SEARCH FOR A HEAVY VECTOR RESONANCE AT THE LHC USING THE ATLAS DETECTOR WITH A INTEGRATED LUMINOSITY OF 36.1 FB−1 √ S OF 13 TEV AND By Forrest Hays Phillips A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Physics—Doctor of Philosophy 2019 ABSTRACT SEARCH FOR A HEAVY VECTOR RESONANCE AT THE LHC USING THE ATLAS DETECTOR WITH A INTEGRATED LUMINOSITY OF 36.1 FB−1 √ S OF 13 TEV AND By Forrest Hays Phillips A search is conducted for a new vector resonance, called the W(cid:48), decaying into a W boson and a 125 GeV Higgs boson with a final state of l±νb¯b, where l can be an electron or muon. The data were collected via pp collisions produced by the Large Hadron Collider running √ at s = 13 TeV and recorded by the ATLAS detector. This data has a total integrated luminosity of 36.1 fb−1. The search is performed by examining the reconstructed invariant mass distributions of the WH candidates for evidence of localized excesses in the mass range 500 GeV to 5 TeV. No significant excess over the background prediction is observed and the results are interpreted in terms of constraints on the production cross-section times branching-fraction of a heavy W(cid:48) resonance. Upper limits are placed on the cross-section times branching-fraction at the 95% confidence level and range between 1 × 10−3 pb and 3 × 10−1 pb depending on the mass of the resonance. In addition, several statistical combinations are performed with searches for a W(cid:48) or Z(cid:48) using bosonic decay modes with final states qqqq, ννqq, lνqq, lνlν, llνν, llνl, llll, and qqbb, as well as leptonic decay modes with final states lν and ll. No significant deviation from the Standard-Model predictions is observed and upper limits on the cross-section are evaluated for a range of increasingly model dependent combinations of search results. These limits are also expressed in terms of constraints on the couplings of a heavy vector resonance to quarks, leptons, and bosons. Dedicated to my wife, Claire, and parents, Wesley and Debra. iii ACKNOWLEDGMENTS The road to getting my Ph.D. has been a long one that was full of forks that would have lead me elsewhere in life. I have only made it this far thanks to the people I’ve met along the way. While there were many such people and I am grateful to all of them, I am especially thankful for my parents, my special education teachers, my undergraduate advisors, my graduate advisor, my wife, and our Lansing family. As someone with at least two learning disabilities, I know how well my life could have turned out. I grew up with many friends/classmates who also had learning disabilities and watched them drop out of school, get expelled, or have a very hard time continuing. I owe it to my parents, Wesley and Debra Phillips, that I ever had a chance in the first place. Thanks to the combination of their love for me and the resources they had at their disposal, I was able to get the help I needed at a young age to eventually overcome, or cope with, my learning disabilities. In the same vane, I’d like to thank my special education teachers: Mrs. Gratzer, Mrs. Junko, and Mrs. Johnston. As backwards as it sounds, without them I would never have outgrown the need for them. I thank them for not only teaching me in a way that fit my needs, but also for their patience when handling a distraught, angry child whose mental state had been disrupted by the medicine he needed to pay attention. There are many things they did for me that I will not forget, even if they have. I’d also like to thank my undergraduate advisors: Dr. Jay Dittmann and Dr. Kenichi Hatakeyama. Thanks to them I was able to find a field in physics I was very interested in and then study it. It was through their funding and support that I was able to research at Fermilab for two summers and at CERN for nearly a month. They were also very patient iv with me and very kind, even if I didn’t catch on as quickly as I would’ve liked. A special thanks goes out to Guro T. Kent Nelson and the KSK family. Until we met you, Claire and I never really felt like Lansing was home and we were eager to leave, but that has since changed and we both hope we are able to stay once all of this (motioning to the dissertation) is finished. Without all of you we would still feel trapped in this usually frozen wasteland. My advisor, the knowledgeable and ever-patient Dr. Wade Fisher, deserves a huge thank you. Without him I would never have gotten into graduate school at MSU to begin with and would have missed out on all the great life events that followed. I honestly don’t think I could have had a better advisor and was very lucky that he took me on as a graduate research assistant. His patience through this whole process has been more appreciated than I have ever told him and I hope I’ve been at least somewhat helpful/useful to him. Finally, I’d like to thank my wife, Claire, for a long list of things. She was willing to move far away from her home and family to let me pursue my dream of getting a Ph.D. She put up with the fact that we didn’t really know a lot of people for a long time, so it took awhile before she felt at home. When she got her first job as a full-time meteorologist in Saginaw, she was willing to put up with the long commute so we could stay in Lansing. But most importantly she stuck with me and was mostly patient through this whole, seemingly-never- ending process. I don’t think I could ever verbalize how much her support has meant to me, but if I had to estimate I’d say it’s at least several Smaugs worth. Speaking of Smaug, his cuddles and excitement to see me when I got home everyday definitely made the hard days better. Since it was Claire’s idea that we get a dog, I should probably thank her for that too. v TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Discovery of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Success of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Problems with the Standard Model . . . . . . . . . . . . . . . . . . . . . . . 1.4 Motivation for a Heavy Vector Triplet . . . . . . . . . . . . . . . . . . . . . . 1.5 Search for a Heavy Vector Triplet . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Standard Model Chapter 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Group Theory and Gauge Invariance . . . . . . . . . . . . . . . . . . 2.1.2 Relativistic Quantum Field Theory . . . . . . . . . . . . . . . . . . . 2.1.3 Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 The Weak Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Electroweak Unification . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6 Electroweak Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7 Yukawa Couplings 2.1.8 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . 2.1.9 The Standard-Model Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Problems with the Standard Model . . . . . . . . . . . . . . . . . . . 2.2.2 Potential Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 A Heavy Vector Triplet . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Beyond the Standard Model Subsystems Chapter 3 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The ATLAS Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Particle Interactions with Matter . . . . . . . . . . . . . . . . . . . . 3.2.2 Detector Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Object Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.1 Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.2 Electrons & Photons . . . . . . . . . . . . . . . . . . . . . . 3.3.1.3 Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.5 Missing Transverse Energy . . . . . . . . . . . . . . . . . . . Jets ix xi 1 2 4 6 7 8 10 10 12 13 14 16 18 21 23 25 27 28 28 31 32 37 38 41 41 43 44 56 58 58 59 59 61 61 64 vi 3.3.2 Calibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jet Scales and Resolutions . . . . . . . . . . . . . . . . . . . 3.3.2.1 4.2.2 Chapter 4 Data Selection and Modeling . . . . . . . . . . . . . . . . . . . . . 4.1 Analysis Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Signal Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Background processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Data and Simulation Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1.1 Triggers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1.2 Pile-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulating Signals & Backgrounds . . . . . . . . . . . . . . . . . . . 4.2.2.1 Signal Samples . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2.2 Background Samples . . . . . . . . . . . . . . . . . . . . . . 4.3 Object Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Small-R Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Large-R Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Common Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Resolved Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Merged Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Event Categorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 4.5.2 Categorization Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Categorization Studies . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Observed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal and Control Regions 65 65 68 69 71 72 78 79 79 79 81 82 82 83 84 84 84 85 85 85 86 87 89 89 90 92 93 Chapter 5 97 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 Nuisance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2.1 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2.2 Averaging and Symmetrization . . . . . . . . . . . . . . . . . . . . . 105 5.2.3 Pruning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3 Conditional & Unconditional Fits . . . . . . . . . . . . . . . . . . . . . . . . 106 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Limit Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.4.1 Log Likelihood Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.4.2 Preparing to Calculate ps+b and pb . . . . . . . . . . . . . . . . . . . 115 5.4.3 Expected Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.4.4 Observed Limits 5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3.1 Fit Regions 5.3.2 Example Fit Results 5.5.1 vii Chapter 6 Extended Interpretations . . . . . . . . . . . . . . . . . . . . . . . 120 6.1 Semi-Leptonic VH Combination . . . . . . . . . . . . . . . . . . . . . . . . . 121 0-Lepton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.1.1 2-Lepton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.1.2 6.1.3 Combined Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2 Combination with Other Final States . . . . . . . . . . . . . . . . . . . . . . 130 Semi-Leptonic and Hadronic VH Combination . . . . . . . . . . . . . 130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 . . . . . . . . . . . . . . . . . . . . . 133 6.2.3 Dilepton Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.2.4 Combined Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2.1 6.2.2 VV Channels 6.2.2.1 Orthogonality Studies Chapter 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 APPENDIX A Multi-jet Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 150 APPENDIX B Extra Plots from Data Selection . . . . . . . . . . . . . . . . . . . 157 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 viii LIST OF TABLES Table 2.1: A table documenting the Standard-Model particles and their important . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . properties. Table 4.1: The name of the triggers used to select data, as well as the time period they were used and what physics object they used for selection. . . . . . . 11 80 Table 4.2: The criteria that are common to the resolved and merged event selections. 86 Table 4.3: The criteria unique to the resolved event selection. . . . . . . . . . . . . . 88 Table 4.4: The criteria unique to the merged event selection. . . . . . . . . . . . . . 88 Table 4.5: Categories used in this analysis. They are based on whether the event is resolved or merged, inside the Higgs mass window or not, whether there are one or two b-tags, and whether there are b-tags outside of the leading large-R jet or not (for the merged events only). . . . . . . . . . . . . . . . 90 Table 5.1: The nuisance parameters included in our search that come from MC sys- tematic uncertainties, grouped by physics process. The value column either lists the prior of each NP or lists if the NP is left to float (F) or is a shape (S) systematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Table 5.2: The regions included in the fit are shown above. All the signal regions were included, while only the control regions from the resolved portion of the analysis were included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Table 6.1: The event selection for the 0-lepton (ZH → ν ¯νb¯b) channel. It is designed in the final state, as well as jets which to select a signal with large Emiss are potentially b-tagged. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 T Table 6.2: The event selection for the 2-lepton (ZH → l¯lb¯b) channel. It is designed to select a signal with two leptons in the final state, as well as jets which are potentially b-tagged. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Table 6.3: A table summarizing which signal samples should be passed through the 0-, 1-, and 2-Lepton event selections for the combinations. The 2-lepton signal is passed through all event selections, the 1-Lepton signal is passed through the 0- and 1-Lepton selections, and the 0-Lepton signal is only passed through the 0-Lepton selection. . . . . . . . . . . . . . . . . . . . . 127 ix Table 6.4: The various Diboson channels included in this combination. On the top are the W(cid:48) decay modes and on the bottom the Z(cid:48) decay modes. The only diboson state a W(cid:48) can decay to is W W , while Z(cid:48) can decay to W W . From there a W can decay to lv or qq, while a Z can decay to νν, ll, or qq. . . . 132 Table 6.5: Lower mass limits set on W(cid:48) exclusive production, Z(cid:48) exclusive production, and simultaneous production with degenerate mass for the HVT Model A (top) and Model B (bottom) benchmark models. It lists these limits for the 1-Lepton analysis this dissertation focused on, the combination of the semi-leptonic 0-, 1-, and 2-Lepton analyses, the semi-leptonic and hadronic VH combination, the VV combination, the dilepton combination, and the final VV+VH+dilepton (VV+VH for Model B) combination. . . . . . . 144 Table 7.1: The lower limits on V (cid:48) mass set by the combinations described in this dissertation. ∗Note that the final combination for HVT-B did not include the dilepton search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 x LIST OF FIGURES Figure 2.1: The vertex made available by QED is the photon (γ) interacting with a charged fermion (fq). This vertex can be rotated or flipped to create other combinations, such as particle anti-particle annihilation. . . . . . . . . . Figure 2.2: Vertices made available by the weak force. The Z boson acts like a heavy photon, while the W± cause flavor changes. Here f represents any fermion, charged or neutral, q2/3 is any quark with charge +2/3, q−1/3 is any quark with charge -1/3, l− is any lepton with charge -1, and νl are the corresponding neutrino to l−. The vertices can be rotated or flipped. 16 20 Figure 2.3: Vertices made available by electroweak symmetry breaking. . . . . . . . . 23 Figure 2.4: Vertices made available by the Yukawa couplings. Here fm represents any fermion with non-zero mass. . . . . . . . . . . . . . . . . . . . . . . . . . 24 Figure 2.5: Vertices made available by the strong force. Here q represents any quark. 27 across normal matter, dark matter, and dark energy [1–8]. Figure 2.6: A pie chart depicting how the energy density of the universe is distributed . . . . . . . . Figure 2.7: On the left are the 8 TeV limits with 20.3 f b−1 and on the right are the 13 TeV limits with 3.2 f b−1. On the left and right, HVT Model A is the red line, while on the right HVT Model B is the magenta line. There is no HVT Model B on the left, instead a Minimal Walking Technicolor model (blue dashed line) was used. . . . . . . . . . . . . . . . . . . . . . Figure 3.1: The LHC ring can be seen on the top [9] and a cross section of it can be . . . . . . . . . . . . . . . . . . . . . . . . . . . seen on the bottom [10]. Figure 3.2: This diagram depicts the accelerator chain used to accelerate protons. The energy that the protons reach for each accelerator is also show, as well as the four detectors along the LHC. . . . . . . . . . . . . . . . . . . 29 36 39 41 Figure 3.3: The ATLAS detector [11]. It is made of four main subsystems; the inner detector, the calorimetry, the muon spectrometer, and the magnet system. 42 Figure 3.4: The ATLAS coordinate system is on the left, while the transformation from θ to pseudo-rapidity is demonstrated on the right. . . . . . . . . . . 45 xi Figure 3.5: The Inner Detector [12] is made of four systems, of which three are shown; the pixel detector, the semi-conductor tracker, and the transition radiation tracker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.6: On the left is a histogram showing the radiation lengths of the various Inner Detector components as a function of pseudo-rapidity. On the right is the same, but for nuclear interaction lengths [13]. . . . . . . . . . . . . Figure 3.7: The calorimetry [14] consists of the Liquid Argon Calorimeter and the Tile Calorimeter. Each of these is divided into smaller parts. . . . . . . . Figure 3.8: On the top left and right are the radiation lengths of the electromag- netic barrel and endcap calorimeters, respectively, as a function of pseudo- rapidity. On the bottom are the nuclear interaction lengths for each com- ponent of the calorimetry as a function of pseudo-rapidity. The tan on the bottom plot is the nuclear interaction length of everything before the calorimeters and light blue is everything after. [13] . . . . . . . . . . . . Figure 3.9: The muon spectrometer [15] is made of four parts; the MDTs, the CSCs, the RPCs, and the TGCs. The magnet system is made of the barrel toroids and end-cap toroids. . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.10: A representation of how different particles interact with the different sub- detectors [16]. It shows a fragment of a longitudinal slice of the detector. Figure 4.1: Feynman diagram of our signal. It shows a W(cid:48) or Z(cid:48) decaying to a Higgs and W or Z, respectively. The Higgs then decays to a b¯b pair while the W/Z decay leptonically. . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 4.2: Feynman diagram for the production of t¯t. Both top quarks decay to a b-quark and W boson. One of the W bosons decays hadronically while the other decays leptonically. . . . . . . . . . . . . . . . . . . . . . . . . Figure 4.3: The Feynman diagrams for the three single-top channels. The t-channel is on the top-left, the s-channel is on the top-right, and the Wt-channel is on the bottom. Each channel has one charged lepton, one neutrino, and at least one b-quark in its final state. In addition, each channel has one or two extra quarks, which may or may not be b-quarks. . . . . . . . . . 46 47 49 50 53 60 72 73 75 Figure 4.4: The Feynman diagram for the simplest V+jets process. In general, a V+jets event has two leptons (charged or neutral) and some number of jets. 75 xii Figure 4.5: The Feynman diagram for the diboson channel/final state that contributes most to our total background. This final state has one charged lepton and one neutrino (MET) that come from a W boson and two jets that come from a Z boson. Other final states for this channel could have two leptons coming from the Z boson and two jets coming from the W boson. The other channel would have two W bosons instead of a W and Z, where one W would decay leptonically and the other would decay hadronically. . . 77 Figure 4.6: The Feynman diagram for the SM Higgs background. It is almost identical to our signal, except for the collision producing a W boson instead of a W(cid:48). 77 Figure 4.7: Sensitivity plots for the categorization studies explored for this analysis. The plot on the left compared the resolved and merged strategies to the SimpleMerge500 strategy. It is clear that while the resolved strategy does well at low masses, it starts to do poorly around 1100 GeV. The opposite can be said of the merged strategy, which does great at higher masses, but poorly below 1100 GeV. This graph makes it clear that combining the two approaches was the right thing to do. The plot on the right compares all the different strategies considered to the SimpleMerge500 strategy. . . . . Figure 4.8: The signal and control regions of the resolved selection. On the left are the Higgs mass control regions and on the right are the Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 4.9: The signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass control regions and on the right are the Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . Figure 4.10: The control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass control regions and on the right are the Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . 93 94 95 96 Figure 5.1: The mV H distributions of the signal regions included in the unconditional fit for the 1.5 TeV signal. On the top are the resolved signal regions. On the bottom are the merged signal regions where there are no additional b-tags outside the leading large-R jet. On the left are the 1 b-tag regions and on the right are the 2 b-tag regions. Note that the signal is not shown because it is essentially zero. . . . . . . . . . . . . . . . . . . . . . . . . . 109 xiii Figure 5.2: The mW H distributions of the control regions included in the uncondi- tional fit for the 1.5 TeV signal. Both are resolved control regions. On the left is the 1 b-tag control region and on the right is the 2 b-tag control region. Note that the signal is not shown because it is essentially zero. . 110 Figure 5.3: Ranked NP pulls for an unconditional fit to the 500 (top-left), 1000 (top- right), 1500 (bottom-left), and 2000 (bottom-right) GeV signals. The yellow bands correspond to the pre-fit impact of each NP. . . . . . . . . 111 Figure 5.4: Ranked nuisance parameter’s covariance matrix for an unconditional fit to the 1500 GeV signal. Deep red means the NPs are highly correlated while deep blue means the NPs are highly anti-correlated. Note that this matrix is symmetrical about the diagonal and that not all NPs are shown. 112 Figure 5.5: This figure shows the limits from the limit setting procedure. The dashed, blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yellow bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid, black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the solid, red and magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Figure 5.6: This figure shows the p-values for the background only hypothesis. This plot is usually used as a measure of discovery, where a lower p-value means we can reject the background hypothesis. In this field, a p-value equivalent to 3σ is necessary to claim we’ve seen evidence and a value equivalent to 5σ is needed to claim discovery. Clearly this search reached neither of those values, so instead we set limits on the benchmark models instead. . 119 Figure 6.1: The post-fit mV H distributions of the 0-Lepton analysis. On the left are the 1 b-tag regions and on the right are the 2 b-tag regions. On the top are the resolved signal regions and on the bottom are the merged signal regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Figure 6.2: Upper limits on σ × BR for the 0-lepton analysis, as a function of signal mass. The dashed blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yellow bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the solid red and magenta lines are the theory cross- sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 xiv Figure 6.3: The post-fit mV H distributions of the 2-Lepton analysis. On the left are the 1 b-tag regions and on the right are the 2 b-tag regions. On the top are the resolved signal regions and on the bottom are the merged signal regions. It should be noted that the mH ranges indicated on the resolved region plots are incorrect, instead they should read 110 GeV < mH < 145 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Figure 6.4: Upper limits on σ × BR for the 2-lepton analysis, as a function of signal mass. The dashed blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yellow bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the solid red and magenta lines are the theory cross- sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Figure 6.5: Upper limits on σ × BR for the semi-leptonic W(cid:48) combinations, as a func- tion of signal mass. The dashed blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yel- low bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. . . . . . . . . . . . 127 Figure 6.6: Upper limits on σ×BR for the semi-leptonic Z(cid:48) combinations, as a function of signal mass. The dashed blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yellow bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. . . . . . . . . . . . . . . . 128 xv Figure 6.7: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the semi-leptonic VH combinations, as a function of signal mass. The dashed blue line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. . . . . . . . . . . . . . . . 129 Figure 6.8: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the VH semi-leptonic and hadronic combination, as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross- section value for the relevant statistic at a given signal mass point. . . . 131 Figure 6.9: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the VV channels as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Figure 6.10: A representation of the kinematic phase space of the resolved dijet mass and merged large-R jet mass. The VV mass windows are shown in the top left and the VH mass windows in the top right. The two are shown on the same plot with the areas they overlap highlighted in gray on the bottom. Please note that in order to fit labels in the appropriate places, the figure is not to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 xvi Figure 6.11: Comparison of the upper limits on the cross-section before (solid red) and after (dashed blue) the orthogonality cut on the VV mass window for the 0-Lepton VH search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Figure 6.12: Comparison of the upper limits on the cross-section before (solid red) and after (dashed blue) the orthogonality cut on the VV mass window for 1-Lepton VH search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Figure 6.13: Comparison of the upper limits on the cross-section before (solid red) and after (dashed blue) the orthogonality cut on the VV mass window for the 2-Lepton VH search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Figure 6.14: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the dilepton channels as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red line is the theory cross-section for our benchmark model per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point.140 Figure 6.15: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for Z(cid:48) production for the combination of the VV, VH, and dilepton searches, as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red line is the theory cross-section for our benchmark model per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point.141 xvii Figure 6.16: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for W(cid:48) production for the combination of the VV, VH, and dilepton searches, as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red line is the theory cross-section for our benchmark model per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. . . . . . . . . 142 Figure 6.17: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the simultaneous production of W(cid:48) and Z(cid:48) with degenerate mass for the combination of the VV, VH, and dilepton searches, as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in in Sec. 5.4.4. Finally, the dashed red line is the theory cross-section for our benchmark model per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. . . . . . . . . . . . . . . . 143 Figure 6.18: Observed 95% CL exclusion contours in the HVT parameter spaces (gH , gF ) (left) and (gq, gl) (right) for resonant masses of 3, 4, and 5 TeV for the final V (cid:48) combination of VV, VH, and dilepton. Areas outside the curves are excluded, as are the colored regions that show the constraints from precision EW measurements. The parameters of the HVT benchmark models (Model A and Model B) are also shown where applicable. . . 143 Figure 6.19: Observed 95% CL exclusion contours in the HVT parameter spaces (gH , gF ) (left) and (gq, gl) (right) for a 4 TeV resonant mass. The exclusion contours of the VV+VH combination (dashed magenta), dilepton com- bination (dotted blue), and VV+VH+dilepton combination (solid red) are shown. Comparing the VV+VH and dilepton combinations to the VV+VH+dilepton combination allows us to make an overall tighter ex- clusion contour. Areas outside the curves are excluded, as are the colored regions that show the constraints from precision EW measurements. The parameters of the HVT benchmark models (Model A and Model B) are also shown where applicable. . . . . . . . . . . . . . . . . . . . . . . 144 xviii Figure A.1: The W boson transverse mass distributions of the data and non-multi-jet backgrounds in the inverted-isolation region. After the fit to MET > 200 GeV is done, the backgrounds will be subtracted from the data to create the multi-jet estimate in the inverted-isolation region. Note that for this figure and all other figures in this section, the distributions are split into electron events (left) and muon events (right). . . . . . . . . . . . . . . . 151 Figure A.2: The missing transverse energy distributions of the data and non-multi-jet backgrounds in the inverted-isolation region. The backgrounds are fit to MET > 200 GeV to correct their normalizations. . . . . . . . . . . . . . 152 Figure A.3: The post-fit W boson transverse mass distributions of the data and non- multi-jet backgrounds in the inverted-isolation region. These distributions will be used to derive the shape of the multi-jet estimate in the inverted- isolation region by subtracting the backgrounds from the data. . . . . . . 152 Figure A.4: The missing transverse energy distributions for the data (black), multi- jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Figure A.5: The heavy vector resonance mass distributions for the data (black), multi- jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Figure A.6: The dijet mass distributions for the data (black), multi-jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). . . . . . . 154 Figure A.7: The W boson transverse mass distributions for the data (black), multi- jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Figure A.8: The W boson transverse momentum distributions for the data (black), multi-jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Figure A.9: The lepton transverse momentum distributions for the data (black), multi- jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 xix Figure B.1: Transverse momentum of the leading small-R jet for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Figure B.2: Transverse momentum of the sub-leading small-R jet for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Figure B.3: Transverse momentum of the leading large-R jet for the signal and con- trol regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . 160 Figure B.4: Transverse momentum of the leading large-R jet for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . 161 Figure B.5: Transverse momentum of the lepton for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . 162 Figure B.6: Transverse momentum of the lepton for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . 163 Figure B.7: Transverse momentum of the lepton for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Figure B.8: Missing Transverse Energy for the signal and control regions of the re- solved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . 165 xx Figure B.9: Missing Transverse Energy for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . 166 Figure B.10: Missing Transverse Energy for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Figure B.11: Transverse momentum of the W boson for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . 168 Figure B.12: Transverse momentum of the W boson for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . 169 Figure B.13: Transverse momentum of the W boson for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . . . 170 Figure B.14: Azimuthal angle between the lepton and MET for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Figure B.15: Azimuthal angle between the lepton and MET for the signal and control regions of the merged selection with no additional b-tags outside the pri- mary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . 172 xxi Figure B.16: Azimuthal angle between the lepton and MET for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . 173 Figure B.17: Transverse mass of the W boson for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . 174 Figure B.18: Transverse mass of the W boson for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . 175 Figure B.19: Transverse mass of the W boson for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Figure B.20: Dijet mass for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . 177 Figure B.21: Higgs mass for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Figure B.22: Higgs mass for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 179 xxii Chapter 1 Introduction This dissertation covers the search for a new particle that offers an extension to an existing force of nature. Before covering the details of the search I find it necessary to discuss the importance of doing arcane studies such as this. I have been asked questions similar to “What practical use does your research have?” enough times that I feel it’s necessary to address it here. Let’s consider the story of Heinrich Hertz, who discovered radio waves simply out of curiosity. During the early days of his discovery, Hertz would show off his device that created and detected radio waves as a cool parlor trick. If he was ever asked what practical importance the radio waves he discovered had, he would say there are none. However, just 10 years after Hertz discovered radio waves, Guglielmo Marconi got a patent using them for wireless communications. The moral of this story is the use of something need not be known in advance for it to be useful. In the modern era we have become accustomed to finding new applications for something that has already been discovered. It has gotten to the point that many people seem to forget that one cannot know the uses for something that has yet to be discovered. So sometimes we need to be reminded that it is useful to discover simply for the sake of discovering and understand that the fruits of the discovery might come later. It is this approach that has helped scientists through the ages discover much about the structure/composition of our 1 universe. In this way scientists have come to describe the universe as containing particles that make up the various types of matter. These particles are given the description fundamental, meaning they are indivisible and thus the most elementary form of matter. All of these fundamental particles then interact and bond with each other to create objects like nuclei and atoms. This process continues until we are left with objects of varying size that comprise our macroscopic universe. Physicists who study the quantum world have been trying to nail down what the funda- mental particles are for nearly a century. Though great strides have been made in discovering these fundamental particles, there are still many questions that have not been answered. For those interested, a brief history is covered below. 1.1 Discovery of the Standard Model Around 450 B.C. the Greek philosopher Democritus came up with the idea that if you were to repeatedly cut something up, at some point there would be something that could not be cut further, which he called atomos. This idea was revived by John Dalton around 1800. He hypothesized that all matter was made up of atoms, each atom of the same element was exactly the same, and atoms joined together to form compounds. This theory eventually brought about Dimitri Mendeleev’s periodic table in 1869. It wasn’t until J. J. Thompson discovered the electron in 1897 that scientists realized the atom had substructure. This new knowledge started a revolution that led to the discovery of many new particles. In 1911, Ernest Rutherford discovered that the atom also had a nucleus, which carried 2 the positive charges of the atom which he called protons. Then in 1932 James Chadwick found that the nucleus also contained neutral neutrons, which were tightly bound together with the protons. Around the time the substructure of the atom was being probed, several manifestations of the photon were also observed: infrared light, ultraviolet light, radio waves, X-rays, and gamma rays∗. The key difference between these manifestations being the energy of the photon, which caused different phenomena to occur. Originally scientists didn’t realize all these phenomena were attributed to the same particle, but they pieced that together over time. In 1928, Paul Dirac unwittingly predicted the existence of anti-matter by trying to create a relativistic wave equation for fermions. Just four years after his prediction, anti-matter was uncovered in the form of the positron, which is a positively charged version of the electron. The next discovery made was the muon in 1936, which is essentially a heavier version of the electron. The finding of the mesons soon followed that, starting with the pions and kaons in 1947. Then the baryons were observed, starting with the lambda baryon in 1950†. Soon there was an entire zoo of mesons and baryons, which together were called hadrons. Symmetries within this particle zoo led physicists to believe that the mesons and baryons must be made of even smaller particles. The model describing these smaller particles was proposed in 1964. This model called these particles quarks and said they were bound together by gluons, which were both discovered in 1968. While the hadrons were being discovered the electron and muon neutrinos were discovered in 1956 and 1962, which were so light they were originally thought to be massless. About ∗Visible light had of course been discovered and studied long before these manifestations of the photon. †Later physicists realized that the proton and neutron were also baryons. 3 10 years later an even heavier version of the electron, the tau particle, was found. In 1983 the weak force carriers, the W and Z bosons were discovered. Only 10 years after that the top quark, the heaviest particle we know of, was discovered. Then there was about a 20 year gap before the observation of the Higgs boson was announced in 2012, about 60 years after it was predicted. Altogether we now know of 6 quarks (the up, down, charm, strange, top, and bottom), 6 leptons (the electron, muon, tau, and their respective neutrinos), 5 gauge bosons (the photon, gluon, W +, W−, and Z), and one Higgs boson. The description of these particles and the way they interact with each other have come to be known as the Standard Model (SM). 1.2 Success of the Standard Model The success of the SM can be seen through how well it describes the phenomena humanity knows about. The more phenomena it describes, the more successful it is said to be. So let’s consider what humanity knows about and compare it to what the SM describes. Humanity knows of four fundamental forces; the electromagnetic (EM) force, the strong nuclear force, the weak nuclear force, and gravity. The SM describes how the first three of these forces work at a fundamental level. Someone might argue that’s only 75% already and that it can only go down from there, but I would say that the forces don’t contribute to humanities knowledge equally. Although all forces are necessary for our universe to exist, the sheer volume of knowledge, and its versatility, contributed by the electromagnetic force is enormous compared to the others. This knowledge of the electromagnetic force and how we’ve harnessed it has also propelled 4 humanity to much greater heights than the knowledge of the other forces. The particle that mediates the electromagnetic force is the photon. The EM force is a long-range force, but it is typically screened∗. It is responsible for binding atomic nuclei and electrons together to create the various elements, then it binds those elements together to create molecules and chemicals. It’s therefore responsible for the sciences that come from chemicals and molecules interacting with each other, such as chemistry and biology. Contact forces, such as pushing or pulling on an object, are just macroscopic versions of the electromagnetic force. The foundations of engineering are the contact forces, so the electromagnetic force is fundamental to engineering. Nearly all of our past and present technology is based on the electromagnetic force. Modern technology uses electricity to power and operate all of our devices. Even technology before the discovery of electricity used electromagnetism, but in the form of contact forces. The list for this force just goes on and on, so I will only mention one more important thing it does. Electromagnetism is responsible for all of the human senses. For example, the light we see is electromagnetic waves, i.e. photons, emitted or reflected off the surfaces of objects. Sound is the vibration of air molecules, and those vibrations are caused by electric repulsion between electrons at the surface of colliding molecules. Our sense of touch is just another contact force, caused by electrons on the surface of our skin being repelled by electrons on the surfaces of objects we touch. Even our taste and smell can be attributed to the electromagnetic force. The next force is the strong nuclear force, and the particle responsible for it is the gluon. In contrast to the EM force, the strong nuclear force is a short-range force. It is ∗Meaning that the force is typically zero because atoms and molecules typically have a net charge of zero. 5 responsible for binding the up and down quarks into the protons and neutrons. It also binds the protons and neutrons together to form atomic nuclei, which are bound together with electrons by the electromagnetic force to create atoms. The major technology that came from our understanding of the strong nuclear force was the development of nuclear reactors. The third force the SM accounts for is the weak nuclear force, and the particles responsible for it are the W± and Z bosons. This force is responsible for the radioactive beta decay of atoms and part of the nuclear fusion process of converting hydrogen into helium in stars∗. Without the weak force, the lifetime of the sun would be drastically different and reach much higher temperatures. Lastly, one thing the four fundamental forces do not do is explain how the particles of the SM get their mass. Although there has been a proposed mechanism for this since the 1960s, called the Higgs mechanism, it was not proven until 2012. So far, the discovery of the Higgs boson is probably the biggest success of the SM in the 21st century. 1.3 Problems with the Standard Model As good as the Standard Model is, it is not complete. There are many phenomena in the universe that it cannot explain. For instance, the particles in the Standard Model only account for about 4% of the universe’s total energy. The rest is made up of dark matter and dark energy, which have only been observed on a cosmic level, i.e., no known particle can explain it. In addition, while it describes electromagnetism, the weak nuclear force, and the strong nuclear force well, the Standard Model does not account for gravity. This is ironic considering ∗Specifically, the weak force converts a proton into a neutron in proton-proton collisions in stars 6 gravity was the first force for which humanity created a model. We also know that the visible universe is made primarily of matter, but in the Standard Model matter and anti-matter are produced equally. This means we should expect equal amounts of matter and anti-matter in the universe, but physicists have no proven explanation as to why this is not what we observe. Physicists want to be able to not only explain these phenomena, but many more that the Standard Model does not explain in its current form. Theorists have put forth many hypotheses that address these problems, but so far none have been proven accurate. 1.4 Motivation for a Heavy Vector Triplet This dissertation explores solutions to a problem known as the Higgs naturalness problem. While I discuss what this is in detail in the next chapter, the basic idea is that the observed Higgs boson mass is too small to explain other observables in the SM. There have been many explanations proposed as to why the Higgs mass is so small, but so far none have been confirmed. One commonality among many of the solutions are new heavy particles that can interact with a Higgs boson. It would be very time consuming to explore all of the various models individually though. So we consider a simplified model called the Heavy Vector Triplet (HVT) that breaks down potentially complicated parameter spaces into a generalized one and introduces two particles called the W(cid:48) and Z(cid:48) (or the V (cid:48) collectively). Constraints set on parameters of this simplified model can then be used to set constraints on parameters specific to the more complex models. 7 1.5 Search for a Heavy Vector Triplet To search for a V (cid:48), or any fairly heavy particle, we actually search for its decay products. The V (cid:48) has many ways that it can decay and there are many groups searching for different decay paths, or channels. The group I worked with was specifically looking for a V (cid:48) that decayed to a Higgs boson and weak vector boson, but my focus was the search for a W(cid:48) that decayed to a Higgs boson and W boson, where the Higgs then decayed to a b-quark and anti-b-quark while the W decayed to a lepton and lepton neutrino. This whole process starts with a particle collider and a particle detector, called the Large Hadron Collider (LHC) and ATLAS detector respectively. The LHC collides two protons together at very high energies, which leads to an “explosion”∗ of particles that potentially includes a W(cid:48). This explosion can also contain many other particles, such as leptons, quarks, and guage bosons. In order to form a picture of what particles are in this explosion, a particle detector is made to encompass the collision point. The particle detector we used is called the ATLAS detector, which stands for A Toroidal LHC AppartuS. As particles from the explosion traverse the detector they leave various digital signatures in it that help us identify what they are. Since the W(cid:48) is rarely produced, we must create many such explosions. In addition, the decay we are looking for can look very similar to other SM processes. This means that as we create more explosions and record them, many of them look very similar to the W(cid:48) decay of interest. Before we can search for anything, we need some way of separating the W(cid:48) decay from the SM processes. The way we do this is by doing a counting experiment, which means modeling our W(cid:48) ∗The word explosion is used here only for the sake of evoking the imagery of an outward burst of material. There is no chemical reaction involved in this process. 8 decay and the SM processes, simulating them, and then comparing the simulations to the data taken by the detector. As is, this approach is very ineffective since the SM processes are orders of magnitude more common than the W(cid:48) decay for the majority of the kinematic phase space. However, we can make it much more effective if we can find portions of phase space where the W(cid:48) decays are less rare and the SM processes are less common. How this phase space is found and chosen, especially the parts I took a major or leading part in, is covered in Chapter 4. In Chapter 5, a statistical procedure is performed to set constraints on parameters of the HVT model. Upper limits on the cross-section times branching-fraction of the W(cid:48) → W H → lνb¯b process are set. These limits on the cross-section times branching- fraction are then used in conjunction with the predicted cross-section to set lower limits on potential masses of the W(cid:48). Finally, in Chapter 6 several statistical combinations with other V (cid:48) decay channels are performed. First a combination with the Z(cid:48) → ZH → ννb¯b and Z(cid:48) → ZH → llb¯b channels is performed. Then that combination is combined with the V (cid:48) → V H → q ¯qb¯b channels. Afterwards this is combined with the V (cid:48) → V V and V (cid:48) → ll/lν channels. Each of these combinations sets upper limits on the cross-section times branching-fraction of the channels involved, as well as lower limits on the mass of the W(cid:48) and Z(cid:48) (or V (cid:48) in the case of degenerate mass). In addition, the final combination sets constraints on the coupling strengths of the HVT model for several V (cid:48) masses. 9 Chapter 2 Theory Over the last century, many different particles were discovered. These particles were put into groups called leptons, mesons, baryons, and force carriers. Physicists found a symmetry within the mesons and baryons that led them to understand these were made up of even smaller particles, which were called quarks. Other symmetries between the different particles were found and a theory was derived that exploited all of them. This theory is known as the Standard Model (SM). Below I will give an explanation of this theory, its apparent flaws, and possible solutions. To do this, I will use a mathematical notation derived from References [17] and [18]. 2.1 The Standard Model As can be seen in Tab. 2.1, there are 17 fundamental particles in the SM: 6 quarks, 6 leptons, 4 force carriers, and one mass generator. Included in this model are three of the four known fundamental forces: the electromagnetic force, the weak force, and the strong force. The photon, γ, mediates the electromagnetic force and can interact with all electrically charged particles. The gluon, g, mediates the strong force and can interact with any colored particles. The weak vector bosons, W +, W−, and Z, mediate the weak force and can interact with any particles that have weak isospin. The Higgs boson is unique: it is not a force carrier, but in the process of interacting with other particles, they acquire mass. 10 Mass Spin Charge Isospin Hypercharge Colored Electric Weak Weak 173.1 GeV 4.18 GeV 0.511 MeV < 2.2 eV 1/2 1/2 1/2 1/2 1/2 1/2 2.2 MeV 4.7 MeV 1.28 GeV 96 MeV Name Quarks up down charm strange top bottom Leptons 1/2 electron 1/2 e neutrino muon 105.66 MeV 1/2 µ neutrino < 0.17 MeV 1/2 tau 1/2 τ neutrino < 18.2 GeV 1/2 Gauge Bosons gluon photon Z boson W boson Scalar Bosons Higgs 91.19 GeV 80.39 GeV 124.97 GeV +2/3 -1/3 +2/3 -1/3 +2/3 -1/3 -1 0 -1 0 -1 0 0 0 0 ± 1 0 +1/2 -1/2 +1/2 -1/2 +1/2 -1/2 -1/2 +1/2 -1/2 +1/2 -1/2 +1/2 0 0 0 ± 1 -1/2 +1/3 +1/3 +1/3 +1/3 +1/3 +1/3 -1 -1 -1 -1 -1 -1 0 0 0 0 1 yes yes yes yes yes yes no no no no no no yes no no no no 1.7768 GeV 0 0 1 1 1 1 0 Table 2.1: A table documenting the Standard-Model particles and their important properties. Table 2.1 also lists the properties of the SM particles. The quarks all have electric charge (which is a linear combination of weak isospin and weak hypercharge), color, and weak isospin, so they interact via all three forces. The leptons are split into charged and neutral leptons. The charged leptons have both electric charge and weak isospin, so they interact via the electromagnetic and weak forces. The neutral leptons, called neutrinos, only have weak isospin, so they interact only via the weak force. Since the W± bosons have charge, they not only mediate the weak force, but also interact electromagnetically. The Higgs boson only interacts directly with particles that have mass. 11 2.1.1 Group Theory and Gauge Invariance In order to explore the SM we will first need to develop an understanding of group theory and gauge invariance. This is because groups are important for physics, as they offer a compact way of writing the fundamental laws of nature. A group is a set of elements, A, plus an operation, R, such as addition or multiplication, which obeys the following rules: • Closure: If a1, a2 ∈ A, then a1Ra2 ∈ A. • Identity: There is an element I such that IRa = a for a ∈ A. • Inverse: For each element a ∈ A there is a unique inverse a−1 such that aRa−1 = I. • Associative: aR(bRc) = (aRb)Rc, where a, b, c ∈ A. A group can consist of discrete or continuous elements. The operation can be commuta- tive or not, groups with commutative operations are called Abelian groups. The elements of the group can be differentiable, in which case the group is called a Lie group. An important property of the groups we will be dealing with is gauge invariance. A gauge transformation is any transformation that alters intrinsic properties of a system, such as potentials, in a way that does not change the extrinsic properties of a system. A group that does not change under such transformations is said to be gauge invariant. Over the course of the next few sections we will see several groups, such as U (1), SU (2), and SU (3). The U (1) group consists of all complex numbers with absolute value 1 under multiplication∗. The SU (2) group consists of all 2× 2 unitary matrices† with determinant 1. The SU (3) group is similar to SU (2), but with 3 × 3 matrices. ∗This is the unit circle in the complex plane. †The hermitian conjugate of a unitary matrix is also its inverse. 12 2.1.2 Relativistic Quantum Field Theory A useful tool used in many areas of physics is the Lagrangian, which holds information about the dynamics of a system. In classical mechanics, it is often dependent on an object’s position and velocity and is used to find the equations of motion for a system. Likewise we can use it to describe how a quantum system of particles interact with each other. To do so, we can make the Lagrangian dependent on the particles’ wave functions and their derivatives instead of their positions and velocities. However, a more useful approach is to gather all particles of the same type into one field, which in this case is a scalar or vector function that has a value for each point in space. In this representation, a non-zero value of the field represents a particle of the fields type and its kinematic properties. This type of approach is that of a field theory, specifically a quantum field theory (QFT). In a quantum field theory, the Lagrangian for a field must depend on the field itself ψ(x), and its co-variant derivative ∂µψ(x). However what we truly want is a relativistic quantum field theory (RQFT), which also requires that the equations of motion derived from the Lagrangian must be the same for all inertial reference frames, i.e., they must be Lorentz invariant. This means the Lagrangian can only depend on combinations of ψ and its adjoint ¯ψ, as well as derivatives, such as ¯ψγµ∂µψ or ∂µ ¯ψ∂µψ. The gamma matrices, γµ, are equal to, 13    ,  , 0 0 1 0 0 0 1 0 0 0 −1 0 0 −1 0 0 −i 0 0 0 0 0 i 0 i 0 −i 0 0 0 0 0 γ0 = γ2 =   0 0 0 1 0 0 1 0 0 −1 0 0 −1 0 0 0 0 0 1 0 0 0 −1 0 −1 0 0 0 1 0 0 0  ,  . γ1 = γ3 = (2.1) 2.1.3 Quantum Electrodynamics Now we will turn our attention to building up a relativistic quantum field theory that rep- resents the SM. We will begin with the electromagnetic force, work through the weak force, and end with the strong force, while also covering some other important topics in between. To get started, we require that our Lagrangian gives us an equation of motion that is the Dirac equation, iγµ∂µψ − mψ = 0 , (2.2) which is essentially the relativistic version of the Shr¨odinger equation for a freely moving particle. If we assume there is some fermion field ψ(x) with no external potential, then the Lagrangian that produces the Dirac equation is L = ¯ψ(iγµ∂µ − m)ψ . (2.3) Where the first term represents the kinetic energy/momentum of the field and the second 14 term represents its mass. We know from classical electrodynamics that there are ways to change the electromagnetic potential such that the electric and magnetic fields are unaltered. In this language the transformation would be an element of a U (1) group and would look like, ψ(x) → ψ(cid:48)(x) = eiqβ(x)ψ(x) , (2.4) where β(x) is an arbitrary scalar field. The Lagrangian needs to be invariant under this transformation, otherwise there would be an observable effect in the physical world. Plugging this transformation into Eq. 2.3 we get, L → L(cid:48) = ¯ψ(cid:48)(iγµ∂µ − m)ψ(cid:48) = ¯ψ(iγµ∂µ − m)ψ − q ¯ψγµ∂µβ(x)ψ , (2.5) so we can see that the Lagrangian is not invariant. However, we can redefine our covariant derivative as ∂µ → ∂(cid:48) Aµ(x) → A(cid:48) µ = ∂µ + iqAµ(x) µ = Aµ(x) − ∂µβ(x) , (2.6) where Aµ(x) is a vector field that also changes under the transformation. Now the Lagrangian is invariant under the transformation above and takes on the form, L(cid:48)(cid:48) = ¯ψ(iγµ∂(cid:48) µ − m)ψ = ¯ψ(iγµ∂µ − m)ψ − q ¯ψγµψAµ . (2.7) This vector field Aµ(x) is actually the electromagnetic potential that was mentioned before and can be interpreted as the photon. Feynman diagram vertices made available by Quantum Electrodynamics (QED) are shown in Fig. 2.1. 15 γ fq fq Figure 2.1: The vertex made available by QED is the photon (γ) interacting with a charged fermion (fq). This vertex can be rotated or flipped to create other combinations, such as particle anti-particle annihilation. 2.1.4 The Weak Force When it comes to the weak force there are two types of bosons and therefore two types of interactions. The simple interaction is when a particle interacts with a Z boson, which is electrically neutral and has no weak isospin. The more complex interaction is when a particle interacts with a W boson, which is electrically charged and has weak isospin one. In this case a particle that interacts with a W boson will switch both charge and flavor. So an up quark will become a down quark while an electron will become an electron neutrino. Another distinction that needs to be made between particles is their helicity, which is whether a particle’s spin is parallel or anti-parallel to its velocity. A particle that has spin parallel to its velocity is said to be right-handed while a particle with spin anti-parallel is said to be left-handed. From experiment we observe that all particles except the neutrinos can be right-handed or left-handed. In the case of neutrinos, neutrinos are left-handed while anti-neutrinos are right-handed. From experiment we also know that the W bosons only interact with left-handed particles. So now we have two sets of particles. The left-handed weak isospin doublets, (2.8) , L u  d c  s t  b νe  e νµ  µ ντ  τ L L L L L 16 which are defined to express the pairings above and the right-handed weak isospin singlets, uR cR tR eR µR τR . dR sR bR (2.9) Where the W bosons act as a rotation of the left-handed doublets such that the flavors are changed in an interaction. This new notation also changes our Lagrangian, which is now L = LL + LR LL = ¯χL(iγµ∂µ − m)χL − q ¯χLγµχLAµ LR = ¯ψR(iγµ∂µ − m)ψR − q ¯ψRγµψRAµ , (2.10) where χL is an element of Eq. 2.8 and ψR is an element of Eq. 2.9. The definition of weak isospin in the left-handed doublets is completely arbitrary though. We’ve given the top row weak isospin 1/2 and the bottom row weak isospin -1/2, when we could have easily done the opposite. So the weak force is invariant under transformations that rotate these in such a fashion. This rotation is an element of a SU (2) group, and takes the form of χL(x) → χ(cid:48) L(x) = eigαa(x)T a χL(x) , (2.11) where g is explained below, αa(x) is an arbitrary vector field, and T a are the Pauli matrices. Plugging this transformation of the left-handed doublets into our Lagrangian gives LL → L(cid:48) = ¯χL(iγµ∂µ − m)χL − q ¯χLγµχLAµ − g ¯χLγµ∂µαa(x)T aχL , (2.12) which reveals that the Lagrangian is not invariant under such a transformation. However, 17 we can redefine the covariant derivative on the left-handed doublets as ∂µ → ∂(cid:48) µ → W(cid:48)a W a µ = W a µ = ∂µ + igW a µ T a µ − ∂µαa , (2.13) where W a µ are three vector fields that also change under the transformation. Now the Lagrangian is invariant under the transformation of left-handed doublets and takes the form L(cid:48)(cid:48) = L(cid:48)(cid:48) L + LR L(cid:48)(cid:48) L = ¯χL(iγµ∂µ − m)χL − q ¯χLγµχLAµ − g ¯χLγµW a µ T aχL (2.14) LR = ¯ψR(iγµ∂µ − m)ψR − q ¯ψRγµψRAµ . Since we get one gauge boson for each vector field and none of the vector fields have mass terms, this leaves us with three massless gauge bosons (W1, W2, W3), two charged and one neutral, with coupling strength g to the left-handed doublets. Experimentally this isn’t quite what we observe. We do find that we have three weak gauge bosons, two charged and one neutral, but the neutral boson has a different coupling than the charged bosons. In addition, we know that the neutral boson can also interact with the right-handed fermion singlets, but that is not possible here. 2.1.5 Electroweak Unification Leaving things in sec. 2.1.4 as they are, there is no way to recover what we observe from experiment. However, there is a fairly straight forward solution: combine the electromagnetic and weak forces into one force called the electroweak (EWK) force. Let’s do this by combining group into the SU (2)×U (1)Y group, the electromagnetic U (1)Q group and the weak SU (2)I3 18 where Q is a particle’s electric charge, I3 is a particle’s weak isospin, and Y = 2(Q− I3) is a particle’s weak hypercharge. Such a transformation would take the form of Eq. 2.15 for the left-handed fermion doublets and Eq. 2.16 for the right-handed fermion singlets. χL(x) → χ(cid:48) L(x) = eig1Y β(x)eig2αa(x)T a χL(x) ψR(x) → ψ(cid:48) R(x) = eig1Y β(x)ψR(x) (2.15) (2.16) Taking the Lagrangian with freely-moving, relativistic left-handed doublets and right- handed singlets, L = LL + LR LL = ¯χL(iγµ∂µ − m)χL LR = ¯ψR(iγµ∂µ − m)ψR , we can plug in these transformations, LL → L(cid:48) L = ¯χL(iγµ∂µ − m)χL − ¯χL(g1Y γµ∂µβ(x) + g2γµ∂µαa(x)T a)χL LR → L(cid:48) R = ¯ψR(iγµ∂µ − m)ψR − g1Y ¯ψRγµ∂µβ(x)ψR , (2.17) (2.18) to see this Lagrangian is not invariant under them. To make this Lagrangian invariant under such a transformation we need to redefine the derivative, µ T a µ = ∂µ + ig1Y Bµ + ig2W a µ(x) = Bµ(x) − ∂µβ(x) ∂µ → ∂(cid:48) Bµ(x) → B(cid:48) µ (x) → W(cid:48)a W a µ (x) = W a µ (x) − ∂µαa(x) , (2.19) 19 where Bµ and W a µ vector fields that also change under the transformation. This creates a gauge invariant Lagrangian, L(cid:48)(cid:48) = L(cid:48)(cid:48) L + L(cid:48)(cid:48) R L(cid:48)(cid:48) L = ¯χL(iγµ∂µ − m)χL − g1Y ¯χLγµBµχL − g2 ¯χLW a L(cid:48)(cid:48) R = ¯ψR(iγµ∂µ − m)ψR − g1Y ¯ψLγµBµψL , µ T aχL (2.20) with four massless gauge bosons; W 1, W 2, W 3, and B. The W 1 and W 2 bosons mix together to become charged W± bosons, while the W 3 and B mix together to become the neutral Z and γ bosons. Due to hypercharge mixing the electric charge and isospin, the Z now has a different coupling from the W± and can also interact with the right handed singlets. Feynman diagram vertices made available by the EWK force are shown in Fig. 2.2. W− l− γ W± νl W± W + q2/3 Z W± q−1/3 W± Z f f Z Z W± W± Figure 2.2: Vertices made available by the weak force. The Z boson acts like a heavy photon, while the W± cause flavor changes. Here f represents any fermion, charged or neutral, q2/3 is any quark with charge +2/3, q−1/3 is any quark with charge -1/3, l− is any lepton with charge -1, and νl are the corresponding neutrino to l−. The vertices can be rotated or flipped. 20 2.1.6 Electroweak Symmetry Breaking In the previous section we unified the electromagnetic and weak forces and were left with four massless gauge bosons. We know from experiment that there are actually three massive weak bosons and one massless photon. In order to reconcile experiment and theory we must again modify the theory, this time by breaking the gauge symmetry of the EWK force. To break the symmetry we start by introducing an SU (2) doublet of complex scalar fields, φ(x) = φ+(x)  , where φ0(x) √ φ+(x) = (φ1(x) + iφ2(x))/ √ φ0(x) = (φ3(x) + iφ4(x))/ 2 2 , (2.21) where φ1−4 are real-valued, scalar fields. This doublet has a kinetic term in the Lagrangian, T (φ) = (∂µ ¯φ)(∂µφ), and we give it the lowest order, non-trivial potential term, V (φ) = µ2 ¯φφ + λ( ¯φφ)2. Specific choices of µ and λ will allow this “Higgs Mechanism” to give the weak bosons their mass. To start the potential needs to be bounded below as φ1−4 go to infinity, so λ must be greater than 0. If µ2 > 0 then there is only one minimum of the potential and it is at 0, this is a trivial case because nothing interesting happens and the weak bosons remain massless. But if µ2 < 0, then the minimum is on the surface of a four dimensional sphere with radius ¯φφ = −µ2/2λ ≡ v2/2, where we call v the vacuum expectation value (VEV). This non-zero VEV breaks the gauge symmetry of the EWK force, i.e., the gauge symmetry associated with the transformations of Eqs. 2.15 and 2.16. To explore the effects of a non-zero VEV, we will pick (φ1, φ2, φ3, φ4) = (0, 0, v, 0) to be 21 the vacuum. The vacuum of the field is now φvac = 1√ 2  . 0 v (2.22) The φ+ component of the field is zero which makes the vacuum neutral, so it conserves charge. In addition, there are no imaginary numbers, so it is real valued. The consequences of this choice can be seen by considering perturbations of the vacuum,  0  , v + H(x) φ(x) = 1√ 2 (2.23) where we call H(x) the Higgs field. The choice of our vacuum also sets the weak hypercharge of the Higgs field, YH = 2(Q − I3), to 1, since Q = 0 and I3 = −1/2. We now plug φ(x) into our Lagrangian from the previous section, Eq. 2.20. There are many terms that come from this, but the ones of interest are, L = µ2H2 + 1 8 v2g2 2((W 1 µ)2 + (W 2 µ)2) + v2(g1Bµ − g2W 3 µ)2 + H.O.T . 1 8 (2.24) (cid:113) First, we see a mass term for the Higgs field, m2 H = 2|µ|2. In order to see the masses of the W and Z bosons, we need to take the physical field definitions given in Eq. 2.25 into account. We now see that mW = vg2/2 and mZ = v g2 1 + g2 2/2, while the photon remains massless. By introducing this complex scalar field doublet and breaking its potential’s symmetry, our result now explains why the weak vector bosons have mass while the photon does not. 22 Feynman diagram vertices made available by EWK symmetry breaking are shown in Fig. 2.3. (W 1 µ ± W 2 µ) 1√ 2 W± µ = 1(cid:113) 1(cid:113) Zµ = Aµ = (g2W 3 µ − gqBµ) (g1W 3 µ + g2Bµ) (2.25) g2 1 + g2 2 g2 1 + g2 2 H W± W± H Z Z H H H H H H V H V H H Figure 2.3: Vertices made available by electroweak symmetry breaking. 2.1.7 Yukawa Couplings Up until now, a mass term for each fermion field (m ¯ψψ) has been included without explana- tion. There is no inherent reason that the fermion fields should have mass, but we observe that they do. The mass of these fermion fields can be explained by their interaction with the Higgs field introduced in Sec. 2.1.6. Here I will give an example of how this is done for the electron, but a similar explanation can be done for the quarks and other charged leptons. First, we assume that the leptons can interact with the Higgs field, the interaction term 23 added to the Lagrangian is Lint = ge( ¯χe Lφψe R + ¯φ ¯ψe Rχe L) . (2.26) The parameter ge is some arbitrary coupling of the electron to the Higgs field, χe L is the electron component of the left-handed doublet, φ is the Higgs doublet, and ψe R is the electron component of the right-handed singlet. After inserting the values of the fields and expanding the Higgs doublet around the VEV, the interaction term becomes Lint = gev√ 2 (¯e− L e− R + ¯e− Re− L ) + . . . . (2.27) It can be seen that through the electron interacting with the Higgs field, it has obtained a √ mass me = gev/ 2. However, since the electron mass is already known, what has really been uncovered is how strongly the electron couples to the Higgs field. Feynman diagram vertices made available by the Yukawa couplings are shown in Fig. 2.4. H fm fm Figure 2.4: Vertices made available by the Yukawa couplings. Here fm represents any fermion with non-zero mass. 24 2.1.8 Quantum Chromodynamics The last force we need to introduce is the strong force, for which we first need to establish the concept of color. Color is a type of charge that represents when there are three charges instead of two, as in the electromagnetic force. Similar to electric charges, particles with the same color repel each other while particles with different colors attract each other. Whether it be electric charge or color, in order for an object to be neutral it needs equal amounts of the relevant charges. In the case of electric charge, the simplest, non-trivial, neutral state is one positive and one negative particle. However, in the case of color, the simplest, non-trivial, neutral state is to have one particle of each color, i.e. three particles. It was for this reason that this “color charge” got its name, because similar to light which has three base colors, red, green, and blue, that combine in equal parts to become white light, colors combine together in equal parts to become neutral. For simplicity, the three color charges were given the names red, green, and blue and when they are combined together in equal parts they create a neutral, white object. One more detail when it comes to color is that each color has an associated anti-color, i.e., anti-red, anti-green, and anti-blue. The anti-colors introduce another level of complexity to the situation because not only do the anti-colors behave similarly to the colors amongst themselves, but every color is attracted to its anti-color and vice versa. This gives us another simple, non-trivial, neutral state, which is one particle with color and another with the associated anti-color. Objects made of these two simple, non-trivial, neutral, color states are called hadrons and mesons respectively. Getting back to the math, the strong force can be included by making the Lagrangian invariant under rotations that are elements of a SU (3) group. In this case the rotations 25 are through the different colors: red, blue, and green. There’s nothing implicitly different between these, so permuting them should not change the physics. Since there is only one gluon in Tab. 2.1 one might assume the transformation only has one generator, but this is not the case. Since the gluon is exchanging color between two differently colored particles it’s actually carrying two colors at the same time: one color and one anti-color. Since there are three colors, one might expect this means there are 3 × 3 = 9 gluon color states, but this is not the case either. Instead there are only eight gluon color states, for which one representation is, √ √ −i(r¯b − b¯r)/ (r¯b + b¯r)/ 2 2 √ √ 2 −i(r¯g − g¯r)/ (r¯g + g¯r)/ √ √ −i(b¯g − g¯b)/ (b¯g + g¯b)/ 2 2 √ √ 2 (r¯r + b¯b − 2g¯g)/ (r¯r − b¯b)/ 2 6 . (2.28) √ A question might be raised about the possibility of (r¯r + b¯b + g¯g)/ 3, but such a gluon would be a color singlet. A color singlet gluon would be a free particle and would act as a long range strong mediator. Since we do not observe a long range strong mediator in our universe, there must be no such state for the gluon. This leaves eight gluon color states, so our transformation needs eight generators. These transformations take the form, ψ(x) → eigsαa(x)λa ψ(x) , (2.29) where the index a runs from 1 to 8 and the λa are the eight base rotations available in this space. 26 As in the previous sections, the Lagrangian is not invariant under such rotations. Re- defining the derivative as, ∂µ → ∂µ + igsGa µ → Ga Ga µλa µ − ∂µαa − fabcGc µαb , (2.30) makes the Lagrangian invariant under these rotations, which I will not write here since it is fairly complex and the relevant terms will be included in the next section. This introduces eight new vector fields Ga µ, which are interpreted as the eight gluon color state fields. Feyn- man diagram vertices made available by Quantum Chromodynamics (QCD) are shown in Fig. 2.5. q g q g g g g g g g Figure 2.5: Vertices made available by the strong force. Here q represents any quark. 2.1.9 The Standard-Model Lagrangian Now that we have defined all three forces in this language, showed how electroweak symmetry breaking gives the weak bosons their mass, and showed how the fermions get their mass, we have enough to put together the more important terms of the Standard-Model Lagrangian. 27 The full Standard-Model Lagrangian, with important terms not discussed included as text, is, L = i ¯ψf γµ∂µψf (cid:88) (cid:88) (cid:88) f + + f + (∂µ ¯φ)(∂µφ) − µ2 ¯φφ − λ( ¯φφ)2 f √ 2( ¯ψf vgf / Rψf L + ¯ψf Lψf R) ¯ψf (ig1Y Bµ + ig2W a µ T a + igsGb µλb)ψf the fermion kinetic terms the fermion mass terms the fermion-boson interaction terms the Higgs kinetic term the Higgs potential term + ¯φ(ig1Y Bµ + ig2W a µ T a)φ the Higgs-boson interaction terms + boson kinetic terms + chiral terms + H.O.T. (2.31) Where ψ ∈ {χL, ψR}, ψL is the upper or lower component of the relevant weak isospin doublet, W a µ and Bµ are the electroweak fields, Gb µ are the gluon fields, and φ is the Higgs doublet. 2.2 Beyond the Standard Model 2.2.1 Problems with the Standard Model In sec. 2.1, we discussed how to build a Lagrangian that contains the particles we know about and how they interact with each other. While it does a good job of describing what is included in it, there are still phenomena that we know about but don’t fully understand and that could not be put into this Lagrangian. Below I discuss a few such phenomena to introduce them to the reader, but the list is not exhaustive. 28 Figure 2.6: A pie chart depicting how the energy density of the universe is distributed across normal matter, dark matter, and dark energy [1–8]. • Gravity: Perhaps the most glaring flaw to the average person is that we’ve only included three of the four fundamental forces. Despite knowing about gravity and how it affects objects, we still do not have a quantum mechanical description for it that would fit in the SM Lagrangian. • Matter-Antimatter Asymmetry: The SM Lagrangian reflects the knowledge that the universe has both matter and antimatter, but it does not explain why the entire visible universe is made of matter. Since matter and antimatter are created and de- stroyed in pairs, you would expect either equal amounts of matter and antimatter in the universe or neither because it all annihilated. 29 Energy Density of the UniverseDark Energy72%Normal Matter5%Dark Matter23% • Dark Matter: Although the SM describes the known particles very well, those parti- cles only account for about 5% of the universe’s energy density. As shown in Fig. 2.6, about 23% of the universe’s energy density is what we call dark matter. This is matter that doesn’t seem to interact with photons, but does have a gravitational pull. It can be indirectly observed through gravitational lensing, galaxy rotation curves, and other methods, but we do not know what dark matter is or how it fits into the SM. • Dark Energy: The other 72% of the universe’s energy density is made up of what we call dark energy, also shown in Fig. 2.6. This is some sort of energy that causes the universe to expand at an accelerating rate. Evidence for dark energy can be inferred via astronomical observations of the large-scale structure of the universe. • Neutrino Masses: For a long time, the neutrinos were thought to be massless, but recent discoveries have shown that they have very small masses. The Standard Model currently has no mechanism that explains the mass of the neutrinos like in the case of the other fermions. Since there are no right-handed neutrinos, the methodology used in Sec. 2.1.7 is not applicable. • The Higgs Naturalness Problem: Another problem related to mass is that the Higgs mass seems to be much lower than expected [19–21]. Taking loop corrections to the Higgs mass into account, we would expect it to be much larger. These loop corrections are large because they are dominated by the contribution of the top-quark loop; this is because the loop corrections are proportional to the mass squared of the particles inside the loop. As it stands, the bare mass to the Higgs must be very large so that it is comparable to the loop corrections. This means that the corrections and the bare mass must be fine tuned so that they cancel out just right to get the observed 30 mass. Although two large numbers could cancel just right to become a small number, it is unnatural in the minds of many physicists. The issue that I will try to address in this dissertation is mainly the Higgs naturalness problem. 2.2.2 Potential Solutions Although there are problems with the SM that aren’t fully understood, theorists have pro- posed many solutions that potentially address them. Some of these theories are currently unreachable in experiment, such as String theory, but most are not. I will list just a few of the ways to solve the issues covered in sec. 2.2.1. • Supersymmetry (SUSY): One way to solve the hierarchy and naturalness problems, while also creating a dark matter candidate, is if there was sort of a mirror to the SM where all of the bosons became fermions and vice versa. The masses of these supersym- metric particles should be on the order of their SM partners. Since fermions contribute positively to the Higgs mass loop corrections while bosons contribute negatively, this could balance the corrections out so that the overall correction is small. In this case the observed Higgs mass and the bare Higgs mass would be similar in scale. The dark matter candidate would be the lightest supersymmetric particle. • Quantum Gravity: To add gravity to the SM, physicists have tried describing it in the quantum formalism. This often means quantizing the gravitational field which leads to a mediator called the graviton. • Hidden Sectors: Similar to SUSY is the idea of hidden sectors of matter which don’t interact with most of the SM particles. Those that do interact with the SM particles 31 do so very weakly or have masses so large that we would not have been able to create them before. How the particles within the hidden sector interact with each other could vary. These hidden particles would also interact with the Higgs and therefore change the mass loop corrections, possibly addressing the naturalness problem. • Extended Gauge Theories: Another way to potentially solve the naturalness prob- lem is to embed the SM group in another group. Depending on how this is done, new gauge invariances and symmetries can be introduced which then manifest as new particles that can interact with the Higgs boson. When it comes to the naturalness problem, some solutions introduce one to three heavy vector bosons. A common decay channel of these heavy vector bosons is to a weak vector boson and a Higgs boson, although it is not limited to this alone. This dissertation will attempt to address the Higgs naturalness problem by searching for heavy vector resonances decaying to a weak vector boson and a Higgs. These heavy vector resonances are commonly referred to as the W(cid:48)± and Z(cid:48), or collectively as the V (cid:48). 2.2.3 A Heavy Vector Triplet There are many different solutions to the naturalness problem that propose one or more heavy vector bosons decaying to a weak vector boson and a Higgs boson [22, 23]. Each of these models has different parameters and many have free parameters that an experiment is not sensitive to. Doing a search for each of these models over their parameter space would be extremely time consuming. To make this search more generalizable, a group of theorists put together a simplified model that can be searched for by experimentalists and then compared to the more complex models by theorists. This simplified model is known as the Heavy 32 Vector Triplet (HVT) [24], it is called this because it introduces three (i.e., a triplet) heavy resonances with a spin of 1 (i.e., vector spin). The nomenclature for the HVT paper is different than what is used in the rest of this dissertation, but it is convenient for the purposes of explaining the HVT. While explaining the HVT we will use the paper’s nomenclature, then we will switch to the standard convention used in the field. In this nomenclature, W represents the SM weak bosons, V represents the Heavy Vector Triplet, J contains the fermion fields, H represents the Goldstone bosons (i.e., the Higgs boson and the longitudinal part of the weak vector bosons), and D is a more complex version of the covariant derivative. The premise of it is to add to the SM Lagrangian terms for these new heavy vector bosons that only depend on the coupling parameters of the heavy vector bosons to the SM particles. This removes any excess free parameters that the experiment has no way of searching for and makes the coupling parameters very general. Experimentalists can set limits on these coupling parameters and then theorists can translate them into the language of their own models. Using V to represent the heavy vector fields and W to represent the weak vector fields, the terms added to the SM Lagrangian are, LV = −1 4 m2 V 2 g2 gV V a µ V µa cF V a µ J µa F D[µV a ν]D[µV ν]a + µ H†τ a ↔ D µ H + + igV cH V a + gV 2 cV V V V a µ V νbD[µV ν]c + g2 V cV V HH V a µ V µaH†H − g 2 cV V W abcW µνaV b µ V c ν , (2.32) 33 where the fields combine to create two charged and one neutral vector boson, V ± µ = µ ∓ iV 2 V 1 µ√ 2 , µ = V 3 V 0 µ . (2.33) Despite this being a “simplification” of other models, this is still fairly complex and I will skip details that aren’t particularly important for understanding this dissertation. The first line of Eq. 2.32 contains the kinetic energy∗† and mass terms of the V bosons. It should be noted that mV is not the mass of the heavy vector bosons since the V a µ fields mix with the W a µ fields after electroweak symmetry breaking. The third line of Eq. 2.32 contains a triple self interaction term, a two-V two-Higgs interaction term, and a two-V one-W interaction term. These are not important for this dissertation, so I will not cover them further. The second line of Eq. 2.32 contains Higgs and fermion interaction terms. The fermion fields are in the second term of this line, cF V a µ J µa F , g2 gV J µa F = (cid:88) ¯fLγµτ afL. (2.34) f This term has three parameters, g–the coupling strength of the SM weak interactions, gV –the coupling strength of this new interaction, and cF –the dimensionless coupling of the HVT to fermions. Here the fermions are all given the same cF , but they could have individual couplings as well. ∗D[µV a †This kinetic energy term also contains trilinear and quadrilinear interaction terms with the W a ν − Dν V a µ , DµV a = DµV a ν = ∂µV a ν + gabcW b µV c ν ν] µ fields, but these are not important for this dissertation. 34 What is most important for this dissertation is the first term of line two, the bosonic interaction term, iH†τ a µ ↔ D µ ↔ D µ H†τ a igV cH V a H = iH†τ aDµH − iDµH†τ aH. H, (2.35) This term has two parameters, gV and cH , where cH is the dimensionless coupling of the V bosons to the Higgs/Goldstone bosons. Since the Goldstone bosons are the longitudinal part of the weak vector bosons, this term leads to vertices involving the Higgs boson as well as the weak vector bosons. Specifically, this means that the HVT can decay to W H, V H, W W , and W Z. At this point we will abandon the nomenclature of the paper for the standard nomencla- ture used by most high energy physicists. For now on V is a reference to the weak vector bosons (W± or Z) and V (cid:48) represents the HVT, which can also be broken into W(cid:48)± and Z(cid:48). This dissertation primarily covers a W(cid:48)± decaying to a Higgs boson and a W± boson. Specifically, we will search for W(cid:48) → W H → lνb¯b, where it is the Higgs that decays hadroni- cally and the W that decays leptonically. The decay of a Z(cid:48) to a Higgs boson and a Z boson will be briefly included in the extensions chapter. Two benchmark models will be considered, HVT Models A and B. HVT Model A represents weakly coupled extensions to the SM gauge group [22]. HVT Model B repre- sents strongly coupled scenarios of electroweak-symmetry-breaking, which are also known as Composite Higgs Models [23]. In both cases the coupling constants are fixed and the free parameters are gV and mV (cid:48). The coupling constants are set so that the branching fractions for fermions and bosons are comparable for Model A, while the fermion branching fractions are suppressed for Model B. Previous searches of this channel, Fig. 2.7, have been performed at center-of-mass energies 35 Figure 2.7: On the left are the 8 TeV limits with 20.3 f b−1 and on the right are the 13 TeV limits with 3.2 f b−1. On the left and right, HVT Model A is the red line, while on the right HVT Model B is the magenta line. There is no HVT Model B on the left, instead a Minimal Walking Technicolor model (blue dashed line) was used. of 8 TeV and 13 TeV with 20.3 and 3.2 fb−1 of integrated luminosity, respectively. The 8 TeV results [25] only cover HVT Model A and exclude W(cid:48) masses up to 1470 GeV, while the 13 TeV results [26] exclude Model A up to mW(cid:48) = 1750 GeV and Model B up to mW(cid:48) = 2220 GeV. The goal of this dissertation is to extend these searches. 36 [GeV]±R1m40060080010001200140016001800 bb) [pb]→ (H × WH) → ±1 BR (R× σ(cid:173)310(cid:173)210(cid:173)110110ATLAS(cid:173)1L dt = 20.3 fb∫ = 8 TeVs=2g~ WH → ±2,R±1MWT R=1v HVT Benchmark model A g±V’Observed 95% Upper LimitExpected 95% Upper Limit 1 Sigma Uncertainty± 2 Sigma Uncertainty± [GeV]W’m100015002000250030003500400045005000) [pb]c,cb b→ BR(H × WH) → W’ →(pp σ2−101−101 ATLAS (cid:173)1 = 13 TeV, 3.2 fbsObserved limitExpected limit1 s.d.±Expected 2 s.d.±Expected =1VHVT Model A, g=3VHVT Model B, g Chapter 3 Experimental Apparatus The visible matter in our universe is mostly made of up quarks, down quarks, and electrons with the gluon and photon mediating the forces that hold the various types of matter to- gether. The rest of the SM particles, as well as most hypothetical particles, are short-lived. This means they will eventually decay to the light particles of ordinary matter. In order to study heavy particles we must produce them in a high energy interaction, whether that be collisions from man-made colliders, in the atmosphere, or from some astronomical object such as a pulsar. Particles with larger mass need a higher collision energy if they are to be produced. The particle search described in this dissertation corresponds to a very large mass, so a very high energy collision is needed. In order to create a high energy collision, we must first accelerate a lot of particles to energies about fourteen orders of magnitude higher than room temperature. How do we accelerate particles to such energies? There are natural particle accelerators in the universe, but none of them are very useful for what we study. These natural particle accelerators either have a high rate of producing particles or produce high energy particles, but not both†, which is what we need. So what high energy physicists do is design man-made particle accelerators that produce †This is not quite true, in the case of astronomical objects like pulsars there are many high energy particles being produced, however, only a small fraction reach earth. 37 high energy beams of particles with a high flux and then collide them. The most power- ful man-made particle accelerator available today is the Large Hadron Collider (LHC). It has four experiments placed around it; ATLAS, CMS, ALICE, and LHCb. Each of these experiments is a particle detector that was built for specific purposes. ATLAS and CMS are both general purpose detectors made to study many different types of reactions, ALICE was designed to study heavy-ion collisions, and LHCb was made specifically for the pur- pose of studying b-mesons. This dissertation covers an analysis that looked at collision data collected by the ATLAS detector. 3.1 The Large Hadron Collider The LHC, depicted in Fig. 3.1, is the most powerful man-made particle accelerator and collider available. It is the latest in a chain of accelerators used to accelerate particles up to the highest energy technologically possible within an achievable budget. The LHC is a 27 km ring made of two pipes for separate clockwise and counter-clockwise moving beams of protons. There are 8 areas cut out around the ring where radio frequency cavities used to accelerate particles are placed. Elsewhere, the pipes are surrounded by superconducting magnets used to bend the trajectory of the proton beams. The LHC was designed to collide bunches of protons 40 million times per second at an energy of 14 TeV with a luminosity∗ of 1034 cm−2 s−1. Each bunch has up to 1011 protons in it and these bunches collide every 25 nanoseconds. Each collision of bunches yields many proton collisions, which was 25 for the data used in this work. The LHC was built between 1998 and 2008† at the France-Switzerland border near ∗Luminosity is the ratio of the collision rate and the effective cross section of the proton beams. †I was in 2nd grade when this started and a junior or senior in high school when it ended. 38 Figure 3.1: The LHC ring can be seen on the top [9] and a cross section of it can be seen on the bottom [10]. 39 Geneva. In 2009 it was turned on with an energy of 7 TeV. The period it ran at 7 and 8 TeV was known as Run-1, which lasted from 2009 to 2013. Then there was a shutdown between 2013 and 2015, called Long Shutdown 1 (LS1), used to upgrade the LHC and de- tectors. The LHC started Run-2 in 2015 and ran until the end 2018, at which point Long Shutdown 2 (LS2) started. LS2 is being used to upgrade the LHC and detectors further and is planned to last until early 2021. After LS2, it is planned that the LHC will run at an energy of 14 TeV. There are plans for it to go to even higher energies with higher luminosities in the far future. A brief description of the process of accelerating particles up to the needed energy is depicted in Fig 3.2. First, an electric field is applied to a hydrogen gas so that electrons are stripped from the hydrogen atoms and only protons are left. These protons are put into Linac 2, which accelerates them to an energy of 50 MeV. The beam of protons that forms is then injected into the Proton Synchrotron Booster, which accelerates them to an energy of 1.4 GeV and also causes the protons to gather up into bunches. Then the beam is injected into the Proton Synchrotron, which accelerates it to an energy of 25 GeV. The beam is then injected into the Super Proton Synchrotron, which accelerates it to an energy of 450 GeV. Finally, the beam is split and injected into the two LHC beam pipes, at which point the two beams are each accelerated to an energy of 6.5 TeV. 40 Figure 3.2: This diagram depicts the accelerator chain used to accelerate protons. The energy that the protons reach for each accelerator is also show, as well as the four detectors along the LHC. 3.2 The ATLAS Detector The ATLAS detector [13], depicted in Fig. 3.3, was designed and made by the collaboration of the same name. It is a general purpose detector made to measure many different physics signals and is made up of several specialized sub-detectors to help do this. Construction of ATLAS finished in 2008, but it has had several of its subsystems upgraded since. Because this dissertation covers data taken in Run-2 of the LHC, I will describe the Run-2 configuration of ATLAS. 3.2.1 Particle Interactions with Matter When a particle traverses a material it will often interact with the material’s atomic structure. The ATLAS detector makes use of two types of interactions: electromagnetic interactions 41 Figure 3.3: The ATLAS detector [11]. It is made of four main subsystems; the inner detector, the calorimetry, the muon spectrometer, and the magnet system. and strong interactions. Electromagnetic interactions in the detector primarily take the form of bremsstrahlung radiation at higher energies, which is the radiation of photons as an electrically charged particle is slowed down by an atomic nucleus, and ionization at lower energies, which is the loss of atomic electrons. Strong interactions take the form of hadronic particles undergoing an inelastic nuclear collision, which produce showers of more hadronic particles. Both types of interactions cause a particle to lose its energy and eventually be stopped in the detector. A particle’s properties determine which ways it can interact with a material. Electrically charged particles with no color or colored components will only interact electromagneti- cally. Particles with color or colored components and zero net electric charge will only interact strongly. Particles with color and electric charge will interact strongly and elec- tromagnetically. Electrically neutral particles with no color will not interact strongly or 42 electromagnetically. How often a particle undergoes one of these interactions is dependent on the properties of the particle and matter involved. The mean distance a particle travels before going through one of these interactions are called the radiation and nuclear interaction lengths, for electromagnetic and strong interactions respectively. The sub-detectors of ATLAS are designed with materials that elicit specific types of interactions from the particles they are meant to detect, as well as a size that gives them enough radiation or interaction lengths to fully stop said particles and absorb their energy. The tracking elements of the detector are designed to create as few interactions as possible so that they barely reduce a particles energy or cause scattering. The electromagnetic and hadronic calorimeters are made to maximize these interactions in order to absorb the entirety of their respective particles energy. In both cases, the interactions create some form of radiation that is turned into an electronic signature that is output by the detector. These signatures can then be used to reconstruct a sort of digital image of the event. 3.2.2 Detector Coordinates Discussing the components of the ATLAS detector requires understanding the coordinate system used to describe it. The detector has a cylindrical geometry, but the kinematics of collisions makes them more spherical. As a result we use a coordinate system that borrows from both of these, shown in Fig. 3.4, but has additions for physics purposes. For the purposes of defining this coordinates system, we will start with Cartesian coordinates. The x-axis of the detector points towards the center of the LHC ring, while the y-axis points up towards the sky. This leaves the z-axis to point along the length of the detector, in the direction of the counter-clockwise beam (looking down from above). The azimuthal 43 angle, φ, is then in the x-y plane and starts from the x-axis rotating towards the y-axis. The radius of any given point is the transverse distance from the z-axis to said point. Since the detector is symmetric in z and φ, the origin is placed at its center. The polar angle, θ, is then the angle between the z-axis and any given vector that starts at the origin, which is zero when the vector is parallel to the z-axis and increases to π when it is anti-parallel. A coordinate useful for physics purposes called the pseudo-rapidity, η, is frequently used. The pseudo-rapidity and its approximation are defined as, (cid:16)|p| + pz |p| − pz (cid:17) , η = 1 2 ln (cid:17)(cid:105) , (3.1) (cid:104) (cid:16) θ 2 η ≈ −ln tan note that p is the three-momentum of a particle. The pseudo-rapidity starts at zero for θ = π/2 and goes to ±∞ as it approaches the ± z-axes, i.e., the beam. Two useful aspects of pseudo-rapidity are that particle production is uniform as a function of it and it is Lorentz invariant under boosts along the z-axis. For these reasons, pseudo-rapidity will usually be used instead of the polar angle. 3.2.3 Subsystems There are four main ATLAS subsystems; the inner detector, the calorimeters, the muon spectrometer, and the magnet system. The inner detector is used for tracking the paths of charged particles, the calorimeters are used for measuring the energies of particles, the muon spectrometer is used for tracking muons, and the magnet system is used to bend the trajectories of charged particles so that their momenta and charge can be calculated using information from the inner detector and muon spectrometer. Each of these is made of smaller components that contribute to the tasks listed above. 44 Figure 3.4: The ATLAS coordinate system is on the left, while the transformation from θ to pseudo-rapidity is demonstrated on the right. In addition to these four main subsystems, there are three additional auxiliary/very- forward detectors: LUCID (LUminosity measurement using Cerenkov Integrating Detector), ALFA (Absolute Luminosity for ATLAS), and the ZDC (Zero-Degree Calorimeter). LUCID and ALFA are used for measuring luminosity while the ZDC is used for heavy-ion collisions. These are not directly used in the data analyzed in this dissertation, so I will not cover them further. Inner Detector The Inner Detector is used for tracking charged particle locations and can be used in con- junction with the magnet system to measure a particle’s momenta. It’s made of three subsystems (see Fig. 3.5): the Silicon Pixel Detector, the Semi-Conductor Tracker (SCT), and the Transition Radiation Tracker (TRT). Together these subsystems provide precision measurements of charged particle trajectories. Figure 3.6 shows the radiation and nuclear interaction lengths for each subsystem as a function of pseudo-rapidity. 45 Figure 3.5: The Inner Detector [12] is made of four systems, of which three are shown; the pixel detector, the semi-conductor tracker, and the transition radiation tracker. The primary function of the Pixel Detector is taking measurements of collision vertices, i.e. where the collisions occurred. It was originally made of three barrel layers and three disk layers on each end, with a fourth layer added in Long Shutdown 1. The three barrel layers have radii of 50.5 mm, 88.5 mm, and 122.5 mm. These layers have 22, 38, and 52 staves respectively∗, where each stave consists of 13 pixel modules. Each pixel module has a layer of silicon on top of 16 front-end (FE) chips and one Module Control Chip (MCC). One FE chip contains 160 rows by 18 columns of pixel cells. On each side of the pixel detector three disk layers are placed at |z| of 495 mm, 580 mm, and 650 mm. Each of the six disks has 8 sectors, with 6 pixel modules per sector. In Long Shutdown 1 a fourth layer, the Insertible B-layer (IBL) [27], with radius 33.4 mm was added. The IBL has 960 pixel sensors, with 50 × 250 micron pixels per sensor. ∗These staves are tilted 20 degrees so they can overlap, this way there are no gaps in φ 46 Figure 3.6: On the left is a histogram showing the radiation lengths of the various Inner Detector components as a function of pseudo-rapidity. On the right is the same, but for nuclear interaction lengths [13]. When a charged particle moves through the modules of the Pixel Detector, it ionizes the silicon that is above the pixels. The electrons from the ionization are then read out as a current by the closest micron pixel below them. The Semi-Conductor Tracker is mainly used for measuring particle momenta. It is made of four barrel layers and nine forward disk layers on each side. The four barrel layers have radii 284 mm, 355 mm, 427 mm, and 498 mm respectively, and each layer is 1530 mm long. Each layer has 384, 480, 576, and 672 modules respectively∗. Each module is made up of four 80 µm pitch micro-strip sensors, plus thermal, mechanical, and electronic structures. When a charged particle moves through the modules they create hits the same way they do in the Pixel Detector. The nine disks in each end-cap are placed at |z| of 853.8 mm, 934.0 mm, 1091.5 mm, 1299.9 mm, 1399.7 mm, 1771.4 mm, 2115.2 mm, 2505.0 mm, and 2720.2 mm. Each of these disks is split into three distinct regions; inner, middle, and outer. All nine disks have 52 modules in their outer region. Disks 1-8 have 40 modules in their middle region, while disk 9 has no middle region. Disks 2-6 have 40 modules in their inner region, while disks 1, 7, 8, ∗The modules of each layer have tilt angles of 11.00, 11.00, 11.25, and 11.25 degrees respectively. 47 |h|00.511.522.533.544.55)0Radiation length (X00.511.522.5|h|00.511.522.533.544.55)0Radiation length (X00.511.522.5ServicesTRTSCTPixelBeam-pipe|h|00.511.522.533.544.55)lInteraction length (00.10.20.30.40.50.60.7|h|00.511.522.533.544.55)lInteraction length (00.10.20.30.40.50.60.7ServicesTRTSCTPixelBeam-pipe and 9 have no inner region. Each module in the middle and outer regions is made of four micro-strip sensors, plus thermal, mechanical, and electronic structures. The modules in the inner regions are made of only two micro-strip sensors. The Transition Radiation Tracker’s primary function is easing the pattern recognition process. It is made up of three barrel layers, and two wheels on each side. Each of the barrel layers have a different thickness described by an inner and outer radius. The inner and outer radii of each barrel layer are 554–694 mm, 697–860 mm, and 836–1066 mm. Each of these layers has 32 modules, where each module is in a carbon-fiber laminate shell which contains an array of straws embedded in a matrix of 19 µm-diameter polypropylene optical fibers. There are 329 straws per module in the inner ring, 520 straws per module in the middle ring, and 793 straws per module in the outer ring. The wheels on each side also have differing longitudinal thicknesses, they cover |z| of 848–1705 mm and 1740–2710 mm and have inner and outer radii of 644–1004 mm. The inner wheel has 12 modules and the outer wheel has 8 modules. Each of these modules has 8 layers of straws in the z-direction. Each straw layer has 768 radially-oriented straws. All of the straws are polyimide drift tubes filled with a gaseous mixture of xenon, carbon- dioxide, and oxygen. A copper wire anode is at the center of each straw. A charged particle moving through the TRT emits Cherenkov light in the matrix of fibers; these photons then pass into the straw and ionize its gas. The electrons from the ionization create a current picked up by the anode wire. 48 Figure 3.7: The calorimetry [14] consists of the Liquid Argon Calorimeter and the Tile Calorimeter. Each of these is divided into smaller parts. Calorimetry The calorimetry, Fig. 3.7, is an array of sampling detectors∗ used for measuring the energy of particles moving through them. There are two main calorimeters; the Liquid Argon Calorimeter and the Tile Calorimeter. Both have full φ-symmetry and coverage around the beam axis. Together these measure the energy of electrically charged particles and hadronic matter. Figure 3.8 shows the radiation and nuclear interaction lengths for each subsystem as a function of pseudo-rapidity. The Liquid Argon (LAr) Calorimeter is split between measuring the energy of charged particles and hadronic matter. It is made up of the LAr electromagnetic barrel, ∗A sampling calorimeter is made up of two distinct materials, a material that produces particle showers and a material that absorbs/measures the energy of those showers. Sampling calorimeters are usually made up of alternating layers of these materials. 49 Figure 3.8: On the top left and right are the radiation lengths of the electromagnetic barrel and endcap calorimeters, respectively, as a function of pseudo-rapidity. On the bottom are the nuclear interaction lengths for each component of the calorimetry as a function of pseudo- rapidity. The tan on the bottom plot is the nuclear interaction length of everything before the calorimeters and light blue is everything after. [13] 50 Pseudorapidity00.20.40.60.811.21.40X0510152025303540Pseudorapidity00.20.40.60.811.21.40X0510152025303540Layer 3Layer 2Layer 1Before accordionPseudorapidity1.61.822.22.42.62.833.20X051015202530354045Pseudorapidity1.61.822.22.42.62.833.20X051015202530354045Layer 3Layer 2Layer 1Before accordion00.511.522.533.544.5502468101214161820Pseudorapidity00.511.522.533.544.55Interaction lengths02468101214161820EM caloTile1Tile2Tile3HEC0HEC1HEC2HEC3FCal1FCal2FCal3 the LAr electromagnetic end-cap (EMEC), the LAr hadronic end-cap (HEC), and the LAr forward (FCal) calorimeters. The barrel and EMEC are mainly for measuring the energy of charged particles, the HEC is mainly for measuring the energy of hadronic matter, while the FCal is made to do both. The LAr electromagnetic barrel is made of two half barrels, one for z > 0, the other for z < 0, together they cover a pseudo-rapidity of |η| < 1.475. Each half of the barrel is made of 1,024 accordion shaped absorbers. These absorbers are split among three layers; a strip cell layer and two square cell layers. There is a small pre-sampler layer just before the readout electronics of the strip cell layer. The EMEC is made of two wheels, one on each side of the detector, that cover 1.375 < |η| < 3.2. Each of these wheels is made of two co-axial wheels, for which the boundary between inner and outer is at |η| ≈ 2.5. The outer wheel contains 768 absorbers while the inner wheel contains 256 absorbers. There is a pre-sampler in front of the EMEC covering 1.5 < |η| < 1.8. The barrel and EMEC were given an accordion geometry to give full φ coverage with no gaps. The absorbers of the LAr barrel and EMEC are made of alternating layers of lead sheets filled with LAr. The HEC is a copper and LAr sampling calorimeter that covers the range 1.5 < |η| < 3.2. It’s made of two wheels, one on each side of the detector. Each wheel has two longitudinal sections, with each section having 32 wedge-shaped modules. The modules of the front wheels are made of 24 copper plates while the modules of the rear wheels are made of 16 copper plates. There are gaps between each of these plates filled with LAr. The FCal is made of two wheels, one on each side of the detector, that cover 3.1 < |η| < 4.9. Each wheel is made of three modules: one electromagnetic module (FCal1) and two 51 hadronic modules (FCal2 and FCal3). FCal1 uses copper as its absorber while FCal2 and FCal3 use tungsten. FCal1 is made of copper plates stacked one behind the other with 12,260 holes drilled through them. Electrodes made of a co-axial copper rod and copper tube are embedded into these holes. FCal2 and FCal3 are each made of two copper end-plates separated by electrodes made of a co-axial tungsten rod and copper tube. The empty space in FCal2 and FCal3 is filled with tungsten slugs. A charged particle passing through the LAr calorimeter will shower as it passes through the absorbers. Particles in the showers will then ionize the LAr and the electrons from the ionization create a current in an anode immersed in the LAr. The Tile Calorimeter’s (TileCal) primary function is measuring the energy of hadronic matter. It is a sampling calorimeter that uses steel as an absorber and scintillator as an ac- tive medium. It is split into the central barrel and two extended barrels, together these cover |η| < 1.7. Each barrel is 64 wedges made of steel plates and scintillating tiles. The tiles and steel plates are oriented radially and normal to the beam pipe. Each wedge is made of layers that alternate the stacking of the steel plates and tiles. Each tile has an optical fiber attached to it that leads to a photomultiplier tube (PMT). A hadron passing through the TileCal starts to shower when it passes through a steel plate. Particles from this shower emit photons in the scintillating tile. These photons are then passed through the optical fiber to the PMTs, which converts the photons to an electrical current. 52 Muon Spectrometer The Muon Spectrometer, Fig. 3.9, is specifically designed to detect and track muons, which often won’t be stopped by the calorimeters like electrons or photons will. It is the outer shell of the detector and is made of Monitored-Drift Tubes (MDT), Cathode-Strip Chambers (CSC), and Trigger Chambers. Together the MDTs and CSCs track muons and help measure their momenta, while the Trigger Chambers create a fast trigger system on muons. Figure 3.9: The muon spectrometer [15] is made of four parts; the MDTs, the CSCs, the RPCs, and the TGCs. The magnet system is made of the barrel toroids and end-cap toroids. The Monitored-Drift Tubes are for measuring muon momenta in the barrel and end- cap regions, |η| < 2.7. In the barrel, they are arranged in three concentric cylindrical shells at radii of about 5 m, 7.5 m, and 10 m. In each end-cap, they are arranged in four wheels located at |z| ≈ 7.4 m, 10.8 m, 14 m, and 21.5 m. Due to high levels of radiation near the 53 beam, 2 < |η| < 2.7, the inner wheel only covers |η| < 2. Across all three shells and eight wheels there are 1,088 chambers. There are 18 primary types of chambers, but there are also some specialized chambers built to minimize acceptance losses in the regions around the magnet coils and support structures. Each chamber has three to eight layers of MDTs. Each MDT is a 29.97 mm diameter, pressurized tube filled with Ar/CO2 and a tungsten-rhenium wire through the center. A muon passing through a drift tube will ionize the gas inside, the resulting ions will be read as a current by the wire at the center. In the very forward region, 2 < |η| < 2.7, the MDTs of the inner wheel are replaced with Cathode-Strip Chambers. This is because the counting rate at |η| > 2 near the first layer exceeds 150 Hz/cm2, which is the limit for safe operation of MDTs. There is one wheel of CSCs on each side of the detector. Each wheel is made of an inner disk and outer disk for full φ coverage, with eight chambers per disk. The inner disk is made of small chambers while the outer disk is made of large chambers. These chambers are trapezoidal prisms filled with Ar/CO2 gas with anode wires placed periodically through them. A muon passing through will ionize the gas, which will be read as a current by the anode wires. While the MDTs offer precision tracking of muons, they are not fast enough for the pur- poses of triggering. Therefore faster Trigger Chambers are used specifically for triggering on muons. There are two types of Trigger Chambers; Resistive-plate Chambers (RPC) and Thin-gap Chambers (TGC). The Resistive-Plate Chambers are used in the barrel, |η| < 1.05. There are three concentric cylindrical layers, the two inner layers envelop the second shell of MDTs, while the third layer is just inside the third shell of MDTs. Each layer is made of alternating small and large chambers for full φ coverage. An individual RPC is a gaseous, parallel electrode- 54 plate detector. A muon moving through the gas ionizes it, which is then read as a current by the plates. The Thin-Gap Chambers are used for fast triggering in the end-caps, 1.05 < |η| < 2.4. There are four wheels on either side of the detector; one just inside the inner wheel of MDTs, two enveloping the third wheel of MDTs, and one just behind those. These are multi-wire proportional chambers with a gas mixture of CO2 and n-C5H12 (n-pentane). Magnets The ATLAS Magnet system, which can be seen in Fig. 3.9, is made of three parts; the Central Solenoid, the Barrel Toroid, and the End-cap Toroids. Together these cover a cylindrical volume that is 22 m in diameter and 26 m in length, and provide a magnetic field that covers a volume of about 12,000 m3. The magnets are used for bending the trajectory of charged particles which allows us to find the charge of a given set of tracks (see sec. 3.3.1.1) and calculate their momenta. The Central Solenoid envelops the tracking system and provides a 2 T axial field. It is a single layer coil made from an Al-stabilized NbTi conductor wound about an aluminum alloy support cylinder. Its flux is returned by the steel of the calorimetry and its support structure. The central solenoid has a length of 5.8 m and inner and outer diameters of 2.46 m and 2.56 m. The Barrel Toroid is embedded in the barrel of the Muon Spectrometer and provides a 0.5 T toroidal field. It consists of eight coils encased in individual stainless-steel vacuum vessels. The coils are made of a pure Al-stabilized Nb/Ti/Cu conductor wound into a disk. The barrel toroid has an overall length of 25.3 m and inner and outer diameters of 9.4 m and 20.1 m. 55 The End-cap Toroids are embedded in the end-caps of the Muon Spectrometer and provide a 1 T toroidal field. Each end-cap is made of a single mass built from eight flat, square coil units and eight keystone wedges. The coils are made from the same material as the coils in the barrel and shaped the same way. 3.2.4 Data Acquisition Collisions in the detector happen every 25 ns (or a frequency of 40 MHz). Writing event data to disk cannot be done anywhere near that quickly, which means that not every event can be saved. This requires reducing the number of events written to disk to some small fraction of the total rate. The event rate needs to be reduced from about 40 MHz to about 200 Hz in order to satisfy our needs. There is a three part triggering system [28] to do this; the Level-1 (L1) trigger, Level-2 (L2) trigger, and event filter. Together the L2 trigger and event filter form the High-Level trigger (HLT). The L1 trigger uses low-level information from several sub-detectors to reduce the event rate from 40 MHz to 75 kHz. The Data Acquisition system (DAQ) handles the data coming out of the detector at the L1 trigger rate and passes it to the L2 trigger. The L2 trigger then studies information tagged as interesting by the L1 trigger at a more detailed level and reduces the event rate to below 3.5 kHz. The DAQ then passes that information to the event filter which uses offline analysis procedures on full-granularity events to reduce the event rate to about 200 Hz. At this point the event data is written to disk. 56 Level 1 Trigger The L1 trigger looks for signatures of high transverse-momentum objects, as well as events with large missing transverse energy (MET or Emiss T ) and large total transverse energy. It is split into three parts; the L1 Calorimeter trigger (L1Calo), the L1 muon trigger, and the Central Trigger Processor (CTP). The L1Calo uses information gathered from the calorimeters to identify high transverse- energy objects such as electrons, photons, jets, and τ -leptons decaying to hadrons. It also searches for events with high Emiss T or high total transverse energy. The L1 muon trigger uses information gathered from the muon trigger chambers to search for patterns of hits consistent with high transverse-momentum muons originating from the collision point. The CTP uses information from the L1Calo and L1 muon trigger to make the final decision of whether an event is selected or not. This decision, as well as the event information, is then read out by the DAQ. The High-Level Trigger The HLT and DAQ consist of detector readout, the L2 trigger, event-building, the event filter, configuration, control, and monitoring. After the L1 trigger accepts an event, it tags regions of interest (ROI) in the detector. Event information is then read out of the detector and sent to the L2 trigger. The L2 trigger uses the ROI to process information from the entire detector in that region, as opposed to just one component of the detector. If an event is accepted its data is built into a single data structure, which is then sent to the event filter. The event filter takes this data structure and applies standard event reconstruction 57 and analysis applications to it. The event filter will decide to accept or reject this event and appends important information used to make the decision to the data structure. This final data structure is then saved to disk. 3.3 Particle Identification There are many types of particles that traverse the detector and leave signatures in the various sub-detectors. However these sub-detector level signatures do not always identify what kind of object left them when considered on their own. Instead it is necessary to use information from across the detector to identify these particles and objects. In addition there are detector effects to take into account. For instance, particles in- teracting with the detector will not always fully deposit their energy inside it. Also the instrumentation of the detector is not uniform. These inaccuracies mean that the data we receive from the detector needs to be calibrated. The magnitude of these inaccuracies varies for different particles, so calibrations are derived for each type of object. 3.3.1 Object Reconstruction Different SM particles and their derivatives can interact with the individual sub-detectors in very similar ways, this makes it hard to distinguish them with information from just one sub-detector. However, if we consider the detector as a whole then these different objects create fairly unique signatures, as demonstrated in Fig. 3.10. Some examples of the signatures different particles leave in the detector and its subsys- tems are shown in Fig. 3.10. Reconstructing these particles is a multi-step process since 58 there are multiple sub-detectors. It often begins by building up objects in each sub-detector and then using different combinations of those to create more complex objects. 3.3.1.1 Tracks When charged particles move through the Inner Detector they create hits in the different layers. Two algorithms are then used to chain these hits together to create tracks. The first algorithm starts at the center of the Inner Detector and travels outward to create tracks. The second algorithm supplements the first by taking leftover hits and creating tracks with them by starting at the outside of the Inner Detector and traveling inward. These tracks represent the trajectory of a single particle moving through the Inner De- tector. Because of the magnetic field produced by the solenoid and toroid magnets, charged particles will have a curved trajectory. This means that tracks will also be curved, which can be used to calculate a particle’s charge and momentum. 3.3.1.2 Electrons & Photons Electrons pass through the Inner Detector creating tracks and then deposit all of their energy in the electromagnetic calorimetry. The first step of reconstructing electrons is creating representations of their energy, which we call electromagnetic clusters. To create these, the radial layers of the electromagnetic calorimetry are summed together to create “towers” of energy in η-φ space. A sliding window algorithm [29] is then used on these towers to find areas where the energy is at a maximum. Some fraction of the towers inside the window then have their energy summed to create an electromagnetic cluster. After the electromagnetic clusters are created, they are matched to the closest tracks within η-φ space to create electron candidates. The momentum of the track is then 59 Figure 3.10: A representation of how different particles interact with the different sub- detectors [16]. It shows a fragment of a longitudinal slice of the detector. rebuilt taking the energy of the cluster into account. The electron candidates are then split into two identifications: loose and signal [30]. The loose identification accepts more candidates but has a higher fake rate, while the signal identification accepts less candidates but has a lower fake rate. Photons behave similarly to electrons in the electromagnetic calorimetry, however they do not create hits in the Inner Detector. Therefore reconstructing a photon is almost the same as reconstructing an electron, the major difference is that electromagnetic clusters are not matched to tracks from the Inner Detector. 60 3.3.1.3 Muons Due to the muon’s relatively large mass and lack of colored components, it is the only charged particle that pierces through the entire detector uninhibited. This means that they leave hits in both the Inner Detector and Muon Spectrometer, while possibly depositing a small fraction of their energy in the calorimetry. Therefore a muon in the detector looks like tracks in both the Inner Detector and Muon Spectrometer and maybe some energy in the calorimetry. This combination of signatures is almost unique to the muon, the exception being very high energy particles that manage to get all the way through the detector. There are four main muon reconstruction algorithms [31] used at ATLAS which use different combinations of the Inner Detector, Muon Spectrometer, and calorimetry. Which of these algorithms is used changes as a function of |η| due to the geometry of the detector. 3.3.1.4 Jets If a q ¯q pair coming out of a collision separates quickly enough, color confinement dictates that it is energetically favorable to create new quarks so that hadrons can form. However, it’s often the case that the momenta of the initial quarks is so large that even the quarks created from this process will eventually get too far away. This leads to a repeating/cascading process of q ¯q creation, called hadronization. In the end, there are showers of hadronic particles where there was once a q ¯q pair coming from the collision. It’s also possible for these showers to yield photons, electrons, and muons. These showers of particles are called Jets and they come in many forms. Topological Clusters The first step of reconstructing jets is to reconstruct the energy of the showers described above. Because some hadrons have non-zero electric charge and the 61 showers contain some photons and electrons, energy from the showers can be deposited in the electromagnetic and/or hadronic calorimetry. Therefore energy in all of the calorimetry is used to construct representations of a jet’s energy, which are called topological clusters [32]. Unlike the electromagnetic clusters described in Sec. 3.3.1.2, which are 2D objects of fixed size, topological clusters are 3D objects with variable size. They are built up by clustering neighboring calorimeter cells whose energy exceeds some threshold determined by the cell’s expected noise [29]. Jet Clustering After the topological clusters have been made they are clustered into jets. There are many clustering algorithms available, but the three most commonly used at ATLAS are the Kt, Anti-Kt [33], and Cambridge-Aachen algorithms. These are essentially three variations of the same algorithm, shown below: 1. Define two ”distances”, the distance between a pair of topo-clusters, Dij = min(p2p T i, p2p T j)∆R2/R2, and the distance between a topo-cluster and the beam, Di = p2 T i. 2. Calculate Di for each topo-cluster and Dij for all pairs of topo-clusters. 3. Find the minimum of these and call it d. 4. If d is one of the Dij, then combine the corresponding topo-clusters. 5. If d is one of the Di, then set it aside and call it a jet. 6. Repeat until all of the topo-clusters have been combined into jets. This algorithm has two variables set by the user, the exponent (p) and jet-radius (R), and two variables determined by the event, the transverse momentum (pT ) and the angular 62 difference between two topo-clusters (∆R). The three variations mentioned above use dif- ferent values of p; Kt uses p = 1, Anti-Kt uses p = −1, and Cambridge-Aachen uses p = 0. The analysis presented in this dissertation uses jets clustered with the Anti-Kt algorithm. Two jet radii were used in this analysis, R = 0.4 and R = 1.0, which are described below. Small-R Jets Until fairly recently, the standard type of jet used for analyses was what we now call a small-R jet. The standard radius for these jets is 0.4 radians. These jets usually represent one quark or gluon that came out of the collision. Large-R Jets For reasons specified in sec. 4.1, the use of what we call Large-R jets has increased. The standard radius for these jets is 1.0 radians. These jets usually represent a very heavy particle, i.e. a W boson or heavier, or some fraction of a heavy particle’s decay products. Track Jets The last type of jet that has become common to use are Track jets, which are jets formed from tracks instead of topological clusters. The standard radius for these jets is 0.2 radians. One or more of these are usually grouped together with a large-R jet to represent the jet’s substructure. These are mostly useful for the purposes of b-tagging large-R jets (described below). Jet Substructure A topic that has gained a lot of attention since 2010/2011 is using a jet’s substructure to help determine what kind of particle it represents. The energy distribution within a jet is dependent on the particle that generated it. For instance, if a jet originally came from a single gluon or light-quark then it will likely have all of its energy concentrated in one spatial location. Whereas if the jet came from a W, Z, or Higgs boson, its energy will 63 be concentrated in two separate locations. The energy of a jet that came from a top-quark, which often decay into a quark and W boson, is usually distributed among three separate locations. Over the last few years, several variables have been developed that help us glean information about a jets substructure. b-tagging Due to the suppression of bottom quarks decaying to up and charm quarks, b-mesons have a longer lifetime than other hadrons commonly produced in collisions. As a result b-mesons decay further away from the collision point than other particles. This is a unique signature and can be identified algorithmically. The algorithms used to do this are called b-tagging algorithms [34, 35] and are useful for both small-R and large-R jets. These algorithms search for tracks within a jet that have a large impact parameter, which is a measure of how far away from the collision point the track starts. The Higgs boson primarily decays to two b-quarks, so b-tagging is very useful when searching for events with a Higgs in them. 3.3.1.5 Missing Transverse Energy Neutrinos and other particles that don’t interact with the electromagnetic or strong nuclear forces don’t leave a trace in the detector. The only way to infer their presence is by adding up the transverse momenta of every other object. If you’ve added up the transverse momenta of everything in the detector, seen or unseen, then the total would be zero. So if the total transverse momenta of everything seen in the detector is non-zero, that means what’s left must be an effect of what’s unseen. Taking the negative of that total yields the Missing Transverse Energy (MET) [36, 37]. 64 3.3.2 Calibrations Due to various effects in the detector, there are differences between the energy collected and that deposited. These inaccuracies are addressed by calibrating the data to account for such effects. The calibrations vary because there is inherent randomness in how particles decay and interact with matter. The calibrations are accurate, but have some systematic uncer- tainties associated with them. Because the calibrations are accurate, it is these systematic uncertainties that we are primarily concerned with. Some systematic uncertainties are more important than others, see for example Sec. 5.1. Both data and Monte-Carlo simulations are independently calibrated; the differences between those calibrations are the primary source of uncertainty. Because of their systematic uncertainties, the Jet Energy Scale (JES), Jet Energy Reso- lution (JER), Jet Mass Scale (JMS), and Jet Mass Resolution (JMR) are some of the more important calibrations for this analysis. This is because their uncertainties are larger than that of photons and leptons, due to the complex nature of reconstructing jet momentum and energy, as well as MET. The complex nature of reconstructing jets is due to hadronization. Put simply, the process of hadronization is a very large set of stochastic processes, which means jets have more variation in their appearance than simpler objects like electrons and photons. 3.3.2.1 Jet Scales and Resolutions While the randomness of hadronization is what causes the large systematics of the jet calibra- tions, it’s actually other effects that necessitate the calibrations themselves. For instance, as the momentum of a particle increases, it travels further through the calorimeters be- fore stopping. If a particle’s momentum is large enough, it can actually traverse the entire 65 calorimeter and exit out the back. When this happens, not all of its energy is deposited in the calorimeters. In addition, the materials of the detector aren’t perfectly uniform; this leads to problems like some portions of the detector absorbing energy better than others. These effects can alter the energy and mass of jets as seen by the detector; as a result we require pT -, η-, and φ-dependent calibrations. These calibrations, mentioned in the previous section, are the JES, JER, JMS, and JMR. The Jet Energy Scale corrects the absolute energy scale of energy deposition in the detector, while the Jet Energy Resolution corrects the variance of the MC energy calibration to match the variance of the data energy calibration. Similarly, the Jet Mass Scale corrects the absolute mass scale of reconstructed jets, while the Jet Mass Resolution corrects the variance of the reconstructed jet mass. The jet energy and mass calibrations are derived in nearly the same fashion: calibrate the EM Energy (Mass) Scale and Resolution, use those to calibrate a portion of the Jet Energy (Mass) Scale and Resolution, then extend the Jet Energy (Mass) Scale and Resolution to the rest of the detector. To do this, start with all the well reconstructed e+e− events in the detector by taking a subset of the data and selecting events with criteria like the following: the event contains two leptons with opposite charge, the leptons’ energy is deposited in the EM calorimeters (i.e., not a muon), and there are no jets in the event. The Z peak should appear in the mass distribution of these events, but it will not be at the correct mass and the variance in the MC will not match that of the data. So we create a calibration that scales the energy (mass) so that the Z peak has the correct mass, as well as a calibration that transforms the variance of the Z peak in MC to that of the data. Next, take back-to-back photon-jet events using the proper criterion. Instead of using a known resonance (like the Z peak) to calibrate the Jet Energy (Mass) Scale and Resolution, we will use the fact that the total pT of an event should be zero. We will do this in a region 66 of the detector that has very high statistics, i.e. η < 1. Since the EM Energy (Mass) Scale and Resolution are set, we can now be sure that jet energy mismeasurements are the cause of non-zero total pT . So we create a calibration that scales the energy (mass) of the jet so that the total pT is brought back to zero, as well as a calibration that transforms the MC resolution to that of the data. Now that the Jet Energy (Mass) Scale and Resolution are defined in a high statistics portion of the detector, extend them to the rest of the detector by using events with two back-to-back jets. Since we are using pT to make these calibrations, the only requirement is that the two jets are back-to-back in φ. This means we can use the calibrations from the photon-jet step, η < 1 in our example, to calibrate a different portion of the detector, say 1 < η < 2. Simply find events where there are two jets back-to-back in φ but where one jet has η < 1 and the other has 1 < η < 2, since the jet with η < 1 has already been calibrated we can create a new calibration for 1 < η < 2. After doing this extension we can repeat this step until the full pT , η, and φ dependent Jet Energy (Mass) Scale and Resolution have been derived. 67 Chapter 4 Data Selection and Modeling When searching for new decay processes we must account for SM processes that have similar decay products and kinematics. The new decay processes are labeled signal processes, for which this search has only W(cid:48) → W H → lνbb, and the similar SM processes are labeled background processes. Differentiating these two types of processes is difficult because the detector only records information about the decay products, not the intermediary particles. To get information about the intermediary particles, we must “reverse” the decay process by adding up the decay products in a specific order. This isn’t a perfect approach though because there is still quite a bit of overlap in kinematic phase space and we are only making an educated guess as to the order in which the products should be added together. To rectify this we must do a counting experiment, which means we take some repre- sentation of what we think our signals and backgrounds should look like and compare this to the data that came from the detector. Specifically, we model our signal and background processes with Monte Carlo (MC) simulations, taking into account how often we expect each to be produced in a collision, and compare these to how many similar events the detector recorded. Another subtlety is that the rate at which the signal process occurs is much lower than that of the background processes. This signal process rate is so low that it’s nearly impossible to see without somehow reducing the amount of background first. To do this we try to find a region of kinematic phase space where the background has 68 been greatly reduced while the signal is mostly intact. This is done by reducing the consid- ered range of one kinematic variable at a time and seeing how the signal and background distributions change across the other variables. Reducing the range of a kinematic variable is referred to as cutting it. A favorable cut is one in which the background has been reduced but the signal has not. After the first cut is picked and applied, the process is repeated until there is no substantial benefit to adding more cuts. Once all the cuts have been picked and applied we compare the data and background distributions of a special variable, which will be introduced later. If there is more data than background in certain ranges of this distribution, then the SM might not account for all the events in those ranges. This means there could be new physics in this region of phase space. In order to test how significant any excesses of data might be a statistical analysis, described in chap. 5, is done. 4.1 Analysis Strategy In previous searches for our signal, the energy of the LHC was lower or the data-set had fewer statistics. The size of the data-set matters because more statistics gives us more definitive results and more probing power. Fundamentally, we are trying to determine if the signal is present in the data or not. There are two sides to this, both of which are related to the rarity of the signal. First, we expect that most of the data we collect are background events. This means that the more data we collect, the better we can represent our backgrounds∗. Second, we expect to see very few signal events, so we increase the chances of this rare process appearing in our data-set by collecting more data. ∗This is done by scaling our background to the data in areas of phase space where no signal is expected. 69 The energy of the LHC is important because the mass of the W(cid:48) is unknown. This means that we need the ability to search a fairly large range of values. What range is accessible for searching is dependent on the energy of the LHC. The low end of this range is trivial; it starts at zero regardless of the energy of the LHC. It’s the high end of this range that the LHC’s energy determines; as the energy increases, the high end of this range also increases. Thus a higher energy LHC can probe a larger range of masses. To get a good representation of what the signal looks like across this range, we pick several mass points for which we simulate the signal. In addition, the energy also changes the kinematics of the signal’s final state. A low mass W(cid:48) produces a lower momentum W and Higgs, while a high mass W(cid:48) produces a high momentum W and Higgs. Low momentum W s and Higgs produce nearly back-to-back lv and b¯b pairs, while high momentum W s and Higgs produce extremely collimated lv and b¯b pairs. As a result, one of two search methods was used in the previous analyses depending on the COM energy of the LHC. When the LHC was at lower COM energies it could only produce lower mass W(cid:48)s. This meant the Higgs had low momentum in the lab frame and the b¯b pairs they produced were nearly back-to-back. The hadronic showers that came from the b-quarks could then be clustered into two separate jets with radii of 0.4 radians. Events like this are called “resolved” because the two jets are separated from each other enough to tell them apart. This was the approach used by the 7 and 8 TeV searches mentioned in Sec. 2.2.3. Since the COM energy of the LHC was increased, it can now produce higher mass W(cid:48)s. This meant the Higgs could have high momentum in the lab frame, so the b¯b pairs they produced could be highly collimated. The hadronic showers coming from the b-quarks would then overlap and become irresolvable. Therefore it is best to consider these as one large-R 70 jet instead of two small-R jets. Events like this are called “merged” because the two small-R jets have merged together and are no longer resolvable. This was the approach used by the previous 13 TeV analysis mentioned in Sec. 2.2.3. While each of these approaches describe their appropriate extremes very well, there are a range of intermediate masses where an event can be described as both resolved and merged. Such an event would have showers from b-quarks that overlap just enough to be clustered together as one large-R jet, but also far enough apart that they could be clustered as two small-R jets. The search presented here combines these two different approaches together. Doing this adds another level of detail to describe events, which means there is more informa- tion to separate the signal from the backgrounds, thereby increasing the search’s sensitivity to the signal. 4.1.1 Signal Characteristics In order to remove background events from the data-set, we must know how they are different from signal events. This means we must first understand the characteristics of our signal, shown in Fig. 4.1. This diagram includes the possibility of a W(cid:48) or a Z(cid:48) being produced. In both cases they will decay to a Higgs and the associated Weak Vector Boson, then the Higgs will decay to a b¯b pair. How the Weak Vector Boson decays is dependent on its flavor. In our case we are specifically interested in the decay of the W(cid:48), which means the Weak Vector Boson is a W boson. While the W boson has several decay modes, the one we were specifically interested in was a decay to a charged lepton and neutrino. This means our final state has one charged lepton, one neutrino, and two b-quarks. In the data this looks like an electron or muon, missing transverse energy, and two small-R jets that are potentially b-tagged or one large-R jet with up to two b-tagged track-jets inside it. 71 The following observations about our signal process can be made. The lepton and MET come from a W boson, which can be used to find the z-component of the neutrino using kinematic restraints. The two b-quarks come from a Higgs boson, so their invariant mass distribution (from the two small-R jets or one large-R jet) should be peaked around the Higgs mass. Finally, the W and Higgs bosons come from a W(cid:48), so they will have correlated four-momenta. q ¯q H V (cid:48) V ¯b b l ¯l Figure 4.1: Feynman diagram of our signal. It shows a W(cid:48) or Z(cid:48) decaying to a Higgs and W or Z, respectively. The Higgs then decays to a b¯b pair while the W/Z decay leptonically. 4.1.2 Background processes The backgrounds we consider are: t¯t, single-top, V+jets, diboson, SM Higgs, and QCD multijet production. Each of these backgrounds mimics the signal in one or more ways. While most of these backgrounds are unique enough that applying cuts will remove a significant portion of them, there are some that are so similar that they can only be taken into account∗. ∗Fortunately, these backgrounds also happen to contribute very little to the total number of events. 72 Irreducible QCD Production of Top Quark Pairs One of the largest backgrounds in this search is the irreducible QCD production of top quark pairs (t¯t), which is illustrated in Fig.4.2. Like the signal it has one charged lepton, one neutrino (MET), and two b-quarks in its final state. However, it also has two additional jets, or a lepton and neutrino, from the second W boson. While the lepton and MET come from a W boson, the two b-quarks do not come from a Higgs boson. Instead, the two b-quarks come from two separate top decays. For a merged event, this means that the b-tagged track jets will almost never be in the same large-R jet. For a resolved event, this means that the invariant mass of the two b-jets will be fairly random. In both cases, the invariant mass of the b-jets and the mass of the large-R jet will not have peaks, let alone peaks near the Higgs mass. g g g t ¯t W + b ¯b W− ¯q q ¯ν l− Figure 4.2: Feynman diagram for the production of t¯t. Both top quarks decay to a b-quark and W boson. One of the W bosons decays hadronically while the other decays leptonically. 73 Electroweak Production of Single Top Quarks Another large background is the EW production of single top quarks (single-top), which has three channels as illustrated in Fig. 4.3: t-channel, s-channel, and Wt-channel. All three channels have one charged lepton, one neutrino (MET), and at least one b-quark in the final state. In addition the s-channel has an additional b-quark, the t-channel has an additional quark, and the Wt-channel has two additional quarks. Like the signal, the lepton and MET in each of these channels come from a W boson. However, the various jets come from different decays, none of which include a Higgs boson. This means that the invariant mass distribution of the small-R jets and the mass distribution of the large-R jet will not have peaks. V+jets The last large background is the Drell-Yan W /Z production in association with jets (V+jets), which is illustrated in Fig. 4.4. Its final state has two leptons and one or more jets. Depending on the flavor of the weak vector boson, the two leptons can be a charged lepton and a neutrino (MET) or two oppositely charged leptons of the same flavor. In the case of W +jets, the charged lepton and MET come from a W boson, just like the signal. However, the jets do not come from a Higgs boson, so the invariant mass distributions of the two small-R jets and the mass distribution of the large-R jet will not have peaks. When it comes Z+jets, the situation is a bit more complex. Because there are two charged leptons in the final state and we only accept events that have exactly one (discussed later), only Z+jets events where one of the charged leptons has not been reconstructed will enter our analysis. In this case, the “lost” lepton will look like MET in the event; however the charged lepton and MET will come from a Z boson instead of a W boson. 74 q ¯q b b b t W + t W + W + ν W + ν l+ ¯b l+ ¯q b ¯q q ¯q t W + g b t W− l− ¯ν Figure 4.3: The Feynman diagrams for the three single-top channels. The t-channel is on the top-left, the s-channel is on the top-right, and the Wt-channel is on the bottom. Each channel has one charged lepton, one neutrino, and at least one b-quark in its final state. In addition, each channel has one or two extra quarks, which may or may not be b-quarks. V q l q ¯l q g Figure 4.4: The Feynman diagram for the simplest V+jets process. In general, a V+jets event has two leptons (charged or neutral) and some number of jets. Diboson Diboson is a background that contributes moderately to the total background and consists of two channels: WW or WZ. The Feynman diagram for the channel/final state that contributes 75 most to our total background is illustrated in Fig. 4.5. Both of these have two jets in the final state, but have either one or two charged leptons and one or zero neutrinos depending on the channel. In the case of WW, there is always one lepton and one neutrino. The WZ channel on the other hand can have two final states; the first would have one lepton and one neutrino coming from the W boson, while the second would have two charged leptons coming from the Z boson. For the final state with two charged leptons, one of the two would need to be “lost” somehow. In both cases the leptons and/or MET would come from a W or Z, which look fairly similar. A distinct difference from the previous backgrounds is that the two jets decay from the same particle, although it is not a Higgs boson. This means that the invariant mass distribution of the small-R jets and mass distribution of the large-R jet would have one or two peaks, but these peaks would be at the W and Z masses instead of the Higgs mass. SM Higgs One of the smallest backgrounds in our analysis is the SM Higgs background, which is illus- trated in Fig 4.6. This background looks almost exactly like the signal, the key difference being that a W∗∗ is produced in the collision instead of a W(cid:48). This means that this back- ground is fairly hard to remove, and really just needs to be accounted for in the statistical analysis. ∗The symbol W∗ refers to an off-shell W , which simply means the mass is not the W peak mass. 76 q ¯q W− W− Z l− ¯ν ¯q q Figure 4.5: The Feynman diagram for the diboson channel/final state that contributes most to our total background. This final state has one charged lepton and one neutrino (MET) that come from a W boson and two jets that come from a Z boson. Other final states for this channel could have two leptons coming from the Z boson and two jets coming from the W boson. The other channel would have two W bosons instead of a W and Z, where one W would decay leptonically and the other would decay hadronically. W− W− H q ¯q l− ¯ν b ¯b Figure 4.6: The Feynman diagram for the SM Higgs background. It is almost identical to our signal, except for the collision producing a W boson instead of a W(cid:48). Multijet The final background, which is also the smallest for our analysis, is the production of QCD multijets (multijet). The multijet background is the combination of many random QCD 77 processes, which means the final state is fairly diverse. However, because of our requirements, the final states that enter our selection have one lepton, MET, and jets. Because it is colliding hadrons, this is actually the largest background produced by the LHC. However, any leptons that appear in a multijet event are faked, meaning that they are actually a jet that was reconstructed as an electron. The fake rate of jets being misidentified as leptons is so small that requiring one lepton in an event reduces the multijet background significantly. In addition, MET in a multijet event usually comes from jets whose energy has been slightly mismeasured, not neutrinos. That means there can be multiple contributions to the MET that are distributed randomly, these usually combine in a way that makes the total MET quite small. We can reduce the multijet background further by requiring a large amount of MET in our events. 4.2 Data and Simulation Samples The first step of our analysis after all the data has been collected is to decide which events in it are useful for our search. This is done by picking a few key characteristics of our signal and then picking which events to include based on their trigger information. Once that has been decided we must simulate signal and background samples in the same phase space. We can then start making decisions based on the simulations while comparing them to the data in areas with low signal content. Doing this comparison ensures that any changes we make affect the data and simulations the same way. 78 4.2.1 Data This analysis used 3.21 fb−1 of detector, as well as 32.9 fb−1 collected in 2016. In total 36.1 fb−1 of √ √ s = 13 TeV data collected in 2015 from the ATLAS s = 13 TeV data was used. This data was required to meet criteria that ensures the ATLAS detector was in good operating condition. 4.2.1.1 Triggers The amount of data that comes out of the High-Level Trigger is still fairly large and contains many events that are not useful for our search. In order to reduce the amount of data we need to run over, we select a portion of it that looks fairly similar to what we expect the signal to look like. This is done using the trigger information. We know signal events should meet certain requirements, we look for triggers that have those requirements, and then we select data that passed those triggers. In our case we expect one lepton and neutrino (MET), so single lepton and MET triggers are used to select our data. We only require the data to have passed one of these triggers. The triggers used can be seen in Tab. 4.1. 4.2.1.2 Pile-Up A complication we face when looking at the data is the concept of pile-up. There are two types of pile-up: in-time and out-of-time. In-time pile-up happens because we are colliding bunches of protons together and not just two. What happens is that some fraction of the protons in one bunch collide with some fraction of protons in the other. As a result, there is more than one collision per bunch crossing. While the fraction of protons that collide in a bunch is very small (the average is about 25 out of about 1011 protons), we still end up with more than one collision per crossing. This means we must figure out which collision is 79 Data-set 2015 2016 - Period A-D3 2016 - Period ≥ D4 Type Triggers xe70 Lepton e24 lhmedium L1EM20VH Lepton e60 lhmedium Lepton Lepton e120 lhloose Missing Energy xe90 mht L1XE50 Lepton e26 lhtight nod0 ivarloose e60 lhmedium nod0 Lepton Lepton e60 medium Lepton e140 lhloose nod0 e300 etcut Lepton Missing Energy xe110 mht L1XE50 Lepton e26 lhtight nod0 ivarloose Lepton e60 lhmedium nod0 e60 lhmedium Lepton Lepton e140 lhloose nod0 e300 etcut Lepton Table 4.1: The name of the triggers used to select data, as well as the time period they were used and what physics object they used for selection. the most interesting and which particles come from that collision. To do this we start by selecting a subset of tracks from the Inner Detector that meet very tight requirements, such as needing an impact parameter very close to the bunch crossing point. Then we figure out where all the different collisions happened by iteratively creating seed positions and seeing how the various tacks fit them. From this we will be left with several different collision points with each track being associated to one of these points. Each of these collisions can then be classified as matched, merged, or split. From there the event can be classified as clean, low pile-up, high pile-up, or split. In the case of a clean event, the most interesting collision has the most tracks originating from it. Out-of-time pile-up is when particles from one bunch crossing are still traversing the detector when the next bunch crossing occurs. This results in both a loss and gain of activity. You lose activity from the event of interest but gain some from an event you are not interested in. This is corrected by using calibrations derived from Monte-Carlo studies. 80 4.2.2 Simulating Signals & Backgrounds To simulate the signals and backgrounds, the Monte Carlo method was used. The Monte Carlo method uses random sampling to produce numerical values that can describe a wide variety of variables. The basic idea for our purposes looks something like the following: 1. Assume there are N fundamental variables associated with the production of an event, such as the 4-vectors of the relevant particles. 2. For each of those N variables a probability distribution function (p.d.f.) is assigned∗. 3. For each event, a random number is sampled from each variables p.d.f. 4. Variables that depend on these fundamental variables can then be calculated†. For a single collision, there are multiple steps where this needs to be done. First, how much energy is carried by the colliding partons‡, and what those partons are, need to be determined. This is done using what we call Parton Distribution Functions (PDFs), which are probability distribution functions that quantify the fractional energy of different partons within a proton. These fractional energies depend on the energy of the proton and each type of parton has its own p.d.f.. Second, what we call the generator level process or hard scatter is simulated. This simply takes in partons from the protons being collided and creates the final state particles for any given signal or background. The simulation will calculate the final state particles’ four-momenta and other kinematic variables. ∗It’s important to note that any restrictions of a particular process are either baked into these p.d.f.s or †If there is a variable in the calculation that is random, it will also have a p.d.f. to be randomly sampled into the calculations of step 4. from.‡A parton is any particle within a proton, i.e., quarks and gluons. 81 Third, any quarks in the final state will be hadronized. This means that new quarks will be created and paired together with the final state quarks to create hadrons. This is necessary because quarks cannot exist on their own; they must be part of a colorless object. Finally, the generator level process with hadronized quarks will be put through a sim- ulation of the detector. During this step, particle interactions with matter are simulated. These interactions cause any charged or hadronic particles to shower electromagnetically or hadronically, respectively. By the end of the full simulation, we should have a description of the signal or background event that is in the same format as the data events. 4.2.2.1 Signal Samples For the signal’s generator level events, Madgraph5 aMC 2.2.2 [38] was used with the NNPDF 2.3 LO [39] PDFs. For parton showering and hadronization, Pythia 8.186 [40] with the A14 tune [41] was used. Samples with signal masses from 500–5000 GeV were produced at varying incremental values; 500–2000 GeV in steps of 100 GeV, 2000–3000 GeV in steps of 200 GeV, and 3000–5000 GeV in steps of 500 GeV. 4.2.2.2 Background Samples Several tools were used to produce the different background samples. For generator level events, MG5 aMC 2.3.2, Powheg-Box [42–44], Powheg-Box v2, Sherpa 2.2.1 [45], and Pythia 8.186 were used. For parton showers and hadronization, Sherpa 2.2.1, Pythia 6.428 [46], Pythia 8.186, and Pythia 8.210 [47] were used. The t¯t background was generated using Powheg-Box v2 with the CT10 [48] PDFs. It was showered and hadronized using Pythia 6.428 with the Perugia 2012 tune [49]. Its cross-section was calculated up to NNLO+NNLL (next-to-next-to-leading-order+next-to-next-to-leading- 82 log) [50–56]. Finally, the predicted transverse momentum spectra of top quarks and the t¯t system were reweighted to the corresponding NNLO parton-level spectra [57]. All three single top channels were generated using Powheg-Box with the CT10 PDFs. They were showered and hadronized using Pythia 6.428 with the Perugia 2012 tune. The s- and t-channel cross-sections were calculated up to NLO [58, 59], while the Wt-channel was done at approximately NNLO [60]. The V+jets background was generated, showered, and hadronized using Sherpa 2.2.1 with the NNPDF 3.0 NNLO [61] PDFs and the default tuning of Sherpa. Its cross-section was calculated up to NNLO [62]. Both the Diboson channels were generated and showered/hadronized using Sherpa 2.1.1 with the CT10 PDFs and default tuning of Sherpa. Their cross-sections were calculated up to NLO. The SM Higgs background was generated using Pythia 8.186 with the NNPDF 2.3 LO PDFs and A14 tuning. Its cross section was calculated up to NNLO+NLO [63]. Multijet was estimated using data-driven techniques described in App. A. This was done because multi-jet MC simulations suffer from limited statistics and difficult modeling in the phase space of this analysis. 4.3 Object Selection The first step in reducing the number of background events is to define what objects we are interested in. Based on the information about the signal discussed above, interesting events will have electrons, muons, small-R jets, and large-R jets in them. Therefore, events are only considered if they contain robust versions of these objects. The following sections define the 83 criteria for being a robust version of each object. 4.3.1 Electrons Electrons are required to have at least 27 GeV of transverse momentum. They must be within an |η| of 2.47 radians. They must meet the ATLAS LH Tight electron ID requirement, which basically reduces the fake rate at the cost of selecting fewer electrons. The significance of the transverse impact parameter, dsig longitudinal impact parameter, |∆zbeam 0 , must be less than 5 standard deviations. The sin θ|, must be less than 0.5 mm. Finally, they 0 must meet ATLAS’ tight isolation requirements. 4.3.2 Muons Muons are also required to have at least 27 GeV of transverse momentum. However, they must be within an |η| of 2.5 radians. They must meet the ATLAS Medium quality muon ID requirement, which is more balanced in terms of reducing fake rates while selecting more muons. The significance of the transverse impact parameter, dsig standard deviations. The longitudinal impact parameter, |∆zbeam 0 , must be less than 3 sin θ|, must be less than 0 0.5 mm. Finally, they must meet ATLAS’ tight isolation requirements. 4.3.3 Small-R Jets Small-R jets are put into two categories, signal or forward, based on their pseudo-rapidity. They are categorized as signal jets if they have |η| < 2.5 and forward jets if they have 2.5 < |η| < 4.5. Signal jets are required to have a minimum transverse momentum of 20 GeV, while forward jets are required to have a minimum transverse momentum of 30 GeV. 84 Signal jets with |η| < 2.4 and pT < 60 GeV are also required to have a jet vertex tagger∗, JVT, less than 0.59. 4.3.4 Large-R Jets Large-R jets are required to have a minimum transverse momentum of 250 GeV. They must be within an |η| of 2.0 radians and have at least one associated track jet. The associated track jets need to have a minimum transverse momentum of 10 GeV, |η| < 2.5, and have at least two track constituents. 4.4 Event Selection Because we are combining two very different methods of clustering jets used by the previous analyses, this analysis has two sets of event selections. One selection considers if the event has resolved jets while the other considers if the event has merged jets. Each event is passed through both selections to maximize the potential of selecting a signal event. Tables 4.2, 4.3, and 4.4 provide a summary of these selections. 4.4.1 Common Selection There are three selection criteria that are common to both the resolved and merged event selections. First, the final state of the signal has only one lepton, so we require there to be exactly one lepton in the event. This lepton can be an electron or a muon, but must meet the object selection criteria defined above. Second, it was found that there was mis-modeling if the invariant mass of the two leading jets, mbb, or the mass of the leading large-R jet, ∗The fraction of a jet’s momentum coming from the primary vertex, specifically in terms of the jet’s tracks. 85 mJ , deviated far from the Higgs mass peak, 125 GeV. So the restriction that these must be within 50–200 GeV for their respective event selections was applied to reduce this effect. Third, there was also mis-modeling at high transverse W mass, mT W , so events are required to have mT W less than 300 GeV. All three of these criteria are summarized in Tab. 4.2. Cut Variable # leptons mbb, mJ mT W Cut Value exactly 1 lepton as defined by object definition 50 < mbb, mJ < 200 GeV mT W < 300 GeV Table 4.2: The criteria that are common to the resolved and merged event selections. 4.4.2 Resolved Selection The following are the criteria unique to the resolved event selection, which is designed around events where the two b-jets from the Higgs are separated enough to tell them apart. Some of these criteria will look similar to those in the merged event selection; these criteria are tuned, however, to the kinematics of events with resolvable jets. The neutrino is one of the signal’s four final state products, so we expect it to carry a fair portion of the event’s energy. Therefore we apply a cut on MET to select events with high energy neutrinos. In addition, we increase the strength of this cut to reduce the multijet background. If the event contained an electron then an 80 GeV cut was applied, while a 40 GeV cut was applied to events with a muon. We found that the pT of the W in the signal as a function of the invariant mass of the WH system behaved differently than that of the backgrounds. So an mW H dependent pW T cut, pW T > −3.26 ∗ 105 GeV2 mW H [GeV] 86 + 709.6 GeV, (4.1) with a lower bound of 150 GeV was applied. We also expect that the two jets in the final state should carry a large fraction of the event’s energy. Therefore, a pT cut of 40 GeV was applied to the leading jet in each event. At tree level the signal has only two jets, which means that we expect signal events to have at least two jets. However, one of the largest backgrounds, t¯t, has a large number of jets. Because t¯t is so large in the case of events with four or more jets, it becomes overwhelmingly dominant. So the allowed number of signal jets in an event has been narrowed to two or three, but no restrictions were made on the number of forward jets. The signal has exactly two b-jets in its final state, but this does not mean they are always tagged as such. It’s possible that one, both, or neither of the b-jets is tagged. However, the signal should not have three or more b-tagged jets. So events with three or more b-tags are vetoed from the selection. The invariant mass of the W(cid:48), mW H , is calculated by combining the W and Higgs objects in the event and the W is created by combining the lepton and MET. While the MET ideally describes the neutrino, it only describes the transverse part, which means there is information missing from the W(cid:48) mass. In order to better reconstruct the invariant mass of the W(cid:48), the pZ of the neutrino is approximated by imposing the W mass as a constraint on the lepton-neutrino system. This doesn’t work perfectly since the W mass has some width, but it at least causes the signal to become more peaked around the W(cid:48) mass while it affects the backgrounds randomly. 4.4.3 Merged Selection The following are the criteria unique to the merged event selection, which is designed around events where the two b-jets from the Higgs are close enough together to consider them as 87 Cut Variable Emiss T pW T Jets Leading jet pT b-tags Cut Value electron events: Emiss T > 80 GeV, muon events: Emiss T > 40 GeV if pW T > 150 GeV, then pW T > −3.26∗105GeV2 mW H [GeV] + 709.60 GeV 2 or 3 signal jets and ≥ 0 forward jets > 45 GeV No more than 2 b-tags Table 4.3: The criteria unique to the resolved event selection. one large, Higgs jet. Like above, some of these criteria look similar to those of the resolved selection, however these criteria are tuned to the kinematics of events with merged jets. A MET cut of 100 GeV is applied for two reasons. The first is because there is a high energy neutrino in the final state, while the second is to heavily reduce the multijet background. For the same reason as the resolved selection, an mW H dependent pW T cut is applied. T > 394 GeV2 × ln(mWH[GeV])− 2350 GeV, with However, the cut takes a different form, pW a lower bound of 150 GeV. In the merged case, the signal has exactly one Higgs jet in its final state, so at least one large-R jet is required. This jet should contain two b-quark showers in it, but both might not always be tagged or they might combine into one shower. Therefore the leading large-R jet is required to have at least one b-tagged track jet associated with it. Similar to the resolved selection, the pz of the neutrino is approximated by imposing the W mass as a constraint on the lepton-neutrino system. This causes the mW H distribution of the signal to peak, while the same distribution of the backgrounds is affected randomly. Cut Variable Emiss T pW T Jets Cut Value if pW T > 150 GeV, then pW ≥ 1 large-R jet, ≥ 1 track-jet associated to leading large-R jet Emiss T > 394 GeV2 × ln(mW H [GeV]) − 2350 GeV T > 100 GeV Table 4.4: The criteria unique to the merged event selection. 88 4.5 Event Categorization Once the events have been selected, they must be categorized as either resolved or merged. A small fraction of events will only pass one selection or the other, making it clear how to categorize them, but most events will pass both selections. For those events that passed both selections, a method of picking which category these events should go into needed to be devised. How this method was devised is the topic of this chapter. 4.5.1 Signal and Control Regions Before explaining the studies done to categorize events, it is important to discuss another level of categorization beyond resolved and merged. While the resolved and merged categories are associated to an event’s jet topology, the phase space of this analysis can also be split into two regions based on where the signal lives. These two regions are called signal (SR) and control regions (CR) and they serve two purposes. First, during the portion of the analysis where cuts were being studied and decided on, we purposefully did not look at the data in the signal regions. Doing so prevents us from biasing the search by forcing the data to peak at a particular point. Instead, only the MC simulations could be seen in the signal regions and any comparisons between the data and MC were done in the control regions. Second, both of these regions will be used in the statistical analysis of Chap. 5 to fit the MC to the data and set limits on the W(cid:48) mass. This split between signal and control regions is based on a Higgs mass window. This mass window spans the range 110 < mbb < 140 GeV for the resolved selection and 75 < mJ < 145 GeV for the merged selection. If an event falls within the resolved or merged Higgs mass window then it is in the signal region for that category, otherwise it is in the appropriate 89 control region. In addition, there are two more splits used by the analysis. The first is based on the number of b-tags in an event, and it has two categories: 1 or 2 b-tags. The second is solely for the merged selection and considers whether there are additional b-tags outside of the leading large-R jet. The purpose of these categories is to further separate the signal from the backgrounds. For instance, the t¯t background contributes more to the merged regions with additional b-tags outside the leading large-R jet than the merged regions without additional b-tags. To summarize, there are three to four categories depending on the event selection. An event can be categorized as resolved or merged, in the signal region or the control region, having 1 or 2 b-tags, and whether there are additional b-tags outside the leading large-R jet or not∗. Table 4.5 visualizes these categories. Selection Resolved Merged Mass Window SR (110 < mbb < 140 GeV) CR SR (75 < mJ < 145 GeV) CR # b-tags # add. b-tags 1 b-tag 2 b-tags 1 b-tag 2 b-tags 0 add. b-tags. 1+ add. b-tags. Table 4.5: Categories used in this analysis. They are based on whether the event is resolved or merged, inside the Higgs mass window or not, whether there are one or two b-tags, and whether there are b-tags outside of the leading large-R jet or not (for the merged events only). 4.5.2 Categorization Strategies As mentioned earlier, we needed some way of deciding whether an event should be categorized as resolved or merged if it passed both selections. Several methods of categorizing events were explored, as described below. ∗This last category is only considered if the event is first categorized as merged 90 1. Resolved: Do not consider the merged selection, every event is classified as resolved. This is the strategy used by the 7 and 8 TeV analyses and was included purely for comparison purposes. 2. Merged: Do not consider the resolved selection, every event is classified as merged. This is the strategy used by the 3.2 fb−1 13 TeV analysis and was included purely for comparison purposes. 3. SimpleMerge500: In the case of the signal, the W and H are created approximately back to back with equal energies. Since the jets coming from the Higgs become more collimated as it is boosted, a good test to see if they are close to merging is to check the pT of the W . Via the kinematics of this problem, it was found that the two small-R jets will start merging when pW T is around 500 GeV. This strategy categorizes events as resolved if they are below that threshold and merged if they are above. 4. PriorityResolved (PR): If an event passes the resolved and merged selections, cat- egorize it as resolved. 5. PriorityMerged (PM): If an event passes the resolved and merged selections, cate- gorize it as merged. 6. PriorityResolvedSR (PRSR): Similar to PriorityResolved, but also checks if it falls into their respective signal or control regions. Prioritize in the following way; Resolved SR > Merged SR > Resolved CR > Merged CR. 7. PriorityMergedSR (PMSR): Similar to PriorityMerged, but also checks if it falls into their respective signal or control regions. Prioritize in the following way; Merged SR > Resolved SR > Merged CR > Resolved CR. 91 8. PriorityResolvedSRbtag (PRSRbtag): Similar to PriorityResolvedSR, but also prioritizes based on number of b-tags in the event. Prioritize in the following way; Resolved SR 2 b-tag > Merged SR 2 b-tag > Resolved SR 1 b-tag > Merged SR 1 b-tag > Resolved CR > Merged CR. 9. PriorityMergedSRbtag (PMSRbtag): Similar to PriorityMergedSR, but also pri- oritizes based on number of b-tags in the event. Prioritize in the following way; Merged SR 2 b-tag > Resolved SR 2 b-tag > Merged SR 1 b-tag > Resolved SR 1 b-tag > Resolved CR > Merged CR. While there are more complex ways of deciding which category best describes an event, for the sake of time it was important to choose a simpler solution for this analysis. Perhaps in the future there will be a group that explores more detailed methods. 4.5.3 Categorization Studies A log-likelihood ratio was used to test which strategy had the best sensitivity to the signal. Looking to Fig. 4.7, it can be seen that PRSRbtag and PMSRbtag had the best sensitivities. However, using these would have made deriving certain systematics too complicated and would’ve taken much more time than was available. Since these both have a marginal advantage over PRSR and PMSR, it was decided to use one of the latter instead. Looking back to Fig. 4.7, it can be seen that PRSR does better than PMSR below signal masses of 1600 GeV and is comparable above that mass. Therefore, PRSR was the categorization strategy used in this search. 92 Figure 4.7: Sensitivity plots for the categorization studies explored for this analysis. The plot on the left compared the resolved and merged strategies to the SimpleMerge500 strategy. It is clear that while the resolved strategy does well at low masses, it starts to do poorly around 1100 GeV. The opposite can be said of the merged strategy, which does great at higher masses, but poorly below 1100 GeV. This graph makes it clear that combining the two approaches was the right thing to do. The plot on the right compares all the different strategies considered to the SimpleMerge500 strategy. 4.6 Observed Results Figures 4.8, 4.9, and 4.10 show the signal mass plots for the final selection; plots of additional variables used in the analysis can be viewed in App. B. An underestimation of V+jets is observed in all regions. This was also observed in other analyses and was attributed to Sherpa 2.2.1. Therefore the Data and MC do not agree well in certain areas before the fit done in Chap. 5. 93 10002000300040005000 [GeV]VHm50100150200250300LLRD vs. Signal MassLLRDResolvedMergedSM500 vs. Signal MassLLRD10002000300040005000 [GeV]VHm00.20.40.60.81ratio10002000300040005000 [GeV]VHm50100150200250300LLRD vs. Signal MassLLRDSM500PRPMPRSRPMSRPRSRbtagPMSRbtag vs. Signal MassLLRD10002000300040005000 [GeV]VHm0.80.850.90.9511.051.11.15ratio Figure 4.8: The signal and control regions of the resolved selection. On the left are the Higgs mass control regions and on the right are the Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 94 Figure 4.9: The signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass control regions and on the right are the Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 95 Figure 4.10: The control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass control regions and on the right are the Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 96 Chapter 5 Statistical Analysis The Data/MC comparison plots shown in Sec. 4.6 are not enough to make a concrete dis- covery or exclusion. The statistical likelihood that these distributions were attained via a signal+background or background only hypothesis must also be taken into account. We will do this by evaluating limits on the cross-section of the W(cid:48) using the CLs method. 5.1 Likelihood Function Before we can calculate limits, we must first pick two test statistics: one that tells us how well our MC agrees with the data and another that helps us differentiate our hypotheses. This section will derive a test statistic that tells us how well the MC and data agree, which will then be used later to derive the test statistic that differentiates hypotheses. Start by assuming there is some p.d.f., f (x|−→ some parameters −→ θ . The probability of observing any value of x within a finite interval θ ), that describes a random variable x given [x, x + dx], for given values of −→ θ , is P (x|−→ θ ) = f (x|−→ θ )dx . (5.1) To extend this to multiple observations of x, under the assumption that each observation is independent, we simply multiply the probability of each observation together to get the 97 total probability (Eq. 5.2). P (−→x |−→ θ ) = (cid:89) i f (xi|−→ θ )dxi, where −→x = (x1, x2, ...) (5.2) This equation gives us the probability of observing outcomes, −→x , given parameter values, −→ θ . How this equation is setup allows us to calculate the probability of getting certain observations if we already know the true values of the parameters. What we have, however, is the observations and what we want to know are the values of the parameters that were likely to have produced said observations. So instead of the probability function, P (−→x |−→ −→ θ |−→x ), that gives the likelihood that some unknown what we really want is a function, L( θ ), parameters produced the observed outcomes. This would allow us to start testing different hypotheses by plugging in different sets of parameters and comparing their likelihoods for the same set of observations. Looking back to Eq. 5.2, if f (x, −→ θ ) has the correct form and its parameters are the true values, then the value of f for the observed outcomes should be high. Alternatively, if the parameters are far from the true values, then the value of f for the observed outcomes should be low. Therefore the function we are looking for, called the Likelihood function, is simply the probability function without the outcome intervals, (cid:89) −→ θ |−→x ) = L( f (xi|−→ θ ). (5.3) i Note that there are no intervals about the parameters here because they have one true value each, as opposed to the outcomes which can take on multiple values. For our purposes the xi of −→x are the contents of each histogram bin in each of the mW H 98 histograms included in the fit. Each of these bins is treated as if their contents are sampled from a Poisson distribution, f (n|ν) = νne−ν/n!, where n is the bin content and ν is the Poisson mean. The parameters fall into two categories: parameters of interest and nuisance parameters. Nuisance parameters are parameters we aren’t necessarily interested in, but affect the like- lihood in some way. For this analysis there is one parameter of interest, the signal rate, but for the combinations mentioned later there could be more. The rest of the parameters are nuisance parameters and are generally related to some experimental or theoretical systematic uncertainty. The final version of our likelihood function is thus: regions(cid:89) bins(cid:89) −→ θ |−→ N ) = L(µ, f (Nij|µ × s( −→ θ )ij + i j k Here µ is a scale factor for the signal rate, bkgs(cid:88) −→ θ )ijk) × b( −→ θ(cid:89) n p(θnom n , θn) (5.4) −→ θ are the nuisance parameters, Nij ∈ −→ N are −→ θ )ij is a function that −→ θ )ijk is a function that describes the amount of the number of data events in a particular bin (j) of a region (i), s( describes the amount of signal in that bin, b( a particular background (k) in bin (i) from region (j), and p(θnom n , θn) is a penalty function. The penalty function reduces the likelihood if a nuisance parameter is moved away from its nominal value and takes the form, p(θnom n , θn) = (θnom n − θn)2/2σ2. This likelihood function, Eq. 5.4, will be our test statistic that informs us how good the agreement between the MC and data is within the uncertainties of the MC, which we will get back to talking about in Sec. 5.3. It will also be used in Sec. 5.4.1 to build our second test statistic for differentiating hypotheses. Before we get to either of those topics, the next section will explore the nuisance parameters of this search. 99 5.2 Nuisance Parameters The nuisance parameters (NP) of this search come from its experimental and model uncer- tainties. The experimental uncertainties are related to how we reconstruct and define various physics objects, i.e. jets, leptons, and missing transverse energy. The modeling uncertainties are related to how we chose to do our parton showers and hadronization, which PDF set we used, etc. This search’s largest experimental uncertainties are from the scale and resolution of the jet energy and mass (as mentioned in Sec. 3.3.2.1), as well as the jet b-tagging efficiency and misidentification rate. The jet scales and resolutions affect what a jet’s energy and mass are, so if these change this can cause migration between not only bins of the mV H distributions but also the signal and control regions. The b-tagging efficiency affects whether a b-jet is b-tagged, while the misidentification rate affects whether a non-b-jet is b-tagged; therefore changing the b-tagging algorithms can affect whether an event enters the one or two b- tagged regions. Uncertainties in the small-R jet energy scale come from in situ calibration studies, dependency on pile-up conditions, and the flavor composition of jets [64, 65]. The uncertainties in the scale and resolution of the large-R jet energy and mass is derived from comparing the calorimeter-based measurements to the track-based measurements [66, 67]. The b-tagging efficiency uncertainty for b-jets and c-jets is derived from t¯t and W +c-jet events, while the light-jet misidentification rate uncertainty is derived from dijet events [35, 68–70]. Smaller experimental uncertainties are those coming from the lepton energy and momen- tum scales, lepton reconstruction and identification efficiency, and trigger efficiencies. The lepton energy and momentum scales are similar to the jet energy and mass scales; as such 100 they affect the lepton’s energy and momentum. If a lepton’s energy and momentum change then the overall event might not make it into the final event selection or it could migrate be- tween bins in a particular mV H distribution; however, it could not migrate between regions. The lepton reconstruction and identification efficiency affect whether a lepton is recognized as a lepton or a jet is misidentified as a lepton, which can increase or decrease the total number of events that enter the final event selection. The trigger efficiencies affect how many events make it into our initial selection and therefore our final selection. The missing transverse energy of an event is calculated by summing up the four-vectors of the small-R jets and lepton and then taking the energy of the new four-vector, so it’s easy to understand that any adjustments to, or uncertainties on, the small-R jets or leptons will propagate to the MET. For this reason, the MET has similar NPs to the small-R jets and leptons. Finally, any uncertainties in the luminosity will affect the overall normalization of the data. A global normalization uncertainty is defined due to the luminosity measurement from a preliminary calibration of the luminosity scale, which followed a similar methodology to Ref. [71]. All of these experimental uncertainties affect the shape of the mV H distributions, can cause migration of events across different regions and histogram bins, and can cause events to migrate in and out of the final event selection. The modeling uncertainties of this search are each assigned to a given signal or back- ground process and can cause variations in both the normalization and shape of said signals and backgrounds. In addition, the statistical uncertainty of each MC sample is considered by introducing shape variations derived from the uncertainty of each bin of the mV H distri- butions. These modeling uncertainties are described below and summarized in Table 5.1. How much signal our search accepts into its final selection affects how well the search 101 Process Signal SM VH Diboson Multijet W +(bb, bc, cc) W +(bl, cl) W +l acceptance Quantity/source Value 1-3% 50% 11% 50% norm. norm. norm. template method norm. resolved/merged mjj SR / mjj CR mJ SR / mJ CR gen., PDF, scale norm. resolved/merged mjj SR / mjj CR mJ SR / mJ CR gen., PDF, scale norm. resolved/merged mjj SR / mjj CR mJ SR / mJ CR gen., PDF, scale S F 28% 2% 6% S F 15% 1% 1% S 30% 16% 7% 3% S Process Single-top t¯t Z+(bb, bc, cc) Z+(bl, cl) Z+l norm. Quantity/source Value 19% 24% 7% 5% S F resolved/merged mjj SR / mjj CR mJ SR / mJ CR PS, ISR/FSR, ME norm. resolved/merged mjj SR / mjj CR mJ SR / mJ CR PS, ISR/FSR, ME pT reweight norm. resolved/merged mjj SR / mjj CR mJ SR / mJ CR norm. resolved/merged mjj SR / mjj CR mJ SR / mJ CR norm. resolved/merged mjj SR / mjj CR mJ SR / mJ CR 26% 7% 5% S S 31% 19% 6% 8% 20% 28% 4% 14% 19% 23% 6% 5% Table 5.1: The nuisance parameters included in our search that come from MC systematic uncertainties, grouped by physics process. The value column either lists the prior of each NP or lists if the NP is left to float (F) or is a shape (S) systematic. can probe a given model, which will change the limits we set later. The uncertainties in the acceptance of the signal process were derived by varying the renormalization and factorization scales by a factor of two. In addition, the nominal PDF set was changed to the MSTW2008 LO PDF set and the tuned parameters were varied according to the eigentune method [41]. The total variation for the signal is less than 3% for all signal masses. Most of the backgrounds have several NPs related to their normalization as well as several modeling uncertainties that affect the background’s shape. The idea behind several of these is fairly similar, so we will talk about what several of them mean here and discuss how they 102 are derived in their own paragraphs. In addition, any unique uncertainties will be left to that background’s paragraph. Each background is given an overall normalization NP that controls its total normaliza- tion across all regions simultaneously. Adjusting this does not cause migration between re- gions or bins, it simply makes the overall size of a background larger or smaller. In addition, most backgrounds are given normalization uncertainties that control the ratio of resolved events to merged events and the ratio of events in the resolved (merged) SR to events in the resolved (merged) CR. Increasing (decreasing) the resolved to merged ratio simply causes the background in the merged region to become smaller (larger), while increasing (decreasing) the SR to CR ratio causes the background in the SR to increase (decrease). The modeling uncertainties for the backgrounds come from the fact that different pro- grams for creating MC give the backgrounds different normalizations and shapes. These uncertainties are derived by changing which program we used to do certain parts of MC creation and comparing the results to the nominal background. The parts of MC creation that we change are the generator (gen.), the PDF set, the cross-section calculator (scale), the parton showering and hadronization (PS), the initial- and final-state radiation (ISR/FSR), and the matrix element calculation (ME). Note that we don’t change these for every back- ground, only backgrounds where we know there are significant differences in the results of the programs. The t¯t and single-top modeling uncertainties were derived as follows [72]. The parton shower, hadronization, and underlying-event model were varied by replacing Pythia 6.428 with Herwigg++ [73] with the UE-EE-5 tune and the CTEQ6L1 PDF set [74]. Differences in the matrix element calculation were assessed by comparing a sample where Powheg was re- placed by MG5 aMC. Variations to the amount of initial- and final-state radiation (ISR/FSR) 103 were made by changing the renormalization and factorization scales by a factor of two and changing to the associated Perugia 2012 tunes. The variations from the pT reweighting of the t¯t samples (Sec. 4.2.2.2) were included as a symmetrized shape uncertainty. The V+jets modeling uncertainties were derived similarly. The PDF set used in the nominal samples were replaced by alternative sets: the hundred NNPDF 3.0 NNLO replicas, the MMHT2014 NNLO set, and the CT14 NNLO set. Variations to the amount of ISR/FSR were made by scaling the renormalization and factorization scales by a factor of two. Finally, a variation was made using MG5 aMC v2.2.2 interfaced to Pythia 8.186 using the A14 tune together with the NNPDF 2.3 LO PDF set. A 50% uncertainty to the normalization of the multijet background was estimated from the fit to the MET distribution described in App. A. A shape variation was also included due to uncertainties in the determination of the template fit in the anti-isolated lepton region. Normalization uncertainties for t¯t, single-top, and V+jets were derived by summing the relative yield variations between the different regions in quadrature. Finally, a normalization uncertainty is assigned to the remaining small backgrounds. For the diboson background this was 11% [75], while for SM VH this was 50%. Due to the nature of some systematics, they must be smoothed or averaged and sym- metrized before entering the fit. Other systematics might need to be removed from the fit entirely, a process referred to as pruning. These processes are described in the next few sections. 5.2.1 Smoothing Systematic uncertainties that are derived by changing attributes of an event instead of chang- ing an event’s weight can cause migration between bins. This effect on events with large 104 weights or samples with low statistics can introduce noise into the evaluation of system- atic uncertainties. To minimize this noise, two smoothing algorithms are applied to these systematic uncertainties. The first algorithm has three steps. The first step is to locate the extrema in the ratio of the nominal to systematic variation distributions. The second step is to merge bins between the extrema together until only one extrema is found in the ratio. Finally, the smoothed ratio is applied to the nominal distribution to create the smoothed systematic uncertainty. The second algorithm has two steps. The first step takes the smoothed systematic dis- tribution and, starting from the right, merges bins together until the statistical uncertainty of the bin is less than 5%. In each of these bins, the integrals of the nominal and systematic variation are compared to create the ±1σ variations. 5.2.2 Averaging and Symmetrization The majority of systematic uncertainties have up and down variations, but some only have one or the other. An example would be the jet ID efficiency, which determines how many jets we have in an event. At run-time we can lower this efficiency to lose jets, but we can- not increase it to gain jets. For systematic uncertainties like these, we assume the nominal value lies symmetrically between the up and down variations. This is confirmed for system- atic variations from samples with a significant number of simulated events. Therefore we symmetrize systematic uncertainties with only one variation around the nominal value, i.e. σdown = nom − σup. For systematic variations with low sample sizes, or modeling uncertainties that don’t have well defined up and down variations, the up and down variations may not be symmetric. This can be problematic for the stability of the fits mentioned in Sec. 5.3, so it would be 105 best to symmetrize these systematic uncertainties. To do this we first average the up and down variations, σavg up = (σup − σdown)/2 + nom, then symmetrize them. 5.2.3 Pruning Some systematic uncertainties have a negligible effect on the fit, while others derived from low statistics regions can introduce noise into the fit and fit stability/speed. These systematic uncertainties are removed by applying the following criteria. For normalization systematic uncertainties, we remove them if the variation is 0.5% or less, or if both up and down variations have the same sign. For shape systematic uncer- tainties, we remove them if fewer than two bins have a deviation over 1% after the overall normalization is removed, or if only one of the up/down variations is non-zero. 5.3 Conditional & Unconditional Fits While setting limits it is important to fit the MC to the data as best it can within the uncer- tainty of the MC. For each signal mass, we will iteratively scale the signal rate up and down to derive limits. So while producing limits for this search, we needed to continually fit the MC to the data. Our limit setting procedure required two types of fits to be done: conditional and unconditional fits. Both of these fits used the likelihood function from Eq. 5.4, Lconditional = L(µ, Lunconditional = L(ˆµ, ˆˆ−→ θ ) ˆ−→ θ ) . (5.5) For the conditional fit we fixed the signal scale factor, µ, and allowed the nuisance parameters, −→ θ , to float, while for the unconditional fit we allowed both to float. To perform the fits, 106 we maximized the appropriate likelihood functions (shown in Eq. 5.5), where the hat and double-hat notations represent estimators for the parameters that were allowed to float. At the end of these fits ˆµ and ˆ−→ θ will be the post-fit values of the signal scale factor and nuisance parameters for the unconditional fit, while ˆˆ−→ θ will be the post-fit values of the nuisance parameters for the conditional fit. For the conditional fit, the value of µ is set by the experimenter and does not change during the fit. 5.3.1 Fit Regions There are many regions in this analysis, but not all of them are used in the fits. All of the signal regions are included, but only two of the control regions are used. The regions included are summarized in Table 5.2. In the resolved regime, both the one and two b-tag signal regions are used. In the merged regime, both the one and two b-tag signal regions with no additional b-tags outside the leading large-R jet were used. The control regions included in the fits are the resolved one and two b-tag regions. These were included to constrain the W+jets background in the signal regions. Resolved Merged 1 b-tag SR/CR 2 b-tags SR/CR SR w/ 0 add. b-tags SR w/ 0 add. b-tags Table 5.2: The regions included in the fit are shown above. All the signal regions were in- cluded, while only the control regions from the resolved portion of the analysis were included. 5.3.2 Example Fit Results While the conditional and unconditional fits are done multiple times per signal mass point, we like to visually confirm that the fit is working correctly at least once. Figs. 5.1 and 107 5.2 show the post-fit mW H distributions of the signal and control regions included in the unconditional fit for the 1.5 TeV signal. Comparing the ratios in these post-fit plots to those of Sec. 4.6, we can see that the agreement between data and MC has improved. In the case of the resolved control regions, we can see that the bump from approximately 200–800 GeV, which was caused by Sherpa’s underestimation of the V+jets background, is now gone. Overall, the agreement between the data and the MC is very good after this fit, so we can confirm that the fitting algorithms are working correctly. We also like to check how the fit has changed the mean and variance of the NPs (Fig. 5.3), as well as the correlation between the NPs (Fig. 5.4). From the NP ranking plots, we can see that the mean of most NPs were not affected much, the exception being some of the normalization nuisance parameters which was expected. The covariance matrix for these ranked NPs show us that most of them are not highly correlated or anti-correlated∗. ∗The high correlation or anti-correlation of some nuisance parameters can be explained, but will not be here. 108 Figure 5.1: The mV H distributions of the signal regions included in the unconditional fit for the 1.5 TeV signal. On the top are the resolved signal regions. On the bottom are the merged signal regions where there are no additional b-tags outside the leading large-R jet. On the left are the 1 b-tag regions and on the right are the 2 b-tag regions. Note that the signal is not shown because it is essentially zero. 109 Events / 100 GeV2−101−10110210310410510610710810910datattsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfmultijetSM VHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36.1 fbs [GeV]VHm310Data / Pred0.811.2Events / 100 GeV110210310410510datattsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfmultijetSM VHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36.1 fbs [GeV]VHm310Data / Pred0.811.2Events / 100 GeV1−10110210310410510610710datattsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfSM VHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36.1 fbs [GeV]VHm310Data / Pred0.811.2Events / 100 GeV1−10110210310410510datattsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfSM VHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36.1 fbs [GeV]VHm310Data / Pred0.811.2 Figure 5.2: The mW H distributions of the control regions included in the unconditional fit for the 1.5 TeV signal. Both are resolved control regions. On the left is the 1 b-tag control region and on the right is the 2 b-tag control region. Note that the signal is not shown because it is essentially zero. 110 Events / 100 GeV1−10110210310410510610710810datattsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfmultijetSM VHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36.1 fbsmBB [GeV]310Data / Pred0.811.2Events / 100 GeV1−10110210310410510610datattsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfmultijetSM VHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36.1 fbsmBB [GeV]310Data / Pred0.811.2 Figure 5.3: Ranked NP pulls for an unconditional fit to the 500 (top-left), 1000 (top-right), 1500 (bottom-left), and 2000 (bottom-right) GeV signals. The yellow bands correspond to the pre-fit impact of each NP. 111 2−1−012alpha_SysMUON_EFF_SYSalpha_IFSRalpha_XS_STopalpha_SysFT_EFF_Eigen_B_0_AntiKt4EMTopoJetsalpha_SysJET_21NP_JET_EtaIntercalibration_NonClosurenorm_ttbar_L1alpha_ttbar_MerResRatio_L1alpha_Luminosityalpha_XS_VH125alpha_SysMODEL_TTbar_Herwigalpha_SysNNLORWalpha_ttbar_ResSRmBBcrRationorm_Wclblalpha_SysFT_EFF_Eigen_C_0_AntiKt4EMTopoJetsalpha_SysMODEL_TTbar_aMcAtNloµ∆0.2−0.15−0.1−0.05−00.050.10.150.21.5−1−0.5−00.511.5θ∆)/0θ (cid:173) θPull: (Normalisationµ Postfit Impact on σ+1µ Postfit Impact on σ(cid:173)1ATLAS = 13 TeVs(cid:173)1 Ldt = 36.1 fb∫=500 GeVVHm4−3−2−1−012345alpha_SysJET_21NP_JET_EtaIntercalibration_NonClosurealpha_ttbar_ResSRmBBcrRatioalpha_STop_ResSRmBBcrRatioalpha_Whf_MerResRationorm_ttbar_L1alpha_IFSRalpha_XS_STopnorm_Wclblalpha_SysMODEL_TTbar_Herwigalpha_Luminosityalpha_SysMODEL_WHFJets_MadGraphalpha_SysNNLORWalpha_SysMODEL_TTbar_aMcAtNlonorm_Whfalpha_XS_VH125µ∆0.3−0.2−0.1−00.10.20.32−1−012θ∆)/0θ (cid:173) θPull: (Normalisationµ Postfit Impact on σ+1µ Postfit Impact on σ(cid:173)1ATLAS = 13 TeVs(cid:173)1 Ldt = 36.1 fb∫=1000 GeVVHm8−6−4−2−02468alpha_SysMODEL_Stop_radnorm_Wclblalpha_SysMODEL_WhlJets_MadGraphalpha_SysMODEL_TTbar_radalpha_STop_MerResRatioalpha_XS_VH125alpha_ttbar_ResSRmBBcrRatioalpha_IFSRalpha_SysFT_EFF_Eigen_C_0_AntiKt2PV0TrackJetsalpha_SysFATJET_Medium_JET_Comb_Baseline_Kinalpha_SysMODEL_TTbar_aMcAtNloalpha_ttbar_MerResRatio_L1alpha_Luminosityalpha_Whf_MerResRationorm_Whfµ∆0.2−0.1−00.10.20.34−3−2−1−012345θ∆)/0θ (cid:173) θPull: (Normalisationµ Postfit Impact on σ+1µ Postfit Impact on σ(cid:173)1ATLAS = 13 TeVs(cid:173)1 Ldt = 36.1 fb∫=1500 GeVVHm15−10−5−051015alpha_SysMODEL_TTbar_aMcAtNloalpha_XS_VH125alpha_IFSRalpha_XS_Wlalpha_SysMODEL_WhlJets_MadGraphalpha_SysNNLORWnorm_Whfalpha_SysMODEL_Stop_radalpha_SysFATJET_Medium_JET_Comb_Baseline_Kinalpha_SysFT_EFF_Eigen_Light_0_AntiKt2PV0TrackJetsalpha_Whf_MerResRatioalpha_Luminosityalpha_ttbar_MerResRatio_L1alpha_SysMODEL_WlJets_MadGraphalpha_SysMODEL_TTbar_radµ∆0.4−0.2−00.20.40.68−6−4−2−0246810θ∆)/0θ (cid:173) θPull: (Normalisationµ Postfit Impact on σ+1µ Postfit Impact on σ(cid:173)1ATLAS = 13 TeVs(cid:173)1 Ldt = 36.1 fb∫=2000 GeVVHm Figure 5.4: Ranked nuisance parameter’s covariance matrix for an unconditional fit to the 1500 GeV signal. Deep red means the NPs are highly correlated while deep blue means the NPs are highly anti-correlated. Note that this matrix is symmetrical about the diagonal and that not all NPs are shown. 112 norm_Wclblnorm_Whfnorm_ttbar_L1MODEL_TTbar_HerwigMODEL_TTbar_aMcAtNloFT_EFF_Eigen_B_0_AntiKt4EMTopoJetsFT_EFF_Eigen_C_0_AntiKt2PV0TrackJetsFT_EFF_Eigen_C_0_AntiKt4EMTopoJetsFT_EFF_Eigen_C_1_AntiKt2PV0TrackJetsFT_EFF_Eigen_C_1_AntiKt4EMTopoJetsJET_21NP_JET_Flavor_CompositionJET_21NP_JET_Pileup_PtTermJET_21NP_JET_Pileup_RhoTopologySTop_ResSRmBBcrRatioFATJET_Medium_JET_Comb_Baseline_KinMODEL_WHFJets_MadGraphNNLORWWhf_MerResRatioXS_STopttbar_MerResRatio_L1ttbar_ResSRmBBcrRatiottbar_ResSRmBBcrRatiottbar_MerResRatio_L1XS_STopWhf_MerResRatioNNLORWMODEL_WHFJets_MadGraphFATJET_Medium_JET_Comb_Baseline_KinSTop_ResSRmBBcrRatioJET_21NP_JET_Pileup_RhoTopologyJET_21NP_JET_Pileup_PtTermJET_21NP_JET_Flavor_CompositionFT_EFF_Eigen_C_1_AntiKt4EMTopoJetsFT_EFF_Eigen_C_1_AntiKt2PV0TrackJetsFT_EFF_Eigen_C_0_AntiKt4EMTopoJetsFT_EFF_Eigen_C_0_AntiKt2PV0TrackJetsFT_EFF_Eigen_B_0_AntiKt4EMTopoJetsMODEL_TTbar_aMcAtNloMODEL_TTbar_Herwignorm_ttbar_L1norm_Whfnorm_Wclbl1−0.8−0.6−0.4−0.2−00.20.40.60.81 5.4 Limit Calculation We are finally ready to evaluate an upper limit on the parameter of interest (POI), which is related to the cross-section of the observed signal. For comparison purposes, we will produce two sets of limits. One set of limits will represent what is expected if the only physics was that of the SM, while the other set will be from what was actually observed. These are called the expected and observed limits respectively. In traditional statistics, the upper limit is calculated by finding the value of the POI that matches the following criterion. Given a value of the POI and the observed value of some test statistic, calculate the probability of getting this observed value or a more extreme one. This is called the p-value or ps+b. Search for the POI that has this probability equal to α, which is some percentage set by the experimenter. In our case the test statistic is the Log-Likelihood-Ratio (described in Sec. 5.4.1), our parameter of interest is the signal rate (denoted as µ), and the value of α is 5%. In addition, our field changes the way the upper limit is calculated by finding the value of the POI for which the CLs [76] is equal to α. The CLs is a modification of the p-value, given by the equation CLs = ps+b 1 − pb . (5.6) Here ps+b is the p-value of the signal+background hypothesis, which is also the p-value mentioned above, and pb is the p-value of the background-only hypothesis. The added benefit of using CLs instead of ps+b, is that it protects us from both significant excesses of data and significant deficits, whereas ps+b only protects us from excesses. If we were only using ps+b, then an excess of data relative to our backgrounds would increase the limits we set later. However, if we had a deficit then ps+b would set a limit that was lower 113 than it should be. By dividing by 1 − pb, we can protect ourselves from deficits as well, i.e., pb approaches one for more significant deficits so 1 − pb approaches zero and CLs blows up. This behavior helps take potential mis-modeling in the background and fluctuations in the data into account. It can also give us a more conservative upper limit that hopefully matches better across different experiments, i.e. different data or potentially different backgrounds. 5.4.1 Log Likelihood Ratio To derive our test statistic for differentiating hypotheses, we will start with the likelihood function (Eq. 5.4). To start we simply divide the likelihood of a conditional fit by that of an unconditional fit, which we call the likelihood ratio, LR = L(µ, L(ˆµ, N ) ˆˆ−→ θ |−→ ˆ−→ θ |−→ N ) . (5.7) The behavior of this function is nice because it is bounded between zero and one. For mathematical and computational purposes we use the negative log-likelihood ratio, LLR = −2 × ln(LR). , (5.8) which we simply call the Log-Likelihood-Ratio. Taking the log is beneficial because it gets rid of exponents, turns division into addition, and turns the function into a χ2 distribution, each of which speeds up the rate of computation. Taking the negative of the log simply means we search for minima instead of maxima, which is necessary for using gradient descent. The Log-Likelihood-Ratio is the test statistic we have chosen to use for differentiating our signal+background and background only hypotheses. We will use it in the next section 114 to calculate the various p-values which are used to calculate CLs. 5.4.2 Preparing to Calculate ps+b and pb As suggested in the previous section, before we can calculate CLs we first need ps+b and pb. To calculate these we need to produce p.d.f.s for the Log-Likelihood-Ratio of our two hypotheses, signal+background and background only, for each value of µ. The former will be denoted∗ as fs+b(LLR, µ) and the latter as fb(LLR, µ). Normally to create these distributions, we would run pseudo-experiments for each value of µ by sampling −→ N with −→ θ set by the conditional fit. This would be done for each hypothesis independently. However, running pseudo-experiments like this takes a lot of time. Instead, we can do an asymptotic approximation of these two distributions by simply calculating the Log-Likelihood-Ratio for each hypothesis. We happen to know that our LLR distributions, fs+b(LLR, µ) and fb(LLR, µ), are both Gaussian distributed with their mean and variance approximately equal to the value of the LLR for each hypothesis. This approximation is much faster than running tens-of-thousands of pseudo-experiments and will yield a final result that is not significantly different. ∗While denoting these distributions as fs+b(LLR) and fb(LLR) would be sufficient (since the LLR depends on µ), I will include µ in the list of variables to be explicit that we have one distribution for each value of µ. 115 5.4.3 Expected Limits To calculate the expected limit for one value of µ, first we found the median of the fb(LLR, µ) distribution, LLRexp. Then we calculated pb by integrating everything in the fb(LLR, µ) distribution that is below the median, pb = (cid:82) LLRexp −∞ fb(LLR, µ)d(LLR). Because we are integrating to the median, pb should be close to 50%. Next we calculated ps+b by integrating everything in the fs+b(LLR, µ) distribution that is above LLRexp, ps+b = (cid:82) ∞ LLRexp fs+b(LLR, µ)d(LLR). Finally, we calculated CLs = ps+b 1−pb . If CLs was not 5%, then we adjusted µ until it was using gradient descent. Increasing µ decreases CLs while decreasing µ increases CLs. The expected upper limit is whichever µ gave us a CLs = 0.05. To create the one and two σ bands of the expected limit, repeat the steps above, but using the ±1σ and ±2σ values of fb(LLR, µ) instead of the median. We did this for each signal mass point to find the expected upper limit for each. The final product of this step is a signal mass dependent curve of the expected limit. We added one and two σ bands to this by repeating the steps above using the ±1σ and ±2σ values of fb(LLR, µ) instead of the median. 5.4.4 Observed Limits To calculate the observed limit for one value of µ, we first found the observed value of the LLR, LLRobs. This is simply the value of the LLR that came from an unconditional fit of the MC to the data. Then we calculated ps+b and pb using the same integrals as in Sec. 5.4.3 but integrating to LLRobs instead of LLRexp. After which we calculated the value of CLs = ps+b 1−pb . We then 116 adjusted µ using gradient descent until CLs = 0.05, which gave us the observed upper limit on µ. We did this for each signal mass point to produce an observed limit curve. 5.5 Results We can take the expected and observed limit curves produced in the last two sections and express them in several ways. The simplest way would be to plot the values of µ per mass point, but this doesn’t tell us much since µ is just a scale factor. A more informative parameter would be the cross-section, so we plot the cross-section limits as a function of signal mass. This is done by taking the upper limits on µ and multiplying them by the relevant signal cross-section per mass point. The results of the limit setting procedure can be seen in Fig. 5.5. Looking to the limits, there is a visible 1σ deficit at about 800 GeV, a 1σ excess from 1600–1700 GeV, a 2σ deficit from 1800–2200 GeV, and a 1σ excess from about 3–4 TeV. Looking to the p-values, there is a 2σ excess at about 3 TeV. 5.5.1 Interpretation In order to claim discovery of a W(cid:48), we need at least a 5σ excess in ps+b. If instead we’d like to claim we found evidence of a W(cid:48), we only need an excess of 3σ. However, it is clear there are no 3 or 5σ excesses in Fig. 5.5. This means we have not discovered or found evidence of a W(cid:48). So instead, we will simply extend the range that the previous analyses excluded HVT Models A and B. Looking to Fig. 5.5, we can see both the observed cross-section limits and the theoretical cross-sections of HVT Models A and B. If Models A or B were true, then we would 117 Figure 5.5: This figure shows the limits from the limit setting procedure. The dashed, blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yellow bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid, black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the solid, red and magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. expect the observed limits to meet or exceed them at the true mass of the W(cid:48). So masses where the observed cross-section limit is below the theoretical value are said to be excluded. This means that HVT Model A is now excluded up to 2.6 TeV, while HVT Model B is excluded up to 2.8 TeV. 118 (GeV)W’m500100015002000250030003500400045005000) [pb]c,cb b→ BR(H × WH) → BR(W’ × σ3−102−101−101 InternalATLAS Analysisbb+Jνl(cid:173)1 L dt = 36.1 fb∫ = 13 TeV sObservedExpected1 s.d.±Expected 2 s.d.±Expected HVT Model AHVT Model B Figure 5.6: This figure shows the p-values for the background only hypothesis. This plot is usually used as a measure of discovery, where a lower p-value means we can reject the background hypothesis. In this field, a p-value equivalent to 3σ is necessary to claim we’ve seen evidence and a value equivalent to 5σ is needed to claim discovery. Clearly this search reached neither of those values, so instead we set limits on the benchmark models instead. 119 (GeV)W’m1000150020002500300035004000450050000Local p3−102−101−10110210σ1σ2σ3 InternalATLAS Analysisbb+Jνl(cid:173)1 L dt = 36.1 fb∫ = 13 TeV sObservedExpected (W’ Signal) Chapter 6 Extended Interpretations In the previous chapters, I only considered the W(cid:48) → W H → lνb¯b process. However this is not the only particle an HVT model can predict or the only decay mode available. For instance, a W(cid:48) could also decay to W H → q ¯qb¯b. The potential existence of a Z(cid:48) was also mentioned in Sec. 2.2.3 and line two of Eq. 2.32 introduced interactions with both the electroweak bosons and the fermions. This means that a V (cid:48) can decay to more than just a V and H; it can also decay to two electroweak bosons, or two fermions. These other HVT decay modes were the search targets of analyses performed by other groups at ATLAS. While it’s easier to design these analyses around a search for a specific decay mode, this lowers the sensitivity to the total signal. In order to increase that sensitivity, there are methods to perform a statistical combination of the results of different analyses after they have been performed individually. These combinations can be done in incremental steps, combining the most similar decay modes first and the least similar decay modes last. Combining the searches in this format causes the combinations to become increasingly model dependent. This chapter covers the combinations that include our search for the W(cid:48) → W H → lνbb decay mode. An important consideration when doing a combination is whether the analyses being combined are orthogonal or not. In this context orthogonal means that there are no events that are common between the analyses being combined. If the analyses are orthogonal, 120 nothing extra needs to be done. If they are not orthogonal, the overlapping events need to be removed from all but one of the included analyses, creating mutually exclusive event selections. 6.1 Semi-Leptonic VH Combination The analysis described in this dissertation was designed with the intention of being com- bined with two other analyses: searches for the Z(cid:48) → ZH → ννbb and Z(cid:48) → ZH → llbb decay modes. These three analyses were designed together to maximize the sensitivity of the combined result. Because of this, these three analyses have similar selections and are orthogonal by design. The orthogonality condition is the exclusive number of charged lep- tons∗ in the final state, which motivated their labels of 0-lepton (ννbb), 1-lepton (lνbb), and 2-lepton (llbb). Each of these analyses are described as being “semi-leptonic” because their final states include both leptons and jets. A quick description of the 0- and 2-lepton channels, as well as their individual limits, follows. 6.1.1 0-Lepton The 0-lepton analysis is designed to search for the Z(cid:48) → ZH → ννbb decay mode, so the data and MC sample selection require exactly zero charged leptons per event. As a result, instead of event selection cuts based on one lepton and Emiss T , they are based on there being a larger amount of Emiss T . Additional cuts are added and/or modified to account for the different topology of the signal. The event selection for this analysis is summarized in Table 6.1 and the post-fit mV H distributions are shown in Fig. 6.1. ∗It is important to note that while neutrinos are leptons, ATLAS cannot detect them directly. So for the purposes of naming these analyses, the term lepton is reserved for counting the number of charged leptons. 121 Resolved ≥ 2 small-R jets Merged ≥ 1 large-R jet > 45 110–140 > 150(120∗) < 7π/9 > 150 > 250 75–145 – – > 200 > 30† < π/3 > 2π/3 Variable Number of Jets Leading jet pT [GeV] mjj, mJ [GeV] (cid:80) pjet T [GeV] T ∆φ(j, j) Emiss [GeV] pmiss −→ [GeV] ,−→p miss T E miss −→ ∆φ( E miss −→ , h) ∆φ( T E miss min[∆φ( T T ) ,small-R jets)] > π/9 (2 or 3 jets), > π/6 (≥ 4 jets) T Table 6.1: The event selection for the 0-lepton (ZH → ν ¯νb¯b) channel. It is designed to select a signal with large Emiss in the final state, as well as jets which are potentially b-tagged. T The upper limits from the 0-lepton analysis are shown in Fig. 6.2. In general there is a 1–2σ deficit starting at low signal masses and extending to high signal masses. The results yield a lower mass limit for the HVT Model A of 2.6 TeV, while Model B has been excluded up to about 2.8 TeV. 6.1.2 2-Lepton The 2-lepton analysis is designed to search for the Z(cid:48) → ZH → llbb decay mode, so the data and MC sample selections require exactly two same flavor, charged leptons per event. As a result, the event selection cuts are based around having two charged leptons in an event. The event selection for this analysis is summarized in Table 6.2 and the post-fit mV H distributions are shown in Fig. 6.3. The upper limits from the 2-lepton analysis are shown in Fig. 6.4. Notable features are a 2σ deficit at 800 GeV and a 1σ deficit starting at about 3 TeV. The results yield a lower mass limit for the HVT Model A of 2 TeV, while Model B has been excluded up to about 2.2 TeV. 122 Figure 6.1: The post-fit mV H distributions of the 0-Lepton analysis. On the left are the 1 b-tag regions and on the right are the 2 b-tag regions. On the top are the resolved signal regions and on the bottom are the merged signal regions. 6.1.3 Combined Results Before combining these three channels, it is important to consider that leptons are not always reconstructed in an event. So it is possible for events that are truly llbb, which would normally be selected by the 2-lepton analysis, to migrate to the 1- or 0-lepton analyses if 123 Events / 100 GeV4−103−102−101−101102103104105106107108109101010111012101310data1.5 TeV HVT x 10ttsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfSM VHttVttHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36 fbs < 140 GeVH110 GeV < m0 lep., 1 b(cid:173)tag resolved [GeV]T,VHm310Data / Pred00.511.52Events / 100 GeV4−103−102−101−1011021031041051061071081091010101110data1.5 TeV HVT x 10ttsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfSM VHttVttHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36 fbs < 140 GeVH110 GeV < m0 lep., 2 b(cid:173)tags resolved [GeV]T,VHm310Data / Pred00.511.52Events / 100 GeV4−103−102−101−1011021031041051061071081091010101110data1.5 TeV HVT x 10ttsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfSM VHttVttHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36 fbs < 145 GeVH75 GeV < m0 lep., 1 b(cid:173)tag merged [GeV]T,VHm310Data / Pred00.511.52Events / 100 GeV4−103−102−101−10110210310410510610710810910data1.5 TeV HVT x 10ttsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfSM VHttVttHuncertaintypre(cid:173)fitATLASInternal(cid:173)1 = 13 TeV, 36 fbs < 145 GeVH75 GeV < m0 lep., 2 b(cid:173)tags merged [GeV]T,VHm310Data / Pred00.511.52 Figure 6.2: Upper limits on σ × BR for the 0-lepton analysis, as a function of signal mass. The dashed blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yellow bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the solid red and magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. Variable Number of Jets Leading jet pT [GeV] mjj, mJ [GeV] Leading lepton pT [GeV] Sub-leading lepton pT [GeV] pT ll [GeV] √ mll [GeV] Emiss / HT [GeV] T Resolved ≥ 2 small-R jets > 45 Merged ≥ 1 large-R jet > 250 75–145 > 27 100–145 > 27 > 7 > 20 + 9(cid:112)mV H /(1 GeV − 320 25 [max[40 GeV, 87 − 0.030 · mV H ], 97 + 0.013 · mV H ] < 1.15 + 8 × 10−3 · mV H /(1 GeV) Table 6.2: The event selection for the 2-lepton (ZH → l¯lb¯b) channel. It is designed to select a signal with two leptons in the final state, as well as jets which are potentially b-tagged. one or both of the leptons are mis-reconstructed. It is also possible for events that are truly lνbb, which would normally be selected by the 1-lepton analysis, to migrate to the 0-lepton analysis if the lepton is mis-reconstructed. It is for these reasons that some of the Z(cid:48) signal 124 (GeV)Z'm500100015002000250030003500400045005000) [pb]b bfi BR(H · ZH) fi Z' fi(pp s3-102-101-10110 InternalATLAS-1 L dt = 36.1 fb(cid:242) = 13 TeV sObserved limitExpected limit1 s.d.–Expected 2 s.d.–Expected =1VHVT Model A, g=3VHVT Model B, g Figure 6.3: The post-fit mV H distributions of the 2-Lepton analysis. On the left are the 1 b-tag regions and on the right are the 2 b-tag regions. On the top are the resolved signal regions and on the bottom are the merged signal regions. It should be noted that the mH ranges indicated on the resolved region plots are incorrect, instead they should read 110 GeV < mH < 145 GeV. can “leak” into the 1-lepton analysis, while some of the W(cid:48) signal can “leak” into the 0-lepton analysis. This leakage is dealt with by passing the 2-Lepton signal through the 0- and 1-Lepton analyses and the 1-Lepton signal through the 0-lepton analysis. Table 6.3 summarizes which signals should be passed through the 0-, 1-, and 2-Lepton event selections for the semi- leptonic W(cid:48), Z(cid:48), and V (cid:48) combinations. However, the amount of 2-Lepton signal that enters the 0- and 1-Lepton analyses is negligible, so including them does not increase our sensitivity 125 Events / 100 GeV1−10110210310410510610710810data1.5 TeV HVT x 10ttsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfSM VHttVttHuncertaintypre-fitATLASInternal-1 = 13 TeV, 36 fbs < 145 GeVH75 GeV < mllJ, 1 b-tag resolved [GeV]VHm310Data / Pred00.511.52Events / 100 GeV110210310410510610data1.5 TeV HVT x 10ttsingle topdibosonW+hlW+hfZ+lZ+hlZ+hfSM VHttVttHuncertaintypre-fitATLASInternal-1 = 13 TeV, 36 fbs < 145 GeVH75 GeV < mllJ, 2 b-tags resolved [GeV]VHm310Data / Pred00.511.52Events / 100 GeV1−10110210310410data1.5 TeV HVT x 10ttsingle topdibosonW+lW+hlW+hfZ+lZ+hlZ+hfSM VHttVttHuncertaintypre-fitATLASInternal-1 = 13 TeV, 36 fbs < 145 GeVH75 GeV < mllJ, 1 b-tag merged [GeV]VHm310Data / Pred00.511.52Events / 100 GeV1−10110210data1.5 TeV HVT x 10ttsingle topdibosonW+hfZ+lZ+hlZ+hfSM VHttVttHuncertaintypre-fitATLASInternal-1 = 13 TeV, 36 fbs < 145 GeVH75 GeV < mllJ, 2 b-tags merged [GeV]VHm310Data / Pred00.511.52 Figure 6.4: Upper limits on σ × BR for the 2-lepton analysis, as a function of signal mass. The dashed blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yellow bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the solid red and magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. to the signal. For this reason, the 2-Lepton signal was not passed through the 0- or 1-Lepton selections for the following combinations. The upper limits on the cross-section as a function of V (cid:48) mass for the semi-leptonic Z(cid:48) combination and the semi-leptonic W(cid:48) combination are shown in Figs. 6.5 and 6.6. It shows that HVT Model A has been excluded up to 2.6 TeV for both the W(cid:48) and Z(cid:48) separately. In addition, HVT Model B has been excluded up to about 2.8 TeV for both the W(cid:48) and Z(cid:48) separately. Up until now we have presented limits on the production cross-section of a W(cid:48) or Z(cid:48) times the branching-ratio of the channel we are interested in (Z(cid:48) → ZH or W(cid:48) → W H). We did this because it was easy to translate the limits on our parameter of interest from the statistical 126 (GeV)Z'm500100015002000250030003500400045005000) [pb]b bfi BR(H · ZH) fi Z' fi(pp s2-101-101 InternalATLAS-1 L dt = 36 fb(cid:242) = 13 TeV sObserved limitExpected limit1 s.d.–Expected 2 s.d.–Expected =1VHVT Model A, g=3VHVT Model B, g Z(cid:48) → ZH → ννbb W(cid:48) → W H → lνbb Z(cid:48) → ZH → llbb 0-Lepton 1-Lepton 2-Lepton yes no no yes yes no yes yes yes Table 6.3: A table summarizing which signal samples should be passed through the 0-, 1-, and 2-Lepton event selections for the combinations. The 2-lepton signal is passed through all event selections, the 1-Lepton signal is passed through the 0- and 1-Lepton selections, and the 0-Lepton signal is only passed through the 0-Lepton selection. Figure 6.5: Upper limits on σ × BR for the semi-leptonic W(cid:48) combinations, as a function of signal mass. The dashed blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yellow bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. analysis to this parameter of the model and it was more meaningful to theorists/third-parties to have a limit on the cross-section times branching-ratio than on some arbitrary scaling factor. This translation was only possible because we had only one channel or the channels being combined had the same production cross-section and branching-ratio. However, from this point forward we will be combining channels that have different 127 [GeV]W’m500100015002000250030003500400045005000) [pb]c,cbb→B(h⋅Wh)→W’→(ppσ4−103−102−101−10110 ATLAS-1 = 13 TeV, 36.1 fbs95% CL limitObserved (CLs)Expected (CLs)σ1±Expected σ2±Expected =1VHVT Model A, g=3VHVT Model B, g Figure 6.6: Upper limits on σ × BR for the semi-leptonic Z(cid:48) combinations, as a function of signal mass. The dashed blue line is the curve of the expected limits as a function of mass produced in Sec. 5.4.3. The green and yellow bands are the 1 and 2σ bands of the expected limits produced in the same section. The solid black line is the curve of the observed limits as a function of mass produced in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. production cross-sections and branching-ratios, so this translation is no longer possible for a limit that is solely a function of signal mass. If we want a limit that is solely a function of signal mass, we will have to make some model-dependent assumptions about the different cross-sections and branching-ratios. For now on, we are fixing the ratios of the cross-sections and branching-ratios involved to the HVT Model A predictions and scaling the total signal all at once. Stated another way, we are not scaling the channel’s production cross-sections and branching-ratios independently of each other, so we cannot set a limit on any one in particular. Instead, we are setting a limit on some arbitrarily picked linear combination of the various production cross-sections and branching-ratios. Since this new value is somewhat arbitrary 128 [GeV]Z’m500100015002000250030003500400045005000) [pb]c,cbb→B(h⋅Zh)→Z’→(ppσ4−103−102−101−10110 ATLAS-1 = 13 TeV, 36.1 fbs95% CL limitObserved (CLs)Expected (CLs)σ1±Expected σ2±Expected =1VHVT Model A, g=3VHVT Model B, g and we don’t want it interpreted in the wrong way, we simply call it the 95% CL Upper Limit and divide it by the total cross-section for all channels involved. So for now on, our limits will be expressed as the 95% CL Upper Limit divided by the total cross-section for all channels involved, as a function of signal mass. The 95% CL Upper Limits on σ × BR/σHVT Model A as a function of V (cid:48) mass of the semi-leptonic V (cid:48) combination are shown in Fig. 6.7. It shows that HVT Model A has been excluded up to about 2.8 TeV, while HVT Model B has been excluded up to about 2.9 TeV. Figure 6.7: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the semi-leptonic VH combinations, as a function of signal mass. The dashed blue line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. 129 [GeV]V’m500100015002000250030003500400045005000Vh) →V’→(ppσ95% CL Upper Limit / 2−101−10110210310 ATLAS-1 = 13 TeV, 36.1 fbs95% CL limitObserved (CLs)Expected (CLs)σ1±Expected σ2±Expected =1VHVT Model A, g=3VHVT Model B, g 6.2 Combination with Other Final States In addition to the semi-leptonic VH combination, there were other combinations performed that include the search described in this dissertation. Many of the analyses that were in- cluded in these combinations started at different times, so the full set were not designed in conjunction with each other. This means that the searches were not orthogonal with each other by default, an issue described below. The first “extended combination” was with the fully-hadronic VH search channel (V (cid:48) → V H → qqbb), which is similar to the semi-leptonic channels except the final state is only composed of jets. Then there was the combination of the VH search channels with the VV search channels (V (cid:48) → V V ), which are referred to as the diboson channels collectively. The last combination was the VV and VH search channels with the dilepton search channels (V (cid:48) → ll, lν). These combinations are explored in the following sections. 6.2.1 Semi-Leptonic and Hadronic VH Combination The most similar decay channels to the semi-leptonic VH search channels are the hadronic VH search channels. The main difference between the two is that the W or Z decays to quarks instead of leptons. Because of this the cuts applied are quite different. This channel is already orthogonal to the semi-leptonic channels because it vetoes charged leptons and significant MET in its final state. The 95% CL upper limits on σ × BR/σHVT Model A as a function of V (cid:48) mass for the semi-leptonic and hadronic VH combination are shown in Fig. 6.8. Most notably, there is an ∼ 2σ deficit around 2.2 TeV and a ∼ 2σ excess around 3 TeV. It shows that HVT Model A is excluded up to about 2.8 TeV, while HVT Model B is excluded up to about 3 TeV. 130 Compared to the semi-leptonic VH combination, HVT Model A has not changed but the mass exclusion on HVT Model B has increased by 0.1 TeV. Figure 6.8: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the VH semi- leptonic and hadronic combination, as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. 6.2.2 VV Channels Up to this point of this dissertation, only decays of V (cid:48) → V H have been considered, but one should also expect V (cid:48) → V V decays in the HVT model with a roughly equal rate. For example, the W(cid:48) could decay to W Z or the Z(cid:48) could decay to W W . The W s and Zs could then decay leptonically or hadronically, so the final states could be fully leptonic, semi- 131 0.511.522.533.544.55m(V’) [TeV]2−101−10110210310HVT model Aσ/σObserved 95% CL limitExpected 95% CL limitσ 1±Expected σ 2±Expected HVT model AHVT model B(cid:173)1 = 13 TeV, 36.1 fbsATLAS WH + ZH→DY HVT V’ leptonic, or fully hadronic. All of these final states were included in the final combination of this dissertation, and a summary of the channels included is shown in Table 6.4. V (cid:48) Bosons Fermions W(cid:48) WZ Z(cid:48) WW ννqq lνqq llqq qqqq lνll lνqq qqqq lνlν Table 6.4: The various Diboson channels included in this combination. On the top are the W(cid:48) decay modes and on the bottom the Z(cid:48) decay modes. The only diboson state a W(cid:48) can decay to is W W , while Z(cid:48) can decay to W W . From there a W can decay to lv or qq, while a Z can decay to νν, ll, or qq. It is worth noting that these channels are not orthogonal with the VH channels, which have similar final state signatures, i.e., leptons, MET, and quarks. An explanation of how the VV and VH channels were made orthogonal, which is a study I performed, can be found in Sec. 6.2.2.1. The 95% CL upper limits on σ × BR/σHVT Model A as a function of V (cid:48) mass for the combination of the VV channels alone are shown in Fig. 6.9. While not significant, there are several 1σ excesses and deficits across the full range of signal masses. The figure shows that resonance masses up to about 3.6 TeV are excluded for HVT Model A, while they are excluded up to about 3.9 TeV for HVT Model B. Compared to the full VH combination, the VV combination sets more stringent limits that exclude HVT Model A at a mass 1.2 TeV higher than the VH limits and HVT Model B at a mass 0.6 TeV higher. 132 Figure 6.9: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the VV channels as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red and solid magenta lines are the theory cross-sections for our benchmark models per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. 6.2.2.1 Orthogonality Studies Among the channels included in the final combination of this dissertation, the only two that are not orthogonal are the VV and VH channels. This is because both the VV and VH channels require 0, 1, or 2 leptons, in addition to jets. Specifically, the analyses with the same number of charged leptons between VV and VH are not orthogonal while analyses with differing numbers of charged leptons are. To introduce orthogonality we must consider where the jets come from in V V as opposed to V H, or vice versa. In the case of the V V channels, the jets are coming from either a W 133 0.511.522.533.544.55m(V’) [TeV]2−101−10110210310HVT model Aσ/σObserved 95% CL limitExpected 95% CL limitσ 1±Expected σ 2±Expected HVT model AHVT model B(cid:173)1 = 13 TeV, 36.1 fbsATLAS WW + WZ→DY HVT V’ or Z boson, while in the V H channels, the jets come from a Higgs boson. To help us, we can make use of our knowledge that the W boson mass is peaked at about 80 GeV, the Z boson mass is peaked at about 91 GeV, and the Higgs boson mass is peaked at about 125 GeV. Looking back to the 1-lepton analysis (Sec. 4.4), there is a jet mass window that defines the signal region from 110-140 GeV in the resolved selection and 75-145 GeV in the merged selection. The 0- and 2-Lepton VH channels have similar mass windows to define their signal regions. The V V channels also use jet mass windows to define their signal regions, which include the W and Z masses. A fairly conservative estimate for the V V mass windows would be 66-106 GeV in the resolved selection and 64-106 GeV in the merged selection. Upon first inspection, one might think that making the merged selection mass windows orthogonal between V V and V H would be enough. However, the resolved dijet mass (mjj) and the merged large-R jet mass (mJ ) are related, so it is a bit more complicated. For instance, Chapter 4.5 mentions it is possible for an event to pass both the resolved and merged event selections, so if we only make the merged selection mass windows orthogonal then such an event could simply go into the resolved selection instead of being completely removed from the analysis. To visualize the problem, the mass windows have been depicted in Fig. 6.10. Looking to this figure, we can see an area of overlap between the V V and V H mass windows. Events in this area of overlap can potentially enter the signal regions of both channels. Please note the way these channels overlap is actually more complicated, as the VH search has regions for b-tags while the VV search has regions for vector boson tags (V-tags). So while the figure only shows 2 dimensions (mjj, mJ ), there are actually 2 more to be considered (# b-tags, # V-tags). For these channels to be orthogonal, the minimal 134 Figure 6.10: A representation of the kinematic phase space of the resolved dijet mass and merged large-R jet mass. The VV mass windows are shown in the top left and the VH mass windows in the top right. The two are shown on the same plot with the areas they overlap highlighted in gray on the bottom. Please note that in order to fit labels in the appropriate places, the figure is not to scale. requirement is no overlap in the 4D space (mjj, mJ , # b-tags, # V-tags). However, including V-tags and b-tags would have been very complicated and time consuming, so it was better to consider the more stringent (and overall simpler) 2D space (mjj vs. mJ ). Since this requirement is more stringent, it also meets the condition for orthogonality. The final decision on how to make these channels orthogonal, was for the VH channel to apply a new cut to its event selection. This cut simply removes any events inside the VV mass windows, which are depicted in red in Fig. 6.10. The effects of this cut on the upper limits in the VH channels can be seen in Figures 6.11, 6.12, and 6.13. 135 6610610664mjj (GeV)mJ (GeV)VV Resolved SRVV Mer. SR11014014575mjj (GeV)mJ (GeV)VH Resolved SRVH Mer. SR661061101401451067564mjj (GeV)mJ (GeV)VV Resolved SRVH Resolved SRVV Mer. SRVH Mer. SRAreas of Overlap The upper limits for the 0-Lepton VH analysis remain mostly unchanged at low masses except for a ∼10% decrease/improvement around 1.5 TeV, but the limits begin to in- crease/deteriorate starting at about 3 TeV. The upper limits for the 1-Lepton VH analysis increase/deteriorate by ∼10% from 500-1000 GeV, but then decrease/improve by ∼5-25% from 1-4 TeV before increasing/deteriorating again. The upper limits for the 2-Lepton VH analysis increase/improve about ∼5-10% from 500-1250 GeV, decrease/deteriorate by ∼5% from 1250-2100 GeV, and then increase/deteriorate again beginning at about 2.8 TeV. These changes to the analyses upper limits were deemed acceptable for a few reasons. Any deterioration of the limits at very low masses would not affect the mass exclusions which take place at higher masses. The 2-Lepton analysis is the smallest contributor of the VH channels, so its deterioration at high masses is negligible to the combined limits. The deterioration of the 0- and 1-Lepton limits at high masses is far above their own exclusion points. Events removed from the VH selection are not necessarily removed from the overall selection of the combination. The purpose of this cut was to remove overlapping events between the VV and VH selections, so it makes sense that a fraction of the events removed from the VH selection will be in the VV selection. 6.2.3 Dilepton Channels The last channels included in the final combination of this dissertation are the dilepton decays of V (cid:48). These are the channels where the V (cid:48) decays directly to two leptons (charged or neutral, but not τ s or tau-neutrinos). For the W(cid:48) this means a decay to lν, while for the Z(cid:48) it means a decay to ll (where the leptons must be of the same flavor.). While this channel is not explicitly made orthogonal with the VV and VH channels (for which a jet veto would be needed), there was no overlap between the data and negligible overlap between the MC 136 Figure 6.11: Comparison of the upper limits on the cross-section before (solid red) and after (dashed blue) the orthogonality cut on the VV mass window for the 0-Lepton VH search. samples. The 95% CL upper limits on σ × BR/σHVT Model A as a function of V (cid:48) mass for the dilepton channels are shown in Fig. 6.14. Because the fermion couplings are small in HVT Model B, dilepton decays are suppressed and therefore the HVT Model B curve does not appear in these plots. The limits show several 1-2σ deficits at masses below 3 GeV and that HVT Model A has been excluded up to resonance masses of about 5 TeV. 6.2.4 Combined Results The 95% CL upper limits on σ×BR/σHVT Model A as a function of V (cid:48) mass from combining all of the channels mentioned are shown in Figs. 6.15 - 6.17. The limits are split into scenarios of W(cid:48) and V (cid:48) exclusive production, as well as simultaneous production with degenerate mass. 137 Figure 6.12: Comparison of the upper limits on the cross-section before (solid red) and after (dashed blue) the orthogonality cut on the VV mass window for 1-Lepton VH search. Because the dilepton channel is part of this combination, we assume Model A ratios of σi/σtot. The observed Z(cid:48) limits are for the most part within 1σ of the expected limits, the excep- tion is a 2–3σ deficit at about 1.9 TeV. They show that HVT Model A Z(cid:48) is excluded up to resonance masses of about 4.3 TeV. The observed W(cid:48) limits are for the most part also within 1σ of the expected limits, the exception is a 2–3σ deficit at about 2.2 TeV. They show that HVT Model A W(cid:48) is excluded up to about resonance masses of 4.6 TeV. The observed V (cid:48) limits are mostly within 1σ of the expected limits, the exception is a 1–2σ deficit at masses above 4 TeV. They show that HVT Model A is excluded up to resonance masses about 5.3 TeV. 138 Figure 6.13: Comparison of the upper limits on the cross-section before (solid red) and after (dashed blue) the orthogonality cut on the VV mass window for the 2-Lepton VH search. Table 6.5 lists the lower mass limits set on W(cid:48) exclusive production, Z(cid:48) exclusive pro- duction, and simultaneous production with degenerate mass for the HVT Model A and Model B benchmark models. It shows these limits for the 1-Lepton analysis this disser- tation focused on, the combination of the semi-leptonic 0-, 1-, and 2-Lepton analyses, the semi-leptonic and hadronic VH combination, the VV combination, the dilepton combination, and the final VV+VH+dilepton (VV+VH for Model B) combination. Looking to this ta- ble we can see that the final VV+VH+dilepton combination improved upon the 1-Lepton analysis’ W(cid:48) exclusive production limit by 2.0 TeV for the HVT Model A benchmark model. We can also see that the final VV+VH+dilepton (VV+VH) combination improved the V (cid:48) (simultaneous production with degenerate mass) lower limits by 500 GeV for the HVT Model A (Model B) benchmark models. 139 Figure 6.14: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the dilepton channels as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red line is the theory cross-section for our benchmark model per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. Finally, we can look at some limits on different parameters of the HVT model, specifically the coupling strengths gH , gf , gl, and gq (see Sec. 2.2.3). These coupling strengths determine the branching-ratios and production cross-sections of the W(cid:48) and Z(cid:48). The coupling strength gH = gV cH determines the decay rate of a V (cid:48) to weak vector bosons and the Higgs boson. The coupling strength gf = g2 cf , determines both the decay rate to fermions as well as gV the production cross-section, σ(pp → V (cid:48)). The HVT model allows for the fermions to have different coupling strengths, such as the quarks having different couplings from the leptons or each generation of the fermions having different couplings. We decided to explore the former by splitting gf into gl and gq, which determine the decay rate to leptons and the 140 12345m(V’) [TeV]4−103−102−101−10110210310410HVT model Aσ/σObserved 95% CL limitExpected 95% CL limitσ 1±Expected σ 2±Expected HVT model A(cid:173)1 = 13 TeV, 36.1 fbsATLAS lv + ll→DY HVT V’ Figure 6.15: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for Z(cid:48) production for the combination of the VV, VH, and dilepton searches, as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red line is the theory cross-section for our benchmark model per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. production cross-section respectively. Figure 6.18 shows the observed 95% CL exclusion contours for gf (= gl = gq) vs. gH and gl vs. gq (with gH fixed) for resonance masses of 3, 4, and 5 TeV for the combination of the VV, VH, and dilepton channels. These exclusion contours are created by adjusting the parameters involved, which affects the cross-sections of VV+VH and dilepton separately. Once the parameters have been adjusted the lower limits on the mass can be evaluated for that particular point. Up until this point, all of our limits have been dependent on our benchmark models, but plotting these parameter spaces allows us to draw model conclusions that are more model independent. For instance, it can now be said that any model with 141 0.511.522.533.544.55m(Z’) [TeV]4−103−102−101−10110210310410HVT model Aσ/σObserved 95% CL limitExpected 95% CL limitσ 1±Expected σ 2±Expected HVT model A(cid:173)1 = 13 TeV, 36.1 fbsATLAS WW + ZH + ll→DY HVT Z’ Figure 6.16: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for W(cid:48) production for the combination of the VV, VH, and dilepton searches, as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in Sec. 5.4.4. Finally, the dashed red line is the theory cross-section for our benchmark model per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. a fermion coupling equal to 0.2 is excluded for resonant mass below 4 TeV but not 5 TeV. This allows theorists to test their own models without experimentalists rerunning all of the analyses involved. Figure 6.19 shows the observed 95% CL exclusion contour of a 4 TeV resonant mass for the VV+VH, dilepton, and VV+VH+dilepton combinations separately. It shows that while the dilepton combination had better exclusion power of gf for values of gH around zero, the VV+VH combination had better exclusion power of gf for values of |gH| > 0.4. This demonstrates why the combination of the VV+VH and dilepton combinations was important. 142 0.511.522.533.544.55m(W’) [TeV]4−103−102−101−10110210310410HVT model Aσ/σObserved 95% CL limitExpected 95% CL limitσ 1±Expected σ 2±Expected HVT model A(cid:173)1 = 13 TeV, 36.1 fbsATLAS WZ + WH + lv →DY HVT W’ Figure 6.17: The 95% CL upper limit on cross section times branching fraction divided by the cross section times branching fraction predicted by HVT Model A for the simultaneous production of W(cid:48) and Z(cid:48) with degenerate mass for the combination of the VV, VH, and dilepton searches, as a function of signal mass. The dashed black line is the curve of the expected limits as a function of mass, as described in in Sec. 5.4.3. The green and yellow bands correspond to the 1 and 2σ bands of the expected limits described in the same section. The solid black line is the curve of the observed limits as a function of mass, as described in in Sec. 5.4.4. Finally, the dashed red line is the theory cross-section for our benchmark model per signal mass point. Each of these curves represent the cross-section value for the relevant statistic at a given signal mass point. Figure 6.18: Observed 95% CL exclusion contours in the HVT parameter spaces (gH , gF ) (left) and (gq, gl) (right) for resonant masses of 3, 4, and 5 TeV for the final V (cid:48) combination of VV, VH, and dilepton. Areas outside the curves are excluded, as are the colored regions that show the constraints from precision EW measurements. The parameters of the HVT benchmark models (Model A and Model B) are also shown where applicable. 143 12345m(V’) [TeV]4−103−102−101−10110210310410HVT model Aσ/σObserved 95% CL limitExpected 95% CL limitσ 1±Expected σ 2±Expected HVT model A(cid:173)1 = 13 TeV, 36.1 fbsATLAS VV + VH + lv + ll→DY HVT V’ 3−2−1−0123HHiggs and vector boson coupling g1−0.8−0.6−0.4−0.2−00.20.40.60.81fFermion coupling gATLAS 3 TeVATLAS 4 TeVATLAS 5 TeVEW fits 3 TeVEW fits 4 TeVEW fits 5 TeVATLAS Preliminary = 13 TeVs(cid:173)136.1 fb VV+VH+lv+ll→V’ Observed A B1−0.8−0.6−0.4−0.2−00.20.40.60.81qQuark coupling g1−0.8−0.6−0.4−0.2−00.20.40.60.81lLepton coupling gATLAS 3 TeVATLAS 4 TeVATLAS 5 TeVEW fits 3 TeVEW fits 4 TeVEW fits 5 TeVATLAS Preliminary = 13 TeVs(cid:173)136.1 fb = (cid:173)0.56Hg VV+VH+lv+ll→V’ Observed A HVT Model A W(cid:48) Z(cid:48) V (cid:48) 1-Lepton VH Semi-leptonic VH VH VV Dilepton VV/VH/ll/lν 2.6 TeV – – 2.6 TeV 2.7 TeV 2.8 TeV 2.6 TeV 3.6 TeV 4.6 TeV 2.7 TeV 2.9 TeV 4.5 TeV 2.8 TeV 3.7 TeV 5.0 TeV 4.6 TeV 4.3 TeV 5.5 TeV HVT Model B W(cid:48) Z(cid:48) V (cid:48) 1-Lepton VH Semi-leptonic VH VH VV VV/VH 2.8 TeV – – 2.8 TeV 2.8 TeV 2.9 TeV 2.8 TeV 3.9 TeV 2.8 TeV 3.6 TeV 3.0 TeV 4.0 TeV 4.5 TeV – – Table 6.5: Lower mass limits set on W(cid:48) exclusive production, Z(cid:48) exclusive production, and simultaneous production with degenerate mass for the HVT Model A (top) and Model B (bottom) benchmark models. It lists these limits for the 1-Lepton analysis this dissertation focused on, the combination of the semi-leptonic 0-, 1-, and 2-Lepton analyses, the semi- leptonic and hadronic VH combination, the VV combination, the dilepton combination, and the final VV+VH+dilepton (VV+VH for Model B) combination. Figure 6.19: Observed 95% CL exclusion contours in the HVT parameter spaces (gH , gF ) (left) and (gq, gl) (right) for a 4 TeV resonant mass. The exclusion contours of the VV+VH combination (dashed magenta), dilepton combination (dotted blue), and VV+VH+dilepton combination (solid red) are shown. Comparing the VV+VH and dilepton combinations to the VV+VH+dilepton combination allows us to make an overall tighter exclusion contour. Areas outside the curves are excluded, as are the colored regions that show the constraints from precision EW measurements. The parameters of the HVT benchmark models (Model A and Model B) are also shown where applicable. 144 3−2−1−0123HHiggs and vector boson coupling g1−0.8−0.6−0.4−0.2−00.20.40.60.81fFermion coupling gEW fits 3 TeVEW fits 4 TeVEW fits 5 TeVATLAS = 13 TeVs-136.1 fbm(V’) = 4 TeVObservedll+lvVV+VHVV+VH+ll+lv A B1−0.8−0.6−0.4−0.2−00.20.40.60.81qQuark coupling g1−0.8−0.6−0.4−0.2−00.20.40.60.81lLepton coupling gEW fits 3 TeVEW fits 4 TeVEW fits 5 TeVATLAS = 13 TeVs-136.1 fb = -0.56Hgm(V’) = 4 TeVObservedll+lvVV+VHVV+VH+ll+lv A Chapter 7 Conclusion This dissertation describes a search for a hypothetical charged vector resonance (W(cid:48)) with a mass in the range [500 GeV, 5000 GeV] that decays to a W boson and a Higgs boson. Specifically it described a search for the final state W(cid:48) → W (→ lv)H(→ b¯b), where l is an electron or muon. The existence of a W(cid:48) boson could provide a solution to the naturalness problem, which is associated with the relatively-low mass of the Higgs boson. The reasoning behind this is that the observed Higgs boson mass is a combination of two things: its bare mass and quantum loop corrections from its interactions. These loop corrections are expected to be proportional to the masses of the particles the Higgs can interact with, to some positive power. So particles with larger masses will make larger contributions to the observed Higgs mass. In addition, whether the interacting particle is a fermion or boson determines whether it makes a positive or negative contribution to the observed mass. As it stands, the top quark makes the largest known contribution to the loop corrections. In fact it is so large that an almost equally large bare mass is required to counter-balance its contribution, even though the observed mass is not particularly large. This means that the loop corrections and bare mass must be finely tuned so that they together yield the observed mass. Essentially, two very large numbers need to be adjusted just right so they sum to a very small number (by comparison). While this is certainly possible, without a mechanism to explain it, it is 145 unnatural in the minds of many physicists. There are several proposed modifications to the Standard Model that would add a mech- anism that accounts for the scale of the Higgs mass. This dissertation explores the concept of extended gauge theories, which embed the electroweak SU (2)L × U (1)Y group within a larger group, such as the SO(5)× U (1)Y group. Breaking the symmetry of this group would yield an SO(4)×U (1)Y group, as well as a naturally lighter Higgs boson and three Goldstone bosons. This SO(4) × U (1)Y group could then be expressed as SU (2)R × SU (2)L × U (1)Y , i.e. the electroweak group times an SU (2)R group. This SU (2)R group would yield three gauge bosons that gain mass from the Goldstone bosons produced by breaking the symmetry of the SO(5) × U (1)Y group. This example is just one of many ways an extended gauge group with a naturally light Higgs boson could be achieved. We employ a generic triplet model (Heavy Vector Triplet [24]) to describe the expected couplings to SM particles. This is done to simplify the phenomenology space, which means it can cover the example given above as well as other complex formulations in a simple manner. Two benchmark models, HVT-A and HVT-B, describe scenarios of enhanced couplings to fermions and bosons, respectively. This dissertation covers the search for the predicted gauge bosons of the HVT model, the W(cid:48) and Z(cid:48), which are collectively known as the V (cid:48) in the case of degenerate mass. It initially focuses on the W(cid:48) → W H → lνb¯b decay channel, using 36.1 fb−1 of 13 TeV center- of-mass energy LHC collision data collected by the ATLAS detector. We design a cut-based analysis to remove Standard-Model backgrounds by selecting events with exactly one lepton, large missing transverse energy, and 2-3 small- or 1+ large-radius jets. Whether an event contains small-R or large-R jets determines if it is considered a resolved or merged event respectively, with the possibility that it could be considered both. To further reduce the 146 amount of background, events are also sorted into 2–3 regions based on the number of b- tagged small-R or track jets they contain. Signal and control regions are established by using a Higgs mass window to separate events that could have included a Higgs boson from those that most likely did not. Events that meet the criteria for more than one region are then categorized using the priority-resolved-signal-region algorithm. This results in 20 regions based on whether an event was in a signal or control region, resolved or merged region, and the number of b-tagged jets. The remaining SM backgrounds that enter these regions are ttbar, single-top, W/Z+jets, diboson, SM WH production, and QCD multijet production. After the event selections are applied to the data, signal, and backgrounds, and all the events are categorized into the appropriate regions, histograms plotting the number of events as a function of W(cid:48) mass are produced for each region. A statistical analysis was done to determine how statistically significant any excesses in the data were over the background. The p-values of this test were calculated per signal mass point and plotted as such. Unfortunately, no significant excesses were observed, so instead 95% confidence level upper limits on the signal cross section as a function of signal mass were produced. These upper limits were compared to the theory cross-sections of the HVT-A and HVT-B benchmark models to produce lower limits on the signal mass of the benchmark models. The lower limits on the mass of the W(cid:48) excluded masses below 2.6 TeV for HVT-A and 2.8 TeV for HVT-B. At this point, attention was turned to the combination of several searches with differing W(cid:48) and Z(cid:48) decay modes. First, the search described above (W(cid:48) → W H → lνb¯b) was combined with searches for Z(cid:48) → ZH → ν ¯νb¯b and Z(cid:48) → ZH → l¯lb¯b, this was called the semi-leptonic VH combination. Then the semi-leptonic VH combination was combined with the fully hadronic VH channel (V (cid:48) → V H → q ¯qb¯b), which we called the full VH combination. 147 Finally, the full VH combination was combined with the diboson (V (cid:48) → V V ) and dilepton (V (cid:48) → lν, l¯l) channels. The resulting lower limits on the V (cid:48) mass for each of these searches and combinations are shown in Table 7.1. Search/Comb. Semi-leptonic VH Full VH Diboson (VV) Dilepton (lν, ll) VV+VH+dilepton HVT-A Limits HVT-B Limits 2.8 TeV 2.8 TeV 3.7 TeV 5.0 TeV 5.5 TeV 2.9 TeV 3.0 TeV 4.0 TeV 4.5 TeV∗ – Table 7.1: The lower limits on V (cid:48) mass set by the combinations described in this dissertation. ∗Note that the final combination for HVT-B did not include the dilepton search. Finally, 2D exclusion contours on the HVT coupling strength parameters (gH , gl, and gq) were produced for the 3, 4, and 5 TeV masses so that theorists could more easily compare their own models. Two cases were studied, one where gl and gq were set to be equal and the resulting gf was varied along with gH to produce contours of gf vs. gH , and another where gH was set to its HVT-A value (−0.56) while gl and gq were varied to produce contours of gl vs. gq. The primary search of this dissertation was performed with 36.1 fb−1 of data. No signif- icant excess was observed and it set the best limits on heavy vector resonance production to date. Future searches will add more data (up to 3000 fb−1) and probe more couplings, such as the 3rd generation quark and lepton couplings. Thus, we expect these future searches to produce better cross-section upper limits and signal mass lower limits, as well as more coupling information for theorists to utilize. 148 APPENDICES 149 APPENDIX A Multi-jet Estimation Nearly all of our backgrounds are derived as Monte-Carlo simulations. The outlier is the multi-jet background, which is derived using a data-driven technique. This is done because multi-jet Monte-Carlo simulations suffer from limited statistics and difficult modeling in the phase space of this analysis. It should be noted that the multi-jet background is derived separately for the electron and muon channels. For those familiar with the LHC, the idea of multi-jet suffering from limited statistics might seem a little bizzarre. Multi-jet is by far the most common SM process produced by proton-proton collisions, so you might expect it to be one of our largest backgrounds. However it is actually one of our smallest, it only contributes 1–2% to the total background in the resolved regime and is negligible in the merged regime to the point that we don’t consider it. This is because the lepton in our signal comes from the collision process and it is well isolated from other physics objects†, whereas the lepton in a multi-jet process either comes from a jet or is faked by a jet or photons from pion decays. We derive the multi-jet estimate by exploiting a key difference between it and our signal. By inverting our isolation requirements, we end up with an area of phase space with a large amount of multijet and little to no signal. While we expect the normalization of the multi-jet background to be different in this region, we expect the shape to be extremely similar to †This is only true for the resolved regime, the lepton can get close to the MET in the merged regime. 150 Figure A.1: The W boson transverse mass distributions of the data and non-multi-jet back- grounds in the inverted-isolation region. After the fit to MET > 200 GeV is done, the backgrounds will be subtracted from the data to create the multi-jet estimate in the inverted- isolation region. Note that for this figure and all other figures in this section, the distributions are split into electron events (left) and muon events (right). the non-inverted region. We simply subtract the other SM backgrounds from the data in this region to get the shape of the multi-jet background for any given distribution. Any uncertainty in this shape is captured in a shape systematic we define. An example of the data and MC distributions for this region are shown in Fig. A.1. Before we do this subtraction we must take into account that the total contribution from the other backgrounds does not match the data well in certain regions of phase space. To fix this we will fit these backgrounds to the data in a region where we expect the multi-jet contribution to be negligible. Specifically, we will fit these backgrounds MET distribution to the data’s for MET greater than 200 GeV, as shown in Fig. A.2. This fixes the agreement between the data and other backgrounds (Fig. A.3) so that the multi-jet shape can be derived. The final step is to create the multi-jet estimate for the non-inverted regions by adjusting the shape and normalization of the multi-jet background from the inverted-isolation region. 151 Entries/5 GeV050100150200250300350400450 InternalATLAS InternalATLASdataDibosonZ+jW+lW+cW+bsingle topttbarelectron event mTW0100200300400500 MC-Data 050100150200Entries/5 GeV0100200300400500600 InternalATLAS InternalATLASdataDibosonZ+jW+lW+cW+bsingle topttbarmuon event mTW0100200300400500 MC-Data 0100200300400 Figure A.2: The missing transverse energy distributions of the data and non-multi-jet back- grounds in the inverted-isolation region. The backgrounds are fit to MET > 200 GeV to correct their normalizations. Figure A.3: The post-fit W boson transverse mass distributions of the data and non-multi- jet backgrounds in the inverted-isolation region. These distributions will be used to derive the shape of the multi-jet estimate in the inverted-isolation region by subtracting the back- grounds from the data. 152 Entries/10 GeV0100200300400500600700800900 InternalATLAS InternalATLASdataDibosonZ+jW+lW+cW+bsingle topttbar [GeV]missTelectron event E050100150200250300350400450500 MC-Data 0200400Entries/10 GeV050100150200250300350400 InternalATLAS InternalATLASdataDibosonZ+jW+lW+cW+bsingle topttbar [GeV]missTmuon event E050100150200250300350400450500 MC-Data 0100200Entries/5 GeV050100150200250300350400450 InternalATLAS InternalATLASdataDibosonZ+jW+lW+cW+bsingle topttbarelectron event mTW0100200300400500 MC-Data 050100150200Entries/5 GeV0100200300400500600 InternalATLAS InternalATLASdataDibosonZ+jW+lW+cW+bsingle topttbarmuon event mTW0100200300400500 MC-Data 0100200300400 This is done by taking the multi-jet background from the inverted-isolation region and ad- justing it bin-by-bin using lepton and jet efficiencies. Essentially, we know how the shape of any distribution changes when we apply certain cuts and we can use that knowledge to adjust the shapes and normalizations of the multi-jet background in the inverted-isolation region to produce the multi-jet background in the non-inverted region. The multi-jet estimates for the non-inverted region are show in Figs. A.4–A.9. Figure A.4: The missing transverse energy distributions for the data (black), multi-jet esti- mate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). 153 Entries/10 GeV010002000300040005000600070008000 InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusive [GeV]missTelectron event E050100150200250300350400450500MC MC-Data 0.4-0.2-00.20.4Entries/10 GeV0100020003000400050006000700080009000 InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusive [GeV]missTmuon event E050100150200250300350400450500MC MC-Data 0.4-0.2-00.20.4 Figure A.5: The heavy vector resonance mass distributions for the data (black), multi-jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). Figure A.6: The dijet mass distributions for the data (black), multi-jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). 154 Entries/40 GeV020004000600080001000012000 InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusiveVHelectron event m0200400600800100012001400MC MC-Data 0.4-0.2-00.20.4Entries/40 GeV05000100001500020000 InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusiveVHmuon event m0200400600800100012001400MC MC-Data 0.4-0.2-00.20.4Entries/10 GeV01000200030004000500060007000 InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusivebbelectron event m50100150200250MC MC-Data 0.4-0.2-00.20.4Entries/10 GeV0200040006000800010000 InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusivebbmuon event m50100150200250MC MC-Data 0.4-0.2-00.20.4 Figure A.7: The W boson transverse mass distributions for the data (black), multi-jet esti- mate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). Figure A.8: The W boson transverse momentum distributions for the data (black), multi-jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). 155 Entries/5 GeV050010001500200025003000350040004500 InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusivemTW_El050100150200250300MC MC-Data 0.4-0.2-00.20.4Entries/5 GeV0100020003000400050006000 InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusivemTW_Mu050100150200250300MC MC-Data 0.4-0.2-00.20.4Entries/20 GeV020004000600080001000012000140001600018000 InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusive [GeV]VTelectron event p100200300400500600700800MC MC-Data 0.4-0.2-00.20.4Entries/20 GeV050001000015000200002500030000 InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusive [GeV]VTmuon event p100200300400500600700800MC MC-Data 0.4-0.2-00.20.4 Figure A.9: The lepton transverse momentum distributions for the data (black), multi-jet estimate (pink), and other backgrounds (blue) in the non-inverted resolved, 1 b-tag region, split into events containing electrons (left) and muons (right). 156 Entries/25 GeV05000100001500020000 InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ1.9931e-03Norm. non-MJ:1.0081e+001tag2pjet0ptvmBBInclusivelepTelectron event p050100150200250300350400MC MC-Data 0.4-0.2-00.20.4Entries/25 GeV0500010000150002000025000 InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusive InternalATLASDatamultijetnon-multijetNorm. MJ8.9024e-01Norm. non-MJ:1.0176e+001tag2pjet0ptvmBBInclusivelepTmuon event p050100150200250300350400MC MC-Data 0.4-0.2-00.20.4 APPENDIX B Extra Plots from Data Selection In Sec. 4.6, the signal mass plots of the cut-based analysis were shown; other distributions from the cut based analysis are shown below. The distributions included are the jet trans- verse momentum, lepton transverse momentum, missing transverse energy, vector boson transverse momentum, azimuthal angle between the lepton & MET, vector boson mass, and the dijet/Higgs candidate mass. Note that these plots show the distributions before the fits described in Sec. 5.3, so they communicate the level of agreement between data and MC before the fit. Any differences between the data and MC in these plots is therefore absorbed by said fit if they are within the statistical and systematic uncertainties. 157 Jet Transverse Momentum Figure B.1: Transverse momentum of the leading small-R jet for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 158 Figure B.2: Transverse momentum of the sub-leading small-R jet for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 159 Figure B.3: Transverse momentum of the leading large-R jet for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 160 Figure B.4: Transverse momentum of the leading large-R jet for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 161 Lepton Transverse Momentum Figure B.5: Transverse momentum of the lepton for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 162 Figure B.6: Transverse momentum of the lepton for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 163 Figure B.7: Transverse momentum of the lepton for the control regions of the merged selec- tion with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 164 Missing Transverse Energy (MET) Figure B.8: Missing Transverse Energy for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 165 Figure B.9: Missing Transverse Energy for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 166 Figure B.10: Missing Transverse Energy for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 167 Vector Boson Transverse Momentum Figure B.11: Transverse momentum of the W boson for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 168 Figure B.12: Transverse momentum of the W boson for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 169 Figure B.13: Transverse momentum of the W boson for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 170 Azimuthal Angle Between Lepton & MET Figure B.14: Azimuthal angle between the lepton and MET for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 171 Figure B.15: Azimuthal angle between the lepton and MET for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 172 Figure B.16: Azimuthal angle between the lepton and MET for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 173 Vector Boson Transverse Mass Figure B.17: Transverse mass of the W boson for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 174 Figure B.18: Transverse mass of the W boson for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 175 Figure B.19: Transverse mass of the W boson for the control regions of the merged selection with additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 176 Dijet & Higgs Masses Figure B.20: Dijet mass for the signal and control regions of the resolved selection. On the left are the Higgs mass side band control regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 177 Figure B.21: Higgs mass for the signal and control regions of the merged selection with no additional b-tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass signal regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 178 Figure B.22: Higgs mass for the control regions of the merged selection with additional b- tags outside the primary large-R jet. On the left are the Higgs mass side band regions and on the right are the central Higgs mass regions. On the top are the 1 b-tag regions and on the bottom are the 2 b-tag regions. 179 BIBLIOGRAPHY 180 BIBLIOGRAPHY [1] G. Hinshaw, J. L. Weiland, R. S. Hill, N. Odegard, D. Larson, C. L. Bennett, J. Dunkley, B. Gold, M. R. Greason, N. Jarosik, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, E. Wollack, M. Halpern, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, and E. L. Wright. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Data Processing, Sky Maps, and Basic Results. ApJS, 180:225–245, February 2009. [2] R. S. Hill, J. L. Weiland, N. Odegard, E. Wollack, G. Hinshaw, D. Larson, C. L. Bennett, M. Halpern, L. Page, J. Dunkley, B. Gold, N. Jarosik, A. Kogut, M. Limon, M.R. Nolta, D. N. Spergel, G. S. Tucker, and E. L. Wright. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Beam Maps and Window Functions. ApJS, 180:246–264, February 2009. [3] B. Gold, C. L. Bennett, R. S. Hill, G. Hinshaw, N. Odegard, D. N. Spergel, J. Weiland, J. Dunkley, M. Halpern, N. Jarosik, A. Kogut, E. Komatsu, D. Larson, S. S. Meyer, M.R. Nolta, E. Wollack, and E. L. Wright. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Galactic Foreground Emission. ApJS, 180:265–282, February 2009. [4] E. L. Wright, X. Chen, N. Odegard, C. L. Bennett, R. S. Hill, G. Hinshaw, N. Jarosik, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, J. L. Weiland, E. Wollack, J. Dunkley, B. Gold, M. Halpern, A. Kogut, D. Larson, M. Limon, S. S. Meyer, and G. S. Tucker. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Source Cat- alog. ApJS, 180:283–295, February 2009. [5] M. R. Nolta, J. Dunkley, R. S. Hill, G. Hinshaw, E. Komatsu, D. Larson, L. Page, D. N. Spergel, C. L. Bennett, B. Gold, N. Jarosik, N. Odegard, J. L. Weiland, E. Wollack, M. Halpern, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, and E.L. Wright. Five- Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Angular Power Spectra. ApJS, 180:296–305, February 2009. [6] J. Dunkley, E. Komatsu, M. R. Nolta, D. N. Spergel, D. Larson, G. Hinshaw, L. Page, C. L. Bennett, B. Gold, N. Jarosik, J. L. Weiland, M. Halpern, R. S. Hill, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, and E. L. Wright. Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from the WMAP data. ApJS, 180:306–376, February 2009. [7] E. Komatsu, J. Dunkley, M. R. Nolta, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. Limon, L. Page, D. N. Spergel, M. Halpern, R. S. Hill, A. Kogut, S. S. Meyer, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright. Five-Year Wilkin- 181 son Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation. ApJS, 180:330–376, February 2009. [8] M. Limon, E. Wollack, M. R. Greason, C. L. Bennett, M. Halpern, G. Hin- shaw, N. Jarosik, A. Kogut, S. S. Meyer, L. Page, D. N. Spergel, G. S. Tucker, E. L. Wright, R. S. Hill, E. Komatsu, M. Nolta, N. Odegard, and J. L. Weiland. Wilkinson microwave anisotropy probe (wmap): Five year explanatory supplement. http://lambda.gsfc.nasa.gov/data/map/doc/MAP supplement.pdf, 2008. [9] AC Team. The four main LHC experiments. Jun 1999. [10] AC Team. Diagram of an LHC dipole magnet. Schma d’un aimant diple du LHC. Jun 1999. [11] Joao Pequenao. Computer generated image of the whole ATLAS detector. Mar 2008. [12] Joao Pequenao. Computer generated image of the ATLAS inner detector. Mar 2008. [13] The ATLAS Collaboration. The ATLAS Experiment at the CERN Large Hadron Col- lider. JINST, 3:S08003, 2008. [14] Joao Pequenao. Computer Generated image of the ATLAS calorimeter. Mar 2008. [15] Joao Pequenao. Computer generated image of the ATLAS Muons subsystem. Mar 2008. [16] Joao Pequenao and Paul Schaffner. How ATLAS detects particles: diagram of particle paths in the detector. Jan 2013. [17] Gordan Kane. Modern Elementary Paticle Physics. Adison-Wesley Publishing Com- pany, 1993. [18] Wade C. Fisher. A Search for Anomalous Haevy-Flavor Quark Production in Associa- tion with W Bosons. 2005. [19] Steven Weinberg. Gauge hierarchies. Phys. Lett. B, 82:387–391, 1979. [20] M. J. G. Veltman. The infrared - ultraviolet connection. Acta Phys. Polon. B, 12:437, 1981. [21] C. H. Llewellyn Smith and Graham G. Ross. The real gauge hierarchy problem. Phys. Lett. B, 105:38–40, 1981. [22] Vernon D. Barger, Wai-Yee Keung, and Ernest Ma. A Gauge Model With Light W and Z Bosons. Phys. Rev., D22:727, 1980. 182 [23] Roberto Contino, David Marzocca, Duccio Pappadopulo, and Riccardo Rattazzi. On the effect of resonances in composite Higgs phenomenology. JHEP, 10:081, 2011. [24] Duccio Pappadopulo, Andrea Thamm, Riccardo Torre, and Andrea Wulzer. Heavy Vector Triplets: Bridging Theory and Data. JHEP, 09:060, 2014. [25] The ATLAS Collaboration. Search for a new resonance decaying to a W or Z boson and a Higgs boson in the (cid:96)(cid:96)/(cid:96)ν/νν + b¯b final states with the ATLAS detector. Eur. Phys. J., C75(6):263, 2015. [26] The ATLAS Collaboration. Search for new resonances decaying to a W or Z boson and s = 13 TeV a Higgs boson in the (cid:96)+(cid:96)−b¯b, (cid:96)νb¯b, and ν ¯νb¯b channels with pp collisions at with the ATLAS detector. Phys. Lett., B765:32–52, 2017. √ [27] The ATLAS Collaboration. ATLAS Insertable B-Layer Technical Design Report. 2010. [28] Morad Aaboud et al. Performance of the ATLAS Trigger System in 2015. Eur. Phys. J., C77(5):317, 2017. [29] W Lampl, S Laplace, D Lelas, P Loch, H Ma, S Menke, S Rajagopalan, D Rousseau, S Snyder, and G Unal. Calorimeter Clustering Algorithms: Description and Per- formance. Technical Report ATL-LARG-PUB-2008-002. ATL-COM-LARG-2008-003, CERN, Geneva, Apr 2008. [30] ATLAS Collaboration. Electron reconstruction and identification efficiency measure- ments with the ATLAS detector using the 2011 LHC proton–proton collision data. Eur. Phys. J. C, 74:2941, 2014. [31] ATLAS Collaboration. Muon reconstruction performance of the ATLAS detector in √ s = 13 TeV. Eur. Phys. J. C, 76:292, 2016. proton–proton collision data at [32] Walter Lampl et al. Calorimeter Clustering Algorithms: Description and Performance. ATL-LARG-PUB-2008-002, 2008. [33] Matteo Cacciari, Gavin P. Salam, and Gregory Soyez. The anti-kt jet clustering algo- rithm. JHEP, 04:063, 2008. [34] ATLAS Collaboration. Performance of b-jet identification in the ATLAS experiment. JINST, 11:P04008, 2016. [35] Optimisation of the ATLAS b-tagging performance for the 2016 LHC Run. Technical Report ATL-PHYS-PUB-2016-012, CERN, Geneva, Jun 2016. [36] ATLAS Collaboration. Performance of algorithms that reconstruct missing transverse s = 8 TeV proton–proton collisions in the ATLAS detector. Eur. Phys. √ momentum in J. C, 77:241, 2017. 183 [37] ATLAS Collaboration. Expected performance of missing transverse momentum recon- √ s = 13 TeV. ATL-PHYS-PUB-2015-023, 2015. struction for the ATLAS detector at [38] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro. The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations. JHEP, 07:079, 2014. [39] Richard D. Ball, Valerio Bertone, Francesco Cerutti, Luigi Del Debbio, Stefano Forte, Alberto Guffanti, Jose I. Latorre, Juan Rojo, and Maria Ubiali. Impact of Heavy Quark Masses on Parton Distributions and LHC Phenomenology. Nucl. Phys., B849:296–363, 2011. [40] Torbjorn Sjostrand, Stephen Mrenna, and Peter Z. Skands. A Brief Introduction to PYTHIA 8.1. Comput. Phys. Commun., 178:852–867, 2008. [41] ATLAS Run 1 Pythia8 tunes. Technical Report ATL-PHYS-PUB-2014-021, CERN, Geneva, Nov 2014. [42] Paolo Nason. A New method for combining NLO QCD with shower Monte Carlo algorithms. JHEP, 11:040, 2004. [43] Stefano Frixione, Paolo Nason, and Carlo Oleari. Matching NLO QCD computations with Parton Shower simulations: the POWHEG method. JHEP, 11:070, 2007. [44] Simone Alioli, Paolo Nason, Carlo Oleari, and Emanuele Re. A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX. JHEP, 06:043, 2010. [45] T. Gleisberg, Stefan. Hoeche, F. Krauss, M. Schonherr, S. Schumann, F. Siegert, and J. Winter. Event generation with SHERPA 1.1. JHEP, 02:007, 2009. [46] Torbjorn Sjostrand, Stephen Mrenna, and Peter Z. Skands. PYTHIA 6.4 Physics and Manual. JHEP, 05:026, 2006. [47] Torbj¨orn Sj¨ostrand, Stefan Ask, Jesper R. Christiansen, Richard Corke, Nishita Desai, Philip Ilten, Stephen Mrenna, Stefan Prestel, Christine O. Rasmussen, and Peter Z. Skands. An introduction to PYTHIA 8.2. Comput. Phys. Commun., 191:159, 2015. [48] Hung-Liang Lai, Marco Guzzi, Joey Huston, Zhao Li, Pavel M. Nadolsky, Jon Pumplin, and C. P. Yuan. New parton distributions for collider physics. Phys. Rev., D82:074024, 2010. [49] Peter Z. Skands. Tuning monte carlo generators: The perugia tunes. Phys. Rev. D, 82:074018, Oct 2010. 184 [50] M. Beneke, P. Falgari, S. Klein, and C. Schwinn. Hadronic top-quark pair production with NNLL threshold resummation. Nucl. Phys. B, 855:695–741, 2012. [51] Matteo Cacciari, Michal Czakon, Michelangelo Mangano, Alexander Mitov, and Paolo Nason. Top-pair production at hadron colliders with next-to-next-to-leading logarithmic soft-gluon resummation. Phys. Lett. B, 710:612–622, 2012. [52] Peter B¨arnreuther, Michal Czakon, and Alexander Mitov. Percent level precision physics at the Tevatron: First genuine NNLO QCD corrections to q ¯q → t¯t + X. Phys. Rev. Lett., 109:132001, 2012. [53] Michal Czakon and Alexander Mitov. NNLO corrections to top-pair production at hadron colliders: the all-fermionic scattering channels. JHEP, 12:054, 2012. [54] Michal Czakon and Alexander Mitov. NNLO corrections to top pair production at hadron colliders: the quark-gluon reaction. JHEP, 01:080, 2013. [55] Michal Czakon, Paul Fiedler, and Alexander Mitov. Total top-quark-pair-production cross section at hadron colliders through O(α4 S). Phys. Rev. Lett., 110:252004, 2013. [56] Michal Czakon and Alexander Mitov. Top++: A Program for the Calculation of the Top-Pair Cross-Section at Hadron Colliders. Comput. Phys. Commun., 185:2930, 2014. [57] Michal Czakon, David Heymes, and Alexander Mitov. Dynamical scales for multi-TeV top-pair production at the LHC. JHEP, 04:071, 2017. [58] M. Aliev, H. Lacker, U. Langenfeld, S. Moch, P. Uwer, and M. Wiedermann. HATHOR: HAdronic Top and Heavy quarks crOss section calculatoR. Comput. Phys. Commun., 182:1034–1046, 2011. [59] P. Kant, O. M. Kind, T. Kintscher, T. Lohse, T. Martini, S. M¡F6¿lbitz, P. Rieck, and P. Uwer. HatHor for single top-quark production: Updated predictions and uncer- tainty estimates for single top-quark production in hadronic collisions. Comput. Phys. Commun., 191:74–89, 2015. [60] Nikolaos Kidonakis. Two-loop soft anomalous dimensions for single top quark associated production with a W− or H−. Phys. Rev. D, 82:054018, 2010. [61] Richard D. Ball et al. Parton distributions for the LHC Run II. JHEP, 04:040, 2015. [62] Kirill Melnikov and Frank Petriello. Electroweak gauge boson production at hadron colliders through O(α2 s). Phys. Rev. D, 74:114017, 2006. [63] LHC Higgs Cross Section Working Group. Handbook of LHC Higgs cross sections: 4. Deciphering the nature of the Higgs sector. 2016. 185 [64] Jet Calibration and Systematic Uncertainties for Jets Reconstructed in the ATLAS De- s = 13 TeV. Technical Report ATL-PHYS-PUB-2015-015, CERN, Geneva, √ tector at Jul 2015. [65] Jet global sequential corrections with the ATLAS detector in proton-proton collisions at sqrt(s) = 8 TeV. Technical Report ATLAS-CONF-2015-002, CERN, Geneva, Mar 2015. [66] Identification of boosted, hadronically-decaying W and Z bosons in s = 13 TeV Monte Carlo Simulations for ATLAS. Technical Report ATL-PHYS-PUB-2015-033, CERN, Geneva, Aug 2015. √ √ [67] Jet mass reconstruction with the ATLAS Detector in early Run 2 data. Technical Report ATLAS-CONF-2016-035, CERN, Geneva, Jul 2016. [68] Georges Aad et al. Performance of b-Jet Identification in the ATLAS Experiment. JINST, 11(04):P04008, 2016. [69] Calibration of b-tagging using dileptonic top pair events in a combinatorial likelihood ap- proach with the ATLAS experiment. Technical Report ATLAS-CONF-2014-004, CERN, Geneva, Feb 2014. [70] Calibration of the performance of b-tagging for c and light-flavour jets in the 2012 ATLAS data. Technical Report ATLAS-CONF-2014-046, CERN, Geneva, Jul 2014. [71] Morad Aaboud et al. Luminosity determination in pp collisions at the ATLAS detector at the LHC. Eur. Phys. J., C76(12):653, 2016. s = 8 TeV using [72] Simulation of top quark production for the ATLAS experiment at sqrt(s) = 13 TeV. Technical Report ATL-PHYS-PUB-2016-004, CERN, Geneva, Jan 2016. [73] M. Bahr et al. Herwig++ Physics and Manual. Eur. Phys. J., C58:639–707, 2008. [74] J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, Pavel M. Nadolsky, and W. K. Tung. New generation of parton distributions with uncertainties from global QCD analysis. JHEP, 07:012, 2002. [75] J. M. Campbell and R. Ellis. Mcfm for the tevatron and the lhc. Nucl. Phys. Proc. Suppl., pages 205–206, 2010. [76] Alexander L. Read. Presentation of search results: The cls technique. J. Phys. G, 28:2693–2704, 2002. 186