"Study of doubly- charmed B→ D(*)D(*)K decays at Belle"

Study of doubly-charmed B → ¯D

(?)

_D

(?)

_{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+

→ ¯D0D0K+and B0 →

D−_D0_K+_{. A candidate for a new c ¯s state at the mass 2.72GeV/c}2_{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:

B0_{→ ¯D}0_D0_K+_{i B}+

→ D−D0K+przeprowadzona została analiza diagram´ow Dalitza. Zaobserwowano strukture w układzie D0_K+_{przy masie około 2.72GeV/c}2_{, 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

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 B0 → ¯D(?)D(?)K0s . . . 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_bcdistributions 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+ → ¯D0D0K+ _{. . . 57}

VI.4.2 Angular distributions . . . 64

VI.4.3 Dalitz plot for B0_{→ D}−_D0_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 R2distributions . . . 26

V-2 CMS momentum spectra for mesons produced in B → ¯D(?)_D(?)_{K . . . 27}

V-3 ¯D0_{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 LRD(MD) for D0and D+decay modes . . . 33

V-8 LRD?±(∆M(D?)) shapes . . . 33

V-9 LRDvs LR¯Dfor B+→ ¯D0D0K+and B0→ D−D0K+ . . . 34

V-10 B+ → ¯D0D0K+_{∆E distributions for various LR} Bcuts . . . 35

V-11 B0 → D−D0K+∆E distributions for various LR_Bcuts . . . 35

V-12 Background components for B+ → ¯D0D0K+_,B0 → D−D0K+_{and B}0 → D?−D0K+ _{. . . 35}

V-13 M(D) and M( ¯D) distributions for B+ → ¯D0D0K+_{and B}0 → D−_D0_K+ _{. . . 36}

V-14 ∆E and M_bcdistributions for B+_{→ ¯}D0D0K+signal MC events. . . 37

V-15 ∆E and Mbcdistributions for B0→ D?−D0K+signal MC events . . . 37

V-16 ∆E and M_bcdistributions for B+_{→ ¯}D?0D0K+signal MC events . . . 38

V-17 ∆E vs Mbcdata distribution and two-dimensional likelihood fit result . . . 40

VI-1 ∆E and Mbcdistributions for B+→ ¯D0D+K0s . . . 42

VI-2 ∆E and M_bcdistributions for B0_{→ D}−D0K+ . . . 42

VI-3 ∆E and Mbcdistributions for B+→ ¯D0D?+K0s . . . 42

VI-4 ∆E and M_bcdistributions for B+_{→ ¯}D?0D+K0_s . . . 42

VI-5 ∆E and Mbcdistributions for B0→ D−D?0K+ . . . 43

VI-6 ∆E and M_bcdistributions for B0_{→ ¯}D?−D0K+ . . . 43

VI-7 ∆E and M_bcdistributions for B+_{→ ¯}D?0D?+K0_s . . . 43

VI-8 ∆E and Mbcdistributions for B0→ D?−D?0K+ . . . 43

VI-9 ∆E and M_bcdistributions for B+_{→ D}−D+K+ . . . 44

VI-10 ∆E and M_bcdistributions for B0_{→ ¯}D0D0K_s0 . . . 44

VI-11 ∆E and M_bcdistributions for B+_{→ D}−D?+K+ . . . 44

VI-12 ∆E and M_bcdistributions for B+_{→ D}?−D+K+ . . . 44

VI-13 ∆E and Mbcdistributions for B0→ ¯D0D?0Ks0+c.c . . . 45

VI-14 ∆E and M_bcdistributions for B+_{→ D}?−D?+K+. . . 45

VI-15 ∆E and M_bcdistributions for B0_{→ ¯}D?0D?0K_s0 . . . 45

viii L  F

VI-17 ∆E and M_bcdistributions for B0_{→ D}−D+K0_s . . . 46

VI-18 ∆E and M_bcdistributions for B0_{→ D}−D?+K0_s+c.c. . . 46

VI-19 ∆E and Mbcdistributions for B+→ ¯D0D?0K+ . . . 46

VI-20 ∆E and M_bcdistributions for B+_{→ ¯}D?0D0K+ . . . 46

VI-21 ∆E and M_bcdistributions for B+ → ¯D?0D?0K+ _{. . . 47}

VI-22 ∆E and Mbcdistributions for B0→ D?−D?+K0s . . . 47

VI-23 Dalitz plot and projections for B+ → ¯D0D0K+ . . . 56

VI-24 Efficiency map and K+_{momenta for B}+ → ¯D0D0K+ _{. . . 56}

VI-25 B+ → ¯D0D0K+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+ → ¯D0X(2.7) for spin-1 X(2.7) . . . 59

VI-29 Dalitz plot and mass spectra for B+ → ¯D0X(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(D0_¯D0_{) with Ψ(4160) and the reflection in M(D}0_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 θKdistributions for 2.5 < M(D0K+) < 2.9 GeV/c2 . . . 64

VI-35 cos θDdistributions for 3.95 < M(D0¯D0) < 4.25 GeV/c2 . . . 65

VI-36 cos θDdistributions for M(D0¯D0) < 3.85 GeV/c2 . . . 65

VI-37 cos θKand cos θDdistributions for the DsJ(2700) and the Ψ(4160) mass regions . . . 65

VI-38 Dalitz plot and projections for B0_{→ D}−_D0_K+ _{. . . 66}

VI-39 B0_{→ D}−_D0_K+_{signal yield versus two-body invariant-mass spectra . . . 67}

VI-40 Fit result to M(D0_K+_{) distribution for B}0 → D−D0K+_{. . . . 68}

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

VI-42 cos θKdistributions for 2.5 < M(D0K+) < 2.9 GeV/c2 . . . 69

A-1 ∆E_{B accepted− ∆EB rejected}vs ∆E_{B accepted}distribution . . . 73

A-2 ∆E_{B accepted− ∆EB 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+ → ¯D0D0K+ _{. . . 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+→ ¯D0D0K+ . . . 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/c2_{), 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/ψK0_{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/ψK0

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 ≡ φ1parameter is defined further, in SectionII.1.1} †_{Throughout this thesis, B = {B}0_,B+

}, D(?)=_{D0,D+_,D?0_,D?+_{} and K = {K}0_,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 IIprovides 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+

→ ¯D0D0K+and B0 → D−D0K+ channels. Summary and concluding remarks are contained in ChapterVII.

3

C II

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.4is 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)Cgroup denotes QCD which governs strong interactions between quarks. The S U(2)L× U(1)Yelectroweak 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

d0  L _c s0  L _t b0 

L. The weak eigenstates (d0_{, s}0_{, b}0_{) 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:           d0 s0 b0           =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 = −√g2 2  ¯u ¯c ¯t  Lγ µ_V CKM           d s b           L W†

µ +Hermitian conjugation, (II.2)

where Wµ denotes W boson, and g2 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_{(ρ − iη)} −λ 1 − λ2/2 Aλ2

Aλ3_{(1 − ρ − iη) −Aλ}2 ₁

          +_O(λ4), (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 ¯νldecays [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:

ACP = 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: ACP(t) = d dtΓ( ¯B0(t) → ¯f) − dtdΓ(B0(t) → f ) d dtΓ( ¯B0(t) → ¯f) + dtdΓ(B0(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 ¯B0_{’s or B}0_{’s. If the final state f can be accessed only by B}0_{and ¯f by ¯B}0_,

the only source of CP violation is B0_{− ¯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 B0_{and ¯B}0_{, CP violation can occur through interference between decay}

without mixing: B0_{→ f , and with mixing: B}0_{→ ¯B}0_{→ 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:

ACP=Sfsin(∆m t) − Cfcos(∆m t), (II.6)

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

∗_{An evidence of direct CP violation in B decays was found in B}0_{→ K}+_π− _{[9]. In this process, involving VubV}?

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 λfCP is defined as:

λfCP = q p ¯AfCP AfCP . (II.8)

In the equations above, ∆m is the mass difference between the BHand BLmass eigenstates‡and corresponds to the frequency of oscillations between B0_{and ¯B}0_{(' 0.5 ps}−1_{), and q/p describes B}0_{− ¯B}0 _{mixing. Af}

CP and ¯AfCP

respectively denote the amplitudes of B0

→ fCPand ¯B0→ fCPdecays.

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:

ACP=_Im(λfCP) 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_tb?=0. (II.10) The triangle is conveniently rescaled by a real factor |VcdV_cb?|= Aλ3, 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_ub? ! , γ ≡ φ3=arg − 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/ψK0

s [11]

and similar modes (J/ψK0

L, J/ψK?0, ψ(2S )K0s, χc1Ks0, ηcKs0), which are dominated by the ¯b → ¯cc¯s transitions. The unitarity triangle together with the current status of our knowledge of VCKMis 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 Bsmeson systems. All these measurements are very consistent.

II.2 Quark diagrams for B → ¯D

_D

_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 TableII-1, where they are classified according}

to the underlying W-emission processes. For clarity’s sake, a convention in notations of entries in TableII-1has 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 BLand heavy BH states being states of definite masses and lifetimes, are defined as follows: | BLi = p | B}0_{i + q | ¯B}0_i,

6 C II. P  B → ¯D(?)_D(?)_{K } -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 |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

FigureII-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+

B0



, _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+

→ ¯D0D+_K0_)/τ

B+ = BF (B0 → D−D0K+)/τ_B0, where τ_B0 and τB+ are the B0and 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, I3) = (1₂, ±1₂). 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 (A0) and I = 1 (A1) (TableII-2). As can be seen in TableII-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 TableII-2can be presented in the form of triangle relations:

− A(B0→ D−D0K+) = A(B0→ D−D+K0) + A(B0→ ¯D0D0K0) (II.13) −A(B+→ ¯D0D+K0) = A(B+→ ¯D0D0K+) + A(B+→ D−D+K+) (II.14) The isospin relations, identical to those presented in TableII-2hold for various classes of decays: B → ¯DDK,

B → ¯DD?_{K, B → ¯D}?_{DK and B → ¯D}?_D?_{K. The amplitudes A}

0and A1are 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 | A0 |, | A1 | and their relative phase δ = arg(A1A?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

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

*

8 C II. P  B → ¯D(?)_D(?)_{K } u B0 D( )0

*

*

*

*

*

*

*

*

*

*

*

*

FigureII-3: Suppressed quark diagrams for B → ¯D(?)_D(?)_{K 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

TableII-1: B → ¯D(?)_D(?)_{K decay modes}

decays through external W-emission

B+ → ¯D0D+K0 B0→ D−D0K+ B+ → ¯D0D?+K0 B0→ D−D?0K+ B+ → ¯D?0D+K0 B0→ D?−D0K+ B+ → ¯D?0D?+K0 B0→ D?−D?0K+

decays through internal W-emission

B+ → D−D+_K+ _B0 → ¯D0D0K0 B+ → D−D?+K+ B0→ ¯D0D?0K0+c.c B+ → D?−D+K+ B+ → D?−D?+K+ _B0 → ¯D?0D?0K0 decays through external+internal W-emission

B+ → ¯D0D0K+ B0→ D−D+K0 B+ → ¯D0D?0K+ _B0 → D−D?+K0+c.c B+ → ¯D?0D0K+ B+ → ¯D?0D?0K+ B0→ D?−D?+K0

TableII-2: Isospin decomposition for B → ¯DDK

Channel Decay amplitudes

B0_{→ ¯DDK A(B}0_{→ D}−_D0_K+_{) = √}1 6A1− 1 √ 2A0 A(B0_{→ D}−_D+_K0_{) = √}1 6A1+ 1 √ 2A0 A(B0_{→ ¯D}0_D0_K0_{) = −}q2 3A1 B+ → ¯DDK A(B+→ ¯D0D+K0) = √1 6A1− 1 √ 2A0 A(B+ → ¯D0D0K+) = √1 6A1+ 1 √ 2A0 A(B+ → D−D+K+) = − q 2 3A1

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/ΛQCD

§_{. 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/mQeffects). 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?hC1(µ) sγµ(1−γ5)¯c
cγµ(1−γ5)¯b
+ C2(µ) cγµ(1−γ5)¯c
sγµ(1−γ5)¯b

i

, (II.15) where C1(µ) and C2(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 C1 =1.13 and C2 = _{−0.3. The term with C}1corresponds to a

color-allowed transition, whereas the term with C2to 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 | sγµ(1−γ5)¯c
|0i × hD(?)| cγµ(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)| sγµ(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

ldecays. 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 ffrom Eq.II.15in terms of factorizable part and non-factorizable corrections in the following way:

He f f∼GFVcbVcs? sγµ(1−γ5)¯c
cγµ(1−γ5)¯b
h C1+C2/Ncolor
1 + 1
+2C28i. (II.18)

(C1+C2/Ncolor) ≡ a1is 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/N2

color [19].

If (C1+C2/Ncolor)  2C2, 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? cγµ(1−γ5)¯c
sγµ(1−γ5)¯b
h C2+C1/Ncolor
1 + ˜1
+2C1˜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 2C1 multiplying ˜8 is much larger than the (C2 +

C1/Ncolor) ≡ a2factor 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¯qji , 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 Λχ ' 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/Λχ. 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 SectionIII.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 B0

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

→ ¯

D

D

K

B0

→ ¯D(?)D(?)K0s 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 B0 _{→ 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. B0

→ D(?)−D??+s , where D??s denotes an excited c ¯s meson decaying to D(?)_K0

s. In such a case, the B0and ¯B0decay 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 B0_{→ D}(?)−_D(?)+_K0

s Dalitz-plot analysis. The D(?)−_D(?)+_K0

s final state is described by two independent Dalitz-plot variables: s+ = p_D(?)++p_K0 s
2 , s−= p_D(?)−+p_K0 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(B0→ D(?)−D(?)+Ks0), ¯A(s+,s−) ≡ A( ¯B0→ D(?)−D(?)+Ks0). (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 B0_{(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

2hG0(s

+_,s−_{) + G}

c(s+,s−) cos (∆m t) − Gs(s+,s−) sin (∆m t)i, (III.4) with

G0(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, ¯B0_{(t) → D}(?)+_D(?)−_K0

s, is defined by the transformations: s+ ↔ s−, A ↔ ¯A and β ↔ −β, that give:

|A(s−,s+; t)|2= 1

2hG0(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 G0(s+,s−) is symmetric under s+ ↔ s−, whereas Gc(s+,s−) is antisymmetric. In the case of real A(s+_,s−_{), Gs(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-1shows). 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 B0_{→ D}(?)−_D(?)+_K0

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 B0

→ D?−D(?)+Ks0 are more useful to probe cos 2β. Evaluating the sensitivity of this method requires a better determination of branching fractions for

B0_{→ D}(?)−_D(?)+_K0_{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}

SectionII.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 dq2Γ(B → ¯D(?)l+νl) |q2_=m2 D(?)_s =6π2f2 D(?)s |a1| 2 |Vcs|2, (III.9)