Measurement of 혾홋 violation in the decay 혽0→→혿∗∗±±혿∓ with data from the LHCb experiment
Dissertation zur Erlangung des akademischen Grades Dr. rer. nat.
vorgelegt von Margarete Schellenberg geboren in Dortmund
Lehrstuhl für Experimentelle Physik V Fakultät Physik Technische Universität Dortmund
Dortmund, 23. Oktober 2019 Der Fakultät Physik der Technischen Universität Dortmund zur Erlangung des akade- mischen Grades eines Dr. rer. nat. vorgelegte Dissertation.
1. Gutachter: Prof. Dr. Bernhard Spaan 2. Gutachter: Prof. Dr. Kevin Kröninger
Datum des Einreichens der Arbeit: 23.10.2019 Datum der mündlichen Prüfung: 28.11.2019 Abstract
This thesis presents the first LHCb measurement of 퐶푃 violation in 퐵0 → 퐷∗±퐷∓ decays. Due to the 푏 → 푐푐푑 transition, 퐵0 → 퐷∗±퐷∓ decays give access to higher-order Stan- dard Model contributions. The decay-time-dependent measurement is performed on a dataset corresponding to an integrated luminosity of 9 fb−1 recorded by the LHCb detector in proton-proton collisions at centre-of-mass energies of 7, 8 and 13 TeV. The 퐶푃 observables are measured with an unbinned maximum likelihood fit as
푆퐷∗퐷 = −0.861 ± 0.077 (stat) ± 0.019 (syst) ,
Δ푆퐷∗퐷 = 0.019 ± 0.075 (stat) ± 0.012 (syst) ,
퐶퐷∗퐷 = −0.059 ± 0.092 (stat) ± 0.020 (syst) ,
Δ퐶퐷∗퐷 = −0.031 ± 0.092 (stat) ± 0.016 (syst) ,
퐴퐷∗퐷 = 0.008 ± 0.014 (stat) ± 0.006 (syst) .
To date, the analysis is the most precise single measurement of 퐶푃 violation in this decay channel. All parameters are consistent with their current world average values.
Kurzfassung
In dieser Arbeit wird die erste LHCb-Messung von 퐶푃-Verletzung in Zerfällen von 퐵0 → 퐷∗±퐷∓ vorgestellt. Aufgrund des 푏 → 푐푐푑-Übergangs, besteht Zugang zu Stan- dardmodellbeiträgen höherer Ordnung. Die zerfallszeitabhängige Messung wird auf einem Datensatz durchgeführt, der einer integrierten Luminosität von 9 fb−1 entspricht, und der vom LHCb-Detektor bei Proton-Proton-Kollisionen mit Schwerpunktsener- gien von 7, 8 und 13 TeV aufgenommen wurde. Die 퐶푃-Observablen werden in einem ungebinnten Maximum-Likelihood-Fit bestimmt zu
푆퐷∗퐷 = −0,861 ± 0,077 (stat) ± 0,019 (syst) ,
Δ푆퐷∗퐷 = 0,019 ± 0,075 (stat) ± 0,012 (syst) ,
퐶퐷∗퐷 = −0,059 ± 0,092 (stat) ± 0,020 (syst) ,
Δ퐶퐷∗퐷 = −0,031 ± 0,092 (stat) ± 0,016 (syst) ,
퐴퐷∗퐷 = 0,008 ± 0,014 (stat) ± 0,006 (syst) .
Die Analyse stellt die bisher präziseste Einzelmessung von 퐶푃-Verletzung in diesem Zerfallskanal dar. Alle Parameter sind konsistent mit ihren aktuellen Weltmittelwerten.
Contents
1 Introduction 1
2 Fundamentals of Flavour Physics 5 2.1 The Standard Model of particle physics ...... 5 2.1.1 Elementary particles ...... 6 2.1.2 Fundamental interactions ...... 7 2.2 Origin of 퐶푃 violation in the Standard Model ...... 7 2.3 Neutral meson system ...... 11 2.4 Classes of 퐶푃 violation ...... 15 2.4.1 Direct 퐶푃 violation ...... 15 2.4.2 Indirect 퐶푃 violation ...... 16 2.4.3 퐶푃 violation in the interference of decay and decay after mixing 16 2.5 퐶푃 violation in the decay 퐵0 → 퐷∗±퐷∓ ...... 18 2.5.1 퐵0–퐵0 mixing phenomenology ...... 18 2.5.2 Decay amplitudes in 퐵0 → 퐷∗±퐷∓ ...... 20
3 The LHCb Experiment at the LHC 27 3.1 The LHC ...... 27 3.2 The LHCb detector ...... 29 3.2.1 Track reconstruction system ...... 31 3.2.2 Particle identification system ...... 33 3.2.3 Trigger ...... 35 3.3 LHCb Software ...... 37 3.3.1 Event reconstruction ...... 37 3.3.2 Stripping ...... 39 3.3.3 Simulation ...... 39 3.4 Prospects of data processing at LHCb ...... 40
4 Data Analysis Techniques 43 4.1 Maximum likelihood method ...... 43
iii Contents
4.2 sPlot technique ...... 44 4.3 Boosted decision tree ...... 45 4.4 Bootstrapping method ...... 47 4.5 Decay-chain fitting ...... 47 4.6 Flavour tagging ...... 47 4.6.1 Same-side flavour tagging ...... 49 4.6.2 Opposite-side flavour tagging ...... 50
5 Datasets and Selection 51 5.1 Reconstruction ...... 51 5.2 Stripping selection ...... 52 5.3 Preselection ...... 55 5.4 Types of backgrounds ...... 56 5.5 Vetoes of physical background ...... 57 5.6 Multivariate selection ...... 61 5.6.1 Training of the boosted decision tree ...... 61 5.6.2 Classifier cut optimisation ...... 64 5.7 Final selection ...... 67
6 Modelling of the invariant Mass Distribution in the Signal Region 69 6.1 Extraction of parameterisations from simulated events ...... 69 6.1.1 Signal candidates ...... 70 6.1.2 Mis-identified background ...... 72 6.1.3 Partially reconstructed background ...... 72 6.2 Extended maximum likelihood fit to LHCb data ...... 75
7 Measurement of 혾홋 Violation in 혽ퟬ → 혿∗±혿∓ Decays 79 7.1 Decay-time parameterisation ...... 79 7.2 Flavour tagging calibration ...... 81 7.2.1 Strategy ...... 81 7.2.2 Single tagger calibration ...... 82 7.2.3 Tagger combination calibration ...... 83 7.3 Decay-time resolution ...... 88 7.4 Decay-time-dependent efficiency ...... 90 7.5 Instrumental asymmetries ...... 92 7.5.1 Strategy ...... 92 7.5.2 Measurement of the detection asymmetry ...... 94
iv Contents
7.6 Determination of 퐶푃 parameters ...... 97
8 Fit Validation and Studies of systematic Effects 103 8.1 Model for fit validation and estimation of systematic uncertainties . 103 8.2 Cross-checks ...... 104 8.2.1 Fit validation using pseudo-experiments ...... 105 8.2.2 Fit validation using simulated signal events ...... 105 8.2.3 Cross-checks on subsets of the data ...... 106 8.2.4 Estimation of statistical uncertainties ...... 107 8.3 Estimation of systematic uncertainties ...... 107 8.3.1 Mass model ...... 107 8.3.2 Decay-time-dependent efficiency ...... 109 8.3.3 Decay-time resolution ...... 110 8.3.4 External inputs to the decay-time fit ...... 110 8.3.5 Flavour tagging calibration ...... 111 8.3.6 Systematic uncertainty on the determination of the detection asymmetry ...... 111 8.4 Total systematic uncertainties ...... 114
9 Conclusion and Outlook 117
Bibliography 121
Acknowledgements 133
v
1 Introduction
Over the last decades, the Standard Model of particle physics (SM) has been intensively evaluated through experiments and is established as an accurate description of the constituents of nature and their fundamental interactions. Although the SM is an extraordinary successful theory, especially in the interplay between theoretical predic- tions and experimental verifications, there are still weaknesses and unsolved questions: For example, the neutrinos are assumed to be massless in the SM, while experimental observations of neutrino oscillations revealed that mass differences between neutrino generations exist [1, 2]. Moreover, cosmological observations indicate that the SM describes only about 5% of the energy in today’s universe [3]. Additional 26% consist of so-called dark matter, while the remaining 69% comprise so-called dark energy. Dark matter becomes evident through gravitational interactions, as it is needed to explain the rotational speed of stars, gases and dust clouds in galaxies [4, 5]. The dark energy on the other hand is postulated to explain the observed accelerated expansion of the universe. Both, the dark matter and the dark energy, are not described by the SM. In addition, gravity, best known from our daily lives, is not explained by the SM, since the integration of gravity into this theory has turned out to be a complicated challenge. Then, there is the biggest question that the SM cannot answer: What happened to the antimatter that was created after the Big Bang? The Big Bang took place about 13.8 billion years ago [3] and led to an equal production of matter and antimatter in the early universe. However, inspecting our universe today, from the smallest scales on Earth to the huge stellar formations, reveals an almost exclusive composition of matter. Hence, it is a major challenge in physics to discover what happened to the antimatter resulting in the baryonic asymmetry observed today. The most common theory to explain the baryonic asymmetry is baryogenesis, which implies the fulfilment of three conditions that were proposed by Andrei Sakharov in 1967 [6]: Firstly, there must have been a deviation from the thermal equilibrium at the early time of the universe. Secondly, the baryon number and thirdly, the charge symmetry (퐶) as well as the combination of charge and parity symmetry (퐶푃) must be violated. More than 50 years ago, 퐶 violation and shortly afterwards also 퐶푃 violation
1 1 Introduction were discovered in particle decays and successfully incorporated in the SM [7]. To date, measurements show that the size of the 퐶푃 violation is compatible with the SM predictions, however, it is an order of magnitude too small to account for the observed matter-antimatter asymmetry [8]. In the SM, 퐶푃 violation is manifested in the Cabibbo- Kobayashi-Maskawa (CKM) matrix, which is unitary and describes the probabilities of quark-flavour transitions. The unitarity leads to conditions that can partly be illustrated as triangles in the complex plane, of which the most common one is called the CKM triangle. In order to probe the SM, high precision measurements of the side lengths and angles of the CKM triangle are performed. These measurements are then combined to reveal, if the CKM triangle is a closed triangle in nature and, hence, if the CKM matrix is indeed a unitary matrix. One angle of the CKM triangle is represented by the parameter 훽. It can be extracted from the quantity sin 2훽, which can be measured with high precision in so-called golden 0 0 plated modes like 퐵 → 퐽/휓퐾S . Such decays allow for a very clean prediction from the theoretical and for an excellent signature in the detector from an experimental point 0 0 0 ∗± ∓ of view. In contrast to 퐵 → 퐽/휓퐾S decays, the decay 퐵 → 퐷 퐷 allows to measure the quantity sin 2훽eff, as further hadronic contributions and phases exist. Knowledge of these contributions is needed to separate them from effects of physics beyond the SM, also called New Physics. In this thesis, a measurement of 퐶푃 violation in 퐵0 → 퐷∗±퐷∓ decays is presented. The analysed data was taken in proton-proton collisions accelerated to a centre-of- mass energy of up to 13 TeV by the world’s most powerful particle accelerator, the Large Hadron Collider (LHC). The data has been recorded by the LHCb detector and corresponds to an integrated luminosity of 9 fb−1. The 퐷∓ meson is reconstructed ( ) as 퐾± 휋∓ 휋∓, while the 퐷∗± meson is reconstructed as 퐷∗± → 퐷 0휋±, where for the ( ) 퐷 0 decays the final states 퐾∓ 휋± and 퐾∓ 휋∓ 휋± 휋± are considered. Due to the full hadronic final state, one big challenge in the presented analysis are physical background contributions. Hence, for a precise measurement it is necessary to develop a selection, which provides a strong separation between the signal and the background components. The measurement of 퐶푃 violation in 퐵0 → 퐷∗±퐷∓ decays was performed in collabor- ation between the LHCb physics groups from Dortmund, CERN, Ferrara and Milan. In order to present the complete measurement, contributions from proponents of the 퐵0 → 퐷∗±퐷∓ analysis group are shown. The corresponding results are indicated in this thesis. The outline of the thesis is as follows: the fundamentals of flavour physics are presen- ted in Ch. 2 and are followed by an introduction to the LHCb experiment in Ch. 3,
2 including details on the sub-detectors and the software framework. In Ch. 4, the data analysis tools and techniques utilised in this thesis are explained. While Ch. 5 discusses the datasets and their preparations, Ch. 6 outlines the mass fit to the invariant 퐷∗±퐷∓ mass distribution, which enables a statistical subtraction of the remaining background from the signal candidates. This is followed by the description of the decay-time- dependent fit performed to measure the 퐶푃 parameters and the studies concerning systematic uncertainties in Ch. 7 and Ch. 8, respectively. Finally, the thesis concludes in Ch. 9 with a discussion of the results and an outlook on further measurements.
3
2 Fundamentals of Flavour Physics
In this chapter, the physical motivation of this thesis is presented. Firstly, the standard model of particle physics is introduced in Sec. 2.1, where the elementary particles and their interactions are discussed. Furthermore, 퐶푃 violation and neutral-meson mixing are described in Sec. 2.2 to Sec. 2.4. Finally, in Sec. 2.5 the theoretical background of the signal channel investigated in this thesis, 퐵0 → 퐷∗±퐷∓, is explained. The information in this chapter are taken from Refs.[9, 10].
2.1 The Standard Model of particle physics
The Standard Model of particle physics (SM) is a relativistic quantum field theory that comprises the elementary particles and fundamental interactions occurring among them. In total, the SM includes twelve fermions and five bosons, where the latter represent the force carriers. The interactions are explained in two theories that are part of the SM. Firstly, there is the Glashow-Weinberg-Salam (GWS) model [11–13], which describes the electroweak interactions, and secondly Quantum Chromodynamics (QCD) [14, 15] defining the strong interaction. A fundamental aspect of the SM is the validity of the 퐶푃푇 theorem [16, 17], which was developed by Wolfgang Pauli in 1955. It states that after the concurrent execution of the transformations charge conjugation 퐶, space reflection 푃 ( ⃗푥→ −) ⃗푥 and time reversal 푇 (푡 → −푡), the physical laws retain their validity. Until 1956 it was assumed that the individual operations are preserved as well. This assumption was experimentally refuted by Wu and Garwin [18]. A year later, Landau postulated that the weak interaction was invariant under 퐶푃 transformation. The opposite was proven in 1964, when 퐶푃 violation was discovered for the first time in decays of long-lived kaons into two pions[7]. Outside the kaon system, 퐶푃 violation was firstly discovered in 퐵 meson decays in 2001 [19, 20]. Just recently 퐶푃 violation was observed by the LHCb experiment additionally in the 퐷 meson system [21].
5 2 Fundamentals of Flavour Physics
2.1.1 Elementary particles
The SM contains twelve elementary particles, six quarks (푞) and six leptons (푙). They are known as fermions, as they carry an intrinsic angular momentum of 1/2 (spin). For each particle an anti-particle (푞, 푙) exists, which has the same properties as its partner except for the charge like quantum numbers, which have an opposite sign. Ascending by mass, they are divided into three families, see Fig. 2.1.
1. 2. 3. mass 2.2 MeV/푐2 1275 MeV/푐2 173.0 GeV/푐2 charge +2/3 푢 +2/3 푐 +2/3 1 1 1 푡 spin /2 /2 /2 125 GeV/푐2 up charm top 0 4.7 MeV/푐2 95 MeV/푐2 4.2 GeV/푐2 0 퐻 Higgs boson −1/3 −1/3 푠 −1/3 1/2 푑 1/2 1/2 푏 down strange bottom quarks
0.511 MeV/푐2 105.7 MeV/푐2 1777 MeV/푐2 0 91.2 GeV/푐2 −1 푒 −1 휇 −1 휏 0 푔 0 1/2 1/2 1/2 1 1 푍 electron muon tau gluon 푍 boson
<2 eV/푐2 <0.19 MeV/푐2 <18.2 MeV/푐2 0 80.4 GeV/푐2 0 휈 0 휈 0 휈 0 훾 ±1 1/2 푒 1/2 휇 1/2 휏 1 1 푊 photon
푒 neutrino 휇 neutrino 휏 neutrino 푊 boson gauge bosons leptons
Figure 2.1 – Overview of the fermions and the gauge bosons in the SM. The values are taken from Ref. [22].
The up-type quarks (푢, 푐, 푡) carry a fractional electric charge of +2/3 푒, while the electric charge of down-type quarks (푑, 푠, 푏) is −1/3 푒, with 푒 being the elementary charge. In addition to the electrical charge, (anti-)quarks carry a(n) (anti-)colour charge, whose possible states are called (anti-)red, (anti-)blue and (anti-)green. The combination of colour and anti-colour as well as the combination of red, green and blue results in neutral colour-charge. If a particle carries a colour charge it can only occur in bound states that are colour-neutral. This phenomenon is called confinement. The bound states of quarks are called hadrons and can be classified further as mesons (푞푞), baryons (푞푞푞, 푞푞푞), tetraquarks (푞푞푞푞) and pentaquarks (푞푞푞푞푞). Resonances compatible with the latter two types of bound states were observed by LHCb in 2015 and 2016 [23–25]. The six leptons are divided into three generations as well. Each generation consists of a charged lepton (electron 푒−, muon 휇−, tau 휏−) and an associated neutral lepton, the neutrino (휈푒, 휈휇, 휈휏). The charged leptons carry a charge of −1푒 and in case of an anti-particle 1푒.
6 2.2 Origin of 퐶푃 violation in the Standard Model
2.1.2 Fundamental interactions
The construction of the SM results from symmetries under transformations of a gauge group. Local invariance of the gauge symmetries is considered to establish the interac- tions and the number of gauge bosons corresponding to the generators of the group. The bosons, which are particles of integer spin, are the mediators of interactions in the SM. The symmetry group of the SM is given as 푆푈(3) ⊗ 푆푈(2) ⊗ 푈(1). As a part of it, the gauge group 푆푈(3), representing the QCD, describes the strong interactions and is associated with eight gluons (푔). Gluons are massless and couple to the colour charge of the quarks. Since gluons carry colour and anti-colour, they can also interact with themselves. The weak and electromagnetic interaction are unified in the electroweak theory and are associated with the gauge group 푆푈(2) ⊗ 푈(1). The exchange particle of the electromagnetic interaction is the photon 휸, which couples to the elementary charge. The photon couples to all fermions, except for the charge-neutral neutrinos. As the photon itself carries no charge, it cannot interact directly with itself like the gluons. The mediators of the weak interaction are the charged 푊 ± bosons and the neutral 푍0 boson. The range of the weak interaction is strongly limited by the masses of their 2 2 exchange particles 푚푊 = 80.4 GeV/푐 and 푚푍 = 91.2 GeV/푐 , while the electromagnetic interaction has theoretically an infinite range, since the photon 훾 is massless. All fermions interact weakly. In addition to the 푊 ± and 푍0 bosons that carry a spin of 1, the Higgs boson, which is spinless, exists. It was predicted in 1964 [26–28] and has been discovered in 2012 by the ATLAS and CMS experiments [29, 30] and is a result of the Higgs mechanism [26, 27] that comprises a formalism of spontaneous symmetry breaking to allow for the 푊 ± and 푍0 bosons masses and enables photons to be massless.
2.2 Origin of 혾홋 violation in the Standard Model
The fermion masses and 퐶푃 violation have a common origin in the SM, the Yukawa coupling to the Higgs field. It requires local gauge invariance between the left- and right- handed fermion fields and the Higgs field. Through spontaneous symmetry breaking, the fermions acquire their masses by their specific coupling constants to the Higgs field. These coupling constants are arbitrary matrices when allowing multiple generations. To obtain the physically observable mass eigenstates, the matrices are diagonalised. In the case of leptons, the matrices can be diagonalised for both 푒, 휇, 휏 and 휈푒, 휈휇, 휈휏 in a way that matrices without mixing are created, due to the fact that neutrinos are considered
7 2 Fundamentals of Flavour Physics to be massless in the SM. In contrast, this is not possible in the quark sector, where only one of the matrices can be diagonalised. By convention, it is performed for the up-type quarks, while for down-type quarks the mass eigenstates (푑, 푠, 푏) are converted into weak interaction eigenstates (푑′, 푠′, 푏′) by the so-called CKM matrix 푉CKM [31, 32]:
⎛푑′⎞ ⎛푑⎞ ⎛푉ᵆ푑 푉ᵆ푠 푉ᵆ푏⎞ ⎛푑⎞ ⎜푠′⎟ = 푉 ⋅ ⎜푠⎟ = ⎜푉 푉 푉 ⎟ ⋅ ⎜푠⎟ . (2.1) ⎜ ⎟ CKM ⎜ ⎟ ⎜ 푐푑 푐푠 푐푏 ⎟ ⎜ ⎟ ⎝푏′⎠ ⎝푏⎠ ⎝푉푡푑 푉푡푠 푉푡푏 ⎠ ⎝푏⎠
The CKM matrix, established by Nicola Cabibbo, Makoto Kobayashi and Toshihide Maskawa, is a complex, unitary 3 × 3 matrix. A unitary matrix 푉 fulfils the relation
푉 †푉 = ퟙ . (2.2)
Due to its non-zero off-diagonal elements, 푉CKM enables quark transitions between up- and down-type quarks of different families. The transition probability of aquark 2 푞푖 into a quark 푞푗 is proportional to |푉푞푖푞푗| . The fact that the CKM-matrix contains a complex phase allows for the violation of the 퐶푃 symmetry. The CKM matrix can be represented in many different forms. One parametrisation is referred to as the Chau-Keung parametrisation [33], which can be seen as a complex rotation matrix 휋 with three Euler mixing angles 휃12, 휃13, 휃23 ∈ [0, /2] and a phase 훿 ∈ (−휋, 휋]. With the abbreviations 푐푖푗 = cos 휃푖푗 and 푠푖푗 = sin 휃푖푗 it is given by
−i훿 ⎛ 푐12푐13 푠12푐13 푠13e ⎞ 푉 = ⎜−푠 푐 − 푐 푠 푠 ei훿 푐 푐 − 푠 푠 푠 ei훿 푠 푐 ⎟ . (2.3) CKM ⎜ 12 23 12 23 13 12 23 12 23 13 23 13 ⎟ i훿 i훿 ⎝ 푠12푠23 − 푐12푐23푠13e −푐12푠23 − 푠12푐23푠13e 푐23푐13 ⎠
Another common way to define the CKM matrix 푉CKM is the Wolfenstein parametrisa- tion [34], which uses a series expansion in the parameter 휆:
휆2 1 − 휆 퐴휆3(휌 − i휂) ⎛ 2 ⎞ 2 휆 2 4 푉CKM = ⎜ −휆 1 − 퐴휆 ⎟ + 풪(휆 ). (2.4) ⎜ 2 ⎟ 3 2 ⎝퐴휆 (1 − 휌 − i휂) −퐴휆 1 ⎠
The Wolfenstein parametrisation illustrates the presence of the complex phase in the
CKM matrix elements of 푉ᵆ푏 and 푉푡푑 in the first order of expansion, represented by the 퐶푃 violating phase 휂. Hence, especially these transitions provide a good access to 퐶푃 violation, which gives an essential motivation for studies of 퐶푃 violation in 퐵0 (푏푑)
8 2.2 Origin of 퐶푃 violation in the Standard Model decays. The relation between the parameters of Eq. (2.3) and Eq. (2.4) can be expressed as
2 3 −i훿 휆 = 푠12 , 퐴휆 = 푠23 , 퐴휆 (휌 − i휂) = 푠13e . (2.5)
+0.018 The four parameters of the Wolfenstein parameters are measured to be 퐴 = 0.810−0.024, +0.00068 +0.013 +0.011 휆 = 0.22548−0.00034, 휌 = 0.145−0.007 and 휂 = 0.343−0.012 [35], where the following relations hold
2 2 휌 = 휌 (1 − 휆 /2) and 휂 = 휂 (1 − 휆 /2) . (2.6)
Moreover, in the Wolfenstein parametrisation, the hierarchical structure of the CKM matrix becomes visible: With increasing distance from the main diagonal, the mag- nitudes of the CKM matrix elements decrease. Due to the fact that the main diagonal elements are of 풪 (1), quark transitions occur most likely within a family, as transitions 2 are proportional to |푉푖푗| . A transition from a generation into a neighbouring genera- tion (Cabibbo or CKM suppression) is more likely (풪(휆2)) than skipping a generation (풪(휆3)), which is referred to a doubly Cabibbo suppressed transition. Exploiting the unitarity requirement (see Eq. (2.2)), nine conditions can be constructed. Six of them are equal to zero and can be represented as triangles in the complex plane. The area of these so-called unitarity triangles is constant for all triangles and is corresponding to the half of the Jarlskog parameter 퐽퐶푃 [36], which is defined as
∗ ∗ 퐽퐶푃 = ±Im(푉푖푘푉푗푙푉푖푙 푉푗푘 ) with 푖 ≠ 푗, 푙 ≠ 푘 . (2.7)
The Jarlskog parameter can be interpreted as a measure of the amount of 퐶푃 violation allowed in the SM, implying that 퐶푃 symmetry is only violated if 퐽퐶푃 ≠ 0. One of the triangles, which can be studied particularly effectively, as its side lengths have the same orders in 휆, fulfils the following equation
∗ ∗ ∗ 푉ᵆ푑푉ᵆ푏 + 푉푐푑푉푐푏 + 푉푡푑푉푡푏 = 0 . (2.8)
This triangle is also known as the 푏푑-triangle or the Unitarity Triangle, as all CKM- matrix elements include transitions of 푏 or 푑 quarks. In Fig. 2.2, the triangle from Eq. (2.8) is shown. Here Eq. (2.8) is divided by the experimentally best-known value ∗ 푉푐푑푉푐푏 [22]. Hence, it has one side of unit length that is aligned along the horizontal axis. The corner points of the triangle have the coordinates (0, 0), (1, 0) and the apex
9 2 Fundamentals of Flavour Physics positioned at (휌, 휂), defined by
푉 푉 ∗ ᵆ푑 ᵆ푏 휌 + i휂 = − ∗ . (2.9) 푉 푐푑푉푐푏
The angles of the Unitarity Triangle are given by ∗ ∗ ∗ 푉푡푑푉푡푏 푉푐푑푉푐푏 푉ᵆ푑푉ᵆ푏 훼 = arg (− ∗ ) , 훽 = arg (− ∗ ) and 휸 = arg (− ∗ ) . (2.10) 푉ᵆ푑푉ᵆ푏 푉푡푑 푉푡푏 푉푐푑푉푐푏 One major goal of experimental flavour physics is to perform multiple measurements of the angles and the side lengths to overconstrain the Unitarity Triangle, which allow to verify, if it closes in its apex. In this way, an indirect search for New Physics is carried out. If experiments show that the sides do not converge at one point, it would be an indication of physics beyond the SM, as the unitarity is a fundamental building block of the SM. The current experimental results of the parameters and their uncertainties are combined by a fit of the CKM fitter group [37] and are in good agreement with each other and the SM. The CKM triangle and the experimental results are shown in Fig. 2.2.
0.7
∆m & ∆ms CKM γ d ε f i t t e r 0.6 ∆md K Summer 18
0.5 sin 2β sol. w/ cos 2β < 0 (excl. at CL > 0.95) 0.4 excluded area has CL > 0.95 > CL has area excluded η 0.3 α α 0.2
Vub 0.1 γ β α 0.0 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 ρ