me: u RETURN BOXto mm min chockout from y'our record. To AVOID FINES mum on or baton dd. duo. DATE DUE DATE DUE DATE DUE usu IsAn mum mun/Emu oppommuy Institution WM‘I A MEASUREMENT OF THE INCLUSIVE DRELL-YAN e+e- CROSS SECTION IN THE INVARIANT MASS RANGE OF 30-60 GEV/C2 FROM pp COLLISIONS AT \/§ = 1.8 TEV By James Walter Thomas McKinley A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1996 ABSTRACT A MEASUREMENT OF THE INCLUSIVE DRELL-YAN e+e- CROSS SECTION IN THE INVARIANT MASS RANGE OF 30-60 GEV/C2 FROM p13 COLLISIONS AT \/§ = 1.8 TEv By James T. McKinley We present a measurement of the inclusive Drell—Yan e+e‘ cross section measured using the DO detector at Fermi National Accelerator Laboratory. 14.7 pb‘1 of data were collected during the first data taking run of the DO detector which was used to measure the invariant mass, photon rapidity, and photon transverse momentum distributions in the invariant mass range of 30-60 GeV/cz. These distributions are compared to the resummed theoretical predictions. For My Parents. iii ACKNOWLEDGMENTS I would first like to thank my thesis advisor Dr. James T. Linnemann for his guidance throughout my graduate career and especially during the analysis and pre- paration of this dissertation. I learned a great deal about computing and statistics as well as physics under his tutelage. Also his support of my research was invaluable in the current tight financial environment of governmentally funded physics research. I would also like to thank Dr. C.-P. Yuan and Csaba Balazs of Michigan State University for adding the photon couplings to their RESBOS Monte Carlo program. This made my analysis and comparison to the current theory much easier. I would also like to thank them for many valuable theoretical discussions and guidance during the preparation of this dissertation. My DO colleagues deserve my thanks as well, but are too numerous to list here. Some who deserve special thanks are: Mike Tartaglia for giving me guidance and training when I was first getting my feet wet in experimental high energy physics. Jan and Joan Guida and Dean Schamberger whose tireless efforts kept the DO experiment running smoothly even during its first days online and who taught me a great deal about the technical aspects of DO. Meena N arain, Ulrich Heintz, Eric Flattum and Ian Adam with whom I had many valuable discussions regarding electron identification in DO. Scott Snyder for his vast computing knowledge, some of which he shared with me. Also Richard Genik III, for helping me keep my sense of humor as well as for some interesting physics conversations. I am thankful to my girlfriend Rachel A. Postema for putting up with me during the final days of my dissertation work and for keeping my outlook bright. Finally, I wish to thank my parents Jacqueline L. Curry and Michael J. Curry, whose unwavering support throughout my university career, as well as my whole life, allowed me to reach this goal. iv Contents LIST OF TABLES LIST OF FIGURES 1 Introduction 2 Theory 2.1 Introduction ........... 2.2 Lowest Order Drell-Yan Process 2.3 Higher Order Drell-Yan Processes 2.4 Kinematics ............ 2.4.1 Lowest Order Kinematics . 2.4.2 Higher Order Kinematics . 2.5 Summary ............. 3 APPARATUS ooooooooooooooooooo oooooooooooooooooooo 3.2.2 The Central Detector (CD) ................... 3.1 The Accelerator ......... 3.2 The DC Detector ......... 3.2.1 Overview ......... 3.2.3 The Calorimeters . . . . 3.2.4 The Muon Detectors . . . 3.2.5 The Trigger And Data Acquisition System .......... 4 Data Sample And Event Selection 4.1 Luminosity ....... _ ..... 4.2 Level 0 Trigger Selection Criteria 4.3 Level 1 Trigger Selection Criteria 4.4 Level 2 Trigger Selection Criteria oooooooooooooooooooo OOOOOOOOOOOOOOOOOOO 0000000000000000000 viii 15 23 24 29 32 35 35 40 40 42 50 59 65 4.5 Offline Data Reconstruction ....................... 4.5.1 Energy Reconstruction ...................... 4.5.2 Track Reconstruction ....................... 4.5.3 Muon Reconstruction ....................... 4.5.4 Vertex Reconstruction ...................... 4.6 Offline Electron Identification ...................... 4.6.1 H-matrix .............................. 4.6.2 Isolation Fraction ......................... 4.6.3 Track Match Significance ..................... 4.6.4 dE/da: ............................... 4.7 Summary ................................. Data Analysis 5.1 Selected Data Sample ........................... 5.2 Background Estimation .......................... 5.2.1 Dijet Background Estimation Method .............. 5.2.2 Z—) 7*?” —> e+e‘ Background Estimation Method ....... 5.2.3 Background Subtraction ..................... 5.3 Online Efficiencies ............................. 5.4 Offline Efficiencies ............................. 5.5 F iducial Acceptance, Kinematic Corrections And Unsmearing ..... 5.6 Cross Section ............................... 5.7 Systematic Errors ............................. Results And Comparison To Theory 6.1 Statistical Compatibility Tests ...................... 6.1.1 Pearson X2 Test .......................... 6.1.2 Smirnov-Cramér—Von Mises Test ................ 6.2 Mass Distribution ............................. 6.3 Photon p1 Distribution .......................... 6.4 Photon Rapidity Distribution ...................... 6.5 Effect Of Parton Distributions ...................... 6.6 Conclusions ................................ 6.7 Possible Future Improvements ...................... The Level 2 Electromagnetic Filter A.1 Filter Scripts ............................... vi 90 90 91 92 97 103 107 117 119 127 127 127 141 146 146 149 155 157 167 170 177 177 178 179 180 182 192 193 200 201 204 A2 The L2_EM Algorithm .......................... 207 A21 Longitudinal Algorithm ..................... 208 A22 Transverse Algorithm ...................... 209 A.2.3 Cone Algorithm .......................... 211 A3 Energy And 1) Dependence Of Cuts ................... 212 AA Cut Tuning ................................ 213 A41 Tuning Of EC Shower Shape Cuts ................ 214 A.4.2 Tuning Of CC Shower Shape Cuts ................ 216 A.5 Internal Organization Of The L2-EM Filter ............... 218 A51 Correspondence Between RCP And Internal Variables ..... 220 A6 Calorimeter Geometry .......................... 222 A.7 L2_EM filter cut values .......................... 225 LIST OF REFERENCES 235 vii List of Tables 2.1 Table of elementary particles. ...................... 2.2 Some symmetry operations and related conservation laws ........ 2.3 Definition of mathematical notation .................... 4.1 Table of electron selection cuts. ..................... 4.2 Table of single electron efficiency parameterizations ........... 5.1 Drell-Yan signal + background events per invariant mass bin ...... 5.2 Drell—Yan signal + background events per pair rapidity bin ....... 5.3 Drell-Yan signal + background events per pair pT bin .......... 5.4 Dijet background events per invariant mass bin. ............ 5.5 Dijet background events per pair rapidity bin. ............. 5.6 Dijet background events per pair pT bin. ................ 5.7 Z—) 7+7” —> e+e" background events per invariant mass bin ...... 5.8 Z—+ 7+7" -—> 6+6- background events per pair rapidity bin ....... 5.9 Z—-) T+T‘ —+ e+e’ background events per pair p7 bin .......... 5.10 Drell-Yan events per invariant mass bin. ................ 5.11 Drell-Yan events per photon rapidity bin ................. 5.12 Drell-Yan events per photon p1 bin .................... 25 120 128 128 129 132 132 134 138 139 139 147 147 149 5.13 Efficiency corrections for Drell-Yan e+e‘ events per invariant mass bin. 157 5.14 Efficiency corrections for Drell-Yan e+e" events per photon rapidity bin.157 5.15 Efficiency corrections for Drell-Yan 6+6- events per photon p1 bin. . . 5.16 Total Efficiency, Kinematic, Fiducial Acceptance and Smearing correc- tion for Drell—Yan e+e‘ events per invariant mass bin .......... 5.17 Total Efficiency, Kinematic, Fiducial Acceptance and Smearing correc— tion for Drell-Yan e+e" events per photon rapidity bin. ........ 5.18 Total Efficiency, Kinematic, Fiducial Acceptance and Smearing correc- tion for Drell-Yan e+e" events per photon pT bin. ........... 5.19 Integrated Drell-Yan cross section per invariant mass bin ........ 5.20 Integrated Drell-Yan cross section per photon rapidity bin. ...... viii 162 167 168 168 170 171 5.21 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 Integrated Drell—Yan cross section per photon p1 bin. ......... Cumulative distribution functions from the do/dm distribution used in the Smirnov—Cramer-Von Mises test. .................. Coodness-of-fit test results from comparing the experimental da/dm distribution to the RESBOS and ISAJET Monte Carlo distributions. . Smirnov-Cramer-Von Mises test on the da/dm distribution combined with a X2 test on the integrated cross section (normalization) ...... Cumulative distribution functions from the da/de distribution used in the Smirnov-Cramer-Von Mises test .................. Goodness-of-fit test results from comparing the experimental da/de distribution to the RESBOS and ISAJET Monte Carlo distributions. . Smirnov-Cramér-Von Mises test on the da/de distribution combined with a x2 test on the integrated cross section (normalization) ...... Cumulative distribution functions from the da/dy distribution used in the Smirnov—Cramér-Von Mises test. .................. Goodness-of-fit test results from comparing the experimental da/dy distribution to the RESBOS and ISAJET Monte Carlo distributions. . Smirnov-Cramer-Von Mises test on the do/dy distribution combined with a X2 test on the integrated cross section (normalization) ...... Difference in integrated cross sections due to input parton distribution functions. ................................. Goodness-of-fit test results from comparing the experimental da/de 171 181 182 182 187 188 188 192 193 193 distribution to the RESBOS Monte Carlo distributions using the CTEQ3M and MRSDO' input parton distribution functions. ........... Integrated cross section summary ..................... ix 198 200 List of Figures 2.1 Lowest order Drell-Yan Feynman diagram. ............... 8 2.2 Drell-Yan O(a,) correction Feynman diagrams .............. 16 2.3 Drell-Yan 0(a3) correction Feynman diagrams. ............ 16 2.4 Drell-Yan virtual correction Feynman diagrams. ............ 16 2.5 Resummed Drell-Yan da/dm. ...................... 22 2.6 Resummed Drell-Yan do/de. ...................... 22 2.7 Resummed Drell-Yan da/dy ........................ 23 2.8 The Collins-Soper 0’ reference frame ................... 24 2.9 Gluon Bremsstrahlung .......................... 29 2.10 Comparison of RESBOS and ISAJET pT spectra. ........... 34 3.1 The Fermilab Tevatron Collider ...................... 36 3.2 An isometric cut—away view of the DO detector. ............ 41 3.3 An overview of the DO calorimeters ................... 51 3.4 The pseudo-projective geometry of the DO calorimeter towers. . . . . 53 3.5 A calorimeter unit cell ........................... 55 3.6 A schematic of the DO Data Acquisition System ............ 72 3.7 A schematic diagram of a Level 2 node ................. 74 4.1 The Level 1 trigger efficiency vs. input ET ............... 85 4.2 The Level 2 trigger efficiency vs. input ET ............... 88 4.3 PELC/PPHO efficiency vs. ET, E, and IETA. ............. 96 4.4 H-matrix X2 for electrons. ........................ 98 4.5 H-matrix X2 for background ........................ 98 4.6 CC H-matrix X2 < 100 Efficiency vs. input |IETA|. .......... 101 4.7 EC H-matrix X2 < 100 Efficiency vs. input IIETAI. .......... 102 4.8 Isolation fraction for electrons ....................... 103 4.9 Isolation fraction for background. .................... 104 4.10 CC Isolation fraction I 50 < 0.15 Efficiency vs. input |IETA|. . . . . 105 X 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 EC Isolation fraction ISO < 0.15 Efficiency vs. input |IETA|. Track match significance for electrons ................... Track match significance for background. ................ Central calorimeter 052 vs. input energy and IETA .......... Central calorimeter 036.1, Vs. input energy and IETA. ......... End calorimeter 05,, vs. input energy and IETA. ............ End calorimeter 035,, vs. input energy and IETA. ........... dE/dcr for electrons and background. .................. Level 1 and Level 2 electron efficiency vs. input ET after calorimeter based offline cuts .............................. CC H—matrix x2 < 100 + isolation fraction I 50 < 0.15 efficiency vs. input |IETA|. ............................... EC H—matrix X2 < 100 + isolation fraction [50 < 0.15 efficiency vs. input |IETA|. ............................... Track match significance + dE/da: cut Efficiency vs. input |IETA|. . . Drell-Yan + background events vs. invariant mass. .......... Drell-Yan + background events vs. pair rapidity ............. Drell-Yan + background events vs. pair pT ................ Dijet background events vs. Drell—Yan signal + background events. Direct photon background events vs. Drell—Yan signal + background events .................................... W + jet background events vs. Drell-Yan signal + background events. Z—) 7'1“!" —+ e+e‘ background events vs. Drell-Yan signal + back- ground events ................................ CC PELC rejection vs. cluster ET, IETA. ............... EC PELC rejection vs. cluster ET, [IETA] ................ Drell-Yan (signal + background) - (dijet + Z—> 7+7“ ——> 6+6“) back— ground events ................................ Level 1 EM ET > 7.0 GeV efficiency for Drell—Yan 8+€_ events after offline cuts. ................................ Level 2 EM ET > 10.0 GeV + Level 2 ISO < 0.15 efficiency for Drell- Yan e+e‘ events after offline cuts ..................... PPHO efficiency for Drell-Yan e+e" events ................ PELC/PPHO efficiency for Drell—Yan 6+6“ events. .......... H-matrix x2 < 100 + ISO < 0.15 efficiency for Drell-Yan 6+6“ events. Track match significance + dE/da: cuts efficiency for Drell-Yan 6+6— events .................................... xi 106 108 108 113 114 115 116 118 121 123 124 126 129 130 130 133 136 137 140 144 145 148 153 154 158 159 160 161 5.17 5.18 5.19 5.20 5.21 5.22 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 Comparison of two smearing correction methods for Drell-Yan 6+6— events. ................................... Kinematic, Fiducial Acceptance and Smearing correction for Drell-Yan e+e‘ events ................................. Total Efficiency, Kinematic, Fiducial Acceptance and Smearing correc- tion for Drell-Yan e+e" events ....................... Drell-Yan e+e" differential cross section da/dm vs. invariant mass. Drell—Yan e+e‘ differential cross section da/dy vs. photon rapidity. Drell—Yan e+e" differential cross section da/de vs. photon pT. . . . . Drell-Yan e+e' differential cross section da/dm vs. invariant mass compared to the RESBOS Monte Carlo. ................ Drell-Yan 6+6“ differential cross section da/dm vs. invariant mass compared to the ISAJET Monte Carlo. ................. Drell-Yan e+e" differential cross section m3d20/dmdylyzo vs. invariant mass compared to theory .......................... Drell-Yan 6+6" differential cross section da/dm vs. invariant mass compared to the CDF 1988-1989 cross section .............. Drell-Yan e+e— differential cross section da/de vs. photon pT com- pared to the RESBOS Monte Carlo .................... Drell-Yan e+e— differential cross section da/de vs. photon pT com- pared to the ISAJ ET Monte Carlo. ................... Drell-Yan e+e— differential cross section da/de vs. photon p1 com- pared to the RESBOS and ISAJ ET Monte Carlos plotted on a linear scale ..................................... Drell—Yan e+e— differential cross section da/dy vs. photon rapidity compared to the RESBOS Monte Carlo. ................ Drell-Yan e+e—- differential cross section da/dy vs. photon rapidity compared to the ISAJET Monte Carlo. ................. Drell—Yan e+e— differential cross section differences between using the MRSDO’ and CTEQ3M parton distributions as input to the RESBOS Monte Carlo. ............................... Drell-Yan e+e— differential cross section differences between the ISA- JET(CTEQ2L) and RESBOS(CTEQ3M) Monte Carlos ......... xii 166 169 172 173 174 185 186 189 190 191 194 195 199 Chapter 1 Introduction The subject of this dissertation is the measurement of the inclusive e+e' (Drell-Yan) cross section in p17 collisions at \fS : 1.8 TeV. I have measured the virtual photon mass, rapidity, and transverse momentum (QT) spectra using data from the DO de- tector collected during DOS’ first collider data run. The DO experiment was originally proposed in 1983. The name is derived from the Tevatron Collider’s D0 interaction region at the Fermi National Accelerator Laboratory (F NAL or Fermilab) in Batavia, Illinois in which the detector resides. Installation was completed in early 1992. The first collisions occurred in the DO detector on May 12, 1992. Several test beam runs were also conducted using the F NAL fixed target facilities during the fabrication and assembly of DO to study its various components. This thesis will focus on the 14.7 pb‘l of data taken during DOs' first run which occurred during the 14 month period between May, 1992 and July, 1993 which is also known as run 1A. The Drell—Yan lepton pair production mechanism was first described by Sidney D. Drell and Tung-Mow Yan in their paper titled ”Massive Lepton-Pair Production in Hadron-Hadron Collisions at High Energies” [1]. This model is often called the “naive” Drell-Yan model since it does not take into account the transverse momentum of the incoming hadrons and thus predicts a zero transverse momentum for the virtual photon. The Drell—Yan process, though rare in proton-antiproton interactions, has the vir- tue of being unaffected by complex final state interactions and is directly comparable to theoretical calculations in a way that few processes involving the strong nuclear force are. It serves as an important test-bed for perturbative QCD (Quantum Chro- modynamics) calculations. Much theoretical work has been done to describe this process more accurately within the framework of the Standard Model and take into account higher order (QCD) corrections to the basic Drell-Yan model. Recent work by C.-P. Yuan and G. A. Ladinsky of Michigan State University, in which the non-perturbative functions in the Collins-Soper-Sterman resummation formalism were studied using fixed-target and collider Drell-Yan data, resulted in parameterizations which yield better agreement with with CDF (Collider Detector at Fermilab) Z boson data than previously found in the literature. CDF has recently (Jan. 1994) published their Drell-Yan cross section for the rapidity range IyI < 1.0 from 4.13 pb’l of data collected in the 1989 collider run at Fermilab. I take advantage of the DO detector’s large rapidity coverage to measure the Drell-Yan cross section in the rapidity range Iyl < 2.5. I also compare measurements of the DO Drell-Yan mass, rapidity, and QT spectra to that predicted by the resummed cross section. Chapter 2 Theory 2. 1 Introduction The Standard Model (SM) of elementary particles describes the interactions between the three basic types of elementary particles which are leptons, quarks, and mediators (force carrying particles). There are six types each of leptons and quarks which are further grouped in pairs into three generations. Table 2.1 shows the currently known leptons and quarks grouped by generation and “flavor”. All these elementary particles have corresponding anti-particles as well, for a total of 12 leptons. In addition, quarks come in three “colors” for a total of 12 X 3 = 36 quarks. All quarks and leptons are fermions with spin 3' The force carrying particles are the photon (7), W+, W‘, and Z vector bosons and the gluon (g) which all have spin 1. The photon mediates the electromagnetic force, the W and Z bosons mediate the weak nuclear force, and the gluon carries the strong Table 2.1: Table of elementary particles. [I generation I] I [ II 1 IIIJ] quarks u c t d s b leptons e p T l/e up u, Table 2.2: Some symmetry operations and related conservation laws. symmetry conservation law time translation H energy conservation space translation H momentum conservation rotation <—> angular momentum conservation gauge transformation H charge conservation nuclear force. There are eight different types of gluons which, when combined with the vector bosons, give a total of 12 force carrying particles in the Standard Model (SM). In addition, the Glashow-Weinberg-Salaam model requires the existence of at least one Higgs boson whose coupling strength to the other particles brings about the difference in their masses. The Standard Model is based on the symmetries that exist in nature. Noether’s Theorem states that symmetries imply conservation laws and vice versa. For example, Table 2.2 lists some symmetries in nature and the physical conservation laws associated with them. The definition of a symmetry is an operation that can be performed (at least conceptually) on a system that leaves it invariant. The systematic mathematical study of symmetries is called group theory. The defining properties of a group are exactly the set of symmetry operations on a system that must hold true, namely 0 Closure. If R,- and R,- are members of the set, then the product RR, = Rk must also be a member of the set. 0 Identity. A member I of the set must exist such that I R, = 12,-I = Hg. 0 Inverse. Every member of the set R,- must have an inverse R,” such that H.112," = I. o Associativity. R;(Rij) = (RiRj)Rk. The most often used groups in elementary particle physics are unitary groups U (n) Unitary groups are groups whose members have the property 12'1 : RI. A unitary group with determinant 1 is called a special unitary group or S U (n) A unitary group with only real elements is known as an orthogonal group 0(n). Finally, a unitary group with only real elements that has determinant 1 is called special orthogonal or 50(n). The SM is composed of three groups which describe the internal symmetries of the theory, namely hypercharge, weak isospin, and color. The hypercharge symmetry is represented by the group U (1), weak isospin by SU (2);, (where the subscript L denotes that only left-handed particles obey this symmetry), and color by S U (3) Thus, the standard model is represented by S U (3) <8) 5' U (2) L (8) U (1) The number of gauge bosons that mediate the forces in the SM is equal to the number of generators in the symmetry group that represents the force. The U (1) group has one generator so the electromagnetic force has one gauge boson, the photon. The S' U (2) group has 2 <8) 2 = 22 — 1 = 3 gauge bosons which are the W+, W‘, and Z0 bosons. Finally, the SU(3) group has 3 (8) 3 = 32 — 1 = 8 gauge bosons which are the gluons. The internal symmetries of the SM refer to the behavior of the SM Lagrangian under U (1), SU (2)1” and SU (3) gauge transformations; the SM Lagrangian is left unchanged under these transformations. An Abelian gauge theory is one in which the fields which represent the gauge bosons commute. In Non-Abelian gauge theories the gauge fields do not commute. Thus, gauge fields do not directly interact with each other in Abelian gauge theories and do directly interact with one another in Non- Abelian theories. Quantum Electrodynamics (QED) which describes electromagnetic interactions is an Abelian gauge theory and photons do not directly interact with themselves. Quantum Chromodynamics (QCD) on the other hand, is a non-Abelian theory and gluons do directly interact with each other. A simple description of this is that photons are uncharged so they cannot couple to each other, whereas gluons carry color charge so they can and do couple to one another. This difference in the behavior of the gauge boson force mediators is the major difference between QED and QCD. In QED, an electric charge polarizes the vacuum due to the virtual electron-positron pairs which surround it. The charge density is higher near the charge and results in an effective coupling constant OE given by 0101) 1— (%.‘-.‘:‘1)1n(9;23) OE: where Q is related to the energy of the probe and ,u is a lower cutoff energy. In QCD, a quark is surrounded by not only virtual quark-antiquark pairs, but by virtual gluon pairs as well. The virtual gluon pairs decrease the effective strong coup- ling constant near the quarks, whereas the quark-antiquark pairs increase the effective coupling. The gluon pairs’ effect dominates and a, is decreased near the quarks. The strong coupling constant has the form 121r 03(6)) _ (33-2n1)1n(95;) where n! is the number of quark flavors and A is the QCD scaling parameter. At lower Q2 values, the strong coupling becomes large which explains why colored particles are confined in color neutral combinations. As Q2 becomes large, a, approaches zero. This is known as asymptotic freedom. This is the reason that perturbative methods can be used for high momentum transfer QCD calculations (known as “pQCD”). Un- fortunately the other side of the coin is that for “sof ” processes, perturbative methods break down, and little is known about “non-perturbative” QCD. At this time the most productive method for studying non—perturbative QCD is “lattice gauge theory” where the goal is to make progress in finding solutions by working with a minimum distance scale so the theory is cut off in momentum transfer and then introducing a variety of techniques such as statistical mechanics for handling complicated systems. Non- perturbative QCD calculations are still very much “work—in-progress”. Consequently, when describing the hadronic collisions which inevitably involve these non-perturbative interactions, one must rely on measured and parameterized parton momentum distri- butions for the initial state hadrons and fragmentation functions which describe how the final state partons evolve into hadron jets. The technique of separating the “hard- scattering” from the “sof ” processes is called factorization. It is not obvious that this approach is valid, however John Collins, Davison Soper, and George Sterman [4] have shown that factorization is valid to all orders in Drell-Yan cross sections for leading twist. 2.2 Lowest Order Drell-Yan Process To calculate the cross section for the lowest order Drell-Yan interaction we first cal- culate the hard scattering cross section. The Feynman diagram of the lowest order Drell-Yan interaction is shown in Figure 2.1. The hard scattering cross section is defined in terms of the matrix element for the process of interest m 1 —IM'I2(qa—> 7* —> 5?) dcosa 2 3211's Using the Feynman Rules we can write down the matrix element for this process -iM = H(1)3)(2'92 sin(9w)7“)v(p4)(%%5)'U(P2)(-iexgz sin(9w)7")U(p1) where e f is the quark charge fraction eq/e. A bit of simplification yields M .__. W[a(p3)7“v(p4)l[U(pzmum)l Figure 2.1: The Lowest order Drell-Yan Feynman diagram. Now, we wish to find the Hermitian conjugate of the matrix element. First we note that 7‘“ = 7°7“7° so the hermitian conjugate of the matrix element is -ie 2sin2 0 .. — M’r = —!—92;.—‘—W—’[v(p4wu(p3)nu(mmom Thus we have IMP = Mw = mIvvvfivv(p2mu(p1)1 Since the initial spins and colors of the incoming quarks are unknown, we must av— erage over these quantities and then sum over the final spins of the leptons which gives ———2 1 1 1 1 M = o O -<—> O - M I I 3 qcolors 3 acolors 2 qulIl 2 aspin X: Z | I colors spins ———2 1 e g2 sin2(0w) lMl : 36 . f 2 4 ' Z Z 7:67cfid7aa6760d . q i=colors abcd=SplnS 5am)vd(p4)ub(p3)fic(pa)27a(zn)ud(p1)vb(p2)vc(p2) - WI The sum over colors gives —2 1 egzsinzfl lMl ___ _. f2 (W) Z 12 (14 7gb7fd70ab75cd ' abcdzsplnS E1014)vd(P4)ub(p3)Hc(P3)Ua(P1)Ud(P1)vb(p2)5c(P2) From the Dirac equation we have Zfiu=p+m and Zfivzp-l—m which yields 2 _ _1_ ejg; sin4(0w) 12 abcd ‘14 [(174 — "Vlad/(’3 " "label 1’2 _ mqlad( [’1 _ mqlbc'YEch/Bd’YaaWHbCI = (i) (elf’gé4 sin4(0w)) . 12 q“ Tflf 174 - mzhafi’s - mzl‘rfll ' Trlfl’2 — mql’Yaf P1 — mqml The Q scale we are interested in is 30-60 GeV, which is orders of magnitude greater that ”the lepton and quark masses (we are well above the bottom quark mass and well 10 below the top quark mass) so we set the mg and mg in the above equation to zero which gives W = (.15) (i——“’—’) Trwn“ AflTrim. m1 Now we need to evaluate the above traces which can be readily accomplished using the following three identities "Fri/w“ 15'7"] = PiapjfiTrl’l'a’l'u'Yfi'll/l Trlvav‘WBV‘J = 4[9°’“9"" - 9069“” + WW] 9“”guu = 4 so the product of the two traces is Tr[,f)47“ p37”]Tr[/>1’Yu P2711] 16P4aP3fiPng' 19"“9” V97u95u — ga”g”“g~aguu + g"“g"”gwga.. — g“”9’“’gwgau + 9°”g“”gwagpu - 9"” 9“”971196” + ow Bu 01’ 311 g g g’w95u-g g gvsgw+gwgw 97.95“] = 161(p4-p1)(p3-p2)-(104-me -p2)+ (P4‘P2)(P1'Psl-(P4'P3)(P1'P2)+ 4(P4'P3)(P1'P2l—(P4'P3)(P1'P2)+ (P4'P2)(P1‘P3)‘(P4'P3)(P1'P2)+ (p4 'P1)(P3 -P2)l SO we have Tr[1’41” [’3’Ylel'l/f’l’7u PM] = 321(1’4 'P1)(P3 'P2) + (P4 °P2)(P1 'Psll Which finally gives us —— e2 33in‘ 9w fMl2 = (2) ('19—1(—‘))[(194‘P1)(P3'P2)+(P4°P2)(P1'Psll q We are now in a position to use what we know about the kinematics of the inter- action, namely that the momentum and energy are conserved in the process. In the 11 rest frame of the vector boson E E E E 0 0 E sin(0) —E sin(0) p1 = 0 a P2 = 0 a P3 = 0 a P4 = 0 E —E Ecos(0) —E cos(0) where 0 is the angle between the leptons and the beam axis in the cm frame and E = x/sT / 2 where S is the cm energy of the quark—antiquark interaction. Using these definitions we have (P4 'P1) = (P3 'P2) = E2(1 " C05(9)) (P4'P2)=(P1'P3)= E2(1+ C03(9)) so (P4 'P1)(P3 'P2) = E4(1— 2C08(9) + C082(9)) (P4 ' P2)(P1 “P3) = E4(1+ QCOSW) + cosz(9)) and (P4 'P1)(P3 'P2) + (P4 'P2)(P1 'P3) = 2E4(1 + C052(9)) hence E442 W = (e) (—i———‘—’) (1+ 442(4)) Now that we have evaluated the summed and averaged square of the matrix ele— ment, we can write down the lowest order Drell-Yan hard scattering cross section d&(q'q’ -> 7‘ -+ if) _ 1 —2 dcos(0) _ (32wé) lMl = (241;) (51—4) (e34; sin4(9W))(1 + com» To put this in more familiar notation we note that 92 sin(0w) = e 2__ 4:4 ~; a —161r2~l37 12 q4 = 16E4 which upon substitution gives d&(qfi—+7‘-+[Z) _ ej‘; 1roz2 2 dcos(0) _ 3 2.; (”COS 0) Integration over 0 = 0 to 7r gives the familiar 00(qfi —> 7* ——> if) total hard scattering amplitude 1r 1 / dc030(1+ cos2 0) = / da:(1+ $2) 2% o —1 This is the hard scattering cross section for a specific quark flavor in the cm frame of the interacting partons (the virtual photon rest frame) where S is the cm energy squared of the partons. The cross section for proton-antiproton scattering is the sum over all flavors, which requires knowledge of the incoming hadron types as well as the longitudinal momentum distributions fa(:z:a) and fb(a:b) of the interacting partons where :13“ and an, are the usual longitudinal momentum fractions of the interacting partons relative to the momenta of the their parent hadrons. The Factorization Theorem states that d0 _ 4W202 1 (16,1 1 d6}; dQdedezT ‘ 9Q25§/H/’5 Xfa/A(€A; #lTab(QT, Q, SEA/€44 (BB/58; 9(1“), P)fb/B(€B; I1) 13 where Tab is the infra-red safe perturbative hard—scattering cross section 00 a N Tab(QT,Q4IPA/{AJB/{BWWLM = Z [M] N=0 71' XTclfr)(QT4QJA/EAJB/ésm) and fa/A(§A,]u), fb/B({B,p) are the parton distribution functions for partons of type (1,6 in hadrons of type A,B mum) = 6446(1—64)+§(9,f)"f.‘734 n=1 This states that the parton remains itself in the absence of interactions. The vari- able p here is the factorization scale, which is arbitrary and determines the energy at which the parton distributions are evaluated. It is typically chosen to be p = Q. Factorization allows one to use perturbation theory to calculate the hard scattering cross section and remove the divergences which are then absorbed into the parton distributions. The parton distributions are non-perturbative quantities, but are uni- versal; they are the same in deep inelastic scattering (DIS) as they are in Drell-Yan. Consequently, one can measure the parton distributions in DIS, and then apply the results to make predictions about Drell-Yan. A heuristic argument for the idea of a parton density was given by Feynman [11] [12]. If we consider electron-proton scattering (DIS) where the proton is assumed to be made up of constituent partons, the partons interact with one another, and exist in purely virtual states. A typical state has a lifetime 1' in this frame. In the rest frame of the electron, 'r is dilated to 7(Ep/mp), while the proton radius r,, 14 is Lorentz-contracted to rp(m,,/Ep). Thus, during the short time it takes for the proton to pass over the electron in this frame, the partons appear to be stationary, because their self-interactions act on dilated time scales that are much longer than the time for the electron-parton collision. Since parton-parton interactions and electron— parton scattering take place on such different time scales, they cannot interfere in a quantum mechanical sense. Consequently, the quantum mechanical amplitudes for the distributions of partons exhibit incoherence relative to the electron-parton cross section, as if they were classical quantities. Thus it makes sense to talk about the probability of finding a parton with a given momentum in a proton and to treat it separately from the hard scattering. This probability is the parton density f. . . . . . . . . - ‘- [2 Given the parton distribution functions, the differential cross section W can be written as (120 81m2 flaw" dexF = 9M3(:ca+:cb) if: eflf;(xa)f%($b)+f70($a)f;($b)] This is the naive Drell-Yan result. In terms of the Q and rapidity of the vector boson (ma = fiey’i‘) the total cross section can be written as — flavor 1 1 7' da(q‘q' —> 7* —-> (T) - ‘ re = f f d ad a . U(PP-+7 -> ) £1: 0 0 :v asbf(:P )fb(Pb)/0 dcosa flavor El: [01 A: “Tdyw'fah/Fey” )fb(\/;€_“"')' /" da(q2j -+ ’7 ——> ”adcosfl o dcosfl where T = .§/ S, S is the center of mass energy squared of the colliding proton- antiproton beams and 31.,— is the rapidity of the virtual photon. Thus, the 15 differential cross section relative to the lepton angular distribution in the cm frame and T = 6/5 = Q2/S = MZ/S is dams —> 7‘- —> e?) “W -— °° 4.. -y,. d&(qfi-+7‘ —> 8?) dcosgd, — ; Lodz/«fah/fe mm. ) dcosfl In the naive Drell-Yan model, the QT of the virtual photon is identically zero. Examination of experimental Drell-Yan data however, clearly shows that this is not the case in nature. Also, the overall event rate predicted by the lowest order calcu- lation is too low by roughly a factor of two when compared to the measured cross section (the so—called “K” factor). This large difference is due to the absence of the higher order Drell-Yan processes which contain large logarithmic terms in (Q2/A2). Consequently, one must go beyond leading order for an accurate comparison of theory and experiment. 2.3 Higher Order Drell-Yan Processes The calculation of higher order Drell-Yan interactions is significantly more complicated than the lowest order calculation. One complication is that we now have a multi-body final state instead of a two-body final state. This complication is overcome in the next-to—leading order (NLO) calculation by splitting the calculation into two two-body pieces: one calculates the process q + E —> 7" + g for example, and then calculates the decay of the virtual photon into a lepton pair. Another problem is that the perturbative NLO result (and higher orders) is singular as QT —-> 0 since it contains both infra—red (i.e. very soft gluon radiation) and collinear (i.e. gluon radiation along the quark direction) divergences. The infra-red and collinear singularities can be removed by the dimensional regu- larization technique [3] combined with factorization. Dimensional regularization 16 Figure 2.2: Drell-Yan 0(a3) correction Feynman diagrams. Figure 2.3: Drell-Yan 0(03) correction Feynman diagrams. Figure 2.4: Drell-Yan virtual correction Feynman diagrams. 17 removes the divergence of an integral by allowing one to evaluate the integral in n dimensional space and then analytically continue back to the desired dimensionality. This technique respects gauge invariance and Lorentz invariance provided the integ- rand is well defined in n dimensions. When calculating cross sections, one prefers to do the calculation as generally as possible. For example, it is desirable to generalize Drell-Yan to the general vector boson cross section since the differences in the cross section for different vector boson types are due mainly to the differences in the 7, Z, and WE couplings. Unfortunately, the W coupling contains 75 which is ill defined in n dimensions. However, a canonical '75 prescription exists that allows one to calculate the anti-symmetrical part of the matrix element in n dimensional spaceetime [5] [6] [7]. Thus it is possible to perform the general vector boson calculation as desired. The transverse momentum QT distribution of the vector boson cannot be described by the N LO calculation in the low QT region. This is because the convergence of the perturbative expansion of the Drell-Yan cross section d0 2 3 — = ava.(ui+aSU2+a,U3+a,U4+-~) dQ’I‘ deteriorates as QT —> 0. _ At first order in a, the final state gluon or quark balances the QT of the vector boson. At second order, an additional jet may be produced and the interference of the one-loop corrections with the first order diagrams appears. It is this interference which, when evaluated at all orders, prevents the divergence of the cross section and yields a physical result. The dominant contributions to the perturbative expansion at low QT have the form d s 2 2 2 43. ~ “4:”1n(a.:) l(%-)(%;)+~l 18 where Q2 is the square of the vector boson mass. This is known as the leading- logarithm approximation to 3%? The convergence of this series is governed by T asln2(Q2/Q§4) instead of 01,. Thus at low QT, a,ln2(Q2/Q2T) will be large even if a, is small. The logarithms in the above expression result from the infra-red and collinear singularities inherent in each addition of either a real or virtual gluon to the diagrams at each successive order. Both singularities are logarithmic and are effect- ively cut off by the total QT. In addition, the overall factor of ln(Q2/Q2T) produces a singularity at QT = 0. This divergence is formally canceled by a negative delta function at the origin. However, one can produce an arbitrarily large cross section by performing an arbitrarily small cut on QT. This unphysical result is due to the finite order of the conventional perturbative expansion. At first, it may seem that this would preclude the possibility of performing the calculation to any order, since any order would require the calculation to all orders! However, this is not the case. The coefficients v.- in the leading—log approximation are not independent and may all be expressed in terms of v1. The summation of this series removes the divergence as QT —> 0. This prescription is called “resummation”, and allows one to perform the calculation to arbitrary order. The Collins-Soper resummation formalism [8] [9], basically consists of separating the hard-gluon emission and soft-gluon emission pieces of the cross section and “re- summing” the soft non-perturbative pieces to all orders in a, while only calculating the perturbative hard piece to a given order n. The resummation is facilitated by the realization that the soft pieces of the cross section all have a similar a, and Q2 depend— ence which is raised to higher powers at each order. Thus, the sum of the soft pieces can be represented by an exponential function whose argument is called the Sudakov form factor 5 (b, Q). Here 6 is the “impact parameter” which is the Fourier transform of theQT of the interaction. Thus we see that this problem has two scales, namely QT 19 and Q. Only the result of this calculation will be presented here; the details may be found in the above references. The fully differential inclusive cross section for vector boson production and decay to lepton pairs in hadron-hadron collisions was recently published by C. Balazs, J. Qiu, and C.-P. Yuan [10]. The kinematics of the vector boson V can be expressed in terms of its mass Q, rapidity y, transverse momentum QT, and azimuthal angle qbv. The kinematics of the leptons from the vector boson decay can be described in terms of the polar angle 0 and azimuthal angle 45 in the Collins-Soper frame [2]. The resummed fully differential cross section is then given in terms of these quantities by da(A+B—+V(—+£?7)+X) _ 1 Q2 szdde§~d¢vdcos Odd) es — 967r25 (Q2 — M3,)2 + M3; Ff, 1 ., 4 - xi-2—7r—2 / dzbe'qr'bzwjubtQ,x...43,0,¢)F;‘;P(b,o,xi,xai jlc +Y(QTa Q7 $443 $3, 0145)} where ij is W.4(b.Q,x.,xB,o.¢) = epr—Sw, Q)}|ka X{[(Cja ® fa/A)($A)(Crb ‘59 fb/B)($B) + (CR. 8’ fa/A)($A)(Cjb ® fb/B)($B)l >4in + 9%)(f2‘3 + f§)(1+ 0) +[(C.. e f./A)(4:4)(C;. e mains) — (C... s Mischa.» e fb/Bxxaii X(g§. - 9%)(f2 - ram COS 9)} and <8) denotes the convolution defined by 20 d :1: (Cja ® fa/A)($A) : A: gfa/A(€A7#)Cja (fish/1) The matrix ij is the Cabbibo—Kobayashi-Masakawa matrix in the case of V = Wit or the identity matrix in the case of V = Z,'y where j represents quark flavors and If represents antiquark flavors. Summation over the dummy indices a and b which rep- resent quarks, antiquarks, or gluons is implied in the above expressions. The Sudakov form factor 3(1), Q) is given by The A and B functions and the Wilson coefficients 6344,21,?an are given in [9]. After fixing the arbitrary renormalization constants C1 = 60 = 26"” (71; is the Euler con- stant) and 02 = 1, All), B“), A(2) and 8(2) may be obtained from Eqs. (3.19) to (3.22) in [9]. If the renormalization scale [I is chosen such that ab = C3 = 26‘”, the Wilson coefficients Cg) from [9] eqs. (3.23) to (3.26) for the parity-conserving part of the resummed result are greatly simplified, and are given by 2 1 Chi) = jk {3(1 - Z) + g(712 — 815(1 — 2)} and 01(91): J 1 i2“. — Z) In addition, the same Wilson coefficients C 1(2) are found to apply to the parity-violating part of the resummed result as well [10]. The integration limits on the impact parameter b are from 0 to 00 in the expres- sion for the differential cross section. However, for b 2 bmax (where 6m“. 2 0.5 GeV"1 21 here), the QCD coupling becomes so large that perturbation theory can no longer be FNP used. Therefore the non-perturbative function is necessary and has the form 2 The Q exams)— — exp[— 22(22 )h 1(1)) - (Ii/4004, 5) - his/3W8» 5) where 12;, hj/A, and fig/B cannot be calculated perturbatively and so must be meas- ured empirically and fit. Also, W is evaluated at b... where b — ¢1+(b/bm..)2 so that 5. never exceeds 6",”. The Y-term in the differential cross section is given by Y(QT)Q3$A,$B,0 d”): 1 d_§A/ 1 _d_§B°° 37A é—a/ 1'3 {B N=0 as( H[ W] fa/A( CA; QlRlIIIRQnQazA,28,9,Plfb/B(§B;Q) where the functions 125,2” are less singular than 517x (logs or 1) as QT —-> 0. Figures T 2.5, 2.6, 2.7 show the resummed virtual photon cross section vs. mass, transverse momentum, and rapidity. 22 Resummed Drell—Yon (photon) cross section vs. moss 3: 10 r- 0 9 [- ..\ 8 ' 2’. 7 . 3 6 L E :0»- Q 5 + D u 4 . 3 .. + + 2 - ++ + ++ + ++ + I ’— ++ 0.9 *- _+_ 0.8 [- + 0,7 » '1’ + 06 [ ++ ‘ + 0.5 [ ++ 0.4 ~ ++ 0.3 *- l J A_ A_ L A A L l l I J L J I L I A A l I l 1 1 l A L A L 30 35 40 45 50 55 60 m (GeV/c’) 7' mass Figure 2.5: Resummed Drell-Yan da/dm. Resummed Drell—Yon (photon) cross section vs. transverse momentum ’>‘ 65 10 r \ l- U h 3 P ”I. V. ~ '4 o + o 3 ” *4. 8 ”I ’4. ‘4 L. on ’0 l 1 r— I... F +4 : ’ 9 . f + r- I.” >- .N *4» _ 4 HA‘ ‘ I N h o- O 9“” 7‘... —i 4 IO '— IT... : w ., . t H, » ft 9 t + D- * +4” . +44, Hut A] ALALL‘LJ LAALAJLJAAAJAAJILLLAIA lLll‘lAJ—LILL—LJIJLLA 0 5 10 15 20 25 30 35 4O 45 50 , p: (GeV/C) 7 Pv Figure 2.6: Resummed Drell-Yan da/de. 23 Resummed Drell-Yon (photon) cross section vs. rapidity E v IO '5' ‘ ~0”~‘~0“" u w . .m.’o""lhl ,. N. 6” h + T. w . MT¢ : 4 * o I» + 1 E- + +‘l’ 5 [+ + I- + + —i 10 5- I -2 10 .=— t -3 '- 10 =- _4 b 10 pl -5 10 Fl l A A A A l A A A A A A J._LA 1_A A A A l A A A A l A A A A 14 A L A l A A LA —4 -3 —2 —I O 1 2 3 4 . . . Y 7 rapidity Figure 2.7: Resummed Drell-Yan da/dy. 2.4 Kinematics The Collins—Soper 0’ reference frame [2] is the dilepton (virtual boson) rest frame defined as follows: In general, the parton momenta 1.6,, and If], are not collinear, hence the z’-axis is chosen such that it bisects the angle flab between 13,, and —I:b. The polar angle 0 is the angle between the lepton momentum l”; and the z’-axis. The azimuthal angle 45 is measured relative to the transverse unit vector (3T that lies in the (160,131,) plane and is anti-parallel to the direction of (I60 + EDT. Consequently, [77 - (IT = 6’ sin0cos <15. Since the x and y axes are not specified, this notation is covariant under rotations in the 0’ frame. An illustration of this reference frame is shown in Figure 2.8. Note that the definition of the z’-axis is somewhat arbitrary. Its orientation is chosen by assuming that on the average, the incoming partons have equal transverse momenta, which should be roughly correct when many events are averaged. However, 24 Figure 2.8: The Collins-Soper 0' reference frame. since the transverse momentum of the incoming partons is unknown, this reference frame does not coincide with the center of momentum reference frame of the partons on an event-by-event basis. The notation used here follows the notation used in [2] and is shown in Table 2.3. 2.4.1 Lowest Order Kinematics The kinematics of the lowest order Drell-Yan process are very straight-forward to calculate. In this “naive” Drell-Yan model, the transverse momenta of the incoming partons is zero, and thus the Lorentz transformation between the lab and center of momentum frames is well defined and is along the z = z’ axis. Thus the 0’ frame does coincide with the cm frame of the partons in this special case. The parton longitudinal momentum fractions are defined as 25 Table 2.3: Definition of mathematical notation. Notation lab Center of momentum frame of P: and Pb“ 0' Collins-Soper Frame P: proton beam 4-momentum in lab frame Pfr proton beam 4-momentum in 0’ frame Pb“ antiproton beam 4-momentum in lab frame Pb“, antiproton beam 4-momentum in 0’ frame k: parton from hadron a 4-momentum in lab frame hf," parton from hadron a 4-momentum in 0’ frame k: parton from hadron b 4—momentum in lab frame 145’ parton from hadron b 4—momentum in 0’ frame 2“ lepton 4-momentum in lab frame I?“ lepton 4—momentum in 0’ frame 8—" antilepton 4—momentum in lab frame 2;, antilepton 4-momentum in 0’ frame q“ = (qo, QT, 0, q3) 7* 4-momentum in lab frame q‘“ = (m,0,0,0) 7“ 4—momentum in 0’ frame VS center of mass energy of beams in lab frame m mass of 7* vT transverse component of vector 13' U; longitudinal component of vector 13' 0 angle betweerfi7 relative to z’ axis in 0’ frame Gab angle between I”; and P]: in 0’ frame xa fraction of proton momentum carried by interacting parton 2:5 fraction of antiproton momentum carried by interacting parton T unit-less “mass fraction” parameter 26 if. x : a P2 k3 35:75:: and it can be shown that m2 = :13“be (13 = {51251.21 = pom = z boost from 0’ to lab 2 q0 = [gig—329)- = energy of vector boson in lab ,8 :: $°_xb = .0 cm (30+3b) 9° _ (150+be 7cm — 24/xaxb Using the above values for 76m and 5m we can write down the Lorentz transform- ations between the lab and 0’ frames. 7cm 0 0 _7cmflcm 0 1 0 0 "vcmficm 0 O 7cm 7cm 0 0 7cmflcm 0 1 0 O A0’-+lab = 0 0 1 0 7cmflcm 0 0 7.44. We can also write down the 4-vectors of the vector boson in the lab and 0’ frames 0 q 0 0 m 0 M_ It’_ q _ a q _ 0 0 (13 and using energy and momentum conservation, we know that 5T, = —E’ e3’ = J?" 110’ = W By definition, 27 0' _ _ q—m—Q SO [0’ 26’ m 1 If we assume that the leptons are massless (this is certainly a valid approximation for electrons at FNAL) then (4032 = (44')? + (43')? = (W + (43’)? I 23 4 I 1311(0) = g—r = _ “ll? 0) so (33,12 tan2(6’) = L";- - ((3')2 which gives I m2 _ m2 tan2]9] (63 )2 = 4(i+tan2(9)) ([T’)2 — 4(1+tan (6)) so we have 63/ = _Z§’ : mcos]9] 2 [TI : _E"’ : msin(0] 2 putting m in terms of Ta, (125 and 5 yields €31 = —25’ = %mcos(fl) (T' = —7T-’ = %msin(fl) Since QT = 0 for the lowest order Drell-Yan process, the lepton angular distri- bution has no O dependence (the distribution is flat in O). Consequently, we may arbitrarily choose O = 0 since the Lorentz transformation between lab and 0’ frames leaves O invariant. Thus, using this choice of O, the lepton 4-vectors in the 0’ frame are 3‘" = %\/S:ra:rb 0 28 7‘" = 955‘" where gf,‘ is the Bjorken-Drell metric. Given the above lepton 4-vectors in the 0’ frame and the Lorentz transformation from the 0’ frame to the lab frame we can also write down the lepton 4-vectors in the lab frame in terms of the 0’ frame variables, namely (:30 + 331,) + (L, -— 3:5) cos(0) 3“ = fl 2,/:raa:6 sin(0) (11:0 — £135) + (:ra + .735) cos(0) (:ra + $1,) — ((Ba — x5) cos(0) —2\/msin(0) 0 (ma — $1,) — (37,, + $5) cos(0) "SI ll "‘ o: Other useful kinematic quantities include the lepton rapidities ye, y; and vector boson rapidity 31,- in the lab frame. The rapidity of a particle is defined as E z 31: %1n(E—1—£:) Since we now know the lepton and vector boson 4-vectors we can immediately write down their rapidities yw‘ — %ln(% 3!! = 311‘ + %ln(jT:::(g)) —cos]6] y? = 3’7. + 2- III( I+cos(9)) Given the above rapidities we see that we can neatly write the vector boson rapidity in terms of the lepton rapidities 314‘ = %(w + y?) In addition, it is evident from the above that the angle 6 in the 0' frame may be conveniently expressed in terms of the difference of the lepton rapidities Aye; = ye — y; cos(0) = tanh(Ay(;) 29 Figure 2.9: Gluon Bremsstrahlung Finally, we wish to write the parton momentum fractions we and Tb in terms of the lepton momenta in the lab frame. This can be readily accomplished using the various kinematic quantities given in the above discussion :ra = ;}§(£°+FO'+£3+F¥) T], = 71§(€°+26—€3 —€—3) we can also put To and at], in terms of the vector boson momenta are = ;%§(q° + C13) Pb = 723-01" - 4“) Similarly, the angle 0 may be expressed in terms of the lepton momenta cos(0) = g:% 2.4.2 Higher Order Kinematics The process shown in Figure 2.9 is an example of a higher order Drell-Yan process. 30 One would like to be able to calculate the initial (Ba and :13], of the incoming quarks as was done for the lowest order Drell-Yan process, but this requires knowledge of the energy carried off by the gluon. Before the gluon bremsstrahlung the quark and antiquark momenta are x9213 rag/S ké‘= 3 , ké‘= 3 x92¥3 _:rh\2/S- The quarks are assumed to have no intrinsic kT here. After the gluon bremsstrahlung, we can write the fraction of the proton momentum carried away by the gluon as _ lP 13ml _ E luon $9 ‘ IPZI+IP2I ‘ 378 Since we assume here that the gluon is the sole source of the quark and antiquark trans— verse momentum and that the quarks are massless and on shell, then if we choose the QT axis along the :1: axis (O = 0) the quark momenta are given by 'i'($a l—Q-Tglfl a“ Tng)\/§ ~ - T ~ - T 14:: 2 0 , 14;; = 2 0 Mac. — RPS - 62% 4w. — 4,)25 — Q21 where we have followed the Collins-Soper prescription of dividing the QT equally among the quark and antiquark. The momentum of the vector boson is then %(xa + $5 — 2mg)\/§ QT 0 %\/(:Ba — xglz — QT - %\/(17b — 2:9)2 _ Q” and the mass of the vector boson is given by q“: 1 1 m2 = ._(x: — Iva-739 — xbxg - $a$b)S — 5Q;a + g (x. — comm. — :4.)st + Q4 - ((4. — mg)? + (an. — 4.)?)Qas The above parameterization in terms of mg is fairly general although it was derived from the specific case of a single gluon bremsstrahlung. If we instead define mg as the fraction of energy radiated prior to the quark-antiquark annihilation, the above equations still hold true. In principle one can use the above quantities to solve for To and an, provided one knows 2:9. Unfortunately, it is very difficult to measure the energy of the initial state radiation since it is often small and the radiated particles escape down the beam pipe. Also, the above does not take into account any intrinsic transverse momentum of the quark or antiquark or final state interactions. The kinematics of the vector boson and its subsequent decay into lepton pairs are predicted by the resummed cross section given in Section 2.3 and are calculated in 31 [10]. They are given in the lab frame by [1‘ where X!1 YI‘ Z11 and qi = 715(qu q3), MT = ‘/Q2 + Q%~, n” = 715(1,0,0,1), and fi" = 715(1,0,0,—1). u %(%+X“sin6cosO+ Y“sin6’sinO+ Z“cos0) (MT cosh y, QT cos O, QT sin O, MT sinh y) M2 (4444—) ewaf’zg—“zaxfl 1( n" n”) MT (1+ 9— 32 2.5 Summary Annihilating quarks radiate gluons just as electrically charged particles radiate photons when accelerated. Gluon radiation increases as the time ( 1 / Q) available for the anni- hilation decreases. Consequently, since gluons carry away transverse momentum, the width of the QT distribution of the vector boson must increase. Hence, for an accurate comparison of theory and experiment for the QT distribution of the Drell—Yan inter- action it is necessary to take into account the effect of multiple soft-gluon emission on the QT distribution. The resummation prescription provides a theoretical means to this end. The resummation calculation is necessary to properly describe the low QT regime of the Drell-Yam process. At high QT, the standard perturbation method is adequate. An overlap region exists however, where it is necessary to match the low QT and high QT results. The energy boundaries of this overlap region depend on the Q of the interaction. One could arbitrarily choose some QT cut in this region and use one result above and the other below the cut, but the resulting relative error in this method is formally 0(cr2 1n4(1 / 013)). By properly matching the resummation calculation with the conventional perturbative result however, the relative error can be reduced to 0(03) [13]. The theoretical techniques required to perform the resummation calculation are fairly complex, but are available in the literature. Fortunately for experimentalists wishing to test the theory, C.-P. Yuan and C. Balazs have recently written a Monte Carlo event generator called RESBOS [15] which includes the resummation calcu— lation. This makes comparison between theory and experiment significantly easier since it allows one to easily make the cuts required by experimental analysis on the theoretical results. 33 Since the RESBOS Monte Carlo program has only recently been available, other Monte Carlo event generators were also used for various aspects of this analysis. The ISAJ ET [14] event generator was used extensively. ISAJ ET includes the NLO perturbative calculation but produces the low QT portion of the distribution in a more empirical manner. The basic method is to calculate the hard scattering and then “evolve backwards” by radiating quarks and gluons and adjusting the momenta of the initial state particles up to some cutoff supplied by the user. The choice of this cutoff makes a noticeable effect on the Drell-Yan QT distribution. A comparison between the RESBOS resummed QT distribution and the ISAJET result is shown in 2.10. The QT distributions from RESBOS and ISAJ ET are clearly different, however the integrated cross sections from both Monte Carlos agree to within a few percent. The parton distributions used in the RESBOS MC are the CTEQ3M distributions; for ISAJET, the CTEQZL distributions were used (the CTEQ3 parton distributions are not yet available in ISAJ ET). However, the parton distribution differences are not the source of the large differences between the QT spectra of these two Monte Carlos, rather it is the empirical manner in which ISAJ ET generates the QT distribution. It may be possible to tune the ISAJET parameters which control the QT distribution to reproduce the RESBOS result, but this has not been done. Thus it is preferable to use the RESBOS Monte Carlo where possible since it uses the full resummation formalism to produce the QT distribution. 34 do/dp, (pb c/GeV) O RESBOS vs. ISAJET Drell—Yon Monte Carlo p, distributions pT distributions _ —A— —- _A_ l— L -A- _ -¢- 0 —> RESBOS _ -¢-- A -—> ISAJET _ «p. _ . P- -.-- -A-. . L .- rA- -A-.i . P—‘o —A— -... "-A- '0" n— _. _ _._.. . A A— .-._, . -A-_A_ 4; . A‘—_A_ '.'*." i— _A__ __ —-—X--. l l l l l l l I l l l L] l l l l l l l l l l i l l l l 1 l J I l l l 1 L4; 0 2 4 6 8 10 12 14 16 20 p,(GeV/c) Figure 2.10: Comparison of RESBOS and ISAJ ET pT spectra. Chapter 3 APPARATUS 3. 1 The Accelerator The accelerator facility at Fermi National Accelerator Laboratory is currently the highest energy accelerator in the world, capable of colliding protons (p’s) on anti- protons (fi’s) with a center of mass energy (fl) of 1.8 trillion electron volts (1.8 TeV). It is comprised of several stages: 0 The Cockcroft-Walton. o The Linac. The Booster. 0 The Main Ring. The Tevatron. The Antiproton Storage Ring. The accelerator is capable of operating in two modes, fixed target mode and collider mode. D0 has used both modes: fixed target mode for test beam studies of our detector components in the NWA (Neutrino West A) experimental hall and collider 35 36 H \ TARGET HALL .. \‘r; snags?“ 1‘ <=' "" \6 aooern 'e‘ -‘ "- 0’ UNAO '2‘;:“..; ‘——- ' . F COCKROFr-WALTON Figure 3.1: The Fermilab Tevatron Collider. .\ 1' V mode for actual physics data taking. A schematic of the accelerator systems can be seen in Figure 3.1. The operation of the accelerator in fixed target mode is as follows. Electrons are added to hydrogen atoms to make negative hydrogen ions and accelerated to an energy of 750 thousand electron volts (750 keV) in the Cockcroft-Walton accelerator. The negatively charged hydrogen ions are then injected into a 500 foot long linear accelerator called the Linac. Here an alternating electric field is applied to nine drift tubes which are spaced further and further apart as the ions travel down the Linac. The fields are varied such that the ions are hidden in the drift tubes when the field is in a direction that would slow them down and emerge into the gaps between the tubes when the field is in the proper direction to accelerate them. The Linac accelerates the ions to an energy of 400 million electron volts (400 MeV). After leaving the Linac the hydrogen ions pass through a carbon foil which strips off the electrons leaving a bare positively charged proton. 37 The 400 MeV protons are then injected into the Booster. The Booster is a rapidly cycling synchrotron 500 feet in diameter. In it, the protons are accelerated by electric fields many times while being forced to travel in a circular path by a magnetic field. The magnetic field is ramped up to maintain the protons’ orbit within the Booster since as the protons gain energy from the electric fields they require a stronger magnetic field keep them contained in the Booster. The protons travel around the Booster approximately 20,000 times which accelerates them to an energy of 8 billion electron volts (8 GeV). The Booster typically cycles twelve times in rapid succession injecting twelve bunches of protons into the Main Ring for the next stage of acceleration. The Main Ring, like the Booster, is also a synchrotron, but is approximately 4 miles in circumference. A ten foot diameter tunnel buried 20 feet below the Illinois prairie west of Chicago houses the 1,000 conventional copper coil dipole magnets that make up the Main Ring. The p’s travel through the main ring and are accelerated by radio frequency cavities (RF cavities) as the magnetic fields are ramped up to maintain the orbit. The RF cavities contain RF electromagnetic (EM) fields which are synchronized such that when the proton bunch enters a cavity, the EM pulse builds up behind it and the protons “surf” on the EM wave. The Main Ring accelerates the protons to 150 GeV. The 150 GeV protons are then extracted from the Main Ring and injected into the Tevatron. The Tevatron is housed in the same underground tunnel that holds the Main Ring and is the same diameter, but it is made up of 1,000 superconducting dipoles. The Tevatron resides directly underneath the Main Ring and gets its name from its ability to accelerate protons to nearly 1 TeV. There are also quadrupole magnets in the Tevatron and Main Ring which focus the beam to maintain the protons bunches’ transverse dimensions. The superconducting magnets must be cooled to about -450 F in order to operate, which requires a vast cryogenic system. If a superconducting -~'~.--- 38 magnet drops out of the superconducting phase while in operation, the large currents necessary to create the 2 Tesla fields that steer the proton bunches around the ring create an immense amount of heat. Dumping all this heat into liquid helium causes it to immediately turn to gas which must be vented. This is known as a quench, and makes a very loud whoosh if one is standing near one of the vents when it occurs. Thankfully it does not occur very often. Under normal operation the protons are accelerated to 900 GeV in the Tevatron. The Tevatron is the last stage of acceleration for the protons. During fixed target operation the 900 GeV protons are extracted from the Tevatron via a switch-yard that steers the bunches down the various experimental beam—lines. These beam—lines contain additional transfer apparatus such as dipoles and quadrupoles, as well as other elements such as targets and mass selectors, to create secondary and even tertiary beams of electrons, pions, muons, neutrinos, etc. . . which the experimentalists use to perform various physics experiments or for calibration and testing purposes. Collider mode operation at Fermilab is the same as fixed target mode up to the Tevatron stage. Running in collider mode requires a source of antiprotons which are created by extracting 120 GeV protons from the Main Ring and slamming them into a target to create antiprotons in the same fashion that other desired particles are created in the experimental beam—lines in fixed target mode. The antiprotons produced are then collected and injected into the Debuncher ring where they are reduced in size by a method known as stochastic cooling. The antiprotons are then transferred to the Ac- cumulator for storage. The combination of Debuncher and Accumulator make up the Antiproton Storage Rings. Accumulating the antiprotons is known as “stacking” and when the antiproton stack is large enough, six bunches of antiprotons are accelerated via the Main Ring and Tevatron to 900 GeV. The protons and antiprotons circulate in opposite directions in the Main Ring and Tevatron due to their opposite charges. 39 The antiproton injection phase of colliding mode operation is the most critical since it takes many hours to accumulate the antiprotons and if they are lost much time and money is wasted. Approximately 107 13’s can be produced from each batch of 1.8 x 1012 9 ps. In collider mode operation, six bunches of protons and anti-protons circulate around the ring simultaneously. The particle bunches are focused into head-on colli- sions at interaction regions which are surrounded by detectors that measure properties of the particles produced in the collisions. One advantage of a collider is that much higher center of momentum energies are obtainable than in a fixed target accelerator. At the Tevatron, each particle beam (proton and antiproton) is a 900 GeV beam, giving a center of mass energy of \/§ = 1.8 TeV. If instead, a 900 GeV 13 beam were incident on a fixed p target, the center of mass energy would be only 42 GeV. Thus the energy available for producing new particles in fixed target mode is much lower than in collider mode. Another advantage of a p13 collider is that if the p and f) bunches can be kept separated, the same accelerator can be used to accelerate both types of particles simultaneously, thus avoiding the need for a separate accelerator for each type of particle. This method reduces the overall cost of construction and operation of the collider and is in fact what is used at Fermilab. The the number of interactions of a given type that can be produced in a given time is directly proportional to the luminosity of an accelerator, where the constant of proportionality is the cross 'section of the given interaction. N; = 0'; f Ldt Thus for a fixed length data run, the luminosity of the accelerator determines how many reactions of a given type will occur since the cross section for the reactions is 40 fixed by nature (and is often what is to be determined). The instantaneous luminosity of a p15 collider is given by the formula NN L=fn-"z*"- where f is the revolution frequency, n is the number of proton (antiproton) bunches in the collider, A is the cross-sectional area of the beams, Np is the number of protons per bunch and N5 is the number of antiprotons per bunch. The Fermilab Accelerator Division is responsible for the optimization of these variables in order to provide the highest possible luminosity. The instantaneous luminosity record for run 1A was L z 1 x 1031cm’2s‘1. There are four interaction regions available: B0, C0, D0, and E0, two of which are currently in use. B0 is home to CDF (Collider Detector Facility at Fermilab) and D0 is home to the D0 Detector. 3.2 The D0 Detector 3.2.1 Overview The DO detector [16] is a general purpose detector. The design goals were to provide excellent calorimetric energy and position resolution, good electron and muon iden- tification, good measurement of parton jets, and good missing ET and scalar ET measurement. The primary physics goals of the DO experiment are to study high mass states and high p1 phenomena. The design of the experiment was based on the fact that new phenomena usually have relatively large branching ratios into leptons whereas the background processes do not. Also parton jets are generally of greater interest in studying the underlying physics processes than are the individual hadrons of which they are comprised. A cut-away isometric view of the DO detector is shown in Figure 3.2. 41 D¢ Detector Figure 3.2: An isometric cut—away view of the D0 detector. 42 The D0 detector consists of three major detector components: 0 A highly hermetic, finely segmented calorimeter constructed of depleted uranium and liquid argon with unit gain, which is thick and radiation hard. 0 A compact tracking system which has fairly good spatial resolution and no central magnetic field. 0 Muon detectors surrounding a thick magnetized iron toroid which allow adequate momentum measurement while minimizing backgrounds from hadron punch- through. In addition, a programmable, high performance triggering and data acquisition system provides a means of reducing the overall event rate by selecting the most interesting events; detecting beam crossings and monitoring the luminosity at D0; and providing facilities for writing the selected event data to magnetic tape. The DO coordinate system is a right-handed coordinate system with the positive z-axis pointing the proton direction and the positive y-axis pointing upward (away from the center of the earth). The angles 45 and 0 are the azimuthal and polar angles respectively with 0 = 0 along the proton direction. The cylindrical r-coordinate is the perpendicular distance from the z-axis (beams). The pseudo—rapidity, n = -ln(tan(0/2)), is approximately equal to the rapidity y = %In((E + pz)/(E — pz)), in the limit (m/E) -> 0. 3.2.2 The Central Detector (CD) The DC central detector is made up of the tracking detectors and the transition radiation detector. The separate detectors are, moving radially outward: (i) the vertex tracking chamber (VTX), (ii) the transition radiation detector (TRD), (iii) the central 43 drift chamber (CDC), and (iv) two forward drift chambers (FDC) which cap the CD on either end. The VTX, TRD, and CDC detectors cover the large angle region of roughly —1.2 S 17 S 1.2, and are oriented parallel to the beam-line. The FDCs are ori- ented perpendicular to the beam. The volume of the CD suite of detectors is bounded by r = 78 cm and z = i135 cm and is surrounded by the calorimeters. The trans- ition between the VTX-TRD-CDC cylinder and the FDC detectors is matched to the transition between the central and end cap calorimeters. The FDC detectors cover the small angle regions of approximately 1.5 S In] S 2.5. Due to the absence of a central magnetic field in D0 , the primary design goals for the CD were resolution of closely spaced tracks, high tracking finding efficiency, and good ionization energy measurement to allow differentiation between single charged particles and photon conversions. The purpose of the TRD was to allow further discrimination between charged hadrons and electrons. The size of the CD drift cells were chosen so that the drift time matched the Tevatron bunch crossing time interval of 3.5 us. A flash analog—to—digital conversion (FADC) system is used for signal digitization with a charge sampling time interval of ~ 10 us. This provides for good two track resolving power and gives an effective detector segmentation of 100-350 mm. The vertex 2 position is measured in D0 using a combination of methods in the CD detectors. These include charge division in the VTX, helical cathode pads in the TRD, and delay lines in the CDC and FDC. The Vertex Drift Chambers (VTX) The innermost tracking detector in D0 is the vertex chamber [18, 20]. The inner radius is 3.7 cm and the outer radius is 16.2 cm. It is comprised of three concentric, mechanically independent cell layers made of carbon fiber tubes. Eight sense wires 44 measure the r — <15 coordinate in each cell. The innermost cell layer consists of 16 cells and the outer two layers are made up of 32 cells each. The carbon fiber tubes whose volumes define the gas volumes have 1 mil thick Al traces on a multi-layer epoxy / Kapton laminate on their surfaces (carbon fiber tube at ground) which provide coarse field shaping for the cells. A coat of resistive epoxy covering the traces prevents charge buildup. The cells are defined by grid of field shaping wires held at ground on either side of the sense wires and which line up with the coarse field shaping traces. Together with the field shaping wires, planes of cathode wires provide a uniform drift field region. Left-right ambiguities were resolved by staggering adjacent wires by :l: 100 pm in each cell. The three radially adjacent cells are offset in (f) to help in pattern recognition and calibration. The sense wires are made of 25 pm NiCoTin [21] at 80 g tension and are read out at both ends to measure the z coordinate of a hit via charge division. The resistivity of the sense wires is 1.8 k9 / m. The grid and cathode wires are made of 152 ,um gold- plated aluminum at a. tension of 360 g. A more detailed description of the electrostatic properties of the VTX may be found in [17]. To obtain good spatial resolution and track pair resolving power, the gas mixture chosen for the VTX was 95% C02 plus 5% ethane at 1 atm with a small admixture of H20. The average drift velocity under normal DO operating conditions (< E >m 1 kV/cm) is about 7.3 pm/ns. Gas gain at the sense wires is about 4 x 10“. An addition of 0.5% H20 to the gas helps stabilize VTX operation in a high radiation environment. 45 The Transition Radiation Detector (TRD) Highly relativistic particles (7 > 103) produce X-ray transition radiation when cross- ing boundaries between materials with differing dielectric constants [22]. The amount of energy produced by these particles depends on the Lorentz 7 which provides a means to discriminate between electrons and other heavier charged particles such as pions. The DC TRD is made up of three separate units, each containing a radiator and a detection chamber. The radiator section of each unit is composed of 393 layers of 18 pm thick polypropylene foil in a volume filled with nitrogen gas. The mean distance between the foil layers is 150 pm with a variation of about 150 pm. The gaps between foil layers are produced by a pattern embossed on the foil. The foil is wrapped around a cylindrical support to produce the gaps. The energy spectrum of the X-rays produced is determined by the thickness of the radiator foil and the gaps. The DC TRD X-rays have a distribution which is peaked at 8 keV with most X-rays having an energy less than 30 keV [23]. The transition radiation X-rays are detected in a two—stage time-expansion radial- drift proportional wire chamber (PWC) located behind each radiator unit. The X-rays typically convert in the first stage of the PWC and the charge drifts radially outward to the sense wires where amplification occurs. The radiator stack and PWC sections of each TRD unit are separated by a pair of 23 pm mylar windows separated by a distance of 2 mm. The outer mylar window is aluminized and serves as a high voltage cathode for the conversion stage of the PWC. Dry C02 gas flows between the mylar windows to prevent the nitrogen gas in the radiator stack from leaking into the PWC and contaminating the 91% Xe, 7% CH4, and 2% C2H4 gas mixture circulating therein. The cylindrical shape of the mylar windows is maintained by a small pressure 46 difference between the radiator, gap, and detector volumes. In addition to the charge produced by the transition radiation, all charged particles which pass through the conversion and amplification gaps produce ionization. The charge clusters produced arrive at the sense Wires over the full 0.6 ps drift interval. Thus, both the magnitude of charge produced and the arrival time of the charge are useful in differentiating between electrons and charged hadrons. The outer support cylinder for each TRD unit is a 1.1 cm thick plastic honeycomb covered by fiberglass skins. Kevlar end rings support the cathode structures. The radiator stack is enclosed by a carbon-fiber tube with end flanges made of Rohacell with carbon-fiber skins. The 15 mm conversion stage and 8 mm amplification stage of the PWC section of each TRD unit are separated by a cathode grid of 70 pm gold-plated tungsten wires. The outer cathode of the amplification stage of each PWC section are constructed of helical copper strips deposited on Kapton foil. The amplification stage anodes of each PWC section are 30 pm gold-plated tungsten wires separated by 100 ,um gold-plated beryllium/ copper potential wires. Each TRD unit has 256 anode readout channels and 256 helical cathode strips with pitch angles between 24 and 46 degrees. The Central Drift Chamber (CDC) Beyond the TRD are the four cylindrical, concentric layers of the CDC [19]. The CDC provides coverage for large angle tracks. The CDC is a cylindrical annulus 184 cm in length with inner and outer radii of 49.5 and 74.5 cm respectively. The CDC is made up of four concentric rings of 32 azimuthal cells each. The high voltage for each cell is individually instrumented to allow it to be turned off remotely. . Each CDC cell contains 7 sense wires made of 30 ,um gold—plated tungsten which 47 are read out at one end and two delay lines which are read out on both ends. The delay lines are situated one on either side of the sense wires. The sense wires are staggered in <15 by $200 pm to remove left-right ambiguities. Radially alternate cells are offset by one-half cell to further aid in pattern recognition. The maximum drift distance is about 7 cm. The delay lines consist of a wire coil wound around a carbon fiber epoxy core. The delay line propagation velocity is about 2.35 mm/ns with a delay to rise time ratio of about 32:1. A pair of potential wires is situated between each anode sense wire with an additional grounded potential wire between the outermost sense wires and the other sense wires to minimize the signal induced on the delay lines by the inner sense wires. The z-coordinate of a hit is determined via the delay lines by measuring the arrival time of the pulse at each end of the delay line. The current is monitored on the grounded potential wires to generate a voltage trip if abnormal conditions arise. The CDC is constructed of 32 identical modules. Each module is made of 4 Rohacell “shelves” covered with epoxy-coated Kevlar cloth and wrapped with two layers of 50 pm Kapton tape. Each shelf contains grooves at the sense wire locations to accommodate a Teflon tube containing a delay line. Field shaping is accomplished by resistive strips screen-printed onto the cathode surfaces. The gas mixture used in the CDC is 92.5% Ar, 4% CH4, 3% C02, and 0.5% H20. The CDC is stable for collected charges on the anode wires of up to 0.35 C/m. The drift velocity in the CDC is about 34 pm/ns for a drift field of 620 V/m in the region where dvdm’ft/dE is negative. The voltage on the inner sense wires is 1.45 kV. The outer sense wire voltage is raised to 1.58 kV to induce larger delay line signals. The gas gain for the inner sense wires is 2 x 104 while the outer sense wire gas gain is 6 X 104. A single layer scintillating fiber detector was installed between the CDC and the central calorimeter which covers about 1 / 32 of the full azimuth. The 128 individual 48 1 mm diameter fibers are aligned parallel to the beam and are read out with a multi- anode photomultiplier tube. This detector is used in conjunction with the CDC drift time to better understand the drift time vs. distance relationship and to quickly de— termine the CDC calibration constants if the operating conditions are changed. The Forward Drift Chambers (FDC) The two FDCs [19] cap the concentric VTX-TRD—CDC cylinders on either end and provide detection of small angle tracks. The F DCs’ inner radius is r S 61 cm which is somewhat larger than that of the VTX chamber to allow passage of cables from the large angle tracking detectors. Each FDC detector is composed of three separate modules: A (I) module to meas- ure the 43 coordinate sandwiched between two 9 modules (which are rotated relative to each other by 45 degrees in 45) to measure the 0 coordinate. The (I) module is con- structed of 36 sectors which contain 16 anode wires each along the z—coordinate. Each 9 module consists of 4 mechanically separate quadrants each containing 6 rectangular cells at increasing radii which contain 8 anode wires along the z-coordinate. The sense wires of the three inner cells are at one edge of the cell so that the ionization elec- trons drift in a single direction to remove left-right ambiguity. Each 9 cell contains one delay line which is identical to the CDC delay lines to measure the orthogonal coordinate. The adjacent anode wires in both the O and (I) chambers are staggered by i200 pm to resolve ambiguities. The (I) chamber electrostatic properties are formed by a single grounded guard wire between anodes. The cell walls are covered with 25 pm aluminum strips on 125 pm G-10 to provide field shaping. The front and back surfaces are Kevlar-coated Nomex honeycomb with copper traces on Kapton. The electrostatics of the O modules are formed from two grounded guard wires between adjacent anodes as in the CDC. The 49 front and back surfaces are Kevlar-coated Rohacell with copper traces on Kapton for field shaping. The side walls are 200 pm aluminum foil on Nonex honeycomb. The FDCs employ the same gas as that used in the CDC and have similar gas gain and drift fields. The maximum drift time at the full radius of the (I) chamber is 1.5 ps. The Central Detector Electronics The readout electronics are almost identical for all CD devices. They consist of three signal processing stages: the preamplifiers, the shapers, and the flash ADC digitizers. The preamplifiers for the sense wires, TRD cathode strips, and CDC / FDC delay lines are based on the Fujitsu MB43458 quad common base amplifier [24]. The CD requires 6080 readout channels. The preamplifier gain is 0.3 mV/fC. Rise and fall times are 5 and 34 ns respectively. Input noise is 2300 electrons for a detector input capacitance of 10 pF. Calibration is accomplished via test pulse charge injection into the preamplifier inputs. The preamplifier output signals travel over 15 m coaxial cables to the shaping circuits [25]. The shaper consists of a video amplifier, a two-zero three—pole shaping circuit, and a cable driver. The shaper output signals travel over 45 m coaxial cables to the FADC digitizers. Gain corrections and voltage offsets occur in an analog buffer amplifier circuit [26]. The dynamic range is increased by using one of two different gains depending upon the amplitude of the signal which results in an expansion of the dynamic range by about a factor of 3. This improves the dE/da: measurement quite a bit. The gain corrected signals then enter the FADC section which is based on SONY CX20116 8-bit FADC which operates at 106 MHz. The digitized data are then stored in a FIFO until a pass/fail decision is made by the Level 1 and Level 1.5 trigger. 50 Due to the long drift times relative to the FADC sampling rate and the high Level 1 output bandwidth, zero suppression of the CD signals is required in order not to exceed the capabilities of the data paths. Zero suppression is accomplished in the last digitization stage via an ASIC designed at F NAL [27] and manufactured by Intel, Inc. The zero suppression circuit examines the sequence of digitized charges and adjacent FADC bucket charge differences. Operating at 26.5 MHz on 4 byte words, it is able to process the data in real time. Digitized data are saved between leading and trailing signal edges where leading and trailing edges are defined by one of several algorithms based on digitized charges or charge differences over threshold [28]. 3.2.3 The Calorimeters The DO calorimeters are the most important D0 detector component for electron, positron and photon detection. In addition to providing the only energy measurement of electrons and positrons (since DC has no central magnetic field), they also provide the majority of the quantities used in electron and photon identification. They also are important for energy measurement and identification of jets and muons and for meas— uring the scalar ET and missing ET. ET is the transverse energy of a cluster defined as ET = m where E; and E,, are gotten by multiplying the cluster energy by the direction cosines of the cluster position with the :1: and y—axes respectively. The DC calorimeters use liquid argon (LAr) as the active medium to sample the ionization produced by electromagnetic and hadronic showers. LAr was chosen for its unit gain (low electro—negativity), simplicity of calibration, radiation hardness, and flexibility in segmenting the calorimeter. The downsides to using LAr are a complicated cryogenic system, uninstrumented regions due to the bulk of the cryostats, and inability to access the calorimeter modules during operation. The calorimeter layout can be seen in Figure 3.3. 51 Dy! LIQUID ARGON CALORIMETER END CALORIMETER Outer Hadronic (Coarse) Middle Hadronic (Fine & Coarse) CENTRAL CALORIMETER Electromagnetic Inner Hadronic , Fine Hadronic (Fine & Coarse) Coarse Hadronic Electromagnetic Figure 3.3: The D0 calorimeters. The various parts of the calorimeters are labeled on the figure. 52 The three DO calorimeters, North End Calorimeter (ECN), Central Calorimeter (CC), and South End Calorimeter (ECS), each reside in separate cryostats in order to provide access to the central detectors which they surround. The rapidity coverage is roughly —1.0 S 17 _<_ 1.0 for the CC. The end calorimeters (EC) extend the coverage to [77] z 4.0. The gaps between the EC and CC are roughly perpendicular to the beam which reduces the missing ET degradation relative to having the ECs nested within the CC shell with gaps parallel to the beams. The DO calorimeters are pseudo—projective; the separate modules are arranged in order to simulate a projective geometry as shown in Figure 3.4. The centers of the cells at increasing depth lie on rays projecting from the center of the interaction region, but cell boundaries are perpendicular to the absorber plates. The Tevatron beam pipe passes through the EC cryostats at the center. The main ring beam pipe passes through all three cryostats near the outer radius. Bellows are used to accommodate the thermal and differential pressure motion of the cryostats to which they are welded. The CC and EC each contain three different types of modules: the electromagnetic (EM), fine hadronic, and coarse hadronic arranged as shown in Figure 3.3. The EM sections use nearly pure depleted uranium [29] absorber plates, the CCEM plates are 3 mm thick and the ECEM plates are 4 mm thick. The fine hadronic sections use uranium(98%)-niobium(2%) alloy [29] absorber plates with a thickness of 6 mm. The coarse hadronic absorber plates are 46.5 mm thick and are made of copper in the CC and stainless steel in the EC. Electrical connections to the absorber plates were made by percussive welding of thick niobium wires to the edges of the plates. The EM sections of the CC and EC are made of four depth layers. The first two layers (EMl and EM2) help differentiate between neutral pions (which usually decay 53 ; , saws“ \\;.V ms: sv ;, . 8 // Tr . \‘s‘f‘S' . 0 ‘ 1,7147%4/4/6/1' ‘ 2 , .4 ‘ ]. ll-i: ' 6 Ii 5 I l . s! i: s ' ""ifimillgi ! El: :0 mmrmezzéé‘iég: E 325555 '2 Mwa «6”?» ' «as -7 1., 3;; ~ , 5 Figure 3.4: A side view of the DO calorimeter towers showing the pseudo- projective geometry. into two photons) and single photons due to the greater conversion probability of the pair of photons. The region of EM shower maximum is covered by the third (EM3) layer, which has finer transverse readout segmentation than the other EM layers. The fourth EM layer (EM4) completes the EM section. The fine hadronic modules are ganged into three or four layers and the coarse hadronic modules are ganged into one or three layers. The depth of the EM plus hadronic layers is 7.2 nuclear absorption lengths (AA) at 17 = 0 in the CC and 10.3 AA at the smallest angles in the EC. The transverse size of the readout cells was chosen to be comparable to the size of EM showers. The typical transverse size of the EM and hadronic cells is An = 0.1 and Aqb = 27r/64 a; 0.1. The EM3 layer’s segmentation is twice as fine (A7} = 0.05 and A¢> = 27r/128 z 0.05) to improve the measurement of the EM shower centroid. Inter—module gangings are made prior to signal digitization in the front end electronics to join segments of cells which cross the CC and EC boundaries. 54 The calorimeter signal boards (excluding the ECEM and small angle EC hadronic) were constructed by laminating two 0.5 mm G-10 [30] sheets together. One of the G—10 sheets is copper clad with the segmented readout pattern milled into it. The other plain G-10 sheet covers the copper readout pattern. The outer surfaces of the G-10 laminate were coated with high resistivity carbon loaded epoxy [31]. Several signal boards at approximately the same 17 and qf) are ganged together in depth to form a readout cell. Differences in the signal ganging cause the readout cells to vary from module to module. The signal board ganging connections are made using insulation displacement connectors and solid wires which has been very reliable. The signal boards for the ECEM and two smallest angle EC hadronic modules were made from multi-layer printed circuit boards. The outer surfaces of these signal boards were coated with the same epoxy as the other signal boards. The segmentation is produced by etched patterns on the interior surfaces. Signal traces on another interior surface bring the signals to the outer edges of the boards. The signal and trace layers are connected by plated-through holes. These signal boards are ganged together in depth via solder—tail header connectors and Kapton printed circuit lines. The electric field in a typical calorimeter cell is created by grounding the absorber plate and connecting the resistive coat of the signal board to a positive high voltage of 2-2.5 kV. The electron drift time across the 2.3 mm gap is approximately 450 ns. The gap thickness is chosen to be large enough to measure minimum ionizing particles. A schematic of a typical calorimeter unit cell is shown in Figure 3.5 The Central Calorimeter (CC) The central calorimeter is comprised of three concentric cylindrical shells. The inner shell is the EM section which is made up of 32 separate modules. The middle shell is the fine hadronic (FH) section which contains 16 separate modules. The outer shell U1 U1 ( Absorber Plate Pad Resistive Coat Liquid Argon \ N 610 Insulator J, / Gap 7 T ’/ _ _ // f k // 1% __ .2 _L I k <—-——- Unit Cell ——> ) Figure 3.5: Calorimeter unit cell schematic. is the coarse hadronic (CH) also made up of 16 modules. The three shells are rotated relative to one another so that particles encounter no more than one intermodule qb gap. The CC modules are made by loosely stacking alternating absorber plates and readout boards in a stainless steel box structure. Delrin spacers pass through holes in the readout boards and extend 2.3 mm on one side and 1.5 mm on the other side. Adjacent spacers have opposite long and short sides to provide room for the signal boards to flex. The CCEM calorimeter is longitudinally segmented into four layers by gauging the readout signals. The first two layers (EMl and EM2) are each 2 X0 thick. The third CCEM layer (EM3) is 6.8 X0. The fourth CCEM layer (EM4) is 9.8 X0. The CCFH is longitudinally segmented into three layers (FHl, FH2, FH3) of thickness 1.3, 1.0, and 0.9 AA. The CCCH modules comprise a single longitudinal segment of thickness 3.2 AA. The total weight of the CC modules and their support structure is 305 metric tons with an additional weight of LAr of 26 metric tons. 56 The End Calorimeters The two end calorimeters (ECN and ECS) each contain four types of modules. The ECs each contain one EM module and one inner hadronic (IH) module to avoid the gaps that would be produced by multiple modules. The absorber plates and readout boards for these modules form disks with no azimuthal gaps. The ECEM is longitud- inally segmented into four layers of thickness 0.3, 2.6, 7.9, and 9.3 X0 (EM1, EM2, EM3, EM4). The cryostat wall increases the thickness of EM1 to about 2 X0. The alternating absorber plates and readout boards are stacked on either side of a stainless steel support whose thickness in radiation lengths is equivalent to a uranium plate. Tie rods and spacers position the plates and readout boards. The ECIH module is longitudinally segmented into four fine hadronic sections each 1.1 AA thick built of uranium absorber plates and one coarse hadronic section 4.1 AA thick made of stainless steel absorber plates. The construction of the ECIH is similar to the ECEM. Outside the ECEM and ECIH modules are concentric rings of the middle hadronic (MH) and outer hadronic (OH) modules. The MH and OH modules are offset relative to each other to prevent particles from penetrating through the azimuthal gaps between adjacent modules. The ECMH modules are longitudinally segmented into four fine hadronic sections and one coarse hadronic section. The fine hadronic sections use uranium absorber plates and are each 0.9 AA thick. The coarse hadronic section is 4.4 AA thick and has absorber plates made of stainless steel. The ECOH modules use stainless steel absorber plates which are at an angle of about 60 degrees relative to the beam. The total weight of the EC calorimeter is approximately 238 metric tons. 57 The Intercryostat Detectors And Massless Gaps In the transition region between the central and end calorimeters (roughly 0.8 S [77] S 1.4) there exists a large amount of uninstrumented material in the form of cryostat walls and support structures. In an attempt to measure the energy deposited in this region a pair of scintillation counter arrays called the intercryostat detector (ICD) was installed on the front surface of the ECs. Each ICD is made up of 384 scintillator tiles of size 517 = dd) 2 0.1. Inside the CC and EC cryostats, additional single cell structures called massless gaps were installed. One ring with standard segmentation was mounted on the CCFH end plates and additional rings were mounted on the front plates of the ECMH and ECOH. These two detectors provide a fair approximation of the standard DO sampling of hadronic showers. The ICD readout uses 1.3 cm diameter phototubes [32] which were extensively tested to prevent failures. Calorimeter Electronics Calorimeter signals are read out via insulated 30 Q coaxial Tefzel [33] cables connected to multi-layer printed circuit feed-through boards which pass through four cryostat feed-through ports located above the liquid argon level. The eight 27-layer T—shaped feed-through boards reorder the signals from the module oriented inputs to an output 77-4) ordering to facilitate easier analysis. The feed-through board outputs travel over short cables to charge—sensitive hybrid preamplifiers [34]-[37] mounted on the outer cryostat surfaces. A single 23K147 Toshiba j-F ET with 9M 0059’1 is used at each preamplifier input. Two different preamplifiers are used with equivalent full scale outputs of 100 and 200 GeV to provide full dynamic range response. The gain variation among the preamps is about 0.5%. The preamplifier output signals travel over 30 m twisted pair cables to baseline 58 subtracter (BLS) shaping and sampleand-hold hybrid circuits. The input signals are integrated (RC = 433 ns) and differentiated (RC = 33 ,us). Readout cells which cross the CC-EC boundary are merged at the BLS input. A trigger pickoff at the BLS input extracts a portion of the signal with a rise time of about 100 ns which is added into trigger towers of An = Aqb = 0.2 for use in event selection. The calorimeter signals are sampled just before a bunch crossing and 2.2 ps after. The difference between the two samples is proportional to the collected charge. Two storage capacitors for each channel provide analog double buffering. To prevent event pile-up, fast baseline restoration occurs within a few ps. The BLS output is amplified by 1 or 8 (depending upon the signal size) to reduce the dynamic range required in the ADCs. A bit records which gain was used. A specific gain may be chosen for calibration purposes. The BLS outputs are multiplexed 16—fold onto the crate backplane and travel over 50 m twisted pair cables in serial time slices to the 24-channel 12—bit ADCs [34]-[37] in the moving counting house (MCH). The ADCs combined with the variable gain (X1 or X8) provide a dynamic range of 215. Each time slice of each channel is digitized in about 10 us yielding a total digitization time of 160 us for 384 signals. The gain parameters are set so that an energy deposition of about 3.75 MeV corresponds to at least one count. A minimum ionizing particle deposits between 8 (EM1) and 90 (FHl) MeV including noise. An adjustable threshold allows channels with (signal-pedestal) below the threshold to be suppressed from the readout buffer (zero suppression). Both single-channel random noise (electronic noise and uranium radioactivity) and multi-channel coherent noise have been measured. The single-channel noise can be represented by 2000 + 3000 x C (nF) electrons. For N channels (N large), the total random noise varies as \/N. The total coherent noise varies as N. The point at which 59 the coherent noise becomes larger than the random noise is on the order of 5000 channels. Calibration of the DO calorimeter electronics is accomplished by a precision pulser [38] which injects charge into the front ends of the preamplifiers via a large resistor. A pulse distribution system delivers equal pulses to each input. A programmable attenuator allows calibration over the full dynamic range. Precision and stability of the calibration system have been measured to be better than 0.25%. 3.2.4 The Muon Detectors The DO muon detection system consists of five iron toroidal magnets surrounded by proportional drift tubes (PDTs) to measure the direction of muon tracks (and thus their momentum) down to an angle of 3 degrees from the beam. Since most hadronic particles comprising parton jets are stopped inside the calorimeter, it is possible to detect muons within the jets much more easily than electrons in jets. The magnetic field direction produced in the toroids is approximately along the qS—coordinate which results in the muon tracks bending roughly in the r-z plane. Given the width of the interaction region at D0 (02 z 30 cm), it is necessary to measure the muon track direction both before and after the bend. The entrance point of the muon track into the toroid is determined by a closely spaced set of measurements before the toroid and the track direction after leaving the toroid is gotten by a set of measurements separated by 1-3 m. The direction of the track before the toroid is determined by combining the primary event vertex, the track measured by the central detector and the first muon chamber track vector (before the toroid) . Muon energy deposition can also be seen in the calorimeter. From the incident and exit tracks on either side of the toroid, the bend angle may be found and used to calculate the momentum. Multiple Coulomb scattering in the toroid limits the relative momentum resolution to greater 60 than 18%. A measurement of the sign of the muon is possible for PT S 200 GeV/c at 17 = 0 and PT S 30 at [77] = 3.3 with 99 The central toroid (CF) spans the region [17] S 1 and the two end toroids (EF) cover the region 1 < [17] S 2.5. The two small angle muon system (SAMUS) toroids fit within the central hole of the EF toroids and cover the region 2.5 < In] S 3.6. The layout of the muon system can be seen in Figure 3.2. The muon system toroids are very thick and thus provide a very clean environment (at least from particles originating from the interaction, cosmic rays and beam loss must be removed by other means) for muon identification. The muon system is fairly hermetic apart from the gaps caused by the CF-EF transition, various support structures, and detector access requirements. The minimum muon momentum necessary to pass through the toroids varies from about 3.5 GeV at n = 0 to around 5 GeV at smaller angles. The wide angle muon system (WAMUS) detects all muon tracks which pass through the CF toroids and most of those which pass through the EF toroids. The WAMUS system contains 164 PDTs of varying sizes. The PDT wires are oriented along the direction of the magnetic field to accurately measure the bend coordinate. The SAMUS system contains three stations with three planes per station on each end. The CF toroid is a 109 cm thick square annulus centered on the beam pipe and weighs 1973 metric tons. Twenty coils of ten turns each carry currents of 2500 A and produce internal magnetic fields of 1.9 T. The two EF toroids are positioned at 447 S [2] S 600 cm and are centered on the beams. The main ring beam pipe passes through a 25 cm hole in the EFs. Eight coils of eight turns each carry 2500 A and produce an internal field of about 2 T. Each EF toroid weighs 800 metric tons. Inside the inner hole of each EF toroid is a SAMUS toroid. The SAMUS toroids each weigh 32 metric tons and are centered on the beams. Two coils of 25 turns each carry currents of 1000 A producing a field around 2 T. 61 WAMUS The wide angle muon system [39] PDTs are arrayed in three layers called A,B, and C layers that surround the toroids. The A layer is just before the toroid and the B and C layers are after the toroid and are separated by 1—3 m to provide a long track lever arm after the magnet. The A layer contains four planes of PDTs and the B and C layers are comprised of three PDT planes each. All WAMUS PDT cells are the same. The differences between chambers are depth in number of cells (3 or 4), width in number of cells (14 to 24), and length (191 to 579 cm). The WAMUS PDT cells are built from aluminum extrusions which are cut to length and pressed together and then sealed with epoxy. The extrusions are shaped such that adjacent planes are offset to remove left-right ambiguities. The cathode pads are made from copper clad Glasteell [40] sheets which have a repeating diamond shaped pattern milled into them using a computer controlled router and are then cut into strips to form the individual cathode pads. The cathode strips are inserted into channels in the top and bottom of the unit cells. The cathode pad surfaces facing the active portion of the cell are covered with 50 pm Kapton tape to ensure electrical isolation from the extrusion. A 50 pm gold plated tungsten anode wire is strung through the center of the cell and is held at 300 g tension by a plug mounted in the aluminum cap extrusion which seals the ends of the cells. The maximum drift distance is 5 cm. The wire sag over 610 cm is 0.6 mm. The aluminum extrusions which form the cells are grounded with the cathode pads held at +2.3 kV and the anode wires held at +4.56 kV. The coordinate along the wire direction (.5) is measured by the cathode pad signals and timing information from the anode wires. Adjacent cell anode wires are jumpered 1This material outgasses and deposits on the wires which reduces the efficiency 62 at one end and the signals for the pair are read out at the other end. A rough measurement of E can be made by measuring the time difference at the ends of the paired wire. Two hits per wire pair are accommodated to allow for 6 rays. This method produces a 6 measurement with a precision of 10 to 20 cm along the wire. A finer .5 measurement is made using the cathode pad signals. The upper and lower cathode strips are made from two independent electrodes which form the inner and outer parts of a repeating diamond pattern. The repeat distance of the diamond pattern is 61 cm. The two inner parts of the diamond pattern (top and bottom) in a given cell are added and read out independently of the sum of the two outer parts of the diamond pattern. Calculation of the sum and difference of the inner and outer signals provides a measurement of C modulo the approximately 30 cm half—wavelength of the diamond pattern. The correct cathode pad solution is determined by the At measurement. The cathode diamond pattern is offset by about 1 / 6 of the repeat length between adjacent planes of PDTs to reduce the ambiguities near the extrema of the diamond pattern. The overall 5 resolution for a given chamber is i3 mm. The WAMUS PDT chambers use a gas mixture of 90% Ar, 5% CF4, and 5% C02. The drift velocity is on the order of 6.5 cm / us but varies across the cells with changing E [41]. Tests of this gas mixture show a nearly linear time to distance relationship. Typical leak rates of WAMUS chambers are about 0.005 cubic feet per hour. SAMUS The three small angle muon system stations at either end of the DO detector are called A, B and C stations. The A station precedes the SAMUS toroid and the B and C stations lie between the SAMUS toroid and the low-beta quadrupole magnet for the D0 insertion. The SAMUS stations are perpendicular to the beams and cover an area 'of 312x312 cm2. Each SAMUS station consists of three doublets of 29 mm internal 63 diameter cylindrical PDT chambers [42]. The orientation of the doublets are along the x, y, and u coordinates (the u coordinate is at an angle of 45 degrees with respect to the x coordinate). The PDTs that make up the doublets form a close packed array with adjacent tubes offset by one-half a tube diameter. The SAMUS PDTs are made from 3 cm diameter stainless steel tubes with indi- vidual end plugs which allow for the gas and electrical connections. A 50 pm gold plated tungsten anode wire at 208 g tension runs through the center of the tube. The wire sag over 3.1 m is 0.24 mm. The gas mixture used is 90% CF4 and 10% CH4. The drift velocity is 9.7 cm / us with a time to distance relationship that is approximately linear. The position resolution for the small angle system is about 300 pm [43]. Muon System Electronics The DO muon system is spread over a large area and consequently the readout elec- tronics for the PDTs are mounted on the chambers. Signal shaping, time to distance conversion, hit latching, monitoring, and signal multiplexing for efficient signal trans- port are all performed locally. Only the digitization and triggering electronics are loc~ ated in the moving counting house (MCH). The electronics boards for each WAMUS chamber are located in an enclosure mounted on the side of the chamber body. The individual boards are: fast signal shaping “motherboards” for each six cells of each chamber plane, one hardware-status “monitor” board, and one multiplexing and signal driver “corner-board”. The motherboard contains a set of hybrid circuits which perform signal shaping and time-information encoding. The cathode pad signal sums are brought to a hybrid circuit charge—sensitive preamplifier (CSP) which is very similar to the calorimeter preamplifiers. The CSP output enters a baseline subtracter (BLS) circuit (similar to the calorimeter BLS) which performs pre— and post-sampling of the signal and stores 64 the difference output on one of two output capacitors. Signals from the jumpered anode wire pairs are amplified and discriminated in a pair of hybrid circuits (2WAD). The 2WAD outputs enter two hybrid time—to-voltage and two time-difference to voltage hybrid circuits which provide time information and time difference for up to two hits per wire pair. The final circuit on the motherboard is a pad latch which latches hits on the cathode pads based on information in the pad BLS hybrids. The corner-board collects the information from the motherboards, multiplexes the pad latch information and sends it across long cables to the MCH. The latch bits are also ORed on the corner-board to provide information on muon activity and majority logic indicating 1, 2, 3, or 4 hits in the chamber. The motherboard analog signals from the pad BLSs and the time and time-difference hybrids is multiplexed and sent in 96 time slices to digitizers in the MCH. The corner boards also contain circuitry for the addition of a cosmic ray veto scintillator array mounted on the outer surface of the detector. Finally, the corner-boards contain pulsers for front end electronics calibration. Information from the monitor board on each chamber passes over a token ring network to the general detector monitoring system. This board monitors the temper- ature, currents, voltages, and gas flow in each chamber. The monitor board is also used to set pad latch thresholds and pulser amplitudes. The SAMUS front end electronics consist of a card containing an amplifier, dis- criminator, time-to—voltage converter, and a latch for each PDT. A SAMUS control board supervises the multiplexing of these signals which are sent to the MCH. The SAMUS monitor boards are the same as the WAMUS boards. The latched SAMUS hits are used to form the SAMUS triggers. Muon chamber cathode pad signals and voltage encoded time information are 65 digitized in the MCH using 12-bit ADCs which are similar to the calorimeter ADCs. The muon system contains a total of 50,920 analog elements. 3.2.5 The Trigger And Data Acquisition System The DO trigger and data acquisition system is used to select interesting physics events and events used for calibration purposes. The DO trigger consists of four main decision levels of increasing sophistication in event selection. The Level 0 trigger is an array of scintillators mounted on the surface of each end calorimeter and is used to signal an inelastic collision in D0. At a luminosity of 5 X 1030 cm‘2s'l, the Level 0 rate is about 150 kHz. The Level 1 trigger is a programmable architecture which uses information from the various DO detector components to select events. Level 1 trigger decisions can be made in the 3.5 us time interval between bunch crossings incurring no deadtime. Level 1.5 triggers require more time. The Level 1 trigger output rate is on the order of 200 Hz. The level 1.5 trigger further reduces the Level 1 output rate to about 100 Hz. The events which pass the Level 1 (and possibly Level 1.5) trigger are transported via the DO data acquisition system to a farm of 48 DEC VAXstation 4000/60 and 4000 / 90 microcomputers known as the Level 2 system. Level 2 assembles the raw event data and runs a combination of sophisticated filters to perform further event selection. The Level 2 output rate is about 2 Hz which coincides with the speed events can be written to the host computer system. The Level 0 Trigger The Level 0 (L0) trigger is used to signal the occurrence of an inelastic collision at D0 and to monitor the luminosity at D0. It consists of two hodoscopes built of scintillation counters which are mounted on the outer surface of each end calorimeter. Two planes of scintillators are rotated 90 degrees with respect to one another to form 66 a checkerboard pattern in each hodoscope. The hodoscopes provide partial coverage for the pseudo-rapidity range 1.9 < [77] < 4.3 and almost full coverage for the range 2.3 < In] < 3.9. The 77 coverage is governed by the requirement that a coincidence of both L0 detectors be 2 99% efficient for detection of non-diffractive inelastic collisions. Each hodoscope contains 20 short (7 x 7 cm2 squares) read out by a photomultiplier tube (PMT) at one end and 8 long (7 x 65 cm2 rectangles) read out at both ends. To provide good timing, 1.6 cm thick Bicron BC-408 PVT scintillators and Phillips XP-2282 photomultiplier tubes are used. Optical fibers distribute UV laser pulses to each PMT for monitoring and calibration purposes. The Level 0 trigger is also used to provide a fast determination of the event vertex for use in the Level 1 and Level 2 triggers. Because the vertex distribution is so large at the Tevatron (02 = 30cm) it is necessary to measure it to provide more accurate ET values for use in the Level 1 and Level 2 triggers. A “fast” L0 vertex with a resolution of 15 cm is computed within the time limit imposed by the Level 1 trigger and a “slow” L0 vertex with a resolution of 3.5 cm is provided to the Level 2 system. The L0 vertex z-coordinate is determined by the arrival time difference between particles which hit opposite hodoscopes. The Level 0 detectors also provide information about multiple interactions in a single bunch crossing which has an appreciable probability at higher luminosities. If a multiple interaction occurs, the L0 time difference is ambiguous and a flag is set to identify such events to subsequent trigger levels. The L0 PMT signals are amplified and split into two readout paths. Along one path, an analog sum of the small counter signals for each hodoscope is computed and a fast vertex position measurement for Level 1 ET corrections is made using a GaAs-based digital TDC [44]. A [2,,th < 100 cm cut is made to discriminate between beam-beam interactions and beam-halo or beam—gas interactions. The other readout 67 path digitizes the time and integrated charge for each counter. A more accurate slower determination of the event vertex is computed by applying full calibration and charge slewing corrections to this data and using the mean time for each hodoscope to find the vertex position. The rms deviation in the time difference is also computed and used to flag multiple interactions. All L0 computations are done in hardware. The Tevatron luminosity is found by measuring the non-diffractive inelastic colli- sion rate. These events are selected by requiring a L0 coincidence and that [2,,th < 100 cm. Scalars are used to count live crossings, coincidences which satisfy the vertex cut, and single hits in groups of counters both with and without valid coincidences. These scalars allow the luminosity to be measured independently for each beam bunch and provide feedback to the accelerator operators. The Level 1 And Level 1.5 Trigger During Run 1A, the Tevatron was operated with six bunches of protons and antipro- tons which results in a time interval of about 3.5 ps between bunch crossings. Trigger decisions which can be made within this time interval incur no deadtime. The hard- ware calorimeter trigger satisfies this requirement, as does part of the muon trigger, however the remaining muon trigger requires several bunch crossings to complete and is logically incorporated as a veto on event transmission. The various Level 1 (L1) trigger components are managed by the L1 framework, which also controls the inter- face to subsequent trigger levels. The L1 framework collects digital information from each L1 specific trigger device and decides whether to pass the event on to the next trigger level. It also coordinates vetoes which inhibit triggers, provides prescaling for each trigger (if needed), correlates the trigger and readout functions, manages com- munication with the front end electronics and trigger control computer (TCC), and ' provides several scalars which allow measurement of trigger rates and deadtimes. 68 Trigger selection is performed using a two dimensional AND-OR network. The 256 latched bits called AND-OR input terms form one set of inputs and contain specific pieces of detector information (e.g., 2 EM clusters with ET > 10 GeV). The 32 orthogonal AND-OR lines are the outputs from the AN D-OR network and correspond to 32 specific Level 1 triggers. The firing of one or more specific L1 triggers results in a readout request by the data acquisition system, provided there are no front end busy restrictions or other vetoes. If a Level 1.5 confirmation is required for a specific Level 1 trigger, the L1 framework forms the L1.5 decision and passes the result to the data acquisition system. In addition, the L1 framework builds a block of information called the trigger block that contains a summary of all the conditions which led to a positive L1 decision (and L1.5 confirmation if required). The L1 trigger data block is passed on to the data logging stream to allow subsequent processors to recompute the input information and confirm the L1 decision. The trigger control computer (TCC) provides for convenient interaction with the L1 trigger system. Configurations for active specific triggers are downloaded to the TCC from the host computer. The large tables of information necessary for program- ming and verification of the hardware memories in specific L1 triggers are downloaded and stored on the TCC’s local disk. The TCC provides access to scalers and registers to allow trigger system programming, diagnostics and monitoring. The TCC software is based on the DEC VAXELN multi-tasking real—time operating system. The TCC software uses a low-level hardware database which contains aliases and descriptions of specific hardware components for easy reference, and a high-level object oriented database for defining and recording trigger configurations. Trigger configurations are continuously monitored for validity and an alarm is set in the general DO alarm system if a configuration becomes invalid. The Level 1 system can trigger on energy deposited in the calorimeter and on 69 tracks in the muon detector. Only the calorimeter trigger was used to collect the data for this thesis so only the calorimeter trigger is described here. For a description of the muon trigger see [16]. The Level 1 calorimeter trigger uses the trigger pickoffs from the calorimeter BLSs which are summed into trigger towers of size An = A45 2 0.2 out to [n] = 4.0. Separate trigger inputs exist for the EM and FH sections of the calorimeter. The summed energy (input voltage) in the trigger towers is converted to ET by weighting it using a sin0 lookup table that assumes an interaction vertex of z = 0. The ET signal in each trigger tower is then digitized in a fast 8—bit FADC (20 us from input to output) and clocked into latches allowing pipeline synchronization of all calorimeter information. The latches can also be supplied with test signals for diagnostic study of all subsequent trigger functions. The 8—bit FADC information provides part of the address for several lookup memories. An additional 3 bits from the L0 trigger provides a rough measure of the interaction vertex z—coordinate. The lookup memories provide EM and FH trans- verse energies for each trigger tower above a fixed out (based on electronics noise and physics considerations) which are corrected for the vertex position (if known). The sum of the EM and FH ET for each trigger tower is formed and stored in a 9-bit register for use in future, more powerful hardware triggers. The lookup memories also provide the EM and FH ET for each trigger tower without the cut and vertex correc- tion. The global sums of the six energy variables returned from the lookup memories (EM, FH, and total transverse energy, corrected and uncorrected) are computed for all trigger towers by pipelined adder trees. The adder trees are arranged such that geographically contiguous regions are kept together thus allowing intermediate sums over large areas (bigger trigger towers) to be used if so desired. 70 The missing ET for the event is computed from the x- and y-components of the global ET. The corrected and uncorrected global total ET are formed from the cor- rected and uncorrected EM and FH ET. These seven energy variables (corrected and uncorrected EM, F H, and total ET and missing ET) are compared to up to 32 pro- grammable thresholds and the results of the comparisons are provided as AND-OR input terms to the Level 1 framework. In addition, the EM ET for each trigger tower is compared to four different programmable reference values. A bit is set for each EM reference value that is exceeded by each trigger tower’s EM ET provided that the FH ET for the trigger tower does not exceed a corresponding hadronic reference value (hadronic veto). The total ET is also compared to four different reference values for each trigger tower producing an additional four bits for each trigger tower. These 12 individual reference values are separately programmable for each trigger tower. The global count of the number of trigger towers whose ET exceeds their reference values is computed by summation over all trigger towers in the pipelined counter trees for each of the four EM ET and total ET reference value sets. The global counts for each reference set are then compared to up to 32 programmable count thresholds and the results of the comparisons are provided to the Level 1 framework as input AND-OR terms. The Data Acquisition System The DO data acquisition system and Level 2 trigger hardware are tightly coupled together and must be described simultaneously. The data acquisition system and Level 2 trigger hardware [45]-[50] is based on a farm of 48 parallel nodes connected to the detector electronics and triggered by a set of eight 32-bit wide high-speed data cables. The nodes consist of a DEC VAXstation 4000/ 60 or 4000 / 90 running the DEC - VAXELN realtime operating system connected via a VME bus adaptor to multi-port 71 memory (for receiving data), and an output memory board. The Level 2 system (L2) collects data from all relevant detector elements and trigger blocks for events which pass the Level 1 triggers. Various software algorithms (L2 filters) are then applied to the events to reduce the L2 input rate from 100 Hz to an output rate of around 2 Hz sent to the host computer system for storage. The data for each event that passes the L1 trigger is sent over parallel data cables to a non-busy node selected to receive the event. The node puts the event data into a final format and runs a combination of L2 filter algorithms on the event. A block diagram of the Level 2 system is shown in Figure 3.6. Approximately 80 VME crates hold the calorimeter and muon chamber ADCs and the FADCs for the central detector components. Fully digitized data appears in the output buffers of these VME crates about 1 ms after a L1 or L1.5 trigger. The L0 and L1 trigger hardware also produce data blocks containing information about the trigger decision. Each VME crate contains a 512 kB memory module with two data buffers. Each VME buffer driver board [50] (VBD) uses list processors to control data transfer from locations in the VME crate onto an output data cable highway. Internal crate transfer bandwidth is about 30 MB /s. The VBD outputs for each sector of the detector are sequentially connected to a high speed data cable. The data cables consist of 32 twisted pair lines for data and 13 twisted pair lines for parity and control. The data cables, clocked at 100 ns intervals, transfer data at 40 MB / 8/ cable. Eight data cables correspond to the eight detector sections (VTX,TRD,CDC,FDC, north and south halves of the calorimeter, L1 trigger, and L0 trigger). Readout control and arbitration for the VBDs is performed using a token passing scheme. Upon receipt of a token the external port processor of the VBD compares the token bits with the crate buffers; if they match, the VBD transfers pending buffers to the data cable. The ' tokens circulate at a clock rate of 1 MHz. 72 @0 DETECTOE READOUT READOUT LEVEL-l SECTION . . ' SECTION TRIGGER L is? j} 31"."- . 5372C--?.hité.cfitité}oil-iii 5:7-‘1 .- -_ ’ .‘ i:»»:;:’.«LT7.-..§>l [ 5‘}. :PTTL mm; COLTS? If‘f‘. ' ' " g j 1 ‘ V l DatuCabk-M i L2node#l L2nodc#2 o o o L2node# Li I t I [71 I 70mm; Cable ~. . : .5 Supervisor 2» i VMS HOST .4 Machine : Ethernet 82:33:: Figure 3.6: A schematic of the DO Data Acquisition System. 73 A Level 2 supervisor processor controls the realtime operation of the data acquis- ition system. A sequencer processor controls data transfer over the data cables via a set of sequencer control boards (one for each data cable). When a valid hardware trigger occurs, the Level 1 system sends an interrupt to the supervisor containing the 32-bit pattern of specific triggers that fired, as well as a 16-bit event number. When the supervisor receives a Level 1 trigger it assigns a Level 2 node for that event and interrupts the sequencer. The sequencer creates readout tokens for the list of crates necessary for the specific trigger pattern, tokens include the low-order bits of the event number to ensure readout integrity. Token circulation and data readout are managed in parallel by the separate sequencer control boards on each data cable. Any combin- ation of data cables, and thus any combination of detector elements, may be readout, providing for flexible debugging and calibration. The eight data cables are connected to all 48 L2 nodes. The L2 nodes are located in a fixed counting area in the DO hall, while the VME crates and L1 trigger hardware are located in the MCH. To connect these two separate counting areas, the data cables pass through an optical isolator circuit which decouples the electrical grounds of the detector and the fixed counting areas. Each L2 node contains multi-port memory modules which receive the data from the data cables, a VAXstation 4000 / 60 or 4000 / 90 processor, a VBD for buffering data that is sent to the host. An integral part of the L2 nodes is the multi-port memory [51] (MPM) which is accessed by the data cables, and the output VBD. Each L2 node contains four MPMs which each contain two channels of 2 MB multi-ported memory to provide inputs for the eight data cables which pass through each node. The total input rate for each L2 node is 320 MB / s. The MPMs appear as contiguous I/ O space memory to the CPU. The incoming data is directly mapped into a single raw data ZEBRA tree structure based on the CERN ZEBRA ‘ memory management package, which is used extensively in DO. No copy operations 74 Layout of Level-2 Node (TU Memory VAXstation 4000/60 VS40 Data-cables to next Level-2 node I] 1] Output cable to next Level-2 node M M P P M M VME-Crate xv—on 3'63 Nuw< >=fl99% efficient for non—diffractive inelastic collisions. By comparing the arrival times of the signals from the two arrays, the approximate position of the interaction vertex may be found. A fast vertex determination with a resolution $15 cm is available within 800 ns after the collision. A more accurate determination with a resolution of :l:3.5 cm is available within 2.1 ps. The vertex position is available as several Level 1 trigger terms; it can also be used in Level 2 processing. The Level 0 system can also identify events which are likely to contain multiple interactions. 82 4.3 Level 1 Trigger Selection Criteria The Level 1 trigger used in this analysis is based mainly on the DO Level 1 calorimeter trigger. The other trigger terms used are the L0_FASTZ_GOOD and MRBSLOSS terms described below. A description of the Level 1 (L1) calorimeter trigger hardware can be found in Chapter 3. A passing L0-FASTZ.GOOD trigger term indicates 1) that an inelastic collision has occurred in the DO detector and 2) that the fast vertex 2 position is within :l:97 cm of the nominal vertex position. Since the Main Ring passes through the DO detector, losses from the Main Ring will show up in the detector and must be rejected. The largest losses occur when beam is injected to the Main Ring every 2.4 s, and again 0.3 3 later when the beam passes through transition. Transition is the point in the acceleration cycle at which the energy of the particles is sufficient to require a change between a non-relativistic model and a highly relativistic model for the behavior of the particles; the energy at which it occurs depends both on the mass of the particles being accelerated and the size of the accelerator ring. When a bunch of non-relativistic particles are traveling in a circular orbit the particles with a larger than average momentum will also have a larger than average velocity and will pull ahead of the rest of the bunch. In order to keep the bunch from blowing up longitudinally, the particles near the front of the bunch must be decelerated relative to the mean momentum of the bunch, and those near the tail must be accelerated relative to the mean momentum of the bunch. Highly relativistic particles, however, must be treated differently. In this energy region, the velocity of a particle is nearly c and constant regardless of its momentum, however, the path length is not constant. A particle with larger than average momentum will have a larger 83 than average bending radius and will thus fall behind the rest of the bunch. So in this situation, one must accelerate the head of the bunch more than the tail. Properly rearranging the accelerating fields when passing through transition is difficult, and accelerators often produce extra losses at that point. These losses are dealt with by vetoing on the MRBSLOSS trigger term. This term is asserted as a possible veto during a 0.4 3 window starting at injection, continuing through transition, and allowing time for the calorimeter and muon high voltage to recover from the large losses. This results in a dead time of about 0.4/2.4 a: 17%. The granularity of the L1 calorimeter trigger is An 2 Ad) 2 0.2 out to [1]] = 4.0. The energy deposited in the calorimeter layers is summed into EM, hadronic, and EM+hadronic towers according to DOS? pseudo-projective geometry and converted to ET using sin0 lookup tables as described in Chapter 3. The L1 trigger used in this analysis required 2 EM towers with ET > 7.0 GeV and was known by the mnemonic EM_2-MED. Ideally, a single EM tower trigger would have been better suited for a highly efficient L1 Drell-Yan trigger due to the electron energy asymmetry from the z boost caused by :30 aé xb. Unfortunately, the L1 to L2 event rate for such a trigger would much too high unless the ET threshold was raised to values unsuitable for lower mass Drell-Yan kinematics. In addition, since L1 only has four available EM thresholds, they must be shared among all physics triggers, so compromise is necessary. Consequently, it was not possible to trigger on tower energy instead of ET (thus allowing a sharp mass turn—on) nor to lower the L1 ET threshold for the EM_2_MED trigger and cut harder in L2, since all users of this L1 trigger would have to agree to the change. For these reasons, the trigger used in this analysis was also used for the DO Z boson . analysis, and the thresholds were set to provide a very efficient L1 and L2 Z trigger 84 while still maintaining a low event rate. The efficiency of this trigger for Drell-Yan events is lower due to the difference between Drell-Yan and Z kinematics, especially for low mass Drell—Yam events. The L1 efficiency turn-on is slower than one might naively expect since no clus— tering is being performed in L1. Since L1 has fixed towers of size An = Ad) 2 0.2, an electron may impact near the boundary of two towers (or even 4 towers in the worst case) sharing its energy between them. The result of this is that the L1 calorimeter trigger is not 100% efficient until the electron ET is four times the threshold value! However, it is about 99% efficient at two times the L1 threshold value. A plot of the L1 efficiency turn-on for a 7 GeV L1 ET threshold vs. the offline ET of the cluster is shown in Figure 4.1. The offline ET threshold was raised to 11.0 GeV to avoid the large uncertainty at ET :2 10.0 GeV. The efficiency of the EM_2_MED trigger is then the product of the efficiencies for the ET of each EM cluster. A description of the Monte Carlo data used to measure this efliciency may be found in section 4.6.1. 85 Efficiency —-A 0.8 0.6 0.4 0.2 Level 1 E, > 7 GeV Efficiency Xz/ndf 1.185 / .3 P1 3.511 :1: .6237 [.— t Fit —> 0.5 + 0.5‘tonh((E,—7)/P,) l l l l l l 1 l l l l l 1 l l l O 5 10 15 20 25 E, (GeV) £(L1 E. > 7 GeV) Figure 4.1: The Level 1 trigger efficiency vs. input ET 86 4.4 Level 2 Trigger Selection Criteria Events passing the EM_2_MED L1 trigger are passed on to the Level 2 system for fur- ther processing. The data used in this analysis were required to pass the ELE_2.HIGH filter script. A detailed description of the algorithm used by the L2.EM filter tool to select good electron candidates is given in Appendix A of this dissertation. The reader should refer to Appendix A for a description of the selection cuts referred to below. The L2_EM filter algorithm uses the full segmentation of the DO calorimeter to identify candidate electron showers. The trigger towers which fired the L1 EM _2_MED trigger are used as seeds for a simple clustering algorithm in the L2-EM filter. The highest ET EM3 cells in the passing L1 trigger towers are found and the cells contained in a An x Ad) = 0.3 x 0.3 square in each EM layer and the first FH layer around the EM3 seed cells are used to construct the majority of the L2-EM shower shape selection variables. The L2_EM filter algorithm is based mainly on whether the EM+FH1 candidate cluster shapes are consistent with the shape of a typical electron shower in the DO calorimeter. L2_EM uses the cluster cell ET to form various shower shape variables which are compared to values determined from real electrons in the DO test beam. The cell ET is used instead of cell energy for the shower shape variables because the Level 2 unpacking algorithm provides cell ET for fast cluster ET calculation. Since the sin0 variation over a typical EM cluster is small and the majority of the L2_EM cut variables are ratios, the effect of using cell ET instead of cell E is negligible. The first L2_EM cut performed on candidate electrons from L1 is a cluster EM ET cut. The sum of the ET in the 4 EM calorimeter floors is corrected for the event vertex (Level 0 slow vertex 2, resolution of :1:3.5 cm) and for energy leakage outside of the 3X3 readout tower cluster and compared to etmin_cc or etmin_ec depending upon 87 the cluster location within the DO calorimeter. Candidates which pass the ET cut can then be subjected to longitudinal and trans- verse shower shape cuts. It was determined early in Run 1A that an acceptable event rate could be achieved with an ET cut of 10 GeV on both electron candidates without using these cuts. The increase in the efficiency of the ELE_2_HIGH filter by not using these cuts was deemed preferable to a lower L1 / L2 ET threshold in combination with them for Z—iee events, thus they were not used in this analysis either. These cuts are however used for single electron triggers and provide an additional rejection of approximately a factor of three. A track match cut may also be performed after the shower shape cuts, but again it was not needed. The final cut performed by L2_EM is an E7 isolation cut. The difference of the sum of the ET in a R = v772 + <15” = 0.4 radius cone of readout towers and the cluster ET (R = 0.15) is calculated and divided by the cluster ET to form an isolation fraction. The electron candidate fails if its isolation fraction is greater than 0.15. The ELE.2_HIGH filter required two electron candidates to pass ET and isolation fraction cuts. The Level 2 efficiency vs. ET for single electrons are in shown Figure 4.2. The offline ET threshold was raised to 11.0 GeV to avoid the large uncertainty at ET = 10.0 GeV. This plot was produced by overlapping single electron Monte Carlo events (plate level DOGEANT) with real DO minimum bias events to simulate the electronic and uranium noise and underlying event activity in a typical DO EM event and then running them through the Level 1 and Level 2 simulators. A description of the Monte Carlo data used to measure this efficiency may be found in section 4.6.1. 88 Efficiency —5 0.8 0.6 0.4 0.2 Level 2 EY > 10 GeV + ISO < 0.15 Efficiency f X’/ndf 6.925 / 3 P1 1.509: .5921 Fit —> 0.5 + O.5ttonh((E.-10)/P.) 1 1 l 1 l 1 J L L i l 1 1 1 l O 5 10 15 20 25 E, (GeV) £(L2 E, >10 GeV +130 < 0.15) Figure 4.2: The Level 2 trigger efficiency vs. input ET 89 4.5 Offline Data Reconstruction The raw event data from the DO detector is in the form of digitized counts of charge deposited in a calorimeter cell, counts per time bin for a tracking chamber wire etc. These quantities must be converted into meaningful data such as EM cluster energy or E7, track position, and event vertex position to facilitate physics analysis. This conversion process is called “reconstruction”, and is performed by a program known as DORECO. DORECO reads in all the calibration data and conversion constants for all the DO sub-detectors and applies this data to all the measured quantities for an event to produce kinematic and particle selection variables. Event reconstruction can be divided into three main stages: Hit Finding the raw detector data is unpacked and converted into hits, which consist of energy deposits in calorimeter cells, pulses on tracking chamber wires etc. Clustering and Tracking hits whose spatial separation is small are combined to form clusters in the calorimeter or tracks in the tracking chambers. Particle Identification calorimeter energy clusters and tracking chamber tracks as well as other information are combined to identify the sources of the tracks and clusters as electron, photon, jet, or muon candidates. The criteria used to identify the particle candidates are quite loose at this stage to guarantee high particle acceptance so as not to lose any candidates. In addition, DORECO computes many selection variables to be used in further analysis, where much tighter selection cuts are generally used to construct a final data sample for a given analysis. 90 4.5.1 Energy Reconstruction Calorimeter hit finding basically consists of converting the charge deposited in a calor- imeter cell to energy in GeV. The conversion factors are determined primarily from test beam measurements where the calorimeter response to an incident particle of known energy and position is found. The conversion factors, known as “sampling fractions”, are then used to reconstruct the unknown energies of particles in DO. Additional cell-by-cell corrections are made for variations in electronics gain and ped— estals. These corrections are periodically measured during times in which there are no collisions and are stored in a database which is accessed by DORECO during re- construction. However, due to various differences between the fixed target test beam setup and the collider installation of the DO detector, this calibration is slightly low and must be corrected. Fortunately, the mass of the Z boson has been measured very precisely at LEP [56], and so serves as an additional calibration point. The measured electron energies are scaled up so that the Z boson peak in the 6+6“ mass distribution matches the LEP measurement. This correction is about 5% in the CC, and 1—2% in the EC. After finding the calorimeter cell energies in GeV, cells with the same 77 and d) coordinates are summed together for the EM and hadronic layers of the calorimeter to form readout “towers”. These towers are then used in subsequent clustering al- gorithms which attempt to reconstruct the total energy or ET of the incident particles. 4.5.2 Track Reconstruction Hit finding in the DO tracking chambers is begun by unpacking the digitized charge deposition versus time. Individual pulses are identified by looking for leading and trailing edges of pulses. Each pulse is integrated to find the total deposited charge. 91 This integrated charge is then used to compute the energy deposition per unit g/cm2 traveled through the tracking chamber, known as dE/dm. The arrival time of the pulse at the front end electronics is used to find the position of the pulse. The time required for the ionization created by the particle traversal to drift to the sense wire measures the radial distance of the hit from the sense wire, and the arrival time of the pulse on the delay line gives its location along the sense wire. Due to left / right ambiguities, there may be two possibilities for the location of a hit which are both used as input to the tracking phase. The staggering of the sense wires usually guarantees that only the correct solutions will yield a good track. In central detector tracking, the object is to identify groups of hits which lie along a straight line. Tracking is first done for each individual layer of the detector to produce track segments. The track segments are then matched between the layers of each detector to form tracks. Finally, these tracks are matched between the vertex chamber, TRD, and outer tracking chambers (CDC and FDC). 4.5.3 Muon Reconstruction A similar procedure is used to find muon tracks in the DO muon detector. However, due to differences in geometry and electronics, the details are quite different. A basic description of DO muon momentum measurement is as follows: A muon track is found in the central detectors as described above and is then matched to a track in the A layer of the muon detector. This gives the track direction before the toroidal magnet. A track in the muon detector B and C layers is then found, and the angle between this track and the track from the central detector and muon A layer is computed. This angle in combination with the value of the magnetic field in the toroid is then used to compute the muon momentum. 92 4.5.4 Vertex Reconstruction In order to compute the transverse energy or momentum of a particle it is necessary to measure the origin of the particle in the lab frame which is known as the event “vertex” . The :r and y positions of the vertex are well known due to the fact that the cross- section of the beam is made very small in these dimensions in order to maximize the luminosity. The typical cross-section of the beam was about 50 pm and is positioned about 3-4 mm from the center of the detector with a drift of less than 50 pm over the length of a data run. Thus, the (11:, y) position of the vertex can be taken as a constant, and for many purposes can be set to (0,0) (the geometrical center of the detector). The z-coordinate of the vertex, however, is less well constrained. Each bunch of particles in the Tevatron has some extent along the beam direction, and the resulting width of the vertex z-coordinate distribution in the detector is about 30 cm. Thus, it is necessary to measure the z-position of the vertex for each event individually. This is done using tracks found in the CDC. The vertex finding method is as follows: 0 Project the tracks found in the CDC back towards the center of the detector. 0 For each track, calculate the impact parameter (the minimum distance between the track and the z-axis of the detector). Discard all tracks with an impact parameter larger than a given cutoff. (This eliminates low-momentum tracks which have undergone a large amount of multiple scattering.) 0 Project each track into the (r,z) plane, and compute the intersection with the z-axis. Plot the z—positions of the intersections. 0 Fit a Gaussian around the peak of the resulting distribution. The mean is the estimate of the z—position of the vertex. The tails of the distribution are also 93 searched for any secondary peaks. The resulting resolution for the vertex z-coordinate is about 1-2 cm. Multiple vertices can typically be separated if they are at least 7 cm apart. 4.6 Offline Electron Identification A standard DO electron candidate, called a PELC, is constructed in the following manner: 0 Candidate electron clusters are formed from calorimeter towers using a “nearest neighbor” algorithm [55]. Towers adjacent to the highest ET tower are added to the cluster if they are above a given ET threshold, and the cluster size is not too large. 0 A PELC must have at least 90% of its energy in the EM calorimeter, and at least 40% of its energy must be contained in a single tower. 0 Using the finer resolution cells in the EM3 layer, the cluster centroid is found by computing the log-weighted weighted sum of the cell positions F .d _ Z; wiF' centro: — Z; w: where the weights w,- are given by w--max(0w +ln( E” )) I a O ZjEj The parameter wo is chosen to minimize the centroid uncertainty and the sums 94 are over all EM3 cells in the cluster. The resulting position resolution is about 1.5—4 mm. o The reconstruction program then searches for a central detector track pointing from the event vertex to the calorimeter cluster within a “road” of An = Ad) 2 $0.1. If such a track is found, the cluster is identified as an electron candidate “PELC”; otherwise, it becomes a photon candidate “PPHO”. The above electron candidate (PELC) criteria are intentionally very loose so as not to lose any real electrons. Hence, the PELC sample is generally intended to be used as starting sample for electron final states and therefore to undergo further selection cuts for a given analysis. The starting data sample used for this analysis consisted of events containing 2 PELC objects with ET > 10 GeV that passed the L1 EM_2.MED trigger and L2 ELE..2_HIGH filter. The additional cuts performed to select the final data sample are described below. The PELC criteria are a superset of the PPHO criteria since it is simply an additional requirement of a track within a An = A4) = i0.1 road of PPHO candidate. Although the PPHO and PELC requirements are intended to be loose, and thus very efficient for real electrons, they are not 100% efficient and so the efficiency of these cuts on real electrons must be measured. Since the PELC criteria are a superset of the PPHO criteria, we may measure PPHO efficiency and then measure the PELC/PPHO efficiency (add the track requirement to a set of PPHOs). To measure the PPHO efficiency we generate single Monte Carlo electrons and then measure the number of these Monte Carlo electrons that are reconstructed as PPHOs by the DORECO reconstruction program. The Monte Carlo electrons are described in more detail in the following section. There is little energy or IETA dependence in the PPHO efficiency so we simply take the average of the PPHO efficiency for 10, 15, 25 and 50 GeV electrons in the CC, and the average PPHO efficiency for 25, 50, 95 100 and 200 GeV electrons in the EC. The CC PPHO efficiency is 97.7% and the EC PPHO efficiency is 99.4%. The tracking portion of the DO detector simulation has been shown to exceed the actual performance of the DO tracking chambers [58], consequently Monte Carlo electrons are not used to measure the PELC efficiency. Instead of measuring the PELC efficiency directly, we measure the PELC/PPHO efficiency, namely the additional efficiency loss incurred by the PELC track requirement. Since the energy dependence of the DO tracking resolution is minimal above 10 GeV (see section 4.6.3), it is possible to use high energy electrons to measure the PELC/PPHO efficiency. Thus we use electrons from Z decays to measure this efficiency. We start with a sample of PELC—PELC, PELC-PPHO, PPHO-PPHO events and require that the ET of each PELC or PPHO be greater than 11.0 GeV. We then require that the invariant mass of the PELC-PELC, PELC-PPHO, PPHO-PPHO pair be within a 80—100 GeV mass window. We then perform all of the electron identification (id) cuts listed in Table 4.1 on one PELC/PPHO object, and if it passes, we fill histograms of energy, ET, and IETA of the other PELC/PPHO object. This process is done again, but with the further requirement that the other object be a PELC. We then switch which object has the electron id cuts made on it and which goes into the histograms and repeat the process to double our statistics. The electron id cuts and the mass cut provide a fairly background free sample of real electrons from Z decays, which allows us to measure the PELC/PPHO efficiency by simply dividing the set of histograms with the additional PELC requirement by the histograms without this requirement. Figure 4.3 shows the resulting PELC/PPHO efficiency. Since there is no obvious energy, ET, or IETA dependence we fit a flat line to each of these distributions and take the average as the PELC/PPHO efficiency. The PELC/PPHO efficiency is 86.9%, which is then multiplied by the PPHO efficiency to give the total PELC efficiency for electrons. 96 PELC/PPHO efficiency from Z —> ee decoys >~ .. U _ 9‘3 1 it; , - A . + I 2075 E ' ' I7 in 5 05 _— 025 :_ Xz/ndf 2.797 / 9 ' E A0 .8615i .10205—01 O ”rillliiiiliiiidLirililiiliiiililiililiiliiiiIiiii 1O 2O 30 4O 5O 60 7O 80 90 100 1 10 . E,(CeV) PELC/PPHO Efficnency from Z electrons 5‘ c 1 ‘2 30.75 —]— m 0.5 025 x/ndf18.85 19 ' A0 .8732i .9790E-O2 Y Efficienc 0 20 40 60 80 100 120 140 160 180 200 Energy (GeV) PELC/PPHO Efficiency from Z electrons 1 0.75 I 0.5 l 0.25 x /ndf 19.70 / 9 A0 .8726 :t .9696E—02 0 —25 —20 -15 -10 -5 0 5 10 15 20 25 IETA PELC/PPHO Efficiency from Z electrons Figure 4.3: PELC/PPHO efficiency vs. ET, E, and IETA. 97 4.6.1 H-matrix The primary tool used for quantifying the information contained in the shape of the electromagnetic shower is the “H-matrix” x2 [57]. The H-matrix x2 does not follow a X2 distribution since the observables from which it is constructed are not Gaussian, but its definition is similar. Given a set of N observations of events of a given type, where each observation forms a vector of M variables 0" = [0'] , - - - , 014}, it is possible to form an estimate of the covariance matrix V IN V=NZ(‘)To-# (o-iu) i=1 where p is the mean of the N observations, namely, [1 = {111,- - - ,pM}. The inverse of the covariance matrix is the “H-matrix” H = V-1 For any subsequent observation 0’ one can define a X2 that is a measure of how likely 0’ came from the same distribution as p X2 = (0' - u)H(0’ - MT Figures 4.4 and 4.5 show the H-matrix X2 distributions for electrons and background respectively. A total of M = 41 variables were used in the construction of the H-matrix (41 degrees of freedom): the fraction of the total cluster energy contained in the EM1, EM2, and EM4 calorimeter layers (longitudinal shower shape), the fraction of the 98 £350 t .500 1:] 250 :- i 200 - 150 L 4]» t , a l 50 t- + i + : +++++ O LILIIrnxrlwm_-J~-mL_4hLadu_Larthrm 0 50 100 150 200 250 300 350 400 450 . 500 H—Motrix X. — Electrons H—motnx x‘ Figure 4.4: H-matrix X2 for electrons. *3; L 6500 [- L i 400 [m] [l 300 — [l], 200 '- ++++ » H+ ++ + + ++++ t 100 [- I + ++ + _ +++ ++++ + + > +++ :t 0 [.rrrll.r.l...rl....l111.1..rrllr..11111L11111...r 0 50 100 1 50 200 250 300 350 400 450 500 H—motrix ‘3 H—Motrix x‘ - Background Figure 4.5: H-matrix X2 for background. 99 cluster energy contained in the 36 EM3 layer cells making up a 6 x 6 array around the highest ET tower in the cluster (transverse shower shape), the logarithm of the cluster energy (to account for the energy dependence of the shower shape), and the z-position of the event vertex (to account for the change in the shower shape due to the impact angle of the track with calorimeter). A separate H-matrix is constructed for each ring in IPHI at each calorimeter [IETA] index. In order to construct the H—matrix and the mean vector p it is necessary to have a data sample which is believed to accurately represent properties of an electron in- teracting with the DO detector over the full range of energies and angles. Due to lack of adequate test beam data (uniform illumination of each calorimeter IETA index at many energies) and the diflerences in the energy scale between the test beam and DO described above, it was decided that Monte Carlo data would be used to tune the H-Matrix. A detailed representation of the DO detector geometry was used in combination with the GEANT 3.14 Monte Carlo detector simulator from the CERN program library to simulate electron tracks passing through DO. Ideally, one would construct an H—Matrix using input events which reflect the jet and underlying event activity as well as the kinematics of the physics signal one wished to extract. Unfor- tunately the amount computing time this would require is immense since propagating all the hadrons created in a typical p13 event is very CPU intensive (not to mention the time required to perform the H-matrix construction for each analysis). Consequently, single electrons were used as the input to the DOGEANT simulation. One would expect then, that the H-Matrix would be less efficient at selecting real DO electrons for a given X2 cut than it is at selecting single MC electrons. In an attempt to simulate the effect of the actual DO environment, single MC electrons were combined event by event with random minimum bias events taken at a luminosity of 2.5 x 103°cm'2s'1. Minimum bias events taken at this luminosity were 100 chosen because the average instantaneous luminosity over all of Run 1A was around 2.5 - 3.0 x 103°cm'2s'1. The difference in the efficiency measured without adding the minimum bias to the Monte Carlo electrons is on the order of 5%, so it is not a very large correction. The minimum bias events contain the effects of uranium noise and pile—up as well as some amount of hadronic activity that roughly approximates the underlying event activity in a typical DO Drell-Yan event. The energy in the calorimeter cells of the single MC electron events is converted into equivalent ADC counts and added to the cell ADC counts from the raw minimum bias data before the zero suppression cut is performed. The events are then reconstructed as if the combined event originated in DO. This data is then used to measure the efficiency of the H-matrix X2 cut. A plot of the efficiency of this cut versus input electron energy and calorimeter IETA is shown in Figures 4.6 and 4.7. 101 Efficiency Efficiency 1.4 1.2 0.8 0.6 0.4 0.2 1.4 1.2 0.8 0.6 0.4 0.2 CC H—motrix x2 Efficiency vs. llETAl llllllllll[ITTIIIIIIIIITIIIH {/nm 16.55 7 9 P1 .6762: .7727E-02 P2 —.531;_E-02t .11876-02 7% Ma“ _+__+_ 1J1l1411d111111111lL1 2.5 5 7.5 10 META] 10 GeV MC cc 5(x’<100) vs. lETA ,_ {/11693315 / 9 1— Pt 8348* .5481E-02 : P2 .2367E-O4t .5066E-03 L L Lh___..__._—==_,_==_.,.__._.__o—r L E [I 7L11l1141l1111l1111111 2.5 5 7.5 10 [IETA] 25 GeV MC CC 5(x2<100) vs. IETA Efficiency Efficiency 1.4 1.2 1 0.8 0.6 0.4 0.2 O 1.4 1.2 0.8 0.6 0.4 0.2 .. {/ncinii / 9 __ Pt .9211: same-02 *- PZ -.27¢7E-021 .QGBOE-OJ b *7 d_ —,_‘ C L C Plllllllllll1JLlLLLlli 2.5 5 7.5 10 2 |IETA| 15 GeV MC CC e(x (100) vs. IETA L. {Aldrin—é / 9 P1 3673* .‘2575-02 E P2 -.1373E-02i .5595E-03 [— E [- L— r- _ILIlllllllllllllillll 1 T 50 GeV MC cc £(x’< 100) vs. IETA 2.5 5 7.5 10 IE AI Figure 4.6: CC H-matrix X2 < 100 Efficiency vs. input |IETA|. 102 Efficiency Efficiency 1.4 1.2 0.8 0.6 0.4 0.2 1.4 1.2 0.8 0.6 0.4 0.2 EC H—motrix x2 Efficiency vs. llETAI Efficiency _ {/m 10.47 / a __ in 1.173: .4674E—01 i: P2 -.2426€—01: awn-02 E _H—N ; +—+— E- :lllllILlJlllllilllll 15 17.5 20 22.5 25 |IETA| 25 GeV MC EC 5(X’<1oo) vs. IETA _ {/nm 3.13.1 T a _ P1 .9995; sens-01 1. P2 —.1761E—02t .17735-02 l.— Ej—A —%—+ 1— —9— C. [- Pliiiiliiillimiiliiiil 15 1 7.5 20 22.5 25 |IEIA| 100 GeV MC EC s(x’<100) vs. IETA Efficiency 1.4 1.2 1 0.8 0.6 0.4 0.2 O 1.4 1.2 0.8 0.6 0.4 0.2 1. {/ndf4.228 / B _ Pt .95781 .4219E-01 '- PZ -.73‘6E-021 .2058E-02 E 'a—W t [— E— L L_liiiiliiiiliriiliiiil 15 17.5 20 22.5 25 HETAI 50 GeV MC EC 5(X2<1oo) vs. IETA _ {#415290 / 9 — P1 .8294: 29185—01 - P2 _.2_831E—02t .14505-02 :7: + * ++ F +74— C E PlillllLtllllAllllllll 15 17.5 20 22.5 25 |IETAI 200 GeV MC EC 8(x2< 100) vs. IETA Figure 4.7: EC H-matrix x2 < 100 Efficiency vs. input |IETA|. 103 Events N o o I 150 —[ 100 ’- .. [ll 50’- C ”flit 0 ALJJJJlJfilth‘JLllL‘Allll‘llllllILJIIJUJILLJ—llllll 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Isolation fraction Isolation Fraction - Electrons Figure 4.8: Isolation fraction for electrons. 4.6.2 Isolation Fraction This variable is a measure of how isolated the PELC cluster is in the calorimeter. The definition of the isolation fraction is E52? — E55?! 150 = EEM COf'C ETotal where cone is the total energy (sum of all calorimeter layers) contained in a cone of radius AR 2 x/Ani + 3452 = 0.4 and E53}: is the EM energy (sum of the four EM calorimeter layers) in a cone of radius AR = 0.2. The electron candidate is required to have I S 0 < 0.15. The isolation fraction for electrons and background is shown in Figures 4.8 and 4.9 respectively. Given the definition of this cut, one would expect that its efficiency increases with increasing electron energy. Since the amount of activity increases nearer the beam 104 331000 0 1.); r i It 800 [ l 7 il l i l 600 1- ] l i l L 400 _ i t 1 1+1 11+ H 200 f 7 1*] “1 > +11 t ”if“ M O o .1 11 r . r .1. . . . 1 . 1.44 mnii’tfi’f'rfimfln 4.. .11..- _ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Isolation fraction Isolation Fraction - Background Figure 4.9: Isolation fraction for background. pipe, one would also expect that the efficiency of this cut decreases as [17] increases. The efficiency plots in Figures 4.10 and 4.11 bear out these expectations. 105 Efficiency Efficiency 1.4 1.2 0.8 0.6 0.4 0.2 1.4 1.2 0.8 0.6 0.4 0.2 Illllllllllllll IIIIIITVITTIIIIIIIITTIITITTI IIIIIIIIIIIFII .8417 t .8708E -02 .1345E-02 lL1LLlL111l1111111 10 HETAI 10 GeV MC CC £(lSO<0.15) VS. IETA L .9925: .27a9E-02 : -.2qssc—02: woos-03 E E— 1. Priiliiirliiiiliiirlii 10 llETAI 25 GeV MC CC 5(ISO<0.15) vs. IETA Efficiency 1.4 1.2 1 0.8 0.6 0.4 0.2 1.2 Efficiency 0.8 0.6 0.4 0.2 CC lsolotion Fraction Efficiency vs. |IETAI / 9 .9681 : -.7290£—02 1 .5267E—02 .8900E-OJ 11111141—1111LLJL11 1_1 l 7.5 10 IlETAI 15 GeV MC cc £(ISO L” 700 600 500 400 I 300 fi+ 200 100 1111+ 1111 I 1 2 3 4 5 dE/dx FDC dE/dx - Electrons +-+ + + .p O . 111l111111111‘1‘1-v-L—1—LL14L1 IIITTIIIITTIIITTITITTTIIIIIIIIIIIIIITITTTIIIT 1 2 3 4 5 dE/dx CDC dE/dx - Background 1L+ +1 + + ++++++ + + + [T1IITTIIIIIIIIIIIIIII1111111111111]!1 3: + 4. *+*++ [IT llllllllllllllnTr‘P‘ *llll 1 2 3 4 5 dE/dx FDC dE/dx - Background Figure 4.18: dE/dsc for electrons and background. 119 Table 4.1: Table of electron selection cuts. 1] Cut Variable I Cut H Level 1 ET ET > 7 GeV Level 2 ET ET > 10 GeV Level 2 isolation fraction I S 0 < 0.15 Online fiducial cut |IETA| S 12 or 14 S |IETA| S 32 EM energy fraction fEM > 0.9 Single tower energy Ehot > 0.4 x Eda, Track in road Anczuktrk < 0.1 and Adam-” < 0.1 Offline ET ET > 11 GeV Offline fiducial cut |IETA| S 11 or 16 S |IETA| S 25 H-matrix X2 x2 < 100.0 Isolation fraction I S 0 < 0.15 Track match significance am, < 10.0 CDC dE/da: (|IETAI S 11) dE/a'a: <1.6 or dE/da: > 3.0 FDC dE/dx (16 S |IETA| S 25) dE/da: <1.5 or dE/da: > 2.6 4.7 Summary A summary table of all the cuts used to select the data sample for this analysis is given in Table 4.1. An event was required to contain 2 electron candidates which passed these cuts to be included in the data sample. Table 4.2 summarizes the single electron efficiency parameterizations for all efficiency corrections made on the data in the following chapters. The Level 1 and Level 2 trigger efficiency curves shown in 4.1 and 4.2 had no additional cuts made on the input Monte Carlo electrons. Since the cuts made in the Level 1 and Level 2 triggers are correlated with the calorimeter based offline electron id cuts, it is necessary to make these cuts on the Monte Carlo electrons in order to be able to simply multiply the resulting Level 1 and Level 2 efficiencies with the efficiencies of the offline cuts when making the overall electron efficiency corrections to the data. Figure 4.19 shows the Level 1 and Level 2 electron efficiency after making the ET, H—matrix, and isolation fraction cuts given in Table 4.1. 120 Table 4.2: Table of single electron efficiency parameterizations. [I Cut Variable I Efficiency Parameterization Level 1 ET 6 = % + §tanh (Egg) w = 3.255 :1: 0.927 Level 2 ET + [50 c = %+ §tanh (L—JETJIO) w = 0.701 :1: 0.072 PPHO Efficiency 600 = 0.971 61.30 = 0.997 PELC/PPHO Efficiency 6 = 0.869 CC H-matrix X2 + ISO c=a+bx |IETA| (0944:0016) “(1035550) : (157—E?) C(E>50) = 0.972 00035525) = (—0.017 :1: 0.002) + (0.0006 i 0.0001)E b(E>25) = —0.0017 EC H-matrix x2 + ISO €=a+bx |IETA| «255652001 = (0.582 :E 0.055) + (W) a(E>200) = 0.85 11125313900) 2 (0.024 :1: 0.003) + (to-076mm) (157—20)} b(E>200) = 0.0028 Utrk + dE/dx c = a + b x (IETA)2 a = 0.934 :1: 0.010 b = —0.00047 :1: 0.00006 121 Level 1 and Level 2 trigger efficiency vs. input E, (after offline cuts) 5‘ C 1 c a) IQ 1 «WW 5 0.8 E r Fit —> 0.5 + O.5*tanh((E,—7.0)/w) f- O.6 i:- 0.4 :— 0.2 _— Xz/ndf 6.277 / 5 ; w 3.255 :1: 926613-01 O 1 1 J l 1 1 1 l 1 1 1 1 l L 14 ; L1 1 1 1 l 0 5 10 15 20 25 E n 5(L1 E. > 7 GeV) vs. 15...... W" >s o 1- :§ - Ca 08 c. C Fit —> 0.5 + O.5*tanh((E,—10.0)/w) 0.6 — 1- 0.4 r- C 5 1—1 0.2 f Xz/ndf 6.862 / 5 ” w .7010 :1: .7198E-01 O L #1 g L L L L l 0 l l l l 1 4L 1 1 l I l l l l l O 5 1O 15 20 25 E inpu 5(12 E.>10 GeV + |SO 10 GeV which pass the Level 2 ELE_2_HIGH filter) used for the Drell-Yan analysis. The Z electron sample is then collected as follows: One PELC is required to pass all the cuts described in Table 4.1. The other PELC is then required to pass all the cuts described in Table 4.1 except the track match significance and dE/da: cuts. The invariant mass of the pair is then required to be within a 80-100 GeV mass window. This is sufficient to provide a relatively background-free di-electron sample which is used as the denominator of the tracking cut efficiency distribution. 123 CC H—rnatrix x2 + Isolation Fraction Efficiency vs. llETAl 1011612686 / 9 1011111129? / 9 >5 >s U _ 0 _ QC, 1-4 35 -10353331: 1:33:35 5 1-4 F 51 -.m1‘?3§§: 132233 g3 1.2 L $2 1.2 t— 1 :- l _— M t 0.6 _— 0.6 :— 0.4 :— 0.4 :— t : 0.2 :- 0-2 I" t 1 1 1 1 C 1 1 1 1 O 14441111111111111 O Ill 1111 [LI] [111 [I 2.5 5 7.5 10 2.5 5 7.5 10 IIETAI |IETA| 10 GeV MC cc 5(X’<100 + ISO3 _ {(11111 4231 / '9 5‘ __ {170111503437 9 ‘ .— . . 1 - ._ . . - QC) 1'4 ~_- :2 -.1662E9302: 5:62:35 QC, 1'4 : P2 -.1711_1_:::2: 5737535 :g—f 1.2 E- fg’ 1.2 :— L1J - L1J _ 1 :- S 1 :— _— —— __ _ 0.8 :— 0.8 E— 0.6 :— 0-6 E- 0.4 E- 0-4 E- 0.2 :— 0-2 E OClllllllllllllllJlllll O:lllllllllllllllllllll 2.5 5 7.5 10 2.5 5 7.5 10 ”ETA! |1ETAI 25 GeV MC CC £(X2<1OO + |SO PELC 2) and repeating this procedure. The tracking cut efficiency versus IETA is shown in Figure 4.22 along with a quadratic fit of the form 6 = p1 + p2 X IETAZ. 126 Efficiency —§ 0.8 0.6 0.4 0.2 Track match signfifcance + dE/dx cut efficiency from Z electrons “ XZ/ndf51.63 / 42 — P1 .9344 :t .977OE-02 _ P2 —.4650£—03 :t .55705—04 1 1 ‘ 4 L H 41- 1H 1 - N ./E f -0 \ .111 ” '1 W Fit —> 6 = p, + pZXIETAz L I I I I #1 I I I I I I I I I I I I I I I I I I -20 -1O 0 10 2O IETA 8(0111 + dE/dx cuts) vs. IETA Figure 4.22: Track match significance + dE/dm cut efficiency vs. input |IETA|. Chapter 5 Data Analysis 5.1 Selected Data Sample The initial data sample for this analysis consisted of events which passed the ELE.2-HIGH trigger/filter combination outlined in Chapter 4. These events were further required to contain two PELC objects (see Chapter 4) with ET > 10.0 GeV. This sample contained 18,749 events. The offline electron identification cuts were then applied to both PELC objects in each event in this data sample and the invariant mass of the PELC-PELC pair was required to be between 30 and 60 GeV/c2. This selection criteria yielded a signal + background sample of 143 events. Figures 5.1, 5.2, 5.3 show the resulting signal + background distributions vs. invariant mass, pair rapidity, and pair p1. Tables 5.1, 5.2, 5.3 show the number of signal + background events per pair mass, rapidity, and pr bin respectively. 5.2 Background Estimation There are several sources of background contamination in the Drell-Yan 6+6— signal sample. They are: 6 Dijet events in which the jets were mis-identified as electrons. 127 128 Table 5.1: Drell—Yan signal + background events per invariant mass bin. II Mass Bin (GeV/c2) Events I Error II 30.0 - 35.0 37.00 :1: 6.08 35.0 - 40.0 33.00 :1: 5.75 40.0 - 45.0 23.00 :1: 4.80 45.0 - 50.0 21.00 :1: 4.58 50.0 - 55.0 16.00 :1: 4.00 55.0 - 60.0 13.00 :t 3.61 II Total I 143.00 I :1: 11.96 II Table 5.2: Drell-Yan signal + background events per pair rapidity bin. Rapidity Bin Events I Error -2.5 - -2.0 4.00 :1: 2.00 -2.0 - -1.5 8.00 :1: 2.83 -1.5 - -l.0 13.00 :1: 3.61 -1.0 - -0.5 21.00 :1: 4.58 -0.5 - 0.0 16.00 :1: 4.00 0.0 - 0.5 21.00 :1: 4.58 0.5 - 1.0 21.00 :1: 4.58 1.0 - 1.5 20.00 :1: 4.47 1.5 - 2.0 13.00 :1: 3.61 2.0 - 2.5 6.00 :1: 2.50 Total 1 143.00 [ i 11.967] 129 Table 5.3: Drell-Yan signal + background events per pair pT bin. [1 PT Bin (GeV/c) I Events I Error II 0.0 - 1.0 4.00 :t 2.00 1.0 - 2.0 14.00 :1: 3.74 2.0 - 3.0 15.00 :1: 3.87 3.0 - 4.0 15.00 :1: 3.87 4.0 - 5.0 12.00 :1: 3.46 5.0 - 6.0 9.00 :1: 3.00 6.0 - 8.0 14.00 :1: 3.74 8.0 - 10.0 13.00 :1: 3.61 10.0 - 12.0 12.00 :t 3.46 12.0 - 16.0 12.00 i 3.46 16.0 - 24.0 12.00 i 3.46 24.0 — 32.0 5.00 i 2.24 32.0 - 40.0 2.00 :1: 1.41 40.0 - 50.0 3.00 i 1.73 II Total 142.00 :1: 11.92 Drell-Yon signal 4» background sample Events 45~ YIIr‘TTl'IjY 'III V Y7 O 04 OT'leITY p b— _- — )— lnvoriont moss (CeV/c’) 1A, - 30<11,<60 Figure 5.1: Drell-Yan + background events vs. invariant mass. 130 Drell—Yon signal «4» backgrOund sample Events 25*- l 7 7 2O IfT hi I 1 V ] T—Y T ‘r o A A A L I A L A A l l A I A l L A L l l I L A L I L J L l l l. L A L l l. 1 l J —4 -3 -2 -l 0 1 2 3 4 Pair rapidity y, — 30- 5 2.5 OEAJAIIILIIJIAAIIA VIAJ‘IIALJIIl‘llllllllllrllllAll 0 5 10 15 20 25 30 35 4O 45 50 Pair 9, (GeV/c) ph-JO 61/ + jets events in which a jet is mis-identified as an electron. 6 Z—> 7+7“ —> e+e‘ events. The largest background contribution comes from particle jets which pass the elec- tron identification cuts described in Table 4.1. The probability that a jet “fakes” an electron in D0 is on the order 1 x 10“, and since we require two electrons in an event, the probability that a dijet event fakes a Drell-Yan event is the square of the probability that a single jet produces a fake electron. Although the probability that a dijet event fakes a Drell-Yan event is very small, the dijet cross section is several orders of magnitude larger than the Drell-Yan e+e‘ cross section and thus the dijet background is a significant background. Figure 5.4 shows the dijet background events relative to the Drell-Yan signal + background events. This background is 30—40% of the total Drell-Yan signal + background sample. The method used to estimate this background is described in the next section. Tables 5.4, 5.5, 5.6 show the number of dijet background events per bin in pair mass, rapidity, and p1 respectively. The reason that the integrated dijet background is smaller in the rapidity, and p7 distri- butions compared to the mass distribution (45.16, 45.17, vs. 45.61 events) is that all distributions have the 30 < M < 60 GeV/c2 requirement made on them, but the rapidity and p71 distributions also have the further constraint of the histogram bounds and we do not include overflow and underflow bins in the integral. The reason the error on the integrated dijet background p1 distribution is smaller than the error on the mass and rapidity distributions is due to a smaller systematic variation in the pT distribution when the rejection fit parameters are varied. 132 Table 5.4: Dijet background events per invariant mass bin. Mass Bin (GeV/c2) Events Error II 30.0 - 35.0 10.85 :1: 0.70 35.0 - 40.0 9.21 i 0.58 40.0 - 45.0 8.04 i 0.51 45.0 - 50.0 6.73 :1: 0.44 50.0 - 55.0 5.78 i 0.38 55.0 - 60.0 5.00 :1: 0.34 II Total 45.61 :1: 1.24 Table 5.5: Dijet background events per pair rapidity bin. I Rapidity Bin Events Error —2.5 - -2.0 2.77 :1: 0.35 -2.0 - -1.5 4.81 :1: 0.38 -1.5 - -1.0 4.87 :1: 0.33 -1.0 - -0.5 4.11 :1: 0.30 -0.5 - 0.0 3.46 :1: 0.29 0.0 - 0.5 3.86 :1: 0.33 0.5 — 1.0 4.66 :1: 0.34 1.0 - 1.5 6.51 i 0.42 1.5 - 2.0 5.99 :1: 0.43 2.0 - 2.5 4.14 :1: 0.47 II Total r4516 I :1: 1.16 II 133 Dijet background events vs. Drell—Yon signal + background sample E C ' ———+———— °’ - -——+——___+_ J Li """""" l 1 10 E‘ _________________________________ . ______ 1 1 10—1E1 l l l i l l l l l l l l 1 l l J l L l l l l 1 l l 1 1 l 30 35 4o 45 50 55 50 Invariant mass (GeV/c2) M, — 30 6+6” and Z -> e+e‘ since the final state is identical. Fortunately, the Z —4 e+e" cross section is small in the invariant mass region 30-60 GeV/c2 of this analysis. However, in the invariant mass region of the Z resonance (around 91.17 GeV), the Z——> [+£- production cross section is several orders of magnitude larger than the virtual photon cross section, and thus if a mechanism exists for an on-shell Z boson to produce a pair of electrons with an invariant mass in the 30-60 GeV/c2 range, these events will contribute to the Drell-Yan background. It turns out that such a mechanism does exist, namely the decay chain 136 Events Direct photon background compared to Drell—Yon signal + background E 4' Afi 1 10 if T _____*—_‘——‘ _1 -------------------------------------------------- 1O -2 10 10- i 1 1 1 1 1 1 1 1 1 l l 1 1 1 1 1 1 1 1 1 L L L 1 L 1_ 1 1 30 35 4O 45 5O 55 60 Invariant mass (GeV/c2) M, — 30 w 1 m + T t i- 4— J— d— T _— _1 _ bbbbb -b—_,. .__h 10 '"M'- ._2 [wk-[:r-r-J. 1O i-.-'L_:_ . . _L -L_.. ‘ . . -r -+-+-F_L . , , - : .—L ‘t 1O_3 1 1 1J11 LL11111 11111111 L'LL11111'111 1.1.111 rlLLLILLLi-11 1 1 1-p11-1-17111JJ 111 O 5 1O 15 20 25 30 35 4O 45 50 Pah'p,(GeV/c) p77 - 3O e+e‘ background events per invariant mass bin. Mass Bin (GeV/c2) Events Error 30.0 - 35.0 0.64 :1: 0.06 35.0 - 40.0 1.01 :t 0.08 40.0 - 45.0 1.17 i 0.09 45.0 - 50.0 1.03 :l: 0.08 50.0 - 55.0 0.89 :l: 0.07 55.0 - 60.0 0.71 :l: 0.07 II Total 1 5.44 l :I: 0.182 n Z—) 7+7” —-> e+VVe‘I/V. It is easy to see how this decay can produce an electron pair with an invariant mass in the 30-60 GeV/c2 range using the following argument: if we assume that energy (mass) of the Z boson is divided equally between the 7+ and 7" , then each 1' will have an energy of around 45 GeV. The T leptons can then decay to an electron and two neutrinos, and if the 7' energy is equally shared between the electron and neutrinos, the resulting electrons will each have an energy around 15 GeV, thus if the final state electrons are roughly back to back in the lab frame, the resulting invariant mass of the pair will be on the order of 30 GeV/c2. Obviously this argument ignores the detailed kinematics of the decays, but it does illustrate how an on-shell Z boson can produce an electron pair in the 30—60 GeV/c2 invariant mass region. The size of this background contribution is reduced by the branching ratio of r ——> am and the details of the kinematics, but the result is that this mechanism produces background that is as much as few percent of the Drell-Yan + background sample in the mass range of 30—60 GeV/c2. Figure 5.7 shows the background events contribution from Z—+ T+T" —> e+e" events relative to the Drell-Yan signal + background sample. Once again, the apparent difference between the size of the background amongst the plots is due to the binning. Tables 5.7, 5.8, and 5.9 show the Z—> 71'7"” —> 6+6- background events in each pair mass, rapidity, and p1 bin respectively. 139 Table 5.8: Z—> T+r' —+ e+e‘ background events per pair rapidity bin. I Rapidity Bin Events Error -2.5 - -2.0 0.05 :l: 0.01 -2.0 - -1.5 0.18 :t 0.03 -1.5 - -1.0 0.39 :I: 0.04 -1.0 - -0.5 0.80 :t 0.07 -0.5 - 0.0 1.30 :I: 0.09 0.0 — 0.5 1.28 :1: 0.09 0.5 - 1.0 0.77 :I: 0.07 1.0 - 1.5 0.42 :l: 0.05 1.5 - 2.0 0.21 :t 0.03 2.0 - 2.5 0.05 :1: 0.01 Total 5.44 :t 0.182 Table 5.9: Z—) 7+7“- -—> e+e‘ background events per pair p1 bin. p1 Bin (GeV/c) Events Error 0.0 - 1.0 0.12 :I: 0.03 1.0 - 2.0 0.34 :l: 0.04 2.0 - 3.0 0.37 i 0.05 3.0 - 4.0 0.44 :l: 0.05 4.0 - 5.0 0.34 :t 0.05 5.0 - 6.0 0.33 :1: 0.05 6.0 - 8.0 0.74 :1: 0.07 8.0 - 10.0 0.51 :1: 0.06 10.0 - 12.0 0.53 :t 0.06 12.0 - 16.0 0.82 :l: 0.07 16.0 - 24.0 0.74 :l: 0.07 24.0 - 32.0 0.13 :t 0.03 32.0 - 40.0 0.03 i 0.02 40.0 - 50.0 0.01 :I: 0.01 Total 5.43 i 0.182 140 Z —>'r'r-—>ee background events vs. Drell—Yon signal + background sample U) I __*__— L5 10 E- 1 E: -------------------------------------------------------- ~ .................. _1E 10 10—2 #1 L1 1 J 1 1 1 1 1 1 1 1 L LJ 1 1 1 1 1 1 J 1 1 1 1 3O 35 4O 45 5O 55 60 Invariant mass (GeV/c2) M,— 30J —t_ ——1— 1 ........................... ..--_ _1 --_-+ ------- v- ------- v- ------- h____ 10 ____L___ 1O_211L1111111111111111111111111J111LL1L11111L1111111 -2.5 -2 -1.5 -1 —O.5 O 0.5 1 1.5 2 2.5 Photon rapidity y, - 30 “J *1 “ii W I 1 1C)_1 - - +++++++++ -1- ++ 10— J11J1111111111111L111J 1J—i-L1-F'F1-1-1-1-1-1'1 111111'117111 111711111111 5 10 15 20 25 30 35 4O 45 50 O Photon pI (GeV/c) p., — 30 7+7” —> (3+6— background events (dashed) vs. Drell-Yan signal + background events (solid). 141 5.2.1 Dijet Background Estimation Method The probability that a jet is mis—identified as an electron in DO has been studied extensively [63] [62]. However, due to the lack of a sufficiently large sample of dijet events, and the small probability that a jet fakes an electron, the error on the fake elec- tron probability measured in these studies is large (as much as 50%). Consequently these probability estimates are not used to estimate the dijet background contribu- tion to Drell-Yan. Rather, a less error prone method is used to estimate the dijet background. Since the initial sample of events from which the Drell-Yan signal sample is selected is large and is almost entirely composed of dijet (or multijet) events, it is possible to use this sample to estimate the dijet background. We start by selecting a sub-sample of our initial data sample by requiring that the event contain 2 PELC objects each with ET > 11 GeV, that are in the fiducial region |IETA] < 11 or 16 < [IETA] < 25, with an invariant mass between 30 and 60 GeV/c2. Since the dijet cross section is so large relative to any other cross section in this invariant mass range, the PELC-PELC sample is comprised almost entirely of dijet events which pass the (rather loose) PELC selection requirements. Two sets of histograms of PELC energy, ET and IETA are then produced from this sample. One set of histograms contains entries of PELC objects with |IETA] < 11 (CC) and the other set contains entries of PELC objects with 16 < |IETA] < 25 (EC). We fill these histograms with both PELC objects in the event in order to double our statistics, but a given PELC only gets entered into a single histogram. The data is split into two sets because the rejection of the offline electron id cuts is different in the CC and EC. Two more sets of histograms of the same quantities are then filled with PELC objects which pass the electron id cuts. Since we only make the additional electron id cuts on a single PELC in this case 142 (which then gets entered into a histogram if it passes), this sample is still mostly background. We then switch which PELC has the additional electron id cuts made on it (and consequently gets entered into a histogram if it passes) again to double our statistics and remain consistent with the histograms without the additional cuts. Since these histograms contain Drell-Yan events we should subtract these signal events since we want to our PELC-PELC histograms to measure the rejection of our electron id cuts on jets which passed the PELC cuts. In order to do this, we use the RESBOS Monte Carlo to estimate the Drell-Yan cross section given our fiducial and kinematic cuts. We fill two sets of histograms (CC and EC again) of ET, energy and IETA with both Drell-Yan Monte Carlo electrons if the event passes our kinematic and fiducial cuts, where each electron is weighted by the PELC efficiency given in Chapter 4 times our integrated luminosity and the Monte Carlo event weight. We then fill another two sets of histograms of ET, energy and IETA of both electrons where each is additionally weighted by the product of the efficiencies of the all the electron id cuts to estimate the number of signal events in our dijet PELC-PELC histograms. We then subtract these signal estimates. Other real electron contamination in these background samples is ignored since it is very small relative to the mostly dijet PELC- PELC sample. We then divide the resulting histograms of PELCs (which now have the Drell- Yan signal events subtracted out) that passed the additional cuts by the histograms without the additional cuts to produce plots of the rejection of these cuts vs. energy, ET and IETA. These plots are then fit in order to parameterize the E, ET or IETA dependence (if any) of the electron id cuts’ rejection. This yields CC and EC single PELC —> electron rejection in functional form. Figures 5.8 and 5.9 show the fitted rejection for CC and EC PELCs respectively. To aid the reader in understanding the foregoing description of the rejection estimation method, a “cartoon” of the rejection 143 in a given bin is e—DY f(PELC —> e) = PELC Now that we have an estimate of the background rejection of the electron id cuts on a single PELC in our sample, we make yet another set of histograms of the variables we are interested in measuring (namely pair mass, rapidity, and p1) using our initial PELC-PELC sample. Each event is required to pass our kinematic and fiducial cuts and is weighted by the product of the rejections we expect on each PELC based on each PELC’s ET or IETA which is input to our rejection fits (depending on whether it is in the CC or EC). These histograms then contain our bin-by-bin dijet background estimate which we will subtract from our signal sample. A cartoon of the dijet back- ground in a given bin is 13,-, = (PELC-PELC) x f(PELCl —->e) x f(PEL02 —>e) In order to estimate the systematic error induced by using our rejection fits, this background estimate is then recomputed 100 times by varying the rejection fit para- meters. The EC rejection fit is varied by varying the two fit parameters according to correlated Gaussians with mean zero and widths equal to the errors on the parameters from the fit. Correlated Gaussians are generated using the CERN program library routine CORGEN and are used because the fit parameters are very anti-correlated. The correlation is given by the covariance matrix from the fit. The CC rejection fit has only one parameter which is also varied by an independent Gaussian random number (gotten from the CERN program library routine RN ORML) with mean zero and width equal to the standard deviation on this fit parameter. The standard devi- ation of these 100 background estimates is then computed bin—by-bin in an attempt 144 C(PELC —-> e) 0.8 0.6 0.4 0.2 0.06 0.04 0.02 Central PELC —) electron rejection xz/ndf 7.257 / 7 ; AO .6319E-01i .2609E-02 F :— Fit -> 5 = constant C —"— E 11A1Jr11v11rlLv1111T$L1111111111111111111111111JJ_111111 1O 20 3O 4O 50 60 7O 80 90 100 1 10 E,(Gev) CC PELC Eh cuts I X’/ndf33.58 / 21 :— AO .6089E—O12t .2551E—02| I. I+ .. HH III H IHI E— Fit —> £=constant 11111111111111111111111111111111JL1J_11111111_L —10 —7.5 —5 —2.5 O 2.5 5 7.5 10 IETA CC PELC IETA, cuts Figure 5.8: CC PELC rejection vs. cluster ET, IETA. 145 s(PELC-—>e) F3 a) _. .0 an .0 is 0.2 Forward PELC —> electron rejection Xz/ndf 15.29 / 6 A0 .1108 :I: .4295E—02 Fit —> 5 = constant —-<>—— A _J_ IIII’IYIITIIIITIIflTTT 1111 L i v Y 1 T 11111111111111111111111111J11111L111111111111 0 2O 3O 4O 50 60 7O 80 90 100 1 10 E, (Cev) EC PELC ET. cuts xz/ndf 5.750 / 8 P1 P2 .3765E—012t .8060E—02 .4727E-O6zl: .554OE-O7 ; i Fu—>a=m+pmmmr E 1 l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16 18 20 22 24 IIETAI EC PELC IETA, cuts Figure 5.9: EC PELC rejection vs. cluster ET, |IETA]. 146 to estimate the systematic error in this background measurement. The standard devi- ation of these measurements is then added in quadrature with the statistical error of the background computed using the central value of the fit parameters. The resulting dijet background estimate is shown in Figure 5.4. For further clarification on how this systematic error is computed see Section 5.3 (Online Efficiencies) where the Level 1 efficiency calculation is described. The systematic error estimation procedure is es- sentially the same, but the PELC-PELC event sample is used as the input rather than the RESBOS Drell-Yan Monte Carlo, and two rejection fits (CC and EC) are used for the background so more random numbers are required. Of course no systematic error due to Monte Carlo is performed here since no Monte Carlo events were used. 5.2.2 Z—) r+r‘ ——> e+e‘ Background Estimation Method The Z—> 7+7” —+ 6+6“ background is measured using the ISAJET Monte Carlo to generate the Z—> T+T" -) e+e" cross section and kinematics. This cross section is then multiplied by the integrated luminosity for the ELE_2_HIGH trigger/filter combination and the online and offline electron id efficiencies computed using the electron kinematics of the Z—> 7+7” —+ e+e' events. This yields an estimate of the number of background events in each mass, rapidity, or pT bin of the data. The resulting background is shown in Figure 5.7. 5.2.3 Background Subtraction The dijet and Z—) T+T' —> e+e‘ background is subtracted bin-by-bin from the signal + background distributions. Figure 5.10 shows the Drell-Yan distribution after sub- tracting both the dijet background shown in Figure 5.4 and the Z—-> 7+7” —> e+e‘ background shown in Figure 5.7 from the signal + background sample in Figures 5.1, 5.2, and 5.3. The number of events remaining per photon mass, rapidity, and pT bin 147 Table 5.10: Drell-Yan events per invariant mass bin. [[Mass Bin (GeV/c2) IEvents I Error II 30.0 - 35.0 25.51 :1: 6.12 35.0 - 40.0 22.78 i 5.77 40.0 - 45.0 13.79 a: 4.82 45.0 - 50.0 13.24 d: 4.60 50.0 - 55.0 9.34 :l: 4.02 55.0 - 60.0 7.29 :1: 3.62 II Total 91.95 I :I: 12.02 Table 5.11: Drell-Yan events per photon rapidity bin. Rapidity Bin Events Error -2.5 - -2.0 1.18 :1: 2.03 -2.0 - -1.5 3.01 i 2.85 -1.5 - —1.0 7.74 :l: 3.62 -1.0 - —0.5 16.09 :1: 4.59 -0.5 - 0.0 11.24 :I: 4.01 0.0 - 0.5 15.86 :1: 4.60 0.5 - 1.0 15.57 :I: 4.60 1.0 - 1.5 13.07 :1: 4.49 1.5 - 2.0 6.80 :1: 3.63 2.0 - 2.5 1.81 :I: 2.49 If Total 92.40 |i12.01]] after background subtraction are given in Tables 5.10, 5.11, and 5.12 respectively. The difference in the event totals between the mass, rapidity, and m distributions are given by da/dm -+ 91.95 = 143849.].ka — 45.61jj — 5.442.," dU/dy -> 92.40 = 1433gg+bk9 - 45.16jj — 5.442.”, da/dm —> 91.40 2' 142sig+bkg — 45.17jj -- 5.432.,“- 148 Events Events Events P P i t —+— r 1 1 m :_ l r _I_ :1 l l 1 l l 1 l l l l l l I I l I l l I l 1 1J l 1 [L1 30 35 4O 45 5O 55 60 Invariant mass (GeV/c2) (DATA—8K6) “ + I 1 —+— 10 E' ___I___ '-+-—' T I C __,,_ I F hllllllJlJllilllIlllllllLllkLllllllIlJlLlllllllllll —2.5 -2 -1.5 -—1 -O.5 O 0.5 1 1.5 2 2.5 Pair rapidity (DATA—8K6) 10 i++++ji 1 1 _____+_____ —1 I 10 IIIIJIIllllllllijlllllllllllLlllllJ llllllll IJLJ O 5 1O 15 2O 25 3O 35 4O 45 5O Drell—Yon (signal + background) — (dijet + Z—>'rT—>ee) background Pair pt (GeV/c) (DATA—JJBKG) Figure 5.10: Drell—Yan (signal + background) - (dijet + Z—> 7+7" —> e+e‘) back- ground events. 149 Table 5.12: Drell-Yan events per photon pT bin. II 137 Bin (GeV/c) I Events I Error II 0.0 - 1.0 3.08 :1: 2.00 1.0 - 2.0 11.28 :t 3.75 2.0 - 3.0 11.33 :1: 3.88 3.0 - 4.0 10.86 :1: 3.88 4.0 - 5.0 7.83 :l: 3.48 5.0 - 6.0 4.94 :l: 3.01 6.0 - 8.0 7.21 :1: 3.76 8.0 - 10.0 7.85 :1: 3.62 10.0 - 12.0 7.73 :l: 3.47 12.0 - 16.0 6.30 :1: 3.47 16.0 - 24.0 6.64 :l: 3.47 24.0 - 32.0 2.87 :l: 2.24 32.0 - 40.0 0.98 :1: 1.42 40.0 - 50.0 2.49 :l: 1.73 II Total I 91.40 | :I: 11.95 H 5.3 Online Efficiencies Figure 5.11 shows the Level 1 trigger efficiency vs. invariant mass, photon rapidity and photon p1. These plots were generated using the Level 1 efficiency fit vs. elec- tron ET shown in 4.19. RESBOS Drell-Yan Monte Carlo events were used to provide the event kinematics which were input to the efficiency fit for each electron in each event. The event efficiency was then taken to be the product of the individual electron efficiencies. The general method for the efficiency in bin i is 1 ”i 5i : I—V‘I'IZIC cut( “)X€cut(ej) where N,- is the number of Monte Carlo events in bin i and 6631(631‘) is the efficiency of the cut evaluated using the electron kinematics from the Monte Carlo. The various em functions are given in Table 4.2. Profile histograms were made of the invariant 150 mass, photon rapidity, and photon m- using the event efficiency as the y-axis variable. A profile histogram is a histogram in which each bin contains the average of the y-values of the entries in that bin. These profile histograms contain the average bin- by-bin efficiency in mass, photon rapidity, and photon pT. The standard deviation of the mean in each bin gives the statistical variation of the efficiency estimate in each bin due to the variations of the electron kinematics within a mass, rapidity, or p1 bin. In order to estimate the systematic error associated with this efficiency measurement, these profile histograms were varied using two different methods which correspond to the systematic error induced by using the fit, and the systematic error induced by using the RESBOS Monte Carlo to furnish the electron kinematics. To estimate the systematic error due to the fit, 100 sets of these profile histograms were generated by varying the fit parameter according to a Gaussian distribution with mean zero and width equal to the error on the fit parameter. The mean of each bin from each set of these 100 profile histograms were then used to fill another set of profile histograms of invariant mass, photon rapidity, and photon m. The bin-by-bin spread of these profile histograms gives an estimate of the bin-by-bin variation of the efficiency due to the variation of the Level 1 efficiency fit. The systematic efficiency variation in bin 7' due to the fit is given by 100 N ”i fi=t 1110—: Z€wt(ei ,P I’lkl pi'Zka i“) X ecut(e£-7P’1k11pi2ka - ° ]) k=lj=l 1' fi=t 100;. 2(6 6"“ 151 where N; :2 Monte Carlo events ccut(e+,p’l[,p’2, . . .]) = modified efficiency Pink = Pn + 071an on = error on fit parameter pn an 2 random Gaussian If there is more than one parameter in a fit, the an are correlated; the correlation is specified by the covariance matrix from the fit. To measure the systematic effect of the particular Drell-Yan kinematics used, the profile histograms were filled using the ISAJ ET Monte Carlo to provide the kinematics of the events. A different set of parton distributions was used to generate the ISAJ ET events as well. The bin-by—bin difference between the RESBOS and ISAJ ET Monte Carlo generated histograms gives a bin-by-bin estimate of the systematic variation due to the input Monte Carlo Systematic error from MC in bin 1 —-> a,-_Mc = |p;_RESBos - Iii—ISAJET] The plots shown in Figure 5.11 contain the average bin-by-bin event efficiency vs. mass, rapidity, or p1 using the central value of the Level 1 efficiency vs. electron ET fit with kinematics provided by the RESBOS Monte Carlo. The error in each bin in each plot is the standard deviation of the bin mean added in quadrature with the spread in- duced by varying the fit and the variation produced by using the ISAJ ET Monte Carlo __ 2 2 Ui—total — \/0.'—u '1” Ui—m + 0.12—MC 152 Figure 5.12 shows the Level 2 filter efficiency vs. invariant mass, photon rapidity and photon p7. These plots were generated using the Level 2 efficiency fit vs. electron ET shown in 4.19. The bin-by-bin mean efficiency and statistical + systematic error were produced in exactly the same fashion as for the Level 1 efficiency plots described above. 153 Efficiency Efficiency Efficiency Level 1 efficiency for Drell—Yon eie' events 1 4 ¢ vP ? r T OBE- t I 0.6 H [.1 111111111111111111111 1111111 30 35 4O 45 5O 55 50 Invariant mass (GeV/c2) £(L1E,>7.0GeV)vs.m L. 1 1 I r *1 I 1 1 I 0.8 I:- C 0.6 - L-1111111111111111111111111111111111111LL1111111J11 -2.5 -2 -1.5 -1 -O.5 O 0.5 1 1.5 2 2.5 Photon rapidity £(L1ET>7.OCeV)vs.y I‘IffiaHIIIHHnHmimimmflnmin‘rrrrnm 0.8 — 0.6 *— 1LJ1L11J41111J111J11111J11111111111114J411L1_111111 O 5 1O 15 2O 25 3O 35 4O 45 5O Photon p, (GeV/c) 5(L1 E, > 7.0 GeV) vs. pT Figure 5.11: Level 1 EM ET > 7.0 GeV efficiency for Drell-Yan 6+e- events after offiine cuts. 154 Level 2 efficiency for Drell—Yon eie’ events >\ 8 '9 1 I Y .9 _ L“0.8 1 0.6 1 b.11LL1J1P11L11111 111111 1111 1 11 50 35 40 45 50 55 6o Invariant mass (GeV/c2) £(L2 E,>10.0 GeV + ISO\ U 8 1 I I r r f fl r r r I I9 I L“0.8L C 0.6 :- b.11111ML1_1_1LL1111111141J_1111111111J11111_1_L1111LL11 -2.5 -2 —1.5 —1 —o.5 o 0.5 1 1.5 2 2.5 Photon rapidity £(L2 EI>10.O GeV + |SO\ 8 '5 1 1111iiiiiiiirrirriiililiiHriirHHHHIHIHIm l*-’o.8[— o.6 1 111111111111111J_1_11111J11144111111111111111L111111 0 5 10 15 2o 25 3o 35 40 45 50 Photon pI (GeV/c) £(L2 E1>1O.O GeV + |SO 10.0 GeV + Level 2 ISO < 0.15 efficiency for Drell-Yan e‘ events after offline cuts. 8 + 155 5.4 Offline Efficiencies The single electron offline electron identification cut efficiencies are given in Table 4.2 in Chapter 4 of this dissertation. These efficiencies were parameterized as functions of ET or energy and / or IETA. In order to be able to make bin-by-bin efficiency correc- tions in invariant mass, photon rapidity, or photon m, we must determine the overall event efficiency for each cut on Drell-Yan e+e‘ events as we did for the online cut efficiencies. To do this, we again use the RESBOS Monte Carlo to provide the Drell- Yan electron kinematics which we input to our efficiency fits for each cut variable to calculate the efficiency for each electron. The event efficiency is then the product of the efficiencies for each electron 1’“ Ci — Ccut(e N1 i=1 )x Ecut(6j—) where N,- is the number of Monte Carlo events in bin 7 and ecut(e}t) is the efficiency of the cut evaluated using the electron kinematics from the Monte Carlo. The systematic efficiency variation in bin 2' due to the single electron efficiency fit is given by 100N.- - cu a a X cu T, I a I 1°“ ”1 fit 'll—OOEIZEC t( 6’1 +[1,1911111’211 l) ‘5 t(e. p1k1p2k l) 1 100 _ _ . _ . 2 01’ fit _ 100 k=1(eik I") where N,- 2 Monte Carlo events ccut(e+, p’l[, p'2, . . .]) = modified efficiency 156 p’nk = p11 + 011an on = error on fit parameter pn an = random Gaussian If there is more than one parameter in a fit, the an are correlated; the correlation is specified by the covariance matrix from the fit. The systematic effect of the particular Drell-Yan kinematics used is again estim— ated using the ISAJET Monte Carlo to provide the kinematics of the events. The bin-by-bin difference between the RESBOS and ISAJ ET Monte Carlo generated his- tograms gives a bin-by-bin estimate of the systematic variation due to the input Monte Carlo Systematic error from MC in bin i —-> 0,-_MC = Ip;_RESBos - Hi—ISAJETI The total error in each bin in each plot is the standard deviation of the bin mean added in quadrature with the spread induced by varying the fit and the variation pro- duced by using the ISAJ ET Monte Carlo _ 2 2 2 ai—total —' \/0i—p + ai—fit + ai—MC Figure 5.13 shows the PPHO efficiency vs. invariant mass, photon rapidity, and photon m. Figure 5.14 shows the PELC/PPHO efficiency vs. the same variables. Figures 5.15 and 5.16 show the H-matrix X2 + Isolation fraction cuts and the track match significance + dE/dx cuts efficiencies vs. mass, rapidity, and p7 of the photon respectively. The only significant efficiency dependence appears in the efficiency vs. 157 Table 5.13: Efficiency corrections for Drell-Yan e+e‘ events per invariant mass bin. H Mass Bin l 6L1 l (Sufi éxgmuso l CPPHO I (PELC/PPHO Ca1.1+dE/d11 I Total H 30.0 - 35.0 0.984 0.998 0.788 0.966 0.743 0.741 0.412 35.0 — 40.0 0.987 0.998 0.796 0.965 0.743 0.741 0.417 40.0 - 45.0 0.990 0.999 0.803 0.966 0.743 0.735 0.419 45.0 - 50.0 0.992 0.999 0.812 0.966 0.743 0.740 0.427 50.0 - 55.0 0.994 0.999 0.820 0.966 0.743 0.734 0.430 55.0 - 60.0 0.995 0.999 0.829 0.966 0.743 0.736 0.435 Table 5.14: Efficiency corrections for Drell—Yan 6+6“ events per photon rapidity bin. Rapidity Bin 6L1 6L1 6X31M+Iso €PPHO EPELC/PPHO 6a.,1.+dE/dx Total -2.5 - -2.0 0.993 0.999 0.741 0.988 0.743 0.509 0.275 -2.0 - -1.5 0.989 0.999 0.746 0.980 0.743 0.613 0.330 -1.5 - -1.0 0.987 0.998 0.770 0.970 0.743 0.696 0.381 -1.0 - -0.5 0.989 0.999 0.834 0.958 0.743 0.802 0.470 -0.5 - 0.0 0.991 0.999 0.852 0.955 0.743 0.849 0.508 0.0 - 0.5 0.991 0.999 0.851 0.955 0.743 0.849 0.508 0.5 — 1.0 0.988 0.998 0.836 0.958 0.743 0.804 0.472 1.0 - 1.5 0.986 0.998 0.766 0.971 0.743 0.693 0.377 1.5 - 2.0 0.989 0.999 0.746 0.980 0.743 0.613 0.329 2.0 - 2.5 0.993 0.999 0.742 0.988 0.743 0.509 0.275 photon rapidity plots. This dependence is due to the difference in the efficiencies between the forward and central regions of the detector. 5.5 Fiducial Acceptance, Kinematic Corrections And Unsmearing The stipulations imposed by the online trigger/filter bandwidth combined with the fact that the D0 detector serves many physics analyses, required that the trigger for this analysis trigger on cluster ET with a higher threshold than was desired. One consequence of this fact was that the data does not have a sharp mass turn-on. Also the DC detector contains gaps at |IETA| = 13 where there is no EM calorimeter, as well as regions at which only some EM signals are present (|IETAI = 12, 14). In addition, 158 Y PPHO efficiency for Drell—Yon eie' events 2 1:- Q.’ l- govs ” ta 5 0.5 :- o.25 E +- 0’1...I11.111111141111111111111 3O 35 4O 45 5O 55 60 Invariant mass (GeV/c2) e(a,,,<10 + dE/dx cut) vs. m K); .. C 1: <13 _ £0.75 E 35 E 0.5 _— O.25 :— O :llLlllllLllJ—LlillllllllliLlJlilJlllIIlJiIllllllll —2.5 -2 -1.5 —1 -O.5 O 0.5 1 1.5 2 2.5 Photon rapidity 5(am<10 + dE/dx cut) vs. y >\ .— 8 1 e .93 $50.75 E 333 E 0.5 :- O.25 E- t— 0 h-ll_llalLll_l_l_1¥llllllJlilllilllllllll_LilllllllllilJtll O 5 10 15 20 25 3O 35 4O 45 5O Photon p, (GeV/c) 5(om<10 + dE/dx cut) vs. pT Figure 5.13: PPHO efficiency for Drell-Yam e+e’ events. 159 Efficiency Efficiency Efficiency PELC/PPHO efficiency for Drell—Yon e’e- events 0.8 f 7 i a] i i i i 0.7} 0.6:— 0'5 I.11LLIII l J_Ll l l l l l LLLI l l l 111 J 11 3O 35 4O 45 5O 55 6O . . Invariant mass (GeV/c2) PELC/PPHO EffICIency 0.8:— —I i J l l l l l I l I l I l l l ._j I I I I I I I I I I I I I T I 0.7 :- 06 L— L. 0'5 llllllLllllLJlLilLllllIllllilLllllillll — -3 -2 -1 O 1 2 3 4 . . Photon rapidty PELC/PPHO EffICIency 0.8:— 411411]lllllllllllllllllJllllllllllllllllJllIllllI JTTIIIITTIIIIIIFIIIIIIIIIIWTIITIIIIIIIIIITIIIIIII O.7 :- 0.6:— +— 0.5 lLlllllllllLllIllllllllllllllllllllllIllIll]illl O 5 10 15 20 25 3O 35 4O 45 5O Photon p1(GeV/c) PELC/PPHO Efficiency Figure 5.14: PELC/PPHO efficiency for Drell-Yan 6+6" events. 160 H—matrix X2 < 100 + ISO < 0.15 efficiency for Drell—Yon eie' events ._l 0.8 0.6 0.4 0.2 Efficiency -ii— IWTIrI—IIITFII TII l l l l [IPLJILI I L] 11 l l l I II 1] O 3 1 0.8 0.6 0.4 0.2 Efficiency 0 —2.5 Efficiency 9.0.0.0 m4>auoo 0 0TH 55 60 Invariant mass (GeV/c2) 5(x’<1oo + |SO 25). In order to easily compare the results of this experiment with theory, it is necessary to correct for the kinematic inefficiencies induced by the ET thresholds and the excluded rapidity regions imposed by the fiducial cuts. Another fact of experimental life is that no measurement is exact; even a perfect experimental apparatus is limited to some quantum mechanically imposed resolution. The energy and position resolution of the DO detector, while very good (although nowhere near the quantum limitl), results in a “feed—down” or “smearing” of most kinematic quantities one would like to measure. This smearing cannot be removed on an event-by-event basis, but can be corrected for statistically given some number of events. The method used to make these kinematic, fiducial, and smearing corrections is straight forward, provided one can model one’s apparatus accurately using computers. 163 It turns out that very detailed simulation of a large complicated detector such as DO is very difficult and CPU intensive, however it can be done provided enough CPU cycles are available. This is exactly what the DOGEANT simulation attempts to do. It tracks each input particle through a precision computer representation of DO using the physics of particle interactions with matter to determine the tracks, scattering angles, calorimeter shower shapes, etc. . .using a Monte Carlo method to simulate what that particle will look like in DO. If enough CPU cycles were available, a full plate-level DOGEANT simulation could produce a very accurate representation of how events look in DO. Unfortunately, even with today’s high-speed computers, it still takes far too long to do a plate-level simulation of more than a few particles in DO so further approximation of the DO geometry is necessary. This is done by replacing the full description of all the uranium plates, readout boards, etc. . .by a homogeneous mixture of the various materials in DO that presents the same number of radiation lengths to an incoming particle. This approximation works fairly well for broad quantities such as cluster energy, but not so well for quantities such as detailed shower shape information, since particles cannot fly off at large angles at the plate edges if there are no plate edges. It is likely that for the kinematic, fiducial, and smearing corrections needed for this analysis, that a homogeneous mixture simulation is accurate enough, and so it is used. To calculate these corrections, we use the ISAJ ET Monte Carlo to generate Drell-Yam 6+6“ events which are fed into the homogeneous mixture DOGEANT simulation to provide the DO raw data banks. The raw data is then reconstructed with DORECO to provide all the typical kinematic quantities available in a DO event. We then make histograms of the pair mass, rapidity, and 197 after making our kinematic and fiducial cuts. The DOGEANT simulation includes the inherent DO energy and position res- olution that causes the smearing. One then divides these histograms by histograms of 164 the same quantities of the input ISAJ ET events. So the correction for a given bin is NMCout NMCin CIx'iruematic,Fiducia.l,Smcaring : This gives the bin—by-bin kinematic, fiducial, and smearing correction necessary to statistically correct for these effects. Since the event sample is rather small (#2000 events), we fit the resulting histograms to produce a smoother correction. We then vary the fit parameters according to the errors on the fit to estimate the errors induced by the fit just as we did for the background, online efficiencies and offline efficiencies. Since the energy and angular resolutions in DO are known to some approximation (AE / E % 15%/\/E for example), it is possible to use an even simpler method to estimate the kinematic, fiducial and smearing corrections. One simply smears the Monte Carlo energies and angles of the Drell-Yan electrons according to Gaussian distributions with mean zero and width equal to the energy and angular resolutions and then recomputes the other kinematic variables using these smeared quantities. This method is likely less accurate than the detailed DOGEANT simulation (even using a homogeneous mixture) but it is also orders of magnitude faster. This method then can be used as check of the DOGEANT simulation and as a means to estimate the systematic error of using the ISAJ ET Monte Carlo as an input to the DOGEANT simulation by using a different input Monte Carlo. Figure 5.17 shows a comparison of this fast smearing method to the DOGEANT simulation. The RESBOS Monte Carlo was used as the input to the fast method. Figure 5.18 shows the combination of the kinematic, fiducial and smearing corrections vs. pair mass, rapidity, and pT. The errors in Figure 5.18 include the difference between these two methods added in quadrature to the estimated fit error. 165 Kinematic, Fiducial, and Smearing Correction for Drell-Yon e‘e' events 0.6 MGM/MC”, Correction g 0.6 _ L 1:1 - .----1---1 I 9 04 L 5 . _ --—--I_--—-----1- _______ r___=_—.+.—: 0: 3 ——1— ———+—— g 0.2 -——+ ----- " """" * ------ 1--- =-—+—— —-+-——- \. _ a _ L2) 0 IIILIIIIIIIIIIIIIIIIIIJILIJIIILIIIIIIIIIIIJIIIJII -2.5 —2 —1.5 —1 —O.5 O 0.5 1 1.5 2 2.5 Photon rapidity C 0.6 .9 7 E E t 04 *— -L_L-‘--I--L-}- 8 2:11 11-1111111111111111111-11-11-1-1-1411—1-1'14'11-1-1‘1‘1‘11111'11 _S _' + 'I -+-FL+++++-1-—1-++_,_+-|-‘l”- -._ 'I‘ U .— 2 0.2 _ \ _ 3 _ L2) 0 IIIIIIIIIILIIIIIIIIIIIIIIIJIIILJIILIIIIIIIIIIIII O 5 1O 15 2O 25 3O 35 4O 45 5O Photon pT (GeV/c) Figure 5.17: Comparison of RESBOS Monte Carlo fast kinematic, fiducial, and smear— ing correction (dashed) to the DOGEANT simulation (solid). 166 Kinematic, Fiducial, and Smearing Correction for Drell—Yon e’e" events 5 0.6 _ '5 L 1 a) " I 1 t 0.4 o I‘— 1 i I L) - I I .s - I o __ 2 0.2 F \_ 1 LE; 0 ,- I I I I I L I I III I L L I I L I I I I I I I I I I I I I 30 35 4O 45 5O 55 60 Invariant mass (GeV/c2) Kinematic. Fiducial, and Smearing correction (c) 0.6 ‘13 I 1 l 8 o 4 f— 3 l —I‘— 5 . ’— ___+_ ____+___ o 1 _I— 0‘ 0.2 — { _=-—I—- i 3 L2) Or-IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIJLIIILLILIIIII —2.5 -2 -1.5 -1 -O.5 O 0.5 1 1.5 2 2.5 Photon rapidity Kinematic, Fiducial, and Smearing correction C 0.6 .9 7 g : t 0.4 ~ 0 _lllIIILIlllllllllllllILIIIIIIIIIIIJIIILILILIIIIIIIIll 0: :VII'VIIIHIIIIIIIIIII‘Il I ["71 'I'IIIII (5 _ 2 0.2 _ \_ _ a C2) 0 IIIIJLIJIIIIIIIIIIIIIIIIIIIIIIIIILJIIIIIIIIIIIIII O 5 10 15 2O 25 3O 35 4O 45 50 . . . . Photon pT (GeV/c) Kinematic, Fiducial, and Smearing correction Figure 5.18: Kinematic, Fiducial Acceptance and Smearing correction for Drell-Yan e+e‘ events. 167 Table 5.16: Total Efficiency, Kinematic, Fiducial Acceptance and Smearing correction for Drell-Yan 6+8“ events per invariant mass bin. II Mass Bin (GeV/?) I Correction I Error II 30.0 - 35.0 0.109 :I:0.023 35.0 - 40.0 0.130 :I:0.012 40.0 - 45.0 0.149 :I:0.012 45.0 - 50.0 0.171 :I:0.019 50.0 - 55.0 0.191 :I:0.023 55.0 - 60.0 0.214 :I:0.058 I Average 0.161 :I:0.029 5.6 Cross Section Figure 5.19 shows the combination of all efficiency, kinematic, fiducial, and smearing corrections to be applied to the Drell—Yan signal sample shown in 5.10. Tables 5.16, 5.17, and 5.18 give the bin-by-bin total correction vs. photon mass, rapidity, and 137 respectively. The efficiencies given in Tables 5.13, 5.14, and 5.15 were measured in such a way that they are all independent and also independent of the kinematic, fidu- cial, and smearing corrections (some cuts were made before the efficiency of the others were measured). Consequently, the total efficiency, kinematic, fiducial, and smearing correction is just the product of all these corrections given by 610111 = 6L1 X 61.2 X 5x2+ISO >< €PPHO >< EPELC/PPHO X €a1111+dE/d11 X €Kinematic+F'iduciaH-Smcaring To produce a cross section we divide each bin by the integrated luminosity accu- mulated by the ELE_2.HIGH trigger and the total efficiency, kinematic, fiducial, and smearing correction according to the formula 168 Table 5.17: Total Efficiency, Kinematic, Fiducial Acceptance and Smearing correction for Drell-Yan e+e‘ events per photon rapidity bin. Rapidity Bin Correction Error -2.5 - -2.0 0.044 :I: 0.012 -2.0 - -1.5 0.093 :I: 0.021 -1.5 - -1.0 0.144 i 0.014 -1.0 - -0.5 0.206 :I: 0.045 -0.5 - 0.0 0.238 :t 0.026 0.0 - 0.5 0.238 :I: 0.040 0.5 - 1.0 0.207 :I: 0.041 1.0 - 1.5 0.142 :L- 0.013 1.5 - 2.0 0.093 :I: 0.024 2.0 - 2.5 0.044 :I: 0.009 II Average 0.145 :t 0.028 Table 5.18: Total Efficiency, Kinematic, Fiducial Acceptance and Smearing correction for Drell-Yan 6+6- events per photon p1 bin. I [17 Bin (GeV/c) Correction Error I 0.0 - 1.0 0.128 :I: 0.020 1.0 - 2.0 0.126 :I: 0.016 2.0 - 3.0 0.128 :I: 0.024 3.0 - 4.0 0.128 :I: 0.012 4.0 - 5.0 0.129 :t 0.015 5.0 - 6.0 0.129 :I: 0.013 6.0 - 8.0 0.129 :I: 0.015 8.0 - 10.0 0.129 :I: 0.015 10.0 - 12.0 0.131 :I: 0.018 12.0 - 16.0 0.130 i 0.014 16.0 - 24.0 0.131 :I: 0.009 24.0 - 32.0 0.132 :t 0.006 32.0 - 40.0 0.135 :I: 0.006 40.0 - 50.0 0.137 d: 0.008 II Average 0.130 I :t 0.014II 169 Total Kinematic, Fiducial, Smearing, And Efficiency Correction C 0.3 .9 p U 13 0.2 I— _+_ 8 f 1 ——+_————+"‘ a I S 0.1 I '— E O 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 30 35 4O 45 5O 55 60 Invariant mass (GeV/c2) Total Efficiency C 0.3 9 E : f I C 0.2 *— 0 _ 2 C —+— —+—— O _ O IIIIILIIIIIIIIIIIIIJ_1_IIILII_ILIIIIIIIJIIIIIIIIIIII -2.5 -2 -1.5 —1 —O.5 O 0.5 1 1.5 2 2.5 Photon rapidity Total Efficiency 0.3 .5 r g t t 0.2 r O r— 8 IIIII I II I I ¢ ¢ ¢ 0 ~TTTYTY Y T T T # +2 0.1 — o 1 1— L 1. O TLLLIIIIIILLILIIILIIIIIIIIIIIILIIIIIIIIILLIIIIIIIIIIII O 5 10 15 20 25 3O 35 4O 45 5O . Photon p1(GeV/c) Total Efficiency Figure 5.19: Total Efficiency, Kinematic, Fiducial Acceptance and Smearing correction for Drell—Yan e+e‘ events. 170 Table 5.19: Integrated Drell-Yan cross section per invariant mass bin. II Mass Bin (GeV/c2) I Integrated Cross Section (pb” Error (pb) II 30.0 - 35.0 15.89 :I: 5.10 35.0 - 40.0 11.96 :I: 3.22 40.0 - 45.0 6.31 i 2.26 45.0 - 50.0 5.28 :I: 1.93 50.0 - 55.0 3.33 :I: 1.49 55.0 - 60.0 2.33 :I: 1.32 II Total 45.07 :I: 7.01 The bin contents are also divided by the bin width to give the correct units. dabin/dxbin = Ubin/Abin Figures 5.20, 5.21, 5.22 show the measured inclusive Drell-Yam e+e‘ cross section vs. invariant mass, photon rapidity, and photon p1 respectively. The inner error bars are the statistical errors only, the outer error bars are the total statistical + systematic errors. Tables 5.19, 5.20, 5.21 list the actual integrated cross section per bin vs. photon mass, rapidity, and p1 along with the associated total error. 5.7 Systematic Errors We have attempted to estimate some of the systematic errors associated with our cross section (which we described earlier) and have included those estimates in our error bars. However, since DO is a relatively new experiment and this data was taken during its first data run, some systematic errors have been neglected since we are still learning about our apparatus. Others such as the smearing induced by the energy and position resolutions described in Section 5.5 are being meticulously studied by groups 171 Table 5.20: Integrated Drell-Yan cross section per photon rapidity bin. Rapidity Bin Integrated Cross Section (pb) Error (pb) -2.5 — -2.0 1.83 :I: 3.20 -2.0 - -1.5 2.20 :t 2.15 -1.5 - -1.0 3.67 :I: 1.80 -1.0 - -0.5 5.32 :I: 1.99 -0.5 — 0.0 3.22 :I: 1.25 0.0 — 0.5 4.54 :t 1.60 0.5 - 1.0 5.13 :I: 1.90 1.0 - 1.5 6.27 :I: 2.33 1.5 - 2.0 4.96 :I: 2.99 2.0 - 2.5 2.81 :I: 3.92 II Total I: 39.93 I :I: 7.71 II Table 5.21: Integrated Drell—Yan cross section per photon p7 bin. pT Bin (GeV/c) Integrated Cross Section (pb) Error (pb) 0.0 -1.0 1.64 i 1.10 1.0 - 2.0 6.07 :I: 2.19 2.0 - 3.0 6.01 :I: 2.36 3.0 - 4.0 5.77 :I: 2.15 4.0 — 5.0 4.14 :t 1.92 5.0 - 6.0 2.61 :I: 1.62 6.0 - 8.0 3.81 :I: 2.04 8.0 - 10.0 4.14 :I: 1.98 10.0 - 12.0 4.03 :I: 1.91 12.0 - 16.0 3.30 :t 1.86 16.0 - 24.0 3.47 :I: 1.84 24.0 - 32.0 1.48 :I: 1.16 32.0 - 40.0 0.50 i 0.72 40.0 - 50.0 1.24 :t 0.87 Total 48.20 i: 6.61 172 Run 1A Drell—Yon Cross Section da/dm7 (pb cz/CeV) 0.3 ~ O1 IllllllllllIlllIIIIIIIILllllll 3O 35 4O 45 50 55 60 Invariant mass (GeV/c2) (DATA—BKC)/(£mflso‘8m*£pfl.c*su*8u*L*K*A*S*£ppHo) Figure 5.20: Drell-Yan 6+6" differential cross section da/dm vs. invariant mass. Inner error bars are statistical error only, outer error bars are statistical + systematic errors. 173 Run 1A Drell—Yon Cross Section A 20 .D O. V. 1 >‘ .. '0 Il- \ b r 1: U 1—0—1 :1 HD—I 10 — 1. “H HH 9 7 :1 " AH 8 - 1 7 HH 3. 0 I-0-1 1 6 '- L I—O—l 5 1. HH 1? 4 ~ .. 1 . HH .5 3 1. 2 1 1 J JLI I I I I l l 1 [LI 1 l I l l l I l I l I I l l I I I I l I l 1 LI I I l —4 —3 -2 —1 O 1 2 3 4 Photon rapidity (DATA— BKG)/(1:.,,..so¢1:11.1112,111,911:L1 *eutLthtSnm) Figure 5.21: Drell-Yan 6+6_ differential cross section da/dy vs. photon rapidity. Inner error bars are statistical error only, outer error bars are statistical + systematic errors. 174 Run 1A Drell—Yon Cross Section ’9 . a) P (D l- \ 1 o _8- P ‘1. f I (i l- .1 U 001 \ 8 HH 1 1.“ p .- 0—4}—" .. .l _‘I L 10 :- : r——I>———c - ‘ IL 1O_2llLLlllllllllllllIllllllllllllllJll lllllllllllll O 5 10 15 20 25 3O 35 4O 45 5O Photon pt (GeV/c) (DATA— BKG)/(5,....so*5m*5mcteutaut-UKtAthtepm) Figure 5.22: Drell-Yan e+e‘ differential cross section da/de vs. photon pT. Inner error bars are statistical error only, outer error bars are statistical + systematic errors. 175 in DO, most notably by the W mass group. One area in which the energy and position resolutions can have a large effect is in the background subtraction. Fortunately we were able to devise a background estimation method that used our data sample for our dijet background which is by far our largest background. Since data was used to estimate the dijet background, the only systematic error we should incur by using it is due to the rejection fits which we varied in an attempt to quantify this error. The Z—1 7+7” —> e+e’ background, on the other hand will induce systematic errors since these events were only smeared using an estimate of the energy and position resolutions, but this background is small and thus cannot have a very large systematic effect. Another area in which we incur systematic errors are the various efficiency correc- tions for the trigger and calorimeter based electron identification cuts. In an attempt to minimize these systematic errors we used the full plate level DOGEAN T simulation in our generation the single Monte Carlo electrons used to measure these efficiencies. In addition these events were overlapped with real DO minimum bias events before reconstruction as described in Chapter 4. The addition of the minimum bias events adds uranium and electronics noise to the events which is absent from the DOGEANT simulation. It also adds some small hadronic activity such as may be due to the un- derlying event in a typical Drell-Yam event. Many groups in DO are currently using this method since it has been shown to mimic real DO electrons well. It would be preferable to use real electron data to measure these efficiencies, but no source of low energy electrons exists without removing the substantial dijet background by using the very cuts one wishes to analyze, so we must rely on Monte Carlo data. Real Z ——> 66 data was used for the tracking cut efficiencies since they should be energy in- dependent, so the systematic error incurred here should be mostly due to background contamination (which should be small since we can cut hard on one of the electrons, 176 and use the calorimeter cuts on the other). Finally, all of these systematic errors are small compared to the statistical errors on this measurement as can be seen by the difference in size of the statistical and statistical + systematic errors shown in Figures 5.20, 5.21, and 5.22. Consequently, it is probably not worthwhile to work on most of the systematic errors any further for this measurement. Even with the factor of two reduction in the statistical error that may be possible by combining the Run 1B data sample with this data, the statistical error would still be significantly larger than the systematic error. The one area of systematic error that would likely benefit from further work is the kinematic, fiducial, and smearing correction. Chapter 6 Results And Comparison To Theory 6.1 Statistical Compatibility Tests The two most important properties of any statistical test are the power of the test and the confidence level of the test. The purpose of a statistical test is to distinguish between a null hypothesis H0 (in our case H0 is the the statement that the experimental and theoretical distributions are compatible) and the alternative hypothesis H1 (the distributions are not compatible). The power of the test is the probability of rejecting the null hypothesis when the alternative is true. In our case the alternative H1 is not well-defined since it is the ensemble of all hypotheses except the null hypothesis Ho. Thus it is not possible to determine whether one test is more powerful than another in general, but only with respect to certain particular deviations from the null hypothesis. The confidence level of a test is the probability of rejecting Ho when it is in fact true. That is to say, if one accepts the null hypothesis whenever the confidence level is greater than 0.05, then truly compatible distributions will fail the test 5% of the time if the experiment were performed many times. Thus the confidence level is the probability that the distributions are compatible. A confidence level near 1.0 indicates that that the two distributions are very similar, values near 0.0 indicate that it is very unlikely that the two distributions came from the same parent distribution. 177 178 Several goodness-of—fit tests exist to help determine if our measured cross section agrees with the theoretical prediction. The tests we will use are the Pearson X2 test [64] and the Smirnov-Cramér-Von Mises test [64]. The Pearson X2 test is the easiest to perform but does not take into account the sign of the deviations so it is less powerful in determining whether the shapes of two distributions are compatible, especially if the data has large errors associated with it. The Smirnov-Cramer-Von Mises test uses the average squared difference of the cumulative distribution functions (integrated probability distributions) to calculate the test statistic and thus is a powerful test of whether the shapes of distributions are compatible. However, it is intended for unbinned data, so some of its power is lost when it is used on binned data since the information about the events’ position within the bins is lost. The Smirnov- Cram‘er-Von Mises test is not sensitive to any difference in the relative normalizations between two distributions since it acts on the cumulative distribution functions which are normalized to 1.0 by definition. However, it can be combined with a X2 test on the normalizations using the formula P(shape + normalization) = P(shape)P(normalization)(1 — ln(P(shape)P(normalization))) The reader should refer to [64] for a detailed descriptions of these tests; we simply define the tests here. 6.1.1 Pearson X2 Test The X2 test uses the sum of the squared bin-by-bin deviation between the two distri- butions being compared divided by the probability content of the each bin under the null hypothesis H0 as its test statistic T 179 (t,- — N______jr)__.-)2 Pi 11: 1:111:11 = 77 §( where N is the integrated contents of the measured distribution, p,- is the probability content of bin i of the measured distribution p,- = d,/N (where d,- is the content of bin i of the measured distribution, d,- = N pg), and t,- is the content of bin i of the theoretical distribution which we want to compare. Given these definitions we can rewrite this test statistic as kzbins (ti _ di)2 71:: 2 1:1 01‘ where we have used the Poisson definition of the variance 0,12 = d;. Thus, if H0 is true, T is distributed as a X21 since it is the sum of the squares of standard scores Z = :r/a. The confidence level P(T) of this test is then given by the upper tail integral of a X2 distribution with k degrees of freedom where k is the number of bins being compared between the two distributions 2k 1 °° a __ Pm: Pu Ik)= 77—111/21/6 emda: 6.1.2 Smirnov-Cramer-Von Mises Test The Smirnov-Cramer-Von Mises test uses the test statistic 12,. I 11.122 where 1.12 = £:(F(a:)—F’(a:))2dF(x) 180 and ”eff is the number of effective events since we are using weighted data. F *(x) is the cumulative probability distribution function of the measured distribution and F (:r) is the cumulative probability distribution function of the theoretical distribution we wish to compare. For binned data we replace the integral by a sum 72.,” = —————2N 2 The asymptotic characteristic function of this test is given by - \/2't lim E(e't"°”“’2) = , z , "elf—m sm \/2zt By inversion of this equation one can compute the confidence level associated with the test statistic neffwz. 6.2 Mass Distribution The integrated Drell-Yam e+e‘ cross section in the 30-60 GeV/c2 mass range computed from the da/dm distribution is 45.1 :I: 7.0(stat. + sys.) pb. The integrated RESBOS cross section is 52.4 :I: 0.4 pb and the integrated ISAJET cross section is 52.2 :I: 0.2 pb. These measurements are consistent within the errors. Figures 6.1 and 6.2 show a comparison between our experimental da/dm distribution and the RESBOS and ISAJ ET Monte Carlo distributions respectively. The RESBOS Monte Carlo was Table 6.1: Cumulative distribution functions from the da/dm distribution used in the Smirnov-Cramer-Von Mises test. 181 Bin FDATA(m) FRESBos(m) FISAJET(m) 1 0.352 0.398 0.383 2 0.618 0.638 0.618 3 0.758 0.788 0.768 4 0.875 0.887 0.869 5 0.948 0.954 0.943 6 1.000 1.000 1.000 run using the CTEQ3M parton distributions, whereas the ISAJ ET Monte Carlo was run using the CTEQ2L parton distributions. Figure 6.3 shows a comparison of the m3d20/dmdy|y=o distribution between the data and the RESBOS Monte Carlo. The bin centers were used for the mass values. Figure 6.4 shows a comparison of our measured cross section to the CDF cross section from the 1988-1989 Fermilab collider run [68]. The :1:-coordinates of the points are the CDF mass centroid. The CDF integrated luminosity for this run was 4.13 pb’l. Table 6.1 shows the cumulative distribution functions used in the Smirnov-Cramer- Von Mises test on the da/dm distributions. Table 6.2 shows the value of the test statistics for the Pearson X2 and Smirnov—Cramér-Von Mises goodness-of-fit tests along with the respective confidence levels of these test statistics computed between the data and RESBOS Monte Carlo and ISAJ ET Monte Carlo. The test statistics for the X2 and Smirnov-Cramér-Von Mises tests were computed as described in Section 6.1. The X2 test confidence levels were calculated using the CERN program library function PROB [67]. The Smirnov-Cramér—Von Mises test confidence levels were gotten by looking up the value in a Smirnov-Cramer—Von Mises test significance table [65]. The results of the Pearson x2 and Smirnov-Cramer-Von Mises tests indicate that invariant mass distributions from both the RESBOS and ISAJ ET Monte Carlo 182 Table 6.2: Goodness-of-fit test results from comparing the experimental da/dm dis- tribution to the RESBOS and ISAJ ET Monte Carlo distributions. Theory Test Test Value Confidence I RESBOS Pearson X2 X7 = 1.458, k = 6 0.962 RESBOS Smirnov-Cram‘er-Von Mises 0.045 0.876 ISAJET Pearson X2 X2 = 1.482, k = 6 0.961 ISAJ ET Smirnov-Cramér—Von Mises 0.016 0.997 Table 6.3: Smirnov-Cramer-Von Mises test on the da/dm distribution combined with a X2 test on the integrated cross section (normalization). Theory Test Confidence RESBOS Smirnov-Cramer-Von Mises 0.611 ISAJET Smirnov-Cramér-Von Mises 0.668 generators are compatible with our experimental measurement. This is not very sur- prising since ISAJ ET and RESBOS agree well in terms of the overall event rate they predict. Table 6.3 shows the result of combining the Smirnov-Cramer-Von Mises test with a x2 test on the integrated cross—section (normalization test) according to P(shape) + P(X2(normalization)) = PshapanormU — ln(P,hapePno,.m)) where P(shape) is the confidence level of the Smirnov-Cram‘er—Von Mises test. The resulting combined probability indicates good agreement (less than 1 standard deviation) between the distributions. 6.3 Photon pT Distribution The integrated Drell-Yan 6+€_ cross section in the 30-60 GeV/c2 mass range using the (117/de distribution is 48.2 :I: 6.6 pb. The integrated RESBOS cross section is 51.8 :I: 0.4 pb and the integrated ISAJET cross section is 52.0 i 0.7 pb. These integral 183 Run 1A Drell—Yon vs. RESBOS do/dm, (pb cz/Gev) u (19 ~ (18 . 0.7 "' ————-0.__.. (16 - (15 - 0.4 f 0J5 h 02 - O1 [LIIIIIIIII IILILJIIIIILLIIIIJ 3O 35 4O 45 5O 55 50 Invariant mass (GeV/c2) RESBOS Figure 6.1: Drell-Yan e+e' differential cross section da/dm vs. invariant mass (filled circles) compared to the resummed theoretical da/dm distribution (open circles) from the RESBOS Monte Carlo. 184 Run 1A Drell—Yon vs. ISAJET dU/dm, (pb cz/Gev) LN 0.9 r 0.8 r ———<)—— 0.7 " —-1h-——— O.6 ~ 0.5 *- O.4 L 0.3 - 0.2 - O1 llllllllllLLllllll[IA—llllllLl 3O 35 4O 45 5O 55 60 lnvoriont moss (GeV/c2) RESBOS Figure 6.2: Drell—Yan e+e' differential cross section da/dm vs. invariant mass (filled circles) compared to the theoretical da/dm distribution (open circles) from the ISA- JET Monte Carlo. 185 Run 1A Drell—Yon vs. RESBOS {y 35 ~ N\ r- > .- O.) O n .2 .- 1 .. 3 . ‘0 E i- Q r- ,p 25 -— P .- E ' in 20 — 15 —' 10 L 5 L— r r L k J I I I I I I I I I I I_ I I I I I I J I LI I I I l 30 35 40 45 5O 55 60 lnvoriont moss (GeV/c2) "1058 Figure 6.3: Drell-Yan e+e‘ differential cross section m3d20/dmdy|y=o vs. m3d20/dmdyly=o invariant mass (filled circles) compared to the resummed theoretical distribution (open circles) from the RESBOS Monte Carlo. 186 Comparison of DO and CDF Drell—Yon cross sections >9»- a) 08— >7» 0 136' Q V5... E \4" 3 3.. 9.0.0.0 auxiaoLo—s 0.4 0.3 ~ O1 4 l l l l l ALA L l l l l l l l l l l l 1 l l 1 l l l l 1 3O 35 4O 45 5O 55 60 Invariant mass (GeV/c2) DO VS. CDF dotO Figure 6.4: D0 Drell-Yan e+e‘ differential cross section da/dm vs. invariant mass (filled circles) compared to the CDF da/dm distribution (open circles) from the 1988- 1989 collider run. 187 Table 6.4: Cumulative distribution functions from the da/de distribution used in the Smirnov-Cramer-Von Mises test. Bin FDATA(PT) FRESBOS(PT) FISAJET(PT) 1 0.034 0.042 0.049 2 0.160 0.140 0.207 3 0.285 0.257 0.392 4 0.404 0.364 0.527 5 0.490 0.454 0.617 6 0.544 0.525 0.681 7 0.623 0.640 0.769 8 0.709 0.723 0.828 9 0.793 0.784 0.870 10 0.861 0.870 0.921 11 0.933 0.945 0.968 12 0.964 0.975 0.987 13 0.974 0.991 0.995 14 1.000 1.000 1.000 cross section measurements also agree within one standard deviation. The differences between the Monte Carlo predictions is due to the difference in the shape of the photon pT distributions and the integration limits of O to 50 GeV/c. One may (correctly) wonder why the experimental integral cross section measured using the pT distribution differs at all from the mass distribution measurement since we start with the same events. This is due to the method used to perform the efficiency, kinematic, fiducial acceptance, smearing and background corrections. Since we use an average bin-by-bin correction in mass and p1, it means that the average total correction will be different between these two variables. Table 6.4 shows the cumulative distribution functions used in the Smirnov-Cramer- Von Mises test on the da/de distributions. Table 6.5 shows the results of the statist- ical tests described in Section 6.1. We see that in this case the Smirnov-Cramer-Von Mises test indicates a strong similarity between the experimental distribution and the RESBOS resummed pT distribution, and a very low compatibility between the data 188 Table 6.5: Goodness-of—fit test results from comparing the experimental da/de dis— tribution to the RESBOS and ISAJ ET Monte Carlo distributions. Theory Test Test Value Confidence RESBOS Pearson X2 X2 = 3.615, k = 14 0.997 RESBOS Smirnov-Cramer-Von Mises 0.028 0.976 ISAJET Pearson X2 X2 = 7858,16 = 14 0.897 ISAJ ET Smirnov-Cramér—Von Mises 0.551 0.030 Table 6.6: Smirnov-Cramér—Von Mises test on the da/de distribution combined with a x2 test on the integrated cross section (normalization). Theory Test Confidence RESBOS Smirnov-Cramer-Von Mises 0.894 ISAJ ET Smirnov-Cramer-Von Mises 0.086 and the ISAJ ET pT distribution. The X2 test on the other hand, suggests compatibil- ity with both RESBOS and ISAJ ET distributions, although with a greater probability that the RESBOS distribution is compatible. We again combine the Smirnov—Cramer—Von Mises test with a X2 test of the nor- malization. The results are shown in Table 6.6. Combining the shape and normal- ization tests does not significantly change the conclusion of the shape test alone, the combined test still strongly favors the RESBOS distribution and rejects the ISAJ ET distribution. This was one of the goals of this measurement, to be able to clearly dis- tinguish between the resummed pT distribution and the artificially produced ISAJ ET pT distribution. The shape of the ISAJ ET pT distribution can be changed by varying the QTW input parameter, but this also affects the total rate and so it is not clear that one can produce the correct pT distribution and the correct integrated cross section simultaneously using ISAJ ET. 189 Run 1A Drell—Yon vs. RESBOS 9 .. Q) l- O l- } r .o r 0‘ l- ‘1 4»- s r .m- \ 8 -¢— 1 —{}'_ —"— ' —-—o———— )— _'-"0—— -1 1o :— ——<>—— C __..__ 10—2IIIIIIIIIlIIIILIIJIIIIIIIIIILIIIIII IIIIIIIIIIIII O 5 10 15 20 25 3O 35 4O 45 5O Photon p, (GeV/c) RESBOS Figure 6.5: Drell-Yan e+e— differential cross section da/de vs. photon pT (filled circles) compared to the resummed theoretical da/de distribution (open circles) from the RESBOS Monte Carlo. 190 Run 1A Drell-Yon vs. ISAJET 9 _ o- 0) h 0 .. \ _ U D P 3 P ,t D- 8 b » a + «o —o— 1 _4p._ ——{’—_ : ——¢—— .. _—+__ . __0__ _1 ——-0——— 10 r _ —4v-—— t —-0-—— 10—2 IJIIIIJIIIIIIIILJIIIIJIIILLIIIIIIII IIIIIIIIIIIII O 5 1O 15 20 25 30 35 4O 45 5O Photon p,(GeV/c) RESBOS Figure 6.6: Drell-Yan e+e— differential cross section da/de vs. photon {)7 (filled circles) compared to the theoretical da/de distribution (open circles) from the ISA- JET Monte Carlo. 191 00 Run 1A Drell—Yon da/dpT distribution compared to ISAJET and RESBOS O + 0 00 Run 1A A RESBOS Monte Carlo (resummation) I ISAJET Monte Carlo do/dp, (pb c/Gev) -‘l _q q k- 6 >— L'— 5 + 4 —1r— '1!“- 4»— 3 -—-u -0— _._u.._ 2 —1 fli— .__.,__ 1 lllllllllllllllIILJJJIJIIIIIIlllllllllltllllll O 2.5 5 7.5 10 12.5 15 17.5 20 22.5 Photon p, (GeV/c) (DATA—BKG)/(8Hm*5m*£mc*au*8u*L*K‘A*S*8m) Figure 6.7: Drell-Yan e+e— differential cross section da/de vs. photon p7 (circles) compared to the resummed theoretical da/de distribution (triangles) from the RES- BOS Monte Carlo and the theoretical da/de distribution (squares) from the ISAJ ET Monte Carlo. 192 Table 6.7: Cumulative distribution functions from the da/dy distribution used in the Smirnov—Cram‘er-Von Mises test. Bin FDATA(y) FRESBOS(y) FISAJET(y) 1 0.046 0.086 0.078 2 0.101 0.190 0.177 3 0.193 0.292 0.285 4 0.326 0.393 0.392 5 0.407 0.496 0.499 6 0.520 0.598 0.605 7 0.649 0.700 0.714 8 0.806 0.810 0.821 9 0.930 0.912 0.922 10 1.000 1.000 1.000 6.4 Photon Rapidity Distribution The integrated Drell-Yan €+e_ cross section in the 30—60 GeV/c2 mass range using the da/dy distribution is 39.9:l:7.7 pb. The integrated RESBOS cross section is 45.6:i:0.4 pb and the integrated ISAJET cross section is 47.4 :1: 0.7 pb. The experimental cross section agrees with both Monte Carlo estimates within the quoted error. The two Monte Carlo predictions now differ by 2.4 standard deviations. This is due to the difference in shape of the photon rapidity distributions and the fact that we are only integrating the region from -2.5 to 2.5 rapidity units. Also the reason the Monte Carlo integrated da/dy cross section is lower than the integrated da/dm cross section is that the integration limits were only -2.5 - 2.5 as imposed by the histogram bounds. There were no Drell-Yam events which passed our cuts with a photon rapidity outside of the y = i2.5 window. The experimental integrated cross section from the rapidity distribution is lower than the measured value from the mass distribution. The origin of this difference was explained in Section 6.3. Table 6.7 shows the cumulative distribution functions used in the 193 Table 6.8: Goodness-of-fit test results from comparing the experimental da/dy distri- bution to the RESBOS and ISAJET Monte Carlo distributions. [r Theory Test Test Value Confidence RESBOS Pearson X2 X2 = 4.816, k = 10 0.939 RESBOS Smirnov-Cramer-Von Mises 0.110 0.538 ISAJET Pearson X2 X2 = 5.020, k = 10 0.890 ISAJ ET Smirnov—Cramer-Von Mises 0.112 0.538 Table 6.9: Smirnov-Cramer-Von Mises test on the da/dy distribution combined with a x2 test on the integrated cross section (normalization). H Theory Test 1 Confidence RESBOS Smirnov—Cramer-Von Mises 0.597 ISAJ ET Smirnov-Cramér-Von Mises 0.486 Smirnov-Cramer-Von Mises test on the da/dy distributions. The results of the stat— istical compatibility tests between the experimental and theoretical photon rapidity distributions are shown in Table 6.8. Both tests indicate good agreement between the data and both theoretical rapidity distributions. The ISAJET and RESBOS photon rapidity distributions are very similar, being very flat in the rapidity range -2.0 to 2.0. Consequently, it is unlikely that given the large errors on this measurement, we can make a significant distinction between the two. The Smirnov-Cramer-Von Mises test is once again combined with the normaliza— tion test. Table 6.9 shows the results of the combined tests. Again, we are unable to make a significant distinction between the ISAJET and RESBOS compatibility with data. 6.5 Effect Of Parton Distributions The experimental parton distribution functions used in the factorization theorem can effect the rate and shape of the theoretical Drell-Yan distributions since they specify 194 Run 1A Drell—Yon vs. RESBOS A 20 a 3 >’: "a \ b D ___‘P__ 10 - : '—‘*—— 9 - 1' ° =Q.___—<>—— ——0—— 8 . 7 ——<>—— -—__0.__ 6 1. 5 . __0._ 4 n. 3 1- 2 - 1 IIIIIIIIIJILIIIIIIIIIIlllllllIllllIllllIllIlIJlll -2.5 -2 -1.5 -1 —O.5 O 0.5 1 1.5 2 2.5 Photon rapidity RESBOS Figure 6.8: Drell-Yan e+e— differential cross section da/dy vs. photon rapidity (filled circles) compared to the resummed theoretical da/dy distribution (open circles) from the RESBOS Monte Carlo. 195 Run 1A Drell—Yan vs. ISAJET A20 .0 3 >2 '0 \ b '0 —.0— 0 .__o_ ¥_0__ 10? i— =1: 9- —JI—— 8.. .__0— 7. ___0.— 6... 5.. —{}_ 4- 3»- 21- 1 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIJIILIIIIIIII -2.5 -2 -1.5 —1 -O.5 O 0.5 1 1.5 2 2.5 Photon rapidity RESBOS Figure 6.9: Drell-Yan e+e— differential cross section da/dy vs. photon rapidity (filled circles) compared to the resummed theoretical da/dy distribution (open circles) from the ISAJ ET Monte Carlo. 196 Table 6.10: Difference in integrated cross sections due to input parton distribution functions. H Distribution Data RESBOS(CTEQ3M) RESBOS(MRSDO’) f da/dm 45.1 :1: 7.0 pb 52.4 :1: 0.4 pb 52.6 :1: 0.4 pb f da/de 48.2 :I: 6.6 pb 51.8 a: 0.4 pb 52.1 i 0.8 pb f da/dy 39.9 :1: 7.7 pb 45.6 :1: 0.4 pb 46.9 :1: 0.8 pb the longitudinal momentum fractions of the partons in the parent hadrons (proton and antiproton in our case). In particular, the gluon distributions tend to vary quite a bit in the low :5 = Q/JS region between different parton distribution function sets. Usually this causes only slight variations between the cross sections at higher a: values since most recent parton distribution functions agree in this range. However, in the case of resummation, we have a two scale problem, namely the Q of the hard scattering and the QT of the multiple soft gluon radiation. Thus, even though our .7: = Q / \fg is generally in a region where different parton distributions agree well, we also are resumming the soft gluon emission. The resummation prescription convolutes the parton distributions with the Wilson coefficients which integrates over the whole a: range (0—1) which may still cause large cross section differences, so we should compare the effect of using different parton distributions in the resummation results. Table 6.10 shows a comparison of the integrated RESBOS cross sections using these two different parton distributions. Figure 6.10 shows the fractional change in the invariant mass, photon rapidity, and photon pT distributions between the RES- BOS Monte Carlo evaluated using the CTEQ3M and MRSDO’ parton distribution functions. We see that both the shape and overall rate is affected by changing the input parton distribution functions. The change in the integrated cross section is small, but Figure 6.10 shows that the shape can be significantly affected, especially the m distribution. The shape of the change in the pT < 20 part of the pT distri— bution is due to two effects. One effect is the convolution of the parton distributions 197 with the Wilson coefficients. The other effect is due to the fact that one should re-fit the non-perturbative function F N P (that accounts for the non-perturbative m < 0.5 GeV/c cutoff in the resummation) using the new parton distributions which was not done. So we have two combined effects that change the shape of the low M part of the m distribution. It is possible that this variation would be reduced (or even go away altogether) if F N P (MRSDO’ ) were available, but it is not. So the change in the shape of the low p1 part of the pp distribution can be thought of as an upper limit on this effect [69]. We also see that the change in the high p7 portion of the pp distribution is very small. This is what we would expect since RESBOS matches the low pT portion (resummation) and high p1 portion (NLO result or “Y” piece) at 20 GeV/c and we do not expect the NLO result to change much in our z = Q / J5; range with the input parton distributions. As a comparison, Figure 6.11 shows the same fractional difference plot between RESBOS(CTEQ3M) and ISAJET(CTEQ2L). Here we see a much larger difference in the shape of the pp distribution which we claim is due to the difference between the resummation calculation and the empirical method used by ISAJET. Consequently, we need to perform the goodness-of-fit tests against the RESBOS Monte Carlo da/de distribution using the MRSDO’ parton distribution functions to check that the large compatibility difference we see between the ISAJ ET and RESBOS Monte Carlo do/de distributions and the data is not simply due to the input parton distributions. Table 6.11 shows a comparison of the goodness-of—fit tests between the data and RESBOS Monte Carlo using the CTEQ3M and MRSDO' parton distribu- tions. We see that the data is very compatible with the RESBOS da/de distribution using the MRSDO’ parton distribution functions as well as the CTEQ3M parton dis- tribution functions, with MRSDO' being slightly more favored. 198 0.2 A(dq/dm)/(da/dm) o (MRSDOP — CTEQSM)/CTEQS’>M —:L— p— '—-4 [ITIII i1 1 i 0.2 J L J l l Lgl J l l l l l l l l l l l l l l l l L J l l 30 35 4O 45 50 55 60 .lnvariant mass (GeV/c2) Change induced by using MRS DO prime pdf ”>7 + "O 0.2 - 5 C l _,_—1——i——1— E 0 - _}__ i a 1 Q [E__+_ ] F b 3,—0.2 1" q llllllLlilllLlllllllllllllllllllllllJLllllJll1111 —2.5 —2 -1.5 -1 —O.5 O 0.5 1 1.5 2 2.5 . Photon rapidity Change induced by using MRS D0 prime pdf A 1 F (5: >— 3 I g 0.5 :— 9 E E—Of) _ \2-1:JJLLlllll111111llllllilJlJlJPllllll[1141411111111 O 5 1O 15 2O 25 30 35 4O 45 50 . _ _ Photon pT (GeV/c) Change induced by usmg MRS DO prime pdf Figure 6.10: The change in the Drell-Yan e+e—- differential cross sections induced by using the MRSDO’ and CTEQ3M parton distributions as inputs to the RESBOS Monte Carlo. Table 6.11: Goodness-of-fit test results from comparing the experimental da/de dis- tribution to the RESBOS Monte Carlo distributions using the CTEQ3M and MRSDO’ input parton distribution functions. Theory Test Test Value Confidence CTEQ3M Pearson X2 X2 = 3.615, [c = 14 0.997 CTEQBM Smirnov-Cramer-Von Mises 0.028 0.976 MRSDO’ Pearson X2 X2 = 7.858, k = 14 0.998 MRSDO' Smirnov-Cramer-Von Mises 0.551 0.997 199 (lSAJET(CTEQZL)—RESBOS(CTEQZ>M))/RESBOS(CTEQI’>M) .___+— .0 M ___*_— ____+_.___ _—+____ llllllf A(dq/dm)/(da/dm) o _+__—_ C 0.2 r l l l l l L l L l l I; l l I l I L I l l I I I I l l i l 30 35 4O 45 5O 55 60 Invariant mass (GeV/c2) Normalized difference between RESBOS and ISAJET ’9. $0.2 :- 3 r —+——+— 1. . “*- 3: 0 , —+—- s ~ + 3,-0.2 #- < _llllJLJlllllllll]lllllllllIJ_LllLLlllllleLlllfllll -2.5 —2 -1.5 -1 —O.5 O 0.5 1 1.5 2 2.5 Photon rapidity Normalized difference between RESBOS and ISAJET g: E F05 5” r 3 a— + >\ 0 r *1. c3 t -°-_.__,_ b 3_ i:lll1111llLllllll4llilLllllllllJllllJ_L4_llL.l_lllllll <1 1 O 5 10 15 20 25 3O 35 4O 45 5O Photon pT (GeV/c) Normalized difference between RESBOS and ISAJET Figure 6.11: The change in the Drell-Yan e+e— differential cross sections between the ISAJET(CTEQ2L) and RESBOS(CTEQ3M) Monte Carlos. 200 Table 6.12: Integrated cross section summary. ”Distribution] Data 1 RESBOS l ISAJET I] f da/dm 45.1 i 7.0 pb 52.4 :I: 0.4 pb 52.2 :I: 0.2 pb f da/de 48.2 :I: 6.6 pb 51.8 :I: 0.4 pb 52.0 :I: 0.7 pb f da/dy 39.9 :I: 7.7 pb 45.6 :1: 0.4 pb 47.4 :I: 0.7 pb 6.6 Conclusions We have calculated the integrated cross sections using the experimental da/dm, da/de, and da/dy distributions and compared these integral cross sections to the integrated cross sections from the RESBOS and ISAJET Monte Carlo distributions. All of our experimental measurements agree with the theoretical cross sections within one stand- ard deviation. Table 6.12 shows a summary of the integrated cross sections. We also have performed goodness-of-fit tests between the experimental da/dm, da/de, and da/dy distributions and the RESBOS and ISAJET Monte Carlo distri- butions. One of the goals of this experiment was to be able to differentiate between the resummed theoretical da/de distribution calculated in the RESBOS Monte Carlo and the more empirical distribution used in the popular ISAJET Monte Carlo. The Smirnov-Cramer-Von Mises test very strongly favors the resummed pT distribution over the ISAJET distribution. Thus we believe that we have succeeded in this goal, even though our statistical errors are large. The significance of the difference in com- patibility between the shape of the data and RESBOS Monte Carlo and data and ISAJ ET Monte Carlo is greater than two standard deviations. The effect of changing the input parton distributions to the RESBOS Monte Carlo from CTEQ3M to MRSDO’ was also evaluated. Although the shapes of the distribu- tions and the total rate are affected by the input parton distributions, the goodness-of- . fit tests still indicate very good agreement between the do/de distributions regardless 201 of the input parton distribution functions, with the MRSDO’ parton distributions being slightly favored. The shape and shape plus normalization tests also indicate good agreement between the experimental mass and rapidity distributions and both ISAJET and RESBOS Monte Carlo distributions. This is not surprising since ISAJET and RESBOS agree within a few percent and the errors on the experimental data are large. 6.7 Possible Future Improvements The best possible improvement one could make in this measurement would be to increase the statistics since our large errors allow us to make only rather weak com- parisons with theory. Fortunately, this should be achievable relatively easily since the second run of D0 (Run 1B) has already been completed with an accumulated luminos- ity 6 times larger than the integrated luminosity from Run 1A (on which this analysis is based), waiting to be analyzed. It is possible that another D0 graduate student will analyze this data, and it is our hope that the progress made in this analysis will serve her/ him well. If it is possible to reduce the statistical error enough that the systematic errors become more significant, the most important improvement to the analysis would be a better kinematic, fiducial acceptance, and smearing correction. This is the single largest correction made in this analysis, and the difference between the correction from ISAJ ET events run through the DOGEANT simulation compared to the smeared RESBOS events is fairly large. A comparison of smearing both ISAJ ET and RESBOS events with our energy and position resolutions shows a difference of 5-10% due to the input kinematics alone, but the shapes of the corrections are similar. Thus, a full - plate level DOGEANT simulation using RESBOS Monte Carlo Drell-Yan events as 202 the input would be preferable. No correction was made for electrons lost in the recoil jets that are present in most Drell-Yan events. This was studied using Monte Carlo by failing an event if one of the electrons fell within a cone of AB = 0.5 of the recoil direction of the jet (the opposite of the momentum vector of the photon) but was found to have a negligible effect. Since these recoil jets are usually soft, the electrons would not necessarily fail even if they were within AR = 0.5 of the jet center, and since failing them if they fell this close had almost no effect, this correction was ignored. Also since we added minimum bias data to our single Monte Carlo electrons when we measured our trigger and electron id cut efficiencies, if there is any systematic error associated with ignoring the recoil jets in the event, it is probably at least somewhat compensated for in our other efficiency measurements. An upgrade to the DO detector is currently underway, and is slated to be completed in 1998. This upgrade includes adding a central magnetic field to D0 to allow determination of the sign of charged particles. This will be a great addition to DO from the point of view of electron final states. It will not only allow one to perform measurements such as the forward-backward asymmetry in Z decays which cannot currently be done with DO but can aid in analyses such as the Drell-Yan e+e" analysis, by allowing one to compare same sign pair vs. opposite sign pair distributions, thus aiding in the background determination. Of course better understanding of the DO detector will greatly improve the quality of any D0 analysis. Much progress has been made since the data sample for this ana- lysis was frozen due to time constraints. New versions of the reconstruction program promise to provide better energy and position measurements and several versions of q the reconstruction program have passed during the preparation of this dissertation. 203 The error induced by the corrections used in this analysis (evidenced by the differ- ences in the integrated cross sections from the mass, rapidity, and pT distributions) is unfortunate, but it is due to the large kinematic corrections that are necessary. The only way to improve this would be to use a more efficient Drell-Yan e+e‘ trigger, which was not possible. Finally, there are some sources of systematic errors not thoroughly studied in this dissertation. These include a more detailed understanding of the energy scale. However, the effect of the energy scale variation is rather small for this analysis. Appendix A The Level 2 Electromagnetic Filter This appendix describes the L2_EM Level 2 Electromagnetic Filter algorithm, event selection cuts, cut tuning, and call tree as well as other details that are necessary to fully understand the filter. The cuts are described functionally and the tuning methods are briefly described. The sections below include: Filter Script What the parameters are and what they mean. Algorithm What is being done. 0 Longitudinal algorithm 0 Transverse algorithm 0 Cone algorithm Energy And 7] Dependence Of Cuts Cut Tuning Brief description. 0 CC cut tuning 0 EC cut tuning ’ Code Organization 204 205 Calorimeter Geometry L2.EM Selection Cuts Input parameters for the L2_EM filter come from 2 sources: the filter script, which supplies the typical user input, and the RCP (Run Control Parameters) file, which contains the tuned selection cuts as well as other control parameters. A.1 Filter Scripts As an example, here is the filter script for the ELE_2.HIGH filter used in Run 1A to trigger on 2 isolated EM candidates both with ET > 10 GeV and which passed the primary electron shape cuts. ! ele_2_high.filt ! Generated from ‘ofln_v73.g1b-triglist;1’ by trigparse 1.19. ! filter_bit e1e_2_high pass_1_of 0 speed 1.0 must_try l2_em num_em 2 etmin_cc 10.0 etmin_ec 10.0 track_match ’IGNDRE’ de1_eta_track 0.03 del_phi_track 0.03 shape_cuts ’ELECTRUN’ do_isolation true cone_de1ta_r 0.4 206 cone_fract_max 0.15 script_end Description of script parameters: num_em For the filter to pass, this many candidates must pass all other cuts etmin_cc ET threshold if the candidate is in the CC etmin_ec ET threshold if the candidate is in the EC track_match string describing type of tracking to do: ‘IGNORE’ = do not require a matching track ‘REQUIRE’ 2 require a track to calorimeter cluster match (CDC or FDC) ‘CDC-ONLY’ = require a track match IF in |IETA| < 13 (CC) ‘FDC-ONLY’ 2 require a track match IF in |IETA| > 13 (EC) ‘VETO’ = Fail if a track points to EM candidate cluster (CDC or F DC) ‘VETO_CDC’ = veto if find a track match and in IIETAI < 13 (CC) ‘VETO_FDC’ = veto if find a track match and in |IETA| > 13 (EC) del_eta_track 1) road size in which to look for matching track del_phi_track 45 road size in which to look for matching track shape..cuts string describing what shower shape cuts to do: ‘ELECTRON’ = uses primary longitudinal and transverse cuts only ‘PHOTON’ 2 drops cuts on EM1 and EM2 ‘E_LONG’ 2 does longitudinal only for electron ‘E.TRANS’ = does transverse only for electron ‘EJGNORE’ = or anything else not among the above does no shape cuts but calls it an electron 207 ‘G.LON G’ = does longitudinal only for photon ‘G_TRANS’ = does transverse only for photon ‘GJGN ORE’ = or anything else not among the above does no shape cuts but calls it a photon ‘P_. . . ’ = also works same as ‘G-.. .’ ‘xx_TIGHT’ = use ALL variables; default is only 4 main variables: FHl, EM3, 05 — 03 or AE5 X5 / E3 X3. All variables means all variables which are turned on in the RCP file, some are turned off. [This option no longer includes EM1 or EM2 cuts] do.isolation If true pass only candidates whose fractional energy difference between a cone of radius 0.15 and a cone of radius cone_delta_r is less than cone_fract_max. cone_delta_r radius of isolation cone in \/A772 + Ad)2 units conefractmax actual cut on fractional energy in isolation (cone-core) /core < conefractmax A.2 The L2_EM Algorithm This algorithm description explains ALL the possible cuts. All the cuts are done if the E.TIGHT parameter option is selected. But generally all cuts have not been used in the past, neither for electrons nor for photons. Cut values not specified directly as tool parameters come from D03LEVEL2$L28IM2L2_EM.RCP, which is downloaded as an STP (Static Parameters) file to Level 2. 1. Find candidate(s) from Level 1 2. Find peak EM3 cell in a trigger tower which triggers at Level 1 208 3. Unpack EM + FHl energy in 3X3 readout towers around peak EM3 cell 4. Find centroid of shower 5. Get vertex 2 position 6. Correct Et for vertex position and leakage out of nominal cluster size. 7. Cut on Level 2 EM Et 8. Cut on longitudinal shape 9. Cut on transverse shape 10. Cut on track match 11. Cut on isolation of candidate The peak EM3 cell is the one with the largest single EM3 energy deposit inside the original candidate trigger tower. A.2.1 Longitudinal Algorithm The cuts are divided into two groups: primary and secondary. The primary cuts are FHl/SUMEM and EM3/SUMEM (f5 and f3 below) and are used unless the SHAPE-CUTS field in the filter script is set to IGNORE. The secondary cuts are not used unless the E.TIGHT option is selected in the filter script SHAPE_CUTS field. For the P_TIGHT or PHOTON options, f1 and f2 below are not done. The tested region for the longitudinal algorithm is 3X3 readout towers around the readout tower containing the peak EM3 cell. This size is independent of eta. This 3X3 region is also the core region used in the cone isolation algorithm. 209 Floor fractions, f; = Eg/SUMEM, i = 1 — 5, are calculated. SUMEM is the sum of the energy in the 4 EM floors. Only FHl participates in floor 5. The cuts are made on f5, f3, f1,(f1+f2), f., (in that order) [NOTE: because of the offset introduced in the energy scale, the f, and f; + f2 cuts are no longer performed even for E-TIGHT] There are lower and upper cuts on f1 through f4; f5 has only an upper side cut performed on it. However, the low side cuts on f1, (f1 + f2), f4 are turned off in the RCP file at this writing. All longitudinal cuts depend on both energy and n. Presently the f3 cuts are quite loose, so that they effectively only serve to eliminate ‘hot’ single cells in the EM layers which trigger. A.2.2 'Iransverse Algorithm The primary cuts are: 0 CC: 05 — 0‘3 (99% efficient value) 0 EC(IETA < 31): AE5x5/E3X3 (99% efficient value) 0 EC(IETA = 31,32): AE7x7/E5x5 (100% efficient value) The other cuts are used only in the E-TIGHT option of the shape_cuts filter script parameter. Some of the cuts described below are turned off even if the E-TIGHT option is selected. IETA MAX 3X3 0'3 AE5X5 E3X3 0'5 0’5—0'3 AE4X4 ngg AE7x7/E5X5 1-12 x x x x x x 13 14—25 26-30 31-32 x 33-37 210 There is no EM calorimeter at IETA=13, so there are no EM triggers there. The candidate automatically fails at IETA =33-37 (no trigger here now). The above diagram shows what cuts the L2-EM filter tool will attempt under the E-TIGHT option, however if the cut values are set very large in the RCP file which contains the cut values, it is as if no cut was made. The descriptions below note those cuts which are turned off in the RCP file at this writing. Transverse Cut Variable Definitions: 03 and 05 are defined as energy-weighted < r > (NOT rms) in units of EM3 cells computed using a 3X3 or 5X5 grid respectively around the peak EM3 cell. 0 MAX/3X3 2 cut Peak EM3 cell energy divided by the sum in a 3X3 EM3 region around the peak cell (no E, 77 dependence), turned off in RCP file. a cut < 03 < cut cut = —(Ag(peak)2 + Bl(peak) + Co) i A (peak) is the energy in the peak EM3 cell, turned off in the RCP file. 0 AE5X5/E3X3 < cut cut on (E5x5 — E3X3)/E3X3, primary in EC, secondary in CC, SUMEM and IETA dependent. E5X5, E3 x3 are energy sums in 5X5 and 3X3 EM3 cell regions around the peak EM3 cell respectively. 0 05 < cut, (independent of E, 77) o 05 — 03 < cut depending on SUMEM, IETA, primary in CC, secondary in EC (turned off in the RCP file above IETA=25). 211 o min,< AE4X4/E2x2 < max From peak EM3 cell, find 7], 45 neighbors with highest energies. Build a 2X2 EM3 array including these 3 cells and the cell which fills out the 2X2 square. Then make a surrounding 4X4 cell square. Call the energies in these two E2X2 and E4)“. Seconary only in CC. Not done in EC, turned off in the RCP file. No energy or n dependence. 0 AE7X7/E5X5 < cut cut an (Egg — E5x5)/E5X5, independent of energy and 17. Since candidate is far forward, it will always be high energy if it passes the ET threshold. Done only for IETA=31,32. A.2.3 Cone Algorithm The core is a sum over the layers selected in D0$ LEVEL2$LZSIMzL2_EM.RCP: Core sums from LO_GAMMA_FLOOR to HI_GAMMA_FLOOR (as of Run 1A, EM 1—4), over the 3X3 Readout Towers centered about the highest EM3 cell. The cone is a sum over LO_CONE-LAYER to HI-CONE.LAYER, with possibly the ICD/MG turned off by CONE_USE_ICD (as of Run 1A, the sum is over all layers except the ICD / MG). The lateral extent is chosen by the CONE_DELTA-R parameter for the script: cells with centers within this radius of the central readout tower are included. 212 A.3 Energy And 77 Dependence Of Cuts Energies are broken up into 4 ranges: O < L(1) < ETHl ETHl < M(2) < ETH2 ETH2 < H(3) < ETH3 ETH3 < X(4) < 00 The L,M,H,X notation is used below in the description of the RCP file contents. The actual energy boundaries ETH1,ETH2,ETH3 are different for CC and EC: CC EC ETHl 14.0 GeV 29.0 GeV ETH2 35.0 GeV 70.0 GeV ETH3 65.0 GeV 150.0 GeV There are 8 eta regions: |IETA| L2_EM eta region 1-2 CC index 1 3-6 CC index 2 7-12 CC index 3 13 no EM, index 4 14—15 EC index 5 16-18 EC index 6 19—24 EC index 7 25-31 EC index 8 32-37 no trigger These indices are used below in the description of input bank contents. 213 AA Cut Tuning The 99% efficient point is defined as that cut value which passes 99% of the test sample (TB for test beam data, MC for Monte Carlo). The ‘100%’ point is defined a little differently, its precise meaning varies and is described below. PRIMARY CUTS: O 0.1< f3 < 0.9 0 f5 < 1.5 x 99% value 0 05 — 03 < cut (CC) or AE5X5/E3x3 < cut (EC). Main cuts set to 99% value (after TB selection cuts!) For very low energy electrons (ET < 10 GeV) the efficiency of these cuts is reduced due to changes in longitudinal shower profile. At 5 GeV these cuts are approximately 95% efficient, decreasing with decreasing energy. At IETA=31,32 the AE7X7/E5X5 < cut selection is used. SECONDARY CUTS: chosen at ‘100%’ efficiency point defined as follows 0 EC: ‘100%’ = 99%(actual) x 1.25 cuts used — added cuts for the EC: * f1 1-3, EC ——> 4-6). J _ETA_BIN index to cut array based on 77 in ETABOUNDS. A2 parameters for central value of 03 cut. 221 B1 parameters for central value of 03 cut. C0 parameters for central value of 03 cut. SS3 difference between min and max of 03 cut. EM3L min size of MAX/E3X3. ESIZE various transverse cuts: 1,2 - not used 3,4 - not used 5,6 - AE7X7/E5X5 7 - not used 8 - max 05 (CC) 9,10 - not used All the energy and 1) dependent RCP cut arrays below get mapped into the L2-EM internal array EMCUTS(j,i,x), where x=1,2,3,4=L,M,H,X below and the IETA index is as described in the energy and 17 binning description section. RCP cut arrays: ELCxi(j) x = L,M,H,X energy index i = 7] index for CC: 1,2,3 table above gives ranges. j 2 cut number: 0 1,2 min,max for floor fraction f1 0 3,4 min,max for floor fraction (f1 + f2) 0 5,6 min,max for floor fraction f3 222 7,8 min,max for floor fraction f4 9 max for floor fraction f5 10 max for 05 — 03 11 max for AE5X5/E3X3 12 max for AE4X4/E2x2 ELExi(j) x = L,M,H energy index i = 7] index for EC: 1,2,3,4 see table (5,6,7,8 in L2_EM) j : cut number: see above for CC E15xCT(j) as above, for IETA=15 in EC EL12xCT(j) as above, for IETA=13, never used since no EM A.6 Calorimeter Geometry Summary Of EM Signal Availability in D0: IETA Comments IETA.TT 1-11 All EM signals present 1-5, i— of TT 6 12 Lose some EM signals % of TT 6 13 No EM signals % of TT 7 14 Lose some EM signals % of TT 7 15-26 All EM signals present 8-13 27-32 No EM3 subdivision 14-16 33-35 Cells coarsen 17-19 Only odd IPHI exist No EM3 subdivision 36-37 No EM signals 20, (21 does not exist) 223 The signal naming convention is as follows: Layer Number I Layer Name I L2_EM limit parameter EM1 (MNLYEM = 1) EM2 EM3a EM3b EM3c EM3d EM4 (MXLYEM = 7) CC Massless gap (MNLYMG = 8) ICD EC Massless gap (MXLYMG = 10) FHl (Fine Hadronic) (MNLYFH = 11) 355:5:goooqazmawmw FH2 FH4 (MXLYFH = 14) CH1 (Coarse Hadronic) (MNLYCH = 15) CH2 CH3 (MXLYCH = 17) The definition of the EM3 sub-layer order is: The same orientation is used in all parts of the calorimeter, with respect to the (,1) and +2 directions. The EM3 layer is indexed effectively with sub-indices in the 45 and 77 directions, with 43 running fastest. An auxiliary system is also sometimes used, substituting a different coordinate, floor, for the depth dimension, and summing all the EM3 signals into a single floor: EM3b 4 EM3d 6 T 45 EM3a 3 EM3c 5 VII Z ._) 1 1 EM1 2 2 EM2 Floor Definitions: 3 3,4,5,6 EM3 4 7 EM4 5 8-14 FH+MG+ICD 6 15—17 CH [I Floor Wayers [ Floor Name [I 224 The layers in each n bin in the offline system are listed below: An x indicates that these regions are instrumented, a g indicates two or more layers which are ganged together. F1uu ECMG I.nb nu 9 CCMG 8 234567 X EM 1 X X _) __) _) _.) IETA 1 2 3 4 5 6 7 8 9 10 11 +12* -12* 13 +14* -14* 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 CAL Type Layer Number 0 range 0.0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9 1.0 1.01.1 1.11.2 -1.1 -1.2 1.21.3 1.31.4 -1.3 -l.4 1.41.5 1.51.6 1.61.7 1.7 1.8 1.81.9 1.9 2.0 2.0 2.1 2.1 2.2 2.2 2.3 2.3 2.4 2.4 2.5 2.5 2.6 2.6 2.7 2.7 2.8 2.8 2.9 2.9 3.0 3.0 3.1 3.13.2 3.2 3.42 3.42 3.7 3.7 4.1 4.1 4.45 4.45 ** 225 A.7 L2_EM filter cut values Below are the actual RCP files used with the L2_EM filter. The first is D0$LEVEL2$L2SIM:L2_EM.TB05.RCP which contains the cuts used for normal data taking in D0. The second is D0$LEVEL2$L2SIM:L2-EM_MC05.RCP which contains cuts intended for use with the Level 2 simulator (L2SIM) when analyzing homogeneous mixture Monte Carlo (instead of plate level DOGEANT). The efficiency of these MC tuned cuts on mixture MC should compare closely to the efficiency of the test beam tuned cuts on real DO data. The test beam cuts should be used for plate level MC since the shower shapes agree well between plate MC and real data. Test Beam Cuts Used Online: \START L2_EH_BCP \SIZB 534 161 I e ELECTROI SRCP bank for ELECTED] analysis. Created 28-l0V-1991 by Yi Xia Updated 12-OCT-1992 by James T. McKinley, Hirek Fatyga, Peter Grudberg, t Janos T. Linnemann Updated 14-l0V-1993 by James T. Linnemann add ETHII_CELL THIS IS L2_RH_TBO5.RCP Order of binned cuts in EXXXXX arrays (except ELECT3 which uses Ul-binned cuts) 3H1(nin) El1(nax) EI12(nin) El12(max) 883(Iin) flI3(nax) El4(min) Efl4(nax) PHl(nax) SIGSI3 ETH3 GEV ELCCXl .000 0.11 .000 0.79 > ETH3 68V ELCCX2 000 0.185 > ETH3 ORV ELCCX3 .000 0.38 .000 0.45 > ETH3 05V 812XCT 000 10000 .000 10000. < ETH1 08V 315LCT 000 0.345 0.315 < ETHI 03V ELECLI 000 0.345 0.365 < ETHl GEV ELECL2 000 0.345 0.440 < BT31 GEV ELECLS .000 0.345 0.365 12 0.0 - 0.2 12 '10000.00 0.33 0.11 0.057 0.034 0.3 - 0.6 12 -10000.000 0.45 0.072 0.061 0.7 - 1.2 12 -10000.00 0.75 0.015 0.065 0.088 0.15 1.3 12 .000 -10000.000 000 10000.000 1.4 - 1.5 12 “10000.000 0.037 1.6 - 1.9 12 '10000.000 0.037 2.0 - 2.5 12 -10000.000 0.037 2.6 - 3.2 12 '10000.000 0.037 ~10000.000 10000.000 10000.000 '10000.000 10000.000 10000.000 0.100 0.900 0.06 0.100 0.900 0.054 0.08 0.100 0.900 10000.000 -10000.000 10000.000 10000.000 0.715 0.100 0.240 0.116 0.715 0.100 0.240 0.115 0.715 0.100 0.315 0.402 0.715 0.100 10000.000 0.205 10000. 10000. 10000. 10000. 10000. 10000. 10000. 10000. 000 000 000 000 .900 000 .900 000 .900 000 .900 000 afruwrf... 228 \EID ! ECEM ETH1 < B < ETH2 03V 1.4 - 1.5 \13317 a15ncr 12 -10000.000 0.295 -10000.000 0.625 0.100 0.900 0.005 0.415 0.041 0.200 0.072 10000 000 \alD ! ECEH ETHl < R < ETH2 GEV 1.6 - 1.9 \ABRIY ELECHl 12 -10000.000 0.295 '10000.000 0.625 0.100 0.900 0.020 0.465 0.041 0.200 0.072 10000.000 \BID ! ECEI ETHl < 3 < ETH2 08V 2.0 - 2.5 \ARRAY ELECH2 12 -10000.000 0.295 -10000.000 0.625 0.100 0.900 0.040 0.540 0.041 0.275 0.189 10000.000 \EID ! ECEH ETH1 < E < ETH2 08V 2.6 - 3.2 \ARRAY ELECH3 12 -10000.000 0.295 -10000 000 0.625 0.100 0 900 0.060 0.465 0 041 10000 000 0.125 10000.000 \aln 1 1 acan aruz < a < arna Gav 1.4 - 1.5 \12317 a15ncr 12 -10000.000 0.210 -10000.000 0.490 0 100 0 900 0.005 0.525 0.073 0.150 0.061 10000 000 \aln 1 1 acan ETH2 < a < aras oav 1.6 - 1.9 \ARBAY aLacn1 12 -10000.000 0 210 -10000 000 0.490 0.100 0.900 0 020 0.575 0 073 0.150 0 061 10000.000 \aln 1 1 acan ETH2 < a < aru3 CEV 2.0 - 2.5 \ARRAY aracn2 12 ~10000.000 0 210 -10000 000 0.490 0 100 0 900 0.040 0.660 0 073 0.225 0.136 10000 000 \aun I 1 acan aru2 < a < arns cav 2.6 - 3.2 \ARRAY aLacua 12 -10000 000 0 210 -10000 000 0 490 0.100 0 900 0.060 0.675 0.073 10000 000 0.118 10000 000 \aln 1 1 acan a > arn3 Gav 1.4 - 1.5 \12317 aisxcr 12 -10000.000 0.135 -10000.000 0.390 0.100 0.900 0.000 0.575 0.093 0.075 0.063 10000.000 \nln I 1 acaa a > araa oav 1.6 - 1.9 \ABRAY aLacx1 12 -10000 000 0.135 -10000.000 0.390 0.100 0.900 0.015 0.626 0.093 0.075 0.063 10000.000 \aln 1 1 scan a > aras Gav 2.0 - 2.5 \ARRAY aLacx2 12 -10000.000 0.135 -10000 000 0.390 0 100 0.900 0.030 0.700 0.093 0.150 0.148 10000.000 \EID 229 ! ECflI B > BT33 03V 2.6 - 3.2 \ARRAY ELECX3 12 -10000.000 0.135 -10000.000 0.390 0.100 0.900 0.050 0.625 0.093 10000.000 0.126 10000.000 \EID ! ! ! SPARE (EH3CUT) \ARRAY ELECTS 10 -10000.000 10000.000 -10000.000 10000.000 -10000.000 0.08 10000.000 10000.000 -10000.000 10000.000 \EID ECCSL--CCSH--EC5L--835--EC7L--EC7H--83--35--4L--QH --------- !tho .08 cut on 7x7-6x5 H1 is from Y1 Xia's tuning \STOP 230 Monte Carlo Cuts: \START L2_fll_RCP \SIZE 564 191 ELECTROI SRCP bank for BLECTROI analysis. Created 28-IOV-1991 by Yi Xia Updated 28-SEP-1992 using DAT_ICP program written by James T. HcKinley Updated 14-l0V-1993 James T. Linnemann add ETHII_CELL THIS IS L2_EI_HCOS.RCP Cuts used: PRIIARY CUTS: EHS cuts set to .1 .9 (10, hi) Ffl1 cuts set to 1.5.99! value SIGHAS-SIGHA3 or 5x5-3x3 as main cuts set to 99% value (after HC selection cuts!!) for IETACI31,32 use Yi Xia’s value of cut on 717-5x5 SECOIDABY CUTS: set at "1001 values" where 100% is defined differently for CC and EC 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 EC: 100% ' 99% t 1.25 1 added cuts for the EC: 1 EH1(high), El12(high),Efl4(low, high),(4:4-2x2)/2x2(high) 1 also SIG$H3 as secondary transverse variable (for ieta<26) I 1 CC: 1001 = 99% + (992-901) I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 EH1(high), BH1+2(high), Rl4(10w, high) also 515-313 as secondary transverse variable All other cuts turned off by setting cuts to +-10000 Cuts tuned and tested by l. Tartaglia and J. HcKinley, using single track electrons generated by I. Graf with GEAIT 3.11, version I, full showering, low cutoff, honogeneous mixture lonte Carlo; events are distributed with Zvertex smeared (sigma-+-30cn), unifornly illuminating each eta bin fron ieta-2 to 32, (eta-1 events were bad), and uniformly spread in phi across (half of) two modules and a phi crack. lo noise or underlying event was included. CC efficiency was determined using a PHI I BHTOT < 0.1 out to eliminate electrons in phi cracks (no ISAJET info kept in this version of cut-tuning ntuple); no selection cuts were imposed on EC electrons. CUTS ARE II THE FOLLOHIIG ORDER II THE BXXXXX ARRAYS: El1(min) BH1(max) EH12(nin) Bl12(max) El3(ain) EH3(max) El4(min) Bfl4(nax) FH1(nax) SIGSH3(max) 515-3X3/3X3(nax) I 1Format version (old format is implicitly version 0) L2EH_VBRSIOI 3 1 (add ETHII_CELL) I 1 For unpacking ETIII-CELL 0.0 1if ETnoninal < this, exclude from ALL sums and cone ET_II_CAEP .7308. I 1 Divider of eta bins \ARRAY BTA_BOUIDS 8 2 6 12 13 15 19 25 32 \EID ! 1 Divider of energy bins 231 \ARRAY R_BOUIDS 6 14.000 35.000 65.000 29.000 70.000 150.000 \EID 1 1 SIOHAB cut S3A2 2.1408 5331 -2.07643 S3C0 -0.01391 SSIOS 10000.000 1 1 RHBHAX cut I RP3 -10000.000 I L0_GAIHA_FLOOR 1 HI_OAHHA-FLOOR 4 14 for Rl4; 5 for Ffil; 6 for all PH; 7for CH; 8 for ICDHG 1 L0_COIE_LAYRR 1 HI_COIE_LAYER 17 17 for EH4 11 for Ffll 14 for PH 17 f0 CH COIE-USE_ICD .FALSE. Einclude ICD/HO in COIE ? 1 warning: COIB_USB-ICD is not fully implemented I 1 CORE R < RTR1 ORV 0.0 - 0.2 \ARRAY RLCCL1 12 -10000.000 0.264 -10000.000 0.676 0.100 0.900 -0.012 0.630 0.113 0.099 0.082 0.155 \RID 1 CCRI R < RTH1 ORV 0.3 - 0.6 \ARRAY RLCCL2 12 -10000.000 0.340 -10000.000 0.728 0.100 0.900 -0.004 0.653 0.108 0.127 0.097 0.188 \fiflD 1 1 CORE R < R731 ORV 0.7 - 1.2 \ARRAY RLCCL3 12 -10000.000 0.476 -10000.000 0.902 0.100 0.900 '0.008 0.625 0.093 0.182 0.151 0.249 \EID 1 ccnn a < 3731 05v 1.3 \ARBAY E12LCT 12 -10000.000 10000.000 -10000.000 10000.000 -10000.000 10000.000 -10000 000 10000.000 10000.000 10000 000 10000 000 10000.000 \RID 1 OCR! RTH1 < R < ETH2 ORV 0.0 - 0.2 \ARRAY RLCCII 12 '10000.000 0.151 -10000.000 0.510 0.100 0.900 '0.007 0.750 0.127 0.058 0.041 0.146 \EID 1 CORR RTHl < R < RTH2 ORV 0.3 - 0.6 \ABBAY chcnz 12 -10000.000 0.184 -10000.000 0.593 0.100 0.900 -0.014 0.726 0.097 0.073 0.062 0.201 \EID 1 1 005! BTn1 < a < 51u2 05v 0.7 - 1.2 \ABRAY ELCCH3 12 -10000 000 0.361 -10000.000 0.843 0 100 0.900 -0.008 0.776 0.123 0.096 0.109 0 210 \EID ‘ 1 ccan ETHi < s < ETH2 03v 1.3 \13311 s12ncr 12 -10000.000 10000 000 -10000 000 10000.000 -10000.000 10000.000 10000 10000 10000 10000 10000 -10000 000 10000.000 10000.000 \EID 1 1 ccsn ETH2 < E < ETH3 057 0.0 - 0.2 \ARRAY ELCCH1 12 -10000.000 0.079 -10000 000 0.008 0.679 0.086 \EID 1 1 corn ETH2 < a < ETH3 05v 0.3 - 0.6 \ARRAY ELCCH2 12 -10000.000 0.119 -10000 000 -0 004 0.656 0.107 \EID 1 1 ccan ETH2 < a < ETH3 03v 0.7 - 1.2 \ARRAY ELCCH3 12 -10000.000 0.217 -10000.000 0.003 0.761 0.120 \EID 1 1 ccsn ETH2 < a < ETH3 05v 1.3 \ARRAY 512301 12 -10000.000 10000.000 -10000.000 -10000.000 10000.000 10000.000 \EID 1 1 0083 3 > ETH3 ORV 0.0 - 0.2 \ARRAY BLCCXl 12 -10000.000 0.060 ~10000.000 -0 011 0.763 0.114 \EID 1 1 ccsn B > 3133 ORV 0.3 - 0.6 \18311 chcx2 12 -10000.000 0.093 -10000.000 -0 028 0.665 0.097 \EID 1 1 ccsn s > ETH3 65v 0.7 - 1.2 \ARRAY chcxa 12 -10000.000 0.166 -10000.000 0.005 0.738 0.120 \EID 1 1 corn 8 > ETH3 05v 1.3 \ARRAY 812XCT 12 -10000.000 10000.000 -10000.000 -10000.000 10000 000 10000.000 \EID 1 1 scsn a < 5731 05v 1.4 - 1.6 \ABRAY EISLCT 12 -10000.000 0.080 -10000 000 0.030 0.419 0.034 \EID 1 1 scan 3 < 3131 05v 1.6 - 1.9 \ARBAY ELECL1 12 -10000.000 0.084 -10000.000 0.040 0.500 0.063 \610 1 1 scan 3 < BT31 ORV 2.0 - 2.6 \ARRAY ELECL2 12 -10000.000 0.059 -10000.000 0.047 0.576 0.066 \EID .000 .378 .038 .480 .048 .683 .072 .000 .000 .321 .031 .482 .036 .643 .068 .000 .000 .574 .134 .553 .161 .489 .345 232 10000. -10000 '10000 10000 000 .100 .031 .100 .045 .100 .074 .000 10000. 000 .100 .024 .100 .043 .100 .072 .000 .000 .100 .051 .100 .076 .100 .132 10000. 10000. 10000. 10000 10000 10000 10000 10000 000 .900 .147 .900 .202 .900 .213 000 000 .900 .124 .900 .202 .900 .217 .000 .000 .900 .000 .900 .000 .900 .000 ‘1 filifil’.’ 9" «.5. {"151 fiflmflflflm’” 1 RCRH R < RTR1 ORV 2.6 - 3.2 \ARRAY ELECLS 12 -10000.000 0.044 -10000.000 0 065 0.896 0 072 \slo 1 scsn sTn1 < s < ETH2 osv 1.4 - 1.6 \ARRAY s1anT 12 -10000.000 0.030 -10000.000 0.074 0.416 0.038 \sln 1 1 scsn sTn1 < s < ETH2 osv 1.6 - 1.9 \ARRAY sLscn1 12 -10000.000 0.040 -10000.000 0.091 0.529 0.064 \810 1 1 scsn s7u1 < s < s7n2 osv 2.0 - 2.6 \ARRAY ELECH2 12 -10000.000 0.030 -10000.000 0.116 0.589 0.089 \510 1 RCRI RTRI < R < ETH2 ORV 2.6 - 3.2 \ARRAY sLscna 12 -10000 000 0.036 -10000.000 0.075 0.619 0 059 \RID 1 1 scsn ETH2 < s < stua Gsv 1.4 - 1.6 \ARBAY s15ncr 12 -10000.000 0 020 -10000 000 0 108 0.619 0.048 \sln 1 1 scsn srnz < s < sTua csv 1.6 - 1.9 \ARRAY sLscn1 12 -10000 000 0.026 -10000 000 0.101 0.554 0.076 \510 1 1 scsn s7n2 < s < srna asv 2.0 - 2.5 \13311 ELECH2 12 -10000.000 0.020 -10000 000 0.146 0.611 0.079 \sln 1 RCRH ETH2 < R < RTH3 ORV 2.6 - 3.2 \ARRAY RLRCH3 12 -10000.000 0.020 -10000.000 0.121 0.613 0.138 \EID 1 1 RCRH R > RTH3 ORV 1.4 - 1.5 \ARRAY R15XCT 12 '10000.000 0.018 -10000.000 0.111 0.533 0.049 \EID 1 RCRH R > RTR3 ORV 1.6 - 1.9 \ARRAY RLRCXl 12 -10000.000 0.024 '10000.000 0.109 0.615 0.083 \EID 1 RCRH R > RTHS ORV 2.0 - 2.5 .415 .244 .360 .055 .355 .095 .301 .349 .390 .245 .291 .045 .304 .086 .246 .106 .271 .246 .256 .050 .278 .129 233 .100 .107 .100 .022 .100 .044 .100 .091 .100 .145 .100 .027 .100 .065 .100 .091 .100 .077 .100 .034 .100 .091 10000 .900 10000. 000 .900 10000. 000 .900 10000. 000 .900 10000. 000 .900 10000. 000 .900 10000. 000 .900 10000. 000 .900 10000. 000 .900 10000. 000 .900 .000 .900 10000. 000 234 \ARRAY RLRCX2 12 -10000.000 0.016 -10000.000 0.207 0.100 0.164 0.648 0.102 0.446 0.147 \EID 1 RCRH R > RTK3 ORV 2.6 - 3.2 \ARRAY RLRCX3 12 -10000.000 0.020 '10000.000 0.271 0.100 0.121 0.613 0.138 1.246 0.077 \310 l 1 SPARE (mascot) \Assnv ELECTS 10 -10000.000 10000.000 -10000 000 10000.000 -10000.000 10000 000 10000.000 -10000 000 10000 000 \sln 1CC6L--c05H--sc5L--336--sc7L--sc7n--63--ss--4L--4n --------- \STOP 0.900 10000.000 0.900 10000.000 0.004 Bibliography [1] Sidney D. Drell and Tung-Mow Yan, Phys. Rev. Lett. 25, 5 (1970). [2] John C. Collins and Davison E. Soper, Phys. Rev. D 16, 7 (1977). [3] George Sterman, An Introduction To Quantum Field Theory, Cambridge Uni- versity Press, 252 (1993). [4] John C. Collins, Davison E. Soper, and George Sterman, Phys. Lett. B 263, 134 (1984). [5] G. ‘t Hooft and M. Veltman, Nucl. Phys. B 246, 189 (1972); P. Breitenlohner and D. Maison, Comm. Math Phys. 52, 11 (1977). [6] J.G. K6rner, G. Schuler, G. Kramer, and B. Lampe, Z. Phys. C 32, 181 (1986). [7] J.G. K6rner, E. Mirkes, G. Schuler, Internat. Jour. Of Modern Phys. A 4, 7 (1989) 1781. [8] J. Collins and D. Soper, Nucl. Phys. B 193, 381 (1981); Nucl. Phys. B 197, 446 (1982); Nucl. Phys. B 213, 545 (1983) (E). [9] J. Collins, D. Soper, and G. Sterman, Nucl. Phys. B 250, 199 (1985). [10] C. Balazs, J. Qiu, C.—P. Yuan, Phys. Lett. B 355, 548 (1995). [11] R.P. Feynman, Phys. Rev. Lett. 23, 1415 (1969). [12] R.P. Feynman, High energy Collisions, ed. C.N. Yang, J .A. 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