Measurement of the Jet Energy Scale in the CMS experiment ... - IIHE
Measurement of the Jet Energy Scale in the CMS experiment ... - IIHE
Measurement of the Jet Energy Scale in the CMS experiment ... - IIHE
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Vrije Universiteit BrusselFaculteit WetenschappenDepartement NatuurkundeIsfahan University <strong>of</strong>TechnologyDepartment <strong>of</strong> Physics<strong>Measurement</strong> <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong><strong>in</strong> <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> with <strong>the</strong> FirstLHC Proton CollisionsMaryam Ze<strong>in</strong>aliPromotoresPr<strong>of</strong>. Dr. Jorgen D’HondtPr<strong>of</strong>. Dr. Mansour HaghighatThesis submitted <strong>in</strong> order to obta<strong>in</strong> <strong>the</strong>academic degree Doctor <strong>of</strong> ScienceSeptember 2011
ContentsIntroduction 11 The Standard Model <strong>of</strong> Particle Physics 31.1 Elementary Particles and Their Interactions . . . . . . . . . . . . . . . 41.1.1 Electroweak Theory . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Strong Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 81.2 Top Quark Sector <strong>of</strong> <strong>the</strong> Standard Model . . . . . . . . . . . . . . . . . 91.2.1 Indirect Evidence for <strong>the</strong> Existence <strong>of</strong> <strong>the</strong> Top Quark . . . . . . 91.2.2 Indirect Constra<strong>in</strong>ts on <strong>the</strong> Mass <strong>of</strong> <strong>the</strong> Top Quark . . . . . . . 101.2.3 Direct Searches <strong>of</strong> <strong>the</strong> Top Quark and <strong>the</strong> Discovery Era . . . . 111.2.4 Top Production and Decay at Hadron Colliders . . . . . . . . . 131.3 Problems <strong>of</strong> <strong>the</strong> Standard Model . . . . . . . . . . . . . . . . . . . . . . 132 The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHC 152.1 The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.1 The Proton Accelerator Cha<strong>in</strong> . . . . . . . . . . . . . . . . . . . 162.1.2 Particle Detectors at <strong>the</strong> LHC . . . . . . . . . . . . . . . . . . . 192.2 The <strong>CMS</strong> Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.1 Tracker System . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Calorimetry System . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.3 Muon System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.4 Trigger<strong>in</strong>g and Data Acquisition . . . . . . . . . . . . . . . . . . 273 Object Reconstruction 293.1 Electron Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.1 Electron Cluster<strong>in</strong>g Algorithms . . . . . . . . . . . . . . . . . . 303.1.2 Electron Track Reconstruction . . . . . . . . . . . . . . . . . . . 303.1.3 Electron Momentum Determ<strong>in</strong>ation . . . . . . . . . . . . . . . . 313.2 <strong>Jet</strong> Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 <strong>Jet</strong> Cluster<strong>in</strong>g Algorithms . . . . . . . . . . . . . . . . . . . . . 333.2.2 <strong>Jet</strong> Reconstruction Performance . . . . . . . . . . . . . . . . . . 353.2.3 <strong>Jet</strong> <strong>Energy</strong> Calibration . . . . . . . . . . . . . . . . . . . . . . . 363.2.4 <strong>Jet</strong> <strong>Energy</strong> Resolution . . . . . . . . . . . . . . . . . . . . . . . 413.2.5 <strong>Jet</strong>-Electron Clean<strong>in</strong>g . . . . . . . . . . . . . . . . . . . . . . . . 443.3 MET Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45iii
IntroductionThe Standard Model <strong>of</strong> particle physics describes <strong>the</strong> elementary particles and <strong>the</strong>ir<strong>in</strong>teractions, with <strong>the</strong> exception <strong>of</strong> gravity which is not formulated with<strong>in</strong> <strong>the</strong> model.The model has been tested <strong>in</strong> diverse <strong>experiment</strong>s and its parameters have been measuredwith great precision <strong>in</strong> high energy physics laboratories. However, <strong>the</strong>re is stilla miss<strong>in</strong>g particle which has not yet been detected, be<strong>in</strong>g <strong>the</strong> Standard Model Higgsboson. Also <strong>the</strong>re is evidence which make <strong>the</strong> Standard Model only an effective <strong>the</strong>oryand motivate <strong>the</strong> need to have new physics beyond it. Therefore, efforts are made tobuild <strong>the</strong> most fundamental <strong>the</strong>ory describ<strong>in</strong>g nature. To search for <strong>the</strong> Higgs particleand test <strong>the</strong> new physics ideas beyond <strong>the</strong> Standard Model, <strong>the</strong> Large Hadron Collider(LHC) was constructed.The LHC, located at <strong>the</strong> CERN laboratory near Geneva, has been collid<strong>in</strong>g beams<strong>of</strong> protons at a center <strong>of</strong> mass energy <strong>of</strong> 7 TeV s<strong>in</strong>ce March 2010. The proton-protoncollisions are detected with <strong>the</strong> use <strong>of</strong> detectors <strong>in</strong>stalled at <strong>the</strong> <strong>in</strong>teraction po<strong>in</strong>ts, suchas <strong>the</strong> <strong>CMS</strong> (Compact Muon Solenoid) <strong>experiment</strong>, which is one <strong>of</strong> <strong>the</strong> two generalpurposedetectors at <strong>the</strong> LHC. The energy and momentum <strong>of</strong> particles which are produced<strong>in</strong> <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> proton-proton <strong>in</strong>teractions, are measured by <strong>the</strong> detectors.While <strong>the</strong> energy <strong>of</strong> leptons can be measured with high precision, <strong>the</strong> measurement <strong>of</strong><strong>the</strong> energy <strong>of</strong> <strong>the</strong> quarks, which hadronize to make so-called jets, is quite challeng<strong>in</strong>g.While strategies are developed to calibrate <strong>the</strong> jet energies with<strong>in</strong> <strong>the</strong> <strong>CMS</strong> collaboration,search<strong>in</strong>g for new physics benefits from a more precise measurement <strong>of</strong> <strong>the</strong> jetenergies than <strong>the</strong> ones obta<strong>in</strong>ed by apply<strong>in</strong>g <strong>the</strong> basel<strong>in</strong>e methods. In this analysis, amethod to calibrate <strong>the</strong> energy <strong>of</strong> <strong>the</strong> jets is applied on <strong>the</strong> proton collisions conta<strong>in</strong><strong>in</strong>gtwo top quarks <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state, pp → t¯t. The high rate <strong>of</strong> <strong>the</strong> production <strong>of</strong> <strong>the</strong>top quark pairs <strong>in</strong> proton-proton collisions provides a huge amount <strong>of</strong> statistics. Thisallows <strong>the</strong> top quark pairs to be used as a tool to perform <strong>the</strong> calibration studies.This <strong>the</strong>sis is organized as follows. In Chapter 1, <strong>the</strong> Standard Model <strong>of</strong> particlephysics is reviewed with emphasis on <strong>the</strong> top quark sector followed by a motivationto construct <strong>the</strong> LHC. The description <strong>of</strong> <strong>the</strong> Large Hadron Collider and <strong>the</strong> <strong>CMS</strong><strong>experiment</strong>, is given <strong>in</strong> Chapter 2. The <strong>CMS</strong> <strong>experiment</strong> detects <strong>the</strong> emerg<strong>in</strong>g particlesfrom <strong>the</strong> proton collisions through electronic signals which are subsequently used toreconstruct <strong>the</strong> physics objects as described <strong>in</strong> Chapter 3. The simulation <strong>of</strong> <strong>the</strong> protoncollisions is described <strong>in</strong> Chapter 4 followed by <strong>the</strong> description <strong>of</strong> <strong>the</strong> procedure to selecttop quark events on which <strong>the</strong> calibration method is based. The method to estimate<strong>the</strong> residual jet energy corrections is described <strong>in</strong> detail <strong>in</strong> Chapter 5. The estimatedresults <strong>of</strong> <strong>the</strong> method are obta<strong>in</strong>ed based on both <strong>the</strong> simulated proton collisions aswell as <strong>the</strong> real proton collisions collected <strong>in</strong> 2010 by <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>. F<strong>in</strong>ally, a1
2 INTRODUCTIONsummary <strong>of</strong> <strong>the</strong> analysis and a conclusion are given <strong>in</strong> Chapter 6.
Chapter 1The Standard Model <strong>of</strong> ParticlePhysicsElementary particle physics aims to understand <strong>the</strong> structure <strong>of</strong> matter at <strong>the</strong> microscopiclevel and expla<strong>in</strong> <strong>the</strong> <strong>in</strong>teractions between <strong>the</strong>m. The most thorough <strong>the</strong>orycurrently known which describes particles and forces is <strong>the</strong> Standard Model <strong>of</strong> elementaryparticle physics [1, 2]. Despite <strong>of</strong> some unanswered questions, <strong>the</strong> StandardModel has been able to describe <strong>the</strong> <strong>experiment</strong>al data s<strong>in</strong>ce it was proposed <strong>in</strong> sixties.Three out <strong>of</strong> four <strong>in</strong>teraction forces which have been observed <strong>in</strong> nature are formulatedwith<strong>in</strong> <strong>the</strong> Standard Model. The particles predicted with<strong>in</strong> <strong>the</strong> model have alreadybeen discovered <strong>in</strong> research laboratories, except for a miss<strong>in</strong>g particle which is <strong>the</strong>mass generator <strong>of</strong> <strong>the</strong> model. The heaviest discovered elementary particle is <strong>the</strong> topquark which was observed <strong>in</strong> 1995 [3–7] at <strong>the</strong> Tevatron collider [8]. The parameters<strong>of</strong> <strong>the</strong> model have been measured with great precision and efforts are made <strong>in</strong> order todiscover <strong>the</strong> last predicted particle <strong>of</strong> <strong>the</strong> model, be<strong>in</strong>g <strong>the</strong> Higgs particle. The mostrecent <strong>experiment</strong> which has been constructed to search for <strong>the</strong> miss<strong>in</strong>g piece <strong>of</strong> <strong>the</strong>Standard Model as well as provid<strong>in</strong>g answers to o<strong>the</strong>r formulated <strong>the</strong>oretical modelsbeyond <strong>the</strong> Standard Model, is <strong>the</strong> Large Hadron Collider (LHC). The two <strong>in</strong>com<strong>in</strong>gproton beams are currently collided at <strong>the</strong> center <strong>of</strong> mass energy <strong>of</strong> 7 TeV to provide<strong>the</strong> <strong>experiment</strong>al data for fur<strong>the</strong>r analysis.In this chapter, <strong>the</strong> Standard Model <strong>of</strong> particle physcis is reviewed. The way <strong>of</strong> formulat<strong>in</strong>g<strong>the</strong> particles and forces <strong>in</strong>to <strong>the</strong> model is described <strong>in</strong> Section 1.1. Specialemphasis is given to <strong>the</strong> top quark sector <strong>of</strong> <strong>the</strong> Standard Model s<strong>in</strong>ce <strong>the</strong> top quarkis <strong>the</strong> heaviest elementary particle discovered so far with dist<strong>in</strong>ct features compared to<strong>the</strong> o<strong>the</strong>r known elementary particles. Hence, Section 1.2 is dedicated to <strong>the</strong> physics<strong>of</strong> <strong>the</strong> top quark and elaborates its properties.3
4 CHAPTER 1: The Standard Model <strong>of</strong> Particle Physics1.1 Elementary Particles and Their Interactions1.1.1 Electroweak TheoryThe fundamental particles, namely quarks and leptons, which are fermions with half<strong>in</strong>tegersp<strong>in</strong> are represented by Dirac sp<strong>in</strong>ors ψ, that fulfill <strong>the</strong> Dirac equation 1iγ µ ∂ µ ψ = 0.In <strong>the</strong> presence <strong>of</strong> an external electromagnetic field A µ , <strong>the</strong> fermions couple to <strong>the</strong>quantum <strong>of</strong> <strong>the</strong> field, which is called <strong>the</strong> photon γ, and <strong>the</strong>ir <strong>in</strong>teraction are describedby a Lagrangian term expressed asL <strong>in</strong>te.m = ∑ f¯ψ f iγ µ D µ ψ f ,where <strong>the</strong> sum is over all fermions and D µ is <strong>the</strong> covariant derivative def<strong>in</strong>ed as D µ =∂ µ + ieQA µ . In <strong>the</strong> language <strong>of</strong> group <strong>the</strong>ory, <strong>the</strong> dirac sp<strong>in</strong>ors are representations <strong>of</strong><strong>the</strong> U e.m (1) group s<strong>in</strong>ce <strong>the</strong> Lagrangian is <strong>in</strong>variant under <strong>the</strong> gauge transformationψ → e ieQ ψ.Accord<strong>in</strong>g to Noe<strong>the</strong>rs’s <strong>the</strong>orem, <strong>the</strong>re is a conservation law that can be associatedto this gauge symmetry. Therefore <strong>the</strong> electric charge Q, which is expressed <strong>in</strong> units<strong>of</strong> quantum number e, is constant dur<strong>in</strong>g <strong>the</strong> electromagnetic <strong>in</strong>teractions. The Diracsp<strong>in</strong>ors associated to <strong>the</strong> elementary particles and <strong>the</strong>ir correspond<strong>in</strong>g charge are listed<strong>in</strong> Table 1.1.As expressed <strong>in</strong> Table 1.1, <strong>the</strong> leptons as well as <strong>the</strong> quarks appear <strong>in</strong> three generationswhose constituents are equal <strong>in</strong> all quantum numbers but differentiated <strong>in</strong> terms<strong>of</strong> mass. The most stable particles are <strong>the</strong> ones listed <strong>in</strong> <strong>the</strong> first generation for each <strong>of</strong><strong>the</strong> two category <strong>of</strong> fermions. The elementary particles <strong>in</strong>teract with <strong>the</strong> electromagneticfield via <strong>the</strong>ir charge, hence no <strong>in</strong>teraction vertex is present between <strong>the</strong> photonand <strong>the</strong> neutr<strong>in</strong>o <strong>in</strong> <strong>the</strong> Standard Model s<strong>in</strong>ce <strong>the</strong> neutr<strong>in</strong>os are neutral fermions.In addition to <strong>the</strong> electromagnetic <strong>in</strong>teractions which are formulated <strong>in</strong> a simple Lagrangianterm and can fully describe <strong>the</strong> <strong>in</strong>teraction <strong>of</strong> <strong>the</strong> charged particles, <strong>the</strong>re isano<strong>the</strong>r k<strong>in</strong>d <strong>of</strong> <strong>in</strong>teraction describ<strong>in</strong>g <strong>the</strong> phenomena like beta decay p → n + e + ν e ,which corresponds to <strong>the</strong> physics processes conta<strong>in</strong><strong>in</strong>g a neutr<strong>in</strong>o as one <strong>of</strong> <strong>the</strong> components<strong>of</strong> <strong>the</strong> <strong>in</strong>teraction. This k<strong>in</strong>d <strong>of</strong> <strong>in</strong>teraction, known as <strong>the</strong> weak <strong>in</strong>teraction,conta<strong>in</strong>s a heavy particle as propagator. Because <strong>of</strong> <strong>the</strong> ra<strong>the</strong>r massive property <strong>of</strong><strong>the</strong> propagator <strong>of</strong> <strong>the</strong> weak <strong>in</strong>teraction, <strong>the</strong>y are difficult to be produced dur<strong>in</strong>g an<strong>in</strong>teraction. Therefore, <strong>the</strong> weak <strong>in</strong>teractions are rare with<strong>in</strong> <strong>the</strong> Standard Model.In order to describe <strong>the</strong> physics <strong>of</strong> <strong>the</strong> weak <strong>in</strong>teractions, an appropriate term shouldbe added to <strong>the</strong> Lagrangian. Accord<strong>in</strong>g to <strong>the</strong> <strong>experiment</strong>al data, where only lefthandedfermions contribute to <strong>the</strong> weak <strong>in</strong>teraction, <strong>the</strong> weak current ¯ψ νe γ µ (1 − γ 5 )ψ e1 The mass term is dropped from <strong>the</strong> Dirac equation for simplicity.
CHAPTER 1: The Standard Model <strong>of</strong> Particle Physics 5fermionsfirst generationQ( )ψe -1ψ νe 0leptonssecond generation ( )ψµ -1ψ νµ 0third generationfirst generation( )ψτ -1ψ ντ 0(ψu)+23ψ d − 1 3quarkssecond generation (ψc)+23ψ s − 1 3third generation(ψt)+23ψ b − 1 3Table 1.1: The elementary particles, be<strong>in</strong>g quarks and leptons, predicted by <strong>the</strong> StandardModel and observed <strong>in</strong> <strong>the</strong> <strong>experiment</strong>. The electric charge Q, associated to eachparticle is also listed.
6 CHAPTER 1: The Standard Model <strong>of</strong> Particle Physicswas proposed, as <strong>the</strong> operator (1 − γ 5 ) is <strong>the</strong> chirality projector that only keeps <strong>the</strong>left-handed part <strong>of</strong> <strong>the</strong> Dirac sp<strong>in</strong>or, for example one can haveP L ψ e ≡ (1 − γ5 )ψ e = ψ e,L .2In order to keep <strong>the</strong> Lagrangian <strong>in</strong>variant under <strong>the</strong> gauge transformation, <strong>the</strong> SU(2) Lsymmetry group is proposed for which <strong>the</strong> new doublets Ψ, constructed from <strong>the</strong> lefthandedDirac sp<strong>in</strong>ors, are <strong>the</strong> representations <strong>of</strong> <strong>the</strong> gauge group which are expressedexplicitly for <strong>the</strong> first generation <strong>of</strong> <strong>the</strong> fermions asΨ Lepton =( )ψνe,Lψ e,Land Ψ Quark =( )ψu,L,ψ d,Land <strong>the</strong> Pauli matrices τ i , are <strong>the</strong> generators <strong>of</strong> <strong>the</strong> group. The three massless gaugebosons which are <strong>in</strong>troduced <strong>in</strong> this group are not able to expla<strong>in</strong> <strong>the</strong> <strong>experiment</strong>aldata, be<strong>in</strong>g <strong>the</strong> three massive particles W ± and Z. Hence <strong>the</strong> idea <strong>of</strong> unification<strong>of</strong> <strong>the</strong> electromagnetism and weak <strong>in</strong>teration was made to comb<strong>in</strong>e <strong>the</strong> two type <strong>of</strong>physics <strong>in</strong>teractions <strong>in</strong>to one unified <strong>the</strong>ory called electroweak which is expla<strong>in</strong>ed <strong>in</strong><strong>the</strong> follow<strong>in</strong>g.In order to comb<strong>in</strong>e <strong>the</strong> electromagnetic and weak <strong>in</strong>teractions <strong>in</strong>to a s<strong>in</strong>gle formalism,a new symmetry group U(1) Y , called hypercharge, is <strong>in</strong>troduced and <strong>the</strong> Lagrangianis modified and expressed asL electroweak = ∑ [¯Ψf iγ µ D µ Ψ f + ¯ψ]f,R iγ µ D µ ψ f,R + K<strong>in</strong>etic terms.,fwhere <strong>the</strong> sum is over all fermions and R refers to <strong>the</strong> right-handed Dirac sp<strong>in</strong>orswhich are obta<strong>in</strong>ed by apply<strong>in</strong>g <strong>the</strong> right-handed chirality operator (1+γ5 ), on <strong>the</strong> Dirac2sp<strong>in</strong>ors. The def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> covariant derivative D µ , is different depend<strong>in</strong>g on whichoperand it is act<strong>in</strong>g. The covariant derivatives <strong>of</strong> doublets and s<strong>in</strong>glets are respectivelyexpressed asτ iD µ = ∂ µ + ig Y Y B µ + ig W2 W µ i ,andD µ = ∂ µ + ig Y Y B µ ,where Y is <strong>the</strong> hypercharge operator. The constants g Y and g W represent respectively<strong>the</strong> coupl<strong>in</strong>g strengths <strong>of</strong> <strong>the</strong> hypercharge and weak <strong>in</strong>tractions. The three W i and oneB gauge bosons are comb<strong>in</strong>ed to express <strong>the</strong> observed physical <strong>in</strong>termediate bosons γ,W ± and Z, as expla<strong>in</strong>ed below.As a consequence <strong>of</strong> <strong>in</strong>troduc<strong>in</strong>g <strong>the</strong> new hypercharge symmetry group, <strong>the</strong> electroweakpart <strong>of</strong> <strong>the</strong> Standard Model can be described on <strong>the</strong> basis <strong>of</strong> <strong>the</strong> SU(2) L ⊗U(1) Y gaugegroup. Therefore, <strong>the</strong> left-handed fermions <strong>in</strong> doublets and right-handed fermions <strong>in</strong>s<strong>in</strong>glets are representations <strong>of</strong> this new symmetry group. The quantum numbers whichare assigned to <strong>the</strong> hypercharge operator Y , as well as <strong>the</strong> weak isosp<strong>in</strong> operators,be<strong>in</strong>g <strong>the</strong> Pauli matrices, are determ<strong>in</strong>ed <strong>in</strong> such a way that <strong>the</strong> appropriate electric
CHAPTER 1: The Standard Model <strong>of</strong> Particle Physics 7charge <strong>of</strong> fermions is recovered. As an example, only <strong>the</strong> first generation <strong>of</strong> <strong>the</strong> leptonsis considered. The Lagrange equation can be expanded as(¯ψνe,L ¯ψ)e,L γµ[τ 3 ] ( )− g Y Y B µ − g W2 W µ3 ψ νe,L+ψ ¯ψ[e,R γ µ − g Y Y B µ]ψ e,R ,e,Lwhere only <strong>the</strong> part responsible to re-produce <strong>the</strong> electromagnetic <strong>in</strong>teraction is kept.The two gauge bosons B µ and Wµ 3 are comb<strong>in</strong>ed to make <strong>the</strong> new physics gauge bosonsA µ and Z µ , accord<strong>in</strong>g to( ) ( ) ( )Aµ cosθW s<strong>in</strong> θ=W BµZ µ − s<strong>in</strong> θ W cos θ W Wµ3 ,where θ W is <strong>the</strong> We<strong>in</strong>berg mix<strong>in</strong>g angle. Therefore <strong>the</strong> Lagrangian expression is simplifiedand given by(¯ψνe,L ¯ψ)e,L γ µ A µ[−g Y cosθ W Y − 1 ( ) 1 0 ] (2 g ψνe,LW s<strong>in</strong> θ W0 −1With <strong>the</strong> assumption <strong>of</strong> e = g Y cosθ W = g W s<strong>in</strong> θ W and assign<strong>in</strong>g( )ψνe,LY = − 1 ( ) ( ) 1 0 ψνe,L,ψ e,L 2 0 1 ψ e,LandY ψ e,R = −1ψ e,R ,<strong>the</strong> Lagrangian expression would be simplified to+eA µ[¯ψe,L γ µ ψ e,L + ¯ψ e,R γ µ ψ e,R]= +eA µ ¯ψe γ µ ψ e .ψ e,L)+ ¯ψ e,R γ µ A µ[−g Y cosθ W YTherefore, <strong>the</strong> electromagnetic <strong>in</strong>teraction term with <strong>the</strong> right electric charge is recoveredgiven that a correct value for <strong>the</strong> hypercharge quantum number is chosen.The problem which arises is that <strong>the</strong> observed weak bosons are massive while an explicitmass term cannot be <strong>in</strong>troduced <strong>in</strong>to <strong>the</strong> Lagrangian as it spoils <strong>the</strong> gauge symmetry.The Higgs mechanism [9–11], which is described <strong>in</strong> detail <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g, is responsiblefor giv<strong>in</strong>g mass to <strong>the</strong> particles <strong>in</strong> <strong>the</strong> Standard Model.Electro-weak Symmetry Break<strong>in</strong>gS<strong>in</strong>ce <strong>the</strong> observed weak bosons are massive, <strong>the</strong> electroweak symmetry must be broken.The formalism <strong>of</strong> symmetry break<strong>in</strong>g is <strong>in</strong>troduced with <strong>the</strong> use <strong>of</strong> <strong>the</strong> Higgsmechanism. A new doublet conta<strong>in</strong><strong>in</strong>g scalar fields is <strong>in</strong>troduced as( ) h1φ =h 2whose hypercharge quantum number is chosen to be + 1 . The correspond<strong>in</strong>g Lagrangian2term which describes <strong>the</strong> dynamic <strong>of</strong> <strong>the</strong> scalar field φ is written asL Higgs = D µ φ † D µ φ − V (φ),]ψ e,R .
8 CHAPTER 1: The Standard Model <strong>of</strong> Particle Physicswhere V (φ) is <strong>the</strong> scalar potential and is given by −µ 2 φ 2 + λφ 4 . Thy symmetry isspontaneously broken when <strong>the</strong> scalar field takes a value <strong>of</strong> v at <strong>the</strong> m<strong>in</strong>imum <strong>of</strong> <strong>the</strong>potential, def<strong>in</strong>ed as v 2 ≡ µ2. Due to <strong>the</strong> quantum fluctuations around <strong>the</strong> m<strong>in</strong>imumλ<strong>of</strong> <strong>the</strong> potential, one can have( )0φ = ,v + h(x)where h(x) corresponds to <strong>the</strong> Higgs boson field, for which <strong>the</strong>re is no <strong>experiment</strong>alevidence yet. Expand<strong>in</strong>g <strong>the</strong> Lagrangian expression around <strong>the</strong> vacuum state, <strong>the</strong> massterms for <strong>the</strong> physical observables W ± and Z are obta<strong>in</strong>ed to bem W = 1 2 g Wv; m Z = 1 √gY 2 2+ g2 W v,while <strong>the</strong> photon rema<strong>in</strong>s massless, as aimed for. All <strong>the</strong> parameters g W , g Y and v arefree parameters <strong>of</strong> <strong>the</strong> model 2 .The Higgs mechanism results <strong>in</strong> giv<strong>in</strong>g mass to <strong>the</strong> weak gauge bosons <strong>of</strong> <strong>the</strong> StandardModel. As a result, <strong>the</strong> fermions are still massless because an explicit mass term wouldviolate <strong>the</strong> gauge <strong>in</strong>variance <strong>of</strong> <strong>the</strong> electroweak Lagrangian as it mixes <strong>the</strong> left andright-handed components <strong>of</strong> <strong>the</strong> Dirac sp<strong>in</strong>orsm f ¯ψf ψ f = m f[¯ψf,L ψ f,R + ¯ψ f,R ψ f,L].Therefore <strong>the</strong> procedure <strong>of</strong> giv<strong>in</strong>g mass to <strong>the</strong> fermions proceeds by <strong>in</strong>troduc<strong>in</strong>g <strong>the</strong>Yukawa <strong>in</strong>teraction terms which make use <strong>of</strong> <strong>the</strong> Higgs doublet <strong>in</strong> order to construct<strong>the</strong> gauge <strong>in</strong>variant expressions out <strong>of</strong> Dirac sp<strong>in</strong>ors. For example, <strong>the</strong> first generation<strong>of</strong> <strong>the</strong> leptons acquire mass when <strong>in</strong>troduc<strong>in</strong>g <strong>the</strong> follow<strong>in</strong>g gauge <strong>in</strong>variant termsL Y ukawa = −G e[¯ΨL φψ R + ¯ψ R φ † Ψ L],where G e is <strong>the</strong> Yukawa coupl<strong>in</strong>g constant for <strong>the</strong> electron field. The Yukawa coupl<strong>in</strong>gconstants which are <strong>in</strong>troduced to <strong>the</strong> <strong>the</strong>ory through L Y ukawa , are aga<strong>in</strong> arbitraryparameters <strong>of</strong> <strong>the</strong> Standard Model.As a consequence, <strong>the</strong> Higgs particle couples to a fermion with a strength proportionalto <strong>the</strong> mass <strong>of</strong> that fermion.1.1.2 Strong InteractionsIn addition to <strong>the</strong> electromagnetic and weak <strong>in</strong>teractions <strong>of</strong> <strong>the</strong> particles with<strong>in</strong> <strong>the</strong>Standard Model, <strong>the</strong>re is ano<strong>the</strong>r k<strong>in</strong>d <strong>of</strong> <strong>in</strong>teraction, called <strong>the</strong> strong <strong>in</strong>teraction,which are specific to <strong>the</strong> quarks carry<strong>in</strong>g color charges. There are three color quantumnumbers which resulted <strong>in</strong> propos<strong>in</strong>g a new gauge group SU(3) c to be responsible for2 The three best measured electroweak parameters, be<strong>in</strong>g <strong>the</strong> electromagnetic coupl<strong>in</strong>g constant α,<strong>the</strong> Fermi constant G F and <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Z boson m Z , are used to determ<strong>in</strong>e <strong>the</strong>se three parametersg W , g Y and v. The parameter µ, represent<strong>in</strong>g <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Higgs boson rema<strong>in</strong>s underterm<strong>in</strong>ed <strong>in</strong><strong>the</strong> <strong>the</strong>ory.
CHAPTER 1: The Standard Model <strong>of</strong> Particle Physics 9<strong>the</strong> strong <strong>in</strong>teractions. The quarks are triplets under <strong>the</strong> gauge transformations <strong>of</strong><strong>the</strong> SU(3) c symmetry group and <strong>the</strong> eight gauge boson, G a , which are <strong>in</strong>troducedaccord<strong>in</strong>gly are <strong>the</strong> massless gluons, <strong>the</strong> <strong>in</strong>termediate bosons <strong>of</strong> <strong>the</strong> strong <strong>in</strong>teractions.The covariant derivative which conta<strong>in</strong>s <strong>the</strong> <strong>in</strong>formation <strong>of</strong> <strong>the</strong> strong <strong>in</strong>teraction is<strong>the</strong>n expressed asD µ = ∂ µ + ig Y Y B µ + ig Wτ i2 W i µ + ig Sλ a2 Ga µ,where <strong>the</strong> g S , be<strong>in</strong>g ano<strong>the</strong>r free parameter <strong>of</strong> <strong>the</strong> model, is <strong>the</strong> strong coupl<strong>in</strong>g constantand λ a are <strong>the</strong> Gelman matrices which are <strong>the</strong> generators <strong>of</strong> <strong>the</strong> color SU(3) cgroup.The Higgs mechanism and <strong>the</strong> Yukawa <strong>in</strong>teractions are as before and <strong>the</strong> SU(3) c rema<strong>in</strong>sunbroken after <strong>the</strong> electroweak symmety break<strong>in</strong>g.1.2 Top Quark Sector <strong>of</strong> <strong>the</strong> Standard Model1.2.1 Indirect Evidence for <strong>the</strong> Existence <strong>of</strong> <strong>the</strong> Top QuarkBefore its discovery <strong>in</strong> 1995, <strong>the</strong> top quark was predicted by <strong>the</strong> Standard Model. Inorder to obta<strong>in</strong> a renormalisable gauge <strong>the</strong>ory, <strong>the</strong> anomalies arris<strong>in</strong>g from <strong>the</strong> so-calledtriangle diagrams <strong>of</strong> which an example can be found <strong>in</strong> Figure 1.1, should cancel. S<strong>in</strong>ceeach triangle is proportional to c f A Q2 f , where Q f is <strong>the</strong> charge <strong>of</strong> fermion and c f Ais <strong>the</strong>axial coupl<strong>in</strong>g <strong>of</strong> <strong>the</strong> weak neutral current, <strong>the</strong> total anomaly can be obta<strong>in</strong>ed byN∑=i=1( 12 × (0)2 − 1 2 × (−1)2 + 1 2 × 3 × (+2 3 )2 − 1 2 × 3 × (−1 3 )2 ),where N denotes an equal number <strong>of</strong> lepton and quark doublets. Therefore <strong>in</strong> order tohave a vanish<strong>in</strong>g anomaly, it was required that <strong>the</strong> quark generations come <strong>in</strong> pairs tocompensate <strong>the</strong> divergencies from <strong>the</strong> fermion loops.Figure 1.1: The triangle diagram conta<strong>in</strong><strong>in</strong>g a fermion loop which produces divergences<strong>in</strong> <strong>the</strong> quantum field <strong>the</strong>ory.Ano<strong>the</strong>r <strong>in</strong>direct evidence for <strong>the</strong> existence <strong>of</strong> <strong>the</strong> third generation <strong>of</strong> quarks, hence<strong>the</strong> top quark as one <strong>of</strong> <strong>the</strong> components, was to provide a natural way to suppress
10 CHAPTER 1: The Standard Model <strong>of</strong> Particle Physics<strong>the</strong>oretically flavour chang<strong>in</strong>g neutral current phenomenon (FCNC) which is not <strong>experiment</strong>allyobserved. Based on <strong>the</strong> GIM mechanism [12] which was orig<strong>in</strong>ally proposedfor two quark generations, <strong>the</strong> same argument which forbids flavour chang<strong>in</strong>gtransitions among down-type quarks, applies for three quark doublets.Also <strong>the</strong>re is lots <strong>of</strong> evidence for <strong>the</strong> existence <strong>of</strong> <strong>the</strong> top quark. There have been somedetailed studies <strong>of</strong> <strong>the</strong> Zb¯b vertex us<strong>in</strong>g <strong>the</strong> data collected at <strong>the</strong> e − e + colliders LEPand SLC, result<strong>in</strong>g <strong>in</strong> <strong>the</strong> measurement <strong>of</strong> <strong>the</strong> isosp<strong>in</strong> <strong>of</strong> <strong>the</strong> b quark [13]. It has beenmeasured [14] that <strong>the</strong> b quark has a left-handed third component <strong>of</strong> <strong>the</strong> isosp<strong>in</strong> <strong>of</strong> − 1 2which implies that <strong>the</strong> b quark must have a weak isosp<strong>in</strong> partner. In o<strong>the</strong>r words, <strong>the</strong>top quark must exists as a counterpart <strong>of</strong> <strong>the</strong> isosp<strong>in</strong> doublet <strong>of</strong> <strong>the</strong> b quark.1.2.2 Indirect Constra<strong>in</strong>ts on <strong>the</strong> Mass <strong>of</strong> <strong>the</strong> Top QuarkAll electroweak quantities <strong>of</strong> <strong>the</strong> Standard Model depend, at lead<strong>in</strong>g order, on threeparameters, namely two gauge coupl<strong>in</strong>g constants g Y , g W and vacuum expectation value<strong>of</strong> <strong>the</strong> Higgs boson field v. At higher order <strong>of</strong> corrections, two extra parameters, be<strong>in</strong>g<strong>the</strong> top quark mass m t and <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Higgs boson m H , contribute to <strong>the</strong> radiativeloop corrections and <strong>the</strong>refore can affect <strong>the</strong> values <strong>of</strong> <strong>the</strong> electroweak observables.Hence, <strong>the</strong> precision electroweak measurements can provide <strong>in</strong>direct constra<strong>in</strong>ts on <strong>the</strong>mass <strong>of</strong> both <strong>the</strong> top quark and <strong>the</strong> Higgs boson.While <strong>the</strong> dependency <strong>of</strong> <strong>the</strong> electroweak observables such as <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Z bosonon <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark is quadratic, <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Higgs boson appears <strong>in</strong>to <strong>the</strong>radiative corrections through logarithmic terms. Therefore, <strong>the</strong> <strong>in</strong>ferred constra<strong>in</strong>ts onm t are stronger than those on m H and that is why <strong>the</strong>re has been a good prediction on<strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark before its discovery. A global fit <strong>of</strong> <strong>the</strong> Standard Model to<strong>the</strong> precision data successfully predicts a value for <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark as [15, 16]m t = 179.4 12.1−9.2 GeV 3 ,which is <strong>in</strong> good agreement with <strong>the</strong> measured value <strong>of</strong> <strong>the</strong> top quark [17]m t = 173.2 ± 0.9 GeV.The successful prediction <strong>of</strong> <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark provided by <strong>the</strong> precision electroweakdata <strong>in</strong>dicates <strong>the</strong> predictive power <strong>of</strong> <strong>the</strong> radiative corrections which persuadesone to obta<strong>in</strong> constra<strong>in</strong>ts on <strong>the</strong> o<strong>the</strong>r parameters <strong>of</strong> <strong>the</strong> model, such as Higgs bosonmass. As already mentioned, because <strong>of</strong> <strong>the</strong> logarithmic dependence <strong>of</strong> <strong>the</strong> radiativecorrections on <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Higgs boson, it is not possible to derive str<strong>in</strong>gent predictionson <strong>the</strong> Higgs boson mass. In <strong>the</strong> follow<strong>in</strong>g, <strong>the</strong> results <strong>of</strong> <strong>the</strong> Standard Model fitto <strong>the</strong> precision electroweak data <strong>in</strong> order to extract constra<strong>in</strong>ts on m H are discussed.Us<strong>in</strong>g <strong>the</strong> measured value <strong>of</strong> <strong>the</strong> top quark mass, <strong>the</strong> Standard Model fit to <strong>the</strong> electroweakprecision data is performed <strong>in</strong> order to <strong>in</strong>fer <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Higgs boson [18].The result <strong>of</strong> <strong>the</strong> fit, expressed <strong>in</strong> terms <strong>of</strong> ∆χ 2 , is shown <strong>in</strong> Figure 1.2. The most likelyvalue <strong>of</strong> <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Standard Model Higgs boson is determ<strong>in</strong>ed from <strong>the</strong> m<strong>in</strong>imum3 Throughout <strong>the</strong> <strong>the</strong>sis, <strong>the</strong> natural units are used.
CHAPTER 1: The Standard Model <strong>of</strong> Particle Physics 11<strong>of</strong> <strong>the</strong> ∆χ 2 curve to be m H = 91 +45−32 GeV , which means <strong>the</strong> data prefers a light Higgsboson. The direct searches for <strong>the</strong> Higgs boson at LEP have exculded a Higgs bosonwith a mass below 114.4 GeV [19], <strong>the</strong>refore <strong>the</strong> preferred value is slightly above <strong>the</strong>exclusion limit.Figure 1.2: The <strong>in</strong>direct determ<strong>in</strong>ation <strong>of</strong> <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Higgs boson obta<strong>in</strong>ed from<strong>the</strong> Standard Model fit to <strong>the</strong> electroweak data assum<strong>in</strong>g <strong>the</strong> world average mass <strong>of</strong><strong>the</strong> top quark [18]. Also <strong>the</strong> exclusion region on m H obta<strong>in</strong>ed from direct searches for<strong>the</strong> Higgs boson at LEP is also shown.Figure 1.3 conta<strong>in</strong>s <strong>the</strong> <strong>in</strong>formation <strong>of</strong> <strong>the</strong> direct as well as <strong>the</strong> <strong>in</strong>direct measurements<strong>of</strong> both <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark and <strong>the</strong> W boson. Also <strong>the</strong> l<strong>in</strong>es <strong>in</strong>dicat<strong>in</strong>g<strong>the</strong> various Higgs boson masses are overlaid. The upper limit <strong>of</strong> 1000 GeV on <strong>the</strong> mass<strong>of</strong> <strong>the</strong> Higgs boson comes from <strong>the</strong> <strong>the</strong>oretical calculation [20]. The direct searches for<strong>the</strong> Standard Model Higgs boson at <strong>the</strong> Tevatron, result <strong>in</strong> an exclusion <strong>of</strong> <strong>the</strong> mass <strong>of</strong><strong>the</strong> Higgs particle <strong>in</strong> a narrow band specified on <strong>the</strong> plot [21].As it can be seen from Figure 1.3, <strong>the</strong> direct and <strong>in</strong>direct measurements <strong>of</strong> m t and m Ware <strong>in</strong> good agreement, confirm<strong>in</strong>g that <strong>the</strong> Standard Model <strong>of</strong> particle physics is notobviously wrong. It should also be noted that <strong>the</strong> contours <strong>of</strong> <strong>the</strong> W and top quarkmasses are calculated at 68% confidence level. In order to make stronger limits on <strong>the</strong>measured values, more data should be provided. Therefore, a more precise determ<strong>in</strong>ation<strong>of</strong> ei<strong>the</strong>r <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark or <strong>the</strong> mass <strong>of</strong> <strong>the</strong> W boson, would makeit possible to accept or reject <strong>the</strong> existence <strong>of</strong> <strong>the</strong> Higgs particle with<strong>in</strong> <strong>the</strong> StandardModel.1.2.3 Direct Searches <strong>of</strong> <strong>the</strong> Top Quark and <strong>the</strong> Discovery EraAs mentioned before, <strong>the</strong> top quark was f<strong>in</strong>ally discovered by <strong>the</strong> CDF [22] and D0 [23]<strong>experiment</strong>s at <strong>the</strong> Tevatron. The latest world average value measured for <strong>the</strong> mass <strong>of</strong>
12 CHAPTER 1: The Standard Model <strong>of</strong> Particle Physics80.5July 2011LEP2 and TevatronLEP1 and SLD68% CLm W[GeV]80.480.3 m H[GeV] ∆α114 300 1000155 175 195m t[GeV]Figure 1.3: The direct (dotted ellipse) and <strong>in</strong>direct (solid ellipse) measurements <strong>of</strong> m Wand m t determ<strong>in</strong>de from <strong>the</strong> data collected at <strong>the</strong> Tevatron and LEP colliders. Thesolid l<strong>in</strong>es represent <strong>the</strong> various configurations <strong>of</strong> <strong>the</strong> Higgs boson mass.<strong>the</strong> top quark can be found <strong>in</strong> Figure 1.4. The <strong>in</strong>direct measurements from precisiondata are also shown <strong>in</strong> <strong>the</strong> same plot. Besides <strong>the</strong> good agreement between <strong>the</strong> directand <strong>in</strong>direct measurements <strong>of</strong> <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark, it can be also seen that <strong>the</strong>mass <strong>of</strong> <strong>the</strong> top quark is measured with an uncerta<strong>in</strong>ty <strong>of</strong> 0.5% on <strong>the</strong> mean valuewhich represents a precise measurement <strong>of</strong> <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark. Even with moredata available from <strong>the</strong> LHC, it is possible to improve <strong>the</strong> accuracy on <strong>the</strong> quoted valuefor <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark.In addition to <strong>the</strong> mass, <strong>the</strong> top quark has o<strong>the</strong>r quantum numbers which shouldbe measured directly from <strong>the</strong> data. The electric charge, for example, is one <strong>of</strong> <strong>the</strong>important quantum numbers that must be checked. S<strong>in</strong>ce <strong>the</strong> center <strong>of</strong> mass energyprovided by <strong>the</strong> LEP collider was around 200 GeV, much less than <strong>the</strong> energy neededto produce a pair <strong>of</strong> top quarks via <strong>the</strong> <strong>in</strong>teraction e + e − → γ ∗ → t¯t, it was not possibleto use <strong>the</strong> collected data from <strong>the</strong> electron-positron collider to measure <strong>the</strong> electriccharge associated to <strong>the</strong> top quark. At a proton-proton collider it is still possible tomeasure <strong>the</strong> electric charge. Because <strong>the</strong> electric charge measurement requires <strong>the</strong> topquark to <strong>in</strong>teract with a photon, <strong>the</strong>n it was suggested to search for <strong>the</strong> <strong>in</strong>teractions <strong>in</strong>which a top is produced toge<strong>the</strong>r with a radiative photon [24]. Accord<strong>in</strong>g to <strong>the</strong> studiesperformed based on <strong>the</strong> data accumulated from <strong>the</strong> Tevatron, it has been shown <strong>the</strong>possibility that <strong>the</strong> top quark is an exotic charge Q t = − 4 quark can be ruled out at3about 95% confidence level [25]. Hence <strong>the</strong> top quark which is produced at hadroncolliders and decays to a W boson and a b quark t → Wb, cannot only have Q t = − 4,3but also it is required to have Q t = + 2.3
CHAPTER 1: The Standard Model <strong>of</strong> Particle Physics 13Top-Quark Mass [GeV]CDF 172.5 ± 1.00D∅ 174.9 ± 1.4Average 173.2 ± 0.90χ 2 /DoF: 6.1 / 10LEP1/SLD 172.6 + 13.5 172.6 − 10.4LEP1/SLD/m W/Γ W179.7 + 11.7 179.7 − 8.7160 170 180 190m t[GeV]July 2011Figure 1.4: The latest results <strong>of</strong> <strong>the</strong> direct measurement on <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quarkprovided by both <strong>experiment</strong>s at <strong>the</strong> Tevatron collider toge<strong>the</strong>r with world average <strong>of</strong><strong>the</strong> measured value <strong>of</strong> m t . The <strong>in</strong>direct measurement from <strong>the</strong> Standard Model fit to<strong>the</strong> electroweak data is also quoted.1.2.4 Top Production and Decay at Hadron CollidersAt hadron colliders, top quarks are produced predom<strong>in</strong>antly via <strong>the</strong> strong <strong>in</strong>teraction.Depend<strong>in</strong>g on <strong>the</strong> center <strong>of</strong> mass energy, ei<strong>the</strong>r gluon-gluon fusion or quark-antiquarkannihilation is <strong>the</strong> ma<strong>in</strong> channel which is responsible for top quark pair production.Due to its huge mass, <strong>the</strong> top quark has a very short life time, shorter than <strong>the</strong> timescale <strong>of</strong> <strong>the</strong> hadronization process. As a result, <strong>the</strong> top quark decays before gett<strong>in</strong>ghadronized. Accord<strong>in</strong>g to <strong>the</strong> Cabibbo-Kobayashi-Maskawa (CKM) matrix [26, 27]which accounts for <strong>the</strong> possibility <strong>of</strong> quark mix<strong>in</strong>g states, <strong>the</strong> probability <strong>of</strong> a topquark decay<strong>in</strong>g <strong>in</strong>to its counterpart is close to 100% [28]. Hence top quarks alwaysdecay to a W boson and a b quark, t → Wb. The W boson can subsequently decayto quarks or leptons, which mark <strong>the</strong> various decay channels <strong>of</strong> a t¯t system. This isexpla<strong>in</strong>ed <strong>in</strong> more detail <strong>in</strong> Section 4.2.1.1.3 Problems <strong>of</strong> <strong>the</strong> Standard ModelThe Standard Model <strong>of</strong> particle physics is a self-consistent gauge field <strong>the</strong>ory which is<strong>in</strong> good agreement with <strong>the</strong> observed <strong>experiment</strong>al data. However, as <strong>the</strong>re are so manyfree parameters with<strong>in</strong> <strong>the</strong> <strong>the</strong>ory, whose values cannot <strong>in</strong> pr<strong>in</strong>ciple be determ<strong>in</strong>ed, thismakes it hard to state that <strong>the</strong> Standard Model is <strong>the</strong> most fundamental <strong>the</strong>ory whichdiscribes nature. In o<strong>the</strong>r words, <strong>in</strong> order to understand why <strong>the</strong> masses <strong>of</strong> <strong>the</strong> quarks,leptons and <strong>the</strong> W and Z gauge bosons have <strong>the</strong>ir observed values, <strong>the</strong>re might be a
14 CHAPTER 1: The Standard Model <strong>of</strong> Particle Physics<strong>the</strong>ory beyond <strong>the</strong> Standard Model from which <strong>the</strong> Standard Model Lagrangian canbe derived.Ano<strong>the</strong>r question that is brought to m<strong>in</strong>d is <strong>the</strong> follow<strong>in</strong>g. The mass <strong>of</strong> <strong>the</strong> top quark,which is <strong>the</strong> heaviest observed elementary particle <strong>of</strong> <strong>the</strong> model, is so close to <strong>the</strong> scale<strong>of</strong> <strong>the</strong> electroweak symmetry break<strong>in</strong>g v ∼ 246 GeV . This opens <strong>the</strong> question if <strong>the</strong>top quark becomes massive dur<strong>in</strong>g <strong>the</strong> spontaneous symmetry break<strong>in</strong>g mechanism or<strong>in</strong>teract<strong>in</strong>g with ano<strong>the</strong>r gauge field results <strong>in</strong> giv<strong>in</strong>g a mass to <strong>the</strong> top quark.Also, due to <strong>the</strong> so-called hierarchy problem, <strong>the</strong>re is a huge gap between M Z ∼ 90 GeV ,<strong>the</strong> energy scale <strong>of</strong> <strong>the</strong> weak <strong>in</strong>teraction <strong>of</strong> <strong>the</strong> model, and <strong>the</strong> Plank mass scaleM Pl ∼ 10 19 GeV , which is <strong>the</strong> fundamental scale <strong>of</strong> gravity. Then, <strong>the</strong>re have been proposalsthat <strong>the</strong> current Standard Model is only an effective version <strong>of</strong> a unified <strong>the</strong>orywith an energy scale <strong>of</strong> <strong>the</strong> order <strong>of</strong> TeV. Ano<strong>the</strong>r question which rema<strong>in</strong>s unansweredwith<strong>in</strong> <strong>the</strong> Standard Model is <strong>the</strong> lack <strong>of</strong> a candidate for dark matter. Therefore, withso many open questions <strong>in</strong> <strong>the</strong> model, some new extention to <strong>the</strong> Standard Modelhave been proposed, such as supersymmetry. The M<strong>in</strong>imal Supersymmetric StandardModel (MSSM), predicts a candidate for dark matter and also provides an answer to<strong>the</strong> hierarchy problem. In order to confirm <strong>the</strong> validity <strong>of</strong> such extentions to <strong>the</strong> model,colliders with higher energies are required.Besides <strong>the</strong> above mentioned challenges <strong>of</strong> <strong>the</strong> model, <strong>the</strong>re is ano<strong>the</strong>r important oneto be noted. S<strong>in</strong>ce <strong>the</strong>re has not yet been any <strong>experiment</strong>al evidence for <strong>the</strong> existence<strong>of</strong> <strong>the</strong> Higgs particle, <strong>the</strong> procedure <strong>of</strong> giv<strong>in</strong>g mass to <strong>the</strong> weak bosons is not <strong>experiment</strong>allyconfirmed. Given that a Higgs boson is not discovered, <strong>the</strong> Standard Model<strong>of</strong> particle physics is still correct but needs some mechanism to expla<strong>in</strong> <strong>the</strong> orig<strong>in</strong> <strong>of</strong>mass <strong>in</strong> nature. That is why <strong>the</strong> next generation <strong>of</strong> <strong>the</strong> accelerator mach<strong>in</strong>es, such as<strong>the</strong> LHC, is constructed to run at higher center <strong>of</strong> mass energies. One <strong>of</strong> <strong>the</strong> importantgoals <strong>of</strong> <strong>the</strong> LHC project, <strong>in</strong> addition to provide <strong>in</strong>formation on <strong>the</strong> presence <strong>of</strong> any newphysics beyond <strong>the</strong> Standard Model, is to search for <strong>the</strong> only miss<strong>in</strong>g particle predictedby <strong>the</strong> Standard Model. If a Higgs particle exists with a mass less than 1 TeV, <strong>the</strong>LHC would be able to discover it.
Chapter 2The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHCThe Standard Model <strong>of</strong> particle physics is <strong>the</strong> most precise <strong>the</strong>ory describ<strong>in</strong>g <strong>the</strong> matterand <strong>the</strong> forces, which has been widely tested at high energy physics colliders. Despite<strong>of</strong> <strong>the</strong> good agreement presented between <strong>the</strong> <strong>the</strong>ory and <strong>the</strong> <strong>experiment</strong>, <strong>the</strong>re isstill some part for which <strong>the</strong>re has not yet been any <strong>experiment</strong>al evidence, be<strong>in</strong>g<strong>the</strong> mechanism <strong>of</strong> <strong>the</strong> electroweak symmetry break<strong>in</strong>g. The discovery <strong>of</strong> <strong>the</strong> Higgsboson would confirm <strong>the</strong> way <strong>of</strong> giv<strong>in</strong>g mass to <strong>the</strong> particles described by <strong>the</strong> <strong>the</strong>ory.Therefore, <strong>the</strong> idea <strong>of</strong> construct<strong>in</strong>g <strong>the</strong> Large Hadron Collider (LHC) was <strong>in</strong>troducedto ma<strong>in</strong>ly search for <strong>the</strong> miss<strong>in</strong>g Higgs particle at higher energy scales <strong>of</strong> physics whichhad not yet been reached.The giant LHC mach<strong>in</strong>e is described <strong>in</strong> more detail <strong>in</strong> Section 2.1 while <strong>the</strong> description<strong>of</strong> one <strong>of</strong> <strong>the</strong> general-purpose detectors <strong>of</strong> <strong>the</strong> LHC, namely <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>, isgiven <strong>in</strong> Section 2.2.2.1 The Large Hadron ColliderThe Large Hadron Collider [29] (LHC) is a proton-proton accelerator <strong>in</strong>stalled <strong>in</strong> atunnel <strong>of</strong> 26.7 km circumference which was orig<strong>in</strong>ally built for <strong>the</strong> Large Electron-Positron collider [30] (LEP). The tunnel is constructed underground, on average about100 m below <strong>the</strong> surface and is located on <strong>the</strong> border between Switzerland and France,near Geneva. The LHC, employed by <strong>the</strong> European Organisation <strong>of</strong> Nuclear Research(CERN), is currently runn<strong>in</strong>g at <strong>the</strong> center <strong>of</strong> mass energy <strong>of</strong> √ s = 7 TeV which is afactor <strong>of</strong> two below <strong>the</strong> design value <strong>of</strong> √ s = 14 TeV . Accelerat<strong>in</strong>g beams <strong>of</strong> particlesup to several TeV, <strong>the</strong> LHC has become <strong>the</strong> most powerful collider mach<strong>in</strong>e. The firstproton-proton collisions at a center <strong>of</strong> mass energy <strong>of</strong> 7 TeV happened <strong>in</strong> March 2010.In addition to collid<strong>in</strong>g beams <strong>of</strong> protons, <strong>the</strong> LHC also provides ion-ion collisions [31].Lead nuclei are used as <strong>the</strong> heavy ion <strong>in</strong> <strong>the</strong>se collisions.The idea <strong>of</strong> construct<strong>in</strong>g a hadron collider was approved at 1995 by CERN. It wasdecided to use <strong>the</strong> same pre-accelerat<strong>in</strong>g and underground facilities which had alreadybeen built to serve <strong>the</strong> LEP <strong>experiment</strong> [32]. The choice <strong>of</strong> collid<strong>in</strong>g protons <strong>in</strong>stead<strong>of</strong> electrons is due to some physics reasons. An important source <strong>of</strong> energy loss <strong>of</strong> a15
16 CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHCrotat<strong>in</strong>g particle when mov<strong>in</strong>g around a circle <strong>of</strong> radius R, is synchrotron radiationwhich is expressed as∆E ∝ 1 ( E) 4,R mwhere E is <strong>the</strong> particle energy and m is <strong>the</strong> particle mass. Therefore, it is obvious tha<strong>the</strong>avier particles loose a smaller amount <strong>of</strong> energy compared to less heavier particles.As <strong>the</strong> bend<strong>in</strong>g radius was already fixed by choos<strong>in</strong>g <strong>the</strong> same tunnel as LEP, us<strong>in</strong>gprotons <strong>in</strong>stead <strong>of</strong> electrons would yield to obta<strong>in</strong> a more stable beam <strong>of</strong> rotat<strong>in</strong>gparticles and consume less power.Despite <strong>of</strong> provid<strong>in</strong>g beams <strong>of</strong> particles with less energy loss per turn, proton-protoncolliders have <strong>the</strong>ir own problems as well. S<strong>in</strong>ce protons are composite particles, <strong>the</strong>nto achieve physics goals up to an energy scale <strong>of</strong> s 1 , <strong>the</strong> collider needs to provide acenter <strong>of</strong> mass energy <strong>of</strong> s 2 which must be higher, hence s 2 ≫ s 1 . This is due to that<strong>the</strong> proton constituents carry a fraction <strong>of</strong> <strong>the</strong> energy <strong>of</strong> <strong>the</strong> proton and <strong>the</strong> <strong>in</strong>teractiontakes place between <strong>the</strong>se constituents. The decision <strong>of</strong> operat<strong>in</strong>g <strong>the</strong> LHC at a center<strong>of</strong> mass energy <strong>of</strong> a few TeV is to make it possible to scan <strong>the</strong> energy scale <strong>of</strong> physics<strong>in</strong>teractions up to 1 TeV scale. S<strong>in</strong>ce one <strong>of</strong> <strong>the</strong> ma<strong>in</strong> goals <strong>of</strong> <strong>the</strong> construction <strong>of</strong> <strong>the</strong>LHC is to search for <strong>the</strong> Standard Model Higgs boson, <strong>the</strong> <strong>the</strong>ory implies an upperlimit on <strong>the</strong> mass <strong>of</strong> <strong>the</strong> Higgs boson to be less than one TeV. Hence a hadron colliderwith a center <strong>of</strong> mass energy <strong>of</strong> a few TeV would allow to discover that miss<strong>in</strong>g piece,if it exists.Prior to <strong>the</strong>ir path <strong>in</strong> <strong>the</strong> LHC, <strong>the</strong> proton beams go through <strong>the</strong> CERN pre-acceleratormach<strong>in</strong>es which are expla<strong>in</strong>ed <strong>in</strong> Section 2.1.1. The LHC has various detectors, each<strong>of</strong> which is specialized for particular physics purposes that are discussed briefly <strong>in</strong>Section 2.1.2.2.1.1 The Proton Accelerator Cha<strong>in</strong>Before be<strong>in</strong>g <strong>in</strong>jected <strong>in</strong>to <strong>the</strong> LHC, protons are pre-accelerated through <strong>the</strong> CERNaccelerator complex [33] depicted <strong>in</strong> Figure 2.1.The current acceleration facilities have served for decades when provid<strong>in</strong>g beams <strong>of</strong>electrons and positrons to <strong>the</strong> LEP collider. For <strong>the</strong> LHC operation, <strong>the</strong>y have beenupgraded to provide beams <strong>of</strong> protons for collisions at unprecedent energies.A duoplasmatron, where emitted electrons from a cathode filament ionize <strong>the</strong> hydrogengas, provides <strong>the</strong> source <strong>of</strong> protons for <strong>the</strong> LHC. The produced protons are <strong>the</strong>ntransfered to <strong>the</strong> first stage <strong>of</strong> <strong>the</strong> accelerator cha<strong>in</strong>, <strong>the</strong> l<strong>in</strong>ear accelerator called L<strong>in</strong>ac,which is based on radio frequency technology. In a L<strong>in</strong>ac mach<strong>in</strong>e, <strong>the</strong> accelerationoccurs by oscillat<strong>in</strong>g an electric field with an appropriate frequency. Due to this reason,particles traverse through <strong>the</strong> accelerator discont<strong>in</strong>uously while mak<strong>in</strong>g so-calledbunches, each <strong>of</strong> which consists <strong>of</strong> thousands <strong>of</strong> particles. When leav<strong>in</strong>g <strong>the</strong> L<strong>in</strong>ac, <strong>the</strong>protons reach an energy <strong>of</strong> 50 MeV.Energetic protons are fed to a booster prior to be <strong>in</strong>jected <strong>in</strong>to <strong>the</strong> Proton Synchrotron(PS) for fur<strong>the</strong>r acceleration. The booster, which is <strong>the</strong> first circular accelerator <strong>in</strong> <strong>the</strong>acceleration cha<strong>in</strong>, enhances <strong>the</strong> energy <strong>of</strong> <strong>the</strong> protons to 1.4 GeV. Subsequently, <strong>the</strong>protons are more accelerated while circulat<strong>in</strong>g <strong>in</strong> <strong>the</strong> 630 m circumference <strong>of</strong> <strong>the</strong> PS r<strong>in</strong>g
CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHC 17Figure 2.1: The CERN accelerator complex compris<strong>in</strong>g <strong>the</strong> full particle acceleratorcha<strong>in</strong>. Start<strong>in</strong>g on <strong>the</strong> left hand side, protons are accelerated while mov<strong>in</strong>g through <strong>the</strong>l<strong>in</strong>ear accelerator (L<strong>in</strong>ac), <strong>the</strong> booster, <strong>the</strong> Proton Synchrotron (PS), and <strong>the</strong> SuperProton Synchrotron (SPS). F<strong>in</strong>ally, <strong>the</strong> protons are <strong>in</strong>jected <strong>in</strong>to <strong>the</strong> Large HadronCollider to reach <strong>the</strong> desired energy.to reach an energy <strong>of</strong> 26 GeV, and are <strong>the</strong>n fed to <strong>the</strong> Super Proton Synchrotron (SPS).The SPS, which is <strong>the</strong> f<strong>in</strong>al stage <strong>in</strong> <strong>the</strong> succession <strong>of</strong> <strong>the</strong> pre-accelerator complex, providesprotons with an energy <strong>of</strong> 450 GeV and <strong>in</strong>ject <strong>the</strong>m <strong>in</strong>to <strong>the</strong> LHC mach<strong>in</strong>e forf<strong>in</strong>al acceleration and subsequently collisions.The proton beams f<strong>in</strong>ally enter <strong>the</strong> LHC tunnel which conta<strong>in</strong>s two beam pipes. The<strong>in</strong>jected beams <strong>of</strong> particles move <strong>in</strong>side <strong>the</strong> pipes <strong>in</strong> opposite directions and cross eacho<strong>the</strong>r at four po<strong>in</strong>ts <strong>of</strong> <strong>the</strong> r<strong>in</strong>g, where <strong>the</strong> detectors are located. The beams are guidedto travel on a circular path with <strong>the</strong> use <strong>of</strong> a strong magnetic field, which is obta<strong>in</strong>ed us<strong>in</strong>gsuperconduct<strong>in</strong>g electromagnets. Each superconduct<strong>in</strong>g electromagnet, made fromcoils <strong>of</strong> superconduct<strong>in</strong>g wire, has a length <strong>of</strong> about 15 m and a weight <strong>of</strong> around 27tonnes. To be operational, <strong>the</strong> temperature must be lowered to values <strong>of</strong> <strong>the</strong> order <strong>of</strong>a few Kelv<strong>in</strong>. The dipole magnets could provide a magnetic field <strong>of</strong> about 8.3 T whichkeeps <strong>the</strong> beams to move around <strong>the</strong> r<strong>in</strong>g with a designed energy <strong>of</strong> about 7 TeV perbeam.While more than 1200 dipole magnets are used to make <strong>the</strong> proton beams circulated,an additional 400 quadrupole magnets are used <strong>in</strong> order to make <strong>the</strong> proton beamsfocused. It is important to have very narrow beams when <strong>the</strong>y are cross<strong>in</strong>g, as it <strong>in</strong>creasesan important parameter <strong>of</strong> <strong>the</strong> mach<strong>in</strong>e, which is called <strong>the</strong> lum<strong>in</strong>osity L anddef<strong>in</strong>ed as [34]L = f N AN B4πσ x σ y,where σ x and σ y represent <strong>the</strong> beam pr<strong>of</strong>iles <strong>in</strong> horizontal and vertical directions at<strong>the</strong> <strong>in</strong>teraction po<strong>in</strong>ts. Also, N A and N B are <strong>the</strong> number <strong>of</strong> particles per bunch <strong>in</strong> <strong>the</strong>beams <strong>of</strong> type A and B, respectively. The f parameter is <strong>the</strong> frequency <strong>of</strong> cross<strong>in</strong>g <strong>of</strong><strong>the</strong> bunches <strong>of</strong> <strong>the</strong> two beams. It is obvious that an <strong>in</strong>creased number <strong>of</strong> particles perbunch or more focused beams would result <strong>in</strong>to a larger lum<strong>in</strong>osity.
18 CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHCThe lum<strong>in</strong>osity <strong>of</strong> <strong>the</strong> mach<strong>in</strong>e is important s<strong>in</strong>ce it represents how many physics eventsoccur <strong>in</strong> an specific time period T. It is related to <strong>the</strong> production cross section σ,accord<strong>in</strong>g toN =∫ T0Ldtσ,where ∫ TLdt represents <strong>the</strong> collected data <strong>in</strong> a period <strong>of</strong> time equivalent to T. Hence,0for rare processes such as <strong>the</strong> <strong>in</strong>teractions where a Higgs particle is produced, a higherlum<strong>in</strong>osity would <strong>in</strong>crease <strong>the</strong> probability <strong>of</strong> be<strong>in</strong>g observed <strong>in</strong> <strong>the</strong> <strong>experiment</strong>.The LHC has been designed to operate with a lum<strong>in</strong>osity <strong>of</strong> L ∼ 10 34 cm −2 s −1 which isat least two orders <strong>of</strong> magnitude higher than <strong>the</strong> lum<strong>in</strong>osities which have been reachedby o<strong>the</strong>r hadron colliders. In order to run at this high lum<strong>in</strong>osity, it is required toobta<strong>in</strong> a number <strong>of</strong> protons per bunches to be <strong>of</strong> <strong>the</strong> order <strong>of</strong> 10 11 . The bunches <strong>of</strong>protons are collided every 25 ns, correspond<strong>in</strong>g to a bunch cross<strong>in</strong>g frequency <strong>of</strong> 40MHz. This would <strong>in</strong>troduce <strong>the</strong> need for high-speed processors to select <strong>the</strong> <strong>in</strong>terest<strong>in</strong>gphysics events and also large storage facilities to collect <strong>the</strong> selected data.S<strong>in</strong>ce <strong>the</strong> start <strong>of</strong> <strong>the</strong> LHC operation <strong>in</strong> March 2010, it has been provid<strong>in</strong>g protonprotoncollisions at a center <strong>of</strong> mass energy <strong>of</strong> 7 TeV. On November 2010, <strong>the</strong> mach<strong>in</strong>estopped tak<strong>in</strong>g proton collision data and for <strong>the</strong> next one month, it has been collid<strong>in</strong>gheavy ions. Dur<strong>in</strong>g <strong>the</strong> 2010 runs, <strong>the</strong> LHC mach<strong>in</strong>e delivered about 47.0 pb −1 <strong>of</strong>data to each <strong>of</strong> its <strong>experiment</strong>s [35], while <strong>the</strong> data which have been recorded by <strong>the</strong><strong>CMS</strong> <strong>experiment</strong> correspond to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 43.2 pb −1 , <strong>of</strong> which only36.1 pb −1 is certified to be used for analyses. The distribution <strong>of</strong> <strong>the</strong> total <strong>in</strong>tegratedlum<strong>in</strong>osity delivered by <strong>the</strong> LHC and recorded by <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> as a function<strong>of</strong> time, is shown <strong>in</strong> Figure 2.2. As quoted on <strong>the</strong> plot, <strong>the</strong> total <strong>in</strong>tegrated lum<strong>in</strong>ositycorresponds to <strong>the</strong> proton-proton collisions taken <strong>in</strong> 2010.Figure 2.2: The total <strong>in</strong>tegrated lum<strong>in</strong>osity delivered by <strong>the</strong> LHC and recorded by <strong>the</strong><strong>CMS</strong> <strong>experiment</strong> dur<strong>in</strong>g <strong>the</strong> proton-proton collisons <strong>in</strong> 2010.
CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHC 19The results <strong>of</strong> analyz<strong>in</strong>g <strong>the</strong> 36.1 pb −1 <strong>of</strong> <strong>the</strong> proton-proton collision data <strong>in</strong> <strong>the</strong>framework <strong>of</strong> this <strong>the</strong>sis are given <strong>in</strong> detail <strong>in</strong> Section 5.6.Currently <strong>the</strong> LHC mach<strong>in</strong>e is operat<strong>in</strong>g at a center <strong>of</strong> mass energy <strong>of</strong> 7 TeV and <strong>the</strong><strong>CMS</strong> <strong>experiment</strong> has already been able to record an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> protoncollisions equivalent to more than 2000 pb −1 = 2 fb −1 <strong>in</strong> 2011, hence 55 times moredata than what had been collected <strong>in</strong> 2010. The LHC has been designed to operateat an even higher center <strong>of</strong> mass energy, be<strong>in</strong>g two times larger than <strong>the</strong> one achieved.It has been decided [36, 37] that <strong>the</strong> mach<strong>in</strong>e cont<strong>in</strong>ues runn<strong>in</strong>g at a center <strong>of</strong> massenergy <strong>of</strong> 7 TeV at least until <strong>the</strong> end <strong>of</strong> 2011.2.1.2 Particle Detectors at <strong>the</strong> LHCFour ma<strong>in</strong> <strong>experiment</strong>s detect collisions which occur at <strong>the</strong> LHC. Two <strong>of</strong> <strong>the</strong>m, be<strong>in</strong>g<strong>CMS</strong> (Compact Muon Solenoid) [38] and ATLAS (A Toroidal LHC ApparatuS) [39],are general-purpose detectors. They are designed to address a wide variety <strong>of</strong> physicsquestions. Search<strong>in</strong>g for <strong>the</strong> Higgs particle, explor<strong>in</strong>g physics at TeV scale and look<strong>in</strong>gfor evidence <strong>of</strong> physics beyond <strong>the</strong> Standard Model such as supersymmetry are some<strong>of</strong> <strong>the</strong> goals which are followed by <strong>the</strong> multi-purpose <strong>experiment</strong>s <strong>of</strong> <strong>the</strong> LHC.The o<strong>the</strong>r two, ALICE (A Large Ion Collider Experiment) [40] and LHCb (LargeHadron Collider beauty) [41], are special-purpose detectors. While <strong>the</strong> former is dedicatedto heavy ion collisions, <strong>the</strong> latter is specialized to <strong>the</strong> studies <strong>of</strong> b quark physics.The location <strong>of</strong> <strong>the</strong>se ma<strong>in</strong> <strong>experiment</strong>s around <strong>the</strong> ma<strong>in</strong> r<strong>in</strong>g <strong>of</strong> <strong>the</strong> LHC is shown <strong>in</strong>Figure 2.3.In addition to <strong>the</strong> four ma<strong>in</strong> <strong>experiment</strong>s, <strong>the</strong> LHC hosts two more detectors. Afirst one, which is named LHCf (Large Hadron Collider forward) [42], is <strong>in</strong>tended tostudy <strong>the</strong> particles that are produced <strong>in</strong> <strong>the</strong> forward region <strong>of</strong> <strong>the</strong> <strong>in</strong>teraction po<strong>in</strong>t<strong>of</strong> proton collisions. It consists <strong>of</strong> two pieces which are <strong>in</strong>stalled on ei<strong>the</strong>r side <strong>of</strong> <strong>the</strong>ATLAS detector, 140 m away from <strong>the</strong> <strong>in</strong>teraction po<strong>in</strong>t. A second one, which is namedTOTEM (TOTal Elastic and diffractive cross section <strong>Measurement</strong>) [43], is designedto measure <strong>the</strong> total proton-proton cross section. It is located on <strong>the</strong> forward region<strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector.2.2 The <strong>CMS</strong> ExperimentDetectors are constructed <strong>in</strong> order to observe and analyze <strong>the</strong> collisions which are providedby <strong>the</strong> collider. Explor<strong>in</strong>g <strong>the</strong> vast physics opportunities <strong>in</strong> <strong>the</strong> proton-protoncollisions at high energy scales reachable at <strong>the</strong> LHC, requires to build general-purposeparticle detectors such as <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>. Particle detectors are generally designedto be symmetric <strong>in</strong> space to make <strong>the</strong> physics analysis easier. The particle isdetected when it <strong>in</strong>teracts with matter. Accord<strong>in</strong>g to <strong>the</strong> various k<strong>in</strong>d <strong>of</strong> particle <strong>in</strong>teractions,<strong>the</strong> detectors conta<strong>in</strong> appropriate materials, each <strong>of</strong> which is aimed to identifysome special type <strong>of</strong> particles or to measure particular properties <strong>of</strong> <strong>the</strong> particles. Thesub-systems are arranged <strong>in</strong> such a way which yields <strong>the</strong> detector to have an onion-likestructure. A slice <strong>of</strong> <strong>the</strong> transverse view <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector is shown <strong>in</strong> Figure 2.5.
20 CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHCFigure 2.3: The location <strong>of</strong> <strong>the</strong> various detectors around <strong>the</strong> LHC ma<strong>in</strong> r<strong>in</strong>g. Alsoshown are <strong>the</strong> <strong>in</strong>jection po<strong>in</strong>ts which are used to enter <strong>the</strong> proton beams <strong>in</strong>to <strong>the</strong> LHCcollider and <strong>the</strong> emergency exit po<strong>in</strong>ts which are used to extract <strong>the</strong> beams from <strong>the</strong>mach<strong>in</strong>e.Emerg<strong>in</strong>g particles from <strong>the</strong> hard <strong>in</strong>teraction first enter <strong>the</strong> track<strong>in</strong>g system. Thecharged particles, when mov<strong>in</strong>g through <strong>the</strong> tracker material leave hits which are f<strong>in</strong>allyused to build <strong>the</strong> trajectory <strong>of</strong> that particle. The presence <strong>of</strong> a magnetic fieldwith parallel field l<strong>in</strong>es to <strong>the</strong> beam axis, bends <strong>the</strong> trajectories <strong>of</strong> <strong>the</strong> charged particles.The bend<strong>in</strong>g radius is subsequently used to determ<strong>in</strong>e <strong>the</strong> momentum <strong>of</strong> <strong>the</strong> particlesas well as <strong>the</strong>ir charges.Right after <strong>the</strong> tracker, <strong>the</strong> calorimetry system exists which comprises <strong>the</strong> electromagneticcalorimeter (ECAL) and <strong>the</strong> hadronic calorimeter (HCAL), aimed to measure <strong>the</strong>energy <strong>of</strong> <strong>the</strong> particles which <strong>in</strong>teract via <strong>the</strong> electromagnetic and strong <strong>in</strong>teractions,respectively. A high energy electron or photon, hits an absorber and <strong>in</strong>itiates a cascade<strong>of</strong> secondary electrons and photons via bremstrahlung and pair production as depicted<strong>in</strong> Figure 2.4. Such an electromagnetic shower can be characterized with ma<strong>in</strong>ly twoparameters, namely <strong>the</strong> radiation length X 0 and <strong>the</strong> Moliere radius R M , which areexpla<strong>in</strong>ed <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g.The longitud<strong>in</strong>al development <strong>of</strong> an electromagnetic shower is described by <strong>the</strong> parameterX 0 which is def<strong>in</strong>ed as <strong>the</strong> distance over which <strong>the</strong> electron loses about 2 3 <strong>of</strong>its energy. This parameter is also a characteristic length scale which determ<strong>in</strong>es <strong>the</strong>thickness <strong>of</strong> <strong>the</strong> ECAL. The lateral development is described by <strong>the</strong> parameter R M .An <strong>in</strong>f<strong>in</strong>ite cyl<strong>in</strong>der centered on <strong>the</strong> electromagnetic shower <strong>in</strong>itiator with a radius <strong>of</strong>∼ 1 R M , conta<strong>in</strong>s ∼ 90% <strong>of</strong> <strong>the</strong> shower energy.Similar to <strong>the</strong> electromagnetic showers, hadronic showers are formed by <strong>the</strong> strong
CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHC 21Figure 2.4: A typical electromagnetic shower which is <strong>in</strong>itiated by a photon and isdevelopp<strong>in</strong>g by bremstrahlung and pair production processes.<strong>in</strong>teraction <strong>of</strong> hadrons with <strong>the</strong> heavy nuclei <strong>of</strong> <strong>the</strong> material used <strong>in</strong> <strong>the</strong> HCAL subsystem.Analogous to X 0 , <strong>the</strong> typical length scale parametriz<strong>in</strong>g <strong>the</strong> thickness <strong>of</strong> <strong>the</strong>hadronic calorimeter is <strong>the</strong> nuclear <strong>in</strong>teraction length λ which is def<strong>in</strong>ed as <strong>the</strong> meanfree path length <strong>of</strong> <strong>the</strong> hadron before <strong>in</strong>teract<strong>in</strong>g with <strong>the</strong> material.At <strong>the</strong> outermost part <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector, <strong>the</strong> muon system is used which is designedto detect and identify muons. Due to <strong>the</strong> ra<strong>the</strong>r large mean free path, muons are only<strong>in</strong>teract<strong>in</strong>g weakly with <strong>the</strong> calorimeter material and deposit a small amount <strong>of</strong> <strong>the</strong>irenergy <strong>in</strong> <strong>the</strong> calorimeter system and penetrate <strong>the</strong> last layer <strong>of</strong> <strong>the</strong> detector, which isaimed to detect muons. S<strong>in</strong>ce one <strong>of</strong> <strong>the</strong> important discovery channels <strong>of</strong> <strong>the</strong> StandardModel Higgs particle is via pp → H → ZZ ∗ → µµµµ, a sub-detector with a high resolutionhas been dedicated for muon identification and measurements. The importance<strong>of</strong> this has affected <strong>the</strong> nam<strong>in</strong>g <strong>of</strong> <strong>the</strong> detector.Between <strong>the</strong> calorimetry and muon system, <strong>the</strong>re is <strong>the</strong> solenoid magnet which generatesa powerful magnetic field <strong>of</strong> 3.8 T. The magnetic field is conf<strong>in</strong>ed us<strong>in</strong>g <strong>the</strong> ironyokes implemented outside <strong>of</strong> <strong>the</strong> magnet coil <strong>in</strong> <strong>the</strong> muon system.The Compact Muon Solenoid <strong>experiment</strong> [44] is an apparatus with 21 m length,15 m width and 15 m height, hav<strong>in</strong>g a total weight <strong>of</strong> about 12500 tonnes, most <strong>of</strong>which is due to <strong>the</strong> iron yokes. Compar<strong>in</strong>g with <strong>the</strong> o<strong>the</strong>r general-purpose detector<strong>of</strong> <strong>the</strong> LHC, ATLAS with 46 m length, 25 m width and 25 m height, <strong>CMS</strong> is muchmore compact. The <strong>CMS</strong> apparatus is <strong>the</strong> only detector <strong>of</strong> <strong>the</strong> LHC <strong>experiment</strong> whichwas constructed on <strong>the</strong> surface <strong>in</strong> fifteen sections, before be<strong>in</strong>g lowered underground<strong>in</strong>to <strong>the</strong> cavern for assembly and <strong>in</strong>stallation. A schematic view <strong>of</strong> <strong>the</strong> Compact MuonSolenoid can be found <strong>in</strong> Figure 2.6.The four-momenta <strong>of</strong> <strong>the</strong> particles which emerge <strong>in</strong> <strong>the</strong> proton-proton <strong>in</strong>teractionsare described us<strong>in</strong>g a spatial coord<strong>in</strong>ate which is centered on <strong>the</strong> <strong>in</strong>teraction po<strong>in</strong>t. They-axis po<strong>in</strong>ts vertically upward while <strong>the</strong> x-axis is chosen to po<strong>in</strong>t to <strong>the</strong> center <strong>of</strong> <strong>the</strong>LHC. The z-axis is <strong>the</strong>n selected along <strong>the</strong> beam direction <strong>in</strong> such a way which yieldsto obta<strong>in</strong> a righthanded coord<strong>in</strong>ate. Based on this choice <strong>of</strong> cartesian coord<strong>in</strong>ates,
22 CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHCFigure 2.5: A slice <strong>of</strong> <strong>the</strong> transverse view <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector show<strong>in</strong>g several particletrajectories while travers<strong>in</strong>g <strong>the</strong> detector material. The tracker system, electromagneticand hadron calorimeters toge<strong>the</strong>r with <strong>the</strong> muon system are constituents <strong>of</strong> <strong>the</strong> <strong>CMS</strong>detector. The location <strong>of</strong> <strong>the</strong> superconduct<strong>in</strong>g magnet is also shown.Figure 2.6: The overal size <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector, exhibit<strong>in</strong>g <strong>the</strong> structure <strong>of</strong> <strong>the</strong> barreland endcaps, is compared with <strong>the</strong> typical size <strong>of</strong> a human. Also shown are <strong>the</strong> varioussub-detectors <strong>of</strong> <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>.
CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHC 23<strong>the</strong> spherical coord<strong>in</strong>ates can also be <strong>in</strong>troduced which are commonly used at hadroncolliders. In addition to <strong>the</strong> polar angle θ and azimuthal angle φ, a new variable is <strong>of</strong>high importance at hadron colliders, namely <strong>the</strong> pseudorapidity η def<strong>in</strong>ed asη ≡ 1 ( E +2 ln pz) ( θ= −ln tan ,E − p z 2)where <strong>the</strong> equation is true <strong>in</strong> <strong>the</strong> massless limit <strong>of</strong> <strong>the</strong> particle with an energy <strong>of</strong> Eand a z component <strong>of</strong> <strong>the</strong> momentum vector <strong>of</strong> p z . Therefore, <strong>in</strong> order to describe <strong>the</strong>angular coord<strong>in</strong>ates <strong>of</strong> <strong>the</strong> particle, it is common to use (η, φ) parameters <strong>in</strong>stead <strong>of</strong>(θ, φ) variables.In <strong>the</strong> follow<strong>in</strong>g sections <strong>the</strong> sub-detectors <strong>of</strong> <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> are briefly expla<strong>in</strong>ed.2.2.1 Tracker SystemThe tracker system [45, 46] <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector is made up <strong>of</strong> silicon based detectorsto provide a high resolution detection required by <strong>the</strong> huge flux <strong>of</strong> charged particlesproduced <strong>in</strong> <strong>the</strong> high lum<strong>in</strong>osity regime <strong>of</strong> <strong>the</strong> mach<strong>in</strong>e. The overall structure <strong>of</strong> <strong>the</strong>track<strong>in</strong>g system comprises a central cyl<strong>in</strong>drical part, called tracker barrel, and twoseparate discs placed at ei<strong>the</strong>r side <strong>of</strong> <strong>the</strong> barrel, called tracker endcaps. In total 13layers <strong>in</strong> <strong>the</strong> central region and 14 layers <strong>in</strong> <strong>the</strong> endcaps form <strong>the</strong> <strong>CMS</strong> silicon tracker.They overall exhibit a diameter <strong>of</strong> 2.4 m and a length <strong>of</strong> 5.4 m result<strong>in</strong>g <strong>in</strong> a track<strong>in</strong>gcoverage <strong>of</strong> |η| ≤ 2.5. Closest to <strong>the</strong> <strong>in</strong>teraction po<strong>in</strong>t is a silicon pixel detector,comprises <strong>the</strong> <strong>in</strong>nermost 3 layers <strong>in</strong> <strong>the</strong> barrel and 2 layers <strong>in</strong> <strong>the</strong> endcaps. A total<strong>of</strong> 66 million pixels with a size <strong>of</strong> 100 × 150 µm 2 cover a distance <strong>of</strong> 4 to 11 cm from<strong>the</strong> beam l<strong>in</strong>e on a cyl<strong>in</strong>drical barrel and endcaps structure with total length <strong>of</strong> 92cm. Besides <strong>the</strong> silicon pixel detectors, silicon strip detectors contribute <strong>in</strong> <strong>the</strong> trackersystem with a variable size from 10 cm × 150 µm to 25 cm × 150 µm. There are 9.6million strip channels <strong>in</strong> total. A longitud<strong>in</strong>al view <strong>of</strong> <strong>the</strong> tracker system can be found<strong>in</strong> Figure 2.7, where only <strong>the</strong> silicon strip layers surround<strong>in</strong>g <strong>the</strong> pixel detector areshown.The transverse momentum <strong>of</strong> a charged particle with an energy <strong>of</strong> about 100 GeVproduced <strong>in</strong> <strong>the</strong> barrel part <strong>of</strong> <strong>the</strong> detector, is measured by <strong>the</strong> tracker system with aprecision <strong>of</strong> about (1-2)%.2.2.2 Calorimetry SystemECALThe electromagnetic calorimeter (ECAL) [47, 48] is aimed to measure <strong>the</strong> energy <strong>of</strong>electrons and photons. Similar to <strong>the</strong> tracker system, <strong>the</strong> ECAL is also charactrized by<strong>the</strong> electromagnetic barrel (EB) and <strong>the</strong> electromagnetic endcap (EE) structure. Leadtungstate PbWO 4 crystals are used to construct <strong>the</strong> ECAL due to its high density andradiation hardness. It has a small radiation length <strong>of</strong> X 0 = 0.89 cm and a Moliere radius<strong>of</strong> R M = 2.2 cm. Each barrel crystal covers an area <strong>of</strong> ∆η ×∆φ = 0.0174 ×0.0174with a length <strong>of</strong> 23 cm which corresponds to 25.8 X 0 . The arrangement <strong>of</strong> <strong>the</strong> barrel
24 CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHCFigure 2.7: The silicon strip layers used <strong>in</strong> <strong>the</strong> tracker barrel (horizontal l<strong>in</strong>es) and <strong>the</strong>tracker endcaps (vertical l<strong>in</strong>es) <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector is shown. The distances from <strong>the</strong><strong>in</strong>teraction po<strong>in</strong>t are quoted <strong>in</strong> units <strong>of</strong> mm. Also shown are <strong>the</strong> various pseudorapidityregions covered by <strong>the</strong> tracker system.crystals is quasi projective, which means <strong>the</strong>y po<strong>in</strong>t almost to <strong>the</strong> <strong>in</strong>teraction po<strong>in</strong>tto m<strong>in</strong>imize <strong>the</strong> cracks. Each endcap crystal is 24.7 X 0 long. The angular coverage<strong>of</strong> <strong>the</strong> EB and EE is def<strong>in</strong>ed to be |η| < 1.479 and 1.479 < |η| < 3.0, respectively.Also each endcap is partially covered by a preshower detector (ES) which extends <strong>the</strong>range <strong>of</strong> 1.653 < |η| < 2.6. The ES is aimed to clearly dist<strong>in</strong>guish π 0 from photons byidentify<strong>in</strong>g two closely spaced photons from π 0 decays. S<strong>in</strong>ce pp → H → γγ is one <strong>of</strong><strong>the</strong> important decay channels <strong>of</strong> <strong>the</strong> Standard Model Higgs boson, π 0 decay<strong>in</strong>g to twophotons could mimic <strong>the</strong> f<strong>in</strong>al products <strong>of</strong> a hard scatter<strong>in</strong>g <strong>in</strong>clud<strong>in</strong>g a Higgs boson as<strong>the</strong> <strong>in</strong>termediate particle. The high granularity <strong>of</strong> <strong>the</strong> preshower would <strong>the</strong>n improve<strong>the</strong> search for <strong>the</strong> Higgs particle.The ECAL is calibrated us<strong>in</strong>g test beam electrons [49], result<strong>in</strong>g <strong>in</strong> a parametrizedenergy resolution which is expressed as [50]σ(E)E = 2.8% √E/GeV⊕ 0.3%.Therefore, an electron with an energy <strong>of</strong> 100 GeV can be measured by <strong>the</strong> ECAL witha precision <strong>of</strong> about 0.4%.
CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHC 25HCALThe hadronic calorimeter (HCAL) [51] is designed to measure <strong>the</strong> energy <strong>of</strong> hadrons.The hadronic barrel (HB) is located between <strong>the</strong> ECAL and <strong>the</strong> solenoid coil and iscomplemented by <strong>the</strong> outer part (HO) placed outside <strong>the</strong> magnet. The HO is designedto absorb <strong>the</strong> remnant <strong>of</strong> <strong>the</strong> hadronic shower which has not been fully absorbed <strong>in</strong><strong>the</strong> HB. The HB cover<strong>in</strong>g <strong>the</strong> pseudorapidity range up to |η| = 1.3, provides a granularity<strong>of</strong> ∆η × ∆φ = 0.087 × 0.087. The hadronic endcap (HE) covers <strong>the</strong> region1.3 < |η| < 3.0 with <strong>the</strong> same granularity as HB up to |η| = 1.6 and a decreas<strong>in</strong>g granularityto ∆η × ∆φ = 0.17 × 0.17 for <strong>the</strong> rest <strong>of</strong> HE coverage. The forward hadroniccalorimeter (HF) provides an extended coverage <strong>of</strong> <strong>the</strong> pseudorapidity from |η| = 3.0up to |η| = 5.0. The HCAL sub-detector is a sampl<strong>in</strong>g calorimeter where layers <strong>of</strong>absorber, result <strong>in</strong> produc<strong>in</strong>g hadron showers, <strong>in</strong>terleav<strong>in</strong>g with layers <strong>of</strong> active material,which measure <strong>the</strong> energy <strong>of</strong> <strong>the</strong> showered particles. In <strong>the</strong> HB and HE, brassplates are used as absorber material while <strong>the</strong> plastic sc<strong>in</strong>tillators constitute <strong>the</strong> activematerial. The HF is constructed from steel as absorber and as active material, quartzfibers are used.The <strong>in</strong>itial calibration <strong>of</strong> <strong>the</strong> HCAL is performed us<strong>in</strong>g test beam data and a genericparametrization as <strong>of</strong> <strong>the</strong> ECAL can be obta<strong>in</strong>ed for <strong>the</strong> HCAL energy resolution. Thecomb<strong>in</strong>ed energy resolution <strong>of</strong> <strong>the</strong> HCAL and <strong>the</strong> ECAL <strong>in</strong> <strong>the</strong> barrel region <strong>of</strong> <strong>the</strong>detector is parametrized as [52]σ(E)E = 84.7% √E/GeV⊕ 7.4%.Therefore, a reconstructed jet with an energy <strong>of</strong> 100 GeV can be measured by <strong>the</strong>calorimeter system with a precision <strong>of</strong> about 11.2%.The f<strong>in</strong>al calibration <strong>of</strong> <strong>the</strong> HCAL is achieved with <strong>the</strong> use <strong>of</strong> physics events based on<strong>the</strong> k<strong>in</strong>ematical properties <strong>of</strong> <strong>the</strong> objects produced <strong>in</strong> <strong>the</strong> hard <strong>in</strong>teraction which areexpla<strong>in</strong>ed <strong>in</strong> more detail <strong>in</strong> <strong>the</strong> next chapter.A quarter <strong>of</strong> <strong>the</strong> longitud<strong>in</strong>al view <strong>of</strong> <strong>the</strong> ECAL and HCAL sub-detectors can befound <strong>in</strong> Figure 2.8 where <strong>the</strong> pseudorapidity boundaries are also overlaid.2.2.3 Muon SystemThe muon chambers [53], <strong>the</strong> outermost sub-detector <strong>of</strong> <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>, is aimedto identify muons and measure <strong>the</strong>ir momenta. Three types <strong>of</strong> detectors are used <strong>in</strong><strong>the</strong> muon system, be<strong>in</strong>g drift tubes (DT), cathode strip chambers (CSC) and resistiveplate chambers (RPC). While <strong>the</strong> first and second ones provide precise trajectorymeasurements, <strong>the</strong> last one is characterized for its fast signal process<strong>in</strong>g. The DTs areused <strong>in</strong> <strong>the</strong> barrel part <strong>of</strong> <strong>the</strong> detector, while <strong>the</strong> CSCs are used <strong>in</strong> <strong>the</strong> endcaps. Thelayers <strong>of</strong> RPCs are <strong>in</strong>stalled <strong>in</strong> both <strong>the</strong> barrel and <strong>the</strong> endcaps.
26 CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHCFigure 2.8: The longitud<strong>in</strong>al view <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector, where <strong>the</strong> pseudorapiditycoverage <strong>of</strong> <strong>the</strong> electromagnetic barrel (EB), <strong>the</strong> electromagnetic endcap (EE), <strong>the</strong>hadronic barrel (HB), <strong>the</strong> hadronic endcap (HE), <strong>the</strong> forward hadronic calorimeter(HF) and <strong>the</strong> outer part (HO) are shown. Also shown is <strong>the</strong> muon system, <strong>the</strong> lastlayer <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector.
CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHC 272.2.4 Trigger<strong>in</strong>g and Data AcquisitionIn one hand, <strong>the</strong> LHC, while work<strong>in</strong>g at <strong>the</strong> design lum<strong>in</strong>osity <strong>of</strong> L = 10 34 , collidesbunch <strong>of</strong> protons every 25 ns result<strong>in</strong>g <strong>in</strong> a bunch cross<strong>in</strong>g rate <strong>of</strong> 40 MHz as mentionedbefore. This corresponds to obta<strong>in</strong> 10 9 proton-proton <strong>in</strong>teractions per second when <strong>the</strong>mach<strong>in</strong>e is operat<strong>in</strong>g at <strong>the</strong> center <strong>of</strong> mass energy <strong>of</strong> 14 TeV. In <strong>the</strong> o<strong>the</strong>r hand, <strong>the</strong>current technology is limited to process only about 100 <strong>in</strong>teractions per second, hence10 7 times less than what is produced. S<strong>in</strong>ce most <strong>of</strong> <strong>the</strong> occured <strong>in</strong>teractions are s<strong>of</strong>tcollisions <strong>of</strong> protons, <strong>the</strong>n one does not lose processes <strong>of</strong> physics <strong>in</strong>terest by us<strong>in</strong>g anoptimal rejection procedure. The process <strong>of</strong> onl<strong>in</strong>e event selection is handled with <strong>the</strong>use <strong>of</strong> <strong>the</strong> trigger system [54, 55], which is performed <strong>in</strong> two stages. A first stage,which is called first level trigger (Level-1), is aimed to reduce <strong>the</strong> rate <strong>of</strong> <strong>the</strong> datatak<strong>in</strong>g down to 100 kHz. Because <strong>the</strong> decision made by <strong>the</strong> Level-1 trigger should beas fast as possible, <strong>the</strong> basic <strong>in</strong>formation from calorimeters and <strong>the</strong> muon system areused to decide whe<strong>the</strong>r an event has to be kept or not. A second stage, which is calledHigh-Level Trigger (HLT), is designed for a fur<strong>the</strong>r reduction <strong>of</strong> <strong>the</strong> rate down to 100Hz. The <strong>in</strong>teractions selected by <strong>the</strong> Level-1 trigger are <strong>the</strong>n fed to <strong>the</strong> HLT for additional<strong>in</strong>vestigation and mak<strong>in</strong>g a more robust decision. The HLT s<strong>of</strong>tware conta<strong>in</strong>smore sophisticated algorithms which are very close to <strong>the</strong> ones applied <strong>in</strong> <strong>the</strong> <strong>of</strong>fl<strong>in</strong>eanalysis. The decision which is made at each stage <strong>of</strong> <strong>the</strong> trigger to keep or rejecta specific <strong>in</strong>teraction is based on <strong>the</strong> reconstruction and identification <strong>of</strong> <strong>the</strong> physicsobjects at <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> that particular <strong>in</strong>teraction. Because processes <strong>of</strong> physics<strong>in</strong>terest, such as production and decay <strong>of</strong> rare particles like Higgs bosons, conta<strong>in</strong> highenergetic objects, such as electrons, photons, muons and jets, various trigger streamsare def<strong>in</strong>ed, each <strong>of</strong> which requires some configurable physics conditions. An <strong>in</strong>teractionwhich fires at least one <strong>of</strong> <strong>the</strong> trigger streams is archived permanently and keptfor <strong>the</strong> <strong>of</strong>fl<strong>in</strong>e analysis.The data which pass <strong>the</strong> HLT filters are stored <strong>in</strong> comput<strong>in</strong>g centers distributed allaround <strong>the</strong> world. Accord<strong>in</strong>g to <strong>the</strong> <strong>CMS</strong> comput<strong>in</strong>g model [56], <strong>the</strong> storage resourcesare connected and managed through <strong>the</strong> Worldwide LHC Comput<strong>in</strong>g Grid(WLCG) [57]. The WLCG project makes it possible for <strong>the</strong> users to access <strong>the</strong> <strong>in</strong>formationand data which have been distributed over <strong>the</strong> storage resources with <strong>the</strong> use<strong>of</strong> <strong>the</strong> Grid network. Comput<strong>in</strong>g centers are grouped <strong>in</strong> three categories referred toas Tier-0, Tier-1 and Tier-2, each <strong>of</strong> which provides some particular tasks. The rawdata which are provided by <strong>the</strong> <strong>CMS</strong> detector are first moved to <strong>the</strong> Tier-0, which islocated at CERN, <strong>in</strong> order to run <strong>the</strong> <strong>in</strong>itial reconstruction algorithms and process<strong>in</strong>g<strong>of</strong> <strong>the</strong> data. The reconstructed data are <strong>the</strong>n transferred to <strong>the</strong> computer resources <strong>of</strong><strong>the</strong> Tier-1 centers. Also <strong>the</strong> simulated proton collisions, which are produced at Tier-2 comput<strong>in</strong>g resources, are kept <strong>in</strong> Tier-1 storage services. In case it is needed, <strong>the</strong>re-reconstruction <strong>of</strong> <strong>the</strong> data is also performed at Tier-1 centers. The several Tiersare kept <strong>in</strong> a hierarchial level. The collected data by <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> are <strong>the</strong>nstored <strong>in</strong> <strong>the</strong> form <strong>of</strong> ROOT files [58] and subsequently analyzed with some dedicateds<strong>of</strong>tware, called <strong>CMS</strong>SW. The <strong>CMS</strong> comput<strong>in</strong>g model provides various data formats t<strong>of</strong>acilitate <strong>the</strong> data transfer and process<strong>in</strong>g. The data which come out <strong>of</strong> <strong>the</strong> detector,conta<strong>in</strong><strong>in</strong>g <strong>the</strong> electronic signals and hits, is called RAW data. They are huge andcompris<strong>in</strong>g <strong>the</strong> low level objects which are not suitable for <strong>the</strong> physics analaysis. After
28 CHAPTER 2: The <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHCrunn<strong>in</strong>g reconstruction algorithms, <strong>the</strong> RAW data format become RECO data formatwith a reduced size which conta<strong>in</strong>s <strong>the</strong> reconstructed physics objects, hence <strong>the</strong> highlevelobjects relevant for <strong>the</strong> analysis. Moreover, a sub-set <strong>of</strong> <strong>the</strong> RECO data, which is<strong>the</strong> Analysis Object Data (AOD) represents <strong>the</strong> third data format layer and is designedto improve <strong>the</strong> data transfer rate among <strong>the</strong> comput<strong>in</strong>g resources with a limited size<strong>of</strong> data events. In addition, a fourth data format is <strong>in</strong>troduced by <strong>the</strong> Physics AnalysisToolkit (PAT) [59] which is aimed to facilitate <strong>the</strong> physics analysis by <strong>in</strong>clud<strong>in</strong>g <strong>the</strong>advanced properties to <strong>the</strong> reconstructed objects. Fur<strong>the</strong>rmore, it is still possible toimprove <strong>the</strong> speed <strong>of</strong> <strong>the</strong> analysis by <strong>the</strong> use <strong>of</strong> Ntuples [58], which are k<strong>in</strong>d <strong>of</strong> Treeobjects <strong>of</strong>ficially produced <strong>in</strong> various collaborat<strong>in</strong>g <strong>in</strong>stitutes result<strong>in</strong>g to enhance <strong>the</strong>analysis speed. This <strong>the</strong>sis is based on <strong>the</strong> process<strong>in</strong>g <strong>of</strong> <strong>the</strong> objects provided <strong>of</strong>ficiallyby <strong>the</strong> <strong>CMS</strong> Brussels team [60], called TopTrees. The version <strong>of</strong> <strong>the</strong> s<strong>of</strong>tware whichhas been used to produce <strong>the</strong> PAT objects is <strong>CMS</strong>SW_3_6_1_patch3 followed by <strong>the</strong><strong>CMS</strong>SW_36X_v1_patch2 version <strong>of</strong> <strong>the</strong> s<strong>of</strong>tware used for <strong>the</strong> production package <strong>of</strong> <strong>the</strong>TopTrees.
Chapter 3Object ReconstructionThe elementary particles, namely quarks and leptons, emerg<strong>in</strong>g from a hard <strong>in</strong>teractionare observed <strong>in</strong> a detector <strong>experiment</strong> when <strong>the</strong>y <strong>in</strong>teract with <strong>the</strong> material <strong>of</strong><strong>the</strong> detector, <strong>the</strong> neutr<strong>in</strong>os be<strong>in</strong>g an exception because <strong>the</strong>y do not <strong>in</strong>teract with <strong>the</strong>matter and <strong>the</strong>ir presence is understood by <strong>the</strong> miss<strong>in</strong>g amount <strong>in</strong> a balance <strong>of</strong> energybetween <strong>in</strong>itial and f<strong>in</strong>al state. Accord<strong>in</strong>g to <strong>the</strong> known physics processes [61], particles<strong>in</strong>teract with matter and release <strong>the</strong>ir energy which subsequently is collected by <strong>the</strong>data acquisition (DAQ) system <strong>of</strong> <strong>the</strong> detector. The <strong>in</strong>formation that comes out <strong>of</strong><strong>the</strong> detector <strong>in</strong>clud electronic signals which are used when reconstruct<strong>in</strong>g <strong>the</strong> physicsobjects. Different objects have a dist<strong>in</strong>ct behaviour while travers<strong>in</strong>g <strong>the</strong> detector. Thedetector has been designed <strong>in</strong> such a way that, for example, <strong>the</strong> electron objects arestopped <strong>in</strong> <strong>the</strong> ECAL and loose almost all <strong>the</strong>ir energy before reach<strong>in</strong>g <strong>the</strong> HCAL. S<strong>in</strong>ceelectrons are charged particles, <strong>the</strong>y leave hits <strong>in</strong> <strong>the</strong> tracker. Therefore an electron isreconstructed by assign<strong>in</strong>g a track <strong>in</strong> <strong>the</strong> tracker to a collection <strong>of</strong> energy deposit <strong>in</strong><strong>the</strong> ECAL. A more precise description <strong>of</strong> <strong>the</strong> electron reconstruction algorithm used <strong>in</strong><strong>the</strong> <strong>CMS</strong> collabration, is given <strong>in</strong> Section 3.1.In case <strong>of</strong> quarks, as <strong>the</strong>y are not observables before <strong>the</strong> hadronization process, <strong>the</strong>ypass through <strong>the</strong> detector as a cascade <strong>of</strong> hadrons, called jets. <strong>Jet</strong> reconstruction isnot as simple as reconstruct<strong>in</strong>g an electron and much more sophisticated algorithmsare needed. A detailed explanation on how to reconstruct <strong>the</strong> jet objects is <strong>the</strong> subject<strong>of</strong> Section 3.2. In addition to <strong>the</strong> electrons and <strong>the</strong> jets, <strong>the</strong> reconstruction <strong>of</strong> <strong>the</strong>neutr<strong>in</strong>os is <strong>of</strong> great importance and is described <strong>in</strong> Section 3.3. Muons can also bereconstructed <strong>in</strong> <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> [62]. As muons are not processed <strong>in</strong> this analysis,no dedicated part is added to expla<strong>in</strong> <strong>the</strong>ir reconstruction.3.1 Electron ReconstructionIn order to reconstruct an electron <strong>in</strong> <strong>CMS</strong>, <strong>the</strong> <strong>in</strong>formation from both <strong>the</strong> trackerand <strong>the</strong> electromagnetic calorimeter is used. The first step is <strong>the</strong> cluster<strong>in</strong>g <strong>of</strong> <strong>the</strong>energy deposits <strong>in</strong> <strong>the</strong> ECAL. It has been shown that most <strong>of</strong> <strong>the</strong> energy <strong>of</strong> a s<strong>in</strong>gleelectron or an unconverted photon reach<strong>in</strong>g <strong>the</strong> ECAL, can be collected <strong>in</strong> a clusterwhich is made <strong>of</strong> a few ECAL crystals. For <strong>in</strong>stance, electrons with an energy <strong>of</strong> 120GeV imp<strong>in</strong>g<strong>in</strong>g at <strong>the</strong> center <strong>of</strong> an ECAL crystal, deposit about 97% <strong>of</strong> <strong>the</strong>ir energy29
30 CHAPTER 3: Object Reconstruction<strong>in</strong> a 5×5 crystal w<strong>in</strong>dow around it [63]. Nom<strong>in</strong>ally electrons radiate photons whiletravers<strong>in</strong>g <strong>the</strong> tracker material. About 35% <strong>of</strong> <strong>the</strong> electrons radiate more than 70% <strong>of</strong><strong>the</strong>ir <strong>in</strong>itial energy before reach<strong>in</strong>g <strong>the</strong> ECAL. The radiated photons reach<strong>in</strong>g <strong>the</strong> ECALspread <strong>the</strong>ir energy <strong>in</strong> <strong>the</strong> φ direction and this makes <strong>the</strong> situation more complicatedto cluster <strong>the</strong> energy <strong>of</strong> such electrons. The idea is to collect <strong>the</strong> electron energy bymak<strong>in</strong>g a cluster <strong>of</strong> clusters, <strong>the</strong> so called supercluster, along a φ road which conta<strong>in</strong>s<strong>the</strong> bremsstrahlung photons emitted along <strong>the</strong> electron trajectory <strong>in</strong> <strong>the</strong> tracker.3.1.1 Electron Cluster<strong>in</strong>g AlgorithmsThe basic Hybrid and Island cluster<strong>in</strong>g algorithms can be used for electrons <strong>in</strong> <strong>the</strong>ECAL barrel and endcap respectively [64]. The ma<strong>in</strong> step for both <strong>of</strong> <strong>the</strong>se algorithmsis to make clusters. A cluster is created by look<strong>in</strong>g for a bump <strong>of</strong> deposited energyaround a local energy maximum called <strong>the</strong> seed crystal. For <strong>the</strong> barrel electrons, <strong>the</strong>energy is collected <strong>in</strong> dom<strong>in</strong>oes made from 3 or 5 crystals <strong>in</strong> η. Dom<strong>in</strong>oes centeredaround <strong>the</strong> seed crystal make a cluster. For <strong>the</strong> endcap electrons, <strong>in</strong>stead <strong>of</strong> mak<strong>in</strong>gdom<strong>in</strong>oes, <strong>the</strong> search for energy deposited around <strong>the</strong> seed crystal is started cell by cell.The energy <strong>of</strong> each crystal is added to <strong>the</strong> cluster until a rise <strong>in</strong> <strong>the</strong> energy is observed.The f<strong>in</strong>al step <strong>in</strong> <strong>the</strong> cluster<strong>in</strong>g algorithms is simply to collect <strong>the</strong> nearby clusters <strong>in</strong><strong>the</strong> φ direction <strong>in</strong> order to recover <strong>the</strong> radiation emitted along <strong>the</strong> trajectory <strong>of</strong> <strong>the</strong>electron.3.1.2 Electron Track ReconstructionHav<strong>in</strong>g clustered <strong>the</strong> energy <strong>of</strong> an electron, <strong>the</strong> next step is to reconstruct and associatea track to it. The start<strong>in</strong>g po<strong>in</strong>t for reconstruct<strong>in</strong>g a track <strong>in</strong> <strong>the</strong> tracker is to f<strong>in</strong>d twohits <strong>in</strong> <strong>the</strong> pixel detector. The two pixel hits that are found serve as a seed to build anelectron track <strong>in</strong> <strong>the</strong> silicon tracker detector.In order to reduce <strong>the</strong> number <strong>of</strong> possible hit comb<strong>in</strong>ations that can form a candidateseed for reconstruct<strong>in</strong>g <strong>the</strong> electron track, <strong>the</strong> search for seeds is restricted to a regionwhich is compatible with a supercluster <strong>in</strong> <strong>the</strong> ECAL. This approach <strong>of</strong> superclusterdrivenpixel seed f<strong>in</strong>d<strong>in</strong>g has <strong>the</strong> advantage <strong>of</strong> <strong>in</strong>creas<strong>in</strong>g <strong>the</strong> purity <strong>of</strong> <strong>the</strong> sample <strong>of</strong>candidate electron tracks. Hits <strong>in</strong> <strong>the</strong> pixel layers are predicted by backward propagation<strong>of</strong> <strong>the</strong> position <strong>of</strong> <strong>the</strong> supercluster through <strong>the</strong> magnetic field towards <strong>the</strong> pixeldetector. A first compatible hit is <strong>the</strong>n looked for <strong>in</strong> <strong>the</strong> <strong>in</strong>nermost pixel layer. Thepredicted trajectory is <strong>the</strong>n propagated to look for a second pixel hit <strong>in</strong> <strong>the</strong> next pixellayer. When two hits are found, a seed is generated.Start<strong>in</strong>g from <strong>the</strong> seed and us<strong>in</strong>g a Be<strong>the</strong> Heitler model<strong>in</strong>g <strong>of</strong> <strong>the</strong> electron energy losses,a trajectory is created and compatible hits on <strong>the</strong> next silicon layers are searched for.The procedure is stopped if no hit is found <strong>in</strong> two subsequent layers. The compatibilityamong <strong>the</strong> predicted trajectory and <strong>the</strong> measured hits is checked us<strong>in</strong>g a fit methodcalled <strong>the</strong> Gaussian Sum Filter (GSF) and is def<strong>in</strong>ed <strong>in</strong> terms <strong>of</strong> a χ 2 value. F<strong>in</strong>ally<strong>the</strong> best track candidate with <strong>the</strong> smallest χ 2 and hav<strong>in</strong>g a m<strong>in</strong>imum <strong>of</strong> five hits is keptand associated to <strong>the</strong> electron.Besides to <strong>the</strong> supercluster-driven pixel seed f<strong>in</strong>d<strong>in</strong>g method, <strong>the</strong>re is a tracker-driven
CHAPTER 3: Object Reconstruction 31seed<strong>in</strong>g algorithm which has been developed for non-isolated electrons [65]. In a trackerdrivenseed<strong>in</strong>g procedure, <strong>the</strong> reconstruction <strong>of</strong> <strong>the</strong> track <strong>of</strong> an electron starts from <strong>the</strong>tracker and <strong>the</strong>n a supercluster <strong>in</strong> <strong>the</strong> ECAL is matched to <strong>the</strong> reconstructed track.The performance plot which compares <strong>the</strong> efficiency to reconstruct <strong>the</strong> track <strong>of</strong> <strong>the</strong>electron is shown <strong>in</strong> Figure 3.1.Figure 3.1: Electron seed<strong>in</strong>g efficiency (solid l<strong>in</strong>e) as a function <strong>of</strong> <strong>the</strong> generated electronη (left) and generated electron p T (right) for electrons <strong>in</strong> a simulated sample <strong>of</strong> Z →e + e − events [66]. The <strong>in</strong>dividual contributions from <strong>the</strong> supercluster-driven (dashedl<strong>in</strong>e) and from <strong>the</strong> tracker-driven (dotted l<strong>in</strong>e) seed<strong>in</strong>g algorithms are also shown.As it is seen from Figure 3.1, <strong>the</strong> electron seed<strong>in</strong>g efficiency obta<strong>in</strong>ed from <strong>the</strong>tracker-driven approach is higher for <strong>the</strong> low-p T electrons, as aimed for. Also twodrops observed <strong>in</strong> <strong>the</strong> η distribution <strong>of</strong> <strong>the</strong> electron seed<strong>in</strong>g efficiency refer to <strong>the</strong>transition region between <strong>the</strong> ECAL barrel and <strong>the</strong> ECAL endcaps. The lower efficiencyobta<strong>in</strong>ed from <strong>the</strong> supercluster-driven approach compared to <strong>the</strong> efficiency derived from<strong>the</strong> tracker-driven approach is ma<strong>in</strong>ly due to an <strong>in</strong>efficiency <strong>in</strong> <strong>the</strong> reconstruction <strong>of</strong>superclusters <strong>in</strong> <strong>the</strong> transition region.3.1.3 Electron Momentum Determ<strong>in</strong>ationThe f<strong>in</strong>al electron momentum is obta<strong>in</strong>ed by comb<strong>in</strong><strong>in</strong>g <strong>the</strong> energy measurement providedby <strong>the</strong> electromagnetic calorimeter with <strong>the</strong> momentum measurement at <strong>the</strong>vertex provided by <strong>the</strong> tracker. If <strong>the</strong> two measurements are comparable, <strong>the</strong>n a comb<strong>in</strong>ation<strong>of</strong> both energy and momentum measurements is used as an estimate <strong>of</strong> <strong>the</strong>f<strong>in</strong>al electron momentum. O<strong>the</strong>rwise, if <strong>the</strong>y disagree significantly, <strong>the</strong> energy measurementfrom <strong>the</strong> ECAL sub-detector is taken as <strong>the</strong> f<strong>in</strong>al estimation given that itexceeds 15 GeV. In case that <strong>the</strong> ECAL measures an energy below 15 GeV, <strong>the</strong>n <strong>the</strong>momentum measured by <strong>the</strong> tracker is used and assigned to <strong>the</strong> electron. The idea beh<strong>in</strong>dthis is that <strong>in</strong> particular at energies <strong>of</strong> around 15 GeV and below, <strong>the</strong> momentumestimation from <strong>the</strong> tracker is more precise than <strong>the</strong> energy measurement provided by
32 CHAPTER 3: Object Reconstruction<strong>the</strong> ECAL.In order to evaluate <strong>the</strong> performance <strong>of</strong> <strong>the</strong> electron reconstruction algorithm <strong>in</strong><strong>CMS</strong>, one can compare <strong>the</strong> various properties <strong>of</strong> reconstructed electrons with <strong>the</strong> generatedelectrons. Therefore a match<strong>in</strong>g <strong>in</strong> (η, φ) space is performed to assign a reconstructedelectron to a generated one. The efficiency <strong>of</strong> f<strong>in</strong>d<strong>in</strong>g a reconstructed electronclose to a generated electron as a function <strong>of</strong> <strong>the</strong> p T <strong>of</strong> <strong>the</strong> generated electron, is shown<strong>in</strong> Figure 3.2. The plot, made for electrons <strong>in</strong> a simulated Z → e + e − sample afterapply<strong>in</strong>g some quality criteria, has been extracted from [66].Figure 3.2: Electron reconstruction efficiency as a function <strong>of</strong> <strong>the</strong> p T <strong>of</strong> <strong>the</strong> generatedelectron [66]. Electrons are from a simulated sample <strong>of</strong> Z → e + e − decays.It is obvious from Figure 3.2 that <strong>the</strong> efficiency <strong>of</strong> reconstruc<strong>in</strong>g electrons <strong>in</strong> <strong>the</strong><strong>CMS</strong> detector is larger than 99%, which means <strong>in</strong> almost all <strong>the</strong> time an electron canbe reconstructed very close to <strong>the</strong> generated electron.3.2 <strong>Jet</strong> ReconstructionQuarks and gluons appear <strong>in</strong> <strong>the</strong> f<strong>in</strong>al states <strong>of</strong> many processes <strong>of</strong> <strong>in</strong>terest <strong>in</strong> <strong>the</strong> protoncollisions observed by <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>. They are carry<strong>in</strong>g color charge, hencedue to <strong>the</strong> quark conf<strong>in</strong>ement pr<strong>in</strong>cipal <strong>the</strong>y cannot exist <strong>in</strong> a free form. Accord<strong>in</strong>gto <strong>the</strong> <strong>the</strong>ory, partons get hadronized to make some colorless hadrons which <strong>the</strong>n areable to be detected. Thus <strong>the</strong> signature <strong>of</strong> a quark or gluon is <strong>the</strong> presence <strong>of</strong> manyparticles mov<strong>in</strong>g <strong>in</strong> <strong>the</strong> direction <strong>of</strong> <strong>the</strong> orig<strong>in</strong>al parton. With a good approximation,<strong>the</strong> hadrons which arise <strong>in</strong> <strong>the</strong> hadronisation process can be collected <strong>in</strong> a cone around
CHAPTER 3: Object Reconstruction 33<strong>the</strong> parton’s direction that characterizes a jet.Different types <strong>of</strong> jets are reconstructed at <strong>CMS</strong> depend<strong>in</strong>g on <strong>the</strong> <strong>in</strong>formation <strong>of</strong> whichsub-detectors is used [67, 68]. As only calorimeter jets are processed <strong>in</strong> this analysis,<strong>the</strong>re is a dedicated section to describe <strong>the</strong> details <strong>of</strong> this k<strong>in</strong>d <strong>of</strong> jets, followed bydescriptions <strong>of</strong> <strong>the</strong> divers jet cluster<strong>in</strong>g algorithms currently available at <strong>CMS</strong>.Calorimeter jets, hereafter called Calo<strong>Jet</strong>s, only use <strong>the</strong> <strong>in</strong>formation <strong>of</strong> <strong>the</strong> <strong>CMS</strong> electromagnetic(ECAL) as well as <strong>the</strong> hadronic (HCAL) calorimeters. The build<strong>in</strong>g blocksfor Calo<strong>Jet</strong>s are calotowers which are made <strong>of</strong> <strong>the</strong> HCAL towers and <strong>the</strong> correspond<strong>in</strong>gECAL crystals. The energy associated to a calotower is simply obta<strong>in</strong>ed by collect<strong>in</strong>g<strong>the</strong> energy deposits <strong>in</strong> those components which pass <strong>the</strong> so-called “Scheme B” energythresholds [69]. This helps to suppress <strong>the</strong> electronic noise and not to count <strong>the</strong>m <strong>in</strong><strong>the</strong> energy <strong>of</strong> <strong>the</strong> calotower. In order to fur<strong>the</strong>r reduce <strong>the</strong> additional contributionfrom pile-up collisions, calotowers are required to have a transverse energy exceed<strong>in</strong>g0.5 GeV to be used for jet cluster<strong>in</strong>g algorithms.3.2.1 <strong>Jet</strong> Cluster<strong>in</strong>g AlgorithmsSeveral methods to def<strong>in</strong>e a jet exist [70–72]. Two <strong>of</strong> <strong>the</strong>m with great importance <strong>in</strong><strong>the</strong> <strong>CMS</strong> colaboration are described below. It will become clear that a first method,<strong>the</strong> iterative cone algorithm, is used <strong>in</strong> <strong>the</strong> onl<strong>in</strong>e trigger<strong>in</strong>g program while a secondone, <strong>the</strong> anti-k T algorithm, is important as it is highly recommended for <strong>the</strong> <strong>of</strong>fl<strong>in</strong>eanalysis.Iterative Cone AlgorithmThis simple algorithm merges <strong>the</strong> energy deposits located <strong>in</strong> a cone around <strong>the</strong> mostenergetic calotowers. The steps taken to create <strong>the</strong> iterative cone jets are listed below.• an E T -ordered list <strong>of</strong> <strong>in</strong>put objects is created;• a cone <strong>of</strong> size R <strong>in</strong> (η, φ) space is cast around <strong>the</strong> seed, <strong>the</strong> object with <strong>the</strong> largestE T ;• objects <strong>in</strong>side <strong>the</strong> cone are used to calculate <strong>the</strong> properties <strong>of</strong> a protojetE T = Σ i E T i , η = Σ i(η i × E T i )E T, φ = Σ i(φ i × E T i )E T;• this protojet is used to seed a new protojet;• <strong>the</strong> procedure is repeated until <strong>the</strong> energy and direction <strong>of</strong> <strong>the</strong> protojet does notchange between iterations;
34 CHAPTER 3: Object Reconstruction• <strong>the</strong> stable protojet becomes a jet and <strong>the</strong> correspond<strong>in</strong>g objects which made <strong>the</strong>protojet, are removed from <strong>the</strong> list to look for o<strong>the</strong>r protojets.This algorithm has a fast response and is <strong>the</strong>refore usually used <strong>in</strong> <strong>the</strong> trigger<strong>in</strong>gs<strong>of</strong>tware <strong>of</strong> <strong>the</strong> onl<strong>in</strong>e trigger system. R is a free parameter <strong>of</strong> <strong>the</strong> algorithm.Anti-k T AlgorithmThe detailed description <strong>of</strong> mak<strong>in</strong>g anti-k T jets is provided here.• start<strong>in</strong>g from an <strong>in</strong>put list <strong>of</strong> objects, <strong>the</strong> algorithm looks for <strong>the</strong> smallest distanceamong all d i and d ij whose def<strong>in</strong>itions areand R ij = √ (η i − η j ) 2 + (φ i − φ j ) 2 ;d i ≡ 12E T id ij ≡ m<strong>in</strong>{ 1 12,2} × ( R ijE T i E T jR )2 ,,• if <strong>the</strong> smallest is a d i , particle i is removed from <strong>the</strong> list <strong>of</strong> particles and called a jet;• if <strong>the</strong> smallest is a d ij , particles i and j are recomb<strong>in</strong>ed to form one s<strong>in</strong>gle newparticle accord<strong>in</strong>g toE T = Σ i E T i , η = Σ i(η i × E T i )E T, φ = Σ i(φ i × E T i )E T;• <strong>the</strong> procedure is repeated until no particles are left.Also here, R is a free parameter <strong>of</strong> <strong>the</strong> algorithm.This algorithm is currently highly recommended with<strong>in</strong> <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>. The reasonis generally that, with respect to <strong>the</strong> iterative cone, <strong>the</strong> anti-k T algorithm is welldef<strong>in</strong>ed <strong>in</strong> <strong>the</strong> sense that it does not have <strong>the</strong> defects such as Infrared and Coll<strong>in</strong>earunsafety which means that <strong>the</strong> algorithm is stable with respect to <strong>the</strong> s<strong>of</strong>t radiationand coll<strong>in</strong>ear decomposition, respectively. In o<strong>the</strong>r words <strong>the</strong> output <strong>of</strong> <strong>the</strong> anti-k Tjet f<strong>in</strong>d<strong>in</strong>g algorithm does not change if any m<strong>in</strong>or reconfiguration <strong>in</strong> <strong>the</strong> <strong>in</strong>put list <strong>of</strong><strong>the</strong> objects, such as add<strong>in</strong>g a s<strong>of</strong>t radiation, occures. However, <strong>the</strong> iterative cone jetf<strong>in</strong>d<strong>in</strong>g algorithm has a fast response and is currently used <strong>in</strong> <strong>the</strong> s<strong>of</strong>tware part <strong>of</strong> <strong>the</strong>
CHAPTER 3: Object Reconstruction 35onl<strong>in</strong>e trigger<strong>in</strong>g system.In this study, only anti-k T jets are processed for which <strong>the</strong> R parameter is set to be0.5.3.2.2 <strong>Jet</strong> Reconstruction PerformanceIn order to evaluate <strong>the</strong> performance <strong>of</strong> <strong>the</strong> jet reconstruction algorithms, <strong>the</strong> createdCalo<strong>Jet</strong>s are compared to <strong>the</strong> particle level jets, <strong>the</strong> so-called Gen<strong>Jet</strong>s, which are <strong>in</strong>dependent<strong>of</strong> <strong>the</strong> response <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector. Gen<strong>Jet</strong>s are made by apply<strong>in</strong>g <strong>the</strong>same jet reconstruction algorithm on <strong>the</strong> stable particles which are com<strong>in</strong>g out <strong>of</strong> <strong>the</strong>hadronisation process. The Gen<strong>Jet</strong>s are not observable jets, because <strong>the</strong>y can only beaccessed from <strong>the</strong> simulated <strong>in</strong>formation. Equivalently, <strong>the</strong> actual detected jets areCalo<strong>Jet</strong>s which are Gen<strong>Jet</strong>s right after <strong>the</strong>y reach <strong>the</strong> detector. The jet match<strong>in</strong>g efficiencywhich is def<strong>in</strong>ed as <strong>the</strong> fraction <strong>of</strong> <strong>the</strong> Gen<strong>Jet</strong>s that match to <strong>the</strong> Calo<strong>Jet</strong>s with<strong>in</strong>a ∆R cone <strong>of</strong> 0.5, is shown <strong>in</strong> Figure 3.3 for various jet cluster<strong>in</strong>g algorithms [69].Figure 3.3: Efficiency <strong>of</strong> match<strong>in</strong>g a reconstructed Calo<strong>Jet</strong> to a Gen<strong>Jet</strong> as a function<strong>of</strong> p Gen<strong>Jet</strong>T for various jet cluster<strong>in</strong>g algorithms [69].Accord<strong>in</strong>g to Figure 3.3, <strong>the</strong> probability <strong>of</strong> f<strong>in</strong>d<strong>in</strong>g calorimeter jets that match togenerated jets with, for example, p T = 30 GeV is larger than 95%. The drop <strong>in</strong> <strong>the</strong>efficiency at low p T is due to <strong>the</strong> presence <strong>of</strong> <strong>the</strong> magnetic field. S<strong>in</strong>ce low-p T chargedhadrons have a smaller bend<strong>in</strong>g radius, <strong>the</strong>refore <strong>the</strong>y escape from <strong>the</strong> cone which iscast arount <strong>the</strong> jet and results <strong>in</strong> a wrong determ<strong>in</strong>ation <strong>of</strong> <strong>the</strong> direction <strong>of</strong> <strong>the</strong> jet.Ano<strong>the</strong>r parameter that describes how well a jet cluster<strong>in</strong>g algorithm works, is <strong>the</strong>
36 CHAPTER 3: Object Reconstructioncalorimeter response to jets R(η, p Gen<strong>Jet</strong>T ) which is def<strong>in</strong>ed asR(η, p Gen<strong>Jet</strong>T) = pCalo<strong>Jet</strong> T.p Gen<strong>Jet</strong>TThe plot <strong>of</strong> <strong>the</strong> jet response as a function <strong>of</strong> <strong>the</strong> p T <strong>of</strong> <strong>the</strong> Gen<strong>Jet</strong>, can be found <strong>in</strong>Figure 3.4. It is seen that <strong>the</strong> response is not equal to one which means that, forexample, for a Calo<strong>Jet</strong> with true p T = 100 GeV , only about 60% <strong>of</strong> its energy can becollected. The effect <strong>of</strong> out-<strong>of</strong>-cone shower<strong>in</strong>g can expla<strong>in</strong> partly why <strong>the</strong> non-unitybehaviour <strong>of</strong> <strong>the</strong> jet response function is observed. Due to that <strong>the</strong> low-p T hadronswhich are produced <strong>in</strong> <strong>the</strong> hadronization step are mostly affected by <strong>the</strong> magneticfield and have usually a small bend<strong>in</strong>g radius, hence <strong>the</strong>y escape from <strong>the</strong> cascade <strong>of</strong>hadrons and do not contribute <strong>in</strong> <strong>the</strong> cone <strong>of</strong> <strong>the</strong> cluster<strong>in</strong>g algorithm and results <strong>in</strong>less collected energy compared to <strong>the</strong> energy <strong>of</strong> <strong>the</strong> <strong>in</strong>itiat<strong>in</strong>g parton.Figure 3.4: The detector response to calorimeter jets as a function <strong>of</strong> p Gen<strong>Jet</strong>Tjet cluster<strong>in</strong>g algorithms [69].for variousIn order to solve this issue and make <strong>the</strong> jet response equal to one, <strong>the</strong> jet calibrationidea is proposed. The calibration <strong>of</strong> <strong>the</strong> jet energy, which is <strong>the</strong> subject <strong>of</strong> <strong>the</strong> nextsection, corrects <strong>the</strong> energy <strong>of</strong> a Calo<strong>Jet</strong> back to <strong>the</strong> true energy <strong>of</strong> its matched Gen<strong>Jet</strong>.3.2.3 <strong>Jet</strong> <strong>Energy</strong> CalibrationA factorized multi-level jet correction has been developed <strong>in</strong> <strong>CMS</strong> [73, 74]. The firstrequired level <strong>of</strong> correction called Level 1 <strong>of</strong>fset, is to apply noise and pile-up suppressionbefore <strong>in</strong>vok<strong>in</strong>g <strong>the</strong> jet algorithm with ei<strong>the</strong>r a set <strong>of</strong> cuts on <strong>the</strong> calotower energies
CHAPTER 3: Object Reconstruction 37or to apply a pile-up subtraction algorithm. The rema<strong>in</strong><strong>in</strong>g required correction levelsare tw<strong>of</strong>old. A Level 2 (L2) relative η correction which corrects for variations withη relative to a control region and ano<strong>the</strong>r Level 3 (L3) absolute p T correction whichcorrects on average <strong>the</strong> Calo<strong>Jet</strong> p T back to Gen<strong>Jet</strong> p T . Both L2 and L3 corrections aredescribed below.L3 Absolute p T Correction• Method with SimulationCalo<strong>Jet</strong>s from simulated QCD dijet events are considered <strong>in</strong> order to derive <strong>the</strong>simulation based calibration factors to be used to scale <strong>the</strong> Calo<strong>Jet</strong> Lorentz vectors.A Calo<strong>Jet</strong> is required to be located <strong>in</strong> <strong>the</strong> control region def<strong>in</strong>ed to be <strong>the</strong>barrel part <strong>of</strong> <strong>the</strong> HCAL with |η| < 1.3 and also it is required to match with aGen<strong>Jet</strong> <strong>in</strong> ∆R = √ (∆η) 2 + (∆φ) 2 < 0.25. For <strong>the</strong> various b<strong>in</strong>s <strong>of</strong> Gen<strong>Jet</strong> p T andfor each pair <strong>of</strong> matched jets, <strong>the</strong> quantity ∆p T = p Calo<strong>Jet</strong>T − p Gen<strong>Jet</strong>T is calculatedand <strong>the</strong> correspond<strong>in</strong>g distribution is fitted with a Gaussian. A typical ∆p T distributioncalculated for Gen<strong>Jet</strong>s with 150 GeV < p Gen<strong>Jet</strong>T<strong>in</strong> Figure 3.5.< 200 GeV , is shownFigure 3.5: The distribution <strong>of</strong> ∆p T for jets reconstructed with <strong>the</strong> iterative conealgorithm and located <strong>in</strong> <strong>the</strong> control region for <strong>the</strong> specified p Gen<strong>Jet</strong>T b<strong>in</strong> [75].It is seen from Figure 3.5 that <strong>the</strong> distribution <strong>of</strong> a generic ∆p T is peaked at negativevalues, as expected due to <strong>the</strong> underestimation <strong>of</strong> <strong>the</strong> p T <strong>of</strong> <strong>the</strong> calorimeterjets. The calorimeter response to jets which is already def<strong>in</strong>ed <strong>in</strong> Section 3.2.2,can be obta<strong>in</strong>ed from <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> ∆p T variable byR(η, p Gen<strong>Jet</strong>T ) = 1 + < ∆p T >,p Gen<strong>Jet</strong>Twhere < ∆p T > is <strong>the</strong> mean value <strong>of</strong> <strong>the</strong> fit result <strong>in</strong> each b<strong>in</strong> <strong>of</strong> <strong>the</strong> p T <strong>of</strong> <strong>the</strong>Gen<strong>Jet</strong>. F<strong>in</strong>ally, <strong>the</strong> absolute jet response is <strong>in</strong>verted numerically to obta<strong>in</strong> <strong>the</strong>absolute jet correction C abs. (η, p Calo<strong>Jet</strong>T ), henceC abs. (η, p Calo<strong>Jet</strong> 1T ) =R(η, p Gen<strong>Jet</strong>T) .
38 CHAPTER 3: Object ReconstructionIt should be noted that <strong>the</strong> absolute jet correction is expressed <strong>in</strong> terms <strong>of</strong> p Calo<strong>Jet</strong>T .Both <strong>the</strong> absolute jet response and <strong>the</strong> absolute jet calibration functions areshown <strong>in</strong> Figure 3.6.Figure 3.6: The jet response as a function <strong>of</strong> p Gen<strong>Jet</strong>T (left) and L3 correction factors asa function <strong>of</strong> p Calo<strong>Jet</strong>T (right) derived from <strong>the</strong> simulated <strong>in</strong>formation [73].Accord<strong>in</strong>g to <strong>the</strong> absolute calibration curve shown <strong>in</strong> Figure 3.6, a calorimeter jetwith a measured transverse momentum <strong>of</strong> about 100 GeV should be multiplied bya factor <strong>of</strong> 1.5 to represent <strong>the</strong> true transverse momentum, result<strong>in</strong>g <strong>in</strong> a jet withp T = 150 GeV . Also it can be understood that <strong>the</strong> high-p T jets receive smallercalibration factors compared to <strong>the</strong> low-p T jets which require larger correctionfactors.• Data Driven MethodIn a data driven method, γ+jet or Z+jet events selected <strong>in</strong> <strong>the</strong> collision data, canbe used to extract <strong>the</strong> jet correction factors. Assum<strong>in</strong>g <strong>the</strong> conservation <strong>of</strong> <strong>the</strong>energy <strong>in</strong> <strong>the</strong> transverse plane, a p T balance technique can be used to compare<strong>the</strong> reconstructed Calo<strong>Jet</strong> p T with <strong>the</strong> transverse momentum <strong>of</strong> <strong>the</strong> producedphoton or Z boson. S<strong>in</strong>ce photon reconstruction only relies on <strong>the</strong> <strong>in</strong>formation<strong>of</strong> <strong>the</strong> ECAL which has a better precision on <strong>the</strong> energy resolution compared to<strong>the</strong> HCAL, <strong>the</strong> jet p T measurement <strong>of</strong> <strong>the</strong> HCAL is corrected for <strong>the</strong> photon p Tmeasurement <strong>of</strong> <strong>the</strong> ECAL. The γ+jet events suffer from huge QCD backgroundswhich can be suppressed by ask<strong>in</strong>g for isolated photons.Complementary a collection <strong>of</strong> Z+jet events <strong>of</strong> which <strong>the</strong> Z boson decays to apair <strong>of</strong> muons, provides a clean signature <strong>of</strong> <strong>the</strong> reconstructed Z boson and issuitable to be used to derive <strong>the</strong> jet energy calibration factors. In this case, <strong>the</strong>HCAL sub-detector is compared to <strong>the</strong> muon chamber <strong>of</strong> <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>.The calibration curve for <strong>the</strong> jet energy which has been derived from simulation iscompared <strong>in</strong> Figure 3.7 with one <strong>of</strong> <strong>the</strong> data driven methods, namely <strong>the</strong> Z+jetbalance. A good agreement between <strong>the</strong> two procedures for obta<strong>in</strong><strong>in</strong>g <strong>the</strong> jetcalibration factors can be observed.
CHAPTER 3: Object Reconstruction 39Figure 3.7: Comparison between <strong>the</strong> L3 calibration curve derived from <strong>the</strong> simulationand <strong>the</strong> Z+jet balance method applied on collision data, and <strong>the</strong> ratio <strong>of</strong> <strong>the</strong> twocorrections [73].L2 Relative η Correction• Method with SimulationThe purpose <strong>of</strong> <strong>the</strong> L2 correction is to correct jets at arbitrary η relative to acontrol region where <strong>the</strong> absolute calibration is easier. As a consequence <strong>of</strong> <strong>the</strong>L2 correction, <strong>the</strong> jet response becomes flat as a function <strong>of</strong> <strong>the</strong> pseudorapidity.In order to derive <strong>the</strong> L2 correction factors from <strong>the</strong> simulation, it is essential tocalculate <strong>the</strong> η dependent L3 absolute correction, C abs. (η, p Calo<strong>Jet</strong>T ), as <strong>in</strong>troducedabove. This is achieved by repeat<strong>in</strong>g exactly <strong>the</strong> same procedure described <strong>in</strong> <strong>the</strong>“L3 Absolute p T Correction” part for multiple η regions.The relative correction is def<strong>in</strong>ed asC rel. (η, p Calo<strong>Jet</strong>T) = pcontrol T,p Calo<strong>Jet</strong>Twhere p controlT and p Calo<strong>Jet</strong>T are <strong>the</strong> measured transverse momenta <strong>of</strong> a Gen<strong>Jet</strong> <strong>of</strong><strong>the</strong> same p Gen<strong>Jet</strong>T <strong>in</strong> <strong>the</strong> control region and <strong>in</strong> an arbitrary η region, respectively.Start<strong>in</strong>g from a reconstructed Calo<strong>Jet</strong> with arbitrary η and with p Calo<strong>Jet</strong>T , <strong>the</strong>absolute correction obta<strong>in</strong>ed for that specific η region can be used to f<strong>in</strong>d <strong>the</strong>true value <strong>of</strong> <strong>the</strong> Calo<strong>Jet</strong> p T , be<strong>in</strong>g p Gen<strong>Jet</strong>Tp Gen<strong>Jet</strong>T= p Calo<strong>Jet</strong>T× C abs. (η, p Calo<strong>Jet</strong>T ).Then with <strong>the</strong> use <strong>of</strong> <strong>the</strong> response <strong>of</strong> <strong>the</strong> detector <strong>in</strong> <strong>the</strong> control region, one canf<strong>in</strong>d <strong>the</strong> value <strong>of</strong> <strong>the</strong> reconstructed Calo<strong>Jet</strong> p T <strong>in</strong> <strong>the</strong> control region for <strong>the</strong> samep Gen<strong>Jet</strong>Tp controlT= R(control, p Gen<strong>Jet</strong>T ) × p Gen<strong>Jet</strong>T .
40 CHAPTER 3: Object ReconstructionHav<strong>in</strong>g determ<strong>in</strong>ed <strong>the</strong> p controlT , it is possible to calculate <strong>the</strong> relative jet correctionas=C rel (η, p Calo<strong>Jet</strong>TR(control, pGen<strong>Jet</strong>R(η, p Gen<strong>Jet</strong>T )T)) = pcontrol Tp Calo<strong>Jet</strong>T== C abs. (η, p Calo<strong>Jet</strong>TR(control, pGen<strong>Jet</strong> Tp Calo<strong>Jet</strong>T) × p Gen<strong>Jet</strong>)×R(control, C abs. (η, p Calo<strong>Jet</strong>TT)×p Calo<strong>Jet</strong>T ).Figure 3.8 shows <strong>the</strong> jet response as a function <strong>of</strong> pseudorapidity for Calo<strong>Jet</strong>sbefore and after apply<strong>in</strong>g L2 corrections. The response <strong>of</strong> <strong>the</strong> detector to <strong>the</strong>jets reconstructed <strong>in</strong> various parts <strong>of</strong> <strong>the</strong> detector becomes flat after apply<strong>in</strong>g L2relative η correction. Also <strong>the</strong> same figure conta<strong>in</strong>s <strong>the</strong> corrected jet responseconsider<strong>in</strong>g both L2 and L3 correction steps.Figure 3.8: The detector response to Calo<strong>Jet</strong>s as a function <strong>of</strong> η without correction(filled circles), with L2 correction (open squares) and with <strong>the</strong> comb<strong>in</strong>ed L2 & L3corrections (open circles) [74].• Data Driven MethodQCD dijet events for which one <strong>of</strong> <strong>the</strong> reconstructed Calo<strong>Jet</strong>s is located outside<strong>the</strong> control region, def<strong>in</strong>ed to be <strong>the</strong> barrel part <strong>of</strong> <strong>the</strong> HCAL with |η| < 1.3, areused. In <strong>the</strong> limit when no o<strong>the</strong>r jet is present <strong>in</strong> <strong>the</strong> dijet topology, <strong>the</strong> p T <strong>of</strong> <strong>the</strong>two lead<strong>in</strong>g jets should be <strong>the</strong> same. Any difference between <strong>the</strong> measurements<strong>of</strong> <strong>the</strong> p T <strong>of</strong> <strong>the</strong> two jets is translated <strong>in</strong>to a calibration factor which is needed to
CHAPTER 3: Object Reconstruction 41be applied on <strong>the</strong> Calo<strong>Jet</strong> p T reconstructed outside <strong>the</strong> barrel. This makes <strong>the</strong>calorimeter response to jets flat as a function <strong>of</strong> η and equal to <strong>the</strong> value obta<strong>in</strong>edfor <strong>the</strong> jets which are reconstructed <strong>in</strong> <strong>the</strong> control region.The result<strong>in</strong>g relative response function obta<strong>in</strong>ed from dijet balanc<strong>in</strong>g is comparedwith <strong>the</strong> one derived from <strong>the</strong> simulation study, which is shown <strong>in</strong> Figure3.9. The agreement between <strong>the</strong>se two confirms <strong>the</strong> validity <strong>of</strong> <strong>the</strong> proposeddata driven method.Figure 3.9: The relative jet response as a function <strong>of</strong> η from both <strong>the</strong> dijet balancemethod and <strong>the</strong> simulation for 200 GeV < p dijetT< 300 GeV [73].3.2.4 <strong>Jet</strong> <strong>Energy</strong> ResolutionNow that jets are corrected, it is <strong>the</strong> time to check how successful <strong>the</strong> correction stepsL2 and L3 are. <strong>Jet</strong> energy resolution is <strong>the</strong> uncerta<strong>in</strong>ty on <strong>the</strong> measured jet energy afterapply<strong>in</strong>g <strong>the</strong> required corrections. Different methods for extract<strong>in</strong>g <strong>the</strong> resolutions on<strong>the</strong> energy <strong>of</strong> <strong>the</strong> jets exist and are discussed below [76].• Method with SimulationSo-called Monte Carlo resolutions are obta<strong>in</strong>ed from simulated data by plott<strong>in</strong>g<strong>the</strong> distributions <strong>of</strong> <strong>the</strong> ratio between <strong>the</strong> corrected Calo<strong>Jet</strong> p T and <strong>the</strong> correspond<strong>in</strong>gGen<strong>Jet</strong> p T as a function <strong>of</strong> <strong>the</strong> p T <strong>of</strong> <strong>the</strong> Gen<strong>Jet</strong>. The Calo<strong>Jet</strong>s arespatially matched to <strong>the</strong> Gen<strong>Jet</strong>s with ∆R = √ (∆η) 2 + (∆φ) 2 < 0.25. Eachdistribution is subsequently fitted with a Gaussian and <strong>the</strong> width <strong>of</strong> <strong>the</strong> fit isused as <strong>the</strong> resolution for that particular p T b<strong>in</strong>.Figure 3.10 shows a generic distribution <strong>of</strong> <strong>the</strong> jet response after apply<strong>in</strong>g L2 andL3 corrections, overimposed with a fitted Gaussian function.• Data Driven MethodThe data driven method for calculat<strong>in</strong>g <strong>the</strong> jet p T resolutions is based on QCD
42 CHAPTER 3: Object ReconstructionFigure 3.10: The jet response after apply<strong>in</strong>g L2 and L3 corrections for a particularp Gen<strong>Jet</strong>T b<strong>in</strong> specified <strong>in</strong> <strong>the</strong> plot. The width <strong>of</strong> <strong>the</strong> fitted function is used to extract <strong>the</strong>jet energy resolutions.dijet events. The parameter A, which is called “Asymmetry”, is used to derive<strong>the</strong> jet resolutions and def<strong>in</strong>ed to beA = (pjet1 T(p jet1T− p jet2T)+ p jet2T) ,where p jet1Tand p jet2Tare <strong>the</strong> transverse momenta <strong>of</strong> <strong>the</strong> two lead<strong>in</strong>g jets <strong>in</strong> eachQCD dijet event. The variance σ A can be measured and simply written asσ 2 A = | ∂A∂p jet1T| 2 σ 2 p jet1T+ |∂A∂p jet2T| 2 σ 2 .p jet2TUs<strong>in</strong>g <strong>the</strong> idea <strong>of</strong> <strong>the</strong> momentum conservation <strong>in</strong> <strong>the</strong> transverse plane <strong>of</strong> <strong>the</strong> dijetevents, one can def<strong>in</strong>e p T ≡ p jet1T= p jet2T. The two lead<strong>in</strong>g jets are fur<strong>the</strong>r requiredto be <strong>in</strong> <strong>the</strong> same η region, so that any dependency on <strong>the</strong> resolution parametersorig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong> differences <strong>in</strong> <strong>the</strong> pseudorapidity regions is removed and onecan assume that σ pT ≡ σp jet1T= σp jet2T. Hence <strong>the</strong> fractional jet p T resolution canbe obta<strong>in</strong>ed from <strong>the</strong> width <strong>of</strong> th <strong>the</strong> Asymmetry parameter <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g way(σ pTp T)=√2σA .The resolutions derived from <strong>the</strong> Asymmetry method are only valid <strong>in</strong> <strong>the</strong> limit<strong>of</strong> exactly two jets <strong>in</strong> <strong>the</strong> QCD dijet events. In <strong>the</strong> case when o<strong>the</strong>r jets arepresent, <strong>the</strong> assumption <strong>of</strong> <strong>the</strong> equality <strong>of</strong> <strong>the</strong> transverse momenta <strong>of</strong> <strong>the</strong> twolead<strong>in</strong>g jets is no longer valid. Hence <strong>in</strong> order to elim<strong>in</strong>ate <strong>the</strong> radiated jetswhich spoil <strong>the</strong> assumption <strong>of</strong> p T -balanc<strong>in</strong>g for <strong>the</strong> two lead<strong>in</strong>g jets, a cut on <strong>the</strong>transverse momentum <strong>of</strong> <strong>the</strong> third lead<strong>in</strong>g jet is applied, p jet3T< p cutT . Therefore<strong>the</strong> resolution values are calculated for different threshold cuts <strong>in</strong> b<strong>in</strong>s <strong>of</strong> p avrT =
CHAPTER 3: Object Reconstruction 43(p jet1T+p jet2T ). As an example, <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> Asymmetry variable for <strong>the</strong>2calorimeter jets with 125 < p avrT < 160 GeV and p jet3T< 15 GeV , is shown <strong>in</strong>Figure 3.11.Figure 3.11: The distribution <strong>of</strong> <strong>the</strong> Asymmetry variable for calorimeter jets with125 < p avrT < 160 GeV and |η| < 1.1. The third lead<strong>in</strong>g jet is required to have a p Tless than 15 GeV [76].The Asymmetry distributions for various threshold cuts on <strong>the</strong> p jet3Tare fitted witha Gaussian. The jet resolutions for each b<strong>in</strong> <strong>of</strong> p avrT can <strong>the</strong>n be plotted as a function<strong>of</strong> p cutT and fitted with a l<strong>in</strong>e. The true jet resolution value is obta<strong>in</strong>d by extrapolat<strong>in</strong>g<strong>the</strong> fitted l<strong>in</strong>e to p cutT = 0 which corresponds to a QCD event with exactly two jets.For <strong>the</strong> same p avrT as <strong>the</strong> one shown <strong>in</strong> Figure 3.11, <strong>the</strong> correspond<strong>in</strong>g jet resolution isshown <strong>in</strong> Figure 3.12.Figure 3.12: The fractional energy resolution as a function <strong>of</strong> <strong>the</strong> p jet3Tthreshold <strong>in</strong> <strong>the</strong>same range <strong>of</strong> p avrT and η as Figure 3.11 derived from Asymmetry method [76].
44 CHAPTER 3: Object ReconstructionThe validation <strong>of</strong> <strong>the</strong> determ<strong>in</strong>ation <strong>of</strong> <strong>the</strong> jet energy resolutions is done by compar<strong>in</strong>g<strong>the</strong> performance <strong>of</strong> both <strong>the</strong> simulation driven and data driven dijet Asymmetrymethods and is shown <strong>in</strong> Figure 3.13.Figure 3.13: Comparisons <strong>of</strong> <strong>the</strong> jet energy resolutions from <strong>the</strong> simulation and Asymmetrymethod for calorimeter jets with |η| < 1.2 [76].There is a good agreement between <strong>the</strong> simulation based and <strong>the</strong> data driven methodsto determ<strong>in</strong>e <strong>the</strong> relative resolution on <strong>the</strong> transverse momentum <strong>of</strong> <strong>the</strong> reconstructedjets, as can be observed from Figure 3.13. A first observation is that <strong>the</strong>resolution decreases for <strong>the</strong> jets with higher transverse momenta. This means <strong>the</strong> moreenergetic jets are measured with a better precision compared to <strong>the</strong> less energetic jets.It can also be understood that a calorimeter jet with an energy <strong>of</strong>, for example, about100 GeV, can be determ<strong>in</strong>ed with an accuracy <strong>of</strong> about 15%.3.2.5 <strong>Jet</strong>-Electron Clean<strong>in</strong>gS<strong>in</strong>ce <strong>the</strong> same detector <strong>in</strong>formation like energy deposits <strong>in</strong> <strong>the</strong> calorimeters is used toreconstruct several physics objects, <strong>the</strong>re is always a possibility to double count <strong>the</strong>physics objects. In <strong>the</strong> analyses that conta<strong>in</strong> both electrons and jets <strong>in</strong> <strong>the</strong>ir f<strong>in</strong>al states,specific attention should be paid to <strong>the</strong> clean<strong>in</strong>g <strong>of</strong> <strong>the</strong> jet collection from electrons aselectrons are always re-reconstructed as jets. The idea <strong>of</strong> <strong>the</strong> clean<strong>in</strong>g method is to lookfor electrons <strong>in</strong> <strong>the</strong> event and <strong>the</strong>n remove any adjacent reconstructed jet. This is <strong>the</strong>default clean<strong>in</strong>g method that is also used <strong>in</strong> this analysis. Also <strong>the</strong>re are some o<strong>the</strong>rstudies which propose to start <strong>the</strong> clean<strong>in</strong>g even before jet reconstruction step [77]. Thisway <strong>the</strong> clean<strong>in</strong>g is done by look<strong>in</strong>g at <strong>the</strong> more low-level objects, ra<strong>the</strong>r than <strong>the</strong> highlevelphysics objects such as jets, and remov<strong>in</strong>g any calotower which is reconstructedclose to <strong>the</strong> electrons, followed by re-runn<strong>in</strong>g <strong>the</strong> jet cluster<strong>in</strong>g algorithm over <strong>the</strong>
CHAPTER 3: Object Reconstruction 45cleaned collection <strong>of</strong> calotowers. The results between both <strong>of</strong> <strong>the</strong>se proposed clean<strong>in</strong>gmethods are comparable. This issue is studied <strong>in</strong> more detail <strong>in</strong> <strong>the</strong> next chapter.3.3 MET ReconstructionNeutr<strong>in</strong>os which appear <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> many physics processes, rema<strong>in</strong> undetecteddue to <strong>the</strong>ir weak <strong>in</strong>teractions with matter. They are <strong>the</strong> only Standard Model particlesthat escape detection without deposit<strong>in</strong>g any energy <strong>in</strong> <strong>the</strong> <strong>CMS</strong> detector. S<strong>in</strong>ce <strong>the</strong><strong>in</strong>itial proton beams are supposed to collide heads-on, <strong>the</strong> energy is conserved <strong>in</strong> <strong>the</strong>plane perpendicular to <strong>the</strong> beam l<strong>in</strong>e. The miss<strong>in</strong>g transverse energy, referred to asETmiss , is <strong>the</strong>refore understood as <strong>the</strong> miss<strong>in</strong>g amount <strong>of</strong> energy deduced from <strong>the</strong> energybalanc<strong>in</strong>g <strong>in</strong> <strong>the</strong> transverse plane.The miss<strong>in</strong>g transverse energy is def<strong>in</strong>ed as <strong>the</strong> transverse part <strong>of</strong> <strong>the</strong> vectorial sum <strong>of</strong>all <strong>the</strong> energy deposits collected <strong>in</strong> calotowers⃗E missT= −Σ n (E n s<strong>in</strong>θ n cosφ n ⃗x + E n s<strong>in</strong>θ n s<strong>in</strong>φ n ⃗y),where θ and φ are <strong>the</strong> polar and azimuthal angles <strong>of</strong> <strong>the</strong> calotowers, respectively. Bydef<strong>in</strong>ition <strong>the</strong> uncorrected calotowers are used to calculate <strong>the</strong> ETmiss . The performance<strong>of</strong> ETmiss has been fur<strong>the</strong>r improved when corrections due to leptons, such as muon, andjets are applied [78]. For example as muons are also weakly <strong>in</strong>teract<strong>in</strong>g particles, <strong>the</strong>ydo not deposit <strong>the</strong>ir energy <strong>in</strong> <strong>the</strong> calorimeter system. Hence for events that conta<strong>in</strong>muons, <strong>the</strong> transverse momentum <strong>of</strong> <strong>the</strong> reconstructed muons should be subtractedfrom <strong>the</strong> above formula. Figure 3.14 shows <strong>the</strong> resolution <strong>of</strong> <strong>the</strong> miss<strong>in</strong>g transverseenergy for <strong>in</strong>clusive t¯t events as a function <strong>of</strong> <strong>the</strong> reconstructed ET miss .It is seen from Figure 3.14 that <strong>in</strong> t¯t events, <strong>the</strong> resolution on <strong>the</strong> miss<strong>in</strong>g transverseenergy grows with <strong>the</strong> reconstructed ETmiss . Hence <strong>in</strong> order to avoid large systematicuncerta<strong>in</strong>ties that can enter <strong>in</strong>to <strong>the</strong> analysis by cutt<strong>in</strong>g on ETmiss , <strong>the</strong> reconstructedmiss<strong>in</strong>g transverse energy is not processed <strong>in</strong> this study.
46 CHAPTER 3: Object ReconstructionFigure 3.14: The resolution <strong>of</strong> <strong>the</strong> miss<strong>in</strong>g transverse energy before (open circles)and after jet corrections (filled circles) for <strong>in</strong>clusive t¯t events as a function <strong>of</strong> <strong>the</strong>reconstructed ET miss [44].
Chapter 4Reconstruct<strong>in</strong>g and Select<strong>in</strong>gPhysics Event4.1 Monte Carlo Event SimulationIn <strong>the</strong> absence <strong>of</strong> <strong>experiment</strong>al data and to model <strong>the</strong> available physics <strong>the</strong>ories, <strong>the</strong> toyMonte Carlo data is produced. The first step <strong>in</strong> <strong>the</strong> Monte Carlo production cha<strong>in</strong> isgenerat<strong>in</strong>g physics “events”, i.e. sets <strong>of</strong> outgo<strong>in</strong>g particles produced <strong>in</strong> <strong>the</strong> <strong>in</strong>teractionsbetween two <strong>in</strong>com<strong>in</strong>g particles, followed by simulat<strong>in</strong>g <strong>the</strong> detector response based on<strong>the</strong> <strong>in</strong>teractions <strong>of</strong> particles produced <strong>in</strong> <strong>the</strong> event with <strong>the</strong> material <strong>of</strong> <strong>the</strong> detector.Hence a very precise model<strong>in</strong>g <strong>of</strong> physics and a good understand<strong>in</strong>g <strong>of</strong> <strong>the</strong> detectorshould result <strong>in</strong> simulated events similar to those from real collision.The whole generation cha<strong>in</strong> <strong>of</strong> an event can be factorized <strong>in</strong>to several stages as illustrated<strong>in</strong> Figure 4.1.Figure 4.1: Illustration <strong>of</strong> <strong>the</strong> event generation cha<strong>in</strong> <strong>in</strong> proton-proton collisionsadapted from [79].47
48 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics EventThe first step which is <strong>the</strong> core <strong>of</strong> <strong>the</strong> generation procedure is to produce <strong>the</strong>hard scatter<strong>in</strong>g process which is a result <strong>of</strong> <strong>the</strong> <strong>in</strong>teraction <strong>of</strong> two high-energy partonsaris<strong>in</strong>g from <strong>the</strong> two collid<strong>in</strong>g protons. The next step is to consider <strong>the</strong> possibility <strong>of</strong>branch<strong>in</strong>g <strong>of</strong> a parton to o<strong>the</strong>r partons or radiation <strong>of</strong> a gluon result<strong>in</strong>g <strong>in</strong> a cascade<strong>of</strong> partons referred to as parton shower<strong>in</strong>g. After produc<strong>in</strong>g a list <strong>of</strong> partons carry<strong>in</strong>gcolor-charges, due to <strong>the</strong> quark conf<strong>in</strong>ement pr<strong>in</strong>ciple, <strong>the</strong>y hadronize and comb<strong>in</strong>e<strong>in</strong> such a way to make colorless hadrons. The produced hadrons that are not stableparticles, decay subsequently <strong>in</strong> <strong>the</strong> last step <strong>of</strong> <strong>the</strong> event generation procedure. Thef<strong>in</strong>al particles <strong>in</strong>teract with <strong>the</strong> detector material and can be reconstructed as described<strong>in</strong> Chapter 3.Some <strong>of</strong> <strong>the</strong>se generation steps such as <strong>the</strong> hard scatter<strong>in</strong>g can be described fromfirst pr<strong>in</strong>ciples, while for o<strong>the</strong>rs like <strong>the</strong> hadronization stage require phenomenologicalmodels to be modeled. The different generation steps are discussed <strong>in</strong> more detail <strong>in</strong><strong>the</strong> follow<strong>in</strong>g sections.4.1.1 Hard Scatter<strong>in</strong>g EventIn a proton-proton collision, along with so many s<strong>of</strong>t elastic <strong>in</strong>teractions, hard <strong>in</strong>elasticscatter<strong>in</strong>g occurs which can <strong>in</strong>clude any new physics such as Higgs bosons or h<strong>in</strong>ts forsuper-symmetric phenomena. In order to describe <strong>the</strong> physics at high-energy hadroncolliders, <strong>the</strong> knowledge <strong>of</strong> Quantum Chromodynamics (QCD) is needed [80]. Accord<strong>in</strong>gto <strong>the</strong> parton model, a proton is not an elementary particle and comprises quarksand gluons, generally called partons. When two protons collide heads on, actually <strong>the</strong>collision happens between <strong>the</strong>ir constituents. Partons with<strong>in</strong> <strong>the</strong> proton are describedby <strong>the</strong> parton distribution functions (pdfs) which are represented as a function <strong>of</strong> <strong>the</strong>momentum fraction carried by <strong>the</strong> parton <strong>in</strong>side <strong>the</strong> proton and also with <strong>the</strong> Q 2 energyscale at which <strong>the</strong> proton is probed. At high center <strong>of</strong> mass energies where <strong>the</strong> strongcoupl<strong>in</strong>g constant α s is small, perturbative calculations are feasible and <strong>the</strong> collisionbetween <strong>the</strong> two <strong>in</strong>com<strong>in</strong>g protons can be described accord<strong>in</strong>gly. A hard scatter<strong>in</strong>gprocesses is schematically depicted <strong>in</strong> Figure 4.2.Figure 4.2: A typical hard scatter<strong>in</strong>g process <strong>in</strong> proton-proton collision.For <strong>the</strong> hard scatter<strong>in</strong>g <strong>in</strong>teractions, <strong>the</strong> factorization <strong>the</strong>orem which predicts <strong>the</strong>characteristics <strong>of</strong> <strong>the</strong> event, is used. As a result, <strong>the</strong> total hadronic cross section can
CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event 49be obta<strong>in</strong>ed asσ p+p→X =∫ 10∫ 1dx 1 dx 2 f1(x p 1 , Q 2 )f2(x p 2 , Q 2 )σ 1+2→X .0Hence <strong>the</strong> evaluation <strong>of</strong> <strong>the</strong> total rate for a hard scatter<strong>in</strong>g <strong>in</strong>teraction, reduces to <strong>the</strong>calculation <strong>of</strong> <strong>the</strong> sub-process cross section σ 1+2→X by identify<strong>in</strong>g <strong>the</strong> lead<strong>in</strong>g-orderpartonic processes that contribute to 1 + 2 → X. For example, for <strong>the</strong> top quark pairproduction <strong>the</strong> Feynman diagrams contribut<strong>in</strong>g to <strong>the</strong> f<strong>in</strong>al state X = t¯t at lead<strong>in</strong>gorder <strong>of</strong> perturbation <strong>the</strong>ory are shown <strong>in</strong> Figure 4.3.Figure 4.3: Lead<strong>in</strong>g order Feynman diagrams for <strong>the</strong> production <strong>of</strong> top quark pair viaquark-antiquark annihilation and gluon-gluon fusion.The distribution <strong>of</strong> <strong>the</strong> various pdfs, extracted from deep-<strong>in</strong>elastic scatter<strong>in</strong>g data,are provided by different collaborations such as CTEQ [81] and can be found <strong>in</strong> Figure4.4. They are obta<strong>in</strong>ed at Q = 100GeV which is about <strong>the</strong> order <strong>of</strong> <strong>the</strong> energyneeded to produce a pair <strong>of</strong> heavy particles like top quark. It is seen that for highcenter <strong>of</strong> mass energies correspond<strong>in</strong>g to small distances, gluons are more energeticcompar<strong>in</strong>g to <strong>the</strong> o<strong>the</strong>r quarks. Therefore it can be deduced that at <strong>the</strong> LHC <strong>the</strong> production<strong>of</strong> t¯t via gluon-gluon fusion is <strong>the</strong> dom<strong>in</strong>ant process compared to <strong>the</strong> Tevatronwhere quark-antiquark annihilation is <strong>the</strong> ma<strong>in</strong> channel for top pair production.4.1.2 Parton-Parton Interaction and Event GenerationAccord<strong>in</strong>g to <strong>the</strong> factorization <strong>the</strong>orem described <strong>in</strong> <strong>the</strong> previous section, <strong>the</strong> calculation<strong>of</strong> <strong>the</strong> partonic sub-process cross section σ 1+2→X which is <strong>the</strong>n convoluted with <strong>the</strong>appropriate pdfs is <strong>the</strong> major task <strong>in</strong> evaluat<strong>in</strong>g <strong>the</strong> cross section <strong>of</strong> <strong>the</strong> hard scatter<strong>in</strong>gevent. The partonic cross section can be fur<strong>the</strong>r factorized as followsdσ 1+2→X = 1 F × |M|2 × dcosθdφ,where M is <strong>the</strong> <strong>in</strong>variant amplitude <strong>of</strong> <strong>the</strong> <strong>in</strong>teraction which depends on <strong>the</strong> phasespace variables such as θ or φ and F is proportional to <strong>the</strong> center <strong>of</strong> mass energydsquared. The differential cross section σ dcosθdφ 1+2→X(θ, φ) provides <strong>the</strong> probabilitydensity functions which are <strong>the</strong>n fed <strong>in</strong>to a generator to generate events accord<strong>in</strong>gly.Monte Carlo techniques are used <strong>in</strong> order to generate events randomly accord<strong>in</strong>g to
50 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics EventFigure 4.4: Parton distribution functions xf(x) for quarks, antiquarks and gluons <strong>in</strong><strong>the</strong> proton at Q = 100GeV . These distributions are obta<strong>in</strong>ed from a fit to deep <strong>in</strong>elasticscatter<strong>in</strong>g data performed by <strong>the</strong> CTEQ collaboration (CTEQ6L1) [82].<strong>the</strong>ir differential cross section. Matrix Element event generators are <strong>the</strong> ma<strong>in</strong> s<strong>of</strong>twaretools <strong>in</strong> <strong>the</strong> procedure <strong>of</strong> generat<strong>in</strong>g events as <strong>the</strong>y are responsible for produc<strong>in</strong>g <strong>the</strong>hard scatter<strong>in</strong>g event based on Monte Carlo methods. They provide <strong>the</strong> lead<strong>in</strong>g Feynmandiagrams contribut<strong>in</strong>g to <strong>the</strong> hard process and calculate <strong>the</strong> <strong>in</strong>variant amplitudeM for that hard <strong>in</strong>teraction. Although <strong>the</strong> contributions from higher order correctionsup to Next-to-Lead<strong>in</strong>g Order (NLO) <strong>in</strong> perturbative calculations can also be <strong>in</strong>cluded<strong>in</strong> <strong>the</strong> Matrix Element methods, only <strong>the</strong> effects <strong>of</strong> <strong>the</strong> dom<strong>in</strong>ant terms <strong>in</strong> <strong>the</strong> <strong>in</strong>variantamplitude are usually taken <strong>in</strong>to account. For example, NLO corrections to <strong>the</strong> hardelementary scatter<strong>in</strong>g amplitude is <strong>in</strong>cluded <strong>in</strong> a Matrix Element by consider<strong>in</strong>g <strong>the</strong>emission <strong>of</strong> a hard gluon from <strong>the</strong> <strong>in</strong>itial or f<strong>in</strong>al partonic legs <strong>in</strong> <strong>the</strong> Feynman diagrmswhile ignor<strong>in</strong>g <strong>the</strong> contributions <strong>in</strong>clud<strong>in</strong>g a loop with a virtual gluons. In this sense,s<strong>in</strong>gularities are not present <strong>in</strong> <strong>the</strong> Matrix Element generators. MadGraph [83] andALPGEN [84] are two examples <strong>of</strong> Matrix Element generators which are widely used <strong>in</strong><strong>the</strong> <strong>CMS</strong> collaboration.With a Matrix Element generator, at <strong>the</strong> end one would have events with particles at<strong>the</strong> f<strong>in</strong>al state which were produced based on <strong>the</strong> Feynman rules presented <strong>in</strong> <strong>the</strong> StandardModel <strong>of</strong> particle physics. The partons <strong>in</strong> ei<strong>the</strong>r <strong>in</strong>itial or f<strong>in</strong>al state <strong>of</strong> an evententail <strong>the</strong> probability <strong>of</strong> radiat<strong>in</strong>g a gluon q → qg, referred to as Initial State Radiation(ISR) or F<strong>in</strong>al State Radiation (FSR) respectively. The emited gluon is subsequentlydecomposed <strong>in</strong>to ei<strong>the</strong>r a pair <strong>of</strong> quark-antiquark g → q¯q or a pair <strong>of</strong> gluons g → gg.Fur<strong>the</strong>r <strong>the</strong> radiated partons entail <strong>the</strong> possibility <strong>of</strong> additional radiations result<strong>in</strong>g <strong>in</strong>a shower <strong>of</strong> partons mov<strong>in</strong>g along <strong>the</strong> direction <strong>of</strong> <strong>the</strong> shower <strong>in</strong>itiator parton. Thesplitt<strong>in</strong>g <strong>of</strong> a parton to o<strong>the</strong>r partons is controled by <strong>the</strong> DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) equations [85–87]. The dedicated s<strong>of</strong>twares which perform <strong>the</strong>
CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event 51parton shower<strong>in</strong>g are called Parton Showers. In a Parton Shower approach, <strong>the</strong> emitedgluon can be radiated coll<strong>in</strong>ear with respect to <strong>the</strong> orig<strong>in</strong>al parton or be very s<strong>of</strong>twhich would yield ma<strong>the</strong>matical s<strong>in</strong>gularities. These k<strong>in</strong>d <strong>of</strong> s<strong>in</strong>gularities are handledby apply<strong>in</strong>g a lower cut-<strong>of</strong>f on <strong>the</strong> appropriate parameters. Some <strong>of</strong> <strong>the</strong> multi-purposeevent generators which besides be<strong>in</strong>g a physics event generator, <strong>in</strong>clude Parton Showerprograms are PYTHIA [88] and HERWIG [89].In order to complement <strong>the</strong> event generation cha<strong>in</strong>, <strong>the</strong> output <strong>of</strong> a Matrix Elementgenerator is <strong>in</strong>terfaced with a Parton Shower program to implement <strong>the</strong> splitt<strong>in</strong>g effectson <strong>the</strong> produced partons <strong>of</strong> <strong>the</strong> generated event. S<strong>in</strong>ce Parton Shower algorithms canalso produce hard partons <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state, it is important to resolve <strong>the</strong> ambiguitiesthat can arise <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state configurations. For example, a given event with N+1jets <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state can be obta<strong>in</strong>ed by ei<strong>the</strong>r a Matrix Element generator with N+1partons <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state or a Matrix Element generator with N partons <strong>in</strong> <strong>the</strong> f<strong>in</strong>alstate where an additional parton is produced dur<strong>in</strong>g <strong>the</strong> parton shower<strong>in</strong>g. Thereforesome match<strong>in</strong>g algorithms have been developed to perform <strong>the</strong> match<strong>in</strong>g between jets<strong>of</strong> hadrons produced <strong>in</strong> a Parton Shower program and hard partons produced <strong>in</strong> a MatrixElement generator so that a probable double count<strong>in</strong>g <strong>of</strong> events is avoided. TheCKKW algorithm [90, 91] and <strong>the</strong> MLM algorithm [92, 93] are two important examples<strong>of</strong> match<strong>in</strong>g schemes used <strong>in</strong> <strong>the</strong> event generation procedure.4.1.3 Detector SimulationThe generated Monte Carlo simulated events provide physically observable particlesonly if <strong>the</strong>y undergo a detector simulation. Equivalently a generated particle is observedwhen it <strong>in</strong>teracts with matter. The simulation <strong>of</strong> <strong>the</strong> <strong>CMS</strong> detector is basedon <strong>the</strong> GEANT4 package which simulates all possible electromagnetic and hadronic<strong>in</strong>teractions <strong>of</strong> particles with matter with<strong>in</strong> a magnetic field. For example, <strong>in</strong> <strong>the</strong> case<strong>of</strong> a generated electron, <strong>the</strong> possible <strong>in</strong>teractions result <strong>in</strong> ionization or radiation. Theemitted photons which present <strong>the</strong>mselves by releas<strong>in</strong>g heat and energy, are collected<strong>in</strong> <strong>the</strong> read-out. This would yield <strong>the</strong> so called <strong>experiment</strong>al “hits” which are <strong>the</strong>deposited energy <strong>of</strong> <strong>the</strong> particles while travers<strong>in</strong>g <strong>the</strong> detector material and constitue<strong>the</strong> build<strong>in</strong>g blocks for <strong>the</strong> roconstruction <strong>of</strong> <strong>the</strong> physics objects. It is worth to mentionthat <strong>the</strong> same reconstruction algorithms are run on ei<strong>the</strong>r Monte Carlo simulatedevents or events com<strong>in</strong>g from a real proton-proton collision so that one could make afair comparison between <strong>the</strong>ory and <strong>experiment</strong>.4.2 The Event TopologyWith ei<strong>the</strong>r Monte Carlo simulated samples or <strong>experiment</strong>al data, physics objects canbe reconstructed. Electronic signals collected <strong>in</strong> <strong>the</strong> read-out <strong>of</strong> <strong>the</strong> detector are translated<strong>in</strong>to position parameters along <strong>the</strong> particles trajectory and are used to reconstruct<strong>the</strong> physics objects and <strong>the</strong>ir k<strong>in</strong>ematic properties as described <strong>in</strong> detail <strong>in</strong> previouschapter. After <strong>the</strong> object reconstruction, it is <strong>the</strong> time to reconstruct <strong>the</strong> physics event<strong>of</strong> <strong>in</strong>terest which is called <strong>the</strong> signal event hereafter. Any o<strong>the</strong>r k<strong>in</strong>d <strong>of</strong> event with <strong>the</strong>same observed f<strong>in</strong>al state as <strong>the</strong> signal but not orig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong> pp → t¯t process, is
52 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Eventcalled generally background and should be removed <strong>in</strong> order not to enter <strong>the</strong> analysisand spoil <strong>the</strong> physics results.The description <strong>of</strong> <strong>the</strong> configuration <strong>of</strong> <strong>the</strong> full signal event <strong>in</strong> this study is <strong>the</strong> subject<strong>of</strong> <strong>the</strong> next section.4.2.1 The Signal Signature <strong>in</strong> <strong>the</strong> DetectorTop quark decays almost exclusively to a W boson and a b quark which subsequentlyhadronizes and produces a jet <strong>in</strong> <strong>the</strong> detectort(¯t) → W + (W − ) + b(¯b).Hence <strong>the</strong> decay products <strong>of</strong> a pair <strong>of</strong> top quarks only depend on <strong>the</strong> various W bosondecay channels as listed <strong>in</strong> Table 4.1 [94, 95].W decay modeBranch<strong>in</strong>g RatioW → lν e ; l = e, µ, τ. ∼ 33%W → q¯q; q¯q = u ¯d, c¯s. ∼ 66%Table 4.1: Various W decay channels and <strong>the</strong>ir branch<strong>in</strong>g ratio.If both <strong>of</strong> <strong>the</strong> Ws com<strong>in</strong>g out <strong>of</strong> <strong>the</strong> top quarks decay <strong>in</strong>to leptons and <strong>the</strong>ir correspond<strong>in</strong>gneutr<strong>in</strong>os, it is called a full leptonic decay mode <strong>of</strong> t¯t system. If only one <strong>of</strong><strong>the</strong> Ws decays leptonically and <strong>the</strong> o<strong>the</strong>r goes to hadrons, this makes a semi-leptonicdecay mode <strong>of</strong> a top quark pair. O<strong>the</strong>rwise, a full hadronic t¯t decay mode correspondsto <strong>the</strong> situation when both <strong>of</strong> <strong>the</strong> W bosons decay hadronically. The branch<strong>in</strong>g ratios<strong>of</strong> <strong>the</strong> different decay modes <strong>of</strong> t¯t event are summarized <strong>in</strong> Table 4.2.t¯t decay modeBranch<strong>in</strong>g RatioFull leptonicSemi leptonicFull Hadronict¯t → W + bW −¯b → l + l − b¯b + E missT ∼ 11%t¯t → W + bW −¯b → lq¯qb¯b + EmissT ∼ 45%t¯t → W + bW −¯b → q¯qq¯qb¯b ∼ 44%Table 4.2: Various t¯t decay channels and <strong>the</strong>ir branch<strong>in</strong>g ratio.
CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event 53In this analysis, <strong>the</strong> semi-leptonic decay channel <strong>of</strong> a t¯t system with <strong>the</strong> lepton be<strong>in</strong>gan electron is considered as signal and a typical signal event is depicted <strong>in</strong> Figure 4.5.Figure 4.5: Observed topology <strong>of</strong> a generic e+jets decay <strong>of</strong> a t¯t system.The f<strong>in</strong>al state <strong>of</strong> a e+jets t¯t decay, conta<strong>in</strong>s an isolated electron, at least four jetsas a result <strong>of</strong> <strong>the</strong> hadronization <strong>of</strong> <strong>the</strong> four quarks <strong>of</strong> which two <strong>of</strong> <strong>the</strong>m orig<strong>in</strong>ate fromheavy partons and miss<strong>in</strong>g transvers energy due to <strong>the</strong> presence <strong>of</strong> a neutr<strong>in</strong>o among<strong>the</strong> decay products.4.3 Simulated Signal and Background Monte CarloSamplesThe simulation <strong>of</strong> all signal and background events was performed us<strong>in</strong>g <strong>the</strong> MadGraphgenerator except for QCD samples which were produced us<strong>in</strong>g PYTHIA. The MLMmatch<strong>in</strong>g scheme was used to perform <strong>the</strong> match<strong>in</strong>g between <strong>the</strong> partons generatedby MadGraph and <strong>the</strong> parton shower<strong>in</strong>g from PYTHIA. In addition to <strong>the</strong> production <strong>of</strong>top quark pairs via <strong>the</strong> strong <strong>in</strong>teraction, s<strong>in</strong>gle top quarks which are produced viaelectro-weak <strong>in</strong>teractions have similar signatures as <strong>the</strong> signal events and are consideredas backgrounds. Also W and Z bosons <strong>in</strong> association with extra jets, where <strong>the</strong> vectorboson decays leptonically, can mimic <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> semi-leptonic t¯t events. Due to<strong>the</strong> large cross section <strong>of</strong> <strong>the</strong> W+jets process as well as <strong>the</strong> <strong>in</strong>clusive production <strong>of</strong> aprompt electron <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state, <strong>the</strong>se events constitute <strong>the</strong> ma<strong>in</strong> backgrounds to <strong>the</strong>signal.The o<strong>the</strong>r ma<strong>in</strong> background are QCD multijet events for which due to <strong>the</strong> huge crosssection, it is hard to produce <strong>the</strong> enough amount <strong>of</strong> statistics which can cover <strong>the</strong> wholephase space. Also because <strong>of</strong> <strong>the</strong> fact that normally no prompt electron is produced<strong>in</strong> a generic QCD event, QCD events are suppressed and one would encounter a lack<strong>of</strong> QCD statistics <strong>in</strong> <strong>the</strong> analysis. Therefore <strong>in</strong> order to obta<strong>in</strong> reliable results, <strong>the</strong>reare some filters applied at generator level namely BCtoE and EMEnriched result<strong>in</strong>g <strong>in</strong>events which are suitable for <strong>the</strong> analyses conta<strong>in</strong><strong>in</strong>g electrons. The first filter tries tosave those events conta<strong>in</strong><strong>in</strong>g electrons with at least 10 GeV <strong>of</strong> transverse energy thatare produced <strong>in</strong> <strong>the</strong> tracker acceptance. These electrons are asked to orig<strong>in</strong>ate from ab or c quark so that events which are able to pass this filter are enriched with electrons
54 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Eventappear<strong>in</strong>g <strong>in</strong> <strong>the</strong> decay products <strong>of</strong> hadrons conta<strong>in</strong><strong>in</strong>g heavy quarks. The second filteraims to ga<strong>the</strong>r those events with isolated electrons at generator level. It looks for s<strong>in</strong>gleelectrons with at least 20 GeV <strong>of</strong> transverse energy with<strong>in</strong> <strong>the</strong> tracker acceptance thatare isolated <strong>in</strong> both <strong>the</strong> tracker and calorimeter sub-detectors. Both filters are made<strong>in</strong> such a way that <strong>the</strong> QCD events surviv<strong>in</strong>g each filter should be used toge<strong>the</strong>r. Forsimplicity QCD events are produced <strong>in</strong> different b<strong>in</strong>s <strong>of</strong> ˆp T , which is def<strong>in</strong>ed as <strong>the</strong>center <strong>of</strong> mass energy <strong>of</strong> <strong>the</strong> two collid<strong>in</strong>g partons. The detailed <strong>in</strong>formation <strong>of</strong> <strong>the</strong>Monte Carlo simulated sets which are processed <strong>in</strong> this study, are given <strong>in</strong> Table 4.3.Process Name <strong>of</strong> Monte Carlo Sample σ[pb] #Eventst¯tW+jets (W → lν)Z+jets (Z → l + l − )tQCD/TTbar<strong>Jet</strong>s-madgraph 157.5 1483404/W<strong>Jet</strong>s-madgraph 31314 10068895/Z<strong>Jet</strong>s-madgraph 3048 1084921/S<strong>in</strong>gleTop_tChannel-madgraph 20.93 528593/S<strong>in</strong>gleTop_tWChannel-madgraph 10.6 459589/S<strong>in</strong>gleTop_sChannel-madgraph 1.4 402055/QCD_BCtoE_Pt20to30 108330 2491921/QCD_BCtoE_Pt30to80 138762 2475597/QCD_BCtoE_Pt80to170 9422.4 1119546/QCD_EMEnriched_Pt20to30 1719150 33355445/QCD_EMEnriched_Pt30to80 3498700 39479587/QCD_EMEnriched_Pt80to170 134088 5494911Table 4.3: Simulated signal and background samples that are processed <strong>in</strong> this study.All samples have <strong>the</strong> end<strong>in</strong>g “/Spr<strong>in</strong>g10-START3X V26 S09-v1/GEN-SIM-RECO”which has been dropped <strong>in</strong> <strong>the</strong> table. The mentioned cross sections correspond to<strong>the</strong> Next-to-Lead<strong>in</strong>g Order (NLO) values except for QCD samples where <strong>the</strong> quotedcross sections have been calculated at Lead<strong>in</strong>g Order (LO) [96]. The numbers are takenfrom a common reference page <strong>of</strong> <strong>the</strong> top quark analysis group <strong>in</strong> <strong>CMS</strong> [97].
CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event 554.4 Event SelectionToge<strong>the</strong>r with signal events, at a proton-proton collider, <strong>the</strong>re are many o<strong>the</strong>r k<strong>in</strong>d <strong>of</strong>events produced <strong>of</strong> which some mimic <strong>the</strong> signal. For example, QCD multijet eventswhere a jet fakes an electron can produce a similar f<strong>in</strong>al state as e+jets t¯t events. Thesame reason<strong>in</strong>g can be made for a W+jets event when <strong>the</strong> W decays leptonically. AlsoZ → e + e − events <strong>of</strong> which one electron escapes detection would be ano<strong>the</strong>r possiblebackground. Hence, <strong>in</strong> order to suppress all <strong>the</strong>se background events, one can benifitfrom <strong>the</strong> k<strong>in</strong>ematics <strong>of</strong> <strong>the</strong> signal and cut on <strong>the</strong> observables that differ between <strong>the</strong>signal and its backgrounds. A detailed procedure <strong>of</strong> select<strong>in</strong>g e+jets t¯t events amongall o<strong>the</strong>r backgrounds is described <strong>in</strong> <strong>the</strong> next section.4.4.1 Selection CutsExpected Number <strong>of</strong> Events at <strong>the</strong> Center <strong>of</strong> Mass <strong>Energy</strong> <strong>of</strong> 7TeVAt 7TeV center <strong>of</strong> mass energy, <strong>the</strong> cross sections for signal and o<strong>the</strong>r relevant backgroundsare listed <strong>in</strong> Table 4.3. Accord<strong>in</strong>g toN = σLwhere L is <strong>the</strong> <strong>in</strong>tegrated lum<strong>in</strong>osity and σ is <strong>the</strong> cross section for a specific k<strong>in</strong>d <strong>of</strong>process, N is <strong>the</strong> number <strong>of</strong> events expected after accumulat<strong>in</strong>g <strong>the</strong> data equivalent toL. The current study is based on L = 100pb −1 and <strong>the</strong> derived results are valid at thisregime <strong>of</strong> lum<strong>in</strong>osity.Pre-SelectionIn order to reduce <strong>the</strong> size <strong>of</strong> <strong>the</strong> various simulated samples, a set <strong>of</strong> pre-selectioncuts is applied. All events are required to have at least one reconstructed electronand four reconstructed jets. Reconstructed electron and jets are asked to have a p Texceed<strong>in</strong>g respectively 15 GeV and 20 GeV and to be with<strong>in</strong> <strong>the</strong> tracker acceptance <strong>of</strong>|η| < 2.4. The p T and η distributions <strong>of</strong> all reconstructed electrons and jets for thoseevents surviv<strong>in</strong>g <strong>the</strong> pre-selection <strong>in</strong>clud<strong>in</strong>g all signal and backgrounds are shown <strong>in</strong>Figure 4.6.As it can be seen from Figure 4.6, <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> pseudorapidity <strong>of</strong> <strong>the</strong>electrons drop down <strong>in</strong> two parts which are represent<strong>in</strong>g <strong>the</strong> cracks between <strong>the</strong> ECALbarrel and <strong>the</strong> ECAL endcap. Also it can be understood that <strong>the</strong> jets <strong>in</strong> <strong>the</strong> semielectronict¯t events, are ma<strong>in</strong>ly reconstructed <strong>in</strong> <strong>the</strong> central part <strong>of</strong> <strong>the</strong> detector.Primary VertexIn order to make sure that <strong>the</strong> event which is under process<strong>in</strong>g, comes out <strong>of</strong> a hardscatter<strong>in</strong>g and has not orig<strong>in</strong>ated from pile-up <strong>in</strong>teractions, some quality cuts are appliedon <strong>the</strong> reconstructed vertex <strong>of</strong> <strong>the</strong> event. S<strong>in</strong>ce one <strong>of</strong> <strong>the</strong> characteristics <strong>of</strong> pile-upcollision is a displacement primary vertex with respect to <strong>the</strong> nom<strong>in</strong>al <strong>in</strong>teraction vertex,one can ask each event to have a reconstructed primary vertex very close to <strong>the</strong>
56 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event#Events610510410tt+jets semi-ele (2776.0 entries)tt+jets o<strong>the</strong>r (7252.3 entries)t+jets, (all channels) (724.1 entries)Z+jets (8577.8 entries)W+jets (14933.0 entries)Multi-jets (7206195.2 entries)#Events510410310310210210101011-1100 20 40 60 80 100 120 140 160 180 200Electron P T(GeV)-110-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Electron η#Events71061051041031021010#Events6105104103102101011-110-1100 20 40 60 80 100 120 140 160 180 200<strong>Jet</strong>s P T(GeV)-2 -1 0 1 2<strong>Jet</strong>s ηFigure 4.6: The p T and η distributions <strong>of</strong> all reconstructed electrons and jets for <strong>the</strong>events surviv<strong>in</strong>g <strong>the</strong> pre-selection step.<strong>in</strong>teraction region. In this analysis, each event is required to have a good reconstructedprimary vertex which is def<strong>in</strong>ed with nd<strong>of</strong> > 4, where <strong>the</strong> number <strong>of</strong> degrees <strong>of</strong> freedomis a measure <strong>of</strong> <strong>the</strong> number <strong>of</strong> tracks used for <strong>the</strong> construction <strong>of</strong> <strong>the</strong> primary vertex.Also a good primary vertex fulfills |z| < 24 cm and ρ < 2 cm, where ρ ≡ √ x 2 + y 2with x and y <strong>the</strong> coord<strong>in</strong>ates <strong>of</strong> <strong>the</strong> reconstructed primary vertex <strong>in</strong> <strong>the</strong> transversepla<strong>in</strong>. The multiplicity <strong>of</strong> good primary vertices for events which have already passed<strong>the</strong> pre-selection cuts is shown <strong>in</strong> Figure 4.7.#Events710610510410310210101-110tt+jets semi-ele (1917.0 entries)tt+jets o<strong>the</strong>r (5439.3 entries)t+jets, (all channels) (565.1 entries)Z+jets (4907.7 entries)W+jets (12952.6 entries)Multi-jets (6655855.5 entries)0 2 4 6 8 10Nr. <strong>of</strong> good primary verticesFigure 4.7: Distribution <strong>of</strong> <strong>the</strong> good primary vertices <strong>in</strong> an event after apply<strong>in</strong>g preselectioncuts.
CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event 57Exactly One ElectronS<strong>in</strong>ce one <strong>of</strong> <strong>the</strong> signal signatures is <strong>the</strong> presence <strong>of</strong> exactly one isolated electron <strong>in</strong> <strong>the</strong>f<strong>in</strong>al state, cutt<strong>in</strong>g on this property <strong>of</strong> <strong>the</strong> event will reduce <strong>the</strong> huge amount <strong>of</strong> backgroundsspecially QCD multijet events where no prompt electron is usually produced.The selected electron is required to pass some more tighter cuts than <strong>the</strong> ones alreadyapplied at <strong>the</strong> pre-selection step. A detailed set <strong>of</strong> cuts that <strong>the</strong> selected electron isasked to pass, is listed below.• The selected electron for each event, if <strong>the</strong>re exists one, is required to pass atighter cut on p T namely p T > 30 GeV . Also, it is asked not to be reconstructed<strong>in</strong> <strong>the</strong> transition region between <strong>the</strong> ECAL barrel and <strong>the</strong> ECAL endcap.As a result, if <strong>the</strong> η <strong>of</strong> <strong>the</strong> supercluster <strong>of</strong> a candidate electron fulfills1.4442 < |η supercluster | < 1.5660, <strong>the</strong>n it is failed to be labeled as selected.• Ano<strong>the</strong>r powerfull observable that is widely used to dist<strong>in</strong>guish between promptelectrons com<strong>in</strong>g out <strong>of</strong> <strong>the</strong> hard scatter<strong>in</strong>g and those which are produced <strong>in</strong> jetstoge<strong>the</strong>r with lots <strong>of</strong> hadrons, is <strong>the</strong> isolation variable. The isolation value, as<strong>the</strong> name says, is a measure <strong>of</strong> how much an electron is isolated with respect to<strong>the</strong> o<strong>the</strong>r activity <strong>in</strong> <strong>the</strong> event and def<strong>in</strong>ed as <strong>the</strong> sum <strong>of</strong> <strong>the</strong> energy deposits <strong>in</strong> acone with a specific radius around <strong>the</strong> electron. Instead <strong>of</strong> an absolute isolationvalue, a relative isolation variable is mostly used which is def<strong>in</strong>ed asRelIso = caloIsoECAL+HCAL + trackIso,ETewhere caloIso ECAL+HCAL and trackIso refer to <strong>the</strong> absolute observed transversemomentum or energy obta<strong>in</strong>ed <strong>in</strong> a cone <strong>of</strong> 0.3 around <strong>the</strong> electron axis <strong>in</strong> <strong>the</strong>calorimeter and <strong>the</strong> tracker, respectively. The distribution <strong>of</strong> <strong>the</strong> relative isolationvariable <strong>of</strong> all reconstructed electrons for those events that already passed <strong>the</strong>primary vertex cut, is plotted <strong>in</strong> Figure 4.8. The distribution <strong>of</strong> <strong>the</strong> relativeisolation variable <strong>of</strong> <strong>the</strong> electron with <strong>the</strong> highest p T <strong>in</strong> <strong>the</strong> event is also shown<strong>in</strong> Figure 4.9. An electron is selected if its relative isolation value is below 0.1.• In order to be selected, <strong>the</strong> electron candidate is asked to pass <strong>the</strong> identificationcuts. The parameters used for electron identification aim to make a differencebetween <strong>the</strong> shape <strong>of</strong> <strong>the</strong> shower<strong>in</strong>g <strong>of</strong> a reconstructed electron compared to ajet. They are briefly reviewed below. The cut values are chosen so that an identificationefficiency equal to 70% can be reached [98].– h/e which is def<strong>in</strong>ed as <strong>the</strong> ratio <strong>of</strong> <strong>the</strong> energy deposited <strong>in</strong> <strong>the</strong> HCAL towerjust beh<strong>in</strong>d <strong>the</strong> electromagnetic seed cluster over <strong>the</strong> energy <strong>of</strong> <strong>the</strong> seed cluster.The selected electron is required to have an h/e value less than 0.025.
58 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event#Events510410tt+jets semi-ele (2717.3 entries)tt+jets o<strong>the</strong>r (7215.5 entries)t+jets, (all channels) (712.5 entries)Z+jets (8215.9 entries)W+jets (14446.5 entries)Multi-jets (7191770.9 entries)#Events510410tt+jets semi-ele (1866.6 entries)tt+jets o<strong>the</strong>r (5423.5 entries)t+jets, (all channels) (554.9 entries)Z+jets (4721.2 entries)W+jets (12499.7 entries)Multi-jets (6668437.6 entries)310310210210101011-1100 1 2 3 4 5Electron RelIsoFigure 4.8: Relative isolation variable <strong>of</strong>all reconstructed electrons for <strong>the</strong> eventspassed primary vertex cut.-1100 1 2 3 4 5Lead<strong>in</strong>g Electron RelIsoFigure 4.9: Relative isolation variable <strong>of</strong><strong>the</strong> electron with <strong>the</strong> maximum p T for <strong>the</strong>events passed primary vertex cut.– ∆η <strong>in</strong> which is def<strong>in</strong>ed as <strong>the</strong> η difference between <strong>the</strong> supercluster and <strong>the</strong>correspond<strong>in</strong>g track extrapolated to <strong>the</strong> ECAL start<strong>in</strong>g from <strong>the</strong> <strong>in</strong>teractionvertex. Depend<strong>in</strong>g on <strong>the</strong> position <strong>of</strong> <strong>the</strong> reconstructed electron, a barrelelectron is asked to have a ∆η <strong>in</strong> less than 0.004 while for <strong>the</strong> endcap electronthis value is set to be 0.005.– ∆φ <strong>in</strong> which is def<strong>in</strong>ed as <strong>the</strong> φ difference between <strong>the</strong> supercluster and <strong>the</strong>correspond<strong>in</strong>g track extrapolated to <strong>the</strong> ECAL start<strong>in</strong>g from <strong>the</strong> <strong>in</strong>teractionvertex. Depend<strong>in</strong>g on <strong>the</strong> position <strong>of</strong> <strong>the</strong> reconstructed electron, a barrelelectron is asked to have a ∆φ <strong>in</strong> less than 0.03 while for <strong>the</strong> endcap electronthis value is set to be 0.02.– σ iηiη is a measure <strong>of</strong> <strong>the</strong> shower width <strong>in</strong> a direction which is very largelyunaffected by shower<strong>in</strong>g <strong>of</strong> electrons <strong>in</strong> <strong>the</strong> tracker material and so has beena key variable <strong>in</strong> electron identification. Depend<strong>in</strong>g on <strong>the</strong> position <strong>of</strong> <strong>the</strong>reconstructed electron, a barrel electron is asked to have a σ iηiη less than0.01 while for <strong>the</strong> endcap electron this value is set to be 0.03.• F<strong>in</strong>ally, <strong>the</strong> selected electron is required to have d 0 < 0.02 cm where d 0 is def<strong>in</strong>edas <strong>the</strong> distance <strong>of</strong> <strong>the</strong> reconstructed track <strong>of</strong> <strong>the</strong> electron <strong>in</strong> <strong>the</strong> transverse pla<strong>in</strong>with respect to <strong>the</strong> beam spot. This variable is useful <strong>in</strong> differentiat<strong>in</strong>g betweenprompt electrons and those which are produced <strong>in</strong> jets. The electrons appear<strong>in</strong>g<strong>in</strong> jets usually have larger d 0 values. Hence cutt<strong>in</strong>g on this variable would rejectmost <strong>of</strong> <strong>the</strong> QCD events.A summary <strong>of</strong> <strong>the</strong> cut values applied on <strong>the</strong> various identification variables <strong>of</strong> <strong>the</strong>selected electron is given <strong>in</strong> Table 4.4.
CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event 59h/e ∆η <strong>in</strong> ∆φ <strong>in</strong> σ iηiηbarrel endcap barrel endcap barrel endcap barrel endcapcut value 0.025 0.025 0.004 0.005 0.03 0.02 0.01 0.03Table 4.4: Required cut values on <strong>the</strong> identification variables <strong>of</strong> <strong>the</strong> selected electron.The distributions <strong>of</strong> <strong>the</strong> shower shape variables toge<strong>the</strong>r with <strong>the</strong> d 0 parameter <strong>of</strong>all reconstructed electrons <strong>in</strong> those events that already passed <strong>the</strong> primary vertex cut,can be found <strong>in</strong> Figure 4.10.As it is seen from Figure 4.10, <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> ∆η <strong>in</strong> variable is sharper comparedto <strong>the</strong> ∆φ <strong>in</strong> distribution, represent<strong>in</strong>g <strong>the</strong> radiated electrons which bend <strong>in</strong> <strong>the</strong>φ-direction due to <strong>the</strong> magnetic field <strong>in</strong> <strong>the</strong> z-direction.Also <strong>the</strong> multiplicity <strong>of</strong> <strong>the</strong> selected electrons per event which already passed <strong>the</strong> primaryvertex cut, is shown <strong>in</strong> Figure 4.11.It can be seen from Figure 4.11 that most <strong>of</strong> <strong>the</strong> QCD multijet events do notconta<strong>in</strong> a selected electron while two selected electrons can be found <strong>in</strong> Z+jets eventsor t¯t → o<strong>the</strong>r, as expected.Those events with Nesel = 1, correspond<strong>in</strong>g to <strong>the</strong> situation that exactly one selectedelectron is present per event, are accepted for fur<strong>the</strong>r analysis.Second Electron VetoSome <strong>of</strong> <strong>the</strong> backgrounds such as Z+jets events where Z → e + e − or backgroundscom<strong>in</strong>g from full leptonic decay <strong>of</strong> t¯t events <strong>of</strong> which both <strong>of</strong> leptons are electrons,conta<strong>in</strong> more than one electron <strong>in</strong> <strong>the</strong>ir f<strong>in</strong>al states. Hence veto<strong>in</strong>g <strong>of</strong> <strong>the</strong> presence <strong>of</strong> asecond electron can suppress <strong>the</strong>se k<strong>in</strong>d <strong>of</strong> backgrounds. For <strong>the</strong> second electron, <strong>the</strong>reare some looser cuts applied so that <strong>the</strong> background rejection efficiency is <strong>in</strong>creased.The p T cut is lowered to 20 GeV compared to <strong>the</strong> selected electron. The cut thresholdson <strong>the</strong> different identification parameters are also loosen and are listed <strong>in</strong> Table 4.5.No isolation requirement nei<strong>the</strong>r d 0 cut is applied on <strong>the</strong> second electron.The multiplicity <strong>of</strong> <strong>the</strong> second loose electron per event, exclud<strong>in</strong>g <strong>the</strong> selected electron,is shown <strong>in</strong> Figure 4.12.The event is rejected if it has at least one second loose electron as def<strong>in</strong>ed above.
60 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event#Events610510410tt+jets semi-ele (2775.2 entries)tt+jets o<strong>the</strong>r (7245.3 entries)t+jets, (all channels) (723.6 entries)Z+jets (8573.7 entries)W+jets (14925.5 entries)Multi-jets (7198962.9 entries)#Events610510410310310210210101011-110-1100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Electron h/e-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5Electron ∆η<strong>in</strong>#Events610510410310210#Events610510410310210101011-110-110-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5Electron ∆φ<strong>in</strong>-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5Electron σ iη iη#Events610510410310210101-110-0.1 -0.05 0 0.05 0.1Electron d 0(cm)Figure 4.10: Shower shape variables and d 0 distribution <strong>of</strong> all reconstructed electrons<strong>in</strong> <strong>the</strong> events pass<strong>in</strong>g <strong>the</strong> primary vertex cut.#Events710610510410310210101-110tt+jets semi-ele (1916.9 entries)tt+jets o<strong>the</strong>r (5434.6 entries)t+jets, (all channels) (564.8 entries)Z+jets (4907.9 entries)W+jets (12959.9 entries)Multi-jets (6655832.6 entries)-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Nr. <strong>of</strong> selected ElectronsFigure 4.11: Multiplicity <strong>of</strong> <strong>the</strong> selected electrons per event which passed <strong>the</strong> primaryvertex cut.
CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event 61h/e ∆η <strong>in</strong> ∆φ <strong>in</strong> σ iηiηbarrel endcap barrel endcap barrel endcap barrel endcapcut value 0.15 0.07 0.007 0.01 0.8 0.7 0.01 0.03Table 4.5: Required cut values on <strong>the</strong> identification variables <strong>of</strong> <strong>the</strong> selected electron.#Events410310tt+jets semi-ele (1053.2 entries)tt+jets o<strong>the</strong>r (360.7 entries)t+jets, (all channels) (154.2 entries)Z+jets (2121.1 entries)W+jets (5370.5 entries)Multi-jets (12280.1 entries)210101-110-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Nr. <strong>of</strong> Loose O<strong>the</strong>r ElectronsFigure 4.12: Multiplicity <strong>of</strong> secondary loose electrons per event which has already hadexactly one selected electron.Muon VetoThe background events from µ+jets t¯t decays conta<strong>in</strong> one muon <strong>in</strong> <strong>the</strong>ir f<strong>in</strong>al states.Also a muon can be reconstructed <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> a W+jets event where <strong>the</strong>W boson decays to a muon and correspond<strong>in</strong>g neutr<strong>in</strong>o. Hence veto<strong>in</strong>g muons <strong>in</strong> <strong>the</strong>f<strong>in</strong>al state <strong>of</strong> <strong>the</strong> event would reject most <strong>of</strong> <strong>the</strong> backgrounds conta<strong>in</strong><strong>in</strong>g a reconstructedmuon. The distribution <strong>of</strong> <strong>the</strong> muon muliplicity for those events that already passed <strong>the</strong>second electron veto cut, is shown <strong>in</strong> Figure 4.13. The reconstructed muons are askedto have a p T exceed<strong>in</strong>g 20GeV and |η| less than 2.1. Also <strong>the</strong>y are required to fulfilla relative isolation value, which is def<strong>in</strong>ed as <strong>the</strong> sum <strong>of</strong> <strong>the</strong> energy deposits <strong>in</strong> a cone<strong>of</strong> 0.3 <strong>in</strong> <strong>the</strong> tracker and <strong>the</strong> calorimeter divided by <strong>the</strong> muon transverse momentum,below 0.05. It can be seen that most <strong>of</strong> <strong>the</strong> t¯t events o<strong>the</strong>r than signal occupy <strong>the</strong>second b<strong>in</strong> <strong>of</strong> <strong>the</strong> distribution correspond<strong>in</strong>g to those events with one reconstructedmuon <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state. Hence <strong>the</strong>se k<strong>in</strong>d <strong>of</strong> backgrounds are suppressed by veto<strong>in</strong>g<strong>the</strong> existence <strong>of</strong> a reconstructed muon <strong>in</strong> <strong>the</strong> event.Only those events appear<strong>in</strong>g <strong>in</strong> <strong>the</strong> first b<strong>in</strong> <strong>of</strong> <strong>the</strong> plot shown <strong>in</strong> Figure 4.13,correspond<strong>in</strong>g to <strong>the</strong> events with N selµ = 0, are processed <strong>in</strong> <strong>the</strong> rest <strong>of</strong> analysis.
62 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event#Events410310tt+jets semi-ele (1022.1 entries)tt+jets o<strong>the</strong>r (310.1 entries)t+jets, (all channels) (149.1 entries)Z+jets (1045.4 entries)W+jets (5314.5 entries)Multi-jets (12144.0 entries)210101-110-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Nr. <strong>of</strong> Loose MuonsFigure 4.13: Muon multiplicity per event which passed <strong>the</strong> Second Electron Veto cut.At Least Four <strong>Jet</strong>sAt <strong>the</strong> pre-selection step, all events are required to have at least four reconstructed jets<strong>in</strong> <strong>the</strong>ir f<strong>in</strong>al states. By tighten<strong>in</strong>g <strong>the</strong> thresholds on <strong>the</strong> jet k<strong>in</strong>ematics and also by apply<strong>in</strong>gsome identification criteria, <strong>the</strong> probability <strong>of</strong> select<strong>in</strong>g <strong>the</strong> four jets orig<strong>in</strong>at<strong>in</strong>gfrom <strong>the</strong> quarks com<strong>in</strong>g out <strong>of</strong> <strong>the</strong> hard scatter<strong>in</strong>g process becomes higher. The eventis required to have at least four selected jets with p T greater than 30GeV. Also jets,<strong>in</strong> order to be labeled as selected, are asked to pass identification cuts that are brieflydescribed below.• f EM which is called <strong>the</strong> electromagnetic energy fraction and is def<strong>in</strong>ed as <strong>the</strong>fraction <strong>of</strong> <strong>the</strong> jet energy measured <strong>in</strong> <strong>the</strong> ECAL. This parameter is useful <strong>in</strong>differentiat<strong>in</strong>g between electrons and jets s<strong>in</strong>ce jets with a high value <strong>of</strong> f EMprobably are electrons with energy deposits also used <strong>in</strong> <strong>the</strong> jet reconstructionalgorithm. The upper limit <strong>of</strong> this variable is set to be 0.9. Also <strong>in</strong> order tosuppress those jets which are actually pure electronic noise, a lower limit equalto 0.05 is put on this parameter.• n90Hits is def<strong>in</strong>ed as <strong>the</strong> m<strong>in</strong>imal number <strong>of</strong> reconstructed hits conta<strong>in</strong><strong>in</strong>g 90%<strong>of</strong> <strong>the</strong> jet energy and is ano<strong>the</strong>r observable that is used for <strong>the</strong> jet identification.The selected jets are required to have a n90Hits variable exceed<strong>in</strong>g 4.• f HPD is def<strong>in</strong>ed as <strong>the</strong> fraction <strong>of</strong> energy <strong>in</strong> <strong>the</strong> hottest HPD readout. The selectedjets are asked to fulfill f HPD < 0.98.The distributions <strong>of</strong> <strong>the</strong> various identification variables <strong>of</strong> all reconstructed jets forthose events that already passed <strong>the</strong> Muon Veto cut, are shown <strong>in</strong> Figure 4.14.Look<strong>in</strong>g at <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> electromagnetic energy fraction, <strong>the</strong>re is a clearpeak at ∼1 represent<strong>in</strong>g <strong>the</strong> electrons which are re-reconstructed as jets. Ano<strong>the</strong>r plotthat shows <strong>the</strong> jet collection <strong>in</strong>cludes electrons and <strong>the</strong>refore needs to be cleaned, is<strong>the</strong> distance between <strong>the</strong> selected electron and <strong>the</strong> closest jet among all reconstructed
CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event 63#Events410310210tt+jets semi-ele (5507.8 entries)tt+jets o<strong>the</strong>r (1102.6 entries)t+jets, (all channels) (669.4 entries)Z+jets (4506.7 entries)W+jets (22945.5 entries)Multi-jets (52361.7 entries)#Events310210101011-110-1100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<strong>Jet</strong>s EMF0 10 20 30 40 50<strong>Jet</strong>s n90Hits#Events410310210101-1100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1<strong>Jet</strong>s fHPDFigure 4.14: <strong>Jet</strong> identification variables <strong>of</strong> all reconstructed jets for <strong>the</strong> events whichpass <strong>the</strong> Muon Veto cut.jets and is shown <strong>in</strong> <strong>the</strong> left plot <strong>of</strong> Figure 5.26. Hence an extra criterion is appliedon each selected jet requir<strong>in</strong>g it not to be located very close to <strong>the</strong> selected electron.All selected jets are asked to be reconstructed <strong>in</strong> a ∆R = √ (∆η) 2 + (∆φ) 2 cone <strong>of</strong> 0.3from <strong>the</strong> selected electron. For comparison, <strong>the</strong> same plot <strong>of</strong> <strong>the</strong> distance between <strong>the</strong>selected electron and <strong>the</strong> closest jet after clean<strong>in</strong>g is also shown <strong>in</strong> <strong>the</strong> right plot <strong>of</strong>Figure 5.26.#Events410310210tt+jets semi-ele (1019.4 entries)tt+jets o<strong>the</strong>r (221.7 entries)t+jets, (all channels) (146.0 entries)Z+jets (1039.5 entries)W+jets (5308.6 entries)Multi-jets (12129.0 entries)#Events210101011-110-1100 0.5 1 1.5 2 2.5 3 3.5 4Electron M<strong>in</strong>imal ∆R(ele, jet)0 0.5 1 1.5 2 2.5 3 3.5 4Electron M<strong>in</strong>imal ∆R(ele, jet)Figure 4.15: M<strong>in</strong>imum ∆R distance between selected electron and <strong>the</strong> closest reconstructedjet before (left) and after (right) clean<strong>in</strong>g. Left plot is filled for <strong>the</strong> eventspass<strong>in</strong>g <strong>the</strong> Muon Veto cut while <strong>the</strong> events with at least four selected jets populate<strong>the</strong> right plot.
64 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics EventThe multiplicity <strong>of</strong> <strong>the</strong> selected jets per event which already passed <strong>the</strong> Muon Vetocut, is shown <strong>in</strong> Figure 4.16. Those events with more than or equal to four selectedjets, are accepted for fur<strong>the</strong>r analysis.#Events410310tt+jets semi-ele (1019.3 entries)tt+jets o<strong>the</strong>r (221.7 entries)t+jets, (all channels) (146.0 entries)Z+jets (1039.5 entries)W+jets (5308.2 entries)Multi-jets (12129.1 entries)210101-1100 1 2 3 4 5 6 7 8 9Nr. <strong>of</strong> Selected <strong>Jet</strong>sFigure 4.16: Multiplicity <strong>of</strong> <strong>the</strong> selected jets per event which pass <strong>the</strong> Muon Veto cut.At Least One B-Tagged <strong>Jet</strong>As <strong>the</strong>re are two b jets produced <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> <strong>the</strong> signal topology, it can beused as a discrim<strong>in</strong>ator between signal and backgrounds. In order to identify b quarkjets, various b-tagg<strong>in</strong>g algorithms have been def<strong>in</strong>ed with<strong>in</strong> <strong>the</strong> <strong>CMS</strong> collaboration. Asimple one is called <strong>the</strong> “Track Count<strong>in</strong>g High Efficiency” algorithm [99] which exploits<strong>the</strong> long lifetime <strong>of</strong> <strong>the</strong> b mesons produced <strong>in</strong> <strong>the</strong> hadronization process <strong>of</strong> <strong>the</strong> b quarks.Because <strong>of</strong> this property, <strong>the</strong> tracks which are associated to a b quark jet, are usuallydisplaced with respect to <strong>the</strong> primary vertex.The algorithm to tag b jets starts by select<strong>in</strong>g good quality tracks and associat<strong>in</strong>g<strong>the</strong>m to a jet by a match<strong>in</strong>g procedure. In a first step, <strong>the</strong> track impact parameter IPis computed by <strong>the</strong> algorithm, which is def<strong>in</strong>ed as <strong>the</strong> distance between <strong>the</strong> primaryvertex and <strong>the</strong> l<strong>in</strong>earised track <strong>in</strong> <strong>the</strong> po<strong>in</strong>t <strong>of</strong> m<strong>in</strong>imum distance between <strong>the</strong> trackand <strong>the</strong> jet axis. This is followed by <strong>the</strong> calculation <strong>of</strong> <strong>the</strong> track impact parametersignificance IP S , which is def<strong>in</strong>ed as <strong>the</strong> impact parameter divided by <strong>the</strong> resolution <strong>of</strong><strong>the</strong> impact parameter σ IP , hence IP S = IPσ IP. In <strong>the</strong> next step, <strong>the</strong> tracks associatedto a jet are ordered <strong>in</strong> decreas<strong>in</strong>g IP S . F<strong>in</strong>ally, <strong>the</strong> “Track Count<strong>in</strong>g High Efficiency”algorithm returns <strong>the</strong> IP S assigned to <strong>the</strong> second track <strong>in</strong> <strong>the</strong> list <strong>of</strong> ordered tracks as<strong>the</strong> b-tagg<strong>in</strong>g discrim<strong>in</strong>ant. The distribution <strong>of</strong> <strong>the</strong> b-tag discrim<strong>in</strong>ator value returnedby this algorithms, is shown <strong>in</strong> Figure 4.17. The events with at least four selected jetsare taken <strong>in</strong>to account.Due to <strong>the</strong> fact that <strong>the</strong> efficiency <strong>of</strong> b-tagg<strong>in</strong>g is not high, and <strong>in</strong> order not to loosetoo many signal events, <strong>the</strong> event is asked to have at least one b-tagged jet among<strong>the</strong> four selected jets. A jet is def<strong>in</strong>ed as a b jet if <strong>the</strong> tagg<strong>in</strong>g value returned by <strong>the</strong>“Track Count<strong>in</strong>g High Efficiency” algorithm exceeds 4. The multiplicity <strong>of</strong> b-taggedjets among <strong>the</strong> four lead<strong>in</strong>g jets for those events that have at least four selected jets is
CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Event 65#Events310210tt+jets semi-ele (1581.3 entries)tt+jets o<strong>the</strong>r (191.9 entries)t+jets, (all channels) (74.2 entries)Z+jets (105.9 entries)W+jets (939.5 entries)Multi-jets (1439.5 entries)101-110-10 -5 0 5 10 15 20 25 30 35 40B-Tag ValueFigure 4.17: The distribution <strong>of</strong> <strong>the</strong> b-tagg<strong>in</strong>g discrim<strong>in</strong>ator <strong>of</strong> <strong>the</strong> four lead<strong>in</strong>g jets,calculated by <strong>the</strong> “Track Count<strong>in</strong>g High Efficiency” algorithm, for <strong>the</strong> events whichhave at least four selected jets.shown <strong>in</strong> Figure 4.18.#Events310210tt+jets semi-ele (405.9 entries)tt+jets o<strong>the</strong>r (50.5 entries)t+jets, (all channels) (19.1 entries)Z+jets (28.1 entries)W+jets (241.0 entries)Multi-jets (379.3 entries)101-110-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Nr. <strong>of</strong> B-Tagged <strong>Jet</strong>sFigure 4.18: Multiplicity <strong>of</strong> <strong>the</strong> b-tagged jets, which is def<strong>in</strong>ed as a jet with a b-tagg<strong>in</strong>gvalue returned by <strong>the</strong> “Track Count<strong>in</strong>g High Efficiency” algorithm exceed<strong>in</strong>g 4, for <strong>the</strong>events which have at least four selected jets.Most <strong>of</strong> <strong>the</strong> W+jets backgrounds as well as <strong>the</strong> QCD multijet events do not conta<strong>in</strong>a b-tagged jet and are suppressed when requir<strong>in</strong>g <strong>the</strong> event to have at least one b jetcandidate.A summary <strong>of</strong> all selection cuts and <strong>the</strong> number <strong>of</strong> events surviv<strong>in</strong>g each step for varioussimulated signal and background samples is listed <strong>in</strong> Table 4.6.Hav<strong>in</strong>g applied <strong>the</strong> above-mentioned selection cuts, one would get a signal overbackground ratio <strong>of</strong> about 2.5. Hence <strong>the</strong> f<strong>in</strong>al sample <strong>of</strong> selected events is dom<strong>in</strong>atedby signal events and <strong>the</strong> purity <strong>of</strong> select<strong>in</strong>g <strong>the</strong> signal events can be fur<strong>the</strong>r improvedby apply<strong>in</strong>g additional cuts which are described <strong>in</strong> more detail <strong>in</strong> <strong>the</strong> next chapter.
66 CHAPTER 4: Reconstruct<strong>in</strong>g and Select<strong>in</strong>g Physics Eventt¯t t¯t W+jets Z+jets t QCD→ e + jets → o<strong>the</strong>r → l + jets → l + l − + jets → l + jets multijetsInitial 2333.0 13417.0 3131400.0 304800.0 3293.0 5.6E+8Pre-Selection 1917.3 5449.0 12961.9 4909.2 565.0 6681710.1PV 1916.5 5446.5 12958.5 4907.6 564.8 6680208.7One Electron 1053.0 360.6 5369.9 2121.1 154.2 12280.22 nd Electron Veto 1021.9 310.0 5313.9 1045.4 149.1 12135.3Muon Veto 1019.2 221.7 5308.0 1039.5 146.0 12120.3Four <strong>Jet</strong>s 405.9 50.5 241.0 28.1 19.1 379.2One B <strong>Jet</strong> 307.8 39.1 30.2 4.5 13.5 35.6Selection Efficiency (%) 13.2 0.3 0.0 0.0 0.4 0.0Table 4.6: Table <strong>of</strong> event selection cuts. The numbers are equivalent to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 100pb −1 . Also shown <strong>in</strong><strong>the</strong> last row <strong>of</strong> <strong>the</strong> table, is <strong>the</strong> efficiency <strong>of</strong> <strong>the</strong> selection cuts which is def<strong>in</strong>ed as <strong>the</strong> survived number <strong>of</strong> events divided by <strong>the</strong><strong>in</strong>itial number <strong>of</strong> events before apply<strong>in</strong>g any cuts. The selection cuts ma<strong>in</strong>ly keep <strong>the</strong> e+jets t¯t events while reject<strong>in</strong>g <strong>the</strong> diversebackgrounds.
Chapter 5Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong>Calibration FactorIn <strong>the</strong> previous chapter, <strong>the</strong> selection cuts, which are applied to enhance <strong>the</strong> purity <strong>of</strong><strong>the</strong> e+jets signal events <strong>in</strong> <strong>the</strong> f<strong>in</strong>al selected sample, were <strong>in</strong>troduced. At <strong>the</strong> end when<strong>the</strong> survived event sample is enriched <strong>in</strong> e+jets t¯t events, one can try to reconstruct<strong>the</strong> W boson and <strong>the</strong> top quark four-vectors by comb<strong>in</strong><strong>in</strong>g <strong>the</strong> <strong>in</strong>dividual objects andreconstruct backwards <strong>the</strong> decay cha<strong>in</strong>. We divide <strong>the</strong> t¯t system <strong>in</strong>to two separateparts depend<strong>in</strong>g on <strong>the</strong> decay products <strong>of</strong> <strong>the</strong> W boson, namely hadronic and leptonic,t h → W h b h → q¯qb h ,t l → W l b l → eν e b l ,where <strong>in</strong>dexes “h” and “l” refer to <strong>the</strong> hadronic and <strong>the</strong> leptonic parts, respectively.S<strong>in</strong>ce <strong>the</strong> four-vector <strong>of</strong> ν e is not reconstructed, <strong>the</strong> four-vectors <strong>of</strong> <strong>the</strong> leptonic Wboson and subsequently <strong>the</strong> leptonic top quark can not be made from <strong>the</strong>ir correspond<strong>in</strong>gdecay products. While <strong>the</strong> leptonic part would rema<strong>in</strong> non-reconstructed,<strong>the</strong> hadronic part can be fully reconstructed. In <strong>the</strong> hadronic part, <strong>the</strong> light quarks qand ¯q, hadronize and produce <strong>the</strong> light jets. Given that <strong>the</strong> light jets are known among<strong>the</strong> collection <strong>of</strong> <strong>the</strong> reconstructed jets, <strong>the</strong> correspond<strong>in</strong>g four-vectors are summedand make <strong>the</strong> four-vector <strong>of</strong> <strong>the</strong> hadronic W boson. Hence <strong>in</strong> order to reconstruct <strong>the</strong>hadronic W boson, one needs to determ<strong>in</strong>e which jets arise from <strong>the</strong> W boson decay.Consequently, <strong>in</strong> order to reconstruct <strong>the</strong> hadronic top quark, one has to determ<strong>in</strong>ewhich jet is <strong>the</strong> hadronic b jet. Although <strong>the</strong> leptonic top quark would not be fullyreconstructed, <strong>in</strong> order to benefit from <strong>the</strong> k<strong>in</strong>ematics <strong>of</strong> <strong>the</strong> leptonic b quark, onemay be <strong>in</strong>terested <strong>in</strong> associat<strong>in</strong>g a jet to <strong>the</strong> leptonic b quark, as well. The task <strong>of</strong>assign<strong>in</strong>g reconstructed jets to partons aris<strong>in</strong>g from t¯t → eν e q¯qb¯b, is a key topic whichis expla<strong>in</strong>ed <strong>in</strong> more detail <strong>in</strong> Section 5.1.Hav<strong>in</strong>g associated <strong>the</strong> jets to <strong>the</strong> partons, one can measure <strong>the</strong> <strong>in</strong>variant masses <strong>of</strong><strong>the</strong> hadronic W boson and <strong>the</strong> hadronic top quark. In Section 5.1.6, it is shown that<strong>the</strong> measured <strong>in</strong>variant masses <strong>of</strong> <strong>the</strong> W boson as well as <strong>of</strong> <strong>the</strong> top quark, deviatefrom <strong>the</strong>ir nom<strong>in</strong>al values that are used <strong>in</strong> <strong>the</strong> generation step <strong>of</strong> <strong>the</strong> simulation. Thisobservation motivates to correct <strong>the</strong> energy <strong>of</strong> <strong>the</strong> jets so that <strong>the</strong> measured masses <strong>of</strong>both <strong>the</strong> W boson and <strong>the</strong> top quark are peaked at <strong>the</strong>ir generated values. Apply<strong>in</strong>g67
68 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor<strong>the</strong> mass constra<strong>in</strong>ts <strong>of</strong> both <strong>the</strong> W boson and <strong>the</strong> top quark on t h → W h b → q¯qb h andrequir<strong>in</strong>g <strong>the</strong> reconstructed jets to fulfill <strong>the</strong> constra<strong>in</strong>ts, while allow<strong>in</strong>g <strong>the</strong> measuredenergies to change with<strong>in</strong> <strong>the</strong>ir resolutions, would result <strong>in</strong> obta<strong>in</strong><strong>in</strong>g jets with modifiedenergy. The method <strong>of</strong> apply<strong>in</strong>g mass constra<strong>in</strong>ts, which makes use <strong>of</strong> <strong>the</strong> LagrangeMultipliers, is called a K<strong>in</strong>ematic Fit and is <strong>in</strong>troduced <strong>in</strong> Section 5.2.In addition to <strong>the</strong> basel<strong>in</strong>e event selection cuts described <strong>in</strong> <strong>the</strong> previous chapter, someextra cuts are applied on <strong>the</strong> selected events which are <strong>in</strong>troduced <strong>in</strong> Section 5.3. Themethod to estimate <strong>the</strong> residual jet energy calibration factors is described <strong>in</strong> Section 5.4and <strong>the</strong> statistical properties <strong>of</strong> <strong>the</strong>se estimators are discussed <strong>in</strong> Section 5.5. F<strong>in</strong>ally,<strong>the</strong> method is applied on <strong>the</strong> proton-proton collision data which has been collected by<strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> dur<strong>in</strong>g <strong>the</strong> physics runs <strong>in</strong> 2010. The plots which compare <strong>the</strong>simulation versus data are shown <strong>in</strong> Section 5.6.5.1 Multi-Variate Analysis Method (MVA)In <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> <strong>the</strong> e+jets t¯t decay, t¯t → e¯ν e q¯qb¯b, <strong>the</strong>re are four quarks producedwhich should result <strong>in</strong> at least four reconstructed jets <strong>in</strong> <strong>the</strong> detector. If no tagg<strong>in</strong>g toolsare used to dist<strong>in</strong>guish between light jets, orig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong> quarks which appear <strong>in</strong><strong>the</strong> decay products <strong>of</strong> <strong>the</strong> hadronic W boson, and b jets, orig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong> b quarkswhich appear <strong>in</strong> <strong>the</strong> decay products <strong>of</strong> both <strong>the</strong> hadronic and <strong>the</strong> leptonic top quarks,one can not differentiate between <strong>the</strong>m. Therefore <strong>in</strong> order to label <strong>the</strong> jets and assign<strong>the</strong>m to <strong>the</strong> four quarks, <strong>the</strong>re are 4! = 24 different ways to do <strong>the</strong> assignment giventhat only four reconstructed jets are taken <strong>in</strong>to account. S<strong>in</strong>ce <strong>the</strong> order <strong>of</strong> <strong>the</strong> jetswhich are assigned to <strong>the</strong> light quarks is not important when reconstruct<strong>in</strong>g <strong>the</strong> Wboson momentum, <strong>the</strong>y can be used <strong>in</strong>terchangably. Hence <strong>the</strong> number <strong>of</strong> differentways to assign four reconstructed jets to <strong>the</strong> four quarks is divided by 2 and reduces to12. The quarks which are produced <strong>in</strong> <strong>the</strong> decay products <strong>of</strong> e+jets t¯t event, <strong>in</strong>itiatefrom heavy particles such as W boson and top quark. Hence <strong>the</strong> four lead<strong>in</strong>g jets thatare <strong>the</strong> jets with <strong>the</strong> maximum p T among <strong>the</strong> reconstructed jet collection, could begood candidates to represent <strong>the</strong> four quarks and thus are used to do <strong>the</strong> assignment.It should be noted that <strong>the</strong>re are usually more than four jets reconstructed <strong>in</strong> <strong>the</strong> eventdue to radiations, hence <strong>the</strong> four lead<strong>in</strong>g jets are not always <strong>the</strong> correct jets as willbe discussed <strong>in</strong> Section 5.1.1. Ano<strong>the</strong>r important note to be emphasized is that <strong>the</strong>electron fakes a jet but is removed from <strong>the</strong> list <strong>of</strong> reconstructed jets.To summarize <strong>the</strong> above discussion, <strong>the</strong>re exist 12 different comb<strong>in</strong>ations <strong>in</strong> order toassociate four lead<strong>in</strong>g jets to <strong>the</strong> four quarks arris<strong>in</strong>g from <strong>the</strong> e+jets t¯t decay. Among12 possible comb<strong>in</strong>ations, only one comb<strong>in</strong>ation represents <strong>the</strong> correct assignment.S<strong>in</strong>ce <strong>the</strong> determ<strong>in</strong>ation <strong>of</strong> <strong>the</strong> correct comb<strong>in</strong>ation <strong>of</strong> jets is essential for <strong>the</strong> rest <strong>of</strong>analysis, specific efforts should be made <strong>in</strong> order to select <strong>the</strong> correct comb<strong>in</strong>ation.In order to select <strong>the</strong> correct jet-parton comb<strong>in</strong>ation among 12 comb<strong>in</strong>ations, when nosimulation <strong>in</strong>formation is available, it is possible to take randomly one comb<strong>in</strong>ation.The probability <strong>of</strong> select<strong>in</strong>g <strong>the</strong> correct jet comb<strong>in</strong>ation randomly is 1 ∼ 8% as only12one among twelve comibnations is <strong>the</strong> right candidate <strong>in</strong> each trial. The probability <strong>of</strong>select<strong>in</strong>g <strong>the</strong> right jet comb<strong>in</strong>ation is fur<strong>the</strong>r improved by consider<strong>in</strong>g <strong>the</strong> k<strong>in</strong>ematics<strong>of</strong> some variables that can differentiate between <strong>the</strong> right and wrong jet comb<strong>in</strong>ations.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 69This is <strong>the</strong> subject <strong>of</strong> <strong>the</strong> methods, generally called Multi-Variate Analyses (MVA),which comb<strong>in</strong>e <strong>the</strong> <strong>in</strong>formation <strong>of</strong> different variables and extract a value that candiscrim<strong>in</strong>ate between <strong>the</strong> right and wrong jet comb<strong>in</strong>ations.There are many MVA methods implemented <strong>in</strong> order to separate signal, which is <strong>in</strong>this case a right jet comb<strong>in</strong>ation, from backgrounds, which are <strong>the</strong> o<strong>the</strong>r eleven wrongjet comb<strong>in</strong>ations [100]. The Likelihood ratio method is used <strong>in</strong> this analysis <strong>in</strong> orderto label <strong>the</strong> jets and select <strong>the</strong> correct jet comb<strong>in</strong>ation.5.1.1 <strong>Jet</strong>-Parton Match<strong>in</strong>g AlgorithmWhen analyz<strong>in</strong>g simulated samples for which <strong>the</strong> <strong>in</strong>formation from <strong>the</strong> generator levelcan be accessed, a Monte Carlo truth jet-parton match<strong>in</strong>g can be <strong>in</strong>troduced. Even<strong>in</strong> case <strong>of</strong> access to <strong>the</strong> full <strong>in</strong>formation <strong>of</strong> <strong>the</strong> partons produced <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state,<strong>in</strong>clud<strong>in</strong>g <strong>the</strong>ir four-vectors, no unambiguous def<strong>in</strong>ition <strong>of</strong> match<strong>in</strong>g is present. Thereare many algorithms to associate <strong>the</strong> reconstructed jets to <strong>the</strong> generated partons [101].In general <strong>the</strong>y match jets to partons based on <strong>the</strong> spatial distance between <strong>the</strong>m. Theone, which is used <strong>in</strong> this analysis, is called ptOrderedM<strong>in</strong>Dist and expla<strong>in</strong>ed brieflyhere.The algorithm starts by mak<strong>in</strong>g a p T -ordered list <strong>of</strong> partons produced <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state.Then it looks for <strong>the</strong> closest reconstructed jet to <strong>the</strong> lead<strong>in</strong>g parton by match<strong>in</strong>g <strong>the</strong>m<strong>in</strong> (η, φ) space. A reconstructed jet is assigned to <strong>the</strong> lead<strong>in</strong>g parton if∆R = √ (η parton − η jet ) 2 + (φ parton − φ jet ) 2 < 0.3.In case such a jet is found, <strong>the</strong> jet is matched to that parton and is subsequentlyremoved from <strong>the</strong> list <strong>of</strong> <strong>the</strong> reconstructed jets. The algorithm cont<strong>in</strong>ues by tak<strong>in</strong>g <strong>the</strong>second lead<strong>in</strong>g parton <strong>in</strong> <strong>the</strong> event. In this procedure, if <strong>the</strong>re is no jet found close to<strong>the</strong> parton, <strong>the</strong> parton rema<strong>in</strong>es unmatched.Apply<strong>in</strong>g <strong>the</strong> above mentioned jet-parton match<strong>in</strong>g algorithm, it is found that <strong>in</strong> only26% <strong>of</strong> <strong>the</strong> selected e+jets t¯t events, a correct jet-parton match<strong>in</strong>g exists. This relativelylow fraction <strong>of</strong> selected events, for which <strong>the</strong> four lead<strong>in</strong>g jets are matched to <strong>the</strong> fourquarks arrise <strong>in</strong> t¯t → eν e q¯qb¯b, can be expla<strong>in</strong>ed as follows. In most <strong>of</strong> <strong>the</strong> e+jets signalevents, one can f<strong>in</strong>d energetic jets orig<strong>in</strong>at<strong>in</strong>g from Initial State Radiation (ISR) whichf<strong>in</strong>ally appear <strong>in</strong> <strong>the</strong> list <strong>of</strong> four lead<strong>in</strong>g jets. The appearance <strong>of</strong> <strong>the</strong> ISR jets <strong>in</strong> <strong>the</strong> list<strong>of</strong> four lead<strong>in</strong>g jets consequently spoils <strong>the</strong> procedure <strong>of</strong> match<strong>in</strong>g <strong>of</strong> <strong>the</strong> hard-scatterpartons to <strong>the</strong> four lead<strong>in</strong>g jets. This physics result can be understood by look<strong>in</strong>g at<strong>the</strong> plot <strong>in</strong> Figure 5.1.Accord<strong>in</strong>g to Figure 5.1, which shows <strong>the</strong> multiplicity <strong>of</strong> <strong>the</strong> generated ISR jets withp T > 30 GeV per selected signal event, <strong>the</strong> fraction <strong>of</strong> <strong>the</strong> selected e+jets t¯t eventsthat conta<strong>in</strong> at least one ISR with a p T exceed<strong>in</strong>g 30 GeV would reach to 74%.Also it can be calculated how <strong>of</strong>ten an ISR jet is reconstructed among <strong>the</strong> four lead<strong>in</strong>gjets. As understood from Figure 5.2, which shows <strong>the</strong> number <strong>of</strong> ISR jets that can bematched to <strong>the</strong> four lead<strong>in</strong>g jets per selected signal event, <strong>in</strong> 56% <strong>of</strong> all selected e+jetst¯t events, <strong>the</strong>re is at least one ISR that is matched to a jet among <strong>the</strong> four lead<strong>in</strong>g jets.Therefore, <strong>the</strong> presence <strong>of</strong> ISR jets among four lead<strong>in</strong>g reconstructed jets <strong>in</strong> signalevents can partially expla<strong>in</strong> why <strong>in</strong> a large fraction <strong>of</strong> <strong>the</strong> selected events, <strong>the</strong> four
70 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor# events12000 Entries 2898810000Mean 1.224RMS 1.013800060004000# events1200010000800060004000Entries 28988Mean 0.662RMS 0.6669200000 1 2 3 4 5 6 7 8 9# ISR jetsFigure 5.1: The number <strong>of</strong> ISR jets withp T > 30 GeV per selected signal event.20000-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5# ISR jetsFigure 5.2: The number <strong>of</strong> ISR jets thatcan be matched to <strong>the</strong> four lead<strong>in</strong>g jetsper selected signal event.lead<strong>in</strong>g jets can not be matched to <strong>the</strong> four quarks <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> t¯t system.Ano<strong>the</strong>r reason that can expla<strong>in</strong> why <strong>the</strong> partons produced <strong>in</strong> <strong>the</strong> hard scatter<strong>in</strong>g t¯tprocess, do not match to <strong>the</strong> four lead<strong>in</strong>g reconstructed jets, is due to <strong>the</strong> acceptancerequirement. S<strong>in</strong>ce reconstructed jets have already passed <strong>the</strong> cuts on η and are notallowed to be located <strong>in</strong> <strong>the</strong> forward region, <strong>the</strong> fraction <strong>of</strong> events where at least oneparton is produced out <strong>of</strong> acceptance region, would not have matched jets. F<strong>in</strong>al StateRadiation (FSR) can spoil <strong>the</strong> procedure <strong>of</strong> match<strong>in</strong>g <strong>of</strong> partons to reconstructed jets,too. S<strong>in</strong>ce FSR can split <strong>the</strong> <strong>in</strong>itial parton which consequently yields two separate jets,<strong>the</strong>refore <strong>the</strong> jets might be reconstructed far from <strong>the</strong> direction <strong>of</strong> <strong>the</strong> orig<strong>in</strong>al partonand can not be found <strong>in</strong> <strong>the</strong> match<strong>in</strong>g algorithm.5.1.2 Likelihood Ratio MethodThe normalized distributions <strong>of</strong> <strong>the</strong> various discrim<strong>in</strong>at<strong>in</strong>g variables, which make <strong>the</strong>Probability Density Functions (PDFs) <strong>of</strong> <strong>the</strong>se variables, are used to def<strong>in</strong>e <strong>the</strong> signaland background likelihoods, L S and L B respectively. A likelihood function is def<strong>in</strong>edas <strong>the</strong> product <strong>of</strong> <strong>the</strong> PDFs P k (x k ) <strong>of</strong> all <strong>in</strong>put variables x k , which constitute n varvariables and can be expressed for signal and background asandL S =L B =n∏vark=1n∏vark=1P S,k (x k ),P B,k (x k ),respectively. For each comb<strong>in</strong>ation among twelve possible comb<strong>in</strong>ations, <strong>the</strong> Likelihoodratio y L is def<strong>in</strong>ed as <strong>the</strong> signal likelihood divided by <strong>the</strong> sum <strong>of</strong> <strong>the</strong> signal andbackground likelihoodsL Sy L = .L S + L B
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 71The ratio y L is <strong>the</strong>n calculated for every jet comb<strong>in</strong>ation <strong>in</strong> each event. Per event, <strong>the</strong>jet comb<strong>in</strong>ation whose y L is <strong>the</strong> largest, is chosen and returned by <strong>the</strong> MVA method.Almost all <strong>the</strong> MVA methods which <strong>in</strong>cludd a Likelihood Ratio method need to betra<strong>in</strong>ed before apply<strong>in</strong>g <strong>the</strong> method on data. In <strong>the</strong> tra<strong>in</strong><strong>in</strong>g phase, one <strong>in</strong>troduces<strong>the</strong> signal and background PDFs to <strong>the</strong> MVA method. Then <strong>the</strong> MVA method tra<strong>in</strong>sitself and learns what k<strong>in</strong>d <strong>of</strong> behaviours can be extracted from both <strong>the</strong> signal andbackgrounds PDFs. In <strong>the</strong> application phase, us<strong>in</strong>g <strong>the</strong> <strong>in</strong>formation which has beencollected <strong>in</strong> <strong>the</strong> tra<strong>in</strong><strong>in</strong>g phase, <strong>the</strong> MVA method returns a value per event and categorizesthat event as signal or background. In case <strong>of</strong> a jet comb<strong>in</strong>ation study, where onesignal aga<strong>in</strong>st eleven backgrounds is present per event, <strong>the</strong> signal is def<strong>in</strong>ed as <strong>the</strong> jetcomb<strong>in</strong>ation with <strong>the</strong> highest value returned by <strong>the</strong> MVA method, although <strong>in</strong> somecases <strong>the</strong> MVA method is not able to return <strong>the</strong> “true” jet comb<strong>in</strong>ation, which correspondsto <strong>the</strong> correct jet comb<strong>in</strong>ation that is matched with <strong>the</strong> hard-scatter partons.5.1.3 Likelihood Concept <strong>in</strong> Bayesian StatisticsAccord<strong>in</strong>g to <strong>the</strong> Bayes’ <strong>the</strong>orem, posterior probability p(Y |X), is related to priorprobability p(Y ), with <strong>the</strong> follow<strong>in</strong>g equationp(Y |X) =p(X|Y )p(Y ),p(X)where p(X|Y ) is <strong>the</strong> conditional probability and is also referred to as <strong>the</strong> Likelihood. In<strong>the</strong> above equation, p(X|Y ) is <strong>the</strong> probability distribution <strong>of</strong> <strong>the</strong> parameter X, whichis usually a cont<strong>in</strong>uous variable <strong>of</strong> <strong>the</strong> event that belongs to <strong>the</strong> class Y and p(Y |X) is<strong>the</strong>n def<strong>in</strong>ed as <strong>the</strong> probability <strong>of</strong> assign<strong>in</strong>g a new observed event to <strong>the</strong> class Y giventhat <strong>the</strong> value X is measured for that particular event. Therefore <strong>in</strong> order to obta<strong>in</strong> <strong>the</strong>posterior probability p(Y |X), <strong>in</strong> addition to <strong>the</strong> prior probability p(Y ), p(X|Y ) shouldalso be known. In <strong>the</strong> language <strong>of</strong> multi-variate techniques, p(X|Y ) is obta<strong>in</strong>ed <strong>in</strong> <strong>the</strong>tra<strong>in</strong><strong>in</strong>g phase. Dur<strong>in</strong>g tra<strong>in</strong><strong>in</strong>g, <strong>the</strong> classes Y with <strong>the</strong>ir properties X are <strong>in</strong>troducedto <strong>the</strong> tra<strong>in</strong>er and subsequently <strong>the</strong> correspond<strong>in</strong>g PDFs are extracted.In order to clarify a bit more how a generic multi variate method works, a simplifiedexample based on Bayes’ <strong>the</strong>orem is expla<strong>in</strong>ed here. Consider a space with one variableX that can take only two values, namely “X = 1” or “X = 2”. In this space, eventsare categorized <strong>in</strong> ei<strong>the</strong>r signal or background classes, hence “Y = S” or “Y = B”. Anevent is said to be measured when <strong>the</strong> X value <strong>of</strong> that particular event is determ<strong>in</strong>ed.Assume 10 such events are selected and fed to an MVA method. The results <strong>of</strong> tra<strong>in</strong><strong>in</strong>gover 10 events, are summarized <strong>in</strong> Figure 5.3.The <strong>in</strong>formation, conta<strong>in</strong><strong>in</strong>g <strong>the</strong> conditional as well as <strong>the</strong> prior probabilities, thatcan be derived from Figure 5.3, is listed below.p(1|S) = 3 5 , p(1|B) = 1 5 ,p(2|S) = 2 5 , p(2|B) = 4 5 ,
72 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration FactorFigure 5.3: A simple example to show <strong>the</strong> basic concepts <strong>of</strong> <strong>the</strong> MVA method. In thisexample, <strong>the</strong> circles, which are represent<strong>in</strong>g events, are observed <strong>in</strong> two different b<strong>in</strong>s,be<strong>in</strong>g ei<strong>the</strong>r “1” or “2” and can belong to one <strong>of</strong> “S” or “B” classes.p(S) = 5 10 ,p(B) = 510 .From <strong>the</strong> above elements, p(1) and p(2) can also be extracted, accord<strong>in</strong>g to <strong>the</strong> “sumrule” <strong>of</strong> <strong>the</strong> probability <strong>the</strong>ory, as followsp(1) = p(1|S)p(S) + p(1|B)p(B) = 2 5 ,p(2) = p(2|S)p(S) + p(2|B)p(B) = 3 5 .Now that all <strong>the</strong> <strong>in</strong>put <strong>in</strong>formation is complete, one can calculate <strong>the</strong> posterior probabilitiesas expressed belowp(S|1) = p(1|S)p(S)p(1)= 912 ,p(S|2) = p(2|S)p(S)p(2)p(B|1) = p(1|B)p(B)p(1)p(B|2) = p(2|B)p(B)p(2)= 412 ,= 312 ,= 812 .The above numbers can be <strong>in</strong>terpreted as follows. Any new event, whose measured Xvalue yields to X = 1, would be assigned to class S s<strong>in</strong>ce <strong>the</strong> probability <strong>of</strong> be<strong>in</strong>g atype “S” is three times more than be<strong>in</strong>g a type “B” event when a value equals 1 ismeasured. With <strong>the</strong> same reason<strong>in</strong>g, any new event is grouped to <strong>the</strong> class B giventhat <strong>the</strong> meaurement <strong>of</strong> <strong>the</strong> X property <strong>of</strong> that event results <strong>in</strong> X = 2.The above example shows <strong>the</strong> basic idea <strong>of</strong> assign<strong>in</strong>g a new observed event to a specificclass <strong>in</strong> a one-dimensional phase space. In most cases, usually more than one <strong>in</strong>putvariable is used. As a result, <strong>the</strong> problem <strong>of</strong> label<strong>in</strong>g an event as signal or background,would not be that simple. Def<strong>in</strong><strong>in</strong>g a s<strong>in</strong>gle variable out <strong>of</strong> many <strong>in</strong>put variables canprovide a possible solution. Hence <strong>the</strong> Likelihood function L(⃗x), which comb<strong>in</strong>es <strong>the</strong>
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 73<strong>in</strong>formation <strong>of</strong> all <strong>in</strong>put variables ⃗x = (x 1 . . .x n ) as expla<strong>in</strong>ed <strong>in</strong> Section 5.1.2, is <strong>in</strong>troduced<strong>in</strong> an n-dimensional phase space.By def<strong>in</strong>ition, L can take values <strong>in</strong> a range <strong>of</strong> [0,1]. Therefore, <strong>in</strong> <strong>the</strong> language <strong>of</strong> LikelihoodRatio, an event is assigned as signal if <strong>the</strong> measurement <strong>of</strong> its X property wouldhappen <strong>in</strong> <strong>the</strong> b<strong>in</strong> whose Likelihood Ratio value is greater than 0.5. Then <strong>the</strong> event ismore signal-like compared to background-like. O<strong>the</strong>rwise, it is labeled as background.In case <strong>of</strong> f<strong>in</strong>d<strong>in</strong>g <strong>the</strong> correct jet comb<strong>in</strong>ation, where eleven backgrounds, be<strong>in</strong>g wrongcomb<strong>in</strong>ations, versus one signal, be<strong>in</strong>g <strong>the</strong> true comb<strong>in</strong>ation, exist, <strong>the</strong> chosen comb<strong>in</strong>ationis def<strong>in</strong>ed as <strong>the</strong> comb<strong>in</strong>ation whose Likelihood Ratio value is <strong>the</strong> maximumamong <strong>the</strong> o<strong>the</strong>r comb<strong>in</strong>ations.5.1.4 Input Variables for Tra<strong>in</strong><strong>in</strong>gVarious observable variables can be used to tra<strong>in</strong> <strong>the</strong> Likelihood Ratio method. Also acomb<strong>in</strong>ation <strong>of</strong> two or three variables is allowed. Among so many candidate variableswhich can be used <strong>in</strong> <strong>the</strong> tra<strong>in</strong><strong>in</strong>g, <strong>the</strong> most discrim<strong>in</strong>at<strong>in</strong>g variables that differentiatemaximally between signal and backgrond, are desired. The discrim<strong>in</strong>at<strong>in</strong>g power<strong>of</strong> a generic variable can be def<strong>in</strong>ed us<strong>in</strong>g different methods. As an example, <strong>the</strong>“separation” S, which can be a measure <strong>of</strong> discrim<strong>in</strong>at<strong>in</strong>g power between signal andbackground, is def<strong>in</strong>ed asS = 1 ∫ (PS (X) − P B (X)) 22 (P S (X) + P B (X)) dX,where P S (X) and P B (X) are <strong>the</strong> PDFs <strong>of</strong> <strong>the</strong> observable X for signal and background,respectively. By def<strong>in</strong>ition, for identical signal and background shapes, <strong>the</strong> separationis zero. If <strong>the</strong> signal and background PDFs do not overlap, <strong>the</strong>n <strong>the</strong> separation takesa value equal to one. Therefore higher values <strong>of</strong> <strong>the</strong> separation, <strong>in</strong>dicate <strong>the</strong> variablehas a higher probability to be a good discrim<strong>in</strong>ant candidate.S<strong>in</strong>ce <strong>the</strong> four vector <strong>of</strong> <strong>the</strong> neutr<strong>in</strong>o is not known, hence <strong>the</strong> leptonic W boson andconsequently <strong>the</strong> leptonic top quark can not be fully reconstructed. Therefore <strong>the</strong> <strong>in</strong>putvariables which can be chosen to tra<strong>in</strong> <strong>the</strong> MVA method with, should be <strong>in</strong>dependentfrom <strong>the</strong> reconstructed neutr<strong>in</strong>o. This makes <strong>the</strong> <strong>in</strong>put candidates to be limited to asmaller collection. Different k<strong>in</strong>ematics <strong>of</strong> <strong>the</strong> objects such as transverse momentump T , pseudorapidity η, polar angle θ and azimuthal angle φ can be used to compare<strong>the</strong> candidates <strong>in</strong> t¯t system. In this analysis, <strong>the</strong> Θ variable, which is def<strong>in</strong>ed as <strong>the</strong>space angle between two vectors <strong>in</strong> a three-dimensional phase space is used which isexpressed asΘ(⃗v 1 ,⃗v 2 ) = cos −1 ( ⃗v1 .⃗v 2|⃗v 1 ||⃗v 2 |)= cos −1 (s<strong>in</strong> θ 1 s<strong>in</strong> θ 2 (cos(φ 1 − φ 2 )) + cosθ 1 cos θ 2 ).The two-component variables <strong>in</strong> <strong>the</strong> t¯t system, exclud<strong>in</strong>g those variables related to <strong>the</strong>k<strong>in</strong>ematics <strong>of</strong> neutr<strong>in</strong>o, which are considered <strong>in</strong> this analysis, are listed below.• Θ(t h , W h ) which is <strong>the</strong> space angle between hadronic top quark and hadronic Wboson,
74 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor• Θ(t h , b h ) which is <strong>the</strong> space angle between hadronic top quark and hadronic b,• Θ(t h , b l ) which is <strong>the</strong> space angle between hadronic top quark and leptonic b,• Θ(t h , e) which is <strong>the</strong> space angle between hadronic top quark and electron,• Θ(W h , b h ) which is <strong>the</strong> space angle between hadronic W boson and hadronic b,• Θ(W h , b l ) which is <strong>the</strong> space angle between hadronic W boson and leptonic b,• Θ(W h , e) which is <strong>the</strong> space angle between hadronic W boson and electron,• Θ(b h , b l ) which is <strong>the</strong> space angle between hadronic b and leptonic b,• Θ(b h , e) which is <strong>the</strong> space angle between hadronic b and electron,• Θ(b l , e) which is <strong>the</strong> space angle between leptonic b and electron,• Θ(q, ¯q) which is <strong>the</strong> space angle between light quarks appear<strong>in</strong>g <strong>in</strong> <strong>the</strong> decay <strong>of</strong>hadronic W boson.Most <strong>of</strong> <strong>the</strong> above mentioned variables are not <strong>in</strong>dependent and conta<strong>in</strong> <strong>the</strong> same<strong>in</strong>formation. This can be seen by look<strong>in</strong>g at <strong>the</strong>ir l<strong>in</strong>ear correlations which are shown<strong>in</strong> Table 5.1. S<strong>in</strong>ce <strong>the</strong> masses <strong>of</strong> <strong>the</strong> W boson and <strong>the</strong> top quark are go<strong>in</strong>g to be usedas constra<strong>in</strong>ts <strong>in</strong> this analysis, as will be expla<strong>in</strong>ed <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g sections, it is alsorequired that <strong>the</strong> selected variables are not highly correlated with <strong>the</strong> W boson andtop quark masses. The last two rows <strong>of</strong> Table 5.1 conta<strong>in</strong> <strong>the</strong> correlation factors amongall considered space angle variables with <strong>the</strong> masses <strong>of</strong> <strong>the</strong> W boson and top quark.The space angle variables are required to have a mutual correlation factor among<strong>the</strong>mselves less than 40%. They are also asked to be correlated with <strong>the</strong> W boson and<strong>the</strong> top quark masses less than 20%. S<strong>in</strong>ce <strong>the</strong> separation powers <strong>of</strong> each <strong>in</strong>dividualvariable are <strong>of</strong> <strong>the</strong> same order, tak<strong>in</strong>g <strong>in</strong>to account <strong>the</strong> requirement on <strong>the</strong> correlationfactor, only 5 variables among <strong>the</strong> above mentioned space angle variables are selectedto be used <strong>in</strong> <strong>the</strong> tra<strong>in</strong><strong>in</strong>g <strong>of</strong> <strong>the</strong> MVA method. In addition, <strong>the</strong>re are two morevariables which are selected for tra<strong>in</strong><strong>in</strong>g. A first one, which is def<strong>in</strong>ed as <strong>the</strong> transversemomentum <strong>of</strong> <strong>the</strong> hadronic top quark candidate relative to <strong>the</strong> sum <strong>of</strong> <strong>the</strong> transversemomenta <strong>of</strong> all hadronic top quark candidates pth TΣp T. S<strong>in</strong>ce with <strong>the</strong> four lead<strong>in</strong>g jets,<strong>the</strong>re are four different ways to comb<strong>in</strong>e three <strong>of</strong> <strong>the</strong>m to reconstruct a top quarkcandidate, <strong>the</strong>n <strong>the</strong> sum <strong>in</strong> <strong>the</strong> denom<strong>in</strong>ator is performed over <strong>the</strong> transverse momenta
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 75Θ(t h , W h ) Θ(t h , b h ) Θ(t h , b l ) Θ(t h , e) Θ(W h , b h ) Θ(W h , b l ) Θ(W h , e) Θ(b h , b l ) Θ(b h , e) Θ(b l , e) Θ(q, ¯q)Θ(t h , W h ) +100.0 - - - - - - - - - -Θ(t h , b h ) -10.0 +100.0 - - - - - - - - -Θ(t h , b l ) +5.6 +6.9 +100.0 - - - - - - - -Θ(t h , e) +5.2 +5.8 +40.8 +100.0 - - - - - - -Θ(W h , b h ) +32.4 +67.9 +7.9 +8.7 +100.0 - - - - - -Θ(W h , b l ) +2.0 +7.3 +60.1 +36.5 +8.0 +100.0 - - - - -Θ(W h , e) +0.5 +4.7 +24.0 +61.1 +4.4 +35.2 +100.0 - - - -Θ(b h , b l ) +3.6 -4.4 +43.2 +32.1 -1.4 +32.7 +33.6 +100.0 - - -Θ(b h , e) +3.8 -3.7 +18.3 +43.5 -0.1 +25.6 +31.5 +31.1 +100.0 - -Θ(b l , e) +15.0 +11.8 +1.2 +0.5 +15.9 +1.3 -1.3 -1.0 +3.1 +100.0 -Θ(q, ¯q) +16.8 -1.8 +2.2 +4.5 +23.6 +0.7 +1.3 +4.3 +5.8 +4.4 +100.0m W -0.7 +2.0 +0.9 -0.1 +1.4 -1.3 -0.5 +3.2 +1.2 -2.5 +21.3m top +3.6 +2.0 +1.7 +2.1 +7.1 -0.4 +1.6 +0.9 +1.2 -1.5 +10.7Table 5.1: The mutual correlation factors <strong>of</strong> some space angle variables and <strong>the</strong>ir correlation coefficients with <strong>the</strong> masses <strong>of</strong> <strong>the</strong>W boson and <strong>the</strong> top quark. All numbers are quoted <strong>in</strong> %.
76 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor<strong>of</strong> <strong>the</strong> four possible top quark candidates. A second variable is derived from <strong>the</strong> b-tagg<strong>in</strong>g discrim<strong>in</strong>ant (β) <strong>of</strong> each <strong>of</strong> <strong>the</strong> two b jet candidates which exist <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state.The variable which is used <strong>in</strong> <strong>the</strong> tra<strong>in</strong><strong>in</strong>g is def<strong>in</strong>ed as <strong>the</strong> b-tagg<strong>in</strong>g discrim<strong>in</strong>ator <strong>of</strong><strong>the</strong> hadronic b jet candidate multiplied by <strong>the</strong> b-tagg<strong>in</strong>g discrim<strong>in</strong>ator <strong>of</strong> <strong>the</strong> leptonicb jet candidate, hence β bh × β bl .The f<strong>in</strong>al selected variables which are used to tra<strong>in</strong> <strong>the</strong> Likelihood Ratio method, arelisted <strong>in</strong> Table 5.2. The mutual correlation factors between <strong>the</strong> selected variables and<strong>the</strong> top quark and <strong>the</strong> W boson masses are also quoted. The separation power <strong>of</strong> each<strong>in</strong>dividual variable is calculated and given <strong>in</strong> <strong>the</strong> last row <strong>of</strong> Table 5.2.Θ(t h , W h ) Θ(t h , b h ) Θ(t h , b l ) Θ(t h , e) Θ(b l , e)p thTΣp Tβ bh × β blm W -0.7 +2.0 +0.9 -0.1 -2.5 +0.4 -0.7m top +3.6 +2.0 +1.7 +2.1 -1.5 +2.5 +0.9Separation Power 0.08 0.06 0.20 0.09 0.30 0.10 0.40Table 5.2: The correlation coefficients <strong>of</strong> <strong>the</strong> variables, which are selected to tra<strong>in</strong> <strong>the</strong>MVA method, with <strong>the</strong> masses <strong>of</strong> <strong>the</strong> W boson and <strong>the</strong> top quark. The separationpower <strong>of</strong> each variable is also shown <strong>in</strong> <strong>the</strong> last row. All numbers are quoted <strong>in</strong> %.The distribution <strong>of</strong> <strong>the</strong> selected variables, which are shown separately for <strong>the</strong> correctand <strong>the</strong> wrong comb<strong>in</strong>ations, can be found <strong>in</strong> Figure 5.4 and Figure 5.5. All selectedevents <strong>in</strong> which a correct jet-parton comb<strong>in</strong>ation exists, are taken <strong>in</strong>to account.5.1.5 The MVA PerformanceAfter tra<strong>in</strong><strong>in</strong>g <strong>the</strong> Likelihood Ratio method us<strong>in</strong>g <strong>the</strong> <strong>in</strong>put variables listed <strong>in</strong> Table 5.2,one can check <strong>the</strong> performance <strong>of</strong> <strong>the</strong> method by look<strong>in</strong>g at some control plots. Forexample, <strong>the</strong> fraction <strong>of</strong> e+jets t¯t events for which <strong>the</strong> chosen jet comb<strong>in</strong>ation returnedby <strong>the</strong> Likelihood Ratio method corresponds to <strong>the</strong> true jet-quark comb<strong>in</strong>ation, couldbe a possible measure to check how well <strong>the</strong> MVA method works.Figure 5.6 shows <strong>the</strong> Likelihood Ratio response, which is def<strong>in</strong>ed to be <strong>the</strong> maximumvalue among twelve possible values, per event. Only those selected e+jets events forwhich <strong>the</strong> hadronic jets are matched to <strong>the</strong> hadronic quarks <strong>in</strong> t → Wb → q¯qb, contributeto this plot. In association with <strong>the</strong> Likelihood Ratio response, <strong>the</strong> chosenjet-parton comb<strong>in</strong>ation is also returned by <strong>the</strong> Likelihood Ratio method, which waschosen ei<strong>the</strong>r rightly or wrongly. The response for <strong>the</strong> events with ei<strong>the</strong>r right orwrong jet comb<strong>in</strong>ation chosen by <strong>the</strong> MVA is shown <strong>in</strong> Figure 5.6. Here, “Good” jetcomb<strong>in</strong>ation, as labelled <strong>in</strong> <strong>the</strong> plot, is def<strong>in</strong>ed as <strong>the</strong> event where <strong>the</strong> correct comb<strong>in</strong>ationis chosen as <strong>the</strong> one where <strong>the</strong> hadronic jets are matched to <strong>the</strong> hadronic quarkst → Wb → q¯qb. Therefore even if <strong>the</strong> MVA is not able to assign <strong>the</strong> fourth jet among<strong>the</strong> four lead<strong>in</strong>g jets to <strong>the</strong> leptonic b quark arris<strong>in</strong>g from <strong>the</strong> leptonic top quark, <strong>the</strong>
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 77a.u.0.0350.030.025Correct Comb<strong>in</strong>ationWrong Comb<strong>in</strong>ationa.u.0.0180.0160.0140.020.0150.010.0050.0120.010.0080.0060.0040.00200 0.5 1 1.5 2 2.5 3 3.5θ(hadronic top, hadronic b)00 0.5 1 1.5 2 2.5 3 3.5θ(hadronic top, leptonic b)a.u.0.020.018a.u.0.0250.0160.0140.020.0120.010.0080.0060.0040.0150.010.0050.00200 0.5 1 1.5 2 2.5 3 3.5θ(hadronic top, electron)00 0.5 1 1.5 2 2.5 3 3.5θ(leptonic b, electron)Figure 5.4: The distributions <strong>of</strong> <strong>the</strong> selected <strong>in</strong>put variables to tra<strong>in</strong> <strong>the</strong> MVA method.
78 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factora.u.0.07 Correct Comb<strong>in</strong>ationa.u.0.030.06Wrong Comb<strong>in</strong>ation0.0250.050.020.040.030.020.0150.010.010.00500 0.5 1 1.5 2 2.5 3 3.5θ(hadronic top, hadronic W)00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8hadronic top 3jetsp /Σ pTTa.u.0.0450.040.0350.030.0250.020.0150.010.00500 0.2 0.4 0.6 0.8 1hadronic b leptonic bb-tag× b-tagFigure 5.5: The distributions <strong>of</strong> <strong>the</strong> selected <strong>in</strong>put variables to tra<strong>in</strong> <strong>the</strong> MVA method.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 79chosen jet-parton comb<strong>in</strong>ation returned by <strong>the</strong> MVA method would still be taken ascorrect. S<strong>in</strong>ce <strong>the</strong>re is no <strong>in</strong>terest <strong>in</strong> <strong>the</strong> reconstruction <strong>of</strong> <strong>the</strong> leptonic part <strong>of</strong> t¯t systemfor this analysis, <strong>the</strong>re is no force applied on <strong>the</strong> MVA method to associate a jetcorrectly to <strong>the</strong> leptonic b quark <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state. The “Bad” jet comb<strong>in</strong>ation <strong>the</strong>n is<strong>the</strong> comb<strong>in</strong>ation for which <strong>the</strong> MVA cannot associate, correctly, at least one hadronicquark <strong>in</strong> <strong>the</strong> t → Wb → q¯qb decay topolology, to a reconstructed jet <strong>in</strong> <strong>the</strong> four lead<strong>in</strong>gjets found <strong>in</strong> <strong>the</strong> event. Although <strong>in</strong> some cases <strong>the</strong> chosen jet-parton comb<strong>in</strong>ationby <strong>the</strong> MVA, is <strong>the</strong> true comb<strong>in</strong>ation, <strong>in</strong> about half <strong>of</strong> <strong>the</strong> cases <strong>the</strong> Likelihood Ratiomethod is not able to return <strong>the</strong> true jet-parton comb<strong>in</strong>ation.# events900800700Good <strong>Jet</strong> Comb<strong>in</strong>ationsBad <strong>Jet</strong> Comb<strong>in</strong>ations60050040030020010000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Likelihood ResponseFigure 5.6: The maximum among twelve possible values returned by <strong>the</strong> LikelihoodRatio method per event. Only those selected e+jets events for which <strong>the</strong> hadronic jetsare matched to <strong>the</strong> hadronic quarks <strong>in</strong> t → Wb → q¯qb, are taken <strong>in</strong>to account.Consider<strong>in</strong>g those selected events with a correct jet-parton comb<strong>in</strong>ation, <strong>the</strong> plotwhich conta<strong>in</strong>s <strong>the</strong> purity versus efficiency for <strong>the</strong> different cut values on <strong>the</strong> Likelihoodresponse, is shown <strong>in</strong> Figure 5.7. Efficiency is def<strong>in</strong>ed as <strong>the</strong> fraction <strong>of</strong> <strong>the</strong> selectede+jets events with a right jet comb<strong>in</strong>ation that pass <strong>the</strong> cut on <strong>the</strong> Likelihood Ratioresponse. Purity is def<strong>in</strong>ed to be <strong>the</strong> number <strong>of</strong> <strong>the</strong> selected e+jets events with a rightjet comb<strong>in</strong>ation that pass <strong>the</strong> cut on <strong>the</strong> Likelihood response relative to <strong>the</strong> multiplicity<strong>of</strong> all selected e+jets events, <strong>in</strong>clud<strong>in</strong>g ei<strong>the</strong>r a right or wrong jet comb<strong>in</strong>ation, thatpass <strong>the</strong> cut on <strong>the</strong> Likelihood Ratio response.Accord<strong>in</strong>g to <strong>the</strong> Monte Carlo truth, <strong>in</strong> only 34% <strong>of</strong> <strong>the</strong> selected e+jets t¯t events,<strong>the</strong> three reconstructed jets among <strong>the</strong> four lead<strong>in</strong>g jets are matched, truly, to <strong>the</strong>hadronic quarks <strong>in</strong> t → Wb → q¯qb. As it is understood from Figure 5.7, <strong>the</strong> LikelihoodRatio method is able to successfully f<strong>in</strong>d <strong>the</strong> true comb<strong>in</strong>ation <strong>in</strong> 51% <strong>of</strong> those eventswith a true jet-parton match<strong>in</strong>g, given that no cut is applied on <strong>the</strong> Likelihood Ratioresponse. Compar<strong>in</strong>g to a random selection <strong>of</strong> <strong>the</strong> true comb<strong>in</strong>ation that yields aprobability <strong>of</strong> about 8% <strong>of</strong> select<strong>in</strong>g <strong>the</strong> right jet comb<strong>in</strong>ation, <strong>the</strong> great improvementwhich is obta<strong>in</strong>ed when us<strong>in</strong>g <strong>the</strong> MVA method is obvious.
80 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration FactorPurity10.90.80.70.60.50.40.30.20.10Likelihood0.2 0.4 0.6 0.8 1EfficiencyFigure 5.7: The purity versus efficiencyplot when cutt<strong>in</strong>g on <strong>the</strong> MVA output for<strong>the</strong> selected e+jets t¯t events with a correctjet-parton comb<strong>in</strong>ation.Figure 5.8: The purity versus efficiencyplot when cutt<strong>in</strong>g on <strong>the</strong> MVA output forall selected e+jets t¯t events.Also shown <strong>in</strong> Figure 5.8 is <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> purity versus efficiency when cutt<strong>in</strong>gon <strong>the</strong> Likelihood response which is plotted for all selected e+jets t¯t events, <strong>in</strong>clud<strong>in</strong>gthose with no correct jet-parton comb<strong>in</strong>ation. Consider<strong>in</strong>g <strong>the</strong>se k<strong>in</strong>d <strong>of</strong> events, <strong>the</strong>efficiency <strong>of</strong> select<strong>in</strong>g <strong>the</strong> correct comb<strong>in</strong>ation with <strong>the</strong> use <strong>of</strong> <strong>the</strong> Likelihood method,when no cut is applied on <strong>the</strong> Likelihood response, reduces to about 17% which isobta<strong>in</strong>ed as 34% × 51% ∼ 17%.5.1.6 Reconstruct<strong>in</strong>g W Boson and Top Quark MassesNow that <strong>the</strong> reconstructed jets are associated to <strong>the</strong> hadronic quarks which areproduced <strong>in</strong> <strong>the</strong> decay products <strong>of</strong> <strong>the</strong> hadronic top quark <strong>of</strong> an e+jets t¯t event,t → Wb → q¯qb, one can reconstruct <strong>the</strong> W boson four-vector by summ<strong>in</strong>g <strong>the</strong> fourvectors<strong>of</strong> <strong>the</strong> two light jets. The four-vector <strong>of</strong> <strong>the</strong> top quark can be subsequentlyreconstructed by add<strong>in</strong>g <strong>the</strong> four-vector <strong>of</strong> <strong>the</strong> hadronic b jet to <strong>the</strong> reconstructedhadronic W boson. Fur<strong>the</strong>rmore, one can measure <strong>the</strong> <strong>in</strong>variant masses <strong>of</strong> <strong>the</strong> W bosonand top quark from <strong>the</strong>ir reconstructed four-vectors. In Figure 5.9, <strong>the</strong> masses <strong>of</strong><strong>the</strong> W boson and <strong>the</strong> top quark are shown for those events where <strong>the</strong> MVA successfullyreturned <strong>the</strong> true jet-parton comb<strong>in</strong>ation. It is observed that <strong>the</strong> reconstructedW boson mass is peaked around 89 GeV while <strong>the</strong> parameter which has been usedfor <strong>the</strong> W boson mass when simulat<strong>in</strong>g <strong>the</strong> events was chosen to be 80.4 GeV. Also<strong>the</strong> mean value <strong>of</strong> <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> top quark mass is shifted from <strong>the</strong> valueused <strong>in</strong> <strong>the</strong> simulation procedure, be<strong>in</strong>g 172.5 GeV, to a reconstructed value which isfound to be about 181 GeV. One should note that <strong>the</strong> events which are used to fill <strong>the</strong>mass distributions with, are those events where <strong>the</strong> MVA successfully chose <strong>the</strong> truejet comb<strong>in</strong>ation. If all selected events were taken <strong>in</strong>to account, tails would appear onboth histograms.This observation leads to <strong>the</strong> fact that, <strong>the</strong> jet energies which are used to build <strong>the</strong> Wboson and top quark four-vectors are over-estimated which yields <strong>the</strong> obta<strong>in</strong>ed higher<strong>in</strong>variant mass values compared to <strong>the</strong> <strong>in</strong>put masses. Therefore this motivates to build
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 81a method to correct <strong>the</strong> jet energies so that <strong>the</strong> known measured values <strong>of</strong> <strong>the</strong> W bosonand <strong>the</strong> top quark masses can be re-obta<strong>in</strong>ed.#eventsEntries 5006350Mean 89.15300RMS 13.452502001501005000 20 40 60 80 100 120 140 160 180 200m W(GeV)#events240 Entries 5006220200Mean 180.9180RMS 20.171601401201008060402000 50 100 150 200 250 300 350 400m top(GeV)Figure 5.9: The distributions <strong>of</strong> <strong>the</strong> reconstructed W boson mass (left) and <strong>the</strong> topquark mass (right) for <strong>the</strong> selected events where a true jet-parton match<strong>in</strong>g is returnedby <strong>the</strong> MVA method.5.2 K<strong>in</strong>ematic FitFor a given event, four-vectors <strong>of</strong> <strong>the</strong> reconstructed objects are measured. The sameevent may conta<strong>in</strong> particles, such as neutr<strong>in</strong>os, for which a direct measurement is notpossible. With <strong>the</strong> assumption <strong>of</strong> certa<strong>in</strong> hypo<strong>the</strong>ses, <strong>the</strong> measured as well as <strong>the</strong> unmeasuredquantities are supposed to fulfill <strong>the</strong> k<strong>in</strong>ematic constra<strong>in</strong>ts which are usuallydeduced from <strong>the</strong> event hypo<strong>the</strong>sis. S<strong>in</strong>ce <strong>the</strong>re are uncerta<strong>in</strong>ties on <strong>the</strong> measuredquantities, <strong>the</strong> constra<strong>in</strong>ts are not exactly satisfied. The constra<strong>in</strong>ts can <strong>the</strong>n be usedto slightly change <strong>the</strong> measured values with<strong>in</strong> <strong>the</strong>ir uncerta<strong>in</strong>ties. The concept <strong>of</strong> anevent-by-event least square fitt<strong>in</strong>g toge<strong>the</strong>r with <strong>the</strong> application <strong>of</strong> Langrange Multiplierscan be utilized to <strong>in</strong>sure that <strong>the</strong> measured as well as <strong>the</strong> unmeasured quantitiesfulfill <strong>the</strong> k<strong>in</strong>ematic constra<strong>in</strong>ts. This procedure is referred to as a k<strong>in</strong>ematic fit.The k<strong>in</strong>ematic fit method has been frequently used <strong>in</strong> various physics analyses performed<strong>in</strong> different collaborations [15]. S<strong>in</strong>ce at an electron-positron collider, <strong>the</strong> <strong>in</strong>itialenergy and momentum <strong>of</strong> <strong>the</strong> <strong>in</strong>com<strong>in</strong>g particles are well known, it is possible to applyconstra<strong>in</strong>ts like energy and momentum conservation via a k<strong>in</strong>ematic fit method to extractfitted values for <strong>the</strong> unmeasured quantities <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state. Also <strong>the</strong> k<strong>in</strong>ematicfit technique can be used for calibration purposes. For example, <strong>the</strong> electromagneticcalorimeter <strong>of</strong> <strong>the</strong> BaBar <strong>experiment</strong> is calibrated by means <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit technique[102]. Us<strong>in</strong>g <strong>the</strong> radiative Bhabha events e − e + → e − e + γ, given that <strong>the</strong> momenta<strong>of</strong> <strong>the</strong> <strong>in</strong>com<strong>in</strong>g and outgo<strong>in</strong>g electrons and positrons, as well as <strong>the</strong> photon’s angularposition are measured, <strong>the</strong> photon energy can be obta<strong>in</strong>ed via a k<strong>in</strong>ematic fit. This fitresults <strong>in</strong> an absolute measurement <strong>of</strong> <strong>the</strong> photon energy which <strong>the</strong>n can be comparedto <strong>the</strong> measured photon energy to obta<strong>in</strong> calibration constants.At hadron colliders like <strong>the</strong> LHC, it is not possible to apply energy-momentum constra<strong>in</strong>tss<strong>in</strong>ce <strong>the</strong> <strong>in</strong>itial k<strong>in</strong>ematic properties <strong>of</strong> <strong>the</strong> collid<strong>in</strong>g partons arris<strong>in</strong>g from
82 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factorprotons can not be known. However, <strong>the</strong> usage <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit method is still feasibleby apply<strong>in</strong>g <strong>the</strong> mass constra<strong>in</strong>ts <strong>in</strong> <strong>the</strong> specific event topologies. For example, <strong>in</strong><strong>the</strong> processes where a W boson is produced and subsequently decays to a pair <strong>of</strong> quarksW → q¯q, apply<strong>in</strong>g <strong>the</strong> mass constra<strong>in</strong>t <strong>of</strong> <strong>the</strong> W boson and forc<strong>in</strong>g <strong>the</strong> four-momenta<strong>of</strong> its decay products to fulfill <strong>the</strong> mass equation, be<strong>in</strong>gm 2 W = (E jet 1+ E jet2 ) 2 − ( ⃗ P jet1 + ⃗ P jet2 ) 2 ,would improve significantly <strong>the</strong> resolution on <strong>the</strong> reconstructed jets. In <strong>the</strong> currentstudy, <strong>the</strong> mass constra<strong>in</strong>ts <strong>of</strong> both <strong>the</strong> W boson and top quark are applied <strong>in</strong> <strong>the</strong>hadronic branch <strong>of</strong> e+jets t¯t system, namely t → Wb → q¯qb. The reconstructedobjects <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state, be<strong>in</strong>g <strong>the</strong> light jets arris<strong>in</strong>g from <strong>the</strong> W boson decay and <strong>the</strong>b quark jet arris<strong>in</strong>g from <strong>the</strong> top quark decay, are asked to fulfill <strong>the</strong> mass constra<strong>in</strong>tswhich are expressed asm 2 W = (E l 1+ E l2 ) 2 − ( ⃗ P l1 + ⃗ P l2 ) 2 ,m 2 top = (E l 1+ E l2 + E b ) 2 − ( ⃗ P l1 + ⃗ P l2 + ⃗ P b ) 2 ,where <strong>in</strong>dex l i refers to one <strong>of</strong> <strong>the</strong> two light jets orig<strong>in</strong>at<strong>in</strong>g from light quarks, namelyup, down, charm and strange. Also <strong>in</strong>dex b <strong>in</strong> <strong>the</strong> above formula, refers to <strong>the</strong> b jetarris<strong>in</strong>g from <strong>the</strong> heavy quark, namely <strong>the</strong> bottom quark.The k<strong>in</strong>ematic fit results are used to estimate <strong>the</strong> residual jet energy correction factorsfor both light as well as b quark jets as will be described <strong>in</strong> detail <strong>in</strong> Section 5.4. Thema<strong>the</strong>matical concept <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit technique is reviewed <strong>in</strong> <strong>the</strong> next section.5.2.1 General Ma<strong>the</strong>matical ConceptA physics problem consists <strong>of</strong> measured quantities, ⃗y = (y 1 , y 2 , . . .,y n ), and unmeasuredvalues, ⃗a = (a 1 , a 2 , . . .,a p ). Also a certa<strong>in</strong> hypo<strong>the</strong>sis orig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong> physics pr<strong>in</strong>cipalsuch as conservation <strong>of</strong> <strong>the</strong> energy and momentum may be <strong>in</strong>troduced. Therefore<strong>the</strong> observed events, conta<strong>in</strong><strong>in</strong>g n measured and p unmeasured values, are constra<strong>in</strong>edto fulfill <strong>the</strong> hypo<strong>the</strong>sis. Due to <strong>the</strong> presence <strong>of</strong> uncerta<strong>in</strong>ties on <strong>the</strong> measured values,usually <strong>the</strong> hypo<strong>the</strong>sis is not respected <strong>in</strong> <strong>the</strong> physics problem which is under consideration.The hypo<strong>the</strong>sis, which is expressed via constra<strong>in</strong>t equations ⃗ f = (f 1 , f 2 , . . .,f m ),is satisfied for only <strong>the</strong> true parameters, namely ȳ and ā as written below.f 1 (ā 1 , ā 2 , . . .,ā p , ȳ 1 , ȳ 2 , . . ., ȳ n ) = 0,f 2 (ā 1 , ā 2 , . . .,ā p , ȳ 1 , ȳ 2 , . . ., ȳ n ) = 0,.f m (ā 1 , ā 2 , . . .,ā p , ȳ 1 , ȳ 2 , . . .,ȳ n ) = 0.In order to obta<strong>in</strong> <strong>the</strong> correction to <strong>the</strong> measured values, ∆⃗y, which yields <strong>the</strong> constra<strong>in</strong>tsto be fulfilled for <strong>the</strong> corrected measurements, <strong>the</strong> extrema <strong>of</strong> <strong>the</strong> above equationsshould be found. Hence <strong>the</strong> Lagrange Multipliers method is used to determ<strong>in</strong>e<strong>the</strong> true values, ⃗y ′ = ⃗y + ∆⃗y, for which <strong>the</strong> constra<strong>in</strong>ts are fulfilled. At <strong>the</strong> same
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 83time <strong>the</strong> calculated corrections to <strong>the</strong> measured values should be m<strong>in</strong>imized which isgoverened with <strong>the</strong> use <strong>of</strong> a χ 2 term. Toge<strong>the</strong>r with <strong>the</strong> Lagrange term, one could writea comb<strong>in</strong>ed likelihood expression, L(⃗y,⃗a, ⃗ λ), as followsL(⃗y,⃗a, ⃗ λ) = S(⃗y) + 2Σ m k=1 λ kf k (⃗y,⃗a),where ⃗ λ are <strong>the</strong> Lagrange Multipliers and S(⃗y) is <strong>the</strong> χ 2 term and is expressed explicitlyasS(⃗y) = ∆⃗y T V −1 ∆⃗y,where V is <strong>the</strong> covariance matrix <strong>of</strong> <strong>the</strong> measured parameters.In order to f<strong>in</strong>d a local m<strong>in</strong>imum for <strong>the</strong> likelihood equation, one has to m<strong>in</strong>imize S(⃗y)under <strong>the</strong> constra<strong>in</strong>ts f k (⃗y,⃗a) = 0. Due to <strong>the</strong> non-l<strong>in</strong>earity <strong>of</strong> <strong>the</strong> physics constra<strong>in</strong>tswhich are used, one has to solve <strong>the</strong> constra<strong>in</strong>ts iteratively.S<strong>in</strong>ce <strong>the</strong> constra<strong>in</strong>ts are fulfilled <strong>in</strong> <strong>the</strong> limit <strong>of</strong> true values, (⃗y ′ ,⃗a ′ ), <strong>the</strong>n f k (⃗y,⃗a) canbe expanded arount a desired value, for example could be <strong>the</strong> value obta<strong>in</strong>ed dur<strong>in</strong>g<strong>the</strong> previous iteration represented by (⃗y ∗ ,⃗a ∗ ), as followsf k (⃗y ′ ,⃗a ′ ) ≈ f k (⃗y ∗ ,⃗a ∗ ) + Σ p ∂f kj=1 .(∆a j − ∆a ∗∂aj) + Σ n ∂f ki=1 .(∆y i − ∆yi ∗ ) ≈ 0,j ∂y iwhere ⃗ X, ⃗ X ∗ and ⃗ X ′ , with X could be ei<strong>the</strong>r y or a, are respectively <strong>the</strong> start value,<strong>the</strong> value after <strong>the</strong> previous iteration and <strong>the</strong> value after <strong>the</strong> current iteration. Also<strong>the</strong> ∆ ⃗ X and ∆ ⃗ X ∗ are def<strong>in</strong>ed as ∆ ⃗ X = ⃗ X ′ − ⃗ X and ∆ ⃗ X ∗ = ⃗ X ∗ − ⃗ X. One can write<strong>the</strong> k<strong>in</strong>ematic constra<strong>in</strong>ts <strong>in</strong> vector notation⃗f ∗ + A(∆⃗a − ∆⃗a ∗ ) + B(∆⃗y − ∆⃗y ∗ ) ≈ 0,or equivalentlywithA∆⃗a + B∆⃗y −⃗c = 0,⎛ ⎞f 1 (⃗a ∗ ,⃗y ∗ )⃗c = A∆⃗a ∗ + B∆⃗y ∗ − f ⃗ ∗ ; f ⃗ ∗ f 2 (⃗a ∗ ,⃗y ∗ )= ⎜ ⎟⎝ . ⎠f m (⃗a ∗ ,⃗y ∗ )In <strong>the</strong> vector notaion, <strong>the</strong> matrices A and B are def<strong>in</strong>ed as A = ∂ f ⃗ and B = ∂ f ⃗∂⃗awhich <strong>the</strong>ir components are expressed explicitly below.⎛⎞∂f 1 ∂f 1 ∂f∂a 1 ∂a 2. . . 1∂a p∂f 2 ∂f 2 ∂fA =∂a 1 ∂a 2. . . 2∂a p⎜⎟⎝ .⎠∂f m ∂f∂a 2. . . m∂a p⎛B = ⎜⎝∂f m∂a 1∂f 1 ∂f 1∂y 1∂f 2 ∂f 2∂y 1.∂f m∂y 1∂y 2. . .∂y 2. . .∂f m∂y 2. . .∂f 1⎞∂y n∂f 2∂y n⎟⎠∂f m∂y n∂⃗y for
84 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration FactorAll <strong>of</strong> <strong>the</strong> partial derviatives <strong>of</strong> <strong>the</strong> constra<strong>in</strong>ts, <strong>in</strong> each step <strong>of</strong> <strong>the</strong> iteration, needto be obta<strong>in</strong>ed at <strong>the</strong> value <strong>of</strong> <strong>the</strong> parameters at <strong>the</strong> previous iteration X ∗ . S<strong>in</strong>ce <strong>the</strong>constra<strong>in</strong>ts depend directly on <strong>the</strong> four-vector components P ⃗ = (⃗p, E), <strong>the</strong> cha<strong>in</strong> ruleis used to obta<strong>in</strong> <strong>the</strong> partial derivatives ∂ f ⃗ = ∂ f ⃗∂⃗y ∂ ⃗P .∂ P ⃗ . As four-momenta <strong>of</strong> <strong>the</strong> particles∂⃗ycan be parametrized <strong>in</strong> different ways, <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> ∂ P ⃗ is not unique. Various∂⃗yparametrization for <strong>the</strong> reconstructed objects exist which are described <strong>in</strong> more detail<strong>in</strong> <strong>the</strong> next section.In <strong>the</strong> new notation, one has to m<strong>in</strong>imize <strong>the</strong> likelihood expression which is written asL = ∆⃗y T V −1 ∆⃗y + 2λ T (A∆⃗a + B∆⃗y −⃗c).The conditions for a local m<strong>in</strong>imum are obta<strong>in</strong>ed, after differentiat<strong>in</strong>g <strong>the</strong> above expressionwith respect to ⃗y, ⃗a and ⃗ λ, as listed belowV −1 ∆⃗y + B T ⃗ λ = 0,A T ⃗ λ = 0,B∆⃗y + A∆⃗a = c.Collect<strong>in</strong>g <strong>the</strong> above three equations <strong>in</strong> a compact form with partitioned matrices⎛ ⎞ ⎛ ⎞ ⎛ ⎞V −1 0 B T ∆⃗y 0⎝ 0 0 A T ⎠ ⎝∆⃗a⎠ = ⎝0⎠B A 0 λ cThis matrix equation will be solved iteratively for <strong>the</strong> unknown values ∆⃗y, ∆⃗a and∆ ⃗ λ.The iteration procedure is repeated until some predef<strong>in</strong>ed convergence criteria are fulfilled.The first one requires that <strong>the</strong> change <strong>in</strong> χ 2 expression between <strong>the</strong> currentiteration, n, and <strong>the</strong> previous iteration, n-1, is smaller than a given value ǫ S , and iswritten asS(n − 1) − S(n)< ǫ S ,ndfwhere ndf is <strong>the</strong> difference between <strong>the</strong> number <strong>of</strong> constra<strong>in</strong>ts and <strong>the</strong> number <strong>of</strong>unmeasured quantities, be<strong>in</strong>gndf = m − p.The second convergency criterion which is checked <strong>in</strong> each iteration, requires that <strong>the</strong>constra<strong>in</strong>ts are fulfilled better than a given value ǫ F , and is expressed asF = Σ m k=1 f(n) k(⃗y,⃗a) < ǫ F .A more complete description <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit technique can be found <strong>in</strong> [103].
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 855.2.2 Four-Vector ParametrizationSeveral parametrizations <strong>in</strong> <strong>the</strong> k<strong>in</strong>ematic fit package are implemented, two <strong>of</strong> whichare commonly used <strong>in</strong> hadron colliders and are <strong>in</strong>troduced briefly here. A first one,called TFitParticleEtThetaPhi, uses E T , θ and φ variables for parametrization. Themomentum vector and <strong>the</strong> energy can be written <strong>in</strong> this parametrization as⎛ ⎞E T cosφ⃗p f = ⎝E T s<strong>in</strong> φ⎠ ,E T cot θE = E Ts<strong>in</strong> θ .In a second parametrization, called TFitParticleEtEtaPhi, where E T , η and φ variablesare used as parameters, <strong>the</strong> four-vector can be expressed as⎛ ⎞E T cosφ⃗p f = ⎝ E T s<strong>in</strong> φ ⎠,E T s<strong>in</strong>h ηE = E T cosh η.In <strong>the</strong> two parametrizations expla<strong>in</strong>ed above, <strong>the</strong> objects are considered as massless.In this study, TFitParticleEtThetaPhi parametrization is used with an additionalrequirement. Dur<strong>in</strong>g <strong>the</strong> fit procedure, it is crucial to keep <strong>the</strong> ratio <strong>of</strong> <strong>the</strong> measuredenergy <strong>of</strong> a jet to its measured momentum constant. Hence, <strong>in</strong> <strong>the</strong> k<strong>in</strong>ematic fit methodwhich is used <strong>in</strong> <strong>the</strong> current study, this additional requirement is applied which isexpressed as followsE fitted|⃗p fitted | = E measured|⃗p measured | .5.2.3 Resolutions on <strong>the</strong> <strong>Jet</strong> K<strong>in</strong>ematicsThe uncerta<strong>in</strong>ties on <strong>the</strong> measured parameters enter <strong>the</strong> k<strong>in</strong>ematic fit package via <strong>the</strong>covariance matrix, as mentioned before. In this analysis where <strong>the</strong> measured objects arereconstructed jets, <strong>the</strong> resolutions on <strong>the</strong> jet properties are required to be calculated.These resolutions are obta<strong>in</strong>ed from simulated e+jets t¯t events <strong>in</strong> which a correct jetpartoncomb<strong>in</strong>ation, accord<strong>in</strong>g to <strong>the</strong> generator level <strong>in</strong>formation, is found. Accord<strong>in</strong>gto <strong>the</strong> jet-parton match<strong>in</strong>g algorithm described <strong>in</strong> Section 5.1.1, <strong>the</strong> reconstructedjets are matched to <strong>the</strong> partons if ∆R difference between <strong>the</strong>m is less than 0.3. Thereconstructed jets are grouped <strong>in</strong> ei<strong>the</strong>r light jets, which are matched to <strong>the</strong> partonsaris<strong>in</strong>g from <strong>the</strong> decay <strong>of</strong> <strong>the</strong> W boson, or b jets, which are matched to <strong>the</strong> b flavoredpartons orig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong> decay <strong>of</strong> <strong>the</strong> top quarks. S<strong>in</strong>ce <strong>the</strong> obta<strong>in</strong>ed resolutionsare found to be dependent on <strong>the</strong> transverse momentum and <strong>the</strong> pseudorapidity <strong>of</strong> <strong>the</strong>reconstructed jets, jets are fur<strong>the</strong>r categorized <strong>in</strong> various b<strong>in</strong>s <strong>of</strong> p T and η. In thisanalysis, eleven b<strong>in</strong>s <strong>of</strong> transverse momentum and twelve b<strong>in</strong>s <strong>of</strong> pseudorapidity areconsidered. For those jets which populate a certa<strong>in</strong> p T -b<strong>in</strong> and η-b<strong>in</strong>, <strong>the</strong> resolutionsare calculated as expla<strong>in</strong>ed <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g. The resolution on <strong>the</strong> X property <strong>of</strong> a
86 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factorjet is computed as <strong>the</strong> width <strong>of</strong> <strong>the</strong> Gaussian function fitted on <strong>the</strong> distribution <strong>of</strong> <strong>the</strong>difference between <strong>the</strong> value <strong>of</strong> <strong>the</strong> X parameter <strong>of</strong> <strong>the</strong> reconstructed jet and <strong>the</strong> value<strong>of</strong> <strong>the</strong> X parameter <strong>of</strong> <strong>the</strong> matched parton to that jet. The resolutions on <strong>the</strong> variousk<strong>in</strong>ematics <strong>of</strong> <strong>the</strong> jets, namely transverse energy E T , polar angle θ and azimuthal angleφ as a function <strong>of</strong> <strong>the</strong> transverse momentum <strong>of</strong> <strong>the</strong> jets are shown <strong>in</strong> Figure 5.10 andFigure 5.11, for <strong>the</strong> light quark and <strong>the</strong> b quark jets, respectively. In each figure, <strong>the</strong>left plots represent <strong>the</strong> typical jets reconstructed <strong>in</strong> <strong>the</strong> barrel while <strong>the</strong> right plots areobta<strong>in</strong>ed for typical jets reconstructed <strong>in</strong> a narrow region <strong>in</strong> <strong>the</strong> endcaps. The obta<strong>in</strong>edresolution plots are fitted with functions which are specified on <strong>the</strong> plot. For example,<strong>the</strong> distribution <strong>of</strong> <strong>the</strong> resolution on <strong>the</strong> transverse energy <strong>of</strong> <strong>the</strong> jets is fitted with afunction which is parametrized as a+b √ p T +cp T . The free parameters <strong>of</strong> <strong>the</strong> functionare determ<strong>in</strong>ed from <strong>the</strong> fit and quoted on each plot.From Figure 5.10, it can be seen that <strong>the</strong> absolute resolution on <strong>the</strong> transverseenergy for jets with higher transverse momenta is worse, while <strong>the</strong> relative resolutionis worse for <strong>the</strong> jets with lower transverse momenta, as discussed <strong>in</strong> Section 3.2.4. Alsoit can be deduced that <strong>the</strong> resolution on <strong>the</strong> angular distributions is better determ<strong>in</strong>edcompared to <strong>the</strong> resolution on <strong>the</strong> transverse energy. This is because <strong>of</strong> <strong>the</strong> match<strong>in</strong>grequirement applied between <strong>the</strong> reconstructed jets and <strong>the</strong> partons <strong>in</strong> (η × φ) space.The resolutions on <strong>the</strong> spatial coord<strong>in</strong>ates <strong>of</strong> <strong>the</strong> reconstructed jets are decreased forjets with higher transverse momenta, because <strong>the</strong> jets with higher transverse momentaare less affected by <strong>the</strong> magnetic field. The polar angle <strong>of</strong> reconstructed jets can bedeterm<strong>in</strong>ed with higher precision compared to <strong>the</strong> azimuthal angle. This is due to<strong>the</strong> bend<strong>in</strong>g <strong>of</strong> charged particle <strong>in</strong> <strong>the</strong> direction <strong>of</strong> φ. Compar<strong>in</strong>g with <strong>the</strong> plots <strong>in</strong>Figure 5.11, it can be understood that <strong>the</strong> resolution on <strong>the</strong> transverse energy <strong>of</strong> bquark jets is worse compared to <strong>the</strong> resolution on <strong>the</strong> transverse energy <strong>of</strong> light quarkjets. This is due to <strong>the</strong> presence <strong>of</strong> muons or neutr<strong>in</strong>os <strong>in</strong> <strong>the</strong> decay products <strong>of</strong> bquark jets which yields to cluster less energy for b quark jets as muons and neutr<strong>in</strong>osare weakly <strong>in</strong>teract<strong>in</strong>g particles. There is no big difference observed when compar<strong>in</strong>g<strong>the</strong> resolutions on <strong>the</strong> spatial k<strong>in</strong>ematics <strong>of</strong> b quark and non-b quark jets.5.3 Extra Event Selection CutsSo far, events have been required to pass <strong>the</strong> basic event selection cuts that are summarized<strong>in</strong> Table 4.6. The selected events are subsequently fed to <strong>the</strong> MVA package. Theoutput <strong>of</strong> <strong>the</strong> MVA method is a match<strong>in</strong>g between <strong>the</strong> reconstructed jets <strong>in</strong> <strong>the</strong> f<strong>in</strong>alstate <strong>of</strong> <strong>the</strong> e+jets t¯t event and <strong>the</strong> partons appear<strong>in</strong>g <strong>in</strong> <strong>the</strong> hadronic branch <strong>of</strong> <strong>the</strong> t¯tsystem t → Wb → q¯qb, as described <strong>in</strong> Section 5.1. The chosen jet-parton comb<strong>in</strong>ationreturned by <strong>the</strong> MVA method, can be ei<strong>the</strong>r right or wrong. As already mentioned, <strong>the</strong>MVA method which has been tra<strong>in</strong>ed us<strong>in</strong>g <strong>the</strong> variables listed <strong>in</strong> Section 5.1.4, is notable to choose <strong>the</strong> correct jet-parton match<strong>in</strong>g <strong>in</strong> about 50% <strong>of</strong> <strong>the</strong> cases. This is partlydue to <strong>the</strong> ISR/FSR effects which makes it possible to get <strong>the</strong> radiation jets among<strong>the</strong> four lead<strong>in</strong>g jets and <strong>the</strong>n at least one <strong>of</strong> <strong>the</strong> four lead<strong>in</strong>g jets can not be matchedto <strong>the</strong> hard-scatter partons. The e+jets t¯t events, for which a wrong jet comb<strong>in</strong>ationis recognized by <strong>the</strong> MVA or <strong>the</strong> event itself is badly reconstructed, contribute to <strong>the</strong>
2χ / ndf18.76 / 7p0 9.492e-08 ±9.472p1 1.185±0.07715p2 -0.01065±0.0091122χ / ndf14.59 / 8p0 0.01114 ±0.001156p1 1.46±0.05471χ 2/ ndf9.294 / 8p0 0.01116±0.002424p1 2.676 ±0.085742χ / ndf18.59 / 7p0 5.956 ±0.5679p1 0±1.719p2 0.03635 ±0.0091672χ / ndf5.119 / 8p0 5.548e-09 ±0.004184p1 0.7435±0.027122χ / ndf26.14 / 8p0 -2.627e-15±0.003853p1 2.629 ±0.09587CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 87) (GeV)Calo<strong>Jet</strong>Tσ(E16141210864200.000≤|η|
2χ / ndf44.49 / 7p0 7.384 ±0.3928p1 0±0.9449p2 0.03934 ±0.0048132χ / ndf15.37 / 8p0 0.003055±0.002043p1 1.587 ±0.047242χ / ndf49.09 / 8p0 9.171e-10 ±0.002244p1 2.997 ±0.05726χ 2/ ndf18.22 / 7p0 7.408 ±0.6552p1 0±4.879p2 0.04002±0.01012χ / ndf27.18 / 8p0 3.981e-08 ±0.002249p1 0.7362±0.023272χ / ndf33.39 / 8p0 4.39e-15±0.004039p1 2.679 ±0.0966588 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor) (GeV)Calo<strong>Jet</strong>Tσ(E16141210864200.000≤|η|
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 89comb<strong>in</strong>atorial backgrounds. In order to reduce <strong>the</strong> contribution <strong>of</strong> <strong>the</strong> comb<strong>in</strong>atorialbackgrounds, additional event selection cuts are applied.In <strong>the</strong> topology <strong>of</strong> <strong>the</strong> e+jets t¯t event, <strong>the</strong>re are four quarks produced <strong>in</strong> <strong>the</strong> f<strong>in</strong>al statewhich subsequently contribute to at least four reconstructed jets <strong>in</strong> <strong>the</strong> detector. Inaddition to <strong>the</strong> four jets which represent <strong>the</strong> quarks generated <strong>in</strong> <strong>the</strong> hard scatter<strong>in</strong>gprocess, any o<strong>the</strong>r jet which is reconstructed <strong>in</strong> <strong>the</strong> event is <strong>the</strong> consequence <strong>of</strong> <strong>the</strong>radiation <strong>of</strong> a parton <strong>in</strong> <strong>the</strong> <strong>in</strong>itial or f<strong>in</strong>al state, generally known as ISR or FSR effect,respectively. As already discussed <strong>in</strong> Section 5.1.1, most <strong>of</strong> <strong>the</strong> e+jets t¯t events, conta<strong>in</strong>at least an ISR jet which is among <strong>the</strong> four lead<strong>in</strong>g reconstructed jets. Then <strong>the</strong>sek<strong>in</strong>d <strong>of</strong> events, where <strong>the</strong> four lead<strong>in</strong>g jets can not be matched to <strong>the</strong> four hard-scatterpartons, would contribute to <strong>the</strong> comb<strong>in</strong>atorial backgrounds. S<strong>in</strong>ce <strong>the</strong>re is no truejet-parton match<strong>in</strong>g <strong>in</strong> such events, <strong>the</strong> MVA method would always return a wrongjet comb<strong>in</strong>ation. Therefore, any event with at least one extra jet is vetoed and is notprocessed <strong>in</strong> <strong>the</strong> rest <strong>of</strong> analysis. In addition to <strong>the</strong> four lead<strong>in</strong>g jets <strong>in</strong> <strong>the</strong> selectedevents, any o<strong>the</strong>r jet with p T > 30 GeV with<strong>in</strong> <strong>the</strong> tracker acceptance is def<strong>in</strong>ed as aradiation jet and <strong>the</strong> events conta<strong>in</strong><strong>in</strong>g at least one radiation are rejected.In order to fur<strong>the</strong>r reduce <strong>the</strong> multiplicity <strong>of</strong> badly reconstructed events <strong>in</strong> <strong>the</strong> f<strong>in</strong>alselected sample, <strong>the</strong> events are constra<strong>in</strong>ed to fulfill some k<strong>in</strong>ematic requirements. Allevents which pass <strong>the</strong> radiation veto cut, are fed to <strong>the</strong> k<strong>in</strong>ematic fit package. Thethree reconstructed jets <strong>in</strong> <strong>the</strong> e+jets t¯t event, which are now associated to <strong>the</strong> partons<strong>in</strong> <strong>the</strong> hadronic branch t → Wb → q¯qb by means <strong>of</strong> <strong>the</strong> MVA method, are forced t<strong>of</strong>ulfill <strong>the</strong> mass constra<strong>in</strong>ts as described <strong>in</strong> Section 5.2. The masses <strong>of</strong> <strong>the</strong> W bosonand top quark used <strong>in</strong> <strong>the</strong> fit package are taken to be equal to those used <strong>in</strong> <strong>the</strong> simulationstep, be<strong>in</strong>g 80.4 GeV and 172.5 GeV respectively. The energy <strong>of</strong> <strong>the</strong> three jetsassociated to <strong>the</strong> hadronic top quark are shifted <strong>in</strong> certa<strong>in</strong> steps. A three-dimensionalspace, which is spanned by <strong>the</strong> changes <strong>in</strong> <strong>the</strong> energies <strong>of</strong> <strong>the</strong> three reconstructed jets(∆E l1 , ∆E l2 , ∆E b ), is obta<strong>in</strong>ed, where ∆E l1 and ∆E l2 represent <strong>the</strong> correction factorson <strong>the</strong> light jets from <strong>the</strong> hadronic W boson decay and ∆E b corresponds to <strong>the</strong> changes<strong>in</strong> <strong>the</strong> energy <strong>of</strong> <strong>the</strong> b jet arris<strong>in</strong>g from <strong>the</strong> hadronic top quark decay. In each po<strong>in</strong>t<strong>in</strong> <strong>the</strong> three-dimensional space, <strong>the</strong> k<strong>in</strong>ematic fit is performed. The fit is applied on agrid <strong>in</strong> this space <strong>in</strong> steps <strong>of</strong> 2% with<strong>in</strong> a w<strong>in</strong>dow <strong>of</strong> ±40% around <strong>the</strong> reconstructedenergies <strong>of</strong> both light as well as b jets, <strong>in</strong>dividually. As a result, <strong>the</strong> energy <strong>of</strong> a genericjet with, for example, E=100 GeV is shifted from E=60 GeV to E=140 GeV <strong>in</strong> steps <strong>of</strong>2 GeV and <strong>in</strong> each step <strong>the</strong> fit is performed, hence 41 times per jet per event. S<strong>in</strong>ce <strong>the</strong>light jets arris<strong>in</strong>g from <strong>the</strong> W boson are <strong>in</strong>dist<strong>in</strong>guishable, <strong>the</strong>ir correction factors canbe assumed to be equal ∆E l1 = ∆E l2 = ∆E l . Therefore <strong>the</strong> k<strong>in</strong>ematic fit is performed<strong>in</strong> a two-dimensional space which is obta<strong>in</strong>ed by shift<strong>in</strong>g <strong>in</strong> <strong>the</strong> energy <strong>of</strong> <strong>the</strong> light jets,for which <strong>the</strong> shifts are taken to be identical, and <strong>in</strong> <strong>the</strong> energy <strong>of</strong> <strong>the</strong> b jet (∆E l , ∆E b ).As a result, <strong>the</strong> k<strong>in</strong>ematic fit is performed 41×41 times per event.In each po<strong>in</strong>t <strong>of</strong> <strong>the</strong> two-dimensional grid, <strong>the</strong> k<strong>in</strong>ematic fit returns a probabilityP K<strong>in</strong>Fit (∆E l , ∆E b ), which represents how likely, for that particular po<strong>in</strong>t <strong>of</strong> <strong>the</strong> grid,<strong>the</strong> hypo<strong>the</strong>sis <strong>of</strong> orig<strong>in</strong>at<strong>in</strong>g <strong>the</strong> three jets <strong>in</strong> e+jets t¯t event from <strong>the</strong> decay <strong>of</strong> <strong>the</strong> topquark <strong>in</strong> <strong>the</strong> topology t → Wb → q¯qb, is correct. At <strong>the</strong> same time, <strong>the</strong> fit probabilitystates how much <strong>the</strong> mass hypo<strong>the</strong>ses are correct for <strong>the</strong> three jets <strong>in</strong> e+jets t¯t event
90 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factorfrom <strong>the</strong> decay <strong>of</strong> <strong>the</strong> top quark <strong>in</strong> <strong>the</strong> topology t → Wb → q¯qb. For a typical event,<strong>the</strong> distribution <strong>of</strong> <strong>the</strong> P K<strong>in</strong>Fit (∆E l , ∆E b ) is shown <strong>in</strong> Figure 5.12.P K<strong>in</strong>Fit(∆E l,∆E b)10.80.60.40.21.4 01.31.21.1∆E b10.90.80.70.60.6∆E l0.7 0.8 0.9 1 1.1 1.21.3 1.4Figure 5.12: The distribution <strong>of</strong> <strong>the</strong> P K<strong>in</strong>Fit (∆E l , ∆E b ) for an e+jets t¯t event.The central po<strong>in</strong>t <strong>of</strong> <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> P K<strong>in</strong>Fit (∆E l , ∆E b ), which is expressedby P K<strong>in</strong>Fit (∆E l = 1, ∆E b = 1), corresponds to <strong>the</strong> situation when no correction isapplied. It is understood from Figure 5.12 that <strong>the</strong> maximum <strong>of</strong> <strong>the</strong> fit, for thatparticular event, does not happen at <strong>the</strong> center <strong>of</strong> <strong>the</strong> grid, which shows <strong>the</strong> need for<strong>the</strong> residual jet energy corrections. In order to justify <strong>the</strong> need for <strong>the</strong> residual jetenergy corrections, one has to <strong>in</strong>crease <strong>the</strong> statistics and may look at <strong>the</strong> distribution<strong>of</strong> <strong>the</strong> values <strong>of</strong> <strong>the</strong> P K<strong>in</strong>Fit (∆E l = 1, ∆E b = 1), which is shown <strong>in</strong> Figure 5.13.There is a clear peak at zero which can be partially <strong>in</strong>terpreted as follows. Thehypo<strong>the</strong>sis that <strong>the</strong> three jets <strong>in</strong> e+jets t¯t event orig<strong>in</strong>ate from <strong>the</strong> top quark is notfulfilled and <strong>the</strong> imposed mass constra<strong>in</strong>ts are not converged, when no correction isapplied. This observation justify <strong>the</strong> nead for <strong>the</strong> residual jet energy correction. Also<strong>the</strong> events, where <strong>the</strong> MVA method is not able to return <strong>the</strong> correct jet-parton comb<strong>in</strong>ation,contribute to <strong>the</strong> peak at zero as will be discussed <strong>in</strong> what follows.For a given event, <strong>the</strong> po<strong>in</strong>t with <strong>the</strong> maximum probability <strong>in</strong> <strong>the</strong> distribution <strong>of</strong> <strong>the</strong>P K<strong>in</strong>Fit (∆E l , ∆E b ), which is expressed by PK<strong>in</strong>Fit max (∆E l, ∆E b ), shows <strong>the</strong> po<strong>in</strong>t where<strong>the</strong> imposed constra<strong>in</strong>ts are maximally fulfilled. The distribution <strong>of</strong> <strong>the</strong> PK<strong>in</strong>Fit max (∆E l, ∆E b )value for all events that already pass <strong>the</strong> radiation veto cut, is shown <strong>in</strong> Figure 5.14.It is seen from Figure 5.14 that for most <strong>of</strong> <strong>the</strong> events, a maximum probabilityexceed<strong>in</strong>g 0.9 is returned by <strong>the</strong> k<strong>in</strong>ematic fit method, represent<strong>in</strong>g those events forwhich <strong>the</strong> maximum happens somewhere <strong>in</strong> <strong>the</strong> region that is scanned. Some <strong>of</strong> <strong>the</strong>events are peaked around zero which represent those events where no maximum is found<strong>in</strong> <strong>the</strong> w<strong>in</strong>dow <strong>of</strong> ±40% around <strong>the</strong> measured values <strong>of</strong> <strong>the</strong> jet energies or events that<strong>the</strong> fit is not converged for <strong>the</strong> given jet-parton comb<strong>in</strong>ation. This can be understoodby look<strong>in</strong>g at <strong>the</strong> two-dimensional histogram which is made by plott<strong>in</strong>g <strong>the</strong> distribution
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 91#Events210tt+jets semi-ele (180.4 entries)tt+jets o<strong>the</strong>r (22.5 entries)t+jets, (all channels) (9.9 entries)Z+jets (2.5 entries)W+jets (21.5 entries)Multi-jets (27.4 entries)10110-10 0.2 0.4 0.6 0.8 1K<strong>in</strong>Fit Probability without correctionsFigure 5.13: The distribution <strong>of</strong> <strong>the</strong> fit probability P K<strong>in</strong>Fit (∆E l = 1, ∆E b = 1) whenno correction is applied. All events surviv<strong>in</strong>g <strong>the</strong> radiation veto cut, are taken <strong>in</strong>toaccount.#Events21010tt+jets semi-ele (180.4 entries)tt+jets o<strong>the</strong>r (22.5 entries)t+jets, (all channels) (9.9 entries)Z+jets (2.5 entries)W+jets (21.5 entries)Multi-jets (27.4 entries)110-10 0.2 0.4 0.6 0.8 1max Probability <strong>of</strong> <strong>the</strong> K<strong>in</strong>FitFigure 5.14: The distribution <strong>of</strong> <strong>the</strong> maximum fit probability P maxK<strong>in</strong>Fit (∆E l, ∆E b ) among<strong>the</strong> fit values which are obta<strong>in</strong>ed <strong>in</strong> <strong>the</strong> scanned range around <strong>the</strong> uncorrected energies.All events survived <strong>the</strong> radiation veto cut, are taken <strong>in</strong>to account.
92 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor<strong>of</strong> P K<strong>in</strong>Fit (∆E l = 1, ∆E b = 1) versus P maxK<strong>in</strong>Fit (∆E l, ∆E b ) as shown <strong>in</strong> Figure 5.15.P K<strong>in</strong>Fit (∆E l =1,∆E b =1)10.80.60.40.200 0.2 0.4 0.6 max0.8 (∆E l,∆E b)1P K<strong>in</strong>FitFigure 5.15: The distribution <strong>of</strong> P K<strong>in</strong>Fit (∆E l = 1, ∆E b = 1) versus P maxK<strong>in</strong>Fit (∆E l, ∆E b ).Only e+jets t¯t events survived <strong>the</strong> radiation veto cut, are taken <strong>in</strong>to account.It is clear from Figure 5.15 that for <strong>the</strong> events with a maximum fit probabilitybelow 0.9, <strong>the</strong> probability returned by <strong>the</strong> fit method when no correction is applied,is populated around zero. This means that <strong>the</strong> jets are assigned wrongly to <strong>the</strong> hardscatter partons by <strong>the</strong> MVA method or <strong>the</strong> event is not correctly reconstructed due to<strong>the</strong> radiations.Each selected event is required to have a PK<strong>in</strong>Fit max exceed<strong>in</strong>g 0.98. This cut aims to keeponly those events for which <strong>the</strong> hypo<strong>the</strong>sis is correct and <strong>the</strong> maximum occurs somewhere<strong>in</strong> <strong>the</strong> space which is be<strong>in</strong>g looked at around <strong>the</strong> uncorrected energies <strong>of</strong> <strong>the</strong> jets.No cut is applied on <strong>the</strong> P K<strong>in</strong>Fit (∆E l = 1, ∆E b = 1), s<strong>in</strong>ce cutt<strong>in</strong>g on this parameteryields to reject those events where a maximum is found <strong>in</strong> <strong>the</strong> w<strong>in</strong>dow <strong>of</strong> <strong>the</strong> residualcorrection and <strong>the</strong>y have to be used for <strong>the</strong> f<strong>in</strong>al estimation <strong>of</strong> <strong>the</strong> jet corrections. Inaddition to <strong>the</strong> radiation veto cut and cutt<strong>in</strong>g on <strong>the</strong> PK<strong>in</strong>Fit max (∆E l, ∆E b ) variable, athird one which is <strong>the</strong> last <strong>in</strong> <strong>the</strong> sequence <strong>of</strong> extra event selection cuts, is <strong>in</strong>troduced.There are some events for which <strong>the</strong> fit procedure does not converge <strong>in</strong> some po<strong>in</strong>ts <strong>in</strong><strong>the</strong> space <strong>of</strong> P K<strong>in</strong>Fit (∆E l , ∆E b ). It is desired to have a fit probability which is smoothlydistributed. Therefore, <strong>the</strong> events which conta<strong>in</strong> at least one po<strong>in</strong>t <strong>in</strong> <strong>the</strong> grid where<strong>the</strong> k<strong>in</strong>ematic fit does not converge, do not contribute <strong>in</strong> <strong>the</strong> rest <strong>of</strong> analysis.The additional analysis specific cuts, which are applied on top <strong>of</strong> <strong>the</strong> basic event selectioncriteria as already <strong>in</strong>troduced <strong>in</strong> Table 4.6, are summarized <strong>in</strong> Table 5.3 and<strong>the</strong> number <strong>of</strong> events pass<strong>in</strong>g each step are also mentioned <strong>in</strong>dividually for signal andrelevant backgrounds.It can be understood from Table 5.3 that <strong>the</strong> t¯t → o<strong>the</strong>r events constitute <strong>the</strong>dom<strong>in</strong>ant background to e+jets t¯t signal events. S<strong>in</strong>ce <strong>the</strong> simulated sample <strong>of</strong> <strong>the</strong>QCD multijet events are statistically limited, <strong>the</strong> uncerta<strong>in</strong>ty on <strong>the</strong> 9.4 events is largecorrespond<strong>in</strong>g to about 8.6 events. Therefore, <strong>the</strong> t¯t → o<strong>the</strong>r is still <strong>the</strong> dom<strong>in</strong>antbackground to <strong>the</strong> analysis.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 93t¯t t¯t W+jets Z+jets t QCD→ e + jets → o<strong>the</strong>r → l + jets → l + l − + jets → l + jets multijetsSelection Table 4.6 307.8 39.1 30.2 4.5 13.5 35.6Radiation Veto 180.4 22.5 21.4 2.5 9.9 27.4PK<strong>in</strong>Fit max75.2 6.3 3.4 0.3 2.5 9.4Gap Veto 66.8 5.5 2.8 0.3 2.2 9.4Table 5.3: Table <strong>of</strong> extra event selection cuts. The numbers are equivalent to an<strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 100pb −1 .Hav<strong>in</strong>g applied <strong>the</strong> extra event selection cuts given <strong>in</strong> Table 5.3, one could reach a signalover background ratio <strong>of</strong> about 3.3 which yields an even more purified f<strong>in</strong>al sample <strong>of</strong>selected events compared to <strong>the</strong> basel<strong>in</strong>e event selection cuts. It is worth to note that <strong>in</strong>a fraction <strong>of</strong> selected events, surviv<strong>in</strong>g <strong>the</strong> basel<strong>in</strong>e cuts, which corresponds to 17%, <strong>the</strong>MVA method is able to return <strong>the</strong> correct jet-parton comb<strong>in</strong>ation while after apply<strong>in</strong>g<strong>the</strong> additional event selection cuts, this fraction is enhanced to 52%. This means that<strong>the</strong> extra event selection cuts are able to reject <strong>the</strong> badly reconstructed events yieldsto suppress <strong>the</strong> comb<strong>in</strong>atorial backgrounds.Those events which are survived <strong>the</strong> additional selection requirements, are acceptedfor <strong>the</strong> f<strong>in</strong>al analysis.5.4 Extract<strong>in</strong>g <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration FactorsThe estimation <strong>of</strong> <strong>the</strong> residual jet energy calibration factors is based on <strong>the</strong> output<strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit. For each selected event, <strong>the</strong> k<strong>in</strong>ematic fit technique returns atwo-dimensional fit probability P K<strong>in</strong>Fit (∆E l , ∆E b ) <strong>in</strong> <strong>the</strong> dimensions <strong>of</strong> <strong>the</strong> residuallight and b jet energy corrections, as expla<strong>in</strong>ed <strong>in</strong> <strong>the</strong> previous section. The jet energycorrection factors are <strong>the</strong>n determ<strong>in</strong>ded by maximiz<strong>in</strong>g <strong>the</strong> fit probabilty returned by<strong>the</strong> fit procedure as described below.The k<strong>in</strong>ematic fit method is applied on <strong>the</strong> three reconstructed jets <strong>in</strong> e+jets t¯t event.The jets, which were associated to <strong>the</strong> three quarks orig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong> hadronicbranch <strong>of</strong> <strong>the</strong> top quark decay t → Wb → q¯qb by means <strong>of</strong> <strong>the</strong> MVA method, areforced to fulfill <strong>the</strong> mass constra<strong>in</strong>ts imposed on <strong>the</strong> system. The four-vectors <strong>of</strong> <strong>the</strong>three reconstructed jets, which are fed to <strong>the</strong> fit package, are parametrized us<strong>in</strong>g <strong>the</strong>TFitParticleEtThetaPhi parametrization for which <strong>the</strong> ratio <strong>of</strong> <strong>the</strong> energy over momentumE is kept constant and equal to <strong>the</strong> measured value dur<strong>in</strong>g <strong>the</strong> fit procedure,|p|as already expla<strong>in</strong>ed <strong>in</strong> Section 5.2.1. The parameters <strong>of</strong> <strong>the</strong> fit are chosen as follows.The parameter ǫ S is set to be 5 × 10 −5 . Also it is chosen to have ǫ F = 10 −4 . Theiteration procedure is repeated 30 times. If <strong>the</strong> fit is not converged after 30 times, <strong>the</strong>fitt<strong>in</strong>g is stopped and a value equal to zero is assigned to <strong>the</strong> probability <strong>of</strong> <strong>the</strong> fit for
94 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factorthat po<strong>in</strong>t <strong>in</strong> <strong>the</strong> two-dimensional space <strong>of</strong> (∆E l , ∆E b ). These events are not selectedas mentioned before.The correction factors on <strong>the</strong> light jet energy ∆E l , and on <strong>the</strong> b quark jet energy ∆E b ,can be <strong>in</strong>terpreted equivalently as <strong>the</strong> residual shifts on <strong>the</strong> light jet energy ∆ε l , andon <strong>the</strong> b quark jet energy ∆ε b , by means <strong>of</strong>and∆E l ≡ (1 + ∆ε l ),∆E b ≡ (1 + ∆ε b ).Therefore, <strong>in</strong> each po<strong>in</strong>t <strong>of</strong> <strong>the</strong> two-dimensional grid (∆E li , ∆E bj ), <strong>the</strong> shifts on <strong>the</strong>light and <strong>the</strong> b quark jet energy are applied by means <strong>of</strong> multiply<strong>in</strong>g <strong>the</strong> follow<strong>in</strong>gfactors(1 + ∆ε li ),and(1 + ∆ε bj ),measby <strong>the</strong> <strong>in</strong>itial measured light jet energy, called E l <strong>in</strong>it , and <strong>the</strong> <strong>in</strong>itial measured b jetmeasenergy, called E b <strong>in</strong>it , respectively. In <strong>the</strong> multiplication factors, 0 ≤ i, j < 41 and <strong>the</strong>start po<strong>in</strong>ts are taken to be∆ε l(i=0) = −0.4,and∆ε b(j=0) = −0.4.Hence, <strong>the</strong> region which is scanned by <strong>the</strong> fit procedure <strong>in</strong>cludes <strong>the</strong> follow<strong>in</strong>g po<strong>in</strong>tsmeas measmeasmeas meas0.60E l <strong>in</strong>it , 0.62E l <strong>in</strong>it , . . .(1 + ∆ε li )E l <strong>in</strong>it . . .,1.38E l <strong>in</strong>it , 1.40E l <strong>in</strong>it ;meas measmeasmeas meas0.60E b <strong>in</strong>it , 0.62E b <strong>in</strong>it , . . .(1 + ∆ε bj )E b <strong>in</strong>it . . .,1.38E b <strong>in</strong>it , 1.40E b <strong>in</strong>it .In each po<strong>in</strong>t <strong>of</strong> <strong>the</strong> scanned region, <strong>the</strong> k<strong>in</strong>ematic fit package returns a probabilitywhich is obta<strong>in</strong>ed from <strong>the</strong> χ 2 <strong>of</strong> <strong>the</strong> fit. Therefore, <strong>in</strong>stead <strong>of</strong> a two-dimensionaldistribution <strong>of</strong> <strong>the</strong> fit probability P K<strong>in</strong>Fit (∆E l , ∆E b ), one can have a two-dimensionaldistribution <strong>of</strong> <strong>the</strong> χ 2 values χ 2 (∆E l , ∆E b ). In each po<strong>in</strong>t <strong>of</strong> <strong>the</strong> two-dimensional grid,<strong>the</strong> χ 2 values <strong>of</strong> all selected events are summed which leads to obta<strong>in</strong> a two-dimensionalparabola <strong>in</strong> <strong>the</strong> dimensions <strong>of</strong> <strong>the</strong> residual light and b jet energy corrections. Thedistribution <strong>of</strong> <strong>the</strong> χ 2 (∆E l , ∆E b ) which is obta<strong>in</strong>ed by <strong>the</strong> sum <strong>of</strong> <strong>the</strong> χ 2 disribution <strong>of</strong><strong>the</strong> <strong>in</strong>dividual selected event, is shown <strong>in</strong> Figure 5.16.The estimated residual light and b jet energy correction factors are determ<strong>in</strong>ed at<strong>the</strong> po<strong>in</strong>t with <strong>the</strong> m<strong>in</strong>imum χ 2 fit probability, be<strong>in</strong>g <strong>the</strong> most central po<strong>in</strong>t <strong>in</strong> <strong>the</strong> plotshown <strong>in</strong> Figure 5.16.The two-dimensional χ 2 distribution is projected <strong>in</strong> <strong>the</strong> directions <strong>of</strong> ∆E l and ∆E bto obta<strong>in</strong> two separate one-dimensional distributions which are subsequently used toestimate <strong>the</strong> f<strong>in</strong>al residual corrections. The projection <strong>in</strong>to <strong>the</strong> ∆E l axis is done us<strong>in</strong>gthose events that participate <strong>in</strong> m<strong>in</strong>imiz<strong>in</strong>g <strong>the</strong> χ 2 distribution <strong>in</strong> <strong>the</strong> direction <strong>of</strong> ∆E b .Also, <strong>the</strong> events which contribute to <strong>the</strong> b<strong>in</strong> <strong>of</strong> <strong>the</strong> m<strong>in</strong>imum χ 2 distribution <strong>in</strong> <strong>the</strong>
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 951.41.31.21.1∆E b10.90.80.70.60.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4∆E lFigure 5.16: The distribution <strong>of</strong> <strong>the</strong> χ 2 (∆E l , ∆E b ) which is obta<strong>in</strong>ed by <strong>the</strong> sum <strong>of</strong> <strong>the</strong>χ 2 distributions for all selected events.direction <strong>of</strong> ∆E l , are taken <strong>in</strong>to account to make <strong>the</strong> projected distribution <strong>in</strong> <strong>the</strong>direction <strong>of</strong> ∆E b axis.Each <strong>of</strong> <strong>the</strong> two projected distributions is fitted with a second-degree polynomial around<strong>the</strong> m<strong>in</strong>imum as shown <strong>in</strong> Figure 5.17.2∆ χ120100806040203×1000.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4∆ E l2∆ χ4500040000350003000025000200001500010000500000.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4∆ E bFigure 5.17: The projected distribution <strong>of</strong> <strong>the</strong> χ 2 (∆E l , ∆E b ) <strong>in</strong>to <strong>the</strong> (left) ∆E l and(right) ∆E b axis which are fitted with a second-degree polynomial. The results areobta<strong>in</strong>ed for 100pb −1 <strong>of</strong> simulated proton collisions.The parameters <strong>of</strong> <strong>the</strong> fitted parabolas are used to extract <strong>the</strong> estimated light aswell as <strong>the</strong> b jet energy corrections, expressed as ∆ε l est and ∆ε b est , respectively. Them<strong>in</strong>imum <strong>of</strong> <strong>the</strong> parabolas refer to <strong>the</strong> f<strong>in</strong>al estimation <strong>of</strong> <strong>the</strong> jet energy correctionswhich are found to be∆ε l est = −10.12 ± 1.11%,and∆ε b est = −1.31 ± 1.72%,
96 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factorwhere <strong>the</strong> quoted uncerta<strong>in</strong>ties are <strong>the</strong> statistical uncerta<strong>in</strong>ties and correspond to <strong>the</strong>deviation <strong>of</strong> 1σ around <strong>the</strong> m<strong>in</strong>imum.The 5σ contour around <strong>the</strong> m<strong>in</strong>imum <strong>of</strong> <strong>the</strong> χ 2 (∆E l , ∆E b ) is obta<strong>in</strong>ed and shown<strong>in</strong> Figure 5.18. There is no significant correlation between light and b jet energycalibration factors observed.1.41.31.21.1∆E b10.90.80.70.60.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4∆E lFigure 5.18: The 5σ contour around <strong>the</strong> m<strong>in</strong>imum <strong>of</strong> <strong>the</strong> χ 2 (∆E l , ∆E b ). The plot isobta<strong>in</strong>ed for an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 100pb −1 .The quoted statistical uncerta<strong>in</strong>ties on <strong>the</strong> estimated jet energy corrections arecalculated us<strong>in</strong>g a number <strong>of</strong> events correspond<strong>in</strong>g to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> about9400pb −1 but are rescaled to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 100pb −1 . Then <strong>the</strong> number<strong>of</strong> simulated events are <strong>in</strong>deed more than what is expected <strong>in</strong> 100pb −1 <strong>of</strong> accumulateddata. Therefore, <strong>the</strong> statistical uncerta<strong>in</strong>ties are actually less than what have beenquoted. The statistical uncerta<strong>in</strong>ties correspond<strong>in</strong>g to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong>about 9400pb −1 are evaluated and reported as <strong>the</strong> uncerta<strong>in</strong>ties on <strong>the</strong> estimated lightand b jet energy corrections as followsand∆ε l est = −10.12 ± 0.13%,∆ε b est = −1.31 ± 0.20%.The various sources for <strong>the</strong> systematical uncerta<strong>in</strong>ties on <strong>the</strong> estimated results arediscussed <strong>in</strong> <strong>the</strong> Section 5.4.2.In a study based on <strong>the</strong> simulated data, <strong>the</strong> residual jet energy corrections can bedeterm<strong>in</strong>ded us<strong>in</strong>g <strong>the</strong> Monte Carlo truth <strong>in</strong>formation, too. A comparison between <strong>the</strong>expected residual corrections from Monte Carlo truth <strong>in</strong>formation and <strong>the</strong> estimatedresidual corrections derived from <strong>the</strong> k<strong>in</strong>ematic fit procedure, can <strong>the</strong>n demonstrate howwell <strong>the</strong> estimation method works. If <strong>the</strong> four-vectors <strong>of</strong> <strong>the</strong> partons can be accessedfrom <strong>the</strong> simulation <strong>in</strong>formation, <strong>the</strong> reconstructed jets can be matched to <strong>the</strong> partons
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 97us<strong>in</strong>g a jet-parton match<strong>in</strong>g algorithm, as described <strong>in</strong> Section 5.1.1. Therefore for apair <strong>of</strong> matched jet-parton, <strong>the</strong> expected residual correction can be obta<strong>in</strong>ed byE jet (1 + ∆ε expjet ) = E parton.Hence, <strong>in</strong> <strong>the</strong> topology <strong>of</strong> <strong>the</strong> hadronic top quark decay t → Wb → q¯qb, one cancalculate <strong>the</strong> expected corrections for <strong>the</strong> light and b jets, asand∆ε expl∆ε expb= E light quark − E light jetE light jet,= E b quark − E b jetE b jet.The distributions <strong>of</strong> <strong>the</strong> expected light as well as <strong>the</strong> b jet energy corrections, which arefitted with a Gaussian function <strong>in</strong> a range <strong>of</strong> ±1.5 times <strong>the</strong> Root Mean Square (RMS)<strong>of</strong> <strong>the</strong> distribution around <strong>the</strong> mean, are shown <strong>in</strong> Figure 5.19. They are obta<strong>in</strong>edus<strong>in</strong>g those selected events for which a true jet-parton match<strong>in</strong>g exists.#events500 Entries 6527χ 2/ ndf5.95 / 12400Constant 482.3Mean -0.08673Sigma 0.1239300#events250200150Entries 3263χ 2/ ndf20.25 / 12Constant 230.2Mean -0.02078Sigma 0.1314200100100500-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1exp∆ε l0-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1exp∆ε bFigure 5.19: The distributions <strong>of</strong> <strong>the</strong> expected residual jet energy corrections for (left)light and (right) b jets. They are fitted with a Gaussian <strong>in</strong> <strong>the</strong> range <strong>of</strong> ±1.5 times <strong>the</strong>RMS around <strong>the</strong> mean.For each <strong>of</strong> <strong>the</strong> plots shown <strong>in</strong> Figure 5.19, <strong>the</strong> expectation value <strong>of</strong> <strong>the</strong> fittedGaussian and its uncerta<strong>in</strong>ty are taken as <strong>the</strong> expected residual jet energy correctionswhich are found to be∆ε l exp = −8.67 ± 0.23%,and∆ε b exp = −2.08 ± 0.36%.The fit results strongly depend on <strong>the</strong> range which is used to perform <strong>the</strong> Gaussian fit.For example, <strong>the</strong> expected residual corrections, obta<strong>in</strong>ed for various ranges <strong>of</strong> <strong>the</strong> fit,are listed <strong>in</strong> Table 5.4.As seen from Table 5.4, <strong>the</strong> expectation values <strong>of</strong> <strong>the</strong> fitted Gaussian functions for<strong>the</strong> light quark jets varies from -8.83% to -7.43% when <strong>in</strong>creas<strong>in</strong>g <strong>the</strong> fit range from
98 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor±1.0 RMS ±1.5 RMS ±2.0 RMS ±3.0 RMS∆ε exp∆ε expl(%) -8.83±0.31 -8.67±0.23 -8.17±0.19 -7.43±0.18b(%) -2.93±0.42 -2.08±0.36 -0.88±0.29 0.47±0.30Table 5.4: The expected residual corrections for light and b jets which are derived from<strong>the</strong> Gaussian fit over <strong>the</strong> various ranges <strong>of</strong> <strong>the</strong> fit.±1RMS to ±3RMS, which results <strong>in</strong> obta<strong>in</strong><strong>in</strong>g a small variation <strong>of</strong> about 1.4%. For<strong>the</strong> case <strong>of</strong> b jets, this variation is larger and reaches to about 3.4% when enlarg<strong>in</strong>g <strong>the</strong>fit range. This is due to <strong>the</strong> fact that, <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> expected b jet residualcorrection, can not be fitted <strong>in</strong> <strong>the</strong> tail with a Gaussian function and <strong>the</strong> expected value<strong>of</strong> <strong>the</strong> fitted Gaussian would change when <strong>in</strong>creas<strong>in</strong>g <strong>the</strong> fit range. This observation<strong>of</strong> <strong>the</strong> non-Gaussian effects <strong>in</strong> <strong>the</strong> tail, which is specific to <strong>the</strong> b jet distribution, canbe expla<strong>in</strong>ed as follows. S<strong>in</strong>ce <strong>the</strong> b jets can decay to a lepton and <strong>the</strong> correspond<strong>in</strong>gneutr<strong>in</strong>o with a probability <strong>of</strong> 20%, hence <strong>the</strong> energy <strong>of</strong> <strong>the</strong> b jet is underestimateddue to <strong>the</strong> escap<strong>in</strong>g neutr<strong>in</strong>o. Hence, us<strong>in</strong>g <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> expected b jet energycorrection, one can have∆ε expb= E b quarkE b jet− 1 > 0,for those events where <strong>the</strong> b jet energy is underestimated. Therefore, <strong>the</strong>se k<strong>in</strong>d <strong>of</strong>events contribute to <strong>the</strong> tail <strong>in</strong> <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> expected b jet energy correctionas shown <strong>in</strong> Figure 5.19. Therefore, <strong>in</strong> order to avoid <strong>the</strong> non-Gaussian effects <strong>in</strong><strong>the</strong> tail, <strong>the</strong> fit range is chosen to be ±1.5 times <strong>the</strong> RMS around <strong>the</strong> mean <strong>of</strong> <strong>the</strong>histogram. However, <strong>the</strong> dependency <strong>of</strong> <strong>the</strong> fit results on <strong>the</strong> considered range <strong>of</strong> <strong>the</strong>Gaussian fit can be taken <strong>in</strong>to account as systematical uncerta<strong>in</strong>ty on <strong>the</strong> expectedvalues for <strong>the</strong> light and b jet energy corrections. Hence, for each <strong>of</strong> <strong>the</strong> expected lightand b jet distributions, <strong>the</strong> maximum variation among <strong>the</strong> various fit ranges comparedto <strong>the</strong> chosen fit range, be<strong>in</strong>g ±1.5RMS, is taken as <strong>the</strong> systematical uncerta<strong>in</strong>ty forthat particular distribution. The result<strong>in</strong>g systematical uncerta<strong>in</strong>ties for <strong>the</strong> expectedlight as well as b jet energy corrections, are listed <strong>in</strong> Table 5.5.Sys. Unc.δ sys. (∆ε expl)(%) ±1.2δ sys. (∆ε expb)(%) ±2.5Table 5.5: The systematical uncerta<strong>in</strong>ties on <strong>the</strong> expected light and b jet energy corrections.In order to compare <strong>the</strong> expected versus <strong>the</strong> estimated results, a new parameter,called <strong>the</strong> bias, is <strong>in</strong>troduced which is def<strong>in</strong>ed as <strong>the</strong> difference between ∆ε estl/b and
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 99∆ε expl/b. The estimation <strong>of</strong> <strong>the</strong> residual jet energy corrections would be well done if<strong>the</strong> bias would be equal to zero. The estimated as well as <strong>the</strong> expected results <strong>of</strong> <strong>the</strong>residual jet energy corrections are collected and <strong>the</strong> correspond<strong>in</strong>g bias for <strong>the</strong> lightand <strong>the</strong> b jets is calculated which can be found <strong>in</strong> Table 5.6.∆ε expl(%) ∆ε estl(%) ∆ε estl− ∆ε expl(%)-8.67±0.23 -10.12±0.13 -1.45±0.26∆ε expb(%) ∆ε estb(%)∆ε estb− ∆ε expb(%)-2.08±0.36 -1.31±0.20 0.77±0.41Table 5.6: The f<strong>in</strong>al estimated residual jet energy corrections obta<strong>in</strong>ed for 100pb −1 .The correspond<strong>in</strong>g expected residual corrections and <strong>the</strong> possible bias are also shown.Tak<strong>in</strong>g <strong>in</strong>to account <strong>the</strong> systematical uncerta<strong>in</strong>ties on <strong>the</strong> expected light and b jetcorrections, mentioned <strong>in</strong> Table 5.5, would result to obta<strong>in</strong> a bias on, for example, <strong>the</strong>estimated b jets to be compatible with zero. O<strong>the</strong>r systematical uncerta<strong>in</strong>ties on <strong>the</strong>estimated residual light and b jet energy corrections are calculated <strong>in</strong> Section 5.4.2.5.4.1 Performance <strong>of</strong> <strong>the</strong> MethodAs discussed <strong>in</strong> Section 5.1.6, <strong>the</strong> distributions <strong>of</strong> <strong>the</strong> reconstructed W boson and topquark masses <strong>in</strong> <strong>the</strong> topology <strong>of</strong> <strong>the</strong> e+jets t¯t events are shifted from <strong>the</strong> values that wereused <strong>in</strong> <strong>the</strong> simulation which consequently motivates <strong>the</strong> need for residual jet energycorrections. S<strong>in</strong>ce <strong>in</strong> <strong>the</strong> hadronic branch <strong>of</strong> <strong>the</strong> e+jets t¯t event, t → Wb → q¯qb, bothlight jets, aris<strong>in</strong>g <strong>in</strong> <strong>the</strong> decay <strong>of</strong> <strong>the</strong> hadronic W boson, and b jets, orig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong>decay <strong>of</strong> <strong>the</strong> hadronic top quark, are present. Then by impos<strong>in</strong>g <strong>the</strong> mass constra<strong>in</strong>ts<strong>of</strong> both <strong>the</strong> W boson and <strong>the</strong> top quark by means <strong>of</strong> a k<strong>in</strong>ematic fit approach, onecould estimate <strong>the</strong> residual light as well as <strong>the</strong> b jet energy corrections as described <strong>in</strong>Section 5.2. The aim <strong>of</strong> <strong>the</strong> method was to correct <strong>the</strong> energies <strong>of</strong> <strong>the</strong> reconstructedjets <strong>in</strong> <strong>the</strong> hadronic branch <strong>of</strong> <strong>the</strong> e+jets t¯t topology so that <strong>the</strong> result<strong>in</strong>g reconstructedW boson and top quark masses are peaked around <strong>the</strong> nom<strong>in</strong>al parameters which havebeen used to generate <strong>the</strong> event.The plots shown <strong>in</strong> Figure 5.20 are <strong>the</strong> <strong>in</strong>variant masses <strong>of</strong> <strong>the</strong> corrected four-vector <strong>of</strong><strong>the</strong> W boson and top quark which are reconstructed from <strong>the</strong> corrected four-vectors <strong>of</strong><strong>the</strong> reconstructed jets <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> e+jets t¯t events. The events for which a truejet-parton comb<strong>in</strong>ation found by <strong>the</strong> MVA method, are used to fill <strong>the</strong> distributions.Compared to <strong>the</strong> plots obta<strong>in</strong>ed <strong>in</strong> Figure 5.9, it is obvious that <strong>the</strong> corrected massdistributions are peaked around <strong>the</strong>ir nom<strong>in</strong>al values as it was aimed. The reconstructedmass values <strong>of</strong> <strong>the</strong> W boson and <strong>the</strong> top quark can be fur<strong>the</strong>r improved byapply<strong>in</strong>g a second iteration which is expla<strong>in</strong>ed <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g. The estimated residual
100 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor#events250200Entries 3263Mean 78.95RMS 10.22#events180160140120Entries 3263Mean 169.7RMS 15.8150100100806050402000 20 40 60 80 100 120 140 160 180 200m W(GeV)00 50 100 150 200 250 300 350 400m top(GeV)Figure 5.20: The distributions <strong>of</strong> <strong>the</strong> reconstructed W boson mass (left) and <strong>the</strong> topquark mass (right) after apply<strong>in</strong>g <strong>the</strong> estimated corrections on <strong>the</strong> reconstructed jets.The e+jets t¯t events that pass <strong>the</strong> extra selection cuts are taken <strong>in</strong>to account.jet energy corrections which are estimated by <strong>the</strong> method <strong>in</strong> a first iteration are appliedon <strong>the</strong> energy <strong>of</strong> <strong>the</strong> reconstructed jets and <strong>the</strong> method is performed for a second time.Ideally <strong>the</strong> estimated residual jet energy corrections <strong>in</strong> <strong>the</strong> second iteration would beequal to zero if <strong>the</strong> bias <strong>in</strong> <strong>the</strong> first iteration was exactly zero. The sum <strong>of</strong> <strong>the</strong> estimatedjet energy corrections obta<strong>in</strong>ed from <strong>the</strong> first and <strong>the</strong> second iterations will be a betterestimate <strong>of</strong> <strong>the</strong> residual jet energy corrections. Ano<strong>the</strong>r th<strong>in</strong>g which can be done toobta<strong>in</strong> a more precise mass distribution for ei<strong>the</strong>r <strong>the</strong> W boson or <strong>the</strong> top quark, is toapply correction factors which varies <strong>in</strong> terms <strong>of</strong> p T <strong>of</strong> <strong>the</strong> jets. The method is appliedon <strong>the</strong> events which are grouped accord<strong>in</strong>g to <strong>the</strong> p T <strong>of</strong> <strong>the</strong> jets to obta<strong>in</strong> an estimatedcorrection for <strong>the</strong> jets contribut<strong>in</strong>g to that particular b<strong>in</strong> <strong>of</strong> p T . The obta<strong>in</strong>ed resultscan <strong>the</strong>n be applied on <strong>the</strong> reconstructed jets belong<strong>in</strong>g to that p T -b<strong>in</strong>. In additionto <strong>the</strong> p T , o<strong>the</strong>r k<strong>in</strong>ematic variables such as η can also be used. Apply<strong>in</strong>g this k<strong>in</strong>d<strong>of</strong> differential estimation <strong>of</strong> <strong>the</strong> jet energy corrections is expected to result <strong>in</strong> a massdistribution <strong>of</strong> both <strong>the</strong> W boson and <strong>the</strong> top quark with an improved resolution.5.4.2 Systematic Uncerta<strong>in</strong>tiesVarious sources can affect <strong>the</strong> f<strong>in</strong>al estimated results <strong>of</strong> <strong>the</strong> jet energy scale correctionfactors. Some <strong>of</strong> <strong>the</strong> most important sources that can <strong>in</strong>fluence <strong>the</strong> results, are discussedbelow.The Value <strong>of</strong> m top as Input Variable for <strong>the</strong> K<strong>in</strong>ematic FitThe k<strong>in</strong>ematic fit package requires <strong>the</strong> event topology to fulfill <strong>the</strong> mass constra<strong>in</strong>ts.The W boson and <strong>the</strong> top quark mass values that are used <strong>in</strong> <strong>the</strong> package may varywith<strong>in</strong> <strong>the</strong>ir uncerta<strong>in</strong>ties. While <strong>the</strong> mass <strong>of</strong> <strong>the</strong> W boson is measured with a highprecesion <strong>of</strong> σ mW = 0.03%, <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark is measured with an accuracy <strong>of</strong>σ mtop = 0.52%, as discussed <strong>in</strong> 1.2. To be conservative, a larger uncerta<strong>in</strong>ty equivalentto 2.5σ mtop is considered on <strong>the</strong> value <strong>of</strong> <strong>the</strong> top quark mass, hence 1.3%. This results<strong>in</strong> an uncerta<strong>in</strong>ty equals to ±2.3 GeV on <strong>the</strong> top quark mass value <strong>of</strong> 172.5 GeV which
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 101has been used <strong>in</strong> <strong>the</strong> k<strong>in</strong>ematic fit procedure. Therefore, <strong>the</strong> analysis is performedwith different top quark mass values, be<strong>in</strong>g <strong>the</strong> larger value 172.5+2.3 GeV and <strong>the</strong>smaller value 172.5-2.3 GeV, as a constra<strong>in</strong>t <strong>in</strong> <strong>the</strong> k<strong>in</strong>ematic fit package. The resultsare shown for <strong>the</strong> expected and estimated residual corrections <strong>in</strong> Table 5.7.∆ε expl(%) ∆ε estl(%) ∆ε estl− ∆ε expl(%)nom<strong>in</strong>al -8.67±0.23 -10.12±0.13 -1.45±0.26larger m top -8.65±0.23 -10.07±0.13 -1.42±0.26smaller m top -8.64±0.23 -10.39±0.13 -1.75±0.26∆ε expb(%) ∆ε estb(%)∆ε estb− ∆ε expb(%)nom<strong>in</strong>al -2.08±0.36 -1.32±0.20 0.76±0.41larger m top -2.08±0.35 0.58±0.20 2.66±0.40smaller m top -1.92±0.37 -2.84±0.20 -0.92±0.42Table 5.7: The expected and estimated residual jet energy corrections and <strong>the</strong> bias on<strong>the</strong> estimations for different top quark masses as constra<strong>in</strong>ts <strong>in</strong> <strong>the</strong> fit package.What is important to note from Table 5.7, is that <strong>the</strong> estimated results on <strong>the</strong> lightjet energy correction do not change when us<strong>in</strong>g a top quark mass as <strong>the</strong> constra<strong>in</strong>twhich is different from what has been used to generate <strong>the</strong> physics events. This observationis due to that <strong>the</strong> light quark jet energy calibration factors are determ<strong>in</strong>ed by<strong>the</strong> mass <strong>of</strong> <strong>the</strong> W boson which is used <strong>in</strong> <strong>the</strong> k<strong>in</strong>ematic fit package and are <strong>in</strong>dependentfrom <strong>the</strong> value <strong>of</strong> <strong>the</strong> top quark mass which is used to constra<strong>in</strong> <strong>the</strong> event topology.The estimated residual b jet energy correction varies from -2.84% to 0.58% when <strong>in</strong>creas<strong>in</strong>g<strong>the</strong> top quark mass value from 170.2 GeV to 174.8 GeV <strong>in</strong> <strong>the</strong> fitt<strong>in</strong>g procedure.Therefore, compared to <strong>the</strong> nom<strong>in</strong>al estimated b jet energy correction, a shift <strong>of</strong> 1.52%or 1.9% is observed when us<strong>in</strong>g <strong>the</strong> smaller or <strong>the</strong> larger mass <strong>of</strong> <strong>the</strong> top quark, respectively.Hence, <strong>the</strong> maximum difference between <strong>the</strong> larger and <strong>the</strong> nom<strong>in</strong>al or <strong>the</strong>smller and <strong>the</strong> nom<strong>in</strong>al, is taken as <strong>the</strong> systematic uncerta<strong>in</strong>ty due to <strong>the</strong> uncerta<strong>in</strong>tyon <strong>the</strong> top quark mass. It should be noted that <strong>the</strong> uncerta<strong>in</strong>ties obta<strong>in</strong>ed from Table5.7 are overestimated s<strong>in</strong>ce a larger deviation <strong>of</strong> <strong>the</strong> top quark mass from its meanvalue has been considered, as mentioned above.Pile-UpToge<strong>the</strong>r with <strong>the</strong> hard scatter<strong>in</strong>g event <strong>in</strong> <strong>the</strong> proton-proton collision, <strong>the</strong>re are manys<strong>of</strong>t processes com<strong>in</strong>g from <strong>the</strong> next bunch cross<strong>in</strong>g that might affect <strong>the</strong> topology<strong>of</strong> <strong>the</strong> event, specially <strong>in</strong> <strong>the</strong> high lum<strong>in</strong>osity phase where <strong>the</strong> time between <strong>the</strong> twosuccessive bunch cross<strong>in</strong>gs is so small that <strong>the</strong> electronic signals from <strong>the</strong> current eventmay not be fully collected yet. Hence <strong>the</strong> pile-up events may result to have moretracks <strong>in</strong> <strong>the</strong> track collection and would yield to reconstruct additional objects <strong>in</strong> <strong>the</strong>f<strong>in</strong>al state. This can vary <strong>the</strong> f<strong>in</strong>al estimation results. In Table 5.8, <strong>the</strong> expected and
102 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factorestimated jet energy corrections for <strong>the</strong> events with and without <strong>in</strong>clud<strong>in</strong>g pile-up aregiven.∆ε expl(%) ∆ε estl (%) ∆ε estl − ∆ε expl(%)nom<strong>in</strong>al -9.38±0.25 -8.75±0.14 0.63±0.29pile-up -9.52±0.24 -8.68±0.13 0.84±0.27∆ε expb(%) ∆ε estb (%)∆ε estb− ∆ε expb(%)nom<strong>in</strong>al -2.66±0.36 -2.81±0.20 -0.15±0.41pile-up -2.82±0.36 -3.30±0.20 -0.48±0.41Table 5.8: The expected and estimated residual jet energy corrections and <strong>the</strong> bias on<strong>the</strong> estimations <strong>in</strong> <strong>the</strong> presence <strong>of</strong> pile-up events. The nom<strong>in</strong>al refers to <strong>the</strong> situationwhen no pile-up is present.As it is seen from Table 5.8, <strong>the</strong> method <strong>of</strong> estimat<strong>in</strong>g <strong>the</strong> residual jet energy correctionsis stable with respect to add<strong>in</strong>g <strong>the</strong> pile-up collisions. Obviously <strong>the</strong> expected aswell as <strong>the</strong> estimated corrections are affected by <strong>the</strong> pile-up collisions. If <strong>the</strong> pile-up iswell simulated, <strong>the</strong> bias should be stable. The estimated residual light and b jet energycorrections do not change much <strong>in</strong> <strong>the</strong> presence <strong>of</strong> <strong>the</strong> pile-up events. The estimatedlight jet energy correction changes from -8.75%, correspond<strong>in</strong>g to <strong>the</strong> nom<strong>in</strong>al value,to -8.68%, which is obta<strong>in</strong>ed when <strong>in</strong>clud<strong>in</strong>g pile-up events. Thus, <strong>the</strong> shift <strong>of</strong> 0.07%is considered as <strong>the</strong> systematic uncerta<strong>in</strong>ty on <strong>the</strong> estimated residual light jet energycorrection due to <strong>the</strong> pile-up effects. The deviation <strong>of</strong> <strong>the</strong> estimated b jet energy correctionfrom <strong>the</strong> nom<strong>in</strong>al value corresponds to 0.49% which is taken as <strong>the</strong> systematicuncerta<strong>in</strong>ty on <strong>the</strong> estimated residual b jet energy correction due to <strong>the</strong> pile-up events.ISR/FSRWhen simulat<strong>in</strong>g physics events, various parameters are set to control <strong>the</strong> productionand decay products <strong>of</strong> <strong>the</strong> hard scatter<strong>in</strong>g <strong>in</strong>teractions. Chang<strong>in</strong>g any <strong>of</strong> <strong>the</strong>se parameters<strong>in</strong>dividually, would yield to obta<strong>in</strong> a different topology <strong>of</strong> <strong>the</strong> event. Therefore,<strong>the</strong> sensitivity <strong>of</strong> <strong>the</strong> estimated results with respect to <strong>the</strong> changes <strong>of</strong> <strong>the</strong>se parameters,has to be studied and <strong>the</strong> possible deviations are taken as <strong>the</strong> systematic uncerta<strong>in</strong>tieson <strong>the</strong> f<strong>in</strong>al estimation. For example, <strong>the</strong> amount <strong>of</strong> ISR/FSR that are allowed to begenerated <strong>in</strong> <strong>the</strong> simulation procedure can affect <strong>the</strong> results. Two different sampleswith a smaller and larger amount <strong>of</strong> radiation compared to <strong>the</strong> nom<strong>in</strong>al situation areprocessed. The estimated residual jet energy corrections are obta<strong>in</strong>ed for each <strong>of</strong> <strong>the</strong>sesamples which are given <strong>in</strong> Table 5.9. The estimated and expected results for light andb jet energy corrections are listed <strong>in</strong>dividually for <strong>the</strong> various samples.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 103∆ε expl(%) ∆ε estl(%) ∆ε estl− ∆ε expl(%)nom<strong>in</strong>al -9.38±0.25 -8.75±0.14 0.63±0.29larger ISR/FSR -8.85±0.26 -8.12±0.14 0.73±0.29smaller ISR/FSR -9.62±0.25 -9.13±0.14 0.49±0.29∆ε expb(%) ∆ε estb(%)∆ε estb− ∆ε expb(%)nom<strong>in</strong>al -2.66±0.36 -2.81±0.20 -0.15±0.41larger ISR/FSR -2.54±0.37 -2.68±0.20 -0.14±0.42smaller ISR/FSR -1.85±0.39 -2.49±0.20 0.64±0.44Table 5.9: The expected and estimated residual jet energy corrections and <strong>the</strong> bias on<strong>the</strong> estimations for <strong>the</strong> samples with larger and smaller amount <strong>of</strong> radiation comparedto <strong>the</strong> nom<strong>in</strong>al.S<strong>in</strong>ce <strong>the</strong> bias on <strong>the</strong> estimated light and b jet energy corrections is small and,<strong>in</strong> some cases, is compatible with zero with<strong>in</strong> <strong>the</strong> statistical uncerta<strong>in</strong>ties, it can bededuced that <strong>the</strong> method is stable with respect to changes <strong>in</strong> ISR/FSR. The estimatedlight jet energy correction changes from -8.12%, correspond<strong>in</strong>g to <strong>the</strong> sample withlarger ISR/FSR, to -9.13%, which is obta<strong>in</strong>ed when consider<strong>in</strong>g smaller amount <strong>of</strong>ISR/FSR <strong>in</strong> <strong>the</strong> simulation procedure. Compared to <strong>the</strong> nom<strong>in</strong>al value which equals-8.75%, <strong>the</strong> maximum deviation <strong>of</strong> <strong>the</strong> estimated light jet energy correction is takenas <strong>the</strong> systematic uncerta<strong>in</strong>ty due to <strong>the</strong> ISR/FSR, be<strong>in</strong>g 0.63%. In case <strong>of</strong> b jets, <strong>the</strong>maximum deviation would reach to 0.32% which is quoted as <strong>the</strong> systematic uncerta<strong>in</strong>tyon <strong>the</strong> estimated b jet energy correction due to <strong>the</strong> ISR/FSR.<strong>Scale</strong> Factor Q 2As already mentioned, <strong>the</strong> <strong>in</strong>itial and f<strong>in</strong>al state radiations are governed by <strong>the</strong> use <strong>of</strong><strong>the</strong> DGLAP equation which describes <strong>the</strong> evolution <strong>of</strong> <strong>the</strong> partonic distribution functionas a function <strong>of</strong> <strong>the</strong> factorization scale Q 2 . Accord<strong>in</strong>g to <strong>the</strong> DGLAP equation, <strong>the</strong> f<strong>in</strong>alstate radiation <strong>in</strong> an event can be started from a maximum energy state Q 2 max, whichis chosen to be <strong>the</strong> squared mass <strong>of</strong> <strong>the</strong> parton shower <strong>in</strong>itiator, down to smaller values<strong>of</strong> <strong>the</strong> Q 2 scale. S<strong>in</strong>ce <strong>the</strong> chosen value used for <strong>the</strong> Q 2 max parameter might not be <strong>the</strong>optimal choice for <strong>the</strong> description <strong>of</strong> observed proton-proton collision data, <strong>the</strong> effects <strong>of</strong>alter<strong>in</strong>g <strong>the</strong> Q 2 value up and downwards are taken <strong>in</strong>to account. Therefore, additionalsamples are simulated for which <strong>the</strong> <strong>in</strong>itial <strong>in</strong>put variable <strong>of</strong> <strong>the</strong> Q 2 max parameter ischanged with<strong>in</strong> an <strong>in</strong>terval. Then, <strong>in</strong> order to check <strong>the</strong> effects <strong>of</strong> <strong>the</strong> variation <strong>of</strong><strong>the</strong> Q 2 scale, <strong>the</strong> method <strong>of</strong> estimat<strong>in</strong>g <strong>the</strong> residual jet energy corrections is appliedon both simulated samples with <strong>the</strong> various sett<strong>in</strong>gs <strong>of</strong> <strong>the</strong> Q 2 parameters and <strong>the</strong>nom<strong>in</strong>al sample which is simulated with <strong>the</strong> pre-def<strong>in</strong>ed sett<strong>in</strong>g <strong>of</strong> <strong>the</strong> Q 2 parameter.The results <strong>of</strong> <strong>the</strong> estimated and expected residual jet energy corrections on differentsett<strong>in</strong>gs <strong>of</strong> <strong>the</strong> Q 2 scale are summarized and a possible bias <strong>of</strong> <strong>the</strong> method is calculated
104 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factorand shown <strong>in</strong> Table 5.10.∆ε expl(%) ∆ε estl(%) ∆ε estl− ∆ε expl(%)nom<strong>in</strong>al -9.38±0.25 -8.75±0.14 0.63±0.29Q 2 up -9.30±0.32 -8.76±0.18 0.54±0.37Q 2 down -10.14±0.32 -9.20±0.17 0.94±0.36∆ε expb(%) ∆ε estb(%)∆ε estb− ∆ε expb(%)nom<strong>in</strong>al -2.66±0.36 -2.81±0.20 -0.15±0.41Q 2 up -2.52±0.49 -2.57±0.26 -0.05±0.55Q 2 down -3.41±0.47 -3.00±0.25 0.41±0.53Table 5.10: The expected and estimated residual jet energy corrections and <strong>the</strong> biason <strong>the</strong> estimations for <strong>the</strong> samples with higher and lower values <strong>of</strong> <strong>the</strong> Q 2 parameter.As seen from <strong>the</strong> results shown <strong>in</strong> Table 5.10, <strong>the</strong> effects <strong>of</strong> chang<strong>in</strong>g <strong>the</strong> <strong>in</strong>putQ 2 parameter on <strong>the</strong> f<strong>in</strong>al estimated results are m<strong>in</strong>or and <strong>the</strong> estimation follows <strong>the</strong>expected value s<strong>in</strong>ce <strong>the</strong> bias on <strong>the</strong> estimated results is compatible with zero. Also <strong>the</strong>effect <strong>of</strong> Q 2 on <strong>the</strong> bias is compatible with zero. The systematical uncerta<strong>in</strong>ties thatcan be assigned due to <strong>the</strong> changes <strong>in</strong> <strong>the</strong> Q 2 scale can be evaluated as follows. Theestimated light jet energy correction changes from -8.76%, correspond<strong>in</strong>g to <strong>the</strong> samplewith larger Q 2 scale, to -9.20%, which is obta<strong>in</strong>ed when consider<strong>in</strong>g <strong>the</strong> simulatedsample with smaller value <strong>of</strong> <strong>the</strong> Q 2 scale. Compared to <strong>the</strong> nom<strong>in</strong>al value which is-8.75%, <strong>the</strong> maximum deviation <strong>of</strong> <strong>the</strong> estimated light jet energy correction is taken as<strong>the</strong> systematic uncerta<strong>in</strong>ty due to <strong>the</strong> changes <strong>in</strong> <strong>the</strong> Q 2 scale, be<strong>in</strong>g 0.45%. In case <strong>of</strong>b jets, <strong>the</strong> maximum deviation would reach 0.24% which is quoted as <strong>the</strong> systematicuncerta<strong>in</strong>ty on <strong>the</strong> estimated b jet energy correction due to <strong>the</strong> variation <strong>of</strong> <strong>the</strong> Q 2 scale.It should be noted that, when a more precise estimation <strong>of</strong> <strong>the</strong> jet energy correctionsis aimed for, <strong>the</strong>se systematic uncerta<strong>in</strong>ties should be estimated with larger simulatedsamples.Match<strong>in</strong>g Matrix Element to Parton ShowerIn <strong>the</strong> procedure <strong>of</strong> simulat<strong>in</strong>g physics collisions, <strong>the</strong> events are fed to a parton showerprogram to apply <strong>the</strong> shower<strong>in</strong>g process on <strong>the</strong> partons which are generated <strong>in</strong> <strong>the</strong>collision, as already expla<strong>in</strong>ed <strong>in</strong> Section 4.1.2. Subsequently a match<strong>in</strong>g scheme isapplied <strong>in</strong> order to avoid double count<strong>in</strong>g <strong>of</strong> <strong>the</strong> radiation. Therefore, <strong>the</strong> jets producedby <strong>the</strong> parton shower method are compared with <strong>the</strong> partons produced with <strong>the</strong> matrixelement method, given that <strong>the</strong> jets exceed<strong>in</strong>g a pre-def<strong>in</strong>ed threshold on <strong>the</strong> transverseenergy. The choice <strong>of</strong> this threshod might have some effects on <strong>the</strong> f<strong>in</strong>al results <strong>of</strong> <strong>the</strong>physics analysis. Hence alternative samples are simulated with a higher and lowerthreshold on <strong>the</strong> transverse energy <strong>of</strong> <strong>the</strong> jets used <strong>in</strong> <strong>the</strong> match<strong>in</strong>g scheme. The effects
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 105<strong>of</strong> <strong>the</strong> variation <strong>of</strong> <strong>the</strong> match<strong>in</strong>g threshold is <strong>in</strong>vestigated by runn<strong>in</strong>g <strong>the</strong> method <strong>of</strong>estimat<strong>in</strong>g <strong>the</strong> residual jet energy corrections on both simulated samples with <strong>the</strong>various sett<strong>in</strong>gs <strong>of</strong> <strong>the</strong> match<strong>in</strong>g thresholds and <strong>the</strong> nom<strong>in</strong>al sample which is simulatedwith <strong>the</strong> pre-def<strong>in</strong>ed sett<strong>in</strong>g <strong>of</strong> <strong>the</strong> match<strong>in</strong>g threshold. The results <strong>of</strong> <strong>the</strong> estimatedand <strong>the</strong> expected residual jet energy corrections with different sett<strong>in</strong>gs <strong>of</strong> <strong>the</strong> match<strong>in</strong>gthreshold are summarized <strong>in</strong> Table 5.11.∆ε expl(%) ∆ε estl(%) ∆ε estl− ∆ε expl(%)nom<strong>in</strong>al -9.38±0.25 -8.75±0.14 0.63±0.29match<strong>in</strong>g up -9.19±0.28 -9.31±0.16 -0.12±0.32match<strong>in</strong>g down -9.74±0.30 -9.37±0.17 0.37±0.34∆ε expb(%) ∆ε estb(%)∆ε estb− ∆ε expb(%)nom<strong>in</strong>al -2.66±0.36 -2.81±0.20 -0.15±0.41match<strong>in</strong>g up -3.07±0.41 -2.77±0.24 0.30±0.47match<strong>in</strong>g down -3.17±0.43 -2.57±0.25 0.60±0.50Table 5.11: The expected and estimated residual jet energy corrections and <strong>the</strong> biason <strong>the</strong> estimations for <strong>the</strong> samples with higher and lower threshold <strong>in</strong> <strong>the</strong> match<strong>in</strong>gprocedure.It is seen from <strong>the</strong> results shown <strong>in</strong> Table 5.11 that <strong>the</strong> effects <strong>of</strong> chang<strong>in</strong>g <strong>the</strong>match<strong>in</strong>g threshold on <strong>the</strong> f<strong>in</strong>al estimated results are m<strong>in</strong>or and <strong>the</strong> estimation follows<strong>the</strong> expected value s<strong>in</strong>ce <strong>the</strong> bias on <strong>the</strong> estimated results is compatible with zero.The systematical uncerta<strong>in</strong>ties that can be assigned due to <strong>the</strong> changes <strong>in</strong> <strong>the</strong> match<strong>in</strong>gthreshold can be evaluated as follows. The estimated light jet energy correctionchanges from -9.31%, correspond<strong>in</strong>g to <strong>the</strong> sample with larger match<strong>in</strong>g threshold, to-9.37%, which is obta<strong>in</strong>ed when consider<strong>in</strong>g <strong>the</strong> simulated sample with smaller value<strong>of</strong> <strong>the</strong> match<strong>in</strong>g threshold. Compared to <strong>the</strong> nom<strong>in</strong>al value which is -8.75%, <strong>the</strong> maximumdeviation <strong>of</strong> <strong>the</strong> estimated light jet energy correction is taken as <strong>the</strong> systematicuncerta<strong>in</strong>ty due to <strong>the</strong> changes <strong>in</strong> <strong>the</strong> match<strong>in</strong>g threshold, be<strong>in</strong>g 0.62%. In case <strong>of</strong>b jets, <strong>the</strong> maximum deviation would reach to 0.24% which is quoted as <strong>the</strong> systematicuncerta<strong>in</strong>ty on <strong>the</strong> estimated b jet energy correction due to <strong>the</strong> variation <strong>of</strong> <strong>the</strong>match<strong>in</strong>g threshold.Comb<strong>in</strong>ed SystematicsThe various systematic uncerta<strong>in</strong>ties on <strong>the</strong> estimated results <strong>of</strong> light and b jet energycorrections are summarized <strong>in</strong> Table 5.12. The total systematical uncerta<strong>in</strong>ty is calculatedas <strong>the</strong> square root <strong>of</strong> <strong>the</strong> sum <strong>of</strong> <strong>the</strong> squared uncerta<strong>in</strong>ties from <strong>the</strong> diversesources which are considered <strong>in</strong> <strong>the</strong> table.
106 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factorm top pile-up ISR/FSR Q 2 match<strong>in</strong>g Total Sys. Unc.δ sys. (∆ε estl )(%) ±0.1 ±0.1 ±0.6 ±0.4 ±0.6 ±0.9δ sys. (∆ε estb )(%) ±0.8 ±0.5 ±0.3 ±0.2 ±0.2 ±1.0Table 5.12: The systematical uncerta<strong>in</strong>ties on <strong>the</strong> estimated light and b jet energycorrection orig<strong>in</strong>at<strong>in</strong>g from various sources. The total uncerta<strong>in</strong>ty is obta<strong>in</strong>ed by <strong>the</strong>square root <strong>of</strong> <strong>the</strong> sum <strong>of</strong> <strong>the</strong> squred <strong>of</strong> <strong>the</strong> <strong>in</strong>dividual errors.As already noted, <strong>the</strong> estimated uncerta<strong>in</strong>ty on <strong>the</strong> light and b jet energy correctionsdue to <strong>the</strong> variation <strong>of</strong> <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark are overestimated. Therefore,<strong>the</strong> uncerta<strong>in</strong>ties <strong>of</strong> 0.3% on <strong>the</strong> light jets and <strong>of</strong> 1.9% on <strong>the</strong> b jets are probably conservativeestimates. They are divided by a factor <strong>of</strong> 2.5 to represent <strong>the</strong> uncerta<strong>in</strong>tieson <strong>the</strong> estimated jet energy corrections result<strong>in</strong>g from one sigma deviation <strong>of</strong> <strong>the</strong> topquark mass value from its world average value. Ano<strong>the</strong>r po<strong>in</strong>t is that, some <strong>of</strong> <strong>the</strong>estimated systematical uncerta<strong>in</strong>ties quoted <strong>in</strong> Table 5.12, recieve large statistical uncerta<strong>in</strong>tiesbecause <strong>the</strong>y are limited with small simulated samples. For example, <strong>the</strong>estimated systematical uncerta<strong>in</strong>ty on <strong>the</strong> light jet energy correction due to pile-upsource, is dom<strong>in</strong>ated by <strong>the</strong> statistical uncerta<strong>in</strong>ty, s<strong>in</strong>ce from Table 5.8 one can haveδ sys. (∆ε est∣ (%) = 0.1 ± 0.2(stat.).pile−upl )The estimated systematical uncerta<strong>in</strong>ty on <strong>the</strong> b jet energy correction due to ei<strong>the</strong>r<strong>the</strong> Q 2 or <strong>the</strong> match<strong>in</strong>g source, is dom<strong>in</strong>ated by <strong>the</strong> statistical uncerta<strong>in</strong>ty, too.5.5 Statistical Properties <strong>of</strong> <strong>the</strong> Estimator5.5.1 L<strong>in</strong>earity <strong>of</strong> <strong>the</strong> EstimatorThe l<strong>in</strong>earity <strong>of</strong> <strong>the</strong> method is checked by look<strong>in</strong>g at <strong>the</strong> calibration curve, which isdef<strong>in</strong>ed as <strong>the</strong> variation <strong>of</strong> <strong>the</strong> estimated versus <strong>the</strong> expected jet energy corrections whileapply<strong>in</strong>g an <strong>in</strong>clusive shift on <strong>the</strong> energy scales <strong>of</strong> <strong>the</strong> reconstructed jets. Apply<strong>in</strong>g anyrelative <strong>in</strong>clusive shift on <strong>the</strong> energy <strong>of</strong> <strong>the</strong> reconstructed jets followed by perform<strong>in</strong>g <strong>the</strong>k<strong>in</strong>ematic fit method to constra<strong>in</strong> <strong>the</strong> event topology to fulfill <strong>the</strong> mass requirements,would yield to obta<strong>in</strong> <strong>the</strong> estimated results which are l<strong>in</strong>early dependent to <strong>the</strong> expectedjet energy corrections. This can be understood by look<strong>in</strong>g at <strong>the</strong> distributions <strong>of</strong> <strong>the</strong>versus ∆E expl,b<strong>the</strong> e+jets t¯t events.∆E estl,bwhich are shown <strong>in</strong> Figure 5.21. The plots are made us<strong>in</strong>g onlyIt can be seen that, when apply<strong>in</strong>g <strong>the</strong> L2 and L3 corrections on <strong>the</strong> <strong>in</strong>put energies<strong>of</strong> <strong>the</strong> reconstructed jets, <strong>the</strong> estimator is able to predict <strong>the</strong> expected jet energycorrections. When mov<strong>in</strong>g away from <strong>the</strong> po<strong>in</strong>t <strong>of</strong> nom<strong>in</strong>al L2 and L3 corrections, adeviation <strong>of</strong> <strong>the</strong> estimated results from <strong>the</strong> expected corrections is observed result<strong>in</strong>g
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 107(%)est∆ε l-2-4(%)est∆ε b42-60-8-2-10-4-12-14-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 exp∆ε l (%)-6-8-10 -5 0 5 10 exp∆ε b (%)Figure 5.21: The calibration curves represent<strong>in</strong>g <strong>the</strong> estimated residual jet energycorrections as a function <strong>of</strong> <strong>the</strong> expected jet energy corrections when apply<strong>in</strong>g several<strong>in</strong>clusive shifts on <strong>the</strong> <strong>in</strong>put energy <strong>of</strong> <strong>the</strong> reconstructed jets. A l<strong>in</strong>e with a unity slopeis overimposed for comparison.<strong>in</strong> a calibration curve with a non-unity slope. This can be fixed as expla<strong>in</strong>ed below.The non-unity slope <strong>of</strong> <strong>the</strong> calibration curves can be solved by <strong>the</strong> use <strong>of</strong> successiveiterations, which means <strong>the</strong> estimated jet energy corrections found <strong>in</strong> a first iterationare applied on <strong>the</strong> <strong>in</strong>itial jet energy scales. In a second iteration, <strong>the</strong> residual jet energycorrections can aga<strong>in</strong> be estimated, start<strong>in</strong>g with <strong>the</strong> corrected jets <strong>of</strong> <strong>the</strong> first iteration.Ideally, <strong>in</strong> a second iteration no residual jet energy correction should be needed, giventhat <strong>the</strong> estimated jet energy correction is equal to <strong>the</strong> expected jet energy correction<strong>in</strong> <strong>the</strong> first iteration. O<strong>the</strong>wise, <strong>the</strong> iteration procedure should be repeated until aresidual jet energy correction equals to zero is estimated. This means after apply<strong>in</strong>gseveral iterations, <strong>the</strong> estimation converges to <strong>the</strong> l<strong>in</strong>e ∆ε estl,b = ∆εexp l,b, means <strong>the</strong> slop<strong>of</strong> <strong>the</strong> calibration curve becomes close to <strong>the</strong> unity.5.5.2 Pull DistributionIn order to evaluate if <strong>the</strong> statistical uncerta<strong>in</strong>ties on <strong>the</strong> estimators have been quotedcorrectly, one can start look<strong>in</strong>g at <strong>the</strong> pull distributions which are def<strong>in</strong>ed as(∆ε i l,b − < ∆ε l,b >),δ∆ε i l,bwhere ∆ε l,b and δ∆ε l,b , as already used, are <strong>the</strong> estimated residual jet energy correctionsand <strong>the</strong>ir statistical errors, respectively. The <strong>in</strong>dex i is expla<strong>in</strong>ed as follows.The <strong>in</strong>itial simulated e+jets t¯t sample conta<strong>in</strong>s about 180000 events. In an analysiscorrespond<strong>in</strong>g to 100pb −1 , one would expect to obta<strong>in</strong> 2333 signal events, as it canbe deduced from <strong>the</strong> cross section <strong>of</strong> <strong>the</strong> e+jets t¯t process. Hence <strong>in</strong>stead <strong>of</strong> giv<strong>in</strong>g aweight to <strong>the</strong> <strong>in</strong>itial simulated events, one can repeat <strong>the</strong> whole analysis on <strong>the</strong> smallerbulks <strong>of</strong> simulated events, so-called pseudo-<strong>experiment</strong>s, where each conta<strong>in</strong> 2333 signalevents. Therefore, start<strong>in</strong>g with <strong>the</strong> <strong>in</strong>itial number <strong>of</strong> simulated signal events, onewould end with about 77 uncorrelated pseudo-<strong>experiment</strong>s. Here no correlation meansthat a s<strong>in</strong>gle signal event is not processed <strong>in</strong> no more than one pseudo-<strong>experiment</strong>.
108 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration FactorOnly e+jets t¯t events are taken <strong>in</strong>to account <strong>in</strong> order to make <strong>the</strong> pull distributions.No background is considered because not enough simulated events are available.Runn<strong>in</strong>g <strong>the</strong> method to estimate <strong>the</strong> residual jet energy callibration factors on each<strong>of</strong> <strong>the</strong> pseudo-<strong>experiment</strong>s, one can obta<strong>in</strong> 77 different values as <strong>the</strong> estimation <strong>of</strong> <strong>the</strong>light and b jet energy corrections, correspond<strong>in</strong>g to each <strong>of</strong> <strong>the</strong> pseudo-<strong>experiment</strong>s.The obta<strong>in</strong>ed results are used to make <strong>the</strong> pull distribution.Now it can be expla<strong>in</strong>ed that <strong>the</strong> <strong>in</strong>dex i <strong>in</strong> <strong>the</strong> def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> pull variable, refers to<strong>the</strong> i-th pseudo-<strong>experiment</strong>. Also <strong>the</strong> sign represents <strong>the</strong> average <strong>of</strong> <strong>the</strong> estimatedresults over all pseudo-<strong>experiment</strong>s.By def<strong>in</strong>ition, <strong>the</strong> pull distribution is peaked at zero and can be fitted with a Gaussianfunction. If <strong>the</strong> estimation <strong>of</strong> <strong>the</strong> statistical uncerta<strong>in</strong>ties is correct, <strong>the</strong>n <strong>the</strong> fittedGaussian would have a variance equal to one. O<strong>the</strong>rwise <strong>the</strong>re would have been an underestimationor overestimation <strong>of</strong> <strong>the</strong> estimated statistical uncerta<strong>in</strong>ties, if <strong>the</strong> width<strong>of</strong> <strong>the</strong> fitted Gaussian would have a larger or smaller width, respectively.The distribution <strong>of</strong> <strong>the</strong> pull variable for <strong>the</strong> estimated light and b jet energy correctionsis shown <strong>in</strong> Figure 5.22.Entries 77Entries 77# pseudo <strong>experiment</strong>s1210862χ / ndf14.29 / 14Constant 7.512 ±1.446Mean -0.0703 ±0.1968Sigma 1.383 ±0.232# pseudo <strong>experiment</strong>s1086χ 2/ ndf6.115 / 14Constant 7.917 ±1.264Mean -0.05784 ±0.18518Sigma 1.458 ±0.17844220-10 -8 -6 -4 -2 0 2 4 6 8 10ii(∆ε l -)/δ∆ε l0-10 -8 -6 -4 -2 0 2 4 6 8 10ii(∆ε b -)/δ∆ε bFigure 5.22: The pull distribution <strong>of</strong> <strong>the</strong> estimated (left) light and (right) b jet energycorrections. The distributions are fitted with a Gaussian function.As it is shown <strong>in</strong> <strong>the</strong> statistical boxes <strong>of</strong> <strong>the</strong> pulls <strong>in</strong> Figure 5.22, <strong>the</strong> width <strong>of</strong> <strong>the</strong>fitted Gaussian functions is about 1.4 for both estimators. These non-unity widths canbe used to correct <strong>the</strong> statistical uncerta<strong>in</strong>ties <strong>of</strong> <strong>the</strong> light and b jet energy corrections,which have been found for 100pb −1 <strong>of</strong> <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>in</strong> Section 5.4, result<strong>in</strong>g <strong>in</strong>and∆ε l est = −10.1 ± 1.5%,∆ε b est = −1.3 ± 2.4%.Therefore, at 7 TeV center <strong>of</strong> mass energy with <strong>the</strong> use <strong>of</strong> a sample <strong>of</strong> simulatedevents correspond<strong>in</strong>g to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 100pb −1 , <strong>the</strong> residual jet energycorrections can be estimated with a statistical precision <strong>of</strong> about 1.5% and 2.4%, forlight quark and b quark jets respectively. While us<strong>in</strong>g <strong>the</strong> full sample <strong>of</strong> simulatedevents which corresponds to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> about 9400pb −1 , <strong>the</strong> residualjet energy corrections can be estimated with a statistical precision <strong>of</strong> about 0.18% and
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 1090.28%, for light quark and b quark jets respectively. These numbers are corrected for<strong>the</strong> non-unity width <strong>of</strong> <strong>the</strong> pull distributions, too. Therefore, <strong>the</strong> results <strong>of</strong> bias onboth <strong>the</strong> light and <strong>the</strong> b jet energy corrections, which were quoted <strong>in</strong> Table 5.6, arere-estimated us<strong>in</strong>g <strong>the</strong> corrected statistical errors and summarized <strong>in</strong> Table 5.13.∆ε expl(%) ∆ε estl(%) ∆ε estl− ∆ε expl(%)-8.67±0.23(stat.)±1.2(sys.) -10.12±0.18(stat.) -1.45±0.29(stat.)±1.2(sys.)∆ε expb(%) ∆ε estb(%)∆ε estb− ∆ε expb(%)-2.08±0.36(stat.)±2.5(sys.) -1.31±0.28(stat.) 0.77±0.46(stat.)±2.5(sys.)Table 5.13: The f<strong>in</strong>al estimated residual jet energy corrections obta<strong>in</strong>ed for 100pb −1 .The correspond<strong>in</strong>g expected residual corrections and <strong>the</strong> possible bias are also shown.The quoted estimated results are corrected for <strong>the</strong> non-unity width <strong>of</strong> <strong>the</strong> pull distributions.The systematical uncerta<strong>in</strong>ties on <strong>the</strong> expected results are mentioned s<strong>in</strong>ceone cannot fit <strong>the</strong> distributions <strong>of</strong> ∆ε expl,bbetter than <strong>the</strong> quoted values which shouldbe taken <strong>in</strong>to account when estimat<strong>in</strong>g <strong>the</strong> bias.Also <strong>in</strong> Table 5.13, <strong>the</strong> systematical uncerta<strong>in</strong>ty on <strong>the</strong> expected residual jet energycorrections are mentioned. S<strong>in</strong>ce one cannot fit <strong>the</strong> distributions <strong>of</strong> ∆ε expl,bbetter than<strong>the</strong> quoted uncerta<strong>in</strong>ties which have to be considered <strong>in</strong> <strong>the</strong> f<strong>in</strong>al estimation <strong>of</strong> <strong>the</strong> bias.5.6 A First Application on <strong>the</strong> 2010 Collision DataSo far, <strong>the</strong> method to estimate <strong>the</strong> residual jet energy scale calibration factors has been<strong>in</strong>troduced and <strong>the</strong> performance <strong>of</strong> <strong>the</strong> method on simulated data has been described.In this section, <strong>the</strong> method is applied for <strong>the</strong> first time on <strong>the</strong> collision data whichis accumulated <strong>in</strong> 2010 and correspond<strong>in</strong>g to 36pb −1 <strong>of</strong> proton-proton collisions at acenter <strong>of</strong> mass energy <strong>of</strong> 7 TeV. The event selection cuts are slightly different comparedto those <strong>in</strong>troduced <strong>in</strong> Section 4.4, and are listed <strong>in</strong> <strong>the</strong> next section. Also <strong>in</strong> order toillustrate how well <strong>the</strong> simulated data predict <strong>the</strong> collision data, some control plots areshown <strong>in</strong> Section 5.6.2 to compare <strong>the</strong> simulated versus collision data. The simulatedsamples which are compared to <strong>the</strong> collision data, are different from what have beenused <strong>in</strong> previous sections because <strong>the</strong>y need to match with <strong>the</strong> collision and detectorparameters <strong>in</strong> <strong>the</strong> low data tak<strong>in</strong>g period. They are listed <strong>in</strong> Table 5.14. Also <strong>the</strong>names <strong>of</strong> 2010 collision data sets are given <strong>in</strong> <strong>the</strong> same table.The method to estimate <strong>the</strong> residual jet energy correction factors has been alreadyapplied on <strong>the</strong> simulated data and <strong>the</strong> results are obta<strong>in</strong>ed for 100pb −1 <strong>of</strong> <strong>in</strong>tegratedlum<strong>in</strong>osity. Although <strong>the</strong> performance <strong>of</strong> <strong>the</strong> method was checked us<strong>in</strong>g a huge amount<strong>of</strong> statistics compared to <strong>the</strong> accumulated collision data, it is still <strong>in</strong>terest<strong>in</strong>g to <strong>in</strong>ves-
110 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration FactorName <strong>of</strong> Data SampleCollision Data/Electron/Run2010B-Nov4ReReco_v1/AOD/EG/Run2010A-Nov4ReReco_v1/AODSimulated Data/TT<strong>Jet</strong>s_TuneD6T_7TeV-madgraph-tauola/Fall10-START38_V12-v2/AODSIM/W<strong>Jet</strong>sToLNu_TuneD6T_7TeV-madgraph-tauola/Fall10-START38_V12-v1/AODSIM/DY<strong>Jet</strong>sToLL_TuneD6T_M-50_7TeV-madgraph-tauola/Fall10-START38_V12-v2/AODSIM/TToBLNu_TuneZ2_s-channel_7TeV-madgraph/Fall10-START38_V12-v1/AODSIM/TToBLNu_TuneZ2_t-channel_7TeV-madgraph/Fall10-START38_V12-v2/AODSIM/TToBLNu_TuneZ2_tW-channel_7TeV-madgraph/Fall10-START38_V12-v2/AODSIM/QCD_Pt-20to30_BCtoE_TuneZ2_7TeV-pythia6/Fall10-START38_V12-v1/AODSIM/QCD_Pt-30to80_BCtoE_TuneZ2_7TeV-pythia6/Fall10-START38_V12-v1/AODSIM/QCD_Pt-80to170_BCtoE_TuneZ2_7TeV-pythia6/Fall10-START38_V12-v1/AODSIM/QCD_Pt-20to30_EMEnriched_TuneZ2_7TeV-pythia6/Fall10-START38_V12-v1/AODSIM/QCD_Pt-30to80_EMEnriched_TuneZ2_7TeV-pythia6/Fall10-START38_V12-v1/AODSIM/QCD_Pt-80to170_EMEnriched_TuneZ2_7TeV-pythia6/Fall10-START38_V12-v1/AODSIMTable 5.14: The list <strong>of</strong> collision data samples and simulated data sets which are usedto evaluate <strong>the</strong> performance <strong>of</strong> <strong>the</strong> simulation.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 111tigate if <strong>the</strong> method works on <strong>the</strong> smaller amount <strong>of</strong> collected events. This is checked<strong>in</strong> <strong>the</strong> follow<strong>in</strong>g.5.6.1 Event Selection CutsThe basel<strong>in</strong>e event selection cuts developed <strong>in</strong> <strong>the</strong> Top Quark Analysis Group with<strong>in</strong><strong>the</strong> <strong>CMS</strong> collaboration are used for this part <strong>of</strong> <strong>the</strong> study. They are collected <strong>in</strong> [104]and are summarized below.• The collision data has been collected us<strong>in</strong>g different trigger requirements depend<strong>in</strong>gon <strong>the</strong> various runs performed dur<strong>in</strong>g <strong>the</strong> data tak<strong>in</strong>g period. Collision eventsare asked to have at least one reconstructed electron with a m<strong>in</strong>imal cut on <strong>the</strong>transverse momentum exceed<strong>in</strong>g 15 GeV. The cut on <strong>the</strong> electron transverse momentumbecomes tighter for some <strong>of</strong> <strong>the</strong> runs. Besides apply<strong>in</strong>g a cut on <strong>the</strong>transverse momentum <strong>of</strong> <strong>the</strong> reconstructed electron, some identification criteriaare also checked for <strong>the</strong> reconstructed electron vary<strong>in</strong>g run by run. The event iskept if <strong>the</strong>re is a reconstructed electron pass<strong>in</strong>g <strong>the</strong> p T cut and <strong>the</strong> identificationrequirements. No trigger criteria are applied on <strong>the</strong> simulated data samples.• In order to reduce <strong>the</strong> huge amount <strong>of</strong> statistics, a set <strong>of</strong> pre-selection cuts isapplied. All events are required to conta<strong>in</strong> at least one reconstructed electronwith p T > 15 GeV and four reconstructed jets, each <strong>of</strong> which has a transversemomentum exceed<strong>in</strong>g 20 GeV . The reconstructed electron and jets are asked tobe with<strong>in</strong> <strong>the</strong> tracker acceptance <strong>of</strong> |η| < 2.4.• The next cut <strong>in</strong> <strong>the</strong> series <strong>of</strong> <strong>the</strong> event selection cuts, is to apply quality cuts on<strong>the</strong> reconstructed vertex <strong>of</strong> <strong>the</strong> event. The primary vertex <strong>of</strong> <strong>the</strong> event is askedto be reconstructed close to <strong>the</strong> nom<strong>in</strong>al <strong>in</strong>teraction po<strong>in</strong>t. At least five tracksshould contribute to reconstruct <strong>the</strong> vertex, nd<strong>of</strong> > 4, and also it is required tohave |z| < 24 cm and ρ < 2 cm relative to nom<strong>in</strong>al centre <strong>of</strong> <strong>the</strong> detector. Thedef<strong>in</strong>itions <strong>of</strong> <strong>the</strong>se parameters were given <strong>in</strong> Section 4.4.1.• The event is asked to have exactly one isolated electron pass<strong>in</strong>g <strong>the</strong> criteria listedbelow.– In order to reject <strong>the</strong> electrons which are reconstructed with<strong>in</strong> jets, <strong>the</strong>candidate electron is required to be reconstructed very close to <strong>the</strong> primaryvertex. The absolute value <strong>of</strong> <strong>the</strong> difference between <strong>the</strong> z-component <strong>of</strong> <strong>the</strong>reconstructed electron and <strong>the</strong> z-component <strong>of</strong> <strong>the</strong> primary vertex <strong>of</strong> <strong>the</strong>event is required to be less than one centimeter.– The threshold cut on <strong>the</strong> transverse energy <strong>of</strong> <strong>the</strong> candidate electron is <strong>in</strong>creasedcompared to <strong>the</strong> cut applied <strong>in</strong> pre-selection step. The reconstructedelectron is needed to have E T > 30 GeV .– The threshold cut on <strong>the</strong> pseudorapidity <strong>of</strong> <strong>the</strong> candidate electron is notchanged with respect to <strong>the</strong> cut applied <strong>in</strong> pre-selection step. As an extrarequirement, <strong>the</strong> electron is asked to be reconstructed outside <strong>of</strong> <strong>the</strong>transition region between <strong>the</strong> ECAL barrel and <strong>the</strong> ECAL endcap.
112 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor– Subsequently <strong>the</strong> candidate electron is required to have an absolute impactparameter d 0 less than 0.02 cm.– Also <strong>the</strong> electron is asked to fulfill <strong>the</strong> identification criteria which weredescribed <strong>in</strong> Section 4.4.1. The cut values are <strong>the</strong> same as before so that anidentification efficiency <strong>of</strong> about 70% can be obta<strong>in</strong>ed.– The f<strong>in</strong>al criteria that <strong>the</strong> candidate electron should fulfill is applied on <strong>the</strong>relIso variable. The def<strong>in</strong>ition <strong>of</strong> <strong>the</strong> relIso variable was already given <strong>in</strong>Section 4.4.1. The electron is asked to have a relIso value less than 0.1.• The event is rejected if it conta<strong>in</strong>s any isolated muon, which is def<strong>in</strong>ed as a globalmuon with a p T exceed<strong>in</strong>g 10 GeV and located with<strong>in</strong> |η| < 2.5. The isolationcriteria is applied on <strong>the</strong> relative isolation variable and <strong>the</strong> candidate muon isdef<strong>in</strong>ed as isolated if its relIso variable is less than 0.2. In case such muon existswith<strong>in</strong> <strong>the</strong> event, that event is rejected.• In this step, <strong>the</strong> events which happen to be <strong>in</strong> <strong>the</strong> peak <strong>of</strong> <strong>the</strong> mass distribution<strong>of</strong> <strong>the</strong> Z boson are removed. The <strong>in</strong>variant mass <strong>of</strong> <strong>the</strong> selected isolated electronand an o<strong>the</strong>r loose electron is calculated. If <strong>the</strong> result<strong>in</strong>g <strong>in</strong>variant mass is with<strong>in</strong><strong>the</strong> range <strong>of</strong> 76 GeV to 106 GeV, <strong>the</strong> event is rejected. The loose electron isdef<strong>in</strong>ed us<strong>in</strong>g <strong>the</strong> same variables as given <strong>in</strong> Section 4.4.1.• There is ano<strong>the</strong>r additional requirement which is applied on <strong>the</strong> selected electron.Electrons can be <strong>in</strong>itiated from various sources <strong>in</strong> <strong>the</strong> event. For example, itis possible that <strong>the</strong> electron orig<strong>in</strong>ates from a converted photon. Photons canproduce an electron-positron pair while travers<strong>in</strong>g <strong>the</strong> tracker material. Therefore<strong>the</strong>re is a probability that <strong>the</strong> selected electron <strong>in</strong>itiates from a photon conversion.In order to reject <strong>the</strong> electrons which come out <strong>of</strong> a conversion process, somealgorithms are developped with<strong>in</strong> <strong>the</strong> <strong>CMS</strong> collaboration. The algorithm startsby look<strong>in</strong>g for a partner track reconstructed very close to <strong>the</strong> selected electron.S<strong>in</strong>ce <strong>the</strong> electron and positron <strong>in</strong>itiated from a converted photon bend <strong>in</strong> oppositedirection due to <strong>the</strong> presence <strong>of</strong> <strong>the</strong> magnetic field, it is required that <strong>the</strong> partnertrack has an opposite charge with respect to <strong>the</strong> electron track. If <strong>the</strong> distancebetween <strong>the</strong> two tracks is less than a pre-def<strong>in</strong>ed value, <strong>the</strong> selected electron isflagged as conversion. The detailed <strong>in</strong>formation <strong>of</strong> <strong>the</strong> technical implementationcan be found <strong>in</strong> [105].• The event is required to have at least 4 reconstructed calorimeter jets. The jetsare reconstructed us<strong>in</strong>g <strong>the</strong> anti-k T algorithm where <strong>the</strong> cone radius is set to be0.5 as before. It should be noted that <strong>the</strong> jets are L2 and L3 corrected. Theselected jets should fulfill <strong>the</strong> follow<strong>in</strong>g requirements.– A tighter cut on <strong>the</strong> transverse momentum <strong>of</strong> <strong>the</strong> reconstructed jets is appliedcompared to <strong>the</strong> cut which has been applied <strong>in</strong> <strong>the</strong> pre-selection step.The jets are asked to have p T > 30 GeV . The threshold cut on <strong>the</strong> pseudorapidityvariable <strong>of</strong> <strong>the</strong> jets is <strong>the</strong> same as <strong>the</strong> one applied <strong>in</strong> <strong>the</strong> pre-selectionstep, |η| < 2.4.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 113– There is a lower limit on <strong>the</strong> f EM variable, def<strong>in</strong>ed <strong>in</strong> Section 4.4.1, which isset to be 0.01. No upper limit is put on this property <strong>of</strong> <strong>the</strong> reconstructedjets.– The selected jets are required to have at least two reconstructed hits whichconta<strong>in</strong> 90% <strong>of</strong> <strong>the</strong> jet energy, n90Hits > 1. The def<strong>in</strong>ition <strong>of</strong> n90Hits wasgiven <strong>in</strong> Section 4.4.1.– The f<strong>in</strong>al cut <strong>in</strong> <strong>the</strong> series <strong>of</strong> jet identification cuts is applied on f HPD variablewhich is def<strong>in</strong>ed <strong>in</strong> Section 4.4.1. The selected jets are asked to have f HPD
114 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factort¯t t¯t W+jets Z+jets t QCD→ e + jets → o<strong>the</strong>r → l + jets → l + l − + jets → l + jets multijetsInitial 951.7 5473.2 1582609.6 154045.9 1188.8 1.9762447E+8 8.5413535E+7Pre-Selection 777.2 2076.4 7607.7 2953.6 221.0 2119019.0 1065204PV 777.1 2076.0 7606.2 2952.8 221.0 2118614 1065116One Electron 425.0 149.1 3234.6 1255.4 64.6 3384.8 8485Muon Veto 423.8 111.6 3232.6 1252.7 63.4 3384.2 8441Z Veto 420.9 106.6 3228.2 693.8 62.9 3381.2 8011Conversion Rejection 397.0 100.1 3023.2 648.4 59.3 1230.5 5118Four <strong>Jet</strong>s 194.0 30.5 159.2 38.6 10.6 65.8 402One B <strong>Jet</strong> 146.8 23.4 15.1 3.5 7.5 4.6 176Table 5.15: Table <strong>of</strong> event selection cuts. The number <strong>of</strong> events are equivalent to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 36.1pb −1 .Data
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 115#Events80706050403020tt+jets semi-ele (146.8 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)#Events3530252015101050 20 40 60 80 100 120 140 160 180 200Selected Electron P T(GeV)-2 -1 0 1 2Selected Electron ηFigure 5.23: The distributions <strong>of</strong> <strong>the</strong> selected electron transverse momentum and pseudorapidity.The collision data events are overimposed to <strong>the</strong> simulation for comparisonpurpose.#Events706050tt+jets semi-ele (144.1 entries)tt+jets o<strong>the</strong>r (23.0 entries)S<strong>in</strong>gleTop+jets (7.4 entries)Z+jets (3.4 entries)W+jets (14.3 entries)QCD (4.6 entries)Data (36.1 pb-1) (170.0 entries)403020100 0.02 0.04 0.06 0.08 0.1Selected Electron RelIsoFigure 5.24: The distribution <strong>of</strong> <strong>the</strong> relative isolation <strong>of</strong> <strong>the</strong> selected electron. Thecollision data events are overimposed to <strong>the</strong> simulation for comparison purpose.
116 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration FactorAlso, one can look at <strong>the</strong> k<strong>in</strong>ematics <strong>of</strong> <strong>the</strong> selected jets. In Figure 5.25, <strong>the</strong> distributions<strong>of</strong> <strong>the</strong> transverse momentum and pseudorapidity <strong>of</strong> <strong>the</strong> selected jets are shown.The events rema<strong>in</strong><strong>in</strong>g after apply<strong>in</strong>g all event selection cuts are taken <strong>in</strong>to account.#Events300250200150100tt+jets semi-ele (656.1 entries)tt+jets o<strong>the</strong>r (103.3 entries)S<strong>in</strong>gleTop+jets (31.9 entries)Z+jets (15.1 entries)W+jets (64.1 entries)QCD (19.5 entries)Data (36.1 pb-1) (789.0 entries)#Events14012010080604050200 50 100 150 200 250 300Selected <strong>Jet</strong>s P T(GeV)-2 -1 0 1 2<strong>Jet</strong> ηFigure 5.25: The distributions <strong>of</strong> <strong>the</strong> transverse momentum and pseudorapidity <strong>of</strong><strong>the</strong> selected jets. The collision data events are overimposed to <strong>the</strong> simulation forcomparison purpose.As it is seen from Figure 5.25, <strong>the</strong> data events follow <strong>the</strong> same behaviour as predictedby <strong>the</strong> simulation. In o<strong>the</strong>r words, <strong>the</strong> available <strong>the</strong>oretical models are able to describe<strong>the</strong> collision data for top quark topologies with this small <strong>in</strong>tegrated lum<strong>in</strong>osity.As ano<strong>the</strong>r property <strong>of</strong> <strong>the</strong> selected events, one can look at <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> ∆Rdistance between <strong>the</strong> selected electron and <strong>the</strong> closest jet among <strong>the</strong> four selected jets.S<strong>in</strong>ce <strong>the</strong> jet-electron clean<strong>in</strong>g has been already applied dur<strong>in</strong>g <strong>the</strong> event selection cuts,<strong>the</strong>n <strong>the</strong> selected jets are expected to be reconstructed <strong>in</strong> a distance greater than 0.3<strong>in</strong> (η, φ) space with respect to <strong>the</strong> selected electron. This can be clearly seen fromFigure 5.26. Aga<strong>in</strong> <strong>the</strong>re is a good agreement between collision data and simulatedevents which can be observed from <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> m<strong>in</strong>imal distance between<strong>the</strong> selected electron and a jet among <strong>the</strong> selected jets.#Events80706050tt+jets semi-ele (146.8 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)403020100 0.5 1 1.5 2 2.5 3 3.5 4Selected Electron M<strong>in</strong>imal ∆R(ele, jet)Figure 5.26: The distribution <strong>of</strong> <strong>the</strong> m<strong>in</strong>imal distance between <strong>the</strong> selected electronand a jet among <strong>the</strong> four selected jets. The collision data events are overimposed to<strong>the</strong> simulation for comparison purpose.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 117The next observable <strong>of</strong> <strong>the</strong> event which is <strong>in</strong>terest<strong>in</strong>g to check, is <strong>the</strong> distribution <strong>of</strong><strong>the</strong> miss<strong>in</strong>g transverse energy ETmiss . S<strong>in</strong>ce <strong>the</strong> source <strong>of</strong> ETmiss <strong>in</strong> <strong>the</strong> e+jets t¯t events,is <strong>the</strong> neutr<strong>in</strong>o which appears <strong>in</strong> <strong>the</strong> decay product <strong>of</strong> <strong>the</strong> W boson, <strong>the</strong>n it is expectedto obta<strong>in</strong> a clear peak around 40 GeV, as it is clearly seen from Figure 5.27.#Events605040tt+jets semi-ele (146.9 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)3020100 20 40 60 80 100 120 140 160 180 200missSelected Events E T(GeV)Figure 5.27: The distribution <strong>of</strong> <strong>the</strong> miss<strong>in</strong>g transverse energy ETmiss . The collisiondata events are overimposed to <strong>the</strong> simulation for comparison purpose.It can be concluded from Figure 5.27 that <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> miss<strong>in</strong>g transverseenergy <strong>in</strong> <strong>the</strong> collision data is comparable with <strong>the</strong> one <strong>in</strong> <strong>the</strong> simulated data.Although <strong>the</strong> four-vector <strong>of</strong> <strong>the</strong> W boson can not be fully reconstructed due to <strong>the</strong>unknown z component <strong>of</strong> <strong>the</strong> neutr<strong>in</strong>o momentum, it is still possible to reconstructano<strong>the</strong>r variable that conta<strong>in</strong>s <strong>the</strong> <strong>in</strong>formation <strong>of</strong> <strong>the</strong> mass <strong>of</strong> <strong>the</strong> W boson. Thetransverse mass <strong>of</strong> <strong>the</strong> W boson, M T , is def<strong>in</strong>ed as√M T = 2p e T Emiss T(1 − cos φ eν ),where e denotes <strong>the</strong> selected electron and φ eν is <strong>the</strong> azimuthal angle between <strong>the</strong>selected electron and <strong>the</strong> miss<strong>in</strong>g transverse energy which is measured <strong>in</strong> <strong>the</strong> transverseplane. The distribution <strong>of</strong> <strong>the</strong> M T is shown <strong>in</strong> Figure 5.28.By def<strong>in</strong>ition, if <strong>the</strong> longitud<strong>in</strong>al component <strong>of</strong> <strong>the</strong> momentum <strong>of</strong> <strong>the</strong> W boson isequal to zero, which means <strong>the</strong> W boson is produced <strong>in</strong> <strong>the</strong> transverse plane, <strong>the</strong>n <strong>the</strong>transverse mass <strong>of</strong> <strong>the</strong> W boson, as def<strong>in</strong>ed above, becomes <strong>the</strong> <strong>in</strong>variant mass <strong>of</strong> <strong>the</strong>W boson. In o<strong>the</strong>r words, <strong>the</strong> mass <strong>of</strong> <strong>the</strong> W boson is <strong>the</strong> upper limit for <strong>the</strong> transversemass variable, M T < 80.4. As it can be seen from Figure 5.28, <strong>the</strong>re is a clear peakaround <strong>the</strong> mass <strong>of</strong> <strong>the</strong> W boson. Also aga<strong>in</strong> a good agreement between <strong>the</strong> collisiondata and <strong>the</strong> simulation can be observed.There is ano<strong>the</strong>r important observable, called M3, that can be used to compare <strong>the</strong>behaviour <strong>of</strong> <strong>the</strong> data versus simulation. The M3 variable can also be used as anestimator for <strong>the</strong> mass <strong>of</strong> <strong>the</strong> reconstructed top quark. Before apply<strong>in</strong>g <strong>the</strong> MVAmethod, which makes it possible to label <strong>the</strong> selected jets and associate <strong>the</strong>m to <strong>the</strong>hadronic quarks com<strong>in</strong>g from <strong>the</strong> hadronic decay <strong>of</strong> <strong>the</strong> top quark t → Wb → q¯qb, itis still possible to construct an estimator that can be a measure <strong>of</strong> <strong>the</strong> <strong>in</strong>variant mass
118 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor#Events3025tt+jets semi-ele (146.9 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)20151050 20 40 60 80 100 120 140 160 180 200Selected Events W M T(GeV)Figure 5.28: The distribution <strong>of</strong> <strong>the</strong> transverse mass <strong>of</strong> <strong>the</strong> W boson M T . The collisiondata events are overimposed to <strong>the</strong> simulation for comparison purpose.<strong>of</strong> <strong>the</strong> top quark. The M3 variable is def<strong>in</strong>ed as <strong>the</strong> <strong>in</strong>variant mass <strong>of</strong> <strong>the</strong> three jetsamong all selected jets for which <strong>the</strong> transverse momentum <strong>of</strong> <strong>the</strong> result<strong>in</strong>g vectorialsum is <strong>the</strong> maximum. The distribution <strong>of</strong> M3 variable for all selected events can befound <strong>in</strong> Figure 5.29.#Events353025tt+jets semi-ele (146.9 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)20151050 100 200 300 400 500 600Selected Events M3 (GeV)Figure 5.29: The distribution <strong>of</strong> <strong>the</strong> M3, def<strong>in</strong>ed as <strong>the</strong> <strong>in</strong>variant mass <strong>of</strong> <strong>the</strong> three jetswhich yield <strong>the</strong> maximum transverse momentum <strong>of</strong> <strong>the</strong> vectorial sum. The collisiondata events are overimposed to <strong>the</strong> simulation for comparison purpose.The peak <strong>of</strong> <strong>the</strong> M3 distribution happens around 180 GeV which is close to <strong>the</strong>mass <strong>of</strong> <strong>the</strong> top quark that was used when simulat<strong>in</strong>g <strong>the</strong> physics events. Then it canbe concluded that <strong>in</strong>deed <strong>the</strong> M3 variable represents a good estimator <strong>of</strong> <strong>the</strong> <strong>in</strong>variantmass <strong>of</strong> <strong>the</strong> top quark. Aga<strong>in</strong> a good agreement between data and simulated eventscan be observed.Now that a good compatibility between <strong>the</strong> collision data and <strong>the</strong> simulation is observedconsider<strong>in</strong>g various properties <strong>of</strong> <strong>the</strong> selected events, one can run <strong>the</strong> MVA package <strong>in</strong>order to reconstructed <strong>the</strong> topology <strong>of</strong> <strong>the</strong> e+jet t¯t event. The same variables as<strong>in</strong>troduced <strong>in</strong> Section 5.1.4, are used as <strong>in</strong>put <strong>in</strong> order to tra<strong>in</strong> <strong>the</strong> Likelihood Ratiomethod. As already mentioned, <strong>the</strong> Likelihood Ratio method assigns a value to eachjet-parton comb<strong>in</strong>ation and returns <strong>the</strong> chosen jet-parton comb<strong>in</strong>ation correspond<strong>in</strong>g to<strong>the</strong> maximum Likelihood Ratio value among <strong>the</strong> 12 possible jet-parton comb<strong>in</strong>ations.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 119The distributions <strong>of</strong> some variables, which were used as <strong>in</strong>put to tra<strong>in</strong> <strong>the</strong> MVA method,are shown for <strong>the</strong> collision data and <strong>the</strong> simulated events <strong>in</strong> Figure 5.30. They arecalculated us<strong>in</strong>g <strong>the</strong> chosen jet-parton comb<strong>in</strong>ation which is returned by <strong>the</strong> MVAmethod.Look<strong>in</strong>g at <strong>the</strong> high-level variables which are shown <strong>in</strong> Figure 5.30, <strong>the</strong> good agreementbetween data and simulation can be observed.With <strong>the</strong> jets labeled to <strong>the</strong> partons <strong>in</strong> <strong>the</strong> t → Wb → q¯qb decay us<strong>in</strong>g <strong>the</strong> MVAmethod, it is time to reconstruct <strong>the</strong> W boson and <strong>the</strong> top quark four-vectors. Thedistributions <strong>of</strong> <strong>the</strong> mass <strong>of</strong> <strong>the</strong> W boson and <strong>the</strong> top quark are shown <strong>in</strong> Figure 5.31.Although <strong>the</strong> chosen jet comb<strong>in</strong>ation by <strong>the</strong> MVA method does not correspond to<strong>the</strong> true comb<strong>in</strong>ation, what is observed from Figure 5.31 is that <strong>the</strong> data events follow<strong>the</strong> same behaviour as <strong>the</strong> simulation and a good agreement can aga<strong>in</strong> be obta<strong>in</strong>ed.In <strong>the</strong> follow<strong>in</strong>g <strong>the</strong> selected events are fed to <strong>the</strong> k<strong>in</strong>ematic fit package. As before, <strong>the</strong>k<strong>in</strong>ematic fit is performed 41×41 times per event while scann<strong>in</strong>g a region correspond<strong>in</strong>gto a w<strong>in</strong>dow <strong>of</strong> ±40% around <strong>the</strong> non-corrected energies <strong>of</strong> <strong>the</strong> light as well as <strong>the</strong> bjets. The fit is performed <strong>in</strong> steps <strong>of</strong> 2% <strong>in</strong> both <strong>the</strong> light and b jet energy correctionsto make a two-dimensional plot <strong>of</strong> <strong>the</strong> fit probabilities. The output <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematicfit method for a typical collision data event is shown <strong>in</strong> Figure 5.32. The <strong>in</strong>formation<strong>of</strong> <strong>the</strong> event, <strong>in</strong>clud<strong>in</strong>g <strong>the</strong> run number and <strong>the</strong> event number, are quoted on <strong>the</strong> plot.It can be seen from Figure 5.32, that <strong>the</strong> fit has been converged <strong>in</strong> each po<strong>in</strong>t <strong>in</strong><strong>the</strong> two-dimensional space which is spanned <strong>in</strong> <strong>the</strong> direction <strong>of</strong> <strong>the</strong> residual light and bjet energy corrections. Also it is clearly observed that <strong>the</strong> maximum <strong>of</strong> <strong>the</strong> fit happenssomewhere <strong>in</strong> <strong>the</strong> scanned region which confirms that <strong>the</strong> mass constra<strong>in</strong>ts are maximallyfulfilled at that po<strong>in</strong>t. Therefore, <strong>the</strong> observed collision data event can representan e+jets t¯t event.The distribution <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit probability <strong>in</strong> <strong>the</strong> selected events when no correctionis applied P K<strong>in</strong>Fit (∆E l = 1, ∆E b = 1), is shown <strong>in</strong> Figure 5.33.The peak at zero justifies <strong>the</strong> need for <strong>the</strong> residual jet energy corrections. It isalso <strong>in</strong>terest<strong>in</strong>g to look at <strong>the</strong> distribution <strong>of</strong> <strong>the</strong> maximum probability returned by <strong>the</strong>k<strong>in</strong>ematic fit PK<strong>in</strong>Fit max (∆E l, ∆E b ) per event, which is shown <strong>in</strong> Figure 5.34.Aga<strong>in</strong> what can be seen from Figure 5.34 is <strong>the</strong> compatibility <strong>of</strong> <strong>the</strong> collision dataand <strong>the</strong> simulated events with<strong>in</strong> <strong>the</strong> statistical uncerta<strong>in</strong>ties.In order to apply <strong>the</strong> method <strong>of</strong> estimat<strong>in</strong>g <strong>the</strong> residual jet energy corrections on <strong>the</strong>data, <strong>the</strong> same extra event selection cuts, as described <strong>in</strong> Section 5.3, are applied on<strong>the</strong> collision data. These additional event selection cuts are summarized <strong>in</strong> Table 5.16.Accord<strong>in</strong>g to <strong>the</strong> results that were obta<strong>in</strong>ed us<strong>in</strong>g 100pb −1 <strong>of</strong> <strong>the</strong> <strong>in</strong>tegrated lum<strong>in</strong>osityshown <strong>in</strong> Table 5.3, <strong>the</strong> number <strong>of</strong> e+jets t¯t events survived, for example, <strong>the</strong>cut on <strong>the</strong> maximum probability <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit PK<strong>in</strong>Fit max > 0.98, is around 75.Therefore, <strong>in</strong> an analysis which corresponds to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 36.1pb −1 ,
120 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor#Events140120100tt+jets semi-ele (146.8 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)80604020#Events80706050403020100 0.5 1 1.5 2 2.5 3AngleThWh#Events50403020100 0.5 1 1.5 2 2.5 3AngleThBh0 0.5 1 1.5 2 2.5 3AngleThBl#Events5040#Events706050304020301020100 0.5 1 1.5 2 2.5 3AngleThEl0 0.5 1 1.5 2 2.5 3AngleBlEl#Events10080#Events454035603025402015201050 0.2 0.4 0.6 0.8 1ThPtOverSumPt0 0.2 0.4 0.6 0.8 1btagFigure 5.30: The distribution <strong>of</strong> <strong>the</strong> observables which are calculated us<strong>in</strong>g <strong>the</strong> fourvectorschosen by <strong>the</strong> Likelihood Ratio method. These variables were used for tra<strong>in</strong><strong>in</strong>g<strong>the</strong> Likelihood Ratio method. The collision data events are overimposed to <strong>the</strong> simulationfor comparison purpose.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 121#Events605040tt+jets semi-ele (146.9 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)#Events353025tt+jets semi-ele (146.9 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)30202015101050 50 100 150 200 250 300m W (GeV)0 100 200 300 400 500 600(GeV)m topFigure 5.31: The mass distributions <strong>of</strong> <strong>the</strong> W boson (left) and <strong>the</strong> top quark (right)us<strong>in</strong>g <strong>the</strong> four-vectors returned by <strong>the</strong> MVA method. The collision data events areoverimposed to <strong>the</strong> simulation for comparison purpose.∆E b1.41.31.21.110.90.80.7Data_143827_440_6070423750.60.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40∆E lFigure 5.32: The k<strong>in</strong>ematic fit output for a typical collision data event. The thirddimension conta<strong>in</strong>s <strong>the</strong> probability <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit P K<strong>in</strong>Fit (∆E l , ∆E b ).0.90.80.70.60.50.40.30.20.1#Events200180160140120tt+jets semi-ele (146.9 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)#Events21010100806014020-1100 0.2 0.4 0.6 0.8 1K<strong>in</strong>Fit Probability without corrections0 0.2 0.4 0.6 0.8 1K<strong>in</strong>Fit Probability without correctionsFigure 5.33: The distribution <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit probability when no correction isapplied shown <strong>in</strong> normal scale (left) and logarithmic scale (right). The collision dataevents are overimposed to <strong>the</strong> simulation for comparison purpose.
122 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factort¯t t¯t W+jets Z+jets t QCD→ e + jets → o<strong>the</strong>r → l + jets → l + l − + jets → l + jets multijetsDataSelection Table 5.15 146.8 23.4 15.1 3.5 7.5 4.6 176Radiation Veto 96.6 16.1 12.2 2.5 5.9 3.7 115PK<strong>in</strong>Fit max41.9 4.8 2.2 0.5 1.3 0.9 40Gap Veto 36.7 4.0 1.9 0.4 1.1 0.9 34Table 5.16: Table <strong>of</strong> extra event selection cuts. The numbers are equivalent to an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 36.1pb −1 .
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 123#Events1008060tt+jets semi-ele (146.8 entries)tt+jets o<strong>the</strong>r (23.4 entries)S<strong>in</strong>gleTop+jets (7.5 entries)Z+jets (3.5 entries)W+jets (15.1 entries)QCD (4.6 entries)Data (36.1 pb-1) (176.0 entries)#Events2101040120-1100 0.2 0.4 0.6 0.8 1max Probability <strong>of</strong> <strong>the</strong> K<strong>in</strong>Fit0 0.2 0.4 0.6 0.8 1max Probability <strong>of</strong> <strong>the</strong> K<strong>in</strong>FitFigure 5.34: The distribution <strong>of</strong> <strong>the</strong> maximum probability returned by <strong>the</strong> k<strong>in</strong>ematicfit shown <strong>in</strong> normal scale (left) and logarithmic scale (right). The collision data eventsare overimposed to <strong>the</strong> simulation for comparison purpose.one may expect to obta<strong>in</strong> a number <strong>of</strong> e+jets t¯t events which survive <strong>the</strong> PK<strong>in</strong>Fit max tobe about 36.1 × 75 ∼ 27, given that <strong>the</strong> same event selection cuts are applied. As it100can be seen from Table 5.16, <strong>the</strong>re are about 42 e+jets t¯t events rema<strong>in</strong>ed at <strong>the</strong> samelevel <strong>of</strong> <strong>the</strong> selection procedure, hence more events than what is expected. This canbe expla<strong>in</strong>ed s<strong>in</strong>ce a looser set <strong>of</strong> cuts has been applied on <strong>the</strong> reconstructed jets whenrunn<strong>in</strong>g on <strong>the</strong> data.With <strong>the</strong> 34 events <strong>in</strong> <strong>the</strong> collision data pass<strong>in</strong>g <strong>the</strong> full event selection, one can startestimat<strong>in</strong>g <strong>the</strong> residual light and b jet energy correction factors. For each <strong>of</strong> <strong>the</strong> surviveddata events, <strong>the</strong> twoi-dimensional grid <strong>of</strong> po<strong>in</strong>ts which is filled with <strong>the</strong> k<strong>in</strong>ematicfit probabilities, P K<strong>in</strong>Fit (∆E l , ∆E b ), is transformed to a χ 2 denoted as χ 2 (∆E l , ∆E b ).The χ 2 values <strong>in</strong> each po<strong>in</strong>t <strong>of</strong> <strong>the</strong> grid are summed for all <strong>of</strong> <strong>the</strong> f<strong>in</strong>al selected dataevents. The results can be found <strong>in</strong> Figure 5.35.The f<strong>in</strong>al estimated light and b jet energy corrections are determ<strong>in</strong>ed at <strong>the</strong> m<strong>in</strong>imumpo<strong>in</strong>t <strong>of</strong> <strong>the</strong> χ 2 (∆E l , ∆E b ) distribution which is shown <strong>in</strong> Figure 5.35. Theprojected χ 2 distribution <strong>in</strong> <strong>the</strong> direction <strong>of</strong> <strong>the</strong> ∆E l and ∆E b , shown <strong>in</strong> Figure 5.36,are fitted with a second-degree polynomial and <strong>the</strong> po<strong>in</strong>ts with <strong>the</strong> m<strong>in</strong>imum valuesare quoted as <strong>the</strong> light and b jet energy calibration factors, respectively. The f<strong>in</strong>alestimated residual jet energy corrections, which are obta<strong>in</strong>ed us<strong>in</strong>g <strong>the</strong> 36.1pb −1 <strong>of</strong> <strong>the</strong>accumulated collision data, are listed <strong>in</strong> Table 5.17.In order to compare <strong>the</strong> results derived from <strong>the</strong> collision data events with <strong>the</strong> resultsobta<strong>in</strong>ed from <strong>the</strong> simulation, <strong>the</strong> same procedure to determ<strong>in</strong>e <strong>the</strong> residual jetenergy corrections can be applied us<strong>in</strong>g only <strong>the</strong> f<strong>in</strong>al survived simulated events whichare listed <strong>in</strong> Table 5.16. The results <strong>of</strong> <strong>the</strong> residual jet energy corrections estimatedbased on <strong>the</strong> simulation are also shown <strong>in</strong> Table 5.17.It should be noted that <strong>the</strong> statistical uncerta<strong>in</strong>ties on <strong>the</strong> estimated jet energy correctionslisted <strong>in</strong> Table 5.17, are corrected for <strong>the</strong> non-unity width <strong>of</strong> <strong>the</strong> pull distributions.The systematical uncerta<strong>in</strong>ties on <strong>the</strong> estimated residual light and b jet energy correctionsderived from <strong>the</strong> simulation are also quoted. Clearly <strong>the</strong> uncerta<strong>in</strong>ties on <strong>the</strong>estimated results are dom<strong>in</strong>ated by <strong>the</strong> statistics. The statistical uncerta<strong>in</strong>ties can be
124 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor1.41.31.21.1∆E b10.90.80.70.60.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4∆E lFigure 5.35: The distribution <strong>of</strong> <strong>the</strong> χ 2 (∆E l , ∆E b ) which is obta<strong>in</strong>ed by <strong>the</strong> sum <strong>of</strong> <strong>the</strong>χ 2 distributions for all selected collision data events.2∆ χ7006005002∆ χ2502004001503002001001005000.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4∆ E l00.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4∆ E bFigure 5.36: The projected distribution <strong>of</strong> <strong>the</strong> χ 2 (∆E l , ∆E b ) <strong>in</strong>to <strong>the</strong> (left) ∆E l and(right) ∆E b axis which are fitted with a second-degree polynomial. The results areobta<strong>in</strong>ed for 36.1pb −1 <strong>of</strong> real proton collision data.
CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor 125simulationdata∆ε est∆ε estl(%) -8.5±2.4(stat.)±0.9(sys.) -8.6±2.7(stat.)b(%) -3.2±3.5(stat.)±1.0(sys.) -5.0±3.9(stat.)Table 5.17: The f<strong>in</strong>al estimated residual jet energy corrections obta<strong>in</strong>ed <strong>in</strong> <strong>the</strong> 36.1pb −1data set. The results found on <strong>the</strong> simulation and <strong>the</strong> collision data events, are listedseparately. The statistical uncerta<strong>in</strong>ties are corrected for <strong>the</strong> non-unity width <strong>of</strong> <strong>the</strong>pull distributions.reduced by us<strong>in</strong>g a larger sample <strong>of</strong> data events, as will be discussed <strong>in</strong> more detail <strong>in</strong><strong>the</strong> next chapter. Obviously <strong>the</strong> residual corrections are found to be equal <strong>in</strong> data andsimulation. They are <strong>in</strong> reasonable agreement with<strong>in</strong> <strong>the</strong> quoted uncerta<strong>in</strong>ties.Summary <strong>of</strong> <strong>the</strong> AnalysisThe method to estimate residual jet energy corrections has been applied on 36.1pb −1<strong>of</strong> <strong>in</strong>tegrated real proton collisions collected by <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> <strong>in</strong> 2010. The residualcorrections are estimated based on <strong>the</strong> e+jets t¯t candidates. Event selection cutshave been applied to enhance <strong>the</strong> signal over background ratio followed by impos<strong>in</strong>g <strong>the</strong>mass constra<strong>in</strong>ts <strong>of</strong> <strong>the</strong> W boson and <strong>the</strong> top quark with <strong>the</strong> use <strong>of</strong> a k<strong>in</strong>ematic fit technique.The k<strong>in</strong>ematic fit procedure provides a χ 2 value used to fill a two-dimensionaldistribution <strong>in</strong> <strong>the</strong> directions <strong>of</strong> ∆E l and ∆E b , on which <strong>the</strong> f<strong>in</strong>al estimation is based.As it is seen from Table 5.17, <strong>the</strong> estimated residual jet energy corrections which areobta<strong>in</strong>ed us<strong>in</strong>g <strong>the</strong> collision data events are compatible with <strong>the</strong> results derived basedon <strong>the</strong> simulation. Then it can be claimed that <strong>the</strong> method works reasonably on <strong>the</strong>36.1pb −1 <strong>of</strong> accumulated 2010 collision data taken by <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> at <strong>the</strong> LHC.
126 CHAPTER 5: Estimat<strong>in</strong>g <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> <strong>Scale</strong> Calibration Factor
Chapter 6ConclusionThe Standard Model <strong>of</strong> particle physics is able to describe successfully <strong>the</strong> elementaryparticles and <strong>the</strong>ir <strong>in</strong>teractions. The parameters <strong>of</strong> <strong>the</strong> model have been widely measured<strong>in</strong> diverse <strong>experiment</strong>s which made <strong>the</strong> Standard Model an excellent candidateto be a reference <strong>the</strong>ory for <strong>the</strong> description <strong>of</strong> <strong>the</strong> nature at <strong>the</strong> energy scale <strong>of</strong> Electroweak<strong>in</strong>teractions. Despite all <strong>the</strong> successes <strong>of</strong> <strong>the</strong> model, <strong>the</strong>re is a part for whichno <strong>experiment</strong>al evidence has been observed yet, be<strong>in</strong>g <strong>the</strong> Higgs mechanism which isresponsible for giv<strong>in</strong>g mass to <strong>the</strong> elementary particles. This was discussed <strong>in</strong> detail <strong>in</strong>Chapter 1.A possible solution would be provided by <strong>the</strong> Large Hadron Collider (LHC) as itma<strong>in</strong>ly aimes to search for <strong>the</strong> miss<strong>in</strong>g particle predicted by <strong>the</strong> mechanism, namely<strong>the</strong> Higgs particle. S<strong>in</strong>ce March 2010, <strong>the</strong> LHC has been collid<strong>in</strong>g beams <strong>of</strong> protons ata center <strong>of</strong> mass energy <strong>of</strong> 7 TeV and <strong>the</strong> collision data has been recorded by detectors<strong>in</strong>stalled at four po<strong>in</strong>ts around <strong>the</strong> LHC ma<strong>in</strong> r<strong>in</strong>g, such as <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>(Compact Muon Solenoid). The full description <strong>of</strong> <strong>the</strong> LHC and <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong>were given <strong>in</strong> Chapter 2.When protons collide, various objects can be produced at <strong>the</strong> f<strong>in</strong>al state, such asisolated leptons and quarks. Accord<strong>in</strong>g to <strong>the</strong> <strong>the</strong>ory, quarks carry color-charges and<strong>the</strong>refore can not be observed <strong>in</strong> free forms. They hadronize and make a cascade <strong>of</strong>hadrons, which is referred to as jets. One <strong>of</strong> <strong>the</strong> ma<strong>in</strong> challenges at hadron collidersis how to def<strong>in</strong>e a jet and how to measure its energy. A simple def<strong>in</strong>ition <strong>of</strong> a jet isto collect all <strong>the</strong> particles produced <strong>in</strong> <strong>the</strong> hadronization process with<strong>in</strong> a cone around<strong>the</strong> <strong>in</strong>itial quark. This works as a first guess but one should note that due to <strong>the</strong>existence <strong>of</strong> <strong>the</strong> magnetic field, <strong>the</strong> charged particles produced <strong>in</strong> <strong>the</strong> cascade are bentand move away from <strong>the</strong> direction <strong>of</strong> <strong>the</strong> <strong>in</strong>itial quark. This yields to a measurement <strong>of</strong><strong>the</strong> energy <strong>of</strong> <strong>the</strong> jets lower than its true energy, as <strong>the</strong> energy deposits <strong>of</strong> <strong>the</strong> escap<strong>in</strong>gparticles are not taken <strong>in</strong>to account <strong>in</strong> <strong>the</strong> jet area considered <strong>in</strong> <strong>the</strong> jet reconstructionalgorithm. Therefore, jets need to be calibrated so that <strong>the</strong>ir energies are correctedback to <strong>the</strong>ir true values. The various jet def<strong>in</strong>itions and calibration strategies whichare <strong>in</strong>troduced with<strong>in</strong> <strong>the</strong> <strong>CMS</strong> collaboration, were expla<strong>in</strong>ed <strong>in</strong> Chapter 3.S<strong>in</strong>ce jets are produced <strong>in</strong> f<strong>in</strong>al states <strong>of</strong> many channels <strong>of</strong> <strong>in</strong>terest, it is desired todeterm<strong>in</strong>e <strong>the</strong>ir energy as precise as possible. Start<strong>in</strong>g with <strong>the</strong> basel<strong>in</strong>e calibrationrecipe, <strong>the</strong> energy <strong>of</strong> <strong>the</strong> jets can be fur<strong>the</strong>r calibrated with <strong>the</strong> use <strong>of</strong> constra<strong>in</strong>ts127
128 CHAPTER 6: Conclusionwhich are imposed on <strong>the</strong> system. In this <strong>the</strong>sis, <strong>the</strong> e+jets decay channel <strong>of</strong> t¯t eventsis chosen as <strong>the</strong> signal t¯t → W ¯Wb¯b → eν e q¯qb¯b, which conta<strong>in</strong>s an isolated electron thatcan be used to suppress <strong>the</strong> QCD multijets background and at least four reconstructedjets where two <strong>of</strong> <strong>the</strong>m orig<strong>in</strong>ate from heavy quarks. Several cuts are applied to reject<strong>the</strong> huge backgrounds and to select <strong>the</strong> signal events which were discussed <strong>in</strong> detail<strong>in</strong> Chapter 4. Hav<strong>in</strong>g applied <strong>the</strong> event selection cuts listed <strong>in</strong> Table 4.6 on a sample<strong>of</strong> simulated events correspond<strong>in</strong>g to 100pb −1 <strong>of</strong> <strong>in</strong>tegrated lum<strong>in</strong>osity, <strong>the</strong>re are about308 e+jets t¯t events rema<strong>in</strong>ed result<strong>in</strong>g <strong>in</strong> a signal over background ratio <strong>of</strong> about 2.5.As <strong>the</strong>re are at least four reconstructed jets <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> e+jets t¯t events,which are <strong>in</strong>dist<strong>in</strong>guishable if no tagg<strong>in</strong>g tools are used, a first task is to label <strong>the</strong>reconstructed jets and associate <strong>the</strong>m to <strong>the</strong> hard scatter partons. This can be donewith <strong>the</strong> use <strong>of</strong> Multi-Variate Analysis techniques (MVA). A Likelihood Ratio methodis used to return a chosen jet comb<strong>in</strong>ation based on <strong>the</strong> <strong>in</strong>formation which is obta<strong>in</strong>eddur<strong>in</strong>g <strong>the</strong> tra<strong>in</strong><strong>in</strong>g <strong>of</strong> <strong>the</strong> MVA method. The chosen jet comb<strong>in</strong>ation by <strong>the</strong> MVAmethod is not all <strong>the</strong> time <strong>the</strong> true comb<strong>in</strong>ation which represents <strong>the</strong> comb<strong>in</strong>ationwhere jets are matched to <strong>the</strong> partons. The performance <strong>of</strong> <strong>the</strong> MVA method stronglydepends on <strong>the</strong> <strong>in</strong>put variables which are chosen for <strong>the</strong> tra<strong>in</strong><strong>in</strong>g. A full discription <strong>of</strong><strong>the</strong> MVA method and a list <strong>of</strong> variables used for <strong>the</strong> tra<strong>in</strong><strong>in</strong>g <strong>of</strong> <strong>the</strong> Likelihood Ratiomethod were given <strong>in</strong> Chapter 5. It was found that <strong>in</strong> around 17% <strong>of</strong> <strong>the</strong> selected e+jetst¯t events, <strong>the</strong> Likelihood Ratio method is able to return <strong>the</strong> correct comb<strong>in</strong>ation where<strong>the</strong> three reconstructed jets match to <strong>the</strong> quarks from <strong>the</strong> hadronic top quark decayt → Wb → q¯qb. This is a great improvement compared to <strong>the</strong> random choice <strong>of</strong> <strong>the</strong>correct jet-parton comb<strong>in</strong>ation which is equal to 8%.Consider<strong>in</strong>g <strong>the</strong> hadronic branch <strong>of</strong> <strong>the</strong> signal t → Wb → q¯qb, <strong>the</strong> masses <strong>of</strong> <strong>the</strong>W boson and <strong>the</strong> top quark can be used to constra<strong>in</strong> <strong>the</strong> system by requir<strong>in</strong>g <strong>the</strong>four-vectors <strong>of</strong> <strong>the</strong> reconstructed jets <strong>of</strong> <strong>the</strong> chosen jet comb<strong>in</strong>ation to fulfill <strong>the</strong> massconstra<strong>in</strong>ts. The procedure <strong>of</strong> apply<strong>in</strong>g constra<strong>in</strong>ts is performed with <strong>the</strong> use <strong>of</strong> ak<strong>in</strong>ematic fit, which was expla<strong>in</strong>ed <strong>in</strong> detail <strong>in</strong> Chapter 5. The t¯t events are excellentcandidates for <strong>the</strong> jet calibration purpose because, <strong>in</strong> addition to <strong>the</strong> high rate <strong>of</strong> toppair production, <strong>the</strong>y conta<strong>in</strong> both <strong>the</strong> light and <strong>the</strong> b jets <strong>in</strong> <strong>the</strong>ir f<strong>in</strong>al state whichyields to obta<strong>in</strong> calibration factors for different flavours <strong>of</strong> <strong>the</strong> quarks. Start<strong>in</strong>g with<strong>the</strong> three jets <strong>in</strong> <strong>the</strong> hadronic branch <strong>of</strong> <strong>the</strong> selected event t → Wb → q¯qb, which arenow assigned to <strong>the</strong> hard scatter partons by means <strong>of</strong> <strong>the</strong> MVA method, <strong>the</strong> energy<strong>of</strong> <strong>the</strong> jets are altered with<strong>in</strong> a w<strong>in</strong>dow <strong>of</strong> ±40% around <strong>the</strong> reconstructed energies.Changes are made <strong>in</strong> steps <strong>of</strong> 2%. As a result, a two-dimensional space is spanned<strong>in</strong> <strong>the</strong> directions <strong>of</strong> <strong>the</strong> light jet energy scale calibration factor ∆E l and <strong>the</strong> b jetenergy scale calibration factor ∆E b . For each selected event and <strong>in</strong> each po<strong>in</strong>t <strong>of</strong> <strong>the</strong>two-dimensional grid, <strong>the</strong> k<strong>in</strong>ematic fit returns a probability P K<strong>in</strong>Fit (∆E l , ∆E b ), whichreflects how likely <strong>the</strong> hypo<strong>the</strong>sis <strong>of</strong> <strong>the</strong> three jets to orig<strong>in</strong>ate from <strong>the</strong> top quark <strong>in</strong> <strong>the</strong>chosen jet comb<strong>in</strong>ation, is true. The probability <strong>of</strong> <strong>the</strong> k<strong>in</strong>ematic fit can be transformed<strong>in</strong>to a χ 2 (∆E l , ∆E b ) space. The <strong>in</strong>formation <strong>of</strong> all <strong>the</strong> selected events is comb<strong>in</strong>ed by<strong>the</strong> sum <strong>of</strong> <strong>the</strong> χ 2 distribution <strong>of</strong> <strong>the</strong> <strong>in</strong>dividual events, which makes a two-dimensionalparabola. The resulted parabola can be projected <strong>in</strong> each <strong>of</strong> <strong>the</strong> two dimensions and<strong>the</strong> m<strong>in</strong>imum <strong>of</strong> each <strong>of</strong> <strong>the</strong> two projected parabolas, refers to an estimation <strong>of</strong> <strong>the</strong>residual jet energy scale calibration factors for both light and b quark jets.
CHAPTER 6: Conclusion 129The method to estimate <strong>the</strong> residual jet energy scale calibration factors has beentested on simulated proton collisions and <strong>the</strong> performance <strong>of</strong> <strong>the</strong> method has beendiscussed <strong>in</strong> Chapter 5. The same method has also been applied on <strong>the</strong> real protoncollisions which have been collected by <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> <strong>in</strong> 2010, correspond<strong>in</strong>gto an <strong>in</strong>tegrated lum<strong>in</strong>osity <strong>of</strong> 36pb −1 <strong>of</strong> proton-proton collisions at a center <strong>of</strong> massenergy <strong>of</strong> 7 TeV which was described <strong>in</strong> detail <strong>in</strong> Chapter 5. The estimated residualjet energy corrections result<strong>in</strong>g from <strong>the</strong> application <strong>of</strong> <strong>the</strong> method on <strong>the</strong> simulationas well as on <strong>the</strong> real proton collisions are summarized <strong>in</strong> Table 6.1. The statisticaluncerta<strong>in</strong>ties quoted on <strong>the</strong> estimated results are corrected for <strong>the</strong> non-unity width<strong>of</strong> <strong>the</strong> pull distributions. The complete description <strong>of</strong> <strong>the</strong> statistical properties <strong>of</strong> <strong>the</strong>estimator <strong>in</strong>clud<strong>in</strong>g <strong>the</strong> pull variable can be found <strong>in</strong> Chapter 5.simulationdata∆ε est∆ε estl(%) -8.1±2.4(stat.)±0.9(sys.) -8.6±2.7(stat.)b(%) -3.2±3.5(stat.)±1.0(sys.) -5.0±3.9(stat.)Table 6.1: The f<strong>in</strong>al estimated residual jet energy corrections obta<strong>in</strong>ed <strong>in</strong> <strong>the</strong> 36.1pb −1data set. The statistical uncerta<strong>in</strong>ties are corrected for <strong>the</strong> non-unity width <strong>of</strong> <strong>the</strong> pulldistributions.Obviously <strong>the</strong> uncerta<strong>in</strong>ties on <strong>the</strong> estimated results are dom<strong>in</strong>ated by <strong>the</strong> limitedstatistics which can be solved when larger samples <strong>of</strong> data events become available. TheLHC runs also <strong>in</strong> 2011 at a center <strong>of</strong> mass energy <strong>of</strong> 7 TeV. So far, more than 2fb −1<strong>of</strong> proton collisions have been collected by <strong>the</strong> <strong>CMS</strong> detector. One can <strong>the</strong>n apply <strong>the</strong>method on this larger amount <strong>of</strong> real proton collisions to improve <strong>the</strong> results, where <strong>the</strong>√statistical uncerta<strong>in</strong>ties would be reduced by a factor <strong>of</strong>362000∼ 0.1, as <strong>the</strong> statisticaluncerta<strong>in</strong>ty δ(stat.), decreases while <strong>the</strong> number <strong>of</strong> events N, <strong>in</strong>creases δ(stat.) ∼ √ 1N.This can be understood by look<strong>in</strong>g at <strong>the</strong> plots <strong>in</strong> Figure 6.1, where <strong>the</strong> statisticalas well as <strong>the</strong> systematical uncerta<strong>in</strong>ties on <strong>the</strong> light and b jet energy corrections areshown as a function <strong>of</strong> an <strong>in</strong>creas<strong>in</strong>g <strong>in</strong>tegrated lum<strong>in</strong>osity.In Figure 6.1, <strong>the</strong> straight l<strong>in</strong>es represent <strong>the</strong> systematical uncerta<strong>in</strong>ties on <strong>the</strong> lightand b jet energy corrections. It is clear that, at some po<strong>in</strong>t which <strong>in</strong> case <strong>of</strong> for example<strong>the</strong> light jet energy correction corresponds to ∫ Ldt ∼ 300pb −1 , <strong>the</strong> systematical uncerta<strong>in</strong>tystarts to be dom<strong>in</strong>ated. Therefore, <strong>the</strong> total uncerta<strong>in</strong>ty would reach a certa<strong>in</strong>amount which is determ<strong>in</strong>ed by <strong>the</strong> systematical limits and <strong>the</strong> f<strong>in</strong>al estimation <strong>of</strong> <strong>the</strong>jet energy corrections cannot be determ<strong>in</strong>ed better than this precesion. This is <strong>the</strong>limit <strong>of</strong> <strong>the</strong> applied calibration technique where <strong>the</strong> precision cannot be improved evenif larger samples <strong>of</strong> collision data are recorded and used <strong>in</strong> <strong>the</strong> analysis. Also <strong>the</strong> bestestimate <strong>of</strong> m top from o<strong>the</strong>r measurements provides an uncerta<strong>in</strong>ty which translates<strong>in</strong>to a limit on <strong>the</strong> precision <strong>of</strong> ∆E b . The bias <strong>of</strong> <strong>the</strong> estimators has an <strong>in</strong>tr<strong>in</strong>sic uncerta<strong>in</strong>tydue to <strong>the</strong> fits which are applied to determ<strong>in</strong>e <strong>the</strong> expected jet energy correction
130 CHAPTER 6: ConclusionUnc.(∆ε l )(%)765Unc.(∆ε b )(%)10846324120 100 200 300 400 500 600 700 800 900 1000 -1Lum<strong>in</strong>osity (pb )0 100 200 300 400 500 600 700 800 900 1000 -1Lum<strong>in</strong>osity (pb )Figure 6.1: The absolute statistical (curved l<strong>in</strong>e) as well as <strong>the</strong> systematical (straightl<strong>in</strong>e) uncerta<strong>in</strong>ties on <strong>the</strong> (left) light and (right) b jet energy corrections as a function<strong>of</strong> <strong>the</strong> lum<strong>in</strong>osity. From some po<strong>in</strong>t onward <strong>the</strong> systematical uncerta<strong>in</strong>ties dom<strong>in</strong>ate<strong>the</strong> statistical uncerta<strong>in</strong>ties.values on simulation, also this is a limit <strong>of</strong> <strong>the</strong> precision <strong>of</strong> ∆E.Although <strong>the</strong> method is able to estimate <strong>the</strong> jet energy corrections <strong>in</strong>clusively, ithas also <strong>the</strong> capability to provide a differential estimation <strong>of</strong> <strong>the</strong> jet energy corrections.When larger amounts <strong>of</strong> statistics are recorded, <strong>the</strong> jets can be categorized <strong>in</strong> terms<strong>of</strong> <strong>the</strong>ir k<strong>in</strong>ematics, such as pseudorapidity η or transverse momentum p T . Then, <strong>the</strong>same method can be applied on <strong>the</strong> events which populate a particular b<strong>in</strong> <strong>of</strong> <strong>the</strong>sek<strong>in</strong>ematic variables and consequently <strong>the</strong> estimated results would be obta<strong>in</strong>ed for thatregion <strong>of</strong> <strong>the</strong> phase space. This yields to obta<strong>in</strong> a differential estimation <strong>of</strong> <strong>the</strong> jetenergy corrections. With 36pb −1 where a low number <strong>of</strong> events survived <strong>the</strong> selectioncuts, a differential estimation is not feasible. Us<strong>in</strong>g a higher amount <strong>of</strong> statistics <strong>of</strong> forexample 2fb −1 , <strong>the</strong>re are sufficient events rema<strong>in</strong><strong>in</strong>g <strong>in</strong> <strong>the</strong> various regions <strong>of</strong> <strong>the</strong> phasespace that makes <strong>the</strong> differential method applicable and would result <strong>in</strong> mean<strong>in</strong>gfulestimations. More detailed <strong>in</strong>formation on this aspect can be found <strong>in</strong> [106].The calibration method expla<strong>in</strong>ed <strong>in</strong> this <strong>the</strong>sis will help <strong>the</strong> search for new physicsphenomena at <strong>the</strong> LHC, because when jets are better calibrated, <strong>the</strong> k<strong>in</strong>ematic topology<strong>of</strong> <strong>the</strong> new physics events can be better reconstructed and differentiated from <strong>the</strong>backgrounds. Moreover, s<strong>in</strong>ce <strong>in</strong> almost all <strong>the</strong> analyses, <strong>the</strong> jet energy scale calibrationis <strong>the</strong> ma<strong>in</strong> source <strong>of</strong> systematical uncerta<strong>in</strong>ty, a precise determ<strong>in</strong>ation <strong>of</strong> <strong>the</strong> energy<strong>of</strong> <strong>the</strong> jets would also reduce <strong>the</strong> uncerta<strong>in</strong>ties orig<strong>in</strong>at<strong>in</strong>g from <strong>the</strong> jet energy scalecalibration factors and makes it possible to perform precision tests even at hadroncolliders like <strong>the</strong> LHC.
Bibliography[1] M. E. Pesk<strong>in</strong> and D. V. Schroeder, An Introduction to Quantum Field Theory,USA: Addison-Wesley (1995) 842 p.[2] F. Halzen and A. D. Mart<strong>in</strong>, Quarks and Leptons: an Introductory Course <strong>in</strong>Modern Particle Physics, New York, Usa: Wiley (1984) 396p.[3] CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 73 (1994) 225–231.[4] CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 74 (1995) 2626–2631.[5] D0 Collaboration, S. Abachi et al., Phys. Rev. Lett. 74 (1995) 2632–2637.[6] CDF Collaboration, F. Abe et al., Phys. Rev. D51 (1995) 4623–4637.[7] CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 75 (1995) 3997.[8] The Tevatron Collider, http://www-bdnew.fnal.gov/tevatron/.[9] F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321–322.[10] P. Higgs, Phys. Rev. Lett. 13 (1964) 508–509.[11] G. Guralnik, C. Hagen, and T. Kibble, Phys. Rev. Lett. 13 (1964) 585–587.[12] S. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2 (Oct, 1970)1285–1292.[13] ALEPH Collaboration, Phys. Rept. 427 (2006) 257–454.[14] D. Schaile and P. M. Zerwas, Phys. Rev. D 45 (May, 1992) 3262–3265.[15] LEP Collaboration, A Comb<strong>in</strong>ation <strong>of</strong> prelim<strong>in</strong>ary electroweak measurementsand constra<strong>in</strong>ts on <strong>the</strong> standard model, hep-ex/0412015.[16] ALEPH Collaboration, J. Alcaraz et al., A Comb<strong>in</strong>ation <strong>of</strong> prelim<strong>in</strong>aryelectroweak measurements and constra<strong>in</strong>ts on <strong>the</strong> standard model,hep-ex/0612034.[17] T. E. W. Group, CDF, and D. Collaborations, Comb<strong>in</strong>ation <strong>of</strong> CDF and DOresults on <strong>the</strong> mass <strong>of</strong> <strong>the</strong> top quark us<strong>in</strong>g up to 5.8 fb-1 <strong>of</strong> data, 1107.5255.131
132 BIBLIOGRAPHY[18] The LEP Electroweak Work<strong>in</strong>g Grouphttp://lepewwg.web.cern.ch/LEPEWWG/.[19] LEP Work<strong>in</strong>g Group for Higgs boson searches Collaboration, R. Barate et al.,Phys. Lett. B565 (2003) 61–75.[20] S. F. Novaes, Standard model: An Introduction, hep-ph/0001283.[21] T. CDF, D. Collaborations, t. T. N. Phenomena, and H. W. Group, Comb<strong>in</strong>edCDF and D0 Upper Limits on Standard Model Higgs Boson Production with upto 8.6 fb-1 <strong>of</strong> Data, 1107.5518.[22] The CDF <strong>experiment</strong>, http://www-cdf.fnal.gov/physics/physics.html.[23] The D0 <strong>experiment</strong>, http://www-d0.fnal.gov/.[24] M. Beneke et al., Top quark physics, hep-ph/0003033.[25] U. Baur, M. Buice, and L. H. Orr, Phys. Rev. D64 (2001) 094019.[26] N. Cabibbo, Phys. Rev. Lett. 10 (1963), no. CERN-TH-342, 531–533.[27] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652–657.[28] Particle Data Group, http://pdg.lbl.gov/.[29] O. S. Brun<strong>in</strong>g, P. Collier, P. Lebrun, S. Myers, R. Ostojic, J. Poole, andP. Proudlock, LHC Design Report. CERN, Geneva, 2004.[30] S. Myers and E. Picasso, Contemp. Phys. 31 (1990) 387–403.[31] <strong>CMS</strong> Collaboration, D. G. d’Enterria, (Ed. ) et al., J. Phys. G34 (2007)2307–2455.[32] L. Evans and P. Bryant, JINST 3 S08001 (2008).[33] M. Benedikt, P. Collier, V. Mertens, J. Poole, and K. Sch<strong>in</strong>dl, LHC DesignReport. CERN, Geneva, 2004.[34] Particle Data Group Collaboration, K. Hagiwara et al., Phys. Rev. D66 (2002)010001.[35] <strong>CMS</strong> Lum<strong>in</strong>osity - Public Results,https://twiki.cern.ch/twiki/b<strong>in</strong>/view/<strong>CMS</strong>Public/LumiPublicResults2010.[36] CERN, Chamonix 2010 Workshop on LHC Performance. CERN, Geneva, 2010.[37] CERN, Evian 2010 Workshop on LHC Commission<strong>in</strong>g. CERN, Geneva, 2010.[38] The <strong>CMS</strong> Collaboration, JINST 3 S08004 (2008).[39] The ATLAS Collaboration, JINST 3 S08003 (2008).
BIBLIOGRAPHY 133[40] The ALICE Collaboration, JINST 3 S08002 (2008).[41] The LHCb Collaboration, JINST 3 S08005 (2008).[42] The LHCf Collaboration, JINST 3 S08006 (2008).[43] The TOTEM Collaboration, JINST 3 S08007 (2008).[44] M. Della Negra, L. Foà, A. Hervé, and A. Petrilli, <strong>CMS</strong> Physics TechnicalDesign Report Volume I: Detector Performance and S<strong>of</strong>tware. Technical DesignReport <strong>CMS</strong>. CERN, Geneva, 2006.[45] V. Karimaki, The <strong>CMS</strong> tracker system project: Technical Design Report.Technical Design Report <strong>CMS</strong>. CERN, Geneva, 1997.[46] The <strong>CMS</strong> tracker: addendum to <strong>the</strong> Technical Design Report. Technical DesignReport <strong>CMS</strong>. CERN, Geneva, 2000.[47] The <strong>CMS</strong> electromagnetic calorimeter project: Technical Design Report.Technical Design Report <strong>CMS</strong>. CERN, Geneva, 1997.[48] P. Bloch, R. Brown, P. Lecoq, and H. Rykaczewski, Changes to <strong>CMS</strong> ECALelectronics: addendum to <strong>the</strong> Technical Design Report. Technical Design Report<strong>CMS</strong>. CERN, Geneva, 2002.[49] F. Ferri and P. Govoni, The <strong>CMS</strong> Electromagnetic Calorimeter Pre-calibrationwith Cosmic Rays and Test Beam Electrons, Tech. Rep. <strong>CMS</strong>-CR-2007-012.CERN-<strong>CMS</strong>-CR-2007-012, CERN, Geneva, 2007.[50] P. Adzic et al., JINST 2 (2007) P04004.[51] The <strong>CMS</strong> hadron calorimeter project: Technical Design Report. TechnicalDesign Report <strong>CMS</strong>. CERN, Geneva, 1997.[52] <strong>CMS</strong> HCAL Collaboration, S. Abdull<strong>in</strong> et al., Eur. Phys. J. C55 (2008)159–171.[53] The <strong>CMS</strong> muon project: Technical Design Report. Technical Design Report<strong>CMS</strong>. CERN, Geneva, 1997.[54] G. L. Bayatyan et al., <strong>CMS</strong> TriDAS project: Technical Design Report; 1, <strong>the</strong>trigger systems. Technical Design Report <strong>CMS</strong>.[55] S. Cittol<strong>in</strong>, A. Rcz, and P. Sphicas, <strong>CMS</strong> trigger and data-acquisition project:Technical Design Report. Technical Design Report <strong>CMS</strong>. CERN, Geneva, 2002.[56] G. L. Bayatyan, M. Della Negra, Fo, A. Herv, and A. Petrilli, <strong>CMS</strong> comput<strong>in</strong>g:Technical Design Report. Technical Design Report <strong>CMS</strong>. CERN, Geneva, 2005.Submitted on 31 May 2005.[57] C. Eck et al., LHC comput<strong>in</strong>g Grid: Technical Design Report. Version 1.06 (20Jun 2005). Technical Design Report LCG. CERN, Geneva, 2005.
134 BIBLIOGRAPHY[58] I. Antcheva et al., Comput. Phys. Commun. 180 (2009), no. 12, 2499–2512.[59] Physics Analysis Toolkit (PAT),https://twiki.cern.ch/twiki/b<strong>in</strong>/view/<strong>CMS</strong>Public/WorkBookPAT.[60] <strong>CMS</strong> Belgian T2 Wiki, http://mon.iihe.ac.be/trac/t2b/wiki.[61] T. S. Virdee, Calorimetry. oai:cds.cern.ch:529421, Tech. Rep.<strong>CMS</strong>-CR-1998-026, CERN, Geneva, 1999.[62] <strong>CMS</strong> Collaboration, The <strong>CMS</strong> Collaboration, <strong>CMS</strong> AN-2008/097 (2008).[63] S. Baffioni, C. Charlot, F. Ferri, D. Futyan, P. Meridiani, I. Puljak, C. Rovelli,R. Salerno, and Y. Sirois, Electron reconstruction <strong>in</strong> <strong>CMS</strong>, Tech. Rep.<strong>CMS</strong>-NOTE-2006-040. CERN-<strong>CMS</strong>-NOTE-2006-040, CERN, Geneva, Feb,2006.[64] E. Meschi, T. Monteiro, C. Seez, and P. Vikas, Electron Reconstruction <strong>in</strong> <strong>the</strong><strong>CMS</strong> Electromagnetic Calorimeter, Tech. Rep. <strong>CMS</strong>-NOTE-2001-034, CERN,Geneva, Jun, 2001.[65] M. Pioppi et al., Electron Pre-identification <strong>in</strong> <strong>the</strong> Particle Flow framework,<strong>CMS</strong> AN-2008/032.[66] W. Adam et al., Electron Reconstruction <strong>in</strong> <strong>CMS</strong>, <strong>CMS</strong> AN-2009/164.[67] The <strong>CMS</strong> Collaboration, <strong>Jet</strong> Performance <strong>in</strong> pp Collisions at 7 TeV, <strong>CMS</strong> PASJME-10-003.[68] V. Chetluru et al., <strong>Jet</strong> Reconstruction Performance at <strong>CMS</strong>, <strong>CMS</strong>AN-2009/067.[69] The <strong>CMS</strong> Collaboration, Performance <strong>of</strong> <strong>Jet</strong> Algorithms <strong>in</strong> <strong>CMS</strong>, <strong>CMS</strong> PASJME-07-003.[70] A. Heister et al., <strong>Measurement</strong> <strong>of</strong> <strong>Jet</strong>s with <strong>the</strong> <strong>CMS</strong> Detector at <strong>the</strong> LHC, <strong>CMS</strong>AN-2005/053.[71] P. Schieferdecker et al., Performance <strong>of</strong> <strong>Jet</strong> Algorithms <strong>in</strong> <strong>CMS</strong>, <strong>CMS</strong>AN-2008/001.[72] M. Cacciari, G. P. Salam, and G. Soyez, JHEP 04 (2008) 063.[73] The <strong>CMS</strong> Collaboration, Plans for <strong>Jet</strong> <strong>Energy</strong> Corrections at <strong>CMS</strong>, <strong>CMS</strong> PASJME-07-002.[74] S. Esen et al., Plans for <strong>Jet</strong> <strong>Energy</strong> Corrections at <strong>CMS</strong>, <strong>CMS</strong> AN-2007/055.[75] R. Harris et al., MC Truth L2 & L3 Factorized <strong>Jet</strong> Corrections at <strong>CMS</strong>, <strong>CMS</strong>AN-2008/003.
BIBLIOGRAPHY 135[76] The <strong>CMS</strong> Collaboration, <strong>Measurement</strong> <strong>of</strong> <strong>the</strong> <strong>Jet</strong> <strong>Energy</strong> Resolutions and <strong>Jet</strong>Reconstruction Efficiency at <strong>CMS</strong>, <strong>CMS</strong> PAS JME-09-007.[77] S. Beauceron et al., <strong>Jet</strong>s-Electron Clean<strong>in</strong>g via CaloTowers Selection, <strong>CMS</strong>AN-2009/149.[78] S. Esen et al., E missTPerformance <strong>in</strong> <strong>CMS</strong>, <strong>CMS</strong> AN-2007/041.[79] M. Dobbs and J. B. Hansen, Computer Physics Communications 134 (2001),no. 1, 41 – 46.[80] R. K. Ellis, W. J. Stirl<strong>in</strong>g, and B. R. Webber, QCD and Collider Physics,Cambridge University Press (2003) 435 p.[81] The Coord<strong>in</strong>ated Theoretical-Experimental Project on QCD,http://www.phys.psu.edu/ ∼ cteq/.[82] Onl<strong>in</strong>e Plott<strong>in</strong>g and Calculation <strong>of</strong> Parton Distributions,http://hepdata.cedar.ac.uk/pdf/pdf3.html.[83] T. Stelzer and W. F. Long, Comput. Phys. Commun. 81 (1994) 357.[84] M. Mangano et al., ALPGEN, a generator for hard multiparton processes <strong>in</strong>hadronic collisions, arXiv:hep-ph/0206293.[85] Y. L. Dokshitzer, Sov. J. Phys. JETP 46 (1977) 641.[86] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438.[87] G. Altarelli and P. G, Nucl. Phys. B126 (1977) 298.[88] T. Sjöstrand, S. Mrenna, and P. Skands, JHEP 05 (2006) 026.[89] G. Corcella et al., JHEP 01 (2001) 010.[90] S. Catani, F. Krauss, R. Kuhn, and B. R. Webber, JHEP 0111 (2001) 063.[91] F. Krauss, JHEP 0208 (2002) 015.[92] S. Hoche, F. Krauss, N. Lavesson, L. Lönnblad, M. Mangano, A. Schalicke, andS. Schumann, Match<strong>in</strong>g Parton Showers and Matrix Elements,arXiv:hep-ph/0602031.[93] M. Mangano, M. Moretti, F. Picc<strong>in</strong><strong>in</strong>i, and M. Treccani, Match<strong>in</strong>g matrixelements and shower evolution for top-quark production <strong>in</strong> hadronic collisions,arXiv:hep-ph/0611129.[94] P. Perez, W W cross section and W branch<strong>in</strong>g ratios at √ s = 183 GeV.<strong>Measurement</strong> <strong>of</strong> WW cross section and W branch<strong>in</strong>g ratios at LEP2 with <strong>the</strong>ALEPH detector, Tech. Rep. CERN-OPEN-99-310. CERN-ALEPH-98-019,CERN, Geneva, Jul, 1998.
136 BIBLIOGRAPHY[95] A. Ealet, WW cross sections and W branch<strong>in</strong>g ratios,.[96] Standard Model Cross Sections for <strong>CMS</strong> at 7 TeV,https://twiki.cern.ch/twiki/b<strong>in</strong>/viewauth/<strong>CMS</strong>/StandardModelCrossSections.[97] Reference Page for Top-Quark Analyses: Samples and Cross-Sections,https://twiki.cern.ch/twiki/b<strong>in</strong>/view/<strong>CMS</strong>/CrossSections 3XSeries.[98] Electron Identification Based on Simple Cuts,https://twiki.cern.ch/twiki/b<strong>in</strong>/view/<strong>CMS</strong>/SimpleCutBasedEleID.[99] A. Rizzi, F. Palla, and G. Segneri, Track impact parameter based b-tagg<strong>in</strong>g with<strong>CMS</strong>, Tech. Rep. <strong>CMS</strong>-NOTE-2006-019. CERN-<strong>CMS</strong>-NOTE-2006-019, CERN,Geneva, Jan, 2006.[100] TMVA Toolkit for Multivariate Data Analysis with ROOT,http://tmva.sourceforge.net/.[101] Top Quark Analysis Framework page: <strong>Jet</strong>-Parton Association Algorithms,https://twiki.cern.ch/twiki/b<strong>in</strong>/view/<strong>CMS</strong>Public/SWGuideTQAFLayer2#<strong>Jet</strong> Parton Asso[102] J. M. Bauer, K<strong>in</strong>ematic Fit for <strong>the</strong> Radiative Bhabha Calibration <strong>of</strong> BaBar’sElectromagnetic Calorimeter, Tech. Rep. hep-ex/0011019. SLAC-PUB-8650.UMS-HEP-2000-09, SLAC, Stanford, CA, Nov, 2000.[103] J. D’Hondt, S. Lowette, et al., Fitt<strong>in</strong>g <strong>of</strong> Event Topologies with ExternalK<strong>in</strong>ematic Constra<strong>in</strong>ts <strong>in</strong> <strong>CMS</strong>, <strong>CMS</strong> Note-2006/023.[104] Reference Selection for <strong>the</strong> t¯t → e + jets Channel,https://twiki.cern.ch/twiki/b<strong>in</strong>/viewauth/<strong>CMS</strong>/TopLeptonPlus<strong>Jet</strong>sRefSel el.[105] F. Bostock et al., Plans for measurement <strong>of</strong> t¯t cross section <strong>in</strong> <strong>the</strong> semileptonicelectron channel with 20 pb −1 <strong>of</strong> data at √ s <strong>of</strong> 10 TeV, <strong>CMS</strong> AN-2009/075.[106] P. Van Mulders and J. D’Hondt, Calibration <strong>of</strong> <strong>the</strong> jet energy scale us<strong>in</strong>g topquark events at <strong>the</strong> LHC. PhD <strong>the</strong>sis, Vrije U., Brussels & Antwerp U., Brussels& Antwerp, 2010. Presented on Jun 2010.
Summary<strong>Jet</strong>s appear <strong>in</strong> <strong>the</strong> f<strong>in</strong>al state <strong>of</strong> many processes <strong>of</strong> <strong>in</strong>terest at <strong>the</strong> Large Hadron Colliderand a precise measurement <strong>of</strong> <strong>the</strong>ir energies is <strong>of</strong> great importance when search<strong>in</strong>g fornew physics phenomena. S<strong>in</strong>ce <strong>the</strong> reconstruction <strong>of</strong> <strong>the</strong> jet objects at hadron collidersis a challeng<strong>in</strong>g task, calibration strategies are developed to compensate for <strong>the</strong> nonperfection<strong>of</strong> <strong>the</strong> jet energy measurements. Historically, jets at hadron colliders arecalibrated us<strong>in</strong>g QCD dijet events. The resolution on <strong>the</strong> jet energy measurement canbe fur<strong>the</strong>r improved consider<strong>in</strong>g <strong>the</strong> physics constra<strong>in</strong>ts which can be imposed on anobserved top quark pair system. Due to <strong>the</strong> precise measurement <strong>of</strong> <strong>the</strong> top quarkmass at <strong>the</strong> Tevatron collider, top quarks can be used as a calibration tool to improve<strong>the</strong> performance <strong>of</strong> <strong>the</strong> diverse analyses. Impos<strong>in</strong>g <strong>the</strong> top quark and W boson massconstra<strong>in</strong>ts on <strong>the</strong> system conta<strong>in</strong><strong>in</strong>g a top quark decay<strong>in</strong>g to jets, and alter<strong>in</strong>g <strong>the</strong> jetenergies with<strong>in</strong> <strong>the</strong>ir uncerta<strong>in</strong>ties, one can obta<strong>in</strong> <strong>the</strong> residual jet energy calibrationfactors.In this <strong>the</strong>sis <strong>the</strong> method to estimate <strong>the</strong> residual jet energy corrections is appliedon <strong>the</strong> events where top quark pairs are produced <strong>in</strong> proton-proton collisions and subsequentlydecay to an isolated electron and at least four reconstructed jets and miss<strong>in</strong>gtransverse energy, hence pp → t¯t → W + W − b¯b → eν e q¯qb¯b processes. The leptonic side<strong>of</strong> <strong>the</strong> event topology is ma<strong>in</strong>ly used to suppress <strong>the</strong> QCD multijet background eventsand <strong>the</strong> hadronic side is used for <strong>the</strong> calibration purpose. Event selection cuts areapplied to enhance <strong>the</strong> ratio <strong>of</strong> signal to background events to about S/B ≃ 2.5, whichis obta<strong>in</strong>ed based on a sample <strong>of</strong> simulated proton collisions correspond<strong>in</strong>g to 100pb −1<strong>of</strong> <strong>in</strong>tegrated lum<strong>in</strong>osity at a center <strong>of</strong> mass energy <strong>of</strong> 7 TeV. In order to reconstruct<strong>the</strong> topology <strong>of</strong> <strong>the</strong> hadronic decay <strong>of</strong> <strong>the</strong> top quark <strong>in</strong> e+jets t¯t events, it is requiredto assign <strong>the</strong> three jets aris<strong>in</strong>g from <strong>the</strong> hadronic top to <strong>the</strong> three partons. A Multi-Variate Analysis method is used to perform <strong>the</strong> association <strong>of</strong> <strong>the</strong> jets to <strong>the</strong> quarks. Inabout 17% <strong>of</strong> e+jets t¯t events, <strong>the</strong> MVA method is able to return <strong>the</strong> correct comb<strong>in</strong>ation,where a match<strong>in</strong>g between jets and partons is found. Apply<strong>in</strong>g an event-by-eventk<strong>in</strong>ematic fitt<strong>in</strong>g technique with Lagrange multipliers, <strong>the</strong> masses <strong>of</strong> <strong>the</strong> W boson and<strong>of</strong> <strong>the</strong> top quark are used to constra<strong>in</strong> <strong>the</strong> e+jets t¯t events <strong>in</strong> <strong>the</strong> hadronic branch. Thek<strong>in</strong>ematic fit is applied <strong>in</strong> a two-dimensional range <strong>of</strong> residual jet energy corrections(∆E l , ∆E b ). The po<strong>in</strong>ts (∆ε l est , ∆ε b est ) m<strong>in</strong>imiz<strong>in</strong>g <strong>the</strong> χ 2 <strong>of</strong> <strong>the</strong> fit are <strong>the</strong> estimatedresidual jet energy corrections. The precision <strong>of</strong> <strong>the</strong> method to estimate <strong>the</strong> residualjet energy corrections is limited to a systematical uncerta<strong>in</strong>ty <strong>of</strong> about 0.9% for lightquark jets and <strong>of</strong> about 1.0% for b quark jets. The ma<strong>in</strong> source <strong>of</strong> <strong>the</strong> systematicaluncerta<strong>in</strong>ty on <strong>the</strong> estimation <strong>of</strong> <strong>the</strong> b quark jet energy correction orig<strong>in</strong>ates from <strong>the</strong>uncerta<strong>in</strong>ty on <strong>the</strong> measured top quark mass provided by <strong>the</strong> Tevatron collider.137
138 SUMMARYAlso <strong>the</strong> method <strong>of</strong> estimat<strong>in</strong>g <strong>the</strong> residual jet energy calibration factors has beenapplied on <strong>the</strong> real proton collisions recorded by <strong>the</strong> <strong>CMS</strong> <strong>experiment</strong> <strong>in</strong> 2010. Theestimated results on <strong>the</strong> simulation and real proton collisions are summarized below.There is a good agreement between simulation and data and <strong>the</strong> residual correctionsare found to be equal.simulationdata∆ε est∆ε estl(%) -8.1±2.4(stat.)±0.9(sys.) -8.6±2.7(stat.)b(%) -3.2±3.5(stat.)±1.0(sys.) -5.0±3.9(stat.)Table 6.2: The f<strong>in</strong>al estimated residual jet energy corrections obta<strong>in</strong>ed <strong>in</strong> <strong>the</strong> 36.1pb −1data set. The statistical uncerta<strong>in</strong>ties are corrected for <strong>the</strong> non-unity width <strong>of</strong> <strong>the</strong> pulldistributions.Compared to <strong>the</strong> residual calibration factors provided by o<strong>the</strong>r data driven methodssuch as QCD dijet events, <strong>the</strong> ga<strong>in</strong> that can be obta<strong>in</strong>ed by apply<strong>in</strong>g <strong>the</strong> calibrationmethod <strong>of</strong> this <strong>the</strong>sis is tw<strong>of</strong>old. A first one is that <strong>the</strong> calibration method described <strong>in</strong>this analysis is able to provide <strong>the</strong> residual jet energy corrections which is differentiatedwith respect to <strong>the</strong> flavour <strong>of</strong> quarks, namely light quark jets and b quark jets. A secondone is that, while <strong>the</strong> methods based on QCD dijet events determ<strong>in</strong>e <strong>the</strong> residual jetenergy corrections and correct <strong>the</strong> energy <strong>of</strong> <strong>the</strong> jets back to <strong>the</strong> generator level jets,<strong>the</strong> method which is described <strong>in</strong> this <strong>the</strong>sis is able to calibrate <strong>the</strong> energy <strong>of</strong> <strong>the</strong> jetsback to <strong>the</strong> energy <strong>of</strong> <strong>the</strong> <strong>in</strong>itiat<strong>in</strong>g quarks, <strong>in</strong>stead <strong>of</strong> <strong>the</strong> generator level jets.