Doctoral thesis
Katarzyna Senderowska
CP symmetry violation in B
s
0
decays to
CP eigenstates in LHCb experiment
Supervisor: dr hab. Mariusz Witek
The Henryk Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, Cracow
Declaration of the author of this dissertation:
Aware of legal responsibility for making untrue statements I hereby declare that I have written this dissertation myself and all the contents of the dissertation have been obtained by legal means.
data, podpis autora
Declaration of the thesis Supervisor: This dissertation is ready to be reviewed.
I would like to express my deepest thanks to my supervisor, dr hab. Mariusz Witek, for his guidance, attention and excellent tutoring through all the years of my Ph.D. studies, for the continuous help with scientific and technical advice and important suggestions and comments to this thesis.
I am very grateful to prof. dr hab. Bogdan Muryn for his support, for a deep insight into the theoretical aspects of the study and a lot of valuable comments on the manuscript of this thesis. I would like to thank all members of Cracow LHCb group for creating a friendly atmosphe-re, many fruitful discussions concerning particle physics and for the help with all programming and software issues.
Last but not least, I wish to express my gratitude to the members of the Department of Particle Interaction and Detection Techniques at the Faculty of Physics and Applied Computer Science AGH and the Division of Particle Physics and Astrophysics of the Henryk Niewodni-czański Institute of Nuclear Physics PAN, who served the advice in selecting research topics, were always willing to discuss the results and helped me when I encountered any technical problems.
This work has been partially supported by National Science Centre pre-doctoral grant.
Celem pracy doktorskiej jest badanie zjawiska łamania kombinowanej symetrii przestrzenno-ładunkowej CP w rozpadach mezonów B0
s do stanów końowych będących stanami własnymi CP.
W Modelu Standowym zjawisko łamania symetrii CP opisywane jest przez macierz Cabbibo-Kobayashi-Maskawa (CKM), ktorej elementy są proporcjonalne do prawdopodobieństw przejść pomiędzy poszczególnymi rodzinami kwarków. Wartości parametrów macierzy wyznaczane są eksperymentalnie. Jednym z parametrów jest faza słaba mieszania φs, określająca stopień
łama-nia symetrii CP w rozpadach mezonów B0
s . Jest to równocześnie jeden z najsłabiej zmierzonych
parametrów. Faza słaba mieszania wyznaczana jest na podstawie analizy rozpadów mezonów Bs0, w których kwark b rozpada sie zgodnie ze schematem b → ccs . Rozpady takie zachodzą głównie poprzez diagramy drzewiaste (wkład diagramów pingwinowych jest zaniedbywalny). W związku z tym amplituda rozpadu zależy od pojedynczej amplitudy macierzy CKM, a war-tość φs może być wyznaczona bezpośrednio z pomiarów symetrii CP. Łamanie symetrii CP w
takich rozpadach jest wynikiem interferencji pomiędzy dwiema amplitudami: pierwsza z nich związana jest z bezpośrednim rozpadem mezonu B0
s do określonego stanu końcowego, druga
natomiast: z oscylacją B0
s - ¯Bs0 i rozpadem powstałego w wyniku oscylacji mezonu ¯Bs0 do tego
samego stanu końcowego. W zależności od zawartości stanu końcowego, rozpady z przejściem kwarkowym b → ccs dzielone są na dwie grupy. Pierwsza z nich to rozpady, w których stan końcowy jest superpozycją stanów o różnych CP. Przykładem takich rozpadów jest proces B0
s → J/ψφ, nazywany, ze względu na bardzo duży stosunek rozgałęzienia, złotym kanałem dla
pomiaru φs. Druga grupa obejmuje rozpady, w których stan końcowy ma określoną parzystość
przestrzenno-ładunkową - innymi słowy, jest stanem własnym CP. Taki stan końcowy złożony jest z dwóch mezonów pseudoskalarnych (jak w przypadku rozpadu B0
s → Ds+Ds−) lub z
jed-nej cząstki wektorowej i jedjed-nej pseduoskalarjed-nej (np. dla rozpadów B0
s → J/ψη, Bs0 → J/ψη0,
Bs0 → J/ψf0 , Bs0 → ηcφlub Bs0 → χc0φ). Analiza rozpadów do stanów własnych CP jest
prost-sza, ponieważ nie wymaga dodatkowego kroku związanego z rozseparowanaiem stanów o róż-nych parzystościach CP. Ze względu jednak na mniejsze stosunki rozgałęzień, pomiar wielkości
φs wymaga większej statystyki danych.
Pomiar fazy φs jest czułym testem na istnienie tzw. Nowej Fizyki, obejmującej zjawiska
wy-kraczające poza zakres Modelu Standardowego. W Modelu Standardowym φs ma bardzo małą
wartość, wyznaczoną dodatkowo z bardzo dużą precyzją. Zaobserwowanie nawet niewielkiego odchylenia od przewidywania Modelu Standardowego może być przesłanką do istnienia Nowej Fizyki.
Eksperyment LHCb, zbierający dane przy Wielkim Zderzaczu Hadronów w ośrodku CERN pod Genewą, jest w chwili obecnej najlepszym na świecie miejscem do przeprowadzania analiz rozpadów mezonów B0
s. Ogromna statystyka przypadków, dedykowany charakter eksperymentu
oraz wysoka jakość aparatury umożliwiają bardzo szczegółowe przetestowanie teorii łamania symetrii CP opartej na macierzy CKM, poszukiwanie rzadkich rozpadów mezonów B oraz zjawisk Nowej Fizyki, wykraczającej poza zakres Modelu Standardowego. Pomiar fazy słabej mieszania φs jest jednym z głównych celów badawczych eksperymentu LHCb.
W ramach pracy doktorskiej wykonano analizy rozpadów B0
s → J/ψf0 oraz Bs0 → Ds+D − s,
w oparciu o dane zebrane w 2011 roku. Kanały te wybrane zostały ze względu na fakt, że dobrze reprezentują klasę rozpadów mezonów B0
s do stanów własnych CP - pierwszy z nich zawiera
miony w stanie końcowym, natomiast drugi jest czysto hadronowy. Ponadto, ich stosunki roz-gałęzień i wydajności detekcji i rekonstrukcji są stosunkowo wysokie na tle innych rozpadów do stanów własnych CP. Oszacowana została także precyzja pomiaru parametru φs na podstawie
1 Theoretical introduction 18
1.1 CP violation within the Standard Model . . . 18
1.2 Unitarity triangles . . . 20
1.3 Measurements of CP violation parameters . . . 21
1.4 Oscillations of neutral mesons . . . 25
1.5 Time-dependent decay rates of B0 s mesons and CP asymmetries . . . 27
1.6 Types of CP violation . . . 28
1.6.1 CP violation in mixing (indirect) . . . 28
1.6.2 CP violation in decay (direct) . . . 29
1.6.3 CP violation in the interference of mixing and decay (mixing induced CP violation) . . . 29
1.7 Measurement of CP violation phase from B0 s decays into CP eigenstates . . . 30
1.8 Current experimental status of φs measurements . . . 32
2 The Large Hadron Collider 34 2.1 CERN accelerating complex and LHC parameters . . . 35
2.2 LHC experiments . . . 36
3 The LHCb experiment 38 3.1 LHCb geometry . . . 38
3.2 Luminosity . . . 38
3.3 LHCb detector . . . 39
3.3.1 The tracking system . . . 41
3.3.2 Particle identification . . . 45
3.4 Trigger . . . 48
3.4.1 Level-0 Trigger . . . 49
3.4.2 High Level Trigger . . . 50
3.5 The Online System . . . 51
3.6 Reconstruction and Analysis Software . . . 52 6
3.7 Data Processing . . . 52 3.8 LHCb data taking in 2011 . . . 53 4 Selection of Bs0 → J/ψf0 decays 55 4.1 Data samples . . . 55 4.2 Decay reconstruction . . . 56 4.3 Trigger . . . 56 4.4 Stripping selection . . . 57 4.5 Offline selection . . . 58
4.6 Optimization of selection criteria with multivariate analysis method . . . 59
4.7 Selection results for Monte Carlo events . . . 63
4.8 Selection results for data . . . 70
4.9 Conclusion . . . 71 5 Selection of Bs0 → D+ sD − s decays 74 5.1 Data samples . . . 74 5.2 Decay reconstruction . . . 74 5.3 Trigger . . . 75
5.4 Stripping and offline selection . . . 75
5.5 Optimization of selection criteria with multivariate analysis methods . . . 75
5.6 Selection results for Monte Carlo events . . . 78
5.7 Selection results for data . . . 83
5.8 Conclusion . . . 84
6 Sensitivity to φs phase - Monte Carlo study 87 6.1 Likelihood function description . . . 88
6.1.1 Invariant mass distributions . . . 88
6.1.2 Proper time distributions . . . 88
6.1.3 Total likelihood function . . . 93
6.2 Monte Carlo input and fit strategy . . . 95
6.3 Toy Monte Carlo results . . . 95
6.4 φs precision dependence on integrated luminosity and mistag rate . . . 97
6.5 Conclusion . . . 98
1.2 a) db unitarity triangle in the Wolfenstein parametrization. b) bs unitarity trian-gle in the Wolfenstein parametrization. Both triantrian-gles are normalised to the baseline of db triangle, i.e. VcdVcb∗ is real and all lengths are divided by |VcdVcb∗|. . 22
1.3 Constraints in the (¯ρ− ¯η) plane (as explained in the text). The red hashed region of the global combination corresponds to 68% CL. . . 23 1.4 Constraints in the ( ¯ρs− ¯ηs) plane. The red hashed region of the global combination
corresponds to 68% CL. The (almost horizontal) thin blue lines corresponds to the 68% and 95% CL constraints on βs. . . 23
1.5 Feynman diagrams contributing to B0
d → J/ψKs0 decay. Left: the tree-type
dia-gram. Right: the penguin diadia-gram. . . 24 1.6 Feynman box diagrams for B0
q − ¯Bq0 mixing (q = s, d). The charge-conjugated
process is obtained by substituting all quarks with antiquarks and taking the complex-conjugates of the CKM matrix elements. . . 25 1.7 Feynman diagrams illustrating B0
q meson decay through a b → ccs quark-level
process. . . 32 1.8 Likelihood confidence regions in the ∆Γs− φsplane. A result obtained from 0.37
fb−1 of data collected at LHCb experiment. . . 33
2.1 Layout of the CERN accelerator complex. . . 35 3.1 The angular correlation between the polar angles of the b and ¯b quarks produced
at LHCb. . . 39 3.2 A schematic view of the LHCb detector. . . 40 3.3 Instantaneous luminosity at the LHC experiments during an example fill.
Lumi-nosity levelling mechanism at the LHCb experiment ensures constant lumiLumi-nosity throughout entire fill. . . 40 3.4 The scheme of R −φ geometry of the VELO sensors. Only a portion of the strips
is showed for clarity. . . 42 3.5 Resolution of primary vertex in x, y and z coordinate as a function of the Monte
Carlo (MC) multiplicity of PV. The MC vertex multiplicity denotes the number of reconstructed tracks corresponding to MC particles coming from the PV. . . . 42
3.6 Impact parameter resolution as a function of log(1/pt) for the TDR design with
25 VELO stations and for a setup with 20 stations. . . 43 3.7 Front view of an x detection layer in the second IT station. . . 44 3.8 Arrangement of OT straw-tube modules in layers and stations. . . 44 3.9 Top picture: the evolution of the strength of the magnetic field along the z-axis.
Bottom picture: various types of track at LHCb: VELO, downstream, upstream, long and T-tracks. . . 45 3.10 θcangle as a function of particle momentum for relativistic charged kaons, pions
and protons for different Cherenkov radiators. The saturated (v << c) angles are shown for various radiators used in the RICH system: aerogel, C4F10 and CF4. 46
3.11 Left: side view layout of the RICH1 detector. Right: top view of the RICH2 detector. . . 47 3.12 Left: side view of the LHCb muon system. Right: front view of a quadrant of a
muon station. Each rectangle corresponds to the chamber. The granularities of four chambers belonging to the four regions of M1 station are also presented. . . 48 3.13 The structure of the LHCb trigger system. . . 49 3.14 Performance of merged π0 removal procedure: the distribution of output
discrimi-nant variable for signal (photons from B0
s → φγ decay) and background (merged
π0). . . 51
3.15 The total integrated luminosities delivered by the LHC machine and recorded by the LHCb detector in 2011. . . 54 4.1 B0
s mass distribution for data events passing the trigger, the stripping and offline
cut-based selection. Events from the signal sidebands region (region between the red dashed lines is excluded) were used as a background sample for the TMVA training. . . 59 4.2 Correlation coefficients matrix for signal events. The relation between the
tech-nical names of variables in Figure and variables symbols used in the text are collected in Table 4.6. . . 62 4.3 Correlation coefficients matrix for background events. The relation between the
technical names of variables in Figure and variables symbols used in the text are collected in Table 4.6. . . 62 4.4 Comparison of signal and background distributions for the variables chosen for
multivariate discrimination. The area of every distribution is normalized to one. 64 4.5 Comparison of ROC curves for selected discrimination methods. . . 65 4.6 The distributions of the BDT discriminant for signal and background samples. . 65 4.7 Signal and background efficiencies as a function of BDT cut value. The
signifi-cance √S
S+B is also plotted. . . 66
4.8 The resolution of B0
s invariant mass for MC events passing the selection. The
distribution is fitted with a sum of a single Gaussian and Crystal Ball function. The histogram is plotted in wide B0
s mass window. . . 66
4.9 The resolution of B0
s proper time for Bs0 → J/ψf0 MC events passing the
4.10 Mass distribution for B0
s → J/ψη
0 background passing the event selection. The
line shows the result of the fit to the Crystal Ball function. . . 69 4.11 Mass distribution for B0
d → J/ψK
∗(892)0background passing the event selection.
Crystal Ball fit is applied. . . 69 4.12 Mass distribution for Bu,d,s → J/ψX background passing the event selection.
The distribution is fitted with a single Gaussian to account for signal (red dashed line) and several background components. The cyan dashed line shows the B0
d →
J/ψπ+π− background, the dashed purple line: Bd0 → J/ψK∗(892)0 background,
the dashed green line: the combinatoric background. Very few B0
s → J/ψη0
background events are present (dashed yellow line). . . 70 4.13 Invariant mass distributions for f0 and J/ψ mesons passing the selection, within
±50MeV/c2 of the B0
s mass. The f0 mass distribution is fitted with a sum of a
single Gaussian and an exponential function. The J/ψ mass spectrum is fitted with a double Gaussian. . . 71 4.14 B0
s invariant mass distribution for data after the Bs0 → J/ψf0 selection. The
distribution is fitted with a single Gaussian for signal (red dashed line) and several background components. The cyan dashed line shows the B0
d → J/ψπ+π −
background, the dashed purple line: B0
d → J/ψK
∗(892)0 background, the dashed
yellow line: the B0
s → J/ψη0 background, the dashed green line: the combinatoric
background. . . 72 4.15 The distribution of B0
s proper time for the selected B0s → J/ψf0 data events. . . 72
5.1 B0
s mass distribution for data after the trigger, the stripping and offline cut-based
selection. . . 76 5.2 B0
s mass distribution for data after the trigger, the stripping, offline cut-based
selection and additional cut on K mesons. . . 76 5.3 Correlation coefficients matrix for signal events. The relation between the
tech-nical names of variables in Figure and variables symbols used in the text are collected in Table 5.5. . . 79 5.4 Correlation coefficients matrix for background events. The relation between the
technical names of variables in Figure and variables symbols used in the text are collected in Table 5.5. . . 79 5.5 Comparison of signal and background distributions for the variables chosen for
multivariate discrimination. The integral of every distribution is normalized to one. . . 80 5.6 The distributions of the BDT discriminant for signal and background samples. . 81 5.7 Signal and background efficiencies as a function of BDT cut value. The
signifi-cance √S
S+B is also plotted. . . 81
5.8 The resolution of B0
s invariant mass for MC events after the selection. . . 82
5.9 Mass distribution for B0
s → D
∗+
s D
∗−
s background passing the selection. A Crystal
Ball fit is applied. . . 83 5.10 Mass distribution for B0
s → D
∗±
s D
±
s background passing the selection. A Crystal
5.11 Invariant mass distributions for D−
s and D+s mesons passing the selection. Both
distributions are fitted with a sum of a single Gaussian and exponential function. 85 5.12 B0
s invariant mass distribution for data passing the B0s → D+sD −
s selection. The
distribution is fitted with a double Gaussian for signal (red dashed line) and several background components. The cyan dashed line shows the B0
s → D
∗+
s D
∗− s
background, the dashed yellow line: B0
s → D
∗±
s D
∓
s the background, the dashed
green line: the combinatoric background. . . 85 6.1 Projection of the PDF for the mass distribution for B0
s → J/ψf0 decay channel.
The dotted green line represents the combinatorial background and the dashed red line is the signal. The solid blue line is the sum of signal and background events. . . 89 6.2 Initial decay rates as a function of the proper time for B0
s and ¯Bs0 decaying to
J/ψf0 final state. No detector effects are included. . . 89
6.3 The decay rates as a function of the proper time for B0
s and ¯B0s decaying to
J/ψf0 final state diluted by a wrong-tag inefficiency. . . 90
6.4 The decay rates as a function of the proper time for B0
s and ¯B0s decaying to
J/ψf0 final state diluted by a wrong-tag inefficiency and convoluted with the
proper time Gaussian resolution. . . 91 6.5 The decay rates as a function of the proper time for B0
s and ¯Bs0decaying to J/ψf0
final state diluted by a mistagging, convoluted with the proper time Gaussian resolution and multiplied by an acceptance function. . . 92 6.6 Decay rates for combinatorial background, taking into account the effect of
Gaus-sian proper time resolution and acceptance function. . . 93 6.7 The projection of proper time distribution for B0
s decays passing the Bs0 → J/ψf0
event selection. The dashed red line corresponds to the signal events and the dot-ted green line is a background contribution. The upper plot shows the expecdot-ted rates for an initially tagged B0
s mesons, the bottom plot- for an initially tagged
¯ B0
s mesons. . . 94
6.8 Toy Monte Carlo results for value φs = -0.04 rad. Left: distribution of φs
me-an fitted value. Middle: φs error distribution. Right: φs pull distribution. All
distributions were fitted with a single Gaussian. . . 96 6.9 Toy Monte Carlo results for φs = -0.04 rad. Left: distribution of ∆Γs mean
fitted value. Middle: ∆Γs error distribution. Right: ∆Γs pull distribution. All
distributions were fitted with a single Gaussian. . . 96 6.10 The φs sensitivity as a function of the wrong tag fraction ωtag. φs starting value
equal to -0.04 was assumed. Error bars are of order ∼ 10−4, thus are not visible
in the plot. . . 97 6.11 The φs sensitivity as a function of integrated luminosity. φs starting value equal
to -0.04 was assumed. Error bars are of order ∼ 10−4− 10−3, therefore they are
and LHCb experiments. . . 33
2.1 The LHC parameters. . . 37
4.1 Muon and di-muon trigger lines efficiencies for B0 s → J/ψf0 events. . . 57
4.2 The summary of stripping and offline selection criteria for B0 s → J/ψf0 . In the stripping phase the f0 candidate is reconstructed in f0 → π+π−and f0 → K+K− decay modes, using the same selection cuts. For the sake of simplicity only cuts for f0 → π+π− are listed. . . 60
4.3 Variables ranking according to the signal/background separation power. The top variable has the best discrimination power. Variables chosen for the final optimi-zation are marked as Y in the last column. In the third column the corresponding notation used in Figure 4.2 and 4.3 is given. . . 61
4.4 Signal efficiencies for the selection steps for B0 s → J/ψf0 simulated events. . . . 67
4.5 Various factors contributing to efficiency of the B0 s → J/ψf0 decay selection. The numbers were estimated from MC sample. . . 68
4.6 Data retentions at the subsequent steps of B0 s → J/ψf0 event selection. . . 73
5.1 The summary of stripping and offline selection criteria for B0 s → D+sD−s decay. . 77
5.2 Variables ranking according to the signal/background separation power. The top variable is the best ranked. Variables chosen for the final optimization are marked as Y in the last column. In the third column the corresponding notation used in Picture 5.3 and 5.4 is given. . . 78
5.3 Signal efficiencies for the selection steps for B0 s → Ds+D − s simulated events. . . . 82
5.4 Various factors contributing to efficiency of the B0 s → Ds+D − s decay selection. The values were determined using MC sample. . . 82
5.5 Data retentions at the subsequent steps of B0 s → Ds+D − s event selection. . . 86
6.1 Input values of likelihood parameters used in toy MC. . . 95
6.2 Toy Monte Carlo results for φs= - 0.04 rad. . . 97
6.3 Toy Monte Carlo results for φs= - 0.16 rad. . . 97
ALICE A Large Ion Collider Experiment ATLAS A Toroidal Large ApparatuS BDT Boosted Decision Tree
BR branching fraction
CKM matrix Cabbibo-Kobayashi-Maskawa matrix CL confidence level
CMS Compact Muon Solenoid DAQ Data Acquisition System
DLLxy difference of the log-likelihoood between particle x and particle y hypotheses
DOCA distance of closest approach DST Data Summary Tape
ECS Experiment Control System ECAL Electronic Calorimeter EFF Event Filter Farm
FD flight distance
HCAL Hadronic Calorimeter HLT High Level Trigger IP impact parameter
IT Inner Tracker L0 Level-0 (of trigger) L0DU L0 Decision Unit LEP Large Electron-Positron LHC Large Hadron Collider
LHCf Large Hadron Collider forward
LLx log-likelihood of the particle x hypothesis
M1-M5 Muon Stations (1-5) MC Monte Carlo
MoEDAL Monopole and Exotics Detector At the LHC NDF number of degrees of freedom
OT Outer Tracker
PDF probability density function PDG Particle Data Group
PS Proton Synchrotron PID Particle Identification PV primary vertex
RICH Ring Imaging Cherenkov Detector SM Standard Model
SPD/PS Scintillator-Pad Detector/Pre-Shower SPS Super Proton Synchrotron
SV secondary vertex
TDR Technical Design Report
TFC Timing and Fast Control System TIS Triggered Independent Of Signal TOS Triggered on Signal
TOTEM TOTal cross-section, Elastic scattering and diffraction dissociation Measurement at the LHC
TMVA Toolkit for Multivariate Data Analysis TT Tracker Turicensis
One of the main fundamental puzzles connecting particle physics with cosmology is related to the non-observation of antimatter in the Universe. The elementary interactions were believed to proceed in the same way for particles and antiparticles. In consequence, the same amounts of baryons and antibaryons should have been created after the Big Bang. However, a clear domination of matter over antimatter is observed in the current Universe, indicating that the corresponding symmetry has to be broken. This symmetry is called a CP symmetry. It consists of two discrete symmetry operations (Ref. [1]): charge conjugation (C) and parity operation (P). Charge conjugation inverts all charges and internal quantum numbers. Parity operation is the space reversal - it reverses the sign of all polar vectors. As a result, combined CP operation transforms matter into antimatter. Until the 1960s it was believed that despite the fact that weak interactions are not invariant under P and C symmetries (Ref. [2]), the combined CP symmetry is preserved. In 1964 James Cronin and Val Fitch analysed the experimental results concerning neutral kaon decays and discovered that weak processes are not invariant under CP transformations as well. (Ref. [3]).
Just three years after the discovery of CP violation, in 1967 Andrei Sakharov claimed that CP violation is one of three conditions necessary to explain the matter-antimatter asymmetry in the Universe (Ref. [4]).
In the Standard Model, CP violation is described in terms of Cabbibo-Kobayashi-Maskawa (CKM) matrix, which brought Makoto Kobayashi and Toshihide Maskawa the Nobel Prize in 2008 ’for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature’. However, the level of CP violation predicted within the Standard Model is too low to explain the advantage of matter over antimatter in the Universe. Thus is it likely that New Physics phenomena might be observed in CP violation process.
One of the main points in the research programme of the LHCb (Large Hadron Collider beauty) experiment is to measure the CP violation parameter in the B0
s sector - the weak
mixing phase φs. This is currently one of the least precisely measured parameters of the CKM
matrix. According to the Standard Model, φs value is tiny and hence any deviation from the
theoretical prediction would be an indication of New Physics. φs is determined from B0s decays
proceeding through b → ccs quark-level process. The final state of such decays can consist of the superposition of CP eigenstates, as in the case of B0
s → J/ψφ , or be a pure CP eigenstate
(for example in B0
s → J/ψf0 , Bs0 → D+sD −
s Bs0 → χc0φ decays).
This dissertation focuses on measuring the weak mixing phase from B0
s decays to CP
eigen-states. First experimental results from 2011 data are given. The precision of the measurement of weak mixing phase is also estimated.
In the first chapter a theoretical introduction concerning the basic concepts and formalism of CP symmetry violation within the Standard Model is presented. Special emphasis is placed to the description of CP violation in B0
s mesons system. A short presentation of the current
experimental results is given.
In the second chapter the specification of the Large Hadron Collider (LHC) and its perfor-mance are described. The research programme of the LHC experiments is also mentioned.
In the third chapter the design and performance of the LHCb experiment are discussed and all its subdetectors are briefly described. Moreover, the main elements of the LHCb software are presented.
In the forth and fifth chapters, the analyses performed on two B0
s decays to CP eigenstates,
Bs0 → J/ψf0 and B0s → Ds+D−s , are described. These decays are good representatives of Bs0
decays since the first contains muons in the final states and the latter is purely hadronic. They were chosen from among other decays to CP eigenstates because of relatively high branching fractions and reconstruction efficiencies. Both analyses are carried out in four steps. The first step occurs on-line. The trigger enriches the B meson content of the data sample. The second step is the stripping where for each decay mode a dedicated stripping line has been prepared. Then the offline cut-based selection of events is performed. In the last phase of the analysis, the optimization of selection criteria using multivariate analysis is made. The most important selection results are discussed.
In the sixth chapter, the LHCb potential of weak mixing phase measurement with B0
s decays
to CP eigenstates is described. Mass and proper time distributions for signal and background events are modelled, taking into account the theoretical dependencies and experimental ef-fects. The estimation of the experimental precision of φs measurement from B0s decays to CP
eigenstates is performed for various configurations of input Monte Carlo parameters. The last chapter contains the summary and outlook.
Theoretical introduction
The process of CP violation was discovered in 1964 in neutral kaon decays (Ref. [3]). The first evidence of CP violation outside the kaon system was found in 2001 in the Belle experiment and it concerned neutral B mesons system, as described in Ref. [5]. This discovery started the new era of research on CP violation process and great progress has been achieved since that time. Light B mesons systems (neutral Bu mesons and charged Bd mesons) were explored
precisely at B-factories: Belle (e.g. Ref. [6]) and BaBar experiments (e.g. Ref. [7]). LHCb is a second generation experiment. High b¯b cross section in pp interactions gives the opportunity to explore CP violation in B0
s system, study rare decays and search for New Physics effects
(Ref. [8]).
Still, the combined transformation of charge conjugation, parity operation and time reversal (T) is an exact symmetry of any interaction, which is the essence of Schwinger-Luders-Pauli theorem, also known as the CPT theorem.
1.1
CP violation within the Standard Model
In the SM framework massive fermions are grouped together in three generations. Each of them consists of a doublet of left-handed particles (weak isospin ±1/2) and singlets of right-handed particles (weak isospin 0) (Ref. [9]). The weak interaction occurs for the left-right-handed particles only, thus the three families of quarks being the eigenstates of the weak interaction are as follows: u d0 ! L , c s0 ! L , t b0 ! L ,
where d0, s0 and b0 denote the eigenstates of the weak interaction. They are linear combinations
of mass eigenstates. The transitions between them are described in terms of
Maskawa (CKM) matrix: d0 s0 b0 = VCKM d s b = Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb d s b .
The CKM matrix is a 3 × 3 unitary matrix. Its elements Vij describe the couplings between the
ith up-type quark and the jth down-type one. The unitarity of CKM matrix ensures the absen-ce of elementary flavour-changing-neutral-current (FNCN) vertiabsen-ces. CP violation is naturally accommodated in the charged-current induced interactions with the structure:
D → U W−,
where D and U denote down- and up-quark flavours respectively and W− is the SU(2)
L gauge
boson.
Non-leptonic charged-current interaction Lagrangian can be expressed in terms of CKM matrix elements: LCC int = − g √ 2(¯u, ¯c, ¯t)Lγ µV CKM d s b L Wµ†+ h.c.,
where g denotes the weak coupling constant which is related to the gauge group SU(2)L and
Wµ† is a field corresponding to the charged W bosons. Each element of the CKM matrix is
related to the strength of the given charged-current process, i.e. the square of |Vij| element is
the probability of a transition between i and j quarks. Values of the CKM elements are not predicted by the SM. They have to be determined experimentally.
The Lagrangian describing charged-current processes is invariant under CP transformation only if CKM matrix is real. The description of CP violation requires complex phases in the CKM matrix. It can be shown that the general unitary n × n quark-mixing matrix (n denoting the number of generations) can be described by (n−1)2 parameters, n(n−1)
2 angles and
(n−1)(n−2) 2
complex phases. Therefore, the CKM elements can be expressed in terms of three real mixing angles and one complex phase (Ref. [10], [11]).
Many CKM matrix parametrizations have been proposed in the literature. One of them is the Keung-Chau parametrization (the so called ’standard’ parametrization) (Ref. [12]):
VCKM = c12c13 s12c13 s13e−iδ13 −s12c23− c12s23s13eiδ13 c12c23− s12s23s13eiδ13 s23c13 s12s23− c12c23s13eiδ13 −c12s23− s12c23s13eiδ13 c23c13 ,
where cij = cos θij and sij = sin θij. The angles θi,j describe the mixing between the three
gene-rations (i, j = 1, 2, 3) and δ13 is the CP violation phase. This parametrization is recommended
especially for numerical evaluations.
The parametrization which makes the hierarchy of the mixing between quark families explicit is more useful for phenomenological applications. This is ensured in the Wolfenstein parame-trization described in Ref. [13]. Each CKM element can be expanded as a power series of the λ = sin θc which has a value |Vus| = 0.22.
VCKM = 1 −12λ2 λ Aλ3(ρ − iη) −λ 1 − 12λ2 Aλ2 Aλ3(1 − ρ − iη) −Aλ2 1
+ O(λ
where parameters are defined as: A ≡ s23 s2 12 , ρ ≡ s13cos δ13 s12s23 , η ≡ s13sin δ13 s12s23 .
This parametrization exposes the smallness of Standard Model CP-violation since it contains a non-zero imaginary component at the third order in λ, specifically in the Vub and Vtd elements.
In the LHC era, the experimental precision will increase significantly, thus higher-order terms (up to O(λ5)) in the Wolfenstein expansion have to be taken into account, in particular in the
study of the B0 s - ¯Bs0 mixing: VCKM = 1 − 12λ2− 1 8λ 4 λ Aλ3(ρ − iη) −λ + A2λ5(1 2 − ρ − iη) 1 − 1 2λ 2− 1 8λ 4(1 + 4A2) Aλ2
Aλ3(1 − ¯ρ − i¯η) −Aλ2+ Aλ4(1
2 − ρ − iη) 1 − 1 2A 2λ4 + O(λ 6), where: ¯ ρ = ρ 1 −1 2λ 2 , η = η¯ 1 −1 2λ 2 .
1.2
Unitarity triangles
The unitarity constraint of the CKM matrix implies six orthogonality conditions. Each of them requires the sum of three complex numbers to vanish. Unitarity conditions can be represented as triangles in the complex plane called ’unitarity triangles’. The triangles differ in shapes which depend on the values of CKM elements. Nevertheless, the area of each triangle is the same and equals to (|JCP|/2 = A2λ6|η|), where JCP is the Jarlskog (Ref. [14]) parameter
defined as:
JCP = ±Im VikVjlVil∗V ∗
jk , (i 6= j, k 6= l).
In terms of Wolfenstein parametrization, JCP ∼ O(10−5), which reflects the level of CP violation
within the SM.
Six unitarity conditions are:
VudVub∗ + VcdVcb∗ + VtdVtb∗ = 0 (db), (1.1) VusVud∗ + VcsVcd∗ + VtsVtd∗ = 0 (sd), (1.2) VusVub∗ + VcsVcb∗ + VtsVtb∗ = 0 (sb), (1.3) VduVdc∗ + VsuVsc∗ + VbuVbc∗ = 0 (uc), (1.4) VdcVdt∗ + VscVst∗+ VbcVbt∗ = 0 (ct), (1.5) VdtVdu∗ + VstVsu∗ + VbtVbu∗ = 0 (tu). (1.6)
The triangles corresponding to these conditions are shown in Figure 1.1. Among them, only two have sides of the same order O(λ4): (db) and (ut) ones. The triangle (db) is related to
observables of the B0
d decays. Its sides differ in O(λ
5) only, thus the triangle is very useful
for the studies of CP violation. Based on the Equation 1.1, three interior angles, illustrated in Figure 1.2a, can be defined as:
α ≡ arg VtdV ∗ tb VudVub∗ , β ≡ arg VcdV ∗ cb VtdVtb∗ , γ ≡ arg VudV ∗ ub VcdVcb∗ .
Figure 1.1: CKM unitarity triangles in the Wolfenstein parametrization. The other interesting triangle is the (sb) one. It describes the CP violation in B0
s decays. At
the order of O(λ5)the angle χ (Figure 1.2b) is related to V
ts element by the following formula:
χ ≡ βs≡ arg VcbVcs∗ VtbVts∗ ' λ2η ' arg(V ts− π).
Other four triangles are degenerated - having one angle significantly smaller than others (Figure 1.1).
The direct measurement of angles and sides of unitarity triangles is a crucial test of the SM. These parameters can be measured in several different ways and any discrepancy appearing between these measurements could give a sign of the presence of New Physics processes beyond the SM.
1.3
Measurements of CP violation parameters
The current combined experimental results on the (¯ρ, ¯η) and ( ¯ρs, ¯ηs) plane are presented in
Figures 1.3 and 1.4. The plots were taken from Ref. [15]. They illustrate the current precision of CPV measurement in the framework of the SM.
A list of the main quantities used to constrain the CKM picture is the following:
• Ru - ratio of |Vub/Vcb|. It can be determined from b→u and b→c decays in the (¯ρ, ¯η) plane.
Ru is constrained as a circle of radius:
Ru = λ 1 − λ22 p ¯ ρ2+ ¯η2,
Figure 1.2: a) db unitarity triangle in the Wolfenstein parametrization. b) bs unitarity triangle in the Wolfenstein parametrization. Both triangles are normalised to the baseline of db triangle, i.e. VcdVcb∗ is real and all lengths are divided by |VcdVcb∗|.
Figure 1.3: Constraints in the (¯ρ − ¯η) plane (as explained in the text). The red hashed region of the global combination corresponds to 68% CL.
Figure 1.4: Constraints in the ( ¯ρs− ¯ηs) plane. The red hashed region of the global combination
corresponds to 68% CL. The (almost horizontal) thin blue lines corresponds to the 68% and 95% CL constraints on βs.
B0 d J/ψ K0 → Ks d s c b W c Vcb∗ Vcs B0 d J/ψ K0 → K s d s c g c b W t, c, u
Figure 1.5: Feynman diagrams contributing to B0
d → J/ψKs0 decay. Left: the tree-type diagram.
Right: the penguin diagram.
• Rt - ratio of |Vtd/Vcb|. Rt can be extracted from the ratio of mass difference in the Bd0
system to the mass difference in the B0
s system. Since ∆Md ∝ |VtdVtb∗|2 and ∆Ms ∝
|VtsVtb∗|2, it can be derived that:
∆Ms ∆Md ∝ MB0s MB0 d ξ|Vts| 2 |Vtd|2 , ξ = O(1),
ξbeing a SU(3) flavour-symmetry breaking factor obtained from lattice QCD calculations. The Rt bounds correspond to a circle with a radius:
Rt= λ
p
(1 − ¯ρ)2+ ¯η2.
From global fits its value is equal to 0.921 ± 0.028.
• k - related to CP violation in Kd0 system. Its bounds correspond to a hyperbola in the
(¯ρ, ¯η) plane. The value of kwas measured with a high precision from neutral kaon decays
to be (2.02+0.53
−0.52) · 10−3.
• α - its value can be determined through the triangle relation α = π − β − γ (because of relatively high contribution of a penguin decays it it not possible to provide a direct and theoretically clean measurement). From global fits to the CKM parameters: α = (91.1 ± 4.3)◦.
• β - can be determined using b → ccs transitions in B0
d decays, from the time dependent
asymmetry of the decay rates. The ’gold-plated’ channel for β measurement is Bd→J/φKs0
(Figure 1.5). From global fits: sin(2β) = 0.691 ± 0.020.
• βs - can be determined as a counterpart of β measurement using b → ccs transitions in
Bs0 decays. The ’golden-plated’ channel for βs (and thus for φs = −2βs) measurement is
B0
B0 s,d B 0 s,d s, d s, d t, c, u W b W b t, c, u B0 s,d B 0 s,d s, d s, d t, c, u W b b t, c, u W
Figure 1.6: Feynman box diagrams for B0
q− ¯Bq0 mixing (q = s, d). The charge-conjugated process
is obtained by substituting all quarks with antiquarks and taking the complex-conjugates of the CKM matrix elements.
• γ - can be measured from pure tree decays (e.g. Bs→DsK) (measurement insensitive
to New Physics), from decays such as B→DK or even from the comparison between Bd→π+π− and Bs→K+K−. From global fits: γ = (67.1 ± 4.3)◦.
1.4
Oscillations of neutral mesons
In the SM, the mixing appears in the second order weak interaction through the box dia-grams shown in Figure 1.6. In these diadia-grams, the t quark contribution is a dominant one, which is the result of the hierarchy of the CKM elements and because the amplitude is proportional to the mass of a particle exchanged in the loop.
There are four pairs of neutral mesons that may exhibit the mixing phenomenon, K0(¯sd) − ¯K0(s ¯d),
D0(c¯u) − ¯D0(¯cu), Bd0(¯bd) − ¯Bd0(b ¯d), Bs0(¯bs) − ¯Bs0(b¯s).
The time evolution of the state a(t)|X0i + b(t)| ¯X0i is determined by the Schrödinger
equ-ation: i∂ ∂t a(t) b(t) ! = H a(t) b(t) ! = M11− i 2Γ11 M12− i 2Γ12 M12∗ − i 2Γ ∗ 12 M22− 2iΓ22 ! a(t) b(t) ! , (1.7)
where |X0i denotes a neutral meson state, | ¯X0iis the corresponding anti-meson and H stands
for the effective 2 × 2 Hamiltonian describing the mixing. H, being a non-hermitian operator, is a linear combination of two hermitian matrices M and Γ:
H = M − i 2Γ.
The CPT theorem indicates that masses and lifetimes of particles and corresponding antipar-ticles are identical (M11 = M22 ≡ M0 and Γ11 = Γ22 ≡ Γ0). The eigenstates of H are the linear
combinations of flavour eigenstates:
|X1i = p|X0i + q| ¯X0i,
|X2i = p|X0i − q| ¯X0i.
where p and q are complex numbers related through the normalization condition |p|2+|q|2 = 1.
Since the eigenstates of weak interactions are characterised by their definite masses, we can define the heavy |XHi and light |XLi states. The energy eigenvalues are the solutions of the
characteristic equation: EH,L= M − i 2Γ ± r (M12− i 2Γ12)(M ∗ 12− i 2Γ ∗ 12) = (M ± ∆M 2 ) − (Γ ± ∆Γ 2 ) = MH,L− i 2∆Γ. (1.8)
where MH,Land ΓH,Ldenote masses and widths of heavy and light mass eigenstates respectively
(the plus sign in the Equation 1.8 corresponds to the heavy eigenstate, while the minus sign -to the light one). The mass difference and the width difference can be written as:
∆M = MH − ML> 0,
∆Γ = ΓH − ΓL> 0.
The average values are:
M = MH + ML 2 , Γ = ΓH + ΓL
2 . From the eigenvector equations, q/p can be derived as:
q p = − s M12q∗− i 2Γ q∗ 12 M12q − i 2Γ q 12 . (1.9)
Therefore the time evolution of heavy and light eigenstates can be written as: |XH,L(t)i = |XH,Lie−i(MH,L−
i
2ΓH,L)t, (1.10)
and the time evolution of the initial particle and antiparticle states, |X0i and | ¯X0i can be
rewritten as: |X0(t)i = f +(t)|X0i + q pf−(t)| ¯X 0i, (1.11) | ¯X0(t)i = f+(t)| ¯X0i + p qf−(t)|X 0i, (1.12) where: f± ≡ 1 2 h e−i(ML−12ΓL)t± e−i(MH−12ΓH)t i . (1.13)
1.5
Time-dependent decay rates of B
s0mesons and CP asymmetries
Two decay amplitudes to the common final state f can be defined as:
Af = hf |T |X0i, (1.14)
¯
Af = hf |T | ¯X0i. (1.15)
In order to compute the time dependent rates for neutral mesons decaying into the same final state, one has to derive the amplitudes of the processes in which an initially created |X0imeson
will follow the time evolution given in Equation 1.11 and decay into the final state f: A[X0→f ] = hf |T |X0(t)i = f+(t)Af +
q
pf−(t) ¯Af. (1.16) Similar formula can be written for | ¯X0i which decays to the final state f:
A[ ¯X0→f ] = hf |T | ¯X0(t)i = f+(t) ¯Af +
p
qf−(t)Af. (1.17) The corresponding decay rates are given as the squares of the amplitude modules:
Γf(t) ≡ Γ(X0(t)→f ) = |Af|2|f+(t)|2+ |λ|2|f−(t)|2+ 2Re[λff+∗(t)f−(t)], (1.18) ¯ Γf(t) ≡ Γ( ¯X0(t)→f ) = |Af|2 p q 2 |f−(t)|2+ |λ|2|f+(t)|2+ 2Re[λff+(t)f−∗(t)], (1.19) where λf denotes: λf ≡ q p ¯ Af Af , (1.20)
and |f±(t)|2 and f+∗(t)f−(t) are factors defined as:
|f±(t)|2 = 1 4e −ΓLt+ eΓHt± 2eΓtcos(∆mt) = e −Γt 2 cosh ∆Γt 2 ± cos(∆mt) , (1.21) f+∗(t)f−(t) = 1 4e ΓHt+ e−ΓLt+ 2ie−Γtsin(∆mt) = e −Γt 2 sinh ∆Γt 2 + i sin(∆mt) . (1.22) Analogically, the decay rates to the CP conjugated state, ¯f, can be derived as:
Γ(X0(t)→ ¯f ) = | ¯Af¯|2 q p |f−(t)|2+ |¯λf¯|2|f+(t)|2+ 2Re[¯λf¯f+(t)f−∗(t)], (1.23) Γ( ¯X0(t)→ ¯f ) = | ¯Af¯|2|f+(t)|2+ |¯λf¯|2|f−(t)|2+ 2Re[|¯λf¯|2f+∗(t)f−(t)], (1.24) where: ¯ λf¯≡ p q Af¯ ¯ Af¯ . (1.25)
1.6
Types of CP violation
There are three types of CP symmetry violation in the SM: • CP violation in mixing (indirect)
It measures the asymmetry between X0→ ¯X0 and ¯X0→X0 processes. In terms of SM
description this means:
q
p 6= 1. (1.26)
• CP violation in decay (direct)
It occurs when decay amplitudes for some process and its CP conjugate are different, i.e: ¯ Af¯ Af 6= 1. (1.27)
• CP violation in the interference of mixing and decay (mixing induced CP violation)
It happens when the asymmetry between the following processes appears: 1. Neutral meson decay to a final state f (X0→f).
2. Oscillation of the neutral meson into a corresponding antimeson followed by a decay into the same final state: (X0→ ¯X0→f).
In terms of SM terminology: Im q p ¯ Af Af 6= 0. (1.28)
1.6.1 CP violation in mixing (indirect)
The genesis of indirect CP violation are differences between the rates of neutral meson mixing: X0→ ¯X0 and ¯X0→X0. In the Schrödinger equation this fact is reflected when the
magnitudes of the off-diagonal elements in the effective Hamiltonian differ: M12− i 2Γ12 6= M12∗ − i 2Γ ∗ 12 . (1.29)
The only difference comes from a phase shift between M12 and Γ12. From the Equation 1.9 it
can be derived that:
q p 2 = M12∗ − i 2Γ ∗ 12 M12−2iΓ12 , (1.30)
thus the inequality of the magnitudes of the off-diagonal elements induce: q p 6= 1. (1.31)
This kind of CP violation can be measured in semileptonic decays B0
q→Xl+ν. For events
where the B0
q has mixed before decaying the final state contains a "wrong-charge" lepton l −.
1.6.2 CP violation in decay (direct)
Direct CP violation is manifested in a difference between decay rates of X→f and its CP-conjugate ¯X→ ¯f. It appears due to the interference between at least two decay amplitudes1, which requires at least two types of phases in the Hamiltonian:
• the weak phases (Φi) from the CKM matrix - such phases change the sign for charge
conjugated process,
• the strong phases (δi) related to the strong interactions of the final state - they do not
change the sign for the conjugate.
Given the above, the amplitudes for the process and its CP-conjugate can be written as a product of the magnitude and two phases:
Af = X i AieiΦieiδi, (1.32) ¯ Af¯= X j Aje−jΦjejδj. (1.33)
The ratio of decay amplitudes is given by a formula: ¯ Af¯ Af = P jAje jδj−φj P iAieiδi+φi . (1.34)
Therefore CP violation in decay occurs if: ¯ Af¯ Af 6= 1. (1.35)
Direct CP violation is the only possible source of CP violation for decays of charged mesons. It occurs also in e.g. B0
d→K+π
− channel, where the interference between the tree and penguin
amplitudes leads to a CP asymmetry of ∼ 10%.
1.6.3 CP violation in the interference of mixing and decay (mixing induced CP violation)
Mixing induced CP violation is caused by an interplay between the mixing and decay am-plitudes. This can happen when the neutral meson X0 has two ways of decaying into the same
final state f: either directly (X→f) or preceded by an oscillation into the corresponding anti-meson (X0→ ¯X0→f).
The ratios of mixing induced CP violation are defined as: λ = q p ¯ Af Af , ¯ λ = q p ¯ Af¯ Af¯ ,
where q/p factor is related to oscillations and ¯Af/Af - to decay. All these quantities are phase
convention independent and thus have a physical meaning.
Mixing induced CP violation requires that Imλ, Im¯λ 6= 1. Even if neither direct nor indirect CP violation occur, the interference between the decays with and without mixing (i.e. interfe-rence of the phases of q/p and ¯A/A) can be the only source of the CP asymmetry giving an imaginary part Imλ, ¯λ 6= 1.
Measurement of CKM parameters from mixing induced CP violation is relatively straightfor-ward and theoretically clean because one does not have to worry about hadronic uncertainties (as in the case of indirect CP violation).
Mixing induced CP violation appears for B0
s decays proceeding through b → ccs quark-level
process, such as B0
s → J/ψφ or decays to CP eigenstates, the latter being the focus of this
thesis.
1.7
Measurement of CP violation phase from B
s0decays into CP
eigen-states
In this chapter, the method of measuring φsphase from Bs0 decays to CP eigenstates through
b → ccs quark-level process is discussed. For such decays, the final state consists of one vector and one pseudo-scalar meson (e.g. B0
s → J/ψη, Bs0 → J/ψη0, Bs0 → J/ψf0, Bs0 → ηcφ and
Bs0 → χc0φ) or two pseudo-scalar mesons (in the case of Bs0 → Ds+Ds−). Both Bs0 and ¯Bs0 decay
to a common final state f with well-defined CP eigenvalue ηCP.
Oscillations in the B0
s system (Ref. [16], [17]) are illustrated by the two box diagrams shown
in Figure 1.6. The amplitude in the loop is proportional to the square of the mass of a particle exchanged in the loop. Moreover Vtb ' 1, hence Bs0 - ¯Bs0 mixing is dominated by exchanging the
t quark. Thus the transition of B0
s into ¯Bs0 is proportional to (V ∗ tbVts)2 ≈ Vts2 and analogically for ¯B0 s→Bs0, to (VtbVts∗)2 ≈ V ∗2
ts . In the case of Bs0 system: |Γ12| << |M12|, thus:
q p = − H21 H12 1/2 ≈ − M21 M12 1/2 = − M ∗ 12 M12 1/2 = − M ∗ 12 |M12| . (1.36) The matrix element M12can be written as M12= |M12|eiθM, where θM = 2 arg(Vts∗Vtb+ π − φCP)
and φCP is defined through CP |Bs0i = eiφCP| ¯Bs0i.
For the B0 s system: 2 arg [V ∗ tsVtb] = φM, where φM = −2ηλ = −2δγ thus: q p ' e iθM = eiφM−φCP. (1.37) and q p = 1. (1.38)
The decay rates can be rewritten as: Γ[Bs0(t)→f ] = |Af| 2 + | ¯A f|2 2 e −Γst cosh ∆Γst 2 + Cfcos (∆Mst) +Dfsinh ∆Γst 2 + Sf sin (∆Mst) , (1.39)
Γ[ ¯Bs0(t)→f ] = |Af| 2+ | ¯A f|2 2 e −Γst cosh ∆Γst 2 − Cfcos (∆Mst) +Dfsinh ∆Γst 2 − Sfsin (∆Mst) , (1.40) where Df ≡ − 2Re(λf) 1 + |λ|2 ≡ A∆Γ, (1.41) Cf ≡ − 1 − |λ|2 1 + |λ|2 ≡ A dir CP, (1.42) Sf ≡ − 2Im(λf) 1 + |λ|2 ≡ A mix−ind CP , (1.43) and (A∆Γ)2+ AdirCP 2 + Amix−indCP 2 = 1. (1.44)
Since penguin contributions to B0
s decays through b → ccs process are doubly-Cabbibo
sup-pressed (Figure 1.7), the decay is governed by a single tree mechanism and therefore the CP violation in decays can be neglected. The hadronic matrix elements and strong phases vanish:
¯ Af
Af
= −ηCPe−2iφD, (1.45)
where φD is a pure weak phase and φD ≡ arg[VcbVcs∗] ≈ 0. This implies:
λf = q p ¯ Af Af = −ηCPeiφCKM ≈ −ηCPeiφs, (1.46)
where CKM phase is defined as: φCKM = φs− 2φD ≈ φs. The weak mixing phase φs can be
written is terms of CKM elements as:
φs ' −2λ2η ∼ −0.036 rad. (1.47)
Therefore, under the assumption that no CP violation is present in mixing and a single decay mechanism is dominant, it can be written:
|λf| = 1,
Cf = 0,
Sf = ηCPsin(φCKM),
Df = ηCPcos(φCKM).
Given the above assumptions, the B0
s decay rates to CP eigenstates can be simplified to:
Γ[Bs0(t)→fCP] = |ACP|2e−Γst cosh ∆Γst 2 − ηCPcos(φs) sinh ∆Γst 2 +ηCPsin(φs) sin(∆Mst)} , (1.48)
B0 q q s c b W c B0 q q s c g c b W t, c, u
Figure 1.7: Feynman diagrams illustrating B0
q meson decay through a b → ccs quark-level
pro-cess. Γ[ ¯Bs0(t)→fCP] = |ACP|2e−Γst cosh ∆Γst 2 − ηCPcos(φs) sinh ∆Γst 2 −ηCPsin(φs) sin(∆Mst)} . (1.49) The time-depended CP asymmetry can be calculated as:
ACP ≡ Γ[ ¯Bs0(t)→fCP] − Γ[ ¯Bs0(t)→fCP] Γ[ ¯B0 s(t)→fCP] + Γ[ ¯B0s(t)→fCP] = −ηCP sin(φs) sin(∆mst) cosh ∆Γst 2 − ηCP cos(φs) sinh ∆Γst 2 . (1.50) Therefore φs phase can be directly extracted from CP asymmetry.
1.8
Current experimental status of φ
smeasurements
The βs phase (related to φsby equation: βs = −φ2s) is believed to be a sensitive probe of New
Physics. Its value is not only very small within the SM, but also determined with a relatively small theoretical uncertainty: βs = 0.01817+0.00087−0.00083rad (Ref. [15]). The new postulated particles
can contribute to the B0
s - ¯B0s box diagram and modify the SM prediction.
The best sensitivity of CP violation measurement is expected to come from B0
s → J/ψφ
decay. Due to a large branching fraction of (1.4 ± 0.5) · 10−3 (Ref. [18]), it is called the golden
channel. However, its final state consisting of two vector mesons is an admixture of CP eigensta-tes and angular analysis is needed to disentangle different CP contributions. In the case of B0 s
decays to CP eigenstates, the measurement can be performed directly. Such measurements can be a valuable cross-check to the main one coming from B0
s → J/ψφ analysis or even combined
together to improve the overall sensitivity.
The first measurements reported by the CDF and D0 experiments indicated large values of φs (Ref. [19] and [20]). This was a limit of New Physics contribution; however, the uncertainty
Figure 1.8: Likelihood confidence regions in the ∆Γs− φs plane. A result obtained from 0.37
fb−1 of data collected at LHCb experiment.
The most recent measurements of CP violating phase from B0
s → J/ψφdecays are described
in Ref. [21] (from D0 experiment), [22] (from CDF) and [23] (from LHCb). The obtained results are summarized in Table 1.1. Additionally, ∆Γs− φs confidence regions measured at LHCb are
presented in Figure 1.8.
Current βsvalue reported in 2012 by Particle Data Group (Ref. [24]) is equal to 0.08+0.05−0.07 rad.
CDF result D0 result LHCb result integrated 5.2 fb−1 8.0 fb−1 0.37 fb−1
luminosity
βs [rad] [0.02, 0.52] ∪ [1.08, 1.55] -
-at 68% CL
φs = −2βs [rad] - −0.55+0.38−0.36 0.15 ± 0.18(stat) ± 0.06(sys)
∆Γs [ps−1] 0.075 ± 0.035(stat) ± 0.006(sys) 0.163+0.065−0.064 0.123 ± 0.029(stat) ± 0.011(sys)
Table 1.1: Most recent results of CP violating phase in B0
s decays performed by D0, CDF and
The Large Hadron Collider
The world’s largest laboratory for particle physics, CERN (Conseil Europeen pour la Re-cherche Nucleaire), was founded in 1954, following a proposal by Louis de Broglie. In 2012 it consists of 20 member states and involves about 8000 scientists from all over the world. Among many other accelerators, LEP (Large Electron Positron), the forerunner of the LHC, worked at CERN from 1989 until 2000. LEP collided electrons and positrons at a centre of mass energy of √
s ' 88 − 209 GeV, and allowed precise measurements of the Z and W± boson masses. Still, one brick in the Standard Model picture, the Higgs boson, was missing. For this reason, after LEP’s experiments success, it was settled to start a new experiment with an increased energy in order to probe the structure of matter more deeply. Physicists not only hoped to observe the Higgs boson, but also to discover new particles, for example SUSY (super-symmetric) particles. The first proposal of the Large Hadron Collider (LHC) comes from 1984. The project was started 10 years later, in 1994. The LHC is located in the same tunnel which was used by the LEP accelerator. The ring of 27 km in circumference is placed 100 m underground across the French and Swiss border. Inside the LHC ring, two opposite proton beams are being collided with a (nominal) center-of-mass energy of √s of 14 TeV. The usage of protons instead of electrons and positrons allows to reduce the synchrotron radiation, but on the other hand proton-proton collisions introduce high background due to many hadronic interactions. The LHC is also able to collide heavy ions (lead nuclei) (Ref. [25], [26]).
The first collisions were observed in the LHC detectors in November 2009 at a center-of-mass energy of 0.9 TeV. The energy of 7 TeV was reached in March 2010, 8 TeV - in 2012. The operation at the nominal energy of 14 TeV is foreseen for 2014.
Figure 2.1: Layout of the CERN accelerator complex.
2.1
CERN accelerating complex and LHC parameters
The Large Hadron Collider itself is the last component in the process of accelerating par-ticles. Before injecting into the LHC, the particles have to be initially speed up through the accelerating chain. A schematic diagram of the CERN accelerator complex is illustrated in Figure 2.1.
Protons, which are extracted from hydrogen gas, are injected into a linear accelerator (LI-NAC2) and accelerated to the energy of 50 MeV. The energy of 1.4 GeV is achieved as the protons pass through the Proton Synchrotron Booster and further, the energy of 25 GeV is reached in Proton Synchrotron (PS). At the next stage, protons are accelerated to 450 GeV in the Super Proton Synchrotron (SPS). Finally, they are injected as two counter-rotating beams into the LHC. At nominal LHC working conditions, a single beam consists of 2808 proton bun-ches of about 1011 protons each. The time separating two consecutive bunches is 25 ns, which
corresponds to a bunch-crossing rate of 40 MHz. The opposite bunches of protons circulate in two beam pipes and are accelerated to the final collision energy.
The acceleration of particles in a forward direction is provided through an oscillating elec-tric field, which is ensured by Radio-Frequency cavities. Circular trajectories are maintained using 1232 dipole magnets, which can generate a magnetic field of up to about 8.3 Tesla. The magnets are kept in a superconducting state by cooling their inner core with liquid helium at the temperature of 1.9 K. The frequencies and magnitudes of both fields are being increased in a controlled manner in order to enlarge the beam energy while still keeping it in a stable orbit. After the acceleration process, the beams are directed into a collision trajectories. At each
interaction point, the beams are focused by quadrupole magnets in order to increase the pro-bability of the collision. The LHC is divided into octants. The collisions take place at the rate of 40 MHz in four interaction points around the LHC ring (Figure 2.1).
The number of inelastic interactions per second generated due to proton-proton collisions depends on the LHC luminosity L and the inelastic proton-proton cross section σinel
pp :
Nppinel = Lσppinel. (2.1) The value of σinel
pp was extrapolated based on SPS and Tevatron data and at
√
s = 14 TeV is equal to approximately 80 mb.
The machine (instantaneous) luminosity depends only on the beam parameters and can be expressed as: L = N 2 bkbf γF 4πβ∗ , (2.2) where:
Nb ≈ 1011 - number of protons per bunch,
kb - number of bunch crossings at the considered interaction point,
f = 11.25 kHz - the revolution frequency γ - relativistic boost,
F ≈ 0.9 - factor associated with a crossing angle of two beams,
β∗ = 0.5 m - magnets quantity related to their ability to focus the beam at the interaction point,
= 3.75µm · rad - normal transverse emittance (compactness of the beam).
For ATLAS and CMS (see Section 2.2), the design luminosity is 1034 cm−2s−1, which makes
it possible to obtain a rate of ∼ 20 inelastic collisions per event. In LHCb, for specific physics analysis needs the luminosity is locally reduced to 2·1032cm−2s−1, thus the number of multiple
interactions per bunch crossing is decreased. Initially, the LHCb experiment was designed to operate at the average value of visible interactions per bunch crossing of 0.4, but currently this number is approximately 4 times larger (Section 3.2).
The design LHC parameters are summarized in Table 2.1 (source: Ref. [26]):
2.2
LHC experiments
The LHC project consists of the accelerator itself and four big experiments with their own research program:
• ATLAS: A Toroidal LHC ApparatuS (Ref. [27]) - one of two general purpose experi-ments, searching for new discoveries in the head-on collisions of protons and heavy ions. The list of possible ATLAS discoveries includes the origin of mass (Higgs boson), extra dimensions of space, unification of fundamental forces, evidence for supersymmetric par-ticles or dark matter candidates in the Universe.
• CMS: Compact Muon Solenoid (Ref. [28]) - another general-purpose experiment desi-gned to investigate a wide range of physics topics. Although the scientific goals of CMS and ATLAS overlap, these experiments use different technical and design solutions.
Injection Collision Beam data
Proton energy [GeV] 450 7000
Relativistic gamma 479.6 7461
Number of particles per bunch 1.15 ·1011 1.15 ·1011
Number of bunches 2808 2808
Longitudinal emittance (4σ) (eVs) 1.0 2.5 Transverse normalised emittance (µm·rad) 3.5 3.75 Circulating beam current (A) 0.584 0.584 Stored energy per beam (MJ) 23.3 362 Peak Luminosity Related Data
RMS bunch length (cm) 11.24 7.55
RMS beam size at IP1 and IP5 (µm) (β∗ = 0.55) 375.2 16.7
RMS beam size at IP2 and IP8 (µm) (β∗ ≈ 10) 279.6 70.9
Geometric luminosity reduction factor F - 0.836 Peak luminosity in IP1 and IP5 (cm−2s−1) - 1.00 · 1034
Luminosity in IP8 (cm−2s−1) - ≈ 2.00 · 1032
Interaction data
Inelastic cross section [mb] 60.0 60.0
Total cross section [mb] 100 100
Events per bunch crossing - 19.02
Beam current lifetime [h] - 44.86
Table 2.1: The LHC parameters.
• ALICE: A Large Ion Collider Experiment (Ref. [29]) - a dedicated heavy-ion experiment to exploit the physics of strongly interacting matter at extreme energy densities - namely the formation of the quark-gluon plasma. Moreover, ALICE is studying proton-proton collisions both in physics areas where ALICE is competitive with other LHC experiments and as a comparison with lead-lead collisions.
• LHCb: Large Hadron Collider beauty experiment (Ref. [30]) - a dedicated b-physics experiment, more widely described in Chapter 3.
There are three additional experiments focused on the specific goals:
• LHCf: Large Hadron Collider forward (Ref. [31]) - designed to study forward production of neutral particles in proton-proton collisions.
• TOTEM: Total Cross Section, Elastic Scattering and Diffraction Dissociation Measurement (Ref. [32]) - to study forward particles produced very close to the LHC beams.
• MoEDAL: Monopole and Exotics Detector at the LHC (Ref. [33]) - designed to explore exotic massive stable (or pseudo-stable) particles.
The LHCb experiment
The main aim of the LHCb experiment is to study CP violation in the B meson system, search for indirect evidence of the New Physics and explore rare decays of b-hadrons. The proposed physics programme requires not only large data samples but also a precise spectro-meter able to detect and measure a variety of quantities of b-hadrons and their decay products (Ref. [34], [35]).
3.1
LHCb geometry
The dominant mechanism of b¯b pairs production in a proton-proton collision is gluon-gluon fusion and quark-antiquark annihilation. b¯b are produced mostly in forward and background directions in the centre of mass frame. The boost further enhances the angular correlation between b and ¯b (Figure 3.1).
In order to access the b¯b pairs, LHCb has been designed to explore the forward region, with an angular acceptance from approximately 10 to 300 (250) mrad in the bending (non-bending) plane of the LHCb magnet. In such a geometry, comparing to 4π geometry, the vertex detector can be located closer to the interaction point, which is crucial for b-hadrons measurements. Moreover, the forward geometry simplifies the mechanical design and ensures access for maintenance and future upgrades. A general layout of the LHCb detector is presented in Figure 3.2.
3.2
Luminosity
The nominal luminosity of the LHC machine is equal to 1034cm−2s−1. The luminosity at
LHCb’s point is reduced to 2 · 1032cm−2s−1 in order to reduce the number of proton-proton
interactions per bunch crossing to ensure clean environment for precise reconstruction of multi-38
body decay chains. This is achieved by changing the beam focus and can be done independently from the other interaction points.
The beam conditions of the LHC in 2010 and 2011 turned out to be significantly different from the nominal ones. Due to a more focused beams, the instantaneous luminosity of about 2 · 1032cm−2s−1 was achieved with a smaller number of bunches and bigger time interval (50
ns instead of 25 ns). In effect, the average number of proton-proton interactions increased by a factor of 4. All affected elements of the LHCb experiment (like trigger, offline reconstruction) were successfully adjusted in order to fulfil these demanding conditions. It was checked that the quality of physics results was not degraded.
In addition, a luminosity levelling mechanism was provided to account for the luminosity change during the fill. It was achieved by the displacement of the beams in y coordinate, and lowering this displacement with the decreased luminosity. In this way a constant instantaneous luminosity could be assured during the whole fill as is shown in Figure 3.3. The luminosity levelling mechanism provides constant occupancies and trigger rates throughout the fill.
The LHC is the world’s most plentiful source of hadrons containing b quarks. The cross section of b¯b pair production from pp collision at the energy√s = 7 TeV was measured by the LHCb experiment to be σ(pp→b¯bX) = 284 µb (Ref. [36]). This corresponds to approximately 5 · 1011 pairs of b¯b produced in 107 s of running (∼ one year of data taking) at nominal LHCb
luminosity L = 2 · 1032cm−2s−1.
3.3
LHCb detector
The spectrometer, located at LHC’s point 8, has been optimized to study the beauty and charm hadrons. The detector, 20 m long and 5 m high, is composed of three main subsystems fulfilling specific requirements (Ref. [30], [37]):
• tracking system - a set of tracking detectors to reconstruct tracks and measure their mo-menta via deflection in the magnet. The primary and secondary vertices can be searched to identify hadron decays,
Figure 3.1: The angular correlation between the polar angles of the b and ¯b quarks produced at LHCb.
Figure 3.2: A schematic view of the LHCb detector.
LHCb
ATLAS/CMS
Figure 3.3: Instantaneous luminosity at the LHC experiments during an example fill. Luminosity levelling mechanism at the LHCb experiment ensures constant luminosity throughout entire fill.
• particle identification system - the number of dedicated subdetectors used to identify charged hadrons, muons, electrons and photons,
• trigger system - a two-level system to select the charm and beauty decays out of the light quarks background.
3.3.1 The tracking system
The LHCb tracking systems consist of Vertex Locator, the detector placed around the interaction point, and the tracker, composed of Tracker Turicensis located upstream of the Magnet and the Inner and Outer Tracker placed downstream. The presence of the magnet, bending charged particles trajectories, is necessary for the measurements of their momenta and transverse momenta.
Vertex Locator
The main purpose of Vertex Locator (VELO) is to provide track reconstruction close to the interaction point for primary and secondary vertices search. This feature is essential for b-hadrons measurements, where the relatively long lifetime and large boost cause the displacement of primary and secondary vertices of O(1 cm). Moreover, VELO provides excellent decay time resolution essential for measuring fast B0
s oscillations and related CP asymmetries.
The detector is built of 21 stations of silicon microstrip detectors located perpendicular to the beam pipe, each with 2048 strips of r − φ geometry. Each disc is divided into two equal and partially overlapping parts, corresponding to R and φ geometry respectively. R sensors are arranged in concentric semi-circles around the beam axis, while φ are laid radially. The structure of VELO sensors is presented in Figure 3.4. Such geometry ensures that the number of tracks within the acceptance, crossing less than four stations is smaller than 0.1%. VELO stations can be mechanically retracted in the vertical direction when beams are injected into the LHC. Due to this feature the sensors can be located only 8 mm away from the beam axis, ensuring the excellent vertex resolutions (Ref. [38]).
In order to reduce the material density between the proton-proton collisions point and VELO detector and protect the electronics from radio frequency current, the stations are kept in a vacuum and are separated from the LHC beam by an aluminium foil of 200 µm thickness. The silicon technology has to be extremely resistant to radiation damages and the detector is kept in temperature below 0◦C (≈ −5◦) to slow down ageing processes.
The resolutions of primary vertex (PV) along x, y and z direction as a function of the PV multiplicity are presented in Figure 3.5 (Ref. [39]). In the z direction, the resolution of PV is equal to σP V
z = 50 µm. The secondary vertex resolution along the beam direction is dependent
on the type of B decay and typically equal to σSV
z = 200 µm. The average impact parameter
resolution is equal to σIP = 30 µm. The dependence between impact parameter resolution and
transverse momentum for Monte Carlo is shown in Figure 3.6. This agrees with the resolution measured in the data. The typical B0
