Measurement of Spin Correlations in t t ¯ Events from pp Collisions at s √ = 7 TeV in the Lepton + Jets Final State with the ATLAS Detector
MMeasurement of Spin Correlations in t ¯ t Events from pp Collisions at √ s = 7 TeV in the Lepton + Jets Final Statewith the ATLAS Detector Dissertationzur Erlangung des mathematisch-naturwissenschaftlichen Doktorgrades”Doctor rerum naturalium“der Georg-August-Universit¨at G¨ottingenim Promotionsprogramm ProPhysder Georg-August University School of Science (GAUSS)vorgelegt vonBoris Lemmeraus GießenG¨ottingen, 2014 a r X i v : . [ h e p - e x ] O c t etreuungsausschussProf. Dr. Ariane FreyProf. Dr. Kevin Kr¨oningerProf. Dr. Arnulf QuadtMitglieder der Pr¨ufungskommission:Referent: Prof. Dr. Arnulf Quadt II. Physikalisches Institut, Georg-August-Universit¨at G¨ottingen
Koreferentin: Jun.-Prof. Dr. Lucia Masetti
Institut f¨ur Physik/ETAP, Johannes Gutenberg-Universit¨at Mainz
2. Koreferentin: Prof. Dr. Ariane Frey
II. Physikalisches Institut, Georg-August-Universit¨at G¨ottingen
Weitere Mitglieder der Pr¨ufungskommission:PD Dr. J¨orn Grosse-Knetter
II. Physikalisches Institut, Georg-August-Universit¨at G¨ottingen
Prof. Dr. Hans Hofs¨ass
II. Physikalisches Institut, Georg-August-Universit¨at G¨ottingen
Prof. Dr. Wolfram Kollatschny
Institut f¨ur Astrophysik, Georg-August-Universit¨at G¨ottingen
Jun.-Prof. Dr. Steffen Schumann
II. Physikalisches Institut, Georg-August-Universit¨at G¨ottingen
Tag der m¨undlichen Pr¨ufung: 10.07.2014Referenz: II.Physik-UniG¨o-Diss-2014/02
So eine Arbeit wird eigentlich nie fertig,man muß sie f¨ur fertig erkl¨aren,wenn man nach Zeit und Umst¨andendas m¨oglichste getan hat.“
Goethe easurement of Spin Correlations in t ¯ t Events from pp Collisions at √ s = 7 TeV in the Lepton + Jets Final State with the ATLASDetector Abstract
The top quark decays before it hadronises. Before its spin state can be changed in a process ofstrong interaction, it is directly transferred to the top quark decay products. The top quarkspin can be deduced by studying angular distributions of the decay products. The StandardModel predicts the top/anti-top quark ( t ¯ t ) pairs to have correlated spins. The degree issensitive to the spin and the production mechanisms of the top quark. Measuring the spincorrelation allows to test the predictions. New physics effects can be reflected in deviationsfrom the prediction. In this thesis the spin correlation of t ¯ t pairs, produced at a centre-of-massenergy of √ s = 7 TeV and reconstructed with the ATLAS detector, is measured. The datasetcorresponds to an integrated luminosity of 4 . − . t ¯ t pairs are reconstructed in the (cid:96) + jetschannel using a kinematic likelihood fit offering the identification of light up- and down-typequarks from the t → bW → bq ¯ q (cid:48) decay. The spin correlation is measured via the distribution ofthe azimuthal angle ∆ φ between two top quark spin analyzers in the laboratory frame. It isexpressed as the degree of t ¯ t spin correlation predicted by the Standard Model, f SM . Theresults of f SM (∆ φ (charged lepton, down-type quark)) = 1 . ± .
14 (stat.) ± .
32 (syst.) ,f SM (∆ φ (charged lepton, b -quark)) = 0 . ± .
18 (stat.) ± .
49 (syst.) ,f SM (∆ φ (combined)) = 1 . ± .
11 (stat.) ± .
22 (syst.) , are consistent with the Standard Model prediction of f SM = 1 . essung von Spin-Korrelationen in t ¯ t -Ereignissen aus pp -Kollisionenbei √ s = 7 TeV im Lepton + Jets Endzustand mit dem ATLASDetektor Zusammenfassung
Das Top-Quark zerf¨allt, bevor es hadronisiert. Bevor die Spin-Konfiguration des Top-Quarksdurch Prozesse der Starken Wechselwirkung ge¨andert werden kann, wird sie direkt an dieZerfallsprodukte des Top-Quarks weitergegeben. R¨uckschl¨usse auf den Spin des Top-Quarksk¨onnen ¨uber Winkelverteilungen der Zerfallsprodukte gezogen werden. Die Spins vonTop-/Anti-Top-Quark ( t ¯ t ) Paaren sind, gem¨aß der Vorhersage durch das Standardmodell,korreliert. Der Grad der Korrelation ist sensitiv auf den Spin und die Produktionsmechanismendes Top-Quarks. Die Messung der Spin-Korrelation bietet einen Test der Vorhersagen. Effektevon Physik jenseits des Standardmodells k¨onnen sich in Abweichungen der vorhergesagtenSpin-Korrelation manifestieren. In dieser Arbeit wird die Spin-Korrelation von Top-QuarkPaaren, die bei einer Schwerpunktsenergie von √ s = 7 TeV produziert und mit dem ATLASDetektor rekonstruiert wurden, gemessen. Der Datensatz entspricht einer integriertenLuminosit¨at von 4 . − . Die Top-Quarks wurden im Lepton+Jets Zerfallskanal mittels eineskinematischen Likelihood-Fits, der eine Trennung der leichten up- und down-Typ Quarks ausdem t → bW → bq ¯ q (cid:48) Zerfall erlaubt, rekonstruiert. Die Spin-Korrelation wird ¨uber dieVerteilung des Azimutalwinkels ∆ φ zwischen zwei Top-Quark Spin-Analysatoren imLaborsystem gemessen. Sie wird als Grad f SM der Spin-Korrelation, wie sie im Rahmen desStandardmodells berechnet wird, angegeben. Die Messungen ergeben f SM (∆ φ (geladenes Lepton, down-Typ Quark)) = 1 . ± .
14 (stat.) ± .
32 (syst.) ,f SM (∆ φ (geladenes Lepton, b -Quark)) = 0 . ± .
18 (stat.) ± .
49 (syst.) ,f SM (∆ φ (kombiniert)) = 1 . ± .
11 (stat.) ± .
22 (syst.) . Die Ergebnisse stimmen mit der Berechnung im Rahmen des Standardmodells, f SM = 1 . ontents
1. Preface 12. Standard Model, Top Quarks and Spin Correlation 5 t ¯ t Events . . . . . . . . . 302.5. Sensitivity of t ¯ t Spin Correlation to Physics Beyond the Standard Model . 422.6. Recent Measurements of t ¯ t Spin Correlation . . . . . . . . . . . . . . . . . 44
3. Experimental Setup 51
4. Analysis Objects 61 τ Leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5. Dataset, Signal and Background Modelling 73 t ¯ t Signal Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3. MC Driven Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4. Data Driven Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 77I ontents
6. Event Selection and Reconstruction 81 t ¯ t Selection in the Lepton+Jets Channel . . . . . . . . . . . . . . . . . . . 816.2. Data/MC Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3. Mismodelling of the Jet Multiplicity . . . . . . . . . . . . . . . . . . . . . 856.4. Reconstruction of t ¯ t Events with a Kinematic Likelihood Fit . . . . . . . 906.5. Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.6. KLFitter Extension for Up/Down-Type Quark Separation . . . . . . . . . 976.7. Reconstruction Efficiencies and Optimizations . . . . . . . . . . . . . . . . 1006.8. KLFitter Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.9. Comparison to Other Reconstruction Methods . . . . . . . . . . . . . . . 105
7. Analysis Strategy 113
8. Systematic Uncertainties 127
9. Results 151
10. Summary, Conclusion and Outlook 173
Danksagung 181Bibliography 183Index 205 II ontents Appendices 209A. Spin Correlation Matrices 211B. Used Datasets 213C. Pretag Yields 219D. KLFitter Likelihood Components 221E. Down-Type Quark p T Spectrum in
POWHEG + PYTHIA ∆ φ for Different MC Generators 235J. Alternative t ¯ t Modeling 239K. Jet Charge 241
CV 245
III
Preface
Curiosity is one of the fundamental driving forces of human kind. Without it, we wouldnot have reached the high level of development in technology and health that we havenowadays and that we do not want to miss. Every little kid is equipped with curiosityand can decide how much it wants to know. Playing the game of asking “Why is that?”again and again will finally end up in asking: “What are we made of?”, “Where do wecome from?” and “Why is everything working the way it does?”.The field of particle physics is addressing these questions. During the last two cen-turies, the knowledge of the fundamental building blocks of nature has developed rapidly,leading to changing ideas of what is really fundamental. The current understanding ofthe elementary particles and their interactions is reflected in the
Standard Model ofParticle Physics ( SM ). This theory framework classifies the particles of matter – thefermions – in groups of quarks and leptons, and it describes the interactions among themvia the exchange of gauge bosons. The power of the Standard Model has been more thanjust the description of particles and forces that are known so far. It also allows precisiontests to check its self-consistency and to search for unknown physics effects.Only very few particles are stable and can be observed and analysed in the laboratory.The more massive the particles are, the earlier they decay into lighter ones. During thevery first moments after the creation of our universe, the environment of very high energydensity allowed a balanced production and decay of such heavy particles. The balancebetween creation and decay got lost during the expansion and cooling of the universe.Sufficient energy for the creation was no longer available.Recreation of such very high energy densities in laboratories on earth is possible byaccelerating particles, colliding them and using their kinetic energy to recreate massiveparticles. The more massive they are, the more energy is required. As technology keptevolving, more and more particles of the Standard Model were discovered.Being the most massive of all quarks, the top quark has been discovered as the last1 . Preface missing quark in 1995 by the two experiments D0 and CDF, located at the Tevatronproton/anti-proton accelerator at Fermilab [1, 2]. Before its discovery, the existence wasalready suggested to complete the third generation of quarks as a partner for the b -quark.Precision measurements of the parameters of the Standard Model allowed to constrainthe top quarks mass. Figure 1.1(a) shows the prediction and, after its discovery, themeasured mass of the top quark as a function of time. (a) [GeV] t m
140 150 160 170 180 190 200 [ G e V ] W M = G e V H M = . H M = G e V H M = G e V H M σ ± Tevatron average kint m σ ± world average W M = G e V H M = . H M = G e V H M = G e V H M
68% and 95% CL fit contours measurements t and m W w/o M68% and 95% CL fit contours measurements H and M t , m W w/o M (b) Figure 1.1.: (a) Indirect determinations of the top quark mass via fits to electroweakobservables, results of direct measurements as well as lower bounds fromdirect searches and W boson width analyses [3]. (b) Measured masses ofthe W boson ( M W ) and the top quark ( m t ), shown in green bands. Theseare compared to electroweak fit results excluding the direct measurementsof m t and M W (blue area) and excluding m t , M W and the measured Higgsboson mass M H (grey area) [4].Indirect searches and limit settings were not only performed for the top quark. Anotherimportant example is the search for the Higgs boson. Before its discovery, the Higgsboson’s role in corrections to the masses of the W boson and the top quark ( m W , m t )was powerful enough to constrain the Higgs boson mass m H via electroweak fits. Figure1.1(b) shows the directly measured masses of the W boson and the top quark comparedto electroweak fit results excluding the direct measurements. Measuring m t and m W shows the preferences for certain Higgs boson masses (diagonal lines). The particleunder study in this thesis is the top quark. As the heaviest of all quarks it offers uniqueopportunities of physics studies. With a lifetime of about 5 · − s, which is shorterthan the time scale of forming bound hadronic systems, hadronisation , the top quarktransfers its spin to its decay products before the spin information is diluted. This makesthe top quark the only quark whose spin is directly accessible.According to the Standard Model, top quarks produced via the strong interaction arealmost unpolarized, but have correlated spins. The degree of correlation depends onthe initial state of the production and the involved production processes. The degree ofcorrelation which will be measured depends on the decay mechanisms as well.In this thesis, the degree of correlation is measured. This addresses the following2uestions: Does the top quark carry a spin of ? Does the production of top/anti-topquark ( t ¯ t ) pairs follow the rules given by the Standard Model? And in particular: Are thespins of top/anti-top quark pairs correlated as they are expected to be? Modificationsof the Standard Model due to new physics effects can be reflected in deviations from thepredicted spin correlation of t ¯ t pairs. This allows the analysis presented in this thesis toconstrain physics effects beyond the Standard Model in the same way that the massesof the top quark and the Higgs boson were constrained before their discovery.It is not only the result and the following conclusions that leave a message. Thedetailed studies of top quark reconstruction and the impact of systematic uncertaintiesguide the way to future measurements of the t ¯ t spin correlation. 3 Standard Model, Top Quarks and Spin Correlation
What are we made of? What does the Universe consist of? And why does naturebehave as it actually does? Physicists observe nature and analyse the underlying lawsand principles. Particle physicists in particular study nature on the elementary level.The actual meaning of “elementary” has developed in time. It started with the elements ,the smallest units of a certain type of matter with unique properties. Dmitri Mendeleevand others started grouping these into the periodic system of elements [5]. Accordingto the approval of the International Union of Pure and Applied Chemistry (IUPAC)114 elements are presently known [6]. Along with the search for the truly fundamentalbuilding blocks of nature comes the search for underlying symmetries. Not only matteris, in terms of size, supposed to be fundamental. Laws of nature can also have morefundamental principles. For the latter, the unification of electricity and magnetism tothe electromagnetic force serves as an example [7]. Symmetries refer to such unified ormore fundamental laws.A first important step in the simplification of the set of elements was made by J. J.Thomson who discovered the electron as being a constituent of all atoms [8]. H. Geigerand E. Marsden made important steps in their scattering experiments [9], depictingthe atom structure as heavy nuclei surrounded by light electrons. These measurementsstrengthened the idea of W. Prout who found the atomic masses being multiples of thehydrogen atom mass [10, 11]. The picture of atomic nuclei as a composition of hydrogennuclei objects was established. The only flaw, the neutrality of some of these components,was finally resolved when J. Chadwick discovered the neutron in 1932 [12].The set of elementary particles seemed to be reduced from 114 elements to the proton,neutron and electron. This small set of building blocks of nature did not last very long.Not only that the discovery of the positron [13] introduced anti-particles – particles with The discoveries of further elements have been reported, but not yet confirmed. At the time of the discovery of the neutron, the number of known elements was smaller. . Standard Model, Top Quarks and Spin Correlation equal masses but quantum numbers such as the electric charge multiplied by − eightfold way [18, 19]. The idea came up that in fact quarks , a new type of particle,are the real fundamental building blocks of which protons, neutrons and several othernewly discovered particles, are made of [20]. The experimental proof for the theory camealong with the results of deep-inelastic scattering ( DIS ) experiments. Results from thesescattering experiments with electrons off protons were compatible with a model of point-like constituents, namely the quarks [21–23].Today we have a consistent set of elementary particles including the building blocks ofmatter, the fermions , as well as three of the four fundamental forces and their mediating gauge bosons : the Standard Model. It will be explained in Section 2.1. For a long timethe mechanism of mass generation of the bosons and fermions has been a mystery. Itwas resolved in 2012 by the discovery of the Higgs boson by the ATLAS and CMSexperiments [24, 25] confirming the Higgs mechanism [26–31] as the process responsiblefor electroweak symmetry breaking and mass generation. Section 2.1.3 explains thisprocess and highlights the important role of the SM’s most massive fermion, the topquark in its study.The production and decay mechanisms of the top quark as well as its discovery and thestudy of most of its properties are explained in Section 2.3. A property of each elementaryparticle is its spin. During the production and decay of particles the spin informationis propagated according to the rules of the conservation of angular momentum. Theknowledge of the spin configuration of a final state demands the knowledge of the initialstate, its spin configuration and the whole dynamics of the scattering process. Hence,measuring the spin configurations and comparing them to the predictions made by theSM leads to a validation of the latter one or to necessary extensions. The fact that thetop quark is the only quark whose spin configurations can be probed directly and theway how a corresponding measurement can be realized is explained in Section 2.4.The measurement of the spin correlation of top and anti-top quark pairs might indicatephysics beyond the SM (BSM) in case of observing an incompatibility between predictionand measurement. Possible BSM scenarios and their effects on the t ¯ t spin correlationare discussed in Section 2.5. As the t ¯ t spin correlation depends on the kinematics of theproduction process, variations of the initial state composition and its kinematics changethe predicted correlation. Thus, measurements presented at the Tevatron [32] colliderand its two experiments D0 [33] and CDF [34] are complementary to the measurementsat the LHC. The results of t ¯ t spin correlation measurements at both the Tevatron andthe LHC will be presented in Section 2.6. Gravity is missing in the SM without breaking the self-consistency of the SM as it can be neglectedat the mass scale of elementary particles. Even though the same idea was brought up by Brout, Englert, Guralnik, Hagen, Higgs and Kibble atabout the same time, the name
Higgs mechanism has manifested. .1. The Standard Model of Particle Physics At the end of this chapter the reader is equipped with all necessary information aboutthe motivation and the idea of a measurement of the t ¯ t spin correlation. The Standard Model of particle physics contains the present knowledge about elementaryparticles and their interactions. Fermions as matter particles with a spin interact viathe mediation of gauge bosons (with spin 1). The underlying mathematical formulationof the SM is a renormalizable quantum field theory based on a local SU (3) × SU (2) × U (1)gauge symmetry [35–45]. While the SU (3) subgroup describes the interaction with thegluon fields ( Quantum Chromodynamics , QCD , also called strong interaction ), SU (2) × U (1) is the representation of the electroweak interaction, unifying the electromagneticand the weak interaction.The main properties of the strong and the weak interaction are described in Sections2.1.1 and 2.1.2. All fermions and gauge bosons are introduced in Figure 2.1.Depending on how the fermions interact, they can be grouped into quarks (interactingvia the strong interaction) and leptons (not interacting via the strong interaction). Anadditional colour charge is assigned to particles interacting via the strong interaction. Left-handed fermions have T = and are arranged in doublets of the weak isospin T ,right-handed ones are singlets with T = 0. Only left-handed fermions interact via theweak interaction. Thereby, the third component of the weak isospin, T , is conserved.Quarks with T = + carry an electric charge of + in terms of the positron charge | e | , quarks with T = − carry a charge of − . In contrast to quarks, leptons with T = + carry a charge of 0. The ones with T = − do carry a charge of −
1. Allweak isospin doublets appear in three generations. Their properties are the same withincreasing masses as the only difference. Since heavy generations will decay into lightones, stable matter on earth is composed of u - and d -quarks as well as electrons. Forall fermions corresponding anti-particles exist. The quantum numbers of the latter haveopposite sign. Table 2.1 lists the fermion properties. The mediating gauge bosons for the interactions are the gluons for the strong interac-tion, the W and Z bosons for the weak interaction and the photons for the electromag-netic interaction. The gauge bosons with their most important properties are listed inTable 2.2.Despite the fact that the SM is a powerful framework to calculate strong and elec-troweak interactions at high precision, it does not describe gravity. The former interac-tions are described in the following sections. Colour charge is the equivalent preserved quantity in QCD as is the electric charge in electrodynamics.Colour is just an additional degree of freedom needed to describe quarks. There is no relation tocolour in the literal sense. From now on all electric charges are quoted in terms of | e | . In this thesis natural units ( (cid:126) = c = 1) are used if not stated otherwise. In particular, this concernsthe units of masses which are quoted as MeV instead of MeV /c , for example. . Standard Model, Top Quarks and Spin Correlation Fermion
Q T Colour Charge Mass [MeV]Up Quark ( u ) +2 / / d ) − / − / c ) +2 / / s ) − / − / t ) +2 / / b ) − / − / ν e ) 0 +1 / < · − Electron ( e ) − − / ν µ ) 0 +1 / < . µ ) − − / ν τ ) 0 +1 / < . τ ) − − / Q and the thirdcomponent of the weak isospin T . The values refer to left-handed fermions.Right-handed ones have T = 0 and Q as the left-handed. The mass valuesare taken from [46, 47].Boson Interaction Q T Gauge Coupling g Charges Mass [MeV] W + Weak +1 +1 / √ πα/ sin θ W (cid:2) √ πα (cid:3) w [e] 80385 W − Weak − − / √ πα/ sin θ W (cid:2) √ πα (cid:3) w [e] 80385 Z Weak 0 0 √ πα/ (sin θ W cos θ W ) w 91188 γ Electromag. 0 0 √ πα — < − g Strong 0 0 √ πα s c 0Table 2.2.: The gauge bosons of the Standard Model with their corresponding interac-tion, electric charge Q , third component of the weak isospin T and couplingconstant g . The mass values are taken from [46]. The couplings refer to theinteraction strengths and the charges to colour ( c ), weak ( w ) and electric ( e )charge. The values α , α s and θ W are explained in Sections 2.1.1 and 2.1.2.8 .1. The Standard Model of Particle Physics u g g W + W - Z Hde cs m tb t +2/3 00+1 -10 00-1/3-1 +2/30-1/3-1 +2/30-1/3-1 Up Quark GluonPhotonW + Boson W - BosonZ Boson Higgs BosonElectron NeutrinoDown QuarkElectron Charm QuarkMuon NeutrinoStrange QuarkMuon Top QuarkTau NeutrinoBottom QuarkTau n e n m n t Fermions Q u a r k s L e p t o n s Bosons
Figure 2.1.: Fermions (quarks and leptons) and gauge bosons of the Standard Modeland some of their basic properties. The small boxes indicate the fields towhich the particles couple: colour (c), electromagnetic (e) and weak (w).The number in the upper right corner represents the electric charge.
The strong interaction and its field theory, QCD, are based on an SU (3) gauge group.The eight generators of the group are represented by eight gluons. As the gauge groupof QCD is non-Abelian, each gluon carries a colour and an anti-colour, allowing it tocouple to other gluons.The strong interaction plays an important role in the regime of high energy physics.In particular, it is the main interaction responsible for t ¯ t pair production process athadron colliders (see Section 2.3) and thus responsible for the spin configuration of the t ¯ t pair. One should be careful not to take the word strong too seriously. The actualstrength of the strong coupling √ πα s depends on the energy scale Q of the process of9 . Standard Model, Top Quarks and Spin Correlation interest, making the strong coupling constant α s everything but a constant. For valuesof α s which are significantly smaller than unity, QCD can be treated perturbatively.Corrections of higher orders lead to the modified effective coupling, calculated at aspecific renormalization scale µ R .The dependence of α s on the energy scale Q and squared renormalization scale µ R isgiven by [48] α s (cid:0) Q , µ R (cid:1) = α s (cid:0) µ R (cid:1) α s ( µ R ) π (11 n c − n f ) ln (cid:0) Q /µ R (cid:1) . (2.1)Both corrections of fermionic and bosonic loops are included, giving a different sign tothe change of α s : n c refers to the number of colours, n f to the number of light quarkflavours ( m q (cid:28) µ R ). The equation can also be reformulated as α s (cid:0) Q , Λ (cid:1) = 12 π (11 n c − n f ) ln ( Q / Λ ) (2.2)by the introduction of the cut-off parameter Λ, which is chosen in a way that it definesthe scale where QCD cannot be calculated using perturbation theory. Depending on thenumber of fermions n f included in the renormalization the values of Λ n f are [46]Λ = 213 ± = 296 ±
10 MeV (2.4)Λ = 339 ±
10 MeV . (2.5)For a renormalization scale µ R set to the energy scale Q of the process of interest,Equation 2.1 describes the energy scale dependence of α s . As n c = 3 and n f < α s (cid:0) Q → (cid:1) with quark confinement as a consequence ofthe coupling increasing with distance. On the other hand, for short ranges and highenergy scales, asymptotic freedom of QCD holds as α s (cid:0) Q → ∞ (cid:1) = 0 [41].Experimental determinations of α s show good agreement with the predicted behaviour.Figure 2.2 summarizes the measurements by the H1 [49, 50], ZEUS [51], D0 [52, 53] andCMS [54] collaborations.A common reference for quoting the value of α s is the mass of the Z boson. The worldaverage value was determined in [46] as α s (cid:0) m Z (cid:1) = 0 . ± . . (2.6) While in nature , the electromagnetic and the weak interaction appear as separate in-teractions with quite different properties, their underlying theoretical framework is the Q refers to the absolute value of the squared four-momentum transferred at a vertex ( Q = (cid:12)(cid:12) q (cid:12)(cid:12) ). Or more precisely: on energy scales we do observe in nature. .1. The Standard Model of Particle Physics Figure 2.2.: The measured dependence of α s on the energy scale (cid:112) Q [54].same. Due to the major contributions of S. Glashow, S. Weinberg and A. Salam [35–37]it is often referred to as the Glashow-Weinberg-Salam ( GWS ) model.The electroweak (EW) symmetry, manifested in the SU (2) × U (1) gauge group, isspontaneously broken via the Higgs mechanism which will be described in detail inthe next section. As a consequence of the EW symmetry breaking, the four masslessbosons W , W , W and B , generators of the SU (2) and U (1) gauge groups, mix tothe observable gauge bosons W + , W − , Z and γ : (cid:18) γZ (cid:19) = (cid:18) cos θ W sin θ W − sin θ W cos θ W (cid:19) (cid:18) BW (cid:19) (2.7) (cid:18) W + W − (cid:19) = (cid:32) √ − i √ √ i √ (cid:33) (cid:18) W W (cid:19) (2.8)The photon as the mediator of the electromagnetic force remains massless, unlike themassive W + , W − and Z of the weak interaction. The mixing angle θ W , or rather itssquared sine, is determined experimentally. The quoted value depends on the renormal-ization scheme and ranges from sin θ W = 0 . θ W = 0 . Electromagnetic Interaction
The quantum field theory describing the electromagnetic part of the GWS model iscalled
Quantum Electrodynamics . It is based on the U (1) part of the SU (2) × U (1)gauge symmetry of the electroweak interaction. Unlike QCD, QED is an Abelian gauge A direct mass term is forbidden to preserve the local gauge invariance. . Standard Model, Top Quarks and Spin Correlation group. As a consequence, no photon-photon couplings exist. Thus, the QED equivalentto Equation 2.1 has no bosonic loop contribution with opposite effect as the fermionicones [48] : α (cid:0) Q , µ R (cid:1) = α (cid:0) µ R (cid:1) − α ( µ R ) π ln (cid:0) Q /µ R (cid:1) . (2.9)Depending on the corrections considered the factor in front of the logarithm maychange, but the Q dependence is the same: α increases with lower Q and vice versa.Equation 2.9 holds for Q (cid:29) µ R only. In the limit of Q → α takes the numericalvalue of , also known as the fine-structure constant . The variations of α by Q arerather low ( α ( M Z ) − = 127 . ± .
014 [46]).
Weak Interaction
Particles taking part in a weak interaction process are members of the same weak isospindoublet (see Figure 2.1). This means that the weak interaction does not cross differentgenerations. However, it is observed in nature that weak interactions across quarkgenerations do occur, for example in the decay of Kaons [46].This is possible as the weak doublet partners of the T = + quarks are in factsuperpositions of mass eigenstates ( d, c, b ). The linear combinations are described bythe unitary CKM matrix [55]: d (cid:48) s (cid:48) b (cid:48) = V ud V us V ub V cd V cs V cb V td V ts V tb dsb . (2.10)The unitarity requirement reduces the nine parameters V ij to three mixing angles and acomplex phase responsible for CP violation . As a consequence of the CKM matrix mixing, the weak interaction allows interactionsacross quark generations. It should be stressed that only left-handed fermions are partof the isospin doublets while the right-handed ones are singlets. Thus, right-handedparticles do not interact via the charged weak interactions involving a W ± boson. To account for the maximal parity violation of the weak interaction – as observed innature [57] – the weak interaction vertex has a vector − axialvector ( V − A ) structure: Here, only electron/positron loops are considered. This corresponds to n f = 1. Named after the editors of [55], M. Kobayashi and T. Maskawa, as well as N. Cabbibo on whoseideas [55] is footing [56]. The combined charge and parity symmetry is broken. As Z bosons are linear combinations of the W and B and only the former one requires particles fromthe isospin doublet, neutral weak interactions do not have this restriction. .1. The Standard Model of Particle Physics W ± : − ig W √ γ µ (cid:0) − γ (cid:1) V ij (2.11) Z : − ig Z γ µ (cid:0) c V − c A γ (cid:1) (2.12)Here, γ µ ( µ = 1 ..
4) represent the Dirac matrices, γ = i (cid:81) j =0 .. γ j , g W/Z the weakcoupling constants as in Table 2.2, c V = T − Q sin θ W the vector and c A = T the axialvector part of the coupling. The V − A structure is manifested in the term γ µ (1 − γ )with the vector component γ µ and the axialvector component γ µ γ . The (1 − γ ) canalso be interpreted as a projection operator for the left-handed components of a fermionwave function. The V − A structure of the weak interaction is of great importance forthe propagation of the top quark’s spin to its decay products (see Section 2.4).The CKM matrix is clearly diagonally dominant, stressing the favoured inter-isospindoublet interactions. The values for V ij are determined experimentally and can be foundin [46]. Flavour changing neutral currents ( FCNC ) would lead to a change of quarkflavour without changing the charge, such as a c → u transition. In the SM, flavourchanging neutral currents are only possible at higher orders (double W exchange) andare strongly suppressed by the GIM mechanism [37]. As predicted and also observed experimentally, the W ± [58,59] and Z [60,61] bosons aremassive. But in order to preserve the local gauge invariance of the SM, masses may notbe attributed to the gauge bosons explicitly. A dynamic mass generation mechanism isneeded, such as the Higgs mechanism [26–31].Before electroweak symmetry breaking the situation is the following: The gauge fields W , , belong to the SU (2) group and couple with a strength g W . The quantity whichis invariant under SU (2) transformations is the weak isospin. B is the correspondinggauge field of the U (1) group with a coupling g (cid:48) and the weak hypercharge Y as theconserved quantity. All four fields are massless.The GWS theory makes use of the Higgs mechanism by adding four scalar fields φ i withspecial properties [36]. Arranged in a complex isospin doublet with hypercharge Y = 1 itpreserves the SU (2) × U (1) gauge invariance. By assigning a vacuum expectation value(VEV) v to the real neutral component, the symmetry operations of the electroweakinteraction are broken and their corresponding bosons get massive. U EM (1) as subgroupof SU (2) × U (1) remains invariant (as the Higgs field with a VEV is neutral). Theconserved quantity is the electric charge Q , related to the third component of the weak The equivalent right-handed projection operator is (1 + γ ). Named after S.L. Glashow, J. Iliopoulos and L. Maiani. . Standard Model, Top Quarks and Spin Correlation isospin T and the weak hypercharge by the Gell-Mann-Nishijima formula [62, 63] Q = T + Y . (2.13)The gauge boson of the U EM (1) group (the photon) remains massless while the others( W ± and Z ) obtain masses. The masses of the gauge bosons depend on the weakcouplings g W and g (cid:48) and the VEV v of the Higgs field [48]: m W = 12 v g W (2.14) m Z = 12 v (cid:113) g W + g (cid:48) (2.15)The weak couplings g W and g (cid:48) are related to the couplings of the gauge bosons via g γ = g W · sin θ W = g (cid:48) · cos θ W = g Z · sin θ W · cos θ W (2.16)Knowing m W , m Z and their couplings allows to predicting the VEV of the Higgs field.The field itself acquires mass, depending on its VEV but also depending on the shapeof its potential. There is no fixed choice for the Higgs potential, but a potential such as V ( φ ) = − µ φ † φ + λ (cid:16) φ † φ (cid:17) (2.17)serves all needs. The parameters µ and λ determine the VEV v via v = (cid:114) µ λ (2.18)but they are in principle free. The field quantum of the Higgs field is the scalar Higgsboson . No prediction on its mass, given by m H = (cid:114) λ v, (2.19)can be made unless the shape of the Higgs potential is known [64].Next to the gauge boson mass terms, the mass terms of the fermions would alsobreak the local gauge invariance of the theory. Thus, also for fermion masses the Higgsmechanism can be used to take a workaround via symmetry breaking, but in a differentway than for the gauge bosons. For each massive fermion f – excluding neutrinos [64]– an additional Yukawa coupling y f to the Higgs field is introduced. This relates thefermion masses m f to the Higgs field VEV: m f = y f √ v (2.20) In its original version it was relating the electric charge to the hadronic isospin I , the baryon number B and the strangeness S via Q = I + ( B + S ). Higher orders in φ break renormalizability [36]. Within the SM, neutrinos are massless. .2. Proton Structure By using the relation v = 1 √ G F (2.21)and the value for the Fermi constant G F = 1 . · − GeV − (determined ex-perimentally via measurement of the muon lifetime [46]), the Higgs VEV v turns out tobe 246.22 GeV. The more massive a fermion, the higher its coupling to the Higgs fieldis.The Higgs mechanism serves well in the GWS model. About 50 years after the proposalof the mechanism it could be experimentally confirmed in 2012 by the observation ofthe missing Higgs boson. The ATLAS [24] and CMS [25] experiment reported theobservation of a new boson having the expected properties of the Higgs particle. Firstdetails could already be studied, leading to evidence of the spin-0 property and a strongpreference to its positive parity [65], as expected. The exact couplings to fermions willhave to be studied in detail in the future.Equation 2.20 shows that the higher a fermion’s mass, the higher its Higgs couplingis. Using the world average top quark mass of m t = 173 .
34 GeV [47] yields a Yukawacoupling of y t = 0 . The Standard Model is a powerful theory, providing the description of a broad varietyof natural phenomena at high precision. However, observed limitations of the StandardModel indicate that it needs to be extended or embedded in a larger theory. Such atheory could unify the strong and the electroweak interaction and also include gravity,which is not described by the SM. Astrophysical observations show distributions of non-baryonic matter interacting via the gravitational force, which cannot be explained withthe matter particles contained in the SM (
Dark Matter , see e.g. [66]). As observed inthe context of neutrino oscillations [67], neutrinos have a non-zero mass. This is alsocontradicting the SM assumption of massless neutrinos.One example for a SM extension is supersymmetry ( SUSY ), introducing a symmetrybetween fermions and bosons [68–76]. Such BSM scenarios include modifications of SMpredictions. The t ¯ t spin correlation, analysed in this thesis, is a possible way to probeBSM physics. The relation between BSM scenarios and t ¯ t spin correlation is explainedin Section 2.5. For the prediction of final state configurations it is important to know production anddecay mechanisms of the process of interest in detail. In the case of t ¯ t production anddecay, the process under study in this thesis, the details are explained in Section 2.3.But furthermore, each process needs a well-defined initial state. 15 . Standard Model, Top Quarks and Spin Correlation Using a proton-proton collider such as the LHC introduces an undetermined initialstate. The machine parameters provide a value for the momenta of the incoming pro-tons. But the initial state of the hard scattering process requires two of the protons’constituents, namely either quarks or gluons (in general: partons ). The density of quarksand gluons within the proton depends on two parameters: the fraction x of the longi-tudinal proton momentum that the parton carries as well as the energy scale Q of thescattering process. As the partons inside the proton interact via the strong interaction,gluon radiations are allowed as well as gluon to quark/antiquark and gluon to gluonsplittings. Hence, the total quark density is a sum of the three valence quark densitiesand the virtual quarks from gluon splittings. In general, the density of a parton a insidea proton is given by the Parton Distribution Function ( PDF ). QCD does not providean a-priori prediction of quark ( q i ) and gluon ( g ) PDFs. The evolution of a PDF with Q for a fixed value of x is described by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi(DGLAP) equations [77–79]: dq i ( x, Q ) d ln Q = α s ( Q )2 π (cid:90) x (cid:20) q i ( y, Q ) y · P qq (cid:18) xy (cid:19) + g ( y, Q ) y · P qg (cid:18) xy (cid:19)(cid:21) dy (2.22) dg ( x, Q ) d ln Q = α s ( Q )2 π (cid:90) x (cid:88) j q j ( y, Q ) y · P gq (cid:18) xy (cid:19) + g ( y, Q ) y · P gg (cid:18) xy (cid:19) dy (2.23)The splitting functions P ab (cid:16) xy (cid:17) describe the probability for a parton b to emit a parton a with a momentum fraction xy . PDFs are determined experimentally via hadron-hadronand lepton-hadron collider measurements. For a parameterization in x and an exampleof PDF determination see for instance [80].Several collaborations are performing fits of f a to data and provide the respective PDFsets. These are for example HERAPDF [80],
CTEQ [81],
NNPDF [82] and
MSTW [83].An example plot for PDFs( x, Q = m t ) is shown in Figure 2.3.Quark PDFs q i contain the sea quark distribution increasing for lower x and – in caseof up and down quark PDFs – a valence quark distribution which peaks at about . Forhigh values of x , quark densities are dominating while gluon densities dominate for lower x . This has an important consequence which should be kept in mind in the context of t ¯ t spin correlation analyses. If t ¯ t pairs are produced, the minimum amount of energyneeded is E = 2 m t . With the assumption that each of the incoming (anti-)protonsprovides a parton with the same energy, the minimum x for t ¯ t production at the Tevatron( E beam = 0 .
98 TeV) and the LHC ( E beam = 3 . x Tevatron = 173 . .
98 TeV = 0 .
18 (2.24) x LHC = 173 . . .
05 (2.25)These two values of x are also indicated in Figure 2.3. For the production of t ¯ t pairs twodifferent mechanisms exist: quark/antiquark annihilation and gluon fusion (see Section16 .3. The Top Quark Q = (m top ) CT10 (central) updownup (valence)down (valence)gluon
LHC (7 TeV)Tevatron (2 TeV)
Figure 2.3.: Parton distribution function of the nominal CT10 set at Q = m t . Theminimal average proton momentum fractions x for t ¯ t production are shownfor Tevatron ( √ s = 2 TeV) and the LHC ( √ s = 7 TeV).2.3). The parton with the higher density defines the dominating production mechanism.This has significant implications on the spin configuration of the t ¯ t pair. In particular,measurements at the Tevatron and the LHC are complementary as different productionmechanisms dominate. How this configuration is determined is discussed in Section 2.4. Several hints suggested the existence of a top quark well in advance, before it wasobserved as the last quark of the SM. As V. Fitch and J. Cronin observed CP violationin 1964 [84], the need for a theoretical explanation came up. One way to establishthis was suggested by Kobayashi and Maskawa in 1973 [55] by the proposal of a thirdquark generation. This idea was strengthened by the discovery of the τ lepton [85],increasing the number of lepton generations to three. As there were only two quarkgenerations, the GIM mechanism broke. With the discovery of the Υ – a meson consistingof a b - and a ¯ b -quark – by the E288 experiment [86], the door to a third generationopened. The need of a weak isospin partner of the b -quark was finally satisfied in 1995with the discovery of the top quark at the Tevatron accelerator by the D0 [1] and theCDF [2] collaborations. Electroweak precision measurements had already constrainedthe top quark’s mass before it was finally measured. By fitting electroweak precision17 . Standard Model, Top Quarks and Spin Correlation data without using direct top quark mass measurements, the top quark mass can todaybe determined as 175 . +2 . − . GeV [4]. Former predictions were summarized in Figure1.1(a). Indirect measurements have a great prediction power as this example shows.A whole set of unique measurement possibilities comes along with the properties of thetop quark. It is the by far heaviest fermion with a Yukawa coupling close to unity (seeEquation 2.20), making it a good probe for Higgs physics studies. In [87] the top quarkdecay width Γ t was calculated at NLO. Approximations for β ≡ m W m t were provided asΓ β → t = Γ t (cid:20) − α s π (cid:18) π − (cid:19)(cid:21) (2.26)Γ β → t = Γ t (cid:20) − α s π (cid:18) (cid:18) − m W m t (cid:19) + 4 π − (cid:19)(cid:21) (2.27)using Γ t = G F m t π √ | V tb | (cid:18) − m W m t (cid:19) (cid:18) m W m t (cid:19) (2.28)with the Fermi constant G F and the CKM matrix element V tb . As the ratio Γ t / Γ t of NLO to LO top decay width versus the ratio m W /m t stays almost constant for0 < m W /m t < . m t = 173 .
34 GeV [47], m W = 80 .
385 GeV [46] and α s (cid:0) m Z (cid:1) = 0 . β → t = 1 .
36 GeV.Using (cid:126) = 6 . · − eV s [88] leads to a predicted top quark lifetime of τ t = (cid:126) Γ t = 4 . · − s (2.29)Comparing the top quark lifetime to the time scale needed for hadronisation [89] givenby t had = (cid:126) Λ QCD = (cid:126)
213 MeV ≈ · − s (2.30)shows one order of magnitude difference. Thus, the top quark decays before formingbound states. This statement was also made in [89] with a quoted t had ≈ − s. Thequoted value of t had depends on the used cutoff parameter Λ QCD .It is important to realize the implications of this relatively short lifetime. In case thetop quark decays before it hadronises, its spin properties would directly be transferredto the decay products. Measurements of the top quark decay width indicate that this isindeed the case (see Section 2.3.2). In the literature, the time scale for hadronisation isused in many cases as the relevant quantity to compare the top quark lifetime to whenarguing about the spin transfer to the decay products. However, the spin decorrelationtime is in fact even longer than the hadronisation time as explained e.g. in [90]. In [91]the depolarization time t depol = (cid:126) m t Λ ≈ · − s (2.31)18 .3. The Top Quark is quoted, which is longer than t had = (cid:126) Λ QCD ≈ · − s.In the following sections a description of the production and the decay mechanisms ofthe top quark is given, followed by an overview of its properties. This provides the basisfor discussing the spin correlation of t ¯ t pairs and the access to it via measurements inSection 2.4. At hadron colliders – to which this discussion will be limited – top quarks can beproduced in two ways: as single top quarks via the electroweak interaction or in pairsvia the strong interaction.In both cases, the production process can be factorized into two components: Theinitial state prescription via the PDFs of a parton i in a proton p , f i/p , and the crosssection ˆ σ of the partonic hard interaction process. This separation is called factorizationtheorem and is described in [92, 93]. In order to factorize, two energy scales need to bedefined. The first one is called factorization scale µ F , separating the perturbative fromthe non-perturbative part. The second one, the renormalization scale µ R , has alreadybeen introduced in Section 2.1.1.For inclusive top quark pair production in proton-proton collisions the factorized crosssection at a centre-of-mass enery √ s reads [94] σ pp → t ¯ tX ( s, m t ) = (cid:88) i,j = q, ¯ q,g s (cid:90) m t d ˆ s L ij (ˆ s, s, µ f ) ˆ σ ij → t ¯ t (ˆ s, m t , µ f , µ r ) . (2.32)with the partonic density L ij (ˆ s, s, µ f ) = 1 s s (cid:90) ˆ s d ˜ s ˜ s f i/p (cid:18) µ f , ˜ ss (cid:19) f j/p (cid:18) µ f , ˆ s ˜ s (cid:19) . (2.33)For t ¯ t production one usually sets µ R = µ F = m t , so to the mass scale of the process ofinterest. t ¯ t Production via Strong Interaction
At hadron colliders top quarks are dominantly produced in pairs via the strong interac-tion. Figure 2.4 shows the different ways of t ¯ t production at leading order.The PDFs determine the initial state and also the contributions of the different dia-grams. By grouping into quark-antiquark annihilation (Figure 2.4(a)) and gluon fusion(figures 2.4(b) - 2.4(d)), two statements can be made, which are of importance for theanalysis of t ¯ t spin correlation: So far, no lepton collider has sufficient energy to produce top quarks. . Standard Model, Top Quarks and Spin Correlation (a) (b)(c) (d) Figure 2.4.: t ¯ t production via strong interactions. (a) Quark/antiquark annihilation,(b)-(d) gluon fusion. • The higher √ s , the lower the needed x for t ¯ t production. The dominating partonsfor low x are gluons. Hence, for high √ s , in particular for LHC energies, theprocess gg → t ¯ t is dominating. In contrast to the LHC, q ¯ q → t ¯ t is the dominatingprocess at the Tevatron. Figure 2.3 illustrates this. • As antiquarks are only available as sea quarks in the case of the LHC, q ¯ q → t ¯ t issuppressed. In the case of the Tevatron antiquarks are present as valence quarksin the anti-proton.In [95], the t ¯ t cross sections have been computed at next-to-next-to-leading order(NNLO) using the MSTW2008nnlo68cl PDF set [83] and assuming a top quark massof m t = 173 . t ¯ t cross sections for up to √ s = 8 TeV, listed inTable 2.3, can be compared to measured values. An overview of all t ¯ t cross sectionmeasurements and a comparison to the theory predictions is shown in Figure 2.5. Themeasurements are in good agreement with the predictions. Single Top Production Via Weak Interaction
Single top quarks can be produced in several ways as illustrated in Figure 2.6: via the s - or the t -channel or in association with a W boson ( W t -channel). In contrast to theproduction of t ¯ t pairs, the single top production channels can be measured individually.Predictions of the cross sections at NNLO were made in [108–110] and are listed in Table2.4.20 .3. The Top Quark Accelerator √ s [ TeV ] σ t ¯ t ± scale unc. ± PDF unc. [ pb ]Tevatron 2 7 . +0 . − .
200 +0 . − . LHC 7 172 . +4 . − . . − . . +6 . − . . − .
14 953 . +22 . − . . − . Table 2.3.: t ¯ t production cross sections at NNLO+NNLL for different accelerators andcentre-of-mass energies calculated for a top quark mass of m t = 173 . s -channel t -channel W t -channelAccelerator √ s [ TeV ] σ t [pb] σ ¯ t [pb] σ t [pb] σ ¯ t [pb] σ t [pb] σ ¯ t [pb]Tevatron 2 0.52 1.04 —LHC 7 3.17 1.42 41.7 22.5 7.814 7.93 3.99 151 91.6 41.8Table 2.4.: Calculated single top production cross sections at NNLO+NNLL for differentaccelerators and centre-of-mass energies for m t = 173 . s - and t -channel cross sections are symmetric for t and ¯ t at the Tevatron.The same is true for the W t cross section at the LHC. The quoted symmetriccross sections refer to both the t and the ¯ t cross sections, not the sum. 21 . Standard Model, Top Quarks and Spin Correlation [TeV] s c r o ss s e c t i on [ pb ]tI n c l u s i v e t
10 ATLAS+CMS PreliminaryTOPLHCWG
Feb 2014 * Preliminary -1 Tevatron combination* L = 8.8 fb -1 ATLAS dilepton L = 0.7 fb -1 CMS dilepton L = 2.3 fb -1 ATLAS lepton+jets* L = 0.7 fb -1 CMS lepton+jets L = 2.3 fb -1 TOPLHCWG combination* L = 1.1 fb -1 ATLAS dilepton* L = 20.3 fb -1 CMS dilepton L = 5.3 fb -1 ATLAS lepton+jets* L = 5.8 fb -1 CMS lepton+jets* L = 2.8 fb
NNLO+NNLL (pp))pNNLO+NNLL (pCzakon, Fiedler, Mitov, PRL 110 (2013) 252004 uncertainties according to PDF4LHC S α ⊕ = 172.5 GeV, PDF top m [TeV] s c r o ss s e c t i on [ pb ]tI n c l u s i v e t Figure 2.5.: Comparison of the measured t ¯ t cross sections at the Tevatron and the LHCusing input values from [97–106] and the predictions from [95]. The figureis taken from [107].A variety of cross section measurements at both the LHC and the Tevatron exist,briefly summarized in Table 2.5. All measurements are in good agreement with theSM prediction at NNLO precision. Events where a single top quark is produced are oneof the main backgrounds for the analysis of t ¯ t spin correlation. Top Quark Decay
The top quark decays via the weak interaction. While decays via the weak interactionusually take place at larger time scales than via the strong interaction, top quark decaysare still at a short time scale due to the top quark’s high mass. As the decay rates ofa top quark into a W boson and a quark q are proportional to the squared absolutevalue of the CMK matrix element | V tq | and | V tb | = 0 . +0 . − . [46], the top quarkcan be considered as decaying uniquely via t → W + b . Hence, the final state of a topquark decay is determined by the decay of the W + boson. In 67.7 % of the cases, the W + boson decays into hadrons [46], leading to a hadronically decaying top quark. Ifnot decaying into hadrons, the W + boson decays into a charged anti-lepton ¯ (cid:96) and theaccording neutrino (¯ eν e , ¯ µν µ , ¯ τ ν τ ). For each of the charged leptons the probabilities arealmost equal [46]. At first order the t ¯ t decay channels can be grouped into the following22 .3. The Top Quark (a) (b) (c) Figure 2.6.: Single top quark production via electroweak interactions in (a) the s -channel,(b) the t -channel and (c) produced in association with a W boson ( W t channel). σ t + σ ¯ t [pb] √ s [ TeV ] Experiment s -channel t -channel W t -channel2 CDF 1 . +0 . − . [111] 1 . +0 . − . [112] —D0 3 . +0 . − . [113] —7 ATLAS < . +20 − [115] 16 . ± . . ± . +5 − [118]8 ATLAS — 82 . ± . . ± . < . . ± . . ± . (cid:110) t ¯ t → b ¯ bW + W − → b ¯ bq ¯ q (cid:48) q (cid:48)(cid:48) ¯ q (cid:48)(cid:48)(cid:48) (2.34)Dilepton : (cid:110) t ¯ t → b ¯ bW + W − → b ¯ b ¯ (cid:96)(cid:96) (cid:48) ν (cid:96) ¯ ν (cid:96) (cid:48) (2.35)Lepton + jets : (cid:40) t ¯ t → b ¯ bW + W − → b ¯ b ¯ (cid:96)ν (cid:96) q ¯ q (cid:48) t ¯ t → b ¯ bW + W − → b ¯ b (cid:96) ¯ ν (cid:96) q ¯ q (cid:48) (2.36)Each decay channel of the t ¯ t pair has advantages and disadvantages. The choice ofthe decay channel for an analysis is thus always a trade-off and there is no ad-hocrecommendation. Practical issues that should be considered when analysing t ¯ t pairs are: • All jets channel
Due to the high branching fraction of W → hadrons this channelleads to a large sample of selected events. However, the signal-to-background ratiowill be quite low as the sample will be contaminated by multijet background to Even though a channel usually refers to a single particle, this expression is commonly used. . Standard Model, Top Quarks and Spin Correlation a large extent. Furthermore, the resolution of the jets is worse than the oneof leptons. Complex event reconstructions will suffer from this fact. A correctassignment of the six jets (two b -jets and four light jets from the hadronic W decay) will be very difficult and the combinatorial background quite large andhard to suppress. • Dilepton channel
Two leptons with a good reconstruction efficiency and en-ergy/momentum resolution can easily be reconstructed as t ¯ t decay products. Eventhe charge of these objects can be determined, allowing for a correct assignmentto the top and anti-top quark. Furthermore, the requirement of two charged lep-tons suppresses multijet background to a large extent. The dilepton channel hasa smaller event yield than the all jets channel. Reconstructing the full events is anon-trivial issue as the two neutrinos cannot be reconstructed. Using momentumconservation in the transverse plane and calculating the missing transverse mo-mentum (see Section 4.4) allows measuring the vectorial sum of the two neutrino’stransverse momenta indirectly. Still, the longitudinal neutrino momenta stay un-determined. Complex reconstruction algorithms such as Neutrino Weighting [125],demanding kinematic assumptions as input, are required. • Lepton+jets channel
The lepton+jets ( (cid:96) +jets) channel is a compromise betweenthe other two. Multijet background is suppressed due to the charged lepton in theevent. The suppression is not as powerful as in the dilepton channel. As only oneneutrino is present, the event kinematics are no longer underconstrained as in thecase of dilepton events. This allows for a full event reconstruction. A light up-typequark jet replaces the second neutrino (compared to the dilepton channel) and canbe reconstructed. But the hadronic equivalent to the second charged lepton, adown-type quark jet, suffers from worse reconstruction efficiency and a less precisemeasurement.The (cid:96) + jets channel was chosen for the analysis presented in this thesis. It offers afull event reconstruction, a high event yield and a sufficiently low multijet background.The technique used for full event reconstruction is described in Section 6.4. Figure 2.7shows the decay of a t ¯ t pair in the (cid:96) + jets channel. The relative yields for each of thethree channels is shown in Figure 2.8(a). It should be mentioned at this point that thedistinction into three different channels is idealized. When analysing events on the basisof reconstructed quantities, migration effects between the channels have to be taken intoaccount. A crucial role is played by the τ lepton. It will decay into either anothercharged lepton and a neutrino or into hadrons within τ τ = 2 . · − s [46]. In thelatter case this leads to a wrong assignment of a leptonically decaying top quark to ahadronically decaying top quark. Thus, Figure 2.8(b) gives a more realistic picture ofthe channels that are actually reconstructed, including migration effects. These are: This lifetime is too short for a direct detection in the detector but allows for the reconstruction of asecondary vertex. .3. The Top Quark gg g t ¯ t W − ¯ bl − ¯ ν l b ¯ duW + Figure 2.7.: The decay of a t ¯ t pair in the (cid:96) + jets channel. • Events from the dilepton channel can migrate to the (cid:96) + jets channel in case theycontain one τ decaying into hadrons, misidentified as jet. • Events from the (cid:96) + jets channel can be identified as such, but the reconstructedlepton does not stem directly from a W boson, but from a leptonically decaying τ . The lepton properties do not match the expectations. This causes a part of the t ¯ t signal being – from the physics point of view – in fact a background. • In case the lepton from the (cid:96) + jets channel is a hadronically decaying τ , recon-structed as a jet, the event will not pass the (cid:96) + jets event selection. Its properties make the top quark unique: As it is the heaviest fermion – and evenelementary particle – its lifetime is too short to create any bound states. Hence, thespin configuration is directly transferred to its decay products. Further, the high massimplies a Yukawa coupling to the Higgs field of about one, making it an important probefor Higgs physics. This section gives an overview of several important top quark propertymeasurements.
Mass
The mass of the top quark has been of great interest since the very beginning. Onthe one hand, loop corrections of the W and Z boson mass allowed for its predictionwithout a direct measurement. On the other hand, a direct measurement could providean important input for electroweak fits and loop correction calculations.Today, combinations of individual top mass measurements of the two Tevatron exper-iments [126], the two LHC experiments [127] as well as a world combination exist [47].25 . Standard Model, Top Quarks and Spin Correlation (a)(b) Figure 2.8.: (a) Decay channels of t ¯ t pairs and their relative rates. (b) A more detailedview of (a) including migration effects caused by τ leptons.The latter one yields m t = 173 . ± .
76 GeV. Figure 2.9 gives an overview of the worldcombination and its input values taken from the individual measurements [128–138].
Charge
By using methods of measuring the charge of jets via the charges of their associatedtracks, the top quark’s charge has been studied extensively. Scenarios of exotic topquarks with a charge of − were excluded and a good agreement with the SM predictionof q t = was found by the CDF [139], D0 [140] and CMS experiments [141]. ATLAShas quoted a direct measurement of q t = 0 . ± .
08 [142]. All analyses were carried outin the (cid:96) + jets channel.26 .3. The Top Quark [GeV] top m165 170 175 180 185117
LHC September 2013 – – (0.23 – Tevatron March 2013 (Run I+II) – – (0.51 – prob.=93% c / ndf =4.3/10 c World comb. 2014 – – (0.27 – -1 = 3.5 fb int L CMS 2011, all jets – (0.69 – -1 = 4.9 fb int L CMS 2011, di-lepton – (0.43 – -1 = 4.9 fb int L CMS 2011, l+jets – – (0.27 – -1 = 4.7 fb int L ATLAS 2011, di-lepton – (0.64 – -1 = 4.7 fb int L ATLAS 2011, l+jets – – (0.23 – -1 = 5.3 fb int L D0 RunII, di-lepton – – (2.36 – -1 = 3.6 fb int L D0 RunII, l+jets – – (0.83 – -1 = 8.7 fb int L +jets missT CDF RunII, E – – (1.26 – -1 = 5.8 fb int L CDF RunII, all jets – – (1.43 – -1 = 5.6 fb int L CDF RunII, di-lepton – (1.95 – -1 = 8.7 fb int L CDF RunII, l+jets – – (0.52 – -1 - 8.7 fb -1 = 3.5 fb int combination - March 2014, L top Tevatron+LHC mATLAS + CDF + CMS + D0 Preliminary ) syst. iJES stat.total ( P r e v i ou s C o m b . Figure 2.9.: World combination of the top quark mass [47] together with the input results[128–138] and the dedicated Tevatron [126] and LHC combinations [127].
Top Charge Asymmetry In t ¯ t production the kinematic distributions p T , η and φ of the top and the anti-topquark are equivalent [143] at LO. Interference terms at NLO, however, cause a differenceof the top and anti-top rapidities y in case of production via q ¯ q annihilation [143]. TheTevatron as p ¯ p collider has well-defined directions of the annihilating q and ¯ q . Hence,an asymmetry A FB = N ( y t > y ¯ t ) − N ( y t < y ¯ t ) N ( y t > y ¯ t ) + N ( y t < y ¯ t ) (2.37)can be calculated [144] and measured. A tension between the measurement and theSM prediction with a significance of more than 2 σ has been observed by the CDFcollaboration [145]. In a similar fashion an asymmetry of leptons from top and anti-top quark decays A lepFB was measured to be also more than 2 σ above the SM prediction[146]. The D0 measurements of A FB [147] and A lepFB [148] were, in contrast to the CDFmeasurements, compatible with the SM expectation.At the LHC, a measurement of A FB is not possible due to the symmetric pp produc-tion. However, as the valence quarks have on average a higher momentum than the seaantiquarks (see Figure 2.3) a non-vanishing charge asymmetry A C = N ( | y t | − | y ¯ t | > − N ( | y t | − | y ¯ t | < N ( | y t | − | y ¯ t | >
0) + N ( | y t | − | y ¯ t | <
0) (2.38) This is true for the dominating case in which these quarks are valence quarks. . Standard Model, Top Quarks and Spin Correlation is predicted [144]. The overall effect is expected to be small due to the charge symmetric gg fusion process dominating at the LHC. The measurements of A C by ATLAS andCMS at √ s = 7 TeV were combined in [149] and reported to be consistent with the SMprediction. In [150, 151] BSM scenarios modifying both A FB and A C were calculated.Figure 2.10 shows a comparison of these BSM predictions to the measurements of AT-LAS, CMS, CDF and D0. For the latter measurement an older result [152] was used forwhich also a tension to the SM prediction was observed. Considering the updated D0 A FB -0.0200.020.040.060.08 A C CMSATLAS CD F D ATLAS SM φ W ′ Models from: ω Ω G µ PRD 84 115013,arXiv:1107.0841
Figure 2.10.: Comparison of the measurements of A FB and A C [145, 152–154] to BSMscenarios from [150, 151] as shown in [153].result [147], a good agreement with the SM and an exclusion of a large parameter spacefor certain BSM scenarios was observed. Branching Fractions / Couplings
The assumption that | V tb | ≈ | V tb | can be measuredas a ratio of the branching fractions B : B ( t → W b ) / (cid:88) q = d,s,b B ( t → W q ) = | V tb | / (cid:88) q = d,s,b | V tq | = | V tb | . (2.39)By assuming unitarity of the CKM matrix and the existence of three quark generationsCDF measures | V tb | = 0 . ± .
05 in the (cid:96) + jets decay channel and | V tb | = 0 . ± . | V tb | > . s + t -channel single top cross section measurement. At28 .3. The Top Quark the LHC, CMS combined the 7 and 8 TeV results of the t -channel single top results to | V tb | = 0 . ± .
038 (exp.) ± .
016 (theo.) [122] while ATLAS measured | V tb | = 0 . +0 . − . in the 8 TeV single top t -channel measurement [119].To probe the weak and electromagnetic couplings of the top quark, t ¯ tV processes –where V represents an additional vector boson such as a photon, a W or a Z boson – needto be investigated. It has to be stated that such studies need a very good understandingof the signal and background modelling. The inclusive measurements of t ¯ tV includeinitial and final state radiation of V from other partons than the top quark as well asinterference terms.At the Tevatron, CDF observed evidence of the t ¯ tγ process by measuring σ t ¯ tγ =0 . ± .
08 pb [155], also consistent with the SM prediction. Consistency with theSM was also measured by ATLAS ( σ t ¯ tγ = 2 . ± . σ t ¯ tγ = 2 . ± . σ t ¯ tγ /σ t ¯ t [157].Heavy gauge bosons in association with t ¯ t events were investigated at the LHC byboth CMS and ATLAS. ATLAS set an upper limit of σ t ¯ tZ < .
71 pb @95 % CL bymeasuring final states with three leptons [158]. In the same final state CMS measured σ t ¯ tZ = 0 . +0 . − . pb [159]. In the same publication the inclusive cross section σ t ¯ tZ + σ t ¯ tW was also measured as σ t ¯ tZ + σ t ¯ tW = 0 . +0 . − . pb. W Boson Helicity
One of the main properties of the top quark decay, that is used for spin correlationanalyses, is the weak V − A structure of its decay vertex. It requires b -quarks to beleft-handed since m b (cid:28) m t . Hence, at LO the W boson must either be left-handed aswell or longitudinally polarized. At LO and by neglecting the b -quark mass, the fractions F of longitudinal, left- and right-handed W polarizations read [160, 161] F : F L : F R = 11 + 2 x : 2 x x : 0 (2.40)with x = m W /m t . Anomalous couplings caused by BSM physics will be reflected indeviations of the W helicity fractions. Table 2.6 summarizes the measurements at theTevatron and the LHC and compared to NLO SM predictions. All measurements agreewith the SM predictions.Comb. CDF and D0 Comb. ATLAS and CMS NNLO SM F . ± .
081 0 . ± .
059 0 . ± . F L — ∗ . ± .
035 0 . ± . F R − . ± .
046 0 . ± .
034 0 . ± . W polarization fractions at the Tevatron [162] and theLHC [163] together with the NLO SM predictions [161] ( ∗ No results of F L were quoted for the Tevatron combination). 29 . Standard Model, Top Quarks and Spin Correlation Width / Lifetime
As stated in Section 2.3, the large mass of the top quark, implying a large decay width,leads to a short top quark lifetime. As it is predicted to be shorter than the timescale ofhadronisation, the direct measurement of the top quark spin, reflected in its polarizationand t ¯ t spin correlation, is possible. A short top quark lifetime is required to perform topquark spin measurements. Vice versa, dilutions in the spin measurements can be a signfor a top lifetime longer than the prediction.Concerning measurements at the Tevatron, neither CDF nor D0 have observed devia-tions from the NLO SM prediction of Γ t = 1 .
36 GeV. D0 measured Γ t = 2 . +0 . − . GeV( τ t = 3 . +0 . − . · − s) via the partial decay width Γ t ( t → bW ) taken from the t -channel single top cross section measurement and the branching fraction B ( t → bW )from t ¯ t events [164]. While this measurement assumed SM couplings, CDF performeda direct measurement and obtained the 68 % CL interval of 1 . < Γ t < .
04 GeV(1 . · − < τ t < . · − s). At the LHC, CMS has recently published a result withimpressive precision. By combining a measurement of the ratio B ( t → W b ) /B ( t → W q )with the results from the single top t -channel cross section measurement [117] theymeasured Γ t = 1 . ± .
02 (stat.) +0 . − . (syst.) GeV [165].These results justify measurements involving top quark spin and the transfer to itsdecay products. The following sections report expectations of the top quark polariza-tion and the top quark spin correlation. Furthermore, they provide a prescription foraccessing these quantities as well as an overview of their measurements. t ¯ t Events
The spin of the top quark is determined by its production process and transferred tothe decay products via the decay process. As Γ t (cid:28) m t , the leading pole approximation [166, 167] can be used to factorize the production and the decay process. By averagingover spin and colour configurations of the initial states, the squared matrix element canbe expressed [168] as (cid:16) or 8 (cid:80) colours (cid:17) (cid:16) (cid:80) spins (cid:17) (cid:12)(cid:12) M ( q ¯ q/gg → t ¯ t → ( f ¯ f (cid:48) b ) ( ¯ f f (cid:48) ¯ b ) ) (cid:12)(cid:12) = λ ab ρ ab, ¯ a ¯ b ¯ λ ¯ a ¯ b , (2.41)where f i represent the fermions of the W boson decay, a, b the top quark spins and λ and ρ the spin density matrix for the production and the decay, respectively. The bar ontop of the variables indicates the corresponding values for the anti-top and the numberused for colour averaging varies for q ¯ q annihilation (3) and gg fusion (8). Using the Pauli30 .4. Top Quark Polarization and Spin Correlation in t ¯ t Events matrices σ the production density matrix can be expressed [168] as ρ ab, ¯ a ¯ b ≡ (cid:16) or 8 (cid:80) colours (cid:17) (cid:16) (cid:80) initial spins (cid:17) M ( q ¯ q/gg → t a ¯ t ¯ a ) M ( q ¯ q/gg → t b ¯ t ¯ b ) ∗ (2.42)= 14 M µ ¯ µ σ µab σ ¯ µ ¯ a ¯ b = 14 (cid:16) M δ ab δ ¯ a ¯ b + M i σ iab δ ¯ a ¯ b + M i δ ab σ ¯ i ¯ a ¯ b + M i ¯ i σ iab σ ¯ i ¯ a ¯ b (cid:17) ≡ M (cid:16) δ ab δ ¯ a ¯ b + P i σ iab δ ¯ a ¯ b + ¯ P ¯ i δ ab σ ¯ i ¯ a ¯ b + (cid:98) C i ¯ i σ iab σ ¯ i ¯ a ¯ b (cid:17) (2.43)Here, M represents the total, spin independent production rate, P i = (cid:104) S i (cid:105) the po-larization of the top quark and (cid:98) C i ¯ i = (cid:104) S i ¯ S ¯ i (cid:105) the correlation between the top and theanti-top quark spin, using the top quark spin operators S . Examples for the spin correla-tion matrix (cid:98) C i ¯ i were calculated in [168] and are shown in Appendix A. The spin densitymatrix λ of a top quark can be simplified by integrating the decay phase space exceptone decay product i , which serves as spin analyser of the top. With (cid:126)e i as its directionof flight in the top quark rest frame one obtains [168]˜ λ ( (cid:126)e ) ab ∼ δ ab + α i (cid:126)e i · (cid:126)σ ab . (2.44)The degree to which the top quark spin is transferred to the decay product i is quantifiedby the spin analysing power α i . This quantity, and in particular its numerical value forseveral spin analyser candidates, is further discussed in Section 2.4.2. The analysingpowers for the decay products of the anti-top have the same magnitude, but oppositesign [168].By choosing one spin analyser for each top quark of a t ¯ t event, i from t and j from ¯ t ,Equation 2.41 leads to dσd (cid:126)e i d(cid:126)e j ∼ α i (cid:126)P (cid:126)e i + α j (cid:126) ¯ P (cid:126)e j + α i α j (cid:126)e i (cid:98) C(cid:126)e j (2.45)Moving from these generalized quantities to measurable ones requires the definition ofa spin quantization axis. One can define this spin axis as z-direction and use polar co-ordinates. Differential distributions of cos θ allow to access the top quark polarization P : 1 σ dσd cos θ i = 12 (cid:0) α i · P · cos θ (cid:1) . (2.46)Here, θ denotes the angle of the spin analyser with respect to the spin basis in thetop quark rest frame. In publications motivating spin correlation measurements (suchas [169]) the following equation is often quoted for the double differential t ¯ t cross section:1 σ d σd cos θ i d cos θ j = 14 (1 + α i B cos θ i + α j B cos θ j + α i α j C cos θ i cos θ j ) (2.47) In polar coordinates, (cid:126)e = (cos φ sin θ, sin φ sin θ, cos θ ). . Standard Model, Top Quarks and Spin Correlation where B i are said to describe the polarization of the top and the anti-top quark and CC = N ( ↑↑ ) + N ( ↓↓ ) − N ( ↑↓ ) − N ( ↓↑ ) N ( ↑↑ ) + N ( ↓↓ ) + N ( ↑↓ ) + N ( ↓↑ ) (2.48)the term used for t ¯ t spin correlation. It is sensible to call C , the relative difference between like and unlike spin configura-tions, correlation term. As − ≤ C ≤ C = − C = 0) and full correlation ( C = 1) are possible. But one should keepin mind that in fact B ∼ P and C ∼ (cid:98) C , using the polarization and spin matricesfrom Equation 2.43. Hence, only parts of these matrices are described by the angulardistributions of Equation 2.47. For reasons of simplicity it is from now on referred to B i and C if not stated otherwise.What degree of polarization and spin correlation can be expected at the LHC? Incase of single top quark production the weak interaction with its V − A structure leadsto a strong polarization [170]. In contrast to single top quark production, the stronginteraction is the dominating production process for t ¯ t pairs. Parity invariance of QCDleads to almost unpolarized t ¯ t pairs – and thus the coefficients B i in Equation 2.47vanish at leading order [166].The remaining question is: What degree of t ¯ t spin correlation can we expect for aQCD production and a weak decay according to the SM? Of course, the answer dependson the choice of spin quantization axis. The measured spin correlation depends on the choice of the spin basis. Choosing a basissuch that the correlation C is maximal increases the separation between correlated anduncorrelated t ¯ t pairs, which is certainly desirable. A first hint to a proper choice is givenby kinematic limits of the two t ¯ t production mechanisms: gg fusion and q ¯ q annihilation. Beam Line Basis t ¯ t production at the kinematic threshold ( m t ¯ t = 2 m t ) leads to a S t ¯ t spin state inthe q ¯ q → t ¯ t channel (due to chirality conservation) and to a S configuration in the g ¯ g → t ¯ t channel [172–174]. This spin configuration is determined by the initial state.It is therefore useful to define the spin quantization axis for the top and the anti-topquark as the directions of the incoming partons. This is illustrated in Figure 2.11. Thisparticular spin basis is referred to as beam line basis [166,172,175]. In this basis the spins In [166], Equation 2.47 has a minus sign in front of C . The reason is that the authors assign the samesign to the spin analysing power values for top and anti-top quarks and introduce the different signin the spin decay density matrix [166]. In some definitions of C the spin analysing powers are alsoalready included, so careful reading is required. In fact, a polarization transverse to the production plane is still allowed [168], but expected to be verysmall [171]. .4. Top Quark Polarization and Spin Correlation in t ¯ t Events (a) (b)
Figure 2.11.: t ¯ t spin configuration at the production threshold limit for (a) q ¯ q annihila-tion and (b) gg fusion.of top and anti-top are aligned in opposite directions ( ↑↓ , due to different bases) for q ¯ q → t ¯ t and in the same direction ( ↑↑ ) for gg → t ¯ t for top quark velocities β →
0. Thebeam line basis is useful in particular at the t ¯ t production threshold where no additionalangular momentum is added. Helicity Basis
Another basis of interest is the helicity basis . Here, the top quark direction of flight in thecentre-of-mass frame is taken as top spin axis (see Figure 2.12(a)). The anti-top spinaxis is defined accordingly. In the ultrarelativistic limit of β → ↑↓ configurationsare allowed for both q ¯ q → t ¯ t and gg → t ¯ t events in the helicity basis due to chiralityconservation of QCD.Kinematic limit considerations allow for a straightforward expression of the spin cor-relation C in the helicity basis. Using Clebsch-Gordan coefficients (see for example [46]for a list) leads to three representations of the S spin state for q ¯ q annihilation at theproduction threshold [176]: | ++ (cid:105) (2.49)1 √ | + −(cid:105) + |− + (cid:105) ) (2.50) |−−(cid:105) (2.51) This notation includes ↓↑ and represents opposite alignment of the spins. The same holds true for ↑↑ and ↓↓ for parallel alignments. Velocities are quoted as fraction β = v/c of the speed of light. Also referred to as centre-of-momentum frame or zero-momentum frame (ZMF). It is the frame wherethe t ¯ t pair is at rest. . Standard Model, Top Quarks and Spin Correlation (a) (b)(c) Figure 2.12.: Illustration of (a) the helicity basis and (b) the off-diagonal basis.(c) The angle θ between a spin analyser (here: the charged lepton) andthe spin basis in the top quark rest frame. The blue arrows indicate thespin quantization axes.The ± signs represent the spin eigenstates in a common basis. Same sign spin states ofthe top and anti-top quark imply opposite sign helicity states and vice versa. Thus,for β → C = − = − . In the ultrarelativistic limit of β → q ¯ q pair and the t ¯ t pair to haveopposite helicities, so C = − m t ¯ t of the t ¯ t system [176]: N ( ↑↓ ) + N ( ↓↑ ) N ( ↑↑ ) + N ( ↓↓ ) = 2 m t ¯ t m t (2.52)Concerning gg fusion, the β → S state and its √ ( | + −(cid:105) − |− + (cid:105) )configuration implies same sign helicity. The limit of β → gg fusion process is the dominant t ¯ t production mode at the LHC, it leads to a particularly high spin correlation C usingthe helicity basis. The t ¯ t cross section as a function of the invariant mass of the t ¯ t ± signs were used not to confuse e.g. | ++ (cid:105) with |↑↑(cid:105) . In the former case, a common basis is usedwhereas in the latter case the top and anti-top quark have individual bases. .4. Top Quark Polarization and Spin Correlation in t ¯ t Events system is shown in Figure 2.13 for both the LHC ( pp collisions at √ s = 14 TeV) andthe Tevatron ( p ¯ p collisions at √ s = 2 TeV) [176]. d (cid:109) _____ d M ( tt )-( pb / G e V ) M(tt) (GeV)- t t R R - t t L L - + t t R R - t t L L - + t t R L - t t L R - + t t R L - t t L R - + m = 175 GeV t LHCTevatron10 -5 -4 -3 -2 -1 (a) (b) Figure 2.13.: (a) t ¯ t production cross section as a function of the invariant mass M t ¯ t of the t ¯ t system for production at the LHC ( pp collisions at √ s = 14 TeV) andthe Tevatron ( p ¯ p collisions at √ s = 2 TeV) [176]. (b) Decomposition of theLHC cross section ( √ s = 14 TeV) into gg fusion and q ¯ q annihilation [172]. Off-Diagonal Basis
The spin configurations are purely of opposite sign in the case of q ¯ q annihilation for boththe beam line basis for β → β →
1. Hence, a basisinterpolating between these two limits can lead to pure oppositely signed spin states ofthe t ¯ t pairs. Such a basis exists [177, 178] and is named off-diagonal basis . With θ ∗ as the top quark production angle and Ψ as the angle between the beam axis and theoff-diagonal basis (see Figure 2.12(b)) [177], the interpolation is done viatan Ψ = β sin θ ∗ cos θ ∗ − β sin θ ∗ . (2.53)The limits β → β → q ¯ q annihilation but not the gg fusion, it is a preferredchoice for measurements at the Tevatron, but not at the LHC. The exact expression ofthe spin correlation matrix (cid:98) C i ¯ i in the off-diagonal basis is shown in Appendix A. Maximal Basis
Optimizations with respect to the helicity basis can be found at the LHC, as well.The off-diagonal basis does not only lead to purely oppositely signed t ¯ t spins for q ¯ q annihilation, but also for gg fusion with unlike-helicity gluons [91]. However, this is notthe case for like-helicity gg fusion, which is dominating at the LHC [91, 172] for low35 . Standard Model, Top Quarks and Spin Correlation invariant masses of the t ¯ t system. A way of defining a maximal basis for the general caseof gg fusion is described in [91, 179]. Expected Spin Correlations for Different Bases
As mentioned, the explicit value of C as defined in Equation 2.48 depends on the pro-duction and the spin analysing basis. For several experimental setups and bases, thesevalues have been calculated using two charged leptons as spin analysers. The resultsfrom [166, 169, 175, 180] are summarized in Table 2.7. The extent of higher order cor-rections differs but does not change the overall picture. The numbers indicate that aproper choice of basis is crucial.Mode √ s C beamline C off-diagonal C helicity C maximal p ¯ p pp pp pp
14 TeV -0.07 [166] -0.09 [166] 0.33 [175] —Table 2.7.: Expected spin correlation C as in Equation 2.48. Two charged leptons servedas spin analysers. Each decay product of the top quark carries information of its parent’s top quark spin.The degree, the spin analysing power α , is determined by the weak interaction and its V − A structure. Large values of α lead to larger differences in the angular distributions of thetop quark spin analysers between the scenarios of SM t ¯ t spin correlation/polarization andvanishing spin correlation/polarization (see equations 2.46 and 2.47). Non-SM couplingsand possible V+A structures will directly be reflected in changes of the predictions for α [181]. Examples for such modifications are shown in Section 2.5.2.In this section the numerical values of the top quark spin analysing powers are intro-duced and explained. Following the convention made in the previous sections, the spinanalysing power of the corresponding anti-top decay products have a reversed sign. To understand the spin analysing power of the b -quark, α b , at leading order, one canboost into the rest frame of the top quark and see easily that the b -quark spin statedepends on the helicity of the W boson. For longitudinally polarized W bosons the b and t spin are parallel. For left-handed W bosons they are anti-parallel. The directionof flight of the b -quark is anti-parallel to its spin. Hence, by using Equation 2.40, α b is One might argue that the same derivation could be repeated for the anti-top, leading to the samesign. But in this case the definition of the angular distribution from Equation 2.46 would need thereversed sign for the anti-top quark. .4. Top Quark Polarization and Spin Correlation in t ¯ t Events determined as α b = F L − F F L + F = 2 (cid:16) m W m t (cid:17) − (cid:16) m W m t (cid:17) + 1 . (2.54)As the b -quark and the W boson are emitted back-to-back in the top quark rest frame,the angle between the W and the top spin axis is π subtracted by the angle between thespin axis and the b . This leads to α W = − α b when comparing Equation 2.46 for bothspin analysers.It is a remarkable feature of the V − A structure of the top decay that leads to amaximal spin analysing power of α l = 1 for charged leptons as shown in [182, 183]. The down-type quark as the T = − component of the weak isospin doublets is the analogueto the charged lepton in terms of weak interactions. Hence, the same value of α isderived at leading order: α d = α s = +1 for down and strange quarks. This makes thedown-type quark the most powerful hadronic analyser. Since it is much more challengingto identify the down-type quark jets, advanced reconstruction techniques are necessary.These reconstruction techniques are described in Sections 6.4 and 6.6.The last analysers to be studied are the T = + decay products from the W boson,namely neutrinos, u - and c -quarks. The analytic form of the analysing power depends on m W m t and is listed in [172]. Given the small values of | α | ≈ . u - and c -quark jets, these are no alternatives to charged leptonsand down-type quarks in this analysis. All spin analysing powers at LO and NLO arelisted in Table 2.8. b -quark W + l + ¯ d -quark or ¯ s -quark u -quark or c -quark α i (LO) -0.41 0.41 1 1 -0.31 α i (NLO) -0.39 0.39 0.998 0.97 -0.32Table 2.8.: Standard Model spin analysing power at LO and NLO for the decay productsof the top quark from the decay t → bW + and for the decay products of the W boson [172, 181, 183–185].While for the spin correlation analyses in the dilepton channel the choice of analyseris quite obvious (the two charged leptons), the (cid:96) + jets channel offers two attractivepossibilities: the down-type quarks due to their high spin analysing power and the b -quarks as being relatively easy to reconstruct. Both will be studied and used forindividual measurements. A combined fit with both analysers will also be performed.In [176] such a combination is suggested. It is further justified by specific checks for thisanalysis (see Section 7.6). 37 . Standard Model, Top Quarks and Spin Correlation t ¯ t Spin Correlation
The most natural way to measure the t ¯ t spin correlation C is via the angular distributionsof the decay products with respect to the corresponding beam axis, such as in Equation2.47. However, there are several other kinematic distributions which are sensitive to thespin correlation. These will be briefly described in the following. Distributions of cos θ i cos θ j To start with the distributions discussed in Section 2.4.1, Figure 2.14 shows the partonlevel distributions of cos θ i cos θ j using two charged leptons in the helicity basis andthe LHC maximal basis [180]. For these distributions the MC@NLO generator was used,simulating both the SM spin correlation of t ¯ t events as well as uncorrelated t ¯ t events.Details about the signal sample are given in Section 5.2. helicity ) - q )cos( + q cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 he li c i t y ) - q ) c o s ( + q d N / d c o s ( N (SM)tt (no corr.)tt ATLAS
Simulation
RWIGE + H @NLOC
M = 7 TeVs dilepton (a) maximal ) - q )cos( + q cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 m a x i m a l ) - q ) c o s ( + q d N / d c o s ( N (SM)tt (no corr.)tt ATLAS
Simulation
RWIGE + H @NLOC
M = 7 TeVs dilepton (b)
Figure 2.14.: Distributions of cos θ i cos θ j at parton level using (a) the helicity and the(b) maximal basis [180]. The charged leptons from the dilepton channelserved as spin analysers. MC@NLO was used generating events with SM spincorrelation ( A = SM) and uncorrelated t ¯ t events ( A = 0). The notation A used in the figure corresponds to the spin correlation C used in this thesis. S-Ratio
This observable makes use of the fact that at the LHC the like-helicity gluons dominatethe production (see Figure 2.13). The
S-Ratio of squared matrix elements for SM spincorrelation and uncorrelated t ¯ t spins, S = ( | M | + | M | ) corr ( | M | + | M | ) uncorr (2.55)= m t { ( t · l + )( t · l − ) + (¯ t · l + )(¯ t · l − ) − m t ( l + · l − ) } ( t · l + )(¯ t · l − )( t · ¯ t ) , (2.56)38 .4. Top Quark Polarization and Spin Correlation in t ¯ t Events is calculated by the four-momentum vectors of the top ( t ) and the anti-top quark (¯ t ) aswell as two analysers, in this case charged leptons ( l ± ) [91]. A comparison between thedistributions of S for a spin correlation as predicted by the SM as well as for uncorrelated t ¯ t pairs at parton level is shown in Figure 2.15(a) [180]. As the cos θ i cos θ j distributions,the S -Ratio demands reconstruction of the full event kinematics. In particular in thecase of the dilepton channel, this is challenging. S-Ratio0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 d N / d S - R a t i o N (SM)tt (no corr.)tt ATLAS
Simulation
RWIGE + H @NLOC
M = 7 TeVs dilepton (a) ) [rad] - ,l + (l fD fD d N / d N (SM)tt (no corr.)tt ATLAS
Simulation
RWIGE + H @NLOC
M = 7 TeVs dilepton (b)
Figure 2.15.: Distributions sensitive to t ¯ t spin correlation [180]: (a) S-Ratio. (b) Az-imuthal angle between the two analysers in the laboratory frame. Thecharged leptons from the dilepton channel served as spin analysers. MC@NLO was used to generate events with SM spin correlation ( A = SM) and uncor-related t ¯ t events ( A = 0). The notation A used in the figure correspondsto the spin correlation C used in this thesis. ∆ φ in the Laboratory Frame The S -Ratio as shown in Equation 2.56 can be expressed in the ZMF [91] as S = (cid:18) − β β (cid:19) (cid:18) (1 + β ) + (1 − β ) cos ( l + , l − ) − β cos ( t, l + ) cos (¯ t, l − )(1 − β cos ( t, l + ))(1 − β cos (¯ t, l − )) (cid:19) . (2.57)The angle between the lepton momenta suggest checking the angular separation of thetwo leptons in ∆ η , ∆ φ and ∆ R = (cid:113) ( φ i − φ j ) + ( η i − η j ) . While ∆ η shows no sep-aration power between the scenario of SM-like t ¯ t spin correlation and uncorrelated t ¯ t pairs [91], ∆ φ does. The ∆ φ distributions were studied in [91] for an LHC setup with √ s = 14 TeV and showed impressive separation power. Before these studies were made,the sensitivity of ∆ φ was already known and checked in the context of Tevatron stud-ies [174, 186]. In Figure 2.16, the ∆ φ distribution for a Tevatron setup and an under-estimated top quark mass of m t = 120 GeV is shown. It can be noticed that the q ¯ q production is insensitive. As q ¯ q production is dominating at the Tevatron, ∆ φ has no39 . Standard Model, Top Quarks and Spin Correlation large sensitivity to the t ¯ t spin correlation. The situation is reversed at the LHC wherethe gg fusion is dominating, in particular in the like-helicity mode as shown in Figure2.13(b). This makes ∆ φ a very interesting observable, uniquely at the LHC. For the I n t . J . M od . P hy s . A . : - . D o w n l o a d e d fr o m . w o r l d s c i e n ti f i c . c o m by E U R O P E AN O R GAN I Z A T I ON F O R NU C LE A R R E S E A RC H ( C E R N ) on / / . F o r p e r s on a l u s e on l y . Figure 2.16.: Azimuthal angle between the two charged leptons from t ¯ t decays producedin p ¯ p collisions at √ s = 2 TeV [186]. The SM prediction (solid line) iscompared to the scenario of uncorrelated t ¯ t pairs (dashed line). A topmass of 120 GeV was assumed.reasons given above, ∆ φ was not only utilized in the publication about the observationof t ¯ t spin correlation at the LHC [187], it is also the observable used for the analysispresented in this thesis. Here, the additional complication with respect to the dilep-ton analyses performed by ATLAS and CMS [180, 187–189] is the identification of thehadronic spin analyser. Further Angular Variables
While the ∆ φ distribution was an obvious candidate for the dilepton channel, an alter-native for the (cid:96) + jets channel was investigated in [91] as well. As the (cid:96) + jets channelbasically allows for a full event reconstruction, boosts into other rest frames than thelaboratory frame are allowed. A sensitive quantity is the cos θ angle between two spinanalysers in the ZMF. This variable provides access to the trace of the spin correlationmatrix [168].By using the down-type quark as analyser for the hadronically decaying top quarkand by placing an additional cut on the invariant mass of the t ¯ t pair ( m t ¯ t <
400 GeV), a good separation power between SM-like and uncorrelated t ¯ t pairs can be achieved, The cut on the invariant mass was motivated by a larger separation between the sample with SM spincorrelation and the uncorrelated sample. See also [169] for discussion on m t ¯ t . .4. Top Quark Polarization and Spin Correlation in t ¯ t Events as shown in [91]. However, this method is limited: the boost requires a fully correctassignment of all involved decay products and furthermore a good energy resolution, inparticular for the m t ¯ t cut. Up to now, this variable has not yet been utilized in a spincorrelation measurement.In [168] it is suggested to measure the sum and the difference of the two analysers’polar angles. As spin basis, the helicity basis should be used but with the top spin axisas common z-axis. This provides access to different linear combinations of the spinmatrix elements (cid:98) C , (cid:98) C , (cid:98) C and (cid:98) C . Accessing the remaining elements of the spinmatrix (cid:98) C is possible by measuring distributions of the spin analysers azimuthal angleshifted by a phase depending on the spin analysers’ polar angle [168]. f SM The degree of t ¯ t spin correlation, C , can be predicted as shown in Section 2.4.1. C andadditional information of the full spin density matrix (cid:98) C influence the shape of severalangular distributions which were introduced in the last sections. For each of theseobservables, two distributions were shown: a SM distribution with an underlying t ¯ t spincorrelation as calculated in 2.4.1 as well as a distribution with uncorrelated t ¯ t pairs, C = 0. Equation 2.48 shows that the spin correlation is a linear function of the amountof parallel and anti-parallel t ¯ t spins. This allows to create templates T corresponding toarbitrary values X of the spin correlation C : T X = f SM · T C = C SM + (cid:0) − f SM (cid:1) T C =0 (2.58)By performing template fits of a distribution which is sensitive to the t ¯ t spin correlation,it is possible to measure the mixing fraction f SM . This immediately leads to a measuredspin correlation C via C = f SM · C SM (2.59)Sometimes f SM is referred to as “fraction of SM spin correlation”. This is reasonableas it defines the amount of a SM sample mixed into a linear combination, but might bemisleading. In particular, it does not have the properties of a fraction in the literal sense(e.g. being bound by 0 and 1). While − ≤ C ≤ f SM is bound by the limits of C . The interpretation of f SM is the following: • f SM < Instead of a correlation, an anti-correlation was observed. • f SM = 0 t ¯ t pairs are uncorrelated. • < f SM < The t ¯ t pairs are less correlated than predicted. • f SM = 1 The observed t ¯ t spin correlation matches the SM prediction. In contrast, separate z-axes are used for the top and anti-top quark in the definition of the helicitybasis. In case an anti-correlation was predicted, a correlation was observed. . Standard Model, Top Quarks and Spin Correlation • f SM > The t ¯ t spin correlation is higher than predicted.BSM physics can have different effects on different observables. Hence, it is possible tomeasure different values of f SM for different observables.It is possible to quote f SM and translate it into C for a particular basis. However,such statements should only be made if the translated result corresponds to the measureddistribution. For example, if f SM was extracted via a fit of cos θ i cos θ j using the helicitybasis, it can be translated into C helicity . A translation into C beamline will be misleading.The quantity f SM was utilized in several measurements which will be presented inSection 2.6.2. It is also used in the analysis presented in this thesis to extract theamount of spin correlation from a distribution of ∆ φ . t ¯ t Spin Correlation to Physics Beyond theStandard Model
The Standard Model allows to calculate the spin correlation C of t ¯ t pairs as shown inTable 2.7. Next to the correlation itself, the SM also provides information about howthe top quark spins are transferred to the decay products. This information is includedin the spin analysing power α , listed in Table 2.8.Possible new physics can affect both C and α . In the following, examples for BSMphysics and their implication on t ¯ t spin correlation are explained. The new physicsprocesses are split into two classes: those affecting the spin correlation C via a modified t ¯ t production and those that modify the top quark decay and hence α . t ¯ t Production
An example for new physics in the t ¯ t production is shown in Figure 2.17(a): the virtualgluon in the t ¯ t production is replaced by other particles like a heavy scalar φ . gg φ t ¯ t W − ¯ bl − ¯ ν l b ¯ duW + (a) gg g t ¯ t W − ¯ bl − b ¯ τν τ H + ν ¯ l (b) Figure 2.17.: (a) An example for modified t ¯ t production. A heavy scalar φ is replacingthe virtual gluon. (b) An example for a modified t ¯ t decay. The W bosonis replaced by a scalar charged Higgs boson.42 .5. Sensitivity of t ¯ t Spin Correlation to Physics Beyond the Standard Model
There are several new physics models which include new particles that are able toreplace the virtual gluon in the t ¯ t production. One example which is in particularinteresting to explain the deviation of A FB as measured by the CDF collaboration [145,146] is the existence of an axigluon [190, 191]. It is part of theories which embed QCDinto a more general SU (3) × SU (3) gauge group [192].Other theories embedding the full SM into a larger gauge group predict the existenceof a heavy neutral gauge boson Z (cid:48) , affecting the t ¯ t spin correlation [193]. Furthermore, Kaluza-Klein gravitons G [194] as part of the Randall-Sundrum model [195] lead todifferent spin correlation coefficients C as calculated in [196]. In [190] a general overviewof spin correlation modifications caused by Spin-0, Spin-1 and Spin-2 resonances is given.A rather model-independent approach to look for new physics in the production is thesearch for non-vanishing top quark chromomagnetic- and chromoelectric dipole momentsof which the latter would lead to a CP violation in QCD and modify the t ¯ t spin correlation[168, 197].Figure 2.18(a) shows the modifications of A FB (as in Equation 2.37) and C helicity byseveral BSM scenarios: axigluons G (cid:48) , scalar colour triplets ∆, scalar colour sextets Σand neutral components of a scalar isodoublet φ [198]. The 68 and 95 % CL resultsfrom ATLAS [187] are included as yellow band and grey dashed line. While the ATLASresults indicate the exclusion of a large parameter space of the BSM models, it shouldbe kept in mind that updated ATLAS results [180] lower the yellow band and reducethe exclusion. This is due to the change of the central value of C helicity = 0 .
40 [187] to C helicity = 0 .
37 (via ∆ φ ) and C helicity = 0 .
23 (via cos θ l + cos θ l − ) [180].Such BSM interpretations need to be handled with care. The ∆ φ distribution and thecos θ i cos θ j distributions are sensitive to different elements of the t ¯ t spin density matrix.Hence, BSM models can have a different impact on both. Treating different distributionsas equivalent, translating the results via f SM and interpreting one in terms of the otheris at least delicate.Another aspect in the context of t ¯ t spin correlation is the search for a scalar partnerof the top quark, stops or top squarks . Such particles are predicted in the context ofSUSY and can mimic the decay signature of top quarks. As these particles are scalar,they will have a large impact on the ∆ φ distribution. This was calculated in [199] andis shown in Figure 2.18(b). t ¯ t Decay
Access to the t ¯ t spin correlation is possible via the top quark decay products serving asanalysers. It is explicitly assumed that the top decay vertex is of pure V − A structure.The implications of a V+A mixture on the spin correlation was evaluated in [181].Next to this rather general test also specific models have been checked. The Two-Higgs-Doublet Model 2HDM , required for example in many SUSY models, includes twoadditional charged Higgs bosons. If their mass is close to the one of the W boson, it willbe hard to identify them directly in top quark decays. But as a charged Higgs boson is ascalar particle, it will modify the t ¯ t spin correlation [172]. In [200] the modifications of thespin analysing power α of the top decay products (Figure 2.18(c)) and the modifications43 . Standard Model, Top Quarks and Spin Correlation of the ∆ φ shape (Figure 2.18(d)) were calculated. In these studies different ratios β ofthe VEVs of the two Higgs doublets were evaluated. It should be mentioned that largevalues of tan β would also lead to significant changes in the W helicity observables [200]. G' Φ (cid:68)(cid:83) (cid:45) (cid:45) (cid:68) A FB (cid:68) C h e l (cid:72) L H C (cid:76) (a) !Φ ! ! ! $ " F r ac ti ono f E v e n t s ! ! $ Azimuthal AngleTopsTops, No Spin Corr.Stop, RH, 200 GeVStop, LH, 200 GeV (b) + H m80 90 100 110 120 130 140 150 160 i (cid:95) -1-0.500.51 = 50 (cid:96) tan + l + W l (cid:105) bb l (cid:105) + l + H (c) bb (cid:113)(cid:54) bb (cid:113) (cid:54) dd N N (cid:96) tan = 1 (cid:96) tan (d) Figure 2.18.: (a) Modifications of A FB and C helicity by several BSM scenarios [198].(b) ∆ φ ( l + , l − ) distributions for top squarks in the dilepton channel, sim-ulated at √ s = 8 TeV [199]. (c) Spin analysing powers in top decays viaa charged Higgs H + [200]. (d) Modifications of ∆ φ (cid:0) b, ¯ b (cid:1) by the decay t → H + b [200]. t ¯ t Spin Correlation
In this section the most recent results concerning the measurement of top quark polar-ization and t ¯ t spin correlation are presented. Top quarks produced in pairs via the strong interaction are predicted to be unpolarized[166]. This yields to vanishing coefficients B in Equation 2.47 and P in Equation 2.46,44 .6. Recent Measurements of t ¯ t Spin Correlation respectively. The D0 collaboration analysed 5 . − of data, taken at √ s = 1 .
96 TeV.They found a good agreement between the cos θ distributions and the SM expectationwithout quoting an explicit value B for the polarization [201]. Instead, a Kolmogorov-Smirnov (KS) test of the SM prediction of cos θ , using the helicity basis, was performed.It lead to a KS test probability of 14 % in the dilepton channel and 58 % in the (cid:96) + jetschannel. The charged leptons were used as analysers.The strategy of fitting templates to the cos θ distributions was also followed by ATLAS.As in the D0 measurement, the helicity basis was used. With 5 . − of data takenat √ s = 7 TeV ATLAS measured B in two scenarios [202]. In the first case, a non-vanishing polarization was assumed to stem from a CP conserving process. This lead to B CP c = − . ± .
014 (stat.) ± .
037 (syst.). In case of a maximally CP violating pro-cess causing the polarization it was measured as B CP v = 0 . ± .
016 (stat.) +0 . − . (syst.).Figure 2.19(a) shows the measured and fitted distribution of cos θ using the CP conserv-ing hypothesis. (a) ) l q cos( -1 -0.5 0 0.5 1 )) l q / d ( c o s ( s d s / ) unfolded kg.b - Data (Syst. uncertaintyMC@NLO parton level = 7 TeVs at -1 CMS, 5.0 fb (b)
Figure 2.19.: (a) Distributions of the angles between the charged leptons and the helicityspin basis. The data was fit with a CP conserving polarization hypothesis[202]. (b) Unfolded distribution of the angle between the charged leptonand the helicity spin basis [188].The CMS collaboration used 5 . − of data taken at √ s = 7 TeV to measure theasymmetry A P = N (cos θ l > − N (cos θ l < N (cos θ l >
0) + N (cos θ l <
0) (2.60)of unfolded cos θ distributions (see Figure 2.19(b)). CP invariance was assumed andthe charged leptons were used as analysers. This asymmetry is directly related to thepolarization B via B = 2 · A P .CMS measured B = 0 . ± .
026 (stat.) ± .
040 (syst.) ± .
016 (top p T ).All top polarization results are in good agreement with the SM prediction of B = 0.45 . Standard Model, Top Quarks and Spin Correlation At the Tevatron, the full dataset of 5.4 fb − was analysed by the CDF and D0 col-laborations. In the dilepton channel D0 performed a fit of t ¯ t signal templates to thecos θ l + cos θ l − distributions mixing uncorrelated events with events correlated as pre-dicted by the SM [203].Their result was a spin correlation C beam = 0 . +0 . − . (stat.) ± .
11 (syst.) which agreeswith the SM prediction C SMbeam = 0 .
78. For the same dataset and t ¯ t channel the spincorrelation was measured via f SM , explained in Section 2.4.4, by using a matrix-element-based approach [204]. This leads to f SM = 0 . +0 . − . (stat.) +0 . − . (syst.) which is in goodagreement with the SM prediction of f SM = 1 . t ¯ t spins at the 97.7 % CL. In the (cid:96) + jets channel D0 measured with the same approach f SM = 1 . +0 . − . (stat.) ± .
18 (syst.). Figure 2.20(a) shows the likelihood discriminant R for the (cid:96) + jets channel. The combination with the result in the dilepton channel leadsto an evidence for t ¯ t spin correlation by excluding the uncorrelated scenario at the 3 σ level ( f SM = 0 . ± .
29) [205].The CDF collaboration fitted two-dimensional distributions of (cos θ l + , cos θ l − ) and(cos θ b , cos θ ¯ b ) between the two charged leptons and the two b jets in the dilepton channelusing 5.1 fb − of data and the beam line basis [206]. The templates were parameterizedas a function of the spin correlation C . As a result, C = 0 . +0 . − . (stat. ⊕ syst.)was obtained. In the (cid:96) + jets channel CDF fitted two-dimensional distributions of(cos θ l cos θ d , cos θ l cos θ b ) using both the beam line and the helicity basis [207] with5.3 fb − of data. In Figure 2.20(b) the fitted result for the cos θ l · cos θ d distribution isshown. The two-dimensional fit lead to C beam = 0 . ± .
64 (stat.) ± .
26 (syst.) and C helicity = 0 . ± .
48 (stat.) ± .
22 (syst.).
R0.4 0.45 0.5 0.55 0.6 / b i n eve n t s N Data SM spin corr.tt no spin corr.tt tmeasured tOtherW+jetsMultijet -1 DØ, L=5.3 fb
R0.4 0.45 0.5 0.55 0.6 / b i n eve n t s N / b i n eve n t s N / b i n eve n t s N / b i n eve n t s N / b i n eve n t s N / b i n eve n t s N (a) ) d θ )*cos( l θ cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 E ve n t s d θ )*cos( l θ cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 E ve n t s ) d θ )*Cos( l θ Beam Basis Bilinear Cos(
Opposite Spin Same Spin Backgrounds Data : 0.72 +/- 0.64 +/- 0.26 κ -1 CDF Run II preliminary L=5.3 fb (b)
Figure 2.20.: Results of measured t ¯ t spin correlations at the Tevatron in the (cid:96) + jetschannel. (a) Likelihood discriminant R for the analysis in the t ¯ t (cid:96) + jetschannel of the D0 analysis. (b) Result of the fitted cos θ l · cos θ d (beam linebasis) distribution of the CDF analysis [207].Moving to the LHC shows that new approaches have to be chosen. On the one46 .6. Recent Measurements of t ¯ t Spin Correlation hand, the helicity basis offers a higher expected value of C compared to the beamline basis. On the other hand the unique opportunity of using ∆ φ distributions isgiven. As this requires no full event reconstruction, except the identification of thespin analysers, the ATLAS and CMS experiments both measured ∆ φ ( l + , l − ) in thedilepton channel. Already 2.1 fb − provided sufficient statistics, allowing to exclude thescenario of uncorrelated t ¯ t pairs at the 5 σ level by ATLAS [187]. Two templates of∆ φ ( l + , l − ) distributions were used: SM prediction of spin correlation and uncorrelated t ¯ t events. A value of f SM = 1 . ± .
14 (stat) +0 . − . (syst.) was measured and translatedto C helicity = 0 . ± .
04 (stat.) +0 . − . (syst.) and C maximal = 0 . ± .
06 (stat.) +0 . − . (syst.),respectively. The sum of ∆ φ distributions for the ee , µµ and eµ channel is shown inFigure 2.21(a) for data and the two distributions of SM-like and uncorrelated t ¯ t events.The full dataset taken at √ s = 7 TeV with an integrated luminosity of 4.6 fb − was φ∆ E v en t s data (SM)tt (uncorrelated)ttsingle top*+jets γ Z/dibosonfake leptons ATLAS -1 Ldt = 2.1 fb ∫ (a) (radians) -l+l fD ) - l + l fD / d ( s d s / ) unfolded kg.b - Data (Syst. uncertaintyMC@NLO parton level) t m = m (SM, Si G.-Bernreuther & Z. W. ) t m = m (uncorrelated, Si G.-Bernreuther & Z. W. = 7 TeVs at -1 CMS, 5.0 fb (b)
Figure 2.21.: LHC measurements of the ∆ φ ( l + , l − ) distributions in t ¯ t events decayingin the dilepton channel. (a) ATLAS result leading to observation of spincorrelation [187]. (b) Unfolded distribution with subtracted background asmeasured by CMS together with NLO predictions [188].also analysed by ATLAS [180] fitting SM-like and uncorrelated t ¯ t signal templates inthe dilepton channel and extracting f SM . The results for the ∆ φ , the S -Ratio (Figure2.23(a)), and the cos θ l + cos θ l − (Figure 2.23(b) for the maximal basis) distributionsare shown in Figure 2.22. All results agree with the SM predictions. However, withthe exception of ∆ φ , all other distributions result in slightly lower values of f SM than47 . Standard Model, Top Quarks and Spin Correlation predicted. The results of [180], shown in Figure 2.22, include measurements of thedilepton channel and the results of this thesis. The results of this thesis are the first LHCmeasurements in the (cid:96) + jets channel. CMS performed an unfolding of the ∆ φ ( l + , l − ) Standard model fraction0 0.5 1 1.5 21.68.4 maximal basis ) - q ) cos( + q cos( – – helicity basis ) - q ) cos( + q cos( – – S-ratio – – (l+jets) fD – – (dilepton) fD – – ATLAS = 7 TeV s, -1 Ldt = 4.6 fb (cid:242) spin correlation measurementstt SM f (stat) – (syst) – Figure 2.22.: Overview of ATLAS results of measurements of the t ¯ t spin correlation forthe 4.6 fb − θ l + cos θ l − ≡ c · c distributions using 5.0 fb − of 7 TeV data. Asymmetries A , A ∆ φ = N (∆ φ (cid:96) + (cid:96) − > π/ − N (∆ φ (cid:96) + (cid:96) − < π/ N (∆ φ (cid:96) + (cid:96) − > π/
2) + N (∆ φ (cid:96) + (cid:96) − < π/ , (2.61) A c c = N ( c · c > − N ( c · c < N ( c · c >
0) + N ( c · c < , (2.62)which are related to the spin correlation, were measured and compared to the predictionsof the SM. Table 2.9 shows the results. Here, A c c is directly related to the spin corre- Data (unfolded)
MC@NLO
NLO (SM) NLO (uncorr.) A ∆ φ . ± . ± . ± .
012 0 . ± .
001 0 . +0 . − . . +0 . − . A c c − . ± . ± . ± . − . ± . − . ± .
006 0
Table 2.9.: Asymmetries related to the t ¯ t spin correlation measured by CMS [188]. Theuncertainties are statistical, systematic and an additional uncertainty fromtop p T reweighting. For MC@NLO the statistical uncertainty is quoted. TheNLO calculation includes the uncertainty of a variation of the renormalizationand factorization scale by a factor of two.lation C helicity as defined in Equation 2.48 [169] via C helicity = − A c c . It is remarkable48 .6. Recent Measurements of t ¯ t Spin Correlation that the ∆ φ distributions agree very well with the SM predictions as seen in Figure2.21(b) while the translated value of C helicity = 0 . ± .
15 does not ( C SMhelicity = 0 .
31, seeTable 2.7). E v en t s / . fit result (SM)tt (no corr.) ttdatabackground ATLAS -1 L dt = 4.6 fb (cid:242) = 7 TeVsdilepton
S-Ratio0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 R a t i o (a) E v en t s / . fit result (SM)tt (no corr.) ttdatabackground ATLAS -1 L dt = 4.6 fb (cid:242) = 7 TeVsdilepton maximal ) - q )cos( + q cos(-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 R a t i o (b) Figure 2.23.: ATLAS measurements using 4.6 fb − of data selecting t ¯ t events in thedilepton channel and fitting (a) the S -Ratio and (b) the cos θ l + cos θ l − forthe maximal basis [180].To conclude, several bases and quantities sensitive to the t ¯ t spin correlation have beenmeasured to be consistent with the SM predictions. The uncertainties, however, arestill too large to allow tight exclusion limits of BSM models. A small but not significanttrend was observed, namely that the cos θ i cos θ j distributions consistently lead to valuesbelow the SM prediction for all Tevatron and LHC measurements. This was not the caseusing ∆ φ .Measurements of the top quark polarization [188, 202] have not shown a deviationfrom zero, agreeing with the LO SM prediction.With the exception of the results presented in this thesis, no measurement of t ¯ t spincorrelation at the LHC in the (cid:96) + jets channel has been published. This motivated thechoice of the (cid:96) + jets channel for the analysis presented in this thesis. This analysis willinvestigate t ¯ t spin correlation via hadronic analysers and test aspects that a measurementin the dilepton channel cannot access. 49 Experimental Setup
Producing t ¯ t pairs requires large production energies. As presently no lepton collideris able to deliver a centre-of-mass energy of √ s = 2 m t , only the two most powerfulhadron colliders are able to produce top quarks. The first one is the proton/anti-protoncollider Tevatron located at Fermilab close to Chicago with √ s = 1 .
96 TeV. The topquark’s discovery was made at the two Tevatron experiments CDF and D0 in 1995[1, 2]. The second accelerator is the
Large Hadron Collider (LHC) colliding protonswith √ s = 7 TeV for the data taking period in 2011 before moving to √ s = 8 TeVfor the 2012 dataset [208]. Currently, the machine is being upgraded to operate with √ s = 13 TeV and √ s = 14 TeV for the time after the Long Shutdown 1 (LS1) [209].Apart from the high centre-of-mass energy, a high luminosity L is the key to maximizethe total number of observed t ¯ t events. With an integrated luminosity (cid:82) L dt of 4 . − and a t ¯ t production cross section of σ t ¯ t (7 TeV) = 173 . ± . ± . t ¯ t pairs were produced for the 2011 dataset.The ATLAS detector was used to take data analysed in this thesis. This chapter willintroduce the accelerator and detector used.
The Large Hadron Collider [208] is the world’s most powerful particle accelerator. It isbased at the international particle research laboratory
CERN (Organisation europ´eennepour la recherche nucl´eaire) near Geneva, Switzerland. Based in a 26 . [208] between the Jura mountains and the Geneva The Tevatron stopped operation in 2011. The LHC tunnel has an inclination of 1.4 %, leading to a variation of its altitude of about ±
60 m.See [212] for details. . Experimental Setup Lake it can operate in three modes: proton/proton, proton/ion and ion/ion collision. TheLHC uses protons for both beams in order to reach the design value of the instantaneousluminosity of L = 10 cm − s − . The design centre-of-mass energy of 14 TeV will bereached after the LS1 will be completed. So far, the LHC was running with √ s = 7 TeV(2011 dataset) and √ s = 8 TeV (2012 dataset). Such high energies require the usage ofsuperconductive magnets. 1232 superconducting dipole magnets use niobium-titaniumcoils operating at 1.9 K and producing a magnetic field of up to 8.3 Tesla. A sophisticatedmagnet design allows housing both beam pipes in the same cryostat. Additional 392superconducting quadrupole magnets are used to focus and stabilize the beam, supportedby further multipole magnets of higher order. Reaching the design values, the LHC willcontain 2808 bunches per beam, each consisting of 1 . · protons and spaced with adistance of 25 ns. Details about the LHC machine parameters for the design values andthe dataset analysed in this thesis are given in Section 5.1.The LHC is fed with protons from the CERN accelerator chain: Hydrogen atoms areionized and accelerated up to 50 MeV at the LINAC II before they are injected to the
Booster (1.4 GeV) which then fills the
Proton Synchroton (PS, 25 GeV). From here theprotons are lead to the
Super Proton Synchrotron (SPS, 450 GeV) before they reachtheir final destination, the LHC. Ions start being accelerated at the
LINAC III andthe
LEIR (Low Energy Ion Ring) before being filled to the PS (450 GeV). The CERNaccelerator complex is illustrated in Figure 3.1.The two proton beams are crossed and brought to collision at four interaction points.Each of them is surrounded by an experiment:
ATLAS [213] and
CMS [214] are twogeneral purpose detectors. They both cover almost the full solid angle and aim forhigh luminosities and low β ∗ to discover rare events. The main physics goals of theseexperiments are the search for a Higgs boson, Dark Matter candidates and signaturesfor supersymmetry. The ALICE [215] experiment focuses on the analysis of heavy ioncollisions searching for signatures of the quark gluon plasma ( QGP ) and analysing thebehaviour of hadronic matter at high densities and temperatures. The
LHCb [216]experiment focuses on B-physics and physics at low scattering angles. It is asymmetricand covers only a part of the phase space. Its physics goals are the study of CP violationand BSM physics involving heavy flavours.Next to the four big experiments several smaller ones are located close to the interac-tion point, such as
MOEDAL [217] searching for magnetic monopoles,
LHCf [218] thatstudies hadron interaction models used in cosmic ray analyses and
TOTEM [219] forelastic and diffractive cross section measurements.
The ATLAS detector is a general purpose detector covering almost the full solid angle.It consists of several layers of tracking, calorimetry and muon chamber devices. ATLAS Beam shapes can be modelled by Gaussian distributions in the transverse plane. The β functiondescribes how the constant beam emittance is reduced to the beam width σ during collimation via σ = √ εβ . The value of β at the interaction point is indicated with β ∗ . .2. The ATLAS Detector Figure 3.1.: The CERN accelerator complex c (cid:13)
CERN.is capable of dealing with event rates of up to 40 million events per second resulting fromthe high luminosities provided by the LHC. As up to 50-140 (for the design values of theLHC, depending on the chosen filling scheme [209]) hard scattering events can pile upduring a bunch crossing, an excellent tracking system is required in order to associate thereconstructed physics objects with different interaction processes. The tracking devicesare also used for tagging jets as b -jets . Such jets emerged from an initial B-meson leadingto a secondary vertex within the tracking system with a probability large enough to beutilized for b -jet tagging.The calorimeters are needed to determine the energy of electrons, photons and jetsprecisely. With a good spatial resolution the calorimeter system is able to provide a highmass resolution.An additional muon system combined with a high magnetic field is the basis for muonreconstruction, triggering and high precision measurement.Before describing individual components of ATLAS in the next sections, some con-ventions about the coordinate system will be explained as they are used throughout thewhole thesis. ATLAS uses a right-handed coordinate system with the beam directiondefining the z-axis. The x/y-plane is transverse to the beam axis with the x-axis pointingfrom the interaction point in the centre of ATLAS to the centre of the LHC ring. The53 . Experimental Setup y-axis points upwards. The azimuthal angle φ is used in the x/y-plane. The polar angle θ is measured from the beam axis. The rapidity y ≡ ln (cid:16) E − p L E + p L (cid:17) (using the longitudinalmomentum component p L ) is preferred to θ as its intervals and corresponding differentialcross sections are invariant under Lorentz boosts along the z-axis [46]. Instead of therapidity y the pseudo-rapidity η ≡ − ln (cid:0) tan (cid:0) θ (cid:1)(cid:1) is often used as an approximation for p (cid:29) m [46]. For massless objects both expressions are equivalent.Many parts of the ATLAS detector are split into a central part with a barrel structureand a forward part with an end-cap structure. An overview of the ATLAS detector withits components is shown in Figure 3.2.Figure 3.2.: The ATLAS detector with its components [213]. Close to the interaction region the particle flux is quite high as up to 1,000 particlesare expected to be created in each bunch crossing [213], depending on the luminositydelivered by the LHC. Those particles, which are charged, will leave tracks in the innerdetector. The number of tracks depends on the instantaneous luminosity and the averagenumber of interactions per bunch crossing, (cid:104) µ (cid:105) . In [220] this number of tracks wasmeasured as N tracks ≈ / (cid:104) µ (cid:105) at √ s = 7 TeV for tracks with p T >
400 MeV.There are several requirements on the devices measuring the tracks of these particles:As the particle density and the production rates are very high, the measurement needsto be made with very high granularity for two reasons. On the one hand, only a high54 .2. The ATLAS Detector granularity enables a separation of all the particle’s tracks and a reconstruction of thecorresponding vertices. On the other hand, the high granularity implies a high numberof readout channels. The higher that number is, the lower the rate per channel gets.Lowering this rate per channel is mandatory for the high event rates. Next to therequirement of performing the measurement as precisely as possible, the detector mustalso minimize the disturbance of the particle’s trajectory. The material – quoted in termsof radiation lengths X – needs to be minimal. This reduces the possibility of trackdeflections and photon conversions. The latter effect is leading to a misidentification ofphotons as charged particles.The Inner Detector ( ID ) of ATLAS consists of tracking systems using three differenttechniques. These are all enclosed in a 2 T solenoidal field for momentum determinationand charge separation. They all make use of the fact that charged particles ionize ma-terial and leave charges that can be kept as signals. The highest resolution is providedby the Silicon Pixel Detector having the smallest distance of R ≥ .
55 cm to the inter-action point. Three layers in the barrel and three discs on each end-cap provide about80.3 million readout channels with an accuracy of 10 × µ m ( R/φ × z for the barrel, R/φ × R for discs).The Pixel Detector is surrounded by the Silicon Microstrip Tracker ( SCT ). Insteadof pixels it uses small-angle (40 mrad) stereo silicon strips with an intrinsic resolutionof 17 × µ m ( R/φ × z for barrel, R/φ × R for discs) for about 6.3 million readoutchannels. The SCT is also split into a barrel and a disc part.The outermost part of the ID system is the Transition Radiation Tracker ( TRT ).Straw tubes covering the range up to | η | = 2 . R/φ information only, usingabout 351,000 channels with an intrinsic accuracy of 130 µ m per straw. The strawsare filled with a Xe/CO /O mixture. Transition radiation is emitted when chargedparticles pass through the material with different dielectric constants [221]. The intensityof the emitted transition radiation depends on the relativistic γ factor of the particlepassing through the TRT. For a given momentum, this allows separating heavy fromlight particles, so for example electrons and pions.The whole tracking system as shown in Figure 3.3 covers a range of | η | < . σ p T /p T = 0 .
05 % · p T [GeV] ⊕ Insertable B-Layer ( IBL ) [222] will be the new innermostcomponent of the ATLAS detector.
By inducing electromagnetic and hadronic showers and measuring their electromagneticcomponents, the calorimeter system is able to determine the particles’ energies. Theseelectromagnetic and hadronic showers must be fully contained in the calorimeter system. The radiation length X is defined as the average distance after which an electron loses its energydown to the fraction of e via Bremsstrahlung [46]. . Experimental Setup Figure 3.3.: The ATLAS Inner Detector [213].Hence, it needs to provide enough material in terms of the radiation length X or thenuclear interaction length λ to stop particles up to energies of several hundreds of GeV.As muons are too heavy to radiate a sufficient amount of energy via Bremsstrahlung, theydo not induce electromagnetic showers. Hence, they propagate through the calorimetersystem, leaving traces of ionized particles. Even though this is insufficient for a reliableestimate of the muon energy, it still allows adding information for muon tracking.The ATLAS calorimeter system has a sampling structure including active materialfor the readout of the signal and also passive material for the shower induction. Thecalorimeter system shown in Figure 3.4 is divided into the Electromagnetic Calorimeter ( ECal ), starting right after the solenoid magnet surrounding the ID system, and the
Hadronic Calorimeter ( HCal ) behind the ECal. While the ECal has a sufficient sizeto stop most electrons and photons via electromagnetic showers, the HCal is neededin addition to stop hadronically showering particles. The measured energy needs to bedetermined with a high precision. Next to a good energy resolution, analyses involvingphotons need another feature provided by the calorimetry. As photons leave no trackin the Inner Detector, their direction can only be determined by the point of impactin the calorimeter. Thus, the ECal provides a very high granularity in particular inits first layer. It also contains a presampler to determine the energy loss in the partsin front of the calorimetry. The ECal uses liquid argon (LAr) as active material andlead/stainless steel as passive material. It has an accordion shape to ensure full φ coverage at high granularity. The η coverage for the ECal is | η | < .
475 for the barrelpart and 1 . < | η | < . η × ∆ φ varies as The nuclear interaction length λ is defined analogously to X , but for hadronic interactions [46]. .2. The ATLAS Detector Figure 3.4.: The ATLAS calorimeter system [213].a function of | η | between 0 . × .
025 and 0 . × . | η | < .
7. In contrast, the
Hadronic End Cap ( HEC )uses a LAr/Copper combination and extends the HCal to | η | < .
2. The forward partwith 3 . < | η | < . Forward Calorimeter ( FCal ) using LAr as activeand copper (EM part) and tungsten (hadronic part) as absorbers.The total resolution of the calorimetry is σ E /E = 10 % / √ E [GeV] ⊕ . σ E /E = 50 % / √ E [GeV] ⊕ σ E /E = 100 % / √ E [GeV] ⊕
10 % FCalThe total thickness of the calorimeter system is ≈
20 X ( | η | < .
4) and ≈
30 X (1 . < | η | < .
2) for the ECal and ≈ λ for the combined ECal, HCal and FCal [213]. As the calorimeter system stops all detectable particles except muons, the muon spec-trometer ( MS ) is placed in the outermost region of ATLAS. A toroidal magnetic field,described in Section 3.2.4, is placed outside the calorimeters. This additional magneticfield, and the caused curvature of the muon tracks, allow for an additional momentummeasurement for muons. The information of the outer muon spectrometer shown inFigure 3.5 is combined with the track information provided by the ID to a combined57 . Experimental Setup muon track. Different techniques are used in the MS. Monitored drift tubes ( MDTs ) areused for precision tracking in both the barrel part of the MS ( | η | < .
4) as well as theend-cap part (1 . < | η | < . CathodeStrip Chambers ( CSCs ) with a high granularity in the region of 2 . < | η | < . Resistive plate chambers ( RPCs ) are used forFigure 3.5.: The ATLAS muon system [213].the barrel and
Thin-gap chambers ( TGCs ) are used for the end-cap part. Both systemsoffer a fast readout. Next to triggering, the RPCs and TGCs are also used to providesecondary tracking information. The whole MS provides about one million channels anda total resolution of σ p T /p T = 10 % at 1 TeV [213]. The ATLAS magnet system consists of four components. They create a magnetic fielddeflecting the particles in order to allow for momentum measurements of the trackingdevices. All of them use superconducting NbTi conductors (+Cu for the toroid) whichare stabilized with Al. The first one, a solenoid providing a 2 T magnetic field at thecentre of the detector, surrounds the ID and is aligned parallel to the beam axis. Thematerial budget of the solenoid is kept low with ≈ .
66 X [223] in order to avoid particleinteractions which disturb the calorimeter measurements.A second set of magnets provides the toroidal field for the MS. It consists of an air-corebarrel magnet with eight racetrack shaped coils [224] and two air-core end-cap magnetswith eight squared coils each. The magnetic field varies between 0.15 T and 2.5 T for58 .2. The ATLAS Detector the barrel, with an average of 0.5 T. The field of the end-cap part varies between 0.2and 3.5 T (1 T average) [213]. The event rate of about 40 MHz (at design value) is far too high to allow for the storage ofall the corresponding collision data. The
Trigger and Data Acquisition ( TDAQ ) systemneeds to filter interesting events at a rate of 200 Hz. This is done via a three level triggersystem, divided into the L1, L2 and Event Filter (EF) trigger. The system of L2 andHL triggers is also referred to as
High Level Trigger ( HLT ).The first one, L1, reduces the rate from 40 MHz to 75 kHz. In contrast to L2 and theEF, which are software based, it is hardware based as it needs to be extremely fast. Byusing low granularity information from the calorimeters and the MS it defines
Regionsof Interest ( ROI ) within the detector. These ROIs contain objects which are definedin the trigger menus . Such objects can be muons, jets, electrons, photons or τ -leptonswith a high transverse momentum. Also, events with a high amount of totally depositedtransverse momentum or missing transverse momentum ( E missT , defined in Section 4.4)can be triggered.Based on the L1 information, the L2 trigger reads out the full detector information.Only a certain part of the detector, the ROI, is read out at this stage. In contrast tothe L1 trigger, the L2 is also able to add information from the ID. After the L2 decisionthe event rate is reduced further below 3.5 kHz.If an event is stored or not is decided by the EF trigger which uses the full detectorinformation available for each event. It reduces the event rate below 200 Hz. In many analyses the data is compared to a prediction. In order to predict the expectednumber of events the luminosity of the dataset under study must be known.By knowing the number of average inelastic interactions per bunch crossing, µ (or thenumber of visible ones µ vis ), the number of bunches n b , the revolution frequency f r andthe production cross section of inelastic proton/proton reactions σ inel (and the efficiency ε to actually observe them) the luminosity can be calculated as L = µn b f r σ inel = µ vis n b f r εσ inel (3.1)according to [225]. ATLAS uses several detectors for an online measurement of theluminosity during data taking. The most important one is LUCID [226], a Cherenkovdetector placed at ±
17 m from the interaction point, 10 cm away from the beam line. Itconsists of 16 Al tubes filled with C F and attached photomultipliers that are used tocollect the Cherenkov light.At ±
140 m from the interaction point the
Zero-Degree Calorimeters ( ZDC ) [227] arelocated right behind the place where the common beam line is split into two. The final-triplet quadrupoles of the LHC deflect all charged particles out of the acceptance of the59 . Experimental Setup
ZDC [225]. It measures events with mesons decaying into photons and neutrons emittedat very forward angles. Such events play an important role in centrality measurementsof heavy ion collisions [227].The main device used for beam loss monitoring is the
Beam Conditions Monitor ( BCM ) [228]. It consists of radiation hard diamond sensors located at ±
184 cm fromthe interaction point. For low luminosity runs before the 2011 dataset, the
MinimumBias Trigger Scintillators ( MBTS ) [229, 230] located at ±
365 cm from the interactionpoint have also been used. For offline luminosity measurements the ATLAS ID and partsof the EMCal (inner wheel of the EMEC and first layer of FCal [225]) are also used.All ways of measuring the luminosity described above are relative measurements with aneed for an absolute calibration.In the future, an absolute calibration of the luminosity will be possible with the
ALFA [231] detector. Located at ±
240 m from the interaction point in one of the
Roman Pots [231] it can be moved as close as 1 mm to the proton beam. For runs with special beamsettings (low β ∗ , low emittance) the measurement of elastic proton/proton scatteringat low angles can be used to calculate the luminosity. This is possible as the totalproton/proton cross section is proportional to the imaginary part of the elastic scatteringamplitude in the limit of zero momentum transfer, as stated by the optical theorem [231].Until ALFA is fully operational, van der Meer (vdM) scans [232] are used to measurethe horizontal and vertical beam profiles Σ x and Σ y by scanning the two proton beamsacross each other horizontally and vertically [233]. These profiles are translated to theabsolute luminosity via L = n b f r n n π Σ x Σ y . (3.2)Based on these vdM scans the luminosity for the dataset used in this analysis ismeasured with an uncertainty of 1.8 % [230].60 Analysis Objects
Within the field of particles physics, the laws of nature are studied at the fundamentallevel. At this level, elementary forces interact with elementary particles. Most objects ofinterest are, however, not accessible by experiments at that level. All quarks except thetop quark are immediately bound within confinement and only observable as compositeobjects. Others such as the top quark, τ leptons, W , Z and Higgs bosons will immedi-ately decay before they can be observed by a detector. Physics objects can be describedat different stages, illustrated in Figure 4.1. The hard interaction process, described atleading order, is often referred to as the parton level . After the hard scattering theprocess of parton showering takes place, forming bound states. These are observable asparticles which are in principle detectable. This level is called the particle level . Reach-ing the detector, the particles will interact and leave signatures. The detection of thesesignatures takes place at the detector level . Each object at parton level has a distincttype of signature that it leaves in the detector. This allows reconstructing objects fromthese signatures.
Reconstruction level is a synonym for the detector level.The goal of the event reconstruction, described in Chapter 6, is to map the objects atthe detector level to the initial objects at parton level. This links the measurement tothe analysis of the physics process of interest.In this chapter the objects recorded in the detector are described. Figure 4.2 shows a t ¯ t candidate event as it is measured in the detector. The detector objects are highlighted.This example of a t ¯ t decaying in the dilepton channel includes all objects of interest forthe analysis presented in this thesis.All of the objects described in the following will be called candidates, as their detectorsignature does not necessarily need to be produced by the expected particle. Jets ,bunches of particles stemming from a hadronisation process, for instance might also be Even though the class of partons only includes quarks and gluons, also leptons and bosons can bedescribed at the same level. . Analysis Objects Figure 4.1.: Illustration of a particle detection process and the different levels of objectdescriptions.mis-reconstructed as electrons in case they deposit a large fraction of their energy in theEM calorimeter. Section 5.4.1 discusses the derivation of a data-driven estimate of suchjets mimicking leptons (referred to as fake leptons ). From now on antiparticles will notbe mentioned explicitly but being included in the name of their corresponding particles.
Being charged, electrons are supposed to leave a track in the ID and to be stopped inthe ECal by the process of an electromagnetic shower [46]. The reconstruction startswith the search of tracks pointing at clusters of energy deposition in the ECal with E T > . Such seed clusters, consisting of 3 × η × φ in the central ECallayer, are matched to tracks with p T > . × × The transverse energy E T is defined as E T = E cluster / cosh ( η track ). .1. Electrons MET b-jetsElectronb-jet b-jetMuon
Figure 4.2.: An ATLAS event display showing a t ¯ t candidate event with both top quarksdecaying leptonically, one into an electron, the other one into a muon. Theevent is shown in the r/φ plane (upper left) and the r/z plane (lower left).The electron is represented by the green track pointing to the correspondingclusters in the ECal. The signature of the muon is visible as red trackpassing the muon spectrometer. The two jets in the event were both taggedas b -jets. Their secondary vertices are visible in the zoomed view of theprimary vertex (upper right). Picture translated from [234], original versionfrom [235].the ECal) energy leakage. The four-momentum vector of the electron is built by takingthe energy from the cluster and the momentum direction from the track. This choice ismotivated by the higher precision with which the quantities can be measured: trackingensures a precise determination of the direction and the calorimeter is more precise in thedetermination of the energy. Due to the acceptance of the ID only electron candidateswith | η cluster | < .
47 are considered. In case the energy cluster of an electron candidateis affected by a malfunctioning front-end board or high voltage supply, it is rejected.The selection of electrons and the rejection of jets mis-identified as electrons is madeby a cut-based classification. These cuts take information from both the calorimeterand the ID into account. The sets used for the present analysis base on the original This is not the case for low-energetic electrons, which will not be studied in this thesis. . Analysis Objects classification into loose , medium and tight electrons in the order of decreasing efficiencyand increasing purity [236, 237]. The original tight definition has been updated andoptimized for the data taking period in 2011 to the tight++ . While the tight++ electronsare used for the event selection in this analysis the medium++ (looser identificationcuts as tight++ [236, 237]) electrons are needed for the estimation of the fake leptonbackground (see Section 5.4.1). The electron identification efficiency for the differentclassifications are shown as a function of pile-up in Figure 4.3. Number of reconstructed vertices2 4 6 8 10 12 14 16 18 20 E l e c t r on i den t i f i c a t i on e ff i c i en cy [ % ] ATLAS
Preliminary -1 ≈ L dt ∫ MC Loose++Data Loose++ MC Medium++Data Medium++ MC Tight++Data Tight++
Figure 4.3.: Identification efficiency for different electron types as a function of pile-up [238].Selected electrons need to pass additional isolation requirements to further suppresscontributions from misidentified jets. The amount of energy (momentum) in a cone witha size of R = 0 . R = 0 .
3) around the electron cluster (track) that does not belong tothe actual electron object is summed. The values, referred to as
EtCone20 and
PtCone30 may not pass a certain value, depending on the E T and the η of the object. These valuesare corrected for energy leakage and additional depositions from pile-up events to ensurea constant isolation efficiency of 90 %. A tag-and-probe method (T&P method) [236,237]is used to determine the efficiency. To avoid a double counting of a detector signatureas both electron and jet, the reconstructed jet (see Section 4.3) which is closest to theelectron is removed if it is within a distance ∆ R = 0 . R = 0 .
4. This removal also ensures a proper isolation of the electron. In order toreach the efficiency plateau of the electron triggers (see Section 6.1) a cut of E T > z -axis between the reconstructed primary vertex and the electron needsto be less than 2 mm.As the simulated events do not mimic the real performance of objects perfectly, cor-rection factors have to be applied. A scale factor (SF) is defined as the ratio of theefficiency in data ( ε data ) and in the simulation ( ε MC ). Such scale factors are appliedto the simulation to properly model the performance. Table 4.1 shows an overview ofcorrections for electron objects. These corrections were mostly studied by evaluating64 .2. Muons Corrected Quantity Applied to ParameterizationEnergy correction for crack region Data, MC η , E T Electron energy scale Data η cluster , φ cluster , E T Electron energy resolution MC η , E Trigger efficiency MC η , E T , data taking periodSelection and isolation efficiency MC η , E T Reconstruction efficiency MC η Table 4.1.: Corrections and scale factors for electrons. Z → ee , J/ψ → ee and W → eν events using a T&P method [236, 237]. For the lattertype of events the ratio of electron energy in the calorimeter to electron momentum in thetracker was studied. Details about the electron energy calibration methods are providedin [239]. Due to the large amount of passive material in front of the transition regionbetween barrel and end-cap calorimeters ( crack region ) with 1 . < | η cluster | < . . < | η cluster | < .
52 are fully rejected for this analysis due tothe low reconstruction efficiency and low resolution.
Muons leave tracks in the ID and the MS system. The
MuId algorithm [240] starts fromthe outermost layer with track segments of the MS. It matches tracks to the ID tracksegments, takes energy losses in the calorimeter into account and refits the full muontrack. The muon momentum is required to be larger than 20 GeV. The acceptancesof the ID and MS require | η | < .
5. As for the electrons, the muon’s distance on the z -axis to the reconstructed primary vertex needs to be less than 2 mm. Similar to theelectron classification, muons are classified as loose and tight. The former classificationis used for the signal selection and the latter for the fake lepton estimation. Loose muonsdemand a certain track quality: A hit in the B-layer of the ID is required if expected.At least one hit in the pixel detector must be recorded. Non-operational pixel sensorsare automatically counted. The sum of hits in the SCT plus the number of crossednon-operational sensors must be at least six. The total number of non-operational pixeland SCT sensors must not be larger than two. With n TRT = n TRT hits + n TRT outliers asthe sum of the TRT track hits and the TRT outliers (as defined in [240]) it is requiredthat n TRT > n TRT outliers /n TRT < . | η | < . n TRT outliers /n TRT < . n TRT > | η | ≥ . PtCone30 and
EtCone20 definitions have been used to65 . Analysis Objects ensure tracking and calorimeter isolation.
EtCone20 < PtCone30 < . R < . p T >
25 GeV and | JVF | > . η , p T Momentum resolution (both ID and MS) MC η , p T Trigger efficiency MC η , φ , data taking periodIsolation efficiency MC data taking periodReconstruction efficiency MC p T , η , φ Table 4.2.: Corrections and scale factors for muons.
After production or scattering, quarks will form hadrons or (in case of the top quark)decay, making it impossible to detect them as free partons. Particles being producedin this process of hadronisation will deposit energy in the ECal and HCal and leavetracks in the ID. These energy depositions are collimated and can be reconstructed asjets. Only the energy depositions induced by electromagnetic processes will be visible.This includes both electromagnetic showers as well as ionization processes from hadronicshowers. As no method of hadronic compensation [46] is used at ATLAS and certain jetcomponents (e.g. neutrinos and slow neutrons) don not leave electromagnetic signatures,the total jet energy must be deduced from the electromagnetic component.Energy depositions in topological clusters [242] are reconstructed as jets using the anti- k t algorithm [243] with a distance parameter R = 0 . FASTJET [244] software.The first level of energy calibration is called
EM scale , as only detectable electromag-netic energy depositions are taken into account. At this stage, the energy is correctedfor contributions arising from the in-time and out-of-time pile-up. These correctionsdepend on the number n PV of reconstructed primary vertices, the average number ofinteractions per bunch crossing, (cid:104) µ (cid:105) , for the specific luminosity block and the pseudo-rapidity η of the jet [246]. The four-momentum of the jet is corrected for the positionof the primary vertex [246]. Jets suffer from a bias of their reconstruction direction In-time pile-up refers to energy depositions from objects not belonging to the hard scattering process,but to the same bunch crossing. Out-of-time pile-up describes energy-depositions from up to 12 (2011dataset) preceding and one following bunch crossings [245]. .3. Jets towards better instrumented regions with respect to poorer detected regions. Thus, anadditional η correction is applied. It is significant in the calorimeter transition regionsonly (see Figure 4.4(a)) [246].After the reconstruction on the level of the calibrated EM scale, jets need to becalibrated to the hadronic scale. A variety of calibration methods is described in [246].For this analysis jets are calibrated using the EM+JES calibration . For this calibrationMonte Carlo simulations are used to calculate the jet response R jetEM = E jetEM /E jettruth . Itis defined as the ratio of the jet energy visible via EM depositions divided by the total(true) jet energy. The inverse of R , parameterized in p T and η , is then used to scalethe jet energy on the EM scale up to the calibrated jet energy. This level of calibratedjet energy is referred to as the EM+JES scale. Figure 4.4(b) shows the jet response asa function of pseudo-rapidity for different bins of jet energy calibrated to the EM+JESscale. | det η |0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 | o r i g i n η | | t r u t h η | E = 30 GeVE = 60 GeVE = 110 GeV E = 400 GeVE = 2000 GeV = 0.6, EM+JES R t Anti k
ATLAS simulation (a) | det η |0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 J e t r e s pon s e a t E M sc a l e E = 30 GeVE = 60 GeVE = 110 GeV E = 400 GeVE = 2000 GeV
ForwardEndcap ForwardTransitionEndcapBarrel EndcapTransitionBarrel = 0.6, EM+JES R t Anti k
ATLAS simulation (b)
Figure 4.4.: (a) Difference between the origin corrected η jet and the true η jet in bins ofthe calorimeter jet energy calibrated with the EM+JES scheme as a functionof η detector [246]. (b) Jet response R jetEM = E jetEM /E jettruth [246].On top of the EM+JES calibration, which is only based on MC simulations, in-situ calibration techniques have been applied [245]. In the central detector region with | η | < . Z bosons serve asreference objects for jets with p T <
800 GeV. For larger transverse momenta, multijetsystems are also used for balancing. The forward region of the detector is calibratedwith respect to the central part using di-jet p T balancing.Jets are required to have | η | < . p T >
25 GeV. If a jet has a distance of∆
R < . p T >
20 GeVwhich did not pass the quality criteria. 67 . Analysis Objects
With increasing instantaneous luminosity the average number of interactions perbunch-crossing rises. Hence, the contribution of jets which are reconstructed but donot belong to the main interaction of interest increases as well. To cope with this prob-lem, a discriminating variable called
Jet Vertex Fraction ( JVF ) is introduced whichseparates jets belonging to the primary vertex of interest from other background jets.This variable is introduced in the following section. To be accepted, a jet needs to passthe criterion of | JVF | ≥ . Jets taken into account for the event reconstruction are all supposed to stem from thesame primary vertex. However, also jets from pile-up events are selected and can pass thequality criteria. Hence, an additional discrimination between primary and pile-up jetsis needed. The discrimination bases on a procedure introduced by the D0 collaboration[247]. It is realized by checking all tracks corresponding to the jet for their primaryvertex. Figure 4.5 illustrates the principle. Each track is associated with at least oneFigure 4.5.: Illustration of the principle of the Jet Vertex Fraction variable. While forthe left jet all tracks originate from PV and none from PV , the right jethas only a certain fraction f of tracks from PV and another fraction 1 − f from PV .primary vertex. In Figure 4.5 all tracks of jet originate from the primary vertex PV ,and none from PV . In the case of jet , only the fraction f of tracks originates fromPV . Such a fraction can be calculated for each pair of jet and primary vertex. Thehigher this fraction f is, the more likely it is that the jet-to-vertex assignment was donecorrectly. Cutting on this fraction, the jet vertex fraction (JVF), allows discriminatingprimary jets from pile-up jets.Figure 4.5 is just a simplification of the real JVF determination. In fact, a track mightbe allocated to several vertices depending on the size of a variable distance window. The68 .3. Jets JVF is computed as shown in Equation 4.1:JVF(jet , vertex) = (cid:80) tracks of jet (vertex ∈ track.vertexlist) · p track, jetT (cid:80) tracks of jet p track, jetT (4.1)The expression (vertex ∈ track.vertexlist) is 1 if the vertex is contained in the list of allvertices assigned to the track and 0 if not.Figure 4.6(b) shows the effect of the pile-up suppression in Z + jets events. As in caseno track could be matched to a jet a JVF of − Jet vertex fraction (JVF)0 0.2 0.4 0.6 0.8 1 F r a c t i on o f j e t s R=0.4 t anti k , 25ns pile up s cm ≤ | η
20 GeV, | ≥ T p Hard scatter jetsJets from pile up
ATLAS
Simulation
PYTHIA QCD dijets (a)
Number of primary vertices1 2 3 4 5 6 7 8 9 10 >10 > j e t s < N ATLAS
Preliminary
R=0.4 EM+JES t Anti k | < 2.5 η > 25 GeV, | jetT p L dt = 2 fb ∫ =7 TeV, s Data 2011Z ALPGEN MC10Data 2011 (|JVF| > 0.75)Z ALPGEN MC10 (|JVF| > 0.75) (b)
Figure 4.6.: (a) Distribution of the JVF variable for jets from the hard scattering processand for pile-up jets [248]. (b) Pile-up suppression by the application of theJVF cut, shown as stability of the number of reconstructed jets against thenumber of primary vertices for Z + jets events [248].not perfectly modelled by the Monte Carlo simulation, scale factors have to be applied.These are derived using a T&P method on a sample of Z → ee and Z → µµ eventsusing specific selections to get samples of hard-scattering jets and pile-up jets. Furtherdetails of the JVF studied with data from the 2012 dataset can be found in [249]. Hadronising b -quarks offer the unique opportunity to tag their jets experimentally. Astop quarks decay to almost 100 % into a b -quark [46], their identification is of specialinterest. B-hadrons inside a b -jet are massive ( m B ≈ cτ ≈ . . Analysis Objects ATLAS ID. Furthermore, B-hadrons have a significant branching fraction of ≈
10 % for B → lνX [46], where l represents either an electron or a muon (10 % per lepton, so 20 %for both). The high p T of the B-hadron’s decay products, caused by the B-hadron’shigh mass, leads to a high track multiplicity inside the b -jet. The b -tagging algorithmsIP3D, SV1 and JetFitterCombNN [250], which are used at ATLAS, make use of thesediscriminative jet properties and provide a quantity called b -tag weight . Cutting onthis weight allows tagging b -jets with a certain purity and efficiency. The fraction ofreal b -jets contained in a b -tagged sample is given as purity. The efficiency gives theprobability to tag a b -jet as such.In this analysis the MV1 tagger is used. It is based on a neural network using theoutput weights of the
JetFitter , IP3D and
SV1 algorithms [250] as input. This taggerwas set up at a working point with a b -tagging efficiency of 70 % and a purity of 91 %for simulated t ¯ t events by requiring the b -tag weight w > . b -taggedjet multiplicity (Figure 4.7(a)). (a) (b) Figure 4.7.: Distribution of the b -jet multiplicity using (a) the default calibration basedon di-jet events ( pTrel + System8 ) [251] and (b) the combined calibration( pTrel + System8 +dileptonic t ¯ t ) [252] for t ¯ t pairs decaying into an electronand at least four jets.The default calibration was known to become unreliable for b -jets with a high p T [251].A modified calibration is needed to perform the spin correlation analysis, as a proper70 .4. Missing Transverse Momentum b -tag multiplicity modelling is essential. The default calibration is combined with a t ¯ t calibration derived from dileptonic t ¯ t events [252]. Taking the dileptonic calibrationensures statistical independence of the calibration and the analysed data in the (cid:96) + jetschannel. The default calibration scale factors ( pTrel + System8 ) are compared to thescale factors using a combined default and t ¯ t calibration (KinSelDL) in Figure 4.8. Allscale factors agree within uncertainties. By comparing the b -jet multiplicity using default [GeV] T Jet p50 100 150 200 250 300 S c a l e F a c t o r pTrel+system8 (stat.+syst.)TagCount SL (stat.+syst)TagCount DL (stat.+syst)KinSel SL (stat.+syst.)KinSel DL (stat.+syst.)KinFit SL (stat.+syst.) ATLAS
Preliminary MV1 70% Ldt= 4.7 fb ∫ = 7 TeVs Figure 4.8.: Overview of b -tag scale factors using different calibration methods [252].calibration (Figure 4.7(a)) to the one calibrated with the improved combined calibration(4.7(b)) one can clearly see the improvement. The disadvantage of hadron colliders compared to lepton colliders is the unknown initialstate. While the momenta of the colliding protons at the LHC are known, their collidingconstituents’ momenta are only known at the level of probability distributions accordingto the PDFs.Particles not leaving a signature in the ATLAS detector, such as neutrinos, can bereconstructed indirectly by the application of the laws of momentum conservation. As theinitial state of the pp collisions is not known, full momentum conversation cannot be used.Nevertheless, momentum conservation can be applied to the transverse plane. Here thetotal momentum before and after the collision is zero. This allows reconstructing the sumof momenta for particles not leaving a signature. It is referred to as Missing TransverseMomentum . E missT expresses its magnitude. It is calculated from the energy depositions The term E missT is misleading and sometimes also referred to as Missing Transverse Energy . Usingthe argument of momentum conservation, the directly missing quantity must be a vector – themomentum. . Analysis Objects in the calorimeters and the tracks in the muon spectrometer which are grouped to theobjects to which they are assigned and calibrated accordingly. The remaining energydepositions are grouped into the CellOut term : E miss x ( y ) = E miss ,electronx ( y ) + E miss,jets x ( y ) + E miss,softjets x ( y ) + E miss,muon ,µx ( y ) + E miss,CellOut x ( y ) (4.2)The magnitude of the E missT is given by E missT = (cid:113) ( E miss x ) + (cid:0) E miss y (cid:1) . (4.3)As the object definitions may vary from analysis to analysis so may the definition of the E missT . In this analysis the following E missT object definition was used: • Electrons as defined in Section 4.1 are required to have p T >
10 GeV. All energyscale correction factors except the out-of-cluster correction are applied. • Jets as defined in Section 4.3 with p T >
20 GeV are used at the EM+JES calibra-tion level. • Jets with 7 GeV < p T <
20 GeV are included by the softjets term. They are beingcalibrated to the EM scale only. • Combined muons (see Section 4.2) with | η | < . . < | η | < .
7, no combined muons can be reconstructed.Still, muons reconstructed using the MS only are included for this η region. Themuon energy deposited in the calorimeters is added to the cellout term in case themuon is isolated (∆ R (muon, jet) > .
3) and added to the jets term in case it isnot. • All further energy cells are added to the CellOut term.A detailed description of the MET performance on the 2011 dataset is given in [253]. τ Leptons
Both top quarks of a produced t ¯ t pair decay into a b -quark and a W boson of whichabout 11 % decay into a τ lepton [46]. These can contribute to the selected signalsample depending on their decay channel (and the decay channel of the other W boson).As ≈
35 % of all τ leptons directly decay into an electron or muon via τ → e ¯ ν e ν τ or τ → µ ¯ ν µ ν τ [46] the detector signature will look very similar to a prompt W → e ¯ ν e or W → µ ¯ ν µ decay. A larger E missT and changed kinematics of the electrons and muonsstemming from decayed τ leptons make the difference. This analysis does not make use ofan explicit τ lepton reconstruction. Events including τ leptons might pass the selectionand the τ decay products will be reconstructed as (non-prompt) electrons, muons orjets.72 Signal and Background Modelling
Physics results in high energy physics experiments can often only be obtained by thecomparison of simulated to measured events. This includes the signal and backgroundevents of which some are estimated by data-driven methods. The used dataset as wellas the modelled signal and background events are described in this chapter. For acomparison of simulated and measured event properties a well-defined event selection isneeded. Hence, this comparison is shown in the following Chapter 6.The simulated events include PDFs, the hard scattering process, parton showering aswell as a simulation of the energy depositions of the particles in the ATLAS detector.The latter was achieved using the GEANT4 [254] toolkit.
Data taken with the ATLAS detector at √ s = 7 TeV are analysed for the measurementof t ¯ t spin correlation. This dataset will be called “the 2011 dataset” or “the 7 TeVdataset”. Each time the ATLAS TDAQ records data , a run number is assigned to thedata taken. Each run is then separated into luminosity blocks ( LBs ). Further, only datapassing certain data quality criteria are taken into account. The data quality can bedegraded if certain components of the detector are not available or are providing baddata due to misconfiguration or malfunction. Physics analyses use a
Good Runs List ( GRL ) including the run numbers and LBs of the data to be analysed. The GRL of the2011 dataset includes the run numbers 178044-191933. These runs were taken in the timebetween the 22nd of March 2011 and the 30th of October 2011. After a technical stop inSeptember 2011 the β ∗ of the proton beams was reduced from 1.5 m to 1.0 m, leading toan increased instantaneous luminosity. The evolution of the instantaneous luminosity is A block of data taking usually starts with the beginning of collisions and end with a beam dump. . Dataset, Signal and Background Modelling shown in Figure 5.1. Figure 5.2(a) shows the evolution of the total integrated luminosityof the 2011 dataset, separated in luminosity delivered by the LHC, recorded by ATLASand accepted by the GRLs. The average number of interactions per bunch crossing, Month in 2010 Month in 2011 Month in 2012 J a n A p r J u l O c t J a n A p r J u l O c t J a n A p r J u l O c t ] s c m P ea k Lu m i no s i t y [ = 7 TeVs = 7 TeVs = 8 TeVs ATLAS
Online Luminosity
Figure 5.1.: Evolution of the instantaneous luminosity delivered to ATLAS during datataking in 2010-2012 [255]. µ , also increased due to the lowering of β ∗ (see Figure 5.2(b)). With increasing (cid:104) µ (cid:105) the so-called pile-up rises: energy depositions reconstructed as objects not belonging tothe hard scattering process are also included in the event and have to be vetoed. Theprocedure to veto such pile-up objects during the event selection was described in Section4.3.1. During the data taking in 2011 the LHC beam parameters had not yet reached Day in 2011 f b T o t a l I n t eg r a t ed Lu m i no s i t y = 7 TeVs ATLAS
LHC DeliveredATLAS RecordedGood for Physics Total Delivered: 5.46 fb Total Recorded: 5.08 fb Good for Physics: 4.57 fb (a)
Mean Number of Interactions per Crossing0 2 4 6 8 10 12 14 16 18 20 22 24 ] R e c o r ded Lu m i no s i t y [ pb
10 =7 TeVsATLAS Online 2011, Ldt=5.2 fb ∫ > = 11.6 µ * = 1.0 m, < β > = 6.3 µ * = 1.5 m, < β (b) Figure 5.2.: (a) Evolution of the integrated luminosity (cid:82) L dt delivered, recorded andaccepted by the GRL for the 2011 dataset [255]. (b) The mean number ofinteractions per bunch crossing for the 2011 dataset [255].their design values. Table 5.1 lists the beam parameters at the beginning and the end74 .2. t ¯ t Signal Samples of the 7 TeV run and compares them to the LHC design values. The total integratedluminosity accepted by the GRLs corresponds to an integrated luminosity of (cid:82) L dt =4 . − . The datasets for both the taken data and the used simulations are listed inAppendix B.Protons/Bunch Bunches Bunch Spacing β ∗ Peak Luminosity2011 1 . · . · cm − s − Nominal 1 . · . · cm − s − Table 5.1.: LHC beam parameters at the end of the 7 TeV run as well as the designvalues [256]. t ¯ t Signal Samples
Top quark pair production is simulated using the NLO generator
MC@NLO v4.01 [257–260]assuming a top quark mass of 172.5 GeV and using the NLO PDF CT10 [81]. For par-ton showering and hadronisation,
HERWIG
JIMMY
ATLAS Un-derlying Event Tune
AUET2-CT10 [263].Samples without spin correlation are generated by setting the
MC@NLO parametersIL = IL = 7 [260]. In this case the top and anti-top quark decay is not performed by MC@NLO , but by
HERWIG . As the top spin information is not propagated to
HERWIG , topquark spins are effectively decorrelated. As a side effect, the sample with uncorrelated t ¯ t pairs has a top width of Γ t = 0. This effect was studied to have no impact on the actualanalysis. Samples including Standard Model spin correlation are generated by settingthe variables IL = IL = 0.The t ¯ t cross section for pp collisions at a centre-of-mass energy of √ s = 7 TeV is σ t ¯ t = 177 +10 − pb for a top quark mass of 172 . /c . It has been calculated atnext-to-next-to leading order (NNLO) in QCD including resummation of next-to-next-to-leading logarithmic (NNLL) soft gluon terms with top++2.0 [96, 264–268]. The PDFand α S uncertainties were calculated using the PDF4LHC prescription [269] with theMSTW2008 68 % CL NNLO [83,270], CT10 NNLO [81,271] and NNPDF2.3 5f FFN [82]PDF sets, added in quadrature to the scale uncertainty. The NNLO+NNLL value, asimplemented in Hathor 1.5 [272], is about 3 % larger than the exact NNLO prediction.All t ¯ t final states except the full hadronic ones are included in these two samples. Thefull hadronic t ¯ t final states are included in the fake lepton estimation, as described inSection 5.4.1.It should be mentioned that despite the fact that a NLO generator, MC@NLO , was usedto model the signal, the t ¯ t spin correlation is not implemented at full NLO [273]. Theimplementation of the spin correlation treatment in MC@NLO , as described in [259], doesnot include spin dependent virtual corrections [273]. The actual effects on the analysis75 . Dataset, Signal and Background Modelling can be neglected. The described missing NLO contributions affect only part of thedifference between LO and NLO. Even the total difference between NLO and LO spincorrelation is low ( ≈ For the background arising from single-top production in the s - and the W t -channel,
MC@NLO + HERWIG is used with NLO PDF CT10, invoking the diagram removal scheme [274] to remove overlaps between the single-top and t ¯ t final states. For the t -channel, AcerMC [275] interfaced to
PYTHIA [276] 6.452 with modified LO PDFs (MRST LO ∗∗ ,LHAPDF 20651) [277, 278] is used. Background contributions arising from
W W , ZZ and W Z production ( diboson back-ground ) are simulated by the
HERWIG generator with modified LO PDFs (MRST LO ∗∗ ,LHAPDF 20651). As HERWIG is a LO generator, the cross sections of the dibosonprocesses are scaled to match the NLO prediction. W +Jets W boson production in association with multiple jets is the dominating source of back-ground events. To simulate these, ALPGEN v2.13 [279] is used. It implements the exactLO matrix elements for final states with up to five partons using the LO PDF setCTEQ6L1 [280]. To simulate parton showering, hadronisation and multi-parton inter-actions, the
HERWIG and
JIMMY generators are used as for the simulation of the t ¯ t signal.Dedicated samples are used for the production of heavy flavour samples ( W + c +jets, W + c ¯ c +jets and W + b ¯ b +jets). The MLM [281] matching scheme of the ALPGEN genera-tor is used to remove overlaps between the n and n + 1 parton samples with parameters RCLUS =0.7 and
ETCLUS =20 GeV. Also, phase space overlaps across the different flavoursamples are removed. While MC simulations are used to determine the kinematic shapesof the W + jets background, its normalization and flavour composition is derived by adata-driven approach, described in Sections 5.4.2 and 5.4.3. Z +Jets For the estimation of the background contribution caused by the Drell-Yan production
Z/γ ∗ → (cid:96) + (cid:96) − plus additional jets, ALPGEN + HERWIG with the LO PDF set CTEQ6L1is used as for the W +jets background. Additional jets are simulated with up to fiveadditional partons on matrix element level. Even though this simulation takes into In most of the other cases
HERWIG served as parton shower and hadronisation generator and JIMMYfor the underlying event model. .4. Data Driven Backgrounds account interferences between Z and γ ∗ bosons, it is briefly called Z +jets background.Two sets of samples were used for Z +jets background: inclusive Z +jets samples and inaddition dedicated Z + b ¯ b samples. Overlapping phase spaces are removed. The crosssections are scaled to match the NNLO predictions. The objects reconstructed as isolated leptons can in fact also be either jets with a highelectromagnetic component or hadrons from jets decaying into leptons that seem to beisolated. This fake lepton background is caused by QCD induced multijet events. Suchevents would demand a large amount of MC statistics and suffer from a limited abilityof proper MC modelling. Thus, a data-driven approach is chosen to estimate the fakelepton background. It is based on the matrix method which is introduced before thechannel specific estimates are explained.
Matrix Method
For the matrix method [282] two dedicated samples are produced by applying differentselection criteria to the taken data. Two different lepton isolation criteria, loose andtight (see sections 4.1 and 4.2), are used. Of these two, the tight definition has morestringent isolation requirements. The total number of events passing each of the criteriais a sum of real and fake leptons: N loose = N loosereal + N loosefake , (5.1) N tight = N tightreal + N tightfake . (5.2)As the tight sample is a subset of the loose, selection efficiencies for real and fake leptonscan be defined as ε real = N tightreal N loosereal , ε fake = N tightfake N loosefake . (5.3)The number of fake leptons within the tight sample can be expressed as N tightfake = ε fake ε real − ε fake (cid:16) ε real N loose − N tight (cid:17) (5.4)Hence, by knowing the real and fake efficiencies the selected loose and tight samples canbe used to obtain a sample of tight fake leptons. Technically this is done by applyingthe weights w loose = ε fake ε real ε real − ε fake , w tight = ε fake ( ε real − ε real − ε fake , (5.5) Hence it is also called multijet background or misidentified lepton background. . Dataset, Signal and Background Modelling to the events, leading to positive weights for loose and negative weights for tight events,and merging them. In the following the determination of ε real and ε take for both the µ + jets and the e + jets channel are described. µ + jets Channel
For the µ + jets channel, two different approaches (A and B) are used and the resultingfake lepton background estimation is averaged. For both methods loose muons are de-fined the same way as tight ones but without the requirement on the isolation ( PtCone30 and
EtCone20 ). In method A, both ε real and ε fake are parameterized in | η | and p T ofthe muon. In method B, ε real is found to be constant as a function of both p T and η while ε fake is parameterized in | η µ | . The fake dominated control region for methodA is defined by cutting on the transverse W boson mass W m T <
20 GeV and E missT + W m T <
60 GeV. Method B uses E missT <
20 GeV and E missT + W m T <
60 GeV instead.Further, the signal efficiencies are obtained from Monte Carlo simulation (which agreeswithin 1 % with the values derived by the T&P method on data [283]). The fake efficien-cies are obtained by an extrapolation of an impact parameter significance d /σ ( d ) [250]dependent tight to loose ratio. This method makes use of the fact that fake muonsoriginate from heavy flavour jets and hence have a larger impact parameter significance. e + jets Channel
In the e + jets channel, ε real is derived via a tag-and-probe method for Z → ee events.As the topologies of t ¯ t and Z → ee events are different and also affect ε real , a correctionfactor derived from Monte Carlo samples is applied to account for that. The fake effi-ciencies are derived in a control region with E missT <
20 GeV in which the contribution offurther backgrounds, estimated via Monte Carlo samples, is subtracted. Both efficienciesare parameterized as functions of | η e | and ∆ R ( e, closest jet). The analysis described inthis thesis demands in particular a good angular distribution of the background. Forthis reason, the ∆ R parameterized fake lepton estimate was validated and implemented.The improvement gained by the new parameterization for the fake lepton backgroundcan be seen by comparing Figure 5.3(a) to 5.3(b). In particular, events with leptons andjets having a close distance are more accurately modelled. W +jets Normalization While MC simulations are used to estimate the W +jets background, a data-driven ap-proach estimates its normalization with lower uncertainty than that from the MC pre-diction. The approach makes use of the charge asymmetry (CA) of W ± production atthe LHC with its pp collision mode. It leads to r = σ ( pp → W + ) /σ ( pp → W − ) ≈ . u - and d -quarks (see Figure 2.3). While the totalnormalization of W +jets is not well modelled by the MC, the ratio r is. It can be used The transverse W boson mass is defined as W m T = (cid:112) p l T p ν T (1 − cos( φ l − φ ν )). .4. Data Driven Backgrounds E v en t s / . Datatt (dil.)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty -1 (cid:48) L dt = 4. fb (cid:42) (cid:42) (lep, jet) min dR p r ed ./ da t a (a) E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty -1 L dt = 4.6 fb (cid:48) (cid:42) (cid:42) (lep, jet) min dR p r ed ./ da t a (b) Figure 5.3.: Improvement by the choice of a new parameterization for the fake leptonestimate. Mismodelling is visible particularly in the ∆ R (lepton, jet) distri-butions: (a) Old parameterization in p T and η of the lepton. The templatestatistics uncertainty is shown for the prediction. (b) New parameterizationwith an additional ∆ R (lepton , jet) dependence. The template statistics un-certainty, the normalization uncertainty for the data driven yields and thetheory uncertainty on the cross sections are shown for the prediction andpropagated to the error band in the ratio.to determine the total yield of W +jets events N W by measuring the number of eventsincluding a positive ( D + ) or negative ( D − ) lepton in data. N W = N W + + N W − = (cid:32) N MC W + + N MC W − N MC W + − N MC W − (cid:33) (cid:0) D + − D − (cid:1) (5.6)= (cid:18) r MC + 1 r MC − (cid:19) (cid:0) D + − D − (cid:1) . (5.7)The common event selection as described in Section 6.1, except cuts on the b -tagging,is used to measure the difference ( D + − D − ). t ¯ t events, fake lepton background and Z -jets events are produced symmetrically (with respect to the lepton charge) to a goodapproximation. Hence, the assumption that the difference ( D + − D − ) is caused by the W -asymmetry is valid after the background from single top production is subtracted.To obtain the normalization for the selection of n jets of which at least one jet is79 . Dataset, Signal and Background Modelling b -tagged the equation W n ≥ = W n pretag f f → n tag (5.8)is used. It includes the fraction f of tagged to untagged jets ( tagging fraction ) forthe 2-jet selection and the ratio f of tagging fractions between the 2-jet and the n -jetselection, which is derived by MC simulation. W +jets Flavour Composition Next to the total normalization of W +jets events, the flavour composition of the sampleneeds to be determined by a data-driven approach. The total number of W +jets eventscan be divided into the jet flavour types bb , cc , c and light with corresponding fractions F (summing up to one). An event of each subclass has a probability P to be b -tagged.Using this classification, the total number of W +jets events in a b -tagged sample with i jets is given by N W ± , tag = N W ± , pretag ( F bb,i P bb,i + F cc,i P cc,i + F c,i P c,i + F light ,i P light ,i ) (5.9)= N W ± , pretag ( F bb,i P bb,i + k ccbb F bb,i P cc,i + F c,i P c,i + F light ,i P light ,i ) . (5.10)Instead of determining F cc,i individually it is expressed by F bb,i and the ratio of F cc,i /F bb,i ,which is derived from MC simulation. Also, the tagging probabilities P are estimatedfrom MC. The flavour compositions are determined for subsamples with positive andnegative leptons separately. By using N W ± , pretag from the CA normalization and re-quiring that N W ± , tag matches between simulation and data, the flavour fractions arerescaled by factors K accordingly. The rescaled flavour fractions are then put back intothe CA normalization method iteratively until convergence is reached. As a baseline,events with two jets are used. Applying the scale factors K to other jet multiplicitysamples can cause the sum of fractions to deviate from one: K bb, F MC bb,i + K cc, F MC cc,i + K c, F MC c,i + K light , F MClight ,i = A. (5.11)A correction is applied by rescaling each fraction K with A : K xx,i = K xx, A . (5.12)80
Event Selection and Reconstruction
With the (cid:96) +jets channel as the t ¯ t final state under study, advantages and challenges comealong. Every event provides the necessary information for full reconstruction of all theobjects of interest. The dilepton channel suffers from the ambiguity of two undetectedneutrinos, making it hard to reconstruct both of them correctly. In contrast, usingassumptions such as the masses of the top quark and the W boson, the whole t ¯ t eventcan be reconstructed in the (cid:96) + jets channel. This includes the single neutrino from theleptonic top decay. The challenge of the (cid:96) + jets channel reconstruction is the separationof the two T = ± non- b -jets.In this chapter the applied selection cuts are explained before the data is compared tothe prediction. The agreement of the MC generator to the measured data is discussedbefore the reconstruction of the events is explained in detail, in particular the separationof the two light jets from the t → W b → bqq (cid:48) decay. The correct selection of top spinanalysers is validated. Finally, reconstruction efficiencies are presented and compared toother reconstruction methods. t ¯ t Selection in the Lepton+Jets Channel
As no dedicated τ lepton reconstruction is used, the chosen (cid:96) + jets channel splits into an e + jets and a µ + jets channel. Still, τ +jets events with leptonically decaying τ leptonsare part of the signal as well. Also dileptonic events including a hadronically decaying τ are selected. In Section 2.3.1 it was explained how the ideal (cid:96) + jets decays compareto the actual ones.Events from the e + jets and µ + jets channel are selected from single lepton triggerstreams. For each period the selected trigger is the unprescaled trigger with the lowest A trigger prescale of p randomly drops a fraction (1 − p ) of events that had passed the trigger chain . Event Selection and Reconstruction p T threshold. To avoid a loss of trigger efficiency, the p T cut of the selected lepton ischosen such that it is well above the p T thresholds of the trigger leptons. With thischoice the reconstructed objects are said to be ‘within the trigger efficiency plateau’.The chosen trigger streams are listed in Table 6.1 and depend on the data taking period.
Electron Trigger Muon TriggerEF e20 medium (before period K)EF e22 medium (period K)EF e22vh medium1 or EF e45 medium1 (period L-M) EF mu18 (before period J)EF mu18 medium (starting period J)
Table 6.1.: Used trigger streams depending on the data taking period.In the following sections, the expression ’good’ refers to objects passing the qualitycriteria as defined in Chapter 4. e + jets Selection
1. The electron trigger must have fired.2. The event must contain at least one primary vertex with at least five tracks.3. Exactly one good electron is found.4. No good muon is found.5. The good electron must match the object that fired the trigger.6. No jet with p T >
20 GeV failing the quality cuts may be included in the event.7. At least four good jets with p T >
25 GeV, | η | < .
5, and a jet vertex fraction | JVF | > .
75 are found.8. E missT >
30 GeV is required.9. The transverse W mass W m T must be larger than 30 GeV.10. At least one b -jet must be identified using the MV1 tagger at the 70 % efficiencyworking point.The cuts on E missT and W m T suppress the multijet background containing fake leptons. to reduce the data rate. Unprescaled triggers have p = 1, so no events are dropped. p µ T >
20 GeV and E e T >
25 GeV. .2. Data/MC Agreement µ + jets Selection
1. The muon trigger must have fired.2. The event must contain at least one primary vertex with at least five tracks.3. Exactly one good muon is found.4. No good electron is found.5. The good muon must match the object that fired the trigger.6. Electrons and muons must not share a track.7. No jet with p T >
20 GeV failing the quality cuts may be included in the event.8. At least four good jets with p T >
25 GeV, | η | < .
5, and a jet vertex fraction | JVF | > .
75 are found.9. E missT >
20 GeV is required.10. W m T + E missT must be larger than 60 GeV.11. At least one b -jet must be identified using the MV1 tagger at the 70 % efficiencyworking point.The cuts on E missT and W m T suppress the multijet background containing fake leptons.As this particular background contamination is lower in the µ + jets channel comparedto the e + jets channel, these cuts were chosen less stringent. The number of events after the event selection and after the application of scale factors forthe signal and background MC, scaled to the integrated luminosity of the data, is shownin Table 6.2. The number of observed events in the data is also shown. The uncertaintiescome from the uncertainties on the cross sections for the MC driven backgrounds andby the variation of the real and fake efficiencies according to their uncertainties for thefake lepton background. The numbers of selected events before the application of the b -tagging cut are listed in Table C.1 in the Appendix.The yield agreement as a function of the run period is shown in Figure 6.1 for aselection with at least four jets. In this section, the agreement between data and prediction is shown. They are in goodagreement. The uncertainties on the prediction (propagated into the yellow uncertaintyband in the ratio) are given by the uncertainties on the calculated cross sections for83 . Event Selection and Reconstruction n jets ≥ n b-tags ≥ e + jets µ + jets W +jets (DD/MC) 2320 ±
390 4840 ± Z +jets (MC) 450 ±
210 480 ± ±
420 1830 ± ±
60 1980 ± ± ± t ¯ t ) 4830 ±
620 9200 ± t ¯ t (MC, l+jets) 15 130 ±
900 25 200 ± t ¯ t (MC, dilepton) 2090 ±
120 3130 ± ± ± n jets ≥ E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ Data Period K p r ed ./ da t a IB-D E-H J L-M (a) E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ Data Period p r ed ./ da t a B-D E-H I J K L-M (b)
Figure 6.1.: Yield for data (points) and the different Monte Carlo contributions (filledhistograms) split into the different run periods for (a) the e + jets channeland (b) the µ + jets channel. The default selection of at least four jets withat least one b -tagged jet was used.84 .3. Mismodelling of the Jet Multiplicity the MC driven backgrounds and the normalization uncertainty on the lepton fake back-ground. These uncertainties are treated as uncorrelated.All of the following plots require the default selection to be passed. The inclusive n jets ≥ b -tagrequirement as well as a split into subsamples of n jets = 4, n jets ≥ n b-tags = 1 and n b-tags ≥ ?? . E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ [GeV] T Lepton p p r ed ./ da t a E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ η Lepton
2 1 0 1 2 p r ed ./ da t a
3 2 1 0 1 2 3 E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ φ Lepton p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ [GeV] T Lepton p p r ed ./ da t a E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ η Lepton
2 1 0 1 2 p r ed ./ da t a
3 2 1 0 1 2 3 E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ φ Lepton p r ed ./ da t a Figure 6.2.: Control distributions for the lepton p T , η and φ distribution of the e + jets(top) and µ + jets (bottom) channel ( n jets ≥ , n b-tags ≥ The jet multiplicity it not well described by the
MC@NLO generator. As shown in Figure6.6, the
MC@NLO generator predicts less jets than observed in data. Furthermore, thejet p T spectrum is modelled softer (see figures 6.4 and 6.5). This fact was extensivelystudied, also independently of this thesis. An extensive study of the jet multiplicity in t ¯ t events can be found in [284].Figure 6.6 shows the jet multiplicity mismodelling of MC@NLO . 85 . Event Selection and Reconstruction E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ [GeV] T Jet p p r ed ./ da t a E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ η Jet
2 1 0 1 2 p r ed ./ da t a
3 2 1 0 1 2 3 E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ φ Jet p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ [GeV] T Jet p p r ed ./ da t a E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ η Jet
2 1 0 1 2 p r ed ./ da t a
3 2 1 0 1 2 3 E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ φ Jet p r ed ./ da t a Figure 6.3.: Control distributions for the jet p T , η and φ distribution of the e + jets (top)and µ + jets (bottom) channel (all selected jets, n jets ≥ , n b-tags ≥ .3. Mismodelling of the Jet Multiplicity E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ [GeV] T Leading jet p p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ [GeV] T p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ [GeV] T p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ [GeV] T Leading jet p p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ [GeV] T p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ [GeV] T p r ed ./ da t a Figure 6.4.: Control distributions for the jet p T of the three highest p T jets of the e + jets(top) and µ + jets (bottom) channel ( n jets ≥ , n b-tags ≥ . Event Selection and Reconstruction E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ [GeV] T p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ [GeV] T Missing E p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ [GeV] T,W m p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ [GeV] T p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ [GeV] T Missing E p r ed ./ da t a E v en t s / G e V Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ [GeV] T,W m p r ed ./ da t a Figure 6.5.: Control distributions for the p T of jet with the 4th highest p T , the missingtransverse momentum and W m T of the e + jets (top) and µ + jets (bottom)channel ( n jets ≥ , n b-tags ≥ .3. Mismodelling of the Jet Multiplicity E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ Number of jets p r ed ./ da t a (a) E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ Number of jets p r ed ./ da t a (b) Figure 6.6.: Jet multiplicity using
MC@NLO as t ¯ t signal generator. (a) e + jets channel.(b) µ + jets channel.A better modelling of the jet multiplicity is provided by other generators, such as POWHEG . This is shown in Figure 6.7, where
POWHEG + PYTHIA was chosen as MC generatorfor the t ¯ t signal. E v en t s / DatattW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ Number of jets p r ed ./ da t a (a) E v en t s / DatattW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ Number of jets p r ed ./ da t a (b) Figure 6.7.: Jet multiplicity using
POWHEG + PYTHIA as t ¯ t signal generator. (a) e + jetschannel. (b) µ + jets channel.Up to the time that this analysis was finalized, only MC@NLO samples with both SMspin correlation and uncorrelated t ¯ t pairs were available. Hence, no alternative gener-ator could be used. Therefore, the effect of mismodelling was included as a source ofuncertainty which will be discussed in Section 8.1.6. Furthermore, an additional pa-rameter was introduced in the analysis that allowed for an in-situ correction of the jet89 . Event Selection and Reconstruction multiplicity (explained in Section 7.3.4). t ¯ t Events with a Kinematic LikelihoodFit
The challenging part of studying the t ¯ t spin correlation in the (cid:96) + jets channel is thecorrect assignment of the reconstructed quantities to the spin analysers, in particu-lar the different jets. To reconstruct the selected events, a kinematic likelihood fitter( KLFitter ) [285] based on the
Bayesian Analysis Toolkit ( BAT ) [286] is used.
KLFitter is a well-established tool used in several analyses, for example [153, 202, 287–290]. Itallows reconstructing events back to leading order parton level. It is based on a cer-tain model of the final state ( t ¯ t in this case) and uses known constraints to map re-constructed objects (jets, charged leptons and E missT ) to the corresponding LO partonobjects. These constraints are e.g. the masses of the W bosons and the (anti)top quarks.Whereas the angles of reconstructed objects are assumed to be correctly reconstructed,their energies and momenta are allowed to vary within their detector resolutions. Thepossible variations of the energies and momenta are given by transfer functions (TF) W (cid:16) ˜ E object | E LO parton (cid:17) , describing the probability for a LO parton (electron) with en-ergy E to be reconstructed as jet (electron) with an energy ˜ E . For muons, energy isreplaced by transverse momentum. Different sets of TFs were derived for different ac-ceptance regions of the detector using MC truth information. The likelihood L used in KLFitter can be written as: L = B { m ( q q ) | m W , Γ W } · B { m ( lν ) | m W , Γ W } · B { m ( q q b had ) | m t , Γ top } · B { m ( lνb lep ) | m t , Γ top } · W (cid:16) ˜ E jet | E b had (cid:17) · W (cid:16) ˜ E jet | E b lep (cid:17) · W (cid:16) ˜ E jet | E q (cid:17) · W (cid:16) ˜ E jet | E q (cid:17) · W (cid:16) ˜ E missx | p x,ν (cid:17) · W (cid:16) ˜ E missy | p y,ν (cid:17) · (cid:40) W (cid:16) ˜ E l | E l (cid:17) , e + jets channel W (˜ p T ,l | p T ,l ) , µ + jets channel (6.1) W are the transfer functions and B Breit-Wigner distributions of the top and W masswhich were fixed to 172.5 GeV and 80.4 GeV, respectively. All functions are normalized togive an integral of 1.0. The maximization of the likelihood is called kinematic fitting.It is performed for all possible combinations within the permutation table mappingreconstructed particles to a model of truth particles. The number of permutationsdepends on the number of reconstructed jets which are passed to KLFitter . In theideal case the four reconstructed jets represent the number of model partons. Then,only the correct jet-to-parton assignment needs to be found. In case of t ¯ t events thenumber of permutations is 4!=24 (two b -quark and two light quark jets are mapped tothe corresponding quarks) with 4!/2=12 maximized likelihood values. The number of Technically, the negative logarithm of the likelihood is minimized. .5. Transfer Functions different likelihood values is reduced by a factor 2 as the kinematics are invariant underexchange of the jets from the hadronically decaying W boson.One possible setup of KLFitter is to use the permutation with the highest likelihoodvalue as best permutation. A more sophisticated approach is calculating a quantitycalled event probability p i for each permutation i . It is based on a normalized likelihoodand additional extensions ∆ p i,j : p i = L i (cid:81) j ∆ p i,j (cid:80) i L i (cid:81) j ∆ p i,j (6.2)One possible extension is the usage of b -tagging information. The simplest approachassigns a weight of zero to a permutation in case a b -tagged jet is on the position of alight quark model particle:∆ p i, veto = (cid:40) , . (6.3)Another way of using a veto method is to veto permutations where the jet assigned toa b -quark is not tagged or a combination of both veto methods. A more sophisticatedextension takes the efficiencies ε b of b -tagging and the mistag rate ε l [291] of light jetsinto account: ∆ p i, tag = (cid:26) ε b , b had was b -tagged(1 − ε b ) , b had was not b -tagged (cid:27) · (cid:26) ε b , b lep was b -tagged(1 − ε b ) , b lep was not b -tagged (cid:27) · (cid:26) ε l , q was b -tagged(1 − ε l ) , q was not b -tagged (cid:27) · (cid:26) ε l , q was b -tagged(1 − ε l ) , q was not b -tagged (cid:27) (6.4)A b -tag algorithm can be used with a certain cut on the b -tag weight, required to taga jet. This cut defines a working point that comes along with certain values of ε b and ε l . As both ε b and ε l depend on the working point used in an analysis, the method isalso referred to as working point method . In Section 6.6 an extension is introduced thatallows separating light up- and down-type jets. This is necessary in order to successfullymap all spin analysers to their reconstructed objects. Transfer functions W model the relation between energies and momenta of reconstructedobjects and the partons of the LO decay signature (labelled as ’truth’). Such a mappingis needed to account for detector resolution effects. The resolution is modelled by adouble-Gaussian function: W ( E reco , E truth ) = 12 π ( p + p p ) (cid:32) e − (∆ E − p p + p e − (∆ E − p p (cid:33) (6.5) Normalized with respect to all permutations. The mistag rate ε l is the probability to tag a non- b -jet as b -jet. . Event Selection and Reconstruction where ∆ E = E truth − E reco E truth . The double-Gaussian functions account for the detector reso-lution and higher order effects. Transfer functions exist for all reconstructed quantities:Jets, Muons, Electrons and E missT . For objects mainly based on calorimeter information(jets, electrons), the parameters p i are functions of E truth . For muons the energies E arereplaced with the transverse momentum p T . From now on all references to an energyinclude the corresponding formulation for transverse momentum.For the determination of the transfer functions a dedicated tool, TFTool , was devel-oped.
TFTool uses a sample of selected t ¯ t events, produced with the MC@NLO generator.For each event every model object (two b -quarks, two light jets and a charged lepton) istried to be matched to a reconstructed object. Objects i and j are matching in case thedistance is ∆ R < .
3. Furthermore, the matching was to be bi-uniquely, meaning thatwithin a distance ∆ R = 0 . b -jets, light jets, electrons or muons. The b -jet requirementis checked on truth level, meaning that the jet must emerge from a b -quark. No explicitrequirement on the tagging was made. This is of course just one specific classificationchoice. Instead of classifying the model, also the reconstructed object could be classifiedin b -tagged and untagged jets. Furthermore, it is possible to separate jets containing asoft muon and those that do not. For the used dataset this option is excluded as muonsare removed from the jets during the event selection process as described in Section 6.1.The resolution of the reconstructed objects varies for different parts of the detectorand is non-uniform in | η | . Hence, different transfer functions are derived for individualobject types and | η | regions. TFTool assumes individual parameterizations of p i for eachobject type and keeps the type of parameterization fixed across | η | . The parameterization as a function of E truth is motivated by the underlying physicseffects which determine the detector resolution. In the case of calorimeter energy, theresolution for higher energies is σ E E ∼ √ E (see Section 3.2.2). In contrast to this, themuon resolution decreases linearly: σ p T p T ∼ p T . Other parameters which are not relatedto the resolution are estimated as a linear function of E truth : p i = a i + b i E truth . Forbins of E truth and | η | , histograms of E truth − E reco E truth are created and filled with the values ofmatched objects. The binning in | η | is fixed and follows the detector structure by usinga dedicated bin for the calorimeter transition region, for instance (see Table 6.3). Thebinning in E truth is variable to allow for sufficient statistics in each bin. Light Jets [0 . , .
8] [0 . , .
37] [1 . , .
52] [1 . , .
5] [2 . , . . , .
8] [0 . , .
37] [1 . , .
52] [1 . , .
5] [2 . , . . , .
8] [0 . , .
37] [1 . , .
52] [1 . , . . , .
11] [1 . , .
25] [1 . , . Table 6.3.: | η | binning used for the transfer functions.For each of the filled histograms, a double Gaussian function as defined in Equation Only the parameterization type is kept fixed, not the numerical parameter values. .5. Transfer Functions p i are not parameterized globally as function of E truth . In Figure 6.8 local fits for truthenergies of about 100 GeV are shown. truth ) / E reco - E truth (E -1 -0.5 0 0.5 1 E v en t s L ight J etsE: 95-105 GeV | < 0.8 h MCSmall GaussianBig GaussianDouble GaussianGlobal Fit (a) truth ) / E reco - E truth (E -1 -0.5 0 0.5 1 E v en t s h MCSmall GaussianBig GaussianDouble GaussianGlobal Fit (b) truth ) / E reco - E truth (E -0.15 -0.1 -0.05 0 0.05 0.1 0.15 E v en t s E lectronsE: 95-105 GeV | < 0.8 h MCSmall GaussianBig GaussianDouble GaussianGlobal Fit (c) truth ) / pt reco - pt truth (pt -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 E v en t s M uonsp T : 95-125 GeV 0.0 < | h | < 0.8 MCSmall GaussianBig GaussianDouble GaussianGlobal Fit (d)
Figure 6.8.: Transfer function fits for (a) light jets, (b) b-jets, (c) electrons and (d)muons for truth energies / transverse momenta of about 100 GeV. Theentries derived from a signal MC sample are fitted locally with a doubleGaussian function, composed of a small and a big Gaussian function. Theglobal TF fit is also shown.The local fits are a sum of a small and a big Gaussian function. Jets have on av-erage significantly worse resolution than leptons. Furthermore, jets have a trend to bereconstructed with less energy than the original parton has, while the lepton transferfunction is symmetric. This can be explained by final state radiation and out-of-conecontributions. For b -jets this effect is even larger, visible in the more dominant tail onthe right side of Figure 6.8(b) compared to Figure 6.8(a). In the next step, all valuesof p i are plotted against E truth for each of the | η | bins and object types. These param-eters are then fitted with an approximated dependence on E truth . This stage is calledparameter estimation. As an example for light jets in the region 0 . ≤ | η | < . . Event Selection and Reconstruction the uncertainties on the local fits. They are increased by a scaling factor to stabilize theparameter estimation fit. One can see that the approximations work reasonably well. E [GeV]100 200 300 400 500 p _0 -0.02-0.0100.010.02 E [GeV]100 200 300 400 500 p _1 p _2 -0.4-0.200.20.40.6E [GeV]100 200 300 400 500 p _3 p _4 Figure 6.9.: Estimation of parameters p i as a function of E truth for light jets in theregion 0 . ≤ | η | < .
8. The parameters were estimated as functions of E truth according to Table 6.4.The jet energies of E truth <
100 GeV are cut off for the approximation and are extrapo-lated for the parameterization. This decision was made to keep the fit stable. Removingthis cutoff would cause the parameter estimation fits to fail. Correlations among theparameters cause jumps between several local fit options. These need to be kept undercontrol to find a common and global parameterization.Table 6.4 lists the chosen parameterizations for the TF parameters p i . The transfer Light Jets B-Jets Electrons Muons p a + b E truth a + b E truth a + b E truth a + b E truth p a / √ E truth + b a / √ E truth + b a / √ E truth + b a + b E truth p a + b E truth a + b E truth a + b E truth a + b E truth p a + b E truth a + b E truth a + b E truth a + b E truth p a + b E truth a + b E truth a + b E truth a + b E truth Table 6.4.: Parameterizations of the transfer function parameters p i as a function of theLO parton energy E truth .functions have been continuously adapted to data taking conditions and object defi-94 .5. Transfer Functions nitions. New running conditions, calibrations and event selections made the changesnecessary. Since then, the parameterizations have changed. Table 6.4 reflects the statusat the time this analysis was performed.Using the parameter estimation fit results for a i and b i as starting values, a globalfit of the transfer functions W for all bins of E truth is performed. The results of thisglobal fit, shown in Figure 6.8, is then taken as set of transfer function parameters andimplemented to KLFitter .One remark on the transfer functions should be made: It is not expected that the∆ E distributions follow a Gaussian distribution. The reason is that the resolution∆ E = E truth − E reco E truth ∼ √ E truth . Hence, within a bin of E truth the resolution is only constantin the limit of a vanishing bin width.In Figure 6.10 the implemented set of TFs for light jets in the central | η | region isshown. The illustration was chosen such that the distribution of possible reconstructedjet energies is shown for a given value of E truth = E parton . The vertical lines indicatethe latter value. While for ( b -)jets, electrons and muons TFTool was used to deriveFigure 6.10.: Transfer functions for light jets with 0 . ≤ | η | < .
8. Vertical lines indicatethe parton energies.the transfer functions, the E missT transfer functions had a dedicated procedure and tool.For early studies the TF was set as constant for all events by fitting the distributionsof E miss x,y − p νx,y with a Gaussian function. It has been optimized at a later stage asfollows. The basic idea is that the TFs map the x and y components of the neutrinomomentum to the x and y components of the measured missing transverse momentum.95 . Event Selection and Reconstruction Since it is known that the E missT depends on the scalar sum of deposited energy inthe calorimeters [292], Σ E T , this quantity has been used to parameterize the width of E miss x,y − p νx,y . The parameterization function was heuristically chosen to be of the Sigmoidtype σ (Σ E T ) = p + p e − p (Σ E T − p ) (6.6)Figure 6.11 shows that the dependence on Σ E T is not negligible.Figure 6.11.: Transfer functions for Neutrinos / E missT , parameterized as a function ofΣ E T .The E missT TF leads to a width of the E miss x,y − p νx,y distribution of ≈
18 GeV forΣ E T = 400 GeV. This is higher than the value of about 10 GeV quoted as E missT resolution in [292]. As the t ¯ t topology is more complex than the di-jet events usedin [292], this is not surprising. It was checked that the TFs for the x and y componentsof the neutrinos are equivalent, as expected.Studies on uncertainties of the TFs can be found in [293]. Concerning the evalua-tion of uncertainties of the spin correlation analysis, no dedicated TF uncertainties havebeen derived. The TFs assume a true model and deviations from this in the simulatedevents are evaluated via the common systematic uncertainties evaluation, such as detec-tor modelling (see Chapter 8). Furthermore, the fit uncertainties of the TF parameterswere found to be small.However, the used TFs are not a perfect model of the mapping of measured to partonicenergies. This can be observed in particular at low jet energies where pile-up effects playa crucial role and disturb the expected proportionality of the jet energy resolution to96 .6. KLFitter Extension for Up/Down-Type Quark Separation √ E . This can be seen in Figure 6.12 and leads to the cutoff for the parameter estimationfit. At low energies two things are observed: The reconstructed energy is not lower, but truth ) / E reco - E truth (E -1 -0.5 0 0.5 1 E v en t s L ight J etsE: 35-45 GeV 0.0 < | h | < 0.8 MCSmall GaussianBig GaussianDouble GaussianGlobal Fit (a) truth ) / E reco - E truth (E -1 -0.5 0 0.5 1 E v en t s L ight J etsE: 25-35 GeV | < 0.8 h MCSmall GaussianBig GaussianDouble GaussianGlobal Fit (b)
Figure 6.12.: Description of the detector resolution effects for light jets at low energies.(a) 35 ≤ E jet <
45 GeV, (b) 25 ≤ E jet <
35 GeV.larger than the corresponding parton energy. Pile-up contributions can lead to suchoverestimations. Furthermore, the global fit is no longer able to properly model the TFsat low energies. This imperfect modelling is of no concern for the presented analysissince
KLFitter provides both fitted jet energies as well as a mapping of reconstructed tomodel objects. In this analysis only the latter property of
KLFitter was used, since forthe chosen spin correlation observable only angular distributions, which are not modifiedby
KLFitter anyway, are used.For analyses using the fitted outputs from
KLFitter possible biases have to be consid-ered. They can be eliminated either by a more advanced parameterization of the transferfunction parameters p i or by a more advanced calibration of the jets. Another approachwould be to map the reconstructed energies to particle level instead of parton level forwhich the description of objects (particles) is better justified than for LO partons. Adetailed discussion goes beyond the scope of the thesis but it is worth to be studied inthe future. By default,
KLFitter is not able to separate the two light jets from the W boson decay aspermuting them keeps the likelihood L invariant. The same holds true for the b -taggingextensions presented so far where only b -jets and non- b -jets are separated.A dedicated extension to the likelihood has been developed, taking into account quan-tities with separation power between the two light jets. There are two facts that help toseparate these: their flavour and the V-A structure of the weak decay vertex.The V-A structure of the W decay vertex predicts differences in the energies betweenthe two light jets [181, 183]. While the suggested frame for the energy determination is97 . Event Selection and Reconstruction the top quark rest frame, a difference in p T is also visible in the laboratory frame whichis easier to determine. Figure 6.13(a) shows the different p T distributions for the prompt b -quark from the top decay and the light up- and down-type jets from the W decay. [GeV] T p0 20 40 60 80 100 120 140 160 180 200 P r obab ili t y light up-type jetlight down-type jetb-jet (a) MV1 Weight0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P r obab ili t y -3 -2 -1 updowncharmstrange (b) MV1 Weight0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P r obab ili t y -3 -2 -1 light up-type jetlight down-type jetb-jet (c) Figure 6.13.: (a) p T spectra for the three jet types from the hadronic top quark decay.(b) Weight of the MV1 b -tagger for different light quark types of the W decay. (c) Weight of the MV1 b -tagger for the three jet types from thehadronic top quark decay.Another possibility is to access the flavour of the two light jets. At first sight both areof ’non- b -type’. But while for a decay of W + → u ¯ d both quarks are clearly light, thedecay W + → c ¯ s contains a charm quark, heavier than all other light quarks. Techniquesused for b -tagging can in principle be used to develop dedicated c -taggers. As on theone hand these are not yet well established and on the other hand a dedicated taggingis not desired, the output of the MV1 b -tagger is sufficient to see differences between thedifferent light jet types. Figure 6.13(b) confirms the expectation that down and strangequarks have almost identical MV1 tag weight distributions while up and charm quarks donot. The minimal difference for the former pair arises from the different p T distributionswhich are correlated to the tagger weight.98 .6. KLFitter Extension for Up/Down-Type Quark Separation As the W + boson decays in 50 % of the cases into a c ¯ s pair, the MV1 tagger weightdistributions can be grouped into (prompt) b -, up- and down-type quarks. The up- anddown-type separation power is shown in Figure 6.13(c). (a) (b) (c) Figure 6.14.: Distribution of the b -tagging weights of the MV1 tagger vs. the p T of a jetfor three types of jets: (a) b -jets coming directly from the top quark decay,(b) down-type quarks coming from the W boson and (c) up-type quarkscoming from the W boson.In order to consider both the differences in p T and in the b -tagging weight w as wellas the correlation between the quantities, two-dimensional, normalized, distributionsfor the three jet types ( b , light up-type and light down-type) are created and used tocalculate event probability extensions according to∆ p i, u/d sep. = P b-type2D ( p T (jet blep ) , w jet blep ) · P b-type2D ( p T (jet bhad ) , w jet bhad ) · P u-type2D ( p T (jet uQ ) , w jet uQ ) · P d-type2D ( p T (jet dQ ) , w jet dQ ) (6.7)By using the extension of the KLFitter event probability, up- and down-type quarkscan be separated to a larger extent than with more simple approaches. Such a simplealternative could be the choice of the least energetic jet as down-type quark jet [294]as it is also used in other reconstruction methods (see Section 6.9). Table 6.5 comparesthe simple approach used on top of a
KLFitter reconstruction to the event probabilityextension with dedicated up- and down-type separation.The reconstruction efficiency is defined as the fraction of real (cid:96) +jets events passing theselection for which the model parton matches the jet it is assigned to within a distanceof ∆
R < . ε reco = N match N selected . (6.8)A detailed discussion about the reconstruction efficiencies and possible optimizationsvia cuts on certain event variables is discussed in the next section. 99 . Event Selection and Reconstruction default ∆ p extensiondown-type quark 31 % 35 % b -quark 50 % 55 %Table 6.5.: Reconstruction efficiencies for the down-type quark and b -quark obtained viaselection and jet-to-parton mapping via KLFitter . For the default
KLFitter setup the jet with the lower energy is taken as down-type quark candidate.
A good reconstruction algorithm always tries to map the reconstructed detector objectsto physics objects with the highest efficiency. What exactly defines a high efficiency isdefined by the individual analyses. In the case of t ¯ t analyses it is desirable to know whichof the reconstructed objects stems from which parton of the t ¯ t decay. One part of thereconstruction concerns the mapping of partons to reconstructed objects. Sometimesan emphasis on a certain type of parton is made while another type might be evenignored. KLFitter reconstructs the full t ¯ t topology and even fits the reconstructedkinematic quantities to best estimates of the underlying parton properties. Only themapping capabilities of KLFitter are used in this thesis, not the actual fitting. Theoverall performance of
KLFitter is illustrated in Figure 6.15. An
MC@NLO sample of t ¯ t events passing the µ + jets events selection is used. The bin label should be read asfollows: • all All events passing the event selection. • l+jets Real (cid:96) + jets events. The rest of about 10 % is given by t ¯ t events decayinginto the dilepton channel. • τ present A τ lepton is present. • all jets present All model partons match a jet which was successfully recon-structed. • all jets within first 4 The jets matching the four partons are also the highestfour in p T and hence given to KLFitter . • W matched
The two jets assigned to the hadronically decaying top quark matchbut might be exchanged. • all partons matched Each jet was correctly assigned to the corresponding parton. • multimatch In at least one case more than one jet matched a parton. • l+Jets+UniqueMatch+AllJetsInFirst4 The event is a real (cid:96) +jets event, eachparton matches a jet and there is no multiple matching. The four matched jetsmust be the jets with the highest p T and hence passed to KLFitter .100 .7. Reconstruction Efficiencies and Optimizations • t had matched The reconstructed hadronically decaying top quark matches thepartonic top quark within ∆
R < . • t lep matched The reconstructed leptonically decaying top quark matches thepartonic top quark within ∆
R < . • t had & t lep matched Both the reconstructed hadronically and leptonically de-caying top quark match the corresponding partonic top quark within ∆
R < . • ν matched The reconstructed neutrino matches the partonic real neutrino.It is important to understand that the relatively low reconstruction efficiencies of35 % (down-type quark) and 55 % ( b -quark) are mainly a problem of acceptance, notof the internal reconstruction performance. To disentangle the two effects, Figure 6.16lists reconstruction efficiencies normalized to two different references. In the first caseefficiencies with respect to all selected (cid:96) + jets events are shown. In the second casematched events are normalized to only those where all four jets passed to KLFitter werebi-uniquely matched to partons. Hence, the second case reflects the internal performanceof the reconstruction algorithm as it considers only those events for which the algorithmhad a chance to correctly reconstruct all jets.The most powerful spin analysers for hadronically decaying top quarks are the down-type quark and the b -quark. While the down-type quark has the same full analysingpower of | α | ≈ b -quark still has a lower but still reasonablespin analysing power of | α | ≈ . b -quark will be studied in detail in this section.It is obvious that the reconstruction efficiencies of 35 % for the down-type quark and55 % for the b -quark are inclusive quantities and depend on the event properties. Theobject definition and event selection gained stability against pile-up, especially by theintroduction of the JVF cut (see Section 4.3). While for early studies the efficiency wasdropping significantly for an increasing number of primary vertices, it could be stabilizedwith the current object definition and event selection as shown in Figure 6.17(a). This isimportant as the analysis needs to be independent of the running conditions. In partic-ular, future spin analyses for which the average number of interactions will increase withthe higher luminosity setups need to be stable with respect to pile-up. An ideal situationwould be that each of the four LO partons hadronises into a single, non-overlapping jet(“model jet”), which is within the detector acceptance and the selection criteria. As theselection requires at least four jets and initial and final state radiation even increase thisnumber, additional jets lead to additional combinatorial background. It is obvious thata larger number of jets increases the probability that a pile-up jet mimics the kinematicsof a jet stemming from a model parton. In particular, the probability that the fourmodel jets are the ones with the highest p T – and thus the ones which are passed to KLFitter – decreases. Figure 6.17(b) illustrates this.In Figure 6.17(c) the reconstruction efficiency as a function of the number of jetspassed to
KLFitter , which were b -tagged, is shown. The result reflects the reconstruc-101 . Event Selection and Reconstruction tion method: Maximal b -quark reconstruction efficiency can be reached for exactly two b -tagged jets as this clearly separates the b -jets from the light jets. Higher b -tag multi-plicities increase the chance of interchanging light and b-jets. For the down-type quark,two b -tags separate light from b -jets. Three tags are likely in the case of a W → cs decayfor which the c -quark was mistagged. Its high b -tag weight helps KLFitter to separateit from the up-type quark. Four tags increase the similarity of the up- and down-typequark properties and decreases the reconstruction efficiency.The p T dependence of the reconstruction efficiency is opposite for down-type quarksand b -quarks. In general, a high p T reduces the probability of a jet to be interchangedwith a low- p T pile-up jet. This is observed for the b -quark as shown in Figure 6.17(d).For the down-type quark, the trend of increasing reconstruction efficiency reverses forhigh p T . This fact is due to the dedicated down-type quark reconstruction, in particularthe up-/down-type separation. It is based on the – on average – lower p T of the down-type quark (due to the V-A structure of the weak decay vertex). In case the p T ofthe down-type quark is too high, the kinematics of the parent W boson force the p T ofthe up-type quark to be low. The assumption that the p T of the down-type quark islower than the p T of the up-type quark is wrong in these cases and leads to a wrongassignment.The main quantities describing the quality of the reconstruction are the KLFitter out-put values for the likelihood (Figure 6.17(e)) and the event probability (Figure 6.17(f)).As the likelihood is a complex quantity, the reconstruction efficiency dependence onit is not intuitive. In general, the argument holds that an event that has a proper t ¯ t topology leads to high values of the likelihood. This implies a high reconstruction ef-ficiency. What is surprising at first sight is the fact that for low values of a likelihood(log(likelihood) < −
55) the reconstruction efficiency for the b -quark increases.The likelihood has several components as shown in Equation 6.1. During the fittingprocess the jet and lepton energies are varied within the transfer functions to allow thejet combinations to match the masses of the W boson and the top quark. By plottingthe individual components of the likelihood (see Appendix D for details), one can seethat the tail of low likelihood values is caused by low values of the W boson and topquark mass Breit-Wigner functions. As the fit maximizes the total likelihood, the fittingprocess is always a trade-off between varying the transfer function value or the Breit-Wigner value. While the width of the Breit-Wigner functions is fixed and given bythe width of the decaying W boson and top quark, the width of the transfer functionsvaries. It is in particular a function of the jet energy. As described in Section 6.5, the jetresolution increases with higher jet energies due to the intrinsic calorimeter resolution.Hence, high energetic jets cause less flexibility of the transfer function. In case the eventhas no good t ¯ t topology, the Breit-Wigner functions will then have low values. This isthe first conclusion about the low values of the likelihood.The second conclusion is: If low values of the likelihood are dominated by high en-ergetic jets, this leads to a high reconstruction efficiency for the b -quark and a lowreconstruction efficiency for the down-type quark, as it was shown in Figure 6.17(d).102 .7. Reconstruction Efficiencies and Optimizations What remains is a discussion of the dependence of the reconstruction efficiencies onthe event probability. It is shown in Figure 6.17(f). In case of a high event proba-bility one particular permutation exists for which – and only for this permutation –the assumptions of a t ¯ t event topology match the observations. This is demonstratedby high correlation of event probability and reconstruction efficiency. The peak of the b -quark reconstruction efficiency around 0.5 is also plausible. Assuming the case of in-distinguishable up- and down-type quarks, the two permutations for which the up- anddown-type jets are permuted have equal event probabilities. In the best case wherethese permutations are clearly favoured, the event probability is about 0.5. In the caseof indistinguishable up- and down-type quarks these are likely up and down quarks, notcharm and strange quarks. If the pair of W jets does not contain a heavy flavour charmquark, there is less risk that the charm quark is interchanged with the hadronic b -quark.This fact connects the high reconstruction efficiency with the similarity of the up- anddown-type quark signatures and the peak of the event probability at 0.5.Following this thought, a high event probability implies a clear separation between thelight up- and down-type quark. This can only be reached if it is a charm/strange quarkpair and the b -tag weight can be utilized. The consequence of the third jet carrying ahigh b -tag weight is a higher probability of interchanging it with the b -quark. Hence,the b -quark reconstruction efficiency will drop. This is exactly what is observed.In terms of future improvements and for a proper understanding of the reconstructiondetails it is useful to know what went wrong in case of a misreconstruction of the down-type quark or the b -quark. Figure 6.18(a) shows to which parton a reconstructed modeljet of a certain type ( b had. , b lep. , light up-type, light down-type) could be matched. Eachjet can be matched to one of the model partons or to none at all. The success of thereconstruction is reflected in the fact that the most probable option is that a jet wasassigned to the corresponding model parton. It is also visible that the light jets withtheir p T being lower than the ones of the b -jets suffer most from acceptance and selectioncuts. As expected the two light quarks are interchanged quite often. But it is remarkablethat it is more likely for a down-type quark to be not reconstructed at all than to beinterchanged with the up-type quark. The fact that the down-type quark has on averagea lower p T explains the higher probability that its jet is not within the four jets which areused for reconstruction (as these have the highest p T ). Figure 6.18(b) shows the η and p T distribution of the down-type quark before a selection is applied. The horizontal andvertical lines indicate the cuts applied on detector level ( p T >
25 GeV and | η | < . b -quark. The following cuts have been evaluated in terms of efficiency gain and statisticsloss: In the latter case the quark jets could be separated by the b -tag weight. . Event Selection and Reconstruction • Top Quark Mass Cut
The invariant mass of the three jet system allocated tothe decay products of the hadronically decaying top quark is required to satisfy | m t − m jjj | <
35 GeV. • W Boson Mass Cut
The invariant mass of the two jet system allocated to thedecay products of the W boson of the hadronically decaying top quark are requiredto satisfy | m W − m jj | <
25 GeV. • Top Quark and W Boson Mass Cut
Both of the former two requirementshave to be satisfied. • Likelihood Cut
The logarithm of the likelihood output of
KLFitter must belarger than -50. • Event Probability Cut
The event probability of
KLFitter has to be larger than0.5. • Jet Multiplicity Cut
Exactly four jets must be present in the selected event. • B-Tag Multiplicity Cut
At least two of the reconstructed jets have to be taggedas b -jets.The results are visualized in Figure 6.19. The distributions of the likelihood and the eventprobability is shown in Figure 7.5 in the context of the validation of the reconstructionsprocess.A combination of the W boson and top quark mass cut is found to be the most powerfulcut for both spin analysers. Other cuts can be considered if an individual analyser isgoing to be optimized. In this case also the specific cuts can be further tuned.In this analysis no cut was placed. Instead, the selected samples were split intosubsamples according to their jet multiplicity and b -tag multiplicity. This choice isfurther discussed in Section 7.3.1. KLFitter can be used with several options, which were presented in the last sections.The actual setup which was used to perform the analysis presented in this thesis is brieflypresented.The number of jets passed to
KLFitter was set to four. These four jets were selectedby a p T ordering of all reconstructed jets. The jets with the highest p T were used. Thiswas tested to lead to the highest reconstruction efficiency. Even though this excludesa proper event reconstruction where the jets matching the four model partons are notwithin the four jets with the highest p T , it also decreases combinatorial background.The latter effect was found to be dominating.No b -jet veto method was used. Instead, the b -tagging information was taken intoaccount via the working point mode and the additional up/down-type quark separation.104 .9. Comparison to Other Reconstruction Methods The top quark mass in used in the
KLFitter likelihood was fixed to 172.5 GeV. Analternative option is to leave it as a free parameter. The former option was chosen as itleads to a higher reconstruction efficiency. Using the free parameter leads to a benefitonly if the fitted top quark mass is utilized in an analysis and needs to be unbiased.This was not the case in the present studies.
It was shown that the
KLFitter with the up-/down-type quark separation extensionprovides good reconstruction efficiencies. As several other t ¯ t reconstruction methodsexist, it is worth to check their reconstruction efficiencies. Two alternatives to KLFitter will be introduced and compared in the following section. The introduction will belimited to the description of the reconstruction of the hadronically decaying top quark. Ithas to be stressed that several variations of these methods exist and that the descriptionsprovided here only represent one specific choice. Furthermore, none of these methodswas optimized for b -quark or down-type quark reconstruction. P T Max
The basic idea of the P T Max method is that the three-jet system with the highesttransverse momentum corresponds to the three jets from the hadronically decaying top.All reconstructed jets are considered. Out of the three selected top quark jets, the jetpair with an invariant mass closest to the W mass is selected as W boson jet pair. Aveto on b -tagged jets is set which can cause the reconstruction method to provide nosolution. Both of the W boson jets are then boosted into the hadronic top quark restframe. The jet with the lower energy is used as down-type quark jet and the top quarkjet not associated with a W boson as b -quark jet. At least two b -tagged jets are required for the topological method. Out of all non-taggedjets the pair with an invariant mass closest to the W mass is selected as W boson jetpair. The invariant mass of all three-jet systems consisting of the two W boson jets anda third tagged jet is calculated. The three-jet system with the invariant mass closestto the top quark mass defines the b -quark jet as non- W boson jet. Within the two W boson jets the down-type quark jet is chosen the same way as for the P T Max method.
Both the P T Max and the topological method were compared to
KLFitter under thefollowing conditions: The event passes the selection as described in Section 6.1 and mustbe a real t ¯ t → (cid:96) + jets event. The results of matched down-type quark and b -quark jetsare provided in Table 6.6. As some reconstruction methods have additional constraintson the event, they only provide solutions for a certain fraction of the selected events (last105 . Event Selection and Reconstruction column). The reconstruction efficiencies are provided both with respect to the numberof events for which a solution exists and with respect to all selected events. Rec. Efficiency (w.r.t. all Events) Remaining Stat. b -quark down-type quark KLFitter
55 % (55 %) 35 % (35 %) 100 %P T Max 44 % (36 %) 32 % (26 %) 82 %Topological Method 42 % (21 %) 29 % (15 %) 51 %
Table 6.6.: t ¯ t reconstruction efficiency for different algorithms. The efficiencies are givenwith respect to all selected t ¯ t events (in parentheses) and to events passingadditional criteria given by the algorithm. The loss of statistics due to thesecriteria is also quoted.The comparison shows that the used setup of KLFitter does not only perform bestbut does furthermore not cause a reduction of statistics.106 .9. Comparison to Other Reconstruction Methods a ll l + j e t s p r e s en t τ a ll j e t s p r e s en t a ll j e t s w i t h i n f i r s t W m a t c hed a ll pa r t on s m a t c hed m u l t i m a t c h l + J e t s + U n i que M a t c h + A ll J e t s I n F i r s t m a t c hed had t m a t c hed l ep t m a t c hed l ep & t had t m a t c hed ν P r obab ili t y Figure 6.15.: Event properties and reconstruction efficiencies using
KLFitter in the µ + jets channel. Numbers are given with respect to all events passingthe event selection. 107 . Event Selection and Reconstruction l + J e t s e v en t s l ep t on m a t c hed m a t c hed had b m a t c hed l ep b u Q m a t c hed d Q m a t c hed W m a t c hed a ll pa r t on s m a t c hed f u ll m a t c h , W i n v a r i an c e m a t c hed had t m a t c hed l ep t m a t c hed l ep & t had t m a t c hed ν P r obab ili t y all l+jets eventsl+jets events with four uniquely matched jets in fitter Figure 6.16.: Reconstruction efficiencies using
KLFitter in the µ +jets channel. Numbersare given with respect to all events passing the event selection (solid line)and to such events where all four jets passed to KLFitter were bi-uniquelymatched to partons (dashed line).108 .9. Comparison to Other Reconstruction Methods
Number of Primary Vertices0 2 4 6 8 10 12 14 16 18 20 R e c . E ff i c i en cy had blight down-type quark (a) Number of Jets4 5 6 7 8 9 R e c . E ff i c i en cy had blight down-type quark (b) Number of b-tagged Jets0 1 2 3 4 R e c . E ff i c i en cy had blight down-type quark (c) [GeV] T p0 50 100 150 200 250 300 R e c . E ff i c i en cy had blight down-type quark (d) Log(Likelihood)-80 -75 -70 -65 -60 -55 -50 -45 -40 R e c . E ff i c i en cy had blight down-type quark (e) Event Probability0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R e c . E ff i c i en cy had blight down-type quark (f) Figure 6.17.: Reconstruction efficiencies of b -quarks and down-type quarks as a func-tion of (a) the number of reconstructed primary vertices, (b) the numberof reconstructed jets, (c) the number of b -tagged jets, (d) the transversemomentum of the jets assigned to the down-type quark and the b -quark,(e) the logarithm of the KLFitter likelihood and (f) the
KLFitter eventprobability. 109 . Event Selection and Reconstruction no par tn er had b lep b light up-type quark light down-type quark P r obab ili t y lep b had blight up-type quarklight down-type quark (a) [GeV] T p0 10 20 30 40 50 60 70 80 90 100 η -4-3-2-101234 Down-Type Quark (Simulated Parton)
Acceptance Cuts on Reconstruction Level (b)
Figure 6.18.: (a) Jet allocated to model partons by
KLFitter (coloured lines) and thepartons to which they were matched (x-axis). (b) η and p T of down-typequarks for the full phase space, before event selection. The red lines indicatethe cuts applied on reconstruction/detector level.110 .9. Comparison to Other Reconstruction Methods ) had . R e c o E ff i c i en cy ( b no cutW mass cuttop quark mass cutn jets == 4EventProb. cutW and top quark mass cutlikelihoodcutn tags > 1 Fraction of Remaining Statistics (a) R e c o E ff i c i en cy ( do w n - t y pe qua r k ) no cutW mass cuttop quark mass cutEventProb. cutn jets == 4W and top quark mass cutlikelihood cutntags > 1 Fraction of Remaining Statistics (b)
Figure 6.19.: Cuts on event properties and their effects on the statistics and reconstruc-tion efficiencies for the (a) b -quark and (b) down-type quark. 111 Analysis Strategy
Many ways to measure the spin correlation in t ¯ t events exist. They access differentcomponents of the t ¯ t spin matrix (cid:98) C , explained in Section 2.4. For dileptonic decaysof t ¯ t pairs there is a clear preference at the LHC. Charged leptons have the best spinanalysing power and can be measured with a high precision. As further the measurementof the azimuthal angle between the two charged leptons in the laboratory frame does notneed any event reconstruction, ∆ φ ( l + , l − ) was the observable of choice for the first LHCresults [187, 188] and also lead to the observation of spin correlation in t ¯ t events [187].In the (cid:96) + jets channel, however, the choice is not straightforward. The selection ofboth, the observable and the spin analyser are motivated in the next session, followed bycross checks of the correct reconstruction of the spin analyser as explained in sections 6.4and 6.6. After that, the measurement of the observable is explained, including the usageof nuisance parameters for systematic uncertainties as well as a correction dedicated tomismodelling of the jet multiplicity in the MC signal generator. The chapter concludeswith a validation of the linearity of the method, a discussion about the correlation ofdown-type quark and b -quark, and the effects of the correlation on the measured results. The fact that the measurement of the azimuthal angle between two spin analysers inthe laboratory frame is suitable for t ¯ t production at the LHC is also valid for the (cid:96) +jets channel. In principle, there exists also a hadronic analyser with the same spinanalysing power as a charged lepton, namely the down-type quark (see Table 2.8). Eventhough its measurement has slightly worse energy and angular resolution compared toa charged lepton, the true challenge lies in its correct identification. In Chapter 6,an advanced reconstruction technique was introduced, which is able to reconstruct the113 . Analysis Strategy down-type quark analyser. As this method reconstructs the full t ¯ t event, even themore complex observables could in principle be studied. However, as these require thecorrect reconstruction of all spin analysers, the separation power between signal with spincorrelation according to the SM and with uncorrelated spins are significantly diluted.By using simulated t ¯ t events, the separation power of a SM spin correlation sampleand a sample of uncorrelated t ¯ t pairs was tested at the detector level for different spincorrelation observables. It was found to be significant only for the ∆ φ distributions.Thus, ∆ φ was chosen as observable for the (cid:96) + jets channel.Next to the most powerful hadronic analyser, the down-type quark, the second mostpowerful analyser, the b -quark, was evaluated at parton level. The respective distribu-tions for the full phase space, without any object cuts on p T and η , are shown in Figure7.1. The larger spin analysing power of the down-type quark in 7.1(a) with respect to (l,d) [rad] fD fD d N / d N (SM)tt (no corr.)tt ATLAS
Simulation
RWIGE + H @NLOC
M = 7 TeVs l+jets (a) (l,b) [rad] fD fD d N / d N (SM)tt (no corr.)tt ATLAS
Simulation
RWIGE + H @NLOC
M = 7 TeVs l+jets (b)
Figure 7.1.: Distribution of the azimuthal angle between the charged lepton and the (a)down-type quark and the (b) b -quark in the laboratory frame [180]. Thedistributions show the full phase space on parton level without usage of cutson p T and η of the objects.the b -quark in 7.1(b) is clearly visible. Also, the reversed sign on the analysing power isreflected in the fact that the deviations of SM expectation and uncorrelated t ¯ t pairs go inopposite directions. This fact plays an important role in this analysis. A mismodellingof t ¯ t kinematics in a MC generator needs to be distinguished from a deviation in the t ¯ t spin correlation. Such a distinction is possible due to the opposite shape changingeffects of spin correlation on the two analysers.The quantities of interest are the ∆ φ distributions on the detector level. Due to thelimitations of the reconstruction efficiency and the acceptance, the separation powerbetween the scenarios with SM spin correlation and uncorrelated t ¯ t pairs are diluted.The ∆ φ distributions at the detector level are shown in Figure 7.2. It is remarkable that While for the down-type quark the ∆ φ distribution is flatter for correlated t ¯ t pairs than for uncorrelatedones, it is steeper for the b -quark. .2. Spin Analyser Validation MC@NLO (SM)MC@NLO (no spin) +jets) µ reco level, after selection ( (l, d) [rad] φ ∆ ( no s p i n ) / ( S M ) N o r m a li z ed N u m be r o f E v en t s (a) MC@NLO (SM)MC@NLO (no spin) +jets) µ reco level, after selection ( (l, b) [rad] φ ∆ ( no s p i n ) / ( S M ) N o r m a li z ed N u m be r o f E v en t s (b) Figure 7.2.: Distribution of the azimuthal angle between the charged lepton and the jetassigned to the (a) down-type quark and the (b) b -quark in the laboratoryframe. The distributions use reconstructed quantities in the µ + jets channelafter the event selection is applied. The shown uncertainties (barely visible)represent the MC statistics uncertainty.both quantities have comparable separation power. This shows that the reduced spinanalysing power of the b -quark (as shown in Figure 7.1) is at least partially compensatedby the larger reconstruction efficiency as summarized in Table 6.5.As a consequence, both the down-type quark and the b -quark are used as analysers. Itis shown in Section 7.6 that treating both analysers as uncorrelated quantities is valid.From now on, the expression “down-type quark” will refer to the jet associated to thedown-type quark and accordingly for the b -quark, if not stated otherwise. The reconstruction of the hadronic analysers as explained in Section 6.6 is an advancedreconstruction technique and based on several input quantities. The performance ofthe reconstruction must be the same on data and on simulated events. Otherwise, areconstruction efficiency, which is different in data and simulation, can mimic a higher orlower spin correlation. The validation of the reconstruction procedure via a comparisonof the input and output quantities of the reconstruction algorithm is the subject of thissection.As b -tagging is important for both the explicit tagging of jets as b -jets as well as forthe continuous distribution of the weights for up/down-type separation, these quantitiesneed to be checked for a good agreement of prediction and data. In the first two columnsof Figure 7.3, the data/MC agreement of the b -tag weight (logarithmic y-axis) and thenumber of b -tagged jets is shown. Here and in the following validation plots, the upperrow shows results of the e +jets channel and the lower row those of the µ +jets channel. No115 . Analysis Strategy significant deviation of the data from the prediction can be observed. The slight slope inthe b -tag multiplicity is corrected during the fit by using the b -tag efficiency uncertaintiesas nuisance parameters. Another quantity of interest is the p T of the down-type quarkjet. In the third column of Figure 7.3, the p T spectrum of the reconstructed down-type quark jet is shown. The kinematic mismodelling of MC@NLO as seen in Figures 6.4and 6.5 propagates to the down-type quark jet. No additional mismodelling or artificialcorrection by the reconstruction is observed. A good agreement of the jet p T spectrum isobserved using POWHEG + PYTHIA (see Figure E.1 in the Appendix). Closely related to the E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ btag weight (MV1) p r ed ./ da t a E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ Number of b tags (MV1 @ 70%) p r ed ./ da t a
50 100 150 200 250 E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ (reco) t dQ p
50 100 150 200 250 p r ed ./ da t a E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ btag weight (MV1) p r ed ./ da t a E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ Number of b tags (MV1 @ 70%) p r ed ./ da t a
50 100 150 200 250 E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ (reco) t dQ p
50 100 150 200 250 p r ed ./ da t a Figure 7.3.: The weight of the
MV1 b -tagger (left), the number of tagged jets (centre)and the p T of the down-type quark candidate (right) for the e + jets channel(upper row) and the µ + jets channel (lower row). p T distribution is the jet index corresponding to the b -quark and down-type quark. Thisindex is assigned during the p T ordering of the jets, starting with zero for the jet withthe highest p T . The first two columns of Figure 7.4 show good agreement for the down-type quark and the b -quark, respectively. As this analysis studies angles between spinanalysers, angular variables of the reconstructed quantities also need to be investigated.The azimuthal angle between the b -quark and the down-type quark was chosen as a116 .2. Spin Analyser Validation E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ Jet Index of dQ p r ed ./ da t a E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ Jet Index of bQhad p r ed ./ da t a E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ (bQhad, dQ) φ ∆ p r ed ./ da t a E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ Jet Index of dQ p r ed ./ da t a E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ Jet Index of bQhad p r ed ./ da t a E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ (bQhad, dQ) φ ∆ p r ed ./ da t a Figure 7.4.: The p T ranking index of the down-type quark jet (left), the index of b -quarkjet from the hadronic top (middle) and the ∆ φ angle between the recon-structed hadronic b -quark and the down-type-quark (left) for the e + jetschannel (upper row) and the µ + jets channel (lower row). 117 . Analysis Strategy control distribution and is shown in the third column of Figure 7.4.An overall summary of the reconstruction via KLFitter is given by the likelihood andthe event probability. Both are shown in Figure 7.5.
80 75 70 65 60 55 50 45 40 E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ Log ( Likelihood )
80 70 60 50 40 p r ed ./ da t a E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ ≥ Event
Probability p r ed ./ da t a
80 75 70 65 60 55 50 45 40 E v en t s / Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ Log ( Likelihood )
80 70 60 50 40 p r ed ./ da t a E v en t s / . Datatt (dil.)tt (no SC)ttW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.6 fb ∫ ≥ + µ ≥ Event
Probability p r ed ./ da t a Figure 7.5.: The logarithm of the KLFitter likelihood (left) and the event probability(right) for the e + jets channel (upper row) and the µ + jets channel (lowerrow).In general, no mismodelling is observed in the reconstruction of the down-type quarkand b -quark spin analysers. The data is considered validated and the measurement isperformed as described in the next sections. The spin correlation measurement must deduce the t ¯ t signal – as well as its spin proper-ties – and the background contribution. This is realized with a template fit, based on theprinciple of a binned likelihood fit. Templates are created for each signal and backgroundcomposition. MC simulation is used for all templates except the fake lepton background,which is derived from data. The measured dataset is split into several channels, which118 .3. Binned Likelihood Fit are explained in the next section. By using the templates from the prediction for signaland background events as well as the measured data distribution, it is possible to definea likelihood L = C (cid:89) j =1 N (cid:89) i =1 e − ( s ij + b ij ) ( s ij + b ij ) n ij n ij ! , (7.1)where C is the number of channels, N is the number of bins per template, and n ij , s ij and b ij are the number of events in the i’th bin and the j’th channel in the data, signal andbackground distribution, respectively. The signal distribution is a linear combinationof the two available signal samples: t ¯ t pairs with spins correlated as predicted by theSM and t ¯ t pairs with uncorrelated spins. The fraction f SM of SM like spin correlationdefines the mixing: s ij = ε signal j N t ¯ t · ( f SM · p SM t ¯ tij + (1 − f SM ) · p unc. t ¯ tij ) (7.2)where p SM t ¯ tij and p unc. t ¯ tij are the entries in bin i of the normalized template for the SMand the sample with uncorrelated spins, respectively, in channel j . The total t ¯ t yield isgiven by the parameter N t ¯ t . It can also be reformulated as the expected t ¯ t yield N exp. t ¯ t scaled by a factor c , N t ¯ t = c · N exp. t ¯ t . The efficiency ε j denotes the fraction of the total t ¯ t yield that is reconstructed in channel j .The background contribution for bin i and channel j breaks down to: b ij = (cid:88) k =1 N k · ε kj · p ijk (7.3)summed over the different background contributions k , each having its own efficiency ε : W +jets, fake lepton background and remaining backgrounds. N k represents the totalnumber of events from the background type k . ε kj is the relative contribution of thebackground events of type k in the channel j to the total number of background events N k . p ijk is the entry of bin i of the normalized background template for the backgroundtype k in the channel j .Technically, this fitting is implemented by transforming the two normalized signaltemplates p SM t ¯ t and p unc. t ¯ t into p sum t ¯ t and p diff t ¯ t via p sum t ¯ t = 12 (cid:16) p SM t ¯ t + p unc. t ¯ t (cid:17) , (7.4) p diff t ¯ t = 12 (cid:16) p SM t ¯ t − p unc. t ¯ t (cid:17) . (7.5)The signal contribution is a linear combination of p sum t ¯ t and p diff. t ¯ t : s ij = ε signal j ( N sum t ¯ t · p sum t ¯ t,ij + N diff t ¯ t · p diff t ¯ t,ij ) (7.6)with the parameter values for the total yield N sum t ¯ t and the parameter for the scaling ofthe difference of SM and uncorrelated events N diff t ¯ t . The values of p sum t ¯ t,ij and p diff t ¯ t,ij represent the entries of bin i of the templates p sum t ¯ t and p diff t ¯ t in the channel j . 119 . Analysis Strategy As these are just linear transformations, the fitted parameter values N sum t ¯ t and N diff t ¯ t can be easily translated into the parameters of interest, namely the cross section scalefactor c and the spin correlation fraction f SM .The linear transformation is introduced to add numerical stability to the fit. With-out the transformation the two signal parameters would be fully anti-correlated. Thetransformation resolves this issue.In addition to the basic fit parameters, further parameters are added. One set ofparameters accounts for the systematic uncertainties, taking them into account as ad-ditional nuisance parameters (NPs) by fitting their effects on the template to the dataand thus constraining their impact. This procedure is described in Section 7.3.3.After the choice of channels is explained in the following section, the treatment of fitparameters is described in Section 7.3.5. Each channel is a subset of the whole data available and has distinct properties: signalto background ratio, reconstruction efficiency, impact of systematic uncertainties andstatistical uncertainty. Some channels are pre-defined as the e + jets and the µ + jetschannel are reconstructed in different, orthogonal data streams. Others are definedby the analysis strategy. In the combination of down-type quark and b -quark results,the different analysers are treated as separate channels. Section 7.6 is dedicated tothe question if the treatment of the down-type quark and the b -quark as independentvariables is justified.Further splitting of the e + jets and the µ + jets data is possible and reasonable. Thefirst splitting divides the data into a channel with exactly four jets and one with at leastfive jets. As shown in Section 6.7, this creates a subsample with a higher reconstructionefficiency for both the down-type quark and the b -quark, namely for n jet = 4. Anothermotivation for this splitting is the jet multiplicity mismodelling of the MC@NLO generator.It is possible to introduce an additional parameter to the fit correcting the efficienciesof the n jet = 4 with respect to the n jet > b -tagged jets is also a criterion to split the data sample into subsetsof higher and lower reconstruction efficiency. As the b -tag multiplicity is also not per-fectly modelled (see Section 7.2), the introduced nuisance parameters dedicated to the b -tagging uncertainties can correct this mismodelling in-situ.All channels used in the analysis are defined and listed in Table 7.1. Results areobtained for the individual channels, combinations with the same analyser and a fullcombination. As the fitting framework
BAT is using the Bayesian approach, a-priori information aboutthe parameters can be included in the fit. This is realized by the addition of priors to each parameter p i . These are multiplied to the likelihood. Different types of priors120 .3. Binned Likelihood Fit Channel Analyser Lepton Flavour Jet Multiplicity B-Tags1 down-type quark electron = 4 =12 > > >
15 muon = 4 = 16 > > > b -quark electron = 4 =110 > > >
113 muon = 4 = 114 > > > This kind of prior is used to constrain the background yieldsaccording to their normalization uncertainties. The Gaussian priors used in this analysisare explained in Section 7.3.5.
Systematic uncertainties affect the results by changing both the shape and the yieldof the measured distributions. Next to the more traditional way of evaluating theireffect via ensemble tests, the option of including them already in the fit can be a goodalternative if certain requirements are fulfilled. If for an uncertainty both the ± σ variations lead to a well-defined template the effects on each bin can be quantifiedand linearly interpolated. This allows to assign an additional fit parameter (nuisanceparameter) β i to the uncertainty i . Within the fit, the effect is considered via modifiedefficiencies used in Equations 7.2 and 7.3:˜ ε = ε + (cid:88) unc. β i ∆ ε i . (7.7) The expression ’width’ of Gaussian distributions refers to the standard deviation σ and may not beconfused with the ’Full Width at Half Maximum’, FWHM. ’Well-defined’ in a sense that the variations are not simply random fluctuations. . Analysis Strategy Here, ∆ ε i is the relative change of yield per bin caused by the systematic effect i .Values of β = ± ± σ deviations. By including systematiceffects as nuisance parameters they can hence improve the data/MC agreement causedby miscalibration covered by systematic uncertainties. Systematic uncertainties includedas nuisance parameters propagate their uncertainties into the fit uncertainty.A complete list of nuisance parameters used in this analysis is provided in Section 8.2. The division into subsets of n jets = 4 and n jets > MC@NLO . Such a correction needs to be applied to thesignal samples as the predicted signal yield in the n jets > n jets > • The backgrounds will fill the gap and will thus be overestimated. • As the efficiencies are deduced from the MC including deviating jet multiplicities,they will be incorrect. By filling the gap in the n jets ≥ n jets = 4 channels. The fit will end up in a compromise betweenan overestimation of the n jets = 4 channels and an underestimation of the n jets > n jets > ±
10 % per integer value ofthe parameter. The method was tested first by performing a combined fit of all down-type quark and all b -quark channels for which the effect of jet multiplicity mismodellingis the same but the ∆ φ shapes are different. With a starting value of 0 and a constantprior, the correction parameters were fitted to be 0 . ± .
16 (down-type quark) and0 . ± .
17 ( b -quark). This means the efficiencies are corrected by 1.088 and 1.086, re-spectively. As a cross check, a second procedure was tested: Before fitting, the t ¯ t yield inthe n jets > Table 7.2 lists all parameters used in the fit. Used priors are set according to theuncertainty on the estimation (see Chapter 8).Further parameters are added for systematic uncertainties (when applicable, see Sec-tion 8.2).122 .4. Method Validation
Parameter Description Gaussian PriorWidth / Mean p ( N SM + N UC ) — p ( N SM − N UC ) — p N rem. BG ,e +jets + N rem. BG ,µ +jets p N W +jets ,n jets =4 ,e +jets p N W +jets ,n jets ≥ ,e +jets p N QCD,n jets =4 ,e +jets p N QCD,n jets ≥ ,e +jets p N W +jets ,n jets =4 ,µ +jets p N W +jets ,n jets ≥ ,µ +jets p N QCD,n jets =4 ,µ +jets p N QCD,n jets ≥ ,µ +jets p ε ( n jet ≥
5) correction —
Table 7.2.: The parameters used for the fit, priors used and their corresponding widthsdivided by the mean values. The mean value of each Gaussian prior is theexpectation value of the corresponding fit parameter. The widths are repre-senting the normalization uncertainties as described in Chapter 8.
A fit works linearly if pseudo data with a given f SM is also fitted as such. For theevaluation of the linearity, pseudo data are created for 16 values of f SM between − . .
0. 100,000 ensembles are created by applying Poissonian fluctuations to each binof signal templates mixed to a given f SM . The mean of the fit output for each f SM isthen fitted with a linear function. It is required that the slope is unity and the offsetzero. The linearity test is successfully passed (see Figure 7.6(a)). No deviations of theexpected slopes and offsets are observed up to two decimal places.The pull p is defined as the difference between the fitted value of f SM and the valueof f SM used to create the pseudo data which was fitted, divided by the uncertainty on f SM of a fit: p ≡ f out SM − f in SM ∆ f out SM . (7.8)For each value of f SM the pull distribution is plotted and the mean and RMS valuesare compared to the expectations of zero and unity, respectively. No deviations fromthe expectations are observed. The pull mean and RMS values are checked over the fullrange of − . < f SM < .
0. The pull mean values are distributed around zero within0.01 for the down-type quark, the b -quark, and the full combination (see Figure 7.6(b)).The pull RMS values were distributed around unity within 0.02 for the down-type quark,the b -quark, and the full combination. 123 . Analysis Strategy out in f-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 ou t S M f -2-1.5-1-0.500.511.522.53 offset: -0.00 +/- 0.00slope: 1.00 +/- 0.00 (a) f SM out-1 -0.5 0 0.5 1 1.5 2 i n ) /f s m _e rr S M ou t - f S M ( f -0.02-0.015-0.01-0.00500.0050.010.0150.02 (b) Figure 7.6.: (a) Linearity test for the combination of down-type quark and b -quark anal-ysers. (b) Pull mean distribution for the combination of down-type quarkand b -quark analysers. The expected statistical uncertainty is evaluated via ensemble tests. Pseudo data arecreated from the signal templates for t ¯ t (at f SM = 1 .
0) and background events accordingto the integrated luminosity of 4 . − . A Poissonian fluctuation is applied to theexpected distribution before a fit is applied. For this test, 100,000 ensembles are createdand fitted. A distribution of the fit output value for f SM was fitted with a Gaussiandistribution and its width is taken as expected statistical uncertainty.Table 7.3 shows the expected statistical uncertainty for each of the eight channels and twoanalysers as well as for the combinations of all down-type quark and b -quark channels.The gain of sensitivity due to the combination of channels is visible. Even thoughthe separation of the down-type quark and b -quark is comparable, it is still on averagelower for the ∆ φ distributions utilizing the b -quark. Two b -tags increase the purity (lessbackground) and the reconstruction efficiency. The latter is also increased in the case offour jets, due to a decreased combinatorial background. The azimuthal angles ∆ φ between the lepton and the down-type quark and the lep-ton and the b -quark are used in this analysis as two independent variables for the fullcombination fit. Their independence is cross-checked by three main points: • The two observables must obtain their spin analysing power from different effects.This is true as the analysing power of the b -quark arises from longitudinally polar-ized W bosons and is degraded by transversely polarized W bosons. The analysingpower of the down-type quark arises from both the longitudinally and the transver-sally polarized W bosons. Independent treatment and combination is suggested124 .6. Analyser Correlation Analyser Lepton Flavour Jet Multiplicity B-Tags Expected ∆ f SM (stat.)down-type quark electron = 4 =1 0.37 > > > > > > b -quark electron = 4 =1 0.70 > > > > > > b -quark all all all 0.18Combination all all all 0.11Table 7.3.: Expected statistical uncertainty for the eight individual channels for bothanalysers as well as for the combination of analysers. 125 . Analysis Strategy in [176]. • At parton level, the two observables need to be uncorrelated. • At detector level, the two observables need to be uncorrelated.These last two points are addressed in this section. The correlation of the two observables∆ φ ( l, d ) and ∆ φ ( l, b ) is evaluated by plotting the two quantities in two-dimensionalhistograms. Table 7.4 lists the correlation coefficients on the detector level for the signal,the backgrounds and the data. For the signal, also parton level results are shown.Correlation SM Sample Uncorr. Sample BG DataParton Level Rec. Level Parton Level Rec. Level Rec. Level e + jets -0.04 -0.12 -0.05 -0.12 -0.10 -0.12 µ + jets -0.11 -0.12 -0.11 -0.12Table 7.4.: Linear correlation coefficients between ∆ φ ( l, d ) and ∆ φ ( l, b ) on the detectorlevel in the e + jets and the µ + jets channel for the SM sample, the samplewithout spin correlation, the background (BG) and in data.The observed correlation is small and consistent between data and prediction. In casethe correlation affects the measurement it is expected to show up in a wrong estimationof the expected statistical uncertainty and deviations in the linearity check. Hence, thelinearity checks from Section 7.4 are repeated taking the correlation between down-typequark and b -quark into account. This is realized by drawing real ensembles as subsetsof the full MC sample instead of applying Poissonian fluctuations.For each generated ensemble, both the ∆ φ ( l, d ) and ∆ φ ( l, b ) values are filled intohistograms to create pseudo data. This procedure replaces the application of Poissonianfluctuation, which is applied independently to each bin, leading to a vanishing correlationbetween the analysers. For each value of f SM , 50 ensembles are created.The new linearity check leads to a slope of 0 . ± .
01 and an offset of − . ± .
01. Within uncertainties, no deviation is observed. An equivalent cross check of thepull distribution for correlated ensembles is not meaningful. Using the procedure ofreal ensembles creates a bias in the estimation of the statistical uncertainty. This is aconsequence of the way the pseudo data templates are created. To obtain a distributionof a certain f SM requires a linear combination of events from the SM t ¯ t sample andthe sample of uncorrelated t ¯ t pairs. Thus, the number of drawn events is larger thanthe expected t ¯ t events. Hence, only for the cases of f SM = 1 . f SM = 0 . Systematic Uncertainties
No simulation including modelling of the underlying physics is expected to be perfect.Limited knowledge and simplified models cause systematic uncertainties, which affectthe precision of the measurement. This concerns both the physics processes under studyas well as the modelling of the detector. The systematic uncertainties relevant for thisanalysis are introduced in the next section. Two different approaches for the evaluationof the uncertainties will be discussed.The classical way to evaluate systematic uncertainties is to perform ensemble tests:systematic templates are created by varying the default templates according to the sys-tematic uncertainty by 1 σ up and down respectively. Poissonian fluctuations of the binsof the modified templates are applied to create a set of ensembles. The fit output distri-bution of this pseudo data follows a Gaussian distribution. Comparing the mean of thisdistribution to the nominal fit result gives the size of the systematic uncertainty. This isdone for all those systematic uncertainties where either a special prescription is needed(as “taking the largest of effects A, B and C and symmetrize”’), where no systematicup and down variation is available (as switching a setting off which is on by default) orwhere a continuous interpolation between a default and a systematic variation makes nosense (as for the comparison of two different MC generators).Another approach is the introduction of nuisance parameters. Instead of performingensemble tests with the templates varied by 1 σ up and down, these templates are usedto calculate the modification of the signal template as a linear function of the parametervalue of the respective uncertainty. This procedure was described in Section 7.3.3Both procedures have been used depending on the type of systematic uncertainty andthe results are provided after a detailed list of all uncertainties in the following section.The discussion of the effects on the measurement is presented at the end of this chapter.127 . Systematic Uncertainties In this section all evaluated uncertainties are described in detail. They are groupedinto detector related uncertainties affecting jet and lepton reconstruction, E missT andluminosity, background and signal modelling as well as method specific uncertaintiesthat are caused by the limited template statistics. Jet Energy Scale
Uncertainties on the different in-situ JES calibration techniques, as discussed in Section4.3, are combined and assigned to categories, depending on their source [245]. The totalnumber of 54 uncertainties are further reduced via combination into groups. In the end,the following numbers of uncertainties remain: • Detector description (2) • Physics modelling (4) • Statistics and method (3) • Mixed category (2)On top of these eleven in-situ JES uncertainties additional sources of uncertainties aredetermined [245]: • η -Intercalibration (2) Statistical and modelling (dominated by
PYTHIA vs.
HERWIG difference in forward region) uncertainties. • Pile-Up (2)
Effects of the number of primary vertices (in-time pile-up) and aver-age number of interactions per bunch crossing (out-of-time pile-up) on the JES. • High p T Jets
Difference between the high p T single hadron response in-situ andin test beam measurements. • MC Non-Closure
Difference between the MC generators used in the calibrationand in the analysis. • Close-By Jets
Uncertainty on the effect of varied jet energy response due toclose-by jets. • Flavour Composition
Uncertainty on the fraction of gluon jets leading to adifferent jet response. • Flavour Response
Uncertainty on the particular gluon and light quark jet re-sponses. • b -JES Uncertainty on the jet response difference for b -jets. It replaces uncertaintieson flavour composition and response in case a jet is tagged as b -jet.128 .1. List of Systematic Uncertainties In total, 21 components of the JES uncertainty are available and evaluated. An overviewof the total JES uncertainty as a function of the jet p T is shown in Figure 8.1(a) andFigure 8.1(b). [GeV] jetT p
20 30 40 · · F r a c t i ona l J ES un c e r t a i n t y ATLAS -1 dt = 4.7 fb L = 7 TeV, s Data 2011, (cid:242) correction in situ = 0.4, EM+JES + R t Anti-k = 0.5 h Total uncertaintyZ+jet+jet g Multijet balance-intercalibration h Single particle (a) [GeV] jetT p
20 30 40 · · F r a c t i ona l J ES un c e r t a i n t y Total uncertainty JES in situ
Baseline decaytFlav. composition, semileptonic t decaytFlav. response, semileptonic tPileup, average 2011 conditions = 0.7 R D Close-by jet,
ATLAS -1 dt = 4.7 fb L = 7 TeV, s Data 2011, (cid:242) correction in situ = 0.4, EM+JES + R t Anti-k = 0.5 h (b) Figure 8.1.: JES systematic uncertainty as a function of jet p T . (a) Total in-situ contri-bution and components [245]. (b) Total JES uncertainty (without b -jet JESuncertainty) with t ¯ t ( (cid:96) + jets channel) specific components [245]. Jet Energy Resolution
The jet energy resolution was measured using the bisector method [295] and di-jet p T balance [296]. The energy resolution determined in data and MC agree within 10 %.This difference is covered by the uncertainties of the resolution measurement. Hence, nocorrection of the MC resolution is applied. The jet energy resolution uncertainty on theanalysis is evaluated by smearing the jets in the MC according to the uncertainties ofthe resolution measurement in an updated version of [296] using the full 2011 dataset. Jet Reconstruction Efficiency
By comparing track jets to calorimeter jets, a difference in the jet reconstruction effi-ciency between data and Monte Carlo simulation is found [297]. The efficiency in datais slightly smaller. For the evaluation of the jet reconstruction efficiency, jets were ran-domly rejected according to the mismatch in efficiency. The jet reconstruction efficiencyin MC is lowered by 0.23 % for jets with a p T between 20 and 30 GeV. Jets with a higher p T are not affected. Jet Vertex Fraction
As described in Section 4.3, scale factors are applied to the jet selection efficiency andinefficiency for both jets emerging from the hard scattering process as well as pile-upjets. Scale factors are applied to the hard scatter jet selection efficiency ε HS and themistag rate I HS . The p T dependence of the JVF SFs is parameterized and fitted. The129 . Systematic Uncertainties uncertainties on these fits are taken as one contribution to the JVF SF uncertainty.Another contribution comes from effects of varied selection cuts applied to the Z +jetsample, which is used to determine the JVF SF. The total JVF uncertainty is obtained bypropagating the uncertainties of the JVF scale factors (∆ ε HS ≈ . I HS ≈ B-Tagging Scale Factors
The scale factors for tagged b -jets, c -jets and mistagged jets are derived by combiningseveral calibration methods as described in Section 4.3.2. The uncertainties on the cor-responding scale factors are indicated in Figure 4.8. For this analysis the uncertaintiesare accessed using the eigenvector method . The covariance matrices of all uncertain-ties are summed. The square roots of the corresponding eigenvalues are then used ascomponents of the total uncertainty. These components are available for the efficienciesof b -jets (9), c -jets (5) and the mistag rate (1). A similar approach of the eigenvectormethod is used in the context of PDF uncertainties, described for example in [298]. Lepton Trigger Scale factor
Uncertainties on the trigger scale factors are derived for both electrons ( ≈ η and E T (electrons) or η and φ (muons). The uncertainties contain components from limited Z boson sample statistics and systematic uncertainties for different T&P selections. Lepton ID and Isolation Efficiency
The uncertainties on electron ID and isolation efficiency scale factors (2-3 %) depend onthe η and E T of the electron. Statistical limitations, a pile-up dependence, the modellingof the underlying events as well as a difference between the isolation efficiency in W/Z and top quark events contribute to the uncertainty.For muons, the isolation efficiency uncertainty ( ≈ . Lepton Reconstruction Efficiency
The uncertainty on the electron reconstruction efficiency (0.6-1.2 %) depends only on | η | of the electron, while uncertainties on the muon reconstruction efficiency ( ≈ . η and φ of the muon. For the muon, thestatistical and systematic uncertainty components are added linearly. Electron Energy Resolution
The electron energy resolution is smeared in the Monte Carlo simulation in order tomatch the resolution in data. Each smearing factor has a relative uncertainty of ≤
10 %130 .1. List of Systematic Uncertainties for electron energies up to 50 GeV and up to 60 % for high energetic electrons. For theevaluation of the uncertainty of the electron resolution, the smearing is performed withthe systematic variation of the smearing factors.
Electron Energy Scale
Before the energy resolution smearing is applied, the energy of the electron is scaled upand down by the corresponding uncertainty. The uncertainties of up to 1.5 % dependon the E T of the electron as well as on the η of φ of the corresponding energy cluster.Dominating contributions result from the modelling of the detector material and thepresampler energy scale. Muon Momentum Scale
A muon momentum scale correction (up to 1.5 %) is applied to the MC simulation bydefault. For the evaluation of the corresponding uncertainty, it is completely switchedoff. The caused effect is quoted as symmetrized uncertainty.
Muon Momentum Resolution
The muon momentum resolution is varied separately for the ID and the muon spec-trometer components according to their uncertainties. Uncertainties on the resolutionsmearing factors vary between 2-12 % (muon spectrometer) and 4-27 % (ID), respectively.The largest difference of the two up and the two down variations is taken as uncertainty.
Two different types of uncertainties affect the E missT . On the one hand, the uncertaintiesof the objects used to calculate the E missT are propagated. On the other hand, dedicated E missT uncertainties exist: The pile-up uncertainty takes into account effects of additionalenergy in the calorimeter coming from pile-up events. The uncertainties on the CellOut term (11-14 %) for calorimeter energy outside reconstructed objects and the
SoftJets term (9-11 %) for soft underlying events are 100 % correlated and evaluated together.The effects of both the pile-up uncertainty (6.6 % effect on both the CellOut and theSoftJets term) and the combined
CellOut/SoftJets uncertainty are added in quadratureto obtain the total E missT uncertainty. The total luminosity of 4 . − for the full 2011 dataset has an uncertainty of 1.8 %,measured via van der Meer scans [230]. To account for this, the expected yields werechanged in the priors accordingly and the fit was repeated with the priors modified upand down by 1.8 %. 131 . Systematic Uncertainties Fake Lepton Normalization
The uncertainty of the QCD fake estimation is evaluated by varying the real and fakeefficiencies according to their uncertainties and adding their effects in quadrature. Thisyields to an normalization uncertainty of 50 % in the e + jets channels and 20 % inthe µ + jets channels. These uncertainties are then used for the prior widths on thebackground yields as described in Section 7.3.5. The same priors were used for the n jets = 4 and the n jets ≥ Fake Lepton Shape
For the e + jets channel the effects on the shape arising from the efficiency uncertaintiesfor real and fake electrons are added in quadrature and taken as systematic uncertainty.In the µ + jets channel, two different methods were used and averaged. Their differenceis taken as systematic uncertainty. W +Jets Normalization The W +jets background was determined using MC samples. A data-driven approach,described in Section 5.4.2, is used to correct the normalization and the heavy flavourcomposition.The factor r MC (Equation 5.7), used to determine the normalization of the W +jetsbackground, will vary with modifications of the chosen MC generator parameters, theJES, the PDF, lepton ID misidentification and b -tagging scale factor uncertainties. Theresulting W +jets normalization uncertainties are used as width for the W +jets priorsin the fit, as described in Section 7.3.5. Different priors were used for the n jets = 4 andthe n jets ≥ W +Jets Shape Uncertainties on the W +jets shape are assigned to the flavour and jet multiplicity de-pendent scale factors as described in Sections 5.4.2 and 5.4.3. The jet multiplicity binswere treated as uncorrelated. The uncertainties contain components addressing the mod-elling, reconstruction and dedicated W +jets generator settings for the factorization andparton matching scales.Details about the W +jets shape and normalization uncertainties can be found in [289]. Remaining Background Sources Z +jets, diboson and single top backgrounds are varied according to the uncertaintieson the theoretical prediction. For Z +jets events the uncertainty is determined usingBerends-Giele scaling [299] to be 48 % for events with exactly four jets. For each addi-tional jet, 24 % additional uncertainty is added in quadrature. The uncertainties on the132 .1. List of Systematic Uncertainties single top cross section are 3 % for the t -channel [109], 4 % for the s -channel [108] and8 % for the W t -channel [110].The uncertainty on the diboson background is 5 % plus 24 % per additional jet notoriginating from a hadronically decaying boson.The total effect on the remaining background sums up to 19 % for the e + jets channeland 15 % for the µ + jets channel, conservatively covered by 20 % on the total remainingbackground. As in the other cases of normalization uncertainties, the uncertainty ispropagated to the prior used in the fit. t ¯ t Modelling Uncertainties
A good modelling of the signal is necessary to correctly interpret the results. As thespin correlation is measured via kinematic distributions, all sources of uncertaintiesaffecting them are of particular interest. This section is dedicated to uncertainties onthe modelling of the t ¯ t signal. Parton Distribution Functions
The PDF used for the t ¯ t signal generator MC@NLO is the CT10 NNLO set [81, 271]. Byusing the LHAPDF framework [300] weights depending on the initial partons’ protonmomentum fractions x i and the scale Q can be obtained to rescale the samples todifferent PDF sets. For the evaluation of the PDF uncertainty, three different PDF setsincluding their nominal and error sets are compared: MSTW2008nlo68cl [83,270], CT10and NNPDF2.3 [82].Pseudo data is generated from the reweighted samples and used for ensemble testing.The fit output values are plotted in Figure 8.2. Each bin contains one up- and downvariation of the error set, except the nominal one in the first bin. For each PDF set, anerror band is drawn. According to the definition of the PDF errors, these bands are theRMS (NNPDF), the asymmetric Hessian (MSTW) and the symmetric Hessian (CT10).The outer edges of the error bands define the total PDF uncertainty, as indicated inthe plots. One can see that the modifications due to different PDF sets have effectsgoing in opposite directions for the two spin analysers. Thus, the combination of thetwo analysers can reduce the PDF uncertainty significantly. Top Quark Mass
Samples with varied masses for the top quark are used for ensemble testing. Figure 8.3shows the mean fit output values for both analysers and the combination. To evaluatethe dependence on the top quark mass, linear fits are performed. The slope s is used tocalculate the uncertainty of the fitted f SM values due to limited knowledge of the topquark mass: ∆ f SM = s · ∆ m t (8.1)133 . Systematic Uncertainties PDF error set S M f ∆ CT10MSTWNNPDF (a)
PDF error set S M f CT10MSTWNNPDF
PDF = 0.085 ∆ (b) PDF error set S M f CT10MSTWNNPDF
PDF = 0.018 ∆ (c) Figure 8.2.: Fitted f SM values for the MSTW2008nlo68cl, CT10 and NNPDF2.3 PDFset and their corresponding error sets. The results are shown for the (a)down-type quark, the (b) b -quark and (c) the full combination of the fit.134 .1. List of Systematic Uncertainties [GeV] top m166 168 170 172 174 176 178 ou t S M f dQbQcombination Figure 8.3.: Fit values for f SM for different values of the top quark mass. For eachanalyser, a linear fit is performed.Several options for choosing ∆ m t exist. Examples are the uncertainty on the worldcombination (∆ m t = 0 .
76 GeV [47]) or the LHC combination (∆ m t = 0 .
95 GeV [127]) aswell as the deviation between the mass used in the generator and the world combination( (cid:12)(cid:12) m MC t − m world t (cid:12)(cid:12) = | . − .
34 GeV | = 0 .
84 GeV). The uncertainty on theLHC combination was used in order to be conservative and cover the deviation of m t used in the generator.While the down-type quark is relatively stable against variations of the top mass, the b -quark is not. This comes from the fact that the spin analysing power for the down-typequark is always 1, independent of the kinematics of the top decay. As the spin analysingpower of b -quark depends on the W boson polarization state, which itself depends onthe top mass (see Equation 2.54 or Figure 7 in [172]), a dependence of the b -quarkas analyser is expected. The obtained values for the slopes s are s dQ < .
01 GeV − , s bQ = 0 .
05 GeV − and s comb. = 0 .
02 GeV − . Top p T uncertainty Recent measurements of the differential top quark cross section [289, 301] showed thatthe top p T spectrum of MC@NLO and the unfolded measurement in data agree withinuncertainties. But especially for the high values of the top p T the agreement is at theedge of the uncertainties. Furthermore, a slope in the ratio is visible in Figure 8.4(a). Itis not only the MC generator itself, which causes the top p T differences. Also the usedPDF set plays an important role, as shown in Figure 8.4(b).The effect of a mismodelled p T spectrum of the top quark is investigated. First, the135 . Systematic Uncertainties - G e V t T dp σ d σ -4 -3 DataALPGEN+HERWIGMC@NLO+HERWIGPOWHEG+HERWIG
ATLAS Preliminary -1 L dt = 4.6 fb ∫ = 7 TeVs [GeV] Tt p0 100 200 300 400 500 600 700 800 D a t a M C (a) [GeV] tT p T heo r y / D a t a DataCT10nloMSTW2008nloNNPDF 2.3HERAPDF 1.5
ATLAS
Preliminary -1 Ldt = 4.6 fb ∫ = 7 TeVs (b) Figure 8.4.: (a) Measured differential top quark cross section as a function of the topquark transverse momentum [301]. The unfolded data is compared to dif-ferent generators. (b) Ratios of the NLO QCD predictions to the measurednormalized differential cross sections for different PDF sets [301].results of [301] had to be reproduced. For that, top quark p T distributions for the fullphase space were compared. MC@NLO and
POWHEG + HERWIG are used as generators andare compared to the unfolded measurement of [301].The distributions, shown in Figure 8.5, reproduce the results from [301]. The ratio ofthe hadronic top p T spectrum measured in data and the one of MC@NLO for the full phasespace is used to reweight the signal sample. The provided combined top p T spectrumof the e + jets and µ + jets channels is used to calculate the scale factors as it had thesmallest uncertainties and no discrepancies between the e + jets and µ + jets numbersare observed. Table 8.1 shows the scale factors. The reweighted sample is then used toperform ensemble tests. The difference between the fitted value of f SM and f SM = 1.0is then quoted as uncertainty for the top p T .top p T [GeV] 0-50 50-100 100-150 150-200 200-250 250-350 350-800Scale Factor 1.01 1.02 1.02 0.99 0.92 0.88 0.86Table 8.1.: Scale factors used to reweight the top p T spectrum of MC@NLO to the onemeasured in data.As the top p T uncertainty is found to be large it was also split into a low and high The
HERWIG status code 155 was used to access the top quarks. .1. List of Systematic Uncertainties N o r m . E n t r i e s DataMC@NLO+HerwigPowHeg+Herwig lep +jets, t m top status 155 (Herwig), (top) [GeV] T p0 100 200 300 400 500 600 700 800 P r ed ./ D a t a (a) N o r m . E n t r i e s DataMC@NLO+HerwigPowHeg+Herwig had +jets, t m top status 155 (Herwig), (top) [GeV] T p0 100 200 300 400 500 600 700 800 P r ed ./ D a t a (b) Figure 8.5.: The p T spectrum of (a) the leptonic and (b) the hadronic top quark.The unfolded data measurement of [301] is compared to MC@NLO and
POWHEG + HERWIG . Here, results from the µ + jets channel are shown.top p T part to check the effect of each region. Table 8.2 shows the top p T uncertaintiesfrom reweighting of the full top p T spectrum (default, quoted as final uncertainty), the“low top p T region” ( ≤
200 GeV) and the “high top p T region” ( >
200 GeV).Top p T Region down-type quark b -quark CombinationFull 0.17 0.24 0.01Low 0.03 0.05 < f SM by reweighting the top p T spectrum of MC@NLO to theone measured in data. The uncertainty was evaluated by reweighting the fullspectrum, the “low top p T region” ( ≤
200 GeV) and the “high top p T region”( >
200 GeV).The high top p T region has a higher influence on the total uncertainty. This fact is nottrivial as the bulk of events is at the low top p T spectrum. Figure 8.6(a) shows the ∆ φ distribution for the Standard Model expectation normalized to one. The blue and redline represent the fraction of the total spectrum for the low and high top p T region. InFigure 8.6(b), the ∆ φ distributions for the low and high top p T regions are normalized toone, which helps to illustrate the different ∆ φ shapes. When pseudo data is constructedfrom the samples reweighted in top p T , the following effects can be observed: The steepcontribution (high top p T ) is scaled down and the total distribution gets flatter. Thiswill be interpreted as a higher spin correlation if the templates used for fitting stayunrescaled. Figure 8.6(c) compares the SM and the uncorrelated t ¯ t spin sample to theSM sample, which is reweighted in top p T . Figure 8.7 shows the same plots but for the137 . Systematic Uncertainties b -quark as analyser. One can see the same effect by the top p T reweighting but theopposite interpretations in terms of f SM . Colour Reconnection
The colour of quarks and gluons is conserved. Colour strings, propagating from theinitial state to the final hadronisation products can break and need to be reconnectedafter factorization. In order to estimate the effect due to the imperfect modelling of thecolour reconnection [302], templates are created with
POWHEG + PYTHIA using the P2011CTEQ5L
PYTHIA tune [303]. Another sample, using the same tune but with the colourreconnection completely turned off (using the
NOCR setting), is used for a second set oftemplates. Via ensemble tests, the difference between the fitted f SM results is evaluated,symmetrized and taken as the uncertainty for colour reconnection. Underlying Event
To study variations of the underlying event, the P2011 CTEQ5L
PYTHIA tune is set to amode ( mpiHi ) where the production rate of semi-hard jets coming from multiple parton-parton reactions is increased. It is compared to the nominal P2011 CTEQ5L
PYTHIA tune sample [303]. Via ensemble tests, the difference between the fitted f SM results isevaluated, symmetrized and taken as uncertainty for underlying events. Parton Showering / Hadronisation
In contrast to spin correlation analyses in the dilepton channel, the measurement in the (cid:96) + jets channel depends significantly on the jet properties. On the one hand these areused to map the jets to the corresponding partons. On the other hand the jet kinematicsare interpreted as representations of the t ¯ t spin correlation.To study the effect due to uncertainties of parton showering and hadronisation, acommon generator, POWHEG , is interfaced to
PYTHIA and to
HERWIG . The
POWHEG + PYTHIA and
POWHEG + HERWIG samples are both used for ensemble testing. The difference betweenthe fit results using
POWHEG + PYTHIA and
POWHEG + HERWIG is taken as uncertainty on theparton showering.
PYTHIA and
HERWIG represent two different physics models for the calculation of partonshowering. While
PYTHIA is based on the Lund string model [276],
HERWIG uses the clusterfragmentation model [261].In order to only account for the showering differences, t ¯ t events including a W → τ ν decay needed to be vetoed. The reason is that τ polarization information is mistakenlyneglected by TAUOLA [304], responsible for τ decays, in POWHEG + HERWIG . For the eval-uation of the parton showering, this τ veto is applied to all samples used for creatingpseudo data and fitting. Furthermore, both samples suffered from a bug in the top spincorrelation for antiquark-gluon and gluon-antiquark production channels. As both gen-erators suffer from the same error and only the difference between them is studied, thereis no residual effect on the determined uncertainty of the measured t ¯ t spin correlation.138 .1. List of Systematic Uncertainties (l, d) φ ∆ T ≤ 200 GeV SM, top p T > 200 GeV parton level, each normalized to total N o r m a li z ed E v en t s (a) (l, d) φ ∆ top p T ≤ 200 GeV SM, top p T > 200 GeV parton level, each normalized to 1 N o r m a li z ed E v en t s (b) (l,d) [rad] fD fD d N / d N (SM)tt (SM Reweighted)tt (no corr.)tt ATLAS = 7 TeVs m + 5 jets, 2 b-tags RWIGE + H @NLOC M (l,d) [rad] fD R a t i o (c) Figure 8.6.: ∆ φ distributions using the down-type quark as analyser. (a) The ∆ φ dis-tribution for the Standard Model expectation normalized to one. The con-tributions from the low and high top p T region are indicated. (b) Samedistributions as (a), but with the low and high top p T region also normal-ized to one. As an example, the effect of reweighting is shown in (c): The SMand the uncorrelated t ¯ t sample are compared to the SM prediction, whichis reweighted in top p T [180]. The ratio plot shows the ratios uncorrelatedover SM (dashed) and reweighted over SM (red dash-dotted). 139 . Systematic Uncertainties (l, b) φ ∆ top p T ≤ 200 GeV SM, top p T > 200 GeV parton level, each normalized to total N o r m a li z ed E v en t s (a) (l, b) φ ∆ top p T ≤ 200 GeV SM, top p T > 200 GeV parton level, each normalized to total N o r m a li z ed E v en t s (b) (l,b) [rad] fD fD d N / d N (SM)tt (SM Reweighted)tt (no corr.)tt ATLAS = 7 TeVs m + 5 jets, 2 b-tags RWIGE + H @NLOC M (l,b) [rad] fD R a t i o (c) Figure 8.7.: ∆ φ distributions using the b -quark as analyser. (a) The ∆ φ distributionfor the Standard Model expectation normalized to one. The contributionsfrom the low and high top p T region are indicated.(b) Same distributionsas (a), but with the low and high top p T region also normalized to one.As an example, the effect of reweighting is shown in (c): The SM andthe uncorrelated t ¯ t sample are compared to the SM prediction, which isreweighted in top p T [180]. The ratio plot shows the ratios uncorrelatedover SM (dashed) and reweighted over SM (red dash-dotted).140 .1. List of Systematic Uncertainties Another large difference between the generators is the top p T modelling. As this quan-tity is a large uncertainty itself (influencing the jet kinematics and the ∆ φ shape), the POWHEG + PYTHIA sample is reweighted to match the top p T spectrum of POWHEG + HERWIG .This is necessary for two reasons: As the top p T uncertainty is quoted explicitly, it shouldnot be double-counted in the showering uncertainty. Furthermore, it must be avoidedthat the difference in top p T cancels a showering uncertainty effect. Indeed, this is ob-served when evaluating the uncertainty without reweighting: While the uncertainty forthe down-type quark combination is very large, there is no effect on the b -quark combi-nation. The reweighting of the top p T removes that effect. The difference between thetop p T spectra of the generators is shown in Figure 8.8. As a cross check the top p T reweighting is also applied the other way around ( POWHEG + HERWIG to POWHEG + PYTHIA ).The resulting uncertainty is the same. The uncertainties on parton showering and hadro-Figure 8.8.: The top p T spectrum on parton level for different generators. The used topstatus codes were 155 for HERWIG and 3 for
PYTHIA . The ratios are withrespect to
MC@NLO .nisation as shown are listed in Table 8.3. Two sets of uncertainties are provided: Thedefault values without reweighting and the results using
POWHEG + PYTHIA reweighted tothe top p T spectrum of POWHEG + HERWIG . For a comparison to the default determina-tion of the parton shower/hadronisation uncertainty (without top p T reweighting), thetop p T uncertainty and the top p T reweighted PS/hadronisation uncertainty, added inquadrature, are also listed. 141 . Systematic Uncertainties Procedure down-type quark b -quark CombinationPS (no reweighting) 0.33 0.02 0.20PS (top p T reweighted) 0.08 0.29 0.16PS (top p T reweighted) ⊕ top p T unc. 0.19 0.38 0.16Table 8.3.: The uncertainties on f SM originating from the difference between POWHEG + PYTHIA and
POWHEG + HERWIG . The first row provides the uncertain-ties using the default
POWHEG +X samples without any reweighting. The sec-ond row provides the uncertainties using
POWHEG + PYTHIA reweighted to thetop p T spectrum of POWHEG + HERWIG . The third row shows results with addingthe dedicated top p T uncertainty as it is evaluated in Section 8.1.6 to the un-certainties in the second row in quadrature. Renormalization/Factorization Scale Variation
Effects of a varied renormalization/factorization scale on the predicted value of the spincorrelation C were studied in [175]. The MC@NLO sample used for the t ¯ t signal is alsoavailable with varied values for the renormalization and the factorization scale µ . Thedefault value µ is varied by a factor 0.5 and 2.0, as in [175]. Ensemble tests are performedfor both the up and down variation of µ . The difference is quoted as uncertainty. Initial and Final State Radiation
ALPGEN + PYTHIA samples with dedicated modifications of the P2011 CTEQ5L
PYTHIA tune [303] are used to increase ( radHi setting) and decrease ( radLo setting) the amountof initial and final state radiation. They are compared with ensemble tests. Half of thedifference between the two samples is symmetrized and taken as uncertainty.
Template Statistics
The precision of the measurement is limited by the available template statistics. Toaccount for this limitation, a corresponding uncertainty contribution is evaluated.A common procedure to evaluate the effect is applying Poissonian fluctuations to thebins of the templates used for fitting. The fluctuations are based on the MC statistics.The width of the output distributions of f SM is supposed to be taken as uncertainty.Evaluated uncertainties ∆ f SM are expected to be independent of the input value of f SM , f SM, in , and the average deviation between fit input and output is expected to bezero: ∂ (cid:0) ∆ f SM (cid:1) ∂f SM, in = 0 (8.2) (cid:104) f SM, in − f SM, out (cid:105) = 0 (8.3)142 .1. List of Systematic Uncertainties
Both expectations are found to be violated, leading to a close investigation of theobserved effect. Several tests are performed: • Only the signal templates are fluctuated. • The MC statistics of the SM spin correlation sample are reduced to the one withoutspin correlation to check for an effect due to different MC statistics of two templatesused for mixing. • Both signal templates are fluctuated individually.While the first tests did not change the situation, the last one helped to understand theeffect. It is illustrated in Figure 8.9. W i d t h o ff S M o u t pu t f SM input Fluct. SMFluct. NoSpinFluct. Both (a) -1.5-1-0.500.511.522.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 f S M o u t p u t f SM input fsmFluct. SMFluct. NoSpinFluct. Both (b) Figure 8.9.: (a) Width and (b) mean of the output f SM distributions after fluctuationof the t ¯ t signal templates bins according to their MC statistics uncertainty.Poissonian fluctuations were applied to either only the SM t ¯ t spin correlationsample, the uncorrelated t ¯ t spin sample or both.Figure 8.9(a) shows the dependence of the MC statistics uncertainty on f SM, in . Threescenarios are evaluated: Poissonian fluctuations are applied to the template with SM t ¯ t spin correlation, to the uncorrelated t ¯ t sample and to both. No effect is expected forpseudo data with f SM = 1.0 if only the template without spin correlation is modified.The signal has no contribution of the uncorrelated template and the non-fluctuatedsample can fully describe the pseudo data. Vice versa, modifying only the StandardModel spin correlation template leaves the fit output unchanged for pseudo data with f SM = 0.0. Using pseudo data with f SM (cid:54) = 0 ( f SM (cid:54) = 1) adds a contribution of the f SM = 1 ( f SM = 0) linearly with f SM .In Figure 8.9(b) the mean values of the f SM output distributions is shown. This effectonly shows up when the MC statistics is low with respect to the total separation power of The sample with uncorrelated t ¯ t pairs contains only of the statistics of the SM correlation sample. . Systematic Uncertainties the signals. The effect vanished for artificially increased separation powers and reducedMC statistics uncertainties.As the fit loses separation power for fluctuated signal templates, it has problems toassign the data to one of the two scenarios and the fit tends towards values for whichthe sum of both MC statistics uncertainties is the smallest ( f SM ≈ .
5, as seen in Figure8.9(a)). For the template statistics uncertainty, both effects are taken into account: thedeviation of the mean and the width of the f SM distribution were added in quadrature.All templates are included in the uncertainty, not just the t ¯ t templates. Only such uncertainties for which a well-defined template for both a ± σ variation ofa systematic effect exist are taken as candidates for using a nuisance parameter. Sucha template is called “well-defined” if an uncertainty has a continuous and symmetriceffect on a template. This excludes two-point uncertainties (e.g. smearing on/off orcomparison of two generators) and uncertainties which need a dedicated evaluation (e.g.checking several effects and quoting the largest).Uncertainty candidates passing these criteria, listed in Tables 8.4 and 8.5, are testedfor statistical relevance. This means that their systematic effect has to be larger thanthe Monte Carlo statistical uncertainty of the corresponding template. The testingprocedure is as follows: • If at least for two bins for either the up or the down variation the difference tothe nominal is larger than the statistical uncertainty of that particular bin, thesystematic uncertainty is defined as significant. • If the total deviation from the nominal sample is larger than the total statisticaluncertainty, it is called significant.This test is done for all systematic uncertainties, all eight channels, both analysers andall template types. It is included as a nuisance parameter only where it is relevant. Forexample, uncertainties only affecting the signal are only used as nuisance parameters forthe signal templates. The used templates are the W +jets background, the fake leptonbackground, the remaining background as well as two signal templates: the sum and thedifference of the SM t ¯ t spin correlation template and the uncorrelated t ¯ t signal template.An exception is made for the signal templates. If either the sum or the difference of thetemplates is significantly affected, both were linked to the nuisance parameter.Figure 8.10 shows an example NP which was tested for all channels. Three differentchannels with different outcome are shown: In the first channel (Figure 8.10(a)), thesystematic variations (red and blue lines) show clear significance with respect to theMC statistics (green) for all bins. In the second channel (Figure 8.10(b)) no bin itselfis statistically significant, but the sum of differences is and in the third case (Figure8.10(c)), the uncertainty is clearly not significant.Tables 8.4 and 8.5 show a list of all systematic uncertainties that are tested and thechannels on which they have an effect.144 .2. Test for NP Inclusion φ ∆0 0.5 1 1.5 2 2.5 3 a b s . d i ff e r e n c e -40-30-20-10010203040 dQ_5jin_1bex_muTemplate affected!atleast2sign. bins (a) φ ∆0 0.5 1 1.5 2 2.5 3 a b s . d i ff e r e n c e -8-6-4-202468 bQ_5jin_2bin_muTemplate affected!total deviation sign. (b) φ ∆0 0.5 1 1.5 2 2.5 3 a b s . d i ff e r e n c e -4-2024 dQ_5jin_2bin_el (c) Figure 8.10.: An example nuisance parameter, tested on one template type and for sev-eral channels. Three channels are shown as example. (a) The systematicvariations (red and blue) show clear significance with respect to the MCstatistics (green) for all bins. (b) No bin itself is statistically significant,but the sum of differences is. (c) The uncertainty is clearly not significant.145 . Systematic Uncertainties
Uncertainty Has effect on template with...(SM + Unc.) (SM - Unc.) Rem. BG W +JetsJES/EffectiveNP Stat1 yes no yes —JES/EffectiveNP Stat2 no no no —JES/EffectiveNP Stat3 no no no —JES/EffectiveNP Model1 yes no yes —JES/EffectiveNP Model2 no no no —JES/EffectiveNP Model3 no no no —JES/EffectiveNP Model4 no no no —JES/EffectiveNP Det1 yes no yes —JES/EffectiveNP Det2 no no no —JES/EffectiveNP Mixed1 no no no —JES/EffectiveNP Mixed2 yes no no —JES/Intercal TotalStat yes no yes —JES/Intercal Theory yes yes yes —JES/SingleParticleHighPt no no no —JES/RelativeNonClosureMC11b yes no no —JES/PileUpOffsetMu yes no yes —JES/PileUpOffsetNPV yes no no —JES/Closeby yes yes yes —JES/FlavorComp yes yes yes —JES/FlavorResponse yes yes yes —JES/BJES yes no yes —btag/break0 no no no —btag/break1 no no no —btag/break2 no no no —btag/break3 no no no —btag/break4 no no no —btag/break5 yes no no —btag/break6 yes no no —btag/break7 no no no —btag/break8 yes no yes —ctag/break0 yes no no —ctag/break1 yes no no —ctag/break2 no no no —ctag/break3 yes no no —ctag/break4 yes no yes —mistag yes no yes —Table 8.4.: List of candidates for the nuisance parameter fit (part 1).146 .3. Evaluation of Non-Profilable Uncertainties Uncertainty Has effect on template with...(SM + Unc.) (SM - Unc.) Rem. BG W +JetsJVF yes no yes —MET/CellOut+SoftJet no no no —MET/PileUp no no no —el/Trigger yes no no —el/Reco yes no no —el/ID yes no yes —el/E scale yes no no —el/E resolution no no no —mu/Trigger yes no yes —mu/Reco yes no no —mu/ID yes no no —WJets/bb4 — — — yesWJets/bb5 — — — yesWJets/bbcc — — — yesWJets/c4 — — — yesWJets/c5 — — — yesTable 8.5.: List of candidates for the nuisance parameter fit (part 2). The uncertainties that cannot be treated as nuisance parameters in the fit are evaluatedwith ensemble tests. 100,000 ensembles are drawn. Either the difference of the output f SM to the expected f SM = 1.0 is quoted or the special procedures are applied as ex-plained in Section 8.1. Table 8.6 shows the list of these additional uncertainties for thecombined down-type quark and b -quark fits as well as for the full combination. This section is dedicated to explain effects observed for certain sources of uncertaintiesand their implications on spin correlation measurements. In particular, cancellationeffects are explained.One important point in the discussion of the systematic uncertainties is the correlationbetween the two analysers. In most of the cases, the effect of a systematic variation isantisymmetric in terms of the resulting value of f SM : A systematic variation leadingto a higher result for f SM with the down-type quark as analyser leads to a lower resultof f SM using the b -quark as analyser. The reason is quite illustrative - after switchingoff spin correlation in t ¯ t events, the ∆ φ distribution becomes steeper for the down-typequark and flatter for the b -quark as shown in Figure 7.1.A higher top quark p T serves as good example for such an effect. The decay products147 . Systematic Uncertainties Uncertainty ∆ f SM down-type quark b -quark CombinationQCD shape ( e + jets) +0 . − .
010 +0 . − .
028 +0 . − . QCD shape ( µ + jets) +0 . − .
004 +0 . − .
006 +0 . − . PDF ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . ± . p T ± . ± . ± . ± . ± . ± . t ¯ t spin analysers and hence to a larger azimuthaldistance. The result is a shift to higher values of ∆ φ , independent of the analyser. Toillustrate this effect, which is consistent for down-type quark and b -quark analysers, asample of uncorrelated t ¯ t pairs is used. This decouples the influences of kinematics andspin configurations. Figure 8.11 shows the shift to higher values of ∆ φ for higher topquark p T . Parton level quantities are used and no phase space cuts are applied.While for higher top quark p T the ∆ φ distribution becomes steeper, it becomes flatterfor higher p T of the t ¯ t pair. In this case the transverse boost of the t ¯ t system collimatesboth top quarks and their decay products, preferring lower values of ∆ φ . Figure 8.12shows the effect.Interpreting such an effect in terms of spin correlation is opposite for down-type quarkand b -quark spin analysers. Corresponding effects were shown in the context of the evalu-ation of the PDF uncertainty. The same holds true for the renormalization/factorizationscale variation, which is also one of the most significant uncertainties.In case a mismodelling is observed in data it will be reflected in deviations of f SM intodifferent directions.148 .4. Important Aspects of Systematic Uncertainties (l, dQ) f D N o r m . E n t r i e s < 20 GeV T T
20 GeV < top p < 60 GeV T
40 GeV < top p < 80 GeV T
60 GeV < top p < 100 GeV T
80 GeV < top p uncorr. spins (a) (l, bQ) f D N o r m . E n t r i e s < 20 GeV T T
20 GeV < top p < 60 GeV T
40 GeV < top p < 80 GeV T
60 GeV < top p < 100 GeV T
80 GeV < top p uncorr. spins (b)
Figure 8.11.: Effects of increased top quark p T on the azimuthal angle between the leptonand the (a) down-type quark and (b) b -quark. The MC@NLO sample withuncorrelated t ¯ t pairs is used to decouple spin and kinematic effects on the∆ φ shape. Parton level quantities are used. (l, dQ) f D N o r m . E n t r i e s ) < 40 GeVt(t T T
40 GeV < p ) < 120 GeVt(t T
80 GeV < p ) < 160 GeVt(t T
120 GeV < p ) < 200 GeVt(t T
160 GeV < p uncorr. spins (a) (l, bQ) f D N o r m . E n t r i e s ) < 40 GeVt(t T T
40 GeV < p ) < 120 GeVt(t T
80 GeV < p ) < 160 GeVt(t T
120 GeV < p ) < 200 GeVt(t T
160 GeV < p uncorr. spins (b)
Figure 8.12.: Effects of increased p T of the t ¯ t system on the azimuthal angle between thelepton and the (a) down-type quark and (b) b -quark. The MC@NLO samplewith uncorrelated t ¯ t pairs was used to decouple spin and kinematic effectson the ∆ φ shape. Parton level quantities were used. 149 Results and Discussion
This chapter presents the results of the t ¯ t spin correlation analysis. First, the fit resultsfor the eight individual channels are presented for the down-type quark as well as for the b -quark analyser. The individual channels are fitted without nuisance parameters. As anext step, the channels are combined for both spin analysers and a full combination isperformed. Nuisance parameters are added to check the effect on the fitted f SM results.Finally, the results are presented with their full uncertainties.After the presentation of the results the chapter concludes with a consistency checkof the fit and a discussion of the effects due to systematic uncertainties. Individual fit channels were neither for the down-type quark nor for the b -quark spinanalyser expected to show a significant difference between the scenarios with SM t ¯ t spin correlation and uncorrelated t ¯ t pairs. As furthermore deviations between the eightchannels are not expected to be motivated by spin correlation effects, the analysis aimsfor a combination. For a cross check, the individual results are still listed in this section.Figure 9.1 shows the results for the fitted values of f SM using the ∆ φ distribution betweenthe charged lepton and the down-type quark and b -quark, respectively. For this fittingsetup no nuisance parameters are used. The quoted uncertainties are purely statistical.The following observations are made: • While for the down-type quark the individual results lie consistently above the SMexpectation of f SM = 1 .
0, the opposite is observed for the b -quark. • The jet multiplicity mismodelling of
MC@NLO (see Section 6.3) does not significantlydisturb the measurement. Otherwise, a tension between the n jet = 4 and the n jet ≥ . Results (l, d) f D via SM f-5 -4 -3 -2 -1 0 1 2 3e+jets,4 jets excl., 1 b-tag excl. e+jets,4 jets excl., 2 b-tag incl. e+jets,5 jets incl., 1 b-tag excl. e+jets,5 jets incl., 2 b-tag incl. +jets,4 jets excl., 1 b-tag excl. m +jets,4 jets excl., 2 b-tag incl. m +jets,5 jets incl., 1 b-tag excl. m +jets,5 jets incl., 2 b-tag incl. m PredictionCombinationData (a) (l, b) f D via SM f-5 -4 -3 -2 -1 0 1 2 3e+jets,4 jets excl., 1 b-tag excl. e+jets,4 jets excl., 2 b-tag incl. e+jets,5 jets incl., 1 b-tag excl. e+jets,5 jets incl., 2 b-tag incl. +jets,4 jets excl., 1 b-tag excl. m +jets,4 jets excl., 2 b-tag incl. m +jets,5 jets incl., 1 b-tag excl. m +jets,5 jets incl., 2 b-tag incl. m PredictionCombinationData (b)
Figure 9.1.: Comparison of single channel fit results using (a) the down-type quark andthe (b) b -quark as spin analyser. The fits were performed without nuisanceparameters. The quoted uncertainties are statistical. Next to the individualfit results the SM expectation (green line) and the result of the combinedfit (yellow band) are shown.152 .2. Combined Fits without Nuisance Parameters • A systematic effect b -quark results in the µ + jets channels can be observed: Theresults of f SM are higher in the n b-tags ≥ n b-tags = 1 channels. For both the down-type quark and the b -quark, the eight channels were combined. Thecombinations have a statistically significant tension. It is indicated by the yellow bandsin Figure 9.1. A full combination of both spin analysers is also performed. It leads toa good agreement with the SM prediction. Table 9.1 shows the results of the individualcombinations including the statistical uncertainty from the data as well as the normal-ization uncertainties from the background sources. The latter is accounted for by theusage of priors in the fit.Combination f SM ± statistical uncertaintydown-type quark 1 . ± . b -quark 0 . ± . . ± . f SM with statistical uncertainties and background normaliza-tion uncertainties. Adding nuisance parameters to the fit has two effects: The fit uncertainty includes thecomponent arising from the systematic uncertainties evaluated via NPs. Furthermore,the central values might change. This is because changes in the shape of the ∆ φ distri-butions can be fitted with either a modified signal composition – and hence a modified f SM – or with systematic variations. In the second case the corresponding nuisanceparameters are fitted to non-zero values. The addition of the nuisance parameters leadsto the fit results shown in Table 9.2.Combination f SM ± (statistical ⊕ NP) uncertaintydown-type quark 1 . ± . b -quark 0 . ± . . ± . f SM including statistical uncertainties and uncertaintiesdue to nuisance parameters. 153 . Results The final results also include those systematic uncertainties that are evaluated via en-semble tests. This evaluation has no effect on the central values. For quoting thefinal result the uncertainty due to NPs is separated from the statistical component via∆ f SM, NP = (cid:113) (∆ f SM, stat.+NP ) − (∆ f SM, stat ) . The results are shown in Table 9.3.What can be noticed, in particular when comparing the final fit result to the resultCombination f SM ± stat. ± NP syst. ± add. syst.down-type quark 1 . ± . ± . ± . b -quark 0 . ± . ± . ± . . ± . ± . ± . f SM including statistical uncertainties, uncertainties dueto nuisance parameters as well as additional systematic uncertainties.without NPs, is that • both the down-type quark and the b -quark combinations are now consistent withthe SM expectation of f SM = 1 . • the nuisance parameters affect the central value of the b -quark combination to alarge extent. • the combination significantly reduces the uncertainties.In the following section the fit output is investigated closely and checked for consis-tency. Several quantities need to be checked to evaluate the consistency of the fit output. Thisis the purpose of this section.The ∆ φ distributions after fitting allow checking for a proper modelling of the data.The distributions of posterior probability density functions of the fit give a hint ifsignificant changes of the assumed background yields were fitted. The measured f SM should not result from significant changes in the background normalization but rather ina mixing of the two signal samples. As a last check, the fit values of the nuisance param-eters need to be investigated. Significant deviations and constraints of their expecteduncertainties would need to be well justified. The following plots show the prediction of each of the eight channels for both the SMspin correlation and uncorrelated t ¯ t pairs. This is compared to the data and the best-fit154 .5. Fit Consistency Checks results including the uncertainties from the fit. The best-fit results are from the fullcombination fit. SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc = 1 b tags = 4, n jets e+jets, n (l, d) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc > 1 b tags = 4, n jets e+jets, n (l, d) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc = 1 b tags > 4, n jets e+jets, n (l, d) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc > 1 b tags > 4, n jets e+jets, n (l, d) [rad] φ ∆ R a t i o E v en t s Figure 9.2.: Prediction of the SM spin correlation and uncorrelated t ¯ t pairs (black dashedand dotted) compared to data (black dots) and the best-fit result (red line)including uncertainties (yellow area). The ratios of SM and uncorrelatedprediction (black line) as well as best-fit to data (red line) are shown. Theseplots show the four e + jets channels using the down-type quark as analyser.It can be noticed that, in general, the fit is able to properly describe the data. Thesegood fit results need a modification of the predicted yields for the signal and the back-ground processes. This is achieved by using the degrees of freedom provided by thepriors on the background estimation and the nuisance parameters.One trend that is useful, in particular for the discussion of the consistency, is a smallslope visible for both the down-type quark and the b -quark post-fit ratios between post-fit results and data. It comes along with the fact that the individual fit results for thedown-type quark and the b -quark tend to deviate from f SM = 1 . . Results SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc = 1 b tags = 4, n jets +jets, n µ (l, d) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc > 1 b tags = 4, n jets +jets, n µ (l, d) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc = 1 b tags > 4, n jets +jets, n µ (l, d) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc > 1 b tags > 4, n jets +jets, n µ (l, d) [rad] φ ∆ R a t i o E v en t s Figure 9.3.: Prediction of the SM spin correlation and uncorrelated t ¯ t pairs (black dashedand dotted) compared to data (black dots) and the best-fit result (red line)including uncertainties (yellow area). The ratios of SM and uncorrelatedprediction (black line) as well as best-fit to data (red line) are shown. Theseplots show the four µ + jets channels using the down-type quark as analyser.156 .5. Fit Consistency Checks SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc = 1 b tags = 4, n jets e+jets, n (l, b) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc > 1 b tags = 4, n jets e+jets, n (l, b) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc = 1 b tags > 4, n jets e+jets, n (l, b) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc > 1 b tags > 4, n jets e+jets, n (l, b) [rad] φ ∆ R a t i o E v en t s Figure 9.4.: Prediction of the SM spin correlation and uncorrelated t ¯ t pairs (black dashedand dotted) compared to data (black dots) and the best-fit result (red line)including uncertainties (yellow area). The ratios of SM and uncorrelatedprediction (black line) as well as best-fit to data (red line) are shown. Theseplots show the four e + jets channels using the b -quark as analyser. 157 . Results SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc = 1 b tags = 4, n jets +jets, n µ (l, b) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc > 1 b tags = 4, n jets +jets, n µ (l, b) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc = 1 b tags > 4, n jets +jets, n µ (l, b) [rad] φ ∆ R a t i o E v en t s SM ttbarnoSC ttbarFake LeptonsW+JetsRem. Bkg.DataPostFitPostFitUnc > 1 b-tags > 4, n jets +jets, n µ (l, b) [rad] φ ∆ R a t i o E v en t s Figure 9.5.: Prediction of the SM spin correlation and uncorrelated t ¯ t pairs (black dashedand dotted) compared to data (black dots) and the best-fit result (red line)including uncertainties (yellow area). The ratios of SM and uncorrelatedprediction (black line) as well as best-fit to data (red line) are shown. Theseplots show the four µ + jets channels using the b -quark as analyser.158 .5. Fit Consistency Checks Priors are set on the background yield estimations to constrain the fit. These are chosento be Gaussian with a width corresponding to the evaluated normalization uncertainty.In this section the probability density functions (PDFs) of the priors are compared tothose of the posteriors. The comparison allows checking if the fit either constrains theprediction (resulting in a narrower posterior) or if it prefers a normalization differentthan the predicted (leading to a shifted mean). Both effects are expected to be small.The posteriors for the full combination fit are shown in Figure 9.6. In addition tothe priors and posteriors of the background yields the posterior of the jet multiplicitycorrection factor (see Section 7.3.4) is shown. The corresponding posterior distributionsfor the down-type quark and b -quark combinations can be found in the Appendix F. Nosignificant deviation of the mean and width values of the priors are observed.There is only a constant prior on the t ¯ t cross section. It was tested that a Gaussianprior corresponding to the theory uncertainty on the cross section does not lead to animproved precision of the measurement. Hence, it can be extracted directly from the fitwithout a bias. The results of the t ¯ t cross section scaling parameter c (introduced inSection 7.3) are shown in Table 9.4.down-type quark b -quark Combination c . ± .
04 0 . ± .
04 0 . ± . t ¯ t cross section scaling parameter c . Uncertainties includestatistical uncertainties and uncertainties due to nuisance parameters.The results are compatible with the SM expectation of c = 1 . Nuisance parameters are implemented via Gaussian priors. Their central values are setto zero as the current modelling is the best estimate by definition. The width of the NPpriors is set to one, corresponding to one standard deviation. It is expected that the fitis able to constrain the uncertainties used as NPs. To estimate the possible constraintthe fit was performed replacing the data with the simulated SM expectation (
Asimovdataset [305]).The expected constraint of the systematic uncertainties is indicated by the grey barsin Figure 9.7 for the full combination fit. Results for the individual down-type quarkand the b -quark combinations can be found in the Appendix G.In Figure 9.7, all expected constraints are shifted. Their mean values are set to theones of the NP post-fit results. This allows comparing the expected (grey band) to themeasured (black line) constraints of the NPs. Before shifting the expected constraints, A NPs is constrained if its posterior width in smaller than one. . Results
RemBkg0 5 10 15 20 25 × p ( R e m B k g | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_4_el1000 2000 3000 4000 5000 6000 p ( W j e t s _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_5_el0 500 1000 1500 2000 p ( W j e t s _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_4_el0 500 1000 1500 2000 2500 3000 p ( Q C D _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_5_el0 200 400 600 800 1000 1200 1400 1600 p ( Q C D _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_4_mu4 6 8 10 12 × p ( W j e t s _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_5_mu1000 2000 3000 4000 p ( W j e t s _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_4_mu1000 2000 3000 4000 p ( Q C D _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_5_mu400 600 800 1000 1200 1400 1600 1800 p ( Q C D _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
JetScaleNP-4 -2 0 2 4 p ( J e t S c a l e N P | d a t a ) PriorprobabilityPosteriorprobability
Figure 9.6.: Prior and posterior distributions for the fit parameters describing the back-ground yields and the jet multiplicity correction for the full combination ofthe down-type quark and the b -quark analysers.160 .5. Fit Consistency Checks J ES _ E ff e c t i v e N P _ S T A T J ES _ E ff e c t i v e N P _ M O D E L1 J ES _ E ff e c t i v e N P _ D E T J ES _ E ff e c t i v e N P _ M I XE D J ES _ E t a I n t e r c a li b r a t i on_ T o t a l S t a t J ES _ E t a I n t e r c a li b r a t i on_ T heo r y J ES _ R e l a t i v e N on C l o s u r e_ M C J ES _ P il eup_ O ff s e t M u_up J ES _ P il eup_ O ff s e t N PV J ES _ c l o s eb y J ES _ f l a v o r _ c o m p J ES _ f l a v o r _ r e s pon s e J ES _ B J e s U n c b t ag_b r ea k t ag_b r ea k t ag_b r ea k c t ag_b r ea k c t ag_b r ea k c t ag_b r ea k c t ag_b r ea k m i s t ag j v f s f e l _ t r i g_ S F m u_ t r i g_ S F e l _ I D _ S F m u_ I D _ S F e l _ r e c o_ S F m u_ r e c o_ S F e l _ E _ sc a l e W J e t s _bb4 W J e t s _bb5 W J e t s _bb cc W J e t s _ c W J e t s _ c N P V a l ue -3-2-10123 Figure 9.7.: Postfit values of the nuisance parameters (black lines) for the full combina-tion. The grey vertical bands behind the lines show the expected uncertain-ties on the nuisance parameters. 161 . Results it was checked that the fit of the SM expectation leads to central values of zero for allNPs.The observed constraints are compatible with the expected ones. They are slightlyhigher in case of JES close-by, JES flavour composition and JES flavour response andslightly lower for the W +jets uncertainties. These differences arise from the down-typequark combination (JES components) and the b -quark combination ( W +jets compo-nents), shown in figures G.1 and G.2.Some of the NPs are highly correlated as seen in the correlation matrix in Figure9.8 for the full combination fit. The values for the correlation coefficients vary between − .
58 and +0 .
45. The matrix includes, next to the nuisance parameters, all other fitparameters. They are listed in Table 9.5. -1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0 -1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0
Figure 9.8.: Correlations between the fit parameters listed in Table 9.5 for the full com-bination of both analysers.Good examples are the anticorrelation between the dominating b -tag SF NP (27) andthe yield as well as the jet multiplicity correction NP (11) and the JES components(12-22). Uncertainties play a crucial role in this measurement. They limit the precision of theresult and and can give a clear hint to further improvements. This section concludesthe chapter of results by discussing the dominating uncertainties and explaining theireffects.162 .6. Discussion of Uncertainties
Parameter Name p ( N SMt ¯ t + N unc.t ¯ t ) p ( N SMt ¯ t − N unc.t ¯ t ) p N rem. backg. ,e +jets + N rem. backg. ,µ +jets p N W +jets ,n jets =4 ,e +jets p N W +jets ,n jets ≥ ,e +jets p N QCD,n jets =4 ,e +jets p N QCD,n jets ≥ ,e +jets p N W +jets ,n jets =4 ,µ +jets p N W +jets ,n jets ≥ ,µ +jets p N QCD,n jets =4 ,µ +jets p N QCD,n jets ≥ ,µ +jets p Jet Multiplicity Correction p JES/EffectiveNP Stat1 p JES/EffectiveNP Model1 p JES/EffectiveNP Det1 p JES/EffectiveNP Mixed2 p JES/Intercal TotalStat p JES/Intercal Theory p JES/RelativeNonClosureMC11b p JES/PileUpOffsetMu p JES/PileUpOffsetNPV p JES/Closeby p JES/FlavorComp p JES/FlavorResponse p JES/BJES p btag/break5 p btag/break6 p btag/break8 p ctag/break0 p ctag/break1 p ctag/break3 p ctag/break4 p mistag p JVF p el/Trigger p mu/Trigger p el/ID p mu/ID p el//Reco p el/E scale p mu//Reco p WJets/bb4 p WJets/bb5 p WJets/bbcc p WJets/c4 p WJets/c5
Table 9.5.: List of all fit parameters. 163 . Results
Uncertainties From Ensemble Testing
A summary of all uncertainties evaluated via ensemble tests is listed in Table 8.6. Thedominating uncertainties are the renormalization/factorization scale, the top quark p T (all affecting both down-type quark and b -quark analysers), the PDFs, as well as theparton showering and the initial and final state radiation (affecting the b -quark a lotmore than the down-type quark).All these uncertainties affect the kinematic configuration of the t ¯ t pair and the spinanalysers. Hence, the impact of the measured spin correlation is expected to be large.This is confirmed in both the measurements of CMS [188, 189] and ATLAS [180]. ThePDF uncertainty can be highlighted as it affects not only the kinematics but also theinitial state composition and the production mechanism. The relation of gluon fusionto quark/antiquark annihilation directly changes the spin configuration. In Figure 9.9the effect of varied PDFs is illustrated. Two default PDF sets (CT10 and HERAPDF)are compared as well as their spread due to evaluation of the error sets. Both sets areplotted at the scale of the top quark mass ( Q = m t ). Q2 = mt2 (a) (b)
Figure 9.9.: (a) PDF distribution as a function of the momentum fraction x for gluons.For both the CT10 and the HERPDF set the variations within the error setsare indicated by the two lines. (b) Relative deviations to the central valueof the CT10, caused by the variations of the CT10 and the HERAPDF errorsets.Two of the uncertainties should be emphasized as they have large effects which do notcancel in the combination. The first one is the initial / final state radiation. As seen inTable 8.6, the b -quark is affected to much larger extent. In Figure 9.10 the effect on theISR/FSR variation on the ∆ φ distributions is shown.The down-type quark is not affected while the b -quark shows a slope in the ISR/FSRup/down ratio. This slope is interpreted by the fit as a deviation in the spin correlation164 .6. Discussion of Uncertainties (l, d) f D N o r m . E n t r i e s ISR/FSR upISR/FSR down ‡ b-tags
4, n ‡ jets +jets, n m (l, d) f D up / do w n (a) (l, b) fD N o r m . E n t r i e s ISR/FSR upISR/FSR down ‡ b-tags
4, n ‡ jets +jets, n m (l, b) f D up / do w n (b) Figure 9.10.: (a) ∆ φ ( l, d ) distribution in the µ + jets channel for the samples with in-creased and decreased initial and final state radiation. (b) ∆ φ ( l, b ) distri-bution in the µ + jets channel for the samples with increased and decreasedinitial and final state radiation.and leads to a large uncertainty. It is expected that the b -quark is affected by the FSRto a much larger extent. The reason is the larger phase space available for FSR radiationdue to the b -quark’s larger p T (see Figure 7.4).Next to ISR/FSR, the modelling of the parton shower has a large impact on themeasured f SM using the b -quark. The compared showering generators, HERWIG and
PYTHIA , base on different showering models (cluster fragmentation vs. string model).Not only kinematics are affected, but also the flavour composition of the b -jets. Figure9.11 shows the number of b -tagged jets for POWHEG + HERWIG and
POWHEG + PYTHIA . Aclear difference is visible. For this plot, the distributions of the different generators arereweighted to the same top quark p T spectrum. Also, τ leptons are vetoed as theirpolarization was not properly handled by the generators. Uncertainties From Nuisance Parameters
There are different ways of classifying the most significant nuisance parameters. Thefive most significant NPs are presented. In the first type of ranking, shown in Table9.6, the NPs with the largest effect on the measured value of f SM are listed for the fullcombination fit. The ranking is created by performing the fit with all NPs included. Afterthat, each NP under test is taken out of the fit. The different f SM values are compared.The sign of the value represents the relative change when taking the nuisance parameterout of the fit.Another ranking can be created by evaluating the effect on the total uncertainty, notthe central value. Table 9.7 shows the NPs with the largest effect on the total uncertainty(which might become either larger or smaller) for the full combination fit. The most165 . Results (l, b) fD N o r m . E n t r i e s PowHeg+HerwigPowHeg+Pythia ‡ b-tags
4, n ‡ jets +jets, n m b-tagged jets1 2 30.97511.0251.05 H e r w i g / P y t h i a Figure 9.11.: Number of b -tagged jets for POWHEG + PYTHIA and
POWHEG + HERWIG in the µ +jets channel. Events containing τ leptons were vetoed and both sampleswere reweighted to the same top quark p T spectrum.NP relative change of f SM JES/BJES +2 . . . − . f SM ) for the fullcombination fit. NP relative change of ∆ f SM JES/FlavorComp − . − . − . . − . .6. Discussion of Uncertainties significant uncertainties for the individual combinations of the down-type quark and b -quark analysers can be found in the Appendix H.As the measurement depends a lot on the modelling of jets, the large contributionof the JES components is expected. The large impact of the b -tagging uncertainty is aconsequence of the utilization of the b -jets as analysers as well as of the dependence ofthe down-type quark reconstruction on the b -tag weight. The title of this section might be misleading. In case a deviation is really expected, it canbe calibrated. To allow for reweightings and calibrations, the preceding measurementsmust have sufficiently high precision. Until changes in the top quark modelling areestablished, indications can be noticed. Such indications are listed in the following,concluding this chapter. The question is: Where did independent measurements indicatea preference of the data to a different modelling than the one implemented in thisanalysis? And if such deviations are observed: What would be the effect on the currentmeasurement? Would it cause further tension between the down-type quark and the b -quark analysers? Or would it bring the results closer together? Top Quark p T As shown in Figure 8.4(a), the top quark p T seems to be modelled imperfectly in MC@NLO .The data prefers a softer p T spectrum, shifted to lower values. As shown in Figure 8.11,this would lead to a flatter ∆ φ distribution. The fit templates based on a harder top p T spectrum will interpret this as a higher spin correlation for the down-type quark and alower spin correlation for the b -quark, respectively (see Figure 7.1). This is exactly whatwas observed in data by measuring f SM : For the down-type quark, the measured f SM is higher than the SM prediction and for the b -quark it is lower. PDF
During the discussion of the PDF uncertainties their large impact and opposite effecton down-type quark and b -quark analysers was stressed (Section 8.1.6). The questionis: Does the data have a preferred PDF? In the ATLAS measurement of the differentialtop quark cross section [301] the impact of the PDF on the top quark p T is checked. Asshown in Figure 8.4(b), HERAPDF is preferred by the data. In particular, this is thecase for large values of top quark p T . Furthermore, the worst modelling seems to be givenby the CT10 PDF. This motivated to check how HERAPDF would affect the measuredvalues of f SM . To answer this question, the LHAPDF reweighting was repeated usingHERAPDF. The results are shown in Figure 9.12. The shown fit values correspond topseudo data created with distributions that are reweighted to a modified PDF.In case the data was modelled with HERAPDF, a larger value of f SM was fitted forthe down-type quark and a smaller value for the b -quark. This means that if the dataprefers HERAPDF – and the indications for that were shown in Figure 8.4(b) – f SM is167 . Results PDF set0 10 20 30 40 50 S M f CT10MSTWNNPDFHERAPDF (a)
PDF set0 10 20 30 40 50 S M f CT10MSTWNNPDFHERAPDF (b)
Figure 9.12.: Fit results for f SM using pseudo data reweighted to different PDF sets anderror sets. (a) Combined fit using the down-type quark. (b) Combined fitusing the b -quark.expected to be fitted with f SM > . f SM < . b -quark. Indeed, this is what is measured. Generator Variation
Concerning the uncertainties assigned to the t ¯ t generator, effects coming from the partonshowering, the renormalization/factorization scale, the underlying event, the ISR/FSRvariation and the colour reconnection are evaluated. There has been no direct comparisonof MC@NLO to other t ¯ t generators. The reasons are the following: • The spin correlation is different for LO and NLO generators. Thus, only NLOgenerators should be used for comparison. • Samples using
POWHEG + HERWIG suffer from a bug in the τ lepton polarization. • All available
POWHEG samples suffer from an additional bug concerning spin corre-lation in the antiquark-gluon and gluon-antiquark production channel. • Samples with uncorrelated t ¯ t spins are not available for any generator other than MC@NLO .However, it should be mentioned that fitting pseudo data created using
POWHEG + HERWIG leads to values of f SM deviating from the expectation of f SM = 1.0. The results are shownin Table 9.8.One of the main differences between POWHEG + HERWIG and
MC@NLO is the top quark p T spectrum. Hence, POWHEG + HERWIG is reweighted to match the top p T spectrum of MC@NLO .Reweighting the top p T spectrum reduces the deviations to the expected fit values of f SM = 1 .
0, but does not fully remove them. One aspect is the non-perfect reweighting.168 .6. Discussion of Uncertainties
Sample f SM down-type quark b -quark Combination MC@NLO
POWHEG + HERWIG (nominal) 1.26 0.64 1.02
POWHEG + HERWIG (top p T reweighted) 1.15 0.73 0.99Table 9.8.: Results from ensemble tests for pseudo data created with MC@NLO , POWHEG + HERWIG (nominal) and
POWHEG + HERWIG reweighted to the top p T spectrum of MC@NLO .Reweighting techniques are not expected to replace event generations with a modifiedmodelling. Furthermore, the
POWHEG + HERWIG sample is known to suffer from bugs affect-ing the τ polarization and a small part of the spin correlation. To eliminate the effectsof these bugs and the effects coming from the reconstruction, an additional comparisonon parton level is done.Figure 9.13 shows the differences in the shape of the ∆ φ distributions for both gener-ators. Additional comparisons of the ∆ φ distributions on parton level before and afterreweighting in top quark p T are shown in Appendix I. MC@NLO (SM)PowHeg+HerwigPowHeg+Herwig (top p T rew . ) full phase space, parton level (l, dQ) φ ∆ R a t i o N o r m . E v en t s (a) MC@NLO (SM)PowHeg+HerwigPowHeg+Herwig (top p T rew . ) full phase space, parton level (l, bQ) φ ∆ R a t i o N o r m . E v en t s (b) Figure 9.13.: Comparison of the ∆ φ distribution between MC@NLO (black) and
POWHEG (red) using (a) the down-type quark and (b) the b -quark as analyser. Thedashed distribution shows the POWHEG spectrum reweighted to match thetop p T distribution of MC@NLO .The ∆ φ distribution is flatter for POWHEG + HERWIG . As shown in Figure 7.1, this resultsin different interpretations of a fitted f SM : higher values of f SM for the down-type quarkand lower ones for the b -quark analyser. Reweighting POWHEG + HERWIG to the top p T spectrum of MC@NLO leads to good agreement for the down-type quark distribution, butnot for the b -quark distribution. However, an improvement is observed. The reason169 . Results for the remaining difference lies in the b -quark energy spectrum. This difference is stillpresent after reweighting, even though the W kinematics of the two generators do agree.To conclude, POWHEG + HERWIG and
MC@NLO lead to different top p T and b -quark energyspectra, both affecting the fitted value of f SM . In the case that POWHEG + HERWIG is ableto describe the data better than
MC@NLO – and Figure 8.5 as well as the jet multiplicitydistribution give indications for this assumption – it could replace
MC@NLO for the fit todata. In this case the measured f SM is expected to be fitted lower using the down-typequark and higher using the b -quark. This would lead to a better compatibility of thedown-type quark and b -quark combinations. The difference between the results of the down-type quark and the b -quark combinationimmediately triggers the question whether the results are consistent. Two types ofconsistency are checked in this section: the one between the e + jets and the µ + jetschannel combinations as well as the one between the down-type quark and the b -quarkcombinations.The fit results for the lepton flavour comparison are shown in Table 9.9 for a fit withoutnuisance parameters. The quoted uncertainties include the statistical and backgroundnormalization uncertainties.Lepton Channel down-type quark b -quark Combination e + jets 1 . ± .
21 0 . ± .
29 1 . ± . µ + jets 1 . ± .
19 0 . ± .
23 1 . ± . . ± .
14 0 . ± .
18 1 . ± . f SM including statistical uncertainties. The fit has been splitinto lepton flavours for this cross check.The results are in good agreement across the different lepton flavours. The consistencyof the down-type quark and the b -quark results are evaluated as well. It is important toconsider the correlation between the spin analysers. Studying the effect of the top quark p T , as done in Section 8.1.6, demonstrates the anticorrelation of certain uncertainties.Changes affecting the shape, which are common for both the down-type quark and the b -quark channels, are interpreted differently in terms of spin correlation.The consistency check for the down-type quark and the b -quark channels is done thefollowing way: • For each uncertainty i , a random number s i according to a Gaussian distribution,centred at 0 with a width 1, is drawn. • Each bin of each template of the signal and background predictions is modified bythe relative change r i expected by the corresponding uncertainty i multiplied with170 .7. Spin Analyzer Consistency Checks the random number s i . For s i >
0, the systematic up variation is taken and for s i <
0, the down fluctuation. • After each systematic variation a Poissonian fluctuation of the template bins isapplied on top to take the statistical uncertainty into account. • Ensembles using the same variations of systematic effects are produced for alldown-type quark and b -quark templates. This ensures the correct propagation ofthe uncertainties’ correlation to the spin analysers. • The ensemble tests are performed and for each ensemble, f SM (down-type quark)is plotted against f SM ( b -quark). • The result from the fit on data is added to the two-dimensional distribution andcompared to the spread of results from the ensemble tests.The 100,000 ensembles were fitted without nuisance parameters. Hence, the result of f SM from the fit to data, for which no nuisance parameters were used in the fit, is shownas a comparison.By using only the statistical uncertainty, the nuisance parameter uncertainties, therenormalization/factorization scale uncertainty and the top p T uncertainty, the resultsof the two analysers are already consistent within the 99.5 % confidence level intervalas shown in Figure 9.14(a). The test was repeated without the top p T uncertainty butwith the uncertainty coming from ISR/FSR and PS/fragmentation instead. This resultis shown in Figure 9.14(b). The compatibility of the two results for the single analysersis also confirmed using this set of uncertainties. 171 . Results via bQ SM f 1 0.5 0 0.5 1 1.5 2 2.5 3 v i a d Q S M f smallest 99.7% interval(s)smallest 95.5% interval(s)smallest 68.3% interval(s)Fit (Data) (a) via bQ SM f 1 0.5 0 0.5 1 1.5 2 2.5 3 v i a d Q S M f smallest 99.7% interval(s)smallest 95.5% interval(s)smallest 68.3% interval(s)Fit (Data) (b) Figure 9.14.: (a) Compatibility check for the results of the down-type quark and the b -quark combination. Only statistical uncertainties, the nuisance parameteruncertainties, the renormalization/factorization scale and the top p T wereused as uncertainties (left). (b) As a cross check the test was repeatedwithout the top p T uncertainty, but with ISR/FSR and PS/hadronisationuncertainty added.172 Summary, Conclusion and Outlook
The aim to measure the t ¯ t spin correlation at √ s = 7 TeV in the (cid:96) + jets channel wasambitious. Hadronic spin analysers are hard to identify, in particular in events with highjet multiplicities, such as at the LHC.Motivated as a precision test of the Standard Model and a search for hints suggestingnew physics, uncertainties needed to be kept low.This chapter concludes the thesis by presenting the results, comparing them to othermeasurements, and by drawing conclusions. Finally, a discussion about future mea-surements of t ¯ t spin correlation provides ideas about what to do next. The presentedresults are interesting by themselves and give a glance on the spin properties of thetop quark: Does it interact as a particle with a spin of , produced by gluon fusion andquark/antiquark annihilation, decaying via the weak interaction before bound states canbe formed? The answer is: yes.The detailed studies of systematic effects, and in particular the comparison of theresults that were measured to results that are expected by motivated changes in the topquark modelling, give a straight-forward recipe for a next-generation t ¯ t spin correlationmeasurement. The t ¯ t spin correlation was measured in the (cid:96) + jets decay mode. By performing atemplate fit of the distributions of the azimuthal angle ∆ φ between the charged leptonand hadronic analysers, the degree of t ¯ t spin correlation, as predicted by the SM, f SM ,was measured. Two different hadronic analysers were used: The down-type quark andthe b -quark. A kinematic fit was utilized to assign jets to the model partons, whichinduced them. To separate the up- and down-type quark jets from the hadronically173
0. Summary, Conclusion and Outlook decaying W boson, b -tagging weight distributions and transverse momenta were utilized.The data was split into events from the e + jets and from the µ + jets channel, into jetmultiplicity bins of n jets = 4 and n jets ≥ n b-tags = 1 and n b-tags ≥ b -quark. A com-bination of the analysers was also performed, constraining the systematic uncertaintiesto a large extent. The results obtained are f SM (down-type quark) =1 . ± .
14 (stat.) ± .
32 (syst.) f SM ( b -quark) =0 . ± .
18 (stat.) ± .
49 (syst.) f SM (comb.) =1 . ± .
11 (stat.) ± .
22 (syst.)The results for both analysers are found to be consistent with the SM and with eachother. This is possible due to the large systematic uncertainties and the asymmetriceffects of the uncertainties on both analysers which are highly anti-correlated: whilethe down-type quark will fit values with f SM > .
0, the b -quark will fit f SM < . φ distributions is shown in Figure 10.1. p E v en t s / . · fit result (SM)tt (no corr.) ttdatabackground ATLAS -1 L dt = 4.6 fb (cid:242) = 7 TeVsl+jets (l,d) [rad] fD R a t i o (a) p E v en t s / . · fit result (SM)tt (no corr.) ttdatabackground ATLAS -1 L dt = 4.6 fb (cid:242) = 7 TeVsl+jets (l,b) [rad] fD R a t i o (b) Figure 10.1.: Distributions of the stacked (a) ∆ φ ( l, d ) and 10.1(b) ∆ φ ( l, b ) distributionsfor the combined fit [180]. The result of the fit to data (blue) is comparedto the templates for background plus t ¯ t signal with SM spin correlation(red dashed) and without spin correlation (black dotted). The ratios of thedata (black points), of the best fit (blue solid) and of the uncorrelated t ¯ t prediction to the SM prediction are also shown.174 To compare the results from this thesis to other measurements of t ¯ t spin correlation,other measurement’s results were transformed into f SM by dividing the measured spincorrelation by the SM expectation. An overall summary is given in Figure 10.2. Itincludes results from [180, 188, 203–207], using different t ¯ t decay modes, observables ofcentre-of-mass energies. SM SM Spin Correlation Fraction f-5 -4 -3 -2 -1 0 1 2 3
Spin Correlation Measurementstt
CDFCDFCDFD0D0D0CMSCMSATLASATLASATLASATLASThis thesisThis thesisThis thesis 2 TeV2 TeV2 TeV2 TeV2 TeV2 TeV7 TeV7 TeV7 TeV7 TeV7 TeV7 TeV7 TeV7 TeV7 TeV l+jetsl+jetsdileptonl+jetsdileptondileptondileptondileptondileptondileptondileptondileptonl+jetsl+jetsl+jets beam basis C hel. basis C beam basis C MEM beam basis C MEM(l,l) f D hel. basis C (l,l) f D S-Ratio hel. basis C max. basis C (l,d) f D (l,b) f D (l,d/b) f D s Group Channel ObservableSM PredictionData Stat. Unc.Total Unc.
Figure 10.2.: Comparison of t ¯ t spin correlation measurements. The results of [180, 188,203–207] using different observables have been divided by their SM expec-tations to compare a common f SM .It can be noticed that all measurements of C in the dilepton channel consistently ob-served less spin correlation than predicted. A second notice concerns the ∆ φ measure-ments. At ATLAS, ∆ φ lead to f SM > (cid:96) +jets resultusing the down-type quark as analysers. The deviation of the down-type quark combi-nation is likely to be caused by a mismodelling of the t ¯ t kinematics as it was discussedin detail. Such a mismodelling, in particular concerning the top quark p T , would lead toa deviation of the dilepton result in the same direction as for the down-type quark. Thiswas indeed observed as ATLAS measured f SM = 1 . ± .
09 (stat.) ± .
18 (syst.) [180].The following section is dedicated to the question: To which conclusions do the f SM values, measured in this thesis, lead? Only advanced methods of down-type quark reconstruction allowed a measurement ofthe t ¯ t spin correlation in the (cid:96) + jets channel. It is the first published measurement of t ¯ t
0. Summary, Conclusion and Outlook spin correlation in the (cid:96) + jets channel at the LHC [180].The obtained results are consistent with the SM prediction. Both utilized spin analy-sers, the down-type quark and the b -quark, suffer differently from the effects of system-atic uncertainties. The measurement helped to understand these effects and to build thebasis for future measurements. A combination of the results leads to a significant reduc-tion of the systematic uncertainties. It allows disentangling effects due to an imperfectmodelling from effects caused by a modified spin configuration.The results of the down-type quark and b -quark analyser combinations show devia-tions from the SM expectation in different directions. Given the uncertainties and theirasymmetric effects on both spin analysers, the results were found to be compatible withthe SM and with each other.It was found by independent measurements that the data prefers a t ¯ t modelling whichis different than the one implemented in this thesis. This concerns the used PDF, thegenerator and the top quark p T spectrum. The effects of these alternative models on thespin correlation measurement were checked. All suggested modifications, preferred bydata in independent measurements, lead to a lower result of f SM for the down-type quarkand a higher for the b -quark. This is a clear indication that motivated modifications inthe t ¯ t modelling lead to a better agreement of the measured values of f SM – for boththe down-type quark and the b -quark combination – with the SM. Furthermore, theindividual down-type quark and b -quark results would be even more consistent in caseof the changed modelling. Details on these tests are shown in the Appendix J.The results presented in this thesis are in good agreement with the SM predictionand other t ¯ t correlation results. Before strong implications on BSM physics can bededuced, the systematic uncertainties need to be reduced further. A trend of a higherspin correlation for the down-type quark and a lower for the b -quark was observed.It can be checked if the measured results give a first indication for new physics phe-nomena. Concluding the spin physics requires reducing both the mismodelling effectsand the uncertainties. BSM modifications in the t ¯ t production with a SM t ¯ t decay wouldlead to a coherent modification of the f SM results for both the down-type quark and the b -quark. This was not observed. Instead, the down-type quark result was higher andthe b -quark result lower than the expectation.BSM physics in the decay would affect the two analysers differently. A popular modelfor a modified top quark decay is t → H + b , so the replacement of the W vector bosonby a scalar charged Higgs boson [306]. In the following it is assumed that such a decaymode occurs in only one of the two top quarks. Such a decay modifies the spin analysing power α of the associated b -quark from − . . φ ( l, b ) would look more like the distributionof uncorrelated t ¯ t pairs. At first sight, this matches the measured f SM result using the b -quark. However, the b -quark under study was the one from the hadronically decayingtop quark. This would require the H ± to belong to the model of a top quark decayinginto three jets. It is also possible that both top quarks decay via a charged Higgs boson, but unlikely as the effect isnot at leading order.
Such a decay of the H ± into two jets would not be the preferred one. Instead, couplingsto τ leptons would be preferred due to the large mass of the τ [200]. This would notmatch the decay signature used for the reconstruction in this thesis. Hence, implicationsof a t → H + b process would not be visible in ∆ φ ( l, b ), but in a measurement of ∆ φ ( b, ¯ b ).Instead of affecting the b -quark and the down-type quark in the ∆ φ distributions, thedominant effect of a charged Higgs boson would be on the side of the charged leptondue to the large τ coupling. Neither were τ leptons reconstructed explicitly, nor are theeffects on the secondary leptons from decayed τ leptons known. Hence, no conclusionson a possible charged Higgs boson in t ¯ t decays can be drawn. This is true in particularin the context of the large systematic uncertainties on the individual down-type quarkand b -quark spin analysers results. This thesis concludes with proposals concerning future t ¯ t spin correlation measurementsat hadron colliders. The LHC provided a large number of t ¯ t pairs, already for the 2011 dataset. Systematicuncertainties limited the presented analysis. Reducing them should have highest prioritybefore repeating the measurement at √ s = 8 TeV. As for many analysis, the jet energyscale uncertainty had a strong impact, too. Improving the calibration would be a greatbenefit.A clear dependence on the modelling of the kinematics of the t ¯ t pairs was shown.Recent measurements of the differential t ¯ t cross section showed a preference of the topquark p T spectrum and a parton distribution function which are different from to thecurrently implemented ones. Hence, a change of the default generator is suggested. Thisalso concerns the jet multiplicity mismodelling of MC@NLO which made it necessary toadd another degree of freedom to the fit in order to deal with this problem. Next tochanging the default MC generator setup, the uncertainties on the t ¯ t modelling shouldbe further investigated and tried to be reduced, too.A larger dataset allows the application of further cuts to improve the purity of thesample. It was also shown that the reconstruction is stable in terms of pile-up whichwill increase with higher luminosities. t ¯ t Event Suppression
A common principle in t ¯ t analyses is to optimize the event selection in a way that thebackground is reduced and the signal contribution maximized. For t ¯ t analyses in the (cid:96) + jets channel, where the properties of the t ¯ t topology assuming a (cid:96) + jets topologyare analysed, t ¯ t contribution from the dilepton channel is a non-negligible backgroundcomponent. As shown in Table 6.2, the dileptonic t ¯ t events represent more than 10 % ofthe t ¯ t signal. 177
0. Summary, Conclusion and Outlook
A reduction of the dilepton contribution should be considered. Some quantities havedifferent distributions for t ¯ t events in the dilepton and in the (cid:96) + jets channel and allowto separate the signatures. As an example, Figure 10.3(a) shows the missing transversemomentum, E missT . The transverse W boson mass (Figure 10.3(b)) and the likelihoodfrom KLFitter (Figure 10.3(c)) show a good discrimination between the two t ¯ t decaychannels as well. missT E0 20 40 60 80 100 120 140 160 180 200 N o r m . E n t r i e s l+jetsdilepton (a) WT m0 20 40 60 80 100 120 140 160 180 200 N o r m . E n t r i e s l+jetsdilepton (b) Log ( Likelihood ) -80 -75 -70 -65 -60 -55 -50 -45 -40 N o r m . E n t r i e s l+jetsdilepton (c) Figure 10.3.: Normalized distributions of (a) the E missT , (b) transverse W boson massand the (c) logarithm of the likelihood from KLFitter . The distributionsare shown for reconstructed simulated quantities of t ¯ t pairs decaying intothe (cid:96) + jets channel and the dilepton channel.It should also be considered to veto a second lepton with a looser cut than the onesused in this analysis (see Section 4.1 and 4.2). t ¯ t Reconstruction
The jet p T distributions and flavour composition was used to separate jets originatingfrom light up- and down-type quarks. This allowed to reconstruct the down-type quarkas spin analyser. During the optimization studies for the down-type quark reconstructionother methods were checked, too.178 A promising utility for jet discrimination is the jet charge . It makes use of the factthat the charge of the quark is propagated to the hadrons to which a jet fragments. Bydetermining the hadron charges one can infer back on the original quark charge [307].Such a method was successfully used in [139–142] to measure the charge of the top quark.As both the up- and down-type quark from the W boson have a charge of the same sign,the jet charge technique has not been used in this thesis. However, future measurementscould benefit from an improved t ¯ t reconstruction due to the usage of jet charge. Inparticular, the correct assignment of the two b -quarks to their parent top quarks canbe improved. While studies in the (cid:96) + jets channel can make use of other supportivereconstruction techniques, the dilepton channel could benefit a lot.In the following the feasibility of the jet charge is briefly demonstrated. Two methodsof jet charge are used. For the “MaxPtTrackCharge” the jet charge corresponds to thecharge of the track within the jet that has the highest p T . For another approach aweighted sum of charges of the tracks within a jet is created. In Figure 10.4 the jetcharges of jets matched to up and anti-up quarks is shown. A clear separation is visible. quark q-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 N o r m . E n t r i e s Matched Up QuarkMatched Anti-Up Quark
Jet Charge Via MaxPtTrackCharge (a) quark q-4 -3 -2 -1 0 1 2 3 4 N o r m . E n t r i e s Matched Up QuarkMatched Anti-Up Quark
Jet Charge Via Weighted Jet Charge (b)
Figure 10.4.: Charge of a jet matched to an up and anti-up quark using the (a) chargeof the jet track with the highest p T and (b) the weighted charge using alltracks.The description of the weighted jet charge as well as further reconstruction optimiza-tion tests can be found in the Appendix K. Studies of the jet charge technique in thecontext of t ¯ t pairs produced in association with a Higgs boson can be found in [308]. t ¯ t Spin Correlation
A strategy for future measurements of t ¯ t spin correlation in the (cid:96) + jets channel issuggested.The Monte Carlo generator used to produce the t ¯ t signal should be chosen such thatno known mismodelling is included. In cases with a clear preference of the data, it should179
0. Summary, Conclusion and Outlook be followed. This also concerns the parton distribution functions and the parton showermodelling.The t ¯ t reconstruction can be further improved by adding information from the jetcharges. As several quantities are available to properly map jets from the t ¯ t decay tothe model partons, a multi-variate reconstruction algorithm is a promising way.Both the production and the decay of t ¯ t pairs need further studies to carefully probethe Standard Model and to look for new physics effects beyond it. Studies in the (cid:96) +jets channel will keep playing an important role. A larger dataset will allow choosingsubsets with a high signal purity. Also, promising differential analyses will be possible.Furthermore, moving to higher centre-of-mass energies allows to probe new productionmechanism compositions due to the increasing dominance of the gluon fusion.In this thesis the recipe for a powerful reconstruction in the (cid:96) + jets channel was givenand will help to establish the next-generation t ¯ t spin correlation measurement.180 anksagung Alles selbst geschrieben? Aber klar doch. Im Ernst. Man hat ja schon so seine Anspr¨uchean sich selbst. Nur heißt ”selbst geschrieben“ nicht gleich ”alles alleine hinbekommen“.Denn w¨ahrend so einer Promotion muss der Mensch auch mal Maschine sein. Und daf¨urbraucht er Unterst¨utzung von Freunden und Familie. Die hatte ich stets, und daf¨ur binich sehr dankbar. Ob vor Ort oder aus der Ferne, sie waren immer f¨ur mich da. Auch,wenn wir uns in der letzten Zeit nicht allzu oft sehen konnten. Ganz besonderer Dankgilt meiner Familie. Ihr habt mir immer Kraft und Unterst¨utzung gegeben. Mit demLemmer-Clan im R¨ucken kann einem nichts passieren!Meine wunderbaren Mitbewohner Joana, Johannes, Steffi, Jarka, Alex, Pascal, Kon-rad, Andrea und Jan machten meine WGs zu einem richtigen zu Hause. W¨art ihr nichtgewesen, h¨atte ich in der ein oder anderen schweren Stunde vielleicht schon die Koffergepackt. Ich hatte das Gl¨uck, in G¨ottingen nicht nur richtig schnell Anschluß, sondernauch richtig gute Freunde zu finden. Danke Folkert, Jan, Joana, Johanna, Lena, Lena,Maike, Marie, Sebastian und Steffi f¨ur die wundersch¨one Zeit!Ohne Doktorvater kein Doktorsohn. Ich danke Arnulf Quadt, dass er mich als Quer-einsteiger in die Teilchenphysik aufgenommen hat. Er hat mich viel gelehrt und ließ mirdie Freiheit, sowohl am CERN unter optimalen Bedingungen zu forschen als auch beimScience Slam einen etwas unkonventionellen Weg der Wissensvermittlung zu gehen.Kevin Kr¨oninger und Lisa Shabalina sorgten f¨ur eine ausgezeichnete t¨agliche Betreu-ung, waren jederzeit hilfsbereit zur Stelle und hielten die Stimmung im B¨uro immer ganzweit oben. Mein Institut war w¨ahrend all der Zeit eine richtig starke Truppe und großeUnterst¨utzung. Heidi Afshar, Heike Ahrens, Lucie Hamdi, Gabriela Herbold, BernadetteTyson und Christa Wohlfahrt sorgten daf¨ur, dass hinter den Kulissen alles reibungslosablief. J¨org Meyer hielt die IT am Leben und versorgte mich mit wertvollem Wissen zurPhysik und zu Computing.Meinen Freunden und Kollegen aus dem II. Physikalischen Institut danke ich f¨ur allden klugen Rat (ob zur Physik oder dar¨uber hinaus) und die Unterhaltung bei derArbeit und vor allem auch drumherum. Danke insbesondere an Andrea, Anna, Chris,181
0. Summary, Conclusion and Outlook
Cora, Fabian, Johannes, Katha, Martina, Matze und Philipp f¨ur Bier, Wein und fr¨ohlichsein! Danke Andrea, du gute Seele des Instituts, dass du mir dabei geholfen hast, meineAnalyse die ersten Schritte gehen zu lassen.Mein herzlicher Dank gilt auch der gesamten ATLAS Kollaboration und dem LHCBeschleuniger-Team. Nur durch eine gewaltige Teamleistung unter Mitwirkung vielerfleißiger Menschen konnte ein so großartiges Experiment entstehen.Nicht zu vergessen sind meine alten Lehrmeister aus Schul- und Uni-Zeiten. Ganzbesonders danken m¨ochte ich Volker Kreuter, Joachim Steinm¨uller und Volker Metag,vor allem auch f¨ur die vielen M¨oglichkeiten, die sie mir geboten haben.Danke Andrea, Joahnnes, Katha und Kevin, dass ihr euch am Ende nochmal Zeitgenommen habt, ¨uber meine Arbeit zu schauen.Ein herzliches Dankesch¨on auch an die Unfallchirurgie der Uniklinik G¨ottingen, diemich in der Nacht vor Abgabe dieser Arbeit noch zusammengen¨aht hat. Gut gemacht,sieht fast wieder so aus wie vorher.Zu guter Letzt auch ein großer Dank an dich, Dana. Du bereicherst t¨aglich mein Leben.Ich bin froh, dass ich dich an meiner Seite habe.182 ibliography [1] D0 Collaboration,
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EtCone20, 64Event probability, 91Factorization scale, 19Factorization theorem, 19Fake lepton background, 77Fake leptons, 62FASTJET, 66FCal , see Forward Calorimeter 57FCNC , see Flavour changing neutralcurrent 13Fermion, 6Fine-structure constant, 12Flavour changing neutral currents, 13Forward Calorimeter, 57Gauge boson, 6Gell-Mann-Nishijima formula, 14GIM mechanism, 13Glashow-Weinberg-Salam model, 11Good Runs List, 73GRL , see Good Runs List 73GWS model , seeGlashow-Weinberg-Salammodel 11Hadronic Calorimeter, 56Hadronic compensation, 66Hadronic decay, 22Hadronic End Cap, 57Hadronisation, 2HCal , see Hadronic Calorimeter 56HEC , see Hadronic End Cap 57Helicity basis, 33HERAPDF, 16Higgs boson, 14High Level Trigger, 59HLT , see High Level Trigger 59IBL , see Insertable B-Layer 55ID , see Inner Detector 55Inner Detector, 55Insertable B-Layer, 55Jet, 61 Jet charge, 179Jet Vertex Fraction, 68JVF , see Jet Vertex Fraction 68Kaluza-Klein gravitons, 43KLFitter, 90LB , see Luminosity block 73Leading pole approximation, 30LEIR, 52Lepton, 7LHC, 51LHCb, 52LHCf, 52LINAC II, 52LINAC III, 52LUCID, 59Luminosity block, 73Matrix method, 77Maximal basis, 36MBTS , see Minimum Bias TriggerScintillators 60MDT , see Monitored drift tubes 58Minimum Bias Trigger Scintillators, 60Missing Transverse Momentum, 71MOEDAL, 52Monitored drift tubes, 58MS , see Muon spectrometer 57MSTW, 16MuId algorithm, 65Muon spectrometer, 57Neutrino oscillation, 15Neutrino Weighting, 24NNPDF, 16Nuisance parameters, 120Off-diagonal basis, 35Optical theorem, 60Particle level, 61Parton, 16Parton Distribution Function, 16Parton level, 61206
NDEX
PDF , see Parton DistributionFunction 16Periodic System of Elements, 5Pile-up, 74Prior, 120Proton Synchroton, 52PS , see Proton Synchrotron 52Pseudo-rapidity, 54PtCone30, 64QCD , see Quantum Chromodynamics7QGP , see Quark Gluon Plasma 52Quantum Chromodynamics, 7Quantum Electrodynamics, 11Quark, 6quark confinement, 10Quark Gluon Plasma, 52Randall-Sundrum model, 43Reconstruction level, 61Region Of Interest, 59Renormalization scale, 10Resistive place chambers, 58ROI , see Region Of Interest 59Roman Pot, 60RPC , see Resistive place chambers 58Run number, 73S-Ratio, 38Scale factor, 64SCT , see Silicon Microstrip Tracker 55Silicon Microstrip Tracker, 55SM , see Standard Model of ParticlePhysics 1Spin analysing power, 31 Standard Model of Particle Physics, 1Stop, 43Strong Interaction, 7Super Proton Synchrotron, 52Supersymmetry, 15SUSY , see Supersymmetry 15Tag-and-probe method, 64Tagging fraction, 80TDAQ , see Trigger and DataAcquisition 59Tevatron, 6TFTool, 92TGC , see Thin-gap chamber 58Thin-gap chamber, 58Top quark, 6Top squark, 43Topological Clusters, 66TOTEM, 52Transfer function, 90Transition Radiation Tracker, 55Trigger and Data Acquisition, 59Trigger menu, 59TRT , see Transition RadiationTracker 55Two-Higgs-Doublet Model, 43 V − A structure, 12Van der Meer scan, 60Weak hypercharge, 13Working point method, 91Yukawa coupling, 14ZDC , see Zero-Degree Calorimeters 59Zero-Degree Calorimeters, 59 207 ppendices Spin Correlation Matrices
The spin correlation matrix (cid:98) C i ¯ i was introduced in Section 2.4. In [168] these matriceswere calculated at leading-order QCD. They are expressed in terms of the top productionvelocity β and production angle Θ in the t ¯ t centre-of-mass frame. The special propertyof the off-diagonal basis is the (cid:98) C element, which is equal to unity, independent of theproduction kinematics. q ¯ q → t ¯ t off-diagonal basis : (cid:98) C = β cos Θ2 − β sin Θ − β sin Θ2 − β sin Θ
00 0 1 helicity basis : (cid:98) C = (2 − β ) sin Θ2 − β cos Θ − sin Θ γ (2 − β cos Θ) − β cos Θ2 − β cos Θ − sin 2Θ γ (2 − β cos Θ) Θ+ β cos Θ2 − β cos Θ gg → t ¯ t . Spin Correlation Matrices helicity basis: (cid:98) C = − β (2 − β )(1+sin Θ)1+2 β cos Θ − β (1+sin Θ) − β sin 2Θ cos Θ γ (1+2 β cos Θ − β (1+sin Θ)) − β − β (1+sin Θ)1+2 β cos Θ − β (1+sin Θ) − β sin 2Θ cos Θ1+2 β cos Θ − β (1+sin Θ) − β ( β (1+sin Θ)+(sin / ) β cos Θ − β (1+sin Θ) Used Datasets . Used Datasets D a t a s e t F un c t i o n d a t a117 T e V . ∗ . ph y s i c s E ga mm a . m e r g e . N T U P T O PE L . ∗ p p p D a t a f o r e + j e t s d a t a117 T e V . ∗ . ph y s i c s M u o n s . m e r g e . N T U P T O P M U . ∗ p p p D a t a f o r µ + j e t s m c T e V . . T M c A t N l o J i mm y . m e r g e . N T U P T O P . e s s r r p t ¯ t s i g n a l ( S M s p i n c o rr . ) m c T e V . . T M c A t N l o J i mm y . m e r g e . N T U P T O P . e s s r r p t ¯ t s i g n a l ( n o s p i n c o rr . ) m c T e V . . A l p g e n J i mm y W e nu N p p t . m e r g e . N T U P T O P . e s s r r p W → e ν + p m c T e V . . A l p g e n J i mm y W e nu N p p t . m e r g e . N T U P T O P . e s s r r p W → e ν + p m c T e V . . A l p g e n J i mm y W e nu N p p t . m e r g e . N T U P T O P . e s s r r p W → e ν + p m c T e V . . A l p g e n J i mm y W e nu N p p t . m e r g e . N T U P T O P . e s s r r p W → e ν + p m c T e V . . A l p g e n J i mm y W e nu N p p t . m e r g e . N T U P T O P . e s s r r p W → e ν + p m c T e V . . A l p g e n J i mm y W e nu N p p t . m e r g e . N T U P T O P . e s s r r p W → e ν + p m c T e V . . A l p g e n J i mm y W m unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W m unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W m unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W m unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W m unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W m unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W t a unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W t a unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W t a unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W t a unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W t a unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W t a unu N p p t . m e r g e . N T U P T O P . e s s r r p W → µ ν + p m c T e V . . A l p g e n J i mm y W bb F u ll N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + b ¯ b + p m c T e V . . A l p g e n J i mm y W bb F u ll N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + b ¯ b + p m c T e V . . A l p g e n J i mm y W bb F u ll N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + b ¯ b + p m c T e V . . A l p g e n J i mm y W bb F u ll N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + b ¯ b + p m c T e V . . A l p g e n W cc F u ll N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + c ¯ c + p m c T e V . . A l p g e n W cc F u ll N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + c ¯ c + p m c T e V . . A l p g e n W cc F u ll N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + c ¯ c + p m c T e V . . A l p g e n W cc F u ll N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + c ¯ c + p m c T e V . . A l p g e n W c N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + c + p m c T e V . . A l p g e n W c N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + c + p m c T e V . . A l p g e n W c N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + c + p m c T e V . . A l p g e n W c N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + c + p m c T e V . . A l p g e n W c N p p t . m e r g e . N T U P T O P . e s s r r p W → e / µ τ ν + c + p T a b l e B . .: D a t a s e t s u s e d f o r t h e a n a l y s i s ( d a t a , t ¯ t a nd W + j e t s ) . a t a s e t F un c t i o n m c T e V . . A l p g e n J i mm y Z ee bb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → ee + b ¯ b + p m c T e V . . A l p g e n J i mm y Z ee bb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → ee + b ¯ b + p m c T e V . . A l p g e n J i mm y Z ee bb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → ee + b ¯ b + p m c T e V . . A l p g e n J i mm y Z ee bb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → ee + b ¯ b + p m c T e V . . A l p g e n J i mm y Z m u m ubb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → µµ + b ¯ b + p m c T e V . . A l p g e n J i mm y Z m u m ubb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → µµ + b ¯ b + p m c T e V . . A l p g e n J i mm y Z m u m ubb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → µµ + b ¯ b + p m c T e V . . A l p g e n J i mm y Z m u m ubb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → µµ + b ¯ b + p m c T e V . . A l p g e n J i mm y Z t a u t a ubb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → ττ + b ¯ b + p m c T e V . . A l p g e n J i mm y Z t a u t a ubb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → ττ + b ¯ b + p m c T e V . . A l p g e n J i mm y Z t a u t a ubb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → ττ + b ¯ b + p m c T e V . . A l p g e n J i mm y Z t a u t a ubb N p n o fi l t e r . m e r g e . N T U P T O P . e s s r r p Z → ττ + b ¯ b + p m c T e V . . A l p g e n J i mm y Z ee N p p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p m c T e V . . A l p g e n J i mm y Z ee N p p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p m c T e V . . A l p g e n J i mm y Z ee N p p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p m c T e V . . A l p g e n J i mm y Z ee N p p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p m c T e V . . A l p g e n J i mm y Z ee N p p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p m c T e V . . A l p g e n J i mm y Z ee N p p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p m c T e V . . A l p g e n J i mm y Z m u m u N p p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p m c T e V . . A l p g e n J i mm y Z m u m u N p p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p m c T e V . . A l p g e n J i mm y Z m u m u N p p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p m c T e V . . A l p g e n J i mm y Z m u m u N p p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p m c T e V . . A l p g e n J i mm y Z m u m u N p p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p m c T e V . . A l p g e n J i mm y Z m u m u N p p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p m c T e V . . A l p g e n J i mm y Z t a u t a u N p p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p m c T e V . . A l p g e n J i mm y Z t a u t a u N p p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p m c T e V . . A l p g e n J i mm y Z t a u t a u N p p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p m c T e V . . A l p g e n J i mm y Z t a u t a u N p p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p m c T e V . . A l p g e n J i mm y Z t a u t a u N p p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p m c T e V . . A l p g e n J i mm y Z t a u t a u N p p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p T a b l e B . .: D a t a s e t s u s e d f o r t h e a n a l y s i s ( Z + j e t s ) . . Used Datasets D a t a s e t F un c t i o n m c T e V . . A l p g e n J i mm y Z ee N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z ee N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z ee N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z ee N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z ee N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z ee N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ee + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z m u m u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z m u m u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z m u m u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z m u m u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z m u m u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z m u m u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → µµ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z t a u t a u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z t a u t a u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z t a u t a u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z t a u t a u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z t a u t a u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p G e V < m ll < G e V m c T e V . . A l p g e n J i mm y Z t a u t a u N p M ll t o40 p t . m e r g e . N T U P T O P . e s s r r p Z → ττ + p G e V < m ll < G e V m c T e V . . WW H e r w i g . m e r g e . N T U P T O P . e s s r r p WW m c T e V . . ZZ H e r w i g . m e r g e . N T U P T O P . e s s r r p ZZ m c T e V . . W Z H e r w i g . m e r g e . N T U P T O P . e s s r r p W Z m c T e V . . s tt c h a n e nu A ce r M C . m e r g e . N T U P T O P . e s s r r p s i n g l e t o p , t - c h a n , e ν m c T e V . . s tt c h a n m unu A ce r M C . m e r g e . N T U P T O P . e s s r r p s i n g l e t o p , t - c h a n , µ ν m c T e V . . s tt c h a n t a unu A ce r M C . m e r g e . N T U P T O P . e s s r r p s i n g l e t o p , t - c h a n , τ ν m c T e V . . s t s c h a n e nu M c A t N l o J i mm y . m e r g e . N T U P T O P . e s s r r p s i n g l e t o p , s - c h a n , e ν m c T e V . . s t s c h a n m unu M c A t N l o J i mm y . m e r g e . N T U P T O P . e s s r r p s i n g l e t o p , s - c h a n , µ ν m c T e V . . s t s c h a n t a unu M c A t N l o J i mm y . m e r g e . N T U P T O P . e s s r r p s i n g l e t o p , s - c h a n , τ ν m c T e V . . s t W t M c A t N l o J i mm y . m e r g e . N T U P T O P . e s s r r p s i n g l e t o p , W t - c h a n , e ν T a b l e B . .: D a t a s e t s u s e d f o r t h e a n a l y s i s ( Z + j e t s , s i n g l e t o p a ndd i b o s o n ) . a t a s e t F un c t i o n m c T e V . . TT b a r P o w H e g P y t h i a . m e r g e . N T U P T O P . e s r p P o w H e g + P y t h i a m c T e V . . TT b a r P o w H e g J i mm y . m e r g e . N T U P T O P . e s r p P o w H e g + H e r w i g m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n qq N p . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n qq N p . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n qq N p . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n qq N p . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n qq N p I N C . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n l n N p . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n l n N p . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n l n N p . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n l n N p . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d H i K T F a c CT E Q L tt b a r l n l n N p I N C . m e r g e . N T U P T O P . e s r p I S R / F S R up m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n qq N p . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n qq N p . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n qq N p . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n qq N p . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n qq N p I N C . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n l n N p . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n l n N p . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n l n N p . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n l n N p . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . A l p G e n P y t h i a P r a d L o K T F a c CT E Q L tt b a r l n l n N p I N C . m e r g e . N T U P T O P . e s r p I S R / F S R d o w n m c T e V . . TT b a r P o w H e g P y t h i a P . m e r g e . N T U P T O P . e s r p P o w H e g + P y t h i a ( P t un e ) m c T e V . . TT b a r P o w H e g P y t h i a P n o C R . m e r g e . N T U P T O P . e s r p P o w H e g + P y t h i a ( P , n o C R ) m c T e V . . TT b a r P o w H e g P y t h i a P m p i H i. m e r g e . N T U P T O P . e s r p P o w H e g + P y t h i a ( P , m o r e U E ) m c T e V . . M c A t N l o J i mm y CT tt b a r m ud o w n L e p t o n F il t e r . m e r g e . N T U P T O P . e s r p R e n . / F a c t . S c a l e U p m c T e V . . M c A t N l o J i mm y CT tt b a r m uup L e p t o n F il t e r . m e r g e . N T U P T O P . e s r p R e n . / F a c t . S c a l e D o w n T a b l e B . .: D a t a s e t s u s e d f o r t h e a n a l y s i s ( m o d e lli n g un ce r t a i n t i e s ) . Pretag Yields n jets ≥
4, pretag e + jets µ + jets W +jets (DD/MC) 12 930 ± ± Z +jets (MC) 2860 ± ± ± ± ±
70 2420 ± ±
10 370 ± t ¯ t ) 19 780 ± ± t ¯ t (MC, l+jets) 17 280 ± ± t ¯ t (MC, dilepton) 2360 ±
140 3530 ± ± ± n jets ≥ b -tagging requirement. For data driven backgrounds statisticaluncertainties are quoted. The uncertainties on the cross sections determinethe uncertainties for the Monte Carlo driven backgrounds. 219 KLFitter Likelihood Components
The value of the (logarithm of the) likelihood (LH) of
KLFitter is a useful quantityto judge the quality of an event reconstruction. However, the likelihood is a complexquantity and needs to be understood properly prior to any interpretation.Instead of checking the likelihood for a global event quality, also its individual com-ponents (Breit-Wigner functions for the masses and transfer functions for the energyand momentum resolutions) can be checked. This allows judging the quality of certainobjects.Figure D.1(a) shows the values of the down-type quark transfer function component.In case the down-type quark does not match, the fit needs to vary the energy of thedown-type quark candidate up the the tails to reach a proper t ¯ t event topology. Thiseffect gets even larger in case the whole W boson (including the up-type quark) doesnot match.A similar effect can be observed for the Breit-Wigner function of the hadronic W boson mass (Figure D.1(b)). The gap between the peak at log( LH ) ≈ −
10 and the tailstarting at log( LH ) ≈ −
15 illustrates the interplay of the TFs and the Breit-Wignerfunctions. Due to the narrow width of the Breit-Wigner functions, the fit prefers the topquark and the W boson masses to be on the resonance. As a consequence, the transferfunctions get values off their peak. This is shown in Figure D.1(c), where the sum of theTF values of the two light quarks from the W boson decay plotted for different valuesof the Breit-Wigner component. Far away from the Breit-Wigner peak, the slope of theGaussian TFs dominate the Breit-Wigner peak: while the Breit-Wigner values are faraway from their peak, the TFs stay in theirs.The interplay between the hadronic top quark and W boson mass is shown in FigureD.1(d). Different areas in the plot indicate the misreconstruction of the hadronicallydecaying W boson, the b -jet of the hadronically decaying top quark or of both.How the shape of the likelihood distributions is affected by the (mis)match of certain221 . KLFitter Likelihood Components model partons is shown in figure D. log( TF ( dQ )) -3 -2 -1 0 1 2 N o r m . E n t r i e s -1 alldQ matcheddQ not matchedW matchedW not matched (a) ) ) had. log (BW (W-30 -25 -20 -15 -10 N o r m . E n t r i e s -2 -1 allW matchedW not matched (b) log(TF (uQ)) + log(TF (dQ))-3 -2 -1 0 1 2 3 N o r m . E n t r i e s -1 ) < -10 had. W -11 < log(BW ) < -11 had. W -15 < log(BW ) < -15 had. W -20 < log(BW ) < -20 had. W -30 < log(BW (c) LogLikelihoodComp_BW_Whad-30 -25 -20 -15 -10
LogL i k e li hood C o m p_ B W _ T had -30-25-20-15-10 (d) Figure D.1.: Components of the
KLFitter likelihood. (a) down-type quark TF for(un)matched down-type quark jets and W bosons. (b) Distribution of theBreit-Wigner function of the hadronic W boson. (c) Sum of the logarithm ofthe light quark jets’ TF values for different values of the Breit-Wigner func-tion of the hadronic W boson. (d) Breit-Wigner functions for the hadronictop quark and W boson mass.222 og(Likelihood)-80 -75 -70 -65 -60 -55 -50 -45 -40 N o r m a li z ed E n t r i e s All matched matched lep
All but b matched had
All but bAll but W matchedOnly W matchedAll matched, b-jets interchanged
Figure D.2.: Distribution of the logarithm of the
KLFitter likelihood for the permutationwith the highest event probability. Different (mis)matching scenarios areshown. 223
Down-Type Quark p T Spectrum in
POWHEG + PYTHIA
In contrast to
MC@NLO , POWHEG + PYTHIA is able to properly model the jet p T spectra.Figure E.1 shows the p T spectra of the down-type quark jet. 225 . Down-Type Quark p T Spectrum in
POWHEG + PYTHIA
50 100 150 200 250 E v en t s / DatattW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.7 fb ∫ ≥ ≥ (reco) t dQ p
50 100 150 200 250 p r ed ./ da t a
50 100 150 200 250 E v en t s / DatattW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.7 fb ∫ ≥ + µ ≥ (reco) t dQ p
50 100 150 200 250 p r ed ./ da t a
50 100 150 200 250 E v en t s / DatattW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.7 fb ∫ ≥ ≥ (reco) t dQ p
50 100 150 200 250 p r ed ./ da t a
50 100 150 200 250 E v en t s / DatattW+jetsZ+jetsDibosonSingle topMisid. leptonsUncertainty L dt = 4.7 fb ∫ ≥ + µ ≥ (reco) t dQ p
50 100 150 200 250 p r ed ./ da t a Figure E.1.: p T spectrum of the reconstructed down-type quark jet with the selectionrequirement of n jets ≥ n b-tags = 1 (upper row) and n b-tags ≥ e + jets (left) and the µ + jets channel (right). POWHEG + PYTHIA was used as generator.226
Posterior Distributions of Fit Parameters . Posterior Distributions of Fit Parameters
RemBkg0 2 4 6 8 10 12 × p ( R e m B k g | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_4_el500 1000 1500 2000 2500 3000 p ( W j e t s _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_5_el0 200 400 600 800 1000 p ( W j e t s _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_4_el0 200 400 600 800 1000 1200 1400 1600 p ( Q C D _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_5_el0 200 400 600 800 p ( Q C D _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_4_mu2000 3000 4000 5000 6000 p ( W j e t s _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_5_mu500 1000 1500 2000 p ( W j e t s _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_4_mu500 1000 1500 2000 p ( Q C D _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_5_mu200 400 600 800 p ( Q C D _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
JetScaleNP-4 -2 0 2 4 p ( J e t S c a l e N P | d a t a ) PriorprobabilityPosteriorprobability
Figure F.1.: Prior and posterior distributions for the fit parameters describing the back-ground yields and the jet multiplicity correction for the combination of thedown-type quark analysers.228 emBkg0 2 4 6 8 10 12 × p ( R e m B k g | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_4_el500 1000 1500 2000 2500 3000 p ( W j e t s _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_5_el0 200 400 600 800 1000 p ( W j e t s _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_4_el0 200 400 600 800 1000 1200 1400 1600 p ( Q C D _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_5_el0 200 400 600 800 p ( Q C D _ _ e l | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_4_mu2000 3000 4000 5000 6000 p ( W j e t s _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
Wjets_5_mu500 1000 1500 2000 p ( W j e t s _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_4_mu500 1000 1500 2000 p ( Q C D _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
QCD_5_mu200 400 600 800 p ( Q C D _ _ m u | d a t a ) -3 × PriorprobabilityPosteriorprobability
JetScaleNP-4 -2 0 2 4 p ( J e t S c a l e N P | d a t a ) PriorprobabilityPosteriorprobability
Figure F.2.: Prior and posterior distributions for the fit parameters describing the back-ground yields and the jet multiplicity correction for the combination of the b -quark analysers. 229 Postfit Values of Nuisance Parameters J ES _ E ff e c t i v e N P _ S T A T J ES _ E ff e c t i v e N P _ M O D E L1 J ES _ E ff e c t i v e N P _ D E T J ES _ E ff e c t i v e N P _ M I XE D J ES _ E t a I n t e r c a li b r a t i on_ T o t a l S t a t J ES _ E t a I n t e r c a li b r a t i on_ T heo r y J ES _ R e l a t i v e N on C l o s u r e_ M C J ES _ P il eup_ O ff s e t M u_up J ES _ P il eup_ O ff s e t N PV J ES _ c l o s eb y J ES _ f l a v o r _ c o m p J ES _ f l a v o r _ r e s pon s e J ES _ B J e s U n c b t ag_b r ea k t ag_b r ea k t ag_b r ea k c t ag_b r ea k c t ag_b r ea k c t ag_b r ea k c t ag_b r ea k m i s t ag j v f s f e l _ t r i g_ S F m u_ t r i g_ S F e l _ I D _ S F m u_ I D _ S F e l _ r e c o_ S F m u_ r e c o_ S F e l _ E _ sc a l e W J e t s _bb4 W J e t s _bb5 W J e t s _bb cc W J e t s _ c W J e t s _ c -3-2-10123 N P V a l ue Figure G.1.: Postfit values of the nuisance parameters (black lines) for the down-typequark combination. The grey bars areas behind the lines show the expecteduncertainties on the nuisance parameters. 231 . Postfit Values of Nuisance Parameters J ES _ E ff e c t i v e N P _ S T A T J ES _ E ff e c t i v e N P _ M O D E L1 J ES _ E ff e c t i v e N P _ D E T J ES _ E ff e c t i v e N P _ M I XE D J ES _ E t a I n t e r c a li b r a t i on_ T o t a l S t a t J ES _ E t a I n t e r c a li b r a t i on_ T heo r y J ES _ R e l a t i v e N on C l o s u r e_ M C J ES _ P il eup_ O ff s e t M u_up J ES _ P il eup_ O ff s e t N PV J ES _ c l o s eb y J ES _ f l a v o r _ c o m p J ES _ f l a v o r _ r e s pon s e J ES _ B J e s U n c b t ag_b r ea k t ag_b r ea k t ag_b r ea k c t ag_b r ea k c t ag_b r ea k c t ag_b r ea k c t ag_b r ea k m i s t ag j v f s f e l _ t r i g_ S F m u_ t r i g_ S F e l _ I D _ S F m u_ I D _ S F e l _ r e c o_ S F m u_ r e c o_ S F e l _ E _ sc a l e W J e t s _bb4 W J e t s _bb5 W J e t s _bb cc W J e t s _ c W J e t s _ c N P V a l ue -3-2-10123 Figure G.2.: Postfit values of the nuisance parameters (black lines) for the b -quark com-bination. The grey bars areas behind the lines show the expected uncer-tainties on the nuisance parameters.232 Most Significant Uncertainties
This section contains lists of the most important uncertainties used as nuisance param-eters for the combinations of the individual spin analysers.NP relative change of f SM btag/break8 + 1.8 %JES/BJES + 1.3 %JES/Intercal TotalStat + 1.1 %JES/EffectiveNP Model1 + 1.0 %JES/EffectiveNP Stat1 + 0.9 %Table H.1.: Most significant nuisance parameters (in terms of change of f SM ) for thecombined fit of down-type quark analysers.NP relative change of f SM JES/RelativeNonClosureMC11b − − − f SM ) for thecombined fit of b -quark analysers. 233 . Most Significant Uncertainties NP relative change of fit uncertaintyJES/FlavorComp − − − − − − − − − b -quark analysers.234 ∆ φ for Different MC Generators The SM prediction of the ∆ φ distributions varies for different generators. To separatethe effects of reconstruction and modeling of the hard scattering process, parton levelresults are shown in Figure I.1 without any selection cuts. The same plots without thesample of uncorrelated t ¯ t pairs and a zoomed ratio is shown in Figure I.2. It is knownthat the top quark p T distributions, which varies for the generators (see Figure 8.8).Hence, the samples were reweighted to the top quark p T spectrum of MC@NLO . The resultis shown in Figure I.3. The good agreement between the top quark p T spectrum of POWHEG + PYTHIA and
MC@NLO + HERWIG is visible as well as a residual effect, independentof the top quark p T . 235 . ∆ φ for Different MC Generators (l,d) f D N o r m . E n t r i e s MC@NLOPowHeg+HerwigPowHeg+PythiaMC@NLO (uncorr.) (l, dQ) f D R a t i o (a) (l,b) f D N o r m . E n t r i e s MC@NLOPowHeg+HerwigPowHeg+PythiaMC@NLO (uncorr.) (l, bQ) f D R a t i o (b) Figure I.1.: SM predictions of the ∆ φ distributions on parton level using different MCgenerators. The charged lepton was used as spin analyzer together with the(a) down-type quark and (b) b -quark. The variations are compared to the MC@NLO sample using uncorrelated t ¯ t pairs. (l,d) fD N o r m . E n t r i e s MC@NLO
PowHeg+Herwig
PowHeg+Pythia (l, dQ) f D R a t i o (a) (l,b) fD N o r m . E n t r i e s MC@NLOPowHeg+HerwigPowHeg+Pythia (l, bQ) f D R a t i o (b) Figure I.2.: SM predictions of the ∆ φ distributions on parton level using different MCgenerators. The charged lepton was used as spin analyzer together with the(a) down-type quark and (b) b -quark.236 (l,d) fD N o r m . E n t r i e s MC@NLOPowHeg+Herwig (rew.)PowHeg+Pythia (rew.) (l, dQ) f D R a t i o (a) (l,b) fD N o r m . E n t r i e s MC@NLOPowHeg+Herwig (rew.)PowHeg+Pythia (rew.) (l, bQ) f D R a t i o (b) Figure I.3.: SM predictions of the ∆ φ distributions on parton level using different MCgenerators. The charged lepton was used as spin analyzer together withthe (a) down-type quark and (b) b -quark. The samples were reweighted tomatch the top quark p T spectrum of MC@NLO . 237
Alternative t ¯ t Modeling
Table J.1 shows several “what if?” scenarios. The modifications are applied to thepseudo data. The effects on the fitted f SM results are shown.Change in Pseudo Data Fitted f SM down-type quark b -quark— 1.00 1.00Replacing MC@NLO with
POWHEG + HERWIG p T to measured spectrum 1.17 0.76Reweighting PDF from CT10 to HERAPDF 1.09 0.83Table J.1.: Effects of changes on the default signal MC. The quoted results of f SM wereobtained by fitting to pseudo data created with the default MC and appliedchanges.The changes must be read as ’If the data would be more like the suggested change, thefollowing results for f SM are expected’. It is remarkable that all tested changes wouldexplain a larger f SM for the down-type quark and a lower for the b -quark. This meanson the other hand: replacing the fitting templates (and not the pseudo data) with amodified version would lead to a lower f SM for the down-type quark and a higher forthe b -quark when fitting the data. 239 Jet Charge
Jets consist of tracks leaving signatures in the ID. There are several options to assign acharge to a jet. For example, the charge of the track with the highest p T can be chosen.A track is part of a jet if ∆ R (track, jet) < .
25. It is also possible to create a weightedcharge using all tracks and their momentum contribution to the total jet momentum.The weighted jet charge is determined via q jet = (cid:80) i q i (cid:12)(cid:12)(cid:12) (cid:126)j · (cid:126)p i (cid:12)(cid:12)(cid:12) k (cid:80) i (cid:12)(cid:12)(cid:12) (cid:126)j · (cid:126)p i (cid:12)(cid:12)(cid:12) k (K.1)using the jet momentum vector (cid:126)j , the momentum vectors (cid:126)p i of all tracks of the jet, thetrack charges q i and a weighting factor k . The weighting factor was set to 0.5 as it wasdone in [142].Tracks taken into account for the charge determination need to pass certain qualitycriteria. These are the following: • Transverse momentum of the track must be at least 1 GeV. • The absolute value of the impact parameter in the transverse plane, d , must notbe larger than 2 mm. • The absolute value of the distance to the primary vertex in z-direction, z , multi-plied with the sine of the track angle to the z-axis must not be larger than 10 mm( | z · sin ( θ ) | ≤ • The track fit quality must be sufficient ( χ / nDOF ≤ . • The track must have at least one hit in the pixel detector. 241 . Jet Charge • The track must have at least six hits in the silicon tracker.In the case where a jet contains no track, a jet charge of zero is assigned.The separation of up and anti-up quarks was demonstrated in Figure 10.4. Anothertest is supposed to check the correct assignment of a b -jet to the leptonically decayingtop quark. The charges of the b -jet and the charged lepton should be of opposite sign.The distributions of the squared sum of the lepton and the presumed b -jet from theleptonically decaying top is shown in Figure K. A preference of opposite sign chargesin case of a matched jet is visible. Finally, a test concerning the correct reconstruction ) lep. b + q lep. (q-1 0 1 2 3 4 5 N o r m . E n t r i e s quark matched lep b quark not matched lep b Jet Charge Via MaxPtTrackCharge (a) ) lep. b + q lep. (q0 1 2 3 4 5 N o r m . E n t r i e s quark matched lep b quark not matched lep b Jet Charge Via Weighted Jet Charge (b)
Figure K.1.: Squared sum of the charges of the charged lepton and the b -quark from theleptonically decaying top quark using (a) the charge of the jet track withthe highest p T and (b) the weighted charge using all tracks.of the hadronically decaying top quark was made. The quantity χ ≡ ( q dQ + q uQ ) · ( q dQ + q uQ − q had.b ) is plotted in Figure K.2. It uses the charges of the light up- anddown-type jets as well as the charge of the b -jet of the hadronically decaying top quark.A correct assignment of all jets and a correct jet charge determination should lead to amaximized χ . This is due to the same sign of the light jet charges and the opposite signof the light jet and b -jet charges. For fully matched hadronically decaying top quarks atrend to high values of χ is clearly visible.242 had. bQ - q dQ + q uQ )*(q dQ + q uQ (q-1 0 1 2 3 4 5 6 7 N o r m . E n t r i e s Decay Products Matched had t Decay Products Not Matched had t Jet Charge Via MaxPtTrackCharge (a) ) had. bQ - q dQ + q uQ )*(q dQ + q uQ (q-1 0 1 2 3 4 5 6 7 N o r m . E n t r i e s Decay Products Matched had t Decay Products Not Matched had t Jet Charge Via Weighted Jet Charge (b)
Figure K.2.: Distribution of the quantity ( q dQ + q uQ ) · ( q dQ + q uQ − q had.b ) involving thecharges of the light up- and down-type quark jet and the b -jet from of thehadronically decaying top quark. The used jet charges were (a) the chargeof the jet track with the highest p T and (b) the weighted charge using alltracks. 243 oris Lemmer Curriculum Vitae
Date of Birth 16.06.1984Place of Birth GießenNationality German
Education
CERN , Meyrin , Research Stay.2010-2014
Georg-August-Universität , Göttingen , PhD Studies, Supervisor: Prof. A. Quadt.
Qualification: Dr. rer. nat.
Winter2007/08
Umeå Universitet , Umeå , Semester Abroad.2006-2008
Justus-Liebig-Universität , Gießen , Mathematics Diploma Program, Minor subjects:Numerical Analysis and Experimental Physics.2004-2009
Justus-Liebig-Universität , Gießen , Physics Diploma Program, Minor subjects:Mathematics and Nuclear Physics.
Qualification: Dipl.-Phys.
Landgraf-Ludwigs-Gymnasium , Gießen , High School.
Qualification: Allgemeine Hochschulreife
Diploma Thesis
Title
Measurement of the Excitation Function of ω Photoproduction on Carbon andNiobium
Referees Prof. V. Metag, Prof. U. MoselDescription Indications for possible mass and width modifications of the ω meson embeddedin a hadronic medium were investigated by measuring the photoproduction crosssection as a function of the incoming beam energy. Springstraße 2 – 37077 Göttingen B [email protected] • ˝˝
Referees Prof. V. Metag, Prof. U. MoselDescription Indications for possible mass and width modifications of the ω meson embeddedin a hadronic medium were investigated by measuring the photoproduction crosssection as a function of the incoming beam energy. Springstraße 2 – 37077 Göttingen B [email protected] • ˝˝ wards and Scholarships Teaching
Lecturer / Dozent , Teilchenphysik , B. Lemmer, CERN.
As part of the German Teachers Programme
WS 2010/11
TA / Übungsgruppenleiter , Einführung in die Kern- und Teilchenphysik , A. Frey,Göttingen University.SS 2010
TA / Übungsgruppenleiter , Atomphysik , A. Quadt, Göttingen University.WS 2009/10
Lecturer / Dozent , Physik für Medizinisch-Techn. Assistenten , M. Thiel / B.Lemmer, Gießen University Medical Center.WS 2009/10
Lab Assistant / Praktikumsbetreuer , Physikpraktikum für Mediziner , R.Novotny, Gießen University.SS 2009
Lab Assistant / Praktikumsbetreuer , Physikpraktikum für Mediziner , R.Novotny, Gießen University.SS 2009
TA / Korrekteur , Analysis II für Lehramtsstudenten , M. Väth, Gießen University.WS 2008/09
Lab Assistant / Praktikumsbetreuer , Physikpraktikum für Mediziner , R.Novotny, Gießen University.WS 2008/09
TA / Korrekteur , Mathematik für Physiker I , B. Lani-Wayda, Gießen University.SS 2008
TA / Übungsgruppenleiter , Theorie der höheren Mechanik , U. Mosel, GießenUniversity.SS 2008
TA / Übungsgruppenleiter , Analysis II , B. Lani-Wayda, Gießen University.WS 2007/08
TA / Korrekteur , Analysis I , B. Lani-Wayda, Gießen University.SS 2007
TA / Tutor , Experimentalphysik II , V. Metag, Gießen University.SS 2007
TA / Korrekteur , Analysis IIIB , B. Lani-Wayda, Gießen University.WS 2006/07
TA / Korrekteur , Analysis IIIA , B. Lani-Wayda, Gießen University.
Schools, Conferences and Workshops
Springstraße 2 – 37077 Göttingen B [email protected] • ˝˝
Springstraße 2 – 37077 Göttingen B [email protected] • ˝˝
011 CTEQ Summer School, Madison2010 Talk at the Spring Meeting of the German Physical Society, Bonn2009 Quarkonium Working Group Meeting, Nara2009 Talk at the Lecture Week of the Graduate School Hadrons and Nuclei, Copenhagen2009 Talk at the Crystal Ball Collaboration Meeting, Edinburgh2009 Summer School of the German National Academic Foundation, Rot an der Rot2009 Lecture Week of the Graduate School Hadrons and Nuclei, Turin2009 DESY Summer School, Hamburg
Publications { The ATLAS Collaboration, Measurements of spin correlation in top-antitop quarkevents from proton-proton collisions at √ s = 7 TeV using the ATLAS detector,Submitted to Phys. Rev. D { J. Erdmann et al., A likelihood-based reconstruction algorithm for top-quarkpairs and the KLFitter framework, Nuclear Instruments and Methods in PhysicsResearch Section A 748 (2014) 18 { B. Lemmer, Bis(s) ins Innere des Protons, Springer Spektrum (2013) { M. Thiel et al., In-medium modifications of the ω meson near the productionthreshold, Eur. Phys. J. A (2013) 49: 132 { The ATLAS Collaboration, Observation of Spin Correlation in top/antitop Eventsfrom pp Collisions at √ s = 7 TeV Using the ATLAS Detector, Phys. Rev. Lett.108 (2012) 212001 { V. Metag et al., Experimental approaches for determining in-medium propertiesof hadrons from photo-nuclear reactions, Progress in Particle and Nuclear Physics,Volume 67, Issue 2 (2012) 530
Miscellaneous
Languages
German
Native speaker
English
Fluent
Swedish
Basic communication skills
Springstraße 2 – 37077 Göttingen B [email protected] • ˝˝