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UBC Theses and Dissertations

Track impact parameter resolution in the BaBar detector Asgeirsson, David J. 2005

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Track Impact Parameter Resolution in the BABAR Detector by David J . Asgeirsson B . S c , Simon Fraser University, 1998 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in The Faculty of Graduate Studies (Physics) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Apr i l 21, 2005 © David J . Asgeirsson, 2005 f Abstract This thesis contains a study of track impact parameter resolution in the BABAR detector using lepton pair events from e+e~~ -» and from 7*7* —>• where I is either e or p,. The high number of these events in the data set and Monte Carlo simulations allows the tails of the resolution to be studied in detail. The Gaussian core of the resolution is consistent within 20% with the track-by-track errors returned by the track fitting software for both data and Monte-Carlo simulations. Beyond about three standard deviations (a) the non-Gaussian tail approximately obeys power laws. A simple parametrization is presented which fits the data well to beyond 10<r. The tail shape is consistent with that expected from a large-angle Coulomb scattering Toy Monte-Carlo. The GEANT4-based BABAR Monte Carlo software reproduces the core out to approximately 2a but the behaviour of the tails further out disagrees with the data and the predictions of Moliere theory. i i i Contents A b s t r a c t i i Conten ts i i i L i s t o f Tables v L i s t o f F igures vi Acknowledgemen t s x i 1 I n t r o d u c t i o n 1 1.1 C P violation in the Standard Model 1 1.2 B° - B° Mixing and the C K M Matr ix 2 1.3 Measuring C P Violation in B decays 3 1.4 Motivation for Track Impact Parameter Resolution Study . . . . 4 2 M u l t i p l e C o u l o m b Sca t te r ing 6 2.1 Theory of Multiple Coulomb Scattering 6 2.2 Monte-Carlo Generation of M C S Distribution 7 2.3 Parametrization of the Resolution Distribution 8 2.4 A Closer Look at the Tails 12 2.5 Dependence on Number of Scatters . . . 12 3 L e p t o n P a i r P r o d u c t i o n Processes 17 3.1 Bhabha Scattering and Muon Pair Production 17 3.2 Two Photon Process 19 4 T h e BABAR De tec to r 23 4.1 Overview 23 4.2 Silicon Vertex Tracker 25 4.3 Mult i -Wire Drift Chamber 25 4.4 Detector of Internally Reflected Cerenkov Light 26 4.5 Electro-Magnetic Calorimeter 27 4.6 Instrumented Flux Return 28 Contents iv 5 Data Acquisition and Event Reconstruction 30 5.1 Overview 30 5.2 Hardware System 31 5.3 L3 Trigger 32 5.4 Reconstruction 33 5.5 Conversion from Helix Parameters 34 6 Event Selection 37 6.1 Overview 37 6.2 NTuple Creation 37 6.3 Cosmic Rays 38 6.4 Detector Acceptance 42 6.5 Beam-Gas and Beam-Wall Tracks 42 6.6 Transverse Momentum Balance 44 6.7 Cut Selection Efficiency 47 6.8 The BABAR Particle ID System 48 7 Analysis 51 7.1 Variable for do Studies 51 7.2 Variable for z0 Studies 52 7.3 Variable for Momentum and Angular Dependence 53 7.4 Comparison of Data and G E A N T M C in Transformed Variable . 61 7.5 Comparison of Data and M C Fitted Parameters 63 7.6 Correlation of Track Impact Parameters 69 8 Conclusions 72 8.1 Results of this Study 72 8.2 Future Plans 73 A Tail Shapes in D A T A and M C Simulation: Az0 74 B Tail Shapes in D A T A and M C Simulation: Sd0 82 Bibliography 90 V List of Tables 6.1 Criteria of the track, photon and neutral lists used in this analysis. 38 6.2 Event selection statistics of the cuts used in this analysis 47 6.3 Cut-based selectors available in the PidElectronMicroSelector. . . 49 6.4 Cut-based selectors available in the PidMuonMicroSelector. . . . 49 vi List of Figures 1.1 Unitarity triangles constructed from the C K M matrix in a) stan-dard quark-mixing parameters and b) Wolfenstein parameteriza-tion 2 2.1 Angles generated from 9~3 distribution in the range 0.005 < 9 < oo. 8 2.2 Distribution of final deflections caused by scattering 20,000 times according to the screened Rutherford cross-section 9 2.3 Plot of the function (Equation 2.4) used to describe the track vertex resolution in this study. The solid line is the function value, the dashed line is the tail function T(x), and the dotted line is the Gaussian core G(x) 10 2.4 Plot of the Toy Monte Carlo distribution. The solid line is a fit to Equation 2.4 11 2.5 Plot of the pulls (data-fit / \Jdata) for the fit in Figure 2.4. . . . 11 2.6 Histogram of u = x~2 for x generated by the M C S distribution, and the transformed version of the fitted function to the data (solid line). The Gaussian core (dashed) and power-law tail (dot-ted) components are also shown 13 2.7 Plot of the mean of the Multiple Coulomb Scattering distribution, jU, as a function of the number of scatters, N 13 2.8 Plot of the fitted parameter representing the width of the core Gaussian, a, as a function of the number of scatters, N 14 2.9 Plot of the fitted parameter representing the fraction of events in the tail, / , as a function of the number of scatters, N 15 2.10 Plot of the fitted parameter representing the power in the inner power-law tail, pa, as a function of the number of scatters, N . . . 15 3.1 Bhabha-scattering of the incoming electron and positron to pro-duce an outgoing lepton pair in the a) annihilation channel and b) exchange channel 18 3.2 Distribution of momenta in the C M frame for Bhabha-scattered electrons produced with the PEP- I I beam energies 19 3.3 Feynman diagrams for a) exchange and b) annihilation diagrams with real photons radiated from all initial and final state particles. Normally only one of the four particles would emit a real photon. 20 List of Figures v i i 3.4 Two-Photon production of a) generic fermion pair, b) lepton pair via a lepton hairpin 21 3.5 Distribution of momenta in the C M frame for muons produced in the Two-Photon process with the initial beam energies of the BABAR experiment 22 4.1 Schematic view of the PEP- I I accelerator at S L A C 23 4.2 Cross-sectional view of the BABAR detector at the P E P - I I accel-erator. A l l dimensions are in mm 24 4.3 Cross-sectional view of a quartz radiator in the D I R C , showing the internal reflections of the Cerenkov photons 27 5.1 Schematic view of BABAR Data-Acquisition system 30 5.2 Schematic view of BABAR Front-End Electronics and Level-1 Trig-ger system 31 5.3 Projection of a charged track into x-y plane. Track parameters do, 4>o a n d P—\, a r e shown 34 5.4 Projection of track momentum into y-z plane. Track parameters PTi PZ and A are shown 35 6.1 Track pr for all events with exactly two charged tracks 39 6.2 The quantity 2~2 tan(A) for a small subset of events in the ntuple. The dashed lines indicate a cut of | ^ t a n ( A ) | < 0.1 40 6.3 Track for events with exactly two charged tracks and passing the Cosmic Ray Loose selection 40 6.4 Track p? for events with exactly two charged tracks and passing the Cosmic Ray Tight selection 41 6.5 Track px for events left with exactly two charged tracks after applying the Cosmic Ray Loose cut 41 6.6 Track px for events passing requirements for exactly 2 tracks, Cosmic Ray Loose cut, and detector acceptance cuts 42 6.7 Profile of track vertices in x-y plane, the sinusoidal curves are created by events originating from the beam-pipe walls 43 6.8 Distribution of the distance r between the track origin and one of the hotspots on the wall of the beampipe. A cut at r > 0.3 cm is indicated by the dashed line 44 6.9 Profile of track vertices in x-y plane, after removing events orig-inating from the beam-pipe walls. . . 45 6.10 Angle between the two tracks in the x-y plane \A<j> — ir\. The dashed line represents a cut at 0.2 radians 46 6.11 Track pT, after requiring they be back-to-back in the x-y plane. . 47 6.12 Track momenta for all events remaining at loose px balance cut. 48 7.1 Plot of Edo resolution for muon pairs in data, with 0.0 < 1/Q2 < 0.1. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function 55 List of Figures vi i i 7.2 Plot of Sdo resolution for muon pairs in data, with 10 < l / Q 2 < 50. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function 55 7.3 Plot of AZQ resolution for muon pairs in data, with 0 < l / Q 2 < 0.2. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function 56 7.4 Plot of A20 resolution for muon pairs in data, with 10 < l / Q 2 < 50. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function 56 7.5 Plot of Sdo resolution for electron pairs in data, with 0.0 < l/Q2 < 0.1. The fit (solid line) is to the function given by Equa-tion 2.4. The dashed line depicts the tail function 57 7.6 Plot of Sdo resolution for electron pairs in data, with 10 < l/Q2 < 50. The fit (solid line) is to the function given by Equa-tion 2.4. The dashed line depicts the tail function 58 7.7 Plot of Azo resolution for electron pairs in data, with 0 < l/Q2 < 0.2. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function 58 7.8 Plot of Azo resolution for electron pairs in data, with 10 < l/Q2 < 50. The fit (solid line) is to the function given by Equa-tion 2.4. The dashed line depicts the tail function 59 7.9 Plot of Sdo resolution for muon pairs in M C simulation, with 0.25 < l / Q 2 < 0.5. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function 59 7.10 Plot of Azo resolution for muon pairs in M C simulation, with 0.25 < l / Q 2 < 0.5. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function 60 7.11 Plot of Sdo resolution for electron pairs in M C simulation, with 0.25 < l / Q 2 < 0.5. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function 60 7.12 Plot of Azo resolution for electron pairs in M C simulation, with 0.25 < l / Q 2 < 0.5. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function 61 7.13 Plot of u for muon-pair Sdo resolution in Data (open histogram) and Monte-Carlo (shaded histogram) in the bin with 0.0 < l / Q 2 < 0.2. The solid curve is a transformed fit to /x-pair data. The dashed curve is a transformed fit to /i-pair Monte-Carlo. The dotted curve is the transformed tail function of the ^i-pair data, and the dot-dash curve is the transformed tail function of the /z-pair Monte-Carlo 62 7.14 Plot of the the mean of Sdo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of l / Q 2 . 64 7.15 Plot of the core width of the Sdo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of l / Q 2 64 List of Figures ix 7.16 Plot of the tail fraction of the Edo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of 1 /Q 2 . 65 7.17 Plot of the power in the inner power-law tail (|p a|) of the Edo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of 1 /Q 2 66 7.18 Plot of the the mean of AZQ resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of 1/Q2. 67 7.19 Plot of the core width of the Azo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of 1 /Q 2 67 7.20 Plot of the tail fraction of the AZQ resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of 1/Q 2 68 7.21 Plot of the power in the inner power-law tail (pa) of the Azo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of 1 /Q 2 69 7.22 Histogram of for muon events in data. Solid line is all muon-pair events. Dashed line is muon-pair events with Azo > 3<T. Dotted line is muon-pair events with Azo > 5a 70 7.23 Histogram of for muon events in data. Solid line is all muon-pair events. Dashed line is muon-pair events with Edo > 3a. Dotted line is muon-pair events with Edo > 5<r 71 A . l Plot of u = (o-(Azo)' V A z ° i | for ^u-pair events with 0 < 1 / Q 2 < 0 . 2 . . . 75 A.2 Plot of u = (o-(Azo)N \ A z ° / . 2 | for ^-pair events with 0.2 < 1/Q 2 < 0.25. 75 A.3 Plot of u = / CT(AZ0) ' \ A z ° / 2 | for /x-pair events with 0.25 < 1 /Q 2 < 0.5. 76 A.4 Plot of u = / CT(AZO) ' \ A z ° / 2 | for |i-pair events with 0.5 < 1/Q 2 < 1.0. . 76 A.5 Plot of u = /<r(Az 0)' \ A 2 ° / . 2 | for /j-pair events with 1.0 < 1 /Q 2 < 2.5. . 77 A.6 Plot of u = (a(Azo)' V A z ° / . 2 | for /u-pair events with 2.5 < 1/Q 2 < 5.0. . 77 A.7 Plot of u = / < T ( A Z 0 ) ' V A z ° / 2 | for /u-pair events with 5 < 1/Q 2 < 10. . . 78 A.8 Plot of u = (o-(Azo)' V A z ° / . 2 | for ^i-pair events with 10 < 1 /Q 2 < 50. . . 78 A.9 Plot of u = / cr(Az 0 ) ' 2 | for e-pair events with 0 < 1/Q 2 < 0.2. . . 79 V A z ° / A . 10 Plot of u = / (j(Az 0)' \ A Z ° J 2 | for e-pair events with 0.2 < 1 /Q 2 < 0.25. . 79 A . l l Plot of u = ( CT(AZO) ' \ A z ° / 2 | for e-pair events with 0.25 < 1 /Q 2 < 0.5. . 80 A.12 Plot of u = /cr(Azo)s \ A z ° , 2 | for e-pair events with 0.5 < 1 /Q 2 < 1.0. . 80 List of Figures x A.13 Plot of u = ( £ ^ 1 ) " for e-pair events with 1.0 < l / Q 2 < 2.5. . 81 A . 14 Plot of u = ( £ ^ l i ) 2 for e-pair events with 2.5 < l / Q 2 < 5.0. . 81 B . l Plot of u = (^§jf)2 for M-pair events with 0 < l / Q 2 < 0.1. . . 83 B.2 Plot of u = {^§^f for ^-pair events with 0.1 < l / Q 2 < 0.25. 83 B.3 Plot of u = ( £ ^ l i ) 2 for ^-pair events with 0.25 < l / Q 2 < 0.5. 84 B.4 Plot of u = ( ^ ^ ) 2 for yu-pair events with 0.5 < l / Q 2 < 1.0. . 84 B.5 Plot of u = ( £ ^ i ) 2 for ^-pair events with 1.0 < l / Q 2 < 2.5. . 85 B.6 Plot of u = ( 2 ^ l i ) 2 for yu-pair events with 2.5 < l / Q 2 < 5.0. . 85 B.7 Plot of u = ( ^ ^ ) 2 for ^-pair events with 5 < l / Q 2 < 10. . . 86 B.8 Plot of u = for /i-pair events with 10 < l / Q 2 < 50. . . 86 B.9 Plot of u = ( ^ f 1 ) 2 for e-pair events with 0 < l / Q 2 < 0.1. . . 87 B.10 Plot of u = {^§^)2 for e-pair events with 0.1 < l / Q 2 < 0.25. . 87 B . l l Plot of u = {^§^j2 for e-pair events with 0.25 < l / Q 2 < 0.5. . 88 B.12 Plot of u = ( £ j g a i ) 2 for e-pair events with 0.5 < l / Q 2 < 1.0. . 88 B.13 Plot of u = {^§jf)2 for e-pair events with 1.0 < l / Q 2 < 2.5. . 89 B.14 Plot of u = {^§^)2 for e-pair events with 2.5 < l / Q 2 < 5.0. . 89 xi Acknowledgements I would first like to thank my supervisor, Thomas Mattison, whose guidance and support made this research possible. I look forward to continuing to work with him in the future. I would also like to thank all of the other members of the BABAR group at U B C , Janis, Chris, Neil, Nasim, Song, Bryan and Doug. Thanks to all of you for your comments and advice during the past two years. Special thanks to Doug Maas for supporting the computing resources that were essential for this work. Finally, I want to thank Elissa for all of her love and support. This work would have been so much more difficult without you. Chapter 1 Introduction i 1.1 C P violation in the Standard Model The primary motivation of the BABAR Experiment is the measurement of C P -violation phenomena in 5-mesons - mesons containing at least one b (bottom or beauty) quark [1]. C P is a quantum-mechanical operator, corresponding to changing the sign of the charge, and applying the 3-dimensional parity or mirror-reversal operator. We know that if a symmetry is respected in nature, there must be a corresponding quantity which is conserved; a result proved by Emmy Noether in 1918 [2]. It was believed that both C and P should be good symmetries of nature, until Madame Wu [3] observed parity violation in 1957. After this discovery, it was seen that the nuclear weak force is maximally parity violating, or that parity is never conserved in a purely weak interaction. The subsequently developed V - A theory of the weak interaction restored C P as a good (unbroken) symmetry, but CP-violation was observed experimentally at a level of 2 parts in 1000 in Kaon decay by Cronin and Fitch in 1964 [4]. Since that time, CP-violation has been one of the most heavily studied topics in particle physics, particularly due to Sakharov's realization that it is a necessary ingre-dient in any explanation of the matter-antimatter asymmetry of the universe It was later shown by the theorists Kobayashi and Moskawa that C P viola-tion is allowed in the Standard Model of particle physics if there are at least three generations of quarks that are mixed through the weak interaction [6]. If there are exactly three generations of quarks, then this weak mixing between quark flavours can be described mathematically by a 3x3 matrix normally written as: where the subscripts u, d, s, c, t, b stand for the "up", "down", "strange", "charm", "top" and "bottom" species of quarks, respectively. In principle each element of the so-called C K M (Cabibbo, Kobayashi, Moskawa) matrix can be complex, but the matrix must be unitary if there are only three generations of quarks in nature. Requiring V to be unitary reduces the number of free parameters from 18 (9 real and 9 imaginary numbers) to 9 real and imaginary numbers. W i t h six flavours of quarks in the Standard Model, there are five relative phases between the wavefunctions of the quarks. These five phases reduce the number of free parameters in V from 9 to 4. A 3x3 matrix containing [5]. (1.1) Chapter 1. Introduction 2 Figure 1.1: Unitarity triangles constructed from the C K M matrix in a) standard quark-mixing parameters and b) Wolfenstein parameterization. four parameters can always be written as the product of three rotations plus a phase. It is this imaginary phase that is responsible for the phenomena of CP-violation. The parameterization of Wolfenstein [7], which is based on experimental measurements, is frequently used when discussing the latest results of tests on the unitarity of the C K M matrix. His parametrization is given by: / l - A 2 / 2 A X3A(p-ir)) \ V=\ - A l - A 2 / 2 A 2 .4 (1.2) \ \3A(l-p-iri) -X2A 1 / where A is the Cabibbo angle and A, p and r] may be experimentally measured. In this parameterization, all experimentally observed CP-violation is related to the imaginary phase TJ in the bottom-left and top-right elements. It is also clear that as written, this form of the C K M matrix is not unitary at the level of A 4 , but it still provides a convenient method of comparison for experimental results. By measuring whether or not the C K M matrix is truly unitary, we can test for new particles beyond the three generations of quarks thus far observed, or physics beyond the laws of the ElectroWeak interaction. One convenient way of testing unitarity is by multiplying one column by a row of the conjugate matrix to obtain an equation with 6 elements: vudv:b + vcdv;b + vtdv;b = o (1.3) which can also be interpreted as the requirement that the sum of three complex quantities vanishes. This can be represented in the Argand plane as the so-called Unitarity Triangle, shown in Figure 1.1a. If the three sides are divided by VCdV*b, the triangle can also be expressed in terms of the Wolfenstein parameters, as shown in Figure 1.1b. 1.2 B° -W Mixing and the C K M Matrix In any system in which the mass (energy) eigenstates are not identical to the flavour eigenstates of the particles, mixing between the different flavour states Chapter 1. Introduction 3 wil l occur. For example, the kaons KS and KL are mass eigenstates, while the K° and K° are flavour eigenstates. In the case of the neutral B mesons, the mass eigenstates are not the same as the flavour eigenstates so the B ° and B° also mix quantum mechanically. Unlike the kaon system, however, the mass difference is large and the lifetime difference is negligible. The oscillation amplitude, A(t), has the same time-dependence in any two-level system: A(t) oc cos(Am t) (1.4) where A m is the mass difference between the two flavour eigenstates. The mixing parameter for the Bd mesons (consisting of a b and d quark-antiquark pair) is then A m j . In the absence of CP-violation in mixing, the mass difference A m j = 2 | M i 2 | where M\i is the off-diagonal matrix element of the mass and decay matrix for the neutral B system. This matrix element is directly related to the C K M matrix elements Vtd and Vtb by the following equation: A/T GFmwVBmBdfBdBBd , 2 / 2 W T / * T / N 2 f, r\ Mu = ^ S0(mt/mw)(VtdVtb) (1.5) where GF is the Fermi constant of the weak interaction, mw is the W boson mass, rriBd, }Bd and Bsd are the mass, weak decay constant and Q C D bag parameter of the Bd meson. The Q C D bag parameter is proportional to the ratio between the nonperturbative coupling of the neutral B-mesons and the weak decay constant, and it is calculated using lattice Q C D . The function So(xt) can be approximated quite well by 0.784:rj' 7 6. For more details on this relation, please consult references [8] or [9]. By carefully measuring the time-dependent distribution of neutral B decays, physicists can determine these values of A m and M i 2 and thereby test the C K M matrix and the Standard Model. 1.3 Measuring CP Violation in B decays The BABAR experiment is using the physics of B-decays and .D-decays to mea-sure the angles a, /? and 7 of the Unitarity Triangle. More details on the physics objectives of the BABAR experiment are given in reference [1]. The cleanest possible process for producing pairs of B mesons is to collide electron-positron pairs at the right energy to produce the T(4s) meson. The T(4s) decays into a pair of B mesons, either B ° , B ° , or B+,B~. In the BABAR experiment electrons and positrons are annihilated to produce the T(4s) meson, which decays almost instantly into a pair of B-mesons. The two B mesons are very nearly at rest in the centre-of-mass frame of the T(As). If the beam energies were symmetric, and the T(As) was at rest in the lab frame, time-dependent analyses would not be feasible because the B lifetime is too short, approximately 10~ 1 2 seconds. At PEP-I I , however, electrons are accelerated to an energy of 9 GeV, and positrons are accelerated to an energy of 3.1 GeV, Chapter 1. Introduction 4 before they collide. The collision creates a system with a C M (centre-of-mass) energy of -y/i = 10.58 GeV, corresponding to the mass of the T(4s) meson, and a Lorentz boost in the lab frame of /?7 = 0.58 because of the asymmetric beam energies. This means that the T(4s) is moving along the beam axis, in the direction of the e~~ beam, with a velocity of roughly one-half the speed of light. This Lorentz boost converts timescales of order 10~ 1 2 seconds into distances on the order of 10~ 4 metres, or roughly 100 / im. Clearly, excellent position resolution along the beam axis of the detector is crucial to any time-dependent analysis. Furthermore, knowledge of this resolution often constitutes one of the dominant sources of systematic uncertainty in time-dependent BABAR results. The cleanest measurement of C P violation in B decays is the measurement of interference between direct and mixed decays to C P eigenstates. As an example, consider the neutral B decay to J/X/JKS, where the decay can proceed through two channels, either B° —• J/ipKs, or B° -> J/ipKs. The initial state of the B can be "tagged" by identifying the flavour of the other B meson. By looking for a decay to a non CP-eigenstate, for example B —> Dlv, the flavour of the products, (or charge of the primary lepton) can be used to identify the parent B as either a B° or B°. The time-independent asymmetry between the two channels (direct and mixed) vanishes because the "tag" B meson wil l decay before the signal B roughly half the time. If the tag is before the signal, the sign of the asymmetry switches, and the time-integrated asymmetry wil l cancel to zero. If, however, the decay vertices of the B ' s are reconstructed, then the time-dependent asymmetry can be measured, and C P violation can be observed. In this way, knowledge of the position resolution of the BABAR detector is crucial for precise measurements of C P violation in B decays. The B E L L E experiment [10], which is nearly identical to BABAR provides an important source of competition as well as an important check for the consis-tency of physics quantities which are being measured for the first time in these B-factory experiments. 1.4 Motivation for Track Impact Parameter Resolution Study Track impact parameter measurement is central to studies of both C P violation and neutral B mixing. Both are dependent on fitting to a time-dependent asymmetry, where the times have been determined by the difference in position along the beam direction between the particle tracks or the vertices formed by the particle tracks. The fit to an asymmetry includes models of the track impact parameter resolution. The resolution model parameters are in turn determined by the application of the model to "control samples", where the physics is known. For example, the track resolution in the BABAR dilepton B-mixing analysis is taken from a fit to the track impact parameter difference in J/ip —> decays, where both tracks must come from a single point in space [11]. These events provide a relatively small sample for fitting, concentrating only on the central Chapter 1. Introduction 5 part of the distribution. Finite resolution moves events from the true At between the B decay times to a different At value. This increases the statistical error in the measurements. If the measurements are not compensated for the resolution it can also introduce systematic shifts in the results. Typically, the fits are repeated with several different models for the resolution, and the difference between them is taken as the systematic error. The tails of the resolution can be particularly difficult to deal with. Events with large reconstructed At values are very important for CP , mixing and life-time measurements, but a significant fraction of such events are actually events with small values of At and large systematic errors. The goal of this thesis is to measure the track impact parameter resolu-tion. A high-statistics control sample of BABAR data events is selected. A parametrization of the resolution is developed that works well out into the tails of the distribution. The resolution is comparable between data and the fully re-constructed BABAR Monte-Carlo simulations. The theory of Multiple Coulomb Scattering, its effects on track impact parameter resolution, and a corresponding resolution model are developed in Chapter 2. The physics processes responsible for creation of the lepton-pair events selected for this analysis are explained in Chapter 3. The BABAR detector is described in Chapter 4. The BABAR data-acquisition electronics and software are described in Chapter 5. The process through which our control sample is selected from all of the BABAR data is doc-umented in Chapter 6. The analysis of the track impact parameter resolution for this control sample is detailed in Chapter 7. Finally, a summary of the findings of this thesis is presented in Chapter 8. 6 Chapter 2 Multiple Coulomb Scattering 2.1 Theory of Multiple Coulomb Scattering Coulomb scattering is the scattering of a charged particle by an atomic nucleus through the electrostatic force, or in the language of quantum field theory, through the exchange of a virtual photon. Scattering from an exponentially-screened Coulomb potential is given by the screened Rutherford cross-section: da n (2Zze2\2 1 — = 2tt 5 (2.1) dfi \ pv J ( s i n 2 ( 0 ) + X 2 ) 2 k ; where Z is the atomic number of the target material, z is the charge in units of e the electron charge, p is the 3-momentum magnitude, and v is the velocity magnitude of the incident particle. The parameter \a is known as the screening angle. Equation 2.1 does not include the relativistic spin-dependence, but this effect is washed out after many small-angle scatters. After integrating over the azimuthal angle <p, and making the small-angle approximation (sin(#) ~ 8), we arrive at the following relation: de a (0 2 + xl)2 a 0* ( } in which we can clearly see that the cross-section is proportional to 8~3. By re-peatedly applying the above cross-section, for each atomic layer, it is possible to calculate the distribution of final angles after an arbitrary path-length has been traversed. Williams [12], [13] was able to qualitatively describe the scattering distribution successfully with the sum of a central Gaussian and a single-scatter tail function. Several years later, Moliere [14] was able to give more exact an-alytical results, for a wide range of angles, although for practical applications this required the tabulation of numerical values. In the limit of small angles the resultant distribution for the angle of deviation 9 can be described by three separate terms to better than 1% accuracy [15]. The first term is a Gaussian core. Simply using this Gaussian term alone is accurate to better than 1% when considering relatively modest angles, 6 < 2a, and material thicknesses of more than 1000 atomic layers (which describes all practical experiments). The sec-ond term oscillates in the core and asymptotically approaches the Rutherford distribution at large angles. The third term is a higher-order correction. Chapter 2. Multiple Coulomb Scattering 7 The width of the central Gaussian peak scales roughly as the square-root of the thickness traversed, but there is an additional logarithmic dependence. The tabulated values calculated using the theory of Moliere have been fit to the following empirical relation: 90 = 1 3 - 6 M e V z v ^ [1 + 0.038ln(x/X0)} (2.3) pep in which x is the thickness in terms of the radiation length of the material, XQ. The Particle Data Group [9] recommends the use of this formula as an approximation to the width of the central Gaussian of the Moliere distribution. As the angle 9 increases, the distribution approaches that of the single scatter, i.e. oc 9~3. The popular detector simulation software package, G E A N T , used the Moliere formulation of Multiple Coulomb Scattering in Version 3 [16]. Version 4 of G E A N T [17] is based on the theory of Lewis [18]. This theory is formulated in terms of the moments of the distributions rather than the distributions themselves. Approximate distributions are chosen which have the correct moments to within certain tolerances. This is identical in the core to the Moliere theory but is not necessarily the same in the far tails. The BABAR collaboration uses G E A N T 4 for all detector simulations. When simulating Multiple Coulomb Scattering (MCS), it is much more effi-cient to select the final angle from the complicated Moliere or Lewis distributions rather than to simulate single scatters millions or billions of times. Using the Moliere distribution directly to describe experimental data is rather awkward, but it is possible to create a parametrization that describes the distribution reasonably well out into the tails. 2.2 Monte-Carlo Generation of MCS Distribution In order to generate a sample of events having the correct distribution, we re-peatedly sample angles from a simple distribution. This is more time consuming than using the Moliere or Lewis distributions, but is certain to provide us with the correct distribution far out into the tails. Figure 2.1 shows the distribution of scattering angles generated by a single throw of a random number following the simple 9~3 distribution, in the range 0.005 < 9 < oo. A second uniform random number cos(cp), is selected from a flat distribution between -1 and 1. The original value of 9 is then multiplied by cos(0) to give a displacement centered on zero. Twenty thousand of these individual displacements were then added together to give the final deflection for a single particle. The entire process was repeated for 100,000 particles and the resultant distribution is shown in Figure 2.2. The Gaussian core of Multiple Coulomb Scattering is clearly visible, along with the power-law tail correspond-ing to the original 9~3 distribution used. This simple "Toy" Monte-Carlo model can be employed later to compare to the track impact parameter resolution dis-Chapter 2. Multiple Coulomb Scattering 8 Figure 2.1: Angles generated from 9 3 distribution in the range 0.005 < 9 < oo. tributions in the experimental and GEANT-simulated data, as well as to help choose an appropriate parameterization of the resolution distribution. 2.3 Parametrization of the Resolution Distribution In order to help us determine the best lineshape for describing the track impact parameter resolution we look more closely at the power-law behaviour of the tails of the distribution. We know that the inner 2a of the distribution is reasonably described by a Gaussian, and we will concentrate on the outer portions of the tail . Far enough out in the tails, the shape must asymptotically approach that of the Rutherford differential cross-section, a power-law of the form 9~3. The transition between the Gaussian core and the 9~3 power-law tail is usually described only approximately. The resolution due to M C S can be accurately described by a Gaussian core and a tail function which eventually becomes a power-law tail at large deflec-tions [15]. We will choose a tail which contains two separate power-laws. We construct a general formulation of this type of function using the following equa-tion: y(x)=N(l-f)G(x)+NfJ^ (2.4) AT Chapter 2. Multiple Coulomb Scattering 9 20 30 Deflection Arb. Units Figure 2.2: Distribution of final deflections caused by scattering 20,000 times according to the screened Rutherford cross-section. G(x) = e i° T(x) = a~p-T(x) = x — p T(x) = x — p ab 2-KO-x — p,\ < aa aa < \x — p\ < ba -Pb AT = 2[ H ' 1 - " - ) + — ) + 2 f 6("'-"-) 1 - Pa x — p\ > ba fc(i-Pb) l-Pb (2.5) (2.6) (2.7) (2.8) (2.9) where N is the total number of events in the histogram, G{x) is the Gaussian core, T(x) is the tail function, AT is the area under the tail , and / is the fraction of the total events under the tail function. The parameters for the Gaussian core G(x) are the mean of the distribution p, and the width of the Gaussian a. The parameters for the tail function T(x) are: the inner breakpoint between the Gaussian core and the first power-law tail a, the power in the first power-law tail pa, the outer breakpoint between the first power-law tail and the second 6, and the power in the second power-law tail p0. Figure 2.3 is a plot of the function with the following values used to generate the plot, N = 10 6, p, = 0.0, a = 1.0, a = 3.0, b = 5.0, pa = 5.0 and pb = 3.0. Chapter 2. Multiple Coulomb Scattering 10 Figure 2.3: Plot of the function (Equation 2.4) used to describe the track vertex resolution in this study. The solid line is the function value, the dashed line is the tail function T(x), and the dotted line is the Gaussian core G(x). The function is dominated by the Gaussian core (dotted line in the Figure) in the range -2< x <2. We see that the tail function (dashed line in the Figure) is flat in the centre of the distribution, and falls off with two distinct power-laws to either side. We note the the function is not smooth at either of the breakpoints, and this can present practical difficulties in achieving convergence when this function is used for curve-fitting. These problems are resolved when two of the four parameters a, b, pa and pb are held fixed. We know that the M C S distribution must eventually vary as 9~3 so we can fix Pb=3. We found that fixing the two breakpoints a, and b allowed the fitting algorithm to easily converge in all cases. As a first test of the ability of Equation 2.4 to describe track impact param-eter resolution distributions, we used it to fit the results of a Toy Monte-Carlo generated by the method described in Section 2.1. One hundred thousand events were generated with increasing numbers of scatters (N), ranging from N=2 up to N=20,000. A l l of the fits described in this section were performed with the two breakpoint parameters fixed at a = 3.0 and b = 5.0. These values were chosen based on the inspection of the tails described in the next section. The outer power-law was fixed at p0 = 3.0, the same power-law dependence as the Rutherford differential cross-section. Figure 2.4 is a plot of the fit to the same distribution shown in Figure 2.1, with N=20,000. We observe reasonable agree-ment well out into the tails of the distribution. Figure 2.5 is a plot of the pulls of the same fit. The pulls are calculated by taking the difference between the Chapter 2. Multiple Coulomb Scattering 11 20 30 Deflection (Arb. Units) Figure 2.4: Plot of the Toy Monte Carlo distribution. The solid line is a fit to Equation 2.4. 3 a. Figure 2.5: Plot of the pulls (data-fit / \Jdata) for the fit in Figure 2.4. Chapter 2. Multiple Coulomb Scattering 12 actual and fitted values, and dividing by the expected statistical error of yjn. We see that the data and fit function agree to within approximately 3 standard deviations over the range between -10 and +10. It is difficult, however, to ver-ify that our fitted function accurately describes the data well into the tails by looking at a log-plot covering several orders of magnitude, such as Figure 2.4. The pulls could also contain subtle systematic variations in the tails. 2.4 A Closer Look at the Tails To more closely examine the agreement in the tails of the distribution we can re-plot the data using a transformed variable, along with the suitably transformed fitting function. We now consider the transformed variable u = (x/a)~2, where x is the displacement generated by the method outlined above, and a is the width of the Gaussian core. If the distribution of x follows the power-law a ; - 3 then u will be flat. Any function transformed from f(x) to f(u) wil l be multiplied by a Jacobian factor of u _ 3 / 2 / 2 . We know that as x goes to infinity the power-law goes to -3. This means that u wil l be flat as it approaches zero. Figure 2.6 shows a histogram of u when x has been generated according to the method described in the previous Section, as well as the transformed fit-function (solid line), transformed Gaussian core (dotted line) and transformed power-law tails (dashed line). We see that in terms of the transformed variable, u, the agreement between the Toy M C and fitting function is quite good over almost the entire range. We can see that there is more than one power describing the data between u=0 and u=0.1, or, equivalently, between x = oo and x =3. The graph is nearly flat between u=0 and u=0.04, meaning that a power-law exponent of-3 between 5a and infinity wil l describe the shape well. Between 3<r and 5cr we see that the data must follow a faster fall-off, so the exponent must be larger. There is a small disagreement around u=0.05, corresponding to somewhere between 4 and 5cr. This disagreement occurs when the tail function peaks, and disappears as the tail decreases steadily. There also appears to be a small disagreement in the region of u < 0.01, corresponding to greater than 10CT. This discrepancy was also visible in the untransformed plot. Overall, we conclude that the parametrization given in Equation 2.4 describes the Toy M C quite well out to approximately lOc. 2.5 Dependence on Number of Scatters The fitting process was repeated for different values of N , and the fitted param-eters tabulated as a function of N . The number of events returned by the best fit was consistent with the actual number N=100,000. Figure 2.7 is a plot of the fit results for the mean of the distribution (^i) versus the number of scatters (N). The mean of the distribution was generally consistent with zero, but var-ied between -0.007 and +0.004. This suggests that we can expect some small Chapter 2. Multiple Coulomb Scattering 13 Figure 2.6: Histogram of u = x~~2 for x generated by the M C S distribution, and the transformed version of the fitted function to the data (solid line). The Gaussian core (dashed) and power-law tail (dotted) components are also shown. Figure 2.7: Plot of the mean of the Multiple Coulomb Scattering distribution, /U, as a function of the number of scatters, N . Chapter 2. Multiple Coulomb Scattering 14 1 10 102 103 104 N scatters Figure 2.8: Plot of the fitted parameter representing the width of the core Gaus-sian, cr, as a function of the number of scatters, N . variation of the mean to occur simply as a result of the statistical fluctuations in the fitting procedure when we apply this model to experimental data. Figure 2.8 is a plot of the fit results for the width of the central Gaussian (cr) versus the number of scatters (N). We can clearly see a logarithmic dependence on the number of scatters. This is in general agreement with the dependence of the core width given earlier in Equation 2.3. We don't expect to see as much change in the experimental data since the thickness of material traversed in the detector doesn't vary by as many orders of magnitude. Figure 2.9 is a plot of the fraction of the events in the tail function, / , as a function of N . When we only have 2 scatters, the purported tail fraction is very high because the distribution is not at all Gaussian. By the time we have increased to 10 5 scatters, the tail fraction has dropped to approximately 2%. We can compare these fractions to the results found when we fit to the data, however, we note that the value of / is not really the number of events beyond 3o\ It is largely determined by the area of the fiat tail function T(x) under the core Gaussian. A change in the width of the core Gaussian wil l therefore change the value of / substantially even if the other parameters describing the tail shape are unchanged. This behaviour must be borne in mind when we examine the value of / in our experimental data. Figure 2.10 is a plot of the exponent in the inner-portion of the power law tail, pa, as a function of N . It ranges between 3.0 and 4.0, increasing rapidly with N . Again, when N=2 the distribution is closer to that of the original x~3 than to the combination of a Gaussian and tail predicted by M C S theory. It appears that the value of pa asymptotically approaches a value of approximately 4.0, and Chapter 2. Multiple Coulomb Scattering 15 Figure 2.10: Plot of the fitted parameter representing the power in the inner power-law tail, pa, as a function of the number of scatters, N . Chapter 2. Multiple Coulomb Scattering 16 we can check to see if the same values are obtained in our fits to experimental data. If the fitting function did not take into account the value of the core width, i.e. if it used x~Pa instead of {^)~ P a we would expect the inner tail power to depend strongly on the core width. As the breakpoint moved into the core it would raise the value of pa and vice-versa. If we divide the track-impact parameters by the core width in the experimental data then we should not see this effect, and the inner power-law should be approximately 4.0 if the tails in the track impact parameter resolution distributions are really due to Multiple Coulomb Scattering. A l l of the behaviour seen in this Toy Monte-Carlo simulation should be clearly visible in the experimental data if the track impact parameter resolution tails are dominated by Multiple Coulomb Scattering. By dividing the devia-tions by the errors calculated by the tracking software we expect to obtain a distribution which closely resembles one of the Toy Monte-Carlo distributions. In real data we are unable to measure exactly the number of scattering layers a track has passed through, but we can construct a quantity that should be proportional to the amount of M C S and examine the resolution as a function of that quantity. On the other hand, if there are other significant contributions to the track impact impact parameter resolution which are not described by either the geometric detector resolution or the effects of Multiple Coulomb Scattering, then the experimental data should display markedly different behaviour than these Toy Monte-Carlo simulations. Chapter 3 17 Lepton Pair Production Processes In this chapter we examine the physical processes responsible for the events which we wil l use to study the track impact parameter resolution. In order to fully explore the tails of the impact parameter resolution functions, it is necessary to use a very large sample of dilepton events so that the distributions will be populated with a significant number at many a away from the central value. The easiest way to study the actual resolutions is to use the Q E D events that are already present in large quantities in the data to characterize the track impact parameter resolution functions. There are two main categories of events containing lepton pairs: the first is "Bhabha" scatters and the related muon pairs, and the second are "Two-Photon" events. 3.1 Bhabha Scattering and Muon Pair Production This section describes the physical processes by which Bhabha scattering and muon pair-production occur. This interaction was named after H . J . Bhabha, who first described it in detail for electrons [19]. The process can be described by the two Feynman diagrams 1 in Figure 3.1a and b, known as the annihilation or t diagram and exchange or s diagram, respectively. The differential cross-section of the annihilation diagram is given by: £ = £ ( 1 + (3.1) where s = E2cm is the square of the total energy in the C M frame. Muon pairs are only produced through the annihilation diagram. The angular distribution for electrons in the final state is more complicated, because it depends on the amplitude for both diagrams, including interference between them. The cross-section for electron production strongly peaks at 6=0 in the C M frame, while the cross-section for muon production is more isotropic in the C M frame. The two final state leptons (either electrons or muons) will have nearly equal momentum magnitude in the C M frame. We expect these leptons should all have one-half ' •Al l Feynman diagrams produced with the JaxoDraw [20] software package. Chapter 3. Lepton Pair Production Processes 18 Figure 3.1: Bhabha-scattering of the incoming electron and positron to produce an outgoing lepton pair in the a) annihilation channel and b) ex-change channel. the total C M energy, or approximately 5.2 GeV. A simulated momentum spec-trum for electron pairs in the C M frame is shown in Figure 3.2. The spectrum was obtained using the " B H W I D E " event generator software package [21] and the standard BABAR reconstruction software described in Chapter 5 of this doc-ument. The spectrum for muon pairs is virtually indistinguishable from that of the electrons, except that the peak momentum occurs about 105 M e V lower due to the muon's larger mass. Lepton pairs scattered through the exchange diagram wil l still have most of their momentum along the beam axis, since the cross-section for that diagram peaks at 0 = 0, and their px spectrum will fall off rapidly. Lepton pairs which interact through the annihilation diagram will have a more isotropic distribution in the C M frame and a momentum spectrum peaking at about 5 GeV. The pr spectrum for leptons produced through annihilation also peaks at roughly 5 GeV with a low-energy tail . The Feynman diagrams shown in Figure 3.1 can be modified by the emis-sion of real photons from either the initial or final state particles, as shown Chapter 3. Lepton Pair Production Processes 19 4500 F 4000; 3500; 3000; 2500; 2000 ; 1500; IOOO ; 500: ~i—i—i—1—i—i—i—i—I—i—r ^ i i i i — i—i i i i i i— ° o t -_J I I I I I I I I I I LiaJ I I I I I L p CM (GeV) Figure 3.2: Distribution of momenta in the C M frame for Bhabha-scattered electrons produced with the PEP- I I beam energies. in Figure 3.3. We expect a low-energy tail in the lepton spectrum, made up of leptons which have lost some of their initial energy by radiating a photon. In both initial and final state radiation, these processes wil l create final state leptons with less than 5 GeV of momentum in the C M frame, and the tracks will not be entirely back-to-back in the C M frame either. Initial state photons wil l tend to be emitted parallel to the incident beams, and wil l therefore es-cape down the beampipe without detection. These events wil l have only two visible charged tracks in them and nearly balanced transverse momentum, but missing energy. If the final state particles emit photons then the photons wil l normally be detected and could be included in the event reconstruction. If they are not included in the event reconstruction the events wil l be rejected because the charged tracks' px will not balance exactly. 3.2 Two Photon Process In order to obtain a large number of dilepton events which have transverse momenta between 0 and 4 GeV we also consider the so-called Two-Photon events, produced via the following reaction: e + e~ -> e + e - 7 * 7 * e+e~l+r (3.2) Chapter 3. Lepton Pair Production Processes 20 e" e Figure 3.3: Feynman diagrams for a) exchange and b) annihilation diagrams with real photons radiated from all initial and final state particles. Normally only one of the four particles would emit a real photon. The general two photon process producing a fermion pair is shown in the Feyn-man diagram of Figure 3.4a. If the final state is hadronic the vertex can only be treated approximately, but if the final state is a lepton pair, as shown in Figure 3.4b, then the process can be treated exactly through a lepton hairpin. These two photon events have momentum spectra peaking at zero and falling off steeply with energy. Their transverse momenta are basically equal and can range up to 5 GeV. They represent the vast majority of dilepton events in which the C M momenta are less than 4 GeV. In addition, any one of the photons in Figure 3.3 can also produce a fermion pair. A simulated spectrum for muon pairs produced through the two-photon process is shown in Figure 3.5. The spectrum was produced using the " G A M G A M " event generator software package which is currently the two-photon physics standard for the BABAR collaboration, al-though it was originally written for the C L E O - I I experiment [22]. Chapter 3. Lepton Pair Production Processes 21 Figure 3.4: Two-Photon production of a) generic fermion pair, b) lepton pair via a lepton hairpin. Chapter 3. Lepton Pair Production Processes 22 Figure 3.5: Distribution of momenta in the CM frame for muons produced in the Two-Photon process with the initial beam energies of the BABAR experiment. 23 Chapter 4 The BABAR Detector 4.1 Overview The BABAR detector is located at the PEP- I I accelerator at the Stanford Linear Accelerator Centre (SLAC) , shown schematically in Figure 4.1. Electrons are accelerated to an energy of 9 GeV, and positrons are accelerated to an energy of 3.1 GeV, before they collide in the interaction region IR2. These energies create a system with a C M energy of 10.58 GeV, and a Lorentz boost in the lab frame of /?7 = 0.58. The BABAR detector, shown in Figure 4.2, is comprised of P E P II Figure 4.1: Schematic view of the PEP- I I accelerator at S L A C . five subsystems. The innermost is the Silicon Vertex Tracker, based on silicon micro-strip detectors. Next is a multi-wire drift chamber filled with an Argon-Isobutane gas mixture, followed by a detector of internally-reflected Cerenkov light used for particle identification. A n electromagnetic calorimeter, composed of CsI(Tl) crystals, measures the photon energies and angles. A 1.5T supercon-ducting magnet encloses the inner four detector layers and the axial magnetic field forces charged particles to follow curved trajectories in the detector. The magnetic flux return is instrumented with resistive plate chambers to detect neutral hadrons and identify muons by penetration. Measurement of the track curvature determines the track momentum and combining the momentum with the velocity measured by the DIRC allows the determination of the particle mass. Full details of the BABAR detector design and performance can be found in [23]. B A B A R C o o i d i n a t e S y s t e m y | ^ " x C r y o g e n i c C h i m n e y \ z I n s t r u m e n t e d F lux R e t u r n ( IFRJ) B a r r e l C h e r e n k o v D e t e c t o r ( D I R C ) M a g n e t i c S h i e l d for D I R C S u p e r c o n d u c t i n g C o i l E l e c t ID m a g n e t i c C a l o r i m e t e r ( E M C ) Dri f t C h a m b e r ( D C H J S i l i c o n V e r t e x T r a c k e r ( S V T J Figure 4.2: Cross-sectional view of the BABAR detector at the PEP- I I accelerator. A l l dimensions are in mm. Chapter 4. The BABAR Detector 25 4.2 Silicon Vertex Tracker The micro-strip silicon vertex tracker (SVT) is composed of five double-sided layers. The layers are segmented into strips used to determine a passing parti-cle's location along both the z and <j> axes. In total, there are roughly 150,000 individual strips read out from the S V T . When a charged particle passes through the active regions of silicon, it deposits energy which leads to the creation of electron-hole pairs. These pairs are swept apart by an applied voltage and the resulting current pulse is detected as a "hit" in one of the strips. The locations of hits are combined to infer the location of the track at each layer of the S V T . When attempting to measure the vertex of dilepton events, the resolution is primarily determined by the S V T , since the hits in the silicon are the clos-est points along the track to the beamspot and their spatial resolution is the best. For tracks with high transverse momentum (pr), the S V T can resolve the point of closest approach to the beamspot to within 50 pm, and sometimes as accurately as 10 pm. This resolution is primarily determined by the following factors: the strip size in the S V T , the angle at which the track traverses the S V T layers and the track momentum. Values are calculated for each track as they are reconstructed in the software, providing resolution estimates on a track by track basis. In addition, the alignment of the S V T with respect to the beams is calibrated on a continual basis, and contributes another systematic error to the vertex measurements. 4.3 Multi-Wire Drift Chamber The multi-wire drift chamber (DCH) is filled with a mixture of helium and isobutane (C4H10) gases kept at 4 mbar above atmospheric pressure. The sense wires are kept at high-voltages ranging between 1900-2000V. There are a total of 28768 wires, and they are arranged into forty different concentric layers. Twenty-four of the forty layers are placed at a small angle relative to the z axis to measure the 8 angle of passing tracks. When a charged particle passes through the D C H , some of the gas is ionized along the path of the track and the ionized gas molecules are attracted to the ground wires while the liberated electrons are drawn toward the sense wires. The electrons collide frequently with gas molecules on their way to the sense wire, resulting in a constant drift velocity. When the electrons are very close to the wires, the increased electric field gives them enough energy to knock loose additional electrons from each gas atom. This creates a large gain in the signal picked up by the sense wire, on the order of 50,000 for this particular gas mixture and voltage setting. The resulting voltage pulses on the sense wires are read out and their arrival times are converted into distances from the wire, using the constant drift velocity. The position resolution ranges between 0.1 and 0.4 mm, depending on how far away from the wire the original track was located. This resolution is ultimately the limiting factor in determining how well the track curvature parameter u>, and hence the momentum, can be measured. Chapter 4. The BABAR Detector 26 The other important measurement made by the D C H is of energy loss per unit distance, or dE/dx. The total charge deposited in each cell of the cham-ber is used to calculate the energy loss over the chamber width. There are a number of corrections applied to improve the measurement accuracy, including continuous updating of the gas pressure, temperature and gain, wire-by-wire geometrical constants and variation of the energy loss with the angle of the in-cident particle. Ultimately the accuracy of dE/dx measurements is roughly 7%. The measurement of dE/dx within the D C H can be used for particle identifi-cation in conjunction with the Detector of Internally Reflected Cerenkov Light (DIRC), or by itself for tracks which are within the acceptance of the D C H but miss the DIRC. 4.4 Detector of Internally Reflected Cerenkov Light A charged particle travelling with velocity /3 = « /c in a medium of refractive index n produces Cerenkov light if n/3 > 1. For relativistic charged particles, a cone of Cerenkov radiation is emitted with a characteristic angle given by: cos(0c) = l/(n/3) (4.1) Measurement of the Cerenkov angle, in conjunction with knowing the particle momentum from the drift chamber, allows a determination of the particle mass and hence, the particle type. The D I R C sub-detector is a novel device used for the first time in the BABAR experiment. It is a ring imaging Cerenkov detector based on total internal re-flection and uses quartz bars as both the radiator and the light guide. The D I R C Cerenkov radiators are 4.9 m long rectangular quartz bars oriented par-allel to the z axis of the detector. The quartz has an index of refraction n=1.473. Through internal reflections, the Cerenkov light from the passage of a particle through the DIRC is carried to the ends of the bar, as shown in Figure 4.3. This radiation is captured by internal reflection in the bar and transmitted to the photon detector array located at the backward end of the detector (forward-going light is reflected by a mirror located on the end of the bar). The high optical quality of the quartz preserves the angle of the emitted Cerenkov light during total internal reflection. A n advantage of the DIRC for an asymmetric collider, like PEP- I I , is that the high momentum tracks are boosted forward, causing a much higher light yield than for particles at normal incidence. This is due to two effects: the longer path length in the quartz and a larger fraction of the produced light being internally reflected in the bar. The photon detector array consists of about 11,000 conventional 2.5 cm-diameter photomultiplier tubes. They are organized in a close-packed array at a distance of about 120 cm from the end of the radiator bars. The photo-tubes, together with modular bases, are located in a gas-tight volume as protection against helium leaks from the drift chamber. Chapter 4. The BABAR Detector 27 Figure 4.3: Cross-sectional view of a quartz radiator in the D I R C , showing the internal reflections of the Cerenkov photons. To maintain good photon transmission for all track dip angles, the standoff region is filled with water to minimize the change in index of refraction. The water seal occurs at a quartz window that is glued to the quartz wedges. The standoff box is surrounded by a steel box which provides adequate magnetic shielding for the photo-tubes. The main design goal for the DIRC was to be able to distinguish pions and kaons at momenta greater than 2 G e V / c . The statistical separation between the two species ranges from about 10a at 2 G e V / c to approximately 2>a at 4 G e V / c , easily meeting the specified design requirements. 4.5 Electro-Magnetic Calorimeter The Electromagnetic Calorimeter (EMC) was designed to measure electromag-netic showers with high efficiency, high angular resolution, and high energy resolution over the range from 20 MeV to 9 GeV. The lower energy limit is de-termined by the need to reconstruct 7r°'s resulting from B decays, and the higher energy limit is necessary to measure Bhabha-scattered electrons and muons. The E M C is also helpful in identifying electrons. This sub-detector is divided into two parts, a cylindrical barrel surround-ing the inner detector systems, and a conical forward end-cap. The barrel is composed of 5880 Thallium-doped Caesium-Iodide crystals, arranged into 280 separate modules, while the end-cap contains 20 modules of 41 crystals each. CsI(Tl) was chosen for the crystals primarily because of its high light yield and small Moliere radius which allow for excellent energy and angular resolution. High-energy photons and electrons travelling through the material of the Chapter 4. The BABAR Detector 28 calorimeter create electromagnetic showers and the resulting photons are ab-sorbed by the crystals and re-emitted as visible light. These scintillation pho-tons are then detected by high-speed silicon photo-diodes mounted on the outer surfaces of the crystals. Silicon photo-diodes were chosen because of their ability to function properly inside the 1.5T magnetic field of the BABAR solenoid and because it is possible to match their absorption spectrum to that of the crystal light output. The measured energy resolution of the E M C ranges from 5% with a 6.13 M e V radioactive source, down to 1.9% for Bhabha-scattered electrons at 7.5 GeV. The angular resolution varies from 12 mrad at low energies to 3 mrad at high energies. Finally, the energy measured in the E M C is used with the momentum determined by the drift chamber in order to identify electrons with an efficiency of between 88-94%, and a pion misidentification probability ranging from 0.15-0.3%. 4.6 Instrumented Flux Return The Instrumented Flux Return (IFR) was designed to identify muons with high efficiency and low contamination, as well as to detect long-lived neutral hadrons over a wide range of angles and momenta. The main objectives for the I F R are large coverage angles, good efficiency and good rejection of muon background down to momenta lower than 1 GeV/c . For the neutral hadrons, high efficiency of detection and good angular resolution are the most important criteria. The IFR is divided into three separate sections: a cylindrical barrel and forward and backward end-caps. Each section is composed of layers of iron absorber material interspersed with active detector layers. The iron absorbs energy from highly-penetrating muons and long-lived neutrals, as well as acting as the flux return for the solenoidal superconducting magnet. The number of layers of steel, and their thickness, were chosen using simulations to optimize the detection of muons and neutral hadrons. The steel absorber consists of 18 steel plates with thicknesses ranging from 2-10 cm. The active detector layers were originally Resistive Plate Chambers (RPCs) [24], that provide two-dimensional coordinates and precise timing information for passing particles. The barrel originally contained 19 R P C layers and the end-caps had 18 active layers. A n R P C consists of two bakelite plastic plates separated by a 2 mm gap containing a gas mixture of argon, freon and isobutane. The bakelite has a volume resistivity of approximately 10 1 1 — 10 1 2 i?cm. The inner surfaces of the gas-filled gap are coated with linseed oil in order to reduce high-voltage breakdown. A voltage of roughly 8 k V is applied between the two plates and the passage of a particle leads to the creation of a limited streamer between the plates. The plates are read capacitively with strips running along both the z axis and in the <j> direction. There are a total of approximately 50,000 channels. The efficiency of individual R P C modules can be determined using cosmic rays. Initially more than 75% of the modules had an efficiency greater than 90%, but at least 50 modules had already failed by the summer of 1999. Between 1999 Chapter 4. The BABAR Detector 29 and 2004 the overall efficiency of this detector system continued to degrade from 98% to roughly 75%. The angular resolution of the system continued to meet the design criteria. Due to the continuing problems with R P C s , the BABAR collaboration de-cided in 2004 to undertake their replacement with another technology known as Limited Streamer Tubes (LSTs) [25]. These use extruded plastic tubes with wires inside, instead of the flat gaps. They have a long successful history, and are manufactured with a high degree of quality and reliability. The first two sextants of the I F R were disassembled and the LSTs were installed during the summer shutdown of 2004. The rest of the R P C modules wil l be replaced during the summer shutdown of 2006. The I F R is primarily used for particle identification of high energy muons and K^s. The efficiency of muon ID has continually decreased due over the past 5 years due to the degradation of the R P C s , but this does not affect this analysis in any way. This analysis studies the vertex resolution of tracks identified as muons and is not concerned with the absolute number of muons, or the efficiency of their detection. Chapter 5 Data Acquisition and Event Reconstruction 5.1 Overview There are three distinct stages of reconstruction between the raw electronic pulses in the detector elements and the calculation of physics quantities of in-terest such as branching fractions and resonance masses. A schematic diagram of the BABAR Data-Acquisition system (Figure 5.1) depicts the main stages of the data reconstruction and storage process. Full details on the BABAR Data-Acquisition system can be found in Reference [23]. There are two levels of Detector Element Front-End '^Electronics | ReadOut "A~*l Module Online Reconstructionr Event Store Figure 5.1: Schematic view of BABAR Data-Acquisition system. triggering in the BABAR experiment. They are commonly referred to as the Level-1 (LI) Trigger and the Level-3 (L3) Trigger. The Level-2 Trigger was not implemented. The Level-1 Trigger uses fast electronics along with a pipeline buffer in order to accept or reject events within 11 ^secs. The Level-3 Trigger is implemented in software and utilizes calculated physics values to decide whether to accept events for storage in the database, or reject them. Chapter 5. Data Acquisition and Event Reconstruction 31 5.2 Hardware System The Front-End Electronics and Level 1 trigger system of the BABAR detector are shown schematically in Figure 5.2. In Step 1 of the process shown in Figure 5.2, Figure 5.2: Schematic view of BABAR Front-End Electronics and Level-1 Trigger system. events are initially just voltage pulses in various detector elements. In Step 2, these voltage pulses are readout by attached Analog-to-Digital Converters (ADCs) which digitize the voltage, and in some cases, the waveform of the raw detector pulses. In Step 3, these digital signals are then passed into a circular buffer, which is being continually overwritten. This buffer is necessary to allow time for the L I trigger to consider a group of signals, and if desired, pass them along, before processing the next signals. In Step 4, the L I trigger fires within a window of 11-12 ps after an e + e~ crossing. If an event of interest is detected, all of the data channels are transferred from the detector to output buffers in Step 5. In Step 6, the detector signals make their way through the buffer, until reaching the top in Step 7. Once at the top of the buffer they are sent from the buffer to the ReadOut Modules (ROMs) in Step 8. The Readout Modules are responsible for sending information from the detector electronics to the BABAR computing system. The A D C and buffer are both part of the Front End Electronics (FEEs) that accompany each detector subsystem. Typically a single F E E card is designed to handle many detector elements in parallel, and R O M s are designed to gather the information from several F E E cards in parallel. The maximum rate for L I acceptance is approximately 2.5 kHz, and BABAR has normally operated at approximately 1 kHz, well within the available bandwidth. The standard L I trigger decision is based on three signal types: charged tracks in the D C H which have higher pr than a threshold value, clusters in the Chapter 5. Data Acquisition and Event Reconstruction 32 E M C which have more than a minimum amount of energy, and muon tracks detected by the IFR. In order to calculate these quantities, a series of Field-Programmable-Gate-Arrays (FPGAs) are used to perform basic reconstruction tasks like constructing track segments from D C H hits, adding energy in adjacent crystals of the E M C and determining where in the IFR muons have passed. The output of the F P G A s are then combined in hardware to generate the overall Level 1 Trigger signals. A n implementation of the same algorithms in software would not be fast enough to process events within the 12 ps window. If an L I accept is issued, then the data is passed on to the next stage, the Level-3 Trigger. 5.3 L3 Trigger From the ROMs , the digital data representing signals in the detector elements pass through an Ethernet connection to a farm of 32 computers running the Level-3 Trigger software. The purpose of the Level-3 Trigger (L3) is to deter-mine whether or not the "event", made up of a collection of detector signals, represents an e+e~ collision, or a collection of random hits. The L3 trigger is essentially a refined and augmented software implementation of the L I trigger logic. For example, in the L3 D C H algorithm the track segments identified in the L I system are refined and refit, adding and subtracting adjacent hits and track segments in order to improve the quality of the fit. This allows a much better resolution on track parameters, and better rejection of tracks resulting from beam-gas or beam-wall interactions. The tracks are combined and averaged by the L3 software to determine an event time, *o-The L3 E M C algorithm forms clusters of E M C hits just as the L I system did, but with added filters applied. The time at which the hit occurred is required to be within 1 ^s of t0, and the energy deposited in the crystal is required to be greater than or equal to 20 MeV. In addition to simply forming clusters, the centroid and lateral moment of each cluster is calculated and used later for particle identification. The event stream at this stage is dominated by Bhabha events. To economize on data storage costs, only a fraction of the Bhabha events are retained. Events are rejected based on one-prong and two-prong topologies, in which either only the positron, or both the positron and electron are seen in the detector. The rejection is scaled to produce a flat distribution in polar angle. Even after this rejection, approximately 30% of the stored events are either radiative Bhabhas passing the filter or Bhabhas stored for use in calibration. This analysis will focus on these "background" events and ignore events containing B mesons. If events pass the L3 trigger, they are stored on tape. The L3 trigger passes events at a rate of approximately 120 Hz, a rate chosen to balance the desire for completeness of the data sample and the burden of database storage and "offline" processing. Chapter 5. Data Acquisition and Event Reconstruction 33 5.4 R e c o n s t r u c t i o n The reconstruction software can be thought of as a bridge between the data-readout in the electronics and the end-user analysis system. One needs to re-construct enough physics information in order to make an informed decision about whether or not we have a signal event, or just background detector hits. Besides applying some of the calibration constants (particularly for the E M C ) , the reconstruction also repeats the preliminary track-finding, taking individual detector hits and associating them with each other into tracks and calorimeter clusters. The software used in reconstruction is built upon some of the same C + + classes as the end-user analysis software in order to improve reliability and consistency. Essentially, raw information from the detector is reconstructed into the object-oriented events that are used in the offline analysis software. For the purposes of the analysis in this thesis the primary task of the re-construction software is to construct three-dimensional trajectories for charged particles out of the individual hits in the S V T and D C H . This is done by first searching for a cluster of hits located in adjacent layers of the D C H , and con-structing a simple track segment from those hits. The charged particle tra-jectories can be described in terms of a 5-parameter helix and the measured value of the magnetic field in the detector. Figure 5.3 depicts the projection of a charged track into the x-y plane. The point-of-closest-approach ( P O C A ) is the point along the track helix closest to the z-axis. In the BABAR set of helix parameters, do is the perpendicular distance between between the P O C A and the z-axis, <po is the angle between the track momentum at the P O C A and the x-axis, Zo is the z-coordinate of the P O C A , p is the radius of curvature of the track, and the curvature ui = 1/p. The two parameters do and zo are referred to as the track impact parameters. Figure 5.4 depicts the projection of track momentum into the y-z plane. The momentum of the track is divided into two components, pr in the x-y plane, and pz along the z-axis. The track parameter tan(A) = pz/pr is the "dip" of the track away from the xy-plane. For the initial track fits, the hypothesized mass is equal to that of a pion, and if particle identification later shows that the track is an electron it is necessary to repeat the helix fit with the correct mass value. Note that determining the absolute origin of a helical track is not possible; we can only locate the point-of-closest-approach to the beam crossing and assume that primary particles have been produced at or near that point. Locating the origin of secondary particles is a more difficult problem, and wil l not be considered in this thesis. The group of hits associated initially with a track are used to perform a x2 fit in order to determine the best values for the track parameters. This is repeated iteratively, adding and removing hits in order to improve the fit results until convergence is achieved. Chapter 5. Data Acquisition and Event Reconstruction 34 Figure 5.3: Projection of a charged track into x-y plane. Track parameters do, (f>o and p = are shown. 5.5 Conversion from Helix Parameters There is one remaining task that is performed in the offline reconstruction code, the calculation of physics 4-vectors from the track helix parameters. In math-ematical terms, the conversion is from five helix parameters plus the magnetic field to three position (x,y,z) and three momentum (px, py, pz) values. The calculations are first performed at the point-of-closest-approach to the z-axis. The transverse momentum, px, is directly proportional to the radius of cur-vature of the helix, p, and the strength of the magnetic field, B: pT (GeV/c) = 0.2997 \B\ p (Tm) (5.1) with a factor of 0.2997 if the units are G e V / c for momentum, Tesla for the magnetic field and metres for the radius of curvature. From pr we can obtain Chapter 5. Data Acquisition and Event Reconstruction 35 y Figure 5.4: Projection of track momentum into y-z plane. Track parameters pr, pz and A are shown. the x and y components of momentum: px =pT cos((p0) (5.2) Py =PTsin(0 o) (5.3) The z component of momentum can be calculated using the relation: pz =pT tan(A) (5.4) The (x,y,z) coordinates of any point along the track can be obtained from the values of do, 4>o, a n d the distance along the helix tp: x = -(do + —) sin(</>0) + — cos(V>) (5.5) CO id y = (d0 + —) cos(0o) + - sin(r/>) (5.6) LO LO z = ZQ + — tan(A)V; (5.7) cu Once these laboratory quantities have been calculated, and the mass is known, a relativistic 4-vector can be constructed for each track. Each event also has a beam-beam C M 4-vector associated with it. These are calculated as part of the offline calibration process. The beam-beam C M 4-vector can be used to rotate and boost the track 4-vectors into the C M frame. It is worth noting that the origin of the detector and the beamspot are not perfectly aligned. In fact the beam-crossings occur in a cigar shaped region of approximately 5 pm width in the x-direction, 100 pm width in the y-direction, Chapter 5. Data Acquisition and Event Reconstruction 36 and a length of about 10 mm in the z-direction. This interaction region is located roughly at the detector coordinates of (x,y) = (-0.25, +2.8) where x and y are in mm. In addition, the z-axis of the detector is not perfectly aligned with the beam axis. It is rotated in the 9 and 0 directions by 20 and 10 milliradians, respectively. Tracks are reconstructed with respect to the detector origin and need to be rotated and shifted appropriately when we are calculating distances and angles with respect to the beam crossing. Chapter 6 Event Selection 37 6.1 Overview There are several fundamental requirements that our signal events must satisfy. The first is that they should contain two charged tracks emanating from the same point in space. The second is that they should contain tracks resulting from an interaction at the beamspot, and with no lifetime, meaning that the observed tracks were created at the point where the incoming beam particles collided. We need to exclude continuum uds events, charm events, B mesons and r pairs because all of these have significant lifetimes. By imposing cuts on the multiplicity and pr balance of the events we can remove nearly all of these sources of background, since they tend to have more final state particles than our events of interest. We also need to exclude events containing cosmic rays, beam-wall and beam-gas interactions. These events have the right number of tracks, but don't origi-nate from the beam-crossing location in the center of the detector. These events can be removed by imposing cuts on the event geometry to distinguish them from our signal events. 6.2 NTuple Creation To begin with, we load all recorded events which passed any of the triggers in the Data Acquisition System. These events contain tracks from e + e~ scatters, B and D decays, r pairs, continuum uds production, cosmic rays, beam-gas, and beam-wall interactions. I used database collections representing approximately 19.2n6 _ 1 of data acquired during the year 2003. There are several different lists of charged tracks, photons and neutrals which are constructed from the raw data and could be used to fill an ntuple. Each list has a different set of requirements which a candidate must pass in order to be included. For this analysis I chose to use the following lists: • Charged Tracks = "GoodTracksLoose" • Photons = "GoodPhotonDefault" • Neutrals = "GoodNeutralLooseAcc" Chapter 6. Event Selection 38 The charged tracks are selected from all candidate charged tracks, based on hits in the S V T and D C H . The photons and neutral hadrons are selected from single-peaked clusters in the E M C which don't correspond to any of the charged track candidates. E M C clusters which are multiply-peaked are used in other processes searching for Ks's and 7To's, and are ignored in my analysis. The criteria used to fill each of the lists used in this analysis are given in Table 6.1. GoodTracksLoose GoodPhotonDefault GoodNeutralLooseAcc Track p r > 0.1 GeV Raw E > 0.1 GeV Raw E > 0.03 GeV Track p < 10 GeV n Crystals > 0 n Crystals > 0 nHits D C H > 12 Lat Moment < 0.8 Lat Moment < 1.1 D O C A in xy < 1.5cm 0.41 < 6 < 2.409 D O C A along z < 10cm Table 6.1: Criteria of the track, photon and neutral lists used in this analysis. The first cuts we impose in our selection process are on event multiplicity: • ^Tracks = 2 • 0 < Nphotons < 1 • 0 < N N e u t r a i s < 1 These cuts are implemented in the "BABAR Micro" code, before the generation of an ntuple file. This results in a considerable saving of computing resources. After these cuts were applied to the data sample, the resultant ntuple contained slightly more than 10.6 million events. Each additional cut improved the purity of the sample, and reduced it in size. The reduction in the number of events caused by each of the cuts I applied is detailed in Section 6.7 of this chapter. The resulting ntuple contains information on all charged tracks, photons and neutrals in each event satisfying the above criteria. In Figure 6.1, a plot is shown of the correlation between the transverse momenta (pr) of the two tracks in a small subset of the data. We can see a diagonal band representing events in which the transverse momentum of the two tracks balances quite exactly. We also see events which have more total energy than the beam-beam C M energy, and must be removed. 6.3 Cosmic Rays The events in Figure 6.1, with two tracks both having px above 5.5 M e V are physically impossible for the initial beams to produce. They must have an origin outside of the detector, and we identify them as cosmic rays. The events with balanced pr less than 5.5 GeV are composed of both cosmic rays A N D our desired signal events, so we wil l need to use a measured quantity other than pr to discriminate between them. We will try to avoid using the track parameters which ultimately determine our quantities of interest ZQ and do. Chapter 6. Event Selection 39 Figure 6.1: Track px for all events with exactly two charged tracks. Cosmic rays traverse the detector more or less vertically, and are recon-structed as two separate tracks, one for each hemisphere. These two tracks have nearly the same values of do and ZQ, and come from locations uniformly distributed throughout the detector. Since we expect cosmic rays to traverse the detector more or less vertically, we can use the angle between the track mo-mentum and the x-y plane, the dip angle, to identify them. This angle is given by the helix parameter tan(A), as shown in Figure 5.4. This angle wil l be equal and opposite in sign for the two tracks created by a cosmic ray traversing the de-tector. Figure 6.2 shows a plot of of the quantity ^ t a n ( A ) for a small subset of events. The dashed lines in the figure represent a cut of | 2^tan(A)| < 0.1. This cut will remove a relatively small fraction of signal events who happen to have balanced values of tan(A). Figure 6.3 shows a plot of the transverse momenta of the two tracks with the requirement that ^ t a n ( A ) < 0.1 and Figure 6.4 shows the same quantities with a cut of 0.01. As we expect, the tracks labelled as cosmic rays do indeed have balanced pr, confirming our identification. In the future, I wil l refer to these two cuts as the Cosmic Ray Loose and Tight cuts. Finally Figure 6.5 shows track pr for those events left after applying the Loose cosmic ray cut. This is the cut chosen for the analysis, and after applying it to the initial data sample of 10.6 million events, I was left with 9.85 million events. We can clearly see that nearly all the events with unphysically high transverse momenta (pr > 5.5 GeV) have been removed simply by identifying them as cosmic ray events based on their dip-angle parameters. The few (< 10) remaining events with unphysically high px wil l be removed by further cuts. Chapter 6. Event Selection 40 Figure 6.2: The quantity ^ t a n ( A ) for a small subset of events in the ntuple. The dashed lines indicate a cut of | £ ] t an (A) | < 0.1 10 i i i | i i i i | i i i i | i i i i | i i i ! | i i i i [ i i i i [ i i i i ] i i i i | i i i rz a> - .-. ^ O 9 r_ _= * 8 - -4^-IS*' - I I i f'-l:-1 r I ' i I i i i i I i i i i I i i i i I i i i t I i i i i I i i i i I i i i i I i i i 1 1 2 3 4 5 6 7 8 9 10 Trk[0] pt (GeV) Figure 6.3: Track px for events with exactly two charged tracks and passing the Cosmic Ray Loose selection. Chapter 6. Event Selection 41 10| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | I I I | I I I I | I I I I | I I I I. > o (3 9 •I ' l I I I I "I I I I I I t 1 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I l" 0 1 2 3 4 5 6 7 8 9 10 Trk[0] pt (GeV) Figure 6.5: Track px for events left with exactly two charged tracks after ap-plying the Cosmic Ray Loose cut. Chapter 6. Event Selection 42 Figure 6.6: Track PT for events passing requirements for exactly 2 tracks, Cos-mic Ray Loose cut, and detector acceptance cuts. 6.4 Detector Acceptance In the next step, I apply a cut to my data sample to remove events which have passed through the very edges of the acceptance of the detector and may not have information associated with them from all of the subsystems. These cuts also remove some events which were very poorly reconstructed. By ensuring that the tracks actually pass through the active regions of the S V T and D C H , we can eliminate these events. The acceptance of the BABAR detector in <f> is essentially complete, while the dip angle acceptance is restricted by the openings for the beampipe to be: —1.0 < tan(A) < 2.2. This cut wil l be referred to as the dip-angle cut. Figure 6.6 shows the effect this cut has on the px spectra of a small sample of events. As we can see, many of the events with unbalanced pr are removed, especially for values of px < 2 G e V / c . Relatively few px balanced events are removed. 6.5 Beam-Gas and Beam-Wall Tracks Electrons or positrons which strike an atom of the residual gas in the vacuum can be measured in the detector. These beam-gas events wil l be px-balanced, but wil l not originate from the beam crossing region. They should have a relatively uniform distribution throughout the volume of the beampipe. We impose a rather loose cut on the reconstructed value of ZQ for each of the two tracks, requiring them to be within -3.0 cm < ZQ < 3.0 cm. This cut should include 95% of all signal events, since the length of the beam bunches is roughly 1.0 cm. Chapter 6. Event Selection 43 The next class of events to be removed resulted from an electron or positron scattering off the material in the walls of the beam-pipe. These beam-wall events must be removed because they didn't originate from the beam crossing. Beam-wall events are clustered at two hot-spots located on the sides of the beam-pipe. By plotting the x-y coordinates of the point-of-closest-approach ( P O C A ) of each track to the beamspot, (see Figure 6.7) we can clearly observe a cluster of tracks at the origin corresponding to the particles created at the beamspot. We also see two sinusoidal arcs in the plot which were created by the particles leaving the hot-spots on the walls. We can calculate the distance r between the track E < o O Q. 2'-1.5E-i\-0.5 E--0.5 E-- l j --1.5 E--2~--2 ||§§I§!' -1.5 -0.5 0.5 1 1.5 2 POCA x (cm) Figure 6.7: Profile of track vertices in x-y plane, the sinusoidal curves are cre-ated by events originating from the beam-pipe walls. origin and the hot spots on the walls: r = | d 0 ±2.7sin(0o) ->ipcos(0o)| (6.1) where Yip is the nominal y-coordinate of the interaction point, as determined by the BABAR offline analysis system (See Section 5.4) and the value of 2.7 cm is the measured distance along the x-axis from the beamspot to the hotspots on the walls. Figure 6.8 shows the distribution of r for one of the hotspots, corresponding to the positive case. We can see a large peak at r=2.7 cm, corresponding to the beamspot, and a small peak at r=0, corresponding to the hotspot on the wall. A cut requiring r > 0.3 cm, as indicated by the dashed line, wil l clearly remove most of the tracks coming from the hotspot. The same distance is calculated individually for each of the two tracks in the event, and for both hotspots, corresponding to the positive and negative signs in Equation 6.1. Chapter 6. Event Selection 44 r (cm) Figure 6.8: Distribution of the distance r between the track origin and one of the hotspots on the wall of the beampipe. A cut at r > 0.3 cm is indicated by the dashed line. Figure 6.9 shows the x-y locations of the P O C A s for the tracks, after removing the beam-wall events using the cut r > 0.3 cm: The diffuse background is primarily due to a combination of poorly-reconstructed tracks, i.e. tracks which have been Coulomb scattered through a large angle. Since the objective is to study in detail these distribution tails, we must carefully select any further cuts so that we don't remove any signal. 6.6 Transverse M o m e n t u m B a l a n c e In this section several different cuts on px balance are explored, all using lab-frame quantities. The values of px are slightly different in the C M frame because of the relative rotation between the beam axis and detector axis, but the effect should be less than the width of the cuts used here. Pairs of tracks resulting from the scattering of an electron or a positron by the residual gas in the beampipe will appear to come from the beamspot location in the x-y plane, but wil l have a flat distribution along the z-axis. These two tracks wil l not have balanced px, since one energetic particle is scattering off an atomic electron. They wil l also not be back-to-back in the x-y plane. By requiring that the two charged tracks are back-to-back in the x-y plane of the detector, we can eliminate many of the beam-gas events from our sample. Figure 6.10 shows a plot of the angle between the two tracks in the x-y plane, \A<f> — n\. A l l events in which the tracks are back-to-back in the x-y plane should be located near 0. Chapter 6. Event Selection 45 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x-axis (cm) Figure 6.9: Profile of track vertices in x-y plane, after removing events originat-ing from the beam-pipe walls. The dashed line indicates a cut of \A<fi — ir\ < 0.2 radians. Figure 6.11 shows the transverse momenta of the tracks after requiring that \A(f> — TT\ < 0.2 for each event. This cut doesn't remove many events in the region where they are px-balanced, but it does significantly lower the number of events in which one of the two tracks has very low pr and the other is substantial. We now make a final requirement based on transverse momentum balance. Based upon inspection of the plot in Figure 6.11, the first choice is a relatively loose cut of |Apr\ < 100 MeV. This significantly cuts into the signal region, but likely wil l still include events with an unobserved photon in the final state. This cut removed approximately 40% of the events in the sample. A second, tighter cut was tested on the same sample of data, requiring \Apr\ < 10 M e V . This cut is stringent enough to eliminate virtually all events with unobserved photons in the final state, but by reducing the number of events in the sample by about 85%, it would severely limit the statistics available for analysis, particularly in the regime of 2 < pr < 4 GeV, between the peaks due to the Bhabha and Two-Photon processes. For the rest of the analysis, the looser cut on pr balance is used. Two-Photon Events are pr balanced and have low momenta for the two tracks, while the Bhabhas and muon pairs are also pr balanced but have higher momenta. A theoretical spectrum for the momentum of muons was shown in Figure 3.5. The lab momentum spectrum for all data tracks is shown in Figure 6.12. The Two-Photon peak is clearly visible just below 1 GeV in the data. Bhabhas and mu-pairs created through the annihilation diagram have momenta equal to one half the initial C M energy of the beams. Initial and Chapter 6. Event Selection 46 Figure 6.10: Angle between the two tracks in the x-y plane | A<f>—ir\. The dashed line represents a cut at 0.2 radians. final-state radiation produces a low-energy tail in the spectrum of electron and muon energies, as shown in the theoretical plot Figure 3.2. These events are clearly seen in the region of 3-5 GeV in the data plotted in Figure 6.12. For the purposes of this analysis, it is not necessary to separate the two contributions in the data; instead we will use samples of Monte-Carlo events produced by both processes. This wil l allow us to compare data to simulation over the full momentum range from 50 M e V to 5.5 GeV. Chapter 6. Event Selection 47 7 p r i I i i: i]vi' , i , , v i i , i , i , , , i I i , i i I i i i , I 0 1 2 3 4 5 6 7 Trk[0] pt (GeV) Figure 6.11: Track px, after requiring they be back-to-back in the x-y plane. 6.7 Cut Selection Efficiency A summary of the selection performance of the applied cuts is listed in Table 6.2. The relative efficiency of each cut is obtained by dividing the number of events Cut Description Nevents Remaining Relative Efficiency e AllEvents 1.0e9 Only2Trks 1.05e7 0.01 CosmicLoose 9.85e6 0.94 Acceptance 6.48e6 0.66 Hot Spots 5.87e6 0.90 back2back 5.18e6 0.88 pt lab loose 3.00e6 0.58 pt lab tight 4.50e5 0.15 Table 6.2: Event selection statistics of the cuts used in this analysis. passing that cut, by the number of events passing the previous series of cuts. We can see that there are two cuts which remove the most events, the initial requirement on the numbers of different particles in the event, and the tightest requirement on transverse momentum balance. By placing the requirements on particle counts into the first step of my analysis procedure (the creation of an ntuple file), the use of local computing resources is much more efficient. The Chapter 6. Event Selection 48 Figure 6.12: Track momenta for all events remaining at loose px balance cut. final requirement on transverse momentum balance is too restrictive at 10 MeV, and so the requirement was set at 100 MeV, to keep a good level of statistics. 6.8 The BABAR Particle ID System In the BABAR experiment, particle identification is a process which is performed in high-level physics analysis code, once tracks and energy clusters have already been found. These procedures are applied to every track and cluster in an event to give them membership in the various lists of electrons, muons, pions, kaons, protons and photons. For example, a charged track might be found which passes the criteria for membership in both the "loose" and "tight" electron categories, but not for the "very tight" list. This track would be listed as a member of both the "tight" and "loose" electron samples. In general, the identification schemes use either a series of cuts on di-rectly observed quantities such as dE/dx, the lateral moment of an electromag-netic shower in the E M C , the number of layers of the I F R that the candidate traversed, the number of associated photons detected in the D I R C , and the Cerenkov angle measured in the DIRC, or the identification schemes can use a neural network, Fisher discriminant or log-likelihood ratio involving combina-tions of the above variables. There are also some schemes which combine cuts on variables and the use of neural networks or discriminants. For my analysis, I have used the standard BABAR electron identification Chapter 6. Event Selection 49 system known as the PidElectronMicroSelector. There are several variations on this selector, eMicroVeryLoose, eMicroLoose, eMicroTight and eMicroVery-Tight. The cuts used in each of these selectors are listed in Table 6.3 Selector dE/dx E / p L A T Efficiency MisID eMicroVeryLoose 500-1000 0.5-5.0 -10-10 > 98% < 20% eMicroLoose 500-1000 0.65-5.0 -10-10 > 97% < 10% eMicroTight 500-1000 0.75-1.3 0-0.6 94-97% < 7% eMicro VeryTight 540-860 0.89-1.2 0-0.6 75-95% <2% Table 6.3: Cut-based selectors available in the PidElectronMicroSelector. The quantities used for the PID selection of electrons are: • dE/dx - the average energy loss per cm in the drift chamber • E / p - the energy measured in the calorimeter divided by track momentum • L A T - the lateral moment of the associated cluster in the calorimeter • Efficiency - percentage of electrons in control samples passing the selector • MisID - percentage of hadrons in control samples passing the selector In the next chapter of this analysis I use the "tight" electron selector. It has very good discrimination against pion contamination and is highly efficient for electrons with moderate momentum. The efficiency drops and MisID fraction rises for very low momentum electrons. In an analysis to determine absolute efficiency, the variations as a function of polar angle and momentum need to be taken into account on a track-by-track basis, but no such corrections are required for the present work. The muon selectors are applied in an identical way in software, though the criteria used for the cuts are somewhat different. In particular, information from the I F R was not used for electrons, but is crucial for good muon identification. The cuts used in each of the muon selectors are given in Table 6.4. Selector ECAL A X 2 Track Efficiency MisID muMicroVeryLoose < 0.5 > 2 N A 90% 20% muMicroLoose < 0.5 > 2 < 7 85% 10% muMicroTight 0.05-0.4 > 2.2 < 5 75% 3% muMicro VeryTight 0.05-0.4 > 2.2 < 5 70% 2% Table 6.4: Cut-based selectors available in the PidMuonMicroSelector. The quantities used for the PID selection of muons are: • ECAL - the total calibrated energy deposited in the calorimeter • A - the number of interaction lengths traversed by the track \ Chapter 6. Event Selection 50 • x 2 Track - The quality of the helix fit to the hits in the I F R • Efficiency - percentage of muons in control samples passing the selector • MisID - percentage of hadrons in control samples passing the selector In the next chapter of this analysis I use the loose muon selector, which has fairly good discrimination against pion and kaon contamination and is very efficient for muons with sufficient momentum. The efficiency drops and MisID fraction rises for muons with momentum less than about 1 GeV. As with the electron ID, absolute efficiency values are not important for the present work. 51 Chapter 7 Analysis We have now selected the data sample necessary for analysing the track impact parameter resolutions and what remains is to calculate the resolution for each event in this sample. Once the resolution has been calculated it can be fit with an appropriate lineshape. Events can also be binned by relevant quantities like lab momenta and angles, to create a parametrization that is a function of track or event properties. The resolution of the track impact parameters is made up of three contribu-tions: the geometric resolution of the detector, the Gaussian core of the Multiple Coulomb Scattering distribution, and the tail of the Multiple Coulomb Scatter-ing distribution. The geometric resolution depends only on detector properties and on the angle of the track. The M C S contributions to the resolution depends on both track momentum and polar angle. The errors assigned to each track by the BABAR tracking software include the geometric and M C S core contributions, but don't take into account the tail of the M C S distribution. The calculated values for the difference in the track impact parameters wil l be divided by the errors assigned by the BABAR tracking software. If the software calculation of the tracking errors is correct, then we expect the distribution of differences to have a Gaussian core and tails created by M C S . The core width of the resolution distribution should be about la as long as the tracking errors are accurate. If the individually assigned tracking errors are larger than they should be, then the width of the core of the resolution distribution wil l be less than la, and conversely the core wil l be wider than la if the assigned tracking errors are too small. In this way, our data can provide a good crosscheck of the tracking system error calculations. 7.1 Variable for d 0 Studies When considering the resolution in the x-y plane, the track impact parameter to use is do- While we know that the tracks originate within the beamspot, the horizontal size of the beam is larger than the resolution for high PT tracks. We obtain a more precise measure of the track resolution by comparing the do values of the two tracks to each other, rather than comparing each separately to the beamspot location. Since do is a signed quantity, two tracks which are back-to-back in the x-y plane should have Sd0=0- We take the sum of d 0 values for the two tracks in the event as our variable of interest. Our two tracks are not exactly back to back in the <fi angle due to the Chapter 7. Analysis 52 beams being rotated relative to the z axis of the detector. This makes Edo depend somewhat on the beam position relative to the detector z axis. The default method of determining the track parameters for each charged track uses the detector origin as a reference point. By subtracting the location of the beamspot from the track-impact parameter values we can obtain measurements with respect to the beamspot, rather than the detector origin. We directly calculate do with respect to the beamspot in the following equa-tion: do = (XPOCA - xip) sin(</>0) - {ypocA - Vip) cos(0o) (7-1) where ( X P O C A , V P O C A ) is the point-of-closest approach of the track to the beamspot, in the x-y plane, and {xiP,yiP) is the nominal location of the in-teraction point, or beamspot in the x-y plane for each event. The error in the quantity Ed0, which we shall denote by a (Edo) is just the quadature sum of the do errors for each individual track. The track-impact parameter resolution in the x-y plane is now given by the expression: a^la\ • By dividing the separation by the error, we can determine whether or not the errors assigned by the BABAR tracking software are accurate and attempt to identify any tails of the distribution. 7.2 Variable for ZQ Studies If the two tracks were exactly back to back in both 9 and <p, then they would have the same ZQ parameter, independently of the detector axes or production point. Two-photon events are typically not back to back in 9. In the BABAR experiment, even Bhabha and /x-pair events are not back to back in 9 due to the beam energy asymmetry. So rather than comparing the raw zo track parameters, we need to find the apparent production point of each track in the beamspot and compare those z coordinate values. First, I perform a \ 2 n t separately for each of the two tracks to locate its production point within the beampsot. The code for this fit procedure is contained in the default BABAR vertexing software known as GeoKin. I wil l denote this production point as afj, and it's associated covariance matrix by o~\. Second, in order to determine the best possible value of the production point, x'i, I take weighted averages of the two individual production points, using only the x-y information and weights W, given by the inverse of the covariance matrices, as in the following equation: 4 = a'x ( W i x l + W a t r u n c a t e d x j ) (7.2) 4 = „'2 (waxa + W l t r u n c a t e d x - i ) (7.3) ( ^^xx ^Txy 0 \ Wyx Wyy 0 so that the x-y coordinates 0 0 0 / of the two points are averaged and the z coordinate is recalculated at the position Chapter 7. Analysis 53 of the x-y average. Applying this transformation yields the best possible estimate of the true production points, while still retaining the freedom for drastically different lo-cations along the z axis. Allowing this freedom along the z axis is crucial if we wish to use the results of this study to understand the resolution of the At distributions in the neutral B mixing analysis. The covariance matrices for the improved vertex coordinates x[ are now given by the following equations: o>x = [Wi + W 2 r u n c a t e d ] - 1 (7.4) 4 = [Wj + wt1r u n c a t e d ] - 1 (7.5) Finally, we use the elements of the improved covariance matrices o\ to obtain the errors in the individual tracking parameters d'0 and z'0 through the usual expressions: = (7-6) where <Tj'2 is the [i,i] element of the covariance matrix cr-. Since the two tracks have independent sources of error, we add their errors in quadrature. In a similar manner to the expression used for do, we construct the quantity g (^ z ° 0 ) where the difference is momentum-ordered, i.e. Azo = zo(high p) - z 0(low p). This is done to eliminate the biases associated with the order in which tracks are recorded in an event. The software algorithm tends to locate tracks in a clockwise fashion, scanning through the r — (f> plane, and systematically identifies smaller track impact parameters before larger ones. The momentum-ordered quantity doesn't suffer from this systematic effect. The error in the denominator is given by the quadrature sum of the two errors calculated with Equation 7.6. Again, by dividing the difference by the assigned error value, we expect to obtain a central Gaussian distribution of width approximately 1, and wide tails consistent with Multiple Coulomb Scattering. 7.3 Variable for Momentum and Angular Dependence The track impact parameter resolution can be broken down into two compo-nents. The first is the geometric resolution of the detector which is independant of track momentum. The other component of track impact parameter resolu-tion is the Multiple Coulomb Scattering which does depend on track momentum. Both of these contributions also depend on the track polar angle, 9. By con-structing a new variable which captures the momentum and angular dependence of M C S , we can use it to quantify the amount of scattering and compare the resolution distributions with the Toy M C predictions. Looking back to Equation 2.3, the core width of the M C S distribution, 9Q, is proportional to ^ X ^ X ° , where X is the thickness of the material traversed, Chapter 7. Analysis 54 Xo is the radiation length of the material, and p is the momentum. The thick-ness of material traversed, X, is proportional to l / s in# , so the core width 9o scales as l / ( p s i n 1 / / 2 8). The lever-arm, or distance from the scattering point to the beamspot is rj sin#, so the do resolution due to M C S is proportional to l / ( p s i n 3 / 2 8). We can also use the relation px = psm9 to state that the do resolution is proportional to 1/(PT s i n 1 / 2 8). The same scattering angle changes the measurement of ZQ by an additional factor of 1/ sin (9, so the ZQ resolution due to M C S varies as l/(psin 5 / 26>) or l/(pTsin3/28). We can now write down the two relations governing the core width of the M C S contribution to the tracking resolutions: <*(Zdo) « . = - 7 7 ^ 7 7 7 ( 7 J ) (7.8) where cr(Sdo) is the core width of the Sdo resolution and <j(£s.Zo) is the core width of the Azo resolution. If we calculate the quantity on the right-hand side for each track, we can estimate the importance of M C S to the resolution distribution. But since the track impact parameter resolutions are calculated using two tracks, each with a different value of 9, we must modify the above expressions to use the 9 values from each of the two tracks. We define the following two quantities: 1 1 psin 3 / 2(0) ~ p T s i n 1 / 2 ( 0 ) 1 1 psin 5 / 2(0) ~~ p T s in 3 / 2 (0 ) 2 sin(9isin6»2 , . Qd = PT1PT2 • O , • O (7-9 sin 0i + sin 02 rfl s in 3 6>! sin 3 92 . . Qz = PT1PT2 . 3 . 3 (7.10) sin 9\ + sin 92 where the subscripts 1 and 2 refer to each of the two tracks. We can use l / Q 2 as a measure of the importance of the M C S contributions to the resolution in each event. Small values of l / Q 2 represent events which are dominated by the geometric resolution and large values of l / Q 2 represent events which are dominated by Multiple Coulomb Scattering. Figure 7.1 shows the excellent results when fitting the functional form to the Sdo resolution data in the bin of events with 0.0 < l / Q 2 < 0.1. This range of l / Q 2 corresponds to events with large values of pr and sin#. This is the regime in which Multiple Coulomb Scattering is least important compared to the geometrical resolution. Higher values of l / Q 2 correspond to the regime in which M C S dominates the resolution. Figure 7.2 shows the results when fitting the functional form to the Sdo resolution data in the bin of events with 10 < l / Q 2 < 50. The agreement is equally good. Figure 7.3 shows that the same functional form works equally well on the other track impact parameter, A^o, for data events in the low l / Q 2 bin; i.e. /i-pairs with 0 < l / Q 2 < 0.2. Figure 7.4 shows the same good agreement for AZQ /i-pair data in the high l / Q 2 Chapter 7. Analysis 55 dO Resolution Figure 7.1: Plot of Sd0 resolution for muon pairs in data, with 0.0 < l/Q2 < 0.1. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. dO Resolution Figure 7.2: Plot of Sd0 resolution for muon pairs in data, with 10 < l/Q2 < 50. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. Chapter 7. Analysis 56 zO Resolution Figure 7.3: Plot of Azo resolution for muon pairs in data, with 0 < 1/Q2 < 0.2. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. zO Resolution Figure 7.4: Plot of Azo resolution for muon pairs in data, with 10 < 1/Q2 < 50. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. Chapter 7. Analysis 57 dO Resolution Figure 7.5: Plot of Ed0 resolution for electron pairs in data, with 0.0 < 1/Q2 < 0.1. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. bin, 10 < 1/Q2 < 50. We see that the distributions have a very similar shape for both track impact parameters. As one would expect, we also observe that the track impact parameter resolutions in data look nearly identical for both the muons and electrons. Figure 7.5 is a plot of the fit to e-pair Edo data in the low 1/Q2 (0.0 < 1/Q2 < 0.2) bin. Figure 7.6 is a plot of the fit to e-pair Edo data in the high 1/Q2 (10 < 1/Q2 < 50) bin. Figure 7.7 is a plot of the fit to e-pair Az0 data in the low 1/Q2 (0.0 < 1/Q2 < 0.2) bin. Figure 7.8 is a plot of the fit to e-pair A 2 0 data in the high 1/Q2 (10 < 1/Q2 < 50) bin. The fits for other values of 1/Q2, which are not shown here, also gave good quality results for both electron and muon pair data events. By repeating the fitting procedure for several bins of 1/Q2, we can tabulate the parameterization as a function of 1/Q2. We perform the same fitting procedure on the G E A N T - 4 based Monte-Carlo events created with the BABAR simulation software as a way to check whether the simulation has the correct treatment of track impact parameter resolution errors and of Multiple Coulomb Scattering. Figure 7.9 is a plot of Edo resolution for muon Monte-Carlo events, and Figure 7.10 is a plot of AZQ resolution for the same sample of M C events. They both look reasonable. We can fit them with the same functional form used for the data, and compare the results. Figure 7.11 is a plot of Edo resolution in electron-pair Monte-Carlo events: it is qualitatively different from the other plots, and we conclude there is a problem with the simulation of this category of events. The BABAR simulation software is a large, Chapter 7. Analysis 58 dO R e s o l u t i o n Figure 7.6: Plot of Sd0 resolution for electron pairs in data, with 10 < 1/Q 2 < 50. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. zO R e s o l u t i o n Figure 7.7: Plot of Az0 resolution for electron pairs in data, with 0 < 1/Q 2 < 0.2. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. Chapter 7. Analysis 59 -30 -20 -10 0 10 20 30 zO Resolution Figure 7.8: Plot of Az0 resolution for electron pairs in data, with 10 < 1/Q2 < 50. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. d0 R e s o l u t i o n Figure 7.9: Plot of Sd0 resolution for muon pairs in M C simulation, with 0.25 < 1/Q2 < 0.5. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. Chapter 7. Analysis 60 r- F zO Resolution Figure 7.10: Plot of Azo resolution for muon pairs in M C simulation, with 0.25 < l/Q2 < 0.5. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. dO Resolution Figure 7.11: Plot of Edo resolution for electron pairs in M C simulation, with 0.25 < l/Q2 < 0.5. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. Chapter 7. Analysis 61 zO Resolution Figure 7.12: Plot of Azo resolution for electron pairs in M C simulation, with 0.25 < 1/Q2 < 0.5. The fit (solid line) is to the function given by Equation 2.4. The dashed line depicts the tail function. centrally administered project, so it was not possible for this problem to be corrected during the timescale of this analysis. Figure 7.12 is a plot of the AZQ resolution for the same electron M C events. It looks reasonable, but since there is a problem with the Sdo resolution, we choose not to use the Azo resolution for M C simulated electrons any further. 7.4 Comparison of Data and G E A N T M C in Transformed Variable To examine the agreement in the tails of the distributions more closely is difficult because the number of events in each bin decreases so rapidly as we move away from zero. By examining the inverse-square of the resolution, u = o r u = ( Azo j ' w e c a n m o r e closely at the agreement between data and Monte-Carlo simulations in the tails of the distributions. This variable transformation has the useful property of mapping the horizontal range from ICT out to oo into the range between 1 and 0. It also rescales the height of the distribution from spanning 4 or more orders of magnitude onto a linear scale. Figure 7.13 is a plot of u for /x-pair 27do resolution in Data (open histogram) and Monte-Carlo (shaded histogram) in the the bin with 0.0 < 1/Q2 < 0.2. The Chapter 7. Analysis 62 Figure 7.13: Plot of u for muon-pair Sdo resolution in Data (open histogram) and Monte-Carlo (shaded histogram) in the bin with 0.0 < l/Q2 < 0.2. The solid curve is a transformed fit to ^i-pair data. The dashed curve is a transformed fit to /u-pair Monte-Carlo. The dotted curve is the transformed tail function of the /i-pair data, and the dot-dash curve is the transformed tail function of the /u-pair Monte-Carlo. solid curve is a transformed fit to p-p&ir data. The dashed curve is a transformed fit to ^j-pair Monte-Carlo. The dotted curve is the transformed tail function of the ^i-pair data, and the dot-dash curve is the transformed tail function of the /u-pair Monte-Carlo. We can clearly see kinks in the fit function where the tail function changes slope between the flat inner core and the inner power-law tail, as well as between the inner and outer power-law tails. We also observe that the fit function appears to be slightly higher than the data at the inner breakpoint, and slightly lower than the data on either side of this. From this, we conclude that for this sample of events, the parametrization used to fit the data works quite well, but not perfectly. The same plot was generated for both track impact parameters, Sdo and Azo, and for both /x-pairs and e-pairs, over the entire observed range of l/Q2 • The plots are presented in Appendices A ( A z 0 ) and B (Sdo). For ^-pairs there appear to be consistently fewer events in the tails of the Monte-Carlo distributions beyond 2a for both track impact parameters. We expect this deficit to result in lowered values for the tail-fraction parameter / , when fitting to the Monte Carlo events, compared to the data events. Chapter 7. Analysis 63 The e-pair Monte-Carlo is clearly flawed in the calculation of the do reso-lution, and we are therefore suspicious of the zo resolution for those events as well, even though it appears more reasonable than do. By simply inspecting these plots we determine that the parametrization used to fit the data works well for the ZQ track impact parameter, out to roughly 10a or u=0.01. 7.5 Comparison of Data and M C Fitted Parameters We now present comparisons between experimental data and BABAR Monte-Carlo simulation, of the fitted parameters for muon samples, in both Sdo and Azo resolution. The comparison is made between our data which contains Bhab-has, /u-pairs and Two-Photon events, and fully-reconstructed Monte-Carlo sim-ulations of Bhabha and /Li-pair events. At the time this analysis was performed, only a tiny sample of several thousand Two-Photon events had been gener-ated and reconstructed for BABAR analysis. It was decided to use the tails of the Bhabha and /x-pair events in the low px regime to perform the analysis. This means that the fits to M C in the range of large 1/Q2 have quite large errors. The analysis could be repeated, at a future date, once a larger sample of Two-Photon lepton-pair events has been generated and reconstructed. No comparisons were made of electron-pair data and M C due to the strange shapes of the Sdo distributions in M C , and the likely errors in the generation of those simulated events. It is possible to allow all of the parameters in our model to float during the fits, but practical difficulties of convergence are often encountered in that case. If we look only at the most extreme edges of the tails we expect the power to be exactly 3.0 due to the Rutherford differential cross-section, so we fix the power in the outer tail at 3.0. By fixing the inner break-point at 3.0 we effectively assign almost all events more than 3a away to be primarily the result of M C S . Finally, based on what we saw by looking at Toy Monte-Carlo, we also choose to fix the breakpoint between the inner and outer power-law tails at 5.0. This fitting procedure allows us to examine the behaviour of the mean, core width, tail fraction, and in inner power-law in more detail. The number of events returned by the fit were always consistent with the true number in the histogram. Figures 7.14-7.17 depict the fitted parameters versus 1/Q2 for the Sdo reso-lution in muon-pair events. Figure 7.14 is a plot of the mean of the distribution for muon-pair events in both data and Monte-Carlo. We see that the mean is consistent with zero, for both data and simulations, once 1/Q2 is greater than 1.0. This means that for high energy muons, in both the data and Monte-Carlo, the mean of the Sdo resolution is shifted away from zero by between 2-3%. This effect cannot be explained at the present time. Figure 7.15 is a plot of the width of the Gaussian core for the Sdo resolution, in both data and Monte-Carlo muon-pair events. We see the core width steadily Chapter 7. Analysis 64 1/Q2 Figure 7.14: Plot of the the mean of Sdo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of l/Q2-I 1.05 o o 0.95 0.9 0.85 10"' 10 1/Q2 Figure 7.15: Plot of the core width of the Sdo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of l/Q2-Chapter 7. Analysis 65 Figure 7.16: Plot of the tail fraction of the Sdo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of l/Q2-decreasing as the value of l/Q2 increases, in both the data and Monte-Carlo samples. This cannot be directly compared with the results we obtained fitting to Toy Monte-Carlo in Chapter 2. In that case we varied the number of scatters over several orders of magnitude, and kept everything else fixed. Here, we expect to see a fixed value of 1.0 if the tracking errors returned by the BABAR software are accurate. For the muon-pairs in the data, the tracking software systematically underestimates the errors by about 6% for low values of l / Q 2 . A t the other end of the spectrum in the data, the tracking software gets the errors about right when l / Q 2 is large. In the Monte-Carlo muon-pair events the same general trend occurs, the core width decreases as l / Q 2 increases. The Monte-Carlo resolution has a core width consistently less than the data. This means that at low values of l / Q 2 , the tracking errors are estimated correctly in the Monte-Carlo, but at high values of l / Q 2 , the tracking errors are overestimated by about 12%. Figure 7.16 is a plot of the fraction of events assigned to the tail of the Sdo resolution, in both data and Monte-Carlo muon-pair events. We see that it varies between roughly 3-6%. This is a reasonable figure based on the results of fitting the Toy M C distributions, and indicates that M C S is the dominant source of the tails in the track impact parameter resolutions. The fraction quoted here is significantly larger than the fraction of events which lie outside of the inner 3a of the resolution distribution due to the tail function being non-zero and flat beneath the core Gaussian. We cannot simply say that 3-6% of the events wil l be in the tail of the distribution function, but we can use the parameter / as Chapter 7. Analysis 66 3— ; — i i i i i • 1 1 I i i i i i • • • I i i i i i i ' i i t" 10"' 1 10 1/Q2 Figure 7.17: Plot of the power in the inner power-law tail (\pa\) of the Sdo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of 1/Q2. a line-shape parameter. We note also that the Monte-Carlo appears to have significantly fewer events in the tail portion for all but the highest values of 1/Q2 or the lowest values of track PT- This agrees with our expectation based on an examination of the transformed resolution variables. Figure 7.17 is a plot of the absolute value of the power pa in the inner power-law of the tail of the Sdo resolution, in both data and Monte-Carlo muon-pair events. In the data, the power decreases from about 5.5 down to 3.5 as 1/Q2 increases. There is much less change in the power in the Monte-Carlo, it remains between 4.0 and 4.5 over the entire range of 1/Q2. We compare the results of fitting to the data and Monte-Carlo distributions in the other impact parameter, ZQ, in a similar way. Figures 7.18-7.21 depict the fitted parameters versus 1/Q2 for the A^o resolution in muon-pair events. Figure 7.18 is a plot of the mean of the distribution for muon-pair events in both data and Monte-Carlo. For both data and simulations, the mean of the Azo resolution distribution is within ±0.08 from being exactly zero. The mean of the data distribution is generally negative, while that of the M C distribution is always positive. These values are not consistent within two standard deviations. Figure 7.19 is a plot of the width of the Gaussian core for the Az0 resolution, in both data and Monte-Carlo muon-pair events. In the data muon-pair sample, the core width ranges between 1.07 and 1.21, consistently larger than 1.0. This indicates that the BABAR tracking software consistently underestimates the ZQ tracking errors by 7-21%. The Monte-Carlo zo resolution has a core width consistently less than the data. It ranges from approximately 0.93 (in a bin with relatively few events) to 1.08. The tracking errors in zo are generally underestimated in the Monte-Carlo simulation, except in the vicinity of very Chapter 7. Analysis 67 10"1 1 10 1/Q2 Figure 7.18: Plot of the the mean of A z 0 resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of l/Q2-J J i i I , i n 10"1 1 10 1/Q2 Figure 7.19: Plot of the core width of the Azo resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of l/Q2. Chapter 7. Analysis 68 1/Q2 Figure 7.20: Plot of the tail fraction of the AZQ resolution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of l/Q2-small values l / Q 2 , where they are overestimated. Figure 7.20 is a plot of the fraction of events assigned to the tail of the Azo resolution, in both data and Monte-Carlo muon-pair events. We see that the tail fraction parameter / varies between roughly 3-6% for the ZQ resolution in the /i-pair data. This is essentially the same fraction seen in the do resolution for these events. Again we note that the fraction quoted here is significantly larger than the true fraction of events which lie outside of the inner 3cr of the resolution distribution, and should mostly be interpreted as a line-shape parameter. We note that the Monte-Carlo appears to have between 3-10% of its events in the tail portion, somewhat higher than the data. This does not agree with what was expected, based on inspection of the transformed resolution variable. This disagreement illustrates the difficulty of interpreting the parameter / as a true representation of the fraction of events in the tail . At the highest values of l / Q 2 the data and M C are consistent due to the large M C errors. Figure 7.21 is a plot of the absolute value of the power pa in the inner power-law of the tail of the Azo resolution, for both the data and Monte-Carlo muon-pair events. In the data events, the power ranges between 4.2 and 5.5, decreasing as l / Q 2 increases. This is the same behaviour seen in the Sdo data where the power also decreased as l / Q 2 increases. The Monte-Carlo events yield inner tail powers that are generally consistent with the data in the range of larger values of l / Q 2 , but are significantly smaller for the smallest values of l / Q 2 . This indicates poor agreement between the tail shape in data and M C Chapter 7. Analysis 69 a. 5 6 o a. 10 1 10 1/Q2 Figure 7.21: Plot of the power in the inner power-law tail (pa) of the Azo reso-lution for muon-pair events in the data (squares), and Monte-Carlo (circles) as a function of 1/Q2. for tracks with high pr-The poor statistics in the Monte-Carlo sample of high 1/Q2 events lead to some large errors in the estimated parameters. These estimates should not be considered reliable until samples of Monte Carlo events with large numbers of low pr, Two-Photon e-pairs and /x-pairs can be generated. 7.6 Correlation of Track Impact Parameters If the large displacements of events in the track impact parameter resolution tails really are due to Multiple Coulomb Scattering, then the quantities Edo and Az0 will be strongly correlated in each event. If an event is in the tail of the distribution of one track-impact parameter, it is more likely to be in the tail of the other as well. This is consistent with what we expect from Multiple Coulomb Scattering, that if a track is deflected through a large angle in the x-y plane, it is likely also deflected through a large angle with respect to the z-axis. Other potential sources of large track impact parameters, such as detector misalignment, or errors in the tracking software, should mostly be uncorrelated in the two track impact parameters. Figure 7.22 (solid-line histogram) is a plot of the Sdo resolution for all muon pair events in the data. The dashed-line histogram represents all muon pair events in which AZQ is greater than 3<r. The dotted-line histogram represents Chapter 7. Analysis 70 dO Resolution (Sigma) Figure 7.22: Histogram of a ^ d o ^ for muon events in data. Solid line is all muon-pair events. Dashed line is muon-pair events with AZQ > 3cr. Dotted line is muon-pair events with Azo > 5cr. all muon pair events in which Azo is greater than 5<r. We can see that as we cut out the events in the core of the Azo distribution we also remove events from the core of the Sdo distribution. Equivalently, events which are in the tails of the Azo distribution are much more likely to also be in the tails of the Sdo distribution. Figure 7.23 (solid-line histogram) is a plot of the Azo resolution for all muon pair events in the data. Again, the dashed-line histogram represents all muon pair events in which Sdo is greater than 3u. The dotted-line histogram rep-resents all muon pair events in which Sdo is greater than 5<r. We see that as we cut out the events in the core of the Sdo distribution we also remove events from the core of the Azo distribution. Events which are in the tails of the Sdo distribution are more likely to also be in the tails of the Azo distribution. These correlations provide additional evidence to suggest that the tails of the track impact parameter resolutions really are due to Multiple Coulomb Scattering. Chapter 7. Analysis 71 zO Resolution (Sigma) Figure 7.23: Histogram of a^l0^ for muon events in data. Solid line is all muon-pair events. Dashed line is muon-pair events with Sdo > 3c. Dotted line is muon-pair events with Sdo > 5cr. 72 Chapter 8 Conclusions 8.1 Results of this Study After analyzing the resolution distributions of the track impact parameters for both data and Monte-Carlo events of the BABAR experiment, there are a number of conclusions we can draw. Firstly, the parameterization of Equation 2.4, using two power-law tails, describes Toy Monte-Carlo events generated with the Multiple Coulomb Scat-tering distribution quite well. The same parameterization also describes both of the track impact parameter distributions in BABAR data quite well out to values of approximately 20<r. The Monte-Carlo simulated sample of muon-pair events is fit well by the functional form of Equation 2.4 for both track impact parameters. The sample of Monte-Carlo simulated electron-pair events has a distinctly different distribution in the Sdo parameter, and we therefore conclude that there was an error somewhere in the simulation process for those events. For both data and Monte-Carlo muon-pair events, the width of the Gaussian core determined through fitting to Equation 2.4 is within 20% percent of being equal to the errors assigned by the BABAR tracking software. This demonstrates that the tracking software correctly models the contributions to the resolution from detector geometry and from the Gaussian core of the Multiple Coulomb Scattering distribution to the level of 20%, but that significant corrections be-tween 5-20% need to be applied to improve the level of agreement. The core widths in the experimental data and Monte-Carlo simulations are also system-atically different, with the Monte-Carlo distributions tending to have smaller widths by about 10% in both d0 and z0. Results of the fits to track impact parameter resolution show that the power-law of the inner tail ranges between x~35 and x~55. This agrees reasonably well with what we observed in the M C S Toy Monte-Carlo. In Section 2.3 we saw that by fitting Equation 2.4 to Toy Monte-Carlo events generated with the exact multiple scattering distribution we obtained inner tail powers pa of approximately 4.0. This leads us to conclude that the tails in the track impact parameter resolution are dominated by Multiple Coulomb Scattering and not by effects like misalignment or failures of the track-fitting software. This also provides a physical motivation for the shape of our parameterization. We found additional evidence that the tails are mostly due to Multiple Coulomb Scattering by looking at the distribution of Sdo for events with large values of Azo and vice-versa. When tracks are widely displaced in one pa-rameter due to a large angle scatter, we expect them on average to be widely Chapter 8. Conclusions 73 displaced in the other parameter as well. Other possible sources of poor impact parameter agreement, like misalignment or software errors should be largely independent in do and 20 • We do indeed see a strong correlation between the two track impact parameters, as shown in Section 7.6. This correlation provides more direct evidence that M C S is the dominant source of the tails in the track impact parameter resolution. 8.2 Future Plans Any experiments which require highly precise knowledge of track impact param-eter resolution wil l need to take into account the effects of Multiple Coulomb Scattering, especially the long tails. The parameterization developed here may prove useful for other researchers, but most importantly it helps increase aware-ness of the importance of Multiple Coulomb Scattering for track impact param-eter resolutions, by including the tails explicitly in the formulation. More directly, we see the possibility of using these results in several time-dependent analyses in the BABAR collaboration. We may be able use this tech-nique to reduce the systematic uncertainty due to the track impact parameter resolution in future measurements of neutral B-mixing at BABAR . The ad-ditional effects of particle misidentification and confusion between direct and cascade leptons also contribute to the At resolution. This means any improve-ment in the measured results wil l ultimately depend on the relative importance of the tracking compared to the other contributions to the At resolution. In order to use the results more directly in other studies, it would be more convenient if the analysis were performed in terms of the variables pr and 9 for the tracks, rather than the combined quantity 1/Q2- This would make it easier to apply the parameterization on a track-by-track basis to the chosen data set. The problem with this approach is that the two tracks don't have the same values of 9, and we don't know which one of the two to blame for a large-angle scatter. The results of this thesis will have to be used in a less direct manner. Finally, it is clear that there are many areas in which the track impact parameter resolution in BABAR Monte-Carlo simulations doesn't agree with the actual data. These discrepancies wil l need to be either resolved or compensated for if the extensive collection of Monte-Carlo generated events is going to be used in future studies sensitive to the tails of the track impact parameter resolution. 74 Appendix A Tail Shapes in DATA and M C Simulation: AZQ B y using the transformed variable u = 1/x2, where x is either a ^ o - j or a ^ 0 ^ , we can more closely compare the shape of the tails between the data and Monte-Carlo events. Figures A.1-A.8 are plots of the quantity u = {j^*^ ) for the muon-selected data and Monte-Carlo events. Figures A.9 -A. 14 are plots of the same quantity u for the electron-selected data and Monte-Carlo events. There are insufficient electron Monte-Carlo events at higher values of l/Q2 to make comparison possible. We don't trust the resolution for the electron M C in ZQ because it is obviously incorrect in do- See Section 7.5 for more details. Appendix A. Tail Shapes in DATA and MC Simulation: Az0 75 Appendix A. Tail Shapes in D A T A and MC Simulation: AZQ 76 I- , i i t t ( i • > . .>> , A 0 0.05 0.1 0.15 0.2 0.25 Figure A.3 : Plot of u = ( £ ^ i ) for ^-pair events with 0.25 < l / Q 2 < 0.5. Figure A.4: Plot of u = (Z^L) for /u-pair events with 0.5 < l / Q 2 < 1.0. Appendix A. Tail Shapes in DATA and MC Simulation: Az0 77 Appendix A. Tail Shapes in D A T A and MC Simulation: Az0 78 Appendix A. Tail Shapes in DATA and MC Simulation: Az0 79 Appendix A. Tail Shapes in DATA and MC Simulation: Az0 80 Appendix A. Tail Shapes in DATA and MC Simulation: Azo 81 82 Appendix B Tail Shapes in DATA and M C Simulation: Ed,Q Figures B.1-B.8 are plots of the quantity u = (^^^ ) for the //-pair data and Monte-Carlo events. Figures B.9-B.14 are plots of the same quantity u for the e-pair data and Monte-Carlo events. Note that the e-pair M C resolution is obviously seriously flawed. The large spike at u=0 corresponds to a significant surplus of M C events at greater than 10a. Appendix B. Tail Shapes in DATA and MC Simulation: Sd0 83 Appendix B. Tail Shapes in DATA and MC Simulation: Sd0 84 Appendix B. Tail Shapes in DATA and MC Simulation: Sd0 85 Appendix B. Tail Shapes in DATA and MC Simulation: 2Jd0 86 Appendix B. Tail Shapes in DATA and MC Simulation: Sd0 87 Appendix B. Tail Shapes in DATA and MC Simulation: Sd0 350 300 250 200 150 ' ' 1 I I I e-pair Data B l l e-pair MC Fit-Data Fit-MC Fit tail-Data Fit tail-MC 0.25 Figure B . l l : Plot of u = {^§^)2 for e-pair events with 0.25 < l/Q2 < 0.5. 1600 1400 1200 1000 800 600 400 200 ' 1 1 1 I I I e-pair Data I I e-pair MC Fit-Data Fit-MC Fit tail-Data Fit tail-MC h — i — i — | — i — i — r 0.25 Figure B.12: Plot of u = ( ) for e-pair events with 0.5 < l/Q2 < 1.0. Appendix B. Tail Shapes in DATA and MC Simulation: Sd0 89 3500 3000 2500 2000 1500 1000 500 - i — i — i — i — i — n I I e-pair Data B # l e-pair MC Fit-Data Fit-MC Fit tail-Data Fit tatl-MC • / ' ' w " ^ 7 : : : : : : : : : : : a ^ m 0.05 0.1 0.15 0.2 0.25 Figure B.13: Plot of u = {^§^)2 for e-pair events with 1.0 < 1 /Q 2 < 2.5. 2500 2000 1500 1000 500 I I e-pair Data I B H e-pair MC Fit-Data Fit-MC Fit tail-Data Fit tail-MC 0.1 0.15 0.2 0.25 Figure B.14: Plot of u = \ ^§jsl) for e-pair events with 2.5 < 1/Q 2 < 5.0. 90 Bibliography [1] P. F . Harrison, H . R. . Quinn [BABAR Collaboration], "The BaBar physics book: Physics at an asymmetric B factory," SLAC-R-0504, (1998) [2] E . Noether, Nachr. v.d. Ges.d.Wiss.zu Gottingen, 235 (1918). [3] C. S. Wu, et al. , Phys. Rev. 105, 1413 (1957). [4] J . H . Christenson, J . W . Cronin, V . L . Fitch and R. Turlay, Phys. Rev. Lett. 13, 138 (1964). [5] A . D. Sakharov, Sov. Phys. Usp. 34, 417 (1991). [6] M . Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973). [7] L . Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983). [8] C. Gay, Ann. Rev. Nucl. Part. Sci. 50, 577 (2000) [9] S. Eidelman et al. [Particle Data Group], Phys. Lett. B 592, 1 (2004). [10] K . Abe et al. [Belle Collaboration], Phys. Rev. Lett. 87, 091802 (2001) [11] B . Aubert et al. [BABAR collaboration], Phys. Rev. Lett. 88, 221803 (2002) [12] E . J . Will iams, Proc. Roy. Soc. A 169, 531 (1939) [13] E . J . Williams, Phys. Rev. 58, 292 (1940) [14] G . Moliere, Z. Naturforsch. 3a, 78 (1948) [15] H . A . Bethe, Phys. Rev. 89, 1256 (1953) [16] R. Brun, et al. , CERN-DD-78-2 -REV [17] S. Agostinelli et al. [ G E A N T 4 Collaboration], Nucl. Instrum. Meth. A 506, 250 (2003). [18] H . W . Lewis, Phys.Rev. 78, 526 (1950) [19] H . J . Bhabha, Proc. Roy. Soc. A154, 195 (1936) [20] D. Binosi and L . Theussl, Comput. Phys. Commun. 161, 76 (2004) [21] S. Jadach, W . Placzek and B . Ward, Phys. Lett. B 390, 298 (1997) Bibliography 91 [22] Y . Kubota et al. [CLEO Collaboration], Nucl. Instrum. Meth. A 320, 66 (1992). [23] B . Aubert et al. [BABAR Collaboration], Nucl. Instrum. Meth. A 479, 1 (2002) [24] R. Santonico and R. Cardarelli, Nucl. Instrum. Meth. 187, 377 (1981). [25] G . Battistoni et al. Nucl. Instrum. Meth. 164, 57 (1979). 

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