with its mean, signifies the nominal B mass,while the width is due to the Ebeam resolution, as discussed in section 3.1.3.

The efficiencies and PDF parameter values determined using signal MC may, in gen-eral, differ from those for real data, depending upon how well MC samples represent thedata. These values are corrected using B0 → D∗+s D− control sample in real data. Due toits kinematical similarity with B0 → D∗+s π− and B0 → D∗−s K+ decays, the Mbc distribu-tion for the control events is expected to be very close to that in signal events. Yet anothersource of discrepancy between the efficiencies for MC and real data comes from differentPID efficiencies for each track. These are corrected by usingD∗+ → D0(K−π+)π+ control,where the kaon or pion track can be identified without invoking PID (see Belle Note 779).Table 3.6 summarizes the signal efficiency (ε), expected signal (Nsig) and background(Nbkg) yields in Mbc signal-region. The background is parametrized as an ARGUS func-tion [62], with the yield and shape allowed to vary. The BB and rare background modeswhich show peaking behaviour are parametrized as additional Gaussian functions withboth yield as well as shape fixed to their value obtained in MC studies.

Figure 3.20 show the result of final fits done to the real data in B0 → D∗+s π− (top)and B0 → D∗−s K+ (bottom) fit-regions. The fit is performed simultaneously to the threeD+s modes in each signal mode. Table 3.7 enlists the corresponding observations. In order

to check the consistency in the simultaneous-fit procedure, each sample is also fitted indi-vidually. The obtained branching fractions are consistent with the previous measurements,implying completeness of our understanding of the background as well as signal processesanalyzed in the Mbc fit-region, with respect to the previous measurements. Though con-sistent with the previous estimates, the method used is not fully reliable and hence, we donot pursue it further.

75



Table 3.6: Signal Efficiency (ε), signal (Nsig) and background (Nbkg) yields inMbc signal-region.

B Mode D+s mode ε(%) Nsig Nbkg

D+s → φπ+ 12.11± 0.11 45.9± 0.4 27± 1

B0 → D∗+s π− D+s → K∗0K+ 6.90± 0.09 29.8± 0.4 38± 2

D+s → K0

SK+ 7.07± 0.09 31.6± 0.4 22.5± 0.7

D+s → φ π+ 10.91± 0.12 29.7± 0.3 11.6± 0.6

B0 → D∗−s K+ D+s → K∗0K+ 5.98± 0.08 18.4± 0.3 9.9± 0.8

D+s → K0

SK+ 6.10± 0.09 20.3± 0.4 10.7± 0.8

)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

ts /

( 0.

0025

GeV

/c

0

2

4

6

8

10

12

14

16

18

20

22 3.3±BR = 22.8

11±argPar_0 = -23.6

8.4±argPar_1 = -37.09

13±argPar_2 = -4.7

12±nBkg_0 = 138

15±nBkg_1 = 219

9.7±nBkg_2 = 91.6

)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

ts /

( 0.

0025

GeV

/c

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22

)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

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0025

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/c

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24

)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

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0025

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/c

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)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

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0025

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/c

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)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

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( 0.

0025

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/c

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)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

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( 0.

0025

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/c

0

2

4

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8

10

12

14

16

18 3.0±BR = 18.1

16±argPar_0 = -17.2

16±argPar_1 = -48.2

19±argPar_2 = -37.7

8.7±nBkg_0 = 66.1

8.4±nBkg_1 = 61.1

6.7±nBkg_2 = 40.5

)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

ts /

( 0.

0025

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/c

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)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

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0025

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/c

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)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

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( 0.

0025

GeV

/c

0

2

4

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)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

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( 0.

0025

GeV

/c

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2

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)2 (GeV/cbcM5.2 5.215.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.3

)2E

ven

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( 0.

0025

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/c

0

2

4

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8

10

Figure 3.20: Fit to the real data in B0 → D∗+s π− (top) and B0 → D∗−s K+

(bottom) Mbc fit-regions. We fit the data simultaneously in the φπ mode (left),the K∗(892)0K mode (middle) and the K0

SK mode (right). The red curves rep-resent signal, dotted-blue the combinatorial background and the magenta curvesrepresent modes which show peaking behaviour (fixed to yields and PDFs from

MC).

76


Table 3.7: Results of the simultaneous as well as individual fits to the three D+s

modes in B0 → D∗+s π− and B0 → D∗−s K+ decays. The first uncertainty in thebranching fractions is statistical, the second comes from experimental uncertain-ties and the third comes from uncertainties in the D+

s decay branching fractions.Only statistical errors are quoted for the individual 1-D fits.

B Mode D+s mode

Signal Yield Efficiency Significance branchingNsig ε(%) Σ fractions (10−5)

B0 → D∗+s π−

simultaneous . . . . . . 11 2.28± 0.33± 0.13± 0.19D+s → φπ+ 30± 7 12.01 4.2 1.7± 0.4

D+s → K∗0K+ 45± 10 6.46 4.7 4.2± 0.9

D+s → K0

SK+ 25± 6 7.03 4.1 2.6± 0.7

BaBar [46] . . . . . . 6.8 2.6+0.5−0.4 ± 0.3

B0 → D∗−s K+

simultaneous . . . . . . 9.9 1.81± 0.30± 0.12± 0.18D+s → φπ+ 34± 7 10.62 5.4 2.2± 0.4

D+s → K∗0K+ 21± 6 5.44 3.6 2.5± 0.7

D+s → K0

SK+ 11± 4 5.86 2.7 1.3± 0.5

BaBar [46] . . . . . . 7.4 2.4± 0.4± 0.2

3.3.2 Mbc-∆E 2D fit

Due to the peaking behaviour just under the signal peak, Mbc is found to be unreliable.Due to its insensitivity to the reconstructed momentum, Mbc can not effectively discrimin-ate signal from signal-like events. ∆E on the contrary, shares exactly this characteristics,as seen earlier. Combining these two variables an effective signal-to-background separa-tion with least possible uncertainties. Consequently, a 2-D fit can yield observations withhigher significance.

We need to analyse signal and background events on the ∆E-Mbc 2-D plot. Note, thatthese events were studied in the two fit-regions independently: studying events in the fit-regionof one, while constraining them to come from the signal-region of the other. The signal eventshave already been visited once on the ∆E-Mbc plot as shown in figure 3.8. For a detailedstudy we again examine the signal events which satisfy all the selection criteria and fallinside the ∆E-Mbc fit-region.

While generating the signal MC, only one of the B mesons from the Υ(4S) decay isfurther decayed into the signal mode4 and hence, each generated entry in signal MC hasonly one signal decay event. If the selection criterion allows for 100% purity - if it selectsonly a correct signal event every time, rejecting any background with 100% efficiency- then each signal MC entry would contribute a single B-meson to the ∆E-Mbc plot.However, in reality 100% background rejection can not be achieved, due to non-idealbehaviour of the detectors and limited efficiency of PID and the signal purity obtained isnever 100%. In effect, one finds more than one B-mesons satisfying the selection processand falling inside the ∆E-Mbc fit-region, in some of the reconstructed events. In suchcases, choice of the signal B-meson among these multiple candidates is ambiguous within

4the other is allowed to decay generically into all possible decay modes of B, which are included in thestandard decay procedure - used to generate BB MC samples.

77


the existing selection process and further technique is required to remove the ambiguity,whenever needed. It is evident, that any additional procedure developed to remove thisambiguity will not necessarily be 100% efficient itself and can cause further loss in thesignal efficiency. In cases, where branching fraction measurement is not needed, i.e. CPanalyses or if the fraction of events with multiple candidacy is low compared to othersources, which can add huge uncertainties in the measurement, one can decide to acceptall the B candidates for further analysis. This prevent further loss of signal efficiency. Theuncertainty due to multiplicity can either be ignored completely or can be added to theoverall error a posteriory. However, in cases where signal as well as background statisticsis limited and the errors introduced due to multiple candidacy can not be ignored - asin case of rare B decay searches, to produce the statistical errors correctly removing themultiplicity prior to carrying out measurement is required.

Multiple Candidates

Poor photon identification and the huge low momentum photon background are the mainsources of large multiplicity observed in decays involving D∗+s meson. Figure 3.21 showsthe multiplicity observed in the ∆E-Mbc fit-region of B0 → D∗+s π− decays: in φπ mode(left), in the K∗(892)0K mode (middle) and the K0

SK mode (right). In about 15% of the

# of B / event0 1 2 3 4 5 6 7 8 9 10

# of

eve

nts

0

2

4

6

8

10

12

14

16

310×

modeφ

ave. multi = 1.17 (15% events)

# of B / event0 1 2 3 4 5 6 7 8 9 10

# of

eve

nts

0

1

2

3

4

56

7

8

9

310×

mode*0

K


# of B / event0 1 2 3 4 5 6 7 8 9 10

# of

eve

nts

0

1

2

3

4

5

6

7

8

9310×

mode0SK


-π*+s D→ 0B multiplicity : B

Figure 3.21: Number of B mesons observed per event inside the ∆E-Mbc fit-region of the B0 → D∗+s π− decays in three modes: (left to right) φπ mode, theK∗(892)0K mode, and the K0

SK mode. A similar proportion is observed also forB0 → D∗−s K+.

events, more than one B mesons are found inside the fit-region. The average multiplicity,i.e. average number of B mesons in an event, is observed to be about 1.17. The similaramounts of multiplicity observed in the three D+

s modes is a consequence of the fact thatthe photon from D∗+s is the cause of it.

Among the multiple B events, the one which has properties closest to that shown by ageneric signal decay in ideal conditions is usually treated as the best choice and is retainedfor further analysis. This is achieved through constructing a χ2-like variable based on all orsome of the measured quantities used in reconstructing the signal event, which measuresthe deviation of the observed values from the nominal or theoretical expectations andchoosing the one with minimum χ2 value. The choices available in this case are (a) theinvariant masses of φ, K∗(892)0, K0

S , D+s and D∗+s meson, (b) PID values for the tracks,

78


and (c) the value of ∆M . The ∆E orMbc values can not be added to this list, since they areto be used for extracting signal branching fraction measurements and selecting candidatesfavoring some specific values can bias the sample with respect to these fit variables. Also, asimultaneous choice of D+

s mass, D∗+s mass and ∆M can turn out to be redundant. Sincethe multiplicity comes from the photon background, a χ2 designed to optimally select thecorrect photon candidate, and hence the correct B meson, is desired. Non of the variables,except D∗+s mass or ∆M satisfies this requirement. Figure 3.22 shows the ∆M -∆E 2-Ddistribution for signal events. The two variables have a significant correlation, since after

)2M (GeV/c∆0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2

E (

GeV

)∆

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2M∆E - ∆

(Correlation = 2.80e-01)

Covariance = 1.81e-04

Figure 3.22: ∆M against ∆E distribution for the signal events. The two vari-ables can be seen to be correlated.

the D+s is kinematically mass-constrain-fitted it is the same photon which dominates the

shapes of ∆M and ∆E. Including the mass difference ∆M into the χ2 variable thusbrings in a bias in the measurement signal via this correlation and we do not include themass difference ∆M into the χ2 for the best candidate selection. On the other hand, anyother variable defined in order to include D∗+s qualitatively into the χ2 is not found to bevery efficient, which tends to compensates the gain in significance achieved in a 2-D fitcompared to a 1-D fit. As a result, though desirable, we do not find this strategy muchadvantageous over doing a simpler 1-D fit to the ∆E.

3.3.3 ∆E

The main advantage of performing a 1-D fit to the ∆E is that now Mbc can be usedto remove the ambiguity in B candidate, without having to invoke any other χ2. Mbc

is almost uncorrelated to ∆E and is found to be 92% efficient in selecting the true Bcandidate among the multiple choices. Among the multiple candidates, the one with theMbc value closest to the nominal B mass of 5.28 GeV/c2 is chosen, rejecting the others.Only events in the Mbc signal-region, i.e. with Mbc between 5.27 GeV/c2 and 5.29 GeV/c2

are considered for further analysis. Figure 3.23 shows fit to the ∆E distribution in B0 →D∗+s π− (left) and B0 → D∗−s K+ (right) fit-regions in the φπ mode.

79


-0.20 -0.10 0.00 0.10 0.20∆E (GeV)

0

1000

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4000

5000

# of

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MINUIT Likelihood Fit to Plot 16&1B0 → Ds*+π-

File: *dsstpi-sb0.hbk 20-SEP-2008 23:34Plot Area Total/Fit 14720. / 14720.Func Area Total/Fit 14709. / 14709.


Likelihood = 141.7χ2= 148.3 for 40 - 5 d.o.f., C.L.=0.673E-13%Errors Parabolic MinosFunction 1: CB Line ShapeAREA 13118. ± 127.2 - 0.000 + 0.000MEAN 2.25663E-03 ± 1.0811E-04 - 0.000 + 0.000SIGMA 1.08165E-02 ± 1.0000E-04 - 0.000 + 0.000ALPHA∗ 2.0000 ± 0.000 - 0.000 + 0.000N∗ 1.4000 ± 0.000 - 0.000 + 0.000Function 2: Gaussian (sigma)AREA 1936.7 ± 65.85 - 0.000 + 0.000MEAN∗ 0.0000 ± 0.000 - 0.000 + 0.000SIGMA 8.07865E-02 ± 2.2498E-03 - 0.000 + 0.000

φ mode

-0.20 -0.10 0.00 0.10 0.20∆E (GeV)

0

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*-K+

File: *dsstK-sb0.hbk 20-SEP-2008 23:41Plot Area Total/Fit 13560. / 13560.Func Area Total/Fit 13549. / 13549.


Likelihood = 149.4χ2= 158.3 for 40 - 5 d.o.f., C.L.=0.132E-14%Errors Parabolic MinosFunction 1: CB Line ShapeAREA 12110. ± 111.7 - 0.000 + 0.000MEAN 2.28627E-03 ± 7.7987E-05 - 0.000 + 0.000SIGMA 1.05687E-02 ± 9.8611E-05 - 0.000 + 0.000ALPHA∗ 2.0000 ± 0.000 - 0.000 + 0.000N∗ 1.4000 ± 0.000 - 0.000 + 0.000Function 2: Gaussian (sigma)AREA 1754.5 ± 64.98 - 0.000 + 0.000MEAN∗ 0.0000 ± 0.000 - 0.000 + 0.000SIGMA 8.06433E-02 ± 2.6186E-03 - 0.000 + 0.000

φ mode

Figure 3.23: ∆E distribution for the B0 → D∗+s π− (left) and B0 → D∗−s K+

(right) events in the φπ mode.

Signal PDF

The signal is fitted with a Crystal Ball (CB) function [67] and a Gaussian with a broadwidth. The CB function is defined as

f(x) =

(N|α|

)Nexp−

1s α

2(N|α|−|α|−x

)N . . . x < −|α|

exp(−1

2

(x−ms

)2). . . x > −|α|

(3.9)

The kinematical fits performed to D+s and D∗+s allows us to define a single PDF with

common parameter values of the CB for the three D+s modes. The parameters α and N

of the CB, signifying the extent of the central Gaussian and the order of the polynomialaccounting for the tail on the lower ∆E values, respectively, are fixed to the values whichachieve the best possible fits to the distributions5. The ratio of area under the CB to thetotal under CB and Gaussian is always found to be closer to 87% and to maintain a minimalset of free parameters we fix this fraction to the observed value in MC. Table 3.8 showsthe parameters of the signal PDF.

Reconstruction Efficiencies

It was mentioned previously that the total data intended to be used in the obtaining thebranching fraction measurements has been collected over years of experimental runs. Dur-

5It has been observed that the Crystal Ball function is highly sensitive to α and N variations making thefit tougher to converge. Also, it is difficult to predict their most probable value just on physics principles.

80


Table 3.8: The signal PDF parameters obtained from MC.

ValueParameter

B0 → D∗+s π− B0 → D∗−s K+

CB line-shape

Mean (MeV) 2.4± 0.1 2.37± 0.12σ (MeV) 10.9± 0.1 10.9± 0.1α 2.0 2.0N 1.4 1.4

GaussianMean (MeV) 0.0 0.0σ (MeV) 78± 1 82± 2

Total CB Fraction 0.87 0.87

ing this period the beam conditions as well as detector performance can not be expectedto be constant. At the same time, attempts to improve the quality of the collected datais continuously made by upgrading the experimental setup or applying analysis softwarepatches. As a consequence the reconstruction efficiencies varies with every experimentalrun. The correct effective reconstruction efficiency to be used while using the whole data-set is then either a weighted average of the efficiencies obtained from signal MC samplesgenerated for each experiment separately, or obtained from a single MC sample, whichcontains data generated in all experiments according to these weights. In both these,weights are the shares from each experiment to the total luminosity. In principle, both

expN10 20 30 40 50

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%)

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weighted signal MC

weighted average

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Figure 3.24: Variation in reconstruction efficiency for B0 → D∗+s π− decay inφπ mode with experimental runs. The red hatched histogram shows efficiencydetermined from a single weighted signal MC, while the red line indicates the

weighted average of the distribution.

81


these approaches are equivalent. A third, somewhat crude, approach is to select two ex-periments each from SVD1 and SVD2 and weigh the two efficiencies according to SVD1and SVD2 share of the total luminosity. Figure 3.24 demonstrates the efficiency variationwith the experimental runs at KEKB for B0 → D∗+s π− decay in the φπ mode. The hatched-histogram shows the value for the efficiency obtained from a single MC sample, generatedwith proper weights in each experiment. The value obtained so is found to be consistentwith a weighted average of the individual efficiencies, represented by red line. We use theformer approach to obtain signal reconstruction efficiencies summarized in table 3.9.

Table 3.9: Signal Efficiencies.

D+s mode

Efficiency (%)

B0 → D∗+s π− B0 → D∗−s K+

D+s → φπ+ 14.07± 0.19 13.55± 0.18

D+s → K∗0K+ 8.29± 0.15 7.05± 0.13

D+s → K0

SK+ 8.07± 0.15 7.33± 0.13

82

4Analysis:

MC to data

10 cm

BELLE The techniques to translate the study performed on the MC samples in theprevious chapter to a more realistic form, which can then be used directly toanalyze the real data, are discussed in this chapter. This involves methodsof validating background shapes and correcting the parameters fixed insignal as well as rare B background PDFs. An effort is made to correctvarious reconstruction efficiencies determined from the MC samples.

4.1 Data Driven Techniques

OUR study has so far been limited to MC samples generated according to the currentunderstanding of the physical processes governing various decays and the detector

performances. As previously pointed out, MC samples allow one to have deeper under-standing of various physical processes by offering separability according to their origin andhence by allowing one to concentrate on one process, while effect of any other process be-ing muted. However, their viability is bound not only by limitations in current knowledgeor modeling as well as computational errors but sometimes also plagued by human errors.The inferences drawn in the previous chapter on the basis of observations of MC studiesneed to be verified as well as consolidated prior to extrapolating to the real data analyses.

Large background MC samples have been used to determine shapes of various back-ground processes; especially, to parametrize the distributions which have peaking struc-tures in the fit-region. A discrepancy between these processes generated in MC samplesand observed in real data can lead to erroneous results. Similarly, an error in the signal

83

ANALYSIS: MC TO DATA

MC samples used for parametrizing the signal can introduce bias in the measurement. Inaddition, a correct representation of both signal as well as background shapes is crucial inorder to obtain the correct optimal point on the FoM curves discussed in the last chapter.The other source of defect can come from the efficiencies determined from MC samples, incase of signal and rare modes. The measures taken to validate and correct these factors,to be used for carrying out the signal branching fraction measurements in real data, arediscussed here.

4.2 Data Sideband studies

The most convenient approach of testifying the background shapes and yields is to choosesome region in data, where all background processes contribute but no signal events areexpected to be present. For example, since a signal event is always expected to have a D+

s ,if the data is chosen from a region which lies far away, say 5σ, from the nominal D+

s mass,it will not contain any signal event. This region is called a D+

s -sideband. Though it quali-fies for the lack of signal, a D+

s sideband can not truly represent the complete backgroundunder the signal in ∆E, since continuum background and many BB processes with a D+

s

or a D∗+s present will not be available in the sideband. And, hence D+s -sideband does not

prove to be a good choice. Though desirable, as in this example, it is usually unfeasible tofind a region in data, which will be completely stripped-off signal events without loosingany of the background events and hence, one is always forced to compromise with a choiceof variable to define the sideband, which will have least possible loss in the background.Also, in order to have a reasonable comparison the size of a sideband must be chosen suchthe background yield approximately matches with that in the signal region.

4.2.1 ∆M sidebands

The signal events satisfy 128 MeV/c2 < ∆M < 162 MeV/c2 as shown in figure 4.1 andhence the data chosen from outside this region can be treated as a ∆M -sideband. To

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Figure 4.1: The ∆M sidebands defined. Distributions shown are from signal MC(left) and in the real data (right)

84

4.2. DATA SIDEBAND STUDIES

maintain the background levels equal to that under the signal peak, four statistically ex-clusive sideband samples are chosen: Left (94 MeV/c2 < ∆M < 128 MeV/c2), Right1(162 MeV/c2 < ∆M < 196 MeV/c2), Right2 (196 MeV/c2 < ∆M < 230 MeV/c2), Right3(230 MeV/c2 < ∆M < 264 MeV/c2) as depicted in figure 4.1.

While analysing the data in the sidebands, the reconstructed D∗+s candidates (mostlyfake) are kinematically mass-contra in-fitted to the central values of respective sidebands,mimicking the procedure in signal-region events. The slope of the ∆M distribution is usedto normalize the statistics in each sideband to that in signal-region. It must be noted, thatthe ∆M -sidebands are devoid of any process involving a D∗+s .

Figure 4.2 shows comparison between the background predicted by MC samples withthose in the ∆M -sidebands. The blue points with statistical errors represent the data,

Ds*+π-: Data - MC comparison


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Figure 4.2: Comparison between ∆E fit-region backgrounds in MC samples(red) and in data ∆M -sidebands (blue) in B0 → D∗+s π− (top row) and B0 →D∗−s K+ (bottom row). The solid-filled histogram signifies contribution from the

rare B decays.

while the red histogram is the background prediction from the MC samples. The contribu-tions due to rare B decays is shown in solid-filled histogram. The ∆M -sidebands show asystematically underestimated background level due to the absence of the processes witha D∗+s meson. The effect is more pronounced in the φπ mode, where the signal efficiency

85


(and equivalently the D∗+s reconstruction efficiency) is highest. No harmful peaking struc-ture, other than those already predicted by the MC samples, is observed in the sidebands.Note, the apparent peaking structure seen in the φπ mode of B0 → D∗+s π− decays is dueto an accidental fluctuation in the continuum MC sample and not a systematic effect.

In conclusion,

1. The ∆M -sidebands show a systematic underestimation of the background, which isattributed to the processes with a D∗+s , present in the MC samples but absent fromthe data sidebands. Particularly, in continuum samples about 10% of the eventscontain a D∗+s candidate.

2. After accounting for the mis-match due toD∗+s and allowing for a 10% disagreement,as mentioned in section 2.3.2, between the data and MC, the background yields areobserved to be consistent within statistical errors.

3. More importantly, we do not observe any fatal peaking structures in data, not pre-dicted by MC samples.

4.2.2 Mbc sidebands

To include processes with a D∗+s , especially from the continuum background, we alsostudied Mbc-sidebands. These sidebands, on the other hand, lack those processes whichtend to peak under the signal in Mbc. The data events with Mbc values between 5.20GeV/c2 and 5.26 GeV/c2 are selected for this purpose. This region is divided into threeslices of widths equal to the signal-region and in each of them, among the multiple Bcandidates, one with Mbc closest to the center is retained (this mimics the best candidateselection procedure in the signal region). We normalize each sideband to the statistics inthe signal-region.

The background distributions from MC samples in the ∆E fit-region are comparedwith the Mbc-sidebands in figure 4.3. It can be inferred, that

1. Mbc-sidebands show statistics consistent with that in MC samples,

2. no systematic underestimation is observed in the data, though the effect of modeswhich peak under the signal in Mbc is slightly lower in the sidebands,

3. again, no additional peaking structure is been observed.

4.2.3 Off-resonance data studies

The validity of the optimal point on the Rtotal FoM curves in figure 3.15 with respect tothe ∆M data sidebands have already been tested. The green curves in these plots cor-respond to the FoM curves obtained using data sidebands. It is inferred from the sameplots, that within the statistical uncertainties, the three curves on each plot are consistentand an optimal point is chosen on the FoM curves based on MC samples (red curves infigure 3.15)). To reaffirm the correctness of the above inference, which is based on the as-sumption that theRtotal calculated using continuum MC samples works for the continuumevents in the real data too, it is necessary to analysis a real data sample, which dominantly

86

4.2. DATA SIDEBAND STUDIES



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Figure 4.3: Comparison between background from the data Mbc-sidebands(blue) and MC (red) samples in B0 → D∗+s π− (top rows) and in B0 → D∗−s K+

(bottom row). The solid-filled histogram signifies contribution from the rare Bdecays.

contains only continuum e+e− → qq events. This is achieved the most easily by analyz-ing data in the Υ(4S) off-resonance region, or equivalently Υ(4S) sidebands. Table 2.2in chapter 2 summarizes the size of the off-resonance data collected at KEKB. Figure 4.4shows the Rtotal distribution for the off-resonance data. It can be clearly seen that theRtotal performs equally well for the off-resonance data events and does not hint towardany discrepancy between MC continuum samples and real data continuum samples. Theoff-resonance data collected is only 10% in size compared to the real data, as opposed tothe continuum MC sample which contains statistics 5 times that in real data and hencecan not itself be used for the optimization purposes.

From the overall study performed on the ∆M - and Mbc-sidebands, it can be concludedthat the MC samples used for background study are sufficient for accounting qualitativelyfor any possible background structure one would expect under the signal peak. On top ofthis, the off-resonance data study supports the validity of the optimization. And hence, a

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Figure 4.4: Rtotal distribution for the off-resonance data.

reasonable signal extraction procedure - a fitting algorithm - can now be devised based onMC studies. The same observations, however, can not be used to make any quantitativeestimates. In order to be able to fix the shapes and yields of the background components,which has prominent peaking structures, an independent approach must be followed.

4.3 Fitting Algorithm for signal Extraction

Based on our knowledge of various background processes studied a procedure to extractsignal by fitting the data in ∆E fit-region can now be devised. It is worthwhile summar-izing the key observations from the MC study which would serve as the guiding principlesbehind constructing the fitting algorithm.

• Signal PDF is a CB and a Gaussian (see section 3.3.3). A common PDF for three D+s

modes is used. PDF parameters are fixed to their values obtained from signal MCsamples and need to be corrected to be useful in real data.

• Dominant BB processes do not show any peaking behaviour and are included inthe combinatorial (flat) background. The sidebands validate this prediction qualitat-ively. The yields and shapes (slopes of the linear functions) are not known precisely.(sections 3.2.1)

• Rare BB processes with a true D+s or a D∗+s , which show prominent peaking struc-

tures are accounted for by adding fixed-yield, fixed-shape PDFs. The yields andshapes, fixed from MC samples, need to be corrected prior to fit the real data. (sec-tion 3.2.2)

• Signal cross-feeds are accounted by fixed-shape PDFs. The yields are proportionalto the branching fractions which are to be measured in the fit. (section 3.2.2)

• continuum processes are included in the flat combinatorial distributions.

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4.3. FITTING ALGORITHM FOR SIGNAL EXTRACTION

4.3.1 Method

If the B0 → D∗+s π− signal yields in the three D+s modes were completely uncorrelated

with each other, one would be doing three individual fits to these three statistically inde-pendent samples and obtaining three distinct signal branching fractions. On the contrary,occurrence of the signal events in the three modes is consequence of a single physical fact,existence of B0 → D∗+s π− decay and the three branching fraction measurements corres-pond to the single physical quantity: branching fraction for B0 → D∗+s π−. Hence, thethree D+

s samples can not be treated as statistically independent and it is customary toperform a fit simultaneously in the three samples, with a single free parameter measuringthe branching fraction of B0 → D∗+s π− decay. The same is true for the B0 → D∗−s K+

decay. Following a more conventional path, it is possible to do three individual fits tothe three D+

s samples and take the weighted average to obtain signal branching fraction.However, in a simultaneous fit number of parameters with which the fit is to be performedcan be reduced, utilizing correlations among various physical quantities. This can be real-ized via an effective gain (a) in the statistical significance, (b) more stability in the fitconvergence, and (c) less sensitivity to systematic uncertainties pertaining only to one ofthe three D+

s modes, but not to all. As a result, we perform a simultaneous fit in the threeD+s modes for the same signal mode.

Due to the prompt track misidentification, the two signal modes cross-feed each othersfit regions, as discussed in section 3.2.2. Due to the prompt pion misidentified as a kaon,some fraction of the B0 → D∗+s π− signal disappears from the B0 → D∗+s π− signal-regionand appears as a background in the B0 → D∗−s K+ fit-region and vice verse. The amountof B0 → D∗+s π− cross-feed into the B0 → D∗−s K+ fit-region depends upon the pionmisidentification probability or pion fake-rate and the signal yield of B0 → D∗+s π−. Asa result, the cross-feed background in one’s fit-region is proportional to the signal yieldin other’s signal-region. Hence, we do a simultaneous fit to the two signal modes, B0 →D∗+s π− and B0 → D∗−s K+, where cross-feed background in one is proportional to thebranching fraction of the other, and vice verse.

All the plots studied in the previous chapter (except in figure 3.20) were produced byappropriately binning the data-set in the respective variables. Representing data this wayhas a strong dependence on the binning details: a coarse binning tends to flatten any stat-istical variation, however significant, while a super-fine binning blows up stochastic fluc-tuations. To surpass this binning-sensitivity and subsequent uncertainties in the fit-results,we adopt the unbinned maximum-likelihood technique [68, 69]. Using Baysian approach,if some test statistic is collected with a priori faith of P(theory), then the a posterior prob-ability P(theory/data) for how much the collected statistic supports the faithfulness of theassumed theory can be calculated as,

P(theory/data) = P(data/theory)× P(theory) (4.1)

where, P(data/theory) is called the likelihood of obtaining the statistic if assumed theorywas true. It is clear from equation (4.1), how crucial the likelihood of the data is inconfirming or rejecting the a priori faith in theory, achieved through all theoretical as wellas previous experimental attempts made. With given theory, a unified PDF P, composed ofPDFs for signal and background, can be designed, which in turn determines the likelihood

89


as,L(data) = P(data/theory) =

∏i

Pi (4.2)

where, i runs over all the data-points collected. If the PDF has free parameters, the best-fitto the data is obtained by maximizing the likelihood with respect to the free parameters.Because logarithm is a monotonous function of its argument, the maximization of log-likelihood is scanned, which is easier to handle and usually offers faster and more stableconvergence of the fit.

In addition, since the whole data-set, irrespective of its distribution among variousyields for signal and backgrounds, should correspond to the total of 657× 106 BB eventscollected throughout the experiment: not all the yields are statistically independent. Thisis achieved by performing an extended maximum-likelihood fit.

In short, an extended unbinned maximum likelihood fit is performed simultaneouslyover six statistically exclusive samples: three D+

s modes in each of B0 → D∗+s π−andB0 → D∗−s K+.

4.3.2 Number of fit-parameters

• The background processes, which fail to show any peaking structure are represen-ted by linear functions with slopes and yields allowed to vary. There are six suchfunctions and hence 12 free parameters.

• The backgrounds with peaking structures are not allowed to vary, but the shapes andyields (via reff ’s as defined in equation (3.5)) are fixed to the values obtained fromMC samples. These parameters need to be corrected for data.

• The cross-feed backgrounds are parametrized with models obtained from the MCsamples and need to be corrected for data. The yields are not fixed, but allowed tovary in proportion to their branching fractions obtained from the fit.

• The signal is modeled according to the MC sample observations and again need tobe corrected. The yields in all the three D+

s modes in a signal mode correspondto a single free parameter: signal branching fraction. Overall, there are two freeparameters: the two signal branching fractions.

In summary, we have 14 free parameters in the overall fit.What remains unresolved now is to formulate a procedure by which various PDF para-

meters as well as efficiencies fixed from MC samples can be correctly translated to the realdata.

4.4 Control studies

To make quantitative estimates about possible differences existing between MC predic-tions and the real data, one chooses to perform a dummy measurement - in present casebranching fraction measurement - of some well-known B decay, in the same MC and datasamples. The B decay process to be used as a control sample is chosen such that,

90

4.4. CONTROL STUDIES

• being a Cabibbo-favored decay, sizable statistic is available in the fit-region, allowingprecise measurements with small statistical uncertainties,

• all input parameters, including the one being estimated, are known precisely in MCsamples as well as in real data, reducing the systematic uncertainties.

• has considerable kinematical overlap with the actual signal mode, so that the sameparametrization, used for modelling signal PDF, is applicable.

Under such controlled conditions, if some parameters are allowed to vary, while keepingothers fixed, in the fits to MC sample and data, the differences in the values for theseparameters can be obtained directly from the fit-results.

4.4.1 Correcting for PDF parameters: Fudge factors

As mentioned in section 3.2.2, care has been taken while choosing a PDF model for eachbackground mode with peaking structure, such that the PDF parameters play roles sim-ilar to those played by the mean and the width of a Gaussian. Also, because all suchbackgrounds are observed to be processes with a D+

s or a D∗+s , systematic shifts occurredduring MC generation to their attributes, like means and widths of the distributions, areexpected to be nearly equal. This applies also to the signal PDFs and hence we use thesame correction factors, termed as fudge factors, for all the PDFs, including signal PDFs.

We perform branching fraction measurement of B0 → D∗+s D− decays as a control-study. This process is Cabibbo-favored with a D∗+s . Being a b → c transition, sizablesignal is available in real data and MC samples and due to the D∗+s , the PDF in ∆E isexpected to share structural similarities with that of B0 → D∗+s π− and B0 → D∗−s K+

decays. Also, it requires minimal amendments to the signal selection criteria developedfor B0 → D∗+s π− and B0 → D∗−s K+ decays before applying to reconstruct B0 → D∗+s D−

signal events1. The same fit-procedure is employed which is to be used for B0 → D∗+s π−

and B0 → D∗−s K+ branching fraction measurements, except now we do not have twosignal modes cross-feeding each other and the mean and width for the CB line-shape,representing central peak of the signal PDF, are allowed to vary. The differences in theirfit-result values are used as the fudge factors for correcting the PDFs in B0 → D∗+s π− andB0 → D∗−s K+ branching fraction measurements.

Figure 4.5 (left) shows distribution of all the BB decay events entering the ∆E fit-region. It can be seen that not only the signal but the most prominent background pro-cesses show structural similarities with those seen in the fit-regions of B0 → D∗+s π− andB0 → D∗−s K+ decays. This is an additional advantage of using the B0 → D∗+s D− controlsample: the same fitting procedure developed for B0 → D∗+s π− can be used. From fig-ure 4.5 (right) it is evident how well the MC samples reproduce the expectations in realdata, considering the scope allowed by the small statistical fluctuations, in this case.

Simultaneous Fit

Figure 4.6 shows fits performed to the MC samples (top) and the real data (bottom). TheMC samples used in this study contained 5 times more statistics compared to the real data,

1A D− is reconstructed in D− → K+π−π− decay.

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Figure 4.5: (left) BB decay events entering ∆E fit-region of B0 → D∗+s D−

decays. (right) Comparison between ∆E fit-region distributions from MC samples(red) and real data (blue).

which effectively reduces the statistical errors on the fudge factors calculated from the fitresults. Since these MC samples were generated at different times, during which the worldaverage for the branching fraction of B0 → D∗+s D− decays changed significantly (seeParticle data group), different runs of MC generation used different branching fractions.To have a fair comparison between branching fraction measurements, we performed fit toa single MC stream as shown in figure 4.6 (middle).

Since the B0 → D∗+s D− branching fractions are already well measured quantitiesin both the samples, recovering their correct values can be used as a cross-examinationfor correctness in the (a) D∗+s skimming applied to reduce the size of the data to beanalyzed (see section 2.3.3), (b) signal reconstruction and selection procedure, and (c)fitting algorithm employed to analyze the distributions in ∆E fit-region.

Table 4.1 compares the branching fraction values obtained from the fits to MC samplesand real data. The expected branching fractions are recovered reasonably well from the

Table 4.1: Branching fraction values obtained from the fits to the MC samplesand the data and their expected values.

Branching Fractions (×10−3)

measured expected

MC5 streams a (7.68± 0.10) . . .1 stream (8.28± 0.24) 8.28

Data (7.51± 0.25) (7.6± 1.6)

a signifies statistics 5 times that in real data.

92



)2E (GeV/c∆-0.2 -0.15 -0.1 -0.05 -0 0.05 0.1 0.15 0.2

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Figure 4.6: Fits performed on MC sample with statistics 5 times data (top),equal to data (middle) and on the real data (bottom). The mean and width of CB

line-shape in signal PDF are allowed to vary.

fits in both the samples and do not indicate any discrepancy neither in the skimmed datanor in any of the analysis techniques used. The control study results appear to be faithfuland fudge factors obtained from the above fits can be considered correct.

Fudge Factors

Table 4.2 shows the values obtained for the signal PDF parameters from the fits. Weobserve a difference of 1.3 ± 0.3 MeV in positions of the mean, while the resolution ofthe CB line-shape signal peak is observed to be 8% wider than in the MC samples. Thesevalues are consistent with the expected 10% discrepancy between MC samples and realdata, as per quoted in section 2.3.2.

93


Table 4.2: Values of the CB line-shape parameters in the signal PDF obtainedfrom the fits to MC samples and the real data. Fudge factors are calculated as∆µ = (µMC − µdata) and rσ = σdata/σMC. Statistical uncertainties on fit valuesare maintained up to an additional significant digit to avoid possible rounding

errors.

µ (MeV) σ (MeV)

MC (0.59± 0.10) (6.41± 0.09)Data (−0.72± 0.25) (6.93± 0.24)

Fudge Factors (1.3± 0.3) (1.08± 0.05)

Individual Fits

A further demonstration of the consistency of the simultaneous fit performed previouslycan be realised by doing individual fits to the B0 → D∗+s D− samples in three D+

s modes.Table 4.3 shows the B0 → D∗+s D− signal yields obtained by fitting the samples in thethree D+

s modes and corresponding branching ration calculated from the yields. Thefirst uncertainty is statistical, while the second comes from uncertainties in the D+

s decaybranching fractions.

Table 4.3: Results of the fits performed on individual D+s modes.

D+s mode

Branching Fraction Significance (Σ)Yield

(%) (stat only) (stat + syst)

D+s → φπ+ (565± 28) (6.6± 0.3± 1.0)× 10−3 22σ 9.7σ

D+s → K∗(892)0K+ (463± 26) (8.4± 0.5± 1.3)× 10−3 17σ 4.7σ

D+s → K0

SK+ (289± 20) (8.4± 0.6± 0.5)× 10−3 14σ 10.6σ

simultaneous (stat only) . . . (7.51± 0.25)× 10−3 56σ . . .

In conclusion,

• We found no serious mistakes: neither in the skimming nor in the reconstruction aswell as fitting procedures,

• The fudge factors calculated are consistent with the 10% discrepancy expected betweenMC samples and real data. These fudge factors are used to correct the parametersobtained from MC samples in B0 → D∗+s π− and B0 → D∗−s K+ study.

• The uncertainties in the fudge factors are considered as an additional source of sys-tematic uncertainty in the fit results.

4.4.2 Correcting for signal Efficiency

The overall signal reconstruction efficiency can be considered as a product of efficiencyof the selection criteria and reconstruction or detection efficiencies of every final-state

94


particles, orε(B0 → D∗sh) = εselect × εγ ×

∏i

εhi (4.3)

where, εselect includes effect of selection by invariant masses, requirements on PID, op-timization procedures, etc., while εh includes PID efficiencies for a hadron itself. Each ofthese factors in equation (4.3) may differ in MC samples and real data, and can propagateinto the overall reconstruction efficiency being different in MC samples from that in realdata. Whenever possible, an attempt is made to find the correction factors via some suit-able control study. However desirable, it is not always possible to find a control samplefor estimating these correction factors, since they are in general functions of particle mo-mentum, which differs from process to process. When no such possibility exists, the overallcorrection factor estimated is add to the systematic uncertainty in the measurement.

In case of PID efficiencies, it is possible to estimate the correction factors as a functionof the hadron momentum precisely. The same process of D∗+ → D0π+ and D0 → K−π+,which was used to obtain the PID efficiency in figure 2.12, can be used to obtain thecorrection factors. As mentioned previously, due to the very well-defined invariant masseswith extremely narrow widths of D∗+ and D0 mesons, it is possible to identify a hadronjust by its energy-momentum and without invoking PID. This property was used to obtainthe PID efficiencies in MC samples and can be trivially extended to real data samples witha D∗+ present. From the to PID efficiency data-sets, the correction factors can now becalculated.

The uncorrected reconstruction efficiencies shown in table 3.9 are recalculated - apply-ing the PID correction factors to all the charged hadrons - and are summarized in table 4.4.

Table 4.4: Signal efficiencies after applying PID correction factors.

D+s mode

Efficiency (%)

B0 → D∗+s π− B0 → D∗−s K+

D+s → φπ+ 15.2± 0.2 13.4± 0.2

D+s → K∗0K+ 7.9± 0.2 6.4± 0.1

D+s → K0

SK+ 8.0± 0.1 6.9± 0.1

Apparently, all the machinery required to carry out the intended branching fractionmeasurements has been developed and the measurements can now be performed. Though,it is worth reiterating a crucial fact, which has been mentioned at early stages, but nevervisited during the analysis so far. While entire fit-region in MC samples is been explored,in order to avoid introducing experimenter’s bias, the data in and around the signal-regionhas never been looked at and is always blinded. The only incidence of visiting the data inthe signal-region was when the branching fractions for the signal modes were obtained byfitting the data in Mbc fit-region, i.e. in figure 3.20. However, it must be noted that (a) theunblinding was done only to check completeness of background study via reproducing theprevious results, (b) that in a variable orthogonal or uncorrelated with ∆E, and more im-portantly (c) no re-tuning of the analysis program or selection criteria was carried; ratherthe entire technique was abandoned for reasons foreseen prior to the measurement. As a

95


consequence, very little knowledge is available about the data in the ∆E signal-region andcan comfortably be considered blinded even at this stage. As a consequence, no enoughinformation on signal-region data is available for one to be able to re-tune parameters, sothat the end results match with previous measurements and hence we do not expect a biasdue to retuning.

However, a bias present in an experiment need not always be intentional or known,but can be result of a biased estimator [55], erroneous inputs, unreasonable assumptions,computational or rounding errors, negligence or simply a mistake in the reconstruction orfitting program. Although a biased measurement is prevalent, every effort should be madeto reduce these sources of systematic uncertainties.

The blind-analysis technique adopted here prevents addition of any bias due to par-tial knowledge about the branching fractions through previous measurements. Thoughit does not guarantee absence of a mistake in the analysis or fitting program. The con-sistency in the results of control study confirms absence of any noticeable mistake in thereconstruction program and the only source of unknown systematic uncertainty remainsto be checked is the fitting program or the fitter.

4.5 Bias in the fitting program

The bias in the fitter will not be evident from a single measurement performed on any con-trol sample, for resulting difference in the measured value from the expected may simplybe a result of statistical fluctuation and not a systematic shift. If the same measurementis performed on a large ensemble of samples, the measured values are expected to be nor-mally distributed around the expected or true value and the spread must correspond tothe statistical uncertainty in each of them, in case with no bias. In general, in an unbiasedexperiment the pull Pi defined as

Pi =(xobs)i − xtrue

(σstat)i(4.4)

- where (xobs)i and xtrue are measured and true values of the measured quantity x, respect-ively, and (σstat)i the statistical uncertainty in the ith measurement - must be normallydistributed around zero with unit width, i.e. N (0, 1). Any deviation from this behaviorcan be attributed to a bias in the measurement, i.e. in the fitter here. Note, in case ofasymmetric errors, if the observed value is found to be deviated on the higher side of the truevalue, we take the negative error and vice verse.

4.5.1 Ensemble Check

To estimate bias in the fitter, we conduct an ensemble test of the fitter. The ensembles areprepared by randomly selecting slices of the BB, continuum as well as rare MC samples.Every ensemble prepared this way contains statistics equal to the real data. To mimic thereal data more realistically, we remove the signal events already present in the rare MCsamples and re-embed a known amount from the separately prepared signal MC samples.This allows us to control the size of the signal events in the total ensemble. Once prepared,we fit the ensemble to extract the signal branching fractions. This procedure is repeated500 times to obtain the error distributions for the experiment.

96

4.5. BIAS IN THE FITTING PROGRAM

Figure 4.7 shows the distributions of the statistical errors and the pull obtained for eachexperiment. The study estimates a typical statistical error of ±0.31 on the B0 → D∗+s π−

B0error2.6 2.8 3 3.2 3.4 3.6 3.8

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Figure 4.7: (Ensemble Study) The distribution of statistical error (left) and pull(right) for theB0 → D∗+s π− (top) andB0 → D∗−s K+ (bottom) branching fraction

measurements on MC ensembles.

decay branching fraction and of about ±0.30 in case of B0 → D∗−s K+ decay. The pulldistribution doesn’t show any significant bias in both the cases. However, due to thelimited MC statistics (only 5 times real data) used to slice-out ensembles, any two samplesmay contain a large fraction of events in common and any hence are expected to haveconsiderable overlap. As a consequence the error as well as the pull distributions obtainedfrom the ensemble study may not be sufficient to reveal a small bias.

97


4.5.2 Toy MC Check

To confirm the observations of the ensemble study, we repeat this study on 5000 inde-pendent toy MC samples, where the MC data-sets are generated using the signal andbackground PDFs with all parameters fixed, while allowing Poissonian variation in yields.Unlike the MC samples generated using the physical processes and detector effects, toy MCsamples are simple statistical artifacts and hence can not reveal biases inherently broughtin due to physical correlations among processes. Though, because of there negligible pro-duction time, thousands of samples, equivalent to ensembles, can be generated in verylittle time and can be used to check bias coming from trivial or minor mistakes very pre-cisely.

Figure 4.8 shows results of the toy MC study. Again, no significant bias is observed.

)-π*+

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Figure 4.8: (Toy MC Study) The distribution of statistical error (left) and pull(right) for theB0 → D∗+s π− (top) andB0 → D∗−s K+ (bottom) branching fraction

measurements on toy MC samples.

98

4.5. BIAS IN THE FITTING PROGRAM

The small deviations in the values of mean observed from the ensemble study - 0.9%in B0 → D∗+s π− and 0.3% in B0 → D∗−s K+ measurement - are added as systematicuncertainties in the respective branching fractions (see section 5.2.11).

As these studied are done using latest known values for the branching fractions fromParticle data group, these studies are repeated posterior to our measurements in real data,in order to demonstrate that the fit to the real data correctly corresponds to one of theimaginary experiments above and represents one incidence on the plots produced by en-semble studies.

99


5Results And Systematics

Results of the fit performed on data are presented. An attempt is madeto account for various possible sources contributing to the systematic un-certainties in the measurements. Signal significance is calculated afterincluding additive uncertainties.

5.1 Results

WITH the tools defined and discussed in the previous two chapters, we perform fit tothe distributions in ∆E fit-regions of B0 → D∗+s π− and B0 → D∗−s K+ decays. As

mentioned previously, we estimate the branching fractions for these decays by doing anextended unbinned maximum likelihood fit simultaneously to six statistically exclusivedata-sets. For comparison study, individual fits to all these samples are also performed.

Simultaneous Fit

We obtain,

B(B0 → D∗+s π−) = (1.75± 0.34 (stat))× 10−5

andB(B0 → D∗−s K+) = (2.02± 0.33 (stat))× 10−5

Figure 5.1 shows the fit results. The statistical significance for the branching fractionsfor B0 → D∗+s π− and B0 → D∗−s K+ decay to be different from zero is 6.8 and 8.8standard deviations, respectively. The errors on the results are consistent with the error

101

RESULTS AND SYSTEMATICS

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Figure 5.1: The simultaneous fit to the B0 → D∗+s π− ((a)-(c): φπ mode,K∗(892)0K mode and K0

SK mode) and B0 → D∗−s K+ ((d)-(e)) signal modes.Signal peaks are shown by the solid curves, while the solid-filled curves representthe cross-feed contributions from the other B0 signal modes. The long-dashedcurves correspond to contribution from the B0 → D+

s π− (B0 → D−s K

+) and thedash-dotted curves to that from B0 → D

(∗)+s ρ− (B+ → D

(∗)−s K+π+). The dotted

curves correspond to the combinatorial background.

distribution obtained in the ensemble study, which is evident from figure 5.2, where theerror distributions obtained from ensemble and toy MC study are reproduced with arrowsshowing the present statistical errors.

Individual Fits

Table 5.1 shows results of fits to the six statistically exclusive samples above. Consist-ent values for branching fractions are obtained from all the measurements and again the

102

5.1. RESULTS

B0error2.6 2.8 3 3.2 3.4 3.6 3.8

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)-!*+

sD(B"

2.6 2.8 3 3.2 3.4 3.6 3.8-610×

Even

ts /

( 1.4

e-08

)

0

20

40

60

80100

120

140

160

180200

220 0.0000000024±mean = 0.0000032244

0.0000000017±sig = 0.0000001693

)-!*+

sD(B"

2.6 2.8 3 3.2 3.4 3.6 3.8-610×

Even

ts /

( 1.4

e-08

)

0

20

40

60

80100

120

140

160

180200

220

-!*+sDError Distr. for BR

)+K*-

sD(B"

2.6 2.8 3 3.2 3.4 3.6 3.8-610×

Even

ts /

( 1.4

e-08

)

0

20

40

60

80

100

120

140

160

180 0.0000000025±mean = 0.0000031858 0.0000000018±sig = 0.0000001762

)+K*-

sD(B"

2.6 2.8 3 3.2 3.4 3.6 3.8-610×

Even

ts /

( 1.4

e-08

)

0

20

40

60

80

100

120

140

160

180

+K*-sD

Error Distr for BR

Figure 5.2: Errors in the branching fraction measurements on the error distribu-tion plots from ensemble (left) and toy MC (right) studies.

advantage of performing a simultaneous fit, discussed in section 4.3.1, is seen from thesignificance values obtained for the measurements from the individual fits compared tothe simultaneous fit.

Table 5.1: Fit results. The first uncertainty is statistical and the second errorcomes from the uncertainties in the branching fractions of D+

s decays.

yield branching significanceD+s mode

Nsig fraction (×10−5) Σ

B0 → D∗+s π−

simultaneous . . . (1.75± 0.34) 6.8σD+s → φπ+ (32.3± 8.1) (1.58± 0.40± 0.24) 3.2σ

D+s → K∗(892)0K+ (29.2± 9.6) (2.30± 0.76± 0.35) 2.6σ

D+s → K0

SK+ (13.1± 6.8) (1.78± 0.92± 0.11) 2.2σ

B0 → D∗−s K+

simultaneous . . . (2.02± 0.33) 8.8σD+s toφπ

+ (32.7± 7.3) (1.81± 0.41± 0.27) 3.2σD+s → K∗(892)0K+ (23.0± 6.8) (2.22± 0.66± 0.34) 2.8σ

D+s → K0

SK+ (13.7± 5.1) (2.14± 0.80± 0.13) 3.1σ

103


Apart from the statistical uncertainties, many additional ones can exist in the meas-ured values due to the limitations in the system used, which include uncertainties in theinput parameters from theory, those from previous experimental measurements or thoseintroduced due to inadequate information about exact detector performance. Unlike thestatistical fluctuations, which can introduce a predictable though indeterministic shiftsin the measurement, the systematic errors are not random and can affect the measuredvalue by adding a bias through constant shifts or scaling factors. Because these factorsoccur systematically and do not vary randomly from experiment to experiment, they canbe pre-determined from control studies. However, as previously mentioned, the task ofchoosing a correct control sample is usually challenging and sometimes turns out to bejust a matter of chance.

5.2 Systematic Uncertainties

The sources of uncertainty which can add systematic shifts in the result can be sub-dividedinto four categories.

• Uncertainties in the information from the previous measurements, such as branch-ing fractions for intermediate decays,

• Uncertainties due to inexact information about the experimental setup,

• Uncertainties involved in the reconstruction procedure, and

• Errors introduced due to the fitting procedure.

Various possible sources affecting the branching fraction measurement are enlistedin table 5.2. This list is reasonably exhaustive and not every source contribute equallystrongly to the overall uncertainty. For example, the largest source of systematic uncer-tainty is expected to come from the branching fractions of the D+

s decays, while thoseintroduced due to selection requirements put on invariant mass distributions can be easilyneglected, as discussed in further sections.

5.2.1 Branching fractions for D+s decays

The uncertainties in D+s decay branching fractions contribute to the overall uncertainty in

two ways:

• directly through yields of the signal and the background processes in which a D+s

meson is present, and

• indirectly via uncertainties in the branching fractions for those background modes,particularly the rare modes, which are calculated using D+

s branching fractions asinputs in previous measurements, similar to the case at hand.

Effect due to former can be estimated by varying the branching fraction value for eachD+s decay used as an input to the fit performed on data and observing the variation in the

fit results, and finally adding the effects in quadrature. Accounting for the effect of D+s

decay uncertainties entering through branching fractions of the rare modes however, is not

104

5.2. SYSTEMATIC UNCERTAINTIES

Table 5.2: Various sources of systematic uncertainties.

Background information (a) branching fractions of D+s decays,

(b) branching fractions of backgroundmodes with peaking structures

Experimental setup (a) charged-track efficiency,(b) photon detection efficiency,(c) PID efficiency

signal reconstruction (a) K0S identification efficiency,

(b) Mφ, MK∗(892)0 , MD+s

selection,(c) ∆M selection,(d) continuum suppression (Rtotal),(e) B multiplicity

fit inputs (a) luminosity or NBB,(b) reconstruction efficiencies due to

limited MC statistics,(c) efficiencies for background modes

with peaking structures,(d) PDF parameters,(e) fit bias

simple to handle, though most of the previous measurements quote a branching fractionvalue with effect of D+

s decay uncertainties separated from the overall error on them. Thisis because, in an involved fitting technique such as simultaneous measurement in variousdecay modes, as desired in the present analysis, it is not obvious usually to interpret andseparate effects due to variations in each D+

s decay, since the overall shift in the globalminimum value at which the simultaneous fit converges is not necessarily linear in thevariation in only one of the branches (or one D+

s decay mode, in this case).Yield of a rare background, with a D+

s meson, can in general be written as

yrare = NBB × εrare × B(B0 → rare)× B(D+s decay)× B(all other) (5.1)

which receives contributions from uncertainties in D+s decay branching fractions twice,

as mentioned above, and also from non-D+s uncertainties in B(B0 → rare), apart from

other sources. Because separating the D+s effect in overall uncertainty in B(B0 → rare) is

not useful when a simultaneous fit is to be performed, we consider the overall uncertaintyin B(B → rare), without separating the D+

s contribution to them, even though the lattermight have been already taken care of, separately. This approach provides the most con-servative way to deal, despite the possibility of double counting the contribution from theD+s decay branching fraction uncertainties and hence leading to an overestimation.

Table 5.3 shows the effect of variation in the D+s decay branching fractions.

One would expect the effect of D+s decay uncertainties to be comparable in the two

signal modes, B0 → D∗+s π− and B0 → D∗−s K+. On the contrary the limited statisticsavailable in data introduces large statistical fluctuations in the fit results, which in turn

105


Table 5.3: Effect of variation in branching fractions of D+s decays on the fit

results.

B modeD+s mode

B(D+s → mode) Branching Variation Cumulative(×10−2) fraction (×10−5) (%) (%)

B0 → D∗+s π−

D+s → φπ+ (2.16− 0.17) 1.84+0.35

−0.34 5.1(2.16 + 0.17) 1.68+0.31

−0.32

D+s → K∗(892)0K+ (2.6− 0.4) 1.79+0.35

−0.33 2.3(2.6 + 0.4) 1.73+0.31

−0.33

D+s → K0

SK+ (1.49− 0.09) 1.77+0.33

−0.34 1.1(1.49 + 0.09) 1.73+0.34

−0.32

5.7

B0 → D∗−s K+

D+s → φπ+ (2.16− 0.17) 2.11+0.35

−0.32 5.4(2.16 + 0.17) 1.91+0.33

−0.28

D+s → K∗(892)0K+ (2.6− 0.4) 2.08+0.35

−0.32 3.9(2.6 + 0.4) 1.94+0.34

−0.28

D+s → K0

SK+ (1.49− 0.09) 2.03+0.36

−0.29 0.5(1.49 + 0.09) 2.02+0.33

−0.31

6.7

manifests in discrepancy in the total uncertainty estimation. To reduce the effect of thefluctuations, we repeat the procedure, using the MC samples with 5 times more statisticsthan that in real data. Table 5.4 shows the variation in the results of the fit to the MC(with BB + continuum + signal + rare : 5 times data) samples, against variations in theD+s decay branching fractions.

Table 5.4: Effect of variation in D+s decay branching fractions on the fit results,

similar to previous table, but using MC samples of size ≈ 5 times data.

B modeD+s mode

B(D+s → mode) branching Variation Cumulative(×10−2) fraction (×10−5) (%) (%)

B0 → D∗+s π−

central value table 3.1 (1.74± 0.15) . . .

D+s → φπ+ (2.16− 0.17) (1.81± 0.15)

5.1(2.16 + 0.17) (1.65± 0.14)

D+s → K∗(892)0K+ (2.6− 0.4) (1.79± 0.14)

2.9(2.6 + 0.4) (1.70± 0.14)

D+s → K0

SK+ (1.49− 0.09) (1.74± 0.14)

0.6(1.49 + 0.09) (1.73± 0.14)

5.9

B0 → D∗−s K+

central value table 3.1 (2.11± 0.14) . . .

D+s → φπ+ (2.16− 0.17) (2.22± 0.15)

5.2(2.16 + 0.17) (2.08± 0.15)

D+s → K∗(892)0K+ (2.6− 0.4) (2.18± 0.15)

3.3(2.6 + 0.4) (2.07± 0.15)

D+s → K0

SK+ (1.49− 0.09) (2.13± 0.14)

1.0(1.49 + 0.09) (2.11± 0.14)

6.2

We use these values as the systematics due to the D+s decay branching fractions, since

they are less sensitive to the statistical fluctuations than the ones obtained from the data,

106


previously.

5.2.2 Branching fractions for background modes

Table 5.5 summarizes the effect of change in the branching fractions for the rare back-ground modes, discussed in section 3.2.2. It should be noted that the systematic errors onthe branching fractions include the effect of D+

s decay uncertainties.

Table 5.5: Effect of variation in branching fractions of the rare background modeswith peaking structures. The first error on the branching fractions are statisticaland the second comes from the D+

s decay uncertainties, as per quoted in the ori-ginal references. We, however, do not separate the D+

s contributions and consideroverall uncertainties.

BackgroundB(background) B0 → D∗+s π− B0 → D∗−s K+

(×10−5) B(×10−5) ∆B(%) (×10−5) ∆B(%)

B0 → D∗+s ρ−(4.4 + 1.3 + 0.8) 1.76+0.32

−0.34 2.02+0.35−0.32

(4.4− 1.2− 0.8) 1.75+0.32−0.31

0.62.02+0.35

−0.32−

B0 → D+s ρ− (1.1 + 0.9 + 0.3) 1.75+0.32

−0.34 2.02+0.35−0.32

(1.1− 0.8− 0.3) 1.74+0.34−0.32

0.62.01+0.34

−0.320.5

B0 → D+s π− (2.5 + 0.4 + 0.2) 1.76+0.34

−0.32 2.02+0.34−0.32

(2.5− 0.4− 0.2) 1.73+0.32−0.34

1.12.00+0.35

−0.311.0

B+ → D∗−s K+π+ (16.7 + 1.6 + 3.5) 1.75+0.34−0.32 2.02+0.35

−0.32

(16.7− 1.6− 3.5) 1.75+0.34−0.32

−2.02+0.34

−0.32−

B+ → D−s K+π+ (20.2 + 1.3 + 3.8) 1.75+0.32

−0.34 2.02+0.34−0.32

(20.2− 1.3− 3.8) 1.75+0.34−0.32

−2.02+0.34

−0.32−

B0 → D−s K+ (2.9 + 0.4 + 0.2) 1.75+0.34

−0.32 2.05+0.35−0.31

(2.9− 0.4− 0.2) 1.74+0.34−0.32

0.62.01+0.33

−0.311.5

Total 1.5 1.9

5.2.3 Charged-track finding efficiency

A charged track is reconstructed by performing a chi2-fit to the hits in the tracking system,i.e. in CDC and SVD. The reconstruction efficiency may differ in the MC samples from thatin real data and can affect the overall signal reconstruction efficiency determined from theMC samples.

To estimate the effect due to differences in the tracking efficiency, a control study onD∗+ → D0π+ decay samples, withD0 → K0

Sπ+π−, is performed. TheD∗+ is only partially

reconstructed in the charged pion modes, i.e. one of the pions from the K0S is not added

to the reconstruction tree. Instead the D∗+ reconstruction momentum deficit can be usedfor obtaining the track-finding probability. The narrow mass distributions for D0 and K0

S

allows for very precise determination of momentum of the missing pion, while that of theD∗+ meson is used for monitoring the signal-to-noise ratio S/N . A charged track witha momentum matching the calculated value is searched in the list of tracks, in order tocalculate the track-finding probability. Details of this study can be accessed from Belle

107




Note 621. The kinematical constraint on the pion track momentum of 250 MeV/c canbe reduced further down to 80 MeV/c by extending this method to MC-embedded datasamples (see Belle Note 641). In this method, a track of desired momentum is embeddedin MC and real data samples, and effect of this additional track on those already existingin the samples is observed. The retrieval of the embedded track gives the efficiency oftrack finding.

5.2.4 Photon Detection Efficiency

Among all, photon detection efficiency appears to be the toughest source to be correctedfor, mainly because of lack of a suitable control sample accounting for the photon mo-mentum ranges in the current analysis. A previous study (see, Belle Note 499) determinesan uncertainty of 2% for the photon detection efficiency, performing control study on asample of B+ → K∗+γ decays. The photons in this study come from a B0 decay and havemomenta as high as 3 GeV. On the contrary, typical momentum of a photon observed ina D∗+s decay lies much lower, i.e. around 150 MeV, compared to this range, as shown infigure 2.15 and hence can not be used directly. An attempt has been made to approachenergy ranges comparable to a D∗+s decay photon energy by using η → and η → decays(see Belle Note 1106). However, a slightly unrealistic assumption of equal systematicsfor a π0 and a photon, is to be made, in order to extract uncertainty in photon detectionefficiency. This study estimates an uncertainty of 7% for photons with momenta as low as100 MeV/c. We believe this would be a conservative estimate of the uncertainty.

We present an independent attempt to evaluate the systematic uncertainty in photondetection, using the correlation between the helicity distribution and the photon energyobserved in the χc1 → J/ψγ decays. We obtain an uncertainty of 3%.

Principle

In the process B+ → χc1K+, with the χc1 meson further decaying to a J/ψ meson and a

photon (γ), the latter being a two body decay, the decay products (particularly, the photon)have well defined momenta (energies) in the χc1 rest frame. When the χc1 is boosted tothe B+ frame, the forward (backward) photons, emitted in the same (opposite) directionas of the χc1 momentum in the B+ frame receive a boost in the opposite (same) directionof their momentum and hence are shifted higher (lower) in energy in the B+ frame. Asa consequence, the photon energy distribution in the B+ frame of reference is completelycorrelated with the χc1 helicity distribution1. As encountered many times previously, atthe Υ(4S) threshold, the B+ pairs are produced nearly at rest in the lab frame and theabove arguments hold true also for the lab frame energies of the photons.

On the other hand, χc1 → J/ψγ is a special case of a V → V V decay, where one ofthe decay products, namely the photon, despite being a vector particle, can have only the(two) transverse polarizations. Because of this, the helicity distribution shows a 1 + cos2 θbehavior 2.

1The χc1 helicity is defined as the angle between the flight direction of the photon and the directionopposite to B+ flight in the χc1 frame.

2with the individual amplitudes proportional to the Clebsch-Gordon coefficients, d11,1 and d11,−1, the totalPDF can be shown to be ∝ ((d11,1)2 + (d11,−1)2)

108








Figure 5.3 shows the χc1 photon energy distribution against its helicity. Due to thecorrelation, all high (low) energy photons populate the cos θ = 1(cos θ = −1) region andthese different helicity regions can be used to compare the photon detection efficiencies inthe data and MC as a function of the photon energies.

Figure 5.3: The Photon Energy against χc1 helicity distribution. The low (high)energy photons populate the cos θ = 1(cos θ = −1) region.

The symmetry in the helicity distribution, on the other hand can be exploited to

• cancel out the systematic errors in the yields (due to NBB, B(χc1 → J/ψγ), etc.),and

• demonstrate the energy dependence of the photon detection efficiency, as any devi-ation from the symmetric nature of the distribution can be directly attributed to thedifferences in the detection efficiency over the energy of the photon.

Procedural Outline

The number of events of the B+ → χc1K+ decay, involving the subsequent photon decay-

ing with the energy Eγ in the lab frame can be written as,

Y (Eγ) = NBB × B(B+→χc1K+) × B(χc1→J/ψγ)(Eγ)× B(J/ψ→ee(µµ))︸︷︷︸N(Eγ)

×ε(Eγ) (5.2)

109


where, B(χc1→J/ψγ)(Eγ) is the χc1 decay branching ratio expressed in terms of the helicityangle (and hence Eγ) and ε(Eγ) is the reconstruction efficiency for the correspondingdecay.

Let e(E) be an energy near the lower (upper) end of the distribution. Due to thesymmetry in the helicity distribution, if e andE are chosen to have corresponding helicitieslocated symmetrically on the helicity plots, we have, N(e) = N(E) and

Y (e)

Y (E)=

ε(e)

ε(E)(5.3)

The above equation can be written for data as,

Ydata(e)

Ydata(E)=

εdata(e)

εdata(E)=

εMC(e)

εMC(E)× Cγ(e)

Cγ(E)× Crest(e)

Crest(E)(5.4)

where, Cγ and Crest are the MC to data correction factors for the photon detection effi-ciencies and all other particle (charged tracks) detection efficiencies respectively. Againusing equation (5.3) for MC samples,

Ydata(e)

Ydata(E)=

YMC(e)

YMC(E)× Cγ(e)

Cγ(E)× Crest(e)

Crest(E)(5.5)

Assuming Crest(e)Crest(E) ≈ 1 over the entire range of photon energies and rearranging terms,

we get

Cdata/MC(e/E) =Cγ(e)

Cγ(E)=

(Ydata(e)

YMC(e)

)/

(Ydata(E)

YMC(E)

)(5.6)

Calculations

Figure 5.4 shows the helicity distribution comparison between signal MC scaled to thedata and the background subtracted data. To subtract the background, we use χc1 in-variant mass sidebands. We select only those events which fall in 5.27 GeV/c2 < Mbc <5.29 GeV/c2 and |∆E| < 50 MeV.

From the comparison above, we obtain,

YMC(e ∼ 220MeV; cos θ ∼ −1) = (338±√

338

14.55)

= 338± 1

YMC(E ∼ 800MeV; cos θ ∼ +1) = (407±√

407

14.55)

= 407± 2

Ydata(e ∼ 220MeV; cos θ ∼ −1) = 263± 16

Ydata(e ∼ 800MeV; cos θ ∼ −1) = 425± 21

where, the factor 14.55 accounts for the signal MC scale-down to the real data statistics.Putting these values in equation (5.6), we have Cγ(220 MeV)/Cγ(800 MeV) = 0.75± 0.10.

110


B+ → χc1 K+


1 10 1 090401/1544 2786. 3.6186E-02 0.6243

-1 0 1χc1 helicity

0

100

200

300

400

500

# of

Eve

nts

1 7 -71 090401/1542 2918. 7.2610E-02 0.6017

Figure 5.4: Comparison between signal MC scaled down to real data (red histo-gram) and the background subtracted real data (blue points). We use χc1 invari-

ant mass sidebands to subtract the background.

This shows that the MC samples represents the real data worse in the low energyphoton region by a factor 1.3 ± 0.2 than in the high energy region. The previous studyclaims a systematic error of about 2% in the high photon energy region, which implies, as-signment of an error of about 3% for the low photon energy region would be conservativeand safe approach.

Criticism

Even-though the above method seems promising and the best possible approach available,it does not escape the following limitations:

1. The factor 1.3 ± 0.2 compares photons with energies in the range of about 200 MeVwith that around 800 MeV. Though the lower energy range is suitable for our presentstudy, the higher energy range is still significantly away from the ones discussed inthe previous work (∼ 2 − 3 GeV), i.e. in Belle Note 499 and further study is highlysought to feel in this gap.

2. The helicity distribution, though highly correlated with the photon energy, this rela-tion is not one-to-one, leaving space for possible underestimation of the errors.

As a result, we do not pursue the estimate from the above study until further checks aremade and use the more conservative value of 7% as the systematic uncertainty in the photondetection efficiency.

111



5.2.5 PID efficiency

PID efficiencies are calculated from MC samples and further corrected for real data usingthe D∗+ → D0π+

slow with D0 → K−π+ control study, as discussed previously. Thoughthe efficiencies have been corrected prior to the measurement, the correction factors areobtained using statistically limited samples of MC and real data, and hence themselvescarry statistical uncertainties. These uncertainties in the correction factors, though secondorder in significance, can be potentially large enough to cause observable bias in the meas-urement. Falling in the same momentum range, all the D+

s daughter tracks receive cor-rection factors in the same range and are expected to be positively correlated. We treatall D+

s daughters together, while the prompt tracks are treated separately. The effect onthe branching fraction measurements due to these are added in quadrature. Table 5.6summarizes the PID systematics study.

Table 5.6: Systematic uncertainties due to uncertainty in the PID efficiency(data/MC) corrections for charged tracks.

CorrectionB0 → D∗+s π− B0 → D∗−s K+

B(×10−5) ∆B(%) B(×10−5) ∆B(%)

PromptKID

(1.0007 + 0.0073) 1.73+0.32−0.34 2.04+0.35

−0.33

(1.0007− 0.0073) 1.76+0.32−0.34

1.12.01+0.34

−0.311.0

PID(0.9591 + 0.0051) 1.73+0.33

−0.33 2.02+0.34−0.31

(0.9591− 0.0051) 1.76+0.34−0.32

1.12.01+0.33

−0.300.5

RestKID

(1.0001 + 0.0204) 1.72+0.32−0.33 2.00+0.34

−0.32

(1.0001− 0.0204) 1.77+0.34−0.33

1.72.05+0.34

−0.311.5

PID(0.9663 + 0.0044) 1.75+0.32

−0.34 2.00+0.34−0.32

(0.9663− 0.0044) 1.74+0.34−0.32

0.62.02+0.33

−0.311.0

Total 2.4 2.1

5.2.6 K0S reconstruction efficiency

The difference in the reconstruction efficiency for a K0S candidate is determined by com-

paring the ratios of the decay widths for D+ → K0Sπ

+ and D+ → K−π+π+ observedin real data and MC samples. The branching ratios for these decays are known to a verygood precision of less than 4% and with sizable samples containing huge D+ statistics, thisstudy has all the merits required to qualify as a control (see, page 90). The data-to-MCratio for the K0

S reconstruction efficiency thus obtained is about 1.046 (1.005) for SVD1(SVD2) data-sets, respectively. The details about this study can be accessed from BelleNote 901. The relevant portion of this internal note is reproduced in digression 3 below,while updating the calculations in the study to accommodate later improvements in theinput values.

The uncertainty in the K0S reconstruction efficiency obtained from MC samples, calcu-

lated by taking weighted averages for SVD1 and SVD2 data-sets, is about 4.6%.

112




Digression 3. The details about a K0S candidate selection requirements are given in sec-

tion 3.1.1. The efficiency with which a K0S is reconstructed satisfying these requirements can

differ in MC samples from that in real data. To estimate a possible discrepancy in the efficienciesthe ratio between the branching fractions for the D+ → K0

Sπ+ and D+ → K−π+π+ observed

in MC and data are compared. In principle, a comparison between branching fractions for theD+ → K0

Sπ+ decay observed in data and MC samples would suffice for calculating the required

K0S reconstruction efficiency difference. However, in such a study the results will be plagued by the

large uncertainties in the branching fractions for B → D+ processes, required to estimate ND+ .These uncertainties are cancelled when the ratio is taken. Taking ratio also allows one to use thee+e− → cc continuum samples to reconstruct a D+ candidate.

We reproduce the basic procedural and computational details from Belle Note 904, except forthe changes occurred due to later updates from CLEO measurements on the branching fractionsfor D+ → K0

Sπ+ and D+ → K−π+π+ decays. And hence, the value for the ratio, quoted in the

previous work, slightly differs from the values mentioned here.Procedural Outline

The latest CLEO results (now included in the Particle data group summary) estimate the ratioRData, of the decay widths of D+ → K0

Sπ+ to that of D+ → K−π+π+ to a very good precision of

∼ 4%.

BR(D+ → K0Sπ

+)

BR(D+ → K−π+π+)=

(1.526± 0.022± 0.038)× 10−2

(9.14± 0.10± 0.17)× 10−2= 0.167± 0.004± 0.006 (5.7)

The decay table used to generate the MC samples3, on the other hand, assumes values for thebranching fractions, which give RMC of,

BR(D+ → K0Sπ

+)

BR(D+ → K−π+π+)=

1.41× 10−2

8.99× 10−2= 0.157 (5.8)

In each case, RMC can be obtained from the ratio between yields in the corresponding decaymodes, i.e. N(D+ → K0

Sπ+)/N(D+ → K−π+π+) in real data and MC samples, respectively.

These ratios can differ from each other due to three factors

1. The difference between the assumed ratio in MC and that observed in data.

2. The charged track identification (PID) efficiency difference between MC and data, which canbe corrected for using the PID tables (discussed elsewhere).

3. The K0S reconstruction efficiency difference between MC and data.

After applying the appropriate scaling factor for (1) and the PID corrections for (2), any resid-ual difference between the data-MC ratios can be assigned to the only remaining factor yet to becorrected: the K0

S reconstruction efficiency difference (3).Calculations

In general,

RD+ =N(D+ → K0

sπ+)

N(D+ → K−π+π+)=

ND+ × B(D+ → K0Sπ

+)× [εK0Sεπ+επ− ]× επ+

ND+ × B(D+ → K−π+π+)× εK− × επ+ × επ+

(5.9)

3For Belle users: In the decay tables, corresponding to the Belle library version b20070528_1559, used forgenerating the MC samples in Belle, the D+ → K∗(892)0π+, D+ → K∗(1430)0π+ and D+ → K∗(1680)0π+

are excluded from the D+ → K−π+π+ non-resonant decay and need to be added for obtaining the resultabove

113




where, εα denote the corresponding detection efficiency of α and differ in data and MC by thecorrection factors (2) for α = h known from PID study or by that in (3) for α = K0

S . For each ofthem, writing εdataα = εMC

α × Cα, where Cα denotes the MC to data correction factor, we have

RDataD+→K0

Sπ+

RMCD+→K0

Sπ+

=N(D+ → K0

Sπ+)Data

N(D+ → K0Sπ

+)MC × CK0SCπ+Cπ− × Cπ+

(5.10)

and, similarly for the other mode.The values obtained in SVD1 data sample, for the above expression are

In SVD1:46349

118441× CK0S× 0.967× 0.971× 0.9709

=0.429

CK0S

(5.11)

In SVD2:88886

93415× CK0S× 0.9515× 0.9611× 0.9555

=1.089

CK0S

(5.12)

Similarly, for the other decay, we have

In SVD1:327991

833812× 1.0604× 0.9646× 0.9873= 0.3895 (5.13)

In SVD2:664263

664366× 1.0619× 0.9477× 0.9660= 1.0286 (5.14)

Applying the data-MC ratio scale factor (1), (0.167/0.157) × (0.6895/0.6861) = 1.0690, wherethe factors in the second parenthesis are theK0

S → π+π− decay widths, We have, CK0S

= 1.046(1.005)in SVD1(SVD2).

Hence, the uncertainty in K0S detection efficiency is given by adding (a) SVD1-2 wighted aver-

age of CK0S

(∼ 1.5%) and (b) uncertainty in the ratio from CLEO results (∼ 4.3%), which is 4.6%.

5.2.7 Rtotal

The likelihood ratio, Rtotal, is optimised to discriminate between a signal event, whichis a BB event, from a continuum background event using MC samples. The FoM curvesobtained from the MC samples are compared with those obtained using real data side-bands, for possible discrepancies in the performance. No significant difference have beenobserved (figure 3.15). Though only qualitative in nature, a more realistic judgementabout the consistent performance of Rtotal has been reached at using off-resonance datain section 4.2.3. Even then, a possible difference between MC and data samples, generally∼ 10%, may cause a shift in the optimal point.

To study this effect, we vary the Rtotal values for the three D+s modes by 10% around

the optimal values. Since, the same background samples are used to determine optimalpoints for all the these modes, we treat the three Rtotal values for one signal mode aspositively correlated, and the three Rtotal values are varied simultaneously. Table 5.7summarizes the effect of Rtotal variation on the signal branching fractions.

5.2.8 NBB

The luminosity at KEKB is monitored and calculated using the e+e− → e+e− Bhabha scat-tering events. The luminosity and subsequently the number of BB events thus estimated

114


Table 5.7: Effect of Rtotal variation on the fit results

Variation in B0 → D∗+s π− B0 → D∗−s K+

Rtotal for B(×10−5) ∆B(%) B(×10−5) ∆B(%)

B0 → D∗+s π−1.74+0.33

−0.34 2.02+0.34−0.32

1.76+0.33−0.34

0.62.02+0.34

−0.32−

B0 → D∗−s K+ 1.76+0.32−0.34 2.01+0.34

−0.32

1.75+0.32−0.34

−2.02+0.35

−0.320.5

Total 0.6 0.5

have an uncertainty of about 1.39%. More details regarding the online and offline lu-minosity calculations at Belle can be obtained from Belle Note 465 and Belle Note 453,respectively. While the NBB calculations are discussed in Belle Note 296.

5.2.9 MC Statistics

The efficiencies for the signal as well as the background modes, which show peaking struc-tures, are determined from an MC samples generated with one of the B’s decaying to therespective mode, while the other side B decaying generically. These samples contain largebut limited statistics and the efficiencies obtained from them have statistical uncertainties.These uncertainties propagate to those in the reff (defined in section 3.2.2).

Table 5.8 summarizes the effect of varying signal efficiencies on the fit results.

Table 5.8: Effect of signal efficiency variation on the fit results.

D+s mode

Efficiency B0 → D∗+s π− B0 → D∗−s K+

ε(%) B(×10−5) ∆B(%) B(×10−5) ∆B(%)

B0 → D∗+s π−

D+s → φπ+ (15.18 + 0.20) 1.74+0.32

−0.33

2.02+0.34−0.31

(15.18− 0.20) 1.76+0.32−0.34

0.6

D+s → K∗(892)0K+ (7.89 + 0.15) 1.75+0.32

−0.33

(7.89− 0.15) 1.74+0.32−0.34

0.6

D+s → K0

SK+ (7.99 + 0.14) 1.75+0.33

−0.33

(7.99− 0.14) 1.76+0.33−0.34

0.6

−

B0 → D∗−s K+

D+s → φπ+ (13.38 + 0.17)

1.75+0.32−0.34

2.02+0.34−0.31

(13.38− 0.17) 2.00+0.34−0.30

1.0

D+s → K∗(892)0K+ (6.43 + 0.14) 2.01+0.34

−0.30

(6.43− 0.14) 2.02+0.33−0.30

0.5

D+s → K0

SK+ (6.95 + 0.13) 2.01+0.33

−0.32

(6.95− 0.13)

−

2.02+0.34−0.31

0.5

Total 1.0 1.2

In case of background processes, we vary reff within the uncertainties on them. Table 5.9summarizes the effect of these variations on the fit results. For the two signal modes cross-feeding each other, it has been discussed previously how the kaon and pion faking prob-

115





Table 5.9: Effect of variation in reff (defined in section 3.2.2) for backgroundmodes with peaking structures on the fit results.

Background mode reffVariation in

B(B0 → D∗+s π−) B(B0 → D∗−s K+)

B0 → D∗+s π−

B0 → D∗+s ρ− (4.7± 0.3)× 10−2 0.3 0.1B0 → D+

s ρ− (6.8± 0.4)× 10−2 0.1 −

B0 → D+s π− (2.9± 0.2)× 10−1 0.6 −

B0 → D∗+s π− cross-feed (4.2± 0.3)× 10−2 0.6 0.5

B0 → D∗−s K+

B+ → D∗−s K+π+ (4.1± 0.1)× 10−2 − −B+ → D−s K

+π+ (4.3± 0.4)× 10−2 0.3 0.2B0 → D−s K

+ (2.0± 0.3)× 10−1 − 0.2B0 → D∗−s K+ cross-feed (1.6± 0.3)× 10−1 0.4 0.9

Total 1.0 1.1

abilities can contribute additionally to the cross-feeding effects. As a consequence, thecorrection factors for the fake-rates are varied within the statistical uncertainties on them,in addition to varying their reff . Though, the contribution from the former is observed tobe negligible compared to the latter.

5.2.10 PDF Shape

The signal and the background modes are parametrized using MC samples and furthercorrected for data using the fudge factors obtained from the B0 → D∗+s D− control study.The statistical uncertainties on the fudge factors introduce additional systematics to theresults. We vary these factors within the uncertainties to estimate the effect on the fitresults, as tabulated in table 5.10.

Table 5.10: Effect of variation in fudge factors (defined in section 4.4.1) on thefit results.

Fudge factorB0 → D∗+s π− B0 → D∗−s K+

B(×10−5) ∆B(%) B(×10−5) ∆B(%)

∆µ (MeV)(−1.30 + 0.35) 1.75+0.32

−0.34 2.02+0.33−0.31

(−1.30− 0.35) 1.75+0.32−0.34

−2.02+0.33

−0.32−

rσ(1.09 + 0.05) 1.80+0.33

−0.33 2.05+0.33−0.31

(1.09− 0.05) 1.69+0.33−0.34

3.42.01+0.32

−0.311.5

Total 3.4 1.5

As previously mentioned, in order to be able to translate the effect of fudge factorscalculated from B0 → D∗+s D− control study, correctly to all the background modes assoon as applied to signal, we have been selective in the choice of parametrization forthese modes. For the same reason, the positions of means of their distributions are alwaysmeasured with respect to that of the signal. This facilitates auto-correcting the mean

116


positions of all the background modes, once the correction factor is applied to the signalmean.

We vary the positions of the means of the background modes within their statisticaluncertainties. Table 5.11 summarizes the uncertainty introduced in the fit results due tothis effect.

Table 5.11: Effect of variation in the fixed peak positions with respect to thecorresponding signal peak positions.

background modeMean positiona ∆B(%)

(MeV) B0 → D∗+s π− B0 → D∗−s K+

B0 → D∗+s π−

B0 → D∗+s ρ− (−177± 4) − −B0 → D+

s ρ− (−43± 5) − −

B0 → D+s π− (130± 10) 0.6 0.5

B0 → D∗+s π− cross-feed (52± 5) − −

B0 → D∗−s K+

B+ → D∗−s K+π+ (−184± 9) − −B+ → D−s K

+π+ (−163± 10) − −B0 → D−s K

+ (120± 10) − 0.5B0 → D∗−s K+ cross-feed (−51± 5) 0.6 0.5

Total 0.9 0.9

a with respect to signal peak position

5.2.11 Fit Bias

The small bias of 0.9% and 0.3% observed in the study performed on MC ensembles inB0 → D∗+s π− and B0 → D∗−s K+ branching fraction measurements, as mentioned insection 4.5.1, is added to the systematic uncertainties on respective measurements.

5.2.12 Other Negligible Sources

Other sources of systematic uncertainty, which are observed to be negligible are listedbelow.

• The parametrization differences of invariant mass distributions of φ, K∗(892)0, K0S ,

D+s and the ∆M distributions between MC samples and real data. In each case, the

size of the selection window is varied within ±10%. The effect on fit results is foundto be negligible.

• Average multiplicity of B0 candidates observed in data may differ from that in MCsamples, if the backgrounds are not well reproduced in the MC samples. Although,it can be inferred from figure 5.5, that the multiplicity in MC samples matches quiteclosely with that in real data and we observe no discrepancy thereof.

Due to their negligible effect on the fit results, these sources are not included in the totalsystematic uncertainty.

117


B0 → Ds*+π-: Multiplicity

0 1 2 3 4 5 6# of B / event

0

250

500

750

1000

# of

Eve

nts

Mean:

in MC = 1.47 +- 0.02

in data = 1.41 +- 0.05

Figure 5.5: The comparison of the B candidate multiplicity per event in dataand MC samples. The histogram in red shows the distribution in MC (BB + cont+ rare MC samples with 5 times statistics in real data), while the blue points are

for data.

5.2.13 Summary

We summarize all the systematic uncertainties discussed above in table 5.12. As expected,the uncertainties in the D+

s decay branching fractions are one of the dominant sources ofsystematic uncertainties in the measurements. However, they are not the most dominantones to contribute, but are overpowered by the uncertainties in the photon detection effi-ciency, according to present estimate. The photon detection systematic uncertainty of 7%,on the other hand, is a conservative estimate and may not be the most realistic one, andrequires conceptualizing more careful approach.

5.3 Results: revisited

After including the systematic errors, we obtain,

B(B0 → D∗+s π−) = (1.75± 0.34 (stat)± 0.17 (syst)± 0.10 (B))× 10−5

andB(B0 → D∗−s K+) = (2.02± 0.33 (stat)± 0.18 (syst)± 0.14 (B))× 10−5,

where now we assign systematic uncertainties calculated in previous sections to the branch-ing fraction measurements, in addition to the statistical uncertainties quoted in section(5.1). The systematic uncertainties coming from inputs external to this study, which are

118

5.4. SIGNAL SIGNIFICANCE

Table 5.12: Various sources of systematic uncertainty and the correspondingreferences used for their determination.

Source Contribution (%)(B0 → D∗+s π−) (B0 → D∗−s K+)

(a) BR(D+s )

Signal mode 5.9 6.2Background modes 1.5 1.9

Total(B) 6.1 6.5

(b) Experimental setupCharged track efficiency 4 4Photon detection efficiency 7 7PID efficiency 2.4 2.1

(c) Reconstruction procedureK0S identification efficiency 1.1 1.1Rtotal 0.6 0.5

(d) Fitting procedureNBB 1.4 1.4MC Statistics 1.4 1.6PDF shape 3.4 1.5Fit bias 0.9 0.3

Total (rest) 9.4 8.8

essentially the values for the D+s decay branching fractions are separated from the rest, in

order to demonstrate the error-budget of the analysis, more clearly. As a result, the seconduncertainty in the above expression comes from experimental systematics, while the thirdcomes from the D+

s decay branching fractions.It should be noted, that even with the data sample used in this analysis the uncertain-

ties are still dominated statistically and can be further improved using larger data-sets infuture. However, the fact, that the measurements presented here are the most precise onesto date, can not be overemphasized.

5.4 Signal Significance

The fact, that the measurements reported in the previous section are plagued by uncer-tainties, makes it necessary to ask the question, “if the observed results are sufficient toconfirm existence of the B0 → D∗+s π− and B0 → D∗−s K+ decays or only result from fluctu-ations in the background events” or more technically, “what is the significance of observingthe signal?”.

In case of no systematic uncertainty, i.e. when working with an ideal experimental ap-paratus4, the statistical fluctuations in the signal and background yields would be the only

4By ideal apparatus we mean ideal behaviour of an accelerator, a detector, ideal signal reconstruction and

119


source of uncertainties in the branching fractions and looking at the branching fractionmeasurements would be sufficient to answer this question. In a more realistic situationhowever, many sources introduce systematic uncertainties in addition to the already ex-isting statistical fluctuations and hence, finding a concrete answer may be a complicatedtask, usually involving correct identification of those systematic uncertainties which canaffect the yields. For example, a mistake in the signal efficiency determination can affectthe branching fraction values obtained from the yield, but is unlikely to be disturbing theobserved yield itself. On the contrary, even if the signal is reconstructed optimally, but thefitting procedure has a tendency to underestimate the signal yield due to non-negligiblefit-bias, some of the signal yield will be overridden by the background and one wouldmis-interpret the signal for purely background fluctuations. In short, even though manysystematic biases existing in the analysis can affect the yields, not all of them contributeto this effect and it is crucial to separate these sources accordingly.

The sources of systematics which inherently affect the signal yield collection over back-ground are called additive sources, while those occurring only in the branching fractionestimation from the signal yields are multiplicative sources. As the names suggest, the ad-ditive sources appear as a shifting factor in the yields, whereas the multiplicative sourcescontribute as a scaling factor. To calculate the significance of observation, hence only theadditive systematic uncertainties need to be included along with the statistical uncertain-ties. This is done by the marginalization technique [70]. As a first step, the sources ofsystematic uncertainty are separated according to their multiplicative or additive nature5.We can categorize,

• Additive UncertaintiesPDF shape, Fit bias, peaking background yields

• Multiplicative UncertaintiesIntermediate decay branching ratios, Detection efficiencies of the Charged track,photon and K0

S , Particle Identification efficiency, NBB, MC statistics

Only the additive systematic uncertainties, affecting the signal yield, are to be includedin the signal significance estimation. The uncertainties in the PDF shapes, in the reff forthe background modes and the fit bias are the only sources which affect the signal yieldadditively and contribute around 3.8% (2.4%) uncertainty in the determination of theB0 → D∗+s π−(B0 → D∗−s K+) signal yield.

fitting modules.5 As a consequence of their defining property, the additive uncertainties do not depend upon the central

value of the physical variable, being measured, while the multiplicative uncertainties are proportional to it.For example, the error on the branching fraction, B = yS is given by

σBB =

√(σyy

)2

+

(σS

S

)2

σB =

√(σy

S

)2

+

(σS

S2

)2

y2

where, y is the signal yield and S is the scale factor. The first error, which is independent of y can be termedas the additive, while the second, proportional to y is the multiplicative in nature.

120

5.4. SIGNAL SIGNIFICANCE

In the marginalization method, the maximization of log-likelihood logL(data), definedin equation (4.2), is scanned with respect to branching fraction under study. The con-cerned branching fraction is fixed to values ranging from zero to its nominal value fromthe best-fit in regular intervals and for each scan-point, value for −2 log(LB/Lmax) isobtained from the fit, where LB(Lmax) is the value for the maximized likelihood whenbranching fraction is fixed to B (allowed to vary). This would be the log-likelihood max-imization curve giving the signal significance, if only statistical errors were present. Toinclude the additive systematics to the process, the maximization curve is convoluted bythe PDF, W (S, σS), representing the behaviour of the systematic uncertainty around thefixed branching fraction value and then the likelihood L is integrated over the convolu-tion, similar to what would be done in a Bayesian context [71]. For all practical purposes,where the uncertainty σS is small compared to S, W (S, σS) can be assumed to be Gaus-sian, so that the probability for obtaining signal yield of y, given the signal branchingfraction, is

py(B) =

∫ ∞0

e−BS1√

2πσ2S

e−(S−S)2/2σ2SdS (5.15)

For small enough σS ’s, the lower limit of the integration can be extended to −∞. Thelikelihood maximization curve above is thus smeared with the probability distribution,py(B) to generate the likelihood maximization curve again, now including the effect of theadditive systematic uncertainties. Figure 5.6 shows the maximization curves before (blue)and after (red) the inclusion of the systematic uncertainties, in case of B0 → D∗+s π− (left)and B0 → D∗−s K+ (right).

Figure 5.6: The negative log-likelihood minimization curves for B0 →D∗+s π− (left) and B0 → D∗−s K+ (right). The curves in blue (red) colour rep-resent the minimization contours before (after) the inclusion of the systematicuncertainty. The near-y-axis-interception-regions of the curves have been blown-

up in size in the inlays.

We obtain a significance of 6σ (8σ) in case of B0 → D∗+s π− (B0 → D∗−s K+) after theinclusion of systematics.

121

6Conclusion and discussion

ρ-1 -0.5 0 0.5 1 1.5 2

η

-1.5

-1

-0.5

0

0.5

1

1.5

2φ

1φ

3φ

ρ-1 -0.5 0 0.5 1 1.5 2

η

-1.5

-1

-0.5

0

0.5

1

1.5

3φ

3φ

2φ

2φ

dm∆

Kε

Kε

dm∆ & sm∆

ubV

1φsin2

< 01

φsol. w/ cos2(excl. at CL > 0.95)

excluded area has CL > 0.95

excluded at CL > 0.95

Summer 2007

CKMf i t t e r

The aftereffects of the branching fraction measurements carriedin this study are discussed. The ratio, RD∗π is calculated and itseffect on the current estimation of CKM angle φ3 is analysed. Aneffort is made to motivate towards applicability of the measure-ments in estimation of CKM element Vub. Future prospects andimplications are driven to.

6.1 RD∗π

The central goal of the B0 → D∗+s π− and B0 → D∗−s K+ branching fraction measurementsis to estimate the ratio RD∗π, to be used for the time dependent CP analysis of the B0 →D∗∓π± system, as discussed in section 1.8. While the branching fraction for the former isto be inserted into equation (1.64) for estimating RD∗π, the size of latter, when comparedto that of CFD, allows one to validate the substitution made of B0 → D∗+s π− for DCSD inarriving at this equation.

The observed value for the branching fraction of B0 → D∗−s K+ decays is about twoorders of magnitude lower than that of CFD, implying

• absence of evidence for final-state rescattering effects envisaged in [34] and fur-ther reviewed in [35]. The rescattering effect is expected to produce enhancementscomparable to the size of CFD in the B0 → D∗−s K+ branching fraction and hencecan be easily ruled out based on present observations. The branching fraction forB0 → D∗−s K+ hence, can be understood purely in terms of the W -exchange amp-litudes.

123

CONCLUSION AND DISCUSSION

• negligibility of W -exchange amplitudes in B0 → D∗∓π± system, in general and inDCSD, in particular.

As a consequence, RD∗π can be estimated reasonably precisely with B0 → D∗+s π− branch-ing fraction measured in this study. Following additional inputs are used in RD∗π estima-tion:

1. tan θC = 0.2314± 0.0021(ref: Particle data group summary) The Cabibbo angle θC has been extracted veryprecisely from the double-beta decay experimental data.

2. B(B0 → D∗−π+) = (2.76± 0.13)× 10−3

(ref: Particle data group summary)

3. fD+s/fD+ = 1.164± 0.006(stat)± 0.020(syst)

(ref: Follana et al. [36]) It should be noted, that we use the ratio between the pseudo-scalar decay constants instead of that between the vector meson decay constants,mainly due to lack of experimental measurements or reliable (unquenched) latticeQCD estimates of the latter. On the other hand, all alternate efforts made in estim-ating the vector meson decay constants support the phenomenological predictionsfrom heavy quark symmetry (HQS)-heavt quark effective theory (HQET) approxima-tions, as summarized in table 6.1. Under HQS, the hadronic properties are expected

Table 6.1: Latest values for D meson decay constants. The entry in blue is theone used for deriving RD∗π.

fD+ fD+s

fD∗+ fD∗+s

(MeV) (MeV)fD+

s/fD+

(MeV) (MeV)fD∗+

s/fD∗+

Experiment a 205.8± 8.9 273± 10 1.33± 0.07 . . . . . . . . .Unquenched LQCD [36] 207± 1.4± 1.6 241± 1.4± 2.9 1.164± 0.006± 0.020 . . . . . . . . .Quenched LQCD [72] 206(4)+17

−10 229(3)+23−12 1.11(1)+1

−1 8.6(3)+5−9 8.3(2)+5

−5 1.04(1)+2−2

NRQM [73] 243+21−17 241+7

−5 1.41+0.08−0.09 223+23

−19 326+21−17 1.41+0.06

−0.05

Bethe-Salpeter [74, 75] 230± 17 248± 27 1.08± 0.01 340± 22 375± 24 1.10± 0.06

a world averages as per summarized in Particle data group.

b as per averaged by the CkmFitter group for Summer’08 results.

to be spin and flavor independent, and the HQET corrections due to finite heavy-quark mass are predicted to be small due to cancellation when ratios between theconstants are taken. As a consequence, we use prediction from the LQCD calcu-lations, which are the most precise ones, so far. The LQCD is also believed to bethe best formalism suited for QCD calculations, depicting strong physics effects. Weexpect an additional uncertainty of few percent due to break down of the equalitybetween the two ratios.

We obtain,RD∗π = (1.58± 0.15(stat)± 0.10(syst))% (6.1)

where the first error is statistical and the second comes from the systematic uncertainties.It should be noted, that the first uncertainty of 0.5% in the fD+

s/fD+ estimate, even though

124





6.2. φ3 ESTIMATION

statistical in nature, has no relation to the statistic used in this analysis, and hence, isconsidered as a systematic error, in calculating RD∗π.

From the value of RD∗π, we obtained, it can be inferred that,

• it is consistent with the theoretical prediction of 2%, as in equation (1.62),

• the value of RD∗π obtained, though the most precise estimate with respect to theoverall error-budget, it is still dominated by statistical error and can be further im-proved using bigger data samples, in future,

• unfortunately, we do not gain significantly over the previous estimations with respectto the error-estimation, mainly because

– even though our estimation is based on higher statistics, we use ∆E variable forsignal extraction, as opposed to the Mbc, used in the previous measurements,and the background levels may differ in these two mutually almost-orthogonalvariables,

– though the statistical error in the obtained branching fractions forB0 → D∗+s π−

is smaller compared to previous measurements and consistent with the naïveexpectations based on statistical or Poissonian scaling factor of

√NBB, we also

obtained a central value lower compared to the previous measurements, bring-ing the relative error back toward those in earlier estimates,

– due to the relatively poor PID performance, compared to that in BaBar, weexpect higher background levels in Belle.

• more importantly, the value for RD∗π has gone further down, suppressing the sens-itivity to the CP violating effects in the time dependent analysis of evolution ofB0 → D∗∓π± decays.

In summary, we report the most precise measurement of the B0 → D∗+s π− and B0 →D∗−s K+ decay branching fractions. This improves the precision with which the parameterRD∗π can be estimated, and thus the prospect of observing CP violating effects in theD∗±π∓ system. The smallness of RD∗π however, implies that the effect of CP violationseen in this system will be small.

6.2 φ3 estimation

The RD∗π estimated above can be plugged-in to antisymmetric part of the time dependentdecay rates for the B0 → D∗∓π± processes, as in equation (1.60) to calculate most-probable value for the | sin(2φ1 + φ3)|. A procedure similar to that used by the CkmFittergroup group, described in [76], is followed for obtaining the confidence level CL plots,shown in figure 6.1. We define a weighted-χ2 based on the inputs corresponding to theobservables in equation (1.60), where the weights w are derived from respective uncer-tainties σ, i.e. w = 1/σ2. Minimization of this weighted-χ2 variable is run for every fixedvalue of | sin(2φ1 + φ3)| in regular intervals between [−1,+1]. The confidence level CL is

125




)|3

φ + 1

φ|sin(20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1 -

CL

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

)|3

φ + 1


±π±*

D→ 0from B

BaBar

Global fit

)%-0.21

+0.23 = ( 1.81π*

DR

)|3

φ + 1

φ|sin(20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1 -

CL

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

)|3

φ + 1


±π±*

D→ 0from B

Belle

Global fit

0.18)%± = (1.58 π*

DR

Figure 6.1: Constraints on | sin(2φ1+φ3)| usingRD∗π estimation from BaBar [46](left) and Belle (right).

calculated from the minimized χ2 using the standard CERNLIB definition,

CL = Prob(χ2, N)

=1√

2NΓ(N/2)

∫ ∞χ2

exp−t/2 tN/2−1dt (6.2)

where N is the number of degrees of freedom in the χ2 calculation.The following inputs are used for the χ2 calculation above:

1. a = −(A+ +A−)D∗π/2 = −0.040± 0.010c = −(A+ −A−)D∗π/2 = −0.007± 0.012(ref: HFAG averages in Summer’09)

2. ∆MB = (0.507± 0.003( stat)± 0.003(syst))ps−1

(ref:Particle data group)

In conclusion,

• we obtain a value for | sin(2φ1 + φ3)|, which is consistent with the previous estim-ate. Our estimate, however is not the best one can achieve, with all the currentunderstanding of the physics of CP violation, since a more reliable estimate can beusually reached by doing a global fit to the quantities, while using world averagesand employing a more accurate extraction procedure as of CkmFitter group.

• The most-probable value for | sin(2φ1 + φ3)| appears to be 1.0+0.0−0.2, which can be

translated into a constraint on φ3 using the very well measured sin(2φ1) as an addi-tional input to the above CL scan. We obtain a rough, preliminary estimate for φ3 tobe (77+20

−23). It must be noted, that the quantity sin(2φ1) is a very precisely known,despite the 4-fold ambiguity in the value of CKM angle φ1 itself.

• The value estimated for φ3 indirectly in the B0 → D∗∓π± CP analysis is not expectedto be better than that obtained from the direct methods, such as ADS/GLW/GGSZ

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http://www.slac.stanford.edu/xorg/hfag/



6.3. |VUB|

estimations, in general and the current estimate of φ3 is (75+16−19) from these analyses,

allowing the uncertainty to be as small as 25%.

6.3 |Vub|Following the formalism on the nonleptonic decays, in relation with the extraction ofthe CKM element |Vub|, developed in Kim et al. [39] and discussed in section 1.8.2, weconduct a feasibility study, using the B0 → D∗+s π− branching fraction measured here. Asa matter of fact, the semi-leptonic decays of the B meson are considered a more suitablecandidates for this purpose, since they allow extraction of hadronic properties related toB, which are the only hadronic quantities present, in the cleanliest manner. The final-statedecay products being mostly leptonic do not have internal structures, determined by thestrong interactions.

The expression (1.67), in its exact form as derived in Kim et al., looks like

B(B0 → D∗+s π−)

B(B0 → D∗+s D−)= [0.456± 0.038]

∣∣∣∣VubVcb

∣∣∣∣2 (6.3)

Inserting,

1. B(B0 → D∗+s D−) = (7.5± 1.6)× 10−3

(ref:Particle data group summary)

2. |Vcb| = (41.6± 0.6)× 10−3

(ref: Particle data group summary)

we obtain,|Vub| = (2.98+0.27

−0.29(stat)± 0.38(syst))× 10−3 (6.4)

which is not only consistent with the world averages obtained from studies of the semi-leptonic decays, but also competitive in terms of error-budget with them.

In conclusion,

• The value obtained for |Vub| from the non-leptonic B0 → D∗+s π− decay appears com-petitive with those obtained from the semi-leptonic decays. The estimation however,must be strictly understood as preliminary and scope must be assigned to the pos-sibility of changing the central value as well as the uncertainty appreciably with duecourse. We summarize the estimations of |Vub| from the past in table 6.2.

• The dominant source of uncertainty in this case would come from the hadronic formfactors and the factorization process involved, as opposed to the semi-leptonic de-cays.

• The expression derived is based on the generalized factorization scheme. Though,there are incidences of extensive usage of this scheme in many phenomenologicalstudies and some studies have been suggested [44], neither a firm theoretical back-ground nor conclusive experimental evidences exists, so far. It is necessary to put thefactorization scheme on test, before any reliable conclusion can be arrived at fromthis approach.

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Table 6.2: Averages of the |Vub| estimates from semi-leptonic studies. For adetailed account on this topic, reader is encouraged to look at the Particle data

group reviews.

MethodEstimate(10−3)

inclusive 4.12± 0.43

exclusive 3.5+0.6−0.5

average 3.95± 0.35

• The value obtained for |Vub depends strongly on the input used for |Vcb|. Particularly,in the light of the pool of estimations with considerably varying central values existsin literature, the value for |Vub| may change drastically.

• The expression (1.67) used for extracting |Vub|, requires measurement of the ratiobetween the branching fractions for B0 → D∗+s π− and B0 → D∗+s D− decays andnot individual branching fractions. The B0 → D∗+s D− sample has been studiedand its branching fraction has been measured in data in the control study, duringthis analysis itself, implying feasibility of measuring it under the same formalismand setup. The Particle data group average for the same shows that the branchingfraction is dominated by systematic uncertainties, while the statistical uncertaintiesare extremely tiny. Following our experience in B0 → D∗+s π− branching fractionmeasurement, a large part of this uncertainty can be attributed to the D∗+s meson.Performing a measurement for the ratio between B0 → D∗+s π− and B0 → D∗+s D−

simultaneously between the B0 → D∗+s π− and B0 → D∗+s D− samples in data, it ispossible to reduce most of the uncertainties due to a D∗+s meson, which is commonto both. As a consequence, it is possible to improve upon the toy estimate in ()within the existing system and using the already collected data.

• On an independent note, it is worth noting that an update on the B0 → D∗+s D−

branching fraction itself may be desirable.

6.4 B0 → D∗+s ρ−: an immediate prospective

While developing the theoretical background behind the B0 → D∗+s π− analysis, it hasbeen emphasized that the time dependent CP analysis of B0 → D∗∓π±, though offersthe cleanliest method to extract the CKM angle φ1, is plagued by a non-zero strong phaseδ in the expression | sin(2φ1 + φ3 + δ)|. Doing a time dependent CP analysis of B0 →D∗∓ρ±, the ambiguity due to this strong phase can be settled. The latter however is aB → V V decay and has three possible polarization modes available, and as a result afull time dependent analysis in the three polarization modes, i.e. an angular analysissimultaneously with the CP analysis, is desired.

On the similar lines of B0 → D∗+s π− branching fraction could be used in the B0 →D∗∓π± CP analysis, the branching fraction for B0 → D∗+s ρ− can be inserted in B0 →

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6.4. B0 → D∗+S ρ−: AN IMMEDIATE PROSPECTIVE

D∗∓ρ± decays evolution studies1. Because B0 → D∗+s ρ− is also a B → V V decay, polar-ization for which is poorly known, measuring the branching fraction for this decay alongwith polarization determination not only avails the ratio RD∗ρ, but also provides an av-enue to prepare the machinery required for the polarization studies in a B → V V decay.In short, the signal extraction techniques developed for the B0 → D∗+s π− branching frac-tion measurement can be easily extended to include the more challenging B0 → D∗+s ρ−

decay, which provides a testing platform for the techniques devised so far, simultaneouslyoffering an apparatus to construct machinery required for handling the complexities dueto polarization.

Assuming the SU(3) symmetry between D∗ and D∗s , and hence extrapolating the ratiobetween branching fractions for B0 → D∗−ρ+ and B0 → D∗−π+ decays to that betweenB0 → D∗+s π− and B0 → D∗+s ρ−, one would expect the branching fraction of latter to beas high as four times that of the former. Although, the poor reconstruction efficiency ofthe additional π0 to the former reduces the efficiency by a factor of 3.5 in the latter case.The background in B0 → D∗+s ρ− fit-region, on the contrary, is expected to be higher thanin B0 → D∗+s π−. As we plan to measure the polarization, the fake D∗+s background dueto low momentum photons, which was earlier reduced by applying a selection criterionon the D∗+s helicity, can not be controlled the same way in case of B0 → D∗+s ρ− studies.This further increases the background in the B0 → D∗+s ρ− fit-region. Overall, if theB0 → D∗+s ρ− branching fraction is found to be equal to that expected due to SU(3)symmetry, the significance with which the signal is observed in the Belle data will not bebetter than 4σ, if the same set up used for B0 → D∗+s π− is retained. In order to improveupon this situation, we

1. include the full data-set taken till date, which adds another 100 fb−1 to the previ-ously used sample,

2. measure theD∗+s and ρ− helicities simultaneously with the branching fraction: Over-all it will be a 3D fit performed simultaneously on three D+

s modes,

3. in addition, plan to use the new or improved tracking algorithm, which is found toincrease the tracking efficiency by about 30%. This algorithm has been optimized tohandle huge backgrounds in Belle-II environment - an upgraded version of existingBelle setup with luminosities at least 10 times higher than in Belle experiment.

6.4.1 Helicity formalism

To determine the polarization of the B0 → D∗+s ρ− decay, we adopt the helicity formalism,where helicity h of a particle is defined as the projection of the spin angular momentum,~s along its linear momentum, ~p. This approach is convenient in the relativistic regime dueto the invariance of the helicity under rotations and boosts along the direction of the mo-mentum of the particle and is widely used in the systems involving high energetic particles,having relativistic velocities. The helicity formalism is developed in detail elsewhere [61].It can be shown that, the amplitude A for a particle with spin quantum number J and

1A ρ− meson has exactly same quark content as in a π− meson, only differing factor being the angularmomenta and in particular, the spins: later is a spin-1 analogue of the former, which is a spin-0 pseudoscalar.

129


the spin projection M along (an arbitrarily defined) z-axis decaying to two daughters withhelicities λ1, λ2,

A ∝ DJ∗Mλ(φ, θ,−φ)Aλ1λ2 (6.5)

where, λ = λ1 − λ2 and the total angular dependence is summed up into the matricesDJ∗j1j2

(α, β, γ) = eiαj1dJj1j2(β)eiγj2 , which are the representations corresponding to an Eulerrotation (α, β, γ) with the Clebsch-Gordon coefficients dJj1j2(β). For a decay, with morethan one helicity configurations, the differential decay rate can be expressed as [49],

d2Γ

Γd cos θ1d cos θ2= |∑i

Aλ1λ2 × fMλ(cos θ1, cos θ2)|2 (6.6)

which shows, that the angular dependence can be factored out from the dynamics. Fora B → V V decay in particular, with λ1,2 = ±1, the conservation of angular momentumrequires the two daughter helicities to be equal, i.e. λ1 = λ2 = 0,±1, since B isa pseudoscalar. This implies, even if in principle λ can take values from 0,±1,±2, theangular momentum conservation restricts it to a smaller subset, which can be be identifiedwith the value of any one of the λ1,2. As a result, the differential decay, as in equation (6.6),for B → V V involves six real parameters, signifying three complex amplitudes: A0, A±1.The amount of longitudinal polarization is defined as, fL = |A0|2/(

∑ |Ai|2), where i runsover all the helicities.

The three polarization states behave differently under parity and hence the CP violat-ing effects in the overall decay amplitude may suffer dilution due to interference betweenvarious polarization amplitudes. In cases, where observing CP violation is the centralgoal of study, use of a variant of the helicity basis, namely the transversity basis is made inorder to avoid dilution of CP violation effects. Because we do not expect any CP violatingobservations from B0 → D∗+s ρ− analysis, we proceed with the helicity basis and indeed,only the extent of longitudinal polarization fL will be measured.

For the B0 → D∗+s ρ− decay, which is a P → V (Pγ)V (PP ) decay, P being a pseudo-scalar, the cross-section follows [77],

d2Γ

d cos θD∗sd cos θρ∝ [(1− fL)(1 + cos2 θD∗s ) sin2 θρ + 4fL sin2 θD∗s cos2 θρ] (6.7)

and it can be seen, that the angular dependence is now completely disentangled from thedynamical effects governing the decay. As a consequence, fL can be easily extracted, sim-ultaneously with the branching fraction measurements, by including D∗+s and ρ− helicitiesin the signal extraction fit.

6.4.2 Signal Efficiencies

The signal events are reconstructed exactly the same way as for the B0 → D∗+s π− analysis,except:

• no selection criterion applied to D∗+s helicity, covering the full domain of [−1,+1],and

• the ρ− is reconstructed in ρ− → π−π0 decay, while reconstructing a π0 from twophotons.

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6.4. B0 → D∗+S ρ−: AN IMMEDIATE PROSPECTIVE

Figure 6.2 shows a typical 3D fit to ∆E, cos θD∗+s and cos θρ− , where the signal events aregenerated assuming purely transverse polarization, i.e. fL = 0. The red curve represents

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-ρθcos

Figure 6.2: 3D fit to the ∆E, cos θD∗+s and cos θρ− variables for the B0 → D∗+s ρ−

signal event in the φπ mode. The events are generated in purely transverse po-larization state. The red curves signify a truly reconstructed signal, brown curverepresents the D∗+s self cross-feed and the gray curve represents the ρ− self cross-

feed.

the correctly reconstructed signal events, while the brown (gray) curves represent the selfcross-feed background due to a false-reconstructed D∗+s (ρ−) meson.

As evident from the angular distributions, the signal reconstruction efficiencies in gen-eral are functions of the polarization fL. It is also observed that the PDF parameters forthe correctly reconstructed signal do not vary significantly with fL, however the amountof cross-feeds are found to be highly sensitive to fL. Table 6.3 shows a rough trend in theamount of correctly reconstructed signal and the cross-feeds with respect to fL.

Table 6.3: Trend in the amount of correctly reconstructed signal as well as cross-feed amounts with respect to polarizations.

YieldPolarization

signal D∗+s cross-feed ρ− cross-feed

Transverse (fL = 0) 7423± 103 2275± 75 327± 29Flat (fL = 0.33) 7217± 106 2427± 79 1275± 55Longitudinal (fL = 1) 5083± 95 1850± 67 2620± 79

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6.4.3 Background MC Studies

The same steps followed for B0 → D∗+s π− analysis are repeated, this time for the back-ground processes which enter the ∆E fit region of the B0 → D∗+s ρ− decays. For the con-trol study performed to correct the PDF parameters in ∆E variable, we use B0 → D∗−ρ+

decays. The dominating factor determining the PDF shape in the ∆E fit-region of B0 →D∗+s ρ− decays is observed to be the π0 from the ρ− reconstruction and hence the abovecontrol sample is well-suited, as far as ∆E PDF parametrization is concerned. As opposedto the branching fraction, the polarization of B0 → D∗−ρ+ has not been measured veryprecisely, particularly in the data collected at Belle. As a result of this, for obtaining cor-rections for the helicity distributions, we intend to perform an independent control studyon samples suited for polarization measurements. This study is under consideration andrecently measured B0

s → D∗+s ρ− polarization may serve to be a fairly reasonable controlsample for this purpose.

Figure 6.3 shows comparison study between the background events predicted by theMC samples and that in data ∆M -sidebands. It can be seen, that the distributions obtained

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Figure 6.3: Comparison between the MC predictions and data for ∆E (left),D∗+s polarization (middle) and ρ− polarization (right). The data points representthe data sideband and the solid histograms indicate various components of the

background predicted by the MC.

from MC samples in fit-regions of all the three variables are in very good agreement withthose observed in data sidebands.

The final fit to the data is to be performed in the three variables: ∆E, cos θD∗+s andcos θρ− . In short, we intend to perform a 3D unbinned extended maximum likelihood fit,simultaneously to three D+

s modes, in the due course.

132

Epilogue

IT is a high time, that we finally land on the same question of CP violation with referenceto the baryonic asymmetry observed in the universe (BAU), with which we took off this

journey in chapter 0. To reinstate the main theme, we then asked

• if the KM mechanism and hence, the extent of CP violation predicted within the SMis sufficient to explain all the CP violation observed?, and if so

• does the amount of CP violation observed within SM explains the matter-antimatterimbalance, for which the whole endeavour of settling the CP violation study hasbeen pursued with utmost care?

The answer to the former appears to be very much affirmative, given the current know-ledge about the physical processes. Apart from a handful of measurements, which are notconclusive enough to be reliable due to limited statistics, all other innumerable effortsput in to study CP violating effects can be understood purely in terms of the KM mech-anism within the SM. Much bigger data samples are needed to confirm existence of newphysics effects and the CP violating phases not originating from KM mechanism. SuperB-factories and LHC are expected to shed more light on these aspects.

Within the current scenario, answer to the latter however, can be conclusively estab-lished to be negative! As mentioned in chapter 0, the amount of matter-antimatter imbal-ance can be quantified as the asymmetry term ηasym ∼ O(10−10). On the other hand, anysuch imbalance originating from CP violating effects, given the KM mechanism holds,must be proportional to the re-phasing invariant or Jarlskog variable J , as discussed inchapter 1. Within Wolfenstein parametrization of the CKM matrix [15], valid to the orderO(|λ|10), the Jarslkog variable is empirically found to be

J = Aλ6η ∼ 10−5 (6.8)

which is much smaller than the maximum value allowed J < 1/(6√

3) ∼ 0.1. This showsthat the CP violation is highly suppressed within the SM due to the strong hierarchyexhibited by the CKM elements. It is important to bear in mind, that existence of CPviolation requires not only J to be non-zero, but also the existence of non-degeneratequark masses and the overall baryonic asymmetry which can be attributed to the CPviolating effects in SM is given by

ηasym(CP viol.) = JMu ×Md

M12(6.9)

133

EPILOGUE

where, Mu = (m2t −m2

c)(m2t −m2

u)(m2c −m2

u), Md = (m2b −m2

s)(m2b −m2

d)(m2s −m2

d) andthe M ∼ O(100 GeV) is the electroweak scale. This leads to

ηasym(CP viol.) ∼ 10−17 (6.10)

which is much smaller than the observed asymmetry in the universe.In summary, the CP violation observed so far is about seven orders of magnitude lower

than that required to explain the BAU and either CP violating effects external to SM orsources of generating the baryonic asymmetry external to CP violation are highly sought.Leptogenesis, baryogenesis and existence of physics beyond the standard model are few ofthe sources, where a possible answer to the baryonic asymmetry can be found, in future.

134

AStandard Model of Particle Physics

Abrief account of the SM is given here. There are many good texts available for a de-tailed study, though my personal favorite is by Cheng and Li [78], and by Ryder [79].

A.1 Schema of modelling

At the heart of it, standard model of particle physics or SM is an endeavour to construct aunified description of the fundamental forces in nature. We understand four basis forcesof interaction:

1. gravity

2. electromagnetic interaction,

3. weak interaction, and

4. strong interaction

The field theoretic framework used for explaining the gravitational and electromagneticinteractions demands two basic types of elements: elementary matter constituents withreceivers for the forces, equivalent to electric charges, and a mechanism through whichthese constituents can interact, formally called medium. The quantum version of field the-ory formalizes the matter constituents as the quantum states in the field theory, while themedium is imagined as composed of mediators, being continuously exchanged betweenthese quantum states. The energy momentum conservation forbids any random creation-exchange-and-absorption mechanism of these mediators. It is only the uncertainty prin-ciple in quantum mechanics ∆E∆t ∼ ~, which leaves a space for this exchange mechanism

135

APPENDIX A. STANDARD MODEL OF PARTICLE PHYSICS

via the uncertainty ∆E in the energy measurement accommodating the violating excessof energy for a time interval of ∆t, during which this measurement is carried. As a con-sequence, the average energy allowed for a mediator depends upon its average or meanlife time, and vice verse. The electromagnetic interaction is mediated via massless photons,allowing the photon to live for infinite time and hence travel infinite distances. Thus, theelectromagnetic interaction is said to have infinite range. In contrast, if the photon weremassive, the non-zero rest mass putting a lower bound on ∆E would have set a limit onthe range up to which the electromagnetic interaction could extend. As a bi-product of thisuncertainty principle, mediators and virtual particles can be created and annihilated con-stantly all the time, even from vacuum, as long as the uncertainties satisfy the principle.Due to this mute interaction between an elementary particle and the mediators or virtualparticles created in the vacuum surrounding it, the apparent particle properties, such asits mass or charge, differ from their bare values - values they would take if no uncertaintyprinciple existed. In more formal terms, the physical parameters observed in reality differin values from their formal analogue used in constructing the Lagrangian. The relationbetween the two is established via a process called renormalization. Extending the fieldtheoretic ideas to strong and weak forces, one would need to build a theory with all theabove necessary ingredients.

On a parallel note, a deeper theoretical outlook, especially that gained from nöther’stheorem [80], has already underlaid a platform for developing a unified formalism, util-ising the fundamental symmetries respected by the interactions. This demands construc-tion of a field theory, which manifestly supports the symmetries inherent in the nature. Asan effect, the standard model tries to identify the symmetry groups underlying the inter-action while developing the field theory and subsequently exploits the allowed spectrumof states as the elementary constituents. Generalizing the Lagrangian to hold local gaugesymmetries, one can recover the required mediators for the interaction.

A.2 Spectrum of SM

The gauge structure of the elementary particles, which now is broadly known as theSM, was first proposed by Glashow [81] and later enriched with the Higgs mechan-ism [82, 83, 84] by Weinberg [85] and Salam [86] and has gone through many iter-ations of development by a large population of physicists over time. According to thismodel, the elementary matter constituents are described as spin-1/2 fermions, whilethe mediators are necessarily spin-1 vector bosons. The electroweak interaction has aSU(2)L × U(1)Y gauge symmetry, which at energies lower than the electroweak break-ing scale gives rise to electromagnetic U(1)EM interaction. The strong interaction, onthe other hand, has a SU(3)C symmetric structure, governed by the color dynamics orQuantum Chromodynamics. Overall, apart from gravity, other three interactions can beunified into SU(3)C × SU(2)L × U(1)Y , which is known as the standard model.

One can distribute the fermions, which are the elementary matter constituents, intotwo groups, according to their participation in the strong interactions. Leptons do notcarry the strong charges, namely colors, and hence do not respond to strong interaction,while each quark comes with three versions - for three “rgb” colors. For example, the elec-tron e and the corresponding neutrino νe are leptons, while the u and d quark comprising

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A.2. SPECTRUM OF SM

a neutron or a proton are quarks. u and d each come with three colors, while there are nocolored electrons. The electron e, electron neutrino νe, u and d quark comprise of the firstgeneration of standard model particle content. Because of maximal parity violation, theleft and right handed versions of these particles behave differently under the weak interac-tions and hence must be accounted independently. For example, there is no right-handedneutrino found in nature. Overall,

Q = ((eL, νeL), eR, (uL, dL), uR, dR)T

constitutes the first generation of the SU(3)C × SU(2)L × U(1)Y group. Of course, eachparticle above comes with its anti-particle. It is interesting to note, that even thoughthree copies of above spectrum, known as generations, have been identified, only thefirst generation is sufficient to account for all the “everyday” matter at low energies. Theparticles belonging to second and third generation, except possibly the neutrinos, live onlyfor a brief period of time after generation and they are necessary not only to complete thespectrum at high energies, but also to sustain additional, but necessary formal structureto the theory, such as long-living strange particles, or CP violation.

It must be noted that the relativistic field theoretic description does not provide a con-sistent apparatus for introducing masses to fermions and gauge bosons, when the gaugesymmetric nature is invoked. In addition, the renormalizability of the theory requires theboson to be strictly massless. A mechanism needs to be devised to compensate for themissing mass terms in the Lagrangian. Higgs et al. suggested the well-sought mechanismfor introducing masses to the fermions and gauge bosons via spontaneous symmetry break-ing, which does not disturb the renormalizibility. Through this mechanism, all the gaugebosons, except photon acquire masses, which is required to keep the weak interactionsshort-ranged. The only requirement for this mechanism to be valid is existence of a spin-0scalar gauge boson, the Higgs boson. Note, that the Higgs boson is the only fundamentalscalar boson in SM.

This completes the spectrum as far as electroweak interaction is concerned. As men-tioned previously, the strong interaction requires non-zero color charges, present on thequarks. This color dynamics is mediated by eight color carrying generators of SU(3)C ,the spin-1 vector bosons or gluons. The remarkable property of strong interactions, whichmakes it special and at the same time immensely complex and non-perturbative, is itsasymptotic freedom. According to this property, the strength of strong interaction increaseswith distance, somewhat similar to the elastic forces experienced in a stretched spring.Due to this, it is impossible to observe a single isolated quark at low energies and thequarks are always realised as composite particles with total color zero, i.e. compositeswith quarks making up into color singlets. These composites are called hadrons. In simpleterms, there are two ways to make a colorless composition: either add three r, g, b colorsin equal amounts or add a color and subtract the same amount, or what is same as addingan anti-color. The quark composites made using the former method are called baryons,while the latter are the mesons. Since, quarks are never observed free or isolated, it isnot possible to estimate their masses by direct observations and usually depend upon thetheoretical scheme used to account for the interaction energies appearing as masses of thequarks in a composite state. The most successful one is known as the minimal subtractionscheme MS.

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APPENDIX A. STANDARD MODEL OF PARTICLE PHYSICS

A.3 Success of SM

It is a remarkable fact, that however complex, SM has been tremendously successful in en-compassing almost all the experimental observations noted so far, even after four decadesof its birth. There are incidences in history, that a particle was predicted much in advanceto its discovery and that too with the attributes matching perfectly well within those es-timated by the SM. A fundamental particle is identified by its mass and a set of quantumnumbers: its spin , electromagnetic charge, “color” and “flavor”. SM has been success-fully accounting for all the particle phenomenology based on the fundamental particleslisted above, with only one loose-end so far: the Higgs boson. Higgs boson has not yet ob-served, directly or indirectly, and enormous efforts are being put in to establish existenceof this only fundamental scalar in SM, giving masses to all other elements. The elementaryparticle spectrum in SM is summarized in table A.1.

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A.3. SUCCESS OF SM

Table A.1: The particle spectrum in the SM. Each of these particles has an anti-particle. (ref: PDGlive)

Particle Symbol Electric MassChargea (e) (MeV/c2)

Leptons (Spin - 1/2)electron e− -1 0.511

electron neutrino νe 0 < 3 · 10−6

muon µ− -1 105.66muon neutrino νµ 0 < 0.19 (90% C.L.)

tau τ− -1 1776.84± 0.17tau neutrino ντ 0 < 18.2

Quarksb,c (Spin - 1/2)up u +2/3 1.5− 3.3

down d -1/3 3.5− 6.0charm c +2/3 1270± 17strange d -1/3 70− 130

top t +2/3 (171.3± 1.1± 1.2)× 103

bottom b -1/3 4.20+0.17−0.07 × 103

Gauge Bosons (Spin - 1)photon γ 0 0W± W± ±1 (80.398± 0.025)× 103

Z0 Z0 0 (91.188± 0.002)× 103

gluons g 0 0Higgs (Spin - 0)

Higgsd H 0 . . .a charges in multiple of e = 1.6× 10−19 Cb u−, d−, s− quark masses are estimates of so-called “current quark masses” at mass scale µ ≈ 2 GeV, whereas c−, b− quark masses are the “running” masses, all in the MS scheme.c each with three colors: r, g, and b.d yet to be seen.

139


BAnalysis Tools

B.1 Blind Analysis Techniques

What would be the first reaction of a student, measuring the acceleration due to gravityg with a tabletop experimental setup in his lab, if he finds the estimated value comingout to be 15.1 m/s2? From the textbooks he uses, he probably knows that the expectedvalue should be around 10 m/s2 on any random place on earth. Not to loose his grades,he will start suspecting something wrong with the setup and will start retuning it untilhe recovers the value he has been taught in the course. Note, the tendency of this studentto ignore checking with the error-budget of his estimate, while struggling hard to get the“correct” central value!

Even in the professional world, such unexpected or surprising discoveries, althoughdesired, are often feared of, and are very rarely talked of loudly, since they bring in pos-sibilities of the experimenter facing severe criticism. Particularly, when some physicalquantity, for which some estimate already exists from previous measurements, is revisitedafresh, one tends to be overcautious, if some large deviation from the so far “standard”value is observed. Although there exist explicit plans in the literature for tackling observa-tions with apparently large deviation, one unintentionally chooses to put more faith in theprevious estimates and wishes to obtain a result consistent with them, mainly due to lackof avenues for checking the correctness by repeating the experiment and hence, to avoidpossible criticism if found wrong in future. This tendency to be overcautious about knownfacts can translate into an urge for obtaining the same result - consistent with the previousmeasurements - and subsequently add, what is known as experimenter’s bias in the result.

Figure B.1 is taken from a compendium maintained by Particle data group for demon-strating experimenter’s bias in high energy physics experiments. In most of the cases dis-

141


APPENDIX B. ANALYSIS TOOLS

Figure B.1: World averages of various physical quantities against year of Particledata group summary update. The distinct local-clustering of the averages of ex-perimental observations performed during the same period shows possibility ofbiased tuning of the setups. Particularly, the middle plots, upper one for K0

S life-time and the lower one for width of ω meson are alarming.

tinct year-segments can be marked, during which all the experimental results lie close toeach other within the segment, while showing large deviation from typical values in theother. A systematic shift in one direction in all the measurements done during the sameperiod is hard to explain, unless a biased setup to obtain desired result is used. In mostof the plots above, averages during the intermediate period were offset from the latestknown values by about 3 standard deviations.

Blind analysis is a technique used to avoid addition of any experimenter’s bias, whileoptimizing the experimental apparatus. In general, the analysis is done without lookingat the answer, by conveniently blinding that portion of the data which is expected tocontain signal events. The blind analysis technique was first used in bio-medical research,employing the double-blinding method, in which the information is kept hidden fromboth clinical surveyor as well as patients under study. In the context of particle physicsexperiments, the following blind analysis techniques are in fashion, though at times manyvariants of these simple methods are used:

• Signal box blindingthe signal region is explicitly blinded until the analysis and optimization is com-pleted. This technique is very well-suited in analyses intended to observe rare pro-cesses, when the signal characteristics are known in advance. This technique hasbeen applied to the B0 → D∗+s π− and B0 → D∗−s K+ decay branching fractionmeasurements performed in this work.

142



B.2. FISHER DISCRIMINANT

• Answer blindingin cases, where signal parametrization itself is an unknown measurable quantity, i.e.when signal box is not well-defined, the previous technique can not be employed. Insuch cases, blinding the value for the parameter being estimated helps in minimizingany possibility for addition of bias. Th blinding of answer is done by adding anunknown offset to the fit result.

• Asymmetry blindingthis technique is specifically designed for CP violation studies, where the sign of ∆tasymmetry is randomly chosen, in addition to blinding its value by adding a randomoffset.

A little detailed information on blind analysis techniques mentioned above can be ob-tained from [87] and the references therein.

B.2 Fisher Discriminant

One of the major tasks in data analysis, involving rare signal events, is to separate thesignal events from the background ones. This process can be very complicated dependingupon the nature of the signal events and the variables which can be used to differentiatethem from the background events.

Even in the simplest case, where a typical signal event has some physical attributedistinctly different from the background so that this single physical observable suffices todefine a signal event in principle, it is not always easy in practice to separate it from thebackground. For example, the signal and the background distributions can show promin-ent peaking structures with distinct means, but significant overlap. In such cases, choosinga signal selection window, based on a simple probabilistic model, may be not be effective.

B.2.1 Theory of resolving power

Intuitively, if a distribution has contributions from two independent sources resulting intwo separate but overlapping components, the two can be resolved, if the distributionshows a prominent inflection region, if not a clear local minimum. In figure B.2 twoGaussian functions are overlapped to make a third total PDF and the relative distancebetween the two Gaussians is varied. While in the first configuration, it is difficult toargue, looking only at the total, that it is made of two separate underlying components,the other on the right poses no such confusion and one can readily claim without effortsexistence of more than one Gaussian components. In practice, the situation is even moreobscured by the statistical fluctuations, due to limited or finite sampling.

Mathematically, whether two components in a distribution can be resolved or not de-pends upon the distances between their means ∆µ = |µ1 − µ2| and the extent to whicheach of them is spread, i.e. their widths σ1, σ2, or in a more compact form on,

F ∝ D2

S(B.1)

with D = f(|µ1 − µ2|) and S = g(σ1, σ2), i.e. intuitively, more the distance between thetwo means or narrower the two components, better the separability.

143


Figure B.2: Two separate components in a distribution can be separated only ifthere exists at least an inflection point in the total.

Even when a physical quantity being measured has a well-defined Dirac-δ distribution,a realistic detector tends to smear the measured value due to its finite non-zero intrinsicresolution. As a consequence, for a given detector resolution σ, two distinct values for thephysical quantity can not be resolved by the detector if the two values lie closer than alimit set by its resolving power F .

B.2.2 Linear Discriminator in multivariate analysis

It is not always possible to find one physical variable, with respect to which two distinctclasses will have well resolved distributions. On the contrary, events from the two classescan have considerably overlapping values along any single variable and hence, to be ableto efficiently discriminate between the two, it is required to apply number of selectioncriteria on multiple variables. This procedure is not very useful, particularly when numberof variables increase, since every selection requirement put on brings in reduction in re-construction efficiency. More seriously, this procedure does not take into account possiblecorrelations among these variables and end up having unwanted biases with un-optimizedreconstruction of signal.

To overcome these limitations inherent to the multivariate analyses techniques, R. A.Fisher introduced the idea of linear discriminator in his celebrated article on "The use ofmultiple measurements in taxonomic problems" [65]. Figure B.3 illustrates this technique.It can be seen that events from two hypothetical classes are spread reasonably overlappingalong x and y variables and none of them can be used to separate the two. Also, fromthe 2-D plot itself, it is evident that the two variables x and y are correlated, and henceconstraining events in one variable is automatically translated into a constraint on theother via the correlation. Hence, x or y do not offer a good choice of discriminator aloneand one has to apply constraints on both x and y together taking the correlation amongthem into consideration. Contrary to x − y, the set of variables x′ − y′, which is obtainedby a simple rotation of the x− y axes, can yield a very effective avenue for discriminatingthe two classes. It is easy to see that the x′ axis efficiently separates the two classes, orwhat is same as to say that the y′ has the maximum resolving power for the two classes,and applying constraint only on a single variable y′ would suffice. Since x′ − y′ system is

144

B.2. FISHER DISCRIMINANT

x

x′

y

y′

Figure B.3: x − y axes may not be a good choice for discriminating the eventsfrom two classes, though x′ − y′ can provide a clear separation.

obtained by a simple rotation, and sometimes additionally a shift of origin, the variabley′ is a linear combination of the original set of variables: x, y. Hence the name lineardiscriminator.

Though simple to visualize on a 2-D plot, most of the real cases involve more than twovariables and it is necessary to construct a formal procedure on how to obtain the lineardiscriminator, similar to y′ above, in a multivariate case. Digression 4 below illustratesthe procedural details derived from Fisher’s work, using the concept of resolving powerdeveloped above.

Digression 4. Let ~x = x1, x2, ...., xn be the set of physical quantities measured in order todiscriminating between two or more classes. Without loss of generality, the procedure is developedhere, assuming only two classes. The task is to figure out a linear combination of xi’s

y = λ1x1 + λ2x2 + λ3x3 + · · ·+ λnxn = λixi

which will offer the best discriminating power. Equivalently, y is a variable along which the twoclasses will have the best resolved distributions. Note, we assume summation over repeated indices.

Let, µ1i and µ2

i be the means of the distributions for the two classes along x1. We define thedifference between the means as

di = |µ1i − µ2

i | (B.2)

The covariance matrix is defined as,

Cij =∑p

〈(xi)p − µi〉〈(xj)p − µj〉 (B.3)

where, p runs over all events.With respect to y, we have the mean µ, mean difference D and variance S, respectively given

145


by,

µ = λ1µ1 + λ2µ2 + · · ·+ λnµn = λiµi (B.4)

D = λ1d1 + λ2d2 + · · ·+ λndn = λidi (B.5)

S =∑p,q

(yp − µ)(yq − µ)

= λiλjCij (B.6)

where, i, j run over all variables and p, q run over all data-pints.In order to obtain the direction y, along which the two classes are best resolved, one needs to

extremize the resolving power F = D2/S in equation (B.1) with respect to λi’s, i.e.

∂

∂λi

(D2

S

)= 0 ⇒ D

S

[2∂D

∂λi− D

S

∂S

∂λi

]= 0 (B.7)

Inserting from equations (B.5) and (B.6), we have

1

2λiCij =

(S

D

)di (B.8)

It should be noted, that the factor of S/D is a constant for the λi’s satisfying equation (B.8) andthe solution |λ〉 to this equation is proportional to the solution |λ′〉 of

C|λ′〉 = |d〉 (B.9)

And, hence the linear combination of xi’s, along which the two (or more) classes show maximalseparation, is completely known from the covariance matrix C and the differences |d〉 in the meansalong measured physical quantities. The existence of non-trivial solution requires non-singularityof the covariance matrix C, which is a symmetric matrix, as well as on |d〉 being not all-zero.

B.3 Kinematical constrain-fit

While reconstructing the tracks from the hits in the tracking detectors, providing somebackground information regarding the kind of physical process involved helps the trackfitting module not only find the correctly reconstructed tracks, but also improve in themomentum and position determination of a track. Unfortunately, the naïve track findingalgorithm can not be provided with such information, in advance to the higher level eventreconstruction, due to randomness in the nature of particle decays. Because of this, thetrack momenta and positions obtained from the track-fitting algorithm, without using thekinematical relations among tracks, usually carry larger uncertainties than that wouldhave been obtained using these relations. For obvious reasons, the kinematical relationsamong various seemingly uncorrelated tracks can be established only consequently, asparticles are reconstructed backward along the decay evolution - toward initial-states.The kinematical fitter, namely the Kalman fitter or kFitter [60], allows one to use thesekinematical relations among daughter tracks in a decay to minimize uncertainties in thetrack parameters, such as momenta and positions. The working principle is explained in

146

B.3. KINEMATICAL CONSTRAIN-FIT

brief here. For a detailed introduction to the basic fitting algorithm and usage, one canrefer the Belle Note 193 and Belle Note 194.

A detected track is represented by its six phase-space co-ordinates,

PPPtrack = (px, py, pz, E, x, y, z) (B.10)

where, (px, py, pz, p0 = E) is the four momentum of the track and (x, y, z) the closestpoint of approach to the pivot position, which is a reference point around IR, used in helixparametrization. The mass of the track is a fixed parameter. For a photon cluster, thepivot position is usually chosen to be either the geometrical detector centre or the IP ofthe event. Because mass of a track is a fixed parameter, the four momentum co-ordinatesare not all independent, but one of them, say E can be eliminated using the relationE =

√m2 + p2.

In general, the kfitter uses the χ2−minimization technique, where a χ2 is constructed,based on the deviation of the measured quantities of a track, i.e. hits in the tracker, fromthe values calculated using equations of motion, based on PPPtrack above. The constraints,i.e. invariant mass of the composite particle or position of the vertex to a known value,etc., are incorporated into the minimization process using Lagrange multipliers scheme.The typical constraints applied are,

• common vertex constraintalmost in all decays, the tracks produced in a decay can be constrained to come froma common decay vertex,

• common vertex constrain to known pointsometimes, one or more tracks from a decay can determining the decay vertex veryprecisely and the rest of the tracks can be demanded to come from the vertex alreadyknown. For example, in a D∗+ → D+π0 decay, the D+ decay vertex can be preciselydetermined from its charged daughters. The tiny lifetime of D∗+ meson allows oneto assume the D∗+ decay vertex to be nearly concurrent with that of D+ and hence,the π0 can be constrained to come from the D+ vertex.

• invariant mass-constraintif the mother particle has a sharp invariant mass, with negligible width, the daughtermomenta can be constrained to add to the fixed nominal mass of mother.

The mathematical details of the algorithm are developed in the digression 5 below.

Digression 5. Let the parameter vector ααα of measured unknown quantities be

ααα = (α1, α2, . . . , αn)T (B.11)

In principle, these quantities are n functions of PPPtrack obtained using the equation of motion forthe tracks. Let the initial, unconstrained or experimentally measured raw values be ααα0. The rconstraints can be written as HHH(ααα) = 0. Expanding HHH at some conveniently chosen point αααA, oneobtains

0 = HHH(αααA) +∂HHH(αααA)

∂ααα(ααα−αααA) ≡ d+Dδααα (B.12)

147




This linear equation in constraints can be incorporated into the χ2 using r Lagrange multipliers λλλ.The function to be minimized is then,

χ2 = (ααα−ααα0)TV −1ααα0

(ααα−ααα0) + 2λλλT D(ααα−αααA) + d (B.13)

where V is the covariance matrix for the measured quantities ααα.The χ2 is minimized with respect to the λλλ and ααα to give two vector equations, which can be

solved for the r unknown multipliers in λλλ and n unknown measured quantities in ααα:

ααα = ααα0 − Vααα0DTλλλ

VD = (DVααα0DT )−1

λλλ = VD(Dδααα0 + d)Vααα = Vααα0 − Vααα0D

TVDDVααα0

χ2 = λλλTV −1D λλλ = λλλT (Dδααα0 + d)

(B.14)

Due to the kinematical constrain fitting, done posterior to a decay reconstruction, mo-mentum resolutions of not only the daughter tracks, but also the composite mother particleimprove. The amount of improvement however, depends upon the kind of tracks involved,ranging from a few percent for a charged track up to a few tens percent for a neutral track,such as a photon. In case of D∗+s mass-constrain-fit, for example, the ∆E resolution isfound to improve by about 32%, as shown in figure B.4. The blue (red) histogram showsthe distribution of the B0 → D∗+s π− signal events in the ∆E fit-region before (after) theD∗+s daughter tracks are mass-constrain-fitted.

B0 → Ds*+π-: Effect of MDs* Constraint

File: *massConstrained.hbkID IDB Symb Date/Time Area Mean R.M.S.

16 2 1 090616/0005 1.000 -1.0991E-03 3.3876E-02

-0.20 -0.10 0.00 0.10 0.20∆E (GeV)

0.00

0.10

0.20

0.30

0.40

# of

Eve

nts,

Before MDs* constraintAfter MDs* constraint

φ mode

Normalized to unit area

16 1 1 090616/0005 1.0000 -1.8073E-04 3.5285E-02

Figure B.4: The effect of D∗+s mass constrain fit on the ∆E distribution: Theblue (red) histogram shows the ∆E distribution before (after) applying the massconstrain fit. The resolution improves by 32% reducing from 19 MeV to 13 MeV.

148

Glossary

Glossary

Cabibbo-favored decay (CFD) Cabibbo-favored decay, involving most off-diagonal CKMmatrix elements; e.g. a b→ u transition, 18, 19, 23, 24, 123

doubly Cabibbo-suppressed decay (DCSD) doubly Cabibbo-suppressed decay, involvingdiagonal or near-diagonal CKM matrix elements; e.g a b → c transition, 18–21, 23,123, 155

data summary tables (DST) database structures used to store information after raw datahas been processed, 44

high energy (e− beam) ring (HER) High energy (elecrton beam) Ring. Because the positronbeam is derived from electron ring, this is maintained at higher energy of 8 GeV, 32

heavt quark effective theory (HQET) An approximation to correct the heavy quark sym-metry (HQS) quantities in series of 1/mQ, where mQ is the heavy quark mass, 22,124

heavy quark symmetry (HQS) A simplification in the (originally) non-perturbative QCD,where the heavy quarks are assumed infinitely massive, 22, 124, 147

interaction point (IP) the point of interaction of the two beams, 32, 35, 39, 43, 44, 52,145

interaction region (IR) region located in Tsukuba experimental hall, where the two beamsinteract at a finite crossing angle of ±11 mrad, 32, 34, 145

low energy (e+ beam) ring (LER) low energy (positron beam) Ring with energy 3.5 GeV,derived from the high energy e− ring, 32

standard model (SM) standard model of elementary particles, which unifies elecromag-netic and nuclear forces into a gauge theory with structure of SU(3)C × SU(2)L ×U(1)Y , 3–6, 10, 16, 17, 19, 21, 25, 133–135, 137

unitary triangle (UT) an (imaginary) triangle realized of the unitarity property of CKMmatrix. In this work we refer to the relation specific to Bd(u) system as UT, 8, 16

149

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INDEX

Index

D+-mass veto, 73RD∗π

definition, 20from B0 → D∗+s π−

determined, 123theory, 21

theoretical prediction, 20Vub, 25, 127∆M

definition, 54sidebands, 84

φ3

definition, 8extraction, 18from B0 → D∗+s π−, 125

RK/πdefinition, 41efficiency, 41

CP transformationphase invariants, 7violating parameters ng, 6

CP violationBB evolution, 11BB mixing, 9

CP violationclassification, 12

DCSD, doubly Cabibbo-suppressed decay,19

∆E, 79definition, 56

Ebeam, beam energy, 32continuum events

defined, 30scintillation, 43Mbc, 72

definition, 56sidebands, 86

Υ(4S)evolution, 14at e+e− collider, 29

ACC, Aerogel Cerenkov chamber, 40ARGUS function, 67

background processesBB decays, 59–62classification, 59continuum events, 66rare B decays, 62–66

BAU, Baryonic Asymmetry in Universe, 1Belle detector

components, 33Blind analysis techniques, 141

Cabibbo angle θC , 5Cabibbo-favored decays, see Wolfenstein

parametrizationCabibbo-suppressed decays, see Wolfen-

stein parametrizationCDC, Central drift chamber, 36

ionization loss in, 39CFD, Cabibbo-favored decay, 19CKM angles, 8continuum suppressionRtotal, 69

systematics, 114Fisher discriminant, see Fisher linear

discriminant, 68control studies, 90–96

for K0S systematics, 112

for photon systematics, 108for PID systematics, 112for track-finding systematics, 107

decay constants, 22, 124fD+

spuzzle, 25

157

INDEX

DST, data summary tables, 45

ECL, Electromagnetic calorimeter, 43–44extended unbinned maximum-likelihood

fit, 90

Fisher linear discriminant, 143fit-region, 57Fox-Wolfram moments, 68

helicityφ and K∗(892)0 meson, 71fL, fraction of longitudinal polariza-

tion, 131D∗+s meson, 54formalism, 129

IR, interaction region, 33

Jarlskog invariant, seeCP transformation

KEKBasymmetric e+e− collider, 32data at, 31

kFitter, 146D∗+s mass-constrain-fit, 56D+s mass-constrain-fit, 53

kinematical constrain-fit, see kFitterKM, Kobayashi-Maskawa mechanism, 6

MC, Monte Carlo simulations, 45

off-resonance data, 86

PID, particle identification, 37, see recon-struction efficiencies, corrected

systematics, 112

quartets, see CP transformation

reconstruction efficienciescorrected, 94for B0 → D∗+s ρ−, 130in ∆E, 80in Mbc, 75

resolution∆E, 57Mbc, 57impact parameter, 35

tracking, 37

signal significance, 119signal-region, 57SM, Standard model of elementary particle

physics, 135SVD, Silicon vertex detector, 34

SVD1 and SVD2, 34symmetry

charge conjugation C, 4parity P, 3time reversal T , 4

TOF, Time-of-flight detector, 39

Unitarity triangle, 8

Wolfenstein parametrization, 7

158

List of Publications(last updated on February 25, 2011)

Submitted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[1] 2010a (Belle Collab.) with A. Das et al.: Measurements of Branching Fractions for B0 →2010D+

s π− and B0 → D+s K

−. In: [submitted to Phys. Rev. D (rapid)].

[2] 2010b (Belle Collab.) with S. Bahinipati et al.: Measurements of time-depentent CPasymmetries in B0 → D∗∓π± decays using a partial reconstruction technique. In: [(tobe) submitted to Phys. Rev. D (rapid)].

[3] 2010c (Belle Collab.) with A. Bozek et al.: Observation of B+ → D∗0τ+ντ and Evidencefor B+ → D0τ+ντ at Belle. In: [submitted to Phys. Rev. Lett.].

[4] 2010d (Belle Collab.) with S. Esen et al.: Observation of Bs → D(∗)+s D

(∗)−s using e+e−

collisions and a determination of the Bs − Bs width difference ∆Γs. In: [submitted toPhys. Rev. Lett.].

Journal Published . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[5] 2010a (Belle Collab.) with C. P. Shen et al.: Evidence for a new resonance and search for2010the Y(4140) in γγ → φJ/ψ. In: Phys. Rev. Lett. 104, p. 112004.

[6] 2010b (Belle Collab.) with A. Poluektov et al.: Evidence for direct CP violation in thedecay B → D(∗)K, D− → KSπ

+π− and measurement of the CKM phase φ3. In: Phys.Rev. D81, p. 112002.

[7] 2010c (Belle Collab.) with T. Aziz, Karim Trabelsi et al.: Measurement of the branchingfractions for B0 → D∗+

s π− and B0 → D∗−s K+ decays. In: Phys. Rev. D81, p. 031101.

[8] 2010d (Belle Collab.) with S. Uehara et al.: Observation of a charmonium-like enhance-ment in the γγ → ωJ/ψ process. In: Phys. Rev. Lett. 104, p. 092001.

[9] 2010e (Belle Collab.) with R. Louvot et al.: Observation of B0s → D∗−

s π+, B0s → D

(∗)−s ρ+

Decays and Measurement of B0s → D∗−

s ρ+ Polarization. In: Phys. Rev. Lett. 104,p. 231801.

[10] 2010f (Belle Collab.) with H. J. Hyun et al.: Search for a Low Mass Particle Decayinginto mu+mu− in B0 → K∗0X and B0 → ρ0X at Belle. In: .

[11] 2010g (Belle Collab.) with C. C. Chiang et al.: Search for B0 → K∗0K∗0, B0 → K∗0K∗0

and B0 → K+π−K∓π± Decays. In: Phys. Rev. D81, p. 071101.

[12] 2010h (Belle Collab.) with B. R. Ko et al.: Search for CP violation in the decays D+(s) →

K0Sπ

+ and D+(s) → K0

SK+. In: Phys. Rev. Lett. 104, p. 181602.

[13] 2010i (Belle Collab.) with M. Petric et al.: Search for leptonic decays of D0 mesons. In:Phys. Rev. D81, p. 091102.

[14] 2009a (Belle Collab.) with R. Mizuk et al.: Dalitz analysis of B → Kπψ′ decays and the2009Z(4430)+. In: Phys. Rev. D80, p. 031104.

[15] 2009b (Belle Collab.) with S. Uehara et al.: High-statistics study of neutral-pion pairproduction in two-photon collisions. In: Phys. Rev. D79, p. 052009.

[16] 2009c (Belle Collab.) with K. Abe et al.: Improved Measurement of Inclusive RadiativeB-meson decays. In: AIP Conf. Proc. 1078, pp. 342–344.

[17] 2009d (Belle Collab.) with K. Belous et al.: Measurement of cross sections of exclusivee+e− → V P processes at

√s = 10.58 GeV. In: Phys. Lett. B681, pp. 400–405.

[18] 2009e (Belle Collab.) with E. Won et al.: Measurement ofD+ → K0SK

+ andD+s → K0

Sπ+.

In: Phys. Rev. D80, p. 111101.

Typeset in LATEX

[19] 2009f (Belle Collab.) with A. Sokolov et al.: Measurement of the branching fraction forthe decay Υ(4S) → Υ(1S)π+π−. In: Phys. Rev. D79, p. 051103.

[20] 2009g (Belle Collab.) with R. Louvot et al.: Measurement of the Decay B0s → D−

s π+ and

Evidence for B0s → D∓

s K± in e+e− Annihilation at

√s 10.87-GeV. In: Phys. Rev. Lett.

102, p. 021801.

[21] 2009h (Belle Collab.) with P. Pakhlov et al.: Measurement of the e+e− → J/ψcc crosssection at

√s ∼ 10.6 GeV. In: Phys. Rev. D79, p. 071101.

[22] 2009i (Belle Collab.) with A. Zupanc et al.: Measurement of yCP in D0 meson decays tothe K0

SK+K− final state. In: Phys. Rev. D80, p. 052006.

[23] 2009j (Belle Collab.) with S. H. Kyeong et al.: Measurements of Charmless Hadronicb → s Penguin Decays in the ππKπ Final State and Observation of B0 → ρ0K+π−.In: Phys. Rev. D80, p. 051103.

[24] 2009k (Belle Collab.) with Y. W. Chang et al.: Observation of B0 → ΛΛK0 and B0 →ΛΛK∗0 at Belle. In: Phys. Rev. D79, p. 052006.

[25] 2008a (Belle Collab.) with N. Taniguchi et al.: Measurement of branching fractions,2008isospin and CP- violating asymmetries for exclusive b → dγ modes. In: Phys. Rev. Lett.101, p. 111801.

[26] 2008b (Belle Collab.) with K. Abe et al.: Measurement of B(D+s → µ+νµ). In: Phys. Rev.

Lett. 100, p. 241801.

[27] 2008c (Belle Collab.) with T. Lesiak et al.: Measurement of masses of the Ξc(2645) andΞc(2815) baryons and observation of Ξc(2980) → Ξc(2645)π observation of Ξc(2980) →Ξc(2645)π. In: Phys. Lett. B665, pp. 9–15.

[28] 2008d (Belle Collab.) with H. Sahoo et al.: Measurements of time-dependent CP viola-tion in B0 → ψ(2S)KS decays. In: Phys. Rev. D77, p. 091103.

[29] 2008e (Belle Collab.) with J. Brodzicka et al.: Observation of a new DsJ meson in B+ →D0D0K+ decays. In: Phys. Rev. Lett. 100, p. 092001.

[30] 2008f (Belle Collab.) with J. H. Chen et al.: Observation of B0 → ppK∗0 with a large K∗0

polarization. In: Phys. Rev. Lett. 100, p. 251801.

[31] 2008g (Belle Collab.) with V. Bhardwaj et al.: Observation of B± → ψ(2S)π± and searchfor direct CP-violation. In: Phys. Rev. D78, p. 051104.

[32] 2008h (Belle Collab.) with G. Pakhlova et al.: Observation of ψ(4415) → DD∗2(2460)

decay using initial-state radiation. In: Phys. Rev. Lett. 100, p. 062001.

[33] 2008i (Belle Collab.) with M. Iwabuchi et al.: Search for B+ → D∗+π0 decay. In: Phys.Rev. Lett. 101, p. 041601.

[34] 2008j (Belle Collab.) with K. Abe et al.: Search for B0 → Λ+c Λ

−c decay at Belle. In: Phys.

Rev. D77, p. 051101.

[35] 2008k (Belle Collab.) with Y. Nishio et al.: Search for lepton-flavor-violating τ → `V 0

decays at Belle. In: Phys. Lett. B664, pp. 35–40.

[36] 2008l (Belle Collab.) with J. Li et al.: Time-dependent CP Asymmetries in B0 → K0Sρ

0γDecays. In: Phys. Rev. Lett. 101, p. 251601.

[37] 2007a (Belle Collab.) with J. Dalseno et al.: Measurement of Branching Fraction and2007Time-Dependent CP Asymmetry Parameters in B0 → D∗+D∗−KS Decays. In: Phys. Rev.D76, p. 072004.

[38] 2007b (Belle Collab.) with K. F. Chen et al.: Search for B → h(∗)νν Decays at Belle. In:Phys. Rev. Lett. 99, p. 221802.

[39] 2007c (Belle Collab.) with O. Tajima et al.: Search for the CP-violating decays Υ(4S) →B0B0 → J/ψK0

S + J/ψ(ηc)K0S . In: Phys. Rev. Lett. 99, p. 211601.

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Study of the B0 to Ds*X decays at Belle · 2013-09-08 · Study of the B0!D sXdecays at Belle A Thesis Submitted to the Tata Institute of Fundamental Research, Mumbai for the degree