Department of Physics, University of Witwatersrand PhD THESIS Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy. Not a jet all the way - an exploration of the strongly interacting dark sector in ATLAS and beyond Student: Sukanya Sinha Student number: 2296508 Supervisor: Prof. Deepak Kar Department of Physics March 17, 2023 DECLARATION I declare that this thesis is my own, unaided work. It is being submitted for the Degree of Doctor of Philosophy at the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination at any other University. Sukanya Sinha 17th day of March 2023 at Johannesburg 2 Abstract Collider searches for dark matter (DM) so far have mostly focussed on scenarios, where DM particles are produced in association with heavy standard model (SM) particles or jets. However, no deviations from SM predictions have been observed so far. Several recent phenomenology papers have proposed models that explore the possibility of accessing the strongly coupled dark sector, giving rise to unusual and unexplored collider topologies. One such signature is termed as semi-visible jet (SVJ), where parton evolution includes dark sector emissions, resulting in jets interspersed with stable invisible particles. Owing to the unusual MET-along-the-jet event topology this is still a largely unexplored domain within LHC. This thesis presents the first results from a search for SVJ in t-channel production mode in pp collisions for an integrated luminosity of 139 fb−1 at centre-of-mass energy corresponding to 13 TeV at the LHC, based on data collected by the ATLAS detector during 2015-2018. Additionally, studies are performed to explore the use of jet substructure methods to distinguish SVJ from SM jets in the first two scenarios, using observables in a IRC-safe linear basis, and ways forward are proposed for this approach to dark-matter at the LHC, including prospects for estimating modelling uncertainties. Dedicated to my mother, for believing in me and dreaming along with me... Acknowledgements 3 4 5 6 7 Contents 1 Standard Model - background and drawbacks 17 1.1 A “mandatory” overview of the Standard Model . . . . . . . . . . . . . . . . 17 1.2 Standard Quantum Chromodynamics – a precursor . . . . . . . . . . . . . . 23 1.2.1 Strong coupling and asymptotic freedom . . . . . . . . . . . . . . . . 24 1.2.2 A non-arduous tour of the fundamentals of QCD inspired event gen- eration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3 Monte Carlo event generators . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4 The limitations of SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 Strongly interacting dark sector - why and how? 33 2.1 Moving on to... the overview of dark matter . . . . . . . . . . . . . . . . . . 33 2.1.1 Early explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.2 Dark matter relic density . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.1.3 Dark matter particle candidates . . . . . . . . . . . . . . . . . . . . . 35 2.1.4 Interactions between Standard Model and dark matter . . . . . . . . 35 2.2 Simplified Models of dark matter . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 QCD-like dark sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.1 Semi-visible jets theory model . . . . . . . . . . . . . . . . . . . . . . 39 2.3.2 Hidden Valley Shower . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.3 Semi-visible jets topology and extra jets . . . . . . . . . . . . . . . . 43 3 Collider basics and getting to know the ATLAS detector 45 3.1 Large Hadron Collider at CERN . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Give me the coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Collisions – why do we love them? . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Hello ATLAS detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4.1 Inner detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4.2 Electromagnetic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.3 Hadronic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4.4 Muon spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5 Object reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.1 Jets – how do we make them? . . . . . . . . . . . . . . . . . . . . . . 57 3.5.2 Emiss T reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6 Triggering on the objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.7 Detector simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.8 Existing searches for Dark Matter with jets and Emiss T in ATLAS . . . . . . . 67 8 4 Search for non-resonant production of semi-visible jets in ATLAS 69 4.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Standard Model background processes . . . . . . . . . . . . . . . . . . . . . 70 4.3 Object definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Dataset and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.1 Simulated samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.2 Data samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5 Event selection and cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5.1 Triggering strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5.2 Data quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.5.3 Jet Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.5.4 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.6 Analysis strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.6.1 Inclusive distributions and definition of signal and control regions . . 80 4.6.2 Signal region distributions . . . . . . . . . . . . . . . . . . . . . . . . 86 4.6.3 Control region distributions . . . . . . . . . . . . . . . . . . . . . . . 89 4.7 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.7.1 Theoretical systematic uncertainties . . . . . . . . . . . . . . . . . . . 90 4.7.2 Experimental systematic uncertainties . . . . . . . . . . . . . . . . . 92 4.8 Fit strategy and background estimation . . . . . . . . . . . . . . . . . . . . . 93 4.8.1 General strategy and fitting procedure . . . . . . . . . . . . . . . . . 93 4.8.2 Verification of multijet background estimation . . . . . . . . . . . . . 94 4.9 Simulaneous fit of signal region and control region . . . . . . . . . . . . . . . 97 4.9.1 Post-fit distributions in 1LCR . . . . . . . . . . . . . . . . . . . . . . 97 4.9.2 Post-fit distributions in 1L1BCR . . . . . . . . . . . . . . . . . . . . 97 4.9.3 Post-fit distributions in 2L CR . . . . . . . . . . . . . . . . . . . . . . 97 4.9.4 Post-fit distribution for combined fit in control region and signal region 98 4.10 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5 Phenomenological explorations of semi-visible jets 106 5.1 Event generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2 Exploring jet substructure of semi-visible jets . . . . . . . . . . . . . . . . . 107 5.2.1 Analysis strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2.2 Jet substructure observables . . . . . . . . . . . . . . . . . . . . . . . 108 5.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2.4 Understanding the model dependence . . . . . . . . . . . . . . . . . . 111 5.2.5 Origin of the differences . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.3 Exploring new observables for dark sector . . . . . . . . . . . . . . . . . . . 116 5.3.1 Energy flow polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.4 Semi-visible jet production with heavy flavour . . . . . . . . . . . . . . . . . 121 5.4.1 Signal modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.4.2 Signal reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.4.3 Search strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 9 6 Monte Carlo Truth Classifier 127 6.1 Necessity of proper truth definitions . . . . . . . . . . . . . . . . . . . . . . . 127 6.2 Original Monte Carlo Truth Classifier and drawbacks . . . . . . . . . . . . . 128 6.3 New implementation of MCTC . . . . . . . . . . . . . . . . . . . . . . . . . 129 Bibliography 132 A Appendices 148 A.1 Use of reclustered jets in ATLAS SVJ analysis . . . . . . . . . . . . . . . . . 148 A.1.1 Particle level studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.1.2 Detector level studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 A.1.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 A.2 Tests for non-collisional background . . . . . . . . . . . . . . . . . . . . . . . 156 10 List of Figures 1.1 SM particle content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 Shape of the potential of a complex scalar field . . . . . . . . . . . . . . . . . 21 1.3 Running of αs at LO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4 Schematic display of IRC safety . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.5 Schematic diagram of FSR and ISR shower development . . . . . . . . . . . 28 1.6 Schematic diagrams of Lund string and cluster hadronisation steps . . . . . . 29 1.7 Potential between a qq̄ pair, as a function of distance between them. . . . . . 30 1.8 Simplified schematic diagram of a complete event generation, with a Z+jets process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1 Types of DM searches governing the WIMP paradigm . . . . . . . . . . . . . 36 2.2 s-channel and t-channel production modes for DM production. . . . . . . . . 39 2.3 Diagram showing the direction of Emiss T for different rinv values . . . . . . . . 40 2.4 Schematic diagram showing evolution of jets with respect to different rinv scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 An example of a Feynman diagram showing cross contributions during t- channel production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1 A schematic diagram of the CERN accelerator complex . . . . . . . . . . . . 46 3.2 Luminosity deliverance distributions . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Number of Interactions per Crossing . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Diagram showing the coordinate system used in the LHC . . . . . . . . . . . 49 3.5 Schematic illustration of different types of inelastic collisions . . . . . . . . . 50 3.6 Schematic diagram of a generic detector and ATLAS . . . . . . . . . . . . . 51 3.7 Schematic representation of the ATLAS inner detector . . . . . . . . . . . . 52 3.8 Schematic representation of the ATLAS calorimeter . . . . . . . . . . . . . . 55 3.9 Schematic representation of the ATLAS muon spectrometer . . . . . . . . . 56 3.10 Illustration of calorimeter-only against particle-flow configurations . . . . . . 58 3.11 Topoclusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.12 Particle-flow algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.13 anti-kt, kT and Cambridge-Aachen jets . . . . . . . . . . . . . . . . . . . . . 61 3.14 Schematic illustration of successive triggering steps . . . . . . . . . . . . . . 64 3.15 An example trigger turn-on curve plot . . . . . . . . . . . . . . . . . . . . . 64 3.16 The flow of the ATLAS simulation software . . . . . . . . . . . . . . . . . . 66 3.17 Schematic diagram showing how Emiss T is determined in the transverse cross- section of a LHC detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.18 Monojet reinterpretation of semi-visible jet signals . . . . . . . . . . . . . . . 68 4.1 Illustrative Feynman diagram and subsequent production mechanism of semi- visible jets via t-channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 11 4.2 Inclusive kinematic distributions for four benchmark signals with different rinv fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 Inclusive kinematic distributions for four benchmark signals with different mediator masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 Inclusive kinematic distributions for four benchmark signals and background with different rinv fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5 Inclusive kinematic distributions for four benchmark signals and background with different mediator masses . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.6 Correlation between the HT and Emiss T distributions . . . . . . . . . . . . . . 85 4.7 Correlation between the pT balance and maxminphi distributions . . . . . . 87 4.8 Comparisons of shape of pbalT and |ϕmax − ϕmin| distributions . . . . . . . . . 88 4.9 The definition of the 9-bins in |ϕmax − ϕmin| and pbalT , defined identically in SR, VR and in each CR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.10 Comparisons of shape of pbalT and |ϕmax−ϕmin| distributions between the total background and data for 1LCR. . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.11 Comparisons of shape of pbalT and |ϕmax−ϕmin| distributions between the total background and data for 1L1BCR. . . . . . . . . . . . . . . . . . . . . . . . 90 4.12 Comparisons of shape of pbalT and |ϕmax−ϕmin| distributions between the total background and data for 2LCR. . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.13 Comparison of different samples of tt̄ for ME and PS variations . . . . . . . 91 4.14 Comparison for DR and DS scheme for tW process modelling . . . . . . . . 92 4.15 Kinematic distributions for multijet background estimation . . . . . . . . . . 96 4.16 9 bin fitted histogram in Emiss T vs HT in 1LCR . . . . . . . . . . . . . . . . . 97 4.17 9 bin fitted histogram in Emiss T vs HT in 1L1BCR . . . . . . . . . . . . . . . 98 4.18 9 bin fitted histogram in Emiss T vs HT in 2LCR . . . . . . . . . . . . . . . . . 98 4.19 9 bin fitted histogram in Emiss T vs HT in SR . . . . . . . . . . . . . . . . . . . 99 4.20 Dominant systematic uncertainty ranking in CR-SR combined fit . . . . . . 101 4.21 Correlation matrix for CR-SR combined fit . . . . . . . . . . . . . . . . . . . 102 4.22 Postfit distributions of unblinded SR kinematic variables . . . . . . . . . . . 103 4.23 Exclusion limit plots for different signals with rinv fractions. . . . . . . . . . 104 4.24 Upper limit on coupling strength . . . . . . . . . . . . . . . . . . . . . . . . 104 5.1 η − ϕ distributions for large-radius jets . . . . . . . . . . . . . . . . . . . . . 107 5.2 Distributions of the azimuthal angle difference between the leading and sub- leading jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.3 Distributions of Emiss T and leading jet pT for different signals . . . . . . . . . 108 5.4 Comparison of substructure observables . . . . . . . . . . . . . . . . . . . . . 110 5.5 Comparison of substructure observables for checking model dependence . . . 112 5.6 Comparison of substructure observables by clustering dark hadrons in final state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.7 Comparison of substructure observables to interpret behaviour of dark hadrons114 5.8 Comparison of substructure observables with intermediate and final state dark hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.9 Comparison of substructure observables to check the effect of trimming . . . 115 5.10 Comparison of substructure observables clustering only the visible hadrons and clustering also with final dark hadrons. . . . . . . . . . . . . . . . . . . . 115 5.11 EFP construction: vertex and angular connectors . . . . . . . . . . . . . . . 116 5.12 EFP construction: a degree one polynomial . . . . . . . . . . . . . . . . . . 116 5.13 EFP diagram with 4 constituents and 5 angularity connectors . . . . . . . . 117 12 5.14 LLR summary distribution containing 10 EFP diagrams . . . . . . . . . . . 118 5.15 EFP distributions corresponding to spikes in LLR summary plot. . . . . . . 118 5.16 Comparison of known jet substructure observables with selected EFPs . . . . 119 5.17 Comparison of other known jet substructure observables with selected EFPs 120 5.18 b-tagged jet multiplicity in signal . . . . . . . . . . . . . . . . . . . . . . . . 123 5.19 η − ϕ distributions for variable-radius jets . . . . . . . . . . . . . . . . . . . 123 5.20 The correlation of charged lepton pT against the ∆ϕ(closest jet, Emiss T )distance from the closest b-tagged jet . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.21 Kinematic distributions for signal and leading background processes . . . . . 125 5.22 Kinematic distributions for signal and leading background processes after SR selections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.1 Flowchart of the old MCTC classification scheme. . . . . . . . . . . . . . . 129 6.2 Flowchart of the new MCTC classification scheme. . . . . . . . . . . . . . . 130 A.1 Particle level distribution of area normalised HT, for R15 reclustered jets for varying Mϕ and rinv. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.2 Particle level objects plotted in the η − ϕ plane . . . . . . . . . . . . . . . . 149 A.3 Particle level distributions for different mediator mass and rinv fractions . . . 150 A.4 Particle level distributions of average overlap between reclustered jets of dif- ferent radius parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A.5 Reconstruction level objects plotted in the η − ϕ plane for three events . . . 151 A.6 Reconstruction level objects plotted in the η − ϕ plane for three other events. 152 A.7 pmean T , prms T , ηmean and ηrms distributions for truth-matched reco-level jets for signal point rinv= 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 A.8 pmean T , prms T , ηmean and ηrms distributions for truth-matched reco-level jets for signal point rinv= 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 A.9 pmean T , prms T , ηmean and ηrms distributions for truth-matched reco-level jets for signal point rinv= 0.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 A.10 pmean T , prms T , ηmean and ηrms distributions for truth-matched reco-level jets for signal point rinv= 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 A.11 Number of subjets seeding the reclustered jet (left). This leads to weird trend in JMS for low pT and mass regime (right). . . . . . . . . . . . . . . . . . . . 155 A.12 Effect of NCB cleaning on data and MC. . . . . . . . . . . . . . . . . . . . . 156 13 List of Tables 2.1 List of Hidden Valley particles . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 Pythia8 HV parameter choices. . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 Summary of jet reconstruction criteria. . . . . . . . . . . . . . . . . . . . . . 71 4.2 Summary of b-tagging selection criteria. . . . . . . . . . . . . . . . . . . . . 72 4.3 Summary of Emiss T reconstruction criteria. . . . . . . . . . . . . . . . . . . . . 72 4.4 Overview of the overlap removal between objects and the corresponding match- ing criteria, listed according to priority. . . . . . . . . . . . . . . . . . . . . . 73 4.5 Details of signal samples generated with rinv values of 0.2, 0.4, 0.6 and 0.8, with dark hadron mass of 10 GeV, each with full detector simulation. . . . . 74 4.6 Summary of generators used for simulation of background processes, along with the PDF and tune used. . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.7 Cutflow table for four benchmark signals of different mediator masses in and rinv of 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.8 Cutflow table for four benchmark signals of mediator mass 2000 GeV with different rinv fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.9 Summary of event pre-selections for different SR and CRs. . . . . . . . . . . 79 4.10 Signal contamination values for four benchmark signals of different mediator masses and rinv of 0.4, for determining HT threshold for CR . . . . . . . . . 85 4.11 Signal significance values for four benchmark signals of different mediator masses and rinv of 0.4, for determining HT threshold for SR . . . . . . . . . . 86 4.12 Signal contamination values for four benchmark signals of different rinv frac- tions, for determining HT threshold for CR . . . . . . . . . . . . . . . . . . . 86 4.13 Signal significance values for four benchmark signals of different rinv fractions, for determining HT threshold for SR . . . . . . . . . . . . . . . . . . . . . . 86 4.14 Signal significance values for signals of mediator mass 3.5 TeV and rinv of 0.6, for the pbalT bin optimisation for deciding final 9 bin grid. . . . . . . . . . . . 87 4.15 Signal significance values for signals of mediator mass 3.5 TeVand rinv0.6, for the maxminphi bin optimisation for deciding final 9 bin grid. . . . . . . . . . 88 4.16 Signal significance values for signals of mediator mass 3.5 TeVand rinv 0.6, for for the pbalT and maxminphi bin grid used for fitting. . . . . . . . . . . . . . . 88 4.17 Post-fit yields from background-only fit . . . . . . . . . . . . . . . . . . . . . 99 4.18 Scale factors for each background processes . . . . . . . . . . . . . . . . . . . 100 5.1 Hidden Valley model parameters considered in the study . . . . . . . . . . . 111 5.2 Grid formation, translation Figure 5.13 into a set of “particle” pairs. . . . . . 117 5.3 Cutflow table summarising selections used in the study . . . . . . . . . . . . 125 14 Preface Standard Model (SM) of particle physics has seen several successes throughout the past century as discussed in Chapter 1. However, the existence of dark matter as validated by astrophysical studies prompted an increase in the number of search programmes exploring the Beyond Standard Model sector using Large Hadron Collider (LHC) data. The obvious question to ask - Is dark matter being produced at the very high energy collisions produced at the LHC? If so, what is its signature? The only SM particle which does not interact with any of the detector components is the neutrino, since they are colour- and charge-neutral, and have faint weak interactions. Hence, the presence of neutrinos causes an imbalance of transverse momentum, which is termed as missing transverse momentum. However, if a collision only produces DM particles, we would not see anything in the detector. So we probe what is termed as mono-X signatures, where X is any SM particle (or object, like a jet) being produced along with a DM particle. The unbalanced production of X is expected to yield a large missing transverse momentum signature, inconsistent with neutrinos. To date, dark matter searches at the LHC have usually focused on Weakly Interacting Massive Particles (WIMPs), but since the existing searches in colliders have found no evidence of DM so far, several recent phenomenology papers have explored the possibility of accessing the dark sector with unique collider topologies. My primary doctoral research spanned across experimental and phenomenological study- ing a class of dark sector models known as dark Quantum Chromodynamics (dark QCD) models. In these models, dark matter is a composite particle emerging from a new “dark force” that is similar to the strong force in the SM, governed by the theory of Quantum Chromodynamics (QCD). As the strong force is responsible for the interactions between the constituents of the proton (quarks and gluons), the new “dark force” would explain interac- tions between the constituents of dark matter particles, and between these constituents and known matter. The details of this model have been discussed in Chapter 2. Running a multi-purpose detector like ATLAS involves building detector components and monitoring their operation, translating detector signals into physics information, obtaining calibrations, and managing the computing resources, amongst other tasks, and one individual cannot contribute to every component of such a large scale experiment. Chapter 3 discusses the basics of collider physics and the details of the ATLAS detector system. Semi-visible jets (SVJ) are jet-like collider objects where the visible states are interspersed with DM particles. Recent phenomenological studies have targetted a non-WIMP scenario which eventually leads to interesting collider signatures, where the final state consists of a mixture of stable, invisible dark hadrons and visible hadrons from the unstable subset of dark hadrons that promptly decay back to SM particles. The total momentum of the dark matter is hence correlated with the momentum of the visible states, leading to the event missing transverse energy close to a jet. As this is also a signature of jet mismeasurement in a detector due to presence of dead regions, or measurement of fake Emiss T contributions, especially from multijet processes, this class of jets had so far been unexplored in ATLAS, and such events had typically been ignored. However, my main doctoral analysis – exploring SVJ 15 in t-channel production mode, discussed in Chapter 4 – has set the first ever limits on this yet unexplored phase space. This has opened up several avenues of accessing the strongly interacting dark sector, by analysing the phenomenological consequences of the observed results. The SVJ signature was proposed a few years back, however the theoretical papers did not include a detailed collider search strategy. I worked on how best we can identify these unconventional jet signatures from the ones arising from Standard model processes and all the studies have been discussed in Chapter 5. One way of distinguishing SVJ from standard jets, is by looking at the internal structure of the jet, and studying the differences in jet behaviour, when dark matter is assumed to be contributing to their formation. I proposed ways to look for such unique collider signatures, using jet-substructure (JSS) observables. I have studied several jet substructure observables to compare SVJ and light quark or gluon initiated jets. Looking at the different dark hadron fractions allows us to check whether the substructure is created by the interspersing of visible hadrons with dark hadrons, or from certain model dependencies. Similarly, since this class of jets are unconventional by nature, there can be possibilities where new JSS observables might be more sensitive to these jets compared to light quark/gluon jets. As a MCnet early-career researcher, I have explored the option of designing new observables in a IRC-safe linear basis in the RIVET framework, using energy flow polynomials and this study proposed ways forward for this approach to dark-matter at the LHC, including prospects for estimating modeling uncertainties. Searching for new physics is important, however understanding the flaws in present physics process modeling and trying to improve our current understanding is equally im- portant. Hence, apart from the analysis and related performance component, I redeveloped the Monte Carlo Truth Classifier (MCTC) tool as part of my ATLAS Qualification task as discussed in Chapter 6, which involved improving the existing truth level particle definitions used within the collaboration by eliminating generator dependence in the classification of truth particles. List of papers/pre-prints and public results • Towards discrimination and improved modelling of dark sector showers, S. Sinha et al, arXiv:2209.14964 [hep-ph] • 2B or not 2B, a study of bottom-quark-philic semi-visible jets, S. Sinha et al, arXiv:2207.01885 [hep-ph] • Search for semi-visible jets in t-channel production mode using Run-2 ATLAS data, ATLAS collaboration, ATLAS-CONF-2022-038 • Theory, phenomenology, and experimental avenues for dark showers: a Snowmass 2021 report, S. Sinha et al, arXiv:2203.09503 [hep-ph] • Hitting two BSM particles with one lepton-jet: search for a top partner decaying to a dark photon, resulting in a lepton-jet, S. Sinha et al, 10.21468/SciPostPhys.13.2.018 • Exploring Jet Substructure in Semi-visible jets, D. Kar and S. Sinha, 10.21468/Sci- PostPhys.10.4.084 • Constraining the Dark Sector with the monojet signature in the ATLAS experiment, ATLAS collaboration, ATL-PHYS-PUB-2021-020 16 Chapter 1 Standard Model - background and drawbacks The Standard Model (SM) of particle physics [1] has been successful in explaining almost all the experimentally observed sub-atomic phenomena. It intricately explains the existence of 17 fundamental particles in nature, starting from the three generations of fermions (quarks and leptons), to the vector (g , γ, W, Z) and scalar (H) bosons. Despite the fact that SM is highly robust, it is not without its limitations, however it still remains an useful starting point in order to probe for new physics. This chapter intends to summarise the SM and state some of the limitations, which will pave the way for the rest of the thesis. 1.1 A “mandatory” overview of the Standard Model The SM of particle physics describes the fundamental particles that comprise everything in the visible universe and their interactions, barring the gravitational interactions between matter. The first step towards the formal construction of the SM dates back to S. Glashow’s efforts in 1961, to connect two of the fundamental forces of nature – electromagnetic (EM) force and weak nuclear force [2]. The SM explains the existence of massive particles by the Higgs mechanism, where a spontaneously broken symmetry associated with a scalar field (Higgs field) results in the appearance of mass terms for gauge bosons and fermions, originally proposed by P. Higgs, R. Brout, F. Englert, G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble in 1964 [3–5]. This was followed by the contributions of S. Weinberg and A. Salam in 1967, which presents the model in its current form [6, 7]. The different particles of the SM are summarised in Fig. 1.1. According to this model, matter is made up of two types of particles – fermions and bosons. Fermions have half- integer spin and obey the principles of Fermi-Dirac statistics. They are known to contribute to the formation of matter as we know it. Bosons are particles with integer spin, obeying the principles of Bose-Einstein statistics, and act as the mediating forces that control the interactions between fermions. The theory also demands that every particle has an associated anti-particle, which is identical in every quantum property apart from the charges. There are hypothesized ex- ceptions where a particle might be its own anti-particle and such particles are termed as Majorana fermions. 17 Figure 1.1: An overview of SM. The violet and green coloured blocks represent the three generations of quarks and leptons respectively. The orange block is the gluon, the pink block is the photon and the seagreen blocks are the vector gauge bosons. The red block is the scalar Higgs boson. Fermions Fermions are sub-divided into three generations of quarks and leptons, with each generation having a significantly larger mass compared to the previous generation, while the other quantum properties remain unchanged. Every generation consists of a pair of quarks and a pair of leptons, along with their associated anti-particles. The pair of quarks in each generation are termed as up-type and down-type, following the naming convention of the first generation quarks. The up-type quarks (or anti-quarks) have a charge of 2 3 (or -2 3 ), whereas the down-type quarks (or anti-quarks) have a charge of -1 3 (or 1 3 ). According to quantum chromodynamics (QCD), quarks also possess a property called colour charge. There are three types of colour charges – red (r), blue (b), and green (g), with each of them being complemented by three anti-colour charges – anti-red (r̄), anti-blue (b̄), and anti-green (ḡ). Every quark (anti-quark) is assumed to carry a colour (anti-colour). However, due to the presence of colour confinement effect [8], quarks cannot exist as isolated stable particles, and instead combine with other quarks to form colour-neutral states called hadrons. Hadrons can be classified as baryons or mesons, depending on whether they are formed of three quarks or a quark – anti-quark pair. Despite the fact that the probability of particles coupling via strong interactions is orders of magnitude higher than them coupling via weak interactions 1, the range of the strong force is restricted to the approximate size of the proton (i.e. 10−15 m), owing to colour confinement constraints. The pair of leptons in each generation, on the other hand, exist as a charged lepton (anti- lepton) carrying a charge of 1 (-1) and a anti-neutrino (neutrino), which is EM neutral. Leptons also carry weak isospin, and hence interact with themselves and other particles, respectively via the EM and weak forces. The charged lepton masses are again seen to increase with increasing generation, but the case of neutrino masses is a more complicated 1The variation of the coupling with the energy scale will be discussed in detail in the next chapter 18 scenario, since the SM traditionally considers them to be massless. However, it has been inferred from experiments studying neutrino oscillations, that neutrinos have a non-zero, albeit infinitesimally small mass value [9]. Bosons The SM bosons account for the different fundamental forces, with a broad classification of being either a gauge boson or a scalar boson. The gauge bosons are spin one force carriers, mediating interactions between different particles, whereas scalar bosons are spin zero particles. • The gluon is a massless gauge boson which mediates the strong force. It couples to quarks (since they contain a colour charge) and also to itself (since it carries a colour charge as well). Gluons mediate as well as participate in strong interactions, thereby allowing the formation of three/four-gluon vertices. There are eight independent colour combinations of gluons possible, as suggested by QCD. One commonly used list of the colour states are: (rb̄+ r̄b)/ √ 2 −i(rb̄− r̄b)/ √ 2 (rḡ + r̄g)/ √ 2 −i(rḡ − r̄g)/ √ 2 (bḡ + b̄g)/ √ 2 −i(bḡ − b̄g)/ √ 2 (rr̄ − b̄b)/ √ 2 (rr̄ + b̄b− 2gḡ)/ √ 6 • The photon (γ) is another massless gauge boson which mediates the EM force. It couples to all fermions containing a non-zero EM charge (i.e. both quarks and leptons). Unlike gluons, the γ does not contain an EM charge itself. • The W± and Z0 gauge bosons mediate the weak force. The Z0 boson is its own anti- particle and can mediate the weak neutral current. Because of its EM neutral nature, it couples to the same fermions as the photon, but also additionally couples to neutrinos. The W− boson is the anti-particle of the W+ boson and vice-versa. They have a EM charge of ±1 and can mediate generation and flavour changing weak processes. • The “Higgs” boson is the only known scalar boson, which is electrically neutral and is responsible for imparting mass to the other fermions. SM particles acquire mass via their interactions with the Higgs field, and the more massive the particle, the stronger the coupling is with the Higgs boson. The Higgs boson is known to interact with only massive SM particles, and also generates the masses, owing to the fact that W± and Z0 gauge bosons are not massless. Gauge theories The SM is a quantum field theory, where the particles can be represented as excitations of quantum fields. These quantum fields of SM and their interactions can be explained using the Lagrangian densities formalism, which are functions of fields and their derivatives respectively. Within the SM, the interactions are described by gauge theories, i.e. the 19 Lagrangian density is invariant under a set of transformations, which are the symmetries of the theory. These transformations have representation matrices which are the generators of the associated symmetry group. The SM symmetry groups are SU(3)C x SU(2)L x U(1)Y . QCD arises from the SU(3)C component, containing (3)2 - 1 = 8 generators (representing the eight gluons), whereas quantum electrodynamics (QED) and weak interactions arises from the SU(2)L x U(1)Y component of the theory, containing ((2)2 - 1) + 1 = 4 generators (representing the γ, W± and Z0 bosons). Symmetries are important in nature, but symmetry breaking is also important. Explicit symmetry breaking of a theory arises because of an explicit term in the Lagrangian, which breaks the symmetry of the theory actively. In this case the Lagrangian in question contains a symmetry breaking term from which the equations of motion are derived. On the other hand, in the case of spontaneous symmetry breaking (SSB), the vacuum breaks the symmetry. The Lagrangian and corresponding equations of motion still obey the spontaneously broken symmetry. Necessity of SSB in gauge theories SSB has profound applications in the realm of gauge theories of particle physics. Owing to the short range nature of weak interactions, the theory must necessarily have massive force mediators (the gauge bosons). Adding explicit mass terms for the gauge bosons would violate gauge symmetry. Such an explicit symmetry breaking compromises the unitarity of the theory, and hence, we require some other form of symmetry breaking that not only gives us the masses but also renders the theory renor- malizable. To this end the phenomenon of SSB (to be attributed subsequently to Higgs Mechanism) appears as a possible candidate. The Glashow-Weinberg-Salam theory of weak interactions, which relies on the SSB of SU(2)L x U(1)Y , is renormalizible [10]. In 1971, G. ’t Hooft had explicitly showed that gauge theories are renormalizible even in the presence of spontaneous symmetry breaking [11]. This established the importance of spontaneous symmetry breaking in the gauge theory of weak interactions and for that matter in any other gauge theory. To understand the question of how SM particles are imparted with mass, it is necessary to peek into the Higgs mechanism. The SM Lagrangian consists of the following components: LSM = Lgauge + Lfermion + LHiggs + LY ukawa (1.1) The Weinberg-Salam model consists of three SU(2)L gauge bosons, W i µ where i = 1,2,3, and one U(1)Y gauge boson, Bµ. The kinetic terms of the theory are, Lgauge = −1 4 W i µνW iµν − 1 4 BµνB µν (1.2) where, W i µν = ∂µW i ν − ∂νW i µ + gϵijkW j µW k ν (1.3) is the gluonic field-strength tensor, and Bµν = ∂µBν − ∂νBµ (1.4) is the hypercharge field-strength tensor. On introducing a SU(2) doublet of complex scalar fields, ϕ = ( ϕ1 + iϕ2 H0 + iϕ3 ) = ( ϕ+ ϕ0 ) (1.5) 20 Figure 1.2: Shape of the potential of a complex scalar field, with infinite number of degen- erate minima (i.e. vacuum) rotated by the phase rotation, leading to SSB [12]. This is the SU(2) complex Higgs doublet with SU(2) x U(1) invariant scalar potential, V (ϕ) = µ2ϕ†ϕ+ λ(ϕ†ϕ)2 (1.6) with µ2 < 0 and λ > 0. When µ2 < 0, the scalar field develops a non-zero vacuum expectation value and this leads to SSB. Since there are an infinite number of degenerate minima rotated by the phase rotation, the shape of the potential takes the form of a Mexican hat as shown in Fig 1.2. The λ term gives quartic self-interactions among the scalar fields and for vacuum stability λ > 0. Now, on minimizing the potential in 1.6 we get, ⟨ϕ†ϕ⟩ = −µ2 λ (1.7) where ⟨ϕ†ϕ⟩ denotes the vacuum expectation value, VEV. The symmetry of the theory enables us to choose ⟨ϕ⟩ = 1√ 2 ( 0 v ) (1.8) via SU(2) × U(1) transformations. Here, H0 = H + v/ √ 2. With this choice, the scalar doublet has U(1)Y charge (hypercharge) Yϕ = 1. To conserve electric charge, only a neutral scalar field can acquire a VEV. Thus, ϕ0 is interpreted to be the neutral component of the doublet. The electromagnetic charge is Q < ϕ >= 0 and hence EM is unbroken by the scalar VEV to yield the following symmetry breaking scheme: SU(2)L x U(1)Y → U(1)EM The scalar Higgs part of the Lagrangian is, LHiggs = (Dµϕ) †(Dµϕ)− V (ϕ) (1.9) To make the SU(2)L x U(1)Y symmetry local, gauge fields,W i µ for SU(2)L and Bµ for U(1)Y have been introduced. The covariant derivative for this weak doublet is, Dµ = ∂µ + ig τi 2 W i µ + i g′ 2 BµY (1.10) 21 with Y = 1. Here, g and g′ are the SU(2)L and U(1)Y gauge couplings respectively, and τi are the usual Pauli matrices [9]. In unitary gauge, there are no Goldstone bosons 2, thus, only the physical Higgs scalar is present in the spectrum after spontaneous symmetry breaking takes place. Therefore, we can write the scalar doublet in the unitary gauge as follows, ⟨ϕ⟩ = 1√ 2 ( 0 v +H ) (1.11) Here, the VEV of the theory, v = √ µ2/λ and H is the real scalar Higgs boson with mass mH = √ 2λv. It is important to note that the other three degrees of freedom, ϕ1, ϕ2, ϕ3, are absorbed by the weak bosons as shown below. From the kinetic term of the Higgs potential, we have, (Dµϕ) †(Dµϕ) → 1 2 (0 v)|∂µ + ig τi 2 W i µ + i g′ 2 Bµ|2 ( 0 v ) (1.12) On solving the above, (Dµϕ) †(Dµϕ) → v2 8 [ g2 [ (W 1 µ) 2 + (W 2 µ) 2 ] + (gW 3 µ − g′Bµ) 2 ] (1.13) Additionally, the square of the covariant derivative involves three and four-point interac- tions between the gauge and scalar fields. The charged vector boson, W− µ and it’s complex conjugate is defined as, W± µ = 1√ 2 (W 1 µ ∓ iW 2 µ) (1.14) Then the g2 term becomes 1 2 (gv 2 )2W+ µ W −µ and yields the W mass, mW = gv 2 (1.15) The two remaining neutral gauge bosons, Z and A, can be defined as, Zµ = gW 3 µ − g′Bµ√ g2 + g′2 , Aµ = g′W 3 µ + gBµ√ g2 + g′2 (1.16) from diagonalizing the mass matrix, thereby giving masses, mZ = v 2 √ g2 + g′2, mA = 0 (1.17) This is how the W and Z bosons obtain mass from the Higgs mechanism, whereas the photon remains massless. Counting the degrees of freedom, in the original theory, the complex doublet had four degrees of freedom. After the Higgs mechanism, the scalar Higgs field has 1 degree of freedom, and the three massive weak bosons, W± and Z have three, thereby conserving the number of degrees of freedom. In order to find out the fermion masses, it is convenient to write the fermions in terms of their left-handed and right-handed projections, ψL,R = 1 2 (1∓ γ5)ψ (1.18) 2Goldstone bosons appear in theories exhibiting spontaneous breaking of continuous symmetries. Each broken direction of a symmetry gives rise to one such massless scalar. 22 Here, γ5 = iγ0γ1γ2γ3 is the fifth Dirac matrix generally used to project a Dirac field into its left-/right-handed components, with γµ = γ0, γ1, γ2, γ3 being the four contravariant gamma matrices. Let us consider the example of electron and its neutrino. Since the W boson couples only with left-handed fermions, thus, the SU(2)L doublet is constructed as LL = ( νL eL ) (1.19) The hypercharge in this case is, YL = -1. Experimentally, it is known that the W boson does not interact with right-handed fields, thus the right-handed electron must be a SU(2)L singlet, and has YR = -2. Now, considering these hypercharges, the leptons can be coupled to the SU(2)L x U(1)Y gauge fields in a gauge invariant manner as follows, Llepton = iēRγ µ(∂µ + i g′ 2 YeBµ)eR + iL̄Lγ µ(∂µ + i g′ 2 YlBµ + i g 2 τiW i µ)LL (1.20) Similarly, all the fermions can be included in the SM. A Dirac fermion mass term takes the form, Lmass = −mψ̄ψ = −m(ψ̄LψR + ψ̄RψL) (1.21) The gauge invariant Yukawa coupling of the Higgs boson to the up and down quarks are, Ld = ydQ̄LΦdR + yuQ̄LϵΦ ∗uR + h.c. (1.22) where, yu,d are the 3×3 complex matrices, ϵ is the 2×2 antisymmetric tensor, Q̄L are the left-handed quark doublets, Φ is the Higgs field, h.c is the Hermitian conjugate, and uR and dR are the right-handed up-type and down-type quark singlets respectively, in the weak- eigenstate basis. This gives the mass matrices of the down-type and up-type quark as, md = Ydv 2 , mu = Yuv 2 (1.23) Similar couplings can be used to generate the mass terms for other charged leptons, whereas neutrinos remain massless because of the absence of right-handedness. 1.2 Standard Quantum Chromodynamics – a precur- sor As discussed briefly in the previous section, QCD is the gauge theory that explains the strong interactions of coloured quarks and gluons. The QCD Lagrangian is given by, L = ∑ q ¯ψq,a(iγ µ∂µδab − gsγ µtCabAC µ −mqδab)ψq,b − 1 4 FA µνF Aµν (1.24) Here, ψq,a are the quark-field spinors for quark flavour q, colour index a (a = 1,2,3) since QCD is the SU(3) component of SU(3) × SU(2) × U(1) group, and mass mq. γ µ are the Dirac γ-matrices, gs (= √ 4παs) is the QCD coupling constant, and AC µ corresponds to gluon fields with C = 1 - 8 (since, N2 C - 1 = 8), i.e. there are 8 gluons. tCab are eight 3 × 3 matrices which are the generators in the fundamental representation of Lie algebra of SU(3) group, 23 and contain information of how a gluon interacting with a quark can rotate the colour of the quark in SU(3) space. In the above equation, the repeated indices are summed over [13–15]. FA µν is the field-strength tensor given by, FA µν = ∂µAA ν − ∂νAA µ − gsfABCBA µAC ν [tA, tB] = ifABCt C (1.25) with fABC being the structure constants of SU(3) group. Looking into QCD from a hadron collider physics perspective, the immediate consequence is that some observables of QCD are calculable using the fundamental parameters of the Lagrangian. However, there are other observables that have to be expressed through models or functions whose effective parameters can only be constrained by fitting to data. Due to the fact that currently there is limited understanding of non-perturbative effects in hadron collisions, this section has been restricted to focus on some key aspects of perturbative QCD (pQCD) that pertain to collider physics [16]. The hard scattering processes in hadron collisions can be described using the collinear factorisation theorem in QCD [17], whereby colliding protons are treated as a collection of partons (point-like constituents of a hadron), carrying a fraction of the proton energy. Parton distribution functions (PDFs) [18] are probabilities of finding these partons with definite energy fractions, and PDFs are treated to be universal in nature. Partons can interact with each other and produce final state SM particles like leptons, and other partons as well. These partons are assumed to have negligible response to non-perturbative QCD effects, and can be treated as seeds of hadronic energy flows, termed as jets [19]. The experimental construction of jets will be discussed in subsequent chapters. According to the above Lagrangian, gluons interact with quarks and anti-quarks and also with other gluons. However, since there are several generators and structure constants for the gauge group, it can be complicated to evaluate the exact colour charges. The Casimir invariants [13] of the fundamental and adjoint representations are CF =4/3 and CA = 3 respectively, which shows that the colour charge of a gluon is higher than that of a quark. 1.2.1 Strong coupling and asymptotic freedom In pQCD, the predictions for observables are based in terms of αs(µ 2 R), where µR is a renormalisation scale. Considering µR ≈ Q (energy scale) of a certain process, αs can give an indication of the effective strength of the strong interaction in that process [20]. The dependence of αs on the energy scale is termed as running of the coupling, and it can be described using renormalisation groups [21]. The running is logarithmic with Q, and satisfies the following renormalisation group equations (RGE): Q2 ∂αs ∂Q2 = β(αs) (1.26) where the beta function drives the Q dependence, and can be defined as, β(αs) = −α2 s(bo + b1αs + b2α 2 s + ...) (1.27) with b0, b1, b2 being the leading order (1-loop, LO), next-to-leading order (2-loop, NLO), next-to-next-to-leading order (3-loop, NNLO) coefficients respectively. The value of the strong coupling is conventionally specified using the value of Q2 to be equal to M2 Z [22], and that leads to αs(MZ) approximated to 0.12. It is seen that while describing QCD processes at other energy scales, a major portion of the quantum corrections can be absorbed into the running coupling. In particular, at higher 24 Figure 1.3: Running of αs at LO, starting with αs(MZ) ≈ 0.12 [23]. momentum transfers (or energies), the quantum corrections can be described by a smaller coupling constant, as can be seen in Fig 1.3, and this behaviour is termed as asymptotic freedom [24, 25]. It can be described as follows, αs(µ) = 1 β0 µ2 Λ2 QCD β0 ≈ 0.5|Nf=5 (1.28) where ΛQCD is the QCD confinement scale [26] with mass dimensions and Nf are the number of active quark flavours, i.e. quarks having mass less than the scale µ. Asymptotic freedom is a pre-requisite for the success of pQCD, however, it should be noted that although αs is small, it is not negligible, hence often higher order QCD corrections are necessary to obtain higher precision. The knowledge of how scattering amplitudes behave in the soft and collinear limit is important for using pQCD to describe the hard scattering processes at the LHC. The soft limit is a scenario where the energy of an emitted gluon becomes small, whereas the collinear limit is the scenario where at least two particles propagate is the same direction, thereby having a small relative angle between their momenta. Scattering amplitudes tend to infinity when the soft or collinear limit is considered. Infrared and collinear (IRC) dynamics as shown in Fig 1.4 is non-perturbative [19], and cannot be used in an expansion in αs terms, thus, understanding IRC limits of scattering amplitudes is necessary in order to construct observables that can be described using pQCD. Furthermore, soft and collinear emissions dominate high-multiplicity final states, and that is necessary to describe the evolution from hard-scattering at short distances to situations where QCD partons non-perturbatively tran- sition to observable hadrons at large distances. 1.2.2 A non-arduous tour of the fundamentals of QCD inspired event generation The matrix element (ME) level cross-section of a hard scattering process can be calculated using a Feynman diagram for the process of interest. Similar to the coupling, masses in QCD run as well due to quantum effects, and can also be characterised using µR. If one were to 25 6= Infrared safety 6= Collinear safety Figure 1.4: Schematic display of IRC safety. A soft gluon emission (top) between two jets should not result in one merged jet (infrared safety). A soft gluon emission at a small angle (bottom) from a jet should not result in changing the jet configuration (collinear safety) [27]. calculate an observable to all orders of perturbation theory, the dependence on µR would be cancelled. However the assumption of any truncation scheme leads to the possibility of a finite dependence on µR, which is related to higher orders in αs [13]. This dependence on µR is usually quantified as a theoretical uncertainty by varying µR over a range. Considering µ2 R ≈ Q2 of a certain process, the range can be defined as, Q 2 ≤ µR ≤ 2Q (1.29) For most processes the effect of higher order variations are typically absorbed within this range. The energy fraction carried by partons entering a hard process is not known because of quantum mechanics, and hence a PDF becomes necessary. The longitudinal momentum fraction of the proton momentum carried by the parton, x = pparton/pproton, termed Bjorken- x, is defined as f(x, µF ), where µF is the factorisation scale appropriate for the interaction. Thus, the initial state non-perturbative divergences can be absorbed in the factorisation scale. Again, as for µR, a finite dependence on µF is seen to arise, and can also be quantified as a theoretical uncertainty, varying µF similiar to 1.29. The PDF evaluation is usually factorised into two parts. The non-perturbative part is parametrised by fitting to experimental data obtained from deep inelastic scattering exper- iments [28, 29], where electrons scatter off a proton, lose energy and get deflected, leading to the breaking up of the proton. The perturbative component is obtained using Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations [30–32], 26 µ2 F ∂fi(xi, µ 2 F ) ∂µ2 F = ∑ j ∫ 1 xi dz z Pij(z)fj (xi z , µ2 F ) (1.30) which shows that the change of PDFs with the factorisation scale is easily calculable in perturbation theory. In order to make perturbative corrections small, it is sensible to associate the factorisation scale with a hard energy or momentum scale in the process. Thus, choosing µF ∼ Q, the DGLAP equations can point to the variation of PDFs with the energy scale. To obtain the hadronic cross-sections, the partonic cross-section has to be combined with the PDFs, after setting the renormalisation and factorisation scales. However, even the hadronic cross-sections are far from what is observed in a hadron collider, since most processes in a collider are modelled as 2 → N processes, with not all N of the final state (FS) particles being created at the hard scatter stage. Furthermore, the order of pertur- bation theory that is used to calculate the cross-section might not be sufficient for some of the measured observables. The parton shower (PS) [33–35] approach is taken in this scenario, i.e. starting from the 2 → 2 process, which defines the energies and directions of the hardest partons and then successively building up the full structure of the event with additional parton branchings. Often it is convenient to apply the PS approach only for leading logarithmics (LL)3. Additionally, a k-factor is used to correct for the cross-section of the hard process, if a NLO sample is not available. It is calculated using the ratio of the theoretically obtained (ME level) NLO or NNLO cross-section over the LO cross section. This k-factor is then used to reweight the event, without having to regenerate the event altogether. This does pose a problem that the difference of kinematics due to higher-order effects is not taken into account, but it is in general a widely accepted method for faster computations. The branching processes g → gg, g → qq̄ and q → qg are governed by QCD and determine the DGLAP equations, and the probability of a branching occurrence given by Sudakov form factors [36, 37]. Furthermore, the parton branchings can occur before and after the QCD interaction vertex leads to initial-state radiation (ISR) and final-state radiation (FSR). FSR is usually mass, pT or emission angle ordered and termed as a time-like shower [38], whereas ISR is termed as space-like shower which involves backwards evolution [39], as shown in Fig 1.5. PS approach also assures IRC safety effectively till LL accuracy using the resummation approach [40], whereby the IR divergences due to higher-order perturbative contributions from virtual gluons are cancelled by the radiation of undetected real gluons. Electroweak processes have non-negligible contributions at the LHC even though the dominant processes originate from strong interactions, and hence QED effects have to be properly estimated when simulating SM background processes. Since the O(αEW ) ≈ O(αs), it is assumed that NLO EW effects are of similar magnitude compared to NNLO QCD effects, however, in certain phase space regions, their contributions can be non-negligible, i.e. virtual exchange of soft/collinear weak gauge bosons or photon emission. 3While identifying the jet by the initiating parton is most intuitive, it ignores the effect of additional radiation. The definition of born-level process is similarly ambiguous, and tying the definition into parton shower restricts us to LL accuracy. 27 From ME Time evolution Softer emission Softer emission From ME Time evolution Softer emission Figure 1.5: Schematic diagram of FSR (left) and ISR (right) shower development [27]. 1.3 Monte Carlo event generators Monte Carlo event generators (MCEG) like Herwig [41], Pythia [42, 43], Sherpa [44] and Powheg [45] perform detailed hard process calculations of cross sections assuming the underlying theory is the SM or one of its possible extensions. MCEG can model the physics of hadron collisions starting from the short distance scales, up to the usual hadron formation and decay scales, with the short distance physics being primarily based on pQCD convolved with parton distribution functions, since QCD is weakly interacting at distances below a femtometer. There are four major aspects that go into designing a MCEG, namely, • perturbative computation of the primary process of interest, with decays of short-lived particles. Events after this stage are usually termed as parton level • generation of QED and QCD radiation using PS • non perturbative transition from partons to hadrons, i.e. hadronisation. Events after this stage are usually termed as particle level • approximations for soft hadron physics and low-pT interactions, i.e. generation of underlying event. One can select a specific hard subprocess of interest at LO and partonic events are generated according to their matrix elements and phase space. Since the particles entering the hard subprocess, and some of them leaving it are usually partons, gluons are radiated, which can further radiate additional gluons or a quark and anti-quark pair, creating a partonic shower. Moreover, the composite nature of protons implies that multiple parton pair interactions (MPI) [46] can occur during the collision of two protons and that has to be accounted for when simulating the SM background processes. Each MPI is associated with a set of ISR and FSR showers, hence in the interleaved showering, MPI, ISR and FSR compete, with the highest ordering variable being allowed to determine the evolution [47, 48]. PS starts at the hard scale and evolves downwards to energy scales near the QCD confinement scale (ΛQCD. This implies a decrease in the energy scale at every new branching and the evolution continues until the parton energy scale is of the order of ΛQCD and they can transition to colour neutral hadrons, by the process of fragmentation [49] and hadronisation [50, 51]. The hadronisation process ideally occurs out of the proton radius and is non-perturbative in nature. There are two approaches when it comes to hadronisation as shown in Fig 1.6, namely, 28 Time evolution q q Colour string String breakup Meson Baryon Time evolution q q g Unstable hadron Figure 1.6: Schematic diagrams of Lund string (left) and cluster (right) hadronisation steps [27]. • Lund string model [52, 53] - Based on the idea that the potential between a q and q̄ at short distances is alike to the electromagnetism potential, i.e. is inversely proportional to the distance. However, as the distance increases the behaviour changes, since unlike the photon, gluons are self-interacting. The QCD potential is linear in nature as shown in Fig 1.7, and each qq̄ pair is assumed to be connected by a massless relativistic string, having no transverse degrees of freedom. The potential energy stored in the string increases as the pair moves away from the production vertex and can break to form new q′q̄′ pairs, forming two colour-singlet systems of qq̄′ and q′q̄. The gluon field between the quark charges are kinks on the system. The breaking continues as long as there is enough energy to form hadrons. • Cluster model [34, 49, 54] - Based on the property of preconfinement [55], whereby after a PS it is possible to form colour-singlet clusters of radiated partons and the mass distribution of these clusters are calculable from first principles. A standard approach is to split any remainder gluons after the shower into qq̄ or diquark pairs. In the large-Nc approximation 4 [56], it can be shown that the adjacent di-/anti-quarks will have matching (anti-)colours, and can be associated with a single colour-singlet cluster. After the clusters are formed, they are matched to a smooth hadron mass spectrum. Additionally, partons coming from hard process, MPI and beam remnants can interact with each other by exchanging colour, depicted as a connection of colour strings between them, and these interactions can lead to merging of two strings, termed as colour reconnec- tion [57, 58]. Colour reconnection reduces the string length, and hence the multiplicity of the resultant hadrons. The hadronisation of a coloured system has more free parameters, owing to the fact that individual partons do not hadronise independently, but rather collectively if they are colour-connected. Hence, tuning 5 the model to one particular dataset does not restrict its predictive power for other collision types or energies. All the particles in a scattering event which is not originating from hard scatter, and consists primarily of beam-beam remnants (BBR) and MPI is grouped together and broadly 4Despite the fact that QCD is an SU(3) gauge theory, it is frequently useful to assume a generalisation to a theory with Nc colours, SU(Nc). For any Nc, a fundamental colour can be combined with a fundamental anti-colour to form an adjoint colour and a colour-singlet. In simpler terms, the colour of a gluon can be thought of as being that of a quark and an anti-quark, with corrections owing to the fact that gluons do not have colour-singlet components. It is important to note that on using the large-Nc limit, corrections to it are expected to be suppressed by 1/N2 c ∼ 10%, however depending on the topology in question the suppression factor dynamically varies. 5The process of determining parameter values using data distributions sensitive to them, in scenarios where the values cannot be obtained from first principles. 29 Figure 1.7: Potential between a qq̄ pair, as a function of distance between them. termed as underlying event, although there can be subtle dependencies of this physics with the actual process of interest. It is important to account for this “background” phenomena in order to precisely model the hard process behaviour and probe physics beyond SM. In general, ME generators can do multi-loop (only automated at one-loop level) or multi- leg calculations whereas PS generators are accurate till LL and deal with the full event evolution. NLO calculations are usually more robust in terms of scale choices, since there can be kinematic features that seen when calculated at NLO, and hence it makes sense to combine the two to get a more precise result for the process under consideration. However, running them back to back can lead to double counting of the resultant jets, since the PS algorithm will take into account multi-legs/loops independently of that done by the higher- order ME calculation. This situation can be well described following the example of a Z+jets production, where the additional jet can come from NLO calculation at ME level, or from QCD radiation at PS level. In order to avoid this double counting, and get a precise result, it is useful to apply a matching/merging scheme [59, 60], that essentially decides a cut-off scale (based on pT or energy of jets) above which ME is used, and below which PS is used. It should be noted that at LO+PS level, matching can also be an issue, where the double counting arises between the extra emissions from the hard process and the PS. There are several schemes available for this purpose, such as, • CKKW (Catani-Krauss-Kuhn-Webber) [61, 62]: PS is started from the highest possible scale, and if there are shower emissions above the matching scale, they are vetoed. Each event is reweighted based on Sudakov form factors and running coupling from PS to avoid dependence on the matching scale. • MLM (Michaelangelo Luigi Mangano) [63]: Original ME partons are matched to the jets obtained from PS6 and if the event has all the jets from PS matched to a ME parton, it is retained. However, this approach throws away a lot of events, making the event generation process inefficient. • CKKW-L (CKKW-Lonneblad) [64, 65]: Basically CKKW, but the weight calculation is done based on PS histories. • NLO merging: Using POWHEG approach [66], momentum of the hardest emission is treated as a cut-off threshold to avoid double counting. Alternatively in MC@NLO ap- 6usually exploiting the distance parameter discussed in detail in the next chapter 30 proach [67], negative weights are assigned for PS configurations which potentially con- tain radiations already considered in the NLO calculation, and ultimately the weights are summed to obtain the correct normalisation of the sample. A full event generation is depicted schematically in Fig 1.8. Proton Proton HS FSR Hadron decay ISR QED ISR BBR DPS Photon conversion Quarks Gluons Stable hadrons Unstable hadrons Photons Z-boson Leptons Fragmentation /hadronisation Figure 1.8: Simplified schematic diagram of a complete event generation, with a Z+jets process. The hard scatter (grey star) results in a Z boson and a quark. The Z boson decays to oppositely charged same flavour lepton pair, and one of those leptons emit a photon. The quark creates a PS via FSR, resulting in the formation of unstable hadrons which subsequently decay to stable hadrons. There is also QED and QCD ISR which results in a photon, and an independent PS respectively. A double parton scattering (DPS) results in two quarks each creating its own PS, with a gluon also radiating a photon that leads to two leptons. BBR also produces stable hadrons [27]. MCEG are used extensively in collider data analysis, to simulate standard model back- ground and new physics signal events. These events are used to calibrate the physics objects 31 based on the simulation of detector response (as discussed in Chapter 3) or to devise ways to discriminate signal against background and design analysis strategies (as discussed in Chapter 4). It is of critical importance to classify particles from MCEGs based on their origin in analysis setups in a generator-agonistic way, as many MCEGs are used in ATLAS and they have different ways of presenting events records. A theoretically well-motivated way to classify particles originating from MCEG was formulated as a part of this thesis, presented in Chapter 6, which is now the default in ATLAS. 1.4 The limitations of SM While the SM provides a very solid ground for understanding how nature behaves, there are still a few open questions which cannot be addressed by the theoretical framework. Although the discovery of the Higgs boson [68, 69] completed the SM, there lies an unexplained large discrepancy between the electroweak scale (O(100 GeV)) and that of the Planck scale (O(1019 GeV)), and this is termed as the hierarchy problem [70]. From the large Higgs propagator correction, it is expected that the Higgs mass is sensitive to mass scales beyond the SM, e.g. the Planck scale. However that is not the case, so the question remains if nature is really so precise and finely-tuned, or there is a possibility of new physics that stabilises the Higgs boson mass to be approximately 125 GeV, by protecting the Higgs mass from arbitrarily high scales. Traditionally SM treats neutrinos as massless, however, experimentally it is confirmed that neutrinos do have a small non-zero mass. This evidence arises from the observation that a neutrino of a particular type or “flavour”, i.e. a muon neutrino can transform into a neutrino of another flavour, i.e. a τ neutrino. This process of transformation is termed as neutrino oscillations, and implies that neutrinos possess mass [71]. It is natural to assume that like the different generations of quarks and charged leptons, neutrinos have an ascending mass hierarchy through the generations, however there is no conclusive evidence of that scenario, and this behaviour of neutrinos are not accounted for in the SM. Despite the fact that SM creates matter and anti-matter in pairs, there is a visible abun- dance of matter over anti-matter in the observed universe, with no appropriate explanation, and this open problem can only be explained by the requirement of a baryogenesis mecha- nism [72]. The strong-CP problem [73] is yet another flaw of the SM. It ties to the question as to why there is no CP violation in strong interactions. The strong sector naturally allows for CP violation, however it has not yet been experimentally observed. However, the most important drawback of SM which is the focus of this thesis, is its inability to account for dark matter (DM), which is known to comprise 27% of the energy budget of the universe. There is astrophysical evidence that hint at the existence of DM and they will be discussed in length in the next chapter. Up until the 1980’s, SM neutrinos were thought to be a potential DM candidate, due to their non-luminous nature, but the idea was dropped once it was experimentally found that neutrinos had the potential of forming superclusters initially and would then break down to form galaxies, which is contrary to observations [74]. The nature of DM is elusive – it could be yet another particle, and that hints at the possibility that the SM might in fact not be a complete theory of the observable universe. 32 Chapter 2 Strongly interacting dark sector - why and how? In the previous chapter we discussed the basic idea of SM and alluded to why it might not be a complete theory. This chapter will touch upon the idea of dark matter (DM) and the possible candidates of particle DM. However, there is no reason to believe that the particle nature of DM is manifested through the existence of just one additional particle. It can be a plethora of particles in a dark sector (DS), much like its SM counterpart. The focus of the second half of this chapter will be on such a class of models which try to incorporate a strongly interacting DS of particles, building upon the standard QCD knowledge. 2.1 Moving on to... the overview of dark matter Over the years there have been several theories postulating the nature of DM, with records dating back to the early 1930s, which states that the universe is comprised of an unknown form of non-baryonic matter that influences the structure of the cosmos. In the following decades, astrophysical and cosmological studies, including the study of cosmic microwave background (CMB) radiation, have indicated indirect evidences for the existence of DM, some of which will be discussed in this section. Yet, till date, no direct evidence of the particle nature of DM has been found at the Large Hadron Collider (LHC). However, in order to probe the nature of DM, it is necessary to appreciate the different ways in which DM can potentially manifest itself in the observable universe. 2.1.1 Early explorations The pioneering work of astronomer F. Zwicky in 1933 [75] – studying the redshifts of Coma galaxy clusters – pointed out that there is a scatter in the apparent velocities of the eight galaxies within that cluster. These large velocity dispersions meant that the cluster density was higher compared to the one that can be obtained only by visible matter, and in turn led to the pre-emptive prediction of the existence of a non- luminous matter, with much higher density than radiating matter [76, 77]. This initial work triggered a wave of experimental and theoretical studies by different groups of astronomers. Although all of the studies cannot be discussed in great detail, as it is beyond the scope of this thesis, it is important to point out the major landmarks that govern our current understanding of DM. One such group of studies pertains to the rotation curves of galaxies, which is the circular velocity profile of the gas and stars in a galaxy, with respect to their distance from the galactic center. In 1970s, V. Rubin et al. [78] and 33 A. Bosma [79] studied the rotation curves of spiral galaxies using optical images since it is possible to infer the mass distribution of galaxies from their rotation curves. They found almost flat rotation curves, with no decline in the outer reaches of the galaxies, which was in contradiction to the expectation, if only visible matter existed. By early 1980s, substantial work went into measuring the rotation curves of galaxies well beyond their optical radii using 21 cm line of neutral hydrogen gas [80, 81], and confirmed the existence of DM. On the other hand, experimental explorations of cosmic microwave background (CMB) [82, 83] – a uniform background of microwaves surrounding all directions in space – established the modern paradigm of the hot big bang cosmology. CMB radiation is estimated to have originated 3.8 x 105 years after the big bang, when the universe was filled with hot, ionised gas. Subsequently, free electrons combined with protons to form neutral hydrogen atoms (termed as recombination). Before this epoch, the CMB photons were tightly coupled to the baryons. Recombination increased the range of CMB photons from short length scales to long length scales, and led to their decoupling, and freely travelling through space. The CMB photons thus retain information about the state of the universe during the recombina- tion timescale, and consequently, about the general nature of matter in the early universe. The observed small fluctuations in CMB increased with time due to the presence of gravity and ultimately lead to the formation of galaxy clusters that are currently observed. The shape of the amount of fluctuations in the CMB temperature spectrum at different angular scales – termed as the power spectrum – is determined by the primordial fluctuations of the inflaton [84] along with hot gas oscillations in the early universe. The amplitudes and resonant frequencies of these oscillations depend on the composition. Analysing this power spectrum can provide a strong handle towards mapping the density of matter in the early universe, and the latest studies report that the energy budget of the universe is composed of ∼27% DM, ∼5% visible matter and ∼68% Dark Energy [85]. Although there are several sources of indication for the existence of DM, they are merely indirect evidences through the gravitational influence of DM on visible, baryonic matter. However, the question still remains as to what is the nature of DM? It is certainly non- baryonic since the measured abundance of light elements that was produced in the primordial nucleosynthesis is much smaller than the total density of matter. One plausible explanation could be that DM is yet another individual (set of) elementary particle(s), not yet discovered. Furthermore, since visible matter comprises of several particles, there is no obvious indication that the same should not be expected for the dark sector. It is also well-established from astrophysical experiments that DM cannot possibly have EM charge, and in order for it to still be around, DM has to be long-lived compared to the age of our universe. 2.1.2 Dark matter relic density Before diving into further specific discussions of DM, it is imperative to understand the abundance of DM that is expected to remain at this point of the lifetime of our universe, termed as relic density. It is believed that the generation of DM in the early universe can proceed via thermal or non-thermal production, or from a particle-antiparticle asymmetry. There are predominantly two approaches when it comes to understanding the “particle” nature of DM. • Following the standard cosmological model (ΛCDM) [86, 87], that is successful in ex- plaining observations for the CMB, thermal history, and large-scale structure of the universe, DM is considered to be a cold, i.e. non relativistic particle (cold dark mat- ter). In this scenario, DM is thought to be a pressure-less component of matter which was decoupled from the thermal bath (freeze-out) well before recombination timescale. 34 There is an alternative approach, termed thermal (freeze-in) involving a Feebly Inter- acting Massive Particle (FIMP) that interacts so feebly with the thermal bath that it never attains thermal equilibrium. The relic density in this case is a combination of initial thermal distributions together with particle masses and couplings, measured in a controlled experiment or astrophysically. • It can also be assumed that larger halos (superclusters) would form first and then later fragment into smaller halos (galaxies), and the only way that would be possible is if the DM component is considered to be relativistic (i.e. hot) during the beginning of the universe. The former idea is particularly favoured, since there is no reason not to assume that like CMB photons, neutrons and other light elements, DM also originated from a thermal decoupling process. The latter idea is usually disfavoured since hot DM would lead to a very different galactic structure, which is not in tune with current astrophysical observations. 2.1.3 Dark matter particle candidates Cold dark matter is thus considered to be a stable nature. It is long-lived enough to form galaxies first and eventually form superclusters, and the potential candidate for this type of dark matter are the weakly-interacting massive particles (WIMPs) [88]. The idea of WIMPs arises in several theories that aim to resolve other limitations of the SM. The super- symmetric extension of SM (SUSY) [89] which is essentially an additional symmetry between fermions and bosons, that allows for inter-conversion of fermions and bosons, deserves specific mention in this context. According to the SUSY hypothesis, every fermion is said to have an associated superpartner boson and vice versa, effectively doubling the number of particles. Under the SUSY scheme of particles, there are several potential electrically neutral and weakly interacting dark matter candidates, like the neutralino (superposition of neutral partners of Higgs and other gauge bosons), the gravitino (superpartner of graviton, coming from the quantum theory of gravity) and the sneutrino (superpartner of neutrino). All these candidates may be placed under the general class of WIMPs and a comprehensive description of SUSY can be found at Ref [90]. Another possible class of particle candidates of DM are the axions, or axion-like particles (ALPs). As discussed briefly in the previous section, axions were originally introduced as a solution to the strong-CP problem, by virtue of an additional global symmetry of the theory which is spontaneously broken below an energy scale, fa [73, 91]. The axion mass is seen to be inversely proportional to fa and can solve the strong-CP problem for a wide range of values. Apart from this purpose, axions can also be treated as candidates of cold dark matter since they are assumed to be produced in the early universe non-thermally. Similarly, ALPs [92] arise in the spontaneous breaking of a global symmetry, however in this case there is no strict relation between the mass of the ALPs and the energy scale fa, and they can be treated as viable candidates for DM. 2.1.4 Interactions between Standard Model and dark matter The different types of DM searches that can be probed when the WIMP scenario is assumed, are shown in Fig 2.1. 35 Figure 2.1: Types of DM searches governing the WIMP paradigm. Annihilation of DM particles and observing the SM outgoing particles is the indirect detection approach, the scattering of a DM particle off of a SM detector is the direct detection approach, and the annihilation of SM particles to form DM particles is the collider approach. Indirect detection (non-collider) Indirect detection methods focus on looking for products of DM interactions, particularly leading to SM particles, rather than the DM itself. An annihilation cross-section of the order of 10−26 cm3s−1 is expected from the measured cosmological DM density, and hence indirect detection methods try to probe the outcome of the annihilation of DM particles. If DM is assumed to be unstable, it would decay and produce observable decay products, and there are several experiments which have been devised to detect DM decay products using γ-rays e.g., VERITAS [93], HESS [94], MAGIC [95], FermiLAT [96], X-rays e.g., XMM-Newton [97], NuSTAR [98], Suzaku [99], neutrinos e.g., IceCube [100], ANTARES [101], Baikal [102], Baksan [103], Super-Kamiokande [104], cosmic rays e.g., HAWC [105] and detectors in space e.g., DAMPE [106], CALET [107], AMS [108]. Despite the fact that none of the above mentioned experiments have been able to report any signal so far, they have been successful in narrowing down the parameter space for future indirect searches. Several of these experiments are already undergoing upgrades (IceCube- Gen2 [109], KM3NET [110], CTA [111], Baikal-GVD [112], Hyper-Kamiokande [113]) that would set the stage for higher sensitivity of indirect DM searches. A very detailed review of this aspect of DM searches can be found at [114]. Direct detection Direct detection of dark matter involves low energy (sub-MeV) scale scattering events, where the main focus is to try and record the rare events when a DM particle scatters off a target material, with negligible background interactions. The driving idea is that dark matter was produced thermally in the early universe and subsequently decreased via annihilations into ordinary matter, to attain a stable equilibrium. The primary signal of DM scattering 36 is nuclear recoils from Earth-based detectors, and direct detection of WIMPs is a widely recognised research area, originally proposed in [115]. Since WIMPs have no electric charge, they should mostly scatter off the atomic nucleus, and subsequent momentum transfer gives rise to a potentially detectable nuclear recoil. The dominant part of nuclear recoil energy induced by WIMP-nucleon interaction is lost as heat, and this leads to atomic motion, which in turn gives rise to phonons [116] in solid materials. The available electronic energy loss can excite or ionise the target atoms, and this leads to narrow emission spectrum scintillations which can be detected by photosensors. Several direct detection experiments like CRESST-III [117], XENON100 [118], DAMA or LIBRA [119] are searching for DM signal excesses over the background, but have not confirmed any positive results yet [120, 121]. Collider searches When it comes to collider searches, they can broadly be classified into, searches for UV complete models, i.e. SUSY, or searches for simplified models. The search performed in the main body of the thesis uses a simplified model of DM as a benchmark, with the assumption that the mediator connecting SM and dark sector is beyond the scale of interaction, and results in a contact interaction. Since, DM is non-luminous in nature, hence the collider final state is expected to contain a significant amount of missing transverse momentum. But, the more important question to ask is, what do we measure, if everything is invisible? The only SM particle which does not interact with any of the detector components1, is the neutrino, since they are colour- and charge- neutral, and have weak interactions. Hence, neutrinos constitute missing transverse momentum (Emiss T ). This can be realised in accordance with the energy-momentum conservation principle which demands that the total pT of all particles has to be zero after collision, since the initial particles (protons) move along the beam axis. Hence, Emiss T can be treated as the negative sum of scalar pT of all other outgoing objects. In order to make the search for dark matter in colliders relatively model-independent, it is assumed that the recoiling visible particles should be governed by Standard Model interactions. There are a significant number of models which predict the possibility of SM bosons being present in any beyond Standard Model (BSM) process, which contribute to initial state radiation (ISR) from partons, and since hadron colliders are dominated by gluon ISR, hence looking at mono-X (X = γ, h, Z, top, jets etc) final states is one of the most used approaches in DM searches, and this will be further discussed in the next chapter. For the context of this thesis, we will restrict ourselves to the alternative (non-WIMP) collider searches paradigm for DM. The remainder of the chapter will discuss the primary type of DM model that was probed for this thesis, in terms of the theoretical considerations and the model parameters. 2.2 Simplified Models of dark matter Several new theories have been proposed where the DM exists within a new dark sector [122– 126], defined by a new set of particles, forces and interactions, and this dark sector can interact with the SM sector via portal interactions, with the renormalisable examples being the lepton, photon and Higgs portals [127]. In scenarios where the BSM mediator is massive compared to the collision energy, the interaction between SM and DM can be modelled as a 1will be discussed in the next chapter 37 contact interaction, and effective field theories (EFTs) can be used to explain the production of invisible particles. In general, dark matter EFTs follow the simple principle that the DM is a single particle candidate beyond the SM and all the other degrees of freedom are either negligible to have any significant impact in the observable spectrum, or are heavy enough to be integrated out [128]. This is determined by one parameter (dimension-D operator) that controls the production rate, and the corresponding Lorentz structure is seen to have a mild effect on the kinematic distributions of the invisible particles. The interaction Lagrangian contains all the gauge and Lorentz invariant terms and the additional DM field (χ) term can be written as an expansion of dimension-D operators: Lχ = ∑ D,i,f CD i OD i,f (2.1) The Wilson Coefficients (CD i ) [129] are parametrised as CD i = Λ4−D i , where Λi is the cut-off scale for the EFT. EFTs are a more generalised approach for studying the dark sector, since the dimension-D (D=6) operators can range over several mediation schemes, and the complicated high-energy interactions are integrated out in the UV completed contact operator limits. In cases where the mediator mass is much less than the collision energy, simplistic de- scriptions of collider phenomenology can be constructed, not considering the extra physics details of the theory observed at energies greater than those accessible at the collider and irrelevant at the LHC. These descriptions are collectively called simplified models, and can be treated as a transition step between full theories and simplest EFTs. The primary idea of simplified models is based on the fact that only a small number of new particles may be present, and these models can be developed from tree-level interactions (pair production, associated production) of invisible particles, which is mostly sufficient for discussing LO collider phenomenology [130]. 2.3 QCD-like dark sector A non-Abelian dark sector which displays asymptotic freedom similar to SM QCD, can be categorised as a QCD-like DS, and it is assumed that in such scenarios, the SM sector and DS are coupled through a portal [131, 132]. The most general class of such models involve a minimal extension of the SM with new particles and couplings. It can be assumed that the new mediator acts as a portal. In s-channel production mode shown in Fig 2.2, the mediator is a colour singlet and has to couple to a pair of either DM particles or SM particles. Since massive colour-neutral spin-1 bosons with axial-vector or vector couplings are present in many BSM theories, Z ′ bosons can be treated as prospective mediators in most models. Here a Z2 discrete symmetry is assumed, under which the SM fields and the mediator are even, but the DM particle is odd, to guarantee DM stability [133–139]. In t-channel production mode (Fig 2.2), the mediator interacts with DM and one of the SM quarks, and usually simplified models consider the scenario of fermionic DM particle which interacts with SM particles via a scalar mediator coupling only to right-handed quarks. A generic t-channel DM simplified model contains an extension of the SM by two additional fields: a DM candidate, χ and a mediator, ϕ which has its fundamental representation in SU(3)c and a dark non-Abelian gauge group [140–143]. 38 Figure 2.2: s-channel and t-channel production modes for DM production. 2.3.1 Semi-visible jets theory model Semi-visible jets [123, 136, 143] (SVJ), are jet-like collider objects where the visible states in the shower are Standard Model hadrons. It is assumed in these scenarios that the strongly coupled hidden sector contains some families of dark quarks which bind into dark hadrons at energies lower than a dark-confinement scale Λd. Reference [143] targets a non-WIMP scenario which eventually leads to interesting collider signature, where the final state consists of a jet aligned with missing transverse momentum2 (Emiss T ) due to a mixture of stable, invisible dark hadrons (with decay time cτ > 10 mm) and visible hadrons from the unstable subset of dark hadrons that promptly decay back to SM particles. The total momentum of the dark matter is hence correlated with the momentum of the visible states, leading to event Emiss T close to a jet. The model of Ref. [143] uses a simplified parameterisation (an extension of simplified models), where a direct mapping of the Lagrangian parameters to physical observables is not possible since some of the dark sector observables depend on non-perturbative physics. The three parameters of this model in the t-channel production mode are: • Mass of the scalar bi-fundamental mediator (Φ), which is in the fundamental represen- tation under both visible QCD and the dark non-Abelian gauge group (for t-channel production), denoted by Mϕ. • Dark hadron mass, denoted by MD. • The ratio of the number of stable dark hadrons over the total number of dark hadrons in the event, rinv. The first two parameters are set during the ME level event generation stage, whereas the third parameter is set during the dark shower stage of event generation, which helps to achieve the desired collider topology in a simple manner. rinv in its intermediate regime makes the SVJ appear, by controlling the invisible energy fraction as shown in Fig 2.3, hence higher the value of rinv, more Emiss T the event will have. In order to introduce a simple portal, the BSM effects are parametrised in an EFT expansion. In this approach, an effective Lagrangian captures all possible interactions: 2The initial momentum of the colliding partons along the beam axis is unknown because the energy of each hadron is split and constantly exchanged between its constituents, so the amount of total missing energy cannot be determined. However, the initial energy of particles travelling transverse to the beam axis is zero, so any net momentum in the transverse direction indicates missing transverse momentum, Emiss T . More details will be discussed in the following chapter. 39 Figure 2.3: Diagram showing the direction of Emiss T for different rinv values [143]. Leff = LSM + (1/Λd)L1 + (1/Λ2 d)L2 where Li’s are constructed from Standard Model operators that obey the SU(3)C x SU(2)L x U(1) gauge symmetries, and the higher-dimensional Lagrangian terms representing effective (i.e. non-fundamental) couplings, are suppressed by powers of Λd. As the dark matter particles can appear in the final state of this model, a dark matter Effective Field Theory approach is used, in which the DM is the only additional degree of freedom beyond the SM accessible by current experiments, and hence the interactions of the DM particle with SM particles are described by effective operators (of dimension-6 or higher) of the form: Lcontact ⊃ (cijαβ/Λ 2)(qiγ µqj)(χαγµχβ) Here, q are the SM quarks, χ are the dark sector quarks, Λ is the scale of the operator, and cijαβ are O(1) couplings containing the information of possible flavour structure. The Lagrangian containing the interaction and kinetic terms is: Ldark ⊃ −1 2 trGd µνG dµν − χ̄a ( i��D −Md,a ) χa Here, the dark sector is a SU(2)D gauge theory with coupling αd = g2d/4π, containing two fermionic states χa = χ1,2, and cijab are O(1) couplings that encode the possible flavour structures, and (assuming minimal flavour-violation) light-flavour production channels dom- inate. If dark mesons exist, their evolution and hadronisation process are currently not very constrained. They could decay promptly and result in a very SM QCD like jet structure, even though the original decaying particles are dark sector ones; they could result in semi- visible jets; or they could behave as completely detector-stable hadrons, in which case the final state is just the missing transverse momentum. Apart from the last case, which is more like a conventional BSM Emiss T signature, the modelling of these scenarios is a fairly unexplored area. 2.3.2 Hidden Valley Shower Pythia8 [43] is a general purpose MCEG used for simulating high-energy collisions, starting from a few-body process and resulting in multi-hadronic final states. It includes libraries containing tree level calculations for different hard processes and initial- and final-state par- ton shower, multiple parton interactions, beam remnants, fragmentation and hadronisation, and particle decays, and is used to model the SM process outcome in a detector. The tool 40 works well for SM, but when models with BSM interpretations have to be studied, there is no guiding principle for the underlying hadronisation. Many BSM models contain sectors with new gauge groups and matter content, which may decouple from the SM at low energy limits. The Hidden Valley (HV) [123, 144–146] module of Pythia8 was designed in order to study a strongly coupled dark sector. The module tends to simulate a reasonably generic framework for studying DS models, and HV being a light hidden sector, the associated particles may have masses as low as 10 GeV and the spectrum of the valley particles and their dynamics depends on the valley gauge group Gv, their spin and the number of particles contained in the theory, along with their group representations. The construction of the HV sector begins with the specification of the particle content. There are 12 particles which are charged under both the SM and HV symmetry groups, with each particle coupling flavour- diagonally with the corresponding state in SM, but has a fundamental representation in the HV colour symmetry as well. They are listed below: Table 2.1: List of Hidden Valley particles Hidden Valley particle Corresponding SM particle Dv d quark Uv u quark Sv s quark Cv c quark Bv b quark Tv t quark Ev e lepton νEv νe, electron neutrino µv µ lepton νµv νµ, muon neutrino τv τ lepton ντv ντ , tau neutrino These particle states are collectively referred to as Fv. In addition to these states, it is assumed that the HV contains either of the following two: • Abelian U(1) gauge symmetry, broken or unbroken leading to a γv which is the massless gauge boson of the HV U(1) group. • Non-abelian SU(N) gauge symmetry, which is unbroken and leads to N2 − 1 massless gauge boson of the HV SU(N) group gv’s and new massive matter particles qv’s in its fundamental representation. There are three alternative production mechanisms of these particles: • A massive Z ′ can act as a mediator for s-channel production modes: qq̄ → Z ′ → qv q̄v • There is also a possibility of kinetic mixing terms: qq̄ → γ → γv → qv q̄v • The final general possibility is the presence of a Fv which is charged under both the HV and SM groups, and hence can have the following production scheme: gg → FvF̄v and subsequently decay to Fv → fqv, where f is a SM fermion 41 For the broken U(1) symmetry, hidden valley photon acquires mass and then decays to regular photons that make pairs of SM particles. As for the SU(N) symmetry, the dark quark hadronises in hidden sector with full string fragmentation evolution, and produces up to eight qv flavours and 64 qvqv mesons. The masses are assumed to be degenerate, but the meson masses and decay mechanisms can be specified by the user, depending on the model. The radiation off the HV charged particles is allowed, and HV particles are not produced in ISR. Since, all the (anti-)particles Fv and qv have one (negative) positive unit of HV charge, hence the HV radiation is similar to that of QCD, so the showering mechanism for the three radiations (QCD, QED and HV) are interleaved. The emissions are arranged in one common sequence of decreasing emission pT scales and hence it is difficult to separate one kind of radiation from the other. The HV particles with no SM couplings are invisible and their presence can only be detected by observing the amount of Emiss T present in a particular event. There are two possibilities of simulating such a scenario where activity in the hidden sector seeps through to the visible sector and constitutes Emiss T . • If the U(1) symmetry is broken, then γv acquires mass, and subsequently decays back to SM particles either by a Z ′ state or some either mediator, since it has a small but non-zero mixing angle with the SM photon. However, the qv remains absent in this scenario due to the lack of any U(1) charge. • In case of the SU(N) symmetries, the gauge group remains unbroken leading to mass- less gauge bosons gv and there is confinement of partons. In this scenario, after the hadronisation, the qvs'and q̄vs'can be obtained which can either decay back to SM or remain stable, depending upon the mixing of the states. The qvs’ can exist as stable and invisible states if they are off-diagonal and flavour-charged, whereas diagonal ones can decay back to the SM and contribute to formation of visible hadrons. The HV module of Pythia8 allows the existence of visible jets (where the final state is like a regular QCD jet, even though the initial constituents came from the hidden sector), invisible jets (where the HV particles are completely stable and only contribute to missing transverse energy), and semi-visible jets (where some of the dark hadrons decay back to SM whereas the others remain stable within collider timescales) [132]. Additionally, depending on the lifetime of the dark sector particles, i.e. when they are long-lived, they can lead to emerging jets [147]. The different possible final states that can arise from the HV module