Messung der Ladungsasymmetrie in Top-Quark

Transcription

IEKP-KA/2011-33
Messung der Ladungsasymmetrie
in Top-Quark-Paarereignissen
mit dem CMS-Experiment
Christian Böser
Diplomarbeit
von
Christian Böser
An der Fakultät für Physik
Institut für Experimentelle Kernphysik
Referent:
Korreferent:
Prof. Dr. Th. Müller
Prof. Dr. J. H. Kühn
02. November 2011
Zusammenfassung
Das Jahr 2011 hat Teilchenphysiker auf der ganzen Welt in Atem gehalten, da die
größte Maschine der Welt, der Large Hadron Collider (LHC) in der Nähe von Genf,
in seinem zweiten Betriebsjahr durch seine tadellose Leistung sogar die optimistischsten Erwartungen übertroffen hat. Der LHC-Beschleuniger liefert hochenergetische
Proton-Proton-Kollisionen, um schwere Elementarteilchen zu erzeugen, die nicht in
gewöhnlicher Materie vorkommen. Um die Kollisionsereignisse aufzuzeichen, wurden große Detektoren wie das Compact-Myon-Solenoid-Experiment (CMS) um die
Wechselwirkungspunkte herum installiert. Mit den riesigen vom LHC-Beschleuniger
produzierten Datenmengen ist es möglich, die Vorhersagen des Standardmodells
der Elementarteilchenphysik bei einer unter Laborbedingungen nie zuvor erreichten
Energieskala zu testen.
Das Standardmodell (SM) wurde in den Sechziger Jahren des vergangenen Jahrhunderts entwickelt und ist seitdem die präziseste Theorie für die Beschreibung
der fundamentalen Bausteine der Materie und deren Wechselwirkungen untereinander. Es vereinigt die drei über Eichbosonen vermittelten Grundkräfte, d.h. die
starke, die schwache und die elektromagnetische Kraft. Darüber hinaus werden
zwölf Fermionen im SM postuliert, die allesamt experimentell gefunden wurden.
Diese kann man zu sechs schwach und elektromagnetisch wechselwirkenden Leptonen, und zu sechs Quarks, die alle drei Kräfte erfahren, gruppieren. Das schwerste aller bekannten Elementarteilchen, das Top-Quark, wurde 1995 am TevatronBeschleuniger in der Nähe von Chicago entdeckt [1, 2]. Top-Quarks besitzen eine
Masse von (173,2 ± 0,9) GeV/c2 [3], was ungefähr der Masse eines Goldatoms entspricht, und zerfallen in nahezu 100% der Fälle unmittelbar in ein Bottom-Quark
und ein W -Boson, bevor sie gebundene Zustände bilden können. Mit ihrer Masse
in der Größenordnung der Skala der elektroschwachen Symmetriebrechung ist das
Top-Quark prädestiniert dazu, das Standardmodell bei hohen Energien auf seine
Grundfesten zu überprüfen.
Top-Quarks werden überwiegend paarweise mittels der starken Wechselwirkung
produziert, was entweder durch Gluon-Gluon-Fusion oder Quark-Antiquark-Annihilation geschieht. Eine besondere Eigenschaft der Top-Quark-Paarproduktion (tt̄)
über asymmetrische Ausgangszustände, d.h. Quark-Antiquark-Annihilation, ist die
Ladungsasymmetrie, die über Prozesse nächstführender Ordnung innerhalb des SM
erklärt werden kann [4–6]. Diese Prozesse führen dazu, dass das Top-Quark bevorzugt in die Richtung des einlaufenden Quarks ausgesandt wird und das AntitopQuark vorzugsweise in Richtung des einlaufenden Antiquarks fliegt. Am ProtonAntiproton-Beschleuniger Tevatron führt dieses Verhalten zu einem Überschuss an
iii
iv
Top-Quarks in Vorwärtsrichtung und einem Überschuss von Antitop-Quarks in Rückwärtsrichtung im Bezug auf den einlaufenden Protonenstrahl. Da am Proton-ProtonBeschleuniger LHC die Antiquarks nur als Seequarks innerhalb der Protonen vorkommen und damit im Durchschnitt einen kleineren Impuls als die Quarks besitzen,
führt hier die Ladungsasymmetrie zu einem Überschuss an Antitop-Quarks in zentralen Detektorbereichen und einem Überschuss an Top-Quarks in Vorwärts- und
Rückwärtsrichtung im Bezug auf die Richtung der Strahlröhre. Um ein Maß für
die Flugrichtungen der Teilchen zu haben, werden nach der Festlegung einer dem
Verlauf eines einlaufenden Teilchenstrahls entsprechenden Richtung z zwei Größen
eingeführt. Zum Einen ist die Rapidität y eines Teilchens mit der Energie E als
E + pz
1
(1)
y := ln
2
E − pz
definiert, wobei pz den Impuls des Teilchens in z-Richtung bezeichnet. Zum Anderen
kann für ein Teilchen die Pseudorapidität η im Laborsystem über
θ
η := − ln tan
(2)
2
festgelegt werden, wobei θ der Winkel zwischen dem Impulsvektor des Teilchens und
der z-Richtung ist. Für masselose Teilchen sind die beiden Größen y und η identisch.
Die Messung der Ladungsasymmetrie ist von aktuellem Interesse, da neueste
Resultate der Tevatron-Experimente [7–9] große Abweichungen von den vom SM
vorhergesagten Werten [10] aufweisen. Das CDF-Experiment misst zudem eine signifikant stärkere Abhängigkeit der Ladungsasymmetrie von der invarianten Masse
des tt̄-Systems als vorhergesagt. Diese Diskrepanzen, sofern experimentell bestätigt,
können nur durch Erweiterungen des SM erklärt werden, die neuartige Austauschteilchen postulieren. Wenn sich eines dieser Modelle als wahr erweisen würde, sollte die Messung der Ladungsasymmetrie am LHC auch eine Abweichung vom SMWert vorweisen, wobei sich hier die Messung als bei Weitem anspruchsvoller herausstellt. Das liegt daran, dass das Verhältnis von Top-Quark-Paarproduktion über
die Quark-Antiquark-Annihilation gegenüber der keine Ladungsasymmetrie aufweisenden Gluon-Gluon-Fusion am LHC wesentlich kleiner als am Tevatron ist. So
sind auch die Werte der SM-Vorhersagen für den LHC deutlich kleiner und damit
experimentell schwerer zugänglich. Aufgrund des symmetrischen Ausgangszustands
in Proton-Proton-Kollisionen sind auch die Rapiditätsverteilungen von Top- und
Antitop-Quarks symmetrisch um Null verteilt und unterscheiden sich durch den Einfluss der Ladungsasymmetrie nur in ihrer Breite. Schon 2010 wurde eine Messung
mit ersten LHC-Daten unter Verwendung der auf diesen Breitenunterschied sensitiven Größe ∆|η| = |ηt | − |ηt̄ | durchgeführt [11]. Zusätzlich zu der Messung in ∆|η|
wird in dieser Arbeit erstmalig die durch theoretische Aspekte motivierte Variable
∆y 2 = (yt − yt̄ )(yt + yt̄ ) [12] verwendet. Die Ladungsasymmetrie in der Top-QuarkPaarproduktion wird als Asymmetrie in diesen Verteilungen über
AC :=
N+ − N−
N+ + N−
(3)
v
definiert, wobei N + und N − die Anzahl an Ereignissen mit positiven bzw. negativen
Werten für ∆|η| und ∆y 2 bezeichnen.
In dieser Diplomarbeit wurde der vom CMS-Experiment zwischen März und
Juni 2011 aufgezeichnete und einer integrierten Luminosität von 1,09 fb−1 entsprechende Datensatz ausgewertet. Mit Hilfe geeigneter Selektionsschnitte wurden aus
dem Datensatz nur solche Ereignisse ausgewählt, die der Signatur des Lepton+JetsZerfallkanals von Top-Quark-Paaren entsprechen. In diesem Zerfallskanal werden
von einem der beim Top-Quark-Paarzerfall entstehenden W -Bosonen zwei Quarks
ausgesandt und vom anderen ein Neutrino und ein geladenes Lepton, wobei in dieser
Analyse nur Elektronen und Myonen berücksichtigt wurden. Dementsprechend wurden nur genau ein isoliertes geladenes Lepton und mindestens vier Jets aufweisende
Ereignisse selektiert. Die Signal-Reinheit im Datensatz kann durch Anwendung eines
b-Taggers, der von der Hadronisierung eines b-Quarks stammende Jets identifiziert,
deutlich erhöht werden, wobei aber auch die Datenmenge abnimmt. Im Zuge dieser Arbeit wurde ein optimales Verhältnis zwischen Signal-Reinheit und Größe des
Datensatzes für die Forderung nach mindestens einem identifizierten b-Jet gefunden.
Die Signatur im Detektor lässt jedoch nicht direkt auf den zugrunde liegenden physikalischen Prozess schließen. Deswegen wurden computergestütze Simulationsrechnungen benutzt, so genannte Monte-Carlo-Simulationen, mit denen sich
grundsätzlich alle in Proton-Proton-Kollisionen vorkommenden physikalischen Prozesse generieren lassen. Für die Analyse wurden simulierte Datensätze für Signalereignisse und signalähnliche Signatur aufweisende Untergrundprozesse der Selektion
unterzogen. Mit der Anwendung eines Binned-Likelihood-Fits wurden ausgewählte
generierte Monte-Carlo-Verteilungen an die gemessenen Verteilungen in Daten angepasst, um so eine Untergrundabschätzung zu erhalten.
Um auf die Vierervektoren der Top-Quarks schließen zu können, die für die Berechnung der zwei sensitiven Variablen ∆|η| und ∆y 2 benötigt werden, wurde ein
spezieller Rekonstruktionsalgorithmus verwendet, der sukzessive die Vierervektoren
der selektierten physikalischen Objekte addiert. Da die Zuordnung der gemessenen
Objekte zu den Endzuständen im Lepton+Jets-Kanal nicht eindeutig ist, wurde
ein Kriterium entwickelt, das für jede mögliche Hypothese die Wahrscheinlichkeit
angibt, dass diese der bestmöglichen Zuordnung entspricht. Für jedes einzelnes Ereignis wurde die Rekonstruktionshypothese mit der größten Wahrscheinlichkeit ausgewählt. Das Auswahlkriterium basiert auf den invarianten Massen der rekonstruierten Top-Quarks und des hadronisch zerfallenden W -Bosons und auf den b-TaggerAusgabewerten der verwendeten Jets.
Nach der Rekonstruktion wurde zunächst der erwartete Untergrund von den Verteilungen abgezogen. Da aber Rekonstruktion und Selektion die Verteilungen von
∆|η| und ∆y 2 beeinflussen, mussten diese Effekte korrigiert werden um die Resultate
mit Theorievorhersagen vergleichen zu können. Zu diesem Zweck wurde eine regularisierte Entfaltungsmethode auf Grundlage einer generalisierten Matrixinversion angewandt. Um den Entfaltungsalgorithmus auf seine Funktionalität zu untersuchen,
wurden zunächst mit Hilfe von Pseudoexperimenten mehrere Tests durchgeführt,
die allesamt zufriedenstellende Ergebnisse erzielten.
Nach der erfolgreichen Überprüfung der Robustheit der Analyse wurden schließ-
0.3
t
0.4
t
l+jets
0.25
0.2
0.15
0.35
Data
NLO prediction
1.09 fb-1 at s = 7 TeV
A C= -0.012 ± 0.026
l+jets
0.3
0.25
0.2
0.15
0.1
0.1
0.05
0
-4
0.45
t
Data
NLO prediction
1.09 fb-1 at s = 7 TeV
A C= -0.016 ± 0.030
t
0.35
1/σ dσ/d((y -y )(y +y ))
t
0.4
t
1/σ dσ/d(|η |-|η |)
vi
0.05
-3
-2
-1
0
1
2
3
4
|η |-|η |
t
(a)
0
-4
-3
-2
-1
0
1
2
3
4
(yt-y )(yt+y )
t
t
t
(b)
Abbildung 1: Entfaltete Verteilungen in ∆|η| (a) und ∆y 2 (b). Zusätzlich sind die
Standardmodellvorhersagen für die tt̄-Ladungsasymmetrie in nächstführender Ordnung gezeigt [13].
lich die gemessenen Verteilungen von ∆|η| und ∆y 2 entfaltet. Mittels Gleichung 3
lassen sich aus den beiden in Abbildung 1 gezeigten Verteilungen die Werte für die
tt̄-Ladungsasymmetrie zu
∆|η|
= −0,016 ± 0,030 (stat.)+0,025
−0,036 (syst.) ,
(4)
∆y 2
= −0,013 ± 0,026 (stat.)+0,029
−0,031 (syst.)
(5)
AC
AC
bestimmen. Diese Resultate sind im Gegensatz zu den Messungen am Tevatron innerhalb der Unsicherheiten in guter Übereinstimmung mit den vom Standardmodell
vorhergesagten Werten von etwa +1% [10]. Ferner zeigt die rekonstruierte und untergrundkorrigierte Ladungsasymmetrie in beiden Variablen keine Abhängigkeit von
der invarianten Masse des tt̄-Systems, wie in Abbildung 2 zu sehen ist.
Durch die Messungen dieser Arbeit hat der Bereich der tt̄-Ladungsasymmetrie
weiter an Bedeutung gewonnen, da auch das ATLAS-Experiment am LHC eine negative, aber noch mit dem Standardmodell verträgliche Asymmetrie misst [14]. Es
sind viele SM-erweiternde Theorien im Umlauf, die die Tevatron-Ergebnisse zufriedenstellend erklären können, aber nicht vereinbar mit einer negativen Ladungsasymmetrie am LHC sind. Dieses Dilemma kann nur durch weitere Messungen am LHC
aufgeklärt werden, weshalb sowohl experimentelle als auch theoretische Physiker
gespannt auf neue Analysen zu diesem Thema blicken.
Zufünftigen Analysen stehen wesentlich größere Datenmengen zur Verfügung,
wobei der heute verfügbare Datensatz bereits einer integrierten Luminosität von
5 fb−1 entspricht. Diese große Menge an Daten wird ein simultanes Entfalten in der
sensitiven Variable und der invarianten Masse des tt̄-Systems erlauben, um somit
Klarheit bezüglich der aufgeworfenen Fragen im Bereich der tt̄-Ladungsasymmetrie
zu schaffen.
0.2
0.15
-1
1.09 fb at s = 7 TeV
l+jets
tt (Powheg)
data
y
0.1
A C (bg subtr.)
η
A C (bg subtr.)
vii
0.05
0.2
0.15
0.05
0
-0.05
-0.05
-0.1
-0.1
-0.15
-0.15
Mtt,rec [GeV/c2]
(a)
tt (Powheg)
data
0.1
0
-0.2
200 300 400 500 600 700 800 900 1000 1100 1200
1.09 fb-1 at s = 7 TeV
l+jets
-0.2
200 300 400 500 600 700 800 900 1000 1100 1200
Mtt,rec [GeV/c2]
(b)
Abbildung 2: Rekonstruierte, untergrundkorrigierte Asymmetrien in ∆|η| (a) und
∆y 2 (b) für verschiedene rekonstruierte invariante tt̄-Massen.
IEKP-KA/2011-33
Measurement of the Charge Asymmetry
in Top Quark Pair Production
at the CMS-Detector
Diploma thesis
of
Christian Böser
At the Department of Physics
Institut für Experimentelle Kernphysik
Advisor:
Second advisor:
Prof. Dr. Th. Müller
Prof. Dr. J. H. Kühn
November 2, 2011
ii
Introduction
The year 2011 has been very exciting for particle physicists all over the world.
The world’s largest machine, the Large Hadron Collider (LHC) near Geneva, has
entered its second year of operation and has worked so well by now, that even the
most optimistic predictions have been beaten. The LHC provides high energetic
proton-proton collisions in order to artificially produce massive particles, which are
not present in ordinary matter. Large detectors, like the Compact Muon Solenoid
(CMS) experiment, have been constructed around the interaction points to record
the collision events. With the large amount of data the LHC has so far produced,
it is now possible to test the predictions of the Standard Model of particle physics
at an energy, which has never been achieved before under laboratory conditions.
The Standard Model (SM) developed in the 1960s, is the most accurate theory
describing the fundamental building blocks of matter and the interactions between
them. It unifies three fundamental forces, namely the strong, the weak and the
electromagnetic force, which are mediated via gauge bosons. Additionally, there are
twelve fermions predicted by the SM, grouped into six leptons, which interact via
the weak and the electromagnetic force, and six quarks, which interact via all three
forces. The heaviest of all known particles is the top quark, which was discovered
at the Tevatron collider near Chicago in 1995 [1, 2]. Top quarks exhibit a mass of
173.2±0.9 GeV/c2 [3] and decay instantly before forming bound states. With a mass
close to the scale of electroweak symmetry breaking the top quark is predestinated
to probe the Standard Model at high energies and to search for hints of new physics.
Top quarks are mainly produced pairwise via the strong interaction either through
gluon gluon fusion or via quark-antiquark annihilation. One particular feature of the
top quark pair production in the quark-antiquark annihilation process is the slight
preference of the top quark to be emitted into the direction of the incoming quark
and of the antitop quark to fly into the direction of the incoming antiquark, referred
to as charge asymmetry [4–6]. Since the antiquarks only occur as sea quarks in the
protons and therefore carry on average a smaller amount of energy compared to the
valence quarks, at the LHC this charge asymmetry leads to an excess of antitop
quarks in the central region of the detector and an excess of top quarks in forward
and backward directions with respect to the beam pipe. The measurement of the
charge asymmetry is of current interest since recent results from the Tevatron [7–9]
show a large deviation from the values predicted by the Standard Model [10]. Only
extensions of the SM which predict new exchange particles are able to explain these
discrepancies. If these models were true, a deviation from the SM predictions would
also be visible at the LHC.
1
2
In this thesis the measurement of the charge asymmetry in top quark pair production using the lepton+jets decay mode is presented. Based on a dataset recorded
by the CMS experiment from March to June 2011 corresponding to an integrated
luminosity of 1.09 fb−1 , the measurement is sensitive enough to give a tendency
towards either the Standard Model or to new physics.
The outline of the thesis is as follows. In chapter 1 a short theoretical introduction to the Standard Model and the charge asymmetry in top quark pair production
is given. The functionality of the LHC and the different subsystems of the CMS
detector are explained in detail in chapter 2. In chapter 3 the simulation of collision
events based on Monte Carlo techniques and the reconstruction of events with raw
detector information are elucidated. The event selection yielding a signal enriched
dataset and the estimation of the remaining background contribution are defined in
chapter 4. Finally, the reconstruction of the top quarks’ four-momenta, the correction for selection and reconstruction effects using a dedicated unfolding technique
and the final measurement are explained in chapter 5, where also the impacts of
systematic uncertainties are studied.
Contents
1 Theoretical Introduction
1.1 The Standard Model of Particle Physics . . . . . . . . . . . . .
1.1.1 Gauge Bosons and Fermions . . . . . . . . . . . . . . . .
1.1.2 Spontaneous Symmetry Breaking and Higgs Mechanism .
1.2 Top Quark Physics . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Production of Top Quarks . . . . . . . . . . . . . . . . .
1.2.2 Decay of the Top Quark . . . . . . . . . . . . . . . . . .
1.3 The Charge Asymmetry of Top Quark Pair Production . . . . .
2 Experimental Setup
2.1 The Large Hadron Collider . . . . . . . . . . . . .
2.2 The Compact Muon Solenoid Detector . . . . . .
2.2.1 Tracking System . . . . . . . . . . . . . .
2.2.2 Calorimetry System . . . . . . . . . . . . .
2.2.3 Muon System . . . . . . . . . . . . . . . .
2.2.4 Trigger System and Computing Structure .
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3 Generation, Simulation and Reconstruction
3.1 Generation of Events . . . . . . . . . . . . . .
3.1.1 Monte Carlo Generators . . . . . . . .
3.1.2 Detector Simulation . . . . . . . . . .
3.2 Reconstruction of Events . . . . . . . . . . . .
3.2.1 Reconstruction of Tracks . . . . . . . .
3.2.2 Reconstruction of Vertices . . . . . . .
3.2.3 Reconstruction of Electron Candidates
3.2.4 Reconstruction of Muon Candidates . .
3.2.5 Reconstruction of Jets . . . . . . . . .
3.2.6 b Tagging . . . . . . . . . . . . . . . .
3.2.7 Missing Transverse Energy . . . . . . .
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4 Selection of Events
4.1 Modeling of Signal and Background
4.1.1 The Lepton+Jets Channel .
4.1.2 Background Processes . . .
4.1.3 Used Monte Carlo Samples .
4.1.4 Used Data . . . . . . . . . .
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Events
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4
CONTENTS
4.2
4.3
4.1.5 Corrections on Monte Carlo Events . . .
Selection Criteria . . . . . . . . . . . . . . . . .
4.2.1 Definitions of Physical Objects . . . . .
4.2.2 Selection Cuts . . . . . . . . . . . . . . .
Background Estimation . . . . . . . . . . . . . .
4.3.1 Data-Driven Modeling of QCD Multi-Jet
4.3.2 Template Fit . . . . . . . . . . . . . . .
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Production Processes
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5 Measurement of the tt̄ Charge Asymmetry
5.1 Full Reconstruction of tt̄ Events . . . . . . . . . . . . . . . .
5.1.1 Reconstruction of all Possible Hypotheses . . . . . .
5.1.2 Selection of one Reconstruction-Hypothesis per Event
5.2 Background Subtraction and Unfolding . . . . . . . . . . . .
5.2.1 Regularized Unfolding . . . . . . . . . . . . . . . . .
5.2.2 Consistency Checks with Pseudo Experiments . . . .
5.2.3 Linearity Checks . . . . . . . . . . . . . . . . . . . .
5.3 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . .
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion and Outlook
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List of Figures
1
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1.1
1.2
1.3
1.4
1.5
1.6
Feynman diagrams for fundamental interactions . . . . . . . . . . . .
Parton Distribution Function . . . . . . . . . . . . . . . . . . . . . .
Leading order Feynman diagrams for the production of tt̄ pairs . . . .
Top quark decay into b and W . . . . . . . . . . . . . . . . . . . . . .
Origin of the tt̄ charge asymmetry in the SM. . . . . . . . . . . . . .
The tt̄ charge asymmetry in the y distribution for LHC and Tevatron.
12
15
16
17
17
20
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Accelerator chain at CERN . . . . . . . . . . . . . . . .
Luminosity Profile in 2011 proton operation at the LHC.
Schematic overview of the CMS detector . . . . . . . . .
Schematic overview of the tracking system . . . . . . . .
The calorimeter system and the muon system . . . . . .
Schematic overview of the muon system . . . . . . . . . .
Architecture of the CMS data acquisition system . . . .
Architecture of the CMS computing grid . . . . . . . . .
.
.
.
.
.
.
.
.
24
26
27
29
30
32
33
33
3.1
3.2
3.3
3.4
3.5
Schematic overview of the factorization in event generation processes
Evolution of a jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jet resolutions for calorimeter jets, JPT jets and PF jets. . . . . . . .
Illustration of a collision event with a b jet. . . . . . . . . . . . . . . .
Discriminator values and mistag rates for the TCHE b tagging algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
42
45
46
4.1
4.2
4.3
4.4
4.5
4.6
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Predicted cross sections of different physics processes at LHC . . . .
Exemplary Feynman diagram of a tt̄ event in the lepton+jets decay
channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exemplary LO Feynman diagrams of W ± and Drell-Yan production
in association with additional jets. . . . . . . . . . . . . . . . . . . .
Exemplary Feynman diagrams of electroweak production of single top
quarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exemplary Feynman diagrams of QCD multi-jet and photon+jets
processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Trigger turn-on curves of the CentralJet30 trigger. . . . . . . . . . .
5
48
. 52
. 53
. 54
. 54
. 55
. 60
6
LIST OF FIGURES
4.7
QCD background shape comparison between MC based template and
data-driven template in the missing transverse energy and M3. . . . .
4.8 Shape comparison of the missing transverse energy and the M3 distributions for signal and background processes. . . . . . . . . . . . . .
4.9 Data to MC comparison of the missing transverse energy and M3. . .
4.10 Data to MC comparison of certain kinematic distributions. . . . . . .
4.11 Data to MC comparison of pT of the four leading jets. . . . . . . . . .
4.12 Data to MC comparison of the pseudorapidity of the four leading jets.
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
65
66
67
68
69
70
Correlation between the reconstructed masses before and after the
linear decorrelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Decorrelated masses m1 , m2 and m3 for the best possible hypotheses
and for all hypotheses. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Probability distribution for a jet to be assigned to a b quark in the
best possible hypothesis as a function of the b tagger discriminator
value of the jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Data to MC comparison after reconstruction for the electron+jets
channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Data to MC comparison after reconstruction for the muon+jets channel. 78
Data to MC comparison of the kinematic distributions for the reconstructed top quarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Distributions of ∆|η| and ∆y 2 for the electron+jets, muon+jets and
the combined lepton+jets channel. . . . . . . . . . . . . . . . . . . . 81
QCD background shape comparison between MC based template and
data-driven template in ∆|η| and ∆y 2 . . . . . . . . . . . . . . . . . . 82
Migration matrix and event selection efficiency as functions of ∆|η|
and ∆y 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Sketch of the L-curve used to derive an optimal regularization parameter τ for the unfolding. . . . . . . . . . . . . . . . . . . . . . . . . . 84
Relative difference in each bin between the unfolded ∆|η| distribution
in pseudo experiments and the true spectrum. . . . . . . . . . . . . . 86
Relative difference in each bin between the unfolded ∆y 2 distribution
in pseudo experiments and the true spectrum. . . . . . . . . . . . . . 87
Pull distributions for each bin of the unfolded ∆|η| spectrum in
pseudo experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Pull distributions for each bin of the unfolded ∆y 2 spectrum in pseudo
experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Consistency checks for the unfolding routine in pseudo experiments
for ∆|η| and ∆y 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Linearity check for ∆|η| and ∆y 2 . . . . . . . . . . . . . . . . . . . . . 90
Comparison of the ∆|η| and ∆y 2 shapes in the sideband QCD models 94
Unfolded results for ∆|η| and ∆y 2 . . . . . . . . . . . . . . . . . . . . 96
Background-subtracted asymmetries for ∆|η| and ∆y 2 for different
regions in the reconstructed invariant tt̄ mass . . . . . . . . . . . . . 97
Summary of the results of tt̄ charge asymmetry measurements at
Tevatron and LHC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
List of Tables
1.1
1.2
1.3
The gauge bosons of the Standard Model . . . . . . . . . . . . . . . . 10
The fermions of the Standard Model . . . . . . . . . . . . . . . . . . 12
Predicted charge asymmetries at LHC for different models . . . . . . 21
2.1
Proton energies at the different acceleration stages . . . . . . . . . . . 25
3.1
Working points of the TCHE b tagging algorithm. . . . . . . . . . . . 47
4.1
4.2
4.3
Overview of the used Monte Carlo samples. . . . . . . . . . . . . .
Summary of the used HLT trigger paths applied on data. . . . . . .
Event yield for the described event selection for a selected dataset
corresponding to an integrated luminosity of 1.09 fb−1 . . . . . . . .
Fit results for the background estimation. . . . . . . . . . . . . . .
4.4
5.1
5.2
5.3
5.4
. 57
. 58
. 63
. 67
Unfolded charge asymmetry AC for ∆|η| and ∆y 2 compared to different generated charge asymmetries. . . . . . . . . . . . . . . . . . .
Summary of the systematic uncertainties taken into account in the
measurement of the tt̄ charge asymmetry. . . . . . . . . . . . . . . . .
The measured asymmetries for both variables at the different stages
of the analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measured bin by bin charge asymmetries for both sensitive variables.
7
90
95
97
97
8
LIST OF TABLES
Chapter 1
Theoretical Introduction
The Standard Model of Particle Physics [15–24] is the most accurate theory today, which describes all known elementary particles and their interactions. It was
developed between 1960 and 1970 and has been very successful in predicting new
fundamental particles and their properties since then. With the experimental discovery of the bottom quark in 1977 [25], the top quark in 1995 [1, 2] and the tau
neutrino in 2000 [26], the Standard Model gained more and more credence.
Despite all success the Standard Model can not be a complete theory of nature,
as the fundamental force of gravitation is not part of it. Gravitation is described
in Einstein’s theory of General Relativity instead [27]. Every approach to combine
these two fundamental theories into a Theory of Everything has failed so far. Furthermore, the Standard Model gives no explanation for dark matter and dark energy,
for which there is a strong evidence in cosmological observations [28]. Thus, there
are many models extending the theory of the Standard Model and attempting to
explain the missing parts of it.
In the following sections a short overview of the Standard Model is given. The
focus is on the pair production of top quarks and the tt̄ charge asymmetry.
1.1
The Standard Model of Particle Physics
The Standard Model (SM) is based on the combination of quantum mechanics and
special relativity and is formulated as a relativistic quantum field theory, which
provides a theoretical framework where continuous systems are described with an
infinite number of degrees of freedom. This framework contains all known particles:
fermions, which are the fundamental building blocks of matter, and gauge bosons,
which are the mediators of the fundamental forces within the SM. The dynamics
of a system of particles can be described by its Lagrangian, which is required to
be invariant under local SU (N ) transformations. This local gauge symmetry is
one major part of the SM. Noether’s Theorem claims that every symmetry of a
physical system leads to a conserved quantity [29]. Therefore, as a result of the
local symmetries SU (3) × SU (2) × SU (1), there are three conserved charges in the
SM: color charge, weak isospin and electric charge. This yields three types of forces
described by the Standard Model, the strong, the weak and the electromagnetic
9
10
CHAPTER 1. THEORETICAL INTRODUCTION
Name
Symbol
photon
γ
gluon
g
W boson
W±
Z boson
Z0
Force
Electric Charge [e]
electromagnetic
0
strong
0
weak
±1
weak
0
Mass [GeV/c2 ]
≤ 1 · 10−27
0
80.399 ± 0.023
91.1876 ± 0.0021
Table 1.1: List of the gauge bosons of the Standard Model with their mediated
force, electric charge and mass. The gluon mass of zero is taken from the theory
prediction. All other values are taken from [38].
force, each coupling to one of the three charges. In addition to these charges, every
particle of the Standard Model has a quantum number named spin, measured in
units of ~. Fermions have half-integer spin, whereas gauge bosons have integer spin.
Further, every particle has a corresponding antiparticle with the same mass but
opposite electric charge.
1.1.1
Gauge Bosons and Fermions
The gauge bosons of the Standard Model are the mediators of all fundamental forces
and have a spin of 1 ~. The electromagnetic force is mediated via photons, which
are massless and carry no electric charge. Interactions between photons and electrically charged particles are described within the theory of Quantum Electrodynamics
(QED) [30–37] and yield for example atoms, which are the bound states of electrons
and nuclei. The mediators of the strong force are gluons, which are also massless and
have no electric charge. Gluons couple to the color charge, whose three states are
named red, green and blue and the corresponding anticolors, respectively. The interactions between gluons and color charged particles are addressed in the theory of
Quantum Chromodynamics (QCD) [23, 24]. Gluons carry one unit of color and one
unit of anticolor charge, yielding eight linear independent kinds of color states for
gluons. An example for the strong interaction is the binding of protons and neutrons
into nuclei. For the weak force there are three gauge bosons, the electrically neutral
Z boson (Z 0 ), and two W bosons (W ± ), which carry an electric charge of either +1 e
or −1 e, where e is the elementary charge with a value of e ≈ 1.602 · 10−19 C. Within
the gauge bosons Z 0 and W ± constitute an exception, because they are not massless, but have a rather large mass instead. The Z boson has a mass of 91.2 GeV/c2
and the W bosons have a mass of 80.4 GeV/c2 [38]. Due to the massive weak gauge
bosons the weak interaction is limited to sub-nuclear scales. An example for the
weak interaction is the β decay of a radioactive nucleus. In table 1.1 the properties
of the gauge bosons are summarized.
Although there are visible differences between the weak force and the electromagnetic force, especially regarding the mass of the gauge bosons, the electroweak
theory [15, 16, 18, 20] accomplished the combination of QED and the theory of the
weak force as a unified theory. According to this theory the electroweak force is mediated by four massless gauge bosons, with which the massive Z 0 and W ± bosons
1.1. THE STANDARD MODEL OF PARTICLE PHYSICS
11
seem to be in conflict. The Higgs mechanism, described in section 1.1.2, solves
the problem by electroweak symmetry breaking (EWSB), giving Z and W bosons
masses and leaving the photon massless.
The fermions of the Standard Model can be separated into quarks and leptons. In
addition the fermions can be ordered into three generations. Every generation consists of two quarks, up and down type, and two leptons, one with an electric charge
and the corresponding neutrino, which is electrically neutral. The constituents of
one generation differ from their corresponding partners from another generation
only in their masses. The three generations of fermions and their properties are
summarized in table 1.2.
Having color charge, weak isospin and electric charge, quarks interact via the
strong, the weak and the electromagnetic force. They can be further separated
into up type and down type quarks. Up type quarks (up, charm, top) all have an
electric charge of + 23 e, whereas down type quarks (down, strange, bottom) possess
an electric charge of − 31 e. In contrast to gluons quarks carry only one unit of color
and antiquarks one unit of anticolor, respectively. Because the energy in the gluon
field between two quarks increases with their distance there are two limiting cases.
On the one hand, if the distance is short, quarks are quasi-free particles, since the
gluon field strength is very small. This phenomenon is called asymptotic freedom of
quarks. On the other hand, when two quarks are being separated, the energy of the
gluon field between them increases until it gets high enough to create a new quarkantiquark pair. The produced quarks build two new color neutral bound states
with the two initial quarks. In respect to this so-called color confinement only
color neutral bound states of quarks, called hadrons, exist. Mesons are hadrons
consisting of a quark and an antiquark, e.g. pions or kaons. Baryons are hadrons
with three quarks as constituents, like the proton and the neutron. The proton plays
an important role in nature, because it is the only hadron which is stable.
Leptons carry no color charge. Electron, muon and tau, which are the electrically
charged leptons, have an electric charge of −1e. These three charged leptons interact
via the electromagnetic and the weak force. Each charged lepton has a corresponding electrically uncharged weak isospin partner, the neutrino. Neutrinos therefore
only interact via the weak force and were assumed to be massless. However direct
observations of flavor changing neutrino currents (neutrino oscillations) [39] claim
neutrinos to carry mass and there are various extensions of the Standard Model trying to include a neutrino mass generation mechanism [40]. Since all fermions of the
Standard Model are spin- 12 particles, they underlie the Pauli exclusion principle [41],
which claims that no two fermions of the same kind can occupy the same quantum
state simultaneously.
The Standard Model predicts the properties of particles and the strength of interactions between them. Since the theory is based on quantum mechanics, the
predictions only allow to calculate the probability for a process A → B, where A
represents a given initial state and B the wanted final state, to occur. Obtained
from Fermi’s Golden Rule the probability is proportional to the transition amplitude
|M|2 of the process [43]. The integration over all possible initial and final states of
the process yields the so-called cross section σ, which is typically quoted in terms
12
Name
Symbol
up quark
u
down quark
d
electron
e
electron neutrino
νe
charm quark
c
strange quark
s
muon
µ
muon neutrino
νµ
top quark
t
bottom quark
b
tau
τ
tau neutrino
ντ
EC [e]
+ 23
− 13
−1
0
+ 23
− 13
−1
0
+ 23
− 13
−1
0
r,
r,
r,
r,
r,
r,
CC
g or
g or
0
0
g or
g or
0
0
g or
g or
0
0
b
b
b
b
b
b
Mass [MeV/c2 ]
1.7-3.3
4.1-5.8
(510.998910 ± 0.00013) · 10−3
< 2 · 10−6
3
1.27+0.07
−0.09 · 10
101+29
−21
105.658367 ± 0.000004
< 0.19
(173.3 ± 1.1) · 103
3
4.19+0.18
−0.06 · 10
1776.82 ± 0.16
< 18.2
Table 1.2: List of all fermions of the Standard Model ordered by generation [38,42].
The Electric Charge (EC) is given in units of the elementary charge e and the Color
Charge (CC) with the abbreviations r, b, g for red, green and blue, and 0 for color
neutral.
of barn [b], defined as 1 b = 1 · 10−24 cm2 . To visualize a given particle interaction together with the transition amplitude in a simple way, the use of Feynman
diagrams turns out as a very powerful tool. In figure 1.1 basic examples for the
three fundamental interactions within the SM are shown. A set of Feynman rules
translates the diagrams into formulas, from which one can calculate the transition
amplitude.
(a)
(b)
(c)
(d)
Figure 1.1: Examples of leading order Feynman diagrams for the fundamental
interactions. Shown is the electron-positron annihilation via electromagnetic (a) and
weak interaction (b). Another example for the weak interaction is the annihilation
of two quarks via a charged W boson (c). Further, the annihilation of two quarks
via the strong force is displayed (d). Time is evolving from left to right.
1.1.2
Spontaneous Symmetry Breaking and Higgs Mechanism
As mentioned before the weak interaction plays a special role within the SM since
its mediators, Z 0 and W ± , are very heavy. Inside the SM gauge bosons have to be
1.2. TOP QUARK PHYSICS
13
massless, since a mass term in the Lagrangian would violate local gauge symmetry.
Thus, an additional field is introduced to provide Z 0 and W ± with masses and
still conserve local gauge invariance. This so-called Higgs field interacts with the
weak gauge fields and with itself in a way that a spontaneous symmetry breaking
takes place. The Goldstone theorem [44, 45] claims that every symmetry breaking
leads to a massless Goldstone boson. But in case of local gauge symmetries these
Goldstone bosons can be absorbed by the Z 0 and W ± bosons, giving the latter
mass and therefore longitudinal polarization states. This process is called Higgs
mechanism [46–48] and is until now the simplest theory providing particles with
mass, because it has only four degrees of freedoms postulated. Three of them, two
charged components and one neutral, correspond to the observed masses of W ± and
Z 0 . The missing one would be visible as a massive Higgs boson, whose discovery
would prove the theory of the Higgs mechanism. Hence, the discovery of the Higgs
boson is one major aim of modern particle physics.
The introduced Higgs field can also explain how leptons and quarks gain their
mass based on the Yukawa coupling. For leptons the additional coupling term in the
Lagrangian is formed in a way that only the electrically charged leptons interact with
the Higgs field. The neutral neutrinos remain massless. For quarks the situation is
more complicated, because all six quarks are required to gain masses and their weak
eigenstates are not equal to their mass eigenstates. The transformations between
the two bases can be described via the Cabibbo-Kobayashi-Maskawa (CKM) matrix
[49, 50]
 
 0 
 
Vud Vus Vub
d
d
d
s0  =  Vcd Vcs Vcb  s = VCKM s ,
(1.1)
Vtd Vts Vtb
b0
b
b
wherein d0 , s0 and b0 represent the weak and d, s and b the mass eigenstates. As all
transition probabilities sum up to one, the CKM matrix is unitary. The individual
elements Vqi qj are proportional to the coupling strength between W ± and the quarks
qi and qj and therefore experimentally accessible via weak decay rates of mesons
and baryons or cross section measurements. Taking all available measurements and
theoretical constraints into account, a global fit results in the absolute values for the
CKM matrix [38], which are given in the matrix ṼCKM :


0.97428 ± 0.00015 0.2253 ± 0.0007 0.00347+0.00016
−0.00012
.
0.97345+0.00015
0.0410+0.0011
(1.2)
ṼCKM =  0.2252 ± 0.0007
−0.0007
−0.00016
+0.00026
+0.0011
+0.000030
0.00862−0.00020
0.0403−0.0007
0.999152−0.000045
As the diagonal elements are much larger than the off-diagonal elements of the
CKM matrix, flavor transitions inside one generation are preferred. Especially the
heaviest quark, the top quark, decays almost always into a bottom quark, since
|Vtd |, |Vts | |Vtb | and |Vtb | ≈ 1.
1.2
Top Quark Physics
The top quark has been predicted as the weak isospin partner of the bottom quark,
since the latter was discovered in 1977 at the Fermilab in Chicago [25]. But for
14
the discovery of the about 35 times heavier top quark much larger experiments
with higher center-of-mass energies had to be built. So it is not surprising that it
took experimental physicists almost 20 years to detect it. In 1995 finally the two
experiments at the Tevatron, CDF and DØ , observed a clear visible signal, which
was indicated as the top quark [1, 2]. Recent measurements result in a top quark
mass of mt = 173.2 ± 0.9GeV/c2 [3]. Because it is this heavy, the lifetime of the top
quark of τ ∼ 10−25 s is by far shorter then the usual time scale of QCD interactions
of ∼ 10−23 s, so it decays before it can hadronize. Therefore it can be seen as quasifree quark and is very handy to study quark properties. The top quark mass is of
the same order as the vacuum expectation value of the Higgs field (v = 246 GeV)
and hence the top quark has a Yukawa coupling gY = v/m√t 2 close to unity. This fact
underlines the important role of the top quark within the Standard Model.
The Tevatron, a proton-antiproton collider, and the Large Hadron Collider (LHC)
(proton-proton) are the only accelerators capable of producing top quarks, as it takes
a high center-of-mass energy of the order of the top quark mass. When colliding
hadrons at high energy, the hadrons’ substructure have to be taken into account,
because only their constituents, the so-called partons, interact with each other. The
momentum of an accelerated proton is distributed over all its partons, which are the
valence quarks, the sea quarks and the gluons. Valence quarks are the quarks which
define the quantum numbers of a hadron, in case of the proton two up and one
down quark. They are bound via virtual gluon exchange. The sea quarks are the
virtual quark-antiquark pairs which arise from gluon splitting. The parton distribution function (PDF) fi,A (xi , Q2 ) gives the fraction of momentum xi of each parton i
inside the hadron A at a given energy scale Q. By using a factorization ansatz [51]
one can calculate the total hadronic cross section of top quark pair production in
collisions of hadrons A and B via
σ(AB → tt̄) =
XZ
dxi dxj fi,A (xi , µ2F )fj,B (xj , µ2F ) · σ̂ij (ij → tt̄; ŝ, µ2R , µ2F ) . (1.3)
i,j
Herein ŝ = (xi pA + xj pB )2 is the square of the center-of-mass energy of the colliding
partons A and B. Further one has to distinguish between the factorization scale
µF , which is introduced by the factorization ansatz to separate short and long range
interactions, and the renormalisation scale µR , which describes the running of the
strong coupling constant and is introduced to regulate divergent terms in the calculation of σ̂(ij → tt̄; ŝ, µ2R , µ2F ). In most calculations of the total hadronic cross
section of top quark pair production only one scale µ = µF = µR is used. Thus
the total cross section σ is dependent on the partonic cross section σ̂(ij → tt̄; ŝ, µ2 ),
which can be calculated by perturbative QCD, and the PDFs for the initial partons
i and j. Since the PDF can not be theoretically calculated, it has to be extracted
experimentally from predictable processes like the deep inelastic scattering. In figure
1.2 the CTEQ6.6 [52] parametrization is shown as an example.
15
102
up quark
i
xi f i(x , µ2)
1.2. TOP QUARK PHYSICS
down quark
up antiquark
down antiquark
10
charm quark
strange quark
bottom quark
gluon
1
10-1 -4
10
10-3
10-2
10-1
1
xi
Figure 1.2: The CTEQ6.6 [52] proton NLO PDF for gluons and quarks at a scale
of µ2 = (172.5 GeV/c2 )2 . This value for µ2 is commonly used for the simulation of
top quark events.
1.2.1
Production of Top Quarks
There are two different processes for the production of top quarks at hadron colliders.
On the one hand top quarks can be produced singly via electroweak interactions.
These single top quarks have been firstly observed at Tevatron in 2009 [53,54]. Their
production gives the unique opportunity to determine directly the CKM matrix
element |Vtb |, since every production mode contains a W tb vertex. On the other
hand there is the top quark pair production, which is the dominant process of the
two, considering collision energies larger than 2·mt c2 . The top quark pair production
takes place via the strong or the electroweak interaction. In figure 1.3 the leading
order Feynman diagrams for top quark pair production are shown, which result from
quark-antiquark annihilation q q̄ → tt̄ (figure 1.3(a) and (b)) and from gluon fusion
gg → tt̄ (figures 1.3(c), (d) and (e)). Assuming a top quark mass of 173.3 GeV/c2
the latest approximate next-to-next-to-leading order calculation yields a total cross
section of top quark pair production in proton-proton collisions [55] of
+7.4+15.4
σ(pp → tt̄) = 162.6−7.6−14.7
pb .
(1.4)
Herein the center-of-mass energy is 7 TeV and the uncertainty arises from scale
variations and PDF uncertainties [56]. In proton-proton collisions at the LHC the
top quark pair production is dominated by gluon-gluon fusion, whereas in protonantiproton collisions at the Tevatron the dominant subprocess for the top quark pair
production is the quark-antiquark annihilation.
16
(a)
(c)
(b)
(d)
(e)
Figure 1.3: Production of top quark pairs in proton-proton collisions. The leading
order Feynman diagrams of the process are shown: Quark-antiquark annihilation via
strong interaction (a) and via electroweak interaction (b), and gluon-gluon fusion
via strong interactions (c), (d) and (e).
1.2.2
Decay of the Top Quark
As already mentioned top quarks decay into a W boson and a bottom quark in nearly
100% of all cases. The other allowed weak decay modes, into a strange or a down
quark, are suppressed due to the small matrix elements |Vts | and |Vtd | (see equation
1.2). In figure 1.4 the dominant decay of the top quark is displayed. The W bosons
further decay into either a quark and an antiquark (q q̄ 0 ) or a charged lepton and its
corresponding neutrino (lνl ). These subsequent decays allow the categorisation of
top quark pair decays into three classes: full-hadronic, semi-leptonic and di-leptonic.
Firstly, the full-hadronic channel occurs if both W bosons decay into q q̄ 0 . Secondly, if
one W bosons decays into q q̄ 0 and the other into lνl , the decay is called semi-leptonic,
or lepton+jets channel. Thirdly, the case when both W bosons decay into lνl is called
di-leptonic decay channel. In principle the coupling of W bosons does not distinguish
between quarks and leptons. Still hadronic decays of the W boson are favored by
a factor of three compared to any leptonic decay, as there are three different color
states for quarks. This yields a two times higher branching ratio for hadronic decays
than for leptonic decays. In this thesis the measurement of the tt̄ charge asymmetry
is performed in the combined electron+jets and muon+jets channel which together
have a branching ratio of ≈ 29.6%. This combined lepton+jets channel represents
an optimum between clear signature and large branching ratio.
1.3. THE CHARGE ASYMMETRY OF TOP QUARK PAIR PRODUCTION 17
(a)
(b)
Figure 1.4: The top quark decay into a bottom quark and a W boson. The W
boson further decays either into a quark q and an antiquark q̄ 0 (a) or into a charged
lepton and the corresponding neutrino (b).
1.3
The Charge Asymmetry of Top Quark Pair
Production
Understanding the production mechanisms of the top quark is very important for
scrutinizing the Standard Model. One particular feature of the top quark pair
production is the charge asymmetry [4,5], which is the excess of top quarks compared
to antitop quarks in certain kinematic regions and vice versa. This effect arises, if
there are preferential directions for top and antitop quarks within the production
mechanism. Certainly only in asymmetric production mechanisms, which are mainly
the quark-antiquark annihilations, there is a preferential direction. The symmetric
gluon-gluon initial state can not lead to an asymmetric tt̄ final state. In quarkantiquark annihilations within the SM there is a slight preference that the top quark
is emitted into the direction of the incoming quark and the antitop quark into the
direction of the incoming antiquark. This phenomenon is explained via next-toleading order QCD corrections to top quark pair production. The NLO corrections
include the interference between initial state radiation (ISR) and final state radiation
(FSR) and between the leading order Feynman diagram and box diagrams. One
representative Feynman diagram for each named process is shown in figure 1.5.
Further a small contribution to the charge asymmetry is given by the interference
of electroweak quark-antiquark annihilation processes.
(a)
(b)
(c)
(d)
Figure 1.5: The interference of initial state radiation (a) and final state radiation
(b) as well as interference between box diagram (c) and leading order diagram (d)
leads to a charge asymmetry in top quark pair production within the SM. The charge
asymmetry is therefore a next-to-leading order effect. Only representative diagrams
for each process are displayed.
18
To measure the excess of top quarks versus antitop quarks one has to determine
the kinematics of particles in hadron collisions. This is done via rapidity and pseudorapidity after defining a direction z, which is usually the direction of one incoming
particle beam. The rapidity y of a particle is defined in the laboratory frame as
E + pz
1
,
(1.5)
y := ln
2
E − pz
where E is the particle’s energy and pz the momentum of the particle in z direction.
The pseudorapidity η is defined as
θ
(1.6)
η := − ln tan
2
for each particle. Herein θ is the angle between the z direction and the particle’s momentum vector. Rapidity and pseudorapidity are equal for massless particles. The
tt̄ charge asymmetry is measurable using these definitions, since η or y distributions
for top and antitop quarks are not equal. As the yielding tt̄ charge asymmetry
is dependent on the initial state particles of a hadron collider, different variables,
sensitive to the charge asymmetry, are required for LHC and Tevatron.
In quark-antiquark annihilations at the proton-antiproton collider Tevatron the
initial state quarks usually originate from the proton, whereas the antiquarks descend mostly from the antiproton. Hence both quarks are valence quarks in most
cases and carry on average a large momentum fraction of the hadron (see figure 1.2).
Considering top quark pair production, the preference of top quarks to be emitted
into the direction of the initial quark and the preference of antitop quarks to fly into
the direction of the initial antiquark, yield an excess of top quarks in direction of the
incoming proton and an excess of antitop quarks in direction of the incoming antiproton. This leads to a forward-backward asymmetry in the η and y distributions,
respectively. At Tevatron a sensitive variable for the forward-backward asymmetry
is simply the difference in rapidities of top and antitop quark
∆y := yt − yt̄ ,
(1.7)
where yt is the rapidity of the outgoing top and yt̄ of the outgoing antitop quark.
In that case the charge asymmetry ATCev is defined as
ATCev :=
N (∆y > 0) − N (∆y < 0)
.
N (∆y > 0) + N (∆y < 0)
(1.8)
Herein N (∆y > 0) and N (∆y < 0) are the number of events with positive and
negative values for ∆y. The Standard Model prediction for this variable yields a
charge asymmetry of ATCev,SM = 0.087 ± 0.010 [10].
At the proton-proton collider LHC the situation is different, since the initial state
is completely symmetric. Ergo η and y distributions of top and antitop quarks are
symmetric as well. Antiquarks only occur as sea quarks in proton-proton collision
and therefore carry on average a smaller momentum fraction of the proton than the
quarks. Therefore in top quark pair production the antitop quarks, preferentially
flying into the direction of the initial antiquarks, are emitted more centrally, whereas
the top quarks are emitted more likely into forward and backward directions, since
the initial quarks carry on average a larger momentum fraction of the proton. Accordingly the widths of the rapidity spectra of top and antitop quark are different.
The rapidity distributions of the top quarks are broader than the distributions of
the antitop quarks and therefore yield a central-peripheral asymmetry. To be sensitive to the central-peripheral asymmetry in this thesis two different variables are
introduced. On the one hand the difference in the absolute values of η is used
∆|η| := |ηt | − |ηt̄ | .
(1.9)
On the other hand, motivated in [12], ∆y 2 is defined as
∆y 2 := (yt − yt̄ ) · (yt + yt̄ ) = yt2 − yt̄2 .
(1.10)
Herein the Tevatron variable ∆y (first factor of the equation) is boosted into the
top-antitop quark rest frame (second factor) to be sensitive to the central-peripheral
asymmetry at LHC. In both cases the charge asymmetry is defined as
AC :=
N+ − N−
,
N+ + N−
(1.11)
where N + is the number of events with positive and N − those with negative ∆|η|
∆|η|
or ∆y 2 , respectively. The SM prediction for ∆|η| is AC = 0.0136 ± 0.0008 [10]
2
and for ∆y 2 it is A∆y
= 0.0115 ± 0.0006 [10], where the uncertainties arises from
C
scale variations and PDF uncertainties. These values are much smaller than the
SM predictions for Tevatron, because of the large fraction of gluon-gluon induced
top quark pair production, which is charge symmetric. The sketches in figure 1.6
illustrate the effects of the charge asymmetry in the η distributions for LHC and
Tevatron.
The measurement of the tt̄ charge asymmetry gains more and more interest,
because it could give an indication of new physics. Considering the open questions
within the Standard Model, it is clear that there are many approaches extending
the theory and trying to close the gaps. These theories beyond the Standard Model
(BSM) predict for example new types of exchange particles with different coupling
strengths. The top quark itself plays an important role in many BSM theories,
since the hypothetical new particles couple preferably to heavy quarks. According
to several BSM theories the top quark pair production is also possible via exchange
of yet unknown heavy particles, like axigluons [6, 57], leptophobic Z 0 bosons [58] or
Kaluza Klein excitations of gluons [59,60]. The charge asymmetry of top quark pair
production is sensitive to such new theories, since the BSM exchange particles have
different coupling strengths to top and antitop quarks, and measuring a deviation
from the SM value could give a strong hint for their existence.
The first unfolded measurement of the charge asymmetry in hadron collisions
was done by the CDF collaboration at the Tevatron in 2007 [61]. The reported
value of ATCev = 0.24 ± 0.14 for the charge asymmetry in ∆y arouse first interest,
despite its large uncertainty. Recent measurements at the Tevatron with more data
20
events
events
(a)
top quark
antitop quark
top quark
antitop quark
y
y
(b)
Figure 1.6: The charge asymmetry in the y distribution. In (a) the situation
in proton-antiproton collisions, like at the Tevatron, is shown. The preference of
the top quarks to fly into the direction of the incoming proton, and the antitop
quark into the direction of the antiproton leads to a forward-backward asymmetry.
In proton-proton collisions at LHC, the same behavior yields a central-peripheral
asymmetry in the y distribution (b).
and therefore smaller statistical uncertainties yield values for the charge asymmetry
of ATCev = 0.201 ± 0.067 (CDF [7]) and ATCev = 0.196 ± 0.065 (DØ [9]), which are
about 2 σ larger than the value predicted in the Standard Model of ATCev,SM ≈
0.06. Especially in kinematic regions, where top quark pairs have an invariant mass
higher than mtt̄ > 450 GeV/c2 , the CDF collaboration finds a deviation from the
SM prediction of 3.4 σ [8]. Such a large value might be a hint for physics beyond
the Standard Model. The most prominent BSM theory capable of explaining the
deviations of the Tevatron results is the model of chiral color [62, 63]. Within this
theory an extension of the Standard Model strong gauge field at high energies is
introduced, leading to a massive axigluon. In some chiral color models axigluons
have an opposite sign for the coupling to top quarks compared to the coupling to
all other quarks, that leads to a large positive value for the charge asymmetry.
In 2010 the first measurement of the charge asymmetry at LHC was done by
the CMS collaboration using the variable ∆|η| (see equation 1.9) yielding a value of
AC = 0.060 ± 0.14 [11, 64]. This result shows no deviation from the SM prediction
of ASM
= 0.013 ± 0.0011, due to the large uncertainty of the measurement. Table
C
1.3 shows the Standard Model prediction as well as the prediction of chiral-color
models for the two variables. The differences between predicted values for the charge
asymmetry from SM and from BSM theories are solely about 2% and therefore
hardly accessible. Still the precision of the measurement presented in this thesis is
sufficient to give a strong tendency whether the result is in good agreement with the
Standard Model predictions or not.
∆|η|
AC
SM
0.0136 ± 0.0008
Octet A 0.0294 ± 0.0016
Octet B 0.0334 ± 0.0018
2
A∆y
C
0.0115 ± 0.0006
0.0276 ± 0.0011
0.0316 ± 0.0011
Table 1.3: Predicted charge asymmetries at LHC for the Standard Model [10] and
for two different color-octet models [13, 59, 60]. The two color-octet models shown
in this table predict axigluons with an opposite sign for the coupling to top quarks
compared to the coupling to all other quarks. The mass for the axigluon in Octet
A is m = 2 TeV/c2 and in Octet B m = 1.8 TeV/c2 .
22
Chapter 2
Experimental Setup
The matter which influences our everyday life is built up only from particles of the
first generation of the Standard Model. On Earth the heavier particles from the
second or third generation are produced solely in high energetic collisions between
cosmic rays and molecules in the outermost layers of the atmosphere and decay
thereafter into their corresponding first generation partners. In order to produce
heavy particles artificially, to measure their properties and therefore to probe the
complete set of particles within the Standard Model large machines are required.
This so-called colliders accelerate particles to very high kinetic energies and let
them impact on other particles. Considering Einstein’s mass-energy equivalence,
E = m · c2 [65], new particles can be produced if the available energy of the collision,
the so-called center-of-mass energy, is larger than the particles’ rest masses. For the
production of the top quark, the heaviest known elementary particle, high energetic
collisions are required. To study these collision events special detectors are needed,
installed hermetically around the interaction point to identify the produced particles
and to determine their momentum and energy.
Over the past decades many colliders with steadily increasing center-of-mass energies have been constructed, a process culminating in the building of the Large
Hadron Collider (LHC) [66] at the European Organization for Nuclear Research
(CERN) center near Geneva, Switzerland. The LHC is currently the world’s most
energetic collider and
√ started its operation with proton-proton collisions at a centerof-mass energy of s = 7 TeV in March 2010. After a short period providing
collisions of lead ions with an energy of 287 TeV per beam in November and December 2010 the LHC restarted proton-proton collisions in 2011. In the next few years
the fast increasing amount of recorded data will show if the LHC succeeds in the
discovery of the Higgs boson, and gives answers to the open questions in particle
physics and hints for physics beyond the Standard Model.
The following sections give an overview of the main parts of CERN’s acceleration
chain and a detailed description of the different components of the CMS detector,
by which the collision data used in this thesis is recorded.
23
24
CHAPTER 2. EXPERIMENTAL SETUP
Figure 2.1: Schematic overview of the different parts within the accelerator chain at
CERN [67] (not to scale). The protons produced in the proton source are preaccelerated within the RFQ, the LINAC2, the PSB and the SPS. Thereafter the yielding
proton bunches are injected into the LHC main ring via transfer lines TI 2 and TI 8,
leading to two counter-rotating beams. The LHC main ring provides eight possible
collision points, Point 1 (P1) to Point 8 (P8) with particle detectors at four of them.
2.1
The Large Hadron Collider
The Large Hadron Collider has been installed in the already existing tunnel of the
former Large Electron Positron collider (LEP), which describes a 26.7 km long ring
between 45 m and 170 m below surface. The LHC main ring is separated into
eight straight sections and eight arcs assorted to octants depicted in figure 2.1. To
save space the two separate beam pipes within the ring share a common magnetic
and cooling system. In total, 1232 superconducting dipole magnets are installed
providing a magnetic field with a strength of up to 8.33 T to keep the counterrotating particle beams within the two beam pipes, whereas 392 quadrupole magnets
are needed for focusing. Each of the eight straight sections provides a possible
collision point, enumerated Point 1 (P1) to Point 8 (P8), where the two beams can
be crossed. However only at P1, P2, P5 and P8 the beams are forced to collide and
therefore four detectors are installed there in order to record the products of the
collision events. The ALICE (A Large Ion Collider Experiment) detector [68] at P2
is optimized to study collisions of heavy ions like lead nuclei, whereas LHCb [69] at
P8 focusses on rare decays of hadrons, which contain bottom or charm quarks. The
two multi-purpose detectors ATLAS (A Toroidal LHC Apparatus) [70] at P1 and
CMS (Compact Muon Solenoid) at P5 are generally aiming for the discovery of the
Higgs boson and searching evidence for physics beyond the Standard Model.
As mentioned before, the LHC is used mainly for proton-proton collisions. The
protons needed for this purpose are produced in a duoplasmatron source by interactions of high-energetic electrons yielding the ionization of hydrogen atoms. Before
2.1. THE LARGE HADRON COLLIDER
Acceleration stage
RFQ
LINAC2
PSB
PS
SPS
LHC
25
Final proton energy [GeV]
7.5 · 10−4
5.0 · 10−2
1.4
26
450
3500 (7000)
Table 2.1: Proton energies after the different stages of the accelerator chain depicted
in figure 2.1. The LHC main ring is designed to accelerate the proton√beams to an
energy up to 7 TeV yielding collision with a center-of-mass energy of s = 14 TeV.
However in the 2010 and 2011 proton operation at LHC the beams are accelerated
to an energy of 3.5 TeV.
entering the LHC main ring, the proton beams are preaccelerated in several steps.
In figure 2.1 the different parts of this acceleration chain together with the proton
source are shown schematically. After extracting the protons from the source by
applying a high voltage of 90 kV, the beam enters the Radio Frequency Quadrupole
(RFQ) [71] and is accelerated to an energy of 750 keV. As a consequence of the acceleration process using resonant microwave cavities, the protons are grouped into
bunches. With a length of 30 m the LINAC2 [72], a linear accelerator, increases
the energy of the proton bunches up to 50 MeV. The following acceleration steps
are accomplished by circular accelerators with increasing radii. After enhancing
the energy with the Proton Synchrotron Booster (PSB) [73] to 1.4 GeV the proton
bunches enter the Proton Synchrotron (PS) [74] with an circumference of 630 m
and gain energy up to 26 GeV. The Super Proton Synchrotron (SPS) [75], with a
circumference of 6.9 km, is the last preacceleration stage before the main ring and
accelerates the beam to an energy of 450 GeV. At the end of this stage the proton
bunches are divided into two beams, which are transfered via transfer lines TI 2 and
TI 8 in opposite directions into the two pipes in the LHC main ring. In table 2.1 the
different acceleration stages and the corresponding achieved proton bunch energies
are summarized.
The completely accelerated proton bunches with an energy up
√ to 3.5 TeV, are
then crossed, yielding collisions with a center-of-mass energy of s = 7 TeV. The
interaction rate Ṅp of a process p within these collisions is dependent on the cross
section σp of the process and the instantaneous luminosity L of the collider
Ṅp = σp · L .
(2.1)
Herein the luminosity L of two colliding bunches a and b is defined as
L=f·
Na Nb
,
4πσx σy
(2.2)
where Na and Nb denote the numbers of particles per bunch, σx and σy are the
transverse sizes of the gaussian shape bunches at the interaction point and f is the
26
(a)
(b)
Figure 2.2: Luminosity profile in the 2011 proton operation at the LHC until 18th
October during stable beams at 7 TeV center-of-mass energy [76]. On the left hand
side, the maximum instantaneous luminosity per day delivered to CMS is shown.
On the right hand side, the integrated luminosity versus time is displayed. The red
curve denotes the integrated luminosity delivered by the LHC and the blue curve
indicates the integrated luminosity successfully recorded by CMS. In this thesis the
dataset recorded until 28th June is used, denoted by the green lines.
beam revolution frequency. Equation 2.2 is a simplification assuming the area of
beams as well-defined and not taking the crossing angle of the beams into account.
In figure 2.2(a) the peak luminosity per day in the 2011 proton operation is displayed. Further, as a measure for the total amount
of collision data delivered to the
R
experiments, the integrated luminosity Lint = L dt versus time handed over by the
LHC in 2011 and recorded by the CMS experiment is shown in figure 2.2(b). In this
thesis the dataset recorded by CMS until 28th June is used.
2.2
The Compact Muon Solenoid Detector
The Compact Muon Solenoid (CMS) detector [77, 78] is located in a cavern at the
LHC Point 5 about 100 m below surface. Since the CMS apparatus is a typical
multi-purpose detector, it is built as hermetically as possible, being able to detect
all particles produced in a collision event. The main advantages of the CMS detector
are the high momentum resolution and reconstruction efficiency for charged particles
and the good muon identification over a large range of momenta and angles.
With a diameter of 14.6 m, a length of 21.6 m and a weight of about 12500 t the
CMS experiment is more compact and twice as heavy as the other multi-purpose
experiment ATLAS. The typical onion-like structure of CMS environing the beam
pipe is shown in figure 2.3. The beam pipe is surrounded consecutively by the
tracking system, the electromagnetic calorimeter (ECAL), the hadronic calorimeter
(HCAL), the superconducting solenoid and the muon system, which is embedded in
the iron return yoke. The solenoid provides a 3.8 T magnet field, which bends the
trajectories of charges particles traversing the detector system. The combination
2.2. THE COMPACT MUON SOLENOID DETECTOR
27
superconducting solenoid
barrel segments
end cap
beam pipe
tracking system
electromagnetic calorimeter
hadronic calorimeter
muon system
return yoke
Figure 2.3: Schematic overview of the CMS apparatus and its sub-detectors adapted
from [67, 79]. The typical onion-like structure of the collider detector around the
beam pipe in the center is shown. The beam pipe is environed by the tracking
system, the electromagnetic and hadronic calorimeters, the superconducting solenoid
and the muon system, which is embedded in the iron return yoke. CMS is built up
by several barrel segments and two end cap discs.
of the signals and information of all these layers of the CMS detector allows the
reconstruction of almost all final state particles. Of course the subdetector system
with its complexity needs ongoing mechanical support. Therefore the CMS detector
was built in 15 sections over ground, which were singly lowered into the cavern and
then assembled. This modular structure provides the possibility of disassembling
for maintenance and inspection.
To provide a unique mathematical description, the CMS detector is described by
a right-handed coordinate system centered at the nominal interaction point. The x
axis is pointing to the center of the LHC main ring and the y axis is directed to the
sky. Therefore the z axis points counter-clock wise along the main ring. In collider
physics it is common to use the z coordinate together with the azimuthal angle φ
and the pseudorapidity η = − ln(tan θ/2) (see equation 1.6) dependent on the polar
angle θ.
2.2.1
Tracking System
The innermost layer of the CMS detector is the CMS tracker [80, 81] with a length
of 5.8 m and a diameter of 2.5 m. The aim of the tracking system is to measure
the trajectories of charged particles, which are bent by the magnetic field of the
28
solenoid, with a high precision in order to derive a particle’s momentum as well as
the sign of its electric charge from the curvature of the track. In addition, the hits
in the tracker system are used to reconstruct the interaction vertices of an event.
At the LHC design luminosity of Ldesign = 1034 /(cm2 s) an expected number of 1000
particles per bunch crossing traverses the tracking system. Therefore the tracking
system requires high granularity and fast response time on the one hand, but also
needs to withstand the severe radiation damage caused by this high particle flux.
For these reasons semiconducting silicon detectors have been chosen, in which the
electric signal from electron-hole pairs, produced by traversing charged particles, are
read out from radiation-hard sensors.
The whole CMS tracker is embedded in a support-tube which is cooled down to
−20 ◦ C to ensure the detector’s working temperature. Figure 2.4 depicts the layout
of the tracking system, which can be separated into silicon pixel detectors in the
innermost part and silicon strip detectors in the outer regions. The tracking system
covers the pseudorapidity range |η| < 2.5 and is equipped with 1440 pixel detector
modules and over 15000 strip detector modules leading to an active area of 200 m2 .
Silicon Pixel Detector
The silicon pixel tracker with an active area of 1 m2 consists of three barrel layers
with a total length of 53 cm and two end cap disks. A total amount of 66 million
pixels are installed, each with a size of 100 × 150 µm2 yielding up to three high
precision space points for each charged particle with a resolution in the r − φ plane
of 10 µm and in z direction a resolution of 15 µm.
Silicon Strip Detector
The silicon strip detector system which environs the pixel tracker is partitioned into
the barrel tracking device and two end cap discs on each side. The barrel part is
separated into Tracker Inner Barrel (TIB) with four layers of silicon sensors and
Tracker Outer Barrel (TOB) with six layers providing up to ten r − φ measurements
for each charged particle with a single point resolution between 30 µm and 50 µm.
The end cap discs are divided into Tracker Inner Disc (TID) with three disk layers
and Tracker End Caps (TEC) with nine disk layers delivering up to twelve z − φ
measurement with a resolution of 30 µm.
There are also stereo modules installed in the barrel and endcap parts. Stereo
modules consist of two strips back-to-back slightly spread to each other enabling the
measurement of the missing coordinate, which is z in the barrel and r in the endcap
region. The achieved resolution in the additional coordinate is between 230µm and
530µm.
2.2.2
Calorimetry System
The second layer within the CMS detector is the calometry system, whose aim is to
completely absorb almost all types of particles in order to measure their energies.
29
Figure 2.4: Schematic longitudinal overview of the CMS tracking system with the
coordinates z, r and η [77]. The interaction point (black dot) is embedded in the
pixel detector and the strip detectors, partitioned into the barrel tracking device
and two end cap discs on each side. The barrel part is separated into Tracker Inner
Barrel (TIB) and Tracker Outer Barrel (TOB), whereas the end cap discs are divided
into Tracker Inner Disc (TID) and Tracker End Caps (TEC). Each line indicates
one detector module and each double line one stereo module.
Hermetically enclosing the CMS tracking system, the calorimetry system is represented by the Electromagnetic Calorimeter (ECAL) [82, 83] in the inner part and
the Hadronic Calorimeter (HCAL) [84] in the outer region, schematically illustrated
in figure 2.5. In principle the ECAL measures the energy of electrons, positrons and
photons, whereas the HCAL determines the energy of charged and neutral hadrons.
The precise measurement of the total energy of a collision event is also important
to derive the missing energy, which can be allocated to the neutrinos produced in
the collision, since they only carry weak charge and marginally interacting with the
detector material.
When electrons and positrons with the initial energy E0 pass an absorber material of length x their energy is given by
E(x) = E0 · e
− Xx
0
.
(2.3)
Herein X0 is the radiation length, which is the distance a particle has passed when
its energy is reduced to E0 /e. By analogy the mean free path length of hadrons in
matter is given by the hadronic interaction length λI . In the calorimetry system
the length of the absorber material is given in units of X0 and λI , respectively. Alternating samples of absorber material, decelerating the particles gradually to zero
energy, and scintillator material allow to reconstruct the energy of one traversing
particle. In the scintillator material photons emitted by a traversing particle can
be translated into an electric signal and therefore one can determine the particle’s
energy. This sampling structure is used in the HCAL. However, in the ECAL a homogeneous construction is used, where only one material is needed, which represents
absorber and scintillator simultaneously, leading to a better energy resolution.
30
Muon Chambers
Muon Chambers
HO
Superconducting Solenoid
HB
HE
EB
HF
EE
Figure 2.5: Longitudinal view of the calorimeters and the muon system in one
quarter of the CMS detector [78]. The tracker system shown in 2.4 is surrounded
by the Electromagnetic Barrel (EB) and the Hadronic Barrel (HB), and enclosed
from the Electromagnetic End caps (EE) and the Hadronic End caps (HE) on each
side. The superconducting solenoid, the Hadronic Outer calorimeter (HO) and the
muon system, which is embedded in the iron return yoke, complete the onion-like
structure. Close to the beam pipe the Hadronic Forward calorimeter (HF) is located.
The fine dashed lines show the values of the pseudorapidity.
Electromagnetic Calorimeter
The homogeneous buildup of the electromagnetic calorimeter is accomplished with
lead tungstate (PbWO4 ). The ECAL can be separated in the electromagnetic barrel
(EB), consisting of 61200 lead tungstate crystals, and the electromagnetic endcaps
(EE) with 7324 crystals. Lead tungstate as the absorber material has a high density
of 8.3 g/cm3 leading to a short radiation length of X0 = 0.89 cm and still is transparent for visible light. The photons emitted by decelerated electrons and positrons are
collected by avalanche photodiodes in the EB and by vacuum phototriodes in the
EE, which are assembled on each crystal. Around 80% of the photons are emitted
within the bunch crossing time of 25 ns giving the ECAL the feasibility of very fast
energy measurements.
The energy resolution (σECAL /E)2 of the ECAL is given by
σ
ECAL
E
2
=
S
√
E
2
+
N
E
2
+ C2
(2.4)
√
where S represents the stochastic term scaling like E, N is the noise term with a
scaling of E and C is a constant term. S, N and C√were determined via measurements with electron test beams, resulting S = 0.028 GeV, N = 4.1 · 10−3 GeV and
C = 0.003 [85].
31
Hadronic Calorimeter
There are four main parts within the HCAL: Hadronic Barrel (HB), Hadronic Endcap (HE), Hadronic Outer (HO) and Hadronic Forward (HF) shown in figure 2.5.
The absorber material of the HCAL is non-magnetic brass, which has a density of
ρ = 8.53 g/cm3 and an interaction length of λI = 16.42 cm, whereas the detector
material is made of plastic scintillator tiles. The hadronic barrel, covering the region with |η| < 1.3, has an effective thickness of 5.82 · λI in y direction, whereas
the hadronic endcap encloses the range of 1.3 < |η| < 3.0 and provides a thickness
of roughly 10 · λI . Both, HB and HE, are enclosed by the magnetic coil with an
inner radius of r = 2.95 m. Since not all hadrons can be stopped by EB and HB in
the central pseudorapidity region, the HO is installed on the outside of the solenoid
using the magnetic coil as an additional absorber layer. In forward direction the HF
covers the area with 3 < |η| < 5.3.
Similar to the ECAL the energy resolution (σHCAL /E)2 of the HCAL is given by
σ
HCAL
E
2
=
S
√
E
2
+ C2
(2.5)
√
E and C is a constant term.
where S represents the stochastic term scaling like √
Test beam measurements yield the
values
S
=
1.98
GeV and C = 0.09 for the
√
hadronic forward and S = 0.847 GeV and C = 0.074 for HE, HB and HO [86].
2.2.3
Muon System
Muons are minimum ionizing particles which lose only little energy due to bremsstrahlung, because of their large mass compared to electrons. Therefore muons are
the only particles, neglecting neutrinos, which can not be brought to halt in the
calometry system. Detecting the muons and measuring their kinematics with high
precision are major goals of the CMS experiment. Hence, the outermost detector
layer of the CMS apparatus is the muon system [87] with three different types of
detectors integrated into the magnetic return yoke with a magnetic field of 2 T. In
figure 2.6 a schematic overview of the barrel part and the endcap region of the muon
system is depicted.
The 250 aluminium Drift Tubes (DT) chambers, located in the barrel region
with |η| < 1.2, are arranged in four layers. Four layers of Cathode Strip Chambers
(CSC) are used in the endcap modules covering the area with 0.9 < |η| < 2.4. In
the region with |η| < 1.6 the muon rate is very high and therefore Resistive Plate
Chambers (RPC) are installed in barrel and endcap parts to provide fast trigger
information in this area.
2.2.4
Trigger System and Computing Structure
The high bunch crossing frequency of 40 MHz at the LHC design luminosity leads
to a huge amount of data which would be impossible to store. In order to reduce the
data rate, two trigger systems are used at a very early stage, selecting only events
32
DT
η=0.8
MB4
1.0
RPC
1.2
MB3
MB2
1.6
MB1
2.1
߬
2.4
ME2
ME1
ME3
ME4
CSC
Figure 2.6: Longitudinal view of the muon system in one quarter of the CMS
detector [77]. The four barrel muon stations, MB 1 to MB 4, are equipped with
Drift Tubes (DT) and Resistive Plate Chambers (RPC), whereas Cathode Strip
Chambers (CSC) and RPCs are installed in the four endcap muon stations, ME 1
to ME 4.
with an interesting physics content. In the first step the Level-1 (L1) trigger [88]
implemented in hardware is applied, reducing the data rate to 0.1 MHz. Only events
passing the L1 trigger requirements are forwarded to the next stage, which is the
so-called High-Level Trigger (HLT) [89]. In the HLT system, implemented in a
software system running on roughly thousand processors, the events are reduced to
the feasible data rate of roughly 100 Hz, and are written to mass storage thereafter.
The network of L1 trigger, HLT and electronic devices constitutes the CMS Data
AcQuisition (DAQ) system, which is depicted in figure 2.7.
To provide CMS members from all over the world with the data recorded from the
detector, the CMS collaboration adapted the hierarchical LHC computing grid [90],
which is organized in layers with one Tier-0 site at CERN, several Tier-1 sites and
many Tier-2 and Tier-3 sites. In figure 2.8 the architecture of the CMS computing
grid is shown. The so-called RAW datasets, stored at the Tier-0 site, contain the
unworked detector information and therefore need large storage. As a backup the
RAW datasets are also transferred to at least one Tier-1 site. After a first calibration
and reconstruction step, the yielding RECO datasets are transferred to the Tier-1
centers, which are at national computing facilities like the Karlsruhe Institute of
Technology in Germany. At Tier-1 centers also the simulated RAW datasets, socalled Monte Carlo (MC) datasets, see section 3.1, are located. The Analysis Object
Data (AOD) is a subset of the RAW data, containing sufficient information for most
analyses. Copies of the AOD datasets, extracted at Tier-1 sites, are transferred to
the numerous Tier-2 and Tier-3 sites, which provide CPU and storage resources for
each individual analysis.
40 MHz
Level-1
Trigger
33
Detector Front-Ends
Readout
Systems
100 kHz
Event
Manager
Builder Network 100 GB/s
Control
and
Monitor
Filter
Systems
100 Hz
Computing Services
Figure 2.7: Architecture of the CMS data acquisition system [78]. The two stages
of event rate reduction from 40 MHz to 100 Hz managed by the control and monitor
system are shown. In the first stage only events passing the L1 Trigger requirements
are handed over to the readout systems, reducing the data rate to 105 Hz. In the
event builder the information is combined into a single event and forwarded to the
HLT filter system, where reconstruction algorithms decide whether to store the event
or not.
Figure 2.8: Architecture of the CMS computing grid, adapted from [78]. At the
single Tier-0 site at CERN the RAW data format from CMS is stored. The RAW
datasets and the RECO datasets are transferred to the Tier-1 sites. At the Tier-2
and Tier-3 centers providing CPU and storage resources for the analysts only the
AOD datasets are stored.
34
Chapter 3
Generation, Simulation and
Reconstruction
In every collision event at LHC typically hundreds of secondary particles from the
hard interaction process are produced yielding uncountable electric signals in the
subdetectors of CMS. To unravel this clutter dedicated algorithms are required,
which translate the raw detector information into physical objects like electrons or
muons. In order to identify the physical objects and to reconstruct their trajectories,
so-called tracks, the reconstruction algorithms link the energy deposits within the
CMS subdetectors and apply several fitting methods.
Although the secondary particles from the hard scattering event are reconstructed with high precision at CMS, it is not possible to directly compare them
to the underlying theory, the Standard Model. In order to gain a reasonable comparison of the observed data and theory predictions from the SM, simulations of
such collision events according to the theory are performed. The simulation is done
via so-called Monte Carlo (MC) methods which are based on randomly generated
processes according to the probability of such a process. A simulation of the CMS
detector is used to model the detector responses caused by the generated events. A
common reconstruction procedure for observed data and for MC events allows the
confrontation of both in order to provide information of the Standard Model or to
give hints for new physics.
The following sections give an overview of the generation process of simulated
events and the different kinds of Monte Carlo generators. Further the reconstruction
procedure, which is applied to the observed data as well as to Monte Carlo events,
is described in detail.
3.1
Generation of Events
The MC event generators produce simulated proton-proton collision events based on
a factorization of the process. In the first step the initial hard scattering is simulated
yielding typically only a few final state particles. Thereafter the radiation of gluons
in the initial and final state as well as the formation of color-neutral hadrons, the socalled hadronization, is generated. In the last step, the decays of unstable particles
35
36
CHAPTER 3. GENERATION, SIMULATION AND RECONSTRUCTION
Figure 3.1: Schematic overview of the different steps of Monte Carlo event generation processes adapted from [91]. The interaction of partons within the protons are
simulated in the hard scattering process. The parton shower stage generates additional soft radiations from the initial and final state partons. In the hadronization
stage color-neutral particles are formed. The decay of unstable particles is simulated
in the last event generation step.
are simulated. The different steps of the MC event generation, which are described
in the following, are illustrated in figure 3.1.
Hard Scattering Process
The hard scattering is a generation step based on perturbative calculation. The
so-called Matrix Element (ME) method uses the Feynman diagrams of a certain
process and the corresponding transition amplitude to calculate the probability of
such a process to arise. Since the strong coupling constant αS is small at large
Q2 scales, the ME method is the most accurate way to simulate the initial protonproton collision, where one parton of each proton takes part in the hard interaction.
The partons of the colliding protons are characterized by a set of PDFs at a certain
energy scale Q2 . An example for a PDF is shown in figure 1.2. At the end of this
step only a small amount of particles occurs in the final state. In the case that
top quarks or W bosons are generated in the initial hard scattering their decay is
simulated at this early stage and their decay products are already included in the
final state.
3.1. GENERATION OF EVENTS
37
Parton Shower
Within the second step of MC event generation the possible radiation of accelerated
color charges, which are called initial state radiation (ISR) and final state radiation
(FSR), are simulated. Examples for ISR and FSR are shown in figure 1.5(a) and (b).
There are two different approaches for the generation of the radiations considering
the determination of the probability of such processes.
On the one hand the Matrix Element method calculates similar to the hard
scattering step the probability of radiation based on perturbation theory. This perturbative calculation is only valid for small values for αS and therefore the ME
method is limited to interactions with high momentum transfer. On the other hand
the Parton Shower (PS) approach uses simplified models for the kinematics of the interactions. A branching of the partons is applied which is described via the DGLAP
QCD evolution equations [92–94]. Therein the probability of a gluon radiation is
calculated via the Altarelli-Parisi splitting functions. The ISR is approximated as
space-like showers within the PS wherein the scale increases backwards in time until
the initial parton is found. As opposed to this the FSR is approximated by timelike showers starting from the final state partons in the hard process. Sequences of
decreasing Q2 scales are used along the positive time axis. A lower cut-off scale is
required to avoid singularities which arise from soft and collinear gluon emission.
Hadronization and Decay
At low energy scales and large distances between the particles the calculations based
on perturbative theory become invalid since the value of the strong coupling αS of
two particles gets too large due to the color confinement. Therefore the next step
within the generation requires purely phenomenological models. The parameters
within these models, explaining the formation of color-neutral bound states, are
tuned to data. One prominent model used for the generation of the hadronization is
the Lund string model [95]. Herein the strong coupling of two particles is described
by color-flux string tubes tensed between them. Color-singlet partons are converted
into color-neutral via iterative break-ups of the color-flux string tubes, where each
break-up leads to a new q q̄ pair. This process continues until the energy of the
strings is too low to create new pairs. At the end of the hadronization stage the
decay of unstable baryons and mesons is simulated randomly according to the known
branching ratios of the hadrons.
Underlying Event and Pile-Up
For a complete simulation of the events not only the initial hard scattering process
has to be considered but also the contributions from other interactions that occur in
proton-proton collisions at LHC. Such contribution arises from the hadronization of
the color-charged remnants of the protons leading to additional soft radiation in the
final state. Further the possibility of multiple interactions of partons have to be considered. These two effects are referred to as underlying event. Moreover, additional
proton-proton interactions in the same bunch crossing, so-called pile-up events, ap-
38
pend typically soft hadrons to the final state. In order to correctly simulate the
contribution of underlying events dedicated algorithms, based on phenomenological
models which are tuned to data, are required. The influence of pile-up events is
simulated using MC event generators.
3.1.1
Monte Carlo Generators
Considering the calculations of the successive simulation steps in the event generation there are different kinds of Monte Carlo generators available. Parton shower
and hadronization event generators like Pythia [96] and Herwig [97] provide a full
event generation including the modelling of the hadronization and the decay. As opposed to this matrix element generators like MadGraph/MadEvent [98–100] or
Alpgen [101] provide simulations with a perturbative calculation of ISR and FSR
in LO matrix elements, but no simulation of the hadronization step. To obtain a full
simulation the matrix element generators have to be interfaced with the generators
of the first category, using dedicated matching techniques, i.e. CKKW [102, 103]
and MLM [104]. Another class of matrix element generators providing NLO calculations for the hard scattering process is represented by Powheg [105–107] and
MC@NLO [108,109]. The individual MC generators used in this thesis are described
in more detail.
Pythia
The Pythia package is a powerful program to simulate particle collisions at high
energies. It provides full event generation including many models beyond the Standard Model for the hard interaction, as well as many technical implementations
for soft contributions. The parton distributions required for the hard scattering
simulation are already included. The ISR and FSR are generated using the PS approach and the hadronization modelling uses the Lund string tubes. Additionally
the decays of unstable hadrons are also simulated. The Pythia package also includes contributions from underlying events. These different parts of the simulation
from Pythia can be applied to hard scattering events simulated by external matrix
element generators like MadGraph/MadEvent.
MadGraph and MadEvent
The combination of the matrix element generator MadGraph and the tree level
generator MadEvent provides the simulation of various kinds of hard interaction
processes. Herein MadGraph determines the amplitude for a given process, including all subprocesses and passes the outcome to MadEvent, which generates
the Monte Carlo events according to the amplitudes of the subprocesses. The parton
showering and hadronization simulation of Pythia can be applied to the events generated from MadGraph/MadEvent which are stored in the Les Houches event
format [110]. MadGraph/MadEvent are also used for simulations beyond the
theory of the Standard Model. Hypothetical exchange particles can be included
easily.
3.2. RECONSTRUCTION OF EVENTS
39
Tauola
The Tauola [111] program package provides a precise simulation of τ -lepton decays.
It is interfaced with the MC event generators in order to correctly consider the effects
of τ spin correlations. For the majority of the MC events used in this thesis the
Tauola package has been applied.
3.1.2
Detector Simulation
In order to compare the generated events with the observed data, it is necessary to
simulate the interactions of the MC events with the CMS detector material. There
are two approaches for the simulation of the CMS detector. On the one hand a full
detector simulation is obtained using the Geant4 toolkit [112]. Geant4 contains
a detailed description of the CMS experiment and all its subdetectors and simulates
with high precision the trajectories of particles which traverse the detector. Herein
Geant4 takes into account all physical effects like bremsstrahlung and hadronic
and electromagnetic showering processes and yields a true copy of electric responses
of the tracker and of the calorimetry system, as well as the responses in the muon
chambers.
On the other hand a fast simulation [113] of the CMS detector is used, when
large amount of simulated events are needed since the full simulation requires much
computing time. The fast simulation uses simplified models of the interactions of
particles with the detector material and of the CMS detector itself but still delivers
competitive results compared to the full simulation. The required computing time
for fast simulation is almost 100 times smaller compared to the full simulation.
3.2
Reconstruction of Events
In order to reconstruct the underlying physical objects, the raw electric signals of the
CMS detector have to be interpreted using dedicated algorithms. The reconstruction
algorithms within the CMS SoftWare (CMSSW) provide successive steps aiming at
obtaining all particles which arose from the hard scattering event. In the first step
the tracks of charged particles are reconstructed by fitting trajectories to the energy
deposits in the tracking system. These tracks are used to trace back the interaction
vertices of the event. Further the tracks are linked to the calorimeter deposits and
the hits in the muon chambers to receive all physical objects of the event. The
reconstruction algorithms described in the following sections are applied on the
observed data and the simulated Monte Carlo events.
3.2.1
Reconstruction of Tracks
Charged particles produced in the hard scattering are effected by the magnetic
field and traverse the tracking system on bent trajectories. They generate energy
deposits, so-called hits, in the different layers of the tracking system of CMS. For the
reconstruction of the trajectories the Combined Track Finder (CTF) [114] is used,
40
exerting the Kalman Filter (KF) track finder [115] in several successive steps. The
KF uses local fits to estimate the parameters of the trajectories. In the first step
hits in the inner pixel detector are used as seeds for a trajectory candidate. A seed
is defined by a hit triplet or by two hits and one beam spot constraint. Thereafter
the KF subsequently runs from layer to layer and constructs the tracks, taking into
account the motion of charged particles through the magnetic field, the energy-loss
and multiple scattering. Each hit matching the extrapolation is used as a seed for a
new track candidate. If there is no hit in one layer the track will be extrapolated to
the next layer since charged particles do not necessarily generate hits in all layers.
If there are no energy deposits in two consecutive layers within the extrapolation,
the trajectory candidate is rejected. The whole set of trajectory candidates is fitted
to the hits in parallel until the outermost layer of the tracking system is reached.
In the next step the track candidates are compared to each other in order to avoid
overlaps. If two tracks have more than half of their hits in common, the trajectory
with the lower amount of hits is rejected or in the case that both of them have the
same amount of hits the algorithm rejects the candidate with the higher χ2 value
of the fit. After the local fit processes of the KF, a global fit is applied in order to
improve the precision of the track parameters.
3.2.2
Reconstruction of Vertices
The reconstruction of the interaction vertices is important to determine the position
of the initial hard scattering. Moreover, the number of found primary vertices is
essential to estimate the number of pile-up collisions in the event. In principle
the reconstruction of a primary vertex (PV) is based on finding a common origin
of several reconstructed tracks. In the first step the tracks are grouped to vertex
candidates. In the second step the clustered tracks are re-fitted assuming they arise
from a common vertex. The fit can be done using different approaches. In the
high level trigger approach for example, which is required to decide fast whether
an event contains an interesting physics content, the search for PV is reduced to a
one-dimensional problem taking only the z position of the PV into account. The
quality of a vertex is given by the χ2 value of the fit and its compatibility with the
beam spot. The vertex with highest sum of p2T of the associated tracks is tagged
as primary vertex. Higher accuracy for the PV is given when only reconstructed
tracks passing certain quality criteria are used for the vertex fit. Such criteria for
the tracks demand typically at least two hits in the pixel detector, seven hits in the
strip detector and a χ2 divided by the number of degrees of freedom below 5.0.
3.2.3
Reconstruction of Electron Candidates
The typical signature of an electron traversing the CMS detector is used to reconstruct the electron candidates in a collision event. Obviously electrons are charged
particles that generate hits in the inner tracker system and leave energy deposits in
the ECAL. Further, electrons lose a large amount of energy due to bremsstrahlung
because of the large electric-charge-over-mass-ratio. The bremsstrahlung photons
41
are emitted along the electrons’ helical direction yielding energy deposits in ECAL
along the φ direction. The reconstruction is in principle based on combining this
information [116]. Firstly, clusters of energy deposits in the ECAL are formed into
superclusters taking into account the energy spread. Superclusters with an energy
larger than 1 GeV are used as starting point to find seeds in the pixel detector.
The track seeds of electrons require at least two hits in the pixel detector. To fit
the tracks for electron candidates a nonlinear Gaussian sum filter [117, 118] is used,
since the bremsstrahlung leads to nonlinear effects in the propagation of electrons.
The resulting tracks which match a supercluster with the requirements of |η| < 2.5
and pT ≥ 5 GeV/c are tagged as primary electron candidates.
After this reconstruction of electron candidates additional requirements can be
applied in order to avoid misidentifying charged hadrons or other objects with a
similar signature [119]. Typically, the used variables are sensitive to the bremsstrahlung and the electromagnetic showering in the tracking system.
3.2.4
Reconstruction of Muon Candidates
Muons traversing the CMS detector generate hits in the inner tracker system and
reach the muon system. The reconstruction of muons in CMS [77,120] contains three
different classes, i.e. stand-alone muons, tracker muons and global muons. Standalone muons are reconstructed with information of the muon system only, whereas
for tracker muons the standard trajectory reconstruction is applied additionally
using the hits in the muon chambers. The reconstruction of global muons is the
most accurate way, since it combines the information of both systems.
For stand-alone muons the hits in the DT, CSC and RPC are used to reconstruct
the trajectories similar to the track reconstruction process explained above. The
seeds obtained from the hits are extrapolated from layer to layer using the KF
method. The propagation takes into account multiple scattering, energy loss and
the non-uniform magnetic field in the muon system. If one layer provides no hit
the track is extrapolated to the next layer. When reaching the outermost layer a
KF is used backwards and a vertex-constrained fit is applied to estimate the track
parameters.
The advantage of tracker muons lies in the possibility of also reconstructing
muons with a low transverse momentum, since these generate only a small amount of
hits in the muon chambers. The reconstruction starts with the trajectories obtained
from tracker information. These tracks are extrapolated outwards also considering
the magnetic field and the energy loss and thereafter are matched to the hits in the
muon chambers.
The reconstruction of global muons starts with the track of a stand-alone muon.
The track is extrapolated inwards to the tracking system and combined with possible
tracks in the tracking system. In order to avoid a large amount of combinations
only a subset of tracks in the tracker is used, which matches the momentum and
the position of the extrapolated stand-alone muon. Thereafter a global fit to the
combinations of tracker trajectories and the track of the stand-alone muon is applied
and the combination with the lowest χ2 value is tagged as global muon.
42
Calorimeter
Jets
Observable
stable particles
Particle
Jets
hadronisation
Theory
and
Modelling
Partons
Figure 3.2: Evolution of a jet, adapted from [67]. The jet reconstruction algorithm
uses the energy deposits in the calometry system, indicated as light blue ellipses
from the ECAL and dark blue ellipses from the HCAL, as input.
In order to avoid misidentifying high energetic hadrons as muons and to assure
that the identified muons arose from the hard interaction process, additional requirements are applied on the reconstructed muons. Hadrons with a high transverse
energy can be misidentified as muons, since they can also reach the muon system.
Such objects are called punch-through hadrons and provide activity in the ECAL
and the HCAL. Therefore the sum of energy deposits in a specific cone around the
muon track, additional information from the tracker, the χ2 value of the fit and the
number of invalid hits are used as discriminator.
3.2.5
Reconstruction of Jets
Since color-charged quarks and gluons obey the QCD confinement, they can not
be observed directly. Hence, a radiated gluon or quark hadronizes several times
causing a narrow cone of hadrons in the detector, a so-called jet. An illustration of
the evolution of a jet is given in figure 3.2. These jets have to be reconstructed with
dedicated algorithms in order to gain information on the initial particle.
The aim of jet reconstruction algorithms, so-called clustering algorithms, is to
identify the initial particle that induced the jet and to determine its four-vector as
precisely as possible. For this purpose they can use all kind of physical objects as
input for the clustering.
There are two important requirements on the algorithms. On the one hand
43
additional soft radiations should not change the number of jets, a requirement called
infrared safety. This is important, because at high energetic proton-proton collisions
a large amount of soft radiations arise from underlying events and soft gluon emission
in the hadron shower or calorimeter noise. On the other hand the splitting of one
particle into two particles sharing the momentum of the initial particle should not
change the jet outcome. This requirement is called collinear safety and is also
essential since gluon splitting occurs very often in shower processes.
Clustering Algorithms
In order to reconstruct a jet the four-momenta of all kinds of physical objects can
be clustered using dedicated algorithms. There are two popular approaches for jet
clustering, i.e. cone-type algorithms or sequential recombination algorithms. Conetype algorithms, like the Iterative CONE (ICONE) algorithm [77] or the Seedless
Infrared Safe Cone (SISCone) algorithm [121], cluster all objects in a given cone
with the radius R in the η − φ plane. The ICONE algorithm builds a so-called
proto-jet from the object with highest transverse energy ET , which is given by
ET = E · sin θ ,
(3.1)
where E is the energy of the object and θ defines the azimuthal angle of its momentum vector. The proto-jet is iteratively used as seed for the next proto-jet until a
certain criteria is reached, e.g. a maximum number of iterations. Within this algorithm the collinear safety is not given since only seeds are allowed with an energy
over a determined threshold. When applying no requirements at all for the seeds,
the time for testing all subsets of cones would be too long. To solve this problem
the Seedless Infrared Safe Cone (SISCone) algorithm uses an advanced non-iterative
procedure which finds all stable cones in a feasible amount of time.
Sequential clustering algorithms reconstruct jets without fixing their geometrical
shape. Within the reconstruction process these objects are sequentially clustered
using the distances di,j between two objects i and j, given by
∆2i,j
2n
.
di,j = min p2n
,
p
T,i T,j ·
D2
(3.2)
Herein ∆2i,j is the distance of the two objects in the η − φ plane and n indicates a
free parameter for different approaches to the distance calculation. The resolution
parameter D determines the size of the jet. The algorithms calculate all possible
distances di,j . Further, the distances to the beam dk,beam of the single objects k,
defined as
dk,beam = p2n
(3.3)
T,k
are calculated. In every iteration the objects i and j with the minimal distance di,j
are appointed. These two objects are merged into a new object and the objects i and
j are removed from the event thereafter. This sequential clustering is interrupted if
the found minimal distance is larger than the smallest distance dk,beam . In this case
the object k is tagged as a completed jet. If all objects in the event are tagged as
44
jet the algorithm stops. Within the sequential clustering algorithms there are three
common approaches which only differ in the choice of the parameter n in equations
3.2 and 3.3 and therefore in the calculation of the distance between two objects. The
kT algorithm uses n = 1, the Cambridge Aachen algorithm n = 0 and the anti-kT
algorithm n = −1. All sequential clustering algorithms are infrared and collinear
safe and additionally require less computing time than the SISCone approach.
Considering the different input objects for the clustering algorithms one has to
distinguish between several types of reconstructed jets. Using the generated stable
particles after the hadronization stage of MC event simulation, generator jets are
build, which are often used as reference for resolution comparisons. Calorimeter
jets, called CaloJets, are reconstructed using the information of calorimeter towers,
which are the unweighted sum of energy deposits in ECAL and HCAL. The CaloJet
approach has a large disadvantage, because the granularity in the ECAL is about
25 times more precise than the one in the HCAL. When using the unweighted sum
of both, the additional precision, gained by a finer granularity of the ECAL, is
lost. The Jet Plus Tracks (JPT) algorithm [122] uses additionally the information
of the tracking system in order to improve the resolution of the CaloJet’s transverse
momentum. The reconstructed tracks are matched to the CaloJets and dedicated
algorithms correct the energy of the jets.
Particle Flow Approach
The use of Particle Flow (PF) objects [123] as input for the jet reconstruction
is another very powerful approach at the CMS experiment. The PF algorithm
combines the information of all sub-detectors in order to identify all stable particles
within an event and reconstruct each of them individually. In principle, the PF
algorithm connects the hits in the ECAL, the energy deposits in the HCAL and the
individual tracks. Herein, the energy clustering in the calorimeters is done separately
for all sub-detectors to benefit from the finer granularity in certain areas. Further the
PF approach profits from the good momentum resolution of the tracking system.
Interpreting the information after several connecting iterations the PF algorithm
provides a complete list of all stable particles in the event, i.e. electrons, muons,
photons and neutral and charged hadrons. Thereafter, these PF particles can be
clustered with any standard jet algorithm yielding a better resolution of the jet
transverse momentum compared to the JPT and CaloJet approach as shown in
figure 3.3.
A special feature when using PF objects is represented by the PFnoPU algorithm.
The PFnoPU algorithm filters charged particles which originate from pile-up events
before jet reconstruction. Therefore a smaller amount of physical objects has to be
considered for the clustering algorithms.
Jet Energy Corrections
Within the reconstruction of the momenta of initial partons there are several influences in determinating the energy of a jet, which distort the measurement. The Jet
Energy Corrections (JEC) aim to correct for all these effects. At CMS the JEC are
s=7 TeV
CMS Simulation
σ(p /pREF) / <p /pREF>
0.30.3
(Anti-k R=0.5)
T
0 < |η| < 0.5
CaloJets
0.2
JPTJets
T
T
T
σ(p /pREFT ) T/ <p /pTREF
>
45
PFJets
0.2
(point-fit)
[%]
point
T
T
0.1
0.1
10
-10
20 100
200 1000
2000
20 30
20
100
200
100 200
(a)
1000 2000
2000
1000
pREF [GeV]
T
(b)
Figure 3.3: Transverse momentum resolutions of calorimeter, JPT and PF jets as
a function of pREF
for the ranges 0 < |η| < 0.5 in (a), taken from [124]. Herein,
T
pT indicates the transverse momentum of the reconstructed jet and pREF
is the
T
transverse momentum of the matched generator jet. In (b) the measured transverse
momentum resolution for PF jets in the same region is shown, adopted from [125].
accomplished in seven successive steps [126], where each step addresses a certain
effect. These steps are also referred to as Level 1 to Level 7 corrections. Herein,
Level 1 to Level 3 aim to correct all instrumental effects, that are calorimeter responses which are dependent on the transverse momentum or the energy of a jet.
Level 4 to Level 7 represent four additional corrections in order to improve the
estimation of the energy and momentum of the underlying partons.
Level 1: Electronic noise and pile-up collisions yield an increased measured jet
energy. Therefore, the Fastjet corrections are applied. Herein the energy
density ρ per unit area is calculated event-by-event and the median of ρ is
subtracted from the jet energy thereafter.
Level 2: Uninstrumented regions of the detector and the non-compensating behavior lead to a non-uniform calorimeter response in η. Therefore, relative
corrections are applied in order to equalize the jet response as a function of η
to the jet response in the central region |η| < 1.3.
Level 3: The calorimeter response is also dependent on the transverse momentum
of the jet. Hence, absolute corrections as a function of pT are applied for a
uniform jet response in pT .
Level 4-7: In Level 4 the effects from different electromagnetic fractions of the
jets are corrected. Since gluons have a higher probability for energy loss by
radiation than light quarks, they cause a smaller amount of energy deposits
in the calorimeter. Level 5 corrects for these flavor-dependent effects. The
46
Figure 3.4: Illustration of a collision event with a b jet, taken from [127]. The b
quark fragments into B hadrons, which mostly decay via the weak interaction and
therefore have a rather long lifetime in the order of τ ≈ 1.6 ps. The decaying B
hadron yields a jet which has displaced tracks with respect to the primary vertex
determined by an impact parameter (IP) for each track. In this figure the IP is
indicated as d0 . In many cases a secondary vertex with a distance from the primary
hard interaction vertex in the transverse plane Lxy can be reconstructed from the
displaced tracks. Lxy and the IPs of the jets are used in many algorithms in CMSSW
as discriminator values to identify b jets.
energy offset arising from underlying events, which is assumed to be nearly
independent from the direction and energy scale of the hard interaction, is
corrected in Level 6. Eventually, Level 7 corrections account for the possible
differences in the energy of an initial parton and the corresponding jet.
Because of differences between MC simulations and observed data in Level 2 and
Level 3 corrections, there are residual corrections [125] which have to be applied on
data.
3.2.6
b Tagging
The identification of b quark induced jets, so-called b jets, is referred to as b tagging.
The yielding discrimination between jets arising from b quarks and jets originating
from light quarks, like up, down and strange quarks or gluons is of major interest
in many analyses. Since the top quark decays into a b quark and a W boson in
nearly 100% of all cases, applying b tagging algorithms leads to a higher purity of
top quark pairs in analyzed data samples.
Figure 3.4 shows some characteristics of a b jet. In collision events with a b quark
in the final state, the b quark forms a B hadron, which mostly decays via the weak
interaction and has a rather long lifetime compared to other hadrons. Therefore the
Algorithm
TCHE
WP
loose
medium
tight
Discriminator Value Mistag Rate
1.7
0.1
3.3
0.01
10.2
0.001
47
Efficiency
0.735
0.61
0.251
Table 3.1: Working points of the TCHE b tagging algorithm and the corresponding
discriminator value thresholds. Additionally the b tagging efficiency is shown [130].
decay of the B hadron usually takes place at a distance of several millimeters in
respect to the primary vertex. The reconstructed tracks of such a jet are displaced,
resulting in an impact parameter (IP), see figure 3.4. These displaced tracks are used
to reconstruct a secondary vertex, which has a significant distance to the primary
vertex in the same order as the lifetime of B hadrons times the velocity of light
cτB ≈ 480 µm.
The CMSSW provides several b tagging algorithms [128] which use these characteristics. All of the algorithms aim for a high efficiency of identifying b quarks
and a low mistag rate, which is the efficiency of misidentifying a jet not originating
from a b quark as a b jet. As one can see in figure 3.5(a) and (b) decreasing the
mistag rate by demanding a higher discriminator value leads to a lower b tagging
efficiency. For each analysis an optimum has to be chosen. In this thesis a b tagging
algorithm called Track Counting High Efficiency (TCHE) is applied to identify b
jets. The TCHE algorithm uses the second highest IP significance of all tracks as
discriminator value. The IP significance is defined as the IP value of a track divided by its uncertainty. For every b tagger there are three different working points
(WP). These working points are named loose, medium and tight and correspond to
a certain mistag rate which determines the threshold of the discriminator value. In
the case of the loose WP a mistag rate of 0.1 is allowed, i.e. one out of ten jets not
originating from a b quark is falsely identified as a b jet. The medium WP demands
a mistag rate of 0.01, whereas a mistag rate of 0.001 determines the tight WP. For
the TCHE algorithm the corresponding discriminator values and the yielding b jet
efficiencies are summarized in table 3.1. In 2010 the commissioning of TCHE and
other b tagging algorithms was done using first collision data [129].
3.2.7
Missing Transverse Energy
The colliding protons at LHC only possess a momentum longitudinal to the beam
axis. Therefore the transverse momentum for each event must be conserved. Particles in the event which are not detected yield an imbalance of the total transverse
momentum. The momentum conservation is used to determine the amount of the
~ T. The MET is used to identify neutrinos,
missing transverse energy, called MET or6 E
which marginally interact with the detector material. The precise determination of
the transverse momentum of the neutrinos in the event is essential in many analyses.
The missing transverse energy is defined as the negative vectorial sum of transverse
48
Non-b Jet Efficiency
a.u.
CMS Preliminary
1
B
10-1
UDS
10-1
G
-2
10
CMS Preliminary
1
10-2
C
10-3
10-3
DUS
G
10-4
10-4
C
-5
10
-10
-5
0
5
10
15
20
25
10-50
30
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
b Jet Efficiency
discriminator value
(a)
(b)
Figure 3.5: Discriminator values and mistag rates for the TCHE b tagging algorithms [128]. The distribution of the discriminator values for the TCHE b-tagging
algorithms are shown in (a) and the mistag rate vs. the b jet efficiency for different
cuts on the discriminator value is depicted in (b). The distributions are shown for
different flavors, which are up, down and strange quarks (UDS/DUS), gluons (G),
charm quarks (C) and bottom quarks (B).
energies of all deposits in the calorimetry system [131], given by
~T = −
6E
N
X
(En sin θn cos φn x̂ + En sin θn sin φn ŷ) ,
(3.4)
n=1
where x̂ and ŷ are the unit vectors in the x and y direction and the index n runs
over all input objects from the calorimeter system. There are several effects which
influence the determination of the MET and have to be corrected for, for example
particles which are emitted close to the beam pipe, escaping the CMS detector
undetected. Moreover, the influences of damaged detector elements and poorly
instrumented areas have to be considered.
The corrections for the MET are categorized into type-I which addresses effects
of jets and muons and type-II which corrects for the effects of pile-up and underlying events. At the type-I stage, since the MET calculation uses the energy of
the calorimeter system, where muons only deposit a small amount of energy, the
transverse energies of all muons are subtracted from the MET value. Further the
jet energy corrections applied to the reconstructed jets are considered yielding the
~ T with
corrected 6 E
Njets
X
corr
raw
~T
~T −
6E
= 6E
p~T,i
− p~T,i
.
(3.5)
corr
raw
p~T,i
corr
p~T,i
i=1
Herein
and
are the transverse momenta of jet i before and after jet energy corrections, respectively. The corrections in equation 3.5 only corrects for jets
49
with a transverse momentum pT > 10 GeV/c, because for jets with a low pT the
uncertainties are large.
Similar to the jet reconstruction, the particle flow approach yields a better resolution in determinating the missing transverse energy. The calculation of the so-called
~ T uses all identified particle flow objects in the sum in equation 3.4 and the
PF-6 E
type-I and type-II corrections are not required.
50
Chapter 4
Selection of Events
The recorded data from the CMS experiment contains millions of events with different physics processes arising from proton-proton collisions. One major step of every
analysis focusing on a certain process is to find a dedicated selection algorithm in
order to maximize the amount of signal events S in the sample and to minimize the
number of background events B. Therefore MC simulations are used to find suitable
selection steps, often referred to as cuts. The knowledge of the topology of the signal
events and the background processes is essential. In proton-proton collisions at the
LHC only one among over 400 million events contains top quark pair production,
since the total inelastic cross section of proton-proton collision is about nine orders
of magnitudes larger than the determined top quark pair production cross section,
as shown in figure 4.1. Still dedicated cuts in this analysis yield a purity of tt̄ signal
events of about 80% after the selection.
The following sections introduce the modeling of signal and background processes
and give an overview of the used datasets and simulated MC samples. Further,
the explicit selection criteria are defined, which are used to optimize the signalto-background ratio S/B and the procedure to estimate the amount of expected
background events is explained.
4.1
Modeling of Signal and Background Events
The precise simulation of signal and background events using Monte Carlo techniques is pivotal for every analysis. By studying the simulated MC samples one
can calculate the selection efficiencies of all contributing processes and estimate the
predicted number of events in a dataset with a determined integrated luminosity.
After the selection, it is crucial to compare the simulated kinematic distributions
of signal and background processes to the observed data in order to validate the
simulations.
4.1.1
The Lepton+Jets Channel
In this thesis the charge asymmetry in top quark pair production is determined in the
tt̄ lepton+jets decay channel. An exemplary LO Feynman diagram for this process
51
52
CHAPTER 4. SELECTION OF EVENTS
Figure 4.1: Predicted cross √
sections of different physics processes
as a function
√
< 4 GeV the cross
of the center-of-mass energy s, adopted from [132]. For s √
sections in proton-antiproton collisions are depicted, and for s > 4 GeV those
for proton-proton collisions. The solid
√ green line indicates the current operation
center-of-mass energy at the LHC of s = 7 GeV. For this value the total inelastic
proton-proton cross section in black is about nine magnitudes larger than the cross
section of top quark pair√production shown in red. Additionally, the center-of-mass
energy of√the Tevatron s = 1.96 GeV and the LHC center-of-mass energy design
value of s = 14 GeV are indicated with dashed green lines.
is shown in figure 4.2. The charged lepton is typically an electron or a muon, or a
corresponding antiparticle, respectively. The final state of the lepton+jets channel
comprises an isolated lepton with a high transverse momentum, two b jets stemming
from the top quark decays and two light jets arising from the hadronically decaying
W boson. Further, a certain amount of missing transverse energy 6 ET is expected,
since a neutrino is produced. In order to simulate the hard interaction of the tt̄
signal events the MadGraph/MadEvent event generator is used. The matrix
element generator MadGraph takes into account all LO Feynman diagrams of the
top quark pair production in association with up to three additional hard partons.
The top quark mass is set to mt = 172.5 GeV/c2 and the CTEQ6L parametrization
is used as PDF of the protons. By using the parton shower generator Pythia the
hadronization and the decay of unstable particles is simulated. For the tt̄ simulation
in Pythia the Z2 tune [133] is used.
4.1. MODELING OF SIGNAL AND BACKGROUND EVENTS
53
Figure 4.2: Exemplary Feynman diagram of a tt̄ event in the lepton+jets decay
channel. Both top quarks decay into a b quark and a W boson. One W boson
further decays into a charged lepton and a neutrino and the other W boson into two
quarks.
Although the tt̄ charge asymmetry is a NLO effect the use of this LO simulation is justified because a model-independent measurement is performed with the
only assumption that the processes leading to the tt̄ charge asymmetry affect the
production of the top quarks and not their decay.
4.1.2
Background Processes
There are several background processes, which show the same or a similar experimental signature as the tt̄ signal events. The production of W bosons and the
electroweak production of single top quarks can contain a high energetic lepton in
the final state. Further, the production of Z bosons and virtual photons γ ∗ , referred
to as Drell-Yan process [134], has to be considered as background. With additional
radiation processes yielding jets in the CMS detector, these processes have a topology similar to tt̄ signal events. Further, QCD multi-jet production with falsely
identified leptons can mimic the signal signature.
W +Jets and Z/γ ∗ +Jets
The cross sections of W boson and Z boson production are about two to three orders of magnitude larger than the cross section of tt̄ production.The production of
a W boson in association with additional radiations, the W +jets production mode,
is predicted to be the most dominant background contribution in the tt̄ lepton+jets
channel. With a leptonic W boson decay the W +jets channel possesses a topology
very similar to the signature of the signal events, as depicted in figure 4.3(a). The
production of a Z boson or a virtual photon in association with additional jets,
referred to as Drell-Yan production, is shown in figure 4.3(b). The generation of
W +jets and Drell-Yan+jets events is also accomplished using the combination of
MadGraph/MadEvent and Pythia. In order to increase the amount of signal-
54
(a)
(b)
Figure 4.3: Exemplary leading order Feynman diagrams of W ± (a) and Drell-Yan
(b) production. The W +jets production mode shows a similar signature as the tt̄
lepton+jets channel. If one lepton is not detected the Drell-Yan process can mimic
a signal event.
(a)
(b)
(c)
Figure 4.4: Exemplary Feynman diagrams of electroweak production of single top
quarks. The s-channel is shown in (a) the t-channel in (b), and the tW-channel in
(c). When the top quark is decaying leptonically as depicted in the diagrams and
additional radiations are produced, the electroweak production of single top quarks
can mimic tt̄ lepton+jets events.
like events the generation of W +jets and Drell-Yan+jets events is restricted to leptonically decaying W and Z/γ ∗ bosons, i.e. W → lνl and Z/γ ∗ → l+ l− , respectively.
Further, for Drell-Yan+jets events the invariant mass of the lepton-antilepton pair
is required to be larger than 50 GeV/c2 .
Single Top Quark Production
Since possessing a top quark in the final state, the topology of the electroweak
single top quark production is very similar to the top quark pair production. Single
top quarks can be produced via three different production modes. Considering a
leptonically decaying single top quark and additional hard radiations as depicted in
figure 4.4, all three single top quark production modes exhibit the experimental tt̄
lepton+jets signature. Similar to the top quark pair production the matrix element
generator MadGraph/MadEvent is used for the simulation of electroweak single
top quark production. The mass of the top quark is set to mt = 172.5 GeV/c2
and the hadronization and the decay of unstable particles is accomplished by the
(a)
(b)
55
(c)
Figure 4.5: Exemplary Feynman diagrams of QCD multi-jet ((a) and (b)) and
photon+jets (c) processes. In (a) the production of a bb̄ pair via the strong interaction is shown. The b quark hadronizes and further decays via the weak interaction
into a c quark, a lepton and an antineutrino, whereas the b̄ quark radiates a gluon
yielding two jets. In (b) an event containing only quarks and gluons in the final state
is depicted. If one of the particles arising from the produced jets is misidentified
as a lepton, these events have also a signal-like signature. The production of a real
photon in association with additional jets is shown in (c).
interfacing with the Pythia event generator. In order to generate more events
containing a lepton in the final state, the simulation of single top quarks arising
from the s-channel and the t-channel production mode is restricted to leptonic top
quark decays.
QCD Multi-Jet Processes
Considering the large production cross section, the influence of multi-jet events produced via the strong interaction has to be taken into account. The final state of
such QCD multi-jet events contains in principle only jets arising from quarks and
gluons. Herein electrons and muons can be produced by semileptonically decaying hadrons containing b and c quarks. An example for this process is shown in
figure 4.5(a). In events comprising only quarks and gluons in the final state as
depicted in figure 4.5(b), one jet can be misidentified as an electron or a muon.
The generation of QCD multi-jet events is fully accomplished by Pythia using
the D6T tune. The QCD events containing muons and electrons in the final state
are simulated separately. A muon enriched background sample in the muon+jets
channel is produced by requiring the produced muon to have a transverse momentum
larger than 15 GeV/c and to be emitted in the pseudorapidity region with |η| < 2.5.
QCD multi-jet events which can mimic electron+jets events are simulated in two
separate non-overlapping phase space regions yielding two different datasets. On
the one hand the so-called BCtoE dataset considers only electrons arising from bor c-hadron decays. Herein the electrons are required to be emitted in the area with
|η| < 2.5 and to possess a transverse energy larger than 10 GeV. On the other hand
the so-called EMEnriched dataset filters the stable final state partons for candidate
objects which are likely to be misidentified as electrons. For this purpose the stable
partons have to fulfill several requirements. In order to further increase the amount
56
of signal-like events the EMEnriched and the BCtoE
q datasets are subdivided into
three samples, each covering a particular p̂T = t̂ · û/ŝ region. Herein t̂,û and ŝ
denote the Mandelstam variables [135].
Photon+Jets Processes
The process with a real photon in the final state, produced in association with
additional hard radiations, is called photon+jets channel. In the detector material
the photon can convert into an electron-positron pair. If only one of the conversion
electrons is reconstructed and four additional jets are produced, an event in the
photon+jets channel shows the same signature as a tt̄ electron+jets event. An
exemplary Feynman diagram for such a process is shown in figure 4.5(c).
The photon+jets MC sample is generated with the known combination of MadGraph/MadEvent and Pythia using the D6T tune, and provides up to four
additional jets. In order to increase the amount of signal-like events, produced photons are required to posses a transverse momentum larger than 20 GeV/c and to be
produced in the pseudorapidity region with |η| < 2.5. Further, the photons have
to be isolated, which means a distance to final state quarks or gluons of ∆R > 0.3
is required. The photon+jets dataset is subdivided into three samples, which cover
three different HT,had regions. In this case HT,had denotes the scalar sum of the
transverse momenta of all final state quarks and gluons.
4.1.3
Used Monte Carlo Samples
The used Monte Carlo samples have been produced in the “Spring11” and “Summer11” production campaigns. An overview of all samples with the corresponding effective cross sections and the overall number of generated events is shown
in table 4.1.3. For the calculation of the effective cross sections of the tt̄ and the
electroweak single top production the MCFM framework is used. The next-toleading order cross section for tt̄ production has been determined to σtt̄ = 157+23
−24 pb
[136, 137]. The single top quark NLO production cross section multiplied by the
leptonic branching fraction of the W boson has been determined as σt = 21.0+1.1
−1.0 pb
[136, 138–140] and the inclusive single-top-quark associated production (tW) NLO
cross section of σtW = 10.6 ± 0.8 pb [139] has been used. Both cross sections include
the production of single top and single antitop quarks. The s-channel single-top
production cross section is small compared to the t-channel and tW production
cross sections and therefore this process is treated as negligible in this analysis. The
NNLO production cross section for W bosons decaying into leptons has been determined to be σW →lν = 31.3 ± 1.6 nb using the FEWZ [141] framework. Finally, the
cross section for Drell-Yan production decaying into two leptons has been calculated
at NNLO also using FEWZ as σZ/γ ∗ →ll (mll > 50 GeV/c2 ) = 3.05 ± 0.13 nb.
Process
tt̄
W +jets
Z/γ ∗ +jets
single top (t-channel)
single top (tW-channel)
QCD µ-enriched
QCD BCtoE
QCD BCtoE
QCD BCtoE
QCD EMEnriched
QCD EMEnriched
QCD EMEnriched
γ+jets
γ+jets
γ+jets
inclusive
W → lν
Z/γ ∗ → l+ l− , m(l+ l− ) > 50 GeV/c2
t → blν
inclusive
p̂T > 20 GeV/c, pµT > 15 GeV/c
20 GeV/c ≤ p̂T < 30 GeV/c
80 GeV/c ≤ p̂T < 170 GeV/c
80 GeV/c ≤ p̂T < 170 GeV/c
40 GeV/c ≤ HT,had < 100 GeV/c
100 GeV/c ≤ HT,had < 200 GeV/c
HT,had ≥ 200 GeV/c
σeff [pb]
157.5
31314
3048
20.93
10.6
84679.3
132160
136804
9360
2454400
3866200
139500
23620
3476
485
57
Nprod
3701947
56789563
36277961
484060
489417
29434562
2243439
1995502
1043390
36136246
70708892
8069591
2217101
1065691
1142171
Table 4.1: Overview of the used Monte Carlo samples with the number of generated
events Nprod and the effective cross section σeff .
4.1.4
Used Data
The collision data used in this thesis was recorded by the CMS detector within
the LHC pp operation in 2011 until the end of June. As described in section 2.2.4
the recorded events have to pass certain requirements of the HLT system. Trigger
paths which require events to posses similar physical objects with certain kinematic
qualities, e.g. a muon with a large transverse momentum, are collected in primary
datasets. In this thesis the primary datasets EleHad for the electron+jets channel
and MuHad for the muon+jets channel are used. These datasets contain the crosstriggers, which require for each event one lepton as well as a certain amount of jets
passing several criteria.
The recorded data is separated into different runs. Each run has a unique runnumber and indicates a certain time period in which the CMS detector has been
recording data without interruptions. The runs are further subdivided into luminosity sections (LS), in which the instantaneous luminosity is considered constant. The
used datasets in this analysis cover the runs 160404 to 167913. For the measurement
of the tt̄ charge asymmetry only the runs and LS in which all CMS subdetectors performed flawlessly can be used. The Data Quality Monitoring (DQM) system [142]
validates runs and LS, which are considered good for such physics analyses and lists
them into a JSON file. Before further analyzing a primary dataset a filter is applied
based on a certain JSON file in order to keep only runs and LS which are validated.
The JSON file used here is Cert 160404-167913 7TeV PromptReco Collisions11JSON [143] and corresponds to an integrated luminosity of 1.09 ± 0.07 fb−1 . After
the reconstruction the certified data has to be cleaned from anomalous HCAL noise
arising from photo multipliers in the HB and HE. For this purpose the HB/HENoiseFilter [144] is applied. Further, events with more than ten associated tracks
and less than 25% of high purity tracks, so-called beam scraping events, are rejected.
58
Primary Dataset
EleHad
Run Range
160404 - 165969
> 165969
MuHad
160404 - 165969
> 165969
HLT Trigger
HLT Ele25 CaloIdVT TrkIdT TriCentralJet30
HLT Ele25 CaloIdVT CaloIsoT TrkIdT TrkIsoTTriCentralJet30
HLT Mu17 TriCentralJet30
HLT IsoMu17 TriCentralJet30
Table 4.2: Summary of the used HLT paths applied on data. All triggers contain the
hadronic TriCentralJet30 part which asks for at least three jets with ET > 30 GeV
and |η| < 2.5. Only this part is also simulated on the Monte Carlo events.
As shown in figure 2.2 the instantaneous luminosity at LHC increased rapidly
during the 2011 data-taking period. When the production rate for the triggered
final state is too large, the correspondent HLT trigger is prescaled, which means
not every event produced above the thresholds is recorded. The measurement of
the charge asymmetry is dependent on a large amount of signal events and therefore only unprescaled triggers are suitable. The choice of the HLT path depends on
the run number. The trigger HLT Ele25 CaloIdVT TrkIdT TriCentralJet30 for the
electron+jets channel is used up to run 165969. This trigger requires one electron
with a transverse energy larger than 25 GeV. Further, in the event at least three
reconstructed jets with a transverse energy larger than 30 GeV in the region with
|η| < 2.5 are demanded. Moreover, electrons have to pass certain identification
requirements. For later runs a trigger additionally asking for isolation of the triggered electrons is used. In the muon+jets channel the HLT Mu17 TriCentralJet30
trigger is utilized up to run 165969. Herein one muon with a transverse momentum larger than 17 GeV/c and at least three jets with a transverse energy
larger than 30 GeV in the region with |η| < 2.5 are required. For later runs the
HLT IsoMu17 TriCentralJet30 trigger is used, which requires an additional isolation
criterion for the triggered muons. In table 4.2 the used triggers for both channels
are summarized.
4.1.5
Corrections on Monte Carlo Events
In order to simulate events as precisely as possible one has to correct for effects, which
are not included correctly in the event generators, i.e. the simulated distribution of
number of pile-up events and the influence of the HLT triggers. The correction is
accomplished by giving each MC event a weight to match the observed data.
Pile-Up Reweighting
The used Monte Carlo samples for QCD multi-jet events, single top production and
γ+jet events have been produced with a flat distribution with a Poisson tail for the
number of pile-up interactions, whereas the samples for the W +jets, Drell-Yan+jets
and the top quark pair production provide non-flat distributions. Both of these
distributions do not cover well the observed pile-up conditions in the 2011 datataking period at LHC. In order to generate more realistic pile-up distributions in
59
the MC samples a pile-up reweighting is applied [145]. Herein every simulated event
gets an event weight according to its number of pile-up events to match the observed
pile-up distribution from data.
Simulation of Triggers
Since the used HLT triggers in both channels are not simulated in the MC events
their influences have to be modeled separately. The HLT paths from table 4.2 consist
of a leptonic part requiring at least one muon or electron in the event and a hadronic
part asking for at least three jets. The efficiencies of the leptonic part and of the
three jet conditions of the triggers are assumed to factorize.
On the one hand the influence of the leptonic part only affects the overall efficiency and will cancel out in the calculation of the charge asymmetry. Therefore this
path is not simulated in the Monte Carlo events. On the other hand the TriCentralJet30 part of these triggers is based on calorimeter jets, whereas in the analysis
particle flow jets are used. Therefore, there is no guarantee that all selected jets
correspond to a triggered jet above the threshold in the transverse momentum, i.e.
pT > 30 GeV/c. The CentralJet30 path, which triggers only one jet with the same
requirements, is used to solve this problem. The selection efficiency of a single PF
jet to be triggered by the CentralJet30 path is measured and parameterized for
different η regions with a Gompertz function
c·pT
(pT ) = a · eb·e
,
(4.1)
where the parameters a, b, and c are obtained from a likelihood fit in data taken
with a single muon trigger. The results [146] are shown in figure 4.6.
The used triggers require at least three jets, therefore the overall probability
of an event with Njets selected jets to be triggered, is given by a combination of
the probabilities of the single jets to be accepted by the CentralJet30 trigger. As
an example for an event with Njets = 4 there are two cases, i.e. that all four jets
have been accepted by the trigger, and that three jets have been accepted by the
trigger and one jet fails the trigger condition. The probabilities of both cases have
to be added to gain the selection efficiency. According to the calculated selection
efficiency of the TriCentral30 trigger path the MC events in the electron+jets and
in the muon+jets channel are reweighted.
Jet Resolution
The resolution for the transverse momenta of jets in observed data is about 10%
worse compared to MC simulations. To correct for this effect, all jets in the MC
simulated samples have to be scaled accordingly. For this purpose, selected jets are
matched to generator jets. The difference between the transverse momentum of the
reconstructed and of the generated jet is multiplied with 0.1 and propagated to the
jet four vectors.
efficiency
efficiency
60
η<-1.4
1
-1.4<η<0
1
0.5
0.5
0.983*exp( -62.382*exp( -0.168*pT))
0
0
50
100
150
0.987*exp( -43.718*exp( -0.153*pT))
0
0
200
jet
pPF
[GeV/c]
T
50
0<η<1.4
1
0.5
200
η>1.4
1
0.5
0.987*exp( -42.391*exp( -0.153*pT))
0
0
150
jet
pPF
[GeV/c]
T
(b)
efficiency
efficiency
(a)
100
50
100
150
0.986*exp( -57.021*exp( -0.163*pT))
200
jet
pPF
[GeV/c]
T
0
0
50
(c)
100
150
200
jet
pPF
[GeV/c]
T
(d)
Figure 4.6: Trigger turn-on curves for a single PF jet to be accepted by the CentralJet30 trigger as function of the transverse momentum of the jet in the pseudorapidity
regions η < −1.4 (a), −1.4 < η < 0 (b), 0 < η < 1.4 (c) and η > 1.4 (d) [146]. Since
the selected jets in this analysis are required to posses a pT larger than 30 GeV/c the
functions given in the diagrams are used to calculate the probability of a simulated
event to pass the HLT trigger used on data.
4.2
Selection Criteria
After modeling the signal and background events, a dedicated selection is applied
on each dataset. All datasets used in this analysis have been reconstructed with
the CMSSW software using particle flow objects. Further, PFnoPU and the Fastjet
corrections introduced in section 3.2.5 are applied. The missing transverse energy
based on particle flow objects is utilized.
In the following sections the requirements on the physical objects, i.e. electrons,
muons and jets, are explained and thereafter the consecutive selection steps are
introduced.
4.2.1
Definitions of Physical Objects
One important requirement of this analysis is that lepton candidates have to be
isolated. Reconstructed electrons and muons which do not origin from a real W
boson, might be produced for instance in decays of hadrons and show a large amount
of hadronic activity in the detector around the lepton. In order to reduce this kind
4.2. SELECTION CRITERIA
61
of background events the relative isolation of the leptons is used, defined as
`
IRel
=
`
ICH
+ IN` H + IP` h
,
pT,`
(4.2)
`
where ICH
indicates the energy generated by stable charged hadrons in a cone of
∆R = 0.4 around the track of the lepton and IN` H and IP` h denote the respective
energy deposits of neutral hadrons and photons. The relative isolation of a lepton
is based on particle flow objects.
Electron Definition
The electron candidates are required to posses a transverse energy larger than
30 GeV and to lie in the region |η| < 2.5. Candidates in the ECAL endcap-barrel
transition region defined as 1.4442 < |ηsc | < 1.5660, where ηsc is the pseudorapidity
of the electron’s supercluster, are rejected. The transverse impact parameter of the
track of the electron with respect to the beamspot is required to be smaller than
0.02 cm. In order to ensure that the electron arises from the primary interaction,
the z position of the electron vertex is required to lie within 1 cm around the z
e
defined in equation 4.2
position of the primary vertex. The relative isolation IRel
has to be smaller than 0.125. Electron candidates must also pass the VBTF70 [147]
identification, which defines additional requirements on a set of variables considering
the influences of the shower shape, the track cluster matching, the isolation and the
conversion rejection. Different cuts are used for electrons in the ECAL barrel and
the ECAL endcap region.
Muon Definition
The muon candidates are required to be reconstructed as global muons and tracker
muons. Further, muons are required to possess a transverse momentum larger than
20 GeV/c and a pseudorapidity of |η| < 2.1. The χ2 value of the global fit applied
in the reconstruction has to be smaller than 10. The transverse impact parameter
of the muon’s track with respect to the beamspot is required to be smaller than
0.02 cm. Similar to the electron candidates the z component of the muon’s vertex
is required to lie within 1 cm around the z position of the primary vertex and the
µ
relative isolation IRel
has to be smaller than 0.125.
Jet Definition
The jets utilized in this analysis are reconstructed with the anti-kT [148] jet algorithm introduced in section 3.2.5, where particle flow objects are used as input. For
the jet size parameter D in equation 3.2 a value of D = 0.5 is chosen. In order to
minimize the contribution of misidentified electrons, photons, pions or calorimeter
noise all jets have to pass the PF-jet ID, which asks for neutral hadron, neutral
electromagnetic and charged electromagnetic fractions below 0.99. Further the ID
requires jets to have at least two constituents. The jet energy is corrected using the
Level 1, Level 2 and Level 3 corrections.
62
4.2.2
Selection Cuts
The following consecutive selection cuts are applied to the observed data and the
simulated MC events. Deploying the corrections introduced in 4.1.5 the simulated
events are not weighted equally.
Primary Vertex Criteria
For each event one good primary vertex is required. The criterion on the the number
of degrees of freedom ndof > 4 has to be fulfilled, where ndof indicates the weighted
sum of the number of tracks used for the construction of the PV. Further, the PV
has to lie in the central region of the CMS detector, determined by the coordinates
|z| < 24 cm and r < 2 cm with respect to the nominal interaction point.
Lepton Cuts
In both channels the existence of exactly one isolated lepton as defined above is
required for each event.
Further, events containing one or more additional leptons are rejected. These
additional lepton are defined more loosely than the isolated leptons and are therefore called loose muons and loose electrons. Loose muons are all global muons which
possess a transverse momentum larger than 10 GeV/c and lie in the region |η| < 2.5.
For loose electrons all electron candidates, without asking for additional identification criteria, with a transverse energy larger than 15 GeV and with a pseudorapidity
of |η| < 2.5 are considered. The relative isolation of both loose leptons is required
to be smaller than 0.25. This cut reduces the contribution of Drell-Yan production
and the dileptonic tt̄ decay heavily.
Conversion Rejection
The selected isolated electron can also originate from a conversion of a photon into an
electron-positron pair, if the transverse energy of the photon is large enough. In order
to reject such events additional cuts in the electron+jets channel are introduced.
Since photons usually travel through the tracker before they convert, there may be
layers of tracker material between the beam line and the conversion vertex which
do not have hits. Therefore the electrons are required to have no missed layers
before the first hit of their track from the beamline. Further, the selection uses the
fact that photon conversions always produce an electron-positron pair. A dedicated
algorithm [149, 150] searches for secondary tracks with opposite curvature in a cone
with a radius of R = 0.3 around the isolated electron. For these secondary tracks two
parameters are calculated, i.e. ∆ cot θ and dx,y . The parameter ∆ cot θ denotes the
difference in cot θ between the electron’s track and a secondary track. Additionally
the value dx,y , which indicates the distance between the two tracks at the point,
where they are parallel, is calculated. If a secondary track with |∆ cot θ| < 0.02 and
dx,y < 0.02 cm is found, the selected electron is considered as a conversion and the
event is rejected.
4.2. SELECTION CRITERIA
63
process
electron+jets
tt̄
4245+625
−658
single top (t)
101+6
−5
single top (tW)
94+7
−7
W +jets
559+33
−33
Drell-Yan+jets
80+4
−4
QCD
222 ± 58
total
5301+629
−661
observed data
5665
muon+jets
6031+888
−934
151+8
−8
125+10
−10
756+43
−43
86+5
−5
68 ± 15
7217+890
−936
7092
lepton+jets
10276+1087
−1143
252+10
−10
220+12
−12
1315+55
−55
166+6
−6
289 ± 60
12517+1090
−1146
12757
Table 4.3: Event yield for the described event selection for a selected dataset corresponding to an integrated luminosity of 1.09 fb−1 . The numbers for the different
processes have been determined based on simulated samples. The uncertainties account for limited MC statistics and for the uncertainty in the theoretically predicted
cross section. Additionally the observed number of events in data is shown. These
values are well compatible with the uncertainties of the prediction.
Jet Cut
In both channels only events with four or more jets passing the requirements defined
above are selected. This selection step heavily increases the purity of tt̄ signal events
in the sample.
b Tagging Requirements
Since both top quarks decay into a W boson and a b quark, there are two expected
b jets in the final state. However, simulations show that requiring only one b tagged
jet yields the optimum between purity of tt̄ signal events in the sample and large
amount of selected events. For this purpose the TCHE algorithm [129] is used at
the medium working point utilizing a discriminator value of 1.7 as threshold.
Applying all selection cuts to the observed data, 5665 events in the electron+jets
and 7092 events in the muon+jets channel are selected. For the simulated MC
samples for signal and background processes one can easily calculate the predicted
number of events Npred for a certain process with the cross section σeff in a dataset
with an integrated luminosity Lint via
Npred =
Nsel · Lint · σeff
.
Nprod
(4.3)
Here Nprod and Nsel are the number of events in the MC sample before and after
the selection. The event yield shown in table 4.3 summarizes these calculations.
In the electron+jets channel roughly 7% more events than predicted are observed,
whereas in the muon+jets channel about 2% less events are observed compared to
the prediction.
64
4.3
Background Estimation
The measurement of the charge asymmetry requires a precise knowledge of the
amount of background events in the analyzed dataset. Since the influence of the
leptonic part of the HLT paths is not simulated in the MC events, a data-driven
background estimation is crucial for this analysis. For this purpose a binned likelihood fit is performed which exploits the discrimination power of the missing transverse energy and the M3 variable [67, 151]. M3 indicates the invariant mass of the
three jets in an event with the largest vectorially summed transverse momentum.
4.3.1
Data-Driven Modeling of QCD Multi-Jet Production
Processes
Despite the large number of generated events in the QCD samples, there are only a
handful of events which pass the selection criteria. Since the signal region defined
by the event selection is only a small part of the phase space generated in the
QCD samples these events are outliers and therefore the QCD templates can not be
trusted. In order to gain a good modeling of QCD multi-jet background process a
data-driven QCD template is created by inverting the cut on the relative isolation
for leptons in both channels. For the purpose of avoiding signal processes in this
template without moving too far away from the phase space of the signal region, a
l
relative isolation of 0.3 < IRel
< 0.5 is chosen as criterion. The used HLT triggers on
data for runs later than 16596 already apply isolation requirements on trigger level,
which can not be inverted. Therefore these runs are not considered for the selection
of a QCD template. When applying the selection with the inverted isolation on
the simulated events for signal and all background processes the purity in terms of
QCD events in the data can be estimated. In the muon+jets channel a QCD purity
of 92% and in the electron+jets channel a QCD purity of 87% is expected. The
use of the QCD enriched data template has to be validated. For this purpose the
shapes of the 6 ET and M3 distributions of the data-driven template are compared
to the shapes of the Monte Carlo templates in the signal region. Figure 4.7 shows
the comparisons for the electron+jets and muon+jets channel separately. Within
the very large uncertainties of the simulated QCD events in the signal region a
reasonable agreement can be found.
4.3.2
Template Fit
Having the data-driven QCD template at hand the background estimation can be
done [152]. For all other processes the default MC samples listed in table 4.1.3 are
used as fit templates. To illustrate the discrimination power of 6 ET and M3 in figure
4.8 the shapes for all participating processes normalized to the same area are shown.
0.45 e+jets
0.4
QCD template (data)
MC QCD
0.35
0.3
0.25
65
norm. to unit area
norm. to unit area
4.3. BACKGROUND ESTIMATION
0.45
0.25
0.2
0.15
0.1
0.1
0.05
0.05
40
60
0
0
80 100 120 140 160 180 200
MC QCD
0.3
0.15
20
QCD template (data)
0.35
0.2
0
0
e+jets
0.4
100
200
300
400
500
0.7
0.6
QCD template (data)
MC QCD
0.5
700
800
(b)
0.4
0.3
norm. to unit area
norm. to unit area
(a)
µ+jets
600
M3 [GeV/c2]
ET [GeV]
0.5
µ+jets
QCD template (data)
0.4
MC QCD
0.3
0.2
0.2
0.1
0.1
0
0
20
40
60
80 100 120 140 160 180 200
0
0
100
200
300
(c)
400
500
600
700
800
M3 [GeV/c2]
ET [GeV]
(d)
Figure 4.7: QCD background shape comparison between MC based template and
data-driven template in 6 ET and M3. In (a) and (b) the distributions in the
electron+jets channel and in (c) and (d) the distributions in the muon+jets channel
are shown. Considering the very limited number of events which pass the selection
criteria in the QCD MC templates, the templates show a reasonable agreement.
One the one hand the 6 ET distribution is able to discriminate between processes with
real W bosons and thus real neutrinos, i.e. W +jets, single top and tt̄, and processes
with no real neutrinos, i.e. Drell-Yan+jets and QCD multi-jet. On the other hand
the M3 distribution is capable of discriminating between tt̄ production and W +jets,
since the former shows a resonance at the top quark mass. The fit is done for
the electron+jets channel and the muon+jets channel separately. To increase the
discrimination power of 6 ET and M3 both datasets are subdivided into two subsets,
one with 6 ET < 40 GeV and one with 6 ET > 40 GeV. In the former 6 ET is fitted and in
the latter M3. Since the W +jets production is asymmetric at the LHC and more
W + bosons than W − bosons are produced, the datasets are further subdivided into
positive and negative electrically charged leptons. The two channels of single top
production are combined.
In total twelve templates are used for the fit, i.e. tt̄, W + +jets, W − +jets, DrellYan+jets, single top and QCD, each in the e+jets and the µ+jets channel. Each fit
template is normalized to the predicted number of events for this process obtained
from MC simulation. Further, for each fit template a fit parameter β is defined,
which indicates the ratio of the measured number of events for this process and the
0.25
tt
e+jets
Single-top
0.2
W+jets positive Lepton
W+jets negative Lepton
Z+jets
0.15
QCD
norm. to unit area
norm. to unit area
66
0.1
0.4
0.35
tt
e+jets, ET>40 GeV
Single-top
0.3
Z+jets
0.25
QCD
0.2
0.15
0.1
0.05
0.05
0
0
20
40
60
0
0
80 100 120 140 160 180 200
100
200
300
400
500
tt
0.2 µ+jets
700
800
(b)
Single-top
0.18
0.16
0.14
Z+jets
0.12
QCD
norm. to unit area
norm. to unit area
(a)
0.22
600
M3 [GeV/c2]
ET [GeV]
0.4
Single-top
T
0.3
Z+jets
0.25
QCD
0.1
0.2
0.08
0.15
0.06
tt
µ+jets, E >40 GeV
0.35
0.1
0.04
0.05
0.02
0
0
20
40
60
80 100 120 140 160 180 200
0
0
100
200
300
(c)
400
500
600
700
800
M3 [GeV/c2]
ET [GeV]
(d)
Figure 4.8: Shape comparison of the 6 ET and the M3 distribution for signal and
background processes. In (a) and (b) the distribution for the electron+jets channel
and in (c) and (d) the distributions in the muon+jets channel are shown. For the
M3 distributions only events with a missing transverse energy above 40 GeV are
shown. For the W +jets process the two subsets with positively and negatively
charged leptons are depicted.
predicted number of events from MC simulation. Since their contribution to the
selected data set is only very small, for Drell-Yan+jets and single top processes the
β values are constrained to 1.0 with an uncertainty of 30%. All other fit parameters
are left free in the fit. Finally, the binned likelihood fits are performed using the
theta framework [153]. The resulting numbers of events of the different processes
are given by the product of the MC predictions for the different processes and the
respective β parameter. Table 4.4 summarizes the fit results for the electron+jets
and the muon+jets channel together with their statistical uncertainties.
In figure 4.9 the measured 6 ET and M3 distributions in the electron+jets and
muon+jets channel are shown together with the Monte Carlo simulation normalized
to the fit results and in both distribution a very good agreement between data and
MC simulation can be found. The whole dataset demonstrates a signal purity of 78%
in the electron+jets channel and 82% in the muon+jets channel. In order to clarify
the agreement the comparison between data and simulation is made in additional
kinematic distributions, which are illustrated in figures 4.10, 4.11 and 4.12.
67
electron+jets
prediction
fit result
4245
4401 ± 165
196
213 ± 58
335
313 ± 84
223
299 ± 90
80
81 ± 24
222
355 ± 71
5301
5663 ± 226
5665
process
tt̄
single top (t + tW)
W +jets (+)
W +jets (−)
Drell-Yan+jets
QCD
total results
observed data
muon+jets
prediction
fit result
6031
5835 ± 199
276
293 ± 81
465
404 ± 106
291
245 ± 109
86
85 ± 26
68
232 ± 79
7217
7094 ± 276
7092
total
fit result
10236 ± 258
507 ± 99
718 ± 135
544 ± 141
165 ± 35
587 ± 106
12757 ± 357
12757
600
1.09 fb-1 at s = 7 TeV
500 e+jets
Data
tt
Single-top
events
events
Table 4.4: Fit results for the background estimation together with their statistical
uncertainties. The samples are fitted in the electron+jets and muon+jets channel
separately. The W +jets template is divided into two samples, one containing events
with a positively charged lepton and one those with a negatively charged lepton.
The signal purity is 78% in the electron+jets channel and 82% in the muon+jets
channel. As a comparison the predicted event numbers summarized in table 4.3 are
shown. The last column represents the background estimation in the lepton+jets
channel, where the results in both channels are summed.
1200
tt
Single-top
1000 e+jets, ET>40 GeV
W+jets
400
W+jets
800
Z+jets
Z+jets
QCD
300
QCD
600
200
400
100
200
0
0
Data
1.09 fb-1 at s = 7 TeV
20
40
60
0
0
80 100 120 140 160 180 200
100
200
300
400
500
1.09 fb-1 at s = 7 TeV
600 µ+jets
500
Data
tt
Single-top
W+jets
Z+jets
400
700
800
(b)
QCD
events
events
(a)
700
600
M3 [GeV/c2]
ET [GeV]
Data
1600 1.09 fb-1 at s = 7 TeV
1400 µ+jets, E >40 GeV
tt
Single-top
1200
W+jets
T
Z+jets
1000
QCD
800
300
600
200
400
100
0
0
200
20
40
60
80 100 120 140 160 180 200
0
0
100
200
300
(c)
400
500
600
700
800
M3 [GeV/c2]
ET [GeV]
(d)
Figure 4.9: Data to MC comparison of 6 ET and M3 in the electron+jets channel ((a)
and (b)) and in the muon+jets channel ((c) and (d)). All MC templates and the
data-driven QCD template are normalized to the fit result shown in table 4.4.
events
events
68
800
1.09 fb-1 at
e+jets
700
Data
s = 7 TeV
tt
Single-top
600
1.09 fb-1 at
µ+jets
800
700
W+jets
500
900
tt
Single-top
W+jets
600
Z+jets
Z+jets
500
Multijet
400
Multijet
400
300
300
200
200
100
100
0
0
20
40
60
0
0
80 100 120 140 160 180 200
pT,lep [GeV/c]
20
40
60
80 100 120 140 160 180 200
pT,lep [GeV/c]
(b)
events
(a)
events
Data
s = 7 TeV
Data
600 1.09 fb-1 at s = 7 TeV
e+jets
500
tt
Single-top
Data
700 1.09 fb-1 at s = 7 TeV
µ+jets
600
tt
Single-top
W+jets
400
W+jets
Z+jets
500
Z+jets
QCD
400
QCD
300
300
200
200
100
0
100
-2
-1.5
-1
-0.5
0
0.5
1
1.5
η
0
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
lep
2
lep
(d)
800
Data
1.09 fb-1 at s = 7 TeV
700 e+jets
tt
Single-top
600
W+jets
500
Z+jets
Data
-1
1000 1.09 fb at s = 7 TeV
µ+jets
tt
Single-top
800
W+jets
Z+jets
600
QCD
400
events
(c)
events
η
300
QCD
400
200
200
100
0
0
20
40
60
80
100
120
140
mT,W [GeV/c2]
0
0
20
40
60
lep
(e)
80
100
120
140
mT,W [GeV/c2]
lep
(f)
Figure 4.10: Data to MC comparison of the transverse momentum and the pseudorapidity of the charged lepton and the transverse mass of the reconstructed leptonically decaying W boson in the electron+jets channel ((a),(c) and (e)) and in the
muon+jets channel ((b),(d) and (f)). All MC templates and the data-driven QCD
template are normalized to the fit result shown in table 4.4.
400
350
1.09 fb-1 at
e+jets
69
events
events
Data
s = 7 TeV
tt
Single-top
300
W+jets
250
Z+jets
1.09 fb-1 at
µ+jets
Data
s = 7 TeV
tt
Single-top
400
W+jets
Z+jets
300
Multijet
200
500
Multijet
200
150
100
100
50
0
0
50
100
150
200
250
p
T,1st jet
0
0
300
[GeV/c]
50
100
250
p
300
[GeV/c]
(b)
events
events
500
1.09 fb-1 at
e+jets
200
T,1st jet
(a)
600
150
Data
s = 7 TeV
tt
Single-top
700
1.09 fb-1 at
µ+jets
Data
s = 7 TeV
tt
Single-top
600
W+jets
400
800
W+jets
500
Z+jets
Multijet
Z+jets
Multijet
400
300
300
200
200
100
100
0
0
50
100
150
200
p
nd
T,2 jet
0
0
250
[GeV/c]
50
100
events
events
nd
250
[GeV/c]
(d)
1000
1.09 fb-1 at
e+jets
200
p
T,2 jet
(c)
800
150
Data
s = 7 TeV
tt
Single-top
1200
1000
1.09 fb-1 at
µ+jets
Data
s = 7 TeV
tt
Single-top
W+jets
W+jets
800
Z+jets
600
Z+jets
Multijet
Multijet
600
400
400
200
200
0
0
50
100
150
p
200
rd
T,3 jet
0
0
250
[GeV/c]
50
100
1.09 fb-1 at
e+jets
200
rd
250
[GeV/c]
(f)
Data
s = 7 TeV
tt
Single-top
1600
events
events
1800
p
T,3 jet
(e)
2000
150
2500
2000
1.09 fb-1 at
µ+jets
Data
s = 7 TeV
tt
Single-top
W+jets
1400
Z+jets
1200
W+jets
Z+jets
1500
Multijet
Multijet
1000
1000
800
600
500
400
200
0
0
20
40
60
80
100
p
120
th
T,4 jet
(g)
140
[GeV/c]
0
0
20
40
60
80
100
p
120
th
T,4 jet
140
[GeV/c]
(h)
Figure 4.11: Data to MC comparison of pT of the four leading jets in the
electron+jets channel ((a),(c),(e) and (g)) and in the muon+jets channel ((b),(d),(f)
and (h)). The jets are ordered by their transverse momentum from higher to lower
values. All MC templates and the data-driven QCD template are normalized to the
fit result shown in table 4.4.
700
600
Data
1.09 fb-1 at s = 7 TeV
e+jets
tt
Single-top
500
events
events
70
W+jets
Z+jets
400
800 1.09 fb-1 at s = 7 TeV
700 µ+jets
Data
600
W+jets
tt
Single-top
Z+jets
500
QCD
QCD
400
300
300
200
200
100
0
100
-2
-1.5
-1
-0.5
0
0.5
1
1.5
0
2
η st
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
η st
1 jet
1 jet
(b)
Data
-1
600 1.09 fb at s = 7 TeV
e+jets
500
tt
Single-top
events
events
(a)
W+jets
Z+jets
400
800 1.09 fb-1 at s = 7 TeV
700 µ+jets
Data
600
W+jets
tt
Single-top
Z+jets
500
QCD
QCD
400
300
300
200
200
100
0
100
-2
-1.5
-1
-0.5
0
0.5
1
1.5
0
2
η nd
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
η nd
2 jet
2 jet
(d)
Data
600 1.09 fb-1 at s = 7 TeV
e+jets
500
tt
Single-top
events
events
(c)
Data
1.09 fb-1 at s = 7 TeV
700 µ+jets
tt
Single-top
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QCD
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500 e+jets
tt
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400
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700 1.09 fb-1 at s = 7 TeV
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0
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-1
-0.5
4 jet
(g)
0
0.5
1
1.5
2
η th
4 jet
(h)
Figure 4.12: Data to MC comparison of the pseudorapidity of the four leading jets
in the electron+jets channel ((a),(c),(e) and (g)) and in the muon+jets channel
((b),(d),(f) and (h)). All MC templates and the data-driven QCD template are
normalized to the fit result shown in table 4.4.
Chapter 5
Measurement of the tt̄ Charge
Asymmetry
The aim of the analysis documented in this thesis is the measurement of the charge
asymmetry in top quark pair production. In the previous chapter the selection
of the dataset corresponding to an integrated luminosity of L = 1.09 fb−1 in the
lepton+jets channel was explained, yielding a tt̄ signal-enriched sample. The two
sensitive variables ∆|η| and ∆y 2 , which are used to measure the charge asymmetry, are not directly accessible in this dataset, but can be reconstructed utilizing
the topology of the lepton+jets channel and the kinematic quantities of the decay
products.
The reconstructed data is still contaminated with background events in which
no top quark pair is present. Their contributions have to be subtracted according
to their expected amount. Further, there are event selection and reconstruction
effects which influence the measurement and have to be corrected for. For this
purpose a regularized unfolding technique is applied resulting the values of the tt̄
charge asymmetry on parton level, which can be compared with the predictions from
theory.
In the following sections the reconstruction of the tt̄ event kinematics is introduced. Moreover, the unfolding technique and the measurement of the tt̄ charge
asymmetry are described and an overview of the systematic uncertainties is given.
5.1
Full Reconstruction of tt̄ Events
In order to calculate the two sensitive variables ∆|η| and ∆y 2 the four-momenta
of both top quarks have to be reconstructed from the measurable detector objects.
The topology of the lepton+jets channel is used to consecutively assign the selected
71
72
CHAPTER 5. MEASUREMENT OF THE T T̄ CHARGE ASYMMETRY
physical objects, i.e. isolated leptons, jets and the missing transverse energy, to the
final state particles. The four-momenta of both top quarks are received by summing
the four-momenta of the final state particles. The assignment is not unambiguous
and from the list of all possible reconstructions, only one hypothesis is chosen in
each event.
In this chapter the labels thad and tlep are used for the hadronically and the
leptonically decaying top quarks. According to this, the W bosons and bottom
quarks which originate from thad and tlep are labeled Whad , bhad , and Wlep , blep ,
respectively. Further, natural units with c = 1 are assumed in the calculations.
5.1.1
Reconstruction of all Possible Hypotheses
The charged lepton is the starting point of the reconstruction, since it is the only object which can be assigned without any ambiguities. It determines tlep to be the top
quark when the lepton is positively charged or to be the antitop quark when it possesses a negative charge. In order to reconstruct Wlep the missing transverse energy
is assumed to come totally from the neutrino, and therefore the x and y components
of 6 ET are assigned to px,ν and py,ν . The z component of the neutrino’s momentum
can be obtained by assuming the neutrino and the charged lepton originate from
Wlep yielding
pWlep = pl + pν ,
(5.1)
where pl and pν are the four momenta of the charged lepton and the neutrino,
respectively. Neglecting the invariant masses of the neutrino and the charged lepton
and assuming the W boson is produced on-shell, i.e. the invariant mass of pWlep is
set to mW = 80.4 GeV/c2 , a quadratic equation for pz,ν can be derived from equation
5.1 [154]:
m2W
2 q
2
2
2
~ T − (pz,l + pz,ν )2 .
~
= El + 6 ET + pz,ν − p~T,l +6 E
(5.2)
Here El is the energy of the charged lepton and p~T,l indicates its transverse momentum. pz,l and pz,ν denote the z components of the four-momenta of the charged
lepton and the neutrino, respectively. The solution of the quadratic equation 5.2 is
given by
s
µ2 p2z,l El2 p2T,ν − µ2
µpz,l
±
pz,ν = 2 ±
−
,
(5.3)
pT,l
p4T,l
p2T,l
where p2T,ν denotes the transverse momentum of the neutrino and the abbreviation
µ is defined as
m2
µ = W + pT,l · pT,ν · cos ∆φ.
(5.4)
2
Here ∆φ is the azimuthal angle between the charged lepton and the neutrino. When
the discriminant in equation 5.3 is positive, there are two real solutions for pz,ν and
5.1. FULL RECONSTRUCTION OF T T̄ EVENTS
73
both are taken into account. However, in nearly 30% of all cases the discriminant is
negative, since the reconstructed mT,W is larger than the W boson pole mass mW ,
mainly due to the finite resolution of the missing transverse energy. In these cases
the x and y components of the neutrino’s momentum are modified marginally such
that the imaginary part of pz,ν becomes zero.
In the next step all selected jets of an event are assigned to the remaining four
decay products in the tt̄ final state, i.e. blep , bhad and the two light quarks from
the hadronically decaying W boson, labeled q1 and q2 . The four-momenta of q1
and q2 are summed to obtain the four-momentum of Whad . Thereafter the fourmomenta of b quarks and W bosons are added to recieve pthad and ptlep . Again all
hypotheses are taken into account, so for an event with N selected jets there are
N · (N − 1) · (N − 2) · (N − 3) possible assignments of the jets to the four quarks
in the first place. Since solutions, where only the two jets assigned to q1 and q2 are
interchanged, are indistinguishable, the amount of hypotheses is reduced by a factor
two. The total amount of hypotheses Nhyp is given by
ν
Nhyp = Nsol
·
N · (N − 1) · (N − 2) · (N − 3)
,
2
(5.5)
ν
where Nsol
is the number of solutions obtained from equation 5.3 for pz,ν , which is
either one or two. For a typically tt̄ signal event with four selected jets and two
solutions for pz,ν there are 24 hypotheses.
5.1.2
Selection of one Reconstruction-Hypothesis per Event
From the list of possible hypotheses only one assignment is used for each event. For
simulated tt̄ events one can define the hypothesis which matches best the true event
kinematics by comparing the reconstructed objects to the true generated partons.
Considering the four-momenta a discriminator d can be defined as
gen ) + ∆R(p
gen ) + ∆R(p rec , p gen ) + ∆R(p rec , p gen ) , (5.6)
rec , p
rec , p
d = ∆R(pWlep
Whad
tlep
thad
Wlep
Whad
tlep
thad
which represents the distance in the η-φ plane between the reconstructed vectors
of the top quarks and W bosons, labeled with “rec”, and the generated vectors,
labeled with “gen”. The hypothesis with the minimal value d is considered as “best
possible” (bp) hypothesis, since this assignment of reconstructed objects describes
best the generated event.
In order to select an appropriate reconstruction hypothesis for each event based
solely on measured quantities, another criterion has to be introduced, which ideally chooses for each event the bp hypothesis. For this purpose the criterion ψ is
created, which gives the probability for a hypothesis to be the best possible one.
The performance of possible components of ψ are studied using the signal Monte
Carlo samples, where the reconstructed top quark four-momenta can be compared
74
to the generated four-momenta. The masses of both reconstructed top quarks and
the hadronically decaying W boson are sensitive to the jet assignment and therefore
they are good discriminator values. But in order to define ψ as a likelihood for reconstructing the bp hypothesis based on independent probabilities, the correlations
between the three masses have to be studied, as shown pairwise in figures 5.1(a) to
(c). There is a clear correlation between mWhad and mthad and therefore the three
masses are linearly decorrelated [152]. The yielding three new masses m1 , m2 and
m3 are linear combinations of the three physical masses, given by


 

mtlep
m1
1.00 −0.05 0.00
 m2  =  0.05
0.93 0.35  mthad  .
(5.7)
m3
−0.01 −0.35 0.94
mWhad
The pairwise correlation of the three decorrelated masses m1 , m2 and m3 are shown
in figure 5.1(d) to (f). One can clearly see, that m1 , m2 and m3 do not show these
dependences among each other. The distributions of the three masses for the best
possible hypotheses are depicted in figure 5.2(a) to (c). They show a narrow resonance, whereas the distributions of m1 , m2 and m3 for all possible hypotheses are
by far broader. By calculating the ratio of the two distributions, for each of the
decorrelated masses a likelihood ratio L(mi ) is obtained. L(mi ) is a measure for the
probability that a given mass value in a hypothesis corresponds to the best possible
hypothesis.
Another quantity which is sensitive to the assignment of the jets to the final
state quarks is the output value of the b tagger algorithm for a jet. In figure 5.3 the
probability Pb (x) that a jet with a certain b tag discriminator value x is assigned to
one of the b quarks in the best possible hypothesis is plotted against the value of the
b tagger discriminator. The resulting Pb (x) distribution can be parameterized via
Pb (x) =
0.6208
+ 0.2202 ,
1 + exp(−1.3530 · x + 2.9100)
(5.8)
also shown in figure 5.3. Eventually, using the decorrelated masses and the b tagger
information the discriminator ψ is defined as
ψ = L(m1 )L(m2 )L(m3 )Pb (xb,lep )Pb (xb,had )(1 − Pb (xq1 ))(1 − Pb (xq2 )) .
(5.9)
Since ψ gives the probability for a hypothesis to be the best possible in each event,
the hypothesis with the highest value for ψ is selected. In order to validate the
reconstruction method figures 5.4 and 5.5 show comparisons in several quantities
of the reconstructed objects Whad , Wlep , thad and tlep between the measured distributions and the simulations in the electron+jets and the muon+jets channel. The
comparisons show a very good agreement. Since the η and y distributions of the top
and the antitop quarks are of particular importance in this analysis, the comparison
between data and MC is shown in figure 5.6. Again a very good agreement is found.
1400
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75
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(b)
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(a)
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m2 [GeV/c2]
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100
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100
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50 100 150 200 250 300 350 400 450
m1 [GeV/c2]
0
m2 [GeV/c2]
(e)
(f)
18000
16000
14000
12000
hypotheses
hypotheses
hypotheses
Figure 5.1: Correlation between the reconstructed masses before and after the linear
decorrelation. The masses of the reconstructed top quarks mtlep and mthad and the
hadronically decaying W boson mWhad are correlated, as depicted in (a),(b) and (c).
The decorrelated masses m1 , m2 and m3 do not show this dependence, as illustrated
in (d),(e) and (f).
18000
16000
14000
12000
10000
10000
8000
8000
6000
6000
4000
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50
m1 [GeV/c2]
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10000
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100
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300
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500
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500
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0
50 100 150 200
0
50 100 150 200
m3 [GeV/c2]
(c)
×103
hypotheses
×103
500
400
900
×103
800
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500
300
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300
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600
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(b)
hypotheses
hypotheses
(a)
700
25000
20000
0
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100 150 200 250 300 350 400 450
30000
100
50
100 150 200 250 300 350 400 450
m1 [GeV/c2]
(d)
0
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100
200
300
(e)
400
500
600
m2 [GeV/c2]
0
-250 -200 -150 -100 -50
m3 [GeV/c2]
(f)
Figure 5.2: Decorrelated masses m1 , m2 and m3 for the best possible hypotheses in
(a),(b) and (c) and for all hypotheses in (d),(e) and (f).
Pb
76
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
b-tagger output
Figure 5.3: Probability distribution for a jet to be assigned to a b quark in the best
possible hypothesis as a function of the TCHE b tagger discriminator value of the
0.6208
jet. Pb is parameterized with Pb (x) = 1+exp(−1.3530·x+2.9100)
+ 0.2202.
Data
1.09 fb-1 at s = 7 TeV
e+jets
tt
Single-top
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77
events
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[GeV/c]
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T,Whad
[GeV/c]
(f)
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1.09 fb-1 at s = 7 TeV
e+jets
tt
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1.09 fb-1 at s = 7 TeV
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50 100 150 200 250 300 350 400 450 500
p
250
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T,t lep
events
Data
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mWhad [GeV/c2]
1.09 fb-1 at s = 7 TeV
e+jets
(e)
1.09 fb-1 at s = 7 TeV
e+jets
150
300
(d)
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(g)
100
(c)
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50
lep
350
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mt [GeV/c ]
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0
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[GeV/c ]
events
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(a)
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50
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3
η
4
0
-4
-3
-2
-1
(h)
0
1
2
3
η
4
Wlep
Whad
(i)
Figure 5.4: Data to MC comparison after reconstruction for the electron+jets channel. The masses of the hadronically decaying top quark (a), of the leptonically
decaying top quark (b) and of the hadronically decaying W boson (c) are shown.
Further the transverse momenta of the hadronically (d) and leptonically decaying
top quark (e) and of the hadronically (f) and leptonically decaying W boson (g)
are depicted. Finally, the pseudorapidities of the hadronically (h) and leptonically
decaying W boson (i) are illustrated. The simulated distributions are normalized to
the fit results.
1400
tt
Single-top
1000
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500
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1.09 fb-1 at s = 7 TeV
µ+jets
tt
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(b)
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(c)
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(a)
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(e)
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tt
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(d)
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(g)
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3
η
4
0
-4
-3
-2
-1
(h)
0
1
2
3
η
4
Wlep
Whad
(i)
Figure 5.5: Data to MC comparison after reconstruction for the muon+jets channel.
The masses of the hadronically decaying top quark (a), of the leptonically decaying
top quark (b) and of the hadronically decaying W boson (c) are shown. Further
the transverse momenta of the hadronically (d) and leptonically decaying top quark
(e) and of the hadronically (f) and leptonically decaying W boson (g) are depicted.
Finally, the pseudorapidities of the hadronically (h) and leptonically decaying W
boson (i) are illustrated. The simulated distributions are normalized to the fit
results.
600
500
events
events
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1.09 fb-1 at s = 7 TeV
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tt
Single-top
W+jets
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(a)
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300
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0
1
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3
4
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0
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0
1
2
3
4
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t
(h)
Figure 5.6: Data to MC comparison of the kinematic distributions for the reconstructed top quarks. The pseudorapidity and the rapidity of the top quark are depicted for the electron+jets (a),(c) and the muon+jets channel (e),(g). Further the
pseudorapidity and the rapidity of the antitop quark are shown for the electron+jets
(b),(d) and the muon+jets channel (f),(h). The simulated distributions are normalized to the fit results.
80
5.2
Background Subtraction and Unfolding
After the reconstruction of the four-momenta of top and antitop quarks the distributions of the two sensitive variables ∆|η| and ∆y 2 are obtained. In figure 5.7
the distributions in the electron+jets, the muon+jets and the combined lepton+jets
channel are depicted. According to equation 1.11 an uncorrected raw charge asym∆|η|
metry can be calculated for both variables, yielding AC,raw = −0.004 ± 0.009 and
2
A∆y
C,raw = −0.004 ± 0.009. These results are a first subgoal of the analysis, however
they can not be compared to any theory prediction, since several effects distort the
measurement at this stage. Despite the tight selection of tt̄ events, about 20% of all
events are still expected to be background events. Since the background events do
not contain any top quark pair, their contributions have to be subtracted. For this
purpose the background templates are normalized to the expected amount obtained
from the fit result, shown in table 4.4. The template of the QCD multi-jet process
has been obtained by a data-driven method. In figure 5.8 the comparison between
the shapes from the QCD MC templates in the signal region and the data-driven
QCD template for the sensitive variables are depicted. Within the uncertainties a
reasonable agreement can be found, which validates the usage of this sample.
The background processes are considered to be independent and their normalized
templates are subtracted from the reconstructed ∆|η| and ∆y 2 distributions. In this
process a Gaussian error propagation is performed taking the measured uncertainties
on the background rates as input. After this step the calculation of the background2
∆|η|
subtracted (bgs) asymmetries results AC,bgs = −0.009±0.010 and A∆y
C,bgs = −0.007±
0.010.
5.2.1
Regularized Unfolding
After the background subtraction one can assume that the reconstructed distributions for ∆|η| and ∆y 2 contain only signal events, but still the backgroundsubtracted asymmetries can not be compared to the theory predictions. On the
one hand the event selection efficiency of tt̄ events is not flat in the two distributions
as shown in figure 5.9(b) and (d), mainly due to the cut on the isolated lepton and on
four or more jets. This affects the shape of the distributions and therefore the measurement of the charge asymmetry. On the other hand the reconstruction method
explained above has an even larger impact on the measurement due to migration
effects. In many events not directly the best possible hypothesis is chosen and therefore the reconstructed four-momenta of the top quarks are distorted. Further, in
consequence of the possibility that jets from the tt̄ decay lie outside the detector
acceptance and due to the finite energy resolution of the CMS detector system the
four-momenta of the top quarks in the best possible hypothesis are smeared compared to the true four-vectors. The smearing effects are visualized in a migration
matrix for each variable, depicted in figure 5.9(a) and (c), where the reconstructed
600
500
events
events
5.2. BACKGROUND SUBTRACTION AND UNFOLDING
Data
1.09 fb-1 at s = 7 TeV
A C,raw= -0.008 ± 0.013
tt
Single-top
e+jets
900
800
700
W+jets
Data
1.09 fb-1 at s = 7 TeV
A C,raw= -0.008 ± 0.013
tt
Single-top
e+jets
W+jets
600
Z+jets
400
81
Z+jets
500
QCD
300
400
200
300
QCD
200
100
0
-4
100
-3
-2
-1
0
1
2
3
700
1.09 fb-1 at s = 7 TeV
A C,raw= -0.001 ± 0.012
600
µ+jets
-3
-2
-1
0
1
2
Data
tt
Single-top
1000
800
t
Data
1.09 fb-1 at s = 7 TeV
A C,raw= -0.000 ± 0.012
tt
Single-top
µ+jets
W+jets
Z+jets
Z+jets
QCD
400
4
t
W+jets
500
3
(yt-y )(yt+y )
t
events
events
t
800
0
-4
4
|η |-|η |
QCD
600
300
400
200
200
100
0
-4
-3
-2
-1
0
1
2
3
1200
Data
1.09 fb-1 at s = 7 TeV
A C,raw= -0.004 ± 0.009
tt
Single-top
l+jets
-1
0
1
2
2000
1800
4
tt
Single-top
l+jets
W+jets
Z+jets
QCD
1000
600
800
400
600
t
Data
1.09 fb-1 at s = 7 TeV
A C,raw= -0.004 ± 0.009
1200
QCD
3
(yt-y )(yt+y )
1400
Z+jets
800
-2
t
1600
W+jets
1000
-3
t
events
events
t
1400
0
-4
4
|η |-|η |
400
200
0
-4
200
-3
-2
-1
0
1
2
3
4
|η |-|η |
t
t
0
-4
-3
-2
-1
0
1
2
3
4
(yt-y )(yt+y )
t
t
Figure 5.7: Distributions of ∆|η| and ∆y 2 for the electron+jets (a),(b), muon+jets
(c),(d) and the combined lepton+jets channel (e),(f). In the plots additionally the
raw asymmetries are shown which are calculated via equation 1.11. The simulation
is normalized to the Monte Carlo prediction.
norm. to unit area
norm. to unit area
82
0.45
0.4
e+jets
QCD template (data)
0.35
MC QCD
0.3
0.25
0.5
e+jets
QCD template (data)
0.4
MC QCD
0.3
0.2
0.2
0.15
0.1
0.1
0.05
0
-4
-3
-2
-1
0
1
2
3
0
-5
4
|η |-|η |
t
-4
-3
-2
0
1
2
3
4
5
(yt-y )(yt+y )
t
(a)
t
(b)
0.4
0.35 µ+jets
norm. to unit area
norm. to unit area
-1
t
QCD template (data)
0.3
MC QCD
0.25
0.2
0.45
0.4 µ+jets
QCD template (data)
0.35
MC QCD
0.3
0.25
0.2
0.15
0.15
0.1
0.1
0.05
0
-4
0.05
-3
-2
-1
0
1
2
3
4
|η |-|η |
t
0
-5
-4
-3
-2
-1
(c)
0
1
2
3
4
5
(yt-y )(yt+y )
t
t
t
(d)
Figure 5.8: QCD background shape comparison between MC based template and
data-driven template in ∆|η| and ∆y 2 for the electron+jets ((a), (b)) and the
muon+jets ((c), (d)) channel.
values are plotted against the generated true values for the MC simulation. To
correct for these effects a regularized unfolding procedure [155] is applied using a
generalized matrix inversion method.
In general, the smearing of the true spectrum ~x yielding the measured distribution w
~ can be described by a transition matrix A:
w
~ = A~x .
(5.10)
The entries of the vectors w
~ and ~x represent the bin entries of the histograms. For
the purpose of obtaining a robust method it is recommended to use twice the number
of bins for the measured spectrum than for the true spectrum [155]. In this analysis
twelve bins are used for the former and six bins are utilized for the latter. The
widths of the bins have been chosen such, that each bin contains approximately
equal numbers of events. The transition matrix A is the product of the migration
matrix and a diagonal matrix with the event selection efficiencies for the different
bins on the diagonal and all other elements equal to zero.
In order to derive the true spectrum ~x equation 5.10 has to be solved. For this
purpose this equation can be formulated as a least-square (LS) problem
FLS (~x) = (A~x − w)
~ T Vw−1 (A~x − w)
~ ,
(5.11)
4
selection efficiency
0.3
3
0.25
2
0.2
1
0
0.15
t
t
|η |-|η |(reconstructed)
0.06 l+jets
0.05
0.04
0.03
-1
0.01
-3
-4
-4
0.02
0.1
-2
0.05
-3
-2
-1
0
1
2
3
0
-4
4
|η |-|η |(generated)
t
-3
-2
0
1
2
3
4
|η |-|η | (generated)
t
t
(b)
3
0.3
2
0.25
selection efficiency
4
1
0.2
0
t
t
-1
t
(a)
(y -y )(y +y )(reconstructed)
83
0.15
-2
0.1
0.06 l+jets
0.05
0.04
0.03
t
t
-1
0.07
0.01
-3
-4
-4
0.02
0.05
-3
-2
-1
0
1
2
3
0
-4
4
(yt-y )(yt+y )(generated)
t
-3
-2
-1
0
1
t
(c)
2
3
4
(yt-y )(yt+y ) (generated)
t
t
(d)
Figure 5.9: Migration matrix and event selection efficiency as functions of ∆|η|
(a),(b) and ∆y 2 (c),(d). In the migration matrices each column is normalized to
unity. Therefore the entries of the migration matrix correspond to the probability
that a selected tt̄ event with a certain generated value for ∆|η| or ∆y 2 is found with
a specific value after reconstruction.
where Vw is the covariance matrix of the measured distribution w.
~ The solution
vector ~x is found by minimizing equation 5.11. Introducing a generalized inverse
matrix A# via
A# = (AT Vw−1 A)−1 AT Vw−1 ,
(5.12)
the solution vector ~x is given by
~xLS = A# w
~ .
(5.13)
Since the transition matrix A exhibits singular values of different orders of magnitude, the generalized inversion of A will be dominated by the smallest singular
values, which belong to highly fluctuating eigenmodes of w.
~ Therefore the simple
solution in equation 5.13 is unstable and shows huge fluctuations for small changes
in w.
~ The details of the singular value analysis can be found in [156]. In this analysis a more advanced unfolding method is used, where two additional terms are
introduced to regularize the problem and to avoid unphysical fluctuations [157,158]:
2
F (~x, κ) = FLS (~x) + τ ||L~x|| + κ(Nobs −
n
X
i=a
(A~xi ))2 .
(5.14)
log ||Lx||2
84
9.25
(yt-y )(yt+y )
9.2
t
t
9.15
9.1
9.05
9
8.95
8.9
8.85
8.8
0.3
(a)
0.4
0.5
0.6
0.7
0.8
log ||Ax-y||2
(b)
Figure 5.10: Sketch of the L-curve used to derive an optimal regularization parameter τ for the unfolding [160] in (a). As an example from the analysis in (b) the
L-curve obtained by a random pseudo experiment is shown for the ∆y 2 variable. For
different values τ the norm of FLS (~x) is plotted logarithmically versus the logarithm
of ||L~x||2 after the minimization process. The optimal value of τ is found at the
kink with highest curvature.
The first of the two additional terms is the regularization term ||L~x||2 , which is scaled
with the parameter τ . The matrix L is introduced such that L~x is proportional to
the second derivatives of ~x. A proper choice of the τ parameter, which denotes the
strength of the applied regularization, is very important for the method. On the one
hand if the value for the τ parameter is too small, the minimization of equation 5.14
is not affected by the regularization and the result shows unphysical fluctuations.
On the other hand if τ is chosen too large the minimization of F (~x, κ) will be
dominated by the regularization term. This would lead to a flat spectrum, since the
regularization term tries to minimize the second derivatives of ~x and therefore the
curvature of the distribution. The L-curve method [159] is a good way to find an
appropriate choice for the regularization parameter τ . Here the norm of FLS (~x) is
plotted logarithmically versus the logarithm of ||L~x||2 after the minimization process
for several fixed values of τ . Exemplary L-shaped curves are illustrated in figure 5.10.
The optimal value of τ is found at the point with highest curvature, which represents
the region, where the solution is neither dominated by the not regularized solution
from minimizing just FLS (~x) nor from the regularization term.
P
The second additional term κ(Nobs − ni=a (A~xi ))2 in equation 5.14 fixes the
norm of the solution. Here Nobs denotes the number of observed events and κ is
a Lagrangian multiplier. This constraint on the normalization is important for instance for measured distributions with small numbers of observed events, since their
statistical uncertainties are Poisson-like and cannot be approximated by Gaussian
uncertainties [161] as it is usually assumed in the minimization of equation 5.11.
85
The statistical uncertainty on the measurement can be calculated from the output of the unfolding procedure containing the full covariance matrix Vx via
 ∂AC 
∂N1
∂AC
∂AC
 .. 
2
(5.15)
...
Vx  .  .
σ AC =
∂N1
∂N6
∂AC
∂N6
∂AC
∂Ni
in equation 5.15 are given for
2N −
∂AC
=
∂N4,5,6
(N + + N − )2
(5.16)
Considering equation 1.11 the partial derivatives
the bins on the positive side by
and for bins on the negative side by
∂AC
−2N +
=
.
∂N1,2,3
(N + + N − )2
5.2.2
(5.17)
Consistency Checks with Pseudo Experiments
Before applying the unfolding procedure to the observed data distribution the optimal value for the τ parameter has to be determined and the method has to be
validated. For this purpose pseudo experiments are executed. In each pseudo experiment a pseudo dataset is obtained by randomly drawing signal and background
events from the Monte Carlo templates for the reconstructed ∆|η| and ∆y 2 distributions according to the expected amount of events due to the processes. These
numbers are set separately for each pseudo dataset by throwing random values from
Gaussian distributions centered around the expected event rates given in table 4.4
with widths corresponding to the respective fit uncertainties. In order to take statistical fluctuations into account the thrown numbers for each process are further
randomly varied according to a Poisson distribution around the respective number. In the yielding pseudo datasets the expected amount of background events is
subtracted and the unfolding method is applied as explained above.
To find an appropriate regularization parameter the unfolding method is applied
on one pseudo dataset using 100 different values for τ between τmin = 10−5 and
τmax = 10−2 . For the 100 τ values the pairs of variates log ||FLS (~x)|| and log ||L~x||2
are plotted yielding the L-curve for this particular pseudo experiment. The point
on the L-curve with the highest curvature determines the optimal value for τ . This
procedure is repeated in total for 1000 pseudo datasets. For the measurement the
mean value of the 1000 yielding τ parameters is used, which is for the ∆|η| variable
τη = 2.2 · 10−5
(5.18)
τy = 3.4 · 10−5 .
(5.19)
and for ∆y 2
Mean
RMS
0.0006184
0.09049
14000 |ηt|-|η |
t
l+jets
12000
10000
8000
6000
12000
Mean
RMS
-0.0009243
0.1142
|ηt|-|η |
t
10000 l+jets
8000
6000
4000
number of pseudoexp.
86
Mean
RMS
12000 |η |-|η |
t
0.001184
0.1134
t
10000
l+jets
8000
6000
4000
4000
2000
2000
0
0.2 0.4 0.6 0.8
2000
0
-1 -0.8 -0.6 -0.4 -0.2
1
0
relative diff. in bin 1
0.0008585
0.111
l+jets
8000
6000
4000
2000
Mean
RMS
t
Mean
RMS
12000 |η |-|η |
t
-0.0006177
0.1111
t
10000
0.2 0.4 0.6 0.8
1
l+jets
8000
6000
4000
1
14000
Mean
RMS
-0.000971
0.09545
|ηt|-|η |
t
12000 l+jets
10000
8000
6000
4000
2000
0
-1 -0.8 -0.6 -0.4 -0.2
(d)
0.2 0.4 0.6 0.8
(c)
2000
0
0
(b)
12000 |η |-|η |
t
0
-1 -0.8 -0.6 -0.4 -0.2
0
-1 -0.8 -0.6 -0.4 -0.2
1
(a)
10000
0.2 0.4 0.6 0.8
0
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
0
-1 -0.8 -0.6 -0.4 -0.2
(e)
0
0.2 0.4 0.6 0.8
1
(f)
Figure 5.11: Relative difference in each bin between the unfolded ∆|η| distribution
in pseudo experiments and the true spectrum. The root mean square (RMS) in each
distribution is a measure for the expected statistical uncertainty in each bin.
In total 50000 pseudo experiments are performed and unfolded with the fixed
τ parameters from above to check the performance of the method. Each unfolded
spectrum is compared bin by bin to the true distribution from the tt̄ signal sample.
For each bin the relative difference between the unfolded and the true spectrum is
calculated, yielding the distributions in figure 5.11 for ∆|η| and 5.12 for ∆y 2 . As one
can see in the figures all relative differences follow a normal distribution centered
around zero, so there is no bias within the unfolding at this point.
Additionally for each bin i the pull Pi is calculated, defined as
Nitrue − Ni
Pi =
,
σi
(5.20)
where Nitrue is the bin content of the generated spectrum and Ni and σi denote the
unfolded bin content and the uncertainty provided by the unfolding method. Figure
5.13 depicts the pull distributions for ∆|η| and figure 5.14 for ∆y 2 . They all follow
a Gaussian distribution centered around zero with a width of one. The value one
for the widths indicates that the uncertainties in each bin are treated correctly.
0.00429
0.07176
14000
12000
10000
8000
Mean
RMS
-0.01106
0.09028
14000 (yt-yt)(yt+yt)
l+jets
12000
10000
8000
6000
Mean
RMS
18000 (y -y )(y +y )
t t
t
t
16000 l+jets
87
Mean
RMS
16000
14000
0.005309
0.08536
(yt-y )(yt+y )
t
t
l+jets
12000
10000
8000
6000
6000
4000
4000
4000
2000
2000
0
0.2 0.4 0.6 0.8
2000
0
-1 -0.8 -0.6 -0.4 -0.2
1
0
0.004676
0.08403
t
t
12000
10000
8000
6000
Mean
RMS
Mean
RMS
8000
6000
2000
2000
0
0.2 0.4 0.6 0.8
1
Mean
RMS
18000
0.004429
0.07688
l+jets
14000
12000
10000
8000
6000
4000
2000
0
-1 -0.8 -0.6 -0.4 -0.2
1
-0.009575
0.09233
10000
4000
0.2 0.4 0.6 0.8
(c)
l+jets
12000
4000
0
-1 -0.8 -0.6 -0.4 -0.2
0
(b)
(yt-y )(yt+y )
14000 l+jets
0
-1 -0.8 -0.6 -0.4 -0.2
1
(a)
16000
0.2 0.4 0.6 0.8
0
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
0
-1 -0.8 -0.6 -0.4 -0.2
1
0
(d)
0.2 0.4 0.6 0.8
1
(e)
(f)
gaussian fit:
|ηt|-|η |
t
3000 l+jets
mean = 0.016 +- 0.004
sigma = 1.001 +- 0.003
2500
2000
1500
gaussian fit:
3500 |ηt|-|η |
t
l+jets
3000
mean = -0.001 +- 0.004
sigma = 0.996 +- 0.003
2500
2000
1500
2000
1500
1000
500
500
-2
-1
0
1
2
3
4
5
-4
-3
-2
-1
0
1
2
pull in bin 1
4
0
-5
5
gaussian fit:
|ηt|-|η |
mean = 0.013 +- 0.004
t
3000 l+jets
sigma = 1.004 +- 0.003
2500
2000
1500
3500
gaussian fit:
|ηt|-|η |
mean = 0.001 +- 0.004
t
3000 l+jets
sigma = 1.000 +- 0.003
2500
1
2
3
4
5
2000
1500
0
-5
-4
-3
-2
-1
pull in bin 4
(d)
2
4
5
0
1
2
3
4
5
mean = -0.000 +- 0.004
sigma = 1.003 +- 0.003
2000
1500
0
-5
-4
-3
-2
-1
pull in bin 5
(e)
3
2500
500
0
1
gaussian fit:
t
1000
-1
0
3000 l+jets
500
-2
-1
3500 |η |-|η |
t
500
-3
-2
(c)
1000
-4
-3
pull in bin 3
1000
0
-5
-4
(b)
(a)
3500
3
pull in bin 2
-3
sigma = 0.999 +- 0.003
2500
500
-4
mean = 0.016 +- 0.004
t
3000 l+jets
1000
0
-5
gaussian fit:
3500 |η |-|η |
t
1000
0
-5
3500
Figure 5.12: Relative difference in each bin between the unfolded ∆y 2 distribution
in pseudo experiments and the true spectrum. Again the RMS is given for each
distribution.
0
1
2
3
4
5
pull in bin 6
(f)
Figure 5.13: Pull distributions for each bin of the unfolded ∆|η| spectrum in pseudo
experiments. For each bin a Gaussian function is fitted in order to obtain the mean
and the width (σ) of the distribution. The single values for mean and σ are given
in the figures and show a good agreement to the expectation.
3500
gaussian fit:
(yt-y )(yt+y )
t
mean = 0.069 +- 0.004
t
3000 l+jets
sigma = 0.995 +- 0.003
2500
2000
1500
3500
gaussian fit:
(yt-y )(yt+y )
t
t
3000 l+jets
mean = -0.114 +- 0.004
sigma = 1.000 +- 0.003
2500
2000
1500
3500
t
1500
1000
500
500
-2
-1
0
1
2
3
4
5
-4
-3
-2
-1
0
1
2
pull in bin 1
gaussian fit:
(yt-y )(yt+y )
t
mean = 0.060 +- 0.004
t
3000 l+jets
sigma = 0.999 +- 0.003
2500
2000
1500
3500
-1
0
1
2
3
4
1500
0
-5
5
sigma = 1.001 +- 0.003
2000
500
-2
mean = -0.096 +- 0.004
t
2500
500
-3
gaussian fit:
(yt-y )(yt+y )
t
1000
-4
0
-5
5
-4
-3
-2
-1
0
1
2
3
4
5
pull in bin 3
(c)
3000 l+jets
1000
0
-5
4
(b)
(a)
3500
3
pull in bin 2
-3
sigma = 0.996 +- 0.003
2000
500
-4
mean = 0.068 +- 0.004
t
2500
1000
0
-5
gaussian fit:
(yt-y )(yt+y )
3000 l+jets
1000
0
-5
88
3500
gaussian fit:
(yt-y )(yt+y )
t
mean = 0.066 +- 0.004
t
3000 l+jets
sigma = 0.998 +- 0.003
2500
2000
1500
1000
500
-4
-3
-2
-1
0
1
2
pull in bin 4
3
4
0
-5
5
-4
-3
-2
-1
0
1
2
pull in bin 5
(d)
3
4
5
pull in bin 6
(e)
(f)
-0.001388
0.03055
t
6000 l+jets
5000
4000
3000
2000
1000
2000
gaussian fit:
1800 |ηt|-|η |
t
1600 l+jets
mean = -0.008 +- 0.004
sigma = 0.998 +- 0.003
1400
1200
1000
800
-0.2
-0.1
0
0.1
0.2
4000
1000
-4
-3
-2
-1
0
1
2
3
4
0
5
0.026 0.028 0.03 0.032 0.034 0.036 0.038 0.04
σ(A )
pull(A )
C
Mean
RMS
-0.002312
0.02656
t
l+jets
6000
5000
4000
3000
2000
5000
C
(b)
(yt-y )(yt+y )
1800
(c)
gaussian fit:
(yt-y )(yt+y )
t
mean = 0.003 +- 0.004
t
1600 l+jets
1400
sigma = 0.995 +- 0.003
1200
1000
800
600
400
1000
0
-0.3
6000
2000
(a)
8000
0.03049
0.0005732
t
3000
AC
t
Mean
RMS
|ηt|-|η |
8000 l+jets
7000
400
0
-5
0.3
9000
600
200
0
-0.3
7000
Mean
RMS
|ηt|-|η |
7000
Figure 5.14: Pull distributions for each bin of the unfolded ∆y 2 spectrum in pseudo
experiments. The single values for mean and σ are given in the figures and show a
good agreement to the expectation.
Mean
RMS
10000
0.02658
0.0005313
(yt-y )(yt+y )
t
8000
t
l+jets
6000
4000
2000
200
-0.2
-0.1
0
0.1
0.2
0.3
0
-5
-4
-3
-2
-1
AC
(d)
0
1
2
3
4
5
pull(A )
0
0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.034
σ(A )
C
(e)
C
(f)
Figure 5.15: Consistency checks for the unfolding routine in pseudo experiments
for ∆|η| and ∆y 2 . In (a) and (d) the unfolded asymmetries in the pseudo experiments are shown. The mean values are in good agreement with the true values.
The pull distributions and the distribution of the statistical uncertainty in pseudo
experiments are depicted in (b),(e) and (c), (f) respectively.
89
Beside the bin by bin comparison, also the result of the charge asymmetry in
the pseudo experiments is checked. In figures 5.15(a) and (d) the distribution of
the unfolded asymmetries are shown. The mean values are in good agreement with
the generated asymmetries in the sample, i.e. −0.0016 in ∆|η| and −0.0022 in
∆y 2 . Further, the pull distributions shown in figures 5.15(b) and (e) have widths
of one, indicating the statistical uncertainties are neither over- nor underestimated.
Finally, figures 5.15(c) and (f) show the estimated statistical uncertainties of the
measurement resulting ±0.031 using ∆|η| and ±0.027 using ∆y 2 .
5.2.3
Linearity Checks
For all pseudo experiments explained above, the standard tt̄ template possessing a
very small charge asymmetry of about −0.2% is used. In order to check if generally
large charge asymmetries that are predicted for physics beyond the Standard Model
are unfolded correctly a linearity check has to be performed. The linearity check is
accomplished using samples with different generated asymmetries over a large range.
An artificial asymmetry can be obtained by re-weighting the events in the standard
template according to their value in ∆|η| or ∆y 2 , respectively. The factors used in
this analysis are
k · (|ηt | − |ηt̄ |) + 1 ,
(5.21)
k · (yt − yt̄ )(yt + yt̄ ) + 1 .
(5.22)
In both cases k is varied between 0.25 and −0.25. In table 5.1 the different choices
of k are listed together with the thereby generated asymmetry in the sample. Again
50000 pseudo experiments are performed for each of the shifted templates. The
mean values of the unfolded distributions are also given in table 5.1 and show a
good agreement to the generated asymmetries within the uncertainties.
To get a closer view of the results of table 5.1, in figure 5.16 the unfolded asymmetries are plotted against the generated values together with the bisector. Ideally
all points should lie on the bisector, which is the case for the ∆|η| variable. However
the measurements for the ∆y 2 variable show a slope unequal to one. This bias is
understood as an effect from the non-flat event selection efficiencies in the outermost
bins and has to be corrected for by dividing the measured result by the actual slope:
2
2
A∆y
C (corr.)
A∆y (unf.)
.
= C
0.94
(5.23)
90
∆|η|
k
true AC
0.25
0.2172
0.1736
0.20
0.15
0.1298
0.0860
0.10
0.05
0.0422
−0.0454
−0.05
−0.0892
−0.10
−0.15
−0.1329
−0.1766
−0.20
−0.25
−0.2202
2
∆|η|
unfolded AC
true A∆y
C
0.2196 ± 0.0305
0.2691
0.1752 ± 0.0305
0.2194
0.1311 ± 0.0304
0.1656
0.0870 ± 0.0304
0.1098
0.0425 ± 0.0304
0.0538
−0.0456 ± 0.0304
−0.0582
−0.0894 ± 0.0305
−0.1141
−0.1338 ± 0.0307
−0.1698
−0.1776 ± 0.0308
−0.2235
−0.2220 ± 0.0310
−0.2731
2
unfolded A∆y
C
0.2533 ± 0.0272
0.2040 ± 0.0270
0.1526 ± 0.0269
0.1008 ± 0.0269
0.0495 ± 0.0267
−0.0542 ± 0.0268
−0.1059 ± 0.0269
−0.1575 ± 0.0270
−0.2090 ± 0.0273
−0.2584 ± 0.0275
0.4
unfolded AC
unfolded AC
Table 5.1: Unfolded charge asymmetries AC for ∆|η| and ∆y 2 compared to the
∆|η|
different true values. The asymmetry in ∆|η| is denoted with AC and the asym∆y 2
metry in ∆y 2 with AC
. The different asymmetries are generated by reweighting
the events with (k · ∆|η| + 1) and (k · ∆y 2 + 1), respectively. The used values for k
are shown in the table.
0.3 |ηt|-|ηt|
l+jets
0.2
0.1
0
0.4
0.3 (yt-yt)(yt+yt)
l+jets
0.2
0.1
0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.4
-0.4
-0.3
bisector
Fit: y= 1.0085 x + 0.0005
-0.3
-0.2
-0.1
(a)
0
0.1
0.2
0.3
0.4
generated A
-0.4
-0.4
bisector
Fit: y= 0.9354 x + 0.0000
-0.3
-0.2
-0.1
C
0
0.1
0.2
0.3
0.4
generated A
C
(b)
Figure 5.16: Linearity check for ∆|η| (a) and ∆y 2 (b). The shown uncertainties
represents the mean statistical uncertainties for the unfolding in the pseudo experiments. Ideally all points should lie on the bisector. While the different measurements
for ∆|η| are in good agreement with the generated asymmetries, the checks for ∆y 2
show a slope unequal to one and therefore are needed to be corrected.
5.3. SYSTEMATIC UNCERTAINTIES
5.3
91
Systematic Uncertainties
There might be several sources of systematic uncertainties, which influence the measurement of the tt̄ charge asymmetry. In principle, only relative uncertainties in the
tt̄ selection efficiency and the acceptance can alter the shapes of the distributions
of ∆|η| and ∆y 2 . Overall rate uncertainties will only change the normalization of
the unfolded spectrum but have no influence on the result of the charge asymmetry
itself, and are therefore not taken into account. In order to determine the impact of
the systematic uncertainties all affected templates are shifted according to the systematic and 50000 pseudo experiments are performed using the distorted templates
to create pseudo datasets. The pseudo datasets are unfolded with the smearing
matrix obtained from the standard Monte Carlo sample thereafter. To estimate the
influence on the charge asymmetry the difference between the systematically distorted asymmetry and the default value is calculated. In the following the sources
of systematic uncertainties are introduced.
Jet Energy Scale (JES)
In order to estimate the systematic uncertainty due to the imperfect knowledge of
the jet energy scale, all jets in the MC templates are shifted simultaneously by
either +1σ or −1σ within their η and pT dependent uncertainty [162]. The resulting
shifts are propagated to the missing transverse energy. Since the model for the QCD
multi-jet process is obtained from data, the JES shift is not applied on it.
Jet Energy Resolution (JER)
As mentioned in section 4.1.5 the resolution of all jets in the MC events has to
be worsen by 10% in order to match the resolution in data. The result of 10%
arises from jet asymmetry measurements and has an uncertainty of 10% for jets in
the region of |η| < 1.5, 15% for jets with 1.5 < |η| < 2.0 and 20% for jets with
|η| > 2.0 [163]. To account for the uncertainty of the resolution, instead rescaling
the jet energy resolution with the nominal value of 10%, it is shifted by either 0%
or 20% for all jets with |η| < 1.5. For jets with 1.5 < |η| < 2.0 the resolution
is reweighted by either −5% or 25% and for jets with |η| > 2.0 the resolution is
changed by −10% or 30%, respectively.
Factorization and Renormalization Scale Uncertainty
The used Monte Carlo samples for the tt̄ signal and most of the background processes
have been produced with MadGraph, which calculates the LO amplitudes for the
processes dependent on the factorization and renormalization scale. To quantify the
92
influence of the scale two alternative Monte Carlo tt̄ samples are utilized. These
“scale up” and “scale down” samples were re-generated multiplying the event-byevent Q2 -scale by 0.25 for the “scale down” and by 4.0 for the “scale up” sample.
Matching Threshold Uncertainty
The transition between the hard process generated with MadGraph and the parton showering performed with Pythia requires a certain matching threshold. To
estimate the uncertainty due to the choice of this matching threshold two alternative
tt̄ samples are generated, where the jet matching threshold is varied by a factor of
0.5 or 2, respectively, from its default value.
Initial and Final State Radiation (ISR/FSR)
The influences of additional or less ISR and FSR are investigated by utilizing shifted
MC samples for the tt̄ process. The strength of the shift can be adjusted in the
parton shower generator Pythia.
PDF Uncertainties
The measured charge asymmetry might be also dependent on the parametrization
of the PDF. For this purpose in total 44 alternative templates for tt̄, W +jets and
Z+jets are generated using 22 different parametrizations according to the eigenvectors of CTEQ6.6 [52]. The resulting 44 uncertainties of the 22 parametrizations are
independent and summed in quadrature.
b Tagging Uncertainty
Another influence on the charge asymmetry might arise from an η dependent b tagging efficiency. The efficiency for the used TCHE tagger has been measured in [164]
and appears to be flat. However, by varying the efficiency within the uncertainties
of this measurement differently for central and peripheral region a non-flat b tagging
efficiency can be obtained. Therefore events with a b tagged jet in the region with
|η| < 1.2 are weighted up, whereas those with a b jets in the region with |η| > 1.2
are weighted down and reversed.
5.3. SYSTEMATIC UNCERTAINTIES
93
Lepton Identification and Event Selection Efficiency
If the event selection efficiencies are different for positively and negatively charged
leptons this leads to an artificial charge asymmetry. Therefore events with positively
2σ
|ηl | or
and negatively charged leptons are scaled differently either with 1 + σ − ηmax
2σ
with 1 − σ + ηmax |ηl |. Here σ denotes the lepton identification uncertainty and
ηmax = 2.5 for electrons and ηmax = 2.1 for muons.
Lepton Trigger Efficiencies
As discussed in section 4.1.5 the leptonic part of the triggers which are applied on
the observed data is not simulated on the generated Monte Carlo samples. The
overall efficiency of the HLT paths does not affect the charge asymmetry and is
assumed to be one. If the efficiencies are different for electrons and muons there can
be a systematic shift of the measurement. In principle the single overall efficiencies
for electron and muon events are determined by the fit result for tt̄ events. To
account for effects of the uncertainties of the fit shown in table 4.4, the unfolding of
pseudo experiments is performed with re-weighted smearing matrices and selection
efficiencies, where events with electrons are weighted up and events with muons are
weighted down within the uncertainties and vice versa.
W +jets modeling
In the background estimation the rates of W + +jets and W − +jets have been determined separately. Since in both templates the shapes are not necessarily similar,
variations of the rates can lead to an asymmetry in the reconstructed ∆|η| and ∆y 2
distributions for the W +jets background. In order to account for the effects from
the W +jets modeling, in two sets of pseudo experiments the rate of one subsample
is increased and the rate of the other is decreased within the uncertainties of the
background estimation. However, a detailed study shows the shapes of both templates are very similar, and therefore no significant influence on the measurement is
found.
QCD Modeling
In figure 5.17 the shapes of the data-driven QCD template separated into two subsamples according to the lepton charge are shown. The asymmetries in the distributions of the reconstructed sensitive variables are slightly different for both cases,
so if the expected rates of positively and negatively charged leptons differ from the
data in the QCD template, a charge asymmetry arise from this effect. To quantify
the influence of the difference in the QCD templates, the QCD events in the pseudo
0.2 e+jets
norm. to unit area
norm. to unit area
94
positive lepton
0.18
negative lepton
0.16
0.14
0.12
0.1
0.12
0.1
0.06
0.06
0.04
0.04
0.02
0.02
-2
-1
0
1
2
3
positive lepton
negative lepton
0.2
0.15
0.25
0.05
-2
-1
0
1
2
3
4
5
(yt-y )(yt+y )
t
0
1
2
3
t
4
|η |-|η |
µ+jets
t
positive lepton
negative lepton
0.15
0.05
-3
-1
0.2
0.1
-4
-2
t
0.1
0
-5
-3
t
norm. to unit area
0.25 e+jets
0
-4
4
|η |-|η |
t
negative lepton
0.14
0.08
-3
positive lepton
0.16
0.08
0
-4
norm. to unit area
0.2 µ+jets
0.18
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
(yt-y )(yt+y )
t
t
Figure 5.17: Comparison of the ∆|η| and ∆y 2 shapes in the sideband QCD models
with positively and negatively charged electrons ((a) and (c)) and muons ((b) and
(d)).
experiments are either drawn from the template with positively charged leptons or
from the template with negatively charged leptons.
Pile-Up Reweighting Uncertainty
There are also influences on the measurement due to the applied pile-up reweighting
on Monte Carlo events. The pile-up model which has been used for the reweighting
has several sources of systematic uncertainties. The effects are expected to be small,
but to account for them the MC events gain additional weights corresponding to a
variation of the average number of pile-up events by ±0.6 [165].
The shifts of the unfolded asymmetry induced by the systematically distorted
samples compared to the nominal value are summarized in table 5.2. When the
shift of the distorted asymmetry goes in the same direction for up- and downward
fluctuation for one systematic uncertainty, only the higher value is quoted. As one
can see, the influence of the W +jets model and the lepton trigger is negligible. The
overall uncertainty is obtained by quadratically summing the different values. For
both variables the matching threshold uncertainty represents the largest influence on
5.4. RESULTS
95
∆|η|
Source of Systematic
JES
JER
2
Q scale
ISR/FSR
Matching threshold
PDF
b tagging
Lepton ID/sel. efficiency
Lepton trigger
W +jets model
QCD model
Pile-Up
Overall
2
AC
A∆y
C
− Variation + Variation − Variation + Variation
−0.003
–
−0.007
–
−0.002
–
−0.001
+0.001
−0.005
+0.008
−0.013
+0.003
−0.006
+0.003
–
+0.024
−0.034
+0.021
−0.026
+0.014
−0.001
+0.001
−0.001
+0.001
−0.001
+0.003
–
+0.001
−0.002
+0.004
−0.002
+0.003
±0.000
±0.000
±0.000
±0.000
±0.000
±0.000
±0.000
±0.000
−0.008
+0.008
−0.006
+0.006
−0.002
+0.002
±0.000
±0.000
−0.036
+0.025
−0.031
+0.029
Table 5.2: Summary of the systematic uncertainties taken into account in the measurement of the tt̄ charge asymmetry. The positive and negative shifts induced by
2
∆|η|
systematic uncertainties in pseudo experiments for AC and A∆y
are shown. For
C
systematic uncertainties which shift the charge asymmetry in the same direction for
up- and downward fluctuations, only the higher value is quoted. The quadratic sum
of all systematic uncertainties yields the overall systematic uncertainties.
∆|η|
the measurement. A big impact on AC
5.4
arises also from the ISR/FSR uncertainties.
Results
After the study of the systematic influences on the measurement and validating
the unfolding method with several cross checks the unfolding is applied on the reconstructed and background-subtracted ∆|η| and ∆y 2 distributions of the observed
data. In figure 5.18 the resulting spectra are shown with the calculated charge
asymmetry and the statistical uncertainty obtained from equation 5.15. Considering equation 5.23 the unfolded asymmetry in ∆y 2 has to be corrected. In table 5.3
the results at different steps of the analysis are summarized.
The final results of the tt̄ charge asymmetry measurement in proton-proton collision at a center-of-mass energy of 7 TeV with data corresponding to an integrated
luminosity of 1.09 fb−1 are
∆|η|
= −0.016 ± 0.030 (stat.)+0.025
−0.036 (syst.)
(5.24)
A∆y
= −0.013 ± 0.026 (stat.)+0.029
−0.031 (syst.)
C
(5.25)
AC
for ∆|η| and
2
l+jets
0.25
0.2
0.15
t
t
0.4
0.35
Data
NLO prediction
1.09 fb-1 at s = 7 TeV
A C= -0.012 ± 0.026
l+jets
0.3
0.25
0.2
0.15
0.1
0.1
0.05
0
-4
0.45
t
t
0.3
Data
NLO prediction
1.09 fb-1 at s = 7 TeV
A C= -0.016 ± 0.030
t
t
0.4
0.35
1/σ dσ/d((y -y )(y +y ))
1/σ dσ/d(|η |-|η |)
96
0.05
-3
-2
-1
0
1
2
3
4
|η |-|η |
t
0
-4
-3
-2
-1
0
1
2
3
(a)
t
(b)
bin 1
0.007
-0.004
0.002
0.000
0.000
0.001
bin 1
0.005
-0.002
0.001
0.001
0.001
0.001
bin 2
-0.004
0.009
-0.007
0.002
0.000
0.001
bin 2
-0.002
0.008
-0.003
0.000
-0.000
0.001
bin 3
0.002
-0.007
0.018
-0.009
0.001
0.001
bin 3
0.001
-0.003
0.008
-0.004
0.000
0.001
bin 4
0.000
0.002
-0.009
0.015
-0.007
0.002
bin 4
0.001
0.000
-0.004
0.008
-0.003
0.001
bin 5
0.000
0.000
0.001
-0.007
0.014
-0.004
bin 5
0.001
-0.000
0.000
-0.003
0.009
-0.002
bin 6
0.001
0.001
0.001
0.002
-0.004
0.007
bin 6
0.001
0.001
0.001
0.001
-0.002
0.005
bin 1
bin 2
bin 3
bin 4
bin 5
bin 6
bin 1
bin 2
bin 3
bin 4
bin 5
bin 6
(c)
4
(yt-y )(yt+y )
t
t
(d)
Figure 5.18: Unfolded results for ∆|η| (a) and ∆y 2 (b). Additionally the NLO
predictions for the Standard Model charge asymmetry are shown taken from [13].
In (c) and (d), the full covariance matrices for the measured spectra are depicted.
for ∆y 2 . Both measured values are within the uncertainties in agreement with each
2
∆|η|
other and with the theory predictions of AC,SM = 0.014 ± 0.001 [10] and A∆y
C,SM =
0.012 ± 0.001 [10]. To take a closer look at the resulting asymmetries in table 5.4 the
∆|η|
values for different ranges in the sensitive variables are shown. The value of AC
arises from the large negative result in the region with 0.5 < |∆|η|| < 1.1, whereas
the unfolded ∆y 2 distribution is more balanced. This shows that the inclusive
asymmetry is not a product of migration effects in the two central bins, which
would have the largest impact on the measured asymmetry.
The CDF experiment at Tevatron observes a charge asymmetry which is strongly
dependent on the invariant mass of the tt̄ system [8] as discussed in section 1.3. To
get a first hint whether this behavior is also present at LHC or not, the backgroundsubtracted asymmetry is measured as function of mtt̄ as shown in figure 5.19. For
both variables there is no mass-dependence visible, which is in contradiction to the
results of the CDF collaboration. However, these studies allow only for a qualitative
statement, since a simultaneous unfolding in the sensitive variable and mtt̄ is crucial.
97
0.2
0.15
-1
tt (Powheg)
data
1.09 fb at s = 7 TeV
l+jets
y
0.1
A C (bg subtr.)
η
A C (bg subtr.)
5.4. RESULTS
0.05
0.2
0.15
0.05
0
-0.05
-0.05
-0.1
-0.1
-0.15
-0.15
Mtt,rec [GeV/c2]
tt (Powheg)
data
0.1
0
-0.2
200 300 400 500 600 700 800 900 1000 1100 1200
1.09 fb-1 at s = 7 TeV
l+jets
-0.2
200 300 400 500 600 700 800 900 1000 1100 1200
Mtt,rec [GeV/c2]
(a)
(b)
Figure 5.19: Background-subtracted asymmetries for ∆|η| (a) and ∆y 2 (b) for different regions in the reconstructed invariant tt̄ mass. Additionally, the values from
the NLO generator Powheg are shown.
observable
∆|η|
∆y 2
Raw
−0.004 ± 0.009
−0.004 ± 0.009
background-subtracted
−0.009 ± 0.010
−0.007 ± 0.010
Unfolded (and corrected)
−0.016 ± 0.030+0.025
−0.036
−0.013 ± 0.026+0.029
−0.031
Table 5.3: The measured asymmetries for both variables at the different stages of
the analysis from the raw value to the background subtracted value and to the final
unfolded result.
Bin Range in ∆|η|
|∆|η|| < 0.5
0.5 < |∆|η|| < 1.1
1.1 < |∆|η||
∆|η|
AC,bin
Bin Range in ∆y 2
0.024 ± 0.089
|∆y 2 | < 0.4
−0.115 ± 0.048 0.4 < |∆y 2 | < 1.1
0.046 ± 0.081 1.1 < |∆y 2 |
2
A∆y
C,bin
−0.009 ± 0.063
−0.036 ± 0.045
0.002 ± 0.053
Table 5.4: The measured bin by bin charge asymmetries for both sensitive variables.
98
Conclusion and Outlook
In this thesis a measurement of the tt̄ charge asymmetry in pp collisions recorded
with the CMS experiment has been performed. The tt̄ charge asymmetry exclusively
occurs in processes with an asymmetric initial state, i.e. quark-antiquark annihilation. It can be explained within the Standard Model by next-to-leading order effects
yielding a slight preference of the top quark to fly into the direction of the incoming
quark and of the antitop quark to fly into the direction of the incoming antiquark.
At the proton-antiproton collider Tevatron this behavior leads to an excess of top
quarks in forward direction and an excess of antitop quarks in backward direction
with respect to the proton beam. At the proton-proton collider LHC an excess of
antitop quarks in the central region of the detector and an excess of top quarks in
forward and backward directions is expected.
Recent measurements at the Tevatron became the focus of attention, since the
two experiments CDF and DØ see a large deviation from the values predicted by
the Standard Model. There are many extensions to the Standard Model, which are
trying to explain these mismatches. If one of the introduced hypothetical heavy
exchange particles exists, i.e. heavy axigluons or leptophobic Z 0 bosons, a deviation
of the charge asymmetry from the SM value should also be visible at the LHC, where
the measurement is by far more challenging. The ratio of top quark pair production
via quark-antiquark annihilation to the charge symmetric gluon-gluon fusion is much
smaller at the LHC than at the Tevatron and therefore the predicted values for the
SM tt̄ charge asymmetry are considerably closer to zero and less accessible. Due to
the symmetric initial state in proton-proton collisions the rapidity distributions of
top and antitop quarks are symmetrically distributed around zero and differ only in
their widths. In this thesis the measurement is accomplished defining the tt̄ charge
asymmetry as asymmetry in the distributions of the two variables ∆|η| = |ηt | − |ηt̄ |
and ∆y 2 = (yt − yt̄ )(yt + yt̄ ), which are sensitive to the differences in the widths of
rapidity and pseudorapidity distributions of top and antitop quarks.
Before the measurement is executed, a selection of tt̄ signal events has been
performed in the electron+jets and muon+jets channel, where one top quark decays
hadronically into three quarks and the other top quark into a bottom quark, a
charged lepton and a neutrino. The amount of background events in the selected
dataset has been estimated via a binned likelihood fit. Thereafter the four-momenta
of the top quarks, which are essential for the charge asymmetry measurement, have
99
100
been reconstructed exploiting a dedicated reconstruction method. In order to correct
for effects of the event selection and reconstruction on the distributions of ∆|η| and
∆y 2 a regularized unfolding method has been applied. To validate the unfolding
method several consistency and linearity checks have been performed. They have
led to satisfying results putting the analysis on solid ground.
Finally, the charge asymmetry in top quark pair production has been measured
in the corrected distributions of ∆|η| and ∆y 2 yielding
∆|η|
= −0.016 ± 0.030 (stat.)+0.025
−0.036 (syst.) ,
(5.26)
∆y 2
= −0.013 ± 0.026 (stat.)+0.029
−0.031 (syst.) .
(5.27)
AC
AC
The results which are comparable to the measurement from the ATLAS collaboration [14] are very interesting, since they are within the uncertainties in good
agreement with the Standard Model predictions. Also the measurement of the
background-subtracted asymmetry as a function of the invariant mass of the tt̄
system shows no tendency to large asymmetries for high tt̄ masses. As opposed to
the Tevatron results showing on average large positive deviations, the LHC measurements only feature a small negative deviation from the predictions as depicted
in figure 5.20.
The extensions to the Standard Model capable of explaining the discrepancies in
the Tevatron measurements predict also higher values for the charge asymmetries at
the LHC. The presented results have not yet the sensitivity to exclude these models
but give a strong hint to the validity of the Standard Model.
In future analyses using the entire dataset recorded by the CMS experiment
in 2011 the significance of the measurement will be increased. Up to now data
corresponding to an integrated luminosity of 5 fb−1 has been recorded by the CMS
experiment. The large amount of data will allow for a simultaneous unfolding in
the sensitive variables and the invariant mass of the tt̄ system in order to confirm
or refute the Tevatron results.
101
Figure 5.20: Summary of the results of tt̄ charge asymmetry measurements at Tevatron and LHC, adopted from [10]. For each result the deviation from the Standard
Model values is depicted in units of σ. The measurements that are marked with
a red star refer to asymmetries on reconstruction level, which are compared to the
MCFM / MC@NLO predictions multiplied by a factor of 1.5. While the Tevatron
results in grey show on average a rather large positive deviation, the results from
ATLAS and CMS (this thesis) in blue represent a small negative deviation.
102
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Danksagungen
An erster Stelle möchte ich mich bei Herrn Prof. Dr. Th. Müller bedanken, dass er
mich in seine exzellente Top-Gruppe aufgenommen hat und mir die Möglichkeit gab,
vor Ort die Arbeitsweise am CERN kennenzulernen und meine Arbeit auf mehreren
Konferenzen vorzustellen.
Herrn Prof. Dr. J. H. Kühn danke ich für die Übernahme des Korreferats.
Ein besonderer Dank gilt meinem Betreuer Dr. Thorsten Chwalek, vor allem für
die unzähligen Ideen zur Verbesserung der Analyse, die stets konstruktive Art Kritik zu üben und das intensive Korrekturlesen dieser Arbeit.
Ein ganz großes Dankeschön geht an Dr. Jeaninne Wagner-Kuhr für die großartige
Unterstützung und an Dr. Thomas Peiffer, auf dessen Arbeit diese Diplomarbeit
aufbaut, für die geduldige Hilfe während der gesamten Analyse. Beiden möchte ich
auch für das Korrekturlesen dieser Arbeit danken.
Frank Roscher danke ich für die große Mitarbeit an dieser umfangreichen Analyse.
Bei Jochen Ott bedanke ich mich für seine unzähligen Hilfeleistungen im Rahmen
dieser Diplomarbeit. Alexis Descroix, Timo Doll und Stephan Riedel möchte ich
für das angenehme Arbeitsklima in unserem Büro danken. Auch bei Dr. Jasmin
Gruschke, Dr. Jyothsna Komaragiri, Dr. Jan Lück, Dr. Manuel Renz, Jens Hansen,
Hauke Held, Benedikt Maier, Steffen Röcker und Jaakob Voigt bedanke ich mich für
die stets positive und belebende Atmosphäre in der Arbeitsgruppe.
Ich danke allen Mitgliedern des EKP, insbesondere dem Admin-Team und Frau
Bräunling, für den Einsatz das reibungslose Arbeiten am Institut zu gewährleisten.
Ich danke meiner Mutter für die moralische und finanzielle Unterstützung während
des gesamten Studiums, ohne die ein erfolgreicher Abschluss nicht möglich gewesen
wäre. Bei meinen Geschwistern und meinen Freunden möchte ich mich für das stets
offene Ohr bei Problemen bedanken.
Der größste Dank gebührt meiner Freundin Christina Figlus für ihre unermüdliche
Kraft, die sie mir jeden Tag schenkt. Du bist die Beste.
115
Hiermit versichere ich, die vorliegende Arbeit selbständig verfasst
und nur die angegebenen Hilfsmittel verwendet zu haben.
Christian Böser
Karlsruhe, November 2011

Messung der Ladungsasymmetrie in Top-Quark

Transcription

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