A Mathematica based Version of the CKMfitter Package

A Mathematica based Version
of the CKMfitter Package
Diplomarbeit
zur Erlangung des akademischen Grades
Diplom-Physiker
vorgelegt von
Andreas Jantsch
geboren in Dresden
Institut für Kern- und Teilchenphysik
Fachrichtung Physik
Fakultät Mathematik und Naturwissenschaften
der Technischen Universität Dresden
2006

1. Gutachter: Prof. Dr. Klaus R. Schubert
2. Gutachter: Dr. Heiko Lacker
Datum des Einreichens der Arbeit: 11. 09. 2006

Kurzfassung
Das CKMfitter Software-Paket stellt ein Analyseinstrument zur Verfügung, mit dessen
Hilfe die Parameter der Cabibbo-Kobayashi-Maskawa-Matrix im Bereich des
Standardmodells und seinen möglichen Erweiterungen eingeschränkt werden können.
Die verwendete statistische Methode ist ein auf klassischer Statistik basierender
Ansatz, genannt Rfit, in dem theoretische Unsicherheiten durch erlaubte Bereiche
dargestellt werden. Das ursprüngliche Paket ist in FORTRAN programmiert und
verwendet das Minimierungspaket MINUIT. Über die Jahre erhöhte sich die Komplexität
der Fitprobleme und damit der CPU-Zeitverbrauch beachtlich. Mit dem Ziel einer
deutlichen Fitzeitreduzierung initiierte Jérôme Charles eine auf Mathematica
basierende Version des CKMfitter-Pakets. Mit diesem Paket konnte unter Verwendung
von symbolischen Kalkulationen und einer effizienten Minimierungsroutine ein
CPU-Zeitgewinn von mehr als einem Faktor 100 erreicht werden. Thema dieser Diplomarbeit
ist die Implementierung der Theorie der neutralen Meson-Oszillationen
als auch der leptonischen Zerfälle geladener B-Mesonen, jeweils im Standardmodell
und einer möglichen Erweiterung. Weiterhin wurde die Behandlung von Dateien,
welche tabellarische Eingabedaten enthalten, entwickelt. Abschließend werden aktuellste
Resultate der Cabibbo-Kobayashi-Maskawa-Matrix Analyse im Standardmodell
und zwei seiner Erweiterungen gezeigt. Diese wurden von der CKMfitter
Gruppe auf der Internationalen Konferenz für Hochenergiephysik 2006 (ICHEP06)
präsentiert.
Abstract
The CKMfitter package provides an analysis tool to constrain the parameters of the
Cabibbo-Kobayashi-Maskawa matrix in the framework of the Standard Model
and possible extensions. The statistical method used is a frequentist based approach
called Rfit, where theoretical uncertainties are represented by allowed ranges. The
original package is coded in FORTRAN and uses the MINUIT minimization package.
Over the years, the complexity of the fit problems and associated with that
the CPU time consumption have increased considerably. With the goal of a significant
fit time reduction, Jérôme Charles initiated a Mathematica based version
of the CKMfitter package. With this new package a gain in CPU time of more
than a factor 100 has been achieved, using symbolic calculations and an efficient
minimization routine. Subject of this thesis is the implementation of the theory
of neutral meson oscillations as well as leptonic decays of charged B-mesons, both,
for the Standard Model and a possible extension. Furthermore, the treatment of
look-up table input files has been developed. Finally, updated results of the global
Cabibbo-Kobayashi-Maskawa matrix analysis in the Standard Model and two
of its extensions have been provided. They have been presented by the CKMfitter
group at the International Conference on High Energy Physics 2006 (ICHEP06).

Contents
Contents v
List of Figures vii
List of Tables ix
1 Introduction 1
2 Theory 3
2.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 The CKM Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 The Unitarity Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Neutral Meson Oscillation . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 CKMfitter 13
3.1 The Statistical Framework - Rfit . . . . . . . . . . . . . . . . . . . . 13
3.2 Fit Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 The CKMfitter Package Code . . . . . . . . . . . . . . . . . . . . . . 15
4 A Mathematica based Version of the CKMfitter Package 17
4.1 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Package Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 CKMfitter.nb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 The Minimization Routine . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 Theory Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.6 Look-Up Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.7 Performance Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Probing the Standard Model 29
5.1 Fit Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Standard Model Fit Results . . . . . . . . . . . . . . . . . . . . . . . 39
v

vi CONTENTS
6 New Physics Beyond the Standard Model 49
6.1 New Physics in B 0 - ¯ B 0 Oscillations . . . . . . . . . . . . . . . . . . . 49
6.2 Charged Higgs Contributions to Leptonic B ± Decays . . . . . . . . . 54
7 Conclusions and Perspectives 57
A The Inami-Lim Functions 59
B Additional Figures of New Physics in B 0 - ¯B 0 Oscillations 61
C Testjob 65
D Source Code 67
D.1 Tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
D.2 dTableauO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
D.3 LoadLUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
E User Guide 71
E.1 Mathematica Terminology . . . . . . . . . . . . . . . . . . . . . . . . 71
E.2 Datacards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
E.3 Input Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
E.4 Theory Package Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . 77
Bibliography 97
Danksagung 103
Erklärung 105

List of Figures
2.1 The rescaled Unitarity Triangle . . . . . . . . . . . . . . . . . . . . . 7
2.2 Box diagram contribution to K 0 - ¯ K 0 mixing . . . . . . . . . . . . . . 9
2.3 Box diagram contribution to B 0 - ¯ B 0 mixing . . . . . . . . . . . . . . 10
4.1 The fit process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 The minimization routine file system . . . . . . . . . . . . . . . . . . 21
4.3 Comparison plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.1 CL on the UT angles α and γ . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Tree-level contributions to leptonic B + decays in the SM . . . . . . . 33
5.3 The Standard Global CKM Fit in the (¯ρ,¯η) plane . . . . . . . . . . . 40
5.4 CL on A, λ, ¯ρ, ¯η, J, α, β and γ . . . . . . . . . . . . . . . . . . . . . 41
5.5 CL on |Vub|incl and sin 2β . . . . . . . . . . . . . . . . . . . . . . . . 42
5.6 CL on B(B + → τ + ντ ) and B(B + → µ + νµ) . . . . . . . . . . . . . . . 43
5.7 CL in the (¯ρ,¯η) plane obtained from B(B + → τ + ντ ) and ∆md . . . . 43
5.8 SM fit results in the (¯ρ,¯η) plane . . . . . . . . . . . . . . . . . . . . . 45
5.9 CL on ASL including LO and NLO QCD corrections . . . . . . . . . 46
6.1 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β . . . . . 50
6.2 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β 51
6.3 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β, γ 51
6.4 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β,
γ, α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.5 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β,
γ, α, ASL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.6 CL on the NP parameters r 2 d and 2ϑd . . . . . . . . . . . . . . . . . 53
6.7 Tree-level contribution from charged Higgs bosons to B + → τ + ντ . . 54
6.8 Constraints on (tan β,m H +) from B(B + → τ + ντ ) . . . . . . . . . . . 55
B.1 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, cos 2β . . . . 61
B.2 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, α . . . . . . . 62
B.3 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, ASL . . . . . 62
B.4 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β, α 63
B.5 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, ASL . . . . . 63
vii

viii LIST OF FIGURES
B.6 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, α . . . 64
B.7 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, γ, α, ASL 64

List of Tables
2.1 The three generations of fundamental fermions . . . . . . . . . . . . 3
2.2 The fundamental interactions . . . . . . . . . . . . . . . . . . . . . . 4
4.1 Directory structure of the Mathematica based CKMfitter package . . 18
4.2 Version labels in the theory package BBbarKKbarMixing . . . . . . 23
4.3 Version labels in the theory package LeptonicDecay . . . . . . . . . . 23
4.4 LUT column order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5 Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.6 Numerical comparison of A, λ, ¯ρ, ¯η and J . . . . . . . . . . . . . . . 26
4.7 Fit time comparison of test job runs . . . . . . . . . . . . . . . . . . 28
4.8 Hardware and software performance tests . . . . . . . . . . . . . . . 28
5.1 Inputs to the CKM fits . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Inputs to the CKM fits (continued) . . . . . . . . . . . . . . . . . . . 38
5.3 Fit results and errors using the Standard Global CKM Fit observables 47
5.4 Fit results and errors (continued) . . . . . . . . . . . . . . . . . . . . 48
ix

x LIST OF TABLES

Chapter 1
Introduction
“Nature has always looked like a horrible mess, but as we go along we see patterns
and put theories together, a certain clarify comes and things get simpler.”
Richard P. Feynman [1]
An important instrument of research is creating models. A physical model is a theory,
which describes its objects and their interactions using mathematical equations
and makes predictions for them. Usually, not all parameters of such a theory are
fixed by the model itself, therefore some free parameters of a theory can only be
determined by experiments.
The Standard Model (SM) of particle physics is the theory of three generations of fundamental
fermions and the interactions between them, mediated by gauge bosons. 1
The SM is a combination of the Quantum Chromodynamics (QCD), which describes
the strong interaction, and the unified theory of electroweak interactions. These are
theories of massless particles. To generate masses preserving the gauge symmetry
of the SM, it can be accommodated by spontaneous symmetry breaking, also called
the Higgs Mechanism. The associated Higgs particle awaits its discovery though.
Due to the disparity of weak and mass eigenstates, quarks can be transformed into
each other via flavor changing weak interaction transitions. The transformation matrix
is called Cabibbo-Kobayashi-Maskawa (CKM) matrix [2,3] and depends on
four independent parameters, three rotation angles and one phase. A non-vanishing
phase would be a source of CP violation 2 (CPV) in the Standard Model.
The four independent parameters of the CKM matrix are free parameters of the
SM and need to be determined from experiments. They can be overconstrained by
measurements of CKM matrix elements and CP asymmetries. This requires a global
analysis, which probes the consistency between the different measurements and their
SM predictions.
1 Particles with half-integer spin are called fermions, particles with integer spin are called bosons.
2 CP violation means an asymmetric behavior of particles and its corresponding antiparticles.
1

2 Chapter 1. Introduction
Unfortunately, the Standard Model cannot answer all questions, e. g. evidences for
dark matter from the measurement of the cosmic microwave backround radiation imply
the existence of non-SM particles. Furthermore, the observed matter-antimatter
asymmetry of the Universe requires additional CP-violating effects. There are many
New Physics (NP) models, like Supersymmetric Models (SUSY) and Grand Unified
Theories (GUTs), that can solve, in principle these and other problems. Probing
the consistency of these models with experimental data is also a part of the global
CKM matrix analysis.
The CKMfitter package [4] provides a global analysis tool, which allows to constrain
the CKM matrix parameters within the Standard Model and its possible extensions.
In the framework of the frequentist approach Rfit, the consistency of recent experimental
results and its theoretical predictions are probed. The software package has
been mainly written in FORTRAN using the MINUIT minimization package. Over
the years, the complexity of the fit problems has increased substantially. Depending
on the fit problem, a fit can last up to the order of days.
In this thesis, a Mathematica based version of the CKMfitter package is introduced.
It has been initiated and commenced by Jérôme Charles. Beside of Mathematica,
which provides a framework for symbolic calculations, a slim and effective
minimization routine is implemented. During this work, the final structure of the
source code with a userfriendly environment accrued and fundamental theory packages
were coded. They include the theoretical predictions of additional NP models
beyond the SM. Important developments are also the routine for the treatment of
numerical input tables and the possibility of having different theoretical frameworks
in one theory package. A talk on the Mathematica based CKMfitter version has
been presented at the DPG Frühjahrstagung 2006 in Dortmund [5].
After one year of development, a lot of important fits, for instance the Standard
Global CKM Fit, can be performed using the Mathematica based version of CKMfitter.
For example, all SM and some NP fit results presented by Stephane T’Jampens
at the International Conference on High Energy Physics 2006 (ICHEP06) in Moscow,
have been produced with the new Mathematica based CKMfitter version.
The thesis is organized as follows. Chapter 2 gives a brief introduction to the
Standard Model and the CKM matrix. Furthermore, a review of the relevant theory
of neutral meson oscillation and CP violation is presented. After an overview about
the original CKMfitter package and its statistical approach Rfit in Chapter 3, the
development of CKMfitter in a Mathematica environment is described in Chapter 4.
Up-to-date plots and results of the global CKM matrix analysis are presented in
Chapter 5 for the Standard Model and in Chapter 6 in the framework of two New
Physics models. A user guide and a development tutorial for the Mathematica based
version of CKMfitter can be found in the appendices.

Chapter 2
Theory
In the following, a brief introduction to the relevant theoretical background is provided.
If not explicitly stated, the information given is taken from the Refs. [6–10]
which recommended for a more detailed overview.
2.1 The Standard Model
The Standard Model of particle physics is the theory of the fundamental particles and
their interactions. The known fundamental fermions are classified into leptons and
quarks. They are ordered in three generations with ascending masses, as summarized
in Table 2.1.
Fermions Generation Charge Color Weak Isospin (I, I3)
1 2 3 Q/e0 left-handed right-handed
Leptons
νe
e− νµ
µ −
ντ
τ −
0
−1
-
(1/2, +1/2)
(1/2, −1/2)
−
(0, 0)
Quarks
ui ci ti +2/3
−1/3
i=r,g,b
(1/2, +1/2)
(1/2, −1/2)
(0, 0)
(0, 0)
di
si
bi
Table 2.1: The three generations of fundamental fermions
For each fermion a corresponding anti-fermion exists with the same mass, but opposite
electric charge, color 1 and third component of the weak isospin, I3.
Beside of gravity 2 , there are three fundamental interactions which are mediated by
vector gauge bosons. Gluons have color and interact among each other as well as
1 Color is the charge of the strong interaction. It is an additional degree of freedom with three
possible values, usually called red (r), green (g) and blue (b).
2 Due to its weakness (the relative strength compared to the strong interaction is of order 10 −38 ),
gravity is irrelevant for particle physics on energy scales currently accessible in experiments and is
not described by the Standard Model. A unified description of all fundamental interactions is the
goal of Theories of Everything (TOEs).
3

4 Chapter 2. Theory
the bosons of the weak interaction, which have weak charge. The three fundamental
interactions are summarized in Table 2.2.
Interaction Couples to Vector Boson Mass (GeV/c 2 ) J P
strong color 8 gluons 0 1 −
electromagnetic electric charge photon γ 0 1 −
weak weak charge W ± , Z 0 ≈ 10 2 1
Table 2.2: The fundamental interactions
The Standard Model is the combination of Quantum Chromodynamics and the unified
theory of electroweak interactions. It is a renormalizable quantum field theory
and carries the group structure of the gauge group SU(3)C ⊗SU(2)L ⊗U(1)Y , where
C means color, L means left and Y is the electroweak hypercharge.
The QCD, represented by the gauge group SU(3)C, is the theory of strong interactions
between colored quarks and gluons. Its coupling constant αS depends on the
energy scale µ, which leads at high energies to a weak coupling, called Asymptotic
freedom, and a strong coupling at low energies leading to so-called Confinement.
Due to the Confinement, no free quarks or gluons have been observed yet. They
only occur in color-neutral bound states, called hadrons, which can be classified into
baryons and mesons 3 .
The unification of electro-magnetic and weak interaction is described in the model of
S. L. Glashow, S. Weinberg and A. Salam by the gauge group SU(2)L⊗U(1)Y ,
where the fermions are represented by left-handed doublets and right-handed singlets
of the weak isospin. Since gauge invariance requires the absence of explicit mass
terms in the Lagrangian, all particles are initially assumed to be massless. As
a possibility of mass generation without spoiling the gauge invariance, P. Higgs
introduced a complex scalar doublet field Φ = (φ1, φ2), which leads to an additional
term in the SM Lagrangian:
LΦ = � ∂µΦ +� (∂ µ Φ) − V (Φ) (2.1)
with a rotationally symmetric potential V (Φ) = −µ 2 Φ + Φ + λ 2 (Φ + Φ) 2 . The Yukawa
interaction of the quarks with the Higgs field is described by:
LY = −Y d
ij ¯ Q I LiΦd I Rj − Y u
ij ¯ Q I LiεΦ ∗ u I Rj + h.c. , (2.2)
(q = u, d) are complex 3 × 3 matrices, i, j are the labels of the fermion
are the left-handed
represent the right-handed down- and up-type
quark singlets in the basis of weak interaction eigenstates, denoted by I.
where Y q
ij
generation and ε is the 2×2 total antisymmetric tensor. The QI Li
quark doublets, where dI Rj and uI Rj
3 Baryons are composed of three quarks with different color (qiqjqk), mesons are composed of a
quark-antiquark pair (qi ¯q ī), where the indices i, j, k represent the color and ī the anti-color respectively.

2.2. The CKM Matrix 5
After spontaneous symmetry breaking from SU(2)L ⊗ U(1)Y to U(1)Q at lower
energies (∼200 GeV), Φ acquires a non-vanishing vacuum expectation value 〈Φ〉 =
� 0, v/ √ 2 � and equation (2.2) results in Dirac mass terms for the quarks:
M u = v √ 2 Y u , M d = v √ 2 Y d . (2.3)
Furthermore, in contrast to the photon γ, the vector bosons W ± and Z 0 obtain
masses and an additional massive spin-0 boson, the physical Higgs particle H, is
predicted 4 .
The mass matrices M u and M d can be diagonalized by unitary transformations:
M u,diag = UM u Ũ † =
M d,diag = V M d ˜ V † =
⎛
⎝
⎛
⎝
mu 0 0
0 mc
0 0 mt
md 0 0
0 ms
0 0 mb
⎞
⎠ (2.4)
⎞
⎠ , (2.5)
where U, Ũ and V, ˜ V are unitary matrices and the quark masses mq are real. Due
to the fact that the left-handed up- and down-type quarks are members of the same
SU(2)L doublet, they cannot be transformed independently. Choosing the up-type
quarks as mass eigenstates5 , the left-handed isospin doublet transforms to:
Q I L =
� u I Li
d I Li
�
= U †
ij
�
uLj �
UV † �
jk dLk
�
, (2.6)
which leads to a mixing matrix in the down-type quark sector, the so-called CKM
matrix VCKM = UV † .
2.2 The CKM Matrix
2.2.1 General Remarks
The Cabibbo-Kobayashi-Maskawa matrix VCKM, is the quark-mixing matrix. It
connects the weak interaction eigenstates d I Li = (d′ , s ′ , b ′ ) and the corresponding
mass eigenstates dLk = (d, s, b) of the down-type quarks through:
⎛
⎝
⎞ ⎛ ⎞ ⎛
d Vud
⎠ = VCKM ⎝ s ⎠ = ⎝ Vcd
Vus
Vcs
Vub
Vcb
⎞ ⎛
d
⎠ ⎝ s
b
b
d ′
s ′
b ′
Vtd Vts Vtb
⎞
⎠ . (2.7)
4
The observation of the Higgs boson is the main goal of the experiments at the Large Hadron
Collider (LHC), starting in 2007.
5
This is done by convention. It is also possible to choose the down-type quarks as mass eigenstates
which would lead to a mixing matrix in the up-type quark sector.

6 Chapter 2. Theory
VCKM is a complex 3×3 matrix and, hence, has 18 independent parameters: nine real
parts and nine imaginary parts of its nine complex matrix elements. An important
property is its unitarity, given by the relation:
VCKMV † †
CKM = V CKMVCKM = 1 , (2.8)
which ensures the conservation of probability. Due to unitarity and the freedom of
phase redefinition, the number of independent parameters is reduced to four and
the CKM matrix can be parameterized, e. g. by three Euler angles and one global
phase.
2.2.2 The Standard Parameterization of VCKM
The Standard Parameterization of the CKM matrix was proposed by Chau and
Keung [11] and is advocated by the Particle Data Group (PDG) [9]. It is obtained
by the product of three unitary complex rotation matrices, where the rotations are
characterized by Euler angles θ12, θ13 and θ23, which are the mixing angles between
the generations, and one overall CP-violating phase δ. The result is:
VCKM =
⎛
⎜
⎝
c12c13 s12c13 s13e −iδ
−s12c23 − c12s23s13e iδ c12c23 − s12s23s13e iδ s23c13
s12s23 − c12c23s13e iδ −c12s23 − s12c23s13e iδ c23c13
⎞
⎟
⎠ , (2.9)
where cij = cosθij and sij = sinθij for i < j = 1, 2, 3. The cij and sij are positive
for θij > 0. The unitarity relation (2.8) is strictly satisfied.
2.2.3 The Wolfenstein Parameterization of VCKM
As a result of the observed hierarchy between the different CKM matrix elements,
Wolfenstein [12] proposed a parameterization in terms of the four parameters A,
λ, ρ and η. It is an expansion of VCKM in λ � |Vus| and defined to all orders in λ
by [13]:
s12 ≡ λ
s23 ≡ Aλ 2
s13e −iδ ≡ Aλ 3 (ρ − iη) .
(2.10)
Thus, up to order of λ4 , the CKM matrix can be written as:
⎛
⎜
VCKM = ⎜
⎝
1 − λ2
2
λ Aλ3 −λ 1 −
(ρ − iη)
λ2
2
Aλ2 ⎞
⎟ + O
⎟
⎠
� λ 4� . (2.11)
Aλ 3 (1 − ρ − iη) −Aλ 2 1

2.3. The Unitarity Triangle 7
2.3 The Unitarity Triangle
As a result of Equation (2.8), there exist 12 different unitarity relations for the CKM
matrix. The rescaled unitarity relation relevant for the B-meson system is:
VudV ∗ ub
VcdV ∗
cb
+ VcdV ∗
cb
VcdV ∗ +
cb
VtdV ∗
tb
VcdV ∗
cb
= 0 , (2.12)
which can be displayed as a triangle in the complex (¯ρ,¯η) plane. Figure 2.1, taken
from Ref. [9], shows the so-called Unitarity Triangle (UT). Independent from phase
conventions, its apex is given by:
¯ρ + i¯η ≡ − VudV ∗ ub
VcdV ∗
cb
Figure 2.1: The rescaled Unitarity Triangle
The sides Ru and Rt of the triangle, are given by:
Ru =
Rt =
and the UT angles6 are defined as:
α =
�
arg −
β =
�
arg −
�
γ = arg
�
�
�
�
�
�
�
�
VudV ∗ ub
VcdV ∗
cb
VtdV ∗
tb
VcdV ∗
cb
VtdV
∗
tb
VudV ∗ ub
VcdV
∗
cb
VtdV ∗
tb
− VudV ∗ ub
VcdV ∗
cb
�
�
�
. (2.13)
�
�
�
� = � ¯ρ 2 + ¯η 2 (2.14)
�
�
�
� =
�
(1 − ¯ρ) 2 + ¯η 2 (2.15)
�
¯η
= arctan
¯η 2 �
(2.16)
− ¯ρ (1 − ¯ρ)
� �
¯η
= arctan
(2.17)
1 − ¯ρ
� �
¯η
= arctan . (2.18)
¯ρ
The area of the Unitarity Triangle is equal to the half of the Jarlskog invariant J [14],
which is defined by the imaginary part of phase-convention independent CKM matrix
element quartets:
Im � VijVklV ∗
il
� ∗
Vkj = J
3�
m,n=1
εikmεjln . (2.19)
6 In this work, the BABAR notation α, β, γ is used, whereas φ2, φ1, φ3 is used in the Belle
collaboration.

8 Chapter 2. Theory
2.4 Neutral Meson Oscillation
The phenomenon of quark flavor oscillation is predicted in the four neutral meson
systems:
K 0 (¯sd) ↔ ¯ K 0 (s ¯ d) , D 0 (cū) ↔ ¯ D 0 (¯cu) , B 0 d (¯ bd) ↔ ¯ B 0 d (b ¯ d) , B 0 s ( ¯ bs) ↔ ¯ B 0 s (b¯s) .
Since the mean life time of the neutral D mesons is very small compared to their
oscillation frequency, neutral meson mixing has been only observed in the Kaon and
B-meson systems, yet.
The time evolution of these transitions is given by the Schrödinger Equation for
two-state systems of instable particles:
i ˙ ψ(t) = ˆ H ψ(t) , (2.20)
where ψ(t) is the two-state system of the neutral mesons, e. g. for the Bd system:
�
|B0 (t)〉
ψ(t) =
| ¯ B0 �
. (2.21)
(t)〉
The Hamilton Operator ˆ H contains the two hermitian 2 × 2 mass (Mij) and decay
width (Γij) matrices:
⎛
ˆH = ˆ M − i
2 ˆ ⎜
Γ = ⎝
M11 − i
2 Γ11 M12 − i
2 Γ12
M21 − i
2 Γ21 M22 − i
2 Γ22
⎞
⎟
⎠ . (2.22)
Since CPT symmetry requires equal masses and decay rates for a particle and its
anti-particle, the diagonal elements must be equal:
Γ11 = Γ22 = Γ (2.23)
M11 = M22 = m (2.24)
and the eight real parameters can be reduced to six independent parameters. The
hermiticity of ˆ Γ and ˆ M leads to:
M21 = M ∗ 12 and Γ21 = Γ ∗ 12 . (2.25)
Because of an arbitrary global phase, only five observables can be defined
� � � �
Γ12
Γ12
m , Γ , |M12| , Re and Im . (2.26)
M12
The Schrödinger Equation (2.20) has well defined solutions ψ(t) for any ψ(0), but
only two mass eigenstates with time-independent flavor composition for each neutral
meson system.
M12

2.4. Neutral Meson Oscillation 9
2.4.1 The K 0 - ¯K 0 System
The flavor eigenstates of the neutral Kaon system mix via weak interaction through
the box diagrams shown in Figure 2.2. These are effective flavor changing neutral
current (FCNC) processes with |∆S| = 2 7 , where the main contributions in the loop
to the observable |ɛK| as described below come from top- and also from charm-quark
exchange.
s u,c,t
d
0
K
W W
u,c,t
0
K
d
s
s
d
W
u,c,t u,c,t
Figure 2.2: Box diagram contribution to K 0 - ¯ K 0 mixing
The two mass eigenstates of the Kaon system are defined according to their lifetimes:
0
K
|K 0 S〉 = pK|K 0 〉 + qK| ¯ K 0 〉 (2.27)
|K 0 L〉 = pK|K 0 〉 − qK| ¯ K 0 〉 (2.28)
where K0 S is the short living, and K0 L is the long living normalized mass eigenstate,
with |qK| + |pK| = 1. An interesting observable of the neutral Kaon system related
to the CKM matrix analysis is the CP-violating parameter |ɛK|. εK is defined by:
εK = 2
3 η+− + 1
3 η00
where η+− and η00 are the ratios of the K 0 L and K0 S
and neutral pair of pions respectively:
η+− = A � K 0 L → π+ π −�
A � K 0 S → π+ π −�
2.4.2 The B 0 - ¯B 0 System
W
0
K
d
s
(2.29)
decay amplitudes to a charged
η00 = A � K0 L → π0π0� A � K0 S → π0π0� . (2.30)
Neutral B-meson oscillations occur in the SM through a second order FCNC process.
They are mediated by the |∆B| = 2 box diagrams shown in Figure 2.3, where
the loop is dominated by W boson and up-type quark contributions.
According to their mass, the eigenstates of the B-meson system are defined by:
|B 0 L〉 ∼ pB|B 0 〉 + qB| ¯ B 0 〉 (2.31)
|B 0 H〉 ∼ pB|B 0 〉 − qB| ¯ B 0 〉 , (2.32)
7 |∆F | expresses the flavor quantum number difference between the initial- and the final-state of
a transition, e. g. |∆S| for Strangeness and |∆B| for Bottomness.

10 Chapter 2. Theory
b u,c,t
d
0
B
W W
u,c,t
0
B
d
b
b
d
0
B
W
u,c,t u,c,t
Figure 2.3: Box diagram contribution to B 0 - ¯ B 0 mixing
where B0 L is the lighter, and B0 H is the heavier eigenstate. Analogous to the
Kaon system, the oscillation parameters pB and qB are complex and normalized
by |qB| + |pB| = 1.
The mass and decay width differences of the mass eigenstates are defined by convention
as:
∆mB ≡ MH − ML � 2|M12| (2.33)
∆ΓB ≡ ΓH − ΓL � 2 Re (M12Γ ∗ 12 )
|M12|
W
0
B
d
b
(2.34)
where ∆ΓB ≪ ∆mB is assumed. The exact ratio qB/pB is given by:
�
2 M
qB
= −
pB
∗ i
12 −
2 Γ∗ �
12
∆mB − i
2 ∆ΓB
. (2.35)
The physical meaningful quantity, that is independent of the phase convention, is:
�
� � �
� qB �2
�M
� � �
�pB
� = �
�
�
∗ i
12 −
2 Γ∗12 M12 − i
�
�
�
�
�
�
�
. (2.36)
2 Γ12
The phase of qB/pB is convention dependent and, hence, not observable. Similar
definitions can be made in the Bs-meson system by replacing only the d-quark by a
s-quark.
2.5 CP Violation
The CP transformation is a combination of charge conjugation C and parity P.
Under C transformation, a particle transforms into its anti-particle, by conjugating
all internal quantum numbers, e. g. Q → −Q for the electromagnetic charge. P
transformation reflects the space coordinate �x into −�x. The combination of both
transforms a left-handed particle into its right-handed anti-particle, e. g. e −
L
→ e+
R .

2.5. CP Violation 11
Within the Standard Model with three families, CP violation can be generated by
a single, non-vanishing phase (¯η �= 0 ⇔ J �= 0) in the CKM matrix. It is related
to flavor changing charged currents of weak interactions and was discovered in the
neutral Kaon system in 1964 [15]. The effects of CP violation relevant for the CKM
matrix analysis can be classified into three different types [9]:
CP violation in decay results from the interference among different decay amplitudes
of a meson M into a multi-particle final state f and its CP conjugated
decay ¯ M → ¯ f:
Af = 〈f|H|M〉 Ā ¯ f = 〈 ¯ f|H| ¯ M〉 . (2.37)
The CP symmetry is violated, if:
�
�
�
�
�
Ā ¯ f
Af
�
�
�
� �= 1 . (2.38)
�
This type of CP violation occurs in charged and neutral meson decays, but is
the only possible source of CP asymmetries in charged meson decays. The CP
asymmetry is defined by:
A f ± ≡ Γ (M − → f − ) − Γ (M + → f + )
Γ (M − → f − ) + Γ (M + → f + ) =
for charged mesons and:
for neutral mesons.
Af ≡ Γ � ¯ M 0 → ¯ f � − Γ � M 0 → f �
Γ � ¯ M 0 → ¯ f � + Γ (M → f) =
� �
�Āf −/Af + �2 − 1
� �
�Āf −/Af + �2 + 1
� �
�Āf ¯/Af �2 − 1
� �
�Āf ¯/Af �2 + 1
(2.39)
(2.40)
CP violation in mixing of neutral mesons results from the fact that mass eigenstates
are different from the CP eigenstates leading to:
� �
�
�
q �
�
�p
� �= 1 , (2.41)
where q and p are the complex parameters of the neutral meson mixing (see
Section 2.4). They can be measured via the asymmetry of semileptonic neutral
meson decays induced by oscillation:
�
Γ ¯M 0
phys (t) → l
ASL ≡
+ � �
X − Γ M 0 phys (t) → l− �
X
�
Γ ¯M 0
phys (t) → l + � �
X + Γ M 0 phys (t) → l− �
X
� �
�
1 − �
q �4
�
�p
�
= � �
�
1 + �
q �4
. (2.42)
�
�p
�

12 Chapter 2. Theory
CP violation in the interference between decays with and without mixing
occurs from:
Im (λfCP ) �= 0 , (2.43)
where λfCP
is the product of q/p and the ratio of the decay amplitudes of a
meson M 0 and its anti-particle ¯ M 0 into the same final CP eigenstate fCP with
CP eigenvalue ηfCP , given as:
λfCP
q
≡
p
ĀfCP
AfCP
q ĀfCP ¯
= ηfCP
p
AfCP
. (2.44)
The amplitudes Ā ¯ fCP and AfCP differ only in the signs of the weak phase for
each term, while ηfCP = ±1.
CP violation in the interference between decays with and without mixing can
be measured via the asymmetry of neutral meson decays into final CP eigenstates
fCP :
AfCP ≡
� � �
Γ ¯M 0
phys (t) → fCP − Γ M 0 �
phys (t) → fCP
� � �
Γ ¯M 0
phys (t) → fCP + Γ M 0 � . (2.45)
phys (t) → fCP

Chapter 3
CKMfitter
The CKMfitter group is an international group of experimental and theoretical particle
physicists with collaborators from the high energy physics experiments ATLAS,
BABAR, Belle and LHCb. Its goal is a global analysis of the CKM matrix, which
contains:
1. Probing the consistency between the SM theory predictions and the experimental
data.
2. Constraining the CKM matrix and QCD model parameters entering the SM
theory predictions.
3. Predicting observables from the Standard Global CKM Fit.
4. Searching for specific signs of New Physics in an extended theoretical framework
and constraining New Physics parameters.
More detailed information is provided in Ref. [4] and on the CKMfitter website [16].
3.1 The Statistical Framework - Rfit
The statistical analysis performed in CKMfitter is entirely based on the frequentist
approach Range Fit (Rfit) [17]. The experimental input information is a set
of Nexp measurements xexp = {xexp(1), . . . , xexp(Nexp)} described by a set of corresponding
theoretical expressions xtheo = {xtheo(1), . . . , xtheo(Nexp)}. The theoretical
expressions are model-dependent functions of a set of Nmod parameters
ymod = {ymod(1), . . . , ymod(Nmod)}. A subset of Ntheo parameters within these ymod
set are considered as being fundamental and free parameters of the theory model,
e. g. the four Wolfenstein parameters in the SM or the top quark mass. These theory
parameters are denoted as ytheo = {ytheo(1), . . . , ytheo(Ntheo)}. The remaining
NQCD = Nmod − Ntheo parameters, which appear due to our present inability to
compute strong interaction quantities precisely, e. g. fBd , Bd, . . . , are denoted as
yQCD = {yQCD(1), . . . , yQCD(NQCD)}.
13

14 Chapter 3. CKMfitter
The quantity minimized in the fit is
χ 2 = −2 ln L(ymod) , (3.1)
with the likelihood function L(ymod), defined by a product of contributions of two
types:
L(ymod) = Lexp(xexp − xtheo(ymod)) · Ltheo(yQCD) . (3.2)
The experimental likelihood Lexp depends on the experimental measurements xexp,
which are gaussian distributed in general, and their theoretical predictions xtheo,
which are functions of the model parameters ymod. The theoretical likelihood Ltheo
describes the knowledge on the QCD parameters yQCD ∈ {ymod}, where the theoretical
uncertainties σsyst are considered to define allowed ranges:
[yQCD − σsyst, yQCD + σsyst] . (3.3)
In the Rfit scheme, the theoretical likelihoods Ltheo(i) do not contribute to the χ 2
of the fit, as long as the yQCD take on values within their.
3.2 Fit Metrology
is determined with
In a first step, the global minimum of Equation (3.1), χ2 min,global
respect to all Nmod parameters. Due to the experimental and theoretical systematics,
this absolute minimal value does in general not correspond to a unique ymod location.
In a second step, a selected subspace of interest of the parameter space, e. g. a =
{¯ρ, ¯η} is scanned, to determin the local χ2-minimum χ2 min,local (a) for each fixed point
of a grid in the parameter space a, with respect to the remaining parameters. The
offset-corrected χ2 is calculated as follows:
∆χ 2 (a) = χ 2 min,local (a) − χ2 min,global
where its minimum is equal to zero by construction.
, (3.4)
Finally, a confidence level (CL) for a is obtained using the well-known PROB function
from the CERN Program Library [18]:
1 − CL = P rob � ∆χ 2 �
(a), Ndof
=
which assumes gaussian statistics.
1
√ 2 Ndof Γ (Ndof /2)
� ∞
χ 2 (ymod)
(3.5)
e −t/2 t Ndof /2−1 dt , (3.6)

3.3. The CKMfitter Package Code 15
3.3 The CKMfitter Package Code
The source code of the CKMfitter package consists of more than 40000 lines FOR-
TRAN code and ca. 2000 lines C++ code. It is public available on the CKMfitter
website. The minimization routine used is MINUIT, the minimizer from the CERN
Program Library [18]. The analysis is driven by datacards, where running options
and input values for the fit are specified. The output file of the procedure is written
in HBOOK format, which is used by PAW 1 macros for producing one- and twodimensional
plots of the fit results.
Over the years, the fit problems became more and more complex. There are difficult,
non-linear fit problems which contain mirror solutions from trigonometric functions.
This has been led to an increasing CPU time consumption and the time for a single
complex fit can last longer than one day.
The main reason for this is the so-called dictionary, a file, which contains hundreds
of theory predictions for the fit variables. Since they are statically loaded, the
dictionary needs to be browsed in each fit step to obtain the theory prediction needed
in the specific fit. As the dictionary file is a very important part of the CKMfitter
source code, a runtime upgrade is difficult to realize in the original source code.
1 The Physics Analysis Workstation (PAW) is a data analysis and presentation tool. [19]

16 Chapter 3. CKMfitter

Chapter 4
A Mathematica based Version
of the CKMfitter Package
With the goal of a significant fit time reduction, Jérôme Charles initiated a
Mathematica based version of the CKMfitter package. The basic concept is the usage
of symbolic calculations of the fit expressions in a fit preparation phase within a
Mathematica environment, which interacts with a FORTRAN based minimization
routine. An analysis is driven by datacards in ASCII format and the results are
provided as data files and colored reference plots, done by Mathematica. The final
combined plots are currently made by ROOT 1 macros using the exported data files.
The ROOT macros have been coded by Vincent Tisserand.
This thesis makes contributions to the source code development of the Mathematica
based CKMfitter package. Most important are the implementation of the theories
of neutral meson oscillations and leptonic decays of charged B mesons in the SM
and in extended framework. Furthermore, the treatment of look-up-table input files
has been developed. Since the source code development is still under progress, this
work displays the current status of August, 2006.
4.1 Mathematica
The software package Mathematica is a computer algebra system, which facilitates
symbolic math calculations. It provides also a programming language based on termrewriting
2 . Mathematica is a proprietary product of Wolfram Research, Inc. [21].
The most recent version, which has also been used in this thesis, is Mathematica 5.2,
released in July, 2005.
1 ROOT is an object-oriented data analysis framework based on C++ [20].
2 Term-rewriting covers several methods of replacing subterms of a formal formula by other,
e. g. simpler terms.
17

18 Chapter 4. A Mathematica based Version of the CKMfitter Package
The core of Mathematica is the kernel, which interactively performs the calculation.
The user interacts with the kernel via a front-end, where a graphical version is available
as well as a commandline interface.
Since the terminology of Mathematica is used in this work, Appendix E.1 gives a
short overview about relevant terms and commands. A more detailed documentation
is provided on the Mathematica website [22].
4.2 Package Structure
The Mathematica based version of the CKMfitter package is modularly composed of
several Mathematica notebooks and packages. Furthermore, there are FORTRAN
files from the minimization routine and datacards in ASCII format. In Table 4.1,
the content of the most relevant directories is shown.
CKMfitter Directory File Content
analysis/ datacards
fortran/ minimir.f, fit.f, dmnfg.f, specialfunctions.f
inputs/ input datacards, χ 2 -input tables, PDG.m
lib/ AnalysisLib.m, FitLib.m
theories/ theory packages, TheoryTutorial.nb
tools/ CreateInputTable.nb
CKMfitter.nb
Table 4.1: Directory structure of the Mathematica based CKMfitter package
The user interface is the CKMfitter.nb notebook. It is located in the main directory,
but should be copied to another analysis specific directory, outside of the
CKMfitter package. For the sake of clarity, complex functions and subroutines are
sourced out from the user interface to library packages. The AnalysisLib package
contains all relevant subroutines for the data processing inside the Mathematica
environment, e. g. subroutines to load datacards and theory packages. The second
library package is named FitLib and includes all functions for the interaction of
CKMfitter.nb with the FORTRAN minimization routine. It provides for example
a subroutine, which translates the Mathematica expressions to FORTRAN code.
As in the original CKMfitter package, a fit is userfriendly driven by datacards. Since
these are ASCII text files, they can be edited without using Mathematica in a plain
text editor. In a datacard, all flags and options of a selected analysis are set, e. g. the
context name, the fit variables and the scan granularity. A more detailed description
of the relevant fit options and settings is given in Appendix E.2.

4.3. CKMfitter.nb 19
An important flag in the datacards is the specification of the input files. These files
contain the numerical input for each measurement xexp and model parameter ymod.
The values are provided in the “inputs” list, where each element is again a list, one
per measurement or model parameter. According to the different error types of a
measurement, there exist different input types, e. g. “Range” or “GaussRange”. It
is also possible to specify the name of a file, which contains a χ 2 -contour of a measurement
xexp. The different input types are distinguished using explicit syntaxes
in the list. A brief description of the syntax for each possible input type is provided
in Appendix E.3.
Furthermore, the theoretical expressions xtheo depend on fixed inputs, e. g. physical
constants. They are listed in a separate package, called PDG, which is also located
in the inputs directory. Its content is the “fixedinputs” list, where each entry is a
“rule”, which replaces a symbol through its numerical value, e. g. mB → 5.2794 for
the B-meson mass.
4.3 CKMfitter.nb
The CKMfitter analysis notebook is the user front-end of the Mathematica based
version. It sources all relevant information from datacards and theory packages,
translates the χ 2 -function in FORTRAN format, runs the minimization and exports
the fit results. In addition, the CKMfitter notebook provides several checks and the
opportunity to monitor the fit process.
Running a fit, the user only needs to select an analysis datacard, where all job
options are set before. Afterwards the analysis is executed step by step as shown
in Figure 4.1. In the first part, all relevant informations are loaded, e. g. library
packages and physical constants. After loading the theory packages and input files
as specified in the datacard, the χ 2 -function and their partial derivatives are symbolically
calculated during the fit preparation. These expressions are transformed
in FORTRAN code format using the Mathematica intrinsic function FortranForm
and written to the file minimirChi2.f. This file is compiled and linked to the minimization
subroutines and the executable minimir is built. The fit starts through
running minimir, which performes the global minimization as well as the one- and
two-dimensional scans. The fit results are written to the file minimir.output and
then loaded from CKMfitter.nb, where the fit results are plotted. Finally, plots
and data files are exported.
Remark: To interrupt the fit process, there is the possibility to quit the Mathematica
kernel. Since minimir is a stand alone executable, it needs to be aborted separately
by the user!

20 Chapter 4. A Mathematica based Version of the CKMfitter Package
Datacard specification
Fit preparation
Source library packages
Load physical constants
Read datacard
Load theory packages
Load numerical inputs
Write χ 2 -function (minimirChi2.f )
Compile
Run minimir
Write
Read Fit results (minimir.output)
Plot results Export Plot and data file
Figure 4.1: The fit process

4.4. The Minimization Routine 21
4.4 The Minimization Routine
Since FORTRAN is more optimized for numerical calculations, a FORTRAN based
minimization routine is used instead of a Mathematica intrinsic function. The main
file is minimir.f, which calls a set of subroutines and functions from the files fit.f,
dmnfg.f and specialfunctions.f. The minimized quantity χ 2 and its partial derivatives
with respect to all fit parameters are located in the file minimirChi2.f, which
is also linked to the executable minimir after the compilation process. The subroutine,
which manages the minimization process is called POINTFIT. It sets the
fit environment, e. g. the starting point and calls the real minimization subroutine
DMNG, which proceeds the unconstrained minimization of the χ 2 -function using their
exact analytic gradients. The subroutine DMNG was coded by David M. Gay using
a quasi-Newton method and is publicly available at NetLib.org [23]. Further informations
can be found in Ref. [24].
Non-standard FORTRAN functions and subroutines, which belong to the analytic
part, e. g. contributions to the χ 2 -function, are the content of the file specialfunctions.f.
Furthermore, it contains the function TABLEAU, which allows the usage of
look-up tables (LUT) as input files, and the function DDILOG from the CERN Library
Package [18] to calculate di-logarithmic functions.
minimirChi2.f
minimir.f
minimir.input
minimir.aux
fit.f Compile minimir
dmnfg.f
specialfunctions.f minimir.output
Figure 4.2: The minimization routine file system
A schema of the file system of the FORTRAN based minimization routine is shown in
Figure 4.4. The FORTRAN files are compiled and linked together to the executable
minimir. The input information for the fit routine is provided by two files in ASCII
format, which are created by the analysis notebook. The file minimir.input contains
the numerical input of the fit variables and minimir.aux provides the fit options,
e. g. the scan direction. The fit result is finally written to the file minimir.output,
which is read from the analysis notebook.

22 Chapter 4. A Mathematica based Version of the CKMfitter Package
4.5 Theory Packages
Theory packages provide the full physical information on the fit variables. Since no
global variables are used in Mathematica, all variables are initialized only in the context
of their theory package. For each observable, the theoretical expression xtheo
is defined and the partial derivatives ∂xtheo/∂ymod with respect to all model parameters
ymod are symbolically calculated. Finally, all expressions are written into
a list named “theory”, which is loaded from the analysis notebook. Theory packages
are stored in a binary format (.mx), that is optimized for input by Mathematica.
It is possible to define an observable in different theoretical frameworks, e. g. the B 0 -
¯B 0 oscillations in the Standard Model or in a New Physics model (see Chapter 6).
Thus, a theory package can include several theory lists, one for each theoretical
framework or possible different parametrization. Different theory lists in the same
theory package are distinguished by version labels, which have to be specified in the
datacard.
It is advantageous for the fit to store all expressions and derivatives in their simplest
form. This can be obtained using the intrinsic Mathematica functions Simplify and
FullSimplify. For complex expressions, this may lead to a very high CPU time
consumption during the theory package development. More detailed information on
the development of theory packages are given in a tutorial, which accrued during
this work. It is available in Appendix E.4.
In the following, four important theory packages, which are related to this work, are
introduced.
4.5.1 CKMmatrix
The theory package CKMmatrix is the basis of the Mathematica based CKMfitter
package and was coded by Jérôme Charles. It contains all relevant definitions
of CKM matrix and Unitarity Triangle parameters in the exact Wolfenstein
parametrization, e. g. the real and imaginary parts of the CKM matrix elements,
Re (Vij) and Im (Vij), and the UT angles α, β, γ.
An important object is the wolfCKM function. This is a list of rules and provides
the real and imaginary parts of all CKM matrix elements Re (Vij) and Im (Vij).
Its argument is the order of the expansion into Wolfenstein parameters, where “∞”
means the exact expressions up to all orders.
The CKMmatrix package needs to be sourced from each theory package, where the
Wolfenstein parameters are used or the wolfCKM function is evaluated.

4.5. Theory Packages 23
4.5.2 BBbarKKbarMixing
The theory of measurements describing the oscillations in neutral K- and B-meson
systems are provided in the BBbarKKbarMixing theory package. It contains all
theoretical expressions and partial derivatives for the observables:
∆md , ∆ms , |ɛK| and ASL,Bd (4.1)
in two theoretical frameworks. The Standard Model definitions are shown in Chapter
5. The second theoretical framework is a model independent extension of the
SM, which is discussed in Chapter 6. The version labels, which needs to be specified
in the datacard are given in Table 4.2.
framework version label
Standard Model ”SM”
New Physics (model independent) ”NP(r,theta)”
Table 4.2: Version labels in the theory package BBbarKKbarMixing
In addition to the observables in Equation (4.1):
sin(2β + 2ϑd) , cos(2β + 2ϑd) , | sin(2β + 2ϑd + γ)| and αNP , (4.2)
are defined in the New Physics framework, where αNP = π − β − γ − ϑd.
4.5.3 LeptonicDecay
The theory package LeptonicDecay contains currently the branching fractions of
charged B mesons into purely leptonic final states:
B � B + → e + �
νe
, B � B + → µ + �
νµ
and B � B + → τ + �
ντ . (4.3)
They are described in the Standard Model framework, given in Chapter 5, as well
as in Two-Higgs-Doublet Models, which are described in Chapter 6. Table 4.3 lists
the version labels for both frameworks.
framework version label
Standard Model ”SM”
New Physics (charged Higgs) ”NP(H+)”
Table 4.3: Version labels in the theory package LeptonicDecay

24 Chapter 4. A Mathematica based Version of the CKMfitter Package
4.5.4 DecayBagParameters
The Mathematica based version of CKMfitter avoids the usage of global variables.
However, there are some model parameters, e. g. the decay constants and bag parameters
of Bd- and Bs-mesons:
fBs ,
fBs
fBd
, Bs , Bs
Bd
, (4.4)
which are used in different theory packages. Thus, they need to be “globalized”,
through initializing them in a separate package, which will be sourced from the other
theory packages. The DecayBagParameters package is currently loaded from the
BBbarKKbarMixing package as well as from the LeptonicDecay package.
4.6 Look-Up Tables
Since there is not always the possibility to express a measurement in terms of a value
and an error, the experimental likelihood, translated into a χ 2 -contour, can also be
used as an input. It is available in a discrete look-up table, where values in-between
the discrete LUT entries are calculated through interpolation. The Mathematica
based CKMfitter version provides the possibility of a cubic spline interpolation and
uses a separate FORTRAN based function to load the LUTs during the fit.
4.6.1 Cubic Spline Interpolation
Piecewise interpolation using low order polynomials leads to a global continuous
interpolating function, but is in general not continuous at the interval boundaries.
This is avoided by using cubic spline interpolation, which gives back a smooth interpolating
function. The goal of cubic spline interpolation is to get an interpolation
formula that is smooth in the first, and continuous in the second derivative, both
within an interval and at its boundaries.
A cubic spline interpolation function s(x) to the sampling points x0 < x1 < . . . <
xn−1 < xn and the corresponding sampling values yj (j = 0, 1, 2, . . . , n) is defined
by [25]:
• s (xj) = yj for (j = 0, 1, . . . , n);
• s (x) is for x ∈ [xi, xi+1] (i = 0, . . . , n − 1) a polynomial of at most of order 3;
• s (x) ∈ C 2 ([x0, xn]);
• s ′′ (x0) = s ′′ (xn) = 0.

4.6. Look-Up Tables 25
In the Mathematica based CKMfitter version, polynomials of third order
y(x) = a + bx + cx 2 + dx 3
(4.5)
are used as cubic spline interpolation functions, where a, b, c and d are the spline
coefficients. They are calculated in a separate notebook using a public available
routine coded by Joseph M. Herrmann [26]. Its result is a six column look-up
table in ASCII format, which is used as input file for the fit. The column order of
the LUT is given in Table 4.4 and the number of lines is written in the first row.
There is also an intrinsic Mathematica function called SplineFit, which generates
a spline function object. Due to the fact that the generated spline function doesn’t
depend directly on the observable values, it cannot be used in CKMfitter to create
the LUTs.
Column: 1 2 3 4 5 6
Content: value of observable χ 2 a b c d
4.6.2 The Tableau Function
Table 4.4: LUT column order
The numerical contribution to the χ 2 -function of variables with LUT input is calculated
by separate FORTRAN functions in the file specialfunctions.f. The function
TABLEAU provides the χ 2 -contribution, where DTABLEAUO2 calculates the contributions
to its gradients 3 . During the first call of TABLEAU, the different LUTs are
loaded using the separate subroutine LoadLUT.
The functions TABLEAU and DTABLEAUO2 are called during the fit for each prediction
value of the fit parameter. Using the spline coefficients of the next lower entry in the
LUT, the contribution to the χ 2 -function or its gradient is calculated. The source
code of these functions is available in Appendix D.
It is possible that the predicted value of the fit lies outside of the LUT boundaries.
In this case, the TABLEAU function will be constantly continued using the first (last)
value of the LUT, if the prediction is smaller (larger) than the lower (upper) bound.
The gradient, obtained by DTABLEAUO2 is set to zero. This can lead to critical effects
at the boundaries. A possibility to avoid such critical effects is to enlarge the LUTs
periodically, if this is allowed by the variable. Examples for these cases are the UT
angles α and γ (see Section 5.1).
3 DTABLEAUO2 (“O2” means over two) calculates the half of the gradient contribution.

26 Chapter 4. A Mathematica based Version of the CKMfitter Package
4.7 Performance Tests
The Mathematica based version of the CKMfitter package was created with the goal
to achieve a significant fit time reduction compared to the original version. Since
both are complex packages, which include also fit preparation and result processing,
an objective comparison of the pure minimization routines is beyond the scope of this
thesis. However, there is the possibility to compare the full analysis process using
a test job, here the Standard Global CKM Fit (see Chapter 5). The conditions, for
the comparison tests done in this work, are summarized in Table 4.5.
Hardware/Software Test job
CPU: Intel P III Analysis: SM global fit
Frequency: 1266 MHz Scan: (¯ρ,¯η) plane
Memory: 2048 MB RAM Granularity: 200
OS: Scientific Linux 3.0.3 fits per point: 2
Compiler: gnu f77 -O
Table 4.5: Test conditions
Figure 4.3 shows the results of the Standard Global CKM Fit and its single constraints
in the (¯ρ,¯η) plane, produced by both versions of the CKMfitter package.
The results presented numerically in Table 4.6 and graphically in Figure 4.3 are
nearly identical.
Parameter Original CKMfitter Mathematica based package
A 0.2272 +0.0010
−0.0010
λ 0.812 +0.015
−0.015
¯ρ 0.187 +0.025
−0.086
¯η 0.333 +0.038
−0.017
J [10−5 ] 3.02 +0.36
−0.17
0.2272 +0.0010
−0.0010
0.813 +0.015
−0.015
0.187 +0.028
−0.086
0.333 +0.038
−0.017
3.02 +0.36
−0.18
Table 4.6: Numerical comparison of the Wolfenstein parameters A, λ, ¯ρ, ¯η and the Jarlskog
invariant J. The errors are quoted as 1-CL=32 % ranges (1σ).
Possible reasons for the very small discrepancies could be the usage of the different
parametrizations of the Lattice QCD parameters, as explained in Section 5.1.12, the
different coded interplation routines for the LUTs or rounding effects value from the
determination of the central value.

4.7. Performance Tests 27
η
η
0.7
0.6
0.5
0.4
0.3
0.2
excluded area has CL > 0.95
sin 2β
Δm d
ε K
Δm s & Δm d
α
luded luded at at CL CL > > 0.95 0.95
sol. w/ cos 2β < 0
(excl. at CL > 0.95)
0.1
0
|Vub /Vcb |
γ
β
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0.6
0.5
0.4
0.3
0.2
0.1
excluded area has CL > 0.95
γ
sin2β
εK
Δmd
γ
α
ρ
Δms
& Δmd
ρ
α
C K M
f i t t e r
ICHEP 2006
α
Vub/Vcb
0
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
εK
sol. w/ cos2β
< 0
(excl. at CL > 0.95)
β
CKM
γ
f i t t e r
ICHEP 2006
Figure 4.3: Confidence level in the (¯ρ,¯η) plane for the Standard Global CKM Fit. The
plots are produced with the original (top) and the Mathematica based (bottom) CKMfitter
package.
α
γ

28 Chapter 4. A Mathematica based Version of the CKMfitter Package
The test job, used for the fit time comparison, is the Standard Global CKM Fit. Its
Mathematica datacard is shown in detail in Appendix C and the result is the yellow
area around the apex of the Unitarity Triangle in Figure 4.3. The fit time needed
using the original CKMfitter package is about 23 hours, whereas the same result
can be obtained by the Mathematica based version in only ten minutes. This is a
reduction of the fit time of more than a factor 100 for the Standard Global CKM
Fit.
Original CKMfitter package ≈ 23 h
Mathematica based CKMfitter ≈ 10 min
Table 4.7: Fit time comparison of test job runs
Since the fit time depends on the hard- and software conditions, the performance of
the Mathematica based CKMfitter package is tested for several configurations. The
results for the test job are summarized in Table 4.8.
Hardware MHz Compiler Operation System Fit time / min
AMD Opteron 2194 f77 -O Scientific Linux 3.0.7 04 : 42
Intel P M 1497 ifort -O2 SUSE Linux 10.0 05 : 25
Intel P M 1497 f77 -O SUSE Linux 10.0 07 : 06
Intel P III 1266 ifort -O2 Scientific Linux 3.0.3 09 : 10
Intel P III 1266 f77 -O Scientific Linux 3.0.3 10 : 18
Table 4.8: Hardware and software performance tests
The most important conclusion is, that the fit results are independent of the used
hard- and software conditions. Using a high developed hardware system in combination
with an optimized FORTRAN compiler leads to a further fit time reduction.
With a fit time reduction of more than a factor 100, the goal of the Mathematica
based CKMfitter development has been achieved.

Chapter 5
Probing the Standard Model
After a brief discussion of the relevant fit inputs, the most important results of the
global CKM matrix analysis in the framework of the Standard Model are presented.
5.1 Fit Inputs
In this section, the SM predictions and measurement methods of the most relevant
observables are described. If not stated otherwise, the expressions given are used
in the Mathematica based version of the CKMfitter package. The input values are
quoted in form of a central value and its uncertainties, where contributions of gaussian
distributed experimental, statistical and theoretical uncertainties are quadratically
added and labeled with “gauss”. Theoretical systematics, labeled with “theo”,
are added linearly and treated in the Rfit scheme. In the case of constraints with
ambiguous solutions (as for the UT angles α and γ), the experimental likelihood
function is directly used as fit input.
Since a complete review of the input measurements is beyond the scope of this thesis,
more detailed informations can be obtained from the References [4, 8, 9] and references
therein. The values of the fit inputs are the most recent results as presented at
ICHEP 2006 and are summarized in Table 5.1. The final plots have been prepared
by Vincent Tisserand using the macros as described in Chapter 4.
5.1.1 |Vud|
The most precise determination of |Vud| stems from superallowed nuclear beta decays.
Taking the average of the most precise experimental results yields [27]:
|Vud| = 0.97377 ± 0.00027gauss . (5.1)
Superallowed nuclear beta decays are pure weak vector transitions (0 + → 0 + ). Thus,
the extraction of |Vud| is theoretically very clean, since theoretical uncertainties
in electroweak radiative corrections, isospin violating electromagnetic effects and
nuclear structure dependence are under good theoretical control.
29

30 Chapter 5. Probing the Standard Model
5.1.2 |Vus|
The matrix element |Vus| has been extracted from semileptonic Kaon decays (Kl3),
e. g. K + → π 0 e + νe. Its current world average (WA) is [9]:
|Vus|Kl3 = 0.2257 ± 0.0021gauss . (5.2)
|Vus| as well as |Vud| are the main inputs to constrain the Wolfenstein parameter λ.
However, |Vus| depends on large theoretical uncertainties from the determination of
the form factor f+(0). Other possibilities to determine |Vus| involve for instance
leptonic Kaon and Pion decays, semileptonic Hyperon decays or hadronic τ decays.
5.1.3 |Vcb|
The magnitude of |Vcb| is determined from exclusive and inclusive measurements 1
of semileptonic B-meson decays to charmed final states (b → clν). Because of theoretical
uncertainties on the form factor calculation, the exclusive determination is
less precise compared to the inclusive one and thus not used in this work. Due to
the large statistics, the inclusive WA [9]:
|Vcb|incl = (41.7 ± 0.7gauss) · 10 −3
(5.3)
is already below the 2% level. It is the main input to constrain the Wolfenstein
parameter A.
5.1.4 |Vub|
Analogous to |Vcb|, the CKM matrix element |Vub| is determined from exclusive and
inclusive measurements of b → ulν transitions. The inclusive determination suffers
from large B → Xclν background and uses therefore phase space regions where the
charm background is kinematically suppressed. Hadronic effects enter in leading
order (LO) via one non-perturbative shape function, which has been extracted from
the photon energy spectrum in B → Xsγ. For the inclusive WA, the BLNP [28]
calculation is chosen [29] 2 :
|Vub| incl. = (4.48 ± 0.24gauss ± 0.39theo) · 10 −3 , (5.4)
The theoretical error is obtained by adding linearly the contributions from weak
annihilation, subleading shape functions and the Heavy-Quark Expansion (HQE)
uncertainty on the b-quark mass.
Direct measurements of exclusive channels depend significantly on form factor calculations.
The form factors can be calculated using unquenched Lattice QCD (LQCD)
or QCD Sum Rules, which still leads to larger theoretical uncertainties compared to
the inclusive determination.
1 Exclusive means the full reconstruction of a single mode, e. g. B + → ¯ D 0 e + νe, where inclusive
measurements integrate over all possible final states, B → Xclν.
2 The central value has been shifted after the ICHEP06 conference to 4.49 · 10 −3 .

5.1. Fit Inputs 31
1 – CL
1.2
1
0.8
0.6
0.4
0.2
C K M
f i t t e r
ICHEP 06
B → ππ
B → ρπ
B → ρρ
Combined
CKM fit
0
0 20 40 60 80 100 120 140 160 180
α (deg)
WA
1 – CL
1.2
1
0.8
0.6
0.4
0.2
C K M
f i t t e r
FPCP 06
Full frequentist treatment on MC basis
CKM fit
no γ meas. in fit
D ( * ) K ( * ) GLW + ADS
D ( * ) K ( * ) WA
GGSZ Combined
0
0 20 40 60 80 100 120 140 160 180
γ (deg)
Figure 5.1: Left: Confidence level on the UT angle α obtained from the combination
(green shaded) of B → ππ (green curve), B → ρπ (red curve) and B → ρρ (blue curve).
Right: Confidence level on the UT angle γ obtained from the combination (green shaded)
of the GLW, ADS and GGSZ methods (see text).
5.1.5 The UT angle α
The direct constraint on the UT angle α is obtained from charmless B decays.
The best current knowledge comes from a combination of the two body isospin
analysis [10] of B → ππ and B → ρρ decays and the Dalitz-plot analysis [10]
of B → ρπ decays. Figure 5.1 (left) shows the confidence level on α from the
combined constraint of these three measurements. The isospin analysis as well as
the α extraction from B → ρπ have been performed by the CKMfitter group using
the most recent results of BABAR and Belle. Due to its non-gaussian shape, the
χ 2 -contour is used as a LUT input file as described in Chapter 4.
5.1.6 sin 2β
The currently best constraint on (¯ρ,¯η) comes from determination of the CP-violating
parameter sin 2β. It is measured with high precision at the B-factories from the
interference between decays with and without mixing in b → c¯cs transitions. The
current world average is obtained by combining the measurements of B → (c¯c)K 0
(BABAR) and B → J/ψKS,L (Belle) and yields [30]:
sin 2β = 0.675 ± 0.026gauss . (5.5)

32 Chapter 5. Probing the Standard Model
5.1.7 The UT angle γ
The extraction of the UT angle γ stems from measurements of direct CP violation
in B → D (∗) K (∗) decays. It is obtained from a combination of the Gronau-
London-Wyler (GLW) [31,32], Atwood-Dunietz-Soni (ADS) [33,34] and Giri-
Grossman-Soffer-Zupan (GGSZ) [35, 36] methods. Figure 5.1 shows the combined
result using a frequentist method, which is advocated by the CKMfitter
group [16]. Analogous to the UT angle α the corresponding χ 2 -contour is used
as a LUT input file (see Chapter 4).
5.1.8 |ɛK|
In the Standard Model, the absolute value of the CP-violating parameter in the
neutral Kaon system |ɛK| is defined by:
|ɛK| = G2 F m2 W mK
12 √ 2π2 f
∆mK
2 �
KBK
+ 2ηctS(xc, xt)Im[VcsV ∗ ∗
cdVtsVtd ]
ηccS(xc)Im[(VcsV ∗
cd )2 ] + ηttS(xt)Im[(VtsV ∗
td )2 ]
�
, (5.6)
where fK is the Kaon decay constant. The hadronic matrix element of the |∆S| = 2
box diagram is proportional to � √ �2, fK BK where BK is the bag parameter. It
has been obtained from Lattice QCD calculations and is the primary source of the
theoretical uncertainties for the prediction of |ɛK|. The parameters ηqiqj are nextto-leading
order (NLO) QCD corrections to the Inami-Lim functions S, which are
listed in Appendix A. The values of ηct and ηtt are shown in Table 5.2, while due to
large uncertainties, a parameterization [37–39] is used for ηcc. This is also shown in
Appendix A. The used average of |ɛK| is [40]:
5.1.9 ∆md
|ɛK| = (2.221 ± 0.008gauss) · 10 −3 . (5.7)
The B 0 - ¯ B 0 oscillation frequency is expressed through the mass difference ∆md between
the mass eigenstates BH and BL. It is predicted in the Standard Model as:
∆md = G2 F
6π 2 ηB mBd f 2 Bd Bd m 2 W S(xt) |VtdV ∗
tb |2 , (5.8)
where ηB is a perturbative QCD correction
√
to the Inami-Lim function. The hadronic
matrix element is proportional to fBd Bd, where the decay constant fBd and bag
parameter Bd are taken from LQCD (see also Section 5.1.12). ∆md has been measured
to high precision in many experiments. The WA is [29]:
∆md = 0.507 ± 0.004gauss . (5.9)

5.1. Fit Inputs 33
5.1.10 ∆ms
In analogy to ∆md, the mass difference of the two mass eigenstates in the neutral
Bs-system is predicted in the Standard Model as:
∆ms = G2 F
6π2 ηB mBsf 2 Bs Bs m 2 W S(xt) |VtsV ∗
tb |2 . (5.10)
The most recent experimental results, presented at the winter conferences 2006,
are a two-sided limit on ∆ms, determined by D∅ [41] and a measurement with a
significance of 99.5% by CDF [42]:
∆m CDF
s
= � 17.33 +0.42
−0.21
� −1
± 0.07syst ps . (5.11)
As input for the CKM matrix analysis, the χ 2 -contour obtained from the measured
amplitude spectrum of the B 0 s - ¯ B 0 s oscillation from both experiments is used in this
work [29].
5.1.11 The Branching Fraction B(B + → τ + ντ)
Another constraint on the CKM matrix element |Vub| comes from purely leptonic
decays of charged B mesons. The tree-level process is mediated in the Standard
Model by annihilation of the charged B meson into a pure leptonic final state via
a virtual W boson as shown in Figure 5.2. Its branching fraction B is predicted to
be [43]:
B(B + → l + νl) = G2 F mBm 2 l
8π
where the decay constant fBd
f 2 �
2
Bd
|Vub| 1 − m2 l
m2 �2
B
, (5.12)
is taken from Lattice QCD calculations (see also
5.1.12). Due to the smallness of |Vub| 2 and an additional helicity suppression proportional
to m2 l , the SM expectation is rather small. Thanks to the successful
running of the B-factories, the decay B + → τ + ντ is now in reach of the experimental
sensitivity. The fit input is a combination of the most recent experimental
likelihoods from BABAR [44] and Belle [45], where the systematics of the BABAR
result have been neglected.
b
u
+
B
+
W
Figure 5.2: Tree-level contributions to leptonic B + decays in the SM
Since several models predict New Physics contributions to this tree-level process,
purely leptonic decays play also an important role on testing extensions of the Standard
Model. A model which predicts additional contributions from charged Higgs
bosons is discussed in Chapter 6.2.
+
l
νl

34 Chapter 5. Probing the Standard Model
5.1.12 Decay Constants and Bag Parameters
The observables ∆md, ∆ms and B(B + → l + νl) depend on a set of decay constants
and bag parameters:
fBd , fBs , Bd and Bs . (5.13)
They can be calculated using Lattice QCD, which leads to significant theoretical uncertainties
for the predictions of the observables discussed in Chapters 5.1.9 to 5.1.11.
Since some combinations of these parameters can be calculated more precisely, the
parameters:
fBd , Bd and ξ = fBs
√
Bs
√
fBd Bd
, (5.14)
have been used in the original CKMfitter package. Unfortunately, the LQCD calculation
of fBd has large theoretical uncertainties, originating from the chiral extrapolation
of the light lattice quark masses (mu,md) to the physical quark masses.
Hence, the theoretical uncertainties of fBd and ξ are anti-correlated. This is not the
case in the different but mathematically equivalent parametrization:
fBs ,
fBs
fBd
, Bs and Bs
Bd
, (5.15)
which is used in the Mathematica based version of the CKMfitter package. It is
free from the large correlation due to the uncertainty from the chiral extrapolation.
However, possible smaller correlations are neglected since their correlation matrix
has not been published.
5.1.13 ASL
Measurements of semileptonic B-meson decays, i.e. B → Xlν, allow the determination
of the magnitude of CP violation in neutral B-meson mixing.
The flavor specific CP asymmetry ASL as defined in Equation (2.42) is related to
the mixing matrix elements in the SM via:
� �
Γ12
ASL = Im . (5.16)
Since the absolute value of Γ12/M12 is suppressed to the percent level through
� � �
� Γ12 �
� �
m2 b
�M12
� = O
m2 �
, (5.17)
W
ASL is expected to be rather small. It is additionally suppressed by another order
of magnitude through the GIM3 factor z = m2 c/m2 b � 0.1. In the presence of New
3 In the SM, the Glashow-Iliopoulos-Maiani (GIM) mechanism leads to a cancellation of
FCNC processes at tree level.
M12

5.1. Fit Inputs 35
Physics the GIM suppression may be diminished, thus a precise measurement of ASL
is an important constraint to New Physics models. Its input value is obtained from
a weighted average of measurements by CLEO [46], BABAR [47,48] and Belle [49]:
ASL = −0.0005 ± 0.0055gauss . (5.18)
where di-leptonic modes as well as hadronic modes are taken into account. A measurement
by D∅ [50] has not been included since it has also contributions from
Bs-meson mixing.
Since the theoretical prediction on Γ12/M12 still depends on large QCD uncertainties,
it has been calculated beyond leading order in QCD.
Leading Order
The basic expression of Γ12/M12 is given in the SM through [51]:
� �SM Γ12
4πm
= −
M12
2 b
3m2 W ¯ηBS(xt)
�
5
8 (K2 − K1) B′
S
B +
�
K1 + K2
�
2
� �
K1 m2 B − m
+ − K2
2 2 b 1
B − 3z (K1
1 − ¯ρ − i¯η
+ K2)
m 2 b
(1 − ¯ρ) 2 + ¯η 2
�
,
(5.19)
where only contributions of O(z) and O(1/mb) are considered. The parameter ¯ηB
is the scale dependend QCD correction factor to the Inami-Lim functions S(xi) and
K1, K2 are linear combinations of Wilson coefficients.
The hadronic matrix elements of the local |∆B| = 2 contributions are parametrized
through B = B(mb) and B ′
S
= B′
S (mb):
〈 ¯ Bd|( ¯ bidi)V −A( ¯ bjdj)V −A|Bd〉 = 8
3 f 2 Bd m2 Bd B(mb) (5.20)
〈 ¯ Bd|( ¯ bidi)S−P ( ¯ bjdj)S−P |Bd〉 = − 5
3 f 2 Bd m2 Bd B′
S(mb) (5.21)
≡ − 5
3 f 2 Bd m2 Bd
m2 Bd
( ¯mb(mb) + ¯md(mb)) 2 BS(mb) .
The scale-dependence can be separated through the factor [52]:
bB(mb) = [αS(mb)] −6/23
�
1 + αS(mb)
�
5165
, (5.22)
4π 3174
which leads to the scale-independent parameters:
ηB = ¯ηB
bB(mb)
and Bd = B(mb) · bB(mb) . (5.23)
Assuming ¯md(mb) → 0, the parameter BS(mb) is used to parametrize the scalarpseudoscalar
hadronic matrix element instead of B ′
S (mb). It depends one the scale mb.
The input values of all parameters are summarized in Table 5.2.

36 Chapter 5. Probing the Standard Model
Next-to-Leading Order
In NLO, higher corrections in z and 1/mb are taken into account as well as penguin
contributions and QCD corrections. The most recent progress is summarized in
Reference [53] and references therein.
The expansion of Γ12/M12 up to the order of λ 2 u/λ 2 t is given by [53]:
Γ12
M12
= λ2 �
t
−Γ
M12
cc
12 + 2 (Γ uc
12 − Γ cc
12) λu
+ (2Γ
λt
uc
12 − Γ cc
12 − Γ uu
12 ) λ2u λ2 �
t
(5.24)
where λi = V ∗
id Vib, for i = u, c, t. The mixing matrix element M12 is predicted in the
SM as [54]:
λ 2 t
G 2 F
12π 2 mBd ηBB(mb)bB(mb)f 2 Bd m2 W S(xt) . (5.25)
The coefficients Γab 12 are expressed through [53]:
Γ ab
12 = G2 F m2 b
−
24π f 2 Bd MBd
� �
F ab (z) + P ab (z)
�
F ab
S (z) + P ab
�
5
S (z)
3 B′ S(mb)
�
8
3 B(mb)
�
, (5.26)
+ Γ ab
12,1/mb
where the short-distance coefficients F ab
(S) (z) contain the contributions from |∆B| = 1
(z) coefficients contain the contributions from penguin opera-
have been computed in Ref. [53], Γcc 12 is given in Ref. [55]
is derived from Γcc 12 taking the limit z → 0. In addition, they depend on
the Wilson coefficients C1, . . . , C6 and C8, which are also given in Table 5.2. The
contains the corrections of order 1/mb.
operators and the P ab
(S)
tors. The coefficients in Γ uc
12
and Γ uu
12
term Γ ab
12,1/mb
A comparison of the SM predictions of ASL at LO and NLO in QCD is shown in the
Section 5.2. The implementation of the NLO prediction for Γ12/M12 has finished
immediately before the end of this thesis and therefore not all cross checks have been
performed up to this point. As a consequence, the theory prediction for Γ12/M12 in
this thesis is used at LO to produce quantitative results if not stated otherwise.

5.1. Fit Inputs 37
Parameter Value ± Error(s) Reference
Errors
GS TH
|Vud| (nuclei) 0.97377 ± 0.00027 [27] ⋆ -
|Vus| (Kℓ3) 0.2257 ± 0.0021 [9] ⋆ -
|Vub| (incl.) (4.48 ± 0.24 ± 0.39) × 10 −3 [9] ⋆ ⋆
|Vcb| (incl.) (41.70 ± 0.70) × 10 −3 [29] ⋆ -
|εK| (2.221 ± 0.008) × 10 −3 [40] ⋆ -
∆md (0.507 ± 0.004) ps −1 [29] ⋆ -
ASL −0.0005 ± 0.0055 [46–49] ⋆ -
∆ms Amplitude spectrum+CDF -LogL [41, 42] ⋆ -
sin(2β) [c¯c] 0.675 ± 0.026 [30] ⋆ -
S +−
ππ −0.58 ± 0.09 [29] ⋆ -
C +−
ππ −0.39 ± 0.07 [29] ⋆ -
C 00
ππ −0.35 ± 0.33 [29] ⋆ -
Bππ all charges Inputs to isospin analysis [29] ⋆ -
S +−
ρρ,L −0.22 ± 0.22 [29] ⋆ -
C +−
ρρ,L −0.06 ± 0.14 [29] ⋆ -
Bρρ,L all charges Inputs to isospin analysis [29] ⋆ -
B 0 → (ρπ) 0 → 3π Time-dependent Dalitz analysis [56] ⋆ -
B − → D (∗) K (∗)− Inputs to GLW analysis [57] ⋆ -
B − → D (∗) K (∗)− Inputs to ADS analysis [57] ⋆ -
B − → D (∗) K (∗)− GGSZ Dalitz analysis [57] ⋆ -
B(B − → τ − ντ ) Experimental likelihoods [44, 45] ⋆ -
mc(mc) (1.24 ± 0.037 ± 0.095) GeV/c 2 [58] ⋆ ⋆
mt(mt) (162.3 ± 2.2) GeV/c 2 [59] ⋆ -
m K + (493.677 ± 0.016) MeV/c 2 [60] - -
∆mK (3.4833 ± 0.0066) × 10 −12 MeV/c 2 [60] - -
mBd (5.2794 ± 0.0005) GeV/c 2 [60] - -
mBs (5.3696 ± 0.0024) GeV/c 2 [60] - -
mW (80.425 ± 0.039) GeV/c 2 [60] - -
GF 1.16637 × 10 −5 GeV −2 [60] - -
fK (159.8 ± 1.5) MeV [60] - -
Table 5.1: Inputs to the CKM fits. If not stated otherwise: for two errors given, the first
stands for statistical and accountable systematic uncertainties and the second stands for systematic
theoretical uncertainties. The last two columns indicate the Rfit treatment of the
input parameters: measurements or parameters that have statistical errors (we include here
experimental systematics) are marked in the “GS” column by an asterisk; measurements or
parameters that have systematic theoretical errors are marked in the “TH” column by an
asterisk. Upper part: experimental determinations of the CKM matrix elements. Middle
part: CP-violation and mixing observables. Lower part: parameters used in SM
predictions that are obtained from experiment.

38 Chapter 5. Probing the Standard Model
Parameter Value ± Error(s) Reference
Errors
GS TH
BK 0.79 ± 0.04 ± 0.09 [61] ⋆ ⋆
αS(m2 Z )
ηct
0.1176 ± 0.0020
0.47 ± 0.04
[60]
[37]
-
-
⋆
⋆
ηtt 0.5765 ± 0.0065 [37, 39] - ⋆
ηB(MS) 0.551 ± 0.007 [52] - ⋆
fBs (236.5 ± 31.5 ± 1) MeV [62] ⋆ ⋆
Bs 1.37 ± 0.14 [62] ⋆ -
fBs /fBd 1.24 ± 0.04 ± 0.06 [62] ⋆ -
Bs/Bd 1.00 ± 0.02 [62] ⋆ ⋆
mb 4.8 ± 0.1 [51] - ⋆
z 0.085 ± 0.01 [51] - ⋆
α S(mb) 0.22 [51] - -
BS(mb) 0.83 ± 0.03 ± 0.07 [51] ⋆ ⋆
K1 −0.295 [51] - -
K2 1.162 [51] - -
C1 −0.184 [52] - -
C2 1.078 [52] - -
C3 0.013 [52] - -
C4 −0.035 [52] - -
C5 0.009 [52] - -
C6 −0.41 [52] - -
C8 −0.14795 [63] - -
Table 5.2: Inputs to the CKM fits (continued). Upper part: parameters of the SM
predictions obtained from theory. Middle part: parameters of the ASL predictions. Lower
part: Wilson Coefficients.

5.2. Standard Model Fit Results 39
5.2 Standard Model Fit Results
The goal of the CKM matrix analysis is to test the quality of the agreement between
the Standard Model predictions of the relevant observables and their experimental
measurements. Furthermore, the CKM matrix parameters are constrained from fits,
performed in the framework of Rfit, where the input values from Tables 5.1 & 5.2
have been used. The fit results are provided numerically as well as graphically in
one- and two-dimensional representations, e. g. in the (¯ρ,¯η) plane. They are obtained
using the Mathematica based version of the CKMfitter package and have been partly
presented at ICHEP 2006.
The Standard Global CKM Fit includes the observables:
|Vud| , |Vus| , |Vub| , |Vcb| , |εK| , ∆md , ∆ms , α , sin 2β and γ , (5.27)
which can be considered as theoretically and experimentally well understood and
provide significant constraints on the fit parameters. Its result is shown in Figure
5.3, where for each individual constraint the 95% CL allowed belt is indicated
in the (¯ρ,¯η) plane 4 .
The left side of the Unitarity Triangle, Ru, is determined from the ratio of |Vub|
and |Vcb|. Its 95 % CL belt is, to a very good approximation, a ring around
(¯ρ, ¯η) = (0, 0). The right side Rt, is constrained by |Vtd| and |Vtb|, which have
been obtained from measurements of the oscillaton frequencies ∆md and ∆ms. Be-
cause of large theoretical uncertainties on fBd
√ Bd, the constraint from ∆md is very
loose. A much better result is obtained from the combination of ∆md and ∆ms,
where most of the theoretical uncertainties from QCD correction factors and LQCD
parameters cancel. The constraint on (¯ρ,¯η) from the γ determination is very loose
and the UT angle α is ambiguous due to mirror solutions, as shown in Figure 5.1.
The currently best known parameter is the UT angle β, which is well constrained
from measurements of sin 2β. Since cos 2β is measured to be a positive number [64],
its mirror solution is excluded at 95 % CL. A direct constraint on the altitude of the
Unitarity Triangle is provided by the measurements of |ɛK| leading to a hyperboliclike
belt in the (¯ρ,¯η) plane.
The allowed area from the Standard Global CKM Fit constrains the apex of the
Unitarity Triangle in the first quadrant of the (¯ρ,¯η) plane. It is covered consistently
by all 95 % CL belts of the individual constraints. The Standard Global CKM
Fit excludes the possibility of ¯η = 0 and thus a real CKM matrix at a CL of at
most 99.9 %.
4 For sin 2β, the 68 % CL (1σ) and 95.5 % CL (2σ) belts are given instead.

40 Chapter 5. Probing the Standard Model
The results of the Standard Global CKM Fit for the Wolfenstein parameters A, λ, ¯ρ,
¯η and the Jarlskog invariant J are shown in Figure 5.4. Furthermore, the fit results
of the UT angles α, β and γ are compared with their SM predictions and their
direct measurements. In this case, SM prediction means the result of the Standard
Global CKM Fit without the respective parameter in the fit. The numerical results
of the relevant observables and model parameters are summarized in Table 5.3 & 5.4.
η
1.5
1
0.5
0
-0.5
-1
excluded area has CL > 0.95
εK
γ
sin2β
V /V ub cb
α
γ
α
excluded at CL > 0.95
CKM
f i t t e r
ICHEP 2006
γ
sol. w/ cos2β
< 0
(excl. at CL > 0.95)
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
α
Δms
Δmd
& Δm
Figure 5.3: Confidence level in the (¯ρ,¯η) plane obtained from the Standard Global CKM Fit
(red bordered yellow area) and from the individual constraints (colored belts). The shaded
areas indicate 95 % CL allowed regions.
d
εK

5.2. Standard Model Fit Results 41
1 - CL
1 - CL
1 - CL
1 - CL
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
0
0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86
A
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
0
0.05 0.1 0.15 0.2 0.25
ρ
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
-6
× 10
26 28 30 32 34 36
× 10
38
J
CKM
f i t t e r
ICHEP 2006
CKM fit
prediction
measurement
0
0 0.1 0.2 0.3 0.4
β
0.5 0.6 0.7
-6
1 - CL
1 - CL
1 - CL
1 - CL
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
0
0.224 0.225 0.226 0.227 0.228 0.229 0.23 0.231
λ
1
0.8
0.6
0.4
0.2
0
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
0.28 0.3 0.32 0.34
η
0.36 0.38 0.4 0.42
CKM
f i t t e r
ICHEP 2006
CKM fit
prediction
measurement
0
0 0.5 1 1.5
α
2 2.5 3
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
CKM fit
prediction
measurement
0
0 0.5 1 1.5
γ
2 2.5 3
Figure 5.4: Confidence level in the (¯ρ,¯η) plane on the Wolfenstein parameters, the UT
angles and the Jarlskog invariant, obtained from the Standard Global CKM Fit. For α, γ
and sin 2β, also the SM predictions and the direct measurements are shown.

42 Chapter 5. Probing the Standard Model
The quality of the agreement between the SM predictions and the experimental data
, which is a
can be obtained from the interpretation of the test statistics χ2 min,ymod
probe of the goodness-of-fit for the SM hypothesis. The p-value P (χ2 min,ymod |SM)
for the validity of the Standard Model needs to be calculated using Toy Monte Carlo
simulations [4]. Since this is beyond the scope of this thesis, an approximative
method is chosen instead.
Assuming naively a gaussian behavior of the global likelihood function L(ymod) with
Ndof = Nexp − Nmod = 6, the p-value can be approximately obtained from the Prob
function, which is given in Equation (3.6). The global χ2-minimum of the Standard
Global CKM Fit is:
= 4.25 . (5.28)
This leads to a p-value of
χ 2 min,ymod
P (χ 2 min,ymod |SM) ≤ P rob(χ2min,ymod ) = 64.2 % (5.29)
for the validity of the Standard Model.
The main contribution to the global χ 2 -minimum stems from the slight disagreement
of the measurements of |Vub| and sin 2β. The plots in Figure 5.5 show the SM
prediction compared to the direct measurement for both observables, |Vub|incl. and
sin 2β. In addition, the result of the Standard Global CKM Fit is shown.
1 - CL
1
0.8
0.6
0.4
0.2
0
CKM
f i t t e r
ICHEP 2006
CKM fit
prediction
measurement
0.0035 0.004 0.0045 0.005 0.0055
| Vub | incl.
1 - CL
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
CKM fit
prediction
measurement
0
0.5 0.6 0.7 0.8 0.9 1
sin(2 β )
Figure 5.5: Confidence level on |Vub|incl (left) and sin 2β (right) obtained from the Standard
Global CKM Fit (blue curve), the SM predictions (red curve) and the direct measurements
(green shaded).
Another possibility for the determination of the CKM matrix element |Vub| is provided
by purely leptonic decays of charged B-mesons. Since the SM expectations for
the corresponding branching fractions are rather small, only the decay B + → τ + ντ
has been seen yet. Figure 5.6 shows the SM predictions from the Standard Global
CKM Fit for B(B + → τ + ντ ) and B(B + → µ + νµ). In addition, the combined
B + → τ + ντ measurement from BABAR and Belle and the corresponding fit result
is shown.

5.2. Standard Model Fit Results 43
1 - CL
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
CKM fit
prediction
measurement
0
0 0.5 1 1.5 2 2.5 3
+ +
4
BR(B → τ ν τ ) × 10
1 - CL
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
0
0.2 0.3 0.4 0.5 0.6 0.7
+ +
6
BR(B → μ ν μ ) × 10
Figure 5.6: Confidence level on B(B + → τ + ντ ) and B(B + → µ + νµ) obtained from the SM
prediction (red curve). Furthermore, the results of the direct measurement (green shaded)
and the fit (blue curve) are given for the decay B + → τ + ντ .
Since the SM prediction of the branching fraction of purely leptonic B-meson decays
depends also on the decay constant fBd , a precise measurement can help to reduce
theoretical uncertainties in the Standard Global CKM Fit. Figure 5.7 shows the
constraint on (¯ρ,¯η) coming from the individual constraints B(B + → τ + ντ ), ∆md
and a combination of both inputs. Since B(B + → τ + ντ ) depends on the matrix
element |Vub|, its 95 % CL belt is to a very good approximation a ring around
(¯ρ, ¯η) = (0, 0). In the combination with ∆md, the theoretical uncertainties are
reduced due to the cancelation of the decay constant contribution. The remaining
theoretical uncertainty stems from the bag factor Bd in the SM prediction of ∆md.
η
1.5
1
0.5
0
-0.5
-1
excluded area has CL > 0.95
+ + B → τ ντ
CKM fit
γ
α
CKM
+ +
f i t t e r Constraint from B → τ ντ
and Δmd
ICHEP 06
-1.5
-1 -0.5 0 0.5 1 1.5 2
Figure 5.7: Confidence level in the (¯ρ,¯η) plane obtained from B(B + → τ + ντ ) (blue ring),
∆md (yellow ring) and a combination of both (green area).
ρ
β
Δmd

44 Chapter 5. Probing the Standard Model
In addition to the Standard Global CKM Fit, several fits for different classes of
constraints have been performed in this thesis. Each plot of Figure 5.8 shows the
95 % CL belts for the individual constraints and the allowed area from the combined
fits.
In the first row, the fit results, obtained from the CP-violating observables α, sin 2β,
γ and |ɛK| are compared with those obtained from the CP-conserving constraints
|Vub|, ∆md and ∆md & ∆ms. An important observation is that a non-real CKM
matrix with ¯η �= 0 and thus CP violation is also obtained from CP-conserving constraints.
The crucial input in this constraint is the recent measurement of ∆ms by
CDF, presented at the winter conferences 2006.
The second row shows the impact of theoretical uncertainties. The allowed area from
the fit using only the UT angle constraints α, sin 2β and γ is much smaller than the
area obtained from the constraints |Vub|, |ɛK|, ∆md and ∆md & ∆ms which depend
on additional QCD parameters.
The left figure of the third row shows the fit results from constraints which are
dominated by tree-level contributions. Beside |Vub|, a combined constraint from α
and β is used which allows to extract γ. Assuming no New Physics contributions to
the ∆I = 3/2 part of b → d transitions in the extraction of α leads in combination
with β, measured in tree-level B → Mc¯c transitions, to a cancellation of a possible
New Physics mixing phase ϑd (see Chapter 6). That gives a determination of
γ = π − β − α independent from possible New Physics in B 0 - ¯ B 0 mixing. Compared
to that, the right plot shows constraints from the loop-dominated observables sin 2β,
|ɛK|, ∆md and ∆md & ∆ms.
Since all fits are consistent with the Standard Global CKM Fit and the allowed
(¯ρ,¯η) range is quite small, the space for possible New Physics effects is significantly
constrained. A more detailed discussion of possible NP contributions in B 0 - ¯ B 0
mixing is described in the next chapter.

5.2. Standard Model Fit Results 45
η
η
η
0.6
0.5
0.4
0.3
0.2
excluded area has CL > 0.95
γ
sin2β
εK
α
0.1
α
γ
β
0
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0.6
0.5
0.4
0.3
0.2
excluded area has CL > 0.95
γ
sin2β
ρ
εK
sol. w/ cos2β
< 0
(excl. at CL > 0.95)
CKM
f i t t e r
ICHEP 2006
α
γ
η
0.6
0.5
0.4
0.3
0.2
excluded area has CL > 0.95
Δmd
Δms
& Δmd
α
0.1
γ
Vub/Vcb
β
0
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
(a) (b)
α
0.1
α
γ
β
0
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0.6
0.5
0.4
0.3
0.2
excluded area has CL > 0.95
(c)
γ(
α)
α
ρ
ρ
sol. w/ cos2β
< 0
(excl. at CL > 0.95)
CKM
f i t t e r
ICHEP 2006
0.1
γ
Vub/Vcb
β
0
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
(e)
α
CKM
γ
f i t t e r
ICHEP 2006
η
η
0.6
0.5
0.4
0.3
0.2
excluded area has CL > 0.95
εK
Δmd
α
ρ
Δms
& Δmd
ρ
CKM
f i t t e r
ICHEP 2006
CKM
f i t t e r
ICHEP 2006
0.1
γ
Vub/Vcb
β
0
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0.6
0.5
0.4
0.3
0.2
sin2β
εK
Δmd
(d)
Δms
& Δmd
α
0.1
γ
β
0
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
ρ
εK
εK
sol. w/ cos2β
< 0
(excl. at CL > 0.95)
Figure 5.8: Confidence level in the (¯ρ,¯η) plane for several global CKM fits. (a) CP-violating
observables vs. (b) CP-conserving observables, (c) theoretically clean observables vs. (d) observables
with significant theory errors from non-perturbative QCD parameters and (e) tree
dominated observables vs. (f) loop dominated observables. In the plot from pure tree observables,
the constraint on α has been used assuming there are no New Physics contributions
to the ∆I = 3/2 part of b → d transitions. In the combination of this constraint with β
from B → Mc¯cKS modes the New Physics mixing phase ϑd cancels, so that it gives a New
Physics free determination of γ = π − β − α.
excluded area has CL > 0.95
(f)
CKM
f i t t e r
ICHEP 2006

46 Chapter 5. Probing the Standard Model
An important constraint on New Physics in B 0 - ¯ B 0 mixing comes from measurements
of the semileptonic CP asymmetry ASL. Since it is not measured precisely
enough to provide a significant constraint in (¯ρ,¯η) plane, the measurement of ASL
is not included in the Standard Global CKM Fit. The prediction of ASL depends
on short and long distance hadronic matrix elements, which need to be calculated
from Lattice QCD. During this work, ASL has been implemented with LO and NLO
QCD corrections. Figure 5.9 shows the confidence levels for the SM prediction of
ASL, taking LO as well as NLO QCD corrections into account. They are compared
with the results of the Standard Global CKM Fit, including ASL. Its input value is
given in Table 5.1.
1 - CL
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
-3
× 10
0
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1
× 10
-0
ASL LO
-3
1 - CL
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
-3
× 10
0
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1
× 10
-0
ASL NLO
Figure 5.9: Confidence level on ASL from the SM prediction. The QCD corrections are
used up to LO (left) and NLO (right).
-3

5.2. Standard Model Fit Results 47
Observable central ± 1σ ± 2σ ± 3σ
λ 0.2272 +0.0010
−0.0010
A 0.813 +0.015
−0.015
¯ρ 0.187 +0.028
−0.086
¯η 0.333 +0.038
−0.017
J [10 −5 ] 3.02 +0.36
−0.18
sin(2α) −0.25 +0.48
−0.15
sin(2α) (meas. not in fit) −0.28 +0.66
−0.17
sin(2β) 0.701 +0.022
−0.022
sin(2β) (meas. not in fit) 0.777 +0.114
−0.052
α (rad) 1.694 +0.082
−0.242
α (rad) (meas. not in fit) 1.713 +0.090
−0.339
α (rad) (direct meas.) 1.62 +0.18
−0.17
β (rad) 0.389 +0.016
−0.015
β (rad) (meas. not in fit) 0.452 +0.099
−0.046
β (rad) (direct meas.) 0.371 +0.018
−0.017
γ � δ (rad) 1.058 +0.244
−0.076
γ � δ (rad) (meas. not in fit) 1.060 +0.250
−0.079
γ � δ (rad) (direct meas.) 1.06 +0.67
−0.44
Ru
Rt
0.382 +0.014
−0.015
0.878 +0.092
−0.030
∆md (ps−1 ) (meas. not in fit) 0.359 +0.270
−0.077
∆ms (ps−1 ) 17.37 +0.35
−0.23
∆ms (ps−1 ) (meas. not in fit) 20.0 +7.0
−4.7
|ɛK| [10−3 ] (meas. not in fit) 2.34 +1.15
−0.73
B(B + → τ + νµ) [10−4 ] 1.06 +0.16
−0.16
B(B + → τ + νµ) [10−4 ] (meas. not in fit) 1.01 +0.16
−0.24
B(B + → µ + νµ) [10−7 ] (meas. not in fit) 4.01 +0.64
−0.95
+0.0020
−0.0020
+0.030
−0.030
+0.054
−0.119
+0.060
−0.034
+0.55
−0.35
+0.64
−0.29
+0.79
−0.32
+0.044
−0.045
+0.141
−0.093
+0.16
−0.33
+0.18
−0.41
+0.47
−0.28
+0.032
−0.031
+0.130
−0.075
+0.036
−0.034
+0.33
−0.15
+0.33
−0.15
+1.10
−0.70
+0.028
−0.029
+0.126
−0.059
+0.37
−0.10
+0.59
−0.40
+11.8
−7.2
+1.56
−0.90
+0.35
−0.34
+0.36
−0.36
+1.4
−1.4
+0.0030
−0.0031
+0.046
−0.045
+0.080
−0.143
+0.077
−0.051
+0.70
−0.51
+0.75
−0.43
+0.91
−0.45
+0.066
−0.067
+0.16
−0.14
+0.25
−0.39
+0.27
−0.48
+0.62
−0.36
+0.049
−0.045
+0.16
−0.10
+0.056
−0.051
+0.40
−0.23
+0.40
−0.23
+1.58
−0.92
+0.043
−0.043
+0.150
−0.089
+0.46
−0.13
+0.79
−0.57
+15.3
−8.6
+1.9
−1.1
+0.58
−0.45
+0.59
−0.44
+2.4
−1.8
Table 5.3: Fit results and errors using the Standard Global CKM Fit observables. For
results marked with “meas. not in fit”, the measurement of the corresponding observable
has not been included in the fit. The “1 − CL” ranges are defined as 32 % (1σ), 4.5 % (2σ)
and 0.3 % (3σ).

48 Chapter 5. Probing the Standard Model
Observable central ± 1σ ± 2σ ± 3σ
|Vud| 0.97384 +0.00023
−0.00024
|Vus| 0.2272 +0.0010
−0.0010
|Vub| [10−3 ] 3.74 +0.14
−0.15
|Vub| [10−3 ] (meas. not in fit) 3.52 +0.18
−0.17
|Vcd| 0.2271 +0.0010
−0.0010
|Vcs| 0.97298 +0.00023
−0.00024
|Vcb| [10−3 ] 41.94 +0.67
−0.69
|Vcb| [10−3 ] (meas. not in fit) 44.7 +1.1
−1.9
|Vtd| [10−3 ] 8.37 +0.90
−0.31
|Vts| [10−3 ] 41.26 +0.68
−0.66
|Vtb| 0.999113 +0.000029
−0.000029
+0.00047
−0.00047
+0.0020
−0.0020
+0.29
−0.29
+0.37
−0.34
+0.0020
−0.0020
+0.00047
−0.00048
+1.4
−1.4
+2.3
−4.6
+1.32
−0.62
+1.3
−1.4
+0.000057
−0.000058
Table 5.4: Fit results and errors (continued)
+0.00071
−0.00071
+0.0030
−0.0031
+0.44
−0.44
+0.55
−0.57
+0.0030
−0.0031
+0.00071
−0.00072
+2.0
−2.0
+3.5
−7.2
+1.47
−0.92
+2.0
−2.1
+0.000084
−0.000088

Chapter 6
New Physics Beyond the
Standard Model
The Standard Model describes successfully all the present data from flavor physics
experiments. Nevertheless, there are still many reasons to believe in physics beyond
the SM. For example, evidences for unobserved “dark matter” and “dark energy”
from cosmic microwave background radiation measurements are not described by
the SM. Furthermore, the strength of CP violation in the CKM matrix, expressed
through the Jarlskog invariant, is significantly to weak and the electroweak phase
transition is not intens enough to generate the observed baryon asymmetry in the
universe.
In the following, two extensions of the Standard Model are discussed. First of all a
model independent description of NP contributions to B 0 - ¯ B 0 mixing and secondly,
charged Higgs contributions to the leptonic decay width of charged B mesons in
Two Higgs Doublet Models (2HDM).
6.1 New Physics in B 0 - ¯B 0 Oscillations
B 0 - ¯ B 0 oscillation occurs in the SM through |∆B| = 2 transitions. A model independent
description of New Physics contributions introduces two additional parameters,
the relative amplitude r 2 d and the relative phase 2ϑd between the mixing
matrix elements including SM and NP contributions compared to SM contributions
only [65, 66]:
� B 0 � �H SM+NP � � ¯ B 0 �
� B 0 |H SM | ¯ B 0 � = r2 d ei2ϑd . (6.1)
The Standard Model is referred to as r 2 d = 1 and 2ϑd = 0. Additional assumptions
are a unitary 3 × 3 CKM matrix and no NP contributions to tree level dominated
processes (which means Γ12 = ΓSM 12 ). More specifically, decay transitions with four
flavor changes (i.e. b → q1¯q2q3, q1 �= q2 �= q3) are dominated by the SM [4]. The
49

50 Chapter 6. New Physics Beyond the Standard Model
theory predictions of the relevant observables are changed as follows [67]:
γ ⇒ γ (tree level dominated)
|Vud|, |Vus| ⇒ |Vud|, |Vus| (tree level dominated)
|Vub|, |Vcb| ⇒ |Vub|, |Vcb| (tree level dominated)
∆md ⇒ ∆m NP
d
sin 2β ⇒ sin(2β + 2ϑd)
= ∆mSM d · r 2 d
cos 2β ⇒ cos(2β + 2ϑd) (6.2)
α (= π − β − γ) ⇒ α NP (= π − β − γ − ϑd)
� �SM Γ12 sin 2ϑd
ASL ⇒ A NP
SL = −Re
M12
r 2 d
� �SM Γ12 cos 2ϑd
+ Im
M12 r2 d
The tree-level dominated constraints from |Vud|, |Vus|, |Vub|, |Vcb| and γ remain unchanged.
The other observables in Eqn. 6.2 have more important loop contributions
which are sensitive to possible New Physics effects. The only constraint on both NP
parameters r 2 d and 2ϑd comes from the semileptonic CP asymmetry ASL, where the
SM prediction of Γ12/M12 is used here at LO in QCD.
The observables in Eqn. 6.2 provide significant constraints in the (¯ρ,¯η) plane as well
as in the (r 2 d ,2ϑd) plane. The results of a global New Physics fit are shown in the
Figures 6.1 to 6.5, where the used input values are summarized in Table 5.1 & 5.2.
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure 6.1: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and sin 2β.
Figure 6.1 shows the confidence level obtained from a global CKM fit in the (¯ρ,¯η)
and (r 2 d ,2ϑd) planes, assuming New Physics contributions to B 0 - ¯ B 0 oscillations.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
.

6.1. New Physics in B 0 - ¯ B 0 Oscillations 51
The constraints stem from |Vud|, |Vub|, |Vcb|, ∆md and sin 2β. The only significant
constraint in the (¯ρ,¯η) plane comes from the ratio |Vub/Vcb|, which leads, to a very
good approximation, to a ring around (¯ρ, ¯η) = (0, 0). The constraint in the (r 2 d ,2ϑd)
plane is very loose. The input of sin 2β leads to a mirror solution at 2ϑd > π/2. It
vanishes after adding cos(2β + 2ϑd) > 0 to the fit inputs, as shown in Fig. 6.2. It
does only marginally effect the result in the (¯ρ,¯η) plane.
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure 6.2: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β and cos 2β.
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
22
ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure 6.3: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β and γ.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

52 Chapter 6. New Physics Beyond the Standard Model
A much better constraint stems from the observable γ. Since it is tree-level dominated,
it directly constrains the CKM matrix parameters (¯ρ,¯η) and thereby the
allowed range in the (r 2 d ,2ϑd) plane. Figure 6.3 shows the two allowed regions remain
in the (¯ρ,¯η) plane and in the (r 2 d ,2ϑd) plane. Since a one-dimensional scan
in ¯η results ¯η �= 0 at a CL of at most 99.9 %, a real CKM matrix is also excluded
assuming New Physics contributions in B 0 - ¯ B 0 oscillations. The first solution in the
(r 2 d ,2ϑd) plane is in agreement with the SM values (r 2 d = 1, 2ϑd = 0), whereas the
second solution would be a clear sign of New Physics effects. Another major input
stems from measurements of the UT angle α. It decreases the allowed regions in the
(¯ρ,¯η) plane as well as in the (r 2 d ,2ϑd) plane as shown in Figure 6.4.
Finally, the inclusion of ASL in the fit gives an additional hard constraint in both
planes. As shown in Figure 6.5, the non-SM solution almost vanishes, which highlights
the importance of the observable ASL in constraining possible New Physics
contributions to B 0 - ¯ B 0 mixing. Additional fits have been performed to describe the
effects from the individual constraints. The results are shown in the Figures B.1
to B.5 , given in Appendix B.
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure 6.4: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β, γ and α.
Figure 6.6 shows the confidence level obtained from one-dimensional scans of the
New Physics parameters r 2 d and 2ϑd, using constraints from |Vud|, |Vub|, |Vcb|, ∆md,
sin 2β, cos 2β, γ, α and ASL. The results
r 2 d
+0.50 +1.28
= 1.02 −0.42 (1σ)
−0.57 (2σ) and 2ϑd = −0.094 +0.049
−0.123
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
+0.11
(1σ) −0.16 (2σ) , (6.3)
are in good agreement with the Standard Model, nevertheless, the r2 d constraint shows
that NP contributions of order O(100 %) are still possible. The non-SM solution is
0

6.1. New Physics in B 0 - ¯ B 0 Oscillations 53
strongly suppressed at more than 99.5 % CL, the SM-like solution shows a small bias
in 2ϑd, mainly caused by the inputs |Vub|incl. and sin 2β.
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure 6.5: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β, γ, α and ASL.
1 - CL
1
0.8
0.6
0.4
0.2
CKM
f i t t e r
ICHEP 2006
0
0 1 2 3
r 2
d
4 5 6
1 - CL
1
0.8
0.6
0.4
0.2
f i t t e r
ICHEP 2006
0
-3 -2 -1 0
2 ϑ (rad)
1 2 3
Figure 6.6: Confidence level on r 2 d and 2ϑd obtained from the global CKM fit, assuming
possible New Physics contributions to the B 0 - ¯ B 0 mixing amplitude.
CKM
d
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

54 Chapter 6. New Physics Beyond the Standard Model
6.2 Charged Higgs Contributions to Leptonic B ± Decays
6.2.1 Two-Higgs-Doublet Models
Charged Higgs bosons are predicted by multiple-Higgs-doublet extensions of the
Standard Model. The simplest versions are Two-Higgs-Doublet Models (2HDM),
which introduce the two Higgs doublet scalar fields Φi, i = 1, 2, where tan β = v2/v1
is the ratio of their vacuum expectation values vi ∼ 〈Φi〉. This leads to five physical
Higgs bosons, three neutral and two charged ones.
The 2HDM are classified in model I, where all fermions obtain their masses from
the same Higgs doublet, and model II, where u-type quarks obtain their mass from
one, d-type quarks and charged leptons from the other doublet [68]. Since there are
no relevant effects from charged B decays in model I, only model II is considered
here. Model II is also realized in the Higgs sector of the Minimal Supersymmetric
Standard Model (MSSM), where the mass of the charged Higgs boson is additionally
restricted to m H ± > mW . This is not the case for the general model II of 2HDM,
where m H ± can take on arbitrary values [9].
6.2.2 Charged Higgs Contributions to Leptonic B ± Decays
Important constraints on the charged Higgs mass m H ± come from purely leptonic
decays of charged B mesons. In contrast to many other rare decays (e. g. b → sγ)
which are influenced by New Physics at the one loop level, the decays B ± → l ± ν
are sensitive to charged Higgs bosons (H ± ) at tree level [69]. Figure 6.7 shows
the tree-level contribution to the decay B + → τ + ντ , which is, in addition to the
SM-W boson, mediated by a H ± boson.
b
u
+
B
+ +
W , H
Figure 6.7: Tree-level contribution from charged Higgs bosons to B + → τ + ντ
+ τ
ντ

6.2. Charged Higgs Contributions to Leptonic B ± Decays 55
The prediction of the branching fraction B of purely leptonic B ± decays can be
written as [70]:
B (B → lν) = G2 F mBm2 l f 2 B
|Vub|
8π
2
�
1 − m2 l
m2 �2
· rH , (6.4)
B
where the factor rH contains the charged Higgs boson contributions. It is defined
by [69]:
�
rH = 1 − tan 2 β m2 B
m2 H ±
�2
1
1 + ɛ0 · tan β
. (6.5)
The factor 1/(1 + ɛ0 · tan β) combines corrections to the coupling H ± uidj which occur
at large tan β in SUSY models [69, 71]. Since a detailed discussion of this effect
occurring in SUSY models is beyond the scope of this thesis, they are neglected
by fixing ɛ0 to be zero. A discussion of the numerical impact of ɛ0 1 is provided in
Ref. [69].
Figure 6.8 shows the confidence level in the (tan β,m H +) plane obtained from the
constraints |Vud|, |Vus|, |Vcb|, |Vub|incl and B(B + → τ + ντ ). Their input values are the
same as summarized in Table 5.1, where for B(B + → τ + ντ ) the combined likelihood
function from BABAR [44] and Belle [45] is taken.
2
) GeV/c
+
m(H
350
300
250
200
150
100
50
0 10 20 30 40 50 60 70 80 90
tan( β )
CKM
1-CL
f i t t e r
ICHEP 2006
Figure 6.8: Confidence level in the (tan β,m H +) plane obtained from |Vud|, |Vus|, |Vcb|,
|Vub|incl and B(B + → τ + ντ ).
Since the constraint in Fig. 6.8 is very loose, only small m H + are exclude for large
tan β at 95 % CL. An experimental lower limit with m H + > 78.6 GeV/c 2 at 95 %
CL is obtained from the combined LEP results of searches in the decay channels
1 Varied in an interval −0.01 < ɛ0 < +0.01 which has been found in Ref. [71].
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

56 Chapter 6. New Physics Beyond the Standard Model
H + → c¯s and H + → τ + ντ [72]. Another interesting constraint stems from ¯ B → Xsγ
decays where a lower limit of m H + > 350 GeV/c 2 at 99 % CL [73] has been found.

Chapter 7
Conclusions and Perspectives
This thesis is a contribution to the development of a Mathematica based version of
the CKMfitter package. It had been introduced by Jérôme Charles to obtain a
significant fit time reduction compared to the original FORTRAN based CKMfitter
package. Using symbolic calculations during the fit preparation and an efficient
FORTRAN based minimization routine, a gain in CPU time of more than a factor
100 has been achieved for the Standard Global CKM Fit.
Related to the Standard Global CKM Fit, the theory of neutral meson oscillations
has been implemented for the B 0 d , B0 s and K 0 systems as well as the theory predictions
for the branching fraction of purely leptonic B-meson decays. More specifically,
the following observables have been implemented: ∆md, ∆ms, ASL, |ɛK| and
B(B + → l + νl).
For the treatment of look-up-table input files, e. g. for ∆ms, B(B + → τ + ντ ) and the
UT angles α and γ, FORTRAN based subroutines (TABLEAU, dTABLEAUO2, LoadLUT)
have been coded. The interpolation of the look-up tables is performed by a Mathematica-based
subroutine using cubic spline interpolation.
The results of the Standard Global CKM Fit show a good agreement between SM
predictions and recent data. The Wolfenstein parameters have been constraint
at 68 % CL to:
A = 0.813 +0.015
+0.0010
+0.028
+0.038
−0.015 , λ = 0.2272 −0.0010 , ¯ρ = 0.187 −0.086 , ¯η = 0.333 −0.017 , (7.1)
and the Jarlskog invariant, which is related to the strength of CP violation in electroweak
transitions is found to be:
J = � 3.02 +0.36�
−5
−0.18 · 10 . (7.2)
57

58 Chapter 7. Conclusions and Perspectives
Since all the present data from experiments in quark-flavor physics is well described
by the Standard Model, the space for New Physics contributions is significantly constrained.
Nevertheless, there is still enough space for possible New Physics effects,
which have been quantified in this thesis in two different extensions of the Standard
Model.
Possible New Physics contributions to neutral meson oscillations have been implemented
model-independently to the B-meson systems as well as to the Kaon system.
The allowed ranges for the NP parameters (r 2 d ,2ϑd) in the Bd system have been found
to:
r 2 d
= 1.02 +0.50
−0.42 and 2ϑd = −0.094 +0.049
−0.123 (at 68 % CL), (7.3)
which is in good agreement with the Standard Model. The small bias in 2ϑd results
from the slight disagreement of the measurements of |Vub|incl and sin 2β. Since a
significant constraint on possible New Physics in B 0 - ¯ B 0 mixing is obtained from the
observable ASL, its theoretical prediction has been implemented including LO QCD
corrections as well as NLO QCD corrections. A preliminary comparison of their SM
predictions is given in this thesis.
Contributions of charged Higgs bosons to the branching fraction of leptonic B-meson
decays have been implemented in the framework of Two-Higgs-Doublet Models. Since
only the decay B + → τ + ντ is currently experimentally accessible, the constraint
on (tan β,m H +) is still loose. An additional fit input from B → Xsγ transitions
would provide an additional constraint, but has not been implemented yet.
To guarantee the stability of the fit results by using the Mathematica based CKMfitter
package, several comparison tests with the original package have been performed.
As a benefit to the CKMfitter group, a user guide has been written during this thesis
as well as a tutorial on the coding of theory packages.
After more than one year of development, the Mathematica based version of the
CKMfitter package has been used for the first time to produce the fit results for
a large conference (ICHEP 2006). Nevertheless, a lot of features available in the
original CKMfitter package are still missing. A Wolfenstein parameter independent
prediction of B(B + → l + νl) as well as the theory of rare Kaon decays need to be
implemented.
Due to its modular structure, the new Mathematica based version of the CKMfitter
package is not only restricted to the CKM matrix analysis. Further applications,
e. g. a leptonic mixing matrix analysis could be easily added.

Appendix A
The Inami-Lim Functions
The Inami-Lim functions are defined by [74]:
where
S (xi) =
�
1
xi
4 +
9
4 (1 − xi) −
3
2 (1 − xi) 2
�
− 3
S (xi, xj) i�=j =
� �3 xi
ln xi
2 1 − xi
��
1
xixj
4 +
3
2 (1 − xi) −
3
4 (1 − xi) 2
�
1
ln(xi)
xi − xj
�
1
+
4 +
3
2 (1 − xj) −
3
4 (1 − xj) 2
�
−
1
ln(xj)
xj − xi
3
�
1
,
4 (1 − xi) (1 − xj)
xi = m2 i
m 2 W
with i = c, t .
The quark masses are used in the MS scheme, which are perturbatively calculated
in LO from the pole masses by:
�
¯mi(mi) = mi 1 − 4
� ��
αS(mi)
. (A.1)
3 π
The QCD correction factor ηcc to the Inami-Lim functions has been parametrized
through [37]:
� � ��
¯mc(mc)
ηcc � (1.46 ± δcc) 1 − 1.2
− 1 [1 + 52 (αS(mZ) − 0.118)] (A.2)
1.25 GeV/c2 with an uncertainty from higher-order corrections parametrized by:
� � ��
¯mc(mc)
δcc = 0.22 1 − 1.8
− 1 [1 + 80 (αS(mZ) − 0.118)] . (A.3)
1.25 GeV/c2 59

60 Appendix A. The Inami-Lim Functions

Appendix B
Additional Figures of New
Physics in B 0 - ¯B 0 Oscillations
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure B.1: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and cos 2β.
61
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

62 Appendix B. Additional Figures of New Physics in B 0 - ¯B 0 Oscillations
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure B.2: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and α.
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure B.3: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and ASL.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure B.4: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β and α.
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure B.5: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and ASL.
1
0
63
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
0.99
0.98
0.97
0.96
0.95
0.94
0.93

64 Appendix B. Additional Figures of New Physics in B 0 - ¯B 0 Oscillations
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure B.6: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β and α.
η
1.5
1
0.5
0
-0.5
-1
γ
α
-1.5
-1 -0.5 0 0.5 1 1.5 2
ρ
β
CKM
1-CL
f i t t e r
ICHEP 2006
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
(rad)
d
2 ϑ
3
2
1
0
-1
-2
CKM
1-CL
f i t t e r
ICHEP 2006
-3
0 1 2 3
r 2
d
4 5 6
Figure B.7: Constraints within the framework of New Physics contributions to the B 0 - ¯ B 0
mixing amplitude on the (¯ρ,¯η) plane (left) and on the (r 2 d ,2ϑd) plane (right), obtained from
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, γ, α and ASL.
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Appendix C
Testjob
{
(* Setting datacard for global CKM fit *)
analysisContext -> "TESTJOB‘",
theoryPackage -> "BBbarKKbarMixing‘",
inputData -> "globalCKMfit_ICHEP06",
job -> 1,
(***** TESTJOB - SM global Fit - small plane *****)
takeMe[1] -> {
"|Vud|","|Vus|","|Vcb|", "|Vub|",
"All(Deltamd)",
"All(Deltams)",
"All(|epsilonK|)",
"sin2beta",
"alpha",
"gamma"
},
(********** common settings **********)
startRange -> {
"A"->{0.7, 0.9},
"lambda"->{0.22, 0.23},
"rhobar"->{0, 0.26},
"etabar"->{0.28, 0.42}
},
65

66 Appendix C. Testjob
}
globalMinSearches -> 100,
useOneDof -> False,
scanQty -> {"rhobar", "etabar"},
scanMin -> {-0.4, 0. },
scanMax -> {+1.0, 0.7},
granularity -> 200,
nbOfScans -> 2,
nbOfFits -> 1,
scanDirection -> "both",
outputType -> { ".dat", {".png",ImageSize->800}, {".eps",ImageSize->800} },
verbose -> True

Appendix D
Source Code
D.1 Tableau
!-----------------------------------------------------------------------
C
C Tableau Function for LUT inputs by A. Jantsch
C
Double precision Function Tableau( ObsPred, fn )
Implicit None
include ’dimarray.f’
integer k, n, fn, lengthTable(maxLUT)
double precision ObsPred, SplinePred
double precision xmin, xmax
double precision tabsave(maxLUT,0:5,maxGranularity)
common /LUT/ lengthTable, tabsave
n = lengthTable(fn)
xmin = tabsave(fn,0,1)
xmax = tabsave(fn,0,n)
if( ObsPred.le.xmin ) then
Tableau = tabsave(fn,1,1)
elseif( ObsPred.ge.xmax ) then
Tableau = tabsave(fn,1,n)
67

68 Appendix D. Source Code
else
k = int( (dble(n-1)*ObsPred +xmax -dble(n)*xmin)/(xmax -xmin) )
SplinePred = ObsPred -tabsave(fn,0,k)
Tableau = tabsave(fn,2,k) +
> tabsave(fn,3,k) *SplinePred +
> tabsave(fn,4,k) *SplinePred**2 +
> tabsave(fn,5,k) *SplinePred**3
end if
End
D.2 dTableauO2
Tableau = Max(Tableau, 0D0)
!-----------------------------------------------------------------------
C
C Derivatives of Tableau function (divided by 2)
C
Double precision Function dTableauO2( ObsPred, fn )
Implicit None
include ’dimarray.f’
integer k, n, fn, lengthTable(maxLUT)
double precision ObsPred, SplinePred
double precision xmin, xmax
double precision tabsave(maxLUT,0:5,maxGranularity)
common /LUT/ lengthTable, tabsave
n = lengthTable(fn)
xmin = tabsave(fn,0,1)
xmax = tabsave(fn,0,n)
if( ObsPred.le.xmin .or. ObsPred.ge.xmax ) then

D.3. LoadLUT 69
else
dTableauO2 = 0D0
k = int( (dble(n-1)*ObsPred +xmax -dble(n)*xmin)/(xmax -xmin) )
SplinePred = ObsPred -tabsave(fn,0,k)
dTableauO2 = tabsave(fn,3,k) +
> 2D0 *tabsave(fn,4,k) *SplinePred +
> 3D0 *tabsave(fn,5,k) *SplinePred**2
end if
End
D.3 LoadLUT
dTableauO2 = dTableauO2 /2D0
!-----------------------------------------------------------------------
C
C subroutine to load LUT’s for Tableau function
C
subroutine LoadLUT( nbOfLUT, LUTfname)
Implicit None
include ’dimarray.f’
integer nbOfLUT
character*200 LUTfname(maxLUT)
integer i, fn, lengthTable(maxLUT)
double precision tabsave(maxLUT,0:5,maxGranularity)
common /LUT/ lengthTable, tabsave
do fn = 1, nbOfLUT
open(22, file = LUTfname(fn), status = ’old’)
read(22,*) lengthTable(fn)

70 Appendix D. Source Code
do i = 1, lengthTable(fn)
read(22,*) tabsave(fn,0,i), tabsave(fn,1,i),
> tabsave(fn,2,i), tabsave(fn,3,i),
> tabsave(fn,4,i), tabsave(fn,5,i)
enddo
close(22)
enddo
End

Appendix E
User Guide
E.1 Mathematica Terminology
In the following, the Mathematica terminology used in this work, is briefly described.
Further informations are given in the Mathematica Documentation Center [22].
Notebook
A Mathematica notebook is a file format used by the graphical front-end. It contains
input cells, which are evaluated by the Mathematica kernel. The result is displayed
in corresponding output cells.
Package
A Mathematica package is a text file, written in Mathematica language. It can be
sourced from Mathematica notebooks and other Mathematica packages and provides
the included definitions like a library file.
List
Lists are general objects that represent collections of expressions. Each type of expression
can be a list element, e. g. a list of strings or a list of lists, which represents
vectors, matrices or higher tensors.
Example: list = { 1 , ”string” , { 3+x , 3 , y } }
Rule
Evaluating a rule replaces an expression by another one.
Example: x → 5
71

72 Appendix E. User Guide
E.2 Datacards
A datacard describes the analysis and contains the settings for all options of a job.
Datacards are ASCII files, located in the analysis directory and can be edited using
a plain text editor. The only object is a list of rules, where each rule specifies an
option for the analysis job, e. g. the input files or the scan granularity.
It is possible to perform more than one job using one datacard. Therefor, an option
can be referred to a specific job adding the job number in brackets to its end, e. g.
analysisContext
’takeMe[3] → {}’ instead of ’takeMe → {}’.
The analysis context specifies the environment of the fit and is used as the name for
each output file. It is a string, where special characters are forbidden, ending with a
“context mark”. A “context mark” in Mathematica is the backquote or grave accent
character (ASCII decimal code 96).
Syntax: analysisContext → ”Contextname`”
Example: analysisContext → ”TestFit`”
theoryPackage
Here, the theory package(s) used in the analysis are specified. For each package,
its context and the version should be specified in a list. Attention: If no version is
specified, the Standard Model version is used. There is also the possibility to set a
list of these lists to load more than one theory package.
Syntax: theoryPackage → { ”package context`” , version → ”version label” }
Example: theoryPackage → { ”LeptonicDecay`” , version → ”SM” }
Example: theoryPackage → { ”LeptonicDecay`” , { ”BBbarKKbarMixing`” ,
inputData
This option specifies the input file(s) for the analysis.
Syntax: inputData → ”inputfile”
Example: inputData → ”globalCKMfit ICHEP06.data”
version → ”NP(r,theta)” } }

E.2. Datacards 73
jobs
The number of jobs in the settings datacard is set with this flag.
Syntax: jobs → Integer
Example: jobs → 2
takeMe
This is the list, which specifies the fit variables. It is a list of observable and parameter
labels.
Syntax: takeMe → { ”label”, ”label”, . . .}
Example: takeMe → { ”alpha”, ”Deltamd”, ”fBd”}
startRange
This list defines the start ranges for all free parameters, which are not in the takeMe
list.
Syntax: startRange → { ”label” → range list }
Example: startRange → { ”A” → {0.75, 0.85}, ”lambda” → {0.226, 0.228} }
globalMinSearches
Here, the number of reruns of the global minimization is set. To avoid finding a
local minimum, this could be a large number.
Syntax: globalMinSearches → Integer
Example: globalMinSearches → 100
scanQty
This flag selects the parameter space for the scan.
Syntax: scanQty → {”label”, ”label”}
Example: scanQty → ”lambda” for 1D scans
Example: scanQty → {”rhobar”, ”etabar”} for 2D scans

74 Appendix E. User Guide
scanMin
The minimal end point in the parameter space is set with scanMin. It is only a
number for 1D scans or a point for 2D scans.
Syntax: scanMin → Point
Example: scanMin → 0.3 for 1D scans
Example: scanMin → { -1.0 , -1.5} for the large (¯ρ,¯η) plane
Example: scanMin → { -0.4 , 0.0} for the small (¯ρ,¯η) plane
scanMax
In analogy to scanMin, scanMax defines the maximal end point in the parameter
space.
Syntax: scanMax → Point
Example: scanMax → 0.6 for 1D scans
Example: scanMax → { +2.0 , +1.5} for the large (¯ρ,¯η) plane
Example: scanMax → { +1.0 , +0.7} for the small (¯ρ,¯η) plane
useOneDof
This option sets the degree of freedom of the fit problem. If it is equal to one, this
option is “True”, otherwise it is set to “False”.
Syntax: useOneDof → Logical
Example: useOneDof → False
granularity
The scan granularity is specified with this option.
Syntax: granularity → Integer
Example: granularity → 300
nbOfScans
The scan path in the parameter space is set with nbOfScans, which can be also
referred to how often a point is scanned. Only the values “1”, “2”, “4” are defined.
Syntax: nbOfScans → Integer
Example: nbOfScans → 2

E.2. Datacards 75
nbOfFits
The number of fits (minimizations) per single scan point in the parameter space is
set with this flag. It should be a positive number.
Syntax: nbOfFits → Integer
Example: nbOfFits → 2
scanDirection
This option defines the scan direction in the parameter space. The possible settings
are ”vertical”, ”horizontal” or ”both”.
Syntax: scanDirection → String
Example: scanDirection → ”both”
outputType
The analysis notebook exports data files and plots. An output type is chosen by its
file extension. For plots, also the image size can be specified.
Syntax: outputType → { ”extansion”, {”extansion”, ImageSize → Integer} }
Example: outputType → { ”.dat”, ”.eps”, {”.png”, ImageSize → 800} }
verbose
Detailed information during the fit can be obtained with setting this option as
“True”.
Syntax: verbose → Logical
Example: verbose → True

76 Appendix E. User Guide
E.3 Input Types
An input datacard is an ASCII file containing the input values of all observables or
parameters of the fit theory given as a list. Each element of this list is again a list,
with a special syntax for each input type. For example the first element in each list
is a string, which represents the label of the observable or parameter. Attention:
Greek letters have to be written out, e. g. “delta“ for “δ“ or “Delta“ for “∆“. The
inputs datacard needs to be located in the inputs subdirectory.
Fixed
Syntax: { “label“ , value }
Example: { “delta“ , 0.0}
Gauss
Syntax: { “label“ , central value , gaussian error }
Example: { “delta“ , 0.0 , 0.1 }
Asymmetric Gauss
Syntax: { “label“ , central value , pos. gaussian error , neg. gaussian error }
Example: { “delta“ , 0.0 , +0.1 , -0.2 }
Range
Syntax: { “label“ , central value , 0 , pos. theoretical error }
Example: { “delta“ , 0.0 , 0 , 0.1 }
Gauss Range
Syntax: { “label“ , central value , pos. gaussian error , pos. theoretical error }
Example: { “delta“ , 0.0 , 0.1 , 0.1 }
Asymmetric Gauss Range
Syntax: { “label“ , c. value , neg. gauss. error , pos. gauss. error , theo. error}
Example: { “delta“ , 0.0 , +0.1 , -0.2 , 0.1 }
Look Up Table (LUT)
Syntax: { “label“ , “file name“ }
Example: { “delta“ , “delta ICHEP06.dat“ }

E.4. Theory Package Tutorial 77
Upper Limit
Syntax: { “label“ , { pos. value } , number between 0 and 100 }
Example: { “delta“ , { 0.4 } , 50 }
Penalty
Syntax: { “label“ , 0 , neg. }
Example: { “delta“ , 0 , -0.1 }
Short Cut
Syntax: { “label“ , “label“ , “label“ , ... }
Example: { “All(delta)“ , “x“ , “y“ , “z“ }
E.4 Theory Package Tutorial
During this work, a tutorial that describes the development of theory packages has
been written. It explains in detail the structure and relevant commands of a theory
package. It is provided as a PostScript document as well as a Mathematica notebook,
named TheoryTutorial.nb. There is the possibility to use the notebook
version as a template file to create a new theory package.
In addition, a version in HTML format is currently available at:
http://iktp.tu-dresden.de/ jantsch/CKMfitter/Tutorials/TheoryTutorial.html

78 Appendix E. User Guide

E.4. Theory Package Tutorial 79

80 Appendix E. User Guide

E.4. Theory Package Tutorial 81

82 Appendix E. User Guide

E.4. Theory Package Tutorial 83

84 Appendix E. User Guide

E.4. Theory Package Tutorial 85

86 Appendix E. User Guide

E.4. Theory Package Tutorial 87

88 Appendix E. User Guide

E.4. Theory Package Tutorial 89

90 Appendix E. User Guide

E.4. Theory Package Tutorial 91

92 Appendix E. User Guide

E.4. Theory Package Tutorial 93

94 Appendix E. User Guide

E.4. Theory Package Tutorial 95

96 Appendix E. User Guide

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Danksagung
Allen voran möchte ich meinen Eltern danken, die mir weit mehr als nur ein sorgenfreies
Studium ermöglicht haben. Desweiteren danke ich meinem Bruder Matthias
für seine Hilfsbereitschaft, meiner Freundin Carolin für ihre Liebe sowie meiner gesamten
Familie für das Vertrauen, welches sie in mich gesetzt hat. Ich bedanke
mich bei all meinen Freunden für ihre Unterstützung und die Nachsicht, die sie in
letzter Zeit mit mir haben mussten. Besonderer zu erwähnen sind dabei Matthias
und Steffen, die mich nicht nur durch den Studienalltag begleitet haben.
Heiko Lacker danke ich für die Möglichkeit dieses interessante Thema zu bearbeiten,
aber vor allem für seine sehr gute und persönliche Betreuung. Besonderer Dank
gilt der CKMfitter Gruppe, allen voran Jérôme Charles für die fruchtbare Zusammenarbeit,
sowie Stephane T’Jampens und Vincent Tisserand für die großartige
Unterstützung beim Schreiben dieser Diplomarbeit.
Für lehrreiche Diskussionen danke ich Klaus Schubert, sowie Gerhard Buchalla, Zoltan
Ligeti und Michele Papucci.
Ich bedanke mich bei den Mitarbeitern, Doktoranden und Diplomanden des IKTP,
insbesondere bei Rosemarie Krause, Rainer Schwierz und Andreas Petzold, deren
Hilfsbereitschaft nicht hoch genug gewürdigt werden kann. Ein besonderer Dank
geht an Rene Nogowski, dessen hilfreiche Anmerkungen wesentlich zum Gelingen
dieser Diplomarbeit beigetragen haben.
Außerdem möchte ich Gerhard Soff, Bernhard Spaan und Sven Jahnke erwähnen,
die entscheidenden Einfluß auf meinen Studienweg hatten.
103

Erklärung
Hiermit versichere ich, daß ich die vorliegende Arbeit selbständig und ohne Benutzung
anderer als der angegebenen Hilfsmittel angefertigt habe. Die aus fremden
Quellen direkt oder indirekt übernommenen Resultate sind als solche kenntlich gemacht.
Dresden, den 11.09.2006 Andreas Jantsch