A Mathematica based Version of the CKMfitter Package
A Mathematica based Version of the CKMfitter Package
A Mathematica based Version of the CKMfitter Package
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A <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>Version</strong><br />
<strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> <strong>Package</strong><br />
Diplomarbeit<br />
zur Erlangung des akademischen Grades<br />
Diplom-Physiker<br />
vorgelegt von<br />
Andreas Jantsch<br />
geboren in Dresden<br />
Institut für Kern- und Teilchenphysik<br />
Fachrichtung Physik<br />
Fakultät Ma<strong>the</strong>matik und Naturwissenschaften<br />
der Technischen Universität Dresden<br />
2006
1. Gutachter: Pr<strong>of</strong>. Dr. Klaus R. Schubert<br />
2. Gutachter: Dr. Heiko Lacker<br />
Datum des Einreichens der Arbeit: 11. 09. 2006
Kurzfassung<br />
Das <strong>CKMfitter</strong> S<strong>of</strong>tware-Paket stellt ein Analyseinstrument zur Verfügung, mit dessen<br />
Hilfe die Parameter der Cabibbo-Kobayashi-Maskawa-Matrix im Bereich des<br />
Standardmodells und seinen möglichen Erweiterungen eingeschränkt werden können.<br />
Die verwendete statistische Methode ist ein auf klassischer Statistik basierender<br />
Ansatz, genannt Rfit, in dem <strong>the</strong>oretische Unsicherheiten durch erlaubte Bereiche<br />
dargestellt werden. Das ursprüngliche Paket ist in FORTRAN programmiert und<br />
verwendet das Minimierungspaket MINUIT. Über die Jahre erhöhte sich die Komplexität<br />
der Fitprobleme und damit der CPU-Zeitverbrauch beachtlich. Mit dem Ziel einer<br />
deutlichen Fitzeitreduzierung initiierte Jérôme Charles eine auf <strong>Ma<strong>the</strong>matica</strong><br />
basierende <strong>Version</strong> des <strong>CKMfitter</strong>-Pakets. Mit diesem Paket konnte unter Verwendung<br />
von symbolischen Kalkulationen und einer effizienten Minimierungsroutine ein<br />
CPU-Zeitgewinn von mehr als einem Faktor 100 erreicht werden. Thema dieser Diplomarbeit<br />
ist die Implementierung der Theorie der neutralen Meson-Oszillationen<br />
als auch der leptonischen Zerfälle geladener B-Mesonen, jeweils im Standardmodell<br />
und einer möglichen Erweiterung. Weiterhin wurde die Behandlung von Dateien,<br />
welche tabellarische Eingabedaten enthalten, entwickelt. Abschließend werden aktuellste<br />
Resultate der Cabibbo-Kobayashi-Maskawa-Matrix Analyse im Standardmodell<br />
und zwei seiner Erweiterungen gezeigt. Diese wurden von der <strong>CKMfitter</strong><br />
Gruppe auf der Internationalen Konferenz für Hochenergiephysik 2006 (ICHEP06)<br />
präsentiert.<br />
Abstract<br />
The <strong>CKMfitter</strong> package provides an analysis tool to constrain <strong>the</strong> parameters <strong>of</strong> <strong>the</strong><br />
Cabibbo-Kobayashi-Maskawa matrix in <strong>the</strong> framework <strong>of</strong> <strong>the</strong> Standard Model<br />
and possible extensions. The statistical method used is a frequentist <strong>based</strong> approach<br />
called Rfit, where <strong>the</strong>oretical uncertainties are represented by allowed ranges. The<br />
original package is coded in FORTRAN and uses <strong>the</strong> MINUIT minimization package.<br />
Over <strong>the</strong> years, <strong>the</strong> complexity <strong>of</strong> <strong>the</strong> fit problems and associated with that<br />
<strong>the</strong> CPU time consumption have increased considerably. With <strong>the</strong> goal <strong>of</strong> a significant<br />
fit time reduction, Jérôme Charles initiated a <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version<br />
<strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package. With this new package a gain in CPU time <strong>of</strong> more<br />
than a factor 100 has been achieved, using symbolic calculations and an efficient<br />
minimization routine. Subject <strong>of</strong> this <strong>the</strong>sis is <strong>the</strong> implementation <strong>of</strong> <strong>the</strong> <strong>the</strong>ory<br />
<strong>of</strong> neutral meson oscillations as well as leptonic decays <strong>of</strong> charged B-mesons, both,<br />
for <strong>the</strong> Standard Model and a possible extension. Fur<strong>the</strong>rmore, <strong>the</strong> treatment <strong>of</strong><br />
look-up table input files has been developed. Finally, updated results <strong>of</strong> <strong>the</strong> global<br />
Cabibbo-Kobayashi-Maskawa matrix analysis in <strong>the</strong> Standard Model and two<br />
<strong>of</strong> its extensions have been provided. They have been presented by <strong>the</strong> <strong>CKMfitter</strong><br />
group at <strong>the</strong> International Conference on High Energy Physics 2006 (ICHEP06).
Contents<br />
Contents v<br />
List <strong>of</strong> Figures vii<br />
List <strong>of</strong> Tables ix<br />
1 Introduction 1<br />
2 Theory 3<br />
2.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3<br />
2.2 The CKM Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
2.3 The Unitarity Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . 7<br />
2.4 Neutral Meson Oscillation . . . . . . . . . . . . . . . . . . . . . . . . 8<br />
2.5 CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br />
3 <strong>CKMfitter</strong> 13<br />
3.1 The Statistical Framework - Rfit . . . . . . . . . . . . . . . . . . . . 13<br />
3.2 Fit Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14<br />
3.3 The <strong>CKMfitter</strong> <strong>Package</strong> Code . . . . . . . . . . . . . . . . . . . . . . 15<br />
4 A <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>Version</strong> <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> <strong>Package</strong> 17<br />
4.1 <strong>Ma<strong>the</strong>matica</strong> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17<br />
4.2 <strong>Package</strong> Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18<br />
4.3 <strong>CKMfitter</strong>.nb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19<br />
4.4 The Minimization Routine . . . . . . . . . . . . . . . . . . . . . . . . 21<br />
4.5 Theory <strong>Package</strong>s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22<br />
4.6 Look-Up Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24<br />
4.7 Performance Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26<br />
5 Probing <strong>the</strong> Standard Model 29<br />
5.1 Fit Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29<br />
5.2 Standard Model Fit Results . . . . . . . . . . . . . . . . . . . . . . . 39<br />
v
vi CONTENTS<br />
6 New Physics Beyond <strong>the</strong> Standard Model 49<br />
6.1 New Physics in B 0 - ¯ B 0 Oscillations . . . . . . . . . . . . . . . . . . . 49<br />
6.2 Charged Higgs Contributions to Leptonic B ± Decays . . . . . . . . . 54<br />
7 Conclusions and Perspectives 57<br />
A The Inami-Lim Functions 59<br />
B Additional Figures <strong>of</strong> New Physics in B 0 - ¯B 0 Oscillations 61<br />
C Testjob 65<br />
D Source Code 67<br />
D.1 Tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67<br />
D.2 dTableauO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68<br />
D.3 LoadLUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69<br />
E User Guide 71<br />
E.1 <strong>Ma<strong>the</strong>matica</strong> Terminology . . . . . . . . . . . . . . . . . . . . . . . . 71<br />
E.2 Datacards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72<br />
E.3 Input Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76<br />
E.4 Theory <strong>Package</strong> Tutorial . . . . . . . . . . . . . . . . . . . . . . . . . 77<br />
Bibliography 97<br />
Danksagung 103<br />
Erklärung 105
List <strong>of</strong> Figures<br />
2.1 The rescaled Unitarity Triangle . . . . . . . . . . . . . . . . . . . . . 7<br />
2.2 Box diagram contribution to K 0 - ¯ K 0 mixing . . . . . . . . . . . . . . 9<br />
2.3 Box diagram contribution to B 0 - ¯ B 0 mixing . . . . . . . . . . . . . . 10<br />
4.1 The fit process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20<br />
4.2 The minimization routine file system . . . . . . . . . . . . . . . . . . 21<br />
4.3 Comparison plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27<br />
5.1 CL on <strong>the</strong> UT angles α and γ . . . . . . . . . . . . . . . . . . . . . . 31<br />
5.2 Tree-level contributions to leptonic B + decays in <strong>the</strong> SM . . . . . . . 33<br />
5.3 The Standard Global CKM Fit in <strong>the</strong> (¯ρ,¯η) plane . . . . . . . . . . . 40<br />
5.4 CL on A, λ, ¯ρ, ¯η, J, α, β and γ . . . . . . . . . . . . . . . . . . . . . 41<br />
5.5 CL on |Vub|incl and sin 2β . . . . . . . . . . . . . . . . . . . . . . . . 42<br />
5.6 CL on B(B + → τ + ντ ) and B(B + → µ + νµ) . . . . . . . . . . . . . . . 43<br />
5.7 CL in <strong>the</strong> (¯ρ,¯η) plane obtained from B(B + → τ + ντ ) and ∆md . . . . 43<br />
5.8 SM fit results in <strong>the</strong> (¯ρ,¯η) plane . . . . . . . . . . . . . . . . . . . . . 45<br />
5.9 CL on ASL including LO and NLO QCD corrections . . . . . . . . . 46<br />
6.1 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β . . . . . 50<br />
6.2 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β 51<br />
6.3 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β, γ 51<br />
6.4 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β,<br />
γ, α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52<br />
6.5 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β,<br />
γ, α, ASL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53<br />
6.6 CL on <strong>the</strong> NP parameters r 2 d and 2ϑd . . . . . . . . . . . . . . . . . 53<br />
6.7 Tree-level contribution from charged Higgs bosons to B + → τ + ντ . . 54<br />
6.8 Constraints on (tan β,m H +) from B(B + → τ + ντ ) . . . . . . . . . . . 55<br />
B.1 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, cos 2β . . . . 61<br />
B.2 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, α . . . . . . . 62<br />
B.3 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, ASL . . . . . 62<br />
B.4 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β, α 63<br />
B.5 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, ASL . . . . . 63<br />
vii
viii LIST OF FIGURES<br />
B.6 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, α . . . 64<br />
B.7 Constraints on NP from |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, γ, α, ASL 64
List <strong>of</strong> Tables<br />
2.1 The three generations <strong>of</strong> fundamental fermions . . . . . . . . . . . . 3<br />
2.2 The fundamental interactions . . . . . . . . . . . . . . . . . . . . . . 4<br />
4.1 Directory structure <strong>of</strong> <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>CKMfitter</strong> package . . 18<br />
4.2 <strong>Version</strong> labels in <strong>the</strong> <strong>the</strong>ory package BBbarKKbarMixing . . . . . . 23<br />
4.3 <strong>Version</strong> labels in <strong>the</strong> <strong>the</strong>ory package LeptonicDecay . . . . . . . . . . 23<br />
4.4 LUT column order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25<br />
4.5 Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26<br />
4.6 Numerical comparison <strong>of</strong> A, λ, ¯ρ, ¯η and J . . . . . . . . . . . . . . . 26<br />
4.7 Fit time comparison <strong>of</strong> test job runs . . . . . . . . . . . . . . . . . . 28<br />
4.8 Hardware and s<strong>of</strong>tware performance tests . . . . . . . . . . . . . . . 28<br />
5.1 Inputs to <strong>the</strong> CKM fits . . . . . . . . . . . . . . . . . . . . . . . . . . 37<br />
5.2 Inputs to <strong>the</strong> CKM fits (continued) . . . . . . . . . . . . . . . . . . . 38<br />
5.3 Fit results and errors using <strong>the</strong> Standard Global CKM Fit observables 47<br />
5.4 Fit results and errors (continued) . . . . . . . . . . . . . . . . . . . . 48<br />
ix
x LIST OF TABLES
Chapter 1<br />
Introduction<br />
“Nature has always looked like a horrible mess, but as we go along we see patterns<br />
and put <strong>the</strong>ories toge<strong>the</strong>r, a certain clarify comes and things get simpler.”<br />
Richard P. Feynman [1]<br />
An important instrument <strong>of</strong> research is creating models. A physical model is a <strong>the</strong>ory,<br />
which describes its objects and <strong>the</strong>ir interactions using ma<strong>the</strong>matical equations<br />
and makes predictions for <strong>the</strong>m. Usually, not all parameters <strong>of</strong> such a <strong>the</strong>ory are<br />
fixed by <strong>the</strong> model itself, <strong>the</strong>refore some free parameters <strong>of</strong> a <strong>the</strong>ory can only be<br />
determined by experiments.<br />
The Standard Model (SM) <strong>of</strong> particle physics is <strong>the</strong> <strong>the</strong>ory <strong>of</strong> three generations <strong>of</strong> fundamental<br />
fermions and <strong>the</strong> interactions between <strong>the</strong>m, mediated by gauge bosons. 1<br />
The SM is a combination <strong>of</strong> <strong>the</strong> Quantum Chromodynamics (QCD), which describes<br />
<strong>the</strong> strong interaction, and <strong>the</strong> unified <strong>the</strong>ory <strong>of</strong> electroweak interactions. These are<br />
<strong>the</strong>ories <strong>of</strong> massless particles. To generate masses preserving <strong>the</strong> gauge symmetry<br />
<strong>of</strong> <strong>the</strong> SM, it can be accommodated by spontaneous symmetry breaking, also called<br />
<strong>the</strong> Higgs Mechanism. The associated Higgs particle awaits its discovery though.<br />
Due to <strong>the</strong> disparity <strong>of</strong> weak and mass eigenstates, quarks can be transformed into<br />
each o<strong>the</strong>r via flavor changing weak interaction transitions. The transformation matrix<br />
is called Cabibbo-Kobayashi-Maskawa (CKM) matrix [2,3] and depends on<br />
four independent parameters, three rotation angles and one phase. A non-vanishing<br />
phase would be a source <strong>of</strong> CP violation 2 (CPV) in <strong>the</strong> Standard Model.<br />
The four independent parameters <strong>of</strong> <strong>the</strong> CKM matrix are free parameters <strong>of</strong> <strong>the</strong><br />
SM and need to be determined from experiments. They can be overconstrained by<br />
measurements <strong>of</strong> CKM matrix elements and CP asymmetries. This requires a global<br />
analysis, which probes <strong>the</strong> consistency between <strong>the</strong> different measurements and <strong>the</strong>ir<br />
SM predictions.<br />
1 Particles with half-integer spin are called fermions, particles with integer spin are called bosons.<br />
2 CP violation means an asymmetric behavior <strong>of</strong> particles and its corresponding antiparticles.<br />
1
2 Chapter 1. Introduction<br />
Unfortunately, <strong>the</strong> Standard Model cannot answer all questions, e. g. evidences for<br />
dark matter from <strong>the</strong> measurement <strong>of</strong> <strong>the</strong> cosmic microwave backround radiation imply<br />
<strong>the</strong> existence <strong>of</strong> non-SM particles. Fur<strong>the</strong>rmore, <strong>the</strong> observed matter-antimatter<br />
asymmetry <strong>of</strong> <strong>the</strong> Universe requires additional CP-violating effects. There are many<br />
New Physics (NP) models, like Supersymmetric Models (SUSY) and Grand Unified<br />
Theories (GUTs), that can solve, in principle <strong>the</strong>se and o<strong>the</strong>r problems. Probing<br />
<strong>the</strong> consistency <strong>of</strong> <strong>the</strong>se models with experimental data is also a part <strong>of</strong> <strong>the</strong> global<br />
CKM matrix analysis.<br />
The <strong>CKMfitter</strong> package [4] provides a global analysis tool, which allows to constrain<br />
<strong>the</strong> CKM matrix parameters within <strong>the</strong> Standard Model and its possible extensions.<br />
In <strong>the</strong> framework <strong>of</strong> <strong>the</strong> frequentist approach Rfit, <strong>the</strong> consistency <strong>of</strong> recent experimental<br />
results and its <strong>the</strong>oretical predictions are probed. The s<strong>of</strong>tware package has<br />
been mainly written in FORTRAN using <strong>the</strong> MINUIT minimization package. Over<br />
<strong>the</strong> years, <strong>the</strong> complexity <strong>of</strong> <strong>the</strong> fit problems has increased substantially. Depending<br />
on <strong>the</strong> fit problem, a fit can last up to <strong>the</strong> order <strong>of</strong> days.<br />
In this <strong>the</strong>sis, a <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package is introduced.<br />
It has been initiated and commenced by Jérôme Charles. Beside <strong>of</strong> <strong>Ma<strong>the</strong>matica</strong>,<br />
which provides a framework for symbolic calculations, a slim and effective<br />
minimization routine is implemented. During this work, <strong>the</strong> final structure <strong>of</strong> <strong>the</strong><br />
source code with a userfriendly environment accrued and fundamental <strong>the</strong>ory packages<br />
were coded. They include <strong>the</strong> <strong>the</strong>oretical predictions <strong>of</strong> additional NP models<br />
beyond <strong>the</strong> SM. Important developments are also <strong>the</strong> routine for <strong>the</strong> treatment <strong>of</strong><br />
numerical input tables and <strong>the</strong> possibility <strong>of</strong> having different <strong>the</strong>oretical frameworks<br />
in one <strong>the</strong>ory package. A talk on <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>CKMfitter</strong> version has<br />
been presented at <strong>the</strong> DPG Frühjahrstagung 2006 in Dortmund [5].<br />
After one year <strong>of</strong> development, a lot <strong>of</strong> important fits, for instance <strong>the</strong> Standard<br />
Global CKM Fit, can be performed using <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>CKMfitter</strong>.<br />
For example, all SM and some NP fit results presented by Stephane T’Jampens<br />
at <strong>the</strong> International Conference on High Energy Physics 2006 (ICHEP06) in Moscow,<br />
have been produced with <strong>the</strong> new <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>CKMfitter</strong> version.<br />
The <strong>the</strong>sis is organized as follows. Chapter 2 gives a brief introduction to <strong>the</strong><br />
Standard Model and <strong>the</strong> CKM matrix. Fur<strong>the</strong>rmore, a review <strong>of</strong> <strong>the</strong> relevant <strong>the</strong>ory<br />
<strong>of</strong> neutral meson oscillation and CP violation is presented. After an overview about<br />
<strong>the</strong> original <strong>CKMfitter</strong> package and its statistical approach Rfit in Chapter 3, <strong>the</strong><br />
development <strong>of</strong> <strong>CKMfitter</strong> in a <strong>Ma<strong>the</strong>matica</strong> environment is described in Chapter 4.<br />
Up-to-date plots and results <strong>of</strong> <strong>the</strong> global CKM matrix analysis are presented in<br />
Chapter 5 for <strong>the</strong> Standard Model and in Chapter 6 in <strong>the</strong> framework <strong>of</strong> two New<br />
Physics models. A user guide and a development tutorial for <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong><br />
version <strong>of</strong> <strong>CKMfitter</strong> can be found in <strong>the</strong> appendices.
Chapter 2<br />
Theory<br />
In <strong>the</strong> following, a brief introduction to <strong>the</strong> relevant <strong>the</strong>oretical background is provided.<br />
If not explicitly stated, <strong>the</strong> information given is taken from <strong>the</strong> Refs. [6–10]<br />
which recommended for a more detailed overview.<br />
2.1 The Standard Model<br />
The Standard Model <strong>of</strong> particle physics is <strong>the</strong> <strong>the</strong>ory <strong>of</strong> <strong>the</strong> fundamental particles and<br />
<strong>the</strong>ir interactions. The known fundamental fermions are classified into leptons and<br />
quarks. They are ordered in three generations with ascending masses, as summarized<br />
in Table 2.1.<br />
Fermions Generation Charge Color Weak Isospin (I, I3)<br />
1 2 3 Q/e0 left-handed right-handed<br />
Leptons<br />
νe<br />
e− νµ<br />
µ −<br />
ντ<br />
τ −<br />
0<br />
−1<br />
-<br />
(1/2, +1/2)<br />
(1/2, −1/2)<br />
−<br />
(0, 0)<br />
Quarks<br />
ui ci ti +2/3<br />
−1/3<br />
i=r,g,b<br />
(1/2, +1/2)<br />
(1/2, −1/2)<br />
(0, 0)<br />
(0, 0)<br />
di<br />
si<br />
bi<br />
Table 2.1: The three generations <strong>of</strong> fundamental fermions<br />
For each fermion a corresponding anti-fermion exists with <strong>the</strong> same mass, but opposite<br />
electric charge, color 1 and third component <strong>of</strong> <strong>the</strong> weak isospin, I3.<br />
Beside <strong>of</strong> gravity 2 , <strong>the</strong>re are three fundamental interactions which are mediated by<br />
vector gauge bosons. Gluons have color and interact among each o<strong>the</strong>r as well as<br />
1 Color is <strong>the</strong> charge <strong>of</strong> <strong>the</strong> strong interaction. It is an additional degree <strong>of</strong> freedom with three<br />
possible values, usually called red (r), green (g) and blue (b).<br />
2 Due to its weakness (<strong>the</strong> relative strength compared to <strong>the</strong> strong interaction is <strong>of</strong> order 10 −38 ),<br />
gravity is irrelevant for particle physics on energy scales currently accessible in experiments and is<br />
not described by <strong>the</strong> Standard Model. A unified description <strong>of</strong> all fundamental interactions is <strong>the</strong><br />
goal <strong>of</strong> Theories <strong>of</strong> Everything (TOEs).<br />
3
4 Chapter 2. Theory<br />
<strong>the</strong> bosons <strong>of</strong> <strong>the</strong> weak interaction, which have weak charge. The three fundamental<br />
interactions are summarized in Table 2.2.<br />
Interaction Couples to Vector Boson Mass (GeV/c 2 ) J P<br />
strong color 8 gluons 0 1 −<br />
electromagnetic electric charge photon γ 0 1 −<br />
weak weak charge W ± , Z 0 ≈ 10 2 1<br />
Table 2.2: The fundamental interactions<br />
The Standard Model is <strong>the</strong> combination <strong>of</strong> Quantum Chromodynamics and <strong>the</strong> unified<br />
<strong>the</strong>ory <strong>of</strong> electroweak interactions. It is a renormalizable quantum field <strong>the</strong>ory<br />
and carries <strong>the</strong> group structure <strong>of</strong> <strong>the</strong> gauge group SU(3)C ⊗SU(2)L ⊗U(1)Y , where<br />
C means color, L means left and Y is <strong>the</strong> electroweak hypercharge.<br />
The QCD, represented by <strong>the</strong> gauge group SU(3)C, is <strong>the</strong> <strong>the</strong>ory <strong>of</strong> strong interactions<br />
between colored quarks and gluons. Its coupling constant αS depends on <strong>the</strong><br />
energy scale µ, which leads at high energies to a weak coupling, called Asymptotic<br />
freedom, and a strong coupling at low energies leading to so-called Confinement.<br />
Due to <strong>the</strong> Confinement, no free quarks or gluons have been observed yet. They<br />
only occur in color-neutral bound states, called hadrons, which can be classified into<br />
baryons and mesons 3 .<br />
The unification <strong>of</strong> electro-magnetic and weak interaction is described in <strong>the</strong> model <strong>of</strong><br />
S. L. Glashow, S. Weinberg and A. Salam by <strong>the</strong> gauge group SU(2)L⊗U(1)Y ,<br />
where <strong>the</strong> fermions are represented by left-handed doublets and right-handed singlets<br />
<strong>of</strong> <strong>the</strong> weak isospin. Since gauge invariance requires <strong>the</strong> absence <strong>of</strong> explicit mass<br />
terms in <strong>the</strong> Lagrangian, all particles are initially assumed to be massless. As<br />
a possibility <strong>of</strong> mass generation without spoiling <strong>the</strong> gauge invariance, P. Higgs<br />
introduced a complex scalar doublet field Φ = (φ1, φ2), which leads to an additional<br />
term in <strong>the</strong> SM Lagrangian:<br />
LΦ = � ∂µΦ +� (∂ µ Φ) − V (Φ) (2.1)<br />
with a rotationally symmetric potential V (Φ) = −µ 2 Φ + Φ + λ 2 (Φ + Φ) 2 . The Yukawa<br />
interaction <strong>of</strong> <strong>the</strong> quarks with <strong>the</strong> Higgs field is described by:<br />
LY = −Y d<br />
ij ¯ Q I LiΦd I Rj − Y u<br />
ij ¯ Q I LiεΦ ∗ u I Rj + h.c. , (2.2)<br />
(q = u, d) are complex 3 × 3 matrices, i, j are <strong>the</strong> labels <strong>of</strong> <strong>the</strong> fermion<br />
are <strong>the</strong> left-handed<br />
represent <strong>the</strong> right-handed down- and up-type<br />
quark singlets in <strong>the</strong> basis <strong>of</strong> weak interaction eigenstates, denoted by I.<br />
where Y q<br />
ij<br />
generation and ε is <strong>the</strong> 2×2 total antisymmetric tensor. The QI Li<br />
quark doublets, where dI Rj and uI Rj<br />
3 Baryons are composed <strong>of</strong> three quarks with different color (qiqjqk), mesons are composed <strong>of</strong> a<br />
quark-antiquark pair (qi ¯q ī), where <strong>the</strong> indices i, j, k represent <strong>the</strong> color and ī <strong>the</strong> anti-color respectively.
2.2. The CKM Matrix 5<br />
After spontaneous symmetry breaking from SU(2)L ⊗ U(1)Y to U(1)Q at lower<br />
energies (∼200 GeV), Φ acquires a non-vanishing vacuum expectation value 〈Φ〉 =<br />
� 0, v/ √ 2 � and equation (2.2) results in Dirac mass terms for <strong>the</strong> quarks:<br />
M u = v √ 2 Y u , M d = v √ 2 Y d . (2.3)<br />
Fur<strong>the</strong>rmore, in contrast to <strong>the</strong> photon γ, <strong>the</strong> vector bosons W ± and Z 0 obtain<br />
masses and an additional massive spin-0 boson, <strong>the</strong> physical Higgs particle H, is<br />
predicted 4 .<br />
The mass matrices M u and M d can be diagonalized by unitary transformations:<br />
M u,diag = UM u Ũ † =<br />
M d,diag = V M d ˜ V † =<br />
⎛<br />
⎝<br />
⎛<br />
⎝<br />
mu 0 0<br />
0 mc<br />
0 0 mt<br />
md 0 0<br />
0 ms<br />
0 0 mb<br />
⎞<br />
⎠ (2.4)<br />
⎞<br />
⎠ , (2.5)<br />
where U, Ũ and V, ˜ V are unitary matrices and <strong>the</strong> quark masses mq are real. Due<br />
to <strong>the</strong> fact that <strong>the</strong> left-handed up- and down-type quarks are members <strong>of</strong> <strong>the</strong> same<br />
SU(2)L doublet, <strong>the</strong>y cannot be transformed independently. Choosing <strong>the</strong> up-type<br />
quarks as mass eigenstates5 , <strong>the</strong> left-handed isospin doublet transforms to:<br />
Q I L =<br />
� u I Li<br />
d I Li<br />
�<br />
= U †<br />
ij<br />
�<br />
uLj �<br />
UV † �<br />
jk dLk<br />
�<br />
, (2.6)<br />
which leads to a mixing matrix in <strong>the</strong> down-type quark sector, <strong>the</strong> so-called CKM<br />
matrix VCKM = UV † .<br />
2.2 The CKM Matrix<br />
2.2.1 General Remarks<br />
The Cabibbo-Kobayashi-Maskawa matrix VCKM, is <strong>the</strong> quark-mixing matrix. It<br />
connects <strong>the</strong> weak interaction eigenstates d I Li = (d′ , s ′ , b ′ ) and <strong>the</strong> corresponding<br />
mass eigenstates dLk = (d, s, b) <strong>of</strong> <strong>the</strong> down-type quarks through:<br />
⎛<br />
⎝<br />
⎞ ⎛ ⎞ ⎛<br />
d Vud<br />
⎠ = VCKM ⎝ s ⎠ = ⎝ Vcd<br />
Vus<br />
Vcs<br />
Vub<br />
Vcb<br />
⎞ ⎛<br />
d<br />
⎠ ⎝ s<br />
b<br />
b<br />
d ′<br />
s ′<br />
b ′<br />
Vtd Vts Vtb<br />
⎞<br />
⎠ . (2.7)<br />
4<br />
The observation <strong>of</strong> <strong>the</strong> Higgs boson is <strong>the</strong> main goal <strong>of</strong> <strong>the</strong> experiments at <strong>the</strong> Large Hadron<br />
Collider (LHC), starting in 2007.<br />
5<br />
This is done by convention. It is also possible to choose <strong>the</strong> down-type quarks as mass eigenstates<br />
which would lead to a mixing matrix in <strong>the</strong> up-type quark sector.
6 Chapter 2. Theory<br />
VCKM is a complex 3×3 matrix and, hence, has 18 independent parameters: nine real<br />
parts and nine imaginary parts <strong>of</strong> its nine complex matrix elements. An important<br />
property is its unitarity, given by <strong>the</strong> relation:<br />
VCKMV † †<br />
CKM = V CKMVCKM = 1 , (2.8)<br />
which ensures <strong>the</strong> conservation <strong>of</strong> probability. Due to unitarity and <strong>the</strong> freedom <strong>of</strong><br />
phase redefinition, <strong>the</strong> number <strong>of</strong> independent parameters is reduced to four and<br />
<strong>the</strong> CKM matrix can be parameterized, e. g. by three Euler angles and one global<br />
phase.<br />
2.2.2 The Standard Parameterization <strong>of</strong> VCKM<br />
The Standard Parameterization <strong>of</strong> <strong>the</strong> CKM matrix was proposed by Chau and<br />
Keung [11] and is advocated by <strong>the</strong> Particle Data Group (PDG) [9]. It is obtained<br />
by <strong>the</strong> product <strong>of</strong> three unitary complex rotation matrices, where <strong>the</strong> rotations are<br />
characterized by Euler angles θ12, θ13 and θ23, which are <strong>the</strong> mixing angles between<br />
<strong>the</strong> generations, and one overall CP-violating phase δ. The result is:<br />
VCKM =<br />
⎛<br />
⎜<br />
⎝<br />
c12c13 s12c13 s13e −iδ<br />
−s12c23 − c12s23s13e iδ c12c23 − s12s23s13e iδ s23c13<br />
s12s23 − c12c23s13e iδ −c12s23 − s12c23s13e iδ c23c13<br />
⎞<br />
⎟<br />
⎠ , (2.9)<br />
where cij = cosθij and sij = sinθij for i < j = 1, 2, 3. The cij and sij are positive<br />
for θij > 0. The unitarity relation (2.8) is strictly satisfied.<br />
2.2.3 The Wolfenstein Parameterization <strong>of</strong> VCKM<br />
As a result <strong>of</strong> <strong>the</strong> observed hierarchy between <strong>the</strong> different CKM matrix elements,<br />
Wolfenstein [12] proposed a parameterization in terms <strong>of</strong> <strong>the</strong> four parameters A,<br />
λ, ρ and η. It is an expansion <strong>of</strong> VCKM in λ � |Vus| and defined to all orders in λ<br />
by [13]:<br />
s12 ≡ λ<br />
s23 ≡ Aλ 2<br />
s13e −iδ ≡ Aλ 3 (ρ − iη) .<br />
(2.10)<br />
Thus, up to order <strong>of</strong> λ4 , <strong>the</strong> CKM matrix can be written as:<br />
⎛<br />
⎜<br />
VCKM = ⎜<br />
⎝<br />
1 − λ2<br />
2<br />
λ Aλ3 −λ 1 −<br />
(ρ − iη)<br />
λ2<br />
2<br />
Aλ2 ⎞<br />
⎟ + O<br />
⎟<br />
⎠<br />
� λ 4� . (2.11)<br />
Aλ 3 (1 − ρ − iη) −Aλ 2 1
2.3. The Unitarity Triangle 7<br />
2.3 The Unitarity Triangle<br />
As a result <strong>of</strong> Equation (2.8), <strong>the</strong>re exist 12 different unitarity relations for <strong>the</strong> CKM<br />
matrix. The rescaled unitarity relation relevant for <strong>the</strong> B-meson system is:<br />
VudV ∗ ub<br />
VcdV ∗<br />
cb<br />
+ VcdV ∗<br />
cb<br />
VcdV ∗ +<br />
cb<br />
VtdV ∗<br />
tb<br />
VcdV ∗<br />
cb<br />
= 0 , (2.12)<br />
which can be displayed as a triangle in <strong>the</strong> complex (¯ρ,¯η) plane. Figure 2.1, taken<br />
from Ref. [9], shows <strong>the</strong> so-called Unitarity Triangle (UT). Independent from phase<br />
conventions, its apex is given by:<br />
¯ρ + i¯η ≡ − VudV ∗ ub<br />
VcdV ∗<br />
cb<br />
Figure 2.1: The rescaled Unitarity Triangle<br />
The sides Ru and Rt <strong>of</strong> <strong>the</strong> triangle, are given by:<br />
Ru =<br />
Rt =<br />
and <strong>the</strong> UT angles6 are defined as:<br />
α =<br />
�<br />
arg −<br />
β =<br />
�<br />
arg −<br />
�<br />
γ = arg<br />
�<br />
�<br />
�<br />
�<br />
�<br />
�<br />
�<br />
�<br />
VudV ∗ ub<br />
VcdV ∗<br />
cb<br />
VtdV ∗<br />
tb<br />
VcdV ∗<br />
cb<br />
VtdV<br />
∗<br />
tb<br />
VudV ∗ ub<br />
VcdV<br />
∗<br />
cb<br />
VtdV ∗<br />
tb<br />
− VudV ∗ ub<br />
VcdV ∗<br />
cb<br />
�<br />
�<br />
�<br />
. (2.13)<br />
�<br />
�<br />
�<br />
� = � ¯ρ 2 + ¯η 2 (2.14)<br />
�<br />
�<br />
�<br />
� =<br />
�<br />
(1 − ¯ρ) 2 + ¯η 2 (2.15)<br />
�<br />
¯η<br />
= arctan<br />
¯η 2 �<br />
(2.16)<br />
− ¯ρ (1 − ¯ρ)<br />
� �<br />
¯η<br />
= arctan<br />
(2.17)<br />
1 − ¯ρ<br />
� �<br />
¯η<br />
= arctan . (2.18)<br />
¯ρ<br />
The area <strong>of</strong> <strong>the</strong> Unitarity Triangle is equal to <strong>the</strong> half <strong>of</strong> <strong>the</strong> Jarlskog invariant J [14],<br />
which is defined by <strong>the</strong> imaginary part <strong>of</strong> phase-convention independent CKM matrix<br />
element quartets:<br />
Im � VijVklV ∗<br />
il<br />
� ∗<br />
Vkj = J<br />
3�<br />
m,n=1<br />
εikmεjln . (2.19)<br />
6 In this work, <strong>the</strong> BABAR notation α, β, γ is used, whereas φ2, φ1, φ3 is used in <strong>the</strong> Belle<br />
collaboration.
8 Chapter 2. Theory<br />
2.4 Neutral Meson Oscillation<br />
The phenomenon <strong>of</strong> quark flavor oscillation is predicted in <strong>the</strong> four neutral meson<br />
systems:<br />
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) .<br />
Since <strong>the</strong> mean life time <strong>of</strong> <strong>the</strong> neutral D mesons is very small compared to <strong>the</strong>ir<br />
oscillation frequency, neutral meson mixing has been only observed in <strong>the</strong> Kaon and<br />
B-meson systems, yet.<br />
The time evolution <strong>of</strong> <strong>the</strong>se transitions is given by <strong>the</strong> Schrödinger Equation for<br />
two-state systems <strong>of</strong> instable particles:<br />
i ˙ ψ(t) = ˆ H ψ(t) , (2.20)<br />
where ψ(t) is <strong>the</strong> two-state system <strong>of</strong> <strong>the</strong> neutral mesons, e. g. for <strong>the</strong> Bd system:<br />
�<br />
|B0 (t)〉<br />
ψ(t) =<br />
| ¯ B0 �<br />
. (2.21)<br />
(t)〉<br />
The Hamilton Operator ˆ H contains <strong>the</strong> two hermitian 2 × 2 mass (Mij) and decay<br />
width (Γij) matrices:<br />
⎛<br />
ˆH = ˆ M − i<br />
2 ˆ ⎜<br />
Γ = ⎝<br />
M11 − i<br />
2 Γ11 M12 − i<br />
2 Γ12<br />
M21 − i<br />
2 Γ21 M22 − i<br />
2 Γ22<br />
⎞<br />
⎟<br />
⎠ . (2.22)<br />
Since CPT symmetry requires equal masses and decay rates for a particle and its<br />
anti-particle, <strong>the</strong> diagonal elements must be equal:<br />
Γ11 = Γ22 = Γ (2.23)<br />
M11 = M22 = m (2.24)<br />
and <strong>the</strong> eight real parameters can be reduced to six independent parameters. The<br />
hermiticity <strong>of</strong> ˆ Γ and ˆ M leads to:<br />
M21 = M ∗ 12 and Γ21 = Γ ∗ 12 . (2.25)<br />
Because <strong>of</strong> an arbitrary global phase, only five observables can be defined<br />
� � � �<br />
Γ12<br />
Γ12<br />
m , Γ , |M12| , Re and Im . (2.26)<br />
M12<br />
The Schrödinger Equation (2.20) has well defined solutions ψ(t) for any ψ(0), but<br />
only two mass eigenstates with time-independent flavor composition for each neutral<br />
meson system.<br />
M12
2.4. Neutral Meson Oscillation 9<br />
2.4.1 The K 0 - ¯K 0 System<br />
The flavor eigenstates <strong>of</strong> <strong>the</strong> neutral Kaon system mix via weak interaction through<br />
<strong>the</strong> box diagrams shown in Figure 2.2. These are effective flavor changing neutral<br />
current (FCNC) processes with |∆S| = 2 7 , where <strong>the</strong> main contributions in <strong>the</strong> loop<br />
to <strong>the</strong> observable |ɛK| as described below come from top- and also from charm-quark<br />
exchange.<br />
s u,c,t<br />
d<br />
0<br />
K<br />
W W<br />
u,c,t<br />
0<br />
K<br />
d<br />
s<br />
s<br />
d<br />
W<br />
u,c,t u,c,t<br />
Figure 2.2: Box diagram contribution to K 0 - ¯ K 0 mixing<br />
The two mass eigenstates <strong>of</strong> <strong>the</strong> Kaon system are defined according to <strong>the</strong>ir lifetimes:<br />
0<br />
K<br />
|K 0 S〉 = pK|K 0 〉 + qK| ¯ K 0 〉 (2.27)<br />
|K 0 L〉 = pK|K 0 〉 − qK| ¯ K 0 〉 (2.28)<br />
where K0 S is <strong>the</strong> short living, and K0 L is <strong>the</strong> long living normalized mass eigenstate,<br />
with |qK| + |pK| = 1. An interesting observable <strong>of</strong> <strong>the</strong> neutral Kaon system related<br />
to <strong>the</strong> CKM matrix analysis is <strong>the</strong> CP-violating parameter |ɛK|. εK is defined by:<br />
εK = 2<br />
3 η+− + 1<br />
3 η00<br />
where η+− and η00 are <strong>the</strong> ratios <strong>of</strong> <strong>the</strong> K 0 L and K0 S<br />
and neutral pair <strong>of</strong> pions respectively:<br />
η+− = A � K 0 L → π+ π −�<br />
A � K 0 S → π+ π −�<br />
2.4.2 The B 0 - ¯B 0 System<br />
W<br />
0<br />
K<br />
d<br />
s<br />
(2.29)<br />
decay amplitudes to a charged<br />
η00 = A � K0 L → π0π0� A � K0 S → π0π0� . (2.30)<br />
Neutral B-meson oscillations occur in <strong>the</strong> SM through a second order FCNC process.<br />
They are mediated by <strong>the</strong> |∆B| = 2 box diagrams shown in Figure 2.3, where<br />
<strong>the</strong> loop is dominated by W boson and up-type quark contributions.<br />
According to <strong>the</strong>ir mass, <strong>the</strong> eigenstates <strong>of</strong> <strong>the</strong> B-meson system are defined by:<br />
|B 0 L〉 ∼ pB|B 0 〉 + qB| ¯ B 0 〉 (2.31)<br />
|B 0 H〉 ∼ pB|B 0 〉 − qB| ¯ B 0 〉 , (2.32)<br />
7 |∆F | expresses <strong>the</strong> flavor quantum number difference between <strong>the</strong> initial- and <strong>the</strong> final-state <strong>of</strong><br />
a transition, e. g. |∆S| for Strangeness and |∆B| for Bottomness.
10 Chapter 2. Theory<br />
b u,c,t<br />
d<br />
0<br />
B<br />
W W<br />
u,c,t<br />
0<br />
B<br />
d<br />
b<br />
b<br />
d<br />
0<br />
B<br />
W<br />
u,c,t u,c,t<br />
Figure 2.3: Box diagram contribution to B 0 - ¯ B 0 mixing<br />
where B0 L is <strong>the</strong> lighter, and B0 H is <strong>the</strong> heavier eigenstate. Analogous to <strong>the</strong><br />
Kaon system, <strong>the</strong> oscillation parameters pB and qB are complex and normalized<br />
by |qB| + |pB| = 1.<br />
The mass and decay width differences <strong>of</strong> <strong>the</strong> mass eigenstates are defined by convention<br />
as:<br />
∆mB ≡ MH − ML � 2|M12| (2.33)<br />
∆ΓB ≡ ΓH − ΓL � 2 Re (M12Γ ∗ 12 )<br />
|M12|<br />
W<br />
0<br />
B<br />
d<br />
b<br />
(2.34)<br />
where ∆ΓB ≪ ∆mB is assumed. The exact ratio qB/pB is given by:<br />
�<br />
2 M<br />
qB<br />
= −<br />
pB<br />
∗ i<br />
12 −<br />
2 Γ∗ �<br />
12<br />
∆mB − i<br />
2 ∆ΓB<br />
. (2.35)<br />
The physical meaningful quantity, that is independent <strong>of</strong> <strong>the</strong> phase convention, is:<br />
�<br />
� � �<br />
� qB �2<br />
�M<br />
� � �<br />
�pB<br />
� = �<br />
�<br />
�<br />
∗ i<br />
12 −<br />
2 Γ∗12 M12 − i<br />
�<br />
�<br />
�<br />
�<br />
�<br />
�<br />
�<br />
. (2.36)<br />
2 Γ12<br />
The phase <strong>of</strong> qB/pB is convention dependent and, hence, not observable. Similar<br />
definitions can be made in <strong>the</strong> Bs-meson system by replacing only <strong>the</strong> d-quark by a<br />
s-quark.<br />
2.5 CP Violation<br />
The CP transformation is a combination <strong>of</strong> charge conjugation C and parity P.<br />
Under C transformation, a particle transforms into its anti-particle, by conjugating<br />
all internal quantum numbers, e. g. Q → −Q for <strong>the</strong> electromagnetic charge. P<br />
transformation reflects <strong>the</strong> space coordinate �x into −�x. The combination <strong>of</strong> both<br />
transforms a left-handed particle into its right-handed anti-particle, e. g. e −<br />
L<br />
→ e+<br />
R .
2.5. CP Violation 11<br />
Within <strong>the</strong> Standard Model with three families, CP violation can be generated by<br />
a single, non-vanishing phase (¯η �= 0 ⇔ J �= 0) in <strong>the</strong> CKM matrix. It is related<br />
to flavor changing charged currents <strong>of</strong> weak interactions and was discovered in <strong>the</strong><br />
neutral Kaon system in 1964 [15]. The effects <strong>of</strong> CP violation relevant for <strong>the</strong> CKM<br />
matrix analysis can be classified into three different types [9]:<br />
CP violation in decay results from <strong>the</strong> interference among different decay amplitudes<br />
<strong>of</strong> a meson M into a multi-particle final state f and its CP conjugated<br />
decay ¯ M → ¯ f:<br />
Af = 〈f|H|M〉 Ā ¯ f = 〈 ¯ f|H| ¯ M〉 . (2.37)<br />
The CP symmetry is violated, if:<br />
�<br />
�<br />
�<br />
�<br />
�<br />
Ā ¯ f<br />
Af<br />
�<br />
�<br />
�<br />
� �= 1 . (2.38)<br />
�<br />
This type <strong>of</strong> CP violation occurs in charged and neutral meson decays, but is<br />
<strong>the</strong> only possible source <strong>of</strong> CP asymmetries in charged meson decays. The CP<br />
asymmetry is defined by:<br />
A f ± ≡ Γ (M − → f − ) − Γ (M + → f + )<br />
Γ (M − → f − ) + Γ (M + → f + ) =<br />
for charged mesons and:<br />
for neutral mesons.<br />
Af ≡ Γ � ¯ M 0 → ¯ f � − Γ � M 0 → f �<br />
Γ � ¯ M 0 → ¯ f � + Γ (M → f) =<br />
� �<br />
�Āf −/Af + �2 − 1<br />
� �<br />
�Āf −/Af + �2 + 1<br />
� �<br />
�Āf ¯/Af �2 − 1<br />
� �<br />
�Āf ¯/Af �2 + 1<br />
(2.39)<br />
(2.40)<br />
CP violation in mixing <strong>of</strong> neutral mesons results from <strong>the</strong> fact that mass eigenstates<br />
are different from <strong>the</strong> CP eigenstates leading to:<br />
� �<br />
�<br />
�<br />
q �<br />
�<br />
�p<br />
� �= 1 , (2.41)<br />
where q and p are <strong>the</strong> complex parameters <strong>of</strong> <strong>the</strong> neutral meson mixing (see<br />
Section 2.4). They can be measured via <strong>the</strong> asymmetry <strong>of</strong> semileptonic neutral<br />
meson decays induced by oscillation:<br />
�<br />
Γ ¯M 0<br />
phys (t) → l<br />
ASL ≡<br />
+ � �<br />
X − Γ M 0 phys (t) → l− �<br />
X<br />
�<br />
Γ ¯M 0<br />
phys (t) → l + � �<br />
X + Γ M 0 phys (t) → l− �<br />
X<br />
� �<br />
�<br />
1 − �<br />
q �4<br />
�<br />
�p<br />
�<br />
= � �<br />
�<br />
1 + �<br />
q �4<br />
. (2.42)<br />
�<br />
�p<br />
�
12 Chapter 2. Theory<br />
CP violation in <strong>the</strong> interference between decays with and without mixing<br />
occurs from:<br />
Im (λfCP ) �= 0 , (2.43)<br />
where λfCP<br />
is <strong>the</strong> product <strong>of</strong> q/p and <strong>the</strong> ratio <strong>of</strong> <strong>the</strong> decay amplitudes <strong>of</strong> a<br />
meson M 0 and its anti-particle ¯ M 0 into <strong>the</strong> same final CP eigenstate fCP with<br />
CP eigenvalue ηfCP , given as:<br />
λfCP<br />
q<br />
≡<br />
p<br />
ĀfCP<br />
AfCP<br />
q ĀfCP ¯<br />
= ηfCP<br />
p<br />
AfCP<br />
. (2.44)<br />
The amplitudes Ā ¯ fCP and AfCP differ only in <strong>the</strong> signs <strong>of</strong> <strong>the</strong> weak phase for<br />
each term, while ηfCP = ±1.<br />
CP violation in <strong>the</strong> interference between decays with and without mixing can<br />
be measured via <strong>the</strong> asymmetry <strong>of</strong> neutral meson decays into final CP eigenstates<br />
fCP :<br />
AfCP ≡<br />
� � �<br />
Γ ¯M 0<br />
phys (t) → fCP − Γ M 0 �<br />
phys (t) → fCP<br />
� � �<br />
Γ ¯M 0<br />
phys (t) → fCP + Γ M 0 � . (2.45)<br />
phys (t) → fCP
Chapter 3<br />
<strong>CKMfitter</strong><br />
The <strong>CKMfitter</strong> group is an international group <strong>of</strong> experimental and <strong>the</strong>oretical particle<br />
physicists with collaborators from <strong>the</strong> high energy physics experiments ATLAS,<br />
BABAR, Belle and LHCb. Its goal is a global analysis <strong>of</strong> <strong>the</strong> CKM matrix, which<br />
contains:<br />
1. Probing <strong>the</strong> consistency between <strong>the</strong> SM <strong>the</strong>ory predictions and <strong>the</strong> experimental<br />
data.<br />
2. Constraining <strong>the</strong> CKM matrix and QCD model parameters entering <strong>the</strong> SM<br />
<strong>the</strong>ory predictions.<br />
3. Predicting observables from <strong>the</strong> Standard Global CKM Fit.<br />
4. Searching for specific signs <strong>of</strong> New Physics in an extended <strong>the</strong>oretical framework<br />
and constraining New Physics parameters.<br />
More detailed information is provided in Ref. [4] and on <strong>the</strong> <strong>CKMfitter</strong> website [16].<br />
3.1 The Statistical Framework - Rfit<br />
The statistical analysis performed in <strong>CKMfitter</strong> is entirely <strong>based</strong> on <strong>the</strong> frequentist<br />
approach Range Fit (Rfit) [17]. The experimental input information is a set<br />
<strong>of</strong> Nexp measurements xexp = {xexp(1), . . . , xexp(Nexp)} described by a set <strong>of</strong> corresponding<br />
<strong>the</strong>oretical expressions x<strong>the</strong>o = {x<strong>the</strong>o(1), . . . , x<strong>the</strong>o(Nexp)}. The <strong>the</strong>oretical<br />
expressions are model-dependent functions <strong>of</strong> a set <strong>of</strong> Nmod parameters<br />
ymod = {ymod(1), . . . , ymod(Nmod)}. A subset <strong>of</strong> N<strong>the</strong>o parameters within <strong>the</strong>se ymod<br />
set are considered as being fundamental and free parameters <strong>of</strong> <strong>the</strong> <strong>the</strong>ory model,<br />
e. g. <strong>the</strong> four Wolfenstein parameters in <strong>the</strong> SM or <strong>the</strong> top quark mass. These <strong>the</strong>ory<br />
parameters are denoted as y<strong>the</strong>o = {y<strong>the</strong>o(1), . . . , y<strong>the</strong>o(N<strong>the</strong>o)}. The remaining<br />
NQCD = Nmod − N<strong>the</strong>o parameters, which appear due to our present inability to<br />
compute strong interaction quantities precisely, e. g. fBd , Bd, . . . , are denoted as<br />
yQCD = {yQCD(1), . . . , yQCD(NQCD)}.<br />
13
14 Chapter 3. <strong>CKMfitter</strong><br />
The quantity minimized in <strong>the</strong> fit is<br />
χ 2 = −2 ln L(ymod) , (3.1)<br />
with <strong>the</strong> likelihood function L(ymod), defined by a product <strong>of</strong> contributions <strong>of</strong> two<br />
types:<br />
L(ymod) = Lexp(xexp − x<strong>the</strong>o(ymod)) · L<strong>the</strong>o(yQCD) . (3.2)<br />
The experimental likelihood Lexp depends on <strong>the</strong> experimental measurements xexp,<br />
which are gaussian distributed in general, and <strong>the</strong>ir <strong>the</strong>oretical predictions x<strong>the</strong>o,<br />
which are functions <strong>of</strong> <strong>the</strong> model parameters ymod. The <strong>the</strong>oretical likelihood L<strong>the</strong>o<br />
describes <strong>the</strong> knowledge on <strong>the</strong> QCD parameters yQCD ∈ {ymod}, where <strong>the</strong> <strong>the</strong>oretical<br />
uncertainties σsyst are considered to define allowed ranges:<br />
[yQCD − σsyst, yQCD + σsyst] . (3.3)<br />
In <strong>the</strong> Rfit scheme, <strong>the</strong> <strong>the</strong>oretical likelihoods L<strong>the</strong>o(i) do not contribute to <strong>the</strong> χ 2<br />
<strong>of</strong> <strong>the</strong> fit, as long as <strong>the</strong> yQCD take on values within <strong>the</strong>ir.<br />
3.2 Fit Metrology<br />
is determined with<br />
In a first step, <strong>the</strong> global minimum <strong>of</strong> Equation (3.1), χ2 min,global<br />
respect to all Nmod parameters. Due to <strong>the</strong> experimental and <strong>the</strong>oretical systematics,<br />
this absolute minimal value does in general not correspond to a unique ymod location.<br />
In a second step, a selected subspace <strong>of</strong> interest <strong>of</strong> <strong>the</strong> parameter space, e. g. a =<br />
{¯ρ, ¯η} is scanned, to determin <strong>the</strong> local χ2-minimum χ2 min,local (a) for each fixed point<br />
<strong>of</strong> a grid in <strong>the</strong> parameter space a, with respect to <strong>the</strong> remaining parameters. The<br />
<strong>of</strong>fset-corrected χ2 is calculated as follows:<br />
∆χ 2 (a) = χ 2 min,local (a) − χ2 min,global<br />
where its minimum is equal to zero by construction.<br />
, (3.4)<br />
Finally, a confidence level (CL) for a is obtained using <strong>the</strong> well-known PROB function<br />
from <strong>the</strong> CERN Program Library [18]:<br />
1 − CL = P rob � ∆χ 2 �<br />
(a), Nd<strong>of</strong><br />
=<br />
which assumes gaussian statistics.<br />
1<br />
√ 2 Nd<strong>of</strong> Γ (Nd<strong>of</strong> /2)<br />
� ∞<br />
χ 2 (ymod)<br />
(3.5)<br />
e −t/2 t Nd<strong>of</strong> /2−1 dt , (3.6)
3.3. The <strong>CKMfitter</strong> <strong>Package</strong> Code 15<br />
3.3 The <strong>CKMfitter</strong> <strong>Package</strong> Code<br />
The source code <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package consists <strong>of</strong> more than 40000 lines FOR-<br />
TRAN code and ca. 2000 lines C++ code. It is public available on <strong>the</strong> <strong>CKMfitter</strong><br />
website. The minimization routine used is MINUIT, <strong>the</strong> minimizer from <strong>the</strong> CERN<br />
Program Library [18]. The analysis is driven by datacards, where running options<br />
and input values for <strong>the</strong> fit are specified. The output file <strong>of</strong> <strong>the</strong> procedure is written<br />
in HBOOK format, which is used by PAW 1 macros for producing one- and twodimensional<br />
plots <strong>of</strong> <strong>the</strong> fit results.<br />
Over <strong>the</strong> years, <strong>the</strong> fit problems became more and more complex. There are difficult,<br />
non-linear fit problems which contain mirror solutions from trigonometric functions.<br />
This has been led to an increasing CPU time consumption and <strong>the</strong> time for a single<br />
complex fit can last longer than one day.<br />
The main reason for this is <strong>the</strong> so-called dictionary, a file, which contains hundreds<br />
<strong>of</strong> <strong>the</strong>ory predictions for <strong>the</strong> fit variables. Since <strong>the</strong>y are statically loaded, <strong>the</strong><br />
dictionary needs to be browsed in each fit step to obtain <strong>the</strong> <strong>the</strong>ory prediction needed<br />
in <strong>the</strong> specific fit. As <strong>the</strong> dictionary file is a very important part <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong><br />
source code, a runtime upgrade is difficult to realize in <strong>the</strong> original source code.<br />
1 The Physics Analysis Workstation (PAW) is a data analysis and presentation tool. [19]
16 Chapter 3. <strong>CKMfitter</strong>
Chapter 4<br />
A <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>Version</strong><br />
<strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> <strong>Package</strong><br />
With <strong>the</strong> goal <strong>of</strong> a significant fit time reduction, Jérôme Charles initiated a<br />
<strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package. The basic concept is <strong>the</strong> usage<br />
<strong>of</strong> symbolic calculations <strong>of</strong> <strong>the</strong> fit expressions in a fit preparation phase within a<br />
<strong>Ma<strong>the</strong>matica</strong> environment, which interacts with a FORTRAN <strong>based</strong> minimization<br />
routine. An analysis is driven by datacards in ASCII format and <strong>the</strong> results are<br />
provided as data files and colored reference plots, done by <strong>Ma<strong>the</strong>matica</strong>. The final<br />
combined plots are currently made by ROOT 1 macros using <strong>the</strong> exported data files.<br />
The ROOT macros have been coded by Vincent Tisserand.<br />
This <strong>the</strong>sis makes contributions to <strong>the</strong> source code development <strong>of</strong> <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong><br />
<strong>based</strong> <strong>CKMfitter</strong> package. Most important are <strong>the</strong> implementation <strong>of</strong> <strong>the</strong> <strong>the</strong>ories<br />
<strong>of</strong> neutral meson oscillations and leptonic decays <strong>of</strong> charged B mesons in <strong>the</strong> SM<br />
and in extended framework. Fur<strong>the</strong>rmore, <strong>the</strong> treatment <strong>of</strong> look-up-table input files<br />
has been developed. Since <strong>the</strong> source code development is still under progress, this<br />
work displays <strong>the</strong> current status <strong>of</strong> August, 2006.<br />
4.1 <strong>Ma<strong>the</strong>matica</strong><br />
The s<strong>of</strong>tware package <strong>Ma<strong>the</strong>matica</strong> is a computer algebra system, which facilitates<br />
symbolic math calculations. It provides also a programming language <strong>based</strong> on termrewriting<br />
2 . <strong>Ma<strong>the</strong>matica</strong> is a proprietary product <strong>of</strong> Wolfram Research, Inc. [21].<br />
The most recent version, which has also been used in this <strong>the</strong>sis, is <strong>Ma<strong>the</strong>matica</strong> 5.2,<br />
released in July, 2005.<br />
1 ROOT is an object-oriented data analysis framework <strong>based</strong> on C++ [20].<br />
2 Term-rewriting covers several methods <strong>of</strong> replacing subterms <strong>of</strong> a formal formula by o<strong>the</strong>r,<br />
e. g. simpler terms.<br />
17
18 Chapter 4. A <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>Version</strong> <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> <strong>Package</strong><br />
The core <strong>of</strong> <strong>Ma<strong>the</strong>matica</strong> is <strong>the</strong> kernel, which interactively performs <strong>the</strong> calculation.<br />
The user interacts with <strong>the</strong> kernel via a front-end, where a graphical version is available<br />
as well as a commandline interface.<br />
Since <strong>the</strong> terminology <strong>of</strong> <strong>Ma<strong>the</strong>matica</strong> is used in this work, Appendix E.1 gives a<br />
short overview about relevant terms and commands. A more detailed documentation<br />
is provided on <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> website [22].<br />
4.2 <strong>Package</strong> Structure<br />
The <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package is modularly composed <strong>of</strong><br />
several <strong>Ma<strong>the</strong>matica</strong> notebooks and packages. Fur<strong>the</strong>rmore, <strong>the</strong>re are FORTRAN<br />
files from <strong>the</strong> minimization routine and datacards in ASCII format. In Table 4.1,<br />
<strong>the</strong> content <strong>of</strong> <strong>the</strong> most relevant directories is shown.<br />
<strong>CKMfitter</strong> Directory File Content<br />
analysis/ datacards<br />
fortran/ minimir.f, fit.f, dmnfg.f, specialfunctions.f<br />
inputs/ input datacards, χ 2 -input tables, PDG.m<br />
lib/ AnalysisLib.m, FitLib.m<br />
<strong>the</strong>ories/ <strong>the</strong>ory packages, TheoryTutorial.nb<br />
tools/ CreateInputTable.nb<br />
<strong>CKMfitter</strong>.nb<br />
Table 4.1: Directory structure <strong>of</strong> <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>CKMfitter</strong> package<br />
The user interface is <strong>the</strong> <strong>CKMfitter</strong>.nb notebook. It is located in <strong>the</strong> main directory,<br />
but should be copied to ano<strong>the</strong>r analysis specific directory, outside <strong>of</strong> <strong>the</strong><br />
<strong>CKMfitter</strong> package. For <strong>the</strong> sake <strong>of</strong> clarity, complex functions and subroutines are<br />
sourced out from <strong>the</strong> user interface to library packages. The AnalysisLib package<br />
contains all relevant subroutines for <strong>the</strong> data processing inside <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong><br />
environment, e. g. subroutines to load datacards and <strong>the</strong>ory packages. The second<br />
library package is named FitLib and includes all functions for <strong>the</strong> interaction <strong>of</strong><br />
<strong>CKMfitter</strong>.nb with <strong>the</strong> FORTRAN minimization routine. It provides for example<br />
a subroutine, which translates <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> expressions to FORTRAN code.<br />
As in <strong>the</strong> original <strong>CKMfitter</strong> package, a fit is userfriendly driven by datacards. Since<br />
<strong>the</strong>se are ASCII text files, <strong>the</strong>y can be edited without using <strong>Ma<strong>the</strong>matica</strong> in a plain<br />
text editor. In a datacard, all flags and options <strong>of</strong> a selected analysis are set, e. g. <strong>the</strong><br />
context name, <strong>the</strong> fit variables and <strong>the</strong> scan granularity. A more detailed description<br />
<strong>of</strong> <strong>the</strong> relevant fit options and settings is given in Appendix E.2.
4.3. <strong>CKMfitter</strong>.nb 19<br />
An important flag in <strong>the</strong> datacards is <strong>the</strong> specification <strong>of</strong> <strong>the</strong> input files. These files<br />
contain <strong>the</strong> numerical input for each measurement xexp and model parameter ymod.<br />
The values are provided in <strong>the</strong> “inputs” list, where each element is again a list, one<br />
per measurement or model parameter. According to <strong>the</strong> different error types <strong>of</strong> a<br />
measurement, <strong>the</strong>re exist different input types, e. g. “Range” or “GaussRange”. It<br />
is also possible to specify <strong>the</strong> name <strong>of</strong> a file, which contains a χ 2 -contour <strong>of</strong> a measurement<br />
xexp. The different input types are distinguished using explicit syntaxes<br />
in <strong>the</strong> list. A brief description <strong>of</strong> <strong>the</strong> syntax for each possible input type is provided<br />
in Appendix E.3.<br />
Fur<strong>the</strong>rmore, <strong>the</strong> <strong>the</strong>oretical expressions x<strong>the</strong>o depend on fixed inputs, e. g. physical<br />
constants. They are listed in a separate package, called PDG, which is also located<br />
in <strong>the</strong> inputs directory. Its content is <strong>the</strong> “fixedinputs” list, where each entry is a<br />
“rule”, which replaces a symbol through its numerical value, e. g. mB → 5.2794 for<br />
<strong>the</strong> B-meson mass.<br />
4.3 <strong>CKMfitter</strong>.nb<br />
The <strong>CKMfitter</strong> analysis notebook is <strong>the</strong> user front-end <strong>of</strong> <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong><br />
version. It sources all relevant information from datacards and <strong>the</strong>ory packages,<br />
translates <strong>the</strong> χ 2 -function in FORTRAN format, runs <strong>the</strong> minimization and exports<br />
<strong>the</strong> fit results. In addition, <strong>the</strong> <strong>CKMfitter</strong> notebook provides several checks and <strong>the</strong><br />
opportunity to monitor <strong>the</strong> fit process.<br />
Running a fit, <strong>the</strong> user only needs to select an analysis datacard, where all job<br />
options are set before. Afterwards <strong>the</strong> analysis is executed step by step as shown<br />
in Figure 4.1. In <strong>the</strong> first part, all relevant informations are loaded, e. g. library<br />
packages and physical constants. After loading <strong>the</strong> <strong>the</strong>ory packages and input files<br />
as specified in <strong>the</strong> datacard, <strong>the</strong> χ 2 -function and <strong>the</strong>ir partial derivatives are symbolically<br />
calculated during <strong>the</strong> fit preparation. These expressions are transformed<br />
in FORTRAN code format using <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> intrinsic function FortranForm<br />
and written to <strong>the</strong> file minimirChi2.f. This file is compiled and linked to <strong>the</strong> minimization<br />
subroutines and <strong>the</strong> executable minimir is built. The fit starts through<br />
running minimir, which performes <strong>the</strong> global minimization as well as <strong>the</strong> one- and<br />
two-dimensional scans. The fit results are written to <strong>the</strong> file minimir.output and<br />
<strong>the</strong>n loaded from <strong>CKMfitter</strong>.nb, where <strong>the</strong> fit results are plotted. Finally, plots<br />
and data files are exported.<br />
Remark: To interrupt <strong>the</strong> fit process, <strong>the</strong>re is <strong>the</strong> possibility to quit <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong><br />
kernel. Since minimir is a stand alone executable, it needs to be aborted separately<br />
by <strong>the</strong> user!
20 Chapter 4. A <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>Version</strong> <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> <strong>Package</strong><br />
Datacard specification<br />
Fit preparation<br />
Source library packages<br />
Load physical constants<br />
Read datacard<br />
Load <strong>the</strong>ory packages<br />
Load numerical inputs<br />
Write χ 2 -function (minimirChi2.f )<br />
Compile<br />
Run minimir<br />
Write<br />
Read Fit results (minimir.output)<br />
Plot results Export Plot and data file<br />
Figure 4.1: The fit process
4.4. The Minimization Routine 21<br />
4.4 The Minimization Routine<br />
Since FORTRAN is more optimized for numerical calculations, a FORTRAN <strong>based</strong><br />
minimization routine is used instead <strong>of</strong> a <strong>Ma<strong>the</strong>matica</strong> intrinsic function. The main<br />
file is minimir.f, which calls a set <strong>of</strong> subroutines and functions from <strong>the</strong> files fit.f,<br />
dmnfg.f and specialfunctions.f. The minimized quantity χ 2 and its partial derivatives<br />
with respect to all fit parameters are located in <strong>the</strong> file minimirChi2.f, which<br />
is also linked to <strong>the</strong> executable minimir after <strong>the</strong> compilation process. The subroutine,<br />
which manages <strong>the</strong> minimization process is called POINTFIT. It sets <strong>the</strong><br />
fit environment, e. g. <strong>the</strong> starting point and calls <strong>the</strong> real minimization subroutine<br />
DMNG, which proceeds <strong>the</strong> unconstrained minimization <strong>of</strong> <strong>the</strong> χ 2 -function using <strong>the</strong>ir<br />
exact analytic gradients. The subroutine DMNG was coded by David M. Gay using<br />
a quasi-Newton method and is publicly available at NetLib.org [23]. Fur<strong>the</strong>r informations<br />
can be found in Ref. [24].<br />
Non-standard FORTRAN functions and subroutines, which belong to <strong>the</strong> analytic<br />
part, e. g. contributions to <strong>the</strong> χ 2 -function, are <strong>the</strong> content <strong>of</strong> <strong>the</strong> file specialfunctions.f.<br />
Fur<strong>the</strong>rmore, it contains <strong>the</strong> function TABLEAU, which allows <strong>the</strong> usage <strong>of</strong><br />
look-up tables (LUT) as input files, and <strong>the</strong> function DDILOG from <strong>the</strong> CERN Library<br />
<strong>Package</strong> [18] to calculate di-logarithmic functions.<br />
minimirChi2.f<br />
minimir.f<br />
minimir.input<br />
minimir.aux<br />
fit.f Compile minimir<br />
dmnfg.f<br />
specialfunctions.f minimir.output<br />
Figure 4.2: The minimization routine file system<br />
A schema <strong>of</strong> <strong>the</strong> file system <strong>of</strong> <strong>the</strong> FORTRAN <strong>based</strong> minimization routine is shown in<br />
Figure 4.4. The FORTRAN files are compiled and linked toge<strong>the</strong>r to <strong>the</strong> executable<br />
minimir. The input information for <strong>the</strong> fit routine is provided by two files in ASCII<br />
format, which are created by <strong>the</strong> analysis notebook. The file minimir.input contains<br />
<strong>the</strong> numerical input <strong>of</strong> <strong>the</strong> fit variables and minimir.aux provides <strong>the</strong> fit options,<br />
e. g. <strong>the</strong> scan direction. The fit result is finally written to <strong>the</strong> file minimir.output,<br />
which is read from <strong>the</strong> analysis notebook.
22 Chapter 4. A <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>Version</strong> <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> <strong>Package</strong><br />
4.5 Theory <strong>Package</strong>s<br />
Theory packages provide <strong>the</strong> full physical information on <strong>the</strong> fit variables. Since no<br />
global variables are used in <strong>Ma<strong>the</strong>matica</strong>, all variables are initialized only in <strong>the</strong> context<br />
<strong>of</strong> <strong>the</strong>ir <strong>the</strong>ory package. For each observable, <strong>the</strong> <strong>the</strong>oretical expression x<strong>the</strong>o<br />
is defined and <strong>the</strong> partial derivatives ∂x<strong>the</strong>o/∂ymod with respect to all model parameters<br />
ymod are symbolically calculated. Finally, all expressions are written into<br />
a list named “<strong>the</strong>ory”, which is loaded from <strong>the</strong> analysis notebook. Theory packages<br />
are stored in a binary format (.mx), that is optimized for input by <strong>Ma<strong>the</strong>matica</strong>.<br />
It is possible to define an observable in different <strong>the</strong>oretical frameworks, e. g. <strong>the</strong> B 0 -<br />
¯B 0 oscillations in <strong>the</strong> Standard Model or in a New Physics model (see Chapter 6).<br />
Thus, a <strong>the</strong>ory package can include several <strong>the</strong>ory lists, one for each <strong>the</strong>oretical<br />
framework or possible different parametrization. Different <strong>the</strong>ory lists in <strong>the</strong> same<br />
<strong>the</strong>ory package are distinguished by version labels, which have to be specified in <strong>the</strong><br />
datacard.<br />
It is advantageous for <strong>the</strong> fit to store all expressions and derivatives in <strong>the</strong>ir simplest<br />
form. This can be obtained using <strong>the</strong> intrinsic <strong>Ma<strong>the</strong>matica</strong> functions Simplify and<br />
FullSimplify. For complex expressions, this may lead to a very high CPU time<br />
consumption during <strong>the</strong> <strong>the</strong>ory package development. More detailed information on<br />
<strong>the</strong> development <strong>of</strong> <strong>the</strong>ory packages are given in a tutorial, which accrued during<br />
this work. It is available in Appendix E.4.<br />
In <strong>the</strong> following, four important <strong>the</strong>ory packages, which are related to this work, are<br />
introduced.<br />
4.5.1 CKMmatrix<br />
The <strong>the</strong>ory package CKMmatrix is <strong>the</strong> basis <strong>of</strong> <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>CKMfitter</strong><br />
package and was coded by Jérôme Charles. It contains all relevant definitions<br />
<strong>of</strong> CKM matrix and Unitarity Triangle parameters in <strong>the</strong> exact Wolfenstein<br />
parametrization, e. g. <strong>the</strong> real and imaginary parts <strong>of</strong> <strong>the</strong> CKM matrix elements,<br />
Re (Vij) and Im (Vij), and <strong>the</strong> UT angles α, β, γ.<br />
An important object is <strong>the</strong> wolfCKM function. This is a list <strong>of</strong> rules and provides<br />
<strong>the</strong> real and imaginary parts <strong>of</strong> all CKM matrix elements Re (Vij) and Im (Vij).<br />
Its argument is <strong>the</strong> order <strong>of</strong> <strong>the</strong> expansion into Wolfenstein parameters, where “∞”<br />
means <strong>the</strong> exact expressions up to all orders.<br />
The CKMmatrix package needs to be sourced from each <strong>the</strong>ory package, where <strong>the</strong><br />
Wolfenstein parameters are used or <strong>the</strong> wolfCKM function is evaluated.
4.5. Theory <strong>Package</strong>s 23<br />
4.5.2 BBbarKKbarMixing<br />
The <strong>the</strong>ory <strong>of</strong> measurements describing <strong>the</strong> oscillations in neutral K- and B-meson<br />
systems are provided in <strong>the</strong> BBbarKKbarMixing <strong>the</strong>ory package. It contains all<br />
<strong>the</strong>oretical expressions and partial derivatives for <strong>the</strong> observables:<br />
∆md , ∆ms , |ɛK| and ASL,Bd (4.1)<br />
in two <strong>the</strong>oretical frameworks. The Standard Model definitions are shown in Chapter<br />
5. The second <strong>the</strong>oretical framework is a model independent extension <strong>of</strong> <strong>the</strong><br />
SM, which is discussed in Chapter 6. The version labels, which needs to be specified<br />
in <strong>the</strong> datacard are given in Table 4.2.<br />
framework version label<br />
Standard Model ”SM”<br />
New Physics (model independent) ”NP(r,<strong>the</strong>ta)”<br />
Table 4.2: <strong>Version</strong> labels in <strong>the</strong> <strong>the</strong>ory package BBbarKKbarMixing<br />
In addition to <strong>the</strong> observables in Equation (4.1):<br />
sin(2β + 2ϑd) , cos(2β + 2ϑd) , | sin(2β + 2ϑd + γ)| and αNP , (4.2)<br />
are defined in <strong>the</strong> New Physics framework, where αNP = π − β − γ − ϑd.<br />
4.5.3 LeptonicDecay<br />
The <strong>the</strong>ory package LeptonicDecay contains currently <strong>the</strong> branching fractions <strong>of</strong><br />
charged B mesons into purely leptonic final states:<br />
B � B + → e + �<br />
νe<br />
, B � B + → µ + �<br />
νµ<br />
and B � B + → τ + �<br />
ντ . (4.3)<br />
They are described in <strong>the</strong> Standard Model framework, given in Chapter 5, as well<br />
as in Two-Higgs-Doublet Models, which are described in Chapter 6. Table 4.3 lists<br />
<strong>the</strong> version labels for both frameworks.<br />
framework version label<br />
Standard Model ”SM”<br />
New Physics (charged Higgs) ”NP(H+)”<br />
Table 4.3: <strong>Version</strong> labels in <strong>the</strong> <strong>the</strong>ory package LeptonicDecay
24 Chapter 4. A <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>Version</strong> <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> <strong>Package</strong><br />
4.5.4 DecayBagParameters<br />
The <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>CKMfitter</strong> avoids <strong>the</strong> usage <strong>of</strong> global variables.<br />
However, <strong>the</strong>re are some model parameters, e. g. <strong>the</strong> decay constants and bag parameters<br />
<strong>of</strong> Bd- and Bs-mesons:<br />
fBs ,<br />
fBs<br />
fBd<br />
, Bs , Bs<br />
Bd<br />
, (4.4)<br />
which are used in different <strong>the</strong>ory packages. Thus, <strong>the</strong>y need to be “globalized”,<br />
through initializing <strong>the</strong>m in a separate package, which will be sourced from <strong>the</strong> o<strong>the</strong>r<br />
<strong>the</strong>ory packages. The DecayBagParameters package is currently loaded from <strong>the</strong><br />
BBbarKKbarMixing package as well as from <strong>the</strong> LeptonicDecay package.<br />
4.6 Look-Up Tables<br />
Since <strong>the</strong>re is not always <strong>the</strong> possibility to express a measurement in terms <strong>of</strong> a value<br />
and an error, <strong>the</strong> experimental likelihood, translated into a χ 2 -contour, can also be<br />
used as an input. It is available in a discrete look-up table, where values in-between<br />
<strong>the</strong> discrete LUT entries are calculated through interpolation. The <strong>Ma<strong>the</strong>matica</strong><br />
<strong>based</strong> <strong>CKMfitter</strong> version provides <strong>the</strong> possibility <strong>of</strong> a cubic spline interpolation and<br />
uses a separate FORTRAN <strong>based</strong> function to load <strong>the</strong> LUTs during <strong>the</strong> fit.<br />
4.6.1 Cubic Spline Interpolation<br />
Piecewise interpolation using low order polynomials leads to a global continuous<br />
interpolating function, but is in general not continuous at <strong>the</strong> interval boundaries.<br />
This is avoided by using cubic spline interpolation, which gives back a smooth interpolating<br />
function. The goal <strong>of</strong> cubic spline interpolation is to get an interpolation<br />
formula that is smooth in <strong>the</strong> first, and continuous in <strong>the</strong> second derivative, both<br />
within an interval and at its boundaries.<br />
A cubic spline interpolation function s(x) to <strong>the</strong> sampling points x0 < x1 < . . . <<br />
xn−1 < xn and <strong>the</strong> corresponding sampling values yj (j = 0, 1, 2, . . . , n) is defined<br />
by [25]:<br />
• s (xj) = yj for (j = 0, 1, . . . , n);<br />
• s (x) is for x ∈ [xi, xi+1] (i = 0, . . . , n − 1) a polynomial <strong>of</strong> at most <strong>of</strong> order 3;<br />
• s (x) ∈ C 2 ([x0, xn]);<br />
• s ′′ (x0) = s ′′ (xn) = 0.
4.6. Look-Up Tables 25<br />
In <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>CKMfitter</strong> version, polynomials <strong>of</strong> third order<br />
y(x) = a + bx + cx 2 + dx 3<br />
(4.5)<br />
are used as cubic spline interpolation functions, where a, b, c and d are <strong>the</strong> spline<br />
coefficients. They are calculated in a separate notebook using a public available<br />
routine coded by Joseph M. Herrmann [26]. Its result is a six column look-up<br />
table in ASCII format, which is used as input file for <strong>the</strong> fit. The column order <strong>of</strong><br />
<strong>the</strong> LUT is given in Table 4.4 and <strong>the</strong> number <strong>of</strong> lines is written in <strong>the</strong> first row.<br />
There is also an intrinsic <strong>Ma<strong>the</strong>matica</strong> function called SplineFit, which generates<br />
a spline function object. Due to <strong>the</strong> fact that <strong>the</strong> generated spline function doesn’t<br />
depend directly on <strong>the</strong> observable values, it cannot be used in <strong>CKMfitter</strong> to create<br />
<strong>the</strong> LUTs.<br />
Column: 1 2 3 4 5 6<br />
Content: value <strong>of</strong> observable χ 2 a b c d<br />
4.6.2 The Tableau Function<br />
Table 4.4: LUT column order<br />
The numerical contribution to <strong>the</strong> χ 2 -function <strong>of</strong> variables with LUT input is calculated<br />
by separate FORTRAN functions in <strong>the</strong> file specialfunctions.f. The function<br />
TABLEAU provides <strong>the</strong> χ 2 -contribution, where DTABLEAUO2 calculates <strong>the</strong> contributions<br />
to its gradients 3 . During <strong>the</strong> first call <strong>of</strong> TABLEAU, <strong>the</strong> different LUTs are<br />
loaded using <strong>the</strong> separate subroutine LoadLUT.<br />
The functions TABLEAU and DTABLEAUO2 are called during <strong>the</strong> fit for each prediction<br />
value <strong>of</strong> <strong>the</strong> fit parameter. Using <strong>the</strong> spline coefficients <strong>of</strong> <strong>the</strong> next lower entry in <strong>the</strong><br />
LUT, <strong>the</strong> contribution to <strong>the</strong> χ 2 -function or its gradient is calculated. The source<br />
code <strong>of</strong> <strong>the</strong>se functions is available in Appendix D.<br />
It is possible that <strong>the</strong> predicted value <strong>of</strong> <strong>the</strong> fit lies outside <strong>of</strong> <strong>the</strong> LUT boundaries.<br />
In this case, <strong>the</strong> TABLEAU function will be constantly continued using <strong>the</strong> first (last)<br />
value <strong>of</strong> <strong>the</strong> LUT, if <strong>the</strong> prediction is smaller (larger) than <strong>the</strong> lower (upper) bound.<br />
The gradient, obtained by DTABLEAUO2 is set to zero. This can lead to critical effects<br />
at <strong>the</strong> boundaries. A possibility to avoid such critical effects is to enlarge <strong>the</strong> LUTs<br />
periodically, if this is allowed by <strong>the</strong> variable. Examples for <strong>the</strong>se cases are <strong>the</strong> UT<br />
angles α and γ (see Section 5.1).<br />
3 DTABLEAUO2 (“O2” means over two) calculates <strong>the</strong> half <strong>of</strong> <strong>the</strong> gradient contribution.
26 Chapter 4. A <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>Version</strong> <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> <strong>Package</strong><br />
4.7 Performance Tests<br />
The <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package was created with <strong>the</strong> goal<br />
to achieve a significant fit time reduction compared to <strong>the</strong> original version. Since<br />
both are complex packages, which include also fit preparation and result processing,<br />
an objective comparison <strong>of</strong> <strong>the</strong> pure minimization routines is beyond <strong>the</strong> scope <strong>of</strong> this<br />
<strong>the</strong>sis. However, <strong>the</strong>re is <strong>the</strong> possibility to compare <strong>the</strong> full analysis process using<br />
a test job, here <strong>the</strong> Standard Global CKM Fit (see Chapter 5). The conditions, for<br />
<strong>the</strong> comparison tests done in this work, are summarized in Table 4.5.<br />
Hardware/S<strong>of</strong>tware Test job<br />
CPU: Intel P III Analysis: SM global fit<br />
Frequency: 1266 MHz Scan: (¯ρ,¯η) plane<br />
Memory: 2048 MB RAM Granularity: 200<br />
OS: Scientific Linux 3.0.3 fits per point: 2<br />
Compiler: gnu f77 -O<br />
Table 4.5: Test conditions<br />
Figure 4.3 shows <strong>the</strong> results <strong>of</strong> <strong>the</strong> Standard Global CKM Fit and its single constraints<br />
in <strong>the</strong> (¯ρ,¯η) plane, produced by both versions <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package.<br />
The results presented numerically in Table 4.6 and graphically in Figure 4.3 are<br />
nearly identical.<br />
Parameter Original <strong>CKMfitter</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> package<br />
A 0.2272 +0.0010<br />
−0.0010<br />
λ 0.812 +0.015<br />
−0.015<br />
¯ρ 0.187 +0.025<br />
−0.086<br />
¯η 0.333 +0.038<br />
−0.017<br />
J [10−5 ] 3.02 +0.36<br />
−0.17<br />
0.2272 +0.0010<br />
−0.0010<br />
0.813 +0.015<br />
−0.015<br />
0.187 +0.028<br />
−0.086<br />
0.333 +0.038<br />
−0.017<br />
3.02 +0.36<br />
−0.18<br />
Table 4.6: Numerical comparison <strong>of</strong> <strong>the</strong> Wolfenstein parameters A, λ, ¯ρ, ¯η and <strong>the</strong> Jarlskog<br />
invariant J. The errors are quoted as 1-CL=32 % ranges (1σ).<br />
Possible reasons for <strong>the</strong> very small discrepancies could be <strong>the</strong> usage <strong>of</strong> <strong>the</strong> different<br />
parametrizations <strong>of</strong> <strong>the</strong> Lattice QCD parameters, as explained in Section 5.1.12, <strong>the</strong><br />
different coded interplation routines for <strong>the</strong> LUTs or rounding effects value from <strong>the</strong><br />
determination <strong>of</strong> <strong>the</strong> central value.
4.7. Performance Tests 27<br />
η<br />
η<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
excluded area has CL > 0.95<br />
sin 2β<br />
Δm d<br />
ε K<br />
Δm s & Δm d<br />
α<br />
luded luded at at CL CL > > 0.95 0.95<br />
sol. w/ cos 2β < 0<br />
(excl. at CL > 0.95)<br />
0.1<br />
0<br />
|Vub /Vcb |<br />
γ<br />
β<br />
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
excluded area has CL > 0.95<br />
γ<br />
sin2β<br />
εK<br />
Δmd<br />
γ<br />
α<br />
ρ<br />
Δms<br />
& Δmd<br />
ρ<br />
α<br />
C K M<br />
f i t t e r<br />
ICHEP 2006<br />
α<br />
Vub/Vcb<br />
0<br />
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1<br />
εK<br />
sol. w/ cos2β<br />
< 0<br />
(excl. at CL > 0.95)<br />
β<br />
CKM<br />
γ<br />
f i t t e r<br />
ICHEP 2006<br />
Figure 4.3: Confidence level in <strong>the</strong> (¯ρ,¯η) plane for <strong>the</strong> Standard Global CKM Fit. The<br />
plots are produced with <strong>the</strong> original (top) and <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> (bottom) <strong>CKMfitter</strong><br />
package.<br />
α<br />
γ
28 Chapter 4. A <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>Version</strong> <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> <strong>Package</strong><br />
The test job, used for <strong>the</strong> fit time comparison, is <strong>the</strong> Standard Global CKM Fit. Its<br />
<strong>Ma<strong>the</strong>matica</strong> datacard is shown in detail in Appendix C and <strong>the</strong> result is <strong>the</strong> yellow<br />
area around <strong>the</strong> apex <strong>of</strong> <strong>the</strong> Unitarity Triangle in Figure 4.3. The fit time needed<br />
using <strong>the</strong> original <strong>CKMfitter</strong> package is about 23 hours, whereas <strong>the</strong> same result<br />
can be obtained by <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version in only ten minutes. This is a<br />
reduction <strong>of</strong> <strong>the</strong> fit time <strong>of</strong> more than a factor 100 for <strong>the</strong> Standard Global CKM<br />
Fit.<br />
Original <strong>CKMfitter</strong> package ≈ 23 h<br />
<strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>CKMfitter</strong> ≈ 10 min<br />
Table 4.7: Fit time comparison <strong>of</strong> test job runs<br />
Since <strong>the</strong> fit time depends on <strong>the</strong> hard- and s<strong>of</strong>tware conditions, <strong>the</strong> performance <strong>of</strong><br />
<strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>CKMfitter</strong> package is tested for several configurations. The<br />
results for <strong>the</strong> test job are summarized in Table 4.8.<br />
Hardware MHz Compiler Operation System Fit time / min<br />
AMD Opteron 2194 f77 -O Scientific Linux 3.0.7 04 : 42<br />
Intel P M 1497 ifort -O2 SUSE Linux 10.0 05 : 25<br />
Intel P M 1497 f77 -O SUSE Linux 10.0 07 : 06<br />
Intel P III 1266 ifort -O2 Scientific Linux 3.0.3 09 : 10<br />
Intel P III 1266 f77 -O Scientific Linux 3.0.3 10 : 18<br />
Table 4.8: Hardware and s<strong>of</strong>tware performance tests<br />
The most important conclusion is, that <strong>the</strong> fit results are independent <strong>of</strong> <strong>the</strong> used<br />
hard- and s<strong>of</strong>tware conditions. Using a high developed hardware system in combination<br />
with an optimized FORTRAN compiler leads to a fur<strong>the</strong>r fit time reduction.<br />
With a fit time reduction <strong>of</strong> more than a factor 100, <strong>the</strong> goal <strong>of</strong> <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong><br />
<strong>based</strong> <strong>CKMfitter</strong> development has been achieved.
Chapter 5<br />
Probing <strong>the</strong> Standard Model<br />
After a brief discussion <strong>of</strong> <strong>the</strong> relevant fit inputs, <strong>the</strong> most important results <strong>of</strong> <strong>the</strong><br />
global CKM matrix analysis in <strong>the</strong> framework <strong>of</strong> <strong>the</strong> Standard Model are presented.<br />
5.1 Fit Inputs<br />
In this section, <strong>the</strong> SM predictions and measurement methods <strong>of</strong> <strong>the</strong> most relevant<br />
observables are described. If not stated o<strong>the</strong>rwise, <strong>the</strong> expressions given are used<br />
in <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package. The input values are<br />
quoted in form <strong>of</strong> a central value and its uncertainties, where contributions <strong>of</strong> gaussian<br />
distributed experimental, statistical and <strong>the</strong>oretical uncertainties are quadratically<br />
added and labeled with “gauss”. Theoretical systematics, labeled with “<strong>the</strong>o”,<br />
are added linearly and treated in <strong>the</strong> Rfit scheme. In <strong>the</strong> case <strong>of</strong> constraints with<br />
ambiguous solutions (as for <strong>the</strong> UT angles α and γ), <strong>the</strong> experimental likelihood<br />
function is directly used as fit input.<br />
Since a complete review <strong>of</strong> <strong>the</strong> input measurements is beyond <strong>the</strong> scope <strong>of</strong> this <strong>the</strong>sis,<br />
more detailed informations can be obtained from <strong>the</strong> References [4, 8, 9] and references<br />
<strong>the</strong>rein. The values <strong>of</strong> <strong>the</strong> fit inputs are <strong>the</strong> most recent results as presented at<br />
ICHEP 2006 and are summarized in Table 5.1. The final plots have been prepared<br />
by Vincent Tisserand using <strong>the</strong> macros as described in Chapter 4.<br />
5.1.1 |Vud|<br />
The most precise determination <strong>of</strong> |Vud| stems from superallowed nuclear beta decays.<br />
Taking <strong>the</strong> average <strong>of</strong> <strong>the</strong> most precise experimental results yields [27]:<br />
|Vud| = 0.97377 ± 0.00027gauss . (5.1)<br />
Superallowed nuclear beta decays are pure weak vector transitions (0 + → 0 + ). Thus,<br />
<strong>the</strong> extraction <strong>of</strong> |Vud| is <strong>the</strong>oretically very clean, since <strong>the</strong>oretical uncertainties<br />
in electroweak radiative corrections, isospin violating electromagnetic effects and<br />
nuclear structure dependence are under good <strong>the</strong>oretical control.<br />
29
30 Chapter 5. Probing <strong>the</strong> Standard Model<br />
5.1.2 |Vus|<br />
The matrix element |Vus| has been extracted from semileptonic Kaon decays (Kl3),<br />
e. g. K + → π 0 e + νe. Its current world average (WA) is [9]:<br />
|Vus|Kl3 = 0.2257 ± 0.0021gauss . (5.2)<br />
|Vus| as well as |Vud| are <strong>the</strong> main inputs to constrain <strong>the</strong> Wolfenstein parameter λ.<br />
However, |Vus| depends on large <strong>the</strong>oretical uncertainties from <strong>the</strong> determination <strong>of</strong><br />
<strong>the</strong> form factor f+(0). O<strong>the</strong>r possibilities to determine |Vus| involve for instance<br />
leptonic Kaon and Pion decays, semileptonic Hyperon decays or hadronic τ decays.<br />
5.1.3 |Vcb|<br />
The magnitude <strong>of</strong> |Vcb| is determined from exclusive and inclusive measurements 1<br />
<strong>of</strong> semileptonic B-meson decays to charmed final states (b → clν). Because <strong>of</strong> <strong>the</strong>oretical<br />
uncertainties on <strong>the</strong> form factor calculation, <strong>the</strong> exclusive determination is<br />
less precise compared to <strong>the</strong> inclusive one and thus not used in this work. Due to<br />
<strong>the</strong> large statistics, <strong>the</strong> inclusive WA [9]:<br />
|Vcb|incl = (41.7 ± 0.7gauss) · 10 −3<br />
(5.3)<br />
is already below <strong>the</strong> 2% level. It is <strong>the</strong> main input to constrain <strong>the</strong> Wolfenstein<br />
parameter A.<br />
5.1.4 |Vub|<br />
Analogous to |Vcb|, <strong>the</strong> CKM matrix element |Vub| is determined from exclusive and<br />
inclusive measurements <strong>of</strong> b → ulν transitions. The inclusive determination suffers<br />
from large B → Xclν background and uses <strong>the</strong>refore phase space regions where <strong>the</strong><br />
charm background is kinematically suppressed. Hadronic effects enter in leading<br />
order (LO) via one non-perturbative shape function, which has been extracted from<br />
<strong>the</strong> photon energy spectrum in B → Xsγ. For <strong>the</strong> inclusive WA, <strong>the</strong> BLNP [28]<br />
calculation is chosen [29] 2 :<br />
|Vub| incl. = (4.48 ± 0.24gauss ± 0.39<strong>the</strong>o) · 10 −3 , (5.4)<br />
The <strong>the</strong>oretical error is obtained by adding linearly <strong>the</strong> contributions from weak<br />
annihilation, subleading shape functions and <strong>the</strong> Heavy-Quark Expansion (HQE)<br />
uncertainty on <strong>the</strong> b-quark mass.<br />
Direct measurements <strong>of</strong> exclusive channels depend significantly on form factor calculations.<br />
The form factors can be calculated using unquenched Lattice QCD (LQCD)<br />
or QCD Sum Rules, which still leads to larger <strong>the</strong>oretical uncertainties compared to<br />
<strong>the</strong> inclusive determination.<br />
1 Exclusive means <strong>the</strong> full reconstruction <strong>of</strong> a single mode, e. g. B + → ¯ D 0 e + νe, where inclusive<br />
measurements integrate over all possible final states, B → Xclν.<br />
2 The central value has been shifted after <strong>the</strong> ICHEP06 conference to 4.49 · 10 −3 .
5.1. Fit Inputs 31<br />
1 – CL<br />
1.2<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
C K M<br />
f i t t e r<br />
ICHEP 06<br />
B → ππ<br />
B → ρπ<br />
B → ρρ<br />
Combined<br />
CKM fit<br />
0<br />
0 20 40 60 80 100 120 140 160 180<br />
α (deg)<br />
WA<br />
1 – CL<br />
1.2<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
C K M<br />
f i t t e r<br />
FPCP 06<br />
Full frequentist treatment on MC basis<br />
CKM fit<br />
no γ meas. in fit<br />
D ( * ) K ( * ) GLW + ADS<br />
D ( * ) K ( * ) WA<br />
GGSZ Combined<br />
0<br />
0 20 40 60 80 100 120 140 160 180<br />
γ (deg)<br />
Figure 5.1: Left: Confidence level on <strong>the</strong> UT angle α obtained from <strong>the</strong> combination<br />
(green shaded) <strong>of</strong> B → ππ (green curve), B → ρπ (red curve) and B → ρρ (blue curve).<br />
Right: Confidence level on <strong>the</strong> UT angle γ obtained from <strong>the</strong> combination (green shaded)<br />
<strong>of</strong> <strong>the</strong> GLW, ADS and GGSZ methods (see text).<br />
5.1.5 The UT angle α<br />
The direct constraint on <strong>the</strong> UT angle α is obtained from charmless B decays.<br />
The best current knowledge comes from a combination <strong>of</strong> <strong>the</strong> two body isospin<br />
analysis [10] <strong>of</strong> B → ππ and B → ρρ decays and <strong>the</strong> Dalitz-plot analysis [10]<br />
<strong>of</strong> B → ρπ decays. Figure 5.1 (left) shows <strong>the</strong> confidence level on α from <strong>the</strong><br />
combined constraint <strong>of</strong> <strong>the</strong>se three measurements. The isospin analysis as well as<br />
<strong>the</strong> α extraction from B → ρπ have been performed by <strong>the</strong> <strong>CKMfitter</strong> group using<br />
<strong>the</strong> most recent results <strong>of</strong> BABAR and Belle. Due to its non-gaussian shape, <strong>the</strong><br />
χ 2 -contour is used as a LUT input file as described in Chapter 4.<br />
5.1.6 sin 2β<br />
The currently best constraint on (¯ρ,¯η) comes from determination <strong>of</strong> <strong>the</strong> CP-violating<br />
parameter sin 2β. It is measured with high precision at <strong>the</strong> B-factories from <strong>the</strong><br />
interference between decays with and without mixing in b → c¯cs transitions. The<br />
current world average is obtained by combining <strong>the</strong> measurements <strong>of</strong> B → (c¯c)K 0<br />
(BABAR) and B → J/ψKS,L (Belle) and yields [30]:<br />
sin 2β = 0.675 ± 0.026gauss . (5.5)
32 Chapter 5. Probing <strong>the</strong> Standard Model<br />
5.1.7 The UT angle γ<br />
The extraction <strong>of</strong> <strong>the</strong> UT angle γ stems from measurements <strong>of</strong> direct CP violation<br />
in B → D (∗) K (∗) decays. It is obtained from a combination <strong>of</strong> <strong>the</strong> Gronau-<br />
London-Wyler (GLW) [31,32], Atwood-Dunietz-Soni (ADS) [33,34] and Giri-<br />
Grossman-S<strong>of</strong>fer-Zupan (GGSZ) [35, 36] methods. Figure 5.1 shows <strong>the</strong> combined<br />
result using a frequentist method, which is advocated by <strong>the</strong> <strong>CKMfitter</strong><br />
group [16]. Analogous to <strong>the</strong> UT angle α <strong>the</strong> corresponding χ 2 -contour is used<br />
as a LUT input file (see Chapter 4).<br />
5.1.8 |ɛK|<br />
In <strong>the</strong> Standard Model, <strong>the</strong> absolute value <strong>of</strong> <strong>the</strong> CP-violating parameter in <strong>the</strong><br />
neutral Kaon system |ɛK| is defined by:<br />
|ɛK| = G2 F m2 W mK<br />
12 √ 2π2 f<br />
∆mK<br />
2 �<br />
KBK<br />
+ 2ηctS(xc, xt)Im[VcsV ∗ ∗<br />
cdVtsVtd ]<br />
ηccS(xc)Im[(VcsV ∗<br />
cd )2 ] + ηttS(xt)Im[(VtsV ∗<br />
td )2 ]<br />
�<br />
, (5.6)<br />
where fK is <strong>the</strong> Kaon decay constant. The hadronic matrix element <strong>of</strong> <strong>the</strong> |∆S| = 2<br />
box diagram is proportional to � √ �2, fK BK where BK is <strong>the</strong> bag parameter. It<br />
has been obtained from Lattice QCD calculations and is <strong>the</strong> primary source <strong>of</strong> <strong>the</strong><br />
<strong>the</strong>oretical uncertainties for <strong>the</strong> prediction <strong>of</strong> |ɛK|. The parameters ηqiqj are nextto-leading<br />
order (NLO) QCD corrections to <strong>the</strong> Inami-Lim functions S, which are<br />
listed in Appendix A. The values <strong>of</strong> ηct and ηtt are shown in Table 5.2, while due to<br />
large uncertainties, a parameterization [37–39] is used for ηcc. This is also shown in<br />
Appendix A. The used average <strong>of</strong> |ɛK| is [40]:<br />
5.1.9 ∆md<br />
|ɛK| = (2.221 ± 0.008gauss) · 10 −3 . (5.7)<br />
The B 0 - ¯ B 0 oscillation frequency is expressed through <strong>the</strong> mass difference ∆md between<br />
<strong>the</strong> mass eigenstates BH and BL. It is predicted in <strong>the</strong> Standard Model as:<br />
∆md = G2 F<br />
6π 2 ηB mBd f 2 Bd Bd m 2 W S(xt) |VtdV ∗<br />
tb |2 , (5.8)<br />
where ηB is a perturbative QCD correction<br />
√<br />
to <strong>the</strong> Inami-Lim function. The hadronic<br />
matrix element is proportional to fBd Bd, where <strong>the</strong> decay constant fBd and bag<br />
parameter Bd are taken from LQCD (see also Section 5.1.12). ∆md has been measured<br />
to high precision in many experiments. The WA is [29]:<br />
∆md = 0.507 ± 0.004gauss . (5.9)
5.1. Fit Inputs 33<br />
5.1.10 ∆ms<br />
In analogy to ∆md, <strong>the</strong> mass difference <strong>of</strong> <strong>the</strong> two mass eigenstates in <strong>the</strong> neutral<br />
Bs-system is predicted in <strong>the</strong> Standard Model as:<br />
∆ms = G2 F<br />
6π2 ηB mBsf 2 Bs Bs m 2 W S(xt) |VtsV ∗<br />
tb |2 . (5.10)<br />
The most recent experimental results, presented at <strong>the</strong> winter conferences 2006,<br />
are a two-sided limit on ∆ms, determined by D∅ [41] and a measurement with a<br />
significance <strong>of</strong> 99.5% by CDF [42]:<br />
∆m CDF<br />
s<br />
= � 17.33 +0.42<br />
−0.21<br />
� −1<br />
± 0.07syst ps . (5.11)<br />
As input for <strong>the</strong> CKM matrix analysis, <strong>the</strong> χ 2 -contour obtained from <strong>the</strong> measured<br />
amplitude spectrum <strong>of</strong> <strong>the</strong> B 0 s - ¯ B 0 s oscillation from both experiments is used in this<br />
work [29].<br />
5.1.11 The Branching Fraction B(B + → τ + ντ)<br />
Ano<strong>the</strong>r constraint on <strong>the</strong> CKM matrix element |Vub| comes from purely leptonic<br />
decays <strong>of</strong> charged B mesons. The tree-level process is mediated in <strong>the</strong> Standard<br />
Model by annihilation <strong>of</strong> <strong>the</strong> charged B meson into a pure leptonic final state via<br />
a virtual W boson as shown in Figure 5.2. Its branching fraction B is predicted to<br />
be [43]:<br />
B(B + → l + νl) = G2 F mBm 2 l<br />
8π<br />
where <strong>the</strong> decay constant fBd<br />
f 2 �<br />
2<br />
Bd<br />
|Vub| 1 − m2 l<br />
m2 �2<br />
B<br />
, (5.12)<br />
is taken from Lattice QCD calculations (see also<br />
5.1.12). Due to <strong>the</strong> smallness <strong>of</strong> |Vub| 2 and an additional helicity suppression proportional<br />
to m2 l , <strong>the</strong> SM expectation is ra<strong>the</strong>r small. Thanks to <strong>the</strong> successful<br />
running <strong>of</strong> <strong>the</strong> B-factories, <strong>the</strong> decay B + → τ + ντ is now in reach <strong>of</strong> <strong>the</strong> experimental<br />
sensitivity. The fit input is a combination <strong>of</strong> <strong>the</strong> most recent experimental<br />
likelihoods from BABAR [44] and Belle [45], where <strong>the</strong> systematics <strong>of</strong> <strong>the</strong> BABAR<br />
result have been neglected.<br />
b<br />
u<br />
+<br />
B<br />
+<br />
W<br />
Figure 5.2: Tree-level contributions to leptonic B + decays in <strong>the</strong> SM<br />
Since several models predict New Physics contributions to this tree-level process,<br />
purely leptonic decays play also an important role on testing extensions <strong>of</strong> <strong>the</strong> Standard<br />
Model. A model which predicts additional contributions from charged Higgs<br />
bosons is discussed in Chapter 6.2.<br />
+<br />
l<br />
νl
34 Chapter 5. Probing <strong>the</strong> Standard Model<br />
5.1.12 Decay Constants and Bag Parameters<br />
The observables ∆md, ∆ms and B(B + → l + νl) depend on a set <strong>of</strong> decay constants<br />
and bag parameters:<br />
fBd , fBs , Bd and Bs . (5.13)<br />
They can be calculated using Lattice QCD, which leads to significant <strong>the</strong>oretical uncertainties<br />
for <strong>the</strong> predictions <strong>of</strong> <strong>the</strong> observables discussed in Chapters 5.1.9 to 5.1.11.<br />
Since some combinations <strong>of</strong> <strong>the</strong>se parameters can be calculated more precisely, <strong>the</strong><br />
parameters:<br />
fBd , Bd and ξ = fBs<br />
√<br />
Bs<br />
√<br />
fBd Bd<br />
, (5.14)<br />
have been used in <strong>the</strong> original <strong>CKMfitter</strong> package. Unfortunately, <strong>the</strong> LQCD calculation<br />
<strong>of</strong> fBd has large <strong>the</strong>oretical uncertainties, originating from <strong>the</strong> chiral extrapolation<br />
<strong>of</strong> <strong>the</strong> light lattice quark masses (mu,md) to <strong>the</strong> physical quark masses.<br />
Hence, <strong>the</strong> <strong>the</strong>oretical uncertainties <strong>of</strong> fBd and ξ are anti-correlated. This is not <strong>the</strong><br />
case in <strong>the</strong> different but ma<strong>the</strong>matically equivalent parametrization:<br />
fBs ,<br />
fBs<br />
fBd<br />
, Bs and Bs<br />
Bd<br />
, (5.15)<br />
which is used in <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package. It is<br />
free from <strong>the</strong> large correlation due to <strong>the</strong> uncertainty from <strong>the</strong> chiral extrapolation.<br />
However, possible smaller correlations are neglected since <strong>the</strong>ir correlation matrix<br />
has not been published.<br />
5.1.13 ASL<br />
Measurements <strong>of</strong> semileptonic B-meson decays, i.e. B → Xlν, allow <strong>the</strong> determination<br />
<strong>of</strong> <strong>the</strong> magnitude <strong>of</strong> CP violation in neutral B-meson mixing.<br />
The flavor specific CP asymmetry ASL as defined in Equation (2.42) is related to<br />
<strong>the</strong> mixing matrix elements in <strong>the</strong> SM via:<br />
� �<br />
Γ12<br />
ASL = Im . (5.16)<br />
Since <strong>the</strong> absolute value <strong>of</strong> Γ12/M12 is suppressed to <strong>the</strong> percent level through<br />
� � �<br />
� Γ12 �<br />
� �<br />
m2 b<br />
�M12<br />
� = O<br />
m2 �<br />
, (5.17)<br />
W<br />
ASL is expected to be ra<strong>the</strong>r small. It is additionally suppressed by ano<strong>the</strong>r order<br />
<strong>of</strong> magnitude through <strong>the</strong> GIM3 factor z = m2 c/m2 b � 0.1. In <strong>the</strong> presence <strong>of</strong> New<br />
3 In <strong>the</strong> SM, <strong>the</strong> Glashow-Iliopoulos-Maiani (GIM) mechanism leads to a cancellation <strong>of</strong><br />
FCNC processes at tree level.<br />
M12
5.1. Fit Inputs 35<br />
Physics <strong>the</strong> GIM suppression may be diminished, thus a precise measurement <strong>of</strong> ASL<br />
is an important constraint to New Physics models. Its input value is obtained from<br />
a weighted average <strong>of</strong> measurements by CLEO [46], BABAR [47,48] and Belle [49]:<br />
ASL = −0.0005 ± 0.0055gauss . (5.18)<br />
where di-leptonic modes as well as hadronic modes are taken into account. A measurement<br />
by D∅ [50] has not been included since it has also contributions from<br />
Bs-meson mixing.<br />
Since <strong>the</strong> <strong>the</strong>oretical prediction on Γ12/M12 still depends on large QCD uncertainties,<br />
it has been calculated beyond leading order in QCD.<br />
Leading Order<br />
The basic expression <strong>of</strong> Γ12/M12 is given in <strong>the</strong> SM through [51]:<br />
� �SM Γ12<br />
4πm<br />
= −<br />
M12<br />
2 b<br />
3m2 W ¯ηBS(xt)<br />
�<br />
5<br />
8 (K2 − K1) B′<br />
S<br />
B +<br />
�<br />
K1 + K2<br />
�<br />
2<br />
� �<br />
K1 m2 B − m<br />
+ − K2<br />
2 2 b 1<br />
B − 3z (K1<br />
1 − ¯ρ − i¯η<br />
+ K2)<br />
m 2 b<br />
(1 − ¯ρ) 2 + ¯η 2<br />
�<br />
,<br />
(5.19)<br />
where only contributions <strong>of</strong> O(z) and O(1/mb) are considered. The parameter ¯ηB<br />
is <strong>the</strong> scale dependend QCD correction factor to <strong>the</strong> Inami-Lim functions S(xi) and<br />
K1, K2 are linear combinations <strong>of</strong> Wilson coefficients.<br />
The hadronic matrix elements <strong>of</strong> <strong>the</strong> local |∆B| = 2 contributions are parametrized<br />
through B = B(mb) and B ′<br />
S<br />
= B′<br />
S (mb):<br />
〈 ¯ Bd|( ¯ bidi)V −A( ¯ bjdj)V −A|Bd〉 = 8<br />
3 f 2 Bd m2 Bd B(mb) (5.20)<br />
〈 ¯ Bd|( ¯ bidi)S−P ( ¯ bjdj)S−P |Bd〉 = − 5<br />
3 f 2 Bd m2 Bd B′<br />
S(mb) (5.21)<br />
≡ − 5<br />
3 f 2 Bd m2 Bd<br />
m2 Bd<br />
( ¯mb(mb) + ¯md(mb)) 2 BS(mb) .<br />
The scale-dependence can be separated through <strong>the</strong> factor [52]:<br />
bB(mb) = [αS(mb)] −6/23<br />
�<br />
1 + αS(mb)<br />
�<br />
5165<br />
, (5.22)<br />
4π 3174<br />
which leads to <strong>the</strong> scale-independent parameters:<br />
ηB = ¯ηB<br />
bB(mb)<br />
and Bd = B(mb) · bB(mb) . (5.23)<br />
Assuming ¯md(mb) → 0, <strong>the</strong> parameter BS(mb) is used to parametrize <strong>the</strong> scalarpseudoscalar<br />
hadronic matrix element instead <strong>of</strong> B ′<br />
S (mb). It depends one <strong>the</strong> scale mb.<br />
The input values <strong>of</strong> all parameters are summarized in Table 5.2.
36 Chapter 5. Probing <strong>the</strong> Standard Model<br />
Next-to-Leading Order<br />
In NLO, higher corrections in z and 1/mb are taken into account as well as penguin<br />
contributions and QCD corrections. The most recent progress is summarized in<br />
Reference [53] and references <strong>the</strong>rein.<br />
The expansion <strong>of</strong> Γ12/M12 up to <strong>the</strong> order <strong>of</strong> λ 2 u/λ 2 t is given by [53]:<br />
Γ12<br />
M12<br />
= λ2 �<br />
t<br />
−Γ<br />
M12<br />
cc<br />
12 + 2 (Γ uc<br />
12 − Γ cc<br />
12) λu<br />
+ (2Γ<br />
λt<br />
uc<br />
12 − Γ cc<br />
12 − Γ uu<br />
12 ) λ2u λ2 �<br />
t<br />
(5.24)<br />
where λi = V ∗<br />
id Vib, for i = u, c, t. The mixing matrix element M12 is predicted in <strong>the</strong><br />
SM as [54]:<br />
λ 2 t<br />
G 2 F<br />
12π 2 mBd ηBB(mb)bB(mb)f 2 Bd m2 W S(xt) . (5.25)<br />
The coefficients Γab 12 are expressed through [53]:<br />
Γ ab<br />
12 = G2 F m2 b<br />
−<br />
24π f 2 Bd MBd<br />
� �<br />
F ab (z) + P ab (z)<br />
�<br />
F ab<br />
S (z) + P ab<br />
�<br />
5<br />
S (z)<br />
3 B′ S(mb)<br />
�<br />
8<br />
3 B(mb)<br />
�<br />
, (5.26)<br />
+ Γ ab<br />
12,1/mb<br />
where <strong>the</strong> short-distance coefficients F ab<br />
(S) (z) contain <strong>the</strong> contributions from |∆B| = 1<br />
(z) coefficients contain <strong>the</strong> contributions from penguin opera-<br />
have been computed in Ref. [53], Γcc 12 is given in Ref. [55]<br />
is derived from Γcc 12 taking <strong>the</strong> limit z → 0. In addition, <strong>the</strong>y depend on<br />
<strong>the</strong> Wilson coefficients C1, . . . , C6 and C8, which are also given in Table 5.2. The<br />
contains <strong>the</strong> corrections <strong>of</strong> order 1/mb.<br />
operators and <strong>the</strong> P ab<br />
(S)<br />
tors. The coefficients in Γ uc<br />
12<br />
and Γ uu<br />
12<br />
term Γ ab<br />
12,1/mb<br />
A comparison <strong>of</strong> <strong>the</strong> SM predictions <strong>of</strong> ASL at LO and NLO in QCD is shown in <strong>the</strong><br />
Section 5.2. The implementation <strong>of</strong> <strong>the</strong> NLO prediction for Γ12/M12 has finished<br />
immediately before <strong>the</strong> end <strong>of</strong> this <strong>the</strong>sis and <strong>the</strong>refore not all cross checks have been<br />
performed up to this point. As a consequence, <strong>the</strong> <strong>the</strong>ory prediction for Γ12/M12 in<br />
this <strong>the</strong>sis is used at LO to produce quantitative results if not stated o<strong>the</strong>rwise.
5.1. Fit Inputs 37<br />
Parameter Value ± Error(s) Reference<br />
Errors<br />
GS TH<br />
|Vud| (nuclei) 0.97377 ± 0.00027 [27] ⋆ -<br />
|Vus| (Kℓ3) 0.2257 ± 0.0021 [9] ⋆ -<br />
|Vub| (incl.) (4.48 ± 0.24 ± 0.39) × 10 −3 [9] ⋆ ⋆<br />
|Vcb| (incl.) (41.70 ± 0.70) × 10 −3 [29] ⋆ -<br />
|εK| (2.221 ± 0.008) × 10 −3 [40] ⋆ -<br />
∆md (0.507 ± 0.004) ps −1 [29] ⋆ -<br />
ASL −0.0005 ± 0.0055 [46–49] ⋆ -<br />
∆ms Amplitude spectrum+CDF -LogL [41, 42] ⋆ -<br />
sin(2β) [c¯c] 0.675 ± 0.026 [30] ⋆ -<br />
S +−<br />
ππ −0.58 ± 0.09 [29] ⋆ -<br />
C +−<br />
ππ −0.39 ± 0.07 [29] ⋆ -<br />
C 00<br />
ππ −0.35 ± 0.33 [29] ⋆ -<br />
Bππ all charges Inputs to isospin analysis [29] ⋆ -<br />
S +−<br />
ρρ,L −0.22 ± 0.22 [29] ⋆ -<br />
C +−<br />
ρρ,L −0.06 ± 0.14 [29] ⋆ -<br />
Bρρ,L all charges Inputs to isospin analysis [29] ⋆ -<br />
B 0 → (ρπ) 0 → 3π Time-dependent Dalitz analysis [56] ⋆ -<br />
B − → D (∗) K (∗)− Inputs to GLW analysis [57] ⋆ -<br />
B − → D (∗) K (∗)− Inputs to ADS analysis [57] ⋆ -<br />
B − → D (∗) K (∗)− GGSZ Dalitz analysis [57] ⋆ -<br />
B(B − → τ − ντ ) Experimental likelihoods [44, 45] ⋆ -<br />
mc(mc) (1.24 ± 0.037 ± 0.095) GeV/c 2 [58] ⋆ ⋆<br />
mt(mt) (162.3 ± 2.2) GeV/c 2 [59] ⋆ -<br />
m K + (493.677 ± 0.016) MeV/c 2 [60] - -<br />
∆mK (3.4833 ± 0.0066) × 10 −12 MeV/c 2 [60] - -<br />
mBd (5.2794 ± 0.0005) GeV/c 2 [60] - -<br />
mBs (5.3696 ± 0.0024) GeV/c 2 [60] - -<br />
mW (80.425 ± 0.039) GeV/c 2 [60] - -<br />
GF 1.16637 × 10 −5 GeV −2 [60] - -<br />
fK (159.8 ± 1.5) MeV [60] - -<br />
Table 5.1: Inputs to <strong>the</strong> CKM fits. If not stated o<strong>the</strong>rwise: for two errors given, <strong>the</strong> first<br />
stands for statistical and accountable systematic uncertainties and <strong>the</strong> second stands for systematic<br />
<strong>the</strong>oretical uncertainties. The last two columns indicate <strong>the</strong> Rfit treatment <strong>of</strong> <strong>the</strong><br />
input parameters: measurements or parameters that have statistical errors (we include here<br />
experimental systematics) are marked in <strong>the</strong> “GS” column by an asterisk; measurements or<br />
parameters that have systematic <strong>the</strong>oretical errors are marked in <strong>the</strong> “TH” column by an<br />
asterisk. Upper part: experimental determinations <strong>of</strong> <strong>the</strong> CKM matrix elements. Middle<br />
part: CP-violation and mixing observables. Lower part: parameters used in SM<br />
predictions that are obtained from experiment.
38 Chapter 5. Probing <strong>the</strong> Standard Model<br />
Parameter Value ± Error(s) Reference<br />
Errors<br />
GS TH<br />
BK 0.79 ± 0.04 ± 0.09 [61] ⋆ ⋆<br />
αS(m2 Z )<br />
ηct<br />
0.1176 ± 0.0020<br />
0.47 ± 0.04<br />
[60]<br />
[37]<br />
-<br />
-<br />
⋆<br />
⋆<br />
ηtt 0.5765 ± 0.0065 [37, 39] - ⋆<br />
ηB(MS) 0.551 ± 0.007 [52] - ⋆<br />
fBs (236.5 ± 31.5 ± 1) MeV [62] ⋆ ⋆<br />
Bs 1.37 ± 0.14 [62] ⋆ -<br />
fBs /fBd 1.24 ± 0.04 ± 0.06 [62] ⋆ -<br />
Bs/Bd 1.00 ± 0.02 [62] ⋆ ⋆<br />
mb 4.8 ± 0.1 [51] - ⋆<br />
z 0.085 ± 0.01 [51] - ⋆<br />
α S(mb) 0.22 [51] - -<br />
BS(mb) 0.83 ± 0.03 ± 0.07 [51] ⋆ ⋆<br />
K1 −0.295 [51] - -<br />
K2 1.162 [51] - -<br />
C1 −0.184 [52] - -<br />
C2 1.078 [52] - -<br />
C3 0.013 [52] - -<br />
C4 −0.035 [52] - -<br />
C5 0.009 [52] - -<br />
C6 −0.41 [52] - -<br />
C8 −0.14795 [63] - -<br />
Table 5.2: Inputs to <strong>the</strong> CKM fits (continued). Upper part: parameters <strong>of</strong> <strong>the</strong> SM<br />
predictions obtained from <strong>the</strong>ory. Middle part: parameters <strong>of</strong> <strong>the</strong> ASL predictions. Lower<br />
part: Wilson Coefficients.
5.2. Standard Model Fit Results 39<br />
5.2 Standard Model Fit Results<br />
The goal <strong>of</strong> <strong>the</strong> CKM matrix analysis is to test <strong>the</strong> quality <strong>of</strong> <strong>the</strong> agreement between<br />
<strong>the</strong> Standard Model predictions <strong>of</strong> <strong>the</strong> relevant observables and <strong>the</strong>ir experimental<br />
measurements. Fur<strong>the</strong>rmore, <strong>the</strong> CKM matrix parameters are constrained from fits,<br />
performed in <strong>the</strong> framework <strong>of</strong> Rfit, where <strong>the</strong> input values from Tables 5.1 & 5.2<br />
have been used. The fit results are provided numerically as well as graphically in<br />
one- and two-dimensional representations, e. g. in <strong>the</strong> (¯ρ,¯η) plane. They are obtained<br />
using <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong> package and have been partly<br />
presented at ICHEP 2006.<br />
The Standard Global CKM Fit includes <strong>the</strong> observables:<br />
|Vud| , |Vus| , |Vub| , |Vcb| , |εK| , ∆md , ∆ms , α , sin 2β and γ , (5.27)<br />
which can be considered as <strong>the</strong>oretically and experimentally well understood and<br />
provide significant constraints on <strong>the</strong> fit parameters. Its result is shown in Figure<br />
5.3, where for each individual constraint <strong>the</strong> 95% CL allowed belt is indicated<br />
in <strong>the</strong> (¯ρ,¯η) plane 4 .<br />
The left side <strong>of</strong> <strong>the</strong> Unitarity Triangle, Ru, is determined from <strong>the</strong> ratio <strong>of</strong> |Vub|<br />
and |Vcb|. Its 95 % CL belt is, to a very good approximation, a ring around<br />
(¯ρ, ¯η) = (0, 0). The right side Rt, is constrained by |Vtd| and |Vtb|, which have<br />
been obtained from measurements <strong>of</strong> <strong>the</strong> oscillaton frequencies ∆md and ∆ms. Be-<br />
cause <strong>of</strong> large <strong>the</strong>oretical uncertainties on fBd<br />
√ Bd, <strong>the</strong> constraint from ∆md is very<br />
loose. A much better result is obtained from <strong>the</strong> combination <strong>of</strong> ∆md and ∆ms,<br />
where most <strong>of</strong> <strong>the</strong> <strong>the</strong>oretical uncertainties from QCD correction factors and LQCD<br />
parameters cancel. The constraint on (¯ρ,¯η) from <strong>the</strong> γ determination is very loose<br />
and <strong>the</strong> UT angle α is ambiguous due to mirror solutions, as shown in Figure 5.1.<br />
The currently best known parameter is <strong>the</strong> UT angle β, which is well constrained<br />
from measurements <strong>of</strong> sin 2β. Since cos 2β is measured to be a positive number [64],<br />
its mirror solution is excluded at 95 % CL. A direct constraint on <strong>the</strong> altitude <strong>of</strong> <strong>the</strong><br />
Unitarity Triangle is provided by <strong>the</strong> measurements <strong>of</strong> |ɛK| leading to a hyperboliclike<br />
belt in <strong>the</strong> (¯ρ,¯η) plane.<br />
The allowed area from <strong>the</strong> Standard Global CKM Fit constrains <strong>the</strong> apex <strong>of</strong> <strong>the</strong><br />
Unitarity Triangle in <strong>the</strong> first quadrant <strong>of</strong> <strong>the</strong> (¯ρ,¯η) plane. It is covered consistently<br />
by all 95 % CL belts <strong>of</strong> <strong>the</strong> individual constraints. The Standard Global CKM<br />
Fit excludes <strong>the</strong> possibility <strong>of</strong> ¯η = 0 and thus a real CKM matrix at a CL <strong>of</strong> at<br />
most 99.9 %.<br />
4 For sin 2β, <strong>the</strong> 68 % CL (1σ) and 95.5 % CL (2σ) belts are given instead.
40 Chapter 5. Probing <strong>the</strong> Standard Model<br />
The results <strong>of</strong> <strong>the</strong> Standard Global CKM Fit for <strong>the</strong> Wolfenstein parameters A, λ, ¯ρ,<br />
¯η and <strong>the</strong> Jarlskog invariant J are shown in Figure 5.4. Fur<strong>the</strong>rmore, <strong>the</strong> fit results<br />
<strong>of</strong> <strong>the</strong> UT angles α, β and γ are compared with <strong>the</strong>ir SM predictions and <strong>the</strong>ir<br />
direct measurements. In this case, SM prediction means <strong>the</strong> result <strong>of</strong> <strong>the</strong> Standard<br />
Global CKM Fit without <strong>the</strong> respective parameter in <strong>the</strong> fit. The numerical results<br />
<strong>of</strong> <strong>the</strong> relevant observables and model parameters are summarized in Table 5.3 & 5.4.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
excluded area has CL > 0.95<br />
εK<br />
γ<br />
sin2β<br />
V /V ub cb<br />
α<br />
γ<br />
α<br />
excluded at CL > 0.95<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
γ<br />
sol. w/ cos2β<br />
< 0<br />
(excl. at CL > 0.95)<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
α<br />
Δms<br />
Δmd<br />
& Δm<br />
Figure 5.3: Confidence level in <strong>the</strong> (¯ρ,¯η) plane obtained from <strong>the</strong> Standard Global CKM Fit<br />
(red bordered yellow area) and from <strong>the</strong> individual constraints (colored belts). The shaded<br />
areas indicate 95 % CL allowed regions.<br />
d<br />
εK
5.2. Standard Model Fit Results 41<br />
1 - CL<br />
1 - CL<br />
1 - CL<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
0<br />
0.77 0.78 0.79 0.8 0.81 0.82 0.83 0.84 0.85 0.86<br />
A<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
0<br />
0.05 0.1 0.15 0.2 0.25<br />
ρ<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
-6<br />
× 10<br />
26 28 30 32 34 36<br />
× 10<br />
38<br />
J<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
CKM fit<br />
prediction<br />
measurement<br />
0<br />
0 0.1 0.2 0.3 0.4<br />
β<br />
0.5 0.6 0.7<br />
-6<br />
1 - CL<br />
1 - CL<br />
1 - CL<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
0<br />
0.224 0.225 0.226 0.227 0.228 0.229 0.23 0.231<br />
λ<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
0.28 0.3 0.32 0.34<br />
η<br />
0.36 0.38 0.4 0.42<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
CKM fit<br />
prediction<br />
measurement<br />
0<br />
0 0.5 1 1.5<br />
α<br />
2 2.5 3<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
CKM fit<br />
prediction<br />
measurement<br />
0<br />
0 0.5 1 1.5<br />
γ<br />
2 2.5 3<br />
Figure 5.4: Confidence level in <strong>the</strong> (¯ρ,¯η) plane on <strong>the</strong> Wolfenstein parameters, <strong>the</strong> UT<br />
angles and <strong>the</strong> Jarlskog invariant, obtained from <strong>the</strong> Standard Global CKM Fit. For α, γ<br />
and sin 2β, also <strong>the</strong> SM predictions and <strong>the</strong> direct measurements are shown.
42 Chapter 5. Probing <strong>the</strong> Standard Model<br />
The quality <strong>of</strong> <strong>the</strong> agreement between <strong>the</strong> SM predictions and <strong>the</strong> experimental data<br />
, which is a<br />
can be obtained from <strong>the</strong> interpretation <strong>of</strong> <strong>the</strong> test statistics χ2 min,ymod<br />
probe <strong>of</strong> <strong>the</strong> goodness-<strong>of</strong>-fit for <strong>the</strong> SM hypo<strong>the</strong>sis. The p-value P (χ2 min,ymod |SM)<br />
for <strong>the</strong> validity <strong>of</strong> <strong>the</strong> Standard Model needs to be calculated using Toy Monte Carlo<br />
simulations [4]. Since this is beyond <strong>the</strong> scope <strong>of</strong> this <strong>the</strong>sis, an approximative<br />
method is chosen instead.<br />
Assuming naively a gaussian behavior <strong>of</strong> <strong>the</strong> global likelihood function L(ymod) with<br />
Nd<strong>of</strong> = Nexp − Nmod = 6, <strong>the</strong> p-value can be approximately obtained from <strong>the</strong> Prob<br />
function, which is given in Equation (3.6). The global χ2-minimum <strong>of</strong> <strong>the</strong> Standard<br />
Global CKM Fit is:<br />
= 4.25 . (5.28)<br />
This leads to a p-value <strong>of</strong><br />
χ 2 min,ymod<br />
P (χ 2 min,ymod |SM) ≤ P rob(χ2min,ymod ) = 64.2 % (5.29)<br />
for <strong>the</strong> validity <strong>of</strong> <strong>the</strong> Standard Model.<br />
The main contribution to <strong>the</strong> global χ 2 -minimum stems from <strong>the</strong> slight disagreement<br />
<strong>of</strong> <strong>the</strong> measurements <strong>of</strong> |Vub| and sin 2β. The plots in Figure 5.5 show <strong>the</strong> SM<br />
prediction compared to <strong>the</strong> direct measurement for both observables, |Vub|incl. and<br />
sin 2β. In addition, <strong>the</strong> result <strong>of</strong> <strong>the</strong> Standard Global CKM Fit is shown.<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
CKM fit<br />
prediction<br />
measurement<br />
0.0035 0.004 0.0045 0.005 0.0055<br />
| Vub | incl.<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
CKM fit<br />
prediction<br />
measurement<br />
0<br />
0.5 0.6 0.7 0.8 0.9 1<br />
sin(2 β )<br />
Figure 5.5: Confidence level on |Vub|incl (left) and sin 2β (right) obtained from <strong>the</strong> Standard<br />
Global CKM Fit (blue curve), <strong>the</strong> SM predictions (red curve) and <strong>the</strong> direct measurements<br />
(green shaded).<br />
Ano<strong>the</strong>r possibility for <strong>the</strong> determination <strong>of</strong> <strong>the</strong> CKM matrix element |Vub| is provided<br />
by purely leptonic decays <strong>of</strong> charged B-mesons. Since <strong>the</strong> SM expectations for<br />
<strong>the</strong> corresponding branching fractions are ra<strong>the</strong>r small, only <strong>the</strong> decay B + → τ + ντ<br />
has been seen yet. Figure 5.6 shows <strong>the</strong> SM predictions from <strong>the</strong> Standard Global<br />
CKM Fit for B(B + → τ + ντ ) and B(B + → µ + νµ). In addition, <strong>the</strong> combined<br />
B + → τ + ντ measurement from BABAR and Belle and <strong>the</strong> corresponding fit result<br />
is shown.
5.2. Standard Model Fit Results 43<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
CKM fit<br />
prediction<br />
measurement<br />
0<br />
0 0.5 1 1.5 2 2.5 3<br />
+ +<br />
4<br />
BR(B → τ ν τ ) × 10<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
0<br />
0.2 0.3 0.4 0.5 0.6 0.7<br />
+ +<br />
6<br />
BR(B → μ ν μ ) × 10<br />
Figure 5.6: Confidence level on B(B + → τ + ντ ) and B(B + → µ + νµ) obtained from <strong>the</strong> SM<br />
prediction (red curve). Fur<strong>the</strong>rmore, <strong>the</strong> results <strong>of</strong> <strong>the</strong> direct measurement (green shaded)<br />
and <strong>the</strong> fit (blue curve) are given for <strong>the</strong> decay B + → τ + ντ .<br />
Since <strong>the</strong> SM prediction <strong>of</strong> <strong>the</strong> branching fraction <strong>of</strong> purely leptonic B-meson decays<br />
depends also on <strong>the</strong> decay constant fBd , a precise measurement can help to reduce<br />
<strong>the</strong>oretical uncertainties in <strong>the</strong> Standard Global CKM Fit. Figure 5.7 shows <strong>the</strong><br />
constraint on (¯ρ,¯η) coming from <strong>the</strong> individual constraints B(B + → τ + ντ ), ∆md<br />
and a combination <strong>of</strong> both inputs. Since B(B + → τ + ντ ) depends on <strong>the</strong> matrix<br />
element |Vub|, its 95 % CL belt is to a very good approximation a ring around<br />
(¯ρ, ¯η) = (0, 0). In <strong>the</strong> combination with ∆md, <strong>the</strong> <strong>the</strong>oretical uncertainties are<br />
reduced due to <strong>the</strong> cancelation <strong>of</strong> <strong>the</strong> decay constant contribution. The remaining<br />
<strong>the</strong>oretical uncertainty stems from <strong>the</strong> bag factor Bd in <strong>the</strong> SM prediction <strong>of</strong> ∆md.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
excluded area has CL > 0.95<br />
+ + B → τ ντ<br />
CKM fit<br />
γ<br />
α<br />
CKM<br />
+ +<br />
f i t t e r Constraint from B → τ ντ<br />
and Δmd<br />
ICHEP 06<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
Figure 5.7: Confidence level in <strong>the</strong> (¯ρ,¯η) plane obtained from B(B + → τ + ντ ) (blue ring),<br />
∆md (yellow ring) and a combination <strong>of</strong> both (green area).<br />
ρ<br />
β<br />
Δmd
44 Chapter 5. Probing <strong>the</strong> Standard Model<br />
In addition to <strong>the</strong> Standard Global CKM Fit, several fits for different classes <strong>of</strong><br />
constraints have been performed in this <strong>the</strong>sis. Each plot <strong>of</strong> Figure 5.8 shows <strong>the</strong><br />
95 % CL belts for <strong>the</strong> individual constraints and <strong>the</strong> allowed area from <strong>the</strong> combined<br />
fits.<br />
In <strong>the</strong> first row, <strong>the</strong> fit results, obtained from <strong>the</strong> CP-violating observables α, sin 2β,<br />
γ and |ɛK| are compared with those obtained from <strong>the</strong> CP-conserving constraints<br />
|Vub|, ∆md and ∆md & ∆ms. An important observation is that a non-real CKM<br />
matrix with ¯η �= 0 and thus CP violation is also obtained from CP-conserving constraints.<br />
The crucial input in this constraint is <strong>the</strong> recent measurement <strong>of</strong> ∆ms by<br />
CDF, presented at <strong>the</strong> winter conferences 2006.<br />
The second row shows <strong>the</strong> impact <strong>of</strong> <strong>the</strong>oretical uncertainties. The allowed area from<br />
<strong>the</strong> fit using only <strong>the</strong> UT angle constraints α, sin 2β and γ is much smaller than <strong>the</strong><br />
area obtained from <strong>the</strong> constraints |Vub|, |ɛK|, ∆md and ∆md & ∆ms which depend<br />
on additional QCD parameters.<br />
The left figure <strong>of</strong> <strong>the</strong> third row shows <strong>the</strong> fit results from constraints which are<br />
dominated by tree-level contributions. Beside |Vub|, a combined constraint from α<br />
and β is used which allows to extract γ. Assuming no New Physics contributions to<br />
<strong>the</strong> ∆I = 3/2 part <strong>of</strong> b → d transitions in <strong>the</strong> extraction <strong>of</strong> α leads in combination<br />
with β, measured in tree-level B → Mc¯c transitions, to a cancellation <strong>of</strong> a possible<br />
New Physics mixing phase ϑd (see Chapter 6). That gives a determination <strong>of</strong><br />
γ = π − β − α independent from possible New Physics in B 0 - ¯ B 0 mixing. Compared<br />
to that, <strong>the</strong> right plot shows constraints from <strong>the</strong> loop-dominated observables sin 2β,<br />
|ɛK|, ∆md and ∆md & ∆ms.<br />
Since all fits are consistent with <strong>the</strong> Standard Global CKM Fit and <strong>the</strong> allowed<br />
(¯ρ,¯η) range is quite small, <strong>the</strong> space for possible New Physics effects is significantly<br />
constrained. A more detailed discussion <strong>of</strong> possible NP contributions in B 0 - ¯ B 0<br />
mixing is described in <strong>the</strong> next chapter.
5.2. Standard Model Fit Results 45<br />
η<br />
η<br />
η<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
excluded area has CL > 0.95<br />
γ<br />
sin2β<br />
εK<br />
α<br />
0.1<br />
α<br />
γ<br />
β<br />
0<br />
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
excluded area has CL > 0.95<br />
γ<br />
sin2β<br />
ρ<br />
εK<br />
sol. w/ cos2β<br />
< 0<br />
(excl. at CL > 0.95)<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
α<br />
γ<br />
η<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
excluded area has CL > 0.95<br />
Δmd<br />
Δms<br />
& Δmd<br />
α<br />
0.1<br />
γ<br />
Vub/Vcb<br />
β<br />
0<br />
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1<br />
(a) (b)<br />
α<br />
0.1<br />
α<br />
γ<br />
β<br />
0<br />
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
excluded area has CL > 0.95<br />
(c)<br />
γ(<br />
α)<br />
α<br />
ρ<br />
ρ<br />
sol. w/ cos2β<br />
< 0<br />
(excl. at CL > 0.95)<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
0.1<br />
γ<br />
Vub/Vcb<br />
β<br />
0<br />
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1<br />
(e)<br />
α<br />
CKM<br />
γ<br />
f i t t e r<br />
ICHEP 2006<br />
η<br />
η<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
excluded area has CL > 0.95<br />
εK<br />
Δmd<br />
α<br />
ρ<br />
Δms<br />
& Δmd<br />
ρ<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
0.1<br />
γ<br />
Vub/Vcb<br />
β<br />
0<br />
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
sin2β<br />
εK<br />
Δmd<br />
(d)<br />
Δms<br />
& Δmd<br />
α<br />
0.1<br />
γ<br />
β<br />
0<br />
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1<br />
ρ<br />
εK<br />
εK<br />
sol. w/ cos2β<br />
< 0<br />
(excl. at CL > 0.95)<br />
Figure 5.8: Confidence level in <strong>the</strong> (¯ρ,¯η) plane for several global CKM fits. (a) CP-violating<br />
observables vs. (b) CP-conserving observables, (c) <strong>the</strong>oretically clean observables vs. (d) observables<br />
with significant <strong>the</strong>ory errors from non-perturbative QCD parameters and (e) tree<br />
dominated observables vs. (f) loop dominated observables. In <strong>the</strong> plot from pure tree observables,<br />
<strong>the</strong> constraint on α has been used assuming <strong>the</strong>re are no New Physics contributions<br />
to <strong>the</strong> ∆I = 3/2 part <strong>of</strong> b → d transitions. In <strong>the</strong> combination <strong>of</strong> this constraint with β<br />
from B → Mc¯cKS modes <strong>the</strong> New Physics mixing phase ϑd cancels, so that it gives a New<br />
Physics free determination <strong>of</strong> γ = π − β − α.<br />
excluded area has CL > 0.95<br />
(f)<br />
CKM<br />
f i t t e r<br />
ICHEP 2006
46 Chapter 5. Probing <strong>the</strong> Standard Model<br />
An important constraint on New Physics in B 0 - ¯ B 0 mixing comes from measurements<br />
<strong>of</strong> <strong>the</strong> semileptonic CP asymmetry ASL. Since it is not measured precisely<br />
enough to provide a significant constraint in (¯ρ,¯η) plane, <strong>the</strong> measurement <strong>of</strong> ASL<br />
is not included in <strong>the</strong> Standard Global CKM Fit. The prediction <strong>of</strong> ASL depends<br />
on short and long distance hadronic matrix elements, which need to be calculated<br />
from Lattice QCD. During this work, ASL has been implemented with LO and NLO<br />
QCD corrections. Figure 5.9 shows <strong>the</strong> confidence levels for <strong>the</strong> SM prediction <strong>of</strong><br />
ASL, taking LO as well as NLO QCD corrections into account. They are compared<br />
with <strong>the</strong> results <strong>of</strong> <strong>the</strong> Standard Global CKM Fit, including ASL. Its input value is<br />
given in Table 5.1.<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
× 10<br />
0<br />
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1<br />
× 10<br />
-0<br />
ASL LO<br />
-3<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
× 10<br />
0<br />
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1<br />
× 10<br />
-0<br />
ASL NLO<br />
Figure 5.9: Confidence level on ASL from <strong>the</strong> SM prediction. The QCD corrections are<br />
used up to LO (left) and NLO (right).<br />
-3
5.2. Standard Model Fit Results 47<br />
Observable central ± 1σ ± 2σ ± 3σ<br />
λ 0.2272 +0.0010<br />
−0.0010<br />
A 0.813 +0.015<br />
−0.015<br />
¯ρ 0.187 +0.028<br />
−0.086<br />
¯η 0.333 +0.038<br />
−0.017<br />
J [10 −5 ] 3.02 +0.36<br />
−0.18<br />
sin(2α) −0.25 +0.48<br />
−0.15<br />
sin(2α) (meas. not in fit) −0.28 +0.66<br />
−0.17<br />
sin(2β) 0.701 +0.022<br />
−0.022<br />
sin(2β) (meas. not in fit) 0.777 +0.114<br />
−0.052<br />
α (rad) 1.694 +0.082<br />
−0.242<br />
α (rad) (meas. not in fit) 1.713 +0.090<br />
−0.339<br />
α (rad) (direct meas.) 1.62 +0.18<br />
−0.17<br />
β (rad) 0.389 +0.016<br />
−0.015<br />
β (rad) (meas. not in fit) 0.452 +0.099<br />
−0.046<br />
β (rad) (direct meas.) 0.371 +0.018<br />
−0.017<br />
γ � δ (rad) 1.058 +0.244<br />
−0.076<br />
γ � δ (rad) (meas. not in fit) 1.060 +0.250<br />
−0.079<br />
γ � δ (rad) (direct meas.) 1.06 +0.67<br />
−0.44<br />
Ru<br />
Rt<br />
0.382 +0.014<br />
−0.015<br />
0.878 +0.092<br />
−0.030<br />
∆md (ps−1 ) (meas. not in fit) 0.359 +0.270<br />
−0.077<br />
∆ms (ps−1 ) 17.37 +0.35<br />
−0.23<br />
∆ms (ps−1 ) (meas. not in fit) 20.0 +7.0<br />
−4.7<br />
|ɛK| [10−3 ] (meas. not in fit) 2.34 +1.15<br />
−0.73<br />
B(B + → τ + νµ) [10−4 ] 1.06 +0.16<br />
−0.16<br />
B(B + → τ + νµ) [10−4 ] (meas. not in fit) 1.01 +0.16<br />
−0.24<br />
B(B + → µ + νµ) [10−7 ] (meas. not in fit) 4.01 +0.64<br />
−0.95<br />
+0.0020<br />
−0.0020<br />
+0.030<br />
−0.030<br />
+0.054<br />
−0.119<br />
+0.060<br />
−0.034<br />
+0.55<br />
−0.35<br />
+0.64<br />
−0.29<br />
+0.79<br />
−0.32<br />
+0.044<br />
−0.045<br />
+0.141<br />
−0.093<br />
+0.16<br />
−0.33<br />
+0.18<br />
−0.41<br />
+0.47<br />
−0.28<br />
+0.032<br />
−0.031<br />
+0.130<br />
−0.075<br />
+0.036<br />
−0.034<br />
+0.33<br />
−0.15<br />
+0.33<br />
−0.15<br />
+1.10<br />
−0.70<br />
+0.028<br />
−0.029<br />
+0.126<br />
−0.059<br />
+0.37<br />
−0.10<br />
+0.59<br />
−0.40<br />
+11.8<br />
−7.2<br />
+1.56<br />
−0.90<br />
+0.35<br />
−0.34<br />
+0.36<br />
−0.36<br />
+1.4<br />
−1.4<br />
+0.0030<br />
−0.0031<br />
+0.046<br />
−0.045<br />
+0.080<br />
−0.143<br />
+0.077<br />
−0.051<br />
+0.70<br />
−0.51<br />
+0.75<br />
−0.43<br />
+0.91<br />
−0.45<br />
+0.066<br />
−0.067<br />
+0.16<br />
−0.14<br />
+0.25<br />
−0.39<br />
+0.27<br />
−0.48<br />
+0.62<br />
−0.36<br />
+0.049<br />
−0.045<br />
+0.16<br />
−0.10<br />
+0.056<br />
−0.051<br />
+0.40<br />
−0.23<br />
+0.40<br />
−0.23<br />
+1.58<br />
−0.92<br />
+0.043<br />
−0.043<br />
+0.150<br />
−0.089<br />
+0.46<br />
−0.13<br />
+0.79<br />
−0.57<br />
+15.3<br />
−8.6<br />
+1.9<br />
−1.1<br />
+0.58<br />
−0.45<br />
+0.59<br />
−0.44<br />
+2.4<br />
−1.8<br />
Table 5.3: Fit results and errors using <strong>the</strong> Standard Global CKM Fit observables. For<br />
results marked with “meas. not in fit”, <strong>the</strong> measurement <strong>of</strong> <strong>the</strong> corresponding observable<br />
has not been included in <strong>the</strong> fit. The “1 − CL” ranges are defined as 32 % (1σ), 4.5 % (2σ)<br />
and 0.3 % (3σ).
48 Chapter 5. Probing <strong>the</strong> Standard Model<br />
Observable central ± 1σ ± 2σ ± 3σ<br />
|Vud| 0.97384 +0.00023<br />
−0.00024<br />
|Vus| 0.2272 +0.0010<br />
−0.0010<br />
|Vub| [10−3 ] 3.74 +0.14<br />
−0.15<br />
|Vub| [10−3 ] (meas. not in fit) 3.52 +0.18<br />
−0.17<br />
|Vcd| 0.2271 +0.0010<br />
−0.0010<br />
|Vcs| 0.97298 +0.00023<br />
−0.00024<br />
|Vcb| [10−3 ] 41.94 +0.67<br />
−0.69<br />
|Vcb| [10−3 ] (meas. not in fit) 44.7 +1.1<br />
−1.9<br />
|Vtd| [10−3 ] 8.37 +0.90<br />
−0.31<br />
|Vts| [10−3 ] 41.26 +0.68<br />
−0.66<br />
|Vtb| 0.999113 +0.000029<br />
−0.000029<br />
+0.00047<br />
−0.00047<br />
+0.0020<br />
−0.0020<br />
+0.29<br />
−0.29<br />
+0.37<br />
−0.34<br />
+0.0020<br />
−0.0020<br />
+0.00047<br />
−0.00048<br />
+1.4<br />
−1.4<br />
+2.3<br />
−4.6<br />
+1.32<br />
−0.62<br />
+1.3<br />
−1.4<br />
+0.000057<br />
−0.000058<br />
Table 5.4: Fit results and errors (continued)<br />
+0.00071<br />
−0.00071<br />
+0.0030<br />
−0.0031<br />
+0.44<br />
−0.44<br />
+0.55<br />
−0.57<br />
+0.0030<br />
−0.0031<br />
+0.00071<br />
−0.00072<br />
+2.0<br />
−2.0<br />
+3.5<br />
−7.2<br />
+1.47<br />
−0.92<br />
+2.0<br />
−2.1<br />
+0.000084<br />
−0.000088
Chapter 6<br />
New Physics Beyond <strong>the</strong><br />
Standard Model<br />
The Standard Model describes successfully all <strong>the</strong> present data from flavor physics<br />
experiments. Never<strong>the</strong>less, <strong>the</strong>re are still many reasons to believe in physics beyond<br />
<strong>the</strong> SM. For example, evidences for unobserved “dark matter” and “dark energy”<br />
from cosmic microwave background radiation measurements are not described by<br />
<strong>the</strong> SM. Fur<strong>the</strong>rmore, <strong>the</strong> strength <strong>of</strong> CP violation in <strong>the</strong> CKM matrix, expressed<br />
through <strong>the</strong> Jarlskog invariant, is significantly to weak and <strong>the</strong> electroweak phase<br />
transition is not intens enough to generate <strong>the</strong> observed baryon asymmetry in <strong>the</strong><br />
universe.<br />
In <strong>the</strong> following, two extensions <strong>of</strong> <strong>the</strong> Standard Model are discussed. First <strong>of</strong> all a<br />
model independent description <strong>of</strong> NP contributions to B 0 - ¯ B 0 mixing and secondly,<br />
charged Higgs contributions to <strong>the</strong> leptonic decay width <strong>of</strong> charged B mesons in<br />
Two Higgs Doublet Models (2HDM).<br />
6.1 New Physics in B 0 - ¯B 0 Oscillations<br />
B 0 - ¯ B 0 oscillation occurs in <strong>the</strong> SM through |∆B| = 2 transitions. A model independent<br />
description <strong>of</strong> New Physics contributions introduces two additional parameters,<br />
<strong>the</strong> relative amplitude r 2 d and <strong>the</strong> relative phase 2ϑd between <strong>the</strong> mixing<br />
matrix elements including SM and NP contributions compared to SM contributions<br />
only [65, 66]:<br />
� B 0 � �H SM+NP � � ¯ B 0 �<br />
� B 0 |H SM | ¯ B 0 � = r2 d ei2ϑd . (6.1)<br />
The Standard Model is referred to as r 2 d = 1 and 2ϑd = 0. Additional assumptions<br />
are a unitary 3 × 3 CKM matrix and no NP contributions to tree level dominated<br />
processes (which means Γ12 = ΓSM 12 ). More specifically, decay transitions with four<br />
flavor changes (i.e. b → q1¯q2q3, q1 �= q2 �= q3) are dominated by <strong>the</strong> SM [4]. The<br />
49
50 Chapter 6. New Physics Beyond <strong>the</strong> Standard Model<br />
<strong>the</strong>ory predictions <strong>of</strong> <strong>the</strong> relevant observables are changed as follows [67]:<br />
γ ⇒ γ (tree level dominated)<br />
|Vud|, |Vus| ⇒ |Vud|, |Vus| (tree level dominated)<br />
|Vub|, |Vcb| ⇒ |Vub|, |Vcb| (tree level dominated)<br />
∆md ⇒ ∆m NP<br />
d<br />
sin 2β ⇒ sin(2β + 2ϑd)<br />
= ∆mSM d · r 2 d<br />
cos 2β ⇒ cos(2β + 2ϑd) (6.2)<br />
α (= π − β − γ) ⇒ α NP (= π − β − γ − ϑd)<br />
� �SM Γ12 sin 2ϑd<br />
ASL ⇒ A NP<br />
SL = −Re<br />
M12<br />
r 2 d<br />
� �SM Γ12 cos 2ϑd<br />
+ Im<br />
M12 r2 d<br />
The tree-level dominated constraints from |Vud|, |Vus|, |Vub|, |Vcb| and γ remain unchanged.<br />
The o<strong>the</strong>r observables in Eqn. 6.2 have more important loop contributions<br />
which are sensitive to possible New Physics effects. The only constraint on both NP<br />
parameters r 2 d and 2ϑd comes from <strong>the</strong> semileptonic CP asymmetry ASL, where <strong>the</strong><br />
SM prediction <strong>of</strong> Γ12/M12 is used here at LO in QCD.<br />
The observables in Eqn. 6.2 provide significant constraints in <strong>the</strong> (¯ρ,¯η) plane as well<br />
as in <strong>the</strong> (r 2 d ,2ϑd) plane. The results <strong>of</strong> a global New Physics fit are shown in <strong>the</strong><br />
Figures 6.1 to 6.5, where <strong>the</strong> used input values are summarized in Table 5.1 & 5.2.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure 6.1: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and sin 2β.<br />
Figure 6.1 shows <strong>the</strong> confidence level obtained from a global CKM fit in <strong>the</strong> (¯ρ,¯η)<br />
and (r 2 d ,2ϑd) planes, assuming New Physics contributions to B 0 - ¯ B 0 oscillations.<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
.
6.1. New Physics in B 0 - ¯ B 0 Oscillations 51<br />
The constraints stem from |Vud|, |Vub|, |Vcb|, ∆md and sin 2β. The only significant<br />
constraint in <strong>the</strong> (¯ρ,¯η) plane comes from <strong>the</strong> ratio |Vub/Vcb|, which leads, to a very<br />
good approximation, to a ring around (¯ρ, ¯η) = (0, 0). The constraint in <strong>the</strong> (r 2 d ,2ϑd)<br />
plane is very loose. The input <strong>of</strong> sin 2β leads to a mirror solution at 2ϑd > π/2. It<br />
vanishes after adding cos(2β + 2ϑd) > 0 to <strong>the</strong> fit inputs, as shown in Fig. 6.2. It<br />
does only marginally effect <strong>the</strong> result in <strong>the</strong> (¯ρ,¯η) plane.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure 6.2: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β and cos 2β.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
22<br />
ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure 6.3: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β and γ.<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0
52 Chapter 6. New Physics Beyond <strong>the</strong> Standard Model<br />
A much better constraint stems from <strong>the</strong> observable γ. Since it is tree-level dominated,<br />
it directly constrains <strong>the</strong> CKM matrix parameters (¯ρ,¯η) and <strong>the</strong>reby <strong>the</strong><br />
allowed range in <strong>the</strong> (r 2 d ,2ϑd) plane. Figure 6.3 shows <strong>the</strong> two allowed regions remain<br />
in <strong>the</strong> (¯ρ,¯η) plane and in <strong>the</strong> (r 2 d ,2ϑd) plane. Since a one-dimensional scan<br />
in ¯η results ¯η �= 0 at a CL <strong>of</strong> at most 99.9 %, a real CKM matrix is also excluded<br />
assuming New Physics contributions in B 0 - ¯ B 0 oscillations. The first solution in <strong>the</strong><br />
(r 2 d ,2ϑd) plane is in agreement with <strong>the</strong> SM values (r 2 d = 1, 2ϑd = 0), whereas <strong>the</strong><br />
second solution would be a clear sign <strong>of</strong> New Physics effects. Ano<strong>the</strong>r major input<br />
stems from measurements <strong>of</strong> <strong>the</strong> UT angle α. It decreases <strong>the</strong> allowed regions in <strong>the</strong><br />
(¯ρ,¯η) plane as well as in <strong>the</strong> (r 2 d ,2ϑd) plane as shown in Figure 6.4.<br />
Finally, <strong>the</strong> inclusion <strong>of</strong> ASL in <strong>the</strong> fit gives an additional hard constraint in both<br />
planes. As shown in Figure 6.5, <strong>the</strong> non-SM solution almost vanishes, which highlights<br />
<strong>the</strong> importance <strong>of</strong> <strong>the</strong> observable ASL in constraining possible New Physics<br />
contributions to B 0 - ¯ B 0 mixing. Additional fits have been performed to describe <strong>the</strong><br />
effects from <strong>the</strong> individual constraints. The results are shown in <strong>the</strong> Figures B.1<br />
to B.5 , given in Appendix B.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure 6.4: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β, γ and α.<br />
Figure 6.6 shows <strong>the</strong> confidence level obtained from one-dimensional scans <strong>of</strong> <strong>the</strong><br />
New Physics parameters r 2 d and 2ϑd, using constraints from |Vud|, |Vub|, |Vcb|, ∆md,<br />
sin 2β, cos 2β, γ, α and ASL. The results<br />
r 2 d<br />
+0.50 +1.28<br />
= 1.02 −0.42 (1σ)<br />
−0.57 (2σ) and 2ϑd = −0.094 +0.049<br />
−0.123<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
+0.11<br />
(1σ) −0.16 (2σ) , (6.3)<br />
are in good agreement with <strong>the</strong> Standard Model, never<strong>the</strong>less, <strong>the</strong> r2 d constraint shows<br />
that NP contributions <strong>of</strong> order O(100 %) are still possible. The non-SM solution is<br />
0
6.1. New Physics in B 0 - ¯ B 0 Oscillations 53<br />
strongly suppressed at more than 99.5 % CL, <strong>the</strong> SM-like solution shows a small bias<br />
in 2ϑd, mainly caused by <strong>the</strong> inputs |Vub|incl. and sin 2β.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure 6.5: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β, γ, α and ASL.<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
CKM<br />
f i t t e r<br />
ICHEP 2006<br />
0<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
1 - CL<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
f i t t e r<br />
ICHEP 2006<br />
0<br />
-3 -2 -1 0<br />
2 ϑ (rad)<br />
1 2 3<br />
Figure 6.6: Confidence level on r 2 d and 2ϑd obtained from <strong>the</strong> global CKM fit, assuming<br />
possible New Physics contributions to <strong>the</strong> B 0 - ¯ B 0 mixing amplitude.<br />
CKM<br />
d<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0
54 Chapter 6. New Physics Beyond <strong>the</strong> Standard Model<br />
6.2 Charged Higgs Contributions to Leptonic B ± Decays<br />
6.2.1 Two-Higgs-Doublet Models<br />
Charged Higgs bosons are predicted by multiple-Higgs-doublet extensions <strong>of</strong> <strong>the</strong><br />
Standard Model. The simplest versions are Two-Higgs-Doublet Models (2HDM),<br />
which introduce <strong>the</strong> two Higgs doublet scalar fields Φi, i = 1, 2, where tan β = v2/v1<br />
is <strong>the</strong> ratio <strong>of</strong> <strong>the</strong>ir vacuum expectation values vi ∼ 〈Φi〉. This leads to five physical<br />
Higgs bosons, three neutral and two charged ones.<br />
The 2HDM are classified in model I, where all fermions obtain <strong>the</strong>ir masses from<br />
<strong>the</strong> same Higgs doublet, and model II, where u-type quarks obtain <strong>the</strong>ir mass from<br />
one, d-type quarks and charged leptons from <strong>the</strong> o<strong>the</strong>r doublet [68]. Since <strong>the</strong>re are<br />
no relevant effects from charged B decays in model I, only model II is considered<br />
here. Model II is also realized in <strong>the</strong> Higgs sector <strong>of</strong> <strong>the</strong> Minimal Supersymmetric<br />
Standard Model (MSSM), where <strong>the</strong> mass <strong>of</strong> <strong>the</strong> charged Higgs boson is additionally<br />
restricted to m H ± > mW . This is not <strong>the</strong> case for <strong>the</strong> general model II <strong>of</strong> 2HDM,<br />
where m H ± can take on arbitrary values [9].<br />
6.2.2 Charged Higgs Contributions to Leptonic B ± Decays<br />
Important constraints on <strong>the</strong> charged Higgs mass m H ± come from purely leptonic<br />
decays <strong>of</strong> charged B mesons. In contrast to many o<strong>the</strong>r rare decays (e. g. b → sγ)<br />
which are influenced by New Physics at <strong>the</strong> one loop level, <strong>the</strong> decays B ± → l ± ν<br />
are sensitive to charged Higgs bosons (H ± ) at tree level [69]. Figure 6.7 shows<br />
<strong>the</strong> tree-level contribution to <strong>the</strong> decay B + → τ + ντ , which is, in addition to <strong>the</strong><br />
SM-W boson, mediated by a H ± boson.<br />
b<br />
u<br />
+<br />
B<br />
+ +<br />
W , H<br />
Figure 6.7: Tree-level contribution from charged Higgs bosons to B + → τ + ντ<br />
+ τ<br />
ντ
6.2. Charged Higgs Contributions to Leptonic B ± Decays 55<br />
The prediction <strong>of</strong> <strong>the</strong> branching fraction B <strong>of</strong> purely leptonic B ± decays can be<br />
written as [70]:<br />
B (B → lν) = G2 F mBm2 l f 2 B<br />
|Vub|<br />
8π<br />
2<br />
�<br />
1 − m2 l<br />
m2 �2<br />
· rH , (6.4)<br />
B<br />
where <strong>the</strong> factor rH contains <strong>the</strong> charged Higgs boson contributions. It is defined<br />
by [69]:<br />
�<br />
rH = 1 − tan 2 β m2 B<br />
m2 H ±<br />
�2<br />
1<br />
1 + ɛ0 · tan β<br />
. (6.5)<br />
The factor 1/(1 + ɛ0 · tan β) combines corrections to <strong>the</strong> coupling H ± uidj which occur<br />
at large tan β in SUSY models [69, 71]. Since a detailed discussion <strong>of</strong> this effect<br />
occurring in SUSY models is beyond <strong>the</strong> scope <strong>of</strong> this <strong>the</strong>sis, <strong>the</strong>y are neglected<br />
by fixing ɛ0 to be zero. A discussion <strong>of</strong> <strong>the</strong> numerical impact <strong>of</strong> ɛ0 1 is provided in<br />
Ref. [69].<br />
Figure 6.8 shows <strong>the</strong> confidence level in <strong>the</strong> (tan β,m H +) plane obtained from <strong>the</strong><br />
constraints |Vud|, |Vus|, |Vcb|, |Vub|incl and B(B + → τ + ντ ). Their input values are <strong>the</strong><br />
same as summarized in Table 5.1, where for B(B + → τ + ντ ) <strong>the</strong> combined likelihood<br />
function from BABAR [44] and Belle [45] is taken.<br />
2<br />
) GeV/c<br />
+<br />
m(H<br />
350<br />
300<br />
250<br />
200<br />
150<br />
100<br />
50<br />
0 10 20 30 40 50 60 70 80 90<br />
tan( β )<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
Figure 6.8: Confidence level in <strong>the</strong> (tan β,m H +) plane obtained from |Vud|, |Vus|, |Vcb|,<br />
|Vub|incl and B(B + → τ + ντ ).<br />
Since <strong>the</strong> constraint in Fig. 6.8 is very loose, only small m H + are exclude for large<br />
tan β at 95 % CL. An experimental lower limit with m H + > 78.6 GeV/c 2 at 95 %<br />
CL is obtained from <strong>the</strong> combined LEP results <strong>of</strong> searches in <strong>the</strong> decay channels<br />
1 Varied in an interval −0.01 < ɛ0 < +0.01 which has been found in Ref. [71].<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0
56 Chapter 6. New Physics Beyond <strong>the</strong> Standard Model<br />
H + → c¯s and H + → τ + ντ [72]. Ano<strong>the</strong>r interesting constraint stems from ¯ B → Xsγ<br />
decays where a lower limit <strong>of</strong> m H + > 350 GeV/c 2 at 99 % CL [73] has been found.
Chapter 7<br />
Conclusions and Perspectives<br />
This <strong>the</strong>sis is a contribution to <strong>the</strong> development <strong>of</strong> a <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong><br />
<strong>the</strong> <strong>CKMfitter</strong> package. It had been introduced by Jérôme Charles to obtain a<br />
significant fit time reduction compared to <strong>the</strong> original FORTRAN <strong>based</strong> <strong>CKMfitter</strong><br />
package. Using symbolic calculations during <strong>the</strong> fit preparation and an efficient<br />
FORTRAN <strong>based</strong> minimization routine, a gain in CPU time <strong>of</strong> more than a factor<br />
100 has been achieved for <strong>the</strong> Standard Global CKM Fit.<br />
Related to <strong>the</strong> Standard Global CKM Fit, <strong>the</strong> <strong>the</strong>ory <strong>of</strong> neutral meson oscillations<br />
has been implemented for <strong>the</strong> B 0 d , B0 s and K 0 systems as well as <strong>the</strong> <strong>the</strong>ory predictions<br />
for <strong>the</strong> branching fraction <strong>of</strong> purely leptonic B-meson decays. More specifically,<br />
<strong>the</strong> following observables have been implemented: ∆md, ∆ms, ASL, |ɛK| and<br />
B(B + → l + νl).<br />
For <strong>the</strong> treatment <strong>of</strong> look-up-table input files, e. g. for ∆ms, B(B + → τ + ντ ) and <strong>the</strong><br />
UT angles α and γ, FORTRAN <strong>based</strong> subroutines (TABLEAU, dTABLEAUO2, LoadLUT)<br />
have been coded. The interpolation <strong>of</strong> <strong>the</strong> look-up tables is performed by a <strong>Ma<strong>the</strong>matica</strong>-<strong>based</strong><br />
subroutine using cubic spline interpolation.<br />
The results <strong>of</strong> <strong>the</strong> Standard Global CKM Fit show a good agreement between SM<br />
predictions and recent data. The Wolfenstein parameters have been constraint<br />
at 68 % CL to:<br />
A = 0.813 +0.015<br />
+0.0010<br />
+0.028<br />
+0.038<br />
−0.015 , λ = 0.2272 −0.0010 , ¯ρ = 0.187 −0.086 , ¯η = 0.333 −0.017 , (7.1)<br />
and <strong>the</strong> Jarlskog invariant, which is related to <strong>the</strong> strength <strong>of</strong> CP violation in electroweak<br />
transitions is found to be:<br />
J = � 3.02 +0.36�<br />
−5<br />
−0.18 · 10 . (7.2)<br />
57
58 Chapter 7. Conclusions and Perspectives<br />
Since all <strong>the</strong> present data from experiments in quark-flavor physics is well described<br />
by <strong>the</strong> Standard Model, <strong>the</strong> space for New Physics contributions is significantly constrained.<br />
Never<strong>the</strong>less, <strong>the</strong>re is still enough space for possible New Physics effects,<br />
which have been quantified in this <strong>the</strong>sis in two different extensions <strong>of</strong> <strong>the</strong> Standard<br />
Model.<br />
Possible New Physics contributions to neutral meson oscillations have been implemented<br />
model-independently to <strong>the</strong> B-meson systems as well as to <strong>the</strong> Kaon system.<br />
The allowed ranges for <strong>the</strong> NP parameters (r 2 d ,2ϑd) in <strong>the</strong> Bd system have been found<br />
to:<br />
r 2 d<br />
= 1.02 +0.50<br />
−0.42 and 2ϑd = −0.094 +0.049<br />
−0.123 (at 68 % CL), (7.3)<br />
which is in good agreement with <strong>the</strong> Standard Model. The small bias in 2ϑd results<br />
from <strong>the</strong> slight disagreement <strong>of</strong> <strong>the</strong> measurements <strong>of</strong> |Vub|incl and sin 2β. Since a<br />
significant constraint on possible New Physics in B 0 - ¯ B 0 mixing is obtained from <strong>the</strong><br />
observable ASL, its <strong>the</strong>oretical prediction has been implemented including LO QCD<br />
corrections as well as NLO QCD corrections. A preliminary comparison <strong>of</strong> <strong>the</strong>ir SM<br />
predictions is given in this <strong>the</strong>sis.<br />
Contributions <strong>of</strong> charged Higgs bosons to <strong>the</strong> branching fraction <strong>of</strong> leptonic B-meson<br />
decays have been implemented in <strong>the</strong> framework <strong>of</strong> Two-Higgs-Doublet Models. Since<br />
only <strong>the</strong> decay B + → τ + ντ is currently experimentally accessible, <strong>the</strong> constraint<br />
on (tan β,m H +) is still loose. An additional fit input from B → Xsγ transitions<br />
would provide an additional constraint, but has not been implemented yet.<br />
To guarantee <strong>the</strong> stability <strong>of</strong> <strong>the</strong> fit results by using <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> <strong>CKMfitter</strong><br />
package, several comparison tests with <strong>the</strong> original package have been performed.<br />
As a benefit to <strong>the</strong> <strong>CKMfitter</strong> group, a user guide has been written during this <strong>the</strong>sis<br />
as well as a tutorial on <strong>the</strong> coding <strong>of</strong> <strong>the</strong>ory packages.<br />
After more than one year <strong>of</strong> development, <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>the</strong><br />
<strong>CKMfitter</strong> package has been used for <strong>the</strong> first time to produce <strong>the</strong> fit results for<br />
a large conference (ICHEP 2006). Never<strong>the</strong>less, a lot <strong>of</strong> features available in <strong>the</strong><br />
original <strong>CKMfitter</strong> package are still missing. A Wolfenstein parameter independent<br />
prediction <strong>of</strong> B(B + → l + νl) as well as <strong>the</strong> <strong>the</strong>ory <strong>of</strong> rare Kaon decays need to be<br />
implemented.<br />
Due to its modular structure, <strong>the</strong> new <strong>Ma<strong>the</strong>matica</strong> <strong>based</strong> version <strong>of</strong> <strong>the</strong> <strong>CKMfitter</strong><br />
package is not only restricted to <strong>the</strong> CKM matrix analysis. Fur<strong>the</strong>r applications,<br />
e. g. a leptonic mixing matrix analysis could be easily added.
Appendix A<br />
The Inami-Lim Functions<br />
The Inami-Lim functions are defined by [74]:<br />
where<br />
S (xi) =<br />
�<br />
1<br />
xi<br />
4 +<br />
9<br />
4 (1 − xi) −<br />
3<br />
2 (1 − xi) 2<br />
�<br />
− 3<br />
S (xi, xj) i�=j =<br />
� �3 xi<br />
ln xi<br />
2 1 − xi<br />
��<br />
1<br />
xixj<br />
4 +<br />
3<br />
2 (1 − xi) −<br />
3<br />
4 (1 − xi) 2<br />
�<br />
1<br />
ln(xi)<br />
xi − xj<br />
�<br />
1<br />
+<br />
4 +<br />
3<br />
2 (1 − xj) −<br />
3<br />
4 (1 − xj) 2<br />
�<br />
−<br />
1<br />
ln(xj)<br />
xj − xi<br />
3<br />
�<br />
1<br />
,<br />
4 (1 − xi) (1 − xj)<br />
xi = m2 i<br />
m 2 W<br />
with i = c, t .<br />
The quark masses are used in <strong>the</strong> MS scheme, which are perturbatively calculated<br />
in LO from <strong>the</strong> pole masses by:<br />
�<br />
¯mi(mi) = mi 1 − 4<br />
� ��<br />
αS(mi)<br />
. (A.1)<br />
3 π<br />
The QCD correction factor ηcc to <strong>the</strong> Inami-Lim functions has been parametrized<br />
through [37]:<br />
� � ��<br />
¯mc(mc)<br />
ηcc � (1.46 ± δcc) 1 − 1.2<br />
− 1 [1 + 52 (αS(mZ) − 0.118)] (A.2)<br />
1.25 GeV/c2 with an uncertainty from higher-order corrections parametrized by:<br />
� � ��<br />
¯mc(mc)<br />
δcc = 0.22 1 − 1.8<br />
− 1 [1 + 80 (αS(mZ) − 0.118)] . (A.3)<br />
1.25 GeV/c2 59
60 Appendix A. The Inami-Lim Functions
Appendix B<br />
Additional Figures <strong>of</strong> New<br />
Physics in B 0 - ¯B 0 Oscillations<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure B.1: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and cos 2β.<br />
61<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0
62 Appendix B. Additional Figures <strong>of</strong> New Physics in B 0 - ¯B 0 Oscillations<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure B.2: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and α.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure B.3: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and ASL.<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure B.4: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, cos 2β and α.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure B.5: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md and ASL.<br />
1<br />
0<br />
63<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
1<br />
0.99<br />
0.98<br />
0.97<br />
0.96<br />
0.95<br />
0.94<br />
0.93
64 Appendix B. Additional Figures <strong>of</strong> New Physics in B 0 - ¯B 0 Oscillations<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure B.6: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β and α.<br />
η<br />
1.5<br />
1<br />
0.5<br />
0<br />
-0.5<br />
-1<br />
γ<br />
α<br />
-1.5<br />
-1 -0.5 0 0.5 1 1.5 2<br />
ρ<br />
β<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
(rad)<br />
d<br />
2 ϑ<br />
3<br />
2<br />
1<br />
0<br />
-1<br />
-2<br />
CKM<br />
1-CL<br />
f i t t e r<br />
ICHEP 2006<br />
-3<br />
0 1 2 3<br />
r 2<br />
d<br />
4 5 6<br />
Figure B.7: Constraints within <strong>the</strong> framework <strong>of</strong> New Physics contributions to <strong>the</strong> B 0 - ¯ B 0<br />
mixing amplitude on <strong>the</strong> (¯ρ,¯η) plane (left) and on <strong>the</strong> (r 2 d ,2ϑd) plane (right), obtained from<br />
a global CKM fit including |Vud|, |Vus|, |Vub|, |Vcb|, ∆md, sin 2β, γ, α and ASL.<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0
Appendix C<br />
Testjob<br />
{<br />
(* Setting datacard for global CKM fit *)<br />
analysisContext -> "TESTJOB‘",<br />
<strong>the</strong>ory<strong>Package</strong> -> "BBbarKKbarMixing‘",<br />
inputData -> "globalCKMfit_ICHEP06",<br />
job -> 1,<br />
(***** TESTJOB - SM global Fit - small plane *****)<br />
takeMe[1] -> {<br />
"|Vud|","|Vus|","|Vcb|", "|Vub|",<br />
"All(Deltamd)",<br />
"All(Deltams)",<br />
"All(|epsilonK|)",<br />
"sin2beta",<br />
"alpha",<br />
"gamma"<br />
},<br />
(********** common settings **********)<br />
startRange -> {<br />
"A"->{0.7, 0.9},<br />
"lambda"->{0.22, 0.23},<br />
"rhobar"->{0, 0.26},<br />
"etabar"->{0.28, 0.42}<br />
},<br />
65
66 Appendix C. Testjob<br />
}<br />
globalMinSearches -> 100,<br />
useOneD<strong>of</strong> -> False,<br />
scanQty -> {"rhobar", "etabar"},<br />
scanMin -> {-0.4, 0. },<br />
scanMax -> {+1.0, 0.7},<br />
granularity -> 200,<br />
nbOfScans -> 2,<br />
nbOfFits -> 1,<br />
scanDirection -> "both",<br />
outputType -> { ".dat", {".png",ImageSize->800}, {".eps",ImageSize->800} },<br />
verbose -> True
Appendix D<br />
Source Code<br />
D.1 Tableau<br />
!-----------------------------------------------------------------------<br />
C<br />
C Tableau Function for LUT inputs by A. Jantsch<br />
C<br />
Double precision Function Tableau( ObsPred, fn )<br />
Implicit None<br />
include ’dimarray.f’<br />
integer k, n, fn, lengthTable(maxLUT)<br />
double precision ObsPred, SplinePred<br />
double precision xmin, xmax<br />
double precision tabsave(maxLUT,0:5,maxGranularity)<br />
common /LUT/ lengthTable, tabsave<br />
n = lengthTable(fn)<br />
xmin = tabsave(fn,0,1)<br />
xmax = tabsave(fn,0,n)<br />
if( ObsPred.le.xmin ) <strong>the</strong>n<br />
Tableau = tabsave(fn,1,1)<br />
elseif( ObsPred.ge.xmax ) <strong>the</strong>n<br />
Tableau = tabsave(fn,1,n)<br />
67
68 Appendix D. Source Code<br />
else<br />
k = int( (dble(n-1)*ObsPred +xmax -dble(n)*xmin)/(xmax -xmin) )<br />
SplinePred = ObsPred -tabsave(fn,0,k)<br />
Tableau = tabsave(fn,2,k) +<br />
> tabsave(fn,3,k) *SplinePred +<br />
> tabsave(fn,4,k) *SplinePred**2 +<br />
> tabsave(fn,5,k) *SplinePred**3<br />
end if<br />
End<br />
D.2 dTableauO2<br />
Tableau = Max(Tableau, 0D0)<br />
!-----------------------------------------------------------------------<br />
C<br />
C Derivatives <strong>of</strong> Tableau function (divided by 2)<br />
C<br />
Double precision Function dTableauO2( ObsPred, fn )<br />
Implicit None<br />
include ’dimarray.f’<br />
integer k, n, fn, lengthTable(maxLUT)<br />
double precision ObsPred, SplinePred<br />
double precision xmin, xmax<br />
double precision tabsave(maxLUT,0:5,maxGranularity)<br />
common /LUT/ lengthTable, tabsave<br />
n = lengthTable(fn)<br />
xmin = tabsave(fn,0,1)<br />
xmax = tabsave(fn,0,n)<br />
if( ObsPred.le.xmin .or. ObsPred.ge.xmax ) <strong>the</strong>n
D.3. LoadLUT 69<br />
else<br />
dTableauO2 = 0D0<br />
k = int( (dble(n-1)*ObsPred +xmax -dble(n)*xmin)/(xmax -xmin) )<br />
SplinePred = ObsPred -tabsave(fn,0,k)<br />
dTableauO2 = tabsave(fn,3,k) +<br />
> 2D0 *tabsave(fn,4,k) *SplinePred +<br />
> 3D0 *tabsave(fn,5,k) *SplinePred**2<br />
end if<br />
End<br />
D.3 LoadLUT<br />
dTableauO2 = dTableauO2 /2D0<br />
!-----------------------------------------------------------------------<br />
C<br />
C subroutine to load LUT’s for Tableau function<br />
C<br />
subroutine LoadLUT( nbOfLUT, LUTfname)<br />
Implicit None<br />
include ’dimarray.f’<br />
integer nbOfLUT<br />
character*200 LUTfname(maxLUT)<br />
integer i, fn, lengthTable(maxLUT)<br />
double precision tabsave(maxLUT,0:5,maxGranularity)<br />
common /LUT/ lengthTable, tabsave<br />
do fn = 1, nbOfLUT<br />
open(22, file = LUTfname(fn), status = ’old’)<br />
read(22,*) lengthTable(fn)
70 Appendix D. Source Code<br />
do i = 1, lengthTable(fn)<br />
read(22,*) tabsave(fn,0,i), tabsave(fn,1,i),<br />
> tabsave(fn,2,i), tabsave(fn,3,i),<br />
> tabsave(fn,4,i), tabsave(fn,5,i)<br />
enddo<br />
close(22)<br />
enddo<br />
End
Appendix E<br />
User Guide<br />
E.1 <strong>Ma<strong>the</strong>matica</strong> Terminology<br />
In <strong>the</strong> following, <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> terminology used in this work, is briefly described.<br />
Fur<strong>the</strong>r informations are given in <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> Documentation Center [22].<br />
Notebook<br />
A <strong>Ma<strong>the</strong>matica</strong> notebook is a file format used by <strong>the</strong> graphical front-end. It contains<br />
input cells, which are evaluated by <strong>the</strong> <strong>Ma<strong>the</strong>matica</strong> kernel. The result is displayed<br />
in corresponding output cells.<br />
<strong>Package</strong><br />
A <strong>Ma<strong>the</strong>matica</strong> package is a text file, written in <strong>Ma<strong>the</strong>matica</strong> language. It can be<br />
sourced from <strong>Ma<strong>the</strong>matica</strong> notebooks and o<strong>the</strong>r <strong>Ma<strong>the</strong>matica</strong> packages and provides<br />
<strong>the</strong> included definitions like a library file.<br />
List<br />
Lists are general objects that represent collections <strong>of</strong> expressions. Each type <strong>of</strong> expression<br />
can be a list element, e. g. a list <strong>of</strong> strings or a list <strong>of</strong> lists, which represents<br />
vectors, matrices or higher tensors.<br />
Example: list = { 1 , ”string” , { 3+x , 3 , y } }<br />
Rule<br />
Evaluating a rule replaces an expression by ano<strong>the</strong>r one.<br />
Example: x → 5<br />
71
72 Appendix E. User Guide<br />
E.2 Datacards<br />
A datacard describes <strong>the</strong> analysis and contains <strong>the</strong> settings for all options <strong>of</strong> a job.<br />
Datacards are ASCII files, located in <strong>the</strong> analysis directory and can be edited using<br />
a plain text editor. The only object is a list <strong>of</strong> rules, where each rule specifies an<br />
option for <strong>the</strong> analysis job, e. g. <strong>the</strong> input files or <strong>the</strong> scan granularity.<br />
It is possible to perform more than one job using one datacard. Therefor, an option<br />
can be referred to a specific job adding <strong>the</strong> job number in brackets to its end, e. g.<br />
analysisContext<br />
’takeMe[3] → {}’ instead <strong>of</strong> ’takeMe → {}’.<br />
The analysis context specifies <strong>the</strong> environment <strong>of</strong> <strong>the</strong> fit and is used as <strong>the</strong> name for<br />
each output file. It is a string, where special characters are forbidden, ending with a<br />
“context mark”. A “context mark” in <strong>Ma<strong>the</strong>matica</strong> is <strong>the</strong> backquote or grave accent<br />
character (ASCII decimal code 96).<br />
Syntax: analysisContext → ”Contextname`”<br />
Example: analysisContext → ”TestFit`”<br />
<strong>the</strong>ory<strong>Package</strong><br />
Here, <strong>the</strong> <strong>the</strong>ory package(s) used in <strong>the</strong> analysis are specified. For each package,<br />
its context and <strong>the</strong> version should be specified in a list. Attention: If no version is<br />
specified, <strong>the</strong> Standard Model version is used. There is also <strong>the</strong> possibility to set a<br />
list <strong>of</strong> <strong>the</strong>se lists to load more than one <strong>the</strong>ory package.<br />
Syntax: <strong>the</strong>ory<strong>Package</strong> → { ”package context`” , version → ”version label” }<br />
Example: <strong>the</strong>ory<strong>Package</strong> → { ”LeptonicDecay`” , version → ”SM” }<br />
Example: <strong>the</strong>ory<strong>Package</strong> → { ”LeptonicDecay`” , { ”BBbarKKbarMixing`” ,<br />
inputData<br />
This option specifies <strong>the</strong> input file(s) for <strong>the</strong> analysis.<br />
Syntax: inputData → ”inputfile”<br />
Example: inputData → ”globalCKMfit ICHEP06.data”<br />
version → ”NP(r,<strong>the</strong>ta)” } }
E.2. Datacards 73<br />
jobs<br />
The number <strong>of</strong> jobs in <strong>the</strong> settings datacard is set with this flag.<br />
Syntax: jobs → Integer<br />
Example: jobs → 2<br />
takeMe<br />
This is <strong>the</strong> list, which specifies <strong>the</strong> fit variables. It is a list <strong>of</strong> observable and parameter<br />
labels.<br />
Syntax: takeMe → { ”label”, ”label”, . . .}<br />
Example: takeMe → { ”alpha”, ”Deltamd”, ”fBd”}<br />
startRange<br />
This list defines <strong>the</strong> start ranges for all free parameters, which are not in <strong>the</strong> takeMe<br />
list.<br />
Syntax: startRange → { ”label” → range list }<br />
Example: startRange → { ”A” → {0.75, 0.85}, ”lambda” → {0.226, 0.228} }<br />
globalMinSearches<br />
Here, <strong>the</strong> number <strong>of</strong> reruns <strong>of</strong> <strong>the</strong> global minimization is set. To avoid finding a<br />
local minimum, this could be a large number.<br />
Syntax: globalMinSearches → Integer<br />
Example: globalMinSearches → 100<br />
scanQty<br />
This flag selects <strong>the</strong> parameter space for <strong>the</strong> scan.<br />
Syntax: scanQty → {”label”, ”label”}<br />
Example: scanQty → ”lambda” for 1D scans<br />
Example: scanQty → {”rhobar”, ”etabar”} for 2D scans
74 Appendix E. User Guide<br />
scanMin<br />
The minimal end point in <strong>the</strong> parameter space is set with scanMin. It is only a<br />
number for 1D scans or a point for 2D scans.<br />
Syntax: scanMin → Point<br />
Example: scanMin → 0.3 for 1D scans<br />
Example: scanMin → { -1.0 , -1.5} for <strong>the</strong> large (¯ρ,¯η) plane<br />
Example: scanMin → { -0.4 , 0.0} for <strong>the</strong> small (¯ρ,¯η) plane<br />
scanMax<br />
In analogy to scanMin, scanMax defines <strong>the</strong> maximal end point in <strong>the</strong> parameter<br />
space.<br />
Syntax: scanMax → Point<br />
Example: scanMax → 0.6 for 1D scans<br />
Example: scanMax → { +2.0 , +1.5} for <strong>the</strong> large (¯ρ,¯η) plane<br />
Example: scanMax → { +1.0 , +0.7} for <strong>the</strong> small (¯ρ,¯η) plane<br />
useOneD<strong>of</strong><br />
This option sets <strong>the</strong> degree <strong>of</strong> freedom <strong>of</strong> <strong>the</strong> fit problem. If it is equal to one, this<br />
option is “True”, o<strong>the</strong>rwise it is set to “False”.<br />
Syntax: useOneD<strong>of</strong> → Logical<br />
Example: useOneD<strong>of</strong> → False<br />
granularity<br />
The scan granularity is specified with this option.<br />
Syntax: granularity → Integer<br />
Example: granularity → 300<br />
nbOfScans<br />
The scan path in <strong>the</strong> parameter space is set with nbOfScans, which can be also<br />
referred to how <strong>of</strong>ten a point is scanned. Only <strong>the</strong> values “1”, “2”, “4” are defined.<br />
Syntax: nbOfScans → Integer<br />
Example: nbOfScans → 2
E.2. Datacards 75<br />
nbOfFits<br />
The number <strong>of</strong> fits (minimizations) per single scan point in <strong>the</strong> parameter space is<br />
set with this flag. It should be a positive number.<br />
Syntax: nbOfFits → Integer<br />
Example: nbOfFits → 2<br />
scanDirection<br />
This option defines <strong>the</strong> scan direction in <strong>the</strong> parameter space. The possible settings<br />
are ”vertical”, ”horizontal” or ”both”.<br />
Syntax: scanDirection → String<br />
Example: scanDirection → ”both”<br />
outputType<br />
The analysis notebook exports data files and plots. An output type is chosen by its<br />
file extension. For plots, also <strong>the</strong> image size can be specified.<br />
Syntax: outputType → { ”extansion”, {”extansion”, ImageSize → Integer} }<br />
Example: outputType → { ”.dat”, ”.eps”, {”.png”, ImageSize → 800} }<br />
verbose<br />
Detailed information during <strong>the</strong> fit can be obtained with setting this option as<br />
“True”.<br />
Syntax: verbose → Logical<br />
Example: verbose → True
76 Appendix E. User Guide<br />
E.3 Input Types<br />
An input datacard is an ASCII file containing <strong>the</strong> input values <strong>of</strong> all observables or<br />
parameters <strong>of</strong> <strong>the</strong> fit <strong>the</strong>ory given as a list. Each element <strong>of</strong> this list is again a list,<br />
with a special syntax for each input type. For example <strong>the</strong> first element in each list<br />
is a string, which represents <strong>the</strong> label <strong>of</strong> <strong>the</strong> observable or parameter. Attention:<br />
Greek letters have to be written out, e. g. “delta“ for “δ“ or “Delta“ for “∆“. The<br />
inputs datacard needs to be located in <strong>the</strong> inputs subdirectory.<br />
Fixed<br />
Syntax: { “label“ , value }<br />
Example: { “delta“ , 0.0}<br />
Gauss<br />
Syntax: { “label“ , central value , gaussian error }<br />
Example: { “delta“ , 0.0 , 0.1 }<br />
Asymmetric Gauss<br />
Syntax: { “label“ , central value , pos. gaussian error , neg. gaussian error }<br />
Example: { “delta“ , 0.0 , +0.1 , -0.2 }<br />
Range<br />
Syntax: { “label“ , central value , 0 , pos. <strong>the</strong>oretical error }<br />
Example: { “delta“ , 0.0 , 0 , 0.1 }<br />
Gauss Range<br />
Syntax: { “label“ , central value , pos. gaussian error , pos. <strong>the</strong>oretical error }<br />
Example: { “delta“ , 0.0 , 0.1 , 0.1 }<br />
Asymmetric Gauss Range<br />
Syntax: { “label“ , c. value , neg. gauss. error , pos. gauss. error , <strong>the</strong>o. error}<br />
Example: { “delta“ , 0.0 , +0.1 , -0.2 , 0.1 }<br />
Look Up Table (LUT)<br />
Syntax: { “label“ , “file name“ }<br />
Example: { “delta“ , “delta ICHEP06.dat“ }
E.4. Theory <strong>Package</strong> Tutorial 77<br />
Upper Limit<br />
Syntax: { “label“ , { pos. value } , number between 0 and 100 }<br />
Example: { “delta“ , { 0.4 } , 50 }<br />
Penalty<br />
Syntax: { “label“ , 0 , neg. }<br />
Example: { “delta“ , 0 , -0.1 }<br />
Short Cut<br />
Syntax: { “label“ , “label“ , “label“ , ... }<br />
Example: { “All(delta)“ , “x“ , “y“ , “z“ }<br />
E.4 Theory <strong>Package</strong> Tutorial<br />
During this work, a tutorial that describes <strong>the</strong> development <strong>of</strong> <strong>the</strong>ory packages has<br />
been written. It explains in detail <strong>the</strong> structure and relevant commands <strong>of</strong> a <strong>the</strong>ory<br />
package. It is provided as a PostScript document as well as a <strong>Ma<strong>the</strong>matica</strong> notebook,<br />
named TheoryTutorial.nb. There is <strong>the</strong> possibility to use <strong>the</strong> notebook<br />
version as a template file to create a new <strong>the</strong>ory package.<br />
In addition, a version in HTML format is currently available at:<br />
http://iktp.tu-dresden.de/ jantsch/<strong>CKMfitter</strong>/Tutorials/TheoryTutorial.html
78 Appendix E. User Guide
E.4. Theory <strong>Package</strong> Tutorial 79
80 Appendix E. User Guide
E.4. Theory <strong>Package</strong> Tutorial 81
82 Appendix E. User Guide
E.4. Theory <strong>Package</strong> Tutorial 83
84 Appendix E. User Guide
E.4. Theory <strong>Package</strong> Tutorial 85
86 Appendix E. User Guide
E.4. Theory <strong>Package</strong> Tutorial 87
88 Appendix E. User Guide
E.4. Theory <strong>Package</strong> Tutorial 89
90 Appendix E. User Guide
E.4. Theory <strong>Package</strong> Tutorial 91
92 Appendix E. User Guide
E.4. Theory <strong>Package</strong> Tutorial 93
94 Appendix E. User Guide
E.4. Theory <strong>Package</strong> Tutorial 95
96 Appendix E. User Guide
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[4] J. Charles et al., “CP violation and <strong>the</strong> CKM matrix: Assessing <strong>the</strong> impact<br />
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[14] C. Jarlskog, “Commutator <strong>of</strong> <strong>the</strong> quark mass matrices in <strong>the</strong> Standard Electroweak<br />
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[15] J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, “Evidence for <strong>the</strong><br />
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Danksagung<br />
Allen voran möchte ich meinen Eltern danken, die mir weit mehr als nur ein sorgenfreies<br />
Studium ermöglicht haben. Desweiteren danke ich meinem Bruder Matthias<br />
für seine Hilfsbereitschaft, meiner Freundin Carolin für ihre Liebe sowie meiner gesamten<br />
Familie für das Vertrauen, welches sie in mich gesetzt hat. Ich bedanke<br />
mich bei all meinen Freunden für ihre Unterstützung und die Nachsicht, die sie in<br />
letzter Zeit mit mir haben mussten. Besonderer zu erwähnen sind dabei Matthias<br />
und Steffen, die mich nicht nur durch den Studienalltag begleitet haben.<br />
Heiko Lacker danke ich für die Möglichkeit dieses interessante Thema zu bearbeiten,<br />
aber vor allem für seine sehr gute und persönliche Betreuung. Besonderer Dank<br />
gilt der <strong>CKMfitter</strong> Gruppe, allen voran Jérôme Charles für die fruchtbare Zusammenarbeit,<br />
sowie Stephane T’Jampens und Vincent Tisserand für die großartige<br />
Unterstützung beim Schreiben dieser Diplomarbeit.<br />
Für lehrreiche Diskussionen danke ich Klaus Schubert, sowie Gerhard Buchalla, Zoltan<br />
Ligeti und Michele Papucci.<br />
Ich bedanke mich bei den Mitarbeitern, Doktoranden und Diplomanden des IKTP,<br />
insbesondere bei Rosemarie Krause, Rainer Schwierz und Andreas Petzold, deren<br />
Hilfsbereitschaft nicht hoch genug gewürdigt werden kann. Ein besonderer Dank<br />
geht an Rene Nogowski, dessen hilfreiche Anmerkungen wesentlich zum Gelingen<br />
dieser Diplomarbeit beigetragen haben.<br />
Außerdem möchte ich Gerhard S<strong>of</strong>f, Bernhard Spaan und Sven Jahnke erwähnen,<br />
die entscheidenden Einfluß auf meinen Studienweg hatten.<br />
103
Erklärung<br />
Hiermit versichere ich, daß ich die vorliegende Arbeit selbständig und ohne Benutzung<br />
anderer als der angegebenen Hilfsmittel angefertigt habe. Die aus fremden<br />
Quellen direkt oder indirekt übernommenen Resultate sind als solche kenntlich gemacht.<br />
Dresden, den 11.09.2006 Andreas Jantsch