A Measurement of the Top Quark Mass in the Lepton + Jets Channel using the Ideogram Technique at DØ by Vivek S. Parihar M.Sc., IIT Kanpur; India, 2002 A dissertation submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in The Department Of Physics at Brown University PROVIDENCE, RHODE ISLAND May 2013 © Copyright 2013 by Vivek S. Parihar This dissertation by Vivek S. Parihar is accepted in its present form by The Department Of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy Date Professor Ulrich Heintz, Advisor Recommended to the Graduate Council Date Professor David Cutts, Reader Date Professor Meenakshi Narain, Reader Approved by the Graduate Council Date Dean Peter M. Weber Dean of the Graduate School iii Acknowledgments The work in this dissertation has been made possible due to the invaluable guidance of Prof. Ulrich Heintz. I ’am extremely thankful for his constant and untiring supervision. I would also like to thank the Brown University Physics Department for providing me support at various stages. I ’am thankful to my reading committee. Much of the work in this thesis was carried out at Fermilab and I would particularly like to thank several of my DØ colleagues. At the seminal stages of the project, I had received great help from Dr. Michele Weber and Dr. Pieter Houben. Throughout the course of my stay at Fermilab, fellow students, postdocs and officemates have extended their helping hands to me. I would like to thank a few (not in any particular order): Dr. DooKee Cho, Dr. Daniel Boline, Dr. Amitabha Das, Dr. Lidija Zivkovic, Dr. Shabnam Jabeen and Dr. Zhenyu Ye. The facilities provided at DØ were conducive for research and I ’am thankful to the management for that. It would not be justified, if I didn’t thank my teachers and mentors who have taught me in various capacities and a special mention in this regards would go to Prof. Allan Widom and Prof. Yogendra Srivastava. In the same spirit, I would also like to thank the Boston University Physics Department for accomodating me as a graduate student from Jan 2007-Aug 2009. Last but not the least, I ’am grateful to my family members for their love, patience and support. I wish to acknowledge my late father, who passed away last year, for his life has been an inspiration to me. iv Abstract of “ A Measurement of the Top Quark Mass in the Lepton + Jets Channel using the Ideogram Technique at DØ,” by Vivek S. Parihar, Ph.D., Brown University, May 2013 This thesis presents a measurement of the mass of the top quark. The top quarks are √ produced through proton-antiproton collisions with s=1.96 TeV at the Fermilab Teva- tron collider using the DØ detector. The method employed for the measurement is called the Ideogram Technique. The events used in the measurement are selected such that the top quark decay signatures (as seen in the detector) are at least one lepton (electron or muon), four or more jets with at least one of them tagged as originating from a b-quark and missing transverse momentum. These events are then fitted using a kinematic fitter. The event-by-event likelihood is calculated using templates that depend on the kinematically reconstructed top quark mass. The events are weighted by their probability to be signal or background, using topological information. The measurement of the top quark mass is 175.7±1.98 GeV /c2 . CONTENTS Acknowledgments iv 1 Top Quark: Theoretical Perspective 1 1.1 Standard Model of Particle Physics . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Electroweak Theory . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Mass Generation mechanism . . . . . . . . . . . . . . . . . . . . . 7 1.2 Top Quark’s role in precision electroweak analyses . . . . . . . . . . . . . 8 1.3 Top quark production and decay . . . . . . . . . . . . . . . . . . . . . . . 10 2 The Experiment 15 2.1 Fermilab Tevatron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The DØ Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Tracking Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.3 Muon System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.4 Luminosity Monitoring System (LM) . . . . . . . . . . . . . . . . 35 2.3 Trigger System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.3.1 Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.2 Level 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.3 Level 3 and DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3 Particle Reconstruction: Algorithms and Identification 45 3.1 Reconstruction in Central tracker . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Reconstruction in Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Muon Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Vertex Reconstruction : Primary . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5.1 Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5.2 Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 v 3.5.3 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.6 Jet Energy scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.6.1 Sample Dependent JES . . . . . . . . . . . . . . . . . . . . . . . 65 3.7 Missing Transverse Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.8 b-jet identification (tagging) and Secondary Vertices . . . . . . . . . . . . . 70 4 Event Selection 73 4.1 Monte Carlo event generation . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3 Event selection and background modeling . . . . . . . . . . . . . . . . . . 78 4.4 Data and Monte Carlo comparison . . . . . . . . . . . . . . . . . . . . . . 81 4.4.1 Data-MC comparison based on b-tagging . . . . . . . . . . . . . . 85 5 Kinematic Fitting for Top Mass 91 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Parton Level Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.2.1 Light Quark Corrections . . . . . . . . . . . . . . . . . . . . . . . 94 5.2.2 b-quark Corrections . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3 Resolution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3.1 Light Quark Resolution . . . . . . . . . . . . . . . . . . . . . . . 103 5.3.2 b-quark resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3.3 Lepton Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.4 HitFit: The kinematic fitter . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4.1 Performance of the fitter . . . . . . . . . . . . . . . . . . . . . . . 128 6 Ideogram Method 131 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.2 Likelihood Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.2.0.1 Top Mass templates . . . . . . . . . . . . . . . . . . . . 136 6.3 W+jets Background shapes . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.4 Likelihood of a sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.6 Ensemble tests with parton matched t t¯ events . . . . . . . . . . . . . . . . 146 6.7 Ensemble tests and calibration with signal and background events . . . . . 154 6.7.1 Channel-by-channel calibrations . . . . . . . . . . . . . . . . . . . 156 6.7.2 Residual Calibrations for each channel . . . . . . . . . . . . . . . . 162 6.7.3 Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7 Results and Systematic uncertainties 174 7.1 Data Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.2 Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.2.1 Detector Response . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.2.2 Production Systematics . . . . . . . . . . . . . . . . . . . . . . . . 180 vi 7.2.2.1 Signal Modeling . . . . . . . . . . . . . . . . . . . . . . 180 7.2.2.2 Background Modeling . . . . . . . . . . . . . . . . . . . 182 7.2.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 8 Conclusions and Perspectives 187 A Parton Energy distributions for light quarks 192 B Parton Energy distributions for b-quarks 203 C Pull distributions 214 vii LIST OF TABLES 4.1 Composition of the final data sample in e + jets and µ + jets channel . . . . . 81 5.1 Fit parameters for E parton = p0 + p1 × E jet + p2 × E 2jet + p3 × E 3jet + p4 × E 4jet + p5 × E 5jet + p6 × E 6jet for light quarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2 Fit parameters for E parton = p0 + p1 × E jet + p2 × E 2jet + p3 × E 3jet + p4 × E 4jet + p5 × E 5jet + p6 × E 6jet for b-quarks. 99 5.3 Parameters of polynomial fit to σ (E parton ) vs. E jet for light quarks . . . . . . . 103 5.4 Parameters of polynomial fit to σ (η parton − η jet ) vs. E jet for light quarks. . . . . 107 5.5 Parameters of polynomial fit to σ (φ parton − φ jet ) vs. E jet for light quarks. . . . . 108 5.6 Parameters of polynomial fit to σ (E parton ) vs. E jet for b-quarks. . . . . . . . . . 111 5.7 Parameters of polynomial fit to σ (φ parton − φ jet ) vs. E jet for light quarks. . . . . 114 5.8 Parameters of polynomial fit to σ (φ parton − φ jet ) vs. E jet for light quarks. . . . . 115 6.1 Parametrization of combinatorial shapes with respect to generator level top mass.141 6.2 Event yields in different channels . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.3 Offset in JES f itted for three different values of JESinput in the four channels . . 148 6.4 mtf itted peak values for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in e + jets, = 2tags channel. . . . . . . . . . . . . . . . . . . . . . . . . 157 6.5 mtf itted peak values for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in µ + jets, = 2tags channel. . . . . . . . . . . . . . . . . . . . . . . . 157 6.6 mtf itted peak values for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in e + jets, 1tag channel. . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.7 mtf itted peak values for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in µ + jets, 1tag channel. . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.8 JES f itted peak values for six different values of JESinput for an input top quark mass of 172.5 GeV, across the four channels. . . . . . . . . . . . . . . . . . . 160 6.9 First Calibration parameters for mtf itted and JES f itted in the four channels. . . . 161 6.10 mtf itted peak values (second calibration) for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in e + jets, = 2tags channel. . . . . . . . . . . . . 164 6.11 mtf itted peak values (second calibration) for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in µ + jets, = 2tags channel. . . . . . . . . . . . . 164 6.12 mtf itted peak values (second calibration) for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in e + jets, 1tag channel. . . . . . . . . . . . . . . 165 6.13 mtf itted peak values (second calibration) for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in µ + jets, 1tag channel. . . . . . . . . . . . . . . 165 6.15 Residual Calibration parameters for mtf itted and JES f itted in the four channels. . 165 viii 6.14 JES f itted (second calibration) peak values for six different values of JESinput for an input top quark mass of 172.5 GeV, across the four channels. . . . . . . . 166 6.16 ’2D’ fitted mass for the combination of four channels. . . . . . . . . . . . . . . 170 6.18 Fitted JES for the combination of four channels. . . . . . . . . . . . . . . . . . 171 6.17 ’1D’ fitted mass for the combination of four channels. . . . . . . . . . . . . . . 171 7.1 Summary of mt measurements in different channels along with the combination of all 4 channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.2 Various sources of systematic uncertainties to the top quark mass measurement. 185 8.1 Values of mtpole , with their 68% confidence interval uncertainties, extracted for different predictions of σt t¯. The result assumes that the top quark mass in the simulation is equal to the pole mass of the top quark propagator. (Ref. [1]) . . . 191 ix LIST OF FIGURES 1.1 Constituents of matter and the force carriers in the Standard Model. . . . . . . 3 1.2 One loop corrections to gluon propagator. . . . . . . . . . . . . . . . . . . . . 5 1.3 Running of strong coupling constant.(Ref. [2]) . . . . . . . . . . . . . . . . . 6 1.4 Virtual top quark loops contributing to W and Z bosons. . . . . . . . . . . . . . 9 1.5 Virtual Higgs-boson loops contributing to the W and Z bosons. . . . . . . . . . 9 1.6 Lines of constant Higgs mass on a plot of MW vs mt The green ellipse shows 68 % CL direct measurement of the most recent MW and mt . (Ref. [3]) . . . . . 10 1.7 Feynman diagrams for t t¯ production via qq¯ annihilation and gg fusion. . . . . . 10 1.8 t t¯ decay topologies based on the branching fractions of the W boson. . . . . . . 14 2.1 Layout of the accelerator complex at Fermilab . . . . . . . . . . . . . . . . . . 17 2.2 Schematic layout of the DØ detector . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Difference in definition of physics and detector η . . . . . . . . . . . . . . . . 20 2.4 Zoomed in view of the DØ tracking detector . . . . . . . . . . . . . . . . . . . 21 2.5 A representation of Silicon Microstrip Tracker (SMT) . . . . . . . . . . . . . . 23 2.6 rφ view of CFT clear fiber waveguides . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Forward and Central Preshower detectors . . . . . . . . . . . . . . . . . . . . 27 2.8 Cross sectional view of a calorimeter cell . . . . . . . . . . . . . . . . . . . . 29 2.9 View of the central and end cap assemblies of the DØ calorimeter. . . . . . . . 31 2.10 Various components of the central and forward muon system at DØ . . . . . . 34 2.11 DØ luminosity monitoring system showing the location (le f t) and the geome- try of LM counters with PMTs (red dots) . . . . . . . . . . . . . . . . . . . . . 35 2.12 Basic trigger road map (Ref.[4]) . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.13 Detailed structure of trigger framework . . . . . . . . . . . . . . . . . . . . . 37 2.14 STT pattern recognition algorithm . . . . . . . . . . . . . . . . . . . . . . . . 41 2.15 SMT clustering algorithm, implemented by STC . . . . . . . . . . . . . . . . . 42 2.16 Data flow in Level 3 data acquisition system (L3DAQ) . . . . . . . . . . . . . 44 3.1 EˆO as a function of |η detector jet | for jets with cone size of R = 0.5. Different number of multiple interactions (MI) are shown . . . . . . . . . . . . . . . . . 58 3.2 MPF response in CC, for jets with Rcone = 0.5, as a function of E 0 . Also, shown is the relative difference of fit with the data points. . . . . . . . . . . . . . . . . 60 3.3 Relative MPF response, Fη for different values of E 0 in γ + jet events. . . . . . 61 3.4 Internal closure of η − dependent corrections for Rcone = 0.5 jets in γ + jet sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.5 Relative uncertainty on Fη . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 x 3.6 Data-to-MC closure as a function of pT in different |η detector jet | bins for Rcone = 0.5 jets in γ + jet sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.7 Jet Energy Scale correction factor as a function of pseudo-rapidity for a few measured values of ET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.8 Rh tuning in MC for |η detector jet | <0.4 for tight photon selection (left) and reversed track isolation (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.9 Correction factor for MC-data difference in jet response for different flavor of jets. [Light quarks(top), gluon(middle) and b-quarks(bottom)] . . . . . . . . . 69 3.10 Neural network b-tagger output for b-jets and light flavor jets. . . . . . . . . . 72 4.1 Stages of simulation, describing a hadron-hadron collisions by MC event gen- erators [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 0 4.2 Data-MC comparison for topological variables : Aplanarity, Centrality, KTmin , 6ET in e + jets channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Data-MC comparison for lepton pT , leading jet pT , η, φ in e + jets channel . . 83 4.4 W transverse mass and invariant top mass distributions . . . . . . . . . . . . . 83 0 4.5 Data-MC comparison for topological variables : Aplanarity, Centrality, KTmin , 6ET in µ + jets channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.6 Data-MC comparison for lepton pT , leading jet pT , η, φ in µ + jets channel . 85 4.7 W transverse mass and invariant top mass distributions in µ + jets channel . . 85 0 4.8 Data-MC comparison of Aplanarity, Centrality, KTmin ,6 ET , W transverse mass and t t¯ invariant mass in e + jets, ≥ 2tags channel . . . . . . . . . . . . . . . . 87 0 4.9 Data-MC comparison of Aplanarity, Centrality, KTmin ,6 ET , W transverse mass and t t¯ invariant mass in µ + jets, ≥ 2tags channel . . . . . . . . . . . . . . . . 88 0 4.10 Data-MC comparison of Aplanarity, Centrality, KTmin ,6 ET , W transverse mass and t t¯ invariant mass in e + jets, 1tag channel . . . . . . . . . . . . . . . . . . 89 0 4.11 Data-MC comparison of Aplanarity, Centrality, KTmin ,6 ET , W transverse mass and t t¯ invariant mass in µ + jets, 1tag channel . . . . . . . . . . . . . . . . . 90 5.1 E parton vs. E jet & (E parton −E jet ) vs. E jet for light quark jets in Region 1, |η| ∈ [0, 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 E parton vs. E jet & (E parton −E jet ) vs. E jet for light quark jets in Region 2, |η| ∈ [0.5, 1.0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.3 E parton vs. E jet & (E parton −E jet ) vs. E jet for light quark jets in Region 3, |η| ∈ [1.0, 1.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.4 E parton vs. E jet & (E parton −E jet ) vs. E jet for light quark jets in Region 4, |η| ∈ [1.5, 2.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.5 E parton vs. E jet & (E parton −E jet ) vs. E jet for b-quark jets in Region 1, |η| ∈ [0, 0.5) 99 5.6 E parton vs. E jet & (E parton −E jet ) vs. E jet for b-quark jets in Region 2, |η| ∈ [0.5, 1.0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.7 E parton vs. E jet & (E parton −E jet ) vs. E jet for b-quark jets in Region 3, |η| ∈ [1.0, 1.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.8 E parton vs. E jet & (E parton −E jet ) vs. E jet for b-quark jets in Region 4, |η| ∈ [1.5, 2.5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.9 Energy resolution for light quarks in the detector η regions 1 and 2, plotted versus E jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.10 Energy resolution for light quarks in the detector η regions 3 and 4, plotted versus E jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 xi 5.11 Distribution of the difference of angular variables at the parton and jet levels for light quarks. (a) pseudo-rapidity (b) azimuth . . . . . . . . . . . . . . . . . . . . . . . . 106 5.12 Pseudo-rapidity (η) resolution vs. E jet for light quarks in detector |η| regions 1 and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.13 Pseudo-rapidity (η) resolution vs. E jet for light quarks in four detector |η| regions 3 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.14 Azimuthal (φ ) resolution vs. E jet for light quarks in detector |η| regions 1 and 2. 109 5.15 Azimuthal (φ ) resolution vs. E jet for light quarks in detector |η| regions 3 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.16 Energy resolution vs. E jet for b-quarks in the detector η regions 1 and 2. . . . . 111 5.17 Energy resolution vs. E jet for b-quarks in the detector η regions 3 and 4. . . . . 112 5.18 Distribution of the difference in angular variables for b-quark at parton and jet levels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.19 Pseudo-rapidity (η) resolution for b-quarks in detector |η| regions 1 and 2, plotted versus E jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.20 Pseudo-rapidity (η) resolution for b-quarks in detector |η| regions 3 and 4, plotted versus E jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.21 Azimuthal (φ ) resolution for b-quarks in detector |η| regions 1 and 2, plotted versus E jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.22 Azimuthal (φ ) resolution vs. E jet for b-quarks in detector |η| regions 3 and 4, plotted versus E jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.23 Distribution of differences in electron energy, pseudo-rapidity and azimuthal variables between detector and parton levels. . . . . . . . . . . . . . . . . . . . 119 5.24 Resolutions of electron energy, pseudo-rapidity and azimuthal variables with respect to electron energy observed in the detector region 1 for t t¯ → e + jets events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.25 Resolutions of electron energy, pseudo-rapidity and azimuthal variables with respect to electron energy observed in the detector region 1 for t t¯ → e + jets events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.26 Resolutions of muon energy, pseudo-rapidity and azimuthal variables with re- spect to muon energy observed in the detector region 1 for t t¯ → µ + jets events. 124 5.27 Resolutions of muon energy, pseudo-rapidity and azimuthal variables with re- spect to muon energy observed in the detector region 1 for t t¯ → µ + jets events. 125 5.28 Parton level corrected hadronic W mass distributions for various top mass sam- ples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.29 Fitted top mass distributions for top mass samples generated with different in- put mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.1 Comparison of the “topological discriminant” values for signal (red) and back- grounds (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2 Purity versus Discriminant fits for e+jets, µ+jets events in two different tagging bins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.3 Right and wrong combination shapes for different generator level top quark masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.4 Right and wrong combination shapes for different generator level top quark masses or 1 b-tagged events. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.5 Weighted wrong combination shapes fitted with double Gaussian. . . . . . . . 140 6.6 A sample BG shape for µ + jets events at JES = 1 . . . . . . . . . . . . . . . 142 xii 6.7 mtf itted distributions with input top quark mass of 172.5 GeV /c2 at JES f itted for JESinput = 1.0, for different channels (parton matched t t¯ events) . . . . . . . . 149 6.8 JES f itted distribution for JESinput = 0.94 in parton matched t t¯ events with input mass of 172.5 GeV /c2 , in different channels . . . . . . . . . . . . . . . . . . . 150 6.9 JES f itted distribution for JESinput = 1.0 in parton matched t t¯ events with input mass of 172.5 GeV /c2 , in different channels . . . . . . . . . . . . . . . . . . . 151 6.10 JES f itted distribution for JESinput = 1.06 in parton matched t t¯ events with input mass of 172.5 GeV /c2 , in different channels . . . . . . . . . . . . . . . . . . . 152 6.11 mtf itted calibration at JES f itted for the nominal JES as input in the parton matched t t¯ events, for different channels . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.12 JES f itted calibrations with JESinput scaled by 0% (top), −6% (middle) and +6% (bottom) for parton matched t t¯ events with different input masses. The y-axis of all the plots represents (JES f itted − 1.0), while the x-axis represents (mtgenerated − 170) GeV /c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.13 A 2D representation of the sum of −2 ln(Likelihood) for 3000 ensembles in one channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.14 (Fitted mass − 172.5) GeV /c2 vs. (Generated mass − 172.5) GeV /c2 for dif- ferent channels for ensembles consisting of signal and background events. . . . 159 6.15 (Fitted JES − 1.0) vs. (Input JES − 1.0) for different channels . . . . . . . . 161 6.16 (Fitted mass − 172.5) GeV vs. (Generated mass − 172.5) GeV for different channels at JES = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.17 (Fitted JES − 1.0) vs. (Input JES − 1.0) for different channels . . . . . . . . 167 6.18 Interpolation of likelihood values inside a bin through bi-linear transformation or nearest neighbor sampling. {Ref.[6] } . . . . . . . . . . . . . . . . . . . . . 170 6.19 JES f itted , 2D-mtf itted and 1D-mtf itted distributions for nominal input JES and an input top quark mass of 172.5 GeV. . . . . . . . . . . . . . . . . . . . . . . . 171 6.20 Pull distributions of JES f itted , mtf itted (2D) and mtf itted (1D) for nominal input JES and an input top quark mass of 172.5 GeV. . . . . . . . . . . . . . . . . . 172 6.21 Pull width vs. generator level top mass for the ’2D’ and ’1D’ cases (Input JES = nominal JES). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.22 Final mass calibrations after combining the four channels. . . . . . . . . . . . 173 7.1 Residual JES parametrization as a function of pT of jets in various |η det | for the t t¯ MC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.1 The twelve input measurements of the top quark mass from the Tevatron col- lider experiments along with the resulting combined value. The grey region corresponds to ±0.94 GeV /c2 . (Ref. [7]) . . . . . . . . . . . . . . . . . . . . . 190 8.2 Measured σt t¯ and theoretical NLO+NNLL and aproximate NNLO calculations of σt t¯ as a function of top quark pole mass, assuming that the mass of top quark in simulation is equal to the pole mass. (Ref. [1]) . . . . . . . . . . . . . . . . 191 xiii CHAPTER One Top Quark: Theoretical Perspective 1.1 Standard Model of Particle Physics The Standard Model (SM) entails our current understanding of particle physics. It has been laid on foundation of experimental discoveries and theoretical advancement and relates to 1 the observed particles and their interactions. It describes two classes of matter: spin 2 fermions and spin 1 bosons (Fig. 1.1) that carry three fundamental forces: the electro- magnetic force, the weak force and the strong force with the first two forces unified into electroweak force. The SM represents gauge group SU(3) × SU(2) × U(1). A gauge the- ory is one that possesses invariance under a set of local transformations i.e. transformations whose parameters are space-time dependent. The Standard Model has been phenomeno- logically successful but it does not describe a complete theory of fundamental interactions e.g. gravity is not included, neither is the viable dark matter that possesses the properties deduced from observational cosmology etc. Fermions are identified as quarks and leptons as shown in Fig. 1.1. The quarks carry color charge and thus interact via the strong force (as well as electromagnetic and weak force). The six quarks (and corresponding anti quarks) are classified into three genera- 1 2 tions with increasing mass for each generation. Every generation contains one quark with charge (+2/3) e and one quark with charge (−1/3) e, which are ofter referred to as the up type and down type respectively. Leptons are also classified into three generations with each generation having a charged lepton with charge (-e) and a corresponding uncharged neutrino. The masses of the particles (determined experimentally) are also shown in the Fig. 1.1. The top quark is the heaviest of all the particles. The mass of the quarks is dependent on the renormalization scheme. This is discussed at the end of this dissertation. 3 Figure 1.1: Constituents of matter and the force carriers in the Standard Model. 1.1.1 Electroweak Theory The electroweak force is represented by SU(2)×U(1)Y gauge group. This is spontaneously broken and results in separate electromagnetic and weak forces. The process of sponta- neously broken symmetry is discussed in Sec. 1.1.3. The carrier of electromagnetic force is the massless spin-1 particle γ (photon). This feature of the photon makes electromagnetic 4 force, a long range force. The carriers of weak force are the W ± and Z bosons, which are massive with masses 80.385± 0.015 GeV /c2 and 91.1876±0.0021 GeV /c2 respectively. This feature of the carriers makes the weak force short range. Weak force can change the flavor of quarks. The amplitude of mixing between quark flavors mediated by the W boson is described by the CKM matrix (Cabibbo-Kobayashi-Maskawa). The terms in the CKM matrix are complex and the matrix is required to be unitary in the Standard Model. The matrix is parametrized by three mixing angles and a CP-violating phase. 1.1.2 Quantum Chromodynamics Quantum Chromodynamics (QCD) is the theory of strong interactions. It is a non-abelian theory with the gauge group representation of SU(3). The quarks are described by a field ψi where i = 1, 2, 3. The quantum number associated with the label i is called color. The eight gauge bosons which are represented by the generators of the gauge group are called gluons. These are taken to be carriers of the strong force. The coupling for strong interactions is the QCD gauge coupling, gs . It is usually redefined in terms of αs as : g2s αs = (1.1) 4π This coupling is energy dependent and due to the non-abelian nature of QCD, αs de- creases as energy increases. Thus, when higher order perturbative calculations are per- formed in the Feynman formalism, the loop diagrams have the effect of “dressing” the cou- plings. This is shown in Fig. 1.2 for one-loop corrections to the gluon propagator. These diagrams contain ultraviolet divergences and need to be renormalized e.g. by subtracting at some renormalization scale µ. If the squared momenta of all the particles coming into the vertex is Q2 , then 5 αs (Q2 ) = α(µ 2 ) − αs (µ 2 )2 β0 ln(Q2 /µ 2 ) + ....... (1.2) The coefficient β0 is calculated to be : 11Nc − 2n f βo = (1.3) 12π where, Nc is the number of colors (=3), n f is the number of active flavors i.e. the number of flavors whose mass threshold is below the momentum scale Q2 . A more precise analysis shows that the effective coupling obeys the differential equation ∂ αs (Q2 ) 2 = β (αs (Q2 )) (1.4) ∂ ln(Q ) The solution of which depends on the boundary value and is often quoted at the Z boson mass αs (MZ2 ) = 0.1161 ± 0.004 This is one of the free parameters of the SM. The running of αs is shown in Fig. 1.3. Figure 1.2: One loop corrections to gluon propagator. 6 Figure 1.3: Running of strong coupling constant.(Ref. [2]) Gauge invariance requires that the gauge coupling for the interaction between gluons must be exactly the same as gauge coupling for interaction between quarks and gluons. Converse to the situation described above, the coupling grows when we go to larger distances and therefore the complicated system of gluon exchanges, which leads to the binding of quarks (anti quarks) inside hadrons leads to stronger and stronger binding as the quarks are pulled apart. This process is called confinement. Thus the only free particles that can be observed at macroscopic distances from each other are color singlets. This process is not completely understood, nevertheless Monte Carlo programs have been developed that can simulate the hadronization in such a way that the results of short distance perturbative calculations at the level of quarks and gluons can be confronted with experiments. 7 1.1.3 Mass Generation mechanism In 1950’s Yang and Mills extended the idea of gauge invariance to local non-abelian trans- formations such as SU(2). In this case, one needs a set of massless gauge fields. In order that such a gauge theory could be applied to weak interactions, particles that transform into one another under weak interactions were looked at e.g. u and d quark. The three gauge bosons were interpreted as the W ± and Z bosons that mediate the weak interactions. The weak interactions were known to be short range and had to be mediated by massive vector bosons, whereas Yang-Mills fields are required to be massless in order to preserve the gauge invariance. This paradox was resolved by the electroweak symmetry breaking mechanism (Ref. [8, 9, 10]) and is often referred to as the mechanism of spontaneous sym- metry breaking.In this scenario, one starts with a theory that possesses the required gauge invariance but the ground state of the theory is not invariant under gauge transformations. The breaking of the invariance arises in the quantization of the theory whereas the La- grangian contains terms that are gauge invariant. One of the consequences of this is that the gauge bosons acquire a mass. It also lead to a renormalizable theory whereby infinities that arise out of the higher order calculations can be reabsorbed into the parameters of the Lagrangian. Another consequence of this mechanism is the existence of a scalar (spin 0) particle, the Higgs boson. Thus the union of QCD and the electroweak gauge theory is known as the Standard Model. It has nineteen fundamental parameters, most of which are associated with the masses of the gauge bosons, the quarks, leptons and Higgs. These are not all indepen- dent and because of the renormalizable nature of the theory, perturbative calculations can be performed at higher order that can predict the cross sections and decay rates for both strongly and weakly interacting particles. The predictions and data are becoming more and more precise, making the tests of the Standard Model increasingly stringent. 8 1.2 Top Quark’s role in precision electroweak analyses The gauge, matter and Higgs sectors of the Standard Model depend on five parameters: the three gauge couplings, gS , g, g’ and the Higgs-field vacuum expectation value and self interaction, v and λ . At tree level, all electroweak quantities depend on three of the parameters g, g’ and v. The three best measured electroweak quantities to determine these parameters at tree level are : 1 g2 g02 1 α= = 4π g2 + g02 137.03599976 1 GF = √ = 1.16637 × 10−5 GeV −2 2v2 1p 2 MZ = g + g02 v = 91.186 GeV 2 The value of α is extracted from low energy experiments, GF is extracted from the muon lifetime and MZ is measured from LEP. At tree level, the W boson mass can be written as : s ! 2 1 1 4πα MW = g2 v2 = MZ2 1 + 1− √ (1.5) 4 2 2GF MZ2 2 MW 2 ≡ 1− By defining sW , which is also referred to as the on-shell definition of sin2 θW MZ2 , one can write MW at tree level as : √πα 2 2GF MW = 2 (1.6) sW At one loop level, this is modified to : 9 √πα 2 2GF MW = 2 (1 − 4r) (1.7) sW where 4r contains the one-loop corrections. The top quark contributes to 4r via the one loop diagrams shown in Fig. 1.4 which contribute to the W and Z masses: 3GF m2 1 (4r)top ≈ − √ t 2 (1.8) 8 2π 2 tW 2 ≡ tan2 θ . This one-loop correction depends quadratically on the top quark where tW W mass. The Higgs boson also contributes to 4r via the loop diagrams shown in Fig. 1.5 11GF MZ2 cW 2 m2h (4r)Higgs ≈ √ ln 2 (1.9) 24 2π 2 MZ Figure 1.4: Virtual top quark loops contributing to W and Z bosons. Figure 1.5: Virtual Higgs-boson loops contributing to the W and Z bosons. 2 ≡ cos2 θ . This correction depends logarithmically on the Higgs-boson mass. where cW W Hence to predict MW at one loop level, one needs not just α,GF , MZ but also mt and mh . 10 In other words, prediction of Higgs boson mass requires α, GF , MZ and mt , MW . Fig. 1.6 shows the plot of MW vs. mt indicating lines of constant Higgs mass. The green ellipse indicates the 68 % confidence level measurements of MW and mt with mW = 80385 ± 15 MeV and mt = 173.2 ± 0.9 GeV /c2 . Mass of Higgs boson is predicted to be less than 152 GeV /c2 at 95% confidence level. Figure 1.6: Lines of constant Higgs mass on a plot of MW vs mt The green ellipse shows 68 % CL direct measurement of the most recent MW and mt . (Ref. [3]) 1.3 Top quark production and decay √ At s = 1.96 TeV center of mass energy of the Tevatron, the top quark is primarily pro- duced in t t¯ pairs which proceeds through either qq¯ annihilation (85%) or gg fusion (15 %). Figure 1.7: Feynman diagrams for t t¯ production via qq¯ annihilation and gg fusion. 11 The production process for the top pair production proceeds via the strong interaction and at high energies can be described by using perturbation techniques. The structure of the colliding proton and antiproton can be resolved into two parts. The hard scattering pro- cess that takes place between the constituents of the colliding hadrons (quarks/antiquarks or gluons) and the internal structure of the colliding proton and antiproton are responsible for the top quark production. In practice, a factorization scale µF2 is introduced to separate the hard scattering partonic cross section from the modeling of the constituents of proton and antiproton. The modeling of the constituents is independent of the hard-scattering pro- a (x, µ 2 ) are introduced that describe the cess and parton distribution functions (PDFs) fPDF F probability density to find a parton a with a longitudinal momentum fraction x inside a col- liding proton or antiproton. The PDFs are determined from the fits to the experimental data. In this analysis, CTEQ6M (Ref. [11]) with a factorization scale of µF2 = (175 GeV )2 was used. The physical observables do not depend on the factorization scale, the overall depen- dence on it remains if the calculations are not done to an infinite order in the perturbation theory via the PDFs. The leading order Feynman diagrams for the hard-scattering process of t t¯ are shown in Fig. 1.7. If the contributions from higher order diagrams are taken into account, the renormalization of divergent quantities becomes necessary. This leads to the introduction of renormalization scale µR2 . In this analysis, the factorization and renormalization scales were chosen to be the same. The t t¯ production cross section in hadron collisions can be written down by integrating over all possible initial state parton momenta and then summing over all contributing initial state parton species. 12  2 2 Q Z a1 a2 σ (P1 , P2 ) = ∑ dx1 dx2 fPDF1 (x1 , µF2 ) fPDF2 (x2 , µF2 ) σˆ x1 P1 , x2 P2 , αs (µF ), 2 a1 ,a2 µR (1.10) In the expression (1.10), σˆ represents the partonic cross section, P1 and P2 are the mo- menta of the incoming hadrons and a1 , a2 represent the parton species that initiate the hard +0.77 nteraction. The t t¯ cross section measured at DØ is found to be 7.78−0.64 pb (Ref. [12]) and is in good agreement with the theoretical prediction of approximate next-to-next-to leading order (NNLO) value of 7.48+0.56 −0.72 pb (Ref. [13, 14, 15, 16]). The top quark decays weakly into a W boson and a down type quark. The CKM matrix element |Vtb | is nearly equal to one. Since, (t → W b)/(t → W q) = |Vtb |2 /(|Vtb |2 + |Vts |2 + |Vtd |2 ), it can be seen that the top quark produced will decay to a b quark nearly 100 % of the time. When measuring the top mass, we assume that the t t¯ will always decay into W + bW − b¯ . The top quark decay width (Ref. [17]) is much larger than ΛQCD and so the decay process happens before the top quarks can hadronize. Thus, the decay mode is determined by the decays of the two W bosons. Thhe three modes that are commonly studied are dilepton, lepton+jets (l+jets) and all-jets topologies. These do not include events with one or more tauonic W boson decays which are not trivial to reconstruct and provide less information about the mass. Thus the word lepton represents electron aor muon when referring to the decay topologies of the top quarks. The branching fractions of various topologies are shown in Fig. 1.8. • Dilepton mode: About 5% of the t t¯ events have W bosons that decay into an electron or a muon plus the corresponding neutrino. These are called dilepton events and are characterized by two oppositely charged isolated energetic leptons, two b quark jets and missing transverse energy due to the presence of the neutrinos in the W decays. 13 • Lepton+jets topology: The lepton+jets events are thos 29% of the t t¯ events that have one W → eν or W → µν and one hadronically decaying W boson. Thses events are characterized by one isolated energetic lepton, atleast four energetic jets (two of which are b jets) and missing transverse energy. The main physics background comes from the events where a leptonically W boson is produced in association withfour jets. Multijet background where one jet mimicks an isolated electron, also plays a role. In lepton+jets events, the transverse momentum components of the neutrino can be obtained from the missing transverse momentum and the vent kinematics is over- constrained if one assumes equal masses of the top and antitop quarks and invariant lν and qq¯0 masses equal to the W boson mass. • All jets events: In about 46% of the t t¯events, both W bosons decay hadronically and hence the event topology in this case is characterized by 6 energetic jets, no charged leptons and no significant missing transverse energy. 14 Figure 1.8: t t¯ decay topologies based on the branching fractions of the W boson. CHAPTER Two The Experiment 2.1 Fermilab Tevatron Collider The Tevatron collider (Ref.[18]) is located at the Fermi National Accelerator Laboratory in Batavia (IL, USA), where the collisions of the protons and anti protons at a center of mass energy of 1.96 TeV, have taken place. It is a ring of 6.3 km in circumference which is a part of a sophisticated accelerator complex. The layout of the accelerator complex is shown in the Fig.2.1 . The particle acceleration goes through various stages : • 750 keV Cockroft-Walton pre-acceleration: The hydrogen gas is ionized by knocking of the peripheral electron and the H − ions thus produced are accelerated, before being passed through a 150 meter long linear accelerator (LINAC). The ions in this system are accelerated to 400 MeV using oscillating electric fields. After the removal of the negatively charged electrons, the protons, thus produced, are passed on to the Booster. • Booster: This is the first in the series of synchrotons involved in the step-by-step acceleration. The protons are accelerated to 8 GeV by going around in circular orbits. 15 16 • Main Injector: This can perform multiple tasks. – Accelerates protons to 150 GeV. – Accelerates protons to 120 GeV and send them to Antiproton source where the proton beam is targeted on a nickel target that produces, amongst many particles, the anti protons, which are then stored in an accumulator ring and sent back to Main Injector for acceleration. – It injects the protons and anti protons into the Tevatron for the final acceleration. • Tevatron: It accelerates the particles from the Main Injector to 980 GeV. The proton and anti proton beams are accelerated in opposite directions and made to collide at two crossing points, the DØ (Ref.[19]) and the CDF (Ref.[20]) detectors. The net resulting center of mass energy is 1.96 TeV. The particle beams are kept on their trajectory and in focus through a series of dipole and quadrupole magnets. The proton and anti proton beams are counter circulated in 36 bunches each with a time pulse of 396 ns. Each bunch has about 1011 protons and 1010 anti protons resulting in the instantaneous luminosity of about 4x1032 cm-2 s-1 . The last collisions at the Tevatron took place on 30 September 2011 which is when the operations were ceased, due to the fact that the machine has been superseded in the center of mass energy by the Large Hadron Collider, which uses protons on protons for collisions. 2.2 The DØ Detector The DØ detector is a general purpose detector that has been used to identify and record in- teresting collision events. The design of the detector is based on accomplishing the physics goals of the experiment. These include, but are not limited to, study of the top quark, the W and the Z bosons and search for the elusive Higgs boson. 17 Figure 2.1: Layout of the accelerator complex at Fermilab The schematic design of the full detector is shown in Fig. 2.2. Surrounding the beam pipe, the detector has the following main components. • Central Tracker enclosed inside a 2 Tesla magnetic field : This system acts as a mag- netic spectrometer. Interactions with the material are minimal here but just enough to identify the track of the charged particles. The magnetic field enables measure- ment of the energy and sign of the charged particle, which in turn facilitates, electron identification and calorimeter calibration. • Calorimeter : This system consists of three sampling calorimeters which are primar- ily made of uranium/liquid-argon and an intercryostat detector. Maximum interac- tions with the material happen here and most particles produced in collisions are “stopped” in this region except for muons and neutrinos. • The Muon system : A dedicated detector to detect and measure the momentum of the muons. A brief discussion of these components will follow in the next sections. 18 Figure 2.2: Schematic layout of the DØ detector 19 DØ uses a right handed Cartesian coordinate system in which the z-axis (also known as the longitudinal direction) points in the direction of the proton beam, the x-axis is in the plane of accelerator whereby the increasing direction of x is pointing away from the center of the Tevatron ring. The y-axis points vertically upwards. The xy plane is also referred to as the rφ plane in the cylindrical coordinate system. Thus, the xy plane or the rφ plane are both perpendicular to the beam direction. If, referring to the spherical coordinate system, the angle θ is taken as polar angle such that θ = 0, points along the proton beam. Since, most particles of interest are produced in an ultra-relativistic regime, it customary to use the pseudo-rapidity (η) which is related to the polar angle θ as follows :    θ η = − ln tan (2.1) 2 The pseudo-rapidity is a high energy approximation of the rapidity (y) which relates to the energy of a particle (E) and its longitudinal momentum (pz ) as follows :   1 E + pz y = ln (2.2) 2 E − pz η is often a useful approximation, when the mass and momentum of the particles resulting from collision are unknown. The sum of longitudinal momentum of particles, produced in the collision process, is often difficult to measure for each event due to the collision remnants draining down the beam pipe. In such a scenario it is often wiser to refer to the transverse momentum (pT = p sinθ ) , since the total momentum in the transverse plane (xy or rφ ) is known to be zero, balancing the initial state of longitudinally colliding protons and anti-protons. When a collision event is read/interpreted, the point of collision is not usually at the center of the detector. Essentially, the hard scattering process of the partons, which carry some fraction of the momentum of the colliding proton (anti-proton) is usually accessible. 20 The bunches of colliding protons are approximately 30 cm long and thus the collision may √ occur within ∼ 2 × 30 cm in the z−direction. The actual position, where the collision occurs is the origin of physics eta (η phys ). While describing the particle trajectories, it is more convenient to do so in a coordinate system, where the origin of collision coincides with the center of the detector. A correction is usually applied to the z-position in such cases and the pseudo-rapidity here is referred to as ηdet . The difference in the two definitions is shown in Fig. 2.3. Figure 2.3: Difference in definition of physics and detector η 2.2.1 Tracking Detector The tracking proceeds through layers of sub-detectors specialized to track various charged particles passing through it. The central tracking system is enclosed in a solenoidal magnet that produces a uniform magnetic field of 2.0 Tesla. The purpose of this magnet is to bend the trajectory of charged particles so that their momentum could be measured precisely. 21 Figure 2.4: Zoomed in view of the DØ tracking detector Silicon Microstrip Tracker (SMT) The silicon microstrip tracker surrounds the beam pipe and is the detector element closest to the interaction point. Hence, it plays a crucial role in tracking and vertex determination of the charged particles released as the product of collisions. The length of the interaction region is about 25 cm, which dictates the length of this device. Since the tracks emanat- ing from the interaction point are almost perpendicular to the detector surfaces for most of the η det range, it is imperative to have the design of SMT in the form of barrel mod- ules, interspersed with disks in the central region and assemblies of disks in the forward 22 region. This architecture allows for a three dimensional reconstruction of tracks and ver- tices, with the barrels covering low values of η and measuring the r − φ coordinates, while the disks provide r − z coordinates and are useful for high values of η . Fig. 2.5 shows a representative view of the SMT detector. Each of the six barrels in the central region has four silicon readout layers and are referred to as ladders . Layers 1 and 2 have 12 ladders while layers 3 and 4 have 24 ladders each. Each barrel is covered at the end with a disk of 12 double-sided wedge detectors, referred to as “F-disks”. In the far forward regions, not straddling the barrels, are the so called “H-disks” that provide tracking for high values of η . The 4 layers in the barrel detectors are implanted as two double sided layers and two single sided layers with the silicon sensor in each layer, segmented into a series of parallel strips of roughly 50 µm pitch. The two sided layers have the strips parallel to the beam pipe on one side while a stereo geometry of either a 2◦ or 90◦ on the other side. The whole detector sums up to a total of 912 readout modules with close to 800 K channels. The pitch of the strips is sufficient to provide a position resolution of 10 µm. The detectors are made out of n-type silicon wafers that are 300 µm thick, with a p+ implants, along the length of the detector. A dielectric medium between the strips and the aluminum coating provides a capacitive coupling between the detector and the readout electronics. A sensor is a poly silicon resistor that is designed to act as a reverse biased diode. A charged particle creates electron-hole pairs while traversing through the sensor with the electrons produced accelerating towards the p+ implants. The image charge formed on the aluminum coating is stored in an analog pipeline in the readout chips. The signal is digitized, when a Level 1 trigger accept is received and read out by a chip that is bonded to the sensor. More details can be found in (Ref.[21]). 23 Figure 2.5: A representation of Silicon Microstrip Tracker (SMT) A double-sided Silicon sensor. Central Fiber Tracker (CFT) The central fiber tracker works through three components : • Scintillating Fibers, which convert the charged particle hits into scintillating light signal. • Clear Fiber waveguides, which convert the scintillation light signal into visible light signal. 24 • Visible light photon counter cassettes (VLPC), which convert the visible light signal into an electrical signal for readout purposes. Eight concentric cylinders with radii of 20-52 cm from the beam pipe, support the scin- tillating fibers. Each cylinder has one doublet of axial fibers (orientation along the beam pipe) and a second doublet of stereo fibers (orientation of ±3◦ with respect to beam pipe) mounted on it. These fibers are made of polystyrene which is doped with a fluorescent dye called paraterphenyl. Excitations in the polystyrene are transferred to the dye via dipole- dipole interactions. The paraterphenyl has a short fluorescence decay of few nanosec- onds and a short emission wavelength. But the mean free path of the emitted light is also short (∼ few hundred microns) and so a secondary wavelength shifter made out of 3- hydroxyflavone is added as a dopant in low concentrations. This allows for re-emittance of a 340 nm wavelength light to a 530 nm wavelength light, which then is freely transmitted in the polystyrene. Small diameter of the fiber (835 micrometer) allows CFT a resolution of around 100 micrometers. More details of the scintillating fibers can be found in Ref.[22]. The 1.66-2.52 meters of scintillating fibers, mounted on the supporting cylinders, are then connected to clear fiber waveguides. The other end of the clear fibers are coated with reflective aluminium. They are very much akin to the scintillating fibers, in their composition but without the fluorescent dye. The scintillation photons travel upto 12 m inside the clear fibers. On the other end, these fibers are connected to the VLPC cassettes. The rφ view of the fiber layout is shown in Fig. 2.6. The VLPCs are impurity-band silicon wafers that act as photo detectors, capable of detecting single photons. The variations in gain, quantum efficiency and thermal noise requires the VLPC wafers to operate at an optimal signal-to-noise ratio between 6 and 8 Volts. Also housed in the cassettes are the analog front end boards (AFE) which act as preamplifier to the digitized signal received from the VLPCs and also provide discrimina- 25 tor signals, temperature control and bias-voltage control electronics. These optimizations allow CFT to work with quick response, high gain and great quantum efficiency for it to function even in high background environment. The signal from axial fibers is used to make L1 triggering decisions (described later in the chapter) at a pT thresholds of 1.5 GeV /c and higher. These signals are then passed on to a second level of triggering system. A third level of trigger system (L3, described later) uses the full CFT information provided through the readout electronics in VLPCs. There are about 77000 readout channels and the photelectron yield per fiber is determined to be around 8.5 and about 99.5% of the thermal noise is found to be below the the threshold of 1 photoelectron. Figure 2.6: rφ view of CFT clear fiber waveguides 26 Solenoid Magnet A superconducting solenoid magnet is wrapped around the central tracking detector system, providing a uniform central field of 2 Tesla. This provides a momentum measurement of the charged tracks by bending the charged particle trajectory. With about 2.73 m in length and 1.42 m in diameter, the solenoid is wound by two layers of multi-filamentary Cu:NbTi wires, stabilized with aluminum. Good field uniformity is achieved by having larger current density at the end of the coils. The thickness of the magnet system is about 1 radiation length (A radiation length X0 is defined as the mean distance over which an electron loses 1 about e of its energy). The polarity of the solenoid is reversible, which is a boon for many measurements as it helps to remove biases in track momentum determination. Forward and Central Preshower Detectors The central preshower (CPS) is placed just outside the solenoid magnet to supplement the energy and momentum measurement of charged particles. It helps mainly in triggering on an electron and offline electron identification. The electrons lose about 1X0 while traversing through the solenoid magnet material, while the CPS provides up to 2 X0 radiation lengths for the electrons. It consists of 5.5 mm lead and three concentric cylindrical layers of triangular shaped scintillators arranged in a xuv geometry (x = axial, u, v = ±22◦ aligned to beam pipe, in a stereo geometry). Using Monte Carlo simulations, the position resolution of a 10 GeV electron is estimated to be w 1.4 mm for the CPS detector (Ref.[23]). The forward preshower (FPS) covers 1.5 < |η det | < 2.5 and is mounted outside the solenoid magnet and in front of the end cap calorimeter. This is used for electron identifi- cation in the forward region. The material used is similar to that of the CPS but here only the uv stereo geometry is in place with a layer of lead (2X0 ) sandwiched between doublets of stereo fibers. The layout of the preshower detectors is shown in Fig. 2.7. 27 Figure 2.7: Forward and Central Preshower detectors 2.2.2 Calorimeter The calorimeter measures the energy of particles. For highly energetic particles, produced in the collisions, it is necessary to “stop” the particles to measure the incident kinetic en- ergy. For this purpose, usually a material with high atomic number (Z) is used. When the incident particle that interacts electromagnetically, say an electron, is suddenly made to interact with such a medium, it undergoes a process of braking radiation (Bremsstrahlung) due to charge particle interactions with the Coulomb field of the nuclei of the material. The result is the emission of photon. The photons thus produced are usually energetic enough to go through pair creation (γ → e+ e− ), which in turn could go through another Bremsstrahlung process. Hence incident electrons or photons go through a chain of the above described process until the energy losses are sufficient and further loss of energy 28 occurs only through ionization. Such a process is called electromagnetic showering. The loss of energy in this process is gauged by the quantity radiation length (X0 ). Incident hadronic particles also go through the “stopping” process but the energy loss here is mainly due to the strong interactions of the hadrons with the nucleus at the core of the material. Thus a material with high atomic weight (A) is desirable. A chain of such inelastic collisions slows down the incident particles until the stopping process cul- minates into that of the ionization mechanism. The shower thus produced is referred to as hadronic shower. The measure of the loss of energy in this process is gauged by the quantity nuclear interaction length (λI ). DØ uses depleted Uranium (with copper and stainless steel) as the material which has the desired characteristic (high Z and A) as described above. Uranium has a X0 of about 3.2 mm and a λI of about 10.5 cm. The ionizing medium used is Liquid Argon (LAr). A system of interleaved absorber (Uranium) and ionizing medium (LAr) is used to mea- sure energy. As most of the energy is absorbed in the highly dense, inert absorber, only a fraction of incident energy can be measured and hence it is called a sampling calorimeter. This process is statistical in nature and hence the resolution achievable on the energy mea- surement of incident particles is limited. The ratio of the signal measured to the energy of the incident particle is defined as the response of the calorimeter. It is intuitive to expect that the response of the electromagnetic and hadronic showers would be different due to the difference in nature of interactions for the two processes, for example, the neutrinos (ν) produced in the pions (π) and Kaons (K) decays will escape undetected. It is also note- worthy that the hadronic showers have an electromagnetic component owing to photons (γ) produced in the decay of uncharged pions (π 0 ) and η particles. For the resolution of energy to be not affected by convolution of electromagnetic component of hadronic show- ering and the electromagnetic showering itself, it is desired that the ratio of the response of electrons to that of the pions (e/π) be close to 1. At DØ , this ratio ranges from 1.04 to 29 1.11, depending on the energy of the incident particle. The DØ calorimeter is divided into large number of modules, with each module built of interleaved absorber plates (depleted Uranium), the active medium (LAr) and signal boards (to measure the signal). A transverse cross section of one calorimeter cell is shown in Fig. 2.8. The distance between the absorber plates and the signal boards is 2.3 mm. The signal boards are made of copper pads surrounded on each side, by a 0.5 mm thick sheets of G-10 to provide a dielectric medium. The surfaces of the sheets facing the active medium are coated with a resistive coating. While the calorimeter is in operation, the copper pad is grounded while the resistive coats are at 2-2.5 kV positive potential. When the showering occurs, the charged particles that ionize the Argon, induce a signal on the copper pad through capacitive coupling. The drift time of the electrons is roughly 450 ns. The signal from adjacent signal boards in a module are summed to form readout cells. Figure 2.8: Cross sectional view of a calorimeter cell The calorimeter cells are arranged in three assemblies, one central calorimeter (CC) and two end calorimeters (EC). Fig.2.9 shows the cross sectional view and the granular- ity of these assemblies. CC has a coverage of |η det | < 1.2 while the EC covers the re- gion 1.1 < |η det | < 4.5 . The central calorimeter has a toroidal geometry with several layers of the calorimeter modules described above. These are classified as 4 electromag- 30 netic layers (EM) with 32 φ modules, 4 fine-hadronic (FH) layers with 16 φ modules and 1 coarse-hadronic (CH) layer with 16 φ modules. The FH and CH measure mostly the hadronic showering and these are placed beyond the EM layers, due to the larger spatial extent of the nuclear interaction length (λI ). The end cap is similar to CC in ar- chitecture with the geometry differing in the fact that the electromagnetic module is disc shaped with cylindrical shaped fine and coarse-hadronic modules beyond it. In CC, the cells span ∆η × ∆φ = 0.1 × 0.1 , with the exception of the third electromagnetic layer, where maximum electromagnetic showering occurs for particles with pT > 20 GeV. In this layer, the granularity is finer for precise determination of the position of showers with ∆η × ∆φ = 0.05 × 0.05. The position resolution is around 0.8-1.2 mm, varying coarsely as √ 1/ E for the electrons (Ref.[24, 25, 26, 27, 28]). More details of the calorimeter can be found in Ref.[29]. 31 (a) η coverage of CC and EC. (b) granularity of CC and EC Figure 2.9: View of the central and end cap assemblies of the DØ calorimeter. 32 2.2.3 Muon System The muon system measures the momentum and trajectory of muons produced in the colli- sions. Owing to their large mass (≈ 200 times mass of electron), muons traverse most of the central detector without interacting. The dominant process of energy loss is ionization. A dedicated system beyond the the coarse-hadronic (CH) puchthrough region is in place to bend and measure the trajectory of muons. The energy loss of a relativistic heavy particle, such as the muons produced at Tevatron energies, is governed by the Bethe-Bloch equation (Ref.[30]): 2me c2 β 2 γ 2 Tmax   dE 2Z 1 2 δ − = Kz ln −β − (2.3) dx A 2β 2 I2 2 wherein, Z,A characterize the material used for absorption and represent the atomic v number and atomic weight respectively. β = c and represents the relativistic regime of the charged particle that passes through the medium. Tmax is the maximum kinetic energy that can be imparted to the charged particle and I represents the mean excitation energy of the atoms in the absorber. At DØ , the muons lie in the minimum ionizing regime of β γ. The system consists of toroidal magnets, just outside the calorimeter, the drift chambers for position of the hit determination and also for triggering purposes, the scintillation coun- ters for fast muon identification and triggering. A map of the magnetic field of the toroidal magnets can be seen in Fig. 2.10. The central toroidal magnets operate at 1.8 Tesla while the end toroids operate at nearly 2 Tesla. The central part of the muon system provides tracking up to |η| < 1 , while the forward part covers 1 < |η det | < 2. The central muon system has three layers of proportional drift tubes (PDTs) with layer A inside the toroidal magnet and layers B and C outside the magnet. Fig. 2.10 shows the placements of the drift tubes in the detector. The drift chambers have large rectangular cross sections and are formed out of extruded aluminum tubes. The layers have 72-96 cells, 33 which are 10.1 cm in length. Each cell is centered with an anode wire made of gold plated tungsten in the center and diamond shaped cathode pads (copper-clad G10), sandwiching the wire, to provide information on hit position along the wire. The drift tubes contain a gas that is a mixture of 84% argon, 8% methane and 8% tetra-fluoro carbon. This provides an ionizing medium for the charged particles. When in operation, the sense wire at the center is held at a positive voltage of 4.7 kV, while the cathode pads are held at 2.3 kV. A charged particle, passing through the drift tubes ionizes the gas and the electrons accelerate towards the positively charged sense wire. This causes further ionization and the the result is an avalanche of electrons, providing an amplified signal. The drift velocity is approximately 10 cm/µs for a maximum drift time of 500 nanoseconds. Each layer has a certain number of decks of the cells, for which the sense wires are ganged together, which are then readout by the electronics. Along with the electron drift time, the time difference between the arrival of the signal pulse from the cell that got the hit and that of its readout partner also recorded. This, along with the charge deposition on the cathode pad, are used to determine the hit position, along the wire. The drift distance resolution is ∼ 1 mm, while the resolution on the time interval varies between 10 cm to 50 cm. Besides the drift chambers, the central muon system is equipped with Cosmic caps and bottom counters. These are installed on the top, sides and bottom of the outer layer of the PDTs to provide a fast timing signal to associate a muon hit in the PDT with the appropriate bunch crossing and also help discriminate against the cosmic ray background. Another component is the Aφ scintillation counter that covers the layer A of PDTs. This facilitates fast triggering on muons with high pT , by matching the tracks to those triggered in the CFT, and identifying the low pT muons that do not make it to the outside of the toroid. The forward muon system utilizes mini drift tubes (MDTs) which have the characteris- tics of low electron drift time (∼132 ns), good coordinate resolution and high segmentation. These are also arranged in 3 layers with layer A inside the forward toroidal magnet. The 34 ionizing medium is a gaseous mixture of CF4 − CH4 (90%-10%). This is the main dif- ference from the central muon PDTs. The scintillators covering layer A, B and C have a trapezoidal shape. Photo multiplier tubes mounted on the detector collect the signal in the form of light and send it out as an electrical pulse for further readout. / muon system (a) DO (b) Muon toroidal magnet field map (c) Muon drift chambers (d) Muon scintillator skeletal view Figure 2.10: Various components of the central and forward muon system at DØ 35 2.2.4 Luminosity Monitoring System (LM) The luminosity monitor determines the Tevatron luminosity at the DØ interaction region and also provides a fast measurement of the z coordinate of the interaction vertex. The basic idea is to measure the number of inelastic p p¯ collisions in each beam crossing. The luminosity is defined as : f N¯ LM L= (2.4) σLM wherein, f is the beam crossing frequency, σLM represents the effective cross section for the inelastic p p¯ collisions, with the acceptance and efficiency of the luminosity mon- itor taken into account (Ref.[31]), while N¯ LM represents the average number of inelastic collisions. The products of these collisions are detected per beam crossing by two discs of plastic scintillators, segmented in 24 sectors along the φ direction, are placed at the far for- ward ends of the detector. At z = ±140 cm, these detectors span a radial area of beam pipe to the forward preshower detectors while providing an η coverage of [2.7,4.4]. The scintil- lation light produced in the Bicron BC-408 scintillator is detected by photo multiplier tubes (PMT), located at the center of each sector (see Fig. 2.11). The whole system is integrated with the Level 1 trigger (see next section). More details can be found in Ref.[32]. Figure 2.11: DØ luminosity monitoring system showing the location (le f t) and the geom- etry of LM counters with PMTs (red dots) 36 2.3 Trigger System To select the interesting physics events out of the plethora of data available for an opera- tional Tevatron with interaction rates close to 1.7 MHz, a three tier system is in place with each sieving through fewer events than its preceding level but with more detail and com- plexity. A cartoon of such a process, known as triggering , is shown in Fig. 2.12, with the three levels being termed Level 1 (L1), Level 2 (L2) and Level 3 (L3). At the first stage, L1 comprises of triggers based on sub-detectors for tracking, calorimeter and muon. L1 passes a digitized signal of accept/reject upon each beam crossing to the second stage (L2) at a rate of ∼2KHz. Here the hardware engines and embedded microprocessors associated with specific sub-detectors take a decision on whether to accept or reject an event based on the individual objects and object correlations. The accept rate for L2 is ∼ 1KHz. The decision is then passed onto a farm of computers in L3,where the sophisticated algorithms analyze the L2 accepted events and send a select few events to tape for offline reconstruction at a rate of ∼100 Hz. A more detailed overview of the three stages of triggering and elements involved in each stage is shown in Fig. 2.13. Figure 2.12: Basic trigger road map (Ref.[4]) 37 Figure 2.13: Detailed structure of trigger framework 2.3.1 Level 1 In this section, the Level 1 trigger elements are described. L1Cal identifies the patterns in transverse energy deposits in the calorimeter. L1CTT and L1Muon pass on the list of tracks with their characteristics to L1 Global trigger, which take decisions based on prede- fined thresholds. The rate of L1 trigger accepts is limited by the maximum readout rates of the participating sub detector system and by the motivation to minimize the deadtime associated with the readout. L1CTT The Level 1 central tracking trigger uses the information from axial fiber hits in the CFT and identifies the hit pattern with the pre-programmed look up tables (LUTs). These LUTs are digitized information encoded for know possible hit patterns in the CFT. The axial fiber information is divided in eighty φ segments of 4.5◦ each. The signals from these sectors 38 are processed by field programmable gate arrays (FPGA) that search for the patterns in LUTs. If a hit pattern is found in the data, a track candidate is generated. Each of these track candidates are characterized by a relative φ within a trigger sector, track momentum and track curvature. These tracks are then passed onto L1Muon and a second stage silicon track trigger (L2STT, described later) for triggering decisions. L1Muon The inputs to Level 1 Muon triggers come from the scintillation counters, the muon wire chambers and the L1CTT tracks. The scintillation counter is segmented in the same way as L1CTT and the tracks obtained via scintillation counters are processed by the FPGAs that performs a combinatorial logic to about 60,000 muon channels. The tracks thus obtained in L1CTT and scintillation counters are matched and sent to Muon trigger manager (MTM). Up to 480 tracks for every bunch crossing can be processed. The trigger terms from scintil- lation counters and muon wire chambers are summed up by the MTM and sent to the global trigger framework (TFW) for trigger decisions. L1Muon can handle up to 256 trigger terms and filters 32 of these to be sent to the TFW. Those tracks that can be classified as high pT tracks are also made to go through the cosmic ray veto scintillation counters. These cosmic ray muons usually penetrate the DØ detector, but do not pass through its center. The timing information of the tracks left by these muons relative to that of the the bunch crossings can be used to reject cosmic ray background. L1Cal The Level 1 calorimeter trigger is responsible for making trigger decisions based on the information obtained from calorimeter. In this, the the electromagnetic modules and the fine-hadronic modules take part by summing the energies measured in the cells to form energy towers that span 0.2 × 0.2 in ηφ space. Various sums are calculated viz. the total 39 electromagnetic energy (E em ), the total hadronic energy (E had ), the scalar sum of electro- magnetic transverse energy (ETem ), scalar sum of the hadronic transverse energy (EThad ) , the total scalar transverse energy (ET ) and the missing transverse energy (6 ET ). These sums are compared to a set of programmable thresholds that yields a trigger term which is sent to the TFW for decisions. In all there are 16 such thresholds at level 1, including the fine-hadronic modules. Additional trigger terms can be constructed according to the desired calorimetry activity covering |η| < 4. 2.3.2 Level 2 The Level 2 triggering system consists of two components. The preprocessor elements are customized to run specialized software algorithms while the L2 global processor uses the preprocessor results to assert a decision on whether to accept or reject an event. Level 2 calorimeter (L2Cal) is a calorimeter preprocessor that uses clustering algorithms to con- struct basic jets or electron candidates using the information from L1Cal trigger towers. Similarly, L2CTT sorts the tracks in order of their pT , using the information from L1CTT. The Level 2 silicon track trigger (L2STT) is the only preprocessor that combines the infor- mation from different sub-detector systems to run sophisticated algorithms of track finding. It is of great utility in identifying the b-quarks due to its precise reconstruction of impact parameter of tracks. This system is described in details below. Silicon Track Trigger (L2STT) The Level 2 silicon track trigger is the the preprocessor that uses the information from L1CTT and SMT to define and construct charged particle tracks during run time. It uses the axial fiber information for the six overlapping sectors of the CTT φ regions. About 46 tracks from each sector and up to 276 tracks per event, sent by L1CTT can be handled 40 by the STT. It constructs a circular trajectory from the information of the hit position in the outermost CFT layer (H-layer), the hit position in the innermost CFT layer (A-layer) and the average beam position. Thus the track position and momentum in the rφ plane is determined from the L1CTT information. It then combines this information with the silicon detectors (SMT), that are arranged into 12 sectors in φ . It then fits the tracks using the hits in the CFT and SMT layers, removing the constraint that the tracks originated from the point of interaction. This is particularly important for B-hadrons produced in the collisions as they have a typical lifetime of 1.5 ps and travel up to 500 µm before decaying. In such cases, the impact parameter or the distance of closest approach to the point of interaction can be determined to good accuracy. The data is processed via three components of the STT: the Fiber road card (FRC), the silicon track card (STC) and the track fitting card (TFC). Each of these plug into a common motherboard for use in a standardized crate. The communication between the components happens through mezzanine cards that use Low voltage differential signal (LVDS) cables. Each board also communicates with a common daughter board where the buffering and transfer of data to the data acquisition system (DAQ) occurs through PCI buses. Each STT crate processes data for two 30◦ sectors in φ . A brief description of each of these components follows. Fiber Road Card The information from L1CTT track candidates is used to define projective roads inside the silicon (SMT). Only the axial clusters in SMT that are found within the 2 mm wide road of the axial CFT hits are used as track candidates. Fig. 2.14 outlines the above principle. Fiber road card is designed receive the data from L1CTT and reformat it for the other daughter boards. After the data has been processed (fitted for tracks), it also broadcasts the results to the data acquisition on global level 2 accept for an event. 41 Figure 2.14: STT pattern recognition algorithm Silicon Trigger Card (STC) The Silicon track card receives digitized data from the 10 axial strips (including Layer 0) of the SMT, in each φ sector. It then masks out the noisy and dead silicon strip data while performing a strip by strip gain and offset correction.The data, thus filtered is allowed to go through a fast clustering algorithm. This involves successive passing of signal through thresholds. At first, each strip signal is tested by a strip threshold and is passed on to form a cluster only if the signal lies above it. Then the maximum of these adjacent strip signals is compared to a cluster threshold. If above the threshold, the cluster of adjacent strips passes on to next stage. The strip that records the maximum signal, along with its two neighboring strips inside a cluster are used to determine a weighted centroid for the respective cluster. Finally, the centroid is associated with the L1CTT tracks and for a centroid lying withing 2 mm road of the L1CTT track, the track is considered as selected. The list of centroids is then passed on to the track fitting card. Fig. 2.15 demonstrates the clustering algorithm implemented by the STC. 42 Figure 2.15: SMT clustering algorithm, implemented by STC Track Fitting Card (TFC) A circular trajectory is fitted using the SMT clusters and CTT tracks by the track fitting card. Clusters that are passed on by STC, are further required to go through another set of pattern recognition. Due to electronic noise, δ ray production and overlapping of signals from clusters of nearby particles, there is a possibility of finding multiple clusters per layer of SMT in the 2mm road defined earlier. To fit a track, it is desired that only one cluster per SMT layer is determined and hence the road is redefined to 1mm wide about the CTT tracks to filter the clusters further. Also, it is expected that the tracks be linear in the rz plane. The clusters in SMT are made to comply with this. Hence the algorithm looks at the silicon hits associated with the CTT track and in each layer, selects the hit that is closest to the track, under the assumption that the track originated from the interaction point. The track is fitted with a function in the rφ plane : b φ (r) = + κr + φ0 (2.5) r wherein, b represents the impact parameter of the track with respect to the center of the detector, κ represents the track curvature, while φ0 is the azimuthal angle of the tangent to the track at the point of closest approach. The fitting is done using a goodness-of-fit measure defined as : 43  2 2 φi − φ (r) χ =∑ (2.6) hits σi wherein, φi is the azimuthal position of the hit in the respective SMT layer, while σi is the resolution on the azimuthal position of the hit. The beam position is often fed back to the system, to keep the fitting up-to-date. Finally the output of impact parameter significance (b/σb ) , b, φ0 , pT derived from 1/κ, the χ 2 , number of SMT layers used in the fit and the dE/dx derived from the pulse height values of the SMT hits are transmitted to the global r Level 2 processor and the Level 3 processors. The resolution σb is found to be close 54 GeV /c 2  to 402 + pT µ m, including the beam spot resolution. More information of the STT system and hardware can be found in Ref.[33]. 2.3.3 Level 3 and DAQ The Level 3 data acquisition system (L3DAQ) communicates the fully digitized data from all detector subsystems to the level 3 trigger system for final filtering processes that run on a L3 trigger computer farm. When the global Level 2 trigger system issues an accept for an event, a total of up to 63 VME crates are read out. This a amounts to 250 kB of data per event, under normal conditions. A single board computer (SBC) in each crate sends out the accepted event fragment over an Ethernet network to a L3 trigger farm node. Here all the information of the event is combined and a modified version of the full reconstruction takes place in just about 100 ms, before the event is written to tape. The final trigger decisions on high level physics objects such as electrons, muons and jets also take place. Hence the two components, event builder and event filter work in tandem to provide an output rate of ∼ 100 Hz. A data flow structure of the level 3 triggering system is shown in Fig. 2.16. More details of the hardware and sophisticated computing languages involved can be found in Ref.[34] and the references therein. 44 Figure 2.16: Data flow in Level 3 data acquisition system (L3DAQ) CHAPTER Three Particle Reconstruction: Algorithms and Identification The data that is stored on tape is in digitized form, that is collected from various detector elements. This data, also referred to as the raw data is a collection of analogue to digital convertor (ADC) counts. To extract the interesting information of this raw data, various pattern recognition algorithms need to be employed so that the all important kinematic parameters of physics objects such as electron, muons, jets, 6ET can be determined, which would lead to identifying the particles produced in the p p¯ collisions. This process is termed as reconstruction. The three main phases of reconstruction can be classified as finding of hits in the detector elements, forming clusters and eventually tracks of charged particles and identification of particles including discrimination from backgrounds. These process are discussed in the following sections. 3.1 Reconstruction in Central tracker The two sub systems that aid in tracking are SMT and CFT (as described in the previous chapter) . The clustering in SMT proceeds through a series of checks made by comparing the 45 46 signal recorded with thresholds and then constructing a centroid of the cluster that satisfies various requirements. SMT, being the closest tracking system to the point of interaction, the cluster building starts with its innermost layer. SMT strip that records a signal above a certain threshold (≈ 8 ADC counts), is allowed to start a cluster, then the strip that is geometrically adjacent to the previous strip is compared to the threshold. All contiguous strips that are above the threshold form a cluster. With a gap in the above process, a new cluster is born. The position of the cluster is determined by the pulse height weighted average of the strips in a cluster. The axial and stereo hits are then combined to yield three dimensional global coordinates for the hits. CFT clustering also proceeds in the same way. The signal pulse is replaced by the light yield per fiber which is eventually transformed to number of ADC counts. The thresholds in this case varies from sector to sector and between axial and stereo layers, depending upon the distance traveled by the light inside the fibers. The centroid position is just the mean of initial and final points of a cluster, in this case. Similarly, for the muon system, it matters, whether the hit was near the center of the active volume of a PDT or near the edge, which governs the drift time of electrons and hence the ADC thresholds. The clusters formed above,do not yet give an estimate of the trajectory of the charged particle. The clusters could represent a superposition of adjacent tracks of charged particles and hence it is important to combine the clustering information in different layers of the tracking system. Dedicated algorithms are designed to construct a charged particle track and filter out the noisy, spurious clusters, arising due to combinatorial ambiguities. One kind of such an algorithm is called histogramming track finder (HTF) (Ref.[35]), whereby a Hough transformation is employed with the parameter space defined by the track curvature (κ) and the azimuthal angle of helix shaped tracks (φ ). The spectrum of tracks that originate from interaction point and form clusters in various layers of SMT and/or CFT, have a certain trajectory and azimuthal angle in the transverse plane. These are mapped into straight lines 47 by the Hough transformation and for several tracks, the intersection of the transformed lines in κφ space is taken as the transverse coordinates of the track. This corresponds to finding narrow peaks in the histogram of κφ . The appropriate track corresponding to the above coordinates can be selected by applying various criteria such as requiring clusters formed by the tracks in at least 4 layers of SMT or CFT, requiring that the track doesn’t have more than three misses (defined by the absence of clusters in the track trajectory, in layer(s) of SMT or CFT, going outward from the point of interaction), requiring that there be no more than 2 misses inside SMT, number of total clusters found be 5 times more than the total number of misses, etc. Tracks not originating from the point of interaction such as those of long lived particles, might not give a single point of intersection in the κφ space and hence the HTF method cannot find them. To come over this problem, an alternative algorithm (AA) (Ref.[36]), is in place. This method is similar to the goodness-of-fit method, defined in Sec. 2.3.2, whereby the hits in the outward layer of the tracking detector are associated with the track if the χ 2 of the track fitting yields a value less than 16 and if the impact parameter of the track lies within 2.5 cm of the beam spot. To account for loss of energy through ionization when the charged particle traverses the material in the tracking volume, the hits associated with the above two methods are used by a Kalman fitter (Ref.[37]), which acts as a global multidimensional χ 2 minimization by predicting through propagation of the track in dif- ferent layers. This whole procedure, provides a refined way of accurately determining the particle trajectory and its kinematics in the tracking volume. 3.2 Reconstruction in Calorimeter As discussed in Sec.2.2.2, the particles (hadrons, electrons, photons) manifest themselves as showers obtained through ionization of LAr in the calorimeter. A digitized signal, 48 recorded by the readout electronics is converted into ADC counts. The ADC counts are converted into units of energy, taking into account the differences in cell response and readout electronics. The signal is corrected for gain and pedestal based on the calibrations obtained throughthe test beam in which particles of known energy are targeted on differ- ent sections of the calorimeter (Ref.[38]) and also via reconstruction of invariant mass of particles, whose mass is known. These ADC counts are converted to ηφ coordinates for each cell and the four-momentum for the cell is is measured using the cell energy in the direction defined using the primary vertex of interaction as the origin. Pcell ≡ Ecell (1, nˆ cell ) nˆ cell = (sinθ cosφ , sinθ sinφ , cosθ ) In the projective direction of nˆ , a tower consisting of cells with signal above 2.5σ of pedestal mean (cell noise), is constructed, with its associated four-momentum being : Ptower ≡ (Etower ,~ptower ) ≡ ∑ Pi i=cells These tower energies and direction are used in reconstructing the energy and direction of physics objects, such as jets. 49 3.3 Muon Reconstruction Sec.2.2.3, describes the muon system in detail. It was noted that the system is made of three layers of drift tubes (PDTs: central and MDTs: forward) and scintillators to account for the time of arrival of the signal and the position of the hit in the muon system. The muon reconstruction proceeds through linking segments in various layers and track fitting. Since the inner A layer is inside the toroid magnet while the two layers B and C , are placed outside the toroidal magnet, this allows the “segments” in the outer layers to be collinear. Hence a propagation of the track from the B and C layers to the layer A in the form of a circular helix can be made. This gives track parameters of position and momentum in layer A. Following this, a matching to tracks in the central tracking system is done, taking into account the losses in the medium (solenoid magnet, toroid magnet and calorimeter) through the error propagation technique. This gives track fitting measure χ 2 and a distance of closest approach (DCA) to the beam spot. 3.4 Vertex Reconstruction : Primary The Primary Vertex (PV) is the longitudinal position (zPV ) along the beam pipe, where the hard scattering process of p p¯ collisions take place. This is important to identify to define transverse energy and pseudo-rapidity of reconstructed objects as well as efficient identification of long lived particles such as hadrons containing b-quarks that travel some distance before decaying and hence leaving their signatures of displaced vertices. There are several soft inelastic collisions that occur in one beam crossing, resulting in tracks that are not the result of the hard scattering process. Such collisions are termed as minimum bias events and serve as a large instrumental background. This can be tackled using a three step process that involves track selection, vertex finding and vertex selection. Since a hard 50 scattering event is most likely to yield high transverse momentum (pT ) tracks, an algorithm proceeds with selecting tracks that have pT > 0.5 GeV, and have at least two SMT hits, for the construction of a PV. Tracks are also selected on the basis of their impact parameter significance defined as b/σb , where b represents the distance of closest approach (DCA) of the track with respect to the beam pipe while σb is the associated uncertainty. The vertex finding algorithm then proceeds through forming clusters of the selected tracks (Ref.[39]). In the decreasing order of pT , the tracks whose DCAs are within 2 cm of the mean value of an already existing cluster are added to the cluster. The tracks in each non-overlapping cluster is then fitted for a vertex using a Kalman fitter. Tracks contributing more to the χ 2 of the fit, are removed from the cluster and the fitting procedure is redone. This ends up in a list of possible vertices for PV, where the hard scattering process, could have occurred. Then comes the vertex selection procedure to eliminate the vertices associated with minimum bias interactions (Ref.[40]). This is done by assigning a MB (minimum bias) probability to each track, based on expected pT distribution of tracks matched to the MB vertices that is obtained from the Monte Carlo simulations. These probabilities are combined to yield (− ln ξ ) k a MB probability for each vertex (PMB ≡ ξ ∑Ntrk k=1 k! k ). In the end, the ; ξ ≡ ∏Ntrk PMB vertex with lowest value of PMB is identified as the primary vertex. For this analysis, it is required that, |zPV | ≤ 60 cm, within the SMT fiducial region and at least 3 tracks are fitted to the PV. 3.5 Particle Identification 3.5.1 Electrons Electrons are primarily identified via the showering process that takes place in the electro- magnetic calorimeter as described in Sec.2.2.2. There are several sources that can emulate 51 the showering produced by electrons as π 0 decays that overlap with a track, γ participating in pair creation, components of hadronic shower overlapping with electromagnetic show- ers etc. The shape of electron showering can be characterized by a cone of energy towers p with radius R = (∆η)2 + (∆φ )2 < 0.2, around the tower of highest energy. Furthermore, parameters such as electromagnetic fraction, defined as the ratio of the energy deposited in the electromagnetic calorimeter to the total energy deposited in all layers of the calorimeter, characterize the electron showering. Another important parameter is the isolation fraction, defined as : Etot (R < 0.4) − EEM (R < 0.2) fiso = Etot (R < 0.2) wherein, Etot is again the total energy deposited in all layers of calorimeter, while the EEM is the energy deposited in electromagnetic calorimeter. For this analysis, it is required that this fraction is less than 15%. Electrons also interact with the tracking system and hence track matching provides a genuine way to measure E/p for electrons. This is done by extrapolating each track within 0.5 × 0.5 of ηφ space and matching the electromagnetic cluster in the central tracking system to that of the electromagnetic energy deposition in  2  2  2 2 δφ δz (ET /pT )−1 the calorimeter. A matching fit χ is defined as σφ + σz + σE . Another T /pT 2 χHM , defining a goodness of fit to seven variables that define an electron showering, such as fractional energy in the four physical depths of calorimeter, shower width in the azimuthal plane of the third layer of EM calorimeter, ln Eshower and ln |zPV | are input as vectors to define a H-matrix χ 2 (Ref.[41]) : 2 χHM = (~vmeasured −~vMC )T H−1 (~vmeasured −~vMC ) 52 1 7 n [xi − x¯in ] xnj − x¯nj   Hi j = ∑ N=7 N n=1 2 x¯ represent mean values of the Monte Carlo (MC) showers. A χHM of less than 50 is required in this analysis. A still tighter electron definition is employed by construct- ing an electron likelihood. This involves constructing a likelihood in the form of L(x) = Psignal (x) Psignal (x)+Pbackground (x) , such that 0 < L(x) < 1 and x is a combination of parameters described above (Ref.[42]). In this analysis, the electron likelihood value must be greater than 0.85 . Also, only electrons in the central calorimeter with |η| < 1.1 , are kept in this analysis. 3.5.2 Muons Muons are identified in the three layers (A, B and C ) of the muon system. Due to less multiple scattering than in the toroidal magnet, the muon momentum can be more precisely measured in the tracking system. Scintillator hits provide the timing information. which helps to tag the muon that results from hard scattering process. In this analysis, muons with |η| < 2 are accepted. The muons are also required to have at least two A layer hits, at least one A layer scintillator hit, at least two B or C layer (outside toroid) PDT hits and at least one BC scintillator hit. The time interval between the beam crossing and the B or C layer scintillator hit, is required to be less then 10 ns. This also provides cosmic background rejection. The track reconstructed in the muon system must match a track reconstructed in the central tracker with χ 2 /nd f < 4, where ndf represents number of degrees of freedom. It is also required that the distance of closest approach (DCA) in the xy plane, be less than 0.2 cm, for the tracks that do not have any SMT hits, while |DCA(x, y)| < 0.04cm, is required for tracks with SMT hits. A muon is counted as tagged tight, when the energy deposited in the calorimeter, within the annular region of 0.1 < R < 0.4, contains less than 8% of the 53 muon pT and if the momenta of all tracks in the region R < 0.5, except for the one matched, adds up to less than 6% of the reconstructed muon pT . To distinguish a muon resulting from the hard scattering process from the soft ones that result from semi-leptonic decays of B- p hadrons, inside the calorimeter, a separation from jets by a distance of ∆R = ∆η 2 + ∆φ 2 > 0.5 is demanded. 3.5.3 Jets Jets are manifestation of final state quarks and gluons that hadronize to form various par- ticles. The complexities arising from the type of particles that shower in the calorimeter, fraction of energy carried by various flavor of particles (b-quarks or light quarks), and the knowledge of the jet energy, limit the precision with which the underlying hard-scattering process of the partons (inside the protons and anti-protons) can be accessed. Jet algorithms are implemented after running a T42 algorithm (Ref.[43]). First, a zero-suppression algo- rithm rejects any calorimeter cells that record energies less than 2.5σ of the cell specific electronic noise. Then T42 algorithm selects those cells that have energy greater than 4σ and which has atleast one neigboring cell that records energy greater than 4σ . This allows for eliminating those cells that have an isolated energy tower, which is most likely to be a fluctuation. Following this, jets are reconstructed using a RunII legacy cone algorithm (Ref.[44]). In the following discussion, y refers to the rapidity as defined in Eq.2.2 rather than the pseudo-rapidity. This is done, so that the mass of the particles is not disregarded (even in the ultra-relativistic regime), especially for low pT objects. Also, all the direc- tions are defined with respect to the primary vertex (PV) of interaction, as the origin. The algorithm proceeds through iterative procedures of clustering into proto jets, splitting and merging. These processes are described briefly in the following discussion. At first, the energy of the cells are summed up in the ηφ space to form pseudo- 54 projective towers. Then the towers with energy greater than 0.5 GeV are ordered in pT to form what are called seeds for pre-clustering. With the seeds in the center, an annular region of ∆R = 0.3 is constructed and the four-momenta inside this cone is added to the four-momenta of the seed. This defines a pre-cluster for each seed. If the pT of the pre- cluster is found to be above 1 GeV, it is tagged as a cluster. If the direction of the initial seed and that of the cluster agree within a small tolerance, the cluster becomes a proto jet. If the above is found not to be true, the pre-cluster is taken as a new seed and the above process is repeated. An axis alignment of the seeds with the clusters within 10−6 in the ηφ space is necessary to get a list of proto jets. The proto jets thus formed, could be an overlap of two or more particle jets and hence processing them through merging and splitting is essential. The idea is to achieve a stable cone axis. At first, all proto jets that are close to one another are reprocessed through clustering. Then, based on the fraction of shared energy between two or more of the close enough proto jets (within ∆R = 0.1), merging or splitting takes place. If the fraction of energy share is more than 50% of the proto jet with smaller pT , amongst the two proto jets, both are merged to form a new proto jet and the clustering is repeated until a stable cone axis is achieved. Similarly, for the energy share less than 50%, the two are split using the mid-point of separation and treated as new proto jets. The im- portant aspect of jet reconstruction algorithm, from the physics point of view is to achieve Infrared and collinear safety. This implies that the algorithm requires to be least sensitive to the soft gluon emission in the initial state (akin to the bremsstrahlung for accelerating charged particles, albeit color-charged in this case) as well as the collinear gluon radiation in the final state. These aspects could cause an artificial splitting or merging of jets arising from a single parton. The mid-point selected for merging andd splitting is a pT weighted average of the ηφ coordinates of the two proto jets involved. Since a mid-point algorithm is efficient enough to resolve a pair of jets, the loss of sensitivity of this algorithm occurs when more than two jets lie in close proximity of each other. A pT cut is also applied to the 55 final collection of jets, to aid in the process of elimination of jets, contaminated with noise. Jet Identification and Selection To ensure that the jets that are selected in the analysis arise from the fragmentation of quaks and gluons, sources of backgrounds, such as mis-identification of electrons, photons and calorimeter noise, need to be minimized. This is done by placing requirements on jet propertes. A jet is termed a good jet, if it satisfies the following criteria : • Jets that have less than 5% of their total energy in the EM calorimeter are rejected. This is done since the hadronic layers, fine-hadronic (FH) and coarse-hadronic (CH), are expected to be noisier than EM. On the other hand, jets with more than 95% of total energy in EM layer, ought to be from electrons or photons and hence these are also rejected. In other words, some activity in both EM, FH and CH is required. • The coarse-hadronic (CH) calorimeter is the noisiest. It is required that jets deposit, less than 40% of the their total energy here. • To alleviate electronic noise, which manifests itself as a hot tower, during the readout i.e. a single isolated tower of energy, it is required that the ratio of the transverse energy of the most energetic cell to that of the next most energetic cell be less than 10. • Jets that are formed by coherent noise in some particular region of the calorimeter are removed by requiring the jets that have more than 90% of their total energy in one calorimeter tower be eliminated. • Furthermore, all the jets are required to have passed the L1 trigger, in order to ensure that the jets are the result of the collision data. This is done by requiring the ratio pL1 preco T (1−CHF) , be greater than 0.5 for the central region. Here, preco T represents the T 56 reconstructed jet transverse momentum, while pL1 T is that of a matched L1 object and CHF represents the coarse-hadronic fraction. • To mitigate the effect of minimum bias vertices on the reconstruction of jets, tracks in the tracking system are selected that point towards the jet. A requirement that at least two of such tracks originate from the primary vertex (PV) makes the jets vertex-confirmed. 3.6 Jet Energy scale The aim of the jet energy scale correction is to correct the reconstructed jet energy to the particle level. For these corrections to remain as model independent as possible, the effects of non-perturbative QCD viz. hadronization and underlying event are not corrected for. This is because the “particle level” energy corresponds to the incident particles on the calorimeter, which include the effects that are mentioned above. The relation between jet energy and particle energy, can be written as : Emeasured − EˆO E particle = (3.1) R.S wherein, Eo is the offset energy, R is the calorimeter response to the particles and S represents the correction factor, accounting for the migration of particles, inside and out- side of the cone, also termed as showering effect due to scattering with the material of the calorimeter, etc. Hence all of these correction factors correct for the instrumental back- grounds. The offset energy is, on average, the energy that does not originate from hard scattering or underlying event. In principle, it depends on the Luminosity (L) and the num- ber of primary vertices (nPV ). The offset energy can be classified into two categories, the first being that whose source is calorimetry noise and pile up (deposition of energy from 57 previous collisions) . This category is termed as NP. The second one arises from multiple p p¯ interactions within one beam crossing and is termed as MI. The average offset energy is estimated for each calorimeter ring in iη, by summing overall energy towers in iφ (here iη refers to integer value of 10.η). The offsets are estimated by triggering on special kinds of events and are measured directly from the data. The “zero-bias” data is collected with the only requirement that the trigger should fire at the same time as the beam crossing. The “minimum-bias” events (as discussed in Sec. 3.4) are collected when the luminosity mon- itor (LM) receives hits at the same time on both disc arrays. This ensures than an inelastic collision of p p¯ has taken place. This data is collected at a constant rate of 0.5 Hz. The NP offset is estimated using the zero-bias data with the veto on luminosity counter, indicating that no interaction occurred. The MI offset is estimated from the minimum bias events and it depends on the number of primary vertices (as the soft inelastic multiple p p¯ collisions are looked for). This dependence is found to be linear and hence the offset for N number of interactions is estimated from events with N+1 PVs and from this the average energy measured for events with one PV is subtracted. To sum up, the offset energy (as a function of L, nPV ), can be written as : EˆOring (iη, nPV , L) = EˆNP ring ring (iη, L) + EˆMI (iη, nPV , L) EˆOring (iη, nPV , L) = EˆZB ring ring (iη, L) + EˆMB ring (iη, nPV + 1, L) − EˆMB (iη, nPV = 1, L) (3.2) 58 Figure 3.1: EˆO as a function of |η detector jet | for jets with cone size of R = 0.5. Different number of multiple interactions (MI) are shown Jet Response The next step is to determine R in Eq. 3.1 , which represents the detector response to the particles. This can be factorized as RCC (E) × Fη (η, E), where RCC represents the detector response in the central calorimeter, while Fη is the factor that normalizes the response of the calorimeter as a function of jet rapidity. This is derived from a sample of γ + jets events (produced through p p¯ → γq + X , p p¯ → γg + X) with various thresholds on the transverse momentum of at least one electromagnetic cluster, found in the central region of the detector. The γ + jet candidate thus selected is not free from contamination of dijet events, in which one of the partons fluctuates to a leading π 0 that decays into photons and thus the photon gets misidentified. This is not a sizeable background, it nevertheless affects low pmeasured Tγ . Hence a missing transverse energy projection fraction method (MPF) (Ref.[45]) is applied to measure the response from these selected events. This is based on the transverse plane momentum balance. A vector sum of all energy towers is projected, transverse to the beam. This equals the 6ET of the event. At the particle level, the photon is expected to balance the hadronic recoil, with ~pT,γ + ~pT,hadronic = 0. This method is also known as tag and probe method. Here, the photon (electromagnetic cluster) serves as the 59 tag, while the hadronic recoil in the calorimeter serves as the probe. In principle, this method could also be applied to Z + jets or dijet events. To have a clear interpretation of 6 ET , the events in the γ + jet sample are selected such that there is exactly one jet, back to back with the photon, by applying a selection cut on ∆φ to be 3.0. Also, the response measurement is restricted to 1 or 2 PVs to have a lower luminosity environment for such events. Even in the ideal case scenario, the response to the particles is non-unity. One can thus write, Rem .~pT,γ + Rhadronic .~pT,hadronic = −~6 E T (3.3) wherein, Rem & Rhadronic are the electromagnetic and hadronic calorimeter responses, respectively. The electron energy scale, determined from using Z → e+ e− events in the data, can be used to tune Monte Carlo simulations to reproduce the electron energy re- sponse in the data. This allows for the corrected photon energy, Rem = 1. With this into consideration, Rhadronic can be estimated by projecting everything on the photon pT unit vector nˆ γ . Hence, ~6 E T · nˆ γ Rhadronic = 1 + (3.4) |~pT,γ | becomes valid for the back to back photon and the hadronic activity. Rhadronic can be considered synonymous with R jet , for the selection of events, described above. This response is measured in different bins of jet pT , but then the resolution on measured jet energy is found to be poor and hence a scaled variable E 0 = pT,γ · cosh η jet is used to measure the jet response. Owing to the limited statistics for E 0 > 350 GeV , the measured response is extrapolated to up to 600 GeV using Monte Carlo models in which the response to single pions have been tuned to match the data. The response is then fitted with a quadratic function : 60 γ+ jet RMPF,CC (E 0 ) = p0 + p1 log(E 0/E0 ) + p2 log2 (E 0/E0 ) (3.5) with E0 = 100 GeV. The fit parameters are shown in Fig. 3.2. Some of the uncertainties in this estimation arise from determination of photon energy scale, photon identification, fragmentation and parton distribution functions (PDFs) of which the first one is dominant. Figure 3.2: MPF response in CC, for jets with Rcone = 0.5, as a function of E 0 . Also, shown is the relative difference of fit with the data points. With the energy offset correction, accounting for the irregularities arising due to φ dependence (hence summing over iφ in Eq. 3.2), the only directional dependence left is in η . This dependence arises, mainly due to varying radiation lengths (λI ) traversed by the particles in the calorimeter. The jet response incorporates this by extending the central calorimeter (CC) response at forward pseudo-rapidity. This extrapolation is then fitted in the following way : 61 γ+ jet γ+ jet RMPF,η (E 0 ) p0,η + p1,η log(E 0/E0 ) + p2,η log2 (E 0/E0 ) Fη ≡ γ+ jet = γ+ jet (3.6) RMPF,CC (E 0 ) RMPF,CC (E 0 ) Fig. 3.3, shows the Fη for various values of E 0 . Figure 3.3: Relative MPF response, Fη for different values of E 0 in γ + jet events. Showering Corrections The showering corrections include effects of energy profiles of particles that deposit en- ergy in and out of the cone of R = 0.5 . The profiles are created as a function of ∆R = q 2 2 y particle − y jet + φ particle − φ jet , which essentially determines the the separation of incident particle from the jet cone axis. Then the correction factor can be defined as : ∑Rcone Rcone ∆R=0 Ein + ∑∆R=0 Eout S= (3.7) ∑∞ ∆R=0 Ein wherein, Ein includes the sources of energy deposited from within, inside the cone, while Eout represents the sources of energy deposited from outside the cone. The energy profiles in the data are then compared to the Monte Carlo particles (without the offset applied) and the contributions from the NP and MI events, to get an absolute showering correction factor. 62 Closure With all the correction factors of Eq. 3.1, in place, it is desired that the < E coorected jet > particle matches with the < E jet >. This sort of ratio (known as internal closure) is expected to be one at least for the Monte Carlo simulations. This is shown in Fig. 3.4 for |η jet | spanned in intervals of 0.4. The dashed lines show the uncertainties expected and the major source is recognized as that of the η dependent response correction factor, shown in   corrected,DATA Fig. 3.5. A similar ratio of / is evaluated as a function of pT jet in different |η detector jet | regions. The data selected here is similar to that used for deriving MPF response correction, albeit a bit relaxed in the sense that the restriction on number of primary vertices is removed, for the sake of completeness. This is shown in Fig. 3.6. The uncertainties (dotted lines in the figure) are defined as the sum in quadrature of data and MC uncertainties. In summary, the jet energy scale for a few values of jet transverse energy can be seen in Fig. 3.7. For most values of ET , it is found to be less than 1.5. More details can be found in (Ref.[46, 47]). 63 Figure 3.4: Internal closure of η − dependent corrections for Rcone = 0.5 jets in γ + jet sample. Figure 3.5: Relative uncertainty on Fη . 64 Figure 3.6: Data-to-MC closure as a function of pT in different |η detector jet | bins for Rcone = 0.5 jets in γ + jet sample. Figure 3.7: Jet Energy Scale correction factor as a function of pseudo-rapidity for a few measured values of ET . After applying the corrections, this analysis includes jets with pT > 15 GeV , within |η detector jet | < 3.4 . Additionally, jets containing muons within ∆R(µ, jetaxis) < 0.5 are corrected for the momentum that is carried away by the muon and neutrino. Since, these muons could be have been generated, via semi-leptonic decay of B-hadrons (containing 65 b-quarks). The neutrino momentum is considered to be the same as that of the muon, for such cases. 3.6.1 Sample Dependent JES In the previous section, the observed jet energy in data and simulations were corrected to the incident particle level. While, it was seen that such a correction involves a response that depends on the kinematics of the incident particle, no dependence on the flavor of the inci- dent particle was considered. This section, explores such dependence, taking into account, the fact that the incident particle (hadron) could have been generated with parent parton as gluon, light quarks or heavy quarks, which in turn are products of the hard scattering process of the underlying event. The reason for considering this is that the particles of dif- ferent flavor, decay in different topologies and hence the correction involving this aspect is called f lavor dependent jet energy scale or sample dependent jet energy scale. The idea is to look for all spatially matched particles inside a cone of R = 0.5 and add the energies, weighted with their response for each particle. This quantity can then be compared between data and simulations. The measured jet energy taken here is without the offset corrections arising due to the NP and MI effects (as discussed in the previous section). The correc- ∑i Ei ·Rdata tion factor F = i , is then multiplied to (E measured jet − Eo ) , to which the corrections ∑i Ei ·RMC i (briefed in the previous section), such as showering and η dependent response are applied. First, a closure for Monte Carlo simulations is obtained by comparing the measured ener- gies and the truth values for the particles inside a jet. The response is measured for particles such as γ, e± , µ ± , π ± , K ± , K0S , K0L , p± , n & Λ, with each being parametrized as a function of γ pT in the γ + jet sample as follows : 66 " !#    E+p1γ E+p3γ RMC γ = 0.25 · p0γ · 1 + Er f q 1 + Er f √ 2·p2 γ 2·p4γ " !# " !# E+p1 E+p3 RMC = 0.25 · 1 + Er f q e± 1 + Er f q e± f or pT > 0.3 else 0 e± 2·p2e± 2·p4e±   RMC µ ± = p 0 + p1 · E .Landau(E, p2 , p3 ) µ± µ± µ± µ± h i RMC = p0 · 1 − p1 · (E/0.75) p2h −1 h h h wherein, Rγ , Re , Rµ , Rh represent the photon, electron, muon and hadronic response, respectively [p0 , p1 , p2 , p3 being the parameters]. The ratio of the measured jet energy (without offset) and sum of the particle energies, with the above responses is shown in the figure below for |η detector 0 = pγ · cosh η jet .  jet | < 0.4 , as a function of E T As expected, this ratio is close to 1. The responses of photon, e± and µ ± are similar in 67 data and hence to take this up to the next level, the hadron response Rh , needs to be tuned to the data. This is done by introducing 3 extra parameters (A, B, C) in the expression for RMC h above, such that : h i Rdata = C · p0 1 − A · p1 · (E/0.75) p2h +B−1 h h h This is then tuned for data selected, with two different selections. First, a tight photon selection, in which a photon of high purity is demanded in γ + jet events, and another one in which the track isolation requirement is reversed and thus selecting mainly dijet events. This tuning is shown in Fig. 3.8 for both kind of events. Figure 3.8: Rh tuning in MC for |η detector jet | <0.4 for tight photon selection (left) and reversed track isolation (right) The figure below, shows the difference between not tuned and tuned MC for a mixed sample of γ + jet and dijet events with tight photon requirement. This is overlay-ed on the data (solid black dots). A much better description of data is achieved for the tuned MC for 68 |η detector jet | < 0.4 , as shown below. This procedure is carried out up to an η of 2.5 and a similar agreement is seen in all the rapidity bins. (Ref.[48]) Including all the responses, the correction factor F , in various rapidity bins is shown in the figure to follow.Here, the blue dots represent gluon jets, the red ones are the b-quark jets , green represents light quark jets and black is the original γ + jet sample. Including the systematic uncertainties (Ref.[49]), the ratio of the F factor of these 3 flavor of jets to that of the average γ + jet, can be used to correct the jet energies in any sample. The deviation jet of such a ratio from 1, is shown in Fig. 3.9 , as a function of pT , in different rapidity bins. 69 Figure 3.9: Correction factor for MC-data difference in jet response for different flavor of jets. [Light quarks(top), gluon(middle) and b-quarks(bottom)] 3.7 Missing Transverse Energy Due to the initial state of unpolarized beams of p & p¯ at the Tevatron, the momentum bal- ance in the transverse plane is natural, to be asked for. Owing to the complex final states, it is not plausible to track down the energy carried away by the products of the collision, that are drained down the beam pipe. With this limitation, the only signature of particles that escape undetected i.e. neutrinos, exotics, is through the momentum balance in the transverse plane. This information is often contaminated with the calorimeter noise (es- pecially in the coarse-hadronic section), pile up events, clusters in the calorimeter that do not end up in jets (unclustered energy), physics objects resolutions etc. Missing transverse energy (MET) is computed by taking the negative vectorial sum of all the energy recorded by calorimeter cells in the transverse plane (above certain threshold). The coarse-hadronic 70 cells are added only if those were considered to form a jet. Hence one can write : ~6 E T = − ∑ ~ETicell (3.8) ETicell >0.1 GeV ~T ) is a vectorial quantity and can be projected into components It can be seen MET (6 E in any direction in the transverse plane and hence it is also referred to as the missing trans- verse momentum. ~6 E T is then corrected for the energy corrections, that are applied to the reconstructed objects such as (jet energy scale) and for the momentum of all the muons in an event, corrected for their energy loss in the calorimeter. 3.8 b-jet identification (tagging) and Secondary Vertices b-quark identification is an important component of this analysis, since t t¯ decay products contain two b-quark jets and hence serve as an important discriminator of the signal from rest of the background events. b-jet identification, not only requires good jet identification but also requires combining the information of the tracking system. To achieve this, clusters are formed using the tracking system information, to reconstruct, secondary vertices. Due to their relative long lifetime, B hadrons travel on the order of a mm, before decaying. Thus, the tracks originating from B-hadron decays, seem to come from a displaced vertex. The jet is termed ’taggable’ if there are at least two tracks associated with the jet, with at least one hit in the SMT sub-detector system. One of the tracks has a pT > 1 GeV , while the other has pT > 0.5 GeV . The longitudinal distance between the secondary vertex and the distance of closest approach (|z − d.c.a.|) is required to be less than 0.4 cm. 71 As seen from the figure above, secondary vertex taggers (SVT) can be used to iden- tify b-jets (Ref.[50]) and aid in the process of ’b-tagging’. There are other identifiable / several such variables are signatures that associate tracks to a corresponding b-jet. At DO, combined in an artificial neural network (NN) to provide discrimination between b-jets and other light flavor jets. A rank is assigned to the following variables, that enter the NN : • The impact parameter significance, defined in Sec. 2.3.2, which represents the sig- nificance of the distance between the primary vertex and the secondary vertex in the xy plane. • A combined variable based on the number of tracks with specific values of the impact parameter significance • a probability for all tracks to originate from a primary vertex • a goodness-of-fit measure χ 2 /nDF for the secondary vertex with highest impact pa- rameter significance • number of tracks that originate from the secondary vertex with highest impact pa- rameter significance • the invariant mass of the above mentioned secondary vertex • number of secondary vertices inside a jet. 72 Figure 3.10: Neural network b-tagger output for b-jets and light flavor jets. The NN , then combines the above variables to provide an output lying in [0, 1], per jet. Fig. 3.10 shows the NN output for b-quark jets and light flavor jets in a sample. As can be seen, the b-jets have the NN output peaking closer to 1, in contrast to that of the light flavor jets (u, d, s, g). In this analysis, a cut on the NN output value of 0.65 is applied. This is termed as MEDIUM operating point and has an efficiency 60% in most of the kinematic region. The light jets have an efficiency of less than 2% with this operating point. The pT, η based efficiency ,derived from the data is termed as tag-rate functions (T RF). This provides a probability for a particular jet to be “b-tagged”, “c-tagged” or light flavor tagged. Such a parametrization for b-jets, is shown in the figure below. Tag rate function (TRF) for b-jets. CHAPTER Four Event Selection This chapter details the Monte Carlo simulation and data samples used for this analysis. It is important to compare the kinematic quantities between data and simulation. For this reason, a large number of events are simulated to describe a certain process and then filled in a histogram to be able to compare to the data. The data sample used in this analysis was collected between June 2006 and September 2009 and corresponds to a total integrated luminosity of about 4.3 f b−1 of RunIIb running of DO. / There are several detector and physics related steps that need to be taken into account, while simulating a process describ- ing the initial and final states of a collision. This can be broken down into several steps as described below. 73 74 4.1 Monte Carlo event generation Figure 4.1: Stages of simulation, describing a hadron-hadron collisions by MC event gen- erators [5]. The Monte Carlo method generates a large number of simulated collision events, entailing the final state particles and their momenta. It works via random sampling of a multidimen- sional phase space of the final state particles, for a given specified theoretical process such as top-anti top production. The idea is to assign certain probability for N points in the mul- tidimensional phase space for a large N, such that the probability corresponds to the actual event produced in the real world. The method can be considered to follow various steps of evolution, starting from the hard subprocess , which represents the hard scattering of the partons (inside proton, anti proton) such as valence quarks and is shown as the black sphere in Fig. 4.1. The probability of finding the parton inside the proton or the anti proton, is given by Parton density f unctions and for this analysis, CTEQ6L1 parton distribution func- tions are used (Ref.[11]). These hard scattering partons carry a large momentum fraction of the incident hadrons and hence the outgoing fundamental particles (top, anti top or new hypothetical particle produced in the black region) represent high momentum scale (q2 ). The outgoing color charged quarks or gluons radiate heavily in the process of acceleration 75 inside a color charged field and split into collinear partons, represented by the brown wavy lines in Fig. 4.1. The process is called parton showering. The final state radiation (FSR) is the parton showering of the outgoing partons in the hard sub process, while the initial state radiation (ISR) is the same for the incoming partons of the hard sub process. This happens, until the virtuality scale (q2 ) reaches the hadronization scale, represented by yel- low ellipses in the above figure. At this point, quarks get confined into hadrons and this process is modeled by Lund string model, in which the self interacting gluon field, acts as a “string” of color flux. The hadrons thus formed decay further and reach the detector. There are also some partons, that do not take part in the hard scattering process and constitute, what is known as Underlying event, represented by the green region in the above figure. PYTHIA (Ref.[51]), is a leading order generator, which implies that it simulates only the lowest-order terms in the perturbative treatment of the hard sub process under consid- eration. The parton showering method used in PYTHIA, takes care of mostly soft partons, collinear with the original parton by summation of higher order logarithmic terms through a Sudakov f orm f actor, that governs the parton shower evolution. Hence it allows for a good description of the jet substructure but a less accurate modeling of jet multiplicity. ALPGEN (Ref.[52]) is also a leading order generator but the hard sub process, ISR, FSR are modeled by employing a matrix element method. This method successfully describes the well separated partons with large transverse momenta, by including the quantum inter- ferences between included diagrams through matrix elements of hard sub processes. Thus the number of radiated quarks or gluons, associated with the hard sub process is always mentioned, when simulating with ALPGEN. Thus a process such as top-anti top produc- tion is referred to as t t¯ + nl p , where nl p = 0, 1, 2..etc., denotes the number of light partons associated with the hard scattering. ALPGEN models the final state partons pretty well but requires interfacing with another generator such as PYTHIA, to model the formation of jets through showering and hadronization. This process is referred to as ALPGEN+PYTHIA, 76 henceforth. In this process, PYTHIA is bound to generate additional jets through shower- ing, which may have been already modeled by a higher value of nlp for a final state parton in ALPGEN. To avoid this double counting in the phase space, a MLM matching algo- rithm is used to determine whether the jets correspond to original final state partons from ALPGEN (Ref.[53]) In this analysis, ALPGEN+PYTHIA was used to generate t t¯ + 0l p, t t¯ + 1l p, t t¯ + 2l p jets 2   with a top mass of 172.5 GeV. A dynamical factorization scale with µF2 = mt2 + ∑ pT , was used. The cross section for t t¯ production was scaled to its next-to-next to leading +0.56 order (NNLO) value of 7.48−0.72 pb (Ref.[13]). The main physics background of W+jets was also simulated using ALPGEN+PYTHIA. Three different sub samples were generated ¯ cc¯ &W qq¯ + nl p. Here, q stands for light quarks (u,d,s) and for this purpose : W bb,W gluons and nl p = 0, 1, 2, 3. W c sub processes were included in the W+light parton sample with massless charm quarks. The number of W bb¯ and W cc¯ events were increased, relative to W+light partons to match their respective NLO cross sections. The lesser significant backgrounds of Z (→ e+ e− , µ + µ − , τ + τ − ) + jets were generated with ALPGEN+PYTHIA ¯ c¯ & Zqq¯/Zgg plus light partons. The Z samples by breaking up into sub samples of Zbb,Zc were normalized to their NNLO cross section value of 256 pb (Ref.[54]). This is 1.3 times the value provided by leading order ALPGEN. In a similar fashion, the population of Zbb¯ was scaled by a factor of 1.52 and that of Zcc¯ by 1.67. The other backgrounds include single top quark production, which was simulated using COMPHEP-SINGLETOP (Ref.[55]). The top quark mass was set to 172.5 GeV. The cross section for single top quark production was computed to NNLO and NNNLO threshold corrections in the s and t − channels and was found to be 3.3 pb (Ref.[56]). The diboson (WW, W Z, ZZ) were simulated using PYTHIA and scaled to their respective NLO cross sections of 12.3pb, 3.7pb & 1.4pb. (Ref.[57]). 77 4.2 Detector Simulation The stable long-lived particles that are produced as the product of the simulation process, described above, interact with the bulk material of the detector. These interactions and the interactions with the magnetic field, allow the particles to be identified. To simulate these interactions, a detailed model of detector composition and geometry , as well as a precise knowledge of solenoidal and toroidal magnetic fields is necessary. The evolution of particles through the detector is based on a software called GEANT3 (Ref.[58]). It uses Monte Carlo methods, random sampling of phase space that is associated with several distinct interactions such as ionization of bulk material, deflection of charged particle in a magnetic field. These are then translated to detector signals that is used as raw data to reconstruct physical objects such as jets, electrons, muons etc. The Monte Carlo simulation of an event, represents a single collision of proton and anti- proton. In practice, there are multiple p p¯ collisions that take place in each beam crossing. The detector signals, thus produced are to be accounted for using the minimum bias back- ground events, which depend on the instantaneous luminosity. Such events collected (as described in Sec.3.4) are overlay-ed on to the MC events. The distribution, thus obtained of the instantaneous luminosity is called luminosity pro f ile. This accounts for detector oc- cupancy at high luminosity, for residual signals from preceding bunch crossings and the particles present in beam halo. The events in the MC are not selected based on any triggers on the reconstructed objects, that would fire in case of actual data taking. To account for this, trigger probabilities are derived as a function of kinematically reconstructed objects and combined into a global trigger probability of an event. This parametrization of the trigger probabilities, provides only a coarse approximation of the actual trigger criteria. The temporal and luminosity dependence of such parametrization is taken into account through event weights. 78 Following the above procedure, some insignificant effects remain to be corrected to match the distribution of the simulated events to that of the data. In general, it is found that the resolutions of the reconstructed objects in the simulated events, are slightly better than those in the data. This can be attributed to the fact that the detector simulation does not take into account, detector aging and electronic noise arising from radiation effects etc. Some of these are accounted for, by re weighting the distribution of events in the simulation for example that of the zPV , which is not perfectly Gaussian in nature for the data. The other inefficiencies in data, such as that of jet identification (JetID), vertex confirmation of jets are accounted for, by applying scale factors to simulated events, which are interpreted as probabilities that are less than 1. In such cases, the scale factors are applied by randomly rejecting the reconstructed objects, for which the inefficiencies in data were found, from the simulated events. 4.3 Event selection and background modeling This analysis focuses on the semileptonic decay mode of t t¯. The final state has the sig- nature of an isolated lepton with high transverse momentum, several jets, large 6 ET . The following discussion describes the selection based on the reconstructed objects in two sep- arate channels i.e. e + jets and µ + jets. Further, the selection criteria is designed to define a data sample enriched with W + jets and t t¯ events and hence events with 4 or more jets are considered. Selection, common to both the channels is as follows : • Good jets (defined in Sec.3.5.3) emanating from a primary vertex with |zPV | < 60 cm and at least 3 tracks attached to it, are required. These are also referred to as vertex confirmed jets. • At least 4 such jets with pT > 20 GeV and |η| < 2.5 are required, with an additional 79 requirement on the jet with highest pT , also termed as leading jet, is required to have pT. > 40 GeV . For e + jets channel, additional selection requirement is as follows : • An isolated electron with pT > 20 GeV and |η| < 1.1 is required. • The electron should have a |∆z(e, PV )| < 1 cm. • No second isolated lepton (e± , µ ± ) with pT > 15 GeV is allowed. • Missing transverse momentum (6 pT )is required to be greater than 20 GeV • ∆φ (e, 6 pT ) > 2.2 − 0.045× 6 pT is required. This is referred to as traingle cut, hence- forth. The above selection allows 1002 data events in the e + jets channel to pass through. For µ + jets channel, additional selection requirement is as follows : • An isolated muon with pT > 20 GeV is required. • Invariant mass of the selected muon and any other muon in the event must comply with mµ µ < 70 GeV or mµ µ > 110 GeV to reject any Z(→ µ µ) + jets events. • No second isolated lepton with pT > 15 GeV is allowed. • |∆z(µ, PV )| < 1 cm is required as in e + jets channel • 6 pT > 25 GeV is required • The triangle cut takes the form, ∆φ (µ, 6 pT ) > 2.1 − 0.035× 6 pT in this channel. A total of 807 events, survive the above selection cuts in the data. The main two standard model processes that produce events with an isolated lepton, 6 pT and at least four jets are t t¯ production and W + jets production. Single top, Z + jets 80 and diboson production can also give rise to such a final state but with much smaller cross sections. The next most important source of events are the multi jet events in which the energy depositions from a jet is mis-identified as a lepton and the 6 pT is mismeasured. The multijet background model is estimated from the multijet events that enter the final data sample using measured selection efficiencies and a data sample, termed as ’loose’, that is a superset of the final data sample for this analysis. The loose sample is obtained by using less stringent cuts for the identification of the leptons. The number of events in the loose and final data samples are denoted by N 0 and N, respectively. N l j denotes the combined number of events with genuine leptons in the loose sample, N j j corresponds to the number of multijet events in the loose sample, εl is the efficiency for a lepton in the loose sample to also pass the final lepton selection, and ε j is the efficiency for a misidentified jet in the loose sample to also pass the final lepton selection. With these definitions, one can write the following equations : N0 = Nl j + N j j and N = εl N l j + ε j N j j Solving the above system of equations for N jl , N j j yields : N − ε jN0 εl N 0 − N Nl j = and N jj = εl − ε j εl − ε j The efficiency εl for the true leptons is found to be 86.9 ± 2.2% for the e + jets events and 93.9 ± 2.2% for µ + jets events. These numbers were obtained from the corresponding W + jets and t t¯ Monte Carlo samples. The efficiency ε j for jets that get mis-identified as lepton, to pass the isolation selection was measured directly from the data. For this purpose, events with 6 pT < 10 GeV , which are dominated by misidentified leptons were used to calculate ε j , as the ratio of number of events in the final and in the loose data samples. For e + jets events, ε j = 13.0 ± 3.0% and for µ + jets events, ε j = 30.6 ± 3.1% was estimated. 81 The multijet background along with the other backgrounds (except W + jets), based on their respective cross section, is subtracted from the data. The resultant is then compared to the expected number of W + jets events, to make up the resulting composition of the final data sample. The data sample compostion the the 2 channels is shown in Table 4.1 Table 4.1: Composition of the final data sample in e + jets and µ + jets channel 4.4 Data and Monte Carlo comparison Pseudo-data samples are constructed from the MC events to calibrate mass in the analysis. For this purpose, it is important that the distributions of topological and kinematic quan- tities agree between data and simulation. Such comparisons in various channels is shown below, after normalizing various components according to Table 5.1 • e+jets channel Fig. 4.2, shows the Data-MC agreement for distribution of certain topological variables, which are defined in Chapter.6. Fig. 4.3, shows the same for certain kinematic distributions. Fig. 4.4 depicts the dis- tributions of other interesting quantities such as the W boson transverse mass and invariant mass of t t¯ system in the data composed of simulated signals and various backgrounds (color code presented in the legend of the plots). 82 0 Figure 4.2: Data-MC comparison for topological variables : Aplanarity, Centrality, KTmin , 6ET in e + jets channel 83 Figure 4.3: Data-MC comparison for lepton pT , leading jet pT , η, φ in e + jets channel Figure 4.4: W transverse mass and invariant top mass distributions • µ + jets channel Based on the sample composition, described in Table 5.1, the control plots for the topolog- ical and kinematic quantities in µ + jets channel are shown in Fig.4.5-4.7. 84 0 Figure 4.5: Data-MC comparison for topological variables : Aplanarity, Centrality, KTmin , 6ET in µ + jets channel 85 Figure 4.6: Data-MC comparison for lepton pT , leading jet pT , η, φ in µ + jets channel Figure 4.7: W transverse mass and invariant top mass distributions in µ + jets channel 4.4.1 Data-MC comparison based on b-tagging Often times, as in this analysis, it is useful to split the samples in the order of the signal content. Since, t t¯ decay products contain two b-quarks, the two channels under considera- tion (e + jets, µ + jets) are further split according to the b-quark content. In this analysis, 86 the events are further sieved through in two different channels, using the NN b-tagging output (defined in Sec.3.8). For an NN output value greater than or equal to 0.65, the jet in the event is considered b-tagged. This allows to have different sample composition in two channels, characterized as follows : • two or more b-tagged jets found in the event (≥ 2tag) • exactly one b-tagged jet found in the event (1 tag) The signal content in each of the 4 channels [(1, ≥ 2)tags ⊗ (e, µ) + jets] above, is good enough to perform calibrations of top mass in each of them separately. Following plots show the data-MC comparison of a few of the selected topological variables that are used in Chapter 7, to discriminate between signal and background. 87 0 Figure 4.8: Data-MC comparison of Aplanarity, Centrality, KTmin ,6 ET , W transverse mass and t t¯ invariant mass in e + jets, ≥ 2tags channel 88 0 Figure 4.9: Data-MC comparison of Aplanarity, Centrality, KTmin ,6 ET , W transverse mass and t t¯ invariant mass in µ + jets, ≥ 2tags channel 89 0 Figure 4.10: Data-MC comparison of Aplanarity, Centrality, KTmin ,6ET , W transverse mass and t t¯ invariant mass in e + jets, 1tag channel 90 0 Figure 4.11: Data-MC comparison of Aplanarity, Centrality, KTmin ,6ET , W transverse mass and t t¯ invariant mass in µ + jets, 1tag channel CHAPTER Five Kinematic Fitting for Top Mass 5.1 Introduction The top quark event selection, as described in the previous chapter, selects events with four or more jets. In an ideal scenario, two jets arise from a b-quark and an anti b-quark and two from light quarks that are the result of a W boson decay. If the correspondence between jets and partons from the top decay were known, the top quark event could easily be re- constructed from the 4-vectors of the final state products. In practice this is not known and therefore all possible jet-parton assignments have to be considered as the input to a fitter that uses goodness-of-fit χ 2 to give a measure of how far the input quantities have to be moved to respect the constraints of a top quark event. Such a measure can be estimated for each jet-parton assignment. For four jets in an event, this amounts to 12 possible combi- nations per event [choice of two out of the 4 jets arising from W boson decay, one of the remaining two, originating from a b-quark while the other from anti b-quark = 42 × 21 ].   This ambiguity increases if more than 4 jets are used for assignments to the partons from top decay. Only the 4 jets with highest momenta are chosen as input to the fitter. This al- lows one to deal with manageable ambiguities. These 4 jets are often referred to as leading 91 92 jets, in the following discussion. For the reconstruction of the top quark event, one requires the 4-momenta of the partons from top decay rather than the 4-vector of the jets observed. This mapping is defined in the following sections. 5.2 Parton Level Corrections The term Parton Level Corrections (PLC), Ref.[59, 60], is used to describe the relation between initial state partons and the final state partons. The final state partons undergo hadronization, fragmentation and hard gluon radiation. These final state partons manifests themselves as jets of particles whose response can be measured in the DØ calorimeter. The PLC’s, correct the energy of the jets to the energy of the partons, from which they originated. As described in Sec.3.6.1, the heavy-quark hadronization differs from the light-quark hadronization. The PLC are thus derived separately for jets originating from light quarks (u,d,s,c) and b-quarks. Also, the response varies with detector η and hence four η-ranges, crudely classifying central, inter-cryostat and forward end-cap regions are used to derive these corrections. |η| ∈ [0, 0.5) is denoted as Region 1, |η| ∈ [0.5, 1) is denoted as Region 2, |η| ∈ [1, 1.5) as Region 3, and |η| > 1.5 as Region 4. The corrections are derived using Monte Carlo simulation of the process t t¯ → l + jets (l=e, µ) which are generated for top quark masses 150, 160, 165, 170, 172.5, 180, 185 and 190 GeV. The hard scattering process is modeled by the event generator ALPGEN+PYTHIA, as described in Sec.4.1. These generated events are then processed through the whole DØ reconstruction algorithm. Using the Monte Carlo information, the primary partons from t t¯ decay are matched to the jets using a jet cone algorithm with cone size R = 0.5. The jets are selected to be isolated with respect to all other jet objects by at least 4R = 0.5, and also matched to the final state parton within 4R = 0.5. The event selection is similar to the one described in 93 Chapter 5, albeit a bit relaxed. • At least four jets with pT > 15 GeV each with |η| < 2.5 • exactly one isolated charged lepton (e or µ) with pT > 20 GeV and |η|< 1.1 (for electrons), |η| < 2.0 (for muons) • Missing Transverse Energy (6ET ) > 20 GeV The closure tests for the corrections derived using these events are described later in this chapter. The following generator level information is used for closure : 1. The W boson mass formed by the two light quarks is compared to the nominal W boson mass of 80.4 GeV. 2. The top quark mass, which is obtained from the kinematic fit is compared to the generator level mass of the various t t¯ samples. Since, eventually, a simultaneous fit to the jet energy scale factor (αJES , a multiplicative factor by which the energy of all jets in an event are scaled) is done in-situ, it is important to parametrize the corrections with respect to the correct choice of variables. The most likely factor is extracted from the data/simulation by using the hadronically decaying W boson in the event. The output JES is equal to input JES if, for a reconstructed W boson mass: MW (αJES × E1parton , αJES × E2parton ) = αJES × MW (E1parton , E2parton ) (5.1) wherein, E1parton , E2parton are the parton level corrected jet energies of the two jets that are used to reconstruct the W boson. The left hand side of the above equation being the 94 reconstructed W boson mass at the αJES , while the right hand side being the reconstructed W boson mass at JES=1. Since the PLC (as will be seen latter) are nonlinear, we derive them as a function of jet energies so that Eq.5.1 holds true at least approximately. This is also the reason for using a relaxed event selection for deriving these corrections. 5.2.1 Light Quark Corrections Based on the framework, described above, the matched jet-parton pairs for light quarks in the t t¯ events are used to plot parton energy distributions in various jet energy bins. The choice of these bin ranges is made separately in each detector η region according to the statistics available. Appendix A shows the parton energy distribution in different jet energy bins. Each of these distributions is fitted with a Gaussian near the peak and the mean of the Gaussian is taken as the most likely parton energy value (E parton ) for that jet energy bin. The values of E parton are then fitted with a polynomial of up to 6th degree as a function of E jet , as shown in Fig.5.1-5.4. E jet is the mean jet energy of the bin. The fits define the mapping of E jet to E parton for light quark jets. |η det | p0(GeV) p1(GeV) p2(GeV) p3(GeV) p4(GeV) p5(GeV) p6(GeV) [0, 0.5) 7.85 0.75 0.00478 −6.42 × 10−5 4.55 × 10−7 −1.56 × 10−9 1.99 × 10−12 [0.5, 1.0) 6.80 0.86 0.000867 −6.66 × 10−6 3.42 × 10−8 −8.53 × 10−11 0.0 [1.0, 1.5) 8.13 0.85 0.000467 −1.01 × 10−7 −5.79 × 10−9 3.96 × 10−12 0.0 [1.5, 2.5) 15.13 0.75 0.001385 −3.09 × 10−6 0.0 0.0 0.0 Table 5.1: Fit parameters for E parton = p0 + p1 × E jet + p2 × E 2jet + p3 × E 3jet + p4 × E 4jet + p5 × E 5jet + p6 × E 6jet for light quarks. 95 (Ejet-Eparton) vs. Ejet for light quarks in region 1 Ejet -Eparton/GeV 5 0 -5 -10 -15 -20 -25 0 50 100 150 200 250 300 Ejet /GeV Figure 5.1: E parton vs. E jet & (E parton −E jet ) vs. E jet for light quark jets in Region 1, |η| ∈ [0, 0.5). 96 Eparton vs Ejet for light quarks in region 2 Eparton/GeV 240 220 200 180 160 140 χ2 / ndf 1.454 / 34 120 p0 6.801 ± 0.458 100 p1 0.86 ± 0.0159 80 p2 0.0008671 ± 0.0002018 60 p3 -6.658e-06 ± 1.607e-06 40 p4 3.416e-08 ± 9.742e-09 20 p5 -8.535e-11 ± 2.671e-11 0 50 100 150 200 250 Ejet /GeV (Ejet-Eparton) vs. Ejet for light quarks in region 2 Ejet -Eparton/GeV 5 0 -5 -10 -15 -20 -25 -30 -35 0 50 100 150 200 250 Ejet /GeV Figure 5.2: E parton vs. E jet & (E parton −E jet ) vs. E jet for light quark jets in Region 2, |η| ∈ [0.5, 1.0) 97 Eparton vs Ejet for light quarks in region 3 300 Eparton/GeV 250 200 χ2 / ndf 1.002 / 22 150 p0 8.133 ± 1.663 p1 0.8501 ± 0.05459 100 p2 0.0004672 ± 0.001039 p3 -1.016e-07 ± 1.092e-05 p4 -5.792e-09 ± 4.935e-08 50 p5 3.956e-12 ± 7.579e-11 0 50 100 150 200 250 300 350 Ejet /GeV (Ejet-Eparton) vs. Ejet for light quarks in region 3 10 Ejet -Eparton/GeV 0 -10 -20 -30 -40 -50 0 50 100 150 200 250 300 350 Ejet /GeV Figure 5.3: E parton vs. E jet & (E parton −E jet ) vs. E jet for light quark jets in Region 3, |η| ∈ [1.0, 1.5) 5.2.2 b-quark Corrections A similar procedure, as for light quark jets is used for the matched jet-parton pairs of b- quarks in t t¯ events to derive the Parton Level Corrections. The Table 5.2 shows the fit 98 Eparton vs Ejet for light quarks in region 4 Eparton/GeV 300 250 200 χ2 / ndf 2.442 / 23 150 p0 15.13 ± 4.353 p1 0.7504 ± 0.1131 100 p2 0.001385 ± 0.0008915 50 p3 -3.094e-06 ± 2.181e-06 50 100 150 200 250 300 350 400 Ejet /GeV (Ejet-Eparton) vs. Ejet for light quarks in region 4 10 Ejet -Eparton/GeV 5 0 -5 -10 -15 -20 -25 -30 50 100 150 200 250 300 350 400 Ejet /GeV Figure 5.4: E parton vs. E jet & (E parton −E jet ) vs. E jet for light quark jets in Region 4, |η| ∈ [1.5, 2.5). 99 parameters. Appendix B shows the parton energy distributions in jet energy bins for the b-quarks. Fig.5.5-5.8, show the fits to E parton vs. E jet for b-quark jets. |η det | p0(GeV) p1(GeV) p2(GeV) p3(GeV) p4(GeV) p5(GeV) p6(GeV) [0, 0.5) 16.85 0.674 0.00225 −6.33 × 10−6 0.0 0.0 0.0 [0.5, 1.0) 17.03 0.721 0.00167 −4.71 × 10−6 0.0 0.0 0.0 [1.0, 1.5) 12.73 1.047 −0.00286 2.52 × 10−5 −9.52 × 10−8 1.22 × 10−10 0.0 [1.5, 2.5) 10.73 1.409 −0.00876 6.22 × 10−5 −1.92 × 10−7 2.11 × 10−10 0.0 Table 5.2: Fit parameters for E parton = p0 + p1 × E jet + p2 × E 2jet + p3 × E 3jet + p4 × E 4jet + p5 × E 5jet + p6 × E 6jet for b-quarks. Eparton vs Ejet for b- quarks in region 1 Eparton/GeV 240 χ2 / ndf 3.271 / 37 220 p0 16.85 ± 0.5626 200 p1 0.674 ± 0.02587 180 160 p2 0.002254 ± 0.0003578 140 p3 -6.334e-06 ± 1.504e-06 120 100 80 60 40 20 0 50 100 150 200 250 300 Ejet /GeV (Ejet-Eparton) vs. Ejet for b- quarks in region 1 Ejet -Eparton/GeV 15 10 5 0 -5 -10 -15 -20 -25 0 50 100 150 200 250 300 Ejet /GeV Figure 5.5: E parton vs. E jet & (E parton −E jet ) vs. E jet for b-quark jets in Region 1, |η| ∈ [0, 0.5) 100 Eparton vs Ejet for b- quarks in region 2 250 Eparton/GeV 200 150 χ2 / ndf 4.005 / 36 100 p0 17.03 ± 0.3679 p1 0.7212 ± 0.01288 50 p2 0.001675 ± 0.0002016 p3 -4.709e-06 ± 1.05e-06 0 0 50 100 150 200 250 300 Ejet /GeV (Ejet-Eparton) vs. Ejet for b- quarks in region 2 Ejet -Eparton/GeV 15 10 5 0 -5 -10 -15 -20 -25 0 50 100 150 200 250 300 Ejet /GeV Figure 5.6: E parton vs. E jet & (E parton −E jet ) vs. E jet for b-quark jets in Region 2, |η| ∈ [0.5, 1.0) 101 Eparton vs Ejet for b- quarks in region 3 Eparton/GeV 300 250 200 χ2 / ndf 0.6072 / 22 150 p0 12.73 ± 1.284 p1 1.047 ± 0.02825 100 p2 -0.002859 ± 0.0002885 p3 2.517e-05 ± 1.944e-06 p4 -9.516e-08 ± 9.206e-09 50 p5 1.216e-10 ± 2.025e-11 0 50 100 150 200 250 300 350 Ejet /GeV (Ejet-Eparton) vs. Ejet for b- quarks in region 3 Ejet -Eparton/GeV 10 0 -10 -20 -30 0 50 100 150 200 250 300 350 Ejet /GeV Figure 5.7: E parton vs. E jet & (E parton −E jet ) vs. E jet for b-quark jets in Region 3, |η| ∈ [1.0, 1.5) 102 Eparton vs Ejet for b- quarks in region 4 Eparton/GeV 300 250 200 χ2 / ndf 3.774 / 21 p0 10.73 ± 1.394 150 p1 1.409 ± 0.02494 p2 -0.008763 ± 0.000194 100 p3 6.222e-05 ± 1.321e-06 p4 -1.921e-07 ± 9.158e-09 p5 2.109e-10 ± 2.403e-11 50 50 100 150 200 250 300 350 Ejet /GeV (Ejet-Eparton) vs. Ejet for b- quarks in region 4 Ejet -Eparton/GeV 20 10 0 -10 -20 50 100 150 200 250 300 350 Ejet /GeV Figure 5.8: E parton vs. E jet & (E parton −E jet ) vs. E jet for b-quark jets in Region 4, |η| ∈ [1.5, 2.5) 5.3 Resolution Functions The kinematic fitter described later in this chapter, uses resolution functions in the com- putation of χ 2 . These are the uncertainties assigned to the kinematic variables of the re- constructed objects within which they are allowed to vary to reach an optimal solution for an event, respecting the given constraints. In the following sections, these resolution 103 functions are presented for parton level corrected jet energies and the angular jet variables pseudorapidity (η) and azimuthal (φ ), for the light and b-quarks. 5.3.1 Light Quark Resolution Since the jets are parton level corrected, the uncertainty on the parton energy for each jet energy bin is taken as the RMS of the Gaussian peak fitted to the parton energy distribu- tions, shown in Appendix A (for the light quarks). This provides a parton energy resolution parametrized as a function of jet energy. Table 5.3 shows the parameters for the polynomial fit to σ (E parton ) vs. E jet , which is shown in Fig.5.9-5.10. |η det | p0(GeV) p1(GeV) p2(GeV) p3(GeV) p4(GeV) p5(GeV) [0, 0.5) 4.55 0.05646 0.0001228 −4.55 × 10−6 3.033 × 10−8 −5.076 × 10−11 [0.5, 1.0) 5.017 0.08429 −0.0004756 1.854 × 10−6 0.0 0.0 [1.0, 1.5) 0.2849 0.5201 −0.007879 5.652 × 10−5 −1.773 × 10−7 2.033 × 10−11 [1.5, 2.5) −7.266 0.8105 −0.009131 4.847 × 10−5 −1.123 × 10−7 9.323 × 10−11 Table 5.3: Parameters of polynomial fit to σ (E parton ) vs. E jet for light quarks 104 Eparton resolution vs Ejet for light quarks in region 1 )/GeV χ2 / ndf 29.88 / 34 25 p0 4.55 ± 0.2156 parton p1 0.05646 ± 0.01413 σ (E p2 0.0001228 ± 0.0003161 20 p3 -4.55e-06 ± 2.764e-06 p4 3.033e-08 ± 8.798e-09 p5 -5.076e-11 ± 9.55e-12 15 10 5 0 50 100 150 200 250 300 Ejet /GeV Eparton resolution vs Ejet for light quarks in region 2 )/GeV χ2 / ndf 55.75 / 36 30 5.017 ± 0.308 parton p0 σ (E 25 p1 0.08429 ± 0.01281 p2 -0.0004756 ± 0.0001513 20 p3 1.854e-06 ± 5.261e-07 15 10 5 0 50 100 150 200 250 300 Ejet /GeV Figure 5.9: Energy resolution for light quarks in the detector η regions 1 and 2, plotted versus E jet . 105 Eparton resolution vs Ejet for light quarks in region 3 )/GeV χ2 / ndf 35.86 / 23 p0 0.2849 ± 2.291 35 parton p1 0.5201 ± 0.1121 σ (E p2 -0.007879 ± 0.002052 30 p3 5.652e-05 ± 1.711e-05 p4 -1.773e-07 ± 6.483e-08 25 p5 2.033e-10 ± 8.983e-11 20 15 10 5 0 50 100 150 200 250 300 350 Ejet /GeV Eparton resolution vs Ejet for light quarks in region 4 χ2 / ndf )/GeV 45 15.81 / 21 p0 -7.266 ± 1.363 parton 40 p1 0.8105 ± 0.02432 p2 -0.009131 ± 6.666e-05 σ (E 35 p3 4.847e-05 ± 5.979e-07 p4 -1.123e-07 ± 3.873e-09 30 p5 9.323e-11 ± 7.533e-12 25 20 15 10 50 100 150 200 250 300 350 Ejet /GeV Figure 5.10: Energy resolution for light quarks in the detector η regions 3 and 4, plotted versus E jet . The vectorial quantities, pseudo-rapidity (physics η or y) and azimuthal angle (φ ), for the parton matched jets also show some jet energy dependence. The pseudo-rapidity and azimuthal angles are not corrected to parton level. Fig.5.11 shows a typical distribution of (η parton − η jet ) and (φ parton − φ jet ) for light quarks in a particular jet energy bin. This dis- tribution is narrowly centered around zero and hence does not need additional corrections. 106 eta_Light_r1_5 eta_Light_r1_5 900 Entries 28283 Mean 0.000462 800 RMS 0.1118 χ2 / ndf 26.42 / 21 700 Prob 0.191 Constant 846.3 ± 9.6 600 Mean 0.0008155 ± 0.0004330 500 Sigma 0.03656 ± 0.00065 400 300 200 100 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (a) phi_Light_r1_9 phi_Light_r1_9 Entries 29636 Mean 0.0002231 1000 RMS 0.09331 χ2 / ndf 24.31 / 17 Prob 0.1112 800 Constant 1069 ± 11.6 Mean 0.0002488 ± 0.0003704 Sigma 0.03152 ± 0.00056 600 400 200 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (b) Figure 5.11: Distribution of the difference of angular variables at the parton and jet levels for light quarks. (a) pseudo-rapidity (b) azimuth The RMS of the distributions in Fig.5.11 are still parametrized as a function of jet energy in the four detector regions. Table 5.4 and Fig. 5.12-5.13 show the polynomial fits and parameter values for the pseudo-rapidity and Table 5.5 and Fig. 5.14-5.15 show the 107 same for azimuthal angle for light quarks. |η det | p0(GeV ) p1(GeV ) p2(GeV ) p3(GeV ) p4(GeV ) p5(GeV ) p6(GeV ) [0, 0.5) 0.12 −0.00425 8.25 × 10−5 −8.65 × 10−7 4.96 × 10−9 −1.46 × 10−11 1.73 × 10−14 [0.5, 1.0) 0.11 −0.00358 6.11 × 10−5 −5.55 × 10−7 2.71 × 10−9 −6.66 × 10−12 6.41 × 10−15 [1.0, 1.5) 0.066 0.000731 −3.89 × 10−5 5.13 × 10−7 −3.07 × 10−9 8.68 × 10−12 −9.43 × 10−15 [1.5, 2.5) 0.143 −0.00216 1.87 × 10−5 −9.09 × 10−8 2.47 × 10−10 −3.47 × 10−13 1.96 × 10−16 Table 5.4: Parameters of polynomial fit to σ (η parton − η jet ) vs. E jet for light quarks. η Resolution vs Ejet for light quarks in region 1 /GeV χ2 / ndf 61.45 / 33 0.06 p0 0.1116 ± 0.000206 parton p1 -0.004251 ± 4.077e-06 σ Eta 0.05 p2 8.254e-05 ± 2.861e-08 p3 -8.646e-07 ± 1.431e-10 p4 4.963e-09 ± 6.386e-13 0.04 p5 -1.463e-11 ± 2.527e-15 p6 1.725e-14 ± 8.457e-18 0.03 0.02 0.01 0 20 40 60 80 100 120 140 160 180 200 220 Ejet /GeV η Resolution vs Ejet for light quarks in region 2 /GeV χ2 / ndf 35.42 / 33 0.06 p0 0.1068 ± 0.0002751 parton p1 -0.003579 ± 4.1e-06 σ Eta p2 6.11e-05 ± 2.669e-08 0.05 p3 -5.548e-07 ± 2.852e-10 p4 2.711e-09 ± 9.693e-13 0.04 p5 -6.658e-12 ± 1.392e-14 p6 6.405e-15 ± 3.642e-17 0.03 0.02 0.01 0 20 40 60 80 100 120 140 160 180 200 220 Ejet /GeV Figure 5.12: Pseudo-rapidity (η) resolution vs. E jet for light quarks in detector |η| regions 1 and 2. 108 η Resolution vs Ejet for light quarks in region 3 /GeV 0.07 χ2 / ndf 17.94 / 20 parton 0.065 p0 0.06574 ± 0.001767 σ Eta p1 0.0007305 ± 3.441e-05 0.06 p2 -3.889e-05 ± 2.816e-07 p3 5.13e-07 ± 1.58e-09 0.055 p4 -3.066e-09 ± 7.879e-12 0.05 p5 8.684e-12 ± 6.048e-14 p6 -9.428e-15 ± 1.646e-16 0.045 0.04 0.035 0.03 0.025 50 100 150 200 250 Ejet /GeV η Resolution vs Ejet for light quarks in region 4 0.09 χ2 / ndf /GeV 20.1 / 21 p0 0.1428 ± 0.001191 parton 0.08 p1 -0.002164 ± 1.244e-05 σ Eta p2 1.872e-05 ± 4.36e-08 0.07 p3 -9.089e-08 ± 1.581e-10 p4 2.466e-10 ± 4.37e-13 0.06 p5 -3.466e-13 ± 1.031e-15 p6 1.96e-16 ± 2.466e-18 0.05 0.04 0.03 0.02 50 100 150 200 250 300 350 400 Ejet /GeV Figure 5.13: Pseudo-rapidity (η) resolution vs. E jet for light quarks in four detector |η| regions 3 and 4. |η det | p0(GeV ) p1(GeV ) p2(GeV ) p3(GeV ) p4(GeV ) p5(GeV ) p6(GeV ) [0, 0.5) 0.12 −0.0049 1.01 × 10−4 −1.21 × 10−6 6.76 × 10−9 −2.07 × 10−11 2.52 × 10−14 [0.5, 1.0) 0.11 −0.0034 5.85 × 10−5 −5.47 × 10−7 2.84 × 10−9 −7.62 × 10−12 8.23 × 10−15 [1.0, 1.5) 0.12 −0.0019 1.86 × 10−5 −9.87 × 10−8 3.10 × 10−10 −5.53 × 10−13 4.36 × 10−16 [1.5, 2.5) 0.09 −0.00037 −7.01 × 10−6 9.55 × 10−8 −4.80 × 10−10 1.11 × 10−12 −9.69 × 10−16 Table 5.5: Parameters of polynomial fit to σ (φ parton − φ jet ) vs. E jet for light quarks. 109 φ Resolution vs Ejet for light quarks in region 1 /GeV 0.07 χ2 / ndf 38.71 / 33 parton 0.06 p0 0.1212 ± 0.0002497 σ Phi p1 -0.004902 ± 5.344e-06 p2 0.0001012 ± 4.208e-08 0.05 p3 -1.121e-06 ± 2.278e-10 p4 6.762e-09 ± 1.075e-12 0.04 p5 -2.075e-11 ± 4.354e-15 p6 2.524e-14 ± 1.43e-17 0.03 0.02 0.01 0 20 40 60 80 100 120 140 160 180 200 Ejet /GeV φ Resolution vs Ejet for light quarks in region 2 /GeV 0.07 χ2 / ndf 41.94 / 34 parton p0 0.1053 ± 0.0002159 σ Phi 0.06 p1 -0.003428 ± 6.46e-06 p2 5.849e-05 ± 6.744e-08 0.05 p3 -5.469e-07 ± 2.141e-10 p4 2.837e-09 ± 1.793e-12 0.04 p5 -7.618e-12 ± 1.804e-14 p6 8.233e-15 ± 5.524e-17 0.03 0.02 0.01 0 50 100 150 200 250 Ejet /GeV Figure 5.14: Azimuthal (φ ) resolution vs. E jet for light quarks in detector |η| regions 1 and 2. 110 φ Resolution vs Ejet for light quarks in region 3 /GeV χ2 / ndf 6.884 / 21 0.08 0.1148 ± 0.006972 parton p0 -0.001998 ± 0.0002852 σ Phi p1 0.07 p2 1.86e-05 ± 5.806e-06 p3 -9.866e-08 ± 6.816e-08 0.06 p4 3.101e-10 ± 4.2e-10 p5 -5.526e-13 ± 1.241e-12 0.05 p6 4.357e-16 ± 1.383e-15 0.04 0.03 0.02 0.01 0 50 100 150 200 250 300 Ejet /GeV φ Resolution vs Ejet for light quarks in region 4 0.07 /GeV χ2 / ndf 30.28 / 20 parton 0.06 p0 0.08563 ± 1 σ Phi p1 -0.0003667 ± 1 p2 -7.005e-06 ± 1 0.05 p3 9.548e-08 ± 1 p4 -4.801e-10 ± 1 0.04 p5 1.105e-12 ± 1 p6 -9.698e-16 ± 1 0.03 0.02 0.01 50 100 150 200 250 300 Ejet /GeV Figure 5.15: Azimuthal (φ ) resolution vs. E jet for light quarks in detector |η| regions 3 and 4. 5.3.2 b-quark resolution Of the four jets that enter the kinematic fitter, two are hypothesized to be b-jets for each jet-parton assignmen. As in the case of light quarks, the energy and angular resolutions are parametrized as a function of b-jet energy. Again, the parton energy resolution is derived from the parton energy distributions in various b-jet energy bins (shown in Appendix B). Table. 5.6 and Fig. 5.16-5.17 show the b-parton energy resolution parametrization as a 111 function of to b-jet energy in four detector regions. |η det | p0(GeV ) p1(GeV ) p2(GeV ) p3(GeV ) p4(GeV ) p5(GeV ) p6(GeV ) [0, 0.5) 12.5 −0.39 0.0106 1.23 × 10−4 7.07 × 10−7 −1.93 × 10−9 2.01 × 10−12 [0.5, 1.0) 11.54 −0.18 0.0043 −3.92 × 10−5 1.62 × 10−7 −2.35 × 10−10 0.0 [1.0, 1.5) −2.88 0.62 −0.0081 5.20 × 10−5 −1.52 × 10−7 1.67 × 10−10 0.0 [1.5, 2.5) −14.79 1.19 −0.0153 9.23 × 10−5 −2.56 × 10−7 2.67 × 10−10 0.0 Table 5.6: Parameters of polynomial fit to σ (E parton ) vs. E jet for b-quarks. Eparton resolution vs Ejet for b- quarks in region 1 )/GeV 24 χ2 / ndf 45.39 / 34 p0 12.5 ± 0.1445 22 parton p1 -0.3928 ± 0.003785 p2 0.01056 ± 4.924e-05 σ (E 20 p3 -0.0001227 ± 3.73e-07 18 p4 7.066e-07 ± 1.336e-09 p5 -1.931e-09 ± 8.666e-12 16 p6 2.007e-12 ± 2.735e-14 14 12 10 8 6 0 50 100 150 200 250 Ejet /GeV Eparton resolution vs Ejet for b- quarks in region 2 )/GeV 28 χ2 / ndf 45.03 / 34 26 p0 11.54 ± 0.272 parton 24 p1 -0.1839 ± 0.007303 σ (E p2 0.004327 ± 6.367e-05 22 p3 -3.919e-05 ± 4.869e-07 20 p4 1.616e-07 ± 3.417e-09 18 p5 -2.348e-10 ± 9.226e-12 16 14 12 10 8 6 0 50 100 150 200 250 Ejet /GeV Figure 5.16: Energy resolution vs. E jet for b-quarks in the detector η regions 1 and 2. 112 Eparton resolution vs Ejet for b- quarks in region 3 )/GeV χ2 / ndf 22.8 / 22 35 p0 -2.878 ± 0.4182 parton p1 0.6165 ± 0.009114 σ (E 30 p2 -0.00813 ± 8.724e-05 p3 5.203e-05 ± 5.768e-07 25 p4 -1.519e-07 ± 3.503e-09 p5 1.673e-10 ± 8.49e-12 20 15 10 5 0 50 100 150 200 250 300 350 Ejet /GeV Eparton resolution vs Ejet for b- quarks in region 4 )/GeV χ2 / ndf 20.81 / 21 40 p0 -14.79 ± 1.385 parton p1 1.19 ± 0.03218 σ (E 35 p2 -0.01526 ± 0.0002964 p3 9.231e-05 ± 1.529e-06 30 p4 -2.563e-07 ± 5.835e-09 p5 2.673e-10 ± 1.166e-11 25 20 15 50 100 150 200 250 300 350 Ejet /GeV Figure 5.17: Energy resolution vs. E jet for b-quarks in the detector η regions 3 and 4. As in the case of light quarks, the angular variables for b-quarks are not corrected. Fig. 5.18, shows that the distribution of jet angular variables are close to the parton values and hence the RMS of the distribution is parametrized with respect to b-jet energy in different detector regions. This parametrization of the pseudo-rapidity of b-quarks is depicted in Table 5.7 and Fig. 5.19-5.20 while Table 5.8 and Fig. 5.21-5.22 shows the same for the azimuthal variable. 113 eta_B_all_r1_4 eta_B_all_r1_4 Entries 12749 300 Mean 0.00053 RMS 0.1343 χ2 / ndf 15.49 / 26 250 Prob 0.9476 Constant 295.8 ± 5.1 200 Mean -0.0006856 ± 0.0009064 Sigma 0.04761 ± 0.00145 150 100 50 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (a) (η parton − η jet ) for matched b jet-parton pair. phi_B_all_r1_7 phi_B_all_r1_7 Entries 19475 600 Mean -0.0001271 RMS 0.1125 500 χ2 / ndf 19.26 / 20 Prob 0.5053 Constant 594.9 ± 8.1 400 Mean -0.001397 ± 0.000530 Sigma 0.03614 ± 0.00081 300 200 100 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 (b) (φ parton − φ jet ) for matched b jet-parton pair. Figure 5.18: Distribution of the difference in angular variables for b-quark at parton and jet levels. 114 |η det | p0(GeV ) p1(GeV ) p2(GeV ) p3(GeV ) p4(GeV ) p5(GeV ) p6(GeV ) p7(GeV ) [0, 0.5) 0.12 −0.0039 6.49 × 10−5 −5.74 × 10−7 2.77 × 10−9 −6.84 × 10−12 6.73 × 10−15 0.0 [0.5, 1.0) 0.12 −0.0038 5.76 × 10−5 −4.78 × 10−7 2.26 × 10−9 −5.96 × 10−12 7.94 × 10−15 −3.99 × 10−18 [1.0, 1.5) 0.14 −0.0035 5.21 × 10−5 −4.29 × 10−7 1.99 × 10−9 −4.79 × 10−12 4.56 × 10−15 0.0 [1.5, 2.5) 0.17 −0.0028 2.44 × 10−5 1.06 × 10−7 2.14 × 10−10 −1.14 × 10−13 −1.07 × 10−16 0.0 Table 5.7: Parameters of polynomial fit to σ (φ parton − φ jet ) vs. E jet for light quarks. η Resolution vs Ejet for b-quarks in region 1 /GeV 0.08 χ2 / ndf 36.26 / 35 parton p0 0.1156 ± 0.0001694 0.07 σ Eta p1 -0.003921 ± 2.409e-06 p2 6.487e-05 ± 2.455e-08 0.06 p3 -5.742e-07 ± 1.883e-10 0.05 p4 2.772e-09 ± 1.198e-12 p5 -6.841e-12 ± 7.246e-15 0.04 p6 6.727e-15 ± 1.887e-17 0.03 0.02 0.01 0 0 20 40 60 80 100 120 140 160 180 200 220 Ejet /GeV η Resolution vs Ejet for b-quarks in region 1 /GeV 0.1 χ2 / ndf 60.55 / 36 parton p0 0.1256 ± 0.0002164 σ Eta p1 -0.003805 ± 1.375e-06 0.08 p2 5.758e-05 ± 5.926e-08 p3 -4.784e-07 ± 5.777e-10 p4 2.261e-09 ± 8.353e-13 p5 -5.957e-12 ± 2.706e-15 0.06 p6 7.939e-15 ± 7.22e-18 p7 -3.993e-18 ± 3.457e-20 0.04 0.02 0 0 50 100 150 200 250 Ejet /GeV Figure 5.19: Pseudo-rapidity (η) resolution for b-quarks in detector |η| regions 1 and 2, plotted versus E jet . 115 η Resolution vs Ejet for b-quarks in region 3 /GeV 0.08 χ2 / ndf 21.07 / 23 parton p0 0.1409 ± 0.004128 σ Eta p1 -0.00356 ± 0.0001057 0.07 p2 5.211e-05 ± 3.697e-07 p3 -4.294e-07 ± 9.748e-09 0.06 p4 1.992e-09 ± 1.026e-10 p5 -4.786e-12 ± 3.647e-13 p6 4.599e-15 ± 4.444e-16 0.05 0.04 0.03 0 50 100 150 200 250 300 Ejet /GeV η Resolution vs Ejet for b-quarks in region 4 /GeV 0.1 χ2 / ndf 35.66 / 21 p0 0.1721 ± 0.00128 parton 0.09 p1 -0.002832 ± 1.52e-05 σ Eta p2 2.436e-05 ± 7.702e-08 0.08 p3 -1.063e-07 ± 2.323e-10 p4 2.139e-10 ± 8.894e-13 0.07 p5 -1.144e-13 ± 2.9e-15 0.06 p6 -1.072e-16 ± 7.183e-18 0.05 0.04 0.03 0.02 50 100 150 200 250 300 Ejet /GeV Figure 5.20: Pseudo-rapidity (η) resolution for b-quarks in detector |η| regions 3 and 4, plotted versus E jet . |η det | p0(GeV ) p1(GeV ) p2(GeV ) p3(GeV ) p4(GeV ) p5(GeV ) p6(GeV ) p7(GeV ) [0, 0.5) 0.11 −0.0033 4.27 × 10−5 −1.97 × 10−7 −6.43 × 10−10 1.01 × 10−11 −3.65 × 10−14 4.43 × 10−17 [0.5, 1.0) 0.13 −0.0039 5.14 × 10−5 −2.96 × 10−7 1.27 × 01−10 6.46 × 10−12 −2.77 × 10−14 3.64 × 10−17 [1.0, 1.5) 0.15 −0.0031 3.01 × 10−5 −1.12 × 10−7 −2.26 × 10−10 3.14 × 10−12 −9.06 × 10−15 8.73 × 10−18 [1.5, 2.5) 0.11 −0.0012 5.91 × 10−6 −1.26 × 10−8 −2.17 × 10−11 3.73 × 10−13 −1.52 × 10−15 2.05 × 10−18 Table 5.8: Parameters of polynomial fit to σ (φ parton − φ jet ) vs. E jet for light quarks. 116 φ Resolution vs Ejet for b-quarks in region 1 /GeV 0.08 χ2 / ndf 60.02 / 32 parton p0 0.1122 ± 0.0001106 σ Phi 0.07 p1 -0.003321 ± 2.589e-06 p2 4.266e-05 ± 6.584e-09 0.06 p3 -1.972e-07 ± 5.602e-11 p4 -6.429e-10 ± 1.094e-12 0.05 p5 1.011e-11± 1.583e-15 p6 -3.652e-14 ± 1.142e-17 0.04 p7 4.427e-17 ± 3.355e-20 0.03 0.02 0.01 0 50 100 150 200 250 Ejet /GeV φ Resolution vs Ejet for b-quarks in region 2 χ2 / ndf /GeV 0.09 32.08 / 32 p0 0.132 ± 0.002877 -0.003863 ± 0.0001283 parton 0.08 p1 p2 5.138e-05 ± 2.002e-06 σ Phi 0.07 p3 -2.955e-07 ± 1.286e-08 p4 1.27e-10 ± 2.064e-11 0.06 p5 6.455e-12 ± 1.871e-13 p6 -2.774e-14 ± 1.034e-15 0.05 p7 3.639e-17 ± 1.79e-18 0.04 0.03 0.02 0.01 0 50 100 150 200 250 Ejet /GeV Figure 5.21: Azimuthal (φ ) resolution for b-quarks in detector |η| regions 1 and 2, plotted versus E jet . 117 φ Resolution vs Ejet for b-quarks in region 3 0.12 /GeV parton χ2 / ndf 11.8 / 20 0.1 0.1463 ± 0.002834 σ Phi p0 p1 -0.003081 ± 5.704e-05 p2 3.009e-05 ± 2.882e-07 0.08 p3 -1.117e-07 ± 1.006e-09 p4 -2.257e-10 ± 6.777e-12 p5 3.137e-12 ± 2.189e-14 0.06 p6 -9.06e-15 ± 1.894e-16 p7 8.726e-18 ± 3.505e-19 0.04 0.02 0 50 100 150 200 250 300 Ejet /GeV φ Resolution vs Ejet for b-quarks in region 4 /GeV χ2 / ndf 13.29 / 19 0.08 p0 0.1123 ± 0.001096 parton p1 -0.001206 ± 1.255e-05 σ Phi 0.07 p2 5.911e-06 ± 7.276e-08 p3 -1.265e-08 ± 4.389e-10 0.06 p4 -2.169e-11 ± 2.053e-12 p5 3.726e-13 ± 7.388e-15 p6 -1.52e-15 ± 1.993e-17 0.05 p7 2.046e-18 ± 5.387e-20 0.04 0.03 0.02 0.01 50 100 150 200 250 300 Ejet /GeV Figure 5.22: Azimuthal (φ ) resolution vs. E jet for b-quarks in detector |η| regions 3 and 4, plotted versus E jet . 5.3.3 Lepton Resolutions The energy (E), pseudo-rapidity (η) and azimuthal variables (φ ) of charged lepton are compared between detector and parton level . This is done using the same set of events that were used to derive the parton level corrections and resolutions for light and b-quarks in the preceding sections. Fig. 5.23, shows the distributions in the differences of these attributes for electrons. Similar distributions exist for muons. Since the differences are small, there 118 are no further corrections applied to the 4-vector of the charged leptons observed in the de- tector. The RMS of these distributions, are parametrized as a function of the lepton energy to yield the resolutions on the lepton E, η and φ . These are the inputs to the kinematic fitter, described in the next section. 119 pcor_Electron_r1_2 pcor_Electron_r1_2 2200 Entries 15140 Mean -0.9713 2000 RMS 3.958 1800 χ2 / ndf 83.22 / 10 1600 Prob 1.171e-13 Constant 2126 ± 24.0 1400 Mean -0.336 ± 0.014 1200 Sigma 1.566 ± 0.013 1000 800 600 400 200 0 -300 -200 -100 0 100 200 300 (a) (E parton − Eelectron ) distribution in an electron energy bin. pcor_Electron_r1_2 1000 eta_Electron_r1_2 Entries 15140 Mean -5.699e-05 RMS 0.002036 800 χ2 / ndf 111.9 / 17 Prob 5.405e-16 Constant 845.7 ± 11.2 600 Mean -3.577e-05 ± 1.391e-05 Sigma 0.00119 ± 0.00002 400 200 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (b) (η parton − ηelectron ) distribution in an electron energy bin. pcor_Electron_r1_2 5000 phi_Electron_r1_2 Entries 15140 Mean -2.453e-05 RMS 0.0004853 4000 χ2 / ndf 122.6 / 2 Prob 2.403e-27 Constant 4980 ± 59.5 3000 Mean -2.513e-05 ± 2.020e-06 Sigma 0.0002147 ± 0.0000020 2000 1000 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (c) (φ parton − φelectron ) distribution in an electron energy bin Figure 5.23: Distribution of differences in electron energy, pseudo-rapidity and azimuthal variables between detector and parton levels. 120 The electron resolution functions are parametrized in two detector η regions. Region 1 covers |η det | ∈ [0, 1.0) and Region 2 covers |η det | > 1.0. Fig. 5.24-5.25, shows the polynomial fits to these parametrization for electrons. 121 (a) electron energy resolution vs. electron energy for detec- tor |η det | ∈ [0, 1.0) (b) electron pseudo-rapidity resolution vs. electron energy for detector |η det | ∈ [0, 1.0) (c) electron azimuthal resolution vs. electron energy for de- tector |η det | ∈ [0, 1.0) Figure 5.24: Resolutions of electron energy, pseudo-rapidity and azimuthal variables with respect to electron energy observed in the detector region 1 for t t¯ → e + jets events. 122 (a) electron energy resolution vs. electron energy for detec- tor |η det | > 1.0 (b) electron pseudo-rapidity resolution vs. electron energy for detector |η det | > 1.0 (c) electron azimuthal resolution vs. electron energy for de- tector |η det | > 1.0 Figure 5.25: Resolutions of electron energy, pseudo-rapidity and azimuthal variables with respect to electron energy observed in the detector region 1 for t t¯ → e + jets events. 123 The muon resolution functions are also parametrized in two detector η regions. Region 1 covers |η det | ∈ [0, 1.6), and Region 2 covers |η det | > 1.6. Fig. 5.26-5.27, shows the polynomial fits to these parametrization for muons. 124 (a) muon energy resolution vs. muon energy for detector |η| ∈ [0, 1.0) (b) muon pseudo-rapidity resolution vs. muon energy for detector |η| ∈ [0, 1.0) (c) muon azimuthal resolution vs. muon energy for detector |η| ∈ [0, 1.0) Figure 5.26: Resolutions of muon energy, pseudo-rapidity and azimuthal variables with respect to muon energy observed in the detector region 1 for t t¯ → µ + jets events. 125 (a) muon energy resolution vs. muon energy for detector |η| > 1.0 (b) muon pseudo-rapidity resolution vs. muon energy for detector |η| > 1.0 (c) muon azimuthal resolution vs. muon energy for detector |η| > 1.0 Figure 5.27: Resolutions of muon energy, pseudo-rapidity and azimuthal variables with respect to muon energy observed in the detector region 1 for t t¯ → µ + jets events. 126 5.4 HitFit: The kinematic fitter The fitter takes as an input, the kinematic quantities of jets and leptons in the final state and varies them within their resolutions to obtain the best possible top quark mass (also referred to as the fitted top mass), under the following constraints : 1. mthad = mtlep had = mlep = 80.4 GeV 2. mW W The first condition states that the mass of the top and anti-top pair produced in the collision, are equal. This is a simple consequence of the CPT invariance, which is the basis of quan- tum field theory. Since, the semi-leptonic decay mode of top quarks is under consideration, both the top and anti-top decay to a W ± boson, one of which decays hadronically (mW had ), lep while the other decays leptonically to an electron or a muon and a neutrino (mW ). The W boson mass is constrained to the known value. mthad refers to the invariant mass of the system of hadronically decaying W boson and a b-quark (anti b-quark), while mtlep refers to the invariant mass of the leptonically decaying W boson and b-quark (anti b-quark). The kinematic fit, uses the measure of goodness of fit, that requires the measured ob- servables (xO ), the fitted values (x f ) and the errors on the measured values (σ (xO )), to minimize : (x f − xO )2 2 χ =∑ (5.2) σ 2 (xO ) wherein, the summation runs over all observables. The list of the observables is as follows : • The pseudo-rapidity (η), the azimuthal angle (φ ) and the energy of the 4 leading jets. 127 • The pseudo-rapidity (η), the azimuthal angle (φ ) and the energy of the charged lepton (e± or µ ± ) The constraints (1 and 2) are non-linear with respect to the above observables and hence the χ 2 is minimized iteratively, till the fitted values comply with the constraints within an infinitesimal tolerance. In this form of the fitting algorithm, the observables (measured − → quantities) are input as one-dimensional vectors (xm ) while the fitted quantities are repre- → − sented by x f . The χ 2 then takes the following form : → − − → → − − → χ 2 = ( x f − xm )T G ( x f − xm ) (5.3) where, G is the inverse of the error matrix that contains the uncertainties of the measure- ments. At the initial step, the full event must be known to compute the above quantities. The missing transverse energy (6 ET ), only provides the initial estimate for the transverse − → momentum of the neutrino ( pνT ). The longitudinal momentum of the neutrino (pνz ) is esti- mated using constraint 1. This yields a quadratic equation in pνz , ± −→  ± ±  ± (pzl b )2 − (E l b )2 (pνz ) + α pzl b pνz − (E l b pνT )2 + α 2/4 = 0 (5.4) where → −−→ − ± α = (mthad )2 − ml ± b + 2 pνT . plT b (5.5) In the above equation, l ± b, refers to the system of the charged lepton and the b-quark involved in the leptonically decaying top. This procedure is carried out for each jet-parton assignment hypothesis for an event. As discussed before, there are 12 such possible assign- ments for the 4 leading jets scenario and with two solutions to Eq.5.4, there are 24 possible solutions and hence that many fitted top mass value per event. For each hypothesis, the 128 parton level corrections are applied to get the 4-momenta of the partons involved, depend- ing on the flavor of the jets assumed for that hypothesis. For each step, the uncertainty and the 4-momenta of the neutrino are adjusted accordingly. The fitting procedure is based on a SQUAW algorithm. Details can be found in (Ref.[61]). 5.4.1 Performance of the fitter The kinematic fitter, outputs a minimum χ 2 and a fitted top mass for each of the 24 solutions per event. In this chapter, we started out with semi-leptonic decay modes of t t¯ samples simulated with ALPGEN+PYTHIA for various input masses of 150 to 190 GeV. The 4- vectors of the jets were corrected to the parton level using the matched jet-parton pairs in these samples. The credibility of these corrections depends on the closure tests. For these tests, events which have all four of the leading jets matched to their respective partons are used. This way, the contamination due to subsidiary arising from gluon radiation effects is minimal. We also know that for such events, there is at least one solution for the event in which the jet-parton assignment hypothesis was correct. There could be at most two fitted top mass values for such events owing to the possible two neutrino solutions that one gets by solving Eq. 5.4. We refer to these solutions as the “correct solutions”. The closure can be classified into two categories : had ) for the events with correct combinations is compared to • The hadronic W mass (mW the generator level value of 80.4 GeV. Fig. 5.28, shows that the peak values of such distributions are independent of the generated top mass and close to (within ∼ 0.3%) the true value. This provides a check for the light parton corrections and hence the correct mapping of the W → qq¯ process in the top quark events. 129 W mass correct perm for gen top mass 150 GeV W mass correct perm for gen top mass 172.5 GeV W mass correct perm for gen top mass 190 GeV Wmass Wmass Wmass Entries 22900 4000 Entries 32441 Entries 32853 Mean 81.36 Mean 81.25 Mean 80.93 4000 2500 RMS 12.48 RMS 12.53 RMS 11.9 χ2 / ndf 0.924 / 2 χ 2 / ndf 1.722 / 2 χ2 / ndf 7.908 / 2 3500 Prob 0.4227 Prob 0.63 Prob 0.01917 2736 ± 36.7 Constant 4011 ± 44.7 3500 Constant Constant 4252 ± 45.8 Mean 80.82 ± 0.19 Mean 80.86 ± 0.14 Mean 80.82 ± 0.13 2000 Sigma 9.08 ± 0.46 3000 Sigma 8.58 ± 0.32 Sigma 8.092 ± 0.263 3000 2500 2500 1500 2000 2000 1000 1500 1500 1000 1000 500 500 500 0 0 0 20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200 had for correct combination (a) mW had for correct combination (b) mW had for correct combination (c) mW in e+jets events generated with top in e+jets events generated with top in e+jets events generated with top mass = 150 GeV mass = 172.5 GeV mass = 190 GeV W mass correct perm W mass correct perm for gen top mass 172.5 Gev W mass correct perm for gen top mass 190 GeV gen top mass 150 Gev Wmass Wmass 4500 Wmass Entries 25969 4500 Entries 35464 Entries 35054 Mean 81.4 Mean 81.04 Mean 80.87 3000 RMS 12.16 RMS 11.99 RMS 11.66 χ 2 / ndf χ 2 / ndf 4000 χ2 / ndf 0.4625 / 2 4000 0.2779 / 3 0.6112 / 2 Prob 0.7935 Prob 0.9641 Prob 0.7367 Constant 3178 ± 39.6 Constant 4431 ± 42.8 Constant 4316 ± 45.6 Mean 80.75 ± 0.17 Mean 80.87 ± 0.11 3500 Mean 80.48 ± 0.20 2500 Sigma 8.654 ± 0.370 3500 Sigma 8.477 ± 0.189 Sigma 9.948 ± 0.475 3000 3000 2000 2500 2500 1500 2000 2000 1500 1500 1000 1000 1000 500 500 500 0 0 0 20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200 20 40 60 80 100 120 140 160 180 200 had for correct combination in (d) mW had for correct combination in (e) mW had for correct combination in (f) mW µ+jets events generated with top µ+jets events generated with top µ+jets events generated with top mass = 150 GeV mass = 172.5 GeV mass = 190 GeV Figure 5.28: Parton level corrected hadronic W mass distributions for various top mass samples. • The peak distributions of the fitted top quark mass at minimum χ 2 for the correct solutions are shown in Fig. 5.29. These distributions include both the neutrino solu- tions for the correct combination. 130 Fitted Top (both neutrino sols.) for gen top mass 150 GeV in ejets channel Fitted Topmass (both neutrino sols.) for gen top mass 172.5 Gev in ejets channel Fitted top for gen top 190 GeV (both neutrino sols.) in ejets channel mass mass 4000 topmassnu topmassnu topmassnu Entries 45800 Entries 64880 Entries 65706 Mean 150.5 4500 Mean 170.6 4000 Mean 186.3 RMS 13.92 RMS 15.2 3500 RMS 11.93 χ2 / ndf 18.98 / 5 χ2 / ndf 5.762 / 4 χ2 / ndf 3.798 / 5 4000 Prob 0.001941 Prob 0.2176 Prob 0.5789 3500 4221 ± 36.0 Constant 4619 ± 37.2 Constant 3000 Constant 3890 ± 33.6 188.7 ± 0.2 Mean 171.4 ± 0.1 Mean Mean 150.5 ± 0.1 3500 Sigma 9.511± 0.268 Sigma 10.31 ± 0.49 Sigma 7.717 ± 0.162 3000 2500 3000 2500 2000 2500 2000 2000 1500 1500 1500 1000 1000 1000 500 500 500 0 0 0 100 120 140 160 180 200 220 240 260 280 300 100 120 140 160 180 200 220 240 260 280 300 100 120 140 160 180 200 220 240 260 280 300 (a) fitted mtop for correct combina- (b) fitted mtop for correct combina- (c) fitted mtop for correct combina- tion in e+jets events generated with tion in e+jets events generated with tion in e+jets events generated with top mass = 150 GeV top mass = 172.5 GeV top mass = 190 GeV Fitted Top (both neutrion sols.) for gen top 150 GeV in mujets channel Fitted Topmass (both neutrino sols.) for gen top mass 172.5 GeV in mujets channel Fitted Topmass (both neutrino sols.) for gen top mass 190 GeV in mujets channel mas 4500 topmassnu topmassnu topmassnu Entries 51938 5000 Entries 70928 4500 Entries 70108 Mean 150.6 Mean 170.6 Mean 186.3 4000 RMS 11.74 RMS 13.69 RMS 14.98 χ2 / ndf 1.702 / 4 χ 2 / ndf 4.8 / 5 4000 χ2 / ndf 8.9 / 5 Prob 0.7904 Prob 0.4408 Prob 0.1131 3500 Constant 4365 ± 38.0 4000 Constant 5041 ± 39.0 Constant 4532 ± 36.5 Mean 150.2 ± 0.1 Mean 171.7 ± 0.1 3500 Mean 188.5 ± 0.1 Sigma 8.276 ± 0.256 Sigma 10.55 ± 0.36 Sigma 9.598 ± 0.262 3000 3000 2500 3000 2500 2000 2000 2000 1500 1500 1000 1000 1000 500 500 0 0 0 100 120 140 160 180 200 220 240 260 280 300 100 120 140 160 180 200 220 240 260 280 300 100 120 140 160 180 200 220 240 260 280 300 (d) fitted mtop for correct combina- (e) fitted mtop for correct combina- (f) fitted mtop for correct combina- tion in µ+jets events generated with tion in µ+jets events generated with tion in µ+jets events generated with top mass = 150 GeV top mass = 172.5 GeV top mass = 190 GeV Figure 5.29: Fitted top mass distributions for top mass samples generated with different input mass. CHAPTER Six Ideogram Method 6.1 Introduction The performance of the kinematic fitter demonstrated in the previous chapter only delin- eates the accuracy of the resulting fit values given that the correct jet-parton assignments are known. This is not usually the case when dealing with the experimental data. It is important to extract the best possible top mass value, utilizing all of the information about an event through the maximum of 24 solutions that are available for the event. It was also discussed that a simultaneous fit to the Jet Energy Scale (JES) is performed for the events under consideration by scaling all the jets in the event by a JES factor (αJES ). This has been the major source of systematic uncertainty in the top quark mass measurements and the idea behind doing a simultaneous fit is to absorb a part of this uncertainty into a statis- tical one. Since the same events are used to extract the fitted top mass (mt ) and the JES, these measurements are correlated. Also extracted from the kinematic fit is the uncertainty on the fitted top mass value for each possible solution (σi ). The fit is repeated for differ- ent values of the αJES which is varied in steps of 1% in an interval of ±15% around 1. JESinput =1 corresponds to the jet energies that are corrected using the γ+jet events, de- 131 132 scribed in Section 3.6 along with the sample dependent corrections described in Section 3.6.1. These energies are further corrected to parton level using the PLC described in the previous chapter (depending on the flavor of the jet). Only those jet-parton assignments for which the kinematic fit converges at all values of the αJES , are taken into account. The fitted mass mi (αJES ), the estimated uncertainty on the fit σi (αJES ) and the goodness of fit χi2 (αJES ), all depend on the JES parameter. Since the fitter uses a constraint of mW = 80.4 GeV, the χ 2 is expected to be the best, when the invariant mass of the hadronically decay- ing W is closest to the known W mass. Additional sensitivity to the fitted JES may come from the fitted mass distribution in the background (W + jets) events. To better discrim- inate between signal and backgrounds, b − tagging is used. The analysis is calibrated for events that have 1 b-tagged jets and for events that have≥ 2 b-tagged jets separately and then combined at a later stage. The events which have no b-tagged jet among the 4-leading jets have poor mass resolution and hence do not contribute to the precision. b-tagging also helps in distinguishing between correct and wrong jet-parton assignment hypotheses for an event. In order to obtain an optimal separation between t t¯ signal and the predominant back- ground (W + jets), without biasing the top quark mass measurement, a likelihood discrim- inant based on certain topological variables, is constructed. This discriminant is used to determine the “purity” of an event to weigh the signal and background probabilities of the event accordingly. These variables are listed as follows : • Aplanarity, which is defined as 3/2 times the smallest eigenvalue of the normalized laboratory-frame momentum tensor of the jets and the charged lepton. • 6ET , missing transverse energy 0 • Centrality, HT2 ≡ HT2/Hk , where HT2 is the scalar sum of the transverse momenta of the jets excluding the leading jet and Hk is the sum of the magnitudes of the 133 momentum components along the beam line of jets, isolated charged lepton, neutrino. The neutrino parallel component (pνk ) is computed by requiring that the invariant mass of the lepton-neutrino system is equal to the known W mass. In case of two real solutions for the above, the smallest one is taken. 0 lesser j • KTmin , defined as (4Rmin i j . ET )/(E W ). T This quantity is the measure of the jet separation normalized to the transverse energy of the reconstructed W boson. 4Rmin ij is the lesser j smallest distance in η − φ space, between any two of the 4 leading jets. ET is the smaller of the two jet ET . The transverse energy of the W boson is defined as ETW ≡ |plepton T | + | 6 ET |. a) Topological discriminant distribution for b) Topological discriminant distribution for t t¯ and W + jets backgrounds in e + jets t t¯ and W + jets backgrounds in µ + jets channel. channel. Figure 6.1: Comparison of the “topological discriminant” values for signal (red) and back- grounds (blue). The discriminant (formed out of the topological variables) shown in Fig. 6.1 can be S  used to map to the signal purity of the event defined as S+B , where S is the normalized discriminant distribution of the signal while B is the same for backgrounds (mainly, W + jets ). Fig. 6.2 shows such a mapping for different tagging bins in both e+jets and µ+jets channel. 134 (a) Purity versus discriminant for events (b) Purity versus discriminant for events with 1 b-tag in e + jets events. with 1 b-tag in µ + jets events. (a) Purity versus discriminant for events (b) Purity versus discriminant for events with =2 b-tag in e + jets events. with =2 b-tag in µ + jets events. Figure 6.2: Purity versus Discriminant fits for e+jets, µ+jets events in two different tagging bins. 6.2 Likelihood Construction Given an event sample, a most likely top mass is extracted by calculating a likelihood for each event (in the sample) as a function of assumed top quark mass (mtop ), the jet energy scale factor (JES) and the fraction of t t¯ events in the event sample ( ftop ). The signal fraction, ftop is used here as a nuisance parameter. Details of the purity fit can be found in Ref.[62, 63].The event likelihood (Levent ), is composed of two terms, describing the hypotheses that the event was t t¯ signal, or an event originating from a background process : 135 Levent (x; mtop , αJES , ftop ) = ftop × Psgn (x; mtop , αJES ) + (1 − ftop ) × Pbkg (x; αJES ) (6.1) where x denotes the full set of observables that characterize the event, ftop is the signal fraction of the event sample and Psgn , Pbkg are the probabilities for t t¯ and W + jets pro- duction, respectively. The backgrounds here mainly comprise of W+jets, while the other backgrounds viz. QCD multijet are not modeled explicitly. The event observable x, can be thought of as a vector, containing all relevant data about the event topology and kinematics. The topological part is not correlated with the mass and is the probability of the observed discriminant value to occur given the event is signal or background like. The kinematic component (x f it ) contains the mass information, extracted from the constrained kinematic fit which provides the sensitivity to the top quark mass and the jet energy scale. Assuming that the two probabilities, described above, are independent of each other, one can write : Psgn (x; mtop , αJES ) ≡ Psgn (D) × Psgn (x f it ; mtop , αJES ) (6.2) and Pbkg (x; αJES ) ≡ Pbkg (D) × Pbkg (x f it ; αJES ) (6.3) where D (discriminant) is evaluated at JESinput = 1. The normalized probability distri- butions of the discriminant (Psgn (D) & Pbkg (D) ) are obtained from Monte Carlo simula- tions as shown in Fig. 6.2 and are assumed to be independent of JES. The mass information (x f it ), consisting of all the fitted masses mi (αJES ), estimated uncertainty σi (αJES ) and the measure of goodness-of-fit χi2 (αJES )is obtained from the kinematic fitter as explained be- low. 136 6.2.0.1 Top Mass templates In this section, the construction of Psgn (x f it; ; mtop , αJES ) is discussed. The output of the kinematic fitter can be used to calculate the signal probability of an event, as the sum over the 24 solutions corresponding to the jet-parton assignment and the two neutrino solutions. The relative probability for each of the solution i to be correct is estimated through a weight function wi . This weight function can be resolved into two parts. Without b-tagging, wi purely represents the χi2 of the corresponding kinematic fit and is defined with a Gaussian probability as wi = exp(− 21 χi2 ). To weight the combinations which are more likely to be correct due to the presence of b-tagged jets, an additional relative weight of wbtag,i is ap- plied. This relative weight, represents the probability that observed b-tags are compatible with the assumed jet-parton assignment. j wbtag,i = ∏ pi (6.4) j=1,n jet j wherein, pi can assume a value of εl, εb , (1 − εl ), (1 − εb ), depending on the assumed flavor of the jet (light or b) and whether or not the particular jet was tagged. The tagging rates for light and b-quark jets εl and εb are determined from data as parametrized functions of jet pT and η , where the jet pT is based on the reconstructed jet energy for JESinput = 1. n jet , here represents the 4-leading jets that are fed into the kinematic fitter. Hence, the total event permutation weight can be written as : 1 j wi = exp(− χi2 ) ∏ pi (6.5) 2 j=1,n jet The mass dependent signal probability is modeled as : 137 24  Z 300  ntag 0 0 0 Psgn (x f it ; mtop , αJES ) = ∑ wi fcorrect . G(mi , m , σi ) . BW (m , mtop )dm i=1 100 24 ntag ) . Sntag   + ∑ wi (1 − fcorrect wrong (mi , mtop ) (6.6) i=1 In the above expression, the two terms correspond to the shapes of correct solutions and the wrong solutions weighted with a fraction that depends on the number of b-tags in the event and is estimated from the Monte Carlo simulations. These shapes are shown ntag in Fig. 6.3-6.4, while the fcorrect values for e+jets and µ+jets is listed in Table. 6.2.0.1. The term compatible with the correct solutions is modeled as a Gaussian, centered at the fitted top mass value mi and width corresponding to its uncertainty σi , convoluted with a Breit-Wigner line shape of width equivalent to the known top quark width of 2 GeV. The Gaussian part describes the experimental resolution while the relativistic Breit-Wigner part describes the the expected distribution of the line shape of the top and the anti-top quarks in the event, for a given top quark mass mt .The BW is normalized to an interval of 100 to 300 GeV in order for it to not bias the top mass measurement in the region of interest. The second term in the signal probability, represents the shape of those jet-parton as- signments that are not correct combinations. These shapes for a given mt are obtained from the Monte Carlo simulation of t t¯ generated at different mt . Only combinations that are known to be “wrong” are filled in histogram with each entry weighted with its coorespond- ing weight wi . These shapes are then fitted with double Gaussian as shown in Fig. 6.5. The fit parameters show a nice linear dependence on mt . This parametrization is shown in Table. 6.1. 138 # tags e + jets µ + jets 1 fcorrect = 0.2708 fcorrect = 0.2702 ≥2 fcorrect = 0.743 fcorrect = 0.7322 ntag Table A: fcorrect for e + jets and µ + jets channels. The value decreases with lesser tags. 200 ID Entries 1234 140 200 ID Entries 1234 140 200 ID Entries 1234 140 Mean 159.1 Mean 166.1 Mean 169.0 RMS 28.79 RMS 29.10 RMS 29.36 150 150 150 150 GeV 160 GeV 165 GeV 100 100 100 50 50 50 0 0 0 100 200 300 100 200 300 100 200 300 all perms BG shape 2 tags all perms BG shape 2 tags all perms BG shape 2 tags 200 ID 1234 200 ID 1234 200 ID 1234 Entries 140 Entries 140 Entries 140 Mean 172.8 Mean 174.3 Mean 176.0 RMS 29.94 RMS 29.60 RMS 29.52 150 150 150 170 GeV 172.5 GeV 175 GeV 100 100 100 50 50 50 0 0 0 100 200 300 100 200 300 100 200 300 all perms BG shape 2 tags all perms BG shape 2 tags all perms BG shape 2 tags 200 ID Entries 1234 140 200 ID Entries 1234 140 200 ID Entries 1234 140 Mean 179.9 Mean 183.6 Mean 186.8 RMS 30.41 RMS 29.85 RMS 30.59 150 150 150 180 GeV 185 GeV 190 GeV 100 100 100 50 50 50 0 0 0 100 200 300 100 200 300 100 200 300 all perms BG shape 2 tags all perms BG shape 2 tags all perms BG shape 2 tags Sum (black) of weighted right combination shape (green) and wrong combination shape (blue), overlay-ed on the t t¯ MC simulation in e + jets channel for events with =2 b-tags Figure 6.3: Right and wrong combination shapes for different generator level top quark masses. 139 200 ID Entries 1234 140 200 ID Entries 1234 140 200 ID Entries 1234 140 Mean 159.0 Mean 165.0 Mean 167.9 RMS 30.90 RMS 31.94 RMS 32.07 150 150 150 150 GeV 160 GeV 165 GeV 100 100 100 50 50 50 0 0 0 100 200 300 100 200 300 100 200 300 all perms BG shape 1 tag all perms BG shape 1 tag all perms BG shape 1 tag 200 ID 1234 200 ID 1234 200 ID 1234 Entries 140 Entries 140 Entries 140 Mean 170.7 Mean 171.7 Mean 174.0 RMS 32.60 RMS 32.55 RMS 32.95 150 150 150 170 GeV 172 GeV 175 GeV 100 100 100 50 50 50 0 0 0 100 200 300 100 200 300 100 200 300 all perms BG shape 1 tag all perms BG shape 1 tag all perms BG shape 1 tag 200 ID Entries 1234 140 200 ID Entries 1234 140 200 ID Entries 1234 140 Mean 176.8 Mean 179.6 Mean 182.8 RMS 33.72 RMS 33.77 RMS 34.03 150 150 150 180 GeV 185 GeV 190 GeV 100 100 100 50 50 50 0 0 0 100 200 300 100 200 300 100 200 300 all perms BG shape 1 tag all perms BG shape 1 tag all perms BG shape 1 tag Sum (black) of weighted right combination shape (green) and wrong combination shape (blue), overlay-ed on the t t¯ MC simulation in e + jets channel for events with exactly 1 b-tag Figure 6.4: Right and wrong combination shapes for different generator level top quark masses or 1 b-tagged events. 140 ID 1234 ID 1234 ID 1234 Entries 140 60 Entries 140 Entries 140 60 Mean RMS 164.9 35.15 Mean RMS 171.2 36.32 Mean RMS 172.7 37.07 24.27 / 64 31.64 / 64 27.97 / 64 P1 46.23 P1 39.18 P1 41.37 P2 148.5 P2 155.1 P2 156.6 P3 P4 18.67 13.93 P3 P4 20.93 12.57 40 P3 P4 22.25 10.67 P5 186.3 40 P5 192.3 P5 204.5 40 P6 39.97 P6 42.29 P6 40.78 20 20 20 0 0 0 100 200 300 100 200 300 100 200 300 combinatorial BG shape 2 tags combinatorial BG shape 2 tags combinatorial BG shape 2 tags ID 1234 ID 1234 ID 1234 Entries 140 Entries 140 Entries 140 Mean 176.1 Mean 176.9 Mean 178.3 RMS 37.86 RMS 37.55 RMS 37.74 40 P1 15.73 / 64 39.55 40 P1 18.14 / 64 39.22 P1 25.46 / 64 37.42 P2 P3 159.8 23.53 P2 P3 161.7 24.08 40 P2 P3 162.0 23.40 P4 10.45 P4 9.900 P4 11.09 P5 P6 209.7 41.51 30 P5 P6 210.4 41.57 P5 P6 210.0 39.66 20 20 20 10 0 0 0 100 200 300 100 200 300 100 200 300 combinatorial BG shape 2 tags combinatorial BG shape 2 tags combinatorial BG shape 2 tags ID 1234 ID 1234 ID 1234 Entries 140 Entries 140 Entries 140 Mean RMS 181.8 38.87 Mean RMS 184.5 38.48 40 Mean RMS 187.1 39.26 22.69 / 64 40 24.42 / 64 14.17 / 64 40 P1 P2 36.75 166.2 P1 P2 36.41 169.8 P1 P2 35.21 172.4 P3 25.77 P3 26.54 P3 27.94 P4 9.155 P4 8.662 30 P4 8.700 P5 P6 219.2 42.36 30 P5 P6 225.7 39.63 P5 P6 230.9 39.48 20 20 20 10 10 0 0 0 100 200 300 100 200 300 100 200 300 combinatorial BG shape 2 tags combinatorial BG shape 2 tags combinatorial BG shape 2 tags Figure 6.5: Weighted wrong combination shapes fitted with double Gaussian. 141 1tag =2 tags Gauss1 Gauss2 Gauss 1 Gauss 2 p0 p1 p0 p1 p0 p1 p0 p1 a 57.08 −0.5468 13.2 −0.134 38.23 −0.264 10.09 −0.13 mean 159.8 0.5241 207.9 0.8694 163.0 0.5679 212.6 1.06 σ 23.29 0.2058 41.66 0.084 24.33 0.221 40.8 0.0037 Parameters used to describe the combinatorial background shapes (arbitrary normalization). The shape is that of double Gaussian G(m f it ) = a × exp −(mean − m f it )2 /2σ 2 , where the 3 parameters a, mean and σ are  parametrized linearly w.r.t. the generator level top mass mt as p0 + p1 . (mt − 175) GeV for e + jets channel (above) and µ + jets channel (below) 1tag =2 tags Gauss1 Gauss2 Gauss 1 Gauss 2 p0 p1 p0 p1 p0 p1 p0 p1 a 28.54 −0.229 6.19 −0.075 38.52 −0.313 9.584 −0.108 mean 159.7 0.605 208.7 1.075 163.5 0.577 214.8 1.014 σ 23.49 0.20341 42.67 0.072 24.78 0.224 42.40 0.047 Table 6.1: Parametrization of combinatorial shapes with respect to generator level top mass. 6.3 W+jets Background shapes In this section, Pbkg (x f it; JES ) in Eq. 6.3 is discussed. The background probability of an event is estimated from W + jets Monte Carlo simulations. It is taken to be : 24 Pbkg (x f it; JES ) ≡ ∑ wi . BG (mi ) (6.7) i=1 where, BG is the background shape estimated from fitted top mass in W + jets sample. Such a shape is shown in Fig. 6.6. A common shape is used for both 1 and = 2 b-tags. This is done to avoid unscrupulous features arising from statistical fluctuations. The shape is also smoothed by calculating the average value in a sliding window of ±10 GeV around each fitted mass. These shapes are evaluated at every JES factor. 142 Figure 6.6: A sample BG shape for µ + jets events at JES = 1 6.4 Likelihood of a sample Since each event is independent, the combined likelihood for the whole sample is calculated as the product of single event likelihoods. This likelihood is maximized with respect to mt , JES & ftop . Lsamp (mt , αJES , ftop ) = ∏ Levt j (mt , αJES , ftop ) (6.8) j The likelihood (L ) is computed for all values of ftop between 0 and 1, in the steps of 0.05. The parameter mt , varies between 125 GeV /c2 to 225 GeV /c2 in steps of 1 GeV /c2 .The third free parameter, αJES , is varied from +15% to −15% around the nominal value of JES, in the steps of 1%. The range is chosen so that the measurement is not affected by boundary effects. First, for each value of αJES and mt , the likelihood is maximized with respect to ftop . This leads to a likelihood distribution in the two dimensional grid of mt and JES (Fig. 143 f itted 6.13). A fit to the global minimum of this distribution yields extracted values of mt and JES f itted for a sample. This procedure is first applied to the simulated MC events to check the consistency of the input and output values of two of the freely floating parameters (mt and JES ). This is achieved through constructing pseudo-experiments and calibration, that are described in the next chapter. 6.5 Calibration The likelihood expressed in Eq. 6.6 entails only an approximate representation of the true detector resolution effects and the complex physics involved in the production and reconstruction of top quark candidate events. The four leading jet hypothesis, described in the chapter 5 does not fully describe an event, for example those events in which the jets arise from radiation effects. Approximations of this form lead to a loss of sensitivity in the determination of mass and introduces a bias on the estimated parameter (mt ). For this reason, the analysis is first calibrated using Monte Carlo (MC) simulated events. The bias on the measured mass and the verity of the estimated statistical uncertainty can be achieved through ensemble testing. Each ensemble corresponds to a simulated experiment that is conducted to emulate the size and composition of the actual data observed in the detector. Hence, an ensemble is also referred to as a pseudo-dataset. To have maximum sensitivity and to capture the detailed structure of likelihood, the ensemble tests are performed in four different channels separately. These four channels are : • e + jets, ≥ 2tags • µ + jets, ≥ 2tags • e + jets, exactly 1tag • µ + jets, exactly 1tag 144 The fractions of t t¯ , W + jets in each channel are allowed to fluctuate according to the estimated fractions in the actual data sample. These fractions were derived in Sec. 4.4 and have been tabulated in Table. 6.2. 3000 such ensembles are constructed in each channel with the amount of t t¯ and W + jets, as shown in the table. The number of effectively independent ensembles varies from channel to channel and thus affects the uncertainty on each of the measured values. The MC statistics available in e + jets, µ + jets channels (without splitting into different tag bins) varies from approximately, 1 million to 50,000 events for t t¯ and W+jets respectively. Each of these events are weighted with event weights (wi ). The effective number of independent ensembles (Ne f f ) that can be constructed from this pool of MC events (Nsamp ) , is crudely given by : Nsamp Ne f f = (6.9) N In the Eq. 6.9, N represents that number of events that are picked in an ensemble. If the ensemble is an admixture of signal and background events, N = Nt t¯ + NW + jets would be the right approximation. Ne f f is used in assigning the uncertainty on the extracted value of the parameters, for each channel. To make optimal use of the available MC statistics, standard re-sampling techniques are used, allowing for multiple use of MC events when constructing the ensembles. If we represent wi as the event weight of the ith event in the sample, then the re-sampling involves, throwing a random number and comparing it to the wi ∑ wi (sum in the denominator runs over all events in the sample under consideration). The ith event is selected only if the value of this fraction is greater than the random number value. The process is carried until N number of events are selected. Since Nsamp is quite large, compared to N (eg. first two rows in Table. 6.2), it leads to a negligible correlation due to the multiply selected events, within an individual ensemble. 145 Ensemble tests are performed for nine input top quark masses (150, 160, 165, 170, 172.5, 175, 180, 185, 190 GeV /c2 ) . The JES factor (αJES ) is varied by multiplying, all the measured jet energies by a factor and feeding the events back into the kinematic fitter to recalculate the likelihood. The reference JES, which corresponds to jet energy scale corrections described in Section. 3.6 is often referred to as the nominal JES with αJES = 1.0. The two dimensional (2D) likelihood, thus obtained is searched for a global minimum in the values of −2 ln(L ), at which location a two dimensional function is fitted : F (mt , αJES ) = p0 × mt2 + p1 × αJES 2 + p2 × mt + p3 × αJES + p4 × mt × αJES + p5 (6.10) The Eq. 6.10, implies a basic assumption that the distribution of −2 ln(L ) values is Gaus- sian along the mt and JES axes. Since the signal shapes involved in calculating the sig- nal likelihoods have been shown to be of Gaussian nature (Fig. 6.3), it corroborates the above assumption. Any deviations would automatically introduce biases and this provides the purpose of achieving accurate calibrations. The standard deviation is the width of the Gaussian, which is represented by σ for which −2 ln(L ) increases by one unit. This is represented as : f itted f itted −2 ln((L (mt + σ )) + 2 ln(L (mt )) = 1 (6.11) f itted Eq. 6.6 is similar for the other extracted parameter JES f itted . The mt and JES f itted are obtained from minimum of the function F (mt , JES) , represented by Eq. 6.10: 146 f itted 2 × p1 × p2 − p3 × p4 2 × p0 × p3 − p2 × p4 (mt , JES f itted ) ≡ (− ,− ) (6.12) 4 × p0 × p1 − p24 4 × p0 × p1 − p24 The function F is fitted to the bins near the minimum, in the range of ±20 GeV /c2 in p the mt direction and ±0.09 in the αJES factor dimension. σ is scaled by Ne f f to provide, a rough approximation of the statistical uncertainty on the corresponding fitted parameters. In the next sections, another way of determining the uncertainty on the measurement is mentioned, that includes the effect of the slope of calibration curves. In order to gauge the effect on calibrations by increasing the impurities of the sam- ples, it is important to check the method with the purest sample first. In the next section, the calibrations for only parton matched t t¯ events are described, followed by full fledged calibrations of signal and background samples in Section 6.7. source e + jets, ≥ 2tags µ + jets, = 2tags e + jets, 1tag µ + jets, 1tag t t¯ production process 125 87 242 167 W+jets 4 6 37 45 Z+jets 1 0 3 3 WW, WZ, ZZ + jets 0 0 2 2 single t production 2 1 4 3 multijets 2 0 19 6 DATA 128 120 286 240 Table 6.2: Event yields in different channels 6.6 Ensemble tests with parton matched t t¯ events In this study, ensemble tests are performed using only the parton matched t t¯ events. In ad- dition, only the correct solutions corresponding to the right jet-parton assignments are used to construct the likelihood. Hence, only the first term in Eq. 6.6 is considered, while the 147 combinatorial and BG shapes are turned off, while evaluating the likelihood. This section mainly describes, how well the input parameters can be reproduced using the method. f itted Fig. 6.7, shows the distribution of mt when the input mass of 172.5 GeV /c2 is used in the four different channels. The bias , which is the difference between the fitted mass and the generator level mass (input), is plotted in Fig. 6.11. This is done for various values of input top mass and is fit linearly, yielding an offset and a slope. As can be seen from the figure, the dependence of the fitted mass on the input mass is stronger for channels with fewer tags. The offset is close to the ideal value for channels with = 2tags. This is somewhat expected as the probability of finding the correct solution in the events for which f itted two of the leading four jets are known to come from the b-quarks. The value of mt corresponds to the point in 2D likelihood space where −2 ln(L ) is minimized with respect to JES and mt . The value of JES f itted is a few percent away from the input value. The distribution of JES f itted for various channels in which the input value is the nominal value, is shown in Fig. 6.9. To understand the behavior of JES f itted with respect to the input JES, the full 2D fit is performed for two other scenarios : • All the jets in the events are scaled down by 6% of their values and an appropriate adjustment to the 6 ET is made. The event selection is done again and the selected events are reconstructed using the same procedure as for the nominal JES. This cor- responds to JESinput = 0.94. The distribution of JES f itted for various channels, in this scenario is shown in Fig. 6.8. • JESinput = 1.06 corresponds to scaling the jet energies up by 6% and following the above procedure. The distribution of JES f itted for various channels, in this scenario is shown in Fig. 6.10. Putting it all together, JES f itted for different input JES is plotted in Fig. 6.12, for different input top masses. Two things follow from these plots : 148 • JES f itted is independent of input top mass for almost all values of input JES. • Although the offsets in the plots do not exactly come out to be at the true value, the behavior of JES f itted is as expected, i.e. the fitted values scale up, when the JESinput is scaled up and vice versa. This is shown in Table. 6.3. Also, the difference between electron and muon channels is negligible, as expected. JESinput e + jets, ≥ 2tags µ + jets, ≥ 2tags e + jets, 1tag µ + jets, 1tag 0.94 0.922 0.924 0.933 0.935 1 0.97 0.97 0.973 0.974 1.06 1.039 1.04 1.049 1.052 Table 6.3: Offset in JES f itted for three different values of JESinput in the four channels 149 (a) e + jets, ≥ 2tags (b) µ + jets, ≥ 2tags (c) e + jets, 1tag (d) µ + jets, 1tag f itted Figure 6.7: mt distributions with input top quark mass of 172.5 GeV /c2 at JES f itted for JESinput = 1.0, for different channels (parton matched t t¯ events) 150 (a) e + jets, ≥ 2tags (b) µ + jets, ≥ 2tags (c) e + jets, 1tag (d) µ + jets, 1tag Figure 6.8: JES f itted distribution for JESinput = 0.94 in parton matched t t¯ events with input mass of 172.5 GeV /c2 , in different channels 151 (a) e + jets, ≥ 2tags (b) µ + jets, ≥ 2tags (c) e + jets, 1tag (d) µ + jets, 1tag Figure 6.9: JES f itted distribution for JESinput = 1.0 in parton matched t t¯ events with input mass of 172.5 GeV /c2 , in different channels 152 (a) e + jets, ≥ 2tags (b) µ + jets, ≥ 2tags (c) e + jets, 1tag (d) µ + jets, 1tag Figure 6.10: JES f itted distribution for JESinput = 1.06 in parton matched t t¯ events with input mass of 172.5 GeV /c2 , in different channels 153 (a) e + jets, ≥ 2tags (b) µ + jets, ≥ 2tags (c) e + jets, 1tag (d) µ + jets, 1tag f itted Figure 6.11: mt calibration at JES f itted for the nominal JES as input in the parton matched t t¯ events, for different channels 154 (a) e + jets, ≥ 2tags (b) µ + jets, 1tag (c) µ + jets, ≥ 2tags (d) e + jets, 1tag (e) e + jets, ≥ 2tags (f) µ + jets, 1tag Figure 6.12: JES f itted calibrations with JESinput scaled by 0% (top), −6% (middle) and +6% (bottom) for parton matched t t¯ events with different input masses. The y- axis of all the plots represents (JES f itted − 1.0), while the x-axis represents (mtgenerated − 170) GeV /c2 . 6.7 Ensemble tests and calibration with signal and back- ground events In this section, the ensemble tests are conducted for samples that are admixtures of t t¯ and W /Z + jets events. The sample composition for each sample, closely follows the one 155 detailed in Table. 6.2, for each of the four channels. The multijet event fractions are low and the shapes do not affect the likelihood and hence they are not included in the ensemble testing procedure. The combinatorial and the BG shapes are turned on. The BG shapes are also varied with scaling of JES and hence the likelihood in this case, represents full likelihood. The calibrations are carried out in three stages : f itted • Single channel JES f itted and mt calibrations are carried out • A residual calibration is derived for both the parameters after re-assigning the likeli- hood values from the calibrated values to the original 2D grid of mt and JES. • All the four channels are combined by adding the −2 ln L values of all the four chan- nels in each calibrated ensemble and mt and JES are determined from the minimum of the combined likelihood. The calibrations in these three stages are detailed in the following sections. Figure 6.13: A 2D representation of the sum of −2 ln(Likelihood) for 3000 ensembles in one channel 156 6.7.1 Channel-by-channel calibrations Ensemble tests in this category involve, construction of pseudo-datasets with signal (t t¯) and background (W /Z + jets) events in the four channels separately with the likelihood entail- ing the signal and background shapes. Each ensemble is fitted with the two dimensional f itted function, F (mt , αJES ), whose minimum value yields a mt and JES f itted. . The distri- butions of these two quantities for the 3000 ensembles that are constructed, are Gaussian. f itted The peak values of mt distributions for different input top quark masses are tabulated in Tables 6.4-6.7, corresponding to the four channels. Similar to the calibration curves, f itted shown in Fig. 6.11, the mt peak values are plotted against the generator level mass in Fig. 6.14 in each of the four channels. The scale of these plots is zoomed in near 172.5 f itted GeV, for convenience. Also, the RMS of the mt distributions are entered in the table, which are multiplied by √ 1 to represent the approximate statistical error on the peak Ne f f f itted value of the mt distribution. These errors are shown as the error bars in Fig. 6.14. To account for the correlation of statistical uncertainties, a 95% confidence interval band (blue shaded area) is drawn along a linear fit to the representative points. The parameters of these fits represents the mass calibration, which are used in the next section. The linear fit parameters (p0 , p1 ) are tabulated in Table. 6.9. Also, shown in the plots is the ideal scenario where the input mass is equal to the output mass. This is drawn as a dotted blue line in the plots and is closer to the actual calibration for channels with ≥ 2tags. Similarly, the parameter, JES f itted is calibrated by repeating the the whole procedure at six different values of input JES (0.94, 0.97, 1.0, 1.03, 1.06). The mass calibrations, men- tioned in the above paragraph are obtained for JESinput = 1.0. The six values of JESinput are obtained, not just by scaling the jets with the appropriate factor, but also by repeating the event selection and adjustment of the 6 ET . The peak values of the JES f itted distribu- tions for the four channels at the input top quark mass of 172.5 GeV are tabulated in Table. 157 6.8. These are plotted as (JES f itted − 1.0) vs (JESinput − 1.0) in Fig. 6.15, with the 95% confidence interval on the linear fit,shown as the shaded yellow band, while the ideal sce- nario depicted as the red dotted line. The parameters of the linear fits, which is the JES calibration, are tabulated in Table 6.9. Generator Mass Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) GeV /c2 (GeV /c2 ) (GeV /c2 ) 150 148.8 2.083 0.086 0.179138 160 157.1 2.363 0.108 0.255204 165 161.9 2.515 0.109 0.274135 170 166.3 2.705 0.068 0.18394 172.5 168.5 2.805 0.072 0.20196 175 170.9 2.855 0.0796 0.227258 180 175.5 2.803 0.079 0.274694 185 179.9 3.028 0.0978 0.2961384 190 184.4 3.412 0.0728 0.2483936 Table 6.4: mtf itted peak values for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in e + jets, = 2tags channel. Generator Mass Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) (GeV /c2 ) (GeV /c2 ) 150 148.9 2.678 0.0709 0.1898702 160 157.1 2.962 0.0885 0.262137 165 161.7 2.951 0.0896 0.2644096 170 165.6 3.235 0.0563 0.1821305 172.5 167.7 3.305 0.0605 0.1999525 175 169.8 3.39 0.06623 0.2245197 180 174 3.398 0.0819 0.2782962 185 179 3.781 0.0813 0.3073953 190 184.1 3.991 0.0606 0.2418546 Table 6.5: mtf itted peak values for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in µ + jets, = 2tags channel. 158 Generator Mass Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) (GeV /c2 ) (GeV /c2 ) 150 144 2.764 0.1062 0.2935368 160 153 3.091 0.137 0.423467 165 158.1 2.821 0.1392 0.3926832 170 162.6 2.941 0.0842 0.2476322 172.5 164.6 3.276 0.0904 0.2961504 175 166.94 3.019 0.1 0.3019 180 171.19 3.435 0.126 0.43281 185 175.6 3.977 0.125 0.497125 190 181.28 3.587 0.091 0.326417 Table 6.6: mtf itted peak values for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in e + jets, 1tag channel. Generator Mass Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) (GeV /c2 ) (GeV /c2 ) 150 145.7 3.498 0.0905 0.316569 160 155.2 3.494 0.1169 0.4084486 165 158.6 3.634 0.1177 0.4277218 170 162.5 3.649 0.0723 0.2638227 172.5 164.16 3.596 0.0775 0.27869 175 165.9 4.25 0.0858 0.36465 180 170.4 4.32 0.1081 0.466992 185 175.6 5.206 0.1083 0.5638098 190 180.36 4.75 0.07855 0.3731125 Table 6.7: mtf itted peak values for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in µ + jets, 1tag channel. 159 MASS calibration e+jets 2tags MASS calibration µ+jets 2tags 15 15 (Fitted mass-172.5) GeV (Fitted mass-172.5) GeV χ2 / ndf 6.008 / 8 χ2 / ndf 40.51 / 8 10 10 Prob 0.6464 Prob 2.578e-06 5 5 p0 -3.844 ± 0.07274 p0 -4.397 ± 0.07276 0 0 p1 0.8936 ± 0.005638 p1 0.876 ± 0.005752 -5 -5 -10 -10 -15 -15 -20 -20 -25 -25 -25 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 (Generator mass-172.5) GeV (Generator mass-172.5) GeV (a) e + jets, ≥ 2tags (b) µ + jets, ≥ 2tags MASS calibration e+jets 1tag MASS calibration µ+jets 1tag (Fitted mass-172.5) GeV 10 χ2 / ndf 7.102 / 8 (Fitted mass-172.5) GeV 10 χ2 / ndf 29.25 / 8 5 5 Prob 0.5256 Prob 0.0002871 0 0 p0 -7.728 ± 0.1051 p0 -7.753 ± 0.1131 -5 -5 p1 0.9271 ± 0.008464 p1 0.8542 ± 0.009299 -10 -10 -15 -15 -20 -20 -25 -25 -30 -30 -25 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 (Generator mass-172.5) GeV (Generator mass-172.5) GeV (c) e + jets, 1tag (d) µ + jets, 1tag Figure 6.14: (Fitted mass − 172.5) GeV /c2 vs. (Generated mass − 172.5) GeV /c2 for dif- ferent channels for ensembles consisting of signal and background events. 160 JESinput JES f itted RMS JESinput JES f itted RMS 0.94 0.976 0.024 0.94 0.981 0.029 0.97 1.003 0.024 0.97 1.009 0.027 1 1.029 0.023 1 1.034 0.027 1.03 1.055 0.022 1.03 1.061 0.026 1.06 1.081 0.022 1.06 1.084 0.026 (a) e + jets, = 2tags (b) µ + jets, = 2tags JESinput JES f itted RMS JESinput JES f itted RMS 0.94 1.04 0.024 0.94 1.047 0.025 0.97 1.063 0.022 0.97 1.07 0.027 1 1.076 0.018 1 1.087 0.025 1.03 1.101 0.019 1.03 1.107 0.023 1.06 1.122 0.019 1.06 1.126 0.022 (c) e + jets, 1tag (d) µ + jets, 1tag Table 6.8: JES f itted peak values for six different values of JESinput for an input top quark mass of 172.5 GeV, across the four channels. 161 JES calibration e+jets 2tags JES calibration µ+jets 2tags χ2 / ndf 0.215 / 4 χ2 / ndf 3.185 / 4 0.08 0.08 Prob 0.9946 Prob 0.5274 0.06 0.06 p0 0.02873 ± 0.0006921 p0 0.0336 ± 0.000693 (Fitted JES-1.0) (Fitted JES-1.0) 0.04 p1 0.8734 ± 0.01495 0.04 p1 0.8545 ± 0.01511 0.02 0.02 0 0 -0.02 -0.02 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 (Input JES-1.0) (Input JES-1.0) (a) e + jets, ≥ 2tags (b) µ + jets, ≥ 2tags JES calibration e+jets 1tag JES calibration µ+jets 1tag χ2 / ndf 10.57 / 4 0.13 χ2 / ndf 1.821 / 4 0.12 0.12 Prob 0.03185 Prob 0.7687 0.11 0.1 p0 0.08015 ± 0.0007849 p0 0.08725 ± 0.0007947 0.1 (Fitted JES-1.0) (Fitted JES-1.0) p1 0.684 ± 0.01755 p1 0.6491 ± 0.01673 0.09 0.08 0.08 0.06 0.07 0.06 0.04 0.05 0.04 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 (Input JES-1.0) (Input JES-1.0) (c) e + jets, 1tag (d) µ + jets, 1tag Figure 6.15: (Fitted JES − 1.0) vs. (Input JES − 1.0) for different channels f itted Channel (mt − 172.5) = p1 × (mtinput − 172.5) + p0 (JES f itted − 1.0) = p1 × (JESinput − 1.0) + p0 p0 (o f f set) p1 (slope) p0 (o f f set) p1 (slope) e + jets, = 2tags -3.844 0.894 0.0287 0.873 µ + jets, = 2tags -4.397 0.876 0.0334 0.855 e + jets, 1tag -7.728 0.927 0.0802 0.684 µ + jets, 1tag -7.753 0.854 0.0873 0.649 f itted Table 6.9: First Calibration parameters for mt and JES f itted in the four channels. 162 6.7.2 Residual Calibrations for each channel The calibrations obtained in the previous section are used to reproduce the input values of top mass and JES. First, the likelihood values in the slices of the 2D grid along the JES direction is fitted with a sum of quadratic and quartic functions in each of the mass bins for every ensemble. This allows one to re-evaluate the likelihood at JES f itted , using the linear transformation, whose parameters are listed in Table 6.9. The likelihood thus evaluated is re-assigned to the bin in the original 2D grid. This completes the calibration along the f itted JES axis. mt is calibrated in a similar way through the linear transformation of mt , f itted listed in the Table. 6.9. The likelihood value is read off from the bin that contains mt and is re-assigned to the bin that contains mtinput . This is done for all the values of JES in the 2D grid. It is to be noted that the bin width of 1 GeV /c2 , is the limitation in the f itted precise knowledge of mt . Each ensemble likelihood distribution is again fitted with f itted the two dimensional function of F (mt , αJES ) and the peak values of the new mt and JES f itted , turn out to be Gaussian in nature. These are tabulated in Tables 6.10-6.13 for f itted mt and Table 6.14 for JES f itted . The new fitted values are again put on a calibration curve, which is referred to as the residual calibration. The residual calibration for the f itted mt peak values vs. the generator level top quark mass is shown in Fig. 6.16 and the residual JES calibration for different channels is shown in Fig. 6.17. As can been seen from both the figures, the dotted lines that represent the ideal scenario of input being equivalent to the output, are eclipsed by the linear fit to the points, within the 95% confidence interval band. This procedure provides a cross check on the calibrations derived in the previous section. It is also worthwhile to note that the offsets in the residual calibrations are within the limits of the bin size and hence further applying these calibrations would require some interpolation within inside the bin. This is discussed in the next section that also discusses the combination of the four channels. 163 Residual MASS calibration e+jets 2tags Residual MASS calibration µ+jets 2tags (Fitted mass-172.5) GeV (Fitted mass-172.5) GeV 20 χ2 / ndf 20 χ2 / ndf 6.639 / 8 47.89 / 8 15 Prob 0.5761 Prob 1.037e-07 10 10 p0 -0.2575 ± 0.07964 p0 -0.2112 ± 0.08209 5 0 p1 1.002 ± 0.006207 0 p1 1.006 ± 0.006465 -5 -10 -10 -15 -20 -20 -25 -25 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 (Generator mass-172.5) GeV (Generator mass-172.5) GeV (a) e + jets, ≥ 2tags (b) µ + jets, ≥ 2tags Residual MASS calibration e+jets 1tag Residual MASS calibration µ+jets 1tag (Fitted mass-172.5) GeV (Fitted mass-172.5) GeV 20 χ2 / ndf 20 χ2 / ndf 9.768 / 8 26.47 / 8 15 15 Prob 0.2817 Prob 0.0008737 10 10 p0 -0.2628 ± 0.1174 p0 -0.644 ± 0.1344 5 5 0 p1 0.9922 ± 0.01014 0 p1 0.9759 ± 0.01115 -5 -5 -10 -10 -15 -15 -20 -20 -25 -25 -25 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 (Generator mass-172.5) GeV (Generator mass-172.5) GeV (c) e + jets, 1tag (d) µ + jets, 1tag Figure 6.16: (Fitted mass − 172.5) GeV vs. (Generated mass − 172.5) GeV for different channels at JES = 1 164 Generator Mass Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) GeV /c2 (GeV /c2 ) (GeV /c2 ) 150 150 2.326 0.086 0.2 160 159.3 2.363 0.108 0.2552 165 164.7 2.515 0.109 0.2741 170 169.6 3.016 0.068 0.2051 172.5 172.1 3.128 0.072 0.2252 175 174.7 3.174 0.0796 0.2527 180 179.9 3.12 0.098 0.3058 185 184.9 3.339 0.0978 0.3266 190 189.9 3.772 0.0728 0.2746 Table 6.10: mtf itted peak values (second calibration) for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in e + jets, = 2tags channel. Generator Mass Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) GeV /c2 (GeV /c2 ) (GeV /c2 ) 150 150.3 3.020 0.0709 0.2141 160 159.6 3.323 0.0885 0.2941 165 164.9 3.336 0.0896 0.2989 170 169.4 3.670 0.0563 0.2066 172.5 171.8 3.739 0.0605 0.2262 175 174.2 3.864 0.06623 0.25591 180 179 3.830 0.0819 0.31367 185 184.8 4.246 0.0813 0.3452 190 190.7 4.466 0.0606 0.27064 Table 6.11: mtf itted peak values (second calibration) for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in µ + jets, = 2tags channel. 165 Generator Mass Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) GeV /c2 (GeV /c2 ) (GeV /c2 ) 150 150 3.637 0.1062 0.38625 160 159.9 3.161 0.137 0.43306 165 165.2 2.904 0.1392 0.40423 170 169.9 3.004 0.0842 0.2529 172.5 172 3.436 0.0904 0.3106 175 174.2 3.728 0.1 0.3728 180 178.8 4.072 0.126 0.5131 185 184.2 4.132 0.125 0.5165 190 190 4.180 0.091 0.3804 Table 6.12: mtf itted peak values (second calibration) for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in e + jets, 1tag channel. Generator Mass Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) GeV /c2 (GeV /c2 ) (GeV /c2 ) 150 149.9 4.000 0.0905 0.362 160 161 3.755 0.1169 0.439 165 164.4 3.959 0.1177 0.466 170 169.5 4.039 0.0723 0.292 172.5 171 4.723 0.0775 0.366 175 173.4 4.825 0.0858 0.414 180 178.4 5.328 0.1081 0.576 185 184.1 6.085 0.1083 0.659 190 189.7 6.170 0.0786 0.485 Table 6.13: mtf itted peak values (second calibration) for different input top quark mass for ensembles with both t t¯ and W /Z + jets events in µ + jets, 1tag channel. f itted Channel (mt − 172.5) = p1 × (mtinput − 172.5) + p0 (JES f itted − 1.0) = p1 × (JESinput − 1.0) + p0 p0 (o f f set) p1 (slope) p0 (o f f set) p1 (slope) e + jets, = 2tags -0.258 1.002 -2.38×10−4 1.008 µ + jets, = 2tags -0.211 1.006 -3.41×10−4 1.006 e + jets, 1tag -0.263 0.992 -1.67×10−4 0.978 µ + jets, 1tag -0.644 0.976 0.0025 0.982 f itted Table 6.15: Residual Calibration parameters for mt and JES f itted in the four channels. 166 JESinput JES f itted RMS JESinput JES f itted RMS 0.94 0.938 0.024 0.94 0.937 0.029 0.97 0.969 0.024 0.97 0.971 0.027 1 1.001 0.023 1 1.000 0.027 1.03 1.030 0.022 1.03 1.032 0.026 1.06 1.060 0.022 1.06 1.059 0.026 (a) e + jets, = 2tags (b) µ + jets, = 2tags JESinput JES f itted RMS JESinput JES f itted RMS 0.94 0.94 0.024 0.94 0.942 0.025 0.97 0.974 0.022 0.97 0.973 0.027 1 0.997 0.018 1 1.004 0.025 1.03 1.029 0.019 1.03 1.032 0.023 1.06 1.059 0.019 1.06 1.061 0.022 (c) e + jets, 1tag (d) µ + jets, 1tag Table 6.14: JES f itted (second calibration) peak values for six different values of JESinput for an input top quark mass of 172.5 GeV, across the four channels. 167 Residual JES calibration e+jets 2tags Residual JES calibration µ+jets 2tags χ2 / ndf 0.6138 / 4 χ2 / ndf 3.273 / 4 0.06 0.06 Prob 0.9615 Prob 0.5132 0.04 0.04 p0 -0.0002384 ± 0.0008535 p0 -0.0003407 ± 0.0008176 0.02 0.02 (Fitted JES-1.0) (Fitted JES-1.0) p1 1.008 ± 0.01797 p1 1.006 ± 0.01774 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 (Input JES-1.0) (Input JES-1.0) (a) e + jets, ≥ 2tags (b) µ + jets, ≥ 2tags Residual JES calibration e+jets 1tag Residual JES calibration µ+jets 1tag χ2 / ndf 3.451 / 4 χ2 / ndf 0.4388 / 4 0.06 0.06 Prob 0.4853 Prob 0.9792 0.04 0.04 p0 -0.0001676 ± 0.001177 p0 0.002451 ± 0.001325 0.02 0.02 (Fitted JES-1.0) (Fitted JES-1.0) p1 0.9783 ± 0.02557 p1 0.9815 ± 0.02829 0 0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 (Input JES-1.0) (Input JES-1.0) (c) e + jets, 1tag (d) µ + jets, 1tag Figure 6.17: (Fitted JES − 1.0) vs. (Input JES − 1.0) for different channels 168 6.7.3 Combination The combination of four channels proceeds through two stages. First, each channel is cali- brated according to the linear transformation presented in Table 6.15. This procedure was described in the previous section and the only difference lies in the fact that the likelihood f itted value at mt is obtained through bi-linear transformation, demonstrated in Fig. 6.18. f itted For this, the quadrant of the bin containing mt is identified and the likelihood values of its nearest neighboring points are used in a bi-linear transformation to obtain the like- f itted f iited lihood value at mt . Unlike, using the bin center closest to mt (as done in previous calibrations), this procedure gives a more accurate local characterization of the likelihood. Every ensemble in each channel is thus calibrated in the above fashion. The 2D nega- tive log-likelihood (−2 ln L ) distribution of each calibrated ensemble is then added across all channels, before fitting it again with the two dimensional function F (mt , αJES ) . The f itted resulting mt and JES f itted values are tabulated in Tables 6.16-6.18. f itted In the discussion that follows, two mt values are quoted : f itted • The ’2D’ fitted mass refers to the peak value of the mt distribution obtained from the minimum of F (mt , JES) and hence it is the fitted mass at JES f itted . f itted • The ’1D’ fitted mass refers to the peak value of mt distribution obtained from the minimum of the 2D likelihood distribution when sliced at JES=1 f itted When the JES f itted comes out to be equal to one, both the definitions of mt yield the same value but the 2D fitted mass uncertainty has a component of the uncertainty of JES f itted , while the 1D fitted mass uncertainty is only statistical in nature. The errors (σ ) on the fitted parameters are re-estimated using the Eq. 6.11, which in the two dimensional case represents a rectangle bounding the elliptical cross-section of the −2 ln L distribution when its value increases by one, with respect to the minimum value. These errors are used 169 in defining the pull distribution of the estimated parameters. Pull is the deviation of the fitted parameter from its average value divided by the uncertainty. When a large number of pseudo-experiments are used for parameter estimation, the pull distribution is expected to be a Gaussian, centered at zero and with unit width. In this analysis, pulls are defined as follows : f itted mt − mtgenerator pullmass = (6.13) σmass JES f itted − JESinput pullJES = (6.14) σJES The distribution of pulls is shown in Fig. 6.20. σmass and σJES are estimated using Eq. 6.11 and are scaled by a constant factor of 1.24 and 1.62 respectively to allow for the widths of the pulls to be close to one. This is demonstrated in Fig. 6.21 which emphasizes the fact that the pull widths are constant over the entire mass range for both the 2D and 1D fitted mass. The distribution of the fitted parameters for the nominal input JES and an input top quark mass of 172.5 GeV are shown in Fig. 6.19. When put on a calibration curve, the 2D and 1D fitted masses lie within 0.5 GeV of the true (input) value and the 95% confidence interval bands (2σ ) around the linear fit to the points, are used in assessing the systematic uncertainty on the measurement, as shown in Fig. 6.22. More distributions of fitted parameters and their respective pulls at three different input top quark mass and three different input JES factor, are shown in Appendix C. The procedure described in this section is applied to the data and the results with sys- tematic errors are detailed in the next chapter. 170 Figure 6.18: Interpolation of likelihood values inside a bin through bi-linear transformation or nearest neighbor sampling. {Ref.[6] } Generator Mass 2D Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) GeV /c2 (GeV /c2 ) (GeV /c2 ) 150 150.4 1.491 0.05103 0.07608573 160 160.2 1.555 0.065 0.101075 165 165.2 1.576 0.0657 0.1035432 170 170 1.675 0.0405 0.0678375 172.5 172.3 1.782 0.04336 0.07726752 175 174.7 1.83 0.0478 0.087474 180 179.7 1.886 0.05975 0.1126885 185 185 2.001 0.0595 0.1190595 190 190.5 2.149 0.0437 0.0939113 Table 6.16: ’2D’ fitted mass for the combination of four channels. 171 JESinput JES f itted RMS 0.94 0.938 0.017 0.97 0.971 0.016 1 0.999 0.015 1.03 1.030 0.014 1.06 1.059 0.014 Table 6.18: Fitted JES for the combination of four channels. Generator Mass 2D Fitted mass RMS Scale = √ 1 Error Ne f f (GeV /c2 ) (GeV /c2 ) GeV /c2 (GeV /c2 ) (GeV /c2 ) 150 150.1 1.135 0.05103 0.05791905 160 160.2 1.132 0.065 0.07358 165 165.1 1.12 0.0657 0.073584 170 169.8 1.141 0.0405 0.0462105 172.5 172.3 1.185 0.04336 0.0513816 175 175 1.154 0.0478 0.0551612 180 180 1.229 0.05975 0.07343275 185 185.3 1.255 0.0595 0.0746725 190 190.5 1.317 0.0437 0.0575529 Table 6.17: ’1D’ fitted mass for the combination of four channels. f itted f itted Figure 6.19: JES f itted , 2D-mt and 1D-mt distributions for nominal input JES and an input top quark mass of 172.5 GeV. 172 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 Mean -0.04224 Mean -0.1271 Mean -0.1719 350 RMS 1.04 350 RMS 0.9961 RMS 1.038 350 χ2 / ndf 13.65 / 21 χ2 / ndf 24.36 / 21 χ2 / ndf 119 / 28 300 Prob 0.8842 Prob 0.2761 Prob 3.433e-13 300 Constant 344.8 ± 7.7 Constant 359.1 ± 8.0 300 Constant 343.8 ± 7.7 Mean -0.0416 ± 0.0190 Mean -0.1258 ± 0.0183 Mean -0.169 ± 0.019 250 Sigma 1.041 ± 0.013 250 Sigma 0.9997 ± 0.0129 Sigma 1.044 ± 0.013 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 f itted f itted Figure 6.20: Pull distributions of JES f itted , mt (2D) and mt (1D) for nominal input JES and an input top quark mass of 172.5 GeV. Pull width vs. mass (2D) Pull width vs. mass (1D) 2 2 2D Mass pull width 1D Mass pull width 1.5 1.5 1 1 χ2 / ndf 14.33 / 8 0.5 0.5 χ2 / ndf 75.98 / 8 Prob 0.07367 Prob 3.142e-13 p0 0.9903 ± 0.003917 0 0 p0 0.9876 ± 0.004044 p1 0.004184 ± 0.0003211 p1 0.001979 ± 0.0003317 -0.5 -0.5 -1 -1 -25 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 (Generator mass-172.5) GeV (Generator mass-172.5) GeV Figure 6.21: Pull width vs. generator level top mass for the ’2D’ and ’1D’ cases (Input JES = nominal JES). 173 Figure 6.22: Final mass calibrations after combining the four channels. CHAPTER Seven Results and Systematic uncertainties 7.1 Data Results The calibrations deduced in the previous chapter are applied to the data events to obtain the observed fitted mass. The results are tabulated in Table. 7.1. To estimate the uncertainties that are statistical and JES related, three measurements are quoted in the table. The full two dimensional (2D) result includes both statistical and JES components in the quoted uncertainty and is derived from the two dimensional likelihood, that is obtained in the same way as described in the previous chapter. The one dimensional (1D) measurement refers to the resultant fitted mass and its uncertainty for the case when the fitted JES is equal to unity. If the Monte Carlo JES was assumed to be true JES, as observed in data, a 1D measurement would result with an uncertainty that is purely statistical in nature. The sample used to derive the JES, described in Section. 3.6, has an uncertainty associated with it, depicted by 1 σ bands that are shown in Fig. 3.6. In this way, fitted JES could be constrained to unity by addition of a prior to −2 ln(L ) with a width that is compatible with the difference in the fitted JES values for the +1 σ and −1 σ values for the input JES. The resulting measurement is referred to as, 2D with prior. The prior used in this analysis is 174 175 given as follows : (y−1)2 n  o prior = −2 × ln √1 × 1 × exp − 2π 0.012 2×(0.012)2 Channel Fitted Mass Fitted JES GeV /c2 e + jets, = 2tags 170.0±3.12 (stat + JES) 1.06±0.022 µ + jets, = 2tags 177.1±3.17 (stat + JES) 1.00 ± 0.028 e + jets, 1tag 175.0±3.84 (stat + JES) 1.01 ± 0.028 µ + jets, 1tag 168.1±3.96 (stat + JES) 1.07 ± 0.028 All 4 channels (2D) 172.6±1.75 (stat + JES) 1.03 ± 0.013 All 4 channels (2D with prior) 173.6±1.67 (stat + JES) 1.02 ± 0.011 e + jets, = 2tags (1D) 174.8±2.17 (stat) n/a µ + jets, = 2tags (1D) 177.0±2.15 (stat) n/a e + jets, 1tag (1D) 176.0 ± 2.65 (stat) n/a µ + jets, 1tag (1D) 174.8±2.81 (stat) n/a All 4 channels (1D) 175.7±1.19 (stat) n/a Table 7.1: Summary of mt measurements in different channels along with the combination of all 4 channels. 7.2 Systematics The evaluation of systematic uncertainties in the top quark mass measurement, involves ac- counting for insufficient understanding of signal and background modeling through various theoretical processes and the lack of complete knowledge of detector response to various objects used in reconstruction of top quark events. The list of systematics is tabulated in Table 7.2. A detailed description of the systemat- ics can be found in Ref.[64],[7] and is briefly summarized below. The uncertainties corresponding to the detector response are evaluated by shifting the central values of the respective parameters viz. JES, lepton momentum scale, trigger effi- ciency, by ±1 standard deviations. The top events in the MC for mt = 172.5 GeV /c2 are 176 re-evaluated with the shift in these parameter values and the systematic uncertainty on the value of mt from these shifts is defined as ±|mt+ − mt− |/2, unless both shifts are in the same direction, relative to the nominal value, in which case the uncertainty is defined as max{|mt+ − mt |/2, |mt− − mt |/2}. When comparing between two models, the largest differ- ence among the resulting fitted masses is taken as the systematic uncertainty. 7.2.1 Detector Response Jet Energy Scale The uncertainties on the global jet energy scale factor, described in Sec. 3.6, are propagated to the mt measurement. It is found that the obatined top quark mass is shifted by 1.2 GeV /c2 up (1.2 GeV /c2 down) when we vary the carrection factors by one σ up (down). This shift is also taken as the width of the prior, as defined in the previous section. For the ’2D’ top quark mass measurement, the corresponding uncertainty is included in the quoted (stat+JES) uncertainty. Sample Dependent JES The response of the detector to various flavors of partons (b, light, gluon) was described in Sec.3.6.1 and the corresponding uncertainties were depicted in Fig.3.9. A variation of this correction factor to the ±1 σ values does not shift the mt or the fitted JES value. Residual JES The in-situ JES calibration entails only the global scale difference in JES between data and MC. The dependence of this scale factor on the pT , η of the jets can introduce a systematic shift to the mt measurement. To evaluate this, the energy of each jet in a t t¯ MC sample is scaled by a factor that is parameterized as a function of pT and η. This parameterization 177 corresponds to the quadratic sum of the uncertainties of the jet energy scale in data and MC. This parameterization is shown in Fig. 7.1. Re-evaluating mt by correcting all the jets in MC (both t t¯, W+jets) using this parametrized uncertainty as a function or the kinematic quantities, allows for determination of the systematic uncertainty. The correction factors shown in the figure are relative to the average jet transverse energy correction factor. Jet Energy Resolution The modeling of energy resolution for jets can affect the parton level corrections (PLC) and hence a variation in the resolution parameter by one standard deviation is studied for t t¯ events for mt = 172.5 GeV /c2 and the shift in the fitted value is used to assign the final systematic uncertainty in this category. Jet Identification Efficiency The uncertainties associated with the scale factors used to arrive at the data to Monte Carlo agreement in the jet identification efficiencies are propagated to the measurement of mt by decreasing the efficiencies in mtgenerated = 172.5 GeV /c2 t t¯ MC sample. The systematic un- certainty corresponds to one sigma deviation uncertainties associated with the scale factors used to achieve data and MC agreement in jet identification efficiencies. JSSR Shifting In general, the jets have better resolution in simulation than in the data. To account for this effect, simulated jets are randomly removed based on data jet identification. In addidtion, the energy scale of jets in the simulation is shifted to match the mean value of transverse momentum imbalance of jets in data and simulation. This process is called shi f ting and the full procedure is termed as JSSR (jet shifting, smearing and removal, Ref.[65]). In this analysis, the shifting is turned off for the t t¯ MC. A systematic uncertainty is assigned 178 toaccomodatethis effect by extracting mt for a sample of t t¯ → l + jets with the mass of 172.5 GeV. b-tagging efficiency A discrepancy between data and MC in the modeling of the efficiency to tag b-jets can result in a systematic shift of the extracted mt value. This effect is evaluated by varying the tag rate functions (TRF) and mistag rate function for light quarks by 5% Ref.[66] and 20% respectively. Redoing the ensemble tests for the t t¯ MC with mt = 172.5 GeV /c2 shifts the extracted top mass by 0.08 GeV, as quoted in Ref.[64]. The TRFs used in this analysis are a part of the top mass extraction algorithm and the only efficiency of the b-jets that could affect the outcome of this analysis is that of the Neural Network b-tagging output that is used to split the sample in one tag and greater than two tag channels, as descibed in Sec. 4.4.1. Since we combine the channels to obtain the final result, the effect of any small change of number of events in individual channels would only be a second order effect to the assigned systematic uncertainty in this category. Trigger Efficiency The Monte Carlo events used in this analysis are weighted to account for the trigger effi- ciency in accordance with the data. To evaluate the systematics effect in the measurement of top mass, all the event weights corresponding to trigger are set to unity and the ensem- ble tests are repeated to observe any shift in the extracted mass. This shift is found to be 0.05 GeV /c2 and is tabulated in the Table 7.2. Lepton Momentum Scale efficiency Differences in lepton momentum scale in data and MC can introduce systematic shift in mt measurement. To evaluate this, the peak of dilepton invariant mass in J/ψ → ll and Z → ll 179 decays are compared between data and MC and an absolute momentum scale for electrons and muons is determined. A fit to the two mass points of 3.09 GeV /c2 and 91.18 GeV /c2 (corresponding to the J/ψ and Z invariant masses) is performed as a function of the mean value of the transverse momentum of the leptons. The mt is measured for t t¯ MC with a generator level mass of 172.5 GeV /c2 without the rescaling of lepton pT and with lepton pT values rescaled using the fit described above. Half of the largest difference in extracted mt is taken as its systematic uncertainty in this category. 0.5 ≤|η|<1.0 p0 0.01576 ± 0.0003295 p1 4.512e-06 ± 1.47e-06 p2 0.02064 ± 0.0005979 0.028 p3 34.98 ± 1.991 0.026 0.024 0.022 0.02 0.018 0.016 0 50 100 150 200 250 300 350 400 450 1.0 ≤|η|<1.5 p0 0.0211 ± 0.0003302 1.5 ≤|η| p0 0.01407 ± 0.003907 p1 -4.4e-07 ± 1.222e-06 p1 1.281e-05 ± 7.667e-06 p2 0.03061 ± 0.0007124 p2 0.03085 ± 0.003378 0.04 p3 41.35 ± 1.557 p3 141.9 ± 20.28 0.04 0.035 0.035 0.03 0.03 0.025 0.025 0.02 0.02 0 50 100 150 200 250 300 350 400 450 50 100 150 200 250 300 350 400 450 Figure 7.1: Residual JES parametrization as a function of pT of jets in various |η det | for the t t¯ MC. 180 7.2.2 Production Systematics 7.2.2.1 Signal Modeling ISR/FSR The incoming and outgoing partons may give rise to additional jets which are referred to as the initial and final state radiation. These jets may well be misidentified as the prod- uct of t t¯ decay and hence bias the reconstruction of the event. This effect is studied using Drell Yan dilepton events which are produced from a qq¯ initial state similar to t t¯ produc- tion (Ref.[67]). The pT of dilepton pairs as a function of their invariant mass is compared between data and MC and the best value for the hadronization scale (ΛQCD ) and the virtu- ality scale (Q2 ), extrapolated to t t¯ mass region, is extracted. The values obtained for ±1 σ deviations around the mean lepton pair pT are used to evaluate the ISR systematics. The QCD evolution equation is the same for both sets of scales and hence the above obtained sets of ΛQCD and Q2 are used for FSR systematics too. This amounts to ±0.65 GeV /c2 uncertainty on the extracted mt when the ensemble tests are done without the prior. Higher Order QCD effects The mt value is extracted for two models of the hard scattering and shower evolution process. ALPGEN interfaced with HERWIG (for parton showering) is compared with MC@NLO Ref.[68] interfaced with HERWIG. This includes the effects of qq¯/gg fraction in the initial state. The JES f itted for the full two dimensional fit yields 1.02 for ALP- GEN+HERWIG and 1.05 for MC@NLO+HERWIG , corresponding to a systematic uncer- tainty of 2.6 GeV /c2 in the extracted mt . 181 Hadronization and UE effects Keeping the same set of hard-scattering processes through ALPGEN, two different mod- els of hadronization are interfaced with ALPGEN. These are HERWIG and PYTHIA. The difference in the extracted mt for these two models yields systematic uncertainty of ±1.7 GeV /c2 for the ’2D’ measurement. Color reconnection The colored objects (quarks), which are produced as the decay product of t t¯, besides the initial and final state radiations interact with each other and also with the remnants of the p p¯ collisions. This can change the kinematic and topological distributions of jets and the systematic uncertainty corresponding to this effect is termed color reconnection. To evaluate this, t t¯ MC is generated with two PYTHIA tunes that involve angular ordering of showers (similar to the nominal sample). These tunes are APRO Ref.[69] and ACR-PRO. The difference in the extracted mt provides the uncertainty of ±0.7 GeV /c2 for the ’2D’ measurement without a prior. Multiple p p¯ interactions Luminosity effects from additional p p¯ interactions are simulated by overlaying on MC events, the unbiased triggers from random p p¯ crossings. These overlaid events are then reweighted to the number of interaction vertices, to assure that the simulation reflects the instantaneous luminosity profile of the data. The ensemble studies are repeated and mt is extracted to find a shift of ±0.07 GeV /c2 , as quoted in Ref.[64]. 182 Parton distribution functions t t¯ MC generated with PYTHIA for a mass of 172.5 GeV is reweighted to match possible excursions in the PDF parameters, represented by two sets of 20 eigenvectors of CTEQ6M uncertainty PDFs Ref.[11]. Ensemble studies are repeated for each of the variants and uncertainties are added in quadrature as follows : !1/2 20 1 δ mtPDF = ∑ {∆M(Si+ ) − ∆M(Si− )}2 (7.1) 2 i=1 The sum runs over PDF excursions in the positive (Si+ ) and negative (Si− ) directions. 7.2.2.2 Background Modeling W+jets heavy flavor scale factor The normalization of the leading order (LO) ALPGEN MC for the W+jets sample is in- creased by a factor of 1.47 for W+heavy flavor jets contribution to provide agreement with higher order (NLO) calculation of cross-sections, that include NLL corrections based on the MCFM MC generator Ref.[57]. This is done before normalizing the backgrounds to the data. To account for the systematics due to this scaling, the analysis is repeated by taking the scale factors as 1.97 and 0.97. The difference in the extracted mt with the nominal is used as systematic uncertainty on the measurement. Modeling of b-quark The default modeling of b-quark fragmentation used in the t t¯ MC corresponds to that of PYTHIA, which is based on the Bowler scheme Ref.[70], whereby the fragmentation pa- rameters are tuned to the LEP e+ e− data Ref.[71]. To assess the systematics, the events are reweighted to account for the differences between LEP and SLAC data Ref.[72]. The 183 ensemble studies are repeated and the difference in the extracted mt is determined. Also the b-jets calorimeter response is different in jets that decay semileptonically than for those that don’t. The effect on mt due to this was studied in Ref.[72] and was determined to be ±0.05 GeV /c2 . This uncertainty is added in quadrature to the one, derived for fragmenta- tion function, mentioned above. The net uncertainty is tabulated in Table7.2. Factorization and Renormalization scales The W+jets MC samples used in this analysis are generated by employing the identical factorization and renormalization scales such that Q2 = MW 2 + p2 , where the sum runs ∑ T over all the jets in an event. To determine the uncertainty on mt due to the uncertainty on this scale, W+jets samples are generated with the scale variation of (Q2 /2) and (2Q2 ). This affects the transverse momentum distribution of jets while keeping the normalization factor intact. The BG shapes are re-evaluated and the likelihood re-determined for the ensemble studies. Half of the largest excursion in the extracted mt is taken as the systematic uncertainty. 7.2.3 Method MC calibration The statistical uncertainties associated with the slope and offset of the final calibration curve is shown in Fig.6.22. The 68% confidence interval (half of the width of the band shown in the Fig. 6.22 )is used to assign the systematic uncertainty on the extracted mt at 172.5 GeV. 184 Multjet contamination Multijet events are not used in deriving the calibrations. The effect on calibration is studied by selecting a multijet-enriched sample of events from data (obtained by using the lepton reverse isolation criteria in selection). The ensemble studies were repeated and the uncer- tainty was found to be ±0.14 GeV /c2 in the extracted mt as shown in Ref.[73]. 185 Systematics Source Uncertainty (2D) Uncertainty (1D) Uncertainty (2D with prior) GeV /c2 GeV /c2 GeV /c2 DETECTOR RESPONSE Jet Energy Scale (JES) n/a ±1.2 n/a Sample Dependent JES 0 0 0 Residual JES ±0.2 0 ±0.2 Jet Energy Resolution ±0.4 ±0.75 0 Jet Identification efficiency ±0.1 ±0.1 ±0.1 JSSR shifting ±0.3 ±0.2 0 b-tagging efficiency ±0.08 ±0.08 ±0.08 Trigger efficiency ±0.05 ±0.05 ±0.05 Lepton scale efficiency ±0.1 ±0.1 ±0.1 PRODUCTION PROCESS Signal Modeling : ISR/FSR ±0.65 ±0.25 ±0.35 Higher order effects ±2.6 ±0.6 ±1.4 Hadronization, UE effects ±1.7 ±0.1 ±1.0 Color Reconnection ±0.7 ±0.1 ±0.4 Multiple p p¯ interactions ±0.07 ±0.07 ±0.07 Choice of PDF ±0.35 ±0.12 ±0.35 Background Modeling : W+jets heavy flavor scale factor ±0.05 ±0.05 ±0.05 b-quark fragmentation modeling ±0.1 0 ±0.1 Factorization & renormalization scales ±0.05 ±0.05 ±0.05 METHOD MC calibration ±0.12 ±0.09 ±0.12 Multijet contamination ±0.1 ±0.1 ±0.1 TOTAL ±3.32 ±1.59 ±1.86 Table 7.2: Various sources of systematic uncertainties to the top quark mass measurement. To summarize, the measurement of the top quark mass using a full two dimensional likelihood fit with in-situ JES yields : 186 mt = 172.6 ± 1.75 (stat + JES) ± 3.32 (sys) GeV /c2 = 172.6 ± 3.75 GeV /c2 (7.2) Constraining the measured jet energy scale to the MC jet energy scale within a known uncertainty, by incorporating a prior yields : mt = 173.6 ± 1.67 (stat + JES) ± 1.86 (sys) GeV /c2 = 173.6 ± 2.49 GeV /c2 (7.3) Using only the apriori JES from γ + jets events, the one dimensional measurement yields : mt = 175.7 ± 1.19 (stat) ± 1.2 (JES) ± 1.04 (sys) GeV /c2 = 175.7 ± 1.98 GeV /c2 (7.4) All the above measurements are consistent with each other, within the assigned uncer- tainties and emphasize the strong dependence of the precision of top quark mass measure- ment on the hadronic jet energy scale. Radiation and higher order effects dominate the systematic uncertainty on the ’2D’ measurement. CHAPTER Eight Conclusions and Perspectives We have measured the top quark mass using 4.3 f b−1 of RunII data collected at DØ . The dominant systematic uncertainty on the top quark mass measurements done till date have been attributed to the jet energy scale. Hence, two different approaches with complimen- tary techniques for jet energy scale were used for measuring the top quark mass in this dissertation. First, a simultaneous fit of the top quark mass with the jet energy scale is performed and the measurement yields a top quark mass value of 172.6 GeV /c2 with an associated uncertainty of 2.17 %. In the second approach, the top quark mass is measured using the apriori jet energy scale calibration derived from γ + jets events. This method yields a value of the top quark mass to be 175.7 GeV /c2 . The associated uncertainty ob- tained via this approach is 1.12 %. It is thus found that the simultaneous measurement is not as precise as the latter because of the important systematic uncertainties that affect the jet energy scale measurement and thus the top quark mass measurement. So we try a third approach which tries to combine the two techniques by applying a jet energy scale prior to the simultaneous measurement. This approach yields a value of top quark mass to be 173.6 GeV /c2 with an associated uncertainty of 1.43 %. This approach only partially mediates the effect and the measurement is still dominated with significant systematic uncertainties 187 188 in the jet energy scale. Hence the precision on the value of top quark mass is best obtainted with apriori jet energy scale. Comparing the measurement of the most precise value of top quark mass obtained in this dissertation (Eq. 7.4) with the most recent Tevatron com- bination for the top quark mass, which is reportedly found to be 173.18 GeV /c2 with an associated uncertainty of 0.54% (Ref. [7]), it can be concluded that that they are compat- ible with each other. A summary of measurements of top quark mass corresponding to various run periods and different channels is shown in Fig. 8.1. A direct comparison of the measurements in this dissertation with those of Fig. 8.1 is not completely trivial due to the presence of various correlations (statistical and systematics related). Top quark mass interpretation It is still evident that the ’2D’ measurement (Eq. 7.2) does not correspond to the best precision with systematic uncertainties corresponding to gluon radiation and higher order effects dominating the jet energy scale uncertainty and hence the top quark mass measure- ment.This brings us to the topic of interpretation of top quark mass. For a quark, the mass parameter is introduced in the QCD Lagrangian whose value depends on the renormaliza- tion scheme and the renormalization scale. At high energies, the QCD coupling constant (αs ) is small and the observables are typically calculated in perturbation theory, commonly applying MS renormalization scheme (Ref. [74]) . For an observed, non-colored particle, the position of the pole in the propagator defines the mass. However, the pole mass cannot be determined to arbitrary accuracy owing to the non-perturbative effects (long distance physics a.k.a. confinement) which is limited by the hadronization scale ΛQCD . The relation between the MS mass and the pole mass is known to three loops (Ref. [75]). In principle, it is possible to absorb higher order corrections into the pole mass definition via line shape with constant width. This has been done for extracting the Z boson mass (Ref. [76]). 189 It is also to be noted that the top quark mass measurements rely on the comparison of the data with the simulated events. The simulation used in this analysis to obtain the calibration is ALPGEN interfaced with PYTHIA. ALPGEN has fixed width in the quark propagator and PYTHIA uses a factor dependent on αs to approximate the loop corrections. In this way a simulation of t t¯ +nl p, n = 0, 1, 2 is obtained. It has been argued that the Monte Carlo mass (mtMC ) is closer to the pole mass within 1 GeV /c2 (Ref. [77, 78]). Mass extraction for the inclusive t t¯ + X cross section has been reported recently by DØ (Ref. [1]). Fig. 8.2 shows the dependence of σt t¯ on the top quark pole mass in various higher order schemes and soft gluon resummations. This is also tabulated in Table 8.1. Although the mass from extracted cross section would never be as precise as that of the direct measurements, the ’2D’ measurement (Eq. 7.2) is compatible with all of the extracted masses shown in the table and is closest to the Approximate NNLO scheme (Ref. [13, 79]). Top mass measurements at hadron colliders are reaching the level of theoretical uncer- tainties. In future, the top quark mass measurement at the t t¯ threshold including bound state effects (Coulomb summations, Ref. [80]) would be a subject matter of study. Also, it would be desirable to determine the top quark mass that is not limited by the uncertainty on the pole mass. This could possibly be achieved by performing t t¯ threshold scans and doing measurements of the σt t¯ near threshold at a future e+ e− collider. The top quark mass parameter measured this way could be translated to the MS mass with a smaller uncertainty due to a better knowledge of the initial state. This has been discussed in literature (e.g. Ref. [81] and references therein) and the technique is similar to the determination of W boson mass from WW cross-section, which was performed at LEP 2. 190 Figure 8.1: The twelve input measurements of the top quark mass from the Tevatron col- lider experiments along with the resulting combined value. The grey region corresponds to ±0.94 GeV /c2 . (Ref. [7]) 191 Figure 8.2: Measured σt t¯ and theoretical NLO+NNLL and aproximate NNLO calculations of σt t¯ as a function of top quark pole mass, assuming that the mass of top quark in simula- tion is equal to the pole mass. (Ref. [1]) Theoretical Prediction mtpole (GeV /c2 ), given mtMC = mtpole +5.7 NLO [82] 164.8−5.4 +5.5 NLO+NLL [83] 166.5−4.8 +5.1 NLO+NNLL [84] 163.0−4.6 +5.2 Approximate NNLO [13] 167.5−4.7 +5.2 Approximate NNLO [79] 166.7−4.5 Table 8.1: Values of mtpole , with their 68% confidence interval uncertainties, extracted for different predictions of σt t¯. The result assumes that the top quark mass in the simulation is equal to the pole mass of the top quark propagator. (Ref. [1]) APPENDIX A Parton Energy distributions for light quarks The Parton Level Corrections and corresponding resolution functions for light quarks are obtained by using matched jet-parton pairs for (u, d, c, s) quarks in t t¯ → l + jets events. The figures below show the parton energy distribution in various jet energy bins for 4 detector regions. The distributions are close to Gaussian and are fitted with the Gaussian function near the peak. The mean value gives the PLC, while the RMS of the Gaussian provides with the parton energy resolution for that particular jet energy bin. 192 193 • Region 1 partonE_Light_r1_0 600 partonE_Light_r1_0 partonE_Light_r1_1 partonE_Light_r1_1 partonE_Light_r1_2 partonE_Light_r1_2 Entries 22185 1000 Entries 40084 900 Entries 35564 Mean 27.1 Mean 29.29 Mean 31.67 800 500 RMS 12.39 RMS 11.25 RMS 10.35 χ / ndf 2 25.58 / 26 800 χ / ndf 2 30.86 / 27 700 χ / ndf 2 49.5 / 27 Prob 0.4861 Prob 0.2768 Prob 0.005188 400 Constant 549.3 ± 6.7 Constant 969.7 ± 8.9 600 Constant 837.9 ± 8.2 Mean 22.18 ± 0.09 600 Mean 25.34 ± 0.07 Mean 28.26 ± 0.08 Sigma 5.453 ± 0.150 Sigma 5.618 ± 0.114 500 Sigma 6.057 ± 0.151 300 400 400 200 300 200 100 200 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_3 partonE_Light_r1_3 partonE_Light_r1_4 partonE_Light_r1_4 partonE_Light_r1_5 partonE_Light_r1_5 700 Entries 25946 Entries 26821 Entries 28283 600 Mean 33.67 600 Mean 35.44 Mean 37.01 RMS 10.22 RMS 10.2 600 RMS 9.857 χ2 / ndf 44.72 / 29 χ2 / ndf 33.88 / 29 χ2 / ndf 35.47 / 29 500 500 Prob 0.03135 Prob 0.2437 500 Prob 0.1895 Constant 597.4 ± 6.7 Constant 612.9 ± 6.8 Constant 644.1 ± 7.0 400 Mean 30.44 ± 0.09 400 Mean 32.32 ± 0.09 Mean 34.07 ± 0.09 400 Sigma 6.448 ± 0.182 Sigma 6.326 ± 0.171 Sigma 6.338 ± 0.167 300 300 300 200 200 200 100 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_6 partonE_Light_r1_6 partonE_Light_r1_7 700 partonE_Light_r1_7 partonE_Light_r1_8 partonE_Light_r1_8 700 Entries 28901 Entries 29420 700 Entries 29831 Mean 38.7 Mean 40.45 Mean 42.15 RMS 9.826 600 RMS 9.882 600 RMS 9.729 600 χ2 / ndf 30.35 / 30 χ2 / ndf 18.98 / 30 χ2 / ndf 43.05 / 30 Prob 0.4476 500 Prob 0.9404 500 Prob 0.05805 500 Constant 647.2 ± 6.9 Constant 648 ± 6.9 Constant 653.8 ± 6.9 Mean 35.96 ± 0.09 400 Mean 37.78 ± 0.09 Mean 39.66 ± 0.09 400 Sigma 6.563 ± 0.171 Sigma 6.631 ± 0.175 400 Sigma 6.701 ± 0.178 300 300 300 200 200 200 100 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.1: E parton distribution for light quarks {|η| ∈ [0, 0.5)} in E jet bins with bin boundaries 0, 19, 23, 26, 28, 30, 32, 34, 36, 38 GeV 194 partonE_Light_r1_9 partonE_Light_r1_9 partonE_Light_r1_10 partonE_Light_r1_10 partonE_Light_r1_11 700 partonE_Light_r1_11 Entries 29636 700 Entries 29720 Entries 30051 600 Mean 43.79 Mean 45.63 Mean 47.42 600 RMS 9.596 600 RMS 9.772 RMS 9.704 χ2 / ndf 24.48 / 30 χ2 / ndf 38.26 / 30 χ2 / ndf 32.43 / 31 500 Prob 0.7497 Prob 0.1431 500 Prob 0.3959 500 Constant 646.1 ± 6.9 Constant 637 ± 6.8 Constant 629.6 ± 6.7 400 Mean 41.39 ± 0.09 Mean 43.33 ± 0.09 400 Mean 45.1 ± 0.1 Sigma 6.634 ± 0.175 400 Sigma 6.702 ± 0.181 Sigma 7.12 ± 0.20 300 300 300 200 200 200 100 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_12 700 partonE_Light_r1_12 partonE_Light_r1_13 partonE_Light_r1_13 partonE_Light_r1_14 partonE_Light_r1_14 Entries 29468 Entries 29089 Entries 70170 600 1400 Mean 49.24 Mean 51.13 Mean 54.2 600 RMS 9.765 RMS 9.97 RMS 9.995 χ2 / ndf 36.43 / 31 500 χ2 / ndf 27.42 / 32 1200 χ2 / ndf 55.95 / 34 500 Prob 0.2307 Prob 0.6976 Prob 0.01026 Constant 620.3 ± 6.6 Constant 604.7 ± 6.5 1000 Constant 1386 ± 9.5 Mean 47.09 ± 0.09 400 Mean 48.68 ± 0.09 Mean 52.05 ± 0.07 400 Sigma 6.643 ± 0.168 Sigma 6.757 ± 0.168 800 Sigma 7.438 ± 0.127 300 300 600 200 200 400 100 100 200 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_15 partonE_Light_r1_15 partonE_Light_r1_16 partonE_Light_r1_16 partonE_Light_r1_17 partonE_Light_r1_17 1200 Entries 39613 1000 Entries 50404 Entries 58235 800 Mean 57.87 Mean 60.97 Mean 65 RMS 10.01 RMS 10.26 1000 RMS 10.46 700 χ / ndf 2 31.29 / 34 χ / ndf 2 39.91 / 35 χ / ndf 2 42.79 / 37 800 Prob 0.601 Prob 0.261 Prob 0.2366 600 Constant 777.8 ± 7.1 Constant 969.7 ± 7.9 800 Constant 1081 ± 8.1 Mean 55.92 ± 0.09 600 Mean 58.97 ± 0.08 Mean 63.26 ± 0.08 500 Sigma 7.535 ± 0.176 Sigma 7.532 ± 0.149 Sigma 8.048 ± 0.151 600 400 400 300 400 200 200 200 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_18 partonE_Light_r1_18 partonE_Light_r1_19 partonE_Light_r1_19 partonE_Light_r1_20 partonE_Light_r1_20 Entries 52717 900 Entries 47573 800 Entries 42198 1000 Mean 69.62 Mean 74.14 Mean 78.75 RMS 10.59 800 RMS 10.93 700 RMS 11.01 χ / ndf 2 29.09 / 36 χ / ndf 2 45.73 / 38 χ / ndf 2 47.64 / 37 800 700 600 Prob 0.7863 Prob 0.1819 Prob 0.1129 Constant 965.5 ± 7.7 600 Constant 850.4 ± 7.1 Constant 740.4 ± 6.7 Mean 67.78 ± 0.09 Mean 72.45 ± 0.09 500 Mean 77.08 ± 0.10 600 Sigma 8.108 ± 0.172 500 Sigma 8.306 ± 0.176 Sigma 8.394 ± 0.203 400 400 400 300 300 200 200 200 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_21 partonE_Light_r1_21 partonE_Light_r1_22 600 partonE_Light_r1_22 partonE_Light_r1_23 500 partonE_Light_r1_23 Entries 37494 Entries 32649 Entries 27978 600 Mean 83.23 Mean 87.98 Mean 92.43 RMS 11.43 500 RMS 11.67 RMS 12.09 400 χ2 / ndf 37.02 / 40 χ2 / ndf 29.2 / 42 χ2 / ndf 44.69 / 42 500 Prob 0.6052 Prob 0.9327 Prob 0.3597 Constant 635.7 ± 5.9 400 Constant 541.6 ± 5.4 Constant 455.4 ± 4.9 400 Mean 81.6 ± 0.1 Mean 86.56 ± 0.11 300 Mean 90.97 ± 0.13 Sigma 8.804 ± 0.212 Sigma 8.865 ± 0.212 Sigma 9.058 ± 0.245 300 300 200 200 200 100 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.2: E parton distribution for light quarks {|η| ∈ [0, 0.5)} in E jet bins with bin boundaries 38, 40, 42, 44, 46, 48, 53, 56, 60, 65, 70, 75, 80, 85, 90, 95 GeV 195 partonE_Light_r1_24 partonE_Light_r1_24 partonE_Light_r1_25 partonE_Light_r1_25 partonE_Light_r1_26 300 partonE_Light_r1_26 350 400 Entries 24603 Entries 21191 Entries 17832 Mean 97.15 Mean 101.6 Mean 106.4 350 RMS 12.4 300 RMS 12.91 250 RMS 12.98 χ2 / ndf 34.2 / 43 χ2 / ndf 54.77 / 45 χ2 / ndf 44.21 / 45 300 Prob 0.8287 250 Prob 0.1509 Prob 0.5052 Constant 384.1 ± 4.5 Constant 322.8 ± 4.0 200 Constant 270 ± 3.7 250 Mean 95.78 ± 0.15 Mean 100.4 ± 0.2 Mean 105.3 ± 0.2 200 Sigma 9.665 ± 0.305 Sigma 9.747 ± 0.307 Sigma 9.569 ± 0.320 150 200 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_27 partonE_Light_r1_27 partonE_Light_r1_28 partonE_Light_r1_28 partonE_Light_r1_29 200 partonE_Light_r1_29 250 Entries 15615 220 Entries 13637 Entries 11722 Mean 110.7 Mean 115.6 180 Mean 120.2 200 RMS 13.52 RMS 13.86 RMS 14.59 χ2 / ndf 43.52 / 44 180 χ2 / ndf 59.9 / 46 160 χ2 / ndf 80.06 / 46 200 Prob 0.492 160 Prob 0.08185 140 Prob 0.001379 Constant 230.9 ± 3.4 Constant 198.4 ± 3.1 Constant 164.3 ± 2.8 Mean 109.8 ± 0.2 140 Mean 114.8 ± 0.2 120 Mean 119.5 ± 0.2 150 Sigma 9.986 ± 0.404 120 Sigma 10.03 ± 0.41 Sigma 9.943 ± 0.434 100 100 100 80 80 60 60 50 40 40 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_30 160 partonE_Light_r1_30 partonE_Light_r1_31 partonE_Light_r1_31 partonE_Light_r1_32 partonE_Light_r1_32 120 Entries 10167 Entries 8789 Entries 7654 140 Mean 124.7 120 Mean 129.1 Mean 133.6 RMS 14.8 RMS 15.32 100 RMS 15.91 120 χ / ndf 2 51.24 / 48 χ / ndf 2 37.85 / 49 χ / ndf 2 62.89 / 50 Prob 0.3479 100 Prob 0.8763 Prob 0.1043 Constant 141.1 ± 2.6 Constant 120.2 ± 2.4 80 Constant 101.2 ± 2.1 100 Mean 124.3 ± 0.3 80 Mean 128.9 ± 0.3 Mean 133.2 ± 0.3 Sigma 10.44 ± 0.50 Sigma 10.49 ± 0.51 Sigma 10.98 ± 0.62 80 60 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_33 partonE_Light_r1_33 partonE_Light_r1_34 partonE_Light_r1_34 partonE_Light_r1_35 partonE_Light_r1_35 180 Entries 12217 140 Entries 9017 90 Entries 6823 Mean 140.3 Mean 149 Mean 156.9 160 RMS 16.6 RMS 17.97 80 RMS 19.7 120 χ / ndf 2 62.54 / 54 χ / ndf 2 56.52 / 55 χ / ndf 2 71.2 / 54 140 70 Prob 0.199 Prob 0.418 Prob 0.05836 100 120 Constant 151 ± 2.5 Constant 106.5 ± 2.1 60 Constant 78.6 ± 1.8 Mean 140.6 ± 0.3 Mean 149.5 ± 0.3 Mean 158.6 ± 0.4 100 Sigma 12.29 ± 0.58 80 Sigma 12.74 ± 0.72 Sigma 12.44 ± 0.83 50 80 60 40 60 30 40 40 20 20 20 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r1_36 partonE_Light_r1_36 partonE_Light_r1_37 partonE_Light_r1_37 partonE_Light_r1_38 partonE_Light_r1_38 70 Entries 5200 Entries 10701 Entries 4724 100 30 Mean 165.9 Mean 181.6 Mean 222.4 RMS 21.72 RMS 28.53 RMS 46.84 60 χ2 / ndf 57.87 / 60 χ2 / ndf 95.29 / 82 25 χ2 / ndf 129.2 / 127 80 Prob 0.5541 Prob 0.1497 Prob 0.4294 50 Constant 56.21 ± 1.46 Constant 82.67 ± 1.53 Constant 21.81 ± 0.65 Mean 168.8 ± 0.5 Mean 187.6 ± 0.4 20 Mean 228.1 ± 0.8 40 Sigma 13.5 ± 1.0 60 Sigma 17.43 ± 0.84 Sigma 23.49 ± 1.46 15 30 40 10 20 20 10 5 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.3: E parton distribution for light quarks {|η| ∈ [0, 0.5)} in E jet bins with bin boundaries 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 150, 160, 170, 180, 220, 400 GeV 196 • Region 2 partonE_Light_r2_0 partonE_Light_r2_0 partonE_Light_r2_1 partonE_Light_r2_1 partonE_Light_r2_2 partonE_Light_r2_2 Entries 3871 Entries 17985 450 Entries 21046 Mean 27.9 400 Mean 29.94 Mean 32.69 100 RMS 13.78 RMS 12.14 400 RMS 12.37 χ / ndf 2 33.65 / 26 350 χ / ndf 2 58.28 / 31 χ / ndf 2 35.62 / 33 350 Prob 0.144 Prob 0.002141 Prob 0.3461 80 300 Constant 90.78 ± 2.64 Constant 392.8 ± 5.3 300 Constant 429.8 ± 5.4 Mean 22.09 ± 0.38 Mean 25.6 ± 0.1 Mean 28.54 ± 0.11 250 60 Sigma 6.265 ± 0.574 Sigma 6.524 ± 0.204 250 Sigma 6.923 ± 0.201 200 200 40 150 150 100 100 20 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_3 partonE_Light_r2_3 partonE_Light_r2_4 partonE_Light_r2_4 partonE_Light_r2_5 partonE_Light_r2_5 350 Entries 16444 Entries 17627 Entries 18614 Mean 34.62 350 Mean 36.25 350 Mean 37.88 300 RMS 12.1 RMS 11.87 RMS 11.74 χ2 / ndf 51.4 / 33 300 χ2 / ndf 52.34 / 34 300 χ2 / ndf 27.02 / 35 Prob 0.02157 Prob 0.02308 Prob 0.8305 250 Constant 328.4 ± 4.6 250 Constant 346.6 ± 4.8 250 Constant 352.6 ± 4.7 Mean 30.61 ± 0.13 Mean 32.65 ± 0.12 Mean 34.08 ± 0.15 200 Sigma 6.851 ± 0.218 200 Sigma 7.097 ± 0.225 200 Sigma 7.919 ± 0.285 150 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_6 partonE_Light_r2_6 partonE_Light_r2_7 partonE_Light_r2_7 partonE_Light_r2_8 partonE_Light_r2_8 400 Entries 19207 400 Entries 19915 400 Entries 20319 Mean 39.43 Mean 41.02 Mean 42.76 350 RMS 11.96 350 RMS 11.73 350 RMS 11.79 χ2 / ndf 35.02 / 34 χ2 / ndf 46.79 / 36 χ2 / ndf 46.64 / 37 300 Prob 0.4195 300 Prob 0.1076 300 Prob 0.1332 Constant 369.9 ± 4.9 Constant 370.5 ± 4.8 Constant 371.7 ± 4.8 250 Mean 35.93 ± 0.12 250 Mean 37.62 ± 0.13 250 Mean 39.41 ± 0.12 Sigma 6.979 ± 0.205 Sigma 7.809 ± 0.248 Sigma 7.698 ± 0.227 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_9 partonE_Light_r2_9 partonE_Light_r2_10 partonE_Light_r2_10 partonE_Light_r2_11 partonE_Light_r2_11 400 400 400 Entries 20427 Entries 20701 Entries 20694 Mean 44.64 Mean 46.17 Mean 47.96 350 350 350 RMS 11.89 RMS 11.8 RMS 11.69 χ2 / ndf 43.44 / 37 χ2 / ndf 43.04 / 36 χ2 / ndf 39.15 / 37 300 Prob 0.2159 300 Prob 0.1952 300 Prob 0.3738 Constant 372.2 ± 4.7 Constant 372.1 ± 4.8 Constant 369.3 ± 4.7 250 Mean 41.4 ± 0.1 250 Mean 43.14 ± 0.14 250 Mean 44.76 ± 0.13 Sigma 7.924 ± 0.245 Sigma 8.207 ± 0.286 Sigma 7.98 ± 0.25 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.4: E parton distribution for light quarks {|η| ∈ [0.5, 1.0)} in E jet bins with bin boundaries 0, 19, 23, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44 GeV 197 partonE_Light_r2_12 partonE_Light_r2_12 partonE_Light_r2_13 partonE_Light_r2_13 partonE_Light_r2_14 partonE_Light_r2_14 400 Entries 20507 400 Entries 20549 900 Entries 50858 Mean 49.71 Mean 51.54 Mean 54.64 350 350 800 RMS 11.64 RMS 11.92 RMS 12.07 χ2 / ndf 34.51 / 37 χ2 / ndf 32.97 / 37 700 χ2 / ndf 38.36 / 39 300 Prob 0.5864 300 Prob 0.6586 Prob 0.4989 Constant 366.4 ± 4.7 Constant 364.1 ± 4.7 600 Constant 860.1 ± 7.0 250 Mean 46.96 ± 0.12 250 Mean 48.49 ± 0.12 Mean 51.9 ± 0.1 Sigma 7.672 ± 0.228 Sigma 7.66 ± 0.23 500 Sigma 8.595 ± 0.179 200 200 400 150 150 300 100 100 200 50 50 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_15 partonE_Light_r2_15 partonE_Light_r2_16 700 partonE_Light_r2_16 partonE_Light_r2_17 partonE_Light_r2_17 Entries 29279 Entries 38128 800 Entries 44884 500 Mean 58.08 Mean 61.41 Mean 65.36 600 700 RMS 11.93 RMS 12.4 RMS 12.4 400 χ2 / ndf 43.33 / 39 χ2 / ndf 45.08 / 41 χ2 / ndf 52.21 / 41 Prob 0.2917 500 Prob 0.3051 600 Prob 0.1128 Constant 498.1 ± 5.3 Constant 632.2 ± 5.9 Constant 733.4 ± 6.4 Mean 55.31 ± 0.12 400 Mean 58.69 ± 0.10 500 Mean 62.64 ± 0.10 300 Sigma 8.489 ± 0.227 Sigma 8.491 ± 0.184 Sigma 8.643 ± 0.181 400 300 200 300 200 200 100 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_18 700 partonE_Light_r2_18 partonE_Light_r2_19 partonE_Light_r2_19 partonE_Light_r2_20 partonE_Light_r2_20 Entries 41681 Entries 38864 Entries 35684 Mean 69.96 600 Mean 74.63 Mean 79.15 600 500 RMS 12.56 RMS 12.79 RMS 13.13 χ / ndf 2 46.45 / 42 χ / ndf 2 42.72 / 44 χ / ndf 2 42.8 / 44 500 Prob 0.2942 500 Prob 0.5266 Prob 0.5229 400 Constant 655.7 ± 5.9 Constant 604.6 ± 5.6 Constant 539.6 ± 5.2 400 Mean 67.17 ± 0.11 400 Mean 72.12 ± 0.10 Mean 76.58 ± 0.12 Sigma 9.229 ± 0.215 Sigma 8.993 ± 0.187 300 Sigma 9.393 ± 0.223 300 300 200 200 200 100 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_21 partonE_Light_r2_21 partonE_Light_r2_22 450 partonE_Light_r2_22 partonE_Light_r2_23 partonE_Light_r2_23 Entries 31915 Entries 28770 400 Entries 25815 500 Mean 83.73 400 Mean 88.36 Mean 92.73 RMS 13.44 RMS 13.91 350 RMS 13.9 χ / ndf 2 59.02 / 46 350 χ / ndf 2 36.09 / 47 χ / ndf 2 53.89 / 47 400 Prob 0.09424 Prob 0.8761 300 Prob 0.2277 Constant 466.9 ± 4.8 300 Constant 408.9 ± 4.4 Constant 369.5 ± 4.2 Mean 81.12 ± 0.14 Mean 85.99 ± 0.15 250 Mean 90.18 ± 0.15 300 Sigma 10.11 ± 0.27 250 Sigma 10.49 ± 0.31 Sigma 9.999 ± 0.281 200 200 200 150 150 100 100 100 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_24 partonE_Light_r2_24 partonE_Light_r2_25 partonE_Light_r2_25 partonE_Light_r2_26 partonE_Light_r2_26 350 Entries 23413 300 Entries 20683 Entries 18232 Mean 97.29 Mean 102.1 250 Mean 106.5 300 RMS 13.95 RMS 14.59 RMS 14.77 χ2 / ndf 46.82 / 51 250 χ2 / ndf 40.95 / 50 χ2 / ndf 59.12 / 53 Prob 0.6402 Prob 0.8155 200 Prob 0.262 250 Constant 317.6 ± 3.7 200 Constant 279.6 ± 3.6 Constant 238.7 ± 3.2 Mean 95.15 ± 0.18 Mean 99.75 ± 0.18 Mean 104.3 ± 0.2 200 Sigma 11.26 ± 0.35 Sigma 10.66 ± 0.34 150 Sigma 11.37 ± 0.39 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.5: E parton distribution for light quarks {|η| ∈ [0.5, 1.0)} in E jet bins with bin boundaries 44, 46, 48, 53, 56, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110 GeV 198 partonE_Light_r2_27 partonE_Light_r2_27 partonE_Light_r2_28 200 partonE_Light_r2_28 partonE_Light_r2_29 partonE_Light_r2_29 180 220 Entries 15999 Entries 14086 Entries 12335 Mean 111.2 180 Mean 115.7 Mean 120.1 200 160 RMS 15.05 160 RMS 15.6 RMS 15.65 180 χ2 / ndf 46.18 / 52 χ2 / ndf 42.55 / 57 140 χ2 / ndf 59.33 / 54 Prob 0.7011 140 Prob 0.9229 Prob 0.2874 160 Constant 207.7 ± 3.0 Constant 173.2 ± 2.7 120 Constant 155.3 ± 2.6 140 Mean 108.9 ± 0.2 120 Mean 113.9 ± 0.3 Mean 118.5 ± 0.2 120 Sigma 11.06 ± 0.40 Sigma 12.76 ± 0.54 100 Sigma 11.24 ± 0.45 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_30 partonE_Light_r2_30 partonE_Light_r2_31 partonE_Light_r2_31 partonE_Light_r2_32 partonE_Light_r2_32 Entries 10808 140 Entries 9643 120 Entries 8412 140 Mean 124.7 Mean 129.4 Mean 134 RMS 16.45 120 RMS 16.59 100 RMS 16.71 120 χ2 / ndf 66.82 / 55 χ2 / ndf 71.61 / 57 χ2 / ndf 57.13 / 56 Prob 0.1319 Prob 0.09219 Prob 0.4327 100 100 Constant 131.3 ± 2.3 Constant 113.4 ± 2.1 80 Constant 99.03 ± 2.00 Mean 122.6 ± 0.3 Mean 127.2 ± 0.3 Mean 131.9 ± 0.3 Sigma 11.85 ± 0.53 80 Sigma 11.92 ± 0.55 Sigma 12.35 ± 0.67 80 60 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_33 partonE_Light_r2_33 partonE_Light_r2_34 partonE_Light_r2_34 partonE_Light_r2_35 120 partonE_Light_r2_35 Entries 13991 Entries 10824 Entries 8320 160 Mean 140.4 140 Mean 149.3 Mean 158.2 RMS 17.9 RMS 18.51 100 RMS 19.39 140 χ / ndf 2 68.08 / 58 120 χ / ndf 2 66.9 / 59 χ / ndf 2 71.14 / 65 Prob 0.1717 Prob 0.2242 Prob 0.2808 120 Constant 156.9 ± 2.5 Constant 120.2 ± 2.2 80 Constant 84.39 ± 1.72 100 Mean 138.6 ± 0.3 Mean 148 ± 0.3 Mean 157 ± 0.4 100 Sigma 12.42 ± 0.50 Sigma 12.33 ± 0.55 Sigma 13.93 ± 0.73 80 60 80 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r2_36 partonE_Light_r2_36 partonE_Light_r2_37 partonE_Light_r2_37 partonE_Light_r2_38 partonE_Light_r2_38 Entries 6496 140 Entries 14290 50 Entries 8235 70 Mean 167.2 Mean 185.1 Mean 230.5 RMS 20.54 RMS 25.64 RMS 45.75 120 60 χ / ndf 2 76.05 / 67 χ / ndf 2 102.9 / 85 40 χ / ndf 2 120.1 / 123 Prob 0.2101 Prob 0.09068 Prob 0.5579 50 Constant 63.18 ± 1.48 100 Constant 108.5 ± 1.7 Constant 37.53 ± 0.84 Mean 166.5 ± 0.5 Mean 184.8 ± 0.4 Mean 225.5 ± 0.9 30 40 Sigma 15.27 ± 1.04 80 Sigma 19.46 ± 0.91 Sigma 27.82 ± 1.86 30 60 20 20 40 10 10 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.6: E parton distribution for light quarks {|η| ∈ [0.5, 1.0)} in E jet bins with bin boundaries 110, 115, 120, 125, 130, 135, 140, 150, 160, 170, 180, 220, 400 GeV 199 • Region 3 partonE_Light_r3_0 partonE_Light_r3_0 partonE_Light_r3_1 partonE_Light_r3_1 partonE_Light_r3_2 partonE_Light_r3_2 45 Entries 1974 1000 Entries 80378 350 Entries 27622 Mean 33.79 Mean 45.05 Mean 53.81 40 RMS 15.06 RMS 16.16 300 RMS 16.13 χ / ndf 2 50.56 / 40 800 χ / ndf 2 61.14 / 60 χ / ndf 2 51.18 / 61 35 Prob 0.1223 Prob 0.4349 Prob 0.8106 250 30 Constant 31.96 ± 1.31 Constant 972.1 ± 6.1 Constant 324.6 ± 3.5 Mean 28.28 ± 0.42 600 Mean 40.32 ± 0.10 Mean 49.72 ± 0.19 Sigma 7.895 ± 0.671 Sigma 12.64 ± 0.20 200 Sigma 13.31 ± 0.37 25 20 150 400 15 100 10 200 50 5 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r3_3 partonE_Light_r3_3 partonE_Light_r3_4 350 partonE_Light_r3_4 partonE_Light_r3_5 partonE_Light_r3_5 350 Entries 27387 Entries 27262 Entries 26232 Mean 58.23 Mean 62.38 300 Mean 67.03 300 RMS 16.61 300 RMS 16.75 RMS 17.08 χ2 / ndf 51.16 / 61 χ2 / ndf 48.36 / 60 χ2 / ndf 55.54 / 61 250 250 Prob 0.8113 250 Prob 0.8595 Prob 0.6733 Constant 316.8 ± 3.4 Constant 316.4 ± 3.5 Constant 293 ± 3.3 Mean 54.18 ± 0.18 200 Mean 58.55 ± 0.18 200 Mean 62.93 ± 0.21 200 Sigma 12.94 ± 0.35 Sigma 12.57 ± 0.34 Sigma 13.57 ± 0.42 150 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r3_6 partonE_Light_r3_6 partonE_Light_r3_7 300 partonE_Light_r3_7 partonE_Light_r3_8 partonE_Light_r3_8 Entries 25041 Entries 24088 Entries 22529 300 Mean 71.27 Mean 75.97 250 Mean 80.31 RMS 17.2 250 RMS 17.6 RMS 17.92 250 χ2 / ndf 56.06 / 63 χ2 / ndf 79.9 / 62 χ2 / ndf 55.79 / 64 Prob 0.72 Prob 0.06265 200 Prob 0.758 Constant 274.9 ± 3.2 200 Constant 265.1 ± 3.1 Constant 242.1 ± 3.0 200 Mean 67.21 ± 0.20 Mean 72.22 ± 0.20 Mean 76.04 ± 0.21 Sigma 13.48 ± 0.40 Sigma 12.88 ± 0.37 150 Sigma 13.18 ± 0.39 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r3_9 partonE_Light_r3_9 partonE_Light_r3_10 250 partonE_Light_r3_10 partonE_Light_r3_11 220 partonE_Light_r3_11 250 Entries 21118 Entries 19870 Entries 18455 Mean 85.1 Mean 89.22 200 Mean 94.2 RMS 18.05 RMS 18.13 180 RMS 18.53 χ2 / ndf 200 χ2 / ndf χ2 / ndf 200 75.41 / 66 94.44 / 68 58.12 / 68 Prob 0.2004 Prob 0.01869 160 Prob 0.7978 Constant 221.8 ± 2.8 Constant 202.9 ± 2.6 140 Constant 186 ± 2.5 150 Mean 81.03 ± 0.22 150 Mean 85.45 ± 0.24 Mean 90.47 ± 0.26 Sigma 13.56 ± 0.41 Sigma 14.4 ± 0.5 120 Sigma 14.46 ± 0.50 100 100 100 80 60 50 50 40 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.7: E parton distribution for light quarks {|η| ∈ [1.0, 1.5)} in E jet bins with bin boundaries 0, 25, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95 GeV 200 partonE_Light_r3_12 partonE_Light_r3_12 partonE_Light_r3_13 partonE_Light_r3_13 partonE_Light_r3_14 partonE_Light_r3_14 200 Entries 16925 180 Entries 15691 160 Entries 14197 180 Mean 98.73 Mean 103 Mean 107.5 RMS 18.84 160 RMS 19.02 140 RMS 18.88 160 χ2 / ndf 71.63 / 70 χ2 / ndf 64.88 / 67 χ2 / ndf 92.25 / 71 140 140 Prob 0.4236 Prob 0.5506 120 Prob 0.04592 Constant 170.6 ± 2.4 120 Constant 156.1 ± 2.3 Constant 140.3 ± 2.2 120 Mean 95.45 ± 0.25 Mean 98.97 ± 0.29 100 Mean 104.4 ± 0.3 Sigma 14.26 ± 0.47 100 Sigma 14.61 ± 0.58 Sigma 14.43 ± 0.53 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r3_15 partonE_Light_r3_15 partonE_Light_r3_16 partonE_Light_r3_16 partonE_Light_r3_17 220 partonE_Light_r3_17 Entries 13153 Entries 11972 Entries 21022 140 Mean 112.2 Mean 116.5 200 Mean 123.1 120 RMS 19.59 RMS 19.67 180 RMS 20.47 120 χ2 / ndf 81.95 / 71 χ2 / ndf 50.17 / 75 χ2 / ndf 66.07 / 78 Prob 0.1761 100 Prob 0.9878 160 Prob 0.8301 100 Constant 126.2 ± 2.0 Constant 110.3 ± 1.8 140 Constant 189 ± 2.4 Mean 108.7 ± 0.3 80 Mean 113.1 ± 0.4 Mean 120.2 ± 0.3 80 Sigma 14.67 ± 0.58 Sigma 16.93 ± 0.81 120 Sigma 16.02 ± 0.48 60 100 60 80 40 60 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r3_18 partonE_Light_r3_18 partonE_Light_r3_19 partonE_Light_r3_19 partonE_Light_r3_20 partonE_Light_r3_20 180 Entries 17160 140 Entries 14199 Entries 11579 160 Mean 132.9 Mean 141.6 100 Mean 150.2 RMS 21.55 RMS 21.94 RMS 22.31 χ / ndf 2 71.44 / 81 120 χ / ndf 2 111.5 / 81 χ / ndf 2 99.27 / 89 140 Prob 0.7672 Prob 0.01403 80 Prob 0.2142 120 Constant 148.2 ± 2.1 100 Constant 119.6 ± 1.9 Constant 92.09 ± 1.56 Mean 129.6 ± 0.3 Mean 138.5 ± 0.3 Mean 148.1 ± 0.4 100 Sigma 16.88 ± 0.58 80 Sigma 16.25 ± 0.59 60 Sigma 18.76 ± 0.82 80 60 40 60 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r3_21 90 partonE_Light_r3_21 partonE_Light_r3_22 partonE_Light_r3_22 partonE_Light_r3_23 partonE_Light_r3_23 80 Entries 9286 70 Entries 7600 Entries 8610 80 Mean 158.7 Mean 167.7 Mean 178.6 70 RMS 23.03 RMS 24.06 RMS 25.49 60 70 χ / ndf 2 67.48 / 90 χ / ndf 2 75.95 / 90 χ / ndf 2 109.9 / 105 60 Prob 0.9635 Prob 0.8547 Prob 0.3531 60 Constant 73.49 ± 1.38 50 Constant 58.29 ± 1.24 Constant 59.63 ± 1.15 Mean 155.7 ± 0.4 Mean 166.1 ± 0.5 50 Mean 176 ± 0.5 50 Sigma 17.74 ± 0.76 40 Sigma 18.53 ± 0.97 Sigma 21.77 ± 1.05 40 40 30 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r3_24 partonE_Light_r3_24 partonE_Light_r3_25 partonE_Light_r3_25 partonE_Light_r3_26 22 partonE_Light_r3_26 90 Entries 11035 Entries 8471 Entries 3656 Mean 197.2 60 Mean 229.7 20 Mean 287.1 80 RMS 27.81 RMS 33.71 18 RMS 52.69 χ2 / ndf 126.9 / 119 50 χ2 / ndf 146.7 / 139 χ2 / ndf 182 / 168 70 Prob 0.2923 Prob 0.3113 16 Prob 0.2178 Constant 68.92 ± 1.17 Constant 44.34 ± 0.86 14 Constant 12.23 ± 0.42 60 40 Mean 195.5 ± 0.5 Mean 229 ± 0.7 Mean 278.4 ± 1.5 50 Sigma 23.73 ± 0.95 Sigma 27.41 ± 1.23 12 Sigma 34.26 ± 2.90 30 10 40 8 30 20 6 20 4 10 10 2 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.8: E parton distribution for light quarks {|η| ∈ [1.0, 1.5)} in E jet bins with bin boundaries 95, 100, 105, 110, 115, 120, 130, 140, 150, 160, 170, 180, 195, 225, 280, 500 GeV 201 • Region 4 partonE_Light_r4_1 partonE_Light_r4_1 partonE_Light_r4_2 partonE_Light_r4_2 partonE_Light_r4_3 partonE_Light_r4_3 Entries 3723 60 Entries 4427 Entries 5672 50 Mean 53.38 Mean 60 60 Mean 64.32 RMS 18.6 RMS 20.15 RMS 20.55 50 χ / ndf 2 80.03 / 67 χ / ndf 2 86.75 / 76 χ / ndf 2 77.85 / 76 Prob 0.1321 Prob 0.1874 50 Prob 0.4196 40 Constant 39.22 ± 1.16 40 Constant 42.53 ± 1.14 Constant 50.83 ± 1.25 Mean 47 ± 0.5 Mean 53.63 ± 0.52 40 Mean 58.6 ± 0.6 30 Sigma 13.25 ± 0.87 Sigma 15.32 ± 0.96 Sigma 17.94 ± 1.38 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r4_4 partonE_Light_r4_4 partonE_Light_r4_5 partonE_Light_r4_5 partonE_Light_r4_6 partonE_Light_r4_6 70 Entries 6680 Entries 7300 80 Entries 7970 Mean 69.05 70 Mean 73.03 Mean 77.26 60 RMS 21.86 RMS 22.76 70 RMS 23.6 χ2 / ndf 64.93 / 81 60 χ2 / ndf 77.64 / 87 χ2 / ndf 98.76 / 92 Prob 0.9039 Prob 0.7537 60 Prob 0.2961 50 Constant 57.15 ± 1.28 50 Constant 60.25 ± 1.28 Constant 60.76 ± 1.32 Mean 62.87 ± 0.48 Mean 66.42 ± 0.47 50 Mean 70.48 ± 0.58 40 Sigma 16.56 ± 0.89 40 Sigma 17.44 ± 0.86 Sigma 20.39 ± 1.27 40 30 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r4_7 partonE_Light_r4_7 partonE_Light_r4_8 partonE_Light_r4_8 partonE_Light_r4_9 partonE_Light_r4_9 80 Entries 8296 80 Entries 8351 Entries 8386 80 Mean 82.43 Mean 86.51 Mean 91 70 RMS 24.54 70 RMS 25.34 RMS 25.35 70 χ2 / ndf 90.95 / 91 χ2 / ndf 109.6 / 97 χ2 / ndf 104.4 / 95 60 Prob 0.4819 60 Prob 0.1797 Prob 0.2402 60 Constant 63.27 ± 1.27 Constant 60.51 ± 1.20 Constant 61.01 ± 1.23 50 Mean 75.55 ± 0.55 50 Mean 80.13 ± 0.50 50 Mean 85.05 ± 0.51 Sigma 20.09 ± 1.12 Sigma 19.54 ± 0.91 Sigma 19.69 ± 0.99 40 40 40 30 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r4_10 partonE_Light_r4_10 partonE_Light_r4_11 partonE_Light_r4_11 partonE_Light_r4_12 partonE_Light_r4_12 70 Entries 8248 Entries 8279 70 Entries 7931 70 Mean 95.17 Mean 99.94 Mean 104.2 60 RMS 25.97 60 RMS 26.52 60 RMS 27.02 χ2 / ndf 83.72 / 95 χ2 / ndf 93.52 / 97 χ2 / ndf 101.4 / 97 50 Prob 0.7893 Prob 0.5813 Prob 0.3604 50 50 Constant 58.49 ± 1.20 Constant 57.98 ± 1.19 Constant 55.02 ± 1.15 Mean 90.59 ± 0.56 Mean 93.68 ± 0.53 Mean 97.98 ± 0.51 40 Sigma 20.46 ± 1.12 40 Sigma 20.13 ± 1.03 40 Sigma 19.27 ± 0.93 30 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.9: E parton distribution for light quarks {|η| ∈ [1.5, 2.5)} in E jet bins with bin boundaries 25, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 GeV 202 partonE_Light_r4_13 partonE_Light_r4_13 partonE_Light_r4_14 partonE_Light_r4_14 partonE_Light_r4_15 partonE_Light_r4_15 70 Entries 7712 Entries 7243 Entries 6712 Mean 108.3 60 Mean 112.4 50 Mean 117.8 60 RMS 27.77 RMS 27.22 RMS 28.67 χ2 / ndf 109.5 / 98 χ2 / ndf 122.5 / 108 χ2 / ndf 106.9 / 107 50 Prob 0.2015 50 Prob 0.1615 Prob 0.4843 40 Constant 53.07 ± 1.13 Constant 47.76 ± 1.02 Constant 42.81 ± 0.96 Mean 103 ± 0.5 40 Mean 108.1 ± 0.6 Mean 113 ± 0.6 40 Sigma 19.64 ± 0.99 Sigma 21.31 ± 1.05 30 Sigma 22.12 ± 1.23 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r4_16 70 partonE_Light_r4_16 partonE_Light_r4_17 partonE_Light_r4_17 partonE_Light_r4_18 partonE_Light_r4_18 Entries 6646 90 Entries 11949 Entries 10141 Mean 122.6 Mean 129 80 Mean 138.9 60 RMS 28.33 80 RMS 29.16 RMS 29.72 70 χ2 / ndf 109.3 / 106 χ2 / ndf 88.12 / 103 χ2 / ndf 121.1 / 109 50 Prob 0.393 70 Prob 0.8518 Prob 0.2008 60 Constant 43.01 ± 0.99 60 Constant 77.83 ± 1.33 Constant 62.9 ± 1.1 40 Mean 116.8 ± 0.6 Mean 124.5 ± 0.5 50 Mean 134.3 ± 0.6 Sigma 20.97 ± 1.11 50 Sigma 21.23 ± 0.90 Sigma 23.3 ± 1.1 40 30 40 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r4_19 partonE_Light_r4_19 partonE_Light_r4_20 partonE_Light_r4_20 partonE_Light_r4_21 partonE_Light_r4_21 Entries 8933 60 Entries 7655 Entries 6581 70 Mean 147.3 Mean 157.1 Mean 166.6 50 RMS 30.46 RMS 31.21 RMS 32.13 50 60 χ / ndf 2 105.1 / 108 χ / ndf 2 114.9 / 107 χ / ndf 2 142.9 / 119 Prob 0.5618 Prob 0.2843 Prob 0.06688 40 50 Constant 54.61 ± 1.09 40 Constant 45.86 ± 1.00 Constant 37.2 ± 0.8 Mean 143.1 ± 0.6 Mean 152.3 ± 0.6 Mean 163.6 ± 0.8 40 Sigma 22.55 ± 1.14 Sigma 21.97 ± 1.19 30 Sigma 25.16 ± 1.48 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r4_22 partonE_Light_r4_22 partonE_Light_r4_23 partonE_Light_r4_23 partonE_Light_r4_24 partonE_Light_r4_24 Entries 5556 50 Entries 6852 Entries 9520 40 60 Mean 176.8 Mean 187.7 Mean 207.6 RMS 31.79 RMS 33.18 RMS 33.83 35 χ / ndf 2 97.93 / 115 40 χ / ndf 2 114.1 / 113 50 χ / ndf 2 137.5 / 134 Prob 0.8732 Prob 0.4527 Prob 0.3993 30 Constant 30.93 ± 0.79 Constant 38.46 ± 0.89 Constant 50.03 ± 0.93 40 25 Mean 172.3 ± 1.0 30 Mean 184.1 ± 0.8 Mean 204.7 ± 0.7 Sigma 27.43 ± 2.27 Sigma 24.39 ± 1.53 Sigma 27.47 ± 1.26 20 30 20 15 20 10 10 10 5 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_Light_r4_25 partonE_Light_r4_25 partonE_Light_r4_26 partonE_Light_r4_26 60 25 Entries 8393 Entries 4681 Mean 243.1 Mean 315.5 50 RMS 37.7 RMS 55.82 χ2 / ndf 210.8 / 164 20 χ2 / ndf 172.7 / 187 Prob 0.008058 Prob 0.7649 40 Constant 37.1 ± 0.7 Constant 14.7 ± 0.4 Mean 239.6 ± 0.9 15 Mean 298.7 ± 1.6 Sigma 35.46 ± 1.93 Sigma 40.44 ± 3.32 30 10 20 5 10 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure A.10: E parton distribution for light quarks {|η| ∈ [1.5, 2.5)} in E jet bins with bin boundaries 100, 105, 110, 115, 120, 130, 140, 150, 160, 170, 180, 195, 225, 280, 500 GeV APPENDIX B Parton Energy distributions for b-quarks The Parton Level Corrections and corresponding resolution functions for b−quarks are obtained by using matched jet-parton pairs for b−quarks in t t¯ → l + jets events. The figures below show the parton energy distribution in various jet energy bins for 4 detector regions. The distributions are close to Gaussian and are fitted with the Gaussian function near the peak. The mean value gives the PLC, while the RMS of the Gaussian provides with the parton energy resolution for that particular jet energy bin. • Region 1 203 204 partonE_B_all_r1_0 partonE_B_all_r1_0 partonE_B_all_r1_1 partonE_B_all_r1_1 partonE_B_all_r1_2 partonE_B_all_r1_2 140 Entries 8503 200 Entries 11100 180 Entries 9289 Mean 40.72 Mean 41.19 Mean 41.69 120 RMS 17.53 180 RMS 15.84 160 RMS 14.82 χ2 / ndf 43.65 / 34 χ2 / ndf 40.88 / 32 χ2 / ndf 42.49 / 36 160 140 100 Prob 0.1242 Prob 0.135 Prob 0.2116 Constant 116.1± 2.5 140 Constant 177.7 ± 3.4 120 Constant 151.9 ± 3.1 Mean 28.89 ± 1.77 Mean 32.07 ± 0.27 Mean 33.53 ± 0.24 80 120 Sigma 14.51 ± 2.88 Sigma 8.135 ± 0.515 100 Sigma 8.288 ± 0.467 100 60 80 80 60 40 60 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_3 partonE_B_all_r1_3 partonE_B_all_r1_4 250 partonE_B_all_r1_4 partonE_B_all_r1_5 partonE_B_all_r1_5 Entries 11114 Entries 12749 Entries 14667 200 Mean 42.91 Mean 44.06 250 Mean 45.11 180 RMS 14.84 RMS 14.22 RMS 14.05 χ2 / ndf 200 χ2 / ndf χ2 / ndf 34.3 / 33 45.63 / 33 44.12 / 34 160 Prob 0.405 Prob 0.07055 Prob 0.1146 200 140 Constant 187.7 ± 3.5 Constant 223.3 ± 3.9 Constant 253.7 ± 4.0 Mean 35.23 ± 0.19 150 Mean 36.43 ± 0.16 Mean 38 ± 0.1 120 Sigma 7.549 ± 0.383 Sigma 7.042 ± 0.287 150 Sigma 7.258 ± 0.274 100 100 80 100 60 40 50 50 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_6 partonE_B_all_r1_6 partonE_B_all_r1_7 partonE_B_all_r1_7 partonE_B_all_r1_8 partonE_B_all_r1_8 Entries 16183 350 Entries 18262 Entries 19475 300 350 Mean 46.56 Mean 47.87 Mean 49.47 RMS 13.93 300 RMS 13.92 RMS 14.12 250 χ / ndf 2 36.06 / 33 χ / ndf 2 55.08 / 36 300 χ / ndf 2 51.22 / 35 Prob 0.3274 Prob 0.0218 Prob 0.0377 250 Constant 285.6 ± 4.4 Constant 315 ± 4.4 250 Constant 334.9 ± 4.6 200 Mean 39.91 ± 0.14 Mean 41.4 ± 0.1 Mean 42.9 ± 0.1 Sigma 7.234 ± 0.274 200 Sigma 7.499 ± 0.243 200 Sigma 7.606 ± 0.256 150 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure B.1: E parton distribution for b-quarks {|η| ∈ [0, 0.5)} in E jet bins with bin boundaries 0, 19, 22, 24, 26, 28, 30, 32, 34, 36 GeV 205 partonE_B_all_r1_9 partonE_B_all_r1_9 partonE_B_all_r1_10 partonE_B_all_r1_10 partonE_B_all_r1_11 partonE_B_all_r1_11 450 400 Entries 21252 Entries 22411 Entries 23865 400 Mean 50.68 Mean 52.25 400 Mean 53.71 350 RMS 14.13 RMS 14.03 RMS 14.03 350 χ2 / ndf 50.73 / 36 χ2 / ndf 44.8 / 35 350 χ2 / ndf 33.43 / 36 300 Prob 0.05264 300 Prob 0.124 Prob 0.5912 Constant 371.3 ± 4.8 Constant 389.4 ± 5.0 300 Constant 410.2 ± 5.0 250 Mean 44.57 ± 0.12 250 Mean 45.87 ± 0.12 Mean 47.73 ± 0.12 Sigma 7.407 ± 0.215 Sigma 7.426 ± 0.227 250 Sigma 7.555 ± 0.215 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_12 500 partonE_B_all_r1_12 partonE_B_all_r1_13 partonE_B_all_r1_13 partonE_B_all_r1_14 partonE_B_all_r1_14 500 Entries 24803 Entries 26131 Entries 26466 Mean 55.29 Mean 56.95 450 Mean 58.65 RMS 13.87 RMS 13.93 400 RMS 14.02 400 χ2 / ndf 400 χ2 / ndf χ2 / ndf 43.12 / 36 23.55 / 34 31.04 / 34 Prob 0.193 Prob 0.9104 350 Prob 0.6136 Constant 427.8 ± 5.2 Constant 455.9 ± 5.4 Constant 461.6 ± 5.4 300 Mean 49.37 ± 0.12 300 Mean 50.99 ± 0.11 300 Mean 52.94 ± 0.11 Sigma 7.776 ± 0.230 Sigma 7.309 ± 0.209 Sigma 7.236 ± 0.201 250 200 200 200 150 100 100 100 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_15 partonE_B_all_r1_15 partonE_B_all_r1_16 partonE_B_all_r1_16 partonE_B_all_r1_17 partonE_B_all_r1_17 1000 1200 Entries 68789 Entries 41861 Entries 55636 Mean 61.39 700 Mean 64.75 Mean 67.59 RMS 14.13 RMS 14.22 RMS 14.2 1000 χ / ndf 2 51.86 / 38 600 χ / ndf 2 39.7 / 37 800 χ / ndf 2 40.21 / 38 Prob 0.06625 Prob 0.3508 Prob 0.3725 Constant 1146 ± 8.2 500 Constant 699.1 ± 6.5 Constant 924.3 ± 7.4 800 Mean 56.04 ± 0.08 Mean 59.41 ± 0.10 600 Mean 62.43 ± 0.08 Sigma 8.189 ± 0.145 400 Sigma 8.055 ± 0.190 Sigma 8.218 ± 0.164 600 300 400 400 200 200 200 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_18 partonE_B_all_r1_18 partonE_B_all_r1_19 partonE_B_all_r1_19 partonE_B_all_r1_20 partonE_B_all_r1_20 1200 Entries 68234 Entries 65483 1000 Entries 61045 Mean 71.45 Mean 75.72 Mean 80.15 1000 1000 RMS 14.41 RMS 14.49 RMS 14.59 χ / ndf 2 53.34 / 39 χ / ndf 2 45.74 / 40 800 χ / ndf 2 43.92 / 41 Prob 0.06278 800 Prob 0.246 Prob 0.3487 800 Constant 1123 ± 8.0 Constant 1057 ± 7.7 Constant 961 ± 7.3 Mean 66.43 ± 0.07 Mean 71.02 ± 0.08 600 Mean 75.67 ± 0.09 Sigma 8.196 ± 0.139 600 Sigma 8.459 ± 0.148 Sigma 8.779 ± 0.163 600 400 400 400 200 200 200 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_21 partonE_B_all_r1_21 partonE_B_all_r1_22 partonE_B_all_r1_22 partonE_B_all_r1_23 partonE_B_all_r1_23 Entries 56762 Entries 51933 Entries 46513 900 800 700 Mean 84.45 Mean 88.84 Mean 93.23 800 RMS 14.78 RMS 14.99 RMS 15.23 700 χ2 / ndf 50.15 / 41 χ2 / ndf 45.47 / 41 600 χ2 / ndf 37.15 / 43 700 Prob 0.1548 600 Prob 0.2911 Prob 0.7223 Constant 885.4 ± 7.0 Constant 803.9 ± 6.6 500 Constant 701.6 ± 6.1 600 Mean 80.04 ± 0.09 500 Mean 84.57 ± 0.10 Mean 89.17 ± 0.10 Sigma 8.843 ± 0.175 Sigma 8.95 ± 0.19 400 Sigma 9.219 ± 0.198 500 400 400 300 300 300 200 200 200 100 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure B.2: E parton distribution for b-quarks {|η| ∈ [0, 0.5)} in E jet bins with bin boundaries 36, 38, 40, 42, 44, 46, 48, 53, 56, 60, 65, 70, 75, 80, 85, 90 GeV 206 partonE_B_all_r1_24 partonE_B_all_r1_24 partonE_B_all_r1_25 partonE_B_all_r1_25 partonE_B_all_r1_26 partonE_B_all_r1_26 Entries 42014 Entries 37064 500 Entries 32819 600 Mean 97.58 Mean 102.1 Mean 106.7 RMS 15.31 500 RMS 15.69 RMS 15.79 χ2 / ndf 52.21 / 43 χ2 / ndf 37.75 / 44 400 χ2 / ndf 46.44 / 46 500 Prob 0.1584 Prob 0.7353 Prob 0.454 400 Constant 631.2 ± 5.8 Constant 544.2 ± 5.3 Constant 476 ± 4.8 400 Mean 93.68 ± 0.11 Mean 98.24 ± 0.07 300 Mean 103.1 ± 0.1 Sigma 9.142 ± 0.204 300 Sigma 9.425 ± 0.226 Sigma 9.792 ± 0.244 300 200 200 200 100 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_27 partonE_B_all_r1_27 partonE_B_all_r1_28 400 partonE_B_all_r1_28 partonE_B_all_r1_29 partonE_B_all_r1_29 Entries 28623 Entries 25416 Entries 21934 400 Mean 111.2 Mean 115.5 300 Mean 120.1 350 RMS 16.13 RMS 16.29 RMS 16.57 350 χ2 / ndf 26.68 / 45 300 χ2 / ndf 37.4 / 47 χ2 / ndf 56.87 / 46 Prob 0.9864 Prob 0.8405 250 Prob 0.1308 300 Constant 409.6 ± 4.5 Constant 357.2 ± 4.2 Constant 307 ± 3.9 250 Mean 107.6 ± 0.1 Mean 111.8 ± 0.2 200 Mean 116.7 ± 0.2 250 Sigma 9.5 ± 0.3 Sigma 10.13 ± 0.30 Sigma 9.776 ± 0.305 200 200 150 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_30 partonE_B_all_r1_30 partonE_B_all_r1_31 partonE_B_all_r1_31 partonE_B_all_r1_32 220 partonE_B_all_r1_32 300 Entries 19175 Entries 16756 Entries 14483 Mean 124.5 Mean 129.1 200 Mean 133.7 250 250 RMS 16.55 RMS 16.63 180 RMS 16.89 χ / ndf 2 62.05 / 47 χ / ndf 2 90.26 / 48 χ / ndf 2 47.25 / 47 Prob 0.06953 Prob 0.0002152 160 Prob 0.4622 200 200 Constant 265.4 ± 3.6 Constant 229 ± 3.3 140 Constant 198.3 ± 3.1 Mean 121.1 ± 0.2 Mean 126.2 ± 0.2 Mean 130.3 ± 0.2 Sigma 9.92 ± 0.32 Sigma 10.19 ± 0.37 120 Sigma 10.04 ± 0.38 150 150 100 100 80 100 60 50 50 40 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_33 partonE_B_all_r1_33 partonE_B_all_r1_34 partonE_B_all_r1_34 partonE_B_all_r1_35 partonE_B_all_r1_35 200 200 Entries 12524 Entries 19700 Entries 14369 180 Mean 138.1 250 Mean 145.1 180 Mean 154.2 RMS 16.98 RMS 17.89 RMS 18.76 160 χ / ndf 2 29.48 / 47 χ / ndf 2 63.04 / 51 160 χ / ndf 2 50.3 / 57 Prob 0.9787 200 Prob 0.1202 140 Prob 0.7227 140 Constant 170 ± 2.9 Constant 251 ± 3.3 Constant 174.6 ± 2.6 120 Mean 135.1 ± 0.2 Mean 142.2 ± 0.2 120 Mean 151.7 ± 0.2 Sigma 10 ± 0.4 150 Sigma 10.83 ± 0.36 Sigma 11.91 ± 0.44 100 100 80 80 100 60 60 40 50 40 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_36 160 partonE_B_all_r1_36 partonE_B_all_r1_37 partonE_B_all_r1_37 partonE_B_all_r1_38 partonE_B_all_r1_38 Entries 10739 120 Entries 7933 160 Entries 14845 140 Mean 163.2 Mean 172.1 Mean 190.8 RMS 19 RMS 19.42 140 RMS 23.81 100 120 χ2 / ndf 53.22 / 58 χ2 / ndf 52.1 / 53 χ2 / ndf 72.47 / 81 Prob 0.6532 Prob 0.509 120 Prob 0.7396 100 Constant 126.3 ± 2.2 80 Constant 94.66 ± 2.01 Constant 126.2 ± 1.9 Mean 161.3 ± 0.3 Mean 170.8 ± 0.3 100 Mean 188.8 ± 0.3 Sigma 12.21 ± 0.53 Sigma 11.5 ± 0.6 Sigma 16.79 ± 0.62 80 60 80 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r1_39 partonE_B_all_r1_39 45 Entries 5740 Mean 231.6 40 RMS 39.07 χ2 / ndf 101.4 / 99 35 Prob 0.4153 30 Constant 32.18 ± 0.88 Mean 226.9 ± 0.8 25 Sigma 21.48 ± 1.62 20 15 10 5 0 0 100 200 300 400 500 600 Figure B.3: E parton distribution for b-quarks {|η| ∈ [0, 0.5)} in E jet bins with bin boundaries 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 150, 160, 170, 180, 220, 400 GeV 207 • Region 2 partonE_B_all_r2_0 partonE_B_all_r2_0 partonE_B_all_r2_1 partonE_B_all_r2_1 partonE_B_all_r2_2 partonE_B_all_r2_2 Entries 2094 140 Entries 8822 Entries 10837 180 35 Mean 41.11 Mean 42.76 Mean 44.57 RMS 18.18 120 RMS 17.36 160 RMS 16.88 30 χ / ndf 2 29.7 / 32 χ / ndf 2 29.03 / 41 χ / ndf 2 44.07 / 39 Prob 0.5833 Prob 0.9196 140 Prob 0.2658 100 25 Constant 30.09 ± 2.81 Constant 121.7 ± 2.5 120 Constant 160.1 ± 2.9 Mean 24.72 ± 8.15 Mean 33.43 ± 0.44 Mean 35.65 ± 0.23 Sigma 16.16 ± 8.94 80 Sigma 11.55 ± 1.00 Sigma 8.848 ± 0.432 20 100 60 80 15 60 10 40 40 5 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_3 160 partonE_B_all_r2_3 partonE_B_all_r2_4 partonE_B_all_r2_4 partonE_B_all_r2_5 partonE_B_all_r2_5 200 Entries 8832 Entries 9886 Entries 11144 160 140 Mean 46 Mean 47.59 180 Mean 48.97 RMS 16.22 140 RMS 16.61 RMS 16.57 120 χ2 / ndf 41.25 / 38 χ2 / ndf 37.57 / 40 160 χ2 / ndf 64.3 / 45 Prob 0.3305 120 Prob 0.5802 140 Prob 0.0309 100 Constant 133 ± 2.8 Constant 144.9 ± 2.9 Constant 158.7 ± 2.8 Mean 37.58 ± 0.25 100 Mean 39.2 ± 0.2 120 Mean 41.33 ± 0.22 Sigma 8.62 ± 0.50 Sigma 8.63 ± 0.43 Sigma 9.662 ± 0.431 80 100 80 60 80 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_6 partonE_B_all_r2_6 partonE_B_all_r2_7 partonE_B_all_r2_7 partonE_B_all_r2_8 250 partonE_B_all_r2_8 Entries 11988 220 Entries 13157 Entries 13951 200 Mean 50.3 200 Mean 51.76 Mean 52.86 180 RMS 16.15 RMS 16.37 RMS 15.83 180 200 χ2 / ndf 47.48 / 41 χ2 / ndf 44.46 / 40 χ2 / ndf 76.35 / 43 160 Prob 0.2254 Prob 0.2892 Prob 0.001299 160 140 Constant 179.4 ± 3.1 Constant 190.8 ± 3.2 Constant 206.4 ± 3.3 140 Mean 42.85 ± 0.19 Mean 43.66 ± 0.23 150 Mean 45.72 ± 0.18 120 Sigma 8.613 ± 0.353 120 Sigma 9.304 ± 0.452 Sigma 8.926 ± 0.330 100 100 100 80 80 60 60 40 40 50 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_9 250 partonE_B_all_r2_9 partonE_B_all_r2_10 partonE_B_all_r2_10 partonE_B_all_r2_11 partonE_B_all_r2_11 Entries 15118 250 Entries 15755 Entries 16704 Mean 54.78 Mean 56.18 250 Mean 57.94 RMS 16.28 RMS 16.04 RMS 16.4 200 χ2 / ndf 42.62 / 42 χ2 / ndf 59.78 / 43 χ2 / ndf 49.45 / 41 200 Prob 0.4442 Prob 0.04587 200 Prob 0.1714 Constant 220.6 ± 3.4 Constant 230.5 ± 3.5 Constant 245.5 ± 3.7 150 Mean 47.12 ± 0.17 Mean 48.97 ± 0.17 Mean 50.69 ± 0.17 Sigma 8.753 ± 0.317 150 Sigma 9.032 ± 0.325 150 Sigma 8.787 ± 0.321 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure B.4: E parton distribution for b-quarks {|η| ∈ [0.5, 1.0)} in E jet bins with bin boundaries 0, 19, 23, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44 GeV 208 partonE_B_all_r2_12 partonE_B_all_r2_12 partonE_B_all_r2_13 partonE_B_all_r2_13 partonE_B_all_r2_14 partonE_B_all_r2_14 300 Entries 17144 Entries 17803 700 Entries 46966 250 Mean 59.41 Mean 60.89 Mean 63.82 RMS 16.29 250 RMS 16.23 600 RMS 16.54 χ2 / ndf 49.91 / 42 χ2 / ndf 58.22 / 44 χ2 / ndf 66.54 / 45 200 Prob 0.1877 Prob 0.07395 Prob 0.02007 500 Constant 250.3 ± 3.7 200 Constant 257.8 ± 3.7 Constant 666.9 ± 5.8 Mean 52.55 ± 0.17 Mean 54.18 ± 0.17 Mean 57.18 ± 0.10 150 Sigma 9.047 ± 0.334 Sigma 9.256 ± 0.320 400 Sigma 9.399 ± 0.193 150 300 100 100 200 50 50 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_15 partonE_B_all_r2_15 partonE_B_all_r2_16 600 partonE_B_all_r2_16 partonE_B_all_r2_17 partonE_B_all_r2_17 450 Entries 28850 Entries 39304 700 Entries 48442 Mean 67.2 Mean 70.16 Mean 73.94 400 500 RMS 16.56 RMS 16.85 600 RMS 16.84 350 χ2 / ndf 55.77 / 44 χ2 / ndf 60.17 / 45 χ2 / ndf 41.62 / 45 Prob 0.1098 Prob 0.06465 Prob 0.6161 400 500 300 Constant 409.3 ± 4.6 Constant 547.2 ± 5.2 Constant 670.3 ± 5.8 Mean 60.59 ± 0.13 Mean 63.63 ± 0.12 Mean 67.55 ± 0.11 250 Sigma 9.332 ± 0.255 Sigma 9.51 ± 0.22 400 Sigma 9.712 ± 0.213 300 200 300 150 200 200 100 100 100 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_18 partonE_B_all_r2_18 partonE_B_all_r2_19 partonE_B_all_r2_19 partonE_B_all_r2_20 partonE_B_all_r2_20 700 Entries 48113 Entries 45439 600 Entries 43284 Mean 78.21 600 Mean 82.53 Mean 86.91 600 RMS 16.96 RMS 17.33 RMS 17.23 χ / ndf 2 χ / ndf 2 500 χ / ndf 2 42.38 / 47 36.13 / 47 61.02 / 49 500 Prob 0.6641 Prob 0.8752 Prob 0.1163 500 Constant 660.1 ± 5.7 Constant 616.3 ± 5.4 400 Constant 573.7 ± 5.2 Mean 71.98 ± 0.11 400 Mean 76.29 ± 0.11 Mean 81.05 ± 0.12 400 Sigma 9.775 ± 0.201 Sigma 9.954 ± 0.215 Sigma 10.28 ± 0.22 300 300 300 200 200 200 100 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_21 partonE_B_all_r2_21 partonE_B_all_r2_22 partonE_B_all_r2_22 partonE_B_all_r2_23 partonE_B_all_r2_23 Entries 40613 500 Entries 37646 Entries 34532 450 Mean 91.3 Mean 95.79 Mean 100.2 500 RMS 17.53 RMS 17.78 RMS 17.85 400 χ / ndf 2 42.09 / 50 χ / ndf 2 30.91 / 48 χ / ndf 2 46.72 / 48 400 Prob 0.779 Prob 0.9737 350 Prob 0.5252 400 Constant 528.7 ± 4.9 Constant 496.8 ± 4.8 Constant 448.6 ± 4.6 Mean 85.25 ± 0.13 Mean 89.85 ± 0.12 300 Mean 94.54 ± 0.14 Sigma 10.78 ± 0.26 300 Sigma 9.956 ± 0.228 Sigma 10.34 ± 0.27 300 250 200 200 200 150 100 100 100 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_24 partonE_B_all_r2_24 partonE_B_all_r2_25 partonE_B_all_r2_25 partonE_B_all_r2_26 partonE_B_all_r2_26 400 Entries 31714 Entries 28697 350 Entries 25969 400 Mean 104.8 Mean 109 Mean 113.3 350 RMS 18.14 RMS 18.17 300 RMS 18.33 350 χ2 / ndf χ2 / ndf χ2 / ndf 73.1 / 54 59.83 / 53 60.82 / 52 300 Prob 0.04275 Prob 0.2417 Prob 0.1881 300 250 Constant 393.6 ± 4.1 Constant 356.6 ± 3.9 Constant 316.7 ± 3.7 Mean 99.33 ± 0.15 250 Mean 103.5 ± 0.2 Mean 107.7 ± 0.2 250 Sigma 11.47 ± 0.30 Sigma 11.15 ± 0.30 200 Sigma 10.99 ± 0.32 200 200 150 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure B.5: E parton distribution for b-quarks {|η| ∈ [0.5, 1.0)} in E jet bins with bin boundaries 44, 46, 48, 53, 56, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110 GeV 209 partonE_B_all_r2_27 partonE_B_all_r2_27 partonE_B_all_r2_28 partonE_B_all_r2_28 partonE_B_all_r2_29 partonE_B_all_r2_29 300 Entries 23161 Entries 20915 240 Entries 18259 Mean 118 Mean 122.2 220 Mean 126.6 RMS 18.86 250 RMS 18.89 RMS 19.06 250 200 χ2 / ndf 52.91 / 55 χ2 / ndf 68.36 / 59 χ2 / ndf 83.23 / 56 Prob 0.555 Prob 0.1894 180 Prob 0.01057 200 200 Constant 277.7 ± 3.4 Constant 244.5 ± 3.1 160 Constant 213.1 ± 3.0 Mean 112.4 ± 0.2 Mean 117.6 ± 0.2 Mean 122.2 ± 0.2 140 Sigma 11.47 ± 0.34 150 Sigma 12.47 ± 0.39 Sigma 11.85 ± 0.41 150 120 100 100 100 80 60 50 50 40 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_30 220 partonE_B_all_r2_30 partonE_B_all_r2_31 200 partonE_B_all_r2_31 partonE_B_all_r2_32 partonE_B_all_r2_32 Entries 16637 Entries 14718 160 Entries 12956 200 Mean 131.1 180 Mean 135.3 Mean 139.9 180 RMS 19.45 160 RMS 19.35 140 RMS 19.55 χ2 / ndf 75.5 / 58 χ2 / ndf 53.16 / 57 χ2 / ndf 54.75 / 59 160 Prob 0.06112 140 Prob 0.6199 120 Prob 0.6328 140 Constant 189.9 ± 2.7 Constant 169.6 ± 2.6 Constant 145.7 ± 2.4 Mean 126.1 ± 0.2 120 Mean 130.8 ± 0.2 100 Mean 135.1 ± 0.3 120 Sigma 12.16 ± 0.43 Sigma 11.87 ± 0.44 Sigma 12.4 ± 0.5 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_33 partonE_B_all_r2_33 partonE_B_all_r2_34 partonE_B_all_r2_34 partonE_B_all_r2_35 partonE_B_all_r2_35 Entries 21859 200 Entries 16774 160 Entries 12755 250 Mean 146.6 180 Mean 155 Mean 164.3 RMS 20.22 RMS 20.65 140 RMS 21.14 χ / ndf 2 60.38 / 62 160 χ / ndf 2 61.8 / 65 χ / ndf 2 65.52 / 65 200 120 Prob 0.5347 Prob 0.5898 Prob 0.4585 140 Constant 235.1 ± 3.0 Constant 176.6 ± 2.5 Constant 131.5 ± 2.2 Mean 142 ± 0.2 120 Mean 150.9 ± 0.2 100 Mean 160.8 ± 0.3 150 Sigma 13.18 ± 0.42 Sigma 13.55 ± 0.47 Sigma 13.45 ± 0.54 100 80 100 80 60 60 40 50 40 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r2_36 partonE_B_all_r2_36 partonE_B_all_r2_37 partonE_B_all_r2_37 partonE_B_all_r2_38 partonE_B_all_r2_38 120 Entries 9990 180 Entries 20850 Entries 9810 Mean 173.6 Mean 192.4 60 Mean 239.6 RMS 22.57 160 RMS 24.97 RMS 38.16 100 χ / ndf 2 54.66 / 65 χ / ndf 2 82.33 / 90 χ / ndf 2 121.2 / 121 140 50 Prob 0.8161 Prob 0.7049 Prob 0.4784 Constant 101.6 ± 1.9 120 Constant 164.3 ± 2.1 Constant 51.82 ± 1.00 80 Mean 169.6 ± 0.3 Mean 188.9 ± 0.3 40 Mean 230 ± 0.7 Sigma 13.33 ± 0.59 100 Sigma 18.55 ± 0.57 Sigma 25.69 ± 1.29 60 30 80 40 60 20 40 20 10 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure B.6: E parton distribution for b-quarks {|η| ∈ [0.5, 1.0)} in E jet bins with bin boundaries 110, 115, 120, 125, 130, 135, 140, 150, 160, 170, 180, 220, 400 GeV 210 • Region 3 partonE_B_all_r3_0 partonE_B_all_r3_0 partonE_B_all_r3_1 partonE_B_all_r3_1 partonE_B_all_r3_2 partonE_B_all_r3_2 35 Entries 1756 Entries 59321 Entries 22543 Mean 47 600 Mean 57.85 250 Mean 66.03 30 RMS 20.41 RMS 20.57 RMS 20.51 χ / ndf 2 44.95 / 38 χ / ndf 2 67.51 / 65 χ / ndf 2 84.86 / 64 25 Prob 0.2037 500 Prob 0.3915 200 Prob 0.04164 Constant 24.08 ± 1.17 Constant 619.3 ± 4.7 Constant 238.6 ± 2.9 Mean 35.76 ± 0.46 400 Mean 49.91 ± 0.13 Mean 58.44 ± 0.21 20 Sigma 7.197 ± 0.687 Sigma 13.45 ± 0.25 150 Sigma 13.15 ± 0.38 15 300 100 10 200 50 5 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r3_3 300 partonE_B_all_r3_3 partonE_B_all_r3_4 partonE_B_all_r3_4 partonE_B_all_r3_5 partonE_B_all_r3_5 300 300 Entries 24315 Entries 25380 Entries 25811 Mean 70.15 Mean 74.49 Mean 78.91 250 RMS 20.46 RMS 20.59 250 RMS 21.17 250 χ2 / ndf 85.55 / 67 χ2 / ndf 81.35 / 68 χ2 / ndf 75.19 / 66 Prob 0.06284 Prob 0.1284 Prob 0.2054 200 Constant 249 ± 2.9 200 Constant 253.5 ± 2.9 200 Constant 258.2 ± 3.0 Mean 63.14 ± 0.21 Mean 67.41 ± 0.21 Mean 71.75 ± 0.20 Sigma 13.82 ± 0.39 Sigma 14.18 ± 0.40 Sigma 13.63 ± 0.39 150 150 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r3_6 300 partonE_B_all_r3_6 partonE_B_all_r3_7 partonE_B_all_r3_7 partonE_B_all_r3_8 partonE_B_all_r3_8 Entries 26143 Entries 26156 Entries 25182 Mean 83.1 250 Mean 87.79 Mean 92.3 250 250 RMS 21.45 RMS 21.93 RMS 21.98 χ2 / ndf 86.93 / 71 χ2 / ndf 63.29 / 73 χ2 / ndf 75.31 / 71 Prob 0.09623 200 Prob 0.7843 Prob 0.3409 200 200 Constant 253 ± 2.9 Constant 247.8 ± 2.8 Constant 239.8 ± 2.8 Mean 75.9 ± 0.2 Mean 80.79 ± 0.22 Mean 85.05 ± 0.22 Sigma 14.9 ± 0.4 150 Sigma 15.09 ± 0.41 150 Sigma 14.76 ± 0.43 150 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r3_9 partonE_B_all_r3_9 partonE_B_all_r3_10 partonE_B_all_r3_10 partonE_B_all_r3_11 partonE_B_all_r3_11 Entries 24374 Entries 23475 250 Entries 22259 250 250 Mean 96.54 Mean 101 Mean 105.6 RMS 22.1 RMS 22.28 RMS 22.51 χ2 / ndf 99.4 / 76 χ2 / ndf 83.95 / 74 200 χ2 / ndf 78.44 / 73 200 200 Prob 0.03713 Prob 0.201 Prob 0.3106 Constant 225.6 ± 2.6 Constant 217.8 ± 2.6 Constant 206.7 ± 2.6 Mean 89.6 ± 0.2 Mean 94.3 ± 0.2 150 Mean 98.6 ± 0.2 150 Sigma 16.06 ± 0.48 150 Sigma 15.29 ± 0.45 Sigma 15.26 ± 0.47 100 100 100 50 50 50 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure B.7: E parton distribution for b-quarks {|η| ∈ [1.0, 1.5)} in E jet bins with bin boundaries 0, 25, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95 GeV 211 partonE_B_all_r3_12 partonE_B_all_r3_12 partonE_B_all_r3_13 partonE_B_all_r3_13 partonE_B_all_r3_14 partonE_B_all_r3_14 240 Entries 21162 200 Entries 19905 180 Entries 18420 220 Mean 110.1 Mean 114.8 Mean 119.4 RMS 22.62 180 RMS 23.12 RMS 23.13 200 160 χ2 / ndf 96.59 / 78 160 χ2 / ndf 74.44 / 78 χ2 / ndf 74.51 / 77 180 140 Prob 0.07539 Prob 0.5932 Prob 0.5594 160 Constant 188.9 ± 2.4 140 Constant 177.3 ± 2.3 Constant 163 ± 2.2 120 140 Mean 103.5 ± 0.3 120 Mean 108.1 ± 0.3 Mean 112.7 ± 0.3 Sigma 16.12 ± 0.49 Sigma 16.29 ± 0.52 100 Sigma 16.15 ± 0.55 120 100 100 80 80 80 60 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r3_15 partonE_B_all_r3_15 partonE_B_all_r3_16 partonE_B_all_r3_16 partonE_B_all_r3_17 partonE_B_all_r3_17 Entries 16964 Entries 15538 Entries 27813 160 160 250 Mean 123.6 Mean 128.2 Mean 134.9 140 RMS 23.27 140 RMS 23.64 RMS 24.01 χ2 / ndf 92.51 / 81 χ2 / ndf 91.79 / 81 χ2 / ndf 81.72 / 81 Prob 0.1796 Prob 0.1937 200 Prob 0.4567 120 120 Constant 144.6 ± 2.0 Constant 132.4 ± 1.9 Constant 235.5 ± 2.6 100 Mean 117.2 ± 0.3 100 Mean 122 ± 0.3 Mean 128.5 ± 0.2 Sigma 17.54 ± 0.66 Sigma 16.63 ± 0.59 150 Sigma 16.76 ± 0.45 80 80 60 60 100 40 40 50 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r3_18 partonE_B_all_r3_18 partonE_B_all_r3_19 partonE_B_all_r3_19 partonE_B_all_r3_20 partonE_B_all_r3_20 140 200 Entries 23089 Entries 19016 Entries 15759 Mean 143.3 160 Mean 152.4 Mean 161.7 180 RMS 24.48 RMS 24.81 120 RMS 25.88 χ / ndf 2 129.4 / 83 140 χ / ndf 2 104.9 / 87 χ / ndf 2 76.07 / 93 160 Prob 0.0008449 Prob 0.09301 100 Prob 0.899 120 140 Constant 187.5 ± 2.3 Constant 149.4 ± 2.0 Constant 119.2 ± 1.7 Mean 137.3 ± 0.3 Mean 146.3 ± 0.3 Mean 155.8 ± 0.4 120 100 80 Sigma 17 ± 0.5 Sigma 18 ± 0.6 Sigma 19.17 ± 0.68 100 80 60 80 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r3_21 120 partonE_B_all_r3_21 partonE_B_all_r3_22 partonE_B_all_r3_22 partonE_B_all_r3_23 partonE_B_all_r3_23 Entries 12799 Entries 10428 Entries 12018 100 Mean 169.6 Mean 178.5 100 Mean 188.9 100 RMS 25.87 RMS 27.07 RMS 27.84 χ / ndf 2 81.55 / 89 χ / ndf 2 110.1 / 102 χ / ndf 2 91.31 / 100 80 Prob 0.7 Prob 0.2742 80 Prob 0.7211 80 Constant 96.88 ± 1.60 Constant 74.54 ± 1.30 Constant 84.14 ± 1.42 Mean 164.3 ± 0.4 Mean 174.2 ± 0.4 Mean 184.2 ± 0.4 Sigma 17.99 ± 0.71 60 Sigma 20.06 ± 0.79 Sigma 20.58 ± 0.86 60 60 40 40 40 20 20 20 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r3_24 120 partonE_B_all_r3_24 partonE_B_all_r3_25 partonE_B_all_r3_25 partonE_B_all_r3_26 30 partonE_B_all_r3_26 80 Entries 15080 Entries 10861 Entries 3876 Mean 207.2 Mean 239.4 Mean 297.6 70 100 RMS 29.26 RMS 32.9 25 RMS 46.43 χ2 / ndf 108.8 / 109 χ2 / ndf 119.3 / 131 χ2 / ndf 194.3 / 165 60 Prob 0.4885 Prob 0.7599 Prob 0.05911 80 Constant 96.55 ± 1.44 Constant 58.86 ± 1.03 20 Constant 15.08 ± 0.46 Mean 202.9 ± 0.4 50 Mean 235.6 ± 0.6 Mean 287.9 ± 1.3 Sigma 22.41 ± 0.82 Sigma 26.68 ± 1.14 Sigma 33.34 ± 2.47 60 40 15 30 40 10 20 20 5 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure B.8: E parton distribution for b-quarks {|η| ∈ [1.0, 1.5)} in E jet bins with bin boundaries 95, 100, 105, 110, 115, 120, 130, 140, 150, 160, 170, 180, 195, 225, 280, 500 GeV 212 • Region 4 partonE_B_all_r4_1 partonE_B_all_r4_1 partonE_B_all_r4_2 partonE_B_all_r4_2 partonE_B_all_r4_3 partonE_B_all_r4_3 40 Entries 3452 45 Entries 3681 Entries 4675 Mean 66.33 Mean 72.73 50 Mean 77.37 35 RMS 22.52 40 RMS 23.2 RMS 24.75 χ / ndf 2 72.04 / 70 χ / ndf 2 95.88 / 85 χ / ndf 2 69.83 / 79 Prob 0.4101 35 Prob 0.1971 40 Prob 0.7599 30 Constant 32.64 ± 1.04 30 Constant 30.74 ± 0.91 Constant 38.58 ± 1.08 25 Mean 57.87 ± 0.58 Mean 63.51 ± 0.62 Mean 68.8 ± 0.6 Sigma 14.2 ± 1.1 25 Sigma 16.7 ± 1.1 30 Sigma 16.79 ± 1.22 20 20 15 20 15 10 10 10 5 5 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r4_4 partonE_B_all_r4_4 partonE_B_all_r4_5 partonE_B_all_r4_5 partonE_B_all_r4_6 partonE_B_all_r4_6 70 Entries 6129 Entries 6788 Entries 7527 60 Mean 82.65 60 Mean 86.52 Mean 90.84 RMS 26.29 RMS 26.97 60 RMS 27.52 50 χ2 / ndf 100.2 / 80 50 χ2 / ndf 103 / 93 χ2 / ndf 88.02 / 97 Prob 0.06269 Prob 0.2237 50 Prob 0.7315 Constant 48.66 ± 1.20 Constant 50.93 ± 1.13 Constant 54.66 ± 1.14 40 Mean 72.18 ± 0.52 40 Mean 77.55 ± 0.54 Mean 80.74 ± 0.51 40 Sigma 16.45 ± 0.99 Sigma 19.04 ± 1.02 Sigma 19.31 ± 0.92 30 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r4_7 partonE_B_all_r4_7 partonE_B_all_r4_8 partonE_B_all_r4_8 partonE_B_all_r4_9 partonE_B_all_r4_9 70 Entries 8248 Entries 8767 Entries 8946 Mean 95.42 70 Mean 100.1 70 Mean 104.2 60 RMS 28.91 RMS 29.35 RMS 29.97 χ2 / ndf 73.69 / 95 60 χ2 / ndf 85.02 / 98 60 χ2 / ndf 134.3 / 110 Prob 0.9485 Prob 0.8222 Prob 0.05791 50 Constant 58.02 ± 1.19 50 Constant 60.38 ± 1.20 50 Constant 58.37 ± 1.14 Mean 85.73 ± 0.59 Mean 89.73 ± 0.55 Mean 94.96 ± 0.56 40 Sigma 21.09 ± 1.21 40 Sigma 20.86 ± 1.08 40 Sigma 22.54 ± 1.09 30 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r4_10 partonE_B_all_r4_10 partonE_B_all_r4_11 partonE_B_all_r4_11 partonE_B_all_r4_12 80 partonE_B_all_r4_12 Entries 9187 80 Entries 9176 Entries 9227 70 Mean 108.6 Mean 113.2 70 Mean 117.1 RMS 30.3 70 RMS 31.06 RMS 31.19 60 χ2 / ndf 84.97 / 99 χ2 / ndf 98.01 / 96 60 χ2 / ndf 106.3 / 106 Prob 0.8415 60 Prob 0.4237 Prob 0.4748 50 Constant 61.42 ± 1.20 Constant 60.32 ± 1.19 Constant 59.63 ± 1.15 50 Mean 98.76 ± 0.52 50 Mean 101.9 ± 0.6 Mean 108.1 ± 0.5 40 Sigma 20.4 ± 1.0 Sigma 20.86 ± 1.10 Sigma 21.27 ± 0.97 40 40 30 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure B.9: E parton distribution for b-quarks {|η| ∈ [1.5, 2.5)} in E jet bins with bin boundaries 25, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 GeV 213 partonE_B_all_r4_13 partonE_B_all_r4_13 partonE_B_all_r4_14 partonE_B_all_r4_14 partonE_B_all_r4_15 partonE_B_all_r4_15 80 70 Entries 8992 Entries 8464 70 Entries 8358 Mean 120.9 Mean 125.8 Mean 129.4 70 RMS 30.65 60 RMS 31.57 RMS 31.93 60 χ2 / ndf 94.83 / 99 χ2 / ndf 114.1 / 95 χ2 / ndf 89.27 / 105 60 Prob 0.5998 50 Prob 0.08859 Prob 0.864 50 Constant 58.97 ± 1.19 Constant 55.93 ± 1.18 Constant 51.95 ± 1.07 50 Mean 111.5 ± 0.5 Mean 115.8 ± 0.5 Mean 118.5 ± 0.6 40 40 Sigma 19.64 ± 0.91 Sigma 18.72 ± 0.90 Sigma 22.6 ± 1.2 40 30 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r4_16 80 partonE_B_all_r4_16 partonE_B_all_r4_17 partonE_B_all_r4_17 partonE_B_all_r4_18 partonE_B_all_r4_18 Entries 8133 120 Entries 15014 Entries 13683 100 70 Mean 133.5 Mean 140.8 Mean 149.1 RMS 31.83 RMS 32.32 RMS 33.26 60 χ2 / ndf 121.7 / 104 100 χ2 / ndf 93.38 / 107 χ2 / ndf 96.29 / 104 80 Prob 0.1132 Prob 0.8232 Prob 0.6921 Constant 50.89 ± 1.08 80 Constant 93.62 ± 1.44 Constant 84.01 ± 1.37 50 Mean 124.1 ± 0.6 Mean 131.9 ± 0.4 Mean 140.8 ± 0.5 Sigma 21.26 ± 1.12 Sigma 22.08 ± 0.85 60 Sigma 22.1 ± 0.9 40 60 30 40 40 20 20 20 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r4_19 partonE_B_all_r4_19 partonE_B_all_r4_20 partonE_B_all_r4_20 partonE_B_all_r4_21 partonE_B_all_r4_21 90 Entries 12207 Entries 10585 70 Entries 9194 80 Mean 158.3 Mean 167 Mean 176.1 80 RMS 33.25 RMS 34.03 RMS 34.09 70 60 χ / ndf 2 163.2 / 111 χ / ndf 2 131.5 / 115 χ / ndf 2 102.1 / 109 70 Prob 0.0009294 60 Prob 0.1388 Prob 0.6663 50 60 Constant 72.63 ± 1.24 Constant 62.46 ± 1.12 Constant 54.76 ± 1.08 Mean 149.4 ± 0.5 50 Mean 158.9 ± 0.5 Mean 167.5 ± 0.6 50 Sigma 22.29 ± 0.89 Sigma 23.5 ± 1.0 40 Sigma 22.94 ± 1.17 40 40 30 30 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r4_22 partonE_B_all_r4_22 partonE_B_all_r4_23 partonE_B_all_r4_23 partonE_B_all_r4_24 partonE_B_all_r4_24 70 Entries 7850 70 Entries 9665 90 Entries 13740 Mean 184.6 Mean 195.2 Mean 214.7 RMS 34.78 RMS 33.92 80 RMS 35.41 60 60 χ / ndf 2 113.8 / 114 χ / ndf 2 125.7 / 123 χ / ndf 2 138.6 / 135 70 Prob 0.4864 Prob 0.4161 Prob 0.3982 50 50 Constant 45.62 ± 0.97 Constant 54.1 ± 1.0 60 Constant 72.67 ± 1.12 Mean 176.1 ± 0.6 Mean 188.7 ± 0.6 Mean 207.5 ± 0.6 40 Sigma 23.26 ± 1.20 40 Sigma 25.64 ± 1.19 50 Sigma 27.6 ± 1.0 30 30 40 30 20 20 20 10 10 10 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 0 100 200 300 400 500 600 partonE_B_all_r4_25 partonE_B_all_r4_25 partonE_B_all_r4_26 partonE_B_all_r4_26 80 Entries 12328 Entries 6273 Mean 248 35 Mean 312.4 70 RMS 38.14 RMS 50.32 χ2 / ndf 212.2 / 156 30 χ2 / ndf 214.4 / 190 60 Prob 0.001864 Prob 0.1081 Constant 57.38 ± 0.93 25 Constant 21.85 ± 0.52 50 Mean 242.2 ± 0.6 Mean 301 ± 1.2 Sigma 31.16 ± 1.20 20 Sigma 38.59 ± 2.23 40 30 15 20 10 10 5 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Figure B.10: E parton distribution for b-quarks {|η| ∈ [1.5, 2.5)} in E jet bins with bin boundaries 100, 105, 110, 115, 120, 130, 140, 150, 160, 170, 180, 195, 225, 280, 500 GeV 214 215 APPENDIX C Pull distributions • Input JES = nominal JES best jes hbesty best mass hbestx best mass 1D hbest1D Entries 3000 Entries 3000 Entries 3000 450 Mean 0.9948 Mean 150.4 700 Mean 150.1 RMS 0.01451 RMS 1.48 RMS 1.589 400 χ2 / ndf 28.61 / 18 χ2 / ndf 9.233 / 12 χ2 / ndf 20.41 / 17 500 600 Prob 0.0534 Prob 0.6829 Prob 0.254 350 Constant 410.6 ± 9.2 Constant 535 ± 12.0 Constant 701.7 ± 15.7 Mean 0.9948 ± 0.0003 Mean 150.4 ± 0.0 500 Mean 150.1 ± 0.0 400 300 Sigma 0.01457 ± 0.00019 Sigma 1.491 ± 0.019 Sigma 1.135 ± 0.015 250 400 300 200 300 200 150 200 100 100 100 50 0 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 130 140 150 160 170 180 190 200 210 130 140 150 160 170 180 190 200 210 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 Mean -0.3484 Mean 0.2499 Mean 0.06458 400 RMS 0.9776 400 RMS 0.9142 RMS 0.9693 350 χ2 / ndf 43.94 / 20 χ2 / ndf 12.69 / 17 χ2 / ndf 26.14 / 21 350 Prob 0.001532 350 Prob 0.7566 Prob 0.2013 Constant 365.7 ± 8.2 Constant 390.9 ± 8.7 300 Constant 369.1 ± 8.3 300 Mean -0.3501 ± 0.0179 300 Mean 0.2503 ± 0.0168 Mean 0.066 ± 0.018 Sigma 0.9817 ± 0.0127 Sigma 0.9186 ± 0.0119 250 Sigma 0.9727 ± 0.0126 250 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 Figure C.1: Fitted parameters and their respective pulls for input top quark mass of 150 GeV /c2 216 • Input JES = nominal JES best jes hbesty best mass hbestx best mass 1D hbest1D Entries 3000 Entries 3000 Entries 3000 400 Mean 0.9994 Mean 172.3 Mean 172.2 450 700 RMS 0.01537 RMS 1.773 RMS 1.429 350 χ2 / ndf 10.98 / 19 χ2 / ndf 17.43 / 16 χ2 / ndf 24.44 / 17 400 Prob 0.9245 Prob 0.3586 600 Prob 0.1081 Constant 388.6 ± 8.7 Constant 447.7 ± 10.0 Constant 667.4 ± 15.0 300 350 Mean 0.9994 ± 0.0003 Mean 172.3 ± 0.0 Mean 172.3 ± 0.0 500 Sigma 0.0154 ± 0.0002 300 Sigma 1.782 ± 0.023 Sigma 1.185 ± 0.015 250 400 250 200 200 300 150 150 200 100 100 50 100 50 0 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 130 140 150 160 170 180 190 200 210 130 140 150 160 170 180 190 200 210 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 Mean -0.04224 Mean -0.1271 Mean -0.1719 350 RMS 1.04 350 RMS 0.9961 RMS 1.038 350 χ2 / ndf 13.65 / 21 χ2 / ndf 24.36 / 21 χ2 / ndf 119 / 28 300 Prob 0.8842 Prob 0.2761 Prob 3.433e-13 300 Constant 344.8 ± 7.7 Constant 359.1 ± 8.0 300 Constant 343.8 ± 7.7 Mean -0.0416 ± 0.0190 Mean -0.1258 ± 0.0183 Mean -0.169 ± 0.019 250 Sigma 1.041 ± 0.013 250 Sigma 0.9997 ± 0.0129 Sigma 1.044 ± 0.013 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 Figure C.2: Fitted parameters and their respective pulls for input top quark mass of 172.5 GeV /c2 217 • Input JES = nominal JES best jes hbesty best mass hbestx best mass 1D hbest1D Entries 3000 Entries 3000 Entries 3000 400 Mean 0.9991 Mean 190.5 Mean 190.4 600 RMS 0.01553 RMS 2.135 RMS 2.12 350 350 χ2 / ndf 11.77 / 18 χ2 / ndf 34.87 / 21 χ2 / ndf 35.36 / 28 Prob 0.8589 Prob 0.02918 Prob 0.1595 500 Constant 385 ± 8.6 300 Constant 371.2 ± 8.3 Constant 597.9 ± 13.5 300 Mean 0.9992 ± 0.0003 Mean 190.5 ± 0.0 Mean 190.5 ± 0.0 Sigma 0.01554 ± 0.00020 Sigma 2.149 ± 0.028 400 Sigma 1.317 ± 0.017 250 250 200 200 300 150 150 200 100 100 100 50 50 0 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 130 140 150 160 170 180 190 200 210 130 140 150 160 170 180 190 200 210 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 Mean -0.06649 350 Mean 0.2865 350 Mean 0.3654 350 RMS 1.051 RMS 1.08 RMS 1.062 χ2 / ndf 18.96 / 23 χ2 / ndf 19.65 / 23 χ2 / ndf 51.85 / 27 300 300 300 Prob 0.7034 Prob 0.663 Prob 0.002767 Constant 340.8 ± 7.6 Constant 332.1 ± 7.4 Constant 337.7 ± 7.6 Mean -0.0647 ± 0.0192 250 Mean 0.2873 ± 0.0197 250 Mean 0.3646 ± 0.0194 250 Sigma 1.054 ± 0.014 Sigma 1.081 ± 0.014 Sigma 1.063 ± 0.014 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 Figure C.3: Fitted parameters and their respective pulls for input top quark mass of 190 GeV /c2 218 • Input JES = 0.94 best jes hbesty best mass hbestx best mass 1D hbest1D Entries 3000 Entries 3000 Entries 3000 400 Mean 0.9313 Mean 150.3 Mean 145.4 700 1.167 RMS 0.01601 RMS 1.545 RMS 500 χ2 / ndf 15.15 / 20 χ2 / ndf 26.41 / 14 χ2 / ndf 12.3 / 11 350 Prob 0.7677 Prob 0.02297 600 Prob 0.3416 Constant 372.1 ± 8.3 Constant 514.8 ± 11.5 Constant 706.7 ± 15.8 300 Mean 0.9313 ± 0.0003 400 Mean 150.3 ± 0.0 Mean 145.4 ± 0.0 500 Sigma 0.01608 ± 0.00021 Sigma 1.55 ± 0.02 Sigma 1.126 ± 0.015 250 300 400 200 300 150 200 200 100 100 50 100 0 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 130 140 150 160 170 180 190 200 210 130 140 150 160 170 180 190 200 210 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 Mean -0.5593 400 Mean 0.1736 400 Mean -3.885 350 RMS 1.019 RMS 0.955 RMS 0.963 χ2 / ndf 19.99 / 22 χ2 / ndf 34.62 / 21 χ2 / ndf 41.2 / 22 350 350 Prob 0.5836 Prob 0.03107 Prob 0.00781 300 Constant 350.4 ± 7.8 Constant 374.4 ± 8.4 Constant 370.7 ± 8.3 300 300 Mean -0.5576 ± 0.0187 Mean 0.1735 ± 0.0175 Mean -3.884 ± 0.018 250 Sigma 1.025 ± 0.013 Sigma 0.959 ± 0.012 Sigma 0.9675 ± 0.0125 250 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 Figure C.4: Fitted parameters and their respective pulls for input top quark mass of 150 GeV /c2 219 • Input JES = 0.94 best jes hbesty best mass hbestx best mass 1D hbest1D Entries 3000 Entries 3000 Entries 3000 Mean 0.9381 450 Mean 171.6 Mean 166.6 350 RMS 0.01723 RMS 1.827 RMS 1.328 χ2 / ndf 20.01 / 21 400 χ2 / ndf 15.36 / 16 600 χ2 / ndf 11.34 / 14 300 Prob 0.5203 Prob 0.4987 Prob 0.6588 Constant 346.1 ± 7.7 350 Constant 435 ± 9.7 Constant 680.9 ± 15.2 500 Mean 0.9381 ± 0.0003 Mean 171.6 ± 0.0 Mean 166.6 ± 0.0 250 Sigma 0.01729 ± 0.00022 300 Sigma 1.834 ± 0.024 Sigma 1.17 ± 0.02 400 200 250 200 300 150 150 200 100 100 50 100 50 0 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 130 140 150 160 170 180 190 200 210 130 140 150 160 170 180 190 200 210 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 350 Mean -0.1245 Mean -0.5115 Mean -4.845 350 RMS 1.078 RMS 1.022 RMS 0.9948 350 χ2 / ndf 18.8 / 21 χ2 / ndf 32.39 / 20 χ2 / ndf 22.12 / 23 300 Prob 0.598 300 Prob 0.0393 Prob 0.5131 Constant 332.6 ± 7.4 Constant 350 ± 7.8 300 Constant 369.1 ± 8.3 250 Mean -0.1266 ± 0.0197 Mean -0.5123 ± 0.0187 Mean -4.852 ± 0.018 250 Sigma 1.08 ± 0.01 Sigma 1.026 ± 0.013 250 Sigma 0.9722 ± 0.0126 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 Figure C.5: Fitted parameters and their respective pulls for input top quark mass of 172.5 GeV /c2 220 • Input JES = 0.94 best jes hbesty best mass hbestx best mass 1D hbest1D Entries 3000 Entries 3000 Entries 3000 Mean 0.9333 Mean 189.9 600 Mean 183.4 350 RMS 0.01797 RMS 2.235 RMS 2.086 350 χ2 / ndf 18.12 / 23 χ2 / ndf 32.12 / 20 χ2 / ndf 24 / 24 300 Prob 0.7507 Prob 0.04208 500 Prob 0.4618 Constant 332.3 ± 7.4 300 Constant 355.6 ± 8.0 Constant 593.8 ± 13.3 250 Mean 0.9334 ± 0.0003 Mean 189.9 ± 0.0 Mean 183.4 ± 0.0 Sigma 0.01801 ± 0.00023 250 Sigma 2.244 ± 0.029 400 Sigma 1.332 ± 0.017 200 200 300 150 150 200 100 100 100 50 50 0 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 130 140 150 160 170 180 190 200 210 130 140 150 160 170 180 190 200 210 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 350 Mean -0.4307 350 Mean -0.02784 Mean -4.943 RMS 1.119 RMS 1.118 350 RMS 1.095 χ2 / ndf 15.68 / 25 χ2 / ndf 34.29 / 22 χ2 / ndf 42.4 / 29 300 300 Prob 0.924 Prob 0.0459 Prob 0.05169 300 Constant 320.3 ± 7.2 Constant 320 ± 7.2 Constant 352.3 ± 7.9 250 Mean -0.4301 ± 0.0205 250 Mean -0.0285 ± 0.0205 Mean -4.96 ± 0.02 Sigma 1.121 ± 0.014 Sigma 1.122 ± 0.014 250 Sigma 1.017 ± 0.013 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 Figure C.6: Fitted parameters and their respective pulls for input top quark mass of 190 GeV /c2 221 • Input JES = 1.06 best jes hbesty best mass hbestx best mass 1D hbest1D Entries 3000 Entries 3000 Entries 3000 450 Mean 1.056 Mean 150.9 800 Mean 155 RMS 0.01386 500 RMS 1.518 RMS 1.223 400 χ2 / ndf 41.55 / 18 χ2 / ndf 25.97 / 14 700 χ2 / ndf 30.36 / 11 Prob 0.001279 Prob 0.0261 Prob 0.001391 350 Constant 428.6 ± 9.6 Constant 519.1 ± 11.6 Constant 764 ± 17.1 400 600 Mean 1.056 ± 0.000 Mean 150.9 ± 0.0 Mean 155 ± 0.0 300 Sigma 0.01396 ± 0.00018 Sigma 1.537 ± 0.020 Sigma 1.043 ± 0.013 500 250 300 400 200 200 300 150 200 100 100 50 100 0 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 130 140 150 160 170 180 190 200 210 130 140 150 160 170 180 190 200 210 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 Mean -0.2918 Mean 0.5422 Mean 4.422 400 RMS 1 RMS 0.9384 RMS 0.9019 350 χ2 / ndf 24.82 / 21 350 χ2 / ndf 19.83 / 21 χ2 / ndf 22.66 / 20 350 Prob 0.2551 Prob 0.5322 Prob 0.3057 300 Constant 357.8 ± 8.0 Constant 381.5 ± 8.5 Constant 395.4 ± 8.8 300 300 Mean -0.2909 ± 0.0183 Mean 0.5413 ± 0.0172 Mean 4.42 ± 0.02 250 Sigma 1.004 ± 0.013 Sigma 0.9412 ± 0.0122 Sigma 0.9077 ± 0.0117 250 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 Figure C.7: Fitted parameters and their respective pulls for input top quark mass of 150 GeV /c2 222 • Input JES = 1.06 best jes hbesty best mass hbestx best mass 1D hbest1D Entries 3000 Entries 3000 Entries 3000 Mean 1.059 Mean 173 700 Mean 178.2 450 RMS 0.01365 RMS 1.686 RMS 1.305 450 χ2 / ndf 27.02 / 19 χ2 / ndf 7.113 / 15 χ2 / ndf 10.56 / 13 400 600 Prob 0.1042 400 Prob 0.9544 Prob 0.6476 350 Constant 436.9 ± 9.8 Constant 467.8 ± 10.5 Constant 674.2 ± 15.1 Mean 1.059 ± 0.000 350 Mean 173 ± 0.0 500 Mean 178.2 ± 0.0 300 Sigma 0.0137 ± 0.0002 Sigma 1.706 ± 0.022 Sigma 1.182 ± 0.015 300 400 250 250 200 300 200 150 150 200 100 100 100 50 50 0 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 130 140 150 160 170 180 190 200 210 130 140 150 160 170 180 190 200 210 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 Mean -0.0875 Mean 0.3182 Mean 4.665 RMS 0.997 RMS 0.954 RMS 0.9592 350 350 350 χ2 / ndf 30.85 / 23 χ2 / ndf 13.84 / 19 χ2 / ndf 10.64 / 18 Prob 0.1266 Prob 0.793 Prob 0.9091 300 Constant 359.1 ± 8.0 300 Constant 374.3 ± 8.4 300 Constant 373 ± 8.3 Mean -0.0892 ± 0.0183 Mean 0.316 ± 0.018 Mean 4.664 ± 0.018 Sigma 1 ± 0.0 Sigma 0.9594 ± 0.0124 Sigma 0.9627 ± 0.0124 250 250 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 Figure C.8: Fitted parameters and their respective pulls for input top quark mass of 172.5 GeV /c2 223 • Input JES = 1.06 best jes hbesty best mass hbestx best mass 1D hbest1D Entries 3000 Entries 3000 Entries 3000 450 Mean 1.057 Mean 191.7 Mean 197.5 400 600 RMS 0.01366 RMS 1.994 RMS 1.622 400 χ2 / ndf 21.85 / 18 χ2 / ndf 25.3 / 21 χ2 / ndf 18.11 / 20 Prob 0.2386 350 Prob 0.2345 Prob 0.5803 500 350 Constant 435.5 ± 9.7 Constant 398.6 ± 8.9 Constant 597.5 ± 13.4 Mean 1.057 ± 0.000 300 Mean 191.7 ± 0.0 Mean 197.5 ± 0.0 300 Sigma 0.01374 ± 0.00018 Sigma 2.002 ± 0.026 Sigma 1.326 ± 0.017 400 250 250 200 300 200 150 150 200 100 100 100 50 50 0 0 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 130 140 150 160 170 180 190 200 210 130 140 150 160 170 180 190 200 210 pull distribution of jes hpully pull distribution of mass hpullx pull distribution of 1D mass hpull1D Entries 3000 Entries 3000 Entries 3000 Mean -0.2376 Mean 0.8624 Mean 5.659 350 350 RMS 1.022 RMS 1.022 RMS 1.097 χ2 / ndf 16.49 / 21 χ2 / ndf 27.1 / 23 350 χ2 / ndf 72.44 / 33 Prob 0.7415 300 Prob 0.2519 Prob 8.853e-05 300 Constant 350.4 ± 7.8 Constant 349.5 ± 7.8 300 Constant 343.3 ± 7.7 Mean -0.2367 ± 0.0187 Mean 0.863 ± 0.019 Mean 5.673 ± 0.019 250 250 Sigma 1.025 ± 0.013 Sigma 1.027 ± 0.013 Sigma 1.045 ± 0.013 250 200 200 200 150 150 150 100 100 100 50 50 50 0 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 Figure C.9: Fitted parameters and their respective pulls for input top quark mass of 190 GeV /c2 BIBLIOGRAPHY [1] V. 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