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Searching for low-mass dark matter with SuperCDMS Soudan detectors Page, William Alexander 2019

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Searching for Low-Mass Dark Matterwith SuperCDMS Soudan DetectorsbyWilliam Alexander PageB.A., Bowdoin College, 2013M.Sc., The University of British Columbia, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)February 2019© William Alexander Page 2019The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the dis-sertation entitled:Searching for Low-Mass Dark Matter with SuperCDMS SoudanDetectorssubmitted by William Alexander Page in partial fulfillment of the re-quirements for the degree of Doctor of Philosophy in Physics.Examining Committee:Scott Oser, Physics and AstronomySupervisorChristopher Hearty, Physics and AstronomySupervisory Committee MemberDouglas Bonn, Physics and AstronomyUniversity ExaminerJiahua Chen, StatisticsUniversity ExaminerLeslie RosenbergExternal ExamineriiAbstractSuperCDMS is a direct-detection dark matter (DM) experiment that usescryogenically cooled germanium and silicon detectors to search for interac-tions between DM particles and detector nuclei, and in this thesis I describemy contributions to the experiment.I start with a brief review of DM and motivate the possibility of its de-tection in underground laboratories with sensitive detectors, and I reviewthe SuperCDMS detector fundamentals. Then I focus on detector develop-ment for the future generation of the experiment, which will deploy an arrayof detectors at SNOLAB in Sudbury, Canada. Specifically I describe char-acterization of prototype detectors from surface facility testing, and discussmeasurements of critical values that determine the detectors’ sensitivity toDM particles, such as the baseline resolution and the phonon collection ef-ficiency. I also describe analysis techniques developed to measure intrinsicdetector noise in a high radiation environment such as a surface test facility.In the final chapters I describe a DM search analysis using four monthsof data from operation of SuperCDMS detectors in the Soudan Mine innorthern Minnesota. I discuss how a particular detector operating mode,called CDMSlite, lowers the energy threshold of the detectors in order toimprove the sensitivity to low-mass DM particles. I also present new analysistechniques that optimize the sensitivity to low-mass DM particles, includingnoise discrimination with multivariate classifiers, instrumental backgroundmodeling, and a profile likelihood signal and background fitting approach.In this analysis we set an upper limit on the DM-nucleon scattering crosssection in germanium that is a factor of 2.5 improvement over the previousCDMSlite result for a DM mass that is five times the proton rest mass.iiiLay SummaryIt has been well established through astrophysical observations that thereis a form of matter that we cannot see (“dark matter”) that is five timesmore abundant than ordinary matter (i.e. stars, planets, dust, etc.) in theuniverse. Through gravity, dark matter is responsible for the formation ofmost galaxies, and in that sense we owe our existence to it, but little isknown about its character. Leading theories posit that it consists of a haloof particles surrounding most galaxies, and in this model many thousands ofdark matter particles stream through the Earth every second with low inter-action rates with normal matter. In this thesis I describe my contributionsto an experiment designed to detect the rare collisions between dark matterparticles and detector material. In an analysis of data from the experiment,I rule out smaller interaction rates between dark matter and these detectorsthan previous analyses.ivPrefaceChapter 1 and 2 contain numerous citations to recognize the body ofwork of others upon which a thesis such as this is built. These contribu-tions include the establishment of the WIMP dark matter paradigm withcontributions dating back to the 1920s, as well as the development of CDMSdetectors dating back to the early 1990s. These contributions of others arediscussed in order to set the stage for the later chapters.Chapter 3 is based on detector characterization work carried out at theUniversity of California, Berkeley under the supervision of Professor MattPyle and research scientist Bruno Serfass. The derivations presented in thischapter resulted from conversations with Matt Pyle. Bruno Serfass assistedgreatly with software issues that arose when operating the detectors andanalyzing the data. Nicholas Zobrist (now a graduate student at Universityof California, Santa Barbara) took on the majority of the work requiredto maintain stable operation of the dilution refrigerator for the 1–2 weekintervals during which we tested a series of detectors. The data analysispresented in this chapter is my own.Chapter 4 and 5 are based on an analysis that has been submitted forpublication. I was the lead author and responsible for a significant por-tion of the analysis presented, and co-led the coordination of the analysiseffort with D’Ann Barker. We co-chaired weekly working group meetings.Major analysis contributions came from D’Ann Barker and Ryan Under-wood. The “Background Models” section (Sec. 5.1) is largely adapted fromthe manuscript and is based on work carried out by D’Ann Barker. RyanUnderwood carried out the majority of the work in the “Fiducial Volume”section, with conceptual contributions from Wolfgang Rau and some tech-nical contributions from myself. Eleanor Fascione, Dan Jardin, and AndrewScarff also contributed to the data analysis. Rob Calkins, Scott Oser, andWolfgang Rau supervised the analysis. As the lead author I contributedsignificantly to the manuscript composition and ushered it through internalcollaboration review as well as through journal peer review.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxiv1 Detection of Particle Dark Matter . . . . . . . . . . . . . . . 11.1 Early Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Galaxy Clusters . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . . . . . 51.4 The Cosmic Microwave Background . . . . . . . . . . . . . . 71.5 Composition Hypotheses . . . . . . . . . . . . . . . . . . . . 81.5.1 The WIMP Hypothesis . . . . . . . . . . . . . . . . . 91.5.2 Axions . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.3 Dark Sector Particles . . . . . . . . . . . . . . . . . . 111.6 Dark Matter Detection . . . . . . . . . . . . . . . . . . . . . 121.6.1 Direct Detection . . . . . . . . . . . . . . . . . . . . . 121.6.2 Collider Production . . . . . . . . . . . . . . . . . . . 181.6.3 Fixed Target . . . . . . . . . . . . . . . . . . . . . . . 191.6.4 Indirect Detection . . . . . . . . . . . . . . . . . . . . 201.7 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . 21viTable of Contents2 Detectors for the Cryogenic Dark Matter Search . . . . . . 222.1 Semiconductor Detector Physics . . . . . . . . . . . . . . . . 222.1.1 Electron Recoils . . . . . . . . . . . . . . . . . . . . . 232.1.2 Nuclear Recoils . . . . . . . . . . . . . . . . . . . . . 242.1.3 Yield Discrimination . . . . . . . . . . . . . . . . . . 252.1.4 Phonon Generation and Propagation . . . . . . . . . 262.2 Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.1 Measuring the e−/h+ Energy . . . . . . . . . . . . . . 312.2.2 Measuring the Phonon Energy . . . . . . . . . . . . . 322.3 Optimal Filter Event Reconstruction . . . . . . . . . . . . . 332.3.1 Simple 1D Optimal Filter . . . . . . . . . . . . . . . . 342.3.2 Non-Stationary Optimal Filter . . . . . . . . . . . . . 362.3.3 Two Template Optimal Filter . . . . . . . . . . . . . 382.3.4 Use of Different OF Algorithms . . . . . . . . . . . . 382.4 iZIP Interleaved Design . . . . . . . . . . . . . . . . . . . . . 402.5 CDMSlite Detectors . . . . . . . . . . . . . . . . . . . . . . . 422.5.1 CDMSlite Biasing and Readout Configuration . . . . 442.5.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . 452.5.3 CDMSlite History: Run 1 and Run 2 . . . . . . . . . 463 Detector Characterization . . . . . . . . . . . . . . . . . . . . 553.1 Detector Resolution . . . . . . . . . . . . . . . . . . . . . . . 553.1.1 Data-Driven Resolution Measurement . . . . . . . . . 563.1.2 Phonon Amplifier Noise . . . . . . . . . . . . . . . . . 593.1.3 TES Voltage Biasing . . . . . . . . . . . . . . . . . . 603.1.4 Phonon Collection Efficiency . . . . . . . . . . . . . . 623.1.5 Optimal Filter Baseline Resolution . . . . . . . . . . 663.2 Detector Leakage Current . . . . . . . . . . . . . . . . . . . . 673.2.1 Detector Ionization Environment at Surface Facilities 693.2.2 Muon Veto Analysis Techniques . . . . . . . . . . . . 703.2.3 Phonon Noise as a Function of Detector Voltage . . . 734 Data Analysis of CDMSlite Run 3 . . . . . . . . . . . . . . . 854.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.2 Salting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2.1 Determining the Target Number of Salt Events . . . 864.2.2 Selecting Events To Replace . . . . . . . . . . . . . . 884.2.3 CDMSlite R3 Salt Application and “Unsalting” . . . 894.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . 904.3.1 Detector Selection and Configuration . . . . . . . . . 90viiTable of Contents4.3.2 Vibration Monitoring . . . . . . . . . . . . . . . . . . 914.3.3 High Voltage Current Monitoring . . . . . . . . . . . 934.3.4 Charge Triggers . . . . . . . . . . . . . . . . . . . . . 944.4 Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.4.1 Energy Corrections . . . . . . . . . . . . . . . . . . . 964.4.2 Nuclear Recoil Energy Scale . . . . . . . . . . . . . . 1004.5 Quality Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 1024.5.2 Prepulse Noise Cut . . . . . . . . . . . . . . . . . . . 1034.5.3 Bad Series Cut . . . . . . . . . . . . . . . . . . . . . . 1044.5.4 Phonon Pulse-Shape Cuts . . . . . . . . . . . . . . . 1044.6 Low Frequency Noise Cuts . . . . . . . . . . . . . . . . . . . 1124.6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . 1124.6.2 High Energy LF Noise Cut . . . . . . . . . . . . . . . 1134.6.3 Detector-Detector Correlations . . . . . . . . . . . . . 1144.6.4 Motivation for Multiple Templates . . . . . . . . . . . 1164.6.5 Boosted Decision Tree (BDT) . . . . . . . . . . . . . 1194.7 Bifurcated Analysis . . . . . . . . . . . . . . . . . . . . . . . 1214.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1234.7.2 Application to CDMSlite Run 3 . . . . . . . . . . . . 1254.7.3 LF Noise Background Correlation . . . . . . . . . . . 1274.7.4 Box Relaxation . . . . . . . . . . . . . . . . . . . . . 1324.8 Fiducial Volume . . . . . . . . . . . . . . . . . . . . . . . . . 1324.8.1 Radial Pulse Simulation . . . . . . . . . . . . . . . . 1354.8.2 Energy Distribution of Reduced NTL events . . . . . 1364.8.3 Radial Distribution of Reduced NTL events . . . . . 1374.8.4 Optimizing the Radial Cut . . . . . . . . . . . . . . . 1374.9 Livetime and Signal Efficiency . . . . . . . . . . . . . . . . . 1384.9.1 Livetime . . . . . . . . . . . . . . . . . . . . . . . . . 1394.9.2 Trigger Efficiency . . . . . . . . . . . . . . . . . . . . 1404.9.3 Signal Simulation . . . . . . . . . . . . . . . . . . . . 1424.9.4 Quality Cut Efficiency . . . . . . . . . . . . . . . . . 1444.9.5 Fiducial Volume Efficiency . . . . . . . . . . . . . . . 1464.9.6 Combining and Parametrizing the Efficiency . . . . . 1475 Profile Likelihood Dark Matter Search with CDMSlite Run3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.1 Background Models . . . . . . . . . . . . . . . . . . . . . . . 1495.1.1 Energy Resolution Model . . . . . . . . . . . . . . . . 1505.1.2 Cosmogenic Activation . . . . . . . . . . . . . . . . . 152viiiTable of Contents5.1.3 Electron Capture of 71Ge . . . . . . . . . . . . . . . . 1545.1.4 Compton Scattering . . . . . . . . . . . . . . . . . . . 1555.1.5 Surface Backgrounds . . . . . . . . . . . . . . . . . . 1565.2 Likelihood and Limit Setting . . . . . . . . . . . . . . . . . . 1585.2.1 Components Needed for the Likelihood . . . . . . . . 1585.2.2 Likelihood Function . . . . . . . . . . . . . . . . . . . 1625.2.3 Limit-Setting Approach . . . . . . . . . . . . . . . . . 1655.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1685.3.1 Background Model Goodness of Fit . . . . . . . . . . 1685.3.2 Exposure of Salt . . . . . . . . . . . . . . . . . . . . . 1705.3.3 Final Spectrum Fit . . . . . . . . . . . . . . . . . . . 1715.3.4 Background Rates . . . . . . . . . . . . . . . . . . . . 1725.3.5 Sensitivity Estimate . . . . . . . . . . . . . . . . . . . 1755.3.6 DM Limit . . . . . . . . . . . . . . . . . . . . . . . . 1766 Conclusion and Future Outlook . . . . . . . . . . . . . . . . . 178Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181AppendicesA Voltage Scan Calibration Results . . . . . . . . . . . . . . . . 211A.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211A.2 Significance of Calibration Systematics . . . . . . . . . . . . 211ixList of Tables2.1 Electron-hole pair creation energies (in eV) and bandgap en-ergies for germanium and silicon. Values from Ref. [62]. . . . 232.2 Detector amplifier specifications for different generations ofthe germanium CDMS detectors. Adapted partially fromRef. [66]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 Important time and frequency values for charge (e−/h+) andphonon amplifier digitizers from SuperCDMS Soudan. R, N ,T , ∆f , ∆t refer respectively to the digitization rate, the num-ber of bins in the read out time stream, the time length of theread out time stream, the lowest resolvable frequency (1/T ),and the time length of one digitization bin. . . . . . . . . . . 342.4 The primary use of the three different OF algorithms. . . . . 402.5 The energies and probabilities of electron capture for 71Ge.The N -shell is included, though events from this EC havenever been observed in the CDMSlite detectors because ofthe low energy of this process. . . . . . . . . . . . . . . . . . 463.1 Measured resistances and bias currents (Ib) by channel on theS12C detector.) The values below the double horizontal lineare not directly measured, but rather derived based on theschematics of Fig. 3.4, where Vb = IbRs, Rl = Rp + Rs, andthe shunt resistor Rs is taken to be 0.024Ω based on historicalmeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 The leakage current, as well as the effective total ohmic de-tector resistance, derived from the listed excess noise levelsat low frequency on the S12C detector. . . . . . . . . . . . . 815.1 Background components considered in the likelihood fit . . . 1515.2 Reconstructed energies and resolutions of the 71Ge decay peaksand the baseline noise in CDMSlite Run 3. . . . . . . . . . . 1525.3 Best-fit energy resolution parameters of the model in Eq. 5.2for Period 1 and Period 2. . . . . . . . . . . . . . . . . . . . 152xList of Tables5.4 Cosmogenic isotopes that decay via electron capture and arepresent in the measured CDMSlite spectrum. The shell ener-gies µ, given in keV, are from Ref. [130]. The amplitudes Λ,from Ref. [131], are normalized with respect to the K shell. . 1545.5 Compton model parameters for CDMSlite, normalized overthe energy range 0–20 keV. All values have been multipliedby a factor of 103 and are in units of keV−1. . . . . . . . . . 1565.6 Constrained or bound variables in the likelihood fit . . . . . . 1655.7 Average single-scatter event rates for energy regions betweenthe activation lines in Run 2 and Run 3, corrected for ef-ficiency. All errors contain ±√N Poissonian uncertainties,and the lowest energy range values additionally include un-certainty from the signal efficiency. . . . . . . . . . . . . . . . 1756.1 The projected exposures and detector parameters for the fourtypes of SNOLAB detectors: Ge iZIP, Si iZIP, Ge HV, Si HV.The exposures are based on 5 years of operation with 80%live time. The HV detectors do not have ionization sensors.The arrangement will be 4 towers of 6 detectors each. Tablereproduced from Ref. [97]. . . . . . . . . . . . . . . . . . . . 180xiList of Figures1.1 (left) The hydrogen surface densities (azimuthally averaged)for five galaxies and (right) the rotation galactic rotationcurves. The bars under the galaxy names indicate the spa-tial resolution of the measurements. R80 corresponds to thegalactic radius containing 80% of the Hydrogen density de-termined from the 21 cm emission. Plot copied from Ref. [5],with permission. Measurements were performed by Rogstadand Shostak in 1972 from the Owens Valley Radio Observa-tory using 21cm observations. . . . . . . . . . . . . . . . . . . 21.2 1E 0657-56 ( c○ 2004 X-ray: NASA/CXC/CfA/M.Markevitchet al.; Optical: NASA/STScI; Magellan/ U.Arizona/ D.Cloweet al.; Lensing Map: NASA/STScI; ESO WFI; Magellan/U.Arizona/ D.Clowe et al., by permission) [11] [12] . . . . . . 41.3 Y is the 4He mass fraction of all baryons. The blue, red, andgreen bands are the modeled number densities of 2H, 3He,and 7Li, respectively, relative to 1H as a function the baryondensity. Yellow boxes show measurements (95% CL) of thelight element abundances; of note is that only upper limitson 3He abundance have been made, and the measurements of7Li are in slight tension with the model. The pink hatchedband shows the inferred baryon to photon ration from the 2Hmeasurement, while the blue hatched band shows the rangeindicated by CMB measurements. From [14] with permission. 61.4 CMB power spectrum predicted by a Λ-CDM cosmology (i.e.a dark energy (69%) and cold dark matter (26%) dominateduniverse). Data points in red are measurements by the Planckcollaboration [15]. Power spectrum of temperature fluctua-tions in the Cosmic Microwave Background ( c○ 2013 ESA/Planck,and the Planck collaboration, from Planck 2013 results. I.Overview of products and results, by permission). . . . . . . 7xiiList of Figures1.5 Number density of WIMPs in the Universe as a functionof time, where the relic density depends on the WIMP an-nihilation cross section, σχχ¯ ( c○ NASA/IPAC ExtragalacticDatabase (NED) which is operated by the Jet Propulsion Lab-oratory, California Institute of Technology, under contractwith the National Aeronautics and Space Administration, bypermission). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Current (solid) and projected (dashed) 90% limits on theWIMP mass vs. WIMP-nucleon cross section parameter space.The yellow shaded region represents the DM cross section atwhich experiments will observe an “irreducible” backgroundfrom nuclear scattering neutrino events. At lower DM mass,the background from solar neutrinos from 7Be and 8B reac-tions dominate, whereas at higher DM mass atmospheric andthe diffuse supernovae neutrino background (DSNB) domi-nate. Figure from SuperCDMS collaboration approved publicplots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.7 The expected WIMP event rate for the given mχ and spin-independent cross section. σSI=10−41cm2 corresponds to roughlythe cross section reported by DAMA/LIBRA, CRESST, CDMSSi, and CoGent. σSI=10−45cm2 corresponds to a cross sec-tion excluded only by results in the last 2–3 years. InternalCDMS figure, from [46]. . . . . . . . . . . . . . . . . . . . . . 171.8 The depth of different underground sites, shown as muon fluxon the y-axis, and meters of water equivalent (mwe) on thex-axis, from different generations of CDMS experiment. SUFstands for Stanford Underground Facility. . . . . . . . . . . . 181.9 Limits on the DM-nucleon scattering cross section, as a func-tion of DM mass from the CMS (left) and ATLAS (right)experiments for different sets of model assumptions. Figuresfrom [47, 48]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Estimated energy vs. electron range in germanium and sil-icon, based on the electron stopping power from Coulombinteractions. Figure adapted from [61]. . . . . . . . . . . . . 242.2 Measurements of the ionization yield of nuclear recoil events,as a function of energy. The prediction from the Lindhardmodel is overlaid. From Ref. [73]. . . . . . . . . . . . . . . . 26xiiiList of Figures2.3 Ionization yield versus recoil energy for calibration data, froma CDMS II detector. The blue data was acquired during ex-posure of the detector to a neutron 252Cf source producingprimarily NRs, which accordingly have a measured ionizationyield less than 1. The red data is from a 133Ba calibration,which primarily consists of ERs from γ interactions in thedetector. The black bands correspond to the ±2σ area inthe ionization yield vs. recoil energy plane where ERs andNRs occur. These bands provide a visualization of the dis-crimination in this plane, and a cut in this plane would becustomized for a specific analysis (not necessarily at the ±2σlevel). From Ref. [46]. . . . . . . . . . . . . . . . . . . . . . . 272.4 A simple depiction of an event in the detector and the re-sulting prompt, NTL (or Luke), and recombination phononproduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5 (top) A cartoon of the iZIP detector, with the individualphonon channels labeled and uniquely colored. (bottom) Acharacteristic phonon signal (i.e. pulse) from the bulk of aniZIP detector. The different pulse shapes on the individualchannels indicate that this event occurred close to channelDS2 and CS1. . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6 (left) An example simulated pulse with the template (nor-malized to the best fit amplitude as given by the optimal fit).The template is also left-right aligned with the best fit timedelay as given by the optimal fit. (right) The frequency do-main representation (magnitude only) of the template, noise(PSD), and optimal filter. . . . . . . . . . . . . . . . . . . . 352.7 (left) Example 10keV total phonon pulses with the OF tem-plate overlaid in blue. (right) The residuals (phonon pulse− template) showing that the position-dependent part of thepulse can be treated as a non-stationary source of noise. . . 372.8 (top)An example of the 2 template fit to data, where theprompt signal is modeled by the fast template and the ballisticsignal is modeled by the slow template. (bottom) An exampleof the 2 template fit performed on the 4 individual phononchannels for a single event on a CDMSlite detector. . . . . . 392.9 The iZIP design with interleaved charge electrodes (±2V ) andphonon rails (0V). Used with permission, from [81]. . . . . . 40xivList of Figures2.10 (left) The electric field and potential lines produced from thephonon rails (yellow) and charge electrodes (green). Noticethat the unique surface E-field extends ∼1mm into the crys-tal and therefore surface events within this margin shouldexhibit asymmetric charge collection. (right) Data from iZIPdetector T3Z1 showing surface event discrimination (discus-sion in main text). Internal CDMS figure, used with permis-sion, from [81]. . . . . . . . . . . . . . . . . . . . . . . . . . . 412.11 (left) The iZIP phonon channel partition on a single side,with channel names labeled. (right) The px-py partition foruniformly distributed events for an iZIP detector, where thetriangular shape is due to the channel layout as well as thepartition variable definition (Eq. 2.17, 2.19). . . . . . . . . . 432.12 A rough schematic of the custom CDMSlite biasing electron-ics, which shows that only a single side of the detector is readout while the other side is biased using the “High Voltage”power supply (credit: R. Thakur [83]). . . . . . . . . . . . . 442.13 DM recoil spectra for different DM masses. The vertical linesshow example energy thresholds and demonstrate the impor-tance of threshold for sensitivity to a low mass DM-nuclear re-coil scattering. The CDMSlite detectors have achieved thresh-olds of ∼0.5keVnr. (credit: M. Pepin [84]) . . . . . . . . . . . 452.14 The reconstructed total phonon energy of data with 71Ge elec-tron capture peaks present. The colors label data taken atdifferent detector voltages, where the NTL amplification isapparent at the 30V, 50V, and 70V data. The energy scale ofthe data at each voltage is calibrated so that theK-shell peaksappear at the correct energy (i.e. the total phonon energygiven by Eq. 2.9), and additional details and insights gainedfrom these “voltage scan” data are discussed in Sec. A.1. . . 47xvList of Figures2.15 (top) A simple depiction of a regular event from a particleinteraction alongside two leakage events, one from the detec-tor bulk and one at the detector interface. (middle) FromP. Luke’s publication, showing the signal-to-noise benefits ofNTL amplification up to ∼ 140V for the diode detector de-scribed in Ref. [76]. (bottom) The baseline total phonon noiseRMS, labeled as “σOF [keVt ]” for the T5Z2 detector, showingan increase in the noise above ∼ 60V and a severe worseningof the noise above ∼ 70V. The triangles on this plot pro-vide supplemental measurements of the noise RMS withoutpre-biasing, demonstrating the effectiveness of the pre-biasprocedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.16 A simple schematic of the CDMSlite biasing circuit. . . . . . 512.17 The CDMSlite Run 2 energy spectrum after all cuts have beenapplied, with the energy of the events measured by the non-stationary optimal filter, and converted to the nuclear recoilenergy using a Lindhard k parameter of 0.159. One of the en-ergy intervals is highlighted between events as a visualizationof the energy intervals considered by the optimum intervalmethod. The optimal interval considers intervals between allpairs of events (not necessarily adjacent pairs of events) . . . 532.18 Spin-Independent WIMP-nucleon cross section 90% CL up-per limits from Run 1 and Run 2 (black solid with 95% un-certainty band) compared to the other most sensitive resultsin this mass range (at the time of publication of the Run 2result): CRESST [94] below 1.6 GeV/c2 and PandaX-II [43]above 4 GeV/c2. . . . . . . . . . . . . . . . . . . . . . . . . . 543.1 (left) Example simulated random trigger (noise only) with thebest fit OF template overlaid in red. Even though there is zerosignal, noise fluctuations result in the OF algorithm fittinga small non-zero amplitude. (right) The OF amplitude fitresults (from a detector with two readout channels) of manyrandom triggers, when the OF algorithm has no constrainton the time delay search window [96]. . . . . . . . . . . . . . 573.2 (left) Data from 133Ba calibration of the S12C detector, in thetotal phonon integral vs. total phonon OF energy estimates.(right) The S12C detector baseline resolution measurementusing random triggers and measured using the constrainedtime-offset OF, giving a resolution of 52.4 eV. . . . . . . . . 59xviList of Figures3.3 (top) The current noise (amps/√Hz) for each channel , thetotal phonon channel sum, and the quadrature channel sumfor the S12C iZIP detector. (bottom) The S12C power noise(watts/√Hz), converted from the measured current noise us-ing the (dI/dP ) conversion described in Sec. 3.1.4. . . . . . 613.4 (left) Simplified circuit diagram for the TES amplifier, withthe shunt resistor (Rs), parasitic resistor (Rp), inductor (L),and TES element labeled. (right) The Thevenin equivalentvoltage-biased TES circuit, where Vb = Ib · Rs and the loadresistor is given by Rl = Rp +Rs. . . . . . . . . . . . . . . . 613.5 The simplified TES thermal diagram. In this simplified model,the bath represents the collection of the crystal and the di-lution refrigerator. More realistic thermal models are consid-ered in Ref. [77, 95]. . . . . . . . . . . . . . . . . . . . . . . . 633.6 (left) The phonon collection efficiency (phonon collection) calcu-lated for events in the 100–200 keV range, where Ecollected iscalculated using the total phonon signal (the channel sum).The solid (dashed) histogram represents the Etrue measuredby the optimal filter (pulse integral) energy estimate. (right)The phonon collection efficiency measured on a channel-by-channel basis. The individual S12C channels (of which thereare 8 total) on average collect 2–3% of the event’s energy. . . 663.7 The power noise observed on S12C with channel Cs2 removed,with the detector baseline resolution (from Eq. 3.16) providedin the overlaid text. . . . . . . . . . . . . . . . . . . . . . . . 683.8 A simulated representation of the energy deposition in surfacedetectors with iZIP dimensions (76mm diameter) as well asfor larger diameter (100mm) detectors being fabricated forthe SNOLAB experiment. The time window over which anevent is typically digitized (50ms) is given by the time windowshown by the vertical lines around 13.5 sec. . . . . . . . . . . 693.9 Random traces acquired on a prototype iZIP detector oper-ated at a surface test facility, showing the slopes due to muon“tails” evident to varying degrees on top of the underlyingphonon noise. The labels in the top left of the plots corre-spond to the labels in scatter plot shown in Fig. 3.11. . . . . 713.10 The PSD constructed with slopeless random traces (labeledas “underlying noise”). The dashed PSDs are calculated withrandom triggers with varying steepness of slopes (in units ofamps/ms) in the randoms. . . . . . . . . . . . . . . . . . . . 72xviiList of Figures3.11 (top) The distribution of randoms in the DC value vs slopeplane, showing the approximate acceptance region of this pa-rameter space to select the random traces that are representa-tive of the underlying TES noise. The red symbols highlightthe region in this plane that the raw traces in Fig. 3.9 areselected. (bottom) A projection of the data onto just thetrace slope axis, showing that the selection criteria becomeincreasingly inefficient when the detector is operated at highvoltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.12 Examples of time domain noise from an iZIP-style prototypedetector. The data come from two channels: channel A, top,blue and channel B, bottom, orange. The left (right) panelshow noise traces from these two channels, with the detec-tor at 0 V (44V). The traces read out at 44 V clearly haveworse noise. It is also clear that the excess noise is corre-lated between channels, which is expected from a charge leak-age noise source that stochastically generates NTL phononsdistributed as small phonon pulses. These small pulses areabsorbed equivalently to the standard phonon pulses on theiZIP—equally on all channels in the ballistic limit—whichleads to correlated noise. . . . . . . . . . . . . . . . . . . . . 753.13 (top) Power Noise PSDs measured on a phonon channel on theS12C detector (with negative HV applied to side 2, readingout side 1). Overlaid on each PSD is a leakage current modelusing the measured phonon fall time. (bottom) Power NoisePSDs measured on a phonon channel of the G23R prototypedetector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.14 The black data depicts a time series of the shot noise model,where single charge quanta are distributed with inter-arrivaltimes randomly sampled from a e−λt distribution (i.e. a Pois-son process). For visualization purposes the delta functionshave been widened from being infinitely narrow to 0.5 mswide. The average current used in this simulation was 20×10−18amps (corresponding to an average of 6.25 charge quanta per50 ms interval, which fluctuated to 8 charges in this realiza-tion). The blue data depicts the phonon power generated bythe charges traversing the detector, for a 50 V detector bias,and for a phonon absorption time of τF = 175 µs to matchthe S12C detector. . . . . . . . . . . . . . . . . . . . . . . . . 77xviiiList of Figures3.15 Example of the simulation of a single 50 ms data streamof charge leakage shot noise injected onto fundamental TESnoise. The green pulse is amplified by a factor of 10 (in orderto be visible) and its fall time is τf = 175 µs to match theS12C detector. Many of the pulses are Poisson distributedin time and added to the fundamental TES noise, giving theblue curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.16 PSDs are made of the simulated traces using different detectorvoltages in the simulations. The noise shape and magnitudeare consistent with those predicted by the shot noise model(Eq. 3.23), as well as consistent with the noise shape seen inthe data (e.g. Fig. 3.13). . . . . . . . . . . . . . . . . . . . . 813.17 Leakage current measurements, as a function of voltage, fora number of detectors (primarily prototype, primarily iZIPstyle and size). All detectors here were measured at a sur-face test facility, with the exception of the blue “T5Z2” datapoint. This data point is determined using data acquired fromSoudan, but is a rough estimate for a number of reasons, oneof which is that low frequency noise present in all PSDs fromSoudan masked the shot noise signal. . . . . . . . . . . . . . . 823.18 (left) The total phonon baseline resolution (σt) for a num-ber of detectors as a function of detector voltage. The solidlines express ideal (no leakage current) resolution scalings asa function of voltage where the σt value remains constant as afunction of voltage. The data points represent resolution mea-surements at those voltages, or deduced resolutions based onPSD measurements and Eq. 3.16. (right) The electron equiv-alent energy resolution (σt) as a function of voltage, derivedfrom the left hand plot using Eq. 3.26. . . . . . . . . . . . . 834.1 (left) The CDMSlite Run 2 90% CL OI exclusion spin- inde-pendent cross section limit as a function of DM mass. (right)The number of DM events that correspond to the excludedσSI from the left figure. . . . . . . . . . . . . . . . . . . . . . 874.2 Example “salt” added to the CDMSlite Run 2 final spectrumat the magnitude indicated by the excluded 90% CL for aDM signal. The salt spectrum is scaled to a much higherlevel than the exclusion limit to illustrate its shape. . . . . . 88xixList of Figures4.3 The distribution of 252Cf calibration data in the crycooler vi-brational phase variable (labeled as CryoPreTime Mod 0.83s).As the selection criteria moves to higher energy, the correla-tion of events with the phase decreases. . . . . . . . . . . . . 904.4 (left) A depiction of the SuperCDMS Soudan detector array,with 15 iZIPs total arranged in 5 “towers.” (right) A depic-tion of an iZIP detector read out in CDMSlite mode. Copiedfrom Ref. [82], with permission. . . . . . . . . . . . . . . . . 914.5 The low energy event density dependence on the the cry-ocooler vibrational phase is shown in the red histogram. Goodevents from actual energy depositions, shown in the blue his-togram, are not correlated with the vibrational phase. . . . . 924.6 An event density plot of lower energy events (-2 < keVt <5) that are more likely to be instrumental noise events thanevents from particle energy depositions in the detector. Baand Cf calibrations are highlighted. A Cf period when nodata was read out is between May 01 and May 05. The April14 stripe is from a single series (01150414 1430) when the coldtrap was being cleaned. A second 3 hour series on May 05(01150505 1422) was removed from the plot as it was saturat-ing the color map—this was the first series after the late-RunCf calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.7 The RF room temperature (TE FEB RACK) (oF) and theleakage current (iseg hv current b) (nA) as a function of cal-endar time. The x-axis runs from mid-February to earlyApril, 2015. . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.8 Correlation between trigger rate and detector voltage. Thered curve is the best fit of the trigger rate to the detectorvoltage model of Eq. 4.2. . . . . . . . . . . . . . . . . . . . . 944.9 Representative PSDs from R3a (left) and R3b (right). Thecolored curves represent the noise on the individual channels,while the black curve represents the channel sum. . . . . . . . 964.10 The drift in the total phonon energy of the 10 keV line, withthe HV current correction prediction overlaid. The other en-ergy scale corrections have been applied to the data in orderto highlight the corrections from the HV current. . . . . . . 97xxList of Figures4.11 The steps of the linear base temperature correction. After fit-ting a line in the Ecorr2 (y axis) vs. base temperature (x axis)plane, the linear dependence is removed to form a correctedEcorr3 variable. A Gaussian fit of the K-shell events’ energies(as measured by Ecorr3) shows the energy resolution. . . . . . 984.12 Data from single-sided readout of T5Z2 at 0V. (left) Rawtotal phonon traces plotted from 10 keV events. These eventsall have the same energy (see alignment in the tail of thepulse), and the dark green-blue traces are selected by theirpeakiness. (right) The high energy 0V spectrum asymmetryin the 10 keV line, with the peakier 10 keV events highlightedin cyan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.13 The steps of the linear 2-template correction. After fittinga line in the Ecorr3 (y axis) vs. 2-template fast amplitude(x axis) plane, the linear dependence is removed to form acorrected ptNF variable. A Gaussian fit of the K-shell in thecorrected Ecorr4 variable shows improved energy resolution. . 1004.14 (left) Comparison of the correction factor for Y= 1 and Y =0.15 events. Note that the difference in the correction factorsis very small for small IHV, but for Run 3, with a leakagecurrent up to IHV =120nA, it is clear that we should includethis correction. (right) The maximum % error on a NR energyestimate (Enr), if the total phonon energy (Ept) is correctedwithout consideration of the ionization yield, which resultsin the nuclear recoil energy incorrectly set higher than whatit would be when properly keeping track of the ionizationyield in the HV current correction. The k is the Lindhard kparameter for the yield. . . . . . . . . . . . . . . . . . . . . . 1014.15 The energy scale conversions for Run 3. The differences fromthe Run 2 energy scalings, which are negligible only for smallHV leakage current, are circled in red. . . . . . . . . . . . . . 1024.16 (left) The distribution of the prepulse standard deviation ofthe randoms for a single series, with a Gaussian fit to thedistribution overlaid. (right) The trend of fitted Gaussian σfor all Run 3 series. . . . . . . . . . . . . . . . . . . . . . . . 1044.17 Comparison of glitch and square templates with the goodphonon template. While the square template only roughlyresembles a “square” shape, we refer to it as square becauseit is different from the glitch pulse primarily in that it plateausfor 300µs, which it makes it wider. . . . . . . . . . . . . . . . 105xxiList of Figures4.18 (left) An example of a glitch event with the best fits of thegood event and glitch template to the event overlaid. (right)An example of a LF noise event with the best fits of the goodevent and LF noise template to the event overlaid. . . . . . . 1064.19 (left) The distribution of good events in the ∆χ2glitch vs. ptOFplane, and (right) the same data reprocessed with a singlenoise PSD, showing that the variations are due to the PSDsthat change for each series. . . . . . . . . . . . . . . . . . . . 1074.20 (top) The ∆χ2LF vs. ptOF plane for a portion of the Run 3data, showing a high concentration of LF noise events. Rawpulse inspection revealed that only a small fraction of datapoints in this plot (those data point at the lowest ∆χ2LF val-ues) resembled good events. The ∆χ2LF parameter is calcu-lated with “template 1” as shown in Fig. 4.26. (bottom) Theparabolic fit coefficient in the ptOF-∆χ2LF plane vs. 3.5σ con-tour area, with the clusters shown. . . . . . . . . . . . . . . 1084.21 (top) The maximum likelihood estimate best fit of 2 Gaus-sians plus a flat distribution to noise events in a single series.(bottom) The 9 cryocooler “loud” clusters, and 8 cryocooler“quiet” cluster, all highlighted by time. . . . . . . . . . . . . 1104.22 (top) A raw trace for a glitch-like event that fails the cut.(bottom) The tuning of the parabolic portion of the cut basedupon the parabola of good events, for series block 15. . . . . . 1114.23 (left) The total phonon pulse of the square pulses. (right) the∆χ2square parameter formed from the square template. . . . . 1124.24 The simulated signal (blue data points) significantly overlapsthe LF noise background (the majority of the red data points)at low energy in this variable plane. . . . . . . . . . . . . . . 1144.25 (left) Example traces showing detector correlated LF noise onthe three Tower 2 detectors (top:T2Z1, middle: T2Z2, bot-tom: T2Z3). (right) Comparisons of signal and backgrounddistributions in the T2Z1-T2Z2 detector trace correlation pa-rameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.26 (top) Three different LFN templates used to fit different lowfrequency noise shapes on T2Z1. (middle) The averaged am-plitude of the Fourier transform of the LFN traces. In the redcurve, the time domain traces had zeros added to the frontand end of the trace before the Fourier transform, which is atrick to to improve the δf resolution of the Fourier transform.(bottom) The phase of the black LF noise template. . . . . . 117xxiiList of Figures4.27 (left) The ∆χ2LF vs. ptOF plane for the data, shown in red,and the simulated signal, shown in blue, when using template#1 to fit the raw traces. (right) equivalent to figure left butusing template #2, showing improved separation between thedata (which consists primarily of LF noise events), and thesignal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.28 The average of LF noise events showing a coherent shape(since incoherent noise would average to roughly 0 on they-axis) on T2Z1 (left) T2Z2 (center) and T2Z3 (right). . . . 1184.29 (left) A two-level decision tree. (right) A diagnostic plots ofthe error rate, for both training and test data, to help selectthe optimal number of weak decision trees. . . . . . . . . . . 1204.30 (top) The feature discrimination power of different BDT pa-rameters, which is calculated using the amount that each fea-ture split point improves the sample purity, weighted by thenumber of data points in the branch of interest, and summedover many decision branches. (bottom) The Run 3b BDTscore as a function of energy, showing good separation of asmall collection of signal events from the population of noiseevents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.31 The CDMSlite R3 low background WIMP-search spectrumwith a relatively loose cut placed on the LF noise BDT score.We seek to answer how many of the low energy events are LFnoise triggers as a function of the cut value, and then set acut such that there is <1 event leakage. . . . . . . . . . . . . 1234.32 The 4 boxes into which events are divided when two bifur-cated cuts (cut A and cut B) are applied to the data. Thenumber of events in the upper left box (NAB) represents thenumber of events that pass both cuts. . . . . . . . . . . . . . 1244.33 (top) The two branches of the bifurcated BDTs of Run 3adata, colored by the full BDT score. (middle) The setup cutvalue in the full BDT score. (bottom) The distribution ofevents considered for the bifurcated analysis shown in black,with those rejected by the setup cut shown in blue. . . . . . 1284.34 The distribution of events failing the setup cut, characterizedwith a kernel density. Integrating the kernel density into thesignal region gives 0 events. . . . . . . . . . . . . . . . . . . . 129xxiiiList of Figures4.35 (left) A toy model of a background distribution (blue) and asignal distribution (red) with known correlation coefficient inthe two variables. (right) The full signal + background distri-bution, where the correlation of the background distributionis estimated using the method described above, and agreeswith the true correlation coefficient. . . . . . . . . . . . . . . 1304.36 (left) The distribution of simulated signal in the bifurcatedanalysis cut plane. 97.5% of the simulated signal is containedin the passage box, while 0.47% is in the AB¯ box, 1.9% isin the A¯B box, and 0.13% is in the A¯B¯ box. (right) Thedistribution of the DM search data in the bifurcated analysiscut plane. The points in the shaded pink area pass bothbifurcated cuts. . . . . . . . . . . . . . . . . . . . . . . . . . 1314.37 Examples of cut loosening when the cuts are uncorrelated(left) and correlated (right). . . . . . . . . . . . . . . . . . . 1314.38 (top) The number of events in the AB box, as the value ofcut B is loosened. (middle) The estimated number of signalevents outside the AB box, as the value of cut B is loosened.(bottom) The number of new events in the AB box minusthe number of estimated signal events shown in red, as wellas the bifurcated analysis’s leakage estimate in blue. . . . . . 133xxivList of Figures4.39 (top left) Calculated voltage map for high radius events, show-ing the difference in electric potential ∆V between the finalcollection points of the positive and negative charge carriers,as a function of initial position of the pair (plotted as radiussquared vs. vertical position). Here, the top of the crystal isbiased at 75 V and the bottom is grounded. Charge carriersin the outermost (radius > 800 mm2) detector annulus canexperience less than the full detector bias voltage (credit: D.Barker [113]). (top right) A cartoon of the radial vs. energydistribution for a homogenous and monoenergetic backgroundin the CDMSlite detectors. The Pi and PO refer to inner andouter probabilities, and these variable names are used in theradial cut efficiency calculation of Sec. 4.9.5 (credit: M. Pepin[82]). (bottom left) Distribution of the radial parameter ξ vs.energy in the DM search data. An energy-dependent cut onξ defines the fiducial volume below 2keVee, while a stricterconstant cut is used above 2keVee. (bottom right) Resolu-tion (1σ) for ξ (radial parameter) shown as a function of ξand energy. At lower energy, the resolution worsens as theincreased noise affects the reconstruction of the radial pa-rameter. (credit: R. Underwood [114]) . . . . . . . . . . . . . 1344.40 The Monte Carlo distribution of reduced NTL events in the2T radius (i.e. ξ) vs. energy plane. The colored points arethose that would pass a cut such that (after normalizing theMonte Carlo to the Run 3 exposure) only 1/8th of an eventwould pass into the signal region. (credit: R. Underwood) . 1384.41 Livetime contribution of each series block to the R3 exposure,with the livetime removed by the cuts listed in Sec. 4.9.1shown at the top of each bar of live days. . . . . . . . . . . . 1394.42 (top) Trigger calculation showing the number of events thattriggered both T2Z1 and the other detectors (numerator counts)as well as the number of events that triggered just the otherdetectors (denominator counts). (bottom) Binned efficiencyof the trigger as a function of energy. Error bars are binomialstatistical uncertainties. . . . . . . . . . . . . . . . . . . . . . 1414.43 (left) Corner plot for trigger efficiency fit of µ and σ. (right)Best-fit error function with 100 MC sample curves from the µvs. σ posterior. These samples serve as visualizations of themodel uncertainties. . . . . . . . . . . . . . . . . . . . . . . . 142xxvList of Figures4.44 A cartoon depicting the pulse simulation procedure (credit A.Anderson: [120]). . . . . . . . . . . . . . . . . . . . . . . . . 1434.45 (left)The fast/slow amplitude ratio, as a function of energy,for the two-template fit optimal filter fit. (center) (right)Two L-shell pulses showing the variation in pulse peakinesswhere the center pulse has is the peakiest of the L-shell events(fast/slow ratio = 0.4) and the right pulse is the least peakyof the L-shell events (fast/slow ratio= −0.6). . . . . . . . . . 1444.46 (left) The distribution of simulated signal events in the ∆χ2glitchvs energy plane with the cut boundary overlaid. (center) Theefficiency of the ∆χ2glitch cut calculated with data the simu-lated signal data. (right) The efficiency of all DQ cuts calcu-lated using Eq. 4.26. All figures use series block 7 simulateddata and cut boundaries. . . . . . . . . . . . . . . . . . . . . 1454.47 The signal efficiency with successive application of the trig-ger efficiency, quality cuts efficiency, and fiducial volume cutefficiency. The final data is included with statistical and sys-tematic 1σ uncertainty. Fitting the efficiency model to thesedata gives the final (blue) efficiency curve and the correspond-ing ± 1σ uncertainty band. . . . . . . . . . . . . . . . . . . . 1485.1 Fits of a Gaussian + linear background to the energy spectraof zero-energy (baseline) events and events from each 71Geactivation peak. The widths of the Gaussians are the energyresolution σ. Copied from [126] with permission. (credit: D.Barker) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.2 The spectra (normalized to event density) of surface eventsexpected from the three surface background locations (left:germanium; center: housing; right: top lid). For each loca-tion, the solid curve represents the mean of the expected eventdistribution (ρ0). The shaded band shows the 1σ uncertainty,where the top and bottom edges of the bands correspond toρ+ and ρ− in Eq. 5.7, respectively. Copied from [126] withpermission. (credit: D. Barker) . . . . . . . . . . . . . . . . . 1585.3 (left) The distribution expected (with detector efficiency ap-plied) from a 7 GeV/c2 DM particle for different values of theLindhard k parameter. (right) Background distributions atthe magnitude expected in the Run 3 data (estimated priorto unblinding). . . . . . . . . . . . . . . . . . . . . . . . . . . 160xxviList of Figures5.4 (left) Uncertainty on the shape of the housing background.(right) The uncertainty on the shape and normalization ofthe housing background, as controlled by a single morphingparameter that is given by a single color. . . . . . . . . . . . 1615.5 (left) Likelihood fit, to a pseudo-dataset, with WIMP crosssection freely floating. (right) A likelihood fit, to the samepseudo-dataset, with WIMP cross section clamped to σχ =2.3×10−42cm2. On both plots, the best fit number of back-ground events from the different backgrounds are shown inthe legend, and the likelihood values are shown in the plottitle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.6 An example qσχ,obs relative to a MC produced f(qσχ |σχ) dis-tribution, as well as the theoretical (1/2)× (δ(0) +χ2) distri-bution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.7 Crame´r-von Mises Statistic of the fit to the data (red line)and 1000 pseudo experiments. . . . . . . . . . . . . . . . . . . 1705.8 (left) The R3 final spectrum with all cuts applied, and withsalt included and highlighted). (right) The distribution of salt(with no cuts applied) highlighted in the energy vs. ξ radialparameter plane, with the real data shown in the background. 1715.9 (left) The salt passage fraction with sequential application ofcuts (shown in the legend) compared to the signal efficiencyestimate. (center) The energy spectrum of events, in the un-salted dataset, selected to be replaced by salt events. Roughly19% of them were multiples, consistent with the passage frac-tion given by the left hand figure. (right) The salt passagefraction, but without the multiples and muon veto cut. . . . 1725.10 (top) Background best fit amplitudes overlaid on the Run 3unsalted data. “Sf” is an abbreviation for “surface”. (bot-tom) 1D and 2D posterior distributions for the number ofevents (or morphing parameter in the case of surface back-grounds) contributing from each background. By posterior,we do not mean to imply that these are the results of aBayesian analysis; rather, these are the fit results from thelikelihood, with asymmetric uncertainties determined with aMarkov-chain Monte Carlo sampling algorithm. . . . . . . . . 173xxviiList of Figures5.11 Comparing prior constraints with fit results of the maximumlikelihood for the efficiency, morphing, and resolution param-eters. The similar variance of the prior and posteriors indi-cates that the data are not providing additional informationfor these systematic uncertainty parameters, and the similarmean indicates that our signal efficiency and resolution modelparameters are in agreement with the data. . . . . . . . . . . 1745.12 (left) R3 1σ sensitivity band and PLR limit (with R2 OI limitoverlaid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.13 The CDMSlite Run 3 90% CL PLR limit (this result, solidblack) on the spin-independent WIMP-nucleon cross section,along with the ± 1σ and ± 2σ sensitivity bands (green andyellow respectively). The CDMSlite Run 3 optimum inter-val limit (dashed grey) and Run 2 (red) optimum intervallimit [143] are overlaid. Examples of limits from other detec-tor technologies are overlaid: DarkSide-50 2018 No Quench-ing Fluctuations (magenta) [84]; PandaX-II 2016 (blue) [43];PICO-60 2017 (orange) [144]; CRESST-II 2016 (cyan) [94];CDEX-10 2018 (purple) [145]. . . . . . . . . . . . . . . . . . . 1776.1 The dashed lines show the projected 90% exclusion sensitiv-ity of the SuperSCDMS SNOLAB experiment, calculated us-ing the optimum interval method. The y axis is the spin-independent WIMP-nucleon cross section. The solid linesshow the current 90% from the CRESST-II [94], SuperCDMS[126, 143], and LUX [147] experiments. The dotted yellow lineis the region of parameter space where the solar neutrino co-herent elastic scatter rate begins to mask the interaction ratefrom DM particles, as calculated in Ref. [148]. Figure fromSuperCDMS approved public plots. . . . . . . . . . . . . . . 180A.1 (left) The zoomed-in version of the left hand plot highlightsthe discrepancy between the expected, by extrapolation, lo-cation of the K-shell peak at 0 V. (right) The location of theK- and L- shell peaks as a function of detector voltage. Thehigher voltage peaks (30 V, 50 V, and 70 V) are linear towithin uncertainty, but there is a deviation from linearity at0 V. The overall energy scale (y axis) has used a calibrationconstant for the detector obtained at 4 V, though the overallscaling of the y axis is irrelevant when checking for linearity. 212xxviiiList of FiguresA.2 Large systematics in Luke phonon collection (15%) and re-combination phonon collection (50%) would contribute at mostto a ∼3% error in the nuclear recoil energy scale. The recom-bination phonon error becomes negligible at high detectorbias. Also the error decreases at higher detector voltage, Vb,as the total phonon energy becomes more correlated with theionization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213xxixGlossaryξ Two Template Radial Parameter2T Two Template, in relation to the optimal filter fitting algorithmBDT Boosted Decision TreeCL Confidence LevelCMB Cosmic Microwave BackgroundCLs Limit setting method that protects against downward fluctuating back-ground ratesCryoPreTime the cryocooler vibrational phase (see crycooler)CVM Crame´r-von Misescryocooler The most vibrationally intense component of the SuperCDMSSoudan cryogenic equipment that generates noise in the detectorsDAQ Data Acquisition (System)DFT Discrete Fourier TransformDOF Degree of FreedomDM Dark MatterEDELWEISS Expe´rience pour DEtecter Les WIMPs En Site Souterrain(Experiment to Detect WIMPs in Underground Site)xxxGlossaryEC Electron CaptureER Electron RecoilFET Field Effect TransistorG23R A germanium prototype detector, with the same dimensions andchannel layout as the SuperCDMS Soudan iZIP detector, though only in-strumented on one sideGOF Goodness of FitHS Housing (for surface backgrounds)HV High VoltageiZIP interleaved Z-sensitive Ionization and Phonon DetectorKDE Kernel Density EstimatekeVee keV energy of an event, converting the energy to that produced byan electron recoil (ee: “electron equivalent”)keVnr keV energy of an event, converting the energy to that produced by anuclear recoil (nr: “nuclear recoil”)keVt keV energy of the total phonon energy of an eventLF Low FrequencyLFN Low Frequency NoiseLT Live TimeMC Monte CarloNR Nuclear RecoilNTL phonons Neganov-Trofimov-Luke phononsxxxiGlossaryOF Optimal FilterOI Optimum Interval methodPDF Probability Density FunctionPLR Profile Likelihood RatioPSD Power Spectral Densitypt2T Total phonon energy, estimated with the two template optimal filteralgorithm, in units of keVptOF Total phonon energy, estimated with the 1D optimal filter algorithm,in units of keVptOF0 Total phonon energy, estimated with the 1D optimal filter algorithmwith the time delay forced to 0, in units of keVptNF Total phonon energy, estimated with the non-stationary optimal filteralgorithm, in units of keVR2 CDMSlite Run 2, which occurred over the course of 2014 and consistedof operating the T5Z2 detector at 70 VR3 CDMSlite Run 3, which occurred over the course of 2015 and consistedof operating the T2Z1 detector up to 75 VR3a The first portion of CDMSlite Run 3, before the voltage was stabilizedand in which the detector noise performance was betterR3b The second portion of CDMSlite Run 3, after the voltage was stabilizedand in which the detector noise performance was worseRandom Trigger A time series of data that is triggered randomly with theintention of reading out a time series in which no event occurs in order tocharacterize detector noise. Also referred to as just a “random.”RMS Root Mean SquaredxxxiiGlossaryROI Region Of InterestS12C A Silicon prototype detector, with the same dimensions and channellayout as the SuperCDMS Soudan iZIP detectorSHM Standard Halo ModelSN Series NumberSNOLAB Sudbury Neutrino Observatory LaboratorySUSY SupersymmetrySQUID Superconducting Quantum Interference DeviceT c Superconductor critical temperatureT2Z1 An iZIP detector used in the SuperCDMS Soudan experiment, andin CDMSlite Run 3, in Tower 2 and the first (highest) positionT5Z2 An iZIP detector used in the SuperCDMS Soudan experiment, andused in CDMSlite Run 1 and Run 2, in Tower 5 and the second (middle)positionTES Transition Edge SensorTL Top Lid (for surface backgrounds)WIMP Weakly Interacting Massive Particlez4z5 correlation Waveform correlation between the Z4 (T2Z1) detectorand Z5 (T2Z2) detectorz4z6 correlation Waveform correlation between the Z4 (T2Z1) detectorand Z6 (T2Z3) detectorxxxiiiAcknowledgementsFirst I would like to thank Scott Oser for his supervision over the past fiveyears. Scott was always available when I needed it and the quality of theguidance he offered—from detailed analysis help to general organizationaladvice—will always amaze me. Scott gave me opportunities to broaden myresearch experience by allowing me to work on a range of topics related toCDMS, and this made a real difference in my appreciation of such a complexexperiment. One thing that struck me midway through this degree is thatI always left meetings with Scott feeling more focused, and feeling betterabout life in general, and I take this to be one mark of a great advisor.Within the CDMS collaboration I have worked with many kind andintelligent people. Over the past year I worked most closely with D’AnnBarker as we inched the CDMSlite Run 3 analysis towards completion withweekly group meetings, Skype calls, and countless emails. I was very lucky tohave D’Ann as a partner as we pushed through the challenges of coordinatinga large group effort within a collaboration like SuperCDMS. I can only hopethat my future work partners will be as knowledgeable and on top of thingsas her. Mark Pepin was particularly generous with his time whenever I hada question related to CDMSlite, and in multiple ways he paved the wayfor the CDMSlite Run 3 analysis. Wolfgang Rau provided valuable adviceon many topics in this thesis, and we were lucky that one of the foremostdetector experts connected so frequently to our analysis calls to share hisknowledge. Rob Calkins was a great help in coordinating analysis work.Bruno Serfass was an approachable source of knowledge on anything CDMSrelated even when I was new to the experiment and more of a time sink thana source. Matt Pyle was also exceptionally generous with his time to teachme about detector physics while I was visiting Berkeley. His enthusiasm forhis research is something to aspire to. It has also been a pleasure to workalongside and/or learn from Nick Zobrist, Arran Phipps, Noah Kurinksy,Ben Loer, Lauren Hsu, Jorge Morales, Steve Yellin, Tsuguo Aramaki, RyanUnderwood, Nick Mast, Anthony Villano, Dan Jardin, and Matt Fritts.At UBC, I thoroughly enjoyed the two years of Danika MacDonell’scompany in the office, and I will miss summer volleyball at Jericho beachxxxivAcknowledgementswith her. Working with Belina von Krosigk was also a pleasure. To AlonHershenhorn, Emanuele Michielin, and the numerous office mates who havepassed through room 222, thank you for your willingness to put down yourwork for me to bounce my own ideas off of you. I also must state my grati-tude to the UBC physics department as a whole. It has been a stimulating,and at times quite humbling, experience to be around so many impressivescientists over the past five years.Last, thank you also to all my sailing, scrambling, climbing, and ski bud-dies who accompanied me on adventures in Vancouver and the surroundingwilderness.xxxvChapter 1Detection of Particle DarkMatter1.1 Early EvidenceIn 1933 Fritz Zwicky calculated the mass of the Coma galaxy cluster byway of the virial theorem, using the velocity dispersion of galaxies fromthe measurements taken by Hubble and Humason [1], and found a mass-to-luminosity ratio of roughly 500, significantly higher than expected for galaxyclusters [2]. Three years later Sinclair Smith performed a similar analysisof the Virgo galaxy cluster and observed a significant discrepancy, by afactor of about 200, between the expected mass per galaxy from luminositymeasurements and his calculation of the mass per galaxy [3]. The term“dark matter” was being used in the literature at this point, but it was notthought of in today’s paradigm, with Smith asserting that the discrepancywas due to uniformly distributed or low luminosity internebular material [4].After Zwicky and Smith, more accurate measurements were made ofthe mass of galaxies, many of which were based on the rotational speeds ofnearby galaxies. At high galactic radii within which the majority of luminousmatter is enclosed, if all the mass in the galaxy were luminous then Newton’sand Gauss’ laws predict the rotational speed of orbital bodies around thegalactic center to be:v(r) =√GMl√1r(1.1)where Ml is the mass of the luminous matter. Horace Babcock in 1939measured rotational speeds of the Andromeda galaxy (M31) and found risingspeeds out to distances beyond where the luminous matter was enclosed (100arc minutes), in tension with Eq. 1.1. He attributed the measurements tolarge amounts of non-luminous matter in the outer part of the galaxy or tonew dynamics that would account for the high rotational speeds without anew mass component[4, 6].We now know definitively that the rotation speed of objects around the11.1. Early EvidenceFigure 1.1: (left) The hydrogen surface densities (azimuthally averaged) forfive galaxies and (right) the rotation galactic rotation curves. The bars un-der the galaxy names indicate the spatial resolution of the measurements.R80 corresponds to the galactic radius containing 80% of the Hydrogen den-sity determined from the 21 cm emission. Plot copied from Ref. [5], withpermission. Measurements were performed by Rogstad and Shostak in 1972from the Owens Valley Radio Observatory using 21cm observations.21.2. Galaxy Clustersgalaxy does not fall off as r−1/2. In the 1960s and 70s, using the Dopplershift of the Hα (optical) hydrogen emission line, Vera Rubin and Kent Fordmeasured the rotation speeds of hydrogen in M31, showing that the speedsare roughly constant at large radius from the galactic center [7]. Subsequentmeasurements of a host of spiral galaxies by Rubin, Ford, and Thonnard alsoshowed flat rotation curves [8]. Outside of the optical band, Rogstad andShostak analyzed the rotation curves from five galaxies using hydrogen’s21cm photon emission [5]. The flat rotation curves measured for these fivegalaxies are shown in Fig. 1.1.To explain the flat rotation curves, if we instead modeled mass in thegalaxy as a spherically symmetric distribution that varies with distance,M(r), the constant stellar rotational velocity at large distances given byv = C =√GM(r)√1r , can be explained by an enclosed mass that varieslinearly with the distance from the center:M(r) ∝ r.Therefore the enclosed mass has a density proportional to the inverse squareof the distance from the galactic center, ρ(r) ∝ 1r2. Most of this matter mustbe dark because the luminous mass density falls off much more rapidly. Toaccount for this missing matter, the leading model is a spherically symmetric“halo” of dark matter distributed throughout our galaxy and other galaxies,interacting gravitationally and making up the majority of mass in the galaxyand the universe.1.2 Galaxy ClustersMeasurements of galaxy clusters since the work of Zwicky and Smith haveprovided additional evidence for dark matter and important information re-garding the composition of dark matter. Observations of galaxy clusters—the largest gravitationally collapsed astrophysical structures—are particu-larly powerful because different techniques can be used to make independentmeasurements of their mass.The Chandra Observatory measures the x-ray emission from the inter-galactic gas of galaxy clusters. With the clusters’ gravitational potentiallargely due to dark matter, their intergalactic gas gains kinetic energy andheats to ∼ 108 K, and they are therefore among the brightest x-ray sources.By measuring the x-rays, the temperature and pressure of the gas can becomputed. If the cluster is in equilibrium, the gravitational potential can becomputed under the good assumption that gravitational forces cancel with31.2. Galaxy Clusterspressure forces. Recent publications matching models of the interstellar gasand dark matter halo to Chandra Observatory data indicate that ordinarymatter makes up 12-15% of the total mass of the cluster [9].In an independent measurement of the cluster mass, the Hubble SpaceTelescope measures light from background objects that bends around themassive structures. The total gravitational mass of the cluster is computedby the strength of the gravitational lensing, which confirms that the majorityof gravitational mass of the clusters is not due to the luminous matter[10].Figure 1.2: 1E 0657-56 ( c○ 2004 X-ray: NASA/CXC/CfA/M.Markevitchet al.; Optical: NASA/STScI; Magellan/ U.Arizona/ D.Clowe et al.; LensingMap: NASA/STScI; ESO WFI; Magellan/ U.Arizona/ D.Clowe et al., bypermission) [11] [12] .The “Bullet” Cluster is one of the most famous examples of the darkmatter making itself apparent in our universe. The Hubble Space Telescopeand Chandra Observatory have observed the collision of two galaxy clustersand measured the distributions of both the gravitational matter measuredwith strong gravitational lensing and luminous matter measured with x-rayemission. As shown in Fig. 1.2, these two distributions are observed to haveseparated. The luminous matter (pink) is superimposed on the distributionof gravitational matter (i.e. predominantly dark matter) as measured bylensing (purple). The two dark matter distributions have passed througheach other while the luminous distributions lag behind due to the impedanceof their collisions. The nearly non-interacting dark matter streams throughunimpeded.41.3. Big Bang Nucleosynthesis1.3 Big Bang NucleosynthesisMeasurement of the abundances of light elements produced in the earlyuniverse was an important step in constraining the dark matter composi-tion hypotheses. These measurements indicated, using a different set ofobservables (and different systematics) than the measurements discussed inSec. 1.1 and 1.2, that the baryon density of the universe was significantlysmaller than the total matter component. These measurements not onlypointed to the existence of dark matter, but also indicated that dark mattermust be non-baryonic.In the first minutes after the Big Bang the energy of photons droppedbelow the binding energy of deuterium (2.2 MeV), allowing protons andneutrons to fuse into deuterium (p + n → 2H + γ). Because both 3Heand 4He have higher binding energies than deuterium, subsequent reactionsproduced those nuclei. The binding energies of nuclei in the A = 5−8 rangeare lower than that of 4He, and so combined with the fact that particledensities at this stage of the universe forbade 3-body interactions, lightnuclei production stopped (to first order) at 4He. Therefore, only 2H, 3He,and 4He were produced in significant quantities in the early universe (withsmall amounts of 7Li). The abundances of these nuclei are sensitive to thebaryon density in the early universe. For example, the efficiency of theprimary reaction that turns deuterium into helium, 2H + p → 3He + γ,depends on the proton number density. Models of these nuclear processespredict abundances shown as colored bands in Fig. 1.3, and show a strongdependence on the baryon density.The measurements of these abundances in today’s universe are compli-cated by the fact that the high densities within stars do result in 3-bodyinteractions, allowing for nuclei production heavier than 4He. Measurementsof the deuterium abundance, made by observing high redshift quasars andexploiting the difference in the absorption properties of deuterium to hydro-gen, provide the best estimates for the baryon density [13]. This measure-ment is shown by the smallest yellow box in Fig. 1.3. The baryon to photonratio of ∼6×10−10 translates to a ∼5% baryon mass-energy density of theuniverse, which is significantly smaller than the mass density needed to ex-plain galactic rotation curves and to explain measurements of the cosmicmicrowave background, discussed next in Sec. 1.4.51.3. Big Bang NucleosynthesisFigure 1.3: Y is the 4He mass fraction of all baryons. The blue, red, andgreen bands are the modeled number densities of 2H, 3He, and 7Li, respec-tively, relative to 1H as a function the baryon density. Yellow boxes showmeasurements (95% CL) of the light element abundances; of note is that onlyupper limits on 3He abundance have been made, and the measurements of7Li are in slight tension with the model. The pink hatched band shows theinferred baryon to photon ration from the 2H measurement, while the bluehatched band shows the range indicated by CMB measurements. From [14]with permission. 61.4. The Cosmic Microwave BackgroundFigure 1.4: CMB power spectrum predicted by a Λ-CDM cosmology (i.e. adark energy (69%) and cold dark matter (26%) dominated universe). Datapoints in red are measurements by the Planck collaboration [15]. Powerspectrum of temperature fluctuations in the Cosmic Microwave Background( c○ 2013 ESA/Planck, and the Planck collaboration, from Planck 2013 re-sults. I. Overview of products and results, by permission).1.4 The Cosmic Microwave BackgroundThe Cosmic Microwave Background (CMB) provides the most accurate mea-surement of the fraction of the dark matter density of the universe. Approx-imately 400,000 years after the Big Bang, the expanding universe cooled toa point where it became energetically favorable for the plasma of protonsand electrons to fall out of equilibrium with photons and form neutral hy-drogen. At this point of “recombination,” the universe became transparentto the photons which make up the CMB radiation, which matches a black-body spectrum with (currently) a temperature of 2.7 K. Slight differences, oranisotropies, in this temperature across the sky, on the order of 100µK, havebeen a rich source of cosmological information, including the most accuratemeasure of the non-baryonic (dark matter) matter density in the universe.Transforming the spatial anisotropies into spherical harmonics gives theCMB power spectrum, or the variance of the spatial fluctuations as a func-tion of angular scale. The acoustic peaks of the power spectrum show the71.5. Composition Hypothesesangular scales at which the photons were slightly hotter and slightly colderthan average at recombination. The hotter overdensities are regions wherethe photons, coupled to the baryonic matter up to the point of recombi-nation, had clumped together because of gravitational wells into which thebaryonic matter was attracted.Importantly for the case of the non-baryonic dark matter, the amplitudeof the peaks depends on the dark matter density in the universe. This is be-cause the pressure of the photons coupled to the baryonic matter opposes theformation of gravitational wells. In order to accurately model the locationand amplitude of the peaks, a decoupled non-baryonic matter componentmust exist which continues to collapse regardless of the photon restoringforce. The photon pressure restoring force does set up an oscillation of thebaryon-photon plasma, which is highly sensitive to the non-baryonic matterdensity, and which is imprinted on the CMB at last scattering in the formof the peaks in the power spectrum [16]. With the CMB power spectrum weare therefore able to determine the non-baryonic matter density. The CMBpower spectrum is sensitive to numerous other properties of the universe,such as the dark energy fraction, and these cosmological parameters are fitin a multidimensional space to the power spectrum. The best current mea-surement of the cold dark matter density fraction of the universe is 26.8%(with a 68.3% dark energy component) [17]. Reference [17] contains furtherdetails on the power spectrum fitting, including discussion of the covariancesbetween cosmological parameters.1.5 Composition HypothesesDespite overwhelming observational evidence that dark matter does exist,very little is known about its composition. A number of theories have beenput forth.In one effort to account for the dark matter, experiments searched forhidden massive compact halo objects (MACHOs), such as black holes ormassive non-luminous planets. They looked for MACHOs in the Milky Wayby waiting for slight unexpected gravitational lensing of distant luminousgalaxies as a MACHO passed between us and the galaxy. These searchesruled out the possibility of MACHOs constituting any more than 25% ofthe Milky Way’s dark matter halo, and therefore disqualified them as theprimary dark matter candidate [18].Most theories predict that non-baryonic particles make up dark matterhalos around galaxies, but still there exist many possibilities for the type of81.5. Composition Hypothesesparticle. If we assume that dark matter is non-baryonic, it is highly likelythat such dark matter is also non-relativistic, or “cold”, dark matter. Rel-ativistic, or “hot”, dark matter conflicts with the accepted model of galaxyformation [19]. Returning to the discussion of Sec. 1.4, had the dark mat-ter been relativistic then its kinetic energy would have largely prevented itsgravitational collapse. However, the gravitational landscape at recombina-tion is well understood, and not only is it imprinted on the CMB but it alsoexplains the formation of the dense small scale structures (galaxies) seen inthe universe today. That the majority of dark matter is cold rules out ahot relativistic (neutrino-like) species from contributing substantially to the26.8% dark matter component.Theoretical models that predict new particles with these characteristicsinclude, but are not limited to, axions [20, 21] and WIMPs [22]. In thefollowing subsections we briefly review these and other candidates, with afocus on candidates to which CDMS detectors are sensitive.1.5.1 The WIMP HypothesisThe SuperCDMS experiment, along with many competitor experiments,searches for Weakly Interacting Massive Particles (WIMPs). The WIMPhypothesis is intriguing because it fits into a parameter space supported bysupersymmetric (SUSY) theory as well as cosmology. A number of assump-tions regarding matter in the early universe allow cosmologists to estimatethe WIMP cross section. First, WIMPs would have been constantly createdand annihilated until some critical point of the universe’s cooling where thelow temperature would prevent any further WIMP creation. Following this,expansion of the universe would have made it exponentially unlikely thata WIMP would collide with its antiparticle and annihilate [22]. This sec-ond critical moment—when annihilation ceases—is known as thermal relic“freeze out.” The particle abundance resulting from freeze out depends onthe WIMP annihilation cross section as shown in Fig. 1.5, with a detailedderivation given in Chapter 3 of Ref. [16]. In order to account for the darkmatter in the universe, the WIMP annihilation cross section is estimated tobe roughly at the scale of the weak force where yet undiscovered particlesare expected to exist as postulated by SUSY [22].This coincidence is what some refer to as the “WIMP miracle,” sinceSUSY was initially proposed as a solution to other problems with the Stan-dard Model of particle physics, but could naturally solve the dark matterproblem as well. SUSY adds particles to the Standard Model and the light-est of these particles, the neutralino, could be the dark matter WIMP. This91.5. Composition HypothesesFigure 1.5: Number density of WIMPs in the Universe as a function oftime, where the relic density depends on the WIMP annihilation cross sec-tion, σχχ¯ ( c○ NASA/IPAC Extragalactic Database (NED) which is operatedby the Jet Propulsion Laboratory, California Institute of Technology, undercontract with the National Aeronautics and Space Administration, by per-mission).101.5. Composition Hypothesesconvergence of SUSY and cosmology is the primary motivation behind theWIMP hypothesis and has launched the dozens of experiments attemptingto detect WIMPs [22].1.5.2 AxionsThe axion was originally proposed as a solution to the strong charge parity(CP) problem [20, 21] and has since become a leading light mass (10−5 to10−2 eV) DM candidate. The strong CP problem refers to the observationthat there is no violation of CP-symmetry in quantum chromodynamics,despite the fact that the theory contains CP-violation. One manifestationof CP-violation would be an electric dipole moment in the neutron, butno such moment has been observed and upper limits [23] have been placedthat are significantly lower than those na¨ıvely expected from quantum chro-modynamics. Peccei and Quinn [24] proposed a solution to the strong CPproblem that includes a new particle (the axion) that cancels out the quan-tum chromodynamic effects that permit CP-violation.Through interactions with standard model particles, axions would beproduced in stars and provide an additional process for the stars to shed en-ergy. Models of these processes combined with stellar lifetime values predicta 10−2 eV upper limit on the axion mass [25]. The axion can convert to twophotons, and terrestrial experiments search for this conversion in resonantcavities threaded by a strong magnetic field [26].1.5.3 Dark Sector ParticlesThe DM could be a new particle from a “dark sector” that does not interactvia the standard model force mediators. Dark sector theories in particularpostulate the existence of particles in the lower mass range (1–10 GeV/c2)[27–29], whereas the SUSY WIMP favors higher masses. One dark sectortheory postulates a new force mediated by a dark photon, which can kine-matically mix with the standard photon and therefore interact with stan-dard model particles [27]. Interactions of such particles with protons canbe searched for in detectors nominally designed for WIMP detection (whichare discussed in Sec. 1.6.1).A second class of dark sector models goes under the name of “asymmetricdark matter,” and these models propose an asymmetry between the darkmatter and its antiparticle [30–32], analogous to the baryon/anti-baryonasymmetry of the universe. Such models postulate the existence of DM par-ticles in the 5–15 GeV/c2 mass range [32]. Many asymmetric DM models111.6. Dark Matter Detectionexist, some of which fit into the SUSY framework, and some of which aremotivated by new dark force mediators. It is not guaranteed that asymmet-ric DM interacts with standard model particles, but interactions are possibleand could proceed via DM-proton scattering. In this case, again, detectorsnominally designed for WIMP detection would be sensitive to these interac-tions [32].1.6 Dark Matter Detection1.6.1 Direct DetectionWhen DM WIMP particles interact in a detector on earth, the DM can loseenergy by colliding with electrons and nuclei in the detector material. Theamount of energy that the DM can lose is significantly greater for nucleithan electrons (as described in more detail below), and so direct detectionexperiments look for DM-nucleon recoils in terrestrial detectors and em-ploy different targets (e.g. liquid argon, liquid xenon, germanium, silicon,calcium tungstate), background rejection techniques, amplifiers, and/or en-ergy thresholds. Direct detection experiments are optimized to certain DMmasses and cross sections based on their detector technology.For a DM WIMP, the expected elastic scattering energy transfer to atarget particle is given by:Erecoil = (mχmTmχ +mT)2v2mT(1− cos(θχ)) (1.2)where mχ is the WIMP mass, mT is the target particle’s mass, v is theWIMP velocity, and θχ is the WIMP scattering angle.The WIMP velocity is given by v and deserves brief discussion. In theStandard Halo Model (SHM) the DM halo is isothermal and isotropic. Thephase space density for a DM particle in the halo, under these assumptions,is given by Maxwell-Boltzmann statisticsf(~x,~v) = Cexp(−E(~x,~v)/kBT ) (1.3)with an energy given byE(~x,~v) =12mχ|~v|2 +mχΦ(~x) (1.4)where Φ(~x) is the gravitational potential and C is a normalization constant.121.6. Dark Matter DetectionThe velocity distribution function is calculated by integrating over thespatial coordinatesf(~v) =∫d3~xf(~x,~v)= Cexp(−mχv2/2kBT )∫exp(−mχΦ(~x))d3~x= C ′exp(−mχv2/2kBT )= C ′exp(−v2/2σ2v)(1.5)where in the last step we have defined the DM velocity dispersion as σ2v =kBT/mχ. The velocity dispersion is related to the characteristic velocityof the dark matter by σv =√1/2v0. Since DM particles moving too fastwill gravitationally escape the galaxy, f(~v) is generally truncated at theescape velocity (vesc is taken to be 544 km/s in the SHM, as estimated fromhigh velocity stars [33]). Therefore, the final velocity distribution functionis given byf(~v) ={C ′e(−v2/v20) v < vesc0 v > vesc.(1.6)The characteristic velocity of the DM is not a constant as a function of radialdistance from the galactic center, and Ref. [34] shows that at radii compara-ble to the Sun’s location, the characteristic velocity is approximately givenby the local circular velocity at the radius of the Sun’s orbit. Historicallya value of v0=220 km/s is used as estimated by galaxy surveys [35]. Theearth is moving through this halo with a velocity ~vE , the vector sum of thesun’s circular and peculiar velocity as well as a small (6%) annual modu-lation due to the earth’s orbital velocity around the sun. The SHM usesvE = 232 km/s for the average velocity of the earth, and boosts the DMvelocity distribution into this frame with the transformation ~v → ~v + ~vE inEq. 1.6.One key element of direct detection of elastic scatters is made clear fromEq. 1.2—the WIMP cannot efficiently transfer energy to target componentsthat are much less massive than a nucleon. Consider the maximum en-ergy transfer of a WIMP-electron collision (θχ → 180◦, mT ≈ me), givingErecoil ≈ 2mev2 = 0.25eV. Signals of this magnitude are “in the noise”projected for the even the lowest threshold next-generation dark matter ex-periments, although through upgrades detector resolutions for the followinggeneration of experiments are projected to approach these values.131.6. Dark Matter DetectionInstead consider the maximum energy transfer of a WIMP-nucleon col-lision where the dark matter particle is well matched kinematically to a Genucleus target: (θχ → 180◦, mχ ≈ mT ≈ 72mp). In this case Erecoil ≈(1/2)mT v2 ≈ 16.5keV, which is certainly a detectable energy.The local dark matter density is approximately ρdm = 0.3 GeV/cm3 [36].Recent measurements of ρdm combine stellar kinematic data with simula-tion to estimate the local galactic gravitational potential in the vicinity ofthe Sun which is then converted to a dark matter density by subtractingout the contribution from luminous matter [37]. The measurements areprone to significant systematic uncertainties, though most measurementsare within 2σ of ρdm = 0.3 GeV/cm3, albeit with significant error bars [37].Despite ever improving measurements of ρdm from improved stellar kine-matic data and simulations, the DM direct detection community assumesρdm = 0.3 GeV/cm3 for consistency and in order to be able to compare DMresults between competing experiments and with previous results.This density means that many DM particles stream through the detec-tors every day, with lower mass DM particles generating a higher flux, anddirect detection experiments hope to measure this rate of DM events. Theexpected differential scattering rate, as a function of Erecoil, is given by:dRdErecoil=ρdmmTmχ∫ ∞vminv3f(v)[dσχTdErecoil(v,Erecoil)]dv[keV kg day]−1(1.7)where dσχT /dErecoil is the differential cross section, vmin is the minimumWIMP velocity in order to produce recoil energy Erecoil1, and ρdm is thelocal dark matter density [40].Except for the differential cross section dσχT /dErecoil and the WIMPmass, all the parameters of the differential scattering rate are estimated inthe SHM. The differential cross section clearly has large implications for thedetectability of WIMP particles. The total cross section could be the sumof a spin-independent and spin-dependent term. As Witten and Goodman1The revolution of the earth around the sun seasonally adds and subtracts from theWIMP velocity relative to the earth, and detecting a seasonal variation in a possibleWIMP signal would be another sign that the signal is indeed the dark matter halo. A dif-ferent direct detection experiment—the DAMA/LIBRA collaboration—claims that theyare seeing this annual modulation in their data and interpret this as a dark matter signal[38]. As shown in Fig. 1.6, multiple other direct detection experiments exclude the DMinterpretation of the DAMA/LIBRA data. Additionally, the COSINE-100 experimentsearches for DM using the same target (sodium iodide) as DAMA/LIBRA to conduct amodel independent test of DAMA’s claim; COSINE-100 observes no signal above the ex-pected background and is currently searching for evidence of an annual modulation signalin their data [39].141.6. Dark Matter Detectionnoted in their 1984 paper [41], the spin-independent term in the cross sectionscales as the number of nucleons squared, which is sometimes referred to as“coherent rate enhancement.” This effect significantly amplifies the spin-independent term relative to the spin-dependent term, and therefore mostdirect-detection experiments present their results as a sensitivity to the spin-independent cross section and as a function of the dark matter mass mχ, asshown in Fig. 1.6.Spin-dependent interactions would occur if the DM particle has a non-zero spin and the target nucleon has a non-zero spin, although the spin-dependent interaction is more difficult to probe. This is not only because ofthe lack of coherent rate enhancement, but also because of lack of dark mat-ter nuclei targets with a large nuclear spin. The spin-dependent interactionwill cancel for DM scattering off of paired nucleons with equal magnitudebut opposite signed spins, and so experiments hoping to detect this inter-action use light odd-proton or odd-neutron nuclei (e.g. 19F as used in thePICASSO experiment [42]) in order to maximize the nuclear spin per unitmass.We compute the right hand side of Eq. 1.7, and integrate over the Erecoilspectrum to obtain the total number of events expected for different targetsand different cross sections as a function of energy threshold. The resultsare shown in Fig. 1.7. In the left plot we show a relatively light WIMP(10 GeV/c2) with a cross section of σSI = 10−41 cm2, now ruled out atthe 90% confidence level by a number of experiments. In the right plot weshow a 100 GeV/c2 WIMP with a cross section of σSI = 10−45 cm2, closeto the 90% exclusion limit of the xenon-based dark matter searches [43–45].As a rough reference, reading off from Fig. 1.7 (left), a recoil threshold of6 keV gives a rate of 1/10 [events kg−1day−1]. CDMS II had roughly 5 kgof detector bulk, translating to a rate of 0.5 events per day.At these relatively low nuclear-recoil energies and low event rates, onefundamental challenge to direct detection experiments is background dis-crimination. One advantage is that the majority of backgrounds will scatteroff electrons in the detector bulk, and most direct detection technologieshave means to distinguish electron recoils from nuclear recoils and thusreject background events, which is further discussed in Sec. 2.1.1. In addi-tion, all direct detection experiments shield the detector as best as possibleto reduce the background rate. To shield from cosmic-ray muon-inducedneutron scattering in the detector, the detectors are operated undergroundwhich reduces the muon flux with the rock overburden. The flux reduc-tion for underground sites in which the CDMS detectors have operated isshown in Fig. 1.8. In addition to being underground, the SuperCDMS cryo-151.6. Dark Matter DetectionFigure 1.6: Current (solid) and projected (dashed) 90% limits on the WIMPmass vs. WIMP-nucleon cross section parameter space. The yellow shadedregion represents the DM cross section at which experiments will observe an“irreducible” background from nuclear scattering neutrino events. At lowerDM mass, the background from solar neutrinos from 7Be and 8B reactionsdominate, whereas at higher DM mass atmospheric and the diffuse super-novae neutrino background (DSNB) dominate. Figure from SuperCDMScollaboration approved public plots.161.6. Dark Matter DetectionFigure 1.7: The expected WIMP event rate for the given mχ and spin-independent cross section. σSI=10−41cm2 corresponds to roughly the crosssection reported by DAMA/LIBRA, CRESST, CDMS Si, and CoGent.σSI=10−45cm2 corresponds to a cross section excluded only by results inthe last 2–3 years. Internal CDMS figure, from [46].stat was surrounded by layers of shielding (polyethylene to block neutrons,ancient lead2 to block γ-rays), and as a result the dominant source of back-ground particles in the detector comes from radioactivity of the shieldingitself, other apparatus materials, and the detectors themselves. Finally, mostdirect-detection experiments have the capacity to estimate the position ofan event in the detector, which is useful for fiducialization, or removingbackground events that occur close to the surface of the detector (at the ex-pense of removing that outer detector volume from the dark matter searchas well).The above is in principle how CDMS and other direct detection exper-iments hope to discover WIMPs. However, ever since the 1990s when thefirst limits on WIMP cross sections were published by CDMS, no such ratehas been observed. There have been reported detections [38], but they havenot held up to further scrutiny and are not widely accepted by the com-2Lead on Earth naturally contains some amount of 235U, which decays to a radioactivetype of lead, 210Pb, which has a half life of 22 years. Most of the uranium is removedwhen the lead is first processed, and the 210Pb begins to decay which further purifies thelead over time. In ancient lead almost all of the 210Pb has already decayed, resulting inradioactive background rates for ancient lead that are ∼1000 times lower than non-ancientlead.171.6. Dark Matter DetectionFigure 1.8: The depth of different underground sites, shown as muon flux onthe y-axis, and meters of water equivalent (mwe) on the x-axis, from differentgenerations of CDMS experiment. SUF stands for Stanford UndergroundFacility.munity. CDMS has gone through three generations of experiments: CDMS,CDMS II, and SuperCDMS Soudan, and is now preparing for SuperCDMSSNOLAB. Each generation of the experiment has increased the total de-tector mass and implemented improved detector technology. In two of theiterations the detectors were moved to a cosmogenically cleaner site (deeper)and radiogenically cleaner environment. Meanwhile competitor experimentsmade similar improvements and new detection technologies were developedin order to address (1) a low rate of WIMP-nucleon collisions, and (2) back-ground rejection.1.6.2 Collider ProductionDM particles could be produced in collisions of leptons and hadrons in highenergy colliders. Because the dark matter particles are stable and inter-act negligibly with the detector, their production in a collision would beobserved as a missing transverse momentum.Both of the general-purpose Large Hadron Collider detectors, ATLASand CMS, have searched for a DM-like missing momentum in collisions [47,48]. The data have been consistent with expectations from Standard Modelprocesses and limits have been placed on the DM production rates. In ahighly model-dependent fashion, production limits can be converted to limitsin the DM-nucleon cross section vs. DM mass plane and be compared with181.6. Dark Matter DetectionFigure 1.9: Limits on the DM-nucleon scattering cross section, as a func-tion of DM mass from the CMS (left) and ATLAS (right) experiments fordifferent sets of model assumptions. Figures from [47, 48].direct-detection limits. These conversions are shown in Fig. 1.9. At highDM mass, collider experiments are not as sensitive as the direct detectiondetectors because they are limited by the energy available in the particlecollisions. At low DM mass by contrast, the collider experiments are moresensitive than direct detection detectors because they are not as limited bythe low energy thresholds that dictate the mass reach of direct detectionexperiments.1.6.3 Fixed TargetFixed target experiments hope to detect dark matter particles by first pro-ducing them from proton beam collisions with a “fixed target,” and thendetecting the dark matter particles in a downstream detector. For example,the MiniBooNE [49] experiment hopes to produce dark matter particles via8 GeV/c2 proton collisions in steel, either by proton Bremsstrahlung or bypi0 or η meson (produced copiously in the proton beam interactions with thesteel) decay. MiniBoone searches for these dark matter production mech-anisms from a “dark photon” kinematically mixing with standard modelparticles (as mentioned in Sec. 1.5.3). Once the dark matter particles areproduced, their elastic nuclear collisions in a downstream detector could beobserved. The MiniBooNE detector consists of 818 tons of CH2 mineraloil scintillator, and dark matter nuclear collisions would produce a protonor neutron track that would generate a small, but detectable, amount of191.6. Dark Matter Detectionscintillation light. In 2018 new limits were placed on the kinematic mixingamplitude for dark matter masses between 10 MeV/c2 and 1GeV/c2 [49].Many other proposed and ongoing experiments plan similar studies, such asT2K [50] and NOνA [51], and new efforts such as LDMX [52] and HPS [53].1.6.4 Indirect DetectionIndirect detection efforts involve looking for signatures of DM interactionwith standard model particles in observable astrophysical objects. One sig-nificant challenge in these efforts is accurately modeling of all other non-DMastrophysical processes so that if a DM signal exists in the data it can becorrectly attributed as such at a high confidence level.One channel by which to look for dark matter is its annihilation intogamma rays, e.g. χχ¯ → γγ, where χ is a DM particle and χ¯ is an anti-DM particle. Researchers look for gamma ray excesses in regions of the skywhere there is predicted to be a high DM density, such as the Milky Waygalactic center. The Fermi Large Area Telescope has observed an excess ofgamma rays (at roughly GeV energies) from the galactic center [54], and insome publications this excess has been interpreted as consistent with anni-hilation of 36–51 GeV/c2 DM particles [55]. Subsequent analyses [56, 57]have found that models with unresolved point sources account for the excessand are a better fit to the data, with high statistical significance. The un-resolved point sources (e.g. millisecond pulsars) are predicted to generate agamma flux just below the current Fermi Large Area Telescope point sourcedetection threshold. Lowering of this threshold with future measurementsand detecting the point sources will provide additional information aboutthe processes creating the gamma rays in the galactic center.Another indirect detection method is to look for neutrino excesses fromastrophysical bodies where DM particles would gravitationally accumulate,where the probability of their annihilation is higher. One such body is theSun, and while most of the standard model annihilation products such asgammas would be trapped by the Sun, the neutrinos would escape. Boththe IceCube [58] and Super-Kamiokande [59] experiments look for the spec-trum of excess neutrinos, and with the analyses indicating that the dataare consistent with expected backgrounds they have placed limits on DMannihilation in the Sun.201.7. Further Reading1.7 Further ReadingWhile this chapter provides a broad overview of the evidence and searchfor DM, for a deeper background there are numerous other review articles.Reference [55] contains further discussion of the history of dark matter. Ref-erence [60] reviews DM from an observational and theoretical perspective.Reference [22] establishes and reviews the theoretical motivation for WIMPDM. Reference [36] provides more detailed discussion of the DM rate andenergy spectrum expected in terrestrial detectors.21Chapter 2Detectors for the CryogenicDark Matter Search2.1 Semiconductor Detector PhysicsThe CDMS detectors consist of kilogram-scale cylindrically-shaped ultra-pure germanium and silicon crystals. Germanium and silicon have goodcharge transport properties and small band gaps, which are important fora favorable detector response to a particle interaction in the detector, asdiscussed in the following sections. In addition, the availability of bothgermanium and silicon with low concentrations of radioactive contaminantsassists in minimizing the number of background events in the detector bulk,which is critical for increasing sensitivity to a DM signal.The CDMS detectors are operated at low temperatures (∼50 mK) atwhich the germanium and silicon behave as insulators because the e−/h+pairs are frozen out of the conduction band. The low temperature also de-creases the heat capacity of the crystal and helps to lower the noise of thereadout electronics. When a particle (DM signal or background particle)recoils in the germanium or silicon, it loses energy as it interacts with theelectrons and nuclei. For electrons and gammas (i.e. the dominant back-grounds), it is kinematically favorable to interact with and lose energy to theelectrons in the detector material. Recoiling heavier particles (& 1 GeV/c2),such as neutrons or DM WIMPs, find it kinematically efficient to impart sig-nificant energy to nuclei in the detector as well as electrons. The detectorsexhibit a different response to nuclear recoils (NRs) versus electron recoils(ERs), and historically this enabled discrimination between a DM signaland background events. Because of the importance of these two interactiontypes in the CDMS detectors, we now discuss some basic dynamics of thesetwo types of recoils.222.1. Semiconductor Detector PhysicsEcreate EgapGe 3.0 0.74Si 3.8 1.12Table 2.1: Electron-hole pair creation energies (in eV) and bandgap energiesfor germanium and silicon. Values from Ref. [62].2.1.1 Electron RecoilsElectromagnetically-interacting background particles such as β and γ par-ticles recoil off electrons in the detector bulk. For example, when a mediumenergy γ-ray (10keV to 1MeV) passes through the detector bulk, it is likelyto interact with an electron via the photoelectric effect or by Compton scat-tering [61]. A recoiling electron will lose energy by Coulomb interactionswith other electrons in the material, creating an electronic cascade. Thestopping lengths for keV-scale electrons in germanium and silicon are onare the scale of µm and decrease for lower energy electrons, as shown inFig. 2.1. Because the stopping lengths are significantly smaller than thelength scale of the detector, all of the electron’s energy will be depositedwithin the crystal.In germanium, the energy to create an e−/h+ pair (Ecreate) is 3.0 eV.This “ionization energy” is frequently also denoted as . Therefore, the elec-tronic cascade from a fully absorbed 10keV γ-ray spreads its energy out byionizing e−/h+ pairs into the conduction band until the individual electronshave insufficient energy to excite another electron, producing ∼3000 e−/h+pairs in the detector. In general, for an electron recoil, the number of e−/h+pairs is given byNe/h = Er/Ecreate = Er/ (2.1)where Er is the energy of the recoil.With the e−/h+ pairs in the electronic cascade having on average 3.0 eVof energy, they are unable to lose additional energy to other electrons. How-ever, they have an energy Ecreate that is greater than the bandgap en-ergy (Egap = 0.74 eV) in germanium. Therefore, the e−/h+ pairs relaxdown to the bandgap energy and they shed energy as lattice vibrations (i.e.phonons). The corresponding e−/h+ pair creation energies and bandgapenergies for silicon are shown in Table 2.1.The values of Ecreate and Egap dictate the partitioning of the initial recoilenergy, of an electron recoil, between the e−/h+ pairs and the phonons.Specifically, for germanium the fact that the e−/h+ pairs transfer all of theirenergy to phonons as they relax from 3.0 to 0.74 eV means that the fraction232.1. Semiconductor Detector PhysicsFigure 2.1: Estimated energy vs. electron range in germanium and silicon,based on the electron stopping power from Coulomb interactions. Figureadapted from [61].of original recoil energy that goes into phonon creation is 1− Egap/Ecreate,or 3/4. For silicon this fraction is 7/10.2.1.2 Nuclear RecoilsWIMP DM particles and neutrons recoil off germanium and silicon nucleiin the detector bulk. Nuclear recoils are similar to electron recoils in manyways; however, they differ most importantly in that a lower fraction of energygoes into e−/h+ pairs. Lindhard theory, which models the propagation ofions through semiconductors, serves as the basis for the following discussionof semiconductor response to NRs [63–65].When a nucleus recoils, it is capable of transferring energy to othernuclei and other surrounding electrons. Nuclei are capable of this froma purely kinematic standpoint, whereas electrons are not because of theirsmall mass [66]. The more even division of energy transfer to excitations ofelectrons versus energy transfer to excitations of other nuclei is particularlytrue for lower energy NRs (. 1 MeV in Ge), and these low energy recoilsare particularly relevant for low-mass DM interactions. The nuclei are freedfrom the crystal lattice and excite other nuclei in a cascade separate fromthe electron cascade. Once the nuclei’s kinetic energy drops below the latticebinding energy, they have insufficient energy to excite other nuclei and theylose their energy to phonon production. In contrast to the electronic cascade,for nuclei there is no analogous Egap. Therefore, nuclear cascades are moreefficient than electron cascades in phonon production.The derivations of the nuclear recoil energy partitioning between elec-242.1. Semiconductor Detector Physicstrons and phonons are computationally intensive [63–65] and not repeatedhere, but the Lindhard model provides a parametrization for the fraction ofthe nuclear recoil energy that ends up in e−/h+ pairs. This fraction dependson the energy of the nuclear recoil and is called the “ionization yield” (oris sometimes referred to as the “quenching factor”). The ionization yield isnormalized so that the ionization yield of electron recoils is equal to 1. Thenumber of electron-hole pairs generated in a nuclear recoil can be writtenas a function of the ionization yield (Y ) and the nuclear recoil energy (Enr)as:Ne/h = (Y (Enr)× Er)/. (2.2)The ionization yield as a function of nuclear-recoil energy is predicted bythe Lindhard theory to beY (Enr) =k · g(ε)1 + kg(ε), (2.3)where k = 0.133Z2/3A−1/2, g(ε) = 3ε0.15+0.7ε0.6+ε, ε = 11.5Enr(keV)Z−7/3,and Z is the atomic number of the detector material [67]. Measurementsof Y in germanium are generally consistent with a small range of k valuesapproximately centered on the Lindhard model prediction of k = 0.159.Figure 2.2 shows ionization yield (efficiency) as a function of energy for theLindhard model as well as measurements of the yield in germanium [68–72].Section 5.2 discusses how the spread in experimental measurements is in-corporated as a systematic uncertainty on k whenever using the Lindhardmodel to determine how a nuclear recoil signal would appear in a CDMSdetector.2.1.3 Yield DiscriminationThe fact that NRs produce fewer e−/h+ pairs, and therefore a smaller ion-ization signal, than ERs of the same recoil energy offers a powerful means ofdiscriminating between the two types of events. CDMS detectors measurethe ionization yield and the recoil energy for each event independently bymeasuring the signal from both the e−/h+ and the phonons.The expected difference in ionization yield between ERs and NRs is ob-served in 133Ba and 252Cf calibration data that induce the two differenttypes of recoils, and is shown for a CDMS II detector3 in Fig. 2.3. The ER3CDMS II was a generation of the experiment that operated from 2003-2009.252.1. Semiconductor Detector PhysicsFigure 2.2: Measurements of the ionization yield of nuclear recoil events, asa function of energy. The prediction from the Lindhard model is overlaid.From Ref. [73].background events can be removed by only selecting events with lower ion-ization yield. The upper (lower) ±2σ band in Fig. 2.3 defines the ER (NR)region of parameters space, where the width of the bands is determined bythe energy resolution of the e−/h+ and phonon measurements. The resolu-tion of these measurements worsens as the energy decreases, demonstratedby the widening of the bands at lower energy, and the specifics of the mea-surements is discussed in Sec. 2.2. Deciding where to “set the cut” in theionization yield vs. recoil energy plane is analysis- and detector-dependent(some complications of which are discussed in Sec. 2.4), though the lower±2σ bands shown in Fig. 2.3 depict a reasonable selection area for NRs.2.1.4 Phonon Generation and PropagationWhile Sec. 2.1.1 and 2.1.2 outlined electron and nuclear recoil dynamics,special attention should be given to the different production mechanisms ofphonons in the detector. Phonon production and amplification is a partic-ularly important aspect of the CDMSlite detector operation, which is thefocus of Chapter 4 of this thesis. Phonons are produced at three differentstages of an event, as depicted in Fig. 2.4, and we describe each stage inmore detail below.The phonons produced in the initial electron or nuclear recoil cascade arereferred to as prompt phonons. These prompt phonons are high frequency262.1. Semiconductor Detector PhysicsFigure 2.3: Ionization yield versus recoil energy for calibration data, froma CDMS II detector. The blue data was acquired during exposure of thedetector to a neutron 252Cf source producing primarily NRs, which accord-ingly have a measured ionization yield less than 1. The red data is froma 133Ba calibration, which primarily consists of ERs from γ interactions inthe detector. The black bands correspond to the ±2σ area in the ioniza-tion yield vs. recoil energy plane where ERs and NRs occur. These bandsprovide a visualization of the discrimination in this plane, and a cut in thisplane would be customized for a specific analysis (not necessarily at the ±2σlevel). From Ref. [46].Figure 2.4: A simple depiction of an event in the detector and the resultingprompt, NTL (or Luke), and recombination phonon production.272.1. Semiconductor Detector Physics(ν >1 THz) and their total energy is given by:EP = Er −Ne/hEgap. (2.4)Two frequency-dependent scattering processes—anharmonic decay (with ascattering rate, ΓA, proportional to ν5) and isotope scattering (ΓB ∝ ν4)—downconvert the initial high frequency phonons into lower frequency phonons.The initial prompt phonons scatter with mean free paths that are muchsmaller than the size of the detector, but once they reach frequencies ofν ∼ 2 THz, the decrease in the scattering rates leads to intermediate meanfree paths of 10 µm–1 cm [74]. This intermediate propagation regime is re-ferred to as “quasi-diffusive,” and the diffusive phonon ball that spreads outfrom the recoil site carries information about the position of the event. Overtimescales of ∼100µs, the quasi-diffusive phonons continue to downconvertto lower frequency, which in turn lowers the scattering rate and increasesthe mean free path to length scales comparable to the size of the detector.This propagation regime is referred to as ballistic, where the distribution ofphonons is homogenous through the detector and carries no position infor-mation about the initial recoil. These phonons bounce around the detector,reflecting off the detector surfaces, and are absorbed with some probabilitywhen they strike an aluminum collection fin (discussed in Sec. 2.2.2).A second production mechanism of phonons adds to the total phononsignal after the initial electron or nuclear recoil. A voltage is applied acrossthe detector so that once an electron cascade has occurred, the excitede−/h+ pairs do not de-excite back into valence states, but rather drift towardthe surfaces of the detector. As e−/h+ pairs drift across the crystal due tothe electric field, they quickly reach a terminal velocity in the crystal andthe additional work done on the carriers is transferred to the crystal latticein the form of Neganov-Trofimov-Luke (NTL) phonons [75, 76]. For a singlee−/h+ pair generated in the bulk of the detector, the work done to drift thecarriers to the surface of the detector by a distance d is given byWq = e|E|d = eVb, (2.5)where e is the absolute value of the charge of the electron, E is the electricfield through the detector, and Vb is the voltage difference between thesurfaces of the detector4. With the number of e−/h+ pairs generated in an4Non-uniformities in the electric field through the detectors, which complicate therelationship given by Eq. 2.5, are discussed in Sec. 2.5.282.2. Amplifierselectron or nuclear recoil given by Eq. 2.1 and 2.2 respectively, the totalenergy in the NTL phonons is given byENTL = eVbNe/h. (2.6)A third class of phonons, called recombination phonons, also contributeto the total phonon signal. Once the e−/h+ pairs reach the surface of thedetector they recombine at the charge collection electrodes (discussed inSec. 2.2.1) and the bandgap energy of the charge carriers is converted intophonons. The energy of the recombination phonons is given byER = Egap ×Ne/h. (2.7)There is some uncertainty in the detector’s collection efficiency for the re-combination phonons. The possibility of reduced collection efficiency of thistype of phonon, relative to prompt and NTL phonons, is discussed furtherin the context of bias scan calibration data (Appendix A.1).The total phonon energy (Et) is nominally the sum of these three sources(prompt, NTL, and recombination), given byEt =(Er −Ne/hEgap)+(eVbNe/h)+(EgapNe/h), (2.8)which simplifies toEt = Er(1 +Y (Er) e Vb). (2.9)In comparing the energy contribution of prompt (Eq. 2.4), NTL (Eq. 2.6),and recombination (Eq. 2.7) phonons to the total phonon energy (Eq. 2.8),notice that for low detector voltage Vb, the prompt and recombinationphonons dominate. However, once Vb = 3 volts the NTL phonon contri-bution makes up half of the total phonon signal (for electron recoils). Inthe CDMSlite operating regime of Vb = 75 volts, the NTL phonon energydominates the total energy, with 25 times more energy than the prompt andrecombination phonons.2.2 AmplifiersThe detector surfaces are instrumented with sensors designed to measure thee−/h+ pairs and phonon energy such that the energy, ionization yield, andposition of an event can be estimated as accurately as possible. As energy292.2. AmplifiersFigure 2.5: (top) A cartoon of the iZIP detector, with the individual phononchannels labeled and uniquely colored. (bottom) A characteristic phononsignal (i.e. pulse) from the bulk of an iZIP detector. The different pulseshapes on the individual channels indicate that this event occurred close tochannel DS2 and CS1.302.2. AmplifiersCDMS SCDMS SCDMSII Soudan SoudanDetector ZIP iZIP CDMSliteMass per Detector [kg] 0.25 ∼0.62 ∼0.62Number of Detectors 19 15 2Phonon Channels per Det. 4 8 4Phonon Energy Res. [eV] ∼180 ∼200 ∼70Trigger Threshold [eV] ∼2000 ∼3000 ∼50Charge Energy Res. [eV] ∼300 ∼450 –Voltage between faces [V] 3 4 ∼70Table 2.2: Detector amplifier specifications for different generations of thegermanium CDMS detectors. Adapted partially from Ref. [66].resolutions improve, event discrimination improves and detector thresholdscan be lowered. As position resolutions improve, radioactive backgroundsthat occur near the surface of the detector can be removed more efficiently.All of these factors improve sensitivity to a DM signal.With R&D advances in sensor design, the resolutions of CDMS detec-tors have improved over time, and different generations of CDMS detectorsemployed different sensor layouts. Table 2.2 provides some information onthe amplifier specification for three different generations of CDMS detec-tors. We specifically list the “iZIP” and “CDMSlite” style detectors used inthe SuperCDMS Soudan generation of the experiment, which we discuss inmore detail in Sec. 2.4 and Sec. 2.5. The DM search analysis described inChapter 4 of this thesis used the “CDMSlite” style detectors. We also listthe “ZIP” detector from the earlier CDMS II generation of the experiment.2.2.1 Measuring the e−/h+ EnergyAs described previously, a voltage is applied across the CDMS detectors sothat the e−/h+ pairs generated in an event do not recombine at the recoilsite but rather drift to the detector faces. As the ionization drifts to the sur-face, image charges are induced on the top and bottom faces of the detector.In CDMS II and SuperCDMS detectors, FET (Field Effect Transistor) am-plifiers read out the image charge on the electrodes and amplify this signal asa voltage that is further amplified by downstream amplifiers and eventuallydigitized as a function of time by the data acquisition system.In the CDMS II ZIP and SuperCDMS iZIP detectors, the voltage hasbeen tuned to the smallest possible value such that the e−/h+ pairs drift312.2. Amplifiersacross the full length of the detector. If the potential difference betweendetector faces is insufficiently high, the the e−/h+ pairs can trap in localminima created by impurities in the crystal. At the same time, if the poten-tial difference between detector faces is too high, the discrimination betweenERs and NRs suffers because the NTL phonons correlate the e−/h+ pair sig-nal with the recoil phonon signal. Therefore, for ER vs. NR discrimination,the optimal field was found to be ∼1 V/cm; for the CDMSlite approachwhere the detector is operated at greater than 70 V, ER vs. NR discrimina-tion is sacrificed in order to amplify the signal to achieve a lower threshold.The propagation time of e−/h+ pairs to the surface of the detectors isO(ns), the amplifier is “faster” than the propagation time (i.e. respondsto a signal on timescales < 1 ns), and the digitization rate of the amplifiersignal is 0.8 µs; therefore, the risetime of the e−/h+ pair signal is sharp andcontained in 1 or 2 time digitization bins. The falltime of the e−/h+ pairsignal is controlled by electronic components of the FET amplifier, and thisfalltime was selected to be significantly longer than the charge collectiontime; therefore the e−/h+ pair signal pulse assumes a relatively constantshape, regardless of the location of the initial recoil [77].While the pulse shape of the e−/h+ pair signal does not contain infor-mation about the position of the event, the electrodes are divided up into anumber of channels, and the relative amplitude of the charge signal in thosechannels does provide position information. For example, the iZIP detectordesign employs a specific electrode channel layout to identify events thatoccur near the surface of the detector, and this feature is described furtherin Sec. 2.4.2.2.2 Measuring the Phonon EnergyPhonon channels are instrumented on the detector faces, and each chan-nel consists of thousands of Transition Edge Sensors (TESs). The TESsare made from tungsten superconducting material whose transition temper-atures (Tc) are tuned in fabrication to be at cryogenic temperatures, butabove the base temperature of the cryostat (i.e. the temperature of the de-tectors). CDMS has fabricated TESs with Tc between 30mK and 200mK.The voltage-biased TESs are held within the range of their transition suchthat when heat from phonons reaches the sensors their resistance changesrapidly and the current through them decreases. The current through theTES is inductively coupled to a Superconducting Quantum Interference De-vice (SQUID) which further amplifies the reduction in current. TESs areamongst the most sensitive phonon amplifiers in existence, and they are322.3. Optimal Filter Event Reconstructionat the heart of the CDMS detectors’ sensitivity to dark matter. The sensortechnology is reviewed in K. Irwin’s and G. Hilton’s review article (Ref. [78]).Significantly more information than just the phonon energy is encoded inthe signal readout from the TESs. The prompt phonons in the first ∼100µsimpart more power on the sensors closest to the event. In contrast, theNTL and ballistic phonons carry no position information, but do providethe best measure of the event energy. The total absorption time of theballistic phonons is dependent on the aluminum collection fin coverage ofthe detector, and this affects the falltime of the phonon signal.The falltime of the phonon signal is roughly linearly dependent on thearea of the coverage, and the iZIP detector’s ∼5% coverage leads to rela-tively long 750µs falltimes. The design decision of relatively low aluminumcoverage led to some advantageous detector response characteristics, namelya straightforward method for separating event position from event energyin the different time scales of the phonon pulse. Figure 2.5 offers a visual-ization of the phonon propagation and collection in the iZIP detectors. Aposition-dependent signal exists in the first ∼100µs of the pulses, from theprompt phonons, with much larger signals on channel DS2 and CS1. Af-ter ∼300µs the phonons are in the fully ballistic regime and impart equalpower to each phonon channel, as shown by the aligned “tails” of the pulsesat later times. We obtain an estimate of the event location by comparingthe energy deposited in the different phonon channels. The partitioning ofenergy in the different channels allows a weighted average to determine thex, y, and z coordinate of the initial event within the detector[77]. The pulseshape of the phonon pulse (e.g. the peakiness of the pulse) in the individualchannels is also used to obtain more information about the event positionand energy. For example, in the analysis described in Sec. 4 of this thesis,it was particularly important to use phonon pulse shape to estimate theradial position of the event in the detector. This estimation is described inSec. 4.8.2.3 Optimal Filter Event ReconstructionFollowing the discussion of the techniques used to measure e−/h+ andphonon energy, it is important to discuss the techniques used to fit andextract information from the raw data. For every event, the raw data con-sists of a time stream of data, read out for each channel, that has beendigitized from the amplifiers. We list the digitization rates and other im-portant time and frequency values in Table 2.3. Because the analysis in this332.3. Optimal Filter Event ReconstructionR (kHz) N T (ms) ∆f (Hz) ∆t (µs)Charge 1250 2048 1.6384 ∼610.4 0.8Phonon 625 4096 6.5536 ∼152.6 1.6Table 2.3: Important time and frequency values for charge (e−/h+) andphonon amplifier digitizers from SuperCDMS Soudan. R, N , T , ∆f , ∆trefer respectively to the digitization rate, the number of bins in the readout time stream, the time length of the read out time stream, the lowestresolvable frequency (1/T ), and the time length of one digitization bin.thesis focuses on the SuperCDMS Soudan generation of the experiment, welist the digitization rates used in that generation, though the digitizationrates do not change much between generations.After the amplifier signals are digitized and saved, we use fitting algo-rithms to extract information from these raw data. For example, for theraw data pulse in Fig. 2.6 (left), we would like to estimate the amplitude ofthe pulse, the time at which the pulse occurs, the “peakiness” of the first∼500µs of the pulse, as well as a χ2 parameter that indicates the goodnessof fit. We primarily rely on three different types of “optimal filter” (OF)algorithms to fit the raw data pulses and extract these pieces of information.2.3.1 Simple 1D Optimal FilterIn this section we explain the basics of optimal filter (OF) theory. UsingFig. 2.6 for visualization, the OF is a fit of a template (red) to the data(blue) to determine the amplitude and start time (i.e. delay) of the pulse inthe data. The OF is truly the optimal algorithm to determine the amplitudeand delay under a set of assumptions: (1) the template perfectly describesthe shape of the underlying pulse in the data, and (2) the noise is a Gaussianrandom process whose power spectral density (PSD) is known. Even whenthese two characteristics aren’t strictly realized in the real data, the optimalfilter fits perform well.For Gaussian random noise, the frequency components of the noise areuncorrelated. This is why it is beneficial to perform the fit in the frequencydomain—an equivalent fit in the time domain would require accounting allcovariances between the time domain data points. The optimal filter per-forms the fit in the frequency domain where each frequency can be weightedby the signal-to-noise ratio, effectively extracting the signal from the noiseto get the best estimate of amplitude and delay. We will use Jeff Filippini’s342.3. Optimal Filter Event ReconstructionFigure 2.6: (left) An example simulated pulse with the template (normalizedto the best fit amplitude as given by the optimal fit). The template isalso left-right aligned with the best fit time delay as given by the optimalfit. (right) The frequency domain representation (magnitude only) of thetemplate, noise (PSD), and optimal filter.notation from Appendix A of his thesis, where A is the template, S is thesignal, and J is the noise PSD [77]. A discussion of how to calculate thenoise PSD is provided in Sec. 3.2. Additionally, the n subscript representsthe Fourier index, and A˜ and S˜ are the Fourier transforms of the templateand signal, respectively.The χ2 for the fit of the signal amplitude, a, is given by:χ2(a) =N∑n|S˜n − A˜n|2Jn(2.10)and the value of a that minimizes this χ2(aˆ) can be solved for analyticallyand is given byaˆ =∑nA˜∗nS˜nJn/∑n|A˜n|2Jn. (2.11)Typically, as suggested by the χ2 in Eq. 2.10, the optimal filter is thought ofin the context of a fit to a pulse; however, Eq. 2.11 shows that the optimalfilter can also be thought of applying a filter to the data. The denominatorof Eq. 2.11 is simply a normalization constant (that is, it is independent ofthe data S˜n), and so the numerator is the application of a Fourier domainfilter φ˜, given byφ˜n =A˜∗nJn, (2.12)352.3. Optimal Filter Event Reconstructionto the Fourier transform of the signal. A visualization of the amplitude ofthis filter is shown in Fig. 2.6 (right), where it is clear that any particularlynoisy frequency (1.2 kHz in this mock example) is deweighted in the optimalfilter φ˜n.The best fit amplitude of the signal, given by a, and the time offset ordelay, given by t0, are obtained by minimizing the following χ2:χ2(a, t0) =N∑n|S˜n − ae−2piit0fnA˜n|2Jn(2.13)which has used the fact that a shift in the time domain (A(t − t0)) corre-sponds to a phase rotation in the frequency domain (A˜(f)e−2piit0f ). This2D χ2 can be minimized analytically, and therefore OFs can be performedefficiently, with the limiting computational step being a fast Fourier trans-form [77].We frequently refer to best fit quantities from the 1D OF algorithm withthe “OF” suffix. For example, “ptOF” refers to the energy of the totalphonon pulse (the sum of the phonon pulses on all the detector channels)as estimated by the fitted amplitude from the standard 1D OF algorithm.2.3.2 Non-Stationary Optimal FilterIn the previous section, we described how the OF formalism was only optimalif the template perfectly described the underlying shape of the pulse inthe data. However, in Sec. 2.4, we described the position-dependent pulseshapes in the iZIP detectors. The non-stationary optimal filter improvesthe energy resolution of the 1D OF algorithm by treating the pulse shapenon-uniformity as a source of non-Gaussian, non-stationary noise5. Theinsight of treating the position dependence of the signal as noise came fromM. Pyle [79]. The non-uniform pulse shape of 10 keV events in the CDMSliteRun 3 detector is shown in Fig. 2.7 (left) with the residuals shown in Fig. 2.7(right). The position-dependent pulse shape variation noise between bin500 and 600 is clearly highly correlated across time and frequencies. Thisintroduces a complication because, in considering the non-stationary noise,the frequency domain no longer exhibits the convenient noise orthogonalityassumed in Eq. 2.16 for a 1D optimal filter. Therefore the non-stationary5The term “non-stationary” describes a type of noise whose variance changes in time.362.3. Optimal Filter Event ReconstructionFigure 2.7: (left) Example 10keV total phonon pulses with the OF tem-plate overlaid in blue. (right) The residuals (phonon pulse − template)showing that the position-dependent part of the pulse can be treated as anon-stationary source of noise.OF algorithm promotes the noise PSD J to a matrix V˜:χ2(a, t0) =∑f,f ′(S˜∗(f)− aA˜∗(f)e2piit0f )(V˜(f, f ′)−1)(S˜(f ′)− aA˜(f ′)e−2piit0f ′).(2.14)and the covariance between frequency modes is included when this χ2 isminimized.The covariance matrix V˜ is determined using the Fourier transforms ofthe pulse residuals (the phonon pulse minus the best fit phonon template)shown in Fig. 2.7. The complete non-stationary optimal filter algorithmalso accounts for the fact that the magnitude of the stationary noise isindependent of the pulse energy but the non-stationary noise increases as afunction of the pulse energy, and therefore the off-diagonal components of V˜undergo a magnitude scaling as a function of the pulse energy. The detailsof this implementation are beyond the scope of this discussion, and furtherdetails can be found in Ref. [80].We frequently refer to best fit quantities from the non-stationary OFwith the “NF” suffix. For example, “ptNF” refers to the energy of the totalphonon pulse as estimated by the non-stationary OF algorithm.372.3. Optimal Filter Event Reconstruction2.3.3 Two Template Optimal FilterIn contrast to the position-dependent deweighting approach of the non-stationary OF, the two-template OF seeks to fit the position-dependentpart of the pulse with two templates and then use the best template fitamplitudes and delays to extract enhanced energy and position informationabout the event. Motivated by the prompt vs. ballistic phonon propagationdynamics discussed in Sec. 2, the two template OF posits that the signal isa linear superposition of two different pulses each with different time con-stants. A “fast” template (with a fast falltime) is used in addition to thestandard “slow” template, such that we can describe the signal pulse asS(t) = a1A1(t− t0) + a2A2(t− t0) + n(t) (2.15)where the noise term n(t) is modeled as Gaussian noise. The slow template(A1) is identical to that used in the 1D OF and the non-stationary OF. Thenthe signal is fit with the fast and slow template, and Fig. 2.8 shows examplesof such fits. Computationally this is relatively straightforward, where theχ2 rewritten with new index j=1,2 for the two templates isχ2(a1, a2, t0) =N∑n|S˜n − a1e−2piit0fnA˜1,n − a2e−2piit0fnA˜2,n|2Jn(2.16)and the χ2 is minimized for t0 and the two amplitudes, a1 and a2.We frequently refer to best fit quantities from the two template OF withthe “2T” suffix.2.3.4 Use of Different OF AlgorithmsFor the Run 3 analysis, based on the analysis task being performed, weuse the parameter(s) derived from one of the three different OF algorithms.The different optimal filters used for the different analysis tasks is givenin Table 2.4. After applying the corrections discussed in Sec. 4.4.1, theenergy resolution of the non-stationary OF is superior to that of the othertwo algorithms, and therefore we use the non-stationary OF algorithm forenergy estimates. As discussed in Sec. 4.8, the parameters derived from thetwo template OF are superior to other OFs for determining the position ofthe event in the detector. Information from the standard 1D OF is used todistinguish good events from instrumental background events.382.3. Optimal Filter Event ReconstructionFigure 2.8: (top)An example of the 2 template fit to data, where the promptsignal is modeled by the fast template and the ballistic signal is modeled bythe slow template. (bottom) An example of the 2 template fit performed onthe 4 individual phonon channels for a single event on a CDMSlite detector.392.4. iZIP Interleaved DesignOF Type Primary Usesimple 1D instrumental background discrim.non-stationary energy estimatetwo template position estimateTable 2.4: The primary use of the three different OF algorithms.Figure 2.9: The iZIP design with interleaved charge electrodes (±2V ) andphonon rails (0V). Used with permission, from [81].2.4 iZIP Interleaved DesignThe CDMS II detectors suffered primarily from one design flaw. The ion-ization from electron recoils close to the detector surface was more prone totrapping, which reduced the ionization yield of the event [81]. This reducedionization yield caused surface electron recoils to mimic the nuclear recoilsignature and thus leak into the WIMP signal region. CDMS II sensitivitieswere limited by this background [81].A new detector (the iZIP) was designed to provide a solution to discrimi-nate the surface event background. The ionization electrodes are interleavedbetween the phonon sensors, as shown in Fig. 2.9, which allows readout ofboth ionization and phonon energy on each detector face. Just as impor-tantly, the phonon TES rails are maintained at 0V while the ionizationelectrodes are kept at opposite potentials (±2V in standard operation) oneither face. This configuration produces a unique electric field within thecrystal (Fig. 2.10) wherein the ionization from surface events will largely becollected on one side of the detector face. The CDMS and EDELWEISScollaborations have shown that the interleaved design allows for robust re-jection of surface events. An analysis cut on asymmetric charge collection402.4. iZIP Interleaved DesignFigure 2.10: (left) The electric field and potential lines produced from thephonon rails (yellow) and charge electrodes (green). Notice that the uniquesurface E-field extends ∼1mm into the crystal and therefore surface eventswithin this margin should exhibit asymmetric charge collection. (right) Datafrom iZIP detector T3Z1 showing surface event discrimination (discussionin main text). Internal CDMS figure, used with permission, from [81].on side 1 vs. side 2 of the iZIP rejects the surface events.Figure 2.10 (right) shows the ionization yield vs. recoil energy for eventsfrom 900 hours of exposure of a Soudan iZIP detector denoted as T3Z1 (thetop detector in tower 3) to a 210Pb source. The 210Pb source was found, asexpected, to produce ∼130 surface electron recoils per hour via beta decay.These events exhibit reduced ionization yield and fail the symmetric chargecut. They populate the region above the 2σ nuclear recoil band and belowionization yields of ∼1. The events below the germanium nuclear recoil bandbut also failing the symmetric charge cut are surface events from recoiling206Pb nuclei (the end product of the 210Pb decay). The colored blue dots areevents that pass the symmetric charge and accordingly show large ionizationyield corresponding to bulk electron recoils. Out of the 90,000 events in thisplot, two outliers exist that pass the charge symmetry cut but show lowionization yield, which are blue and circled in black [81]. Overall, this studydemonstrates robust surface rejection capability of the iZIP.The partitioning of channels on the iZIP, with 3 inner phonon channelsand an outer annulus channel as shown in Fig. 2.5, provides x, y, and zinformation about the position of the event in the detector. The promptphonon signal leads to a larger pulse in the channel closest to the event anda smaller pulse in the channel furthest from the event, and so we can usea linear combination of the optimal filter amplitude from each channel to412.5. CDMSlite Detectorsmake phonon-based position estimates. We label the calibrated individualchannel amplitudes as pXOF, where X refers to channel A,B,C, D, or thesum of the channels “sum”. Then with the channel layout as shown inFig. 2.11, each side of the iZIP (as indicated by the i index) provides an xposition estimate ofpxpartOFi =cos(30◦) · pdOFi + cos(150◦) · pbOFi + cos(270◦) · pcOFipsumOFi(2.17)and a y position estimate ofpypartOFi =sin(30◦) · pdOFi + sin(150◦) · pbOFi + sin(270◦) · pcOFipsumOFi.(2.18)With the energy partitioned into only 3 channels, the resolution of thesepartition estimates are never interpreted to be exact, and Fig. 2.11 demon-strates this where the distribution of a set of events uniformly distributedin the detector shows up as a “partition triangle,” instead of a circle. De-spite the apparent crudeness of these partition estimates, they are useful forchecking the individual channel calibrations as well as identifying certainelectronic glitch or instrumental events that only cause a signal on one ofthe channels (an example of this type of event is discussed in Sec. 4.6). Az position estimate is given by the normalized difference in phonon energybetween the two sides:pzpartOF =psum1OF− psum2OFpsum1OF + psum2OF. (2.19)The additional fast template information from the two template OF fittinghas been used to enhance position information in particular for the devel-opment of a radial parameter for single-sided iZIP readout, as discussed inSec. 4.8.2.5 CDMSlite DetectorsIn 2012, subsequent to the development of the iZIP, CDMS explored the op-eration of an iZIP detector in an alternative configuration in which a higherbias is applied across the detector. This higher detector voltage amplifies theionization signal by producing NTL phonons. Instead of biasing the detectorso that there is a 4 V potential difference between the faces, roughly −70 Vis applied. This custom biasing configuration required some modifications422.5. CDMSlite DetectorsFigure 2.11: (left) The iZIP phonon channel partition on a single side, withchannel names labeled. (right) The px-py partition for uniformly distributedevents for an iZIP detector, where the triangular shape is due to the channellayout as well as the partition variable definition (Eq. 2.17, 2.19).to the electronics. A single side of the detector was biased to high voltage,while the other side was grounded, and only the phonon channels and chargechannels on the grounded side were read out, as shown in Fig. 2.12.The primary advantage of the CDMSlite mode is that the energy thresh-old of the detectors is lowered. The NTL amplification increases the signal,but not the noise, of events so that low energy recoils are able to be mea-sured and reconstructed. Without NTL amplification these events would be“buried” under readout noise. The advantage of a low detector threshold isincreased sensitivity to a low mass DM signal—kinematically recoils fromlighter DM particles deposit less energy in the detector. Figure 2.13 demon-strates the shifting of the DM spectral shape to lower energy as a functionof DM mass and the importance of a low threshold. For example a detectorwith a 2 keV nuclear recoil threshold has no sensitivity to 2 GeV/c2 DM-nuclear recoil signal. The CDMSlite detectors have achieved thresholds of∼0.5keVnr and are therefore sensitive to DM masses lower than 2 GeV/c2.The primary disadvantage of the CDMSlite operating mode is that ER/NRdiscrimination by using the partitioning of the energy between e−/h+ pairsand phonons is no longer possible. This is the sacrifice that is made for thelower detector threshold: as is clear from Eq. 2.6 and 2.9, a large detec-tor voltage and correspondingly large production of NTL phonons creates astrong correlation between the phonon signal and the e−/h+ pair signal forboth ERs and NRs. The result of this tradeoff between a lower threshold432.5. CDMSlite DetectorsFigure 2.12: A rough schematic of the custom CDMSlite biasing electronics,which shows that only a single side of the detector is read out while the otherside is biased using the “High Voltage” power supply (credit: R. Thakur[83]).and ER/NR discrimination is that the CDMSlite detectors have an “irre-ducible” source of background from the ER events. The three CDMSliteanalyses have therefore been background limited. Each analysis has im-proved upon the methods of the background reduction and modeling tech-niques of the earlier runs, and despite the irreducible ER background someCDMSlite analyses have produced world-leading limits for WIMP massesless than 5 GeV/c2 at the time of publication [82].The higher background rate in the CDMSlite detectors from ERs is re-duced slightly in the low energy region of interest (ROI)—where the detec-tors are most sensitive to low-mass DM—because ER background events areamplified by a larger fraction than NR signal events. This is just a conse-quence of the total phonon energy dependence on the ionization yield, asshown in Eq. 2.9. The effect of the larger ER amplification is to “stretch”ER backgrounds relative to the NR signals, which leads to the reduced ERbackground rate in the ROI.2.5.1 CDMSlite Biasing and Readout ConfigurationThe CDMSlite biasing configuration led to a number of non-ideal featuresof the detector response relative to the iZIP operation. First, with half thechannels being read out, the signal to noise ratio was√2 worse relativeto what it could have been if all channels could be read out (assuminguncorrelated noise between the channels). The single-sided readout also442.5. CDMSlite DetectorsFigure 2.13: DM recoil spectra for different DM masses. The vertical linesshow example energy thresholds and demonstrate the importance of thresh-old for sensitivity to a low mass DM-nuclear recoil scattering. The CDMSlitedetectors have achieved thresholds of ∼0.5keVnr. (credit: M. Pepin [84])required the development of new estimates of the event position, which aredescribed in Sec. 4.8. In addition, the surface events that the iZIP’s uniqueelectric field was designed to remove could not be discriminated againstwith the CDMSlite biasing configuration. We revisit the modeling of thisbackground in the CDMSlite detectors in Chapter 4.The potentially most significant side effect of the CDMSlite biasing con-figuration was the non-uniformity of the electric field in the detectors. Thenon-uniformity arises because the materials surrounding the detector (e.g.the housing) are grounded, and this distorts the electric field. One cross-sectional slice of the detector is shown in Fig. 4.39, where near the detectorsidewall the ∆V can be smaller than the applied voltage. Therefore, forevents at high radii, the voltage drop experienced by an electron-hole pair(and thus the NTL amplification) can be reduced such that the reconstructedenergy is significantly lower than for the same energy events at lower radii.Section 4.8 explains how we model and remove these events when analyzingdata from the CDMSlite detector.2.5.2 CalibrationWe calibrate the CDMSlite detectors by activating the detector’s 70Ge witha 252Cf neutron source. Activation of 70Ge by neutron capture produces452.5. CDMSlite DetectorsShell Energy [keV] Probability %K 10.37 87.6L 1.30 10.5M 0.160 1.78N 0.0015 0.14Table 2.5: The energies and probabilities of electron capture for 71Ge. TheN -shell is included, though events from this EC have never been observedin the CDMSlite detectors because of the low energy of this process.71Ge, which decays by electron capture with a 11.43 day half-life [85]. Thisprocess is given by:70Ge + n→ 71Ge71Ge + e→ 71Ga + νe71Ga→ 71Ga + γ’s + e’s.(2.20)The γ’s + e’s represent the decay products from the electron recoil as 71Gareorders its electron shell into a stable state [67]. These decays producepeaks at the K-, L-, and M -shell binding energies of 71Ga of 10.37, 1.30, and0.16 keV, respectively [86]. The energy spectrum with K-, and L-, shell 71Geelectron capture peaks is shown in Fig. 2.14. The most prominent peak is theK-shell (1s orbital) peak. Decays from the lower energy peaks (L- and M -shell peaks) occur at a lower rate because of smaller electron wavefunction-nucleus overlap as well as electron binding energy factors. Formulae forcomputing electron capture probabilities are computed in Ref. [87, 88], andtabulated in Table 2.5. In the data we observe a ratio between K- and L-shell peaks that is consistent with the expected relative rates.The K-shell peak is used to calibrate the energy scale and correct forany time and position variation in the detector response. The L- and M -shell peaks are used to check the resulting energy scale for linearity. Theserelatively low energy 71Ge activation peaks are critical for the energy scalecalibration of CDMSlite because, unlike the calibration lines from othercommon calibration sources (133Ba and 252Cf), these events have sufficientlylow energy that they do not saturate the TESs in CDMSlite operation.2.5.3 CDMSlite History: Run 1 and Run 2The early commissioning work for CDMSlite surveyed the array of Super-CDMS iZIP detectors (the array pictured in Fig. 4.4) to determine which462.5. CDMSlite DetectorsFigure 2.14: The reconstructed total phonon energy of data with 71Ge elec-tron capture peaks present. The colors label data taken at different detectorvoltages, where the NTL amplification is apparent at the 30V, 50V, and 70Vdata. The energy scale of the data at each voltage is calibrated so that theK-shell peaks appear at the correct energy (i.e. the total phonon energygiven by Eq. 2.9), and additional details and insights gained from these“voltage scan” data are discussed in Sec. A.1.detector was best suited for CDMSlite operation. In this selection process,one of the primary factors was the detector’s leakage current, which imposeda voltage maximum below which the detector exhibited stable operation.While the dominant mechanism of charge leakage in the iZIP detectors isunknown (and is likely from a combination of mechanisms and variable be-tween detectors), the basic idea is simple. When the voltage applied acrossthe detector is sufficiently high, a small current will start to flow across thedetector (i.e. the detector resistance is not infinite). This current could comeeither from the interface of the detector or from charges in shallow-potentialimpurity sites in the bulk of the detector (both of which are depicted inFig. 2.15), or from another source. The leakage charges will produce NTLphonons that will be read out by the TES phonon amplifiers, and becausethe leakage current is a quasi-Poissonian process, the leakage current createsa source of noise in the phonon signal.To first order, the leakage current in the tested iZIP detectors at Soudanwas found to be negligible until a certain “breakdown” voltage was reached.This “breakdown” effect was also observed by P.N. Luke in one of the firstpublications describing the NTL effect [76], as depicted in Fig. 2.15 (cen-ter). The optimal operating voltage before breakdown in the iZIP detectorswas found to be ∼70 V. At these voltages, even relatively large detectorresistances, ∼1 PΩ, will produce leakage currents of 4×105 electrons/sec472.5. CDMSlite Detectorsthrough the detector, which is a significant source of NTL phonon noise inthe phonon amplifier. Leakage current studies are discussed in more detailin Sec. 3.2, and efforts to improve the detector leakage currents for futureCDMS detector are underway.CDMSlite Run 1 operated the detector in the middle of the fifth tower,as depicted in Fig. 4.4 and referred to as T5Z2, that had low leakage cur-rents. The optimal voltage was found to be 69V, where above this voltagethe increase of a leakage current led to elevated noise. The detector wasoperated for a relatively small period of time in 2012 and the DM searchlive time totalled 6.25 kg-days. The analysis achieved an analysis thresh-old of 170 keVee, and the limit placed on DM-nucleon cross section wasworld-leading below 6 GeV/c2 DM mass at the time of publication[82].CDMSlite Run 2 also operated the T5Z2 detector, but did so for signif-icantly longer (70 kg-days over the course of 2014) and biased the detectorto 70V. A number of operational and analysis improvements were made inthis run. The three most significant operational improvements are describedin more detail below.1. Pre-biasing. Run 1 observed that the noise in the minutes followingthe HV biasing of a detector was higher relative to later times. At thetime the hypothesis for this excess noise was a higher leakage currentfrom trapped charge lying in shallow potentials, and the charges werereleased once the detector was biased to high voltage; this hypothesisis made in Appendix B of Ref. [83]. The period of higher noise wasobserved even following the standard CDMS detector neutralizationprocedure, which consists of exposing the detector to 1.31 eV LEDphotons prior to operation. The motivation behind this neutralizationis to nullify any possible trapped charge, as described in Appendix H ofRef. [89]. Unfortunately, there was no evidence that the neutralizationprocedure eliminated the period of elevated noise following HV biasing.It was found that this period of elevated noise could be shortened ifthe detector voltage was raised above the operating voltage for ∼10minutes, with the hypothesis that the higher voltage emptied the trapsmore efficiently. The noise at later times, after the pre-bias period, wasalso found to be lower than without the pre-bias. Fig. 2.15 bottomshows the baseline RMS baseline noise with and without the pre-bias,with the triangles (without pre-bias) showing a worse resolution thanthe circles (with pre-bias).During the second CDMSlite run, a 80V prebias was used for 10 min-utes prior to the data taking with the detector biased at the 70V482.5. CDMSlite DetectorsFigure 2.15: (top) A simple depiction of a regular event from a particleinteraction alongside two leakage events, one from the detector bulk andone at the detector interface. (middle) From P. Luke’s publication, showingthe signal-to-noise benefits of NTL amplification up to ∼ 140V for the diodedetector described in Ref. [76]. (bottom) The baseline total phonon noiseRMS, labeled as “σOF [keVt ]” for the T5Z2 detector, showing an increasein the noise above ∼ 60V and a severe worsening of the noise above ∼ 70V.The triangles on this plot provide supplemental measurements of the noiseRMS without pre-biasing, demonstrating the effectiveness of the pre-biasprocedure.492.5. CDMSlite Detectorsoperating voltage6.2. Accelerometer Vibration Readings. Analysis of the phonon sig-nal power spectral density (PSD) (e.g. Fig. 4.9) on Soudan detectorsshowed a broadband increase of noise at low frequencies, in contrastwith what is expected from fundamental TES noise, which does not in-crease at low frequencies [78]. A subsequent series of studies at Soudanwas used to trace this LFN excess back to vibrational sources in theSoudan setup [90]. Accelerometers were installed on the cryocooler,known to be the most vibrationally active part of the experiment, andevent rates and LFN noise events were indeed found to be stronglycorrelated with the periods of higher amplitude cryocooler vibrations.In Run 2, the accelerometer readings were used to identify and isolateperiods of particularly high LFN. Then, particularly restrictive cutsto discriminate against LFN induced events were used for those peri-ods with high noise. The accelerometer readings are discussed in thecontext of CDMSlite Run 3 in Sec. 4.3.2.3. High Voltage Current Readings. The schematic diagram shownin Fig. 2.16 depicts the HV biasing scheme of CDMSlite detectors,where the schematic is simplified to not show any of the readout elec-tronics. A critical part of the biasing scheme, which was added priorto Run 2, is to continuously monitor the HV current (IHV ) in order todetermine the detector voltage relative to the applied voltage. Withan effectively infinite detector resistance, the nominal IHV would be0 amps. However, parasitic resistance to ground (shown here as Rp >1 GΩ), enables a small current (up to 120 nA) to be sourced by the HVpower supply. While the exact location of this parasitic resistance (i.e.current leak) is unknown, there is strong evidence that the leak occursat a component of the room temperature electronics, potentially onthe CDMSlite HV biasing board.A large bias resistor (Rb = 196 MΩ) is used to protect against signifi-cant current flow into the cryostat in the case of a short, however thatlarge bias resistor also results in a significant decrease of the voltageat the detector, even for small HV leakage currents. For example, a120 nA leakage current results in a reduction of 24 V at the detector.By reading the HV power supply current over the course of Run 2,the voltage of the detector was calculated on an event-by-event basis.6This same pre-bias procedure was used for CDMSlite Run 3, where a 85V was used asthe pre-bias point for the nominal 75V operating voltage, as described further in Sec. 4.502.5. CDMSlite DetectorsFigure 2.16: A simple schematic of the CDMSlite biasing circuit.The largest current measured in Run 2 was found to be 25nA, whichcorresponds to a 6.5% difference between the applied voltage and thedetector voltage, which was corrected for in the analysis.7In addition to the operational improvements made for CDMSlite Run 2,numerous advances were also made during the analysis and described inM. Pepin’s thesis [67].One significant improvement was the development of a parameter thatidentified events that occurred at high detector radii that would experiencereduced NTL amplification and therefore would be misreconstructed to alower energy [91]. Further improvements to the radial fiducial volume mod-eling were made for the CDMSlite Run 3 analysis, and we save the moredetailed description of the radial parameter to that section of this thesis(Sec. 4.8).The CDMSlite Run 2 analysis also found environmental variables (e.g.cryostat base temperature) and pulse-shape characteristics (e.g. pulse peak-iness) that were correlated with the energy estimate of an event, and theanalysis corrected the energy scale with these variables and significantly im-proved the resolution of the 252Cf calibration peaks. Because the Run 3analysis employs similar energy corrections, we describe the details of theprocedure in Sec. 4.4.1 in the context of the Run 3 data.7In contrast to Run 2, the largest current measured in Run 3 was 120nA, and thecurrent was significantly less stable throughout the run. The incorporation of the HVcurrent readings for the Run 3 analysis is discussed in Sec. 4.3.3.512.5. CDMSlite DetectorsCDMSlite Run 1 and Run 2 DM Exclusion Limits: TheOptimum Interval MethodThe operational and analysis improvement described above led to the Run 2analysis achieving an analysis threshold of 0.056keVee. The addition of theradial cut resulted in a final energy spectrum of events that was largelyunderstood, where different populations of events could be attributed toknown sources. The final Run 2 energy spectrum (converted to the nuclearrecoil energy) is shown in Fig. 2.17, where the two background peaks arefrom M - and L-shell events. Despite an understanding of most backgroundsources, at the lowest energies (0–0.75 keVnr in Fig. 2.17) the ability todistinguish a good event (caused by an energy deposition in the detector)from an instrumental event (caused by a noise fluctuation that was largeenough to generate a trigger and read out the “event”) is diminished. TheRun 2 analysis cuts were defined in such a way that there was a significantuncertainty about whether the events at the lowest energy were good eventsor instrumental events.Because of the potential leakage of instrumental events past the Run 2analysis cuts, the analysis used a conservative technique for setting a DMexclusion limit. The technique is called the Optimal Interval (OI) methoddeveloped by S. Yellin [92, 93], and is succinctly described in the title of his2002 paper: “Finding an Upper Limit in the Presence of Unknown Back-ground.” The method does not require a background model, which sig-nificantly simplifies the analysis. This is because, not only do backgroundmodels not need to be developed, but also instrumental events (for whichthere is no physically motivated background model) can leak past the cutswithout serious consequences for the analysis.The OI method works by interpreting all events as potential signal eventsand then computing a signal size that is incompatible with the data at the90% confidence level (CL). The method to quantify “incompatible” uses theexpected signal shape and the energy intervals between events. An exampleinterval in the CDMSlite Run 2 data is shown in Fig. 2.17. For the DM signalshapes shown in Fig. 2.13, one can find an interval where the backgroundis low and where the signal shape is especially high that will be able tooptimally exclude the signal at some magnitude at 90% CL; this interval isthe “optimal interval.” The OI method finds this interval and applies anappropriate trials factor for selecting an interval that gives the best limit(i.e. it accounts for the “look elsewhere” effect).This approach has a number advantages in that it does not require abackground model and is conservative, but one clear disadvantage is that522.5. CDMSlite DetectorsFigure 2.17: The CDMSlite Run 2 energy spectrum after all cuts have beenapplied, with the energy of the events measured by the non-stationary op-timal filter, and converted to the nuclear recoil energy using a Lindhardk parameter of 0.159. One of the energy intervals is highlighted betweenevents as a visualization of the energy intervals considered by the optimuminterval method. The optimal interval considers intervals between all pairsof events (not necessarily adjacent pairs of events)it cannot be used to discover a signal in the data; it can only be used toset upper limits. The Run 1 and Run 2 analyses used this method andtherefore were not discovery-potential analyses, and in the Run 3 analysiswe explicitly address this shortcoming and employ a likelihood approachto search for a signal in the dataset with discovery potential. Despite thisshortcoming of the OI method, at the time of publication, the Run 2 analysisexcluded new spin independent WIMP-nucleon parameter space between 1.6and 5.5 GeV/c2. Both the Run 1 and Run 2 90% CL upper limits are shownin Fig. 2.18.532.5. CDMSlite DetectorsFigure 2.18: Spin-Independent WIMP-nucleon cross section 90% CL upperlimits from Run 1 and Run 2 (black solid with 95% uncertainty band) com-pared to the other most sensitive results in this mass range (at the timeof publication of the Run 2 result): CRESST [94] below 1.6 GeV/c2 andPandaX-II [43] above 4 GeV/c2.54Chapter 3Detector Characterization3.1 Detector ResolutionThe CDMS detectors’ energy resolution is one of the primary drivers oftheir sensitivity to low-mass DM. This is because, as previously discussedin Sec. 2.5 and depicted in Fig. 2.13, the expected signal from DM particlesof increasingly lower mass is higher at low energy. Better energy resolutionsallow the detectors to resolve lower energy recoils, effectively increasing thedetector’s exposure to a low-mass DM signal. In this chapter we focusonly on the resolution of the phonon amplifiers, as opposed to the e−/h+amplifiers, because (1) the phonon amplifiers are intrinsically more sensitiveand therefore are the primary drivers of the sensitivity, and (2) the capacityto improve the phonon amplifiers with new detector designs is greater andtherefore many CDMS R&D efforts are channeled in this direction.Detector resolutions are typically thought of as varying as a function ofenergy (e.g. see Sec. 5.1.1), but in this chapter we focus on the “baseline”resolution because it is particularly important for the detector’s sensitivityto the lowest energy recoils. The baseline resolution is a measurement of thedetector’s resolution at zero energy. Therefore, a lower baseline resolutioncorresponds directly to a lower detector threshold and the ability of thedetector to observe lower energy recoils above the detector noise.Detector research and development efforts seek to improve the baselineresolution by either increasing the amplifier signal or decreasing the amplifiernoise. Efforts to increase the signal include NTL amplification (discussedin Sec. 3.2) as well as increasing the phonon collection efficiency of theamplifiers (measured in Sec. 3.1.4). Efforts to measure, understand, andreduce the phonon noise are discussed in Sec. 3.1.2 and Sec. 3.2.We use two complementary methods of calculating the baseline resolu-tion. The simpler method is a direct measurement using a combination ofcalibration data and noise data acquired from the detector. The more in-volved method combines characterizations of the detector’s electronic andphonon collection properties with resolution models from signal processingand optimal filter theory [78]. The more detailed analysis of the baseline553.1. Detector Resolutionresolution indicates that the signal-to-noise ratio should be considered as afunction of frequency, which has implications for the preferred pulse shape,or pulse bandwidth, as discussed in Sec. 3.1.5.Below we compare the results from these two different detector base-line resolution calculations for a number of prototype detectors fabricatedas part of the R&D effort for SuperCDMS SNOLAB. These detectors werefabricated with lower TES transition temperatures (Tc) than the SoudaniZIP detectors, which was one of the primary design changes to reduce thephonon noise. To first order, this noise reduction occurs because the pro-jected dominant noise source for SNOLAB, the thermal fluctuation noisebetween the TES and the cooling bath, is reduced at lower temperatures;for further discussion see Ref. [95]. The detector resolution measurementresults demonstrate improved detector performance of the prototype detec-tors relative to the SuperCDMS Soudan detectors, and the achievement ofdetector performance goals for the SuperCDMS SNOLAB experiment.3.1.1 Data-Driven Resolution MeasurementThe direct measurement of a detector’s baseline resolution requires calibra-tion data and “noise” data. The calibration data is used to determine acalibration factor to convert the output of the phonon amplifier (in amps)to an energy (in eV). The use of the “noise” data is less obvious; it consists oftime series data that are digitized and read out in the same way as standardevents, except that the data acquisition system is triggered randomly withthe intention of reading out a time series in which no event occurs. There-fore, noise data are frequently referred to as “random triggers” or simply as“randoms.”The random triggers are fit by the 1D optimal filter (OF) algorithmand the amplitude of the fit is recorded. As shown in Fig. 3.1 (left), eventhough there has been no energy deposition in the detector when the randomtrigger was digitized, the OF algorithm will fit a small non-zero amplitudebecause of the noise fluctuations. The noise fluctuation in this example hascaused the amplitude to fit to a negative value (which is just as likely as apositive value when no signal exists in the data). It is clear that the OFamplitude fit of the random triggers can be used to characterize the detectornoise because larger noise fluctuations will result in a larger span of the fitamplitudes to both positive and negative values. The standard deviation ofthe distribution of OF amplitude fits to a collection of random triggers givesthe baseline resolution.563.1. Detector ResolutionFigure 3.1: (left) Example simulated random trigger (noise only) with thebest fit OF template overlaid in red. Even though there is zero signal, noisefluctuations result in the OF algorithm fitting a small non-zero amplitude.(right) The OF amplitude fit results (from a detector with two readoutchannels) of many random triggers, when the OF algorithm has no constrainton the time delay search window [96].Constraining the Optimal Filter Time OffsetFigure 3.1 (right) shows the histogram of many OF fit amplitudes to randomtriggers (for a two-channel detector), and also reveals a subtlety in the OFfitting of random triggers. The bimodal distribution occurs because of thetime-offset degree of freedom (which minimizes the OF χ2 for left-right shiftsof the time domain template as described in Sec. 2.3), which allows the OFfit to find the largest noise fluctuation that resembles a signal pulse withinthe entire time series of data. The OF then returns the amplitude of thatfluctuation. Because it is likely that the OF finds a noise fluctuation thatis significantly different than zero, the distribution shown in Fig. 3.1 (right)has a deficit of events at reconstructed amplitudes at and around 0 keV.This undesired time-offset degree of freedom of the standard OF is dis-abled when measuring the detector baseline resolution: the time offset (t0in Eq. 2.16) is clamped to t0 = 0 in the fit. With this adjustment, we referto the OF amplitude of the total phonon trace as “ptOF0,” where the 0 in-dicates that the t0 has been forced to 0, and with this adjustment OF fits torandom triggers result in a Gaussian distribution, as shown for example inFig. 3.2, and the 1σ width of this Gaussian distribution defines the baselineresolution.573.1. Detector ResolutionEfficient Calibration of low-Tc DetectorsThe lower Tc TES transition temperature has the primary advantage ofreducing the TES noise, but one negative side effect of the lower transitiontemperatures is that the TESs exhibit a smaller “dynamic range.” Thatis, low-Tc TESs begin to saturate for lower levels of incident power, andthis became problematic in the testing of these prototype detectors becausethe TES began to saturate at the energies used to calibrate the detector. Inparticular, the 356 keV γ events from 133Ba served as an efficient calibrationsource for the SuperCDMS Soudan iZIP detectors, which were fabricatedwith Tc in the 70–110 mK range. In this section, we show results from aprototype detector (referred to as detector S12C, with the same dimensionsand channel layout as the iZIP detectors) with Tc in the 60–68 mK range,and the TESs were found to begin to saturate for the 356 keV γ events. Theefficient nature of the 133Ba 356 keV γ calibration still made it appealing fordetector calibration for the resolution studies presented in this section, butbecause the TES saturation introduced non-linearity into the TES response(and therefore non-linearity into the energy scale of the OF amplitude energyestimate) extra calibration steps were required.The OF underestimates the energy of saturated events, and so the na¨ıvecalibration approach of centering the 356 keV events at that energy resultsin an energy scale that is biased to low energy. The pulse “integral” energyestimate, which has worse energy resolution than the OF energy estimate butis less prone to energy scale non-linearities from saturation effects than theOF energy estimate, was used to correct the OF amplitude energy estimate.The correction assumed that (1) the integral and OF energy estimates wereboth linear at low energies and that (2) the integral energy estimate waslinear up to 356 keV. The latter of these assumptions comes from the factthat the OF estimate is highly sensitive to the pulse shape, whereas theintegral is less sensitive. The saturation correction technique centers the356 keV events at that energy in the integral energy estimate, and alsoforces a one-to-one relationship between the integral and OF at lower energy.This ensures that at low energy, at the energy where the baseline resolutionis measured, the OF energy scale is unbiased. Figure 3.2 left depicts thiscalibration and correction technique.With the OF energy scale calibrated, the baseline resolution was deter-mined in units of eV as shown in Fig. 3.2 right. At the time of measurement,the 52.4 eV resolution measured on S12C was one of the lowest measuredresolutions on an iZIP-style detector. This improvement was one importantpiece of evidence of the detector noise improvements from lowering the Tc583.1. Detector ResolutionFigure 3.2: (left) Data from 133Ba calibration of the S12C detector, in the to-tal phonon integral vs. total phonon OF energy estimates. (right) The S12Cdetector baseline resolution measurement using random triggers and mea-sured using the constrained time-offset OF, giving a resolution of 52.4 eV.of the TES sensors, and the following sections (3.1.2, 3.1.3, 3.1.4) focus onthe TES characterization to improve the understanding of such an energyresolution.3.1.2 Phonon Amplifier NoiseThe noise power spectral density (PSD) is the variance of noise as a functionof frequency, and is calculated using a collection of random triggers that areuncontaminated by events and are representative of the detector noise. Thediscrete Fourier transform (DFT) g˜n of a time series gk is given by:g˜n =1NN2−1∑k=−N2gke−2piifntk (3.1)whereN is the number of samples in the time series and where g˜n is evaluatedfor n = −N2 → N2 − 1 in integer steps. The frequency spacing is given byfn = n1T where T is the time length of the time series in seconds. The timesteps are given by tk = k∆t = kTN and in this notation are symmetricalaround 0. To calculate the PSD J , the squared magnitude of the DFT |g˜n|2is calculated for each random, and the average value of |g˜n|2 is calculatedfor each frequency index n:Jn = 2T × 〈|g˜n|2〉. (3.2)593.1. Detector ResolutionThe normalization convention for PSDs is notoriously inconsistent, inpart because at times it is convenient to consider the positive and negativefrequencies over which the Fourier transform is defined and at times it isconvenient to consider just the positive frequencies. CDMS has historicallyconsidered positive frequencies, and then “folded over” the PSD across they axis and applied the factor of 2.When considering the TES noise, gk is typically a current in amps, andtherefore the units of the PSD, Jn, is amps2/Hz. Very commonly, instead ofconsidering Jn as a representation of the variance of the noise as a functionof frequency, we plot the standard deviation, or√Jn. There are benefits ofboth units, and√Jn is convenient since this quantity is linearly proportionalto the baseline resolution of the detector (Eq. 3.16) as discussed in Sec. 3.1.5.We show multiple PSDs, in units of amps/√Hz, for the iZIP S12C de-tector in Fig. 3.3 top. The individual channels are shown by the coloredlines, while the total phonon (i.e. traces summed in the time domain) PSDis shown by the solid black line. The quadrature sum of the individual chan-nels is shown by the dashed black line, and it is useful because it representsthe total phonon noise that would exist if that noise between the individualchannels was uncorrelated. Since fundamental TES noise is uncorrelatedbetween channels and environmental noise sources are more likely to be cor-related between channels, comparison of the total phonon and quadraturesum PSDs is useful for diagnosing reducible environmental noise sources.Also, as shown in Sec. 3.1.5, the quadrature noise is useful for deriving a“best case” baseline energy resolution that is achievable if the environmentalnoise sources are removed.3.1.3 TES Voltage BiasingThe TES is a very sensitive power to current amplifier, and in this sectionwe discuss how to measure the dynamic current response (dI) of the TESto a change of input power (dP ). The voltage-biased TES schematic isshown in Fig. 3.4 (left) where the shunt resistor Rs is significantly smallerthan the equilibrium TES resistance, an inductor L is included to facilitatereadout of the current through the circuit with a SQUID amplifier (a highlysensitive current to voltage converter), and there is stray parasitic resistanceRp in the circuit. It is critical that the TES be voltage-biased (i.e. Rs Rp + RTES) as opposed to current-biased for stable operation. To firstorder, this stability condition can be explained when considering the power603.1. Detector ResolutionFigure 3.3: (top) The current noise (amps/√Hz) for each channel , the totalphonon channel sum, and the quadrature channel sum for the S12C iZIPdetector. (bottom) The S12C power noise (watts/√Hz), converted from themeasured current noise using the (dI/dP ) conversion described in Sec. 3.1.4.Figure 3.4: (left) Simplified circuit diagram for the TES amplifier, with theshunt resistor (Rs), parasitic resistor (Rp), inductor (L), and TES elementlabeled. (right) The Thevenin equivalent voltage-biased TES circuit, whereVb = Ib ·Rs and the load resistor is given by Rl = Rp +Rs.613.1. Detector Resolutiondissipated in the TES:PJoule =V 2R= I2 ·R. (3.3)When the TES resistance increases upon some incident phonon power froman event in the detector, with a constant voltage across the TES the Joulepower dissipated decreases as indicated by Eq. 3.3. This reduction of Joulepower for an increase in TES resistance returns the TES to equilibrium andis referred to as negative electrothermal feedback. By contrast for a current-biased TES, increasing the resistance increases the Joule power dissipated inthe TES, thereby further increasing the resistance and this positive feedbackloop results in unstable operation. A significantly more thorough consider-ation of stable voltage-biased TES operation is discussed in Ref. [95] andRef. [78].3.1.4 Phonon Collection EfficiencyThe phonon collection efficiency (phonon collection) is the fraction of energyfrom an event that ends up as dissipated power in the TES. It is an impor-tant quantity to measure because the baseline energy resolution is directlyproportional to phonon collection, and because measurements of phonon collectionhelp inform design choices for future detectors. We calculate the energy col-lected in the TES using the current read out by the TES in combinationwith measurements of the electronic components of the TES circuit. Thenwe take the ratio of this collected energy with the known true energy of theevent, and define this as the TES collection efficiency:phonon collection =EcollectedEtrue. (3.4)Measuring the TES collection efficiency is not only a critical step incalculating the theoretical baseline resolution of the detector (Sec. 3.1.5)as well as the detector leakage current (Sec. 3.2), but it is important tomeasure for each prototype detector to determine how changes in the detec-tor fabrication process are affecting the TES collection properties. For theprototype detector S12C, we do the phonon collection measurement on achannel-by-channel basis as well as for the full detector (the channel sum).When the TES is operating in the stable voltage-biased regime, the tem-perature of the TES never increases or decreases appreciably and rather apower flux δP incident on the sensor results in a reduction of current δIthrough the sensor that satisfies PTES,in = PTES,out. In order to compute623.1. Detector ResolutionFigure 3.5: The simplified TES thermal diagram. In this simplified model,the bath represents the collection of the crystal and the dilution refrigerator.More realistic thermal models are considered in Ref. [77, 95].these two terms we start with the simplest versions of the electric differentialequation governing the TES:Vb = IRl − LdIdt+ IRTES (3.5)as well as the thermal differential equation governing the TES:CdTdt= I2RTES − Pbath + δP (3.6)where C and T are the TES heat capacity and temperature respectively,the electronic component names are derived from Fig. 3.4 (right), and thethermal component names are derived from Fig. 3.5. To see how a non-equilibrium power input δP is converted in a change δI from the equilibriumTES current I = I0 + δI, we solve for the voltage across the TES:VTES = I ·RTES = Vb − I(RL) + LdITESdt(3.7)and plug this into the thermal differential equation, Eq. 3.6:CdTdt= I ·(Vb − I ·RL + LdITESdt)− Pbath + δP. (3.8)We now integrate both sides of the equation over time and extend the inte-gral to effectively infinite times (long after the TES has returned to equilib-633.1. Detector Resolutionrium), giving:∫ t→∞0dt CdTdt=∫ t→∞0dt(I ·(Vb − I ·RL + LdITESdt)− Pbath + δP).(3.9)Because T and ITES either do not change or return to their original valuesas the TES returns to equilibrium, the differential terms go to zero whenintegrated. Further rearranging so as to solve to for the circuit response tothe power pulse δP gives:∫ t→∞0dt δP = −∫ t→∞0dt (I · (Vb − I ·RL)− Pbath) . (3.10)We also make the approximation that Pbath stays at its equilibrium valueas long as the TES stays in the stable regime of negative electrothermal feed-back. This is a valid assumption because, when balancing a δP in Eq. 3.6,the decrease in Joule heating (V 2TESRTES) due to the increase in RTES is signif-icantly more efficient than increasing Pbath. This is attributed to the factthat a change in the power dissipation to the bath requires a temperaturechange of the TES:Pbath ∝ (T 5 − T 5bath) (3.11)and as previously stated the TES does not appreciably change temperaturewhen in electrothermal feedback. The equilibrium power dissipation to thebath is equal to the equilibrium Joule power dissipation in the TES, andtherefore:Pbath = PJoule,0 = I0 · VTES = I0(Vb − I0RL). (3.12)By Taylor expanding the TES current through the TES (ITES = I0 +δI)we express the current as an equilibrium component I0 as well as a variablecomponent δI that represents the dynamic signal that is measured. We alsoplug in for Pbath, and Eq. 3.10 becomes:∫ t→∞0dtδP = −∫ t→∞0dt((I0 + δI) · (Vb − (I0 + δI) ·RL)−[I0(Vb − I0RL)]).(3.13)Cancelling and rearranging terms gives a relationship between a displace-ment from the power equilibrium and the displacement from the current643.1. Detector ResolutionAs1 Bs1 Cs1 Ds1 As2 Bs2 Cs2 Ds2R0(Ω) 0.3 0.27 0.29 0.30 0.35 0.29 0.31 0.33Rp(Ω) 0.016 0.016 0.015 0.016 0.018 0.017 0.018 0.017Ib(µA) 82.0 60.5 63.4 63.5 48.8 64.6 64.6 72.2Vb(µV ) 2.0 1.5 1.5 1.5 1.2 1.6 1.6 1.7Rl(Ω) 0.030 0.030 0.029 0.030 0.032 0.031 0.032 0.031Table 3.1: Measured resistances and bias currents (Ib) by channel on theS12C detector.) The values below the double horizontal line are not directlymeasured, but rather derived based on the schematics of Fig. 3.4, whereVb = IbRs, Rl = Rp + Rs, and the shunt resistor Rs is taken to be 0.024Ωbased on historical measurements.equilibrium, δI:∫ t→∞0dt δP = −∫ t→∞0dt[δI · (2I0RL − Vb) + (δI)2RL]. (3.14)We can perform the integral effectively to t → ∞ by integrating to a timeafter the system has returned to equilibrium. For an energy deposition in aniZIP detector, the phonon propagation and phonon collection fin coverage(discussed in Sec. 2.4) result in δP pulse fall times of τF ∼ 750µs. Therefore,as long as we integrate to t & 3ms we will integrate over the full time thatthe phonon power from the event is being dissipated in the sensor, which isequivalent to the energy from the event that is collected by the TES:Ecollected =∫ &3ms0dt δP. (3.15)The combination of Eq. 3.15 and Eq. 3.14 allows us to calculate theenergy absorbed in the TES using the measurable electronic componentsI0, RL, Vb as well as the TES signal δI. Ecollected can be calculated on anevent-by-event basis, and the true energy of the event Etrue can be deter-mined with an auxiliary detector calibration, and so the phonon collectionefficiency can be calculated on an event-by-event basis with Eq. 3.4.Phonon collection efficiency results for the S12C detector are shown inFig. 3.6. A series of <70% efficient phonon absorption processes occur be-tween the TES and the detector which leads to the observed average 22%total collection efficiency. The lossy phonon collection mechanisms are notcovered in this thesis, and are an active area of research within CDMS [95].The final result of 〈phonon collection〉 = 22% is invaluable in evaluating the653.1. Detector ResolutionFigure 3.6: (left) The phonon collection efficiency (phonon collection) calcu-lated for events in the 100–200 keV range, where Ecollected is calculatedusing the total phonon signal (the channel sum). The solid (dashed) his-togram represents the Etrue measured by the optimal filter (pulse integral)energy estimate. (right) The phonon collection efficiency measured on achannel-by-channel basis. The individual S12C channels (of which there are8 total) on average collect 2–3% of the event’s energy.detector performance because it indicates to what extent the detector reso-lution can be attribute to phonon collection properties vs. noise properties,which in turn informs future detector designs.3.1.5 Optimal Filter Baseline ResolutionThe best possible energy resolution of the TES via the optimal filter can bedetermined by evaluating the second derivative of the OF χ2(a) amplitudefit (Eq. 2.16), which gives:σaˆ =[ N/2∑n=0(2|A˜n|2Jn)]−1/2. (3.16)A˜n is the template and normalized such that:N∑n=0An[∆t] = phonon collection. (3.17)Here ∆t = 0.8µs and is the time digitization rate, and we use the measuredcollection efficiency of 22% as shown in Fig. 3.6 left.663.2. Detector Leakage CurrentFor the PSD (Jn), we use power noise as given by Fig. 3.3. This gives σ =51.2 eV. Comparing with measured resolution obtained via the data drivenmethod (the 52.4 eV shown in Fig. 3.1), offers a good cross check becausethese two values were arrived at relatively independently: the data-drivenmethod depended on a calibrated energy scale, while the calculation of thetheoretical resolution of the detector depended on a calibrated energy scale,a measurement of the phonon collection efficiency, as well as an accuratepower noise measurement.The utility of Eq. 3.16 goes far beyond the above example of under-standing the signal and the noise contributing to the data-based resolutionmodel. The combination of a noise estimation and phonon collection effi-ciency estimates with Eq. 3.16 permits projections of detector resolutionsfor future and existing detectors operating in ideal operating conditions. Forexample, in Fig. 3.7 we consider the fundamental limit of the S12C resolu-tion operated in an ideal environment, instead of at the surface test facilitywhere environmental noise sources abound. In this extrapolation to betteroperating conditions, we remove the anomalously noisy channel Cs2 fromthe equation, which decreases phonon collection by78 but overall leads to animproved detector resolution as√J is decreased by a larger fraction. Theremoval of this channel leads to an estimated baseline resolution of 48 eV.The next projection indicates that if we could remove the non-fundamentalnoise sources that generates a correlated noise component on all TESs (seenas the difference between the channel sum and the quadrature sum PSDs)then a 35 eV threshold would be achievable on S12C. Extrapolations such asthis are important for future hardware decisions, where for example we showthat a 15% improvement in the baseline resolution can be achieved withoutany detector design changes but instead with superior environmental noisesuppression strategies.3.2 Detector Leakage CurrentWith the phonon resolution of the prototype detectors demonstrated tonearly 50 eV, our R&D efforts turned to the high voltage (HV) operation ofthe prototype detectors. The CDMSlite detectors, operated as part of theSuperCDMS-Soudan payload, first demonstrated the effectiveness of apply-ing HV (70 V) across CDMS detectors to improve detection sensitivitiesto WIMPs lighter than 5 GeV/c2, as shown in Fig. 2.18. For SuperCDMSSNOLAB, combining high resolution phonon sensors with HV across the de-tector is projected to provide sensitivities to WIMP masses below 1 GeV/c2673.2. Detector Leakage CurrentFigure 3.7: The power noise observed on S12C with channel Cs2 removed,with the detector baseline resolution (from Eq. 3.16) provided in the overlaidtext.[97]. For example, by measuring the phonon signal with a baseline reso-lution of σ=50 eV, and operating the detector at 50 V, the phonons fromrecoils that ionize just a single e−/h+ pair (<30 eV nuclear recoils) will beobserved above the TES noise.The preceding paragraph brings up the following question: since we havedemonstrated that prototype detectors have reached baseline resolutions of50 eV (Sec. 3.1.1) and have demonstrated operation of Soudan IZIP detec-tors up to 70 V (Sec. 2.5), why have we not observed this signal from singlee−/h+ pairs above the TES noise? The signature of this single e−/h+ pairresolution would be a quantized energy spectrum at the lowest energies,which as of yet is unobserved in the iZIP-sized detectors. The shorter an-swer to this question comes in two parts. First, the high sensitivity of theprototype detectors presents challenges when testing them in the high radia-tion environment at a surface test facility as opposed to underground at theSoudan or SNOLAB sites. Second, by operating a smaller detector that hasreduced susceptibility to the radiation levels in surface test facilities, andhas lower intrinsic TES noise, the SuperCDMS collaboration has observedthe signal from single e−/h+ pairs above the TES noise [98]. The longerexplanation of the difficulties of operating of large (iZIP scale) detectors atHV at the surface is discussed below in the following order:• Section 3.2.1 describes the high rate of background particles (primarily683.2. Detector Leakage CurrentFigure 3.8: A simulated representation of the energy deposition in surfacedetectors with iZIP dimensions (76mm diameter) as well as for larger diam-eter (100mm) detectors being fabricated for the SNOLAB experiment. Thetime window over which an event is typically digitized (50ms) is given bythe time window shown by the vertical lines around 13.5 sec.from cosmic ray muons) in the detector. These backgrounds generatea large number of e−/h+ pairs in the detector.• Section 3.2.2 describes the involved process of identifying and removingperiods of time when the detector is affected by muons.• Section 3.2.3 describes measurements of detector leakage currents,specifically:– how the leakage current appears as an excess phonon noise in thedetector– a leakage current model, as well as a complementary simulationof the model, to confirm that the excess phonon noise that isobserved comes from a leakage current– a measurement of the leakage current on a variety of differentdetectors, with implications for the detector resolutions– a discussion of leading hypotheses for the source of the leakagecurrent.3.2.1 Detector Ionization Environment at Surface FacilitiesWe have found that performance tests at the level of single e−/h+ pairsensitivity require significant improvement of our control of background ion-ization in the detector when testing iZIP-sized detectors at the surface. Oneunavoidable background comes from cosmic ray muons that strike iZIP-sizeddetectors at a rate of 0.8 Hz with a Poisson distribution in time. Because themuons are high enough in energy that they stream through the full length of693.2. Detector Leakage Currentthe detector, on average each muon deposits the same amount of energy inthe detector (under the rough assumption that the muons travel verticallythrough the full height of the germanium). Muons traversing through the25.4 mm detector height deposit 18 MeV of energy, which translates to18 MeV3eV per e/h pair= 6× 106 e−/h+ pairs. (3.18)For a 0.8 Hz muon rate, this is equivalent to a muon-induced flow of 4.8 ×106 e−/h+ pairs per second, or a current of 0.8 pA, through the detector.This is an enormous current, especially in the face of trying to observe thesignal from a single e−/h+ pair. Even if the detector has the sensitivity toobserve low energy events that ionize a single e−/h+ pair, in practice it ismade exceptionally difficult because of the muon-induced ionization that is6 orders of magnitude larger.Fortunately, we have the ability to identify periods during which, andshortly after, a muon has passed through the detector. Unfortunately, theiZIP detectors require approximately 250 ms to return to equilibrium fromthe thermal energy generated in the detector from a muon event. Thisrecovery time means that there is very little time that the detector is in anoperational state when high-quality data can be read out.Figure 3.8 shows a simulation of a time series of energy deposited in thedetector from muons, as a function of time, at a surface test facility. No-tice that the thermal decay time from muon events (250 ms) is significantlylonger than the athermal signal from lower energy events, and the time win-dow over which an event is typically digitized at the surface test facilities(50 ms) is given by the time window indicated by the vertical purple linesaround 13.5 seconds. As can be visualized in Fig. 3.8, there are periodsof time sufficiently long after a muon event that the detector has returnedto equilibrium. However, as the detector voltage is increased and the en-ergy from the muon event generated in the detector increases linearly as afunction of voltage, the amplitude of the “muon tails” grows and it becomesincreasingly rare that the detector is operating in equilibrium. Analysis toolsare able to identify when an event or noise trigger has been read on top ofa “muon tail”, and we veto these events using tools described in Sec. 3.2.2.3.2.2 Muon Veto Analysis TechniquesFigure 3.9 shows real data (50 ms waveform) from random triggers from aniZIP operated at the surface, which shows large variation in the slopes ofthe noise traces and a large variation in the DC value of the traces. This703.2. Detector Leakage CurrentFigure 3.9: Random traces acquired on a prototype iZIP detector operatedat a surface test facility, showing the slopes due to muon “tails” evident tovarying degrees on top of the underlying phonon noise. The labels in the topleft of the plots correspond to the labels in scatter plot shown in Fig. 3.11.713.2. Detector Leakage CurrentFigure 3.10: The PSD constructed with slopeless random traces (labeled as“underlying noise”). The dashed PSDs are calculated with random triggerswith varying steepness of slopes (in units of amps/ms) in the randoms.variation is to be expected based on the simulated energy deposited in aniZIP detector as a function of time shown in Fig. 3.8. The challenge posedby data of this nature is that if all the randomly acquired traces are includedin the PSD calculation (Eq. 3.2), large amounts of low frequency noise willbe injected into the PSDs which will be misinterpreted to be the actualunderlying TES noise. This is shown directly in the PSDs in the dashedlines in Fig. 3.10, where we have selected random traces with a particularslope across them to create the PSD. Even for PSDs constructed with tracesonly containing modest slopes across them, for example 1×10−6amps/ms,there is a ×5 increase of the noise at 50 Hz as shown by the dark blue PSDvs. the “underlying” PSD that is constructed with slope-less traces.In order to reliably measure the underlying phonon noise, some selectioncriteria must be applied to the random triggers prior to constructing thePSD. We have developed analysis scripts to reject randoms traces contam-inated with muons, other background events, or that are otherwise anoma-lously noisy, by characterizing the randoms pulses with 4 quantities thatprovide information about the state of the detector at the time when therandom was read out. These 4 quantities are:1. The DC value of the random: if a muon has recently interacted in thedetector, the DC value will be high2. The slope of the random: if the detector is relaxing back to equilibriumafter a muon event, there will be a significant slope across the trace3. The optimal filter best fit amplitude (aˆ): if a pulse exists in the ran-723.2. Detector Leakage Currentdoms, the optimal filter should fit the pulse and the corresponding aˆshould be large relative to the pulse-free random triggers4. The optimal filter χ2 value: if the random is anomalously noisy orcontains multiple pulses, the χ2/DOF will be significantly greater than1Figure 3.11 top shows the distribution of randoms in the DC value vs.slope plane for 5 different voltages. Based on the tolerance of the PSDto sloped randoms, an acceptance region is defined in the 4D parameterspace, which is roughly shown as the boxed region in the DC value vs. slopeplane. At low voltages, the detector is stable enough to apply the selectioncriteria and acquire a sufficient number of randoms (∼20) to construct aPSD that represents the true TES noise. However, at higher voltages thedistribution of random traces in the parameter space changes significantly,as shown clearly in Fig. 3.11 bottom. In particular, the necessary cut onthe slope of the trace (given by the red dashed vertical line in the inset ofFig. 3.11 bottom) has an 80% random acceptance with the 0 V data, but a<5% acceptance for the detector operated at 44 V. With low cut efficienciesat high voltage, these analysis scripts have indicated that sometimes 20minutes of data is required in order to acquire 1 second of high quality data.Despite the waiting time introduced by the muon-veto analysis techniques,they are critical for accurately measuring the TES noise PSD in a surfacetest facility.3.2.3 Phonon Noise as a Function of Detector VoltageEven though the muon-veto analysis scripts are effective at removing peri-ods of high ionization in the detector, surprisingly there is a source of lowerenergy ionization that appears to be a continuous source of current throughthe detector. Up to this point we have been unable to eliminate or reliablyreduce this source of ionization. We observe small numbers of e−/h+ pairsleaking across the detector as a continuous source of excess power noise inthe TES. The leakage becomes increasingly problematic at higher voltagebecause it becomes a large additional source of Luke phonons, which gener-ates excess noise in the TES.The excess noise is visible in the time domain, and Fig. 3.12 explicitlyshows the phonon noise on two channels at 0 V (left) compared to 44 V(right). The excess noise is also clearly visible in the frequency domain, asshown for the S12C detector, as well as the G23R prototype detector (which733.2. Detector Leakage CurrentFigure 3.11: (top) The distribution of randoms in the DC value vs slopeplane, showing the approximate acceptance region of this parameter spaceto select the random traces that are representative of the underlying TESnoise. The red symbols highlight the region in this plane that the raw tracesin Fig. 3.9 are selected. (bottom) A projection of the data onto just the traceslope axis, showing that the selection criteria become increasingly inefficientwhen the detector is operated at high voltage.743.2. Detector Leakage CurrentFigure 3.12: Examples of time domain noise from an iZIP-style prototypedetector. The data come from two channels: channel A, top, blue andchannel B, bottom, orange. The left (right) panel show noise traces fromthese two channels, with the detector at 0 V (44V). The traces read outat 44 V clearly have worse noise. It is also clear that the excess noiseis correlated between channels, which is expected from a charge leakagenoise source that stochastically generates NTL phonons distributed as smallphonon pulses. These small pulses are absorbed equivalently to the standardphonon pulses on the iZIP—equally on all channels in the ballistic limit—which leads to correlated noise.753.2. Detector Leakage CurrentFigure 3.13: (top) Power Noise PSDs measured on a phonon channel onthe S12C detector (with negative HV applied to side 2, reading out side 1).Overlaid on each PSD is a leakage current model using the measured phononfall time. (bottom) Power Noise PSDs measured on a phonon channel of theG23R prototype detector.is also an iZIP-style detector), in Fig. 3.13. The excess noise has a character-istic shape—largest at low frequencies and negligible at high frequencies—which is consistent with the shape of the phonon pulses (recall the pulsetemplate curve shown in Fig. 2.6 right). This makes the excess shot noise aparticularly unpleasant noise source: it has perfect overlap with the signalbandwidth.Every detector tested at the surface exhibits this excess voltage-dependentnoise, and below we motivate a physical noise model that derives the ob-served bandwidth of the shot noise and helps quantify the processes thatare producing the excess noise.Charge Leakage Noise: Derivation of the Effect of Detector ShotNoise on TES NoiseWe refer to charge leakage noise as a “shot noise” because we use WalterSchottky’s 1918 result of vacuum tube noise [99] as a starting point for amodel for how individual electrons and holes traversing the detector appear763.2. Detector Leakage CurrentFigure 3.14: The black data depicts a time series of the shot noise model,where single charge quanta are distributed with inter-arrival times randomlysampled from a e−λt distribution (i.e. a Poisson process). For visualizationpurposes the delta functions have been widened from being infinitely narrowto 0.5 ms wide. The average current used in this simulation was 20×10−18amps (corresponding to an average of 6.25 charge quanta per 50 ms interval,which fluctuated to 8 charges in this realization). The blue data depicts thephonon power generated by the charges traversing the detector, for a 50 Vdetector bias, and for a phonon absorption time of τF = 175 µs to matchthe S12C detector.as phonon noise in the TES. The shot noise model posits that the chargestraversing the detector are delta-function quanta bursts that are Poissoniandistributed in time. We denote these charge bursts as δI(t) × e, whichoccur over a time length Ttrace. Figure 3.14 depicts a time series of chargequanta bursts, where for visualization purposes the delta functions havebeen widened from being infinitely narrow to 0.5 ms wide.Schottky’s result shows that the current noise PSD (SI , in units of[Amps2/Hz]) generated by Poisson distributed charge quanta is linearly pro-portional to the current (I):SI(f) = 2 e |I| = 2 e2 〈N〉Ttrace, (3.19)where 〈N〉 is the average number of charge quanta in a time series of lengthTtrace. Notice that SI is independent of frequency. The units of the righthand side are coulombs2/sec = amps2/Hz, as desired. A full derivation ofthis result is not given here, though the general form of this result makessense in that Fourier transforms of δI(t) give white noise ( |δI(ω)|2 = 1) and773.2. Detector Leakage Currenttherefore the total noise power spectral density of these Dirac-delta pulsesis also white. Additionally, it makes sense that the magnitude of the noiseincreases with the current (i.e. the number of charge quanta fluctuations).A full derivation of this result is given in Ref. [100].The average rate of the dirac-delta functions (i.e. the charge leakagerate) is given by NTtrace = 〈n˙〉 which we use throughout the remainder ofthis derivation of the charge leakage noise model. Note that by consideringjust the average leakage rate, this model excludes even small cascade effects,where the leakage of a single e−/h+ pair increases the breakdown likelihoodof other e−/h+ pairs.Using the shot noise result, Eq. 3.19, for the noise created by delta-function shaped charge quanta, we derive the PSD for phonon pulses with acharacteristic fall time. The delta-functions of current through the detectorrelease NTL phonons that bounce around the detector and are absorbed atthe same rate as phonons from normal pulses, with the detector thereforeeffectively acting as a low pass filter. The filtering effect is depicted inFig. 3.14, where the charge quanta bursts are converted to the blue 175 µs falltime pulses. The detector also acts as a current to phonon power converterby the NTL effect:Phonon Power = I∆V. (3.20)The full current to power transfer function is then given by∣∣∣∣∂P∂I∣∣∣∣ = ∣∣∣∣ ∆V1 + iωτf∣∣∣∣ (3.21)where τf is the absorption time of the NTL phonon signal (also known asthe pulse fall time, and 1/τf is also known as the signal bandwidth). Wethen convert the current noise (units of [Amps2/Hz]) to power noise (unitsof [Watts2/Hz]) by multiplying by the square of the current-power transferfunction, giving:SP,shot =∣∣∣∣∣ ∆V 21 + ω2τ2f∣∣∣∣∣× e2 × 〈n˙〉. (3.22)The final consideration is a correction to Eq. 3.22 for the < 100% phononcollection efficiency. Because only a small fraction of the total phonon energyis absorbed in the TESs, the power noise observed in the TES (√SP,shot, inunits of [Watts/√Hz]) is decreased proportionally to the phonon collection783.2. Detector Leakage Currentefficiency, phonon collection:SP,shot(ω) =∣∣∣∣∣ ∆V 21 + ω2τ2f∣∣∣∣∣× e2 × 〈n˙〉 × 2phonon collection. (3.23)This model predicts a specific frequency spectrum of the shot noise, and wesee that the spectral shape predicted by this model exists in both simulationsof detector noise and the real data, as shown below.Simulation of the Effect of Detector Shot Noise on TES NoiseThe noise model described above can be simulated on top of fundamentalTES noise in order to see if the PSD of the simulated traces matches theprediction from the derivation. A depiction of this simulation is shown inFig. 3.15.We start with a simulation of the fundamental TES noise, without anycharge leakage noise present. We then inject small pulses into the TES noise;the amplitude of the pulses is given by the voltage across the detector whilethe number of pulses is determined by the magnitude of the leakage current.The fall time of the pulse is given by the phonon collection time (i.e. thefall time of the phonon pulses) and in this simulation we use τf = 175 µs tomatch the collection time observed on the S12C detector. In Fig. 3.15, thegreen pulse shows one of the charge leakage pulses for a detector voltage of50 V (multiplied by 10 in order to be visible). We them sum 6250 of thesepulses together, Poisson distributed over the 52 ms simulated time series(which corresponds to the number of charge leakage for a leakage current of0.02 pA), and the result of summing many of these pulses together is givenby the red curve. Adding the fundamental TES noise gives the blue curve.We simulate a collection of the TES plus shot noise time series at aparticular voltage, and then compute their PSD. We repeat this for voltages0 V, 10 V, 20 V, ... 100 V to see how the noise changes in the frequencydomain as a function of voltage, and the results are shown in Fig. 3.16.Notice that the excess noise shape observed in the simulation matches thespectral shape observed in the data for the S12C detector (Fig. 3.13 top)which has the same signal bandwidth used in the simulation.Notice also, in Fig. 3.13, the excess noise spectral shape change is dif-ferent between the S12C and G23R detectors, and this spectral difference ispredicted from the model of Eq. 3.22. The noise moves to higher frequen-cies from G23R (which has a fall time of τf = 875 µs) to S12C (which hasa fall time of τf = 175 µs), which is the trend predicted by the model. The793.2. Detector Leakage CurrentFigure 3.15: Example of the simulation of a single 50 ms data stream ofcharge leakage shot noise injected onto fundamental TES noise. The greenpulse is amplified by a factor of 10 (in order to be visible) and its fall timeis τf = 175 µs to match the S12C detector. Many of the pulses are Poissondistributed in time and added to the fundamental TES noise, giving theblue curve.spectral shape predicted by the model is specifically overlaid as grey linesin Fig. 3.13, which fits the data well except for on S12C at the highest volt-ages. The best explanation for this behaviour only at the highest voltages isthat the excess power from the NTL phonons generated by charge leakage ischanging the dynamic response of the TES and changing the pulse fall timeslightly from τf = 175 µs.Calculation of Leakage CurrentsThe excess TES noise is used to calculate the leakage current for the S12Cdetector at a variety of voltages. We solve for the charge leakage rate, n˙, inEq. 3.23, where for SP,shot(ω, V ) we calculate the contribution of the shotnoise by subtracting the 0 V noise from the total noise:SP,shot(ω, V ) = SP,total(ω, V )− SP,total(ω, V = 0). (3.24)We solve for n˙ and other quantities relating to the detector leakage current,as a function of voltage using the PSD curves shown in Fig. 3.13 top, inTable 3.2.The leakage current is calculated for a variety of detectors on which anexcess noise was found as a function of voltage applied across the detector,as shown in Fig. 3.17. The error bars include uncertainties in the measure-ments of power noise as well as uncertainties in the voltage applied across803.2. Detector Leakage CurrentFigure 3.16: PSDs are made of the simulated traces using different detectorvoltages in the simulations. The noise shape and magnitude are consistentwith those predicted by the shot noise model (Eq. 3.23), as well as consistentwith the noise shape seen in the data (e.g. Fig. 3.13).Voltage (V) SP,total(ω = 0, V ) charges/sec (n˙) I (amps) Rdetector (Ω)0 1.6×10−33 ∼0 ∼0 ∼ ∞10 1.6×10−33 ∼0 ∼0 ∼ ∞21 3.6×10−33 1.2×105 1.5×10−14 1.1×101548 1.0×10−32 9.4×104 1.5×10−14 3.2×101573 4.4×10−32 2.1×105 2.3×10−14 2.2×101581 1.2×10−31 4.7×105 7.6×10−14 1.1×1015Table 3.2: The leakage current, as well as the effective total ohmic detectorresistance, derived from the listed excess noise levels at low frequency onthe S12C detector.813.2. Detector Leakage CurrentFigure 3.17: Leakage current measurements, as a function of voltage, for anumber of detectors (primarily prototype, primarily iZIP style and size). Alldetectors here were measured at a surface test facility, with the exceptionof the blue “T5Z2” data point. This data point is determined using dataacquired from Soudan, but is a rough estimate for a number of reasons, oneof which is that low frequency noise present in all PSDs from Soudan maskedthe shot noise signal.the detector. Examining just the S12C data points, it aligned with expec-tations that at some voltage the current increased significantly (as observedat ∼80 V) based upon the idea of “breakdown,” where above some voltagethe Coulomb barriers of the detector interface are overcome and a currentflows. The more surprising finding from this study was the leakage currentbehaviour at low voltage, showing a nearly voltage-independent magnitude.This I-V relationship was seen on a number of the other tested detectors,and has implications for the detector resolution, which are discussed in thesection below.Effect on Detector ResolutionsThe primary problem with the leakage current is that it negatively affects thedetector baseline resolution. The principle of NTL amplification, as shown inFig. 2.15 center, is that the signal increases linearly with the voltage and the823.2. Detector Leakage CurrentFigure 3.18: (left) The total phonon baseline resolution (σt) for a numberof detectors as a function of detector voltage. The solid lines express ideal(no leakage current) resolution scalings as a function of voltage where the σtvalue remains constant as a function of voltage. The data points representresolution measurements at those voltages, or deduced resolutions basedon PSD measurements and Eq. 3.16. (right) The electron equivalent energyresolution (σt) as a function of voltage, derived from the left hand plot usingEq. 3.26.noise remains constant until some “breakdown” value. With constant noisevs. detector voltage, the baseline resolution will be constant as a functionof voltage when measured in units of total phonon energy. As describedin Sec. 2.5, the NTL amplification of the signal is linear as a function ofvoltage:Et = Eee(1 +e Vb) (3.25)so that when the baseline energy resolution is measured in electron equiva-lent energy units, it is inversely proportional to the voltage bias:σee =σt1 + e Vb/. (3.26)When there is constant noise vs. voltage the σt resolution is constant, andthe solid lines in Fig. 3.18 show this ideal baseline energy resolution scaling.When that resolution is converted to the electron equivalent energy scale(σee), the resolution is inversely proportional to the detector voltage, asshown in Fig. 3.18 (right).When the noise increases as a function of voltage, the σee baseline res-olution will improve more slowly as function of the detector voltage thanindicated by the solid lines of Fig. 3.18. For the data points in Fig. 3.18 left,we use Eq. 3.16 to calculate the total phonon energy detector baseline reso-lution σt, using the observed noise Jn as a function of voltage. For Vdet > 0833.2. Detector Leakage Currentthe data points are above the ideal resolutions because the noise increasesas a function of voltage.Current Source HypothesesUp to this point we have not been able to reduce the shot noise inducedby the leakage current nor identify the source of the ionization. We haverobustly demonstrated that infrared radiation (IR) is not the cause. To elim-inate low energy ionization due to any infrared background at the detector,IR blocking “Bock Black” paste developed by the millimeter astronomy com-munity [101] was applied to the detector lid in between measurements of theleakage current, and no change in the leakage was observed. Other hypothe-ses for the leakage currents are leakage from the instrumented surface ofthe detector, and leakage from the detector bulk from impurities in shallowpotential wells hopping into the conduction band. This bulk leakage modelwould be akin to the leakage that is observed in the prebiasing proceduredescribed in Sec. 2.5.3 and discussed further in the Appendices of Ref. [83].An additional hypothesis relates to the unstable voltage across the detec-tor due to the large amount of ionization generated from muon events. Itis of course possible that the leakage current is a combination of multiplesources, which may make it particularly difficult to diagnose. Subsequentto the measurements made here, additional insights into the combinationof the contributions to the leakage current have been made on the smallerscale quantization detector [98] with improved control of systematic errorsfrom the smaller detector area.Fortunately, many of the mechanisms that would generate the signifi-cant leakage currents shown in Fig. 3.17 would be reduced when operatingthe detector underground. An additional piece of evidence that suggeststhat the leakage current is suppressed for underground detector operationis presented in the following chapter. This chapter contains analysis of datafrom the T2Z1 detector operated at Soudan in 2015 during which the de-tector voltage varied between 50 V and 75 V. While the low frequency noisepresent at Soudan limited abilities to study the voltage dependent shot noise,Sec. 4.3.4 presents a study of the trigger rate on T2Z1 and shows that below72 V the detector operation was stable and the trigger rate effects due tonoise were exponentially decreasing below 72 V.84Chapter 4Data Analysis of CDMSliteRun 34.1 MotivationAs the final CDMSlite run at Soudan, the CDMSlite Run 3 analysis servedas a test-bed for many techniques that are being considered for the SNO-LAB science analyses, and we discuss three of these new techniques in thischapter. First, a blinding technique was used for the first time in CDMSliteas a method to reduce the possibility of analyzer bias. Second, advancedmultivariate discrimination techniques were used to efficiently remove in-strumental backgrounds at the lowest energies measurable by the detector,which were a limiting background for the Run 2 analysis. Third, a MonteCarlo modeling of the detector fiducial volume was used to inform our “ra-dial cut” which removed misreconstructed events at high detector radii.All of the above techniques contributed towards the larger goal of mov-ing away from the optimum interval limit-setting approach, discussed inSec. 2.5.3, towards a more powerful limit-setting approach. Successful im-plementation of the above techniques allowed us to accomplish that goal andthe following chapter, Chapter 5, discusses the details of the limit-setting ap-proach used for the Run 3 analysis. This method had the benefit of movingCDMSlite from an exclusion-only analysis into one with discovery poten-tial, and also the benefit of improving the limit over the optimum intervalmethod.In addition to the new techniques brought to the Run 3 analysis dis-cussed in the following chapter, a number of the techniques employed arethe same as those used in the Run 2 analysis. In particular, we use the samemethod of pulse simulation to measure cut efficiency. We also use the samemethod of discriminating against the more obvious instrumental noise typeswith optimal filter fits using instrumental noise templates. Additionally, thefiducial volume Monte Carlo built on the radial parameter successes fromthe Run 2 analysis.854.2. Salting4.2 SaltingPrior to the start of the CDMSlite R3 analysis, the collaboration decidedto adopt a blinding strategy to prevent analyzers from making biased de-cisions in order to reach a desired result. Because this was the first timethat a CDMSlite analysis would be performed blind, a blinding task forcewas formed to consider a variety of blinding schemes. These included datadivision, where a fraction of the events are used to tune cuts, develop back-ground models, etc. and the remaining events are used to search for a DMsignal. “Box blinding” was also considered, where the region of the parame-ter space where the signal is expected is hidden from the analyzers and onlyopened once analysis decisions are finalized using information outside of theblinded region.Ultimately these two blinding schemes were determined to be ill-suitedfor CDMSlite because of instrumental noise events that exist in the data. In-strumental noise events make up the majority of events and their magnitudeand characteristics change as a function of time. Therefore it is desirableto be able to see all events at each stage of the analysis and so, rather thanhiding events, the blinding task force implemented data “salting” in which afraction of the events in the dataset were replaced with artificial signal-likeevents. This procedure effectively masked the true amount of DM signal inthe data (i.e. it is “blinding with a flashlight”).The salting procedure was developed openly in that the algorithm wasknown to the CDMSlite Run 3 analyzers, and therefore CDMSlite analyzerswere able to contribute to parts of the salting development. In Sec. 4.2.1and 4.2.2, I describe my primary contributions to the salting effort. Thisopen development of the salting algorithm did not violate blinding becausethe salting algorithm was designed such that the salted events had a normal-ization and energy spectrum that was randomized within some pre-definedrange.4.2.1 Determining the Target Number of Salt EventsThere is not an obvious number of artificial events to insert into the datasetin order to accomplish the goal of reducing analyzer bias. This sectionexplains how the number of artificial events was chosen non-arbitrarily.First, there should be enough salt that analyzers will be wary of tuningcuts to remove events, while too much salt would potentially mask instru-mental backgrounds that we want to be able to identify and remove withcuts. The simplest choice of the number of salt events is some fraction of864.2. SaltingFigure 4.1: (left) The CDMSlite Run 2 90% CL OI exclusion spin- indepen-dent cross section limit as a function of DM mass. (right) The number ofDM events that correspond to the excluded σSI from the left figure.the expected number of events in the analysis’s region of interest. How-ever, this option proved to be inadequate for CDMSlite because the vastmajority of events in the region of interest are concentrated in a narrowenergy range from the 71Ge decays. Instead of the 71Ge peaks dictating thenumber of artificial events added, the energy ranges where there is a lowerlevel of expected background should dictate the number of events added.This is because these are the energy ranges where the dataset has the high-est sensitivity to signal and therefore where it’s most important to have anappropriate amount of salt.Instead of using a fraction of the expected number of events in the anal-ysis’s region of interest, the projected optimum interval sensitivity of theanalysis can be used to naturally incorporate the energy region of low back-ground when choosing the number of artificial events. Because the optimalinterval method’s sensitivity generally is determined from an energy rangein the final spectrum where there is a low background rate, this methodwill not be susceptible to the potential problem of a large fraction of eventsappearing in the 71Ge peaks.For salting CDMSlite Run 3, we estimate the sensitivity by using theCDMSlite Run 2 OI cross section limit. The cross section limit can beconverted into a number of excluded events using the exposure (i.e. thekg × days length of the run) as a function of the DM mass, and is shown inFig. 4.1.The 20 GeV/c2 WIMP spectrum covers the full energy range of interestfor the analysis, and therefore we base the salt normalization off of thisWIMP mass, which gives 62 events at the excluded limit. A visualizationof this number of artificial events, given energies drawn from an exponential874.2. SaltingFigure 4.2: Example “salt” added to the CDMSlite Run 2 final spectrumat the magnitude indicated by the excluded 90% CL for a DM signal. Thesalt spectrum is scaled to a much higher level than the exclusion limit toillustrate its shape.distribution with a constant offset, is shown in Fig. 4.2 along with the finalRun 2 spectrum. This visualization shows that inserting events at levelof the 90% OI CL sensitivity gives a reasonable number of events: theartificial event density is sufficient to prevent cut tuning but not so largeas to mask the real events in the regions of low background or potentiallymask an unanticipated or instrumental background in the CDMSlite Run 3dataset. This method of determining the number of salt events based onthe sensitivity was adopted because it reduces the arbitrary nature of thechoice.The number of salt events to use for Run 3 was determined based on the62 events calculated from Run 2, but scaled by the relative duration of theruns, corrected by the estimated passage fraction of the salt to analysis cuts(∼15%), and also randomized within a range of possible values. The finalrange was 280–840 events, and the number of salt events used for Run 3 wasrevealed to be 393 after the dataset was unblinded.4.2.2 Selecting Events To ReplaceThe salting algorithm was designed to replace events with artificial events,rather than adding artificial events, for technical simplicity. This was in partto avoid having to work around the sequential event IDs that are a featureof the CDMS data format. Each CDMS event also contains a significantamount of metadata in addition to the digitized waveforms (e.g. cryocoolerphase information, base temperature information, trigger time stamps, de-884.2. Saltingtector array trigger information). Trying to appropriately generate artificialdata for all these categories would be a challenge, and so instead the existingalgorithm only creates artificial waveform data and otherwise inherits themetadata from the replaced event.While the event replacement approach had many advantages, it requiredparticular attention when selecting which events to replace. Of course thereplaced events were chosen to be uniformly distributed in calendar time(as expected of a DM signal). In other variables it would be less easy toreplace events in such a way that the artificial events resemble the expecteddistributions of DM signal events. For example, the artificial events shouldbe uniformly distributed with the cryocooler vibrational phase, but becauseas shown in Fig. 4.6 the majority of events occur within a certain range ofthe cryocooler vibrational phase, additional analysis selection criteria wererequired when choosing the events to replace.During inspection of the first attempt to salt the CDMSlite Run 3dataset, the salt events were found to be not uniform in cryocooler vibra-tional phase but highly correlated with the cryocooler-induced noise triggers.Events that are reconstructed at higher energy are less likely to be low fre-quency noise, as shown in Fig. 4.3, and so the undesired correlation wasremoved by raising the minimum energy of the replaced events. Fig. 4.3indicates that an energy cut of 2 keVt is sufficient to remove the correlation,and because the low frequency noise was found to get worse after the 252Cfcalibration data shown in the figure, the final energy cut was set at 3 keVt.4.2.3 CDMSlite R3 Salt Application and “Unsalting”In the final application of the salting algorithm, the event energies werechosen from an exponential distribution with a constant offset:P (E) ∝ C + (1/D) exp−E/D; E ∈ [0.05, 5] keVee, (4.1)where the exponential component was chosen to roughly approximate aWIMP spectrum and the constant offset was chosen so that salt existedover the analysis energy region of interest. C and D were randomized hid-den parameters, sampled logarithmically from 1/3 to 3 keV−1ee for C, andfrom 0.5 to 2 keVee for D. The chosen energy was restricted from 0.05 to5 keVee to match the analysis’s region of interest. The randomly selectedparameters used were C = 0.6967 keV−1ee and D = 1.299 keVee, resulting in anearly uniform distribution of salt events over the energy region of interest.The success of the salting procedure was confirmed in a post-unblindinganalysis effort. The salt was inserted at an appropriate magnitude and as-894.3. Experimental SetupFigure 4.3: The distribution of 252Cf calibration data in the crycooler vibra-tional phase variable (labeled as CryoPreTime Mod 0.83s). As the selectioncriteria moves to higher energy, the correlation of events with the phasedecreases.sumed a signal-like distribution in the parameters of interest for the analysis.Details of these post-analysis confirmations are documented in the resultssection of the following chapter, in Sec. 5.3.2.4.3 Experimental SetupA comprehensive review of the CDMSlite experimental setup in the contextof the Run 1 and Run 2 analyses can be found in the theses of R. B. Thakurand M. Pepin [67, 83]. Therefore, this section primarily highlights the ex-perimental setup differences between Run 3 and the earlier CDMSlite runs.4.3.1 Detector Selection and ConfigurationFigure 4.4 shows the SuperCDMS Soudan detector array with the T5Z2and T2Z1 detectors labeled. The decision to switch CDMSlite detectorsfrom T5Z2 to T2Z1 for Run 3 was based on a number of factors. First, abetter noise environment was observed on T2Z1, relative to T5Z2 and mostother detectors, leading to a slightly better baseline resolution than thatobserved with T5Z2. The superior noise performance was primarily becauseof reduced susceptibility to vibrational noise as discussed in Sec. 4.3.2. Weexpected this would enable a lower analysis threshold for Run 3 and animproved sensitivity to low-mass DM. In addition, during Run 3 commis-904.3. Experimental SetupFigure 4.4: (left) A depiction of the SuperCDMS Soudan detector array, with15 iZIPs total arranged in 5 “towers.” (right) A depiction of an iZIP detectorread out in CDMSlite mode. Copied from Ref. [82], with permission.sioning, we saw that the measured leakage current for T2Z1 was less thanthat of other candidate detectors. And finally, the decision to switch fromT5Z2 to T2Z1 for Run 3 was intended to demonstrate reproducibility of theCDMSlite operating technique across multiple detectors.Side 2 of T2Z1 was biased with a 75V applied voltage while the phononand charge channels on side 1 were read out. As discussed in Sec. 4.3.3,and unlike in Run 2, there were times throughout Run 3 that the detectorvoltage differed significantly from the applied power supply voltage. Thedata acquisition readout for Run 3 was also limited to a single tower (Tower2) as opposed to previous CDMSlite runs when the whole detector arraywas read out.The pre-biasing procedure—where the detector voltage is raised abovethe operating voltage for 10 minutes before data taking as discussed inSec. 2.5.3—was found to reduce charge leakage shot noise in the phononsignal at the start of each data series. For Run 3, 85V was used as the pre-bias point for the nominal 75V operating voltage. (In Run 2, a 80V prebiaswas used for the Run 2 70V operating voltage).4.3.2 Vibration MonitoringOne of the central challenges to all low threshold SuperCDMS Soudan anal-yses is the significant contribution of “low frequency noise” (LFN) to thenoise budget of the phonon readout. This noise source affects all Soudandetectors, degrades detector baseline resolutions, forces the setting of higheranalysis thresholds, and as a result reduces sensitivity to low mass WIMPs.Section 2.5.3 describes how, prior to CDMSlite Run 2, accelerometers wereinstalled on the cryocooler, which was the most vibrationally intensive com-ponent of the experiment.914.3. Experimental SetupFigure 4.5: The low energy event density dependence on the the cryocoolervibrational phase is shown in the red histogram. Good events from actualenergy depositions, shown in the blue histogram, are not correlated with thevibrational phase.The most useful cryocooler correlation analysis variable, discovered dur-ing the CDMSlite Run 2 analysis, was the “time since” last cryocooler chirp,where the cryocooler chirps were found to occur at 0.83 second intervals[102]. Because on occasion the accelerometer did not register a cryocoolerchirp and a period lasted n × 0.83 seconds, the 0.83s modulus of the cry-ocooler “time since” variable proved the most beneficial analysis variable forRun 2 and Run 3 analyses.Because the LFN caused triggers that could mimic low energy events(<5 keVt), and because the LFN caused more triggers than any other back-ground source in Run 3, we use the low energy event density as a proxyfor the intensity of the LFN. Figure 4.5 shows the low energy event densitydependence on the time since last cryocooler chirp (also known as Cry-oPreTime Mod 0.83s, or more generally as the cryocooler phase). The LFNvibrations on T2Z1 are strongest 0.1 seconds after a crycooler cycle. Section4.6 describes how we incorporate this information into overall LFN qualitycuts.The vibrational noise from the cryocooler was also found to vary overthe course of the ∼60 day dataset, as shown in Fig. 4.6. In addition to thevariable event rate in the 0–0.2s range of the crycooler phase, an additional924.3. Experimental SetupFigure 4.6: An event density plot of lower energy events (-2 < keVt < 5)that are more likely to be instrumental noise events than events from particleenergy depositions in the detector. Ba and Cf calibrations are highlighted.A Cf period when no data was read out is between May 01 and May 05.The April 14 stripe is from a single series (01150414 1430) when the coldtrap was being cleaned. A second 3 hour series on May 05 (01150505 1422)was removed from the plot as it was saturating the color map—this was thefirst series after the late-Run Cf calibration.noise source that is uncorrelated with the crycooler vibrational phase wasobserved to start on roughly April 1, and is explored below in Sec. 4.3.3.4.3.3 High Voltage Current MonitoringMonitoring of the HV power supply current began with CDMSlite Run 2,and the motivation and basic aspects of this current reading are describedin Sec. 2.5.3. The HV current readings were critical for correcting the recon-structed energy of events in the Run 3 analysis. Referring to the schematicshown in Fig. 2.16, the value of Rp (and therefore IHV ) varies significantlyover time, between an effectively infinite resistance and 625 MΩ. As shownin Fig. 4.7, Rp was found to be strongly correlated with the temperature inthe room that housed the electronics (the RF room). Because this varia-tion is undesirable, the air handler which supplied fresh air to the RF roomwas turned off in early April 2015, raising the temperature in the room andincreasing the parasitic resistance (decreasing the leakage current).934.3. Experimental SetupFigure 4.7: The RF room temperature (TE FEB RACK) (oF) and the leak-age current (iseg hv current b) (nA) as a function of calendar time. Thex-axis runs from mid-February to early April, 2015.Figure 4.8: Correlation between trigger rate and detector voltage. The redcurve is the best fit of the trigger rate to the detector voltage model ofEq. 4.2.The detector bias voltage is the most important factor in CDMSlite’senergy scale, so it is crucial to monitor the HV current in order to knowthe detector voltage over the course of the run. The value of Rp variessignificantly over time, and Sec. 4.4.1 discusses how we correct the variableenergy scale using the HV current readings.4.3.4 Charge TriggersThe HV current readings have also led to an improved understanding ofT2Z1’s noise environment over time. Using the trigger rate as a metric forthe noise environment (with a higher trigger rate corresponding to highernoise), Fig. 4.8 shows the dependence of the trigger rate on the detectorvoltage, after the majority of cryocooler-induced LFN triggers are removed.944.3. Experimental SetupFor |Vdet| < 72 V, there is an exponential dependence of the trigger rateon the detector voltage, and a sharply increasing power law dependence for|Vdet| > 72 V. The inability to fit all the data to one functional form suggeststwo things, the first trivial and the second more important and providinginsights to the detector’s HV behaviour:1. Factors other than the detector voltage influence the charge triggerrate (e.g. May 09, etc.).2. The process of charge trigger generation is different in different voltageregimes:• Below 72 V, there is no detector instability or “breakdown,” andincrease in the noise will be exponentially suppressed at low volt-age.• Above 72 V, a “breakdown” process begins with a roughly power-law trigger rate dependence.We fit for this voltage breakdown point, V0, with the functional formbelow:TrigRate[Hz] ={B × exp[y × V ] : V < V0B × exp[y × V ] +A× (V − V0)x : V ≥ V0 (4.2)and obtained a best fit value of V0=72.3 V.Because of this dependence, when the detector voltage increased above72 V (in magnitude) at the beginning of April, both the trigger rate andthe observed noise became significantly worse. Two representative PSDs areshown in Fig. 4.9, where the left plot comes from a series in March and theright plot comes from a series in May. This figure shows that the noise from0–1 kHz (the signal band) increased by a factor of ∼3 after Vdet increasedabove 72 V, on April 1. In the Run 3 analysis, we consistently separatedata according to these two distinct detector noise environments and referto the two periods as R3a (which includes all series up to and including theApril 1st series 1150401 1411) and R3b. Different detector resolutions andcut efficiencies are evident for the data obtained in R3a and R3b.954.4. Energy ScaleFigure 4.9: Representative PSDs from R3a (left) and R3b (right). Thecolored curves represent the noise on the individual channels, while theblack curve represents the channel sum.4.4 Energy Scale4.4.1 Energy CorrectionsAs explained in Sec. 2.3, we estimate the amplitude of the total phononpulse (the sum of the pulse from the 4 CDMSlite detector phonon channels)using the OF algorithms. Then, as explained in Sec. 2.5, we convert the OFamplitude to an energy using the 71Ge EC events. However, even after ap-plying this simple calibration factor, the OF-based energy estimate requiredcorrection for environmental and detector conditions.In particular, we observed that the amplification of the detector driftedby up to 30% over the course of Run 3. We found that we were able to cor-rect for detector amplification drifts using known environmental variables.Once the corrections were applied, the resolution of the K,L, and M shellswere comparable to expected resolutions from the Run 2 analysis. The fol-lowing subsections discuss the environmental variables and other inputs tothe corrections.Current CorrectionUsing the schematic in Fig. 2.16 we can solve for the detector voltage, relativeto the applied HV voltage, using the measured HV current:Vdet = Vb − IHVRb. (4.3)The main assumption going into the model shown in Fig. 2.16 is that allthe leakage current is downstream of the bias resistor, Rb. We directly showthat this is a reasonable assumption by verifying that the measured phononenergy of electron recoil calibration peaks depends on the detector voltage964.4. Energy ScaleFigure 4.10: The drift in the total phonon energy of the 10 keV line, withthe HV current correction prediction overlaid. The other energy scale cor-rections have been applied to the data in order to highlight the correctionsfrom the HV current.asEt = Er(1 +e Vdet) (4.4)with Vdet defined in Eq. 4.3, and Er is the recoil energy. When we solve forthe expected total phonon energy of the 10 keV calibration line using a Vdetthat has been corrected by the measured leakage current IHV, we see goodagreement with the drift of the 10 keV line seen in the data, as shown inFig. 4.10. The maximum current reached is ∼120 nA which corresponds toan almost 30% correction. The HV current correction is significantly largerthan the other energy corrections discussed below.Base Temperature CorrectionAfter the 10 keV line has been corrected by the HV current, the measuredenergy shows a positive, roughly linear relationship with the cryostat basetemperature. Unlike the HV current correction, the magnitude of this cor-rection has not been determined from first principles. We fit this dependenceto a straight line and then correct the energy variable in order to remove974.4. Energy ScaleFigure 4.11: The steps of the linear base temperature correction. Afterfitting a line in the Ecorr2 (y axis) vs. base temperature (x axis) plane,the linear dependence is removed to form a corrected Ecorr3 variable. AGaussian fit of the K-shell events’ energies (as measured by Ecorr3) showsthe energy resolution.the dependence. The correction is given byEcorr3(Tbase) = Ecorr2 × (1 + (Tbase − T¯base)× Cbase), (4.5)where Ecorr3 is the base temperature-corrected energy, Ecorr2 is the energywith just the HV current correction, Tbase is the base temperature, T¯base isthe mean base temperature over the course of the run, and Cbase is the slopeof the linear fit. A visualization of the base temperature correction steps isshown in Fig. 4.11.Position CorrectionAs discussed in Sec. 2.2.2, the position of the event in the detector affectsthe reconstructed energy of the event. This is referred to as “position depen-dence,” and it is corrected for in a number of ways. The standard methodemployed by most iZIP detector analyses to reduce position dependence is touse the non-stationary optimal filter algorithm (Sec. 2.3), which deweightsthe beginning portion of the pulse where the pulse shape has the strongestdependence on the event’s position. However, the non-stationary optimalfilter best-fit amplitude is still found to contain some residual position de-pendence.Figure 4.12 shows 10 keV events from the T5Z2 detector operated inCDMSlite mode (single-sided readout) but at 0V. In this configuration,position-dependent effects are expected to be particularly prominent be-cause there are no NTL phonons, which spread phonon energy out in a984.4. Energy ScaleFigure 4.12: Data from single-sided readout of T5Z2 at 0V. (left) Raw totalphonon traces plotted from 10 keV events. These events all have the sameenergy (see alignment in the tail of the pulse), and the dark green-bluetraces are selected by their peakiness. (right) The high energy 0V spectrumasymmetry in the 10 keV line, with the peakier 10 keV events highlightedin cyan.vertical column along the e−/h+ track and therefore reduce position depen-dence. These data therefore serve well to demonstrate position-dependenteffects: the peakier pulses (blue-green traces in Fig. 4.12 left) are recon-structed to a higher energy by the non-stationary optimal filter (cyan his-togram in Fig. 4.12 right). Below we describe how we correct for the sameposition-dependent effects in the CDMSlite Run 3 data.The two template (2T) fast amplitude measures the peakiness of thepulse and therefore is one measure of the position of the event. The vari-ations in peaky vs. non-peaky pulse shapes, from the prompt phononsdescribed in Sec. 2.4, correspond to the proximity of the original recoil tothe phonon channels on the single side of the detector.After the base temperature correction, as expected we see a positivelinear relationship between the the non-stationary optimum filter amplitudeand the 2-template fast amplitude. This correlation is shown in Fig. 4.13.We approximate this dependence as linear, fit a line to the data, and performthe following correction:Ecorr4(2Tfast) = Ecorr3 × (1 + (2Tfast − 2T¯fast)× C2Tfast) (4.6)where Ecorr4 is the 2-template fast amplitude-corrected energy, Ecorr3 is thebase temperature-corrected energy, 2Tfast is the 2-template fast amplitudebest fit, 2T¯fast is the mean 2-template fast amplitude best fit, and C2Tfast is994.4. Energy ScaleFigure 4.13: The steps of the linear 2-template correction. After fittinga line in the Ecorr3 (y axis) vs. 2-template fast amplitude (x axis) plane,the linear dependence is removed to form a corrected ptNF variable. AGaussian fit of the K-shell in the corrected Ecorr4 variable shows improvedenergy resolution.the slope of the linear fit. The mean energy of the K-shell events does notchange with the correction, and the resolution improves from 4.21 keVt to2.94 keVt.The corrections discussed above with respect to current and temperatureare also applied to the fast amplitudes from the 2-template optimal filter fit.The fast amplitude can be positive or negative and does not have a directcorrespondence to energy, and so the dependence of the fast amplitude forthe 10 keV events cannot be independently fit to variables (e.g. BaseTemp,HV current) in the same way as the slow amplitude. Instead it is assumedthat the fast amplitude scales the same way as the slow amplitude, so weapply the corrections based on the fits to the slow template distributions. InFig. 4.13, the “corrected 2-template fast amplitude” is corrected accordinglybefore it is used in the final correction for energy correction.4.4.2 Nuclear Recoil Energy ScaleA subtlety in the energy scale corrections arises because we have applied thecorrection to all the data, but used ER events to derive the correction factors.While the base temperature and the 2-template fast amplitude correctionsshould not differ between ERs and NRs, the HV current correction doesdepend on the recoil type because it depends on the ionization yield. Theenergy scale has been corrected under the assumption that the ionizationyield equals 1. With a NR generating fewer free electrons, relative to anER of an equivalent recoil energy, the NTL gain will be less for the NR andso the HV current/voltage correction factor will be smaller for the nuclear1004.4. Energy ScaleFigure 4.14: (left) Comparison of the correction factor for Y= 1 and Y =0.15 events. Note that the difference in the correction factors is very smallfor small IHV, but for Run 3, with a leakage current up to IHV =120nA, it isclear that we should include this correction. (right) The maximum % erroron a NR energy estimate (Enr), if the total phonon energy (Ept) is correctedwithout consideration of the ionization yield, which results in the nuclearrecoil energy incorrectly set higher than what it would be when properlykeeping track of the ionization yield in the HV current correction. The k isthe Lindhard k parameter for the yield.recoils.First we obtain the ER HV current correction in Eq. 4.4, which convertsa measured total phonon energy to a corrected total phonon energy:Ept,corr|er =[Ept,uncorr]/[1− IHV ×Rb/e+ Vnom]. (4.7)The same calculation for nuclear recoils, now including the ionization yieldY (Enr), as described in Sec. 2.1.2, gives:Ept,corr∣∣nr=[Ept,uncorr]/[1− Y (Enr)× IHV ×Rb/e+ Y (Enr)× Vnom]. (4.8)In Fig. 4.14 we estimate the maximum error (by using the maximumcurrent IHV =120 nA) that would be introduced in the NR energy scale ifwe ignore the fact that our original HV correction assumed an ionizationyield of one. This error is the ratio of Eq. 4.7 and Eq. 4.8:Enr,ignoring yield in correctionEnr,including yield in correction=1− Y (Enr)×IHV ×Rb/e+Y (Enr)×Vnom1− IHV ×Rb/e+Vnom. (4.9)1014.5. Quality CutsFigure 4.15: The energy scale conversions for Run 3. The differences fromthe Run 2 energy scalings, which are negligible only for small HV leakagecurrent, are circled in red.In order to account for this systematic error, when converting from cor-rected total phonon or electron equivalent energy to nuclear recoil energy,first the Y = 1 HV current correction is undone, and then the correctionis reapplied with the accurate yield based upon the event energy. The fullenergy calibration and correction flow chart is shown in Fig. 4.15.4.5 Quality Cuts4.5.1 OverviewAs one of the first steps in our goal to look for a dark matter signal in thedata, we must remove events that (1) were not caused by a particle interac-tion in the detector (which we refer to as “instrumental” events), (2) werelikely to be reconstructed improperly by our event fitting algorithms, or (3)were recorded when the detector was behaving anomalously. In the process1024.5. Quality Cutsof removing these “bad events” with cuts, it is inevitable that some goodevents are also removed, and we account for this effect by computing the“signal efficiency” of these cuts. Throughout this section, we seek to definecuts that maximally reject the bad events but minimally reject good events,thereby maximizing the signal efficiency and increasing the DM sensitivityof the analysis.Because this analysis employs profile likelihood methods to search forDM—fitting background and signal models to events that pass all cuts—itis imperative to identify and remove all instrumental noise events whose dis-tributions cannot be modeled with a probability distribution in the fit. Inthe lowest energy range of the analysis, where the experiment is most sensi-tive to DM particles with mass < 10 GeV/c2, the instrumental backgroundis the highest. In this section I describe the different selection criteria toremove the more conspicuous types of instrumental noise events. In the nextsection, Sec. 4.6, I will describe the use of multivariate techniques to reduceinstrumental noise leakage to less than 1 event while maintaining as low ofan energy threshold as possible.4.5.2 Prepulse Noise CutThe raw data traces have been read out such that the bin8 that caused thetrigger is the 500th bin of the trace (out of a total of 4096 bins), which isdepicted in the example pulse in Fig. 2.5. The bins before this triggering binare referred to as the “prepulse.” Because the prepulse precedes the upwardfluctuation that caused the trigger, it is useful for characterizing underlying“baseline” noise that exists in each channel. The standard deviation of theprepulse data points is recorded for every channel for every event. Thedistribution of the prepulse standard deviation of the randoms is fit to aGaussian for each series (a roughly 3 hour period of time in which theSuperCDMS DAQ divides blocks of data), as shown for an example seriesin Fig. 4.16 left. Events whose prepulse standard deviation is outside the 4σrange in any channel are removed by this cut, and the full Run 3 trend ofprepulse baseline noise is shown in Fig. 4.16 right. Events that are removedhave anomalously bad LF noise, electronic noise, or are pileup events. Whenapplying the prepulse noise cut alone, the efficiency of this cut was found tobe high, greater than 99% for nearly all series.8The bin width is set by the digitization rate of the signal and is 0.8 µs, as previouslyprovided in Table 2.3.1034.5. Quality CutsFigure 4.16: (left) The distribution of the prepulse standard deviation of therandoms for a single series, with a Gaussian fit to the distribution overlaid.(right) The trend of fitted Gaussian σ for all Run 3 series.4.5.3 Bad Series CutWe remove series which had particularly poor noise performance or anoma-lous detector behavior with this cut. Shortly after the start of Run 3 therewas a brief attempt to lower the hardware trigger threshold. Because thiseffort was unsuccessful (was met with high trigger rates) the lower thresholdperiod only afflicted 6 series. These series are removed by this cut. Addition-ally, series with high trigger rates due to cold trap cleanings or immediatelyfollowing detector calibrations were also removed. This cut removed only 11series total, out of a total of 600 series. The livetime removed by this cutwas less than 1% of the total Run 3 livetime.4.5.4 Phonon Pulse-Shape CutsInformation from pulse-shape fits can discriminate signal events from in-strumental noise events having a characteristic pulse shape. Six differenttemplates are fit to each event using the optimal filter method: a signaltemplate, a square pulse template, an electronics glitch template with fastrise and falltimes, and three low-frequency noise (LF noise) templates. Thesignal, square pulse, and electronics glitch templates are shown in Fig. 4.17.The glitch and square templates were created iteratively in the followingsteps:1. identifying, by eye, instrumental noise events in the data set2. averaging the noise events to obtain a rough approximation of theinstrumental noise events’ shape1044.5. Quality CutsFigure 4.17: Comparison of glitch and square templates with the goodphonon template. While the square template only roughly resembles a“square” shape, we refer to it as square because it is different from theglitch pulse primarily in that it plateaus for 300µs, which it makes it wider.3. processing the entire dataset with the rough templates4. using χ2 information from the fits of the templates to the data, iden-tifying all instrumental events that resemble the instrumental noisetemplates5. normalizing and averaging all the instrumental noise events resemblingeither a glitch or square pulse into an improved (e.g. higher statistics)templateFor the LF noise, three different templates were created because the LFnoise assumes different pulse shapes, as discussed in Sec. 4.6.4.The OF algorithm outputs a goodness of fit χ2 (that is smaller when thetemplate is a good fit to the raw data trace), and therefore the differenceof χ2, or ∆χ2, between an OF fit with a “good event” template and an“instrumental” event template is a good indication of whether an event is an“instrumental” event. In fact, for consistent shapes of the instrumental noisepulses (i.e. pulse shapes that match the template) and Gaussian noise, this∆χ2 parameter is the optimal way to discriminate instrumental backgroundfrom signal. The ∆χ2 parameter is defined as:∆χ2LF,glitch,square ≡ χ2OF − χ2LF,glitch,square, (4.10)where OF corresponds to the standard signal-template fit, and LF, “glitch”and “square” correspond to the fits using the LF noise, glitch and square1054.5. Quality CutsFigure 4.18: (left) An example of a glitch event with the best fits of thegood event and glitch template to the event overlaid. (right) An example ofa LF noise event with the best fits of the good event and LF noise templateto the event overlaid.pulse templates respectively. Lower values of ∆χ2 indicate events that havea more signal-like shape.Glitch events (for example the event shown in Fig. 4.18) and squareevents have relatively uniform pulse shapes and do not resemble the signalpulse shape. Therefore, a single template for each is sufficient to efficientlydiscriminate against these event types.Every phonon chi-squared based cut boundary was defined not just basedon the ∆χ2 value, but also based on the events’ reconstructed energy (i.e.ptOF). It is important to incorporate the event energy in the determinationof the cut boundary because discrimination between signal and instrumentalbackground is much more difficult at lower energy. The distribution of ∆χ2values for good events varies as a function of ptOF, and has a downwardsloping parabolic shape, as shown in Fig. 4.19 left, so the cut boundarycorrespondingly takes on a parabolic shape in the ptOF vs. ∆χ2 plane.Series BlocksJust as the distribution of ∆χ2 values for good events changes as a functionof ptOF, it also changes as a function of time. This motivates the divisionof the R3 data set into blocks of series, each of which having a custom cutboundary in the ptOF vs. ∆χ2 planesInitially it was not obvious why the ptOF vs ∆χ2 distributions variedin time. The parabolas consist of good events that should not change pulseshape over the course of the run, so why does the parameter that fundamen-tally measures pulse shape (it depends on pulse shape based on χ2 values)vary in time for these good events? The origin of the variation is explored by1064.5. Quality CutsFigure 4.19: (left) The distribution of good events in the ∆χ2glitch vs. ptOFplane, and (right) the same data reprocessed with a single noise PSD, show-ing that the variations are due to the PSDs that change for each series.noting that, while the pulse shape should not change for these good events,the noise PSD that is used in the optimal filter fit changes on a series-by-series basis, and so we hypothesized that the time variations were due to thedifferent PSDs used in the fit. The OF fits for each series are carried outwith a noise PSD calculated from random events from that series. Figure4.19 right shows the ptOF vs. ∆χ2glitch plane for the same group of events asshown in 4.19 left, but processed with a single PSD; the single “good event”parabola in this plane shows that all of the variations are a result of thevarying PSDs.It still makes sense to use different PSDs for each series because incorpo-rating the variable noise environment into the OF fits will obviously resultin better fits. However, this deficiency in the data processing necessitateddividing up the Run 3 dataset into series blocks.In addition to dividing the R3 dataset based on calendar time, it provedbeneficial to divide the dataset based on the cryocooler vibration phasevariable. Because this variable is correlated with the intensity of the lowfrequency noise environment on an event by event basis, data block divisionswere created such that periods with particularly bad noise were isolated fromthe rest of the data set and harsher cuts could be defined for the noisierblocks.To form the division boundaries, each series was first divided into a“loud” and “quiet” section based on the cryocooler vibration phase variable.Then, each series was characterized in three different ways:1. The distribution of events in the ptOF vs. ∆χ2glitch plane was fit to aparabola and the slope of the parabola was recorded.1074.5. Quality CutsFigure 4.20: (top) The ∆χ2LF vs. ptOF plane for a portion of the Run 3data, showing a high concentration of LF noise events. Raw pulse inspec-tion revealed that only a small fraction of data points in this plot (thosedata point at the lowest ∆χ2LF values) resembled good events. The ∆χ2LFparameter is calculated with “template 1” as shown in Fig. 4.26. (bottom)The parabolic fit coefficient in the ptOF-∆χ2LF plane vs. 3.5σ contour area,with the clusters shown.2. The distribution of events in the ptOF vs. ∆χ2LF plane was fit to aparabola and the slope of the parabola was recorded.3. The 3.5σ contour area of the LF noise events in the ptOF vs. ∆χ2LFplane (shown in green in Fig. 4.20 top) was recorded.We designed a method for grouping series together according to the char-acteristic variables above, where the purpose of grouping series was to allowfor the development of a cut boundary specifically for the series that wellinto that group. This method uses the k-mean clustering algorithm, whichclusters series together that are close to eachother in the three dimensionalspace of parameters listed above.The k-means clustering algorithm picks k centroids in an n-dimensionalspace, assigns each data point to a centroid, and finds the distance between1084.5. Quality Cutsthe points and the centroid looping over all centroids [103]. The programthen iterates by moving around the centroids until the distance between thepoints and the centroid are minimized.In deciding how many clusters to use, there is a tradeoff between manyclusters (and capturing smaller changes in the distribution of events in thecut variables) and few clusters (and having good statistics with which todefine the cut). For the cryocooler loud selection, with greater than 9 clustersthe statistics become lacking in the cluster with the largest contour area. Forthe cryocooler quiet selection, using greater than 8 clusters only marginallyshifts the divisions in the higher contour area clusters, and so 8 clusters arechosen for the crycooler quiet selection. The clustering groups are shown forthe cryocooler loud series blocks in Fig. 4.20, where for this visualization thethree dimensions of the clustering are projected into the ∆χ2LF vs. contourarea 2D plane.All the series blocks are also shown in Fig. 4.21 right. Figure 4.21 leftshows events in the cryocooler vibrational phase fit to a 2 Gaussian + flatdistribution. One Gaussian is fit around the region of largest event density(from 0–0.2 seconds), and the 2σ bound of the Gaussian is selected andincluded in the cryocooler “loud” category. If a second peak exists in thecrycooler phase variable (which it does for a number of the series in April),a second Gaussian is fit to the second event density (from 0.3–0.4 seconds),and the 1σ bound of the Gaussian is selected and included in the cryocooler“loud” category.Pulse Shape Glitch EventsThe origin of “glitch events” has been explained as electronic noise in theTES bias circuit that generates spiky pulse triggers with rise and falltimesfaster than typical events, with time constants set by the TES response asopposed to phonon absorption times in the detector, as described in [104](p. 197).A cut boundary with two components was defined in the ptOF vs.∆χ2glitch plane for the 17 Run 3 series blocks. The first component wasa flat cut as a function of energy and was defined using random triggers sothat >90% of randoms passed; this ensured that the cut has roughly 90% orgreater signal efficiency at low energy. The second component was parabolicas a function of energy corresponding to the distribution of good events inthe ptOF vs. ∆χ2glitch plane. First a parabola was fit to all the triggered (i.e.non random) events in the series block, shown in Fig. 4.22 as the “centralfit” parabola. Then the residuals of the ∆χ2glitch values for the parabolic fit1094.5. Quality CutsFigure 4.21: (top) The maximum likelihood estimate best fit of 2 Gaussiansplus a flat distribution to noise events in a single series. (bottom) The 9cryocooler “loud” clusters, and 8 cryocooler “quiet” cluster, all highlightedby time.1104.5. Quality CutsFigure 4.22: (top) A raw trace for a glitch-like event that fails the cut.(bottom) The tuning of the parabolic portion of the cut based upon theparabola of good events, for series block 15.and the data were binned by energy and a Gaussian was fit to the residu-als, depicted as the red error bars in Fig. 4.22. Finally, a second parabolawas fit to the µ+Nσ point of a Gaussian fit to events in each energy bin,where N was tuned for each of the 17 series blocks on the basis of raw traceexamination. For series block 15, we used N=4 and the second parabola isgiven by the green “cut fit” line in figure 4.22, which defines the location ofthe cut. Fig. 4.22 left shows an example glitch event that is rejected by thiscut.1114.6. Low Frequency Noise CutsFigure 4.23: (left) The total phonon pulse of the square pulses. (right) the∆χ2square parameter formed from the square template.Square Pulse CutThis type of glitch has a fast rise and falltime, but its pulse shape is poorlyapproximated by the standard glitch template because the signal plateausfor ∼200µs at a maximum value before steeply falling. The total phonontraces from 10 example square pulse events are in Fig. 4.23. Because theseevents are not efficiently discriminated against with the glitch template, acustom template (which roughly resembles a square shape), was created toidentify these events with a ∆χ2square parameter.When events are plotted in the ptOF vs. ∆χ2square plane, Fig. 4.23 rightshows a clear cluster of square events at high ∆χ2square that are separatedfrom the good event distribution, allowing for a highly efficient cut to bemade in this plane. Note that in this plot, all data quality cuts other thatthe spot cut and multiples cut have been applied. A parabolic cut boundarywas defined in the ptOF vs. ∆χ2square plane for the 17 Run 3 series blocks.4.6 Low Frequency Noise Cuts4.6.1 MotivationThe majority (>75%) of events in the Run 3 dataset are a result of lowfrequency noise fluctuations in the phonon signal that cause the detector totrigger. These events are particularly difficult to discriminate against fortwo reasons:1. The dominant frequencies of good phonon pulses are relatively low(<1.25 kHz) and so their bandwidth overlaps the LFN event band-width. The frequency band overlap can be seen clearly in the time1124.6. Low Frequency Noise Cutsdomain in Fig. 4.18 right. This makes optimal filter pulse fitting lessefficient for discrimination.2. Unlike glitch and square events, the shape of LFN events is inconsistentand so a single LFN template does not produce a good fit for all LFNevents.These challenges become increasingly difficult to overcome as one tries todiscriminate to lower energies, where the signal-to-(baseline) noise becomeslow enough that the signal from an actual energy deposition and the fluc-tuation from a LFN feature are sufficiently buried under baseline noise thatthe small bump in the trace that caused the trigger could be attributed toeither source.Figure 4.24 demonstrates this challenge clearly: when the simulated goodevent signal is plotted on top of the data (which is dominated by LF noisetriggers at low energy) in the ptOF vs ∆χ2LF plane, we see significant overlapof the two distributions. The overlap is especially bad in the energy regionbetween the 50% trigger point (the lowest possible analysis threshold, dis-cussed futher in Sec. 4.9.2) and 2.2keVt. This is problematic because toset a cut with good signal passage near the 50% trigger point, we wouldalso need to pass a large number of LF events. At the same time, a cutthat removes the majority of the LF events has very low signal passage andeffectively moves the analysis threshold above 2.2keVt because of its lowefficiency. In the sections below, we describe the Run 3 campaign to removethis background from the dataset, which involved separate cuts for high andlow energy events, as well as a multivariate boosted decision tree approachto improve discrimination in the low energy range.4.6.2 High Energy LF Noise CutSection 4.6.1 described why LF noise discrimination becomes increasinglydifficult at low energies. Therefore, at higher energies where discriminationis easier, a simple cut based on a ∆χ2LF parameter from a LFN templatewas designed to remove LFN that reconstructed above 5 keVt. This 5 keVtvalue was determined so that a cut could be defined and have nearly 100%efficiency for good phonon events. The high energy LFN cut was definedindividually for the 17 series blocks (following the same procedure as for theglitch and square pulse cut).1134.6. Low Frequency Noise CutsFigure 4.24: The simulated signal (blue data points) significantly overlapsthe LF noise background (the majority of the red data points) at low energyin this variable plane.4.6.3 Detector-Detector CorrelationsTo try to solve the signal-noise overlap problem shown in Fig. 4.24, we lookfor parameters other than the ∆χ2LF to efficiently identify and reject LFnoise near threshold.Because the LF noise originates from vibrations in the experimentalsetup, and because the vibrational sources producing LF noise triggersshould couple to all detectors in a tower, we examined the correlationsbetween the phonon waveforms on the CDMSlite detector and the otherdetectors in the tower.In order to measure the correlation between raw total phonon traces, weuse the Spearman correlation statistic [105], which is the same as the stan-dard correlation statistic, except that the Spearman statistic ranks arrays(which in this application are the time domain traces) before calculatingthe correlation. “Ranking” is a simple transformation of an array where thevalue is replaced by the index of that value when the array is sorted fromlowest to highest. For example, for the array V = [4, 8, 9, 1], the ranked arrayis rgV = [2, 3, 4, 1]. Then the general definition of the Spearman correlationcoefficient between arrays X and Y of length N isrs =cov(rgX, rgY)σrgXσrgY=∑Ni (rgX,i − r¯gX)(rgY,i − r¯gY )√∑Ni (rgX,i − r¯gX)2√∑Ni (rgY,i − r¯gY )2, (4.11)where rgX and rgY are rank variables. σrgX is the standard deviation of the1144.6. Low Frequency Noise CutsFigure 4.25: (left) Example traces showing detector correlated LF noiseon the three Tower 2 detectors (top:T2Z1, middle: T2Z2, bottom: T2Z3).(right) Comparisons of signal and background distributions in the T2Z1-T2Z2 detector trace correlation parameter.ranked variables, and r¯gX is the average of the ranked variable. The Spear-man correlation coefficient, as opposed to standard correlation coefficient,was used because it measures monotonic relationships in general as opposedto just linear relationships. Using the more general statistic was justifiedsince we wouldn’t expect purely linear detector-detector LF noise correla-tions. The correlation coefficient will be a number between −1 and +1, with−1 being maximally anticorrelated, +1 being maximally correlated, and 0being uncorrelated.The Spearman correlation parameter between the time stream wave-forms on the CDMSlite detector and the other detectors in the tower, forthe events between the 50% trigger efficiency (1.29 keVt) and 3 keVt, isshown as the red histogram in Fig. 4.25 right. These events are in the re-gion of signal-background overlap shown in Fig. 4.24, many of which are LFnoise events,In contrast to a LF noise trigger on T2Z1, a good event from a trueenergy deposition on T2Z1 should not be strongly correlated with the LFnoise on the other detectors. We simulate the distribution of correlationcoefficients for good events using the pulse simulation approach describedin Sec. 4.9 for the pulses on T2Z1. On the other detectors, T2Z2 and T2Z3,we leave the randoms unchanged and calculate the correlation parameterbetween the simulated pulse on T2Z1 and the random noise event on theother two detectors. The cyan histogram in Fig. 4.25 shows the T2Z1-T2Z2 correlations when the simulation uses randoms acquired at the endof the series. The event separation seen between the red (primarily LF1154.6. Low Frequency Noise Cutsnoise) and cyan (signal) histograms is insufficient for setting an efficientcut in the correlation variable alone, but the separation between histogramsindicates that the correlation parameters carry discrimination information.Therefore, as described in Sec. 4.6.5, we used the correlation parametersin a multivariate signal-noise separation boosted decision tree algorithm toimprove the LF noise cut.The names we use for the Spearman correlation variables are either“T2Z1-T2Z2 correlation” / “T2Z1-T2Z3 correlation” or, for brevity, “z4z5correlation” / “z4z6 correlation”.4.6.4 Motivation for Multiple TemplatesSuperCDMS Soudan iZIP analyses generated the instrumental noise tem-plates [106], and were able to use a single template for each class of instru-mental noise to remove instrumental backgrounds from the dataset. Becauseof the Run 3 analysis’s goal to search for a DM signal to as low of an energythreshold as possible, and because of the particular importance of removingLF noise at low energies, we explored the use of multiple LF noise tem-plates in order to improve the characterization of the LF noise that assumesdifferent pulse shapes.Development of new templates was also motivated because, when weaverage many low frequency noise total phonon traces on T2Z1 from theRun 3 dataset, the shape of the resulting average trace deviates from thelow frequency noise template developed for the earlier SuperCDMS SoudaniZIP analyses. As shown in Fig. 4.26, the template developed in the iZIPanalyses (magenta) does not have the ∼600Hz oscillation that is seen in theaverage of many low frequency noise total phonon traces (black).Figure 4.27 shows that the ∆χ2LF parameter based on the black template(template #2) improves discrimination between signal and LFN background:with the same data selected, there is increased separation of the blue (signal)and red (background) data.In addition, because the 600Hz mode is seen so strongly in T2Z1 andthe other Tower 2 detectors, as shown in Fig. 4.28, we test a discriminationparameter using the amplitude and ∆χ2LF fit values of a template isolated tothis frequency. Averaging the Fourier transform of the individual LF noisetraces shows a peak at 602.7 Hz with a phase of 114.6 degrees relative to thestart of the trace. This phase offset is observed to be relatively constant,as expected, since the upward fluctuation of the noise causes the trigger.The resulting time-domain template is shown in Fig. 4.26 (left), labeled“template 3.”1164.6. Low Frequency Noise CutsFigure 4.26: (top) Three different LFN templates used to fit different lowfrequency noise shapes on T2Z1. (middle) The averaged amplitude of theFourier transform of the LFN traces. In the red curve, the time domaintraces had zeros added to the front and end of the trace before the Fouriertransform, which is a trick to to improve the δf resolution of the Fouriertransform. (bottom) The phase of the black LF noise template.1174.6. Low Frequency Noise CutsFigure 4.27: (left) The ∆χ2LF vs. ptOF plane for the data, shown in red,and the simulated signal, shown in blue, when using template #1 to fit theraw traces. (right) equivalent to figure left but using template #2, showingimproved separation between the data (which consists primarily of LF noiseevents), and the signal.Figure 4.28: The average of LF noise events showing a coherent shape (sinceincoherent noise would average to roughly 0 on the y-axis) on T2Z1 (left)T2Z2 (center) and T2Z3 (right).1184.6. Low Frequency Noise Cuts4.6.5 Boosted Decision Tree (BDT)A boosted decision tree is a well-suited algorithm to reduce the multipleLF noise parameters into a single BDT parameter (a score between −1 and+1) that maximizes the separation of signal from background. The primary“handles” on LF noise are:1. the three ∆χ2LF parameters from pulse shape fits to the three differentLF noise templates.2. the tˆ− variable, which represents the time since the last cryocoolercycle. The cycle period is ∼0.83 seconds and LF noise causes triggersmore frequently in the ∼0.2 seconds after the start of the cycle, withthe remainder of the cycle containing fewer LFN triggers.3. the correlation of the phonon waveforms between the CDMSlite de-tector and the other detectors in the tower, because the vibrationalsources producing LF noise triggers couple to all detectors in a tower.More specifically, we use a BDT as a LF noise classifier to find a cut byreducing a multidimensional space down to a one dimensional space wheresignal and background are well separated. There are many options anddifferent types of BDTs to use, and because only very small performancedifferences were observed when testing different BDT algorithms, we choosethe Adaboost algorithm [107, 108]. The algorithm reduces overtraining using“ensemble methods,” by forming many trees of low dimensions (2 or 3) andcombining the results of these trees so that the classification of a single eventdepends on all the trees.Optimizer Fine TuningIn the two-level decision tree shown in Fig. 4.29, the bluer boxes are moresignal-like, and the red boxes are more background-like. At every divisionof the tree, the population of events is divided based upon the variable thatwill maximally separate signal from background (computed with the giniindex [109]). When using the Adaptive boosting method, at the lowest levelof the tree, events that have been misclassified are increased in weight andsupplied to additional small decision trees. The number of weak decisiontrees used in the algorithm is set by the analyzer, and should be determinedby examining the error rate (the fraction of misclassified event) as a functionof the number of decision trees. Once the training data sample has flattenedon this plot, additional weak learners will not find additional correlations in1194.6. Low Frequency Noise CutsFigure 4.29: (left) A two-level decision tree. (right) A diagnostic plots ofthe error rate, for both training and test data, to help select the optimalnumber of weak decision trees.the parameter space and therefore not improve discrimination. One strongindication of overtraining is that the test data curve increases. The numberof weak decision trees was chosen to be 8000, which ensured that the errorrate for the test data did not begin to increase.Full BDTIn addition to providing a powerful discrimination parameter, the BDTframework provides flexibility that allowed for avoidance of the series blockdata division procedure described in Sec. 4.7. While the series blocks basedon calendar time and cryocooler vibrational phase were critical for the 2DptOF vs. ∆χ2 based cuts, for the BDT, instead of incorporating cryocoolerinformation by dividing the data set according to cryo-loud and cryo-quiettime periods, it makes sense to to include the cryocooler score as a BDTinput. Similarly, instead of dividing the data set according to calendar time,we can include the event time as a BDT input and the BDT will make a con-tinuous cut based on the event time as opposed to our previous 17 discreteblocks approach.At the same time, dividing the series up into time periods proved veryuseful when monitoring the detector’s noise environment, and so below weconsider the Run 3 data set in two different periods, divided into Run 3aand Run 3b as described in Sec. 4.3.4. The key reason for doing this is notbecause the BDT would be unable to find a LF noise correlation in calendartime and set the cuts accordingly. Instead, the reason for dividing the dataalong Run 3a and Run 3b is because the nature of the background changesbetween the two periods. Whereas LF noise makes up the majority of lowenergy triggers in Run 3a, in Run 3b charge triggers and LF noise eachmake up roughly 50% of the low energy triggers, as shown in Sec. 4.3.4 and1204.7. Bifurcated Analysisin particular Fig. 4.6. It is simple to understand that BDT performancewill be suboptimal if it is designed to discriminate against LF noise events,but the background is made up a combination of charge trigger events andLF noise events. We overcome this issue by training BDTs separately forRun 3a and Run 3b. For the Run 3b data, the analysis threshold is raisedfrom 1.29 keVt to 1.9 keVt in order to eliminate the charge leakage triggerbackground (which reconstructs to energies below 1.9keVt).Therefore, the full BDT score discriminator is based on 8 different param-eters: (1) cryocooler phase (2) z4z5 correlation (3) z4z6 correlation (4)(5)(6)∆χ2LF parameters for 3 different templates, (7) event time, and (8) energy,and the BDT is tuned for the two distinct time periods. The feature discrim-ination power of each of these parameters is shown in Fig. 4.30 for the Run 3bBDT. It is clear that the detector-detector correlations and cryocooler infor-mation improves discrimination, even though the ∆χ2LF parameters providethe best discrimination.Figure 4.30 right also shows the R3b BDT score vs. energy, where apopulation of good events is separated from the high density population ofevents made up of LFN triggers. What is also apparent, however, is thatbelow ∼3keVt the distribution of events at higher BDT score begins to mergeinto the population at lower BDT score, again showing that discriminationis more difficult at low energy. It is not obvious from this plot where toplace a cut on BDT score in order to maintain signal acceptance down toas low of an energy as possible, while simultaneously not accepting a LFnoise event. In Sec. 4.7 below, we describe a procedure for splitting up theBDT score of Fig. 4.30 into two BDTs and setting a cut to ensure LF noisebackground leakage to less than one event.4.7 Bifurcated AnalysisThe bifurcated analysis is a method to measure the number of events from acertain background that is leaking past a set of quality cuts. The frameworkis suited for measuring leakage of a background that is difficult to modelbecause neither a spectral energy shape (a PDF) of the background nor abackground rate estimate is needed to employ it. The method estimates theleakage using side band information (i.e. information outside of the signalregion)[110, 111].Therefore, the bifurcated analysis is attractive when considering theRun 3 analysis goal to understand the background at the lowest energieswhere the dominant source of events is difficult-to-model LF noise triggers.1214.7. Bifurcated AnalysisFigure 4.30: (top) The feature discrimination power of different BDT pa-rameters, which is calculated using the amount that each feature split pointimproves the sample purity, weighted by the number of data points in thebranch of interest, and summed over many decision branches. (bottom) TheRun 3b BDT score as a function of energy, showing good separation of asmall collection of signal events from the population of noise events .1224.7. Bifurcated AnalysisFigure 4.31: The CDMSlite R3 low background WIMP-search spectrumwith a relatively loose cut placed on the LF noise BDT score. We seek toanswer how many of the low energy events are LF noise triggers as a functionof the cut value, and then set a cut such that there is <1 event leakage.In particular, the LF noise cut must have a low (<1 event) leakage of LFnoise triggers past the cut. If LF noise trigger events leak in significantquantities past the cut, the background model will not describe the data.This is because the background model only includes components that de-scribe particle energy depositions in the detector from radioactive decays.Figure 4.31 shows an example of an event spectrum with a relatively looseLF noise cut, where some LF noise events are present in the spectrum. IfLF noise events leak past the final cut, they will be incorrectly incorporatedinto the PDF of a different background, and the likelihood fit for backgroundand DM signal components will be systematically biased.4.7.1 IntroductionThe number of LF noise events leaking past a set of cuts is given by:Nleak = NLF · P (cuts), (4.12)where P (cuts) is the passage fraction of the cuts and NLF is the number ofLF noise events. While both NLF and P (cuts) are unknown, they can beestimated if there exist two uncorrelated sets of cuts that are both sensitiveto LF noise events. The cuts are uncorrelated if cut A removes the samefraction of background regardless of whether cut B has been applied, andvice versa.The two uncorrelated cuts that are both sensitive to the backgroundof interest are labeled as cut A and B. Then events are divided into the1234.7. Bifurcated AnalysisFigure 4.32: The 4 boxes into which events are divided when two bifurcatedcuts (cut A and cut B) are applied to the data. The number of events in theupper left box (NAB) represents the number of events that pass both cuts.4 different categories of passing and failing the cuts, which are shown inFig. 4.32. The AB box, which is the region of the parameters space thatboth cut A and cut B accept, is in the upper left. The number of eventsin the signal box is NAB. Two types of events will be in the signal box:(1) good pulses from real energy depositions in the detector which we labelNSIG and (2) low frequency noise triggers which we label NLEAK .In order to solve for NLEAK , we must know the passage fraction of cut Aand cut B to the background, which we label P (AB). We also must know thesignal efficiency9 of the two cuts, labeled S(AB). Making the uncorrelatedcut efficiency assumption, an assumption that is tested in Sec. 4.7.3 and4.7.4, P (AB) = P (A)P (B) and S(AB) = S(A)S(B). We also define NBKGsuch that NSIG+NBKG is the total number of events in all 4 boxes: NSIG+NBKG = NAB +NA¯B +NAB¯ +NA¯B¯. We also define P (A¯) as the rejectionfraction of the background (i.e. P (A¯) = 1 − P (A)), and use the equivalentnotation for cut B and for the signal efficiency S. The number of events inthe individual boxes is therefore given by:NAB = S(A)S(B)×NSIG + P (A)P (B)×NBKG (4.13)9Throughout this section “signal” refers not necessarily to a DM signal, but rather anyevent that is not a LF noise background event.1244.7. Bifurcated AnalysisNAB +NA¯B = S(B)×NSIG + P (B)×NBKG (4.14)NAB +NAB¯ = S(A)×NSIG + P (A)×NBKG (4.15)NA¯B¯ = S(A¯)S(B¯)×NSIG + P (A¯)P (B¯)×NBKG (4.16)We numerically solve the 4 equations above for the 4 unknowns: P (A),P (B), NSIG, NBKG. Then the number of background events leaking intothe signal box can be solved for, and is given by:NLEAK = P (A)P (B)×NBKG. (4.17)If the bifurcated cuts are 100% efficient to signal the above system ofequations can be analytically solved to give:NLEAK =NAB¯NA¯BNA¯B¯, (4.18)but because we can determine the values of S(A) and S(B) from pulsesimulations of good events, and they are <100% efficiency, we numericallysolve the system of equations and apply Eq. 4.17.4.7.2 Application to CDMSlite Run 3The BDT-based discrimination discussed in Sec. 4.6.5 is particularly wellsuited for the bifurcated analysis application. First, the bifurcated analysisis designed to work in two dimensions (i.e. the two dimensions on which thetwo cuts, A and B, can be placed), and a BDT reduces the dimensionality ofthe discrimination parameters from N inputs to a single output. Therefore,training two BDTs reduces the eight-dimensional discrimination parameterspace to 2 dimensions that can be used for the bifurcated analysis.Second, the bifurcated analysis is designed to work when the A and Bcuts are uncorrelated. For the LF noise cut we have developed parametersfrom 3 different sources that all have LF noise discrimination power butare not necessarily correlated—(1) detector signal correlation information,(2) cryocooler information, and (3) pulse shape information with the ∆χ2LFparameters. To proceed with the bifurcated analysis we split up this infor-mation into two BDTs that each discriminate against LF noise but do sousing parameters in which we would not expect there to be extreme corre-lations.1254.7. Bifurcated AnalysisThe eight LF noise discrimination parameters are used in two differentBDTs with the goal of creating BDT outputs that are good discrimina-tors but uncorrelated. Branch A is primarily ∆χ2 based and Branch B isprimarily cryocooler and detector-detector correlation based, although thedecision of which parameters to use in which branch was ultimately madeempirically. That is, different arrangements of parameters were tested untilan arrangement that yielded uncorrelated BDT outputs was found. For thisreason the template 3 ∆χ2LF parameter is not in the same branch as theother ∆χ2LF parameters, and the division of parameters is given below:• branch A– ∆χ2LF- template 1– ∆χ2LF- template 2– Event Time– energy (ptOF)• branch B– cryocooler phase– T2Z1-T2Z2 correlation– T2Z1-T2Z3 correlation– ∆χ2LF- template 3– Event Time– energy (ptOF).Figure 4.33 left shows the BDT output of the two different branches.There is a population of events in the upper right of the plot for which theoutput of the two BDTs are clearly correlated between the two BDTs. Theseare signal events and it is an indication that the two bifurcated BDTs aresuccessfully discriminating; they are together separating events that the fullBDT (indicated by the color) also separates with high BDT score. Moreimportantly for the bifurcated analysis to work, the LF noise background(at lower BDT score) should be uncorrelated between the two cuts. Thehigh density of events at significantly lower BDT score (<0) is correlatedbetween the two branches. This agrees with intuition, since very bad LFnoise can be identified based on ∆χ2LF information as well as cryocooler ordetector-detector correlation information. In order to apply the bifurcatedanalysis, additional cuts can be defined to select events closest to the cut1264.7. Bifurcated Analysisboundary (where it’s most important to measure event leakage) and as willbe shown robustly below, the cuts are uncorrelated in the region of the BDTvalues where there is risk of LF noise events leaking past the cut.Setup Cut VerificationIn the bifurcated analysis framework, “setup” cuts are those that are appliedto the data before performing the bifurcated analysis. In this case our setupcuts include all of the quality cuts described in Sec. 4.5, as well as a simpleLF noise setup cut that removes particularly bad LF noise. The location ofthe setup cut, which is defined on the full BDT score, is shown in Fig. 4.33center. This location was chosen to remove events that are sufficiently badLF noise triggers that they can be taken out of the bifurcated analysisbecause they will not leak into the signal region. The events surviving thesetup cuts—the events that will be used in the bifurcated analysis—areshown in black in Fig. 4.33 right.By first looking at the population of events that fail the full BDT setupcut, shown in Fig. 4.34, we see that they are removed from the AB box.However, because there are nearly 2× 106 events that make up this popula-tion, we’d like to check that the tails of the distribution do not extend intothe AB box. A 2D kernel density estimate is used to approximate the distri-bution [112]. The bandwidth of the kernel—the width of the Gaussian thatis added to each data point—is chosen automatically by the functions suchthat the density estimate is optimal for normal densities. When the KDE isfound using this default bandwidth, the density is integrated over the signalregion, and 0 events are found to be entering into the signal region.4.7.3 LF Noise Background CorrelationOur full data set contains signal and background events. We would liketo calculate correlation coefficients for the data from the background dis-tribution only, but it is unknown which data points come from which dis-tribution. However, we know the shape of the signal distribution, definedas PS . We also can estimate the magnitude of signal and background datain the full data set, allowing us to define the distribution of all events asPfull = PB +NSNBPS . Using this information we can subtract off the effectsof the signal data and calculate correlation coefficients for the backgroundPB distribution alone.To keep with standard convention for 2D correlations/covariances, we’lllabel branch A as X and branch B as Y. Using Pearson’s correlation co-1274.7. Bifurcated AnalysisFigure 4.33: (top) The two branches of the bifurcated BDTs of Run 3a data,colored by the full BDT score. (middle) The setup cut value in the full BDTscore. (bottom) The distribution of events considered for the bifurcatedanalysis shown in black, with those rejected by the setup cut shown in blue.1284.7. Bifurcated AnalysisFigure 4.34: The distribution of events failing the setup cut, characterizedwith a kernel density. Integrating the kernel density into the signal regiongives 0 events.efficient, ρX,Y =cov(X,Y )σXσY, we need to first calculate the covariance of thebackground distribution and we start with the signal data covariance:covS(X,Y ) =∫ ∫dxdy(x− µx|B)(y − µy|B)PS(x, y). (4.19)Equation 4.19 differs from the standard covariance equation only in that theµx|B and µy|B terms are the means of the background-only variables. Thebackground distribution is unknown but these means can be estimated byusing the full and signal means, µfull and µS , with the relationship:µfull =NSNB +NSµS +NBNB +NSµB. (4.20)When going from continuous distributions to discrete data points, andcalculating the sample covariance, it is important to normalize all the co-variances to the estimated number of background events. With the samplecovariance for the full data set given bycovfull(X,Y ) =1NS +NB − 1NS+NB∑i(xi − µx|B)(yi − µy|B) (4.21)the background covariance iscovB(X,Y ) =NS +NB − 1NB − 1 covfull(X,Y )−NSNBcovS(X,Y ). (4.22)1294.7. Bifurcated AnalysisFigure 4.35: (left) A toy model of a background distribution (blue) and a sig-nal distribution (red) with known correlation coefficient in the two variables.(right) The full signal + background distribution, where the correlation ofthe background distribution is estimated using the method described above,and agrees with the true correlation coefficient.Once the variances for the background-only distributions are calculatedusing the same framework, the correlation coefficients between different vari-ables of the background-only data can be calculated.This method was tested on a toy model. The blue and red points inFig. 4.36 left shows the two distinct distributions, where we seek to deter-mine the correlation coefficient of the blue distribution. The true correlationcoefficients are shown in the legend. The ρbackground is the value that is be-ing estimated using the information in the right figure. The full data setis shown in black and the “contamination” distribution is shown by thecontours, and the relative amount of signal to background is known. Theestimates of ρbackground agrees with the answer given by the toy models.This method was applied to the Run 3 DM search data as well as thesimulated distribution of signal in the branch A and branch B BDT variableplane. The signal is simulated using the methods described in Sec. 4.9,and the distribution is shown in Fig. 4.36 left. We obtain a correlationcoefficient of ρbackground = −0.039 for the LF noise background, as shown inFig. 4.36. This correlation coefficient is relatively small, lending confidenceto the requirement that the bifurcated cuts be minimally correlated, andthe bifurcated analysis is performed with the AB box defined as shown inFig. 4.36, giving a LF noise leakage estimate of 0.3 ± 0.1 events.1304.7. Bifurcated AnalysisFigure 4.36: (left) The distribution of simulated signal in the bifurcatedanalysis cut plane. 97.5% of the simulated signal is contained in the passagebox, while 0.47% is in the AB¯ box, 1.9% is in the A¯B box, and 0.13%is in the A¯B¯ box. (right) The distribution of the DM search data in thebifurcated analysis cut plane. The points in the shaded pink area pass bothbifurcated cuts.Figure 4.37: Examples of cut loosening when the cuts are uncorrelated (left)and correlated (right).1314.8. Fiducial Volume4.7.4 Box RelaxationWhile the correlation coefficient result above is an indication that the LFnoise event distribution passing the setup cuts is uncorrelated, it does notensure accuracy of the bifurcated analysis results. We must also check cor-relation of the bifurcated cut parameters directly in the vicinity of the cutlocation, and this is done with the box-relaxation method.If the bifurcated result is accurate for a set of bifurcated cuts, then asa cut is loosened, new events will enter into the signal box and the newbifurcated leakage estimate will increase by the number of new events inthe box (to within uncertainties). This procedure is complicated by the factthat the bifurcated analysis cuts are <100% efficient to signal, as shown inFig. 4.36 left. This means that as the box is loosened, we must accountfor the fact that an event that is absorbed into the signal box might be asignal event. Since the bifurcated analysis incorporates the signal efficiency,the Nleak estimate should not increase if the absorbed event is likely to bea signal event. We account for the <100% signal efficiency by solving thenon-linear equations of Eq. 4.13, 4.14, 4.15, 4.16, and 4.17.After solving with a particular value of cut A and B, knowing the signalpassage fractions and the total number of signal events NSIG allows oneto calculate the estimated number of signal events outside the box, givenby (1−S(A)S(B)) × NSIG. This value is shown in Fig. 4.38 center as afunction of different values for the branch B cut. In addition, as the branchB cut value is loosened the number of events in the AB box increases, andthis increase is shown in Fig. 4.38 left. To check the bifurcated analysisresult, one would expect the number of new events in the AB box minusthe estimated number of new signal events in the AB box (the differencebetween points in the left and center plot) to increase at the same rate asthe bifurcated analysis’s Nleak estimate. In the right plot, this rate, whichis called the “direct estimate” because it directly counts the data points,is overlaid on top of the bifurcated Nleak estimate, showing agreement, towithin uncertainty. We therefore verified that the number of events enteringthe box matched the bifurcated analysis’s prediction to within uncertainties,which is consistent with the cuts being uncorrelated and therefore supportingthe validity of the leakage estimates.4.8 Fiducial VolumeBecause CDMSlite detectors have non-uniform electric fields, the NTL am-plification (the second term on the right hand side of in Eq. 4.23) and the1324.8. Fiducial VolumeFigure 4.38: (top) The number of events in the AB box, as the value of cutB is loosened. (middle) The estimated number of signal events outside theAB box, as the value of cut B is loosened. (bottom) The number of newevents in the AB box minus the number of estimated signal events shownin red, as well as the bifurcated analysis’s leakage estimate in blue.1334.8. Fiducial VolumeRadial Position[mm2]800 1000 1200 1400VerticalPosition[mm]0510152025∆V[volts]30354045505560657075Figure 4.39: (top left) Calculated voltage map for high radius events, show-ing the difference in electric potential ∆V between the final collection pointsof the positive and negative charge carriers, as a function of initial positionof the pair (plotted as radius squared vs. vertical position). Here, the topof the crystal is biased at 75 V and the bottom is grounded. Charge carriersin the outermost (radius > 800 mm2) detector annulus can experience lessthan the full detector bias voltage (credit: D. Barker [113]). (top right) Acartoon of the radial vs. energy distribution for a homogenous and monoen-ergetic background in the CDMSlite detectors. The Pi and PO refer to innerand outer probabilities, and these variable names are used in the radial cutefficiency calculation of Sec. 4.9.5 (credit: M. Pepin [82]). (bottom left) Dis-tribution of the radial parameter ξ vs. energy in the DM search data. Anenergy-dependent cut on ξ defines the fiducial volume below 2keVee, while astricter constant cut is used above 2keVee. (bottom right) Resolution (1σ)for ξ (radial parameter) shown as a function of ξ and energy. At lower en-ergy, the resolution worsens as the increased noise affects the reconstructionof the radial parameter. (credit: R. Underwood [114])1344.8. Fiducial Volumereconstructed recoil energy vary with the location at which an event takesplace inside the detector:Et = Er + Ere ∆V. (4.23)For most events, ∆V in Eq. 4.23 is equal to the full potential differencebetween the detector faces, resulting in maximal NTL amplification. How-ever, as shown in Fig. 4.39, near the detector sidewall ∆V can be smaller.The voltage drop experienced by an electron-hole pair, and thus the NTLamplification, can be reduced such that the reconstructed energy of somehigh-radius events is significantly lower.The reduced NTL events are a particularly undesirable background sincetheir energy is improperly reconstructed. Figure 4.39 top right shows an ex-ample of this background: for monoenergetic and homogeneously distributedsource of events, for example the K-, L-, and M - shell peaks, the electricfield non-uniformity creates a low energy “tail” at the energies below thepeak, as shown in Fig. 4.39 (top right). We therefore design a parameterthat uses pulse shape information to estimate the radial position of an eventin the detector. Then we place a cut on the this parameter to remove thereduced Luke gain events from the dataset, as shown in the cartoon in thetop right.The radial parameter for CDMSlite R3, which we refer to as ξ, is con-structed using individual channel fit parameters from the two-template Op-timal Filter Phonon algorithm (2TOF), exploiting the layout of the channelmap and in particular the outer annulus of channel A (see Fig. 4.4). Theparameter was developed by W. Rau [91] for CDMSlite Run 2, and we re-peated his steps to calculate the parameter for Run 3 [115]. To first order,the two-template radial parameter (2TRP) takes the difference in the am-plitude of the fast template for channel A with the fast template of the innerchannel with the largest amplitude. This can be thought of qualitatively ascreating a parameter that is correlated with the peakiness of the pulse onchannel A relative to the other inner channels.4.8.1 Radial Pulse SimulationIn order to remove reduced NTL events effectively, we must understandhow these events are distributed in the radial estimator ξ vs. energy plane.In particular, the resolution of ξ degrades at lower energies because thelower signal-to-noise ratio of the pulses inhibits the ability of the 2TOF toextract position information. Fig. 4.39 bottom left shows the “flaring” of the1354.8. Fiducial Volumeradial parameter at lower energies, which represents its worse resolution atlower energy. Below 2keVee we model the ξ vs. energy plane using a pulsesimulation similar to that described in Sec. 4.9.3. Events in the L-shell(1.21-1.45keVee) which pass all other quality cuts are used as model events,as they span the full volume. After the L-shell events are fit using the two-template Optimal Filter (2TOF) algorithm, we take the 2TOF amplitudesand re-scaled them to a lower energy. These templates are then added to anoise trace from end-of-run randoms from the same series as the event. For aconstant input radius, the output radius is found to be Gaussian distributedin ξ, and the width of the Gaussian is shown in Fig. 4.39 bottom right.We use the radial resolution in the ξ vs. energy plane as one step in themodeling of the reduced NTL events.The radial resolution map shown in Fig. 4.39 can be thought of as partof the detector’s radial response function. For inputs to this response func-tion, we will estimate the true distribution of reduced NTL events in the ξvs. energy plane and convolve this distribution with the radial resolutionresponse.4.8.2 Energy Distribution of Reduced NTL eventsFor the energy distribution of reduced NTL events, we use a smoothedhistogram of the effective potential distribution shown in Fig. 4.39 (topleft). Because the amplification of the signal is proportional to voltage, thehistogram represents the perfect-resolution energy response of the detectorto a homogeneously distributed mono-energetic source of events.We define the reduced NTL events to include any event whose recoil lo-cation results in a reconstructed recoil energy that differs from the true recoilenergy by more than the 1σ detector energy resolution. This correspondsto events that see less than 93.3% of the full bias voltage Vdet. For electronrecoils, the measured event energy is reduced from the nominal expectationaccording toEmeasured = Enominal ×1 + ∆Vγ1 + Vdetγ, (4.24)where ∆V is the potential difference experienced by charge carriers producedat the recoil location, and Vdet is the nominal potential difference.The shape of the voltage distribution is a source of systematic uncer-tainty for the distribution of reduced NTLs, and to account for this weperform the same analysis with an alternate voltage distribution containingmore features in the voltage spectrum from the simulation. This predicts1364.8. Fiducial Volumea slightly higher leakage rate of reduced NTLs given the same radial cut,and gives us a handle on the systematic uncertainty on the rate of reducedNTLs we expect to pass our radial cut.4.8.3 Radial Distribution of Reduced NTL eventsThe majority of reduced NTL events are measured only slightly lower inenergy than their true energy, because the distribution of ∆V inside thedetector peaks strongly at the nominal voltage. Thus the energy regions justbelow the strong K- and L-shell 71Ge-decay peaks provide good samples ofreduced NTL events, with high signal to noise which we used to determinethe distribution in the radial parameter ξ.We model the radial distribution of reduced NTLs by defining a regionin the radial parameter (ξ ∈ [−2 × 10−5,+4 × 10−5]) outside of which weobserve no reduced NTLs, and selecting events in this region within a smallenergy range below the L-shell capture peak (0.7–1.2 keVee). Creating acumulative distribution function in ξ for these events gives us an idea ofthe distribution of reduced NTLs in ξ. A systematic uncertainty on thisdistribution is estimated by removing the upper bound in ξ while narrowingthe energy window, which creates a distribution that predicts slightly morereduced NTLs passing the same cut.4.8.4 Optimizing the Radial CutCombining the expected energy distributions of reduced NTL events, thevoltage map model, and the resolution model for ξ as a function of energy,we model the reduced NTL distribution in the energy-ξ plane. We randomlysample from these distributions and thus produce a prediction for the 2Dprobability distribution of the data in these variables, which is shown inFig. 4.40. A cut can then be set on this distribution of events to allow acertain shape and magnitude of leakage past the cut.This specific choice of event leakage was chosen after a detailed sensitiv-ity study. While choosing a looser cut results in a better signal efficiency(improving the analysis’s sensitivity), a looser cut also results in a higherreduced NTL background leakage past the cuts (worsening the analysis’ssensitivity). Monte Carlo simulations indicated that minimal extra sensi-tivity was gained by choosing a looser cut [116]. Therefore, in order toeliminate the need to include a reduced NTL model into the profile likeli-hood background model, we chose a particularly restrictive cut.We set a cut on ξ as a function of energy that results in 0.13± 0.1stat ±1374.9. Livetime and Signal EfficiencyFigure 4.40: The Monte Carlo distribution of reduced NTL events in the2T radius (i.e. ξ) vs. energy plane. The colored points are those thatwould pass a cut such that (after normalizing the Monte Carlo to the Run 3exposure) only 1/8th of an event would pass into the signal region. (credit:R. Underwood)0.44sys reduced NTLs passing the cut. The systematic error is estimatedfrom Monte Carlo simulations with the alternate radial and voltage models(with the radial distribution of reduced NTLs being the larger contributor).The cut boundary was chosen such that the expected distribution of reducedNTLs passing the cut is uniform in energy between 0.07 and 2 keVee. Theradial parameter cut imposes an analysis threshold of 70 eVee, which isdetermined by the lowest well-determined bound of the radial resolutionmodel.4.9 Livetime and Signal EfficiencyWhile the cuts described in Sec. 4.5 are designed to remove backgroundevents, they remove some good events as well. When looking for a DMsignal in the data, we must account for the fact that the cuts make theanalysis less than 100% efficient to the signal. Because signal-backgrounddiscrimination becomes increasingly difficult at lower energies, we calculatethe efficiency as a function of energy, as described in Sec. 4.9.3.In addition to the cuts removing good events that could potentially befrom DM particle interactions, the hardware trigger of the experiment alsoonly reads out events with a sufficiently high signal-to-noise ratio. Therefore,recoils in the detector that are below a certain energy range are never readout as events, and we account for the reduced signal rate at low energy bycalculating the trigger efficiency as a function of energy. This is described1384.9. Livetime and Signal EfficiencyFigure 4.41: Livetime contribution of each series block to the R3 exposure,with the livetime removed by the cuts listed in Sec. 4.9.1 shown at the topof each bar of live days.in Sec. 4.9.2.We also consider the amount of time that the detector was “live” andable to record an event, as described in Sec. 4.9.1. The “livetime” is one com-ponent in the calculation of the magnitude with which a DM signal wouldappear in the detector. Ignoring annual modulation effects, the number ofevents observed from a DM signal increases linearly with the livetime.4.9.1 LivetimeFor the “livetime” of the dataset we consider the raw amount of time overwhich the dataset was acquired, and then account for data acquisition inef-ficiencies as well as removal of certain periods of time from the DM searchdataset, which reduces the raw livetime.When the data acquisition system records an event, it also records theamount of time that it was idle and waiting for that event (i.e. that event’slivetime). This amount of time will be slightly less than the time betweenthe event of interest and the previous event because the data acquisitionsystem requires a small amount of time to read out an event that amountsto some “deadtime.” Simply summing the livetime quantity for each eventgives the cumulative livetime of the dataset. Without any cuts applied, thelow background DM search Run 3 dataset lasted 66.9 days.The cuts described in Sec. 4.5 remove periods of time when the detectorwas behaving sub-optimally. These cuts are the high voltage, NuMI beam,prepulse baseline, charge chi squared, bad series, bad GPS time, bad triggerrate, and the “T2Z1 triggered” cuts. With all R3 LT cuts applied, the totalRun 3 livetime is 60.9 days.1394.9. Livetime and Signal EfficiencyWe compute the livetime removed for these cuts for each of the 17 seriesblocks, and as shown in Fig. 4.41 the cuts retain most of the livetime (thoughit should be noted that some of the bad series are not assigned to a seriesblock, and so visually this plot underestimates the livetime removed by thecuts). After livetime cuts have been applied, the Run 3 total livetime is 60.9days (or 36.4 ± 0.3 kg-days of exposure for the T2Z1 mass of 0.597 ± 0.005kg [117]).4.9.2 Trigger EfficiencyThe central idea of the trigger efficiency is that the detector’s DAQ hardwareonly issues a trigger and reads out events whose energy deposition is largeenough to create a significant increase of the signal above the baseline noise.Therefore, while very low energy events buried in the baseline noise willnever cause a trigger and high energy events will always cause a trigger,there is a middle ground of events that are only slightly distinguishable abovebaseline noise and that the hardware trigger has a non-zero and non-unityefficiency of identifying. Trigger efficiency studies calculate this efficiencyand parametrize it as a function of event energy.The trigger efficiency is typically calculated using 252Cf calibration data,as opposed to low background data, because the method for calculatingthe trigger efficiency improves with increased statistics. For Run 3, thetrigger efficiency was calculated using the three-day 252Cf calibration takenat the beginning of Run 3. The vast majority of the events that occuron the detector of interest (in this case T2Z1) were read out because theycaused a trigger on T2Z1. Therefore measuring the trigger efficiency onT2Z1 with events that are known a priori to have caused a trigger is biased—the measurement will indicate 100% trigger efficiency. Instead, the methodemployed to measure the T2Z1 efficiency operates on the idea that eventsthat have caused a trigger on the other active detectors (in this case T2Z2and T2Z3) are an unbiased sample of events that can be used to measurethe T2Z1 trigger efficiency. This is because for good events, the trigger ofthe other detectors should not depend on the trigger of T2Z1. Of thoseevents that caused a trigger on the other detectors, we calculate, in energybins of 0.2 keVt, the fraction of events that caused a trigger on T2Z1. Thatfraction (labeled “efficiency” on the y axis) is plotted as a function of energyin Fig. 4.42.To select the events for the trigger efficiency calculation, we apply re-strictive cuts [118] to ensure that the measurement is being made with goodevents that were not due to instrumental backgrounds (e.g. low frequency1404.9. Livetime and Signal EfficiencyFigure 4.42: (top) Trigger calculation showing the number of events thattriggered both T2Z1 and the other detectors (numerator counts) as well asthe number of events that triggered just the other detectors (denominatorcounts). (bottom) Binned efficiency of the trigger as a function of energy.Error bars are binomial statistical uncertainties.noise or glitch triggers). The high rate 252Cf calibration data with which thetrigger efficiency is measured also helps to ensure that instrumental noiseevents make up a small fraction of the events being used to measure theefficiency.The trigger efficiency was parametrized as a modified error function givenby Efficiency(E) = 0.5(1 + erf(E−µ√2σ))where µ corresponds to the 50%efficiency point and σ determines the width of the curve.Figure 4.42 shows that there is an energy range of interest between 0.7and 1.5 keVt in which some events have issued a trigger on T2Z1 (i.e. passingevents) and some events have not issued a trigger on T2Z1 (i.e. failingevents). Given the error function trigger efficiency model, at an energy E,the probability of an event passing is 0.5(1 + erf(E−µ√2σ)), and the probabilityof an event failing is 0.5(1− erf(E−µ√2σ)). The data consists of energies of thepassing events (Epass) and failing events (Efail), and so up to a normalization1414.9. Livetime and Signal EfficiencyFigure 4.43: (left) Corner plot for trigger efficiency fit of µ and σ. (right)Best-fit error function with 100 MC sample curves from the µ vs. σ posterior.These samples serve as visualizations of the model uncertainties.factor, the log likelihood of the data is:lnL =N∑i=1ln[0.5(1 + erf(Epass,i − µ√2σ))]+M∑j=1ln[0.5(1− erf(Efail,j − µ√2σ))],(4.25)where N is the number passing events, and M is the number of failing events.Maximizing the likelihood gives best-fit parameters: µ=1.29 keVt, σ=0.20 keVt.To evaluate posteriors, uniform priors were used on µ and σ, and the emcee[119] MCMC package was used to sample the likelihood. The covariance be-tween model parameters, as well as the best fit model, is shown in Fig. 4.43.4.9.3 Signal SimulationThe efficiencies of the glitch, phonon χ2, square, and BDT-based LF noisecut were computed using a simulation of good events, which we refer toas a “pulse simulation.” A cartoon of the simulation procedure is shownin Fig. 4.44, where we use a noiseless pulse template for the shape of thesimulated pulse, and we use a random event to represent the noise for thesimulated pulse.The CDMS BatFaker software package, written by B. Loer as part of thesalting effort, as well as MATLAB tools, are used to create the fake events ina format that can be processed through the CDMS reconstruction software.BatFaker and the MATLAB tools work by replacing events in an existing1424.9. Livetime and Signal EfficiencyFigure 4.44: A cartoon depicting the pulse simulation procedure (credit A.Anderson: [120]).CDMS raw data file with fake data that can be specified by the user. Inparticular, one creates a fake pulse by combining (1) a noise trace from arandom event in the real data, (2) a slow template scaled to a user-definedenergy, and (3) a fast template scaled to a user-defined amplitude.For calculating cut efficiency as a function of energy, fake pulses aresimulated that are uniformly distributed in energy. To first order, this isaccomplished by uniformly varying the amplitude of the noiseless templateshown in Fig. 4.44 while not scaling the noise traces. To second order, thepulse shape varies as a function of energy, which we include in the simulation.The majority of energy-dependent pulse shape variations, as well as thepeaky and non-peaky position-dependent features that exist in individualchannels, are averaged out when the 4 channels are summed to make thetotal phonon pulse; however, there are still pulse shape deviations from thestandard optimal filter template. We seek to simulate these variations inour fake data using a 2-template (1 slow, 1 fast) approach.To capture the variations in peakiness of the pulses as a function of en-ergy, we use the ratio of fast to slow amplitude fits of good events, which isshown in Fig. 4.45 left. We also examine the raw L-shell traces at the ex-trema of the fast/slow amplitude ratio distribution in Fig. 4.45 right. Theseraw traces confirm that the L-shell events at the extreme of the ratio distri-bution are still good events, and the full range of the fast/slow ratio can besampled to create fake pulses that resemble true signal. The slow amplitudeswere uniformly sampled such that the pulse energies would be uniformly dis-tributed between 0 and 60keVt. To determine the relative amplitude of thefast and slow template to use for this simulated data, we use the distribution1434.9. Livetime and Signal EfficiencyFigure 4.45: (left)The fast/slow amplitude ratio, as a function of energy, forthe two-template fit optimal filter fit. (center) (right) Two L-shell pulsesshowing the variation in pulse peakiness where the center pulse has is thepeakiest of the L-shell events (fast/slow ratio = 0.4) and the right pulse isthe least peaky of the L-shell events (fast/slow ratio= −0.6).of the fast/slow template ratio observed in the low background data from0–30keVt, in energy bins of 2keVt. The fast amplitudes were sampled fromthe fast/slow amplitude ratio distributions.4.9.4 Quality Cut EfficiencyAfter creating the fake event datasets we process them through the eventreconstruction software, and then examine the location of our simulatedgood events in the parameter spaces in which we have placed cuts. Anexample of the simulated events in the ∆χ2glitch vs. energy plane is shownin Fig. 4.46 left, where the slope of the parabolic distribution matches thetrends observed in the real data. One complication is that the glitch, phononχ2, and square cuts were tuned on 17 different series blocks of data asdescribed in Sec. 4.6, while the BDT-based cut was tuned separately forR3a and R3b as described in Sec. 4.7. Accordingly, the efficiencies of thesecuts were calculated for the different series block by simulating the fakepulses with randoms selected from that series block.The efficiency of any given cut is simply the passage fraction for a specificenergy bin, the ratio of passing events to all events, where “all events”constitutes all the events that pass the livetime cuts.Because many of the DQ cuts are based on the pulse shape and optimalfilter fitting results, some correlation is expected between the cuts. Forexample, one would hope that clear glitch events would also fail the phononχ2 cut and potentially the square pulse cut. The correlation between thedifferent cuts is accounted for by simply computing the logical AND of allthe cuts. The final efficiency for a group of data quality (DQ) cuts, for each1444.9. Livetime and Signal EfficiencyFigure 4.46: (left) The distribution of simulated signal events in the ∆χ2glitchvs energy plane with the cut boundary overlaid. (center) The efficiency ofthe ∆χ2glitch cut calculated with data the simulated signal data. (right) Theefficiency of all DQ cuts calculated using Eq. 4.26. All figures use seriesblock 7 simulated data and cut boundaries.energy bin, is thenEfficiency = sum[cutsLT&cutsDQ]/sum[cutsLT], (4.26)where the data quality cuts are collectively labeled as cutsDQ and the live-time cuts are collectively labeled as cutsLT. The efficiency of the glitch cutalone and for all DQ cuts, for series block 7, are shown in Fig. 4.46 centerand left respectively.The error bar for each energy bin is the 1σ width of a binomial distri-bution with the central value (probability of passing) p, the number of totalevents for each bin is N , and the number of events passing the cut in thatenergy bin is given by k. This distribution is given by:B (k;N, p) =(Nk)pk(1− p)N−k = N !k!(N − k)!pk(1− p)N−k. (4.27)Using the binomial distribution to estimate the uncertainty breaks downwhen p = 1 and p = 0, where the variance of the binomial distributionvanishes, unrealistically indicating 100% certainty of the efficiency measure-ment. Alternatives for computing sensible uncertainty bands for these caseshave been proposed that employ Bayes’ theorem. Using a uniform prior onp and a binomial distribution as the likelihood, one can analytically solvefor the posterior on p:P (p; k,N) = (N + 1)B (k;N, p) . (4.28)as derived in Ref. [121]. Equation 4.28 does not result in zero variance whenk = 0 and k = N , and so we use this result for the two edge cases.1454.9. Livetime and Signal Efficiency4.9.5 Fiducial Volume EfficiencyThe final fiducial volume efficiency is the fraction of events that are recon-structed at the correct energy that pass the radial cut. In a method devel-oped by W.Rau, R. Underwood, and K. Page [122], the radial cut efficiencyis split up into two parts. The first part, called the “energy efficiency,”is the fraction of events reconstructed at the correct energy. The secondpart, called the “peak efficiency,” is the fraction of events at a given energythat pass the radial cut. The energy efficiency and peak efficiency are givenby the first term and second term, respectively, on the right-hand-side ofEq. 4.29:Efficiency =PiR+ Po + Pi=Po + PiR+ Po + Pi× PiPo + Pi. (4.29)The notation of Equation 4.29 is defined in Fig. 4.39 top right.The “energy efficiency” was measured by finding the ratios of K-, M -, and L- shells events with misreconstructed energy to those events withcorrectly reconstructed energy. The number of misreconstructed K-, M -,and L- shell events was estimated using a time-dependent fit that exploitedthe fact that the rate of the K-, M -, and L- shell backgrounds decay with a11.43 day half-life, the half-life of 71Ge, while other dominant backgroundsdecay over significantly longer timescales or not at all. Details of the energyefficiency calculation can be found in Ref. [82].The “peak efficiency” can be measured with techniques similar to thoseused to construct the reduced NTL model in Sec. 4.8. We use a Monte Carlosimulation based on the radial resolution model to simulate the radial pa-rameter distribution for events having the full NTL amplification. Becausethe DM signal will be uniformly distributed in the detector, we model theξ distribution for uniformly distributed, full NTL amplification events withreconstructed energies in the L-shell line. We statistically subtract the smallcontribution of non-71Ge backgrounds from this distribution and deconvolvethe radial-parameter resolution at the L-shell line energy (1.3 keVee). Thedeconvolved distribution is expected to be the underlying “true” distributionof ξ for events at the L-shell energy. We then use the resolution model ofξ, described in Sec. 4.8, to scale this distribution according to energies from0.07 to 2keVee, thereby creating energy-dependent probability distributionsfor ξ. We then apply the radial cut to these simulated distributions, and bydoing so obtain the efficiency of the fiducial volume cut for events with fullNTL amplification.1464.9. Livetime and Signal Efficiency4.9.6 Combining and Parametrizing the EfficiencyThe trigger efficiency, quality cut efficiency, and the radial cut efficiency arecombined by multiplying the mean value of the efficiencies together and usingMonte Carlo to propagate the uncertainty between the different sources.For the quality cut efficiency, systematic differences between the effi-ciencies from the different series blocks require propagating the uncertaintybetween series blocks to obtain a combined total efficiency curve for allquality cuts and all series blocks. To generate a Monte Carlo (MC) sam-ple efficiency curve, for each energy bin a random number is drawn from abinomial distribution using the central value (probability of passing) p andnumber of total events N for each bin. Visually, this means sampling thedistribution described by the error bars in Fig. 4.46 right independently foreach energy bin. In order to properly weight the efficiency contribution fromeach series block, the number of MC efficiency curves created for each seriesblocks is linearly proportional to that series’ livetime. Many MC curves aregenerated and multiplied together to determine the mean efficiency valuesas well as the 1σ efficiency uncertainty.For the trigger efficiency, we sample the 2D posterior for the trigger effi-ciency model µ and σ. The posterior is shown in Fig. 4.43 left while the MCsampled trigger efficiency curves are shown in Fig. 4.43 right. The trigger ef-ficiency curves are incorporated into the total efficiency by multiplying themby the quality cut MC efficiency curves, although it is clear that the triggerefficiency is a subdominant contribution. That is, the quality cut efficiencyfalls to a low value at 1.75keVt where the trigger has high efficiency.The total efficiency curve is parametrized with an error function givenbyEfficiency(E) = A(1 + erf(E − µ√2σ))(4.30)and the results of the fit of that model are shown in Fig. 4.47.1474.9. Livetime and Signal EfficiencyFigure 4.47: The signal efficiency with successive application of the triggerefficiency, quality cuts efficiency, and fiducial volume cut efficiency. Thefinal data is included with statistical and systematic 1σ uncertainty. Fittingthe efficiency model to these data gives the final (blue) efficiency curve andthe corresponding ± 1σ uncertainty band.148Chapter 5Profile Likelihood DarkMatter Search withCDMSlite Run 3The final chapter of this thesis focuses on the aspects of the Run 3 analysisthat deviated sharply from past CDMSlite analyses and allowed us to employmore powerful statistical methods to look for a DM signal in the Run 3dataset. We start the chapter by discussing background models for all ofthe known backgrounds in the CDMSlite Run 3 region of interest (ROI). Wealso develop a signal model, and then characterize the dominant systematicuncertainties of the analysis. These systematic uncertainties in turn allowthe signal and background models to change shape (within a systematicuncertainty range) in the fit. Then, after unblinding (or “unsalting”) thedata and observing no DM-like component in the data, we employ the profilelikelihood framework that uses a likelihood ratio statistic to set a limit on aDM signal in a way that naturally incorporates systematic uncertainties intothe limit. Our approach of accounting for known backgrounds has significantbenefits over the optimum interval approach (discussed in Sec. 2.5.3) of theprevious CDMSlite analysis. Namely, the background models and profilelikelihood method have the benefits of (1) improving the limit over the OI,and (2) moving CDMSlite from an exclusion-only analysis into one withdiscovery potential.5.1 Background ModelsThe SuperCDMS cryostat was surrounded by layers of shielding that blockedalmost all external radiation, such as γ-rays and neutrons from the cavernwalls, and as a result the dominant source of background particles in thedetector came from radioactivity of the shielding, other apparatus materials,and the detectors themselves. The profile likelihood analysis, discussed inmore detail in Sec. 5.2, requires probability distribution functions (PDFs)1495.1. Background Modelsfor every background component that will contribute events to the DM-search data. In addition to the PDF that defines the spectral shape ofthe background, if the number of events from a background component isknown, the expected number of events contributing from that background(i.e. the normalization) can be included in the profile likelihood fit.The primary backgrounds modeled for this analysis are cosmogenic ac-tivation of the crystal, neutron activation from 252Cf calibration, Comptonscattering from primordial isotopes in the apparatus materials, and 210Pbcontamination on the surfaces of the detector and its copper housing. Weuse Monte Carlo simulations, as well as data-driven fits, to model these back-grounds. Table 5.1 lists the considered background components and whetheror not the information about the normalization was included in the likeli-hood fit. These background models were developed by D. Barker, and belowwe briefly describe each background for completeness. We direct the readerto D. Barker’s thesis [123] for a more thorough discussion of the backgroundmodel development. In particular, D. Barker discusses the applicability ofthe Geant4 simulation to package to the low energies (50 eV–2 keV) of in-terest to this analysis through validation of the low-energy electromagneticphysics lists in Geant4.5.1.1 Energy Resolution ModelWe require a good model of the energy resolution in order to calculate the ex-pected energy spectra for background models. We model the total CDMSliteenergy resolution as in Ref. [82]:σT(Er,ee) =√σ2E + σ2F(Er,ee) + σ2PD(Er,ee) (5.1)=√σ2E +BEr,ee + (AEr,ee)2. (5.2)The energy-independent term σE describes the baseline resolution and ac-counts for electronics noise and any drift in the operating conditions. TheFano term σF accounts for fluctuations in the number of generated charges [125]and is proportional to√Er,ee. The σPD term reflects the position depen-dence of the event within the detector due to the electric field, TES response,etc., and is proportional to Er,ee. Separating out the energy dependence weend up with the three model parameters σE, B, and A.We use several measurements to determine the resolution model forRun 3. We use randomly triggered events to determine the zero-energy noisedistribution. Additionally we use the widths of the K-, L-, and M -shell 71Ge1505.1. Background ModelsComponent Normalization Constrainedvs. Free, and Other CommentsCompton FreeTritium (3H) Free71Ge Free68Ga Free65Zn Free65Fe FreeSurface Germanium ConstrainedSurface Housing ConstrainedSurface Top Lid Constrainedneutrons Not included(contributes <1 event [124])low rate isotopes (60Co,54Mn,49Vn) Not included(contributes <1 event)Instrumental Backgrounds Not included(contributes <1 event (Sec. 4.7))Table 5.1: Background components considered in the likelihood fit1515.1. Background ModelsPeak Energy Resolutionµ [keVee ] σ [eVee ]K shell 10.35± 0.002 108± 2L shell 1.33± 0.003 36.3± 2.0M shell 0.162± 0.002 13.9± 2.0BaselinePeriod 1 0.0 9.87± 0.04Period 2 0.0 12.7± 0.04Table 5.2: Reconstructed energies and resolutions of the 71Ge decay peaksand the baseline noise in CDMSlite Run 3.σE [eVee ] B [eVee ] A (×103)Period 1 9.87± 0.04 0.87± 0.12 4.94± 1.27Period 2 12.7± 0.04 0.80± 0.12 5.49± 1.13Table 5.3: Best-fit energy resolution parameters of the model in Eq. 5.2 forPeriod 1 and Period 2.activation peaks to determine the energy dependence of the resolution. Wefit these peaks with a combination of a Gaussian and linear backgroundmodel in order to determine the width of the peaks.Table 5.2 gives the peak position µ and resolution σ of each 71Ge peak,as given by the Gaussian fits to the peaks shown in Fig. 5.1. Becausethe zero-energy baseline resolution varies with the applied bias voltage andwith environmental conditions, all of which changed between Period 1 andPeriod 2, we calculate separate livetime-weighted average resolutions foreach period. These are given in Table 5.2. The measured widths of the K-,L-, and M -shell peaks are consistent between Period 1 and Period 2, and socommon values are used for both periods.We apply this energy-dependent resolution model when calculating theexpected energy distribution for the background and DM signal compo-nents. We propagate uncertainties in the model parameters as systematicuncertainties in the profile likelihood fit of Sec. 5.2.5.1.2 Cosmogenic ActivationCosmic rays that strike the detectors (during fabrication, storage, and trans-portation above ground) activate the crystals. In germanium detectors, cos-mogenically produced tritium is a significant background, with contributionsfrom other isotopes that decay primarily either by β-decay or electron cap-ture (EC). The isotopes that undergo EC give discrete peaks in the detectors1525.1. Background ModelsFigure 5.1: Fits of a Gaussian + linear background to the energy spectraof zero-energy (baseline) events and events from each 71Ge activation peak.The widths of the Gaussians are the energy resolution σ. Copied from [126]with permission. (credit: D. Barker)below ∼10 keV and were observed in the CDMSlite Run 2 spectrum [127].We describe analytic models for the tritium beta-decay spectrum and theEC lines.TritiumNon-relativistic β-decay theory describes the tritium’s decay spectrum be-cause its endpoint, or Q-value, satisfies the relationship Q  mec2, whereme is the electron mass. The distribution of the electron’s kinetic energyEKE is described byftritium(EKE) =C√E2KE + 2EKEmec2 (Q− EKE)2× (EKE +mec2)F (Z,EKE) , (5.3)where C is a normalization constant and F (Z,EKE) is the Fermi func-tion [128]. The non-relativistic approximation for the Fermi function isgiven byF (Z,EKE) =2piη1− e−2piη , with η =αZ(EKE +mec2)pc. (5.4)1535.1. Background ModelsShell: K L1 M1µ Λ µ Λ µ Λ68Ga 9.66 1.0 1.20 0.1107 0.140 0.018365Zn 8.98 1.0 1.10 0.1168 0.122 0.019255Fe 6.54 1.0 0.77 0.1111 0.082 0.0178Table 5.4: Cosmogenic isotopes that decay via electron capture and arepresent in the measured CDMSlite spectrum. The shell energies µ, given inkeV, are from Ref. [130]. The amplitudes Λ, from Ref. [131], are normalizedwith respect to the K shell.Here Z is the atomic number of the daughter nucleus, α is the fine structureconstant, and p is the electron’s momentum [129]. The analytical descriptiongiven by Eqs. 5.3 and 5.4 describes the tritium background used for thelikelihood analysis.Electron Capture PeaksThe cosmogenic isotopes that decay via EC and are present in the mea-sured CDMSlite spectrum are listed in Table 5.4 with their shell energiesand relative amplitudes, normalized to the K shell. The observed energydistribution is a Gaussian peak at the energy of the respective shell with awidth given by the detector’s energy resolution.In our background model, the amplitude ratio between the K-, L- andM -shell peaks is assumed to be as given in Table 5.4. The contribution ofeach EC isotope to the spectrum is given by an equation of the typefECpeaks(E) =∑i=K,L,MΛiσi√2piexp[−12(E − µiσi)2], (5.5)where Λi are the amplitudes of the respective shells, µi are the shell energies,and σi are the energy resolutions at the respective energies.By modeling the EC peaks with Eq. 5.5, the number of events in theK shell is the only free parameter in the likelihood fit, with the other peakamplitudes determined from the branching ratios.5.1.3 Electron Capture of 71GeThe 71Ge EC peaks, which are used to calibrate the energy scale and havebeen previously discussed in Sec. 2.5, are also a source of background. There-fore this component is included in the background model. They are modeled1545.1. Background Modelsusing the same functional form as the cosmogenic EC peaks (Eq. 5.5) withthe one exception that, due to the large overall number of events, the L2peak is not negligible and is thus included in the fit. This component, omit-ted from Table 2.5, has an energy of 1.14 keV and relative amplitude of0.0011.5.1.4 Compton ScatteringThe dominant contributors to the Compton background are the radiogenicphotons from trace amounts of contamination in the experimental materials.These originate from the shield materials (polyethylene and lead) as well asthe cryostat and towers (copper). Typically the energy distribution of therecoiling electrons is assumed to be uniform, or “flat,” however for the lowenergies of CDMSlite, atomic binding energy effects create a measurabledeviation from a flat Compton energy spectrum. In particular, in bothgermanium and silicon “Compton steps,” which are step-like features createdin the energy spectrum because the detector collects at least the bindingenergy of any freed electron, have been observed. The Monash ComptonModel [132] accounts for the atomic binding energy in the calculation of theenergy of the scattered incident photon and the detector’s recoiling electron.For example, the electrons in the K shell of germanium have a bindingenergy of 11.1 keV. This energy is deposited in the detector due to thereorganization of the electron shells, along with any additional energy thatis given to the freed electron by the incident gamma. Thus, an electronscattered from the K shell can never deposit less than 11.1 keV in thedetector, and likewise for electrons in the other atomic shells.While the expected location of the steps agrees with measurements byother experiments, there is more uncertainty surrounding the size of thesteps. Na¨ıvely we would expect the number of electrons in each shell todetermine the relative size of the steps; however details of the electron wavefunctions can also affect the step size. The Compton steps have been directlyobserved in silicon detectors [133]. In germanium, only the K-shell step hasbeen measured directly, and so other methods must be used to estimate thelower energy steps [134].To estimate the shape of this particular background, we carried out aGeant4 simulation [135–137] of 238U decays. We fit a model consisting ofa sum of error functions,fC(E) = Λ0 +∑i=K,L,M,N0.5Λi(1 + erf[E − µi√2σi]), (5.6)1555.1. Background ModelsΛK ΛL ΛM ΛN5.7± 0.3 15.2± 0.5 9.43± 1.40 18.7± 1.3Table 5.5: Compton model parameters for CDMSlite, normalized over theenergy range 0–20 keV. All values have been multiplied by a factor of 103and are in units of keV−1.to the the simulated events that scatter once in the CDMSlite detector.The location of each step is given by µi, while σi is the energy resolution atthat energy given by the energy resolution model of Sec. 5.1.1. The Λi, theamplitudes of the error functions, are the relative step sizes, and are chosenso that Eq. 5.6 is normalized to one over the energy range 0–20 keV. Theconstant term Λ0 in Eq. 5.6 has a value of 0.005 keV−1 and accounts for aflat background required to fit the simulated spectrum.Table 5.5 gives the final parameters of our Compton model, extractedfrom a fit of Eq. 5.6 to the Geant4 simulation.5.1.5 Surface BackgroundsSurface events are primarily due to the decay of 210Pb, which is a long-liveddaughter of 222Rn. Radon exposure can cause 210Pb to become implantedinto the surfaces of the detectors and their surrounding copper housings.Radiation from the 210Pb decay chain consists primarily of betas, Augerelectrons, 206Pb ions, and alphas which have a small mean free path in Geand will deposit the majority of their energy within a few millimeters of thedetector’s surface. To understand this background and build a model of itsexpected distribution in energy, we use a Geant4 simulation and a detectorresponse function. We normalize the predicted rate of surface backgroundsusing a study of alphas in SuperCDMS iZIP data. Again, this effort wasled by D. Barker, and we direct the reader to D. Barker’s thesis [123] for adetailed description of this work.Simulation of 210Pb ContaminationIn Geant4, we use the Screened Nuclear Recoil physics list [138] to modelthe implantation of 210Pb into the material surfaces along with any recoil ofnuclei by subsequent decays to the stable isotope 206Pb. We consider threelocations from where surface events may originate: the copper directly abovethe detector (“top lid”, TL), the cylindrical housing (H) and the surface ofthe germanium crystal itself (Ge).1565.1. Background ModelsWe simulated energy deposition from the decays of 210Pb, 210Bi, and210Po for the three locations. We then apply a detector response functionto each simulated decay. The detector response model uses the voltage mapof Fig. 4.39 and the resolution model of Eq. 5.2 to approximate the totalphonon energy measured in the detector. The response model outputs theexpected surface background spectra for this analysis, used in the likelihoodfit. The spectral shapes of the three surface backgrounds are shown inFig. 5.10.NormalizationWe normalize the surface background rate with an independent measure-ment of the alpha decay events in the CDMSlite detector, using a dataset with a livetime of ∼380 days taken with the detector operated in iZIPmode. Because this iZIP-mode data set provides more detailed informa-tion on event positions, the observed rates could be attributed to surfaceevent sources originating from parents on the top lid, housing, and detectorsurface. The detector surface rate is deduced from the surface facing theneighboring detector. This rate is then subtracted from the event rate mea-sured on the side wall and the surface facing the top lid to determine therate from the other two locations (H and TL). The single-scatter events thatpass the voltage cut in the simulation are then scaled to the Run 3 livetimeto get the expected number of surface events. The germanium, housing, andtop lid are estimated to respectively contribute 3.4, 6.5, and 17 events from0–2 keVee after signal efficiency cuts have been applied.Systematic UncertaintiesThere are two main sources of systematic uncertainty on the energy spec-tra for surface events: uncertainties in the voltage map that determines thevoltage ∆V for each event, and the location of the fiducial volume cut. Themap in Fig. 4.39 assumes no additional detectors in the tower. Includingthe detector beneath the CDMSlite detector results in a difference of 0.5 Vand 1 V for the top and bottom faces respectively, which we incorporate as asystematic uncertainty. Additionally, we model uncertainties in the fiducialvolume cut (using the voltage cut Vcut as a proxy for the radial parame-ter cut) by varying the voltage cut from roughly Vcut − 2 V to Vcut + 1 V.Figure 5.2 shows the spectra and the 1σ uncertainty from the voltage mapand voltage cut systematics. Section 5.2.1 discusses how these systematicuncertainties are incorporated into the likelihood.1575.2. Likelihood and Limit SettingFigure 5.2: The spectra (normalized to event density) of surface eventsexpected from the three surface background locations (left: germanium;center: housing; right: top lid). For each location, the solid curve representsthe mean of the expected event distribution (ρ0). The shaded band showsthe 1σ uncertainty, where the top and bottom edges of the bands correspondto ρ+ and ρ− in Eq. 5.7, respectively. Copied from [126] with permission.(credit: D. Barker)5.2 Likelihood and Limit SettingThe understanding of backgrounds in the CDMSlite energy range (0–25keVee)was greatly improved in the analysis of the CDMSlite R2 dataset, and withthis improved understanding, background modeling likelihood approacheswere used for the CDMSlite Run 3 WIMP-search analysis. This likelihoodanalysis of Run 3 data improves upon the Run 2 analysis for two mainreasons. First, it provides improved sensitivity over the optimum intervalmethod. Second, the optimum interval method only allows one to excludea signal whereas in likelihood analyses the signal is included in a fit andtherefore a likelihood analysis has discovery potential. For the R3 analysiswe employ the profile likelihood ratio (PLR) method [139], which has theadditional benefit of naturally incorporating systematic uncertainties intosignal and background models and reflecting those systematic uncertaintiesin the sensitivity.5.2.1 Components Needed for the LikelihoodThe two most fundamental components of the likelihood are the data andthe model. The data are the energy (in keVee) of the unblinded events thatpass all quality cuts. The time values of the events are also used to determineif the event is from Run 3a or Run 3b (since the resolution are different forR3a and R3b as described in Sec. 5.1.1), so that events from R3a and R3bwill be treated separately in the likelihood. For testing the likelihood prior1585.2. Likelihood and Limit Settingto unblinding, we also created fake datasets (i.e. pseudo-datasets) that arerepresentative of our expectations for the final dataset. Because this analysisis focused on WIMPs between 1 and 10GeV/c2, and the energy spectrumof a WIMP signal in this mass range above 2keVee is small relative to thesignal below 2keVee, only data in the 0–2keVee range is considered for thefinal likelihood fit. The higher energy 5–25keVee region, where WIMPs of1–10GeV/c2 contribute negligibly to the event rate, is used to generate abackground rate estimation which is used in the sensitivity estimate.The model’s fit parameters allow both the magnitude and the shape ofthese distributions to vary, though for the surface backgrounds the normal-ization will be constrained within a range determined in Sec. 5.1.5. Thereare a total of 20 model parameters that we fit for in the likelihood: thenumber of background events from (1) Compton scatters; beta or EC de-cays from (2) 3H, (3) Ge, (4) Ga,(5) Zn, (6) Fe; (7-9) morphing parametersthat determine the number of events from the surface backgrounds as wellas the the surface background shapes (see discussion below for morphingparameter details); (10-12) parameters describing the signal efficiency; (13-18) parameters describing the detector resolution; (19) the Lindhard k value(see Sec. 2.1.2 for additional details); (20) the WIMP cross section. The first6 parameters are allowed to float freely in the fit. Parameters 7–19 are fitincluding constraint terms from prior information, and the nature of theseconstraints is described below. Parameter 20—the WIMP cross section—will sometimes be freely floating and sometimes be clamped to a specificcross section as is customary in the PLR method, and will be describedbelow.Of the 20 parameters listed above, 13 account for the systematic un-certainties. Explicitly these are the 3 parameters describing the signal effi-ciency, 6 parameters describing the resolution, 1 parameter (the Lindhardk parameter) describing the yield function, and the 3 morphing parametersthat take into account the systematics on the shape and normalization ofthe surface background.In addition to the 20 parameters, the likelihood function also containsbackground distributions and a signal distribution for a given WIMP mass.These distributions are plotted in Fig. 5.3. Figure 5.3 left shows the signaldistribution for 3 different values of the Lindhard k (the Lindhard model isdescribed in Sec. 2.5). Figure 5.3 right shows background distributions withno efficiency corrections applied, in units of differential rate (events/kg/day,or DRU), of the 9 different background components used in the likelihood fit.The rates are the central value background rates extrapolated to 0–2keVeefrom a fit to the backgrounds in the 5–25keVee range.1595.2. Likelihood and Limit SettingFigure 5.3: (left) The distribution expected (with detector efficiency ap-plied) from a 7 GeV/c2 DM particle for different values of the Lindhard kparameter. (right) Background distributions at the magnitude expected inthe Run 3 data (estimated prior to unblinding).Surface Background Morphing ParametersThe surface background components (Surface Germanium, Surface Hous-ing, and Surface Top Lid) deserve additional discussion because they aretreated differently than the non-surface backgrounds in two ways. First, thealpha-count study described in Sec. 5.1.5 provides prior information on therate of this background; we constrain the rate of the surface backgroundswith a Gaussian prior according to the alpha rates study. Second, as shownin Fig. 5.4, there is significantly larger uncertainty on the spectral shape ofthe surface backgrounds than on non-surface backgrounds. As explained inSec. 5.1.5, this is due to systematic uncertainties in the modeling of the volt-age cut and the variations in the voltage map for the surface backgrounds;the voltage cut and map modeling systematics do not arise for the otherbackgrounds which are more homogeneously distributed throughout the de-tector. Whereas the shapes of the non-surface backgrounds have only smallfreedom to change in the likelihood fit (due to efficiency and resolution pa-rameters), we allow the surface background shapes to shift significantly ifthe data pulls the surface distributions away from the mean distributions.This shape-shifting and rate-shifting is implemented with a “morphingparameter,” which acts as a fit parameter in the likelihood. With threecurves giving the mean, 1σ upper, and 1σ lower event densities of a back-ground component, the morphing parameter m is implemented as:ρ(E,m) ={ρmed(E) +m× (ρup(E)− ρmed(E)) m > 0ρmed(E) +m× (ρmed(E)− ρlow(E)) m < 0 (5.7)1605.2. Likelihood and Limit SettingFigure 5.4: (left) Uncertainty on the shape of the housing background.(right) The uncertainty on the shape and normalization of the housing back-ground, as controlled by a single morphing parameter that is given by a singlecolor.where m=0 results in the mean curve, and m=1 and m=−1 result in the1σ upper and 1σ lower curve respectively, and other values of m result in alinear combination of two of the curves. The ρlow, ρmed, and ρup variablesare shown in Fig. 5.4 left as the lower, middle, and upper curves. Figure 5.4right shows the effect of the morphing parameter, where the color legendis yellow: m=1, black: m=0, blue: m=−1. The event density spectra arenormalized to the expected number of events for that background (from thenormalization based on the alpha count study, discussed in Sec. 5.1.5). Forexample, for the Ge surface spectrum:∫ρGe(E) dE = NGe. (5.8)While the morphing parameters are designed to keep track of the uncer-tainty due to the voltage cut and voltage map systematics, we can addition-ally incorporate the uncertainty on the surface background normalizationdue to the alpha counts measurement. Incorporating this extra uncertaintyhas the effect of increasing the uncertainty on the morphing parameters.From the alpha study, the total number of events we expect, with no cutsapplied, from each surface component is NGe = 5.7 ± 1.5, NHS = 11.0 ±0.68, NTL = 28 ± 3.0 with correlations ρGe−HS = −0.35, ρGe−TL = −0.18,ρHS−TL = −0.19. The morphing parameters, because of their connection toan efficiency systematic uncertainty, are 100% correlated. For example, ifthe voltage cut from the surface background modeling procedure were to belooser than its approximated central value value by 1σ, then all the spectrawould shift up together. We use Monte Carlo techniques to combine the1615.2. Likelihood and Limit Settinguncertainty from the alpha study and the morphing parameters, and theresulting matrix describing the covariance between numbers of events fromthe germanium, housing, and top lid is:V = 8.9 28.6 2528.6 149 10725 107 111 . (5.9)With the uncertainties from the alpha study and the morphing parame-ters combined into a covariance matrix for the number of surface backgroundevents from different sources, we can propagate the correlated uncertainty tothe morphing parameters. Following standard error propagation rules, andwith variable names m1 = mGe, m2 = mHS , m3 = mTL; and N1 = NGe,N2 = NHS , N3 = NTL with a covariance matrix V between the three vari-ables, we can determine the covariance/variance between the variables, forexample between m1 and m2, as:Mm1,m2 =3∑i,j∂m1∂Ni∂m2∂Nj|x=µ × Vij , (5.10)where the relationship between a morphing parameter and a correspondingnumber of events is known and displayed in Fig. 5.4.This gives the following covariance matrix between morphing parame-ters:M = 1.24 0.89 0.910.89 1.04 0.870.91 0.87 1.06 . (5.11)With the morphing parameters as fit parameters in the likelihood, theyare constrained with a three-dimensional Gaussian whose variances/covariancesare given by the M matrix above. Explicitly, the constraint is given by:lnLM Constraint = −12[ 3∑i=13∑j=1M−1ij (mi − µmi)(mj − µmj )](5.12)where µm is the mean of the constraint and m are the fit parameters.5.2.2 Likelihood FunctionWe employ the unbinned extended maximum likelihood method to fit oursignal and background models to the data. The signal and background1625.2. Likelihood and Limit Settingmodels discussed above are incorporated as a product of three likelihoods:a term to allow the fitted number of signal and background events to havePoissonian fluctuations around the observed number of events, a term withthe signal and background distributions to discriminate between signal andbackgrounds, and a term that constrains certain parameters with prior in-formation:L = LPoiss. × Lenergy dist. × LConstr.. (5.13)The data input into the likelihood are the number of events passing allcuts, as well as the energies of those events. Taking the logarithm of likeli-hood, and dropping constant terms, which are unnecessary for maximizingthe likelihood, we obtain, for the LPoiss. and Lenergy dist. terms:ln(LPoiss.) + ln(Lenergy dist.) = −[νχ +∑bνb +∑sbνsb]+N∑i=1ln[νχfχ(Ei, ~n)+∑bνbfb(Ei, ~n)+∑sbρsb(Ei,msb, ~n)](5.14)The νχ is the expected number of events from the WIMP component,νsb are expected event numbers from the surface background components,and νb are expected event numbers from the non-surface backgrounds. Thef terms are PDFs for the energy of the WIMP distribution and the non-surface background distributions. The ρ terms are the event densities, as afunction of energy, of the surface backgrounds, which depend on the mor-phing parameters m. The ρ variables have normalization of events/energysuch that if they are integrated over energy the result is a number of eventscoming from that surface background, as described in Eq. 5.8. Therefore,the ρ terms have the same units as the νbfb terms in the same sum. The ~nis a vector of efficiency and resolution parameters that allow the shapes ofthe fb and ρ shapes to shift to within a prior determined range that is rep-resentative of the uncertainty of the efficiency and resolution. The iteratori is over the N events that are being fitted.The full constraint term in the likelihood is shown in Eq. 5.15. Be-cause the Lindhard k constraint is in 1 dimension it is instructive to step1635.2. Likelihood and Limit Settingthrough that constraint, and then apply analogously the same idea in higherdimensions when thinking about the other constraints. The Lindhard k isa parameter in the fit that can vary in order to maximize the likelihood,but this constraint applies a penalty (a negative contribution to the likeli-hood value) if k is different than the central value of the prior on k. Thiscentral value is given by µk = 0.157, as given by Lindhard theory appliedto germanium. For the form of the constraint, we use a Gaussian prior.Therefore, the magnitude of the penalty is determined by the uncertaintyon µk, which is given by σk, and is estimated from auxiliary measurementsof the ionization yield in germanium [73]. Because these measurements donot provide precise information about the NR ionization yield, particularlyat low energy, we use a weak constraint on k by choosing σk = 0.05.ln(LConstr.) =− (k − µk)22σ2k− 12[ 3∑i,j(ei − µei)E−1ij (ej − µej )]− 12[ 6∑i,j(ri − µri)R−1ij (rj − µrj )]− 12[ 3∑i,j(mi − µmi)M−1ij (mj − µmj )].(5.15)We constrain the three parameters describing the signal efficiency, ~e,with a 3D Gaussian prior using the results of Sec. 4.9. The center of the 3DGaussian is given by the best-fit values of the parameters ~µe, and its shapeis determined by the covariance matrix between best-fit values, given by E.We similarly constrain the resolution parameters, ~r, using the 6D Gaussianprior from the resolution model of Sec. 5.1.1, with best-fit resolution modelvalues of ~µr and covariance matrix R. Because the Run 3a and Run 3b de-tector resolutions were modeled independently, R contains zeros in elementslinking the two periods. The morphing parameters, ~m, which incorporatesystematics of the surface backgrounds, are constrained in the final term ofEq. 5.15. The expected values for the morphing parameters, ~µm, as well asthe covariance matrix (M) between them, determine the constraint.1645.2. Likelihood and Limit SettingVariable Constraint Parameter Boundarye (efficiency vector) 3D Gaussian All variables > 0m (morphing vector) 3D Gaussian All event densities > 0r (resolution vector) 6D Gaussian All variables > 0k (Lindhard) 1D Gaussian 0 < k < 0.3 (3σ from µk)σχ (DM cross section) No Constraint σχ > 0Table 5.6: Constrained or bound variables in the likelihood fit5.2.3 Limit-Setting ApproachFor a given WIMP mass we test the hypothesis that a WIMP signal withcross section σχ exists in the data. The PLR method allows us to calculatethe probability of observing a σχ signal strength when the hypothesis istrue. Then we scan over σχ to calculate the probability (p value) of differentsignal strength hypotheses, and then quote a cross section sensitivity whenthe probability of the signal hypothesis equals 0.1. This is equivalent toquoting a cross section when the signal hypothesis is rejected at the 90%confidence level (CL).The first step in the limit setting procedure is to compute the q statisticfor the dataset, for a certain signal strength hypothesis, σχ:qσχ ={−2lnλ(σχ) σˆχ < σχ0 σˆχ > σχ, (5.16)where λ is defined asλ(σχ) =L(σχ,ˆˆθ)L(σˆχ, θˆ) . (5.17)The numerator of λ(σχ) is the likelihood of a fit that has constrained thesignal component to the test hypothesis value σχ, andˆˆθ are the values ofthe nuisance parameters that maximize the likelihood given the constrainton σχ. The denominator of λ(σχ) is the likelihood with no constraints—thecross section σχ is permitted to float, along with the nuisance parameters,and the values that maximize the likelihood are labeled σˆχ and θˆ.To provide some sense for these statistical variables, λ(σχ) is a num-ber between 0 and 1, and λ(σχ)=1 corresponds to the most test-signal-like outcome—the best fit signal value of σχ equals the hypothesis value of1655.2. Likelihood and Limit SettingFigure 5.5: (left) Likelihood fit, to a pseudo-dataset, with WIMP crosssection freely floating. (right) A likelihood fit, to the same pseudo-dataset,with WIMP cross section clamped to σχ = 2.3×10−42cm2. On both plots,the best fit number of background events from the different backgrounds areshown in the legend, and the likelihood values are shown in the plot title.σχ. The λ(σχ)=1 case corresponds to qσχ=0. Signal hypotheses for whichσˆχ > σχ are compatible with the data when calculating upper limits. There-fore qσχ is set to 0 in these cases, which is the value that indicates the highestdegree of compatibility between the signal hypothesis and the data.Using this likelihood ratio statistic all parameters in the likelihood otherthan σχ (i.e. the systematic uncertainty parameters and the numbers ofbackground events) are profiled out as nuisance parameters by maximizingL as a function of these parameters with σχ held constant.An example of the signal-rejection power of the likelihood ratio test isdemonstrated in the following plot which shows fit results of backgroundmodels and a 5GeV/c2 DM signal to a pseudo-dataset. The left plot showsa fit representing the denominator of λ(σχ) with the WIMP cross sectionfloating, giving a lnL value of 2773.577. The right plot shows a fit rep-resenting the numerator of λ(σχ) with the WIMP cross section clamped to2.3×10−42 cm2, giving a lnL value of 2772.737. Therefore, λ(σχ) = ∆ lnL =−0.84 and qσχ = 1.68. Based on this qσχ value we can assign a rejectionconfidence level (CL) to a 5GeV/c2 DM particle with σχ=2.3×10−42 cm2existing in this data set. We now discuss how to calculate the rejection CL.After obtaining the qσχ,obs statistic for a data set, the probability ofobtaining that qσχ,obs value assuming the signal hypothesis is true (i.e. thata signal of size σχ exists in the data) is given bypσχ =∞∫qσχ,obsf(qσχ |σχ) dqσχ . (5.18)1665.2. Likelihood and Limit SettingFigure 5.6: An example qσχ,obs relative to a MC produced f(qσχ |σχ) distri-bution, as well as the theoretical (1/2)× (δ(0) + χ2) distribution.Here f(qσχ |σχ) is the probability distribution function of qσχ when the testhypothesis is true. We reject σχ at 90% confidence when pσχ = 0.1.Computing f(qσχ |σχ) requires significant amounts of computation be-cause the distribution of qσχ values must be approximated by simulating andfitting at least hundreds of pseudo-datasets containing background + signalcomponents. To avoid calculating f(qσχ |σχ), Wilks’ theorem [140] says thatthis distribution asymptotically approaches a mixture of a delta function atzero and a chi-square distribution with one degree of freedom, with the deltafunction δ(0) and the χ2dof=1 distribution each having a weight of 1/2. In thetext below we will refer to this mixture of distributions as (1/2)×(δ(0)+χ2).If it can be shown that this asymptotic behavior is realized for the CDMSliteR3 likelihood function then f(qσχ |σχ) = (1/2)× (δ(0)+χ2) can be assumed.We calculate f(qσχ |σχ) by fitting pseudo-datasets of signal + backgroundand confirm that these distributions converge to the distribution predictedby Wilks’ Theorem. One such of these checks is shown in Fig. 5.6, and sowe use the theoretical distribution in sensitivity and limit calculations.The CLs MethodA slight modification is made to the above formalism of setting a 90% CLlimit. This modification is used in order to protect against the possibilityof the profile likelihood method excluding a WIMP cross section to whichthe experiment is not sensitive in the case that the background statisti-cally fluctuates to a low number of events. The method, called the CLsmethod [141], gives a slightly higher signal cross section for the 90% exclu-sion than would otherwise be obtained, and in this sense it is conservative.1675.3. ResultsThe method adds an additional integral into the denominator of the p-valuethat is the probability of q to be larger than the observed test statistic underthe background-only hypothesis:pσχ,CLs =∞∫qσχ,obsf(qσχ |σχ) dqσχ∞∫qσχ,obsf(qσχ |σχ = 0) dqσχ(5.19)This has the effect of making pσχ,CLs greater than pσχ , which will requirea higher σχ hypothesis to be tested in order to reach pσχ=0.1 and excludethat hypothesis at 90% CL.5.3 ResultsThe best-fit value of σχ for the DM masses considered in this analysis isfound to be well below the experiment’s sensitivity, and so we choose to setan upper limit.5.3.1 Background Model Goodness of FitAfter unsalting the R3 data set, we test whether the background modellikelihood is a good fit to the data. The likelihood fitting procedure describedin Sec. 5.2 provides no information as to the goodness of fit (GOF) of themodel to the data, and therefore we define a procedure to evaluate the GOFthat outputs a probability (i.e. a p-value) for the data on the assumptionthat the model is correct. We use the Crame´r-von Mises GOF statistic [142]because it does not require binning of the data, overcomes some deficienciesof the more common KS test, yet is still relatively simple compared to somealternative GOF metrics. The Crame´r-von Mises statistic CVM is definedas:CVM = n∫ [Fn(x)− F (x)]2P (x)dx (5.20)where Fn(x) is the empirical cumulative distribution function (CDF) of thedata, F (x) is the CDF of the model’s fit to the data, and P (x) is the PDFof the model’s fit to the data. The n variable is the number of data points.Also, since our model is a function of energy, the x variable is energy. Alarger Crame´r-von Mises statistics corresponds to a worse fit.1685.3. ResultsWe use Monte Carlo to calculate the confidence level that the Crame´r-von Mises statistic indicates that the data comes from the backgroundmodel. MC distributions of the goodness-of-fit statistic are calculated usingfits to fake data sets that have been generated according to the backgroundmodel. This custom MC approach allows us to take into account the sys-tematic uncertainties in the background model. The steps of the procedureare:1. Fit the real data and determine best fit values and posterior distribu-tions for the model parameters.(a) This fit and the posterior distributions are shown in Fig. 5.10.2. Calculate the Crame´r-von Mises statistic for the fit to the real data.3. Generate 1000 pseudo-experiments based off the fit to the real data.(a) For the number of events from each background, sample from aPoissonian distribution with mean centered at the best fit valuesfrom step (1).(b) For the other nuisance parameters, sample from the posteriordistributions from step (1).(c) Fit and calculate the Crame´r-von Mises statistic for each pseudo-experiment4. Calculate the fraction of the pseudo-experiments’ Crame´r-von Misesstatistics that are larger than the one for the real data from step (2).This fraction is the p-value.Prior to unblinding, we agreed on a p-value threshold of 0.05, belowwhich we would investigate inaccuracies in the background model, abandonthe limit obtained with the profile likelihood method, and resort to the moreconservative optimum interval limit-setting technique.As shown in Fig. 5.7, the GOF procedure gives a p-value of 0.988 forthe hypothesis that the data comes from the background model, indicatinga particularly good fit. Two varieties of checks were performed to ensurethat the above procedure was unbiased. First, the procedure above wasrepeated but removing step (3b) and instead, when creating the pseudo-datasets, setting the systematic nuisance parameters to the best fit valueswe found in step (1). This changed the p-value only slightly, from 0.988 to0.986, indicating that the high p-value is not a result of an overestimation of1695.3. ResultsFigure 5.7: Crame´r-von Mises Statistic of the fit to the data (red line) and1000 pseudo experiments.our systematic uncertainties. Second, pseudo-experiments themselves weretested as if they were the real data set (i.e. they were fit with the likeli-hood and the Crame´r-von Mises statistic was determined for that fit, andsubsequently a set of pseudo-experiment based on that fit were created andfit, etc.). This procedure, carried out 7 times, did not show any bias in theobtained p-value scores, which were between 0.053 and 0.85.5.3.2 Exposure of SaltWith the analysis (cuts, background models, likelihood function) frozen,the dataset was unblinded by exposing the salt in the dataset. Followupchecks were performed on the salt and confirmed that it was inserted at theintended magnitude and with the intended energy and radial distribution,as shown in Fig. 5.8.The passage fraction of the salt as a function of energy from 0-2keVee wasexamined to check the expectation that the salt passage fraction matched,to within uncertainties, the estimated signal efficiency. The red error barsin Fig. 5.9 left show the salt passage fraction with all cuts (DQ efficiencycuts + multiples + muon veto + radial) applied. Relative to the estimatedsignal efficiency, the passage fraction is for salt is systematically lower. Themultiples cut is the largest reason for the reduced salt passage fraction. Themultiples cut and muon veto cut passage fraction of the salt is the largestdiscrepancy with our estimated signal efficiency, since both of these cuts wereestimated to be >98% efficient for signal, while they are ∼ 80% efficient forsalt.The reason why the multiples cut has a lower passage fraction of salt thanwe estimated for the signal is that the replaced events come from background1705.3. ResultsFigure 5.8: (left) The R3 final spectrum with all cuts applied, and with saltincluded and highlighted). (right) The distribution of salt (with no cutsapplied) highlighted in the energy vs. ξ radial parameter plane, with thereal data shown in the background.(i.e. α, β, γ recoils), and these sources are expected to pass the multiplescut with lower efficiency than a DM signal. This is explicitly shown inFig. 5.9 center, where the unsalted dataset is loaded and the events that werereplaced by salt are selected. 73/393 ' 19% of these events are identifiedby the multiples cut. When the waveform of these events is replaced with asalted waveform (i.e. a good low energy pulse) the event is still likely goingto be identified as a multiple because the waveform information on the otherdetectors is not changed in the salting algorithm. This 19% multiples rateis consistent with Fig. 5.9 left.Figure 5.9 right shows the salt passage fraction with just the DQ effi-ciency cuts and radial cut applied to the salt in pink. Omitting the multiplesand muon veto cuts removes the systematic 1.5–2σ disagreements. This isa confirmation that the salting procedure was successful, as it assumed adistribution in event ntuple parameter space that resembled a DM signal.5.3.3 Final Spectrum FitThe final Run 3 spectrum after application of all selection cuts is shownin Fig. 5.10. The main features are the 71Ge electron-capture L- and M -shell peaks at 1.30 and 0.16 keVee respectively. Events contributed frombackgrounds other than 71Ge exist between the peaks and are well mod-eled. We do not observe a population of events below the M shell, whichis consistent with the steep decrease of the signal efficiency in this rangeand consistent with the expectations from the background model. Poste-riors for the number of events contributing each background are shown in1715.3. ResultsFigure 5.9: (left) The salt passage fraction with sequential application ofcuts (shown in the legend) compared to the signal efficiency estimate. (cen-ter) The energy spectrum of events, in the unsalted dataset, selected to bereplaced by salt events. Roughly 19% of them were multiples, consistentwith the passage fraction given by the left hand figure. (right) The saltpassage fraction, but without the multiples and muon veto cut.Fig. 5.10 right. In the figure axes labels, m1, m2, m3 corresponds to themorphing parameters for the Ge, Housing, and Top Lid surface backgrounds.As expected, a strong covariance is observed between the Compton and 3Hbackground components, which in this energy range do not contain suffi-ciently distinct spectral features to remove their degeneracy in the fit. Thesurface background components are strongly correlated through the priorconstraint covariance matrix, M, described in Sec. 5.2.We find that the surface background component covariances from thelikelihood fit match the prior constraint covariances, shown in the top rowof Fig. 5.11, indicating that these 0.07–2.0 keVee data do not provide anyadditional information on the surface background. Figure 5.11 also showsthat all of the systematic uncertainty parameters that were constrained inthe likelihood (i.e. resolution and efficiency parameters) are within the rangeto which they were constrained by the prior. Additionally, the posteriorshave similar widths as the priors indicating that the data are not providingadditional information for these parameters.5.3.4 Background RatesWe calculate the average background rates of single-scatter events betweenthe 71Ge peaks, corrected for efficiency, as shown in Table 5.7. The Run 3background rate is higher than Run 2, in line with the expectation of thehigher surface background for an endcap detector (T2Z1) relative to T5Z2.T5Z2 was surrounded on both of its faces by the two adjacent detectors,whereas T2Z1 was the highest detector in the tower and therefore had one1725.3. ResultsFigure 5.10: (top) Background best fit amplitudes overlaid on the Run 3unsalted data. “Sf” is an abbreviation for “surface”. (bottom) 1D and 2Dposterior distributions for the number of events (or morphing parameter inthe case of surface backgrounds) contributing from each background. Byposterior, we do not mean to imply that these are the results of a Bayesiananalysis; rather, these are the fit results from the likelihood, with asymmet-ric uncertainties determined with a Markov-chain Monte Carlo samplingalgorithm.1735.3. ResultsFigure 5.11: Comparing prior constraints with fit results of the maximumlikelihood for the efficiency, morphing, and resolution parameters. The sim-ilar variance of the prior and posteriors indicates that the data are notproviding additional information for these systematic uncertainty parame-ters, and the similar mean indicates that our signal efficiency and resolutionmodel parameters are in agreement with the data.1745.3. ResultsRange Run 2 Rate Run 3 Rate[keVee] [keVee kg d]−1 [keVee kg d]−10.2–1.2 1.09± 0.18 1.9± 0.31.4–10 1.00± 0.06 1.3± 0.111–20 0.30± 0.03 0.71± 0.07Table 5.7: Average single-scatter event rates for energy regions between theactivation lines in Run 2 and Run 3, corrected for efficiency. All errorscontain ±√N Poissonian uncertainties, and the lowest energy range valuesadditionally include uncertainty from the signal efficiency.face exposed to the top lid copper. It is also expected from T2Z1’s positionin the tower that the ability to identify and remove multiple scatters isdiminished. Additionally, Run 3 had fewer other active detectors in the fullarray relative to Run 2. Since the active detectors are used to identify themultiple scatter events, this factor would also reduce the efficiency of theRun 3 multiples cut. Therefore, a higher fraction of multiple scatter eventscould be passing the multiples cut and contributing to the event counts inthe table.5.3.5 Sensitivity EstimatePrior to unsalting the data, we calculated the 90% CL sensitivity of theRun 3 analysis to a DM signal based on projected background rates in thisanalysis’s energy region of interest (ROI), 0.07–2.0 keVee. The sensitivitycalculation also uses the likelihood framework presented in Sec. 5.2. To esti-mate the background rates in the ROI, we measure them in the 5–25 keVeerange and extrapolate the rates to lower energy. We choose 5 keVee becausesalt was not inserted above this energy and because the DM signal con-tribution above this energy for DM masses < 10 GeV/c2 is expected to benegligible. Also, because 5 keVee is below the lowest K-shell energy of theEC isotopes considered, all background components are constrained in thisrange. We perform a maximum likelihood fit, using the likelihood definedin Eq. 5.13 but without the DM signal. We also omit the resolution and ef-ficiency systematic uncertainties because those extra terms are unnecessarywhen fitting the 5–25 keVee background spectrum. This fit provides best-fitvalues of, as well as covariances between, background rates in the 5–25 keVeerange for the nine background components. The expected background in theROI can directly be calculated from the best fit in the 5–25 keVee range.1755.3. ResultsThe uncertainty is determined from the covariance matrix of the fit.Background-only pseudo-experiments are then generated by samplingfrom the nine different background distributions. The number of eventsthrown for each background component is randomized, first by samplingfrom the 9D Gaussian distribution provided by the 5–25 keVee maximumlikelihood fit and second by adding a Poissonian fluctuation to the sampledvalue. The 90% CL PLR limit, using the CLs technique, is calculated for500 of these pseudo-experiments, and the resulting ± 1σ bands are shownas the red band in Fig. 5.12.In addition to determining parameters for generating the pseudo-experiments,the 5–25 keVee fit provides constraints on the surface background morphingparameters (the µmi of Eq. 5.15). While this fit used a prior constraintcentered at 0 for all morphing parameters, the respective posteriors peakedat −0.19, −0.2, and −0.25 for the germanium, top lid, and housing surfacebackground locations respectively. This indicates a slightly lower surfacebackground rate than predicted by the alpha decay study. The 5–25 keVeefit also slightly reduced the uncertainty on the morphing parameters, asgiven in the following covariance matrix:M = 0.570 0.419 0.4200.419 0.572 0.3890.420 0.389 0.485 . (5.21)These updated central values for the constraint were used in the likelihoodfor both the sensitivity estimate and the final limit, along with the updatedcovariance matrix for the morphing parameters.5.3.6 DM LimitFigure 5.13 shows the final CDMSlite Run 3 limit calculated with the spec-trum in Fig. 5.10. From 2.5–10 GeV/c2 we find a factor of 2–3 improvementin the excluded DM-nucleon cross section over the CDMSlite Run 2 optimuminterval analysis [143]. This improvement is achieved despite the smallerexposure (36 vs. 70 kg-days) and higher background rate in Run 3, demon-strating the discrimination power of the PLR method. Below 2.5 GeV/c2,we exclude little to no additional parameter space because the effective en-ergy threshold for this analysis is slightly higher than that for CDMSliteRun 2.1765.3. ResultsFigure 5.12: (left) R3 1σ sensitivity band and PLR limit (with R2 OI limitoverlaid)Figure 5.13: The CDMSlite Run 3 90% CL PLR limit (this result, solidblack) on the spin-independent WIMP-nucleon cross section, along withthe ± 1σ and ± 2σ sensitivity bands (green and yellow respectively). TheCDMSlite Run 3 optimum interval limit (dashed grey) and Run 2 (red) op-timum interval limit [143] are overlaid. Examples of limits from other detec-tor technologies are overlaid: DarkSide-50 2018 No Quenching Fluctuations(magenta) [84]; PandaX-II 2016 (blue) [43]; PICO-60 2017 (orange) [144];CRESST-II 2016 (cyan) [94]; CDEX-10 2018 (purple) [145].177Chapter 6Conclusion and FutureOutlookThe ability of the CDMSlite operating mode to improve the SuperCDMSdetectors’ sensitivity to low mass DM is clear from the exclusion limits inFig. 5.13. The CDMSlite Run 1 result first demonstrated the promise ofthe Neganov-Trofimov-Luke (NTL) effect to lower the detector threshold,while CDMSlite Run 2 used an improved understanding of the detectornoise environment and electric field to reduce backgrounds and further lowerthe detector threshold, producing world-leading sensitivity to DM particlesbetween 2 and 6 GeV/c2. Then in this thesis, the CDMSlite approach wastaken a step further.In Chapter 4 and Chapter 5, we introduced new analysis methods thatmoved CDMSlite from an exclusion-only analysis to one with discoverypotential, in addition to improving the low-mass sensitivity of CDMSlite.Specifically, the results demonstrate successful modeling of radioactive back-grounds in CDMSlite detectors down to low energies, as well as the dis-crimination power of a profile likelihood fit to set strong limits on a DMsignal even in the presence of irreducible backgrounds. A number of anal-ysis developments enabled this approach, including improved rejection ofinstrumental backgrounds using detector-detector correlations in a boosteddecision tree, removal of events at high radii with misreconstructed energiesby an improved fiducial volume cut, and Monte Carlo modeling of surfacebackgrounds in the detector.The SuperCDMS collaboration is currently constructing a new experi-ment, SuperCDMS SNOLAB, which will use the NTL effect to reach lowerthresholds in detectors designed specifically for high-voltage operation [97,146]. The analysis developments in the CDMSlite Run 3 analysis, in particu-lar the profile likelihood approach, are planned for future SNOLAB analyses.Because our use of the profile likelihood method for CDMSlite Run 3 wasthe first time it had been used in a search for DM in a CDMSlite detector,our results show that employment of such a method is possible even whenthe analysis is pushing to the limits of the detector threshold.178Chapter 6. Conclusion and Future OutlookIn addition to the CDMSlite Run 3 analysis, the detector characteriza-tion results presented in Chapter 3 provided important information for theprojected sensitivity of the SuperCDMS SNOLAB detectors. Namely, theimproved baseline resolution measured on prototype lower transition tem-perature (Tc) detectors confirmed resolution extrapolations for the lower Tcdetectors being fabricated for SuperCDMS SNOLAB. The number of detec-tors, as well as their projected resolution, is given in Table 6.1.The projected DM sensitivity of the SuperCDMS SNOLAB experimentcan be calculated using the values from Table 6.1 as well as the backgroundrate projections detailed in Ref. [97]. By moving to the deeper laboratoryat SNOLAB, the flux from cosmogenic muons and cosmogenically producedneutrons will be reduced by approximately 2.5 orders of magnitude relativeto Soudan. The neutron background is not expected to be significant relativeto electron recoil backgrounds, and the decays from 3H are expected tocontribute the highest rate in Ge detectors. The projected sensitivities,calculated using the conservative optimum interval method, are shown inFig. 6.1.Looking towards the future of DM direct detection, the dotted yellowline in Fig. 6.1 is the region of parameter space where the solar neutrinocoherent elastic scatter rate is expected to begin to mask the interactionrate from DM particles. These nuclear recoil events would look like DMevents. Of course it is possible that particle DM is discovered before thiscoherent neutrino scattering “floor” is reached, but if it is not then thecoherent neutrino scattering will constitute a new, difficult background todiscriminate against. Pushing down to this floor will mark an importantbenchmark for direct detection, and pushing past it will require innovativenew detector designs.179Chapter 6. Conclusion and Future OutlookGe (iZIP) Si (iZIP) Ge (HV) Si (HV)Number of detectors 10 2 8 4Total exposure (kg·yr) 56 4.8 44 9.6Phonon resolution (eV) 50 25 10 5Ionization resolution (eV) 100 110 - -Voltage Bias (V) 6 8 100 100Table 6.1: The projected exposures and detector parameters for the fourtypes of SNOLAB detectors: Ge iZIP, Si iZIP, Ge HV, Si HV. The exposuresare based on 5 years of operation with 80% live time. The HV detectors donot have ionization sensors. The arrangement will be 4 towers of 6 detectorseach. Table reproduced from Ref. [97].Figure 6.1: The dashed lines show the projected 90% exclusion sensitivityof the SuperSCDMS SNOLAB experiment, calculated using the optimuminterval method. The y axis is the spin-independent WIMP-nucleon crosssection. The solid lines show the current 90% from the CRESST-II [94], Su-perCDMS [126, 143], and LUX [147] experiments. 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PhD thesis, University of California, Berkeley, 2016.URL http://cdms.berkeley.edu/Dissertations/phipps.pdf.210Appendix AVoltage Scan CalibrationResultsA.1 LinearityCalibration studies from CDMSlite Run 1 and Run 2 indicated that theenergy scale of the CDMSlite detectors was more complex than the simpleNeganov-Trofimov-Luke model (Eq. 2.9) and that the total phonon amplifi-cation was not exactly linear as a function of bias voltage, Vb. These studiesalso indicated that the ionization and phonon collection of the CDMSlitedetectors differed from the iZIP detectors and therefore the energy scalediffered between the two detector configurations.Therefore, prior to the CDMSlite Run 3 analysis, we performed a ded-icated study of the CDMSlite detectors’ energy scale as a function of biasvoltage. The 10.4 keVee and 1.3 keVee ER events were used to calibrate theenergy scale of both the T2Z1 and T5Z2 detectors at a variety of voltages.When first processing the data, the calibration used was the iZIP calibra-tion which was determined with the detector at 4V. Fig. 2.14 and A.1 showthe reconstructed energy of the 10.4 keVee and 1.3 keVee peaks for T5Z2detector voltages of 0V, 30V, 50V, and 70V.A.2 Significance of Calibration SystematicsThe bias scan data can be better explained with slight modifications to sim-ple Luke gain scaling assumption: Ept = Er(1 +eVb ). If there is suppressedcollection of recombination phonons in CDMSlite, but we still assume 100%collection of recombination phonons, how does this affect the ER and NRenergy scale? If there is impact ionization10 that is ignored, how does this10Impact ionization is a process by which e−/h+ pairs in the detectors can ionize ad-ditional e−/h+ pairs, thereby creating additional Luke phonons with an energy that isnon-linear as a function of the voltage across the detector. While this is a second-orderprocess, as seen by the fact that the total phonon energy scale is to first order linear with211A.2. Significance of Calibration SystematicsFigure A.1: (left) The zoomed-in version of the left hand plot highlights thediscrepancy between the expected, by extrapolation, location of the K-shellpeak at 0 V. (right) The location of the K- and L- shell peaks as a function ofdetector voltage. The higher voltage peaks (30 V, 50 V, and 70 V) are linearto within uncertainty, but there is a deviation from linearity at 0 V. Theoverall energy scale (y axis) has used a calibration constant for the detectorobtained at 4 V, though the overall scaling of the y axis is irrelevant whenchecking for linearity.affect the ER and NR energy scale? We now introduce a few variablesto model these effects. The true recombination phonon collection fractionis given by fRC,T . The true Luke phonon collection fraction is given byfL,T . Fortunately, neglecting the two effects above will have no effect onthe electron recoil energy scale, as discussed below. In the NR energy scale,neglecting the effects will only introduce a slight bias. The plots in Fig. A.2show the ratio between the measured NR energy, Enr,M , and the true NR en-ergy, Enr,T , as a function of fRC,T and fL,T . We used an example ionizationyield of Y (Er) = 0.2 and assumed  = 3 eV.Electron RecoilsDue to the ratio of Egap = 0.79eV to Ecreate = 3eV in Ge, 74% of anelectron recoil’s energy, Er, goes into primary phonons. The other 26% ofenergy goes into ionization, which either recombines immediately, emittingthe energy as phonons (when Vb = 0), or drifts across the crystal emittingenergy as Luke phonons (Eluke = neheVb), until the ionization recombinesthe voltage, it has been observed in CDMS style detectors[149].212A.2. Significance of Calibration SystematicsFigure A.2: Large systematics in Luke phonon collection (15%) and recom-bination phonon collection (50%) would contribute at most to a ∼3% errorin the nuclear recoil energy scale. The recombination phonon error becomesnegligible at high detector bias. Also the error decreases at higher detectorvoltage, Vb, as the total phonon energy becomes more correlated with theionization.213A.2. Significance of Calibration Systematicsat the electrodes and emits the recombination energy as phonons back intothe crystal.The bias scan data can be better explained with slight modifications tothe above process. One explanation is that recombination phonons are neverbeing collected. At 0V charges could be trapping and never recombining.At higher voltages it’s possible that recombination phonons are not beingcollected because (1) phonons created at the electrodes aren’t emitted backinto the crystal or (2) perhaps the charges trap near the surface and neverrecombine. We can treat this as a recombination phonon suppression, fRC,T ,which is < 1.An equally consistent explanation of the bias scan data is that impactionization is occurring at the higher voltages, where drifting ionization de-neutralizes an impurity, creating more ionization and therefore excess Lukephonons. We can treat this as a preferential collection of Luke phonons inthe energy scaling, where fL, T > 1.These effects modify the total phonon energy scale. With full recombi-nation phonon collection and without impact ionization the total phononenergy scale becomesEpt = Er(1 +eVb) (A.1)while with suppressed recombination phonon collection and impact ioniza-tion it becomesEpt = Er(0.74 + 0.26× fRC + fL eVb) (A.2)At a certain voltage Vb, we measure a calibration constant, aM , thatscales an ADC value to the total phonon energy, Ept. aM is typically esti-mated given an electron recoil Er of known energy and assuming the energyscale of Eq. A.1:Ept,M = aMADC, (A.3)aM =10keV(1 + eVb )ADC|10keV . (A.4)If Eq. A.1 is an incorrect energy scale, and instead the true energy scaleis given by Eq. A.2, then the true calibration constant, aT , would be differ-ent than aM (making Ept,M incorrect). The true calibration is denoted by214A.2. Significance of Calibration Systematicssubscript T :aT =10keV(0.74 + 0.26× fRC + fL eVb )ADC|10keV . (A.5)However, when converting from the total phonon energy scale to theelectron recoil scale, if a mistake is done the calibration, that mistake isundone and the correct electron recoil energy is still obtained. Below wecompare the measured (“M”) electron recoil to the true (“T”) electron recoiland see that they are equivalent:Er,M =Ept,M1 + eVb=aMADC1 + eVb=10keVADC|10keV ADC(A.6)vs.Er,T =Ept,T(0.74 + 0.26× fRC + fL eVb )=aTADC(0.74 + 0.26× fRC + fL eVb )=10keVADC|10keV ADC.(A.7)The bias in the Ept,M energy scale, is “calibrated out,” and does notintroduce a bias in the ER energy scale.Nuclear RecoilsFor the nuclear recoil energy scale a small (∼1%) systematic is introducedwhen a wrong assumption is made about recombination phonon collectionor impact ionization, which does not cancel out, because the CDMSlitecalibration is always done using electron recoils. The full recombinationphonon collection, without impact ionization, energy scale is given byEpt = Enr(1 + YeVb) (A.8)215A.2. Significance of Calibration Systematicswhile suppressed recombination phonon collection efficiency and impact ion-ization is given byEpt = Enr[(1− Y (0.26)) + fRCY (0.26) + fLY eVb]. (A.9)When converting from total phonon energy to nuclear recoil energy, nowthe presence of the ionization yield factor Y (Er) creates the small systematicbias in the measured nuclear recoil Enr,M vs. the true nuclear recoil Enr,T .Enr,M =Ept,M1 + Y eVb=aM ×ADC1 + Y eVb=11 + Y eVb10keV(1 + eVb )ADC|10keV ×ADC(A.10)Enr,T =aT ×ADC(1− Y (0.26)) + fRCY (0.26) + fLY eVb=1(1− Y (0.26)) + fRCY (0.26) + fLY eVb10keV(0.76 + fLeVb )ADC|10keV ×ADC(A.11)The aM and aT are the calibrations from the Electron Recoil section above.The ratio of the measured NR energy to the true NR energy is given by:Enr,MEnr,T=[(1− Y (0.26)) + fRCY (0.26) + fLY eVb][1 + eVb][1 + Y eVb][(0.74 + 0.26× fRC + fL eVb )] . (A.12)The dependence of this ratio on fL and fRC is shown in the contour plots inFig. A.2 at the beginning of this appendix. It shows that not accounting forsuppressed recombination phonon collection and not accounting for impactionization has a small effect on the NR energy scale.Three different cases are shown explicitly here. We use a nuclear recoilwith Y=0.2, a detector voltage bias of 70V, and assume  = 3eV.1. With no impact ionization, fL = 1, but with fRC = 0, the measuredrecoil energy is only 0.2% larger than the true recoil.2. With fL = 1.1 (10% impact ionization), and with fRC = 0 the mea-sured recoil energy is 0.99% smaller than the true energy.3. With fL = 1.1 and with fRC = 1, the measured recoil energy is 1.2%smaller than the true energy.216

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