Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Crook, Daniel Weston Mckay
Master of Science - MSc
University of British Columbia
The nature of dark matter is one of the greatest extant mysteries in physics. Many hypotheses have been proposed to explain dark matter, but none have ever been substantiated. Some of these hypotheses propose dark sectors, i.e., entire families of dark matter particles. One such hypothesis is that of the dark photon, a massive, gauge boson which would be the mediator of the dark electromagnetic force. The dark photon would couple feebly to the photon, and thereby be producible through e⁺e⁻ annihilation at a particle collider. Described here are contributions toward a search for the dark photon at Belle II, an electron-positron collider experiment in Japan. This search focuses on invisible decays of the dark photon, with a single initial-state radiation photon serving as the only evidence of the production and decay. Due to the delicate nature of this event signature, the dark photon search requires a carefully constructed set of event requirements and vetoes that remove as much background as possible without removing a large amount of the signal. To further this goal, two control samples were designed. One control sample is of a clean radiative dimuon sample, used as a signal proxy, while the other is of the radiative Bhabha background. These control samples are then used to design vetoes and selection criteria for the search for the dark photon at Belle II, termed the single-photon analysis.
Signal proxy and radiative Bhabha background studies:contributions to the search for invisibly decaying darkphotons at Belle IIbyDaniel Weston Mckay CrookB.Sc., McGill University, 2021A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)August 2023© Daniel Weston Mckay Crook, 2023The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Signal proxy and radiative Bhabha background studies: contributionsto the search for invisibly decaying dark photons at Belle IIsubmitted by Daniel Weston Mckay Crook in partial fulfillment of the require-ments for the degree of Master of Science in Physics.Examining Committee:Christopher Hearty, Professor, Physics and Astronomy, UBCSupervisorJanis McKenna, Professor, Physics and Astronomy, UBCSupervisory Committee MemberiiAbstractThe nature of dark matter is one of the greatest extant mysteries in physics. Manyhypotheses have been proposed to explain dark matter, but none have ever beensubstantiated. Some of these hypotheses propose dark sectors, i.e., entire familiesof dark matter particles. One such hypothesis is that of the dark photon, a massive,gauge boson which would be the mediator of the dark electromagnetic force. Thedark photon would couple feebly to the photon, and thereby be producible throughe+e− annihilation at a particle collider. Described here are contributions toward asearch for the dark photon at Belle II, an electron-positron collider experiment inJapan. This search focuses on invisible decays of the dark photon, with a singleinitial-state radiation photon serving as the only evidence of the production anddecay. Due to the delicate nature of this event signature, the dark photon searchrequires a carefully constructed set of event requirements and vetoes that removeas much background as possible without removing a large amount of the signal.To further this goal, two control samples were designed. One control sample isof a clean radiative dimuon sample, used as a signal proxy, while the other is ofthe radiative Bhabha background. These control samples are then used to designvetoes and selection criteria for the search for the dark photon at Belle II, termedthe single-photon analysis.iiiLay SummaryThrough many astronomical observations, physicists have shown that the majorityof mass in our universe is not ordinary, visible matter. The majority of mass inour universe appears to be invisible, meaning that it does not interact with light;this mass is thus called dark matter. While decades of physics have produced andtested a model that describes regular matter, it doesn’t incorporate dark matter atall, and there is very little understanding of what dark matter is. As one of thegreatest mysteries in physics, an empirical explanation of dark matter could revo-lutionize the field. This thesis presents part of an analysis that uses data from theBelle II particle collider experiment to search for a potential form of dark matter, ahypothetical particle called the dark photon.ivPrefaceThis thesis is based on data of the Belle II experiment, which is a large, interna-tional collaboration. The result is original and as yet unpublished.The single photon analysis is performed by a collaboration of groups at twouniversities. At UBC, this group consists of my supervisor Prof. Christopher Hearty,and Dr. Michael De Nuccio. At Karlsruhe Institute of Technology, this is Prof. Tor-ben Ferber, and Dr. Giacomo De Pietro. Miho Wakai and Dr. Sam Cunliffe havepreviously worked on this analysis.The design of the analysis was done by the aforementioned group at Belle II.The sections of work presented in the analysis section are entirely the author’sown, as are the figures in that section. The data and Monte Carlo events used inthis thesis were produced by the Belle II collaboration. The beams used by Belle IIwere produced by the SuperKEKB accelerator. The data used here was collectedfrom 2019 to 2022 by the Belle II collaboration, a process in which the author ofthis thesis was not involved.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Dark matter and evidence for its existence . . . . . . . . . . . . . 11.2 Dark sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Searching for dark matter at particle colliders . . . . . . . . . . . 62 The Belle II detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.1 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 14vi2.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.1 Track reconstruction . . . . . . . . . . . . . . . . . . . . 162.3 K0L and muon detector . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Building KLM clusters . . . . . . . . . . . . . . . . . . . 172.3.2 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Particle identification . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Triggers and data acquisition . . . . . . . . . . . . . . . . . . . . 192.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 The dark photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Production and decay . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Parameters and current exclusion limits . . . . . . . . . . . . . . 254 Analysis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Structure of the analysis . . . . . . . . . . . . . . . . . . . . . . . 294.3.1 Study of signal . . . . . . . . . . . . . . . . . . . . . . . 294.3.2 Study of backgrounds and veto construction . . . . . . . . 304.3.3 Data-MC corrections . . . . . . . . . . . . . . . . . . . . 304.3.4 Fitting procedures and signal extraction . . . . . . . . . . 304.3.5 Systematic uncertainty estimation, unblinding, and settingupper limits . . . . . . . . . . . . . . . . . . . . . . . . . 315 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.1 Datasets used . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1.2 MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Radiative dimuon control Sample Selection Criteria . . . . . . . . 345.2.1 Muon requirements . . . . . . . . . . . . . . . . . . . . . 345.2.2 Event requirements . . . . . . . . . . . . . . . . . . . . . 355.2.3 Photon requirements . . . . . . . . . . . . . . . . . . . . 375.2.4 Selection summary and control sample plots . . . . . . . . 46vii5.2.5 Selection of KLM clusters, extra photons, and tracks . . . 485.3 KLM vetoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.1 Number of KLM layers . . . . . . . . . . . . . . . . . . . 605.3.2 KLM timing . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.3 Innermost KLM layer . . . . . . . . . . . . . . . . . . . . 655.3.4 Angular separation from the single photon . . . . . . . . . 675.4 ECL cluster vetoes . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4.1 Neutral ECL clusters . . . . . . . . . . . . . . . . . . . . 705.4.2 Charged ECL clusters . . . . . . . . . . . . . . . . . . . . 765.5 Single photon requirements and vetoes from elsewhere . . . . . . 785.6 Radiative Bhabha control sample . . . . . . . . . . . . . . . . . . 805.7 Track Vetoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.8 Veto Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 966 Conclusion and prospects . . . . . . . . . . . . . . . . . . . . . . . . 99Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101A Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.1 Trigger lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105viiiList of TablesTable 5.1 Summary of the amount of data used, and at what collision en-ergy it was taken . . . . . . . . . . . . . . . . . . . . . . . . . 33Table 5.2 Radiative dimuon control sample criteria . . . . . . . . . . . . 46Table 5.3 Summary of the extra object selection criteria for the radiativedimuon control sample . . . . . . . . . . . . . . . . . . . . . 59Table 5.4 Summary of the KLM vetoes . . . . . . . . . . . . . . . . . . . 63Table 5.5 Summary of the photon vetoes . . . . . . . . . . . . . . . . . 73Table 5.6 Summary of the selection criteria for the radiative Bhabha con-trol sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Table 5.7 Summary of the skim track veto. All three criteria must be sat-isfied to veto the event. . . . . . . . . . . . . . . . . . . . . . 88Table 5.8 Summary of vetoes for the single-photon analysis with MC anddata veto percentages . . . . . . . . . . . . . . . . . . . . . . 97ixList of FiguresFigure 1.1 Rotation curve for the spiral galaxy NGC 3198 . . . . . . . . 3Figure 1.2 The Bullet Cluster with visible and invisible mass centres high-lighted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 2.1 Cross section diagram of Belle II . . . . . . . . . . . . . . . . 9Figure 2.2 The Belle II coordinate system overlaid on the detector . . . . 10Figure 2.3 ECL crystal layout with thetaID labels . . . . . . . . . . . . . 13Figure 3.1 Exclusion limits for the invisibly decaying dark photon . . . . 26Figure 4.1 Feynman diagram for the production and decay channel of thedark photon . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 5.1 Lµ for both muons in the radiative dimuon control sample, forboth MC and data . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 5.2 The square of the mass recoiling against the dimuon systemdistribution for the radiative dimuon control sample . . . . . . 37Figure 5.3 Recoil momentum against the dimuon system for the radiativedimuon control sample . . . . . . . . . . . . . . . . . . . . . 38Figure 5.4 Ratio of the recoil photon momentum and recoil momentum asa function of the angle between the recoil momentum and therecoil photon . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 5.5 A crop of the plots of ratio of the recoil photon momentum andrecoil momentum as a function of the angle between the recoilmomentum and the recoil photon . . . . . . . . . . . . . . . . 40xFigure 5.6 Ratio of the recoil photon momentum to the recoil momentumas a function of the 3D angle between the recoil photon andthe most energetic extra photon with a lab energy of at least250MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 5.7 Adjusted measured photon energy divided by the recoil mo-mentum of the two muons . . . . . . . . . . . . . . . . . . . 43Figure 5.8 Distribution of the lab frame θ of the recoil photon for theradiative dimuon control sample . . . . . . . . . . . . . . . . 44Figure 5.9 Angle between the recoil photon and the closer of the twomuon tracks, measured at the ECL . . . . . . . . . . . . . . . 45Figure 5.10 ZMVA as a function of the 3D angle between the recoil photonand the nearest muon, as measured at the ECL . . . . . . . . . 47Figure 5.11 E∗ as a function of the thetaID for the recoil photon for dataand MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 5.12 Recoil mass squared as a function of the thetaID for the recoilphoton for data and MC . . . . . . . . . . . . . . . . . . . . . 50Figure 5.13 θ angular difference between a KLM cluster and the closermuon plotted for data and MC . . . . . . . . . . . . . . . . . 51Figure 5.14 nK as a function of the θ difference between the KLM clusterand the closer muon . . . . . . . . . . . . . . . . . . . . . . 53Figure 5.15 nK as a function of the φ difference between the KLM clusterand the closer muon . . . . . . . . . . . . . . . . . . . . . . 54Figure 5.16 θ and φ angular separation between the extra photons and theclosest muon of the pair for data and MC . . . . . . . . . . . . 56Figure 5.17 Photon timing for extra photons with angular requirements . . 57Figure 5.18 nK for KLM clusters close to the recoil photon . . . . . . . . . 61Figure 5.19 nK for the radiative dimuon control sample . . . . . . . . . . 62Figure 5.20 KLM cluster timing for 1- and 2-layer clusters . . . . . . . . . 64Figure 5.21 The innermost KLM layer for 1- and 2-layer KLM clusters . . . 66Figure 5.22 3D angle between the recoil photon and KLM clusters for 1-and 2-layer clusters . . . . . . . . . . . . . . . . . . . . . . . 69Figure 5.23 Energy of the extra photons for the radiative dimuon controlsample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71xiFigure 5.24 Energy of the extra photons for the radiative dimuon controlsample split by energy category . . . . . . . . . . . . . . . . 72Figure 5.25 Timing information as a function of the lab energy for the extraphotons in the radiative dimuon control sample . . . . . . . . 74Figure 5.26 The 3D angle between the extra and recoil photons as a func-tion of the extra photon energy for the radiative dimuon controlsample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Figure 5.27 Percentage of the radiative dimuon control sample vetoed bythe mid energy ECL veto as function of the veto requirements . 77Figure 5.28 Highest Et distribution for the radiative dimuon control sample 78Figure 5.29 The distribution of the largest Et in each event for the radiativeBhabha control sample . . . . . . . . . . . . . . . . . . . . . 81Figure 5.30 nOOT for the radiative Bhabha control sample . . . . . . . . . 82Figure 5.31 Angular proximity of single-photon clusters to MC electronsand photons for the radiative Bhabha control sample . . . . . 85Figure 5.32 θ and φ for the MC electrons in the radiative Bhabha controlsample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Figure 5.33 E∗ as a function of the thetaID for the single photon for theradiative Bhabha control sample . . . . . . . . . . . . . . . . 87Figure 5.34 Square of the mass recoiling against the single photon as afunction of the thetaID for the single photon for the radiativeBhabha control sample . . . . . . . . . . . . . . . . . . . . . 88Figure 5.35 ROC curves for the possible track vetoes, using the radiativedimuon control sample as a proxy for signal . . . . . . . . . . 90Figure 5.36 Momentum as a function of |dz| for tracks in both control samples 91Figure 5.37 Percentage of the sample vetoed as a function of the momen-tum bound of the track veto for both control samples . . . . . 93Figure 5.38 E∗ as a function of the thetaID for the single photon for theradiative Bhabha control sample with the track veto applied . 95Figure 5.39 Square of the mass recoiling against the single photon as afunction of the thetaID for the single photon for the radiativeBhabha control sample with the track veto applied . . . . . . 95xiiList of Symbolsα Fine structure constantαµ Angle between the recoil photon and the closer muon, measured at the ECLαD Dark fine structure constantαE Angle between an extra photon and the single photonαK Angle between a KLM cluster and the single photonαR Angle between the recoil photon and recoil momentum vectorsγ Photonε Strength of mixingθ Polar angleθR Theta of the recoil photonµ∓ Muonτ∓ Tauonϒ (4S) Upsilon meson at the 4S resonanceϒ (5S) Upsilon meson at the 5S resonanceφ Azimuthal angleχ Lightest dark matter particlexiiiA′ Dark photonA′µ Dark photon gauge fieldC2M Cluster second momentdr Distance from IP to POCA in the x-y planedt99 Width of the time distribution that 99% of photons of a given energy fallintodz Distance from IP to POCA in ze Elementary chargee∓ Electron or positronE∗ Centre of mass energyEt Energy associated with a track in the ECLF ′µν Dark field strength tensor,IK Innermost KLM layer in a clusterJµEM Standard Model electromagnetic currentLµ Likelihood of a particle being a muonmχ Mass of the lightest dark matter particlemA′ Mass of the dark photonmR Mass recoiling against the dimuon systemnCDC Number of hits in the CDCnK Number of KLM layers in a clusternOOT Number of out-of-time crystals in the ECLp Momentum of a trackxivpµ Momentum of the muonpP Momentum of the recoil photonpR Recoil momentumpT Transverse momentum of a trackR Ratio of the cross sections of e+e− to hadrons and e+e− to muonsr Orbital distancetc Timing of an ECL clustertK Timing of a KLM clusterZMVA Cluster Zernike multivariate analysis (MVA)xvList of AbbreviationsARICH aerogel ring-imaging Cherenkovbasf2 Belle II analysis software frameworkCDC central drift chamberCMB cosmic microwave backgroundCMS centre of massDAQ data acquisitionECL electromagnetic calorimeterEM electromagnetismFPGA field programmable gate arraysgbasf2 grid based Belle II analysis software frameworkGR general relativityHLT high level triggerIP interaction pointISR initial-state radiationKLM K0L and muon detectorL1 level 1MC Monte CarloMIP minimum ionizing particleMOND modified Newtonian dynamicsxviMVA multivariate analysisPOCA point of closest approachPXD pixel detectorROC receiver operating characteristicRPC resistive plate chamberSP single photonSVD silicon vertex detectorTOP time-of-propagationxviiAcknowledgmentsI would like to express my gratitude to numerous people. Firstly, I extend mysincere thanks to Chris for guiding and supporting me throughout this research. Iam truly appreciative of Michael’s guidance and exceptional editing skills, whichsignificantly enhanced the quality of this thesis. I also extend my thanks to Mihofor her constant presence in the office, as well as her assistance and friendship.Additionally, I appreciate the contributions of the rest of the UBC Belle II group,including those who have since left.Thank you to my entire family, with a special mention for my mom, dad, andsister (I’d definitely get called out by my mom if I didn’t thank her specifically) towhom I talked whenever I needed distraction or commiseration. My appreciationalso goes out to all my friends for their camaraderie and unwavering support. Aheartfelt thanks is reserved for my extraordinary partner, Emma.Thank you also to Metric; it is doubtful I could have continued working as lateinto the evening without your music.xviiiChapter 1IntroductionThe Standard Model of particle physics is a theory that describes the fundamen-tal particles and their interactions, aside from gravity. It dates from the secondhalf of the 1900s, and has been extremely successful, especially in predicting theexistence of undiscovered particles. The Standard Model divides particles intocategories based on their spin and how they interact, and reveals the intricate struc-ture through the use of group theory and quantum field theory. However, withthe discovery of the Higgs Boson in 2012, the predictions of the Standard Modelhave been mostly exhausted. The Standard Model has been extensively tested, andhas not yet been convincingly disproven, though there are many questions that itleaves unanswered. Chief among these is the matter-antimatter asymmetry in theuniverse, the existence of neutrino masses, and the existence of dark matter anddark energy. The Standard Model has also thus far been irreconcilable with theo-ries of gravity. As such, searching for new physics, or physics beyond the StandardModel, is a major area of research in particle physics.1.1 Dark matter and evidence for its existenceDark matter is the name we give to the cause of a variety of gravitational effects thatphysicists have observed that are not explained by our theory of gravity, generalrelativity (GR). There are some physicists who advocate for new models of gravityas the solution, instead of dark matter. The modified Newtonian dynamics (MOND)1scheme is one of several proposed alternatives that claim to remove the need fordark matter, however it has serious flaws [4], and there are currently no widelyaccepted or plausible alternatives to GR.There are a great number of empirical observations that indicate or stronglysupport the existence of dark matter. Several of the most famous or relevant piecesof evidence are briefly summarized here:• Rotation curves are the graph of the rotational speed of stars and gas aroundthe centre of the galaxy as a function of their distance from it. The datafor these graphs comes from measurements that use redshifted 21cm light,which corresponds to the emission line of atomic hydrogen, and from pho-tometry measurements of the brightness of the stars [6].Following Newtonian dynamics, Kepler’s third law can be used, which isequivalent to setting the gravitational force equal to the centripetal force. Assuch, theory predicts that the rotational speed of the disk should be propor-tional to 1/√r, where r is the orbital distance, as the visible mass of the diskdecreases. However, when the observational data is compared to theory, itbecomes clear that the two do not agree, because experimental evidence indi-cates the rotational curve becomes roughly flat towards the edge of the disk,and even far beyond as the mass becomes much less dense [6]. Figure 1.1shows the data and theory comparison for the spiral galaxy NGC 3198.The widely accepted solution to this discrepancy is that there must be moremass in the galaxy, mass that we aren’t able to observe. This mass is in theshape of a roughly spherical halo, and is therefore referred to as the darkmatter halo [6].• The cosmic microwave background (CMB) is radiation from photons emit-ted after the cooling and formation of neutral hydrogen in the early universe[12]. These photons have been redshifted to the microwave range as the uni-verse has expanded. The CMB spectrum is a near perfect blackbody witha temperature of 2.73 K [21]. The CMB is isotropic to the 10−5 level, andthe very small anisotropies provide a wealth of knowledge, including thatbaryons account for only a fraction of the mass density of the universe. This21985ApJ. . .295. .305V DISTRIBUTION OF DARK MATTER IN NGC 3198 309 No. 2, 1985 Fig. 4.—Fit of exponential disk with maximum mass and halo to observed rotation curve (dots with error bars). The scale length of the disk has been taken equal to that of the light distribution (60", corresponding to 2.68 kpc). The halo curve is based on eq. (1), a = 8.5 kpc, y = 2.1, p(R0) = 0.0040 M0 pc-3. dark matter to visible matter inside the last point of the rota- tion curve (at 30 kpc) is 3.9. The enclosed halo mass is 0.8 times the disk mass at R25; the enclosed halo mass is 1.5 times the disk mass at the Holmberg radius. The total mass inside 30 kpc isl5xl010Mo. Another property of interest is the mass-to- light ratio of the disk; we find M/Lß(disk) < 3.6 Mq/LBq and M/LF(disk) < 4.4 Mq/LVq. The disk-halo model shown in Figure 4 has the character- istic flat rotation curve over a large part of the galaxy. Beyond 30 kpc it is a mere extrapolation, but the observations inside 30 kpc do not show any sign of a decline, and the extrapolated curve may well be close to the true one. To obtain an estimate of the minimum amount of dark matter at large distances from the center we have also made a fit, shown in Figure 6, with a halo density law whose slope changes from —2 in the inner region to — 4 in the outer region : PhaloW CC (2) Fig. 5.—Cumulative distribution of mass with radius for disk and halo for the maximum disk mass case. Two halo fits are shown. The curve labeled “ normal ” halo is based on eq. (1); the parameters of the fit are the same as those in Fig. 4. The curve labeled “minimum” halo is based on eq. (2); it corresponds to a density distribution whose slope changes from —2 in the inner regions to —3.5 in the outer regions. This curve represents an estimate of the minimum amount of dark matter in NGC 3198 inside 50 kpc. © American Astronomical Society • Provided by the NASA Astrophysics Data System Figure 1.1: The rotation curve for the spiral galaxy NGC 3198. Data is shownwith error bars. The data has curve fit to it, and also displayed are twocurves for the rotational curve for the visible disk, and for the theoreticalhalo component. It is clear that the visible disk curve does not fit t e databy itself. Figure from [30].indicates there must be non-baryonic dark matter, and precise CMB measure-ments can also provide an estimate for the total amount [21].• GR predicts that the curvature of space affects light, not just objects withmass. This means that large concentrations of mass can cause light to curve,and bend light around distant objects or create multiple images of one lightsource. The deflection of the light corresponds to the mass of whatevercaused the space-time curvature, allowing a mass estimate of distant plan-ets, galaxies, and galaxy clusters. By using bright galaxies or point-likesources such as quasars, the masses of galaxies can be precisely calculatedand compared to the observable mass. These comparisons indicate that thereis a lot of dark matter in galaxies, including galaxies where the dark matteris the vast majority of the mass present [3].A particularly compelling case of the gravitational lensing argument is that3of the Bullet cluster. It’s a galaxy cluster that contains two colliding galaxyclusters [25]. The components of the colliding clusters behave differently,with the mass centres (as imaged using gravitational lensing) having contin-ued moving, while the visible matter in the form of plasma clouds havingslowed down due to electromagnetic interactions. These plasma clouds areeasily visible due to their X-ray emission [11]. Without dark matter the cen-tre of mass map would correspond to that of the visible matter, in this case,primarily the plasma. As an image of the visible matter and mass centresshown in Fig. 1.2 demonstrates however, these visible mass and mass cen-tres are significantly different, providing strong evidence for the existence ofdark matter [25]. The Bullet cluster is often considered to refute the MONDtheories for alternative gravity, or at least remove the ability of MOND toavoid the idea of dark matter [10].For more dark matter evidence, or details on the evidence above, see [6] and[3].Despite all the evidence for its existence, little is known about dark matter, andthere are not many qualities we can say with certainty it has. We do know it hasmass, and that there is much more of this mass than there is regular matter. Darkmatter must also be stable [8]. It is an intensively studied subject though, and thereis much we can say dark matter is not. It is generally accepted that dark matteris non-baryonic, and simulations of structure formation of the universe show thathot (relativistic) dark matter is not viable explanation, ruling out neutrinos as darkmatter [4] [5]. Dark matter must also have no electrical charge, and not interact, orat least interact very feebly with the photon and all other Standard Model particles(excluding through gravity). If it interacted more strongly it would it be visibleby other means. These requirements rule out all Standard Model particles, leavingnew particles as the only possible candidates [4]. This of course assumes that it isparticle based, which it might not be, but given the success of the Standard Model,it seems a reasonable assumption [8].Many hypothetical particles have been proposed, with various degrees of the-oretical backing, and as our knowledge has improved, many of these have beenruled out. Physicists have developed a check list of qualities that a dark matter4Figure 1.2: The Bullet cluster as imaged by Chandra, Hubble Space Tele-scope, and gravitational lensing analyses. Image from [1]. This imageis from after the collision of the galaxy clusters. The plasma, detectedby its emission of X-rays, is shown in pink, and the gravitational centresmeasured by lensing are shown in blue. These regions are physicallyseparated, providing evidence for dark matter. The visible matter hasbeen slowed by interactions, while much of the mass of the clusters hascontinued to move.candidate should have to be considered viable [29]. The 10 criteria on that listinclude checking the candidates’ compatibility with current experimental and em-pirical evidence, such as searches that have already been conducted. There’s alsothe question of whether the candidate could ever be probed experimentally. Whilenot an absolute physical requirement, it must be a testable explanation to belong tothe realm of science.1.2 Dark sectorsGiven the total mass of dark matter in the universe relative to that of StandardModel matter, there is every reason to assume that dark matter is of similar com-5plexity in nature to that of ordinary matter [8]. This potential complexity bringsaround the concept of a dark sector. Dark sectors are a hypothetical collectionof dark matter particles and their mediators, analogous to that described by theStandard Model. Dark sectors may have structure and their own fields and gaugeinteractions, but do not interact through standard model gauge groups [20].Dark sectors must interact with regular matter through gravity, otherwise theyfail to explain dark matter. However, for us to be able to detect them through othermeans, there must be some other interaction with regular matter. In dark sectormodels this interaction occurs through a portal [14]. Portal is the general termfor the operator or interaction mechanism that connects the two sectors [27], andis called this because it can be thought of as the gateway between the two sectors.The interaction mechanism takes different forms depending on the model and whattype of particle is mediating the interaction.Portals are an important concept in dark sectors, though they are not withoutissues, because they ignore the gauge principle to at least some degree [14]. Thisis not necessarily a problem, but should not be ignored either, because the gaugeprinciple is the basis for all known fundamental interactions [7].The case which best respects the gauge principle is when the mediator is aspin 1 particle. Here, the interaction is caused by the coupling of a dark sectorand Standard Model gauge boson. While there are many other possible mediators,including spin 1/2 and 0 particles, this thesis is concerned with the vector-mediatorcase, where a dark particle called the dark photon mixes with the photon.1.3 Searching for dark matter at particle collidersDue to the intense interest in the nature of dark matter, the number of searches andsearch techniques has rapidly proliferated in recent years. The search techniquescan be categorized into one of three categories: direct detection, which looks fordark matter from the galaxy colliding with standard model particles; indirect de-tection, which looks for the standard model products of dark matter annihilating;and collider experiments that aim to produce dark matter through the collision ofStandard Model particles [8]. All of these search techniques rely on the assumptionthat dark matter interacts, though feebly, with Standard Model particles.6Particle accelerators are machines that are able to accelerate various chargedparticles up to extraordinarily high energies. The beams of high-energy particlesthey produce can then be used to explore many areas of particle physics. Particlecolliders are particle accelerators that collide two beams of high-energy particles.Particle colliders are a common tool in the search for dark matter because theyare able to cause a large number of particle interactions, or collisions, in a con-trolled manner, and at an often otherwise inaccessible energy. This opens up thestudy of rare decays of Standard Model particles that would not otherwise be ob-servable in large enough numbers to draw useful conclusions. While the dark mat-ter particles themselves are not detectable in such decays, the other products of theinteraction are, and quantities such as the missing mass and missing energy can bemeasured [8]. This allows such events to be studied, and permits the measure ofquantities of the dark matter candidate. As interactions involving dark matter mustbe quite rare, statistics are one of the key strengths of dark matter searches usingparticle colliders.When models propose a dark matter candidate that could feasibly be producedin a particle collider experiment, we attempt to quantify how often we would expectthis decay to occur, what it would look like in our experiment, and what otherdecays and effects may be present that would obscure it. We refer to these ideasand quantities as the cross section, signal, and backgrounds respectively.7Chapter 2The Belle II detectorAll information in this chapter that is not otherwise cited is from the Belle II Tech-nical Design Report [13] and the Belle II Physics Book [22].Belle II is a high-energy particle physics collider experiment based in Tsukuba,Japan. Belle II is the sole experiment that uses the SuperKEKB particle collider,the current holder of the world record for the highest instantaneous luminosityfor a beam colliding experiment. SuperKEKB collides electrons and positrons af-ter accelerating them to 7GeV and 4GeV respectively. The Belle II detector isbuilt around the collision point, and is used to observe the results of the collision.Belle II is the successor to the Belle experiment, one of two original B-factory ex-periments, so called because the collision energies of such experiments are chosenso that they produce a large number of B mesons. Specifically, at Belle II, thecentre of mass energy is 10.58GeV [13], which corresponds to the energy of theϒ (4S) resonance. ϒ (4S) is bottomonium, and decays primarily into two B mesons.Belle II is a next-generation B-factory experiment, and while B physics composesa large part of the Belle II physics program, it is not all Belle II is used for. Belle IIis well suited to dark sector searches because of the precisely understood initialstate, the hermetic nature of the detector, and the triggers designed specifically forsuch searches. Additionally e+e− collisions at this energy are relatively clean, withfewer particles produced by each collision.The Belle II detector is composed of many subsystems and subdetectors. Fig-ure 2.1 is a diagram that shows a cross section view from the top of the detector,8with the various components labelled. The detector is clearly asymmetric, with thepoint of collision off-centre. This is because of the asymmetry in the energy ofthe electron and positron beams, with the electron beam having 3GeV more en-ergy in the lab frame. The centre of mass (CMS) frame, therefore, is not the sameas the lab frame and is boosted relative to it. This boost is in the same directionas the electron beam. This direction is more often called the forward direction,and is useful when discussing parts of the detector because of the aforementionedasymmetry. The detector is elongated in the forward direction and contains moredetection elements, as most particles will tend to travel this way.Figure 2.1: A cross section view from the top of the Belle II detector. Dia-gram from [22].The coordinate system Belle II uses defines the forward direction as the positivez direction. For the Cartesian system, y is then defined as the positive verticaldirection, and x is then horizontal by the right hand rule. However Belle II primarilyuses spherical and cylindrical coordinate systems, with the same z definition. θ is9used to denote the polar axis, and φ for the azimuthal axis. See Fig. 2.2 for adiagram that shows the two coordinate systems in relation to the detector.FIG. 1: Coordinate system of Belle II software. x axis is horizontal and toward outside of the acceleratortunnel, which is roughly northeast. y is vertical upward. z is the Belle solenoid axis, which is bisector of twobeams; roughly toward the direction of electron beam. φ is azimuthal angle around z-axis. φ = 0 is definedfor (x, y, z) = (1, 0, 0). θ is zenith angle with respect to z-axis. θ = 0 is defined for (x, y, z) = (0, 0, 1).A. Numbering in SoftwareNumbering in software should be defined in the following manner prioritizing in this order:1. Sub-components within one physical component are numbered consecutively without goingback and forth to the neighboring components.2. A component at smaller ρ has smaller number.3. A component at smaller φ has smaller number.4. A component at smaller θ has smaller number.The definition is dependent on how we interpret the first item, namely how we regard a singlephysical component is; this should be considered in geometrically-natural way. As an example, inthe case of Belle ECL, we regarded the entire calorimeter as a single body and the body is in thesame ρ. The id for crystals counts up in the order of φ and then θ. If we have regarded that theBelle ECL consists of forward endcap, barrel and backward endcap, the numbering would havestarted from forward endcap, then comes to the backward endcap, followed by the barrel; whichmay be mechanically correct, but geometrically unnatural and results in a complicated numbering.2Figure 2.2: The coordinate systems for Belle II overlaid on the detector. Sil-houetted people included for scale. The z-axis is in the direction of theelectron beam, and y is vertical. θ and φ are defined from the Cartesiancoordinates in the standard way. Image from [18].Inside the core of the Belle II detector, there is a magnetic field that causescharged particl s to follow trajectories that curve in φ . The magnetic field c eatedby the solenoid, shown in Fig. 2.1, and is uniform inside it, with a strength of 1.5 T.The solenoid axis is directly along th z-axis.Belle II is designed to detect the results of the collision of an e+e− pair. Thecollision, and all associated detection information is called an event. Events havean event time, calculated by various subdetectors. All other times in the event arereferenced to this time, with t = 0ns being the same time as the event time. Eventsand their quantities will be referred to frequently in the rest of this thesis. Thelocation in the detector at which the two beams collide is called the interaction10point (IP). The IP is not exactly at the origin of the coordinate system, though it isvery close to it.Due to the high luminosity of SuperKEKB, a significant beam-induced back-ground is present at Belle II. This background, more commonly referred to as beambackground, comprises various effects originating from the high-energy beams. Itis necessary to discuss this background here because it is important to the designof many components of the detector, and to the design of the analysis conducted inthis thesis. The primary causes of the beam background at Belle II include: syn-chrotron radiation, wherein high-energy particles emit radiation due to their curvedtrajectory; beam-gas scattering, wherein the few gas molecules in the beam pipedeflect the beam particles, causing them to interact with accelerator and detectorcomponents, resulting in particle showers; Touschek scattering, a phenomenon inwhich beam particles in one bunch deflect each other, also leading to showers ofparticles from interactions with material; and radiative Bhabhas, in which photonsare emitted and then either shower in the detector, or travel along the beam pipeand interact with the magnets, thereby producing a substantial quantity of neutrons.The details of the most relevant subdetectors and systems of the Belle II de-tector are described below. For a more in-depth description of Belle II, see theTechnical Design Report [13].2.1 CalorimetryThe main purpose of the calorimeter is to detect photons and electrons, and mea-sure their energy and position. It also detects other particles however, and canbe used to help refine their position. This type of calorimeter — one designedprimarily for photons and electrons — is often referred to as an electromagneticcalorimeter. At Belle II, this detector is therefore called the ECL.The Belle II calorimeter largely reuses the calorimeter from the Belle detector,and is composed of over 8000 crystals of CsI(Tl), which are wrapped in teflon,aluminum and mylar. The crystals are scintillators, and when non-photon particlesenter them, they emit light. The photons that enter or are emitted in them remaininside by reflecting off of the materials in which they are wrapped, and the light ismeasured by photodiodes glued to the back of the crystals. The light output of the11scintillator is proportional to the energy of the particle deposited into the crystal,and is calibrated using e+e−→ γγ events [13].The calorimeter crystals are arranged into three sections: the forward endcap,barrel, and backward endcap. Together these sections cover the entire range ofφ in the angular range of 12.4◦ < θ < 155.1◦, except for the gaps between thebarrel and endcaps. These gaps in the calorimeter are for support structure andcable routing, and result in gaps in acceptance at 31.4◦ < θ < 32.2◦ and 128.7◦ <θ < 130.7◦ [2]. The shape and size of the crystals varies throughout the detectorfor two primary reasons: to allow the crystals to nest tightly together, and so thateach row of crystals could be built tilted towards, though not directly at, the IP.These two design features reduce the probability of particles going through thecalorimeter without depositing any energy in it, or for photons, without depositingtheir full energy. When energy ‘leaks’ out through the calorimeter, it is referredto as leakage. Some particles, such as muons, leave a specific amount of energyin the ECL, around 190MeV; a particle that does this is referred to as a minimumionizing particle (MIP).A common way to refer to the polar position of a cluster in the calorimeteris with a quantity called thetaID. ThetaID corresponds to a ring of crystals in thecalorimeter, and is used in place of θ because of its simplicity, and the clear wayin which it relates to the detector. While θ values are available for clusters in thecalorimeter, the determination of more precise location for the cluster than simplythe centre of a crystal relies on a centroid algorithm which has been shown tonot always return the best results. A diagram of the layout of crystals with theircorresponding thetaID is shown in Fig. 2.3.2.1.1 ClusteringIncoming particles do not in general deposit energy in just a single calorimetercrystal, and therefore the energy in the calorimeter must be reconstructed into clus-ters of crystals that are caused by the particles that interacted with it. What followsis a simplified description of the clustering method, containing only the most rele-vant parts.Crystals that have more than 10MeV are all considered as possible starting12FIG. 4: Mapping of ECL cell IDs to the theta IDs0 10 20 30 40 50150−100−50−050100150h2_phiCrate_v03Entries 8736Mean x 0.1541± 24.7 Mean y 1.112± 0.09597 Std Dev x 0.109± 14.4 Std Dev y 0.7862± 103.9 Integral 8736 0 0 0 0 8736 0 0 0 0pos_phi_degrees:crate_idFIG. 5: Plot / mapping of phi in degrees vs crate ID. Each crate encompasses all the crystals froma narrow wedge in phi.4. SOFTWARE STRUCTURE NOTESThe ECL cluster times are stored using 12 bits of data. They are stored as integermultiples of 2000212= 0.48828125 ns and span the range [1000, 1000] ns. The allocation ofthe bits starts from 1000 ns. The way that root stores the number is not by using only 12bits but by using 32 bits, prepending zeros onto the number, and then applying compression.Note that there are actually 212 + 1 numbers between 1000 ns and 1000 ns, so storingthe time +1000 ns actually requires a 13th bit. For further details, see the discussion inQuestions [2]. There is talk of updating the way in which time information is stored (usefewer bits and k won’t be a constant) so the code may have to be updated accordingly.8Figure 2.3: The layout of one line of crystals in the ECL. The right side of thediagram is the forward endcap, while the left is the backward endcap.Each crystal has its corresponding thetaID written on it. The black l nbetween thetaIDs 41 and 42 signifies that the collision point is directlybelow that crystal gap and also indicates the presence of a 1.5mm thickmechanical structure. The tilt of the crystals is illustrated in this dia-gram. Image from [19].points for a group of crystals. These crystals then have all 8 nearest neighbourswith at least 0.5MeV added to their group. The groups merge if there are any crys-tals that are in both. The groups of crystals are then split into clusters, each withexactly one crystal that is both a local maximum, and above 10MeV. The energyfrom each crystal is assigned to the nearest clusters according to its proximity tothe position of the cluster, taken initially as the centre of the local maximum crys-tal. The exact positi n of the clu ter is then c lcul d from th positions of eachcrystals belonging to a cluster using the energy assigned to the cluster from eachcrystal as a weight. The cluster energy and position are refined by iterating theprocess above. The maximum size of a cluster is 21 crystals arranged in a five byfive grid with the corners exclude . The minimum energy of a cluster is 20MeV.Unfortunately, due to a bug in current reconstruction software, when there aremore than 10 cluster in a group, only one cluster is reconstructed. This is muchmore likely to happen when there are many crystals with non-negligible amounts ofenergy in them, which occurs whe the event is subject to high beam backgro ndconditions.132.1.2 VariablesThe calorimeter measures and provides a variety of quantities that can be usedin event reconstruction or analysis. Those quantities that are used directly in thisthesis are described below.First and foremost, every cluster in the calorimeter will have an energy, θ , andφ position, as well as a thetaID value. In addition to this, the time of the cluster,tc, is recorded, as well as a measure of the error on that time, dt99. The error onthe timing is quantified as the width of the time distribution that 99% of photonsof that energy would fall into. However, as this quantity has not been calibratedrecently, and was calculated exclusively using simulated data, it can not be reliedon to that extent. Despite this, it is still useful when considering the timing ofclusters of different energies because the resolution of the cluster time is inverselyproportional to the energy of it. Therefore, when comparing cluster timing acrossa wide range of energy it is better to use the ratio of these quantities, as the widthof the time distribution accounts for the energy correlation, resulting in a roughlyenergy independent measurement of the time.There are also several quantities that characterize the shape of the ECL cluster,such as the cluster Zernike multivariate analysis (MVA), ZMVA, and cluster secondmoment, C2M. These are referred to collectively as shower shape variables. ZMVAcan be used to distinguish photon clusters from other ECL clusters. It is a dimen-sionless quantity that ranges between 0 and 1, with the closer to 1 it is, the morephoton-like the ECL cluster is. C2M also helps to distinguish between ECL clusters,though in a more complicated manner that need not be discussed. It has units ofcm2.There is also a useful calorimeter variable that applies to the event as a whole,as opposed to a particular ECL cluster: the number of out-of-time crystals in theECL, nOOT. This is the number of crystals that measure at least 7MeV, and have anabsolute value of tc of at least 110ns. nOOT is a useful measure of the amount ofenergy and beam background present during and surrounding an event.142.2 TrackingTracking in Belle II relies on several subdetectors, namely the pixel detector (PXD),the silicon vertex detector (SVD), and the central drift chamber (CDC). The purposeof the PXD and SVD is to detect the paths of particles near the IP. The informationobtained is then used to reconstruct particle tracks and the vertices of particles thatdecayed near the IP. Both the PXD and SVD are silicon-based detectors, though ofdifferent designs.The PXD is the innermost detector in Belle II, and is designed to be composedof two cylinders that encircle the beam pipe, though only one full layer and a frac-tion of the second layer has been installed. The cylindrical layers are both com-posed of many pixels of silicon, each of which is able to detect the passage of acharged particle through it. The pixels use DEPFET (DEPleted Field Effect Tran-sistor) technology. DEPFET was chosen because this design has low noise at theexpected operating temperature, and combines detection and amplification, allow-ing the readout electronics to be moved away from this detector. This, combinedwith the low power draw of the sensors and lack of required cooling, allows forthem to be made very thin. This thinness is important because it reduces the amountof material on which particles might scatter, enabling a more accurate reconstruc-tion of the particle tracks. This detector is expensive, and has a large number ofchannels to be readout, but is necessary in such a high-luminosity environment.The SVD consists of four layers of double sided silicon strips, arranged in con-centric cylinders. The outer three layers deviate from cylindrical in the forwarddirection, sloping inwards to cover more of the forward solid angle with the sameamount of detector material. This region is important because of the forward boost,as discussed earlier in Chapter 2, and without sloping would require a longer SVD.The strips are double-sided with one set of strips that run along the z-axis, andone set that wraps around in the φ direction. This setup reduces significantly thenumber of readout channels, while still providing 2D information; when two stripsthat overlap record hits, their overlap must be the location of the hit. However,this assumption is no longer valid when a high percentage of strips are hit duringone event, and this technology can only be used in regions where this is less of aconcern.15The CDC is a drift chamber, which is the main tracking system at Belle II. Itis designed to detect the paths of charged particles and permit their reconstructionin 3D. This track reconstruction allows precise measurements of the momenta ofthe charged particles based on the curvature of those tracks. The CDC is a largegas-filled chamber that contains nearly 15000 sense wires. Each wire is in oneof two orientations, either axial, along the magnetic field lines, or stereo, whichmeans skewed relative to the axial wires. When charged particles travel throughthe CDC they ionize the gas, and the ionization charges drift to the sense wires,which record the pulse in charge. The axial and stereo wires together provideenough information to reconstruct 3D position by forming a grid. The drift timeof the charges to the various wires is also taken into account, which allows precisetracking.2.2.1 Track reconstructionPattern recognition algorithms are used to build track candidates in the PXD andSVD together, and separately in the CDC. These algorithms consider informationsuch as the location and timing of the hits in the various detectors. The track can-didates from the two systems are then extrapolated, and if they appear compatiblebased on this extrapolation, they are merged.The track candidates are then curve fit, allowing the determination of the mo-menta, and the point of closest approach (POCA) to the IP. Other track quantitiesthat are available during analysis include the transverse momentum, and the num-ber of CDC hits, nCDC.2.3 K0L and muon detectorThe K0L and muon detector (KLM) is the outermost detector and is designed to de-tect K0L and muons, hence its name. Similar to the ECL, the KLM is split into threesections: the forward endcap, barrel, and backward endcap. The KLM covers theentire φ range in the angular range of 18◦ < θ < 155◦. Unlike the ECL however,there is no gap in coverage between the barrel and endcaps, and they overlap signif-icantly in coverage to ensure this. The barrel contains 15 layers of detector materialalternating with iron plates arranged in 8 sections, forming in an octagonal prism16around the solenoid and ECL. The endcaps consists of 14 layers of detection mate-rial alternating with iron plates, and these are arranged such that they cap the barrelKLM, as shown in Fig. 2.1.The KLM uses two different detection subsystems, one carried over from Belle,and one that is designed for the higher-luminosity environment of Belle II.The resistive plate chamber (RPC) is the technology used by Belle, and inBelle II it comprises the outer 12 layers of the barrel. They consist of two glass par-allel plate electrodes separated by gas. Particles ionize the gas as they pass throughit, discharging the plates in that location. This discharge is then recorded. Due tothe discharge and the subsequently required charging of the plates, RPC detectorshave a large dead time between particle passages. This means the efficiency ofRPC detectors drops significantly when they are exposed to high particle incidencerates, and they can only be used in areas of the detector that sees fewer particles.The inner layers of the KLM are exposed to a large amount of beam background,and the endcaps see a large neutron background due to e−e− scattering interactingoutside of the detector.For these regions of higher particle incidence, scintillator strips are used. Eachlayer of detector material is comprised of two planes of strips which are orthogonaland layered on top of each other. The strips are read out by silicon photomultipliersin groups of 4, and if more than one strip in the group of 4 registers a hit, all strips inthe group of 4 are considered hit. This is called a multi-strip hit, and this behaviouris caused by the current firmware and is not the intended long-term function.While not the primary purpose of the detector, the KLM does detect photonsthat have passed through the ECL, and can be used as a backup photon detector.This will be discussed further later in this thesis.2.3.1 Building KLM clustersThe first step in making a KLM cluster is to build 1D hits in the scintillator layers.1D hits are composed of hit scintillator strips, and often contain only 1 strip. Theymay contain multiple hit strips if they are neighbouring strips, are not from thesame group of 4 readout strips (otherwise it would be a multi-strip hit as definedearlier), and have times that are close together. At this stage, instead of being used,17all multi-strip hits are entirely ignored by the reconstruction software. This will bechanged in future software releases.After building 1D hits, 2D hits are constructed by combining overlapping 1Dhits. These 2D hits have the position of the location of the overlap of the strips, anda time consisting of the average of the 1D hits.The 2D hits from each layer of the KLM are then used to reconstruct KLMclusters based on their relative positions. Note that in the KLM there is only amagnetic field in the iron plates, as they act as the return flux for the solenoidthat generates the magnetic field for the core of the detector. This means that evencharged particles do not curve between the iron layers, and instead travel in straightlines. The 2D hits must be in a shape consistent with this behaviour to form a KLMcluster.2.3.2 VariablesThe KLM provides a variety of quantities which can be used in event reconstruc-tion or analysis. Those quantities that are used directly in this thesis are brieflydescribed below.KLM clusters contain the following pieces of information: the number of layersinvolved in the cluster, nK , the innermost layer in the cluster, IK , the timing of thecluster, tK , and θ and φ location information. Of these, the timing of the clusterneeds to be discussed further. While KLM timing information has been collectedthroughout the entirety of Belle II running, it has only recently been calibrated.Previous to this calibration, it could not be used for analyses. The timing for aKLM cluster is simply the earliest time that any one of its layers recorded. This isnot ideal, however, as the timing could be more precise if it were a combination ofthe information from all the layers involved in the cluster.2.4 Particle identificationThe particle identification system at Belle II consists primarily of two detectors:one in the barrel, and one in the forward endcap. In the barrel, the time-of-propagation (TOP) counter is located between the CDC and the ECL. The TOPcounter is composed of 16 quartz bars connected to multi-anode photon detectors.18It uses the location and timing of Cherenkov photons (photons emitted by particleswhen they are travelling faster than the local speed of light) as they interact with thephoton detectors. The photons are reflected inside the crystal through total internalreflection until they reach the photon detectors. The photon detectors then measurethe location and incidence time of the photons at the end of the quartz bars. Theangle of the cone of Cherenkov radiation in a medium is dependent on the speedof the particle. Steeper angles require more internal reflections to reach the photondetectors, thus taking longer to reach them.In the forward endcap, the detector is the aerogel ring-imaging Cherenkov(ARICH). It is also located between the CDC and the ECL, with the TOP counterextending around the ARICH until its far end. The ARICH comprises two lay-ers of aerogel with different indices of refraction. The aerogel causes particlesto emit Cherenkov radiation, much as the TOP counter does. The ARICH allows theCherenkov rings to spread out, and land on an array of photon sensors.Both detectors provide a measurement of a particles’ speed. This measurementis then combined with the measurement of its momentum by the CDC to give amass hypothesis, and therefore a hypothesis for the particles’ identity.This information is then combined with the information that other subdetec-tors are able to provide: the SVD and CDC measure specific ionization which isuseful for hadron identification, while the ECL and KLM provide extra informationfor identifying leptons. Neutral particles meanwhile get all their particle identifi-cation information from the ECL and KLM. All of the information gathered is thencombined to give the probability that a particle has a certain identity.2.5 Triggers and data acquisitionThe trigger system is responsible for recognizing events that are of interest, and, intandem with the data acquisition (DAQ) system, recording them. It is a two-levelsystem that utilizes hardware for a fast initial trigger, and software for a slower,more refined check.The level 1 (L1) trigger is the hardware-based trigger, and uses field pro-grammable gate arrays (FPGA) to take information from all the subdetectors anddecide whether to trigger on the event or not. If the L1 trigger activates, the DAQ19system takes the output from all of the subdetectors and sends it to the high leveltrigger (HLT) system. The HLT system is a computing cluster composed of thou-sands of CPU cores, which rapidly performs event reconstruction. This real-timeprocess allows an accurate determination of whether the event is of interest. At thisstage, the event is either discarded, or stored permanently. The L1 trigger systemis designed to be capable of an output rate of 30 kHz, while the HLT reduces therate to a maximum of 1.4 kHz.To determine if an event is of interest, it is compared to a list of triggers. Trig-gers are a list of criteria relating to physics that Belle II is interested in. For thisthesis, there are several triggers of interest, a few of which are often referred to asthe single-photon triggers.For L1 the triggers of interest are:• hie: At least 1GeV total in the ECL crystals with 4≤ thetaID≤ 58. The eventdoes not appear Bhabha-like.• lml6: Exactly one ECL cluster with CMS energy, E∗ > 1GeV within 13 ≤thetaID≤ 58, and no other cluster with laboratory frame energy E > 0.3GeVanywhere in the ECL.• lml13: Exactly one ECL cluster with E∗> 0.5GeV within 21≤ thetaID≤ 44,and no other cluster with E > 0.3GeV anywhere in the ECL.• lml16: Identical to lml13, but there must also be no tracks in the event, asdefined by the L1 trigger, not by reconstruction. This trigger did not existfor some of the earlier data, but lml13 did. As a result, both are used for fulldata coverage, despite lml16 being a subset of lml13, because lml16 is moreuseful in this analysis.The HLT triggers of interest are:• 1: At least one neutral ECL cluster with E∗ > 2GeV in 32◦ < θ < 130◦.Event does not appear to be a e+e−→ γγ or Bhabha event.• 2: Exactly one photon-like ECL cluster with E∗ > 1 in 45◦ < θ < 115◦. Noother E∗ > 0.3GeV ECL clusters.20• 3: Exactly one photon-like ECL cluster with E∗ > 1 in 44◦ < θ < 98◦. Noother E∗ > 0.3GeV ECL clusters.For more details on the HLT and L1 triggers above, see Appendix A.2.6 SoftwareData processing and event reconstruction at Belle II is done with the Belle II Anal-ysis Software Framework (basf2). This framework is written in C++ and Python,and is used to perform a variety of tasks including simulation, reconstruction, andanalysis [23]. For analysis, basf2 is interacted with through Python. ROOT [9] isused for the storing, input, and output of data at all levels.A version of basf2, called gbasf2 (grid basf2) is used on the Belle II grid, a dis-tributed computing system. The grid has a hierarchical structure for data storageand processing, and is designed to handle the large amounts of data and computa-tion required by Belle II [17].21Chapter 3The dark photonIn this thesis, part of a search for dark matter is presented. The dark matter inthis search is described by one particular model, that of the minimal dark photon.In this model, the dark photon, denoted by A′, is a gauge boson of and mediatorof a dark force analogous to the electromagnetic force [16]. This model does notassume that there is only one particle explaining dark matter; on the contrary, itrequires the existence of a dark sector. The lightest dark matter particle will bedenoted by χ . This chapter explains the theory behind the dark photon model, anddiscusses the production and decay of dark photons, as well as the parameter spaceand current exclusion limits.3.1 TheoryThere are several constructions of the dark photon model, with some versions hav-ing massive dark photons while in others the dark photons are massless. Thissection describes only the massive case. Throughout, when choices are made thatdifferentiate this dark photon from other possible choices, it will be noted.In this version of the dark photon model there is a broken dark U(1) gaugesymmetry, and the corresponding dark field strength tensor mixes with the Stan-dard Model U(1) hypercharge field strength tensor [16]. In other dark photon con-structions, the mixing can be with the Standard Model U(1) electric charge groupinstead of the hypercharge group. For the massive dark photon, this changes the22physics of the dark photon [14].In the description of dark sectors in Sec. 1.2, it was mentioned that the waythat dark sectors interacts with Standard Model particles is called a portal. Giventhat in this model, the dark sector mediator that interacts with Standard Modelparticles is a gauge boson, the portal is the vector portal. For the dark photonthe portal is described by the coupling of the dark photon to the Standard Modelelectromagnetic current, JµEM. The Lagrangian formalism for the portal isLγA′ ⊃ eεA′µJµEM, (3.1)where e is the normal electromagnetic coupling constant, ε is the strength of themixing, and A′µ is the dark photon gauge field [22]. Note that this means the cou-pling of the dark and regular photons is not quantized, and is instead the arbitrary,though presumably small, eε . The Lagrangian contains terms in addition to theportal term, and these come — along with the portal term — from the diagonal-ization of the gauge and kinetic terms after electroweak symmetry breaking. Thediagonalization procedure can be seen in [14], starting from the general kineticterms in a Lagrangian for two Abelian gauge bosons. The relevant terms in theLagrangian are thenLγA′ ⊃−14F′µνF′µν +12m2A′A′µA′µ + εeA′µJµEM +LA′χχ , (3.2)where F ′µν is the dark field strength tensor, and mA′ is the mass of the dark photon[16]. This mass is obtained from the Stueckelberg Lagrangian [14]. To keep thismodel more general, the exact form of the interaction of the dark photon with darkmatter is not specified, and those terms are left as LA′χχ .3.2 Production and decayDue to the coupling of the dark and Standard Model photon, dark photons can beproduced by any mechanism that produces a virtual photon with sufficient mass.This of course opens up many potential avenues of production for dark photons atparticle colliders. With Belle II, the best production mechanism is the annihilationof the e+e− pair, and this is the case studied in this thesis.23The dark photon can decay to both dark matter and Standard Model particles,because of its coupling to the electromagnetism (EM) current. The coupling to theStandard Model is suppressed by ε however, so it assumed that if it is kinematicallyallowed (mA′ > 2mχ , where mχ is the mass of the lightest dark matter particle), thedark photon will preferentially decay to dark matter. This statement can be mademore precise by looking at the partial decay widths of the dark photon.For the decay of the dark photon to Standard Model leptons, the partial decaywidth isΓ(A′→ ℓ+ℓ−) = 13αε2mA′√1− 4m2ℓm2A′(1+2m2ℓm2A′), (3.3)[14] where ℓ= e,µ,τ , mℓ is the mass of the corresponding lepton, and α is the finestructure constant. This decay can only occur if mA′ > 2mℓ, for whichever leptonis being considered. The partial decay width of the dark photon to hadrons cannotbe calculated directly, but can be determined through empirical measurements ofthe ratio of the cross sections of e+e− to hadrons and e+e− to muons. This ratio isdenoted as R. With this you getΓ(A′→ hadrons) = Γ(A′→ µ+µ−)R, (3.4)[16] for the partial decay width to hadrons. Just as for the leptons, these decayscan only occur if kinematically allowed.The partial width for the dark photon decaying into dark matter has a similarform [14] to that of decaying into leptons:Γ(A′→ χχ¯) = 13αDmA′√1− 4m2χm2A′(1+2m2χm2A′). (3.5)The key difference is that Eq. 3.5 contains αD, the dark sector fine structure con-stant, instead of ε2 and the Standard Model fine structure constant. These eachquantify the the strength of their respective force between elementary charged par-ticles. The result of this difference in the partial decay width formulas is that thedecay into dark matter will be dominant if mA′ > 2mχ and αD ≫ αε2. The poten-tial values of ε will be discussed more in the next section, but it is expected to besmall, of the order of 10−2 or smaller [14]. The value of αD is unknown, but it is24not of particular interest for this analysis. A simplification will be introduced inthe next section such that αD can be ignored.3.3 Parameters and current exclusion limitsThe analysis presented in this thesis is searching for the invisible decay of the darkphoton at Belle II, and so only the parameters and parameter space relevant to theinvisible decay will be discussed.The minimal model of the dark photon described above is one of the simplestdark sector models, as there are only three free parameters: the mass of the darkphoton, mA′ , the strength of mixing, ε , and the branching fraction of the dark pho-ton into dark sector particles. A simplification can be made, which is that weassume the branching fraction to dark matter is 1 unless the final dark matter stateis kinematically disallowed, where the branching fraction is 0. This assumption isequivalent to assuming that αD≫ αε2. With this simplification there are only twofree parameters, and together these form the parameter space for the dark photon.From analyses of the potential strength of mixing [16], a range of 10−6 < ε <10−2 is the most interesting and attainable region to explore. As for mA′ , it is largelyunrestricted, however, the parameter is usually split into two ranges, divided bywhether the dark photon mass is greater or less than twice the mass of the electron.Only part of the upper range, 10−3 GeV/c2 < mA′ < 10GeV/c2, is accessible atBelle II [16].Many other experiments have conducted searches for dark photons, and havealready excluded regions of the parameter space, as shown in Fig. 3.1. This figurealso shows the predicted search region for this analysis at Belle II.25the use of a proton beam allows for more production processes to be considered. Thus far, limitshave been set by the following experiments: ν-CAL I (57, 58), using pi0 → A′γ decays (59) andproton bremsstrahlung (60); CHARM (61), using η(′) → A′γ decays (62); and NOMAD (63) andPS191 (64), using pi0 → A′γ decays (65). Figure 3 shows that these limits are largely comparableto those obtained from electron beam dumps and exceed them at higher masses. In the near fu-ture, two variations of the SeaQuest experiment, dubbed SpinQuest and DarkQuest, will use the120-GeV main injector proton beam at Fermilab incident on a beam dump to search for A′ →µ+µ− and A′ → e+e− decays (66, 67). As can be seen in Figure 4, this will greatly enhance thesensitivity compared with that of the older proton beam dumps. Further into the future, proposedexperiments like SHiP could provide even greater enhancements (68).4.6. Searches in Meson and Lepton DecaysWhilemeson decay processes contribute to the production of dark photons at hadron colliders andproton beam dumps, experiments like NA48/2, located at the CERN SPS, exclusively exploit suchdecays. Speci!cally, NA48/2 searched for pi0 → A′γ followed by prompt A′ → e+e− decays usingpi0 mesons produced in K+→ pi+pi0 decays (69). Figure 3 shows that the NA48/2 constraints areworld leading for prompt decays in the 10–100 MeV mass region. Soon, the Mu3e experimentlocated at the Paul Scherrer Institute is expected to provide the !rst dark photon sensitivity inlepton decays using stopped muons (70). Figure 4 shows that Mu3e could soon be sensitive tocurrently unexplored parameter space. Future experiments that exploit other meson decays arealso being considered (see, e.g., Reference 71).5. SEARCHES FOR INVISIBLE DARK PHOTONSThe current constraints on invisible A′ → χχ¯ decays are summarized in Figure 5. These resultswere obtained by looking for an excess of events with a consistent missing invariant mass, formed10–610–2 10–1 1 1010–510–410–310–210–11εmA' (GeV)mA' /mX = 3, αD = 0.5mA' /mX = 5, αD = αEMNA64NA62BaBarLEPBelle IINA64Scalar XMajorana XPseudo-Dirac XFigure 5Constraints and proposed sensitivity within the next 5 years on invisible A′ decays. The color scheme is thesame as in Figure 3. The thermal targets for different dark matter scenarios (solid and dashed black lines) arealso shown (72). Figure adapted from Reference 14 (CC BY 4.0) using DarkCast (21).48 Graham • Hearty • WilliamsAnnu. Rev. Nucl. Part. Sci. 2021.71:37-58. Downloaded from www.annualreviews.org Access provided by University of British Columbia on 08/08/23. See copyright for approved use. Figure 3.1: An exclusion plot for the minimal dark photon decaying invisi-bly. The xclusion plot is in the parameter space f rmed by the mixingparameter and the mass of the dark photon. The filled-in regions areexcluded by the labelled experiment. The coloured lines are projectedexclusion limits, and the black dashed and solid lines represent the pa-rameters for which the labelled model is compatible with the amountsof dark matter observed in the universe. These compatibility lines aredependent on the ratio of the dark photon and dark matter masses andαD. The Belle II projection is for 20 fb−1, less that 1/20 of the currentdataset. This projection has however been determined to be overly opti-mistic. Figure from [16].26Chapter 4Analysis structureThe single-photon analysis is an analysis being conducted at Belle II in search ofdark photons. Specifically the analysis is searching for the invisible decays of themassive dark photon described in Chapter 3.4.1 OverviewWhen the dark photon is produced by e+e− annihilation and decays invisibly, theprocess is entirely undetectable if there are no other particles involved. WhileBelle II is able to measure missing energy and mass, this cannot happen for entirelyundetectable processes, because there’s no indication anything happened at all.The events are simply not stored. In order for this to be an event that could bestudied, there must be a detectable particle in the event, such as an initial-stateradiation (ISR) photon. This requirement gives rise to the name of this analysis:the single-photon analysis. The Feynman diagram for this interaction is shown inFig. 4.1.The signal for the invisible decay of a dark photon is a narrow peak in a smoothbackground of the distribution of the missing mass of the event, also called therecoil mass. This recoil mass corresponds to the mass of the dark photon.Belle II is well suited for this analysis because of the large amount of the solidangle it covers, the known initial state, and the dedicated single-photon triggers. Aspreviously discussed in Chapter 2, Belle II has several single-photon triggers that27Chapter 3Search for dark photons3.1 Analysis overviewThis chapter will describe the search for dark photons at Belle II. The searchis for an event signature of e+e ! A0;A0 ! , where the dark photonA0 kinetically mixes with a SM photon, and there is a single recoil photonbeing produced. The dark photon then decays invisibly to two dark mattercandidates . In the final state, the only particle visible in the detector isthe single recoil photon. This is under the assumption that A0 is not thelightest dark sector particle, and if so, it will decay into a SM particle. TheFeynman diagram of this event is presented in figure 3.1. Similar searcheswith the same signature are conducted at the BaBar experiment [40] andNA64 experiment [41].Figure 3.1: Feynman diagram of the invisibly decaying dark photon event.The goal for this analysis in the end is to conduct a “bump” hunt on theinvariant mass of the dark photon, which is measured by the single photonenergy using equation 3.1. mA0 represents the dark photon mass, E⇤ISR isthe center of mass (CoM) energy of the single recoil photon, and E⇤beam isthe CoM energy of the beam.m2A0 = 4E⇤beam(E⇤beam  E⇤ISR) (3.1)25e+e−Figure 4.1: A Feynman diagram of the signal event for the single-photonanalysis. A virtual photon is produced by e+e− annihilation, whichthen kinetically mixes wi h th dark photon with coupling ε . The resul-tant dark photon then decays invisibly to dark matter, represented by χ .Modified from [31].store events where there is a lone, high-energy photon. These triggers are essentialto this analysis.4.2 Backgrou dsEvents that are not caused by dark photons are sometimes able to mimic the ap-pearance of a single-photon event. These events are called background events. Themain sources of background are listed here along with a brief description:• Radiative Bhabhas: Bha ha events are the scattering of the e+e− pair, andare extremely common at Belle II. The term radiative Bhabha means a pho-ton is emitted as well: e+e−→ e+e−γ . If two out of three of the resultingparticles are not detected due to detector inefficiency or detector acceptance,the remaining particle can cause an ECL cluster that could constitute a viablesingle-photon cluster. In this case, the event then imitates a single-photon28event. This happens for only a tiny fraction of radiative Bhabha events, butbecause their cross section is so high, they are an important background.• γγ(γ): these events are simply e+e−→ γγ(γ), and can imitate single-photonevents when all but one of the photons are missed or are out of acceptance.• Cosmic rays: these events are caused by cosmic rays that pass through thedetector. These particles trace one path through the detector, and might leaveonly one ECL cluster of significant energy. The tracks in these events do notgenerally pass through the IP. These events also typically have a KLM clusterwith many layers, which can be useful in rejecting them.• Beam background: this is described in Chapter 2. Since beam background isthe term for many processes there isn’t one particular way that these eventsmimic signal. In general, as in the other cases, only one particle leaves asignificant ECL cluster.The single-photon analysis is complex due to the difficulty of characterizingthese backgrounds [14]. Each background requires careful study and constructionof vetoes to remove as much of it as reasonably possible.4.3 Structure of the analysisDue to its complexity, the single-photon analysis is broken into components whichare then completed by the analysis team. The components are described in thefollowing subsections.4.3.1 Study of signalWhile the event signature for this analysis is an event that contains just one photon,that does not mean events that have absolutely anything else in them will be re-jected. True single-photon events could also contain tracks, extra photons, or KLMclusters caused by beam background, noise, or any one of a number of processes.In order to determine what single-photon events actually look like, it is neces-sary to study a channel that can stand in for the signal, along with signal MonteCarlo (MC). The signal proxy has the advantage of being available for both data29and MC, while the signal MC represents the exact events of interest, but is clearlynot available for data. Radiative dimuon events were chosen as the signal proxybecause they are clean, and well understood. The radiated photon in these eventswill represent the single-photon. Signal studies also allow the investigation of theshape of the distributions that will be used in signal extraction.4.3.2 Study of backgrounds and veto constructionThe major backgrounds listed in Sec. 4.2 all need to be studied in detail. Thepurpose of this is twofold: in combination with the characteristics of the signalevents, vetoes can be constructed to significantly reduce the backgrounds; to studythe shape and scale of the distributions that will be used in signal extraction.4.3.3 Data-MC correctionsMC does not represent data perfectly, and because much of the analysis is con-ducted using MC to keep the analysis blind, correction factors are needed. Thesecorrection factors are particularly important when calculating the inefficiency ofthe ECL and KLM, as small changes in how often these subdetectors fail to detectparticles can have a large impact on how many background events are let through.4.3.4 Fitting procedures and signal extractionIn the single-photon analysis, the signal will be fit and extracted in two regions,each treated slightly differently. Both regions will use the recoil mass squared m2R,which is related to the centre-of-mass energy of the ISR photon, E∗, by the formulam2R =√s(√s−2E∗) , (4.1)where√s is the CMS collision energy. As mentioned earlier, the mass of the darkphoton corresponds to the recoil mass, and so mA′ = mR. This allows Eq. 4.1 to bemodified and rewritten asE∗ =√s2− m2A′2√s, (4.2)where it is clear that E∗ is dependent on mA′ , which is fixed, and√s, which hasseveral different values throughout data taking at Belle II. This makes fitting m2R30superior to fitting E∗.The two fitting regions and their corresponding methods are:• When the single-photon E∗ is below 2GeV, the signal will be fit from the1D distribution of the recoil mass squared, m2R. This was chosen becauseit is extremely difficult to quantify the magnitude of the radiative Bhabhabackground in this energy range. The background distribution is expected tobe smooth however, which should allow the 1D fit to work effectively.• When the single-photon E∗ is above 2GeV, the signal will be fit from the2D distribution of m2R as a function of the thetaID. This allows for excellentseparation of the backgrounds and signal, if the shape and magnitude of thebackground distributions can be quantified, which is expected to be possible.4.3.5 Systematic uncertainty estimation, unblinding, and settingupper limitsThese are the final components of the analysis and are necessary steps. Unblindingrefers to the analysis, specifically the signal analysis, finally being run on the ac-tual data instead of the simulated data. Simulated data is used to prevent accidentalbiases in the analysis, but the the final result must, of course, be based on the data.Since other components need significant work, no specifics of this will be givenhere. It will follow the normal procedures in Belle II, as laid out in [24].The study of the signal proxy and the radiative Bhabha background, and theconstruction of the associated vetoes is presented in the next chapter.31Chapter 5AnalysisThis chapter presents vetoes to be used in the single-photon analysis that involveKLM clusters, ECL clusters, and tracks. These vetoes are designed to veto theevents when we detect potential particles present in the event besides the singlephoton itself. Radiative dimuons are a well-defined event type that both allow usto study the background signals in the detector by looking at what happens asidefrom the photon and muon pair, and act as a proxy for single-photon events in dataand MC. When being used this way, the radiated photon is referred to and treatedas the single photon. Due to the utility of radiative dimuon events, a clean controlsample was constructed of these events to make use of their abilities.A significant area in which data and MC disagree is that of the KLM responseto high-energy photons. This is particularly important because the single-photonanalysis will use the KLM as a backup photon detector. The radiative photonsin the radiative dimuon control sample allow us to study a pure sample of high-energy photons, which is useful in characterizing the response in the KLM and ECLfor both data and MC. While radiative dimuons can be used as a signal proxy,the other control sample used is composed of radiative Bhabhas, one of the majorbackgrounds.In the following sections we will go through the datasets used, the constructionof the control samples, and the vetoes proposed by use of them. Unless otherwisespecified, all quantities discussed in this chapter are measured in laboratory restframe. All histograms in this chapter include an information tab that provides vari-32ous data. This data includes the number of entries in the entire parameter space (notjust those in the subspace displayed), the mean and standard deviation of the rele-vant axes, and the integral of displayed plot entries. Additionally, the tab presentsoverflow and underflow numbers, which for 2D histograms are eight individualnumbers that indicate the axis in which events are in overflow or underflow.5.1 Datasets usedThis analysis uses data and MC simulated data from the Belle II collaboration. Thedata and MC samples used are described in more detail below.5.1.1 DataBelle II has been collecting data since 2019, and has amassed a dataset of almost432 fb−1. However, the single-photon triggers were not active for the first few datataking sessions, and so the dataset available for this analysis is 425 fb−1. All of thedata used was taken in 2020–2022. These data comprise primarily data taken withthe CMS collision energy at the ϒ (4S) resonance, but also contain data taken off-resonance, and some taken at various energies near theϒ (5S) resonance. Table 5.1shows the breakdown of the data used in this analysis by the collision energy atwhich it was collected.Table 5.1: The CMS collision energies at which the data was collected. Foreach energy, the corresponding amount of data used in this analysis islisted. The total amount of data used is also listed.CMS collision energies Size(fb−1)ϒ (4S) 363.4ϒ (4S) off-resonance 42.3Various energies nearϒ (5S) resonance 19.4Total data used 425.25.1.2 MCMC is produced centrally at Belle II for all analysts to use. The MC simulationsimulates the e+e− collisions, the resulting particles, and their interactions with33the detector. The electronic response of the detector to those particles is also fullysimulated and then processed and reconstructed with the same software that isused for data. Non-specialized MC that is intended for broad use may be referredto as generic MC. There is a campaign associated with each basf2 release, with themost current being campaign 15. There are two types of MC: run-independent,and run-dependent. Run-independent means that the backgrounds are simulated,and the MC does not depend directly on the data. Run-dependent means that thebackground is from the data, and the MC is split into sections that correspond towhen in the data that background is from. Run-dependent is the superior typeof MC, as it more accurately represents data compared to run-independent MC.This analysis uses only run-dependent MC from campaign 15, which is denoted asMC15rd. Only the part of MC15rd that corresponds to the data with single-photontriggers is used.Due to its simulated nature, MC is split into event types. In this analysis, thee+e− and µ+µ− samples are used. While run-dependent MC uses the backgroundsfrom data, and corresponds to particular parts of the dataset, it does not generallyhave the same luminosity. The e+e− MC15rd sample has one tenth the luminosityof data, whereas the µ+µ− sample has four times the luminosity of data.As will be noted again later, the regular MC15rd Bhabha samples require thatboth of the scattering MC electrons have θ values that are in ECL acceptance. Thisgeneration requirement is quite relevant for this analysis.5.2 Radiative dimuon control Sample Selection CriteriaThe selection criteria for the radiative dimuon control sample must make sure thatthe events in the sample are actually radiative dimuon events, while preservingsufficient statistics, and not inadvertently biasing the sample by unnecessarily re-moving events that are relevant. The following sections go through the variouscategories of requirements.5.2.1 Muon requirementsBoth muons must pass the criteria listed in Table 5.2. These parameters and valueswere chosen based on their use in another Belle II analysis [28]. Several require-34ments that were included in that analysis were excluded here, due to the differentgoals of this work. One requirement which has been added is a cut on Lµ , which isthe likelihood that the particle is a muon. Lµ is calculated using information fromthe particle identification system. The Lµ cut is loose, however when the Lµ for themuons in the pair is plotted in Fig. 5.1, it can be seen that this requirement is suf-ficient to require that virtually all events have at least one muon with a Lµ greaterthan 0.95. Due to leakage and gaps in the KLM among other reasons, the Lµ forone muon can be low while the other is high, and this pair is with high probabilitya muon pair, which is why these events are kept.5.2.2 Event requirementsThe control sample events should contain only one ISR photon and the muon pair.To achieve this we make cuts on the square of the mass recoiling against thedimuon system, mR, and on the recoil momentum of the dimuon system, pR. Insingle-ISR events, there should be no recoil mass, and the distribution would beentirely concentrated at zero. However, primarily due to finite resolution, whilethe distribution for m2R is peaked at zero, it exhibits a non-zero width. Reconstruc-tion error and the limits of resolution should cause equal smearing in the positiveand negative directions, but situations such as a double-ISR could cause there tobe a small recoil mass due to their non-zero invariant mass. Some events of thistype will be events we want to keep, and as such, the range of acceptable valuesshould be biased in the positive direction. The chosen range of acceptable valuesis −0.25GeV2/c4 < m2R < 1GeV2/c4. Figure 5.2 shows the m2R distribution for thecontrol sample with this cut enacted. This distribution can’t be seen without thiscut since it is implemented at an earlier stage of data analysis, and is one of severalcuts that reduces the sample to a manageable size.To ensure that there is an ISR photon of the energy we are interested in, werequire a recoil momentum of at least 0.5GeV/c. The remaining recoil momentumdistribution can be seen in Fig. 5.3. Ideally this recoil momentum should all bedue to the photon, however this and the m2R criteria are not enough to guaranteethis, and in the next subsection we will add some selection criteria to select eventswhere this is true.350 0.2 0.4 0.6 0.8 10µ for µL00.20.40.60.811µ for µLEntries 5348278Mean x 0.9815Mean y 0.9795Std Dev x 0.09478Std Dev y 0.1005Integral 5.348e+06 0 0 0 0 5348278 0 0 0 0 1102103104105106100 0.2 0.4 0.6 0.8 10µ for µL00.20.40.60.811µ for µLEntries 2.36105e+07Mean x 0.9864Mean y 0.9838Std Dev x 0.0792Std Dev y 0.0888Integral 2.361e+07 0 0 0 0 2.36105e+07 0 0 0 0 110210310410510610710Figure 5.1: Log plot of the Lµ of the two muons, µ0 and µ1, for data (top),and MC (bottom). These plots have the whole radiative dimuon controlsample selection criteria applied, which includes a requirement that bothmuons have Lµ > 0.1. It can be seen that with this requirement there arevery few events in the sample that have a poor Lµ for both muons.360.4− 0.2− 0 0.2 0.4 0.6 0.8 1 1.2 1.4)4c/2 (GeVR2m050100150200250300350400450500310×Entries 5348278Mean 0.1028Std Dev 0.2288Underflow 0Overflow 0Integral 5.348e+06DataMCFigure 5.2: The distribution of the square of the mass recoiling against thedimuon system. MC is normalized to data by luminosity and the infor-mation tab is for data. The distribution is curtailed at −0.25GeV2/c4and 1GeV2/c4, as this is the distribution with all control sample crite-ria applied. The distribution is peaked very near zero as expected, andis somewhat asymmetric with the bins on the positive side consistentlycontaining more entries than their negative counterparts.A summary of the event requirements is given in Table 5.2.5.2.3 Photon requirementsThe criteria applied thus far are not sufficient to achieve a very clean control sam-ple, but this can be greatly helped by checking if there is a photon correspondingto roughly what is expected based on the recoil momentum: a recoil photon. Theterm recoil photon will be used interchangeably with the term single photon SP forradiative dimuon events. If a recoil photon cannot be matched to the recoil mo-mentum, the event will be discarded. The matching and other photon criteria are370 1 2 3 4 5 6 7 8)c (GeV/Rp050100150200250300310×Entries 5348278Mean 2.121Std Dev 1.445Underflow 0Overflow 0Integral 5.348e+06DataMCFigure 5.3: The distribution of the recoil momentum against the dimuon sys-tem for both data and MC. MC is normalized to data by luminosity andthe information tab is for data. The distribution is cut at 0.5GeV/c, asthis is the distribution with all control sample criteria applied. The dis-tribution is peaked near this cutoff, which is expected since there aremore radiative dimuon and other mimicking events with smaller recoil,such as lower energy radiated photons. The local maximum between 4and 5GeV/c is caused by beam energy photons.summarised in Table 5.2.The angle between the recoil photon and recoil momentum vectors, αR, shouldbe near zero when the photon is the main cause of the recoil. The momentum ofthe recoil photon, pp, should also be very close to that of the recoil momentum pR.This means the ratio of the two momenta should be close to 1. Extra ISR photonsand the uncertainties in reconstruction introduce some smearing though, causingboth distributions to be less sharply peaked. Figure 5.4 shows the parameter spacefor the momentum ratio and angular separation between the recoil momentum andrecoil photon.380 20 40 60 80 100 120 140 160 180 )° ( Rα00.20.40.60.811.21.41.61.82Rp/PpEntries 8858134Mean x 10.32Mean y 0.8781Std Dev x 24.53Std Dev y 0.2631Integral 8.858e+06 0 600 0 0 8857534 0 0 0 01102103104105106100 20 40 60 80 100 120 140 160 180 )° ( Rα00.20.40.60.811.21.41.61.82Rp/PpEntries 3.743466e+07Mean x 9.829Mean y 0.8898Std Dev x 24.25Std Dev y 0.2542Integral 3.742e+07 0 16015 0 0 3.741865e+07 0 0 0 0110210310410510610Figure 5.4: The ratio of the recoil photon momentum to the recoil momentumagainst the two muons vs. the 3D angle between the recoil momentumand the recoil photon, plotted for data (top) and MC (bottom). Both plotsare log scaled, and were made with all control sample criteria applied,except for the criteria based on these plots. The vast majority of entriesare in the region near a momentum ratio of 1, and αR < 20◦.390 2 4 6 8 10 12 14 16 18 20 )° ( Rα0.80.850.90.9511.051.11.151.2Rp/PpEntries 8858134Mean x 2.438Mean y 0.9782Std Dev x 2.984Std Dev y 0.05021Integral 7.307e+06 0 19872 7578 0 7307234 123619 0 415774 9840573104105106100 2 4 6 8 10 12 14 16 18 20 )° ( Rα0.80.850.90.9511.051.11.151.2Rp/PpEntries 3.743466e+07Mean x 2.332Mean y 0.9813Std Dev x 2.9Std Dev y 0.04982Integral 3.138e+07 0 81771 52919 0 3.137951e+07 506756 0 1534854 3878853410510610Figure 5.5: The ratio of the recoil photon momentum to the recoil momentumagainst the two muons vs. the 3D angle between the recoil momentumand the recoil photon, plotted for data (top) and MC (bottom). Both plotsare a crop to the area of interest of the plots in Fig. 5.4. The vast majorityof entries are still present in the crop, due to the large concentration at aratio of 1 with small angular separation.40The region bounded vertically by 0.95 and 1.1, and horizontally by 0 and 6◦contains most of the events, and Fig. 5.5 shows a crop to that area. There aretwo main factors that cause the spread of energy near a ratio of 1: leakage of thephotons through the ECL causing an underestimation of the photon energy, andbeam background causing an overestimate. While the leakage effect is strongerthan the beam background one, we don’t necessarily want to include events wherethe leakage of the single photon is significant as this could make the sample lessrepresentative of signal. There is, however, another reason for the asymmetricrange around a ratio value of 1. There is an effect that is primarily present below aratio of 0.95 that is likely caused by a relatively high-energy secondary ISR photon.These events are not desired, and might not get caught by other requirements orvetoes.Figure 5.6 shows a clear separation of the events that likely have a high-energysecond ISR photon. These plots are of the momentum ratio as a function of the3D angular separation of the candidate recoil photon and the most energetic extraphoton, where that extra photon has a lab energy of at least 250MeV.The two populations can be fairly well separated by a line at 0.95 for the mo-mentum ratio. The upper distribution is what is expected from from a radiativedimuon event, where the momentum ratio is close to 1, and the angle of the recoilphoton and most energetic extra photon is not correlated. The lower distributionhas a strong correlation between the momentum ratio and the angular separationof the photons, which is indicative of something else going on in the event. Thesuspected cause of this effect is double-ISR, as most events in this group are sep-arated by a distance too significant for it to be attributed to a clustering issue. Ina clustering issue, a photon cluster gets split into two potential photons; however,this observed separation is more in line with the possibility of double-ISR. To con-firm whether the energies of the photons are consistent with this explanation, theenergies of the recoil photon and the most energetic extra photon in the event aresummed for events within a region of the parameter space shown in Figure 5.6where the momentum ratio is below 0.9, and the 3D angular difference is less than90◦. The momentum ratio is then recalculated with the new combined energy valueused in place of the measured photon energy. A plot of this recalculated momen-tum ratio can be seen in Fig. 5.7, which shows that the sum of these two photons’410 20 40 60 80 100 120 140 160 180 )° ( Eα0.80.850.90.9511.051.1Rp/PpEntries 53769Mean x 78.32Mean y 0.9566Std Dev x 42.15Std Dev y 0.06043Integral 5.377e+04 0 0 0 0 53769 0 0 0 0 0204060801001200 20 40 60 80 100 120 140 160 180 )° ( Eα0.80.850.90.9511.051.1Rp/PpEntries 328313Mean x 82.21Mean y 0.9649Std Dev x 41.41Std Dev y 0.05756Integral 3.283e+05 0 0 0 0 328313 0 0 0 0 0100200300400500600700800Figure 5.6: The ratio of the recoil photon momentum to the recoil momentumas a function of the 3D angle between the recoil photon and the mostenergetic extra photon with a lab energy of at least 250MeV. Top is data,and bottom is MC. These were made with the control sample criteriaexcept with a momentum ratio of 0.8 as the lower bound instead of0.95. In both plots there are two distinct components: one group has astrong correlation between the axes, while the other does not. These twogroups do overlap, though the higher concentration components can bereasonably separated by a line at y= 0.95.42energies produces a momentum ratio with a distribution that peaks near 1. Thisplot therefore supports the theory that these events have double-ISR.0 0.2 0.4 0.6 0.8 1 1.2 1.4Rp/Pp Adjusted 020040060080010001200140016001800200022002400Entries 9274Mean 1.013Std Dev 0.07933Underflow 0Overflow 67Integral 9207DataMCFigure 5.7: The adjusted measured photon energy divided by the recoil mo-mentum against the two muons. Adjusted means that the energy of themost energetic extra photon was added to that of the recoil photon. Thisprocess resulted in a distribution peaked at one, indicating that these twophotons combined contain the expected energy of the recoil photon. MCis normalized to data by luminosity and the information tab is for data.The recoil photon is also required to have a θ value in the range (36.4◦, 120.1◦)which corresponds to a thetaID range of [16, 54]. This is a subset of the barrelcomponent of the ECL, which has a thetaID range of [13, 58]. The reason for therequirement is that the barrel is the most reliable part of the ECL, but more impor-tantly it also increases the chances of detecting the other particles in the event suchas muons or electrons if they are present since they are less likely to both go downthe beam. This is most relevant for radiative Bhabha events as these disproportion-ately have their radiated photons at lower angle, and without this restriction to the43barrel there would be an overwhelming number of low angle radiative Bhabhas.The three crystals from both edges of the barrel are excluded as there is increasedleakage there. An additional crystal is excluded on the high side for reasons thatwill be discussed later. The θ distribution for the recoil photon, θR, in the controlsample is displayed in Fig. 5.8.0 20 40 60 80 100 120 140 160 180 )° ( θ020406080100120140160180310×Entries 5348278Mean 75.47Std Dev 23.98Underflow 0Overflow 0Integral 5.348e+06DataMCFigure 5.8: The distribution of the lab frame θ of the recoil photon. MC nor-malized to data by luminosity and the information tab is for data. Allsample selection criteria have been applied.In order to reduce possible contamination of the shower shape variables in thecontrol sample, a selection is applied that requires that the photon is at least 15◦away from either muon, as measured at the ECL. We’ll use αµ to denote thisquantity.Figure 5.9 shows that there are a significant number of events for which this isa consideration, and that the range that is excluded should be as small as possibleso as to not unnecessarily veto events.This selection is achieved by use of variables that store where the extrapolated440 20 40 60 80 100 120 140 160 180 )° ( µα020406080100120140160180200310×Entries 7104058Mean 51.04Std Dev 39.63Underflow 0Overflow 0Integral 7.104e+06DataMCFigure 5.9: Angle between the recoil photon and the closer of the two muontracks, measured at the ECL. MC normalized to data by luminosity andthe information tab is for data. All control sample selection criteriahave been applied except a cut on the angle between the recoil photonand the closer of the two muon tracks, as measured at the ECL. Thepeak near zero is caused by the preferential emission of the photon inthe direction of a muon. It is not peaked at zero due to the path curvatureexperienced by the muon. The peak at high angular difference is causedby beam energy photons.paths of the muons intersect the ECL. These locations are then compared to thelocation of the photon in the ECL. The extrapolation is necessary since the paths ofmuons curve due to the magnetic field, while those of photons do not. Additionallythe muons are not required to have an associated ECL cluster, and so those cannot beused instead. The reasoning for requiring αµ > 15◦ is based on the geometry of theECL: each crystal is approximately 2.5◦ wide, and a cluster is, at its largest, a 5x5array of crystals centred on the most-energetic crystal, with the corners missing.The cluster size necessitates that there be the space of at least 5 crystals between45the centres of the clusters to prevent all overlap of clusters. 15◦ is 6 crystals widehowever, and was chosen in order to be safe.The appropriateness of this cut can be seen in Fig. 5.10, where it will removethe events with low ZMVA at low angle difference, where there is a noticeable abun-dance of such events.5.2.4 Selection summary and control sample plotsTable 5.2 summarizes the selection criteria for the radiative dimuon control sample.Table 5.2: The full selection criteria for the radiative dimuon control sample.The criteria are split into their three categories corresponding to the sec-tions in which they are described.Category Criteria Explanation of variableMuondr < 2cm Distance from IP to POCA inthe x-y plane|dz| < 4cm Distance from IP to POCA inzLµ > 0.1 Likelihood the particle is amuonpµ > 2GeV/c Momentum of the muonnCDC > 4 Number of CDC hits for themuon trackEvent−0.25GeV2/c4 < m2R < 1GeV2/c4 The square of the massrecoiling against the dimuonsystempR > 0.5GeV/c Momentum recoiling againstthe dimuon systemPhotonαR < 6◦ Angle between the recoilmomentum and photon0.95< pPpR < 1.1 Ratio of the recoil photonand recoil momenta36.4◦ < θR < 120.1◦ Lab θ of the recoil photonαµ > 15◦ Angle between the recoilphoton and the closer muonat the ECLThe first plot of interest made using this control sample is a 2D histogram of460 20 40 60 80 100 120 140 160 180 )° ( µα00.10.20.30.40.50.60.70.80.91MVAZEntries 7104058Mean x 51.04Mean y 0.8348Std Dev x 39.63Std Dev y 0.1663Integral 7.104e+06 0 0 0 0 7104058 0 0 0 0 1102103104100 20 40 60 80 100 120 140 160 180 )° ( µα00.10.20.30.40.50.60.70.80.91MVAZEntries 3.138211e+07Mean x 51.12Mean y 0.8334Std Dev x 39.79Std Dev y 0.1665Integral 3.138e+07 0 0 0 0 3.138211e+07 0 0 0 0 110210310410Figure 5.10: The recoil photon ZMVA as a function of the 3D angle betweenthe recoil photon and the nearest muon, as measured at the ECL. Datais shown at top, while MC is shown in bottom. All control sample se-lection criteria are applied except a cut on the angle between the recoilphoton and the nearest muon, as measured at the ECL. ZMVA rangesfrom 0 to 1, with higher values indicating the ECL cluster is photon-like. On the left of the plot there is a considerable concentration ofrecoil photons that are within a few degrees of a muon at the ECL, andthat have a ZMVA < 0.7. These photon ECL clusters were likely con-taminated by the muon ECL cluster, resulting in lower ZMVA values.47the single-photon E∗, the energy in the CMS frame, as a function of the thetaID ofthe photon. This plot is shown for data and MC in Fig. 5.11.Also of particular interest for the single-photon analysis is the recoil mass ofthe event squared as a function of the thetaID of the photon. For the radiativedimuon control sample the recoil mass is specifically the mass recoiling against theradiative photon, because this is analogous to what the recoil mass is in a single-photon event. This plot is displayed for both data and MC in Fig. 5.12.5.2.5 Selection of KLM clusters, extra photons, and tracksNow that the events of the control sample have been selected, the extra photons,KLM clusters, and tracks that are not part of the radiative dimuon can be selected.The following sections describe the selection of these objects.KLM clustersThe two energetic muons in radiative dimuon events should cause large KLM clus-ters as the KLM is designed to detect muons. However, sometimes the muon KLMclusters can look similar to those caused by high-energy photons, and are clustersthat would not be present in a single-photon event. As such, in order to study theresponse of the KLM to single-photon events we need to ignore the KLM’s responseto the muons.Since the muon-associated KLM clusters range widely in number of layers andlocation, the surest way to ignore them is to ignore all KLM clusters near the muons.Unfortunately the angular resolution of the KLM is far worse than that of the ECL,and so we must exclude a large angular area around each muon track when we wantto look at the KLM response to the rest of the event.To determine the angular range that safely excludes the vast majority of muonsbut that does not overly exclude regions of the KLM, plots of the θ and φ betweenKLM clusters and the closer muon can be made. Figure 5.13 shows these plots, fromwhich we can see the distribution of the muon related clusters relative to the muonmomentum. The angle difference in φ is significantly larger than that in θ due tothe curling of the muons in the magnetic field, which affects only the φ position.This is also why φ and θ are used here instead of the 3D angle difference, because4820 30 40 50 60 70thetaID123456 (GeV)E*Entries 5348278Mean x 35.2Mean y 2.074Std Dev x 10.62Std Dev y 1.382Integral 5.236e+06 0 0 0 0 5235878 0 0 112400 0020004000600080001000012000140001600020 30 40 50 60 70thetaID123456 (GeV)E*Entries 2.36105e+07Mean x 35.51Mean y 2.061Std Dev x 10.7Std Dev y 1.388Integral 2.311e+07 0 0 0 0 2.310689e+07 0 0 503605 0010000200003000040000500006000070000Figure 5.11: The data (top), and MC (bottom), distributions of E∗ as a func-tion of the thetaID for the recoil photon. These are made with theradiative dimuon control sample, and as such only have entries in thethetaID range of [16, 54] which corresponds to the θ requirement ofthe sample. The plots contain two distinct densely populated areas:one at high energy that is composed of beam energy photons, and theother covering most of the parameter space below 2GeV.4920 30 40 50 60 70thetaID020406080100)4c/2 (GeVR2mEntries 5348278Mean x 34.99Mean y 68.62Std Dev x 10.64Std Dev y 29.32Integral 5.348e+06 0 0 0 0 5348278 0 0 0 0010002000300040005000600070008000900020 30 40 50 60 70thetaID020406080100)4c/2 (GeVR2mEntries 2.36105e+07Mean x 35.29Mean y 68.97Std Dev x 10.73Std Dev y 29.45Integral 2.357e+07 0 39181 0 0 2.357131e+07 0 0 11 00500010000150002000025000300003500040000Figure 5.12: The data (top), and MC (bottom), distributions of the square ofthe mass recoiling against the photon as a function of the thetaID forthe recoil photon. Due to relation of the square of the mass recoilingagainst the photon to the CMS energy of the photon, this plot appearsvery similar to E∗ vs thetaID, though reflected around the x-axis andtranslated. However, there are slight other differences because the datawas taken at several different beam energies, as laid out in Table 5.1.500 20 40 60 80 100 120 140 160 180 )° difference between KLM cluster and muon ( θ010002000300040005000600070008000310×Entries 1.181037e+07Mean 9.827Std Dev 14.95Underflow 0Overflow 0Integral 1.181e+07DataMC0 20 40 60 80 100 120 140 160 180 )° difference between KLM cluster and muon ( φ020040060080010001200140016001800310×Entries 1.181037e+07Mean 26.55Std Dev 31.38Underflow 0Overflow 0Integral 1.181e+07DataMCFigure 5.13: The θ (top), and φ (bottom), angular difference between a KLMcluster and the closest muon. Closer here means the muon nearest inθ for the θ difference plot, or nearest in φ for the φ difference plot.MC normalized to data by luminosity and the information tabs are fordata. These are made using the radiative dimuon control sample, butwithout criteria for the selection of the KLM clusters. The θ differenceis very narrowly peaked at 0◦, while the φ difference peak is broaderand peaked near 10◦. This is caused by the curvature of muons in φ .Both plots show excellent agreement between data and MC.51to account for the large spread in φ , we would have to exclude cases where the θvalue is well away from the muon, thereby excluding more clusters unnecessarily.The θ and φ angle difference plots indicate that very generous bounds for theexclusion of KLM clusters would be 10◦ in θ , and 30◦ in φ . This would mean ex-cluding all clusters within 10◦ in θ , and within 30◦ in φ , which would be expectedto be sufficient. However, when the number of layers in a KLM cluster is plotted asa function of the φ or θ difference between the KLM cluster and the nearest muonfor clusters that are relatively in time, it becomes evident that the above cuts wouldlet muon associated clusters through. In time is defined as |tK | < 20ns, a regionwhich contains roughly 70% of entries in the previous histograms. These plots canbe seen in Fig. 5.14 and Fig. 5.15 for the θ and φ differences respectively.These figures suggest that more appropriate values for the exclusion boundsare 20◦ in θ , and 30–40◦ in φ . In order to be more cautious, 40◦ in φ was chosenlest muon related KLM clusters be included in the radiative dimuon control sample.Incidentally, both sets of figures demonstrate the similarity of data and MC for thesequantities.It’s essential to note that because of the way in which KLM clusters are handled,only the four KLM clusters with the highest nK are considered. The muon proximityrequirements are then imposed, and the remaining KLM clusters are the ones usedhereafter.Extra photon selectionWhile neutral ECL clusters are not necessarily photons, in this thesis all neutralclusters will generally be referred to as photons.The recoil photon is not the only photon of interest in the control sample events.We are also interested in the lower energy photons, as they will be useful later indetermining appropriate ECL vetoes. Of these other, lower energy photons, onlythe three highest lab energy ones that satisfy |tc| < 200ns will be considered, andif they pass the other criteria described below, stored and referred to as the extraphotons. The tc requirement is present so that photons that are not related to therest of the event are not included, and was inspired by a similar cut made in [28]. Ifout-of-time photons were included in the extra photons it could result in the event520 20 40 60 80 100 120 140 160 180 )° difference between KLM cluster and muon ( θ24681012141618Kn Entries 8647070Mean x 5.077Mean y 9.471Std Dev x 11.25Std Dev y 5.257Integral 8.641e+06 0 5941 0 0 8641129 0 0 0 01102103104105106100 20 40 60 80 100 120 140 160 180 )° difference between KLM cluster and muon ( θ24681012141618Kn Entries 3.596735e+07Mean x 4.688Mean y 9.349Std Dev x 11.04Std Dev y 4.969Integral 3.594e+07 0 26965 0 0 3.594038e+07 0 0 0 0110210310410510610Figure 5.14: nK plotted as a function of the θ difference between the KLMcluster and the closer (in θ ) muon. Data is shown at the top, and MCon the bottom. These are made using the radiative dimuon controlsample, but without criteria for the selection of the KLM clusters. Theyreveal the correlation of the number of layers in a KLM cluster andthe θ proximity of that KLM cluster to a muon. There is a particularconcentration of high nK clusters below a θ difference of 20◦ in bothdata and MC. This correlation is due to the KLM clusters caused by themuons themselves.530 20 40 60 80 100 120 140 160 180 )° difference between KLM cluster and muon ( φ24681012141618Kn Entries 8647070Mean x 17.57Mean y 9.471Std Dev x 25.37Std Dev y 5.257Integral 8.641e+06 0 5941 0 0 8641129 0 0 0 01102103104105100 20 40 60 80 100 120 140 160 180 )° difference between KLM cluster and muon ( φ24681012141618Kn Entries 3.596735e+07Mean x 17.91Mean y 9.349Std Dev x 27.4Std Dev y 4.969Integral 3.594e+07 0 26965 0 0 3.594038e+07 0 0 0 0110210310410510610Figure 5.15: nK plotted as a function of the φ difference between the KLMcluster and the closer (in φ ) muon. Data is shown on the top, andMC on the bottom. These are made using the radiative dimuon controlsample, but without criteria for the selection of the KLM clusters. Theyreveal the correlation of the number of layers in a KLM cluster andthe φ proximity of that KLM cluster to a muon. There is a particularconcentration of high nK clusters below a φ difference of 35◦ in bothdata and MC, though there is another structure in MC that extends tolarger φ differences as well. This correlation is due to the KLM clusterscaused by the muons themselves.54being vetoed unnecessarily. The rest of the extra photon criteria relate to theirproximity to the muons both in space and time.Just as muons can cause KLM clusters, they can also cause neutral ECL clustersby producing final state radiation (FSR) photons. They can also create photonsthrough interactions with detector material, though FSR is the dominant cause.Again, similarly to the KLM clusters caused by the muons, these particular photonswould not be present in a single-photon event, and thus when using this sampleto design photon vetoes these photons need to be ignored. Plotting the θ and φbetween the extra photons and the closest muon, shown in Fig. 5.16 reveals thatthere is a concentration of photons near the muons. In θ the curve levels out at10◦ separation, and in φ there is a sharp initial spike, followed by a small secondbump ending at 15◦ separation. The differing behaviour between θ and φ may bepartially explained again by the curving trajectory of muons in φ . There is alsoa noticeable difference between MC and data, though the histogram shape is quitesimilar. MC15rd has significantly more extra photons present than data.The criteria for spatial proximity to the muons are established based on thecurves’ behavior, as detailed above. Consequently, the threshold is defined as beingwithin 10◦ in θ and 15◦ in φ . With the spacial proximity requirements set, thetemporal proximity can be considered.When the extra photons are sorted by whether they are close to the muon or notaccording to the θ and φ criteria above, their respective time distributions can bestudied. Figure 5.17 shows these two time distributions, and it can be seen that themajority of proximal photons have a small absolute time, while those away fromthe muon do not. Most photons near muons have a time less than 40ns, while mostfurther away from muons have a time greater than 40ns. This occurs because themajority of photons close to the muons are associated with the muons, whereasthe photons further away are usually not related, caused by beam background orsimilar. As we do not want to study photons that are caused by the muons, we canuse a combination of the angular and temporal proximity to exclude such photons.Therefore, the selection for the extra photons is that the three highest momentumphotons with |tc|< 200ns are stored if they do not also satisfy all three of ∆θ < 10◦,∆φ < 15◦, and |tc|< 40ns. The time requirement ensures that relatively out-of-timephotons are not excluded from the control sample simply because they are spatially550 20 40 60 80 100 120 140 160 180 )° difference between muon and photon ( θ020406080100120140310×Entries 7860825Mean 41.31Std Dev 26.84Underflow 0Overflow 0Integral 7.861e+06DataMC0 20 40 60 80 100 120 140 160 180 )° difference between muon and photon ( φ020406080100120310×Entries 7860825Mean 62.01Std Dev 44.68Underflow 0Overflow 0Integral 7.861e+06DataMCFigure 5.16: The θ (top), and φ (bottom), angular separation between the ex-tra photons and the closest muon of the pair for both data and MC.Closest in this case is determined by 3D angular difference. MC isscaled to data by luminosity and the information tabs are for data.These are made using the radiative dimuon control sample, but with-out criteria for the selection of the ECL clusters. MC and data agree inshape, though not in number, with MC always higher. Peaks are presentat or near zero for both, as expected.560 20 40 60 80 100 120 140 160 180 200| (ns)ct |02000400060008000100001200014000Entries 210850Mean 47.37Std Dev 52.01Underflow 0Overflow 0Integral 2.108e+05DataMC0 20 40 60 80 100 120 140 160 180 200| (ns)ct |0102030405060708090100310×Entries 7649975Mean 88.24Std Dev 56.36Underflow 0Overflow 0Integral 7.65e+06DataMCFigure 5.17: The absolute value of tc for extra photons that fall into the anglecut defined above (top), and extra photons that are not in the angle cut(bottom), for both data and MC. MC is scaled to data by luminosity andthe information tabs are for data. These are made using the radiativedimuon control sample, but without criteria for the selection of the ECLclusters. The spikes that can be seen in both plots are due to binning.There are few photons in the top plot that have poor timing, while thereare many in the bottom plot.57close to the muons. The boundary value was chosen as 40ns because it correspondsto the location that both plots in Fig. 5.17 approach a plateau.Tracks and extra object selection summaryTracks are selected as long as they are not the track of either of the two muons.They are ranked by various quantities, and stored according to what is requiredfor studying vetoes. Table 5.3 summarizes the selection of the KLM clusters, extraphotons, and tracks for the radiative dimuon control sample.58Table 5.3: A summary of the criteria for selecting KLM cluster, extra photons, and tracks for the radiative dimuoncontrol sample. The initial criteria is applied. This means that even if there are more than 4 KLM clusters in theevent, there might not be 4 KLM clusters that are considered part of the radiative dimuon control sample if the 4with the highest number of layers do not pass the secondary requirement.Category CriteriaInitial SecondaryKLM clusters 4 highest nK clusters Not within 20◦ in θ AND 40◦ in φ of amuonExtra photons 3 most energetic (non-recoil) photons with |tc|< 200ns Not satisfying both: within 10◦ in θAND within 15◦ in φ of a muon,|tc|< 40nsTracks Any track that is not one of the muon tracks –595.3 KLM vetoesAs has been mentioned, the KLM can be used to detect photons, and therefore canbe a useful detector for vetoing events that contain extra photons or other particlesthat we do not want. As discussed in the control sample subsection on self con-tamination 5.2.5, we have removed KLM clusters that are near the muons since themuons are well detected by the KLM, but are not of interest since the purpose ofthis control sample is to replicate single-photon events, which would not includethese muons.5.3.1 Number of KLM layersOne of the most useful quantities that the KLM provides is the number of layersof the KLM that are involved in the cluster, which we denote as nK . We use thisquantity to categorize the KLM clusters, allowing us to tailor veto requirements forthe different nK . nK is the natural parameter by which to categorize due to thedistinctly different distributions it has in response to various particles, particularlymuons, and photons. Additionally, for the lowest-layer KLM clusters there is a goodchance that the KLM cluster is caused by non-interesting physics and noise, likeneutrons, and therefore these KLM clusters need to be treated somewhat differently.An important distribution to study when designing KLM based vetoes for thesingle-photon analysis is the number of KLM layers in clusters caused by the single-photons. In this sample, because of the recoil photon and recoil momentum ratiorequirement, the single photon has been well reconstructed, and therefore likely haslittle leakage through the ECL. The leakage matters since high leakage means moreenergy in the KLM, and less in the ECL. For these high-energy (≳ 0.5GeV), low-leakage photons, the distribution of the number of layers of nearby KLM clustersis shown in Fig. 5.18. The KLM clusters that are caused by these photons rarelyexceed 2 layers, with around 99.6% of them being 1 or 2 layer clusters. Thisindicates that any KLM cluster with 3 or more layers is highly unlikely to be causedby a well reconstructed single photon, and we are free to veto on these clusterswithout worrying about vetoing on the signal itself.While noise can cause KLM clusters with a small number of layers, they arerarely larger than even 1 layer, meaning that if there is a KLM cluster of more600 2 4 6 8 10 12 14 16 18 20 22 24Kn110210310410510Entries 103366Mean 1.063Std Dev 0.4107Underflow 0Overflow 0Integral 1.034e+05DataMCFigure 5.18: nK for KLM clusters within 8◦ of the recoil photon for eventsin the radiative dimuon control sample. MC is normalized to data byluminosity and the information tab is for data. The angular differencerestriction is imposed so that this plot contains primarily KLM clustersthat are likely caused by the recoil photon. 8◦ was chosen as the angu-lar bound based on being roughly half the angular spread of photonsrelated KLM clusters, shown later in Fig. 5.22. This figure is intendedto demonstrate that the vast majority of KLM clusters close to the re-coil photon have fewer than 3 layers. MC has considerably more entriesthan data even when normalized, a fact that is obscured by the loga-rithmic scale. The likely cause of this is multi-strip hits.than 2 layers in the event, there is likely something else in the event. As such,we veto events that have a KLM cluster of 3 or more layers regardless of the otherKLM quantities. This does not veto a significant percent of the control sample, andtherefore nor should it for signal. The distribution of nK for the control sample canbe seen in Fig. 5.19, where it is important to note that an event can have more thanone KLM cluster, and thus more than one entry in this plot. While there are only6115 KLM layers in the barrel, it is possible for barrel and endcap layers to both beinvolved in one cluster, thereby allowing nK to exceed 15 on occasion. The vastmajority of KLM clusters are 1 or 2 layers, with less than 1% having more than 2layers.0 2 4 6 8 10 12 14 16 18 20 22 24Kn10210310410510610Entries 4515382Mean 1.086Std Dev 0.6433Underflow 0Overflow 18Integral 4.515e+06DataMCFigure 5.19: nK for the radiative dimuon control sample. MC is normalizedto data by luminosity and the information tab is for data. As events canhave a number of KLM clusters, there are many events with no entriesin this plot and many with multiple. The vast majority of KLM clustershave 1 or 2 layers. The data-MC agreement is considerably better thanwhen looking only within 8◦ of the recoil photon.For the 1- and 2-layer clusters other KLM quantities are taken into considerationsince these clusters don’t necessarily indicate that the event isn’t a single-photonevent. Aside from that, there are far more of these clusters, and overly broad vetoeswould veto much more of the signal proxy than the 3+ layer veto. The quantitiesthat we use to characterize the KLM clusters aside from the number of layers are:the innermost layer of the cluster, IK , the timing of the cluster, tK , and the angular62proximity of the cluster to the single photon, αK .Table 5.4 shows a summary of the KLM vetoes using the parameters just de-scribed. A more detailed description of these vetoes and their motivation can befound below.Table 5.4: Summary of the KLM vetoes for the single-photon analysis, de-veloped using the radiative dimuon control sample. If any KLM clustersatisfies one of the sets of requirements, the event would be vetoed.nK Veto requirements≥ 3 none2−10ns < tK < 30nsIK ≤ 4αK > 15◦1−10ns < tK < 30nsIK ≤ 4αK > 165◦5.3.2 KLM timingKLM clusters that are directly related to the event should all have a tK near 0ns,and those clusters that don’t are likely caused by beam background, cosmic rays,or noise. Therefore, we only want to veto 1- and 2-layer KLM clusters if they aresufficiently in time, and would prefer to err on the side of classifying more clustersas in time to ensure that all event-related clusters are considered in time.The plots in Fig. 5.20 show the tK distribution for MC and data for both 1 and 2layers. The MC and data curves are noticeably different, both in shape and centrelocation. The data histogram is noticeably asymmetric, with a sharp rise at t = 0ns.This behaviour is what is expected since the timing is calibrated. For MC however,the peak is largely symmetric, perhaps indicating that it is not calibrated, or iscalibrated using a different method. This MC-data difference is not so substantialas to give rise to issues, though its cause is not fully understood. Where MC anddata do agree however, is on the long tails present for both the 1- and 2-layer KLMclusters. There are tails on both sides, though, for the 1-layer clusters, the left-sidetail is much more prominent.63100− 80− 60− 40− 20− 0 20 40 60 80 100 (ns)Kt0100200300400500600700310×Entries 4289038Mean 10.22− Std Dev 37.23Underflow 1.523e+06Overflow 2.405e+05Integral 2.526e+06DataMC100− 80− 60− 40− 20− 0 20 40 60 80 100 (ns)Kt0200040006000800010000120001400016000Entries 192931Mean 3.344− Std Dev 46.07Underflow 6.635e+04Overflow 2.657e+04Integral 1e+05DataMCFigure 5.20: tK distributions for 1- and 2-layer clusters (top and bottom, re-spectively) in the radiative dimuon control sample for data and MC. MCis normalized to data by luminosity and the information tabs are fordata. The two plots are similar in shape, though not in scale. Data andMC noticeably do not agree, with MC being more sharply and symmet-rically peaked, with a higher central value, and more entries. However,the two central values are close together, and the peaks nearly entirelyoverlap, settling to background levels at the same place.64A range of −10ns to 30ns encompasses the entire peaking region near 0nsfor data and MC in both plots, plus a small region around the peak. Outside thisrange, the distribution is flat, an indication that those KLM clusters are not collision-related. As such, in time is defined as−10ns< tK < 30ns, and only clusters withinthis time window will be vetoed on.5.3.3 Innermost KLM layerIK is useful particularly in discriminating between photons and other particles, andwhether or not that photon is coming from the IP. Particles in general, but espe-cially photons that are coming from the IP, will register in the inner layers of theKLM, not just in the outer layers. This means if IK is sufficiently high, it is veryunlikely to be caused by a particle coming from inside the detector at all, and par-ticularly not by a photon. This also makes IK useful for ignoring noise and externalinterference in the detector, since it allows us to ignore the KLM cluster if IK isgreater than a certain value.From the plots of IK in Fig. 5.21, it can be seen that most clusters have a highIK , and that at low IK where we are most interested the entries per bin varies signif-icantly. The histograms are relatively flat for IK values of 5–10, and at a level thatis consistent with our expectation of most particles not starting a KLM cluster in themiddle of the layers of the KLM. For the reasons discussed in the last paragraph,we will only veto on clusters with a IK of 4 or lower, as these are likely collisionrelated clusters. 4 was chosen as the boundary for this veto based on where theplots in Fig. 5.21 change behaviours and level out.The high proportion of events that have a IK greater than 10 is caused by ener-getic neutrons that hit the detector from the outside. These neutrons are created bythe accelerator and beams in large numbers, and impact the KLM significantly sinceit is the outer detector. Shielding to limit this effect is being installed in the currentBelle II shutdown (2023). Neutrons produced by the beams interacting with ac-celerator and detector material can also cause some KLM clusters in the innermostfew layers of the KLM, though this is a less prominent effect.Low IK is also where there is the greatest MC-data discrepancy for both 1- and2-layer clusters. The cause of these discrepancies is likely the differing ways that652 4 6 8 10 12 14 16KI02004006008001000310×Entries 4289038Mean 8.97Std Dev 4.541Underflow 0Overflow 0Integral 4.289e+06DataMC2 4 6 8 10 12 14 16KI01000020000300004000050000600007000080000Entries 192931Mean 9.635Std Dev 4.06Underflow 0Overflow 0Integral 1.929e+05DataMCFigure 5.21: IK for 1- and 2-layer clusters (top and bottom, respectively) inthe radiative dimuon control sample. MC is normalized to data by lu-minosity and the information tabs are for data. Data-MC agreement isvery good aside from the first few layers. In the innermost layers MChas significantly more entries, an effect ascribed to the differing treat-ment of multi-strip hits in data and MC. The large number of entrieswith an innermost layer of 12 or higher is predominantly caused byneutrons coming from outside of the detector.66MC and data handle multi-strip hits. Multi-strip hits are particularly prominent inthis analysis because of the way photons shower in the KLM, but they are an issuefor the Belle II collaboration as a whole, including in K0L reconstruction [15]. Fora more detailed description of multi-strip hits see [26].One other potential factor for the MC-data discrepancies could be an overes-timation of the photon detection efficiency in MC. This could occur because thecalibration of detection efficiencies in the KLM is done using muon events. Thedetection efficiency of photons is not necessarily directly linked to that of muons,and could cause a discrepancy between MC and data as a result. This would have tobe a small effect, or one that is only present for the scintillators since the MC-dataagreement is reasonable at higher IK .Another curiosity present in the plots of IK is that there is a notable lack ofclusters with an innermost layer of 2 relative to an innermost layer of 1 or 3. Thisis present among both 1- and 2-layer clusters, and data and MC, though it is moreprominent in 1-layer clusters and in data. Part of the reason for this is may be thatthe innermost two layers of the barrel, which are scintillator, are less sensitive thanthe outer barrel layers. This loss of sensitivity at least in data may be caused in partby the multi-strip hit issue. This means clusters with innermost layer 1 or 2, are lesslikely, particularly if we are looking for only 1 layer clusters. However, since parti-cles lose energy rapidly in the KLM, especially in the steel plates at the beginning,we do see a relatively large number that start in layer 1, particularly consideringtheir poor efficiency for photons. In addition, the innermost layers receive muchmore background radiation due to being closer to the IP, which also contributes toparticularly the higher number of IK = 1 clusters. However, a complete descriptionand understanding of the cause of these IK distributions has thus far eluded us.5.3.4 Angular separation from the single photonThe angle from the single photon requirements are harder to motivate with thissample, but the purpose of them is to not veto on a KLM cluster that is eithercaused by the single photon (2 layer) or that is just noise (1 layer). We must also becareful that we do veto backgrounds though, such as e+e− → γγ(γ) with its back-to-back photon. In Fig. 5.22 it can be seen that 15◦ (3D) away from the single67photon is the extent of the KLM response to a single photon. As such, to veto on a2 layer KLM cluster, αK must be greater than 15◦. We can be reasonably confidentthat the peak present in both plots below 15◦ angular separation is indeed the KLMresponse to the photon itself, because only in this region of the plots does the shapedeviate significantly from that expected if the KLM cluster and photon angles wereuncorrelated.For 1 layer clusters, even with all the other criteria, they are likely to be causedby beam background or noise, and therefore these clusters should not be vetoed onunnecessarily. Events that produce back-to-back photons are a large backgroundfor the single-photon analysis, thus we want to be as sensitive to such events aspossible. The back-to-back photons in these events might cause a KLM cluster, evenif the ECL failed to detect them. As a result, we only veto the event if the 1 layerKLM cluster is within 15◦ of back-to-back, equivalent to greater than 165◦ from thesingle photon. This value was chosen based on the upper plot of Fig. 5.22. This plotdoes not show any interesting behaviour near 165◦ that would indicate it is a goodvalue for such a cut, which is as expected, since in a radiative dimuon sample thereshould be no preference for a KLM cluster back-to-back with the photon. However,as is mentioned above, this plot does show that 1-layer KLM clusters which are theresult of photons are predominantly within 15◦ of that photon. This implies that1-layer KLM clusters caused by back-to-back-photons should be more than 165◦from the recoil photon, hence this bound.Figure 5.22 demonstrates another large MC-data discrepancy, one that is stronglyrelated to the discrepancy shown in the IK plots. This MC-data difference is presentonly at small angular separation of the KLM cluster and radiated photon, and there-fore is related to differing detection and reconstruction efficiencies of photons inthe KLM in MC and data. This means that this large disagreement between MC anddata is caused by the same issue that caused the disagreement in the IK plots: multi-strip hits. This hypothesis is supported by a study conducted by another analysisteam member. In that study, the way that the hits are used in reconstruction in MCand data was reconciled as much as possible. The result of this was that there werea significant number of KLM clusters near photons that were reconstructed whenmulti-strip hits were used, that were not reconstructed with the regular code.Luckily, the way multi-strip hits are handled currently is a recognized issue,680 20 40 60 80 100 120 140 160 180 )° ( Kα0100002000030000400005000060000700008000090000Entries 4289038Mean 81.86Std Dev 39.12Underflow 0Overflow 0Integral 4.289e+06DataMC0 20 40 60 80 100 120 140 160 180 )° ( Kα0500100015002000250030003500400045005000Entries 192931Mean 77.81Std Dev 35.3Underflow 0Overflow 0Integral 1.929e+05DataMCFigure 5.22: The 3D angle between the recoil photon and 1- and 2-layer KLMclusters (top and bottom, respectively) in the radiative dimuon controlsample. MC is normalized to data by luminosity and the informationtabs are for data. Data and MC are in good agreement except for atsmaller (αK). Below 20◦ angular separation there is a spike in entries,as would be expected if some of the photons are detected by the KLM.However, MC indicates that significantly more photons are detected bythe KLM than data suggests. This gap in detection is again most likelycaused by the poor treatment of multi-strip hits in data.69and the reconstruction code has been modified for the next basf2 release, both bymaking better use of multi-strip hits in data reconstruction, and by making MCreconstruction more realistic.5.4 ECL cluster vetoes5.4.1 Neutral ECL clustersThis subsection describes the selection of the vetoes that use neutral ECL clusters.Even though these clusters may not in fact be photons, they will frequently bereferred to as photons as previously mentioned.The presence of extra photons in the event — the definition of which can befound in subsection 5.2.5 — can be indicative of the event not being that of asingle photon. However, lower-energy photons can be produced as a result of beambackground, and do not necessarily mean we should veto the event. These twofactors must again be balanced in deciding on the photon vetoes.For photons, we use their energy in the lab frame to sort them into three cat-egories: low energy (20–100MeV), mid energy (100–300MeV) and high energy(> 300MeV). These ranges are chosen broadly along the lines of both frequencyof occurrence, and an understanding of the detector. 20MeV is the lower bound forthe low energy range because it is the lowest energy for a photon that Belle II willreconstruct. A log scale plot of the full energy range is shown in Fig. 5.23, andlinear scale plots of the energy broken down by category are shown in Fig. 5.24.700 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 (GeV)E110210310410510610710 Entries 7731637Mean 0.03942Std Dev 0.04223Underflow 0Overflow 9Integral 7.732e+06DataMCFigure 5.23: The energy of the extra photons for the radiative dimuon controlsample. MC is normalized to data by luminosity and the informationtab is for data. Events may have multiple entries. Data and MC have asimilar distribution shape, but there are more entries in MC.710.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (GeV)E050010001500200025003000350040004500Entries 20976Mean 0.476Std Dev 0.2585Underflow 0Overflow 361Integral 2.062e+04DataMC0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 (GeV)E020004000600080001000012000140001600018000Entries 283001Mean 0.1492Std Dev 0.04534Underflow 0Overflow 0Integral 2.83e+05DataMC0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 (GeV)E050100150200250300350400310×Entries 7427660Mean 0.0339Std Dev 0.01471Underflow 0Overflow 0Integral 7.428e+06DataMCFigure 5.24: The energy of the extra photons in the high (top left), mid (top right), and low (bottom) energy rangesfor both data and MC for the radiative dimuon control sample. MC is normalized to data by luminosity and theinformation tabs are for data. The high and mid energy plots are similar, with both looking like an exponentialdecay. The low energy plot however has two notable departures from this: first, near 22MeV the distributionreaches a peak and decreases towards 20MeV, the threshold for photons at Belle II. Second, there is a kink justabove 50MeV, which is caused by the requirement of |tc|/dt99 < 1 imposed on photons with E < 50MeV.72The variables we use to characterize photons aside from lab energy are a quan-tity that represents the timing of the cluster, |tc|/dt99, and αE , the angle between thesingle photon and the extra photon. tc is not used directly because 100–300MeV isa wide energy range, and without dt99 the timing is energy dependent. Table 5.5displays a summary of the photon vetoes. The vetoes are described in more detailTable 5.5: Summary of the photon vetoes for the single-photon analysis, de-veloped using the radiative dimuon control sample.Photon Energy Veto requirements> 300MeV none100–300MeV|tc|/dt99 < 3αE > 15◦20–100MeV|tc|/dt99 < 1αE > 176◦below. The values are chosen based on Fig. 5.25, and Fig. 5.26. Figure 5.25 showsthe distributions of |tc|/dt99 as a function of the photon energy, both for data andMC. Figure 5.26 shows the distributions of the angle from the single photon as afunction of the photon energy, both for data and MC.For the high energy range, any event with an extra photon that falls into thisrange (E > 300MeV) will be vetoed, regardless of the photon’s timing or angularproximity to the single photon.In the mid energy range, in order to veto the event, |tc|/dt99 must be less than3 to ensure it likely is a result of the event itself, and it must be more than 15◦ awayfrom the single photon, so that we are sure it is not caused by interactions of thesingle photon with the detector.In the low energy range, it’s important to note that due to the large number ofphotons detected with < 50MeV, there is already a |tc|/dt99 cut. Only photonswith |tc|/dt99 < 1 are kept, and so to ensure consistency in the low energy range,the timing veto requirement is |tc|/dt99 < 1.The angle from the single photon is again harder to motivate with this sample,however we want to veto as many events as possible that are potentially contami-nated, while removing few ’clean’ events. To study the optimal angular proximityrequirement for the middle energy range, a plot was produced of the fraction of the730 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (GeV)E012345678910|/dt99ct|Entries 7731637Mean x 0.03889Mean y 0.5884Std Dev x 0.03377Std Dev y 0.7503Integral 7.724e+06 0 7050 266 0 7724226 95 0 0 01102103104105106100 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (GeV)E012345678910|/dt99ct|Entries 3.687736e+07Mean x 0.0421Mean y 0.5923Std Dev x 0.04117Std Dev y 0.7491Integral 3.682e+07 0 52819 3109 0 3.68206e+07 837 0 0 0110210310410510610710Figure 5.25: |tc|/dt99 as a function of the lab energy for the extra photonsin the radiative dimuon control sample. Data can be seen in the topplot, and MC in the bottom plot. Below 50MeV, photons must have|tc|/dt99 < 1, with the entries above this in the plots being caused byrounding.740 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (GeV)E020406080100120140160180 )° ( EαEntries 7731637Mean x 0.03911Mean y 95.26Std Dev x 0.03431Std Dev y 36.55Integral 7.73e+06 0 0 0 0 7730164 1473 0 0 01102103104100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (GeV)E020406080100120140160180 )° ( EαEntries 3.687736e+07Mean x 0.04229Mean y 95.44Std Dev x 0.03996Std Dev y 36.59Integral 3.685e+07 0 0 0 0 3.6851e+07 26360 0 0 0110210310410510Figure 5.26: The 3D angle between the extra and recoil photons as a func-tion of the extra photon energy. Data can be seen in the top plot, andMC in the bottom plot. The angular distribution corresponds to the ex-pected shape for two uncorrelated vectors. Almost all extra photonsin excess of 100MeV have more than 15◦ angular separation from therecoil photon. As well, there are some extra photons with less than100MeV that have more than 176◦ angular separation from the recoilphoton, though not many, which is as expected with this sample.75sample vetoed as a function of the angle from the single photon requirement. Thisplot, Fig. 5.27, indicates that at 15◦, only 3.4% of the sample is vetoed in data,and 4.4% in MC. 15◦ is consistent with the angular proximity to single photonsused elsewhere in this analysis, and vetoing around 4% of the signal was deemedacceptable. Without knowing how this cut affects all potential backgrounds, thereis no optimization that can be done.For the low energy range the angle requirement is, much like the correspondingKLM veto, about vetoing events with back-to-back photons. Therefore for the eventto be vetoed there must be a photon with αE > 176◦. This value is chosen basedon the work of another collaborator who is studying a e+e−→ γγ(γ) sample.5.4.2 Charged ECL clustersThe ECL vetoes in the previous subsection only dealt with neutral ECL clusters,clusters without an associated track. This of course leaves charged ECL clustersto deal with. One way to deal with this is to make a cut on Et , the energy of anECL cluster that is associated with a track. While Et is corrected for leakage andbackground, it is otherwise simply the energy deposited in the crystal and thereforedoes not depend on a mass assumption. This assumption independence makes thisa good variable for use in a veto.A MIP will deposit around 190MeV in the ECL, whereas energetic electronswill deposit more. Based on this, a 100MeV Et veto should remove events wheretrack reconstruction for electrons failed but the ECL still recorded a large Et , andalso events that contain MIPs. 100MeV should also be a high enough energy thresh-old that it is infrequent that a beam background related cluster overlaps with a track,and is mistaken as associated. This consideration matters since the neutral ECL ve-toes have a different, and in the 100–300MeV energy range, tighter veto than thatof track associated ECL clusters.Figure 5.28 shows the distribution of Et for the track with the highest Et for theradiative dimuon control sample. The majority of tracks with an associated Et havea Et in excess of 100MeV, however, 99.5% of events in this control sample do nothave a track with an associated Et . This keeps the percent of the sample vetoed bythis veto to below 0.5%. It is also why this veto does not amount to vetoing almost760 20 40 60 80 100 120 140 160 180 )° lower bound ( Eα0.511.522.533.5Percentage of control sample vetoed0 20 40 60 80 100 120 140 160 180 )° lower bound ( Eα0.511.522.533.544.5Percentage of control sample vetoedFigure 5.27: The percentage of the radiative dimuon control sample vetoedby the mid energy ECL veto as a function of the minimum angle fromthe single photon. Data is shown at top, and MC at bottom.every event with a track. The impact of this veto on a background sample can beexamined in the following sections to ensure that it serves its intended purpose.770 1 2 3 4 5 6 7 8 9 10 (GeV)tE0200040006000800010000Entries 5348278Mean 0.4358Std Dev 1.11Underflow 0Overflow 5.322e+06Integral 2.596e+04DataMCFigure 5.28: The distribution of the largest Et in each event for the radiativedimuon control sample for both data and MC. MC is normalized to databy luminosity and the information tab is for data. While MC has moreentries than data at very low Et they generally agree. Note that aremore events in overflow than in the plot by several orders of magnitude,as most events do not have any track associated ECL clusters.5.5 Single photon requirements and vetoes fromelsewhereThere are several requirements and vetoes for the single-photon analysis that arebased primarily on work outside this thesis, and that have not yet been mentioned.The most interesting of these are discussed here as they will form part of the se-lection criteria for the next control sample. The veto is on nCDC, while the require-ments concern the single photon itself and use several variables.Cosmics are a major background for the single-photon analysis. Cosmic raysoften leave tracks, however, these tracks do not in general pass through or originatefrom the IP. As such, a veto for these events must not have any location require-78ment, but also must not simply veto on all tracks, lest too high a percentage ofthe signal gets vetoed trying to remove this one background. The nCDC veto waschosen based partly on cosmic ray events.The variable nCDC is the number of CDC hits for a given track. The more hitsthere are, the more confident we are in that track. Tracks may have anywhere from0 to over 100 nCDC. Based on a cosmic study by a collaborator, cosmic tracks havea similar distribution of nCDC to that of other tracks, and if tracks have 10 or moreCDC hits it is extremely likely to be a well-reconstructed track of a charged particle.Therefore, if any track has nCDC > 9, we veto the event. When implementing thisveto in the radiative dimuon sample only tracks aside from those of the muons areconsidered, as specified in Subsection 5.2.5.The photon requirements are designed to ensure that the photon in single-photon events is in a good location, of a sufficient energy, and displaying the qual-ities expected of a photon originating from near the IP.In the selection of the radiative dimuon sample it was discussed why we restrictthe photon to a subset of the barrel, and those reasons still hold for the single-photon analysis. The thetaID range for the single photon is [16, 54] for E∗ >1GeV. For 0.5GeV < E∗ < 1GeV, there is a θ instead of thetaID restriction dueto the range that is part of the trigger that goes down to 0.5GeV. This θ restrictionis (44◦, 98◦), which corresponds to roughly [21, 44] in thetaID.The lower bound for the single photon center of mass energy is 0.5GeV sincethis is around where the lower energy bound for the relevant single-photon triggers,and is also approaching the limit of our sensitivity for single-photon events.Shower shape variables are used to ensure the neutral clusters are consistentwith those of photons by quantifying the shape of the ECL cluster. Using plotsof the various shower shape variables from the radiative dimuon control sampleand from backgrounds, a collaborator determined that ZMVA andC2M have the bestdiscriminating power. For the single photon, the requirements are ZMVA > 0.75,and 0.5cm2 <C2M < 1.5cm2. See Chapter 2 for the definitions of ZMVA and C2M.795.6 Radiative Bhabha control sampleThe radiative dimuon control sample is useful for studying the effects of vetoeson signal, but not for understanding their effects on the background. RadiativeBhabhas are expected to be one of the major sources of background, and are there-fore helpful for studying track cuts. Just as importantly, it is possible to obtaininformation relevant to the fitting for the single-photon analysis. This includes dis-tributions of the single-photon E∗, and the square of the recoil mass against theradiative photon as a function of the single-photon thetaID. However, due to theMC samples available, these plots are not complete. This will be discussed moreprecisely later.The key veto to be studied with this sample is one based on track information,specifically p or pT , and |dz|. This sample will also allow some study of the Etveto. Since radiative Bhabhas are a background, this sample should consist onlyof the events that would be left behind by the other single-photon selection criteriaand vetoes so that we can tune our vetoes to remove as much of this sample aspossible. When comparing to the radmumu sample this radiative Bhabha samplewill allow us to see the effect of the track cut on both signal and background.Once all the vetoes besides the Et veto have been applied, we can examinethe distribution of Et for this background. Figure 5.29 shows the distributions ofthe highest Et associated with a track in each event for this preliminary radiativeBhabha sample. The same plot, but for the radiative dimuon control sample can beseen in Fig 5.28.The distributions themselves are different shapes, with the Bhabha distributiondisplaying significantly more prominent peaks near 4 and 7GeV. However, the keyfeature is the percentage of events that even have a track with a Et : the vast majorityof radiative Bhabha events do, while the inverse is true for radiative dimuon events.As a result, the Et veto significantly reduces the preliminary version of the radiativeBhabha sample, cutting almost 80%, while only removing just over 0.3% of thesignal proxy, the radiative dimuon control sample. This is due primarily to thehigher percentage of events with a track associated Et as most Et values are above100MeV in both samples.One last veto that needs to be discussed in more detail is a veto on the number of80Entries 6232Mean 4.421Std Dev 1.861Underflow 0Overflow 1171Integral 50610 1 2 3 4 5 6 7 8 9 10 (GeV)tE020406080100120140160180200220240Figure 5.29: The distribution of the largest Et in each event for the radiativeBhabha control sample, excluding a cut on Et . There are peaks in theexpected places for a Bhabha event, since Et is a lab quantity, and inthe lab the electron and positron initially have around 7GeV and 4GeVrespectively. Also noticeable is the peak near 0GeV with many trackshaving less than 100MeV.ECL out-of-time crystals (nOOT). This variable, defined in Chapter 2, is indicative ofthe magnitude of the beam background in an event. It was noticed by a collaboratorthat some events had nOOT in excess of 2000, a significant fraction of the totalcrystals in the ECL (O(8000)). These events have degraded clustering and trackingdue to the immense beam background, and are not entirely reliable. In the radiativeBhabha sample, higher energy single-photon events were disproportionately likelyto have a nOOT in excess of 1000, relative to the overall sample. A distribution ofthe nOOT for the otherwise complete radiative Bhabha control sample can be seenin Fig. 5.30.Based on this plot, a veto of nOOT > 400 was chosen, which is also consis-tent with where the collaborator started to notice issues, and with a study beingconducted to calculate the final luminosity values for data.81Entries 1421Mean 147.3Std Dev 115.6Underflow 0Overflow 48Integral 13730 100 200 300 400 500 600 700 800 900 1000OOTn110210Figure 5.30: The number of ECL out-of-time crystals for the radiative Bhabhacontrol sample, excluding a cut on this quantity (nOOT). The majorityof events have a reasonable number of out-of-time crystals, but thereare events in this sample exceeding 500 or even 1000 out-of-time crys-tals.The full radiative Bhabha control sample selection criteria are summarized inTable 5.6. Aside from the p- or pT -by-|dz| veto, which will be determined withthis sample, these are also the current single-photon selection criteria.82Table 5.6: Summary of the selection criteria for the radiative Bhabha control sample. They are split into requirementsthat the events must satisfy, and vetoes which must not be satisfied. These are most of the vetoes and requirementscurrently used in the single-photon analysisType Category CriteriaRequirementsSingle PhotonE∗ > 0.5GeV16≤ thetaID≤ 54If E∗ < 1GeV, 44◦ < θ < 98◦ZMVA > 0.750.5cm2 <C2M < 1.5cm2L1 Trigger At least one of the L1 trigger lines discussed in Chapter 2 is satisfiedHLT At least one of the HLT trigger lines discussed in Chapter 2 is satisfiedVetoesKLMnK ≥ 3nK = 2 and −10ns < tK < 30ns and IK ≤ 4 and αK > 15◦nK = 1 and −10ns < tK < 30ns and IK ≤ 4 and αK > 165◦ECLE > 300MeV100MeV < E < 300MeV and |tc|/dt99 < 3 and αE > 15◦E < 100MeV and |tc|/dt99 < 1 and αE > 176◦Et > 100MeVMiscellaneousnCDC > 9nOOT > 40083This control sample also allows the study of the type of Bhabha events thatmake it through the current criteria. Figure 5.31 is a 2D histogram of the anglesfrom the single photon to the nearest MC electron and nearest MC photon with atleast 400MeV. The 400MeV requirement for the photon is to ensure that it’s aphoton that could conceivably cause a single photon cluster. This plot reveals threedistinct populations: one where the single photon is near an MC photon, one whereit’s near an MC electron and not near an MC photon, and one where single photon isnear neither an MC electron or MC photon. The events where there is an MC photonnear the single photon are what is expected for radiative Bhabha events and occurswhen the radiated photon is in the reduced barrel, and the electron tracks are outof acceptance, or not well reconstructed. As the plot reveals however, the majorityof events do not fall into this category. The events where the single photon is nearan MC electron and not near an MC photon are caused by some of the same issuesas the previous category, but instead the single photon is actually the electron ECLcluster, and tracking failed sufficiently so that the track is not matched to the cluster.In this case, the second electron was out of acceptance or also poorly reconstructed.For the events where the single photon is near both an MC electron and MCphoton, these are likely FSR photons, but the characteristics of the event shouldnot significantly differ from those of the broader two categories. The group wherethere is not an MC electron or MC photon nearby is the smallest of the groupswith only 16 entries. In these events the single-photon cluster must be causedby beam background. There are some important questions and considerations forthese events because of this: are these events that will show up in data, or are theyare an artifact of the way in which MC is constructed? We also need to ensure theseevents are not double counted by showing up in beam background studies andradiative Bhabha studies. The beam background studies are currently underway,and will help us understand how to treat these events.In all the radiative Bhabha control sample events the electrons have been missedto at least some degree, and so it is interesting to see where they are in the detec-tor. Figure 5.32 shows the θ vs φ distribution for the scattered MC electron andpositron. There is a large concentration of events around a θ of 12◦, the smallestθ in ECL acceptance. This occurs because there is an ever increasing number ofBhabha events at lower angle, and the Bhabha events in MC15rd are generated840 20 40 60 80 100 120 140 160 180 )° ( ±e3D angular separation of SP and MC 020406080100120140160180 )° ( γ3D angular separation of SP and MC Entries 1327Mean x 10.92Mean y 67.84Std Dev x 24.24Std Dev y 44.99Integral 1327 0 0 0 0 1327 0 0 0 0020406080100120140160Figure 5.31: The 3D angle between the single photon and the nearest MCphoton over 400MeV is shown as a function of the 3D angle betweenthe single photon and the nearest MC electron. This reveals the truecause of the single-photon clusters, and that in the majority of casesthis an electron, not a photon. There are also 16 events where thesingle photon is not close to either an MC photon or electron. In theseevents the single-photon cluster is caused by beam background.with the requirement that both electrons are within ECL acceptance. Unfortunately,events with lower angle Bhabhas are present in data since they occur in Belle II,and as such MC cannot accurately reflect data in this respect. The potential im-portance of this background indicates, however, that a more useful sample may beimperative to this analysis. This will be discussed further in the next section on thetrack veto.It is key to note that at this point, no veto on tracks with few CDC hits or lowEt has been applied. This means some events where there is a high-energy track— likely caused by an electron — have not yet been vetoed. With that caveat,Fig. 5.33 and Fig. 5.34 display the CMS energy of the single photon as a functionof the cluster thetaID, and the squared mass recoiling against the single photon as85150− 100− 50− 0 50 100 150 )° ( φ020406080100120140160180 )° ( θEntries 2654Mean x 2.054− Mean y 55.12Std Dev x 98.65Std Dev y 46.53Integral 2654 0 0 0 0 2654 0 0 0 005101520253035Figure 5.32: θ plotted against φ for the non-initial MC electrons in eachevent. There are two entries per event. There is a concentration ofelectrons at low θ , at the edge of the ECL acceptance, however thereare also many electrons throughout the barrel θ range. For the low an-gle Bhabhas there is a φ dependency, with more electrons being near0◦ than near ±180◦, a dependency that is not seen elsewhere in thisplot. This is caused by the boost, which is not perfectly along the z-axis, and consequently biases particles toward 0◦ in φ .a function of the thetaID.5.7 Track VetoesVetoes on track information are designed to remove events that likely contain acharged particle, since those are not necessarily caught by previous vetoes. Themere presence of a track is not sufficient to veto events, since many tracks arecaused by the beam background even in signal events, and we don’t want to vetothose events. Beam background is an important consideration, particularly look-ing into the future, and even at the latter part of the current dataset, where thedetrimental effects of an ever increasing instantaneous luminosity are obvious. By8620 30 40 50 60 70thetaID123456 (GeV)E*Entries 1327Mean x 38.43Mean y 1.468Std Dev x 11.5Std Dev y 0.491Integral 1327 0 0 0 0 1327 0 0 0 0024681012141618Figure 5.33: The distribution of E∗ as a function of the thetaID for the sin-gle photon, made using the radiative Bhabha control sample. ThethetaID range narrows below 1GeV because of the triggers that willbe used for the single-photon analysis. Also below 1GeV there arefewer events than there likely should be due to the angle requirementsused in generic MC15rd Bhabha production.using the track information, a more selective veto can be made, one tailored tocatch higher-energy particles that are collision related.To achieve this selection, the parameters used need to relate to the energy of thetrack, and somehow measure its location in the detector. The best available optionsfor energy are the momentum or transverse momentum, p or pT . There are also twomain options for location information, the transverse or longitudinal distance to theIP from the POCA, denoted by dr and dz respectively. pT , dz, and dr are alreadyin use together in a cut in the single-photon skim, a reduced data sample that isused to lessen the demand on computing resources. This cut is already applied inthe single-photon skim that will be used for the single-photon analysis. We cannotavoid this, but we can apply a tighter veto, and veto more events than the skimdoes. The skim track veto is shown in Table 5.7. Since a tighter veto is possible,8720 30 40 50 60 70thetaID020406080100)4c/2 (GeVR2mEntries 1327Mean x 38.43Mean y 80.87Std Dev x 11.5Std Dev y 10.4Integral 1327 0 0 0 0 1327 0 0 0 00246810Figure 5.34: The distribution of the square of the mass recoiling against thesingle photon as a function of the thetaID for the single photon, madeusing the radiative Bhabha control sample. This plot uses the entireradiative Bhabha control sample, and has no track cuts applied.it is entirely possible to have no requirement of dr for the track veto, allowing usto make a 2D pT -by-|dz| veto. Alternatively, we could simply apply a differentveto, using p instead of pT for example, and be cognizant of the skim veto whilestudying it.Table 5.7: Summary of the skim track veto. All three criteria must be satisfiedto veto the event.Veto criteria Explanation of variablepT > 0.15GeV/c Transverse momentum of a track|dz| < 2cm Absolute z distance from POCA to IPdr < 0.5cm Transverse distance from POCA to IPThe p- or pT -by-|dz| cut is meant to catch events where a track was recon-structed that originates from the IP and has sufficient transverse momentum to besomething we don’t want. Radiative Bhabhas, particularly those with higher energy88radiated photons should have tracks like this due to recoil of the photon. However,we are also interested in vetoing events with tracks that do not originate from theIP but that do have a POCA nearby. These tracks could be caused by secondarycharged particles. In the case of Bhabhas, tracks like these might be caused by theBhabhas interacting with the detector just outside of acceptance.While pT is used in the skim veto, there’s no reason to assume that it is nec-essarily better than p for our purposes. In order to avoid needing to look at allpossible combinations of the cut parameters in detail, receiver operating character-istic (ROC) curves were made for a few p or pT and |dz| combinations. These canbe seen in Fig. 5.35.These curves show the fraction of the radiative Bhabha control sample rejectedby the specific veto as a function of the fraction of the radiative dimuon controlsample kept. In these plots the ideal curve would pass through (1,1), and the closerto this point the better. The upper plot uses MC for both samples, while the loweruses data for the radiative dimuon control sample. For the majority of the space,p is significantly better than pT , and where it is not, they are not far apart, andmay in fact be effectively tied in efficacy. While this conclusion is specifically for|dz| < 10cm, the curves for pT -by-|dz| < 20cm is nearly identical to that of pT -by-|dz| < 10cm, and so the conclusion will likely hold for other |dz| values too.Based on these ROC plots, p will be used instead of pT , and a good starting placethis veto would be rejecting somewhere in the neighbourhood of 60–80% of theradiative Bhabha control sample.Now that p has been selected over pT , we can look at the relevant 2D parameterspace for track veto. Plots of this space can be seen in Fig. 5.36.890.9 0.92 0.94 0.96 0.98 1 control sample keptγ-µ+µFraction of 00.20.40.60.81 control sample rejectedγ-e+eFraction of , |dz| < 10 cmTp , |dz| < 20 cmTp , |dz| < 10 cmp0.92 0.94 0.96 0.98 1 control sample keptγ-µ+µFraction of 00.20.40.60.81 control sample rejectedγ-e+eFraction of , |dz| < 10 cmTp , |dz| < 20 cmTp , |dz| < 10 cmpFigure 5.35: ROC curves for the possible track vetoes, using the radiativedimuon control sample as a proxy for signal. On the y-axis is thefraction of the radiative Bhabha control sample vetoed, and on the x-axis is the fraction of the radiative dimuon control sample kept. Theupper plot uses MC for both control samples, while the lower plot usesdata for the radiative dimuons. For a pT -by-|dz| veto there is little dif-ference between a bound of 10cm or 20cm, while a p-by-|dz| veto isgenerally better for the same |dz| bound.900 1 2 3 4 5 6 7 8 9 10|dz| (cm)00.20.40.60.811.21.41.61.82)c (GeV/pEntries 5348278Mean x 3.119Mean y 0.1942Std Dev x 2.636Std Dev y 0.1952Integral 5.031e+05 0 5230 4839920 0 503128 0 0 0 0020004000600080001000012000140000 1 2 3 4 5 6 7 8 9 10|dz| (cm)00.20.40.60.811.21.41.61.82)c (GeV/pEntries 2.36105e+07Mean x 3.116Mean y 0.194Std Dev x 2.617Std Dev y 0.1918Integral 2.279e+06 0 23222 21308462 0 2278814 0 0 0 0010000200003000040000500000 1 2 3 4 5 6 7 8 9 10|dz| (cm)00.20.40.60.811.21.41.61.82)c (GeV/pEntries 1327Mean x 2.738Mean y 0.4239Std Dev x 2.67Std Dev y 0.3821Integral 1193 0 125 0 0 1193 9 0 0 0024681012141618Figure 5.36: Lab momentum as a function of |dz| for the track with the highest momentum that also satisfies |dz| <10cm. The upper two plots are made using the radiative dimuon sample, data on the left, MC on the right. Thelower plot is for the radiative Bhabha control sample. The radiative dimuon plots have most of their entries inthe top right overflow bin because there is no track that satisfies the required |dz|.91These plots are difficult to visually compare due to the differing statistics,but the mean values reveal significantly higher average momentum values, andsomewhat smaller |dz| values in the radiative Bhabha sample than in the radiativedimuon sample. Data and MC are in good agreement for radiative dimuons. It’salso noticeable that the majority of events in the radiative dimuon plots are in thedouble overflow bin, which overwhelmingly corresponds to no track in the event.This is in contrast to the radiative Bhabha sample where there are no events in thatbin.Finally to select the values for the boundaries of the p-by-|dz| veto it is usefulto look at the veto percentage as function of the p value for several different |dz|values, and then compare these percentages from the radiative dimuon and radiativeBhabha samples. Plots of these percentages can be seen in Fig. 5.37. Note thatwhen using |dz| < 10cm, and p > 0GeV/c for the veto, the plot for the radiativeBhabhas indicates that ∼ 100% of events are vetoed — compared to ∼ 10% forthe radiative dimuon sample — which means virtually all events have a track thatoriginates near the IP, which was somewhat unexpected. This is discussed furtherbelow.92Figure 5.37: Percentage of the sample vetoed as a function of the momentum bound. The upper two plots are madeusing the radiative dimuon sample, data on the left, MC on the right. The lower plot is for the radiative Bhabhacontrol sample. The radiative dimuon plots veto at most around 10% of events, because the rest of the eventsdo not have any extra tracks. On the other hand, nearly 100% of radiative Bhabha tracks have tracks, and so arevetoed when using a large |dz| upper bound and a lower p bound of 0GeV/c.93These plots indicate that a veto of p> 0.2GeV/c and |dz|< 10cm vetoes around72% of the radiative Bhabha control sample, and just over 2% of the radiativedimuon control sample. It is also in the region of the plots where, as the p-bound islowered, the percentage of radiative Bhabhas vetoed increases quickly, but is justbefore where the percentage of radiative dimuons vetoed increases rapidly. In otherwords, this is a more conservative veto for now, since there’s still more informa-tion required to justify a stronger veto. Some of that necessary information will, infuture, come from a new radiative Bhabha MC sample.As mentioned above, virtually all radiative Bhabha control sample events havetracks. This is assumed to be a result of how this sample of MC was generated,because the current MC15rd sample requires both electrons be within ECL accep-tance. This results in there being no events in which the electrons are at low enoughangles to not leave tracks, and as such, this sample isn’t enough to decide this veto.Another MC sample has been requested, which would also use MC15rd but havedifferent angle requirements. This sample will require that either an energetic pho-ton or one of electrons be within the barrel, while at least one of the electrons isoutside of the ECL acceptance. The sample will cover an entirely separate set ofBhabha events than the generic MC15rd, and will be used to update the resultsin this section. Full optimization, however, will require all relevant backgroundsto be sufficiently studied so that a more quantitative measure, such as a figure ofmerit, can be used to determine the ideal veto bounds. The study of this trackveto will need to be revisited when the requested sample is available, and the otherbackgrounds are better understood.Despite the temporariness of this vetoes’ boundaries, applying it to the radiativeBhabha control sample is useful because the plots of E∗ and recoil mass squaredas a function of thetaID are demonstrative of the final result for this background.The skim veto is also applied for these plots, though it only vetoes 12 extra events.The aforementioned plots are Fig. 5.38 and Fig. 5.39 respectively.With the track veto applied, the radiative Bhabha cross section above 0.5GeVis 9.1 fb and above 2GeV is 0.77 fb.9420 30 40 50 60 70thetaID123456 (GeV)E*Entries 374Mean x 38.13Mean y 1.427Std Dev x 11.89Std Dev y 0.4927Integral 374 0 0 0 0 374 0 0 0 0012345678Figure 5.38: The distribution of E∗ as a function of the thetaID for the sin-gle photon for the radiative Bhabha control sample with the track vetoapplied. The skim veto is also applied, though its affect is small. Thetrack veto removed almost 75% of control sample events, and particu-larly removed events with a higher single-photon E∗.20 30 40 50 60 70thetaID020406080100)4c/2 (GeVR2mEntries 374Mean x 38.13Mean y 81.68Std Dev x 11.89Std Dev y 10.46Integral 374 0 0 0 0 374 0 0 0 00123456Figure 5.39: The distribution of the square of the mass recoiling against thesingle photon as a function of the thetaID for the single photon, madeusing the radiative Bhabha control sample with the track veto applied.The skim veto is also applied, though its affect is small.955.8 Veto SummaryA summary of the single-photon vetoes discussed in this chapter are shown inTable 5.8. This table also includes the veto percentages of the control sample dueto these cuts individually and combined, for both data and MC. The table also hasthe ratio of the veto percentages, which makes it apparent where MC best representsdata and where it does not.As expected, MC and data do not agree for the few-layer KLM vetoes, whichmakes sense since there was marked disagreement between data and MC for a va-riety of KLM quantities for KLM clusters with few layers. Interestingly, in general,MC tends to underestimate the KLM veto percents, and particularly underestimatesthe 1-layer veto. This is not what might initially be expected, because one of themajor disagreements between MC and data for KLM clusters is caused by multi-strip hits, and tends to result in more clusters in MC than data. The increased num-ber of clusters would be expected to cause higher KLM veto percentages in MC,not lower veto percentages. However, most of the data-MC discrepancy relatedto multi-strip hits for 1- and 2-layer KLM clusters is due to KLM clusters causedby the recoil photon (see Fig. 5.22), and because we intentionally constructed theKLM vetoes such that single-photon-caused KLM clusters do not veto events, themulti-strip hit issue does not factor in very much to the data-MC veto percentagedisagreement.Conversely to the KLM vetoes, MC overestimates the effect the photon vetoeshave, with the higher energy vetoes being significantly worse. This was not entirelyexpected, as MC has many more extra photons present, increasing the chances ofone meeting the veto requirements.The vetoes on Et , nCDC, and p-by-|dz| all have excellent data-MC agreement.The veto percentage for the nOOT veto could not be checked with current ntuples,but will be calculated in future.96Table 5.8: A summary of the vetoes described in this thesis for the single-photon analysis. The percentage of theradiative dimuon control sample vetoed by each veto is tabulated (if known) for data, MC, and their ratio. Theseveto percentages are calculated with each veto acting individually. The total percentage vetoed is also displayed.Category Veto % vetoedMain criterion Other criteria Data MC Data/MCKLMnK ≥ 3 – 0.58 0.54 1.1nK = 2−10ns < tK < 30nsIK ≤ 4 0.15 0.11 1.3αK > 15◦nK = 1−10ns < tK < 30nsIK ≤ 4 0.17 0.081 2.1αK > 165◦ECLE > 300MeV – 0.37 0.62 0.60100MeV < E < 300MeV|tc|/dt99 < 3 3.4 4.4 0.77αE > 15◦E < 100MeV|tc|/dt99 < 1 0.16 0.17 0.96αE > 176◦Et > 100MeV – 0.33 0.32 1.0Tracks nCDC > 9 – 1.6 1.6 1.0p> 0.2GeV/c |dz|< 10cm 2.6 2.7 0.97Miscellaneous nOOT > 400 – – – –Combined – – 8.2 9.4 0.8797Since the mid energy photon veto is the biggest veto, and the other large vetoeshave good data-MC agreement, the overall data-MC disagreement is small, but notinsignificant, with a data/MC ratio of 0.87. It’s also interesting to note that thedifferent vetoes are largely orthogonal. The KLM vetoes, and the extra photon ECLvetoes are fully orthogonal within their category by design, but there is no reasonthat the other vetoes have to be. If all the vetoes were fully orthogonal 9.4%, and11% of data and MC would be vetoed, respectively, instead of the actual 8.2%, and9.4%.98Chapter 6Conclusion and prospectsThe dark photon is a hypothetical particle in the dark sector. It is a massive gaugeboson, and mediates a dark equivalent of the electromagnetic force. The darkphoton mixes with the regular photon, allowing its production and detection. AtBelle II, the dark photon can be produced by the annihilation of an e+e− pair. Ifthe dark photon mass is large enough relative to the mass of other dark matter par-ticles, it can decay invisibly. These events can be detected when they involve theinitial state radiation of a photon. Belle II is well suited for this analysis due to theclean environment of the e+e− collisions, its dedicated single-photon triggers, andits hermiticity.The single-photon analysis is the search for the invisible decay of the darkphoton at Belle II. An overview of the single-photon analysis and its componentswas given. Two components of this analysis were presented, namely the study ofa signal proxy, and the study of the radiative Bhabha background. Vetoes for thesingle-photon analysis designed from these studies were explained, and their vetopercentages were calculated for both data and MC.The next step for the work presented in this thesis is the reproduction of the ra-diative Bhabha control sample with a new MC sample generated specifically for thisstudy. This will permit a more complete study of the radiative Bhabha background,particularly at low angle and low centre of mass energy. With this information, andthe information from the other backgrounds, the track cut can be adjusted so thatit improves the sensitivity of the search. The recoil mass squared as a function of99thetaID plot can also be adjusted, which will be used in the fitting of the radiativebhabha background.In the next version of Belle II analysis and reconstruction software, there aretwo changes that are particularly relevant for the single-photon analysis: multi-strip hits will be used in reconstruction, and the ECL clustering algorithms havebeen fixed.The multi-strip hit change will allow far more photon-caused KLM clusters tobe reconstructed correctly, or reconstructed at all. This will improve the rejectionof background events, particularly radiative Bhabha and γγ(γ) events, as the KLMmay detect photons when the ECL has failed.The fixed ECL clustering will also improve background rejection by correctlyreconstructing more than one photon. This effect will be most prominent in eventswith a larger beam background, like those recorded near the end of data-taking.For the analysis as a whole to progress, the other components need to be com-pleted. Several of the sections have already been completed by other members ofthe analysis team, or are currently underway. The major outstanding componentsare the study of signal MC, and data-MC corrections. After these are completed,and all components finalized, the systematic uncertainty estimations, and unblind-ing can commence.The single-photon analysis will, once complete, explore new parameter spacefor the dark photon. This parameter space includes regions in which the dark pho-ton could explain the relic density of dark matter in the universe. This will con-tribute meaningfully to the search for dark matter. 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Master’sthesis, University of British Columbia, 2021. → page 28104Appendix ASupporting MaterialsA.1 Trigger linesThe full HLT lines required as part of the radiative Bhabha control sample are listedbelow with an explanation of the requirements:• software trigger cut&filter&ge1 Estargt2 GeV neutral clst 32130 notgg2clst ee1leg1clst ee1leg1trk eeBrem– At least one E∗ > 2GeV neutral ECL cluster in θ range [32◦, 130◦]and not meeting the requirements of a γγ event or a partially detectedBhabha event.• software trigger cut&filter&1 photon Estargt1 GeV clust in 45115and no other clust Estargt0.3 GeV– Exactly one E∗ > 1 photon ECL cluster in θ range [45◦, 115◦] and noother E∗ > 0.3GeV ECL clusters• software trigger cut&filter&1 photon Estargt0.5 GeV clust in 4498and no other clust Estargt0.3 GeV– Exactly one E∗ > 0.5 photon ECL cluster in θ range [44◦, 98◦] and noother E∗ > 0.3GeV ECL clusters105Similarly, the L1 trigger lines required as part of the radiative Bhabha controlsample are listed below with an explanation of the requirements:• hie– Total combined energy> 1GeV in the ECL crystals with 4≤ thetaID≤58. The event must not be vetoed for being classified as a Bhabha orby the injection veto• lml6– Exactly one E∗ > 1GeV cluster with 13 ≤ thetaID ≤ 58 and no otherE > 0.3GeV cluster. The event must not be vetoed by the injectionveto.• lml13– Exactly one E∗ > 0.5GeV cluster with 21≤ thetaID≤ 44 and no otherE > 0.3GeV cluster. The event must not be vetoed by the injectionveto.• lml16– Exactly one E∗ > 0.5GeV cluster with 21≤ thetaID≤ 44 and no otherE > 0.3GeV cluster. The event must not have a track (as defined by theL1 trigger, or be vetoed by the injection veto.The requirements for classification as a Bhabha event for the L1 trigger are asfollows:1. There are at least two ECL clusters, one of which has E∗> 3GeV and anotherthat has E∗ > 4.5GeV2. The sum of the CMS frame polar angles for those two clusters is in the range[165◦, 190◦]3. The difference of the CMS frame polar angles for those two clusters is in therange [160◦, 200◦]106
Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Hearty, Christopher
Signal proxy and radiative Bhabha background studies : contributions to the search for invisibly decaying dark photons at Belle II
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