Branching Fraction Measurement of B ± → χc1π +π −K ± andSearch for a Narrow Resonance with the Belle Experiment

Dissertation
zur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades„Doctor rerum naturalium“
der Georg-August-Universität Göttingen
im Promotionsprogramm ProPhys
der Georg-August University School of Science (GAUSS)

vorgelegt von
Elisabeth Patricia Panzenböck
aus Mainz

Göttingen, 2014

Betreuungsausschuss
Prof. Dr. Ariane Frey
II. Physikalisches Institut, Georg-August-Universität Göttingen

Prof. Dr. Arnulf Quadt
II. Physikalisches Institut, Georg-August-Universität Göttingen

Mitglieder der Prüfungskommission:
Referent: Prof. Dr. Ariane Frey
II. Physikalisches Institut, Georg-August-Universität Göttingen

Koreferent: Associate Prof. Kenkichi Miyabayashi (Doctor of Science)
Faculty Division of Natural Sciences, Research Group of Physics, Nara Women’s University, Nara, Japan

Weitere Mitglieder der Prüfungskommission:
Prof. Laura Covi, PhD
Institut für Theoretische Physik, Georg-August-Universität Göttingen

PD Dr. Jörn Große-Knetter
II. Physikalisches Institut, Georg-August-Universität Göttingen

Prof. Hisaki Hayashii (Doctor of Science)
Faculty Division of Natural Sciences, Research Group of Physics, Nara Women’s University, Nara, Japan

Prof Satoru Hirenzaki (Doctor of Science)
Faculty Division of Natural Sciences, Research Group of Physics, Nara Women’s University, Nara, Japan

Tag der mündlichen Prüfung: 2. Dezember 2014

Referenz: II.Physik-UniGö-Diss-2014/06

Branching Fraction Measurement of B ± → χc1π +π −K ± andSearch for a Narrow Resonance with the Belle Experiment

Dissertationin
Research Cooperation
andJoint Doctoral Supervision
betweenthe Faculty of Physics,
Georg-August University School of Science (GAUSS),
Georg-August-University, Göttingen, Germany
andthe School of Interdisciplinary Research of
Scientific Phenomena and Information,
Graduate School of Humanities and Sciences,
Nara Women’s University, Nara, Japansubmitted by
Elisabeth Patricia Panzenböck

Göttingen, 2014

Abstract
The effective QCD colour potential can be derived by studying quarkonia through spectroscopy to determine their masses. In the charmonium spectrum, which is well understoodand precisely measured below the DD̄ production threshold, not all predicted states above thatthreshold have been observed yet and others cannot be assigned to conventional charmonium
states. To fathom the nature of the latter so-called exotic or charmonium-like states motivates
this thesis. A first observation and measurement of the branching fraction of the four-body
B meson decay B ± → χc1 π + π − K ± is presented. In addition, the χc1 π + π − invariant mass
spectrum is studied in order to search for contributions of any intermediate charmonium(-like)resonance. This study
× 106 B B̄ pairs data sample produced at a center√ is based on a 772
of-mass energy of s = 10.58 GeV/c2 corresponding to the Υ(4S) resonance and collectedby the Belle detector at the KEKB asymmetric-energy e+ e− collider located at KEK, Japan.
The signal yield is extracted by an unbinned maximum likelihood fit to the difference-to-beamenergy ∆E distribution and the B decay branching fraction is obtained to beB(B + → χc1 π + π − K + ) = ( 3.89 ± 0.19 (stat) ± 0.20 (syst) ) × 10−4 .
Using the χc1 π + π − mass spectrum, an upper limit including both statistical and systematic
uncertainties is given for a new decay mode of the exotic X(3872) charmonium-like state,
B(B ± → X(3872) K ± ) × B(X(3872) → χc1 (1P )π + π − ) < 1.4 × 10−6 @ 90% C.L.,
as well as a preliminary upper limit for the still undiscovered conventional charmonium χc1 (2P )following a mass assumption of 3920 MeV/c2 ,B(B ± → χc1 (2P )K ± ) × B(χc1 (2P ) → χc1 (1P )π + π − ) < 1.1 × 10−5 @ 90% C.L..
These results favour the hypothesis of X(3872) being an admixture of a D0 D̄∗0 molecule state
and a conventional charmonium.

Kurz-Zusammenfassung
Das effektive QCD-Farbpotential kann aus der Untersuchung von Quarkonia durch Spektroskopie zur Bestimmung ihrer Massen abgeleitet werden. Im unterhalb der DD̄-ProduktionsSchwelle gut verstandenen und präzise vermessenen Charmoniumspektrum wurden noch nicht
alle vorhergesagten Zustände oberhalb der Schwelle beobachtet, und andere können keinen
konventionellen Charmoniumzuständen zugeordnet werden. Das Ergründen der Natur der
letz-teren sogenannten exotischen oder Charmonium-ähnlichen Zuständen stellt die Motivation dieser Arbeit dar. Es wird die erste Beobachtung und Messung des Verzweigungsverhältnisses des Vierkörper-B-Meson-Zerfalls B ± → χc1 π + π − K ± präsentiert. Zusätzlich wird das
invariante Massenspektrum von χc1 π + π − untersucht, um nach Beiträgen von Charmonium(-ähnlichen) Zwischenresonanzen zu suchen. Diese Arbeit basiert√auf einem Datensatz2 von
6772 × 10 B B̄-Paaren, die bei einer Schwerpunktsenergie von s = 10, 58 GeV/c , derΥ(4S)-Resonanz entsprechend, vom Belle-Detektor am energetisch-asymmetrischen e+ e− Beschleuniger KEKB am KEK, Japan, gesammelt wurden. Die Signalausbeute wird aus einem
ungebinnten Maximum-Likelihood-Fit der Differenz-zu-Strahl-Energie ∆E-Verteilung gewonnen und daraus das B-Zerfalls-Verzweigungsverhältnis zu
B(B + → χc1 π + π − K + ) = ( 3, 89 ± 0, 19 (stat) ± 0, 20 (syst) ) × 10−4
bestimmt. Im invarianten Massenspektrum von χc1 π + π − wird eine obere Grenze einschließlich
statistischer und systematischer Unsicherheiten für eine neue Zerfallsmode des exotischen,Charmonium-ähnlichen Zustands X(3872) mit
B(B ± → X(3872) K ± ) × B(X(3872) → χc1 (1P )π + π − ) < 1, 4 × 10−6 @ 90% C.L.,
als auch eine vorläufige obere Grenze des weiterhin unbeobachteten, konventionellen Charmoniums χc1 (2P ), bei einer vorhergesagten Masse von 3920 MeV/c2 , mit
B(B ± → χc1 (2P )K ± ) × B(χc1 (2P ) → χc1 (1P )π + π − ) < 1, 1 × 10−5 @ 90% C.L.
gegeben. Diese Ergebnisse begünstigen die Hypothese, dass X(3872) einen Mischzustand aus
einem D0 D̄∗0 -Molekül und einem konventionellen Charmonium darstellt.

Table of Contents
1. Preface

1

2. Introduction to Particle Physics
2.1. Elementary Particles of the Standard Model . . . . . . . . . . . . . . . . . .
2.2. Interactions of the Standard Model . . . . . . . . . . . . . . . . . . . . . . .
2.2.1. Electroweak Unification and The Flavour Changing Weak Interaction
The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . .
CP Violation and the CKM Matrix . . . . . . . . . . . . . . . . . .
2.2.2. The Strong Interaction and Hadronisation . . . . . . . . . . . . . . .

5
5
6
7
7
9
10

3. Motivation
3.1. The Charmonium Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1. The As Yet Unseen χc1 (2P ) - A Conventional Charmonium . . . . .
3.2. Exotic Charmonium(-like) States . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1. The Puzzling X(3872) - An Exotic Charmonium . . . . . . . . . . .
3.3. B Meson Decays as a Source for Charmonium . . . . . . . . . . . . . . . .

13
13
15
15
18
22

4. The Belle Experiment4.1. The KEKB Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. The Belle Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Silicon Vertex Detector SVD . . . . . . . . . . . . . . . . . . . .
The Central Drift Chamber CDC . . . . . . . . . . . . . . . . . . . .
The Time-of-Flight Counter TOF . . . . . . . . . . . . . . . . . . .
The Aerogel Cherenkov Counter ACC . . . . . . . . . . . . . . . . .
The Electromagnetic Calorimeter ECL . . . . . . . . . . . . . . . . .
The KL0 and Muon Detector KLM . . . . . . . . . . . . . . . . . . . .
Trigger and Data Acquisition . . . . . . . . . . . . . . . . . . . . . .
4.3. Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Data Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25
26
27
28
28
29
29
29
30
30
32
33

5. Event Selection
5.1. Monte Carlo Simulation Dataset . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Reconstruction of χc1 Candidates . . . . . . . . . . . . . . . . . . . . . . . .
5.3. Selection of Light Meson Candidates . . . . . . . . . . . . . . . . . . . . . .
5.3.1. Low Transverse Momentum Pion Selection . . . . . . . . . . . . . .
5.4. Reconstruction of B Candidates . . . . . . . . . . . . . . . . . . . . . . . .

35
35
36
38
38
40

i

5.5. Properties of the reconstructed B Candidate . . . . . . . . . . . . . . . . . .

40

6. Monte Carlo Studies
6.1. Background Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1. Expected Background in ∆E . . . . . . . . . . . . . . . . . . . . . .
6.1.2. Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.3. Expected Background in Mχc1 ππ . . . . . . . . . . . . . . . . . . . .
6.2. Fitting Strategy and Observable Extraction . . . . . . . . . . . . . . . . . . .
6.2.1. ∆E Fit and Efficiency Estimation . . . . . . . . . . . . . . . . . . .
∆E Signal Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Efficiency Estimation for different Masses of an Intermediate State . . . . . .
Background Fit for ∆E . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Total PDF for ∆E in PHSP . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2. Fitting of the Invariant Mass Spectrum Mχc1 ππ . . . . . . . . . . . .
Mχc1 ππ Signal Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Expected Signal Yield in Mχc1 ππ in X(3872) Case . . . . . . . . . . . . . .
Background Fit for Mχc1 ππ . . . . . . . . . . . . . . . . . . . . . . . . . . .
Total PDF for Mχc1 ππ in X(3872) Case . . . . . . . . . . . . . . . . . . . .

45
45
45
46
47
49
49
49
51
53
54
54
54
56
57
57

7. Results and Discussion
7.1. ∆E Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2. Weighted Detection Efficiency Estimation . . . . . . . . . . . . . . . . . . .
7.3. Efficiency Correction and Systematic Uncertainty . . . . . . . . . . . . . . .
7.4. Branching Fraction of B ± → χc1 π + π − K ± . . . . . . . . . . . . . . . . . .
7.5. χc1 π + π − Invariant Mass Spectrum . . . . . . . . . . . . . . . . . . . . . . .
7.6. Upper Limit for X(3872) decaying to χc1 π + π − . . . . . . . . . . . . . . . .
7.7. Upper Limit for χc1 (2P ) decaying to χc1 (1P )π + π − . . . . . . . . . . . . . .

59
59
60
62
64
65
66
67

8. Conclusion

69

A. Cut Selection for π 0 Veto73

B. Estimated ∆E Distribution in the Larger Window

74

C. Expected Signal Yield in the ∆E Distribution in the Phase Space75

D. Upper Limit Calculation

76

E. Other Invariant Mass Combinations

77

Bibliographyi

Acknowledgementsvi

Curriculum Vitæ

viii

ii

1. Preface
The grand questions of mankind, what are the innermost structures of matter, space and time,
and which laws are the basis for the fundamental forces in the universe, have not yet beenanswered completely. But it seems like we are getting closer: The well established theoryof particle physics, the Standard Model, has been proven right over and over again, not least
with the prediction and discovery of the Higgs boson in 2012. However, because of its nature as a renormalisable quantum field theory, many parameters of the Standard Model needexperimental results as input to be determined and cannot be deduced from first principle calculations: When a quantity is found to become finite by the procedure of renormalisation, an
experimental result is required to settle that quantity’s absolute value.
Many weak decay transition rates and CP violation phenomena have been measured to determine the four weak effective couplings in the quark sector as formulated in the CabibboKobayashi-Maskawa (CKM) matrix. This matrix includes one irreducible complex phase that
leads to violation of CP invariance. A comprehensive method to test if the measured effective
couplings match with the CKM framework is provided by the so-called unitarity triangle.
On the other hand, there is only one gauge coupling in the strong interaction, the running
coupling constant αs (q 2 ). Its absolute value has been determined to a precision of a few percent through various measurements at τ lepton hadronic decays, deep inelastic scattering andhadron production at e+ e− colliders. The strong interaction’s characteristics such as asymptotic freedom and the non-Abelian nature of quantum chromodynamics (QCD), the gauge
group describing the strong interaction, are proven and the standard explanation of confinement of quarks in hadrons is widely established. However, even in 2014, 40 years after the
ground-breaking discovery of the J/ψ meson in November 1974 led to the golden age of charmonium spectroscopy, some aspects of QCD are still posing a riddle. In the non-perturbative
regime of αs , two basic questions are not yet solved satisfactorily about the issue of exotic
hadrons: While there is no explicit prohibition rule to form hadrons having unusual constituent configurations like tetraquarks, meson-meson molecules etc, there have been very few
implications or evidence. On the other hand, some of the states in the charmonium spectrum
are predicted, but have not been identified yet; others remain unconfirmed. The suppression
mechanism is unknown. Also, the constituent quarks are not always a good approximation to
be regarded as building blocks to form hadrons. Thus, a more appropriate picture is required
to be deducted, especially for excited states.

1With these exotic hadrons the question arises if there is an exceptional yet unknown mechanism related to excited states, and if so to what consequences such a concept might lead
us in our understanding of the universe. Thus, the rigorous survey of the charmonium andbottomonium spectra is still pursued with much enthusiasm.
It should be noted that a particular excited state may largely affect the evolution of our universe: Carbon forms at the interior of burning stars in what is called the triple-alpha process.
Here, the 12 C as we know it is not produced directly, but rather in an excited state 7 MeV
above the ground state, called the Hoyle state.
The existence of such a state as an argument for stable carbon was predicted by Hoyle in 1954
[34, 43]. This excited state is playing an essential role in the currently observed abundance of
12
C in the universe. And even though there is no obvious link between exotic hadrons and thetime evolution of the universe after the big bang, it offers a great motivation to further explore
excited hadron states and look for something unexpected that might broaden our understandingof the world around us.
As vast a field particle physics may be,
One of the greatest intellectual challenges of modern physics is
to understand confinement not just as a phenomenon but
to comprehend it quantitatively from the theory of the strong force.
So it was written in the Physics Performance Report of the PANDA experiment where I did
my graduate studies. And it is a great way to summarise the goal that motivates the enormous
efforts that have been made in my current field of research - quarkonium spectroscopy. In thisthesis, I will describe and explain the Ph.D. research work which led to the worldwide firstobservation of the B ± → χc1 π + π − K ± final state along with the search for any intermediatecharmonium(-like) state in this particular decay mode. There are two hypotheses pursued fora resonance decaying to χc1 π + π − .
Now, the general structure of this work is as follows: The purpose of Chapter 2 is to provide abrief introduction to the Standard Model of particle physics. In particular, an overview of thefundamental particles and the interactions between these particles is given. Emphasise is puton the flavour changing weak interaction as it is the source to create charmonium in B meson
decays. Another important point is the investigation of hadronisation and thus the quark-quark
potential. The latter is picked up in Chapter 3 which provides a motivation to further explore
the charmomium spectrum and look for so-called exotic charmonium(-like) states.
Here, the main goal of this work is defined, namely the study of a new aspect of the establishedbut still not well understood X(3872) and the search for the yet to be discovered χc1 (2P ).
In Chapter 4 the Belle experiment is introduced as the experimental apparatus to do so. The
KEKB accelerator as well as the Belle detector with its sub-detectors are briefly characterized.
Event selection criteria to reconstruct B meson candidates and a look at their properties aredescribed in Chapter 5. Using a best candidate selection and narrowing down the sample ofreconstructed B candidates to the signal region of interest, Monte Carlo simulation studies are
pursued to estimate potential background and reconstruction effiencies as well as constructing

2

fitting routines for observable extraction. These methods and the calculation of systematic
uncertainties are presented in Chapter 6. Results from this thesis research work and their
discussion can be found in Chapter 7. Chapter 8 concludes this work with a summary andoutlook.
Much of particle physics is concerned with the high-energy interactions of relativistic particles. S.I. based units are used to present e.g. luminosity, its unit is inverse barn where
1 barn = 10−28 m2 . A system of units commonly used in particle physics is the so-called
natural units. To simplify the expression of quantities we conveniently choose: c = h̄ = 1,
where c is the speed of light in vacuum and h̄ the unit of action in quantum mechanics. Therefore, masses are presented in GeV rather than GeV/c2 and momenta in GeV as well instead of
GeV/c. The Einstein energy-momentum relation E 2 = p2 c2 + m2 c4 becomes E 2 = p2 + m2 .

3

2. Introduction to Particle Physics
Particle physics is about exploring, understanding and giving a description of the visible matterin our Universe. It is at the heart of our understanding of the laws of nature. Bringing together avariety of theories and experimental observations throughout the latter half of the 20th century
concluded in the so-called Standard Model of Particle Physics which provides a unified pictureof the fundamental building blocks and the interactions between them.

2.1. Elementary Particles of the Standard Model
While it seems that almost all commonly encountered phenomena can be described in termsof the electron, electron neutrino, proton and neutron, it was observed that at higher energy
scales, further structure becomes apparent. Eventually, in the Standard Model we find 12 spinhalf particles (fermions) which can be divided into leptons and quarks as shown in Tab(2.1).
The first generation consists of the lightest particles, the electron e− , the electron neutrino
νe , the up u and down quark d. As far as we know, those are elementary point-like particles
and not composite structures with a sub-structure. These light fermions represent the basic
components of the low-energy Universe. Each of these particles has two identical siblings that
only differ in their higher masses. The second generation includes the muon µ, muon neutrino
νµ , strange s and charm quark c. The third generation consists of the tau-lepton τ , tau neutrinoντ , top t and bottom quark b. [40] The fact that one fermion doublet forms one generation
originates in the SU (2) gauge symmetry of the electroweak interaction.
Leptons
Particleelectron e−
neutrino νe
2nd generation muon µ−
neutrino νµ
3rd generation tau τ −neutrino ντ

1st generation

QuarksQ (e)Mass (GeV)

Particle

Q (e)Mass (GeV)

−1
0
−1
0
−1
0

0.0005< 10−90.106< 10−9
1.78< 10−9down dup u
strange s
charm cbottom b
top t

−1/3
+2/3
−1/3
+2/3
−1/3
+2/3

0.003
0.005
0.1
1.3
4.5
174

Table 2.1.: Particles of the Standard Model [81].

52.2. Interactions of the Standard Model
The fermions interact with each other through four fundamental forces: gravity, electromagnetism, the strong and the weak force. The first is not yet included in the Standard Modeldescription due to its negligible strength compared to the other three forces at the currently
achievable energy scale. This leads to the many free parameters that can only be determined through experiments and the large difference between electroweak and grand unification scales. The SM could be considered an effective theory only valid at energies far below
the Planck scale. An overview of the forces can be found in Tab(2.2).
Force

Boson

Strong
8 Gluons gElectromagnetic Photon γ
Weak
W Bosons W ±Z Boson Z 0

Spin

Mass (GeV)

1
1
1
1

0
0
80.4
91.2

Table 2.2.: Interactions of the Standard Model.

Each force is described by a Quantum Field Theory (QFT) corresponding to the exchange of
gauge bosons, spin-1 force-carrying particles. In these QFTs, particles are described as fields
that are solutions of the Euler-Lagrange equations. The Euler-Lagrange equation contains
terms for the free particle fields as well as the interactions (vertex factors): L = Lfree +Linteraction . The forces’ nature therefore is determined by the properties of the gauge bosons
and the way in which they couple to the spin-half fermions. The dynamics of each of the 12
fundamental fermions are described by spinors ψ. Their interaction with the field is given by
the Dirac equation of relativistic quantum mechanics, which is equivalent to the Lagrangian
LDirac = ψ̄(x)(iγ µ ∂µ − m)ψ(x).
The so-called gamma matrices γ µ are given by the following, where σ i denotes the Paulimatrices:




1 0
0 σi
0
i
.
γ =
,γ =
−σ i 00 −1
The Lagrange density has to be renormalisable and invariant under local gauge transformations. Its solutions satisfy the proper relativistic dispersion relation.
A particle couples to a force-carrying boson only if it carries the charge of the interaction. The
coupling strength of the gauge bosons to the fermions is described by an associated coupling
constant g for each type of interaction. Often α is used, representing a more convenientassociated dimensionless constant with α ∝ g 2 . A factor of the coupling constant for eachinteraction vertex is included in the quantum-mechanical transition matrix element M for aninteraction process. The Feynman rules can be deduced from the Lagrange equations andallow to calculate the transition probabilities between two quantum mechanical states using
Fermis Golden Rule. They are proportional to the matrix element squared |M|2 .

6

2.2.1. Electroweak Unification and The Flavour Changing Weak
Interaction
The unification of the electromagnetic and the weak force was formulated by Glashow, Weinberg and Salam (GWS) [85]. The electroweak theory is represented by the gauge group
SU(2)L × U(1)Y , where L refers to left-handedness as the charged weak interaction only
couples to fermions of left-handed chirality, and Y is the hypercharge. Y is related to theisospin I and the electric charge Q by the Gell-Mann-Nishijima equation: Q = I3 + Y /2.
The group has four generators that lead to four massless fields that do not directly correspondto observable particles: W1µ , W2µ , W3µ generated by the weak isospin and Bµ0 generated by
the hypercharge. These massless bosons mix to form the physical particles resulting in thefollowing relation for the neutral currents: 0 
 0
Zcos θW sin θWB=γ− sin θW cos θWW3µwith θW being the weak mixing angle (Weinberg angle). This angle can be expressed using g
and g 0 , the coupling constants of the SU(2) and U(1) gauge groups, respectively,gMW
cos θW = p.
=
MZ
g 2 + g 02
The W ± bosons of the charged-current interactions are similarly treated by mixing the W1,2
fields:1
W ± = √ (W1µ ∓ iW2µ ).
2
All 12 fermions of the Standard Model are participating in the weak interaction. The charged
W ± bosons consequently couple together fermions differing by one unit of electric charge.
This charged-current weak interaction is the only known force to introduce a change of quark
and lepton flavour, and is therefore particularly important when considering particle decays.
The weak neutral-current interaction is mediated by the electrically neutral Z 0 boson. It has
been shown that here flavour changing neutral-currents are forbidden in the lowest order.
In the above sketched electroweak theory, the original fields are massless; massive gaugebosons would lead to violation of local gauge invariance. This is in contradiction to measurements of the actually observed gauge bosons with exception of the photon and gluon. The W ±and Z 0 bosons are in fact very massive objects: MW ≈ 80 GeV and MZ ≈ 91 GeV.
The Higgs Mechanism
In the Standard Model, the explicit addition of vector boson and fermion masses to the Lagrangian leads to a breakdown of the local SU (2)L × U (1)Y gauge invariance. The Higgs
mechanism solves this issue assuming that the symmetry is broken spontaneously.

7A scalar field that couples to the fermions and gauge bosons is added. Interaction with thesymmetry breaking vacuum allows the W and Z bosons as well as all the matter fields toacquire masses. The theory was formulated by Brout & Englert and Guralnik, Hagen & Kibble
and Higgs [33, 39, 42, 41] and introduces the Higgs field as a doublet of complex scalar fields
with four real components: 
1 φ1 + iφ2
1 φ+
=√
.
φ= √
2 φ0
2 φ3 + iφ4
The Higgs field Lagrangian’s corresponding potential is represented as
V (φ) = µ2 (φ† φ) + λ(φ† φ)2
where λ needs to be > 0 in order for a stable minimum to exist. Choosing µ2 < 0, thepotential shows a non-zero vacuum expectation value v with v = −µ2 /λ and resembles thefamous "mexican hat" shape, with many possible ground states. This leads to the so-called
spontaneous symmetry breaking of the electroweak theory.
Expanding the field φ in the new minimum leads to an excitation√ of the Higgs field in form of
a new massive scalar, the Higgs boson, with a mass of mH = 2µ. It is the interaction of theinitially massless particles with this non-zero Higgs field that gives them their masses: The
new Lagrangian contains terms that assign mass to the W ± and Z 0 bosons, while the Yukawa
coupling λf to the Higgs field generates the fermion masses
1 p
MW = v g 2 + g 02
2

and

1
MZ = v2

and1
Mf = √ λf v.
2

The fermion flavour difference as well as the heavy quark masses are thought to be due toYukawa coupling with the Higgs boson. The light quarks’ effective masses are mainly comingfrom dynamical effects caused by being surrounded by sea quarks, i.e. due to chiral symmetry
breaking.
After non-significant "observations" of a Higgs boson with a mass around 115 GeV at LEP,
and measurements from Tevatron neither confirming nor discarding these hints [3, 68], a newparticle consistent with the Standard Model Higgs boson was finally reported with a significance of 7σ in July 2012 by the ATLAS and CMS experiments at CERN’s LHC [71, 78, 69].
The Higgs boson decays predominantly into a bb̄ or a W W ∗ → `ν`ν pair, searches have also
been performed in the τ + τ − channel.
A very clean signal, offering the best mass resolution, is provided by the decays H → ZZ ∗ →4` and H → γγ, even though they have relatively small branching ratios. The Higgs mass ismeasured in these channels by both ATLAS and CMS resulting in a combined mass: mH =125.36 GeV based on data collected at 7 and 8 TeV with an integrated luminosity of 25 fb−1
[72].
Further studies show the Higgs boson not being a spin-1 particle due to its decay to two
photons [50, 86] but rather, having positive parity, favouring spin-0 [70].

8CP Violation and the CKM Matrix
To explain the observed matter-antimatter asymmetry in the Universe, Sakharov [61] originally formulated three conditions of which one was the possibility of CP violation in the
weak interactions of quarks and leptons. And indeed, contrary to its conservation in QED and
QCD, parity and charge-conjugation are maximally violated in the weak interaction. It was
found that the weak interaction acts only on left-handed particles and right-handed antiparticles in the weak charged current which is formulated as a V − A interaction, where V and A
are vector and axial vector couplings, respectively. CP violation, the combination of charge
conjugation and parity is not kept under some exceptional conditions, which was confirmed
experimentally in the decay of neutral K mesons in 1964 [29]. It has to date been observed in
the quark sector only.
The coupling strength of the weak interaction for quarks can be determined from the studyof nuclear β-decays and was found to differ at certain quark weak interaction vertices. The
effective coupling for a particular quark-quark transition can be expressed as the product of theuniversal weak coupling and a factor for mixing between the mass and the weak eigenstates.
The weak interactions of quarks are described in terms of the unitary Cabibbo-KobayashiMaskawa (CKM) matrix [25, 48]. The weak eigenstates are related to the mass eigenstatesby  0 
d
Vud Vus Vubds0  =  Vcd Vcs Vcb  s .
Vtd Vts Vtb
b0b
The absolute value of the matrix elements of this unitary matrix were determined experimentally: |Vud | |Vus | |Vub |0.974 0.225 0.004
 |Vcd | |Vcs | |Vcb |  ≈ 0.225 0.973 0.041 .
|Vtd | |Vts | |Vtb |0.009 0.040 0.999
The diagonal entries have been found to be close to one while the off-diagonal elements arerather small which implies that the rotation angles between the quark mass and weak eigenstates are relatively small. Consequently, the weak interactions between quarks of different generations are suppressed the most for the couplings between first and third generation
quarks, ub and td.
Among several possible conventions of the CKM matrix, we often use the Wolfenstein parameterisation, where the matrix is conveniently expressed as an expansion in the small parameter
λ = sin θc = 0.225 written in terms of four real parameters: 
Vud Vus Vub
1 − λ2 /2λAλ3 (ρ − iη) Vcd Vcs Vcb  = 
 + O(λ4 ).
−λ
1 − λ2 /2
Aλ2
Vtd Vts Vtb
Aλ3 (1 − ρ − iη) −Aλ2
1

9In order to let CP violation occur in the quark sector, the matrix needs to contain an irreducible
complex phase corresponding to η 6= 0. Such a complex phase can be accommodated in thequark mixing matrix if there are six quark flavours, equivalent to three generations. Now,
the large numbers of B and B̄ mesons produced at the Belle and BaBar experiments allowed
precise measurements of the branching ratios for semi-leptonic decays as well as various CP
asymmetries and from that determination of the elements of the CKM matrix and the anglesof the unitarity triangle.

2.2.2. The Strong Interaction and Hadronisation
The gauge group of the strong interaction is the non-abelian SU(3)C group with the eight
generators being the force mediating massless gluons. The dynamics of quarks and gluons aredescribed by the gauge invariant Quantum Chromodynamic (QCD) Lagrangian. QCD’s equivalent of electric charge is known as colour, an additional quantum number of the quarks, thattakes three conserved values red, green and blue. It has been introduced to avoid a violationof the Pauli principle which was encountered with the observation of the ∆++ resonance consisting of three up-quarks with parallel spin. If the quarks can be distinguished by colour, thePauli-principle remains valid. Only particles that have non-zero colour charge, namely quarks
and gluons, couple to gluons, which is the reason for leptons not to feel the strong force. As
gluons connect quark states of different colour they must carry colour charge themselves inorder for colour conservation. Quarks have never been observed directly as free, fractionally charged particles despite an abundance of experimental evidence of their existence. The
non-observation is explained by the hypothesis of colour confinement: quarks are always observed to be confined to bound, colourless states. This behaviour is thought to originate fromgluon-gluon self-interaction, since the generators of the SU(3)C group do not commute.
A qualitative understanding of colour confinement, for which there is currently no analytic
proof, can be given by looking at two quarks becoming separated. The interaction between
the quarks can be described in terms of an exchange of attractively self-interacting virtual
gluons. The effect of these interactions is to form a flux tube of interacting gluons of approximately constant energy density and squeeze the colour field between the quarks into thistube. At some point it becomes energetically favourable to create a new quark-antiquark pair,instead of allowing the tube to extend further. Colour confinement therefore places strong
restrictions on the structure of possible hadronic states; the allowed combinations of quarks
and antiquarks are those where a colourless state can be formed: mesons as a pair of quark
and antiquark (q q̄), baryons made of three quarks (qqq) and antibaryons formed from three
antiquarks (q̄ q̄ q̄). Hadronic states thus correspond to and can be labelled by different combinations of quark flavours and different angular momenta states J P , where J is the total angularmomentum, P the parity eigenvalue. In processes such as e+ e− → q q̄, as a consequence ofcolour confinement, quarks do not propagate freely. This process is known as fragmentationand hadronisation: high-energetic quarks and gluons produce jets of colourless particles, thehadrons, with all quark flavours that are energetically accessible, see Fig(2.1).

10Figure 2.1.: Illustration of hadronisation: A flux tube forms and breaks up into colourless final stateparticles [81].

At low-energy scales, QCD processes such as the hadronisation process are currently not calculable from first-principles using traditional perturbation theory as the coupling constant of
QCD becomes large, αS ∼ O(1), and therefore the perturbation expansion does not converge
rapidly. Here, lattice QCD studies are being carried out as a promising alternative.
On the other hand, the value of αS depends on the energy scale of the interaction. Consequently at the typical scale for modern high-energy collider experiments, αS becomes sufficiently small to make the usage of perturbation theory feasible due to the property of QCD
being known as asymptotic freedom. Admittedly, unlike QED, higher-order corrections stillcannot be neglected. But in this regime the quarks can be treated as quasi-free particles, rather
than being strongly bound within hadrons.
It should accordingly be mentioned that hadronic states are often treated as bound states ofsimplified light valence quarks with constituent masses but are, as discussed above, far more
complex, most of the mass of hadrons originating from the energy of the strongly interacting
sea of virtual quarks and gluons within the bound state. While current masses of light quarks
are just a few MeV, the constituent masses can be thought of as the effective masses of thequarks as they move within and interact with the QCD potential. Whilst this is valid forlight quarks, in case of heavy quarks the masses given by the Higgs mechanism and Yukawa
coupling would be dominant.
The non-relativistic QCD potential as a function of the distance r between a heavy quark anda heavy antiquark given as
4 αS
+ κr
Vqq̄ (r) = −3 r
can be used to predict a spectrum of q q̄ bound states. The linear term becomes dominant atlarger radii. It is believed to be responsible and setting the length scale for confinement forheavy quark states.

11

Figure 2.2.: Illustration of the form of the non-relativistic QCD potential for a bound heavy q q̄ state,assuming αS = 0.2 and κ = 1 GeV/fm. [81]12

3. Motivation
On the trail of the QCD colour potential, flavorless meson systems of a heavy quark and its
own antiquark, so-called quarkonia provide a fruitful environment for spectroscopy of theirbound states. Quarkonia, as strongly interacting systems, have been studied in an analogous
manner to hydrogen atoms or positronium; from the measured spectrum and transition probabilities between the states one can derive the effective potential.
When describing charmonium formation from cc̄ pairs that are produced in B decays or ine+ e− annihilation, effective field theories are used. They are found to work quite well becauseof the large masses of the heavier charmonium (cc̄) and bottomonium (bb̄) systems. Thosesystems can be treated by approximation as non-relativistic because of their velocities being
relatively low with β ≡ v/c ∼ 0.3 and β ∼ 0.1, respectively. Large masses are functioning asa natural cut off permitting the use of perturbation theory, most often non-relativistic QCD.

3.1. The Charmonium Spectrum
Charmonium bound states possessing quantum numbers J P C = 1−− are observed as resonances in e+ e− annihilation at centre-of-mass energies between 3 and 4.5 GeV. This series ofresonances can be found in a scan by varying the beam energy. They show as enhancements inthe total hadronic cross section and charmonia having other quantum numbers are identified
in consecutive decay modes. The resultant assignments to the various charmonium states areshown in Fig(3.1). The lowest state with these quantum numbers is the 1S ψ, for historical reasons called J/ψ, with a mass of 3097 MeV [13, 12]. Using methods of spectroscopy, particles
can be named with their spectroscopic notation or their mass, sometimes excitation series areused: e.g. ψ 0 being the ψ(2S) state. At centre-of-mass energies of ∼ 10 GeV, another groupof resonances can be seen, the bound states of the bottomonium system. The bb̄ spectrum isvery similar to that of cc̄ which leads to the important conclusion that the QCD potential isuniversally applicable for different heavy quark flavours.
In general, heavy quarkonia show the following four kinds of decays:
1. Radiative decays, e.g. χc1 → J/ψγ:
The charmonium states have a limited life time and decay mainly through the strong
interaction. Excited states can convert via de-excitation by emitting photons. Throughelectromagnetic transitions, a change in orbital angular momentum or spin, energeticallylower states can be reached. An example would be the decay of the first radially excited
charmonium state ψ(2S) to the 1P triplet system of χcJ states.

13Ψ(4415)cc̄

Ψ(4160)4

Ψ(4040)Mass/GeVηc(2S)

χc2(2P)χc0(2P)Ψ(3770)Ψ'hc(1P)

χc1(1P)χc2(1P)χc0(1P)3J/Ψηc(1S)

0‒+

1‒‒

1+‒

0++
JPC

1++

2++

Figure 3.1: Spectrum with masses and J P C
of the observed and assigned
charmonium bound states; basedon [81]

2. Annihilation of the q q̄ pair, e.g. J/ψ → `+ `− .
3. Feed down by pion pair emission, e.g. ψ 0 → J/ψπ + π − .
4. Addition of light q q̄ pairs, e.g. ψ(3770) → D0 D̄0 :The light q q̄ pairs are taken from vacuum production via formation of light mesons. This
decay mechanism can only take place above a certain energy threshold as the light q q̄pair has to be made from the excitation energy of the quarkonium.
Okubo, Zweig and Iizuka formulated the so-called OZI rule to describe the suppression mechanism in the case if the Q-value is not sufficient to create a q q̄ pair from an open-charm meson
pair. Then, the OZI suppressed amplitude of decaying to a lower charmonium state by emitting a pion pair is the allowed strong decay. In the case where an open-charm meson paircannot be produced, no quark current exists continuing from initial to final state. Such an amplitude cannot be large, and therefore the second decay mechanism is suppressed comparedto the third. Thus the charmonia which are below the so-called open-charm or DD̄ threshold at 3740 MeV that decay only via weak, electromagnetic and strong but OZI-suppressed
processes have narrow decay widths, e.g. Γ = 0.088 MeV in case of J/ψ.
On the other hand, above that threshold, strong decays to a D meson pair predominantly
take place as it is in general the simplest strong decay. Due to these available new decay
possibilities, states decay faster which results in wide decay widths. ψ(3770) is the first state
above threshold and has a 93% probability to decay to DD̄, as expected [58]. Looking at themore strongly bound bottomonium the decay to b quarks and mesons does not take place till
the third excitation, Υ(4S) at 10.58 GeV, which is the first one to decay to B B̄ among theJ P C = 1−− bottomonium states.

14

3.1.1. The As Yet Unseen χc1 (2P ) - A Conventional Charmonium
The χc1 (1P ) is the lightest axial vector meson containing a cc̄ pair with a mass of 3510 MeV.
Its radial excitation with spectroscopic notation 23 P1 , the χc1 (2P ) (or χ0c1 ), is predicted but
still undiscovered. Its quantum numbers are J P C = 1++ and it is part of a spin triplet.
The mass prediction for the χc1 (2P ) lies above the open-charm threshold and is expectedaround 3920 MeV [84]. In this energy region Belle has reported the observation of two resonant structures denoted by Y (3940) and Z(3930) [28, 27]. The temporarily dubbed Z(3930)
was observed as an enhancement in the DD̄ mass spectrum from e+ e− → e+ e− DD̄ events
where γγ → DD̄ takes place and is now interpreted as the χc2 (2P ) [83]. Since 2014, theY (3940) has been identified as the same state as X(3915) and is recognized as the χc0 (2P )[58]. Looking at the total decay widths of these two states, Γtot (χc0 (2P )) = (20 ± 5) MeV
and Γtot (χc2 (2P )) = (24 ± 6) MeV, it comes at a natural choice to assume a width of approximately 20 MeV for χc1 (2P ). In the beginning of this study the comparison with thewell known ψ 0 → J/ψπ + π − as one of the major decay modes led to the speculation that
χc1 (2P ) has a significant branching fraction to a χc1 (1P )π + π − final state. Based on the assumption that B(χc1 (2P ) → χc1 π + π − ) ≈ B(ψ(2S) → Jψπ + π − ), ignoring quantum number
differences, χc1 (2P ) was thought to be a narrow resonance.
It is noteworthy that decays predicted as major decay modes, χc1 (2P ) → J/ψγ and χc1 (2P ) →ψ(2S)γ, have not been observed nor has a peak been seen in the D0 D̄∗0 spectrum at 3920 MeV.
This leaves the χc1 (2P ) → χc1 π + π − as the almost only remaining decay mode to search forχc1 (2P ). The decay χc1 (2P ) → DD̄ is forbidden by parity conservation, while the χc1 (2P )could decay into DD̄∗ , if energetically allowed. The absence of radiative decays and no decay
to a charmed meson pair give a hint that the χc1 (2P ) might be another special case and deviate from the conventional charmonium spectrum. This will become important when looking
at possible interpretations of the X(3872) state later in this chapter.
This thesis, studying the B ± → χc1 π + π − K ± final state, is motivated by a search for the yet
unseen intermediate state of χc1 (2P ) in the invariant mass distribution of χc1 (1P )π + π − .

3.2. Exotic Charmonium(-like) States
So far, all of the predicted cc̄ states below the DD̄ threshold at 3740 MeV have been observed
and measured with high precision. They were found to have properties that are well describedby the charmonium model. In addition, all the J P C = 1−− states with a mass > 2mD wereidentified. But there are still predicted but unconfirmed states above the open-charm threshold.
Charmonium is the best understood hadronic system and is much cleaner with regard to thedense spectrum of light states where mixing of neutral mesons plays an important role becauseof the SU (3) flavour symmetry. So-called exotic states containing cc̄ are expected to be identified easier than the ones predicted in the light spectrum of hadrons comprised of light quarks(u, d, s)15

Strong evidence for meson-like states that do not fit into the simple q q̄ scheme has beensteadily accumulating during the past decade. Especially above the open-charm threshold,several charmonium-like states with unusual characteristics have been reported [59].
These so-called XY Z states are charmonium-like and bottomonium-like hadrons that do not
fit into any of the remaining predicted but still undiscovered states in the cc̄ and bb̄ spectra
as shown in Fig(3.2). X, Y and Z are temporary names usually followed by their mass,indicating that there are still open questions about these particles’ properties and their place inthe spectra.

Figure 3.2: Spectrum with masses and J P C
of established, predicted but
undiscovered, and XYZ states;
based on [59].

To identify exotic states, further studies of the quarkonium model that specifies the allowed
states of a q q̄ system are necessary; if a quarkonium-like hadron decays into a final state withproperties that do not match the expected ones of any of the unfilled levels in the associatedquarkonium spectrum, it is consequently called exotic. Some indications for an exotic hadroncandidate are a large branching fraction to anything but DD̄, an unusually narrow decay widthor a decay mode that cannot be explained as a charmonium state. Since the c and b quarks areheavy, their production from the vacuum in the fragmentation process is heavily suppressed.
Thus, heavy quarks that are seen among the decay products of a hadron must have existed
among its original constituents.
QCD-motivated models predict the existence of hadrons of more complex structure than conventional mesons or baryons. Even though, up-to-date experimental searches have passed
without conclusive evidence for specifically QCD-motivated exotic hadrons, all confirmed
hadrons seem to be either mesons or baryons. Information on the many discovered exotic
charmonium-like states is rather limited. There are four main interpretations to explain theobserved exotic states such as hybrids and multiquark states or molecular configurations.

16

Multiquarks: Tetra- and Pentaquarks
There are two types of multiquark states for which there is no explicit suppression rule
in QCD. Originally, the idea of a tetraquark was introduced to explain the propertiesof light scalar mesons like some of the f0 and a0 states as they seemed to lie outside
the existing quark model classifications [46]. A tetraquark is an exotic meson, a tightly
bound four-quark state, that is suggested as a diquark-diantiquark structure in which
the quarks interact dominantly by a gluon exchange. Strong decay modes are expected
to proceed via rearrangement processes, followed by disassociation. The cc̄ states thatmight be classified as tetraquarks could be easily distinguished from conventional charmonium states as they could have non-zero charge or contain strangeness, for example
the Z(4430)− with a quark content of (cc̄dū) [24, 80].
A pentaquark on the other hand consists of three quarks and one quark-antiquark pairbound together, thus being classified as an exotic baryon. Unfortunately, all pentaquark
candidates so-far have been found by high-statistics experiments to be statistical effects
rather than true resonances. Despite these null results, as of 2009 LEPS results continue
to show the existence of a narrow state with a mass of (1524 ± 4) MeV, with a statisticalsignificance of 5.1 σ [56] which is considered a pentaquark candidate.
So in principle, combinations such as tetraquark (q q̄q q̄) or pentaquark states (qqqq q̄) could
exist, either as bound states in their own right or as hadronic molecules such as (q q̄)-(qqq) or(q q̄)-(q q̄).
Di-Meson Molecular States
In analogy to the formation of deuterons from a proton and a neutron, it is also believedthat mesons can as well interact to form molecules. Such a mesonic molecule would be a
set of two (or more) mesons bound together by the strong force. The interaction within
the molecule is thought to be due to the asymptotic and therefore distance-dependent
nature of the quark potential: the confined one-gluon exchange interaction among thequarks should dominate at short distances, an interaction via pion exchange at large. A
consequence of their loose binding is that they tend to decay as if being free mesons.
However, molecular states should result in distinctive decay patterns which would distinguish them from tetraquarks. The difference could possibly be determined from theirdeviating degrees of freedom and size. Right now, Zb (10610) and Zb (10650) seem tobe the best candidates for such an observation [76].
The other two approaches to explain exotic states are of a different nature.
Quark-Gluon Hybrids and Glueballs
States with an excited gluon in addition to the quarkonium are referred to as hybrid
mesons, and are described by many different models, among them the flux-tube model[44]. The lowest excitations of the potential produced by gluons lead to the lowest
mass hybrids; of which some show exotic quantum numbers like J P C = 0+− , 1−+ or
2+− . Those are not allowed per se in case of conventional charmonium states and theirobservation would indicate the existence of an exotic resonance.

17

The lowest charmonium hybrids are predicted by lattice QCD to have masses of about
4200 MeV [53], whereas their dominant decays are expected to be open-charm decays
including P -wave mesons in their final states, and hadronic transition to charmonium
via emission of light hadrons. In addition, it seems that gluons, the self-interacting
mediator particles of the strong interaction, could also form bound states by themselves,
so-called glueballs [55].
Threshold Effects
Being close to the open-charm threshold can also give rise to structures in cross sections and kinematic distributions. There is in fact a variety of thresholds, depending on
the different types of D mesons, e.g. mD0 D∗0 = 3872 MeV or mD+ D− = 3740 MeV.
S-wave scattering dominates the cross section at the threshold, however in a few cases
higher order waves also play a role. Considering that states in a relative S-wave with low
relative momentum can live long enough on the timescale of the strong interaction tointeract by exchanging pions, a formation of mesonic molecules due to this attractive interaction is thinkable; while any repulsive interaction might suggest a virtual state above
threshold [32]. Thus, there can be a structure in the cross section near the kinematicalthreshold which may or may not be an ordinary, real resonance. In addition, the effectcoming from cc̄ states near a threshold that can interact with the latter and might result ina mass shift of both, the charmonium resonance and the threshold-related enhancement,
could be quite significant in the observed cross section.

3.2.1. The Puzzling X(3872) - An Exotic Charmonium
In the last decade a number of exciting discoveries of new hadron states has challenged
our description of hadron spectroscopy. The first and, since its discovery in 2003 by the
Belle collaboration, most studied exotic hidden-charm state’s existence is the well established, but still mysterious X(3872), see Fig(3.3) [26]. The X(3872) was initially observed
as a narrow peak in the J/ψπ + π − invariant mass distribution in the charged B meson decay
B + → J/ψπ + π − K + , and due to the decay to a J/ψ naturally identified as a new charmonium state. Its existence, in inclusive pp̄ production, too, was soon confirmed by several otherexperiments [10, 6, 2]. It is also the only exotic charmonium-like state that has been observedin several decay modes.
The X(3872)’s mass is (3871.2 ± 0.5) MeV in the current world average [58] which is very
close to the open-charm threshold of D0 D̄∗0 at (3871.8 ± 0.12) MeV and does not match withany predicted cc̄ state. Its exotic nature was in fact first signalled by its unusually narrow widthof only Γ < 1.2 MeV in the best current estimate [58]. This is puzzling as broad structures
are expected above threshold which allows the strong decay to a charmed meson pair.
The first step in order to identify the structure of such a state is the determination of its quantum
numbers such as total angular momentum J, charge conjugation C and parity P . By now,
all early questions concerning quantum numbers and the existence of any charged or neutral
partner states have been resolved:

18

The Belle and Babar collaborations observed the radiative decay X(3872) → J/ψγ which determines the C-parity to be positive, C = +1 [4, 15]. Two recent searches for a C-odd partner
state of X(3872) in the J/ψη and the χc1,2 γ final states were pursued [67, 57]. However, theproduction of such a partner in two-body B decays and its decay to either J/ψη or χc1,2 γ arefound to be considerably suppressed [45, 19].

(a) Unbinned fit of the MJ/ψππ for theX(3872) → J/ψπ + π − signal region.

(b) In addition to a large ψ 0 → J/ψπ + π − peak,X(3872) was seen as a significant spike.

Figure 3.3.: First observation of a narrow charmonium-like state with a mass of (3872±0.6±0.5) MeV
and a statistical significance in excess of 10σ [26].

The decay angular distributions from X(3872) → J/ψπ + π − and the 3π invariant mass spectrum in J/ψπ + π − π 0 studied as well as the associated pions’ invariant mass distributions
agree well with expectations for a 1++ assignment as proposed by Belle [74], CDF [77] andBaBar [73]. This result has been confirmed and established by the LHCb collaboration in2013 as can be seen in Fig(3.4).
Currently, X(3872) is only observed in B meson decays and hadron collisions. But ever since
there has been conclusive evidence for this now well established state, many hypotheses about
its nature have been proposed, especially attempts to treat it as a multiquark state.
The X(3872) has been proposed to be the still undiscovered conventional charmonium stateχc1 (2P ) as they are sharing the same quantum numbers of 1++ and it displays some characteristics of a charmonium-like state. It is however an unlikely assignment disfavoured by
the value of the X(3872) mass leading to a mass difference of ∼ 50 MeV [84, 74]. Detailed
angular analyses also showed that the decay to J/ψρ0 is the dominant one, leading to theconclusion of isospin violation if X(3872) were a conventional charmonium. What was seen
in experiments, is that the di-pion mass distribution in X(3872) → J/ψπ + π − is consistent
with originating from a ρ0 resonance, while there has been evidence that the tri-pion mass
distribution in X(3872) → J/ψπ + π − π 0 favours the decay via a virtual ω resonance.

19

Figure 3.4: Background-subtracted
distribution of cos θX for (top) all
candidates and for (bottom) candidates with |cos θππ | > 0.6 for thedata (points with error bars) compared to the expected distributions
for the J P C = 1++ (red solid
histogram) and J P C = 2−+ hypotheses (blue dashed histogram)at LHCb experiment [1].

The ratio was found to be
B(X(3872) → J/ψω)= 1.0 ± 0.4 ± 0.3.
B(X(3872) → J/ψρ)The sizes of the branching fractions are comparable to each other which makes it necessary
to clarify the mechanism to cause such a large isospin violation of the X(3872) [74]. Thisevidence for isospin breaking contradicts the assignment as χc1 (2P ) is part of an isospin
singlet and can therefore not be matched with the isosinglet X(3872).
Isospin violation can also be shown by isospin partners with differing masses, as well as
different electric charge: Partner states have been predicted for the X(3872) as a compact fourquark state as it was originally theorised as a tetraquark candidate. This explanation predicts¯ The former would bethat a neutral doublet should exist corresponding to [cu][c̄ū] and [cd][c̄d].
identified with the X(3872) and produced in the known charged B + → J/ψπ + π − K + decay,
whereas the additional neutral state should be observed in neutral B 0 → J/ψπ + π − K 0 decays
having a mass differing by a few MeV from the X(3872). However, recent studies have notrevealed any significant separation between the masses of the X states produced in charged
versus neutral B decays [54, 31, 75]. The ratio of the branching fractions
B(B + → X(3872)K + )∼ 10%
B(B 0 → X(3872)K 0 )is predicted, but the experimental value is found to be 0.5 ± 0.3 ± 0.05 [22, 11] speaking infavour of the decay of a single X(3872) state. Also, no evidence for a charged partner of theX(3872) predicted by the tetraquark models in the J/ψπ + π 0 mode has been seen strongly
suggesting isospin zero [9].

20As X(3872) shows to be a 1++ state and its mass being near the D0 D̄∗0 mass threshold, it
appears as a good candidate for a molecule-like D0 D̄∗0 bound state [64]. With L = 0 it would
be a deuteron-like state bound by pion exchange. Supporting this interpretation, the branching
fraction of X(3872) → D0 D̄∗0 has been measured to be one order of magnitude larger thanthat of X(3872) → J/ψπ + π − [58, 14].
As the indications for a molecular state keep on accumulating, QCD Sum Rule calculations
reported a better agreement with the experimental mass of X(3872) when using a current forthe mesonic molecule of the type (D∗0 D̄0 − D0 D̄∗0 ) than for tetraquark calculations [52].
However, an issue appears when looking at the difference in mass between the X(3872)
and the D0 D̄∗0 threshold which corresponds to the available binding energy in a potential
molecule: The D0 D̄∗0 threshold is at (3871 ± 0.27) MeV, leaving only (0.16 ± 0.31) MeV foran exceptionally weak binding. The binding energy can be translated to the distance between
the two D mesons assuming a Yukawa-like potential with pion exchange, resulting in 5 fm
in case of X(3872) [23]. Compared to the J/ψ meson’s size of about 0.4 fm, the differencein volume is so large that the probability for a c and c̄ quark to form a J/ψ and therefore
the significant branching to the latter cannot be explained. At last, such a very loosely boundobject could hardly be produced in high energy proton-antiproton collisions like they’ve been
observed at Tevatron, making it difficult to explain the X(3872)’s production cross section
measured by the CDF experiment.
These (pure) molecule interpretations meet some more issues when looking at radiative decays
like the heavily suppressed decay to ψ(2S)γ which is crucial for understanding the structureof X(3872). BaBar and LHCb found a signal while the Belle experiment did not [15,20]. This
controversial result is reflected in the ratio of the branching fractions
B(X → ψ(2S)γ)= 2.46 ± 0.64 ± 0.29.
B(X → J/ψγ)
However, this ratio is predicted to be of order ×10−3 for a DD̄∗ molecule [30]. This leads
to the conclusion that the relatively large branching fraction for X(3872) → ψ(2S)γ is generally inconsistent with a pure D̄0 D∗0 molecular interpretation of the X(3872), and possibly
indicates mixing with a significant cc̄ component. [79, 63, 15]
Also in QCD Sum Rules calculations it is possible to explain the properties of the X(3872) assuming it is a mixture between a cc̄ state and the D0 D̄∗0 , D∗0 D̄0 , D+ D∗− and D− D∗+ molecular states [52]. The so-called coupled channel approach shows that the X(3872) emerges ina constituent quark model calculation as a mixed state of a DD∗ molecule and χc1 (2P ) state.
This dual structure may explain simultaneously the isospin violation shown by the experimental data and the radiative decay rates [60].
So, an admixture state of a D0 D̄∗0 molecule and a conventional charmonium, possibly thestill undiscovered χc1 (2P ), seems to be the most plausible hypothesis at the moment [66].
The possibility to find such an intermediate state in multi-body B decays as B → χc1 π + π − Kis provided by the so-called B factories and the aim of this thesis.

21

3.3. B Meson Decays as a Source for Charmonium
In the last decade, both an experimental and a theoretical revival in charmonium physics dueto the B factories were seen. In general, e+ e− colliders are an excellent place to study theheavy quark spectra due to their well-defined production of quarks: the QED process is wellunderstood, there is no need to know the initial parton structure functions as e± are pointlike
particles and the production process is experimentally very clean as there are no remnants asin proton-(anti)proton colliders. Although the B factories were primarily constructed to study
the CP -violation in the B meson system and quarkonium studies were not a priority at thestart of the running, these facilities have come to function as heavy meson factories collecting
large enriched charm samples on spectra and decays. The discovery of a large number of newcharmonium-like meson states as described in the previous section became an unexpectedbonus. The B factories also provide useful information on charmonium production rates indifferent processes, as well as kinematic characteristics of produced charmonia, with whichmodels can be tested numerically and charmonium properties be determined. Charmonium
states can be produced in numerous production mechanisms:
The simplest B meson decays yielding charmonium states are B → KXcc̄ . In two-body B
decays the two decay daughters are so quickly getting apart due to the B mesons large massof 5.28 GeV that very little final state interaction takes place. In such a situation, the decay
amplitude can be formulated as the product of two currents into corresponding final state particles. This mechanism is called factorisation. These B → KXcc̄ decays are described by
the Cabibbo-favoured b → cc̄s transition, and thus have large branching fractions, O(10−3 ),assuring a high statistics sample. Decays of this type favour production of charmonia bearing J P C = 0−+ , 1−− and 1++ , while known quantum numbers of the parent B meson allow
the determination of the spin-parity of the produced Xcc̄ by performing angular analyses. B
mesons can decay into almost all possible charmonium states, with typical inclusive branching
fractions ∼ 1%, although some states are dynamically suppressed. The first example of charmonium production in B decays was the decay B → J/ψX. Two-body decays of the type
B → (cc̄)K (∗) have been extensively studied, because of their extremely clean experimental environment and their importance for CP-violation measurements. But even the study ofmultiple-body decays, like this thesis’ target final state χc1 π + π − , is made feasible and benefits
from the B factories’ environment. There is no additional particle production in B B̄ events,
thus controlling combinatorial background is manageable.
In resonant direct production in e+ e− annihilation with Initial State Radiation the initial e±radiates one or more photons lowering the centre-of-mass energy accordingly. If the energyof the emitted γ is large enough the charmonium mass range can be reached, allowing the
production of charmonia with spin-parity J P C = 1−− . Being a higher-order QED process
suppressed by αem is compensated by very high luminosities. In the special case of both
leptons emitting photons that interact, two-photon collisions (γγ fusion) can give access tostates with different quantum numbers. At B factories, two photon collisions produce J P C =0±+ , 2±+ and 3++ charmonia.

22In hadronisation originating from cc̄ pairs produced in e+ e− annihilation, prompt charmonium production was expected to be in e+ e− → cc̄g(g). Here, the gluons carry away enoughenergy so the mass of the cc̄ pair results in the charmonium mass region. Another contribution
from e+ e− → (cc̄)(cc̄) was estimated to be so small, that detection of this process was considered hardly possible. Surprisingly, this double charmonium production, e+ e− → Xcc̄ Ycc̄ ,was observed with a much larger cross section than QCD originally predicted [18].
Thanks to the variety of possible reactions, consequently, the observed spectrum provides aprobe of the QCD potential and coupling strength in the non-relativistic limit.

23

4. The Belle ExperimentThis chapter introduces the KEKB accelerator and the Belle experimental apparatus basedon standard literature on the topic. More detailed information can be found in references[18, 65, 16].
The Belle experiment is a high energy electron-positron collider experiment and one of two
dedicated B physics experiments in the world1 . It is located at the High Energy Accelerator
Research Organization (KEK) in Tsukuba, Japan, and conducted by an international collaboration with members from 18 countries. After the first physics run in 1999, the Belle experiment
took data mainly for a centre-of-mass energy of 10.58 GeV corresponding to the Υ(4S) resonance. The Υ(4S) decays with a branching fraction > 96% almost exclusively to charged
or neutral B meson pairs [58]. Therefore, e+ e− experiments operating at this centre-of-massenergy have earned the name of B factories. A total integrated luminosity of 1052 fb−1 wasaccumulated by the Belle detector; operation at the Υ(4S) stopped in 2008, and including runs
at other energy points, data taking ceased in 2010. KEKB is now turned off to be upgraded toSuperKEKB which is planned to have a 40× higher luminosity performance.
Belle was primarily designed for a detailed study and test of the Kobayashi-Maskawa mechanism for CP asymmetry in the B meson system and to understand the origin of CP violation.
The other physics goals were precise measurements of decays of bottom and charm mesons,τ leptons and the search for rare or forbidden processes in the Standard Model. The baseline
requirements for such an experiment were the following:• high luminosity, namely instantaneous luminosity up to 1034 cm−2 s−1 ,• B 0 B̄ 0 pairs with a boost factor of the centre-of-mass system relative to the laboratory
system sufficient for observing the time evolution of B decays,• high-resolution and large-coverage detector with excellent particle identification.
Two important findings that then led to the construction of the B factories like the KEKB accelerator were the long lifetime of the B meson and a substantial rate for B 0 B̄ 0 mixing due tothe very large top quark mass. Furthermore, extraordinary advancements in the performance ofe+ e− storage rings with order-of-magnitude luminosity improvements contributed. Progressin the capabilities of large solid-angle detectors, especially considering performance of dataacquisition systems to handle the huge event rates associated with the available luminosities,
precision tracking and vertexing devices was made. Last but not least, the advancement insoftware and storage technologies required to deal with these large data samples permitted therealisation of the B factories.
1

The other experiment is BaBar at SLAC in Stanford, USA.

25

4.1. The KEKB Accelerator
KEKB is an asymmetric energy accelerator colliding 8 GeV electron and 3.5 GeV positron
beams. It was designed to produce B mesons with a lab-frame boost sufficient to enable
decay-time-dependent measurements; therefore the Υ(4S) resonance at 10.58 GeV, on which
KEKB was mostly running, was boosted by βγ = 0.425.

Figure 4.1: Schematic view of the KEKB
ring at KEK in Tsukuba,
Japan [18].

On the accelerator side the biggest challenge was how to reach the necessary high luminosity.
It was decided on a setup in which the two beams collide with a small but finite crossing angleof 22 mrad in the horizontal plane at the beam collision point.
Characteristic

Value

Beam energy [GeV]
Beam current [A]Beam size at IP
x [µm]
y [µm]z [mm]
Luminosity [cm−2 s−1 ]Number of beam bunches
Bunch spacing [m]Beam crossing angle [mrad]

8.0 (e− ), 3.5 (e+ )1.2 (e− ), 1.6 (e+ )80
15
2.1 × 1034
1584
1.84
22

Table 4.1.: Machine parameters of KEKB during the last stage of operation [18].

26In this collision scheme, the beams are naturally apart after the beam bunch crossing, thus noadditional magnet is necessary to let beams reenter and circulate two proper rings. In 2007,
the superconducting crab cavity was installed. It realigns the directions of the beam bunches
so they pass through each other head-on, leading to a higher beam-beam interaction resulting
in higher specific luminosity. This treatment eliminates parasitic collisions and reduces synchrotron radiation around the beam collision point. Also shorter bunch spacing is allowed inprinciple and there is more available space for the detector components near the interaction
point. KEKB eventually reached a peak luminosity of 2.1 × 1034 cm−2 s−1 , more than twice aslarge as the design luminosity.

4.2. The Belle Detector
Belle is a large solid angle spectrometer. It shows a compact design with a cylindrical symmetry around the beam axis surrounding the beam crossing point. Its size was optimised to realisea large enough tracking system volume and to make the volume of the calorimeter reasonable.
Due to the asymmetric collider the detector is arranged asymmetrically along the direction of
the boost to maximise acceptance. By convention, the forward and backward sides are defined
relative to the high energy beam. With the boost, more particles are produced on average inthe forward direction.

Figure 4.2.: The Belle detector [17].

27The forward and backward acceptances are constrained by the beamline geometry. The fieldstrength of the superconducting solenoid magnet is 1.5 T with the field uniformity better than
4% in the central tracking volume.
The main requirements for such a detector were a composition of light material for the innerdetector and very good vertexing capability. Also good particle identification over a broad
range of momentum was important, especially π − K separation which is mandatory for CP
studies. Electromagnetic calorimetry to precisely measure the energy of both electrons andphotons is also essentially important. A good KL and muon identification was significant
for CP violation measurements as well. Regarding the huge amount of events produced by
the high luminosity machine, data handling capabilities from the detector system front-end,classified by a trigger system, and storing for subsequent processing were a challenge as well.
An overview of the Belle detector with its major components can be seen in Fig(4.2). Below, abrief description of specifications of the Belle sub-detectors can be found with a summarisingtable at the end of this chapter:

The Silicon Vertex Detector SVD
During the first three years, the so-called SVD1 detector was used. Its design consisted of threelayers of AC coupled double-sided silicon-strip detectors and covered the 23◦ < θ < 140◦ partof the Belle detector acceptance. It was adequate at the beginning of the experiment. However,it was foreseen that limited radiation hardness of the chip and its long shaping time among
other issues with connected Belle read-out electronics made it difficult to keep up with theperformance improvements of the KEKB collider. Consequently, efforts to upgrade the SVD
started before KEKB began operation. The second generation silicon vertex detector, SVD2
was installed in summer 2003. It consists of four layers of double-sided silicon-strip detectors.
It covered the full angular acceptance of 17◦ < θ < 150◦ , showed improved radiation tolerance
and used a newly developed read-out chip. The radii of the SVD2 layers range from 20 mm to
88 mm. For this upgrade, the inner radius of the beam pipe was reduced to 15 mm. The SVD
is mechanically supported by the central drift chamber, CDC. This contributes to a minimal
material accumulation inside the inner radius of the CDC.

The Central Drift Chamber CDC
Several important roles are assigned to the CDC. In the first place, it performs the reconstruction of charged particle tracks and the precise measurement of their hit coordinates in thedetector volume. Together with the magnetic field provided by the superconducting solenoid,
particle momenta can be measured. The reconstructed tracks in the CDC are extrapolated to
the SVD to pick up associated hits in order to precisely determine the event vertex as well as
the B meson vertices. This tracking system combination provides good momentum resolution
without problematic deterioration for low-momentum tracks. Tracks with pT > 70 MeV canbe reconstructed where pT is the transverse momentum with respect to the beam axis.

28

The resolution for tracking by SVD and CDC is measured to be σpT /pT = (0.0030/β ⊕0.0019pT ), where pT is in GeV. Finally, using dE/dx measurements within the gas volume
enables particle identification. A helium-ethane (50 : 50) gas mixture was chosen to minimise
multiple scattering. The CDC also provides efficient and reliable trigger signals for charged
particles. The innermost part of the CDC was modified in 2003 to accommodate for the SVD2.
Since the majority of the particles in B meson decays have momenta < 1 GeV, minimising
the material when building a tracking detector is important.
The CDC is built asymmetrically along the beam direction with an angular coverage of 17◦ ≤θ ≤ 150◦ . Its outer radius is 880 mm and it has 50 cylindrical layers. The chamber has atotal of 8400 drift cells. The maximum wire length is 2400 mm. In all layers, except the three
innermost that correspond to the cathode part, the maximum drift distance is between 8 and
10 mm. The maximum drift time is < 300 ns. Detailed alignment and calibration led to an
overall spatial resolution of ∼ 130 µm and a dE/dx resolution of 7% for minimum-ionizing
particles, which is important for particle identification.

The Time-of-Flight Counter TOF
A barrel of 128 plastic scintillation counters makes up the time-of-flight system, where one
module consists of two counters and one thin trigger counter. This configuration includes an
air gap to remove false triggers from photon-conversion which is coming from a large photon
background due to scattering in the beam pipe. The detector’s acceptance is 33◦ < θ < 121◦
where the minimum transverse momentum to reach a TOF counter is 0.28 GeV. The systemmeasures the time of flight for charged particles whose tracks have been reconstructed by the
CDC distinguishing between kaons and pions for tracks with momenta < 1.2 GeV.

The Aerogel Cherenkov Counter ACC
The detector is built from aerogel modules of several distinct types, varying in refractive index,
number and size of photomultipliers used to detect photons, according to their position in polar
angle. The modules are filled with a silica aerogel radiator and covered with a reflector of highreflectivity. The barrel part is divided into 60 identical sectors in the ϕ direction where there
are 16 modules arranged in each sector. The polar angle acceptance of the forward end cap
part is 13.6◦ < θ < 33.4◦ . This device consists of 12 identical sectors in the ϕ direction with
19 modules in each sector. The polar angle coverage in the barrel is 33.3◦ < θ < 127.9◦ . The
ACC also helps to achieve the necessary K − π separation for tracks with momenta > 1.5 GeV
which are not covered by the TOF.

The Electromagnetic Calorimeter ECL
The ECL is located inside the solenoid coil. It consists of a barrel section and two end caps
made of finely segmented arrays of 8736 CsI(Tl) crystals in total.

29

Each tower-like shaped crystal is 30 cm long with the scintillation light detected with twopieces of silicon photodiodes. The crystals point towards the interaction point but are tilted bya small angle in the θ and ϕ directions to prevent photons escaping through the gaps between
them. The barrel part is 3.0 m in length and has an inner radius of 1.25 m, while the end caps
are located at z = +2.0 m and z = −1.0 m. This scheme results in a total angular coverageof 12.4◦ < θ < 155.1◦ . Calorimeter calibration is performed using cosmic rays, Bhabha
scattering and e+ e− → γγ events. The ECL energy resolution varies from 4% for 100 MeV
particles to about 1.6% at 8 GeV. For low energies, the angular resolution is about 13 mradand for high energies 3 mrad. This system also provides the E/p ratio of shower energy to
track momentum which represents the main parameter for electron-hadron separation.
In addition, the ECL determines both online and offline luminosity by measuring the rate of
Bhabha events using geometrical coincidences of high energy deposits in the forward andbackward parts.
The KL0 and Muon Detector KLM
The iron flux-return yoke outside the solenoid magnet is instrumented with layers of active
detector material. There are 15 detector layers of double-gap glass-electrodes-based resistive
plate counters, and 14 layers of 4.7 cm thick iron plates alternating in the barrel region. Eachend cap is made of 14 detector layers and 14 iron layers. A traversing ionising particle results
in a local discharge in the 2 mm gas gap and initiates a signal in the electrodes. They identify
KL mesons and muons with momenta > 600 MeV with high efficiency. The polar angle
coverage is 45◦ < θ < 125◦ for the octagonal barrel part, while the forward and backward end
caps extend this range to 20◦ < θ < 155◦ . The direction of a KL can be determined from its
hadronic shower where we assume the kaon to originate from the e+ e− interaction point. A
discrimination between muons and hadrons, namely π ± or K ± , becomes possible through thediffering range and transverse deflection of non-showering charged particles. Hit position is
resolved to about 1.1 cm and results in an angular resolution of under 10 mrad.
Trigger and Data Acquisition
The trigger system consists of sub-triggers that are provided by CDC, ECL, TOF, and KLM
signals and the global decision logic (GDL) which makes a logical combination of summarised
information from the sub-triggers. The GDL triggers on hadronic, Bhabha and µ+ µ− pair
events, etc. These make up the so-called Level1 (L1) hardware trigger and are not processedby the Level3 (L3) software trigger run on the online computer farm. Here, good charged
particle tracks are selected to reduce the size of raw data that will be recorded. The typicaltrigger rate is about 300 Hz at beam currents of 200 mA for LER and 100 mA for HER.
Later, a coincidence with sub-triggers from TOF hits or ECL isolated clusters was required aswell to reduce such a high trigger rate while maintaining an appropriate efficiency for physics
events. In the final state, the timing of the Belle detector was dependent on the KEKB Radio
Frequency that controlled the bunch crossing rate of ∼ 509 MHz corresponding to ∼ 2 ns

30

intervals. With the TOF information available, it is capable of picking up the proper beamcrossing signal. By the ECL trigger alone, a timing resolution of 20 ns is achieved and it issufficient to perform an energy measurement for pure neutral events such as e+ e− → γγ.
Sub-detector

Detector typeθ coverage

SVD
CDC
TOF
ACC
ECL
KLM
L1
L3

3 − 4 double-sided Si detectors
Drift chamber
Scintillator tubes
Cherenkov counter with aerogel
CsI(Tl)Resistive plate counters
Hardware
Software[17◦ ; 150◦ ][17◦ ; 150◦ ][34◦ ; 130◦ ][13.6◦ ; 127.9◦ ][12.4◦ ; 155.1◦ ][20◦ ; 155◦ ]full Belle acceptancefull Belle acceptance

Table 4.2.: Main characteristics of the Belle detector [18].

The Belle Data Acquisition (DAQ) was originally required to read out the eight sub-detectors
at a design trigger rate of 500 Hz. It was built in order to record the data read by VME processors after a Belle specific event builder and the L3 data reduction by real time processing.
As an increasing trigger rate was necessary to keep up with the expected luminosity increase,
Belle DAQ systems were continuously upgraded.
Physics process
Υ(4S) → B B̄Hadron production
from continuumµ+ µ− + τ + τ −Bhabha (θ ≤ 17◦ , prescaled by 100γγ (θ‘17◦ , prescaled by 100
two-photon (θ‘17◦ , pT ≤ 0.1 GeV/c
Total

Cross section (nb)

Trigger rate (Hz)

1.2

12

2.8
1.6
44
2.4
∼ 15
∼ 67

28
16
4.4
0.24
∼ 35
∼ 96

Table 4.3.: Total cross-section and trigger rates with a luminosity L = 1034 cm−2 s−1 for some physics
processes at 10.58 GeV [35].

Improvement of the FASTBUS readout and upgrades of five detector sub-systems resulted ina reduction in dead time to less than 1%. The event builder and VME based online computer
farm performing the L3 trigger were replaced with the so-called EFARM and connected via
Fast Ethernet providing sufficient bandwidth. A large scale PC farm for real time full event
reconstruction directly fed by the event builder, the so-called RFARM, was introduced as wellusing parallel processing of events offering processing power for the back-end system to copewith the expected increase in luminosity.

31

The processing results such as the reconstructed IP position were also fed back to the accelerator control, which greatly contributed to realize proper accelerator parameters tuning.

4.3. Particle Identification
Charged hadron identification is accomplished by a so-called likelihood method. Here, theavailable information from detector response is composed from probability density functions(PDF) and is used to give a likelihood. The total likelihood for a given particle type is obtained by combining the sub-detectors’ likelihoods. Information is collected from the silica
aerogel Cerenkov counters ACC giving the number of Cherenkov photons, the time-of-flightmeasurements from the TOF counters and a specific ionization measurement dE/dx in the
CDC together with the measured momentum. The product of likelihoods for the particle beingof type i, Li (i = K, π), is then calculated as Li = LCDC × LTOF × LACC .
Charged pions and kaons are distinguished and selected by their ratio being defined as
 
L(K)
K
,
=
L
π
L(π) + L(K)for kaons and correspondingly L(π/K) for pions. The finally applied cut value is optimized
depending on the analysis.
For lepton identification, a combination of the following parameters is arranged into a likelihood: dE/dx in the CDC, E/p ratio (E being the energy deposited in the ECL and p the
momentum measured in the CDC and SVD), shower shape in the ECL and the number of
Cherenkov photons in the ACC. In case of electron identification, performance is slightly lower
at around θ ∼ 125◦ due to a small gap between the barrel and backward endcap ECL. Muons
are identified based on track penetration depth and hit scatter pattern in the KLM system. Their
reconstructed hits in the KLM are compared to the extrapolation of CDC tracks. The likelihood
ratio
Lµ
L=
Lµ + Lπ + LK
based on the muon, pion and kaon hypotheses is used as a discriminating variable.
As the photon is an electrically neutral particle, it can not be identified directly. For photon
selection, the electromagnetic interaction in the ECL is studied in form of the resultant shower
production. A 5 × 5 crystal matrix surrounding the seed crystal is defined, where the seedis determined as the one crystal with the highest energy deposit among neighbouring crystals
which has to be at least > 10 MeV. In order to retrieve the incident particle energy, thecrystal matrix’s contributions are summed up if they exceed a threshold of typically 0.5 MeV.
The position of such a particle is found through an energy-weightage of the energies of eachcontributing crystal. Still remaining biases for both reconstructed energy and incident position
are corrected by dedicated Monte Carlo studies due to electromagnetic shower behaviour beinga QED process and therefore predictable. Finally, showers without associated charged tracks

32

reconstructed as explained above and extrapolated to the ECL are sought to distinguish photon
from charged particle candidates.

4.4. Data Sample
The Belle data consists of a large number of events which not only come from e+ e− →Υ(4S) → B B̄ but also from several other processes. In fact, non-B B̄ events show a large
cross section in electron-positron colliders operating at the Υ(4S) resonance. The most common is e+ e− elastic scattering known as Bhabha scattering. But also radiative Bhabha reactions, continuum events from e+ e− → q q̄ (where q stands for u, d, s and c), two-photon
processes e+ e− → γγ, lepton pair production e+ e− → `+ `− (where ` stands for µ or τ ) andbeam-gas interactions contribute. The total number of B B̄ pairs in the used high statistics sample of e+ e− → Υ(4S) → B B̄, making up the foundation of this thesis, is (771.6 ± 10.6) × 106
with a total integrated luminosity of 703 fb−1 .
Hadronic events are selected (skimming) by requiring at least three reconstructed charged
tracks with a transverse momentum pT > 0.1 GeV originating in the vicinity of the interaction
point. The total visible energy from charged tracks and reconstructed photons√in the ECL in
the centre-of-mass system (cms) must exceed 20% of the total cms energy s. Hadronic
events furthermore have to show two or more large-angle clusters (−0.7 < cos θ < 0.9 inthe laboratory √frame), with an average cluster energy < 1 GeV and a total cluster energy
0.18 − 0.8 × s. This way τ -pair, beam gas and two-photon events as well as radiative
Bhabha and higher multiplicity QED processes√are suppressed. Also, an absolute value ofthe z component of the cms momentum < 50% s, and a reconstructed primary vertex that is
consistent with the known location of the interaction point have to be satisfied. By establishingthese criteria the selection of hadronic events is found to be 99% efficient for B meson signal
events determined by Monte Carlo simulation.
In this study, we are interested in events coming from those B B̄ events in which one of the
B mesons has J/ψ → `+ `− (` being e or µ) in the final state of its decay. As some ofthe hadronic event skim’s selection criteria are not favourable to inclusive ψ events, eventswith J/ψ and ψ(2S) candidates are explicitly added. One lepton track is required to have aprobability > 0.01 of being an electron or > 0.1 of being a muon and the di-lepton system has
to pass a tight mass cut within 2780 MeV ≤ M`` ≤ 3820 MeV. To suppress continuum twojet non-Υ(4S) background relative to B B̄ events, the ratio of second to zeroth Fox-Wolfram
moments, R2 = H2 /H0 , is required to be < 0.5 [37]. Badly measured charged particle tracks
or those that are not coming from the interaction point are removed by requiring a distance of
closest approach to the interaction point (impact parameters) along the beam direction z < 50
mm and r < 15 mm in the transverse xy-plane.

33

5. Event Selection
The B meson decay studied in this thesis, B ± → χc1 π + π − K ± , is reconstructed by combining
the final state particles. While the pions and the charged kaon are stable, the proper secondary
decays of χc1 and its daughters have to be considered. Monte Carlo simulations are used to
optimise the event selection and reconstruction process.

5.1. Monte Carlo Simulation Dataset
In order to build a reconstruction routine for B → χc1 π + π − K and to estimate the reconstruction efficiency for a decay signal and the possible background contamination, we need tounderstand acceptance and performance of the detector. For this purpose an adequate Monte
Carlo simulation (MC) is performed with an event generator designed for the simulation ofphysics of B decays. The EvtGen program [51] is used to generate the B → χc1 π + π − K
decay signal events as well as possible background events. This program is capable to handle
the key kinematics and angular distributions of the resultant decay products. It uses a genericand an analysis specific decay table as its main input. This way specifically defined B meson
decays can be generated, not only according to the available phase space but also as decay
processes through particular intermediate states. For the description and modeling of the detector material and interactions of particles during their passage through the detector, a Belle
specific GEANT3-based [8] software is used. This simulation contains the particles’ energy
loss in active material, detector response as well as readout electronics and digitisation effects.
Signal MC samples of 0.5 million events have been generated for each of the following categories:• B ± → χc1 π + π − K ± obeying the available phase space for a four-body decay,• B ± → X(3872)K ± followed by the hypothetical decay of X(3872) → χc1 π + π − ,• B ± → χc1 (2P )K ± followed by the hypothetical decay of χc1 (2P ) → χc1 (1P )π + π − .
In the two latter cases, the first two-body B decay and the X(3872) and χc1 (2P ) decays,
respectively, are treated by phase space decay (PHSP) while taking final state radiation into
account. Also, in the quasi two-body B decay case, two more signal MC samples have been
prepared setting the intermediate state mass to 3900 MeV and 3950 MeV to further understandthe reconstruction efficiency as a function of the χc1 π + π − invariant mass as discussed in detaillater in Sec(6.2).

35

5.2. Reconstruction of χc1 Candidates
χc1 meson candidates are reconstructed by combining a J/ψ candidate with a photon.
The J/ψ meson is reconstructed from oppositely charged lepton pairs via its decay mode
J/ψ → `+ `− , where ` stands for e± or µ± . Charged tracks are identified as muons if theysatisfy a muon likelihood > 0.1 or as electrons if the electron likelihood is > 0.01.
A correction for final state radiation or bremsstrahlung in the inner parts of the detector isapplied by including the four-momentum of every photon detected within 0.05 radians of theoriginal electron or positron direction in the e+ e− invariant mass calculation.
Taking into account the still existing small radiative tails, an asymmetric invariant mass requirement of 2950(3030) MeV ≤ Mee(µµ) ≤ 3130 MeV is chosen to define a J/ψ candidatein the electron (muon) channel.
A vertex and a mass-constrained fit are applied to the selected J/ψ candidates. The applicationof kinematic fits reduces the effects of limited detector resolution and improves the mass or
momentum resolutions to determine the decay vertex of a particle candidate. There are two
types of kinematic fitting used in this analysis:• The invariant mass of the candidate is fixed to the known, nominal mass and the momenta of its daughters are re-calculated in the mass-constrained fit. This approach canbe applied when the reconstructed state has a narrow width and the observed invariant
mass distribution is dominated by detector resolution.
• In case of the vertex-constrained fit tracks used to reconstruct a candidate may not converge in the same point. By tuning the momentum and position of each daughter particle
according to its measurement errors the candidate particle’s decay vertex is obtained sothat all tracks pass through a single point. This tuning of momentum or position is done
by a minimisation of the χ2 , using a Lagrange multiplier approach.
A cms momentum cut of p∗`` < 2.0 GeV rejects J/ψ mesons not coming from B decays.
Fig(5.1) shows the di-lepton invariant mass distribution in phase space signal MC, cut requirements are indicated by vertical red lines.
To identify χc1 the invariant J/ψγ mass is used in a mass window of 3467 MeV ≤ MJ/ψγ ≤3535 MeV, see Fig(5.2). Photon selection is requiring Eγ > 100 MeV to reduce combinatorial background. Also, a veto on photons coming from π 0 is applied by rejecting photons
if their combined invariant mass with another randomly picked photon in the same event iswithin 117 MeV ≤ Mγγ ≤ 153 MeV [49]. From the invariant mass of the combined systemwith another photon and the energy and polar angle of the photon of interest, a probability is
calculated for the latter photon to come from a π 0 and we require this parameter to be < 80%
to still accept the photon for χc1 reconstruction. A figure of merit study was performed tofind the best cut and can be found in App(A). A mass-constrained fit is performed on the χc1candidates in order to improve momentum resolution.

36

Events

4000

2000

0

2.95

3

3.05
Me+e- (GeV)

3.1

3.15

(a) Di-electron mass with selection windows indicated by red lines.

Events

6000

4000

2000

0

3.05

3.1

3.15

Mµ+µ- (GeV)(b) Di-muon mass with selection windows indicated by red lines.
Figure 5.1.: Invariant di-lepton mass spectrum to select a J/ψ candidate in phase space MC.

37

4000

Events

3000
2000
1000
0
3.46

3.48

3.5
MJ/ ψγ (GeV)

3.52

3.54

Figure 5.2.: Invariant J/ψγ mass spectrum to select a χc1 candidate in phase space MC with selection
window indicated by red lines.

In the next step, reconstructing the final B candidate requires the selection of an oppositely
charged pion pair. Then, a charged kaon will be added. In the event selection itself no explicit
requirements are applied for a possible intermediate state decaying to χc1 π + π − ; the possibilityof a resonance in the χc1 π + π − spectrum is only implemented in the respective quasi two-bodydecay MC datasets.

5.3. Selection of Light Meson Candidates
For charged pions the likelihood ratio is required to be L(π/K) > 0.6 as well as for charged
kaons L(K/π) > 0.6. Looking at the invariant di-pion mass, there is no obvious backgroundcoming from γ → e+ e− where e± are misidentified as pions, and therefore no need to apply
a mass cut, see Fig(5.3). Still, a vertex-constrained fit is applied to the pion pair and its
convergence is required.

5.3.1. Low Transverse Momentum Pion Selection
In case of the X(3872) → χc1 π + π − mode, the pions’ transverse momenta with respect to thebeam axis, pT , are low. This is because only a small Q-value is available, mX(3872) − mχ −2 × mπ ≈ 80 MeV, and reconstruction efficiency is limited by the Belle detector’s acceptancefor low pT tracks [82]. Such particles curl up in the Central Drift Chamber CDC and caneventually be reconstructed multiple times, ending up forming excess B candidates. Thiseffect results in pions that appear to have same charge with nearly identical three-momenta,
or oppositely-charged pions with nearly opposite three-momenta, as illustrated in Fig(5.4).
Similar considerations can be made for the χc1 (2P ) case.

38

Events

3000

2000

1000

0

0.5

1
Mπ+π- (GeV)

Figure 5.3.: Invariant di-pion mass spectrum in B candidate reconstruction, the red dashed line hints
the possible cut.

Figure 5.4.: Curl up of low pT pions.

For pions with pT < 0.25 GeV, tracks are paired to check if the following criteria is satisfied:
if |pπ1 − pπ2 | < 0.1 GeV the opening angle between those tracks is checked. Duplicatedreconstruction should appear for angles ∼ 0◦ and ∼ 180◦ . MC shows that these tracks are infact concentrated around | cos(θopen )| ≈ 1, see Fig(5.5). Focusing on cos(θopen ) > 0.95 forsame-charged tracks and cos(θopen ) < −0.95 for oppositely-charged tracks, only one of themis regarded as the real and correctly reconstructed track.
Comparing these pion tracks, the one closest to the interaction point is chosen: The particle
with the smaller value of
2
2
dz
dr
+
15mm
50mm
where dr and dz are the impact parameters in mm, is selected based on the knowledge established by previous Belle studies [38, 47].

39

Events

10000

5000

0
-1

-0.5

0
cos(θopen)

0.5

1

Figure 5.5.: cos(θopen ) for multiply reconstructed pions in | cos(θopen )| ≈ 1.

Applying this method, misidentification of a pion as a kaon can be reduced to 5% and multiplicity of B candidates is improved. The pion and kaon selection efficiency is 94% each.

5.4. Reconstruction of B Candidates
The reconstruction of B meson candidates is completed by combining the χc1 π + π − systemwith a charged kaon; the inclusion of the charge conjugate state is from here-on implied.
We make full use of the two-body kinematics of the Υ(4S) → B B̄ decay in the cms byconsidering that the reconstructed B’s energy must consequently be half of the total energy ofthe e+ e− system. The kinematic variables of interest to identify a B meson candidate are thebeam-constrained mass Mbc and the difference-to-beam energy ∆E:
q
2
Mbc =
Ebeam
− (pχc1 ππ + pK )2∆E =(Eχc1 ππ + EK ) − Ebeam ,where Ebeam is half the e+ e− system (beam) energy, p the momenta and E the energies of theparticle candidates in the cms of Υ(4S).

5.5. Properties of the reconstructed B Candidate
Mbc peaks at the nominal B meson mass for correctly reconstructed B candidates and has amuch better resolution than the bare reconstructed mass, hence providing a better separationof signal and background. In an ideal case, ∆E is supposed to peak at zero for properly
reconstructed B mesons.

40The multiplicity is defined as the fraction of the events with more than one B candidate out of
the total number of events having B candidates. So far, multiplicity is at 39.7% for multiple
reconstructed candidates per phase space event found in the signal region. This seems relatively high but is expected for a four-body final state. Multiplicity is increasing in case of thequasi two-body decays and also with increasing mass.
Decay mode
B ± → χc1 π + π − K ± (PHSP)
B ± → X(3872)K ±B ± → χc1 (2P )K ±Multiplicity
39.7%
45.6%
48.2%

Table 5.1.: Multiplicity in |∆E| < 0.12 GeV for Monte Carlo datasets.

This can be understood when looking at the very limited phase space left for pions in the
decay B → X(3872)(→ χc1 π + π − )K or B → χc1 (2P )(→ χc1 π + π − )K: The kaons havehigh momentum but the pions are rather slow, in contrary to the B → χc1 π + π − K phase spacecase where a more kinematically symmetric event topology of kaons and pions can be allowed.
As the detection efficiency for charged particles is a function of the particle’s momentum pT , it
becomes more difficult to only select the proper low pT pions for an event, leading to a higher
probability of wrong combination and consequently higher multiplicity.
Looking at the ∆E distributions, the signal peaks at ∆E = 0 are exhibiting similar resolutions
for all three cases. This is an effect of employing a mass-constrained fit to the χc1 candidate
which includes a photon. A tail appears in the positive ∆E region in the quasi two-bodydecays which is not visible in the phase space case. It is due to a difference in reconstruction
efficiency as explained above. Since both, X(3872) and χc1 (2P ) at 3920 MeV are still close
to the kinematic threshold. Wrongly picked up pions are mainly having higher momenta thantrue ones. On the other hand, all the available kinematical region is averaged in the phasespace case thus no such tendency shows in the ∆E projection.
If more than one B meson candidate is found in an event, the best candidate is selected as theone having the closest beam-constrained mass Mbc compared to the nominal B mass. With
this best candidate selection, BCS, 87.4% of the correctly combined B meson candidates are
selected in the phase space signal MC case.
The ∆E and Mbc distributions for all three MC datasets are shown in Fig(5.6) and Fig(5.7)
for comparison. As the ∆E distributions are showing the desired behaviour of peaking atzero, the invariant mass spectra of χc1 π + π − are presented in the B candidate signal enhanced
region1 . They can be seen in Fig(5.8). The manageable amount of events in the M (χc1 π + π − )tail in signal MC is due to wrong combination. A clear peak around 3.87 GeV indicates thereconstructed X(3872) and the χc1 (2P ) is visible at its predicted mass at 3.92 GeV. In phasespace MC the spectrum is distributed over the entire possible range, as expected.
These plotted distributions verify the proper functionality of the reconstruction routine.
1

see next Chapter for Figure of Merit study and deduction of the signal window and signal enhanced region.

41

B ± → χc1 π + π − K ±Events

10000

5000

0

-0.1

-0.05

0
∆ E (GeV)

0.05

0.1

0
∆ E (GeV)

0.05

0.1

0
∆ E (GeV)

0.05

0.1

6000

B ± → X(3872)K ±Events

4000

2000

0

-0.1

-0.05

8000

B ± → χc1 (2P )K ±Events

6000

4000

2000

0

-0.1

-0.05

Figure 5.6.: ∆E distribution for all three MC datasets; Mbc > 5.27 GeV, BCS applied.

42

3000

B ± → χc1 π + π − K ±Events

2000

1000

0

5.27

5.275
5.28
Mbc (GeV/c2)

5.285

5.29

5.285

5.29

5.285

5.29

B ± → X(3872)K ±Events

1500

1000

500

0

5.27

5.275
5.28
Mbc (GeV/c2)

B ± → χc1 (2P )K ±Events

2000

1000

0

5.27

5.275
5.28
Mbc (GeV/c2)

Figure 5.7.: Mbc distribution for all three MC datasets; |∆E| < 0.12 GeV, BCS applied.

43

Events

1500

B ± → χc1 π + π − K ±

1000

500

0
3.8

4

4.2
4.4
Mχ π+π- (GeV/c2)

4.6

4.8

c1

B ± → X(3872)K ±Events

2000
1500
1000
500
0
3.85

3.86

3.87
Mχ π+π- (GeV/c2)

3.88

3.89

c1

B ± → χc1 (2P )K ±Events

3000

2000

1000

0

3.9

3.91

3.92
3.93
Mχ π+π- (GeV/c2)

3.94

c1

Figure 5.8.: Mχc1 ππ invariant mass distribution for all three MC datasets; |∆E| < 0.12 GeV, Mbc >5.27 GeV, BCS applied.

44

6. Monte Carlo Studies
For the reconstruction procedure to get B ± meson candidates from χc1 π + π − K ± , the background contamination modes have to be identified and their contributions to be estimated.
The ∆E signal is then fitted to study the line shape in order to extract a probability density
function, and ultimately calculate a branching fraction. For this, the reconstruction efficiency
is needed and estimated as a function of the invariant mass of the χc1 π + π − intermediate state(Mχc1 ππ ). Finally, the Mχc1 ππ spectrum is studied and a fitting probability density function is
obtained to hunt for a resonance.

6.1. Background Estimation
We expect the dominant background for the B ± → χc1 (→ J/ψγ)π + π − K ± final state tocome from those B decays that have a real J/ψ → e+ e− or J/ψ → µ+ µ− in their final state.
Therefore, a large B → J/ψX MC sample has been prepared to estimate the major background modes, where X denotes one or more decay products. Decays of higher charmonium
states such as ψ(2S), χc0 , χc1 , χc2 with a J/ψ in the final state are also taken into account. It
provides a B → J/ψX dataset that is 100× larger than the later used Belle data sample andincludes all known and kinematically allowed decay modes having J/ψ in the final state.

6.1.1. Expected Background in ∆E
Fig(6.1) shows the possible contributions to background estimated from the B → J/ψX MC
sample. There is only one mode to point out, all other backgrounds have shown to be smoothin their ∆E distributions in the |∆E| < 0.12 GeV window.
The structures in this distribution can be divided in the signal part, B → χc1 Xsu , whereπ + π − K can be formed from Xs̄u which is the s̄ and u quark system to be hadronised according
to the PYTHIA/JETSET string fragmentation model. It results in the desired signal final state
indicated by ∆E peaking at zero. The other contribution is coming from B ± → χc2 π + π − K ±and represents the background. A small peak can be seen, shifted to the negative ∆E region
because of the χc2 ’s slightly higher mass at 3556 MeV which is being mass-constrained to theχc1 mass of 3510 MeV.

45

Figure 6.1.: ∆E distribution on B → J/ψX MC sample; Mbc > 5.27 GeV and BCS applied.

6.1.2. Figure of Merit
Candidates satisfying Mbc > 5.27 GeV and |∆E| < 0.12 GeV are considered for extractinga B decay signal as shown in Fig(6.1). Due to this ∆E window choice, the obtained ∆E
distribution is adjusted to not include the intricately shaped background coming from B decays
with five-body final states. These backgrounds tend to cluster at −0.14 GeV or +0.14 GeV
for five- or three-body modes, respectively, as a result of an ignored or additional pion.

√
S/ S + B√
S/ S + B

In order to study the χc1 π + π − mass spectrum, the B decay candidates are taken from the tightsignal enhanced window. To validate this requirement, a Figure of Merit study was performed.
Using phase space signal MC and the B → J/ψX MC sample for background we assume thatthe branching of B ± → χc1 π + π − K ± is comparable to B ± → J/ψπ + π − K ± with 4 × 10−4(see below), and estimate the signal yield.

Lower ∆E Limit (GeV)(a) Fixing the upper ∆E limit.

Upper ∆E Limit (GeV)(b) Fixing the optimized lower ∆E limit.

Figure 6.2.: Figure of merit study to optimise the ∆E window for the signal enhanced region.

46

The upper ∆E √limit is fixed while ramping the lower limit. This way the ratio of signal andbackground S/ S + B, where S is the expected signal yield and B the expected background,is checked. After finding the lower limit for ∆E to be −0.02 GeV considering the error,
to give the best result, the procedure is repeated for the upper limit. Looking for the signal
enhanced region, the decision falls on a symmetric ∆E window, fixing the upper and lower
limit to ±0.02 GeV to cope with a potential ∆E resolution difference between data and MC.
A two-dimensional illustration of the signal enhanced region is shown in Fig(6.3).
120

0.02

100

∆E (GeV)

0.01
80

0

60

-0.01

40

20

-0.02
5.27

5.275

5.28
Mbc (GeV/c2)

5.285

5.29

0

Figure 6.3.: Two-dimensional illustration of the signal enhanced region in ∆E and Mbc with cuts
indicated by red lines.

The ∆E distribution in the formerly chosen larger window can be seen in App(B).

6.1.3. Expected Background in Mχc1 ππLooking at the invariant mass spectrum of χc1 π + π − using the B → J/ψX MC sample, theonly peaking component is a reflection from ψ 0 that appears at 4.1 GeV as shown in Fig(6.4a).
If, in the event, a ψ 0 decays to J/ψπ + π − and an additional, unrelated photon is picked up,
a fake χc1 candidate can be formed. Because of the mass-constrained fit applied to the χc1candidates, combining this fake χc1 with the pion pair coming from ψ 0 results in a peak at
4.1 GeV. To have a closer look at the bump in the expected invariant mass spectrum ofχc1 π + π − , it was also plotted on a logarithmic scale as shown in Fig(6.4b). A ψ 0 -veto wasdeveloped in order to suppress this peaking background but found to create an artificial dip andis therefore being ineffective. As this peak is not interfering with the mass region of interest,
which is the χc1 (2P ) mass prediction or the X(3872) mass (between 3.89 − 3.95 GeV), we
leave this reflection untreated.

47

(a) Mχc1 ππ distribution on B → J/ψX MC.

(b) Mχc1 ππ distribution on logarithmic scale.
Figure 6.4.: Expected Mχc1 ππ distribution on the B → J/ψX MC sample; Mbc > 5.27 GeV, |∆E| <0.02 GeV and BCS applied.

In connection with the χc1 π + π − invariant mass, it was further verified that, concerning thebackground, a cut on the di-pion mass is not necessary. There is close to no contamination
below 0.3 GeV. The invariant mass of π + π − was plotted with the B → J/ψX MC sample
and can be seen in Fig(6.5). As the B → J/ψX MC sample consists of all known modes witha J/ψ in the final state, including the signal decay B ± → χc1 π + π − K ± , π + π − might originate
from π − K ∗ , K ∗ , KS0 or ρK ± . These modes are handled as fragmentation processes.

48

1500

Events

1000

500

0

0.5

1
Mπ+π- (GeV)

Figure 6.5.: Mπ+ π− on the B → J/ψX MC sample scaled to data.

6.2. Fitting Strategy and Observable Extraction
For both, the ∆E and Mχc1 ππ spectra, the line shapes are studied. A fitting routine is compiled leading to Probability Density Functions (PDF) which will be used to extract the B →χc1 π + π − K signal yield from the Belle data (see next chapter). In order to make assumptions
about the expected signal yield, the reconstruction efficiency is estimated as a function of themass of an intermediate state for different mass hypotheses. Finally, toy studies verify the
choice of PDFs.

6.2.1. ∆E Fit and Efficiency Estimation∆E Signal PDF
For the extraction of the signal yield, an unbinned maximum likelihood (UML) fit is used on
a 0.5 million events sample for B ± → χc1 π + π − K ± (PHSP). The probability density function(PDF) to describe this B decay signal is composed by summing the following two components:• the peak is parametrised using a sum of two Gaussians Ppeak (∆Ei ) and• the outlier (OL) is described by a second order Chebyshev polynomial POL (∆Ei ).
Thus the likelihood function to be maximised isN

e−N Y[nOL · POL (∆Ei ) + npeak · Ppeak (∆Ei )].
L=
N ! i=1

(6.1)

49

Here
2

2

(∆Ei −µ1 )(∆Ei −µ2 )
−
−
1
1
2
2
2σ1
2σ2
Ppeak (∆Ei ) = (1 − ftail ) √ e+ ftail √ e
σ1 2π
σ2 2π

(6.2)

where N is the total number of candidate events, npeak and nOL denote the number of peak andoutlier events, respectively. ftail is the fraction of the second Gaussian, µ1 , µ2 and σ1 , σ2 are
the mean values and standard deviations for the corresponding Gaussians. In all fits, µ1 = µ2
is assumed throughout this analysis.
Fig(6.6) shows the fitted distribution. The Chebyshev polynomial describing the outlier is
not counted as part of the signal as it is thought to be absorbed as a part of combinatorial
background.

EventsB ± → χc1 π + π − K ±∆E (GeV)Figure 6.6.: Fitted ∆E distribution for B ± → χc1 π + π − K ± in signal phase space MC (blue), signal
part (red) and combinatorial background (green); Mbc > 5.27 GeV and BCS applied.

The ∆E distribution peaks at zero showing a standard deviation of 4.6 MeV, the outlier isdue to wrong combination coming from misreconstruction. A peak yield of 65255 ± 339 is
extract out of 0.5 million generated events.
In all three signal MC cases the same fitting function Eqn(6.1) is used. As expected for signal
MC, also the distributions for X(3872) and χc1 (2P ) as intermediate states of the main B
meson decay peak at zero as shown in Fig(6.7).
A signal yield of 32222 ± 259 is extract with a standard deviation of 5.3 MeV for the X(3872)
case. In the χc1 (2P ) signal MC, the peak yield is found to be 52258 ± 337 with a standard
deviation of 5.6 MeV. Chebyshev polynomials were used to describe the combinatorial background in both cases.
These peak yields will be of interest when calculating the reconstruction efficiency for thedifferent intermediate states.

50

Events

B ± → X(3872)K ±∆E (GeV)

EventsB ± → χc1 (2P )K ±∆E (GeV)Figure 6.7.: Fitted ∆E distribution for B ± → X(3872)K ± and B ± → χc1 (2P )K ± in theirrespective signal MC (blue), signal part (red) and combinatorial background (green);
Mbc > 5.27 GeV and BCS applied.

Efficiency Estimation for different Masses of an Intermediate
State
Here, efficiency is defined as the peak yield extracted from the signal MC fit divided by the
number of generated events. Hence, the reconstruction efficiency estimated for signal extraction in the phase space is ε = 13.05 %. However, the reconstruction efficiency would dependon the mass of an intermediate state.

51

And as this study’s goal is to search for such a resonance we are especially looking at X(3872)
and χc1 (2P ) in their decays to χc1 π + π − . Respective signal MC samples have been prepared
for the quasi two-body decay processes and in order for a general search for resonances, 3900,
3920 and 3950 MeV have been generated in addition to X(3872). The fits to the ∆E distributions for the two other mass hypotheses, namely 3900 and 3950 MeV, can be seen inFig(6.8).

EventsB ± → X(3900)K ±∆E (GeV)(a) Fit of the ∆E distribution for mX = 3900 MeV yields 41361 ± 271
peak events.

EventsB ± → X(3950)K ±∆E (GeV)(b) Fit of the ∆E distribution for mX = 3950 MeV yields 51841 ± 343
peak events.
Figure 6.8.: Individual fits for ∆E for masses 3900 MeV and 3950 MeV of the χc1 π + π − system in
the respective signal MC (blue), signal part (red) and combinatorial background (green);
Mbc > 5.27 GeV, |∆E| < 0.02 GeV and BCS applied .

52

The reconstruction efficiencies are summarised in Tab(6.1). As expected, efficiency increases
with the mass of the intermediate state: as the X(3872) is very close to the kinematical threshold in B ± → χc1 π + π − K ± the Q-value is limited and therefore the transverse momentum of
the pions is so small that a sizable amount of the pions are missed to be detected. As themass of the χc1 π + π − system increases, the pions’ pT becomes larger and thus easier to bereconstructed.
Assumed mass for X (MeV)

Efficiency (%)

3872
3900
3920
3950
PHSP

6.44
8.27
10.45
10.37
13.05

Table 6.1.: Efficiencies for phase space decay and different mass assumptions of the χc1 π + π − system
in B → X(→ χc1 π + π − )K extracted from their respective ∆E distributions.

This feature is taken into account to obtain the B decay branching fraction from the data as
described later in Sec(7.2).

Background ∆E PDF
As seen in the previous section, only B ± → χc2 π + π − K ± decays form a peak in the ∆E background distribution. For this contribution, an individual fit is performed to obtain a probability
density function to describe its line shape.

EventsB ± → χc2 π + π − K ±∆E (GeV)Figure 6.9.: Fit of B → χc2 π + π − K in the B → J/ψX MC sample (blue), signal part (red) and combinatorial background (green); Mbc > 5.27 GeV, |∆E| < 0.02 GeV and BCS applied.

53

Thus the total background ∆E PDF is given by summing the following two components:• B → χc2 π + π − K is parameterized by a sum of two Gaussians, see Fig(6.9),• all other background modes have shown to be smoothly distributed in the ∆E spectrum
and will be described by a first order polynomial.
The same fitting method is applied as described for the signal part of the ∆E spectrum above,
compare Eqn(6.1). The optimum line shape parameters, namely the ratio of the areas of thetwo Gaussians as well as the ratio of their sigmas and mean and sigma, are selected.

Total PDF for the ∆E Distribution
Combining the signal and the background PDFs, the likelihood function L for the fit to the∆E distribution is expressed as:Ne−(npeak1 +npeak2 +nOL ) Y[nOL · POL (∆Ei )npeak1 · Ppeak1 (∆Ei ) + npeak2 · Ppeak2 (∆Ei )].
L=
N!
i=1
(6.3)
where nOL and POL (∆Ei ) are the normalisation and PDF for the outlier component to express
combinatorial background and misreconstructed signal due to picking up a wrong final stateparticle combination, npeak1 and Ppeak1 (∆Ei ) are signal yield and PDF of the peak component for the B ± → χc1 π + π − K ± signal, and npeak2 and Ppeak2 (∆Ei ) are the normalisation
and PDF of the peaking component for the B ± → χc2 π + π − K ± background. As already described, POL (∆Ei ) is expressed as a first order Chebyshev polynomial and both Ppeak1 (∆Ei )and Ppeak2 (∆Ei ) are formulated by a sum of two Gaussians. During the fit with this PDF to
describe the overall distribution, the mean and sigma as well as signal yield of the signal peakcomponent are floated. At the same time, all parameters are fixed for the peaking backgroundof B ± → χc2 π + π − K ± except for its normalisation. The result of the final fit by MC can beseen in Fig(6.10) as a typical pseudo-experiment.

6.2.2. Fitting of the Invariant Mass Spectrum Mχc1 ππ
A fitting routine for the Mχc1 ππ spectrum is prepared to search for any narrow resonance inthe χc1 π + π − final state.

Mχc1 ππ Signal Fit
The invariant mass spectrum in the B ± → X(3872)K ± signal MC is fitted with a PDF composed of• a sum of two Gaussians for the peaking component and• a second order Chebyshev polynomial to describe the outlier component.

54Events∆E (GeV)Figure 6.10.: One typical pseudo-experiment to test the complete PDF for the phase space case (blue)of ∆E distribution with mean and sigma of the χc1 π + π − signal as well as fractions of
the backgrounds floated. Each component is shown as: B → χc2 π + π − K (cyan), signal(red), flat background (lightgreen).

Looking at the signal enhanced region in Mbc > 5.27 GeV and |∆E| < 0.02 GeV a signal
yield of 29062 ± 215 events can be extracted and translated to a reconstruction efficiency
ε = 5.81% for X(3872) as the intermediate state. Its signal is found at mean = 3.871 GeV
with a standard deviation of 1.8MeV. A second order Chebyshev polynomial was used todescribe the combinatorial background.

EventsB ± → X(3872)K ±Mχc1 ππ (GeV/c2 )Figure 6.11.: Fitted Mχc1 ππ distribution in the X(3872) signal MC (blue), signal part (red) and combinatorial background (green); Mbc > 5.27 GeV, |∆E| < 0.02 GeV and BCS applied.

55

Fitting the invariant mass spectrum with the χc1 (2P ) hypothesis, the peak component of thesignal is found to be 9.31% with 46527 ± 226 signal events based on the 0.5 million generated
events. The outlier component is thought to be absorbed into combinatorial background anddescribed with a first order Chebyshev polynomial.

EventsB ± → χc1 (2P )K ±Mχc1 ππ (GeV/c2 )Figure 6.12.: Fitted Mχc1 ππ distribution in the χc1 (2P ) signal MC (blue), signal part (red) and combinatorial background (green); Mbc > 5.27 GeV, |∆E| < 0.02 GeV and BCS applied.

The reconstruction efficiencies in the Mχc1 ππ spectrum as a function of the intermediate state’s
mass are summarised in Tab(6.2).
Assumed Mχc1 ππ (MeV)

Efficiency (%)

3872
3920

5.81
9.31

Table 6.2.: Efficiencies for phase space decay and different mass assumptions of the χc1 π + π − system
in B → X(→ χc1 π + π − )K extracted from their respective Mχc1 π+ π− distributions.

Expected Signal Yield in the Mχc1 ππ Distribution
in the X(3872) Case
To make an estimation for how many events can be expected under the X(3872) hypothesis, we assume the branching ratio B(X(3872) → χc1 π + π − ) to be similar to the branching
B(X(3872) → J/ψπ + π − ), then roughly 15 events can be expected in data.

56

NB B̄ × B(B → X(3872)(→ χc1 π + π − )K) × B(χc1 → J/ψγ) × B(J/ψ → l+ l− ) × ε= 771.581 · 106 × 8 · 10−6 × 0.344 × (0.0593 + 0.0594) × 0.0581≈ 15 events

Background Fit for Mχc1 ππ
The background distribution for the invariant mass spectrum of χc1 π + π − is obtained from the
B → J/ψX MC sample. It is described best by a third order threshold function, where mT h
is the threshold and mx the value of Mχc1 ππ ,135a · (mx − mT h ) 2 + b · (mx − mT h ) 2 + c · (mx − mT h ) 2 .

EventsAs shown in Fig(6.13), this parametrisation is found to be capable to describe the Mχc1 ππdistribution in a non-resonant case.

Mχc1 ππ (GeV/c2 )Figure 6.13.: Fit of Mχc1 ππ in the B → J/ψX MC sample (blue) and signal part (red); Mbc >5.27 GeV, |∆E| < 0.02 GeV and BCS applied.

Total Probability Density Function for the Mχc1 ππ Distribution
in the X(3872) CaseTo describe the overall distribution we use a PDF where the signal part is modelled by the sumof two Gaussians with the mean and standard deviation fixed due to the small expected signal
yield. The signal yield itself is floated though. All parameters for the background which is
described by a third order threshold function are floated as well.

57

Events

The number of background events from the B → J/ψX MC sample has been scaled down tomatch data - the original B → J/ψX MC sample contains 100× the amount of data events.
The result of the final fit can be seen in Fig(6.14).

Mχc1 ππ (GeV/c2 )Figure 6.14.: One typical pseudo-experiment assuming that B(X(3872) → J/ψπ + π − ) =B(X(3872) → χc1 π + π − ) to test the complete PDF for the X(3872) case (blue) ofMχc1 ππ distribution with fractions of the signal and all background parameters floated.
The components are shown as signal (red) and flat background (lightgreen).

The reliability of the fitting procedure for both, the ∆E and Mχc1 ππ distributions has been
confirmed in a linearity test using a Toy MC sample. For ∆E, 2000 pseudo-experiments weregenerated for each dataset at seven different input signal yields between 800 and 2100. For
Mχc1 ππ , six datasets of 2000 samples each were generated with different signal yields ranging
from 0 to 25. In both cases, good linearity and stability of the fitting routine was observed.

58

7. Results and Discussion
The fit procedure to extract a signal yield from the ∆E distribution for B ± → χc1 π + π − K ±has been composed and tested in Monte Carlo datasets. Applying the fit to the data sample
allows to look for the existence of this decay and if found, to obtain the branching fraction.
The invariant mass spectrum is studied in the data sample and by applying the derived fitting
procedures, the decays X(3872) → χc1 π + π − and χc1 (2P ) → χc1 (1P )π + π − are looked for,with the goal to clarify the existence of the decays of such resonances.

7.1. ∆E Distribution

Events/0.0024 MeV

The Belle data is analysed for a sample as described in Section(4.4) and a fit to extract a B
decay signal in the ∆E spectrum is performed. A sum of two Gaussians is used for the signal
part. Its tail part Gaussian’s shape and fraction are fixed according to the MC study PDF but thecore part Gaussian’s mean and sigma of the signal as well as the signal yield are floated. The
background is described by a first order Chebyshev polynomial, its normalisation is floated.
The possible B ± → χc2 π + π − K ± contamination which clusters at ∆E = −0.05 GeV is also
considered; the shape of this component is fixed by the MC and normalisation is floated.

∆E (GeV)Figure 7.1.: Fit of ∆E distribution on data (blue), signal part (red) and χc2 π + π − peaking background(cyan), combinatorial background (green); Mbc > 5.27 GeV, |∆E| < 0.12 GeV andBCS applied.

59

The B ± → χc1 π + π − K ± decay signal yield is found to be 1597 ± 76 events. The statisticalsignificance is 29.6 σ. It was obtained in a likelihood ratio test that is used to compare the fitof two models: the null model is a special case and assumes no signal, while the alternativemodel is represented by the fit above. The fit’s quality is represented by the likelihood ratio
which expresses how much more likely the data is for one model than for the other. Thereby
this result is consistent with the assumption about the expected signal yield made in App(C).
Comparing ∆E resolutions in MC and data, they are found to be consistent within 10%:σdata /σM C = 1.18 ± 0.068. The χc2 π + π − K background contributes with 196 ± 80 events.

7.2. Weighted Detection Efficiency Estimation
Now, that it is obvious that the reconstruction efficiency is a function of the invariant massof χc1 π + π − , using a less straightforward approach to tackle the phase space decay of B ± →χc1 π + π − K ± becomes mandatory. The net ∆E signal yield that will be extracted from a fit to
the Belle data Nobs is a product of the actual signal yield Nphys and the detection efficiency ofthe detector and particle identification ε:
Nobs = Nphys · ε.
In order to give a proper branching fraction in the long term, the number of physical events
Nphys is desired. With the reconstruction efficiency being a function of the χc1 π + π − invariant
mass, we will weight the efficiency per mass bin ε(i) with the ∆E signal yield in data permass bin Nobs (i) in the following way:P10ε(i) · Nobs (i)(7.1)
ε = i=1P10
i=1 Nobs (i)First, the signal yield in ten 100 MeV mass bins in the Mχc1 ππ spectrum in phase space signal
MC is calculated. From that, the respective efficiencies ε(i) are determined. Finally, a total
reconstruction efficiency can be obtained by weighting with the signal yield Nobs in data inthe same mass bins. The result for the efficiencies per mass bin can be seen in Fig(7.2a), the∆E signal yields used for weighting in Fig(7.2b).
Through this method a resultant reconstruction efficiency of ε = 13.41% is obtained for thephase space case, compared to the formerly gained 13.05% without weighting.

60

Efficiency ε (%)

15

10

5

0

4

Mχ ππ (GeV/c2)

4.5

c1

(a) Reconstruction efficiencies from phase space signal MC.

∆ E Signal Yield

400

300

200

100

0

4

Mχ ππ (GeV/c2)

4.5

c1

(b) ∆E signal yield in data.
Figure 7.2.: Weighted detection efficiency for ten bins of Mχc1 ππ .

61

7.3. Efficiency Correction and Systematic Uncertainty
Systematic uncertainty needs to be considered when one does not exactly know the dimensions of the errors that come along with the use of Monte Carlo datasets, imperfect particle
identification on other typical measurements. The following methods describe the estimationof the correction to the reconstruction efficiency caused by this kind of possible differences,for example between data and MC.
Number of B Mesons
The official number of B mesons coming from the decay of Υ(4S) recorded by Belleis given with (771.581 ± 10.566) × 106 . It is determined by counting the recorded
hadronic events and the event shape variables’ distributions to separate continuum andB B̄ events. Due to the uncertainty on this number a systematic uncertainty of 1.37% istaken into account.
Tracking Efficiency
Track finding efficiency has been measured by the number of partially and fully reconstructed D∗ decays in D∗ → πD0 , with D0 → ππKS0 and KS0 → π + π − . By calculating
the ratio of tracking efficiency in data and MC, the systematic uncertainty associatedwith tracking has also been evaluated. That study’s results present (−0.13 ± 0.30 ±0.10)% per track [21], where a systematic uncertainty of 0.10% is given for particles
with momentum higher than 200 MeV. It was also found that tracking charge asymmetry is negligible in data while giving (−0.52 ± 0.20)% in MC. In this analysis, we
do not apply a correction to the reconstruction efficiency as the deviation is small. As
a conservative estimation, the central value and the statistical and systematic errors are
summed in quadrature to be 0.342% per track. This thesis’ targeted final state contains
five charged particles, and as the tracks are correlated the uncertainties from each track
are added linearly. Consequently, 0.342 × 5 = 1.71% is assigned.
Electron ID Efficiency
The electron identification efficiency correction which can arise as a potential difference
between data and MC needs to be estimated. The efficiency correction and its uncertainty have been determined by investigating the ratio of single and double tag yields indata and MC using a J/ψ → e+ e− mode. The correction is estimated to be 0.9755, withan uncertainty of 0.0042 which will be considered as 0.43% per J/ψ → e+ e− decay inthe systematic uncertainty.
Muon ID Efficiency
To determine the muon identification efficiency correction and its contribution to thesystematic uncertainty, e+ e− → e+ e− µ+ µ− events as well as J/ψ → µ+ µ− decays areused. The efficiency correction and its uncertainty are separately checked for positively
and negatively charged muons and the variation depending on the experimental periods are considered. A weighted average of those results gives 0.9654 as the efficiency
correction and 0.0299 as its uncertainty per J/ψ → µ+ µ− decay.

62

The contribution from electrons and muons is averaged. In conclusion, the combined
lepton identification systematic uncertainty is 1.77%.
Kaon and Pion ID Efficiency
For kaon and pion identification efficiency corrections due to differences between data+
and MC, estimations are made based on a D∗+ → D0 (K − π + )πslow
process. Similar to
the muon identification case, variation depending on experimental periods are checked
and the weighted average gives 1.0119 ± 0.0124 for kaon identification correction andits error which will be considered as 1.23% in the systematic uncertainties.
For pions, the identification efficiency correction and its error are determined by thesame process as kaons. As a π + π − pair has to be considered in the final state, the efficiency correction is 0.9798 with its error being 0.0192 per π + π − pair. This corresponds
to a systematic uncertainty of 1.96%.
π 0 Veto
In the event selection for the reconstruction of χc1 candidates a π 0 veto is used in orderto reject background photons coming from π 0 decays. This systematic uncertainty is
estimated by obtaining the ratio of ∆E signal yield when using the cut, Yw , and without
using the cut, Ywo , for data and Monte Carlo, and then dividing those in a double ratio:(Yw /Ywo )data
data
.
=
R
MC
(Yw /Ywo )MC
Based on this conservative approach, the difference between the efficiency in data andMC due to the π 0 veto has been estimated to be 1.22%.
Probability Density Functions
When extracting the signal yield, a fit to the data is performed which is described by aset of parameters. The possible influence of this modeling can be estimated by varying
the fixed parameters by ±1σ. The resultant variation of the respective ∆E signal yieldof each parameter change is added in quadrature. This systematic uncertainty for the B
decay branching fraction is estimated to be 2.96%.
Signal MC Sample Statistics
The extraction of the uncorrected detection efficiency ε by fitting the ∆E distribution insignal MC bears a systematic uncertainty due to the finite number of statistical generated
events. This contribution is found to be 0.51%.
Secondary Branching Fraction
For calculating the branching fraction of B → χc1 π + π − K branching fractions of secondary decays as χc1 → J/ψγ and J/ψ → `+ `− given by the PDG are used [58].
Those secondary decays’ uncertainties are considered as part of the systematic uncertainty in obtaining B(B + → χc1 π + π − K + ): B(χc1 → J/ψγ) = (34.8 ± 1.5)% andB(J/ψ → `+ `− ) = (11.87 ± 0.08)% show an uncertainty themselves. Adding them inquadrature amounts to1.50%.
Q
The total correction to the efficiency is given as corr(ε) = n corrn , with averaged lepton
ID, kaon and pion ID contributing, and will be considered with a factor 0.9622.

63

Therefore, the efficiency difference between data and the MC simulation is less than 4.0%.
A corrected resultant reconstruction efficiency of 13.41% × 0.9622 = 12.90% is acquired for
B ± → χc1 π + π − K ± .
The systematic uncertainty from each source sums up in quadrature to its final total value of
5.10%. Tab(7.1a) summarises all contributions to the systematic uncertainty estimated by themethods described above.
variable

Parameter

uncertainty (%)

Value
1597 ± 76
12.90%(34.8 ± 1.5)%(5.94 ± 0.06)%(5.93 ± 0.06)%(772 ± 11) × 106

PDF
pion ID
lepton ID
tracking
2ndary BF
NB B̄
kaon ID
π 0 veto
signal MC

2.96
1.96
1.77
1.71
1.50
1.37
1.23
1.22
0.52

Nsig
εdet
B(χc1 → J/ψγ)
B(J/ψ → e+ e− )B(J/ψ → µ+ µ− )
NB B̄

total

5.10

(b) Summary of values used in calculating thebranching ratio.

(a) Summary of systematic uncertainties.

Table 7.1.: Parameters and their values.

7.4. Branching Fraction of B ± → χc1π +π −K ±The branching fraction of the decay is then calculated as follows:
B(B + → χc1 π + π − K + ) =Nsig,εdet × B(χc1 → J/ψγ) × B(J/ψ → `+ `− ) × NB B̄

where Nsig is the signal yield, NB B̄ the number of B B̄ pairs and B the secondary branchingratios including B(J/ψ → `+ `− ) = [B(J/ψ → e+ e− ) + B(J/ψ → µ+ µ− )]. The respective
values used in calculating the branching fraction are summarised in Tab(7.1b).
We find the branching ratio of B ± → χc1 π + π − K ± to be 3.89 × 10−4 ; including statistical andsystematic uncertainties, as derived in Sec(7.3), the final result is:B(B + → χc1 π + π − K + ) = ( 3.89 ± 0.19 (stat) ± 0.20 (syst) ) × 10−4 .

64

7.5. χc1π +π − Invariant Mass Spectrum
The invariant mass of χc1 π + π − is studied with the data sample and an intermediate charmoniumlike state is looked for. In the signal enhanced window of Mbc > 5.27 GeV and |∆E| <0.02 GeV no distinguished structure is showing at the masses of interest, 3872 MeV for theX(3872) or 3920 MeV for the χc1 (2P ).

Events

100

50

0
3.8

4

4.2
4.4
Mχ ππ (GeV/c2)

4.6

4.8

c1

(a) Mχc1 π+ π− on data; whole kinematic range.

4

Events

3

2

1

0
3.8

3.85

3.9
Mχ ππ (GeV/c2)

3.95

4

c1

(b) Mχc1 π+ π− on data; region of interest.
Figure 7.3.: Invariant mass distribution of χc1 π + π − on data; Mbc > 5.27 GeV, |∆E| < 0.02 GeV
and BCS applied.

65

7.6. Upper Limit for X(3872) decaying to χc1π +π −The relevant χc1 π + π − invariant mass spectrum around 3871 MeV is shown in Fig(7.4). A 3σ
region is defined from the X(3872) → χc1 π + π − signal MC and in this signal region we findzero events. Including sideband regions, we found 60 events from 3.8 to 4.0 GeV.

4

Events

3

2

1

0
3.8

3.85

3.9
Mχ ππ (GeV/c2)

3.95

4

c1

Figure 7.4.: Invariant mass distribution of χc1 π + π − in data with 3σ signal region marked by the redlines; Mbc > 5.27 GeV, |∆E| < 0.02 GeV and BCS applied.

Because of the very limited statistics around the search window, proper convergence of the
UML fit could not be realized in the Mχc1 ππ distribution to look for a potential X(3872) contribution. Instead, the signal yield upper limit is estimated by counting events with featuring
an approach based on the work of Feldman and Cousins [36]. As there is no signal, the numberof background events in the signal region is assumed to be zero as well. From the Figure andTable provided by [36] that can be found in App(D), one obtains 2.44 events as an upper limit
at 90% C.L.. Taking into account the 5.1% systematic uncertainty, 2.56 events are regarded as
the proper upper limit.
Thus an upper limit of the product of branching fractions, including both statistical and systematic uncertainties is found to beB(B ± → X(3872) K ± ) × B(X(3872) → χc1 (1P )π + π − ) < 1.4 × 10−6 @ 90% C.L..

The values used in calculating the branching fraction are the same as in Tab(7.1b) but for thenumber of signal events Nsig = 2.56 and the efficiency εdet = 5.59%.

66

7.7. Upper Limit for χc1(2P ) decaying to χc1(1P )π +π −In case of a possible decay of χc1 (2P ) → χc1 (1P )π + π − the invariant mass spectrum ofχc1 (1P )π + π − is searched for a signature around the predicted χc1 (2P ) mass at 3920 MeV.

EventsThe χc1 (2P ) signal in the Mχc1 ππ spectrum is expressed by a PDF composed as a convolution of a Breit-Wigner function with a Gaussian. The width of the Breit-Wigner is chosen
as 20 MeV based on the assumptions made in Sec(3.1.1) when comparing with the alreadydiscovered χc0 (2P ) and χc2 (2P ) states. 2 MeV are assumed for the Gaussian’s standard deviation to represent detector mass resolution; they were obtained from the signal MC fits witha very narrow resonance at 3920 MeV.

Mχc1 π+ π− (GeV/c2 )Figure 7.5.: Fit of the invariant mass distribution of χc1 π + π − on data (blue), signal part (red) and combinatorial background (green); Mbc > 5.27 GeV, |∆E| < 0.02 GeV and BCS applied.

The fit result to the experimental data can be seen in Fig(7.5). A signal yield of 12.2 ± 9.1
events can be extracted. As the significance is not sufficient to quote an evidence, a productbranching fraction upper limit is presented. Including statistical and systematic uncertainties
as well as considering the 90% confidence level, the number of observed signal events amounts
to 30.34 events. The derived upper limit is found to be:

B(B ± → χc1 (2P )K ± ) × B(χc1 (2P ) → χc1 (1P )π + π − ) < 1.1 × 10−5 @ 90% C.L..
The values used in calculating the branching fraction are the same as in Tab(7.1b) but for thenumber of signal events Nsig = 30.34 and the efficiency εdet = 8.91%.

67

8. Conclusion
The exploration of the quark potential led us to study the spectra of heavy quarkonia, especially the charmonium system cc̄. While many states have been measured with high precision
and were found to obey the Standard Model predictions, some states above the open-charm
threshold could not yet be seen. Furthermore, new resonances with unusual exotic quantum
numbers have shown up and are waiting to be studied for their structure and nature to become
known.
The Belle experiment at the KEK B factory discovered many of those charmonium-like exotic
hadrons. Among those, the exotic X(3872) was first observed in the charged B meson decay
B + → X(3872)K + followed by X(3872) → J/ψπ + π − by the Belle Collaboration in 2003.
This resonance has a much smaller width, Γ < 1.2 MeV at 90% C.L., than other charmonia above the open-charm threshold. Its mass is very close to the threshold of the charmed
meson pair D0 D̄∗0 and its quantum numbers were finally pinned down to be J P C = 1++ .
Furthermore, not finding a charged partner suggests isospin zero. All this concluded in many
speculations about the nature of this well established but puzzling state. By now, an admixture of a D0 D̄∗0 molecule and a conventional charmonium, most likely the still undiscoveredχc1 (2P ), is one of the most plausible hypotheses. However, the little energy that is left forthe molecule’s binding energy can be translated to a distance between the two D mesons andimplies a size of about a few fm. This seems too large to explain the significant branching to J/ψ. The mass prediction for χc1 (2P ) is around 3920 MeV and this member of theχcJ (2P ) spin triplet has yet to be seen. Even if X(3872) does not contain a χc1 (2P ) component, a significant branching of χc1 (2P ) to the χc1 π + π − final state can be expected, just like
ψ 0 → J/ψπ + π − is one of the major decay modes. Considerations as to its width by comparing with the recently established χc0 (2P ) and χc2 (2P ) states leads to assuming a possibly
relatively broad resonance with Γ ≈ 20 MeV. Summarising the statements above, the Mχc1 ππdistribution in still to be measured B ± → χc1 π + π − K ± decay is also a suitable source to look
for a signature and clarify the existence of the decays of such resonances.
The results presented in this thesis are based on a total of 772 × 106 B B̄ events accumulated
at the Υ(4S) resonance collected by the Belle detector at the KEKB asymmetric-energy e+ e−
collider. Charged pions and kaons were selected using the combined information from tracking and particle identification sub-detectors. Their identification is based upon a likelihood
ratio. Due to the small Q-value in case of X(3872) and χc1 (2P ) decaying to χc1 π + π − additional selection criteria for pions with low transverse momentum became necessary to reducefake tracks and improve multiplicity. J/ψ candidates were reconstructed from their di-lepton
decay mode and paired with a photon to form χc1 . By combining the χc1 π + π − system with acharged kaon, B meson candidates were reconstructed.

69

The kinematic variables of interest, the beam-constrained mass Mbc and the difference-tobeam energy ∆E are defined in the Υ(4S) rest frame. The signal extraction region wasdetermined to be in |∆E| < 0.12 GeV and Mbc > 5.27 GeV. A best candidate selection
was applied in case of multiple B candidates per event based on closest mass compared to thenominal B mass.
Using Monte Carlo samples of 0.5 million events each for the four-body phase space caseand the quasi two-body cases with hypothetical intermediate states decaying to χc1 π + π −possible contributions to the background have been estimated. The dominant backgroundfor B ± → χc1 π + π − K ± decays was expected to come from those B decays with a real
J/ψ → `+ `− in their final state. A large B → J/ψX Monte Carlo sample exposed that
B ± → χc2 π + π − K ± only has a small contribution to form a peak in the ∆E projection; allother backgrounds have shown to be smooth. In the invariant mass spectrum of χc1 π + π − theonly peaking component is found to be a reflection from ψ 0 which was left untreated since
ψ 0 is not interfering with the mass region of interest corresponding to the X(3872) mass orthe χc1 (2P ) mass at 3920 MeV. A fitting routine for ∆E and Mχc1 ππ has been obtained bycomposing the respective Probability Density Functions and was tested in toy Monte Carlostudies. The reconstruction efficiency has been calculated as a function of the invariant massof the intermediate state and probable signal yields have been estimated. These and the study
about contributions to the systematic uncertainty were used to obtain a branching ratio andupper limits.
The B ± → χc1 π + π − K ± signal yield is extracted from an unbinned maximum likelihood fit
to the ∆E distribution. We made a first observation of the decay B ± → χc1 π + π − K ± with astatistical significance of 29.6 σ. The signal yield reconstructed in data is 1597±76 events andvery well comparable to the expected signal yield of 1700 events. We find the ∆E resolution
to be consistent with the Monte Carlo expectation. The B decay branching fraction has beenmeasured to beB(B + → χc1 π + π − K + ) = ( 3.89 ± 0.19 (stat) ± 0.20 (syst) ) × 10−4 .
In order to identify a new X(3872) decay mode and a still unseen χc1 (2P ), the χc1 π + π −invariant mass spectrum has been studied. Under a plausible assumption concerning branchingratios, the signal was expected to be in this study’s sensitivity range about 15 or more events.
However, the data exhibits no signature.
Consequently, an upper limit for the exotic X(3872) charmonium-like state to decay to χc1 π + π −has been calculated for the first time using the method of Feldman and Cousins including both
statistical and systematic uncertainties which we report with
B(B ± → X(3872) K ± ) × B(X(3872) → χc1 (1P )π + π − ) < 1.4 × 10−6 @ 90% C.L.,
Comparison with its sibling triplet states and well thought out assumptions about the propertiesof the still undiscovered conventional charmonium χc1 (2P ) following a mass assumption of
3920 MeV led to a preliminary upper limit with
B(B ± → χc1 (2P )K ± ) × B(χc1 (2P ) → χc1 (1P )π + π − ) < 1.1 × 10−5 @ 90% C.L..

70

These results are compatible with and favour the interpretation of X(3872) as an admixture
state of a D0 D̄∗0 molecule and a conventional state which might be the suspicious and maybe
not so conventional χc1 (2P ). In [66], Takizawa et al. investigate X(3872) as a charmoniumhadronic molecule hybrid suggesting a scenario of a cc̄ core state coupling to the D0 D̄∗0
and D+ D∗− molecular states. This model is capable of explaining many of the observed
properties of the X(3872), such as its large isospin symmetry breaking, the strangely highproduction rate in pp̄ collisions when considering the size of the molecule, and the absenceof a charged partner. The non-observation of the χc1 (2P ) resonance predicted by the quark
model is explained by a strong coupling to DD̄∗ two-meson state which results in a resonancewith a very broad width - being consistent with our assumptions.
The admixture scenario for X(3872) predicts other probable final states to look for a partner
states. To properly see the signature of a broad admixed X(3872) state, search for partner
states and to fully understand its nature more statistics are needed. They will be provided
by the Belle2 experiment at the next-generation flavour-factory SuperKEKB, a collider with adesign luminosity of 8×1035 cm−2 s−1 which is around 50 times as large as the peak luminosity
achieved by the KEKB collider. The Belle2 detector is a general purpose spectrometer toefficiently collect data of e+ e− collisions and consists of several sub-detector components
where the Time-of-Propagation counters and the ring-imaging Cherenkov counters are will
replace the TOF and ACC [5, 62]. The physics reach was investigated and studies at the Belle2
experiment will focus on flavour physics and CP violation measurements [7].

71A. Cut Selection for π 0 Veto————————————-√
S/ S + B

A figure of merit study was performed to select the most appropriate cut value for the π 0 veto.
0.8 was chosen meaning all photons that have a probability > 80% to originate from a π 0 will
be rejected and not used in the reconstruction of χc1 candidates from J/ψγ.

Prob. for γ coming from π 0
Figure A.1.: Figure of merit study to select a veto for photons coming from π 0 in the χc1 candidate
reconstruction.

73

B. Estimated ∆E Distribution in the
Larger Window
There are five modes to point out when looking at |∆E| < 0.2 GeV which can be divided asfollows:• Signal: B ± → χc1 Xsu , where Xsu can form π + π − K ± and result in the signal final state.
In this case ∆E peaks at zero.
• Background: B ± → ψ 0 π + π − K ± , where ψ 0 → χc1 γ. The missing energy from the ψ 0
decay photon, compared to the signal final state, is 80 to 280 MeV, resulting in a broad
bump in the negative ∆E region.
• Background: B ± → χc2 Xsu , where Xsu → π + π − K ± . A small peak can be seen,shifted to the negative ∆E region because of the χc2 ’s slightly higher mass (∆m ≈50 MeV).
• Background: B ± → χc1 K ∗ and B → χc1 Xsd . If one additional, unrelated pion isadded, the signal final state is faked. Here, ∆E is shifted to values > 0.15 GeV. B 0 →χc1 Xsd is kinematically the same as χc1 K ∗ when Xsd → π ∓ K ± happens.

Figure B.1.: ∆E distribution on the B → J/ψX MC sample; Mbc > 5.27 GeV and BCS applied.

74

C. Expected Signal Yield in the ∆E
Distribution in the Phase Space
Based on the estimated efficiency ε an expected peak yield for the yet unseen B ± → χc1 π + π − K ±decay can be calculated assuming that it is obeying a phase space distribution:
NB B̄ × B(B → χc1 π + π − K) × B(χc1 → J/ψγ) × B(J/ψ → l+ l− ) × ε= 771.6 · 106 × 4 · 10−4 × 0.344 × (0.0593 + 0.0594) × 0.1341≈ 1700 events
where, taking possible decay dynamics into account, the assumed branching fraction of B →χc1 π + π − K was derived from the decay of B → J/ψπ + π − K [58] as follows:
B(B → χc1 π + π − K) = B(B → J/ψπ + π − K) ·= 8.1 · 10−4 ·

B(B → χc1 K)
B(B → J/ψK)

4.79 · 10−4
1.02 · 10−3

≈ 4 · 10−4
About 1700 events are expected in data when looking for the B ± → χc1 π + π − K ± decay in its∆E distribution.

75

D. Upper Limit Calculation

Figure D.1.: Upper end µ2 of our 90% C.L. confidence intervals [µ1 , µ2 ], for unknown Poisson signal
mean µ in the presence of expected Poisson background with known mean b. The curves
for the cases n0 from zero through ten are plotted. Dotted portions on the upper left
indicate regions where µ1 is non-zero. Dashed portions in the lower right indicate regions
where the probability of obtaining the number of events observed or fewer is less than
1%, even if µ = 0. [36]Figure D.2.: 90% C.L. intervals for the Poisson signal mean µ, for total events observed n0 , for known
mean background b ranging from zero to five. [36]76

E. Other Invariant Mass
Combinations
While one of the main goals of this analysis is the study of any χc1 π + π − intermediate state,some other invariant mass combinations have been visited as well. However, no unknown
structures can be seen in the following invariant mass spectra of χc1 and the higher mass π,
the kaon and the lower mass π or the Kπ + π − system.

Events

100

50

0
0.8

1

1.2
1.4
MKππ (GeV/c2)

1.6

1.8

Figure E.1.: Invariant mass distribution of Kππ in data; Mbc > 5.27 GeV, |∆E| < 0.02 GeV andBCS applied.

77

Events

100

50

0
3.6

3.8

4
4.2
Mχ π(max) (GeV/c2)

4.4

4.6

c1

(a) Mass distribution of χc1 and the higher mass π in data.

Events

100

50

0
0.6

0.8

1
1.2
MKπ(min) (GeV/c2)

1.4

1.6

(b) Mass distribution of K and the lower mass π in data. The peaking
structure might be coming from a K ∗ (892) decay.
Figure E.2.: Invariant mass distributions of correlated final state particles in data; Mbc > 5.27 GeV,|∆E| < 0.02 GeV and BCS applied.

78

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v

Acknowledgements
There are so many people I would like to thank for helping me being able to finish this "second
book of mine". Maybe I start at the beginning when the idea of not just continuing my university studies but also doing so in part in Japan found its way into my head. I was inspired byMiriam Fritsch to take an interest in charmonium spectroscopy. The Belle experiment offereda great opportunity to combine my plans for a dissertation in charmonium analyses and aninternational exchange with Japan.
Through Miriam I met Soeren Lange, a colleague from the PANDA collaboration who also
happens to be a Belle member, who then introduced me to Ariane Frey in Göttingen and Kenkichi Miyabayashi in Nara. They have been my supervisors and most respected role models
for the last four years. Both have been unbelievably supportive of the still not common concept of a binational joint doctoral supervision. I would not have been able to do my doctorate
under these excellent conditions without their help. Also many others on the administrative
level have been contributing to negotiate and set up contracts, making sure both universities’
requirements will be satisfied for the award of a doctor’s degree, especially Stefan Häußler
and all the officials of Nara Women’s University taking care of formalities.
Even before I started in Göttingen in January 2011, Wolfgang Gradl took me to one of the
B factory legacy book meetings held at the University of Mainz and I met some future Bellecolleagues for the first time which left me very excited to become a part of such an inspiring collaboration. I am much indebted to my German supervisor Ariane Frey for her warm
welcome, support and assistance at any time. She has been a reliable pillar of good will andcare throughout this time. Also a special thank you goes to Philipp Hamer for many hoursof patiently introducing me to the art of C++ and the Belle experiment’s analysis framework
BASF. I thank my colleagues and fellow PhD students in the II. Institute, past and present,
for their support and encouragement, especially Julia Rieger, Martina Schulte-Borchers, Jens
Weingarten und Jörn Große-Knetter.
At Nara Women’s University I would like to thank the staff of the international office for
their help with all the small things and formalities. My thanks go to all colleagues for useful
discussions and to my fellow students in the high energy physics lab for many smiles andcheerfulness. But the biggest impact had the incredible and never-ending support and assistance, patience and motivation I received from my Japanese supervisor Kenkichi Miyabayashi
who I owe my deepest gratitude. His enthusiasm and dedication have been a great motivation.
I am further indebted to Miyabayashi-san for valuable discussions and help in the preperationof this thesis. For a lot of discussions, explanations and advice, and not to forget constantencouragement and guidance through all the different phases of this analysis, I would like towholeheartedly thank Vishal Bhardwaj.

vi

I wish to thank every member of the Belle collaboration and the KEKB accelerator team fortheir hard work to provide the excellent smooth operation of the experiment - no analysis without data. Also, I am incredibly grateful to my Belle Note referees, Seema Bahinipati, Pavel
Krokovny and Eiichi Nakano for their specific comments on a number of the more technicaldetails and for monitoring my progress. For providing critical assessments and suggestions, I
am particularly grateful to Karim Trabelsi, Yoshihide Sakai, Mikihiko Nakao, and all of theBelle physics analysis colleagues and the conveners of the Charm and New Physics Subgroupas well as BAM and BGM meetings.
I would like to thank my colleagues and friends, Jason Mansour, Philipp Hamer, Andrea Knueand Boris Lemmer, for proofreading and for their comments on early drafts of this work, a
lot of encouragement and help with LaTeX and Root issues. I also have to say a big thankyou to my long time friend Lisa Büschelberger for an outsider’s view and valuable advice inall things linguistical (and probably another best-of list of particle physics lingo). A special
thank you to Oliver Steffen, for proofreading in every stage of this thesis, useful comments,
help with any technical questions and for general moral support at any hour. Thank you forbeing there for me when I really needed it.
For moral support and for reminding me to keep a healthy work-life balance I would like tosay thank you to the "Bier & Schnitzel"-Crew and my lovely flatmates, Jason Mansour andFrauke Bergmann. I thank Lucie Hamdi for her patience with all the complicated paperwork
for reimbursement for conferences and coming "home" to Göttingen.
Zuletzt möchte ich meinen lieben Eltern, Herbert und Trudelinde Panzenböck danken. Mein
Leben lang haben sie mich bei all meinen Entscheidungen ermutigt, beim Überwinden derdamit verbundenen Herausforderungen unterstützt und mich immer daran erinnert, mein Bestes
zu geben. Ich bin sehr dankbar für all die aufmunternden Worte während der kurzen Telefonpausen, in den kleinen E-Mails zwischendurch und das Verständnis und die Geduld, mit denen
sie meine Ausschweifungen über technische Analyse-Details ertragen haben. Ihr seid großartig!

vii

Curriculum VitæPersonal Data:
Name:
Date of Birth:
Nationality:

Elisabeth Patricia Panzenböck
February 24, 1985
in Mainz, Germany
GermanEducation:
Jan 2011 - Dec 2014

PhD student under Joint Doctoral Supervisionat Georg August University, Göttingen, Germany
and Nara Women’s University, Nara, Japan
Doctoral Thesis: Branching Fraction Measurement of B ± →χc1 π + π − K ± and Search for a Narrow Resonance with the
Belle Experiment
under supervision of Prof. Dr. Ariane Frey (Göttingen)and Assoc. Prof. Dr. Kenkichi Miyabayashi (Nara)Apr 2012 - Mar 2014

Research period at Nara Women’s University, Nara, Japan

Nov 2010
Apr 2004 - Nov 2010

Graduation (Diplom): grade very good
Study of Physics at Johannes Gutenberg University,
Mainz, Germany
Diploma Thesis: Setup of a Test Station for the PANDA Luminosity Monitor System
under supervision of Prof. Dr. Achim Denig

Mar 2004

Graduation (Abitur): grade very goodviii

Scientific Schools:
Sep 2014
Oct 2012
Sep 2011
Aug 2011
Aug 2010

46. Herbstschule für Hochenergiephysik
in Maria-Laach, GermanyFirst Asia-Europe-Pacific School of High-Energy Physics
in Fukuoka, Japan
Belle Analysis Schoolin Jeju, South Korea
School on Amplitude Analysis in Modern Physics
in Bad Honnef, Germany
Summer School on Hadron Physics
in Bad Honnef, Germany

Presentations:Feb 2014
Nov 2013

Mar 2013
Mar 2013
Mar 2010

Study of B ± → χc1 π + π − K ±Talk at Lake Louise Winter Institute, Lake Louise, Canada
Study of B ± → χc1 π + π − K ±Talk at XV International Conference on Hadron Spectroscopy,
Nara, Japan
Study of B ± → χc1 π + π − K ±Talk at 69th Annual Meeting of JPS, Hiroshima, Japan
Study of B ± → χc1 π + π − K ±Talk at DPG Spring Conference, Dresden, Germany
A Luminosity Monitor for PANDA
Poster presentation at DPG Spring Conference, Bonn, Germany

Scholarships:
Apr 2012 - Mar 2014

Japanese Government (MEXT) & DAAD Scholarshipas partial funding of PhD Thesis

Apr - Sep 2010

Scholarship of Johannes Gutenberg University Mainz
as partial funding of Diploma Thesisix