**Study of doubly-charmed B → ¯D**

**Study of doubly-charmed B → ¯D**

**(?)**

_{D}

_{D}**(?)**

_{K decays}

_{K decays}**at Belle**

**Jolanta Brodzicka**

Henryk Niewodnicza´nski Institute of Nuclear Physics, Polish Academy of Sciences,

Krak´ow, Poland

A thesis submitted for the Doctor degree

prepared under the supervision of Assoc. Prof. Maria R´o˙za´nska

**i**

**A**

*In this thesis, doubly-charmed B → ¯D*(?)* _{D}*(?)

_{K decays are studied. The analysis is based on the data sample}of about 275 × 106 * _{B ¯B pairs recorded by the Belle detector, operating at the KEK-B asymmetric-energy e}*+

_{e}_{−}

*collider. Statisticaly significant signals have been observed in 18 B → ¯D*(?)* _{D}*(?)

_{K decay modes; six of them}*represent first observations. The branching-fraction measurement for the nearly complete set of B → ¯D*(?)* _{D}*(?)

_{K}*decays enabled isospin-amplitude analysis and provides a valuable information on B meson decay dynamics.*
*The Dalitz-plot studies have been performed for the two isospin-conjugated modes B*+

→ ¯*D*0*D*0*K*+*and B*0 →

*D*−* _{D}*0

*+*

_{K}*2*

_{. A candidate for a new c ¯s state at the mass 2.72GeV/c}_{has been found.}

**S**

**TRESZCZENIE**

*Praca dotyczy badania klasy podw´ojnie powabnych rozpad´ow B → ¯D*(?)* _{D}*(?)

_{K. Analiza została przeprowadzona}w oparciu o 275×106_{par mezon´ow B ¯B zarejestrowanych w detektorze Belle pracujacym na akceleratorze }

*KEK-B, kt´ory jest zderzaczem e*+_{e}_{−}_{o asymetrycznych energiach wiazek i całkowitej energii w układzie ´srodka masy}

*r´ownej masie rezonansu Υ(4S ). Statystycznie znaczace sygnały zostały zaobserowane dla 18-tu kanał´ow rozpadu*

*B → ¯D*(?)* _{D}*(?)

_{K; sze´s´c z nich zaobserwowano po raz pierwszy. Stosunki rozgałezie´n zmierzone dla niemal}kompletnej klasy rozpad´ow umo˙zliwiły przeprowadzenie analizy amplitud izospinowych, a tak˙ze dostarczaja
*cennych informacji o dynamice rozpad´ow mezon´ow B. Dla dw´och sprze˙zonych izospinowo kanał´ow rozpadu:*

*B*0_{→ ¯}* _{D}*0

*0*

_{D}*+*

_{K}*+*

_{i B}*→ D*−*D*0*K*+przeprowadzona została analiza diagram´ow Dalitza. Zaobserwowano strukture
*w układzie D*0* _{K}*+

*2*

_{przy masie około 2.72GeV/c}

_{, kt´ora mo˙ze by´c nowym stanem c ¯s.}**iii**

**A**

I would like to thank all these people who helped me somehow in my work and life during last a few years, when I have prepared my thesis. I am grateful for your support and encouragement, inspiration and all the devoted time. Reading this, you will recognize that my warm thanks are exactly for you. I am sure of this.

Thank you all

Chciałabym podziekowa´c wszystkim tym, kt´orzy w jakikolwiek spos´ob pomogli mi w mojej pracy i w ˙zyciu w ciagu ostatnich kilku lat kiedy przygotowywałam ten doktorat. Jestem Wam wdzieczna za Wasza pomoc i wsparcie, inspiracje oraz za czas kt´ory mi po´swiecili´scie. Jestem pewna, ˙ze czytajac to bedziecie wiedzie´c, ˙ze moje gorace podziekowania sa wła´snie dla Was.

**v**

**C**

**Abstract**

**i**

**Acknowledgments**

**iii**

**Contents**

**v**

**Chapter I**

**Introduction**

**1**

**Chapter II** * Phenomenology of B → ¯D*(?)

*(?)*

_{D}

_{K}

_{decays}

_{3}**II.1** Preliminaries . . . 3

**II.1.1** Flavor structure of the SM and the CKM matrix . . . 3

**II.1.2** CP violation . . . 4

**II.2** *Quark diagrams for B → ¯D*(?)* _{D}*(?)

_{K}_{. . . .}

_{5}

**II.3** Isospin relations . . . 6

**II.4** *Hadronic effects in B decays* . . . 8

**II.4.1** Heavy-quark symmetry . . . 9

**II.4.2** Factorization . . . 10

**II.4.3** Color suppression . . . 11

**II.4.4** Chiral symmetry . . . 12

**Chapter III** **Physics motivations and prior studies** **13**
**III.1** *CP violation and measurement of β in B*0
→ ¯*D*(?)*D*(?)*K*0*s* . . . 13

**III.2** Decay process dynamics . . . 14

**III.3** *Spectroscopy of c ¯s and c¯c states* . . . 15

**III.4** *The ’charm-counting’ puzzle and earlier studies of B → ¯D*(?)* _{D}*(?)

_{K}_{. . . 16}

**Chapter IV** **The experimental apparatus** **19**
**IV.1** Specifics of the B-factory environment . . . 19

**IV.2** The KEK-B B-factory . . . 20

**IV.3** Main characteristics of the Belle detector . . . 21

**Chapter V** **Data analysis** **25**
**V.1** Data and Monte Carlo samples . . . 25

**V.2** Selection . . . 25

**vi** **C**

**V.2.2** *Selection criteria for K*±_{, π}±* _{and K}*0

*s*, π0 . . . 26

**V.2.3** *D meson reconstruction* . . . 27

**V.2.4** *D*?_{meson reconstruction} _{. . . 28}

**V.3** *Reconstruction of B → ¯D*(?)* _{D}*(?)

_{K}_{. . . 28}

**V.3.1** *Selection of the best B candidate* . . . 28

**V.3.2** Background suppression . . . 34

**V.3.3** Background study . . . 34

**V.3.4** Reconstruction efficiency . . . 37

**V.3.5** Signal estimation . . . 39

**Chapter VI** **Results** **41**
**VI.1** Branching-fraction measurements . . . 41

**VI.1.1** ∆*E and M _{bc}*distributions and the fit results . . . 41

**VI.1.2** Determination of branching fractions . . . 47

**VI.1.3** Systematic-error studies . . . 50

**VI.2** *Checking isospin relations for B → ¯D*(?)* _{D}*(?)

_{K}_{. . . 53}

**VI.3** Implications for the ’charm-counting’ puzzle. . . 54

**VI.4** Search for resonances . . . 55

**VI.4.1** *Dalitz-plot analysis for B*+
→ ¯*D*0*D*0*K*+ _{. . . 57}

**VI.4.2** Angular distributions . . . 64

**VI.4.3** *Dalitz plot for B*0* _{→ D}*−

*0*

_{D}*+*

_{K}_{. . . 66}

**Chapter VII** **Summary** **71**

**Appendix A** **Best B selection efficiency****73**

**Appendix B** * Signal migration between B → ¯D*(?)

*(?)*

_{D}

_{K}

_{modes}

_{75}**Appendix C** **Physical constants used in the analysis** **77**

**vii**

**List of Figures**

**II-1** Constraints on the CKM unitarity triangle . . . 6

**II-2** *Leading quark diagrams for B → ¯D*(?)* _{D}*(?)

_{K decays}_{. . . .}

_{7}

**II-3** *Suppressed quark diagrams for B → ¯D*(?)* _{D}*(?)

_{K decays}_{. . . .}

_{8}

**IV-1** The KEK-B ring . . . 20

**IV-2** The Belle detector . . . 22

**V-1** *R*2distributions . . . 26

**V-2** *CMS momentum spectra for mesons produced in B → ¯D*(?)* _{D}*(?)

_{K}_{. . . 27}

**V-3** *¯D*0_{invariant-mass distributions}_{. . . 29}

**V-4** *D*−_{invariant-mass distributions} _{. . . 30}

**V-5** ∆*M(D*?−) distributions . . . 31

**V-6** ∆*M( ¯D*?0) distributions . . . 32

**V-7** LR*D(MD) for D*0*and D*+decay modes . . . 33

**V-8** LR*D*?±*(∆M(D*?)) shapes . . . 33

**V-9** LR*D*vs LR*¯Dfor B*+→ ¯*D*0*D*0*K*+*and B*0*→ D*−*D*0*K*+ . . . 34

**V-10** *B*+
→ ¯*D*0*D*0*K*+_{∆}_{E distributions for various LR}*B*cuts . . . 35

**V-11** *B*0
*→ D*−*D*0*K*+∆*E distributions for various LR _{B}*cuts . . . 35

**V-12** *Background components for B*+
→ ¯*D*0*D*0*K*+* _{,B}*0

*→ D*−

*D*0

*K*+

*0*

_{and B}*→ D*?−

*D*0

*K*+

_{. . . 35}

**V-13** *M(D) and M( ¯D) distributions for B*+
→ ¯*D*0*D*0*K*+* _{and B}*0

*→ D*−

*0*

_{D}*+*

_{K}_{. . . 36}

**V-14** ∆*E and M _{bc}distributions for B*+

_{→ ¯}

*D*0

*D*0

*K*+signal MC events. . . 37

**V-15** ∆*E and Mbcdistributions for B*0*→ D*?−*D*0*K*+signal MC events . . . 37

**V-16** ∆*E and M _{bc}distributions for B*+

_{→ ¯}

*D*?0

*D*0

*K*+signal MC events . . . 38

**V-17** ∆*E vs Mbc*data distribution and two-dimensional likelihood fit result . . . 40

**VI-1** ∆*E and Mbcdistributions for B*+→ ¯*D*0*D*+*K*0*s* . . . 42

**VI-2** ∆*E and M _{bc}distributions for B*0

*−*

_{→ D}*D*0

*K*+ . . . 42

**VI-3** ∆*E and Mbcdistributions for B*+→ ¯*D*0*D*?+*K*0*s* . . . 42

**VI-4** ∆*E and M _{bc}distributions for B*+

_{→ ¯}

*D*?0

*D*+

*K*0

*. . . 42*

_{s}**VI-5** ∆*E and Mbcdistributions for B*0*→ D*−*D*?0*K*+ . . . 43

**VI-6** ∆*E and M _{bc}distributions for B*0

_{→ ¯}

*D*?−

*D*0

*K*+ . . . 43

**VI-7** ∆*E and M _{bc}distributions for B*+

_{→ ¯}

*D*?0

*D*?+

*K*0

*. . . 43*

_{s}**VI-8** ∆*E and Mbcdistributions for B*0*→ D*?−*D*?0*K*+ . . . 43

**VI-9** ∆*E and M _{bc}distributions for B*+

*−*

_{→ D}*D*+

*K*+ . . . 44

**VI-10** ∆*E and M _{bc}distributions for B*0

_{→ ¯}

*D*0

*D*0

*K*0 . . . 44

_{s}**VI-11** ∆*E and M _{bc}distributions for B*+

*−*

_{→ D}*D*?+

*K*+ . . . 44

**VI-12** ∆*E and M _{bc}distributions for B*+

*?−*

_{→ D}*D*+

*K*+ . . . 44

**VI-13** ∆*E and Mbcdistributions for B*0→ ¯*D*0*D*?0*Ks*0+*c.c* . . . 45

**VI-14** ∆*E and M _{bc}distributions for B*+

*?−*

_{→ D}*D*?+

*K*+. . . 45

**VI-15** ∆*E and M _{bc}distributions for B*0

_{→ ¯}

*D*?0

*D*?0

*K*0 . . . 45

_{s}**viii** **L F**

**VI-17** ∆*E and M _{bc}distributions for B*0

*−*

_{→ D}*D*+

*K*0

*. . . 46*

_{s}**VI-18** ∆*E and M _{bc}distributions for B*0

*−*

_{→ D}*D*?+

*K*0

*+*

_{s}*c.c*. . . 46

**VI-19** ∆*E and Mbcdistributions for B*+→ ¯*D*0*D*?0*K*+ . . . 46

**VI-20** ∆*E and M _{bc}distributions for B*+

_{→ ¯}

*D*?0

*D*0

*K*+ . . . 46

**VI-21** ∆*E and M _{bc}distributions for B*+
→ ¯

*D*?0

*D*?0

*K*+

_{. . . 47}

**VI-22** ∆*E and Mbcdistributions for B*0*→ D*?−*D*?+*K*0*s* . . . 47

**VI-23** *Dalitz plot and projections for B*+
→ ¯*D*0*D*0*K*+ . . . 56

**VI-24** *Efficiency map and K*+* _{momenta for B}*+
→ ¯

*D*0

*D*0

*K*+

_{. . . 56}

**VI-25** *B*+
→ ¯*D*0*D*0*K*+signal yield versus two-body invariant-mass spectra . . . 57

**VI-26** *Dalitz plot and mass spectra for B*+
*→ Ψ(3770)K*+ . . . 58

**VI-27** *Dalitz plot and mass spectra for B*+
*→ Ψ(4160)K*+ _{. . . 59}

**VI-28** *Dalitz plot and mass spectra for B*+
→ ¯*D*0*X(2.7) for spin-1 X(2.7)* . . . 59

**VI-29** *Dalitz plot and mass spectra for B*+
→ ¯*D*0*X(2.7) for spin-2 X(2.7)* . . . 59

**VI-30** Background-free mass distributions and fits results . . . 60

**VI-31** *The fit to M(D*0* _{¯D}*0

*0*

_{) with Ψ(4160) and the reflection in M(D}*+*

_{K}_{)}

_{. . . 60}

**VI-32** Dalitz-plot projections with the MC predictions superimposed . . . 61

**VI-33** Dalitz-plot projections compared to the MC predictions for various interference models . . . 62

**VI-34** cos θ*Kdistributions for 2.5 < M(D*0*K*+*) < 2.9 GeV/c*2 . . . 64

**VI-35** cos θ*Ddistributions for 3.95 < M(D*0*¯D*0*) < 4.25 GeV/c*2 . . . 65

**VI-36** cos θ*Ddistributions for M(D*0*¯D*0*) < 3.85 GeV/c*2 . . . 65

**VI-37** cos θ*K*and cos θ*Ddistributions for the DsJ*(2700) and the Ψ(4160) mass regions . . . 65

**VI-38** *Dalitz plot and projections for B*0* _{→ D}*−

*0*

_{D}*+*

_{K}_{. . . 66}

**VI-39** *B*0* _{→ D}*−

*0*

_{D}*+*

_{K}_{signal yield versus two-body invariant-mass spectra}

_{. . . 67}

**VI-40** *Fit result to M(D*0* _{K}*+

*0*

_{) distribution for B}*→ D*−

*D*0

*K*+

_{.}

_{. . . 68}

**VI-41** Background-free Dalitz-plot projections for with predicted reflections . . . 68

**VI-42** cos θ*Kdistributions for 2.5 < M(D*0*K*+*) < 2.9 GeV/c*2 . . . 69

**A-1** ∆*E _{B accepted}_{− ∆E}_{B rejected}vs ∆E_{B accepted}*distribution . . . 73

**A-2** ∆*E _{B accepted}_{− ∆E}_{B rejected}vs ∆E_{B accepted}*for various signal-background cases . . . 74

**A-3** *Mbc B accepted− Mbc B rejected* *vs Mbc B accepted* for various signal-background cases . . . 74

**ix**

**List of Tables**

**II-1** *B → ¯D*(?)* _{D}*(?)

_{K decay modes}_{. . . .}

_{9}

**II-2** *Isospin decomposition for B → ¯DDK* . . . 9

**III-1** *BF (B → ¯D*(?)*D*(?)*K) measured by BaBar* . . . 17

**V-1** *Effective efficiencies for B → ¯D*(?)* _{D}*(?)

_{K}_{. . . 39}

**VI-1** *Fitted yields and signal parameters for B → ¯D*(?)* _{D}*(?)

_{K}_{. . . 48}

**VI-2** *Measured BF for B → ¯D*(?)* _{D}*(?)

_{K}_{. . . 49}

**VI-3** *Isospin amplitudes for B → ¯D*(?)* _{D}*(?)

_{K}_{. . . 53}

**VI-4** *Fitted BF ’s for B → ¯D*(?)* _{D}*(?)

_{K}_{. . . 54}

**VI-5** *Yields and parameters of resonances contributing to B*+
→ ¯*D*0*D*0*K*+ _{. . . 61}

**VI-6** *Systematic uncertainty contributions to DsJ*(2700) . . . 62

**VI-7** Systematic uncertainty contributions to Ψ(4160) . . . 63

**VI-8** *BF ’s of quasi-two-body components of B*+→ ¯*D*0*D*0*K*+ . . . 63

**A-1** *Efficiency of the best B candidate selection* . . . 74

**B-1** *Labels of B → ¯D*(?)* _{D}*(?)

_{K decay modes}_{. . . 75}

**B-2** Cross-feed efficiencies . . . 76

**B-3** Cross-feed yields . . . 76

**C-1** Intermediate branching fractions [8] . . . 77

**1**

**C I**

**I**

*B mesons, the lightest hadrons containing b quark, play an important role in testing the Standard Model (SM)*

*and searching for its extensions. B meson decays offer an excellent place to explore the physics of CP violation*
and provide the most direct way to determine many of the SM parameters, in particular the elements of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix: the CP-violating phase and the mixing angles which describe
weak couplings between the third and the other two quark generations. Precise measurements of a big variety of

*B decays aim at checking the internal consistency of the SM’s flavor sector. This provides stringent tests of the*

theory and of the Kobayashi-Maskawa mechanism of CP violation in particular [1]. These studies are closely related to the main unsolved problems of the SM, like the explanation of baryogenesis, number of fermion generations, or fermion mass spectrum.

The extraction of the fundamental parameters from weak decays is non-trivial because of poorly understood
hadronic effects, like the confinement of quarks inside hadrons. There are several theoretical approaches, like
the heavy-quark effective theory, factorization approximation, perturbative QCD, lattice QCD etc, which can be
applied to a variety of specific problems. In addition, the realization of approximate symmetries like SU(3) or
isospin symmetry allows us to make predictions beyond the range of applicability of the perturbation theory.
*Since the b quark mass is large (' 4.5 GeV/c*2_{), these tools provide relatively precise predictions for many}

*observables in B decays. Extensive studies of weak B decays to hadronic final states can validate such approaches.*
Moreover, such studies can provide useful information on some of the hadronic effects in decay mechanism such
as color suppression or final-state interactions.

*The unique role played by B mesons in the precise tests of the Standard Model and in ‘New Physics’ searches*
has led to the construction of two dedicated accelerators called ‘B-factories’: KEK-B in Japan and PEP-II in the
*USA. A B-factory is a high-luminosity asymmetric-energy e*+_{e}_{−}_{collider, providing a clean sample of B ¯B pairs}

*coming from Υ(4S ) decays. Comprehensive studies of B meson decays in the clean environment of B-factories*
provide an ideal opportunity to realize the above-mentioned physics goals, and some of them have already
*been successfully accomplished. Here one should mention the observation of CP violation in several B decay*
modes and precise measurement of the sin 2β ∗ _{parameter (related to the phase of the CKM matrix) from a}

*time-dependent CP asymmetry in B → J/ψK*0_{decay driven by the ¯b → ¯cc¯s quark transition.}

*B → ¯D*(?)* _{D}*(?)

*†*

_{K}_{decays can provide valuable information for the above-mentioned studies. At the level}

*of quark diagrams they are induced predominantly by the Cabibbo-favored ¯b → ¯cW*+

*→ ¯cc ¯s transitions, as*
*in the ‘golden-plated’ B → J/ψK*0

*s* *mode, but with additional light-quark pair q¯q creation from the vacuum.*
Decays of this type are in principle suitable to determine β. In particular, if they proceed via quasi-two-body
intermediate states, they can be used to measure both sin 2β and cos 2β parameters and thus can help to resolve
*discrete ambiguities in the β extraction. Moreover, B → ¯D*(?)* _{D}*(?)

_{K decays are an interesting testing ground for}*phenomenology of B meson decays. The presence of isoscalar non-spectator quarks in the final states implies*
a simple isospin structure of the amplitudes involved. Therefore, these decays represent a rather simple case for

∗_{β ≡ φ}_{1}_{parameter is defined further, in Section}** _{II.1.1}**
†

*0*

_{Throughout this thesis, B = {B}_{,}

*+*

_{B}*}, D*(?)=* _{{D}*0,

*D*+

_{,}

*?0*

_{D}_{,}

*?+*

_{D}*0*

_{} and K = {K}_{,}

*+*

_{K}**2** **C I. I**

theoretical description and thus allow us to study the origin and applicability of the factorization, the nature of
*the color suppression and the effects of the final-state interactions. B decays to the ¯D*(?)* _{D}*(?)

_{K three-body final}*states are also a promising place for spectroscopic studies of c ¯s and c¯c states lying above D*(?)* _{K and D}*(?)

*(?)*

_{¯D}*thresholds, respectively. The unexpected discovery of the DsJ(2317) and DsJ*(2457) mesons [2] shows that our
*understanding of c ¯s states might be incomplete. The recent claim of the observation of the DsJ*(2632) meson
with unusual properties [3*], further calls for exploration of c ¯s multiplets. Experimental data on the c¯c states with*
*open decay channels to D*(?)* _{¯D}*(?)

_{are scarce. The relatively best-known such a meson is Ψ(3770); the properties}

*of the other three known states, observed in e*+_{e}_{−}_{formation experiments, are hardly known.}

*The possibility that a significant fraction of ¯b → ¯cc¯s decays hadronizes as B → ¯D*(?)* _{D}*(?)

_{K was first }sugges-ted in the context of the ‘charm-counting’ problem [4]. This puzzle arose from inconsistency between the
*ex-perimental results on the inclusive charm yield from B decays (giving number of charmed hadrons per B decay)*
*and the inclusive semileptonic rate. Since a substantial amount of charm in B decays is due to the ¯b → ¯cc¯s*
*process, the precise measurements of B → ¯D*(?)* _{D}*(?)

_{K branching fractions could solve the ’charm-counting’}*problem, which is still unsolved. B → ¯D*(?)* _{D}*(?)

_{K decays have previously been studied with small data samples}by CLEO [5] and ALEPH [6], and a comprehensive study has recently been performed by BaBar [7].

The content of this thesis is the following. Chapter **II**provides a description of the phenomenology of the
*studied B decays together with a short overview of selected hadronic effects related to B → ¯D*(?)* _{D}*(?)

_{K decays.}Chapter **III** *contains the motivations for the study of doubly-charmed B decays, and previous knowledge of*
these decays. Chapter **IV** tackles the physics environment of B-factories, describes the KEK-B machine and
main features of the Belle detector, and establishes a short comparison with the BaBar experiment. The data
analysis details, such as the description of a data sample, the selection criteria used in the analysis and the
*method of the B → ¯D*(?)* _{D}*(?)

_{K reconstruction are elaborated in Chapter}

_{V}_{. Chapter}

_{VI}_{contains the results}

*obtained: the measured branching fractions and isospin analysis of B → ¯D*(?)* _{D}*(?)

_{K decays, and the Dalitz-plot}*analyses performed for B*+

→ ¯*D*0*D*0*K*+*and B*0 *→ D*−*D*0*K*+ channels. Summary and concluding remarks are
contained in Chapter**VII**.

**3**

**C II**

**P B → ¯D**

**P B → ¯D**

### (?)

*D*

### (?)

*K *

*In this chapter phenomenological description of B → ¯D*(?)* _{D}*(?)

_{K decays is reviewed. Flavor structure of the}*SM, the concept of the CKM matrix and CP-violation effects in B decays are briefly introduced in Section* **II.1**.
*Quark diagrams for B → ¯D*(?)* _{D}*(?)

_{K are described in Section}

_{II.2}_{, isospin relations and isospin-amplitudes}

decomposition are contained in Section **II.3**. Section **II.4**is devoted to overview of selected hadronic effects
*related to B → ¯D*(?)* _{D}*(?)

_{K decays.}**II.1 Preliminaries**

**II.1.1 Flavor structure of the SM and the CKM matrix**

In the Standard Model of elementary particles there are three generations of leptons and quarks. Their interactions
*are represented by the S U(3)C× S U(2)L× U(1)Ygauge group. The S U(3)C*group denotes QCD which governs
*strong interactions between quarks. The S U(2)L× U(1)Y*electroweak group (generated respectively by the weak
*isospin and weak hyper charge Y) describes the transformation with different properties for fermions with right*
*and left chiralities. The right-handed leptons and quark are singlets under the weak S U(2)L* transformation
(weak isosinglets), whereas the left-handed components transform as doublets. The weak doublets of quarks
are different from those forming the quark generations, and are given by_{u}

*d*0
*L*
_{c}*s*0
*L*
_{t}*b*0

*L*. The weak eigenstates
*(d*0_{,} * _{s}*0

_{,}

*0*

_{b}

_{) are a linear combination of the mass eigenstates (d, s, b) and are related by the }*Cabibbo-Kobayashi-Maskawa (CKM) unitary matrix (VCKM*) as follows:
*d*0
*s*0
*b*0
=*VCKM*
*d*
*s*
*b*
=
*Vud* *Vus* *Vub*
*Vcd* *Vcs* *Vcb*
*Vtd* *Vts* *Vtb*
*d*
*s*
*b*
(**II.1)**

*The charged current mediated by the W boson is described by an interaction Lagrangian:*
L = −√*g*2
2
*¯u ¯c ¯t*
*L*γ
µ_{V}*CKM*
*d*
*s*
*b*
*L*
*W*†

µ +Hermitian conjugation, (**II.2)**

*where W*µ *denotes W boson, and g*2 *is the gauge coupling corresponding to S U(2)L* and being related to the
*Fermi constant (GF*). Since off-diagonal elements of the CKM matrix are different from zero, the charged current
induces transitions between different quark generations. The neutral current conserves flavors due to the unitarity
of the CKM matrix, thus flavor-changing neutral currents (FCNCs) are absent at the tree level in the SM (GIM
mechanism).

In the three-generation SM, there are four independent parameters in the CKM matrix: three mixing angles between different generations and one irreducible imaginary phase which is a source of CP violation. The

**hie-4** * C II. P B → ¯D*(?)

*(?)*

_{D}

_{K }rarchy of the CKM matrix elements is demonstrated in the Wolfenstein parameterization of the CKM matrix:

*VCKM*=
1 − λ2_{/2} _{λ} * _{Aλ}*3

*−λ 1 − λ2/2*

_{(ρ − iη)}*Aλ*2

*Aλ*3_{(1 − ρ − iη)}* _{−Aλ}*2

_{1}

+_{O(λ}4), (**II.3)**

in which λ is an expansion parameter, whereas η represents the phase which is carried by the most off-diagonal
*elements Vub* *and Vtd* *in this convention. λ =| Vus* *|' 0.22 and A =| Vcb*| /λ2 ' 0.85 are estimated from the
*CKM elements being measured directly from the semileptonic tree-level processes: |Vus*|= 0.2205 ± 0.0018 from
*s → u l ¯νl(Kl3) decays and |Vcb|= 0.041 ± 0.003 from b → c l ¯νl*decays [8].

**II.1.2 CP violation**

Weak interactions violate charge-conjugation invariance C, spatial reflection P and, at a much lower level, their product CP. CP symmetry is broken in any theory containing an irreducible phase in the Lagrangian. The neces-sary condition for CP asymmetry to occur in any given process is interference between amplitudes with non-zero relative phase which has opposite signs for the CP-conjugated modes. In the Standard Model the only source of CP violation is the phase in the CKM matrix (Kobayashi-Maskawa mechanism) and the theory offers well-defined constraints on CP violation in weak decays.

*The great advantage of B mesons is a big variety of channels in which CP violation can be studied. The basic*
observables in these studies are asymmetries between partial decay rates:

A*CP* = Γ*( ¯B → ¯f) − Γ(B → f )*

Γ*( ¯B → ¯f) + Γ(B → f )* (**II.4)**

*and, in the case of neutral B mesons, time-dependent asymmetries:*
A*CP(t) =*
*d*
*dt*Γ*( ¯B*0*(t) → ¯f) −* *dtd*Γ*(B*0*(t) → f )*
*d*
*dt*Γ*( ¯B*0*(t) → ¯f) +* *dtd*Γ*(B*0*(t) → f )*
. (**II.5)**

The first observable (Eq.**II.4**) measures the CP violation in decay, called direct CP violation. It occurs when the
total amplitudes for a decay and the CP-conjugated process have different magnitudes. This type of CP violation
requires at least two contributing amplitudes to the decay, which have different weak and strong phases ∗_{.}

In the second observable (Eq.**II.5***), one measures the differences between decays of time-evolving neutral B*
*mesons identified at time zero as pure ¯B*0* _{’s or B}*0

*0*

_{’s. If the final state f can be accessed only by B}*0*

_{and ¯f by ¯B}_{,}

*the only source of CP violation is B*0* _{− ¯B}*0

_{mixing. This type of CP violation, called CP violation in mixing or}

indirect CP violation†* _{is expected to be very small in the B mesons sector (< O(10}*−3

_{)).}

*When the final state f is common to B*0* _{and ¯B}*0

_{, CP violation can occur through interference between decay}

*without mixing: B*0* _{→ f , and with mixing: B}*0

*0*

_{→ ¯B}

_{→ f . This form of CP violation, called time-dependent CP}*asymmetry, can for example be observed in decays to final states which are CP eigenstates ( fCP). For B mesons*
this asymmetry has the particularly simple form:

A*CP*=*Sfsin(∆m t) − Cfcos(∆m t),* (**II.6)**

where
*Cf* = 1− |λ*fCP*|
2
1+ |λ*fCP*|2
, *Sf* = *2Im(λfCP*)
1+ |λ*fCP*|2
, (**II.7)**

∗* _{An evidence of direct CP violation in B decays was found in B}*0

*+*

_{→ K}_{π}−

_{[}

_{9}

*?*

_{]. In this process, involving V}_{ub}_{V}*us*, the color-suppressed
*b → u¯us tree amplitude is comparable to gluonic b → s penguin amplitude. Therefore the large interference effects between these*

amplitudes are expected to give significant CP violation.

* II.2. Quark diagrams for B → ¯D*(?)

*(?)*

_{D}

_{K}

_{5}and λ*fCP* is defined as:

λ*fCP* =
*q*
*p*
*¯AfCP*
*AfCP*
. (**II.8)**

*In the equations above, ∆m is the mass difference between the BHand BL*mass eigenstates‡and corresponds to
*the frequency of oscillations between B*0* _{and ¯B}*0

_{(' 0.5 ps}−1

*0*

_{), and q/p describes B}*0*

_{− ¯B}

_{mixing. A}_{f}*CP* *and ¯AfCP*

*respectively denote the amplitudes of B*0

*→ fCPand ¯B*0*→ fCP*decays.

*If the decay amplitudes fulfill |AfCP*|=| ¯*AfCP*|, the interference between decays with and without mixing is the

only source of CP violation, and we have:

A*CP*=* _{Im(λ}fCP) sin(∆m t).* (

**II.9)**

In this case, CP asymmetry is directly related to the CKM phase and allows us to extract the CKM parameters with negligible corrections due to strong interactions.

*The central target of CP studies in the B meson sector is the well-known unitarity triangle (UT), which is the*
geometrical representation in the complex plane of the unitarity relation applied to the first and third columns of
the CKM matrix:

*VudV _{ub}*? +

*VcdV*? +

_{cb}*VtdV*?=0. (

_{tb}**II.10)**

*The triangle is conveniently rescaled by a real factor |VcdV*?

_{cb}*|= Aλ*3, so that the apex has the coordinate ( ¯ρ, ¯η) = (ρ(1 − λ2

_{/}

_{2), η(1 − λ}2

_{/2)). The angles of the UT are expressed by CKM elements in the following way:}

β ≡ φ1 =arg − *VcdV*
?
*cb*
*VtdV _{tb}*?
!
, α ≡ φ2 =arg −

*VtdV*?

*tb*

*VudV*? ! , γ ≡ φ3=arg −

_{ub}*VudV*?

*ub*

*VcdV*? ! . (

_{cb}**II.11)**

Measuring these three angles and over-constraining the UT are the main goals of B-factory experiments. So far
the angle β has been measured with high precision [10*] from the theoretically cleanest mode B → J/ψK*0

*s* [11]

*and similar modes (J/ψK*0

*L, J/ψK*?0*, ψ(2S )K*0*s*, χ*c1Ks*0, η*cKs*0*), which are dominated by the ¯b → ¯cc¯s transitions.*
*The unitarity triangle together with the current status of our knowledge of VCKM*is summarized in Fig.**II-1**[12].
The constraints on the parameter ( ¯ρ, ¯η) (indicated by the red band), are based on several measurements obtained
*mainly for B mesons, but also for the K and Bs*meson systems. All these measurements are very consistent.

**II.2 Quark diagrams for B → ¯D**

(?)**II.2 Quark diagrams for B → ¯D**

_{D}

(?)_{D}

_{K}

_{K}

*At the quark level, B → ¯D*(?)* _{D}*(?)

*+*

_{K decays are described by the ¯b → ¯cW}*→ ¯cc ¯s transition, and thus are *
*CKM-favored. They proceed through two different processes: external W-emission and internal W-emission. Internal*

*W-emission corresponds to a color-suppressed amplitude, since it can only lead to stable hadrons if the colors of*

combined quarks match appropriately to form a color-singlet state. Depending on the final states, some decays
proceed purely through one of these amplitudes, whereas for others both processes contribute to the transition
*amplitude. These leading (’tree’) quark diagrams for charged and neutral B decays are shown in Fig.***II-2**.

*A gluonic penguin diagram ¯b → g¯s can also contribute to the production of ¯D*(?)* _{D}*(?)

_{K final states (Fig.}

_{II-3}_{).}

However, the penguin diagram contribution is suppressed compared to the tree diagram. The decay modes with

*D*(?)0* _{¯D}*(?)0

*+*

_{K in the final state, can also proceed via the ¯b → ¯uW}*→ ¯uu ¯s transition, with additional c¯c pair*
creation from the vacuum (Fig.**II-3***). This transition, involving two CKM-unfavored weak vertices ¯b → ¯uW and*

*W → u¯s, is suppressed by the factor | VubVus|/| VcbVcs*|' 2 × 10−2*, as well as by another large factor due to c¯c*
pair creation.

*All possible B → ¯D*(?)* _{D}*(?)

_{K decay modes are summarized in Table}

_{II-1}_{, where they are classified according}

to the underlying W-emission processes. For clarity’s sake, a convention in notations of entries in Table**II-1**has
*been adopted, in which D*(?)_{containing a ¯c quark from the ¯b → ¯cW transition is first listed in the final-particles}

list. The inclusion of the charge-conjugated transitions is implied.

‡_{The light B}_{L}_{and heavy B}_{H}* _{states being states of definite masses and lifetimes, are defined as follows: | B}_{L}_{i = p | B}*0

*0*

_{i + q | ¯B}_{i,}

**6** * C II. P B → ¯D*(?)

*(?)*

_{D}*-1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 1.5 2 sin2β γ γ α α ∆md ∆ms&∆md εK ε*

_{K }_{K}|Vub/Vcb| sin2β α β γ ρ η

excluded area has CL>0.95

*C K M*
f i t t e r
CKM 2005

Figure**II-1: Constraints on the CKM unitarity triangle using several observables in Winter 2005**

**II.3 Isospin relations**

*In general, weak interactions do not conserve isospin. However, b → c¯cs involves only isoscalar quarks, thus*
*the transition is an conserving process (∆I = 0). As was argued in the previous section, the *
*isospin-conserving b → c¯cs transition dominates in B → ¯D*(?)* _{D}*(?)

_{K decays. The only isospin-violating quark diagram:}*¯b → ¯uW → ¯uu¯s is at least doubly CKM-suppressed.*

*B meson is an isodoublet state:** _{B}*+

*B*0

, * _{B}¯B*−0, with the isospin determined by a light quark. If the spectator light

*quark does not play a role in B meson decay, the following isospin relation (first noted by Lipkin and Sanda [*13]) holds for the partial decay rates:

Γ*(B*+* _{→ f (¯cc ¯s)) = Γ(B}*0

*(*

_{→ ˜f(¯cc ¯s)),}**II.12)**

*where ˜f(¯cc ¯s) is the isospin mirror of the f (¯cc ¯s) final state obtained by a 180*◦_{isospin rotation. The relation above}

*implies for example: BF (B*+

→ ¯*D*0*D*+* _{K}*0

_{)/τ}

*B*+ = *BF (B*0 *→ D*−*D*0*K*+)/τ* _{B}*0, where τ

*0 and τ*

_{B}*B*+

*are the B*0and

*B*+

_{lifetimes. Testing such simple relations can provide interesting information on a decay process.}

*Irrespectively of the underlying quark diagrams describing the decay, final states of B → ¯D*(?)* _{D}*(?)

_{K can be}*decomposed into states of a definite isospin. Decomposition is straightforward in the case of the ∆I = 0 transition*
[14*], where the final state has (I, I*3) = (1_{2}, ±1_{2}*). Choosing the D*(?)*K subsystem as a base of the decomposition,*

*the total amplitude of the final state is expressed as a linear combination of the amplitudes describing D*(?)_{K of}

*definite isospin I = 0 (A*0*) and I = 1 (A*1) (Table**II-2**). As can be seen in Table**II-2**, color-suppressed decays are

*associated to the I = 1 amplitude. Similar decompositions can also be done in two other possible ¯D*(?)* _{D}*(?)

_{and}

*¯D*(?)_{K subsystems.}

The expressions from Table**II-2**can be presented in the form of triangle relations:

*− A(B*0*→ D*−*D*0*K*+*) = A(B*0*→ D*−*D*+*K*0*) + A(B*0→ ¯*D*0*D*0*K*0) (**II.13)**
*−A(B*+→ ¯*D*0*D*+*K*0*) = A(B*+→ ¯*D*0*D*0*K*+*) + A(B*+*→ D*−*D*+*K*+) (**II.14)**
The isospin relations, identical to those presented in Table**II-2***hold for various classes of decays: B → ¯DDK,*

*B → ¯DD*?* _{K, B → ¯D}*?

*?*

_{DK and B → ¯D}*?*

_{D}

_{K. The amplitudes A}0*and A*1are generally different for each set of

decays. Isospin-amplitudes decomposition is valid not only for the total decay amplitude but also for individual helicity amplitudes, as well in a function of Dalitz-plot variables.

**II.3. Isospin relations** **7**

*The branching fractions measured for the neutral and charged B → ¯D*(?)* _{D}*(?)

_{K decay modes can be used to}*extract the isospin amplitudes | A*0 *|, | A*1 *| and their relative phase δ = arg(A*1*A*?_{0}) for each set of decays. This

can provide an insight into a decay mechanism and into the strength of final-state interactions.

b D( )0

### *

B0 D( )0### *

K0 W+ d c d c u u s B+ D( )+### *

D( )-### *

K+ W+ b u c u c d d s B+ D( )0### *

D( )0### *

K+ W+ b u c u c u u s b D( )-### *

B0 D( )+### *

K0 W+ d c d c d d s K+ B+ D( )0### *

D( )0### *

W+ b u c s u c u u B+ D( )0### *

D( )+### *

K0 W+ b u c s u c d d B0 D( )-### *

D( )0### *

K+ W+ b d c d u u c s B0 D( )-### *

D( )+### *

K0 W+ b d c d d d c s**8** * C II. P B → ¯D*(?)

*(?)*

_{D}*u B0 D( )0*

_{K }### *

D( )0### *

K0 W+ b d d u c c s D( )0### *

D( )+### *

s B+ K0 g b u u c d d c u c t W+ K+ D( )-### *

D( )0### *

s B0_{g}d d c u u c u c t W+ b K0 s u c t D( )-

### *

D( )+### *

B0_{g}d d c d d c W+ b D( )0

### *

D( )0### *

s B+ K+ g b u u c u u c W+ u c t D( )0### *

u B+ D( )0### *

K+ W+ b u u u c c sFigure* II-3: Suppressed quark diagrams for B → ¯D*(?)

*(?)*

_{D}

_{K decays}**II.4 Hadronic effects in B decays**

**II.4 Hadronic effects in B decays**

Free-quark-line diagrams, like those shown in Figs.**II-2**, **II-3**, are a gross oversimplification of the description
*of real B meson exclusive weak decays. A b quark is bound by strong interactions inside a B meson, and the*
non-perturbative nature of these interactions makes theoretical description of a decay considerably more
compli-cated. Additional complications arise from possible interactions between initial and final quarks, or between final
quarks themselves, and from the fact that final-state quarks must recombine to form the observed color-singlet
hadrons. As a result, the phenomomenology of weak hadronic decays is characterized by a complex interplay
between weak and strong forces. Theoretical tools to control that complex interplay have been developed only
recently, in the past decade. They rely on concepts of heavy-quark symmetry, heavy-quark expansion and chiral
symmetry. These concepts are briefly discussed in the following sections.

**II.4. Hadronic effects in B decays****9**

Table* II-1: B → ¯D*(?)

*(?)*

_{D}

_{K decay modes}*decays through external W-emission*

*B*+
→ ¯*D*0*D*+*K*0 *B*0*→ D*−*D*0*K*+
*B*+
→ ¯*D*0*D*?+*K*0 *B*0*→ D*−*D*?0*K*+
*B*+
→ ¯*D*?0*D*+*K*0 *B*0*→ D*?−*D*0*K*+
*B*+
→ ¯*D*?0*D*?+*K*0 *B*0*→ D*?−*D*?0*K*+

*decays through internal W-emission*

*B*+
*→ D*−*D*+* _{K}*+

*0 → ¯*

_{B}*D*0

*D*0

*K*0

*B*+

*→ D*−

*D*?+

*K*+

*B*0→ ¯

*D*0

*D*?0

*K*0+

*c.c*

*B*+

*→ D*?−

*D*+

*K*+

*B*+

*→ D*?−

*D*?+

*K*+

*0 → ¯*

_{B}*D*?0

*D*?0

*K*0

*decays through external+internal W-emission*

*B*+
→ ¯*D*0*D*0*K*+ *B*0*→ D*−*D*+*K*0
*B*+
→ ¯*D*0*D*?0*K*+ * _{B}*0

*→ D*−

*D*?+

*K*0+

*c.c*

*B*+ → ¯

*D*?0

*D*0

*K*+

*B*+ → ¯

*D*?0

*D*?0

*K*+

*B*0

*→ D*?−

*D*?+

*K*0

Table**II-2: Isospin decomposition for B → ¯DDK**

Channel Decay amplitudes

*B*0_{→ ¯}_{DDK}* _{A(B}*0

*−*

_{→ D}*0*

_{D}*+*

_{K}_{) =}

_{√}1 6

*A*1− 1 √ 2

*A*0

*A(B*0

*−*

_{→ D}*+*

_{D}*0*

_{K}_{) =}

_{√}1 6

*A*1+ 1 √ 2

*A*0

*A(B*0

_{→ ¯}

*0*

_{D}*0*

_{D}*0*

_{K}_{) = −}q2 3

*A*1

*B*+ → ¯

*DDK*

*A(B*+→ ¯

*D*0

*D*+

*K*0) = √1 6

*A*1− 1 √ 2

*A*0

*A(B*+ → ¯

*D*0

*D*0

*K*+) = √1 6

*A*1+ 1 √ 2

*A*0

*A(B*+

*→ D*−

*D*+

*K*+) = − q 2 3

*A*1

**II.4.1 Heavy-quark symmetry**

*The symmetry that is especially important in B meson physics is heavy-quark symmetry. It is modeled on *
well-known atomic physics concepts.

*As the quark mass increases, its velocity in a light Q¯q meson rest frame decreases. The *
*heavy-quark Compton wavelength (∼ 1/mQ*) is much smaller than the hadronic radius, which is of the order of 1/Λ*QCD*

§_{. Mesons with one heavy quark become similar to the hydrogen atom. The mass of the heavy quark is irrelevant}

in interactions, only its charge and the light-quark mass determine all levels and transition rates. The properties
of the light meson become governed by the dynamics of the light quark. In the limit of an infinitely
*heavy-quark mass (mQ* → ∞), the heavy quark acts as a static color source of chromoelectric field. The relativistic
*effects such as chromomagnetic interactions (gluon-exchange) vanish as mQ*→ ∞. Therefore, the QCD
*interac-tion cannot distinguish between charm or bottom. Consequently the decay amplitudes and form factors of b and*

*c hadrons are related to each other; this is called ‘heavy-quark flavor symmetry’. Since the heavy-quark spin (~sQ*)
participates in interactions only through such relativistic effects, it decouples from the light quark (‘heavy-quark

**10** * C II. P B → ¯D*(?)

*(?)*

_{D}

_{K }spin symmetry’). Due to heavy-quark spin symmetry, the spin quantum numbers of light degrees of freedom
*(total angular momentum of the light quark ~jq* = ~*L + ~sq, where ~sq* *and ~L respectively denote its spin and *
*or-bital momentum) and those of the heavy quark (~sQ*) are separately conserved, and thus become good quantum
*numbers. The total angular momentum of the meson is: ~J = ~jq*+~*sQ*.

The interaction of light degrees of freedom with the heavy-quark spin (‘hyperfine interaction’) is suppressed
*by 1/mQ*. Since the quark masses are finite, heavy-quark symmetry is only approximate symmetry, and relativistic
*corrections must be taken into account (so-called 1/mQ*effects). Both, the spin and heavy-flavor symmetries are
*violated. The chromomagnetic term (∼ ~sq*~*sQ*) results in the hyperfine splitting of the heavy-light mesons, which
*is the splitting of the states with an equal ~jqbut a different ~J, e.g. D−D*?*and B−B*?.

*In the absence of a rigorous quantitative theory description of non-leptonic B decays, a low-energy effective*
theory has been developed: the Heavy Quark Effective Theory (HQET). The main ideas of HQET [15] are
sketched below.

HQET incorporates the idea of heavy-quark symmetry and postulates an interaction Lagrangian in which
de-grees of freedom of heavy quarks are (partially) ‘integrated out’. Such an effective Lagrangian is non-local. The
locality of interactions is restored by expanding the effective action into infinite series of local terms in an
Opera-tor Product Expansion (OPE) (another popular term is: the Heavy Quark Expansion (HQE)). Roughly speaking,
*this corresponds to an expansion in powers of 1/mQ*. This expansion enables us to separate short-distance and
long-distance physics phenomena. The long-distance effects correspond to interactions at low energy and they
are fully accounted for in the effective Lagrangian, as they would be in a full theory. Short-distance effects arise
*from quantum corrections involving large virtual momenta (O(mQ*)), and they are not described in the effective
theory because heavy particles have been partially integrated out. They are incorporated into effective theory in
a perturbative way, by using renormalization-group techniques. The renormalized effective Hamiltonian for the
*¯b → ¯cc¯s tree transition becomes [*16]:

*He f f∼GFVcbVcs*?*hC*1*(µ) sγ*µ(1−γ5*)¯c cγ*µ(1−γ5*)¯b + C*2*(µ) cγ*µ(1−γ5*)¯c sγ*µ(1−γ5*)¯b*

i

, (**II.15)**
*where C*1*(µ) and C*2(µ) are the Wilson coefficients. These depend on the renormalization scale µ, and at the scale

*relevant to B decays (i.e. µ ' mb) they amount to C*1 =*1.13 and C*2 = * _{−0.3. The term with C}*1corresponds to a

*color-allowed transition, whereas the term with C*2to a color-suppressed one.

Assumption about the factorization of a total amplitude and effects of color suppression are discussed in the following sections.

**II.4.2 Factorization**

One of the crucial assumptions used in HQET calculations is that of a factorization of a total amplitude
describ-ing a weak hadronic decay. In essence this corresponds to a local hadron-parton duality hypothesis, accorddescrib-ing
to which hadronization effects are unimportant in calculation of a decay amplitude. Therefore, it is enough to
consider the short-distance part of the process, with the subsequent hadronization taking place with unit
proba-bility. The most common factorization approach is the ’naive’ factorization model, in which decay amplitudes
are calculated by replacing hadronic matrix elements of four-quark operators by products of current matrix
elements determined by meson decay constants and form factors. ‘Factorizable’ strong-interaction effects are
*parameterized by phenomenological coefficients denoted hereafter as ai*. These depend on the color and Dirac
structure of the operators, but they are postulated to be universal constants. Besides the factorization assumption,
the evaluation of amplitudes for hadronic decays requires the input of hadronic form factors and meson decay
constants. These must be determined experimentally or by lattice QCD calculations. Another consequence of the
factorization approach is that the strong final-state interactions also become calculable.

There is no rigorous proof of the factorization, although arguments for its validity do exist thanks to large

*Ncolor* expansion approach [17]. Its justification for large energy-release decays is sought in the ’color

trans-parency’ phenomenon [18] explained below.

*The final-state quarks from a weak decay (e.g. ¯b → ¯cc¯s process) travel in a medium of gluons and light*

**II.4. Hadronic effects in B decays****11**

remain close together as they move through the colored medium. If they are in a color singlet, they interact with
*the medium not individually but as a color dipole. Then, it is possible that the c ¯s pair will leave the colored*
*environment before its dipole moment grows large enough for its interactions to be significant. In this case, c ¯s*
*will hadronize as D*(?)*s* *. If, by way of contrast, the ¯cs pair has a large invariant mass, then the quarks will interact*
*strongly with the medium and then it is unlikely they will re-assemble into D*(?)*s* .

*For a B decay with a large energy release, the W decay products travel fast enough to leave the interaction*
region without influencing the second hadron. The soft interactions with the remaining decay products is
*sup-pressed by 1/mQ. Therefore, since a b quark mass is much larger than a c quark mass, the factorization hypothesis*
*for B meson decays is expected to be more reliable than for D decays.*

*The factorized amplitude for B → ¯D*(?)* _{D}*(?)

*s* decays, expressed as the product of two independent hadronic
currents is:

*A ∼ GFVcbVcs*?*hD*(?)*s* *| sγ*µ(1−γ5*)¯c|0i × hD*(?)*| cγ*µ(1−γ5*)¯b | Bi.* (**II.16)**

*The first hadron current that creates D*(?)*s* *from the vacuum is related to the meson decay constant f _{D}*(?)

*s*by:

*hDs(pDs)| sγ*µ(1−γ5

*)¯c |0i = i fDsp*µ

*Ds*,

*hD*?

*s(pD*?

*s*,

*D*?

*s)| sγ*µ

_{(1−γ}5

*)¯c|0i = i fD*?

*sp*µ

*D*?

*s*µ

*D*?

*s*(

**II.17)**

*where p*(?)

_{D}*s*

*is the D*(?)

*s*momentum and

*D*?

*s*

*is the D*?

*s* polarization vector. The second hadron current in Eq.**II.16**
*describes the formation of the ¯D*(?) _{meson containing the spectator quark. It is related to hadronic form factors}

*that can be determined from semileptonic B → ¯D*(?)* _{l}*+

_{ν}

*l*decays.
**II.4.3 Color suppression**

*Exchanges and emissions of colored gluons in a B meson decay change the color states of quarks, and can thus*
*rearrange the color structure of the transition amplitude. For three colors of quarks (Ncolor* =3) such color effects
are expected to be suppressed by 1/3.

*In HQET, color effects are formally accounted for by rewriting the He f f*from Eq.**II.15**in terms of factorizable
part and non-factorizable corrections in the following way:

*He f f∼GFVcbVcs*? *sγ*µ(1−γ5*)¯c cγ*µ(1−γ5*)¯bh C*1+*C*2/*Ncolor*1 + 1 +*2C*28i. (**II.18)**

*(C*1+*C*2/*Ncolor) ≡ a*1is the coefficient of factorizable term (as in Eq.**II.16**). The coefficients 1and 8respectively

describe the deviation of the color-singlet amplitude from the naively factorized form, and the admixture of the
*color-octet operator O*(8)_{. This operator cannot generate any c ¯s state and therefore requires the presence of at}

least one extra gluon in the transition.

It is often assumed that amplitudes with the ‘wrong’ color structure to factorize are intrinsically small. There-fore, 1and 8are expected to be small as well, and in the ‘naive’ factorization approach they are equal to zero.

*It can be shown that all non-factorizable contributions are suppressed by 1/N*2

*color* [19].

*If (C*1+*C*2/*Ncolor) 2C*2, the amplitude is said to be color-allowed, if the reverse is true, then the

ampli-tude is classified as color-suppressed. For the decay dominated by the color-suppressed ampliampli-tude, the effective Hamiltonian is as follows:

*He f f∼GFVcbVcs*? *cγ*µ(1−γ5*)¯c sγ*µ(1−γ5*)¯bh C*2+*C*1/*Ncolor*1 + ˜1 +*2C*1˜8

i

, (**II.19)**

with ˜1and ˜8describing the corrections to the factorized form (analogous to Eq.**II.18**). Factorization in

*color-suppressed decays is not very reliable, since the coefficient 2C*1 multiplying ˜8 *is much larger than the (C*2 +

*C*1/*Ncolor) ≡ a*2factor multiplying the factorizable part. Thus, the presence of large non-factorizable color-octet

contribution in such processes is very possible. Experimentally, it is still not obvious whether the factorization
*hypothesis is satisfied in decays which proceed via internal W-emission.*

Factorization can be violated due to the presence of Final-state interactions (FSI) between the decay products. These interactions can generate phases between hadronic amplitudes and induce rescattering into other channels. FSI can also substantially influence decay rates through interferences. Large phase differences would make CP-violation studies especially interesting in such decay modes.

**12** * C II. P B → ¯D*(?)

*(?)*

_{D}

_{K }**II.4.4 Chiral symmetry**

Another important QCD symmetry, complementary to the heavy-quark limit, arises in the limit of vanishing
*light-quark masses. As mu*,*md*,*ms*→ 0, there is no interaction between quarks of left and right helicities; they decouple
*from each other. In this limit, the Lagrangian is invariant under rotations among (uL*,*dL*,*sL) and (uR*,*dR*,*sR*),
*separately. This corresponds to S U(3)L* *× S U(3)R* chiral flavor symmetry. As a consequence, one can expect
*parity-doubling in the low-lying u, d, s spectrum. In the real world, due to non-trivial QCD vacuum this symmetry*
*is spontaneously broken by the quark condensate hqi¯qj*i , 0. Symmetry-breaking results in eight Goldstone
*bosons in the light spectrum (identified with the physical pseudoscalar mesons π, K and η) and in the degeneration*
*of opposite-parity q¯q states. The scale of spontaneous breaking of chiral symmetry is Λ*χ ' 1 GeV, where

Λχ is related to the value of the quark condensate. The Λχ energy scale enables us to construct the effective

*Langrangian describing low-energy interactions of particles with light masses and small momenta (m, p *
Λχ*) by introducing systematic expansion in powers of mq*/Λχ *and p/Λ*χ. This gives the basis for the Chiral

*Perturbation Theory (ChPT) in the u, d, s sector.*

*Despite large masses of heavy-flavor hadrons, the ChPT can also be applied to Q¯q systems simultaneously*
*with heavy-quark symmetry. Due to the existence of the large-energy scale (∼ mQ*), the role of chiral-symmetry
*breaking becomes less important for Q¯q hadrons, and chiral symmetry is effectively restored. This interesting*
*effect, which has important consequences for the spectroscopy of Q¯q mesons, will be discussed in more detail in*
Section**III.3**.

*ChPT is used in non-leptonic B meson decays to calculate amplitudes in the phase-space regions where*
*particles have low momenta; for example it has been employed in calculations of B*0

**13**

**C III**

**P **

*To this day, only less than 40% part of the total B decay width is experimentally known, thus the observation of*
unknown decay modes and measurement of their BF ’s are interesting in themselves. However, there are several
*places where B → ¯D*(?)* _{D}*(?)

_{K can provide especially interesting information.}**III.1 CP violation and measurement of β in B**

0
**III.1 CP violation and measurement of β in B**

### → ¯

*D*

(?)*D*

(?)*K*

*s*0

*B*0

→ ¯*D*(?)*D*(?)*K*0*s* decays are interesting channels for investigating CP violation effects [21] [22] [20]. These
*processes, induced by the same ¯b → ¯cc¯s transition as the gold-plated B*0 * _{→ J/ψK}*0

*s* mode with negligible
penguin pollution, are suitable to investigate the angle β. An important feature of these three-body decays is the
*possibility of occurring resonances in their final states, e.g. B*0

*→ D*(?)−*D*??+*s* *, where D*??*s* *denotes an excited c ¯s*
*meson decaying to D*(?)* _{K}*0

*s. In such a case, the B*0*and ¯B*0*decay amplitudes are different, because in B and ¯B the*
corresponding final states are at different kinematical points of the Dalitz plot and the CP violation appears in
*the interference region between the D*??+

*s* *and D*??−*s* bands. The mixing angle β has to be measured through the
*time-dependent B*0* _{→ D}*(?)−

*(?)+*

_{D}*0*

_{K}*s* Dalitz-plot analysis.
*The D*(?)−* _{D}*(?)+

*0*

_{K}*s* final state is described by two independent Dalitz-plot variables:
*s*+
= *p _{D}*(?)++

*p*0

_{K}*s*2 ,

*s*−=

*p*(?)−+

_{D}*p*0

_{K}*s*2 , (

**III.1)**

*and the CP-eigenstates are those satisfying s*+_{=} _{s}_{−}_{.}

*The amplitudes for unmixed neutral B mesons are defined as:*

*A(s*+

,*s*−*) ≡ A(B*0*→ D*(?)−*D*(?)+*Ks*0*), ¯A(s*+,*s*−*) ≡ A( ¯B*0*→ D*(?)−*D*(?)+*Ks*0). (**III.2)**
*Their strong phases (the Breit-Wigner phases) depend on s*+* _{and s}*−

*+*

_{, and are different in A and ¯A when s}_{,}

*−*

_{s}_{.}

*The time-dependent amplitude (A) for an oscillating B*0* _{(t) state, tagged as B}*0

_{at t = 0, is given by:}*A(s*+,*s*−*; t) = A(s*+,*s*−) cos1

2*(∆m t) + ie−2iβ¯A(s*

+

,*s*−) sin1

2*(∆m t).* (**III.3)**

*For the sake of simplicity, e−Γt* _{and constant phase-space factors have been omitted in Eq.}_{III.3}_{. The square of}

the time-dependent amplitude is:
*|A(s*+,*s*−* _{; t)|}*2

_{=}1

2*hG*0*(s*

+_{,}* _{s}*−

_{) + G}*c(s*+,*s*−*) cos (∆m t) − Gs(s*+,*s*−*) sin (∆m t)*i, (**III.4)**
with

*G*0*(s*+,*s*−*) = |A(s*+,*s*−)|2+| ¯*A(s*+,*s*−)|2, (**III.5)**

*Gc(s*+,*s*−*) = |A(s*+,*s*−)|2− | ¯*A(s*+,*s*−)|2, (**III.6)**

*Gs(s*+,*s*−*) = 2Ime−2iβ¯AA*?

**14** **C III. P **

*The CP-conjugated channel, ¯B*0* _{(t) → D}*(?)+

*(?)−*

_{D}*0*

_{K}*s, is defined by the transformations: s*+ *↔ s*−*, A ↔ ¯A and*
β ↔ −β, that give:

*|A(s*−,*s*+*; t)|*2= 1

2*hG*0*(s*−,*s*

+

*) − Gc(s*−,*s*+*) cos (∆m t) + Gs(s*−,*s*+*) sin (∆m t)*i. (**III.8)**
*If one neglects penguin contribution, then | A(s*+_{,}_{s}_{−}_{) |=| ¯A(s}_{−}_{,}* _{s}*+

_{) | and there is no direct CP violation. In this}

*case, the time-independent term G*0*(s*+,*s*−*) is symmetric under s*+ *↔ s*−*, whereas Gc(s*+,*s*−) is antisymmetric.
*In the case of real A(s*+_{,}* _{s}*−

*+*

_{), G}_{s}_{(s}_{,}

*−*

_{s}*+*

_{) would be symmetric under s}*↔ s*− and only the CP-violating part
*proportional to sin (2β) would survive. However, when the amplitude A(s*+_{,}_{s}_{−}_{) has a non-trivial CP-conserving}

phase, which originates from strong interactions, both the CP-violating sin (2β) and CP-conserving cos (2β) terms
*contribute to Gs(s*+,*s*−). These terms can be disentangled by symmetric and antisymmetric integrations of the
*sin (∆m t) component in the time-dependent Dalitz-plot analysis.*

The measurement of cos (2β) in addition to sin (2β) allows us to solve the β → π

2 − β ambiguity. The overall

constraints on the unitarity triangle already leave no room for such vagueness (as Figure **II-1**shows). However,
the unitarity-triangle constraints only hold within the Standard Model which can be tested in B-factories using
information on the sign of cos (2β).

*To measure the cos (2β) term, knowledge of the CP-even phase of ¯AA*?_{is necessary. Such strong phases are}

unknown in a three-body decay. However, the assumption of the dominance of intermediate resonances allows us to use the Breit-Wigner phase in selected parts of the Dalitz plot. Other phases, such as those due to the final-state interactions, are assumed to be constant or to vary slowly under resonances, whereas the Breit-Wigner phase changes quickly.

*Multi-parameters fit with all possible contributing resonances applied to the time-dependent B*0* _{→ D}*(?)−

*(?)+*

_{D}*0*

_{K}*s*
Dalitz plot allows us to extract both the sine and the cosine of the CKM phase. The properties of the included
resonances have to be known beforehand. The number of cos 2β sensitive events in the Dalitz plot, generated
*by the Breit-Wigner interference (described by ¯AA*?_{term) grows like the resonance width and like the square}

*of its mass. Thus, the broader and heavier resonances contributing to B*0

*→ D*?−*D*(?)+*Ks*0 are more useful to
probe cos 2β. Evaluating the sensitivity of this method requires a better determination of branching fractions for

*B*0* _{→ D}*(?)−

*(?)+*

_{D}*0*

_{K}_{decay modes as well as knowledge of contributing resonances.}

**III.2 Decay process dynamics**

*A study of B → ¯D*(?)* _{D}*(?)

_{K decays can give insight into the phenomenology of B meson decays described in}Section**II.4**. Measurement of BF ’s for unobserved decay modes is interesting in itself. The measured BF ’s can
be a source of information on the nature of color suppression. The size and the relative sign of a color-suppressed
*amplitude can be determined by comparing rates for B → ¯D*(?)* _{D}*(?)

_{K decay modes where both external and }inter-nal diagrams are present and can interfere, and the related modes described by the color-suppressed amplitudes.
The effect of color-suppression can be obscured by the effects of FSI. There is no experimental information on
*such final-state interactions between D mesons yet. The strength of these long-distance effects can be determined*
*by performing an isospin analysis of the related B → ¯D*(?)* _{D}*(?)

_{K decay modes, because FSI can cause an isospin}phase shift.

*Another important problem is verification of the factorization. This approximation works excellently for B*
*decays with a large energy release like B → ¯D*(?)_{π}* _{. Since B → ¯D}*(?)

*(?)*

_{D}*(?)*

_{K decays or B → ¯D}*(?)*

_{D}*s*
*quasi-two-body modes are processes with relatively small Q-value, testing the factorization hypothesis is especially*
valuable to them. In general, such a test can be carried out by comparing measured rates and polarizations for
*semileptonic and hadronic B decays with the predictions of phenomenological models based on the factorization*
*hypothesis. For example, for the color-allowed B → ¯D*(?)* _{D}*(?)

*s* decay, the factorization approach gives:
Γ*(B → ¯D*(?)*D*(?)* _{s}* )

*d*

*dq*2Γ

*(B → ¯D*(?)

*l*+ν

*l*) |

*q*2

_{=}

*2*

_{m}*D(?)*=6π2

_{s}*f*2

*D*(?)

*s*

*|a*1| 2

*|Vcs*|2, (

**III.9)**