Search for Supersymmetric Pseudo-Goldstini at √ S = 13 Tev

Search for Supersymmetric √ Pseudo-Goldstini at s = 13 TeV with the ATLAS Detector

A Dissertation

Presented to The Faculty of the Graduate School of Arts and Sciences Brandeis University Department of Physics Craig Blocker, Martin A. Fisher School of Physics, Advisor

In Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy

by David Michael Dodsworth February, 2021 This dissertation, directed and approved by David Michael Dodsworth’s committee, has been accepted and approved by the Graduate Faculty of Brandeis University in partial fulﬁllment of the requirements for the degree of:

DOCTOR OF PHILOSOPHY

Eric Chasalow, Dean Graduate School of Arts and Sciences

Dissertation Committee: Craig Blocker, Martin A. Fisher School of Physics, Chair James Bensinger, Martin A. Fisher School of Physics Andy Haas, Department of Physics, New York University c Copyright by

David Michael Dodsworth

2021 You are a stolen sigh-between- -naps. A mind, unwrapped You are time’s relative. Kind, and kinder to a faulty space Patient inter-action; conferred—interred Your courage unreserved. Grace is eternal, a kin to gratefulness. Aching to ward the whole

I will always love you.

iv Acknowledgments

I’d first like to thank Craig Blocker, my advisor. This thesis would not be possible without him, both as an active collaborator and as a sanity check for my own ideas. I’m thankful for his patience throughout my PhD and for his thoughtful, insightful, and considerate input at every step. Thank you to Jim Bensinger for being on my committee and for his general support throughout my time at Brandeis—whether that was during my advanced exam, at alignment meetings, or over dinner in Geneva. I’d also like to thank Andy Haas for serving on my committee and his generous mentorship at the beginning of my particle physics journey, all those years ago. Thank you to my parents, Andrew Dodsworth and Shirley Grace Dodsworth, for their unquestioning support (in all its forms) and for giving me the space and resources to find my own path. Thank you to my grandmother Doris Wallis for the happy childhood memories that I remember fondly. Thank you to my siblings Duncan, Emma, Michael, Peter, Steven, and Vicky for being there when it counts, as well as all my family for being a refuge during Christmas breaks. Thank you to Kelsey O’Connor, my colleague and ally throughout our PhDs, who has become a lifelong friend. Thank you for the shared experiences and for being my first friend in an unfamiliar place. Thank you as well to Graham Stoddard for his companionship

v during the summer and for opening up his home to me when I was commuting. Thank you to Lu An and Isabella Soldner-Rembold for their calming perspectives and eﬀorts in keeping me sociable. Thank you to Siyuan Sun for showing me the ropes at CERN, our candid conversations about life and philosophy, and for my initial foray into bouldering. Thank you to Joanna Robaszweski for succor during tough times and to Aparna Baskaran for taking the time to help me with statistical mechanics. Thank you to James Cox, Chris Daly, Michela Grant, Jennifer Hadley, Alex Lˆe,Duncan Leggat, Kevin Lee, Shabria Ray, Tricia Tillman, Hugo Wainwright, Isabel Wheeler, and Caz Yang for their enjoyable company and interesting discussions. Thank you to old friends Jay Blake, Will Bovill, Kathy George, Nikki George, Jen Johnson, Harrison Louca, Carl Shrimpton, and Charlie Shute for keeping me grounded and connected. Thank you to the Brandeis Melee group (Tazio De Tomassi, Sam Ruditsky, Max Everson, Kevin Loew, Kevin Wei, David Heaton, Henry Goodridge, Jon Maeda, Kei Isobe, Dylan Quinn, and Yuhua Ni) for fond memories and a welcome distraction from work. Thank you to my new family in Dallas—Diantha Pyle, Phillip Pyle, Davonda Parker, MaToya Parker, Shayla Parker, Walter Parker, and Mary Lofton—for their support, encouragement, and welcoming me with open arms. Finally, I’d like to thank Marcus Pyle. Thank you for the adventures we’ve had across the globe, thank you for inspiring me and motivating me to go beyond what I thought I could accomplish, and thank you for continuing to surprise and delight me with your love of life and learning. You were there during the hard and uncertain times, and are the source of so much joy and satisfaction. I count myself lucky to have found you and I am proud to share my journey through life with you.

vi Abstract √ Search for Supersymmetric Pseudo-Goldstini at s = 13 TeV with the ATLAS Detector A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University, Waltham, Massachusetts by David Michael Dodsworth

This thesis presents a search for events in a ﬁnal state containing two opposite-sign, same-

miss flavor leptons, two isolated photons, and significant missing transverse momentum, ET . The final state signature is motivated by slepton pair-production in models describing exten- sions to the Minimal Supersymmetric Standard Model, with sleptons decaying to a neutralino and lepton, and neutralinos decaying to a photon and one of two supersymmetric particles— the massless, true goldstino and the massive pseudo-goldstino. This work is a novel analysis and is performed with the full Run-II ATLAS dataset, consisting of 139 fb−1 of proton-proton √ collision data, at a center-of-mass energy s = 13 TeV. No significant excess is observed, and results are presented as exclusion limits at the 95% confidence level as a function of the neutralino-to-pseudo-goldstino branching ratio for each signal sample.

vii Contents

Abstract vii

1 Introduction1

2 Theoretical Motivation3 2.1 A Brief History of Particle Physics...... 3 2.2 The Standard Model...... 6 2.3 Supersymmetry...... 14 2.4 Pseudo-Goldstini and the mssm-goldstini Model...... 18

3 Analysis Search Strategy 24

4 Experimental Apparatus 28 4.1 The Large Hadron Collider...... 28 4.2 The ATLAS Detector...... 33

5 Object Reconstruction at ATLAS 42 5.1 Track and Vertex Reconstruction...... 42 5.2 Electrons and Photons...... 43 5.3 Muons...... 48 5.4 Object Isolation...... 51 5.5 Hadronic Jet Reconstruction...... 53 miss 5.6 ET Reconstruction...... 55 5.7 Overlap Removal...... 58 5.8 The Trigger System...... 60

6 Monte Carlo Event Simulation 63 6.1 Signal Monte Carlo Generation...... 63 6.2 Background Monte Carlo Generation...... 64 6.3 Scale Factor Reweighting...... 66

viii CONTENTS

7 Physics Object Deﬁnitions 68 7.1 Baseline and Signal Electrons...... 68 7.2 Baseline and Signal Muons...... 69 7.3 Photons...... 70 7.4 Baseline Jets...... 72 7.5 Signal b-Jet Veto...... 72 7.6 Baseline Taus...... 73

8 Event Selection 74 8.1 Preselection...... 75 8.2 Channel Selection...... 76

9 Background Estimation 90 9.1 Background Types and Relevance...... 91 9.2 Jets Misidentified as Leptons...... 92 9.3 Different-Flavor Lepton Pair Backgrounds...... 93 9.4 Jets Misidentified as Photons...... 93 9.5 ``γγ Normalization Scaling...... 97

10 Systematic and Statistical Uncertainties 104 10.1 Electron and Photon Uncertainties...... 105 10.2 Muon Uncertainties...... 106 10.3 Jet Uncertainties...... 107 10.4 Flavor-Tagging Uncertainties...... 108 miss 10.5 Pileup, Luminosity, and ET Uncertainties...... 108 10.6 Statistical Uncertainties...... 109

11 Results and Conclusion 111 11.1 Exclusion Limit Calculation on Signal Samples...... 114

ix List of Tables

2.1 Global symmetries, local gauge symmetries, and their resulting conserved quantities...... 11

3.1 Parameter combinations in the mssm-goldstini signal samples...... 27

4.1 List of all accelerators in the CERN complex, with the center-of-mass energies to which proton beams are accelerated...... 29 4.2 List of all experiments present at the LHC, with a brief description. References to the experiments’ technical design reports and/or proposals are included.. 29

5.1 [87] A list of all electron discriminating variables used by the ATLAS identification algorithms. The Rejection column indicates whether the variable offers discriminating power between light-flavor (LF) jets, heavy-flavor (HF) jets, or photon conversions; the Usage column indicates a direct selection cut (C) or use as part of the likelihood function (LH)...... 46 5.2 [88] A list of all photon discriminating variables used by the ATLAS identification algorithms, for both Loose and Tight photon identification...... 47 5.3 List of dielectron triggers used in this analysis for each year of data-taking (NB: The logical OR of two triggers were used for 2017 and 2018, except for runs B5-B8 where the sole trigger denoted by † was used, due to partial pre-scaling)...... 61 5.4 List of dimuon triggers used in this analysis for each year of data-taking... 61 5.5 List of multi-lepton triggers used in this analysis for each year of data-taking. 62

6.1 A summary of all simulated MC samples used in the analysis and their associated modeling programs and parameters, where Generator is the program calculating the interaction matrix elements, Order is the order to which cross- sections are calculated, Showering is the programs simulating hadronization and fragmentation of colored partons, PDF is the parton density function of the proton’s internal structure, and Tune is a set of tunable parameters passed to the Showering program. GEANT4 is used for detector simulation for all samples and has been omitted as a column...... 65

x LIST OF TABLES

6.2 A list of all scale factors (SFs) implemented in this analysis. Pileup and trigger scale factors are single event-level SFs, while electron, muon, photon, and jet SFs are object-level SFs and applied to all relevant objects in the signal regions. Electron SFs are only applied to electron channel events and muon SFs are only applied to muon channel events...... 66

miss 7.1 Baseline electron criteria used in the ET calculation...... 69 7.2 Signal electron criteria used in the signal region definitions...... 69 miss 7.3 Baseline muon criteria used in the ET calculation...... 70 7.4 Signal muon criteria used in the signal region definitions...... 70 miss 7.5 Baseline photon criteria used in the ET calculation...... 71 7.6 Signal photon criteria used in the signal region definitions...... 71 miss 7.7 Baseline jet criteria used in the ET calculation...... 72 7.8 Signal b-jet criteria used to veto events...... 73 miss 7.9 Baseline tau criteria used in the ET calculation...... 73 8.1 Preselection criteria for both signal channels...... 76 8.2 Selection criteria for the electron signal channel...... 89 8.3 Selection criteria for the muon signal channel...... 89

10.1 Number of simulated events for the ``γγ MC samples in√ the electron and miss muon channels, along with their statistical uncertainty ( N) for each ET bin shown in Figures 11.1 and 11.2 (before scaling)...... 110

11.1 Event yields for every selectron signal sample, for the simulated PGLD branching ratio (BR) of 0.5 and the limiting BR= 0.0 and BR= 1.0 cases. Samples where the yields are above the upper limit of expected signal events for all BRs are Excluded; those that cross the upper limit threshold are BR-dependent; and those that are always below it are Not excluded. BR-dependent sample yields as a function of BR are additionally shown in Figure 11.3...... 117 11.2 Event yields for every smuon signal sample, for the simulated PGLD branching ratio (BR) of 0.5 and the limiting BR= 0.0 and BR= 1.0 cases. Samples where the yields are above the upper limit of expected signal events for all BRs are Excluded; those that cross the upper limit threshold are BR-dependent; and those that are always below it are Not excluded. BR-dependent sample yields as a function of BR are additionally shown in Figure 11.4...... 118

xi List of Figures

2.1 [37] Table of all particles predicted by the Standard Model...... 9 2.2 [46] The ‘Sombrero’ Higgs potential V (H), as a function of H ...... 13 2.3 Feynman diagram for slepton pair-production to pseudo-goldstini cascade decays, with supersymmetric particles highlighted in red. Decays occur via neutralinos to a ﬁnal state with two leptons, two photons, and missing transverse energy from the massless goldstini G˜ or massive pseudo-goldstini G˜0 ...... 23

3.1 Visual representation of the 21 signal samples. Diﬀerent colors denote diﬀerent

slepton M`˜ masses; black (500 GeV), green (400 GeV), red (300 GeV), and blue (200 GeV)...... 26

4.1 [65] Diagram of the LHC accelerator complex...... 30 4.2 [66] Diagram showing the eight interaction points along the LHC ring and the location of the ALICE, ATLAS, CMS, and LHCb detectors...... 31 4.3 [77] Cutaway view of the ATLAS detector...... 32 4.4 [77] Cutaway view of the inner detector...... 34 4.5 [77] Cut-away view of the ATLAS calorimeter system...... 36 4.6 [77] Cutaway view of the muon detection system...... 39

5.1 [99] Distribution of the output discriminant of the DL1 b-tagging algorithm for b-jets, c-jets, and light-ﬂavor jets...... 56

8.1 Background estimate of the dominant ``γγ MC sample for leading and sub- leading lepton pT. Four representative signal samples are overlaid and labeled

by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading electron pT on the top-left and sub-leading electron pT on the bottom-left. Muon channel estimates are on the right, with leading muon pT on the top-right and sub-leading muon pT on the bottom-right. Overﬂow bins are visible for all four plots...... 78

xii LIST OF FIGURES

8.2 Background estimate of the dominant ``γγ MC sample for leading and sub- leading lepton identiﬁcation classiﬁcation. Four representative signal samples

are overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading electron identification type on the top-left and sub-leading electron identification type on the bottom-left. Muon channel estimates are on the right, with leading muon identification type on the top-right and sub-leading muon identification type on the bottom- right. Overflow bins are visible for all four plots...... 80 8.3 Background estimate of the dominant ``γγ MC sample for leading and sub- leading lepton isolation classification. Four representative signal samples are

overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading electron isolation type on the top-left and sub-leading electron isolation type on the bottom-left. Muon channel estimates are on the right, with leading muon isolation type on the top-right and sub-leading muon isolation type on the bottom-right. Overﬂow bins are visible for all four plots...... 81 8.4 Background estimate of the dominant ``γγ MC sample for leading and sub- leading photon pT. Four representative signal samples are overlaid and labeled

by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading photon pT on the top-left and sub-leading photon pT on the bottom-left. Muon channel estimates are on the right, with leading photon pT on the top-right and sub-leading photon pT on the bottom-right. Overflow bins are visible for all four plots...... 83 8.5 Background estimate of the dominant ``γγ MC sample for leading and sub- leading photon identification classification. Four representative signal sam-

ples are overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading photon identification type on the top-left and sub-leading photon identification type on the bottom-left. Muon channel estimates are on the right, with leading photon identification type on the top-right and sub-leading photon identification type on the bottom-right. Overflow bins are visible for all four plots...... 84 8.6 Background estimate of the dominant ``γγ MC sample for leading and sub- iso leading photon isolation energies ET . Four representative signal samples are overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD). Electron iso channel estimates are on the left, with leading photon ET on the top-left and iso sub-leading photon ET on the bottom-left. Muon channel estimates are on iso iso the right, with leading photon ET on the top-right and sub-leading ET on the bottom-right. Overflow bins are visible for all four plots...... 85

xiii LIST OF FIGURES

8.7 Background estimate of the dominant ``γγ MC sample for the dilepton invariant mass Four representative signal samples are overlaid and labeled by their

mass parameters (M`˜, Mχ˜, MP GLD). The dielectron invariant mass for the electron channel is on the top, and the dimuon invariant mass for the muon channel is on the bottom...... 87

iso 9.1 The isolation energy ET distributions and Crystal Ball fits for tight identification (red) and pseudo-tight identification (blue) photons in MC ``γγ QCD samples. Events must contain two signal leptons (electrons or muons) but can contain any number of photons as long as at least one is tight or pseudo-tight. The tight photon histogram integral was scaled to the pseudo-tight photon integral to illustrate the relative shape differences of the distributions (a scale factor of 0.08)...... 96 9.2 A combined fit (blue) formed of two separate Crystal Ball functions. The real pseudo-photon fit (red) is constructed from the CBShape pseudo-photon fit result parameters from Figure 9.1, while the fake fit (green) is unconstrained with all parameters free. Events do not need to pass signal lepton requirements, but there must be at least two leptons with VeryLoose identi- miss fication and a pT ≥ 20 GeV. The isolated (ET ≤ 0 GeV) and anti-isolated miss (8 ≤ ET ≤ 80 GeV) integrals from the unconstrained fake fit are used in calculating the fake factor for photons...... 98 9.3 Data and ``γγ MC event counts in the isolation pair control regions, before normalization scaling, for the electron channel. Dilepton invariant masses miss are on the left and ET is on the right, with plots ordered by pair type: Iso-Anti/IA (top), Anti-Iso/AI (middle), and Anti-Anti/AA (bottom). These event counts are used with the fake factor f to determine the number of fake photons...... 99 9.4 Data and ``γγ MC event counts in the isolation pair control regions, before normalization scaling, for the muon channel. Dilepton invariant masses are on miss the left and ET is on the right, with plots ordered by pair type: Iso-Anti/IA (top), Anti-Iso/AI (middle), and Anti-Anti/AA (bottom). These event counts are used with the fake factor f to determine the number of fake photons... 100 9.5 Background estimates in the electron channel for ``γγ and fake photon backgrounds in the Z-mass (81 − 101 GeV) before fitting (above) and post-fit (below). The fit was done iteratively until the fake photon estimates converged to two decimal places...... 101 9.6 Background estimates in the muon channel for ``γγ and fake photon backgrounds in the Z-mass (81 − 101 GeV) before fitting (above) and post-fit (below). The fit was done iteratively until the fake photon estimates converged to two decimal places...... 102

xiv LIST OF FIGURES

miss 11.1 Event counts in the electron channel for the ﬁnal discriminant ET . The total SM background from ``γγ QCD and fake sources is stacked in the solid- color histogram, with their total uncertainty represented by the hashed area. Data events are illustrated by black circle markers, and the data-to-SM ratio is shown at the bottom, with bins containing zero events left empty in both cases. Three representative signal samples are overlaid and labeled by their

mass parameters (M`˜, Mχ˜, MP GLD)...... 112 miss 11.2 Event counts in the muon channel for the ﬁnal discriminant ET . The total SM background from ``γγ QCD and fake sources is stacked in the solid-color histogram, with their total uncertainty represented by the hashed area. Data events are illustrated by black circle markers, and the data-to-SM ratio is shown at the bottom, with bins containing zero events left empty in both cases. Three representative signal samples are overlaid and labeled by their

mass parameters (M`˜, Mχ˜, MP GLD)...... 113 11.3 A plot of the event yields as a function of the PGLD branching ratio (BR) for the four selectron signal samples, whose exclusion is dependent on the choice

of BR. The signal samples are labeled by their mass parameters (M`˜, Mχ˜, MP GLD) and are ordered by their event yields at BR=0. The 95% conﬁdence level (CL) upper limit is shown as a solid red line at 7.83 events, with branching ratios resulting in event yields higher than this value excluded...... 119 11.4 A plot of the event yields as a function of the PGLD branching ratio (BR) for the four smuon signal samples, whose exclusion is dependent on the choice

xv Chapter 1

Introduction

Particle physics is at a turning point, with experimental observations once again outpacing their theoretical underpinning. The discovery of the Higgs boson in 2012 marked the com- pletion of the Standard Model—the most comprehensive formulation of fundamental particle interactions to date—but leaves several questions unanswered. An explanation of neutrinos’ ability to oscillate between ﬂavors and a quantum description of gravity are both missing from the Standard Model. Similarly, the collisions and rotational velocities of galaxies infer a mass distribution inconsistent with their known stellar mass, with a majority coming from invisible, as-yet undiscovered “dark” matter. Many theories have been penned to explain the existence of particles suitable as dark matter candidates, with their detection a focus of many particle-collider experiments. This analysis builds on one such class of models—supersymmetry—to conduct a search for a new, massive, stable particle in a previously-unexplored kinematic region. Chapter 2 provides historical context regarding the current state of particle physics research and explains the theoretical basis for this work, with Chapter 3 describing kinematic details of the signal models. The proton-proton collision data in this thesis were mea-

1 CHAPTER 1. INTRODUCTION sured and recorded by the ATLAS detector; Chapter 4 outlines its design and performance, while Chapter 5 discusses the programs and algorithms that reconstruct the raw data into high-level physics objects. Chapter 6 describes the methodology behind simulating signal and background samples. Chapters 7 and 8 define the selections on physics objects and event-level features respectively, used in determining a signal-sensitive region-of-interest. Background processes relevant to the final selection region are estimated in Chapter 9 and supplemented by the quantification of systematic and statistical uncertainties in Chapter 10. Lastly, Chapter 11 presents the final results along with the statistical analysis used to set the 95% exclusion limits.

2 Chapter 2

Theoretical Motivation

2.1 A Brief History of Particle Physics

Fundamental particle experiments date back to the late 19th century, with the 1897 discovery of the negatively-charged electron by J.J. Thomson in his analysis of cathode ray tube emissions [1]. Historically, gravity was thought to be well-understood since the days of Isaac Newton, and electromagnetism was well on its way since the formulation of James Maxwell’s equations (with a notable exception in the form of the black-body radiation problem). How- ever, the quantization of light, in 1905, by Albert Einstein to explain the photoelectric eﬀect [2], Max Planck’s equation for black-body radiation [3], and advancements by early pioneers such as Niels Bohr [4] and Werner Heisenberg [5] in the early 20th century laid the foundation for quantum mechanics—a new era of modern physics. In 1920, there were thought to be only three fundamental particles—Thomson’s electron, Einstein’s photon, and the proton; the positively-charged particle forming the center of an atom recently discovered by Ernest Rutherford in 1919 [6]. Building upon Erwin Schr¨odinger’squantum-mechanical wave function [7], the formulation of quantum electrody-

3 CHAPTER 2. THEORETICAL MOTIVATION namics (QED) in 1928 by Paul Dirac [8] was a crucial step in mathematically formalizing particle interactions and became a model of the quantum field theories (QFTs) that were to come, positing the existence of the positron, the first antiparticle (later discovered by Carl Anderson in 1932 [9]). Observations of the neutron, in 1932 by James Chadwick [10], intensified contemporary debate on the nature of the force binding atoms together (“strong” enough to counteract electric repulsion between protons), and in 1933 Enrico Fermi proposed a new type of interaction to account for the as-yet-unexplained β decay observed in atoms [11] (later renamed from Fermi’s interaction to the “weak” force). The number of fundamental forces had now doubled from two (electromagnetism and gravity) to four. This proliferation of forces was soon matched with a flurry of new particle discoveries, beginning with the muon in 1937 by Anderson and Seth Neddermeyer [12] (erroneously interpreted as a strongly-interacting pion for a decade until its own discovery in 1947 [13]). The invention of the bubble chamber in 1952 by Donald Glaser [14] precipitated a boom in new particle observations and the list of “fundamental” particles ballooned to well over 100. Theorists were struggling to make sense of this messy “zoo” of particles and began to search for ways to reconcile these particle observations with an underlying structure. The advent of non-Abelian gauge theories in 1954 by Chen-Ning Yang and Robert Mills [15] to generalize QED for strong particle interactions—coupled with symmetry-breaking theories proposed by Yoichiro Nambu [16] and Jeffrey Goldstone [17] to explain how mass generation could occur in massless field theories—was a much-needed theoretical breakthrough. Earlier work by Sheldon Glashow in 1961 [18], on the unification between the electromagnetic and weak forces, gained a foothold with the formulation of a unified gauge theory by both Steven Weinberg [19] and Abdus Salam [20]. This was complemented by several 1964 papers from Robert Brout and Fran¸coisEnglert [21], Peter Higgs [22][23], and fi-

4 CHAPTER 2. THEORETICAL MOTIVATION nally Gerald Guralnik, Carl Hagen, and Tom Kibble [24]—detailing the Higgs mechanism by which the theorized electroweak W +, W −, and Z0 gauge bosons acquired large masses (and additionally explaining the notable absence of any new, massless electroweak bosons). Similar headway was being made in classifying strongly-interacting hadrons. Initial attempts by Eugene Wigner and Heisenberg classified hadrons by charge and isospin [25], fol- lowed by their “strangeness” property by Murray Gell-Mann [26] and Tadao Nakano/Kazuhiko Nishijima [27]. In 1964, Gell-Mann [28] and George Zweig [29] offered quarks as an explanation for these group structures, proposing that baryons were actually a composite particle of three quarks. Deep inelastic scattering experiments at the Stanford Liner Accelerator Center (SLAC) in 1969 [30] confirmed the existence of quarks, along with their point-like nature and theorized fractional charge. Finally, quantum chromodynamics (QCD), with its concept of “color charge” and its corresponding conservation as a gauge theory symmetry, solidified in 1973 [31] and marked the birth of the Standard Model (SM) as we now understand it. Since the 1970s, the SM stands as the crowning achievement of 20th-century particle physics. It is the culmination of decades of study, combining originally disparate theories of QED, quantum flavordynamics (QFD), and QCD in an effort to describe all particle types and interactions known to exist from experimental observations. With the discovery of the Higgs boson in 2012 [32], every particle predicted by the SM has been observed with remarkably good agreement between experiment and theory. While the SM is the most successful theory we currently have, it does have its limitations. Of the four fundamental forces described, gravity remains conspicuously absent. Several quantum gravity theories do exist, but all suffer renormalization issues and diverge beyond first (leading) order. Neutrino flavor oscillations are another strike against the SM. First observed by the Super-Kamioka Nucleon Decay Experiment (Super-KamiokaNDE) in 1998 [33], with additional precision measurements measured at the Sudbury Neutrino Observatory

5 CHAPTER 2. THEORETICAL MOTIVATION in 2001 [34], flavor oscillations are only possible if the neutrinos have a non-zero mass (at-odds the SM’s prediction of explicit zero masses). Finally, the anomalous rotational velocity of galaxies [35] is another piece of evidence for the existence of new, non-SM particles. Galactic rotational velocities are measured to be much higher than the velocities expected from the calculation of a galaxy’s visible stellar mass distribution. To account for this, the majority of a galaxy’s mass must reside in a ring of invisible, stable matter on its outskirts. This “dark matter” is the focus of many beyond-the-SM (BSM) theories, which seek to explain its existence, including the theoretical underpinning and experimental analysis presented in this thesis. Nevertheless, the SM seems to contain some truth in its description of our physical reality. Many non-SM models use the SM as a basis upon which to scaffold additional fields and constraints, and it is a very useful starting point in understanding fundamental particles and their interactions.

2.2 The Standard Model

Mathematically, the SM is often symbolized by the symmetry group product:

SUC (3) ⊗ SU(2) ⊗ SU(1) (2.1)

where SUC (3) signiﬁes the symmetry associated with strong force interactions, and SU(2) ⊗ SU(1) is the uniﬁed electroweak interaction symmetry. Symmetries are an important concept in particle physics and give rise to conserved quantities that dictate which particle interactions are allowed in nature. However, this representation as the tensor product of Lie groups is rather esoteric; instead, it is easier to describe

6 CHAPTER 2. THEORETICAL MOTIVATION the fundamental particles in the SM and the conservation laws that govern their interactions. As a QFT, all particles can be thought of as excitations of an underlying ﬁeld. The SM has four distinct ﬁeld classes:

1. Two fermionic ﬁelds, ΨL and ΨR (representing chirality - see section 2.2.2)

2. Electroweak boson ﬁelds, W1, W2, W3, and B

3. Gluon boson ﬁeld, G

4. Higgs boson ﬁeld, Φ

These ﬁeld classes can be further grouped into two broad types: fermions and bosons.

Fermions and bosons behave rather differently. Fermions are subject to Fermi-Dirac statistics and are forbidden from occupying the same quantum state (i.e., having the exact same quantum numbers as another fermion in the same system). This phenomenon is known as the Pauli exclusion principle [36] and results in phenomena such as distinct electron shells in atoms, as well as the degeneracy pressure that prevents neutron stars from collapsing into a singularity. On the other hand, bosons obey Bose-Einstein statistics and can occupy the same quantum state, with light amplification and lasers a practical consequence of this feature. Fermions can be further divided according to what fundamental forces they experience. All fermions can interact weakly, and all experience gravitational effects. The first distinction appears with the ability to interact strongly; particles that can are called quarks, while those that cannot are named leptons. All quarks have a fractional charge (equal to either +2/3 or −1/3 the fundamental electron charge, q) and are thus subject to the electromagnetic force, whereas half of the leptons are electrically neutral and are supplementarily labeled as neutrinos.

7 CHAPTER 2. THEORETICAL MOTIVATION

Finally, all fermions have distinct antiparticles. Antiparticles have the same mass as their regular matter counterparts, but the opposite electric charge and lepton number or baryon number in the case of quarks (Section 2.3.2). Bosons are typically thought of as force carriers or force mediators, with particles interacting through the relevant associated boson. This can include emission (e.g., bremsstrahlung radiation), kinematic interactions (e.g., two negatively-charged electrons repelled through exchanging a photon), or particle annihilation and creation (e.g., an electron and positron annihilating to create a photon, and vice versa). The six boson ﬁelds listed above are all massless; however, due to the Higgs mechanism (Section 2.2.3) the four massless electroweak ﬁelds “mix” to create physically observable particles:

A = sin θW W3 + cos θW B (2.2)

Z = cos θW W3 − sin θW B (2.3)

± 1 W = √ (W1 ∓ iW2) (2.4) 2

mW where θW is the Weinberg/weak mixing angle, with cos θW = . Here, A is the massless, mZ neutral photon, Z is the massive, neutral weak Z boson, and W ± are the electrically-charged weak W bosons. These four bosons, along with the gluon, all have spin-1 and are vector bosons, while the Higgs is spin-0 and is the only scalar boson in the SM. The only boson to have a distinct antiparticle is the W +/W − boson, with every other boson being its own antiparticle. Figure 2.1 lists all the particles in the SM, along with their mass, charge, and spin.

8 CHAPTER 2. THEORETICAL MOTIVATION

Figure 2.1: [37] Table of all particles predicted by the Standard Model.

9 CHAPTER 2. THEORETICAL MOTIVATION

2.2.1 Symmetries and Conserved Quantities in the SM

Observations of symmetry in a physical system place powerful constraints on its dynamic content. This is concisely expressed in Emmy Noether’s theorem [38] [39], which states any system with a continuous symmetry will have a corresponding conserved current. More speciﬁcally, the equations of motion describing a system are represented by its Lagrangian function (e.g., L = T − V in classical mechanics); if these equations are invariant under a continuous transformation, the invariance leads to a conserved quantity in time (Noether currents). A good classical example of Noether’s theorem is time invariance. The transformation

m P3 2 in this case is Q = d/dt, and L = 2 i=1 x˙ i − V (x). In this instance, Noether’s theorem states:

3 X ∂L j = Q[x ] − L ∂x˙ i i=1 i 3 3 m X m X = 2 · x˙ · x˙ − x˙2 + V (x) (2.5) 2 i i 2 i i=1 i=1 3 m X = x˙2 + V (x) 2 i i=1

where j is the Noether charge, and the right-hand side is the kinetic and potential energy of a system. The time derivative j˙ is the Noether current, and its invariance j˙ = 0 leads to the well-known conservation of energy. Quantum mechanically, the theorem can be similarly applied to ﬁeld theories to derive the same conservation laws (also known as Takahashi-Ward identities [40][41], formulated by Yasushi Takahashi and John Ward). Gauge theories are the foundation of the SM precisely because a gauge theory is a ﬁeld theory that is invariant under local transformations. In

10 CHAPTER 2. THEORETICAL MOTIVATION this context, a global transformation is one which does not change in space or time (e.g., a translation), whereas a local transformation is location-dependent (e.g., the electric scalar potential in relation to the distribution of charges). The three symmetry gauge groups of the SM, along with several other global symmetries important to physical systems, can be seen in Table 2.1 along with their associated conserved currents.

Table 2.1: Global symmetries, local gauge symmetries, and their resulting conserved quantities.

Symmetry Conserved Quantity

Time invariance Energy Translational symmetry Linear momentum Rotational symmetry Angular momentum Lorentz invariance CPT symmetry U(1) gauge invariance Electric charge

SU(2)L gauge invariance Weak isospin SU(3) gauge invariance Color charge

2.2.2 Fermion Chirality

The SM is a chiral theory; the fermion ﬁelds have both a left and right chiral component

that transform diﬀerently under the SU(2)L gauge transformation. Left-handed fermions ΨL form a weak isospin doublet with T3 = ±1/2, while right-handed fermions form a singlet with T3 = 0. The reverse is true for antifermions, with right-handed antifermions forming the weak isospin doublet and left-handed antifermions forming the singlet. This results in left-handed fermions and right-handed antifermions being able to interact via the weak force

+ − (e.g., W → uL dR and W → eL νR are both allowed), but not their counterparts (e.g.,

11 CHAPTER 2. THEORETICAL MOTIVATION

+ − W → uR dL and W → eR νL are both forbidden). Massive fermions are a superposition of both the left and right fermionic ﬁelds, but massless fermions have ﬁxed chirality. Right-handed neutrinos (and left-handed antineutrinos) do not exist in the SM as a consequence of this phenomenon, due to their predicted massless nature.

2.2.3 Spontaneous Symmetry Breaking and the Higgs Field

Unification is another important concept in particle physics, with the unification of the four fundamental forces a much-coveted end-goal. Unification theories attempt to model the fundamental forces as aspects of a single, unified force at high-energy scales, whose symmetry breaks down as the energy scale decreases. A grand unified theory is one that merges the electromagnetic (EM), weak, and strong forces, while a theory of everything accommodates all four. Electroweak unification is currently the only experimentally-observed unification theory, with the discovery of neutral currents in 1973 by Gargamelle [42], and the W and Z bosons in 1983 by the UA1 [43] and UA2 [44] experiments at the Conseil Européenpour la Recherche Nucléaire (CERN).

The Higgs field is a scalar SU(2)L doublet, with four degrees of freedom, and plays a critical role in electroweak unification and its broken symmetry at low energies. The potential energy of the Higgs field is defined as:

2 2 4 V (H) = mH |H| + λ|H| (2.6)

where H is the scalar Higgs ﬁeld. This potential is symmetric and is the familiar “sombrero” potential (reproduced in Figure 2.2). The vacuum expectation value (VEV) is at the

2 minimum of this potential, which occurs when mH < 0 and λ > 0 [45]. The VEV deﬁnes

12 CHAPTER 2. THEORETICAL MOTIVATION

Figure 2.2: [46] The ‘Sombrero’ Higgs potential V (H), as a function of H

the electroweak scale—the energy above which the EM and weak force are uniﬁed—and is

q 2 −mH calculated as the fraction of the two free parameters: ν = 2λ = 246 GeV. At energy scales below the VEV, the Higgs ﬁeld symmetry is spontaneously broken. The Higgs ﬁeld, now in its minimum potential vacuum state, is no longer invariant under gauge

symmetry transformations. With the Higgs field coupling to the electroweak SU(2)L ⊗U(1)Y gauge fields, the W +, W −, and Z bosons each gain a longitudinal degree of freedom from the Higgs field and acquire non-zero masses. Only the photon is left invariant and massless, with the remaining degree of freedom becoming the scalar Higgs boson. The Higgs boson is also responsible for fermion masses through Yukawa coupling [47]— interactions between the scalar Higgs field and the fermionic Dirac fields formulated by Hideki Yukawa. The Higgs field explains the observation of massive W ± and Z bosons, as well as the mechanism by which fermions gain mass. However, the Higgs’s interaction with every massive particle, and 151 of the SM’s 18 free parameters’ dependence on the Higgs field [49],

1Nine Yukawa fermion mass terms, the Higg mass and VEV, and four Cabibo-Kobayashi-Maskawa (CKM) terms [48].

13 CHAPTER 2. THEORETICAL MOTIVATION

2 introduces a new problem: the sensitivity of mH . Self-interaction of the Higgs ﬁeld via

2 2 fermion loops introduces corrections to mH on the order of ΛUV , where ΛUV is the ultraviolet momentum cut-oﬀ used to regulate the momentum integral [45]. It can also be interpreted as the energy for new physics, where the problem comes into focus; if the ultraviolet cut-

18 oﬀ is on the order of the reduced Planck scale MP = 8πGNewton ≈ 10 GeV—the energy scale at which quantum gravitational eﬀects are theorized to become important—then the

2 2 2 corrections to mH are enormous compared to the actual value of mH at (125 GeV) . This is known as the “hierarchy problem,” and is the motivation for many BSM theories.

2.3 Supersymmetry

Supersymmetry is one class of BSM theories that attempts to solve the hierarchy problem with a new, invariant transformation that produces an additional symmetry between fermions and bosons. Supersymmetry models propose that every SM fermion has a boson superpartner, and every SM boson has a fermion superpartner, avoiding the extreme ﬁne-tuning of

2 mH by explicitly canceling higher-order terms with supersymmetric counterterms. The naming convention for fermionic superpartners is to add an s- prefix, with squarks (sup, sdown, scharm, sstrange, stop, and sbottom squarks) and sleptons (selectrons, smuons, staus, and sneutrinos) differing by spin-1/2 to form the new spin-0 scalar sfermions. Bosonic superpartners have an −ino suffix added to the name root, with the electroweak gauge fields

W1,W2,W3, and B having wino and bino (gaugino) superpartners, with the gluons partnered to gluinos and the Higgs to the higgsinos. Bosinos all have intrinsic spin-1/2. Together, this collection of particles and sparticles is known as the Minimally Superysmmetric Standard Model (MSSM). Sparticles are denoted with a tilde to diﬀerentiate them from their SM counterparts, e.g., t˜ for a stop squark.

14 CHAPTER 2. THEORETICAL MOTIVATION

If this fermion/boson symmetry were unbroken, then the sparticles would have the same mass as their particles. This is ostensibly not the case as no sparticles have been experimentally observed. Instead, this symmetry must again be broken at lower energies and push the sparticle masses above those of their SM partners, similar to the mass asymmetry of the photon and weak bosons in electroweak symmetry breaking. Within supersymmetry theories, there are multiple explanations for the broken symmetry, including; the Fayet-Iliopoulos (D-term) mechanism [50]; O’Raifeartaigh (F -term) models [51]; Planck-scale (supergravity) mediation [52]; extra-dimensonal and anomaly-mediated models [53]; and gauge-mediated supersymmetry breaking (GMSB) [54][55][56][57]. This last class of models forms the theoretical basis of this thesis and its experimental search region and is described in more detail below.

2.3.1 GMSB and Hidden Sectors

The MSSM contains a scalar superpotential that is similar to the Higgs potential, including a separate vacuum expectation value that regulates the energy scale of supersymmetry breaking. It is difficult to explain the appearance of an acceptably large superscalar VEV (i.e., precluding the current observation of any sparticles) by adding new gauge fields that interact with the known SM fields at tree-level, as renormalizable supersymmetry does not contain any (scalar)-(gaugino)-(gaugino) couplings [45]. GMSB models solve this issue with the introduction of intermediate messenger fields that couple with both the new supersymmetric fields and SM fields, allowing indirect interactions between the two via loop-level corrections. Gauge-mediated models do this by coupling to the ordinary SM electroweak and QCD gauge interactions, as opposed to gravity (Planck)-mediated models that couple √ to the gravitational field. This can lead to supersymmetry-breaking VEV ( F ) values as

15 CHAPTER 2. THEORETICAL MOTIVATION low as 104 GeV depending on the messenger particle mass, in contrast to 1011 GeV for gravity mediation [45]. These supersymmetry-breaking sectors, interacting indirectly with the visible SM sectors, are known as hidden sectors. Although gravity is not the mechanism that GMSB models use to explain supersymmetry breaking at relatively low energy scales, it is still incorporated into its framework. In standard GMSB, a single hidden sector provides a solitary, massless neutral fermion called the goldstino. This goldstino is absorbed by the gravitino, the superpartner to the theorized graviton, and becomes its longitudinal degree of freedom, analogous to the way in which Nambu-Goldstone bosons are absorbed by the weak bosons in SM theory. This similarly causes the gravitino to gain mass, in what is called the super-Higgs mechanism. [45]. With

Mmess << MP , the gravitino in GMSB models is much lighter than the MSSM sparticles, very nearly massless, and is the lightest supersymmetric particle (LSP) in GMSB. This makes the gravitino the leading dark matter candidate in these models, due to its low mass and neutral electric charge.

2.3.2 R-parity and Gaugino/Higgsino Mixing

To our current knowledge, there are two accidental global symmetries, and their conservation is observed for all known particle interactions. An accidental symmetry is one that is not explicitly conserved, but the terms in the SM Lagrangian function that would violate it are not communicable at everyday energy scales. These two approximately-conserved quantities are the baryon and lepton numbers. In the SM, hadrons formed of three quarks have a baryon number of +1, mesons formed of one quark and anti-quark have a baryon number of 0, and hadrons formed of three antiquarks have a baryon number of -1. Similarly, leptons have a lepton number of +1, while anti-leptons

16 CHAPTER 2. THEORETICAL MOTIVATION have a lepton number of -1. For any given interaction, the baryon/lepton numbers of the particles before must be the same as those after. Proton decay is possible if both the baryon and lepton number are not conserved, and the current lower bound of 1.67 × 1034 years set by Super-Kamiokande [58] [59] is a strong constraint on the weakness of this violation, if it exists. In supersymmetry, the baryon and lepton number are allowed to change, with the focus instead shifting to a quantity known as R-parity, deﬁned as:

3(B−L)+2s PR = (−1) (2.7) where B is the baryon number, L is the lepton number, and s is the intrinsic particle spin. Supersymmetric models are typically classiﬁed by whether they do or do not conserve R- parity, with SM particles having an R-parity of +1 and sparticles an R-parity of −1. Another point of note is that higgsinos and gauginos are also subject to SM electroweak symmetry breaking, and similarly mix together to form particle mass eigenstates from their linear combinations. There are four higgsinos, with two of them being neutral. These

˜0 ˜0 ˜ ˜ 0 neutral higgsinos (Hu and Hd ) mix with the two neutral gauginos (B and W ) to form

˜+ ˜− four mass eigenstates known as neutralinos. The two charged higgsinos (Hu and Hd ) then combine with the two charged gauginos (W˜+ and W˜−) to form two mass eigenstates known as charginos, with charges ±1. The neutralinos are of special interest to many supersymmetric models, since the lightest neutralino N˜1 is often the LSP and a dark matter candidate. In GMSB models this is not the case, with the gravitino being the LSP and the lightest neutralino becoming the next- lightest supersymmetric partner (NLSP). However, the NLSP is still an important role, as the NLSP can only decay to the LSP and an SM particle and does so exclusively. The mass

17 CHAPTER 2. THEORETICAL MOTIVATION and properties of the NLSP then aﬀect the lifetime of this decay and, if suﬃciently short, make direct detection possible within modern collider detectors.

2.4 Pseudo-Goldstini and the mssm-goldstini Model

The theoretical groundwork is finally in place to discuss the mssm-goldstini model. The mssm-goldstini model is a non-minimal GMSB theory with multiple hidden sectors, with each hidden sector coupling to the visible sector by an intermediate messenger field. One linear combination of these messenger fields is the massless goldstino/gravitino, while additional combinations give rise to the massive pseudo-goldstini. The mssm-goldstini framework is somewhat general, and its principles can be applied to scenarios with an arbitrary number of hidden sectors, but for this analysis we are considering the simplest case: Two hidden sectors, with one massless goldstino and one massive pseudo-goldstino. The development of this analysis would not be possible without the work done by Gabriele Ferreti, Alberto Mariotti, Kentarou Mawatari, and Christoffer Peterson, whose paper forms the theoretical basis for the analysis search strategy presented here [60].

2.4.1 Goldstini Masses and the Neutralino Mass Matrix

The addition of multiple goldstini Weyl fermions extends the usual 4 × 4 MSSM neutralino mass matrix to a (4 + n) × (4 + n) symmetric mass matrix M. In the gauge eigenbasis

˜ ˜ (3) ˜ 0 ˜ 0 (B, W , Hd , Hu, η˜1, ...η˜n), where η1 ... ηn are the new goldstini, M takes the form:

! M M M = 4×4 4×n (2.8) Mn×4 Mn×n

where n = 2 for the rest of this analysis. The usual MSSM neutralino matrix M4×4 is given

18 CHAPTER 2. THEORETICAL MOTIVATION by:

  MB 0 −mZ sin θω cos β mZ sin θω sin β  0 M m cos θ cos β m cos θ sin β  W Z ω Z ω  M4×4 =   (2.9) −mZ sin θω cos β mZ cos θω cos β 0 −µ  mZ sin θω sin β −mZ cos θω sin β −µ 0

2 2 2 2 p 2 2 with mZ = (g1 + g2)ν /2, sinθω = g1 (g1 + g2), νd = νcosβ, νu = νsinβ, ν = 246 GeV, and

µ is the supersymmetric higgsino mass. Here, g1 and g2 are the weak isospin SU(2)L and

weak hypercharge U(1)Y gauge couplings respectively. With the visible SUSY sector indirectly coupling to the two SUSY-breaking sectors, the MSSM soft masses gain contributions from all n sectors, i.e., the bino, wino, down/up-type Higgs soft masses, and the soft B-parameter can be written as:

6 6 6 6 X X X X M = M ,M = M , m2 = m2 ,B = B (2.10) B B(i) W W (i) Hd/u Hd/u(i) (i) i=1 i=1 i=1 i=1

The mixing terms between the MSSM neutralinos and the goldstini are then contained within the M4×2 block, which can be expressed in terms of various VEVs as:

 M hD i M hD i  − B√(5) Y − B√(6) Y 2f1 2f2  M hD 3 i M hD 3 i   − W√(5) T − W√(6) T  2f1 2f2  2 2  M4×2 =  ν(m cosβ+B(5)sinβ) ν(m cosβ+B(6)sinβ)  (2.11) − Hd(6) − Hd(6)   f1 f2  2 2  ν(m cosβ+B(5)sinβ) ν(m cosβ+B(6)sinβ)  − Hu(6) − Hu(6) f1 f2

2 2 where hDY i = −g1ν cos2β/2, hDT 3 i = g2ν cos2β/2, and hFii = fi—the SUSY-breaking scale of hidden sector i.

Finally, the M2×2 mass matrix block concerns the masses of the two goldstini. There

19 CHAPTER 2. THEORETICAL MOTIVATION are two ways in which the goldstini acquire masses. The ﬁrst is from diagonal tree-level

2 mass terms for theη ˜s, but all of these contributions are suppressed by a factor of 1/fi and are typically negligible [60]. The second is through radiative corrections, where the diﬀerent

SUSY-breaking sectors interact at the loop level. The leading contributions to M2×2 can be obtained from the SUSY operators (Reference [60], page 4) and integrating out the gauge and Higgs superﬁelds at one-loop (outlined in detail in reference [61]). The exact contributions are model-dependent and are speciﬁed by the dynamics of the hidden/messenger sectors and the two-point function hη˜1η˜2i [60]. However, some general comments on the structure of

M2×2 can be made on the basis of spontaenous SUSY-breaking. The full (4 + 2) × (4 + 2) matrix has a zero eigenstate corresponding to the true goldstini ˜ (GLD) G. With the assumption that the fi values are much greater than the VEVs of the D and F terms of the gauge and Higgs superﬁelds, the true goldstino will, to a good approximation, be aligned with the linear combination only involving the goldstini, i.e.,

˜0 f1η˜1 + f2η˜2. The remaining eigenstate corresponds to the pseudo-goldstini (PGLD) G . The zero eigenvector from the linear combination of the two goldstini imposes additional constraints on the M2×2 matrix. The diagonal entries can now be expressed in terms of the oﬀ-diagonal entry M12, which is taken as a model-dependent parameter, such that

f2M12 ! − M12 M = f1 (2.12) 2×2 f1M12 M12 − f2

2 P2 2 In the simple case where f1 >> f2, the contribution to the vacuum energy f = i=1 fi ≈

2 f1 , and the massless goldstino mode is aligned withη ˜1. In this case, the PGLD is aligned

withη ˜ , with a mass approximately given as M 0 ≈ (f /f )M . Although the mass is 2 G˜ 1 2 12

enhanced by a factor of f1/f2, it cannot be arbitrarily large as the backreaction of the MSSM sector to the second hidden sector becomes too severe and the formalism breaks

20 CHAPTER 2. THEORETICAL MOTIVATION down [61]. The PGLD mass varies from 0 − 375 GeV in this analysis and is always smaller than the lightest observable-sector superysymmetric partner (LOSP). In this context, the PGLD is the next-to-lightest supersymmetric partner (NLSP) and the GLD is the lightest supersymmetric partner (LSP).

2.4.2 The LOSP: Production and Decay Modes

In GMSB, the colored superpartners are signiﬁcally heavier than the uncolored ones as a consequence of the soft mass boundary values and the renormalization group evolution. Stops and gluinos are also required to be in the multi-TeV range in order to accommodate a 125 GeV Higgs boson, leaving the electroweak superpartners, of which the right-handed sleptons and the bino are the only ones not charged under SU(2)L, to become the lightest SM superpartners as a result [60]. This motivates the choice of the bino-like neutralino as the LOSP, with right-handed slepton pair-production as the production mechanism. The sleptons are mass-degenerate at the messenger scale. This is weakly broken at low energies, with the selectron and smuon remaining approximately degenerate while the stau mass eigenstate is somewhat lighter, due to Yukawa couplings in the renormalization group equations [60]. This splitting is still small unless tan β (and the left-right stau mixing) is large—which is not expected to be the case—and the three sleptons are treated as having a common mass that is always heavier than the bino. Direct slepton decay to the PGLD/GLD is strongly suppressed with respect to their gauge neutralino couplings, and so the branching ratio for slepton decays to the neutralino and their corresponding SM lepton is taken to be 100%. The neutralino can decay to either the PGLD or the GLD along with an associated photon. Neutralino decays involving a Z boson e.g., χ˜ → ZG˜0 are possible, but are suppressed

21 CHAPTER 2. THEORETICAL MOTIVATION

2 2 both by phase space (as the Z is relatively massive) and by the factor sin θω/ cos θω, and are not considered further. The partial decay width for the GLD decay mode is:

2 5 cos θωM Γ(˜χ → γG˜) = χ (2.13) 16πf 2

with the partial decay width for the PGLD decay mode:

2 2 5 2 3 K 0 cos θ M M 0 ˜0 G ω χ G Γ(˜χ → γG ) = 2 1 − 2 (2.14) 16πf Mχ

2 enhanced by a factor K 0 , and suppressed by the PGLD-to-neutralino mass ratio. The K- G factor is a free parameter in the model, and its choice greatly affects the relative branching ratio between the two decays. The PGLD and neutralino also affect the branching ratio, as well as the kinematics of the final state particles. The Feynman diagram of the full process for both the GLD and PGLD decay modes can be seen in Figure 2.3.

22 CHAPTER 2. THEORETICAL MOTIVATION

`+ p γ + `˜ 0 R χ˜1 G˜/G˜0

˜ ˜0 ˜− χ˜0 G/G `R 1 p γ `−

Figure 2.3: Feynman diagram for slepton pair-production to pseudo-goldstini cascade decays, with supersymmetric particles highlighted in red. Decays occur via neutralinos to a ﬁnal state with two leptons, two photons, and missing transverse energy from the massless goldstini G˜ or massive pseudo-goldstini G˜0 .

23 Chapter 3

Analysis Search Strategy

Both the GLD and PGLD decay are distinguished by the large number of particles in their

miss ﬁnal state, with two high-pT leptons, two high-pT photons, and signiﬁcant ET suppressing the majority of SM backgrounds. Additionally, the PGLD decay is characterized by softer

miss photon and ET distributions relative to the neutralino-to-PGLD mass ratio chosen. Softer lepton distributions also occur in much of the parameter space, depending on slepton and neutralino mass. Existing searches do not take into account this intermediate PGLD decay, with the signal

miss region kinematic selections reﬂecting this. The inclusive diphoton + ET search at Run II

miss [62] requires photons with pT above 75 GeV for all signal regions, as well as a ET above 150 GeV; similarly, the 2 − 3 lepton search at Run II [63] requires a dilepton invariant mass

1 m`` ≥ 111 GeV and a stransverse mass mT 2 ≥ 100 GeV. Both of these analysis selections are insensitive to the PGLD decay across the majority of the parameter space being considered—

1Stransverse mass is a common discriminating variable in SUSY searches, and is deﬁned as m = T 2 `1 `2 miss `1 `2 min max mT {pT , qT }, mT {pT , pT − qT } , where pT and pT are the transverse momentum vectors qT `1 of the two leptons, and qT is a transverse momentum vector that minimizes the larger of mT (pT , qT ) and `2 miss p mT (pT , pT − qT ), where mT (pT qT ) = 2(pT qT − pT qT ).

24 CHAPTER 3. ANALYSIS SEARCH STRATEGY

only oﬀering sensitivity when the pseudo-goldstini is massless, or if its decay is suﬃciently suppressed by a high mass and low K-factor. The parameter space is dependent on several variables. The choice of slepton mass,

˜0 neutralino mass, and PGLD mass—as well as the K-factor Kν enhancing χ → γG —all affect the kinematic distributions of the final state particles. In order to efficiently probe this space, several informed decisions are made for each of these variables.

The slepton mass M`˜ governs the production cross-section in addition to the kinematic distributions, with the cross-section limiting the total event yields for a given sample. Four diﬀerent slepton mass values were chosen, from 200 GeV based on the upper limit of sensitivity for existing searches, to 500 GeV at which point the expected signal event yield is marginally signiﬁcant.

The neutralino mass Mχ˜ is constrained from above by the slepton mass and the PGLD

mass from below. Its choice impacts the hardness of the lepton and photon pT spectra

miss as well as the ET distribution. Neutralino mass values are chosen according to a mass splitting parameter x, where Mχ˜ = x(M`˜− MP GLD) + MP GLD. Four possible mass splittings x ∈ {0.1, 0.3, 0.5, 0.7} are explored for a combination of slepton and PGLD masses.

The PGLD masses are fractions of the slepton mass (MP GLD ∈ {0, 0.25, 0.5, 0.75} M`˜), with several low-fraction, low-splitting combinations exploring compressed mass spectra.

Finally, the sample dependence on Kν is eliminated. For each generated signal sample, a K-factor is chosen such that the branching ratio is 0.5 for both decays. This ensures enough events are generated for each decay mode, and that any K-factor can be simulated from sampling post-generation. A visual representation of all 21 benchmark signal points is shown in Figure 3.1, and the parameter values for each point are shown in Table 3.1.

25 CHAPTER 3. ANALYSIS SEARCH STRATEGY

Figure 3.1: Visual representation of the 21 signal samples. Diﬀerent colors denote diﬀerent slepton M`˜ masses; black (500 GeV), green (400 GeV), red (300 GeV), and blue (200 GeV).

26 CHAPTER 3. ANALYSIS SEARCH STRATEGY

Table 3.1: Parameter combinations in the mssm-goldstini signal samples.

M`˜ [GeV] x Mχ˜ [GeV] MPGLD [GeV] 200 0.1 20 0 200 0.3 95 50 200 0.5 150 100 200 0.5 175 150 200 0.7 185 150 300 0.1 30 0 300 0.3 142.5 75 300 0.5 225 150 300 0.5 262.5 225 300 0.7 277.5 225 400 0.1 40 0 400 0.3 190 100 400 0.5 200 0 400 0.5 250 100 400 0.5 300 200 400 0.5 350 300 400 0.7 370 300 500 0.5 250 0 500 0.5 312.5 125 500 0.5 375 250 500 0.5 437.5 375

27 Chapter 4

Experimental Apparatus

4.1 The Large Hadron Collider

The Large Hadron Collider (LHC) is the largest circular particle accelerator in the world. Forming a ring 27 km in circumference at the border of the Pays-de-Gex region of France and Geneva, Switzerland, it accelerates two proton beams up to 6.5 TeV to produce collisions at a center-of-mass energy of 13 TeV. These alternately circulating beams cross every 25 nanoseconds at several intersection points along the ring, with approximately 20 collisions per crossing creating 600 million collisions per second [64]. The LHC is the last in a series of accelerators forming the accelerator complex at the Conseil Européenpour la Recherche Nucléaire (CERN), seen in Figure 4.1. The LHC is the only accelerator in which collision and detection occurs, with beam collisions occurring along eight fixed-intersection points along the ring. The proton beams start out as bottled hydrogen gas and are incrementally fed through and accelerated by several smaller synchrotrons before being injected into the LHC (see Table 4.1 for acceleration energies). Seven detector experiments are currently in operation at the LHC, with an eighth (FASER, the “ForwArd

28 CHAPTER 4. EXPERIMENTAL APPARATUS

Table 4.1: List of all accelerators in the CERN complex, with the center-of-mass energies to which proton beams are accelerated.

Accelerator Name Acceleration Energy Linac 2 50 MeV Proton Synchotron Booster 1.4 GeV Proton Synchotron 25 GeV Super Proton Synchotron 450 GeV Large Hadron Collider 6.5 TeV

Table 4.2: List of all experiments present at the LHC, with a brief description. References to the experiments’ technical design reports and/or proposals are included.

Experiment Name Experiment Description ALICE [67] Heavy ion; proton-nucleus and nucleus-nucleus collisions ATLAS [68][69] All-purpose detector; toroidal magnets CMS [70][71] All-purpose detector; central muon solenoid FASER [72] Neutrinos and new light and weakly-coupled particles in forward region LHCb [73] B-quark measurements, CP violation, rare decays LHCf [74] Photons, neutral pions in forward region MOEDAL [75] Magnetic monopoles and highly ionizing Stable Massive Particles (SMPs) TOTEM [76] Elastic and inelastic proton scattering cross-sections and diﬀractive dissociation

Search ExpeRiment”) approved and projected to start taking data in 2021. The four most prominent experiments are ALICE, ATLAS, CMS, and LHCb. ALICE and LHCb are specialized experiments—focusing on heavy-ion and b-quark measurements, respectively—while CMS and ATLAS are designed to be general, all-purpose detectors, comprising multiple specialized layers that are capable of measuring (or inferring) the energy and momentum of all particles in any given event collision (a full list of detectors is given in Table 4.2, along with their positions along the LHC in Figure 4.2.) Operating since 2008, the LHC has overseen various upgrades over its lifetime. Most

29 CHAPTER 4. EXPERIMENTAL APPARATUS

Figure 4.1: [65] Diagram of the LHC accelerator complex.

30 CHAPTER 4. EXPERIMENTAL APPARATUS

Figure 4.2: [66] Diagram showing the eight interaction points along the LHC ring and the location of the ALICE, ATLAS, CMS, and LHCb detectors.

31 CHAPTER 4. EXPERIMENTAL APPARATUS

Figure 4.3: [77] Cutaway view of the ATLAS detector

recently, the Run II program at 13 TeV was completed a the end of 2018, comprising the largest dataset ever recorded at 139 fb−1 at the highest-ever center-of-mass energy. The Run II dataset provides the best opportunity for observation of both rare events and the creation of potential new, exotic particles previously impossible at lower energies. The LHC is currently being upgraded, in preparation for Run III data-taking in autumn 2021, with a projected instantaneous luminosity twice that of Run II at a center-of-mass energy of 14 TeV. Further into the future, the High-Luminosity LHC upgrade is expected to be completed by 2027, with an instantaneous luminosity six times that of Run II.

32 CHAPTER 4. EXPERIMENTAL APPARATUS 4.2 The ATLAS Detector

The event collision data presented in this thesis comes exclusively from ATLAS (A Toroidal LHC ApparatuS). As its name suggests, its deﬁning feature is its large 2.5 T toroidal magnets. The detector in its entirety can be seen in Figure 4.3, with each component being described in more detail below.

4.2.1 ATLAS Coordinate System

The coordinate system used in ATLAS is important in understanding and interpreting the results in this paper, with several kinematic features used to select signal events dependent on the detector’s geometry. In Cartesian coordinates, the orthogonal axes are defined with respect to the interaction point where the two beams collide. For the x-axis, the positive direction points radially inwards towards the center of the LHC ring; for the y-axis, the positive direction points upwards; and for the z-axis, the positive direction points along the beam line, forming a right-handed coordinate system. Spherical coordinates are also frequently used, with the azimuthal φ angle defined in the x − y plane around the beam (z) axis, and the polar angle θ defined as the angle from the same axis. In lieu of θ, the pseudo-rapidity η is often used, defined as η = − ln tan(θ/2). The variable ∆R = p∆η2 + ∆φ2 is often used to distinguish the spatial isolation of physics objects. The ATLAS detector has full coverage in the x − y plane, but not along the beam direction. Experimentally this is not an issue, as the z-direction momentum of the colliding protons is not well-defined (due to the probabilistic parton density functions of their internal quarks) and the transverse x − y plane is useful for defining several discriminating variables.

miss Both the transverse momentum pTand the missing transverse momentum ET —deﬁned as

33 CHAPTER 4. EXPERIMENTAL APPARATUS

Figure 4.4: [77] Cutaway view of the inner detector

the vector required to conserve momentum from all the detected particles in the transverse plane—are well-suited for distinguishing novel physics processes from backgrounds.

4.2.2 Inner Detector

The inner detector (ID) (Figure 4.4) measures the momentum of charged particle tracks and provides both primary and secondary interaction vertex measurements for grouping particles into individual collision/decay events. It is designed to handle a high particle-track density, due to its proximity to the interaction point, with its very ﬁne spatial granularity. The ID is composed of three sub-detectors; the pixel detector (PD), the semiconductor trackers (SCTs), and the transition radiation trackers (TRTs). The PD is the innermost detector, composed of radiation-hard silicon pixel modules of identical size. In the barrel region (the central region around η = 0), the modules are

34 CHAPTER 4. EXPERIMENTAL APPARATUS

arranged in three concentric cylinders around the beam axis, while modules in the end-cap region are arranged into three disks on each side perpendicular to the beam axis. Both pixel regions provide a spatial resolution of 10 µm in the bending φ direction and 115 µm (in z/R for the barrel and end-cap respectively), with a typical particle crossing three layers in order to construct a high-precision track. The SCTs are the middle component of the ID and are also made of silicon, divided into microstrips with a pitch size of 80 µm. Each module consists of two sensors glued back-to- back on a thermal support with a relative angle of 40 mrad between them to provide two- dimensional measurement information. The barrel region contains four concentric cylindrical layers, while the end-cap layers are grouped into disks of nine layers on each side. The intrinsic resolution of an SCT layer is 17 µm (φ) and 580 µm (z/R). The TRTs are the outermost component of the ID; they provide a complementary measurement of the track position in φ, but do not provide z/R position information to which they run parallel (in the barrel and end-cap respectively). The TRT layers are a collection of drift tubes; these 4 mm wide ‘straws’ are made from wound Kapton (a polyimide ﬁlm) reinforced by carbon ﬁber and contain a gold-plated tungsten wire at ground potential. The

tubes are filled with a 70% Xe, 27% CO2, 3% O2 gas mixture, and incoming particles ionize this gas to produce electrons that then “drift” towards the central wire to be detected. The intrinsic resolution of a drift tube is 130 µm, with tubes oriented coaxially in the barrel region and radially in the end-caps. A typical particle interacts with 36 drift tubes. Elec- tron identification efficiency is improved with the inclusion of transition radiation material (polypropylene fibers in the barrel, foil in the end-caps) between the straws, with electrons above 2 GeV creating 7 − 10 additional hits from low-energy photon emissions at these material boundaries.

35 CHAPTER 4. EXPERIMENTAL APPARATUS

Figure 4.5: [77] Cut-away view of the ATLAS calorimeter system

Central solenoid. Finally, the entire inner detector is encapsulated by the central solenoid. The solenoid provides a 2T axial magnetic ﬁeld, bending charged particle tracks in the x − y plane. Its design is also as thin as possible—roughly 0.66 radiation lengths [78]—minimizing particle energy loss within the magnet itself, prior to particles reaching the calorimeters.

4.2.3 Calorimeters

ATLAS contains several calorimeters that sample the energy of incoming particles through “active” components and reduce it through “absorber” materials (Figure 4.5). They are separated into two types—EM calorimeters for electrons and photons interacting through the electromagnetic force and hadronic calorimeters for particles interacting through the strong nuclear force. Particles passing through the calorimeters form “showers”—a cascading chain

36 CHAPTER 4. EXPERIMENTAL APPARATUS

of lesser-energy particles produced from interaction with the dense calorimeter material— whose features can be used to identify the original showering object. The calorimeters provide coverage over the full η range of the ATLAS detector (|η| ≤ 4.9),

miss and are crucial for both individual object identiﬁcation and event-level features such as ET . With a combined thickness of ∼ 11 nuclear interaction lengths, the calorimeters provide suﬃcient containment for both shower types and limit energy leakage to the muon system.

Liquid Argon Electromagnetic Calorimeters

The EM calorimeter is the first component that particles interact with after leaving the ID, divided into a barrel (|η| < 1.475) and two end-cap components (1.375 < |η| < 3.2). It is a lead-liquid argon (LAr) detector, with LAr-filled klapton electrodes sampling particle energies and lead absorber plates collecting the shower energy. These detectors are arranged in an “accordion-like” geometry, providing complete and uniform φ coverage without any azimuthal cracks. The LAr EM calorimeters in both regions are formed of three layers; the first sampling layer consists of long, thin strips with granularity ∆η = 0.0031, ∆φ = 0.098, and a depth of 98 mm; the second layer is where the largest majority of the EM shower is collected, and is composed of square cells with granularity ∆η = 0.025, ∆φ = 0.0245, and a depth of 337 mm; and the final, third layer, where the tail of the EM shower is collected, has a coarser granularity of ∆η = 0.05, ∆φ = 0.0245, and a depth of 42 mm. An additional, thin LAr presampling layer is also present in the barrel at |η| < 1.8, to correct for energy losses in the ID barrel region due to the presence of the central solenoid.

Hadronic Calorimeters

Tile calorimeter. The tile calorimeter envelops the LAr detectors and is entirely coaxial, with no end-caps. It has a central barrel in the region |η| < 1.0 and an extended barrel

37 CHAPTER 4. EXPERIMENTAL APPARATUS region on either side at 0.8 < |η| < 1.7. The active material is a scintillator, made from injection-molded polystyrene mixed with ﬂuorescent dye (paraterphenyl [PTP] and 1,4-bis- (2-(5-phenyloxazolyl))-benzene [POPOP] [79], speciﬁcally), with steel as the absorber material. All regions have three radial layers, and a tile granularity ∆η = 0.1 and ∆φ = 0.1.

LAr hadronic end-cap calorimeter. The Hadronic End-cap Calorimeter (HEC) consists of two wheels on each detector side and is located directly behind the LAr EM calorimeter end-caps. The HEC overlaps with both the tile calorimeter (1.5 < |η| < 1.7) and forward calorimeter (3.1 < η < 3.2) to provide additional absorber material at the calorimeter transition boundaries. Each wheel is composed of 32 wedge-shaped segments at two separate depths. Copper plates of 50 mm thickness act as the absorber material (with the exception of the ﬁrst plate in each wheel, which is half-thickness), with 8.5 mm LAr gaps interleaved between them.

LAr forward calorimeter. The Forward Calorimeter (FCal) covers the forward/backward 3.1 < |η| < 4.9 range. Each side has 3 FCal layers, with the first being made of copper and optimized for EM interactions and the latter two made of tungsten for hadronic showers. Each layer is composed of these densely-packed, coaxial hollow tubes filled with solid rods, with a thin LAr gap acting as the sensitive medium. As the most forward-facing of the detectors, the FCal is subject to a high particle flux and has two distinct design changes in order to cope with this. The first is a recession of its front face of 1.2 m with respect to the LAr EM calorimeter, in order to reduce neutron diffuse reflections back into the ID cavity. The second is a smaller LAr gap within the metal rods (roughly 4 − 10 times smaller, depending on the layer) to prevent ion build-up.

38 CHAPTER 4. EXPERIMENTAL APPARATUS

Figure 4.6: [77] Cutaway view of the muon detection system

4.2.4 Muon System

Toroid Magnets

The eponymous toroidal magnets surround the ATLAS calorimeters and are separated into one barrel and two end-cap regions, as shown in Figure 4.6. The barrel toroid is composed of eight “racetrack”-shaped superconducting coils, with an air-core to minimize the amount of material traversed by particles leaving the barrel calorimeters. The coils are evenly spaced in φ and placed with their long sides coaxial to the beam line, covering the range |η| < 1.4. Each end-cap toroid is formed of eight ﬂat, square coils evenly spaced in φ and separated by keystone wedges. These are placed behind the inner muon end-cap wheel, covering the range 1.6 < |η| < 2.7.

39 CHAPTER 4. EXPERIMENTAL APPARATUS

Muon Spectrometer

The muon spectrometer (Figure 4.6) is the final, outer layer of the ATLAS detector, designed for the detection of muons. It contains four different detector types: In the barrel region, monitored drift tubes (MDTs) are placed in three concentric, cylindrical layers and perform a momentum precision measurement, while three resistive plate chamber (RPC) layers are used for trigger measurements. The end-caps consist of three wheels perpendicular to the beam line, with each wheel containing two spatially-separated disks of eight wedge-shaped chambers. MDTs are once again used for the precision measurements (with the exception of the innermost end-cap section between 2.0 < |η| < 2.7, where cathode strip chambers [CSCs] fulfill this role), and multiple thin gap chamber (TGCs) are placed along the inner and middle wheel for the end-cap trigger measurements. These components provide a com-

bined resolution performance of σpT /pT of 10% for 1 TeV muons.

Monitored drift tubes. MDTs provide the precision momentum measurement for the muon spectrometer over the majority of the η range. A single drift tube is made of Alu- minum and is 29.970 mm in diameter, with a tungsten-rhenium wire in the center held at a potential of 3080 V. The tube is ﬁlled with a 97% Ar, 3% CO2 gas which is ionized by incoming muons. MDTs vary between 496−2177 mm in length, depending on their location.

Cathode strip chambers. CSCs provide additional granularity in the innermost end-cap where particle the ﬂux is highest and are better equipped for higher counting rates. They are multiwire proportional chambers, with each chamber containing ﬁve panels. Each panel is formed of two outer copper-clad laminates (acting as the cathode), a sheet of polyurethane foam, and a series of parallel anode wires made from gold-clad tungsten (with 3% rhenium),

40 CHAPTER 4. EXPERIMENTAL APPARATUS

with the wires immersed in a 80% Ar, 20% CO2 gas mixture [80].

Resistive plate chambers. RPCs are gaseous parallel electrode plates. Two resistive plates are held 2 mm apart by spacers, with the space ﬁlled by a 94.7% C2F2H4, 5% Iso-

C4H10, 0.3% SF6 ionizing gas mixture. An electric ﬁeld of 4.9 kV/mm is applied in the gap, and graphite electrodes on the outside of the resistive plates read out the resulting electron avalanche.

Thin gap chambers. TGCs are multiwire proportional chambers, similar to CSCs in structure. The smaller wire-to-cathode distance of 1.4 mm, and a highly-quenching gas mixture

of 55% CO2, 45% n-C5H12 (n-pentane), allows precise time resolutions for the majority of tracks.

41 Chapter 5

Object Reconstruction at ATLAS

The ATLAS detector is a collection of many smaller components, with each performing a specific, well-defined task on a subset of particle candidates. The ability to combine these measurements into a cohesive picture is where the true power of ATLAS lies, with the process of transforming low-level detector read-out information to event-level particles known as reconstruction. Object reconstruction typically occurs over several steps and differs depending on the object type, but track and vertex reconstruction are crucial for all physics objects.

5.1 Track and Vertex Reconstruction

A track is the path that any charged particle takes through the detector, and track reconstruction is vital for identifying the particles involved in any given physics process. Vertices are where multiple tracks intersect, and are treated as the origin point for particle interactions and decays. Reconstruction begins in the inner detector, with measurement “hits” across multiple ID

42 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS layers reconstructed into an initial track by the ATLAS New Tracking (NEWT) algorithms [81]. These algorithms employ pattern-recognition software to both global (finding possible vertex seeds) and local (using individual cluster measurements for track-building) detector information. An inside-out strategy is applied with the ID information, with adjacent pixel and strip energy measurements clustered into three-dimensional SpacePoint objects. These SpacePoints are then used to find primary and secondary vertex seeds, with tracks built between them. A Kalman fitter technique [82] is used to simultaneously follow the trajectory and include successive hits in the track candidate fit, progressively updating track information throughout the ID layers. Track ambiguities are solved using a track-scoring strategy [83]; tracks are extended out to the TRT layers, then corroborated by a second algorithm sequence seeding tracks in the TRTs from the outside-in.

5.2 Electrons and Photons

5.2.1 Electron and Photon Reconstruction

Electrons and photons interact similarly in ATLAS and are collectively treated as “electrons” by the reconstruction algorithms until the final step. Similar to ID vertex reconstruction, electron seed identification (now in the EM calorimeter) is the first step. A sliding window of 3 × 5 middle EM layer cells—corresponding to 0.075 × 0.125 in η × φ— is used to search for longitudinal “towers” (the window spacing projected through all layers) that contain a combined total transverse energy above 2.5 GeV. These towers are then treated as a seed, and clusters are formed around them by employing a clustering algorithm [84]. Cluster kinematics are reconstructed using an enlarged, spatially-dependent sliding window, with duplicated seeds removed at this stage.

43 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

Electron tracks are then found using the standard pattern recognition algorithms described above, using the pion hypothesis for energy loss due to detector material interaction [85]. Track seeds composed of three different silicon detector hits with a transverse momentum larger than 1 GeV that cannot be extended to a full seven-hit track with the pion hypothesis are candidates for an additional attempt using a specialized electron hypothesis. Such candidates must pass additional shower shape requirements [85], but this additional step improves electron performance with minimal computational overhead. Tracks are then fit to EM clusters based on their middle layer η and φ positions, with loose track- cluster associations refit using an optimized Gaussian Sum Filter (GSF) that takes non-linear bremsstrahlung effects into account [86]. If multiple tracks are matched to the same EM cluster, an ambiguity-solving algorithm is used to choose a best-fit primary track, using cluster-track distance ∆R from various momentum hypotheses and the pixel and first silicon layer hit information [85]. Finally, electron candidates are separated into one of three types: unconverted photons, with a track matched to the primary vertex and no hits in the ID (due to the photon’s lack of an electric charge); converted photons, with a track matched to a secondary vertex arising from a photon decay into an electron-positron pair; and prompt electrons, with a track matched to the primary vertex and hits in the ID.

5.2.2 Electron Identiﬁcation

The reconstructed electron candidates are further processed to improve discrimination between signal- and background-like objects. Electron identiﬁcation algorithms use information such as the electron cluster and track shower shape, track-cluster matching results, and bremsstrahlung eﬀect variables to achieve this. A full list of electron discriminating variables

44 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS is shown in Table 5.1. The identification algorithms are likelihood-based (LH) methods, using multivariate analysis (MVA) techniques to evaluate these discriminating variables when making a selection decision. Probability density functions are created from these variables to assign a probability that any given object is signal or background, affording better background rejection for a given signal efficiency compared to a cut-based algorithm on each variable. Five identification operating points are provided: VeryLoose, Loose, LooseAndBLayer, Medium, and Tight. Each successive operating point is a stricter subset of the previous, reducing signal electron identification efficiency but also reducing the rate of jet misidentification as electrons (“fake” electrons). The choice of operating point for an analysis depends on whether too few signal or too many background events are the limiting factor for sensitivity, and analyzers have flexibility in deciding on the most suitable. This analysis uses Tight identification for signal electrons.

5.2.3 Photon Identiﬁcation

Photon identification is similar to electron identification, with many of the same variables used in both. Generally, prompt photons produce narrower energy depositions in the EM calorimeter, and have smaller leakage to the HCAL compared to background photons from jets. Neutral pion π0 → γγ decays—a significant background source—are also characterized by two separate local energy maxima in the first layer of the EM calorimeter and are removed [88]. A full list of variables used by the photon identification algorithms is given in Table 5.2. Two pre-defined sets of cuts (Loose and Tight) are commonly used by analysis groups and are shown in Table 5.2, but customized combinations are also employed for specific

45 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

Table 5.1: [87] A list of all electron discriminating variables used by the ATLAS identification algorithms. The Rejection column indicates whether the variable offers discriminating power between light-flavor (LF) jets, heavy-flavor (HF) jets, or photon conversions; the Usage column indicates a direct selection cut (C) or use as part of the likelihood function (LH).

46 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

Table 5.2: [88] A list of all photon discriminating variables used by the ATLAS identiﬁcation algorithms, for both Loose and Tight photon identiﬁcation.

47 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

scenarios. This analysis uses one such combination, pseudo-tight photons (Section 7.3.2), for estimating the data-driven fake photon background.

5.3 Muons

5.3.1 Muon Reconstruction

Muon reconstruction is first performed independently in the ID and MS and combined later using the information available for each candidate in an event. Muon reconstruction in the ID is the same as any other charged particle, outlined in Section 5.1. MS muon reconstruction starts with a hit pattern search inside each individual muon chamber to form segments. For the MDTs and nearby trigger chambers, a Hough transform [89] is used to search for aligned hits in the bending η-plane, with MDT segments reconstructed from straight-line fits to the hits in each layer. Segments in the CSC detectors are built using a separate combinatorial search in the η and φ planes, with RPC and TGC hits used to measure the coordinate orthogonal to the bending plane [90]. The segments in the different MS layers are then fit together to form track candidates. The middle MS layers, where more trigger hits are available, are used as the track seed, with segments in the inner and outer layers matched to the middle layer seed based on hit multiplicity and fit quality. Two or more matching segments are required to build a track, except for the barrel-endcap transition region where a single high-quality segment with known η and φ information is required. Overlap removal is then applied to segments that are shared between several track candidates, with special consideration given to close-by muons in specific cases [90]. Finally, a global χ2 is calculated from the hits associated with each track.

48 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

5.3.2 Combined Muon Reconstruction

Information from the ID and MS reconstructions and the calorimeters is used in the ﬁnal muon reconstruction step. Depending on the information present, four reconstructed muon types are available [90]:

• Combined (CB) muons: A muon is combined if it is formed from both ID and MS tracks using a global refitting. Muons are first reconstructed outside-in from the MS, with MS hits added or removed to improve fit quality, then complemented with an inside-out search from the ID.

• Segment-tagged (ST) muons: A muon is segment-tagged if its ID track is associated with at least one local track segment in the MDT or CSC chambers but only crosses a

single MS layer (due to low pT, or presence in a region with reduced MS acceptance).

• Calorimeter-tagged (CT) muons: A muon is calorimeter-tagged if its ID track can be matched to a minimum-ionizing energy deposit in the calorimeter. This type has the lowest purity of all muon types, but is optimized in the region |η| ≤ 0.1 and transverse

momentum range 15 ≤ pT ≤ 100 GeV, where the MS is only partially instrumented due to cabling and services to the calorimeters and ID.

• Extrapolated (ME) muons: A muon is extrapolated if its track only uses MS hits and is loosely associated with the interaction point. Extrapolated muons are mainly used to extend acceptance past ID coverage at |η| ≤ 2.5, in the 2.5 ≤ |η| ≤ 2.7 region.

5.3.3 Muon Quality/Identiﬁcation

Background muons, mainly from pion and kaon decays, are further suppressed for reconstructed muons by applying certain quality requirements. In-ﬂight decays of charged hadrons

49 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS in the ID are characterized by a distinctive “kink” topology in the reconstructed track with poor ﬁt quality in the combined track [90]. For CB tracks, three variables are used in muon identiﬁcation:

• q/p signiﬁcance: the absolute value of the diﬀerence between the ratio of the charge (q) and momentum (p) of the muons measured in the ID and MS, divided by the quadrature sum of their corresponding uncertainties.

• ρ0 : the absolute value of the diﬀerence between the transverse momentum measure-

ments in the ID and MS, divided by the pT of the combined track.

• χ2 ﬁt: the normalized χ2 of the combined track ﬁt.

A requirement on the number of hits is also utilized to ensure a robust momentum measurment. For ID hits, there must be at least one Pixel hit, at least five SCT hits, fewer than three Pixel or SCT holes1, and at least 10% of the original TRT hits of the track included in the final fit (in the region 0.1 ≤ |η| ≤ 1.9). MS hit requirements vary on the quality working point, with a choice of VeryLoose, Loose, Medium, Tight, LowPt, and

HighPt. The LowPt and HighPt working points are optimized for specific muon pT ranges, with LowPt improving efficiency muons in the pT range 3 ≤ pT ≤ 5 GeV [91], and HighPt for muons with a pT above 100 GeV [92]. Final state muons in this analysis are required to have a pT above 20 GeV, with the vast majority below 100 GeV; as such, these specialized working points are not used. Similar to electron identification, the Medium working point is a stricter subset of Loose, and Tight is a stricter subset of Medium, trading lower efficiency for lower fake rates. Tight identification is used for signal muons in this analysis, which have the following requirements:

1A hole is deﬁned as an activer sensor traversed by the track, but containing no hits.

50 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

• Muons must be CB, with hits in at least two MS stations.

• At least 3 hits in two or more MDT layers (except for the |η| ≤ 0.1, allowing for one MDT hit with no more than one MDT hole).

• A q/p signiﬁcance ≤ 7.

• A normalized ﬁt χ2 ≤ 8.

5.4 Object Isolation

The use of isolation criteria is another way to further separate signal-like objects from background processes. An object is isolated if it comprises a significant fraction of the total energy content in a ∆R cone centered on its track, with cone size varying to accommodate different final state scenarios. Two different measurements of the isolation are in use by ATLAS— calorimeter-based isolation and track-based isolation. These can be used separately or in combination to form multiple working points. For calorimeter-based isolation, topological clusters are first seeded by cells with de- posited energies greater than four times their expected noise-level threshold (the quadrature sum of the electronic and pile-up2 noise, dependent on the component’s spatial position and calorimeter type [93][94]). Clusters are then expanded outwards, iteratively adding neighbor- ing cells with energy depositions greater than twice their noise level until no qualifying cells remain. A final, single-cell shell is then added to the cluster and the energy of all positive-

isol energy clusters inside this volume is summed into a raw isolation energy variable, ET,raw [87]. Energy leakage from the candidate outside this spatial cluster is modeled and ﬁt to a Crystal

2Additional hit measurements from unrelated proton-proton collisions within the same temporal resolution interval.

51 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

Ball function, the pile-up and underlying event contribution is estimated from the ambient energy density [95], and the core energy of the candidate in a ∆η × ∆φ = 0.125 × 0.175 central rectangle is used to calculate the fully-corrected, calorimeter-based isolation variable

isol ET,cone:

isol isol ET,cone = ET,raw − ET,core − ET,leakage − ET,pile-up (5.1)

Track-based isolation uses tracks with pT ≥ 1 GeV that satisfy basic track-quality requirements. Tracks must also originate from the primary vertex, enforced through an additional requirement on their longitudinal impact parameter and polar angle of |z0 sin θ| ≤ 3 mm. The momentum of all qualifying tracks is then summed within a ∆R cone around the candidate track, with corrections for bremsstrahlung radiation made by extrapolating tracks to the second EM calorimeter layer and removing any tracks within a ∆η × ∆φ = 0.05 × 0.1 window [87]. Track-based isolation can take advantage of the ﬁner track granularity to use narrower cones sizes, improving separation in boosted (highly-relativistic particles that collimate in the lab frame) and busy (many particles in the ﬁnal state) environments. This is done using

isol a variable-cone-size track isolation, pT,var, with its ∆R deﬁned as:

10 GeV ∆R = min ,Rmax (5.2) pT

where Rmax is the maximum cone size (0.2, 0.3, or 0.4). The value 10 GeV in the numerator is derived from tt sample simulations and is designed to maximize background rejection. The FixedCutLoose isolation working point is used for both electrons and muons in the signal region and uses both types of isolation. For electrons, this corresponds to the selec-

∆R≤0.2 ∆Rmax=0.2 ∆R≤0.2 tion ET,cone /pT ≤ 0.2 and pT,var /pT ≤ 0.15, while for muons ET,cone /pT ≤ 0.3 and

52 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

∆Rmax=0.3 pT,var /pT ≤ 0.15. Signal region photons only pass a calorimeter-based isolation require- ∆R≤0.4 ment, passing the FixedCutTightCaloOnly working point where ET,cone ≤ 0.022pT + 2.45 [GeV] for an isolated photon.

5.5 Hadronic Jet Reconstruction

Due to color conﬁnement, energetic particles from the initial collision carrying colored charge quickly fragment into multiple hadrons.. This results in a conical shower of particles in the ID and calorimeters that is known as a jet. This analysis does not require jets as a signal feature, but their presence is ubiquitous in many QCD background processes and all jets in

miss an event are used in the ET calculation. Jets can also indicate the presence of pile-up in an event. For these reasons, the correct calibration of jet energies, and accurate identification of their flavor, is important for this analysis. Jets are first reconstructed into topo-clusters based on calorimeter energy depositions using the same method described in Section 5.4 for calorimeter-based isolation. These topo- clusters are then grouped into jets using the anti-kt algorithm [96]. This algorithm defines two distance parameters—one between topo-cluster pairs, dij, and one between each topo- cluster and the beam, diB:

∆R2 d = min(k2p, k2p) ij (5.3) ij ti tj R2 2p diB = kti (5.4)

where ∆R is the usual spatial separation, kt is the transverse momentum (pT), R is a radius scale (0.4 in this case), and p governs the relative power of the energy versus geometrical

53 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

∆R scales (−1 for anti-kt algorithms). Initially, these distances are calculated for all topo-clusters and topo-cluster pairs in an

event. The algorithm then selects the smallest distance parameter; if it is a dij distance,

the two topo-clusters are combined into a single cluster, and if it is diB, the topo-cluster is labeled a jet and removed from the list of entries. The distances are then re-calculated and the process repeated until no topo-clusters remain. Pile-up jets are further suppressed through the application of the Jet Vertex Tool (JVT) [97]. The JVT calculates the Jet Vertex Fraction (JVF) in order to remove jets not associated

with the primary hard-scattering vertex. The JVF is a ratio of the scalar sum of the pT of

matched tracks originating from a given primary vertex PVj to the scalar pT sum of all

matched tracks in a given jet, ji:

P ji m pT(trackm, PVj) JVF (j , PVj) = (5.5) i P P ji n l pT(trackl , PVn)

where m runs over all tracks originating from PVj matched to ji, n over all primary vertices in an event, and l over all tracks originating from PVn matched to ji. Several JVF working points are deﬁned; this analysis uses the Tight JVT working point, requiring a JVF ≥ 0.5 for jets with a 20 ≤ pT ≤ 60 GeV and |η| ≤ 2.4. This working point

miss is also used in b-jet tagging, as well as ET reconstruction.

5.5.1 b-jet Tagging

Jets containing relatively long-lived hadrons with a b-quark can be distinguished from other jets using a series of ﬂavor-tagging algorithms. Top quarks in tt production, a major QCD background for this analysis, decay via t → W b 95.7% of the time [98]. In order to minimize this background, a b-jet veto is applied to the signal region.

54 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

The b-jets in this analysis are identiﬁed using the DL1 algorithm [99], a deep feed-forward neural network (NN) trained using Keras [100] with a Theano [101] backend and stochastic optimization by Adam [102]. The NN is trained on a hybrid sample of tt and Z 0 simulated samples, with at least one leptonic W decay in the tt samples to ensure a large fraction of

0 c-jets and hadronic-jet pair decays from Z to optimize performance for high-jet pT.

Four low-level algorithms are fed into DL1 as input, along with a jet’s pT and η. The first calculates the transverse and longitudinal impact parameter significance of tracks; the second reconstructs a best-possible secondary vertex; the third reconstructs a full decay chain using the b-hadron flight path; and the fourth reconstructs a full decay chain for c-hadrons. DL1’s final b-tagging discriminant is defined as:

pb DDL1 = ln (5.6) fc · pc + (1 − fc) · plight where pb, pc, and plight represent the b-jet, c-jet, and light-flavor jet probabilities, and fc is the effective c-jet fraction in the background training sample (chosen as 8% from optimization). This discriminant is divided into five “pseudo-continuous” bins, corresponding to four single- cut operating points. The maximal 85% efficiency operating point was chosen to identify as many b-jets as possible with minimal light-flavor jet impurity (as shown in Figure 5.1), where DDL1 ≥ 0.46 corresponds to the 85% operating point. The b-tagging is valid for jets passing the Tight JVT working point with pT ≥ 20 GeV and |η| ≤ 2.5.

miss 5.6 ET Reconstruction

Long-lived particles that only interact via the weak force and gravity pass through ATLAS without being detected. Neutrinos are the only particles in the SM that match this description, but many BSM models predict additional weakly-interacting particles, including the

55 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

Figure 5.1: [99] Distribution of the output discriminant of the DL1 b-tagging algorithm for b-jets, c-jets, and light-ﬂavor jets.

56 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

massless goldstino and massive pseudo-goldstino proposed by the signal models used in this analysis. Instead of direct detection, the existence of these particles is inferred via momentum conservation. Prior to collision, the total transverse momentum of the colliding partons is zero, so the vector sum of all final state particles’ transverse momentum must also be zero. Any difference required to balance the final state transverse momentum to zero is

3 miss known as the missing transverse momentum, or missing transverse energy (written as ET ).

miss Signiﬁcant ET is a key signature for novel, non-SM physics events, and many analyses use it as the ﬁnal discriminating variable.

miss ET reconstruction depends on the momentum of all hard objects in an event, with an additional soft term:

miss X * X * X * X * X * X * −E = pT + pT + pT + pT + pT + pT (5.7) T e γ τ µ j soft

miss where ET is the magnitude of the object vector sums. All hard objects are fully-calibrated

miss and must pass a baseline selection requirement in order to be included in the ET calculation; these are individual to the object type and are described in Chapter7. The soft term is exclusively reconstructed from ID tracks from the hard-scatter primary

miss vertex that are not associated with any hard object. ET is calculated from all hard objects and jets using a dedicated METMaker tool provided by the Jet/EtMiss working group. This tool uses its own overlap removal (Section 5.7) to remove hard objects sharing the same track

and determine the tracks contributing to the soft term. All soft-term tracks must have pT

≥ 400 MeV, |η| ≤ 2.5, and an impact parameter |z0 sin(θ)| ≤ 1.5 mm relative to the primary

3A potentially confusing nomenclature for newcomers, this momentum/energy interchangeability comes from the use of natural units (c = 1) in the energy-momentum relation E2 = p2c2 + m2c4, for relativistic particles with momenta much larger than their rest mass.

57 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

vertex [103].

5.7 Overlap Removal

Physics objects can be ﬂagged as more than one type (both an electron and photon, for example) if they pass multiple object type criteria in the reconstruction software. In these instances, object overlap removal is applied to prevent inclusion of the same object multiple times in a single event. Overlap removal is performed twice in this analysis—once when

miss reconstructing ET from all the baseline objects in an event, and again on signal objects to satisfy the signal region selection.

miss The overlap removal priority during ET reconstruction follows the same order as the sums in Equation 5.7; reconstructed baseline electrons are highest priority and always included; reconstructed baseline photons are included if they do not overlap with baseline electrons, and so on. The only exception is muon-jet overlap where muons are ﬁrst ghost- associated with jets. Muons are entered into the jet-clustering algorithm with inﬁnitesimally small momentum, and if tagged as part of a jet, the muon momentum is removed from the

miss ET calculation. Jets are then checked for catastrophic energy loss (and removed), or if they are ﬁnal state radiation (and kept) [103]. For the signal region overlap removal, a default recommendation has been applied in accordance with the harmonisation study groups’ ﬁndings. [104]. This corresponds to the following sequential object comparisons and removals:

1. Electron-electron overlap: Any two electrons that share the same track. Electrons with

an ambiguous author are removed ﬁrst, then the lower pT electron of the two if both are unambiguous.

58 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

2. τ-electron overlap: Any τ-leptons within a ∆R cone of 0.2 around any electron candidate.

3. τ-muon overlap: Any τ-leptons within a ∆R cone of 0.2 around any muon candidate.

4. Electron-muon overlap: Any calo-tagged muon that shares the same track as an electron candidate.

5. Photon-electron overlap: Any photons within a ∆R cone of 0.4 around any electron candidate.

6. Photon-muon overlap: Any photons within a ∆R cone of 0.4 around any muon candidate.

7. Electron-jet overlap: Any jets within an inner ∆R cone of 0.2 of an electron candidate, then any electrons within an outer ∆R cone of 0.4 around all surviving jet candidates.

8. Muon-jet overlap: Any jets that have a low track multiplicity (< 3 track hits) or a low

relative pT ratio (where the muon’s pT is larger than the jet’s, and when the muon’s

pT is greater than 0.7 of the total track pT sum) within an inner ∆R cone of 0.2. Afterwards, any muons within a ∆R cone of 0.4 of all surviving jet candidates.

9. Jet-τ overlap: Any non-b-tagged jets within a ∆R cone of 0.2 around any τ-lepton candidate, then any τ-leptons within a ∆R cone of 0.2 around any b-tagged jets.

10. Jet-photon overlap: Any jets within a ∆R cone of 0.4 around any photon candidate.

11. Fat jet 4-electron overlap: Any fat jets within a ∆R cone of 0.4 around any electron candidate. 4Fat jets occur in boosted environments where multiple jets collimate and are no longer separable. They are not relevant to this analysis, but their overlap criteria are included here for completeness.

59 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

12. Jet-fat jet overlap: Any jets within a ∆R cone of 0.4 around any fat jet candidate.

5.8 The Trigger System

The volume of data produced by ATLAS is large. The raw event rate is 40 MHz, corresponding to a data rate of 100 PB/s [105], and it is impossible to store every event being produced. To combat this technical challenge, ATLAS employs an automated triggering system to save only the events most relevant to physics analyses. It does this in two steps. The ﬁrst-level trigger (L1) is hardware-based and uses a subset of the detector information to reduce the event rate from 40 MHz to 100 kHZ. It does this in real-time while the detector is in operation (also referred to as online data processing) where speed and selection sim- plicity is prioritized. These initial events are then held in a readout buﬀer and run through a second, software-based high-level trigger (HLT). The HLT triggers employ stricter object quality cuts to further reduce the event rate down to 1 kHz, corresponding to around 1 − 1.5 GB/s of recorded data [105]. Triggers are critical to any analysis as only events that pass an available trigger are actually recorded. The list of triggers available is also continuously updated as analysis groups make new requests for specialized triggers and as both hardware and software changes come about over the data-taking period. Triggers are typically designed to accept events

miss with high-pT physics objects (electrons, muons, jets, etc.) and events with significant ET in various combinations. The analysis presented here is the first of its kind in both Run I and Run II, and the final state is somewhat specialized; as such, there is no dedicated trigger available to select

miss events with two leptons, two photons, and ET . A three-object muon and diphoton trigger (2g10 loose mu20) is available for the entire Run II period, but the associated electron

60 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS channel trigger (e20 lhmedium 2g10 loose) is not present in 2017 and 2018 data. A dilepton trigger is chosen instead as it is well-supported and existing SUSY derivations frequently incorporate it as a thinning criteria. It is also inclusive of the more complex ﬁnal state with all signal samples at every benchmark point passing the threshold criteria. The lowest, unprescaled dielectron trigger varies over the entire Run II data-taking period; these triggers are listed in Table 5.3

Table 5.3: List of dielectron triggers used in this analysis for each year of data-taking (NB: The logical OR of two triggers were used for 2017 and 2018, except for runs B5-B8 where the sole trigger denoted by † was used, due to partial pre-scaling).

Data-Taking Period Dielectron Trigger

2015 HLT 2e12 lhloose L12EM10VH 2016 HLT 2e17 lhvloose nod0 2017 HLT e17 lhvloose nod0 L12EM15VHI OR HLT 2e24 lhvloose nod0 2017 (B5-B8) HLT 2e24 lhvloose † 2018 HLT e17 lhvloose nod0 L12EM15VHI OR HLT 2e24 lhvloose nod0

For muons, there are only two trigger conﬁgurations, shown in Table 5.4:

Table 5.4: List of dimuon triggers used in this analysis for each year of data-taking.

Data-Taking Period Dimuon Trigger

2015 HLT 2mu10 2016 HLT 2mu14 2017 HLT 2mu14 2018 HLT 2mu14

61 CHAPTER 5. OBJECT RECONSTRUCTION AT ATLAS

Finally, a multi-lepton trigger is used for the diﬀerent-ﬂavor lepton control region requiring one triggered electron and one triggered muon, shown in Table 5.5.

Table 5.5: List of multi-lepton triggers used in this analysis for each year of data-taking.

Data-Taking Period Multi-lepton Trigger

2015 HLT e17 lhloose nod0 mu14 2016 HLT e17 lhloose mu14 2017 HLT e17 lhloose mu14 2018 HLT e17 lhloose mu14

For data, events are required to pass the trigger selection for the lepton channel, year, and period they were taken, while for simulated Monte Carlo (MC) samples a random run number for the appropriate year is applied with the PileupReweightingTool and then treated accordingly. Trigger matching is also applied to the relevant reconstructed objects (e.g., only the objects identiﬁed as electrons are counted for passing the electron trigger). Event-level scale factors and uncertainties on MC simulations for the dielectron triggers are calculated with the TrigGlobalEﬃciencyCorrectionTool provided by the EGamma working group, combining multiple trigger legs and data periods. For muons, the Muon Combined Performance (MCP) group provides a tool to provide event-level scale factors and uncertainties for the dimuon triggers. Further details on scale factors and systematic uncertainty are in 6.3 and 10 respectively.

62 Chapter 6

Monte Carlo Event Simulation

ATLAS analyses use simulated data of signal and background processes to predict, constrain, and validate event estimates in the final signal region before comparison with experimental data. This methodology is known as “blinding” an analysis, preventing experimenter’s bias from overfitting selection cuts due to prior knowledge. A suite of programs is used in generating realistic, accurate simulations of ATLAS particle physics events. Events are first created by an MC event generator, then passed to a parton showering program to simulate hadronization before being given to software simulating the ATLAS detector geometry.

6.1 Signal Monte Carlo Generation

In the analysis presented here, signal MC samples are generated using MadGraph5 aMC@NLO (MG5) [106], which calculates the interaction matrix elements for the mssm-goldstini model. The model is defined by various input files, including: a Lagrangian equation describing all relevant quantum fields and interactions; a list of particle decays, coupling constants, and propagators; and parameter cards specifying the fixed parameter values and free parameter

63 CHAPTER 6. MONTE CARLO EVENT SIMULATION

ranges for each signal sample. MG5 uses these inputs to generate hard collision events to tree-level, leading-order (LO) accuracy. Higher-order corrections for samples are possible and are referred to as one-loop, next-to-leading order (NLO) and two-loop, next-to-next-to- leading-order (NNLO). Cross-sections are calculated to NLO and used to scale the signal samples post-generation. For signal samples, Pythia8 [107] is the parton showering program used to model the fragmentation and hadronization of colored partons at low energy scales. A parton density function (PDF)—modeling the internal structure of the colliding protons—is provided to Pythia8, along with an underlying event (UE) tune (a set of customizable parameters, e.g. the amount of initial and ﬁnal state radiation). The NNPDF2.3 LO parton density function and A14 underlying event tune are used for signal samples. Output from Pythia8 is then passed to EvtGen [108], a program that decays b-hadrons, to improve modeling accuracy of b-jets. GEANT4 [109] takes the output from EvtGen and simulates its passage through, and detection by, the sub-detectors of ATLAS so that it can be reconstructed oﬄine in the same manner as data. The Atlfast-II [110] simulation is used to approximate the full Run- II geometry, achieving good agreement between kinematic distributions while minimizing computational resources.

6.2 Background Monte Carlo Generation

The majority of the background MC samples use the same generator-showering combination, with the exception of the V γγ QCD background. The diboson (WW , WZ, ZZ), top-boson (W t, W t), and tt backgrounds are generated with POWHEG [111] before being passed to Pythia8 and EvtGen. Diboson backgrounds are calculated to NLO order and use the CT10

64 CHAPTER 6. MONTE CARLO EVENT SIMULATION

PDF and CTEQ6L1 tune, while the top-boson and tt backgrounds are calculated to leading order with the NNPDF2.30 LO PDF and A14 tune. The main V γγ QCD background is simulated using Sherpa2.2 [112], a composite program that both generates the hard-collision events and simulates parton showering. Sherpa2.2 contains an inbuilt tune that is used alongside the NNPDF3.0 NNLO PDF to calculate these events to LO accuracy. GEANT4 is used for detector simulation for all background MC samples. A summary of all MC samples used in this analysis is given in Table 6.1.

Table 6.1: A summary of all simulated MC samples used in the analysis and their associated modeling programs and parameters, where Generator is the program calculating the interaction matrix elements, Order is the order to which cross-sections are calculated, Showering is the programs simulating hadronization and fragmentation of colored partons, PDF is the parton density function of the proton’s internal structure, and Tune is a set of tunable parameters passed to the Showering program. GEANT4 is used for detector simulation for all samples and has been omitted as a column.

Sample Name Generator Order Showering PDF, Tune

Signal MG5 LO Pythia8 + EvtGen NNPDF2.3 LO, A14 V γγ (``γγ) Sherpa2.2 LO Sherpa2.2 NNPDF3.0 NNLO, Default WW (`ν`ν) POWHEG NLO Pythia8 + EvtGen CT10, AZNLO CTEQ6L1 WZ (`ν``) POWHEG NLO Pythia8 + EvtGen CT10, AZNLO CTEQ6L1 WZ (qq``) POWHEG NLO Pythia8 + EvtGen CT10, AZNLO CTEQ6L1 ZZ (````) POWHEG NLO Pythia8 + EvtGen CT10, AZNLO CTEQ6L1 ZZ (qq``) POWHEG NLO Pythia8 + EvtGen CT10, AZNLO CTEQ6L1 W t POWHEG LO Pythia8 + EvtGen NNPDF2.3 LO, A14 W t POWHEG LO Pythia8 + EvtGen NNPDF2.3 LO, A14 tt POWHEG LO Pythia8 + EvtGen NNPDF2.3 LO, A14

65 CHAPTER 6. MONTE CARLO EVENT SIMULATION 6.3 Scale Factor Reweighting

ATLAS performance groups compare generated MC samples to data in order to determine their accuracy. Diﬀerences are taken into account by reweighting MC samples to match detector performance in recorded data, with recommendations from these groups taking the form of scale factors. Scale factors are the ratio of MC to data for a series of pT and η bins (“slices”). This ratio of events minimizes correlated systematic uncertainties between MC and data. Scale factors are implemented through specialized performance tools, with a full list of all scale factors used in this analysis shown in Table 6.2.

Table 6.2: A list of all scale factors (SFs) implemented in this analysis. Pileup and trigger scale factors are single event-level SFs, while electron, muon, photon, and jet SFs are object- level SFs and applied to all relevant objects in the signal regions. Electron SFs are only applied to electron channel events and muon SFs are only applied to muon channel events.

Scale Factor Description Performance Tool

Pileup PileupReweightingTool Electron Reconstruction AsgElectronEfficiencyCorrectionTool Electron Identification AsgElectronEfficiencyCorrectionTool Electron Isolation AsgElectronEfficiencyCorrectionTool Electron Trigger TrigGlobalEfficiencyCorrectionTool Muon Reconstruction MuonEfficiencyScaleFactors Muon Isolation MuonEfficiencyScaleFactors Muon Trigger MuonTriggerScaleFactors Photon Identification AsgPhotonEfficiencyCorrectionTool Photon Isolation AsgPhotonEfficiencyCorrectionTool Jet Flavour Tagging BTaggingEfficiencyTool

66 CHAPTER 6. MONTE CARLO EVENT SIMULATION

Scale factors have an associated uncertainty that is also calculated by these performance tools (see Section 10 for more details).

67 Chapter 7

Physics Object Deﬁnitions

Physics analyses are based on reconstructed objects, such as electrons, photons, muons,

miss taus, jets, and ET . For each object, diﬀerent combinations of criteria (working points) are available for reconstructed objects depending on the needs of the analysis. Baseline selections

miss are applied for all objects entering the ET calculation (electrons, photons, taus, muons,

miss and jets). The ET is calculated at the event-level, and the selections are loose in order to include all signiﬁcant-pT objects. Signal selections are stricter and exploit the kinematic diﬀerences between signal and background samples. Signal objects pass the baseline criteria in addition to their own selection requirements.

7.1 Baseline and Signal Electrons

Baseline electrons are calibrated with the EGammaCalibrationAndSmearingTool and pass

VeryLooseLH identiﬁcation. Kinematic requirements include pT ≥ 10 GeV and |η| ≤ 2.47, as well as a veto on the transition region between the barrel and endcap calorimeters (1.37 ≤ |η| ≤ 1.52, known as the “crack” region). Signal electrons must survive overlap removal of

68 CHAPTER 7. PHYSICS OBJECT DEFINITIONS the baseline object collections, pass TightLH identiﬁcation, pass FixedCutLoose isolation, and have a pT ≥ 20 GeV. Summaries of the baseline and signal electron requirements are given in Table 7.1 and Table 7.2 respectively.

miss Table 7.1: Baseline electron criteria used in the ET calculation.

Cut Description Cut Value

Calibration Correction Successfully applied Identiﬁcation VeryLooseLH

Transverse Momentum pT ≥ 10 GeV Central Pseudo-rapidity |η| ≤ 2.47 Crack Veto ! (1.37 ≤ |η| ≤ 1.52)

Table 7.2: Signal electron criteria used in the signal region deﬁnitions.

Cut Description Cut Value

Overlap Removal (OR) Passing standard OR Identiﬁcation TightLH Isolation FixedCutLoose

Transverse Momentum pT ≥ 20 GeV

7.2 Baseline and Signal Muons

Baseline muons are calibrated with the MuonCalibrationAndSmearingTool and pass Very-

Loose identiﬁcation, with kinematic requirements of pT ≥ 10 GeV and |η| ≤ 2.7. Signal muons must survive overlap removal of the baseline object collections, pass Tight identiﬁca-

69 CHAPTER 7. PHYSICS OBJECT DEFINITIONS

tion, pass FixedCutLoose isolation, and have pT ≥ 20 GeV. Summaries of the baseline and signal muon requirements are given in Table 7.3 and Table 7.4 respectively.

miss Table 7.3: Baseline muon criteria used in the ET calculation.

Cut Description Cut Value

Calibration Correction Successfully applied Identiﬁcation VeryLoose

Transverse Momentum pT ≥ 10 GeV Central Pseudo-rapidity |η| ≤ 2.7

Table 7.4: Signal muon criteria used in the signal region deﬁnitions.

Cut Description Cut Value

Overlap Removal (OR) Passing standard OR Identiﬁcation Tight Isolation FixedCutLoose

Transverse Momentum pT ≥ 20 GeV

7.3 Photons

7.3.1 Baseline and Signal Photons

Baseline photons are calibrated using the EGammaCalibrationAndSmearingTool and Iso- lationCorrectionTool and pass Loose identiﬁcation. Similar to baseline electrons, baseline photons must have pT ≥ 10 GeV, |η| ≤ 2.47 and not be in the crack region. Signal photons

70 CHAPTER 7. PHYSICS OBJECT DEFINITIONS must survive overlap removal of the baseline object collections, pass Tight identiﬁcation, pass FixedCutTightCaloOnly isolation, and have a pT ≥ 20 GeV. Summaries of the baseline and signal photon requirements are given in Table 7.5 and Table 7.6 respectively.

miss Table 7.5: Baseline photon criteria used in the ET calculation.

Cut Description Cut Value

Calibration Correction Successfully applied Isolation Correction Successfully applied Identiﬁcation Loose

Transverse Momentum pT ≥ 10 GeV Central Pseudo-rapidity |η| ≤ 2.47 Crack Veto ! (1.37 ≤ |η| ≤ 1.52)

Table 7.6: Signal photon criteria used in the signal region deﬁnitions.

Cut Description Cut Value

Overlap Removal (OR) Passing standard OR Identiﬁcation Tight Isolation FixedCutTightCaloOnly

Transverse Momentum pT ≥ 20 GeV

7.3.2 Pseudo-Tight and Anti-Isolated Photons

Photons are classiﬁed as pseudo-tight if they pass all Tight requirements, except for at least one of four discriminating variables: ws3, fside, ∆Es, and Eratio, listed in Table 5.2. This is

71 CHAPTER 7. PHYSICS OBJECT DEFINITIONS

iso ∆R≤0.4 also known as the LoosePrime4 working point. Photons are isolated if ET = ET,cone −

iso 0.022pT − 2.45 ≤ 0 GeV and anti-isolated if 8 ≤ ET ≤ 80 GeV. Pseudo-tight and anti-isolated photons are used to deﬁne three data-driven control regions, adjacent to the signal region, for estimating the number of jets misidentiﬁed as photons (photon fakes). This estimation method is described in detail in Section 9.4.

7.4 Baseline Jets

There is no kinematic selection for baseline jets. Baseline jets are calibrated with the Jet- CalibrationTool, with a JVF calculated by the JVT and muon ghost association applied. A summary of the baseline jet requirements is given in Table 7.7.

miss Table 7.7: Baseline jet criteria used in the ET calculation.

Cut Description Cut Value

Calibration Correction Successfully applied JVF Calculation Successfully applied Muon Ghost Association Successfully applied

7.5 Signal b-Jet Veto

Jets are identiﬁed as signal b-jets if they are calibrated by the JetCalibrationTool, pass kinematic requirements of pT ≥ 20 GeV and |η| ≤ 2.5, and pass the 85% eﬃciency working point for the DL1 algorithm. An event is rejected (vetoed) if it contains a signal b-jet. A summary of these requirements is given in Table 7.8.

72 CHAPTER 7. PHYSICS OBJECT DEFINITIONS

Table 7.8: Signal b-jet criteria used to veto events.

Cut Description Cut Value

Calibration Correction Successfully applied

Transverse Momentum pT ≥ 20 GeV Central Pseudo-rapidity |η| ≤ 2.5

DDL1 score ≥ 0.46 (85% eﬃciency)

7.6 Baseline Taus

Taus are not a signal region selection object, as stau production is not featured as a slepton production mode in the signal samples, and they are much more likely to appear as hadronic

miss jets than as hard objects. However, a baseline selection is still applied and given to the ET tool in case taus are present.

miss Table 7.9: Baseline tau criteria used in the ET calculation.

Cut Description Cut Value

Calibration Correction Successfully applied Identiﬁcation Loose

Transverse Momentum pT ≥ 20 GeV Central Pseudo-rapidity |η| ≤ 2.5 Crack Veto !(1.37 ≤ |η| ≥ 1.52)

73 Chapter 8

Event Selection

Signal physics object definitions are used in conjunction with event-level features to define the signal regions. The signal regions are designed to maximize signal acceptance and reject background SM events. Two signal regions are defined for this analysis: the first probes

miss a final state of two electrons, two photons, and significant ET and is referred to as the electron channel; the second probes a final state of two muons, two photons, and significant

miss ET and is referred to as the muon channel. The most prominent background by far is V γγ → ``γγ, with fake photons and leptons sub-dominant backgrounds and the diboson, top-boson, and tt negligible in comparison. As a result, discussions of discriminaing selections (and corresponding kinematic plots) in this section are focused on ``γγ. Some selection criteria are also informed by the background estimation studies, with a brief description presented here and an in-depth explanation provided in Chapter9. A common preselection for both channels is described ﬁrst in Section 8.1. The electron and muon channels share many of the same properties and are discussed in tandem in Section 8.2, with explicit diﬀerences mentioned when applicable.

74 CHAPTER 8. EVENT SELECTION 8.1 Preselection

Preselection cuts are used to reject events with incomplete reconstruction information, large amounts of calorimeter noise, and non-collision backgrounds. These cuts are collectively referred to as event and jet cleaning and are implemented to ensure data quality. Events must ﬁrst be contained within the relevant GoodRunsList (GRL), a list of all data- taking periods in a given year, where each detector component is functioning as expected and data quality is good. For data this is a simple check on its run number, whereas for MC samples a random run number must ﬁrst be assigned by applying the PileupReweightingTool. Four GRLs were used in this analysis:

• data18 13TeV.periodAllYear DetStatus-v105-pro22-13 Unknown PHYS StandardGRL All Good 25ns Triggerno17e33prim.xml

• data17 13TeV.periodAllYear DetStatus-v105-pro22-13 Unknown PHYS StandardGRL All Good 25ns Triggerno17e33prim.xml

• data16 13TeV.periodAllYear DetStatus-v105-pro22-13 Unknown PHYS StandardGRL All Good 25ns WITH IGNORES.xml

• data15 13TeV.periodAllYear DetStatus-v105-pro22-13 Unknown PHYS StandardGRL All Good 25ns.xml where each list corresponds to a data-taking year in the Run II dataset (2015-2018). Events can be ﬂagged as incomplete or noisy, either by individual sub-detectors or at the event level. Flags for the SCT, LAr, and Tile sub-detectors are checked, as well as the Core event-level ﬂag, and the event is rejected if any are raised. In addition, the event must contain a primary vertex and not contain any LooseBad jets, where a LooseBad jet indicates the presence of calorimeter noise or pile-up (described in detail in Reference [113]). A summary of these preselection cuts is given in Table 8.1.

75 CHAPTER 8. EVENT SELECTION

Table 8.1: Preselection criteria for both signal channels.

Cut Description Cut Value

Contained in GoodRunsList Yes Number of LAr flagged errors 0 Number of Tile flagged errors 0 Number of SCT flagged errors 0 Number of Core flagged errors 0 Number of primary vertices in event ≥ 1 Number of LooseBad jets 0

8.2 Channel Selection

Leptons and photons must pass several requirements (stated in Chapter7) in order to be labeled as signal objects. These include pT, identification, and isolation selections, with multiple criteria combinations possible. The decisions for each variable are discussed here and illustrated with distribution plots. The number of signal objects for each channel is also a choice: for this analysis, exactly two signal leptons for each channel are required with no different-flavor leptons (i.e., two signal electrons and zero signal muons for the electron channel, and zero signal electrons and two signal muons for the muon channel). The number of signal photons is inclusive; at least two are required, but more are permitted. These have a small overall impact (roughly 1% of events) but discriminate slightly against the ``γγ background. In this section, plot distributions are shown sequentially with all previous kinematic selections applied. For example, the lepton pT plots in Section 8.2.3 have lepton charge and

76 CHAPTER 8. EVENT SELECTION

ﬂavor pair selections and a b-jet veto applied, while the dilepton invariant mass distributions

miss in Section 8.2.10 have every signal selection (except for ET ) applied. The complete list of signal region selections is given in Table 8.2 for the electron channel and Table 8.3 for the muon channel.

8.2.1 Lepton Charge and Flavor

The signal samples are characterized by two opposite-sign, same-flavor leptons (i.e., e+e− and µ+µ− lepton pairs). Both of these features are used to define the signal regions, as almost all signal events pass this selection and both fake lepton backgrounds (from same- sign dilepton pairs), and backgrounds from two independent lepton decays (e.g., WZ, WW , τ +τ−) are suppressed. Signal region requirements on charge and flavor pairs also allow the use of same-sign and different-flavor data-driven control regions for background estimation (described in Sections 9.2 and Section 9.3).

8.2.2 b-Jet Veto

An event is vetoed if it contains a signal b-jet (Section 7.5) as this indicates the presence of top quark backgrounds. The highest eﬃciency working point for the BTaggingEﬃciencyTool is chosen, as top quark backgrounds have comparatively large production cross-sections (e.g., 730 pb for tt, 37.9 pb for W t) and the b-jet veto is highly discriminating—reducing tt backgrounds by a relative factor of 10 when compared to signal.

8.2.3 Lepton pT

Events must ﬁrst pass the year-appropriate dilepton trigger (Section 5.8) to be recorded by

the HLT. For the electron channel, this sets a minimum pT threshold of 12, 17, or 24 GeV

77 CHAPTER 8. EVENT SELECTION

Figure 8.1: Background estimate of the dominant ``γγ MC sample for leading and sub- leading lepton pT. Four representative signal samples are overlaid and labeled by their mass

parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading electron pT on the top-left and sub-leading electron pT on the bottom-left. Muon channel estimates are on the right, with leading muon pT on the top-right and sub-leading muon pT on the bottom-right. Overﬂow bins are visible for all four plots.

78 CHAPTER 8. EVENT SELECTION

depending on the data-taking periods (2015, 2016–2018, and periods B1 − B4 in 2017, respectively). For the muon channel, the minimum pT thresholds are lower at 10 and 14

GeV for the 2015 and 2016-2018 data-taking periods respectively. A modest pT cut of 20 GeV was chosen for both channels as to be several GeV above the trigger thresholds for the vast majority of signal (except periods B1 − B4), and for consistency across channels.

The leading and sub-leading lepton pT are shown in Figure 8.1. A higher pT cut does provide some discriminating power for certain signal samples, but not for those with compressed slepton-neutralino mass splittings (e.g., the [M`˜, Mχ˜, MP GLD] = [300, 262.5. 225] and [200 187.5, 150 samples]). No additional cuts were made in order to deﬁne a signal region inclusive of every signal sample.

8.2.4 Lepton Identiﬁcation

The leading and sub-leading lepton identification classifications are given in Figure 8.5, where leptons are classified by the most restrictive selection they pass (e.g., a lepton labeled as “loose” is one that passes very loose and loose requirements, but not medium or tight). A tight identification requirement for both leptons was chosen; it is marginally discriminating, but uniform across the signal samples.

8.2.5 Lepton Isolation

The leading and sub-leading lepton isolation classiﬁcations are given in Figure 8.3, where leptons are classiﬁed by the most restrictive selection they pass (e.g., a lepton labeled as “FCLoose” is one that passes no isolation requirement and FCLoose, but not FCTight). FCTight is modestly discriminating, and a case can be made for choosing this as the lepton isolation requirement. However, photon isolation was studied more extensively for this anal-

79 CHAPTER 8. EVENT SELECTION

Figure 8.2: Background estimate of the dominant ``γγ MC sample for leading and sub- leading lepton identification classification. Four representative signal samples are overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading electron identification type on the top-left and sub-leading electron identification type on the bottom-left. Muon channel estimates are on the right, with leading muon identification type on the top-right and sub-leading muon identification type on the bottom-right. Overflow bins are visible for all four plots.

80 CHAPTER 8. EVENT SELECTION

Figure 8.3: Background estimate of the dominant ``γγ MC sample for leading and sub- leading lepton isolation classiﬁcation. Four representative signal samples are overlaid and

labeled by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading electron isolation type on the top-left and sub-leading electron isolation type on the bottom-left. Muon channel estimates are on the right, with leading muon isolation type on the top-right and sub-leading muon isolation type on the bottom-right. Overﬂow bins are visible for all four plots.

81 CHAPTER 8. EVENT SELECTION

ysis, and an FCLoose requirement was chosen to ensure leptons were well-separated from signal photons, without further optimization.

8.2.6 Photon pT

The leading and sub-leading photon pT is given in Figure 8.4 for both channels. Photon pT is more discriminating than lepton pT (particularly for leading photons) and a case can be made for extending the leading photon pT requirement up to 40 or 50 GeV. However, this results in a relative eﬃciency of 50% or less for the most compressed neutralino-GLD/PGLD mass splittings (i.e., the [M`˜, Mχ˜, MP GLD] = [200, 175, 150] and [300, 30, 0] samples).

miss This, coupled with the observation that ET is much more discriminating against the ``γγ background, resulted in no additional photon pT cut.

8.2.7 Photon Identiﬁcation

The leading and sub-leading photon identification classifications are given in Figure 8.5, where photons are classified by the most restrictive selection they pass (e.g., a photon labeled as “pseudo-tight” is one that passes loose and pseudo-tight requirements, but not tight). A tight photon identification requirement for both photons is marginally discriminating (removing around 20% of ``γγ events and 5 − 15% of signal) but is crucial in defining data- driven control regions for estimating fake photon backgrounds in Section 9.4. For this reason, both photons are required to pass tight identification for the signal region.

8.2.8 Photon Isolation

iso The leading and sub-leading photon isolation energies, ET , are given in Figure 8.6. The

iso signal samples peak at a slightly lower ET than the ``γγ background, with the separation

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Figure 8.4: Background estimate of the dominant ``γγ MC sample for leading and sub- leading photon pT. Four representative signal samples are overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading photon pT on the top-left and sub-leading photon pT on the bottom-left. Muon channel estimates are on the right, with leading photon pT on the top-right and sub-leading photon pT on the bottom-right. Overﬂow bins are visible for all four plots.

83 CHAPTER 8. EVENT SELECTION

Figure 8.5: Background estimate of the dominant ``γγ MC sample for leading and sub- leading photon identification classification. Four representative signal samples are overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the left, with leading photon identification type on the top-left and sub-leading photon identification type on the bottom-left. Muon channel estimates are on the right, with leading photon identification type on the top-right and sub-leading photon identification type on the bottom-right. Overflow bins are visible for all four plots.

84 CHAPTER 8. EVENT SELECTION

Figure 8.6: Background estimate of the dominant ``γγ MC sample for leading and sub- iso leading photon isolation energies ET . Four representative signal samples are overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD). Electron channel estimates are on the iso iso left, with leading photon ET on the top-left and sub-leading photon ET on the bottom- iso left. Muon channel estimates are on the right, with leading photon ET on the top-right and iso sub-leading ET on the bottom-right. Overﬂow bins are visible for all four plots.

85 CHAPTER 8. EVENT SELECTION

between peaks becoming more pronounced for larger mass separations between the neutralino

iso and GLD/PGLD. Both signal photons must be isolated, with ET ≤ 0 GeV, and the photon isolation requirement is used to deﬁne data-driven control regions for fake photon estimation (similar to photon identiﬁcation) in Section 9.4.

8.2.9 Diphoton Invariant Mass

Several ``γγ events were found to have extremely low diphoton invariant mass; a minimum diphoton invariant mass above 20 GeV was implemented to remove these events with no signal eﬃcency loss.

8.2.10 Z-mass veto

The dilepton invariant mass distribution in both channels is given in Figure 8.7. A prominent peak at the Z-mass (81 − 101 GeV) is observed for both channels and a Z-mass veto was implemented to remove these on-shell Z-boson decays. A secondary, broader peak is also observed for the ``γγ background below 80 GeV, however a general dilepton invariant mass requirement of 101 GeV and above caused a large signal eﬃciency loss in many of the lower slepton mass samples and was not considered further. The Z-mass peak is also used to deﬁne a control region for calculating the normalization scale factor of the ``γγ background in Section 9.5.

miss 8.2.11 ET

miss The ﬁnal discriminating variable is ET . It is highly discriminating, with the ``γγ events

miss miss dropping sharply for increasing ET . The ﬁnal ET cut for each channel was chosen after completing the background estimation studies and is discussed in detail in Section 9.5. The

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Figure 8.7: Background estimate of the dominant ``γγ MC sample for the dilepton invariant mass Four representative signal samples are overlaid and labeled by their mass parameters

(M`˜, Mχ˜, MP GLD). The dielectron invariant mass for the electron channel is on the top, and the dimuon invariant mass for the muon channel is on the bottom. 87 CHAPTER 8. EVENT SELECTION

miss miss ET requirement is the only diﬀerence between the two channels, with ET ≥ 100 GeV

miss miss for the signal electron region and ET ≥ 110 GeV for the signal muon region. The ET distributions for the validation and signal regions can be seen in Figures 11.1 and 11.2.

88 CHAPTER 8. EVENT SELECTION

Table 8.2: Selection criteria for the electron signal channel.

Cut Description Cut Value

Passing dielectron triggers Yes Number of Signal b-jets (b-jet veto) 0 Number of Signal photons 2 Number of Signal electrons 2 Number of Signal muons 0 Sign of electron charges Opposite-sign

Diphoton invariant mass mγ1γ2 ≥ 20 GeV

Dielectron invariant mass (Z-peak veto) !(81 ≤ m`1`2 ≤ 101) GeV

miss Missing transverse momentum ET ≥ 100 GeV

Table 8.3: Selection criteria for the muon signal channel.

Cut Description Cut Value

Passing dimuon triggers Yes Number of Signal b-jets (b-jet veto) 0 Number of Signal photons 2 Number of Signal electrons 0 Number of Signal muons 2 Sign of muon charges Opposite-sign

Diphoton invariant mass mγ1γ2 ≥ 20 GeV

Dimuon invariant mass (Z-peak veto) !(81 ≤ mµ1µ2 ≤ 101) GeV

miss Missing transverse momentum ET ≥ 110 GeV

89 Chapter 9

Background Estimation

The accurate prediction of background processes in the signal region is the core of an analysis and governs its predictive power. Poorly modeled backgrounds can introduce large uncertainties and reduce the signiﬁcance of discovery/exclusion estimates, reducing the upper limit at which conﬁdence intervals can be set. The standard procedure for ensuring good modeling accuracy is the use of control and validation regions adjacent to, but exclusive of, the signal region. Adjacency in this context typically refers to a single signal cut inversion (e.g., a control region with same-sign lepton

miss pairs instead of opposite-sign, or events with ET ≤ 50 GeV when the signal region requires

miss ET > 50 GeV), where the similarity in all other cuts results in kinematic similarity of the final state objects between the regions. Control regions are used to scale and fit MC backgrounds to observed data, while validation regions are used to confirm the control region fit is accurate and applicable to multiple regions, with no additional fitting. This analysis uses multiple control regions for several key backgrounds, along with a validation region for each channel. Backgrounds relevant to the analysis are first discussed in Section 9.1. Fake lepton

90 CHAPTER 9. BACKGROUND ESTIMATION

backgrounds, backgrounds where leptons come from diﬀerent, independent sources (e.g., WW , WZ, tt, τ +τ −), and fake photon backgrounds—along with their estimation using data-driven control regions—are shown in Sections 9.2, 9.3, and 9.4 respectively. Finally, the dominant ``γγ background is described in Section 9.5.

9.1 Background Types and Relevance

Broadly, there are two types of backgrounds: reducible and irreducible. Irreducible backgrounds have final state signatures identical to the signal and are generally difficult to separate from signal events. For this analysis, these backgrounds are small due to the high-multiplicity final state. Diboson, top-boson and tt processes can produce W/Z pairs

miss that decay leptonically to produce high-pT leptons and real ET in the form of neutrinos, but must produce photons radiatively from the leptons. MC samples of these processes were investigated and found to be statistically-limited in the signal region, with fewer than 10 simulated events for all samples from an initial event generation of 107 − 108 events. Such a small number of events makes any inference about these background shapes impossible. Reducible backgrounds are more pertinent to this analysis and several sources are explored. Reducible backgrounds are those where one or more signal objects are fakes (e.g., jets misidentiﬁed as leptons or photons, and QCD backgrounds where hadronic jets are re-

miss constructed as ET ). Monte Carlo simulations are typically incapable of modeling fake objects, since by definition fakes are poorly-reconstructed objects whose mis-modeling goes undetected. Instead, data in data-driven control regions are used to estimate fake lepton and fake photon backgrounds, which are found to be sub-dominant. Diboson, top-boson and tt processes reappear as a potential source for real leptons and are estimated with same-sign and different-flavor lepton pair control regions. Fake photon

91 CHAPTER 9. BACKGROUND ESTIMATION sources are modeled with pseudo-tight and anti-isolated photons in their own data-driven control regions. The ``γγ background is also a reducible background and the dominant background source.

It is capable of producing two real, high-pT leptons and two real, high-pT photons, but not

miss a single, signiﬁcant source of ET (in the form of neutrinos or stable BSM particles). A control region in the Z invariant mass peak is used to normalize the MC simulation yields to

miss data and determine a scale factor, with ET chosen as the ﬁnal discriminant and the scale

miss factor validated in the low-ET region.

9.2 Jets Misidentiﬁed as Leptons

Jets misidentified as leptons are charge-agnostic, with an equal probability of being reconstructed as positively or negatively charged. This was verified in a sample enriched in fake leptons. The charge misidentification rates for real electrons and muons are also small and were verified by looking at Z → `±`± with similar kinematics to the leptons in this search.

F SS F SS The number of fakes, then, are Nee = Nee and Nµµ = Nµµ for the electron and muon

SS SS channel respectively. In this context, Nee and Nµµ are event counts that pass all the signal

miss region cuts, except for the ET requirement and the opposite-charge lepton requirement. Very few events are present in both regions, with a single same-sign, dielectron channel event and zero same-sign dimuon events. No events lie within the Z invariant mass peak and so do not aﬀect the ``γγ scale factor estimation in Section 9.5.

92 CHAPTER 9. BACKGROUND ESTIMATION 9.3 Diﬀerent-Flavor Lepton Pair Backgrounds

Decays involving two bosons (WW , WZ, ZZ), top-quarks (that almost exclusively decay to W b), or τ-leptons can result in two independent leptonic decays. These processes are flavor-agnostic, as the branching ratios for leptonic W , Z, and τ decays are very similar for both lepton generations. These backgrounds can be measured using different-flavor, data- driven control regions to provide a complementary estimate to that of direct MC simulation (previously found to be very small and statistics-limited). Two different-flavor control regions (one with opposite-sign and one with same-sign lepton pairs) are used to estimate the impact of these backgrounds, which is then split equally between the two channels. This results in

DF 1 OS SS DF 1 OS SS background estimates Nee = 2 Neµ −Neµ and Nµµ = 2 Neµ −Neµ where DF denotes the background estimate from the different-flavor lepton pair regions and same-sign events are subtracted to prevent fake lepton double-counting. Once again, very few events are present in these control regions, with three opposite- sign, different-flavor eµ events and one same-sign, different-flavor eµ event. The same-sign,

miss different-flavor event is not subtracted as it is well-separated in ET from the opposite-sign, different-flavor events, leading to a conservative value for this (small) background source of

miss 1.5 events for each channel across the entire ET range.

9.4 Jets Misidentiﬁed as Photons

The signal region requires two signal photons that have tight identiﬁcation and are isolated. The number of fakes in the signal region cannot be directly counted as they are indistin- guishable from real photons. Instead, adjacent control regions—where photons failing the signal region requirements can be counted—are used to create an estimate. Pseudo-tight

93 CHAPTER 9. BACKGROUND ESTIMATION

photons (Section 7.3.2) are used to deﬁne this data-driven region, with the photon isolation energy (and the number of isolated and anti-isolated photons) as the discriminating variable. There were insuﬃcient statistics for the pseudo-tight events in MC simulations and data to

iso determine their ET as a function of pT and η. Instead, data and MC samples with similar distributions in these variables were used to determine average behavior, and create a single estimate for fake photons independent of pT and η. Every real photon has a probability α of being isolated and (1-α) of being anti-isolated. Similarly, every fake photon has a probability β of being isolated and (1-β) of being anti-

iso isolated. Photons between the isolated and anti-isolated definitions (those with 0 ≤ ET ≤ 8 GeV) are ignored and the ratio of isolated-to-anti-isolated events is used for the fake factor calculation. With two photons in an event, four measurement types are possible: An event can have isolated-isolated (II); isolated-anti-isolated (IA); anti-isolated-isolated (AI); or anti-isolated- anti-isolated (AA) photon pairs, where photons in an event are ordered by their pT and classified as leading or sub-leading. There are also four “true” photon pair combinations; real-real (RR), real-fake (RF), fake-real (FR) and fake-fake (FF), where real denotes a truth- level photon and fake denotes any non-photon, truth-level object. The measurement types can be expressed in terms of the true photon pair combinations and the probabilities of a real and fake photon being identified as isolated, giving:

2 2 NII = α NRR + αβNRF + βαNFR + β NFF (9.1)

NIA = α(1 − α)NRR + α(1 − β)NRF + β(1 − α)NFR + β(1 − β)NFF (9.2)

NAI = (1 − α)αNRR + (1 − α)βNRF + (1 − β)αNFR + (1 − β)βNFF (9.3) 2 NAA = (1 − α) NRR + (1 − α)(1 − β)NRF (9.4) 2 +(1 − β)(1 − α)NFR + (1 − β) NFF

94 CHAPTER 9. BACKGROUND ESTIMATION

These equations cannot be solved to give the fake rate in the isolated-isolated category without knowing the number of real-real events, which is not possible in a blind analysis. Instead, the simulated ``γγ samples are used as a reliable estimate of real diphotons, and

subtracted from the measured events in the four regions to remove the dependence on NRR. This modiﬁes the event counts such that:

F 2 NII = αβNRF + βαNFR + β NFF (9.5) C C 2 C = f NIA + NAI − f NAA (9.6)

C C C where f ≡ β/(1 − β) is a fake factor, NIA, NAI , and NAA are the ``γγ-corrected event counts, and the negative last term is a correction for double-counting. The fake factor f is estimated with pseudo-tight photons in MC ``γγ samples and data.

1 iso Crystal Ball functions [114] are ﬁt to the photon ET for pseudo-tight ``γγ events. A

iso combined fit on ET is then performed on pseudo-tight photon data with the constrained, real photon Crystal Ball from the fit to the ``γγ events and an unconstrained fake distribution. Finally, the integrals for isolated and anti-isolated photons from the fake fit are used to calculate the fake factor. The distributions of tight and pseudo-tight photon isolation energies and the Crystal Ball fits for the ``γγ are given in Figure 9.1. The tight photons are re-scaled to illustrate the shape differences between the tight photons (red) and pseudo-tight photons (blue), supporting the assumption that the fake factor for pseudo-tight photons can be applied to the tight photons in the signal region. The fitted parameters for the Crystal Ball functions are shown, with

1Crystal Ball functions combine the features of a Gaussian distribution and a power-law tail. It is parameterized with four variables: the mean value µ at which the Gaussian peaks; the standard deviation σ of the Gaussian; the power n to which the tail decays; and α, the crossover or “stitch” point. For the isolation energy ﬁts, values below α exhibit Gaussian-like behavior and values above obey the power law.

95 CHAPTER 9. BACKGROUND ESTIMATION

iso Figure 9.1: The isolation energy ET distributions and Crystal Ball fits for tight identification (red) and pseudo-tight identification (blue) photons in MC ``γγ QCD samples. Events must contain two signal leptons (electrons or muons) but can contain any number of photons as long as at least one is tight or pseudo-tight. The tight photon histogram integral was scaled to the pseudo-tight photon integral to illustrate the relative shape differences of the distributions (a scale factor of 0.08).

96 CHAPTER 9. BACKGROUND ESTIMATION

the parameters for the pseudo-tight Crystal Ball used in the combined ﬁt. The combined ﬁt

iso is shown in Figure 9.2. The fake ﬁt integrals for the isolated and anti-isolated ET are 122.2 and 462.7, giving a fake factor f = 122.2/462.7 = 0.26. Using this fake factor in conjunction with the data and MC event counts in the IA, AI, and AA control regions yields the ﬁnal fake photon estimates. These are calculated per-bin for the dilepton invariant mass (to estimate the ``γγ normalization scale factor [Section 9.5])

miss and for the ﬁnal discriminating variable, ET . The distributions of the invariant mass and

miss ET before scaling are shown in Figures 9.3 and 9.4 for the electron and muon channels respectively.

9.5 ``γγ Normalization Scaling

The cross-sections of the dominant ``γγ background samples were calculated to leading- order, resulting in an underestimation. To account for this, a control region is used to ﬁt

miss and scale the MC samples to data, defined by a ET ≤ 50 GeV cut and a dilepton invariant mass within the Z mass peak (81 ≤ M`` ≤ 101 GeV). The fit is performed with HistFitter [115], a statistical data analysis framework built on RooFit [116]. Fake lepton sources do not contribute to the Z-peak region, but fake photons are present and depend on the ``γγ normalization scale factor. The fit is initially performed with the unscaled fake photon bin estimates and iteratively refit until the fake photon estimates converge to two decimal places. The likelihood fits in the Z-mass peak for the electron and muon channel are shown in Figures 9.5 and 9.6 respectively. The reweighted ``γγ samples are compared to data by defining a validation with the

miss same ET ≤ 50 GeV requirement with an additional Z-mass veto. This validation region for both channels was gradually extended until the number of observed data events were on

97 CHAPTER 9. BACKGROUND ESTIMATION

Figure 9.2: A combined fit (blue) formed of two separate Crystal Ball functions. The real pseudo-photon fit (red) is constructed from the CBShape pseudo-photon fit result parameters from Figure 9.1, while the fake fit (green) is unconstrained with all parameters free. Events do not need to pass signal lepton requirements, but there must be at least two leptons with miss VeryLoose identification and a pT ≥ 20 GeV. The isolated (ET ≤ 0 GeV) and anti-isolated miss (8 ≤ ET ≤ 80 GeV) integrals from the unconstrained fake fit are used in calculating the fake factor for photons.

98 CHAPTER 9. BACKGROUND ESTIMATION

Figure 9.3: Data and ``γγ MC event counts in the isolation pair control regions, before normalization scaling, for the electron channel. Dilepton invariant masses are on the left miss and ET is on the right, with plots ordered by pair type: Iso-Anti/IA (top), Anti-Iso/AI (middle), and Anti-Anti/AA (bottom). These event counts are used with the fake factor f to determine the number of fake photons.

99 CHAPTER 9. BACKGROUND ESTIMATION

Figure 9.4: Data and ``γγ MC event counts in the isolation pair control regions, before normalization scaling, for the muon channel. Dilepton invariant masses are on the left miss and ET is on the right, with plots ordered by pair type: Iso-Anti/IA (top), Anti-Iso/AI (middle), and Anti-Anti/AA (bottom). These event counts are used with the fake factor f to determine the number of fake photons. 100 CHAPTER 9. BACKGROUND ESTIMATION

Figure 9.5: Background estimates in the electron channel for ``γγ and fake photon backgrounds in the Z-mass (81 − 101 GeV) before fitting (above) and post-fit (below). The fit was done iteratively until the fake photon estimates converged to two decimal places.

101 CHAPTER 9. BACKGROUND ESTIMATION

Figure 9.6: Background estimates in the muon channel for ``γγ and fake photon backgrounds in the Z-mass (81 − 101 GeV) before fitting (above) and post-fit (below). The fit was done iteratively until the fake photon estimates converged to two decimal places.

102 CHAPTER 9. BACKGROUND ESTIMATION

the order of 1 or 0 for several bins. For both selections, background estimates and data were

miss observed up to ET ≤ 80 GeV and the two signal regions were deﬁned with an additional

miss ET separation above 80 GeV to minimize the ``γγ background. For the electron channel,

miss miss ET ≥ 100 GeV was chosen, and ET ≥ 110 GeV was chosen for the muon channel. The

miss validation region background estimates are shown as part of the ﬁnal ET plots in Figure 11.1 for the electron channel and Figure 11.2 for the muon channel.

103 Chapter 10

Systematic and Statistical Uncertainties

Systematic uncertainties arise whenever an experimental measurement is made, due to imper- fect knowledge of the measurement’s true value. The majority of experimental uncertainties in ATLAS analyses result from object reconstruction and calibration, in addition to pileup

miss reweighting, luminosity, and ET uncertainties. Systematic uncertainties are parameterized as independent nuisance parameters that are normally-distributed, small, and symmetric about a nominal measurement value. Perfor-

miss mance group for each physics object (i.e, the E/Gamma, Muon, Jet/ET , and Flavour Tagging groups) proﬁle these uncertainties and provide ±1σ variations for each reconstruction scale factor and energy calibration applied. Scale factor uncertainties are calculated for the relevant object and the two variations are stored as weighted variables in the nominal analysis output ﬁle. Energy scale and resolution uncertainties are applied at the event-level, with a single uncertainty applied to all relevant objects (e.g., the EGammaCalibTool energy scale uncertainty for electrons and photons) in

104 CHAPTER 10. SYSTEMATIC AND STATISTICAL UNCERTAINTIES an event before performing the analysis selection. This output is stored as a separate file and the process is repeated for the remaining uncertainties as variations on the nominal selection. Different correlation models (specific to each performance group) account for possible correlations in pT and η between objects for a given nuisance parameter, to varying degrees of accuracy. The most complete models can have several hundred parameters (representing each η×pT bin, and a dozen or so correlated uncertainties between them), which are useful for minimizing the total uncertainty for precision measurements at the expense of computation time. This analysis is the first of its kind—probing many different signal scenarios with a spread of kinematic distribution on the final state objects—and the estimated SM background in the signal region is very small. As such, it is not particularly sensitive to any single uncertainty and the simplest correlation model for each object performance tool is chosen. These models typically sum each uncertainty source for a given measurement in quadrature and provide a conservative overestimate of the total systematic uncertainty. The systematics sources for specific objects are discussed in Sections 10.1, 10.2, 10.3,

miss and 10.4; pileup, luminosity and ET are discussed in Section 10.5; and the statistical uncertainty for MC samples is described in Section 10.6.

10.1 Electron and Photon Uncertainties

A typical 100 GeV electron/photon can deposit 3 − 20% of its energy before reaching the calorimeters, in addition to 5% of its energy outside of its EM cluster [117]. EM clusters are ﬁrst calibrated to the object’s original energy using a multivariate technique based on MC simulations, using information such as the longitudinal shower shape and ID track features

105 CHAPTER 10. SYSTEMATIC AND STATISTICAL UNCERTAINTIES

[117]. The longitudinal EM layers (i.e., the presampling, first, and second layers) are then calibrated relative to one another using simulated Z → µµ decays, before applying corrections to account for regions with non-optimal high voltages and geometrical defects. Z → ee decays are then used to compare MC simulation to data to apply a final correction, before the calibration is cross-checked with J/ψ → ee and Z → ``γ events in data. Each of these steps provides an uncertainty source that is profiled by the energy calibration tool and applied to the event selection as a total energy scale and total energy resolution uncertainty (using the 1 NP v1 correlation model). Finally, electrons and photons simulated using Atlfast-II are compared to full simulation objects and an additional energy scale uncertainty is calculated. Both electrons in an event are reweighted with a reconstruction, identification, and isolation scale factor, with the ±1σ variations calculated by the separate efficiency correction tools (using the TOTAL correlation model). A similar procedure is done for photons and their identification and isolation scale factors, with an additional isolation correction term factored in to the photons’ isolation scale factor uncertainty. A global trigger scale factor for the electron pair is calculated by the TrigGlobalEfficiencyCorrectionTool to provide an overall trigger scale factor and uncertainty for the event.

10.2 Muon Uncertainties

Muon momentum is calibrated to J/ψ → µµ and Z → µµ events in data, with the pT of individual tracks corrected for inaccuracies in the magnetic ﬁeld description, detector dimensions and calorimeter energy loss. These correction parameters are extracted using a likelihood ﬁt to data with simulated MC samples. Momentum resolution is also estimated using data, with the MC smeared such that the dimuon invariant mass peak for both decays agrees with data. These corrections provide an energy scale and resolution uncertainty for

106 CHAPTER 10. SYSTEMATIC AND STATISTICAL UNCERTAINTIES signal muons (using the default correlation model). MS/ID alignment is additionally studied using special runs with no magnetic field to derive another correction (and uncertainty). Both signal muons must pass Tight identification and have a combined ID and MS track, with the muon reconstruction efficiency tool providing separate ID and MS uncertainties as well as an additional overall uncertainty from the combination procedure (using the default correlation model). Statistical uncertainties from the tag and probe reconstruction method are also incorporated as separate uncertainties on the muons. Finally, isolation scale factors for the muons are calculated, with a systematic and statistical uncertainty provided by the muon isolation efficiency tool.

10.3 Jet Uncertainties

Reconstructed jet energies need to be calibrated in order to account for energy lost in the absorber material, energy leakage outside of the calorimeters, and energy deposits below the energy thresholds of the detector sub-components. Jet calibration occurs in several steps [118]. First, the individual topo-clusters in a jet are calibrated to the energy scale of the EM shower using MC simulations. Next, an origin correction is applied to the jet that changes its direction to point towards the primary vertex. A pile-up energy correction is then applied based on the average energy due to pile-up (from measuring the median energy density in the calorimeter within |η| ≤ 2.0) multiplied by the jet area. A scale factor for the jet energy scale is then calculated by comparing the reconstructed jet energies at this stage with the true jet energies in simulated MC samples. A global sequential correction [119] scheme—using information such as the energy deposit topology, associated ID tracks, and muon spectrometer activity—is applied next to account for the diﬀerent energy responses between gluon and light-quark jets. Finally, several jet

107 CHAPTER 10. SYSTEMATIC AND STATISTICAL UNCERTAINTIES

energy responses are found by measuring the pT balance of jets to reference objects (e.g. photons, Z bosons, and other jets) in data. Each of these steps is a source of systematic uncertainty on the jet energy scale and resolution. These uncertainties are combined into the smallest possible subset of nuisance parameters using the R4 SR Scenario1 SimpleJER.conﬁg correlation model ﬁle, parameter- izing the uncertainties as 6 JES and 8 JER nuisance parameters. This discards substantial correlation information and overestimates the total uncertainty from jet calibration, but minimizes computation time.

10.4 Flavor-Tagging Uncertainties

Uncertainties on the b-tagging eﬃciency are originally estimated by individually varying each systematic in an analysis and comparing the performance of the b-tagging DL1 discriminant, after each variation, to the nominal value. This quickly becomes prohibitive as the number of systematics increase, and a method for reducing the total number of b-tagging uncertainties was developed by the Flavour Tagging group. The total number of uncertainties is reduced (while preserving bin-by-bin correlations) through an eigenvector decomposition on the co- variance matrix sum for each source of uncertainty [99], reducing the number of uncertainty parameters to 16.

miss 10.5 Pileup, Luminosity, and ET Uncertainties

The PileupReweightingTool calculates variations on its scale factor reweighting and are implemented as an event-level uncertainty. The uncertainty in the combined 2015–2018 integrated luminosity is 1.7 % [120], obtained using the LUCID-2 detector [121] for the primary

108 CHAPTER 10. SYSTEMATIC AND STATISTICAL UNCERTAINTIES

luminosity measurements, and is implemented as an overall scale factor.

miss The ET is built from all hard physics objects in an event and an additional soft term.

miss As such, the majority of the uncertainty on ET is already accounted for by the systematic variations on these objects. The only term that is not accounted for is the soft track term;

miss this is calculated by the Jet/ET group from two in-situ methods and implemented as

miss (one-sided) perpendicular and parallel ET resolutions.

10.6 Statistical Uncertainties

For simulated samples, statistical uncertainties are calculated per bin and scaled with the appropriate scale factors. Signal samples that are sensitive to discovery/exclusion have a large number of simulated events and a statistical uncertainty very small relative to their

miss systematic uncertainties, while the statistical uncertainties are larger for the higher ET ``γγ bins.

miss The number of simulated events and their statistical uncertainty per ET bin are shown in Table 10.1 for the ``γγ background in both channels. Statistical uncertainties are summed in quadrature with the total systematic uncertainty to calculate the total uncertainty.

109 CHAPTER 10. SYSTEMATIC AND STATISTICAL UNCERTAINTIES

Table 10.1: Number of simulated events for the ``γγ√ MC samples in the electron and muon miss channels, along with their statistical uncertainty ( N) for each ET bin shown in Figures 11.1 and 11.2 (before scaling).

miss ET Range [GeV] eeγγ MC µµγγ MC 0-10 7102 ±84 7456 ±86 10-20 8409 ±92 8722 ±194 20-30 5062 ±71 5163 ±72 30-40 2540 ±50 2826 ±53 40-50 1255 ±35 1405 ±37 50-60 595 ±24 742 ±27 60-70 269 ±16 343 ±19 70-80 126 ±11 178 ±13 80-90 56 ±7 69 ±8 90-100 24 ±5 53 ±7 100-110 14 ±4 27 ±5 110-120 7 ±3 20 ±4 120-130 4 ±2 9 ±3 130-140 1 ±1 6 ±2 140-150 0 5 ±2 150-160 1 ±1 4 ±2 160-170 1 ±1 2 ±1 170-180 1 ±1 1 ±1 180-190 1 ±1 0 190-200 0 3 ±1 200-210 0 2 ±1 210-220 0 0 220-230 0 4 ±2 230-240 0 0 240-250 0 0 250-260 0 1 ±1 260-270 0 0 270-280 0 1±1 280-290 0 0 290-300 0 0 ≥ 300 0 10 ±3

110 Chapter 11

Results and Conclusion

miss The full signal region ET distributions after unblinding are shown in Figures 11.1 and 11.2 for the electron and muon channels respectively. The total SM background from ``γγ QCD and fake lepton and photon sources are stacked in the solid-color histogram, with the total SM uncertainty represented by a hashed area. Observed data events are shown as black markers with vertical Poisson error bars. Three representative signal sample distributions are displayed as colored-line histograms with their labels corresponding to the mass parameters

(M`˜, Mχ˜, MP GLD). Overflow bins are visible as the final bin at 290 − 300 GeV. The data- to-SM ratio is shown at the bottom of each figure, with empty data bins and ratios above 2.0 not shown. Three events are observed in the electron channel signal region and zero events are ob-

+0.17 served in the muon channel, with 0.06−0.06 expected background events for the electron

+0.73 channel and 0.77−0.33 expected background events for the muon channel.

111 CHAPTER 11. RESULTS AND CONCLUSION

miss Figure 11.1: Event counts in the electron channel for the ﬁnal discriminant ET . The total SM background from ``γγ QCD and fake sources is stacked in the solid-color histogram, with their total uncertainty represented by the hashed area. Data events are illustrated by black circle markers, and the data-to-SM ratio is shown at the bottom, with bins containing zero events left empty in both cases. Three representative signal samples are overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD).

112 CHAPTER 11. RESULTS AND CONCLUSION

miss Figure 11.2: Event counts in the muon channel for the ﬁnal discriminant ET . The total SM background from ``γγ QCD and fake sources is stacked in the solid-color histogram, with their total uncertainty represented by the hashed area. Data events are illustrated by black circle markers, and the data-to-SM ratio is shown at the bottom, with bins containing zero events left empty in both cases. Three representative signal samples are overlaid and labeled by their mass parameters (M`˜, Mχ˜, MP GLD).

113 CHAPTER 11. RESULTS AND CONCLUSION 11.1 Exclusion Limit Calculation on Signal Samples

The excess in the electron channel is not suﬃciently signiﬁcant to be interpreted as a discovery and no excess is present in the muon channel. Instead, exclusion limits are placed on the signal samples and the excess is assumed to only be from a possible signal, as this provides the most conservative estimate for exclusion limits on the number of signal events.

For an expected number of events µ0, the Poisson probability of observing events n events is: e−µ0 µn P (n |µ ) = 0 (11.1) obs 0 n!

If nobs events are observed, an upper limit on µ0 at a conﬁdence level CL is given by:

P (n ≤ nobs|µ0) = 1 − CL (11.2)

In this search, all limits are set to a 95% CL which is the standard prescription for ATLAS exclusion limits. The above formulation is a simpliﬁcation and does not take into account the uncertainty on the value of µ0. The true value of the expected number of signal events µt is µt = (1±δ)µ0, where δ is a nuisance parameter. If δ is small compared to 1, it can be parameterized as a Gaussian distribution with mean 0 and a standard deviation σ equal to the total systematic uncertainty. The original Poisson distribution is convolved with the Gaussian distribution of δ in order to take this into account:

−µ n e t µ 1 −δ2 t 2 P (n|µt, σ) = · √ e 2σ (11.3) n σ 2π

This expression is integrated with respect to δ (the lower limit of integration is extended to

114 CHAPTER 11. RESULTS AND CONCLUSION

-∞ with negligible change for the values of σ considered here), then evaluated for speciﬁc values of n: ∞ −µ n Z e t µ 1 −δ2 ˜ t 2 P (n|µt, σ) = dδ · √ e 2σ (11.4) −∞ n! σ 2π

δ where µt = (1 + δ)µ0. We make the substitution x = σ to simplify the integral to:

n −µ0 Z ∞ 2 µ0 e −σx n − x √ dx e (1 + σx) e 2 (11.5) 2πn! −∞

Solving for individual n values yields:

2 2 µ0σ ˜ −µ0 P (0|µ0, σ): e e 2 (11.6) 2 2 µ0σ ˜ 2 −µ0 P (1|µ0, σ): −µ0(σ − 1)e e 2 (11.7)

µ2σ2 1 0 ˜ 2 2 2 2 −µ0 2 P (2|µ0, σ): 2 µ0 (σ − 1) + σ e e (11.8)

µ2σ2 1 0 ˜ 3 2 2 2 2 −µ0 2 P (3|µ0, σ): − 6 µ0(σ − 1) 3σ + (σ − 1) e e (11.9)

The upper limit on µ0 at conﬁdence level CL is given by:

˜ P (n ≤ nobs|µ0, σ) = 1 − CL (11.10)

These results can now be used to calculate the 95% conﬁdence level upper limits for the expected number of events in the electron and muon channels using the σ values for each channel and solving numerically for µ0 (done with Mathematica [122]). The systematic uncertainties on the signal samples in each channel are relatively uniform as a percentage of the number of events—with a small spread—and are 6.3% and 17.6% for the electron and

115 CHAPTER 11. RESULTS AND CONCLUSION muon channel respectively. For the electron channel, the upper limit is:

˜ ˜ P (0|µ0, 0.063) + P (1|µ0, 0.063) + (11.11) ˜ ˜ P (2|µ0, 0.063) + P (3|µ0, 0.063) = 0.05 (11.12)

e =⇒ µ0 = 7.83 (11.13) and for the muon channel:

˜ P (0|µ0, 0.176e) = 0.05 (11.14)

µ =⇒ µ0 = 3.15 (11.15)

These values can be compared to the event yields for the signal samples in the signal regions to determine which are excluded with 95% conﬁdence. As stated in Chapter3, the signal samples were generated with a K-factor such that the PGLD branching ratio was 0.5. By using the event yields for BR= 0.5—along with the truth-level information of the ﬁnal state decays for each event (double PGLD, double GLD, or asymmetric)—event yields can be calculated for a range of BRs. One of three statements can be made for each signal sample: Excluded if the yields are above the 95% CL upper limit for every BR; BR-dependent if the upper limit threshold is crossed at a certain BR value; and Not excluded if the yields are always below the 95% CL upper limit for every BR. These results are shown for the electron channel in Table 11.1 and in Table 11.2 for the muon channel. There are four BR-dependent samples for each channel; these samples’ event yields are plotted as a function of BR and are shown in Figure 11.3 for the electron channel and Figure 11.4 for the muon channel. For cases where the exclusion is BR-dependent, a lower BR exclusion limit has been calculated from a linear interpolation. An exception to

116 CHAPTER 11. RESULTS AND CONCLUSION

Table 11.1: Event yields for every selectron signal sample, for the simulated PGLD branching ratio (BR) of 0.5 and the limiting BR= 0.0 and BR= 1.0 cases. Samples where the yields are above the upper limit of expected signal events for all BRs are Excluded; those that cross the upper limit threshold are BR-dependent; and those that are always below it are Not excluded. BR-dependent sample yields as a function of BR are additionally shown in Figure 11.3.

Yields

M`˜ [GeV] Mχ˜ [GeV] MPGLD [GeV] BR = 0.5 BR = 1.0 BR = 0.0 Result 200 150 100 42.14 26.17 55.61 Excluded 300 225 150 20.61 15.82 23.68 400 200 0 11.33 11.57 11.77 400 250 100 10.90 10.47 11.67 200 20 0 42.86 42.82 44.31 200 95 50 52.49 47.09 57.55 300 30 0 19.38 19.53 18.68 300 142.5 75 24.55 23.11 25.44 400 40 0 8.81 8.55 9.04 400 190 100 10.83 10.40 11.09 200 175 150 18.68 6.00 29.99 BR ≤ 0.94 excluded 300 262.5 225 10.61 4.71 16.35 BR ≤ 0.74 excluded 400 300 200 8.95 6.90 10.50 BR ≤ 0.77 excluded 200 187.5 150 8.86 3.40 12.85 BR ≤ 0.61 excluded 400 350 300 5.07 2.44 6.97 Not excluded 500 250 0 5.12 4.96 5.13 500 312.5 125 4.94 4.75 5.23 500 375 250 4.17 3.49 4.57 500 437.5 375 2.61 1.53 3.27 300 277.5 225 5.52 2.70 7.28 400 370 300 2.96 1.97 3.48

117 CHAPTER 11. RESULTS AND CONCLUSION

Table 11.2: Event yields for every smuon signal sample, for the simulated PGLD branching ratio (BR) of 0.5 and the limiting BR= 0.0 and BR= 1.0 cases. Samples where the yields are above the upper limit of expected signal events for all BRs are Excluded; those that cross the upper limit threshold are BR-dependent; and those that are always below it are Not excluded. BR-dependent sample yields as a function of BR are additionally shown in Figure 11.4.

Yields

M`˜ [GeV] Mχ˜ [GeV] MPGLD [GeV] BR = 0.5 BR = 1.0 BR = 0.0 Result 200 150 100 31.23 21.44 38.45 Excluded 200 175 150 15.32 4.75 26.41 300 225 150 14.73 11192 17.67 300 262.5 225 7.93 3.51 11.77 400 200 0 7.05 7.56 6.85 400 250 100 7.08 6.74 7.37 400 300 200 6.56 5.67 7.28 200 20 0 29.13 28.45 31.63 200 95 50 36.51 28.24 43.13 200 187.5 150 8.79 3.27 12.49 300 30 0 12.69 12.92 12.49 300 142.5 75 15.80 15.49 16.21 400 40 0 5.28 5.60 5.20 400 190 100 6.59 6.20 6.84 400 350 300 4.00 2.13 5.78 BR ≤ 0.74 excluded 500 250 0 3.14 3.24 3.14 —— 500 312.5 125 3.16 3.11 3.38 —— 300 277.5 225 5.25 2.63 7.08 BR ≤ 0.92 excluded 500 375 250 2.83 2.42 2.99 Not excluded 500 437.5 375 2.02 1.28 2.49 400 370 300 2.51 1.53 3.21

118 CHAPTER 11. RESULTS AND CONCLUSION

Figure 11.3: A plot of the event yields as a function of the PGLD branching ratio (BR) for the four selectron signal samples, whose exclusion is dependent on the choice of BR. The signal samples are labeled by their mass parameters (M`˜, Mχ˜, MP GLD) and are ordered by their event yields at BR=0. The 95% conﬁdence level (CL) upper limit is shown as a solid red line at 7.83 events, with branching ratios resulting in event yields higher than this value excluded.

119 CHAPTER 11. RESULTS AND CONCLUSION

Figure 11.4: A plot of the event yields as a function of the PGLD branching ratio (BR) for the four smuon signal samples, whose exclusion is dependent on the choice of BR. The signal samples are labeled by their mass parameters (M`˜, Mχ˜, MP GLD) and are ordered by their event yields at BR= 0. The 95% conﬁdence level (CL) upper limit is shown as a solid red line at 3.15 events, with branching ratios resulting in event yields higher than this value excluded.

120 CHAPTER 11. RESULTS AND CONCLUSION

this are the (M`˜, Mχ˜, MP GLD) = (500, 250, 0) and (M`˜, Mχ˜, MP GLD) = (500, 312.5, 125) smuon signal samples, which are nearly independent of BR and are always close to the 95% CL limit. A BR exclusion limit is not included for these two samples due to this behavior. All selectron signal samples with a slepton mass of 200 GeV are excluded, with the

exception of (M`˜, Mχ˜, MP GLD) = (200, 175, 150) GeV for BRs above 0.94 and

(M`˜, Mχ˜, MP GLD) = (200, 187.5, 150) GeV for BRs above 0.61. These are the most compressed scenarios at 200 GeV and are experimentally difficult to probe, as their yield efficiencies suffer from even the modest 20 GeV selections on the leading and sub-leading leptons. A similar result in seen for smuon samples with a slepton mass of 300 GeV; all are excluded except for the most compressed scenario of (M`˜, Mχ˜, MP GLD) = (300, 277.5, 225) GeV at BRs above 0.92. Samples with a slepton mass of 400 and 500 GeV have smaller production cross-sections— 0.38 fb and 0.14 fb respectively, compared to 6.58 fb and 1.32 fb for slepton masses of 200 and 300 GeV—and the event yields for many samples were expected to be marginal if a handful of events were observed in data. This was the case in the electron channel and only

the least compressed M`˜ = 400 GeV scenarios were excluded, with no exclusion at all for the

M`˜ = 500 GeV samples. The muon channel is more exclusionary, with the majority of M`˜ =

400 GeV scenarios excluded and two M`˜ = 500 GeV that hover around the 95% upper limit. If the three observed events in the electron channel are not a statistical ﬂuctuation, continued asymmetry between the number of events in each channel as more collision events are recorded would also indicate a mass hierarchy between the selectron and smuon, with the selectron being noticeably lighter. The parameter space for the signal sample models is complex and this analysis cannot place strong, unconditional limits across its multi-dimensional span. However, it does provide exclusion limits on the majority of the generated signal samples and, as the ﬁrst analysis to

121 CHAPTER 11. RESULTS AND CONCLUSION probe these mssm-goldstini models, can be used as a guiding template for future searches.

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