Searches for Dark Matter Production in Two Fixed Target Experiments: HPS and LDMX A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Thomas Brice Eichlersmith IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Jeremiah Mans Jan, 2025 © Thomas Brice Eichlersmith 2025 ALL RIGHTS RESERVED Acknowledgements There are many people that have earned my gratitude for their contribution to my time in graduate school. While I will struggle to name them all, I do wish to specifically thank those people who have helped teach me their knowledge of physics along the way: Tim Nelson, Cam Bravo, Matt Graham, PF Butti, Omar Moreno, Philip Schuster, Natalia Toro, Stefania Gori, Lene Kristian Bryngemark, Einar Elén, Erik Wallin, Christian Herwig, Nhan Tran, Cristina Mantilla Suarez, Lauren Tompkins, Valentina Dutta, Ruth Pöttgen, and Alic Spellman. Of particular note is my advisor Jeremy Mans. Professor Mans has been wonderfully flexible as I have changed course while navigating the uncertain waters of graduate school. Thank you. Finally, there are many friends who I have met while in graduate school or with whom I have gotten closer. Your support outside of physics has been necessary for me over the years and I am in your debt. i Dedication To my friends and family trying to comprehend anything I have done over the past six and a half years, I hope this helps. To a future graduate student desparately trying to understand these experiments and their code, I also hope this helps and I wish you luck. ii Abstract Astrophysical evidence strongly indicates the presence of particulate Dark Matter (DM) within our universe; however, the specific particle nature of DM is still unknown. The wide variety of possible DM particles produces a similar range of experiments focused on probing these different categories of possible DM. This work describes two experiments taking different approaches to search for Light DM residing in the 1MeV-1GeV mass range being produced by electron interactions. The Light Dark Matter eXperiment (LDMX) is a proposed fixed target experiment designed for a missing momentum search with an additional, orthogonal missing energy search channel described here. The Heavy Photon Search experiment (HPS) is another fixed target experiment designed for a displaced vertex search with distances of O(10cm) which are not probed by longer baseline experiments. Specifically, a search in HPS data for a specific Light DM model with a strongly-coupled dark sector enabling a higher expected production rate while keeping the characteristic decay length within HPS’s acceptance is also presented. iii Contents Acknowledgements i Dedication ii Abstract iii Contents iv List of Tables vii List of Figures ix I Laying the Groundwork 1 1 Introduction 2 1.1 Standard Model of Particle Physics . . . . . . . . . . . . . . . . . . . . . 4 1.2 Standard Model Analogies . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Particle Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Displaced Particle Decays . . . . . . . . . . . . . . . . . . . . . . 9 2 Dark Matter 12 2.1 Dark Matter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Evidence for Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Particle Nature of Dark Matter . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Invisible Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 iv 2.5 Visible Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 II LDMX 22 3 Light Dark Matter eXperiment 23 3.1 Missing Momentum Signature . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 The Beam Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Detector Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4 Mid-Shower Simulation 31 4.1 General Data Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.1.1 Data Processing Stages . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.2 Biasing and Filtering Technique . . . . . . . . . . . . . . . . . . 34 4.2 Standard Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.1 Dominant Contributors in this Analysis . . . . . . . . . . . . . . 37 4.2.2 Expanding Production of these Processes . . . . . . . . . . . . . 39 4.3 Dark Matter Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3.1 G4DarkBreM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3.2 MadGraph/MadEvent . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3.3 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Missing Energy Search 46 5.1 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Background Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3 Systematics and Background Uncertainty . . . . . . . . . . . . . . . . . 55 5.4 Reach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 III HPS 61 6 The Heavy Photon Search Experiment 62 6.1 Strongly Interacting Massive Particles . . . . . . . . . . . . . . . . . . . 62 6.2 Detector Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 v 6.2.1 Silicon Vertex Tracker . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2.2 Electromagnetic Calorimeter . . . . . . . . . . . . . . . . . . . . 67 7 Alignment 70 7.1 Alignment Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.2 Detector Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8 Data Set 80 8.1 Collected Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.1.1 Pair 1 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.3 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 8.4 Analysis Pre-Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 9 Displaced Vertex Search 88 9.1 Signal Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 9.2 Reconstruction Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.3 Physics Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.4 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 9.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 9.5.1 Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 9.5.2 Sensitivity and Exclusion . . . . . . . . . . . . . . . . . . . . . . 103 IV Conclusion 106 10 Conclusion 107 10.1 Simulation Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 10.2 Tracking Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.3 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References 110 vi List of Tables 4.1 Parameters of mid-shower biased samples. In all of the samples used in this study, we use the same value for the Sorting and Biasing Thresholds. 36 4.2 Configuration of the simulation samples used in this analysis. Ebeam is the beam energy being studied (4 or 8 GeV in this work). mA is the mass of the A’ in MeV, ϵ is the dark brem mixing strength, EECal Front primary is the energy of the primary electron at the front of the ECal, EA′ is the energy of the generated A’, Etot nuc is the total energy transferred to nuclear interactions during the event, and Etot µ is the total energy of produced muons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1 Cut-flow analysis comparing background and various signal hypotheses for the simple cuts used in this analysis. The event yield for the back- ground sample is calculated using the event weights and represent the number of events out of 1013 EoT equivalent. The signal efficiency is rel- ative to the full simulation sample. The efficiency and event yield values on a given row are for after the analysis stage of that row. The first table is the cutflow for the 4GeV beam, and the second table is for the 8GeV beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.1 Values for cuts used within the Pair 1 Trigger during data collection of the HPS 2016 dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 9.1 Polynomial coefficients for radiative fraction and acceptance functions. 91 9.2 Parameters for Equation (9.14). . . . . . . . . . . . . . . . . . . . . . . . 100 vii 9.3 Region definitions for use in background estimation via sidebands. Region F is the signal region in which we are searching for an excess. mVD is the mass point we are searching for, σm is the detector mass resolution evaluated at mVD , ycut0,min is the optimized cut value evaluated at mVD , and yfloor0,min is the maximum value of y0,min such that region C has at least one thousand events in it. . . . . . . . . . . . . . . . . . . . . . . . . . 102 viii List of Figures 1.1 Comic #2351 from XKCD[1]. If able to be designed from scratch, the names of things in the standard model could be more helpful. . . . . . . 2 1.2 The Standard Model of Particle Physics showing the twelve fermions and five bosons, their various properities (mass, charge, spin), labels (box and circle colors), and interactions (brown loops). Credit to Cush on wikipedia for providing this diagram freely accessible and usable for any purpose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Feynman diagram for the bremsstrahlung process where a charged lepton (ℓ−) interacts with a nucleus (Z) via a photon (γ) and then emits another photon before recoiling. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Energy of the outgoing electron (Ee) for standard processes (dominated by bremsstrahlung) in gray and some dark bremsstrahlung (see Sec- tion 2.4) in colors. Figure 3 from [2]. . . . . . . . . . . . . . . . . . . . 8 1.5 Dipiction of neutral kaon and anti-kaon mixing including the underlying Feynman diagram that results from applying the Standard Model (SM) to this system. This figure was created by user NikNaks on Wikipedia and is licensed under CC Attribution-Share Alike 3.0 Unported. . . . . 10 1.6 Image of CERN’s first liquid hydrogen bubble chamber from 1960 [3] with pink annotations added. The pink circle highlights the location described in the text where a Λ baryon decays into two particles (most likely a proton and a negative pion). The pink dashed line traces out a possible path the Λ traveled before its decay. . . . . . . . . . . . . . . . 11 2.1 Comic #1758 from XKCD[1]. Generally, claiming that either Einstein or Newton was wrong will be a bad time. . . . . . . . . . . . . . . . . . . . 12 ix https://commons.wikimedia.org/wiki/File:Kaon-box-diagram-with-bar.svg https://creativecommons.org/licenses/by-sa/3.0/deed.en 2.2 Depiction of the velocity of stars within a galaxy as a function of their distance from the galactic center. The dotted line is a prediction of this relationship using General Relativity (GR) along with the mass tabulated from the visible starts while the data points (and the solid line fitted to them) are what are actually seen in galaxies today. . . . . . . . . . . . 15 2.3 From [4], the co-moving cosmological number density of Dark Matter (DM) as a function of the universe temperature. As the universe cools, the number density decreases until the DM becomes too sparse to interact with other DM particles, “freezing” to a specific number density until today. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Mass scale of Thermal Relic DM. The regions in red are excluded by ap- plying the thermal relic assumption to our observations of the universe’s early evolution. me is the electron mass and mp the proton mass. . . . 18 2.5 Feynman diagram of how a massive field Φ could allow for a standard photon (γ) to mix with a dark photon (A′). . . . . . . . . . . . . . . . . 18 2.6 Feynman diagram for the dark brem process. . . . . . . . . . . . . . . . 19 3.1 Diagram showing relative rates of background processes within LDMX along with how they motivate various aspects of the design. /E stands for “missing” energy or energy that is “lost” to neutrinos that are extremely unlikely to be detectable within LDMX. . . . . . . . . . . . . . . . . . . 24 3.2 Diagram of LDMX detector apparatus with a representation of a signal event where a dark brem occurs within the target. Credit to Christian Herwig for original development of diagram. . . . . . . . . . . . . . . . 28 3.3 Rendering of LDMX detector apparatus focusing on tracker, target, and ECal. The magnet would fully encompass the tracker, target and trigger scintillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 Diagram of LDMX Electromagnetic Calorimeter (ECal) design showing the longitudinal segmentation (top and bottom right) and the transverse segmentation (bottom left). Credit to Joe Muse. . . . . . . . . . . . . . 30 x 4.1 Flow chart of how data is processed in the Light Dark Matter eXperiment (LDMX) framework. Each processor has the ability to “add” data to the event as well as “get” data from the event. The processors are run in a user-defined sequence. Data can also be loaded from one or more input data files where the event data from those files is loaded into memory before the first processor is run. After all processors are done with an event, it is saved to the output data file. . . . . . . . . . . . . . . . . . . 32 4.2 Diagram of processing stages within LDMX showing the external sources of data or software that are used in those stages as well as which stages are commonly done within the centralized processing framework. . . . . 34 4.3 The total reconstructed energy within the ECal as a fraction of the beam energy separated by the total nuclear energy within the event (the energy going into the photon-nuclear and electron-nuclear processes). . . . . . . 39 4.4 Fraction of hits within the Hadronic Calorimeter (HCal) that were caused by muons or anti-muons. The events lying within the zero bin still have HCal hits caused by other types of particles. . . . . . . . . . . . . . . . . 40 4.5 Longitudinal location (z) where the µ+ was produced within the simu- lation for a variety of biasing factors B for the 4 GeV beam (left) and 8 GeV beam (right). The rate R is the equivalent number of unbiased Electrons on Target (EoT) divided by the CPU time necessary to pro- duce the sample. We can observe that while increasing B also improves the rate R, we eventually over-bias and lose access to the late tail of the distribution (red, puple, and brown in the left plot and gray in the right plot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.6 ECal reconstructed energy EECAL as a fraction of the beam energy EBeam comparing the unbiased and biased samples scaled to the same EoT. The 4 GeV (8 GeV) beam case is shown on the left (right). . . . . . . . . . . 42 4.7 Distributions of simulated signal events with a 4GeV beam. The energy of the produced dark photon is shown on the left while the longitudinal location where the production occurred is on the right. . . . . . . . . . . 44 xi 5.1 The total reconstructed energy in all layers of the ECal (EECal). The sig- nal and background distributions are normalized such that their integral is one. The 4GeV beam is shown on the left and the 8GeV beam is shown on the right. The events falling into bins with EECal > 1.5GeV (3.16GeV) for 4GeV (8GeV) are omitted from this plot but included in efficiency calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 Variables used in signal selection for the ECal as Target (EaT) analysis channel. In each figure, all other cuts are applied except for the variable in question. Some of the bins are empty in which case the upper Poisson limit given the total sample size is drawn with error bars. The grey line shows the selection cut. The background sample is the enriched nuclear and dimuon samples. The top (bottom) row shows the 4GeV (8GeV) beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.3 Studies of how the tracker requirement affect the simulation samples. . . 52 5.4 The total reconstructed energy in all layers of the ECal (EECal) as a fraction of the beam energy (EBeam) for all samples that pass the selection criteria except the selection on ECal energy. The 4GeV beam is shown on the left and the 8GeV beam is shown on the right. The gray lines mark the edges of the analysis bins used to estimate the expected exclusion limit and the black line is the upper limit on the ECal energy which also serves as the upper limit of an analysis bin. . . . . . . . . . . . . . . . . 53 5.5 Exponential fit of the total simulated background distribution. The shaded region is a 95% confidence band on the fit which is interpreted as the uncertainty on the value of the fit. Some bins in the simulated distribution are empty in which case the upper Poisson limit given the sample size is drawn as an error bar in that bin. . . . . . . . . . . . . . 54 5.6 Background prediction within the three final analysis bins (blue) com- pared to the unconstrained simulation prediction in gray. The uncer- tainty on the background prediction is taken from the 95% confidence band shown on the fit. Some bins in the simulation prediction are empty in which case the upper Poisson limit given the sample size is drawn as an error bar in that bin. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 xii 5.7 The total reconstructed energy in the ECal EECAL comparing ten differ- ent smeared calibrations (colors) to the original unsmeared calibrations (black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.8 Effect of smeared ECal calibrations on the ECal hit RMS for events that pass the original trigger. Outside of statistical fluctuations in the lowest- populated bins, the smearing has a less than 5% effect throughout the range of potential cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.9 Reach (left) and maximum allowed yield (right) comparisons for the 4GeV beam case as we vary the cut on maximum HCal photoelectrons (PE). We observe minimal movement in the final reach as a function of this cut; nevertheless, the biggest change occurs around a threshold of 30. . . . . 58 5.10 The sensitivity of the EaT analysis channel compared to other experi- ments led by NA64[5, 6], BABAR [7], and COHERENT[8] (gray), Scalar, Majorana, and Pseudo-Dirac theory expectations (black, top-to-bottom), and LDMX projections (colors). The blue (orange) line corresponds to the 4GeV (8GeV) beam studied in this work. The red line is the nomi- nal LDMX analysis sensitivity for the Missing Momentum (MM) analysis channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.1 Diagram of dark brem production of a dark photon followed by its decay back into an electron-positron pair through Strongly-Interacting Massive Particle (SIMP) composite particle VD with the emission of a lighter SIMP composite particle πD. The two-particle vertex VD −A′ is allowed since they share enough quantum properties to mix. . . . . . . . . . . . 63 6.2 Feynman diagrams for the SM process of trident production. . . . . . . 64 6.3 Full rendering of Heavy Photon Search experiment (HPS). Beam would enter the detector from the left and exit to the right if it does not interact. 66 6.4 Simplified diagram of HPS showing an example displaced decay within the named subsystems. The blue arrow is some DM candidate particle that has a macroscopic lifetime causing the decay vertex to be observably displaced from the production within the target. . . . . . . . . . . . . . 66 xiii 6.5 Figure 10 of [9]. Rendering of the Silicon Vertex Tracker (SVT) with the vacuum enclosure shown in gray, active silicon components in red, and readout electronics in green. Similar to Figure 6.3, the beam would traverse this rendering from left to right. . . . . . . . . . . . . . . . . . 68 6.6 Figure 27 of [9]. Depiction of an event passing the pair trigger in the 2016 data collection run. This depiction also displays the variables used in order to evaluate events and make the trigger decision. . . . . . . . . 69 7.1 Momentum error derived from injecting a certain amount of position error (blue and orange) compared to normal distributions (red and purple) with known momentum error. These distributions are normalized to have unit integrals. The trials were done in a simplified experimental model with quantities similar to HPS: a 2GeV electron curving within a 1T magnetic field, the sensors measuring the position were separated by 5 cm but their separation were also allowed to deviate from the true value by the position error given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.2 Plotting all six global coordinates completely specifying position and ori- entation of each of the sensors in the HPS tracking detector. Two different versions of the in-software detector model are plotted; however, on this scale, they overlap one another. . . . . . . . . . . . . . . . . . . . . . . . 76 7.3 Plotting all six global coordinates completely specifying position and ori- entation of each of the sensors in the HPS tracking detector relative to the “Original” detector from Figure 7.2. We can now observe the quan- titative differences between the different versions. . . . . . . . . . . . . . 77 7.4 The transverse impact parameter of the tracks at the target d0 = √ x20 + y20 which should be centered on zero to align with the known beamspot at the target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.5 The magnitude of the track momenta which should be centered on the known beam value of 2.3GeV. . . . . . . . . . . . . . . . . . . . . . . . . 79 8.1 Relative efficiency of vertex quality pre-selection requirements. . . . . . 85 8.2 Number of vertices passing the quality pre-selection. . . . . . . . . . . . 86 xiv 8.3 Relative efficiency of pre-selection of events. Nvtx is the number of ver- tices passing the vertex pre-selection requirements. Since other triggers for different purposes are present in the data, the Pair 1 Trigger is applied to data but not to simulation. . . . . . . . . . . . . . . . . . . . . . . . . 87 9.1 Depiction of dark photon yield calculation using a single run of the data (∼ 1.6%). The gray vertical lines show the bounds of the search using the nominal ratio mA′/mVD = 1.66. . . . . . . . . . . . . . . . . . . . . . 91 9.2 Expected signal yieldNsig for some rudimentary example efficiencies. Red contour lines are given at 1 and 1.6% (the approximate fraction of this data subsample) to give a sense of scale. Values are “clipped” to the color bar (for example, a signal yield of 2 × 104 is colored yellow which is marked as 1 × 103) so we can focus on the important orders of magnitude. 93 9.3 Diagrams showing L1L1 (left) and L1L2 (right) vertex examples. . . . . 94 9.4 Diagrams of examples for how different types of L1L2 vertices effect the observed values of y0. Truly displaced vertices (left) have both tracks whose y0 is far from 0mm while fake displaced vertices (middle) and not displaced (right) have at least one if not both tracks with y0 ≈ 0mm. The diamonds are the reconstructed vertex, the solid circles are the re- constructed hits, and the empty circles are “missed” hits either because the particle did not pass through those sensors (left) or due to some physical or electronic inefficiencies (center and right). . . . . . . . . . . . 96 9.5 Distributions of cut variables for a few example mass points and ≈ 10% of the full data sample. The vertices are required to be L1L2, have their momentum sum be within the signal region, and their invariant mass within the specified mass window. . . . . . . . . . . . . . . . . . . . . . 97 9.6 y0,min distribution after the other selections (L1L2, Psum in SR, recon- structed mass lying in mass window, Vertex Projection Significance (VPS) < 4, and σy0,max < 0.4mm). . . . . . . . . . . . . . . . . . . . . . . . . 98 9.7 Binomail significance (left) being maximized after the VPS< 4 and σy0,max < 0.4mm cuts leading to cuts (right, blue) which are smoothed into a continuous function (right, orange) as described in the text. . . . . . . 99 xv 9.8 y0,min distribution for the 10% data sample with the ycut0,min function over- layed in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.9 Signal efficiency as a function of z and mass scale mVD for this analysis (left) and L1L1 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.10 Search results for the full 2016 data set. . . . . . . . . . . . . . . . . . . 103 9.11 Signal yield for the L1L2 reconstruction category with the described cuts showing the theoretical expecation (left) and the maximum allowed (right) derived using the Optimum Interval Method (OIM). . . . . . . . 104 9.12 Sensitivity of this analysis with contour drawn including systematic errors surpressing the expected signal yield. . . . . . . . . . . . . . . . . . . . . 105 xvi Part I Laying the Groundwork 1 Chapter 1 Introduction Figure 1.1: Comic #2351 from XKCD[1]. If able to be designed from scratch, the names of things in the standard model could be more helpful. The long and winding road of a dissertation is not always neatly packaged into a template - this can easily be seen in my own experience. While focusing much of my time on the technical aspects of data processing software for a proposed (yet to be built) experiment, I found myself near the end of my journey without real data to analyze from which I can make real conclusions about the physical world. This motivated me to participate in another experiment – extremely similar to the original one – sharing 2 3 theoretical motivations, technological designs, and even people. This experience of participating in two similar experiments has provided an abundant field of learning opportunities (as well as roadblocks) for me and my journey as a new physicist. Physics is a lazy science. I would go even further and state that this reductionist perspective is a main attraction for many physicists. We1 want to avoid memorizing as much as possible; therefore, motivating the idea of condensing sets of observations into “laws” that can be represented in an even more compact mathematical form. We make up these “laws” and the vocabulary surrounding them not to decieve but merely to make communication about our observations and the experiments that make these observa- tions easier. We sometimes debate the origins of these laws and their true philosophical meaning, but often, on a day-to-day basis, the background of them is unimportant. The interesting work comes from testing them, breaking them, and remaking them. For our purposes here, this is what a “theory” is: a package of laws and their mathematical forms with which we can make predictions of the observations of our experiments. While I speak in the context of all physics, in reality, I am residing within a small corner. Primarily concerned with individual particles and how they interact with one another, my field can be described as investigating the foundations of the universe. Our experiments require giving these particles comparatively large amounts of energy, contextualizing the name High Energy Physics (HEP). Giving such small particles such large amounts of energy requires extrodinarily large and complex apparatuses. HEP is filled with experiments many stories tall, collaborations consisting of hundreds of institutions and thousands of people, and observations lasting years if not decades. This grand scale is helpful to keep in mind when I speak of the experiments in this thesis – constructed and operated by only a few institutions (less than 10 in each) and only dozens of collaborators. Contrary to many other sciences and experimental methods, these experiments still last longer than a typical doctoral student career. While I report on these experiments, I will not detail their full operation, design, or capabilities. I will focus on the work that I was able to contribute; hopefully providing a building block for these experiments and HEP to grow in the future, and give only the necessary context for parts with which I am less familiar. 1Here I use the “royal we” as representing my point of view of the culture within the field. It should in no way be construed as scientific or exact statements and does not represent every physicist’s point of view. 4 Before partitioning this thesis according to the two experiments in which I par- ticipated, Section 1.1 summarizes our current laws and its representative theory that best predict our set of observations within HEP, Chapter 2 will explore the theoretical ground on which both of the experiments rest which will also provide necessary vocab- ulary for discussing these two experiments. Part II presents the work of my initial years as a graduate student developing data processing and realistic simulation software for a proposed experiment. This part includes a description of this proposed experiment in Chapter 3, the simulation infrastructure for it in Chapter 4, and an analysis of this simulated data showcasing this experiment’s performance with ∼ 2.5% of its target data set in Chapter 5. Part III describes the work of my last years showing the difficulties of working with data taken in the real world (with all the complexities that implies). Similar to Part II, this part first describes the experimental setup in Chapter 6, how we can tune our detector model to better match how it was constructed in Chapter 7, the simulation and collected data used are detailed in Chapter 8, and the analysis per- formed on this data is described in Chapter 9. We then conclude with Chapter 10 which returns back to this high-level-view from which we can discuss what we have learned about physics via these two experiments. 1.1 Standard Model of Particle Physics What makes up all the stuff around us? This question has been posed, debated, dis- cussed, and studied since the ancient times. Our best current understanding is that stuff is made of particles that interact with one another. While “particle” is difficult to strictly define qualitatively, we have a model for them and their interactions which we creatively named the “Standard Model” (SM). Figure 1.2 displays the known elemen- tary particles, some of their basic properties, and how they interact with one another. Not all of the particles represented within this diagram are of critical importance to this work, but some deserve further description. The two lowest mass quarks – the up and the down – are the fundamental con- stituents of protons and neutrons which make up the nuclei of atoms which make up all the day-to-day stuff we interact with. These stay bound together within these nuclei via the strong nuclear force (mediated and represented in Figure 1.2 by the gluon). These 5 Figure 1.2: The Standard Model of Particle Physics showing the twelve fermions and five bosons, their various properities (mass, charge, spin), labels (box and circle colors), and interactions (brown loops). Credit to Cush on wikipedia for providing this diagram freely accessible and usable for any purpose. 6 quarks and the top row of leptons also have an electric charge enabling them to interact via with electromagnetic force. The electromagnetic force, represented and mediated by the photon, is the force responsible for light and magnetism. The lowest mass charged lepton (electron) is also very abundant – most commonly residing in orbits around nu- clei helping form atoms. Both experiments in this thesis utilize beams of electrons that have been accelerated to high energies and then impinge upon a material of high density to initiate interactions for further study. The mathematical expressions to calculate how these particles interact are compli- cated. As mentioned earlier, physicists are lazy and so we have developed a method to represent the specifics of these interactions graphically in a form where many key aspects of the interaction can be understood at an intuitive level without needing to write down any of the long mathematical formulae. These representations are called “Feynman diagrams” after the 20th century physicist Richard Feynman. Feynman dia- grams allow us to represent interactions by defining a set of “vertices” that are allowed by our theory and then constructing processes from these vertices that include the in- and out- going particles that we wish to study. Such diagrams are more than visual sketches; they can be algorithmically converted into precise mathematical forms which can be used to directly calculate measurable predictions of the model containing this set of vertices. The conversion of these diagrams into mathematical forms and the cal- culation of these forms has been written into various computer programs (I say this to emphasize that these diagrams directly represent the mathematics that can be used to calculate them). Figure 1.3 shows an example Feynman diagram representing the bremsstrahlung process where a charged lepton (ℓ−) interacts with a nucleus (Z) via a photon (γ) and then emits another photon before recoiling. We can see three verticies in this diagram: two “fundamental” vertices where the lepton and photon lines connect and one “effective” vertex where the photon connects with the nucleus. The fundamen- tal vertices are actually strictly defined within the Standard Model, but the effective vertex represents a helpful approximation that is accurate at the energy scales we are studying.2 Due to the complexity of the nucleus itself, the calculation of Figure 1.3 into any 2The distinction between fundamental and effective vertices is not a well defined one. I find it helpful here, but it is not necessarily made elsewhere in physics literature. 7 ℓ− Z ℓ− γ γ Figure 1.3: Feynman diagram for the bremsstrahlung process where a charged lepton (ℓ−) interacts with a nucleus (Z) via a photon (γ) and then emits another photon before recoiling. observables is difficult to do without any approximations. However, one of the initial parametrizations of bremsstrahlung offers a good glimpse of how it behaves [10]. When doing experiments, we often count the number of particles produced with certain criteria. These “rates” can then be translated into estimates of process cross sections (how likely a process is to occur producing particles with the observed properties) with knowledge of how the experiment was conducted. These cross sections are also calculatable from diagrams like Figure 1.3 and the initial parameterization can be summarized as dσ dEγ ∝ 1 Eγ (1.1) This shows the dominant shape of the cross section – it grows as the energy of the produced photon (Eγ) gets smaller. The full expression prevents the cross section from diverging as the photon energy approaches zero; nevertheless, the general inverse relationship between the rate and the energy of the produced photon holds in both the SM calculations and observations from experiments. Figure 1.4 shows another view of this rate: instead of looking at the energy of the produced photon (Eγ), we can inspect the energy of the outgoing electron after it emits the photon (Ee). This is more readily compared to other processes that do not produce a photon and we can observe a large difference between this SM process and other possible processes that we search for. This inverse relationship is somewhat special and is mostly due to the fact that photons do not have any mass; however, this special relationship opens a door towards potential discovery. The vast majority of electrons interacting with a material will do 8 Figure 1.4: Energy of the outgoing electron (Ee) for standard processes (dominated by bremsstrahlung) in gray and some dark bremsstrahlung (see Section 2.4) in colors. Figure 3 from [2]. so softly, only emitting low energy photons, which distinctly separates these electrons from ones that undergo rarer processes (or perhaps so-far undiscovered processes). Both experiments described in this work make heavy use of this fact. As the work carried out for this dissertation is experimental rather than the defi- nition and calculation of a model, I intend to stay within this diagrammatic realm – representing the theoretical calculations being tested with these diagrams and limiting the scope of mathematical formulae presented. Occasionally, the weight of these vertices which can be interpreted as the strength of the interaction is presented along with the vertices in order to give a sense of scale and connect these diagrams to the formulae that are included here. 1.2 Standard Model Analogies The SM has been extremely successful in describing the current world of particle physics, including some intricate behaviors of particles that were incredibly confusing when first encountered. When attempting to describe unknown phenomena (Chapter 2), we often draw on this previous experience to form new ideas and models. This section, while a brief detour, will help give context for two key behaviors that the models we are 9 searching for would give their particles. 1.2.1 Particle Mixing All of the models tested for in this work use the idea of “particle mixing” in order to connect the realm of theoretical/undiscovered particles with the realm of SM particles. In the quantum arena, we treat individual particles in a probabilistic manner – often times when we observe a particle with certain properties, it can be observed again in a different context with different properties. While this sounds odd, several experiments have supported this random nature of the smallest particles over the course of twentieth century physics. As a well-understood example, neutral kaons have been observed to “swap” with their own anti-particle, which we can describe via this concept of mixing. Kaons are particles that contain two quarks, one of which is a strange quark (or its anti-quark). A neutral kaon (K0) and its anti-particle (K0) are nearly indistinguishable – they have the same electric charge (0) and the same mass amongst other shared properties – but we can distinguish them by observing the types of particles they produce when they decay. Describing kaons as consisting of two quarks allows us to understand the mixing ofK0 andK0, diagrammed in Figure 1.5. From a higher-level view, where we are unaware of the constituents of the kaon, it appears like the K0 and K0 are swapping places with some probability. Our experiments can observe this probability by counting the frequency with which we see K0 compared to K0. We can also use our model of the kaons (as composed of quarks) and calculate this same probability (specifically, using the Feynman diagram shown in Figure 1.5). Comparing the two gives us information about the SM and how the quarks interact with one another. 1.2.2 Displaced Particle Decays Ever since particle physicists have started designing detectors, we have also observed particles that are “missed” by our detector mechanisms. Figure 1.6 shows an early example of this where a “bubble chamber” shows the paths of charged particles passing through the liquid (the white lines). In the middle of this image, you can observe two lines emerging seemingly out of nowhere (the red annotations were added later). The mechanism that allows a bubble chamber to observe the path of particles requires the 10 W K0 s du,c,t W + d u,c,t s K0 Figure 1.5: Dipiction of neutral kaon and anti-kaon mixing including the underlying Feynman diagram that results from applying the SM to this system. This figure was created by user NikNaks on Wikipedia and is licensed under CC Attribution-Share Alike 3.0 Unported. particle to have a charge, so a neutral particle (a Λ in this case) does not leave a path itself. The only signal that it was there is the displaced vertex from which two other particles (probably a proton and a negative pion) emerge (emphasized with a red circle in Figure 1.6). This is a “displaced decay” of a particle which was produced by a high energy (16GeV) negative pion interacting with the liquid in the bubble chamber. While particle physicists have developed other detector apparatuses, some of which can observe neutral particles more directly, this more indirect detection mechanism is still extremely useful especially if a model being tested has particles that have no way of directly interacting with SM matter (the stuff we make our detectors out of). Part III of this thesis focuses entirely on an experiment designed to search for these displaced vertices, the displacment of the vertex coming from the presence of an as-yet undiscovered particle being produced and then decaying back into SM particles. https://commons.wikimedia.org/wiki/File:Kaon-box-diagram-with-bar.svg https://creativecommons.org/licenses/by-sa/3.0/deed.en https://creativecommons.org/licenses/by-sa/3.0/deed.en 11 π− Figure 1.6: Image of CERN’s first liquid hydrogen bubble chamber from 1960 [3] with pink annotations added. The pink circle highlights the location described in the text where a Λ baryon decays into two particles (most likely a proton and a negative pion). The pink dashed line traces out a possible path the Λ traveled before its decay. Chapter 2 Dark Matter Figure 2.1: Comic #1758 from XKCD[1]. Generally, claiming that either Einstein or Newton was wrong will be a bad time. New phenomena have puzzled physicists throughout history, but the confusion sur- rounding dark matter – estimated to be roughly 85% of the total matter in the universe – is surely one of the biggest puzzles. While many astronomical observations at various scales have confirmed the existence of dark matter, we have not yet seen any observa- tions of its particle interacting with our detectors. This has ruled out many models for dark matter’s particle nature, but there are still many more available which can both explain the current observations of dark matter gravitational and cosmological effects while skirting the limitations defined by the lack of observation in particle experiments. 12 13 2.1 Dark Matter? The term DM rose out of a brutally honest description of the present level of under- standing. “Matter” originates from comparison to the particles that make ourselves up and with which we are familiar – both this new phenomena and our normal “stuff” interact gravitationally, attracting other groups of “matter” into complex cosmological systems. The adjective “dark” refers to the literal fact that we cannot see it with our telescopes. Other matter out in the universe can be seen via the light that it gives off (or reflects), so the fact that this matter is not visible in this way motivates the adjective. That is it. Dark Matter (DM) is merely a shorthand for this aspect of the universe that is both known to exist within the cosmos and whose particle nature is completely unknown. 2.2 Evidence for Dark Matter The first (replicable) evidence physicists had for an unseen material floating throughout the universe was the observation of galactic rotation curves [11, 12]. These observations measure the speed of different stars within a galaxy and compare this speed to the distance of that star from the center of the galaxy. We can calculate this relationship using GR [13] and the observations differ drastically from this calculation. The stars within galaxies we’ve observed move much faster than GR would predict (Figure 2.2) leaving us with two explanations: either GR is not the correct theory to use in this situation or there is more un-seen mass floating within the galaxy allowing these stars to move faster without leaving the galactic orbit. Other indirect measurements give us additional ways to access information about this odd phenomena. Within the framework of GR, since energy and mass actually warp the fabric of spacetime, we expect to see light itself follow a bent path around massive objects - a phenomenum that is called gravitational lensing and is observed and well modeled by GR’s predictions [14]. The accuracy of GR within this context – a mass and distance scale similar to the rotation curve oddities also observed – put more requirements on any modified theory of gravity that could both explain the rotation curves and gravitational lensing. Additionally, measurements on some of the largest scales and from the early universe display signs of a certain mass density attributed to 14 matter that does not interact in the same ways as our normal matter (called “baryonic” matter). In the early universe, standard matter was compressed into high densities and ex- isted at high temperature creating a sea of plasma. Gravity attracts while pressure from the squeezing of this plasma repulses which produces oscillations in the density of matter in space and time. The repulsive pressure within the plasma originates from particles interacting electromagnetically with each other, so the oscillations would be disrupted by the presence of extra matter interacting gravitationally but not interact- ing electromagnetically. These Baryon Acoustic Oscillations (BAO) are imprinted on our snapshot of this early-universe plasma, the Cosmic Microwave Background (CMB), which we can measure with a high degree of accuracy and fit to various models of what existed at this time of the universe. The best fit of these models corresponds to only ∼ 5% of the mass density being normal matter like we see today (“baryonic”) while the rest is composed of material that only significantly interacts gravitationally [15]. Additional astromomy observations from Type1a supernovae [16], fitting models of big bang nucleosynthesis [17], and constraints on non-particulate theories explaining these phenomena (for example mini black holes [18]) all allow us to conclude pretty comfortably that DM exists as a particle in our universe. 2.3 Particle Nature of Dark Matter The theoretical possibilities explaining DM are broad [19] even when excluding ourselves to the assumption that the DM phenomenum is explained by the existence of a DM particle. The CMB observations are tied to very early on in the existence of the universe, so it is natural to assume that this DM particle has existed since the start of the universe alongside our normal matter particles. With this in mind, we can outline a few criteria that must be met by a proposed DM candidate. • Dark There has been no detection of these particles via the light observed with our telescopes; therefore, the DM has to not interact via the electromagnetic force. • Long Living Measurements of DM’s mass density and presence agree across time (from as early as the CMB era), so DM needs to have a long lifetime. 15 Figure 2.2: Depiction of the velocity of stars within a galaxy as a function of their distance from the galactic center. The dotted line is a prediction of this relationship using GR along with the mass tabulated from the visible starts while the data points (and the solid line fitted to them) are what are actually seen in galaxies today. 16 • Universal Density Since we can indirectly measure a DM mass density on cos- mological scales, we impose the requirement that any DM model needs to allow for this density. • Thermal Relic Both DM and standard matter have similar cosmological densi- ties, so we expect some interaction (even a weak one) should connect their origins to the early universe allowing both to exist from the Big Bang. This criterion is not as firm as the others (DM models correctly describing the current observations while avoiding this criterion do exist); nevertheless, this criterion is well motivated and it is satisfied by all of the models for DM studied in this work. Even with these assumptions and the requirements they imply, there still exist a plethora of theoretical models that can satisfy all of them. The upside is that a thermal-relic assumption closely connects the mass of individual DM particles to the interaction strength it has with standard matter. In this assumption, the DM evolves along with the universe allowing its number density to follow the density of standard matter until the standard matter does not have enough energy to produce more DM. While the universe continues to expand, the DM continues to decay into standard matter until it becomes too sparse to annihilate with itself and is thus “frozen” at a specific number density. Since this “frozen” density changes depending on how easy it is for the DM to interact with standard matter, the “frozen” number density goes down as the interaction strength increases (Figure 2.3). The additional requirement of the observed astronomical mass density connects the mass of individual DM particles to their interaction strength with standard matter. mχ ↔ observed mass density ↔ dN dV ↔ thermal relic hypothesis ↔ ⟨σv⟩ (2.1) This connection allows us to define strict “thermal relic targets” which can help be measuring sticks for how well our experiments search for DM (these targets will appear in plots later). In addition to the connection between interaction strength and particle mass implied by the thermal relic hypothesis, it also puts some loose bounds on mass of individual particles (Figure 2.4). If the mass is too high (≳ 100 TeV), the DM will be too strongly coupled to the standard matter and would be over-produced within the early universe. 17 Figure 2.3: From [4], the co-moving cosmological number density of DM as a function of the universe temperature. As the universe cools, the number density decreases until the DM becomes too sparse to interact with other DM particles, “freezing” to a specific number density until today. 18 ∼ 1 MeV ∼ mp ∼ 100 TeV Too few “Light” DM “WIMPs” Too many Figure 2.4: Mass scale of Thermal Relic DM. The regions in red are excluded by apply- ing the thermal relic assumption to our observations of the universe’s early evolution. me is the electron mass and mp the proton mass. γ A′ Φ Φ Figure 2.5: Feynman diagram of how a massive field Φ could allow for a standard photon (γ) to mix with a dark photon (A′). Highly precise agreement between nuclear abundance observations and predictions from a model of the early universe with only SM particles provides the lower limit; allowing a thermal relic DM mass ≲ 1MeV would then break the consistency between this model and several of its predictions [17, 20, 21]. The mass scale of thermal relic DM is further divided by the mass of the proton (mp). Above this threshold, the DM could be interacting with standard matter through the standard Weak Force (as described in Chapter 1); thus, they are named Weakly Interacting Massive Particles (WIMPs). This phase space was first searched due to its theoretical simplicitity: no new forces, just an extra particle (or two) creating the clouds of DM we see today. Unfortunately, these searches have not found evidence of a WIMP-like signature[22, 23, 24] and the phase space has become tighter and more exluded after many years of searching. Below mp, the DM requires a new force which is even weaker than the Weak Force. In comparison to the heavier WIMPs, this category of DM is called “Light” DM. In general, the definition of a new force of nature is not well constrained; however, we can define a “baseline” model that can represent more concrete theories in the context of our experiments. For this situation, we postulate the existence of a massive gauge boson that represents an additional UD(1) symmetry of nature. Since this additional symmetry has the same structure as the electromagnetic interaction whose gauge boson 19 ℓ− Z ℓ− A′ γ Figure 2.6: Feynman diagram for the dark brem process. is the standard photon, we generally refer to this postulated massive boson as a dark photon. Without any additional assumptions, this new dark photon does not interact with any standard matter, so we also assume that some previously-unknown massive fields are able to interact with both. These other fields need to be sufficiently massive (or sufficiently weakly coupled) so that they only allow the standard and dark photons to weakly mix. Figure 2.5 diagrams how the standard and dark photons mix which can then be effectively represented by a new vertex in our model. A′ ℓ+ ℓ− ⇒ A′ ℓ+ ℓ− γ ∝ ϵ (2.2) where the mixing strength ϵ encapsulates the effect of a massive field Φ at the energies of our experiments (presumably low enough to avoid creation of a Φ directly) and is generally small (≪ 1). This new vertex within our model for the universe enables a new process to occur: so-called “Dark Bremsstrahlung” (dark brem) where a charged particle exchanges a standard photon with a nucleus and then emits a dark photon and recoils. But what happens to the dark photon after it interacts with standard matter via this vertex? It cannot be the long-lived DM we view in the universe today because this vertex allows for it to decay back into standard matter eventually (and in some models, rather quickly). This is where we expand on the idea of a “dark sector”. In this description of DM, we already have a dark photon representing some force and a very 20 heavy field that can interact with it. Suppose this “dark sector” also has other particles (like the standard sector) – one (or more) of which could be long-lived and represent the DM we observe in the universe today (so-called “DM candidates”). We can further partition this category of models depending on what happens to the dark photon after we produce it within an experiment. 2.4 Invisible Signature One of the simplest options is to hypothesize another particle that can take on the role of long-lived DM and only interact “within” this dark sector (i.e. it only interacts with the dark photon from our point of view). Calling this particle χ (and its anti-particle χ), we then have an additional dark sector vertex. A′ χ χ ∝ αD (2.3) where αD is a parameter representing the strength of this dark sector interaction, limited to be below 1 but usually much larger than ϵ so that this interaction is more likely than the one with standard matter. If the mass of the dark photon is greater than twice the mass of the χ (mA′ > 2mχ), then this vertex can occur immediately after a dark photon is produced within an experiment. Since the χ does not interact with our normal SM particles, the energy they have is “lost” from our perspective. Part II focuses on a proposed experiment designed to precisely measure all of the visible energy so that this “invisible signature” of DM being produced could be observed. When comparing analyses across DM search literature, it is common to parameter- ize the DM phase space with an effective interaction strength y and the mass of the candidate DM mχ that would constitute the astronomical DM observed today. The parameter y is related to the parameters of our “baseline” model by y = αDϵ 2 ( mχ mA′ )4 21 where we make standard, benchmark choices αD = 0.5 and mA′ = 3mχ when converting from (ϵ,mA′) to (y,mχ). 2.5 Visible Signature There is no a priori reason for mA′ > 2mχ – the quantum nature of these fundamental particles allows for “virtual” dark photons to have more or less energy than precisely mA′ allowing them to mediate the interactions between χ and SM particles, so we must accomodate the possiblity that mA′ < 2mχ. In this case, there would be a significant probability that the dark photon, after it is produced, would convert back into SM particles. Since these decay products are observable by our detectors, this signature is called visible. These visible signatures are a bit more complex to model due to the fact that both a production process and a decay process need to occur according to the model of DM we are testing. With this in mind, many models that provide more detail about how particles exist within a dark sector have been created, each of which providing a specific estimate for production and decay rates. Part III focuses on an experiment looking for these visible signatures and my work specifically was investigating one of these more intricate dark sector models in detail. Section 6.1 provides more detail on the dark sector model studied in this work. Part II LDMX 22 Chapter 3 Light Dark Matter eXperiment LDMX is a proposed fixed-target experiment aiming to definitively explore the thermal relic light dark matter phase space. Even as a proposed experiment, it has a detailed plan for construction, a beam already in construction, and well established connections with current technologies used within HEP. While LDMX is not yet built, it has a well formulated simulation infrastructure that can realistically model how the detector design responds to various types of interactions happening within it. 3.1 Missing Momentum Signature LDMX aims to search for an invisible signature – with known incident particle kinemat- ics, measuring the outgoing particle kinematics allows for a natural deduction. All of the incoming momentum must exit somehow and so if the detector is unable to observe some momentum (some momentum is “missing”) frequently enough, we can conclude that some other, previously-unknown, process is taking place and carrying momentum away from the experiment. Precisely understanding both the incident and outgoing momenta requires knowledge of the currently known processes and how to detect the particles they produce. The center of Figure 3.1 displays an ordering of known processes based on how frequently they occur given an incident electron with 4GeV of energy. This chart can be broken into three regions. 23 24 K± decay in ECal � ! µ+µ� � ! hadrons � ! 1n/K0 L + soft 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 � ! K± + soft AAACDXicbVA9SwNBEN3zM8avqKXNYhQEIdyJYMqAjWATwaiQi2Fus5cs7t4eu3NKOPIHbPwrNhaK2Nrb+W/cxBSa+GDg8d4MM/OiVAqLvv/lzczOzS8sFpaKyyura+uljc1LqzPDeINpqc11BJZLkfAGCpT8OjUcVCT5VXR7MvSv7rixQicX2E95S0E3EbFggE5ql3bDLigFNDSi20MwRt/Ts5swVfTAaSq3OsZBu1T2K/4IdJoEY1ImY9Tbpc+wo1mmeIJMgrXNwE+xlYNBwSQfFMPM8hTYLXR509EEFLetfPTNgO45pUNjbVwlSEfq74kclLV9FblOBdizk95Q/M9rZhhXW7lI0gx5wn4WxZmkqOkwGtoRhjOUfUeAGeFupawHBhi6AIsuhGDy5WlyeVgJ/EpwflSuVcdxFMg22SH7JCDHpEZOSZ00CCMP5Im8kFfv0Xv23rz3n9YZbzyzRf7A+/gGGWCbiQ==AAACDXicbVA9SwNBEN3zM8avqKXNYhQEIdyJYMqAjWATwaiQi2Fus5cs7t4eu3NKOPIHbPwrNhaK2Nrb+W/cxBSa+GDg8d4MM/OiVAqLvv/lzczOzS8sFpaKyyura+uljc1LqzPDeINpqc11BJZLkfAGCpT8OjUcVCT5VXR7MvSv7rixQicX2E95S0E3EbFggE5ql3bDLigFNDSi20MwRt/Ts5swVfTAaSq3OsZBu1T2K/4IdJoEY1ImY9Tbpc+wo1mmeIJMgrXNwE+xlYNBwSQfFMPM8hTYLXR509EEFLetfPTNgO45pUNjbVwlSEfq74kclLV9FblOBdizk95Q/M9rZhhXW7lI0gx5wn4WxZmkqOkwGtoRhjOUfUeAGeFupawHBhi6AIsuhGDy5WlyeVgJ/EpwflSuVcdxFMg22SH7JCDHpEZOSZ00CCMP5Im8kFfv0Xv23rz3n9YZbzyzRf7A+/gGGWCbiQ==AAACDXicbVA9SwNBEN3zM8avqKXNYhQEIdyJYMqAjWATwaiQi2Fus5cs7t4eu3NKOPIHbPwrNhaK2Nrb+W/cxBSa+GDg8d4MM/OiVAqLvv/lzczOzS8sFpaKyyura+uljc1LqzPDeINpqc11BJZLkfAGCpT8OjUcVCT5VXR7MvSv7rixQicX2E95S0E3EbFggE5ql3bDLigFNDSi20MwRt/Ts5swVfTAaSq3OsZBu1T2K/4IdJoEY1ImY9Tbpc+wo1mmeIJMgrXNwE+xlYNBwSQfFMPM8hTYLXR509EEFLetfPTNgO45pUNjbVwlSEfq74kclLV9FblOBdizk95Q/M9rZhhXW7lI0gx5wn4WxZmkqOkwGtoRhjOUfUeAGeFupawHBhi6AIsuhGDy5WlyeVgJ/EpwflSuVcdxFMg22SH7JCDHpEZOSZ00CCMP5Im8kFfv0Xv23rz3n9YZbzyzRf7A+/gGGWCbiQ==AAACDXicbVA9SwNBEN3zM8avqKXNYhQEIdyJYMqAjWATwaiQi2Fus5cs7t4eu3NKOPIHbPwrNhaK2Nrb+W/cxBSa+GDg8d4MM/OiVAqLvv/lzczOzS8sFpaKyyura+uljc1LqzPDeINpqc11BJZLkfAGCpT8OjUcVCT5VXR7MvSv7rixQicX2E95S0E3EbFggE5ql3bDLigFNDSi20MwRt/Ts5swVfTAaSq3OsZBu1T2K/4IdJoEY1ImY9Tbpc+wo1mmeIJMgrXNwE+xlYNBwSQfFMPM8hTYLXR509EEFLetfPTNgO45pUNjbVwlSEfq74kclLV9FblOBdizk95Q/M9rZhhXW7lI0gx5wn4WxZmkqOkwGtoRhjOUfUeAGeFupawHBhi6AIsuhGDy5WlyeVgJ/EpwflSuVcdxFMg22SH7JCDHpEZOSZ00CCMP5Im8kFfv0Xv23rz3n9YZbzyzRf7A+/gGGWCbiQ== increasingly rare photo-nuclear ... ν (no e–) incoming outgoing 100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13 10-14 10-15 10-16 … relative rate
 (4 GeV) irreducible 
 “invisible” backgrounds ≪ 10-16 “visible” backgrounds 
 (EM interactions) ⌫⌫̄ ⌫ (Møller + CCQE) e� e� EN +µ+µ� +hadrons ... +e+e– trident Veto Handles bremsstrahlung +hard � Track momentum ECal Energy ECal BDT ECal Track Track multiplicity HCal Hits Design driver charged-current Gaussian energy fluctuations Irreducible
 prompt E Rare reactions ➝ products escape ECal and/or anomalous energy deposition / ECal depth, resolution, layout (no projective cracks) Design Drivers HCal depth, segmen- tation, veto energy threshold ECal 3D granularity, MIP-sensitivity SRT acceptance Figure 3.1: Diagram showing relative rates of background processes within LDMX along with how they motivate various aspects of the design. /E stands for “missing” energy or energy that is “lost” to neutrinos that are extremely unlikely to be detectable within LDMX. The blue region at the top are the most frequent types of processes, but are simul- taneously less complicated. The processes in this region (as the right side shows) put requirements on the design of the ECal, making sure it can quickly and faithfully re- construct the outgoing energy in photons and electrons. The ECal must be able to veto the first several orders of magnitude in basic energy fluctuations including simple inter- actions in the target like bremsstrahlung or trident production of an electron-positron pair. Entering the pink region is where the background processes become rarer and more complicated. One of the first ways to partition these backgrounds is whether charged secondary particles are produced within the target. These “prompt” backgrounds help define the recoil tracker’s design and are thus left to be caught by it. More frequently, a photon is produced within the target (which is not observable by the tracker) and then undergoes complicated photon-nuclear interactions within the ECal producing particles that are difficult for the ECal itself to observe. The HCal is able to detect the presence of these hadrons and muons, acting as a “pure” veto in the sense that the signal process 25 should never create significant activity within it. All of these subsystems can collaborate to help LDMX reject known SM backgrounds down to ∼ 10−16 fraction of all incoming electrons where extremely rare known processes that produce invisible products (neutrinos ν) begin to emerge. Detailed study of the missing momentum search strategy[25, 2, 26] shows that LDMX can reject all simulated backgrounds to a relative rate of ≳ 10−14. This performance is accomplished through the basic design drivers outlined above and in Figure 3.1, but also through the additional granularity of the ECal enabling the use of a Boosted Decision Tree (BDT) to distinguish between background and signal events using features of the showers within the ECal. Cycles of detailed simulation and redesign have led to a detailed construction plan for the LDMX detector apparatus. The subsequent sections of this chapter detail this design as well as the beam it is expected to receive. 3.2 The Beam Line LDMX is planned to receive electrons from the superconducting Linear Accelerator (Linac) at SLAC National Accelerator Laboratory. The SLAC Linac can provide high energy, high rate, and low intensity electron beams for the various experiments it hosts. Specifically, the Linac Coherent Light Source (LCLS) is used to guide the beam (of a certain energy) towards the experimental hall - the upgraded phase of LCLS (LCLS II [27]) is currently under construction and is what will be used for running with LDMX. The experiment is hosted in End Station A (ESA) at SLAC which requires an additional upgrade to the accelerator complex in order to recieve its beam. LESA (Linac to ESA) beamline [28] is also currently being constructed and will be ready for test beam in early 2025. Part of the infrastructure that transfers the beam to ESA (Sector 30 Transfer Line – S30XL) is already constructed and LDMX components are expected to participate in a test beam runs with it. 26 3.3 Detector Design As described above, LDMX is a missing momentum experiment and its design is focused on measuring both the incoming and outgoing momenta of charged particles interacting with a thin target such that any momentum given to undetectable (dark) particles can be precisely determined. In addition, the energy of neutral particles must be measured. This design has led to four subsystems each with specialized roles.1 1. Tracker Measure charged particle momenta both before (“Tagger”) and after (“Recoil”) the target, using a dipole magnetic field of 1.4T. 2. Electromagnetic Calorimeter (ECal) Measure the total energy of electrons, positrons, and photons. 3. Hadronic Calorimeter (HCal) Veto additional particles difficult for other sub- systems to measured (muons, pions, hadrons,...). 4. Trigger Scintillator Count the number of electrons incident on the target in time to make a trigger decision (more on what a “trigger” is below). Figure 3.2 displays these subsystems in a diagram along with a representation of a dark brem interaction occurrring within the target. Figure 3.3 shows a rendering of the detector design. The Tracker (purple in Figure 3.2) is a thin silicon strip detector modeled after the HPS tracker. These silicon strip sensors are arranged in layer pairs where one layer is angled slightly askew relative to the other in the pair to enable reconstruction of three dimensional hit locations. The part of the tracker upstream of the target (to the left in Figure 3.2) is named the “tagger” since its purpose is to measure the incident electrons’ momenta, rejecting electrons with momentum below 30% of the expected beam momentum. The tagger is situated within the bulk of the magnetic field enabling highly precise measurement of the high incident momentum. The other part of the tracker located downstream of the target (to the right in Figure 3.2) is named the “recoil” tracker since its job is to measure the momenta of all charged particles recoiling 1These subsystems also take on additional roles when the full breadth of the LDMX physics program is taken into account. This description just focuses on the DM search. 27 from interactions within the target. While it is not located within the magnet volume, it is still situated within the fringe field enabling it to maintain track reconstruction down to the lower-momentum products of interesting processes within the target. In many HEP experiments, a “trigger” system is necessary in order to filter data defined to be interesting by the experiment from the wealth of uninteresting (or normal) data that is expected to be produced at a much higher rate. These trigger systems are the first filter that any data goes through and are designed to help the experiment obtain a statistically large data sample without collecting an overwhelming amount of data. The ECal is the primary detector subsystem responsible for making this trigger decision due to its excellent energy resolution and fast measurement capabilities. As a search for missing momentum, the ECal requires less than 30% of the incoming beam energy to be observed within the first twenty sensitive silicon layers. The Trigger Scintillator (yellow-orange in Figure 3.2) is designed to help inform the trigger decision by counting the number of electrons present within the detector. It is made of layers of vertically segmented bars of plastic scintillator. These layers are arranged in pairs where the layers within each pair are offset from one another to cover any gaps between the bars. These bars are readout in time to be used within a trigger decision combined with information from the ECal and HCal. The HCal (green in Figure 3.2) is a sampling calorimeter made up of alternating layers of steel absorber and plastic scintillator bars. The HCal is further subdivided into the “side” HCal which is situated around the ECal and the “back” HCal down- stream of the ECal. The back HCal has the orientation of the scintillator bars alternate between vertical and horizontal so that clusters and tracks can have three-dimensional coordinates more preceisely identified. The ECal (blue in Figure 3.2), as a primary volume of interest within the analy- sis discussed here, is given its own diagram Figure 3.4. The ECal is also a sampling calorimeter; however, it uses a different absorber material and a different sensing mech- anism to more precisely measure the energy of electrons, positrons, and photons. The calorimeter is constructed out of seventeen layers each consisting of tungsten absorber, service materials, and sensitive silicon sensors. Each of the layers of the detector has two sub-layers of sensitive silicon sensors and each of these sub-layers are built up out of the hexagonal High-Density modules designed for the CMS Phase II High Granularity 28 Figure 3.2: Diagram of LDMX detector apparatus with a representation of a signal event where a dark brem occurs within the target. Credit to Christian Herwig for original development of diagram. Calorimeter upgrade[29]. These hexagons are arranged in a “flower” providing excel- lent transverse resolution of shower location and shower shape. The layers are built and arranged in order to space the sensitive silicon sub-layers to give good longitudinal resolution of showers as well. In total, the designed ECal has more than one hundred thousand channels that can each individually detect particles depositing energies from ≈ 0.1MeV up to ≈ 1GeV. This high granularity calorimeter gives LDMX excellent dis- crimination power since it can measure the amplitude and location of several incident particles. Due to its high performance, the ECal is a primary tool for both designing a trigger decision (with the aid of the Trigger Scintillator’s electron count) as well as downstream analysis separating SM background processes from potential DM signal. 29 Figure 3.3: Rendering of LDMX detector apparatus focusing on tracker, target, and ECal. The magnet would fully encompass the tracker, target and trigger scintillator. 30 Tu ng ste n Ab so rb er Se rv ice s P CB Co oli ng P lat e Se rv ice s P CB 17 ECAL Layers Pre-shower layer Each Layer 7 Silicon Sensors Tu ng ste n Ab so rb er Tu ng ste n Ab so rb er Si lic on S en so r Si lic on S en so r 432 Readout Channels Figure 3.4: Diagram of LDMX ECal design showing the longitudinal segmentation (top and bottom right) and the transverse segmentation (bottom left). Credit to Joe Muse. Chapter 4 Mid-Shower Simulation LDMX (like many other HEP experiments) uses an intricate software stack in order to realistically and efficiently simulate particle interactions with the detector, emulate the electronics that would be used to measure these interactions, and reconstruct the output of these electronics into physically-understandable variables. These software tasks are accomplished by a wide swath of different software packages, some of which custom- written for LDMX, most of which written in C++. This chapter is focused on describing this simulation infrastructure – focusing particularly on parts of the infrastructure I was involved in – while also pointing out areas that are expected to remain constant in the presence of data gathered from a real detector. After a discussion of general data processing, I move into discussing the specific samples used to do this simulation study. 4.1 General Data Process One of the core principles helping organize HEP data is the concept of an “event.” In the context of LDMX, an event is the data collected within a small window of time around the arrival of a beam electron into our detector apparatus. In essence, each event within the software is independent from one another; however, they all share a similar structure to the information they hold. A natural example is a data table: a row has the same variables in each of its columns as all the other rows. Events behave the same way; however, unlike a data table, the structure of an event can be more intricate than simply a series of values corresponding to different column titles. While 31 32 ... Input Data File ... ... Output Data File Processor Event Bus start event end event get add Figure 4.1: Flow chart of how data is processed in the LDMX framework. Each processor has the ability to “add” data to the event as well as “get” data from the event. The processors are run in a user-defined sequence. Data can also be loaded from one or more input data files where the event data from those files is loaded into memory before the first processor is run. After all processors are done with an event, it is saved to the output data file. our basic “unit” of data is an event, we require many events in order to make statistical conclusions about our data; thus, we have developed an “event processing framework” that allows us to unify the various aspects of processing the data held within an event. The event processing framework designed, developed, and maintained by LDMX is designed to allow the flexibility necessary to do the wide range of tasks necessary for the experiment. The C++ framework uses CERN’s ROOT [30] for data serialization, Boost for logging, and Python [31] for dynamic run-time configuration. As diagramed in Figure 4.1, the design is a sequential model: the “event bus” stops at the individual pro- cessors in a certain sequence. The individual processors can inspect the data currently on the bus and board more passengers onto the bus for later processors to use and which can be eventually serialized into the output ROOT file. The processors can be built separately from the framework and then dynamically loaded and created at run-time by 33 an abstract factory. This design choice allows for all of the computationally-intensive software tasks necessary for LDMX (simulation, reconstruction and some analysis tasks) to use this framework and be organized into separate modules which are only loaded into memory when that module is being used. The serialization portion of the framework is similarly dynamic; focusing on enabling users’ code to add data structures from the most simple (e.g. individual booleans) to the most complicated (e.g. containers of custom classes). This wide array of data types is supported by ROOT’s dictionary system during the serialization stage and abstract wrapper classes with partially-specialized template derivatives during run-time. Such complexity within the framework is necessary in order to allow a simple interface – one where the user interacts with simple and complex types in the same way. Combining this highly dynamic serialization library with the sequential-processing model configured at run-time gives a strong foundation for all of the software needs of LDMX. Written in C++, this software framework enables high performance for all of the major data processing tasks necessary for the experiment. Moreover, its design focuses on flexibility and modularity so that seemingly-disparate data processing tasks can be unified under one framework. Everything from simulation to detector emulation to event reconstruction to analysis calculations can be done within this framework, reading and writing files from this framework and enabling our software to be well organized while also centralized in one location. 4.1.1 Data Processing Stages The centralized nature of the LDMX processing framework makes it much simpler to stay unified as a collaboration. Experts are able to work on their specialized area of the software and share those improvements with the entire collaboration with ease. Since the flexibility of the framework allows for arbitrary groupings of these different data processing stages, we can choose to separate them into natural groups that correspond to the different areas on which experts focus (diagrammed in Figure 4.2). In this work (as the chapter title implies), we focus mainly on the detector simula- tion stage where events are produced using random sampling of relevant physical phase spaces in a way geared towards realistically modeling the detector and particles inter- acting with it. LDMX like most HEP experiments use Geant4[32] to help perform this 34 SIM DIGI REC Analysis Variables RAW Real Detector Framework Data Software Library Custom Subsystem Tooling Geant4 Figure 4.2: Diagram of processing stages within LDMX showing the external sources of data or software that are used in those stages as well as which stages are commonly done within the centralized processing framework. complicated detector simulation. The downstream stages, namely electronic emulation and reconstruction, are not described further here; however, it should be emphasized that all subsystems of LDMX have their own custom implementations of these stages in order to realistically emulate and reconstruct the data within their subsystem. 4.1.2 Biasing and Filtering Technique Many of the processes most interesting for a DM search within LDMX are rare relative to processes that are “easier” to reject by normal data processing. In many cases, the rarity of these more interesting processes is a computational roadblock since waiting for the entire detector simulation (hundreds to thousands of particles each with dozens if not hundreds of interactions) to complete before looking for these processes of interest wastes a lot of computer time. Typically, only one out of every ten thousand events actually contains a process we are interested in. To make simulation of these processes more computationally feasible, we turn to the common simulation techniques of bias- ing (artificially increasing the probability of a certain process occurring) and filtering (proactively ending the simulation of events if certain criteria are not met). 35 In the analysis channel studied here, all of our interesting processes occur within the developing shower of particles in the ECal. While more complicated than checking for processes happening to the single beam electron, we can select specific processes occurring within this shower in a computationally efficient way by tuning the order with which the simulation processes particles. The detector simulation is focused only on particle-material interactions and the materials are assumed to not change significantly during these interactions, so the order in which particles are simulated does not change how the physics is modeled. We choose to process particles in two groups – “high” and “low” energy – where the high-energy particles are all processed before the low-energy particles. The border between these two (called the “Sorting Threshold”) is configurable by the user and enables a specific Geant4 function to be called mid-shower after all of the high-energy particles are processed (i.e. simulated until their total energy falls below the Sorting Threshold). Suppose we know from studying other (unbiased or low-bias) samples that showers need to have a minimum amount of energy entering a specific process of interest before the shower becomes “important.” We can set this minimum energy (here called the “Filtering Threshold”) as a requirement on the ECal shower for the simulated event to be kept within the sample and, in order to improve the speed of the filtering, we can apply this requirment during the simulation when the event transitions from the high-energy group to the low-energy group of particles. Now, this infrastructure of determining a minimum simulated energy and having an energy-based sorting of the processing order already helps improve the computational efficiency of these simluation samples (from ∼ 6k CPU-hours to ∼ 30 CPU-hours for the “important” events from ∼ 1B total events). Nevertheless, it is still extremely inefficient (most simulated events do not pass our filter requirements) and so we add biasing in order to artificially increase the rate of the process we wish the sample to focus on. Since the processes we are interested in biasing are connected to particles that are abundant within the electromagnetic ECal shower, we need to require only particles above a certain energy threshold (the “Biasing Threshold”) to avoid over-sampling of the process. In addition, we also need to define the factor that we will multiply the cross section by when a particle is above the Biasing Threshold and in the ECal volumes (the “Biasing Factor”). 36 Artificially increasing the rate of a process relative to its true rate measured in nature is obviously unphysical, but we can account for this by weighting the events depending on this Biasing Factor when counting them. The simulation engine we use does this weight calculation for us and it accounts for both the biasing factor (beginning the weight with a factor of 1/B) and unbiased processes happening to a particle with biasing enabled (increasing the event weight to reflect that physical processes at the natural rate have ocurred). Parameter Dimension Symbol Short Description Sorting Threshold Energy TS Minimum total energy to be pro- cessed in first group Biasing Threshold Energy TB Minimum total energy for a parti- cle to be biased Biasing Factor Dimension-less B Factor to multiply cross section by Filtering Threshold Energy TF Minimum energy transfered into a process for the event to be kept Table 4.1: Parameters of mid-shower biased samples. In all of the samples used in this study, we use the same value for the Sorting and Biasing Thresholds. Table 4.1 summarizes the parameters related to this sample generation technique. Practically, we choose to have the Biasing and Sorting Thresholds share the same value for all of the samples in this work in order to reduce the number of parameters needed to tune and to process all of the biased particles (as well as particles of similar energies but different flavors) before making any filtering decision. Generally, validation of this biasing and filtering technique can be done by checking that the samples generated with this technique match samples generated without this technique and with the filtering applied after the fact. This validation needs to be done on a per-sample basis since the specific aspects of the simulation that need to align may change depending on the specific physics for which we are biasing and filtering. 4.2 Standard Processes As displayed in Figure 3.1, there are a few categories of physics processes that are described by the standard model and we expect to happen within our experiment. These “background” processes – while interesting for other analyses of LDMX data – 37 should be rejected by this search for DM since they are understood to not be DM by the standard model and other experiments. In order to thoroughly and faithfully estimate our ability to reject these backgrounds, we need to realistically simulate them within our detector volume. Geant4 [32] has been developed for precisely this purpose. This study uses a slightly modified copy of Geant4 v10.2.3 which focuses on improving the accuracy of processes relevant to a DM search within LDMX with the following needs. 1. Updating cross section and sampling of γ → µ+µ− process to align it better with collected data and calculations of the standard model. 2. Update nuclear cascade to align with more recent CLAS data specifically focused on the rate of low-multiplicity forward neutrons (Appendix A.A of [25]). 3. Introduction of back scattering π0 in a γp → π0p within a nuclear cascade (Ap- pendix A.B of [25]). These updates enabled us to more reliably study the known and expected background processes interactions with the LDMX detector design and compare this data to simu- lations of a model of a DM production process. 4.2.1 Dominant Contributors in this Analysis First and foremost, the analysis detailed in Chapter 5 is focused on utilizing data col- lected with LDMX that is not used in the nominal Missing Momentum (MM) analysis. This design goal produces two requirements on the samples we will analyze. • Same Trigger Requirement – while designing a separate trigger for different analy- ses is feasible, this specific analysis is focused on working with the same data that is collected for the primary MM analysis. The trigger requirement for the primary MM analysis is one of missing energy within the ECal making it an appropriate trigger for this analysis as well. • Opposite Tracking Requirement – the primary MM analysis enforces the assump- tion that the processes of interest occur within the thin target; thus, the electron track leaving the target (colloquially named the “recoil”) is required to have less 38 than 30% of the beam energy. In this case, we flip this requirement and instead require the electron to arrive at the ECal with more than 87.5% of the beam energy. While the application of this requirement would be done after the trigger requirement in real data, for simulation we apply this requirement first so that the simulation samples we are studying can be more efficiently produced. All of the simulation samples studied here require the primary beam electron to arrive at the ECal with more than 87.5% of its original energy. Once this requirement is imposed, we then turn our attention towards the processes that occur within the ECal. Generally, we focus on processes that would cause the energy in the ECal to be estimated at a significantly smaller value than the known beam energy. This “missing” energy is used as a key piece of evidence that the event in question should be stored for further study and thus is the main ingredient in the infrastructure deciding whether a specific event should be stored (the “trigger”). From prior experience with LDMX (e.g. [25, 2]), we expect certain types of standard processes to dominate the backgrounds that pass this trigger requirement on missing energy: interactions with the nuclei of the detector material producing hard-to-detect hadrons (so-called “nuclear” processes) and conversion of photons into a pair of muons (which are also difficult for the ECal to contain). Nevertheless, we still can use a large, unbiased simulation sample in order to ver- ify these previous conclusions and inform our decisions on how the subsequent biased samples should be produced. Figure 4.3 is a key piece of evidence along these lines where we separate the total reconstructed energy (the estimate from our detector) as a fraction of the known beam energy by the amount of energy going into these nuclear processes (so-called “nuclear” energy). We see that as more energy is given to these nuclear interactions, the reconstructed fraction decreases. Eventually, we reach a point where enough energy has gone to these nuclear interactions that we expect the event to be below the trigger threshold and be kept by the trigger system at a significant rate. Figure 4.3 shows that a significant fraction of events whose total nuclear energy is greater than 62.5% of the beam energy fall below the threshold for the trigger. This figure also shows that there are other types of events that do not have much (if any) nuclear interactions (since their nuclear energy is less than 10% of the beam energy) but still are reconstructed below the trigger threshold. Further investigation reveals 39 0.0 0.2 0.4 0.6 0.8 1.0 EECal/EBeam 100 101 102 103 104 105 106 107 Ev en ts LDMX Preliminary Below Trigger Threshold 109 EoT (4GeV) Nuclear Energy Fraction < 10% Between 10% and 62.5% > 62.5% All 0.0 0.2 0.4 0.6 0.8 1.0 EECal/EBeam 100 101 102 103 104 105 106 107Ev en ts LDMX Preliminary Below Trigger Threshold 109 EoT (8GeV) Nuclear Energy Fraction < 10% Between 10% and 62.5% > 62.5% All Figure 4.3: The total reconstructed energy within the ECal as a fraction of the beam energy separated by the total nuclear energy within the event (the energy going into the photon-nuclear and electron-nuclear processes). that these events are indeed “di-muon” events (Figure 4.4). These two categories of processes – nuclear and di-muon – are the primary background processes since they mimic our missing energy trigger selection at a significant rate. 4.2.2 Expanding Production of these Processes As mentioned in Section 4.1.2, the unfiltered and unbiased simulation is not very effi- cient. In some sense, this shows that our basic trigger selection is doing a good job of rejecting the background processes that our analysis finds uninteresting; nevertheless, we need to efficiently sample events that contain interesting background processes at a large scale so we can analyze them in detail. This is where the biasing and filtering tech- nique described above is employed – our main goal is to improve the speed with which we can generate events originating from these background processes while maintaining faithfulness to the unbiased and unfiltered distributions. The process of tuning these simulation variables is largely a guess-and-check one. We test a variety of different settings and see how they perform both in terms of their computational speed as well as their faithfulness to the underlying distributions. For 40 0.0 0.2 0.4 0.6 0.8 1.0 Muon-Caused Fraction of HCal Hits 100 101 102 Ev en ts / 1B E oT Unbiased Events Passing Trigger with Erec < 0.5Ebeam and Enuc < 0.25Ebeam LDMX Internal Beam 4GeV 8GeV Figure 4.4: Fraction of hits within the HCal that were caused by muons or anti-muons. The events lying within the zero bin still have HCal hits caused by other types of particles. our purposes, the filtering threshold TF can be estimated from the large unbiased and unfiltered sample since that sample contains events with these processes of interest. Specifically, Figure 4.3 displays a value of TF = 0.625EBeam for the nuclear processes. More detailed study of the events residing in the low-side tail of the blue distribution showed that TF = 0.5EBeam was a good choice for the di-muon process; however, since there were fewer events containing the di-muon process within this sample, other choices of TF were surveyed in order to confirm this value. The other threshold TB (which we also set to be TS in this work) is tightly coupled with the biasing factor B and so we test them both together. A rather simple proxy for checking that the events sampled during biasing and filtering matches the underlying distribution is to inspect the longitudinal distribution of where the process ocurred. While the biasing and filtering should improve the speed of the simulation, it should not observably distort this distribution which roughly follows the development of the shower (an example of this is shown in Figure 4.5). We can confirm that our biased and filtered simulations are properly extending the tail of the distributions in which we are interested. Figure 4.6 shows a comparison 41 200 300 400 500 600 Starting Z of + / mm 10 5 10 4 10 3 10 2 10 1 Ev en t F ra ct io n LDMX Internal 4 GeV B = 1e4 R = 4.25MHz B = 5e4 R = 4.81MHz B = 1e5 R = 5.49MHz B = 5e5 R = 6.30MHz B = 1e6 R = 6.86MHz B = 5e6 R = 10.93MHz 200 300 400 500 600 Starting Z of + / mm 10 4 10 3 10 2 10 1 Ev en t F ra ct io n LDMX Internal 8GeV B = 1e2 R = 0.09MHz B = 5e2 R = 0.36MHz B = 1e3 R = 0.58MHz B = 5e3 R = 1.23MHz B = 1e4 R = 1.45MHz B = 5e4 R = 1.94MHz B = 1e5 R = 2.40MHz B = 5e5 R = 5.90MHz Figure 4.5: Longitudinal location (z) where the µ+ was produced within the simulation for a variety of biasing factors B for the 4 GeV beam (left) and 8 GeV beam (right). The rate R is the equivalent number of unbiased EoT divided by the CPU time necessary to produce the sample. We can observe that while increasing B also improves the rate R, we eventually over-bias and lose access to the late tail of the distribution (red, puple, and brown in the left plot and gray in the right plot). between a large unbiased sample that has been scaled to the same equivalent EoT as the biased and filtered samples – there we see that the biased samples properly and smoothly extend the low reconstructed energy fraction tail which corresponds to the signal region in this missing energy search. 4.3 Dark Matter Signal The particular signal process this analysis channel is looking for is the production of a dark photon followed by an invisible decay. In this regime, what happens to the dark photon after it is produced is irrelevant to the analysis since both it and its products are not observable by our detector. With this focus in mind, we developed a dark bremsstrahlung simulation method that allows for the visible particle (the recoiling lepton) to be distributed according to a full matrix element calculation (via MG/ME) while the incident particle can have 42 0.0 0.2 0.4 0.6 0.8 1.0 EECAL/EBeam 101 103 105 107 109 1011 Ev en ts in 1 013 E oT 4GeVLDMX Internal Scaled Unbiased Biased Nuclear Biased Di-Muon Biased Total 0.0 0.2 0.4 0.6 0.8 1.0 EECAL/EBeam 101 103 105 107 109 1011 Ev en ts in 1 013 E oT 8GeVLDMX Internal Scaled Unbiased Biased Nuclear Biased Di-Muon Biased Total Figure 4.6: ECal reconstructed energy EECAL as a fraction of the beam energy EBeam comparing the unbiased and biased samples scaled to the same EoT. The 4 GeV (8 GeV) beam case is shown on the left (right). variable energy and be handled by Geant4 directly. This novel simulation technique allows for the dark bremsstrahlung process to be treated (from Geant4’s perspective) on the same footing as the background processes while maintaining the precision of a matrix-calculator method. 4.3.1 G4DarkBreM To accurately simulate the kinematics of the dark bremsstrahlung process for electrons in thick targets, the process must be included at the level of experimental simulation instead of using initial state event generators to account for the possibility of energy loss through bremsstrahlung or multiple scattering prior to the dark matter interac- tion. Accurate kinematic simulation of the outgoing electron are required for optimal experimental sensitivity measurements and appropriate design of search strategies. For this reason, we utilize G4DarkBreM [33] which performs this embedding of the dark bremmstrahlung process into Geant4. G4DarkBreM calculates the cross section using numerical integrals of the Weizsäcker-Williams approximation, and the kinematics are simulated using a scaling technique of MadGraph/MadEvent event libraries. The accuracy of the total cross section and kinematics is validated using 43 MadGraph/MadEvent samples at a range of incident lepton energies. 4.3.2 MadGraph/MadEvent For this study in LDMX, the code package used to develop and validate G4DarkBreM was also used to generate the input refernece libraries for sample generation. This code is a custom version of MadGraph/MadEvent4 with the following updates. • Introduction of basic dark sector particles (massive boson and spin-1/2 fermion) which act as the representatives of the dark sector interacting with the standard model particles. • Definition of a nuclear particle (electrically neutral, spin-1/2 fermion) with new couplings to the dark photon including the nuclear form factors. • Updating the definition of the electron to include its small (but non-zero) mass to prevent divergence of the cross section at lower energies. These updates, along with some wrapping code, enable the generation of dark bremm- strahlung events for a range of target nuclei, incident energies, and either incident electrons or muons. As suggested by the validation of G4DarkBreM [33], we generate libraries where the incident electron’s energy drops in steps of 10% down from the beam energy. Since the dark photon is required to be simulated with at least 50% of the beam energy, we stop the library generation also at 50% of the beam energy. The primary nuclei that could dark brem within the ECal are tungsten, silicon, oxygen, and copper so those are the nuclei for which we generate libraries. The rest of the nuclei within the simulated ECal (sodium, calcium, carbon for example) have atomic numbers close to those nuclei already sampled and so can be faithfully simulated with these libraries. In order to avoid duplicating the same event, a unique reference library is generated for each simulation run even when keeping the dark photon mass and beam energy the same. 4.3.3 Characterization These samples, as expected, show that the dark bremsstrahlung process can occur fol- lowing normal shower development peaking at ∼ 1X0 into the ECal (with the exception 44 0 1000 2000 3000 4000 Total Energy of A' / MeV 10 3 10 2 W ei gh te d Ev en t F ra ct io n 4GeVLDMX Internal mA / MeV 1 10 100 1000 0 1 5 10 15 20 25 Depth of DB in ECal / X0 0 100 200 300 400 Z Location of DB / mm 10 4 10 3 10 2 10 1 100 W ei gh te d Ev en t F ra ct io n LDMX Internal mA / MeV 1 10 100 1000 Figure 4.7: Distributions of simulated signal events with a 4GeV beam. The energy of the produced dark photon is shown on the left while the longitudinal location where the production occurred is on the right. of higher mass dark photon samples which are kinematically prevented from being gen- erated further into the shower). Figure 4.7 shows some example distributions from these samples. We can observe the normal shower development behavior in the location of the dark bremsstrahlung and we can see the simulation-level cut on the dark photon energy at half the beam energy (in this case 2GeV) which is imposed to improve com- putational efficiency without losing events that would otherwise pass the downstream trigger requirement. 4.4 Summary While this chapter has been a long and winding road through the simulation infras- tructure of LDMX, the key feature that I would like to emphasize is that it is able to efficiently and reliably simulate both background and signal processes within this search for DM. These samples, after being thoroughly studied and validated, were en- larged to provide a sufficiently sized sample for the downstream analysis of Chapter 5. 45 Sample Biasing Factor Biasing Threshold Signal m max(log10(mA),2)) A /ϵ2 0.5EBeam Enriched Nuclear 200 0.375EBeam Di-Muon 105 0.5EBeam Sample Filtering Cuts Unbiased EECal Front primary ≥ 0.875EBeam Signal EECal Front primary ≥ 0.875EBeam & EA′ ≥ 0.5EBeam Enriched Nuclear EECal Front primary ≥ 0.875EBeam & Etot nuc ≥ 0.625EBeam Di-Muon EECal Front primary ≥ 0.875EBeam & Etot µ ≥ 0.5EBeam Table 4.2: Configuration of the simulation samples used in this analysis. Ebeam is the beam energy being studied (4 or 8 GeV in this work). mA is the mass of the A’ in MeV, ϵ is the dark brem mixing strength, EECal Front primary is the energy of the primary electron at the front of the ECal, EA′ is the energy of the generated A’, Etot nuc is the total energy transferred to nuclear interactions during the event, and Etot µ is the total energy of produced muons. Specifically, the background samples were generated to be statistically equivalent to 1 × 1013 EoT and the signal samples were generated to provide ∼ 1M events for each dark photon mass point to study selection efficiencies. Table 4.2 gives a summary of the configuration used to generate these samples. Chapter 5 Missing Energy Search One of the primary strengths of the LDMX detector design is its ability to use the tagging and recoil tracker system to reject a large number of background events by separating a nominal beam with non-standard energy loss from a nominal beam with standard energy loss or even low-energy beam. Moreover, the tagging and recoil tracker system gives LDMX the potential to further suppress backgrounds or potentially study DM properties by studying the transverse momenta of electrons recoiling from dark- bremsstrahlung candidate events. While this design is optimal for a large number of Electrons on Target (EoT), the strategy has some limitations for a low EoT data run. To limit multiple scattering which ruins momentum measurements, the baseline detector configuration requires a thin (≈ 0.1X0) target. In the early stages of LDMX, when the total EoT will be lower, a different analysis strategy and detector configuration may be optimal to probe the largest amount of the y–mχ phase space. The alternative strategy ignores the dedicated target inside the tracker volume and instead uses the ECal as an active target. The EaT analysis channel for LDMX thus has two primary purposes which can be separated by the timeline over which they are relevant. 1. Short Term: In early running, when the number of EoT is relatively small, the nominal MM analysis will not have obtained significant reach into new DM phase space (yet). The EaT channel serves here as a way to obtain world-leading sensitivity early in the lifetime of LDMX and give the collaboration a first look at 46 47 the data the apparatus has collected. 2. Long Term: As LDMX collects data, the MM analysis enters into unexplored phase space and serves as a better discovery mechanism due to its access to the Tagger and Recoil trackers. The EaT channel, while struggling to suppress com- plicated backgrounds with relatively limited analysis handles, can operate “or- thogonally” in the collected data since its primary selection (an approximately beam-energy electron passing through the Recoil tracker) is inverted relative to the MM analysis (the electron passing through the Recoil tracker has significantly less energy than the beam). An initial study of the EaT analysis channel is the primary focus of Part II of this thesis, focusing primarily on the first (short term) purpose. In this regard, we target an EoT that is reasonable to accomplish early in the running of LDMX and avoids particularly intricate backgrounds. A total of 1 × 1013 EoT fits these requirements by avoiding the charged current production of neutrinos and represents ∼ 2.5% of the first full LDMX dataset which is obtainable within approximately a two weeks of nominal beam time1. Since EaT is expected to be the first physics analysis on LDMX data, we want a simple and robust analysis that can withstand the test of time and the complexities of real data. With these design goals in mind, a simple “cut-and-count” analysis has been devel- oped. The simplicity of this analysis is one of its strengths, enabling it to be applicable despite potential surprises arising from first encounters with real data. The bulk of time and effort on this first investigation was focused on making this investigation possible via the introduction of midshower process filtering and a dark bremsstrahlung simulation process described in Chapter 4. 5.1 Selection The core goal of most search analyses is to develop a selection that avoids events known to be standard processes (i.e. backgrounds) while keeping events containing the process 1Assuming the LDMX detector apparatus and beam delivery is operating according to specifications, we expect the beam to be delivered on a frequency of 37.5MHz with a duty cycle of ≈0.5 and the number of electrons within each bunch to be Poisson distributed with µ = 1. 37.5MHz× 0.5× P (µ = 1, 1)× 2week ≈ 1 × 1013 EoT. 48 0 200 400 600 800 1000 1200 1400 EECal / MeV 10 7 10 6 10 5 10 4 10 3 Ev en ts (I nt eg ra l N or m al iz ed to 1 ) 4GeVLDMX Preliminary Background mA = 1MeV mA = 10MeV mA = 100MeV mA = 1000MeV 0 500 1000 1500 2000 2500 3000 EECal / MeV 10 8 10 7 10 6 10 5 10 4 10 3 Ev en ts (I nt eg ra l N or m al iz ed to 1 ) 8GeVLDMX Preliminary Background mA = 1MeV mA = 10MeV mA = 100MeV mA = 1000MeV Figure 5.1: The total reconstructed energy in all layers of the ECal (EECal). The signal and background distributions are normalized such that their integral is one. The 4GeV beam is shown on the left and the 8GeV beam is shown on the right. The events falling into bins with EECal > 1.5GeV (3.16GeV) for 4GeV (8GeV) are omitted from this plot but included in efficiency calculations. being searched for (in this case, the production of DM via dark-bremsstrahlung). Both the EaT analysis channel and the primary MM channel share the dark-bremsstrahlung signature of missing energy within the ECal relative to the known incident beam energy. This allows for the first selection made to be shared between these two channels – a requirement that the sum of the observed energy in the first twenty layers of the ECal is less than 1.5GeV (3.16GeV) for a 4GeV (8GeV) beam. This preliminary selection acts as the primary trigger for both of these analyses – selecting events that resemble dark- bremsstrahlung via their lower-than-average observed energy during data collection – however, the EaT analysis channel in the early-running scenario may not be collecting data with a fully calibrated energy scale at the trigger level. The potential for mis- calibration motivates tightening the analysis missing energy requirement by 400MeV as well as requiring the sum to be performed over all thirty-four layers of the ECal instead of only the first twenty. Figure 5.1 shows the unit-normalized distributions of this total reconstructed energy within the ECal after the trigger has been applied. After this total energy requirement, there are still backgrounds left and those that 49 remain are largely events where a significant fraction of the energy is carried by particles difficult to contain (or even observe) within the ECal (neutrons, KL, or muons for example). To suppress these types of persistent backgrounds, the event is required to have less than 10 PE deposited in any single bar of the HCal – max(PEHCal) < 10 – well below the typical signal of a through-going muon of 80 PE in any single bar. While this selection does a good job of removing events containing long-living parti- cles, there are some events containing several low-energy particles that range out within the ECal itself. These particles (often due to their number) distribute energy over a range of cells, so we can suppress these events by requiring the spatial distribution of energy within the ECal to be small. We choose to measure the spatial distribution using the energy-weighted root-mean-squared spread of the hits within the ECal2 – the ECal Hit RMS – which is typically larger for these background processes than for a single, truncated electromagnetic shower present within a signal event. We therefore require the ECal Hit RMS to be less than 20mm. These two selection variables are shown in Figure 5.2 where the distributions are shown after all other selections are applied (including the trigger selection and the tighter selection on the total ECal energy including all layers). Table 5.1 shows the tabulation of this selection for the studied simulation samples. Figure 5.4 shows the signal and background total ECal energy distributions after these selections are applied where we can still observe difference in the shapes of these distributions which we will exploit in the final statistical analysis of these samples. An additional, sneaky selection has also been applied due to how the simulation samples were constructed. As mentioned in Section 4.2, the simulation samples have a filter requiring the primary electron to arrive at the ECal with at least 87.5% of the beam energy. This simulation filter is attempting to replicate a requirement by the tracking system that would be applied after the trigger requirement when analyzing real data. Studies of other simulation samples without this tracker requirement showed that while the tracker is required to help confirm a near-full-energy electron arrives at the ECal, it keeps over 90% of the events that are interesting for this analysis (i.e. nothing interesting happens in the thin target). Figure 5.3 shows example figures from 2A “hit” is defined as a single cell within the ECal registering a signal corresponding to 50% (or more) of the energy deposited by a typical muon during the event. 50 0 10 20 30 40 >50 max(PEHCal) 10 2 100 102 104 106 Ev en ts 4GeVLDMX Preliminary N 1 D = 0.5 mA = 3m Background mA = 1MeV; y = 1e-13 mA = 10MeV; y = 1e-11 mA = 100MeV; y = 1e-09 mA = 1000MeV; y = 1e-04 0 10 20 30 40 >50 ECal Hit RMS / mm 10 2 10 1 100 101 102 103 104Ev en ts 4GeVLDMX Preliminary N 1 D = 0.5 mA = 3m Background mA = 1MeV; y = 1e-13 mA = 10MeV; y = 1e-11 mA = 100MeV; y = 1e-09 mA = 1000MeV; y = 1e-04 0 10 20 30 40 >50 max(PEHCal) 10 2 10 1 100 101 102 103 104 105 106 Ev en ts 8GeVLDMX Preliminary N 1 D = 0.5 mA = 3m Background mA = 1MeV; y = 1e-13 mA = 10MeV; y = 1e-11 mA = 100MeV; y = 1e-09 mA = 1000MeV; y = 1e-04 0 10 20 30 40 >50 ECal Hit RMS / mm 10 1 100 101 102 103 Ev en ts 8GeVLDMX Preliminary N 1 D = 0.5 mA = 3m Background mA = 1MeV; y = 1e-13 mA = 10MeV; y = 1e-11 mA = 100MeV; y = 1e-09 mA = 1000MeV; y = 1e-04 Figure 5.2: Variables used in signal selection for the EaT analysis channel. In each figure, all other cuts are applied except for the variable in question. Some of the bins are empty in which case the upper Poisson limit given the total sample size is drawn with error bars. The grey line shows the selection cut. The background sample is the enriched nuclear and dimuon samples. The top (bottom) row shows the 4GeV (8GeV) beam. 51 Analysis Stage for 4GeV Beam Background Signal Efficiency (%) Event Yield 1 MeV 10 MeV 100 MeV 1 GeV ECal Trigger (E20 < 1.5 GeV) 4.60 × 107 58 67 71 83 ECal Energy (EECal < 1.1 GeV) 1.95 × 106 35 48 53 72 max(PEHCal) < 10 1.15 × 103 34 47 52 69 ECal Hit RMS < 20 mm 126 28 37 41 33 Analysis Stage for 8GeV Beam Background Signal Efficiency (%) Event Yield 1 MeV 10 MeV 100 MeV 1 GeV ECal Trigger (E20 < 3.16 GeV) 6.10 × 107 66 74 79 89 ECal Energy (EECal < 2.76 GeV) 6.88 × 106 52 63 69 84 max(PEHCal) < 10 31.8 50 61 67 81 ECal Hit RMS < 20 mm 7 43 52 56 52 Table 5.1: Cut-flow analysis comparing background and various signal hypotheses for the simple cuts used in this analysis. The event yield for the background sample is calculated using the event weights and represent the number of events out of 1013 EoT equivalent. The signal efficiency is relative to the full simulation sample. The efficiency and event yield values on a given row are for after the analysis stage of that row. The first table is the cutflow for the 4GeV beam, and the second table is for the 8GeV beam. these studies that confirm the necessity of the tracker while also displaying that it is highly efficient. Due to the tracker requirement’s high efficiency and the large separation between events that are acceptable into this analysis and events that are not, we neglect any systematic error (more on this in Section 5.3) related to this selection and instead plan to set the tracker requirement’s value to whatever threshold is necessary to keep the 90% efficiency when studying real data. 5.2 Background Prediction When performing a search for new processes, a common technique is to develop a well- motivated prediction for the background processes and then look for excesses above this prediction. Before we go through rigourously defining what an “excess” means and how to quantify what types of new processes we would exclude if no excess is observed, we should more precisely define what our background prediction is. We begin with fitting the cumulative background distribution with a simple expo- nential function. Figure 5.5 shows this cumulative distribution, the resulting fit, and the 95% confidence intervals surrounding the fit. This fit is a helpful prediction tool 52 50 100 150 200 250 z / mm 0 500 1000 1500 2000 2500 3000 3500x / m LDMX Internal ptarget T = 0 for all x Difference Between 4.0 GeV - 3.50 GeV 4.0 GeV - 1.50 GeV 8.0 GeV - 7.00 GeV 8.0 GeV - 3.16 GeV (a) Difference in x position along the electron’s path between different energy electrons. This shows that the change in position is smaller than a single ECal cell width, but it is well within a conservaitive alignment of the track- ing detector. 0.0 0.2 0.4 0.6 0.8 1.0 Primary e Energy Fraction at ECal 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Tr ig ge r S um F ra ct io n 107 EoT (8GeV)LDMX Internal 100 101 102 103 104 105 Ev en ts (b) Comparison of trigger sum in the ECal to the primary electron’s energy at the ECal. Below the horizontal gray line are the events that pass the trigger where there is large sepa- ration between events where something inter- esting happens in the target (left) and in the ECal (right). Figure 5.3: Studies of how the tracker requirement affect the simulation samples. 53 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 EECal/EBeam 10 2 10 1 100 101 102Ev en ts 1013 EoT (4GeV)LDMX Preliminary D = 0.5 mA = 3m Background mA = 1MeV; y = 1e-13 mA = 10MeV; y = 1e-11 mA = 100MeV; y = 1e-09 mA = 1000MeV; y = 1e-04 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 EECal/EBeam 10 1 100 101 Ev en ts 1013 EoT (8GeV)LDMX Preliminary D = 0.5 mA = 3m Background mA = 1MeV; y = 1e-13 mA = 10MeV; y = 1e-11 mA = 100MeV; y = 1e-09 mA = 1000MeV; y = 1e-04 Figure 5.4: The total reconstructed energy in all layers of the ECal (EECal) as a fraction of the beam energy (EBeam) for all samples that pass the selection criteria except the selection on ECal energy. The 4GeV beam is shown on the left and the 8GeV beam is shown on the right. The gray lines mark the edges of the analysis bins used to estimate the expected exclusion limit and the black line is the upper limit on the ECal energy which also serves as the upper limit of an analysis bin. 54 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Efrac = EECAL/EBeam 10 1 100 101 102 103 N bk gd b el ow E fra c 4GeVLDMX Preliminary AeB(1 Efrac) A = 1.82e+11 B = -28.9 Cumulative Bkgd MC 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Efrac = EECAL/EBeam 10 1 100 101 N bk gd b el ow E fra c 8GeVLDMX Preliminary AeB(1 Efrac) A = 2.78e+05 B = -16.6 Cumulative Bkgd MC Figure 5.5: Exponential fit of the total simulated background distribution. The shaded region is a 95% confidence band on the fit which is interpreted as the uncertainty on the value of the fit. Some bins in the simulated distribution are empty in which case the upper Poisson limit given the sample size is drawn as an error bar in that bin. in two keys ways. The fit produces a meaningful prediction across the entire range of observable values which is a helpful way to “smooth” out this sample that is limited in size after the selection criteria already applied. Moreover, the fit uses the 400MeV-wide control region between the upper limit on the total ECal energy and the trigger thresh- old applied during readout to help constrain its parameters. This constraint reduces the uncertainty on the background prediction while also serving as an example of what an analysis using real data could do – this same region could be utilized in order to help set the scale of the background in a data-driven way. The fit is then used to provide a background prediction within each of the three analysis bins, shown in Figure 5.6 along with the unconstrained simulation prediction for comparison. 55 0.00 0.05 0.10 0.15 0.20 0.25 Analysis Bin Efrac = EECAL/EBeam 100 101 102 N bk gd / bi n 4GeVLDMX Internal Bkgd MC AeB(1 Efrac) A = 1.82e+11 B = -28.9 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Analysis Bin Efrac = EECAL/EBeam 10 1 100 101 N bk gd / bi n 8GeVLDMX Internal Bkgd MC AeB(1 Efrac) A = 2.78e+05 B = -16.6 Figure 5.6: Background prediction within the three final analysis bins (blue) compared to the unconstrained simulation prediction in gray. The uncertainty on the background prediction is taken from the 95% confidence band shown on the fit. Some bins in the simulation prediction are empty in which case the upper Poisson limit given the sample size is drawn as an error bar in that bin. 5.3 Systematics and Background Uncertainty Besides the statistical uncertainty due to the finite size of the simulation sample (whose estimation is represented by the 95% confidence intervals around the fit), there is ad- ditional sources of uncertainty that originate from the experimental design (so-called “systematic” uncertainty). Most prominently in this early-running analysis, we expect mis-calibration of the ECal to be a potentially major source of systematic uncertainty especially since the calibration of the ECal used when applying the trigger selection will probably differ from the calibration used later on during the final analysis.3 Estimating this systematic uncertainty is rather direct; we simply vary the calibration used at the reconstruction and analysis stage within our data processing and see how that changes the results.4 3This difference is expected since we can improve the calibration of the detector as we collect more data; however, acknowledging this difference helps us put requirements on the accuracy of the intial calibration used at the trigger level in order for this analysis to function as expected. 4The estimate for both the ECal and HCal systematics is done with the 4GeV beam sample since that estimate is expected to be more conservative because the 4GeV beam sample allows more background 56 0 250 500 750 1000 1250 1500 1750 2000 EECAL / MeV 10 2 10 1 100 101 102 103 Ev en ts in 6 × 10 9 E oT Signal Region 4GeVLDMX Internal Fail Trigger (Nele = 1) Smeared Unsmeared (a) Events failing the original trigger. 102 103 Ev en ts in 6 × 10 9 E oT 4GeVLDMX Internal Pass Trigger (Nele = 1) Smeared Unsmeared 0 200 400 600 800 1000 EECAL / MeV 0.90 0.95 1.00 1.05 1.10 Sm ea re d / U ns m ea re d (b) Events passing the original trigger and falling within one of the three final analysis bins over EECAL. Figure 5.7: The total reconstructed energy in the ECal EECAL comparing ten different smeared calibrations (colors) to the original unsmeared calibrations (black). We generated ten different calibrations, smearing the calibration constants by 10% in an uncorrelated fashion and by an additional, correlated 2% (5%) within the central (outer) modules of the ECal. These smearing factors are a conservative estimate of the accuracy of the initial calibrations used by the trigger to make data collection decisions. These ten different calibrations were then used to re-reconstruct the simulated events ten different times yielding Figure 5.7 and Figure 5.8. Figure 5.7a reassures us that our signal region (below 1.1GeV) is not polluted with events that would have failed the trigger selection even after these rather large calibration changes relative to the trigger calibrations (i.e. we would not have overestimated our signal efficiency). In addition, we can estimate the resulting systematic variation on the background event yield within the three final analysis bins to be 5% using Figure 5.7b. The final selection on the ECal Hit RMS is only affected negligibly as shown in Figure 5.8 where the distribution only changes by ≲ 5% within the range of potential cuts. Estimating the systematic uncertainty due to the HCal variable is not as straight yield past its selections compared to the 8GeV beam sample. 57 10 2 10 1 100 101 102 Ev en ts in 6 × 10 9 E oT 4GeVLDMX Internal Pass Trigger (Nele = 1) Smeared Unsmeared 0 5 10 15 20 25 30 35 40 ECal Hit RMS / mm 0.90 0.95 1.00 1.05 1.10 Sm ea re d / U ns m ea re d Figure 5.8: Effect of smeared ECal calibrations on the ECal hit RMS for events that pass the original trigger. Outside of statistical fluctuations in the lowest-populated bins, the smearing has a less than 5% effect throughout the range of potential cuts. forward since we are unsure on how best to directly alter the conditions with which the HCal is reconstructed. We can still estimate the systematic uncertainty due to this cut variable by testing a wide range of cut variables, deducing which ones are “acceptable” and then extracting how the background yield varies between these different cut choices. (What we want to be emulating is the underlying distribution shifting relative to our analysis cut, but we are mimicing this by shifting the cut relative to the distribution.) Figure 5.9a shows how the final reach (see Section 5.4) changes depending on this cut. While not much movement is observed, we can see separation ocurring when the threshold goves above ∼ 30 (more clearly seen in Figure 5.9b). With this in mind, we estimate our “tolerance” to a systematic error on the maximum HCal PE to be 1 − B<10/B<20 where B 0 N vt x < 20.0 0.2 0.4 0.6 0.8 1.0 Ef fic ie nc y 10.7pb 1HPS Internal 20 16Event Pre-Selection Data Sim Bkgd SIMP 30 MeV SIMP 60 MeV SIMP 90 MeV Figure 8.3: Relative efficiency of pre-selection of events. Nvtx is the number of vertices passing the vertex pre-selection requirements. Since other triggers for different purposes are present in the data, the Pair 1 Trigger is applied to data but not to simulation. Chapter 9 Displaced Vertex Search A search for visibly-decaying DM is not novel, many experiments have been purposed- built for such a search; however, HPS is specially focused on DM that decays within shorter distances. While HPS has previously searched for simple dark sectors more similar to the benchmark model used within the LDMX missing energy search [48], no DM was found and the relatively longer decay lengths of this dark sector model harmed the signal efficiency of the analysis and decreased the resulting sensitivity. Section 6.1 introduced the idea of SIMP DM which has additional interest for HPS. Specifically, the increase in complexity of the DM model slightly decouples the production rate from the decay rate meaning the number of events where DM is produced is not as tightly connected to how displaced the particles are expected to be within the detector. The downside of this decoupling is that some energy is lost to the production of the lighter dark meson πD when the dark photon decays within the dark sector; however, this also means HPS has not searched this regime since one of the key selections within its previous displaced vertex search was requiring the total momentum of the vertex to be near the beam energy instead of significantly below it. With the samples presented in Chapter 8, a search for SIMPs within the collected data has been performed. An additional, orthogonal category of data was introduced to be combined with prior work, enabling investigation of a larger portion of the already- collected data. 88 89 9.1 Signal Yield The expected signal yield is a crucial part of any kind of search analysis and this one is no exception. The prior search within LDMX benefited from its missing momentum/energy technique in this regard since the mixing strength ϵ is only connected to the scale of the signal yield and not how the signal events would present themselves in the data. In a visible search, the mixing strength also affects the decay length and thus a more intricate signal yield estimate is required to account for the fact that changing ϵ not only changes the magnitude but also changes which events are more likely to be observed within data. In this analysis, we modify the expected signal yield calculation done previously by HPS [48] to accomodate the two viable decay channels within this SIMP model. The fundamental premise of this calculation is re-weighting events based on their decay length so that they appear to have been drawn from a specific mixing strength’s decay distribution. Specifically, we integrate over a decay weighting factor multiplied by the signal efficiency as a function of the decay length in order to estimate the total signal yield for a specific mixing strength. The details of this calculation are presented in this section since they are relatively novel and not presented fully in another pub- lication; however, this does force me to deviate from my previous commitment to few mathematical equations and less jargony language. Skip to Section 9.2 if this detail does not interest you. The magnitude of this estimate is set by the relationship between dark photon production cross section and the radiative trident differential cross section[49]. σA′ = ϵ2 3π 2α mA′ dσγ∗ dm ∣∣∣∣ m=mA′ (9.1) Multiplying both sides by the dataset’s luminosity then gives us event yields. NA′ = ϵ2 3π 2α mA′ dNγ∗ dm ∣∣∣∣ m=mA′ (9.2) The complexity arises when we remember that specifically radiative tridents are not directly observable – they are intertwined with other standard processes that produce 90 the same outgoing particles (Bethe-Heitler tridents for example). Thus, we estimate the radiative trident differential yield dNγ∗/dm by modifying the observable trident differ- ential yield dN3e,CR/dmreco by two simulation-derived factors. The radiative fraction frad estimates the fraction of trident events that originate from the radiative process. frad = dNγ∗ dm / dN3e dmreco (9.3) And the radiative acceptance times efficiency Arad estimates the fraction of trident events that are within the geometric acceptance of the detector and pass the trigger and preselection requirements. Arad = dN3e,CR dmreco / dN3e,gen dmreco (9.4) where CR stands for the Control Region in Psum (see Section 9.3 for the definition of this variable). Thus, the expected yield of dark photons created within the detector but not necessarily within its acceptance or passing selection requirements is NA′ = ϵ2 3π 2α mA′ frad Arad dN3e,CR dmreco ∣∣∣∣ m=mA′ (9.5) Both frad and Arad use the event and vertex pre-selection before any downstream anal- ysis selections, so they are common amongst all potential analysis channels that share this pre-selection; however, they both vary as a function of the dark photon mass. Poly- nomials were fit to the simulation-derived values for these ratios and the parameters for these polynomials are given in Table 9.1. These ingredients along with the resulting NA′ is shown in Figure 9.1 using a run corresponding to ≈ 1.6% of the entire dataset. In this signal hypothesis, we do not observe the dark photon’s production or decay, instead, the dark photon decays to an unobservable dark pion and the neutral dark vector meson VD which in turn is what decays back into a standard model electron- positron pair. The above calculation of NA′ accounts for the dark photon’s production, but we need to account for the dark photon’s decay and the dark vector meson’s decay. The first decay has a branching ratio BR(A′ → πDVD) which is complicated by the fact that there are actually two different dark neutral vector mesons that fit our requirements. 91 Coefficient frad Arad c0 0.105 -0.489 c1 / MeV −1.17 × 10−3 0.0737 c2 / MeV2 7.44 × 10−6 −4.34 × 10−3 c3 / MeV3 −1.67 × 10−8 1.34 × 10−4 c4 / MeV4 0.0 −2.36 × 10−6 c5 / MeV5 0.0 2.54 × 10−8 c6 / MeV6 0.0 −1.71 × 10−10 c7 / MeV7 0.0 7.03 × 10−13 c8 / MeV8 0.0 −1.62 × 10−15 c9 / MeV9 0.0 1.60 × 10−18 Table 9.1: Polynomial coefficients for radiative fraction and acceptance functions. 101 102 103 104 N 3e ,C R 0.1712pb 1HPS Internal 20 16 0.05 0.10f ra d 0 50 100 150 200 Mass / MeV 0.0 0.2 0.4A r ad (a) Ingredients to the total dark photon yield calculation including the observed tri- dent yield (top), radiative fraction (middle), and radiative acceptance (bottom). 0 50 100 150 200 mA / MeV 106 107 108 109 1010 N A / 2 0.1712pb 1HPS Internal 20 16 (b) Resulting total dark photon yield. Figure 9.1: Depiction of dark photon yield calculation using a single run of the data (∼ 1.6%). The gray vertical lines show the bounds of the search using the nominal ratio mA′/mVD = 1.66. 92 The second process is embedded in the decay rate of the dark vector meson to electron- positron pairs Γ(VD → e+e−).1 Let E(z) be the signal efficiency of the analysis as a function of the z where the VD decayed into the electron-positron pair. Then we can sum over the possible VD and estimate the fraction of NA′ that produce a VD which decays and passes the analysis requirements. Nsig = NA′ ∫ ∞ ztarget ∑ VD DVD (z)E(z)dz (9.6) where DVD (z) = BR(A′ → πDVD) e−(z−ztarget)/(γcτVD ) γcτVD (9.7) The branching ratio BR(A′ → πDVD) and lifetime τVD are taken from [38] (along with the general procedure of this estimate). What is important to remember is that the lifetime is dependent on ϵ2, so while increasing ϵ2 increases NA′ it also makes the lifetime shorter and thus the VD decay “looks” more like standard un-displaced background. The VD energy (and thus the relativisitic γ) used in DVD (z) is only distributed over a small range (within < O(1GeV)) so we replace it with the mean ⟨γ⟩ in order to make the calculation more practical. Additionally, we impose an upper limit on the decay z to be 25 cm after the target since the efficiency of the reconstruction past ∼ 20 cm drops to zero. Nsig = NA′ ∫ 25 cm 0 dz′ ∑ VD BR(A′ → πDVD) e−z′/(⟨γ⟩cτVD ) ⟨γ⟩cτVD E(z′ + ztarget) (9.8) where z′ = z − ztarget. Figure 9.2a shows Nsig with a uniformly perfect signal efficiency E(z) = 1 to give a sense of scale, but a slightly more realistic example is using a step-wise signal efficiency E(z) = 0.1 if 20mm < z < 100mm and 0 otherwise Figure 9.2b. In summary, the key point to make here is that the signal efficiency as a function of true decay length E(z) is the single analysis-dependent ingredient that effects the resulting signal yield. The rest of the ingredients are either fixed by the dataset being searched (e.g. trident differential production, frad, Arad) or are parameters that we vary 1There are other decay processes of the VD! Namely, the so-called “three-prong” decay VD → πDe+e− would greatly increase the potential rate at the cost of preventing the e+e− from reconstructing the mVD mass resonance. 93 20 40 60 80 100 120 Invariant Mass / MeV 10 8 10 7 10 6 10 5 10 42 0.1712pb 1HPS Internal 20 16 10 3 10 2 10 1 100 101 102 103 N si g ( E( z) = 1) (a) Uniformly perfect signal efficiency with yield dominated by prompt decays. 20 40 60 80 100 120 Invariant Mass / MeV 10 8 10 7 10 6 10 5 10 42 0.1712pb 1HPS Internal 20 16 10 3 10 2 10 1 100 101 102 103 N si g ( E( z) = 0. 1 if 20 .0 < z< 10 0. 0m m el se 0) (b) A step signal efficiency mocking a perfect analysis cut in displaced vertex z. Figure 9.2: Expected signal yield Nsig for some rudimentary example efficiencies. Red contour lines are given at 1 and 1.6% (the approximate fraction of this data subsample) to give a sense of scale. Values are “clipped” to the color bar (for example, a signal yield of 2 × 104 is colored yellow which is marked as 1 × 103) so we can focus on the important orders of magnitude. for the search (ϵ2, mVD ). A moderately realistic signal efficiency shows the range of ϵ2 and mVD in which this dataset has potential sensitivity (Figure 9.2b) which informs the ranges these values will take below. 9.2 Reconstruction Categories The design of HPS separates reconstructed tracks into categories that are not necessarily associated with the underlying physics for which we are searching. Specifically, the number of layers (and which layers) that are included within a track has a large effect on the resulting precision of the reconstructed physics variables of that track; thus, we categorize vertices on whether one or both of their tracks contain both sensors in the first layer. Figure 9.3 shows examples of the two categories considered within this work. The L1L1 category whose verticies have both tracks with good precision was first studied for this signal search[47] when optimizing the pre-selection requirements described earlier. 94 L1 L2 Target L1L1 e− e+ L1 L2 Target L1L2 e− e+ Figure 9.3: Diagrams showing L1L1 (left) and L1L2 (right) vertex examples. Extending this search to the L1L2 category has a rather simple motivation. While the vertices may degrade in precision, the additional increase of data both in terms of total volume and the potential to observe higher-displaced vertices is expected to improve the sensitivity of the HPS SIMP search. 9.3 Physics Variables There are many possible variables that could be used to separate candidate signal ver- tices from the standard process backgrounds. While this section is not an exhaustive list of all possible variables, it does explain the ones used within this analysis as well as motivation for their use. The SIMP signal vertex will have significantly less energy than the amount delivered by the beam because of the production of a light dark meson when the heavier vector meson VD is produced. Thus, a selection on the sum of the momentum magnitudes is applied. Psum = |p⃗e− |+ |p⃗e+ | (9.9) Specifically, the Signal Region (SR) used for the actual search requires 1.0GeV < Psum < 1.9GeV and the Control Region (CR) used for determining the trident differential pro- duction rate is 1.9GeV < Psum < 2.4GeV. Since we are searching for the dark vector boson VD via its 2-body decay into an electron and a positron, we expect the invariant mass of the vertex mreco to be within 95 the resolution of the detector σm of the mass we are searching for mVD . pm = |mreco −mVD | σm (9.10) Applying an upper limit on pm is often refered to as a “mass window” since it results in mreco residing within a small range around mVD . For optimizing the other selections, pm < 2.8 is used. The search for a potential excess of SIMP-like events is using sidebands along mreco in data so no selection on pm is imposed when searching. The estimate of the resulting sensitivity uses the signal region selection from the search of pm < 1.5. Each vertex can be projected back to the target using its position and total mo- mentum, and the location of the vertex at the target can be compared to the beam position extracted from 1% of each data-collection run. The separation between the projected position x⃗ = (x, y) and the beam position µ⃗ = (µx′ , µy′) is then measured and normalized by the width of the beam in that direction (σx′ , σy′). The distribution of the beamspot is allowed to be rotated relative to our chosen axes by the angle θbeam, meaning we need to rotate x⃗ before comparing it to the beam spot. The total “ellipti- cal” separation between the reconstructed vertex projected back to the target and the beam spot can be quantified by Nσ = √( x cos θbeam − y sin θbeam − µx′ σx′ )2 + ( x sin θbeam + y cos θbeam − µy′ σy′ )2 (9.11) I refer to this as VPS since it represents how significantly a vertex deviates from origi- nating at the beam spot. Since the detector’s tracking modules are oriented to be most senstive in the vertical direction, the vertical impact parameter y0 has higher precision compared to the hor- izontal impact parameter. For truly-displaced signal vertices, both tracks making the vertex would have y0 far from zero while background vertices would have at least one track with y0 near zero (undisplaced vertices would have both, but mis-reconstructed fake-displaced vertices could have one far from zero). Figure 9.4 shows some diagrams that illuminate this effect. While there are many ways for a vertex to end up being “fake displaced” (for example, a missing or mis-chosen hit in the first layer like the example shown in Figure 9.4), forcing both y0 to be far from zero removes these background 96 L1 L2 Target Truly Displaced e− e+ ye − 0 ye + 0 L1 L2 Target Fake Displaced ye − 0 ≈ 0mm e− e+ True e+ ye + 0 L1 L2 Target Not Displaced ye − 0 ≈ 0mm ye + 0 ≈ 0mm e− e+ Figure 9.4: Diagrams of examples for how different types of L1L2 vertices effect the observed values of y0. Truly displaced vertices (left) have both tracks whose y0 is far from 0mm while fake displaced vertices (middle) and not displaced (right) have at least one if not both tracks with y0 ≈ 0mm. The diamonds are the reconstructed vertex, the solid circles are the reconstructed hits, and the empty circles are “missed” hits either because the particle did not pass through those sensors (left) or due to some physical or electronic inefficiencies (center and right). processes. This motivates selecting vertices based on requiring the minimum of the two absolute value y0 to be above a certain threshold. y0,min = min(|y0,e− |, |y0,e+ |) (9.12) which more sharply distinguishes truly displaced vertices compared to the vertex z often muddled by fake-displaced vertices where one track is mis-reconstructed at high |y0|. The track fit uncertainty of the vertical impact parameter σy0 is a helpful quality parameter measuring how confident the track fit is in the y0 value. Placing an upper limit on this value for both tracks within a vertex effectively requires both tracks to have good vertical resolution, helping remove some highly-displaced vertices presumably arising from mis-reconstructed tracks. σy0,max = max(σy0,e− , σy0,e+) (9.13) Figure 9.5 displays the distributions of these last three variables for a few example mass points after the other, solidified selections (L1L2 vertex, the momentum sum is in the Signal Region, and the invariant mass falls within the chosen mass window). Vertex z is left for late-stage statistical analysis of the results and – being highly correlated with y0,min– is redundant with this variable. 97 0 5 10 15 20 VPS 10 4 10 3 10 2 10 1 100 Ev en t D en si ty 1.07pb 1HPS Internal 20 16L1L2 and Psum SR |mreco mVD|/ m < 2.8 Data (30 MeV) Data (60 MeV) Data (90 MeV) SIMP 30 MeV SIMP 60 MeV SIMP 90 MeV (a) VPS 0 2 4 6 8 10 y0, min / mm 10 5 10 4 10 3 10 2 10 1 100 101 102 Ev en t D en si ty 1.07pb 1HPS Internal 20 16L1L2 and Psum SR |mreco mVD|/ m < 2.8 Data (30 MeV) Data (60 MeV) Data (90 MeV) SIMP 30 MeV SIMP 60 MeV SIMP 90 MeV (b) y0,min 0.0 0.5 1.0 1.5 2.0 2.5 3.0 y0, max / mm 10 4 10 3 10 2 10 1 100 101 102 Ev en t D en si ty 1.07pb 1HPS Internal 20 16L1L2 and Psum SR |mreco mVD|/ m < 2.8 Data (30 MeV) Data (60 MeV) Data (90 MeV) SIMP 30 MeV SIMP 60 MeV SIMP 90 MeV (c) σy0,max Figure 9.5: Distributions of cut variables for a few example mass points and ≈ 10% of the full data sample. The vertices are required to be L1L2, have their momentum sum be within the signal region, and their invariant mass within the specified mass window. 98 0 2 4 6 8 10 y0, min / mm 10 5 10 4 10 3 10 2 10 1 100 101 102 Ev en t D en si ty 1.07pb 1HPS Internal 20 16L1L2 and Psum SR |mreco mVD|/ m < 2.8 VPS < 4 and y0, max < 0.4mm Data (30 MeV) Data (60 MeV) Data (90 MeV) SIMP 30 MeV SIMP 60 MeV SIMP 90 MeV Figure 9.6: y0,min distribution after the other selections (L1L2, Psum in SR, reconstructed mass lying in mass window, VPS < 4, and σy0,max < 0.4mm). 9.4 Selection The selection on these variables was optimized using a subsample of the collected data amounting to ≈ 10% of the full sample. All of the variables except the last, separating variable (y0,min) were varied while keeping the relative signal efficiency high (> 80%), focused on removing the highly-displaced events leading to the long tail in data observed in Figure 9.5b. Figure 9.6 shows the cleaned y0,min distribution after these selections. There are still a few highly-displaced events left; however, removing them incurs a higher penalty on signal efficiency and leads to a lower resulting sensitivity estimated from this subsample. The final variable y0,min was chosen by maximizing the binomial significance of the expected signal at ϵ2 = 1 × 10−6 over the observed data for this 10% subsample. For this optimization, the expected signal was artificially scaled such that it would be on the same order-of-magnitude as the observed data (specifically, a scaling factor of 0.1/ϵ was found to work well although variations in this scaling factor did not vary the chosen selections by much). The y0,min values chosen by this maximization (Figure 9.7) are ragged due to the 99 0 2 4 6 8 10 y0, min Cut 20 40 60 80 100 120 M as s / M eV 1.07pb 1HPS Internal 20 16 -4 -2 0 2 4 6 8 Z B i (a) Binomial significance (ZBi) for a range of masses and y0,min cut choices with the points maximizing ZBi for each mass drawn in red circles and the linear fit with its flat continu- ation as a red line. 20 40 60 80 100 120 Mass / MeV 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 y 0 ,m in C ut 1.07pb 1HPS Internal 20 16 2 = 1.0e-06 Cuts Chosen by Maximizing ZBi Fit to 40 MeV < mVD < 120 MeV and Extend (b) Values maximing the binomial signficance (blue) along with the linear fit and flat exten- sion (orange). Figure 9.7: Binomail significance (left) being maximized after the VPS< 4 and σy0,max < 0.4mm cuts leading to cuts (right, blue) which are smoothed into a continuous function (right, orange) as described in the text. finite nature of the subsample on which they are being chosen. In order to avoid over- biasing the selection to this arbitrarily-chosen subsample, the actual cuts used in the final analysis are smoothed by fitting a line to the area whose maximumim binomial significance is above zero and then continuining this line with flat values outside this region. In summary, the following cuts developed for this analysis are • L1L2: In the vertex, one of the particles must have a hit in both sensors in layer 1 and layer 2 while the other must not satisfy this requirement. • Low Momentum Sum: 1.0GeV < Psum < 1.9mm • Decay after Target: z > ztarget • Vertex Projects Back to Beamspot: VPS < 4 • Good Vertical Resolution: σy0,max < 0.4mm 100 Parameter Value A 1.66mm B 1.86mm C −5.1 × 10−3mm/MeV D 1.25mm Table 9.2: Parameters for Equation (9.14). A search for a statistically-significant excess of highly-displaced events reconstructed within resolution of a specific mass is then performed at this point. The sensitivity is estimated using two additional cuts. • Within Mass Window: pm < 1.5 • Truly Displaced: y0,min > ycut0,min where ycut0,min(mreco) =  A mreco ≤ 40MeV B + Cmreco 40MeV < mreco < 120MeV D mreco ≥ 120MeV (9.14) and the parameters of this function are given in Table 9.2. These selections lead to 6 highly-displaced events left in 10% of the data which can be seen in Figure 9.8. The total signal efficiency of this analysis as a function of reconstructed z Figure 9.9a is of the same magnitude but slightly further displaced than the L1L1 analysis Figure 9.9b. 9.5 Results With the selection fixed, we can analyze the full sample of data without fear of statistical bias. 9.5.1 Search The first stage in this analysis is to perform the search for an excess of highly-displaced events above what could be expected by the background. The signal model we are searching for would have extra events that are both highly-displaced and centered on 101 0 50 100 150 200 250 mreco / MeV 0 1 2 3 4 5 6 y 0 ,m in / m m L1L2 VPS < 4.0 y0, max < 0.4 mm 1.07pb 1HPS Internal 20 16 100 101 102 103 Ev en ts Figure 9.8: y0,min distribution for the 10% data sample with the ycut0,min function overlayed in red. 20 40 60 80 100 120 Invariant Mass / MeV 0 50 100 150 200 Tr ue V er te x z / m m L1L2 SIMP SimHPS Internal 20 16 0.000 0.005 0.010 0.015 0.020 0.025 Si gn al E ffi ci en cy (a) L1L2 20 40 60 80 100 120 Invariant Mass / MeV 0 50 100 150 200 Tr ue V er te x z / m m L1L1 VPS < 2 and y0, min ZBi Plain Max SIMP SimHPS Internal 20 16 0.000 0.005 0.010 0.015 0.020 0.025 Si gn al E ffi ci en cy (b) L1L1 Figure 9.9: Signal efficiency as a function of z and mass scale mVD for this analysis (left) and L1L1 (right). 102 Region mreco Range y0,min Range A (mVD − 4.5σm,mVD − 1.5σm) (ycut0,min,∞) B (mVD − 4.5σm,mVD − 1.5σm) (yfloor0,min, y cut 0,min) C (mVD − 1.5σm,mVD + 1.5σm) (yfloor0,min, y cut 0,min) D (mVD + 1.5σm,mVD + 4.5σm) (yfloor0,min, y cut 0,min) E (mVD + 1.5σm,mVD + 4.5σm) (ycut0,min,∞) F (mVD − 1.5σm,mVD + 1.5σm) (ycut0,min,∞) Table 9.3: Region definitions for use in background estimation via sidebands. Region F is the signal region in which we are searching for an excess. mVD is the mass point we are searching for, σm is the detector mass resolution evaluated at mVD , ycut0,min is the optimized cut value evaluated at mVD , and yfloor0,min is the maximum value of y0,min such that region C has at least one thousand events in it. a specific resonant mass; thus, the search is performed in the y0,min– mreco space. The expected number of background events is estimated using a variation of the ABCD technique with two sidebands in mass (one below and one above the mass window in which we are searching) and one sideband in y0,min (one below the highly-dipsplaced region). The sidebands in mass are defined in terms of the mass resolution of the detector σm and the values defining these edges were optimized within the L1L1 analysis. The sideband in y0,min is defined such that the region within the mass window but at lower displacment (Region C) has at least one thousand events within it to avoid signal contaminating the background estimation. Table 9.3 gives the definition of these regions. Figure 9.10a shows the search results comparing the expected number of events to the observed along with a corresponding p-value estimated using ten thousand toy experiments.2 Figure 9.10b shows an example of the search calculation using the mass point resulting in the smallest p-value observed. A few mass points had an excess of events observed within the signal region; however, these excesses never exceeded a global significance of 3σ and the masses that do show the largest excess in this channel do not display corresponding excess in the L1L1 channel. Thus these excesses are understood as rare but normal statistical fluctuations. 2Each toy experiment is constructed by sampling a toy values for regions C (normally distributed about actual value), B+D (normally distributed about actual value), and A+E (poisson distributed about actual value) and then re-calculating the expected number in region F. The number of toy experiments whose expected value in F was greater than or equal to the observed number in F divided by the total number of toy experiments is then our estimate p-value. 103 0 2 4 6 8 10 12 Ev en ts 1.5 InvM Window 6.5 InvM Sideband L1L2 VPS < 4, y0, max < 0.4 mm y0, min ZBi Max in Core 10.7pb 1HPS Internal 20 16 SR L1L2 Observed Expected 25 50 75 100 125 150 175 200 Invariant Mass / MeV 10 3 10 1 (a) Expected and observed events (top) with their p-values (bottom). 0 50 100 150 200 250 mreco / MeV 0 1 2 3 4 5 6 y 0 ,m in / m m A B C D EF mtrue = 97 MeV A = 13 E = 1 B = 8018 D = 611 C = 1112 Fexp = C × (max(A + E, 0.4)/(B + D)) = 1.8 Fobs = 9 P Value = 4.2e-04 10.7pb 1HPS Internal 20 16 100 101 102 103 104 Ev en ts (b) Search calculation yielding the minimum p-value (with ycut0,min in light red). Figure 9.10: Search results for the full 2016 data set. 9.5.2 Sensitivity and Exclusion With no statistically-significant excess observed, we can then move towards estimating the sensitivity of the analysis and then to exclude certain regions of parameter space. The sensitivity estimate is rather simple; just a ratio between the expected signal yield and the maximum signal yield allowed by the observed data distribution. Section 9.1 already described the expected signal calculation used here as well as previously within the selection optimization. Figure 9.11a shows the final expected signal yield for this selection using the full data sample. The maximum signal yield allowed is calculated using the OIM [50] which leverages the differing shape between signal and background and allows for the presence of unknown background to alter its scale. The OIM result is shown in Figure 9.11b. Figure 9.12 shows the sensitivity for both this analysis of L1L2 and the L1L1 analysis for comparison along with a contour drawn where the sensitivity equals one after being suppressed by potential systematic errors. Systematic errors arising from the radiative fraction estimate, pre-selection cuts, radiative acceptance estimate, uncertainty in the target position, final selection cuts, mass resolution, beamspot resolution, and difference 104 20 40 60 80 100 120 Invariant Mass / MeV 10 8 10 7 10 6 10 5 10 42 L1L2 10.7pb 1HPS Internal 20 16 0 2 4 6 8 10 12 Ex pe ct ed S ig na l (a) Expected 20 40 60 80 100 120 Invariant Mass / MeV 10 8 10 7 10 6 10 5 10 42 L1L2 10.7pb 1HPS Internal 20 16 4 6 8 10 12 14 16 18 M ax S ig na l A llo w ed E st im at e (b) Maximum Allowed Figure 9.11: Signal yield for the L1L2 reconstruction category with the described cuts showing the theoretical expecation (left) and the maximum allowed (right) derived using the OIM. in shape of Psum distributions were all evaluated leading to total systematic error of < 10% for all but the lowest mass points evaluated (rising up to ∼ 18%). This contour encloses the excluded parameter space where the expected yield ex- ceeds what is allowed. In this region of parameter space, BABAR [7] has excluded the parameters above ϵ2 = 10−6, so this analysis does not exclude new parameter space. Nevertheless, this analysis was able to confirm BABAR’s results and is the first of its kind (displaced-vertex search in the low-Psum region) for HPS and thus opens the door to further refinement and investigation with later and larger datasets. 105 20 40 60 80 100 120 Invariant Mass / MeV 10 8 10 7 10 6 10 5 10 42 L1L2 10.7pb 1HPS Internal 20 16 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Ex pe ct ed / M ax A llo w ed (a) L1L2 20 40 60 80 100 120 Invariant Mass / MeV 10 8 10 7 10 6 10 5 10 42 L1L1 10.7pb 1HPS Internal 20 16 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Ex pe ct ed / M ax A llo w ed (b) L1L1 Figure 9.12: Sensitivity of this analysis with contour drawn including systematic errors surpressing the expected signal yield. Part IV Conclusion 106 Chapter 10 Conclusion Both LDMX and HPS are important and required experiments in our search for the par- ticle nature of DM. While neither has progressed far enough to exclude new parameter space (or potentially discover DM that was previously out of reach), both experiments take novel approaches towards this search. These approaches are required because the traditional tactics have not yielded discovery and are unable to reach into these specific regions of parameter space. The LDMX EaT analysis presented here is novel for LDMX not only due to the missing-energy search style, but also because we took care to estimate the magnitude of potential systematic errors, form a quantitative non-zero background prediction with its own statistical uncertainty, and performed the statistical analysis over several bins. Previous LDMX analyses focusing on zero-background searches, while important and interesting in their own right, do not have the flexibility necessary to be an analysis used with data collected form a newly built apparatus. The EaT analysis channel is such an analysis and it is well prepared to be the first analysis of LDMX data making new physics conclusions about our universe. The HPS SIMP L1L2 analysis extended the SIMP search to a previously-unexplored reconstruction category, enabling us to use a larger fraction of the data already collected by HPS. This SIMP search, while not excluding new phase space with the current data set, shows promise. We have been able to show that a displaced-vertex search with the possibility of missing energy is possible. Extending such a search to unexplored phase space, an area where no other experiments have access, is simply awaiting the necessary 107 108 reconstruction studies in order to analyze later and larger data sets already collected by the detector. As newer and smaller experiments compared to other HEP experiments, different excitements and difficulties arise. While participating in the full breadth of a particle physics experiment has taught me a lot and given me opportunity to carve my own path of learning, the lack of larger support structures has revealed some specifically intricate problems that LDMX and HPS share. 10.1 Simulation Validity The design and initial studies of any experiment requires some simulation of the mea- surements it could perform. After the experiment has been constructed and collects data, the simulation is still incredibly useful for studying how known processes present themselves within the observations. The collected data could have a previously-unknown and rare process within it, so we must be careful to avoid biasing our analysis in such a way that would ruin the validity of our results. A reasonable strategy is simply to avoid studying all of the data at once and instead design and optimize the analysis on a particular subset of the data. This process is called “blinding” and was used in Part III; however, this also naturally means the analysis can easily be optimized only for the special events within the subset and the events outside of this subset are unaccounted for. A valid simulation, one which correctly predicts the scale and shape of the observable distributions, is a valuable tool that could partially or completely replace this blinding procedure. In this case, the simulated observations could be used to help design and optimize the analysis; avoiding bias while also allowing the analysis to explore the full breadth of potential events that could be seen. While the “valid simulation” I describe here is not necessarily possible, we can move forward towards it in order to make future analyses better. One of the first studies that will be done with new LDMX data will be comparing how our simulated events compare to the events actually being collected by the real detector. Returning to these comparisons for HPS and its subsequent, larger sets of data is being done and will be beneficial for those analyses as well especially given the separation between data and 109 simulation already being observed within the pre-selection (for example, in Figure 8.1). 10.2 Tracking Reconstruction Most of my work on tracking and the alignment of a tracking detector was done within the context of HPS. This particular tracking detector is somewhat of a “trial by fire” due to its delicate, two-sided nature required by the specific physics for which we are searching. Nevertheless, this experience has given me a solid introduction to charged particle tracking. 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Computing and Software for Big Science, 6, 2022. Acknowledgements Dedication Abstract Contents List of Tables List of Figures I Laying the Groundwork Introduction Standard Model of Particle Physics Standard Model Analogies Particle Mixing Displaced Particle Decays Dark Matter Dark Matter? Evidence for Dark Matter Particle Nature of Dark Matter Invisible Signature Visible Signature II LDMX Light Dark Matter eXperiment Missing Momentum Signature The Beam Line Detector Design Mid-Shower Simulation General Data Process Data Processing Stages Biasing and Filtering Technique Standard Processes Dominant Contributors in this Analysis Expanding Production of these Processes Dark Matter Signal G4DarkBreM MadGraph/MadEvent Characterization Summary Missing Energy Search Selection Background Prediction Systematics and Background Uncertainty Reach III HPS The Heavy Photon Search Experiment Strongly Interacting Massive Particles Detector Apparatus Silicon Vertex Tracker Electromagnetic Calorimeter Alignment Alignment Procedure Detector Visualization Results Data Set Collected Data Pair 1 Trigger Simulation Reconstruction Analysis Pre-Selection Displaced Vertex Search Signal Yield Reconstruction Categories Physics Variables Selection Results Search Sensitivity and Exclusion IV Conclusion Conclusion Simulation Validity Tracking Reconstruction Reflection References