A SEARCH FOR RESONANT Z(cid:48) PRODUCTION IN HIGH-MASS DIELECTRON FINAL STATES WITH THE ATLAS DETECTOR IN RUN-2 OF THE LARGE HADRON COLLIDER By Christopher G. Willis A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics - Doctor of Philosophy 2018 ABSTRACT A SEARCH FOR RESONANT Z(cid:48) PRODUCTION IN HIGH-MASS DIELECTRON FINAL STATES WITH THE ATLAS DETECTOR IN RUN-2 OF THE LARGE HADRON COLLIDER By Christopher G. Willis A search is performed for new resonant high-mass phenomena in the dielectron final state. √ The search uses 36.1 fb−1 of proton-proton collision data, collected at s = 13 TeV by the ATLAS experiment at the Large Hadron Collider during its 2015 and 2016 data-taking runs. The dielectron invariant mass is used as the search variable. No significant deviations from the Standard Model prediction are observed. Upper limits at the 95% credibility level are set on the cross section times branching fraction to dielectron pairs for resonant Z(cid:48) models considered in the search. Lower limits on the resonance pole mass are also presented. For the Z(cid:48) the E6-motivated Z(cid:48) coupling strength γ(cid:48). SSM, masses are excluded up to 4.5 TeV, while masses up to 4.1 TeV are excluded in χ model. Limits are also derived in the Minimal Z(cid:48) Model on the relative In addition, a series of studies are conducted in order to assess and reduce the dominant systematic uncertainty of this analysis, which arises from the imprecise knowledge of the Parton Distribution Functions in regions of very high parton x. While this uncertainty does not limit the discovery potential of the analysis presented here, it has the potential to do so in future searches. A novel approach is developed, and is shown to significantly reduce this systematic uncertainty in the high-mass search region of interest, thereby improving the discovery potential of future analyses. For my parents, Susan and Gibson Willis, my brother, Matthew Willis, and my full time partner in crime, Jone Brunelle. Thanks for always believing in me. iii ACKNOWLEDGMENTS First, I would like to give a huge thanks to my excellent adviser Chip Brock. It is hard to compress 7 years worth of gratitude into a single paragraph. Chip played a formative role in this experience, providing me with enough space to learn and work on my own, but also the necessary guidance if I got stuck, and to him I’ll forever be grateful. I would also like to thank Dan Hayden, the contact of the dilepton analysis and my surrogate adviser at CERN. Dan always took the time to answer the many questions of a new student, and continued to point me in the right direction as we got up and running with the Run-2 dilepton search. From Dan, I learned how to do a real LHC physics analysis, and by the end I actually understood how everything fit together. Thanks to the MSU HEP group for supporting me during my Ph.D., and for funding me as I went to workshops and conferences to present my work. A special thanks to those who were responsible for maintaining the MSU Tier3, especially during analysis crunch time. I would like to thank my committee members, Norman Birge, Joey Huston, Chris Wrede, and C.-P. Yuan, for accepting the job and taking to the time to read and comment on this thesis. Also, a big thanks to Brenda Wenzlick and Kim Crosslan for helping me to schedule my thesis defense, and taking care of the paperwork that followed. Finally, I’d like to thank C.-P. Yuan, Carl Schmidt, Josh Isaacson, and T.-J. Hou for sharing their PDF knowledge and theory insights for the completion of the ePump project, which formed the latter half of this thesis. iv TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Part I The Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetries of the Standard Model Chapter 2 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fundamental Particles and Interactions . . . . . . . . . . . . . . . . . . . . . 2.1.1 Matter Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Forces and Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Particle Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Lagrangian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Gauge Symmetries Standard Model Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 2.3 The Unified Electroweak Theory . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Fermion Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 The Electroweak Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Yukawa Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 The Electroweak Lagrangian Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Asymptotic Freedom and Confinement . . . . . . . . . . . . . . . . . 2.4 Quantum Chromodynamics Chapter 3 Proton-Proton Collisions . . . . . . . . . . . . . . . . . . . . . . . 3.1 Proton Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Proton-Proton Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 QCD Factorization Theorem . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Hadronic Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Parton Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Extracting PDFs from Experimental Data . . . . . . . . . . . . . . . 3.2.2 Assessing PDF Uncertainties . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 4 Two Body Scattering and The Drell-Yan Process . . . . . . . 4.1 The S-Matrix and Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . v 1 2 9 9 10 12 14 14 15 16 17 20 21 21 22 24 26 30 33 36 38 40 40 42 44 45 48 50 53 56 57 57 Scattering Amplitudes 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Two Body Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Drell-Yan Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Leading-Order Cross Section . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 PDF Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Higher-Order Corrections . . . . . . . . . . . . . . . . . . . . . . . . Chapter 5 Physics Beyond the Standard Model . . . . . . . . . . . . . . . 5.1 Limitations of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . 5.2 Theories Beyond the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Models with Extended Gauge Symmetries 5.3.1 Dilepton Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 The Sequential Standard Model . . . . . . . . . . . . . . . . . . . . . 5.3.3 The E6 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 The Minimal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Previous Experimental Limits . . . . . . . . . . . . . . . . . . . . . . . . . . 58 59 60 62 64 69 71 74 74 78 80 81 84 85 88 90 Part II The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Chapter 6 The Large Hadron Collider 94 . . . . . . . . . . . . . . . . . . . . . 94 6.1 The LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.2 CERN Accelerator Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3 LHC Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Pile-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3.2 Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.4 LHC Performance in Run-II . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.5 Experiments at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Chapter 7 The ATLAS Experiment . . . . . . . . . . . . . . . . . . . . . . . 104 7.1 Detector Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.2 Coordinate System and Event Kinematics . . . . . . . . . . . . . . . . . . . 107 7.3 Tracking System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.3.1 Pixel Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.3.2 Semiconductor Tracker . . . . . . . . . . . . . . . . . . . . . . . . . . 112 . . . . . . . . . . . . . . . . . . . . . . 112 7.3.3 Transition Radiation Tracker 7.4 Calorimetry System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.4.1 Electromagnetic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . 115 7.4.2 Hadronic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.5 Muon System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7.6 Trigger and Data-Acquisition System . . . . . . . . . . . . . . . . . . . . . . 120 Chapter 8 Electron Reconstruction in ATLAS . . . . . . . . . . . . . . . . 124 8.1 Track and Vertex Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 124 vi 8.1.1 Track Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.1.2 Vertex Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.2 Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.2.1 Electron Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.2.2 Electron Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.2.3 Electron Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.2.4 Electron Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 8.3 Energy Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.3.1 Energy Scale Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.3.2 Energy Resolution Calibration . . . . . . . . . . . . . . . . . . . . . . 137 8.4 Efficiency Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Chapter 9 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . 141 9.1 Event Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 9.1.1 Parton Shower Generators . . . . . . . . . . . . . . . . . . . . . . . . 142 . . . . . . . . . . . . . . . . . . . . . . . 143 9.1.2 Matrix Element Generators 9.1.3 Generators Used in the Dilepton Analysis . . . . . . . . . . . . . . . 143 9.2 ATLAS Simulation Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 9.2.1 Event Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 9.2.2 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 9.2.3 Digitization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.3 Simulation Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.3.1 Drell-Yan Production . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.3.2 Photon-Induced Production . . . . . . . . . . . . . . . . . . . . . . . 149 t¯t Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.3.3 9.3.4 Single-Top Quark Production . . . . . . . . . . . . . . . . . . . . . . 151 9.3.5 Diboson Production . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 9.3.6 Z(cid:48) Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.3.7 MC Sample Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.4 Corrections to Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Sample Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . 155 9.4.1 . . . . . . . . . . . . . . . . . . . 155 9.4.2 Efficiency Scale Factor Corrections 9.4.3 Energy Scale and Resolution Smearing . . . . . . . . . . . . . . . . . 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.4.4 Pile-Up Profile 9.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 k-Factor Corrections Part III The Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Chapter 10 Electron Channel Dilepton Analysis . . . . . . . . . . . . . . . . 163 10.1 Analysis Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.2 Data Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.2.1 ATLAS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.2.2 Analysis Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 10.2.3 Dataset Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 vii 10.3 Monte Carlo Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 10.4 Analysis Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 10.5 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 10.5.1 Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 10.5.2 Cutflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.5.3 Signal Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 10.6 Fake Background Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 10.6.1 The Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 10.6.2 Real- and Fake-Rate Determination . . . . . . . . . . . . . . . . . . . 182 10.6.3 Background Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . 183 10.7 Comparison of Data to the Standard Model Background Expectation . . . . 184 10.7.1 Event Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 10.7.2 Kinematic Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 186 10.8 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 10.8.1 Experimental Uncertainties . . . . . . . . . . . . . . . . . . . . . . . 193 10.8.2 Theoretical Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 10.8.4 Comment on PDF Uncertainties . . . . . . . . . . . . . . . . . . . . . 202 Chapter 11 Statistical Interpretation and Results . . . . . . . . . . . . . . . 206 11.1 Statistical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 11.2 Signal Reweighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 11.2.1 Signal Template Generation . . . . . . . . . . . . . . . . . . . . . . . 211 11.2.2 Interference Templates . . . . . . . . . . . . . . . . . . . . . . . . . . 215 11.3 Signal Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 11.3.1 Log-Likelihood Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . 219 11.3.2 BumpHunter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 11.4.1 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 11.4.2 Limits on SSM and E6 Model Z(cid:48) Bosons . . . . . . . . . . . . . . . . 229 11.4.3 Limits on Minimal Model Z(cid:48) Bosons . . . . . . . . . . . . . . . . . . 236 11.5 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 11.4 Exclusion Limits Part IV PDF Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Chapter 12 The Drell-Yan Process Revisited . . . . . . . . . . . . . . . . . . 245 12.1 The Drell-Yan Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 12.1.1 Drell-Yan Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 12.1.2 The Collins-Soper Frame . . . . . . . . . . . . . . . . . . . . . . . . . 250 12.1.3 Triple-Differential Cross Section . . . . . . . . . . . . . . . . . . . . . 252 12.1.4 Kinematic Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 254 12.1.5 Explaining the cosθ∗ Asymmetry . . . . . . . . . . . . . . . . . . . . 255 12.1.6 cosθ∗ as a Quark Flavor Enhancer . . . . . . . . . . . . . . . . . . . . 260 viii Chapter 13 PDF Error Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 263 13.1 The ePump Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 13.2 PDF Update Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 13.3 Construction of ePump Templates . . . . . . . . . . . . . . . . . . . . . . . . 267 13.3.1 Kinematic Reach at the LHC . . . . . . . . . . . . . . . . . . . . . . 268 13.3.2 Template Simulation Samples . . . . . . . . . . . . . . . . . . . . . . 269 13.3.3 Template Event Selection . . . . . . . . . . . . . . . . . . . . . . . . 270 13.3.4 Template Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 13.3.5 Pseudo-Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . 272 13.3.6 Finalized Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 13.4.1 Impact on CT14Hera2 PDFs . . . . . . . . . . . . . . . . . . . . . . . 276 13.4.2 Impact on High-Mass Drell-Yan . . . . . . . . . . . . . . . . . . . . . 282 13.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 13.4 Results of the Update Part V The Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Chapter 14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 14.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 14.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Appendix A Properties of Analysis Simulation and Data . . . . . . . . . . . . . . 295 Appendix B Additional Dielectron Cutflow Tables . . . . . . . . . . . . . . . . . . 300 Appendix C Atlantis Event Displays . . . . . . . . . . . . . . . . . . . . . . . . 302 Appendix D ePump Inputs and Supplimental Results . . . . . . . . . . . . . . . . 305 Appendix E Early Warning System . . . . . . . . . . . . . . . . . . . . . . . . . . 334 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 ix LIST OF TABLES Table 2.1: Matter content of the Standard Model. The three generations of quarks and leptons are shown along with their name, symbol, charges under each interaction, and mass [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 2.2: Force carriers and Higgs boson of the Standard Model. The name, symbol, . . . charges under each interaction,and mass of each particle is listed [1]. Table 2.3: Three generations of fermions in the Electroweak theory with their respec- tive quantum numbers. Left-handed particles are grouped into iso-doublets, while right-handed particles exist as iso-singlets. The right-handed neutri- nos are included for reference, but are not formally included as part of the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electroweak theory. Table 2.4: Neutral current couplings of Standard Model fermions. The rows spec- ify the couplings within each respective quark and lepton generation. Al- ternate definitions often are written in terms of gV = 1 2(gL + gR) and 2(gL − gR), which function as coefficients to the vector (γµ) and gA = 1 axial-vector (γµγ5) terms of the neutral current Lagrangian. . . . . . . . . Table 5.1: Standard Model fermion charges under the U (1)ψ and U (1)χ symmetries of the E6 GUT model. Once the mixing angle θE6 is specified, the Z(cid:48) Lagrangian of Eq. 5.2 is uniquely defined, and can be used to make E6- related predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 5.2: E6 models for several motivated choices of mixing angle θE6. From left to right, these are the ψ model, the neutral model, the η model, the inert model, the secluded sector model, and the χ model. . . . . . . . . . . . . Table 5.3: Several Z(cid:48) Min models for well-motivated choices of θMin and γ(cid:48). From left to right, these are the pure Z(cid:48) χ of the E6 model, and the Z(cid:48) 3R of the left-right symmetric model. . . . . . . . . . . . . . . . . . . . . B−L model, the Z(cid:48) Table 5.4: Quantum numbers associated with Standard Model fermions according to Eq. 5.12. Once the mixing angle θMin and relative coupling γ(cid:48) are specified, the Z(cid:48) Lagrangian of Eq. 5.2 is uniquely defined and can be used to make . . . . . . . . . . . . . . . . . . . . . . . . . . Minimal model predictions. x 11 12 24 36 86 87 89 89 Table 5.5: Observed limits at 95% C.L. on the mass of a Z(cid:48) boson from previous experimental searches at hadron colliders. The CMS analyses listed do not provide limits on the Z(cid:48) χ, which would otherwise be slightly higher than what was obtained for the Z(cid:48) ψ. The integrated luminosity for each analysis is rounded to nearest whole number. The (cid:96)(cid:96) channel refers searches that combine individual electron and muon channels. The analysis presented here is denoted with “*.” . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Table 6.1: Main LHC beam parameters under typical running conditions. Shown are values for the 2015 run and the 2016 run through June, along with the LHC design values [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Table 8.1: Run-2 electron identification variables [3]. . . . . . . . . . . . . . . . . . . 132 Table 9.1: Mass-dependent NLO-to-NNLO QCD and higher-order EW k-Factor cor- rections for several values of dielectron invariant mass. These corrections are applied to the enhance the nominal Drell-Yan background estimate with NNLO QCD and higher-order EW effects. . . . . . . . . . . . . . . . . . . 149 Table 9.2: Mass-dependent Photon-Induced k-Factor corrections at several values of dielectron invariant mass. These corrections are applied to the nominal the Drell-Yan background estimate to incorporate Photon-Induced effects. . . 150 Table 9.3: Mass-dependent LO-to-NNLO QCD k-Factor corrections at several values of dielectron mass mee. These corrections are applied to enhance the nom- inal Z(cid:48) signal estimate with NNLO QCD effects. . . . . . . . . . . . . . . 154 Table 9.4: A summary of MC samples used for the background and signal modeling in the electron channel of the dilepton analysis. The columns include the process of interest, the MC generation program, the matrix element order, the order of the k-Factor correction, the parton shower program, and the PDF set used [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Table 10.1: The number of events passing the electron channel event selection in the 36.1 fb−1 dataset obtained by ATLAS during the 2015+2016 physics runs. The first column indicates the absolute number of events passing selection, the middle column depicts the relative efficiency of each cut, and the last column shows the absolute efficiency. . . . . . . . . . . . . . . . . . . . . . 177 Table 10.2: The number of events passing the electron channel event selection in a sam- ple of simulated Z(cid:48) χ events with pole mass MZ(cid:48) = 3 TeV. The sample has all corrections applied that are described in Section. 9.4, and is normalized to an integrated luminosity of 36.1 fb−1 according to the procedure outline in Section 9.4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 xi Table 10.3: Expected and observed event yields in different invariant mass intervals according to the event selection of Section 10.5.1. The errors include both statistical and systematic uncertainties (described in Section 10.8). . . . . 186 Table 10.4: A Summary of the relative systematic uncertainties affecting the expected signal and background event yields at dielectron invariant masses of 2 and 4 TeV. Background values correspond to the relative change in the total number of expected events in the 2 and 4 TeV bins of the mee histogram de- picted in Figure 10.6, while signal values are calculated from changes in the FWHM of the Z(cid:48) χ signal at pole masses of MZ(cid:48) = 2 TeV and MZ(cid:48) = 4 TeV. Here N/A refers to uncertainties not considered on the signal process [5]. . 203 Table 11.1: Expected and observed 95% C.L. lower mass limits for the SSM and E6 Z(cid:48) bosons considered in the analysis. Electron, muon, and combined channel limits are shown [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Table 11.2: Observed and expected 95% C.L. lower mass limits on the Z(cid:48) Min pole mass for the three representative models considered in the text. As a result of including the interference effects and the variation of the decay width ΓZ(cid:48) through the modulation of γ(cid:48), a slightly weaker limit for the Z(cid:48) χ is obtained in the Minimal model approach as compared to the standard approach of Section 11.4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Table 11.3: Range in the observed and expected 95% C.L. lower limits on the Z(cid:48) Min pole mass for several representative values of the relative coupling strength γ(cid:48). Both the electron and muon channels are combined. . . . . . . . . . . 241 Table 11.4: Range in the observed and expected 95% C.L. upper limits on the relative Min boson . . . . . . 241 coupling strength γ(cid:48) for several representative values of the Z(cid:48) pole mass. Both the electron and muon channels are combined. √ Table 13.1: The ranges in parton-x probed in the Drell-Yan process at the s = 13 TeV LHC for various values of dilepton rapidity and four-momentum transfer Q = m(cid:96)(cid:96). The maximum values assumed for the dilepton rapidity are determined from the |y(cid:96)(cid:96)| distributions in Figure 12.4, where the Drell-Yan cross section falls to zero. The minimum and maximum values in x are calculated using Eq. 12.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Table 13.2: Impact of 3000 fb−1 update on the CT14Hera2 uv(x) and dv(x) valence and ¯u(x) and ¯d(x) sea distributions for several values of x using the stan- dard triple-differential templates at Q = 3 TeV. To be compared with Figures 13.4 and 13.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 xii Table 13.3: The estimated PDF uncertainty in several invariant mass bins for the dis- tributions shown in Figure 13.8. The current CT14Hera2 uncertainty esti- mates are shown in the first column, the result of the 3000 fb−1 update is shown next, and the 3000 fb−1 update with an additional cos θ∗ > 0 requirement on the dilepton selection is shown last. Notice the pre-update values for the central-central (CC) selection are consistent with those as- sessed in the dilepton analysis (See Fig. 10.11). . . . . . . . . . . . . . . . 283 Table 13.4: The post-update PDF uncertainty as compared to the experimental and dominant theoretical uncertainties in the electron channel of the dilepton analysis. As the PDF uncertainty will be reduced well below the current ex- perimental uncertainty, attention will be shifted to the reduction of others, such as the “PDFChoice” uncertainty, improving the discovery potential of future iterations of the dilepton analysis. . . . . . . . . . . . . . . . . . . 285 Table A.1: The NLO Powheg+Pythia8 Drell-Yan MC samples used in the modeling of the electron channel [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Table A.2: The LO Pythia8 Z(cid:48) reweighting procedure [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 χ MC samples used to validate the electron channel Table A.3: The LO Pythia8 Drell-Yan MC samples used to construct reweighted Z(cid:48) signal templates for the electron channel search and limit setting of the dilepton analysis [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Table A.4: The NLO Powheg+Pythia6 tt and Single-Top MC samples used in the modeling of the electron channel in the dilepton analysis. A generator-level filtering efficiency of 0.10534 is accounted for in the t¯t sample cross section estimate [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Table A.5: The NLO Sherpa W W , W Z, and ZZ Diboson MC samples used in the modeling of the electron channel in the dilepton analysis [4]. . . . . . . . . 298 Table A.6: LHC Datasets used in the dilepton analysis [4]. . . . . . . . . . . . . . . . 299 Table B.1: The number of simulated Drell-Yan events passing the electron channel event selection of the dilepton analysis. . . . . . . . . . . . . . . . . . . . 300 Table B.2: The number of simulated Top (t¯t + W t) events passing the electron channel event selection of the dilepton analysis. . . . . . . . . . . . . . . . . . . . 301 Table B.3: The number of simulated Diboson (W W + W Z + ZZ) events passing the electron channel event selection of the dilepton analysis. . . . . . . . . . . 301 xiii Table D.1: Impact of 300 fb−1 update on the CT14Hera2 uv(x) and dv(x) valence and ¯u(x) and ¯d(x) sea distributions for several values of x using the standard triple-differential templates at Q = 3 TeV. To be compared with Fig.s D.25 and D.26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 Table D.2: The estimated PDF uncertainty in several invariant mass bins for the dis- tributions shown in Figure D.27. The current CT14Hera2 uncertainty es- timates are shown in the first column, the result of the 300 fb−1 update is shown next, and the 300 fb−1 update with an additional cos θ∗ > 0 requirement on the dilepton selection is shown last. . . . . . . . . . . . . . 332 xiv LIST OF FIGURES Figure 2.1: The QED interaction vertex. The solid lines represent the motion of the charged leptons and the curved line represent the propagation of the photon. The strength of the interaction is determined by the magnitude of electric charge Q in units of fundamental charge e and is labeled at the vertex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 2.2: The shape of the Higgs potential V (Φ) of Eq. 2.19 for the choice of µ2 > 0. The potential V (Φ) obtains a minimum at the non-zero value of Φ†Φ = µ2/2λ ≡ v2/2 [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 2.3: Fundamental interaction vertices with the Higgs boson in the EW the- ory: the Higgs-fermion interaction (a), and the three-point (b) and four- point (c) gauge interactions. . . . . . . . . . . . . . . . . . . . . . . . . . Figure 2.4: Fundamental three-point (a) and four-point (b) Higgs boson self-interaction vertices in the EW theory. . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 2.5: Fundamental three-point (a) and four-point (b)-(c) gauge boson vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . in the EW theory. Figure 2.6: Fundamental interaction vertices with quarks in the EW theory: the charged current interactions (a)-(b) and neutral current interaction (c). . Figure 2.7: Fundamental interaction vertices with leptons in the EW theory: the charged current interactions (a)-(b) and neutral current interactions (c). 19 27 29 30 33 34 35 Figure 2.8: Fundamental interaction vertices in QCD: the quark-gluon interaction (a), the three-point gluon interation (b), and the four-point gluon interation (c). 37 Figure 2.9: The running of the αS coupling constant in QCD as a function of the energy scale Q with a variety of experimental measurements overlaid [1]. Figure 3.1: Substructure of proton depicting interactions between its partonic con- stituents: valence quarks, sea quarks, and gluon [7]. . . . . . . . . . . . . Figure 3.2: Diagram of a typical hard scattering interaction between partons at the LHC [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 41 42 xv Figure 3.3: An inelastic collision at the LHC. The dark red blob depicts the hard process, with its associated red lines the resulting final state radiation. Beam remnants are light blue, and initial state radiation is depicted by the dark blue lines. The purple blob is a secondary hard scatter along with its own final state radiation in purple. The light green blobs de- pict hadronization of any color-charged partons emanating from the hard scatter or the ISR/FSR showering. Subsequent dark green blobs corre- sponds to hadronic decays of unstable particles and yellow lines indicate the presence of electromagnetic radiation [9]. . . . . . . . . . . . . . . . . Figure 3.4: CT14NNLO PDFs provided by the CTEQ collaboration. Depicted are gluon and all quark flavors as a function of x, evaluated at the scales of Q = 2 GeV (a) and Q = 100 GeV (b) [10]. . . . . . . . . . . . . . . . . . . . . Figure 3.5: Parton kinematics for the regions in (x, Q2) probed by the LHC and pre- vious DIS and fixed target experiments. The scale Q2 of a process is often identified with the mass M of a produced resonance. This is indicated by . . . four discrete values on the plot, for the LHC with √ s = 14 TeV [8]. Figure 3.6: PDF uncertainties associated with the g(x), u(x), d(x), s(x), uv(x), dv(x), ¯u(x), and ¯d(x) distributions in the CT14NNLO and CT14Hera2 PDF sets evaluated at a scale of Q = 3 TeV. . . . . . . . . . . . . . . . . . . . . . 46 49 52 54 Figure 4.1: Two body scattering as described by Eq. 4.5 [11]. . . . . . . . . . . . . . 61 Figure 4.2: The Drell-Yan process as observed at the LHC [12]. . . . . . . . . . . . . 62 Figure 4.3: The LO Feynman Diagram for Drell-Yan production mediated by a virtual photon γ∗ (a), and on-shell Z boson (b). The propagator and vertex factors of each diagram are depicted. The initial- and final-state particles are represented by the Dirac spinors u and v, which label their momenta and spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 4.4: The invariant mass distribution of dilepton pairs m(cid:96)(cid:96) produced via the s = 13 TeV. The MC generator ResBos, con- Drell-Yan mechanism at figured with the CT14Hera2 PDF set, was used for the prediction. The PDF error, calculated with the symmetrized form of Eq. 3.10, is indicated in the sub-panel, and steadily increases with dilepton mass. . . . . . . . √ Figure 4.5: Example higher-order contributions to the Drell-Yan process: NLO initial- state gluon radiation qq → γ∗/Z +g (a), NLO initial-state quark radiation qg → γ∗/Z + q (b), NLO virtual gluon correction qq → γ∗/Z (c), NLO final-state photon radiation qq → γ∗/Z + γ (d). . . . . . . . . . . . . . . 66 70 72 xvi Figure 5.1: Leading Order Feynman diagram for Z(cid:48) production at the LHC. The prop- agator and vertex factors are depicted, and the initial- and final-state fermions are represented by the Dirac spinors u and v, which label their momenta and spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.2: Predicted Z(cid:48) SSM event yields for MZ(cid:48) = 1 TeV (a) and MZ(cid:48) = 1 TeV (b). The vertical axis displays the number of predicted Z(cid:48) events per given √ s = 13 TeV unit of integrated luminosity, as would be observed at the LHC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.3: Predicted Z(cid:48) event yields for various choices of θE6 in the E6 model with MZ(cid:48) = 1 TeV (a) and MZ(cid:48) = 1 TeV (b). Starting from the Z(cid:48) ψ value of θE6 = 0, E6 models will exhibit increasing cross sections and decay widths, until the Z(cid:48) χ value of θE6 = π/2 is reached, at which point a decrease is observed. All E6 models fall within the range defined by Z(cid:48) ψ χ. The vertical axis displays the number of predicted Z(cid:48) events per and Z(cid:48) √ given unit of integrated luminosity, as would be observed at the s = 13 TeV LHC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5.4: Predicted Z(cid:48) Min event yields for MZ(cid:48) = 3 TeV according to Eq. 5.12. In (a), θMin is held constant for several distinct values of γ(cid:48). Similarly, in (b), γ(cid:48) is held constant while θMin is varied. The vertical axis displays the number of predicted Z(cid:48) Min events per given unit of integrated luminosity, s = 13 TeV LHC. . . . . . . . . . . . . . . as would be observed at the √ Figure 6.1: Illustration of the 27 km long LHC, its four main experiments (ALICE, ATLAS, CMS, and LHCb), the pre-accelerators used in staging proton s = 13 TeV, and the region sur- acceleration to reach a CM energy of . . . . . . . . . . . . . . . . . . . . . rounding Geneva, Switzerland [13]. √ Figure 6.2: Illustration of the CERN accelerator complex. Proton acceleration is achieved through the following stages: Linac 2, PSB, SP, SPS, and LHC. Accelerators used in other CERN experiments are shown as well [14]. . . Figure 6.3: Superconducting dipole magnets as seen in the LHC tunnel [15]. . . . . . Figure 6.4: Depiction of a Z → µ+µ− event as recorded by the ATLAS detector during Run-1 of the LHC, along with the 25 simultaneous proton-proton interactions during this particular bunch crossing. The two muons orig- inate from a common vertex, due to the decay of the Z boson, and are highlighted in yellow [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . xvii 82 85 87 90 95 97 98 99 Figure 6.5: Cumulative integrated luminosity obtained per week as delivered by the LHC and recorded by the ATLAS detector during the 2015 (a) and 2016 (b) data-taking runs [17, 18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Figure 7.1: Individual particles are identified and measured in the ATLAS detector according to their characteristic signatures left in each detector subsystem. Solid lines represent trajectories of charged particles, while dashed lines indicate neutral particles. Showers are depicted for particles that interact with the dense detector material [19]. . . . . . . . . . . . . . . . . . . . . 105 Figure 7.2: Cut-away view of the ATLAS detector with individual subsystems la- beled [20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Figure 7.3: Cut-away view of the ATLAS Inner Detector. Subsystem components are shown in both the barrel and end-cap regions [21]. . . . . . . . . . . . . . 110 Figure 7.4: Schematic of the barrel region of ATLAS Inner Detector, as crossed by a charged particle [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Figure 7.5: Cut-away view of the ATLAS calorimetry system. Depicted are the Elec- tromagnetic and Hadronic sampling calorimeters, both of which consist of individual barrel and end-cap modules. The forward components are closest to the beam line [22]. . . . . . . . . . . . . . . . . . . . . . . . . . 114 Figure 7.6: Depiction of an ECAL barrel module in the precision region of the ATLAS detector. The accordion shape is shown, along with the three independent sampling layers. The granularity in η and φ of the cells of each layer is labeled [23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Figure 7.7: Sketch of an ATLAS tile calorimeter module [24]. . . . . . . . . . . . . . 117 Figure 7.8: Schematic of the ATLAS detector forward calorimeter systems. The three FCAL modules are depicted [25]. . . . . . . . . . . . . . . . . . . . . . . 119 Figure 7.9: Cut-away view of the ATLAS Muon System [26]. . . . . . . . . . . . . . 119 Figure 7.10: The production cross sections and corresponding event rates for a variety of physics processes observable at the LHC. BSM processes, such as Z(cid:48) production, are expected to occur at extremely small rates compared to the nominalrate of proton-proton inelastic collisions indicated by σtot. In this regard, a trigger is essential to correctlyidentify and select such events for offline analysis [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Figure 8.1: Depiction of ATLAS detector systems used to perform electron recon- struction and identification [3]. . . . . . . . . . . . . . . . . . . . . . . . 125 xviii Figure 8.2: Illustration of the transverse impact parameter d0 (a) and longitudinal impact parameter z0 (b) [27]. . . . . . . . . . . . . . . . . . . . . . . . . 127 Figure 8.3: Illustration comparing the Etcone and topoEtcone isolation methods. The grid corresponds to the η − φ granularity of the middle layer of the EM calorimeter. The electron candidate cluster is mostly contained in the central 5 × 7 window. An isolation cone with radius ∆R = 0.4 is drawn around the candidate. In the Etcone method, all cells (yellow) within the radius ∆R are used, while in the topoEtcone method, only topoclusters (red), whose barycenters fall within ∆R, are used, effectively reducing noise from pile-up [28]. . . . . . . . . . . . . . . . . . . . . . . . 135 Figure 9.1: The LO Feynman diagrams for Photon-Induced Drell-Yan production. Both t-channel (a) and u-channel (b) modes are shown. . . . . . . . . . . 150 Figure 9.2: The LO Feynman diagrams for t¯t production. The gluon initiated s- channel (a) and t-channel (b) modes are depicted, along with quark- antiquark initiated production (c). . . . . . . . . . . . . . . . . . . . . . 150 Figure 9.3: The LO Feynman diagrams (a) and (b) for W t-channel Single-Top pro- duction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Figure 9.4: The LO Feynman diagrams for Diboson production. Depicted are the s-channel (a), t-channel (b), and u-channel (c) production modes. The W Z and ZZ processes can proceed through all modes, while the ZZ process can produced only through the t- and u-channels. In each diagram Vi = W, Z or γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Figure 9.5: Electron efficiency scale factor corrections as a function of dielectron in- variant mass mee. Shown are the scale factors applied to correct the electron reconstruction, identification, isolation, and trigger efficiencies. . 156 Figure 9.6: Invariant Mass distribution for the MZ(cid:48) = 3 TeV Z(cid:48) χ predicted by the simulation with and without the application of the energy scale correction and resolution smearing. The effect is to both shift and broaden the shape of the Z(cid:48) peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Figure 9.7: Mean number of interactions per bunch crossing in the Z → e+e− MC sample with and without the pile-up correction as compared to data. The effect of reweighting is to match the pile-up conditions in data. . . . . . . 159 xix Figure 9.8: The four mass-dependent k-Factor corrections applied in the dilepton anal- ysis. The NLO-to-NNLO QCD correction, LO-to-NLO EW correction, and the LO Photon-Induced QED correction are applied to enhance the description of the Drell-Yan spectrum, while the LO-to-NNLO QCD cor- rection is applied to the Z(cid:48) signal. . . . . . . . . . . . . . . . . . . . . . . 160 Figure 10.1: Day-by-day data-taking efficiency of the ATLAS detector during in the 2015 (a) and 2016 (b) runs. The efficiency over the course of each run was above 92%. Empty bins correspond to days during which the LHC did not deliver stable beams [18]. . . . . . . . . . . . . . . . . . . . . . . 167 Figure 10.2: Average number of proton-proton interactions per bunch crossing during the 2015 and 2016 data-taking periods. The data shown correspond to periods with stable beams delivered to the ATLAS detector [18]. . . . . . 168 Figure 10.3: The Z(cid:48) signal efficiency, according to the dielectron selection defined in Section 10.5.1, as a function of Z(cid:48) χ pole mass. The selection is maximally efficient at high mass, the primary target of the search. The quick “turn- on” at low mass and slow “turn-off” at high mass are due to the electron kinematics at each of these scales. . . . . . . . . . . . . . . . . . . . . . . 179 Figure 10.4: Z(cid:48) χ signals post-selection for several characteristic mass points. A low- mass tail begins to develop as the kinematic limit is reached. The number of events within specific invariant mass intervals is displayed in Table 10.3 for the 4 and 5 TeV pole mass Z(cid:48) χ. . . . . . . . . . . . . . . . . . . . . . 179 Figure 10.5: The data-driven fake background estimate used in the dielectron channel of the dilepton analysis. The estimate is obtained with the matrix method, following the technique outlined in Section 10.6 [4]. . . . . . . . . . . . . 185 Figure 10.6: Dielectron invariant mass distribution for candidate dielectron events se- lected by the full event selection. The stacked sum of Standard Model backgrounds is normalized to 36.1 fb−1 and compared to data. Three Z(cid:48) χ signals are included for reference, overlaid with the Standard Model expectation. No obvious signal-like excess is present. The lower-panel sub-plot shows the “post-fit” result obtained from the statistical analysis. 189 Figure 10.7: Leading (a) and subleading (b) electron ET distributions in the dielectron invariant mass region mee > 120 GeV, after the full event selection. The stacked sum of Standard Model backgrounds is normalized to 36.1 fb−1 and compared to data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 xx Figure 10.8: Leading (a) and subleading (b) electron η distributions in the dielectron invariant mass region mee > 120 GeV, after the full event selection. The stacked sum of Standard Model backgrounds normalized to 36.1 fb−1 and compared to data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Figure 10.9: Leading (a) and subleading (b) electron φ distributions in the dielectron invariant mass region mee > 120 GeV, after the full event selection. The stacked sum of all Standard Model backgrounds normalized to 36.1 fb−1 and compared to data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Figure 10.10:Experimental systematic uncertainties in the electron channel of the dilep- ton analysis. Shown are effects that shift the Standard Model event yield by greater than 1% anywhere in the dielectron invariant mass search spec- trum. The largest experimental uncertainty is due to the W +jets & QCD Multi-jet fake background estimate, which rises to 10% by mee = 2 TeV. See Section 10.8.1 for explanations of the source of each systematic un- certainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Figure 10.11:Theoretical systematic uncertainties in the electron channel of the dilep- ton analysis (a). Shown are effects that shift the Standard Model event yield by greater than 1% anywhere in the dielectron invariant mass search spectrum. The largest uncertainty is due to the PDF Variation uncer- tainty, which is also shown decomposed into its individual eigenvector variations (b). See Section 10.8.2 for an explanation of the source of each systematic uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Figure 10.12:Drell-Yan cross section uncertainty as a function of invariant mass as predicted by several modern PDF sets [4]. . . . . . . . . . . . . . . . . . 204 Figure 11.1: Validation of the signal reweighting procedure. Reweighted signal tem- plates are compared to their dedicated Pythia counterparts at pole masses of 2 (a) and 3 (b) TeV for the Z(cid:48) χ of the E6 model. The effect of inter- ference is not included in the dedicated generation, and is similarly not included in the signal reweighting procedure for this comparison. Per- cent level agreement is seen in the peak region, and the signal shape is effectively captured across the entire invariant mass spectrum. . . . . . . 214 Figure 11.2: Example Minimal model signal template with fixed mixing angle θMin = 0 and Z(cid:48) pole mass MZ(cid:48) = 3 TeV. The template is a function of dielectron mass mee and the fourth power of the Minimal model relative coupling strength γ(cid:48) = gZ(cid:48)/gZ , and is generated from reweighted Pythia Drell- Yan events after passing the full electron channel event selection. . . . . 217 xxi Figure 11.3: Sampling distribution f (q0|0) for the LLR test statistic q0. The distribu- tion is generated from background-only pseudo-experiments and is used to extract the p-value associated with individual Z(cid:48) signal hypotheses. The distribution follows a χ2 function with one degree of freedom according to Wilk’s theorem [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Figure 11.4: The resulting p0 scan in the electron channel search region, derived using Zχ signal templates with pole masses between 0.15 and 3.0 TeV. The local p-values are depicted as the black curve, where the results for individual mass points have been interpolated. The corresponding local and global significance levels are depicted as gray and red dashed lines, respectively. The lack of sufficient statistics at high mass can result in an ill defined PDF and negative probabilities. Therefore, above 1.5 TeV the so-called “capped” p-value is used, while the “un-capped” value is used below. See Ref. [4] for more details. . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Figure 11.5: Results of BumpHunter test for the search range mee ∈ [0.12, 3.00] TeV. Depicted are the local p0 of each interval searched (a), and the global p0 (b), which is extracted from an ensemble of background-only psuedo- experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 Figure 11.6: The dielectron mass distribution, showing the observed data, the Standard Model background prediction, and the corresponding bin-by-bin local sig- nificance. The extremely fine search binning is clearly visible in this plot. The interval identified by the pair of blue lines corresponds to the most significant local excess [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Figure 11.7: The 95% C.L. upper limits on the Z(cid:48) production cross section times branching fraction for the electron channel (a) and the combined elec- tron plus muon channel (b) as a function of Z(cid:48) pole mass. The expected limit is depicted as a black dashed line, with yellow and green bands in- dicating ±1σ and ±2σ variations from the median value. The observed limit is shown as a solid red line and the above region is excluded at 95% C.L. Signal theoretical uncertainties are shown on the SSM cross section prediction as a gray band for reference, but are not included as part of the limit calculation [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Figure 11.8: The LO cross section times branching fraction predictions for Z(cid:48) produc- tion in the SSM and E6 models considered in the analysis. Values were obtained using Pythia8 configured with the NNPDF2.3 PDF set, and are plotted as a function of Z(cid:48) pole mass for s = 13 TeV [5]. √ . . . . . . . . 234 xxii Figure 11.9: Ratio of the observed 95% C.L. upper limit to the theoretical prediction SSM cross section times branching fraction as a function of Z(cid:48) pole for the Z(cid:48) mass for the combined electron and muon channels. Previous iterations s = 7, 8, and 13 TeV, are of the ATLAS dilepton search, performed at compared to the current result obtained here, which is shown in black. The region above each line is excluded at 95% C.L. [5]. √ . . . . . . . . . . 235 Figure 11.10:Observed and expected 95% C.L. upper limits on the relative coupling strength γ(cid:48) as a function of the Z(cid:48) Min pole mass for the electron channel (a) and the combined electron plus muon channel (b). Three representative limit curves are shown, for specific values of the mixing angle θMin, which correspond to the Z(cid:48) B−L models at specific values of γ(cid:48). All Minimal model limits fit within the gray band, which depend on the choice of θMin ∈ [0, π]. The edges of the band are defined by the two models with the minimum and maximum Z(cid:48) Min cross sections. The region above each curve is excluded at 95% C.L. . . . . . . . . . . . . . . . . . . . . . . . . 239 3R, and Z(cid:48) χ, Z(cid:48) Figure 11.11:Observed and expected 95% C.L. upper limits on the relative coupling strength γ(cid:48) as a function of the mixing angle θMin for the electron chan- nel (a) and the combined electron and muon channels (b). Limit curves are set for ten representative values of the Z(cid:48) Min pole mass MZ(cid:48). The struc- ture observed in the limit curves for a given pole mass as a function of θMin originates from the changing shape of the resonance. The resonance with the smallest (largest) width is obtained at θMin (cid:39) 1.0 (θMin (cid:39) 2.2), which define the edges of the gray band in Figure 11.10. The region above each curve is excluded at 95% C.L. . . . . . . . . . . . . . . . . . . . . . 240 Figure 12.1: The Drell-Yan process initiated by a quark-antiquark pair as observed at the LHC [29]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 √ Figure 12.2: The transferred momentum squared Q2 versus the parton momentum frac- s = 7 TeV. The regions probed by previous DIS, fixed-target, tion x at and collider-based experiments are labeled. The accessible regions to the ATLAS and CMS experiments are indicated by the solid blue line, and the shaded region within denotes measurements using the central region of each detector. The high-mass search region of the dilepton analysis can be considered as beyond mee = 1 TeV, where two high-x partons (x (cid:38) 0.1) are required to initiate the Drell-Yan process [30]. . . . . . . . . . . . . . 249 Figure 12.3: The Collins-Soper reference frame. The ˆz axis bisects the angle between the quark momentum p1 and the negative of the antiquark momentum −p2. The ˆy axis is defined perpendicular to the plane spanned by the par- ton momenta, and ˆx axis completes the right handed coordinate system. The polar angle θ∗ is measured from the ˆz axis in this frame [31]. . . . . 251 xxiii Figure 12.4: The absolute dilepton rapidity |y(cid:96)(cid:96)| in several slices in dilepton invariant mass m(cid:96)(cid:96) ranging from 40 GeV to 1 TeV ((a)-(f)). The relative fractions of the u-type and d-type Drell-Yan sub-processes are shown to illustrate the partonic flavor decomposition in this observable. . . . . . . . . . . . 256 Figure 12.5: The lepton angular distribution cos θ∗ in several slices of dilepton invariant mass m(cid:96)(cid:96) ranging from 40 GeV to 1 TeV ((a)-(f)). The relative fractions of u-type and d-type sub-processes are shown as well, which exhibit a strong angular dependency, especially at high mass. . . . . . . . . . . . . 257 Figure 12.6: Drell-Yan asymmetry coefficients of Eq. 12.12. The parameter C0 q ((a)) describes the magnitude of the symmetric component of the Drell-Yan cross section, while C1 q ((b)) describes the magnitude of the asymmetric component. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 q and C1 . . . 260 Figure 12.7: Ratio of Drell-Yan asymmetry coefficients C0 Figure 12.8: The dilepton invariant mass m(cid:96)(cid:96) as selected by forward (cos θ∗ > 0) or backward (cos θ∗ < 0) events in the CS frame ((a)) and the corresponding PDF uncertainty associated with each selection ((b)). Events arising from the forward-forward lepton topology are not included. As with the other kinematic distributions, the CT14Hera2 PDF set is used. The selection has the effect of modulating the PDF systematic uncertainty. q of Eq. 12.12. . . . . . . 261 Figure 13.1: ePump requires two sets of inputs to perform an update: the first consists of a set of binned signal templates, each of which is comprised of a nominal theory prediction and n eigenvector variations (n = 56 in CT14Hera2), and the second consists of an analogous set of binned data (or pseudo-data) templates, along with their statistical uncertainties. The observed reduc- tion in PDF uncertainty associated with the updated PDF set depends on the statistical precision of the data. Datasets with high statistical precision are better able to constrain the PDF uncertainty. . . . . . . . . 264 Figure 13.2: The triple-differential cross section of Eq. 13.2 in the range of m(cid:96)(cid:96) ∈ [381, 525] GeV and 0 < |y(cid:96)(cid:96)| < 1.8, scaled to 3000 fb−1. The specific kinematic region is labeled in each plot (a)-(i). The ResBos prediction is indicated by the solid line, and pseudo-data are shown as solid black markers. In each plot, the sub-panel shows the ratio between the theoret- ical prediction and the measurement. Error bars represent the statistical uncertainty on the pseudo-data and the solid band represents the PDF uncertainty on the dilepton event yield. . . . . . . . . . . . . . . . . . . . 274 xxiv Figure 13.3: The triple-differential cross section of Eq. 13.2 in the range of m(cid:96)(cid:96) ∈ [381, 525] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6, scaled to 3000 fb−1. The specific kinematic region is labeled in each plot (a)-(i). The ResBos prediction is indicated by the solid line, and pseudo-data are shown as solid black markers. In each plot, the sub-panel shows the ratio between the theoret- ical prediction and the measurement. Error bars represent the statistical uncertainty on the pseudo-data and the solid band represents the PDF uncertainty on the dilepton event yield. . . . . . . . . . . . . . . . . . . . 275 Figure 13.4: Impact of the 3000 fb−1 update on the CT14Hera2 uv(x) (a) and dv(x) (b) PDF uncertainties assuming single-differential mass-only templates: “Up- date1D,” double-differential mass and rapidity templates: “Update2D,” and the standard triple-differential mass, rapidity, and cos θ∗ templates: “Up- date3D”at Q = 3 TeV. The utilization of the cos θ∗ dimension in the update provides for a substantial reduction in the PDF uncertainty. . . . 278 Figure 13.5: Impact of the 3000 fb−1 update on the CT14Hera2 ¯u(x) (a) and ¯d(x) (b) PDF uncertainties assuming single-differential mass-only templates: “Up- date1D,” double-differential mass and rapidity templates: “Update2D,” and the standard triple-differential mass, rapidity, and cos θ∗ templates: “Up- date3D”at Q = 3 TeV. The utilization of the cos θ∗ dimension in the update provides for a substantial reduction in the PDF uncertainty. . . . 279 Figure 13.6: Impact of the 3000 fb−1 update on the CT14Hera2 uv(x) (a) and dv(x) (b) distributions using the double-differential mass and rapidity templates: “Update2D Rapidity,” and double-differential mass and cos θ∗ templates: “Update2D Cosine” at Q = 3 TeV. . . . . . . . . . . . . . . . . . . . . . 280 Figure 13.7: Impact of the 3000 fb−1 update on the CT14Hera2 ¯u(x) (a) and ¯d(x) (b) distributions using the double-differential mass and rapidity templates: “Update2D Rapidity,” and double-differential mass and cos θ∗ templates: “Update2D Cosine” at Q = 3 TeV. . . . . . . . . . . . . . . . . . . . . . 281 Figure 13.8: The dilepton invariant mass distribution for central-central dilepton events (a), as would be selected by the dilepton event selection of Section 10.5, and with an additional cos θ∗ > 0 requirement added to the selection (b). The ratio sub-plot depicts the CT14Hera2 PDF uncertainty before and after the 3000 fb−1 update. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Figure 14.1: Expected 95% C.L. upper limits on σB with 300 fb−1 of data (a) and 3000 fb−1 of data (b). Theory curves are shown for the Z(cid:48) LR, and Z(cid:48) SSM models. Mass limits can be extracted from the intersection between the expected limit and the theory prediction. The region above each curve is excluded at 95% C.L. [32]. . . . . . . . . . . . . . . . . . . . . . . . . . 292 ψ, Z(cid:48) xxv Figure C.1: The Atlantis event display for the highest dielectron invariant mass event selected in the 2015 analysis. The “leading” electron has an ET of 373 GeV and an η of -1.03. The “subleading” electron has an ET of 246 GeV and an η of 2.45. The invariant mass of the dielectron pair is 1.78 TeV [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Figure C.2: The Atlantis event display for the highest dielectron invariant mass event selected in the 2016 analysis. The “leading” electron has an ET of 841 GeV, an η of -1.26, and a φ of -1.15. The “subleading” electron has an ET of 655 GeV, an η of 1.32, and a φ of 1.83. The invariant mass of the dielectron pair is 2.90 TeV [5]. . . . . . . . . . . . . . . . . . . . . . 304 Figure D.1: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [40, 66] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . . 306 Figure D.2: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [40, 66] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . . 307 Figure D.3: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [66, 80] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . . 308 Figure D.4: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [66, 80] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . . 309 Figure D.5: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [80, 91] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . . 310 Figure D.6: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [80, 91] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . . 311 Figure D.7: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [91, 102] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . . 312 Figure D.8: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [91, 102] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . . 313 Figure D.9: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [102, 116] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . 314 Figure D.10: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [102, 116] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . 315 Figure D.11: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [116, 145] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . 316 xxvi Figure D.12: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [116, 145] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . 317 Figure D.13: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [145, 200] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . 318 Figure D.14: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [145, 200] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . 319 Figure D.15: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [200, 275] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . 320 Figure D.16: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [200, 275] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . 321 Figure D.17: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [275, 381] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . 322 Figure D.18: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [275, 381] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . 323 Figure D.19: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [381, 525] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . 324 Figure D.20: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [381, 525] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . 325 Figure D.21: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [525, 725] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . 326 Figure D.22: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [525, 725] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . 327 Figure D.23: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [725, 1000] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. . . . . . . . . . . . . . . . . 328 Figure D.24: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [725, 1000] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. . . . . . . . . . . . . . . . 329 Figure D.25: Impact of the 300 fb−1 update on the CT14Hera2 uv(x) (a) and dv(x) (b) PDF uncertainties assuming single-differential mass-only templates: “Up- date1D,” double-differential mass and rapidity templates: “Update2D,” and the standard triple-differential mass, rapidity, and cos θ∗ templates: “Up- date3D”at Q = 3 TeV. The utilization of the cos θ∗ dimension in the update provides for a substantial reduction in the PDF uncertainty. . . . 330 xxvii Figure D.26: Impact of the 300 fb−1 update on the CT14Hera2 ¯u(x) (a) and ¯d(x) (b) PDF uncertainties assuming single-differential mass-only templates: “Up- date1D,” double-differential mass and rapidity templates: “Update2D,” and the standard triple-differential mass, rapidity, and cos θ∗ templates: “Up- date3D”at Q = 3 TeV. The utilization of the cos θ∗ dimension in the update provides for a substantial reduction in the PDF uncertainty. . . . 331 Figure D.27: The dilepton invariant mass distribution for central-central dilepton events (a), as would be selected by the dilepton event selection of Section 10.5, and with an additional cos θ∗ > 0 requirement added to the selection (b). The ratio sub-plot depicts the CT14Hera2 PDF uncertainty before and after the 300 fb−1 update. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Figure E.1: Run-1 dielectron invariant mass distribution (a) and electron+MET trans- verse mass distribution (b), the primary search variables in the dilepton and lepton+MET analyses, respectively [33, 34]. . . . . . . . . . . . . . . 336 Figure E.2: Run-1 95% C.L. lower mass limits in the dilepton analysis (a) and lep- ton+MET analysis (b) [33, 34]. . . . . . . . . . . . . . . . . . . . . . . . 337 Figure E.3: The EWS applied to the full dataset collected during Run-1 of the LHC. The fit is performed in the low pT region of [125, 300] GeV and extrapo- lated to the monitoring region to assess the high-pT data. Three signals are overlaid using simulation samples: a 1.5 TeV Z(cid:48) SSM, and a 3.0 TeV W(cid:48) SSM, each of which displays a characteristic Jacobian peak in the pT spectrum. SSM, a 2.5 TeV Z(cid:48) . . . . . . . . . . . . . . . . . . . . . . . . . . 340 Figure E.4: Operation of the Run-2 EWS in the electron channel: May to June 2016 (a)-(f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 Figure E.5: Operation of the Run-2 EWS in the electron channel: June to July 2016 (a)-(f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Figure E.6: Operation of the Run-2 EWS in the muon channel: May to June 2016 (a)-(f).344 Figure E.7: Operation of the Run-2 EWS in the muon channel: June to July 2016 (a)-(f).345 xxviii Part I The Theory 1 Chapter 1 Introduction The idea that matter is composed of a finite number of elementary constituents is very old and dates back to at least the 5th century BCE in Ancient Greece. It was during this period that the atomic hypothesis was first formulated Democritus and his mentor Leucippus, that all of matter was composed of small, indivisible particles called atoms.1 However, this was one of many competing ideas about the fundamental nature of matter at the time , and it would take another 2400 years for modern experiments to test for and later establish the existence of atoms. In the 19th century, John Dalton proposed the modern form of the atomic hypothesis, that each chemical element found in nature was composed of indivisible atoms, which can combine, separate, and rearrange in chemical reactions, and form the true building blocks of matter. Early into the 20th century, however, it was realized that atoms themselves were not indivisible as originally thought; rather they comprised even more fundamental particles: electrons, protons, and neutrons. J.J. Thompson discovered the electron in 1897 through his study of cathode rays [35] and Ernest Rutherford discovered the atomic nucleus in 1911 and later identified the pro- ton through early α particle scattering experiments [36]. The following discovery of the neutron [37] by James Chadwick in 1932 seemed to complete the picture of atoms being 1The word “atom” derives from the Ancient Greek ατ oµoς, atomos, i.e., “uncuttable,” or “indivisible.” 2 composed of three “elementary particles.” Over the subsequent decades, hundreds of new particles were discovered and each was thought to be elementary in its own right, collectively dubbed the “Particle Zoo.” Attempts at a systematic classification lead to the development of the quark model [38, 39], which proposed that the known strongly interacting particles were actually bound states of truly elementary quarks. The prediction and subsequent discovery of the Ω− baryon in 1964 at Brookhaven [40] was one of the early successes of the quark model. In the late 1960s, deep inelastic scattering experiments at the Stanford Linear Accelerator (SLAC) [41, 42], where highly energetic electrons were scattered off of nuclei, showed that the proton had an internal structure, providing strong evidence for existence of quarks. As technology progressed, increasingly powerful particle accelerators were developed, allowing for ever higher energies to be achieved and ever smaller sub-nuclear distances to be probed. This led to the production and discovery of more new and exotic particles, and, driven by the interplay between experimental observations and theoretical predictions, the Standard Model of Particle Physics began to take shape. The gluon, the force mediator of strong interaction, was discovered in 1979 with the Positron-Elektron Tandem Ring Anlage (PETRA) at the Deutsches Elektronen-Synchrotron (DESY) [43], and four years later in 1983, the W± and Z bosons, the mediators of the weak interaction, were discovered with the Super-Proton-Antiproton Synchrotron (Sp¯pS) at the Conseil Europ´een pour la Recherche Nucl´eaire (CERN) [44, 45, 46, 47]. The top quark was discovered in 1995 by the DØ and CDF collaborations at the Tevatron [48, 49], the tau neutrino in 2000 by the DONUT collaboration [50], and the Higgs boson in 2012 by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) at CERN [51, 52], all 3 of which were predicted by the Standard Model. The Standard Model has been precisely tested over the course of several decades, and is able to describe all current experimental observations in particle physics, making it an extraordinarily successful theory. It explains the structure of matter, classifies the known elementary particles, and accounts for three of four fundamental forces of nature.2 The precision and scope of Standard Model are indeed astonishing; however, despite its enormous experimental success, there are several classes of observed phenomena that it cannot obviously explain. These include the existence and nature of dark matter and dark energy, the baryon asymmetry of the universe, and the fundamental nature of neutrinos, to name a few. The Standard Model also suffers from several theoretical, or conceptual, prob- lems, such as how to consistently incorporate gravity into the theory. Taken together, these observations suggest that the Standard Model is incomplete and some kind of extension or modification is required. If the 20th century was about establishing foundational principles, the 21st century is about discovering what lies beyond. Today, searches are performed for new particles and interactions not predicted by the Standard Model, but by theories beyond that attempt to address the limitations mentioned above by extending it in various ways. Since the Standard Model has been precisely tested for several decades over an enormous range of energies, new physics phenomena are expected to occur very rarely, to be produced only at high energies, or both. However, as the history of particle physics has shown, searches performed at high-energy colliders are a very promising way to discover new particles, and many theories predict new phenomena at the TeV scale, possibly within the reach of the LHC. The LHC, a proton-proton collider located outside Geneva, Switzerland at CERN, holds 2The strong and weak nuclear forces, electromagnetism, and gravity. 4 the record as the highest energy particle collider constructed thus far, and as such, offers a unique opportunity to search directly for the production of new physics. Four specialized particle detectors, such as ATLAS (A Toroidal LHC ApparatuS), are used to detect, record, and reconstruct the assemblage of particles that originate from the high-energy proton-proton collisions provided by the LHC. The focus of this dissertation is the search for a new heavy electrically neutral spin-1 boson,3 known colloquially as a Z(cid:48), which is produced in the proton-proton collisions at the LHC and decays into a pair of charged leptons. For the search considered here, the final-state consists of an electron and a positron, or a dielectron pair; i.e., pp → Z(cid:48) → e+e− + X. The search is performed with the full dataset collected by the ATLAS detector during the √ 2015 and 2016 runs of the LHC at a center-of-mass (CM) energy of s = 13 TeV, allowing for a previously unexplored energy scale to be probed for the first time. As a Z(cid:48) would decay too quickly to be detected directly in the ATLAS detector, events with highly energetic dielectron pairs are searched for instead. The smoking-gun signature of Z(cid:48) production would be the appearance of a narrow resonant peak in the dielectron invariant mass spectrum, where the position of the peak would depend on the mass of the Z(cid:48) boson. Its specific properties, such as its decay width ΓZ(cid:48), production cross section σZ(cid:48), and branching fraction to dielectron pairs Br(Z(cid:48) → e+e−), all depend on the particular model under consideration. While the Z(cid:48) is not predicted by Standard Model, it is predicted by several well-motivated extensions, such as the E6 model [53], a Grand Unified Theory.4 Thus, discovering a Z(cid:48) would contribute significantly to our fundamental understanding of the elementary particles 3A particle with integer spin. 4Grand Unified Theories (GUTs) predict the three gauge interactions of the Standard Model are unified by a single gauge symmetry at very high energy. 5 and their interactions, and would help to address some of the limitations of the Standard Model outlined above. In the case of non-observation, data are used to set exclusion limits on proposed extensions of the Standard Model, thereby restricting their ranges of validity. In order to perform a robust search for new physics like the one presented here, precise predictions of the Standard Model processes that result in dielectron pairs are required. This in turn requires a precise understanding of the Parton Distribution Functions (PDFs) of the proton, which describe its internal structure, and are therefore instrumental in the modeling of the high-energy proton-proton collisions at the LHC. While the discovery potential of the search presented here is not strictly limited by the current precision of the PDFs, the ability to discriminate between different Z(cid:48) models in a discovery scenario and to set more powerful exclusion limits in future iterations of the search will be strongly affected unless the current uncertainty of the PDFs can be sufficiently reduced. In the final chapter of this dissertation, an exploratory study is performed with precisely this goal in mind. The dissertations is organized as follows: Part I: The theoretical foundations and predictions needed for the subsequent chapters are described. • Chapter 1: Introduction • Chapter 2: The particles and forces of the Standard Model are introduced, along with its mathematical formulation, including the Electroweak Theory, the Higgs Mechanism, and Quantum Chromodynamics (QCD). • Chapter 3: The physics of LHC proton-proton collisions are described and PDFs are introduced. • Chapter 4: The primary Standard Model background process to the dilepton search is 6 due to Drell-Yan production. Drell-Yan physics is introduced and its production cross section is calculated using the diagrammatic technique developed by Feynman. The effect of the PDF uncertainty on the predicted high-mass Drell-Yan cross section is qualitatively assessed. • Chapter 5: The limitations of the Standard Model are described. Several theories of new physics that predict high-mass dilepton resonances are introduced. Candi- date models searched for in this dissertation are presented along with their potential experimental signatures. Exclusion limits on Z(cid:48) production are provided for recent experiments. Part II: The LHC and ATLAS detector are described, along with electron reconstruction and identification, and the Monte Carlo simulation used to model the signal and Standard Model background processes present in the search. • Chapter 6: The operation and performance of LHC is presented. • Chapter 7: The ATLAS experiment is introduced with an emphasis on detector sub- systems relevant for measuring electrons. • Chapter 8: The reconstruction and identification of electron candidates from inner detector tracks and calorimeter energy clusters in the ATLAS detector are explained. Corrections applied to electrons in data and simulation are also described. • Chapter 9: Simulation samples used to model the signal and background processes contributing to the analysis are described, along with their various corrections. Part III: The search for new physics in the dielectron final-state is performed and the results of the search are interpreted and presented. 7 • Chapter 10: The search for dielectron resonances is performed. The estimate of the QCD Multijet & W +jets fake background is obtained using a data-driven technique. The dielectron event selection is described and data are compared directly to the Stan- dard Model background estimate. Event yields are quantified and several kinematic distributions are shown, including the dielectron invariant mass. Systematic uncer- tainties are assessed. • Chapter 11: The data in the dielectron invariant mass distribution are scrutinized for the presence of a signal. If no significant excess is found, a Bayesian technique is used to set 95% C.L. upper limits on the cross section times branching fraction for Z(cid:48) production, as well as on several other Z(cid:48) model parameters of interest. Part IV: The systematic uncertainty of PDFs is studied and a novel approach is developed that can be used to reduce it for future analyses. • Chapter 12: The Drell-Yan process is revisited in an attempt to better understand the nature of the high-mass PDF uncertainty that arises in the primary search region of the dielectron invariant mass spectrum. The cos θ∗ variable is identified as potentially useful in a future PDF global analysis. • Chapter 13: A novel technique is developed to reduce the high-mass PDF uncertainty. Results are quantified and considered in the context of a future dilepton search. Part V: The conclusion to the dissertation is given, which includes a summary of the results obtained and an outlook to the future of the dilepton search and LHC program. • Chapter 14: Conclusion 8 Chapter 2 The Standard Model In this chapter, an introduction to the Standard Model of Particle Physics is given. The fundamental particles and interactions are described, its mathematical formulation is pre- sented, and the theories of the Electroweak interaction, the Higgs Mechanism, and Quantum Chromodynamics are explained in detail. 2.1 Fundamental Particles and Interactions The Standard Model of particle physics is a gauge-invariant, relativistic quantum field theory (QFT). It describes the dynamics of and interactions between all currently known elementary particles, and successfully incorporates three of the four fundamental interactions into its theoretical framework.1 Since its inception in the early 1970s, the Standard Model has passed every experimental test and precisely predicted a wide range of phenomena, making it today one of the best established and well-tested theories in the history of science. Broadly speaking, the particles of the Standard Model can be divided two classes: there are the matter particles and the force-carriers, or mediators. The matter particles are fermions,2 which interact through the exchange of force mediating gauge bosons.3 In 1The 4th, Gravity, is explained by Einstein’s General Theory of Relativity as the warping of the geometry of space-time. 2Fermions are particles with half-integer spin. 3Bosons are particles with integer spin. All force mediating particles in the Standard Model are spin-1. 9 addition, the Standard Model contains a spin-0 scalar boson, known as the Higgs boson, as a consequence of the Higgs Mechanism (See Sec. 2.3.4), which is responsible for the generation of all elementary particle masses in the model. 2.1.1 Matter Particles The matter particles, the basic building blocks of matter, can be further subdivided into two categories according to their allowed interactions: these are the leptons and the quarks. Table 2.1 lists the properties of each matter particle in the Standard Model.4 There are six leptons: three charged leptons, the electron (e), the muon (µ), and the tau (τ ), each of which carries an electric charge of Q = −1 e, and three neutral leptons, the electron-neutrino (νe), the muon-neutrino (νµ), and the tau-neutrino (ντ ), each of which carries an electric charge of zero. Thus, the charged leptons can interact electromagnetically, while the neutral leptons can not. Furthermore, each leptons carries a weak isospin t3 and weak hypercharge quantum number Y , and are thus susceptible to the weak interactions as well. Analogous to the leptons, there are six quarks: three up-type quarks, the up (u), the charm (c), and the top (t), each of which carries an electric charge of Q = +2/3 e, and three down-type quarks, the down (d), the strange (s), and the bottom (b), with Q = −1/3 e. Quarks are not found in isolation like the leptons, rather they form composite structures called hadrons, which come in two varieties: two-quark state mesons, and three-quark state baryons, the most well known being the proton (uud), and neutron (ddu). Not only do the quarks have weak isospin t3 and weak hypercharge Y , they also carry one of three 4Natural units will be used throughout this dissertation, such that  = c = 1. As a result energy and mass have the same dimension. 10 Particle Name Symbol EM Charge [e] Weak Charge Strong Charge electron neutrino electron muon neutrino muon tau neutrino tau up down charm strange top bottom νe e νµ µ ντ τ u d c s t b (Q) 0 -1 0 -1 0 -1 +2/3 −1/3 +2/3 −1/3 +2/3 −1/3 (t3) +1/2 −1/2 +1/2 −1/2 +1/2 −1/2 +1/2 −1/2 +1/2 −1/2 +1/2 −1/2 (Color) No No No No No No Yes Yes Yes Yes Yes Yes Mass [MeV] < 2 · 10−6 0.511 < 0.19 106 < 18.2 1777 2.2 4.7 96 1.3 173.1 · 103 4.2 · 103 Table 2.1: Matter content of the Standard Model. The three generations of quarks and lep- tons are shown along with their name, symbol, charges under each interaction, and mass [1]. possible color charges, called “red,” “green,” or “blue,” as opposed to the singular charge familiar in electromagnetism, signifying their possible interaction through the strong force. In addition, pairs of elementary fermions form three distinct generations. The first generations of leptons is defined by the (νe, e) pair, the second by the (νµ, µ) pair, and the third by the (ντ , τ ) pair, with a similar pairing for the quarks. Each particle within a generation has greater mass than the corresponding particle of lower generation, but their possible interactions are identical. Why each generation should be replicated like this in nature is currently unknown. The first generation of matter particles is stable, forming all ordinary matter in the universe. Atoms, for example, are composed of electrons orbiting atomic nuclei, themselves composed of protons and neutrons, which are ultimately made from bound states of up and down quarks. On the other hand, the second and third generations are only seen in high- energy environments like cosmic rays, particle accelerators, and the early universe [54], and 11 Particle Name Symbol EM Charge [e] Weak Charge Strong Charge photon W -boson Z-boson gluon Higgs γ W± Z g H (Q) 0 ±1 0 0 0 (t3) 0 ±1 1 0 -1/2 (Color) No No No Yes No Mass [MeV] 0 80.3 · 103 91.2 · 103 0 125.5 · 103 Table 2.2: Force carriers and Higgs boson of the Standard Model. The name, symbol, charges under each interaction,and mass of each particle is listed [1]. quickly decay into the lower generation particles. Neutrinos of all generations are apparently stable due to their extremely small masses, but they rarely interact with normal matter. Fermions across generations are referred to as flavors, such that, e.g., the electron and muon are distinct flavors of charged leptons. Each quark and lepton has an antimatter counterpart with identical values for mass and spin, but an opposite electric charge, weak charge, and color charge. The most well known antimatter particle is the anti-electron, or positron, which was first theorized by Dirac in 1928 [55] and discovered by Anderson in 1932 [56] from his analysis of cosmic ray tracks. 2.1.2 Forces and Interactions There are four known fundamental forces in nature: gravitation, the electromagnetic force, and the weak, and strong nuclear forces. The Standard Model describes the later three, with the incorporation of gravity into the theory still an open problem.5 Each force can be attributed to the exchange of one of more spin-1 gauge bosons: the photon γ, the massive W± and Z bosons, and the gluon g, whose properties are listed in Table 2.2. The photon γ is the mediator of the electromagnetic force. It is responsible for all forms of electromagnetic 5The influence of gravity at the currently probed subatomic energy scales is entirely negligible, and, as a consequence, its incorporation into the theory not strictly necessary to make accurate predictions. 12 radiation, including heat and light, as well as the phenomena of electricity, and magnetism. The photon is massless and its range of influence is infinite, diminishing in influence as 1/r2 according to Coulomb’s Law. Only particles with an electric charge couple to the photon, which includes all matter particles except the neutrinos. The photon itself does not have a charge. The W +, W−, and Z bosons are mediators of the weak interaction, responsible for radioactive β-decay and any process involving neutrinos. Weak interactions can proceed through a “charged current” interaction, mediated by the W + or W− bosons, or though a “neutral current” interaction, mediated by the Z boson. The β− decay of the neutron n → p e− ¯νe, for example, is mediated by a W− boson, in which the down quark d of the neutron is converted into an up quark u, and the W− subsequently decays into an electron e− and electron antineutrino ¯νe. Likewise, the exchange of a Z boson can occur in neutrino elastic scattering reactions such as νµ e− → νµ e−. The weak bosons are massive. The mass of the W + and W− bosons are identical with MW (cid:39) 80.3 GeV and the mass of the Z boson is slightly larger with MZ (cid:39) 91.2 GeV, which limits their range of influence to roughly 10−18 m, or about 0.1% the diameter of a proton. All matter particles carry weak charge and can therefore interact through the exchange of a weak boson. The mediators of the strong interaction are a set of eight massless, electrically neutral gluons g. The strong force holds matter together, confining quarks into “color neutral” bound-states of hadrons. As a residual effect, the strong force is responsible for the nuclear force that binds protons and neutrons together and as a result allows for the formation of stable atomic nuclei. Gluons themselves carry color charge, and can therefore couple to quarks and to each other. Despite being massless, the fact that gluons can self-couple leads 13 to the observed short-ranged influence of the interaction, a distance of roughly 1 fm, or the size of an atomic nucleus. 2.1.3 Particle Masses In its simplest form, the Standard Model predicts that all particles should be massless, in strong contradiction with experiment. To overcome this obstacle, a new spin-0, or scalar, field is introduced into the theory, the Higgs field. Through the Higgs Mechanism (See Sec. 2.3.4), the masses of the gauge bosons and fermions are generated from spontaneous symmetry breaking induced by the Higgs field. The result is the Higgs boson H, and the Standard Model particles that interact with the Higgs field obtain a mass. The properties of the Higgs boson can be seen in Table 2.2. The Higgs boson has been searched for since its original proposal by Higgs [57], Englert and Brout [58], and Guralnik, Hagen, and Kibble [59] independently in 1964. It was finally discovered in 2012 by the ATLAS and CMS collaborations at the LHC at CERN. 2.2 Mathematical Formulation The Standard Model is formulated as a gauge-invariant quantum field theory (QFT) [60]. The combination of quantum mechanics and special relativity leads to the introduction of quantum fields and their associated field quanta, or particle [54]. All particles in the Standard Model are described as excitations of underlying quantum fields, which are operator valued and defined at each point in space-time. The fermions in the model are defined by fermion, or Dirac, fields ψ(x), gauge bosons by vector fields Aµ(x), and the Higgs field by a scalar field φ(x). The dynamics and evolution of the fields is described using a Lagrangian formalism, 14 which is introduced in the following section. In addition, the Standard Model is a gauge theory6, meaning the Lagrangian can remain invariant with respect to certain symmetry transformation applied to the fields. The gen- erators of the transformations form a group, and the gauge group of the Standard Model is SU (3)C × SU (2)L × U (1)Y . 2.2.1 Lagrangian Dynamics The Standard Model is usually expressed in a Lagrangian formalism. The quantity that encodes the possible dynamics of the fundamental fields is the action S, with (cid:90) (cid:90) S[φ] = L dt = L (xµ, φ, ∂µφ) d4x, (2.1) where L = T − V is the Lagrangian. In quantum field theory, it is customary to work with the Lagrangian density L .7 Here, the Lagrangian is written L = L (xµ, φ, ∂µφ), which is a function of the space-time dependent quantum field φ(xµ), the derivative ∂µφ(xµ), and the standard space-time four-vector xµ, with x0 = ct, and xi = x.8 Possible terms in the Lagrangian include kinetic terms, which describe the propagation of free fields, mass terms, which specify particle masses, and interaction terms, which couple different particles together. The Lagrangian is often said to define the theory, since, using the rules of quantum field theory, all directly observable quantities, such as scattering cross section and particle decay widths, can be calculated from it. 6Additional degrees of freedom exist in the definition of the Lagrangian that do not necessarily correspond to changes in the physical state of the system. 7The Lagrangian density is commonly referred to as the Lagrangian, which will be the case here, and throughout the rest of the dissertation. 8The field theory definition effectively generalizes that from classical mechanics, where the Lagrangian L = L(t, q, ˙q) is a function of time, t, the generalized coordinates q, and its derivative ˙q. 15 A physical system evolves so that the action S is minimized. By requiring δS = 0, the Euler-Lagrange Equation is derived, (cid:32) ∂µ ∂(cid:0)∂µφ(cid:1)(cid:33) ∂L − ∂L ∂φ = 0, (2.2) which can be used to obtain the equations of motion for the space-time evolution of the quantum fields. In the case of electromagnetism, these are known as Maxwell’s equations, and in the case of the weak and strong interactions, these are known as the Yang-Mills equations. 2.2.2 Symmetries of the Standard Model The Standard Model is said to exhibit a symmetry if the form of its Lagrangian is left invariant by a transformation applied to its fields. Symmetries are described by symmetry groups, such as the group of translation or group of rotations. If the symmetry is continuous, Noether’s Theorem [61] states there must be a conserved current, and hence a conserved charge. For example, the conserved “charge” of a rotationally invariant system is its angular momentum. The Standard Model broadly exhibits two kinds of symmetries: external and internal. External symmetries are related to space-time transformations. These can be continuous, as in the Lorentz transformations, rotations, and boosts that form the Poincar´e group, or discrete. The discrete transformations include parity-inversion, P (changing the sign of the spatial coordinates, r → −r), charge-conjugation, C (changing the sign of charges), and the operation of time-reversal, T (changing the sign of the time coordinate, t → −t). The electromagnetic and strong interactions are invariant under, and hence conserve, 16 both P and CP; the weak interaction, on the other hand, conserves neither. It turns out the Standard Model is not invariant under any one of the discrete transformations, nor any successive two, however it is invariant under the combined operation of CPT . The internal, or gauge symmetries, are discussed in the next section. 2.2.3 Gauge Symmetries The Standard Model Lagrangian is invariant under a set of local (i.e. space-time depen- dent) symmetry transformations of the fields, known as gauge transformations. Gauge symmetries are some of the most consequential symmetries for the Standard Model, as they mandate the existence of the force carriers. The following illustrates the consequences of requiring a theory to be locally gauge in- variant in the context of Quantum Electrodynamics (QED). The Lagrangian describing the motion of a free spin-1/2 fermion field ψ(x) of mass m is L = ψ(x)(cid:0)i/∂ − m(cid:1) ψ(x). (2.3) Here, ψ(x) refers to the electron, ψ(x) = ψ†γ0 to the positron, and /∂ ≡ γµ∂µ denotes the Lorentz contraction between γµ and ∂µ, where γµ are the 4 × 4 Dirac γ matrices, 0 1  , 1 0  0 −σi  , σi 0 γ0 = γi = (2.4) which themselves are composed of the 2 × 2 Pauli spin matrices σi. The Lagrangian of Eq. 2.3 is left invariant under a rescaling of the fields by a global phase factor α. To see this explicitly, if ψ(x) → ψ(cid:48)(x) = eiQeαψ(x) is substituted into Eq. 2.3, 17 then L → L (cid:48) = L . Thus, the Lagrangian is globally gauge invariant. The phase factor eiQeα is an element of the symmetry group U (1)Q of rotations in one dimension. The charge operator Q is generator of the group, and returns the magnitude of electric charge associated with the field ψ(x). Taking this a step further and demanding that local gauge invariance holds, i.e., that the Lagrangian of Eq. 2.3 remain invariant under a transformation where the phase α(x) now depends on local space-time coordinates, results in L → L (cid:48) = ψ(cid:48)(cid:0)iγµ∂µ − m(cid:1) ψ(cid:48) = e+iQeα(x)ψ(cid:0)iγµ∂µ − m(cid:1) e−iQeα(x)ψ − ψ(cid:0)Qeγµ∂µα(x)(cid:1) ψ = ψ(cid:0)iγµ∂µ − m(cid:1) ψ (cid:124) (cid:125) (cid:124) (cid:125) (cid:123)(cid:122) (cid:123)(cid:122) original Lagrangian derivative term (2.5) . The Lagrangian of Eq. 2.5 is no longer invariant under the local U (1) transformation due to resulting derivative term associated with α(x). However, gauge invariance can be restored with the following substitution, ∂µ → Dµ = ∂µ − ieQAµ, (2.6) where Dµ is known as the gauge covariant derivative, which is constructed by adding a new spin-1 field Aµ to the standard definition of the four-derivative ∂µ. If the new field Aµ is required to transform like Aµ → A(cid:48) µ = Aµ + ∂µα(x), (2.7) during a gauge transformation, then the ∂µα(x) term in Eq. 2.7 exactly compensates for the derivative term in Eq. 2.5 and thus, the requirement of local gauge invariance is satisfied. 18 −ieQ Aµ(x) ψ(x) ψ(x) Figure 2.1: The QED interaction vertex. The solid lines represent the motion of the charged leptons and the curved line represent the propagation of the photon. The strength of the interaction is determined by the magnitude of electric charge Q in units of fundamental charge e and is labeled at the vertex. The “new” field Aµ is none other than the photon field, the mediator of the electromagnetic force. The insertion of Eq. 2.6 into Eq. 2.5 results in the following Lagrangian, L = ψ(x)(cid:0)iγµDµ − m(cid:1) ψ(x) = ψ(x)(cid:0)iγµ∂µ − m(cid:1) ψ(x) − Qe ¯ψ(x)γµAµψ(x) (2.8) where the second term corresponds to the interaction between the fermion fields ψ(x) and ψ(x) and the photon field Aµ, which is represented pictorially in Figure 2.1. As the photon can propagate in its own right, an additional kinetic term can be added to 2.8. The result is the full QED Lagrangian, F µνFµν + ψ(x)(cid:0)iγµ∂µ − m(cid:1) ψ(x) − Qe ¯ψ(x)γµAµψ(x) (2.9) LQED = −1 4 where F µν = ∂µAν − ∂νAµ is the usual electromagnetic field strength tensor. Incidentally, the photon field Aµ must be massless, since any term of the form Lmass = 1 2 m2AµAµ would violate the local gauge invariance of Eq. 2.9. 19 Thus, the requirement that the free field theory of Eq. 2.3 be invariant with respect to local gauge transformations, forces the introduction of a new massless field Aµ, understood to be the photon, and recovers the complete theory of quantum electrodynamics. In fact, this procedure of “promoting” a free field theory to an interacting theory is exactly how each of the fundamental interactions of the Standard Model are generated. The gauge group of the Standard Model is SU (3)C × SU (2)L × U (1)Y . Invariance under SU (3)C leads to the introduction of the gluons of the strong interaction, while invariance under SU (2)L × U (1)Y leads to the force mediators of the unified Electroweak interaction. 2.2.4 Standard Model Lagrangian The Standard Model Lagrangian can be broken down into four sectors, LSM = LEW + LHiggs + LYukawa + LQCD. (2.10) The following sections describe the Lagrangian of each sector in detail. The dynamics of the unified Electroweak interaction are encoded in LEW, while LHiggs and LYukawa describe the dynamics of the Higgs field, and how fermion masses are generated, respectively. The last term, LQCD, describes the dynamics of the strong interaction. The electromagnetic component, LQED, is included in the unified description of the Electroweak sector, and becomes apparant only after the spontaneous symmetry breaking induced by the Higgs field. 20 2.3 The Unified Electroweak Theory The Electroweak (EW) theory unifies the weak and electromagnetic interactions, which his- torically were treated as two separate theories. The unification of these theories was first described by Weinberg, Glashow, and Salam, earning them the 1979 Nobel Prize in Physics. The Electroweak theory is invariant under simultaneous SU (2)L × U (1)Y gauge transforma- tions, where L refers to “left-handed” particles, with the weak isospin t as the generator of SU (2)L transformations, and the weak hypercharge Y as the generator of U (1)Y transfor- mations. Under the action of the Higgs Mechanism, the Electroweak group is spontaneously broken, such that SU (2)L× U (1)Y → U (1)Q, which recovers the theory of electromagnetism in the low energy limit. 2.3.1 History Enrico Fermi proposed the first theory of the weak interaction in 1934 in order to explain radioactive β decays [62, 63] of nuclear physics. The original theory proposed by Fermi was subsequently generalized over the course of many decades to account the new and varied observations associated with weak interaction physics. C.S. Wu discovered parity violation in 1956 by measuring the beta decay spectrum of Co-60 [64] and around the same time James Cronin and Val Fitch discovered CP violation in their analysis of the neutral Kaon system [65]. The discovery of parity violation suggested Fermi’s theory must be modified. In 1958, Feynman and Gell-Mann proposed a V-A (vector minus axial-vector) Lagrangian for the theory, so that the weak interaction acted only on left-handed particles (and right-handed antiparticles) [66, 67]. The mathematical framework for the modern formulation of the theory was laid down 21 by Yang and Mills in 1956 by extending the classic idea of gauge symmetry to non-abelian field theories [68]. Building on the work of Yang and Mills, in 1961 Glashow linked the weak and electromagnetic forces by formulating a theory with three massless gauge bosons, two charged and one neutral, the progenitors of the W± and Z bosons of the Standard Model, and the photon γ [69]. The only massless field that had been observed in nature at the time was the photon, and the short range of the weak interaction indicated its force carriers must be massive. This problem of massless force carriers was solved when in 1967 Weinberg [70], and in 1968 Salam [71], incorporated the Higgs Mechanism into Glashow’s theory, thereby attributing the gauge bosons mass. The result was a theory that predicted one massless particle, the photon γ, and three new massive W± and Z bosons, the carriers of the weak interaction. The first evidence for this theory came in 1973 when the Gargamelle bubble chamber at CERN provided indirect evidence for the weak neutral current from the elastic scattering of neutrinos in matter [72, 73]. For their work, Glashow, Weinberg, and Salam (GWS) shared the 1979 Nobel Prize in physics. Four year later, direct evidence came in 1983 with the subsequent discoveries of the W± and Z bosons by the UA1 and UA2 collaborations at CERN [44, 45, 46, 47], marking one of the major intellectual achievements of humankind. It would take until 2012 for the Higgs boson, the last particle predicted by the GWS theory, to be discovered at CERN by the ATLAS and CMS collaborations [51, 52]. 2.3.2 Fermion Fields The force mediators of the weak interaction couple with different strengths to left-handed and right-handed fermion fields; and the charged current couples exclusively to left-handed fermions. The left- and right-handed components of a fermion fields can be obtained from 22 the projection operators PL and PR, (cid:16) 1 − γ5(cid:17) ψ, ψR = PRψ = (cid:16) 1 + γ5(cid:17) ψ, 1 2 (2.11) ψL = PLψ = 1 2 where γ5 = iγ0γ1γ2γ3 is a product of the 4× 4 Dirac γ matrices. Because of the asymmetric treatment of particles and antiparticles on the basis of their handedness, the Electroweak interaction is said to be chiral. The Electroweak Theory introduces the weak isospin quantum number t. This allows all left-handed fermions to be arranged into SU (2)L iso-doublets of t = 1/2 with an upper component of t3 = 1/2 and lower component of t3 = −1/2. The right-handed fermions are arranged into SU (2)L iso-singlets with t = 0 and t3 = 0, which is a consequence of their non-participation in the charged-current interaction. The right-handed neutrinos (and left-handed antineutrinos) are not included with the rest of the singlets.9 Thus, in the Electroweak theory, the fermions are described as ui L  , di L Qi = ui R, di R, Li =  , ei R, (2.12) νi L ei L where the index i refers the to individual fermion generation. Furthermore, regardless of their handedness, all fermions in the theory are given an intrin- sic weak hypercharge Y , which is introduced ultimately to accommodate the electromagnetic interaction. Table 2.3 details the fields and their associated quantum numbers. The standard electric charge Q of a fermion can be expressed in terms of the third component of weak 9At the time of the construction of the theory, there was no experimental evidence that neutrinos had mass and they could therefore be neglected from the theory; however the discovery of neutrino oscillations suggests a more complex story. See Section 5.1 for more details. 23 Quarks Leptons II Fermion Generation III I (cid:18)uL (cid:19) (cid:18)cL (cid:19) (cid:18)tL (cid:18)νeL (cid:19) (cid:18)νµL (cid:19) (cid:18)ντ L dL uR dR sL cR sR bL tR bR eL e− R νR µL µ− R νµR τL τ− R ντ R Quantum Number Y Q (cid:19) +2/3−1/3 (cid:19) +2/3 −1/3 0−1 −1 0 t3 +1/2−1/2 0 0 +1/2−1/2 0 0 +1/6 +1/6 +2/3 −1/3 −1/2−1/2 −1 0 Table 2.3: Three generations of fermions in the Electroweak theory with their respective quantum numbers. Left-handed particles are grouped into iso-doublets, while right-handed particles exist as iso-singlets. The right-handed neutrinos are included for reference, but are not formally included as part of the Electroweak theory. isospin t3 and the weak hypercharge Y as Q = t3 + Y, (2.13) which is a consequence of Electroweak unification. 2.3.3 The Electroweak Lagrangian The symmetry group SU (2)L contains three generators ti = 1 2 σi, where σi are the Pauli ma- trices, and the L in this context means “left-handed only.” The gauge bosons corresponding to the generators of the SU (2)L symmetry are called the W µ 1 , W µ 2 , and W µ 3 , which by con- struction only couple to the left-handed fermions of the theory. Furthermore, the symmetry group associated with weak hypercharge is U (1)Y , which contains a single generator Y. The corresponding gauge boson, known as the Bµ, couples equally between fermion chiralities. The procedure of generating the Electroweak interactions follows the same prescription as was carried out for QED in Section 2.2.3. Thus, the requirement of local gauge invariance 24 under the SU (2)L × U (1)Y group results in the following Lagrangian, 3(cid:88) 3(cid:88) 3(cid:88) (cid:0)i /D(cid:1) Li (cid:0)i /D(cid:1) ui LEW = i Q L i L L L + (cid:0)i /D(cid:1) di R i R d (2.14) R + i=1 3(cid:88) 3(cid:88) i=1 i=1 + ei R (cid:0)i /D(cid:1) Qi (cid:0)i /D(cid:1) ei i=1 µνW µνi − 1 W i 4 R − 1 4 L + ui R i=1 BµνBµν where i is the generation index, and Qi L and Li L refer to the left-handed quark and lepton iso-doublets, respectively, and ui R, di R, and ei R to the right-handed up-type and down-type quark and lepton iso-singlets, respectively, according to Table 2.3. The covariant derivative is Dµ = ∂µ + igtiW i µ + ig(cid:48)Y Bµ, ti = σi. 1 2 (2.15) The two coupling constants: g associated with SU (2)L, and g(cid:48) associated with U (1)Y , set the strengths of their corresponding interactions. The field strength tensors are ν − ∂νW i W i µν = ∂µW i Bµν = ∂µBν − ∂νBµ. µ − gijkW j µW k ν (2.16) in the relation (cid:2)ti, tj The SU (2)L group is non-abelian, meaning its generators do not commute, which is codified (cid:3) = iijktk, where ijk is the Levi-Civita symbol. As a results, self- interactions between the W µ i gauge fields become possible, as seen in the third term of the field strength tensor of Eq. 2.16. The U (1)Y group is abelian, and as a result no self- interactions are present between the gauge fields. The Electroweak Lagrangian of Eq. 2.14 is incomplete in one crucial respect: the fermions and gauge fields lack mass terms, and are hence massless fields in direct contradiction with 25 experiment. In fact, it is impossible to add explicit mass terms since the left- and right- handed fields exhibit different transformation properties; i.e., standard mass terms of the form Lmass = mψψ = m(cid:0)ψLψR + ψRψL (cid:1) break SU (2)L gauge invariance. The solution to this puzzle is developed in the next section. Still, the reason nature treats left- and right-handed fields so differently is one of the unresolved mysteries of the Standard Model. 2.3.4 The Higgs Mechanism The solution to the fermion and gauge boson mass paradox is obtained through the spon- taneously breaking of Electroweak symmetry via the Higgs Mechanism [57, 58, 59]. The idea is to introduce a new particle into the Electroweak theory Φ(x), which is an SU (2)L iso-doublet of complex scalar fields, φ+(x)  . φ0(x) Φ(x) = (2.17) The upper t3 = 1/2 component φ+ is electrically charged with Q = +1 and lower t3 = −1/2 component φ0 is electrically neutral. Each component carries a weak hypercharge of Yφ = 1/2. The field Φ(x), known as the Higgs field, can obtain a non-zero value in its vacuum state, which results in the breaking of the SU (2)L × U (1)Y symmetry of the Electroweak Lagrangian of Eq. 2.14 and in the spontaneous generation of gauge-invariant mass terms. The Higgs Lagrangian is LHiggs = |DµΦ|2 − V (Φ) , (2.18) where Dµ is the Electroweak covariant derivative of Eq. 2.15 and the Higgs potential is 26 Figure 2.2: The shape of the Higgs potential V (Φ) of Eq. 2.19 for the choice of µ2 > 0. The potential V (Φ) obtains a minimum at the non-zero value of Φ†Φ = µ2/2λ ≡ v2/2 [6]. defined as V (Φ) = −µ2Φ†Φ + λ (cid:16) (cid:17)2 , Φ†Φ (2.19) where µ and λ are real-valued constants. The Higgs potential of Eq. 2.19 is invariant under SU (2)L×U (1)Y gauge transformations and is constructed in such a way that the vacuum state is degenerate. By requiring µ2 > 0 and λ > 0, the potential V (Φ) is minimized when Φ†Φ = µ2/2λ ≡ v2/2, where v is a non- zero number known as the vacuum expectation value (or vev). The situation is depicted in Figure 2.2. Due to the degeneracy of the V (Φ) vacuum, there are an infinite number of states that satisfy the minimum requirement. A specific choice amounts to choosing a gauge and the vacuum state of the scalar field becomes Φ0 = 1√ 2 ( 0 v ). The Higgs field Φ can now be expressed in terms of an expansion around its vacuum state Φ0 and as a real valued scalar 27 field H(x), which has a zero vev, so that Eq. 2.17 becomes  0  . v + H(x) Φ(x) = 1√ 2 (2.20) In terms of H(x) the SU (2)L×U (1)Y symmetry is spontaneously broken, such that SU (2)L× U (1)Y → U (1)Q.10 The field H(x) describes a neutral scalar, called the Higgs boson. While one degree of freedom of Φ(x) contributes to H(x), the other three contribute to the longitudinal polarizations of the W± and Z bosons. As a result, in terms of the redefined Higgs field of Eq. 2.20, the scalar Lagrangian of Eq. 2.18 becomes (cid:0)∂µH(cid:1)† (v + H)2(cid:12)(cid:12)(cid:12)W 1 LHiggs = −1 2 − g2 8 ∂µH − λv2H2 − λvH3 − λ 4 H4 (v + H)2(cid:0)g(cid:48)Bµ − gW µ 3 (cid:1)2 (2.21) µ − iW 2 µ (cid:12)(cid:12)(cid:12)2 − 1 8 which now contains the desired gauge boson mass terms. To identify the mass terms explic- itly, it helps to redefine the gauge fields in the mass basis in order to eliminate the mixed terms between fields. With that, the physical gauge fields become (cid:17) µ − g(cid:48)Bµ 1√ 2 (cid:16) 1(cid:112) 1(cid:112) µ µ ∓ iW 2 (cid:16) W 1 (cid:16) gW 3 g(cid:48)W 3 g2 + g(cid:48)2 g2 + g(cid:48)2 (cid:17) ≡ W 3 (cid:17) ≡ W 3 W± µ = Zµ = Aµ = µ cos θW − Bµ sin θW (2.22) µ + gBµ µ sin θW + Bµ cos θW where θW is the Weinberg Angle, defined by tan θW = g(cid:48)/g, which parametrizes the 10A better term might be “hidden,” or no longer manifest, since the Lagrangian is still invariant, but the vacuum is not. 28 f W−/Z W−/Z H H H f (a) W +/Z (b) W +/Z (c) H Figure 2.3: Fundamental interaction vertices with the Higgs boson in the EW theory: the Higgs-fermion interaction (a), and the three-point (b) and four-point (c) gauge interactions. degree of mixing between the gauge fields after spontaneous symmetry breaking.11 The gauge couplings g of SU (2)L and g(cid:48) of U (1)Y are related to the electromagnetic coupling e with e = g sin θW = g(cid:48) cos θW , and a coupling for the weak neutral current can be specified by gZ = e sin θW cos θW . (2.23) (2.24) With these substitutions, the scalar Lagrangian of Eq. 2.18 takes on a much simpler form, (2.25) (cid:0)∂µH(cid:1)† 4 LHiggs = −1 2 − g2 (v + H)2 W + ∂µH − λv2H2 − λvH3 − λ 4 (v + H)2 ZµZµ g2 + g(cid:48)2(cid:17) H4 (cid:16) µ W µ− − 1 8 which highlights the three- and four-point interactions between the Higgs field and the other gauge fields, depicted in Figure 2.3, as well as its own self-interactions, depicted in Figure 2.4. 11Said another way, the Weinberg angle relates the weak eigenstates (W 3, B) to the mass eigenstates (Z, A). 29 H H H H H (a) H (b) H Figure 2.4: Fundamental three-point (a) and four-point (b) Higgs boson self-interaction vertices in the EW theory. In this form, Eq. 2.25 readily yields the following mass terms, √ 2λv 1 2 1 2 gv gZ v, MH = MW = MZ = (2.26) which are all proportional to the vev v. The photon remains massless. From the experimental values of MW and MZ , the vev is deduced to be v (cid:39) 246 GeV [1]. It has units of mass, and is the only parameter of the Standard Model that is not a dimensionless number, setting an absolute scale for the Electroweak theory. 2.3.5 Yukawa Interactions With the addition of the scalar field Φ(x) into the model, the fermion masses can similarly be generated through spontaneous symmetry breaking according to Yukawa type interactions.12 12i.e. an interaction between a scalar and fermion fields, reminiscent of Yukawa’s original theory that described the nuclear force between nucleons as mediated by Pions. 30 The general gauge invariant Yukawa Lagrangian is [60] LYukawa = − QjLΦλ (d) jk dkR − 3(cid:88) j,k=1 3(cid:88) j,k=1 QjL(−iσ2Φ∗)λ (u) jk ukR − 3(cid:88) j,k=1 LjLΦλ ((cid:96)) jk ekR + h.c., (2.27) where the generational indices j and k allow inter-generation mixing between the fields, and the λ (f) jk terms are called Yukawa Matrices, each of which depends on 9 complex parameters. After the scalar field Φ(x) acquires a vev as in Eq. 2.20, the Yukawa Lagrangian becomes LYukawa = − 1√ 2 (v + H) 3(cid:88) (cid:16) j,k=1 (d) jk djLdkR + λ (u) jk ujLukR + λ ((cid:96)) jk ejLekR λ (cid:17) + h.c. (2.28) The fermion mass terms can be made explicit by diagonalizing the Yukawa Matrices. This can be accomplished with a suitable set of field transformations such that, for the up- and down-type quarks, † uLλ(u)UuR = U  λu 0 0 λc 0 0 0 0 λt  , U † dLλ(d)UdR =  λd 0 0 λs 0 0 0 0 λb  , (2.29) with an identical definition for the leptons of Eq. 2.28. The unitary matrices Uf L,R are used to transform the fermion fields from the electroweak basis to the mass basis. As a result, distinct mass eigenstates are formed from linear superpositions of electroweak eigenstates. The price of this procedure is that, like before, it mixes the fundamental fields together, in this case of different generations. 31 Thus, the Yukawa Lagrangian of Eq. 2.28 can be re-expressed as (d) i diLdiR + λ λ (u) i uiLuiR + λ ((cid:96)) i eiLeiR (d) i didi + λ λ (u) i uiui + λ ((cid:96)) i eiei , (cid:17) LYukawa = − 1√ 2 (v + H) = − 1√ 2 (v + H) (cid:16) (cid:16) 3(cid:88) 3(cid:88) i=1 i=1 and the fermion mass terms can now be read off explicitly, mf = 1√ 2 λf v, (cid:17) + h.c. (2.30) (2.31) which are also proportional to the vev v, just like the mass terms of the Standard Model gauge fields. Effectively, Eq. 2.27 split into two parts: the first of which provided the required fermion mass terms, and the second of which specified the Yukawa interactions between fermions and the Higgs field (as in Fig. 2.3). The electroweak theory provides no explanation for the empirically determined values of λf , and hence provides no fundamental understanding of the fermion masses [60]. Fur- thermore, as the neutrinos have no right-handed partners νiR in the Standard Model, they cannot acquire mass terms through the Yukawa interaction described here, nor subsequently can they interact with the Higgs field. Again, the neutrinos are strictly massless in the Standard Model. The electroweak interactions with fermions defined in Eq. 2.14 were originally specified between identical fermion generations. However, the fermion field redefinitions of Eq. 2.29 mix the generations together. As fermions propagate through the vacuum, they are perceived as maintaining a distinct identity, known as a mass eigenstate, which is a superposition of flavor eigenstates. Fermions oscillate between their flavor eigenstates as they travel, a 32 W− W− W + W− γ/Z γ/Z W + (a) W + (b) W− W + (c) γ/Z Figure 2.5: Fundamental three-point (a) and four-point (b)-(c) gauge boson vertices in the EW theory. phenomenon called fermion flavor oscillation. 2.3.6 The Electroweak Lagrangian Revisited After spontaneous symmetry breaking, the Electroweak Lagrangian of Eq. 2.14 can be re- expressed in terms of the physical fermion and gauge fields. The substitution of the physical gauge fields of Eq. 2.22 into Eq. 2.14 results in dis- tinct three- and four-point gauge interactions as seen in Figure 2.5. These include W W Z and W W γ interactions, as well as W W W W , W W γγ, W W ZZ, and W W Zγ interactions. Furthermore, the covariant derivative of Eq. 2.15 becomes Dµ = ∂µ + ieQAµ + i g√ 2 (cid:0)t+W + µ + t−W− µ (cid:1) + igZ (cid:16) t3 − Q sin2 θW (cid:17) , (2.32) where the weak isospin raising and lowering operators are defined as t± = 1√ 2 (t1 ± it2). Upon substitution of Eq. 2.32 into the Electroweak Lagrangian of Eq. 2.14, the standard form of charged and neutral current electroweak interactions with fermions can be written down. The charged current interaction between the W boson and the quarks (as in Fig. 2.6) is 33 qi ¯qj W + W− ¯qj (a) qi (b) ¯q q Z (c) Figure 2.6: Fundamental interaction vertices with quarks in the EW theory: the charged current interactions (a)-(b) and neutral current interaction (c). given by LW q = −i g√ 2 3(cid:88) (cid:16) n,m=1 VmnuLmγµdLnW + µ + (Vmn)† dLmγµuLnW− µ (cid:17) . (2.33) Because all quarks are massive, it is natural to express their fields in the mass basis. Here, the matrix V = U † uUd results from the fermion field redefinitions of Eq. 2.29, and is known as the Cabbibo-Kobayashi-Maskawa (CKM) Matrix [74, 75], defined as  Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb 1 − λ2/2   VCKM = = λ Aλ3 (ρ − iη) −λ 1 − λ2/2 Aλ3 (1 − ρ − iη) −Aλ2 Aλ2 1  + O(cid:16) (2.34) λ4(cid:17) . The CKM matrix parametrizes the degree of mixing between the quark flavor eigenstates of the weak interaction and their corresponding mass eigenstates due to spontaneous symmetry breaking. 34 ν(cid:96) ¯ν(cid:96) (cid:96)+/¯ν(cid:96) W + W− Z (cid:96)+ (a) (cid:96)− (b) (cid:96)−/ν(cid:96) (c) Figure 2.7: Fundamental interaction vertices with leptons in the EW theory: the charged current interactions (a)-(b) and neutral current interactions (c). The matrix is expressed in terms of the Wolfenstein parametrization [76], which is valid to fourth order in λ with λ (cid:39) 0.225, A (cid:39) 0.8, ρ (cid:39) 0.13, and η (cid:39) 0.35 [1]. This form nicely highlights the magnitude of each CKM matrix element, and the nearly diagonal structure of the matrix, with off-diagonal terms becoming proportionally smaller. Since VCKM is not strictly diagonal, the W± bosons couple to quark mass eigenstates of different generations. Likewise, the charged current interaction with leptons (as in Fig. 2.7) is given by (cid:0)νLiγµ(cid:96)LiW + µ + (cid:96)LiγµνLiW− µ (cid:1) . (2.35) LW (cid:96) = −i g√ 2 3(cid:88) i=1 The charged current interaction with leptons does not require an analogous mixing matrix of the form V = U † ν U(cid:96), since the neutrinos are massless in the Standard Model by definition.13 Lastly, the interaction between the neutral electroweak gauge bosons and fermions is given by L Z/γψ = −ieQψγµψAµ − igZ ψγµ(cid:16) (cid:17) ψZµ = −ieQψγµψAµ − igZ ψγµ (gLPL + gRPR) ψZµ, t3 − Q sin2 θW (2.36) 13In the minimal extension of the Standard Model with three massive neutrinos, V = VP M N S is known as the PMNS matrix, analogous to the CKM matrix of the quark sector. 35 gV gA gL gR 3 xW −1 4 − 2 3 xW −2 2 − 2 3 xW +1 u +1 4 3 xW −1 d −1 3 xW +1 4 + 1 3 xW +1 2 + 1 4 −1 +1 +1 0 ν(cid:96) 4 4 2 −1 −1 4 + xW +1 2 + xW (cid:96) 4 +xW Table 2.4: Neutral current couplings of Standard Model fermions. The rows specify the couplings within each respective quark and lepton generation. Alternate definitions often are written in terms of gV = 1 to the vector (γµ) and axial-vector (γµγ5) terms of the neutral current Lagrangian. 2(gL − gR), which function as coefficients 2(gL + gR) and gA = 1 where ψ stands for any quark or lepton of the three fermion generations. The first term in the Lagrangian of Eq. 2.36 is identified with the standard electromagnetic interaction induced by the photon γ and the second term with the neutral current interaction of the Z boson. In contrast to the charged current interactions, the neutral current interactions have no flavor changing transitions. The left- and right-handed couplings are defined as gL = t3−Q sin2 θW and gR = −Q sin2 θW , respectively. Therefore, the neutral current interaction is chiral, coupling with different strengths to the left- and right-handed fields. The values of the couplings are listed in Table 2.4. 2.4 Quantum Chromodynamics Quantum Chromodynamics (QCD) is the theory of the strong interaction. It is an unbroken non-abelian gauge theory of gauge group SU (3)C . The quark fields carry one of three possible color charges NC = 3 and are grouped into SU (3)C color-triplets ψj where j refers to “red,” “green,” or “blue.” The leptons are color-singlets and do no participate in the strong interaction. The gluons are the force mediates of the strong interaction and also carry color charge themselves. There are eight color combinations of gluons. 36 q ¯q g (a) g g g (b) g g (c) g g Figure 2.8: Fundamental interaction vertices in QCD: the quark-gluon interaction (a), the three-point gluon interation (b), and the four-point gluon interation (c). The QCD Lagrangian is LQCD = q=1 i,j=1 6(cid:88) 3(cid:88) (cid:0)i /Dij − mqδij (cid:1) ψq j − ψ q i 8(cid:88) a=1 1 4 GµνaGa µν, (2.37) where the summations occur over each quark of flavor q with mass mq and also over the color indices i and j in reference to the three possible color charges. The gauge covariant derivative is (Dµ)ij = ∂µδij + igSta ijGa µ, ta ij = λa ij 2 , (2.38) which introduces the eight Gell-Mann matrices λa trices, as well as the gauge fields for the gluons Ga ij, the SU (3) analogs of SU (2) Pauli ma- C − 1 = 8; i.e., µ with a running from 1 to N 2 there are eight kinds of gluons. The coupling gS sets the strength of the strong interaction and can be expressed as a “fine-structure constant” for QCD by αS = g2 S/4π, often called the strong coupling constant. The gluon field strength tensor is Ga µν = ∂µGa ν − ∂νGa µ − gSf abcGb µGc ν, [ta, tb] = ifabctc, (2.39) 37 which, as was also the case in the SU (2)L Electroweak theory, includes an additional term due to the non-abelian structure of the gauge group. Therefore, the kinetic term for the gluon fields in Eq. 2.37 predicts 3- and 4-point self-interactions, which are depicted in Figure 2.8. The fabc terms are the structure constants of the SU (3)C gauge group. 2.4.1 Asymptotic Freedom and Confinement The strong force is appropriately named due to the large value of its coupling constant αS. By comparison, the electromagnetic coupling αEM is more than an order of magnitude smaller. The values assumed by the coupling constants depend on the energy or momentum-transfer scale µ associated with the given physical process. This “running” of the coupling can be parametrized by the beta function [60], β(α) = µ2 ∂α(µ2) ∂µ2 = βiα(µ2)i+2 (2.40) ∞(cid:88) i=0 whose solution gives the scale dependence of the coupling. When keeping only the leading term in the expansion, the solution is α(µ2) = 1 − α(µ2)β0 ln(cid:0)µ2/µ2 α(µ2) 0 (cid:1) , (2.41) where µ0 is a reference scale where α is known, and the factor β0 is the leading-order coefficient in the perturbative expansion of Eq. 2.40, and can be calculated explicitly. In QED, the coefficient β0 = 1/(3π), which results in a running, such that αEM(µ) → ∞ as µ → ∞; i.e., as the energy scale increases, so does the strength of the electromagnetic interaction. Here, the running is fairly slow, increasing from its low energy value of αEM(0) ≈ 38 Figure 2.9: The running of the αS coupling constant in QCD as a function of the energy scale Q with a variety of experimental measurements overlaid [1]. 1/137 to a value of αEM(MZ ) ≈ 1/128 at the scale of the Z boson mass. However, the behavior of the QCD coupling constant differs significantly from its electro- magnetic counterpart due to non-abelian nature of the theory. The corresponding QCD beta function is negative with β0 = − (33 − 2NF ) /12π, where NF is the number of active quark flavors. Consequently, this leads to a running where αS(µ) → 0 as µ → ∞, a phenomenon known as Asymptotic Freedom [77, 78]. As a result, the quarks do not feel the strong force at high energies and behave essentially as free particles. On the other hand, at low energies the interactions become strong, leading to the confinement of quarks and gluons into bound colorless hadrons. As long as the energy scale is sufficiently large, the smallness of αS makes perturbative calculations of QCD observables, such as hadronic scattering cross sections, possible. How- ever, as αS approaches unity, the perturbative approach breaks down. The scale at which this happens is known as ΛQCD ≈ 220 MeV. At the mass of the Z boson, the value of αS(MZ ) = 0.118 [1]. Figure 2.9 plots the running of αS as a function of the energy scale Q. 39 Chapter 3 Proton-Proton Collisions This chapter provides a foundation for understanding the phenomenology of the proton- proton collisions at the LHC. This is important because the signature of new physics, if it exists, will be identified by performing a detailed analysis on a large dataset of such collisions. In this chapter, the composite nature of the proton is described in detail, along with the dynamics of proton-proton collisions themselves. 3.1 Proton Structure The proton is a composite particle. Unlike the point-like particles of the Standard Model (as in Table 2.1), the proton has an internal structure. By scattering point-like probes off of it, the internal structure of the proton can be investigated. The so-called resolution of the probe can be expressed in terms of the energy scale Q2, the squared momentum transfer between the probe and proton during a scattering process. The proton consists of an ensemble of interacting quarks and gluons, collectively known as partons, whose dynamics are described by QCD, the SU (3) gauge theory of the strong interaction (See Sec. 2.4). Due to the non-abelian nature of the theory, the interaction strength between the quarks and gluons increases (decreases) with distance (energy). As a result, individual quarks and gluons are only ever found as bound hadronic states, such as the proton, a process known as confinement. At low values of Q2, the proton appears point- 40 Figure 3.1: Substructure of proton depicting interactions between its partonic constituents: valence quarks, sea quarks, and gluon [7]. like, and its internal structure cannot be resolved. However, as smaller and smaller distances are probed (larger energy scale Q2), the strength of the QCD interaction decreases, which allows the partons to be treated essentially as a free particles. This is known as asymptotic freedom. The static properties of the proton (spin, electric charge, isospin, etc.) are fully deter- mined by its three valence quarks: two up quarks and one down quark (uud). Along with the three valence quarks, the proton (and all hadrons for that matter) consists of an indefinite number of quark-antiquark pairs known as sea quarks (See Fig. 3.1). Sea quarks arise when a gluon splits, or fluctuates, into a qq pair (as in Fig. 2.8). The constant flux of gluons splitting and recombining results in what is known as “the quark sea,” the set of all off-shell quarks in the proton consistent with the uncertainty principle. 41 Figure 3.2: Diagram of a typical hard scattering interaction between partons at the LHC [8]. 3.1.1 Proton-Proton Collisions The majority of proton-proton collisions at the LHC are elastic, meaning the colliding protons remain intact leaving their substructure unresolved. On the other hand, in an inelastic collision, or hard scatter, the protons break apart. In this case, the momentum transfer squared Q2 is large, and the proton substructure can be resolved. Here, the interaction is not between the protons themselves, but between two of their constituent partons, one from each proton. Since the partons go on to initiate in the interaction essentially act as free particles, perturbative QCD (pQCD) can be applied to calculate the partonic cross section ˆσ of the two body interaction. A sketch of a generic hard scatter at the LHC can be seen in Figure 3.2. Here, the incoming protons A and B each carry a momentum PA = PB = 1 2 s, half of the center-of- √ mass (CM) energy √ s of the LHC. The proton momentum is distributed among its many partonic constituents, and as such, the partons a and b that go on to initiate the hard scatter each carry a momentum pa and pb, some fraction less than the total proton momentum. The 42 momentum fraction x of each colliding parton is defined by xa = xb = pa PA pb PB , , 0 < xa < 1 0 < xb < 1 and the parton four-vectors can be written, √ s 2 pµ a = (xa, 0, 0, xa), pµ b = √ s 2 (xb, 0, 0,−xb). (3.1) (3.2) Since each parton carries a fraction of the total proton momentum, the energy that goes into the collision is only a fraction of the total CM energy of the LHC. The energy scale Q2 of the collision can be expressed as the squared sum of the parton four-vectors, Q2 =(cid:0)pµ a + pµ b (cid:1)2 = xaxbs, (3.3) which is proportional to the momentum fraction x of each parton and the CM energy of the collider. The two partons that initiate the process scatter with a process dependent cross section ˆσ(Q2, xa, xb), which describes the probability for the event to occur, and can be calculated perturbatively as a series in αS in the framework of pQCD (to be introduced in Sec. 4.1). The probability to find a given parton with momentum fraction x at resolution scale Q2 is given by a parton distribution function (PDF) where, for example, the probability to find parton a in proton A with xa at scale Q2 is denoted fa/A(xa, Q2). These functions are non-perturbative and parton dependent, and describe how the proton momentum is distributed among its constituent partons. A detailed discussion starts in Sec. 3.2. 43 Event kinematics depend heavily on how much momenta the initial state partons carry. For example, in the production of a resonance particle, such as a W or Z boson, the degree to which a system is boosted longitudinally depends on the difference between the momentum fractions xa and xb of the incoming partons. Assuming the partons are massless and that the collision occurs head on, the rapidity, a measure of the longitudinal boost can be defined, (cid:19) (cid:18) E + pz E − pz y = 1 2 ln = 1 2 ln xa xb . (3.4) The expression simplifies when the energy E and longitudinal momentum pz are taken to be the summed components of the partonic four-vectors in Eq. 3.3. In general, since xa (cid:54)= xb, Eq. 3.4 predicts a mediating W or Z boson produced in the hard collision will move along the z-axis relative to the CM frame of the proton-proton system. The inversion of Eq. 3.4, combined with Eq. 3.3, results in an expression for the partonic x fractions in terms of measurable kinematic variables, xa = Q√ s e+y, xb = e−y. Q√ s (3.5) Thus, given the fixed CM energy to probe different values of x. √ s of the LHC, different values of Q2 and y can be used 3.1.2 QCD Factorization Theorem The QCD Factorization Theorem greatly simplifies the calculation of hadronic cross sections. It states that the hadronic cross section σAB can be reduced, or factorized, into two separate components. The first is the short-distance part, defined by the partonic cross 44 section ˆσ, which is calculable in pQCD, and the second is the non-perturbative long-distance part, codified by the PDFs f (x, Q2). This factorization is possible for two reasons: first, the partonic cross section is independent of the type of colliding hadrons; and second, the PDFs are universal, meaning they are a property of the colliding hadrons themselves and are therefore independent of the dynamics of the scattering process. According to the factorization theorem, the total hadronic cross section can be written [8], (cid:90) (cid:88) σAB = dxadxbfa/A(xa, µ2 F )fb/B(xb, µ2 F ) × [ˆσLO + αS(µ2 R)ˆσN LO + ...]ab→X (3.6) a,b where the indices a and b run over all relevant parton flavors, and the partonic cross section ˆσ is expressed as a perturbative series in terms of αS. The parameter µ2 F is known as the factorization scale, while µ2 R, on the other hand, is known as the renormalization scale. Each scale is an artifact of the perturbative approach and reflects the uncertainty in the prediction due to unknown higher-order terms. For the Drell-Yan process, each scale is generally characterized by the scale of the collision Q2, so that µ2 R = µ2 F = Q2. Different choices for µ2 R and µ2 F lead to in general different numerical results for the cross section estimate, so up and down variations of the nominal choice are often used as a way to asses the theoretical uncertainty. 3.1.3 Hadronic Events A complete picture of the collision does not end with the calculation of the hadronic cross section of Eq. 3.6. A diagram of a typical hard scatter, and the subsequent evolution of the event, is depicted in Figure 3.3. In such events, hundreds of particles are produced, with energies and momenta spanning several orders of magnitude, which results in numerous 45 Figure 3.3: An inelastic collision at the LHC. The dark red blob depicts the hard process, with its associated red lines the resulting final state radiation. Beam remnants are light blue, and initial state radiation is depicted by the dark blue lines. The purple blob is a secondary hard scatter along with its own final state radiation in purple. The light green blobs depict hadronization of any color-charged partons emanating from the hard scatter or the ISR/FSR showering. Subsequent dark green blobs corresponds to hadronic decays of unstable particles and yellow lines indicate the presence of electromagnetic radiation [9]. complex topologies and distinct final states. In such a collision, each proton breaks apart, leaving behind a remnant of the partons that did not participate in the hard scatter. The debris that results is left in a color-charged state from which a particle cascade, or parton shower, develops. Subsequent radiation and quark and gluon splittings drive the shower evolution. Eventually, the shower stops around scale of ΛQCD in a process known as hadronization, where the partons become confined within hadrons. However most hadrons are unstable and subsequently decay, sometimes multiple times, resulting in additional hadrons and radiation. The complex interplay between 46 the remaining beam remnants, and the additional partons produced in the event from splittings, the shower, and radiation, is referred to as the underlying event, which serves as a background to the original hard scatter. Despite the complexity, the phenomenology of the collision can be factorized into several distinct stages: Hard Scattering Process - the collision between partons a and b in proton A and B. The partonic cross section can be calculated at some fixed-order in perturbation theory. Multi-Parton Interactions (MPIs) - because protons are composite objects, more than one hard scatter can occur from the same proton-proton collision. Underlying Event (UE) - an unavoidable background to the hard scatter. It consists of all event particles that accompany the hard scatter; i.e., beam remnants, MPIs, and those from branchings, as well other in-time proton collisions. Initial State Radiation (ISR) - partons can radiate after the proton breaks apart but before the hard collision itself. Most ISR is soft; i.e., confined to small transverse momenta. Final State Radiation (FSR) - final state partons can undergo radiation as well. If the final state is color-neutral, like a pair of leptons, only electromagnetic radiation is possible. Beam Remnants - the extraction of the partons from the proton to initiate the hard scatter leaves behind a remnant of the proton that is no longer color-neutral. Hadronization - the non-perturbative process of transforming color-charged partons orig- inating from the hard scatter, ISR and FSR branchings, and elsewhere in the event 47 into color-neutral hadrons. Particle Decays - hadrons produced are generally unstable and subsequently decay, some- times multiple times through a decay-chain, until a stable state is reached. If they are charged, additional electromagnetic radiation can be produced. Understanding the evolution of the event in stages like this allows for robust Monte Carlo (MC) simulation samples to be developed to model the proton-proton collisions at the LHC, the subject of Section 9.1. 3.2 Parton Distribution Functions The PDFs fi(x, Q2) determine how the momentum of the proton is shared among its partonic constituents, where fi(x, Q2) is the density of parton species i per unit of momentum fraction. This is a complicated situation because each parton flavor (i.e. valence quarks, sea quarks, and gluons) shares a different fraction x of the total proton momentum. Not only that, but the momentum fraction each parton carries is strongly dependent on the energy scale Q2 of the hard scatter. Figure 3.4 illustrates the situation for the valence quarks uv(x) and dv(x), the sea quarks u(x), d(x), s(x), s(x), c(x), and c(x), and the gluon g(x), at the scales of Q2 = (2 GeV)2 and Q2 = (100 GeV)2.1 The evolution of the PDFs from one scale Q2 to another can be described analytically in the framework of pQCD with the DGLAP equations [79].2 Furthermore, it can be seen that the valence quarks dominate at low Q2 and for x (cid:38) 0.1, 1The total u quark content of the proton is the sum of its valence and sea contributions (i.e. u(x) = uv(x) + u(x)). The same holds for the d quark content (i.e. d(x) = dv(x) + d(x)). For the heavy quarks, like the strange, s(x) = s(x), since there is no valence component. 2Dokshitzer-Gribov-Lipatov-Altarelli-Parisi 48 (a) (b) Figure 3.4: CT14NNLO PDFs provided by the CTEQ collaboration. Depicted are gluon and all quark flavors as a function of x, evaluated at the scales of Q = 2 GeV (a) and Q = 100 GeV (b) [10]. with the u quark more than twice as large as the d quark, and both much larger than the heavy quarks of the sea in this region. At x (cid:39) 1/3, the u(x) and d(x) distributions peak, which corresponds to the valence part of the proton. At low x, the distributions show increasing contributions from the sea and gluon. The flavor symmetry of the quark sea is actively being explored at the LHC, with the goal of understanding how proton momentum is shared by the s(x) and s(x) distributions as compared to the u(x) and d(x) distributions. As the energy scale Q2 is increased, the contributions from the sea and the gluon grow in importance, as can be seen in the rapidly changing shapes of their distributions at very low x. Further, the peaks associated with the valence u(x) and d(x) diminish in significance, as now more of the momentum of the proton is carried by the low-x sea. In particular, the gluon distribution dominates in the region x (cid:46) 0.05. Also, the sea becomes more flavor 49 symmetric, as additional sea quarks arise from qq gluon splittings. Lastly, an important requirement for the PDFs is that they be properly normalized. This is ensured by two sum rules. The first is momentum sum rule where, (cid:90) 1 (cid:88) 0 i xfi(x, Q2)dx = 1, (3.7) which requires the total momentum fraction of each parton within the proton adds up to unity. The second is the quark-number sum rule where, (cid:90) 1 0 (cid:90) 1 0 fuv (x, Q2)dx = 2, fdv (x, Q2)dx = 1, (3.8) which ensures the number of up and down valence quarks agrees with the uud definition of the proton, independent of the scale Q2 of the scattering. 3.2.1 Extracting PDFs from Experimental Data The PDFs are non-perturbative by nature and cannot, therefore, be calculated analytically in the framework of pQCD. Nevertheless, the determination of the PDFs is crucial, as they are required for the estimation of LHC cross sections according to Eq. 3.6. The situation can be overcome by performing a global analysis, in which the PDF for each parton flavor is parametrized by an arbitrary functional form, dependent on x and Q2, and fit to an ensemble of cross section data points, spanning a large range in x and Q2 values from many experiments. The factorization theorem allows cross sections to be predicted for each experimental dataset used. The predicted cross sections are then fit simultaneously to the available experimental data, which allows the fit parameters to be extracted, and hence 50 the PDFs. The entire procedure owes its success to the universal nature of the form of the PDFs, allowing for a wide variety of experimental inputs to be used. Extracting PDFs is a major theoretical industry with multiple collaborations participat- ing. The most prominent collaborations are CTEQ [10], MSTW [80], and NNPDF [81], with each providing semi-regular updates to their PDF parameterizations, as new experi- mental data and theoretical techniques become available. However, there is no unique set of PDFs commonly accepted by the HEP community, as each group strives to provide the best parametrization possible. Differences can arise from, for example, the choice of the PDF fit parameterizations, the use and exclusion of certain input datasets, various theoretical assumptions, and the way in which experimental and theoretical errors are handled. The determination of most global fits relies on data from lepton-nucleon fixed target ex- periments, Deep Inelastic Scattering (DIS) experiments at HERA, and collider-based Teva- tron and LHC experiments [8, 82].3 These sources cover a wide range in the (x, Q2) kinematic phase space, as seen in Figure 3.5, with somewhat considerable overlap as well. In general, the PDF global analyses start by choosing suitable parameterizations for the parton distributions. These are generic functional forms that specify the valence quark distributions uv(x) and dv(x), the gluon g(x), and the qq sea distributions, and can be represented by [8] f (x, Q2 0) = a0xa1 (1 − x)a2 P (x; a3, ...). (3.9) The parameters a1 and a2 are physically motivated, and the reference value Q2 0 is chosen in the 1-2 GeV range. The first two terms alone cannot sufficiently describe the quark and gluon 3The lepton-nucleon data arises from experiments at SLAC, FNAL, and CERN, which includes e, µ, and ν DIS measurements on Hydrogen, Deuterium, and nuclear targets [83]. 51 Figure 3.5: Parton kinematics for the regions in (x, Q2) probed by the LHC and previous DIS and fixed target experiments. The scale Q2 of a process is often identified with the mass M of a produced resonance. This is indicated by four discrete values on the plot, for the LHC with s = 14 TeV [8]. √ distributions, so a third term P (x; a3, ...), is added, which provides the required flexibility to the PDF parametrization during the fit. This is a smooth function depending on one or more parameters, and separate functions exist for each parton flavor.4 The numerical values of the PDF fit parameters are extracted by performing a χ2 min- imization, which compares the measured experimental data to the theoretical expectations obtained with Eq. 3.9. The global minimum of the χ2 function defines the central, or best- fit, values {ai}0, whose uncertainty is given by the variation around the minimum of the 4For example, in the CT14NNLO PDF set, the functions P (x; a3, ...) are taken to be a linear combination of specialized orthogonal polynomials. All together, 8 parameters are associated with the valence distributions uv(x) and dv(x), 5 with the gluon g(x), and 13 with sea distributions ¯u(x), ¯d(x), and ¯s(x) for a total of n = 26 fit parameters. 52 χ2 function. DGLAP evolution then allows the post-fit PDFs to be evolved from the initial scale Q2 0 to higher scales Q2, and to be compared against data to evaluate the performance of the fit. 3.2.2 Assessing PDF Uncertainties In addition to obtaining the best-fit PDFs from the global analysis, it is also necessary as- sess their associated uncertainties (as in Fig. 3.6). Uncertainties on the PDFs arise from the experimental uncertainties of the input measurements and the theoretical assumptions made in the global fit. These uncertainties can be propagated to the PDFs, which allows for a standard PDF uncertainty to be defined. The PDF uncertainty is one of the major contributing factors to the overall theoretical uncertainty on the hadronic cross section calcu- lation of Eq. 3.6, and therefore must be assessed when performing a search or measurement at the LHC. The PDF uncertainties are estimated using the Hessian method [84, 85]. Here, a quadratic expansion of the χ2 function is made around its global minimum, which results in an n × n matrix of second-order partial derivatives known as the Hessian matrix. The Hessian matrix is then diagonalized by a set of n orthonormal eigenvectors, where n is the number of free parameters in the PDF fit (e.g. n = 26 in the CT14NNLO PDF set). The PDFs are varied along the positive and negative direction of each eigenvector, which correspond to unique variations of the nominal PDF parameters. Each variation results in a new PDF, and the set of all 2n (one set f + i for up and one set f− i for down) eigenvector variations is referred to as a PDF error set.5 5Due to the diagonalization of the Hessian matrix,the eigenvectors are admixtures of the n global PDF fit parameters, so an individual eigenvector, for example, can be composed of contributions from all parton flavors. 53 Figure 3.6: PDF uncertainties associated with the g(x), u(x), d(x), s(x), uv(x), dv(x), ¯u(x), and ¯d(x) distributions in the CT14NNLO and CT14Hera2 PDF sets evaluated at a scale of Q = 3 TeV. 54 0.80.91.01.11.210-410-310-210-10.20.50.9PDF Ratio to CT14HERA2xg(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2CT14NNLO/CT14HERA20.80.91.01.11.210-410-310-210-10.20.50.90.80.91.01.11.210-410-310-210-10.20.50.9PDF Ratio to CT14HERA2xu(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2CT14NNLO/CT14HERA20.80.91.01.11.210-410-310-210-10.20.50.90.80.91.01.11.210-410-310-210-10.20.50.9PDF Ratio to CT14HERA2xd(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2CT14NNLO/CT14HERA20.80.91.01.11.210-410-310-210-10.20.50.90.50.60.70.80.91.01.11.21.31.41.510-410-310-210-10.20.50.9PDF Ratio to CT14HERA2xs(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2CT14NNLO/CT14HERA20.50.60.70.80.91.01.11.21.31.41.510-410-310-210-10.20.50.90.80.91.01.11.210-410-310-210-10.20.50.9PDF Ratio to CT14HERA2xuv(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2CT14NNLO/CT14HERA20.80.91.01.11.210-410-310-210-10.20.50.90.80.91.01.11.210-410-310-210-10.20.50.9PDF Ratio to CT14HERA2xdv(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2CT14NNLO/CT14HERA20.80.91.01.11.210-410-310-210-10.20.50.90.80.91.01.11.210-410-310-210-10.20.50.9PDF Ratio to CT14HERA2x–u(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2CT14NNLO/CT14HERA20.80.91.01.11.210-410-310-210-10.20.50.90.80.91.01.11.210-410-310-210-10.20.50.9PDF Ratio to CT14HERA2x–d(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2CT14NNLO/CT14HERA20.80.91.01.11.210-410-310-210-10.20.50.9 With the PDF error set, the PDF uncertainty for LHC observables can be obtained. For example, in the prediction of generic hadronic cross section σ, the PDF uncertainty ∆σPDF, is determined by ∆σ+ PDF = ∆σ− PDF = (cid:118)(cid:117)(cid:117)(cid:116) n(cid:88) (cid:118)(cid:117)(cid:117)(cid:116) n(cid:88) i=1 i=1 i ) − σ(f0), σ(f− (cid:2)max(cid:0)σ(f + (cid:2)max(cid:0)σ(f0) − σ(f + i ) − σ(f0), 0(cid:1)(cid:3)2 i ), 0(cid:1)(cid:3)2 i ), σ(f0) − σ(f− (3.10) , where σ(·) is evaluated using f0, the nominal set of PDFs obtained in the global fit, and i and f− i , the + and − variations of PDF eigenvector i in the error set of n, respectively. f + The ∆σ+ term adds the PDF error contributions in quadrature that lead to an increase in the nominal value of the observable σ, while ∆σ− adds those that lead to a decrease, thus providing a definition for asymmetric errors. The plot in Figure 3.6 shows the PDF uncertainties for several individual parton flavors in the CT14NNLO and CT14Hera2 PDF sets. Generally speaking, the PDF uncertainties increase at the extreme values of x where input data is scarce. The valence distributions uv(x) and dv(x) are least precisely known at very low x, roughly for x < 10−2, and the uv(x) distribution is determined better than dv(x) at high x, roughly x > 0.1. The ¯u(x) and ¯d(x) distributions are both least precisely known at high x, again where input data is difficult to obtain. Each of these distributions plays an important role in the initial-state quark-antiquark annihilation that results in the Drell-Yan process. Each PDF set is used to model Standard Model processes in later sections of the disserta- tion. The CT14NNLO set is used to model the Drell-Yan background in the dilepton analysis, presented in the part III, while the CT14Hera2 set is used to perform exploratory studies in 55 an attempt to reduce of the high-x PDF uncertainty in part IV. 3.2.3 Summary PDFs play an essential part in all measurements and searches performed at the LHC. The state-of-the-art PDF sets provided by the PDF collaborations come with an accuracy of a few to several percent over the wide kinematic range probed by the LHC, a significant accomplishment. However, in the regions where no little or no data are provided for global analyses, such as at very large or small x, the PDF uncertainty can grow considerably and start to impact the accuracy of the theory prediction for the total cross section. This is especially evident in the high-mass dilepton spectrum of the Drell-Yan process. In the final chapter, the results of several studies are presented on how to reduce the high-mass PDF uncertainty for future iterations of the dilepton analysis. In the mean time, the next chapter introduces the Drell-Yan process in detail. 56 Chapter 4 Two Body Scattering and The Drell-Yan Process In this chapter, the Drell-Yan process is introduced and described in detail. The production of a pair of massive leptons through the Drell-Yan mechanism provides for one of the best motivated channels to search for Z(cid:48) production, and a detailed understanding therefore is essential to performing a robust search at the LHC. The goal of this chapter is to calculate the Drell-Yan production cross section, which will shed light on the nature of its production mechanism and kinematics. To accomplish this, the S-matrix and Feynman Rules are introduced, which provide the means to calculate hard scatter cross sections using a diagrammatic technique. With these ideas in place, the Drell-Yan production cross section is calculated to the leading-order in perturbation theory, and the results are discussed, with a focus on the effects of the PDFs at high dilepton mass. Lastly, higher-order diagrams associated with Drell-Yan sub-processes important to the analysis are presented. 4.1 The S-Matrix and Feynman Rules In the last chapter, it was shown that in order to calculate hadronic cross sections, one must first convolute the PDFs with an expression of the fixed-order partonic cross section ˆσ for 57 the scattering process of interest. This section introduces the concepts relevant for actually calculating these rates, given a specific interaction Lagrangian Lint of the Standard Model. 4.1.1 Scattering Amplitudes In quantum field theory, the operator S that connects and evolves an initial-state of particles |i(cid:105) undergoing a scattering process to an outgoing final-state |f(cid:105) is known as the S-matrix. The matrix elements of the S-matrix, Sf i = (cid:104)f| S |i(cid:105) , (4.1) are known as scattering amplitudes, which encode the probability amplitudes for a given initial-state to scatter into a certain final-state.1 In general, the S-matrix can be written, ∞(cid:88) n=0 (−i)n n! S = (cid:90) ∞ −∞ dx4 1 ··· (cid:90) ∞ −∞ dx4 nT [Hint (x1)··· Hint (xn)] , (4.2) which is a sum over all possible interactions. The operator T denotes a time-ordered product, ensuring interactions occur in the correct order, and the particle interactions are codified in terms of an interaction Hamiltonian Hint.2 As the S-matrix contains a complete description of the scattering process, what becomes clear from Eq. 4.2 is that it can be used to calculate scattering amplitudes order-by-order according to the series expansion. This forms the starting point to the perturbative approach 1In QED for example, the initial- or final-state is specified by a definite number of photons, electrons, and positrons with given spins, momenta, and polarizations [86]. 2The Hamiltonian density H is fully equivalent to, and can be derived from, the more standard La- grangian density L . Its use here is due to working in the interaction picture. 58 used in the calculation of all Standard Model scattering processes. A crucial point of this approach is that each Hint term carries with it one power of the coupling constant α(Q2) for the Standard Model interaction under consideration.3 The implication is that the perturbative description is only valid when the value of the coupling constant is small, such that α (cid:28) 1 for the Q2 range under study. If this is the case, the approach is highly predictive, and the calculation of only the first few terms in the expansion can result in an accurate approximation of the full amplitude. For example, the electromagnetic coupling constant is αEM ≈ 1/137, so higher-order corrections to electromagnetic processes quickly become vanishingly small, making QED an extremely precise theory. In this way, the infinite number of possible (and increasingly unlikely) particle interactions can be split up into different orders of probability. In the series expansion of S, a calculation truncated at the first non-zero term is called a calculation at the Leading-Order (LO). A calculation including the next higher-order term is a calculation at the Next-to-Leading- Order (NLO), followed by the Next-to-Next-to-Leading-Order (NNLO), etc. 4.1.2 Feynman Diagrams The calculation of physical cross sections from Eq. 4.2 is a technically challenging task, since it requires the evaluation of complex integrals over a multiple variables. Feynman’s contribution was to recognize regular structures within the integrals and devise a way to represent them graphically as Feynman diagrams. Each part of a diagram has a mathematical interpretation, and the Feynman rules [87] provide a way to translate them into an expression for the scattering amplitude, thereby circumventing the need for a detailed 3Hint is related to Lint through a Legendre transformation, which specifies the relevant coupling. 59 calculation altogether. The diagrammatic approach has the advantage of being intuitive, depicting the scattering process in terms of the exchange of force particles, and the production and decay of short- lived matter particles. The particle interactions are written in terms of propagators and vertex factors (as seen in Fig. 2.6 and Fig. 2.7 for the electroweak interaction).4 Together vertices and propagators can be combined following the Feynman rules to make Feynman diagrams. The scattering amplitude S of Eq. 4.2 can be expressed in terms of the Feynman am- plitude M,5 such that, Sf i = δf i − iMf i (2π)4 δ4(cid:0)kf − pi (cid:1) , (4.3) where ki and kf represent the four-momentum sums of the initial- and final-state particles, respectively. The entire problem boils down to obtaining the amplitude M, which amounts to drawing all topologically distinct Feynman diagrams for a given order n in the S-matrix expansion according to the Feynman rules. 4.1.3 Two Body Scattering With knowledge of the Feynman Rules, and the Lagrangian of Eq. 2.36, which specifies the electromagnetic and neutral current interactions, it is enough to write down the Drell-Yan matrix element. The final step is to combine the matrix element with known kinematic factors to obtain an expression for the cross section. 4The vertex factors are derived from interaction Lagrangian of a given theory. Likewise, the propagators are derived from free field Lagrangian. 5Likewise, the term “matrix element” is often used interchangeably. 60 Figure 4.1: Two body scattering as described by Eq. 4.5 [11]. In many body scattering (i.e. 1 + 2 → 3 + 4 + ... + n), the matrix element M appears as a factor in the differential cross section as follows [87], dˆσ = (2π)4δ4 (p1 + p2 − k3 − ... − kn) (cid:113) (p1 · p2)2 − m2 1 4 n(cid:89) j=3 |M|2 d3kj (2π)3 , 1 2Ef (4.4) 1m2 2 where the four-momenta of the initial- and final-state particles are denoted pi and kj, re- spectively. Equation 4.4 reduces significantly in the special case of two body scattering. If the collision takes place in the center-of-mass (CM) frame where E1 = E2 = E and the colliding partons are massless and have equal and opposite momenta, then [87], dˆσ = 1 64π2 (E1 + E2)2 |kf| |pi| |M|2 dΩ, (4.5) where |kf| and |pi| are the magnitudes of the outgoing and incoming momentum of either particle, respectively (See Fig 4.1). The dynamics of the process are entirely encoded in the amplitude M, with the absolute square proportional to the differential cross section. Once the amplitude M known, the calculation reduces to pure kinematics, independent of the form of the interaction, making 61 Figure 4.2: The Drell-Yan process as observed at the LHC [12]. two body scattering particularly simple. 4.2 The Drell-Yan Process The Drell-Yan process refers to the production of a massive pair of charged leptons (cid:96)+(cid:96)− from qq annihilation (See Fig. 4.2), as predicted by Drell and Yan in 1970 [88].6 In the low-energy Drell-Yan process, a quark and antiquark annihilate through the production of a virtual photon γ∗, which then decays into a pair of charged leptons. The angular distributions of the leptons depend on the precise nature of the interaction, which is specified by the matrix element M. Since present day colliders operate at sufficiently high CM energies, it is possible for the Drell-Yan process to proceed through the exchange of an on-shell Z boson as well. At the LHC, the γ∗ and Z processes interfere, and must, therefore, be considered together in the 6The leptons can be electrons, muons, or taus. The focus of this dissertation will be the e+e− electron channel. 62 calculation of Drell-Yan observables, such as the total cross section. In this way, the high- energy LO process is expressed qq → γ∗/Z → (cid:96)+(cid:96)− + X, where X denotes any additional final-state particles (e.g. radiated partons, the underlying event, etc.). Together, the mass m(cid:96)(cid:96) of the dilepton system and the mass mγ∗/Z of the intermediate state γ∗/Z can be related to the hard scale Q2 of the collision with Q2 =(cid:0)pµ 1 + pµ 2 (cid:1)2 = (cid:16) (cid:17)2 (cid:96)+ + kµ kµ (cid:96)− = m2 γ∗/Z = m2 (cid:96)(cid:96) = x1x2s, (4.6) where kµ (cid:96)+ and kµ (cid:96)− are the four-vectors of the final-state dilepton pair, pµ 1 and pµ 2 are the four- vectors of the initial-state quark-antiquark pair, and s is the CM energy of the collision. √ As a consequence, the dilepton invariant mass will exhibit a spectrum of values consistent with the x values carried by the colliding quark and antiquark in accordance with the PDFs. The Drell-Yan process itself is important for several reasons. First, it provides a means to compare theoretical calculations to experimental measurements, allowing for stress tests of pQCD.7 Second, it provides insight into the nature of the PDFs, allowing for better constraints to be placed on their parametrization.8 Third, it is the dominant background to many searches for new physics, and specifically to the one presented later in this dissertation. For these reasons, significant theoretical effort has been committed to its understanding, with the calculation of differential cross section established at up to NNLO [89, 90]. To gain insight into the dynamics of this process, the LO Drell-Yan cross section cal- culation is now performed using the tools developed thus far in this chapter. Later, the 7Consider the following example: at the LO, the transverse momentum p(cid:96)(cid:96) T of the dilepton system is zero by conservation of momentum, but at higher orders, the initial-state quarks can radiate one or more energetic gluons, which results in a non-zero dilepton p(cid:96)(cid:96) T . Precise measurements of this quantity, then, provide a way to directly test the predictions of pQCD. 8Specifically, since an antiquark is needed to initiate the Drell-Yan process, a cross section measurement provides sensitivity to the u(x) and d(x) sea distributions. 63 results of the calculation are qualitatively extended by discussing the effects of perturbative higher-order QCD and EW corrections. 4.2.1 Leading-Order Cross Section To illustrate the Feynman approach, the LO expression for the Drell-Yan cross section is calculated using Feynman diagrams. The approach is to write down the matrix element M, which encodes the dynamics of the process, and find its absolute square for insertion into Eq. 4.5 to obtain an expression for the total cross section. The calculation proceeds in the CM frame and in the massless limit. The initial- and final-state particle four-vectors are given by pµ 1 = E (1, 0, 0, 1) 2 = E (1, 0, 0,−1) pµ kµ 1 = E (1, sin θ, 0, cos θ) 2 = E (1,− sin θ, 0,− cos θ) , kµ (4.7) where pµ i refer to the quarks and kµ i to the leptons (See Fig. 4.1 for example). For simplicity, it is useful to express the kinematics in terms of three Lorentz invariant quantities, known as the Mandalstam variables, ˆs = (p1 + p2)2 = (k1 + k2)2 = 4E2 ˆt = (p1 − k1)2 = (p2 − k2)2 = − ˆs 2 ˆu = (p2 − k1)2 = (p1 − k2)2 = − ˆs 2 (1 − cos θ) (1 + cos θ) , (4.8) where the four-vectors of Eq. 4.7 have been used to simplify each expression. The parameter 64 ˆs can be identified with the squared CM energy of the q ¯q initial-state, or equivalently, the (cid:96)+(cid:96)− final-state in accordance with Eq. 4.6, and, ˆt and ˆu introduce angular dependencies. The Drell-Yan process can proceed through a virtual photon γ∗ or an on-shell Z boson at the LHC in accordance with the interaction Lagrangian of Eq. 2.36. The total amplitude M can be written as a sum, with M = Mγ∗ + MZ . Figure 4.3 depicts the corresponding Feynman diagrams. The Feynman rules state that each fermion line should be tracked backwards through the diagram, and the associated factors are written down in the direction opposite to the arrows. With this, the matrix element for the virtual photon γ∗ is (cid:104) iMγ∗ = u(k1)(−iQ(cid:96)eγµ)v(k2) (cid:105) −igµν (p1 + p2)2 [v(p2)(−iQqeγν)u(p1)] , (4.9) (4.10) and similarly, the matrix element for the Z boson is iMZ =(cid:2)u(k1)(cid:0)−igZ γµ (gLPL + gRPR)(cid:1) v(k2)(cid:3) × −igµν (p1 + p2)2 − M 2 Z + iMZ ΓZ × [v(p2) (−igZ γν (gLPL + gRPR)) u(p1)] . The initial- and final-state fermions are associated with pairs of Dirac spinors u and v, which label the momenta and spin states of the particles. Writing the spinor indices explicitly allows the total matrix element M to be expressed concisely as (cid:2)uj(k1)γµvj(k2)(cid:3)(cid:2)vi(p2)γµui(p1)(cid:3) , Mij = Gij (4.11) where the subscripts i and j index the helicity of the initial- and final-state fermions, and 65 q v(p2) −iQqeγµ u(p1) q q γ∗ −i gµν ˆs (a) (cid:96)+ v(k2) −iQ(cid:96)eγν u(k1) (cid:96)− (cid:96)+ v(p2) v(k2) −igZ γµ (gLPL + gRPR) u(p1) Z −igµν Z+iMZΓZ ˆs−M2 q (b) −igZ γν (gLPL + gRPR) u(k1) (cid:96)− Figure 4.3: The LO Feynman Diagram for Drell-Yan production mediated by a virtual photon γ∗ (a), and on-shell Z boson (b). The propagator and vertex factors of each diagram are depicted. The initial- and final-state particles are represented by the Dirac spinors u and v, which label their momenta and spin. 66 Gij groups together the couplings, charges, and propagator factors with Gij(ˆs) = e2 Q(cid:96)Qq ˆs + g2 Z i gq g(cid:96) ˆs − M 2 Z + iMZ ΓZ j . (4.12) The electroweak charges (Q, gL, and gR) for the incoming and outgoing fermions are provided in Table 2.4, and the mass and decay width of the Z Boson are denoted MZ and ΓZ , respectively. To enter the expression for the two body cross section of Eq. 4.5, the modulus squared of the matrix element M must be evaluated. As the cross sections at the LHC are unpolarized, an average over initial-state helicity modes and sum over final-state modes must be taken. Since there are two particles in the initial-state, a factor of 1/4 is included for the average, and from angular momentum conservation, the four possible helicity states are MLL, MLR, MRL, MRR. Therefore, an expression for the squared total matrix element can be written, (cid:88) i,j∈{L,R} |M|2 = |Mij|2. (4.13) The evaluation of each term in the sum involves calculating traces of products of Dirac γ matrices, and the result is |MLL|2 = |GLL(ˆs)|2 4ˆu2, |MRL|2 = |GRL(ˆs)|2 4ˆt2, |MLR|2 = |GLR(ˆs)|2 4ˆt2 |MRR|2 = |GRR(ˆs)|2 4ˆu2. (4.14) Inserting the matrix element M of Eq. 4.14 into the expression for the differential cross section of Eq. 4.5, accounting for the spin factor of 1/4, and integrating over solid angle 67 leads to the following partonic cross section, ˆσ(cid:0)q ¯q → γ∗/Z → (cid:96)+(cid:96)−(cid:1) = ˆs 48π (cid:104)|GLL(ˆs)|2 + |GLR(ˆs)|2 + |GRL(ˆs)|2 + |GRR(ˆs)|2(cid:105) (4.15) , which completes the partonic component of the LO calculation. Equation 4.15 has a strong dependence on the partonic CM energy, and hence the dilepton mass by Eq. 4.6. In the high-mass limit of ˆs (cid:29) M 2 Z , far from the Z boson mass peak, the cross section is inversely proportional to the partonic CM energy ˆσ ∼ 1/ˆs, since the amplitudes |Gij|2 ∼ 1/ˆs2. Thus, in this regime, the cross section exhibits a smooth and rapidly falling behavior.9 On the other hand, if ˆs ≈ M 2 Z , a dramatic change to the shape of the smoothly falling spectrum occurs, as it is replaced by a Breit-Weigner resonance due to the exchange of an on-shell Z boson, or a possible Z(cid:48) boson at ˆs ≈ M 2 Z(cid:48). Equation 4.15 also makes manifest the interference effect due to the exchange of either the γ∗ or the Z boson in the intermediate state. For example, cross terms from each of processes appear when squaring the amplitudes |Gij|2. Interference effects like this can play an important role if a Z(cid:48) boson is exchanged as well, which will be discussed further in Section 5.3. The final step of the calculation is to convolute Eq. 4.15 with the PDFs to obtain the LO estimate of the total cross section. This requires two modifications. First, since the colliding qq pair annihilates into a colorless final-state, it must carry one of three possible color combinations (rr, bb, gg), so an average over initial-state colors must be taken as well. This is usually expressed in terms of the QCD color factor with NC = 1/3. Second, since 9As the high-mass region is the main target of the search, any observed discrepancy that is not smoothly falling might then be an indication of new physics. 68 the incoming quark-antiquark system exhibits a spectrum of CM energies due to the PDFs, the replacement ˆs → x1x2s is made, where √ s is the CM energy of the LHC. Folding in the PDFs, as described by Eq. 3.6, results in an expression for the total hadronic cross section, (cid:90) (cid:88) dx1dx2 (cid:2)fq(x1, ˆs)f¯q(x2, ˆs) + fq(x2, ˆs)f¯q(x1, ˆs)(cid:3) (cid:104)|GLL(ˆs)|2 + |GRR(ˆs)|2 + |GLR(ˆs)|2 + |GRL(ˆs)|2(cid:105) σ = q × ˆs 48π 1 NC (4.16) , where the sum runs over the quark flavors q, and fq(x) and f¯q(x) are the associated PDFs, evaluated at the scale Q2 = ˆs = m2 (cid:96)(cid:96) of the event. 4.2.2 PDF Uncertainty Equation 4.16 is the result of the LO calculation. The expression for the cross section is given by a sum over all quark-antiquark flavors, weighted by various coupling and propagator factors. This is significant because it indicates the Drell-Yan process has direct access to individual parton flavors and their PDFs through its initial-state. Specifically, the Drell-Yan cross section receives significant contributions from the less well measured sea quark PDFs. Therefore, the Drell-Yan process can be used to get a handle on the flavor decom- position of the sea, as well as the valence distributions uv(x), dv(x), and the gluon g(x) PDFs.10 Furthermore, the Drell-Yan process at the LHC is somewhat unique in that it provides complimentary information in (x,Q2) not well covered by previous experiments, as well as extending this range significantly (See Fig. 3.5). For the purposes of the dilepton analysis, knowledge of Drell-Yan production is important 10The Drell-Yan processes at NLO is initiated by quark-gluon (and antiquark-gluon) scattering, providing a probe of the gluon distribution g(x) (see Sec. 4.2.3). 69 √ Figure 4.4: The invariant mass distribution of dilepton pairs m(cid:96)(cid:96) produced via the Drell-Yan s = 13 TeV. The MC generator ResBos, configured with the CT14Hera2 mechanism at PDF set, was used for the prediction. The PDF error, calculated with the symmetrized form of Eq. 3.10, is indicated in the sub-panel, and steadily increases with dilepton mass. because it is the dominant background to the dilepton search, so understanding it well is paramount to performing a successful analysis. However, as will be discussed in subsequent chapters, the PDF uncertainty is the dominant theoretical uncertainty at high dilepton invariant mass, the primary region of interest for the search. This uncertainty results from the imprecise knowledge of the valence and sea quark distri- butions at high x, which are required to produce dilepton states with large invariant masses as indicated by m2 (cid:96)(cid:96) = x1x2s. However, at the same time, these are the very distributions probed by the Drell-Yan process via Eq. 4.16, which means dedicated measurements of the Drell-Yan cross section may be used to constrain the PDFs affecting the high-mass search region of the dilepton analysis. 70 Dilepton Invariant Mass [GeV]5010020030010002000+X) [pb]-l+ lfi*/Z g fi(pp s7-106-105-104-103-102-101-10110210310 = 13 TeVs CT14Hera2 ResBosDrell-YanPDF ErrorDilepton Invariant Mass [GeV]50100200300400100020003000 Rel. Error 0.80.850.90.9511.051.11.151.2 The effect of the PDFs on the description of the dilepton spectrum is shown in Figure 4.4. Here, the ResBos [91, 92, 93] MC generator, configured with the CT14Hera2 [94] set of PDFs was used.11 Samples were generated in slices of dilepton invariant mass, weighted, and stitched together. The PDF error is calculated bin-by-bin with the CT14Hera2 error set PDFs using the symmetrized form of Eq. 3.10. At m(cid:96)(cid:96) = 1 TeV, the uncertainty is roughly 5%, but by m(cid:96)(cid:96) = 5 TeV it stands at 30%. Accurate knowledge of the PDFs is crucial for direct searches of new physics (like in this dissertation), but it is also crucial for indirect searches through precision measurements. Future iterations of the dilepton analysis could benefit greatly from new global fit measure- ments, which translate directly into improved discovery potential, since the PDF uncertainty would become further constrained. At the same time, LHC measurements of Drell-Yan might help resolve disagreements between the various global fit datasets offered by the PDF fitting groups, via its enhanced sensitivity to the strange quark and gluon PDFs. In the final chapter, an exploratory study is presented which aims to reduce the PDF uncertainty associated with Drell-Yan production by incorporating new measurements in the PDF global fits. The achieved error reduction is quantified and the results show future LHC measurements can indeed improve the PDF descriptions. 4.2.3 Higher-Order Corrections The Feynman diagrams in Figure 4.3 show the LO contributions to the Drell-Yan process, but countless other sub-processes contribute to qq → γ∗/Z → (cid:96)+(cid:96)− + X as well. These sub-processes include additional final-state particles that arise from initial- or final-state 11MC generators are the subject of Section 9.1. The generator ResBos calculates fully differential cross sections for electroweak processes, such as neutral current Drell-Yan, at up to the NNLO in QCD. 71 ¯q q ¯q g q (a) γ∗/Z (c) g γ∗/Z l+ l− q g ¯q q γ∗/Z (b) q γ γ∗/Z (d) Figure 4.5: Example higher-order contributions to the Drell-Yan process: NLO initial-state gluon radiation qq → γ∗/Z +g (a), NLO initial-state quark radiation qg → γ∗/Z +q (b), NLO virtual gluon correction qq → γ∗/Z (c), NLO final-state photon radiation qq → γ∗/Z +γ (d). radiation (i.e. from photons or gluons), as well as internal particle loops, which arise from the combination of two or more interaction vertices according to the Feynman rules.12 Sub-processes like this are known as higher-order corrections, since they originate from higher-order terms in the S-matrix expansion. For example, at LO that only quark- antiquark annihilation can occur, while at NLO quark-gluon (and antiquark-gluon) scattering becomes possible, and at NNLO gluon-gluon scattering opens up as well. Some examples of this are shown in Figure 4.5. Each sub-processes acts to correct to the LO estimate obtained for the Drell-Yan cross section. In general, higher-order corrections due to the strong interaction are larger than those of EW interaction at the same order in perturbation theory, since the strong coupling αS is 12Each order in the series expansion of M includes one additional vertex factor, and therefore, the number of diagrams at a given order increases like a factorial (due to the possible permutations), resulting in complex multi-loop diagrams. 72 much larger than the electromagnetic αEM . QCD corrections to the Drell-Yan process can be sizeable, and both types are accounted in the description of the Drell-Yan background for the dilepton analysis (as described in Sec. 9.4.5). Higher-order corrections also introduce qualitatively new features to the kinematics of the LO process. For example, at LO the dilepton transverse momentum p(cid:96)(cid:96) T is identically zero, but at NLO and beyond, it is non-zero due to the emission of initial-state radiation, allowing for the γ∗/Z to recoil in the transverse plane of the reaction. Beyond LO calculations can be extremely challenging. It seems like at this point, all that is required to calculate the cross section to any accuracy is to draw the relevant Feyn- man diagrams for the process under consideration, and follow feynman rules to obtain an expression for the matrix element. However, a technicality arises beyond the LO in the S-matrix expansion: loop diagrams are encountered, which when evaluated lead to divergent integrals in the calculation of the cross section. Finite results can only be obtained after the renormalization of the theory, which is a way to systematically handle the aforementioned divergences. 73 Chapter 5 Physics Beyond the Standard Model The Standard Model is an extremely successful theory and one of the best tested in the history of physics. However, despite its outstanding successes, there are multiple experi- mental and theoretical indications that the Standard Model is an incomplete theory, which suggests it is only a part or an aspect of one more fundamental. In the following chapter, the limitations of the Standard Model are discussed and an overview of several Beyond the Standard Model (BSM) theories is presented. The emphasis is placed on models that predict new high-mass dilepton resonances, which are the main target of the search performed in this dissertation. 5.1 Limitations of the Standard Model Over the past several decades, experiments in the fields of cosmology, astronomy, neutrino physics, and collider physics have uncovered phenomena that cannot naturally be explained within the framework of the Standard Model. These phenomena include: Neutrino Masses: Neutrinos are massless in the Standard Model, but recent experiments [95] have established that the three generations of neutrinos mix, or undergo flavor os- cillations, as they travel, and therefore must have tiny but non-zero masses.1 The 1This can only happen if there is a mismatch between between the neutrino mass and flavor eigenstates analogous to the situation in the quark sector. 74 matrix responsible for the mixing is known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, just like the CKM matrix of the quark sector. However, unlike the CKM matrix, the off-diagonal entries of the PMNS matrix are large, indicating there is a much more substantial amount of mixing in the lepton sector. The reason for such asymmetric behavior between the quark and lepton sectors is unknown, as are the origin and nature of the neutrino masses; specifically, why they are so many orders of magnitude smaller than the other fundamental fermions. The current limit on the neutrino masses is mνe + mνµ + mντ < 0.12 eV (95% C.L.) [96]. The neutrinos are the only electrically neutral fermions in the Standard Model, and as such, they could be described with Majorana,2 as opposed to Dirac fields, unlike all other Standard Model fermions. If this is the case, it has profound implications for the mechanism responsible for neutrinos mass generation [97].3 Dark Matter and Dark Energy: Cosmological measurements have determined that only 5% of the universe is comprised of ordinary baryonic matter, as described by the Stan- dard Model, while the rest consists of entirely unknown forms, known as dark matter (≈ 23%) and dark energy (≈ 72%). The first evidence of dark matter came from measurements of the orbital velocities of stars and gas in spiral galaxies [99]. The ob- served dynamics could only be explained if there was some non-luminous component of matter present. Now the existence of dark matter is supported by many other astro- 2A Majorana fermion is its own antiparticle. 3If neutrinos are Dirac particles, their masses can be generated through the Higgs Mechanism by introduc- ing right-handed neutrinos into the Standard Model (analogous to the generation of the up- and down-type quark masses from Eq. 2.27). However, since the neutrino masses are known to be at or below the eV scale, their Yukawa couplings would be unnaturally small. The alternative is to consider them as Majorana particles, in which case, the See-Saw mechanism would be responsible for their mass generation. Left-Right symmetric models [98] can naturally accommodate these ideas, which predict heavy right-handed versions of the W and Z bosons of the Standard Model, making them a natural target for the search performed in this dissertation. 75 physical and cosmological observations, including gravitational lensing measurements and features in the cosmic microwave background [100]. Due to its observed proper- ties, a potential dark matter particle cannot interact via the strong or electromagnetic forces. The neutrino is the only particle in the SM for which this applies, but neutrinos can only account for a small fraction of the observed dark matter because they are so light, so a viable candidate is still missing. Likewise, the nature of dark energy is just as mysterious. Astronomers have measured the rate of expansion of the universe, and determined it is accelerating [101]. This can be accommodated in the framework of general relativity with the addition of a small non-zero cosmological constant; how- ever, attempts to derive its value within the Standard Model lead to predictions vastly different from observation. Matter-Antimatter Asymmetry: A common assumption is that matter and antimatter are created in equal amounts in the Big Bang. However, another set of cosmological observations indicate that the visible fraction of the universe is composed almost en- tirely of matter with comparatively little antimatter present [102]. This imbalance is known as the matter-antimatter asymmetry. One way to account for this asymmetry is to introduce interactions that treat matter and antimatter differently. Indeed, CP violating interactions are firmly established in the Standard Model, but their effect is too small to account for the observed imbalance, suggesting new BSM processes may be responsible. Gravity: The Standard Model provides a successful description of the strong, weak, and electromagnetic forces, but leaves gravity out, as it is currently unknown how to de- scribe gravitational interactions quantum mechanically. Likewise, the Standard Model 76 does not offer an explanation as to why the gravitational force is so many orders of magnitude weaker than the other fundamental forces. Attempts to develop a quantum theory of gravity consistent with the Standard Model are an active area of research. In addition to the above experimental phenomena, there are several theoretical aspects of the Standard Model that are not well understood. The following list highlights some of the areas of concern: Fermion Generations: The Standard Model contains exactly three generations of quarks and three generation of leptons. This is not a consequence of the theory, but rather it is an experimental fact. Each quark and lepton in the Standard Model seems to be replicated three times, differentiated only by their masses. Why there are three gener- ations, and why each generation differs so much in mass is currently unknown. A more compelling model would explain the seemingly arbitrary nature of three generations and why there are not more or less. Hierarchy Problem: Quark and charged lepton masses in the Standard Model are gen- erated through the Higgs mechanism (See Sec 2.3.4). In the Standard Model, the Higgs boson mass receives large quantum corrections through its interaction with vir- tual particles, specifically the top quark. Whereas most other loop corrections in the Standard Model scale by factors of ln Λ, the corrections to the Higgs mass scale addi- tively and quadratically like Λ2. If the Standard Model is valid up to the Plank scale, Λ = MPlank ≈ 1019 GeV, then the size of these corrections is actually much larger than the observed mass of the Higgs boson. The relation between the bare mass and H,0 + O(Λ2), with MH,0 denoting the observed mass of the Higgs boson is M 2 bare mass and O(Λ2) the size of the corrections. Since the observed value of MH is H = M 2 77 around 125 GeV, an extremely precise cancellation must take place between the bare mass term and the correction term. This unnatural fine tuning of the bare Higgs mass due to the discrepant scales involved is known as the hierarchy problem [103]. Free Parameters: The Standard Model has 19 free parameters. With the introduction of neutrino masses, and the phases and mixing angles of the PMNS matrix, the number of free parameters increases to 26. The values of these parameters cannot be explained by the Standard Model, nor do there seem to be any coherent relations between them; rather they have to be determined directly from experiment. With so many free param- eters, the Standard Model seems too ad-hoc to be the fundamental theory, suggesting there are unresolved conceptual issues. Therefore, the search for new physics is moti- vated not just by the experimental observations outlined above, but also by the desire to simplify what seems like an overly complicated and arbitrary theory. Taken together, these experimental and theoretical difficulties suggest the Standard Model is part of a more fundamental theory. Several candidate theories that attempt to extend the Standard Model are presented in the following section. 5.2 Theories Beyond the Standard Model Many Beyond the Standard Model theories have been developed in an attempt to address one or more of the limitations presented in Section 5.1, and some of the most well-motivated ones are presented here. They include: Supersymmetry: In Supersymmetry (SUSY), each particle in the Standard Model is as- signed a “superpartner” particle, whose spin differs by 1/2 compared to its Standard 78 Model counterpart; i.e., each boson receives a fermion partner and vice-versa. Since these partners have not been observed, SUSY must be broken at some higher en- ergy scale, making the partners much heavier than their Standard Model counterparts. SUSY provides a solution to the hierarchy problem. The quadratic divergence in the squared Higgs mass is stabilized through a series of cancellations between loop diagrams involving the superpartners and those involving particles of the Standard Model. Like- wise, in SUSY models that conserve R-parity,4 the lightest SUSY superpartner (LSP) is stable and can function as a viable dark matter candidate. Extra Dimensions: Extra Dimensional models extend the Standard Model with extra spa- tial dimensions in an attempt to solve the hierarchy problem. In these models, the Standard Model particles are localized to the usual (3+1)-dimensions of spacetime, but the graviton, the force mediator of gravitation, is allowed to propagate into the extra spatial dimensions, which explains why gravity seems so much weaker than the other fundamental forces. As a result, the true strength of gravity is comparable to the other forces, and the Plank scale is reduced to the electroweak scale, thus solving the hierarchy problem. Two well-known models are the ADD model [104], proposed by Arkani-Hamed, Dimopoulos, and Dvali, and the Randall-Sundrum (RS) model [105]. Grand Unified Theories: Grand Unified Theories (GUTs) propose that the SU (3)C × SU (2)L×U (1)Y gauge symmetry of the Standard Model originates from a single, larger symmetry group, which provides a deeper relation between the quarks and leptons. In GUTs, all interactions are described by a local gauge theory with a single running coupling constant, effectively unifying the separate forces of the Standard Model into 4The Standard Model particles are given an R-parity of +1 while the supersymmetric particles receive an R-parity of -1. If R-parity is conserved, then the lightest superpartner is necessarily stable. 79 a single fundamental interaction. The scale of this unification is known as the GUT scale with MGUT ≈ 1016 GeV, similar to how the weak and electromagnetic forces unify at the electroweak scale. The original GUT model was based on the gauge group SU (5) and was proposed by Glashow and Georgi in 1974 [106]. This is the simplest group that can contain the Standard Model; examples of other unifying groups include SO(10) and E6 [107, 53]. GUTs are attractive because their phenomenology touches on almost every problem outlined in Section 5.1. However, a problematic feature shared by many GUTs is the prediction that the proton decay, with a lifetime much larger than 1030 years, which has not been observed experimentally [108]. Likewise gravity is not included as part of the unification. 5.3 Models with Extended Gauge Symmetries Many BSM theories, including each of those outlined in Section. 5.2, predict the existence of one or more new high-mass bosons, which can decay into lepton-antilepton, or dilepton, pairs of Standard Model particles. These include such objects as a spin-0 sneutrino of R- parity violating SUSY, a spin-2 graviton as in the RS extra dimensional model, or a spin-1 force mediating gauge boson that arises from a new U (1) symmetry, predicted in many GUTs [109]; and these are just to name a few.5 Because this feature is common to so many extensions of the SM, searches for new high-mass bosons are highly motivated, and are therefore some of the first to be conducted at newly commissioned hadron colliders. The focus of this dissertation will be on the case where the Standard Model gauge group 5Of course there could also exist particles that no one has yet theorized. 80 is extended to include at least one additional U (1)(cid:48) symmetry [107, 110], so that it becomes SU (3)C × SU (2)L × U (1)Y × U (1)(cid:48). (5.1) The breaking of this symmetry results in a new massive neutral spin-1 gauge boson, known as the Z(cid:48) boson, the signature of which would be a resonance in the invariant mass spectrum of its decay products at the pole mass of the new particle. For the analysis presented in this dissertation, the Z(cid:48) decay products are taken to be the electron and positron; i.e., the search is performed in the so-called dielectron final-state, or the “electron channel.” Moreover, as part of the dilepton analysis, a search is also performed in the dimuon final-state, or “muon channel,” but this is not presented here (see Ref. [111] for details). 5.3.1 Dilepton Phenomenology The general Lagrangian for the interaction between a hypothetical Z(cid:48) state and pair of fermions is [110] Z(cid:48) = gZ(cid:48) ¯f γµ(cid:16) L (cid:17) f Z(cid:48) µ, PR PL + zfR where gZ(cid:48) is the new U (1)(cid:48) gauge coupling constant, zfL handed fermion charges under the new U (1)(cid:48) symmetry, and PL and PR are the standard are the left- and right- and zfR zfL (5.2) projection operators. This interaction is phenomenologically identical to the Standard Model neutral current interaction defined in Eq. 2.36, and is only differentiated by the values taken by the new U (1)(cid:48) fermion charges and gauge coupling constant.6 The associated Feynman 6For the search performed in this dissertation, family universality is assumed among the U(1)(cid:48) fermion charges, such that, for example, the values taken by the electron, muon, and tau are identical. 81 q v(p2) −igZ(cid:48)γµ (zLPL + zRPR) u(p1) q Z(cid:48) −igµν Z(cid:48)+iMZ(cid:48)ΓZ(cid:48) ˆs−M2 (cid:96)+ v(k2) −igZ(cid:48)γν (zLPL + zRPR) u(k1) (cid:96)− Figure 5.1: Leading Order Feynman diagram for Z(cid:48) production at the LHC. The propagator and vertex factors are depicted, and the initial- and final-state fermions are represented by the Dirac spinors u and v, which label their momenta and spin. diagram for Z(cid:48) production can be seen in Figure 5.1. The result of extending the electroweak sector with an additional neutral current inter- action leads to mass and kinetic mixing between the Z boson of the Standard Model and the BSM Z(cid:48) boson [109]. This can result in tiny modifications to the nominal mass MZ and width ΓZ of the Standard Model Z boson; however, precision measurements have put stringent constrains on such mixing, and in this analysis it can be safely ignored [109]. The matrix element for the Drell-Yan process in Eq. 4.11 of Section 4.2.1 can be extended to include a term for the new Z(cid:48) boson with M = Mγ∗ +MZ +MZ(cid:48). Applying the Feynman rules to the diagram in Figure 5.1 allows the amplitude of Eq. 4.12 to be extended so that, Gij = e2 Q(cid:96)Qq ˆs + g2 Z igq gl ˆs − M 2 Z + iMZ ΓZ j + g2 Z(cid:48) where the Z(cid:48) mass is MZ(cid:48) and its width is ΓZ(cid:48). ˆs − M 2 j izq zl Z(cid:48) + iMZ(cid:48)ΓZ(cid:48) , (5.3) Despite the addition of this new intermediate state, the general expression for the Drell- 82 Yan cross section remains identical to what was derived in Eq. 4.16. By simply inserting Eq. 5.3 into the expressions for the cross section of Eq. 4.16, the effect of the new Z(cid:48) state is fully included. However, the inclusion of the Z(cid:48) as a possible exchange particle gives rise to new model- dependent interference effects, which can be traced back to the cross terms present in |M|2. The size and nature of these effects depend on the values taken by the new gauge coupling constant gZ(cid:48) and the U (1)(cid:48) fermion quantum numbers zfL,R . The total Z(cid:48) decay width to fermion-antifermion pairs can be written, ΓZ(cid:48)→f ¯f = NC gZ(cid:48) MZ(cid:48) 24π z2 fL + z2 fR , ΓZ(cid:48) = ΓZ(cid:48)→f ¯f , (5.4) (cid:16) (cid:17) (cid:88) f where the color factor NC is 1 for lepton final-states, and 3 for quark final-states, and the Z(cid:48) mass is assumed to be much heavier than its decay products. Often models with extended gauge symmetries require the existence of additional exotic fermions in order to be self consistent. If these exotic decay modes are kinematically acces- sible, then the size of the nominal Z(cid:48) width in Eq. 5.4 is increased, which has the effect of reducing the branching fractions to the conventional Standard Model fermions. In gen- eral, these effects are model-dependent in nature, and are therefore neglected in the dilepton analysis.7 The branching fraction of the Z(cid:48) into fermion-antifermion pairs is 7Likewise, for each of the Z(cid:48) models considered in the analysis, the couplings that specify the cubic and quartic gauge interactions to other Standard Model bosons are set to zero; i.e., only Standard Model decays to fermions are considered in the expression for the Z(cid:48) width. This choice is avoids the introduction of model-dependent degress of freedom into the interpretation of the search results. (5.5) Br(cid:0)Z(cid:48) → f ¯f(cid:1) = Γf ¯f ΓZ(cid:48) . 83 Once the Z(cid:48) mass and fermion couplings are defined for the model under consideration, the relative Z(cid:48) width ΓZ(cid:48)/MZ(cid:48) can be calculated, which functions as a useful metric for comparisons between different models. For the models considered in the search, the relative width varies over a wide range, from less than 1% to upwards of 40%. Among these are The Sequential Standard Model (SSM), the E6 model, and the Minimal Z(cid:48) model, which are now presented. 5.3.2 The Sequential Standard Model In the dilepton analysis, the Sequential Standard Model (SSM) Z(cid:48) [107] is used as a useful benchmark model. Here, the Z(cid:48) SSM is assigned the same couplings to fermions as the Standard Model Z boson, but has a different mass and width (via Eq. 5.4). Similarly, the gauge coupling constant gZ(cid:48) is taken to be identical to gZ . As a result, the Z(cid:48) SSM has a relatively large width relative to its mass, with a value of 3.0%. Strictly speaking, aside from the difference in mass, the Z(cid:48) SSM is identical to the Standard Model Z boson, and can be seen in Figure 5.2 for values of MZ(cid:48) = 1 TeV and MZ(cid:48) = 3 TeV. The fermion charges in the SSM are defined, f − qf sin2 θW = t3 = −qf sin2 θW , zfL zfR (5.6) where t3 is the weak isospin, q is the electric charge, and θW is the Weinberg angle. The values associated with each of these parameters are presented in Table 2.4 of Section 2.3.6. The Z(cid:48) SSM is not a well-motivated model from a theoretical point of view due to its strict replication of Standard Model parameters. However, as an entry point, the Z(cid:48) SSM serves as 84 (a) Figure 5.2: Predicted Z(cid:48) SSM event yields for MZ(cid:48) = 1 TeV (a) and MZ(cid:48) = 1 TeV (b). The vertical axis displays the number of predicted Z(cid:48) events per given unit of integrated luminosity, as would be observed at the √ s = 13 TeV LHC. (b) a useful benchmark for comparisons between experiments, as well as comparisons against other more theoretically motivated models considered in this analysis. 5.3.3 The E6 Model The E6 model [53, 112, 110] is a GUT which predicts the symmetry group at the GUT scale is the gauge group E6. The successive decomposition of this group results in the gauge group of the Standard Model plus two additional abelian U (1)(cid:48) groups, the mixing of which results in a Z(cid:48) boson that could be potentially observable at the TeV scale. At the GUT scale, the E6 symmetry is broken in three stages: first, E6 → SO(10) × U (1)ψ, followed by SO(10) → SU (5) × U (1)χ, and finally SU (5) breaks to the Standard Model group defined by Eq. 5.1. Schematically, this can be written, E6 → SO(10) × U (1)ψ → SU (5) × U (1)χ × U (1)ψ → SM × U (1)θE6 . (5.7) 85 Dielectron Invariant Mass [GeV]900920940960980100010201040106010801100-1Events/pb00.010.020.030.040.050.060.070.08 InternalATLAS = 13 TeVs = 1 TeVSSMZ'MSSMZ'Dielectron Invariant Mass [GeV]260027002800290030003100320033003400-1Events/pb00.020.040.060.080.10.120.140.163-10· InternalATLAS = 13 TeVs = 3 TeVSSMZ'MSSMZ' qψ fL qχ fL qψ fR qχ f fR u +1 −1 −1 +1 d +1 −1 −1 −3 ν(cid:96) +1 +3 −1 +5 (cid:96) +1 +3 −1 +1 Table 5.1: Standard Model fermion charges under the U (1)ψ and U (1)χ symmetries of the E6 GUT model. Once the mixing angle θE6 is specified, the Z(cid:48) Lagrangian of Eq. 5.2 is uniquely defined, and can be used to make E6-related predictions. If a linear combination of the two abelian symmetries U (1)χ and U (1)ψ is broken at the TeV scale, a physical Z(cid:48) state results, where, Z(cid:48)(θE6) = Z(cid:48) ψ cos θE6 + Z(cid:48) χ sin θE6. (5.8) Here, the degree of mixing is determined by the angle θE6 ∈ [0, π], which is a free parameter of the theory. The Z(cid:48) gauge coupling constant gZ(cid:48) is normalized in such a way that the Standard Model gauge groups are unified into SU (5) at the GUT scale, where, (cid:114)5 3 gZ(cid:48) = (cid:114)5 3 gZ sin θW = g tan θW . (5.9) Further, the mixing angle θE6 determines the values of the U (1)(cid:48) fermion charges, qψ √ fL 2 6 qψ √ fR 6 2 cos θE6 − qχ √ fL 2 10 cos θE6 − qχ √ fR 10 2 zfL = zfR = sin θE6 sin θE6, (5.10) where qψ fL,R and qχ fL,R are the left- and right-handed fermion charges under the U (1)ψ and U (1)χ symmetries, which are depicted in Table 5.1. 86 (a) (b) Figure 5.3: Predicted Z(cid:48) event yields for various choices of θE6 in the E6 model with MZ(cid:48) = 1 TeV (a) and MZ(cid:48) = 1 TeV (b). Starting from the Z(cid:48) ψ value of θE6 = 0, E6 models will exhibit increasing cross sections and decay widths, until the Z(cid:48) χ value of θE6 = π/2 is reached, at which point a decrease is observed. All E6 models fall within the range defined χ. The vertical axis displays the number of predicted Z(cid:48) events per given unit by Z(cid:48) of integrated luminosity, as would be observed at the √ s = 13 TeV LHC. ψ and Z(cid:48) Several well-motivated choices for θE6 exist [110], and once its value is specified, associ- ated Z(cid:48) observables can be determined, such as the decay width via Eq. 5.4 and cross section via Eq. 4.16. The names of these well-motivated E6 models are listed in Table 5.2. Several Z(cid:48) signals predicted in E6 model can be seen in Figure 5.3 for different values of θE6. In the case of the Z(cid:48) the Z(cid:48) ψ, the relative width takes on a minimum value of 0.5%, while for χ it is maximized at 1.2%. The values for all other E6 models fall in between. Model Z(cid:48) 0.0 θE6 ψ Z(cid:48) N Z(cid:48) η Z(cid:48) 0.419π 0.710π 0.210π 0.129π 0.5π Z(cid:48) Z(cid:48) I S χ Table 5.2: E6 models for several motivated choices of mixing angle θE6. From left to right, these are the ψ model, the neutral model, the η model, the inert model, the secluded sector model, and the χ model. 87 Dielectron Invariant Mass [GeV]900920940960980100010201040106010801100-1Events/pb00.010.020.030.040.050.060.070.08 InternalATLAS = 13 TeVs = 1 TeVSSMZ'M)p = 0.50 6Eq (cZ')p = 0.71 6Eq (IZ')p = 0.21 6Eq (hZ' = 0)6Eq (yZ'Dielectron Invariant Mass [GeV]260027002800290030003100320033003400-1Events/pb00.020.040.060.080.10.120.140.163-10· InternalATLAS = 13 TeVs = 3 TeVSSMZ'M)p = 0.50 6Eq (cZ')p = 0.71 6Eq (IZ')p = 0.21 6Eq (hZ' = 0)6Eq (yZ' 5.3.4 The Minimal Model The Minimal Z(cid:48) model [113, 114] attempts to describe Z(cid:48) phenomenology according to a minimal set of assumptions. Unlike the various GUT scenarios, in the Minimal model, no exotic fermions are required to generate an anomaly free theory. Here, only three right- handed neutrinos (one for each fermion generation) are required. The Standard Model gauge group of Eq. 5.1 is extended by a U (1)B−L symmetry, so that it becomes SU (3)C×SU (2)L× U (1)Y × U (1)B−L. The new Z(cid:48) Min boson results from the mixing between the U (1)Y symmetry of the Stan- dard Model and the new U (1)B−L symmetry. As a result, the generator of the new U (1)(cid:48) symmetry is a linear combination of Standard Model weak hypercharge Y and baryon-minus- lepton number B − L. The phenomenology of Z(cid:48) production and decay in the Minimal model is characterized by two effective effective coupling constants: gB−L, associated with the B − L current, and gY , with the weak hypercharge Y where, gY = γ(cid:48)gZ sin θMin gB−L = γ(cid:48)gZ cos θMin. (5.11) Here, the mixing angle θMin controls the level of mixing between the U (1)Y and U (1)B−L gauge groups, and the relative coupling γ(cid:48) measures the strength of the Z(cid:48) boson gauge coupling relative to that of the Z boson of the Standard Model. Each of these are free parameters in the Minimal model. Just like in the context of the E6 model, specific choices for γ(cid:48) and θMin characterize well- known Z(cid:48) states, several of which are listed in Table 5.3. These include the Z(cid:48) χ, which belongs 88 Z(cid:48) B−L (cid:113)5 8 sin θW 0 Model γ(cid:48) sin θMin χ Z(cid:48) (cid:113)41 −(cid:113)16 (cid:113)25 41 Z(cid:48) 3R (cid:113)25 −(cid:113)4 (cid:113)1 5 24 sin θW 12 sin θW Table 5.3: Several Z(cid:48) right, these are the pure Z(cid:48) symmetric model. 1 cos θMin Min models for well-motivated choices of θMin and γ(cid:48). From left to 3R of the left-right χ of the E6 model, and the Z(cid:48) B−L model, the Z(cid:48) 41 5 YfL f YfR u +1 6 +2 3 6 −1 d +1 3 ν(cid:96) −1 0 2 −1 2 −1 (cid:96) (B − L)fL (B − L)fR +1 3 +1 3 −1 −1 +1 3 +1 3 −1 −1 Table 5.4: Quantum numbers associated with Standard Model fermions according to Eq. 5.12. Once the mixing angle θMin and relative coupling γ(cid:48) are specified, the Z(cid:48) La- grangian of Eq. 5.2 is uniquely defined and can be used to make Minimal model predictions. to the E6 model of the previous section, the Z(cid:48) and the Z(cid:48) B−L of the pure B − L model [117].8 3R of the left-right symmetric model [115, 116], The left-handed and right-handed fermions charges under the new U (1)(cid:48) symmetry are zfL = gY YfL zfR = gY YfR + gB−L (B − L)fL + gB−L (B − L)fR , (5.12) where Yf = Qf − t3 f is the standard weak hypercharge, and (B − L)f is the fermion Baryon Number minus its Lepton Number, whose values are listed explicitly in Table 5.4. Several Min signal scenarios are depicted in Figure 5.4 as the values of θMin and γ(cid:48) are varied. The Z(cid:48) Z(cid:48) width is sensitive to these choices, and taking the Z(cid:48) B−L models as examples, the 8A common point in Z(cid:48) parameter space is shared between both the Minimal model and E6 model for χ boson. 3R and Z(cid:48) the Z(cid:48) 89 (a) Figure 5.4: Predicted Z(cid:48) Min event yields for MZ(cid:48) = 3 TeV according to Eq. 5.12. In (a), θMin is held constant for several distinct values of γ(cid:48). Similarly, in (b), γ(cid:48) is held constant while θMin is varied. The vertical axis displays the number of predicted Z(cid:48) Min events per given unit (b) of integrated luminosity, as would be observed at the √ s = 13 TeV LHC. width varies from < 1% to up to 12.8% and 39.5% respectively for the γ(cid:48) ranges considered in the dilepton search. 5.4 Previous Experimental Limits Both direct and indirect searches for Z(cid:48) bosons have been conducted at several previous col- lider experiments. Early results were obtained from the LEP experiment at CERN, followed by the DØ and CDF experiments at the Tevatron, and more recently, the ATLAS and CMS experiments at the LHC. Examples of previous limits on the Z(cid:48) mass are shown in Table 5.5. Some of the earliest limits on Z(cid:48) production come from indirect measurements taken at the LEP experiment. LEP searched for Z(cid:48) bosons in several fermionic final-states via e+e− → Z(cid:48) → f ¯f . Here, the Z(cid:48) state would interfere with the intermediate state γ∗/Z of Drell-Yan, modifying the total cross section and angular distributions of the final-state fermions. The 90 Dielectron Invariant Mass [GeV]260027002800290030003100320033003400-1Events/pb00.050.10.150.20.250.30.350.40.450.53-10· InternalATLAS = 13 TeVs = 3 TeVMinZ'M) = -2 Minqtan(' = 0.10)g (MinZ'' = 0.30)g (MinZ'' = 0.50)g (MinZ'' = 0.75)g (MinZ'' = 1.00) g (MinZ'' = 1.50) g (MinZ'Dielectron Invariant Mass [GeV]260027002800290030003100320033003400-1Events/pb00.050.10.150.20.250.30.350.40.450.53-10· InternalATLAS = 13 TeVs = 3 TeVMinZ'MWq sin85' = g = 0)Minq (MinZ'/20)p = 4Minq (MinZ'/20) p = 8Minq (MinZ'/20) p = 12Minq (MinZ'/20) p = 16Minq (MinZ' DØ CDF CMS ATLAS CMS ATLAS CMS 1.96 1.96 8 8 13 13 13 13 5 5 20 20 3 3 36 36 ψ Z(cid:48) 0.903 0.930 Lower Limit on MZ(cid:48) [TeV] Z(cid:48) SSM 1.023 1.071 2.90 2.90 3.18 3.36 4.5 4.5 Z(cid:48) 0.891 0.917 2.57 2.51 2.70 2.74 3.9 3.8 - 4.1 2.62 3.05 χ - - Reference [120] [121] [122] [123] [124] [111] [125] ee µµ (cid:96)(cid:96) (cid:96)(cid:96) (cid:96)(cid:96) (cid:96)(cid:96) (cid:96)(cid:96) (cid:96)(cid:96) Collaboration √ s [TeV] L [fb−1] Channel ATLAS* Sec. 11.4.2 Table 5.5: Observed limits at 95% C.L. on the mass of a Z(cid:48) boson from previous experimental searches at hadron colliders. The CMS analyses listed do not provide limits on the Z(cid:48) χ, which would otherwise be slightly higher than what was obtained for the Z(cid:48) ψ. The integrated luminosity for each analysis is rounded to nearest whole number. The (cid:96)(cid:96) channel refers searches that combine individual electron and muon channels. The analysis presented here is denoted with “*.” strongest indirect limits come from a combined analysis using the data taken by the four LEP experiments, which combines differential cross section and angular measurements from multiple final-states (See. Sec. 3.5 of Ref. [118]). The result is a lower mass limit of 1.787 TeV on the Z(cid:48) SSM, and 0.481 TeV and 0.673 TeV on the Z(cid:48) χ, respectively [119]. ψ and Z(cid:48) Following LEP, the DØ and CDF collaborations at the Tevatron set direct limits on Z(cid:48) production using proton-antiproton collision data at s = 1.96 TeV. These were the √ strongest direct limits set before the operation of the LHC. The DØ limits [120] were set in the electron channel with 5.3 fb−1 of data, while the CDF limits [121] were set with 4.6 fb−1 in the muon channel. For DØ, this resulted in lower mass limits of 1.023, 0.891, 0.903 TeV for the Z(cid:48) χ respectively. Likewise, for CDF the results were 1.071, 0.917, and ψ, and Z(cid:48) SSM, Z(cid:48) 0.930 TeV. The LHC currently holds record as the highest energy particle collider, and as such, its ability to set exclusion limits on Z(cid:48) production is vastly improved compared to what was 91 achievable at LEP and the Tevatron. The most stringent direct limits come from the ATLAS and CMS experiments using proton-proton collision data. Searches have been conducted at √ s = 7, 8 and 13 TeV, with varying amounts of integrated luminosity. Some of the highest direct limits set to date come from the dilepton analysis presented in this dissertation (See √ s = 8 and 13 TeV are displayed in Sec. 11.4.2). Results obtained by ATLAS and CMS at Table 5.5. 92 Part II The Experiment 93 Chapter 6 The Large Hadron Collider This chapter introduces the Large Hadron Collider (LHC) and discusses its operation and performance during the data-taking periods used for the dilepton analysis. The LHC is a synchrotron used to accelerate, store, and collide beams of protons and heavy ions at high energies. The analysis of such collisions provides evidence for the laws governing subatomic particle interactions, and allows for theories Beyond the Standard Model (BSM) to be tested by searching for potentially unobserved particles such as the Z(cid:48) boson. 6.1 The LHC The LHC is currently the world’s largest and most powerful particle accelerator (See Fig. 6.1). It is a hadron collider that accelerates beams of protons (and heavy ions) up to an energy √ of 6.5 TeV, resulting in a total center-of-mass (CM) energy of s = 13 TeV. Operated by The European Organization for Nuclear Research (CERN), the LHC is housed in an excavated tunnel under the Franco-Swiss border outside of Geneva, Switzerland [126]. The tunnel is 26.7 km in circumference, with a depth of 45 m (toward lake Geneva) and 170 m (at the foot of the Jura mountains). It was originally excavated in the 1980s for a dif- ferent machine, the Large Electron Positron Collider (LEP) [126]. LEP was decommissioned in 2000, so that the existing infrastructure could be reused, with adequate modification, for the construction of the LHC and its new experiments [126]. 94 Figure 6.1: Illustration of the 27 km long LHC, its four main experiments (ALICE, ATLAS, CMS, and LHCb), the pre-accelerators used in staging proton acceleration to reach a CM energy of s = 13 TeV, and the region surrounding Geneva, Switzerland [13]. √ The LHC was designed to search physics Beyond the Standard Model (as outlined in Sec. 5.1), and test the predictions of the Standard Model in an energy range inaccessible to previous experiments. Its first research run was conducted from 2010 to 2013 at √ s = 8 TeV. Run-1 saw the discovery of the Higgs Boson, as well as a host of searches and √ s = 7 and measurements consistent with Standard Model predictions. From 2013 to 2015 the machine was shut down for repairs and upgrades, in preparation the second run when the CM energy was increased to √ s = 13 TeV. 95 6.2 CERN Accelerator Complex In order to reach the maximum beam energy of 6.5 TeV in the main ring of the LHC, protons are accelerated through successive stages by smaller accelerators at CERN’s accelerator complex [127] (See Fig. 6.2). Proton acceleration starts with the Linac2 linear accelerator. The proton source is a single bottle of hydrogen gas. The hydrogen is passed through an electric field, ionizing it, and the resulting protons are accelerated to 50 MeV. The second stage occurs when the beam is injected into the Proton Synchrotron Booster (PSB), a circular accelerator made of four superimposed synchrotron rings. The PSB accelerates protons up to 1.4 GeV and injects them into the 628 m ring of the Proton Synchrotron (PS). In the third stage, the PS pushes the beam up to 25 GeV and transfers it to the nearly 7 km ring of the Super Proton Synchrotron (SPS). Finally, in stage five, the SPS further accelerates the beam to 450 GeV and injects it into the LHC [127]. 6.3 LHC Operation Once injected, the LHC is designed to accelerate and store two counter-rotating beams of high-energy particles. The beams travel in opposite directions, and are contained in separate beam pipes held at ultra-high vacuum (10−13 atm) to prevent collisions with extraneous gas molecules [128]. Protons are accelerated by 16 Radio-Frequency (RF) cavities (8 per beam) situated along the straight sections of the LHC main ring [129]. The RF cavities are designed to provide 400 MHz oscillating electric fields, so that a charged particle, as it passes through, will feel an overall force in its direction of motion. For each pass, the proton receives an extra 16 MeV 96 Figure 6.2: Illustration of the CERN accelerator complex. Proton acceleration is achieved through the following stages: Linac 2, PSB, SP, SPS, and LHC. Accelerators used in other CERN experiments are shown as well [14]. of energy, and as a result, it takes about 20 minutes to go from 0.45 TeV at injection to 6.5 TeV for collisions [130]. After this, the main role of the RF cavities is to keep the protons tightly bunched together to ensure the highest possible luminosity. The proton beams are guided through the main ring of the LHC by powerful magnetic fields. The LHC utilizes 1232 superconducting dipole magnets, measuring 15 m in length and weighing 35 tons (See Fig. 6.3). The electromagnet’s coils are threaded with a niobium- titanium (NbTi) alloy and cooled to 1.9 K. With an applied current of 11,080 A, the resulting field strength is 8.3 T. The beams spread out as they travel, so it is also necessary to align and focus them. This is achieved with 392 quadrupole magnets, which alternate between 97 Figure 6.3: Superconducting dipole magnets as seen in the LHC tunnel [15]. squeezing the beams horizontally and vertically as they circulate. Additional higher-order magnets help as well by correcting for small imperfections in the extreme regions of the dipole fields [131]. 6.3.1 Pile-up LHC beams circulate with a frequency of 11.245 kHz,1 and can do so for many hours under normal operating conditions. As a result of the RF acceleration scheme, the beams fall into discrete packets called “bunches.”2 At the LHC design value, each proton beam contains 2808 bunches, with 1.2 × 1011 protons per bunch at the start of data-taking [128]. The bunch-spacing of the LHC is 25 ns, which corresponds to a bunch crossing rate of 40 MHz, the number of times per second bunches cross at each interaction point (IP). The bunch crossing rate is related to the frequency of the RF cavities through fRF = hfrev, where h is an integer, known as the harmonic number. This ensures the proton always receives a 1The proton revolution frequency is derived from frev = βc/2πRLHC. 2A “perfectly timed” proton will have a revolution frequency frev that is a multiple of the 400 MHz RF oscillation frequency, and as a result will experience zero accelerating voltage; however, any proton that exhibits a small variation in frev will be slightly accelerated or decelerated, and will therefore oscillate around the perfectly timed proton. The aggregate of such particles forms a bunch [132]. 98 Figure 6.4: Depiction of a Z → µ+µ− event as recorded by the ATLAS detector during Run-1 of the LHC, along with the 25 simultaneous proton-proton interactions during this particular bunch crossing. The two muons originate from a common vertex, due to the decay of the Z boson, and are highlighted in yellow [16]. positive accelerating voltage from each RF cavity. During a single bunch crossing, there can be multiple collisions between the roughly 2 × 1011 particles (as in Fig. 6.4). These additional interactions are known as pile-up. The collisions in a single bunch crossing, in addition to the one of interest, are known as in-time pile-up. This serves as an additional background to identifying and reconstructing various physics objects in the event. Further, out-of-time pile-up results from collisions that occur in bunch crossing just before or after the collision of interest. This can affect the detector response to the in-time collision, since various detector electronics integrate for more time than the 25 ns between individual bunch crossings. The amount of pile-up present in events can be quantified by the average number of interactions per bunch crossing. In Run-1 for example, this value was (cid:104)µ(cid:105) = 20.7 at √ s = 8 TeV [17]. The reconstruction of every physics object in an event is affected by pile-up in some way, so it must be accounted for when performing and LHC physics analysis (as in Sec. 9.4.4). 99 6.3.2 Luminosity Aside from its CM energy, the most important design feature of an accelerator is its in- stantaneous luminosity (or just the luminosity), the number of protons passing through a given unit area per unit time. In fact, the LHC was chosen to be a pp collider (as opposed to pp like the Tevatron) as a way to maximize its luminosity. Many physics processes of interest at the LHC are only produced rarely, so a large number of collisions is required. The instantaneous luminosity links the event rate of a physics process to its production cross section σ where, = σL, dN dt (6.1) so a higher instantaneous luminosity increases the rate at which rare events are produced, and hence improves the potential for the discovery of new physics. Likewise, the integrated luminosity, Lint =(cid:82) L dt, can be used to determine how many collisions events occurred for a process of interest over a particular data-taking period or run of the LHC.3 The instantaneous luminosity can be expressed in terms of the following accelerator parameters [133], L = N 2nbf γ 4πnβ∗ F, (6.2) where N is the number of particles per bunch, nb is the number of circulating bunches, frev is the proton revolution frequency, and γ is the Lorentz factor. The cross-sectional area of each beam is related to n, the normalized emittance, and β∗ the value of the beta function 3At the LHC, cross sections are measured in barns (1 barn = 10−24cm2). The integrated luminosity is similarly expressed in terms of inverse barns, or more commonly inverse femtobarns, fb−1. 100 Year Ebeam Np [TeV] 6.5 2015 6.5 2016 Design 7.0 1.10 ×1011 1.10 ×1011 1.15 ×1011 nb n [µm] β∗ [m] Bunch 2244 2076 2808 3.5 3.4 3.75 0.8 0.4 0.55 Spacing [ns] 50/25 25 25 Peak lumi. [cm−2s−1] 5.1 × 1033 1.4 × 1034 1.0 × 1034 Table 6.1: Main LHC beam parameters under typical running conditions. Shown are values for the 2015 run and the 2016 run through June, along with the LHC design values [2]. at the interaction point.4 The parameter F is a geometrical factor, expressing the fact that the bunches do not necessarily collide head-on, resulting in a small luminosity reduction, with a design value of around 0.8. Table 6.1 lists the main LHC beam parameters during the 2015 and 2016 data-taking runs, as well as the design values for comparison. 6.4 LHC Performance in Run-II The LHC started its second run in 2015, after its first long shutdown (LS1). During this two year shutdown, the magnet systems were overhauled to prepare for the increase in the CM energy of the collider from √ s = 8 TeV in Run-1 to √ s = 13 TeV in Run-2. After several months of commissioning, the first stable beams were delivered to the experiments on June 3, 2015, and the intensity was ramped up slowly throughout the summer [134]. The bunch spacing was initially set to 50 ns, then was increased to 25 ns, making it possible to fill up to 2244 bunches into the machine. By the end of the proton physics running period, the peak instantaneous luminosity was L ≈ 5 × 1033cm−2s−1, about 50% its design value. The integrated luminosity delivered over the course of the run was just above 4 fb−1, falling below the initial projections, due to various machine related difficulties interrupting the run. The bulk of the data arrived toward the end, reaching production rates 4The beta function is πσ2/, the width of the beam squared divided by the emittence. If beta is small, the beam is narrow, like at the interaction point. 101 (a) (b) Figure 6.5: Cumulative integrated luminosity obtained per week as delivered by the LHC and recorded by the ATLAS detector during the 2015 (a) and 2016 (b) data-taking runs [17, 18]. of 200-250 pb−1/day [134]. On the other hand, the 2016 run was extremely successful. The machine reached its design luminosity in June, for the first time, and then regularly operated at 30% above that [135]. The target integrated luminosity for the year was 25 fb−1, and almost 40 fb−1 was recorded, with the machine spending a record 60% of its time delivering stable beams to the experiments [136]. During the 2015 + 2016 data-taking periods, the LHC delivered a total of 42.7 fb−1, of which 36.1 fb−1 was recorded by the ATLAS detector, and made usable for analysis. The average number of interactions per bunch crossing for this period was (cid:104)µ(cid:105) = 23.7. Figure 6.5 shows the integrated luminosity for the 2015 and 2016 runs as a function of time. 6.5 Experiments at the LHC The four main experiments at the LHC are ATLAS (A Toroidal LHC Apparatus), CMS (the Compact Muon Solenoid), LHCb (the Large Hadron Collider beauty), and ALICE (A 102 Large Ion Collider Experiment). ATLAS and CMS are general purpose detectors, designed to investigate the largest possible range of physics phenomena. LHCb, on the other hand, is optimized to study the production and decay of B hadrons as a way to investigate CP violation5. ALICE specializes in the study of heavy-ion collisions and the physics of the quark-gluon plasma. Each detector is installed in a large underground cavern built around the four collision points of the LHC beam (See Fig. 6.1). In addition, the main experiments are complemented by three specialized smaller ones: TOTEM (the TOTal Elastic and diffractive cross section Measurement), LHCf (the Large Hadron Collider forward), and MoEDAL (Monopole and Exotics Detector at the LHC). TOTEM is installed close to the CMS interaction point, LHCf near ATLAS, and MoEDAL near LHCb. Each of the smaller experiments is designed with enhanced sensitivity to study specific aspects of proton collisions [128]. 5B hadrons contain a bottom quark 103 Chapter 7 The ATLAS Experiment This chapter introduces the ATLAS detector, the experimental apparatus used to record LHC proton-proton collisions and search for Z(cid:48) boson in the dilepton analysis. 7.1 Detector Overview The ATLAS (A Toroidal LHC ApparatuS) detector is one of two general purpose high- energy physics detectors at the LHC. It is the largest particle detector ever constructed, measuring 46 m long, 26 m high, 26 m wide, and weighing over 7000 tons. As a general purpose detector, it is designed to cover the broadest possible range of physics phenomena, rather than focusing on a particular signature or type of process. In- deed, if new physics exists, ATLAS will have a good chance of detecting it and performing dedicated measurements to understand its properties. At the same time, the ATLAS detec- tor is designed to be able to perform a host of Standard Model measurements over a wide range of final-states and complex decay topologies. The experiment is designed to operate at a luminosity of L = 1034 cm−2s−1, one of the highest proton-proton collision rates yet achieved in a collider experiment, while simul- taneously maintaining sensitivity to a wide variety of physics signatures [19]. This puts definite requirements on the design of the detector and helps to establish its physics goals. Specifically, the ATLAS experiment needs to be able to detect, measure, and reconstruct 104 Figure 7.1: Individual particles are identified and measured in the ATLAS detector accord- ing to their characteristic signatures left in each detector subsystem. Solid lines represent trajectories of charged particles, while dashed lines indicate neutral particles. Showers are depicted for particles that interact with the dense detector material [19]. final-states defined by electrons, photons, jets, B hadrons, muons, taus, and/or neutrinos, all over a wide kinematic range. The ATLAS detector is comprised of a series of dedicated subsystems to take advantage of the distinct signatures left by each particle as it interacts with the detector material (See Fig. 7.1). Charged particles are identified by ionization tracks, as their trajectory bends as they pass through a magnetic field. Tracks are formed from a series of ionizing hits in the inner detector, which can be used to estimate charge and momentum. Neutral particles leave no tracks; rather, they interact with dense detector material, creating a shower of secondary particles. The resulting shower profile can be used to estimate the incident particle energy. Muons are highly penetrating, and their path is determined with additional tracking 105 Figure 7.2: Cut-away view of the ATLAS detector with individual subsystems labeled [20]. information provided in the outer detector. Neutrinos are undetectable, but their presence can be inferred by calculating the amount of missing energy in the event. Unstable particles like B hadrons have long lifetimes and decay in flight producing a secondary decay vertex. A secondary (or displaced) vertex like this can be used to identify and reconstruct unstable particles in the event. The ATLAS detector layout can be seen in Figure 7.2. For particle detection, three main subsystem are used: the tracking subsystem, the calorimetry subsystem, and the muon sub- system. The tracking system resides in the inner detector (ID), which is immersed in a 2 T solenoidal magnetic field, and is used to measure the charge and momentum of charged particles. The ID itself us comprised of three subsystems: the Pixel Detector, the Semi- conductor Tracker (SCT), and the Transition Radiation Tracker (TRT), which together are 106 used to achieve high momentum resolution and reconstruction efficiency. The calorimetry system surrounds the ID, and is comprised of electromagnetic and hadronic components, which measures the energy of electrons, photons, and hadrons, and contributes to their identification. The muon system sits outside the hadronic calorimeter and provides stand- alone precision muon-momentum measurements that supplement the tracking measurements of the ID. Lastly, the trigger and data-acquisition (TDAQ) system is responsible for triggering on interesting hard interaction events, and saving them to to permanent storage for later offline analysis. Each subsystem consists of a barrel detector, centered around the interaction point (IP), and two end-caps, for detecting particles in forward regions of the detector. This maximizes the detector acceptance, and is the origin of its cylindrical shape. 7.2 Coordinate System and Event Kinematics The ATLAS detector uses a right-handed coordinate system. The origin is set at the nominal IP at the center of the detector. The z-axis coincides with the beam line, the x-axis points to the center of the LHC ring, and the y-axis points upward to complete the right-handed system. Spherical coordinates are used as well, where R is the radius in the x − y (transverse) plane of the detector, and φ is the azimuthal angle, measured clockwise around the z-axis from the positive x-axis. Instead of the polar angle θ, measured from the positive z-axis, the pseudorapidity η is used, where (cid:20) tan (cid:18) θ (cid:19)(cid:21) 2 . η = − ln 107 (7.1) The advantage is that η is a Lorentz invariant quantity for boosts along the z-axis. In the massless limit, the rapidity y of a particle defined by Eq. 3.4 reduces to the pseudorapidity of Eq. 7.1, which is a good approximation for the final-state electrons of dielectron events at the LHC. It is also useful to define several kinematic variables in the transverse plane of the detector due to conservation of momentum in the collision. The transverse momentum pT and transverse energy ET of a particle are given by (cid:113) (cid:113) pT = ET = x + p2 p2 y, p2 T + m2. (7.2) Again, in the massless limit, the transverse energy ET reduces to the transverse momentum pT . For particles like neutrinos, it is useful to define the missing transverse momentum pmiss , T the negative vector sum of all visible reconstructed transverse momenta in the event, pT,i. (7.3) i A large value of pmiss T indicates that one or more particles in the event escaped detection, the primary candidate being a neutrino. The missing transverse energy (MET) is the magnitude, defined by Emiss T = |pmiss T |. Lastly, the distance ∆R between two objects in the η − φ space of detector can be calculated from ∆R = (∆η)2 + (∆φ)2. (7.4) T = −(cid:88) pmiss (cid:113) 108 7.3 Tracking System The ATLAS Inner Detector (ID) [23] covers the region |η| ≤ 2.5, also known as the precision region of the ATLAS detector, and is designed to measure charged particle trajectories and locate interaction vertices (See Fig. 7.3). A precise measurement of the trajectory can be used to extract the charge and momentum of a charged particle. The ID is located within a 2 T axial magnetic field provided by the external superconducting central solenoid magnet [23].1 The curvature of the particle’s trajectory in the magnetic field is due to the Lorentz Force and is proportional to its momentum. The achieved momentum resolution is σ/pT = 0.05%pT ⊕ 1% [23]. In addition to tracking, the ID helps with vertex reconstruction. Pattern recognition software is used to identify tracks that originate from the primary vertex, as well as those from secondary vertices due to in-flight decays of B hadrons, b-quark jets, or τ leptons. Measuring 7.0 m long and 2.3 m wide, the ID is the detector closest to the beam line. The ID consists of three subsystems: the Silicon Pixel Detector, the Semiconductor Tracker (SCT), and the Transition Radiation Tracker (TRT). The Pixel Detector and SCT contribute to the determination of primary and secondary vertices and precision tracking, while the TRT supplements these momenta measurements and aids in electron identification. A charged particle entering the ID passes through 3 layers of silicon pixels in the Pixel Detector, 8 layers of silicon strips in SCT, and finally through the straw-tubes of the TRT, which provide 36 hits on average (See Fig. 7.4). Pattern recognition software is then used to fit a high precision track through the series of hits provided by each subsystem. In addition, a fourth subsystem, the Insertable B-Layer (IBL), was added in Run-2 to maintain a high 1The solenoid uses a single winding of aluminum-stabilized niobium-titanium (NbTi) coil, operating at a nominal current of 7.73 kA, and temperature of 4.6 K. The casing is designed to be as thin as possible to minimize particle scatterings before they reach the calorimetry system [23]. 109 Figure 7.3: Cut-away view of the ATLAS Inner Detector. Subsystem components are shown in both the barrel and end-cap regions [21]. level of tracking performance in the high luminosity environment of Run-2, where increased pile-up was expected. 7.3.1 Pixel Detector The high granularity silicon Pixel Detector [23] is the innermost part of the ID system and closest to the IP, covering |η| < 2.5. It is designed for the measurement of charged particle tracks to high resolution, and identify and reconstruct primary and secondary decay vertices. When a high energy charged particle passes through the silicon layer of each module, it ionizes the silicon molecules. The resulting current produced from the movement of electron-hole pairs is detected by readout electronics. 110 Figure 7.4: Schematic of the barrel region of ATLAS Inner Detector, as crossed by a charged particle [21]. The pixel detector consists of three concentric layers of silicon pixel modules in the barrel region, measuring 50.5 mm, 88.5 mm, 122.5 mm from the beam axis. In the end-caps, the modules are located on perpendicular disks measuring 495 mm, 580 mm, and 650 mm from the IP. There are 1456 pixel modules in the barrel and 288 in the end-caps. The high- granularity design results in approximately 80 million pixels, each with individual readout channels, about half of the number of channels in ATLAS. The modules measure 2 × 6 cm2 in area, have a thickness of 250 µm, and are composed of roughly 47, 000 silicon pixels. Likewise, the silicon pixels measure 50 × 400 µm2 in area, which is how precision tracking and vertexing is obtained. The position resolution is 10 µm in the R − φ plane and 115 µm in the z (R) direction of the barrel (end-caps). 111 7.3.2 Semiconductor Tracker The SCT [23] surrounds the Pixel Detector and forms the next layer of the ID. It covers the same range as the Pixel Detector with |η| ≤ 2.5, and works in a similar way, making use of the ionization of the silicon for charged particle detection. Like the Pixel Detector, the high granularity of the SCT results in approximately 6.3 million readout channels. In the barrel region, the SCT is comprised of four cylindrical layers of silicon microstrip sensors, and in the forward region, nine perpendicular disks are located in each end-cap. The 285 µm thick silicon microstrips measure 80 µm× 12 cm, allowing the SCT to maintain coverage over a larger surface area, as it is further from the beam line. The silicon microstrips are two sided, allowing for two hits per layer. Each layer is arranged stereoscopically; i.e., angled with respect to the other by 40 mrad, allowing for a position measurement in all three coordinates. In the end-caps, the strips run radially and utilize the two-layer stereo structure as well. The position resolution of each module is 17 µm in the R − φ plane, and 580 µm in the z (R) direction of the barrel (end-caps). 7.3.3 Transition Radiation Tracker The outermost subsystem of the ID is the TRT [23], which covers a range |η| ≤ 2.0. It provides additional tracking, utilizing a large number of hits, up to 36 on average per track. Like the Pixel Detector and SCT, the TRT is composed of separate barrel and end-cap modules. The barrel consists of many layers of gaseous straw drift-tubes, which run parallel to the beam axis, and are interleaved with transition radiation material. The construction of the end-caps is similar, with the straws arranged radially into 18 wheels. The drift tubes are 4 mm in diameter, 144 cm (37 cm) long in the barrel (end-cap) 112 region, and are filled with a Xe-based gas mixture. The interior of each tube is coated with a conducting layer of aluminum, and a gold plated tungsten wire runs along the center. This cathode-anode pair is held at a potential of a few kV. When a charged particle passes through the transition radiation material, X-ray photons are produced. The particle and the photons pass through the drift tube and ionize of the Xe-based gas. The gas electrons are collected at the anode, and the resulting current is stereoscopically read out. The TRT is also used to perform particle identification. The amount of transition radia- tion produced by a charged particle is proportional to the Lorentz factor γ = E/m. The elec- tron, with its small mass, typically produces significant amounts of transition radiation, while the pion, almost 300 times heavier, produces much less. This results in different amounts of electric current collected from the ionized gas. Typically, seven to ten high-threshold hits from transition radiation are expected for electrons with energies above 2 GeV [23]. The TRT has a position resolution of 130 µm per drift tube in the R−φ plane. While the position resolution of the TRT worse than the Pixel or SCT subdetectors, its large number of drift tubes ensure many measurements are taken, especially far from the beam line, which contribute significantly to the momentum reconstruction of the charged particle. The use of the lower resolution drift tubes results in approximately 351, 000 TRT readout channels. 7.4 Calorimetry System The ATLAS calorimetry system [23] surrounds the inner detector, and covers the region |η| < 4.9. It is designed to provide dedicated energy measurements for incident electrons, photons, jets, and hadrons. The calorimetry system consists of three independent subsystems: the Electromagnetic Calorimeter (ECAL), the Hadronic Calorimeter (HCAL), both of which 113 Figure 7.5: Cut-away view of the ATLAS calorimetry system. Depicted are the Electromag- netic and Hadronic sampling calorimeters, both of which consist of individual barrel and end-cap modules. The forward components are closest to the beam line [22]. cover up to |η| < 3.2, and the Forward Calorimeter (FCAL), which covers the forward region 3.2 < |η| < 4.9 (See Fig. 7.5). The ECAL and HCAL consist of both barrel and end-cap components, while the FCAL is entirely forward by definition. Each calorimetry subsystem relies on sampling calorimetry, which makes use of two dis- tinct materials sandwiched together in alternating layers: a dense absorber and an active medium. An incident particle passing through the dense layers of absorbing material initiates a particle cascade, or shower, of lower energy secondary particles. The secondary particles in the shower interact with the active layers, which are connected to readout electronics for measurement. The shower continues to propagate until a minimal energy threshold is 114 reached, at which new particle production stops. In this way, the energy of the incident particle is obtained by measuring the energy deposited by its shower over the multiple active layers of the calorimeter. An energy calibration is required using particles of known energies. 7.4.1 Electromagnetic Calorimeter The Electromagnetic Calorimeter (ECAL) [23] measures the energies of electrons and pho- tons. The EM calorimeter is divided into a central barrel, covering the region |η| < 1.475, and two symmetric end-caps for 1.375 < |η| < 3.2. The EM barrel (EMB) takes the form of two identical half-barrels, separated by a small gap (4 mm) at z = 0, while the EM end-caps (EMEC) are themselves divided into two coaxial wheels (See Fig. 7.5). The outer wheel runs from 1.375 < |η| < 2.5, and the inner one from 2.5 < |η| < 3.2. The region between the barrel and end-caps, 1.37 < |η| < 1.52, is known colloquially as the crack region, which consists of cables and cooling for the ID.2 The ECAL uses lead plates as the absorbing medium and liquid Argon (LAr) as the active medium, which fills the gaps between the absorber layers, to produce a signal, while copper electrodes are used for readout. The alternating layers of active and passive material are arranged in an accordion style geometry, which leads to very uniform performance in terms of linearity and resolution as a function of φ. The design √ resolution is σE/E = 10%/ E ⊕ 0.7% [23]. When a charged particle passes through the dense lead absorbers, it decelerates, resulting in the emission of high energy photons due to bremstrahlung. These photons convert into electron-positron pairs, initiating an electromagnetic cascade, and forming a shower of lower energy particles. The shower particles ionize the LAr, and the ions drift toward electrodes 2Due to the poor energy resolution, particles from this region are usually excluded in ATLAS physics analyses. 115 Figure 7.6: Depiction of an ECAL barrel module in the precision region of the ATLAS detector. The accordion shape is shown, along with the three independent sampling layers. The granularity in η and φ of the cells of each layer is labeled [23]. for readout, which results in a signal proportional to the deposited energy. The precision region of the detector covers up to |η| < 2.5, matching the acceptance of the ID. Here, the ECAL is subdivided into three sampling layers of varying granularities in η and φ (See Fig. 7.6). The first sampling layer has a depth of 4.3 radiation lengths X0,3 and is composed of cells finely segmented in η, with a granularity of ∆η × ∆φ = 0.0031 × 0.098. The fine granularity allows for precision position measurements to be made, and can be used to distinguish photons from converted pions.4 The second sampling layer has a depth of 16 X0, and collects the majority of the energy of the electromagnetic shower. It is also 3The width of the absorber is measured in radiation lengths X0, the distance over which an incident 4The in-flight decay π0 → γγ can result in two highly collimated photons. electron loses 1/e of its initial energy. 116 Figure 7.7: Sketch of an ATLAS tile calorimeter module [24]. coarser, with a granularity of ∆η × ∆φ = 0.025 × 0.0245. The third layer is the coarsest at ∆η × ∆φ = 0.05× 0.0245, and is only reached by the highest energy particles, collecting the tail of the shower. In addition, within the region |η| < 1.8 and situated before the first sampling layer, is a layer of LAr, known as the presampler, which helps correct for energy losses of incident electrons or photons due to uninstrumented material within the inner detector. Outside the precision region, there is less material blockage, so a presampler is not necessary. 7.4.2 Hadronic Calorimeter The hadronic calorimeter (HCAL) [23] forms the next detector layer of the ATLAS detector, covering a region up to |η| < 3.2. Like the ECAL, it is a sampling calorimeter, and consists of two subdetectors: the tile calorimeter and the hadronic end-cap calorimeter (HEC). 117 The tile calorimeter consists of a barrel, covering |η| < 1.0, and two extended barrels for 0.8 < |η| < 1.7, each segmented in three layers. Compared to the ECAL, the granularity ∆η×∆φ is much coarser, measuring 0.1×0.1 in the first two layers, and 0.2×0.1 in the third. As a sampling calorimeter, it uses scintillating tiles as its active material, interleaved with layers of passive steel absorbers (See Fig. 7.7). Hadronic interactions with the scintillator material causes it to luminesce. Light signals are read out by wavelength shifting fibers connected to photomultiplier tubes, which convert the light signals into an electric current to be read out by electronics. The HEC covers the region 1.5 < |η| < 3.2 and is located in front of the EMEC. It is a LAr sampling calorimeter with copper plate absorbers, which are more robust for the high radiation environment of the forward region of the detector. It consists of two wheels per end-cap where each wheel is divided into two segments for a total of four segmented layers. The forward calorimeter (FCAL) covers the range 3.1 < |η| < 4.9. It is located in front of the EMEC and below the HEC (See Fig. 7.8) and is closest to the beam line. It consists of three modules which are used to make electromagnetic and hadronic measurements. The E ⊕ 3%, and for the FCAL √ design energy resolution for the tile and HEC is σE/E = 50%/ √ E ⊕ 10%. it is σE/E = 100%/ 7.5 Muon System The Muon Spectrometer (MS) [23] forms the outermost layer of the ATLAS detector (See Fig. 7.9). It is a tracking detector designed to measure the position and momentum of muons in the range |η| < 2.7, as well as trigger on muons in the range |η| < 2.4, based on their deflection in an applied magnetic field. 118 Figure 7.8: Schematic of the ATLAS detector forward calorimeter systems. The three FCAL modules are depicted [25]. Figure 7.9: Cut-away view of the ATLAS Muon System [26]. 119 Three toroid magnets, one in the barrel and two in the end-caps, produce the neces- sary magnetic field. The superconducting air-core toroid magnet is formed by an eight-fold symmetric set of 25.3 m long flat race-track coils, which operate at a current of 20.5 kA and temperature of 4.6 K to generate an approximately 0.5 T magnetic field for the region |η| < 1.4 of the MS. The two additional end-cap toroids are positioned at each end of the central solenoid magnet (for the ID) and provide a 1.0 T magnetic field for additional bend- ing power in the MS end-caps, 1.6 < |η| < 2.7. The transition region is defined by the gap in coverage between the two η regions, which relies on the fields of each magnet. Muons are highly penetrating particles. They leave energetic hits in the ID, but pass almost uninterrupted through the calorimeters before reaching the MS, which is comprised of many gas-filled chambers. As they pass through, the muons ionize the gas, producing a series of hits, which can then be used to reconstruct its track. Four subdetectors are used: the Monitored Drift Tube chambers (MDT) and the Cathode-Strip Chambers (CSC) perform precision tracking measurements, while the Resistive Plate Chambers (RPC) and the Thin Gap Chambers (TGC) provide triggering capabilities and secondary track measurements. The strong fields and large size of MS allow for the trajectories of energetic muons to be measured with a high level of accuracy; something not possible with the use of the ID subsystems alone. The MS can measure muon momenta up to 3 TeV, and provides a transverse momentum resolution of 10% at 1 TeV. 7.6 Trigger and Data-Acquisition System An essential component of any collider experiment is an efficient trigger and data-acquisition (TDAQ) system. The trigger selects collision events that contain physics processes of interest, 120 Figure 7.10: The production cross sections and corresponding event rates for a variety of physics processes observable at the LHC. BSM processes, such as Z(cid:48) production, are expected to occur at extremely small rates compared to the nominalrate of proton-proton inelastic collisions indicated by σtot. In this regard, a trigger is essential to correctlyidentify and select such events for offline analysis [8]. 121 0.111010-710-610-510-410-310-210-110010110210310410510610710810910-710-610-510-410-310-210-1100101102103104105106107108109σσσσZZσσσσWWσσσσWHσσσσVBFMH=125 GeVWJS2012σσσσjet(ETjet > 100 GeV)σσσσjet(ETjet > √√√√s/20)σσσσggHLHCTevatronevents / sec for L = 1033 cm-2s-1 σσσσbσσσσtotproton - (anti)proton cross sectionsσσσσWσσσσZσσσσtσ σ σ σ ((((nb))))√√√√s (TeV){ and in so doing, reduces the enormous event rate provided by the high luminosity of the LHC to something manageable that can be recorded and permanently stored for subsequent offline analysis (See Fig. 7.10). The ATLAS trigger system consists of two tiers, a hardware-based first-level (L1) trigger, and a single software-based high-level trigger (HLT). With this setup, the 40 MHz bunch crossing rate is reduced to a rate of 100 kHz at the L1 trigger, and then to an average recording rate of about 1 kHz at the HLT, which is compatible with offline computing and storage requirements [137]. The average size of an event accepted by the HLT is on the order of 1 MB. The types of signatures accepted by the trigger are encoded in the trigger menu, which contains several hundred predefined items.5 The L1 trigger is implemented with custom electronics, and is designed to search for regions of interest (RoIs) of high-energy activity within the detector using coarse-grained in- formation from the L1 calorimeter trigger (L1Calo) and L1 muon (L1Muon) trigger systems. The window to make a positive identification is about 2.5 µs, during which time the event is stored in temporary pipeline memories [137]. The L1 trigger decision to keep, and pass the event to the HLT, or to discard it, is made by the by the Central Trigger Processor (CPT). In the case of an L1 trigger acceptance, the event is buffered in the Read-Out System (ROS), and sent to the HLT for subsequent processing. The HLT is purely software based and reduces the L1 trigger output rate of 100 kHz to approximately 1 kHz on average within a processing time of about 200 ms. Software algorithms run over the RoIs identified by the L1 trigger, or if needed over the full detector, in order to reconstruct the event. In addition, finer-granularity calorimeter information, precision measurements from the MS, and tracking information from the ID, are made use 5For example, the HLT 2e17 lhloose trigger, used to select events in the dilepton analysis, requires an event to have at least two electrons with ET > 17 GeV that pass a loose level of likelihood based identification at the HLT level. 122 of, which are not available at the L1 trigger [138]. In order to minimize processing time, most HLT triggers follow a two-stage approach, where a fast first-pass reconstruction is performed to reject events early on, and a slower precision reconstruction is applied in the second-stage for any remaining events. The recon- struction algorithms run on a unique PC farm of about 40,000 processor cores [138]. Events accepted by the HLT are transferred to the CERN computing center, and stored in the RAW (bytestream) data format on magnetic tapes for later reprocessing. With an average of 25 in- teractions per bunch crossing at 40 MHz in 2016, of all the proton-proton collisions observed by the ATLAS detector, only about one in a million is saved for later offline analysis. 123 Chapter 8 Electron Reconstruction in ATLAS The search conducted in this dissertation is focused on a final-state defined by an electron- positron pair. This chapter describes the reconstruction and identification of central (|η| < 2.47) electron candidates in the ATLAS experiment. Electrons are reconstructed from the hits they leave in the Inner Detector (ID) and from the energy clusters they deposit in the EM calorimeter.1 Dedicated algorithms are used to reconstruct these detector level signals into meaningful physics objects usable for analysis. Moreover, identification criteria are applied to the reconstructed objects in order to separate true electron candidates from so-called fakes, or non-electron objects like jets which have been misidentified. 8.1 Track and Vertex Reconstruction As a charged particle, an electron that passes through the central region of the ATLAS detector leaves a series of hits in the ID and an energy deposit in the EM calorimeter as its signature (See Fig. 8.1).2 The reconstruction of central electrons starts with track reconstruction, which aims to reconstruct the trajectories of charged particles according to the hits left in the three ID subsystems. Here, sophisticated track reconstruction algorithms 1Aside from the curvature of their tracks, electrons and positrons share the same signature in the ATLAS detector, so for the remained of the chapter, unless otherwise stated, positrons are also denoted as “electrons.” 2Muons do as well, but leave only a negligible amount of energy in the calorimeter, since they are highly penetrating. 124 Figure 8.1: Depiction of ATLAS detector systems used to perform electron reconstruction and identification [3]. are employed and are discussed below. After each event track has been reconstructed, vertex reconstruction occurs, where the tracks are extrapolated toward the beam line to reconstruct the positions of the event ver- tices, which originate from the multiple proton-proton interactions during a bunch crossing. Likewise, sophisticated vertex finder algorithms are used for the reconstruction. 8.1.1 Track Reconstruction Several track finding strategies exist within the ATLAS experiment [139]. The primary strategy is known as the “inside-out” strategy [140], which is also the baseline for the re- construction of prompt charged particles.3 Here, the reconstruction follows the natural particle trajectory, starting from the silicon hits present in the pixel and SCT detectors, and 3Prompt, or signal-like, electrons originate from an event primary vertex, such as from the decay of a W or Z boson. 125 gradually extends outward to the TRT. The inside-out strategy works in two steps: first pattern recognition is employed, followed by a track fit. In the first step, pixel and SCT hits are transformed into three dimensional “space-points.” Pattern recognition software is used to form track candidates (or track seeds) out of 3 silicon space-points, providing an initial track direction and orientation. This is followed by a track-building process where the directional information is propagated through the detector layers, away from the interaction point, and silicon hits that match the assumed trajectory are included in the updated track candidate definition. The track-building procedure is performed using a Kalman filter algorithm [139], which updates the track fit iteratively with each successive hit, accounting for the material effects of the detector. The tracks are formed according to the “pion hypothesis,” where energy losses due to radiation are assumed to be negligible. However if a track candidate is coincident with an EM cluster and cannot be fully extended using the pion hypothesis, the “electron hypothesis” is used instead. Under this hypothesis, the trajectory is allowed to be affected by energy losses, such as from bremstrahlung, as the particle can interact with detector material at each material surface [3]. In the second step, track candidates are extended to the TRT, and refit using both the silicon and TRT information again assuming either the electron or pion hypothesis. The refitted track is compared to its silicon-only segment and retained only if the two are consistent, and the track contains at least seven silicon hits. Tracks successfully reconstructed with the inside-out algorithm must have a transverse momentum pT > 400 MeV. 126 (a) (b) Figure 8.2: Illustration of the transverse impact parameter d0 (a) and longitudinal impact parameter z0 (b) [27]. 8.1.2 Vertex Reconstruction After all tracks have been successfully reconstructed, vertex finder algorithms [140] are used to assign them to their respective vertices.4 Vertex reconstruction starts by forming vertex seeds out of bunches of closely related tracks. Tracks incompatible with the initial seed go on to form new seeds, and the process repeats until a set of consistent vertices has been identified and all tracks are uniquely matched [140]. Reconstructed vertices are required have at least two associated tracks, and the primary vertex is defined as the one with the highest(cid:80) p2 T of its associated tracks. The position of the primary vertex can be described by the transverse and longitudinal impact parameters of its tracks (See Fig. 8.2). The transverse impact parameter d0 measures the distance of closest approach between the track and the primary vertex in the transverse plane of the detector, while the longitudinal impact parameter z0 measures the distance of the track from the vertex along the beam line (the z-axis). The d0 significance, the value 4Vertices can originate as pile-up vertices from soft-interactions during the bunch crossing, as secondary vertices from in-flight particle decays, and as the primary vertices from event hard-scatters. 127 of d0 divided by its uncertainty, can be used to separate prompt from non-prompt electron candidates, as non-prompt candidates usually have a larger value. 8.2 Electrons Electrons in the central region of the ATLAS detector are reconstructed from energy deposits in the EM calorimeter that are matched to charged particle tracks in the inner detector. The procedure starts with the triggering of a high energy cluster in the EM calorimeter, and is followed by the reconstruction and calibration of the measured cluster energy. Discrimination between signal-like (or true) and background-like (or fake) electrons can be achieved with specific electron identification and isolation requirements. 8.2.1 Electron Trigger Events with prompt electrons are selected by the ATLAS triggering system. All electron triggers are seeded at L1. Here, L1 trigger algorithms search for one or more EM calorime- ter trigger objects above a certain energy threshold, and identify RoIs where clusters are built [141]. Events passing the L1 trigger are sent to the HLT, which employs two further stages of selection. The first stage algorithms are less precise, but are fast to reject events early on and reduce the overall trigger rate. The second stage algorithms utilize more precise offline-like requirements, such as isolation and likelihood-based identification, which function to further reduce the event rate. These requirements are looser than the actual offline algorithms to avoid losses in efficiency at this early stage in selection. Electrons used to perform the electron channel search in the dilepton analysis are selected 128 with the HLT 2e17 lhloose trigger, which requires events to have at least two ET > 17 GeV electrons that satisfy loose likelihood-based identification at the HLT level. 8.2.2 Electron Reconstruction The reconstruction of central electrons relies on two detector subsystems: the inner detec- tor, which provides tracking information, and the EM calorimeter, which provides energy measurements. In a successful reconstruction, energy clusters are built from clusters seeds in the EM calorimeter, and are subsequently matched to at least one reconstructed track in the inner detector. The reconstruction starts by building clusters from the energy deposits in the EM calorime- ter. Here, the η−φ space of the calorimeter is partitioned into a grid with Nη×Nφ = 200×256 cells. Each cell in the grid has a granularity of ∆η × ∆φ = 0.025 × 0.025, the same as the middle layer of the EM calorimeter (as in Fig. 7.6). Within each grid cell, an “energy tower” is formed by summing together the energy of all calorimeter depth layers. Next, a sliding window algorithm [142] scans the grid of energy towers with a fixed 3 × 5 window. A seed cluster is formed from the energy towers if the transverse energy within the window is at least ET > 2.5 GeV, and if it is a local maximum within the scanned region. Finally, a smaller 3× 3 window, less sensitive to detector noise, is used to determine the position of the seed cluster. The energy weighted barycenter of the cells within the resulting seed cluster is used to define its position. In the next step, a search is performed to match reconstructed tracks to seed clusters. Reconstructed tracks are extrapolated to the middle layer of the EM calorimeter, and are considered loosely matched to clusters if they pass through the cluster barycenter within a cone of ∆R < 0.3. If multiple tracks match the cluster, then the one with the most silicon 129 hits and the smallest distance to the cluster barycenter is chosen as the primary. A cluster without any matched tracks is discarded as an electron candidate, and reclassified as an unconverted photon. With a successful match, the cluster is rebuilt using a 3 × 7 window of energy towers in the EM calorimeter barrel, and a 5 × 5 window in the end-caps. As a final step, the cluster energy is calibrated to the original electron energy using multivariate techniques [143], which are based on simulation. The four-momentum of the electron candidate is determined from the calibrated cluster energy and the corresponding track parameters. The energy component E comes from the cluster energy, and the directional components θ and φ are taken from the matched track. The finalized four-momentum of the reconstructed electron is pµ = (E, px, py, pz) = (E, p sin θ cos φ, p sin θ sin φ, p cos θ), (8.1) making it ready for use in physics analysis. 8.2.3 Electron Identification A sample of reconstructed electrons consists of both prompt electrons and objects that are misidentified as electrons, such as hadronic jets or electrons from in-flight photon conversions, which leave similar signatures in the ID and EM calorimeter.5 Therefore, a set of electron identification criteria must be applied to distinguish the real electrons from the fakes. This procedure is known as electron identification. For central electrons in Run-2, electron identification relies on a likelihood-based identification technique. Electron identification algorithms are based on the discriminating variables in Table 8.1. 5In the following discussion, unless otherwise stated, prompt electrons are considered “signal,” while all other objects are considered “background.” 130 These variables describe the lateral and longitudinal shape of EM showers in the calorimeter,6 the properties of reconstructed ID tracks, information from the TRT, and the degree to which tracks and energy clusters are well matched [144]. Together, these variables can be combined into a single menu to obtain the desired level of background rejection. In Run-2, the electron identification algorithms were reoptimized in accordance with the ATLAS detector upgrades. With the commissioning of the IBL, additional hits in the innermost pixel layer could be used to further discriminate between electrons and converted photons. Likewise, modifications to the TRT gas mixture led to an improved likelihood-based method to characterize high-threshold TRT hits. In Run-1, two different electron identification methods were developed within ATLAS. The first method [3] utilized a simple cut-based approach, where selection requirements are applied sequentially to the track and shower shape variables of the electron candidate in order to make a decision. The second method [3] relies on a likelihood function to evaluate multiple shower shape and tracking variables simultaneously in order to make a selection decision. This multivariate analysis (MVA) technique results in an improved background rejection and increased signal acceptance relative to what could be obtained with the Run- 1 cut-based approach. Therefore, in Run-2, the cut-based approach was retired, and the likelihood-based algorithm was made the baseline for electron identification. The likelihood function is obtained from data, and is constructed out of the signal and background probability distributions of the discriminating variables of Table 8.1. The dis- tributions are two dimensional, and are divided into 9 |η| bins and 6 ET bins to balance the available statistics with their evolution in |η| and ET . The probability for a recon- structed electron to be signal or background is calculated, and is then combined into a single 6These often go by the name of “shower shape” variables. 131 (cid:113) (ΣEiη2 Middle layer of EM calorimeter Back layer of EM Calorimeter Description Type Hadronic leakage Ratio of ET in the first layer of the hadronic calorimeter to ET of the EM cluster (used over the range |η| < 0.8 or |η| > 1.37) Ratio of ET in the hadronic calorimeter to ET of the EM clus- ter (used over the range 0.8 < |η| < 1.37) Ratio of the energy in the back layer to the total energy in the EM accordion calorimeter. This variable is only used below 100 GeV because it is known to be inefficient at high energies. i )/(ΣEi) − ((ΣEiηi)/(ΣEi))2, Lateral shower width, where Ei is the energy and ηi is the pseudorapidity of cell i and the sum is calculated within a window of 3 × 5 cells Ratio of the energy in 3 × 3 cells over the energy in 3 × 7 cells centered at the electron cluster position Ratio of the energy in 3 × 7 cells over the energy in 7 × 7 cells centered at the electron cluster position strips in a window of ∆η × ∆φ ≈ 0.0625 × 0.2, corresponding typically to 20 strips in η, and imax is the index of the highest- energy strip Ratio of the energy difference between the largest and second largest energy deposits in the cluster over the sum of these en- ergies Ratio of the energy in the strip layer to the total energy in the EM accordion calorimeter Shower width,(cid:112)(ΣEi(i − imax)2)/(ΣEi), where i runs over all Strip layer of EM Calorimeter Track conditions Number of hits in the innermost pixel layer; discriminates Name Rhad1 Rhad f3 wη2 Rφ Rη wstot Eratio f1 nBlayer TRT Track-cluster matching nPixel nSi d0 d0/σd0 against photon conversions Number of hits in the pixel detector Number of total hits in the pixel and SCT detectors Transverse impact parameter with respect to the beam line Significance of transverse impact parameter defined as the ratio of d0 and its uncertainty Momentum lost by the track between the perigee and the last measurement point divided by the original momentum Likelihood probability based on transition radiation in the TRT ePrHT ∆η between the cluster position in the strip layer and the ex- trapolated track ∆φ between the cluster position in the middle layer and the track extrapolated from the perigee Defined as ∆φ2, but the track momentum is rescaled to the cluster energy before extrapolating the track from the perigee to the middle layer of the calorimeter Ratio of the cluster energy to the track momentum ∆φres ∆p/p ∆φ2 ∆η1 E/p Table 8.1: Run-2 electron identification variables [3]. 132 discriminant dL, on which the final selection decision is made, where, n(cid:89) i=1 dL = LSLS + LB , LS(B)(x) = PS(B),i(xi), (8.2) where PS,i(xi) is the signal distribution and PB,i(xi) is the background distribution for discriminating variable i evaluated at a value of xi. In this approach, three distinct electron identification operating points are defined: Loose, Medium, and Tight. Each operating point offers an increased level of background rejection at the cost of a decreased signal efficiency. The levels are defined in such a way that a sample of electrons obtained from a tighter operating point will be a subset of looser one. For ex- ample, electrons that pass tight will be a subset of those passing medium, etc. The Loose criterion offers a selection that specializes in rejecting light flavor jets, while the Medium and Tight criteria include additional discriminating variables in their definitions and are better suited at rejecting electrons from heavy flavor decays and photon conversions [3]. The signal and background likelihood functions are optimized using Monte Carlo (MC) simulation. For the signal, a sample of Z → e+e− events is used, while the background uses a sample of dijet events. Disagreements are observed between the data and MC in the positions or widths of several variables due to incomplete knowledge of the detector description and the modeling of shower shapes in the detector simulation. Therefore, data-driven corrections are applied to the simulation to match the data during the optimization procedure to ensure similar performance. 133 8.2.4 Electron Isolation In addition to the electron identification requirement, it is possible to further separate signal from background with an electron isolation requirement. Isolation quantifies the amount of detector activity surrounding the electron candidate, either as additional calorimeter energy or as transverse momenta from ID tracks.7 As such, both calorimeter-based and track-based definitions of isolation exist, and both are used to select electrons in the dilepton analysis. In the calorimeter-based definition, the isolation energy Econe∆R T is obtained by summing over the transverse energy deposited in each EM calorimeter cell within a certain cone radius ∆R around the center of the electron candidate cluster, (cid:88) ∆R<0.2 (cid:113) Econe0.2 T = Ecell T , ∆R = (∆η)2 + (∆φ)2. (8.3) The transverse energy ET of the electron cluster itself is not included in the sum. The core of the electron cluster is defined to be within the 5 × 7 window, measuring ∆η × ∆φ = 0.125 × 0.175, centered on the reconstructed electron cluster barycenter. The isolation is corrected on an event-by-event basis for electron energy leakage into the cone, and for pile- up effects. In Run-1 this Etcone method suffered from a high sensitivity to pile-up noise, which decreased its ability to discriminate between signal and background. The situation was improved in Run-2 with the topoEtcone method, which naturally applies some noise sup- pression by using topological clusters, or topoclusters, instead of using individual calorime- ter cells in the isolation calculation [28]. Topoclusters are built from contiguous calorimeter 7Electrons that originate from non-prompt decays or from misidentified hadronic jets exhibit broader energy clusters in the calorimeter and feature multiple ID tracks. 134 Figure 8.3: Illustration comparing the Etcone and topoEtcone isolation methods. The grid corresponds to the η − φ granularity of the middle layer of the EM calorimeter. The electron candidate cluster is mostly contained in the central 5 × 7 window. An isolation cone with radius ∆R = 0.4 is drawn around the candidate. In the Etcone method, all cells (yellow) within the radius ∆R are used, while in the topoEtcone method, only topoclusters (red), whose barycenters fall within ∆R, are used, effectively reducing noise from pile-up [28]. cells whose energy falls above a certain threshold. These are then clustered together and the topoEtcone isolation variable is constructed by summing up the transverse energy of all topoclusters whose barycenters fall into the isolation cone ∆R (See Fig. 8.3). Likewise, the track-based isolation variable pvarcone∆R T is obtained by summing over the transverse momenta pT of all ID tracks within a cone ∆R centered on the track that was matched to the electron candidate cluster, pvarcone0.2 T = ptrack T , X = min(0.2, 10 GeV/pT ). (8.4) (cid:88) ∆R 20 GeV. 146 description of the geometry and material composition of particle detectors. The detailed interactions between the particles and the detector material is simulated, and additional particles that result from bremstrahlung, or from electromagnetic or hadronic showers, are propagated through the detector as well. An accurate description of the detector geometry is crucial for the simulation. Experimental results are continually compared with previous versions of the ATLAS geometric model to help further refine its description [159]. The result is a record of hits and energy deposits for each simulated event. This information is written to an output hit file. 9.2.3 Digitization In the final stage of the simulation chain, the responses of the detector readout electronics are simulated and digitized using the hit files. The effect of pile-up is included in this stage as well. Pile-up is not strictly simulated as part of the event generation; rather, it is included at this stage by reading in various types of simulated events. Hits from these events are “overlaid” with the hits from the generated event, effectively reducing the amount of computation required. Furthermore, the hardware-based L1 trigger is configured to run in a “pass-through” mode on the digitized detector information, and its L1 decisions are emulated. No event is discarded, and the set of L1 trigger hypotheses for each event is evaluated and saved. The digitized events are written in the Raw Data Object (RDO) file format, just like real data. Additional “truth level” information is retained, which includes a detailed record of each particle in the event from the generation stage. ATLAS reconstruction software (as in Sec. 8.2) can be run on the RDO files. The output of reconstruction is an Analysis Object Dataset (AOD) file, which contains both “truth 147 level” and “reconstruction level” information usable for physics analysis. These files contain all the necessary information to interface with the tools provided by the ATLAS combined performance groups, and apply, for example, energy scale calibration to the data (as in Sec. 8.3.1), and resolution smearing (as in Sec. 9.4.3), and efficiency scale factor corrections (as in Sec. 9.4.2) to the simulation. Likewise, they provide the ability to assess the systematic uncertainties associated with these operations (as in Sec. 10.8). 9.3 Simulation Samples Each Standard Model background to the electron channel search is defined by a final-state with at least two prompt electrons. An irreducible component of this arises from the Drell- Yan and Photon-Induced processes, and non-negligible reducible components, in order of largest to smallest, arise from t¯t and Single-Top quark production, Multi-jet & W +jets pro- duction, and Diboson production.6 The Multi-jet & W +jets backgrounds are the result of the misidentification of one or more non-prompt electrons or hadronic jets as a prompt elec- trons, and is estimated using a data-driven approach (See Sec. 10.6). All other backgrounds are described by dedicated MC simulation. 9.3.1 Drell-Yan Production The Drell-Yan process (depicted in Fig. 4.2) is generated at the NLO with the Powheg event generator, configured with the CT10 [160] PDF set. Showering and hadronization is performed with Pythia8 using the ATLAS AZNLO tune [161]. 6Irreducible meaning the Z(cid:48) → e+e− signal has an identical final-state to the γ∗/Z → e+e− background. Reducible backgrounds are identified by other final-state objects in addition to the two electrons. For example, the t¯t and W t processes include at least one additional b-jet in the final-state, and Diboson processes include neutrinos or additional jets. 148 mee [GeV] wQCD = σN N LO/σN LO 1.044 0.985 wEW 500 750 1000 1500 2000 3000 4000 5000 1.044 1.043 1.035 1.018 0.985 0.955 0.844 0.972 0.953 0.927 0.896 0.857 0.834 0.784 Table 9.1: Mass-dependent NLO-to-NNLO QCD and higher-order EW k-Factor corrections for several values of dielectron invariant mass. These corrections are applied to the enhance the nominal Drell-Yan background estimate with NNLO QCD and higher-order EW effects. Powheg is used to generate 19 individual mass-binned samples ranging from 120 GeV to 5000+ GeV in slices of true dielectron invariant mass mee (See Appendix A for details). Generating multiple mass-binned samples ensures a small statistical uncertainty is achieved and results in an adequate number of events to model the kinematic distributions of interest. Below 120 GeV, a dedicated sample of Z → e+e− events is used to model the Z boson mass peak (See Appendix A for details). This sample has a lower mass threshold of 60 GeV, but no upper threshold, so an upper limit of 120 GeV is applied manually to smoothly match it with the lowest mass-binned sample. In order to improve the accuracy of this prediction, Drell-Yan event yields are enhanced by several mass-dependent k-Factors corrections. These include higher-order QCD corrections, derived with Vrap at the NNLO, and higher-order EW corrections derived with Mcsanc at the NLO. Table 9.1 lists the characteristic values of the corrections as a function of dielectron invariant mass. Further detail on the derivation and application of k-Factor corrections is provided in Section 9.4.5. 9.3.2 Photon-Induced Production An additional irreducible component of the background comes from the Photon-Induced process γγ → (cid:96)+(cid:96)− through t- and u-channel modes, as depicted in Figure 9.1. This process generally makes a small contribution to the overall Standard Model background estimate, 149 γ γ (a) (cid:96)− (cid:96)+ γ γ (b) (cid:96)− (cid:96)+ Figure 9.1: The LO Feynman diagrams for Photon-Induced Drell-Yan production. Both t-channel (a) and u-channel (b) modes are shown. but becomes sizeable at high mass, as depicted in Table 9.2. The Photon-Induced process is not included in the dedicated generation of Drell-Yan; instead, it is accounted for with a mass-dependent k-Factor correction, derived using the MRST2004QED [162] PDF set. The photon PDF γ(x, Q2) is included in this specialized PDF set, which enables the initial-state production of QED related processes. mee [GeV] wP I 500 750 1000 1500 2000 3000 4000 5000 1.039 1.045 1.051 1.060 1.075 1.111 1.139 1.263 Table 9.2: Mass-dependent Photon-Induced k-Factor corrections at several values of dielec- tron invariant mass. These corrections are applied to the nominal the Drell-Yan background estimate to incorporate Photon-Induced effects. 9.3.3 t¯t Production g g (a) t ¯t g g (b) t ¯t ¯q q (c) t ¯t Figure 9.2: The LO Feynman diagrams for t¯t production. The gluon initiated s-channel (a) and t-channel (b) modes are depicted, along with quark-antiquark initiated production (c). 150 The t¯t process is generated at NLO using Powheg with the CT10 PDF set. The top quark mass mtop is set to 172.5 GeV. A single sample is produced, and no mass-threshold requirements are made in the generation. The Pythia6 generator, configured with the Perugia 2012 tune [163], is used to model the underlying event and parton showering. A global k-Factor is derived using the Top++ 2.0 generator, which brings the cross section from the NLO to the NNLO in QCD, and includes Next-to-Next-to-Leading-Log (NNLL) terms from soft-gluon resummation. The total t¯t cross section comes to 831.76 pb after the application of the global k-Factor correction, which has a value of 1.1949. The relevant ¯tt processes are shown in Figure. 9.2. The electron channel background originates from t → bW , where each W boson decays leptonically through W → eν. 9.3.4 Single-Top Quark Production b g W (a) t b g (b) W t Figure 9.3: The LO Feynman diagrams (a) and (b) for W t-channel Single-Top production. The majority of top quarks at the LHC are produced in pairs via the strong interaction (See Fig. 9.2). However, the weak interaction allows for the production of a single top quark in association with another particle (See Fig. 9.3. The Single-Top background arises predom- inantly from so-called W t-channel Single-Top production, where a top quark is produced in association with a W boson.7 7Single-Top production also proceeds through s- and t-channel modes. 151 This background is generated at NLO in QCD with Powheg with the CT10 PDF, and showed in Pythia6 using the Perugia 2012 tune. Again, Top++2.0 is used to bring the calculation from NLO to NNLO in QCD, where the total cross section measures 71.7 pb after the application of the global k-Factor correction, whose value is 1.054. 9.3.5 Diboson Production ¯q q V0 (a) V2 V1 ¯q q V2 ¯q (b) V1 q (c) V1 V2 Figure 9.4: The LO Feynman diagrams for Diboson production. Depicted are the s- channel (a), t-channel (b), and u-channel (c) production modes. The W Z and ZZ processes can proceed through all modes, while the ZZ process can produced only through the t- and u-channels. In each diagram Vi = W, Z or γ. The smallest background component arises from Diboson production, or the production of pairs of vector bosons. These pairs form three distinct final-states: W W , W Z, and ZZ, as seen in Figure 9.4. Each process is generated in Sherpa 2.1.1 at NLO with the CT10 PDF, which also includes the parton showering. Because of its small contribution to the overall background, no attempt is made to include a k-Factor correction. As with the Drell- Yan process, a series of mass-thresholds are required in the generation (See Appendix A for details). In the W W channel, the dielectron background originates from the decay W W → eνeν; in the W Z channel, though the decays W Z → (cid:96)νee or W Z → qqee; and in the ZZ channel, though the decays ZZ → (cid:96)(cid:96)ee, ZZ → qqee, or ZZ → ννee. Each set of mass-binned samples covers the entire search region, and provides an adequate statistical uncertainty. The largest 152 contribution to the Diboson background arises from the ZZ → qqee process with a cross section of 16.59 pb. 9.3.6 Z(cid:48) Production Dedicated Z(cid:48) signal samples are generated with Pythia8 at LO using the NNPDF23LO [164] PDF set and the A14 tune [165] for showering and hadronization. Here, the several “bench- mark” samples are generated: four at Z(cid:48) pole masses of 2, 3, 4, and 5 TeV for the Z(cid:48) one at a pole mass of 3 TeV for the Z(cid:48) Model Drell-Yan process is not included in the Z(cid:48) generation. SSM. The effect of interference with the Standard χ, and These benchmark mass-points are used to validate the signal reweighting technique used later for the signal search (See Sec. 11.3) and limit setting (See Sec. 11.4) phases of the analysis.8 Signal reweighting is applied to a set of LO Drell-Yan samples, generated with Pythia8 using the NNPDF23LO PDF set. These samples are generated with an identical range in truth dielectron invariant mass mee as the corresponding NLO Powheg Drell-Yan samples described in Section 9.3.1. In addition, higher-order QCD corrections are applied to the reweighted Z(cid:48) signals. A LO-to-NNLO QCD k-Factor, derived using Vrap, takes the LO Pythia8 Z(cid:48) signal estimates to the NNLO, the same order as the Powheg Drell-Yan background estimate. While the higher-order QCD effects are expected to be the same between signal and background, the higher-order EW corrections are model dependent, and are therefore not applied to the signal. Several values of the QCD k-Factor correction are shown in Table 9.3. 8A Signal reweighting tool was developed to generate Z(cid:48) signal samples for user-defined pole masses and model parameters using the set of LO Drell-Yan samples as input. This avoids the otherwise computation- ally intensive process of generating new samples for each mass point and model under investigation. See Section 11.2 for more details. 153 mee [GeV] wQCD = σN N LO/σLO 1.356 500 750 1000 1500 2000 3000 4000 5000 1.380 1.411 1.435 1.423 1.339 1.264 1.025 Table 9.3: Mass-dependent LO-to-NNLO QCD k-Factor corrections at several values of dielectron mass mee. These corrections are applied to enhance the nominal Z(cid:48) signal estimate with NNLO QCD effects. 9.3.7 MC Sample Summary A summary of the generators and PDF sets used for the electron channel analysis is displayed in Table 9.4. For a comprehensive list of MC samples and their properties, see Appendix A. Generator Process q ¯q → Z/γ∗ → e+e− Powheg v2 Powheg v2 t¯t Powheg v2 W t Sherpa 2.1.1 NLO NLO W W , W Z, ZZ q ¯q → Z/γ∗ → e+e− Pythia 8.186 LO q ¯q → Z(cid:48) → e+e− Pythia 8.186 LO Order Corrected Parton Shower PDF Pythia 8.186 CT10 Pythia 6.428 CT10 Pythia 6.428 CT10 Sherpa 2.1.1 CT10 Pythia 8.186 NNPDF23LO Pythia 8.186 NNPDF23LO NLO NNLO NLO NNLO NLO NNLO NNLO NNLO Table 9.4: A summary of MC samples used for the background and signal modeling in the electron channel of the dilepton analysis. The columns include the process of interest, the MC generation program, the matrix element order, the order of the k-Factor correction, the parton shower program, and the PDF set used [4]. 9.4 Corrections to Simulation A series of corrections are applied to the simulation to achieve the closest possible level of agreement between data and MC. In the ATLAS detector simulation, all major components and material interactions are simulated within the detector volume, and the resulting electron distributions are generally well matched to data. However, the modeling is by no means perfect, and discrepancies can arise, for example, from assumptions made about the exact 154 material distribution within the ATLAS detector. To account for these effects, efficiency scale factor and energy resolution corrections are applied to enhance the simulation. Likewise, corrections to the pile-up distribution assumed in the MC generation are performed, and k-Factor corrections are used to enhance the description of the simulation with the inclusion of higher-order effects absent in the original generation. 9.4.1 Sample Normalization As the amount of integrated luminosity in the final LHC dataset is not known a-priori, the MC samples used in the analysis were generated with integrated luminosities that were in general different from the dataset value of Lint = 36.1 fb−1. Therefore, to match the observed integrated luminosity in data, each background must be correctly normalized. The following event weight is used, wMC = σbkgfilter Nevts Lint, (9.1) where σbkg is the associated background sample cross section, filter is its filtering efficiency, Nevts is the number of events in the sample, and Lint is the integrated luminosity of the dataset. The values used are obtained from the generator outputs listed in Appendix A. 9.4.2 Efficiency Scale Factor Corrections The total efficiency to identify an electron in the ATLAS detector is the product of the electron reconstruction, ID, trigger, and isolation efficiencies (as outlined in Sec. 8.4). When extracted from simulation, these efficiencies show discrepancies, at the level of a few percent, to those measured in data, due to the imperfect modeling of the ATLAS detector. Therefore, 155 Figure 9.5: Electron efficiency scale factor corrections as a function of dielectron invari- ant mass mee. Shown are the scale factors applied to correct the electron reconstruction, identification, isolation, and trigger efficiencies. to mitigate these effects and bring the simulation into the best possible agreement with the data, efficiency scale factor corrections are applied to the simulation. These corrections are obtained from the following efficiency measurements, wReco = data Reco MC Reco , wID = data ID MC ID , wTrigger = data Trigger MC Trigger , wIso = data Iso MC Iso (9.2) The scale factor corrections of Eq. 9.2 are applied on an event-by-event basis as multiplicative event weights. The size of the corrections are sensitive to the event electron ET and η, but in general they are close to unity. In the case of Drell-Yan, with two electrons in the final-state, two sets of scale factor corrections are applied, one for each electron. Figure 9.5 shows the effect of each correction as a function of dielectron invariant mass mee. The ATLAS electron/photon combined performance group (EGP) [166] provides the efficiency scale factors used in the electron channel of the dilepton analysis. Here, the 156 Dielectron Invariant Mass [GeV]200300400100020003000Efficiency Correction0.920.940.960.9811.021.04 InternalATLAS-1 = 13 TeV, 36.1 fbsDilepton Search SelectionIDwIsowRecowTrigw EGP group identifies the Z boson mass peak as a very clean control region and uses the tag-and-probe [3] method, for example, to estimate electron efficiency scale factors, using increasingly coarse bins in ET . The scale factors are extracted from measurements performed in two-dimensional bins of ET and η, which account for changes in efficiency according to the electron energy and ATLAS detector region. Scale factor uncertainties arise due to assumptions about the stability of the measurements out to high-ET , where electrons cannot be measured precisely (See Sec. 10.8.1). The corrections are applied with the Electron- EfficiencyCorrection [167] tool, which is called during the main analysis workflow (See Sec. 10.5.1). The electron ID efficiency correction is larger than the others depicted in Figure 9.5. This is due to the different behaviors of the “leading,” or higher ET , and “subleading,” or lower ET , electrons in the dielectron event, which are used in the construction of the dielectron invariant mass, and due to the assumptions regarding the evolution of the electron shower shapes in ET and η. As a likelihood-based electron identification is used (See Sec. 8.2.3), the measurements performed to extract the electron ID scale factors are particularly sensitive to any lack of knowledge in the ET and η kinematic variables. 9.4.3 Energy Scale and Resolution Smearing The electron energy resolution observed in data differs from what is predicted in simulation, so energy smearing is performed on the simulation to match what is observed in data (See Sec. 8.3.2). This is achieved by smearing the simulated electron energy with Gaussian cor- rection factors, dependent on electron ET and η. These smearing factors are extracted from dedicated measurements comparing the Z boson width between simulation and data using a sample of Z → e+e− events. 157 Figure 9.6: Invariant Mass distribution for the MZ(cid:48) = 3 TeV Z(cid:48) χ predicted by the simulation with and without the application of the energy scale correction and resolution smearing. The effect is to both shift and broaden the shape of the Z(cid:48) peak. In addition, the electron energy scale is calibrated in order to improve the level of agree- ment between data and simulation. The Z boson mass peak is used as a way to estimate the measured electron energy, since it provides a constraint on the invariant mass of dielec- tron pairs, and hence the transverse energy ET of each electron. From this, scale factors can be extracted and used to scale the electron energy. Disagreements between data and simulation can arise due to a number of reasons (See Sec. 8.3.1). The ATLAS EGP group derives these corrections, and includes them in an analysis tool, the ElectronPhotonFour- MomentumCorrection tool [146], which is used to smear event-by-event the electron energy in simulation during analysis (See Sec. 10.5.1). Further, it is used to calibrate the electron en- ergy scale in both simulation and data. The invariant mass distribution of the MZ(cid:48) = 3 TeV Z(cid:48) χ with and without the application of energy resolution smearing and energy scale correc- tions can be see in Figure 9.6. The effect is to shift the Z(cid:48) peak from its nominal value and broaden its shape. 158 Dielectron Invariant Mass [GeV]260027002800290030003100320033003400Events00.511.522.53 InternalATLAS-1 = 13 TeV, 36.1 fbsDilepton Search Selection eefi cZ' ee (corrected)fi cZ' Figure 9.7: Mean number of interactions per bunch crossing in the Z → e+e− MC sample with and without the pile-up correction as compared to data. The effect of reweighting is to match the pile-up conditions in data. 9.4.4 Pile-Up Profile The multiple simultaneous proton-proton collisions that occur during a single bunch crossing at the LHC are known as pile-up, as depicted in Figure 6.4. The simulation samples are usually produced before or during a given data-taking period, and while the effect of pile-up is included, the exact conditions that need to be matched to emulate the data are not known a-priori. Therefore, only a reasonable estimate of the pile-up conditions in data can be folded into the simulation, which must be corrected for later, once the run conditions have been realized. Each simulation sample is generated with a pile-up profile around an average number of proton-proton collisions per bunch crossing (cid:104)µ(cid:105). The (cid:104)µ(cid:105) profile in simulation is reweighted to match the one observed in data using event weights, such that the evolving LHC running conditions are accounted for. Reweighting is performed using the PileupReweighting tool [168], provided by the by 159 >mMean Interactions per Bunch-Crossing <051015202530354045Events0100200300400500600700800310· InternalATLAS-1 = 13 TeV, 36.1 fbsData eefi* gZ/ ee (corrected)fi* gZ/ Figure 9.8: The four mass-dependent k-Factor corrections applied in the dilepton analy- sis. The NLO-to-NNLO QCD correction, LO-to-NLO EW correction, and the LO Photon- Induced QED correction are applied to enhance the description of the Drell-Yan spectrum, while the LO-to-NNLO QCD correction is applied to the Z(cid:48) signal. the ATLAS Analysis Software Group (ASG). The result of the reweighting procedure can be see in Figure 9.7, where the (cid:104)µ(cid:105) profile in the dedicated Z/γ∗ → e+e− sample is reweighted and compared to data. The overall effect of pile-up on the dilepton search is expected to be small, since the signal is characterized by two high ET objects, effectively suppressing pile-up interactions. 9.4.5 k-Factor Corrections Several k-Factor corrections are applied to the simulation in order to improve the nominal descriptions of the Drell-Yan, tt, and W t background estimates, as well as the Z(cid:48) signal estimate. These corrections include higher-order effects, which are absent from the original event generation, and are derived by taking the ratio between a higher-order cross section calculation and a lower-order one. 160 Dielectron Invariant Mass [GeV]200300400100020003000k-Factor Correction0.80.911.11.21.31.41.51.6 InternalATLAS-1 = 13 TeV, 36.1 fbsNLOs/NNLOs: QCDwLOs/NNLOs: QCDwEWwPIw Both mass-dependent and mass-independent, or global, k-Factor corrections are used in the dilepton analysis. For the NLO Drell-Yan background, the following mass-dependent k-Factor corrections include the NLO-to-NNLO QCD correction, the LO-to-NLO EW correc- tion, and the Photon-Induced QED correction. For the LO Z(cid:48) signal, only the LO-to-NNLO QCD correction is applied.9 The mass-dependent NNLO QCD correction is derived with Vrap 0.9, while the NLO EW correction comes from McSanc. Final-state photon radiation is simulated with Pho- tos, so the EW corrections derived with McSanc include only the contributions from initial-state photon radiation and electroweak loops [4]. The set of mass-dependent correc- tions is depicted in Figure 9.8. Similarly, global k-Factor corrections are applied to the ¯tt and W t Single-Top processes, each derived with the Top++ generator, providing a description at the NNLO in QCD. Cor- rections to the Diboson processes are not required, due to their relatively small contribution to the overall background estimate. The LPXKFactorTool [169] incorporates the functional forms of these corrections, and is used to apply them as event weights during the main analysis workflow (See Sec. 10.5.1). Several example weights are listed in Table 9.1 for several values of dielectron invariant mass. Each k-Factor correction comes with an associated systematic uncertainty; these are assessed in Section 10.8. 9For the signal, the precise form of EW k-Factor correction depends on the model-dependent couplings between the Z(cid:48) and the Standard Model W and Z bosons; hence this correction is left out. 161 Part III The Analysis 162 Chapter 10 Electron Channel Dilepton Analysis In this chapter, the dilepton analysis is introduced, and the search for new physics in dielec- tron final-states is performed using data taken during the 2015 and 2016 runs of the LHC √ at s = 13 TeV. This analysis constitutes the contribution to the ATLAS published result by the author. 10.1 Analysis Strategy The limitations of the Standard Model outlined in Section 5.1 all point indirectly to the ex- istence of physics Beyond the Standard Model (BSM), suggesting some kind of modification or extension to the Standard Model must take place. Many BSM theories extend the Stan- dard Model with additional U (1)(cid:48) gauge symmetries (as outlined in Sec. 5.3), the breaking of which lead to one or more massive Z(cid:48) bosons. New Z(cid:48) states can potentially decay into pairs of charged leptons through the Z(cid:48) → (cid:96)+(cid:96)− decay channel, the signature of which would appear as a narrow resonance, or “bump,” in the dilepton invariant mass spectrum.1 To that end, a model-independent analysis is developed and conducted in order to search for signatures of new physics, such as the production of a heavy Z(cid:48) boson, in final-states defined by pairs of same-flavor, opposite-charge leptons, which are produced as a result of proton-proton collisions at the LHC. The analysis is performed in two distinct leptonic 1The decay width ΓZ(cid:48) is model-dependent and can be calculated with Eq. 5.4. 163 channels: the electron channel, in which the final-state consists of an electron-positron pair, and the muon channel, in which the final-state consists of a muon-antimuon pair. Each channel is as its own independent analysis, and later statistically combined to obtain a single dilepton result. The focus of this dissertation, and the following chapter, will be on the search for resonant new physics in the electron channel. In the electron channel, the dielectron invariant mass is used to discriminate the signal from the Standard Model background processes. This is the most direct way to search for new gauge bosons, as the invariant mass spectrum will display a resonance peak at the pole mass of the new particle. The dielectron invariant mass is built according to (cid:113) 2Ee+ T Ee− me+e− = T (cosh ∆η − cos ∆φ), (10.1) where Ee+ T and Ee− T are the electron transverse energies, ∆η is the difference in pseudora- pidities, and ∆φ is the difference in azimuthal angles in the transverse plane of the ATLAS detector. The general strategy of the analysis is to select events recorded in LHC proton-proton collisions with two good quality, high ET electron candidates, in order to reconstruct the dielectron invariant mass. The invariant mass distribution in data is then compared to the background expectation provided by the Standard Model, and any observed deviations are statistically quantified. All signal and background processes are described by Monte Carlo (MC) simulation (See Sec. 9.3), except for the W +jets & QCD Multi-jet fake background, which is obtained through the data-driven matrix method. The leading Standard Model background contri- bution arises from the Drell-Yan process, while subleading backgrounds arise from the Top 164 (t¯t, W t), Diboson (W W , W Z, ZZ), and fake (W +jets & QCD Multi-jet) processes. The signal region of the search is defined by the region in the dielectron invariant mass beyond the Z boson mass peak. Two separate analyses took place in the years 2015 and 2016: the first was performed on the complete 2015 dataset [170], while the second, without any major modifications, was performed on the full 2015+2016 combination [5]. The 2015 analysis was initially “blinded,” such that any event with an invariant mass mee > 500 GeV was rejected.2 After it had been shown that the search strategy was sound and a sufficient level of agreement had been obtained between the data and the MC simulation below the 500 GeV threshold, the analysis was “unblinded” with the approval of the ATLAS collaboration. During this time, this author developed an Early-Warning-System (EWS) [171] (See Ap- pendix E), which ran on slimmed-down datasets produced at the Tier-0 computing facility at CERN, in order to alert the analysis team of the development of any anomalous signatures as data were being taken, which could facilitate a faster unblinding if necessary.3 The dilepton analysis provides excellent sensitivity to signatures of new physics ow- ing to its clean, fully reconstructed final-state, high signal-efficiency, and set of small well- understood backgrounds [5]. For these reasons, it is one of the earliest searches to be con- ducted at newly commissioned hadron colliders. This chapter now presents the 2015+2016 analysis. 2In a blind analysis, the final result is kept hidden (in this case, the observed invariant mass spectrum above a certain mass threshold) until the analysis is essentially complete. The idea is to avoid any “subcon- scious” experimental bias, such as modifying the event selection to enhance a potential signal. 3The search sensitivity of the Run-1 dilepton analysis would be quickly surpassed with only 1-2 fb−1 s = 13 TeV data. The rate of data collection in early Run-2 was so fast that the integrated of Run-2 luminosity would double every few weeks, so a signal really could “jump out” from one week to the next. This is the importance of the EWS, especially since the performance of the main analysis and subsequent unblinding could take months. √ 165 10.2 Data Samples The dilepton analysis uses proton-proton collision data collected at the LHC during the 2015 √ and 2016 data-taking periods at s = 13 TeV. The total integrated luminosity corresponds to 36.1 fb−1, as recorded by the ATLAS detector after excluding data-taking periods where one or more subdetectors were not fully operational. 10.2.1 ATLAS Data ATLAS data-taking is divided into periods. Periods usually last several weeks, and are clas- sified according to consistent configurations of the detector and trigger. Significant changes either to the configuration of the detector or to the trigger usually result in a new period. Periods are listed alphabetically with a single letter, and consist of specific data-taking runs. Each run is assigned a unique run number, and usually corresponds to a single fill of the LHC, which lasts for multiple hours.4 Runs are further broken down into luminosity blocks, collections of data recorded over the course of a few minutes, during which time the instantaneous luminosity, and detector conditions, are approximately constant. Luminosity blocks are tabulated in a file called the Good Runs List (GRL) [172]. A luminosity block is flagged as “good” in the GRL if certain data quality requirements are met, such as the stability of LHC beams, the absence of any hardware malfunctions, and that relevant ATLAS subsystems are operational. Datasets usable for LHC physics analysis consist only of runs and luminosity blocks found within a GRL. 4Some of the longer runs have lasted for slightly more than a day. 166 (a) (b) Figure 10.1: Day-by-day data-taking efficiency of the ATLAS detector during in the 2015 (a) and 2016 (b) runs. The efficiency over the course of each run was above 92%. Empty bins correspond to days during which the LHC did not deliver stable beams [18]. 10.2.2 Analysis Data The search uses LHC proton-proton collisions at √ s = 13 TeV collected in the years 2015 and 2016, as recorded by the ATLAS detector. The 2015-2016 dataset is comprised of events recorded from August through October 2015,5 and from April through October 2016 [173]. During these periods, the LHC delivered a total integrated luminosity of 42.7 fb−1. For the 2015 run alone, this corresponded to 4.2 fb−1, with a total of 3.9 fb−1 recorded by the ATLAS detector (See Fig. 6.5). Likewise, in the following year, the LHC delivered 38.5 fb−1, and of that, the ATLAS detector recorded 36.5 fb−1 (See Fig. 6.5). Among all luminosity blocks recorded, only those identified in a GRL as “good” were analyzed. Two GRLs were used in the analysis, one for each year of data-taking. Thus, the final dataset consisted of an integrated luminosity of 36.1 fb−1 usable for analysis, of which 3.2 fb−1 belonged to 2015, and 32.9 fb−1 to 2016. A 3.2% luminosity uncertainty was 5In June and July of 2015, the LHC operated with a bunch-spacing of 50 ns. In August, it was subsequently reduced to its design value of 25 ns. Only data taken with the 25 ns bunch spacing is analyzed, since the integrated luminosity of the 50 ns run is small and would otherwise require entirely new MC samples. 167 Figure 10.2: Average number of proton-proton interactions per bunch crossing during the 2015 and 2016 data-taking periods. The data shown correspond to periods with stable beams delivered to the ATLAS detector [18]. assessed on the final dataset, as recommended by ATLAS luminosity group [174], by taking a weighted average of the luminosity uncertainty of each year. A complete list of data-taking periods used for analysis can be found in Appendix A. The data-taking efficiency of the ATLAS detector can be seen in Figure 10.1. For the 2015 run, the average efficiency was 92.1%, which was roughly the same in 2016 at 92.4%, despite the average number of pile-up interactions almost doubling (See Sec. 6.3.1). The average number of pile-up interactions was (cid:104)µ(cid:105) = 13.7 in 2015, which increased to (cid:104)µ(cid:105) = 24.0 in 2016. The average over both years was (cid:104)µ(cid:105) = 23.7, and the associated pile-up distributions can be seen in Figure 10.2. 10.2.3 Dataset Derivations Many physics analyses, including this one, produce slimmed down datasets associated with the particular final-state under study. These dataset derivations remove otherwise ex- 168 traneous event or particle level information from the AOD event record, which eliminates overhead, streamlines the analysis, and speeds up the turnaround time for common analysis tasks. This can be especially useful, since the data and simulation need to be run over frequently.6 In Run-1, the situation regarding derivations was untenable. The AOD files (See Sec. 9.2.3) were not Root [175] compatible, and therefore had to be converted into a series of inter- mediate Root-readable formats.7 The production and formatting of these “n-tuples” was left to each analysis team, and without any quality control, a large number of incompatible formats resulted, making it difficult to perform cross-checks, or validate results obtained by other analysis teams working on similar channels. The Run-2 event data model [176] (EDM) solves this by uniformizing all group data- formats. The AOD is replaced by a Root-readable format from the start, called the xAOD, and derived datasets, DxAODs, are produced centrally at CERN from xAODs during event reconstruction (See Sec. 9.2.3) [177]. The DxAODs produced for the dilepton analysis are designed to retain events with two or more reconstructed electrons with ET > 20 GeV or muons with pT > 20 GeV. In addition, the electron has to satisfy the Loose likelihood-based identification criterion (See Sec. 8.2.3). All data and simulation samples analyzed in the analysis are produced in this derived format. As a result, the physical size of the DxAODs is reduced to approximately 1% the size of the input xAODs. This reduction in size was large enough to allow for both the data and simulation samples to be stored locally on the MSU Tier-3 [178] computing cluster, on which 6This occurs for a variety of reasons: to conduct performance studies, to study new kinematic variables, to check the effect of changes to the selection, if bugs are found, if updated CP recommendations are released, if running systematic studies, after a bulk-reprocessing of the data, etc. 7Root is statistical analysis framework commonly used in HEP to process and visualize data, usually in the form of histograms. 169 the analysis was performed by this author. 10.3 Monte Carlo Samples As outlined in the previous chapter, simulation samples provide a way to estimate the Stan- dard Model background expectation, and therefore allows for a direct comparison to be made against events selected in data. The Standard Model backgrounds in the electron channel arise from processes with two real final-state electrons, and are outlined in Section 9.3. Back- ground samples are used to model events arising from the Drell-Yan process, Photon-Induced production, Top (W t-channel Single-Top and t¯t) production, and Diboson (W W , W Z, and ZZ) production. For each sample, the ATLAS detector response is simulated with Geant4, and all events are reconstructed with the same reconstruction software that is used for data (See Sec. 9.2); and like the data, all MC samples are built as DxAOD derivations. The simulation is corrected for a variety of effects. The electron energy resolution is smeared (See Sec. 9.4.3), event level reweighting is performed to account for pile-up effects (See Sec. 9.4.4), and scale factors are applied to correct for differences observed in the reconstruction, identification, trigger, and isolation efficiencies between the MC samples and data (See Sec. 9.4.2). 10.4 Analysis Software In order to conduct the analysis, a dedicated framework is used to “run over”8 the proton- proton collision events stored in the DxAOD derivations. The analysis framework houses the event selection (to be defined in Sec. 10.5.1), and interfaces with the various CP tools, which 8i.e. load, analyze, and accept or reject 170 are used to apply the required event and object level corrections to data and simulation, and to assess systematic uncertainties. The dilepton analysis uses the RootCore framework [179], which is provided and main- tained by the ATLAS collaboration. Using Rootcore, data and MC samples are loaded with the SampleHandler [180] package, and processed using EventLoop [181]. The analysis framework is built within a particular Analysis Base Release (ABR) [182]. The ABR con- tains specific releases of general purpose physics analysis tools, which facilitate coordination between analysis groups, and ensures projects are efficiently organized and up to date. The development and deployment of ABRs is handled by the ATLAS Software Group [183]. The results of the dilepton analysis were obtained under ABR version 2.4.27, the most up-to-date version at the time of publication, which contains the following physics analysis tool versions: • ElectronPhotonFourMomentumCorrection-02-03-00: package used to perform elec- tron energy calibration (See Sec. 8.3.1) and resolution smearing (See Sec. 9.4.3) • ElectronEfficiencyCorrection-00-01-88: package used to apply efficiency scale factor corrections to simulation (See Sec. 9.4.2) • PileupReweighting-00-04-01: package used to correct the pile-up profile in simula- tion (See Sec. 9.4.4) • IsolationCorrection-00-01-27: package used to correct electron candidate isolation energy • ElectronPhotonSelectorTools-00-02-92-18: package used to perform likelihood- based electron identification (See Sec. 8.2.3) 171 • IsolationSelection-00-06-05: package used to calculate the isolation energy of electron candidates (See Sec. 8.2.4) In addition, packages specific to the dilepton analysis include: • LPXKfactorTool-00-01-05 [169]: package used to apply higher-order k-Factor cor- rections to simulation (See Sec. 9.4.5), as well as to normalize signal and background samples with the appropriate cross sections (See Sec. 9.4.1) • LPXSignalReweightingTool-00-00-07 [184]: package used to create Z(cid:48) signal tem- plates from LO Drell-Yan simulation for use in the statistical analysis (See Sec. 11.2) 10.5 Event Selection The electron channel search targets LHC proton-proton events with two or more high-ET electron candidates. For an event to be considered a high-mass dielectron candidate event, each event electron must pass certain quality requirements, defined by the dielectron event selection in the proceeding section. These requirements enhance the probability of selecting a real electron, rather than a misidentified jet or an electron from a photon conversion. Events that retain at least two electrons passing selection are then used to calculate the dielectron invariant mass according to Eq. 10.1. The same set of selection criteria are applied equally to events in data and simulation, including both the signal and background MC. Background events passing the selection are subsequently scaled to the integrated luminosity observed in data (as in Sec. 9.4.1), and combined to provide an estimate for the Standard Model background expectation, which is then used to model the observed event yield in data. This allows for a direct comparison to 172 be made between the data, the overall Standard Model expectation, and the various BSM predictions, allowing for a conclusion to be drawn regarding the presence of a signal. 10.5.1 Selection Criteria A set of selection criteria is applied to obtain a high quality sample of high-mass dielectron events from data and simulation. Events passing the selection are retained for the subsequent statistical analysis (See Sec. 11.1). The selection criteria are applied first at the “event level,” then at the “object level,” and finally at the dielectron “pair level,” moving from broad to narrow in scope. As a preliminary step, the DxAOD preselection is applied to skim the data and MC samples used in the analysis (See Sec. 10.2.3). The event selection starts at the event level with the following requirements: GRL: Data events must originate from luminosity blocks identified as good by the GRL, ensuring the LHC and ATLAS detector are operating under stable conditions (See Sec. 10.2.1). A GRL requirement is not placed on the simulation, since it is generated with nominal detector and beam conditions. Trigger: Events must pass the HLT 2e17 lhloose trigger, which requires at least two elec- trons to satisfy the Loose likelihood-based identification requirement, each with a minimum transverse energy of ET > 17 GeV. Event Cleaning: Event cleaning removes incomplete or corrupt events. The GRL removes entire luminosity blocks affected by detector related problems. To provide additional discrimination, event level detector flags exist to reject events that are incomplete, contain corrupt data, or are inconsistent with LAr, Tile, and SCT error requirements. 173 Two Electrons: At least two electron candidates must originate from the event primary vertex (See Sec. 8.1.2). All electrons considered are reconstructed according to the standard ATLAS algorithm (See Sec. 8.2.2), which reconstructs energy depositions in the EM calorimeter, and attempts to match each with an single inner detector track. Next, the object level selection is applied to select events with at least two good electron candidates. A set of selection criteria is applied to each electron in the event. If an electron fails a cut at any stage in the proceeding selection, then it is excluded from the list of candidate good electrons. If the number of candidates falls below two, then the event is rejected from the analysis. At this stage, the energy scale calibrations (See Sec. 8.3.1) and energy resolution smearing (See Sec. 8.3.2) are applied to the individual electron candidates in data and MC, respectively. The object level selection is: Pseudorapidity: The electron candidate must be reconstructed from an energy cluster in the central region (|η| < 2.47) of the ATLAS detector. Additionally, electrons in the transition region, 1.37 < |η| < 1.52, between the barrel and end-cap EM calorimeters are rejected, due to a degraded energy resolution in this region. Object Quality: The quality of the cluster associated with an electron candidate is checked. The object quality requirement rejects candidates with bad quality or fake clusters re- sulting from calorimeter problems, such as malfunctioning hardware, LAr noise bursts, or data integrity errors [185]. Transverse Energy: An electron candidate must have a transverse energy ET > 30 GeV. The ET cut is chosen to ensure that the HLT 2e17 lhloose trigger is fully efficient. d0 Significance: The track associated with an electron candidate must have a d0 signifi- cance σd0 that satisfies d0/σd0 < 5 (See Sec. 8.1.2). This requirement helps to select 174 prompt electron candidates that originate from the event primary vertex, and not from secondary vertices, or photons conversions. Longitudinal Impact Parameter: Similarly, the track associated with an electron candi- date must have a longitudinal impact parameter z0 with respect to the reconstructed primary vertex, satisfying |∆z0 sin θ| < 0.5 mm (See Sec. 8.1.2). Electron Identification: Each electron candidate must satisfy the Medium likelihood- based electron identification criterion (See Sec. 8.2.3), which provides discrimination based on longitudinal and lateral shower shape variables, tracking variables, and cluster- track matching quantities (See Table 8.1). The electron identification requirement serves to reduce background processes that fake the electron signature. Isolation: Each electron candidate must satisfy the Loose isolation criterion, which re- quires that the energy deposition in the EM calorimeter and inner detector track matched to the electron candidate both be well isolated (See Sec. 8.2.4). The isolation requirement further suppresses backgrounds originating from non-prompt decays and misidentified hadronic jets. The final stage of selection focuses on the construction of the dielectron object: Pair Formation: If two or more electron candidates are retained after the object level selection, then the two with the highest transverse momentum ET are selected to form a dielectron pair. Dielectron Invariant Mass: The invariant mass of the pair is calculated with Eq. 10.1 and must satisfy mee > 80 GeV. 175 Events that satisfy all of the preceding requirements are retained for analysis.9 10.5.2 Cutflows Tables 10.1 and 10.2 list the number of events in the full 36.1 fb−1 dataset and the simulated Z(cid:48) χ signal sample, respectively, that pass the dielectron event selection of Section 10.5.1. These cutflow tables summarize the number of events that pass each stage of selection, along with their relative and absolute selection efficiencies. There are initially 82,684,764 events in data, and after applying the selection, 12,716,634 candidate events remain in use for analysis.10 An identical set of selection criteria is applied to the signal and background simulation, in order to make a meaningful comparison with the data.11 Event and object level corrections are applied to the MC to enhance its accuracy. All corrections described in Section 9.4 are included, and their combined effect on the resulting number of Z(cid:48) χ events can be seen in the final row of Table 10.2, denoted as “Event Weights.” The MC simulation is normalized to an integrated luminosity of 36.1 fb−1 to match the observed value in data. The resulting signal acceptance is approximately 71% for a 3 TeV Z(cid:48) χ signal. Additional cutflow tables for the Drell-Yan, Top, and Diboson background processes can be found in Appendix B. The total background rejection for the Top and Diboson backgrounds is approximately 98% and 88%, respectively. 9In the muon channel, an opposite-sign charge requirement is placed on dimuon candidates. This is not required in the electron channel, however, due to the possible charge misidentification of high-ET electrons, resulting from either bremstrahlung, or from the limited momentum resolution associated with high-pT tracks in the Inner Detector. 10In fact, considering the dataset was initially skimmed, the number of events before preselection totaled 6,327,792,063. Consequently, out of the entire 36.1 fb−1 dataset, only approximately 0.2% of LHC events were selected for use in the electron channel analysis. 11The GRL requirement is not included, however, since the MC is generated under nominal beam and detector conditions. 176 Selection Cut Number Events Relative Eff. [%] Absolute Eff. [%] Post-selection GRL Trigger Event Cleaning 2 Electrons η Object Quality ET d0 Sig. z0 Identification Isolation Invariant Mass 82 684 764 80 525 303 24 639 061 24 601 393 24 601 392 23 784 390 23 722 173 16 035 333 15 774 118 15 597 169 13 756 703 13 241 876 12 716 634 100.00 97.39 30.60 99.85 100.00 96.68 99.74 67.60 98.37 98.88 88.20 96.26 96.03 100.00 97.39 29.80 29.75 29.75 28.77 28.69 19.39 19.08 18.86 16.64 16.01 15.38 Table 10.1: The number of events passing the electron channel event selection in the 36.1 fb−1 dataset obtained by ATLAS during the 2015+2016 physics runs. The first column indicates the absolute number of events passing selection, the middle column depicts the relative efficiency of each cut, and the last column shows the absolute efficiency. Selection Cut Number Events Relative Eff. [%] Absolute Eff. [%] Post-selection GRL Trigger Event Cleaning 2 Electrons η Object Quality ET d0 Sig. z0 Identification Isolation Invariant Mass Event Weights 38.48 38.48 33.12 33.12 33.12 32.54 32.50 30.55 30.48 30.40 28.64 28.17 28.16 27.27 100.00 100.00 86.08 100.00 100.00 98.23 99.87 94.02 99.75 99.74 94.21 98.36 99.99 96.82 100.00 100.00 86.08 86.08 86.08 84.56 84.45 79.40 79.20 78.99 74.42 73.20 73.19 70.86 Table 10.2: The number of events passing the electron channel event selection in a sample of simulated Z(cid:48) χ events with pole mass MZ(cid:48) = 3 TeV. The sample has all corrections applied that are described in Section. 9.4, and is normalized to an integrated luminosity of 36.1 fb−1 according to the procedure outline in Section 9.4.1. 177 10.5.3 Signal Acceptance The pole mass MZ(cid:48) of a hypothetical Z(cid:48) boson is generally a free parameter of the new physics model, as is the case for the models considered in this analysis. Therefore, it is important to quantify the performance of the dielectron event selection as a function of the a-priori unknown Z(cid:48) pole mass, as the kinematics and reconstruction properties of the decay electrons are sensitive to the precise value of MZ(cid:48). The signal acceptance times efficiency is plotted as a function of MZ(cid:48) in Figure 10.3.12 Here, the dielectron event selection is applied to individual Z(cid:48) signal templates, ranging from MZ(cid:48) = 150 GeV to MZ(cid:48) = 5000 GeV in 50 GeV intervals, and the resulting total efficiency for each is quantified. A smooth polynomial curve is fit through each resulting point, providing an estimate of the total efficiency within the 50 GeV intervals. For a Z(cid:48) χ with MZ(cid:48) = 150 GeV, the total efficiency is approximately 32%, which quickly rises to 71% for the 3 TeV Z(cid:48) χ (as can also be seen in Table 10.2), before slowing dropping off again at high mass. The quick “turn-on” at low mass is due to the high-ET and central-η requirements in the dielectron selection. At higher pole masses the Z(cid:48) is more likely to be produced at rest, and as a result, its decay electrons tend to be more central, leading to additional events passing the η requirement. However, above about 2.5 TeV in pole mass, the efficiency starts to drop again due to the strong decrease in parton luminosity at high momentum transfer. The colliding quarks have insufficient energy to produce on-shell Z(cid:48) events, and as a result, the relative fraction of events in the low-mass tail of the spectrum becomes enhanced (See Fig. 10.4). The electrons in the tail are again less central, causing a drop in acceptance. 12The acceptance times efficiency A is the number of selected dielectron candidate events divided by the total number of generated events; sometimes also called the total efficiency, or just the efficiency. 178 Figure 10.3: The Z(cid:48) signal efficiency, according to the dielectron selection defined in Sec- tion 10.5.1, as a function of Z(cid:48) χ pole mass. The selection is maximally efficient at high mass, the primary target of the search. The quick “turn-on” at low mass and slow “turn-off” at high mass are due to the electron kinematics at each of these scales. Figure 10.4: Z(cid:48) χ signals post-selection for several characteristic mass points. A low-mass tail begins to develop as the kinematic limit is reached. The number of events within specific invariant mass intervals is displayed in Table 10.3 for the 4 and 5 TeV pole mass Z(cid:48) χ. 179 [TeV]χZ’M0.511.522.533.544.55Acceptance x Efficiency0.20.30.40.50.60.70.80.91 SimulationATLAS = 13 TeVsDilepton Search Selection ee→Z’ µµ →Z’ Dielectron Invariant Mass [GeV]9010020030010002000Events3-102-101-10110210310410 InternalATLAS-1 = 13 TeV, 36.1 fbsDilepton Search Selection (2 TeV)cZ' (3 TeV)cZ' (4 TeV) cZ' (5 TeV) cZ' 10.6 Fake Background Estimate The dielectron event selection is robust, greatly reducing the probability of selecting events with non-prompt, or fake, electrons. However, because of the overwhelming production rate of Multi-jet events at the LHC, a small but non-negligible number of fake events pass the dielectron event selection. Therefore, this “fake” background exists as a component of the selected data, and must be properly accounted for in the modeling the overall dielectron background estimate. The fake background consists of events that originate primarily from two sources: W +jets production, in which a misidentified jet and electron pass selection, and QCD Multi-jet pro- duction, in which two misidentified jets pass selection. Due to the relatively low probability of this happening, it is prohibitive to generate MC simulation samples with sufficient statis- tics to model this background accurately. Instead, a fully data-driven approach is taken, known as the matrix method [186, 187]. Here, the fake background is fully estimated from a sample of events obtained directly from the 36.1 fb−1 dataset used in the analysis. 10.6.1 The Matrix Method In the matrix method, two comparable event selections are defined: “loose” and “tight.” The tight selection coincides with the nominal dielectron event selection of Section 10.5.1. The loose selection is defined relative to the tight selection, where the electron identification requirement is relaxed from the Medium to the Loose operating point, and the isolation requirement removed entirely. Therefore, the set of objects passing the tight selection forms a subset of those passing the loose selection. The idea is to measure the electron efficiency in each selection in order to identify the signal contamination from misidentified jets. 180 Accordingly, with two unique selections and two final-state electrons, four distinct ob- servables can be defined that measure the number of dielectron pairs passing the tight and loose selections, respectively. These observables are NT T , NT L, NLT , NLL, where, the first (second) index refers to the leading (subleading) electron,13 and the L subscript refers to an electron passing the loose selection, but failing the tight selection, while the T subscript refers to an electron that passes both. Similarly, for any dielectron kinematic distribution, the true background expectation can be decomposed into four components, which are unmeasurable and consist of distinct com- binations of real and fake electrons. These components are NRR, NRF , NF R, NF F , where, again, the first and second indices refer to the leading and subleading electron candidates, and the R and F subscripts define whether the electron candidate is real or fake. One of the four component is due entirely to real electrons, while the other three account for the fake combinations. It is not possible to estimate these true quantities directly, but it is possible to con- struct a system of equations that relates the experimentally measurable quantities to the unmeasurable ones:  =   NT T NT L NLT NT T r2 r(1 − r) (1 − r)r (1 − r)2 rf r(1 − f ) (1 − r)f f r f (1 − r) (1 − f )r (1 − r)(1 − f ) (1 − f )(1 − r)   NRR NRF NF R NF F  . (10.2) f 2 f (1 − f ) (1 − f )f (1 − f )2 Here, the real-rate r and fake-rate f provide the connection between the measurable and 13Of the selected electron pair, “leading” denotes the one with the highest ET , while “subleading” refers to the one with the second highest ET . 181 unmeasurable terms. The real-rate r (fake-rate f ) is the probability that a real (fake) electron selected by the loose selection will also be selected by the tight selection. If r and f can be estimated, then the fake background contribution to the electron channel can be identified, reading off the matrix entries, as Multi−jet & W+jets T T N = rf NRF + f rNF R + f 2NF F (10.3) which, as required, is in terms of dielectron pairs with at least one fake electron. The final step of the method is to re-express the unmeasurable terms on the right hand side of Eq. 10.3 in terms of the measurable ones, which can be obtained simply by inverting the matrix of Eq. 10.2. 10.6.2 Real- and Fake-Rate Determination The real-rate and fake-rate are defined as r = N tight real N loose real , f = N tight f ake N loose f ake , (10.4) where the number of electrons passing the tight selection Ntight forms a subset of those passing the loose selection Nloose. In general, the real-rate and fake-rate of Eq. 10.4 are functions of electron ET and |η|. In order to obtain an estimate of these rates, enriched samples of real and fake electrons are needed. The enriched sample of real electrons can be easily obtained from the Drell-Yan MC sim- ulation, since by definition its events contain two real electrons. The real-rate r is estimated to be greater than 90% for ET > 30 GeV and increases with ET . When binned as a function 182 of |η|, the rate drops by approximately 2%, moving from low to high |η|. An enriched sample of fake electrons is obtained directly from data. Here, a special selection is defined, which combines several single electron triggers that incorporate the likelihood-based Loose or vLoose electron identification requirements. To mitigate contamination from real electrons in the fake enriched sample, three further selection criteria are imposed. First, an /ET > 25 GeV requirement (See Sec. 7.2) is used to veto events that originate from W → eν decays. Second, in order to suppress events that originate from the Drell-Yan process, a veto is placed on events with two electrons that satisfy likelihood-based Medium identification criterion. Lastly, the requirement |MZ − mee| > 20 GeV is used to veto events within 20 GeV of the Z boson mass peak. Any remaining contamination from real electrons is estimated using the Drell-Yan, Top, and Diboson MC samples. Whatever remains of the MC post-selection is subtracted from the data to finally obtain the fake enriched sample, which is used to determine N tight f ake and N loose f ake . For ET > 30 GeV, the fake rate is estimated to be between approximately 16% and 34% depending on the |η| and ET regions considered. 10.6.3 Background Extrapolation The fake background estimate is obtained from the sample of events defined by Eq. 10.3. The invariant mass of each dielectron pair mee in the sample is calculated and binned in a histogram, providing the fake background estimate to be used in conjunction with the MC for the analysis.14 However, at this stage, the resulting background estimate only applies above the region of the Z boson mass peak, due to veto applied in the fake-rate derivation, 14With a sample of fake background events, kinematic variables in addition to the invariant mass can be extracted, the results of which can be seen in Figures 10.7, 10.8, and 10.9. 183 and runs out of statistics at high mass. In order to obtain coverage over the entire dielectron search region, the estimate is ex- trapolated both down to low mass, below the Z boson mass peak, and out to high mass using a series of fits to the mee distribution. The functional form of the fit is not known a priori, but the so-called “dijet function” has been used successfully in previous ATLAS dijet searches [188, 189], and is therefore similarly used here. The form is given by f (x) = p1(1 − x)p2xp3+p4 log(x)+p5 log2(x), x = mee√ s (10.5) where p1 through p5 are free parameters, allowing for flexibility in the resulting fit. To account for any bias in the choice of fit range, the starting point is varied from 125 GeV to 195 GeV in steps of 10 GeV. This results in eight individual fits, and for each, the χ2/ndf falls between 1.2 and 1.6. The resulting central estimate is taken to be the χ2 weighted mean of the set of fits, and the fit uncertainty is characterized by the resulting envelope. The final result can be seen in Figure 10.5. Systematic uncertainties related to the matrix method technique, and to the fit uncertainty, are included as part of the error estimate. 10.7 Comparison of Data to the Standard Model Back- ground Expectation In the following section, several direct comparisons are made between the observed data, selected from the 36.1 fb−1 LHC dataset, and the full Standard Model expectation. All non-negligible backgrounds are taken into account for the comparison. The Standard Model expectation is obtained by applying the dielectron event selection 184 Figure 10.5: The data-driven fake background estimate used in the dielectron channel of the dilepton analysis. The estimate is obtained with the matrix method, following the technique outlined in Section 10.6 [4]. (See Sec. 10.5.1) to the Drell-Yan, Top, and Diboson MC samples. The resulting kinematic distributions are normalized (See Sec. 9.4.1) and summed together. The inclusion of the fake background in the sum completes the estimate. 10.7.1 Event Yields The expected and observed event yields are presented in Table 10.3 in several bins of dielec- tron invariant mass. The expected event yields are subdivided into the various background processes, and the Z(cid:48) χ signal process, with a pole mass of 3 and 4 TeV, is presented as well 185 Events3−102−101−10110210310410 InternalATLAS-113 TeV, 36.1 fbQCD bkg est.QCD bkg unc. upQCD bkg unc. down [GeV]eeM1001000rel. uncer.0510 mee [GeV] Drell–Yan Top quarks Dibosons Fakes Total SM Data Z(cid:48) χ (4 TeV) Z(cid:48) χ (5 TeV) mee [GeV] Drell–Yan Top quarks Dibosons Fakes Total SM Data Z(cid:48) χ (4 TeV) Z(cid:48) χ (5 TeV) 80–120 11 800 000 ± 700 000 28 600 ± 1800 31 400 ± 3300 11 000 ± 9000 11 900 000 ± 700 000 0.00635 ± 0.00021 0.00305 ± 0.00012 12 415 434 120–250 216 000 ± 11 000 44 600 ± 2900 7000 ± 700 5600 ± 2000 273 000 ± 12 000 0.0390 ± 0.0015 0.0165 ± 0.0006 275 711 250–400 17 230 ± 1000 8300 ± 600 1300 ± 140 780 ± 80 27 600 ± 1100 0.0564 ± 0.0025 0.0225 ± 0.0010 27 538 700–900 421 ± 34 94 ± 8 39 ± 4 39 ± 6 590 ± 40 900–1200 176 ± 17 27.9 ± 2.8 16.9 ± 2.1 16.1 ± 2.0 237 ± 17 1200–1800 62 ± 7 5.1 ± 0.7 5.8 ± 0.8 7.9 ± 2.3 81 ± 7 589 0.0585 ± 0.0035 0.0218 ± 0.0013 209 0.074 ± 0.005 0.0295 ± 0.0021 61 0.121 ± 0.011 0.040 ± 0.004 400–500 2640 ± 180 1130 ± 80 228 ± 25 151 ± 2 4150 ± 200 4140 0.0334 ± 0.0027 0.0139 ± 0.0007 1800–3000 8.7 ± 1.3 < 0.001 0.74 ± 0.11 1.6 ± 1.2 11.0 ± 1.8 0.172 ± 0.017 0.040 ± 0.004 10 500–700 1620 ± 120 560 ± 40 146 ± 16 113 ± 17 2440 ± 130 0.064 ± 0.004 0.0275 ± 0.0015 2390 3000–6000 0.34 ± 0.07 < 0.001 0.028 ± 0.004 0.08 ± 0.27 0.45 ± 0.28 2.57 ± 0.27 0.280 ± 0.030 0 Table 10.3: Expected and observed event yields in different invariant mass intervals according to the event selection of Section 10.5.1. The errors include both statistical and systematic uncertainties (described in Section 10.8). for comparison. Coarser binning is used at high mass to account for the lower number of expected events. For each mass bin considered, the observed event yield in data agrees well with the Standard Model prediction, generally falling within ±1σ according to the quoted uncertain- ties.15 The errors include the combined statistical, experimental, and theoretical systematic uncertainties, which are assessed in Section 10.8. 10.7.2 Kinematic Distributions In this section, the kinematic properties of electrons in both data and MC simulation passing the full event selection are compared. The MC includes the Drell-Yan, Top, and Diboson processes, each stacked, summed, and normalized to 36.1 fb−1. Together with the fake 15A deficit exists in the 900-1200 GeV and 1200-1800 GeV bins. Potential sources for this discrepancy were investigated, and no problems were found. 186 background estimate, the overall Standard Model expectation is obtained, and compared directly to data for several kinematic distribution of interest. Figure 10.6 shows the dielectron invariant mass distribution, the main discriminating variable of the search, while Figures 10.7, 10.8, and 10.9 show distributions for the leading and subleading electron transverse energy ET , pseudorapidity η, and azimuthal angle φ, respectively. The electron ET , η, and φ distributions are presented in the signal region region beyond the Z boson mass peak mee > 120 GeV. For each distribution presented, the data, shown with their associated Poisson statis- tical uncertainty, are compared to the stacked Standard Model expectation, and the ratio between the two can be seen in the sub-panel. Here, the turquoise band quantifies total systematic uncertainty on the Standard Model expectation, and it can generally be seen that the background modeling is consistent with the statistical uncertainty of the data. The dielectron invariant mass serves as the primary search distribution for resonant new physics (as outlined in Sec. 10.1). The bin widths are constant in log(mee), and the spectrum is seen to be quickly falling, covering multiple orders of magnitude by the time the TeV scale is reached. Clearly visible is the Z boson mass peak at MZ (cid:39) 91 GeV. Also visible are three resonant Z(cid:48) χ signals from MC with pole masses of MZ(cid:48) = 3 TeV, MZ(cid:48) = 4 TeV, and MZ(cid:48) = 5 TeV, overlaid with the Standard Model expectation. The observed data agree well with the Standard Model expectation over the entirety of the invariant mass spectrum, with no obvious signal-like excesses present. As quantified in the yield table, a slight deficit can be seen in the 1-2 TeV range. The highest dielectron invariant mass event has Mee = 2.90 TeV. A display for this event, as well as the highest mass event from the 2015 run can be seen in Appendix C, along with their associated kinematic information. 187 Like the invariant mass, the electron ET spectrum, coincident with the ET > 30 GeV requirement, falls rapidly over several orders of magnitude with increasing electron ET . The contribution from jets, which are generally “harder” than electrons, increases as high ET is reached. In the ratio sub-panel, a good level of agreement between the data and the Standard Model expectation can be seen, as the statistical error of the data overlaps with the systematic band. The overlaid Z(cid:48) signals exhibit a peak at roughly half their mass value, due two the two body nature of the decay. The electron η distribution is symmetric relative to the beam axis, displaying a maximum at η = 0, and slowly falling off at higher values, until |η| = 2.47 is reached. The interval 1.37 < |η| < 1.52 defines the transition region between the barrel and end-caps of the calorimeter. Events still populate this region, since the η plotted is the so-called “best- estimate,” which utilizes the tracking information (See Sec. 8.2.2). The event selection requirement is placed on the η position of the energy deposition in the EM calorimeter second layer.16 The systematic uncertainty, displayed in the sub-panel, is an approximately constant 5%, and a good level of agreement is seen between data and background expectation. The electron φ distribution exhibits azimuthal symmetry in the x− y transverse plane of the detector, with electrons equally distributed over [−π, π]. The overall level of agreement between the Standard Model expectation and data is good, with data accounted for by the flat 5% systematic band. While no obvious signature of resonant new physics can be seen in the mee distribution, a rigorous statistical analysis is needed to conclude to what degree the data agrees with background expectation, which is the subject of the next chapter. 16Indeed, plotting this shows the transition region devoid of events. 188 Figure 10.6: Dielectron invariant mass distribution for candidate dielectron events selected by the full event selection. The stacked sum of Standard Model backgrounds is normalized to 36.1 fb−1 and compared to data. Three Z(cid:48) χ signals are included for reference, overlaid with the Standard Model expectation. No obvious signal-like excess is present. The lower-panel sub-plot shows the “post-fit” result obtained from the statistical analysis. 10.8 Systematic Uncertainties This final section describes the systematic uncertainties accounted for in the electron channel of the dilepton analysis. Statistical uncertainties affecting the Standard Model background expectation are negligible (< 1%), owing to the high statistics simulation samples used in the analysis, and can therefore be safely neglected.17 The systematic uncertainties affecting the analysis can be divided into two general cat- 17Statistical uncertainties arise from the inherent random fluctuations in a measurement. In a large number of measurements, statistical uncertainties average to zero. 189 9010020010002000Events-210-110110210310410510610710Data*γZ/Top QuarksDibosonMulti-Jet & W+Jets (3 TeV)χZ’ (4 TeV)χZ’ (5 TeV)χZ’ATLAS-1 = 13 TeV, 36.1 fbsDielectron Search SelectionData / Bkg 0.60.811.21.4Dielectron Invariant Mass [GeV]10020030010002000 (post-fit)Data / Bkg0.60.811.21.4 (a) (b) Figure 10.7: Leading (a) and subleading (b) electron ET distributions in the dielectron invariant mass region mee > 120 GeV, after the full event selection. The stacked sum of Standard Model backgrounds is normalized to 36.1 fb−1 and compared to data. 190 3010020010002000Events-210-110110210310410510610710810Data*γZ/Top QuarksDibosonMulti-Jet & W+Jets (2 TeV)χZ’ (3 TeV)χZ’ (4 TeV)χZ’ InternalATLAS-1 = 13 TeV, 36.1 fbsDilepton Search Selection > 120 GeVeem [GeV]TLeading Electron E304010020030010002000Data / Bkg 0.60.811.21.43010020010002000Events-210-110110210310410510610710810Data*γZ/Top QuarksDibosonMulti-Jet & W+Jets (2 TeV)χZ’ (3 TeV)χZ’ (4 TeV)χZ’ InternalATLAS-1 = 13 TeV, 36.1 fbsDilepton Search Selection > 120 GeVeem [GeV]TSubleading Electron E304010020030010002000Data / Bkg 0.60.811.21.4 (a) (b) Figure 10.8: Leading (a) and subleading (b) electron η distributions in the dielectron invari- ant mass region mee > 120 GeV, after the full event selection. The stacked sum of Standard Model backgrounds normalized to 36.1 fb−1 and compared to data. 191 -3-2-10123Events5000100001500020000250003000035000Data*γZ/Top QuarksDibosonMulti-Jet & W+Jets (2 TeV)χZ’ (3 TeV)χZ’ (4 TeV)χZ’ InternalATLAS-1 = 13 TeV, 36.1 fbsDilepton Search Selection > 120 GeVeemηLeading Electron -3-2-10123Data / Bkg 0.80.850.90.9511.051.11.151.2-3-2-10123Events5000100001500020000250003000035000Data*γZ/Top QuarksDibosonMulti-Jet & W+Jets (2 TeV)χZ’ (3 TeV)χZ’ (4 TeV)χZ’ InternalATLAS-1 = 13 TeV, 36.1 fbsDilepton Search Selection > 120 GeVeemηSubleading Electron -3-2-10123Data / Bkg 0.80.850.90.9511.051.11.151.2 (a) (b) Figure 10.9: Leading (a) and subleading (b) electron φ distributions in the dielectron in- variant mass region mee > 120 GeV, after the full event selection. The stacked sum of all Standard Model backgrounds normalized to 36.1 fb−1 and compared to data. 192 -3-2-10123Events2000400060008000100001200014000160001800020000Data*γZ/Top QuarksDibosonMulti-Jet & W+Jets (2 TeV)χZ’ (3 TeV)χZ’ (4 TeV)χZ’ InternalATLAS-1 = 13 TeV, 36.1 fbsDilepton Search Selection > 120 GeVeemφLeading Electron -3-2-10123Data / Bkg 0.80.850.90.9511.051.11.151.2-3-2-10123Events2000400060008000100001200014000160001800020000Data*γZ/Top QuarksDibosonMulti-Jet & W+Jets (2 TeV)χZ’ (3 TeV)χZ’ (4 TeV)χZ’ InternalATLAS-1 = 13 TeV, 36.1 fbsDilepton Search Selection > 120 GeVeemφSubleading Electron -3-2-10123Data / Bkg 0.80.850.90.9511.051.11.151.2 egories: experimental (to be described in Sec. 10.8.1) and theoretical (to be described in Sec. 10.8.2).18 All systematic uncertainties are assessed as functions of dielectron invariant mass mee, and quantify the ±1σ variations in the expected dielectron event yield, according to the Standard Model expectation (See Table 10.3). For each systematic source, the larger of two “upward” or “downward” systematic variations is taken to construct a symmetrized uncertainty estimate.19 The result of this procedure can be seen in Figure 10.10 for the experimental sources and Figure 10.11 for the theoretical sources. These symmetric variations are then accounted for as nuisance parameters in the search and limit setting procedures of the statistical analysis performed in the next chapter (See Sec. 11.1). 10.8.1 Experimental Uncertainties The experimental uncertainties affecting the description of the Standard Model background estimate in the electron channel from several sources. Those relating to the measurement of electrons include: the simulation of the detector response [190] (i.e. the electron efficiencies), and the precision of the electron energy measurement [191] (i.e. the energy scale calibration, and electron energy resolution). Additional experimental uncertainties include those arising from the luminosity estimate, the LHC beam energy, the pile-up profile, and the Multi-jet & W +jets fake background estimate. The magnitude of each can be seen in Figure 10.10 as a function of the dielectron invariant mass mee. There are a total of 11 experimental uncertainties considered in the electron channel. They include: 18Systematic uncertainties arise from consistent systematic effects introduced during the measurement, and by the assumptions, or choices, made in the analysis that affect the interpretation of the data. Systematic uncertainties are the result of experimental bias, and cannot simply be reduced by taking more data. 19An upward or downward systematic variation leads to an increased or decreased estimate on the Standard Model background yield. 193 Figure 10.10: Experimental systematic uncertainties in the electron channel of the dilepton analysis. Shown are effects that shift the Standard Model event yield by greater than 1% anywhere in the dielectron invariant mass search spectrum. The largest experimental un- certainty is due to the W +jets & QCD Multi-jet fake background estimate, which rises to 10% by mee = 2 TeV. See Section 10.8.1 for explanations of the source of each systematic uncertainty. Electron Efficiencies: The electron reconstruction, identification, isolation, and trigger efficiency scale factors account for the difference between the efficiency observed in data and the one predicted by the simulation (See Sec. 9.4.2). Each efficiency correc- tion comes with an associated systematic uncertainty, provided by the ATLAS elec- tron/photon combined performance group (EGP), as an “upward” and “downward” variation from the nominal scale factor estimate. The variations are applied as event weights with the ElectronEfficiencyCorrectionTool when running over the simu- lation samples for analysis. The systematic uncertainties are determined by varying the tag-and-probe methodology used to extract the scale factor corrections [3]. For electrons with an ET > 200 GeV the tag-and-probe technique can no longer be used 194 Dielectron Invariant Mass [GeV]90100200300100020003000Systematic Uncertainty [%]02468101214161820E. ScaleEff. IDEff. IsoLumi.FakesBeam E.Total InternalATLAS-1 = 13 TeV, 36.5 fbs due to lack of statistics, so scale factors and their uncertainties are extrapolated to high ET . The uncertainty in the identification scale factor is determined to be 2.0% for ET > 150 [3]. Similarly, the uncertainty in the isolation scale factor was determined to be 2.0% for ET > 150 GeV and 5.0% for ET > 500 [190]. At an invariant mass of mee = 2 TeV, the propagation of the electron reconstruction and trigger efficiency uncertainties each result in a negligible uncertainty (< 1%) on the Standard Model background estimate, while the ID and isolation efficiencies result in uncertainties of 2.6%, and 9.1%, respectively (See the “Eff. ID” and “Eff. Iso” curves in Fig. 10.10).20 Electron Energy Scale: The measured electron energies are calibrated using a multivari- ate analysis (MVA) and in-situ techniques. Systematic uncertainties arise from the calibration procedure, and are provided by the ATLAS EGP group (See Sec. 8.3.1). As with the nominal correction, the ElectronPhotonFourMomentumCorrection tool is used to apply the “upward” and “downward” systematic variations in the energy scale correction on an event-by-event basis to the associated event electrons. While the calibration is strictly applied to data, in order to assess the systematic uncertainty, it is instead applied to the simulation, which has a negligible statistical uncertainty. There are over 60 systematic variations associated with the energy scale calibration [145]. The dilepton analysis uses a simplified calibration scheme in which all systematic vari- ations are summed together in quadrature, reducing this number down to one, which greatly simplifies the complexity of the statistical analysis, presented in the following chapter. At an invariant mass of mee = 2 TeV, the uncertainty in the electron energy scale leads to a 4.1% uncertainty on the Standard Model background estimate (See the 20Indeed, two “kinks” can be seen in isolation scale factor uncertainty at mee (cid:39) 300 GeV and mee (cid:39) 1 TeV in Figure 10.10, which arise as the scale factor uncertainty estimate abruptly changes at ET = 150 GeV and ET = 500 GeV. 195 “E. Scale” curve in Fig. 10.10). The energy scale in general does not greatly impact the sensitivity of the search, as the main effect would be to shift the precise location of a potential Z(cid:48) peak, while still preserving its shape. Electron Energy Resolution: The electron energy resolution in simulation differs from the one observed in data. The simulation is corrected by smearing the individual electron energies according to data-driven energy smearing factors extracted by the ATLAS EGP group (as explained in Sec. 8.3.2). As was the case with the energy scale calibration, a single systematic uncertainty is used to account for the energy resolution uncertainty [145]. The “upward” and “downward” variations are applied on an event- by-event basis using the ElectronPhotonFourMomentumCorrection tool. For high ET electrons, the energy resolution is dominated by the so-called calorimeter “constant term” (See Sec. 8.3.2), which has an uncertainty at the per-mille level [143]. Therefore, only small shifts are introduced to the width of the energy smearing distribution, and the resultant uncertainty on the Standard Model expectation is negligible. W +jets & Multi-jet Fake Background Estimate: The matrix method is used to es- timate the background contribution that arises from misidentified jets that pass the dielectron selection. The uncertainty on the fake background estimate is ascertained by comparing variations in the methodology of the matrix method. This includes varying the extrapolation methodology, how the electron fake-rate is determined, and comparing the resulting estimate from events with same-charge and opposite-charge electrons. Each uncertainty is summed together in quadrature to produce an overall uncertainty estimate. The extrapolation uncertainty can become very large at high dielectron masses mee (See the “Fakes” curve in Fig. 10.10), but it does not greatly 196 affect the interpretation of the final results of the analysis. Luminosity: A flat 3.2% uncertainty [192] is assessed on the integrated luminosity of the 36.1 fb−1 dataset according to the ATLAS luminosity group [174] (See the “Lumi.” curve in Fig. 10.10). The luminosity uncertainty affects the normalization of all signal and background simulation samples. LHC Beam Energy: The LHC beam energy is estimated to have an uncertainty of 0.65%, which results in an 2.0% uncertainty on the Standard Model background estimate at an invariant mass of mee = 2 TeV (See the “Beam E.” curve in Fig. 10.10) Monte Carlo statistics: While not strictly a systematic uncertainty, the MC statistical uncertainty is assessed and found to have a negligible effect (< 1%) on the Standard Model estimate. Each MC sample used in the electron channel analysis is generated with an adequate number of statistics to ensure accurate modeling of the dielectron invariant mass distribution. Pile-up Reweighting: The PileupReweighting tool is used to reweight the MC simula- tion to account for the changing pile-up conditions in data. The tool provides “upward” and “downward” variations to account for the systematic uncertainty associated with the extraction of the pile-up scale factors from data [193]. Systematic variations are applied on an event-by-event basis as event weights during the analysis. The pile-up uncertainty was determined to have a negligible effect (< 1%) on uncertainty of the overall Standard Model background estimate. 197 10.8.2 Theoretical Uncertainties Similarly, multiple theoretical uncertainties affect the description of the Standard Model background estimate in the electron channel, and can assessed on each individual back- ground. Theoretical uncertainties affecting the Drell-Yan background estimate, the domi- nant background of the analysis, include those arising from the Parton Distribution Functions (PDFs), the specific choice of PDF set used to perform the uncertainty estimate, the PDF scale uncertainty, the high-mass running of αS, and assumptions regarding the calculations of the Electroweak (EW) and Photon-Induced (PI) k-Factor corrections. Uncertainties af- fecting the Top and Diboson background estimates arise from their respective cross section estimates, and hence are treated as normalization uncertainties. The magnitude of each uncertainty can be seen in Figure 10.11 as a function of the dielectron invariant mass mee. There are a total of 8 theoretical uncertainties considered in the electron channel. They include: PDF Variation: The Drell-Yan cross section is corrected from NLO to NNLO in QCD with the mass-dependent QCD k-Factor corrections (See Sec. 9.4.5). The NNLO cross section is calculated with Vrap configured with the CT14NNLO PDF set. The PDF uncertainties are obtained using the 90% C.L. CT14NNLO PDF error set, and propagated to the uncertainty on the Drell-Yan cross section estimate, which can be interpreted as the uncertainty on the QCD k-Factor correction. At an invariant mass of mee = 2 TeV, the PDF variation uncertainty is 8.7% (See the “PDFVar” curve in Fig. 10.11). The CT14NNLO error set contains 26 eigenvector variations (See Sec. 3.2.2), which are used to obtain an estimate of the overall PDF uncertainty with Eq. 3.10. As a simplification to the dilepton search, instead of utilizing the full set of 26 eigenvector variations for the 198 (a) (b) Figure 10.11: Theoretical systematic uncertainties in the electron channel of the dilepton analysis (a). Shown are effects that shift the Standard Model event yield by greater than 1% anywhere in the dielectron invariant mass search spectrum. The largest uncertainty is due to the PDF Variation uncertainty, which is also shown decomposed into its individual eigen- vector variations (b). See Section 10.8.2 for an explanation of the source of each systematic uncertainty. 199 Dielectron Invariant Mass [GeV]90100200300100020003000Systematic Uncertainty [%]02468101214161820AlphaSEW kFPI kFPDFVarPDFChoicePDFScaleTotal InternalATLAS-1 = 13 TeV, 36.1 fbsDielectron Invariant Mass [GeV]90100200300100020003000Systematic Uncertainty [%]02468101214161820PDFVar1PDFVar2PDFVar3PDFVar4PDFVar5PDFVar6PDFVar7PDF Total InternalATLAS-1 = 13 TeV, 36.1 fbs PDF uncertainty calculation, a re-diagonalized set of 7 was used, which was provided by the CTEQ collaboration [194] and obtained with MP4LHC [195].21 This is the minimal number of variations that still captures the necessary mass-dependent correlations, and maintains an accurate description of the PDF error, as it pertains to the Drell-Yan process.22 The forms of each of the 7 eigenvector variations are incorporated into the LPXKFactorTool and applied on an event-by-event basis as individual event weights to account for the PDF uncertainty on the QCD k-Factor correction. The PDF variation uncertainty is the largest theoretical systematic uncertainty in the dilepton search. PDF Choice: In addition to the PDF variation uncertainty on the NNLO QCD calculation, a PDF choice uncertainty is also considered, according to the recommendations of the PDF4LHC forum [195]. The PDF choice uncertainty arises from the specific choice of PDF set used to calculate the NNLO QCD k-Factor correction. At an invariant mass of mee = 2 TeV, the PDF choice uncertainty is < 1.0%, but becomes significant at high mass (See the “PDFChoice” curve in Fig. 10.11). It is derived by comparing the central cross section prediction of several other PDF parameterizations, namely MMHT14 [196] and NNPDF3.0 [81], to the 90% C.L. error envelope prediction of CT14NNLO (See Fig. 10.12). If the central value of an alternative PDF set lies outside the CT14NNLO error envelope, it is taken as an additional uncertainty. Specifically, this occurs roughly at an invariant mass of 3.5 TeV, at which point the nominal NNPDF prediction diverges from the CT14NNLO envelope. PDF Scale: Owing to the finite precision of the fixed-order NNLO QCD calculation, the 21The reduction in the number of nuisance parameters prevents the statistical analysis from becoming computationally intractable. 22This is to say, the quadrature sum of the 7 eigenvector variations matches the CT14NNLO error envelope of the full 26. 200 theoretical uncertainties arising from the renormalization and factorization scale choices are considered (See Sec. 3.1.2). Also derived with Vrap, these uncertainties are ob- tained by simultaneously varying the renormalization scale µR and factorization scale µF of the nominal CT14NNLO prediction up and down by a factor of two (See the “PDFScale” curve in Fig. 10.11). αS Value: The final PDF related uncertainty arises from the value of αS chosen for the Vrap calculations (See Fig. 2.9). The uncertainty on αS is computed by varying the nominal value of 0.118 by ±0.003, and propagating it to the NNLO Drell-Yan cross section prediction (See the “AlphaS” curve in Fig. 10.11). EW k-Factor: A systematic uncertainty on the EW k-Factor correction is considered. It is currently not known how to precisely combine calculation involving higher-order EW effects with those involving higher-order QCD effects. Both additive and multiplicative approaches are considered, and the difference between the two approaches is taken as a systematic uncertainty (See the “EW kF” curve in Fig. 10.11). PI k-Factor: A systematic uncertainty on the PI k-Factor correction is considered and arises from the choice of quark mass values and uncertainty on the photon PDF γ(x, Q2) in the calculation of the PI cross section (See the “PI kF” curve in Fig. 10.11). Top Background Normalization: Like the Drell-Yan background, theoretical uncertain- ties affecting the normalization of the Top background are considered. The systematic effects are accounted for with variations on the renormalization and factorization scales, the PDF variations, and variations on αS. Additionally, variations in association with several top quark specific effects, such as the value of top quark mass, are included, as recommended by the Top working group [197]. 201 Diboson Background Normalization: Like the Drell-Yan and Top backgrounds, sys- tematic effects for the Diboson background are accounted for, which included variations on the renormalization and factorization scales, the PDFs, and αS. 10.8.3 Summary This final section summarizes the effect each systematic has on the expected Z(cid:48) signal and Standard Model background yields. Table 10.4 presents the relative impact each uncertainty has on the nominal event yield, according to the prescribed ±1σ variations. Values presented are obtained directly from the estimates shown in Figures 10.10 and 10.11. Any systematic uncertainty that changes the event yield by less than 1% over the entirety of the dielectron search region is neglected in the statistical analysis, as the final result is not affected. These neglected uncertainties include the uncertainty on the electron trigger and reconstruction efficiencies, the electron energy resolution, the MC statistical uncertainty, the pile-up uncertainty, and the Diboson background normalization. Furthermore, theoretical uncertainties are not considered on the signal processes, since the analysis aims to be as model-independent as possible, and systematics associated with the signal can strongly depend on the particular theory of new physics. This decision does not have a significant effect on the final statistical results. The following chapter discusses the statistical analysis in detail. 10.8.4 Comment on PDF Uncertainties As described in Section 10.8.2, the systematic uncertainty associated with the PDFs gives rise to the dominant source of theoretical uncertainty in the dilepton analysis. While this 202 mee at 2 TeV [%] mee at 4 TeV [%] Signal Background Signal Background 3.2 <1.0 2.0 <1.0 <1.0 8.7 1.0 1.6 2.4 3.4 <1.0 <1.0 <1.0 9.1 <1.0 2.6 4.1 <1.0 10 18 3.2 <1.0 4.1 <1.0 N/A N/A N/A N/A N/A N/A N/A N/A <1.0 9.7 <1.0 2.4 <1.0 <1.0 N/A 11 3.2 <1.0 4.1 <1.0 8.4 19 2.0 2.7 5.5 7.6 <1.0 <1.0 <1.0 9.7 <1.0 2.4 6.1 <1.0 129 132 Source Luminosity MC statistical Beam energy Pile-up effects DY PDF choice DY PDF variation DY PDF scales DY αS DY EW corrections DY PI corrections Top quarks theoretical Dibosons theoretical Reconstruction efficiency Isolation efficiency Trigger efficiency Identification efficiency 3.2 <1.0 2.0 <1.0 N/A N/A N/A N/A N/A N/A N/A N/A <1.0 9.1 <1.0 2.6 Electron energy scale <1.0 Electron energy resolution <1.0 W +jets & Multi-jet Total N/A 10 Table 10.4: A Summary of the relative systematic uncertainties affecting the expected signal and background event yields at dielectron invariant masses of 2 and 4 TeV. Background values correspond to the relative change in the total number of expected events in the 2 and 4 TeV bins of the mee histogram depicted in Figure 10.6, while signal values are calculated from changes in the FWHM of the Z(cid:48) χ signal at pole masses of MZ(cid:48) = 2 TeV and MZ(cid:48) = 4 TeV. Here N/A refers to uncertainties not considered on the signal process [5]. uncertainty does not necessarily prohibit the discovery of a high-mass resonance, such as a Z(cid:48) boson, it can greatly affect the characterization of such a discovery, as the ability to distinguish between different possible models may be diminished. Furthermore, the size of the PDF uncertainty can severely reduce the discovery potential of other theoretical interpre- tations, such as those arising from contact-interaction type models [62], or any kind of new physics that features a broad excess, rather than a distinct resonant peak, over the Standard 203 Figure 10.12: Drell-Yan cross section uncertainty as a function of invariant mass as predicted by several modern PDF sets [4]. Model expectation. As the dilepton final-state is sensitive to a host of new phenomena (See Sec. 5.2), it is crucial to find a way to reduce this theoretical uncertainty; otherwise, despite the increasing size of future LHC datasets, it will become increasingly difficult to exclude non-resonant models of new physics or characterize the Z(cid:48) in case of discovery. Moreover, in accordance with the PDF4LHC recommendation [195], the dilepton analysis includes an ad-hoc “PDF Choice” uncertainty (See Sec. 10.8.2) to account for the fact that the nominal MMHT14 and NNPDF3.0 predictions depart from the CT14NNLO 90% C.L. error band, which is the baseline PDF used in the dilepton analysis (See Fig. 10.12). Folding in the PDF uncertainties from multiple sets results in an overestimate and a loss in discovery potential. If the “PDF Variation” uncertainty can be reduced, such that the modern PDF sets are brought into closer agreement, then the “PDF Choice” uncertainty 204 becomes irrelevant.23 Therefore, attacking the PDF uncertainty is crucial to the future success of the dilepton search. A strategy devised to further reduce this uncertainty is presented in Section 10.1. 23The inclusion of additional high-quality LHC data into future PDF global fits could have the effect of resolving disagreements between the current estimates provided by the PDF global fitting groups, especially at high x where the PDFs are least constrained. 205 Chapter 11 Statistical Interpretation and Results In this chapter, the statistical methods used to search for new high-mass resonances in the electron channel of the dilepton analysis are presented. The search for a signal is performed by scanning the dielectron invariant mass distribution of Figure 10.6 for deviations in the observed event yield relative to the Standard Model expectation; a significant excess might indicate the presence of a signal. The search is performed with a dedicated statistical model, which is constructed in Section 11.1. To facilitate the desired interpretations of the data, the statistical model utilizes Z(cid:48) signal templates, which parametrize the expected Z(cid:48) event yields for different theoretical models and mass hypotheses MZ(cid:48) under investigation. These templates are produced with a dedicated signal reweighting technique, described in Section 11.2. The search for a signal and the results thereof are presented in Section 11.3. Two complimentary methods are used to quantify the significance of any potentially observed excess. In case of a null result, exclusion limits are set at 95% C.L. on several Z(cid:48) model parameters of interest using a Bayesian technique outlined in Section 11.4. The results are interpreted in the context of the SSM, the E6 model, and the Minimal model.1 The chapter closes with a final discussion of results in Section 11.5. 1Additional resonant and non-resonant interpretations are possible as well, but are beyond the scope of this dissertation. See Section 5.2 for a brief discussion. 206 11.1 Statistical Model The following section describes the construction of the statistical model used to perform the statistical analysis in the electron channel of the dilepton analysis. The model is based on the dielectron invariant mass mee, due to its excellent discriminating power between the signal-like excesses of a Z(cid:48) boson and the smoothly falling background predicted by the Standard Model (as discussed in Sec. 10.1). To ensure consistency between the search and limit setting, a common statistical model is developed, which is then used to perform statistical hypothesis tests for the search (as in Sec. 11.3.1), and set upper limits on Z(cid:48) model parameters of interest (as in Sec. 11.4). The model is formulated in terms of a binned likelihood function L(µ, θ), where µ is the parameter of interest, which for this analysis parametrizes either the signal strength, the signal cross section, or a coupling constant, and θ is a set of nuisance parameters, which account for the electron systematic uncertainties (as described in Sec. 10.8). The likelihood itself is constructed from a product of Poisson2 probabilities from the Nbins = 66 invariant mass bins of the dielectron invariant mass distribution of Figure 10.6. The multi-bin approach allows for signal shape information, such as the decay width ΓZ(cid:48) of a Z(cid:48) boson, to be distributed over multiple mee mass bins. Incorporating this additional information into the likelihood function improves the overall sensitivity of the search. The likelihood function is L(µ, θ) = Nbins(cid:89) k=1 λk(µ, θ)e−λk(µ,θ) nk! Nsys(cid:89) i=1 N (θi, 0, 1) (11.1) 2The Poisson distribution P (n|λ) = λne−λ/n! quantifies the probability that exactly n events are ob- served when a mean number of events λ is expected. 207 where the first product is over each mee mass bin k and the second is over the systematic uncertainties. For a given mass bin k, the number of observed events nk is described by a Poisson distribution with expectation value λk(µ, θ). The total number of expected dielectron events λk(µ, θ) in a given mee mass bin k depends on the values of µ and θ. The parameter µ ≥ 0 is known as the signal strength, which acts a multiplicative normalization factor on the expected signal yield, while the parameter θ specifies a set of nuisance parameters, which encode the effect of the systematic uncertainties (defined in Sec. 10.8) on the nominal signal and background expectations. The nominal range of each variation can be seen in Figure 10.10 and Figure 10.11. In order to aid in constraining the best-fit values of the nuisance parameters in the statistical analysis, the likelihood is augmented by a product of unit normal distributions N (θi, 0, 1), one for each nuisance parameter θi. The expected number of events λk(µ, θ) can be decomposed into a sum of expected Z(cid:48) signal events sk(θ) and Standard Model background events bk(θ), where λk (µ, θ) = µsk (θ) + bk (θ) . (11.2) Again, the signal strength µ serves as a floating normalization constant on the signal expec- tation sk (θ). The background expectation bk(θ) is the sum of dielectron events that pass the dielectron event selection (See Sec. 10.5) and go on to populate each mee mass bin k.3 The expected number of signal events sk(θ) can be further expressed as 1 + Nsys(cid:88) i=1  . θi δski sk sk(θ) = sk × (11.3) 3These include the standard Drell-Yan, Top, Diboson, and W +jets & QCD Multi-jet processes. 208 The central value sk is obtained from Eq. 6.1, where sk = σZ(cid:48)Br(Z(cid:48) → e+e−) · A(Z(cid:48) → e+e−) · Lint (11.4) where σZ(cid:48) is the Z(cid:48) production cross section calculated with Eq. 4.16, Br(Z(cid:48) → e+e−) is the Z(cid:48) branching fraction to electron pairs, A(Z(cid:48) → e+e−) is the total efficiency to select a Z(cid:48) signal event, quantified in Figure 10.3 as a function of pole mass, and Lint is the integrated luminosity of the dataset. The ratio δski/ski in Eq. 11.3 expresses the relative shift of the expected signal yield sk in mee mass bin k according to the ±1σ variation induced by systematic uncertainty i.4 These shifts are parametrized by the mass-dependent relative systematic uncertainties of Figure 10.10 and Figure 10.11, whose explicit values are summarized in Table 10.4. Likewise, the expected number of background events can be expressed (11.5) 1 + bk(θ) = bk ×  Nsys(cid:88) i=1 θi δbki bk where the central value bk is the Standard Model expectation in mee mass bin k and includes the contributions from each background process. It similarly follows from Eq. 6.1, accounting for the respective cross sections, branching fractions, and total efficiencies of each individual background process. The relative shift in the background expectation from systematic i is accounted for by δbki/bki, and is due to the same systematics uncertainties in Figure 10.10 and Figure 10.11 as the signal process. Correlations between signal and background system- atic uncertainties are accounted for across all mass bins, since both sk(θ) and bk(θ) are each 4The nuisance parameters θi, associated with each systematic uncertainty i, act as free normalizations, which have the effect of smearing out the nominal Poisson prediction. 209 affected by the same set of systematics. Equations 11.3, 11.4, and 11.5 specify all the required inputs to perform the statistical analysis. However, in order to perform a comprehensive search, multiple Z(cid:48) models and mass hypotheses MZ(cid:48) must be tested. Therefore, the dilepton analysis makes use of a signal reweighting technique, which is used to generate Z(cid:48) signal samples on-the-fly using the Leading-Order (LO) Drell-Yan Monte Carlo (MC) samples (See Sec. 9.3.6). The power in this technique lies in its ability to reweight the Drell-Yan MC to any Z(cid:48) model or pole mass MZ(cid:48) of interest. A reweighted Z(cid:48) signal sample can then be used to construct a Z(cid:48) signal template, a dielectron invariant mass histogram that contains the precise signal expectation sk(θ) for the specific Z(cid:48) model and pole mass MZ(cid:48) under investigation. This formulation allows for the parameter sk(θ) of Eq. 11.3 to be fully specified for a wide variety of Z(cid:48) models or model parameters under consideration, thus enabling the possibility of a comprehensive search. 11.2 Signal Reweighting This is a contribution to the ATLAS dilepton analysis by the author and is described in detail in Reference [198]. The dilepton analysis uses signal templates in order to interpret the level of disagreement between the observed data and the Standard Model expectation as part of the statistical analysis. Signal templates are produced from the LO Drell-Yan MC samples with the LPXSignalReweightingTool. Event-by-event weights are calculated and applied to Drell-Yan events passing the event selection, thereby “reweighting,” or rescaling, them into corresponding Z(cid:48) signal events. Effectively this produces a unique sample of Z(cid:48) signal events, which, for all intents and purposes, matches what would otherwise be produced 210 from a dedicated MC generation, but with orders of magnitude less computing required. As a result, signal templates can be generated for a diverse range of Z(cid:48) models and model parameter choices using only the standard analysis Drell-Yan simulation. This allows for a much more comprehensive statistical analysis to be performed, which in the case of the Minimal model, would otherwise be impossible due to the large amount of signal MC required to effectively scan the model parameter space. The signal templates include the same set of experimental and theoretical corrections as applied to the other simulation samples, including the standard electron efficiency, en- ergy resolution, and pile-up corrections (described in Sec. 9.4). In addition, higher-order QCD corrections are applied to the signal templates, utilizing the same LO-to-NNLO QCD mass-dependent k-Factor as the background MC (See Table 9.3); however, higher-order EW corrections are left out due to the model-dependent nature of the interaction between the Z(cid:48) and the other Standard Model gauge bosons.5 Two distinct reweighting strategies are employed in the dilepton analysis. The SSM and E6 model signal templates neglect the effect of interference (as described in Sec. 5.3.1) between the γ∗/Z and Z(cid:48) processes, while the Minimal model templates include this effect fully. The following sections describe each of these approaches in detail. 11.2.1 Signal Template Generation Signal templates are obtained from the reweighting of the Pythia8 LO Drell-Yan simulation 5Indeed, in the calculation of the Z(cid:48) decay width Γ (See Sec. 9.3.6). The idea is to replace the Drell-Yan differential cross section with the Z(cid:48), only the Standard Model fermion decay channels are considered; i.e., right-handed neutrinos, additional gauge bosons, and exotic fermions are neglected as possible decays. 211 corresponding cross section for Z(cid:48) production, where dσ dˆt (q ¯q → γ∗/Z → e+e−) → dσ dˆt (q ¯q → Z(cid:48) → e+e−). (11.6) This can be accomplished if analytic expressions for the Drell-Yan and Z(cid:48) matrix elements are known. As these expressions were calculated in Section 4.2.1 and Section 5.3.1, they can now be used here. The ratio between these two matrix elements, each evaluated with the truth level kinematic information6 of the dielectron event, defines the associated event weight wRW. This weight can then be used to reweight the specified Drell-Yan event to the corresponding Z(cid:48) event. The event weight, a function of the Z(cid:48) pole mass, takes the form (cid:0)Z(cid:48)(cid:1)(cid:12)(cid:12)2 i,j∈{L,R}(cid:12)(cid:12)Mij (cid:80) i,j∈{L,R}(cid:12)(cid:12)Mij (γ∗/Z)(cid:12)(cid:12)2 , (cid:80) wRW(MZ(cid:48)) = (11.7) where the numerator and denominator define the Z(cid:48) and Drell-Yan matrix elements, respec- tively, and the sum is performed over all helicity states. The matrix element for each process share a common structure and can be expressed as the product of an angular term, and term that factorizes the couplings, such that |Mij|2 = 4(cid:12)(cid:12)Gij (cid:12)(cid:12)2(cid:0)1 + λij cos θ(cid:1)2 , λij = +1, −1, if i = j if i (cid:54)= j, (11.8) 6i.e. with the four-momenta of the initial-state quarks and final-state electrons 212 where the couplings are Gij(ˆs) =  e2 QlQq g2 Z(cid:48) ˆs−M2 Z q gl ig ˆs + g2 j ˆs−M2 Z+iMZΓZ q zl iz j Z(cid:48) , Z(cid:48)+iM Z(cid:48)Γ for γ∗/Z , for Z(cid:48). (11.9) Thus, to evaluate the event weight wRW(MZ(cid:48)) of Eq. 11.7, a set of factors associated with the Drell-Yan process, with the Z(cid:48) model under investigation, and with the kinematics of the event are required. The Drell-Yan matrix element is defined by the EW fermion quantum numbers Q, gL, and gR, which are listed in Table 2.4, by the mass MZ and decay width ΓZ of the Standard Model Z boson, and by its the gauge coupling constant gZ . Likewise, the Z(cid:48) matrix element is defined by the U (1)(cid:48) gauge coupling gZ(cid:48), by the mass MZ(cid:48) and decay width ΓZ(cid:48) of the Z(cid:48) boson, and by the fermion U (1)(cid:48) quantum numbers zL and zR. The analytic expression for decay width of the Z(cid:48) boson is obtained from Eq. 5.4. The associated parameters for the SSM, the E6 model, and the Minimal model can be found in Section 5.3.2, Section 5.3.3, and Section 5.3.4, respectively, although this formulation is fully general, and any set of Z(cid:48) model parameters can suffice. Each matrix element shares several common kinematic factors as well. Both depend on the scale of event ˆs = m2 ee, obtained from the dielectron invariant mass, and both share a dependence on the initial-state quark flavors and final-state electron angular terms cos θ. It is necessary to identify the flavor of the initial-state quark-antiquark pair on an event-by- event, so that the quark quantum numbers can be correctly specified in Eq. 11.9; similarly, it is necessary to identify the four-momenta of the electrons that originate from the Z(cid:48) decay vertex in order to to extract cos θ. The reweighting technique itself is codified in a the LPXSignalReweightingTool [184], a 213 (a) (b) Figure 11.1: Validation of the signal reweighting procedure. Reweighted signal templates are compared to their dedicated Pythia counterparts at pole masses of 2 (a) and 3 (b) TeV for the Z(cid:48) χ of the E6 model. The effect of interference is not included in the dedicated generation, and is similarly not included in the signal reweighting procedure for this comparison. Percent level agreement is seen in the peak region, and the signal shape is effectively captured across the entire invariant mass spectrum. package which interfaces with the Rootcore framework, and is designed to be plugged into the main dilepton analysis workflow in the same way as the other standard analysis tools (See Sec. 10.4). The user specifies a Z(cid:48) model of interest, which determines gauge coupling constant gZ(cid:48) and U (1)(cid:48) fermion quantum numbers, and chooses a Z(cid:48) pole mass in order to obtain the associated event weights. The Standard Model parameters are predefined, and the tool selects the required incom- ing quarks and outgoing leptons from the event, so that their four-momenta and associated quantum numbers can be used to evaluate the matrix elements of Eq. 11.8, and hence return the weight of Eq. 11.7. The operation of the tool can be seen in Figure 5.2, 5.3, and 5.4 in which reweighted Z(cid:48) signal templates for the SSM, the E6 model, and the Minimal model were generated for several model parameters of interest. The reweighting procedure was extensively validated by comparing sets of reweighted 214 -1Events / pb-710-610-510-410-310-210-110110210310 InternalATLAS = 13 TeVsχModel: Z’Reweighted PDYDedicated Z’PDYTruth Dielectron Invariant Mass [GeV]90100200300400100020003000Reweight/Dedicated 0.60.70.80.911.11.21.31.4-1Events / pb-710-610-510-410-310-210-110110210310 InternalATLAS = 13 TeVsχModel: Z’Reweighted PDYDedicated Z’PDYTruth Dielectron Invariant Mass [GeV]90100200300400100020003000Reweight/Dedicated 0.60.70.80.911.11.21.31.4 signal templates from the LPXSignalReweightingTool to their dedicated MC counterparts. Figure 11.1 shows the electron channel validation plots for E6 model Z(cid:48) χ at pole masses of 2 and 3 TeV.7 An excellent level of agreement can be seen between the dedicated and reweighted signal templates, especially in the region surrounding the peak. 11.2.2 Interference Templates The previous strategy can be extended by taking into account the full quantum mechanical interference between the γ∗/Z Drell-Yan and Z(cid:48) signal processes, which results in an inter- ference template. Like before, this amounts to modifying the differential cross section, such that dσ dˆt (q ¯q → γ∗/Z → e+e−) → dσ dˆt (q ¯q → γ∗/Z/Z(cid:48) → e+e−). (11.10) Here, the interference modification applies only to the numerator of the event weight of Eq. 11.7, so that now, wRW(MZ(cid:48)) = i,j∈{L,R}(cid:12)(cid:12)Mij (cid:80) (cid:80) (cid:0)γ∗/Z/Z(cid:48)(cid:1)(cid:12)(cid:12)2 (cid:12)(cid:12)Mij (γ∗/Z)(cid:12)(cid:12)2 i,jL,R . (11.11) The substitution does not affect the expressions for the matrix elements of Eq. 11.8, rather the form of the factorized amplitude is extended, where  Gij(ˆs) = e2 QlQq ˆs + g2 Z e2 QlQq ˆs + g2 Z q gl ig j ˆs−M2 Z+iMZΓZ q gl ig j ˆs−M2 Z+iMZΓZ , + g2 Z(cid:48) q zl iz j Z(cid:48)+iMZ(cid:48)ΓZ(cid:48) , ˆs−M2 for γ∗/Z for γ∗/Z/Z(cid:48). (11.12) 7Similar validation plots were generated for the muon channel, as the reweighting technique is identical. 215 The dilepton analysis applies this extended method to the generation of signal templates for the Minimal model, which provides for a more general interpretation of the search results than the SSM or E6 models. In the later cases, σB is taken as the parameter of interest, while for the Minimal model, the parameters of interest is the relative coupling strength γ(cid:48) for a fixed mixing angle θMin (See Sec. 5.3.4). This means limits on these model parameters can be directly translated into limits on σB for entire classes of Z(cid:48) models owing to the more general phenomenology. The Minimal model Z(cid:48) gauge coupling and fermion quantum numbers are determined by gZ(cid:48) = γ(cid:48)gZ zf = Yf sin θMin + (B − L)f cos θMin (11.13) where zf is the fermion U (1)(cid:48) charge with its left- or right-handed chiral index is suppressed, and Y and B − L are the Standard Model fermion weak hypercharge and baryon number minus lepton number, respectively. With two independent model parameters (γ(cid:48), θMin), it is not sufficient to generate signal templates as a function of the invariant mass mee alone for each Z(cid:48) pole mass of interest. Instead, the template definition is extended to two dimensions, so that the Minimal model templates are generated as a function of both mee and γ(cid:48) for fixed values of Z(cid:48) pole mass MZ(cid:48) and mixing angle θMin. Figure 11.2 shows an example of these combined two-dimensional templates, which ac- count for the interference between the γ∗/Z and Z(cid:48) processes and assume a mixing angle of θMin = 0 and pole mass of MZ(cid:48) = 3 TeV. For each Minimal model template, the decay width ΓZ(cid:48) is recalculated for each value of γ(cid:48) considered during the template construction. 216 Figure 11.2: Example Minimal model signal template with fixed mixing angle θMin = 0 and Z(cid:48) pole mass MZ(cid:48) = 3 TeV. The template is a function of dielectron mass mee and the fourth power of the Minimal model relative coupling strength γ(cid:48) = gZ(cid:48)/gZ , and is generated from reweighted Pythia Drell-Yan events after passing the full electron channel event selection. This ensures the signal cross section evolves with the value taken by the decay width ΓZ(cid:48) in Eq. 11.12, as required when adjusting the coupling strength between the fermions and Z(cid:48) boson. 11.3 Signal Search The search for a signal-like excess is conducted by scanning the dielectron invariant mass spectrum for local deviations in the observed event yield relative to the Standard Model background expectation. Two complementary search methods are used to interpret the significance of any observed excesses in the data. For each method, the significance of 217 1000Dielectron Invariant Mass [GeV]10-109-108-107-106-105-104-103-102-101-1014'g7-106-105-104-103-102-101-101-1Events/pbDilepton Search Selection InternalATLAS = 13 TeVs an observed excess is extracted with a p-value p0, the probability of observing a result equal to or more extreme than the one actually observed, according to the background-only hypothesis.8 In the first method, a log-likelihood ratio (LLR) test is performed, which assesses the compatibility of the observed dataset under a so-called null hypothesis H0 as compared to a specific alternative hypothesis H1. In the dilepton analysis, H0 refers to the prediction of the Standard Model, while H1 refers to the Beyond the Standard Model (BSM) prediction of the SM+Z(cid:48), evaluated for a given Z(cid:48) model and pole mass MZ(cid:48) as described by Eq. 11.2. The LLR test is performed multiple times, once for each Z(cid:48) pole mass under investigation, since the pole mass of a hypothetical Z(cid:48) boson is unknown a-priori. This requires the use of individual Z(cid:48) signal templates (See Sec. 11.2). The LLR test results in a local p0, or a p0 associated with a specific mass hypothesis MZ(cid:48) under investigation. The second search method relies on the BumpHunter (BH) test [199], which scans all possible contiguous intervals in the dielectron invariant mass spectrum, and identifies the one where the data maximally deviates from the Standard Model prediction; i.e., it locates the most significant “bump.” Based on this interval, it extracts the corresponding p-value according to Poisson statistics. Unlike the LLR test, the BH test does not require an explicit signal model to be assumed for the test to be carried out, making it a fully model-independent approach. 8A p-value is the probability of obtaining a test statistic t equal to or more extreme than the one observed in the actual data t0, assuming the null hypothesis H0 is true: p0 = p(t ≥ t0|H0). 218 11.3.1 Log-Likelihood Ratio Test The LLR test is a statistical test which compares how well two different statistical models fit a set of observations. For the dilepton analysis, the comparison is made between the Standard Model (i.e. the null hypothesis H0) to a possible BSM signal model (i.e. the alternative hypothesis H1). The signal hypothesis H1 consists of the combined background plus signal expectation SM+Z(cid:48) according to Eq. 11.2 for a given pole mass of interest, while H0 consists of the Standard Model expectation only. By construction, the Standard Model hypothesis is a special, or nested, case of the BSM hypothesis, since setting µ equal to zero in Eq. 11.2 fully recovers the Standard Model hypothesis H0. The LLR test is performed as a function of Z(cid:48) pole mass; i.e., multiple times in discrete steps, which requires the use of Z(cid:48) signal templates. A set of over 500 Z(cid:48) χ templates is used, which are generated without interference and range in pole mass from 120 GeV to 5 TeV. The Z(cid:48) χ model has a decay width ΓZ(cid:48) that is 1.2% the value of its pole mass MZ(cid:48). The templates are constructed with a specialized dielectron invariant mass distribution, which utilizes an optimized histogram binning based on the dielectron invariant mass res- olution of the ATLAS detector. In this case, each signal hypothesis, indexed by its pole mass MZ(cid:48), is tested in steps that correspond to the spacing between histogram bins. The fine spacing between MZ(cid:48) mass-points ensures an adequate scan is taken over dielectron invariant mass spectrum for potential signal-like excesses, and is what gives rise to the over 500 signal templates used in the scan. The use of the LLR test is based on the Neyman-Person lemma [200], which states that this is most powerful test to reject H0 in favor of H1. To test for excesses in the data, the LLR test is carried out using the Histfactory [201] package, along with the RooStats [202] 219 and RooFit [203] packages. The LLR test statistic q0 is used to extract the probability p0 that the Standard Model background will fluctuate up to its observed value in data or greater.9 The p-value p0 for each mass hypothesis is computed using the following test statistic, the logarithm of the profile likelihood ratio,  q0 = 0, −2 ln (cid:34)L(cid:16) L(cid:16) ˆˆθ0 0, ˆµ,ˆθ (cid:17)(cid:35) (cid:17) for ˆµ < 0 for ˆµ ≥ 0, , (11.14) where L(µ, θ) is the likelihood function of Eq. 11.1, and µ and θ are the signal strength and nuisance parameters, respectively. The numerator defines the Standard Model hypothesis, where ˆˆθ0 is the value of θ that maximizes L under the condition that µ = 0; i.e., in accordance with the background-only hypothesis H0. The best-fit values, or most probable values for a given dataset, are obtained with the Maximum Likelihood (ML) method.10 Likewise, the denominator defines the alternative hypothesis H1, where the ML estimates obtained by the likelihood are ˆµ and ˆθ. As a consequence, higher values of q0 correspond to increasing incompatibility between the data and the null hypothesis H0. For a given dataset, the numerator is always smaller than or equal to the denominator, so the ratio is always less than or equal to one. Thus, the more significant an excess, the larger the difference between the respective likelihoods in Eq. 11.14. In the case of a downward fluctuation in data, the ML estimate for the signal strength ˆµ 9The test statistic is a scalar number that quantifies the difference between hypotheses H0 and H1 for a given dataset. 10This method finds the values of the parameter of interest µ and nuisance parameters θ that that provide the likelihood function with a global maximum. These values are known as maximum likelihood estimators and are denoted with a hat; i.e. ˆµ is the maximum likelihood estimate for µ. 220 Figure 11.3: Sampling distribution f (q0|0) for the LLR test statistic q0. The distribution is generated from background-only pseudo-experiments and is used to extract the p-value associated with individual Z(cid:48) signal hypotheses. The distribution follows a χ2 function with one degree of freedom according to Wilk’s theorem [5]. can become negative. In this case the test statistic q0 of Eq. 11.14 is set to zero by definition, associated with the fact that the number of signal events is a strictly positive quantity. This definition leads to a peak at q0 = 0 in the sampling distribution f (q0|0) of the test statistic, which represents the probability that a background-only dataset will produces a downward fluctuation, a situation that occurs 50% of the time. The sampling distribution f (q0|0) is used to extract the p-value associated with each signal hypothesis under investigation. The sampling distribution f (q0|0) is built by gener- ating background-only pseudo-experiments, or toys, collections of simulated event counts drawn from the Standard Model expectation that corresponds to the integrated luminosity of the actual dataset. An individual pseudo-experiment is obtained by sampling the number of expected background events in each mass bin k of the dielectron invariant mass distri- bution according to Poisson statistics. The test statistic q0 is recalculated and binned in a 221 0Test Statistic q05101520Number of toys110210310410510610=0)m|=0mf(q histogram, thereby building up f (q0|0) one sample at a time, as shown in Figure 11.3. In the large sample limit, the distribution f (q0|0) follows a χ2 distribution for one degree of freedom. This is the result of Wilk’s theorem [204], which states the distribution of the LLR test statistic follows a χ2 k distribution, as the sample size approaches infinity, where k is the difference in the number of degrees of freedom between H1 and H0. Since the two models are nested by construction, the difference in free parameters k is equal to one. The resulting χ2 fit can be seen in Figure 11.3 as the solid line. Thus, the p-value can be extracted directly without the need to generate an ensemble of pseudo-experiments, greatly reducing the required amount of computing [205]. The probability p0 associated with the actual experimental observation qobs is calculated 0 from p0 = p(q0 ≥ qobs 0 |H0) = f (q0|0)dq0, (11.15) (cid:90) ∞ qobs 0 using the ensemble. Further, the p-value p0 is usually translated to significance scale z based on the number of Gaussian standard deviations, z = Φ−1 (1 − p0) . (11.16) The function Φ−1(x) is the inverse of the cumulative Gaussian distribution. If the observed p-value falls below a predefined critical threshold, the data are unlikely to have been realized under H0, and therefore H0 is rejected in favor of H1. In High Energy Physics, a p0 smaller than 1.35 · 10−3 constitutes evidence for a signal, while a p0 less than 2.87 · 10−7 formally constitutes a discovery. These two thresholds are derived from the one sided integral of a unit Gaussian distribution, where evidence corresponds to the value of the integral beyond +3σ and discovery to beyond +5σ. In this sense, the p-value can be interpreted as the 222 false discovery probability [199]. The above machinery is used to extract local p-values, those that correspond to a specific signal hypotheses, localized in Z(cid:48) pole mass. However, as many Z(cid:48) mass hypotheses are tested, it becomes increasingly likely that at least one should produce a significant result. This is known as the look-elsewhere-effect, and can be accounted for by calculating the so-called global p-value. The global p-value can be extracted from an ensemble of background-only pseudo-experiments, where, for each pseudo-experiment, the value of the most significant local fluctuation z0 in the dielectron spectrum is used to build up the sampling distribution n(z0|0) of largest local significances. (cid:90) ∞ Thus, the global p-value can be obtained from pglobal = p(z0 ≥ zobs 0 |H0) = n(z0|0)dz0. zobs 0 (11.17) where zobs 0 is the value of the largest observed local significance in the actual dataset. The data were scrutinized for a signal, but no significant excess was found. Figure 11.4 shows the extracted local and global p-values as a function of Z(cid:48) pole mass for the electron channel search. The observed global p-value is pglobal = 58%, meaning more than half of background-only pseudo-experiments produce an excess as significant as the one observed in data. The smallest local p-value is 0.6%, which translates to an excess of 2.5σ, and is observed for a Z(cid:48) pole mass of 2.37 TeV. When compared to the global significance scale, however, this excess falls below the value obtained by the majority of background-only experiments, and is therefore not significant. As such, the observed data are consistent with Standard Model hypothesis, and, at least in this analysis, the Standard Model cannot be rejected in 223 Figure 11.4: The resulting p0 scan in the electron channel search region, derived using Zχ signal templates with pole masses between 0.15 and 3.0 TeV. The local p-values are depicted as the black curve, where the results for individual mass points have been interpolated. The corresponding local and global significance levels are depicted as gray and red dashed lines, respectively. The lack of sufficient statistics at high mass can result in an ill defined PDF and negative probabilities. Therefore, above 1.5 TeV the so-called “capped” p-value is used, while the “un-capped” value is used below. See Ref. [4] for more details. favor of any Z(cid:48) signal hypothesis. Lastly, it should be noted that a small unavoidable level of model-dependence exists in the definition of p0, since a particular signal scenario must be assumed for the calculation of q0 in Eq. 11.14. In the dilepton search, the theoretically well-motivated Z(cid:48) χ is assumed, as a benchmark, and therefore, the observed p-values are extracted in the context of this model. To account for this model-dependence, the complimentary model-independent Bum- pHunter test is also performed. 224 [TeV]Z’M0.20.30.412340Local p3−102−101−10110210σ0Local significanceσ1σ2σ3σ0Global significance for largest excessσ1ATLAS-1 = 13 TeV, 36.1 fbsee→χ, Z’0Observed p 11.3.2 BumpHunter Test The BumpHunter [199] (BH) is a tool that can be used to find the largest localized excess in data, without the need to assume a particular signal model. Therefore, the BumpHunter is sensitive to a broader range of new physics models, as compared to the LLR test, which is designed to test for resonant Z(cid:48) production. The BumpHunter tool makes use of the BH test, which scans all possible contiguous bins in the dielectron invariant mass distribution and identifies the one with the most sig- nificant disagreement between the data and the Standard Model background expectation. For each window searched, the significance is identified with Poisson p-value by computing the number of events equal to or greater than the observed event count as compared to the expected number under the null hypothesis H0 of the Standard Model, such that p0 = p(n ≥ nobs|H0) = λke−λ k! , (11.18) ∞(cid:88) k=nobs where n is the number of observed events and λ is the Standard Model expectation in the window being tested. In the dilepton search, the smallest window scanned ranges from two bins to half the total number of bins in the invariant mass distribution. The results of the electron channel scan can be seen in Figure 11.5a, where the search intervals range from 120 GeV to 3 TeV. The window with the smallest p-value, or most significant excess, is used to construct the BH test statistic, defined as the negative logarithm of the smallest p-value identified in the scan, (cid:104) (cid:105) . pmin 0 t = − log 225 (11.19) (a) (b) Figure 11.5: Results of BumpHunter test for the search range mee ∈ [0.12, 3.00] TeV. Depicted are the local p0 of each interval searched (a), and the global p0 (b), which is extracted from an ensemble of background-only psuedo-experiments. This statistic is used to extract a global p-value of the search. As with the LLR test, the sampling distribution for the BH test statistic is built up using pseudo-experiments, where pseudo-data is sampled from the H0 background estimate of the Standard Model, and the observed BH statistic is then compared to the ensemble. The global p-value is the fraction of background-only pseudo-experiments where the BH test statistic is greater than or equal to the value observed in data, whose value is denoted by the arrow in Figure 11.5b. As can be seen, the majority of pseudo-experiments have a BH statistic larger than the value observed in data, indicating the observed data are consistent with the Standard Model. As a result of the scan, the global p-value is found to be 71% in the electron channel. In Figure 11.6, the observed data with accompanying Standard Model background prediction is depicted, along with the bin-by-bin local significance in ratio sub-panel. The interval identified as having the most discrepant upward fluctuation is indicated by the pair of blue lines, which corresponds to a set of mass bins around 2.4 TeV, the same region identified by 226 0.20.30.40.5123Dielectron Invariant Mass [TeV]3-103-10·23-10·32-102-10·22-10·31-101-10·21-10·31Local p-value of the Interval InternalATLAS-1 = 13 TeV, 36.1 fbsBumpHunter Statistic2468101214161820Pseudo-Experiments110210310410510 InternalATLAS-1 = 13 TeV, 36.1 fbs-value = 0.71pPseudo-ExperimentsValue in data Figure 11.6: The dielectron mass distribution, showing the observed data, the Standard Model background prediction, and the corresponding bin-by-bin local significance. The extremely fine search binning is clearly visible in this plot. The interval identified by the pair of blue lines corresponds to the most significant local excess [5]. the LLR test. As with the LLR test, the mass spectrum scrutinized according to the BH test exhibits no characteristic signs of new physics. In this case, exclusion limits are placed on various model parameters of interest, such as production cross sections and coupling scenarios for the Z(cid:48) models considered in this analysis, the subject of the following section. 11.4 Exclusion Limits The signal search determined the data were consistent with the Standard Model, which rules out any alternative explanations in terms of new physics. In the absence of a significant excess, exclusion limits11 are set at 95% C.L. using a Bayesian approach on several Z(cid:48) 11An exclusion limit identifies a region in the model parameter space that is still consistent with the data. 227 0.20.30.412345Events3−102−101−10110210310410510610DataBackground-1 = 13 TeV, 36.1 fbs ee→Z’ ATLASDielectron Invariant Mass [TeV]0.20.30.412345Significance3−2−1−012 model parameters of interest. For the SSM and E6 models, limits are placed on σB(Z(cid:48) → e+e−), the Z(cid:48) cross section times branching fraction to electron pairs. Similarly for the Minimal model, limits are placed on the relative coupling strength γ(cid:48), which controls the overall strength of the Z(cid:48) Min interaction. The limit setting procedure, like the signal search, relies on the binned likelihood function of Eq. 11.1. For the limit setting, however, the nominal parameter of interest µ is re-expressed as the cross section times branching fraction σB for the extraction of the SSM and E6 model limits, and as the relative coupling strength γ(cid:48) for the extraction of the Minimal model limits. Furthermore, like the search, the use of Z(cid:48) signal templates is continued here as well. A total of 118 signal templates is considered for the limit setting. Each is generated without interference, and range in pole mass from 150 GeV to 5 TeV in 50 GeV increments. The templates are a function of the Z(cid:48) pole mass, and are defined over the standard log-binned dielectron invariant mass distribution of Figure 10.6, and include the Z boson mass peak in their definition as well. This allows for the nuisance parameters associated with each mass hypothesis to be maximally constrained in the limit setting. 11.4.1 Bayesian Inference In the dilepton analysis, exclusion limits are calculated using Bayes’ theorem [206], p(H|D) = p(D|H)p(H) p(D) , (11.20) which relates p(H|D), the probability that a hypothesis H is true given the observed data D, to p(D|H), the probability of observing the data given that the hypothesis is true, and 228 to the individual probabilities for the hypothesis p(H) and data p(D) separately.12 Each term in Eq. 11.20 has an interpretation. The likelihood p(D|H) is the evidence for the hypothesis provided by the observed data, which is identified with the likelihood function of Eq. 11.1. The prior probability, or just the prior, p(H) is the probability that the hypothesis is true before the data are taken. The evidence p(D) is the total probability of the data, accounting for all possible hypotheses, and acts as a normalization constant. The posterior probability, or just posterior, p(H|D) is the probability that the hypothesis is true after taking the data. The posterior quantifies the impact the data has on the prior, so Bayes’ theorem specifies how to update the probability of a certain hypothesis as new evidence becomes available. Bayesian probability requires the specification of a prior. In the dilepton analysis, the prior is assumed to be flat, or uniform and positive, such that, for example, the prior on the signal cross section is p(σB) = 1 for σB ≥ 0 and 0 otherwise. This choice leaves the statistical interpretation as general as possible. With a well-defined prior and likelihood, a posterior can be constructed for each signal hypothesis under investigation using Bayes’ theorem of Eq. 11.20. Then, the posterior can be integrated to set exclusion limits at 95% C.L. on the Z(cid:48) model parameter of interest; i.e., σB or γ(cid:48).13 11.4.2 Limits on SSM and E6 Model Z(cid:48) Bosons Since the data were found to be consistent with the Standard Model, exclusion limits are set on the existence of Z(cid:48) bosons. The following section presents limits for the SSM and E6 12In the Bayesian statistics, probability is interpreted as a measure of degree of belief in a proposition or hypothesis, such as the existence of a Z(cid:48) boson with a certain pole mass. Frequentist statistics, on the other hand, interprets probability as the relative frequency of a certain event occurring in a large number of trials. 13In the Bayesian approach, a credibility interval is the range of values in the posterior that includes 95% of the total probability. 229 models, where upper limits are placed on the signal cross section times branching fraction σB for Z(cid:48) production.14 Exclusion limits are extracted as a function of Z(cid:48) pole mass, meaning individual Z(cid:48) signal templates are used to model each mass point under investigation. A total of 118 Z(cid:48) mass points are tested using signal templates for the individual SSM and E6 models considered. The templates themselves are functions of the dielectron invariant mass mee (as in Fig. 11.1) and have a Z(cid:48) pole mass MZ(cid:48) that ranges from 150 GeV to 5 TeV in 50 GeV steps. As the limit calculation relies on the binned likelihood function of Eq. 11.1, which is nominally in terms of the parameter of interest µ, a change of variables must be made in order to set limits directly on σB. Therefore, µ is replaced by µ = (σB)/(σB)Z(cid:48), the signal cross section times branching fraction relative to its theoretical expectation, effectively trading one floating signal normalization µ for another (σB). With this change, the posterior is obtained via Bayes’ theorem, p(σB, θ|n) = L(n|σB, θ)p(σB) p(n) . (11.21) The likelihood L(n|σB, θ) is a function of σB and θ, the vector of electron channel nuisance parameters.15 The prior on the signal is p(σB), which is uniform for σB ≥ 0 and is equal to 0 otherwise and the prior on the data is p(n), which is independent of σB. The vector n parametrizes the number of observed data events in each mass bin k of the dielectron invariant mass distribution. In order to formulate the posterior strictly in terms of the parameter of interest σB, the 14Upper limits define the region in Z(cid:48) parameter space that is still consistent with the observed dataset; or, equivalently, they exclude the region that is now inconsistent. 15Systematics that contribute at the level of 1% or less over the dielectron invariant mass search region are not included, since it is computationally expensive and does not affect the final result. 230 nuisance parameters θ of the likelihood function must be integrated over. This dependence can be removed numerically with Markov Chain Monte Carlo (MCMC), which is performed with the Bayesian Analysis Toolkit (BAT) [207]. This process of integrating out the nuisance parameters is known as marginalization, and the marginalized likelihood is (cid:90) L(cid:48)(n|σB) = L(n|σB, θ)dθ. (11.22) With that, the posterior on σB alone becomes p(σB|n) = L(cid:48)(n|σB)p(σB) p(n) . (11.23) The 95% C.L. upper limits on σB can be obtained by integrating posterior p(σB|n); the maximum value corresponds to most likely signal excess in data. Exclusion limits on σB are obtained such that, 0.95 = (cid:82) (σB)95 (cid:82) ∞ p(σB|n)d(σB) 0 p(σB|n)d(σB) 0 (11.24) where the value (σB)95 is the value of the parameter of interest for which the 95% C.L. upper limit is satisfied. This integration is also performed using BAT. The upper limits on σB as a function of Z(cid:48) pole mass are depicted in Figure 11.7, along with the Z(cid:48) production cross sections for the Z(cid:48) χ of the E6 model. The observed limits, or those extracted with the observed 36.1 fb−1 dataset, are represented SSM, and Z(cid:48) ψ and Z(cid:48) by the solid red line. These tend to fluctuate due to the local excesses and deficits in the observed data of the dielectron invariant mass distribution. The parameter space of models that predict a σB above the red line are excluded at 95% C.L. The gray band on the Z(cid:48) SSM theory line depicts the total theoretical uncertainty on the 231 (a) (b) Figure 11.7: The 95% C.L. upper limits on the Z(cid:48) production cross section times branching fraction for the electron channel (a) and the combined electron plus muon channel (b) as a function of Z(cid:48) pole mass. The expected limit is depicted as a black dashed line, with yellow and green bands indicating ±1σ and ±2σ variations from the median value. The observed limit is shown as a solid red line and the above region is excluded at 95% C.L. Signal theoretical uncertainties are shown on the SSM cross section prediction as a gray band for reference, but are not included as part of the limit calculation [5]. 232 [TeV]Z’M0.511.522.533.544.55 B [pb]σ-510-410-310-210-1101Expected limitσ 1±Expected σ 2±Expected Observed limitSSMZ’χZ’ψZ’ATLAS ee→Z’ -1 = 13 TeV, 36.1 fbs [TeV]Z’M0.511.522.533.544.55 B [pb]σ-510-410-310-210-1101Expected limitσ 1±Expected σ 2±Expected Observed limitSSMZ’χZ’ψZ’ATLAS ll→Z’ -1 = 13 TeV, 36.1 fbs signal, but it is not included as part of the limit calculation. This uncertainty can become significant at high mass, where it is due almost exclusively to the imprecise knowledge of the Parton Distribution Functions (PDFs). A strategy to reduce this uncertainty in future iterations of the analysis is presented in Section 10.1. The expected limits, or those extracted under the assumption that the Standard Model is true, are also shown, which are marked by a dotted-dashed black line.16 The expected limits are obtained by generating 10,000 background-only pseudo-experiments (pseudo-data invariant mass distributions) for each Z(cid:48) mass point tested. The pseudo-data are sampled according to the Poisson expectation in each mee mass bin k and the Gaussian parameter- izations of the systematic uncertainties. For each pseudo-experiment, the 95% C.L. upper limit on (σB)95 is recalculated via Eq. 11.22, and the resulting set of values is used build a distribution for (σB)95. The median value of the distribution defines the expected limit, and the green and yellow bands define the ±1σ (68%) and ±2σ (95%) quantiles. This procedure is carried out for each mass point tested and the limit plot is built up point-by-point. Just as the Z(cid:48) χ was used in p0 scan for the LLR test, it is similarly used here as the benchmark model to extract the limits, meaning it is the signal model assumed in the likelihood definition. In addition, the same signal acceptance (as in Fig. 10.3) is assumed for all Z(cid:48) models tested. This approximation works well since interference is neglected and the models here have similar decay widths. Above a mass of roughly 3.5 TeV, the upper limits start to weaken. This is due primarily to the quickly falling production cross section, as the kinematic limit is reached: signal pro- duction is occurring increasingly off-shell in the low-mass tail, as opposed to in the width of 16“Expected” limit is somewhat of a misnomer as it means nothing new or interesting is expected. A better term might be “Standard Model” limit. 233 Figure 11.8: The LO cross section times branching fraction predictions for Z(cid:48) production in the SSM and E6 models considered in the analysis. Values were obtained using Pythia8 configured with the NNPDF2.3 PDF set, and are plotted as a function of Z(cid:48) pole mass for √ s = 13 TeV [5]. the resonance (compare Fig. 10.4), which subsequently leads to a drop in selection efficiency (compare Fig. 10.3). The Z(cid:48) pole mass at which the theoretical value of σB intersects the observed limit defines a lower limit on mass. The theoretical cross section for each model as a function of mass can be seen in Figure 11.8. In the electron channel, the lower mass limits range from 3.6 TeV for the Z(cid:48) SSM. Table 11.1 lists the lower mass limits for the ψ to 4.3 TeV for the Z(cid:48) SSM and each of the E6 models considered in the analysis. Finally, in order to set the strongest exclusion limits possible, the individual electron and muon channels are statistically combined using BAT.17 For this, the likelihood of Eq. 11.1 is extended to include the muon channel, incorporating the observed dimuon data and signal and background expectations in all dimuon invariant mass bins, as well as the muon chan- 17This is done assuming lepton universality, which applies for the signal models considered in this analysis. 234 Z’ Polemass [TeV]0123456B [pb]σ-610-510-410-310-210-110110210310SSM Z’ψE6 Z’NE6 Z’ηE6 Z’IE6 Z’SqE6 Z’χE6 Z’ InternalATLAS ll→Z’ = 13 TeVsPythia 8PDF: NNPDF23LO Figure 11.9: Ratio of the observed 95% C.L. upper limit to the theoretical prediction for the Z(cid:48) SSM cross section times branching fraction as a function of Z(cid:48) pole mass for the combined √ electron and muon channels. Previous iterations of the ATLAS dilepton search, performed s = 7, 8, and 13 TeV, are compared to the current result obtained here, which is shown at in black. The region above each line is excluded at 95% C.L. [5]. Model Width [%] θE6 [rad] Z(cid:48) SSM Z(cid:48) χ Z(cid:48) S Z(cid:48) I Z(cid:48) η Z(cid:48) N Z(cid:48) ψ 3.0 1.2 1.2 1.1 0.6 0.6 0.5 - 0.50 π 0.63 π 0.71 π 0.21 π −0.08 π 0 π Lower limits on MZ(cid:48) [TeV] (cid:96)(cid:96) ee µµ Obs Exp Obs Exp Obs Exp 4.3 3.9 3.9 3.8 3.7 3.6 3.6 4.3 3.9 3.8 3.8 3.7 3.6 3.6 4.0 3.6 3.6 3.5 3.4 3.4 3.3 3.9 3.6 3.5 3.4 3.3 3.3 3.2 4.5 4.1 4.0 4.0 3.9 3.8 3.8 4.5 4.0 4.0 3.9 3.8 3.8 3.7 Table 11.1: Expected and observed 95% C.L. lower mass limits for the SSM and E6 Z(cid:48) bosons considered in the analysis. Electron, muon, and combined channel limits are shown [5]. 235 [TeV]Z’M0.511.522.533.544.55SSMσ/limitσ-510-410-310-210-110110 )-1 36.1 fbµµ; -1 = 13 TeV (ee 36.1 fbsATLAS )-1 3.2 fbµµ; -1 = 13 TeV (ee 3.2 fbsATLAS )-1 20.5 fbµµ; -1 = 8 TeV (ee 20.3 fbsATLAS )-1 5.0 fbµµ; -1 = 7 TeV (ee 4.9 fbsATLAS )-1 42 pbµµ; -1 = 7 TeV (ee 39 pbsATLAS ATLAS ll→Z’ nel systematic uncertainties (See Ref. [5] for a complete list). The multichannel likelihood becomes L(n|σB, θ) = Nchannel(cid:89) Nbins(cid:89) l=1 k=1 λlk(σB, θ)e−λlk(µ,θ) nlk! Nsys(cid:89) i=1 N (θi, 0, 1), (11.25) which, following the same procedure as before, is marginalized, and used to construct a posterior from which the associated combined channel limits are extracted at 95% C.L. The combined upper limits on σB are similarly depicted in Figure 11.7, and the combined lower limits on mass can be seen in Table 11.1. In addition, Figure 11.9 compares the exclusion limits obtained for the Z(cid:48) SSM in this analysis to several previous iterations of the ATLAS dilepton search [111, 123, 208, 209]. 11.4.3 Limits on Minimal Model Z(cid:48) Bosons For the Minimal model, exclusion limits are set directly on the relative coupling strength γ(cid:48) of the Z(cid:48) Min boson relative to the Standard Model Z boson, as a function of both the Z(cid:48) Min pole mass and mixing angle θMin. This requires the use of two-dimensional signal templates (See Sec. 11.2.2), which are generated as a function of both the dielectron invariant mass mee and γ(cid:48) for fixed values for θMin and MZ(cid:48) (See Fig. 11.2). As with the SSM and E6 limit setting of the previous section, 118 signal templates are utilized, corresponding to the individual Z(cid:48) pole masses tested. The use of two-dimensional templates requires a modification to the nominal definition of the likelihood function of Eq. 11.1, since the expected number of events and systematic variations are now extended to depend on both mee and γ(cid:48). Otherwise, the limit setting procedure is identical to the approach of the previous section. 236 In order to modify the likelihood of Eq. 11.1, the signal expectation sk(θ) in each mee (cid:0)γ(cid:48), θ(cid:1), which encodes the Minimal model mass bin k is replaced by a non-linear function fk signal expectation as a function of the relative coupling γ(cid:48). As such, the expected number of dielectron events becomes (cid:0)γ(cid:48), θ(cid:1) = fk (cid:0)γ(cid:48), θ(cid:1) + bk (θ) . λk (11.26) In this formalism, each function fk is defined for a fixed pole mass MZ(cid:48) and mixing angle θMin. Furthermore, this replacement fully incorporates the γ∗/Z/Z(cid:48) interference effect, as well as the functional dependence of the Z(cid:48) decay width ΓZ(cid:48) on the relative coupling γ(cid:48). The functions fk are obtained from a series of interpolations, performed in each mee mass bin k of the two-dimensional signal templates (as in Fig. 11.2). Since the Standard Model expectation bk in Eq. 11.26 already contains the Drell-Yan background estimate, it is isolated and subtracted from each function fk to avoid double counting. Thus, each function fk consists of a pure Z(cid:48) signal component and the γ∗/Z/Z(cid:48) interference component. For specific choices of θMin and γ(cid:48), the signal expectation can be negative, which results from the γ∗/Z/Z(cid:48) interference. In the limiting case γ(cid:48) → 0, the Standard Model expectation is exactly recovered as the function fk → 0.18 With these changes, the marginalized likelihood becomes, (cid:90) L(cid:48)(n|γ(cid:48)) = L(n|γ(cid:48), θ)dθ. (11.27) Bayes’ theorem is used with the prior p(γ(cid:48)), which is constant as a function of γ(cid:48)4 > 0 and 0 otherwise. This choice of γ(cid:48)4 relies on the fact that the pure Z(cid:48) 18Analogously, the Standard Model expectation is exactly recovered in the limit of µ → 0 in Eq. 11.2. Min component of the total 237 production cross section is proportional to γ(cid:48)4 in the Minimal model. Thus, the posterior takes the form, p(γ(cid:48)|n) = L(cid:48)(n|γ(cid:48))p(γ(cid:48)) p(n) . (11.28) The 95% C.L. upper limits on γ(cid:48) are obtained by integrating posterior p(γ(cid:48)|n), as was done in the previous section, to find γ(cid:48) 95 such that (cid:82) γ(cid:48) (cid:82) ∞ p(γ(cid:48)|n)dγ(cid:48) 0 p(γ(cid:48)|n)dγ(cid:48) . 95 0 0.95 = (11.29) As with the SSM and E6 model limits, these calculations are performed with BAT. The resulting expected and observed upper limits can be seen in Figure 11.10. The collection θMin ∈ [0, π] of all Minimal model limit curves fits within the gray band. The values of θMin that correspond to the minimum and maximum Z(cid:48) Min cross sections are used to specify the edges of the band. Limit curves for the three representative models are shown, which for specific values of γ(cid:48) and θMin correspond to the Z(cid:48) χ, Z(cid:48) 3R, and Z(cid:48) B−L, respectively (See Table 5.3 for details). The region above each curve is excluded at 95% C.L. Exclusion limits as a function of θMin are also depicted in Figure 11.11, which provide a complimentary interpretation to the mass limits where θMin takes a fixed value. For each case, following the strategy of extending the likelihood definition to include the muon channel, the combined limits are shown as well. Lower limits on MZ(cid:48) Min for specific Z(cid:48) Min models can be obtained as well, following an approach similar to the one in the previous section. The intersection between the upper limit on γ(cid:48) and the nominal value of γ(cid:48) provided by theory, such as those in Table 5.3, defines a lower limit on Z(cid:48) Min pole mass. The lower mass limits are given in Table 11.2 for the electron, 238 (a) (b) Figure 11.10: Observed and expected 95% C.L. upper limits on the relative coupling strength γ(cid:48) as a function of the Z(cid:48) Min pole mass for the electron channel (a) and the combined electron plus muon channel (b). Three representative limit curves are shown, for specific values of the mixing angle θMin, which correspond to the Z(cid:48) B−L models at specific values of γ(cid:48). All Minimal model limits fit within the gray band, which depend on the choice of θMin ∈ [0, π]. The edges of the band are defined by the two models with the minimum and maximum Z(cid:48) Min cross sections. The region above each curve is excluded at 95% C.L. χ, Z(cid:48) 3R, and Z(cid:48) 239 [TeV]MinZ’M0.511.522.533.544.55’γ2−101−10110Exp. Obs.]π [0, ∈ MinθLimit range for )χ(Z’Minθ)3R(Z’Minθ)B-L(Z’MinθATLAS-1 = 13 TeV, 36.1 fbs ee→ MinZ’ [TeV]MinZ’M0.511.522.533.544.55’γ2−101−10110Exp. Obs.]π [0, ∈ MinθLimit range for )χ(Z’Minθ)3R(Z’Minθ)B-L(Z’MinθATLAS-1 = 13 TeV, 36.1 fbs ll→ MinZ’ (a) (b) Figure 11.11: Observed and expected 95% C.L. upper limits on the relative coupling strength γ(cid:48) as a function of the mixing angle θMin for the electron channel (a) and the combined electron and muon channels (b). Limit curves are set for ten representative values of the Z(cid:48) Min pole mass MZ(cid:48). The structure observed in the limit curves for a given pole mass as a function of θMin originates from the changing shape of the resonance. The resonance with the smallest (largest) width is obtained at θMin (cid:39) 1.0 (θMin (cid:39) 2.2), which define the edges of the gray band in Figure 11.10. The region above each curve is excluded at 95% C.L. 240 Minθ00.511.522.53 ’γ2−101−10110Exp. Obs. = 5.0 TeVMinZ’M = 4.5 TeVMinZ’M = 4.0 TeVMinZ’M = 3.5 TeVMinZ’M = 3.0 TeVMinZ’M = 2.5 TeVMinZ’M = 2.0 TeVMinZ’M = 1.5 TeVMinZ’M = 1.0 TeVMinZ’M = 0.5 TeVMinZ’MATLAS-1 = 13 TeV, 36.1 fbs ee→ MinZ’Minθ00.511.522.53 ’γ2−101−10110Exp. Obs. = 5.0 TeVMinZ’M = 4.5 TeVMinZ’M = 4.0 TeVMinZ’M = 3.5 TeVMinZ’M = 3.0 TeVMinZ’M = 2.5 TeVMinZ’M = 2.0 TeVMinZ’M = 1.5 TeVMinZ’M = 1.0 TeVMinZ’M = 0.5 TeVMinZ’MATLAS-1 = 13 TeV, 36.1 fbs ll→ MinZ’ γ(cid:48) tan θMin Lower limits on MZ(cid:48) ee µµ Min [TeV] (cid:96)(cid:96) Obs Exp Obs Exp Obs Exp 3.7 4.0 4.0 3.7 3.9 4.0 3.4 3.6 3.6 3.3 3.6 3.6 3.9 4.1 4.2 3.8 4.1 4.1 −4 5 -2 0 (cid:113)41 (cid:113)5 (cid:113)25 24 sin θMin 8 sin θMin 12 sin θMin Model χ Z(cid:48) Z(cid:48) 3R Z(cid:48) B−L Table 11.2: Observed and expected 95% C.L. lower mass limits on the Z(cid:48) Min pole mass for the three representative models considered in the text. As a result of including the interference effects and the variation of the decay width ΓZ(cid:48) through the modulation of γ(cid:48), a slightly weaker limit for the Z(cid:48) χ is obtained in the Minimal model approach as compared to the standard approach of Section 11.4.2. γ(cid:48) 0.01 0.02 0.04 0.10 0.20 0.40 Range in limits on MZ(cid:48) Exp. Obs. Min [TeV] 0.2-0.8 0.4-1.6 0.8-2.3 1.7-3.3 2.4-4.0 3.2-4.5 0.2-0.8 0.4-1.5 0.8-2.3 1.6-3.2 2.4-3.9 3.1-4.6 Table 11.3: Range in the observed and expected 95% C.L. lower limits on the Z(cid:48) Min pole mass for several representative values of the relative coupling strength γ(cid:48). Both the electron and muon channels are combined. MZ(cid:48) Min [TeV] 1 2 3 4 5 Range in limits on γ(cid:48) Exp. Obs. 0.011-0.049 0.026-0.11 0.077-0.36 0.21-1.0 0.63-2.1 0.012-0.053 0.032-0.14 0.079-0.37 0.22-1.1 0.55-2.2 Table 11.4: Range in the observed and expected 95% C.L. upper limits on the relative coupling strength γ(cid:48) for several representative values of the Z(cid:48) Min boson pole mass. Both the electron and muon channels are combined. 241 muon, and combined dilepton channels. Furthermore, the ranges in the observed and expected lower limits on MZ(cid:48) for several characteristic values of γ(cid:48) are shown in Table 11.3, and likewise, the ranges of observed and expected limits on γ(cid:48) for several characteristic values of MZ(cid:48) Min are shown in Table 11.4. Min 11.5 Discussion of Results This chapter presented the dilepton statistical analysis, which consisted of distinct search and limit setting phases. The dataset used to conduct these analyses consisted of 36.1 fb−1 of proton-proton collisions at √ s = 13 TeV, as collected at the LHC and recorded by the ATLAS detector in the 2015 and 2016 data-taking runs. The record collision energy of the LHC allowed for a search for new physics to be conducted in a new as yet unexplored high-energy regime. The main result of the dilepton analysis is the non-observation of any statistically sig- nificant resonant-like excesses over the predicted Standard Model background in either the dielectron or dimuon invariant mass distributions. As such, exclusion limits on Z(cid:48) production were set in a variety of models. The models considered were the SSM, the E6 model, and the Minimal model, all of which predict the existence of a heavy spin-1 resonance that couples and decays to pairs of Standard Model fermions. As the search confirmed there was no excess in the data, upper limits were placed on the cross section times branching fraction σB for Z(cid:48) production in the SSM and the E6 model for well-motivated choices of θE6, and on the relative coupling strength γ(cid:48) in the Minimal model. In each case, these results were converted into lower limits on the Z(cid:48) pole mass, which are some of the most stringent such limits to date. The resultant limits do 242 not strictly exclude any particular model, but they set tight constraints on Z(cid:48) production over the invariant mass range searched at the LHC and on the allowed regions of model parameter space. While this dissertation presents exclusion limits on models predicting resonant Z(cid:48) pro- duction, other interpretations of the dilepton results are possible as well, and are included in detail in the most recent ATLAS dilepton search paper (See Ref. [5]). In particular, limits are set on several non-resonant Contact-Interaction (CI) models, whose signature would ap- pear as a broad excess, rather than a resonant peak, over the Standard Model expectation. In addition, so-called generic limits are also provided to facilitate the reinterpretation of the dilepton results in the context of other models that predict narrow dilepton resonances, such as the Randall-Sundrum (RS) model [105], briefly mentioned in Section 5.2. 243 Part IV PDF Uncertainty 244 Chapter 12 The Drell-Yan Process Revisited As described in Section 10.8.2, the dominant systematic uncertainty in the dilepton analysis is purely theoretical and originates with the PDFs. The primary contributions to this theory uncertainty arise from the “PDF Variation” and “PDF Choice” systematic uncertainties, where the former is due to the modeling of the PDFs, and the later is due to a lack of agreement among the modern PDF sets. To see this explicitly, consider the Standard Model prediction at a dielectron mass of mee = 4 TeV, which resides precisely in high-mass search region of interest, and is the primary target of the dilepton analysis. Here, the total theoretical uncertainty is 23%, which is roughly twice as large as the experimental uncertainty of 13%.1 The individual “PDF Variation” and “PDF Choice” uncertainties are 19% and 8.4%, respectively. As mee = 5 TeV is reached, these increase to 29% and 47%, respectively, so that the total theory uncertainty becomes 57%; the total experimental uncertainty, on the other hand, remains largely the same, increasingly only to 15%. A precise understanding of the PDF uncertainty is therefore crucial to future Standard Model measurements, as well as high-mass searches for new physics. Consequently, in this chapter, the physics of the Drell-Yan process is further investigated in order to better un- derstand the nature of the PDF uncertainty with the hope of further constraining it. 1The contribution from the fake background is neglected in this estimate. 245 Figure 12.1: The Drell-Yan process initiated by a quark-antiquark pair as observed at the LHC [29]. 12.1 The Drell-Yan Process The Drell-Yan process was introduced in Section 4.2. At the LO in QCD, the process is initiated by the annihilation of quark and antiquark pair via the s-channel exchange of an electroweak boson, which then decays into a pair of same-flavor, oppositely-charged leptons as in Fig. 12.1. The intermediate boson can be produced either through the annihilation of a valance quark with a sea quark (e.g. u¯u, d ¯d, etc.), or through the annihilation of two sea quarks (e.g. s¯s, c¯c, etc.) in the collision. 12.1.1 Drell-Yan Kinematics The kinematics of the Drell-Yan process can be fully described in terms of three kinematic variables, which are defined below. Although some of the definitions have already been provided in Section 4.2, they are repeated here for convenience. In the Drell-Yan process, the quark q carries a fraction x1 of total proton momentum P1. Likewise, the antiquark ¯q carries a fraction x2 of total proton momentum P2. Thus, the 246 parton four-momenta can be written, p1 = x1P1, p2 = x2P2 (12.1) Using Eq. 12.1, the relativistic invariant ˆs (as defined in Sec. 4.2.1) can be calculated from ˆs = (p1 + p2)2 = x1x2s, (12.2) √ s is the CM energy of the colliding proton-proton (pp) system, and where √ ˆs is the CM energy of the quark-antiquark (q ¯q) system. The squared CM energy ˆs equals the scale of the collision Q2 and invariant mass m(cid:96)(cid:96) of the dilepton system, ˆs = Q2 = m2 (cid:96)(cid:96). (12.3) The dilepton invariant mass m(cid:96)(cid:96) defines the first of the three Drell-Yan kinematic observables of interest. The second variable to be considered is the dilepton rapidity, (cid:19) (cid:18) E + Pz E − Pz y(cid:96)(cid:96) = 1 2 ln , (12.4) where E and Pz are the energy and longitudinal momentum of the dilepton system. The four-momenta of the colliding quark and antiquark can be written, √ s 2 p1 = (x1, 0, 0, x1), p2 = √ s 2 (x2, 0, 0,−x2), (12.5) 247 and upon substitution of Eq.12.5 into Eq. 12.4, the rapidity can be re-expressed as y(cid:96)(cid:96) = 1 2 ln x1 x2 , (12.6) which is exclusively in terms of the parton momentum fractions x1 and x2. Furthermore, the substitution of Eq. 12.2 and Eq. 12.3 into Eq. 12.6 results in an expression for the partonic x fractions in terms of Drell-Yan kinematic observables, x1 = m(cid:96)(cid:96)√ s e+y(cid:96)(cid:96), x2 = e−y(cid:96)(cid:96). m(cid:96)(cid:96)√ s (12.7) With this, the PDFs fq/P1(x1, Q2) and f¯q/P2(x2, Q2) (as in Sec. 3.2) can be extracted from differential measurements of the Drell-Yan cross section in the dimensions of dilepton mass m(cid:96)(cid:96) and rapidity y(cid:96)(cid:96). In the high-mass region of the dilepton analysis, the x values relevant for Drell-Yan production can be calculated from Eq. 12.7. For a central dilepton event with an invariant mass of m(cid:96)(cid:96) = 3 TeV and rapidity of y(cid:96)(cid:96) = 0, the momentum fractions involved are x (cid:39) 0.23. For higher masses, the relevant average x values are of course larger, so any future dilepton search would sample fairly high values of x. These values correspond roughly to where the valence distributions peak (See Fig. 3.4); for larger dilepton masses or rapidities, the x values that result are placed firmly beyond, in the extreme region of the PDF tails. As depicted in Figure 3.6, the uncertainties associated with the valence and sea PDFs in the moderate-x region of 10−3 (cid:46) x (cid:46) 0.1 are fairly well constrained; however, as the high-x region of x (cid:38) 0.1 is reached, these uncertainties become significant. Therefore, this suggests the PDF systematic uncertainty that dominates the high-mass dilepton spectrum 248 √ Figure 12.2: The transferred momentum squared Q2 versus the parton momentum fraction s = 7 TeV. The regions probed by previous DIS, fixed-target, and collider-based x at experiments are labeled. The accessible regions to the ATLAS and CMS experiments are indicated by the solid blue line, and the shaded region within denotes measurements using the central region of each detector. The high-mass search region of the dilepton analysis can be considered as beyond mee = 1 TeV, where two high-x partons (x (cid:38) 0.1) are required to initiate the Drell-Yan process [30]. is precisely due to the unconstrained high-x behavior of the PDFs. The reason for this inherent high-x uncertainty in the modern PDF sets is due to a lack of experimental data, as previous experiments have yet to probe the PDFs in these extreme regions (x, Q2) of kinematic phase space (See Fig. 12.2). At the same time, measurements in these regions can be challenging due to the steeply falling nature of the PDFs at high x (See Fig. 3.4). 249 The valence quark PDFs in the high-x region have been measured in previous experi- ments [30], so the largest uncertainties in these distributions are found at low x and at very high x. For example, in the CT14Hera2 error set, the uv(x) and dv(x) distributions have an uncertainty that reaches 20% at x (cid:38) 0.5 and x (cid:46) 10−3 for Q2 = (3 TeV)2 (See Fig. 3.6). The sea quark and gluon PDFs, on the other hand, are the least constrained at high x due to a lack of experimental data in this region, and due to the much more rapid fall-off that takes place as compared to the valence PDFs. The gluon g(x) uncertainty reaches 20% at x (cid:38) 0.3 at Q2 = (3 TeV)2, and likewise for the ¯u(x) and ¯d(x) distributions, this occurs at x (cid:38) 0.2 (See Fig. 3.6). From this, it can be seen that the high-mass Drell-Yan PDF uncertainty arises precisely from this lack of knowledge in the high-x sea PDFs, and to a lesser extend the high-x valence PDFs, as both a quark and antiquark are required to initiate the LO Drell-Yan process. That a difficult region of x causes a large uncertainty for high-mass Drell-Yan perhaps it is also an opportunity: the LHC may allow for the PDFs to be probed in this previously inaccessible (x, Q2) kinematic regime. The Drell-Yan process is crucial in this regard, as contemporary cross section measurements may help to validate the less-precise datasets used in the PDF global fits and thereby improve their overall quality. Thus, dedicated measurements targeting these precise x regions at the LHC could help to reduce the PDF uncertainty and lead to an overall better understanding of proton substructure. The third observable of interest is defined in the following section. 12.1.2 The Collins-Soper Frame The third observable of interest is the polar angle θ∗ of the outgoing lepton relative to the incoming quark, measured in a special rest frame of the dilepton system known as the 250 Figure 12.3: The Collins-Soper reference frame. The ˆz axis bisects the angle between the quark momentum p1 and the negative of the antiquark momentum −p2. The ˆy axis is defined perpendicular to the plane spanned by the parton momenta, and ˆx axis completes the right handed coordinate system. The polar angle θ∗ is measured from the ˆz axis in this frame [31]. Collins-Soper (CS) frame [210] (See Fig. 12.1.2).2 Starting from the rest frame of the dilepton system, the CS frame can be obtained from the following set of coordinate transformations: 1. The ˆz axis is defined such that it bisects the angle between the momentum vector of the incoming quark q and the negative momentum vector of the incoming antiquark ¯q.3 2. The ˆy axis is defined as the vector normal to the plane spanned by the momenta of the quark-antiquark pair. 3. The ˆx axis is chosen such that a right handed coordinate system is obtained. transverse momenta, the CS frame reduces to the dilepton rest frame. The polar angle θ∗, measuring the direction of the outgoing lepton (cid:96)− relative to ˆz in the 2The CS frame accounts for the non-zero pT of the dilepton system. When the dilepton system has zero 3This definition introduces an inherent sign ambiguity, since it is not clear at the LHC which proton carries the quark. However, the quark direction can be inferred on an event-by-event basis according to the longitudinal boost of the dilepton system y(cid:96)(cid:96), assuming this is in the same direction as that of the initial-state quark. 251 CS frame, can be calculated directly from lab frame quantities, 2(cid:0)p+ (cid:113) 1 p− 2 − p− M 2 1 p+ 2 (cid:96)(cid:96) + P 2 M(cid:96)(cid:96) (cid:1) T cos θ∗ = Pz|Pz| , (12.8) where PT and Pz are the transverse and longitudinal momentum of the dilepton system, respectively and, (cid:0)Ei ± pz,i (cid:1) , p± i = 1√ 2 i = 1, 2, (12.9) where the lepton (antilepton) energy and longitudinal momentum are specified by E1 and pz,1 (E2 and pz,2), respectively. Notice this definition requires the electric charge of each lepton to be correctly identified. Within the CS frame, Drell-Yan events can be classified as forward or backward ac- cording to the lepton decay angle. Events where the outgoing lepton travels in the same direction as the incoming quark (i.e. cos θ∗ > 0) are classified as forward events, while those where it travels in the opposite direction (i.e. cos θ∗ < 0) are classified as backward events. For forward events, the helicity of the final-state lepton generally matches that of the initial-state quark, while for backwards events, the helicities are opposite (See Eq. 4.14). 12.1.3 Triple-Differential Cross Section The Drell-Yan triple-differential cross section can be represented as a function of the three variables outlined in the previous two sections: the dilepton invariant mass m(cid:96)(cid:96), the dilepton rapidity y(cid:96)(cid:96), and the cosine of the polar angle θ∗ of the incoming quark and outgoing lepton measured in the CS frame cos θ∗. At the LO, the Drell-Yan triple-differential cross section 252 can be written, d3σ dm(cid:96)(cid:96)dy(cid:96)(cid:96)d cos θ∗ = √ (cid:88) q πα2 3m(cid:96)(cid:96)s (cid:104) (cid:105) fq/P1(x1, Q2)f¯q/P2(x2, Q2) + (q ↔ q) , Pq (12.10) where s is the CM energy of the LHC, x1 and x2 are the incoming parton momentum fractions, and fq/P1(x1, Q2) and f¯q/P2(x2, Q2) are the PDFs for partons of flavor q and ¯q, respectively. The scale of the collision Q2 is set to the dilepton invariant mass m(cid:96)(cid:96), which is equal to the partonic CM energy as in Eq. 12.3. The term (q ↔ q) accounts for the fact that either proton can carry a sea quark, as the LHC is a proton-proton collider. The function Pq encodes the dynamics with (cid:16) 1 + cos2 θ∗(cid:17) Pq = Q2 (cid:96) Q2 q + Q(cid:96)Qq Z (cid:1) (cid:1)2 (cid:0)m2 (cid:104)(cid:0)m2 (cid:96)(cid:96) − M 2 2m2 (cid:96)(cid:96) (cid:96)(cid:96) − M 2 sin2 θW cos2 θW (cid:105) (cid:1)2 m4 (cid:96)(cid:96) (cid:19)(cid:16) 1 + cos2 θ∗(cid:17) (cid:96)(cid:96) − M 2 (cid:104)(cid:0)m2 (cid:19)(cid:18) Z M 2 A + gq2 gq2 V + Γ2 + Γ2 Z Z Z g(cid:96)2 A + g(cid:96)2 V sin4 θW cos4 θW (cid:20)(cid:18) + × (cid:105)(cid:104) Z M 2 Z V gq g(cid:96) V (1 + cos2 θ∗) + 2g(cid:96) Agq A cos θ∗(cid:105) V cos θ∗(cid:21) + 8g(cid:96) Ag(cid:96) V gq Agq , (12.11) where MZ and ΓZ are the mass and width of the Z boson, Qf is the fermion electric charge, and gf V and gf A are the electroweak vector and axial-vector fermion couplings, respectively, the values of which are listed in Table 2.4. Each term in Eq. 12.11 has an associated interpretation: the first term corresponds to the exchange of a virtual photon γ∗, the second results from γ∗/Z interference, and the third from pure Z boson exchange. The cross section of Eq. 12.10 exhibits a strong mass 253 dependence, falling continuously, and smoothly like 1/m2 (cid:96)(cid:96), except at the mass of the Z boson, where the smoothly falling behavior is replaced by a Breit-Wigner resonance. The dependence on rapidity y(cid:96)(cid:96) provides direct sensitivity to the PDFs, as Eq. 12.6 is expressed in terms of the parton momentum fractions x1 and x2. The dilepton mass m(cid:96)(cid:96) provides further sensitivity, since the relative contribution of u-type and d-type quarks to the total cross section varies with mass.4 Finally, the dependence on cos θ∗ provides additional sensitivity to the terms in Eq. 12.11 that contain axial- and axial-vector prefactors. 12.1.4 Kinematic Distributions The kinematic distributions for the absolute dilepton rapidity |y(cid:96)(cid:96)| and cosine of the polar angle cos θ∗ are presented according to the prediction of the triple-differential cross section of Eq. 12.10. The ResBos MC generator (See Sec. 9.1.2), configured with the CT14Hera2 PDF set, is used for the event generation. The simulation consists of a set of mass-binned √ samples, covering an invariant mass range of m(cid:96)(cid:96) = 40 GeV to m(cid:96)(cid:96) = 1 TeV at s = 13 TeV. For each sample, the lepton pseudorapidity is required to be within |η| < 4.9, which covers the forward region of the ATLAS detector. A simple selection is applied to the events of each sample: first, leptons must have a transverse momentum of ET > 30 GeV, and second, events must consist of either a central-central (|η| < 2.47) or central-forward (2.5 < |η| < 4.9) lepton pair (See Sec. 13.3.1), in order to be detectable within the ATLAS detector. The |y(cid:96)(cid:96)| (See Fig. 12.4) and cos θ∗ (See Fig. 12.5) distributions are depicted in slices of dilepton invariant mass m(cid:96)(cid:96) to illustrate how they evolve as a function of mass. Further, the flavor decomposition of each observable is presented, which shows how the overall Drell- 4At the Z peak, for example, the partons couple almost exclusively through the weak interaction; off-peak, however, the electromagnetic coupling to γ∗ becomes increasingly dominant. 254 Yan event yield decomposes into its individual contributions from the u-type and d-type Drell-Yan sub-processes.5 The cos θ∗ distributions show that the rates associated with each quark-type sub-process are in fact highly dependent on the polar angle θ∗. For example, above the Z boson mass peak and in the forward region of cos θ∗ > 0, an increasing degree of separation can be seen between the rates associated with the u-type and d-type sub-processes as compared to the backward region of cos θ∗ < 0. In this region, and especially at high masses, it can be seen that the contribution to the total cross section is due predominantly to u-type sub-process. The fact that cos θ∗ exhibits such a strong sensitivity to the distinct parton sub-processes suggests it might be useful to include as an additional dimension in future PDF global fits. Along with the more traditional observables of dilepton invariant mass and rapidity, as es- tablished in Section 12.1.1, cos θ∗ might provide additional discriminating power between the individual parton flavors, and thereby function to constrain the PDF uncertainty further than what could be achieved otherwise. Indeed, this idea is explored in detail in Section 12.1.6. The |y(cid:96)(cid:96)| distributions show the contributions of the u-type and d-type sub-processes to the total cross section as well. Below the Z boson mass peak, the maximum rapidity attained is |y(cid:96)(cid:96)| (cid:46) 3.5. By the time m(cid:96)(cid:96) = 1 TeV is reached, the maximum rapidity reduces to |y(cid:96)(cid:96)| (cid:46) 2.5. The relative fraction of the d-type rate is enhanced at higher rapidities. 12.1.5 Explaining the cosθ∗ Asymmetry In order to better understand the asymmetric behavior of the u- and d-type Drell-Yan sub- processes as a function of mass and cos θ∗ (as depicted in Fig. 12.5), the function Pq of 5The u-type sub-processes include initial-states defined by uu, ug, and ug, where u is any up-type quark and g is the gluon. An identical definition applies to the d-type sub-processes, with u replaced by d. 255 (a) (c) (e) (b) (d) (f) Figure 12.4: The absolute dilepton rapidity |y(cid:96)(cid:96)| in several slices in dilepton invariant mass m(cid:96)(cid:96) ranging from 40 GeV to 1 TeV ((a)-(f)). The relative fractions of the u-type and d-type Drell-Yan sub-processes are shown to illustrate the partonic flavor decomposition in this observable. 256 Dilepton Rapidity00.511.522.533.544.55-1Events/pb01234567Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 66 GeVee40 < mDilepton Rapidity00.511.522.533.544.55-1Events/pb01020304050607080Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 116 GeVee66 < mDilepton Rapidity00.511.522.533.544.55-1Events/pb00.20.40.60.811.21.41.61.8Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 250 GeVee116 < mDilepton Rapidity00.511.522.533.544.55-1Events/pb00.020.040.060.080.1Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 400 GeVee250 < mDilepton Rapidity00.511.522.533.544.55-1Events/pb00.0050.010.0150.020.025Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 600 GeVee400 < mDilepton Rapidity00.511.522.533.544.55-1Events/pb00.0010.0020.0030.0040.0050.0060.007Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 1000 GeVee600 < m (a) (c) (e) (b) (d) (f) Figure 12.5: The lepton angular distribution cos θ∗ in several slices of dilepton invariant mass m(cid:96)(cid:96) ranging from 40 GeV to 1 TeV ((a)-(f)). The relative fractions of u-type and d-type sub-processes are shown as well, which exhibit a strong angular dependency, especially at high mass. 257 CS*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81-1Events/pb01234567Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 66 GeVee40 < mCS*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81-1Events/pb01020304050607080Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 116 GeVee66 < mCS*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81-1Events/pb00.20.40.60.811.21.41.61.822.2Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 250 GeVee116 < mCS*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81-1Events/pb00.020.040.060.080.10.120.14Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 400 GeVee250 < mCS*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81-1Events/pb00.0050.010.0150.020.025Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 600 GeVee400 < mCS*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81-1Events/pb00.0010.0020.0030.0040.0050.006Totalu-Typed-Type CT14Hera2ResBos = 13 TeVs < 1000 GeVee600 < m Eq. 12.11 can be recast in a more suitable form, (cid:16) 1 + cos2 θ∗(cid:17) Pq = C0 q + C1 q cos θ∗, (12.12) Q2 (cid:96) Q2 q + 2Q(cid:96)Qqg(cid:96) V gq V χ1(m(cid:96)(cid:96)) + (cid:18) g(cid:96)2 A + g(cid:96)2 V Ag(cid:96) V gq Agq V χ2(m(cid:96)(cid:96)) (cid:19)(cid:18) (cid:105) gq2 A + gq2 V (cid:19) (cid:21) χ2(m(cid:96)(cid:96)) (12.13) where, C0 q = C1 q = (cid:20) (cid:104) χ1(m(cid:96)(cid:96)) = κ χ2(m(cid:96)(cid:96)) = κ2 (cid:1) Agq 4Q(cid:96)Qqg(cid:96) (cid:0)m2 Aχ1(m(cid:96)(cid:96)) + 8g(cid:96) (cid:0)m2 (cid:1)2 (cid:96)(cid:96) − M 2 m2 Z (cid:96)(cid:96) (cid:96)(cid:96) − M 2 + Γ2 (cid:0)m2 (cid:1)2 Z m4 (cid:96)(cid:96) (cid:96)(cid:96) − M 2 Z M 2 Z M 2 + Γ2 Z Z Z κ = 1 sin θW cos θW . Here, χ1 is due to γ∗/Z interference and χ2 is due to Z boson exchange. In this form, Pq is described by an even function6(cid:0)1 + cos2 θ∗(cid:1) and an odd function7 (cos θ∗). The odd part is responsible for inducing the well-known forward-backward asymmetry AF B. At low mass, below the Z boson mass peak, the Drell-Yan process proceeds primarily through γ∗ photon exchange. In this limit, χ1 and χ2 → 0 as m(cid:96)(cid:96) → 0. Therefore, the C1 term of Eq. 12.12 vanishes and C0 q is described purely by the electromagnetic component of q the cross section. As a result, Pq of Eq. 12.12 is described almost exclusively by an even function, and little to no asymmetry is present. This accounts for the behavior of the cos θ∗ distribution of Figure 12.5a. At the Z boson mass peak, the Drell-Yan proceeds almost exclusively though on-shell 6An even function satisfies f(x) = f(−x) 7An odd function satisfies f(−x) = −f(x) 258 (a) (b) Figure 12.6: Drell-Yan asymmetry coefficients of Eq. 12.12. The parameter C0 the magnitude of the symmetric component of the Drell-Yan cross section, while C1 describes the magnitude of the asymmetric component. q ((a)) describes q ((b)) Z boson exchange. Here, χ1(MZ ) = 0 and χ2(MZ ) = κ2 M2 Z Γ2 Z Ag(cid:96) 8g(cid:96) V gq Agq V χ2(MZ ) is indeed non-zero, ensuring Pq has an odd function component, the . While the term C1 q = value of the coefficient 8g(cid:96) V is too small to induce any asymmetric effects. There- fore, the forward-backward asymmetry is appreciably small at the Z peak, as in the cos θ∗ Ag(cid:96) V gq Agq distribution of Figure 12.5b. At masses beyond the Z peak, the contributions from γ∗/Z interference and pure Z boson exchange are non-negligible, and as a result, both χ1 and χ2 functions are non- zero, and are roughly comparable in magnitude. In this limit, both the C0 q and C1 q terms contribute to the total cross section. The dominant contribution to C1 q comes from the term 4Q(cid:96)Qqg(cid:96) Agq Aχ1(m(cid:96)(cid:96)), and it is precisely this term that induces the forward-backward asymmetry observed in the Drell-Yan process. The size of this effect differs for the u- and d-type Drell-Yan sub-processes due to the values of their electroweak couplings and the dilepton mass scale of interest, which results in C1 u > C1 d. 259 [GeV]s50100150200250300)s(0qC0123456789100uC0dC [GeV]s50100150200250300)s(1qC8-6-4-2-02468101uC1dC Figure 12.7: Ratio of Drell-Yan asymmetry coefficients C0 q and C1 q of Eq. 12.12. Consequently, this explains the increasingly asymmetric behavior of the remaining cos θ∗ distributions in Figures 12.5c through 12.5f. Likewise, the fact that C0 u > C0 d explains the increasing separation between the rates of the u- and d-type sub-processes as a function of dilepton mass m(cid:96)(cid:96). To drive this point home, consider the dilepton behavior at m(cid:96)(cid:96) = 3 TeV. Here, the u-type coefficients are C0 u = 0.44 + 0.015 + 0.16 = 0.62 and C1 u = 0.99 + 0.033 = 1.03, while the d-type are C0 d = 0.11 + 0.013 + 0.21 = 0.33 and C1 d = 0.49 + 0.059 = 0.56. The d-type components are each roughly a factor of two smaller than their u-type counterparts. For each quark flavor q, C1 q > C0 q , indicating a sizeable asymmetry is present, as the magnitude of the odd function in Eq. 12.12 dominates. Figure 12.6 displays the Cq terms explicitly as √ a function of ˆs = m(cid:96)(cid:96), and Figure 12.7 plots their ratio. 12.1.6 cosθ∗ as a Quark Flavor Enhancer The cosine of the polar angle cos θ∗ is a highly discriminating variable, sensitive to the kinematics of the different partonic sub-processes in the Drell-Yan process. This fact can 260 [GeV]s50100150200250300)s(d)/Cs(uC01234560d/C0uC1d/C1uC (a) (b) Figure 12.8: The dilepton invariant mass m(cid:96)(cid:96) as selected by forward (cos θ∗ > 0) or backward (cos θ∗ < 0) events in the CS frame ((a)) and the corresponding PDF uncertainty associated with each selection ((b)). Events arising from the forward-forward lepton topology are not included. As with the other kinematic distributions, the CT14Hera2 PDF set is used. The selection has the effect of modulating the PDF systematic uncertainty. be exploited to “sculpt” the PDFs so as to reduce their overall uncertainty. For example, with a forward cos θ∗ > 0 requirement, the d-type contribution to the total cross section will be suppressed, and, as the dv(x) PDFs are less precisely determined than their uv(x) counterparts (as in Fig. 3.6), the overall PDF uncertainty will be reduced.8 This effect can be used even today to modulate the PDF systematic uncertainty as in Figure 12.8. Here, just as with the other kinematic distributions, the state-of-the-art CT14Hera2 PDF set is used in the ResBos event generation. The invariant mass m(cid:96)(cid:96) is shown along with the uncertainty on the event yield due to the PDFs according to forward and backward selections applied to events in the CS frame. In the nominal selection, the PDF uncertainty at m(cid:96)(cid:96) = 3 TeV is 15% and at m(cid:96)(cid:96) = 5 TeV it is 31% (black curves). The forward selection (blue-dashed curves) reduces the PDF uncertainty 8The uv(x) distribution is better constrained than the dv(x) distribution precisely because the dominant contribution to the valence structure of the proton is from the up quark. As as result, the uv(x) distribution is more likely to initiate the Drell-Yan process in high-energy scattering experiments. 261 Dilepton Invariant Mass [GeV]7010020030010002000-1Events/pb7-106-105-104-103-102-101-10110210310 CT14Hera2ResBos = 13 TeVsDrell-Yan*<0qcos*>0qcosDilepton Invariant Mass [GeV]7010020030010002000Systematic Error [%]0510152025303540 CT14Hera2ResBos = 13 TeVsPDF Err.*<0qPDF Err.: cos*>0qPDF Err.: cos to 14% at 3 TeV and to 29% at 5 TeV, while the backward (red-dashed) selection has the opposite effect; it actually increases the overall uncertainty. Therefore, by imposing the requirement of cos θ∗ > 0, the u-type component of the Drell-Yan cross section is enhanced (and the d-type component is suppressed); as such, the overall PDF uncertainty is reduced, which, when propagated to the invariant mass, has the effect of reducing the uncertainty on the high-mass Drell-Yan event yield. This strategy of using cos θ∗ to modulate the flavor content of the Drell-Yan cross section to reduce the overall PDF uncertainty is novel and could, for example, be applied to the dilepton analysis, where, in addition to the nominal selection defined in Section 10.5, an additional requirement could be placed to select only forward events. The advantage of such a selection is that this additional requirement can applied to the search now; there is no need to wait for future PDF sets or additional LHC data. Furthermore, in the case of a non-resonant excess, this method could provide a useful handle in diagnosing the nature of the excess, which would otherwise be stymied by the large PDF uncertainty at high-mass. To reiterate, the fact that a simple selection requirement on cos θ∗ can increase or decrease the PDF systematic uncertainty suggests, again, it can supplement the invariant mass and rapidity distributions as constraints in the PDF global fits, and therefore motivates the followup study presented in the next chapter. 262 Chapter 13 PDF Error Reduction The goal of this chapter is to assess how future LHC measurements of the Drell-Yan cross section can help to constrain the valence and sea PDF uncertainties at high x, and thereby re- duce the PDF systematic uncertainty on the Drell-Yan event yield at high dilepton invariant mass. This is a work proposed by this author. To accomplish this, the ePump [211] package is used, which is able to update an exist- ing PDF set with new experimental data (or pseudo-data) and consequently produce new In the previous chapter, cos θ∗ was established as a potentially best-fit and error PDFs. sensitive discriminating variable, since it separates u-type and d-type parton flavors. In this chapter, ePump is used to update an existing PDF set with Drell-Yan pseudo-data, which takes advantage of cos θ∗ explicitly. Figure 13.1 sketches the planned use of ePump. The pseudo-data are generated for the Drell-Yan process at several characteristic inte- grated luminosities and are incorporated into the CT14Hera2 PDF set using ePump. The updated PDFs are subsequently used to assess any potential improvement in the high-mass dilepton spectrum.1 This chapter is formatted as follows: first the ePump package and its functionality is introduced (Sec. 13.1), then the PDF update analysis is performed (Sec. 13.3), and its effects are assessed (Sec. 13.4). Finally, a brief outlook is presented (Sec. 13.5). 1The CT14Hera2 set incorporates data from fixed target experiments, the Tevatron, LHC Run-1, and HERA [94]. 263 Figure 13.1: ePump requires two sets of inputs to perform an update: the first consists of a set of binned signal templates, each of which is comprised of a nominal theory prediction and n eigenvector variations (n = 56 in CT14Hera2), and the second consists of an analogous set of binned data (or pseudo-data) templates, along with their statistical uncertainties. The observed reduction in PDF uncertainty associated with the updated PDF set depends on the statistical precision of the data. Datasets with high statistical precision are better able to constrain the PDF uncertainty. 13.1 The ePump Package The ePump (Error PDF Updating Method Package) package can be used to update or optimize a PDF set for analysis.2 The package itself package contains two executables: UpdatePDFs and OptimizePDFs. The first executable UpdatePDFs can be used to predict the impact of new experimental data on an already existing PDF set (e.g. the CT14Hera2 set). This is achieved by producing updated best-fit and updated error PDFs given new data by essentially re-performing the χ2 minimization of the global PDF analysis (See Sec. 3.2.1) and Hessian error analysis (See Sec. 3.2.2). However, a full global re-analysis of the data is not required, as the same χ2 minimum found during global analysis is assumed for the update, allowing for results to be obtained extremely quickly. As the global χ2 function is a sum over discrete measurements, the requisite ePump inputs, or templates, are formatted as histograms with a finite binning. The data templates in- 2A PDF set in the context of this discussion includes both the nominal (or best-fit) and hessian error PDFs (see Sec. 3.2.2 for details). 264 ePumpDrell-Yan Pseudo-Data- Stat. Unc.Drell-Yan Templates- Nominal- 56 EigenvectorsUpdated PDF Set clude new experimental data, each bin of which is treated a possible input to the PDF fit (i.e. the “pseudo-data”), and their associated statistical uncertainties. The signal templates3 include theoretical predictions for a given process, and are comprised of a nominal predic- tion and n eigenvector variations (See Fig. 13.1). In the previous chapter, the importance of m(cid:96)(cid:96), y(cid:96)(cid:96), and cos θ∗ was described, and in ePump, these variables are specifically used as parameters. The individual entries of these input templates are appended to the original global χ2 function, in order to extend it with the new data (or pseudo-data). Since the global χ2 minimum is already known from the global fit, the new best-fit parameters can be found algebraically, along with the new eigenvectors of the Hessian matrix, leading to updated best-fit PDFs and updated error PDFs. The updated PDFs can then be used directly to assess the impact of the new data on the PDF uncertainty for various observables of interest, such as the dilepton invariant mass m(cid:96)(cid:96). The uncertainty reduction observed in the PDFs after the update depends on the sta- tistical precision of the “old” experimental inputs, as well as the “new” inputs from the pseudo-data used in the update. All things being equal, a dataset with a larger integrated luminosity (and hence a smaller statistical uncertainty) will be able to better constrain the PDF uncertainty, than one with a smaller integrated luminosity (and hence a larger statisti- cal uncertainty). In the results that follow, the pseudo-data used in the update are generated with Lint = 3000 fb−1 to coincide with the lifetime dataset collected at the LHC. The second executable OptimizePDFs can be used to transform an input set of error PDFs into a “simplified” set, optimized for the description of a specific observable. The 3Here, the ePump “signal templates” are not to be confused with those used for the dilepton signal reweighting procedure developed in Section 11.2. 265 error PDFs of the optimized set are ranked according to their overall contribution to the total PDF error for the observable under study. As a result, this procedure simplifies the calculation of the PDF error, since the sum in Eq. 3.10 can be truncated after the first few optimized eigenvectors. A reduced PDF error set like this was used to estimate the PDF uncertainty in the dilepton analysis (See Sec. 10.8.2 and Fig. 10.11). Because of its versatility, ePump offers a way to quickly and accurately assess how new datasets might impact the best-fit and error PDFs of contemporary PDF sets. This makes ePump the ideal tool to assess the impact of future Drell-Yan measurements at the LHC on the high-mass dilepton PDF uncertainty, the results of which are presented in Section 13.4. 13.2 PDF Update Strategy The proceeding sections of this chapter detail the “PDF Update Analysis,” in which pseudo- data and signal templates for the Drell-Yan process are constructed and used for the PDF update with ePump. The overall strategy is this: two regions in the dilepton invariant mass m(cid:96)(cid:96) are defined and treated differently; the first, a “control region,” encompasses well-measured masses and excludes the potential effects of new physics; the second, a “signal region,” extends beyond the excluded masses and is not used to predict the updated PDF set. The PDF update is performed using templates constructed in the control region, and the updated PDFs are then used to assess the reduction in PDF uncertainty in the signal region, which coincides with the high-mass search region of the dilepton analysis. The analysis presented here is performed at “truth level,” such that the acceptance and efficiency effects associated with the reconstruction and identification of prompt leptons in 266 the ATLAS detector are neglected. However, leptons are very well measured, so this is an acceptable first look at this technique. 13.3 Construction of ePump Templates In order to probe the PDFs in the three dimensions of interest: m(cid:96)(cid:96), y(cid:96)(cid:96), and cos θ∗, the ePump templates are produced in slices of the triple-differential Drell-Yan cross section of Eq. 12.10. The templates cover a finite region of kinematic phase space, consistent with the fiducial volume of the ATLAS detector, in order to simulate an actual future measurement. The control region considered for the present update is designed explicitly to probe the PDFs at high x and is defined by 40 < m(cid:96)(cid:96) < 1000 GeV, |y(cid:96)(cid:96)| < 3.6, −1 < cos θ∗ < 1. (13.1) Here, the range in dilepton mass is limited to a maximum value of 1 TeV. This “Standard Model” control region is defined in order to prevent possible contamination from high-mass new physics processes that would otherwise bias the outcome of a measurement. The final- state leptons must have a transverse momentum p(cid:96) T > 30 GeV, and be produced in either the central-central or central-forward decay configurations (discussed further in Sec. 13.3.1).4 The inclusion of the cos θ∗ observable in the definition is crucial, as it provides the additional discriminating power between the u-type and d-type parton level Drell-Yan sub-processes (as in Fig. 12.5). 4Central leptons are defined by |η| < 2.47 while forward leptons are 2.5 < |η| < 4.9. 267 13.3.1 Kinematic Reach at the LHC The extent to which the PDFs can be probed at high x via the Drell-Yan mechanism is largely dependent on the fiducial region considered in the ATLAS detector. In the Drell-Yan process, for example, three distinct final-state topologies are possible within the ATLAS detector acceptance, depending on the rapidity of the intermediate γ∗/Z boson:5 • The central-central topology results when yγ∗/Z is small, so both decay leptons enter the central region (|η| < 2.47) of ATLAS detector. • The central-forward topology results when yγ∗/Z is larger, such that one lepton enters the central region (|η| < 2.47), and the other is boosted into the forward region (2.5 < |η| < 4.9) of the ATLAS detector. • The forward-forward topology results when the yγ∗/Z is large, so both leptons are boosted into the forward region (2.5 < |η| < 4.9) of the ATLAS detector. The central, or precision, region of the ATLAS detector is covered by both the tracking (See Sec. 7.3) and calorimetry (See Sec. 7.4) systems, providing for a robust electron measure- ment. The forward region beyond is covered only by the calorimetry system, making the reconstruction and identification of forward electrons much more challenging. However, the use of forward leptons allows for higher values of dilepton rapidity to be measured, and hence more extreme values of x to be probed. For example, the dilepton rapidity in the central-central configuration is limited to |yγ∗/Z| < 2.47. On the other hand, the use of central-forward leptons extends this range up to |yγ∗/Z| < 3.6 (See Fig. 12.4). 5The γ∗/Z boson receives a longitudinal boost when there is a difference between x1 and x2 of the colliding quark and antiquark as in Eq. 12.6. As it decays, its longitudinal momentum is transferred to the final-state leptons. 268 m(cid:96)(cid:96) [GeV] Max. |y(cid:96)(cid:96)| 40 - 66 66 - 116 116 - 250 250 - 400 400 - 600 600 - 1000 3.5 3.5 3.5 3.3 2.9 2.5 xMin 9.3 × 10−5 1.5 × 10−4 2.7 × 10−4 7.1 × 10−4 1.7 × 10−3 3.8 × 10−3 xMax 0.17 0.30 0.64 0.83 0.84 0.94 Table 13.1: The ranges in parton-x probed in the Drell-Yan process at the s = 13 TeV LHC for various values of dilepton rapidity and four-momentum transfer Q = m(cid:96)(cid:96). The maximum values assumed for the dilepton rapidity are determined from the |y(cid:96)(cid:96)| distributions in Figure 12.4, where the Drell-Yan cross section falls to zero. The minimum and maximum values in x are calculated using Eq. 12.7. √ Previous ATLAS measurements have been performed in the central region of the detector, but to obtain the highest possible sensitivity to the x values relevant at high mass, the forward region must be considered as well. Thus, the forward region motivates the choice of the fiducial volume considered in Eq. 13.1 for the generation of ePump templates. √ Table 13.1 lists the values of xMin and xMax calculated for the s = 13 TeV LHC in several invariant mass ranges using yMin and yMax for central-forward leptons. Here, it can be seen that the measurement of the Drell-Yan cross section starting at low invariant mass, such as 40 < m(cid:96)(cid:96) < 60 GeV, already probe values of x (cid:38) 0.1, and by m(cid:96)(cid:96) = 1 TeV the values of x probed approach even unity. 13.3.2 Template Simulation Samples Simulated samples for the Drell-Yan process are generated at NNLO in QCD using the ResBos MC generator and the CT14Hera2 PDF set. This provides a precise theoretical prediction of the Drell-Yan signal expectation. Additional background contributions are 269 neglected, but would be similar to those assessed in the dilepton analysis.6 As a technique to deal with the steeply falling cross section, Drell-Yan samples are gen- erated in slices of dilepton invariant mass m(cid:96)(cid:96) to maintain high statistical precision. At the generator level, the final-state leptons are required to have a transverse momentum of pT > 30 GeV and pseudorapidity of |η| < 4.9, and the dilepton system is required to be within a rapidity of |yγ∗/Z| < 4.9. All MC samples are stitched together, and are gener- √ ated at a CM energy of s = 13 TeV to coincide with Run-2 of the LHC. A description of the parton shower and underlying event (See Sec. 3.1.3) that normally accompany the high energy event are not simulated, nor is the effect of QED FSR radiation.7 For the error analysis, ResBos provides a set of n = 57 cross section weights for each generated MC event, which correspond to the n = 1 nominal cross section prediction and the n = 56 eigenvector variations associated with the CT14Hera2 error PDFs (as described in Sec. 3.2.2). This is a crucial feature, since these event weights are used to assess the PDF uncertainties (via Eq. 3.10 in Sec. 3.2.2) in the fiducial volume considered. 13.3.3 Template Event Selection ePump signal templates are constructed using events from simulation that pass a loose event selection. For each dilepton event, the final-state leptons are required to pass a minimum transverse momentum threshold and be produced in either the central-central or central- forward configurations (See Sec. 13.3.1). For events with an invariant mass of m(cid:96)(cid:96) > 80 GeV, the lepton transverse momentum requirement is pT > 30 GeV, while for low-mass events that fall between 40 < m(cid:96)(cid:96) < 80, 6These include t¯t production, W t Single-Top production, Diboson production, and W +jets & Multi-jet production if considering the electron channel. 7These effects are negligible for the present study. 270 the requirement is pT > 15 GeV. In the central-central configuration, both leptons must be within |η| < 2.47. In the central-forward configuration, one lepton must be within |η| < 2.47 while the other must be is within 2.5 < |η| < 4.9. 13.3.4 Template Binning Drell-Yan events passing selection are binned in histograms and used to construct ePump templates for the update. The templates are defined in terms of the triple-differential cross section of Eq. 12.10, which is parametrized as (cid:18) Lint (cid:19) d3σ dm(cid:96)(cid:96)d|y(cid:96)(cid:96)|d cos θ∗ N ijk pseudo−data (∆m(cid:96)(cid:96))i(2∆|y(cid:96)(cid:96)|)j(∆ cos θ∗)k = ijk , (13.2) where i, j, and k correspond to the bin indices of the distributions of interest: the invariant The number of pseudo-data events are given by N ijk mass m(cid:96)(cid:96), the absolute rapidity |y(cid:96)(cid:96)|, and the cosine of the polar angle cos θ∗, respectively. pseudo−data, the integrated luminosity of the pseudo-dataset is Lint, and (∆m(cid:96)(cid:96))i, (2∆|y(cid:96)(cid:96)|)j, and (∆ cos θ∗)k are the bin widths for each of the distributions of interest.8 The factor of two in the denominator accounts for the modulus in the rapidity bin width. The binning used to parametrize Eq. 13.2 is • 40 < m(cid:96)(cid:96) < 1000 : {40, 66, 80, 91, 102, 116, 145, 200, 275, 381, 525, 725, 1000} • 0 < |y(cid:96)(cid:96)| < 2.4 : {0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4} • 2.4 < |y(cid:96)(cid:96)| < 3.6 : {2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6} • −1 < cos θ∗ < 1 : {−1.0, −0.7, −0.4, 0.0, 0.4, 0.7, 1.0}. 8In a realistic measurement, the numerator of Eq. 13.2 would be replaced by N ijk data − N ijk bkg, where the background component arises from the standard set of dilepton backgrounds (as described in Sec. 9.3.) 271 There are 12 bins in mass m(cid:96)(cid:96), 18 equidistant bins in rapidity |y(cid:96)(cid:96)|, and 6 bins in the cosine of the polar angle cos θ∗. The total number of measurement bins is Nbins = 12× 18× 6 = 1296 over the fiducial region considered. Each bin is assigned a unique bin number from 1 to Nbins. The binning scheme is motivated by several previous ATLAS cross section measure- ments [212, 213]. The fine-granularity binning in |y(cid:96)(cid:96)| is crucial, since the parton momentum fraction x is highly dependent on y(cid:96)(cid:96), as in Eq. 12.7. A large number of rapidity bins, therefore, allows for as much PDF shape information to be extracted as possible. Likewise, the cos θ∗ binning separates forward and backward events, providing improved sensitivity to u-type and d-type parton sub-processes. The range considered in dilepton rapidity covers both central-central and central-forward topologies, and the cosine of the polar angle cos θ∗ covers the entire acceptance. 13.3.5 Pseudo-Data Generation The next step is to use the binned signal templates to generate pseudo-data using the Drell- Yan event samples of finite statistics. This step is sufficient to create pseudo-data templates now more sensitive to the high-x PDFs relevant to a dilepton search. Pseudo-data are generated according to the Drell-Yan event yield in each of the Nbins = 1296 template bins defined by Eq. 13.2. For each bin considered, the Drell-Yan cross section estimate σijk Drell−Yan is scaled by a characteristic integrated luminosity Lint to arrive at a definite Drell-Yan event yield N ijk Drell−Yan. The resulting yield is assumed to be the mean of statistics, thereby populating the bin with N ijk a Poisson distribution, which is then used to throw a random number according to Poisson pseudo−data events.9 The characteristic lumi- 9The probability to observe Npseudo−data events in a given bin is P (Npseudo−data|σDrell−YanLint), 272 nosity considered is Lint = 3000 fb−1, coincident with the final dataset of the LHC. Thus, this procedure results in a set of Nbins = 1296 pseudo-data entries, each associated with the Drell-Yan theory prediction over the fiducial region defined by Eq. 13.1. This constitutes a set of pseudo-data templates ready to be used for the update. 13.3.6 Finalized Templates With the requisite ePump signal and pseudo-data templates, the update can be performed as in Figure 13.1). The templates can be represented visually as single dimensional cos θ∗ histograms in slices of m(cid:96)(cid:96) and |y(cid:96)(cid:96)| according to the binning scheme outlined above in Section 13.3.4. As a concrete example, the cos θ∗ distributions in the 381-525 GeV mass slice are depicted in Figure 13.2, and Figure 13.3 for Lint = 3000 fb−1 with both the Drell- Yan and pseudo-data event yields overlaid. The 3000 fb−1 templates constructed in the other mass slices are presented in Appendix D. The MC simulation provides sufficient statistics over the majority of the fiducial volume considered, but there are isolated regions of phase space where the statistical uncertainty on the MC can become large. In order to not bias the update, bins in the MC estimate where the statistical uncertainty is greater than 3% are removed. 13.4 Results of the Update With a prepared set of templates, the UpdatePDFs executable of ePump is used to assess the impact of the 3000 fb−1 pseudo-dataset on the CT14Hera2 PDFs, as well as assess the reduction in the PDF systematic uncertainty in the high-mass dilepton spectrum. where P (n|ν) is the Poisson distribution. 273 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure 13.2: The triple-differential cross section of Eq. 13.2 in the range of m(cid:96)(cid:96) ∈ [381, 525] GeV and 0 < |y(cid:96)(cid:96)| < 1.8, scaled to 3000 fb−1. The specific kinematic region is labeled in each plot (a)-(i). The ResBos prediction is indicated by the solid line, and pseudo-data are shown as solid black markers. In each plot, the sub-panel shows the ratio between the theoretical prediction and the measurement. Error bars represent the statistical uncertainty on the pseudo-data and the solid band represents the PDF uncertainty on the dilepton event yield. 274 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020004000600080001000012000140001600018000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02000400060008000100001200014000160001800020000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02000400060008000100001200014000160001800020000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000180002000022000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000180002000022000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000180002000022000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000180002000022000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure 13.3: The triple-differential cross section of Eq. 13.2 in the range of m(cid:96)(cid:96) ∈ [381, 525] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6, scaled to 3000 fb−1. The specific kinematic region is labeled in each plot (a)-(i). The ResBos prediction is indicated by the solid line, and pseudo-data are shown as solid black markers. In each plot, the sub-panel shows the ratio between the theoretical prediction and the measurement. Error bars represent the statistical uncertainty on the pseudo-data and the solid band represents the PDF uncertainty on the dilepton event yield. 275 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02000400060008000100001200014000160001800020000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020004000600080001000012000140001600018000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02000400060008000100001200014000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600180020002200Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.511.522.5Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 13.4.1 Impact on CT14Hera2 PDFs The results of the ePump update with the 3000 fb−1 pseudo-dataset can be seen in Fig- ures 13.4 and 13.5, which were generated with the PDFigure [214] package and depict the PDF error bands for several parton flavors (the impact of a 300 fb−1 pseudo-dataset can be seen in Appendix D). Of particular importance to description of high-mass Drell-Yan are the ¯u(x) and ¯d(x) sea and uv(x) and dv(x) valence distributions, which are the focus of the following discussion. The sea distributions show a considerable reduction in uncertainty at high x. For exam- ple, the sea uncertainty is reduced to (cid:39) 20% at x (cid:39) 0.5 from the pre-update value of (cid:39) 70%. On the other hand, the improvement in valence distributions at high x is not as dramatic; however, substantial improvement is observed in the ranges of low and intermediate x. The uv(x) distribution remains better constrained than dv(x) at high x after the update, com- paring 2.6% to 11%, respectively. Table 13.2 lists the pre- and post-update uncertainties for several parton flavors and values of x explicitly. Results are also interpreted in terms of templates with reduced dimensionality; i.e., with single-differential or double-differential templates, as compared to the fully triple-differential templates constructed in Section 13.3. In this case, the single-differential templates are produced by integrating over the rapidity and cos θ∗ bins of Eq. 13.2, leaving the invariant mass as the only remaining dimension; likewise, the double- differential templates are produced by integrating over cos θ∗, leaving the both invariant mass and rapidity as the remainders for the update. The results obtained from each set are presented and overlaid in Figures 13.4 and 13.5, so that the relative impact of the PDF up- date can be assessed as each new dimension of Drell-Yan cross section is added. The addition 276 (cid:17) (cid:16) dσ dm(cid:96)(cid:96) (cid:16) (cid:17) d2σ dm(cid:96)(cid:96)d|y(cid:96)(cid:96)| uv(x) dv(x) ¯u(x) ¯d(x) x δpre [%] δpost [%] δpre [%] δpost [%] δpre [%] δpost [%] δpre [%] δpost [%] 3.4 2.6 4.8 12 5.8 7.5 16 45 0.7 0.9 2.6 7.0 0.1 0.3 0.5 0.7 Table 13.2: Impact of 3000 fb−1 update on the CT14Hera2 uv(x) and dv(x) valence and ¯u(x) and ¯d(x) sea distributions for several values of x using the standard triple-differential templates at Q = 3 TeV. To be compared with Figures 13.4 and 13.5. 9.8 30 71 280 11 32 69 250 2.2 8.3 20 77 1.5 3.6 11 30 3.8 11 20 67 of the cos θ∗ dimension provides for a dramatic reduction in PDF uncertainty as compared to what could otherwise be achieved in a single- or double-differential measurement alone. Furthermore, the effect of cos θ∗ itself on the performance of the PDF update is studied. The results are presented in Figures 13.6 and 13.7, which compare the reduction in PDF uncertainty using templates based on the double-differential cross sections dm(cid:96)(cid:96)d cos θ∗ . Just as before, the triple-differential templates of Section 13.3.4 are used and (cid:16) d2σ dm(cid:96)(cid:96)d|y(cid:96)(cid:96)| (cid:17) (cid:16) d2σ (cid:17) to produce the reduced dimensionality templates here by integrating out the rapidity y(cid:96)(cid:96) or the cos θ∗ dimensions explicitly from Eq. 13.2. Each template retains its binning in the mass dimension to accommodate the steeply falling Drell-Yan cross section. The results show the uncertainty on the uv(x) and dv(x) valence distribution is reduced in the low and intermediate x ranges in the cos θ∗ dimension, while the rapidity y(cid:96)(cid:96) dimension provides for a slightly better reduction at high x, especially on the uv(x) uncertainty. On the other hand, the uncertainties on the ¯u(x) and ¯d(x) sea distributions each show a similar level of reduction at both low and high x, independent of the update performed. Taken together, these results indicate that cos θ∗ by itself is able to constrain the PDF uncertainty to a degree comparable with y(cid:96)(cid:96), and in some cases better, confirming its discriminating power that was suspected in the preceding chapter (See Sec. 12.1.6) 277 (a) (b) Figure 13.4: Impact of the 3000 fb−1 update on the CT14Hera2 uv(x) (a) and dv(x) (b) PDF uncertainties assuming single-differential mass-only templates: “Update1D,” double- differential mass and rapidity templates: “Update2D,” and the standard triple-differential mass, rapidity, and cos θ∗ templates: “Update3D”at Q = 3 TeV. The utilization of the cos θ∗ dimension in the update provides for a substantial reduction in the PDF uncertainty. 278 Error bands of uv(x,Q)xuv(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update1DUpdate2DUpdate3D0.80.91.01.11.210-410-310-210-10.20.50.9Error bands of dv(x,Q)xdv(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update1DUpdate2DUpdate3D0.80.91.01.11.210-410-310-210-10.20.50.9 (a) (b) Impact of the 3000 fb−1 update on the CT14Hera2 ¯u(x) (a) and ¯d(x) (b) Figure 13.5: PDF uncertainties assuming single-differential mass-only templates: “Update1D,” double- differential mass and rapidity templates: “Update2D,” and the standard triple-differential mass, rapidity, and cos θ∗ templates: “Update3D”at Q = 3 TeV. The utilization of the cos θ∗ dimension in the update provides for a substantial reduction in the PDF uncertainty. 279 Error bands of –u(x,Q)x–u(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update1DUpdate2DUpdate3D0.80.91.01.11.210-410-310-210-10.20.50.9Error bands of –d(x,Q)x–d(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update1DUpdate2DUpdate3D0.80.91.01.11.210-410-310-210-10.20.50.9 (a) (b) Figure 13.6: Impact of the 3000 fb−1 update on the CT14Hera2 uv(x) (a) and dv(x) (b) dis- tributions using the double-differential mass and rapidity templates: “Update2D Rapidity,” and double-differential mass and cos θ∗ templates: “Update2D Cosine” at Q = 3 TeV. 280 Error bands of uv(x,Q)xuv(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update2D_RapidityUpdate2D_Cosine0.80.91.01.11.210-410-310-210-10.20.50.9Error bands of dv(x,Q)xdv(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update2D_RapidityUpdate2D_Cosine0.80.91.01.11.210-410-310-210-10.20.50.9 (a) (b) Figure 13.7: Impact of the 3000 fb−1 update on the CT14Hera2 ¯u(x) (a) and ¯d(x) (b) dis- tributions using the double-differential mass and rapidity templates: “Update2D Rapidity,” and double-differential mass and cos θ∗ templates: “Update2D Cosine” at Q = 3 TeV. 281 Error bands of –u(x,Q)x–u(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update2D_RapidityUpdate2D_Cosine0.80.91.01.11.210-410-310-210-10.20.50.9Error bands of –d(x,Q)x–d(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update2D_RapidityUpdate2D_Cosine0.80.91.01.11.210-410-310-210-10.20.50.9 The bottom line is this: a global fit of PDFs which includes medium mass Drell-Yan LHC data, with rapidity and cos θ∗ sensitivity, would dramatically improve the precision of the up and down PDFs. No other data inputs are capable of achieving this improvement. 13.4.2 Impact on High-Mass Drell-Yan With an updated set of PDFs, the systematic uncertainty on the high-mass Drell-Yan cross section prediction can now be re-assessed. As part of the update, ePump is able to prop- agate the reduced PDF uncertainty directly to an observable of interest, in this case the dilepton invariant mass. In order to make contact with the dilepton analysis, the invariant mass distribution assessed here utilizes leptons that originate in the central-central decay configuration, as these are precisely the ones selected by the dilepton event selection (See Sec. 10.5). The results are presented in Figure 13.8, which shows the impact of the 3000 fb−1 pseudo- dataset on the high-mass PDF systematic uncertainty (the impact of the 300 fb−1 pseudo- dataset can be seen in Appendix D). The PDF uncertainty is evaluated at several charac- teristic values of dilepton mass, which are listed in Table 13.3. At m(cid:96)(cid:96) = 5 TeV, the PDF systematic uncertainty is reduced from 31% to 8.9%, a reduction of roughly a factor of 3.5. Similarly, at m(cid:96)(cid:96) = 3 TeV, the uncertainty is reduced from 15% to 3.7%, roughly a factor of 4. In each case, a substantial improvement is obtained compared to the current state-of- the-art predictions (as depicted in Fig 12.8). The PDF uncertainty assessed in the dilepton analysis is, for example, 13% and 29% at 3 and 5 TeV, respectively (See Fig. 10.11). Furthermore, Figure 13.8 displays the dilepton invariant mass with the analysis strategy of imposing a cos θ∗ > 0 requirement, where the error is assessed with the updated PDFs (as first explored in Sec. 12.1.6). Such a combination provides for a marginal, but non-trivial 282 m(cid:96)(cid:96) [TeV] CC Selection δP DF post [%] δP DF pre CC+cos θ∗ Selection δP DF [%] pre δP DF post [%] [%] 1 2 3 4 5 5.9 9.6 15 22 31 1.0 2.0 3.7 6.0 8.9 5.6 8.9 13 20 28 0.9 1.7 3.2 5.3 8.0 Table 13.3: The estimated PDF uncertainty in several invariant mass bins for the distribu- tions shown in Figure 13.8. The current CT14Hera2 uncertainty estimates are shown in the first column, the result of the 3000 fb−1 update is shown next, and the 3000 fb−1 update with an additional cos θ∗ > 0 requirement on the dilepton selection is shown last. Notice the pre-update values for the central-central (CC) selection are consistent with those assessed in the dilepton analysis (See Fig. 10.11). further reduction in the PDF uncertainty at high mass at the expense of a reduction in statistics. Table 13.3 quantifies the PDF uncertainty before and after the update in several high-mass bins of interest for each of these selections. Finally, further improvement at high mass could be obtained by extending the nominal 1 TeV control region if, for example, the presence of new physics is definitely ruled out. The bottom line is this: The heretofore enormous PDF uncertainties in any BSM search at high mass can be reduced below the current experimental uncertainties with the use of LHC Drell-Yan inputs from a possible future PDF global fit. 13.5 Outlook The impact of a future Drell-Yan cross section measurement on the CT14Hera2 PDF uncer- √ s = 13 TeV LHC with 3000 fb−1 of tainty was assessed using the ePump package at the Drell-Yan pseudo-data. The phase space region considered for the PDF update was based on three variables: the dilepton mass m(cid:96)(cid:96), the dilepton rapidity y(cid:96)(cid:96), and the cosine of the polar angle cos θ∗. This region was divided into 1296 histogram bins and used to construct ePump 283 (a) (b) Figure 13.8: The dilepton invariant mass distribution for central-central dilepton events (a), as would be selected by the dilepton event selection of Section 10.5, and with an additional cos θ∗ > 0 requirement added to the selection (b). The ratio sub-plot depicts the CT14Hera2 PDF uncertainty before and after the 3000 fb−1 update. 284 Dilepton Invariant Mass [GeV]310Events2-101-10110210310410510Drell-YanPDF Error [Hera2]PDF Error [Update]-1 = 13 TeV, 3000 fbs ResBosCC LeptonsDilepton Invariant Mass [GeV]90010002000300040005000Data/Bkg 0.70.80.911.11.21.3Dilepton Invariant Mass [GeV]310Events2-101-10110210310410510Drell-YanPDF Error [Hera2]PDF Error [Update]-1 = 13 TeV, 3000 fbs ResBos*>0qCC Leptons, cosDilepton Invariant Mass [GeV]90010002000300040005000Data/Bkg 0.70.80.911.11.21.3 m(cid:96)(cid:96) [TeV] CC Selection δP DF post [%] δP DF [%] δChoice [%] δT heory [%] δExp [%] δT otal [%] Dilepton Analysis 1 2 3 4 5 1.0 2.0 3.7 6.0 8.9 5.4 8.7 13 19 29 0.0 0.0 0.0 8.4 47 5.9 9.8 15 23 57 7.7 11 12 13 14 9.7 15 19 26 59 Table 13.4: The post-update PDF uncertainty as compared to the experimental and domi- nant theoretical uncertainties in the electron channel of the dilepton analysis. As the PDF uncertainty will be reduced well below the current experimental uncertainty, attention will be shifted to the reduction of others, such as the “PDFChoice” uncertainty, improving the discovery potential of future iterations of the dilepton analysis. pseudo-data and signal templates, which were designed to probe the PDFs in the extreme kinematic regions of (x,Q2) only accessible at the LHC. The CT14Hera2 PDF set was used for the update. The results showed a significant reduction in the uncertainties associated with all parton flavors, especially ¯u(x) and ¯d(x) sea at high x. Likewise, these reduced PDF uncertainties, when propagated to the high-mass dilepton spectrum, lead to a significantly improved estimate of the event yield. These proof-of-concept results indicate a great deal of improvement can still be obtained from precision PDFs measurements at LHC. The use of cos θ∗ as an additional dimension in future PDF global fits is absolutely crucial, as it supplements the more standard double- differential measurements in invariant mass and rapidity; when used in conjunction, as was done here, the reduction in uncertainty can be dramatic. For these reasons, Drell-Yan cross section measurements could be vital to the success of future searches and measurements at the LHC. Not only will the PDF uncertainty that affects the high-mass dilepton analysis be reduced, improving the discovery potential of many non-resonant new physics models, but the inclusion of new and robust data into the modern PDF global fits will bring the uncertainty estimates of the various global fitting groups into 285 better agreement (See Fig. 10.12). Such an opportunity would call for a re-assessment of the “PDFChoice” uncertainty, which may further improve the dilepton discovery potential, as the largest uncertainty would no longer be due to the PDFs. Table 13.4 compares these uncertainties explicitly.10 There- fore, for the reasons outlined in this chapter, experiments at the LHC should strongly consider performing precision measurements of the Drell-Yan triple-differential cross section in order to further constrain the PDF uncertainties in future PDF global fits. 10The uncertainty on the QCD background estimate is not included in the above calculations. 286 Part V The Conclusion 287 Chapter 14 Conclusion In this final chapter, a conclusion is given by summarizing the results obtained in the dilepton analysis, and a brief outlook to the future of the analysis and the broader LHC physics program is provided. 14.1 Summary The Standard Model is an extremely successful theory, but we know it cannot be the final theory of elementary particles and interactions. As a consequence of addressing the exper- imental and/or conceptual limitations of the Standard Model, Beyond the Standard Model (BSM) theories often predict striking new phenomena, such as new particles or interactions, many of which are potentially observable at the LHC. This search performed in this dissertation focused on a class of BSM theories that extend the SU (3) × SU (2) × U (1) Standard Model gauge group with one or more U (1)(cid:48) gauge symmetries. These theories predict new heavy gauge bosons, known as Z(cid:48) bosons, which potentially interact with and decay into pairs of charged leptons. The smoking-gun signature of Z(cid:48) production at the LHC would be the appearance of a resonant peak in the dilepton invariant mass spectrum. A search for a new high-mass resonance was performed in dielectron final-states using 36.1 fb−1 of proton-proton collision data. The dataset was recorded with the ATLAS detector 288 during the 2015 and 2016 data-taking runs of the LHC at a CM energy of √ s = 13 TeV at CERN in Geneva, Switzerland. The dielectron invariant mass spectrum was measured and the highest-mass event ob- served was 2.9 TeV. The Standard Model background was estimated with MC simulation and data-driven techniques. The dominant irreducible background originates with the pro- duction of high-ET dielectron pairs via the Drell-Yan process. Additional backgrounds arise from t¯t production, Single-Top production, Diboson production, and QCD Multi-jet and W +jets production, in which one or more jets of the event are misidentified as electrons. The dielectron invariant mass spectrum was compared to data, and possible deviations were quantified using a log-likelihood ratio (LLR) test and the BumpHunter package. According to the LLR test, the largest local excess occurred at a mass of MZ(cid:48) = 2.37 TeV with a p-value of 0.6% or a significance of 2.5 σ. The global significance of the excess was negligible, and hence the data were found to be consistent with the Standard Model. A similar result was obtained with the BumpHunter. As no significant excess was observed, 95% C.L. upper limits were set on several Z(cid:48) model parameters of interest. These included the cross section times branching fraction to electrons σB in the SSM and E6 models and the relative coupling strength γ(cid:48) in the Minimal Z(cid:48) model. Limits were set using the electron channel dataset and later statistically combined with the muon channel for maximal sensitivity. For the SSM, masses below 4.5 TeV were excluded, and for the E6 models, lower mass limits were set between 3.8 and 4.1 TeV for the Z(cid:48) ψ and Z(cid:48) χ, respectively. Other E6 motivated models are constrained between these values. For the Minimal model, a range of mass limits were set depending on the assumed value of the relative coupling strength γ(cid:48). In particular, for the Z(cid:48) B−L models, masses below 3.9 and 4.0 TeV were excluded. While no model 3R and Z(cid:48) 289 was definitely ruled out, these limits help to strongly restrict the allowed Z(cid:48) parameter space and are currently some of the most stringent limits set to date (See Table 5.5). The sensitivity of the dilepton search will continue to improve as additional data are collected in subsequent LHC runs. However, as events increasingly populate the high-mass dilepton tail, knowledge of the Parton Distribution Functions (PDFs) will become increas- ingly important. While the current analysis is relatively insensitive to the theoretical un- certainty of the PDFs, the sensitivity of future analyses will eventually weaken and plateau, unless the PDF uncertainty can be sufficiently reduced. To this end, ePump, which can emulate a full PDF global fit, was used to produce an updated set of CT14Hera2 PDFs that incorporate 3000 fb−1 of Drell-Yan pseudo-data in the dimensions of dilepton mass mee, rapidity yee, and cos θ∗. The updated PDFs were then used to assess the reduction in PDF uncertainty in the high-mass search region of interest of the dilepton analysis. The results indicate a significant reduction is possible, such that, for example, at an invariant mass of mee = 5 TeV, the current CT14Hera2 uncertainty of 31% is reduced to 8.9%, a roughly factor of 4 improvement. As a result, the PDFs would no longer be the dominant theoretical uncertainty at high mass, thereby shifting the focus to reduction of the others. Thus, a greatly improved understanding of the PDFs may likely be obtained from future Drell-Yan measurements at the LHC, ensuring that future iterations of the dilepton search are able to maintain a high discovery potential. 290 14.2 Outlook Run-2 of the LHC saw an increase of CM energy from √ s = 8 to 13 TeV. This upgrade dramatically enhanced the sensitivity of the dilepton search to new high-mass phenomena. By the end of 2016, a total of 36.1 fb−1 had been recorded by the ATLAS detector. This allowed some of the strongest exclusion limits on the existence of Z(cid:48) bosons to be set to date. The 2017 run is now complete, which collected an additional 50 fb−1 [215], and by the end of 2018, a total of (cid:39) 150 fb−1 is expected, which will conclude Run-2 at the LHC. As more data are collected, more comprehensive searches can be performed. With in- creased statistics, the cosine of the polar angle cos θ∗ could be used, for example, as a discriminating variable in conjunction with the dilepton invariant mass m(cid:96)(cid:96) to enhance the sensitivity to a potential high-mass signals via interference effects [216]. Likewise, perform- ing the search in exclusive channels, such a Z(cid:48) + n jets also becomes possible, as well as the application of b-jet vetoes or /ET cuts to further reduce the Top and Diboson Standard Model backgrounds. Alternatively, to achieve even more powerful exclusion limits using the existing data, multi-channel combinations are being considered for the first time. Just as the electron and muon channels are statistically combined within the dilepton analysis, search channels can also be combined across analyses. One of the first attempts at this come from the lepton+MET and dilepton searches [217]. This is the first in a series of more ambitious and ongoing grand combinations, which aim to combine not only the (cid:96)ν and (cid:96)(cid:96) leptonic channels, but also the V V and V H Diboson channels1 for maximal sensitivity [218]. Future grand combinations might benefit from 1Here, V denotes a W or Z boson. 291 (a) (b) Figure 14.1: Expected 95% C.L. upper limits on σB with 300 fb−1 of data (a) and 3000 fb−1 of data (b). Theory curves are shown for the Z(cid:48) SSM models. Mass limits can be extracted from the intersection between the expected limit and the theory prediction. The region above each curve is excluded at 95% C.L. [32]. LR, and Z(cid:48) ψ, Z(cid:48) folding in additional search channels as well (e.g. τ τ , tt, tb, bb, jj, etc.). Combinations like this provide a more comprehensive picture of the excluded parameter space, and will greatly help with the interpretation of future data. Looking further ahead, Run-3 starts in 2021, after a two year long shutdown for LHC and detector related upgrades, where for the first time it is anticipated that the machine will s = 14 TeV. The Run-3 goal is to collect an additional 150 fb−1 run with a CM energy of for a total of 300 fb−1. After Run-3, the LHC and detectors will be upgraded once again for √ the High-Luminosity LHC (HL-LHC) program, which will increase the LHC luminosity by a factor of 10 beyond its original design value, with a goal of collecting 3000 fb−1 by the mid-2030s [219]. These Phase-2 upgrades will greatly increase the potential for discovery and the possible study of new exotic phenomena. On the other hand, if no new physics is to be found in the 300 fb−1 (3000 fb−1) dataset, the LHC is expected to exclude the Z(cid:48) ψ below a mass of 5.01 (6.29) TeV, the Z(cid:48) LR below 5.62 292 (7.52) TeV, and the Z(cid:48) SSM below 6.44 (8.55) TeV in the electron channel alone [32]. The inclusion of the muon channel would improve these limits further still. Figure 14.1 depicts the upper cross section limits for these three models given 300 fb−1 and 3000 fb−1 of data collected at the s = 14 TeV LHC. √ Of course the Phase-2 improvements will require the heroic efforts of the LHC collabo- rations, as the tremendous gain in instantaneous luminosity comes with its own challenges, such as an extremely high event pile-up. In Run-3 alone, pile-up is expected to occur at an average rate of 70 interactions per bunch crossing, and 140 is foreseen at the HL-LHC. These high pile-up scenarios call for extensive modifications to almost every aspect of the ATLAS detector, especially in the areas of triggering, tracking, and reconstruction, in order to ac- commodate the higher number of charged tracks and collision vertices per event. Research and development in these areas is currently ongoing to ensure the successful operation of the LHC now and into the foreseeable future. 293 APPENDICES 294 Appendix A Properties of Analysis Simulation and Data Monte Carlo Samples This sections lists the properties of the Monte Carlo (MC) samples used in the modeling of the electron channel of the dilepton analysis. The NLO Drell-Yan samples are listed in Table A.1, the Z(cid:48) samples in Table A.2, the LO Drell-Yan samples in Table A.3, the Top samples in Table A.4, and the Diboson samples in Table A.5. Each table lists the process, the generated range in invariant mass, the ATLAS dataset ID (DSID), the sample cross section σB, and the total number of events associated with each sample. The integrated luminosity of a sample can be calculated from Lint = Nevt/(σB). Data Samples This sections lists the datasets used in the dilepton analysis, as recorded at √ s = 13 TeV by the ATLAS detector during the 2015 and 2016 runs of the LHC. The datasets, along with their period, associated runs, total integrated luminosity, and year recorded, are listed in Table A.6. It should be noted that the central value of 33257 pb−1 for the 2016 dataset was updated 295 Process Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee mee [GeV] DSID - 120-180 180-250 250-400 400-600 600-800 800-1000 1000-1250 1250-1500 1500-1750 1750-2000 2000-2250 2250-2500 2500-2750 2750-3000 3000-3500 3500-4000 4000-4500 4500-5000 5000+ 361106 301020 301021 301022 301023 301024 301025 301026 301027 301028 301029 301030 301031 301032 301033 301034 301035 301036 301037 301038 σB [pb] 1.901 × 10+03 1.748 × 10+01 2.921 × 10+00 1.082 × 10+00 1.955 × 10−01 3.740 × 10−02 1.061 × 10−02 4.258 × 10−03 4.422 × 10−03 5.452 × 10−04 2.299 × 10−04 1.039 × 10−04 4.940 × 10−05 2.445 × 10−05 1.249 × 10−05 1.003 × 10−05 2.934 × 10−06 8.976 × 10−07 2.807 × 10−07 1.265 × 10−07 Nevt [103] 2000 500 250 150 100 150 50 50 50 50 50 50 50 50 50 50 50 50 50 50 Table A.1: The NLO Powheg+Pythia8 Drell-Yan MC samples used in the modeling of the electron channel [4]. Process MZ(cid:48) [GeV] DSID Z(cid:48) χ → ee 301215 Z(cid:48) χ → ee 301246 Z(cid:48) χ → ee 301217 Z(cid:48) χ → ee 301218 2000 3000 4000 5000 Table A.2: The LO Pythia8 Z(cid:48) ing procedure [4]. σB [pb] 8.8463 × 10−03 8.1018 × 10−04 1.0375 × 10−04 1.8298 × 10−05 Nevt [103] 20 20 20 20 χ MC samples used to validate the electron channel reweight- 296 Process γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee γ∗/Z → ee mee [GeV] DSID 70-120 120-180 180-250 250-400 400-600 600-800 800-1000 1000-1250 1250-1500 1500-1750 1750-2000 2000-2250 2250-2500 2500-2750 2750-3000 3000-3500 3500-4000 4000-4500 4500-5000 5000+ 301540 301541 301542 301543 301544 301545 301546 301547 301548 301549 301550 301551 301552 301553 301554 301555 301556 301557 301558 301559 σB [pb] 1.4260 × 10+03 1.3836 × 10+01 2.3332 × 10+00 8.6745 × 10−01 1.5568 × 10−01 2.9710 × 10−02 8.3465 × 10−03 3.3105 × 10−03 1.0982 × 10−03 4.1872 × 10−04 1.7595 × 10−04 7.9961 × 10−05 3.8234 × 10−05 1.9048 × 10−05 9.8535 × 10−06 8.0449 × 10−06 2.4190 × 10−06 7.5769 × 10−07 2.4326 × 10−07 1.1667 × 10−07 Nevt [103] 250 250 250 150 100 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 Table A.3: The LO Pythia8 Drell-Yan MC samples used to construct reweighted Z(cid:48) signal templates for the electron channel search and limit setting of the dilepton analysis [4]. Process mee [GeV] DSID tt → (cid:96)X W t → (cid:96)X W t → (cid:96)X 410015 410016 410009 - - - σB [pb] 7.333 × 10+01 3.584 × 10+00 3.581 × 10+00 Nevt [103] 20000 1000 1000 Table A.4: The NLO Powheg+Pythia6 tt and Single-Top MC samples used in the mod- eling of the electron channel in the dilepton analysis. A generator-level filtering efficiency of 0.10534 is accounted for in the t¯t sample cross section estimate [4]. 297 Process V V → eνeν V V → eνeν V V → eνeν V V → eνeν V V → eνeν V V → eνeν V V → eνeν V V → eνeν V V → (cid:96)(cid:96)ee V V → (cid:96)(cid:96)ee V V → (cid:96)(cid:96)ee V V → (cid:96)(cid:96)ee V V → (cid:96)(cid:96)ee V V → (cid:96)(cid:96)ee V V → (cid:96)(cid:96)ee V V → (cid:96)(cid:96)ee V V → (cid:96)νee V V → (cid:96)νee V V → (cid:96)νee V V → (cid:96)νee V V → (cid:96)νee V V → (cid:96)νee V V → (cid:96)νee V V → (cid:96)νee V V → qqee V V → qqee V V → qqee V V → qqee V V → qqee V V → qqee V V → qqee V V → qqee mee [GeV] DSID 50-150 150-500 500-1000 1000-2000 2000-3000 3000-4000 4000-5000 5000+ 50-150 150-500 500-1000 1000-2000 2000-3000 3000-4000 4000-5000 5000+ 50-150 150-500 500-1000 1000-2000 2000-3000 3000-4000 4000-5000 5000+ 50-150 150-500 500-1000 1000-2000 2000-3000 3000-4000 4000-5000 5000+ 303014 303015 303016 303017 303018 303019 303020 303021 303022 303023 303024 303025 303026 303027 303028 303029 303030 303031 303032 303033 303034 303035 303036 303037 303038 303039 303040 303041 303042 303043 303044 303045 σB [pb] 8.2795 × 10−01 2.3217 × 10−01 9.3993 × 10−03 1.1795 × 10−03 1.2414 × 10−04 2.7399 × 10−05 6.4562 × 10−06 1.3890 × 10−06 1.3096 × 10+00 2.0000 × 10−02 6.4948 × 10−04 4.9253 × 10−05 1.3996 × 10−06 8.2389 × 10−08 6.3062 × 10−09 5.4441 × 10−10 6.8373 × 10−01 8.6295 × 10−02 3.4398 × 10−03 2.7244 × 10−04 7.6162 × 10−06 4.0674 × 10−07 2.3892 × 10−08 1.8118 × 10−09 5.7027 × 10+00 1.7567 × 10−01 4.0750 × 10−03 3.1586 × 10−04 1.0039 × 10−05 6.6015 × 10−07 5.3459 × 10−08 4.8468 × 10−09 Nevt [103] 200 50 50 50 50 50 50 50 200 50 50 50 50 50 50 50 200 50 50 50 50 50 50 50 200 50 50 50 50 50 50 50 Table A.5: The NLO Sherpa W W , W Z, and ZZ Diboson MC samples used in the modeling of the electron channel in the dilepton analysis [4]. 298 to 32862 pb−1 in February 2017 for the Moriond conference, according to the ATLAS lumi- nosity group recommendations [172]. As a result, the total integrated luminosity of the 2015 + 2016 dataset combination becomes 3213 pb−1+ 32862 pb−1 = 36075 pb−1, or 36.1 fb−1 as quoted in each of the kinematic plots of the analysis chapter. The resulting luminosity uncertainty was updated as well, with an estimate of 3.2% for the 2015+2016 combination. Period Runs D E F G H J 276262 - 276954 278880 - 279928 279932 - 280368 280423 - 281075 281317 - 281411 282625 - 284484 Total 276262 - 284484 A B C D E F G I L 297730 - 300279 300345 - 300908 301912 - 302393 302737 - 303560 303638 - 303892 303943 - 304494 305380 - 306451 307126 - 309759 310015 - 311481 Total 297730 - 311481 Total 276262 - 311481 Lint [pb−1] 51.646 435.61 301.69 724.40 264.36 1435.2 3213.0 546.57 1949.5 2884.2 4684.1 1507.1 3449.0 3886.4 8124.4 6325.7 33257 36470 Year 2015 2015 2015 2015 2015 2015 2015 2016 2016 2016 2016 2016 2016 2016 2016 2016 2016 2015+2016 Table A.6: LHC Datasets used in the dilepton analysis [4]. 299 Appendix B Additional Dielectron Cutflow Tables The following section presents the electron channel cutflow tables for the Drell-Yan, Top, and Diboson background processes according to the analysis event selection criteria of Sec- tion 10.5.1. Table B.1 presents the Drell-Yan cutflow, Table B.2 presents the combined Top (t¯t + W t) cutflow, and Table B.3 presents the combined Diboson (W W + W Z + ZZ) cutflow. Each table is generated using the standard set of simulation samples of Appendix A. Selection Cut Number Events Relative Eff. [%] Absolute Eff. [%] Post-selection GRL Trigger Event Cleaning 2 Electrons η Object Quality ET d0 Sig. z0 Identification Isolation Invariant Mass Event Weights 27 792 145.29 27 792 145.29 23 084 040.92 23 084 040.92 23 083 701.84 22 378 782.21 22 325 785.14 15 435 590.20 15 316 004.67 15 166 910.86 14 107 638.84 13 586 918.28 13 133 770.28 12 093 108.99 100.00 100.00 83.06 100.00 100.00 96.95 99.76 69.14 99.23 99.03 93.02 96.31 96.66 92.08 100.00 100.00 83.06 83.06 83.06 80.52 80.33 55.54 55.11 54.57 50.76 48.89 47.26 43.51 Table B.1: The number of simulated Drell-Yan events passing the electron channel event selection of the dilepton analysis. 300 Selection Cut Number Events Relative Eff. [%] Absolute Eff. [%] Post-selection GRL Trigger Event 2 Electrons η Object Quality ET d0 Sig. z0 Identification Isolation Invariant Mass Event Weights 3 432 740.84 3 432 740.84 237 117.65 237 117.65 237 107.39 236 688.49 236 650.55 159 981.32 150 297.89 149 239.60 120 975.13 112 549.93 87 752.50 82 520.12 100.00 100.00 6.91 100.00 100.00 99.82 99.98 67.60 93.95 99.30 81.06 93.04 77.97 94.04 100.00 100.00 6.91 6.91 6.91 6.90 6.89 4.66 4.38 4.35 3.52 3.28 2.56 2.40 Table B.2: The number of simulated Top (t¯t + W t) events passing the electron channel event selection of the dilepton analysis. Selection Cut Number Events Relative Eff. [%] Absolute Eff. [%] Post-selection GRL Trigger Event Cleaning 2 Electrons η Object Quality ET d0 Sig. z0 Identification Isolation Invariant Mass Event Weights 315 370.19 315 370.19 98 956.35 98 956.35 98 863.77 96 974.46 96 814.36 56 665.20 55 965.28 55 287.77 49 027.21 46 306.62 40 228.69 37 328.22 100.00 100.00 31.38 100.00 99.91 98.09 99.83 58.53 98.76 98.79 88.68 94.45 86.87 92.79 100.00 100.00 31.38 31.38 31.35 30.75 30.70 17.97 17.75 17.53 15.55 14.68 12.76 11.84 Table B.3: The number of simulated Diboson (W W + W Z + ZZ) events passing the electron channel event selection of the dilepton analysis. 301 Appendix C Atlantis Event Displays This section presents the event display for the highest invariant mass dielectron events se- lected in the 2015+2016 dilepton analysis, according to the 3.2 fb−1 and 36.1 fb−1 datasets that were used, respectively. The event displays here are produced with Atlantis [220], as part of the ATLAS software framework [156]. Figure C.1 depicts the highest-mass dilepton event in the 2015 search, and Figure C.2 depicts the same in the 2016 search. Energy depositions in the cells of the ECAL and HCAL are indicated by green and red towers, respectively. Reconstructed tracks associated with the selected electron candidates are indicated by red lines. Other tracks in the event with ET > 2 GeV, including those from pile-up, are indicated by cyan lines. 302 Figure C.1: The Atlantis event display for the highest dielectron invariant mass event selected in the 2015 analysis. The “leading” electron has an ET of 373 GeV and an η of -1.03. The “subleading” electron has an ET of 246 GeV and an η of 2.45. The invariant mass of the dielectron pair is 1.78 TeV [5]. 303 Figure C.2: The Atlantis event display for the highest dielectron invariant mass event selected in the 2016 analysis. The “leading” electron has an ET of 841 GeV, an η of -1.26, and a φ of -1.15. The “subleading” electron has an ET of 655 GeV, an η of 1.32, and a φ of 1.83. The invariant mass of the dielectron pair is 2.90 TeV [5]. 304 Appendix D ePump Inputs and Supplimental Results 3000 fb−1 Templates This section presents the complete set of finalized ePump templates used to model the triple- differential Drell-Yan cross section of Eq. 13.2 that were used to perform the update analysis of the final chapter of the dissertation. The cross section is presented in slices of invariant mass according to histogram binning of Section 13.3.4 and is scaled to 3000 fb−1. Pseudo- data are generated according to the procedure of Section 13.3.5. The specific kinematic region is labeled in each plot. The ResBos prediction is indicated by the solid black line and pseudo-data are shown as solid black markers. The sub-panel shows the ratio between the theoretical prediction and the resulting pseudo-data. Error bars represent the statistical uncertainty on the pseudo-data and the solid band represents the PDF uncertainty on the dilepton event yield. 305 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.1: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [40, 66] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 306 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.2: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [40, 66] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 307 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050010001500200025003000350040004500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500300035004000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200400600800100012001400160018002000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020040060080010001200310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 66 GeVll40 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.3: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [66, 80] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 308 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.4: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [66, 80] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 309 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600180020002200310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020040060080010001200140016001800310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700800900310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 80 GeVll66 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.5: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [80, 91] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 310 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.6: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [80, 91] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 311 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0510152025610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0510152025610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events024681012141618202224610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events024681012141618610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0246810121416610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000200030004000500060007000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010000200003000040000500006000070000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 91 GeVll80 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.7: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [91, 102] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 312 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.8: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [91, 102] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 313 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05101520253035610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events051015202530610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0510152025610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0246810121416182022610·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020040060080010001200140016001800200022002400310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 102 GeVll91 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.9: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [102, 116] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 314 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.10: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [102, 116] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 315 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200400600800100012001400310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200400600800100012001400310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020040060080010001200310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350400310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120140310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 116 GeVll102 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.11: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [116, 145] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 316 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700800310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.12: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [116, 145] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 317 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700800310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700800310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700800310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600700310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350400310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120140160310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 145 GeVll116 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.13: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [145, 200] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 318 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.14: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [145, 200] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 319 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120140160180310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010000200003000040000500006000070000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 200 GeVll145 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.15: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [200, 275] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 320 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120140310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.16: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [200, 275] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 321 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120140310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120140310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100310·Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100002000030000400005000060000700008000090000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100002000030000400005000060000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05000100001500020000250003000035000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100020003000400050006000700080009000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 275 GeVll200 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.17: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [275, 381] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 322 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0500010000150002000025000300003500040000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050001000015000200002500030000350004000045000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050001000015000200002500030000350004000045000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000020000300004000050000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000020000300004000050000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000020000300004000050000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000020000300004000050000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000020000300004000050000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100002000030000400005000060000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.18: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [275, 381] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 323 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100002000030000400005000060000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000020000300004000050000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000020000300004000050000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0500010000150002000025000300003500040000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050001000015000200002500030000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02000400060008000100001200014000160001800020000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050010001500200025003000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 381 GeVll275 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.19: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [381, 525] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 324 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020004000600080001000012000140001600018000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02000400060008000100001200014000160001800020000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02000400060008000100001200014000160001800020000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000180002000022000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000180002000022000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000180002000022000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000120001400016000180002000022000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.20: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [381, 525] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 325 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02000400060008000100001200014000160001800020000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020004000600080001000012000140001600018000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02000400060008000100001200014000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200040006000800010000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600180020002200Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.511.522.5Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 525 GeVll381 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.21: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [525, 725] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 326 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100020003000400050006000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000200030004000500060007000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000200030004000500060007000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000600070008000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000600070008000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000600070008000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events010002000300040005000600070008000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events01000200030004000500060007000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.22: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [525, 725] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. 327 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100020003000400050006000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050010001500200025003000350040004500Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020040060080010001200Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events050100150200250300350Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.20.40.60.811.21.4Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 725 GeVll525 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.23: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [725, 1000] GeV and 0 < |y(cid:96)(cid:96)| < 1.8. 328 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0200400600800100012001400160018002000Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 0.2ll0.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600180020002200Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 0.4ll0.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 0.6ll0.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 0.8ll0.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 1.0ll0.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 1.2ll1.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events05001000150020002500Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 1.4ll1.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events02004006008001000120014001600180020002200Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 1.6ll1.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020040060080010001200140016001800Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 1.8ll1.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) (c) (d) (e) (g) (h) (f) (i) Figure D.24: ePump templates (a)-(i) for the triple-differential cross section of Eq. 13.2 in m(cid:96)(cid:96) ∈ [725, 1000] GeV and 1.8 < |y(cid:96)(cid:96)| < 3.6. Impact on CT14Hera2 PDFs with 300 fb−1 The results of the ePump update on the CT14Hera2 PDFs with the 300 fb−1 pseudo-dataset can be seen in Figures D.25 and D.26, and in Table D.1. 329 *qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020040060080010001200Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 2.0ll1.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events0100200300400500600Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 2.2ll2.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events020406080100120140160180200Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 2.4ll2.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.20.40.60.811.21.41.61.82Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 2.6ll2.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 2.8ll2.6 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 3.0ll2.8 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 3.2ll3.0 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 3.4ll3.2 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Events00.10.20.30.40.50.60.70.80.91Pseudo-DataDrell-YanPDF Uncertainty-1 = 13 TeV, 3000 fbs ResBos < 1000 GeVll725 < m| < 3.6ll3.4 < |y*qcos1-0.8-0.6-0.4-0.2-00.20.40.60.81Data/Bkg0.90.9511.051.1 (a) (b) Figure D.25: Impact of the 300 fb−1 update on the CT14Hera2 uv(x) (a) and dv(x) (b) PDF uncertainties assuming single-differential mass-only templates: “Update1D,” double- differential mass and rapidity templates: “Update2D,” and the standard triple-differential mass, rapidity, and cos θ∗ templates: “Update3D”at Q = 3 TeV. The utilization of the cos θ∗ dimension in the update provides for a substantial reduction in the PDF uncertainty. 330 Error bands of uv(x,Q)xuv(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update1DUpdate2DUpdate3D0.80.91.01.11.210-410-310-210-10.20.50.9Error bands of dv(x,Q)xdv(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update1DUpdate2DUpdate3D0.80.91.01.11.210-410-310-210-10.20.50.9 (a) (b) Figure D.26: Impact of the 300 fb−1 update on the CT14Hera2 ¯u(x) (a) and ¯d(x) (b) PDF un- certainties assuming single-differential mass-only templates: “Update1D,” double-differential mass and rapidity templates: “Update2D,” and the standard triple-differential mass, rapid- ity, and cos θ∗ templates: “Update3D”at Q = 3 TeV. The utilization of the cos θ∗ dimension in the update provides for a substantial reduction in the PDF uncertainty. 331 Error bands of –u(x,Q)x–u(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update1DUpdate2DUpdate3D0.80.91.01.11.210-410-310-210-10.20.50.9Error bands of –d(x,Q)x–d(x,Q) at Q =3.0 TeV 90%C.L.CT14HERA2Update1DUpdate2DUpdate3D0.80.91.01.11.210-410-310-210-10.20.50.9 uv(x) dv(x) ¯u(x) ¯d(x) δpre [%] δpost [%] δpre [%] δpost [%] δpre [%] δpost [%] δpre [%] δpost [%] 3.4 2.6 4.8 12 1.4 1.6 3.9 9.7 5.8 7.5 16 45 2.7 5.7 14 41 9.8 30 71 280 4.3 17 43 180 11 32 69 250 6.0 19 41 160 x 0.1 0.3 0.5 0.7 Impact of 300 fb−1 update on the CT14Hera2 uv(x) and dv(x) valence and Table D.1: ¯u(x) and ¯d(x) sea distributions for several values of x using the standard triple-differential templates at Q = 3 TeV. To be compared with Fig.s D.25 and D.26. Impact on High-Mass Drell-Yan with 300 fb−1 The results of the ePump update on the high-mass Drell-Yan spectrum with the 300 fb−1 pseudo-dataset can be seen in Figure D.27. Table D.2 depicts the PDF uncertainty before and after the update in several high-mass bins of interest. m(cid:96)(cid:96) [TeV] CC Selection δP DF post [%] δP DF pre CC+cos θ∗ Selection δP DF [%] pre δP DF post [%] [%] 1 2 3 4 5 5.8 9.4 15 23 32 1.9 3.8 7.1 13 19 5.5 8.8 13 21 29 1.7 3.5 6.1 11 16 Table D.2: The estimated PDF uncertainty in several invariant mass bins for the distributions shown in Figure D.27. The current CT14Hera2 uncertainty estimates are shown in the first column, the result of the 300 fb−1 update is shown next, and the 300 fb−1 update with an additional cos θ∗ > 0 requirement on the dilepton selection is shown last. 332 (a) (b) Figure D.27: The dilepton invariant mass distribution for central-central dilepton events (a), as would be selected by the dilepton event selection of Section 10.5, and with an additional cos θ∗ > 0 requirement added to the selection (b). The ratio sub-plot depicts the CT14Hera2 PDF uncertainty before and after the 300 fb−1 update. 333 Dilepton Invariant Mass [GeV]310Events2-101-10110210310410510Drell-YanPDF Error [Hera2]PDF Error [Update]-1 = 13 TeV, 300 fbs ResBosCC LeptonsDilepton Invariant Mass [GeV]90010002000300040005000Data/Bkg 0.70.80.911.11.21.3Dilepton Invariant Mass [GeV]310Events2-101-10110210310410510Drell-YanPDF Error [Hera2]PDF Error [Update]-1 = 13 TeV, 300 fbs ResBos*>0qCC Leptons, cosDilepton Invariant Mass [GeV]90010002000300040005000Data/Bkg 0.70.80.911.11.21.3 Appendix E Early Warning System This Appendix details the Early Warning System (EWS) used in the monitoring of signatures of new physics in the dilepton and lepton+MET analyses during the early data-taking in Run-2 of the LHC. Physics Motivation Many BSM physics scenarios predict the existence of new force mediating gauge bosons, such as the electrically neutral Z(cid:48) or electrically charged W(cid:48) bosons (See Sec. 5.3). The Run-2 dilepton and lepton+MET Exotics analyses, which search for Z(cid:48) and W(cid:48) production respectively, will surpass their Run-1 of the LHC mass limits [33, 34] with as little as a 1-2 fb−1 of 13 TeV data. As such, if new physics is indeed present, it could start to become apparent very quickly as the data are collected. The electron transverse energy ET and muon transverse momentum pT 1 spectra offer a unique way to search for new model-independent resonant physics, as the signature of a two body decay would appear as a Jacobian peak2 over the smoothly falling Standard Model background. for simplicity. 1For the rest of the section, the “lepton pT ” will be used in reference to the electron ET and muon pT 2A visible feature in the pT spectrum at roughly half the mass of the resonance due to the back-to-back nature of a two body decay. 334 To be prepared for potential new physics scenarios, an EWS was developed to monitor the inclusive lepton pT spectrum for anomalies as the data are recorded by the ATLAS detector. This system runs in parallel with the offline dilepton and lepton+MET analyses, and provides crucial feedback to the question “Are we in discovery or exclusion mode?”, thereby allowing the analyses teams to develop a strategy for quicker unblinding if necessary. Resonant Searches in Run-1 With the onset of Run-2 of the LHC, high energy proton-proton collisions will provide new opportunities to search for BSM physics. The traditional method to search for new resonant phenomena is to reconstruct the dilepton invariant mass m(cid:96)(cid:96) as in the Z(cid:48) search and the transverse mass mT as in the W(cid:48) search. The invariant mass of dilepton pairs is m(cid:96)+(cid:96)− = T p(cid:96)− 2p(cid:96)+ T (cosh ∆η − cos ∆φ), (E.1) where p(cid:96)+ T and p(cid:96)− T are the lepton transverse momenta, ∆η is the difference in their pseudo- rapidities, and ∆φ is the difference in their azimuthal angles in the transverse plane of the ATLAS detector. Likewise, the transverse mass is (cid:113) (cid:113) mT = 2Emiss T T (1 − cos φ(cid:96)ν), p(cid:96) (E.2) where p(cid:96) T is the lepton transverse momentum, Emiss T is the magnitude of the missing trans- verse momentum vector, and φ(cid:96)ν is the angle between the lepton pT and neutrino /ET in the transverse plane of the ATLAS detector. Figure E.1 shows the dielectron invariant Mass and electron+MET transverse mass spec- 335 (a) (b) Figure E.1: Run-1 dielectron invariant mass distribution (a) and electron+MET transverse mass distribution (b), the primary search variables in the dilepton and lepton+MET analy- ses, respectively [33, 34]. tra, the primary search variables in each respective analysis. Because the signature of a new heavy resonance would appear as a resonant peak against the smoothly falling Standard Model background, the above searches have traditionally relied on these two search variables for their respective analyses; however, at the same time, because of the two body nature of each decay, the presence of such a resonance would also be indicated by a Jacobian peak in the lepton pT spectrum. In this way, the inclusive lepton pT spectrum could be used to perform a model-independent search for both the Z(cid:48) and W(cid:48) bosons simultaneously, as well as other resonances decaying into leptons (e.g. a lepton-flavor violating Z(cid:48) boson [221]). Furthermore, the search sensitivity of the Exotics Run-1 dilepton and lepton+MET anal- yses will be quickly surpassed with only 1-2 fb−1 of Run-2 data. The Run-1 combined limits for each analysis are shown in Figure E.2. The dilepton analysis interprets the search results in terms of the Sequential Standard Model (SSM)3 (See Sec. 5.3.2), setting a combined chan- nel limit of 2.90 TeV. Similarly, the lepton+MET analysis interprets their results in terms 3The Z(cid:48) SSM is assigned the same couplings to fermions as the Standard Model Z boson. 336 (a) (b) Figure E.2: Run-1 95% C.L. lower mass limits in the dilepton analysis (a) and lepton+MET analysis (b) [33, 34]. of the SSM W(cid:48) Boson4, setting a combined channel limit of 3.24 TeV. After the first 1-2 fb−1 of Run-2 data is collected, the Run-1 search sensitivity will be exceeded. As the amount of data collected will double every few weeks, and then every few months, a new signal really could “jump out” from one week to the next. This demonstrates the need for a robust EWS, especially since the official analysis and subsequent unblinding procedure could take several months, thereby delaying a possible early discovery. EWS Construction The inclusive lepton pT spectrum is used as the primary variable to discriminate between signal and background in a search for Z(cid:48) and W(cid:48) bosons. The choice of lepton pT is based on two important considerations: first, it naturally accommodates many new physics scenarios in a model-independent way, and second, it mitigates any experimental bias, since it is a 4The W(cid:48) SSM is assigned the same couplings to fermions as the Standard Model W boson. 337 fundamental observable at the LHC. For a two body decay, the signal would appear as a Jacobian peak on the smoothly falling Standard Model background, which would grow with increasing integrated luminosity. The EWS procedure is as follows: the inclusive lepton pT distribution is built and updated as new data are recorded, a functional form is fit on a small subrange of the data at low pT , the form of the fit is extrapolated to high lepton pT (i.e. the “monitoring” region), and the data are compared to the extrapolation, refitting as new data become available. The monitoring will occur in both the electron and muon channels, such that a peak observed in the electron channel could be verified in the muon channel, which would function as an independent cross check. Any sort of persistent shape difference between the extrapolated fit and the data in the monitoring region could indicate some kind of anomalous behavior in the high invariant mass or high transverse mass search regions; in fact it would not be possible to tell a-priori as to which is the source. One possible way to identify the source would be to introduce a progressively stronger missing transverse energy /ET requirement into the EWS event selection. A potential W(cid:48) signal would be insensitive to the /ET requirement, while the Z(cid:48) signal would be strongly diminished. Alternatively, a two lepton requirement could be used to isolate a potential Z(cid:48) signal. EWS Tests on Run-1 Data In order to test the EWS idea, the full Run-1 LHC dataset is used.5. An event selection is developed for the EWS in order to select events from LHC proton-proton collisions that 5At the time of its construction, Run-2 data-taking had not yet started. 338 contain at least one electron or muon. The EWS event selection is modified according to the 8 TeV Run-1 dilepton analysis. The modifications include: an introduction of single electron triggers, a relaxation of the requirement that every event must contain at least 2 electrons, and lastly, the removal of a minimal invariant mass requirement. Furthermore, the EWS selection requires event electrons to pass the same electron identification and isolation criteria as applied to the “leading” electron of the 8 TeV dilepton analysis. The event level selection criteria used for this study are: 1. Event is in the GoodRunsList (GRL) 2. Event has at least one primary vertex, with number of tracks > 2 3. Event satisfies the trigger configuration: EF e24vhi medium1||EF e60 medium1||EF g35 loose g25 loose 4. Event fulfills LArError < 2 for protection against noise bursts and data corruption The following object level selection criteria are used as well: 1. Each electron has author 1 or 3 2. Each electron must be within |η| < 2.47 and exclude the crack region 1.37 < |η| < 1.52 3. Each cluster must pass calorimeter quality requirements 4. Each electron must have pT > 40 GeV 5. Each electron must have at least isEM Medium++ identification 6. Each electron must be isolated using an ET dependent cut: EtCone20 pT NPV corrected < 0.007 × ET + 5.0 GeV 339 Figure E.3: The EWS applied to the full dataset collected during Run-1 of the LHC. The fit is performed in the low pT region of [125, 300] GeV and extrapolated to the monitoring region to assess the high-pT data. Three signals are overlaid using simulation samples: a 1.5 TeV Z(cid:48) SSM, each of which displays a characteristic Jacobian peak in the pT spectrum. SSM, and a 3.0 TeV W(cid:48) SSM, a 2.5 TeV Z(cid:48) In the case that multiple electron candidates satisfy the above EWS requirements, each is selected and retained for analysis, in order to build the fully inclusive lepton pT distribution. With the requisite lepton pT distribution, the fit and extrapolation can be performed out to high lepton pT . Since the shape of the lepton pT spectrum is unknown a priori, a functional form for the fit must be carefully chosen. The so-called dijet function [188, 189], f (x) = a(1 − x)bxc+d log(x), x = pT√ s , (E.3) where a, b, and c are parameters to be fitted, is used, as it demonstrated the best reliability among several alternative fits. Follow-up studies showed it was robust against the chosen low-pT interval of the fit, and the luminosity scale set by pseudo-data.6 The fit is performed 6Pseudo-data were generated at different characteristic integrated luminosities, and the performance of the EWS fitting was evaluated. 340 on the nominal low-pT interval of [125, 300] GeV and is extrapolated to pT = 2 TeV, where the 20.3 fb−1 of Run-1 data runs out. The result of the event selection and fitting strategy can be seen in Figure E.3. With an extrapolated background estimate, the need to compare data to simulation at high lepton pT , where MC statistics start to run out, is relieved. EWS Operation during Run-2 Data-Taking The formulation of the EWS for Run-2 followed directly from its successful test-phase in Run-1. As before, the EWS utilized a modified event selection based on the 13 TeV dilepton analysis (See Sec. 10.5). The dilepton requirements were loosened to select events with at least 1 electron, and the invariant mass requirement is removed entirely. All other selection requirements were identical. If multiple electrons passed selection, each was used to construct the electron ET spectrum, making it fully inclusive. The same strategy was employed in the muon channel, resulting in an inclusive muon pT spectrum. Furthermore, the Run-2 EWS ran on TAG dataset derivations, slimmed-down DxAODs which were produced centrally at the CERN Tier-0 computing center [222], and consequently became available several days before the DxAOD derivations used for offline analysis (See Sec.10.2.3). The use of TAG datasets allowed for some level of data-scouting, as the EWS was consistently about a week “ahead” of the offline analysis. Figures E.4 and E.5, and E.6 and E.7 show several runs of the EWS in the electron and muon channels, respectively, during LHC data-taking from May to July 2016. No signal- like high-pT excesses were observed in either channel during the 2015 or 2016 LHC runs. However, the EWS allowed for the identification and study of several candidate high-pT leptons before subsequent observation in the offline dilepton and lepton+MET analyses. 341 (a) (b) (c) (d) (e) (f) Figure E.4: Operation of the Run-2 EWS in the electron channel: May to June 2016 (a)-(f). 342 (a) (b) (c) (d) (e) (f) Figure E.5: Operation of the Run-2 EWS in the electron channel: June to July 2016 (a)-(f). 343 (a) (b) (c) (d) (e) (f) Figure E.6: Operation of the Run-2 EWS in the muon channel: May to June 2016 (a)-(f). 344 (a) (b) (c) (d) (e) (f) Figure E.7: Operation of the Run-2 EWS in the muon channel: June to July 2016 (a)-(f). 345 BIBLIOGRAPHY 346 BIBLIOGRAPHY [1] Particle Data Group Collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C40 (2016) no. 10, 100001. [2] R. Bruce, G. Arduini, H. Bartosik, R. De Maria, M. Giovannozzi, G. Iadarola, J. Jowett, K. S. B. Li, M. Lamont, A. Lechner, E. Metral, D. Mirarchi, T. Pieloni, S. Redaelli, G. Rumolo, B. 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