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Measuring the velocity and mass of stable massive particles using the ATLAS detector at the LHC Mills, William 2012

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Measuring the Velocity and Mass of Stable Massive Particles using the ATLAS detector at the LHC  by William Mills B.Sc. Simon Fraser University, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Physics)  The University Of British Columbia (Vancouver) August 2012 c William Mills, 2012  Abstract A generic search for new detector-stable high-mass particles at the LHC using the ATLAS detector is conducted, by measuring the velocity of candidate tracks in ATLAS’ Inner Detector. Four novel new velocity estimation techniques are developed, the best of which achieve velocity resolutions of better than 8% in simulation. 2.8 fb−1 of 2011 ATLAS data is analyzed via these methods, yielding no significant signal for new stable massive particles at the sub-TeV scale, and an exclusion limit at the 95% confidence level for the existence of generic stable massive gluinos below a mass of 665 GeV.  ii  Preface This research was produced as part of the ATLAS collaboration at the LHC (CERN). The analysis contains both the author’s work, and the contributions of his supervisor Colin Gay; Colin produced the original prototype code behind the velocityfinding algorithms, after which the author managed its development and validation into its final state. In addition to the central analysis code, the use of the SYNTMaker and D3PDMaker software packages was instrumental in processing standard centralized ATLAS data into a format suitable for end-user analysis. At the time of the author’s involvement, the SYNTMaker package was managed by Christian Ohm (Stockholm) et al, although the author was solely responsible for the expansion of this package to include the information relevant to this analysis (specifically, the TRT suite of variables). For the final 2011 processing, this processing was migrated to the D3PDMaker package managed by Sascha Mehlhase (Niels Bohr Institute).  iii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiii  Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xiv  1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  2  Theory Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4  2.1  Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . .  4  2.2  Universal Extra Dimensions . . . . . . . . . . . . . . . . . . . .  8  2.3  Hyperquarks & Vector Confinement . . . . . . . . . . . . . . . .  8  2.4  Stable 4th Generation Quarks . . . . . . . . . . . . . . . . . . . .  9  2.5  Leptoquark Mediators . . . . . . . . . . . . . . . . . . . . . . . .  9  2.6  Magnetic Monopoles . . . . . . . . . . . . . . . . . . . . . . . .  10  Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  11  3.1  13  3  Inner Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv  3.1.1  Pixel Detector . . . . . . . . . . . . . . . . . . . . . . . .  15  3.1.2  Semiconductor Tracker . . . . . . . . . . . . . . . . . . .  15  3.1.3  Transition Radiation Tracker . . . . . . . . . . . . . . . .  16  Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  17  3.2.1  Liquid Argon Calorimeters . . . . . . . . . . . . . . . . .  17  3.2.2  Tile Calorimeter . . . . . . . . . . . . . . . . . . . . . .  19  Muon System . . . . . . . . . . . . . . . . . . . . . . . . . . . .  19  3.3.1  Monitored Drift Tubes . . . . . . . . . . . . . . . . . . .  20  3.3.2  Cathode Strip Chambers . . . . . . . . . . . . . . . . . .  20  3.3.3  Resistive Plate Chambers . . . . . . . . . . . . . . . . . .  21  3.3.4  Thin Gap Chambers . . . . . . . . . . . . . . . . . . . .  21  The TRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22  3.4.1  Ideal Operation . . . . . . . . . . . . . . . . . . . . . . .  22  Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26  4.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26  4.1.1  Falling Edge β Estimator . . . . . . . . . . . . . . . . . .  26  4.1.2  Rising Edge β Estimator . . . . . . . . . . . . . . . . . .  44  4.1.3  Track Fit Optimization β Estimator . . . . . . . . . . . .  47  4.1.4  dE/dx-Correlate β Estimator . . . . . . . . . . . . . . . .  50  4.1.5  β and Mass Calibration . . . . . . . . . . . . . . . . . . .  53  Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . . .  56  Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  60  5.1  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  60  5.1.1  Falling Edge β Estimator . . . . . . . . . . . . . . . . . .  62  5.1.2  Rising Edge β Estimator . . . . . . . . . . . . . . . . . .  70  5.1.3  Refit β Estimator . . . . . . . . . . . . . . . . . . . . . .  74  5.1.4  dE/dx Correlate β Estimator . . . . . . . . . . . . . . . .  77  5.1.5  Correlations . . . . . . . . . . . . . . . . . . . . . . . . .  82  ABCD Background Estimation . . . . . . . . . . . . . . . . . . .  83  Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  91  6.1  91  3.2  3.3  3.4  4  4.2 5  5.2 6  β Estimators  β Estimators  Signal Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . v  7  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  98  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Appendix A ABCD Background Estimation . . . . . . . . . . . . . . . 105  vi  List of Tables Table 2.1  Mass limits and possible pair-production cross-sections for some interesting SUSY-related CHAMPs . . . . . . . . . . . . . . .  7  Table 4.1  Selection cuts used to make the FE β = 1 PDFs. . . . . . . . .  33  Table 4.2  Selection cuts used to make the RE β = 1 PDFs. . . . . . . . .  46  Table 4.3  Selection cuts used to clean data for the track fit optimization β estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  49  Table 4.4  Selection cuts used to clean data for the dE/dx-driven β estimator. 52  Table 5.1  Cuts applied in constructing FE-driven β width. . . . . . . . .  63  Table 5.2  Cuts applied in constructing RE-driven β width. . . . . . . . .  72  Table 5.3  Cuts applied in constructing refit-driven β width. . . . . . . . .  76  Table 5.4  Cuts applied in constructing dE/dx-driven β width. . . . . . . .  80  Table 5.5  Residual correlations between the four β estimators. . . . . . .  83  Table 5.6  Maximum signal significance found at each simulated mass point after stepping through an array of cuts. . . . . . . . . . . . . .  Table 5.7  89  Tabulation of trials to determine systematic on background estimation due to small dependency between βRE and pT . . . . .  89  Table 6.1  Sample definition cuts as optimized in the Chapter 5. . . . . . .  92  Table 6.2  A few parameters of the tracks observed in our signal region. .  96  vii  List of Figures Figure 2.1  Proton decay via an anti scalar strange sfermion. . . . . . . .  6  Figure 3.1  Schematic of the ATLAS detector. . . . . . . . . . . . . . . .  12  Figure 3.2  Cross-sectional cutaway of 1/4 of the Inner Detector. . . . . .  14  Figure 3.3  Digitization scheme for a hit in the TRT. . . . . . . . . . . . .  24  Figure 4.1  Example of a PDF for falling edge position and a collection of falling edges from a single track. . . . . . . . . . . . . . . . .  29  Figure 4.2  Effect of signal reflections on timing of TRT LT information. .  31  Figure 4.3  Bethe-Bloch ionization curves in the TRT active gas. . . . . .  35  Figure 4.4  Example of PDFs of the falling edge position, for hits produced by tracks crossing near the straw edge. . . . . . . . . .  Figure 4.5  38  Example of PDFs of the falling edge position, all taken at |η| < 0.2, in steps of 0.15 mm in track radius. . . . . . . . . . . . .  40  Figure 4.6  Unprocessed scatterplot of βFE versus βtrue . . . . . . . . . . .  54  Figure 4.7  βtrue spectrum for some gluinos in MC. . . . . . . . . . . . .  55  Figure 4.8  Division of a 2D region into control and signal regions for an ABCD background estimation. . . . . . . . . . . . . . . . . .  57  Figure 5.1  Relative FE-driven β width in simulation. . . . . . . . . . . .  61  Figure 5.2  Scatterplot of βtrue vs. the calibrated βFE . . . . . . . . . . .  61  Figure 5.3  Mass dependence of bias in FE-driven β estimator. . . . . . .  64  Figure 5.4  FE-driven mass resolution for our MC CHAMP sample. . . .  64  Figure 5.5  Mass dependence of bias in FE-driven mass estimator. . . . .  65  Figure 5.6  Reconstructed proton mass peak from simulated minimum bias data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  67  Figure 5.7  As Figure 5.6, stacked by particle ID. . . . . . . . . . . . . .  Figure 5.8  Reconstructed proton mass peak from real 2010 minimum bias  Figure 5.9  68  data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  68  Conditional bimodality of FE PDFs. . . . . . . . . . . . . . .  69  Figure 5.10 Toy simulation qualitatively reproducing the bimodality of Figure 5.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  70  Figure 5.11 Relative RE-driven β width in simulation. . . . . . . . . . . .  71  Figure 5.12 Scatterplot of βtrue vs. the calibrated βRE . . . . . . . . . . . .  71  Figure 5.13 Mass dependence of bias in RE-driven β estimator. . . . . . .  73  Figure 5.14 RE-driven mass resolution for our MC CHAMP sample. . . .  73  Figure 5.15 Mass dependence of bias in RE-driven mass estimator. . . . .  74  Figure 5.16 Relative refit-driven β width in simulation. . . . . . . . . . .  74  Figure 5.17 Scatterplot of βtrue vs. the calibrated βRe f it . . . . . . . . . . .  75  Figure 5.18 Mass dependence of bias in refit-driven β estimator. . . . . .  76  Figure 5.19 Refit-driven mass resolution for our MC CHAMP sample. . .  78  Figure 5.20 Mass dependence of bias in refit-driven mass estimator. . . . .  78  Figure 5.21 Relative dE/dx-driven β width in simulation. . . . . . . . . .  79  Figure 5.22 Scatteplot of βtrue vs. the calibrated βdE/dx . . . . . . . . . . .  79  Figure 5.23 Mass dependence of bias in dE/dx-driven β estimator. . . . .  81  Figure 5.24 dE/dx-driven mass resolution for our MC CHAMP sample. . .  81  Figure 5.25 Mass dependence of bias in dE/dx-driven mass estimator. . . .  82  Figure 5.26 Trigger efficiency for our signal by EF mu22 as a function of true β and transverse momentum. . . . . . . . . . . . . . . .  86  Figure 5.27 Ratio of muons from Z decays to the entire data sample considered from the EF mu22 trigger, as a function of pT . . . . .  87  Figure 5.28 Production cross section as a function of mass for the gluinobased r-hadrons. . . . . . . . . . . . . . . . . . . . . . . . .  88  Figure 5.29 MC signal acceptance for search region. . . . . . . . . . . . .  90  Figure 6.1  Results from ABCD analysis for 2.8 fb−1 of 2011 ATLAS data. 92  Figure 6.2  Reconstructed βRE for the tracks in Figure 6.1. . . . . . . . .  93  Figure 6.3  Stable gluino exclusion limit. . . . . . . . . . . . . . . . . . .  96  ix  Figure 6.4  FE-driven mass estimates for the five events observed in our signal region. . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure A.1  97  Division of a 2D region into control and signal regions for an ABCD background estimation. Note that the variables x and y must be independent for the background in order for such an estimator to be appropriate. . . . . . . . . . . . . . . . . . . . 105  x  Glossary Anomaly Mediated Supersymmetry Breaking  AMSB  A Toroidal Lhc ApparatuS  ATLAS BSM  Beyond the Standard Model  CBF  Crystal Ball Function  CHAMP  Forward Calorimeter  FCAL FE  Falling Edge (of TRT LT pattern)  GMSB  Gauge Mediated Supersymmetry Breaking  LAr Hadronic Endcap Calorimeter  HEC HT  CHArged Massive Particle  High Threshold  LAR  Liquid Argon  LHC  Large Hadron Collider  LSP LT  Lightest Supersymmetric Particle Low Threshold  MSSM NLSP  Minimal Supersymmetric Standard Model Next-to-Lightest Supersymmetric Particle xi  QCD  Quantum Chromodynamics  QED  Quantum Electrodynamics  RE  Rising Edge (of TRT LT pattern)  RPC  Resistive Plate Chamber  SCT  Semiconductor Tracker  SM  Standard Model  SUGRA SUSY  SUperGRAvity  SUperSYmmetry  TRT  Transition Radiation Tracker  UED  Universal Extra Dimensions  xii  Acknowledgments The author would like to acknowledge the tremendous support afforded him by the ATLAS community at large, and particularly that of the stable massive particle working group at CERN, Copenhagen, Stockholm and Indiana, without which the navigation of this project would have been unmanageable.  xiii  To my parents, for making this possible.  xiv  Chapter 1  Introduction This work uses natural units, c = h¯ = 1, everywhere. By the end of the 20th Century, particle physics had a lot to answer for. Galaxies were spinning too quickly for their visible mass, and astrophysicists were demanding heretofore unseen dark matter to make up the difference; the extreme fine tuning forced by the hierarchy problem to separate the weak and gravitational scales in the absence of some new physics was attracting acute skepticism; theories incorporating new spacial dimensions that could support massive excitations of existing particles were proliferating; no compelling theoretical reason for the number of generations of quarks to be limited to three was accepted, and Dirac’s long sought magnetic monopole continued to have far more interest than experimental support; and, perhaps most of all, physicists were still waiting to see if the Higgs, linchpin of the Standard Model, actually existed. All these ideas were to be served by the Large Hadron Collider, but at its outset no one could say if experiment would uncover any of the new particles which attempts to address these problems anticipated. Given this air of possibility, we sought to position the present study to see as many sorts of new physics as possible; with the LHC opening up discovery potential for so much new physics, a study concerning only a very particular decay mode of a very particular new particle didn’t seem in keeping with the scope of new physics potentially available. As we review in a later chapter, many disparate theories include the potential for new detector stable particles with masses on the 1  order of 100 GeV or greater; while absolutely all quasi-stable Standard Model particles will be traveling near the speed of light above an easily-imposed momentum threshold, these new objects, with masses of at least a few percent of the LHC center of mass energy, will have substantial support at speeds on the order of tens of percent of c; therefore, a well-performing velocity estimator will be sensitive to a very generic class of new physics in the essentially background free regime of low β and high momentum. After settling on velocity as our broad-based discriminator of choice, a strategy for measuring it had to be developed. In order to not only estimate velocities, but to also bring something unique to the ATLAS experiment and the field of collider measurements and to showcase the performance of the Transition Radiation Tracker (the TRT, an ATLAS subsystem to which our group at UBC made a significant contribution), we sought to develop some novel velocity estimators using the TRT in the Inner Detector, ATLAS’ precision tracker. At only about a meter in radius and four meters in length, the very compact size of the TRT makes velocity estimation therein seem problematic, compared to using one of the subsystems with much longer lever arms already in use in similar studies, such as the tile calorimeter in the barrel or the resistive plate chamber (RPC) subdetector in the muon system with its excellent timing well suited to a time-of-flight measurement of β . Nevertheless, we decided to press on and investigate the feasibility of using a relatively cheap (compared to the pixel tracker, a semiconducting system which makes an excellent dE/dx estimate from which relativistic γ and β can be extracted), compact (compared to the large systems mentioned above) gas-based detector to make a competitive independent measurement of velocity using some novel techniques not yet widely pursued by the collaboration, so that future such experiments might be able to access similar information even in the absence of ATLAS’ tremendous scale or budget. The advantage that the TRT presents in this study (and which in later chapters we will exploit to great effect), is the relatively very large number of measurements it makes per track; as will be expanded on in the Detector chapter, the TRT makes an average of 30 measurements containing timing and ionization information per track, while the other subsystems employed in measuring velocity in other studies typically make fewer than 5 measurements. By developing the detailed understanding of how signal and background interact 2  with the TRT and its digitization that will be elucidated in the following work, this tremendous volume of information would prove to hold the resolving power for β that we seek.  3  Chapter 2  Theory Overview Many theories [1–13] anticipate the possible emergence of new physics at the LHC that includes new particles stable enough to leave a direct track in the ATLAS detector, while being far more massive than any detector-stable Standard Model particle. Owing to the dramatic new reach of the LHC, no data exists at the time of this writing to refute the possibility of seeing any of these new massive particles at the TeV scale, and as such we have designed as inclusive a search for them as possible. What follows is a very brief survey of just some of the theories that might produce a signal in the present inclusive analysis. For a more thorough menu, see the comprehensive survey provided by [3].  2.1  Supersymmetry  One feature of the Standard Model which is immediately evident and aesthetically irksome is the apparent strict division in spin statistics between the force carriers, and the generations of particles which exchange them. Supersymmetry (SUSY) adds a symmetry between fermions and bosons to the theory which not only removes this unsightly dichotomy, but also can be constructed to effectively address many major problems in particle physics today; dark matter [14–18], the hierarchy problem [19–21], and grand unified theories [19, 22–24] are all well served by the introduction of SUSY. Owing to the plethora of different formulations of SUSY which have been  4  written down in efforts to solve these problems, many different opportunities for detector-stable charged supersymmetric particles (sparticles) exist; excellent reviews of the relevant particle content of these options and the corresponding detector searches and literature are given in [3] and most recently in [1], the highlights of which we summarize here. A feature of general SUSY theories which at first appears to be a serious problem, but can be solved to the great simultaneous benefit of dark matter and charged detector-stable particle searches is considered in [25]. Upon writing down a general superpotential, terms which violate lepton and baryon number conservation immediately appear, allowing a ud quark scattering process into an e+ and u¯ via an anti scalar strange sfermion as illustrated in Figure 2.1, thus jeopardizing the very well tested stability of the proton. The authors go on to illustrate how the width of the proton is proportional to the product of the couplings of the baryon and lepton number violating terms in the superpotential. The obvious, but certainly not the only, strategy for forcing the proton width to zero in order to preserve its stability, is to enforce at least one of baryon or lepton conservation, thereby sending the corresponding coupling to zero; this is readily achieved by constructing a new quantum number R out of the baryon number B, lepton number L and spin S of a particle such that R = (−1)3B+L+2S , called R-parity, and demanding it be conserved. Note that all SM particles have R = +1, whereas all sparticles have R = −1; so, if Rparity is conserved, the lightest supersymmetric particle (LSP) must be absolutely stable as it has no phase space left to decay while still conserving its R-parity. If the LSP is electrically neutral, then it is a dark matter candidate; and if additionally its coupling to the next-to-lightest supersymmetric particle (NLSP) is low enough, the NLSP may be long lived enough to be visible, if charged, to this analysis. Once R-parity is established to eliminate decays with purely SM final states, [1] and [3] describe several scenarios with NLSPs rendered quasi-stable via conditions familiar from the SM. In a typical Minimal Supersymmetric Standard Model (MSSM) theory as well as in Anomaly Mediated Supersymmetry Breaking (AMSB) [26], it is possible to arrange to have the LSP be the neutralino χ˜ 10 (and therefore a valid dark matter candidate), with a nearly mass degenerate chargino χ˜ 1± as the NLSP; the mass gap between the χ˜ 10 and χ˜ 1± of less than a GeV leaves very little phase space for the chargino to decay, rendering it potentially long lived in 5  ✁ d  e+  s¯˜  u  u¯  Figure 2.1: Proton decay via an anti scalar strange sfermion. Enforcing the conservation of r-parity forbids this diagram and preserves the proton’s well-tested stability.  a fashion similar to the long lifetime of the SM free neutron versus beta decay. Nevertheless, in the case of AMSB the mass difference is greater than the pion mass and admits the decay χ˜ 1± → χ˜ 10 π ± , leaving cτ(χ˜ 1± ) < 7cm, thus suppressing, but not completely eliminating, the direct tracks from these charginos in the TRT (which has an outer radius of 1 m). MSSM models can be tuned by hand to make the mass splitting between the chargino and neutralino less than the pion mass, thus forcing the chargino to decay via the suppressed χ˜ 1± → χ˜ 10 e± νe with a travel distance cτ much greater than the scale of the ATLAS detector, providing an excellent candidate for this search from one of the most common formulations of SUSY; similarly, tuning can instead be chosen to make the MSSM stop quark nearly degenerate with the neutralino, to the same effect. In addition to considering possible mass degeneracies between LSPs and NLSPs in order to imagine a detector-stable charged massive particle, another option is to have the LSP and NLSP arbitrarily distant in mass, but very weakly coupled; this is exactly the case in supergravity (SUGRA) and Gauge Mediated Supersymmetry Breaking (GMSB) scenarios, where the LSP is a gravitino which owes its very small coupling to the rest of the sector to the weakness of the gravitational interaction. Similar to the SM µ ± which weakly couples to its kinematically allowed daughters by an off-shell W ± , the NLSPs in these scenarios can have very long lifetimes and leave the sought-after direct tracks in ATLAS. A summary phenomenological breakdown of the theoretical NLSP candidates in GMSB is given in [4], based on the mass orderings of the neutralino and the sleptons; the NLSP may be the neutralino, the stau, or both may decay to the LSP in parallel; or, if 6  the selectron and smuon masses are roughly degenerate in mass and too small to decay into a τ˜1 τ pair, all three sleptons may decay in parallel as co-NLSPs. In the SUGRA case, [5] only presents the stau as a possible charged quasi-stable NLSP. Of course, the production cross-sections for all of these possible detector-stable charged massive particles (CHAMPs, used interchangeably with stable massive particles or SMPs) are very model dependent. Nevertheless, [3] suggests some possible production rates for several of the signal-generating particles they explore at a few different mass points, assuming minimal exotic contributions to their production and neglecting the effect of cascades from other exotics; we summarize some of the relevant numbers in Table 2.1, along with some mass limits collected by [1] from LEP and Tevatron experiments. CHAMP ˜R ˜L ˜χ1± t˜1 g˜  Mass Limit [GeV] 98.0 98.5 171.0 241.5 ∼ 400  σ [fb] 8.2 19 O(100) 370 2.3 ×106  Mass Point [GeV] 200 200 200 500 200  Table 2.1: Mass limits and possible pair-production cross-sections for some interesting SUSY-related CHAMPs. All mass limits are taken from [1] except for the gluino limit reported by the CDF collaboration in [2]; cross-sections as calculated assuming the minimal presence of beyond the Standard Model (BSM) effects and at the stated mass point are all taken from [3], with the exception of the chargino cross-section estimate from [6]. Of particular interest to us in our study will be the tremendous production cross section expected for gluinos. As these massive colored objects move through the detector, they will strongly interact with regular quarks and gluons, forming objects called R-hadrons, so named for the conservation of R-parity required for gluinos to be stable enough to hadronize and form them. These objects could stand out dramatically in several fb−1 of data, even if their production is suppressed by several √ orders of magnitude due to the current s = 7 TeV of LHC collisions compared to √ the calculations performed for the design center of mass energy of s = 14 TeV.  7  2.2  Universal Extra Dimensions  A theory of universal extra dimensions (UEDs) which allows all of the SM fields to propagate in the higher dimensions (as opposed to other extra dimension models which only permit gravitons in the ‘bulk’, or the as-yet undiscovered spacial manifold beyond the usual 3+1 dimensions) is presented in [7, 8]. A pivotal feature of this work is the insistence on separate momentum conservation within the extra dimensions; if this is the case, the momentum conserved in the extra dimension (the authors above considered only a single UED) would appear to be a conserved quantum number in the familiar non-compact dimensions, thus preventing the decay of the Kaluza-Klein (KK) excitations of SM particles, rendering them candidates for our search. In order for loop contributions to electroweak processes to remain consistent with experiment, the masses of the lightest KK states should be greater than about 350 GeV; [8] estimates a cross section for producing a pair of stable KK states at the LHC with contributions from KK quark-quark, gluon-gluon and quark-gluon pairs to be about 1 pb for a KK mass of 1.5 TeV. Cosmological constraints discussed in [9] suggest a minimum mass of about 1 TeV for a stable charged KK object.  2.3  Hyperquarks & Vector Confinement  The authors of [10] entertain the notion of augmenting the Standard Model Lagrangian with a new class of vectorlike quarks, called the hyperquarks, which experience a confining gauge interaction between the hypercolor charge they bear, in very close analogy to how QCD augments the QED Lagrangian. The same reference asserts that vectorlike hyperquarks also avoid constraints imposed by precision electroweak and dilepton and dijet resonance searches, and therefore make a promising candidate for possible new physics at the LHC. The same authors further discuss the analogy between the hypercolor augmentation of the Standard Model and the QCD augmentation of QED in [11], where they introduce the interesting property of particle species, which they define as a quantum number labeling a multiplet representation of the gauge group; flavors are then attributed to each element in these representations. The concept of species does not arise in a simple scenario with a gauge group of U(1), since all repre8  sentations of that group are one-dimensional, resulting in an exact correspondence between flavor and species, obviating one of them; the addition of the hypercolor gauge, however, may allow for more than one representation of the gauge group and thus meaningful species numbers. This analogy is continued in [11] to argue that just as the π ± lifetime is long compared to the net-flavorless π 0 since the charged pions must decay via the weak quark-lepton coupling, so too are the symmetries giving rise to species number in the hypercolor extension nearly conserved and broken only by an analogous 4-fermion operator coupling the pair of hyperquarks to a pair of SM fermions. This near-conservation of species number therefore leads to detector-stable pion analogues π˜ ± , the hyperpions. Tevatron constraints demand only that the mass of the hyperpion be greater than at least 250 GeV; [10] estimates a pair production cross-section at the LHC of O(10) fb at this mass.  2.4  Stable 4th Generation Quarks  The existence of a 4th generation of quarks beyond the Standard Model which may be charged, massive, and the lightest of which (denoted U) stable enough to fall within the reach of this analysis is anticipated by [12], motivated as a candidate for strongly-interacting dark matter. The authors therein claim that such an object is not presently excluded above a mass of 220 GeV, and could be produced at the LHC with a cross-section of O(10) pb at such masses (the cross-section drops to a few fb for masses in the TeV range). These U quarks will of course hadronize into heavy analogues of the pion and other familiar QCD objects, about 60% of which is predicted to be charged and thus visible to ATLAS’ tracker [12].  2.5  Leptoquark Mediators  A theory of leptoquarks which couple to SM quarks only via scalar mediators which may be massive and stable enough to be visible to this analysis is posited in [13]. A typical leptoquark model appends a pair of gauge groups UL (1)  UB (1)  to the Standard Model gauge groups and asserts that these remain unbroken at currently accessed energies in order to explain the observed conservation of the corresponding lepton and baryon number charges. [13] describes a minimal leptoquark 9  scenario that introduces the least number of new particles necessitated by these new gauge groups, beginning with only two leptoquarks (bearing both non-zero lepton and baryon quantum numbers) and a right-handed neutrino. However, the authors of [13] seek to avoid introducing stable colored fermion leptoquarks, and so also incorporate two new scalar doublets denoted (φ 7/6 , φ 1/6 ) and (χ 1/6 , χ −5/6 ), which couple the leptoquarks to the Standard Model quarks. A major upshot of this leptoquark model is that these scalars bear anomalous fractional charges, rendering at least one of them completely stable, charged, and thus eligible to leave a direct track in ATLAS and be detected by our methods1 ; the same model suggests a lower bound for the mass of such a scalar to be about 280 GeV, well within the mass range LHC collisions can access.  2.6  Magnetic Monopoles  At the time of this writing, the magnetic monopole remains the longest-sought unobserved particle, much vaunted by theory since it was originally postulated to explain charge quantization by Dirac in 1931 [27], and reappearing again in many grand unified theories (GUTs) and searches for the same [28–30]. GUT theories produce monopoles with masses > 1015−16 GeV [3, 31]; also, monopole masses in the 107−14 GeV ranges are featured in some models with more sophisticated gauge groups, but both of these classes of models are far outside the reach of current colliders. However, models such as those put forward in [32] may accommodate for a TeV-scale monopole. A recent Tevatron search [33] excludes Drell-Yan pair produced monopoles below masses of 360 GeV; at the LHC, [34] estimates a tremendous production cross section from photon fusion for a 500 GeV monopole of 1 nb; this cross-section falls to 1 pb at 1.3 TeV, and 1 fb at 2.8 TeV. Like the anomalously charged objects present in leptoquark models above, the extremely high effective charge of a Dirac monopole will result in mismeasurement of its velocity and mass; nevertheless, we briefly mention this class of particle here since the uniquely high ionization they produce will be immediately visible with essentially no background in an ionization detector like the TRT. 1 Of course, while we may be able to detect these particles, all our work was done for objects with  unit charge; the anomalous charges born by these particles will likely render mass estimates on them unreliable.  10  Chapter 3  Detector The ATLAS detector consists of a set of three precision tracking subsystems which compose the Inner Detector, surrounded by two calorimeters, and a muon tracking and tagging system at its outermost layer; information from all these subsystems contribute to the trigger and standard reported particle properties, but our velocity estimators rely exclusively on information provided by the Transition Radiation Tracker (TRT). As such, we give a short overview of the other subsystems that make up the ATLAS detector before describing the details of TRT timing and digitization necessary for understanding our subsequent analysis.  11  12 Figure 3.1: Schematic of the ATLAS detector. Image c 2008 CERN [35]  3.1  Inner Detector  The first set of three detectors which particles traverse as they radiate from the interaction point compose the Inner Detector, ATLAS’ precision tracking system, illustrated in 1/4 cross-section in Figure 3.2. Semiconducting elements make up the first two subsystems, the Pixel Detector and the Semiconductor Tracker (SCT), relying on the promotion of charge from valence to conduction bands as charged particles traverse the active elements in these subsystems; the single-hit spatial resolution in these subsystems is determined by the size of the individual semiconducting elements, and the granularity at which they divide their active surfaces. Outside of these lies the TRT, which is responsible for providing approximately 30 space points on a charged track out to a cylindrical radius of 1 m for |η| < 2, as well as providing electron tagging through the production of transition radiation for objects with an extremely high relativistic boost, γ. The entire Inner Detector is immersed in a 2 T longitudinal magnetic field provided by ATLAS’ solenoid, imposed to curve charged tracks in the Inner Detector to allow for a transverse momentum measurement therein good to a relative precision of 0.05% pT GeV ⊕1%, as well as the identification of the sign of their charge. Furthermore, the standard for combined tracking resolution for the Inner Detector for hard1 tracks is a resolution for the transverse impact parameter d0 for a track’s origin of 10 µm [36].  1 ‘hard’  denotes high momentum for tracks.  13  14 Figure 3.2: Cross-sectional cutaway of 1/4 of the Inner Detector. Image c ATLAS Collaboration [37].  3.1.1  Pixel Detector  The Pixel Detector is the first subsystem encountered by particles produced in the ATLAS detector, and is responsible for making the very high precision space measurements at low global radius necessary to locate the vertices and transverse impact parameters of objects produced in the interaction region. The pixel barrel consists of three layers of active elements arranged coaxially with the beam pipe at radii of 48 mm, 110.3 mm and 159.3 mm, while the pixel endcaps consist of three wheels located at |z| = 473 mm, 635 mm and 776 mm respectively. The surfaces facing the interaction region of each of these subelements are covered by collections of pixel readout modules; 1744 such modules instrument an active area of 16.4 mm by 60.8 mm each with 47,232 individual semiconducting pixels per readout module, for a total of approximately 82 million channels in this subsystem; studies presented in [36] demonstrated that the pixel noise occupancy was roughly 1 channel in 5000; after masking, the pixel detector produces less than one expected noise hit per bunch crossing. Most pixels are 50 µm by 400 µm, with the narrower dimension oriented to provide maximum granularity in Rφ (corresponding to an orientation parallel to the beam line in the barrel, and radial from it in the endcaps); longer pixels of about 600 µm as well as pairs of ganged pixels are also used to provide coverage over gaps necessitated by the readout electronics. In order to mitigate noise from thermal promotion of electrons across the semiconducting band gap, a C3 F8 cooling system is used to keep the pixel operating temperature at 0◦ C.  3.1.2  Semiconductor Tracker  At radii larger than the Pixel Detector, the volume sought to be filled with precision tracking elements becomes prohibitively large and costly to instrument with the extremely high granularity of the pixels; therefore, subsequent subsystems must employ coarser detector elements in a fashion sophisticated enough to produce comparable position resolution at higher radius. The first such subsystem is the Semiconductor Tracker (SCT). The SCT is arranged as four cylindrical layers concentric with the beam axis in the barrel, and nine wheels in either endcap. Each of these surfaces is instrumented with silicon-strip detector modules (4088 in total), 15  which in turn consist of two layers of 768 individually instrumented silicon strips each; these silicon strips are the fundamental detector element in the SCT, and are typically about 12 cm by 80 µm. The two layers of strips in each detector module are glued together with a stereo angle of 40 mrad to provide a measure of resolution along the length of the channel, while still only requiring per-strip rather than per-pixel instrumentation; this configuration affords an intrinsic spacial resolution of 17 µm in R − φ and 580 µm in z (barrel) and R (endcaps) [38]. The SCT shares the Pixel Detector’s cooling system, and operates at a design temperature of -7◦ C.  3.1.3  Transition Radiation Tracker  The Transition Radiation Tracker (TRT) is the outermost subsystem in the Inner Detector, filling its volume from radii from the global z axis of 56.3 cm to 106.6 cm with 350,848 individual detector channels. Each TRT channel is a proportional ionization chamber 4 mm in diameter and of varying lengths of up to tens of centimeters; these ‘straws’ are arranged longitudinally in the barrel and radially in 80 endcap wheels so as to intersect a track with |η| < 2 and pT > 500 MeV about thirty times, providing 30 independent measurements of a track in its volume. Individual hit resolution for the TRT is on the order of 130 µm [39]. The straws are filled with an active gas mixture of 70%Xe / 27%CO2 / 3% O2 , and instrumented with an axial anode wire held at a positive voltage relative to the straw wall, in order to attract the ionization electrons liberated by charged particles traversing the straw. Furthermore, in addition to its contribution to tracking, the TRT also serves to tag electrons via the transition radiation (TR) produced by charged particles with γ > 1000 as they cross dielectric boundaries; in order to facilitate this, the TRT straws are packed in polypropylene fiber, so that electrons will traverse many dielectric boundaries in the TRT and so maximize their chances of producing TR and the accompanying high ionization of the detector gas. Regular ionization for tracking is distinguished from high ionization from TR by the two voltage levels that TRT information is simultaneously digitized against; TRT digitization and timing are more thoroughly discussed in the dedicated section at the end of this chapter.  16  3.2  Calorimetry  Outside of the Inner Detector and its solenoid are mounted ATLAS’ several calorimeters, designed to measure the energies of incident photons, electrons, and jets, and prevent these objects from penetrating through to the muon system. The Liquid Argon Calorimeters provide calorimetry for both electromagnetic and hadronic objects, while the Tile Calorimeter provides additional hadronic calorimetry. The combined calorimetric subsystems cover2 up to |η| < 4.9, typically with finer segmentation in the η-region covered by the Inner Detector for precision physics. The combined calorimeters provide at least 22 electromagnetic interaction lengths for electrons and photons, and at least 11 radiation lengths for hadrons, designed as such in order to provide both excellent energy hermeticity and a correspondingly good ETMiss measurement [38].  3.2.1  Liquid Argon Calorimeters  The LAr calorimetric system is composed of three subsystems: the LAr Electromagnetic Calorimeter, the LAr Hadronic Endcap Calorimeter (HEC), and the LAr Forward Calorimeter (FCal). LAr Electromagnetic Calorimeter The LAr Electromagnetic Calorimeter provides electromagnetic calorimetric coverage for |η| < 3.2 via stacks of lead absorber plates each of one interaction length in thickness, interleaved with an active ionization medium of liquid argon. The subdetector is subdivided into three components, a barrel section covering |η| < 1.475, and two endcaps which together cover 1.375 < |η| < 3.2; all of these are seated inside the same vacuum chamber as the inner solenoid. For |η| < 2.5, the electromagnetic calorimeter is subdivided into three radial sections, and two outside of this. Inside these modules, the interleaved lead absorbers and active LAr layers are folded into an ‘accordion’ geometry, whose ridges run parallel to the beam axis in 2 Pseudorapidity  η is defined as η = − ln(tan  θ ) 2  where θ is the angle a track makes with the beam line. For a particle with E >> m, the pseudorapidity of a particle equals its rapidity.  17  the barrel and radially from it in the endcaps; this geometry allows sections of the cylindrical detector to be stacked on top of each other at the joints in φ , thereby eliminating all azimuthal cracks in this subdetector. The electromagnetic calorimeter’s performance in both test beam and later in subsequently updated simulation is illustrated in [38]. The authors therein explain that the electromagnetic calorimeter performs best at low |η|, attributing performance degradation at higher |η| to greater path length through material in the Inner Detector; also, slightly better performance was reported for photons than for electrons. At an energy of 100 GeV and through the η range considered in our study, the electromagnetic calorimeter reconstructs electron and photon energies at the 1-2% level, except for near roughly |η| = 1.5, near where the barrel and endcap components of the detector join, degrading the relative energy resolution to as poor as 7%. Away from the crossover between the barrel and endcap, this performance meets the design specification for the electromagnetic LAr calorimeter’s energy resolution for these objects of σEE = √ 10% ⊕ 0.7% [40]. E[GeV ]  LAr Hadronic Endcap Calorimeter The LAr HEC [38] provides hadronic calorimetry coverage that bridges the η gap between the Tile Calorimeter and the FCal (discussed below) via 2 endcap wheels in each endcap. These wheels cover 1.5 < |η| < 3.2, and are composed of 32 segments in φ and two segments in z. Each segment is a stack of flat copper absorber plates sandwiching the active medium of LAr in 8.5 mm gaps; the front wheels are more finely segmented, employing 24 25 mm absorber plates, while the outer wheels use 16 50 mm plates. A jet energy measurement is sought by this detector to σE /E = 50%/ E[GeV ] ⊕ 3%. LAr Forward Calorimeter ATLAS’ calorimetric reach is extended out to 3.1 < |η| < 4.9 by the final component of the LAr calorimetric system, the FCal [38]. 3 modules in each endcap compose this subdetector, providing about 10 interaction lengths in depth; each arranges a honeycomb lattice of cylindrical channels parallel to the beam line which contain a coaxial rod of the same absorber material, and LAr filling the gaps be18  tween the rods and lattice. The first of these modules uses copper as its absorbing material, and the outer two use tungsten. The design resolution for hadron energy measurements in this subdetector is σE /E = 100%/ E[GeV ] ⊕ 10%.  3.2.2  Tile Calorimeter  The Tile Calorimeter [41], seated at central η just outside the barrel portion of the electromagnetic calorimeter, provides ATLAS with its hadronic calorimetry in the barrel region via stacks of low carbon steel absorbers sandwiching plastic scintillator, covering |η| < 1 with the long barrel portion of the subdetector, and extending this reach over 0.8 < |η| < 1.7 with the extended barrels on either side. These subdetectors are segmented azimuthally into 64 φ sectors; radially into three layers totaling 7.4 proton interaction lengths; and in steps of 0.1 in η in the inner two layers, and 0.2 in the outermost one. Pion test beams incident on tile calorimeter modules at η = 0.35 were measured in energy to σE /E = 56.4%/  3.3  E[GeV ] ⊕ 5.5% [38].  Muon System  ATLAS’ final major subsystem, situated outside the calorimeters and extending to a radius of 10 m, is the Muon System, a collection of four subdetectors and a toroidal magnet system designed to provide precision tracking for a stand-alone measurement of muon pT (via the Monitored Drift Tube and Cathode Strip Chamber subdetectors) as well as muon triggering information (from the Resistive Plate Chambers and the Thin Gap Chambers). Tracking abilities extend out to |η| < 2.7, and the chambers are intended to measure a track sagitta of 500 µm to better than 10% accuracy in order to provide that level of accuracy on the corresponding stand-alone pT measurement [38] of a 1 TeV muon; bending is provided in the r − z plane by a toroidal magnet system. These toroidal magnets consist of 8 loops each in the barrel and both endcaps spaced evenly azimuthally, situated such that their planes extend radially and run in z. The triggering subsystems extend to |η| < 2.4, and provides good enough time resolution to serve the additional role of identifying which bunch crossing an event came from.  19  3.3.1  Monitored Drift Tubes  The MDT component of the muon system is a collection of 354,000 monitored drift tubes mounted transverse to the beam axis in φ , covering |η| < 2.7 and seated in the three barrel muon stations at roughly 5, 7.5 and 10 m radius, and in the muon system endcap wheels at roughly 7, 13, and 21 m in z. The MDT system achieves φ hermeticity in the barrel by arranging for its 8 azimuthal sectors to alternate between large and small module configuration such that the large modules overlap the smaller adjacent ones in φ ; similar azimuthal coverage is achieved in the endcaps by two layers of overlapping radial ‘fan-blade’-like geometry. The individual drift tubes function qualitatively similarly to the TRT straws the rest of this study focuses in, although the MDTs are 29.97 mm in diameter and filled with Ar/CO2 (93/7) as their active gas [38]; ionizing radiation traverses the tube, and the ionization electrons produced drift towards the central anode wire (held at 3080 V), the rising edge in time of the receipt of which corresponds to the distance of closest approach of the track to the anode via the established drift time relationship; the best tangent to all these drift circles informs the reconstructed track. The same reference indicates that the performance in terms of spatial resolution of a single tube degrades with intensity of irradiation, and extremity of crossing radius (either too close to the anode wire or too close to the tube wall), but that in a realistic operating environment a per-hit spatial resolution of σ = 80 µm can be achieved in practice.  3.3.2  Cathode Strip Chambers  The MDTs which the muon system relies on for making its independent measurement of pT will degrade in performance and fail in the high-occupancy environment anticipated in the innermost layer of the muon system endcap for very forward pseudorapidities (|η| > 2); as such, the MDTs are replaced by Cathode Strip Chambers (CSCs) in this region, which can perform in such a high-occupancy environment. The CSCs operate like multiwire proportional chambers filled with Ar / CO2 (80/20) except that instead of reading out the anode sense wires, two layers of nearby cathode strips lie in a plane parallel to the sense wires, wherein a charge is induced and read out as ionization is received by the plane of sense wires (in the 20  CSC, the anode wires run radially, while the two layers of cathode strips run parallel and orthogonal to this in φ respectively); each CSC chamber has four layers of sense wires and cathode strips, allowing 4 position measurements per track. In a photon test beam, the CSC demonstrated a per-layer resolution of 65 µm at 1 kHz/cm2 intensity [38].  3.3.3  Resistive Plate Chambers  The muon system meets its triggering demands in the barrel via the Resistive Plate Chambers (RPCs). Three RPC trigger stations are mounted adjacent to the outer two layers of the MDTs in the barrel (one on either side of the MDTs in the second layer and the third RPC either directly behind or in front of the outer MDT layer in the large and small azimuthal sectors, respectively). Each RPC consists of two parallel plates separated by a 2 mm gap of C2 H2 F4 / Iso-C4 H10 / SF6 (94.7/5/0.3), with an electric field gradient of 4.9 kV/mm between them [38] which will allows an avalanche when ionizing radiation crosses the gap; each RPC station contains two of these gaps. RPC signals demonstrate a 5 ns width with a 1.5 ns jitter, thus fast and narrow enough both for triggering and bunch crossing identification, as desired.  3.3.4  Thin Gap Chambers  The final component of the muon system is the Thin Gap Chambers (TGCs), which both extend the muon system’s triggering capability out to |η| < 2.4, and provide the azimuthal tracking information to complete the measurement made by the MDTs. Each TGC consists of a 2.8 mm gap filled with CO2 / n-pentane (55/45), and is instrumented with a plane of sense wires at 1.8 mm pitch and 2.9 kV operating voltage. The chambers are oriented so that the wire spacing is in φ , in order to provide the azimuthal coordinate in tracking that the MDT tubes are insensitive to. Two sets of these chambers are present on the inner endcap wheel of the muon system, adjacent to the corresponding MDTs, and 7 TGCs are associated with the middle MDT endcap group. The high fields between wires in the TGC yield fast drift velocities everywhere except for tracks passing orthogonally to the gap plane exactly between two wires; despite these areas of weak field, > 99% of hits will 21  arrive within the 25 ns bunch crossing, in time for the trigger and bunch crossing identification [38].  3.4  The TRT  In order to extract a measure of particle velocity from ATLAS, we seek precision timing information by which we can distinguish a slow, late-arriving heavy track from the great majority of β = 1 Standard Model tracks. No subsystem in ATLAS provides a larger average number of independent measurements with precision timing information per track than the TRT; thus not only does the TRT provide the timing information we want, but it provides many measurements of it by which we may seek to suppress the effect of random hit fluctuations on the final measurement of β as extracted from the collection of all hits on a track. In this section we describe the geometry and nominal operation of the TRT.  3.4.1  Ideal Operation  The TRT is the outermost subdetector in the Inner Detector, ATLAS’ precision tracker; it consists of roughly 300,000 proportional-counting ionization chamber channels, and is designed to provide about 30 position measurements on the track of a typical electrically charged particle out to a radius of 1 m from ATLAS’ central axis and within |η| < 2, each of individual resolution of about 130 µm in the plane orthogonal to the channel’s anode wire. The TRT also tags the presence of high levels of ionization in a single hit as indicative of the presence of the transition radiation produced when a charged particle of γ > 1000 crosses the boundary between the channel wall and the polypropylene fiber the gas chambers are packed in. Individual TRT channels are manufactured in a ‘straw’ geometry - a cylindrical tube of 4 mm diameter, running parallel to the beam line in the barrel, and radially in the endcaps; an anode wire runs down the axis of each straw, held at a high voltage compared to the straw wall in order to cause ionization electrons liberated by ionizing radiation from the detector gas (70% Xe, 27% CO2 , 3% O2 ) to drift toward the anode, where they can be registered by the readout electronics. When a particle traverses a straw, an average of about 7 ionization clusters are 22  produced, stochastically distributed along the particle’s path in the gas. Ionization from these clusters drifts towards the anode wire at a roughly3 constant rate, arriving at about 50 ns after ionization when drifting in from the straw wall (2 mm from the anode). Fig 3.3 illustrates how electronics digitize this pulse once received at the readout electronics at the instrumented end of the straw. The voltage received is digitized against two thresholds: a low threshold (LT) meant to capture precision timing information, and a high threshold (HT) which is only exceeded in the presence of high ionization, typically the result of transition radiation. The LT is given 8 3.125 ns bins per 25 ns bunch crossing, while the HT is afforded a single bin each bunch crossing. Ionization information is digitized by assigning a bit set to 1 to each of these bins where ionization exceeded the appropriate threshold, and 0 where it did not. Upon receipt of a trigger, the 27 bits of information from the triggering bunch crossing as well as the following two (in order to accommodate ionization drift time) are packed into a 32-bit integer and read out. The clock in Figure 3.3 is calibrated for each4 straw to start a fixed time t0 before the time of arrival of a β = 1 straight track from the nominal interaction point at the center of the detector. LT bits set at times earlier than t0 indicate the presence of pileup in the event and are typically discarded. This stability in clock start time with respect to the time of arrival of a straight β = 1 track is where the TRT derives its position resolution from at the sub-straw level, and is a key feature that will allow us to extract velocity information as described in the next chapter. Since the TRT only resolves time into 3.125 ns bins, the transit time of a β = 1 particle across a 4 mm diameter straw is negligible, and all the ionization thus produced can be safely approximated as having been produced along the track in the gas within a given straw simultaneously. Under that approximation and recalling the near-constancy over radius of the drift velocity of ionization electrons in the active gas, we conclude that the ionization that is produced closest to the anode wire will arrive there first, and therefore be responsible for the rising edge of the TRT LT information. In this way, the position of the LT rising edge encodes the 3 The  drift time as a function of distance from the anode wire is typically fit in practice to a third-order polynomial, although the departures from linearity are a small correction. 4 Actually, this clock is calibrated per read-out chip, which gang together clusters of 16 neighboring straws.  23  33 34 35 36 37 38  The TRT is made of 420 000 thin drift–tubes (straws) with 4 mm in diameter filled with a gas mixture of 70% Xe, 23% CO2 and 3% O2 , and equipped with a 31 µm diameter gold-plated tungsten wire. Details of the straw design and the detector set-up can be found in [1] and [3, 4] resp. The read-out time for each straw is three bunch–crossings, i. e. 75 ns which is about twice the maximum drift time inside the straw. A signal is called a hit if it exceeds a certain threshold in this time window as described in the next paragraph. Besides the detection of transition radiation, the TRT, like any multi-wire proportional cham1  The upper exclusion limit is about 100 GeV [11].  Pulse–Height  32  1 0 1 HT bits 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 LT bits High Threshold (HT) at 5 keV  Low Threshold (LT) at 300 eV Time-over-Threshold Bunch–Crossing 1  8  Bunch–Crossing 2  16  Bunch–Crossing 3  24  Timebin (∗3.125 ns)  Figure 1: Schematic illustration of a signal and the electronics output. The time Figure 3.3: Digitization scheme for corresponding a hit in the TRT. Ionization is digitized in which the signalacross exceeds the low threshold (Time-over-Threshold) is stored in 24 time bins three bunch crossings versus 24 low threshold bins for tracking of 3.125 ns (cf. bit pattern at the top of the figure). The high threshold is tested only once per information, and 3 high threshold bins which serve as high-ionization bunch–crossing resulting in three HT bits.  flags, nominally for the sake of identifying transition radiation. Note that only a few ionization clusters are produced by a track in a given straw, corresponding to the 2 peaks in the diagram; if such clusters are sufficiently spaced out, the anode voltage may dip below the LT, creating holes in the bit pattern or even fake rising and falling edges, which will be discussed in later chapters. Image c T. Lohse [42].  distance of closest radial approach of a track to the anode wire, thus constraining the track to having crossed the straw tangent to a drift circle centered on the anode and setting an ideal lower limit for the TRT’s per-hit position resolution of about 80 µm. Another upshot of this timing scheme is the nominal position of the falling edge of the TRT LT information. By similar logic to what related the rising edge of the LT pattern to the distance of closest radial approach of a track to the anode wire, the falling edge corresponds to the ionization produced furthest from the anode; but this should always be from the edge of the straw, 2 mm away. By this logic the LT falling edge should always arrive in the same timing bin, given that the assumption that the particle indeed arrived concurrently with a β = 1 track as per 24  the local clock calibration holds. A later falling edge should in principle indicate the presence of a particle traveling with β < 1; this model will serve as the starting point for our most powerful β estimator, which we present in the following chapter. A final note on TRT tracking which will become relevant as we construct our β estimators is how ATLAS’ tracking algorithm treats outlying TRT hits included on a track. When attempting to identify the hits which belong to a track candidate, the tracking algorithm begins with a loose requirement on TRT hit position, considering all hits within a wide ‘road’ around the potential track. After a first-pass track is fit, any TRT hits which sit more than 2.5σ away from the track is labeled a ‘tube hit’, and is deweighted in the second-pass final track fit by attributing an uncer√ tainty of 4 mm/ 12 to the hit’s position, essentially retaining only the information that the straw was struck somewhere in its volume.  25  Chapter 4  Methodology Four different techniques were developed to attempt to measure velocity using TRT information: two which rely on the departure from expectation of the rising and falling edges of TRT low-threshold bit patterns, one which seeks to find a deeper minimum in the χ 2 -value of the track fit when β is allowed to float, and one which estimates dE/dx in the TRT and correlates this estimator to β . Owing to the availability of very clean control regions in data, signal above the Standard Model backgrounds was sought as an excess in an ‘ABCD’-control region search.  4.1  β Estimators  The TRT low-threshold bit patterns contain information about timing, tracking and ionization which can be extracted with a careful and realistic treatment of the LT bits and their behavior. Four different strategies for estimating β based on this information were developed and are presented here.  4.1.1  Falling Edge β Estimator  The nominal TRT behavior described in the Detector chapter suggests that for a β = 1 particle, the falling edges of TRT LT bit patterns should always come in the same timing bin, but the same will not be true for a track with β < 1; this distinction provides our first and what will prove later to be one of our best sources of 26  information about β in the TRT. One of the elegant features of this approach is that it is a simple matter to construct realistic probability density functions (PDFs1 ) of where the falling edges of TRT information arrive for β = 1 tracks from real data, so freeing a key part of the analysis from dependence on simulation. Once these PDFs have been established by collecting the falling edge positions for a large number of Standard Model tracks (predominantly π and µ) of momentum great enough to ensure their velocity is very close to β = 1 and their ionization behavior is consistent with that of high momentum backgrounds, the β of a candidate track can be estimated by trying to quantify the departure of its distribution of falling edges from the β = 1 PDF. An example of a falling edge PDF and a collection of falling edges from a particular track to be compared in this way is illustrated in Figure 4.1. The quantification of the difference between a given track’s distribution of falling edges and the corresponding β = 1 PDFs is made simple by the assumption that the differences between them are due solely to the delay in arrival time of the potentially slow candidate track at each struck channel; thus a hypothesis β for a track will determine how late each hit on the track came, given the helictical path length from the interaction point to the straw each was produced at, with respect to a β = 1 track. This delay is subtracted off of the time of the falling edge for each hit, and the negative log likelihood LFE (β ) that the resulting corrected distribution of N falling edges came from the β = 1 PDF is computed via N  LFE = − ∑ log ri (β )p FEi − f loor i=1  ∆ti (β ) −1 3.125ns  +(1 − ri (β ))p FEi − f loor  ∆ti (β ) 3.125ns  (4.1)  where β is the hypothesis velocity being tested, FEi is the unadjusted falling edge bin of the ith hit on the track being measured, ∆ti (β ) is the delay in arrival of the ith hit with respect to a β = 1 track due to traveling at the hypothesis velocity β , p(n) is the probability weight in bin n of the appropriate PDF for the falling edges 1 To attempt to dispel the confusion: parton distribution functions will be called as such in this work; PDF will always stand for probability density function hereafter.  27  of β = 1 tracks, and ri (β ) =  ∆ti (β )mod3.125ns 3.125ns  Upon careful inspection, Equation 4.1 describes adjusting the falling edges of each hit on our track to be measured back by the appropriate time relative to a β = 1 track, and dividing the weight of each hit across two adjacent bins when this delay isn’t an integer multiple of 3.125 ns, in proportion to the amount by which ∆ti (β ) exceeds such a multiple 2 . In the case where ∆ti (β ) is large enough to adjust the falling edge to before the beginning of the read-out window, the expression inside the logarithm of Equation 4.1 is replaced with a fixed value of 10−15 for this hit, effectively forbidding such a large adjustment; if the unadjusted falling edge came in the final bin of the readout window, then the second term within the logarithm in Equation 4.1 is replaced with the sum of all the weights of the bins following the position of the adjusted position of the falling edge, to account for the fact these bins would be cut off and all add their weight to the final bit in the LT train 3 . When the correct hypothesis β is chosen in this procedure, the delay subtracted should make the collection of falling edges on the track most consistent with the β = 1 behavior, thus minimizing LFE ; the velocity found in this way is thus accepted as the falling edge β estimate for the track. While this simple prescription provides an elegant idea for estimating β , the performance of the estimator is determined by how the data is prepared for analysis, how the β = 1 PDFs are constructed and how the estimator is calibrated, all of which we discuss below. Data Preparation for the FE Estimator The discussion above describes our β estimation technique at its core, but a number of small effects exist which can influence the position of the falling edge of a hit and obscure the effect of β , and these behaviors must be accounted for at the per-hit level. 2 ie,  if ∆ti (β ) is 1.1 timing bins and the uncorrected falling edge comes in bin 20 for the ith hit on track, then 0.1 of the weight of bin 18 of the corresponding β = 1 PDF is added to 0.9 times the weight of bin 19 to form the corresponding term in sum 4.1. 3 ie, while the probability of having a falling edge in any bin except the last is just the probability of the voltage signal falling below the LT in the corresponding timing bin, the probability of getting a falling edge in the final bin is the probability of falling below threshold at this or any later time, past the end of the read-out window.  28  5 6 4  5  Work In Progress  Work In Progress  4  3  3 2 2 1  1  0 0  0 0  2  4  6  8 10 12 14 16 18 20 22 24  5  Last LT Bit PDF  10  15  20  25  30  Last LT Bit  Figure 4.1: An example of a PDF for falling edge position (left) and a collection of falling edges from a single track (right). Equation 4.1 gives a prescription for how to adjust the distribution on the right for a hypothesis β , and extract a likelihood from a spectrum of PDFs like the one on the left.  As mentioned previously, the delay in arrival of a particle at a channel due to a slow velocity depends on the path length to that hit; this path length is taken to be the helictical path from the principle vertex to the struck channel, with the bending radius determined by the track’s reconstructed pT in a 2 T magnetic field. Before any hit is admitted into the β = 1 PDFs or considered in the estimation of β for a candidate, the effects of the local and run-dependent TRT calibration must be scaled out. These calibrations are discussed more completely in the detector chapter, but are essentially the local clock start time t0 (set to correspond to the arrival time of a β = 1 particle from the interaction point), and the r(t) relation (a third-order polynomial in drift time t yielding the drift circle radius r) which describes the drift velocity of ionization electrons in the detector gas. Both t0 and r(t) are available from the conditions database for the run and channel in question during the creation of the flat analysis ntuples. t0 is dealt with just by subtracting t0 − 9.375 ns from the time of the falling edge for each hit, arbitrarily enforcing an effective state of t0 = 9.375 ns everywhere. In the same vein, r(t) is corrected for by calling the raw measured drift time traw for the hit4 , and then computing what this raw drift time tnominal should have been if the straw had one standardized r(t) relationship by solving this nominal r(t) for t given the hit’s reported drift circle 4 Actually, instead of the raw measured drift time, the time t  in the event summary data, and traw is extracted from this.  29  = traw +t0 + 3.125 2 ns−10ns is reported  radius. The difference traw − tnominal is then subtracted from the time of the falling edge; in this way the timing of all hits are corrected to a calibration both consistent across channels and stable in time with respect to detector drift. A final effect which results in timing distortions in both the falling and rising edges of the LT bit pattern is the reflection of signals at the terminated end of each TRT straw. When ionization electrons are received by an anode wire, one voltage pulse travels along the wire to the read out electronics, defining the rising edge of the LT pattern, while a copy of this pulse travels backwards and reflects off of the glass terminating bead at the inner end of each anode wire; the later arrival of this reflection then results in the definition of the signal’s falling edge. However, [43] demonstrates that these reflections can cause more problems than a simple delay due to signal propagation. The recombination of the direct and reflected signals for hits close to the terminated end of the wire produce a voltage pulse that is both larger and differently shaped than that from a hit produced further from the terminating bead; therefore, signals produced at different points along the straw will cross the LT level at different times with respect to their arrival at the electronics. Fig. 4.2 illustrates an example of this effect in the endcaps: signals from hits produced far from the read-out electronics appear to arrive sooner than signals produced closer to the instrumentation, due to the larger voltage pulse resulting from the recombination of the direct and reflected signals crossing the LT earlier than it otherwise would have. In order to account for this nontrivial interaction of signal reflections and LT threshold settings, effective signal propagation velocities were extracted from β = 1 tracks in data as the slope of the straight line fit to plots of average signal propagation time (defined as the time to the middle of the falling edge bin, on average) versus the length of wire the signal had to traverse to reach the read out electronics, separately for barrel and endcap. This effective signal propagation velocity was then used to estimate the delay in arrival of the falling edge of the LT bit pattern for each hit of interest, and this delay was then subtracted off in order to remove the effect on timing of the track’s position along the length of the straw.  30  30 29.5 Work In Progress  29 28.5 28 27.5 27 26.5 0  5  10  15  20  25  30  35  40  distance from readout electronics [cm]  Figure 4.2: Signals from hits far from a straw’s read out electronics appear to arrive slightly earlier than signals from hits produced closer to the instrumented end of the straw; this is attributed to the increased size and altered shape of the voltage pulse produced when direct and reflected signals from hits near the terminating bead overlap and recombine.  Track & Hit Selection The TRT is designed to extract its tracking information from a collection of roughly 30 hits on a given track, where extreme outliers are essentially removed by deweighting from the track fit; in the same vein, the FE β estimator identifies and discards hits showing known pathologies before admitting them to the β = 1 PDFs or the evaluation of the β estimate itself. Selections were made both to ensure good velocity reconstruction, and especially to provide optimal signal significance when using β as a discriminating variable; Table 4.1 shows the evolution in S/  B) as cuts are applied to a sample  of roughly 80,000 simulated generic gluino-based r-hadrons with masses evenly distributed from 100 GeV-1 TeV in steps of 100 GeV representing possible interesting signals, and approximately a half-million simulated Z → µ µ decays as a hard background 5 . Additional motivations for the track- and hit-level cuts used to 5 These  samples were not chosen to realistically represent a possible data sample; they only serve  31  select healthy LT bit patterns for our FE β estimator are also discussed below.  here to provide large samples of signal and background tracks, so we may ensure that the cuts we are applying preferentially remove background tracks. In the Validation chapter, we will construct a realistic combination of signal MC and data-driven backgrounds in order to finalize our search method with optimal real signal significance.  32  Cut Candidate Track p > 50 GeV PDFs require 40 < p < 50 GeV At least 20 good hits on candid. track No tube hits  33  Track radius ∈ [−2, 2] mm No hit radii > 2 mm  No Pileup  No 1-bit noise  No HT hits  Z → ee with HT  Z → ee without HT  BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B  [0,0.1) 0 91 0 68 0 60 0 483 0 351 0 335 0 330 0 320 0 1 0 320 0 1 -  [0.1,0.2) 9 3474 1158 6 3283 1340 1 3097 3097 0 3134 0 3064 0 3013 0 2978 0 2992 0 800 0 2992 0 800 -  [0.2,0.3) 89 7871 834 64 7630 954 3 7037 4063 0 5827 0 5708 0 5667 0 5591 0 5571 0 4070 1 5571 5571 0 4070 -  [0.3,0.4) 331 11076 609 236 10529 685 56 9724 1299 28 8750 1654 27 8521 1640 24 8369 1708 18 8225 1939 15 8137 2101 14 6121 1636 39 8137 1303 8 6121 2164  β range [0.4,0.5) [0.5,0.6) 1439 6023 10974 9226 289 119 933 3909 10432 9050 342 145 528 2901 9644 8200 420 152 391 2249 8884 7280 449 154 376 2178 8772 7185 452 154 376 2177 8631 7005 445 150 346 1963 8386 6721 451 152 295 1811 8303 6770 483 159 245 1485 7019 5984 448 155 734 3723 8303 6770 306 111 200 1104 7019 5984 496 180  [0.6,0.7) 17249 7249 55 11898 7285 67 10183 6742 67 8456 6034 66 8178 5983 66 8256 5970 66 7863 5796 65 7593 5797 67 6612 5264 65 13118 5797 51 4654 5264 77  [0.7,0.8) 36079 5195 27 27364 5167 31 23988 4593 30 21011 4067 28 19962 4032 29 19994 3974 28 19053 3867 28 18728 3877 28 16581 3576 28 22444 3877 26 9502 3576 37  [0.8,0.9) 72731 4407 16 60374 4356 18 55265 3939 17 51477 3509 15 49357 3488 16 49170 3429 15 47322 3292 15 46745 3327 15 41912 3199 16 30742 3327 19 17119 3199 24  √ Table 4.1: Selection cuts used to make the FE β = 1 PDFs, showing S/ B as each subsequent cut is compounded with all the preceding ones. White rows use a sample of MC Z → µ µ for background; gray rows use Z → ee.  [0.9,1.0) 237440 12814 26 266671 14632 28 226727 12404 26 209281 9570 21 211451 9605 21 210885 9618 21 203750 9133 20 203902 8999 20 192671 8490 19 59899 9098 37 48699 8526 39  Sum 371390 72377 371455 72432 319652 65440 292893 57538 291529 56709 290882 56011 280315 54319 279089 54093 259520 44524 130700 54192 81286 44560  Our β estimation technique for the falling edge rests on an understanding of where such falling edges arrive for a β = 1 track, an expectation that can be constructed directly from data. Any trigger stream without track selection cuts applied to it will of course be overwhelmingly dominated by Standard Model backgrounds, and the tracks in such samples will be predominantly from particles of about 1 GeV in mass or less; therefore, by accepting all tracks of momentum greater than a few GeV, we easily ensure an essentially perfectly clean sample of β = 1 tracks from which to draw our β = 1 PDFs for the positions of the falling edge. However, while such a low momentum cut serves the purpose of getting a clean β = 1 sample, a harder cut is necessary for getting stable performance from background muons and pions in an ionization detector like the TRT across the very wide range of momenta we intend to examine; Figure 4.3 illustrates the ionization behavior of muons as a function of momentum, and shows that these backgrounds do not reach their stable plateau in ionization efficiency until momenta of at least about 40 GeV. Were we to only demand momenta high enough to provide β = 1 backgrounds, the β = 1 PDFs would be overwhelmed (since the momentum spectrum of SM backgrounds falls exponentially) by a preponderance of lower momentum SM tracks sitting nearer their minimum ionizing regime; these tracks would have a higher rate of failing to ionize the TRT gas near the straw wall compared to the more highly ionizing tracks sitting atop the relativistic rise and plateau of Figure 4.3, and so would pull the distributions of falling edges earlier. These early-skewed falling edge distributions would then cause the falling edges of hard SM tracks who are atop their relativistic rise to appear relatively late, thus causing the β estimation routine described above to return fake slow estimates of their velocity, thereby pushing SM backgrounds into our intended low-velocity signal region. In order to avoid this major source of fakes, we demand that tracks have at least 40 GeV in momentum to be admitted to the β = 1 PDFs (and also less than 50 GeV, since this momentum marks the lower bound of our signal search region); the suppression of the fake slow background tail and corresponding improvement in signal significance can be seen by comparing the first two rows in Table 4.1. In practice, these PDFs were easily re-generated every time the TRT was recalibrated (very roughly after each month of running). It should be noted, however, that this construction introduces an irreducible systematic on the velocity measurement of CHAMPs, which are not 34  anticipated to be produced at the same highly relativistic values of γ that our backgrounds and β = 1 PDFs are restricted to; signal produced low into the minimum ionizing regime of their Bethe-Bloch curves will have fake early falling edges and thus be reconstructed as moving faster than they truly were, obscuring them under the tails of the high velocity background, and highly ionizing signal particles produced at low momenta far up the kinematic region of their ionization curves may appear even slower than they truly were. In this case, while the β estimate for signal will be systematically biased, it will place that signal in a lower background region of the β spectrum, improving the significance of discrimination based on that variable. The improvement in signal to background tallied in Table 4.1 after enforcing a high minimum momentum for tracks admitted to the β = 1 PDFs emphasizes the improvement in significance gained by suppressing slow background fakes, and motivates our decision to accommodate this systematic in β .  Figure 4.3: Bethe-Bloch ionization curves in the TRT active gas for SM backgrounds (as well as a few mass points of scalar tau). Image c T. Lohse [42].  35  An immediate and obvious danger for a study which looks for signal as atypical timing in the TRT is the presence of TRT-only tracks from low-momentum loopers, cosmics, or other sources of track fragments. To help avoid contamination of our β = 1 PDFs by such objects, tracks were required to possess at least one hit in the SCT for inclusion here. Ionization efficiency for typical tracks in the TRT active gas is understood to be far from perfect, and the ideal of continuous ionization being produced as a track passes through the straw is not borne out in reality; a typical TRT hit is understood to consist of roughly 7 ionization clusters, where a minimum ionizing particle has a mean free path on the order of a few hundred microns (see for example the simulation illustrated in figure 4-11 of [44]). If the ionizing radiation fails to produce ionization near the straw wall, a fake early falling edge will be observed, which as discussed above produces a fast bias on β and obscures our signal; abnormally excessive ionization near the straw wall will pull the estimator in the other direction, creating fake slow particle signatures. Care must be taken to understand how ionization and also voltage digitization affects the placement of the falling edge, and if its behavior changes dramatically in particular variables, then individual β = 1 PDFs must be constructed in bins of these variables, as we explore below. The detector geometry of the TRT is responsible for one simple effect on apparent ionization in the TRT. As described in the Detector chapter, barrel straws run parallel to the axis from which η is measured, while endcap straws extend radially from this axis. In either case, the actual path length of a track through the active gas in the straw is a function of η, with the shortest paths through the straw corresponding to the values of η where the track passes close to transversely through the straw (low η in the barrel and high η in the endcaps). However, the cylindrically symmetric nature of an individual TRT channel means that no information regarding the particle’s transit along the channel axis is preserved, effectively projecting all ionization into a plane transverse to the channel’s anode wire. Therefore, the tracks that passed through a channel at values of η resulting in a large longitudinal displacement within the straw will appear to have an artificially high ionization efficiency, as the ionization produced during their longer path in the straw is projected into the local transverse plane. In order to control for this effect, falling edge PDFs were generated in 10 equal-sized bins in |η|, over 0 < |η| < 2. An example 36  of these is illustrated in Figure 4.4.  37  6  6  6  5  5  5  Work In Progress ave. = 15.361  4  Work In Progress ave. = 15.517  4  3  3  3  2  2  2  1  1  1  0 0  0 0  0 0  2  4  6  8  10 12 14 16 18 20 22 24  2  4  6  8  10 12 14 16 18 20 22 24  Last LT Bit (|η|<0.2)  Work In Progress ave. = 15.768  4  2  4  6  8  10 12 14 16 18 20 22 24  Last LT Bit (0.2<|η|<0.4)  Last LT Bit (0.4<|η|<0.6)  7 6  6  5  38  4  6  5  Work In Progress ave. = 15.947  5  Work In Progress ave. = 16.130  4  3  3  3  2  2  2  1  1  1  0 0  0 0  0 0  2  4  6  8  10 12 14 16 18 20 22 24  2  4  6  8  10 12 14 16 18 20 22 24  Last LT Bit (0.6<|η|<0.8)  Work In Progress ave. = 15.946  4  2  4  6  8  10 12 14 16 18 20 22 24  Last LT Bit (0.8<|η|<1.0)  Last LT Bit (1.0<|η|<1.2)  7 6  6  5  5  Work In Progress ave. = 15.739  4  6 5  Work In Progress ave. = 15.569  4  3  3  3  2  2  2  1  1  1  0 0  0 0  0 0  2  4  6  8  10 12 14 16 18 20 22 24 Last LT Bit (1.2<|η|<1.4)  2  4  6  8  Work In Progress ave. = 15.334  4  10 12 14 16 18 20 22 24 Last LT Bit (1.4<|η|<1.6)  2  4  6  8  10 12 14 16 18 20 22 24 Last LT Bit (1.6<|η|<1.8)  Figure 4.4: Example of PDFs of the falling edge position, for hits produced by tracks crossing near the straw edge, in steps of 0.2 in |η|. As eta increases, a straight track will go from crossing the barrel straws orthogonally, to the barrel / overlap region around |η| ≈ 1 where the straw crossing is at a shallower angle, and back to crossing the endcap straws orthogonally at high |η|. The figures illustrate that in the intermediate region, the falling edge appears slightly later, an effect which we anticipate due to the greater ionization produced in these straws due to a charged particle’s longer path length therein.  Intuitively, the closer a hit included on a track is to the radial edge of the straw it was produced in, the more likely it is to show ionization near the straw wall (and therefore exhibit a falling edge determined by timing rather than ionization efficiency), since in order for there to be a hit recorded at all some ionization must have been deposited at some point along the track’s intersection with the straw, all of which is near the straw wall, in contrast to the case where the track passes through the straw at a smaller radius and has a chance of only producing ionization far from the edge of the tube. Consequently, hits nearer the straw wall will likely show later falling edges, which would drag the β estimate to slower β if this effect is not controlled for. Therefore, in addition to |η|, β = 1 PDFs are also simultaneously binned in 10 equal bins of track radius rT , defined as the absolute value of the distance of a reconstructed track from the anode wire at each hit, for a total of 100 bins of PDF expectations for the position of the falling edge. The effect on the falling edge of producing a hit at various distances from the anode wire is illustrated in Figure 4.5.  39  2.5 3  2.5 2  Work In Progress  2  1.5  2.5  Work In Progress  Work In Progress  2  1.5  1.5 1  1  0.5  0.5  1  0 0  0.5 2  4  6  8  10 12 14 16 18 20 22 24  0 0  2  4  6  8  10 12 14 16 18 20 22 24  Last LT Bit (0.65<|r|<0.8)  4  3  3.5  8  10 12 14 16 18 20 22 24 Last LT Bit (0.95<|r|<1.1)  5 Work In Progress  4  Work In Progress  40  3  2.5 2  1.5  2  1.5  1  1  0.5  1  0.5 2  4  6  8  10 12 14 16 18 20 22 24  0 0  2  4  6  8  10 12 14 16 18 20 22 24  Last LT Bit (1.1<|r|<1.25)  6  0 0  5  5 4  3  3  2  2  2  1  1  1  6  8  10 12 14 16 18 20 22 24 Last LT Bit (1.55<|r|<1.7)  6  8  10 12 14 16 18 20 22 24  6 Work In Progress  4  4  4  Last LT Bit (1.4<|r|<1.55)  3  2  2  Last LT Bit (1.25<|r|<1.4)  6 Work In Progress  4  0 0  6  3  2  5  4  4 Work In Progress  2.5  0 0  2  Last LT Bit (0.8<|r|<0.95)  4.5  3.5  0 0  0 0  2  4  6  8  10 12 14 16 18 20 22 24 Last LT Bit (1.7<|r|<1.85)  0 0  Work In Progress  2  4  6  8  10 12 14 16 18 20 22 24 Last LT Bit (1.85<|r|<2)  Figure 4.5: Example of PDFs of the falling edge position, all taken at |η| < 0.2, in steps of 0.15 mm in track radius. The falling edge is constrained to later times as the track moves to the edge of the straw, as anticipated. Structure in the low radius bins is explored below.  Due to the high variability both in number and in position along the track of ionization clusters produced by radiation incident on a TRT straw, a single or a few hits will not constrain β effectively; it is the large number of independent timing and ionization measurements made by the TRT on each track in its acceptance which provide us with enough information to overcome the effects of ionization inefficiency as well as the compact size of the TRT. The third entry of Table 4.1 demands a minimum of 20 accepted TRT hits on a track before it is considered for β = 1 PDFs or analysis, and tabulates the resulting dramatic improvement of signal to noise, particularly in the low β tails of the distribution. By removing sparsely populated tracks from our analysis, we eliminate those tracks weakly constrained enough to have their β estimate dragged far from the true velocity by the presence of a few bad hits, whether they be from noise, misattribution of hits to the track, or ionization inefficiency, thereby suppressing the fake slow β tail of the SM backgrounds. This cut is by far the most costly in terms of acceptance, but essentially guarantees that our attention is given only to well-measured tracks, particularly once the subsequent cuts help to remove classes of spurious or misleading hits on track. As discussed in the Chapter 3, a TRT track will often include tube hits, which are hits close enough to the track to be considered by the tracking algorithm, but far enough away after first pass fitting (2.5σ ) that they are deweighted in the final √ track fit by being assigned an error of 4 mm / 12 (essentially retaining only the information that the track may have passed anywhere within the 4 mm diameter straw). The tracking algorithm is thus protected from misattributed hits included by its mostly open policy of considering all nearby hits as a potential member of the track, but we must take care not to be misled by such objects in our velocity estimation routine. Table 4.1 shows that the omission of these hits from consideration further suppresses the slow tail from the SM backgrounds. In the same vein as discarding the tubes hits for potentially diluting the pull of more auspicious hits, hits that fail to reconstruct within the straw at both hit and track level are discarded. The cut on track radius (fifth entry in Table 4.1) is implemented to little effect; if a track does not pass within the straw, the drift circle within the straw will necessarily be far from the track, causing the hit to be tagged as a tube hit and discarded by the previous cut in the table. The cut on 41  drift circle radius is in fact slightly costly in terms of significance, but we insist upon it nonetheless since its effect is small and a well-behaved drift circle radius (i.e., one that could have possibly been produced by a struck straw) is necessary for correcting the timing delay due to the r(t) relationship, as previously discussed. Given the long drift time of ions in TRT gas compared to the bunch spacing as well as the high occupancy anticipated for this detector at full LHC luminosity, a substantial amount of pileup from previous bunch crossings is anticipated6 . As the effect on the falling edge of the potential receipt of ionization from multiple hits in the same straw is not simple to resolve, we choose to abandon all hits which appear to be from pileup events, as indicated by the presence of ionization above the LT in the first four bins of a hit. Row 7 of Table 4.1 shows the small significance gains this cleaning cut affects. Additionally, the LT bit pattern must exhibit more than 1 set bit; TRT signal shaping electronics do not respond faster than 1 bit7 [45], and so such patterns must be noise, typically characteristic of clock pickup in the front-end read out boards. Furthermore, a conservative per-hit requirement that should the FE come in either of the last two LT bits, it must not be isolated by an unset bit immediately preceding it. The last (and therefore latest in time) two bits of the LT pattern hold the most statistical power for converging a slow estimate for β , fakes of which are the most serious background when trying to identify the presence of CHAMPS in data. For this reason such powerful hits are only accepted when they are not isolated from the rest of the ionization; isolated LT bits in this position potentially produced by electronics pickup must not be allowed to drag backgrounds to slower apparent β . Signal significance is further improved by this cut in the slow β tails, as demonstrated in the eighth entry in Table 4.1. In order for the FE of a TRT LT pattern to fall within the readout window, little enough charge must be received at the anode that it can dissipate quickly enough; this is typically the case for hits where the ionization received was produced by a single track ionizing the gas without TR. However, in a hit where TR or some other 6 At  present, only every second bucket is filled in the LHC beam, resulting in bunch spacings of 50ns instead of the nominal 25 ns, so this is less of a concern for the early data we are considering. 7 Actually, the reported peaking time is 8 ns, almost 3 bits long, so the dismissal of hits with only a single set LT bit is quite loose.  42  source of abnormally high ionization is present, the signal shaping may not be able to produce a falling edge before the end of the readout window, thus causing the loss of the timing of the FE. These hits will all be interpreted as having a FE in the last LT bin, and so will produce a slow bias on the velocity estimate for that track, when in fact the hit was simply highly ionizing. In order to further suppress the slow fake tail of the SM backgrounds, hits passing the HT were dropped, according to this argument; however, the ninth row of Table 4.1 demonstrates that this is very detrimental in terms of significance at low β . This is due to the fact that the simulated r-hadrons exhibit a higher proportion of HT hits than the tracks in the Z → µ µ sample (probably due to the fact that the cross section for r-hadron production places the bulk of them in the kinematic region of their Bethe-Bloch curves, which is more highly ionizing and thus more likely to lead to HT hits than the backgrounds which sit atop their relativistic rise); while the presence of HT hits influences our velocity estimate in the manner described, the signal tracks were on the whole faking slower more often than the tracks in the Z → µ µ events, thus leaving them better separated if worse measured. However, the last two entries in Table 4.1 (shaded gray) illustrate exactly the opposite behavior when the Z → µ µ sample is replaced with a Z → ee simulation. The first of these rows illustrate the significance of the r-hadron signal versus a background containing many e± when HT hits are considered by the β estimating algorithm; the second shows the same, but with HT hits thrown out. In this case, dramatic gains in significance are made by throwing out the HT hits, contrary to the µ ± -dominated background case. In light of this, we elect to throw out HT hits as originally planned, on grounds that the ionization produced by e± at the momenta generated in Z decay more realistically represents the background environment we anticipate in the signal region of our data; while the simulated µ ± we examined here have momenta of order 10 GeV, our signal search region extends up to 1 TeV, well into the momentum regime where µ ± and π ± begin to TR (at or above about p = 140 GeV for pions and p = 105 GeV for muons) and more closely resemble the ionization behavior of e± produced in Z decay.  43  4.1.2  Rising Edge β Estimator  Techniques exactly analogous to those explored for the falling edge β estimator can be applied to the rising edge of TRT LT information with little adaption. In addition to the low extra conceptual overhead, we can immediately imagine that the information about β in the RE will be relatively uncorrelated to FE β since, at least ideally, the universally stationary position of the FE is completely decoupled from the position of the RE, which is determined by the radial distance of a track within a struck straw to the receiving anode wire. Data preparation for the RE estimator is nearly identical to the considerations described in the corresponding section above regarding FE β ; the only major difference is that the effective propagation speed along an anode wire of the ionization voltage pulse is re-measured in terms of the arrival time of the rising edge instead of the falling edge, to allow for different shaping behavior on either end of the pulse as a result of the recombination of direct and reflected voltage pulses. Track & Hit Selection As per our discussion of hit selection for the FE β estimator, we begin by presenting a summary Table 4.2 of the track- and hit-level cleaning cuts applied, indicating the effect each has on the signal and background samples discussed above in bins of β . The motivation and discussion surrounding Table 4.2 closely follows the preceding discussion of Table 4.1 for the FE-driven β estimator, with a few interesting differences. Like the FE β = 1 PDFs, 100 RE PDFs are populated in 10 bins in |η| by 10 bins in tracking radius; however, in the case of the RE PDFs the tracking radii bins are not simply a necessity of non-ideal ionization resulting in departures from expectation, but a more fundamental requirement of the method. While at least in principle the position of the FE should be stationary for a β = 1 particle regardless of where the track crosses the straw, the same is not true for the RE, by design; the RE can be pushed later not only by a slow β , but also simply by crossing the straw further from the anode wire. Predictably, demanding a track have a minimum of p = 50 GeV for admission into the β = 1 PDFs has the opposite effect on signal significance in RE-driven 44  β than it did for the FE. If lower ionizing tracks were admitted to the RE PDFs, the position of the RE would appear later, since more hits which failed to ionize the gas at their distance of closest approach to the anode wire would be recorded. Then, when looking at samples with p > 50 GeV where SM backgrounds are all the way up their relativistic rise in ionization efficiency, background tracks would typically have collections of rising edges which were on average earlier than the β = 1 PDFs distribution would indicate, causing these background tracks to fake fast (or, perhaps, since they are all moving at β = 1 already anyways, it is more accurate to say their ability to fake slow is relatively suppressed) in the RE-driven estimator (instead of faking slow like low momentum β = 1 PDFs cause in the FEdriven estimator). Table 4.2 illustrates that this demand on the RE PDFs is indeed costly in terms of significance. Nevertheless, the cost of this cut to significance is small compared to the degree by which the RE-driven β estimate outstrips the FE-driven one in terms of signal significance (by as much as an order of magnitude in the β regime we will soon understand to be the most interesting), so we choose to retain this cut for the sake of making a better estimate of β , which will serve our purposes when trying to accurately reconstruct the mass of a track, given its momentum and this estimate of velocity. The merit of the removal of HT hits from the data used by this β estimator is much less obvious than in the case of the FE-driven one. Conceptually, the difference is that naively, the position of the RE does not depend on the amount of ionization deposited in the straw; the drift time for ions from the distance of closest approach should remain the same, unlike the position of the FE which is contingent upon the channel’s ability to dissipate charge and shape the resulting signal; as such, the removal of the HT hits in the ninth row of Table 4.2 does not benefit signal significance versus the Z → µ µ sample. The last two shaded entries in the table are again the same exercise, but versus a background of Z → ee events; the effect of rejecting HT hits varies by velocity bin. In the Z → ee study, rejecting HT hits improves significance in the range 0.5 < β < 0.7; since this regime will prove to be the most significant range for a search given trigger considerations, and due to the overall preferential rejection of background by the HT cut, the HT cut on hits for the RE-driven analysis is maintained despite its shortcomings compared to the FE case. 45  Cut Candidate Track p > 50 GeV PDFs require 40 < p < 50 GeV At least 20 good hits on candid. track No tube hits  46  Track radius ∈ [−2, 2]mm No hit radii > 2mm  No Pileup  No 1-bit noise  No HT hits  Z → ee with HT  Z → ee without HT  BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B  [0,0.1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -  [0.1,0.2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -  [0.2,0.3) 1 0 0 1 0 0 0 0 0 2 0 2 0 1 0 0 0 0 0 0 3 0 0 0 0 -  [0.3,0.4) 1 23 23 1 30 30 0 24 1 250 250 2 231 163 2 189 134 0 174 0 176 1 38 38 21 176 38 3 38 22  β range [0.4,0.5) [0.5,0.6) 8 48 1769 5877 625 848 10 61 1937 6251 613 800 6 23 1848 5912 754 1233 3 34 2726 9171 1574 1573 1 33 2609 9095 2609 1583 1 32 2465 8768 2465 1550 1 24 2331 8335 2331 1701 1 24 2335 8345 2335 1703 0 26 1290 7116 1396 43 166 2335 8345 356 648 15 82 1290 7116 333 786  [0.6,0.7) 494 10003 450 586 10670 441 360 9712 512 804 16186 571 796 15965 566 781 15819 566 727 15308 568 718 15298 571 721 13139 489 431 15298 737 303 13139 755  [0.7,0.8) 3984 14994 238 4748 15529 225 3678 14297 236 6887 15668 189 6857 15461 187 6898 15549 187 6313 15483 195 6305 15474 195 5960 13029 169 1437 15474 408 1499 13029 337  [0.8,0.9) 21993 17793 120 25059 17709 112 21433 16242 111 31394 8045 45 31376 7932 45 32034 7914 44 30036 7786 45 30045 7756 45 29383 6560 38 6916 7756 93 7708 6560 75  √ Table 4.2: Selection cuts used to make the RE β = 1 PDFs, showing S/ B as each subsequent cut is compounded with all the preceding ones. White rows use a sample of MC Z → µ µ for background; gray rows use Z → ee.  [0.9,1.0) 344207 23950 41 340298 22296 38 293637 19320 36 253706 5487 11 252455 5418 11 251126 5320 11 243238 4967 10 243021 4943 10 224533 3585 8 122316 4984 14 72463 3606 13  Sum 370736 74409 370764 74422 319137 67355 292829 57535 291520 56713 290874 56025 280339 54384 280114 54327 260624 44757 131333 54368 82073 44778  4.1.3  Track Fit Optimization β Estimator  TRT timing and the ATLAS tracking algorithms rely on the assumption that tracks are produced by β = 1 particles, as discussed in the Detector chapter; consequently, track fit quality degrades as this assumption is violated. By allowing β to float in a track fitting procedure, a deeper χ 2 minimum should exist for slow tracks which will provide another measure of β . If a particle travels with β < 1, it will arrive later than expected at each TRT straw that it strikes. This late arrival means that the rising edge of the LT bit pattern will be registered correspondingly late, and this longer time to the receipt of the rising edge fakes a longer drift time of ionization electrons in the straw which is reconstructed as an artificially enlarged drift circle, thereby pulling the track position away from the anode wire at each hit. However, if we imagine that a track has an equal number of drift circles tangent to it on either side, then we can approximate that these pulls roughly cancel each other out and leave the track unmoved from its true position. While this approximation may seem very coarse, it allows us to avoid having to invoke the computationally expensive ATLAS tracking algorithms many times (ie, once for every hypothesis beta to be tested) for each track of interest. Given this approximation, we can then search for a deeper minimum in the χ 2 residual of the track fit by stepping through hypothesis values of β , calculating how late this value of β made the particle at each hit compared to the β = 1 assumption in the tracking algorithm, and translating this into a new drift circle at each hit from which we can calculate a new χ 2 contribution. When the correct hypothesis beta is chosen, the drift circles of hits on track will be corrected back to be as close to tangent to the track as possible, thus minimizing the χ 2 of the track fit and defining our track fit optimization β estimator. One serious problem immediately presents itself with a direct application of this prescription. After first pass track fitting, the tracking algorithms deweight any hit on track that appears more than 2.5σ away from the track, by setting that √ hit’s position to the center of the straw and its error to 4 mm / 12 (ie, assigning a flat probability that the track passed anywhere through the 4 mm wide straw); these hits are referred to as ‘tube hits’, and this procedure is intended to deweight the pull of hits that appear to be extreme outliers upon first pass track fitting. However, as  47  discussed above, particles arriving at TRT straws out of time with the assumptions of the tracking algorithm will necessarily make large individual χ 2 contributions, and so may be discarded as tube hits by this procedure even though they contain useful information for the track fit optimizing β estimator. As such, we consider reintroducing all tube hits which fell within a generous 6σ window at their originally reconstructed drift circle radius for use in this β estimator. But in doing so, we reintroduce the problem that the tube hit deweighting procedure was designed to suppress in the first place by also re-including genuine outliers that sit far from the track for reasons other than a slow velocity. Therefore, this β estimator must be performed in two passes, much like the original track fitting procedure; first, a first pass β estimate is made after re-including all the nearby outliers as described above. Then, in order to re-exclude any hits that remain far from the track even after they are adjusted for a floating β , all hit residuals are computed at the current β estimate, and any hit which remains more than 2.5σ from the track after adjustment for β is thrown out again. Finally, χ 2 is then re-minimized with respect to β using only the surviving hits; the resulting second-pass β estimate is what we originally constructed to be the refit β estimate. While the above prescription for reconsidering marginal hits is well-reasoned in principle, Table 4.3 demonstrates that it is in fact very costly in terms of sig√ nificance, reducing S/ B for the study samples used in Tables 4.1 and 4.2 on one case by more than a factor of two near our well-triggered regime; evidently the pull on the backgrounds to fake slow speeds far outstrips what information can be gained from this procedure, so we abandon it in what follows and retain only the same non-tube hits considered by the ATLAS tracking algorithm in its final track fit. Even without this attempt to recover additional information about the track, this β estimator shows superior signal acceptance and significance compared to the β = 1 PDF template driven methods described above.  48  Cut Candidate Track p > 50 GeV At least 20 good hits on candid. track Reinclude tubes, refit & reject far outliers  BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B  [0,0.1) 1 20 20 0 0 0 0 -  [0.1,0.2) 5 278 124 0 0 0 32 -  [0.2,0.3) 27 478 92 0 1 0 216 -  [0.3,0.4) 63 1036 131 1 365 365 1 1387 1387  β range [0.4,0.5) [0.5,0.6) 197 483 4192 12153 299 553 0 26 2835 9464 1856 12 281 4828 13228 1394 789  [0.6,0.7) 2342 21668 448 721 17197 640 2026 20102 447  [0.7,0.8) 12673 18542 165 5960 14412 187 12154 13130 119  [0.8,0.9) 58392 9193 38 29383 6713 39 53316 5102 22  Table 4.3: Selection cuts used to clean data for the track fit optimization β estimator. Each row is compounded with the cuts above it; gray shaded entries are not retained in final analysis.  [0.9,1.0) 295637 6709 12 224610 4217 9 22067 3426 23  Sum 369820 74269 260701 55204 89857 61451  49  4.1.4  dE/dx-Correlate β Estimator  Our final TRT based β estimator focuses on the ionization effects of slow massive particles, rather than their late timing. For p > 40 GeV, the π and µ backgrounds are all the way up the relativistic rises of their Bethe-Bloch ionization curves. However, the massive stable particles we are searching for will be at much lower relativistic γ at these momenta and therefore exhibit dramatically different ionization behavior than their backgrounds; in fact, if produced at p ≈ m, signal particles will inhabit the very highly ionizing kinematic region of their Bethe-Bloch curve, as illustrated in Figure 4.3; at momenta p ≈ 4m, they quickly fall into their minimum ionizing regime. In both these cases, signal ionization behavior may appear very different that that of the SM backgrounds, and a TRT track with roughly 30 ionization chamber hits, each with 24 bits of ionization information in them, may be well-suited to tagging such departures. In order to quantify a departure from background ionization in a single discriminating parameter, a comparison of the average number of set TRT LT bits per hit on track in a CHAMP candidate to that of a high-momentum SM background was made. First, an expectation for this average number of LT set bits per hit was constructed for high-momentum backgrounds in MC; this expectation had to be constructed as a function of η and track-anode radial distance, as these parameters directly influence the number of LT bits set via path and hit distribution in the detector gas as explained earlier, which would influence the dE/dx correlate if left uncontrolled. Then, for a track to be measured, the average difference between the number of set LT bits per hit for the track and the background expectation was calculated over all hits on track. The resulting measured parameter is thus a correlate of the particle’s dE/dx in the detector gas, relative to that of a high momentum background track; and if a sample of particles (such as a generic CHAMP signal sample) remains on one side of the minimum ionizing turn-around of the BetheBloch curve (at about p = 4m), the particle’s dE/dx will correspond to a unique value of relativistic γ and therefore β , allowing us to calibrate this dE/dx correlate into a β measurement, as discussed below. Table 4.4 examines the evolution of signal and background acceptance and distribution across our dE/dx correlate as cleaning cuts are applied, as per tables  50  4.1 - 4.3; recall when examining this table that the slowest β for signal corresponds to the most ionization, so after calibration velocities will decrease left to right in this table, contrary to the previous ones. One immediately obvious potential problem with treating the number of set LT bits in a TRT hit as a direct correlate of the ionization efficiency of the track that produced it, is that the LT readout window is finite; so if a track is so delayed that some of its ionization is not received before the 24th and final bit in the timing window, that ionization cannot be counted by this algorithm and the hit will appear lower ionizing than it truly was, when in fact it was just traveling very slowly (and therefore actually has very high ionization efficiency). In order to combat this problem, hits with the last LT bit set were omitted from the analysis. While this suppressed the systematic fast fakes that are generated in the signal sample by the counting error described, Table 4.4 demonstrates that this is very costly in terms of significance at high values of the dE/dx correlate (and correspondingly low velocities). As the mid- to low-velocity regime is where we hope to be able to identify the presence of new physics, we choose to accept the fast systematic on the slow tail of our signal in exchange for nonetheless heightened sensitivity there. The now-familiar problem with whether or not to retain hits with HT bits set presents itself again in the dE/dx correlate. Significance versus the Z → µ µ background in our most interesting regions is once again lost by omitting these hits; but as in the case of the FE-driven estimator, the last two entries in Table 4.4 indicate a strong improvement in signal significance versus the relatively high TR environment produced by a Z → ee background, which is more representative of an environment with many high momentum µ ± and π ± , and thus motivates the omission of these hits from the analysis.  51  Cut Candidate Track p > 50 GeV At least 20 good hits on candid. track No hits with bit 23 set  No HT hits  Z → ee with HT  52 Z → ee without HT  BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B BKG Signal √ S/ B  < −3.5 2165 294 6 818 17 1 784 11 0 701 8 0 3351 7822 135 7277 8616 101  [-3.5,-2) 215 206 14 43 119 18 35 130 22 34 169 29 16689 7108 55 24726 7098 45  [-2,-0.5) 24964 8407 53 19891 7350 52 21645 7577 51 29681 8333 48 36107 7002 37 36000 6616 35  dE/dx correlate range [-0.5,1.0) [1.0,2.5) [2.5,4.0) 265123 74011 4615 18829 20949 16383 37 77 241 234899 60944 2952 16883 18978 15103 35 77 278 235581 55460 2285 16928 17331 11496 35 74 240 224133 45207 1587 16100 16386 11120 34 77 279 40046 35234 24531 7447 7625 7574 37 41 48 32519 23261 13324 6876 6647 6102 38 44 53  [4.0,5.5) 513 8429 372 242 8004 515 212 3999 275 107 4265 412 15411 6892 56 7188 5079 60  [5.5,7.0) 83 1061 116 43 1016 155 37 413 68 11 252 76 10246 5427 54 3758 3976 65  [7.0,8.5) 8 22 8 4 21 11 4 11 6 5 5 2 6657 4522 55 1842 3009 70  Table 4.4: Selection cuts used to clean data for the dE/dx-driven β estimator. Each of the first four rows are compounded with all the rows before it, except for the dark-gray shaded entry, which was studied but removed for poor performance in significance. The last two rows (light gray) replace the Z → µ µ background with a Z → ee background, as above.  [8.5,10) 6 0 0 5 0 0 5 0 0 1 0 0 19376 6072 44 11806 2619 24  Sum 371703 74580 319841 67491 316048 57876 301467 56638 207648 67491 161701 56638  4.1.5  β and Mass Calibration  The procedures described above produce correlates to β and dE/dx which must be calibrated for β so we can examine the relative velocity resolution achieved in simulation by each estimator as a figure of merit, and then separately for mass so that in the event of the presence of a signal in data we may extract the best estimate of the mass that excess corresponds to. Calibrations were performed on the same sample of generic gluino-based r-hadrons (roughly 80,000 tracks evenly distributed in steps of 100 GeV in mass from 100 GeV - 1 TeV) as was used to understand the hit and track level cuts earlier in the chapter. We sought to calibrate β as simply as possible, by generating a scatterplot from the r-hadron sample of βtrue vs. our reconstructed velocity βest (for tracks with at least 20 good hits in the relevant β estimator, and where βest is any of the raw values returned by the preceding four estimation techniques) and finding the best fit line that maps the raw reconstruction onto the true velocity; an example of one of these scatterplots is given in Figure 4.6. However, a number of difficulties with this simple prescription arose that are of brief note here, due to the nature of the samples used. For this sample, the βtrue spectrum was by no means flat; βtrue is best supported around βtrue ∈ [0.5, 0.8] due to the production cross-section for these objects favoring their production at momenta near their mass, and begins to fall sharply outside this range; Figure 4.7 illustrates the true β spectra for 100, 500, and 1000 GeV gluinos. Tails exist to much slower velocities, but statistically reasonable support runs out for βtrue < 0.1. Even within its well-supported region, the shape of the βtrue spectrum causes biases in calibration; since there is relatively high statistics for the βtrue range noted, slow fakes in reconstruction from this part of the sample will compete with or even dominate the genuinely slower tracks in terms of βest (i.e., a given slow βest will be populated by well measured slow tracks and badly measured fast tracks; the shape of the βtrue distribution may cause these badly measured fast tracks to be populous enough to dominate this part of the βest spectrum, when we would prefer to include only the well-measured tracks when defining our calibration). In order to mitigate this problem, we construct a new calibration sample out of the old, by weighting the βtrue spectrum to be flat across its well-supported region; any bin (taken to be of size 0.01 in βtrue ) with fewer  53  than 10 tracks in it in the original sample is abandoned from the calibration fit on account of ill support. Then, in order to avoid spurious pulls on the calibration in regions not well supported by βest , columns in reconstructed velocity in the scatterplot of βtrue vs. βest with less than three tracks were also abandoned before fitting, thus creating a sample well supported in both βtrue and βest across as much of their ranges as possible. To calibrate, the average βtrue was calculated for each bin of width 0.01 in βest , and a fifth order polynomial was fit to the resulting profile. This polynomial (projected linearly below the well-supported regime in βest ) then serves as our calibration for that β estimator; this procedure was done separately  β  true  for |η| < 1 and |η| > 1 (barrel and endcap), and for each of the four β estimators.  500  1 0.9  400  0.8 300  0.7 0.6  200  0.5 0.4  Work In Progress  100  0.3 0.2  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9  1 β  0  FE  Figure 4.6: Unprocessed scatterplot of βFE versus βtrue . In principle calibration may be conducted by fitting the average βtrue as a function of reconstructed βFE , but several steps of processing must be done first in order to remove sources of bias, as detailed in the text.  A similar procedure was pursued for calibrating 1/β γ for use in extracting a 54  250 200  Work In Progress 100 GeV ~ g ~ 500 GeV g 1000 GeV ~ g  150 100 50  0  0.2  0.4  0.6  0.8  1  Figure 4.7: βtrue spectrum for 100, 500, and 1000 GeV gluinos in simulation. The heaviest objects provide support to the lowest velocities, but even this reach is exhausted by βtrue < 0.1.  mass estimate (via m = p/β γ), to variable success. The difficulty in this case, was that the above method takes the mean of βtrue as the best estimate of the true velocity corresponding to a given value of βest . This being the case, a sample of tracks of a given βest will be mapped onto this average true velocity by the calibration, but this mapping will only produce a peak at the correct peak value of βtrue in this βest bin if the distribution of βtrue in that bin is symmetric about its peak value. This did not appear to be a problem in practice for the β calibrations, but when a βest value with a (roughly) symmetric βtrue distribution is used to calculate the estimate of 1/β γ, substantial skew in the true distribution is generated, and the mean of the distribution now sits far from the mode. Since it is the mode of a distribution that will appear above a background and which any peak finding algorithm will pick out, we would prefer to ensure that the mode and not the mean 55  of the mass distribution sits on the true mass after calibration; as such, we must modify the calibration technique used for β to calibrate 1/β γ for the mode of its distribution. The β and mass resolutions resulting from these calibrations, as well as their biases in mass, will be presented in the next chapter.  4.2  Signal Detection  After constructing correlates to β , we seek to use this information to decide if an excess exists over the SM backgrounds; however, we also want to employ an excess-detection technique that allows us to remain as model-independent as possible, and therefore the typical approach of carefully estimating the amount of SM backgrounds in a given sample and after a set of cuts designed to isolate signal from a single model is not appropriate here. Conveniently, we can be absolutely assured that the β of any stable SM track is approximately 1 at any momentum that a stable CHAMP will be produced; thus if perfectly measured, β provides perfect background rejection while retaining high signal efficiency. In practice, however, we must contend with the β = 1 SM tracks which are mismeasured to possess β < 1, producing a slow fake tail in our measured β spectrum. So long as we restrict our attention to the range of momenta that sit above the plateau of the Bethe-Bloch ionization curve for background π and µ (that is, p > 50 GeV), the ionization of these backgrounds in the detector gas will not change with momentum, and we can look for an excess over this β distribution as evidence of the presence of a sample of very massive stable tracks. The expected constancy in pT of the β distribution for background tracks, taken together with the near-confinement of all CHAMP signals to high momenta and low β , satisfies all the assumptions of an ‘ABCD’ excess search, the details of which are presented in Appendix A. Consider populating a scatterplot between two track parameters for a given sample containing signal plus background; divide this scatterplot into 4 regions by picking one threshold value along each axis, and label the regions A, B, C and D, as illustrated in Figure 4.8. Then, if these two track parameters are independent for  56  the background and Ni is the number of background counts in the ith quadrant, NA ND = NB NC  (4.2)  for samples with enough statistics to well-represent the true background distribution. Thus, if the thresholds in the two parameters can be set such that signal is confined to a single quadrant, evidence for the presence of this signal can be sought as an excess of counts in that region over what is predicted by the above ratio.  ymax
  y
  B  C
  
  control
  control
  y’
  A
  D
  control
  signal
  ymin
  x
 xmin
  x’
  xmax
  Figure 4.8: Division of a 2D region into control and signal regions for an ABCD background estimation. Note that the variables x and y must be independent for the background in order for such an estimator to be appropriate.  Generic CHAMPs satisfy this scenario fairly well, if we choose our discriminating variables to be β and pT , and look at momenta high enough that these variables decouple for the background (ie, momenta where stable Standard Model particles all have β ≈ 1, so that the shape of the measured β distribution is purely a result of mismeasurement, which we take to be independent of momentum be57  ginning at the high momentum plateau of the Bethe-Bloch curve for µ ± and π ± backgrounds). In this scenario, we can imagine the bulk of our CHAMP signal will be mostly confined to a region of high p (since production cross-sections for objects with masses comparable to the center of mass energy highly suppress production of tracks with p << m), and low β (where low essentially means measurably different than 1, again thanks to the similarity of momentum and mass enforced by the production cross sections). Unfortunately, the background regions as defined in this way are not strictly signal free - one could imagine the very high momentum tail of the CHAMP distribution which might become difficult to distinguish from background in β - but we proceed nonetheless, with the assumption that errors introduced by counting the events in the control regions as all background will be much smaller than the size of statistical counting fluctuations there, since we anticipate that the cross-sections for producing background in these regions will outstrip the signal cross-section by many orders of magnitude. Two concerns arise immediately after proposing the use of this ABCD excess detection technique: how to treat systematics arising from small unanticipated correlations between β and pT , and how to decide exactly where to place the control region boundaries in these variables. In order to treat the first problem, we may extract a sample of Z → µ µ events from data by examining the invariant mass of muon pairs; by taking the sample of resulting muons, we construct a very clean sample of SM tracks to which we may apply the above prescription, and check if it correctly estimates the number of tracks present in the high-pT , low-β signal region (region ’D’ in what follows). From this, we may extract the ratio of actual to predicted background tracks in D, and use this as a multiplicative correction factor on the background estimate in D; the statistical width on this correction then represents the systematic width on the background estimate introduced by possible dependencies between the two search parameters. The second problem foreseen above, that of deciding how exactly to divide the pT − β plane into four regions, is resolved by estimating boundaries that we may anticipate will produce the highest signal significance for massive and as-yet nonexcluded signal particles. While we must of course rely on the ten mass points of r-hadron simulations previously examined, we seek a data-driven understanding of our backgrounds via the same Z → µ µ sample extracted from data used to control 58  background correlations. While this muon sample is clean background, it must be scaled to the same data volume as the complete data sample, which we will achieve via a simple re-weighting, detailed in the next chapter. Similarly, the signal simulations must also be scaled to a size appropriate for the data; after collecting the data intended for analysis, the total integrated luminosity may be tabulated, and then, using the cross-sections reported for the r-hadron processes in our simulations, the simulation output may be scaled to the same integrated luminosity as the data, allowing a cogent comparison between our signal MC and a data-driven pure background. Once this has been constructed for each individual r-hadron mass point, we may step through the pT and β boundaries in their plane, extract the signal and predicted background in each resulting signal region, and construct the signal significance associated with that choice of boundaries; the boundaries producing the highest significance at each mass point are then taken as the optimal cuts for that mass, and the optimal set of cuts that produce a significance of about 2 at the highest mass possible are chosen as our final set of cuts, in anticipation of trying to exclude generic detector stable gluino-based CHAMPs at as high a mass as possible to the 95% C.L.  59  Chapter 5  Validation A number of calibration and measurement exercises were done on data and simulation, in order to quantify and demonstrate the performance of the β estimators described in the previous chapter. We present the results of these studies here, before moving on to discuss the data-driven tune used in our final search for the appearance of CHAMPs at ATLAS.  5.1  β Estimators  The procedures described in the preceding chapter for estimating β are not unbiased. For the purposes of searching for new physics in our final data sample, this is a manageable problem; as long as we have a reliable correlate to β , we can use it to distinguish signal from background. Nonetheless, we seek to calibrate our β estimators so that we can quantify their performance across a range of masses and across the entire velocity spectrum. Additionally, we independently calibrate the corresponding mass estimators for the purpose of being able to reconstruct a mass peak, should a signal in excess of background arise. To this end, we calibrated our β and mass estimators on a sample of simulated CHAMPs with masses from 100 GeV - 1 TeV, in steps of 100 GeV as described in detail in the previous chapter. The detailed results of these studies, as well as comments on the acceptance of cuts designed to admit only well-performing tracks for each method, are discussed in this section.  60  1000 800  800  Work In Progress s = 7 TeV  700  µ = 0.0173 ± 0.0006  Work In Progress s = 7 TeV µ = 0.0094 ± 0.0007  600  σ = 0.0830 ± 0.0005  σ = 0.0735 ± 0.0006  500  600  400 400  300 200  200  100 0 -0.5 -0.4 -0.3 -0.2 -0.1 0  0  0.1 0.2 0.3 0.4 0.5  -0.4  -0.2  Relative Residuals for FE β (barrel)  0  0.2  0.4  Relative Residuals for FE β (ec)  FE  0.75  140  0.7  120  0.65 100  0.6 0.55  80  0.5  60  0.45  Corrected β  Corrected β  FE  Figure 5.1: Relative FE-driven β width for well-reconstructed CHAMPS from 100-1000 GeV. Fits are to the Gaussian cores of the distributions; errors reported are statistical only. Notice that the compact TRT is able to make an independent β measurement with resolution on par with the far larger muon system, which reports resolutions around 5%.  0.9  100  0.85 0.8  80  0.75 0.7  60  0.65 0.6  40  0.55  40  0.5  0.4  Work In Progress  20  0.3  0.4  0.5  0.6  0.7  0.8  0.9 β  true  1  20 Work In Progress  0.45  0.35 0  0.4  0.4  0.5  0.6  0.7  0.8  0.9 β  (barrel)  1  0  (endcap)  true  Figure 5.2: The tracks of Figure 5.1, plotted as βtrue vs. the calibrated βFE in the barrel (left) and endcap (right). In both cases, peaks in a slice of βtrue are slightly narrower for slower true velocities, as anticipated by the greater ease of distinguishing slower moving particles from those moving with β ≈ 1.  61  5.1.1  Falling Edge β Estimator  Figure 5.1 illustrates the final resolution1 after calibration of the FE β estimator for well-reconstructed tracks. The endcap (on the right) shows the best performance of any of our β estimators, measuring velocity with a relative width of 7.4%; compare this to the β resolution of roughly 5% reported in [46] for a time of flight measurement done using the RPC in ATLAS, a subsystem which extends out to a radius of 11m compared to the TRT’s 1m, but which only makes 3 measurements of a track rather than the TRT’s roughly 30. Figure 5.2 represents the same tracks in a 2D scatterplot, to illustrate how the width in reconstructed β is narrowest for the slowest true velocities; we anticipate this since these slowest particles are the easiest to resolve from those traveling at the speed of light. In order to fully qualify the performance illustrated in Figure 5.1, Table 5.1 presents the cuts that were applied to our sample of simulated CHAMPS to generate these β resolution peaks, and indicate how many tracks were dropped due to each cut. It should be understood that in Table 5.1 and in the analogous ones for the other β estimators, the numbers presented in the lower half of the table indicate the number of tracks eliminated only due to the cut listed on the left. Table 5.1 shows that for the FE β estimator, requiring a minimum number of hits on track (in this case 20) to pass the hit-level cleaning cuts described in the Methodology chapter is solely responsible for the largest fraction of track rejections by a wide margin, although these rejected tracks are preferentially badly measured and contribute mostly to the background tails as illustrated in the previous chapter. Table 4.1 also from that chapter illustrates the breakdown of hit-level cuts that push the number of acceptable hits on track below the requisite 20; for tracks that don’t start out with less than this minimum cutoff, it is the omission of tube hits and HT hits 1 In  what follows, resolutions are reported as the width of the Gaussian cores of distributions, as fit either to a Gaussian distribution with the typical parameterization, 2  g(x) = Ae−(x−µ)  /2σ 2  or, for distributions with pronounced one-sided tails, a crystal ball function, parameterized as CBF(x) =  g(x) −n A(B − x−µ σ )  62  if (x − µ)/σ > −α otherwise  (5.1)  Total FE Accepted hit quality low reco high reco  |η| < 1 |η| > 1 43971 37835 19887 13757 Tracks lost exclusively by cut: 11081 9092 645 21 3463 3690  Table 5.1: Total CHAMPS available to be measured vs. number accepted into Figure 5.1, the cuts applied in that figure, and how many tracks were dropped from each β resolution peak exclusively due to each cut. that contributes by far the most to the costliness of the cut for minimum number of accepted hits on track. Besides hit quality demands, the only other significant losses to the statistics in Figure 5.1 are due to the upper cutoff in raw reconstructed β , where the algorithm starts to fail to produce a strong correlate to β ; of note for this effect is its greater significance in the endcap compared to the barrel, as show by Table 5.1, but this is just an artifact of the distribution of true β in η. FE-driven β is well supported and calibrated for reconstructed values of velocity in the barrel such that β ∈ [0.2, 0.8] and the endcap such that β ∈ [0.3, 0.98]. These differences are qualitatively consistent with what we expect based on the relative distance of each subsystem to the interaction point; the closer barrel retains slower tracks within its timing window, but the further endcaps have better resolving power and so can reach more effectively to higher velocities. Figure 5.3 illustrates how appropriate the β calibration for the FE is for each mass point, by plotting the bias in relative β resolution at each mass point, minus the bias of the overall sample. As anticipated from the fact that the complete sample was flat in mass from 100 GeV to 1 TeV in steps of 100 GeV, the global calibration performance is most consistent with that of the subsamples in the middle of this mass range. The trend towards faster biases for heavier particles is a remnant of the biasing mostly corrected by the weighting procedure described in the previous chapter; while the r-hadron sample is weighted to be flat in βtrue so that the slow fake tails of the faster bulk of the distribution don’t bias the slowest values of β high, the distribution eventually completely runs out of support and cannot be  63  residuals  residuals  0.01  FE  0.01  β  β  FE  0.005  0.02  0  0  -0.005 -0.01 Work In Progress β Residuals (Barrel)  -0.01 -0.015  Work In Progress β Residuals (Endcap)  -0.02  FE  FE  -0.03 100 200 300 400 500 600 700 800 900 1000  -0.02 100 200 300 400 500 600 700 800 900 1000 mass [GeV]  mass [GeV]  Figure 5.3: Mean relative residual β for each mass point after calibrating FEdriven β across the entire mass spectrum, minus the center of the relative β residuals for the entire mass spectrum, in barrel (left) and endcap (right). In both cases, our calibration biases heavier particles slightly faster.  2000  1000  1800  Work In Progress s = 7 TeV  1600 1400 1200  rHadron MC, 100GeV - 1TeV CBF  600  µ = 0.002 ± 0.006  1000  Work In Progress s = 7 TeV  800  rHadron MC, 100GeV - 1TeV CBF  µ = 0.006 ± 0.001  σ = 0.107 ± 0.005  σ = 0.114 ± 0.001  800  400  600 400  200  200 0  -0.5  0  0.5  1  1.5  0  2  Relative Residuals for FE mass (barrel)  -0.5  0  0.5  1  1.5  2  Relative Residuals for FE mass (ec)  Figure 5.4: FE-driven mass resolution for our MC CHAMP sample, in barrel and endcap. Fit parameters reported are for the Gaussian core of the crystal ball function. Mass is reconstructed here using the true momentum, in order to illustrate the effects on the mass resolution of the β estimator alone.  corrected be reweighting, leaving the extreme slow tail to still be biased slightly faster. Since the heaviest r-hadrons occupy the slowest part of the β spectrum, this fast bias preferentially affects them. In addition to examining the optimal performance of the β estimator, we’d like 64  massFE residuals  massFE residuals  0.06 0.05 0.04  Work In Progress massFE Residuals (Barrel)  0.03 0.02  0.12 0.1 0.08  Work In Progress massFE Residuals (Endcap)  0.06 0.04 0.02  0.01  0  0  -0.02  -0.01  -0.04  -0.02  -0.06  -0.03 100 200 300 400 500 600 700 800 900 1000  100 200 300 400 500 600 700 800 900 1000  mass [GeV]  mass [GeV]  Figure 5.5: Mean relative residual mass for each mass point after calibrating FE-driven 1/β γ across the entire mass spectrum, minus the center of the relative mass residuals for the entire mass spectrum, in barrel (left) and endcap (right).  to anticipate what we might see in terms of a mass spectrum extracted from our velocity estimate via m = p/β γ, should an excess over background be observed in the ABCD analysis. To do this, we begin by making a new calibration of our velocity estimator, this time expressed as 1/β γ to eliminate any biases there, similarly to how we calibrated β above save now on the mode instead of the mean of the truth distributions, as discussed in the calibrations section of the previous chapter; the results of these calibrations via fifth order polynomials can be seen in Figure 5.4, which turns the β measurement of Figure 5.1 into a mass estimate, using the true momentum in simulation in order to illustrate the contribution to the mass spectra of the velocity estimator alone. The mass distributions are fit to a crystal ball function (a Gaussian core with a one-sided power-law tail) in order to model the pronounced high-side mass tail which we observe and anticipate as an artifact of the roughly symmetric errors in β , since the mass goes as 1/β . The Gaussian cores of these fits suggests mass resolutions on the order of 10% ideally. Since we anticipate that momentum will be much better measured by ATLAS than our velocity estimators, we do not anticipate substantial degradation from such factors. Figure 5.5 illustrates the drift in bias of this mass estimate, as a function of mass. Corresponding to the drift in velocity biases, p/β γ returns a slightly lighter mass estimate for the faster-biased heavier tracks. The effect is more pronounced 65  in the endcaps, where as previously noted more of the slowest tracks are either too badly out of time to be reconstructed well, or simply lost outright, thus exacerbating the residual bias to faster velocities and lighter masses. From this information, the systematic width on the FE-driven mass estimate due to calibration can be estimated in the event of a mass peak presenting itself in data. As a final exercise to explore the performance of the FE β estimation methods, we sought to reconstruct the mass of the proton, in MC and early data; it should be noted, however, that in what follows the calibrations were re-done using slow MC protons, in order to make a more appropriate analogy to the discovery situation, which will reconstruct mass peaks using the above calibrations which were based on simulations close to the presumed mass of new physics. Figure 5.6 illustrates the proton peak in MC, where the proton mass is reconstructed to within a single MeV in the barrel at 938 MeV, and to within a few percent in the endcap at 923 MeV. The anomalously superior performance of the barrel over the endcap in this study was attributed to the tight boundaries on usable momentum space; in order to reconstruct a proton’s mass, we had to demand tracks with only a couple of GeV in momentum, so that the proton’s velocity would be readily resolved from c and well calibrated. However, ATLAS’s tracking algorithm begins to falter below about pT = 500 MeV, and tracks begin to loop in the magnetic field at about pT = 350 MeV; therefore, there was very little usable momentum space for calibration and analysis in the endcap especially, since a track with pT > 500 MeV must also have a comparable or larger longitudinal momentum in order to be found in the endcap (which sits at higher |η| than the barrel), thus quickly pushing us up against our β reconstruction limit in momentum. This tight squeeze will of course not be present in data, since we will be examining tracks with hundreds of GeV in p, far above ATLAS’ reconstruction minimum, and yet still within our velocity reconstruction limits for CHAMPs. More interesting features of the MC proton measurement can be seen in Figure 5.7. Here, we take the same information as in Figure 5.6, and plot it on a log scale, stacked by particle ID. The features on either side of the proton peak (especially visible in the barrel plots) are identified as kaons at low mass, and a broad deuteron peak at high mass. The kaons appear as expected, albeit badly suppressed in the endcap due to their even smaller mass compounding the problems with momen66  Figure 5.6: Reconstructed proton mass peak from simulated minimum bias data, in the barrel (left) and endcap (right). The barrel reproduces the proton mass to less than a single MeV.  tum space described above; the deuteron peak appears badly smeared, consistent with the expectation that these objects are not coming from the principle vertex, but rather from inelastic collisions with the beam pipe and perhaps the pixels and SCT, and thus are not as delayed in arrival at the TRT as they would have been, had they been traveling with low velocity all the way from the principle vertex as our β reconstruction assumes. Qualitatively, this reduced delay corresponds to a higher reconstructed β and subsequently a lower mass estimate as born out in the figure, compounded by the additional smearing attributed to the large uncertainty in production location. Finally, Figure 5.8 shows the same reconstruction now applied to an early 2010 data run from the LHC. The proton peak is clearly visible and sits within a few percent of its true mass, validating this method for the first time on real data. Like the simulation, the barrel shows additionally visible kaon and deuteron structure; both barrel and endcap mass measurements show a resolution of about 10%, in line with what we expected from simulation and calibration. Sidebar on Exceptional FE β = 1 PDF Shapes After constructing the β = 1 PDFs for the position of the falling edge, some unanticipated structure was observed in hits passing close to the anode wire. The left of Figure 5.9 shows one such FE PDF, which looks qualitatively bimodal, a fea67  Figure 5.7: As Figure 5.6, now plotted on a log scale and stacked by particle ID. The additional visible structure in the barrel corresponds to the presence of kaons and deuterons in the sample.  Figure 5.8: Reconstructed proton mass peak from real 2010 minimum bias data, in the barrel (left) and endcap (right). The proton mass is extracted from real data to within a few percent.  ture which was not originally anticipated from the model of a stable falling edge position with an early but monotonically decreasing tail from ionization inefficiency. However, this behavior was reproduced qualitatively by a toy simulation that models the baseline recovery response of the TRT electronics. When ionization electrons are received at an anode wire, the dissipation of the accumulated charge causes a negative baseline excursion which is only slowly returned to zero; this baseline restoration does not affect the rising edge of the LT bit pattern and hence has no effect on the tracking performance of the TRT, and so may be set 68  Figure 5.9: Examples of two falling edge β = 1 PDFs. On the right, a PDF for hits passing far from the anode wire looks like what we might expect qualitatively; but on the left, a PDF for hits nearer the anode wire have a pronounced bimodal structure that was originally surprising.  to occur gradually to avoid voltage baseline ringing. However, this long baseline restoration process can affect the timing of the falling edge under the right conditions; if the last ionization cluster is received at the anode while the voltage baseline has not yet fully recovered form the previous ionization cluster in the hit, it will appear to be smaller than it truly is and hence may fall short of the low threshold level, particularly if this LT is tuned to be as high as possible while still accepting the majority of ionization clusters, as it would be to most efficiently suppress noise. A simple toy simulation which modeled a TRT hit as a Poissonian distribution of ionization clusters spaced 200 µm apart on average along a path through a straw passing 200 µm from the anode, assumed a typical drift velocity for ionization electrons of 40 µm/ns and which modeled the anode voltage as being at the LT level for 3 timing bits (3.125 ns each) upon receipt of an ionization cluster (corresponding to the shortest time TRT signal shaping will allow for a physical signal) and below 0 for the following 8 bits (corresponding to 25 ns or 1 LHC bunch crossing, as slow as can be afforded if demanding a good baseline by the next bunch crossing), successfully qualitatively reproduces the behavior seen in data, as shown in Figure 5.10.  69  counts / 3.125 ns  1000 800  Work In Progress  600 400 200 0 0  10  20  30  40  50  60  70  FE time [ns]  Figure 5.10: The results of a very simple toy simulation which models the effect of a long baseline restoration time on the position of the falling edge. The behavior of the left of Figure 5.9 is qualitatively reproduced.  5.1.2  Rising Edge β Estimator  The same calibration exercise performed for the FE β estimator was carried out for the RE method. Figure 5.11 illustrates the relative widths of the best performing tracks under the RE β estimator; the barrel exhibits a β resolution of about 18%, while the endcap resolves velocity to within 10%. These resolutions are comparable but slightly inferior to the corresponding results produced by using the falling edge information; this was anticipated, since the timing of the rising edge is not only determined by the arrival time of the track at a straw (which up to ionization effects completely determines the position of the falling edge), but is conflated with the position the track crosses the straw, which, while accounted for in the analysis, is not perfectly reconstructed and therefore smears the performance accordingly. Figure 5.12 illustrates the spread in reconstructed velocity as a function of true velocity; again as anticipated, the best measurements are possible for the slowest values of β . 70  500  700  Work In Progress s = 7 TeV  400  Work In Progress s = 7 TeV  600  µ = 0.049 ± 0.002 σ = 0.176 ± 0.001  µ = -0.0225 ± 0.0008 σ = 0.0977 ± 0.0006  500 400  300  300 200 200 100 0  100 -0.6  -0.4 -0.2  0  0.2  0.4  0.6  0  0.8  -0.4  -0.2  Relative Residuals for RE β (barrel)  0  0.2  0.4  0.6  0.8  Relative Residuals for RE β (ec)  RE  0.7  60  Corrected β  Corrected β  RE  Figure 5.11: Relative RE-driven β width for well-reconstructed CHAMPS from 100-1000 GeV. Fits are to the Gaussian cores of the distributions; errors reported are statistical only.  0.65 50  0.6  40  0.55  60 0.8 50  0.75 0.7  40  0.65 0.5  30  0.6  20  0.55  0.45 0.4  20  0.5 10  0.35 0.3  0.4  0.5  0.6  0.7  0.8  0.9  β  true  1  10  0.45  Work In Progress 0.3  30  0  0.4  Work In Progress 0.4  0.5  0.6  0.7  0.8  0.9 β  (barrel)  true  1  0  (endcap)  Figure 5.12: The tracks of Figure 5.11, plotted as βtrue vs. the calibrated βRE in the barrel (left) and endcap (right). Resolution as a function of βtrue behaves as in the FE-driven case.  Table 5.2 identifies the cuts responsible for rejecting the RE β measurement for use in Figure 5.11. Similarly to the FE case, it is the hit quality cuts that are responsible for some of the largest losses. Table 4.2 from the previous chapter shows that the same internal components of the definition of hit quality as in the FE case are the leading contributors to RE measurement failure in terms of minimum number of hits surviving after hit-level cuts; dropping tube hits and HT hits produces the bulk of the contraction in acceptance here. Besides the hit quality cuts, Table 5.2 identifies the upper cutoff in RE-reconstructed β where such ve71  Total (|η| < 1) RE Accepted hit quality low reco high reco far tails  |η| < 1 |η| > 1 43971 37835 14121 12065 Tracks lost exclusively by cut: 8997 7816 285 7 9685 4413 7 4  Table 5.2: Total CHAMPS available to be measured vs. number accepted into Figure 5.11, the cuts applied in that figure, and how many tracks were dropped from each β resolution peak exclusively due to each cut. locities become difficult to reliably distinguish from β = 1 as the only other major culprit in rejecting this measurement, again much akin to the FE case. For the REdriven β estimator, the well-correlated regime between βest and βtrue extends over βest ∈ [0.45, 0.7] in the barrel and βest ∈ [0.5, 0.8] in the endcap; the much higher minimum in both these cases compared to the corresponding performance of the FE-driven estimator is largely just an artifact of the fast bias the estimator bares before calibration, but may also be attributed to the RE-driven estimator’s greater reliance on good tracking, with degrades as tracks move further from the tracking algorithm’s assumption of β = 1. Figure 5.13 shows the difference between mass dependent and global biases on the RE-driven β estimate in barrel and endcap. Similarly to the FE case, the overall trend is towards faster biases for heavier particles, for the same reasons as discussed above. The barrel in particular shows a substantial mass-dependent skew in the bias, owing to the very limited range of well calibrated tracks there. Finally, Figures 5.14 and 5.15 illustrate the performance of the 1/β γ calibration when used in conjunction with ptrue to reconstruct the mass of our signal tracks. The wider Gaussian cores and longer power law tails visible in the REdriven mass residual plots compared to the same for the FE-driven estimator are consistent with the substantial bias drift in mass also visible here, both owing to the limited regime in which RE β remains both well supported and well measured.  72  residuals RE  0.02 0  β  residuals RE  β  0.04  -0.02  0.03 0.02 0.01 0  -0.04  -0.01 Work In Progress β Residuals (Barrel)  -0.06  Work In Progress β Residuals (Endcap)  -0.02  RE  -0.08  RE  -0.03  100 200 300 400 500 600 700 800 900 1000  100 200 300 400 500 600 700 800 900 1000  mass [GeV]  mass [GeV]  Figure 5.13: Mean relative residual β for each mass point after calibrating RE-driven β across the entire mass spectrum, minus the center of the relative β residuals for the entire mass spectrum, in barrel (left) and endcap (right).  1000  1200 Work In Progress s = 7 TeV  1000  rHadron MC, 100GeV - 1TeV CBF  800  rHadron MC, 100GeV - 1TeV CBF  600  µ = 0.06 ± 0.02  600  Work In Progress s = 7 TeV  800  µ = 0.050 ± 0.002  σ = 0.16 ± 0.01  σ = 0.175 ± 0.001  400 400 200  200 0  0  1  2  3  0  4  Relative Residuals for RE mass (barrel)  -0.5  0  0.5  1  1.5  2  2.5  3  Relative Residuals for RE mass (ec)  Figure 5.14: RE-driven mass resolution for our MC CHAMP sample, in barrel and endcap. Fit parameters reported are for the Gaussian core of the crystal ball function. Mass is reconstructed here using the true momentum, in order to illustrate the effects on the mass resolution of the β estimator alone.  73  massRE residuals  massRE residuals  0.3 0.25  Work In Progress massRE Residuals (Barrel)  0.2 0.15  0.15  Work In Progress massRE Residuals (Endcap)  0.1 0.05  0.1 0.05  0  0 -0.05  -0.05 -0.1  -0.1  100 200 300 400 500 600 700 800 900 1000  100 200 300 400 500 600 700 800 900 1000  mass [GeV]  mass [GeV]  Figure 5.15: Mean relative residual mass for each mass point after calibrating RE-driven 1/β γ across the entire mass spectrum, minus the center of the relative mass residuals for the entire mass spectrum, in barrel (left) and endcap (right).  1400  1800  Work In Progress s = 7 TeV  1200  µ = -0.005 ± 0.001 σ = 0.1885 ± 0.0003  1000  1600  Work In Progress s = 7 TeV  1400  µ = -0.0726 ± 0.0003 σ = 0.0930 ± 0.0003  1200  800  1000  600  800 600  400  400  200 0  200 -0.5  0  0.5  1  1.5  2  2.5  0  3  Relative Residuals for Refit β (barrel)  -0.5  0  0.5  1  1.5  2  Relative Residuals for Refit β (ec)  Figure 5.16: Relative refit-driven β width for well-reconstructed CHAMPS from 100-1000 GeV. Fits are to the Gaussian core of the distribution; errors reported are statistical only.  5.1.3  Refit β Estimator  The track refit β estimator was put through similar paces to what is presented above for the FE and RE techniques. Figure 5.16 shows the relative β residuals achieved using this technique; the endcap shows particularly good performance, and both it and the barrel are competitive with the RE-driven β estimate. The substantially reduced performance of the barrel compared to the endcap is anticipated by the very  74  Refit  120  0.55  100  0.5 80 0.45  Corrected β  Refit  Corrected β  0.6  0.8 70 0.75 60 0.7 50  0.65  40  0.6 60  0.4 0.35 0.3  30  0.55 40  20  0.5  20  10  0.45  Work In Progress  0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 β  true  Work In Progress 1  0  0.4  0.3  0.4  0.5  0.6  0.7  0.8  0.9 β  (barrel)  true  1  0  (endcap)  Figure 5.17: The tracks of Figure 5.16, plotted as βtrue vs. the calibrated βRe f it in the barrel (left) and endcap (right). Resolution as a function of βtrue behaves as in the previous cases, although a particularly limited regime of the barrel remains well calibrated for this method.  small differences in drift circle radii that would be produced by traveling slower than c close to the interaction point; since LT timing information is binned in 3.125 ns bins, a one-bin difference in DC for a particle traveling with β = 0.5 compared to β = 1 isn’t produced until more than 90 cm from the interaction point - nearly outside the 1 m radius TRT in the barrel. Figure 5.17 illustrates these tracks distributions in reconstructed velocity versus true velocity, and the same anticipated behavior remains evident. Table 5.3 shows the acceptance achieved in making Figure 5.16 and the effects of the cuts that went into it; much the same behavior as the RE case, which relies on the same raw information, is visible there. Of particular note for this estimator is its excellent acceptance for signal, particularly in the endcaps, where it admits substantially more signal than the previous two β estimators into its competitively tight beta resolution peak; furthermore, despite this superior signal acceptance, the final columns of tables 4.1, 4.2 and 4.3 demonstrate that the overall acceptance of the well-performing regime of the refit-driven estimator does not accept substantially more background than the template-based methods, thus providing it with an edge in terms of signal discrimination. Figure 5.18 illustrates the drift of the bias on the velocity residual for this estimator as a function of mass, and shows the typical trend; the calibration is once again optimal for central values of mass and correspondingly β , as limited by the 75  Total Refit Accepted hit quality low reco high reco  |η| < 1 |η| > 1 43971 37835 17100 16720 Tracks lost exclusively by cut: 6093 5214 1287 193 12336 4732  Table 5.3: Total CHAMPS available to be measured vs. number accepted into Figure 5.16, the cuts applied in that figure, and how many tracks were dropped from each β resolution peak exclusively due to each cut.  residuals refit  0.05  0  0.04 0.02  β  β  refit  residuals  well-calibrated regime of the estimator.  0 -0.05 -0.02 -0.1  Work In Progress β Residuals (Barrel)  -0.04  refit  -0.15 100 200 300 400 500 600 700 800 900 1000  Work In Progress β Residuals (Endcap) refit  -0.06 100 200 300 400 500 600 700 800 900 1000  mass [GeV]  mass [GeV]  Figure 5.18: Mean relative residual β for each mass point after calibrating refit-driven β across the entire mass spectrum, minus the center of the relative β residuals for the entire mass spectrum, in barrel (left) and endcap (right).  Finally, we examine the performance of the refit-reconstructed β estimator in terms of mass reconstruction performance. Figure 5.19 shows the relative mass residuals for 1/β γ derived from the refit-driven velocity estimator combined with the true momentum, and Figure 5.20 shows how the bias of this distribution drifts as a function of mass. Much like the RE-driven estimate, the refit-based mass residuals are both quite wide and experience substantial drift in mass compared to the FE-driven estimates. The lightest, and hence on average fastest, r-hadrons in the simulation considered will have their mass overestimated by the refit-driven 76  estimator by as much as 50% in the barrel, compared to only 5% by the FE-driven estimator. In the case of all these estimators, residual calibration bias will make the fastest tracks reconstruct with a slow bias, as discussed; but in the case of the RE and refit-driven β estimates, LT bits at the rising edge of the bit pattern lost to ionization inefficiency will force velocity estimates even slower, exacerbating this problem and leaving the lightest, fastest particles underestimated in velocity and overestimated in mass more severely than in the FE-driven case, where ionization inefficiency at the end of the bit train makes the estimate bias faster instead, mitigating the slow calibration bias for these tracks. Ionization inefficiency creates a problem for the refit-driven estimator in particular, since its construction is less accommodating of this effect than that of the FE and RE driven methods. In the case of the two template driven methods, the loss of bits from the relevant end of the LT bit train is present in the construction of the β = 1 PDFs, and so will pull those β estimates more weakly by design; but in the case of the refit-driven estimate, the algorithm makes no allowances for failure to ionize at the distance of closest approach of the track to the anode wire, allowing such hits to pull the β estimate more strongly. In the barrel, where the timing differences are so small to begin with, these spurious and unmitigated pulls are most harmful. All these effects conspire to anticipate the broader resolution and worsening mass bias present in the refit-driven estimator. However, despite all this, Table 4.3 illustrated in the last chapter that the refit-driven β estimate excels beyond all others at producing signal well in excess of background in the β spectrum, and will thus prove most useful when searching for new signal, even if the mass of a new particle found by the refit-driven analysis may be better reconstructed by the FE-driven one.  5.1.4  dE/dx Correlate β Estimator  The performance tests applied to the other three β estimators were further executed for the dE/dx correlate method of β reconstruction. Figure 5.21 presents the relative residuals for the dE/dx-driven β estimate, illustrating the excellent performance of this estimator in extracting velocity in both barrel (where it far outstrips the performance of the RE and refit-driven estimators) and in the endcap. This β estimator is particularly useful for environments like the TRT barrel, which are  77  1800  1800  1600  1600 Work In Progress s = 7 TeV  1400 1200  1200  rHadron MC, 100GeV - 1TeV CBF  1000  µ = 0.062 ± 0.002  800  σ = 0.1920 ± 0.0035  600  600  400  400  σ = 0.169 ± 0.002  200  200 0  rHadron MC, 100GeV - 1TeV CBF  1000  µ = 0.0895 ± 0.0043  800  Work In Progress s = 7 TeV  1400  -1  0  1  2  3  4  0  5  Relative Residuals for Refit mass (barrel)  -1  0  1  2  3  4  Relative Residuals for Refit mass (ec)  massrefit residuals  massrefit residuals  Figure 5.19: Refit-driven mass resolution for our MC CHAMP sample, in barrel and endcap. Fit parameters reported are for the Gaussian core of the crystal ball function. Mass is reconstructed here using the true momentum, in order to illustrate the effects on the mass resolution of the β estimator alone.  0.5 0.4  Work In Progress massrefit Residuals (Barrel)  0.3  0.3 0.2  Work In Progress massrefit Residuals (Endcap)  0.1  0.2 0.1  0  0 -0.1 -0.1 100 200 300 400 500 600 700 800 900 1000  100 200 300 400 500 600 700 800 900 1000  mass [GeV]  mass [GeV]  Figure 5.20: Mean relative residual mass for each mass point after calibrating refit-driven 1/β γ across the entire mass spectrum, minus the center of the relative mass residuals for the entire mass spectrum, in barrel (left) and endcap (right).  78  1200 1000  Work In Progress s = 7 TeV  1200  Work In Progress s = 7 TeV  µ = 0.017 ± 0.005  1000  µ = 0.0127 ± 0.0007  σ = 0.102 ± 0.005  σ = 0.0979 ± 0.0006  800  800  600  600  400  400  200  200  0  -0.6  -0.4  -0.2  0  0.2  0.4  0  0.6  -0.6  -0.4  Relative Residuals for dE/dx β (barrel)  -0.2  0  0.2  0.4  0.6  Relative Residuals for dE/dx β (ec)  0.8  Work In Progress  140 120  0.7  100 0.6  80  dE/dx  160  Corrected β  Corrected β  dE/dx  Figure 5.21: Relative dE/dx-driven β width for well-reconstructed CHAMPS from 100-1000 GeV. Fits are to the Gaussian cores of the distributions; errors reported are statistical only.  0.9 0.8  120 Work In Progress  100 80  0.7  60  0.6  60 0.5  20  20  0.4 0.2  40 0.5  40  0.3  0.4  0.5  0.6  0.7  0.8  0.9 β  true  1  0.4 0.2  0  0.3  0.4  0.5  0.6  0.7  0.8  0.9  β  (barrel)  true  1  0  (endcap)  Figure 5.22: The tracks of Figure 5.21, plotted as βtrue vs. the calibrated βdE/dx in the barrel (left) and endcap (right). Resolution as a function of βtrue appears stable across the well-calibrated regime, as the ionization level used to estimate velocity in this method is not subject to the influence of the lever arm as the other three were.  79  Total dE/dx Accepted hit quality low reco high reco far tails  |η| < 1 |η| > 1 43971 37835 25444 19994 Tracks lost exclusively by cut: 7755 3361 1624 3748 2017 1580 66 0  Table 5.4: Total CHAMPS available to be measured vs. number accepted into Figure 5.21, the cuts applied in that figure, and how many tracks were dropped from each β resolution peak exclusively due to each cut. so close to the interaction point that timing resolution limits our ability to distinguish between particles moving with even significantly different speeds; since the dE/dx-driven estimator gets its information about velocity from ionization efficiency as per Figure 4.3 and not from timing, we are not limited by the same geometric constraints that handicap all the other velocity estimators. This point is further made by Figure 5.22, which shows comparatively stable widths in reconstructed β across the spectrum of true velocity. The acceptance and losses by the cuts used to make these plots are summarized in Table 5.4. The most costly cut employed is as usual the minimum number of quality hits on track, but despite this and the other constraints on acceptance into Figure 5.21, the dE/dx estimator maintains excellent performance throughout the detector while showing the best acceptance for signal by far into its well calibrated region. Table 4.4, however, also points out that this estimator has difficulty rejecting background, thus limiting its usefulness in identifying new physics for reasons that will be explored later; looking ahead to a possible signal scenario, the dE/dx-driven estimator may be the method of choice for reconstructing a new particle’s mass, particularly in the barrel, after its presence has been detected. The utility of the dE/dx-driven β estimate is further supported in Figure 5.23, which shows the drift in dE/dx driven residual bias as a function of mass; only the FE-driven velocity estimate is more stable across the mass range considered. Finally, figures 5.24 and 5.25 illustrate the mass reconstruction performance using the dE/dx correlate method. The mass residuals depicted there far outperform 80  residuals dE/dx  0.01  β  residuals dE/dx  β  0.02  0  0.03 0.02 0.01 0 -0.01  -0.01  -0.02  Work In Progress β Residuals (Barrel)  -0.02  Work In Progress β Residuals (Endcap)  -0.03  dE/dx  dE/dx  -0.04  -0.03  -0.05 100 200 300 400 500 600 700 800 900 1000  100 200 300 400 500 600 700 800 900 1000 mass [GeV]  mass [GeV]  Figure 5.23: Mean relative residual β for each mass point after calibrating dE/dx-driven β across the entire mass spectrum, minus the center of the relative β residuals for the entire mass spectrum, in barrel (left) and endcap (right).  1600  2200 2000  1400 Work In Progress s = 7 TeV  1800 1600 1400  rHadron MC, 100GeV - 1TeV CBF  1200 1000 800 600 400 200 0  Work In Progress s = 7 TeV  1200  rHadron MC, 100GeV - 1TeV CBF  1000  µ = -0.0161 ± 0.0009  µ = 0.03 ± 0.01  800  σ = 0.1432 ± 0.0002  σ = 0.113 ± 0.006  600 400 200 -0.5  0  0.5  1  1.5  2  0  2.5  Relative Residuals for dE/dx mass (barrel)  -0.5  0  0.5  1  1.5  2  2.5  3  Relative Residuals for dE/dx mass (ec)  Figure 5.24: dE/dx-driven mass resolution for our MC CHAMP sample, in barrel and endcap. Fit parameters reported are for the Gaussian core of the crystal ball function. Mass is reconstructed here using the true momentum, in order to illustrate the effects on the mass resolution of the β estimator alone.  the accuracy achieved in either the RE or refit-driven methods, and approach the level of performance of the FE-based estimator.  81  massdE/dx residuals  massdE/dx residuals  0.08 Work In Progress massdE/dx Residuals (Barrel)  0.06 0.04 0.02 0  0.3 0.2  Work In Progress massdE/dx Residuals (Endcap)  0.1 0  -0.02 -0.1 -0.04 100 200 300 400 500 600 700 800 900 1000  100 200 300 400 500 600 700 800 900 1000  mass [GeV]  mass [GeV]  Figure 5.25: Mean relative residual mass for each mass point after calibrating dE/dx-driven 1/β γ across the entire mass spectrum, minus the center of the relative mass residuals for the entire mass spectrum, in barrel (left) and endcap (right).  5.1.5  Correlations  Given that we have simultaneously developed 4 different β estimators that capitalize on four different aspects of TRT information, we would like to understand the extent to which they are independent and get a better understanding of how much information about velocity we truly have by examining the correlations between the residuals of all these estimators; to wit, Table 5.5 lists the correlation coefficients between residuals for each pair of samples presented in the above β resolution plots. Immediately evident are the very strong correlations between the residuals of the RE- and refit-driven β estimators, present due to the fact that both estimators rely almost entirely on the position of the rising edge of the TRT’s LT information (which as explained above contains all the tracking information as well, hence passing fakes from the RE measurement along to be similarly mismeasured by the refit-driven method). The sign of almost every correlation coefficient can be understood via the effect ionization inefficiency has on each velocity estimator: low ionization inefficiency causes the FE- and dE/dx-driven β estimates to fake fast (bits lost off the end of the LT bit pattern make the FE look earlier and thus from a particle that arrived sooner; and the dE/dx-driven β estimator is calibrated to the kinematic region of the Bethe82  FE RE Refit dE/dx ion.  FE -0.28 -0.40 0.30 -0.58  RE -0.28  Refit -0.40 0.80  0.80 -0.04 0.59  dE/dx 0.30 -0.04 0.05  ion. -0.58 0.59 0.68  0.05 0.68  Table 5.5: Residual correlations between the four β estimators (uncalibrated for FE-, RE- and refit driven estimators), and their correlation to the raw dE/dx ionization correlate. The strong correlations to ionization and the consistent signs indicate ionization inefficiency is a major source of bias and error for the three timing-based β estimators. Bloch curve, so lower ionization also corresponds to higher velocity), and the REand refit-driven estimators to fake slow (bits lost off the front of the LT bit pattern make the RE look later and thus from a slower particle, and also make the drift circle for that hit look larger and thus from a later particle, causing the RE- and refit-driven methods to fake slow respectively). The only sign inconsistency is on the very weak correlation between the dE/dx- and refit-driven velocity estimates. Furthermore, the final row and column in Table 5.5 shows that the residuals of the timing-based β estimators are quite strongly correlated to ionization efficiency in the TRT, indicating that ionization inefficiencies obscuring the true rising and falling edges of the TRT LT bit patterns are a major contributor to mismeasurement here, as anticipated.  5.2  ABCD Background Estimation  In the Methodology chapter, we planned to implement the ABCD background estimation technique described there on β − pT space, in order to detect any excess at high pT and low β that would indicate the presence of a CHAMP signal. Given the performance and correlations of our β estimators detailed in the previous section, we explore the detailed execution of this method here. When we first described our intended method for implementing an ABCD search in the previous chapter, one important decision was intentionally left open so that it could be informed by the performance testing of the velocity estimators 83  above; while we described an ABCD search in a β − pT plane, we did not say which of the four β estimators we would choose. At first blush, we would like to use the falling-edge β estimator or perhaps the dE/dx-driven one, for their superior performance in velocity resolution documented in the previous section. However, recall that our best strategy for constructing a data-driven understanding of our background is to extract a sample of muons from Z 0 → µ + µ − events in data, and use these to model our high momentum background under the auspices of the fact that for roughly p > 50 GeV, the ionization for µ ± and π ± remains constant, atop its relativistic rise. While this is in principle true for the individual background particles, this construction will fail to model the total ionization delivered to the TRT at very high momenta, when transition radiation photons accompany the hard backgrounds (appearing at roughly γ > 1000, corresponding to about p > 105 GeV for muons and p > 140 GeV for pions). This additional ionization appearing at the end of the LT bit train will drag the FE later, as well as increase the estimated ionization in the straw used in the dE/dx β correlate, and hence in both cases push the corresponding β estimate slower, creating a dangerous source of systematic fakes in our intended low-velocity signal region 2 . In order to avoid this systematic and construct a method where our data-driven background can be suitably employed, we therefore choose to use the rising edge driven β estimator in our ABCD search, since additional TR arriving at the ends of hits does not affect the position of the rising edge or its estimate of velocity. Since the TR photons should follow the track within the straw of the radiation that produced them, they will not travel closer to the anode wire than the track, and therefore will not ionize the gas closer than the distance of closest approach of the track to the anode, thus preserving the rising edge despite obscuring the dE/dx estimate and the FE. In fact, in the presence of TR we can be more assured that the LT timing bit that corresponds to the distance of closest approach will be set high, and not missed due to a stochastic failure of the charged particle to ionize the gas at its closest approach, thus suppressing one source of fakes in the RE-driven β estimate and making it the best choice for consideration here. 2 Actually,  a small fake slow tail can be seen in the distribution of relative β residuals in the endcap, on the right of Figure 5.1, where excess ionization has caused a small portion of the sample to appear systematically slow.  84  Now that we’ve made a judicious choice of which velocity estimator will best discriminate against background, we must estimate where to divide pT and βRE to construct signal and control regions with the best chance of setting a strong limit in the absence of a signal excess; we proceed with this as a function of signal mass, on 2.8 fb−1 of 2011 ATLAS data from the EF mu22 trigger (this trigger records any event with a muon-like object of pT > 22 GeV; a generic CHAMP traveling fast enough to trigger the muon station within its bunch crossing will satisfy this trigger. EF mu22’s efficiency for our generic gluino simulations as functions of velocity and momentum is shown in Figure 5.26). Efficiency is lost in all muon triggers below a minimum cutoff velocity, as is clearly indicated on the left of this figure; below this velocity, the CHAMP fails to trigger the muon system before the end of the bunch crossing it was produced in. At velocities high enough to pass the muon system in time, efficiencies of roughly 35% for r-hadrons are observed in simulation; we emphasize that this is actually a worst-case scenario, compared to other CHAMP searches, since this suppression of the trigger efficiency is due to the ability of the bare color charge of a gluino to pick up quarks as it traverses the detector, changing the charge of the r-hadron in flight, and thus possibly rendering it neutral by the time it arrives at the muon system, or allowing it to possess different charge states as it traverses the detector, causing it to fail to reconstruct as a muon. For CHAMPs with more conventional charge states, this efficiency should approach the muon efficiency at sufficient velocity. As discussed in the previous chapter, in order to use the muons from Z 0 decays as a representative sample for the entire background, this muon distribution will have to be scaled so that its tail in pT is of an appropriately representative size for the whole sample. Figure 5.27 shows the ratio of muons from Z 0 decays to the size of the entire data sample, as a function of pT . The very flat tail in this distribution indicates that the shape in pT of the sample of muons from these decays is well-representative of the whole, and we can construct a data-driven pure background sample of an appropriate size for the 2.8 fb−1 dataset simply by scaling up the Z muon distribution by a factor of ten. In the same spirit, the r-hadron simulations must be scaled to this integrated luminosity; Figure 5.28 illustrates the production cross section for each mass point √ at s = 7 TeV along with the systematic uncertainty due to the choice of parton distribution (discussed further in the Analysis chapter), which when taken in con85  Trigger Acceptance  Trigger Acceptance  0.35 0.3 0.25  Work In Progress  0.2  0.28 0.26 0.24 0.22  Work In Progress  0.2 0.18  0.15  0.16  0.1  0.14 0.12  0.05 0 0  0.1 0.2  0.4  0.6  0.8  0.08  1 True β  100 200 300 400 500 600 700 800 900 1000 p [GeV] T  Figure 5.26: Trigger efficiency for our signal by EF mu22 as a function of true β (left) and transverse momentum (right), illustrating the plateau in efficiency we can expect when populating our β − pT search region. Efficiency drops off abruptly for particles moving too slowly to trigger the muon system before the end of the bunch crossing they were produced in.  junction with the number of events presented in the sample defines a scale factor by which we may weight each sample for comparison to our 2.8 fb−1 of data. Now that appropriately weighted signal distributions at each mass point have been made to match our data-driven background construction, we can proceed with determining where best to place the division in pT and βRE at each mass point for the best signal significance. The resulting signal significance for each choice of boundaries produced by stepping βRE from 0 to 1 in steps of 0.05, and pT from 50 GeV to 1 TeV in steps of 10 GeV was determined in the endcap (where our velocity estimators perform best) for each mass point; simultaneously, in order to suppress backgrounds we also stepped through maximum βFE cuts from 0 to 1 in increments of 0.5. Also simultaneously, we demanded that both the FE and REdriven β estimators returned at least 20 good hits on track, in accordance with the positive results of the study on this presented in the Methodology chapter. The significance s of an amount of signal S atop a background B in this case was defined √ as s = S/ S + B; after every possible combination of these cuts was examined for each mass point, the combination resulting in the highest significance was reported as reproduced in Table 5.6. Given these values, we choose the cuts found for the  86  0.18 0.16 Work In Progress  0.14 0.12 0.1 0.08 0.06 0.04 60  80  100  120  140  160  180  200  Figure 5.27: The ratio of muons from Z decays to the entire data sample considered from the EF mu22 trigger, as a function of pT . The shape in pT of the Z muons represents the entire sample very well for almost all of our region of interest where the Z muons are well supported, and a simple scale factor of 10 is appropriate to scale the Z muon sample up to the size of the total background.  600 GeV mass point as our optimal cuts, for this is the heaviest mass point for which the optimal significance remains above 1.96, so we may anticipate the furthest reach for a 2σ exclusion limit with these cuts; we note here that given these cuts and trigger requirement, only 5 out of over 200,000 electrons from simulated Z → ee events survive, further justifying our use of a muon sample to understand our backgrounds here. The acceptance for gluino-based r-hadrons in simulation for these cuts under the EF mu22 trigger and in the TRT endcap (1 < |η| < 2, where our β reconstruction performs best and has smaller slow fake tails, a crucial consideration in the Analysis chapter when we look for slow-moving signal) is illustrated in Figure 5.29; it is of due note that our momentum cut, chosen to optimize signal 87  10  log (σ [pb])  4 3  Work In Progress s = 7 TeV Gluino Production (MC) ~ g production ± 1σ  2 1 0 -1 -2 -3  100 200 300 400 500 600 700 800 900 1000 mass [GeV]  Figure 5.28: Production cross section as a function of mass for the gluinobased r-hadrons considered in this study, fit to a third order polynomial. 1σ width represents the PDF uncertainty systematic. Simulations at each mass point were scaled by this cross section, times the analyzed data volume of 2.8 fb−1 before being added to the data sample when estimating signal significance for control region boundary optimization.  sensitivity for high mass, as yet un-excluded objects, produces low acceptance for the lightest model points (two orders of magnitude suppressed at 200 GeV and zero at 100 GeV); the effect of this on our search will be discussed with the corresponding results in the Analysis chapter. Finally, we observe that the correlation between βRE and pT in our Z 0 decay sample under our cleaning cuts is 0.017, a small but non-negligible residual correlation and therefore departure from the independence required as explained in Appendix A. In order to produce an appropriate correction factor for this effect and quantify the systematic uncertainty associated with this small dependency, we took our Z 0 decay sample under the cleaning cuts defined above, and divided it into as many adjacent pT bands as possible such that there were at least 50 tracks with βRE < 0.8. Then, the number of tracks with βRE in 88  Mass [GeV] 100 200 300 400 500 600 700 800 900 1000  Max s 588 131 40.5 16.1 6.48 3.19 1.51 0.823 0.475 0.264  pT div. [GeV] 60 70 110 130 180 220 240 240 390 390  βRE div. 0.95 0.85 0.85 0.80 0.80 0.80 0.95 0.95 0.95 0.95  βFE cut 0.95 0.95 0.95 0.95 0.95 0.95 0.75 0.75 0.95 0.95  S 363000 19300 2010 311 53.9 13.7 2.27 0.677 0.226 0.0695  B 1850 249 44.7 6.20 1.53 0.485 0 0 0 0  Table 5.6: Maximum significance found at each simulated mass point, after testing every combination of division in pT in steps of 10 GeV from 50 GeV to 1 TeV; βRE in steps of 0.05 from 0 to 1; and cuts for maximum βFE in steps of 0.05 from 0 to 1. pT,min [GeV] 50 60 75  pT ’ [GeV] 55 65 100  pT,max [GeV] 60 75 1000  predicted 112 70.6 25.2  actual 94 68 17  correction 0.84 ± 0.09 0.96 ± 0.17 0.68 ± 0.19  Table 5.7: pT boundaries used to form ABCD regions in the βRE − pT plane along with the βRE division at 0.80, where pT,min , pT ’ and pT,max correspond to xmin , x’ and xmax in Figure 4.8, the corresponding predicted and actual number of counts in the resulting signal region, and the ratio between the two used to codify the systematic error of this background estimation. Errors reported on these ratios are derived from the Poissonian counting errors on the control regions; the correction factors were combined as a weighted average to form the final systematic correction of 0.84 ± 0.04. every second one of these pT bins was predicted via the ABCD method using the pT bin before it, thus allowing us to estimate the multiplicative correction factor needed to being this prediction in line with the actual number of background tracks. In so doing, we produce an independent estimate of this correction for every two pT bins produced in this way, as tabulated in 5.7. The average correction factor and corresponding systematic returned by this method was 0.84 ± 0.04. 89  0.035 0.03 0.025 0.02 0.015 Work In Progress Gluino Acceptance (MC)  0.01 0.005 0  200  400  600  800  1000  Figure 5.29: Acceptance in simulation for gluino-based r-hadrons as a function of mass, under the signal region definition discussed in the text and the EF mu22 trigger, in the endcap.  90  Chapter 6  Analysis After developing and validating our β estimators and establishing how we can use them to be sensitive to the appearance of CHAMPs in data, we examined 2.8 fb−1 of data taken by ATLAS in 2011 using these techniques. In this chapter, we present the results of the ABCD analysis developed in the previous chapters, and the limits we can set on the mass of a particular model of gluino-based supersymmetric hadronic CHAMP.  6.1  Signal Search  As described previously, gluino-based r-hadrons were sought in 2.8 fb−1 of 2011 ATLAS data. The sample was constructed as per the requirements listed in Table 6.1, confined to the TRT endcap where our β reconstruction was best (and therefore more free of tails from slow β fakes). After applying the systematic correction and width for pT and βRE correlation and including the statistical width on the background counts in control regions A, B and C, a total of 4.90 ± 0.45 counts are anticipated in our signal region; an actual 5 counts are observed. The counts in each region are illustrated in Figure 6.1; the βRE spectrum for all these tracks is presented in Figure 6.2. Using our background estimate, we can seek to produce a 95% CL on the maximum mean number of signal events present in our signal region in data using this background model in a CLs analysis [47]. To briefly reiterate the method as  91  Sample Definition EF mu22 trigger from 2011 physics muons stream 50 GeV< pT <1 TeV 1.0 < |η| < 2.0 βFE < 0.95 ≥ 20 good FE hits ≥ 20 good RE hits Table 6.1: Sample definition cuts as optimized in the Chapter 5. After this construction, the pT − β RE plane was divided into control and signal regions at pT = 220 GeV and βRE = 0.8.  βRE  1  B
  C
  control
  control
  59,496  194
  A
  D
  
  0.8  control
  signal
  1,796
  5
 pT  0 50 GeV  220 GeV  1 TeV  Figure 6.1: Results from ABCD analysis for 2.8 fb−1 of 2011 ATLAS data.  detailed in the reference, signal is considered excluded at the 95% CL in CLs when 1 −CLs ≤ 0.95 92  (6.1)  Figure 6.2: Reconstructed βRE for the tracks in Figure 6.1.  where CLs ≡ CLs+b /CLb  (6.2)  where CLb is the probability that the background only hypothesis (a Poissonian distribution with mean 4.9) can produce as much or fewer events in the signal region as were observed, and similarly for CLs+b , the probability that the background plus some signal hypothesis (the aforementioned background distribution with a pull from a Poissonian signal distribution added to it, with signal mean an adjustable parameter) could produce as much or fewer events therein. CLs therefore is similar to a more conventional frequentist CL, but normalized to the p-value of the background hypothesis in order to suppress exclusion when the background and signal plus background hypothesis are poorly separated. From our background estimate detailed above, we can trivially extract CLb = 0.551 . We therefore need to find a signal hypothesis such that CLs+b satisfies the 1 And,  correspondingly, the p-value that our observed 5 counts could have been produced by our  93  constraint 6.1; we proceed by choosing a mean signal hypothesis, and performing pseudoexperiments consisting of a random pull from our background, added to a random pull from a Poissonian of mean equal to the signal hypothesis mean to be tested. After performing 106 of these pseudoexperiments, CLs+b for this signal hypothesis was found as the fraction of the distribution which produced 5 or fewer total events as per its definition. This procedure was repeated for signal hypothesis in the range (0,10] mean events in steps of 0.01, yielding a CLs+b which satisfies constraint 6.1 for a mean signal hypothesis of 6.62 ± 0.48 signal events (where the uncertainty reported represents the propagation of the uncertainty on the mean background estimate). Therefore, we report a CLs -based 95% CL upper limit on the number of signal events detected to be 6.62 ± 0.48 events, and can construct the corresponding mass limit for generic r-parity conserving gluinos by finding the largest cross section that could produce such a result. In principle, this cross section σ (m) is given by, where L is the integrated luminosity examined and A(m) is the acceptance by our cuts of the signal as a function of mass as extracted from simulation, σ (m) =  6.62 A(m)L  (6.3)  The systematic uncertainty on A(m) was estimated via the Poissionian fluctuations of the number of counts before and after the sample cleaning cuts, and the 2011-standard systematic uncertainty on the luminosity of 3.7% was adopted as per [48]; these were combined in quadrature with the systematic uncertainty reported on the upper limit of signal events above to form the uncertainty on our 95% C.L. upper limit on the production cross-section for generic gluinos. Furthermore, the theory cross-sections for the production of these objects were recalculated using two different sets of parton distribution functions, in order to estimate the systematic uncertainty brought about by this choice. Cross sections √ for pair production of gluinos from s = 7 TeV proton collisions were calculated using MadGraph 5 [49], using the CTEQ6M and CTEQ5l parton distribution function sets; at the high masses where we find our exclusion limit in what follows, the parton distribution systematic produced a relative error on the theoretical producestimated background alone is 0.45  94  tion cross-section of roughly 25%. This procedure produces the exclusion plot presented in Figure 6.3; the fine black line derived from equation 6.3 represents our experimental 95% C.L. upper limit on the gluino production cross section as a function of mass, surrounded by its 1- and 2σ uncertainties. The heavy black line represents the calculated cross sections for generic gluino production, and the crosshatched area is the 1σ systematic uncertainty on this calculation from choice of parton distribution function. This cross-section limit intersects the theory cross-section at m = 665 GeV, indicating the absence of generic detector-stable gluinos to the 95% CL below this mass. Figure 6.4 illustrates the FE-driven mass distribution of the 5 tracks admitted to our signal region; given the relative mass resolution of 11% illustrated in Figure 5.4 for the FE-driven estimator in the endcap, no obvious shape is evident at this integrated luminosity. Table 6.2 presents the salient track parameters for these objects; immediately evident is how close four of the five fall to the velocity boundary of βRE = 0.8, indicating that while these objects fell in our signal region, they remain very background-like. Finally, we point out that our zero acceptance for 100 GeV gluinos as illustrated in Figure 5.29 stops us from explicitly excluding these objects at that mass; however, a comparison of our acceptance to the calculated cross sections in Figure 5.28 reveals that the production cross section grows much more quickly than the acceptance drops off as we move to lower masses excluded in previous studies (from 600 GeV to 200 GeV, the cross section grows by three orders of magnitude, while the acceptance only drops by two); therefore, we rely on the previous work to exclude gluinos below 200 GeV in mass.  95  10  log (σ [pb])  -1 ~ g production ± 1σ 95% C.L. upper limit 1σ uncertainty 2σ uncertainty  -1.5 -2 Work In Progress -2.5  s = 7 TeV  ∫ Ldt = 2.8fb  -1  m > 665 GeV -3 600 650 700 750 800 850 900 950 1000 mass [GeV]  Figure 6.3: Stable gluino exclusion limit produced using the RE-driven β estimator as the complimentary discriminator to pT . Heavy black line indicates the reported gluino production cross section at each mass point, fit to a third order polynomial and surrounded by its 1σ uncertainty from choice of parton distribution; the fine black line represents the experimental 95% confidence level upper limit on the production cross section, with its 1- and 2-σ uncertainties.  mFE [GeV] 326 494 600 710 750  βFE 0.900 0.915 0.681 0.750 0.758  βRE 0.790 0.788 0.799 0.793 0.639  Hits (FE) 30 25 20 25 21  Hits (RE) 30 25 20 26 21  pT [GeV] 242 332 335 390 420  η -1.24 -1.39 -1.16 1.31 -1.30  Table 6.2: A few parameters of the tracks observed in our signal region. Hits (FE) refers to the number of hits on track accepted by our hit cleaning requirements, and similarly for Hits (RE). Four of the five signal region tracks sit within error (about 10% as reported in the Validation chapter) of the velocity boundary of the signal region. 96  1  Work In Progress  0.8 0.6 0.4 0.2  300 350 400 450 500 550 600 650 700 750  ×103  massFE [GeV] Figure 6.4: FE-driven mass estimates for the five events observed in our signal region. Mass resolution for this estimator in the endcap is 11% at best.  97  Chapter 7  Conclusion All told, four novel velocity estimators were developed for use in ATLAS’ TRT, each with its own strengths and weaknesses. The falling edge and dE/dx driven β estimators provided the best velocity and mass resolution for tracks with relatively clean ionization in the TRT, approaching that achieved in [46]. Alternatively, while it exhibited slightly worse β resolution, the rising edge driven velocity estimator avoided the systematics from transition radiation induced ionization from SM backgrounds at very high momentum, and was therefore able to produce a strong limit on the generic gluino mass of mg˜ > 665 GeV, outpacing the limits reported in 2011 by [50] of 544 GeV. In addition to setting new lower bounds on the masses of some generic rhadrons, we have also achieved the auxiliary purpose of our research, by demonstrating through several different techniques that competitive velocity measurements can be made using a compact gas-based detector. Similar studies such as[46, 51] successfully employ the large displacement of the muon system or tile calorimeter to extract estimates of β . [46] reports resolutions of 5% in β , outpacing our best of 7% only by a small margin; they also report superior resolutions of 2.6% at the CMS experiment via a dE/dx measurement in their semiconducting tracker. For future experiments where geometric constraints forbid the construction of a detector like the muon system (an order of magnitude bigger in radius than the TRT), or budgetary shortfalls make expensive semiconductor systems like the pixels at ATLAS or the analogous systems at CMS unfeasible, we have demon98  strated that a relatively cheap and compact gas tracker like the TRT which makes many independent timing and ionization measurements on each track can provide an effective alternative velocity measurement, without making major sacrifices in performance. The future of this analysis will benefit predominantly from the integrated luminosity to be delivered in the coming years at ATLAS. One benefit of designing this analysis to be as data driven as possible is that it will improve with added statistics; as data accrues and if TRT operation becomes more stable versus recalibration, the templates that our FE- and RE-driven β estimators rely on will become better constrained, thus suppressing the systematics derived from template construction that are folded into our β widths in these estimators. As the body of Z → µ µ events recorded grows, the exercises to estimate and control our backgrounds will produce better constraints. In addition to this passive convergence, three avenues present themselves for the pursuit of improvements to this project. First, the algorithmic design of the FEand RE-driven velocity estimators could benefit from a comprehensive and carefully validated study of how the relevant conditions change between calibrations of the TRT. Presently, the TRT is recalibrated on a roughly monthly basis, wherein all the timing and alignment constants are updated to keep up with detector drift and aging. At present, we have compensated for any effect this could have simply by re-generating our templates in parallel. However, if a process were developed such that the same set of templates could be constructed and used across all the data, then much higher statistics would be available in our templates, thus better constraining them as anticipated above, and also opening up the opportunity to increase the granularity at which they are constructed in order to more finely capture the local effects their binning attempts to parse. A second potential advance in this analysis would be an improvement of our understanding of very high pT backgrounds. At present, we rely on the stability of the Bethe-Bloch curve atop its relativistic rise to justify using the muons from Z decays to understand the ionization behavior of our backgrounds to arbitrarily high momentum. While this is justified for the ionization directly attributed to SM backgrounds, it does not capture the systematics introduced by transition radiation, and therefore we are forced to rely on our RE-driven β estimator in order to sup99  press and avoid these systematics in our exclusion limit. The RE-driven estimator was illustrated to be substandard to the FE-driven one when transition radiation is not a factor, particularly in the barrel. In time, a large enough sample of very hard muons will be collected and greater insight into the effect of transition radiation on the FE velocity estimator will be gained, possibly allowing the use of our best estimator in our ABCD search as well as giving us direct and unextrapolated control of our highest pT backgrounds. A final course of further study that may shore up this analysis would be continued examination of the small departure from independence between βRE and pT in the final ABCD background control exercise. At present, the small misestimation of the background in our signal region is examined and codified as a systematic to the method, but it is not a-priori clear why the position of the rising edge of the TRT LT pattern (in other words, a characteristic of the data determined by ionization) should have any dependence on momentum past the top of its relativistic rise in dE/dx. It is not impossible that at this early stage, limited statistics have allowed a fluctuation away from the ABCD prediction which is technically exact only at infinite statistics; but as the correction for this effect currently sits 4σ from unity, this possibility alone seems distant, or at least incomplete. In time, a stronger understanding of this systematic may be gained allowing its suppression or excision, thus furthering the reach of this already dominant limit on new physics.  100  Bibliography [1] Are Raklev. Massive Metastable Charged (S)particles at the LHC. arXiv:0908.0315.v2, 2009. [2] W Beenakker, Brensing. S, M D’Onofrio, Kr¨amer. M, A Kulesza, E Laenen, M Matinez, and I Niessen. Improved squark and gluino mass limits from searches for supersymmetry at hadron colliders. arXiv:1106.5647v2, 2011. [3] M Fairbairn, A Kraan, D Milstead, T Sj¨ostrand, P Skands, and T Sloan. Stable Massive Particles at Colliders. arXiv:hep-ph/0611040v2, 2006. [4] S Ambrosanio, G Kribs, and S Martin. Signals for gauge-mediated supersymmetry breaking models at the CERN LEP2 collider. arXiv:hep-ph/9703211v2, 1997. [5] J Feng, S Su, and F Takayama. Supergravity with a Gravitino LSP. arXiv:hep-ph/0404231v2, 2005. [6] A Gladyshev, D Kazakov, and M Paucar. 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Luminosity Determination in pp Collisions at √ s = 7 TeV using the ATLAS Detector in 2011. ATLAS-CONF-2011-116, 2011. [49] J Alwall, M Herquet, F Maltoni, O Mattelaer, and T Stelzer. MadGraph 5: Going Beyond. arXiv:1106.0522v1, 2011. [50] The ATLAS Collaboration. Search for Heavy Long-Lived Charged Particles √ with the ATLAS Detector in pp Collisions at s = 7TeV. arXiv:1106.4495v2, 2011. [51] The ATLAS Collaboration. Search for stable hadronizing squarks and gluinos with the ATLAS experiment at the LHC. arXiv:1103.1984v1, 2011.  104  Appendix A  ABCD Background Estimation  ymax
  y
  B  C
  
  control
  control
  y’
  A
  D
  control
  signal
  ymin
  x
 xmin
  x’
  xmax
  Figure A.1: Division of a 2D region into control and signal regions for an ABCD background estimation. Note that the variables x and y must be independent for the background in order for such an estimator to be appropriate. Consider the arrangement depicted in Figure A.1, in which a sample is rep-  105  resented in a scatterplot of two variables x and y which are independent1 for the background distribution BKG(x, y), the signal can be contained in one quadrant of the plot by some judicious choice of divisions x and y . These conditions being satisfied, independence in x and y allows the background distribution to be factorized as BKG(x, y) = f (x)g(y)  (A.1)  and the number of background events NA in the control region labeled ‘A’ in the figure can be written as y  NA =  x  f (x)g(y)dxdy  ymin xmin y  =  ymin  g(y)dy  x xmin  f (x)dx  (A.2) (A.3)  and similarly for the number of background events NB , NC and ND in regions B, C and D respectively, ymax  NB = NC = ND =  y ymax y y ymin  x  g(y)dy  g(y)dy g(y)dy  xmin xmax x xmax x  f (x)dx  (A.4)  f (x)dx  (A.5)  f (x)dx  (A.6)  Upon inspection of equations A.3-A.6, the popularly applied relationship for the amount of background in each quadrant NA ND = NB NC  (A.7)  is immediately evident, allowing the estimation of the amount of background in the signal region D simply by counting the populations in regions A, B and C.  1 It  is a very popular mistake when using ABCD background estimation to claim that it is enough for the variables to be merely uncorrelated; we remind the reader that this is a necessary but not sufficient condition for independence, and encourage the consideration of the simple illustrative exercise of y = x2 .  106  

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