A Study of W Boson Properties with Four-Jet W⁺W⁻ Events at LEP Citation Shvorob, Alexander (2001) A Study of W Boson Properties with Four-Jet W⁺W⁻ Events at LEP. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5drh-kh53. https://resolver.caltech.edu/CaltechTHESIS:07292014-091223849 Abstract In this thesis I present a study of W pair production in e + e - annihilation using fully hadronic W + W - events. Data collected by the L3 detector at LEP in 1996-1998, at collision center-of-mass energies between 161 and 189 GeV, was used in my analysis. Analysis of the total and differential W + W - cross sections with the resulting sample of 1,932 W + W - → qqqq event candidates allowed me to make precision measurements of a number of properties of the W boson. I combined my measurements with those using other W + W - final states to obtain stringent constraints on the W boson's couplings to fermions, other gauge bosons, and scalar Higgs field by measuring the total e + e - → W + W - cross section and its energy dependence σ(e + e - → W + W - ) = {2.68 +0.98 -0.67 (stat.)± 0.14(syst.) pb, √s = 161.34 GeV {12.04 +1.38 -1.29 (stat.)± 0.23(syst.) pb, √s = 172.13 GeV {16.45 ± 0.67(stat.) ± 0.26(syst.) pb, √s = 182.68 GeV {16.28 ± 0.38(stat.) ± 0.26(syst.) pb, √s = 188.64 GeV the fraction of W bosons decaying into hadrons BR(W →qq') = 68.72 ± 0.69(stat.) ± 0.38(syst.) %, invisible non-SM width of the W boson Γ invisible W less than MeV at 95% C.L., the mass of the W boson M W = 80.44 ± 0.08(stat.)± 0.06(syst.) GeV, the total width of the W boson Γ W = 2.18 ± 0.20(stat.)± 0.11(syst.) GeV, the anomalous triple gauge boson couplings of the W Δg Z 1 = 0.16 +0.13 -0.20 (stat.) ± 0.11(syst.) Δk γ = 0.26 +0.24 -0.33 (stat.) ± 0.16(syst.) λ γ = 0.18 +0.13 -0.20 (stat.) ± 0.11(syst.) No significant deviations from Standard Model predictions were found in any of the measurements. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Physics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Newman, Harvey B. Thesis Committee: Newman, Harvey B. (chair) Hughes, Emlyn Willard Weinstein, Alan Jay Wise, Mark B. Defense Date: 8 September 2000 Record Number: CaltechTHESIS:07292014-091223849 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07292014-091223849 DOI: 10.7907/5drh-kh53 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 8619 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 29 Jul 2014 18:18 Last Modified: 12 Sep 2022 23:15 Thesis Files Preview PDF - Final Version See Usage Policy. 28MB Repository Staff Only: item control page

A Study of W Boson Properties with Four-Jet w+w- events at LEP Thesis by Alexander Shvorob In Partial Fulfilln1ent of the Requirements for t he Degree of Doctor of Philosophy California Institute of Technology Pru:;adena. California 2001 (Defended Septeinb('r 8, 2000) 11 © 2001 Alexander Shvorob All Rights R eserved Acknowledgments Ill Acknowledgments First of alL I would like to express my gratitude for the support that my adviser, Harvey Newman. has given me throughout my stay at Caltech and CERN. My work would have been so much harder without his patient guidance and trust in me. This thesis wouldn't be possible without help from many people in L3. It was indeed a pleasure to work with Alan Button, Chris Tully. 1\Iagda Pedace, :\Iarat Gataullin, Sergey Shevchenko and many other talented and interesting people. I owe special thanks to Martin Grunewald, who never turned down any questions of mine and was always there for me whenever I had a problem. 1\Iy interest in high energy physics was in large part due to the example that outstanding physicists like Vladimir Obraztsov and Giorgio Gratta set for me. I am indebted to them for my very enjoyable and engaging introduction to the field. 1\Iy sincerest thanks go to my friends. near and far away. who gave me the strength to persevere in my work. Daniel van Dierendonck did really a lot to keep my spirits up and make my life in Geneva fun. Kaia and Rego Ostonen never let me forget that there is life outside CER and acquainted me with so many new, exciting things. Ildar 1\Iukhametzhanov and Dima Pavlushko made me remember good old days and always kept me full of hope. Olga and Vasiliy lomokonovy have been such a wonderful company and a great source of support when I needed it. I am grateful for all the help and advice I got from Wenwen Lu, J ayant Shukla and Alexander Zvyagin. Thanks also to Edik, Denis. Joana . Lisbet, ivlyfanwy. Sandra, Tony. Reg and Winston for all the great times I had with them. I am indeed lucky to have friends like these. My parents and brother never left me to feel alone and lost. I am sorry I am so far away a nd don't see them nearly often enough. Last but not least at all. my warmest thanks to Oksana, who made it all worthwhile. IV Acknowledgments Abstract Abstract In this thesis I present. a study of W pair production in e+e- annihilation using fully hadronic w +w- events. Data collected by the L3 dctcctor at LEP in 1996 1998, at collision center-of-mass energies between_ 161 and 189 GeV, was used in my analysis. Analysis of the total and differential V\'+w- cross sections with the resulting sample of 1,932 V\'+V\r- ----. qqqq event candidates allowed me to make precision measurements of a number of properties of the W boson. I combined my measurements with those using other w +w- final states to obtain stringent constraints on thC' W boson's couplings to fermions, other gauge bosons, and scalar Higgs field by measuring the total e+e-----. w+v..;- cross section and its energy dependence 2.68::':K~~(stat.) ± 0.14(syst.) pb, vs 161.34 GeV 12.04::':~:~~(stat.) ± 0.23(syst.) pb, yiS = 172.13 GeV 16...!5 ± 0.67(stat.) ± 0.26(s:yst.) pb, jS = 182.68 GeV 16.28 ± 0.38(stat.) ± 0.26(syst . ) pb, yiS = 188.64 GeV = the fraction of W bosons decaying into hadrons BR(V\'----. qq') 68.72 ± 0.69(stat.) ± 0.38(8yst.) %, invisible non-SM width of thE' W boson rtt.visibtc < 17 MeV at 95% C.L .. Abstract Vl the mass of the V\1 boson 80.44 ± 0.08(stat.) ± O.OG(syst .) GeV, f\Ivv the total width of the vV boson 2.18 ± 0.20(stat.) ± O.ll(syst.) GeV, fw the anomalous triple gaugE' boson couplings of the W 6.gf 0.1G~2:~g(stat.) ± 0.11 (syst .) 6,,..,1 0.2G~gjj(stat.) ± 0.16(syst.) >., 0.18~2: ~g(stat.) ± O.ll(.<>yst.). No significant deviations from Standard Model predictions were found in any of the measurements. Contents Vll Contents 1 Introduction and Overview 1 2 Tests of Electroweak Physics with e+c-----* w +w - Events 3 3 2.1 Physics of the Vv Boson before LEP 2 . . . . . . . 3 2.2 e+e-----* v,r+w - Production in the Standard Ivlodel 9 2.3 e+e-----* w +w - Production a~:; a Probe of Electroweak Physics . 15 2.3.1 Constraints on Higgs Boson Mass . . . . . . . . . 16 2.3.2 Constraints on Extensions of the Standard Model 18 The L3 Detector at LEP 2 21 3.1 The LEP Accelerator . . . 21 3.2 LEP Beam Energy i\ Ieasurement 24 3.3 The L3 Detector . . . . . . 25 3.3.1 L3 Tracking System . 27 3.3.2 Electromagnetic Calorimeter 29 3.3.3 Scintillation Counters . 29 3.3.4 Hadron Calorimeter . 30 3.3.5 i\rluon Chambers 30 3.3.6 L3 Luminosity 1\Icasurement 31 3.3. 7 Trigger and Data Acquisition 31 3.3.8 Event Generation and Detector Simulation 32 Energy Flow Reconstruction in L3 . . . . . . . . . 32 Event Selection and Cross Section Measurement 35 4.1 Event Characteristics and Main Backgrounds . 35 QCD Multijet Events . . . . . . . . . . 37 3.4 4 4.1.1 Contents Vlll 4.1.2 Z 0 Z0 Events 39 4.1.3 /"/ Events . 40 4.1.4 .l\Ionte Carlo Simulation of ~ww - ---+ qqqq Events 41 4.2 Online Trigger . . . . . . . . . . . . . . 42 4.3 Detector and Data Acquisition Status . 42 4.4 Multijet Event PrC'selection 43 4.5 4.6 4. 7 4.4.1 Multiplicity Requirements 44 4.4.2 Visible Energy a11d Energy Imbalance . 45 4.4.3 Lepton and Photon Rejection 46 J et a nd W Boson Reconstruction . 47 4.5.1 J et Construction Algorithm 48 4 .5.2 J et Quality Requirements . 50 4.5.3 Constrained Kinematic Fit of l\Ieasured J et P arameters . 52 4 .5 .4 J et Combination Algorithms 56 Preselection Summary T eural . ... 59 etwork Discrimina tor 60 4.8 Selection of w + ~r----+ qqqq events 65 4.9 l\Ieasurement of the vv + ~r- ---+ qqqq Cross Section 67 4.10 Rarity Analysis . . . . 69 4 .11 Analysis U ncNtaintics 70 4 .11.1 .l\Iodeling of the Detector Response 72 4 .11.2 Modeling of the w +w - Signal Properties 76 4. 11.3 Modeling of the Backgrounds 80 4. 11.4 Luminosity 83 4 .11.5 .l\ IC Statistics 83 4.11.6 Summary of the Systematic Uncertainties 84 4.12 Summary of the Results . . . . . . . . . . . .. . 85 4.12.1 Measurement of the Total vv+w - Cross Section and the W Branching Ratios . . . . . . . . . . . . . . 86 4.12.2 l\Icasurcment of the I Yes I Matrix Element 90 Contents IX 4 .12.3 Limits on the Invisible Decays of the \TV 5 Measurement of the W Mass and Width 93 5.1 Threshold Cross Section Fit 9-! 5.2 Threshold Measurement Uncertainties . 96 5.3 W Invariant l'vlass Spectrum Reconstruction 98 5.3.1 Direct W Invariant l\Iass Reconstruction 99 5.3.2 Jet Combination Algorithm 99 5.3.3 Event Quality Requirements 100 5.4 Monte Carlo Reweighting l\Iethod 104 5.5 Fit of the Invariant l\Iass Spectra 105 5.6 Direct Reconstruction l\Ieasurement Uncertainties 112 5.6.1 Modeling of the Detector Response 113 5.6.2 l'v!odeling of the \\' +\\r- Signal Properties 114 5.6.3 l\Iodeling of the Backgrounds 116 5.6.4 Beam Energy Uncertainty 117 5.6.5 M C Statistics 117 5.6.6 Other Checks 118 5.6.7 Summary of the Systematic Uncertainties 118 5.7 6 91 Summary of the Results 119 Search for Anomalous Triple Gauge Boson Couplings 123 6.1 Parametrization of Trilinear Couplings of the Vv Boson 125 6.2 Reconstruction of the \\T- Direction Distribution 127 6.2.1 Jet and W Boson Charge Estimation 128 6.2.2 Event Selection and Jet Combination 130 6.3 Extended l\Iaximum Likelihood Fit 132 6.4 Analysis Uncertainties 135 6.4.1 l\Iodeling of the Detector Response 136 6.4.2 ~IodC'ling of the w +v.,r- 136 6.4.3 l\Iodeling of the Backgrounds Signal Properties 137 Contents X 6.5 7 6.4.4 l\IC Statistics 138 6.4.5 Other Checks 138 6.4.6 Summary of the Systemat ic Uncertainties 138 Summary of t he Results . . . . . . . . . . . . . . 139 Summary and Conclusions 141 7.1 Summary of Results 141 7.2 Combination with Other Results . 145 7.3 Conclusions . . . . . . . . . . . . 147 List o f Tables xi List of Tables 2.1 Fermions in the Standard 1odcl. 5 2.2 The Cabibbo-Kobayashi-I'daskawa matrix elements. 11 2.3 The W boson decay and w +v..r- final state branching ratios. 12 3.1 LEP beam energy uncertainty and spread. . . . . . . . . . . 25 4.1 S!\l cross sections for e+ E'----+ V..'+vv- ---+ qqqq and background processes. 39 4.2 Integrated luminosity for the data sets used in the analysis . . . . 43 4.3 Efficiency of the jet combination algorithms used in the analysis. 57 4.4 umber of data events and S 1 cross sections acc0pted by the preselection cuts. . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Number of data events and Sl\1 cross sections accepted by the neural network selection . . . . . . . 4.G 67 c+e----+ w +w ----+ qqqq and e+e----+ qq cross sections determined in neural net output fits. 4. 7 60 . . . . . . . . . . . . . . . . . . . . . . . 69 e+e----+ v..:+ vv----+ qqqq and e+e----+ qq cross sect ions determined in rarity fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.8 Integrated luminosity for the calibration data sets used in the analysis. 73 4.9 Systematic errors of the w+w---+qqqq cross section measurement. 84 4.10 l\1easured e+E'---+ w +w ---+ qqqq cross sections. . . . . . . . . . . . 85 4.11 Efficiencies of fvf!v, qqfv selections used by L3 at Js = 189 GeV. 87 4.12 Tumber of selected data events and expected background cross section for evCv, qq[v selections used by L3 at Js = 189 GeV. . . . . . . . . 88 4. 13 Measured W decay branching ratios and total V\1+w - cross sections . . . . . . . . . . . . . . . . . . 90 umber of jet combinations used for t he \rV mass fits. 102 at d ifferent energies. 5.1 List o f Tables xu 5.2 Results of independent i\lw and fw measurements with direct reconstruction of w+w-- qqqq events. . . . . . . . . . . . . . . . . . . . . 5.3 111 Results of simultaneous i\lw and fw measurements with direct reconstruction of w +vi- - qqqq events. . . . . . . . . . . . . 113 5.4 Systematic errors of the measurements of i\Iw and fw . 118 6.1 1 umber of events usee! for triple gauge boson couplings fits. 132 6.2 Systematic errors of the triple gauge boson couplings measurements. 139 xiii List of Figures List of Figures 2.1 A w___.ev event recorded by the UA1 detector .. 8 2.2 Lowest order e+e-___... w +w - Feynman diagrams. 9 2.3 Total cross section of the reaction e+e-___. . w +w - and contributions of the CC03 diagrams as a function of VS· 2.4 10 Polar angle distribution of the V..'- boson and contributions of the CC03 diagrams for c+e-___... w +v..'- production at Js = 189 GeV. . . . 2.5 10 Effects of finite W width and QED radiative corrections on the total c+e- ----7 w +w - cross section. 0 13 0 2.6 e+e-___. . udsc Feynman diagrams. 14 2. 7 The muon decay. 17 2.8 Radiative corrections to the W boson propagator. 17 3.1 Schematic map of the LEP complex. 22 3.2 The LEP injector chain. 23 3.3 The L3 detector. 26 3.4 A side view of t he inner L3 detector. 27 3.5 A schematic r ¢ view of t he L3 central tracker. 28 4.1 End view of a four-jet event recorded by t he L3 detector. 36 4.2 Order a~ QCD diagrams contributing to four-jet fina l states. 38 -!.3 The Z 0 Z0 production diagram. . . . . . . 40 4.4 Two-photon hadronic event production. . 40 4.5 An H CAL noise event. . . . . . . . . . . 44 4.6 Distribution of the number of reconstructed energy flow objects. -!5 4.7 Distribution of t he number of reconstructed t racks. 45 4.8 Distributions of t he normalized visib le energy and longitudinal energy . . . . . . . imbalance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 List of Figures x1v 4.9 Distributions of the maximal E.I\I bump energy and maximal muon . . . . . . . . . . . . . . . . 48 4.10 Distribut ion of the y 34 parameter for selected hadronic events. 50 4.1 1 Distribution of the maximal electromagnetic jet energy fraction. 51 4.12 Distribution of the minimal jet multiplicity. 51 mome~um. 4.13 Distribut ion of the average reconstructed W mass before and after 4C kinematic fit for ~r+w- ---4 qCiqq .1\ Ionte Carlo events .. 54 4.14 Distribution of the 4C kinematic fit probability. . . 54 4.15 Distribution of the jet combination choice variables. 58 4.16 Distribution of the y 34 distance a fter the preselection. 62 4.17 Distribution of the spherocity after t he preselection .. 62 4.18 Distribution of the W mass difference m 4 c 1-m4 c 2 after the preselection. 63 4.19 Distribut ion of the minimal jet-jet angle Bmin after the preselection. 64 4.20 Distribution of the average hemisphere mass after the preselection. 64 4.21 Distributions of the neural network output . 66 4.22 Distributions of the event rarity. . . . . . . 72 4.23 Angular distribution of average jet energy in Z0 ---Tqq events. 74 4.24 Angular distribution of jet energy variance in Z0 ---Tqq events. 74 4.25 Angular dependence of reconstructed jet angle biases in Z0 ---T qq events. 76 4.26 Angular dependence of reconstructed jet angle resolutions in Z0 ---T qq events. . . . . . . . . . . . 77 4.27 The ratio of selected event rates in Z0 data and .1\IC as a function of y 3 4 and neural net output. 82 4.28 e+e- ---4 w +w - cross section measured at different energies. 89 5.1 w+V\r- cross section in the threshold region . . . . . . . . . 95 5.2 w+w- cross section at 5.3 The m 5 c distribution for selected w +w - ---4 qqqq candidates for various vs = 161.34 GcV as a function of .1\Iw. jet combination algorithms. 5...! . . . . . . . . . . . . . . . . . . . . . . . 96 101 Distributions of t he 5C kinematic fit probability P y2(\2 5c; 5) for two best jet combinations. 102 List o f Fig ures 5.5 XV Reconstructed m 5 c invariant mass distributions for selected jet combinations. 103 5.6 Correlation of the best and second best jet combination l\Iw fit results. 108 5.7 Average fitted W mass for small t-- IC samples with different Mw . . . . 5.8 Distributions of the Mw fit results and fit errors for small t-- IC samples. 110 5.9 Average fitted W width for small MC samples with different fw . . . . 109 111 5.10 Distributions of the fw fit results and fit errors for small t--IC samples. 112 6.1 The phase space of the e+e-.....-4 \ V+-vv - ....-4f1f 2 f3 f 4 four-fermion system. 124 6.2 Distribution of the number of tracks used in jet charge measurement. 131 6.3 Distribution of the W boson charge sign estimator. 131 6.4 Reconstructed cos 8w distributions for events selected for triple gauge boson couplings fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.1 T he comparison of the direct and indirect measurements of Mw , t--.ft· . 147 7.2 The indirect measurement of t--. IJ-1. . . . . . . . . . . . . . . . . . . . . 147 XVI List of Figures Introduction and Overview 1 Chapter 1 Introduction and Overview In this thesis I present my study of the pair production ofW bosons in the electronpositron annihilation process e+e-- w +w-. The purpose of the measurements I performed is to determine important properties of the W boson. such as its mass, width. decay branching ratios, and couplings to the photon and the zo boson. Abundant production of reconstructible w +w - events in a clean environment allows one to measure these quantities with a precision unattainable with other techniques. The measurements are compared with precise theoretical calculations to test the standard electroweak theory and search for possible deviations from it. In the absence of significant disagreements with SM predictions, the measurements can be used to constrain yet-unknown parameters of the Standard :tv lodel, such as the Higgs boson mass. My analysis deals specifically with events in which both W bosons decay hadronically W- qq', resulting in four-jet ~T+ \v -- qqqq events. Such events represent about 46% of the total W pair production cross section. Reconstruction of the differential e+e- - w +w- cross sections with these events allows t he most precise measurement of a number of parameters of the ~' boson. I combine my measurements based on four-jet events with those using other types of w +w - events to arrive at the best experimental estimates, to be compared with theoretical predictions. t\Iy analysis was based on the data collected by the L3 detector at LEP in 19961998 in runs with center-of-mass energies from 161 to 189 GeV. Approximately 2,000 multijet w+w- candidate events found in these data were used in my measurements. Below I give a brief description of the following chapters of my thesis. Chapter 2 discusses the status of theW boson in t he Standard .l\Iodel, the properties of the W production as predicted by the S1\I. and observable effects which could Int roductio n and Overv ie w 2 be introduced by new elcctroweak theories being proposed to replace it. Chapter 3 gives an overview of LEP and L3. the experimental facilities used to obtain the data I used. with an emphasis on characteristics especially important for my study. Chapter 4 describes the methods I developed for discriminating w +w ----+ qqqq events from physics background processes and for measuring the w +w- ---+ qqqq production cross section. The methods are applied to the L3 data sets to select events for analysis of the ~, mass and width and triple gauge boson couplings. The results of the cross section measurements are combined with those of semileptonic and leptonic V..'+w - events to measure the total w+w- cross section and ~' branching ratios. Comparison of those with the SI\ I allowed me to put limits on Cabibbo- Kobayashi-Maskawa matrix clements and possible non-standard invisible W decays. Chapter 5 is devoted to my measurements of theW mass, using the dependence of the total w +w - cross section on the W mass in the threshold region, as well as the reconstructed dijet invariant mass distributions. The latter are also used for a direct measurement of the \tV width. I combine my measurements with those obtained with semileptonic events to arrive at a more accurate estimate of the W mass, used to check the consistency of the Standard Model and infer the mass of the Higgs boson. Chapter 6 describes reconstruction of angular distributions in w+ ~'- ---+ qqqq events which are used, together with the total w+w - cross section, to study triple gauge couplings involving W bosons. Limits on possible deviations of the w +w -, j Z0 couplings from their SI\I values are obtained. Chapter 7 summarizes the results obtained in my study. Tests of E lectroweak Physics with e+e- ----t y..,r+ v.,;- Events 3 Chapter 2 Tests of Electroweak Physics with e+e -~ w+w- Events 2.1 Physics of the W Boson b e fore LEP 2 The first experiments whose understanding required the W boson were made, and the study of the weak interactions began, when spontaneous radioactivity of uranium was discovered by Henri Becquerel in 1896 [1]. It was soon realized that several quite different decay processes contributed to it [2] and the one which was accompanied by production of electrons was called t he ,8-decay. Creation of t he theory which explained t his phenomenon. among many others discovered later. was one of the major achievements of physics in the twentieth century. The neutrino [3] was postulated to explain apparent non-conservation of energy in ,6-decays, and t he discovery of two different kinds of nucleons [4] la id the foundation for the t heory of the weak interactions by Enrico Fermi [5]. The t heory tried to describe ,8-decay in terms of point couplings of charged vector currents created by weak isospin doublets. Later t h e conjecture of parity violation by the weak interactions was put forward [6] and confirmed experimentally [7]. This led to replacement of the original vector currents with the currently accepted V A structure [8]. Sever al attempts wer e made to improve on the contact interaction nature of the Fermi theory, which limits it to be a mere low-en ergy. non-renormalizable approximation. Introducing a vector boson mediating the interaction, in analogy with the photon in electromagnetism. would solve this problem. The analogy cannot be complete though, because a massless interm ediate boson would be inconsistent with the 4 Tests of Electroweak Physics with e+c----t Vv+w- Events observed very short range of charged current weak interactions ('"'-' 10- 17 m). Eventually it was realized that short-range forces can be mediated by massive particles [9) and a heavy intermediate vector boson. denoted W. was proposed as a carrier of the weak charged current interaction by Oskar Klein in 1938 [10]. This chain of outstanding theoretical and cxperimental insights culminated in the development by A. Salam, S. Weinberg. and S. Glashow, among others. in the early 1970s of the modern electroweak theory [11), commonly referred to as the Standard l\Iodel. The underlying principle of the theory is the invariance of ature with respect to local transformations of fermion fields. given by the SU(2) group in weak isospin space and the U(1) group in the hypercharge space (non-Abelian gauge theories were first considered by C.N. Yang and R.L. l\Iills [12]) . To reproduce the low-energy phenomenology of the weak interactions, the symmetries have to be spontaneously broken by interactions with au additional scalar field whose ground state is not invariant under the gauge group transformations (the Higgs mechanism [13]). The structure and the mathematical apparatus of the theory are explained in great detail in the literature. e.g., see [14). The theory was proven to be renormalizable [15) and subsequently was shown to possess great prcdictive power over a wide energy range. It explained all the data available at the time it was proposed. and has passed numerous new experimental checks since its inception [16] , invariably with great success. Properties of the intermediate w± and Z0 vector bosons, carriers of the weak interaction, were among most crucial predictions of the Salam-Weinberg theory. The origin of theW boson in the Standard model is local SU(2) gauge symmetry in weak isospin space, postulated for doublets ofleft-handed fermions, listed in Table 2.1. Isospin raising and lowering operators. associated with SU(2) gauge fields. change the electric charge of the fermion they act on and therefore can be associated with particles of charge 1 and -1 respectively. The third SU(2) generator mixes with the generator of the U(1) hypercharge symmetry group upon breaking of the symmetries, giving the photon and the Z0 boson. The couplings of the W boson to fermions are determined by the fermionic kinetic 2.1 Physics of theW Boson before LEP 2 5 T3 = +1/2 T3 = -1/ 2 T3 = 0 Table 2.1: Fermions in the Standard l\1odcl and their weak isospin eigenvalues. Left-handed fermions belong to SU(2) doublets, while right-handed ones are SU(2) singlets. Quark weak eigenstates are mixtures of quark m ass eigcnstates. term of the SM Lagrangian. after introducing t he covariant derivative (2.1) where Y and Bt.t arc hypercharge and the corresponding gauge field. TJ are SU(2) generators (commonly represented by Pauli matrices), 9 1 and 92 arc U(1) and SU(2) couplings respectively, and the vector fields H 'J are related to w ± bosons as w ± = ~( H' l =F i Tr 2 ). The gauge terms of the covariant derivatives arc responsible for the terms in the Lagrangian (2.2) L Jermiomc which give rise to couplings of W boson fields to the charged currents J ±l' - """' ~1. /.t 1 - /5 ± D 'Pl 2 T ljJ . (2.3) 111 Couplings to all left-handed fermions have the same strength ~ Self-couplings of W bosons arc also fully determined by the gauge structure of the covariant derivatives. through its effect on the W field strength tensors Trt.tv = ; 2 [V,., Vv] which enter the bosonic kinetic term Lbosonic = (2 .4) Tests of E lectroweak Physics with e+e----+ w +v.r- Events 6 Quadrat ic terms in t he field st rength tensors introduce triple gauge , vv +v,r- and Z0 \,Y+v,r- couplings, discussed in Cha pter 6. as well as quadruple vertices 11 v.w vv - . 1 zo vv+w - . zozo w +v.r- . \\'+v.'- v,r+ w - . While t he gauge symmetry idea allows one to predict many properties of t h e W b oson, the picture is incomplete' without t he concept of sp ontaneous symmetry breaking. I\Iass terms for the W b osons (of the form m 2 T~v;~ r V~) are not gaugeinvariant , still a (large) W mass is necessary t o account for t he very short range of weak charged current interactions. The Higgs m echa nism provides a solut ion of t his problem by p ostulating the exist ence of a n addit ional (Lorentz) scalar field which is an SU(2) doublet (this is t hC' minimal case: additional doublets and larger multiplets can b e easily incorporated into au extended Higgs model; such a complication is not warranted by any exp erimental da ta yet ). For some choices of the p aram eters 11 2 , ). of the Higgs Lagrangian (2.5) LHiggs t he ground state of the scalar field has a non-t rivial exp ectation value and is not SU(2)-invariant. \V'it hou t loss of generality (after a gauge rotation) pertur bation s of such a ground st a te can be writ ten as a doublet of t he form w(.r) = -1 ( J2 0 v + H (x) (2.6) ) ' where v is the vacuum expectation value. and H (x:) is the scalar field of the Higgs boson - a new part icle predicted by t his mecha nism of elect roweak symmetry breaking. After choosing a particular scalar field ground st a te t he Higgs term becomes L Higgs 9] vv~ vV ~('u + H )2 + ~(g2H'f- g1B11 )(g2H 1 ~- gtBJ-L)(v + H)2 + 1 1 2 ). 2 4 ~ (811 H)(fJ1 , H) - !!_(v + H f - -(v + H )4 . 2 (2 .7) The t erms containing v 2 factors and gauge fields can be inter preted as mass terms for 2.1 Physics of theW Boson before LEP 2 7 two charged W bosons with mass 1 (2.8) -v92 2 and a neutral Z0 boson, given by the ZJ.L = 92 l-V: - 9 1 BJ-L combination. with mass 1 / 912 + 92· 2 -v (2 .9) 2 The orthogonal W~, BJ.L combination A~,= 9 1 W~+92 BJ.L does not acquire a mass term and is associated with the photon. The other terms in the Higgs Lagrangian describe interactions of the Higgs and gauge fields (H Z0 Z0 , H w+w- . H H Z0 Z0 , and H H w +w - vertices) and Higgs boson self-interactions. The angle Bw by which the (A 1,, Z1~) and ( M~~. BJ.L) bases are rotated, called the electroweak mixing angle, is completely determined (to leading order) by the strengths of the hypercharge U(1) and isospin SU(2) interactions: 91 sinBw = -~~~ I 2 'J' v 91 + 92 92 cos Bw = --;::::;:~=::;;: -/9f + 9~ (2.10) This single angle, in turn, determines the ratio of the vector boson masses Mw i\Iz cosBw, (2.11) the ratio of electromagnetic fermion charges e = .J47rO:em and weak charged current couplings 91 cosBw = 92 sinBw , (2.12) and the structure of the neutral current couplings 1ff a nd Z0 {f. By 1980, studies of deep inelastic neutrino scattering produced an accurate measurement of the mixing angle Bw . Combined with the precise knowledge of O:em• t his led to predictions for the interm ediate vector boson masses [1 7]. 8 Tests of Electroweak Physics with e+e-- t w+w- Events These predictions were confirmed in 1983 by the discovery by the UA1 experiment at CER N's SPS collider of both the \",T [18] and Z0 [19] bosons. with masses in agreement with t he Standard :Model. Figure 2.1: A W - t e// event recorded by the UA1 detector. An energetic electron amidst hadronic background can be seen in the lower right quadrant. Five events of this type constituted the discovery of theW boson by the UA1 experiment. Since the discovery of the W boson, precision studies of its properties were performed by the CDF and D 0 experiments at the Tevatron collider at Fermilab [20]. Also information on the triple gauge boson couplings and limits on possible deviations from the Sl\1 were obtained by analysis of LEP 1 electroweak measurements [21]. Still LEP 2 measurements have been able to make an important contribution to and opened a new window on the physics of the W boson. 2.2 c+e- ----t vv+w- Produc tion in t he Standard Mode l 2.2 9 e+e---+ w+w- Production in the Standard Model Production of 'N pairs is possible in <'+<'- collisions at center-of-mass energies above rv2 lw. It proce<'ds through the four diagrams shown in Figure 2.2. e The contribution e e e v Figure 2.2 : Lowest order e+e - ----t w +w - Feynman diagrams. of the s-channel Higgs exchange diagram is very small because of the weakness of the He+<'- coupling and is suppressed by a factor life / JS relative to the other diagrams. This diagram is usually ignored in t heoretical calculations. The remaining three diagrams are commonly referred to as the CC03 set. The 8-channcl photon and Z 0 boson exchange diagrams involve 'Y v..;+v.,r- and Tests of Electroweak Physics with e+c----t w+w- Eve nts 10 zow+w- vertices due to the non-Abelian nature of the SU(2) gauge group and are therefore especia lly interesting. Close to threshold the .<>-channel P-wave amplitudes are suppressed compared to the f-channel neutrino exchange. as shown in Figurc 2.3. But already several GeV abovc the threshold, i.e., at the higher energy points data from which were used in my analysis , all diagrams' contributions as well as their interferences become important (sec Figure 2.4). The complete formulae for the CC03 e+e- ---t V\'+w - matrix element and its helicity amplitudes composition can be found in [22]. 50 r-------------~---------. .....-;;········· g 25 Li. / ?.:.-··· Totol ~ -2: ~~~:_ :~=~~~:~ t:) . ..... ·················.... 180 200 220 a> en r 0 (.) I ....... i Q) + ~ l? ······· 160 40 .-----------------------• ..0 0.. "0 240 -Ts [GeV] Figure 2.3: Total cross section of the reaction e+e- - w+w- and contributions of the CC03 diagrams as a function of .JS. 20 ................. !C............... Total z• 0 . :.~~:.:.:;.:.~~:~.~~:~:.:~.·.:: ::::.:.:.:~~~~:.:~.:.:.~~:~:.:::.:~;:.:.:.:::::/{: ~ r 4 .................. ...YJ:. . .... ······-· ·-··········· ....•.....-·-:::::::/ · · ·· ;;;z -20 '--~---'--~---'--~---'---~--' -1 -0.5 0 0.5 cos ewFigure 2.4: Polar angle distribution of t he w - boson and contributions of the CC03 diagrams for e+c- - vv+w - production at Vs = 189 GeV. Once produced. theW bosons decay into all kinematically allowed SU(2) doublets with the partial width given by (2.13) wh0re the V matrix, relating mass a nd SU(2) eigenstates. is assumed to be diagonal for lepton families and is given by the Cabibbo-Kobayashi-1\Iaskawa matrix for quark 2.2 e+e-____, w+w- Production in the Standard Model 11 doublets. The QCD correction factor is again trivial for leptons and takes into account color SU(3) degrees of freedom and QCD radiative corrections for quarks: 3 ( 1 + as(M~) ). (2.14) 7r The kinematic factor accounting for non-zero fermion masses, 1- [1 (m 1 + mr ) 2 l\.I 2w . 1- (m 1 - mr )2 2 X l\Iw 2 2 )2 _ ( m 12 -mf' ] _ m 12 +mr ? 21\I\v 4 21\Iw ' (2.15) can be considered to be one with good accuracy (better than 0.5%) for all kinematically allowed fermion pairs. Given the structure of the CKl'vi matrix (see Table 2.2) , the two dominant hadronic decay modes of the W arc vv+____, ud and w+ ____, cs (and their charge conjugates), each haYing branching ratio about three times that Of W ----1 fv for a given lepton family. The branching ratios of tll<' v.· boson and the resulting composition of v_r+v,r- events arc given in Table 2.3. The Standard l\1odel predicts about 10% of w+w- events to be of the fv(v type. with the rest shared roughly evenly between qqfv and qqqq events. 0.9742- 0.9757 0.219- 0.225 ( 0.004 - 0.014 0.219- 0.226 0.9734 - 0.9749 0.035 - 0.043 ) 0.002 - 0.005 0.037- 0.043 0.9990- 0.9993 Table 2.2: 90% confidence limits on Cabibbo-Kobayashi-l\Iaskawa matrix elements [23]. Columns in the table correspond to quark mass eigenstates d, s. and b. rows to respective weak isospin eigcnstates. Unitarity of the matrix within the three quark generations is assumed. The total width of theW boson in the SM, r.v Tests of Electroweak Physics with c+e----+ w+w- Events 12 \\.Decay \i\'---+ (v W---+ hadrons BR[%] 32.49 67.51 \V+\i\·- Final State (v(v qq(v qqqq Relative Frequency [WJ 10.56 43.86 45.58 Table 2.3: TheW boson decay and w +w - final state branching ratios predicted by the Standard Model. amounts to 2.067 ± 0.021 GcV using the current knowledge of l\Iw and cannot be ignored in calculations of the c+ e----+ vv+vv- total and differential cross sections. The finite W width leads to large corrections to the zero-width-approximation total w+w- cross section at all LEP 2 energies [24], as shown in Figure 2.5. The effect is particularly strong at the threshold, resulting in a much less sharp rise of the cross section with energy and the possibility of w+w- production below the y8 = 2Mw threshold. The finite W width also introduces a spread of the masses of the produced W bosons, which, ignoring phase space factors, is given by the Breit-Wigner density 1 :;;: (rn 2 - Iviwrw tvl~)2 + M~r~ · (2.17) This spread has important consequences for the precision of the W mass measurement described in Chapter 5. Higher-order EW processes arc another significant source of corrections to e+c----+ w +w - cross section. Initial state radiation decreases the w+w- cross section by 0(10%) at LEP 2 energies [25]. It also leads to an 0(1 GeV) downward shift of the distribution of the produced W boson masses. This shift has to be corrected for in the measurement of the W mass. The Coulomb singularity [26, 27] is another QED process especially important at the threshold. It is due to the electromagnetic attraction between slowl:v moving charged W bosons. The correction is proportional to Gem/ f3w and diverges at the threshold in the limit of zero W width. With finite rw the correction amounts to about 5% right at the threshold and smaller values at 2.2 e+e------.. w + w - Production in the Standard Model 13 ,.........., .0 0.. '---' 20 --?=:: ./;·::·;·;:::.:::.:::::•:·:::··········::··:~·:~:=:::::::::7:.-:_,.,,.,.,.,, ·········------ I s s 10 + I ...:,::::::>' i No ISR , r w=O No ISR , r w = 2.1 GeV )/ :.:; :.:: a> + a> ISR r :!/ - ' ISR, r w = 2.1 GeV / .... :*· t) 0 w= .... ,. ,,,rr::·· 0 150 160 170 180 190 200 210 220 230 240 250 -Vs [GeV] Figure 2.5: Total E'+e------.. w+w- cross section as a function of Vs· Effects of finite W width and QED radiative corrections are shown. higher energies. The total effect of QED radiative corrections on t he e+e------.. w +v.,'cross section is shown in Figure 2.5. QED corrections are explicitly taken into account in the an alytical programs and l\Ionte Carlo event generators I used in my analysis. The bulk of other electroweak radiative corrections is incorporated in the calculations by using energy-scale-dependent (running) values of EW parameters. Considering W bosons as r esonances with finite width, rather t han zero-width (stable) particles, leads to an other complication. To preserve gauge invariance of the e+e-----. v.,r+ w -----. f 1f 2 f3I 1 amplitude one has to consider all Feynman d iagrams leading to the f1f 2 f3 f 4 final state, as opposed to the CC03 set. For w +w - -like four-fermion final states, the number of contributing diagram s ranges from 9 to 56 depending on the final state, e.g., 11 for udsc and 43 for uddu four-quark final states. with both charged and neutral current graphs present as a gen eral rule [28]. Figure 2.6 shows all the Tests o f E lectroweak P hysics with e+e- ~ y.,r+w - Events 14 4 6 6 3 5 5 6 3 3 5 2 4 2 graph 1 graph 2 graph 3 5 5 5 6 6 6 4 4 4 3 2 4 2 3 2 graph 4 graph 5 5 3 2 graph 6 4 4 6 6 6 4 3 2 5 2 graph 7 g raph 8 4 2 graph 10 5 2 graph 9 4 2 graph 11 Figure 2.6: The complete set of e+c- ~ udsr Feynman diagrams. Graphs 1, 6, and 7 are CC03 double- resona nt diagr am s. The rest of the diagrams contain one \ V boson each a nd are single-reson ant . 2.3 e+e----+ w+w- Production as a Probe of Electroweak Physics 15 diagrams contributing to e+e----+ udsc. Fortunately, in the commonly used gauges the single-resonant and non-resonant graphs are suppressed relative to double-resonant CC03 ones by powers of fw / lw ~ 2.5 x 10- 2 . They are ignored in the l'vlonte Carlo programs I use and I have to evaluate the effect of their neglect on my measurements. After taking into account all t he corrections mentioned above, SM calculations predict values of the e+e- ---+ w +w - cross section of approximately 3 pb at the threshold, rising fast to "'16 pb at .JR = 189 GeV (see Figure 2.5). By and large. the precision of the available theoretical calculations is adequate for the measurements I am interested in. The current. understanding of the w +w - cross section is accurate to 2'/t at the threshold and about 1% at higher energies. while the average ISR energy loss. bearing on the 'N mass measurement, is known to about 15 MeV [29]. Those uncertainties are small compared to actual experimental errors. 2.3 e+e-----+ w+w- Production as a Probe of Elec- troweak Physics The sizeable cross section and good theoretical understanding of the reaction e+e----+ w +w - makes it a promising tool for precision studies of the physics of the w boson. It was recognized already in 1961 that "electron-positron collisions would in fact constitute a good experimental means for [intermediate charged vector mesons'] detection.'· [30] The first naive methods of measuring the mass and electromagnetic properties of theW with e+e----+ w +V\r- events were proposed as early as in 1965 [31] . Experimentally, e+e- colliders offers significant advantages over pioneering pp machines. As the backgrounds for c+e----+ w+w- event selections are mostly due to other electroweak processes, they do not dominate and this allows one to study all decay modes of the W . An e+e- collision has a well-defined initial state which enables one to reconstruct the full event kinematics, in contrast to hadron colliders, where only transverse momenta can be used for measurements. Also the importance of triple gauge boson vertices in W pair production makes their st udy easier in e+e- Tests of Electroweak Physics with e+e-- ~r+ w - Events 16 collisions. Soon aft er the Standard .l\Iodcl was developed. e+e- - w +w- annihilation was proposed as a means to test the S.l\I and contrast it with other electroweak models [32], and to measure its triple gauge boson couplings [33]. The reaction is particularly interesting because an intricate interplay (gau ge cancellations) between all four diagrams in Figure 2.2 is required to make the total c+e- - w+w- cross section and its partial waves preserve unitarity at high energies (see. for example [14]). It was found [34] that the requirement of "good" high-energy behavior of e+e- - w+w- amplitudes limits the structure of a prospective clectroweak t heory to the spontaneously broken Yangs-.l\ Iills gauge one of the Standard Iodel, and also puts severe constraints on the mechanism of the symmetry breaking. Thus a lso from the theoretical point of view, c+e- - w +w - production is a sensitive probe of th<' gau ge structure of the SJ\I and its possible extensions. 2.3.1 Constraints on Higgs Boson Mass The m ass of the Higgs boson .l\ 111 is not predicted by the Standard Model and is an object of enormous interest. Finding the Higgs boson and determining its mass arc some of the main goals of current exp erimental programs at LEP and FXAL and of the LHC project . .l\lwdependent r adiative correct ions to well-measured processes involving t h e W boson provide a means of estimating .l\ln , using the available precision electrowcak measurements. The width of the muon decay ,,- - e- Dev11 (sec Figure 2. 7) can be measured very accurately and allows a very precise determin ation of the Fermi constant Gp [35] which is t he strength of weak charged current interactions at zero momentum transfer. This coupling is given by 2 .lh (2. 18) After expressing sin Bvv in terms of .l\ Iw and the well-measured Mz the relation be- 2 .3 e+e- ---t w+w- Production a s a Probe of Ele ctroweak Phys ic s H 17 t w~ w w wvvOvv w b Figure 2.8: Radiative corrections to the 'N boson propagator involving the top quark and the Higgs boson. Figure 2.7: The diagram of the muon decay 11----+ e- fievw comes (2.19) Radiative corrections modify the above tree-level expression. The QED part of the corrections is taken into account by using the running value of electromagnetic coupling O:em at 1\Iz instead of the one at low energies, which is very precisely measured. Elcctroweak corrections due to loop insertions in theW boson propagators. shown in Figure 2.8. are taken into account explicitly, 2 1\ Iw(l - M~ (2.20) M2 ) z The weak radiative correction [36] is dominated by the term which depends on the top quark mass l\I 1. bnt has a non-negligible l\I 11 dependence as well, [ 3 cot 11 l 2 r-- q ( 2 Bv·l 1\.I~"' + 2 cot Bw - !\I~ 3 11 1\I~, + ...t 3 1) l\ Iz 3 ln 1\ I~, - 2 ') 1ncos Bw +cot- Bw + 41]. 18 (2.21) Thus, given measurements of Gp and vector boson and top quark masses and a Tests of Electroweak Physics with e+e- ----. w+v~,r- Events 18 calculation of O.em (f\ I~), the Gp definitiou (2.20) can be interpreted to give an estimate of f\1 11 • The results of this analysis. done with the currently measured value of f..Iw. are reported in Chapter 7. 2.3.2 Constraints on Extensions of the Standard Model Various new theories proposed to replace the SM predict properties of the W bo~on and w+v...r- product ion that differ from those expected in the Standard Model. c+e- ----. w+v...r- production can be used to put them to an experimental test. Supersymmetry is currently the leading '·new physics·' candidate. It predicts a large number of new particles, which introduce additional large radiative corrections to electroweak processes. Equation (2.20), reinterpreted as a calculation of the W mass, predicts directly measurable 0(100 MeV) shifts of lVIw in the Minimal Supersymmetric Standard Modcl [37], depending on the internal parameters of the model. Supersymmetric Sf\! extensions also predict difficult to observe 0(10- 2 ) anomalous triple gauge boson couplings [38]. New particles predicted by tcdmicolor models or obtained by adding more generations to the three currently included in the Standard f\Iodel, also enter W observables through loops in W propagators. and lead to similar deviations [39]. Other more exotic theories predict deviations from the SM in e+e- ----. w +w already at the tree level. For example, expanding the gauge groups of the SM can lead to additional neutral intermcdiatc vector bosons. The existence of such a boson would modify, by its mixing with Z0 . the e+e- ----. zo____. v...r+ w - process with large. modelparameter-dependent effects on the v,r+w - total and differential cross sections [40]. Another extension of the Sl\I attempts to solve the hierarchy problem [41] (naturalness of gravitational radiative corrections to the Higgs boson mass) by postulating the existence of a few additional dimensions and an electroweak-gravitation ·'unification scale'' ills of the order of a few hundred GeV [42]. Being compact with sizes of the order of 1 mm, these extra dimensions do not affect macroscopic Newton's law, yet the low effective gravity scale !11s makes graviton exchange between Standard 2.3 e+e- -+ w+w- Production as a Probe of Electroweak Physics 19 Iodel particles non-negligible. The e+e---+ ~'+ ~r- cross section receives large, up to a few <;(, corrections from additional graviton diagrams and can be used to put bounds on parameters of such models [43]. In addition. in some models, inc-luding supersymmetric ones, non-standard ~~ decays are allowed. Such models may possess regions of the parameter space which do not predict observable effects in lower energy data but significantly affect the W branching ratios, width, or visible <.>+e- --+ w+w- cross sections, if the non-standard W d<.>cays make the ~r+ ~r- events escape detection [44, 45]. 20 Tests of Electroweak Physics with e+e---+ w +w- Events The L3 D e t ect o r at LEP 2 21 Chapter 3 The L3 Detector at LEP 2 1\Iy analysis was based on data collected by t he L3 detector at the LEP accelerator. This chapter gives an overview of these experimental facilities. with an emphasis on the elements important forth<> measurements described in this thesis. 3 .1 The LEP A ccelerator The Large E lectron Positron storage ring (LEP) is an accelerator operated by the European Laboratory for Particle P hysics (CER:\') located near Geneva, Switzerland (see Figure 3.1). It is designed to accelerate and collide electron and positron beams with energies up to approximately 100 GeV and is currently the world's most powerful e+<>- collider. The electron and positron beams are provided by the LEP injector chain [46], shown in F igure 3.2, using the previously built Proton Synchrotron (PS) and Super Proton Synchrotron (SPS) accelerators. Positrons arc created in a tungsten converter target hit by a 200 l\1cV e- beam from LIL (LEP Injecteur Lineare), a high-intensity linac. A second linac accelerates the electrons and positrons up to 600 l\1eV to be accumulated and cooled iu th<' Electron Positron Accumulation Ring (EPA). PS and SPS subsequently accelerate the beams up to 3.5 GeV and 20 GeV respectively. after which they can be injected into LEP [47]. The LEP tunnel. 27.6 km long, consists of eight straight and eight curved sections. The latter house about 3,300 dipole bending magnets which keep the beams in orbit. and about 800 quadrupole and 500 sextupole magnets used for beam focusing. The energy needed to accelerate t he beams and compensate for synchrotron The L3 Detector at LEP 2 22 f - - 1 1 km ALEPH OPA~ .- -· Switzerland Jura L3 SPS France Geneva Airport Figure 3.1: Schematic map of the region near Geneva, showing the location of LEP, the Proton Synchrotron (PS) and Super Proton Synchrotron (SPS) together with the four LEP exp eriments (ALEPH, DELPHI. L3 , and OPAL). radiation losses is provided by 288 superconducting radiofrequency cavities placed in the straight sections. Electron and positron beams, each composed of 4 equidistant bunches, revolve in opposite directions around the ring with a frequency of 11.4 kHz and are collimated a nd brought into collision at four equidistant interaction points equipped with det ectors (ALEPH , DELPHI. L3 , and OPAL). The interaction point size, determined by the bunch size, is about 1 em along the beam direction. a nd 200 nm and 20 nm in the horizontal and vertical directions respectively in t he plane orthogonal to the beam axis. The construction of the LEP tunnel was started in 1983 and the first electron positron collisions were recorded in August 1989. During Phase 1 (until 1995) LEP 23 3.1 The LEP Accelerator LINACS (LIL) ~~-- 200 MeV e- e--? e+ converter-- PS .3.5 GeV LEP Figure 3.2: The LEP injector chain. was operated with .-v45 GeV beams, corresponding to center-of-mass energies close to the zo boson mass. This allowed each of the four experiments to collect about 4 million e+e- ---t Z 0 events and make unrivaled precision measurements of the Z0 boson properties [48]. Phase 2 started in 1995 with superconducting RF cavities replacing the originally installed copper ones, increasing the maximal LEP beam energy from 45 to 70 GeV, and then to 86 GeV and higher in later years. This brought the collision center-of-mass energy above the w +w - pair production t hreshold and allowed precision studies of the W boson. It also greatly extended the scope of searches of new particles predicted by various extensions of the Standard Model. LEP 2 also accomplished a large increase in the luminosity delivered to the exp eriments. due to the lesser importance of beam-beam interactions which limited the beam current at LEP 1. Additionally, for parts of the 1996 data-taking LEP was The L3 D e t ector at LEP 2 24 operated in the bunch train mode, replacing each lepton bunch by two bunchlets separated by ,..,_,250 ns, and thus achieving a higher total beam current. A peak luminosity of 1.2 x 1032 s- 1cm- 2 was achieved in 1998. This. together with increasing overall operational efficiency, a llowed each experiment to collect up to 2 pb- 1 of data per day. 3.2 LEP Beam Energy Measurement Precise knowledge of the LEP beam energy is crucial for some of the most important measurements at LEP, e.g. those of the mass and width of Z0 and the mass of the W boson. At. LEP 1. the resonant depolarization method [49] gave an opportunity to measure the beam energy with very high precision. The method exploits the Sokolov-Ternov self-polarization of an electron beam due to emission of synchrotron radiation [50]. This transverse polarization can be destroyed by exciting the beam with an oscillating transverse magnetic field of a frequency that matches the electron spin precession frequency. Thus the ratio of the well-measured LEP orbital and resonant depolarization frequencies gives an estimate of the beam energy with a precision of about. 200 keY. The beam energy is interpolated between the calibration polarization runs using measurements from NMR probes inserted in 16 of the dipole magnets. Elaborate methods were developed for t his interpolation, accounting for a wide variety of external influences, achieving an energy measurement precision of 5 r-.IeV [51] . The calibration became a much more difficult task at LEP 2. as significant transverse polarization cannot be achieved at energies above around 60 GeV due to depolarizing resonance driven by magnetic field imperfections. Therefore, to get a precise beam energy estimate. Nr-.IR measurements at LEP 2 center-of-mass energies have to be extrapolated over a wide energy interval down to resonant depolarization measurements in the 41- 55 GeV range. The uncertainty of this extrapolation turns out to be the limiting factor of the beam energy measurement at LEP 2. It was evaluated using complementary flux loop probes, with which all LEP dipole magnets are equipped, 3.3 T he L3 D e tector 25 and was found to be in the 20 30 .t\IeV range for the data used in my analysis. as shown in Table 3.1. Data period 96A (July August 1996) 96B (Octob<'r November 1996) 97 (July ovember 1997) 98 (.t\Iay November 1998) CJe1, [ leV J 27 30 25 20 144±7 165±8 219±11 237±12 Table 3.1: Beam energy uncertainty and spread measurements measured for the LEP running periods used in the analysis [52]. A new method to measure the LEP beam energy was employed in 1999. The LEP spectrometer [53] is a system of beam orbit monitors designed to determine the beam energy by measuring the deflection of the beam by a LEP lattice bending dipole. Precision of about 10 .t\IeV is expected to be ultimately achieved with this technique. 3.3 The L3 D etector The L3 detector [54] is designed to study high energy e+e- collisions up to center-ofmass energies '""' 200 GeV with emphasis on reliable identification and precise measurement of energy of electrons, photons, and muons. Figure 3.3 shows a perspective view of the detector. L3 subcletectors are a rranged in layers of increasing size sur- rounding the interaction point. The tracking system and calorimeters are contained in a steel tube (see Figure 3.4). which supports also the muon chambers and maintains the alignment of the detector. Starting from the interaction point radially outwards. t h<' main detector components arc: • the tracking system measuring directions and momenta of charged particles • the Electromagnetic CALorimeter (ECAL), whose purpose is to measure energies and directions of electrons and photons • the Hadron CALorimeter (HCAL), measuring the energies and directions of 26 The L3 D e t ector at LEP 2 Outer Cooling Circuit J,-~...,_- Inner Cooling Circuit Figure 3.3: A perspective view of the L3 detector. hadrons • MUon CHambers (MUCH), measuring the directions and momenta of muons In addition, luminosity monitors used to detect electrons from small-angle e+e--+ e+e- Bhabha scattering, are installed close to the beam pipe at both sides at a distance of 2.8 meters from the interaction point. The entire detector is surrounded by a solenoidal magnet providing a magnetic field of 0.5 T along the beam axis. Coils wrapped around magnet doors create a toroidal magnetic field of 1.5 T which is used to measure momenta of forward muons. In the following sections the subdetectors used in t he present analysis are described in greater detail. 3.3 The L3 Detector 27 ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss< : Muon: Filter Support Tube SOcm HCAL Barrel ~===~==~=:*====~/ Scintillator Counters r---------------------~ ECAL :Barrel I HCAL Endcap I Figure 3.4: A side view of the inner L3 detector. In addition to the main detector components also shown are Active Lead Ring (ALR), and silicon (SLUM) and calorimeter (LUJ\II) components of the luminosity monitor. 3.3.1 L3 Tracking System The aim of the L3 tracking system. shown in Figure 3.5. is to reconstruct charged particle trajectories in the central region of L3. to measure particle charge and momentum, and to reconstruct SC'condary vertices from decays of particles in fli ght. Its principal components arc the Silicon l\Iicrovertex Detector (SMD), Time Expansion Chamber (TEC), Forward Tracking Chambers (FTC), and z-chambers. The S ID [55] consists of two layers of double-sided silicon ladders 35.5 em long, located at distances of G em and 8 em from the z-axis and covering the polar angle's 22° - 158°. The outer silicon surface of each ladder is read out using strips with a 50 ~Lm pitch for the n p coordinate measurements; the inner surface is read out with a 150 J..Llll pitch (central region) or 200 ,,,m pitch (forward regions) for the z coordinate' measurements. The singlc track resolution of the SMD is 6 ~m in n/J and 20- 25 J.Llll in z. The TEC is a drift chamber with inner radius 8.5 em., outer radius 47 em. and The L3 Detector at LEP 2 28 40mm I TEC '\ Figure 3.5: A schematic r¢ view of the L3 central tracker. length 98 em. Radial cathode wire planes divide the TEC into 12 inner and 24 outer sectors, which are in turn subdivided radially by a plane of mixed anode sense wires and additional cathode wires. Planes of closely spaced grid wires on either side of each anode plane provide a homogeneous low electric field in most of the sector (drift region). with a small high-field region near the anode plane (amplification region). The timing of the ionization signal measured at each anode determines the distance to the track along a line perpendicular to the anode plane with an average resolution of'"" 50 J..Lm. The z coordinate of a track at fixed distance from the beam axis is measured by two layers of drift chambers surrounding the cylindrical outer surface of TEC and covering the polar angles 45° < () < 135°. The z-chambers provide single track resolution of'"" 300 J..Lm. Two layers of FTC cover the end of TEC and measure the x-y coordinates of a track at fixed ::: . These proportional chambers cover the polar angles 9.5° < () (180° - ()) < 37.5° and have a spatial resolution of '"" 150 J..Lm. 3.3 The L3 D etector 3.3.2 29 Electromagnetic Calorimeter Most of the electromagnetic calorimeter is made of bismuth germanium oxide Bi 4 Ge 3 0 12 (BGO) crystals. BGO was chosen to be used as both the showering and d etecting medium for electrons and photons because of its very short radiation length and Moliere radius and high scintillation light yield. The BGO calorimeter consists of two symmetrical half barrels (7680 crystals in total) surrounding the central tracking system with polar angle coverage 42° < B < 138°, and two endcaps (1527 crystals each) located behind the FTC with polar angle coverage 10° < B (180° - B) < 37°. The crystals are 24 em long truncated pyramids about 2 x 2 cm 2 at the inner and 3 x 3 cm 2 at the outer end, a ligned with their axis pointing to the area around the interaction point. Two silicon photodiodes are glued to the rear end of each crystal to detect the BGO scintillation light. For electrons and hotons the ener y resolution of the calorimeter is parametrised as [56] ~ = c ( Je + br + d 2 + (f/, where a= 1.54-10- 2 GeV 112 . b = 0.38-10- 2 , = 0.25 · 10- 2 GeV and d = 0.9 · 10- 2 . For electrons and photons of more than 5 GeV energy resolution better than 2% and angular resolution better than 2 mrad are achieved. The angular interval between the barrel and endcap parts of the BGO calorimeter is covered by ECAL GAP filler (EGAP), a lead-scintillator sampling calorimeter which has a worse resolution. The Active Lead Ring (ALR ) is a similar calorimeter covering the forward angular regions between the coverage of BGO endcaps and the luminosity monitor 4.5° < B (180°- B) < 8.8°. A more detailed description of the electromagnetic calorimeter can be found in [57]. 3.3.3 Scintillation Counters A layer of plastic scintillation counters located between the electromagnetic and hadron calorimeters provides time-of-flight measurement of traversing charged particles. with a better than 1 ns resolution. This information is used to reject cosmic The L3 D etector at LEP 2 30 muons piercing the detector (and leaving a pair of scintillator hits separated in time), to identify beam bunchlets when LEP runs in the bunch train mode. and also to provide an independent trigger for events originating from e+e- collisions. 3.3.4 Hadron Calorimeter l\lost hadrons pass through ECAL which has a thickness of about 0.95 nuclear interaction lengths. The hadron calorimeter [58] is designed to absorb and measure th€' energy of the hadrons. La~rers of depleted uranium with total thickness of 4 7 nuclear interaction lengths. interspersed with 10 proportional wire chambers. serve as the absorber material and shower detectors respectively. The wires in successive layers of the wire chambers arC' rotated relative to C'ach other. thus providing 2D shower coordinate measurement. The wires are readout in cells (towers) each covering ""2.5° x 2.5° solid angle in the barrel region. The HCAL barrel i1; divided into 16 modules in </J and 9 modules in :::. with angular coverage 35° < () < 145°. Tllf' HCAL endcaps consist of three rings: an outer ring and two inner rings. covering the polar angle regions 5.5° < () (180° - ()) < 35°. The barrel HCAL is surrounded by the muon filter, a system of brass plates interleaved with layers of proportional chambers, adding about 1 nuclear interaction length to the HCAL. Its aim is to absorb the tails of HCAL showers and prevent them from reaching the muon chambers. 3.3.5 Muon Chambers The barrel muon chamb0rs consist of eight octants. each containing three layers of ··p·· -chambers, measuring the r-</J coordinates, and two " Z"' -chambers. measuring the ::; coordinate ou the muon track. The barrel muon chambers cover the angular range 43° < () < 137°. Three additional layers of drift chambers arC' mounted on the magnet doors on each side of th<' interaction point. extending the angular coverage to 22° < () < 158°. 31 3.3 The L3 D etector The momentum resolution CJ(pr) / Pr for muons varies from approximately 2% in the barrel to approximate!)' 30% in the endcaps. 3.3.6 L3 Luminosity Measurement The luminosity measurC'ment at 13 is based on sm all-angle Bhabha scattering, whose cross section is very large at low polar angles and can bC' calculated with small theoretical uncertainty. The luminosity is derived from the calculated cross section and the number of e+e- eveuts c-ounted by the luminosity monitor [59]. The luminosity detector consists of a BGO calorimeter (L Ul\II) at both sides of the interaction point with polar angle coverage 31 62 mrad. and a silicon tracker (SLUl\I) in front of LUl\IL providing better position measurement for electrons and positrons entering the calorimeter. Bhabha events for the luminosity measurement are selected as back-to-back energy deposits in LUl\II. The precision of the luminosity measurement by 13 is dominated by systematic uncertainties for all data sets I used and is about 0.2o/c . 3.3. 7 Trigger and Data Acquisition AftC'r each beam crossing the trigger system decides whether a n e+e- interaction took place, in which case the dC'tector signals are read out, digitized and written to tape (the event is triggered). Triggering is done in three levels of increasing complexity. The level 1 trigger consists of five independent triggers using signals from calorimeters. TEC, luminosity monitor. muon chambers. and scintillation countC'rs. and either initiates digitisation, or clears the front end electronics before the next beam crossing. After a positive decision the detector data arc stored within 500 J-tS in multievC'nt buffers, and during that time all further data taking is stopp ed (dead time). egative decisions do not induce dead time. as the elC'ctronics are cleared before the next beam crossing. The level 2 trigger combines the fast digitized trigger data from all subdetectors for events which were not automatically accepted by coincidence of at kast two level 1 triggers. 32 The L3 Detector at LEP 2 The level 3 trigger uses ah·ead,\' fully digiti zed signals from a ll subdetectors to refine the decision of the level 2 trigger. The level 1 trigger rate varies between 5- 20 Hz. a nd the final event rate written on tape is about 1 5 Hz. 3.3.8 Event Generation and Detector Simulation The complexity oft he detector makes it impossible to predict analytically the detector response to any particular physics process. l\Ionte Carlo simulations provide a way to predict the properties of the events reconstructed by the detector. Generation of events usu ally proceeds via two steps. First events, i.e.. lists of particle types and energy-momentum four-vectors. are created with a Monte Carlo generator corresponding to a given interaction process or physical model. Then the generated particles are propagated through a detailed representation of the detector using the GEANT 3 [60] detector description a nd simula tion tool. which simulates energy loss. scat tering and showering of the particles in the detector material. as well as the response of each active detector element. Detector imperfections (inactive or noisy BGO crystals. disconnected TEC sectors. etc.) are included in the simulation nsing detector status databases created during data taking. In the end, the simulated events are used as a n input for the same reconstruction and analysis programs that are used on real 13 data. Comparison of this reconstructed Monte Carlo data wit h the originally generated events allows one to determine corrections needed to interpret the data recorded by the detector in terms of underlying physics models. 3.4 Energy Flow Reconstruction in L3 The event record written on tape during data taking consists of "raw d ata," e.g., digitized signals from TEC wires, BGO crystals, HCAL readout towers. etc. A multistep reconstruction procedure is necessary to transform this information into the 3.4 Energy Flow Reconstruction in L3 33 intuitively understandable form of lists of measured parameters (momentum. charge. etc.) of reconstructed particle::;. The first step involves convert ing the digitized raw data into physical quantities such as drift distances or deposited energies, using results of dedicated subdetector calibration procedures. In the second step, active element signals are combined within each subdetector int.o higher level objects. representing traces of detected particles. Charged tracks are reconstructed by performing circular fits in the r¢ plane for TEC and SMD hits. after which the polar angle information for the tracks is obtained from matched :.:chamber hits. l\1uon tracks are found in a similar way from .t\IUCH hits. Neighbouring ECAL crystals with energy deposits larger than 10 .t\IeV arc combined into "bumps.'' Depending on the pattern of the crystal energy deposits, the bumps are classified as electromagnetic or h adronic, and the corresponding bump energy estimates arc calculated. Similarly, HCAL hits with energies more than 9 MeV are combined into dusters. In the next stage. information from all subdetectors is combined into objects which give the closest possible approximation to particles produced in the e+e- collision. ECAL bumps and HCAL clusters are matched according to their angular proximity, and where possible a TEC/SMD track is associated to the resulting object, called ''A Smallest Resolvable Cluster" (ASRC) . Ideally the ASRCs correspond to individual particles. However, due to the finite resolution of the detector. close-by signals in the detectors sometimes overlap, resulting in clusters which correspond to more than one particle. .t\Iuon tracks are matched to characteristic minimum ionising particle deposits in HCAL and to a TEC/ S.t\ID track whenever possible. The last step of the reconstruction starts with the list of reconstructed hadrons. i.e., ASRCs which are not electromagnetic ECAL bumps. and attempts to obtain more precise energy estimates for them than would have been obtained by simply summing up calorimetric energy deposits. The improved energy estimator is a linear function of the energies measured in TEC/S.t\ID, ECAL. and HCAL. The energy of The L3 D etector at LEP 2 34 an ASRC is given by (3.1 ) where index i runs over ECAL. HCAL modules and TEC/Sl\ID. and Ei is the energy measured by the corresponding subdetector. Different correction factors Gi are calculated for different regions of I cos(} I· They are determined with a large sample of Z0--+ qq events in a minimization which varies the Gi parameters, trying to adj ust the sum of energies of all ASRCs and muons in each calibration event to the nominal value of fi [61] . An additional set of correction factors is obtained for about 3% of events which have no reconstructed TEC information. e.g., events recorded during short periods of time in the beginning of each LEP fill when the TEC high voltage is switched off to prevent possible damage to the detector in casC' of a beam loss. Combining in this way the information from the calorimeters, the central tracker, and the muon chambers, the total energy of hadronic Z0 events can be measured with a resolution better than 10o/c. Event Selection a nd Cross Section Measu rement 35 Chapter 4 Event Selection and Cross Section Measurement The first step in my analysis of V\' boson properties is selection of a data sample enriched in w+w- ~ qqqq events. This chapter describes the methods I developed to select and reconstruct such events. and to measure their production cross section. It covers online and offline selection of hadronic multijet events, and Monte Carlo simulations I used to model th<:> signal and background event properties. I describe the multivariate discriminators I used for separating the w +w - signal from potentially high backgrounds. Also addr<:>ssed are the problems of reconstruction of hadronic jets, including the kinematic fits of the reconstructed jet parameters. and combination of the jets into pairs corresponding to the primary W bosons. This chapter also describes the fit of the neural network output distribution for the multijet events selected from the data. which I use to determine the e+ e - ~ w+w-~ qqqq cross section. I also present the results of applying these methods to the L3 data samples I analyzed and discuss the systematic uncertainties of these measurements. The data collected by L3 at .JS of 189 GeV is the most statistically significant dataset used in my analysis. Iu this chapter I use it, together with l\1onte Carlo sets generated at this value of yfi;, for data-1\Ionte Carlo comparisons. 4 .1 Event Charact eristics and Main Backgrounds A four-jet event recorded by the L3 detector, shown in Figure 4.1 , illustrates the event properties I rely on in selecting w +w - ~ qqqq events in th<:> data. 36 Event Selection and Cross Section Measurement Run # 663205 Event# 3974 Total Energy : 173.13 GeV ' / / / 0 0 Transverse Imbalance : .0515 Longitudinal Imbalance : -.0144 Figure 4.1: End view of a typical fom-jet event recorded by the L3 detector. Hadronization of the two high invariant mass quark-antiquark systems produced in e+e- -+ w +-vv- -+ qqqq events, results in events with a large number of charged and neutral hadrons in the final state, the typical number of reconstructed calorimetric dusters being 50-70. Fragmentation preserves the dear multijet topology of the w+w- -+ qqqq events. l\Iost of them have a distinct four-jet signature giving a clear r0presentation of the 4.1 Event Characteristics and Main Backgrounds 37 underlying four quark final state. A small fraction (--,_,10%) of the events appear to consist of more than four jets because of the gluon radiation off the quarks. As the collision center-of-mass energies accessible at LEP 2 are not much higher than theW pair production threshold, the photons radiated off the incoming electrons typically do not carry a large momentum. This. together with t he L3 detector 's hcrmeticity and good momE-ntum rE-solution for hadronic jets, accounts for the low missing energy and momentum values usually observed in qqqq events. Semileptonic decays of c quarks coming from ~r+ ---? cs decays. typically produce low energy leptons well inside jE'ts. As there arc no other significant mechanisms to produce energetic isolated photons or leptons in 't\f+~r- ---? qqqq events, one does not expect signal events to contain such leptons or photons. One more important property of w +w - ---? qqqq decays is due to the smallness of the W boson width (fw ~ 2 GE'V) relative to the W boson mass (Mw ~ 80 GeV). A consequence of this is the existence in a typical four-jet w +w - ---? qqqq event of at least one combination of jets into two pairs with both dijct invariant masses close to 80 GeV. Unfortunately, several Standard l\Iodel processes with non-negligible cross sections produce events which can fake the apparent ch arac teristics of qqqq events. Below I dE-scribe the background sources which will be corrected for in the subsequent analysis and the l\ Ionte Carlo generators used to model them. I also provide a description of the l\1onte Carlo simulatio11 of the signal events, on which my measurements were based. 4.1.1 QCD Multijet Events The most important source of background at all the energies considered is due to QCD radiative corrections to e+ c- ---? qq production. SomE' relevant Feynman diagrams are shown in Figure 4.2. Although production of events with a four-j et topology is suppressed by a factor n1. the qq production cross section is much la rger than the signal cross section. as Event Selection and Cross Section Measurement 3 Figure 4.2: Some order o.~ QCD diagrams contributing to four-jet final states. shown in Table 4.1. and leads to significant accepted background at all the energies used in the analysis. Tln·oughout the thesis I will refer to events of this kind as qq events. An important feature of e+e- ___, qq annihilation, not present at center-of-mass energirs at or below l\Iz, is the prcsrncr of so called "radiativr returns to Z 0 .'' In such events, emission of an energetic ISR photon effE'ctivcly reduces the e+e- collision centrr-of-mass energ~' to the Z0 resonaucc and leads to production of a qq system with an invariant mass ::::::::1\ Iz together with an energetic photon at a typically low polar augk. The PYTHIA 5. 7 [G2]l\Ionte Carlo event generator was u ed to simulate qq events. The electroweak part of the proccss is based on the first order e+c- ___, qq matrix elemeut including the 'Y/Z 0 interference. Initial state radiation is simulated using the JETSET parton shower algorithm. This is expected to provide an imperfect description of events with energetic, high-pr ISR photons, but it does provide an accurate simulation of full energy qq events, which are responsiblr for the bulk of the background coming from this process. Thr JETSET parton shower algorithm is also used to simulatc QCD final-state radiation. To better model the hard gluon emission lrading to QCD multijet events, 4.1 Event Characteristics and Main Backgrounds 39 the first branch of the parton shower is adjusted to approximate the first order matrix clement qq(r) at this branch, thus taking advantage of both leading log and matrix element methods. The validity of this approach to the description of multijet QCD events was extensively tested at LEP 1 [63] and is discussed in Section 4.11. Ftagmentation of the resulting system of quarks and gluons was done with the LUND string fragmentation model [62] with the parameters determined in studies of LEP 1 data [64]. a(e+e -+ w+w -+qqqq) [ph l a(e+e--+qq) [pb] a(e+e--+ Z0 Z0 ) [pb] a(e+e--+c+e- hadrons) [pb] 161.0 1.76 147 0.42 12300 Vf; [GeV] 172.0 182.7 5.67 7.17 121 108 0.43 0.59 12900 13400 189.0 7.59 98 0.97 13700 Table 4.1: Standard f\Iodcl cross section predictions for e+c- -+ v.r+w--+ qqqq and various background processes. The e+e--+ w +V\r--+ qqqq cross section values given are calculated with GE TLE [65] at f'.hv = 80.33 GeV. The prediction for Z0 Z0 cross section corresponds to a lower cut on fermionpair masses of 10 GeV. The prediction for c+c--+e+e- hadrons corresponds to a lower cut on the hadronic mass of 3 GeV. 4.1.2 Z0 Z0 Events At center-of-mass energies above 2 t>.h the diagram shown in Figure 4.3 lwcomes important. The Z 0 Z0 -mediated four-quark production is suppressed by a significant phase space factor and relatively weak coupling of Z0 to ekctrons, hut the resulting four-jet events, denoted in the following as Z0 Z0 events, are indistinguishable from \V+V\r--+ qqqq events on an event-by-event basis and constitute an irreducible, albeit small. background. ~Ionte Carlo events of this type were generated with PYTHIA 5.7 [62], using the NC02 matrix element (the diagram shown in Figure 4.3). Uncertainties in tlw simulation of this electroweak process are expected to be small. A dedicated measurement by L3 [66] confirmed that the f\1onte Carlo correctly describes the observed rat<' and Event Selection and Cross Section Measurement 40 e Figure 4.3: The Z0 Z0 production diagram. Figure' 4.4: Two-photon hadronic event production. properties of Z0 Z0 events. 4.1.3 TY Events The most copious source of hadronic events at LEP 2 is interaction of virtual WeizsackerWilliams photons radiated off the incoming electrons, as schematically shown in Figure 4.4. The electrons are typically scattered by a small angle and escape detection, leading to hadronic events with large missing energy. Requiring a clear multijet structure further reduces the accepted cross section for this process. Nevertheless, due to the very high effective photon flux, the hadronic two-photon cross section is orders of magnitude larger than that of high-energy electroweak processes. Therefore the background coming from this source cannot be neglected and will be taken into account in the analysis. l'd onte Carlo simulation of two-photon hadronic events is not very reliable due to incomplete theoretical understanding of the interaction of virtual photons. The PYTHIA 5. 7 [62] generator, incorporating t he vector dominance model (VDM), r esolved and direct interactions of photons, was used as the main Monte Carlo source of 11 events. Its predictions were cross-checked with those of PHOJET [67]. a dedicated 4 .1 Even t C h aracteristics and Main Back g r o unds 41 two-photon event generator based 011 the dual parton model (DPM) [68]. 4 .1.4 Monte C a rlo Simulation of w+w ----*qqqq Eve nts I rely on l\Ionte Carlo simulations to <'Stimate the effects of my event reconstruction and selection on w+w- events, the purity of the event samples I usc in my analysis. and, most importantly. to infer the \ \' boson parameters under study from the observed four-jet event properties that these parameters affect. Special attention should be paid to the choice of the w+v.,r- l\lonte Carlo model and the uncertainties that it implies for this inference. The KORAL\\' 1.33 [69] event generator was th<' main source of \\'+y.,·- ;\Iont<' Carlo ev<'nts for my analysis. It incorporates the CC03 subset of e+e- -t f 1f 1 f:lf 1 diagrams with finite v\' width. The (11<'glected) interference with other diagrams leading to the same four-quark states. e.g.. through Z0 Z0 . breaks tiH' gauge invariance of the procedure in principle. but this effect is small numerically (in the unitary gauge) [70]. All possible w+w- four-fermion fina l states are produced and th<' full CKM matrix and QCD radiative corrections to f(W -tqq') are used. l\Iultiple ISR photons with finite transverse momenta are generated according to the O(a;71 J Yennie-Frautschi-Suura exponentiation technique [71]. Coulomb corrections [72] are taken into account as well. JETSET parton showers and the LU D string model, with the naive singlet color connection. are us<'d to simulate the QCD radiation off and hadronization of the quarks produced. Bose-Einstein correlations in the resulting multihadron system arc modeled with the LUBOEI algorithm [73]. Dependence on thes<' particular choices has to be investigated and will be discu. sed in Section 4.11. The EXCALIBUR event generator [74]. implementing the full set of e+e- -tf1f2f:3ft diagrams and not limited to CC03 ones. has been used in L3 for studies of v\'+vv- -t qqCv events, where this limitation is important. As this is not the case for the four-jet channel, and because EXCALIBUR 's ISR schenl<' is significantly simplified. I do not use it for my analysis. For e+e- -t v,r+ w - -t qq(v , good agre<'ment was obserV<'d Event Selection and C r oss Section Measurement 42 between the KORALW and EXCALIBUR predictions. 4.2 Online Trigge r The conditions required by the L3 online trigger to be satisfied by an event to be written into the L3 data stream, constitute the first step in t he selection of w+w- ---t qqqq events. A positive decision from the logical OR of three online Level 1 triggers is required to select a high energy hadronic event: • The energy trigger requires either a total energy in the calorimeters of 25 GeV, or a minimum energy in the barrel region of 15 GeV in both HCAL and ECAL or 8 GeV in ECAL only. This trigger alone has an efficiency more than 99.9% for high energy h adronic events. • The scintillator trigger requires at least 5 scintillator hits within 30 ns from a bunch crossing. covering at least a 90° angle in ¢. • The TEC trigger demands at least two tracks with minimum Prof 150 MeV and acoplanarity less than 60°. The latter two triggers have efficiencies in excess of 95% for high energy hadronic events each. Higher levels of the trigger system reduce the accepted rate of correlated calorimetric noise, beam-gas and cosmic events by applying tighter cuts on events selected by only one of the Level 1 triggers, without affecting genuine high energy events. The overall efficiency of the L3 trigger system for w +w- ---t qqqq events is more than 99.9% and is assumed to be 1 in the following analysis. 4.3 Detector and Data Acquisition Status Severe malfunctioning of the detector components crucial for my analysis, such as the calorimeters or the ener gy trigger, can have a very large effect on the characteristics and rate of accepted events. Moreover, such time-dependent hardware problems are 4.4 Multijet Eve nt Preselection 43 difficult to incorporate in the Monte Carlo production. Therefore, to reduce systematic uncertainties related to the performance of the detector, I reject both data and Monte Carlo events in the runs during which any of the relevant L3 subsystems: the hadronic calorimeter, the electromagnetic calorimeter, the energy trigger, the luminosity monitor and t he global data acquisition system, were not operating normally according to the online database. As summarized in Table 4.2, this requirement leads to only a small reduction in the integrated luminosity I use in my analysis. Data period 96A (July - August 1996) 96B (October - November 1996) 97 (July - November 1997) 98 (May - ovember 1998) L recorded [pb- J ] 10.90 10.25 55.46 176.35 L selected [pb - l ] 10.82 10.22 55.30 176.34 (vs) [Gev J 161.34 172.13 182.68 188.64 Table 4.2: Total recorded and selected integrated luminosity for the data sets used in the analysis. The accepted-luminosity averaged center-of-mass energy is also given. 4.4 Multijet Event Preselection The online trigger does a good job in rejecting noise and cosmic events, but the output rate of the combined L3 trigger is still as high as a few Hz. The bulk of the accepted cross section at this stage, of the order of 50 nb, consists of events which are dearly incompatible with the w+w- ---t qqqq hypothesis, such as low-effective-..fi rr events, dilepton events and qq events with a large energy missing due to ISR. My multijet event preselection aims to reject such events and leave a clean sample of multijet hadronic events suitable for my measurements. This section describes the criteria I use for this purpose. Event Selection and Cross Section Measurement 44 4.4.1 Multiplicity Requirements I require the selected event s to h ave at least 30 ASRCs with energies above 100 1\leV. This cut does not affect sign al events, as shown in Figure 4.6. while reducing the L3 d ata by a factor of 125. In addition I require the event to have at least 10 tracks passing the criteria d escribed in Section 3.4. The purpose of this cut is to remove event s with correlated electronics noise faking large spurious energy d epositions in the HCAL. An example of such an event is shown in Figure 4.5. This cut also has a negligible effect on t h e signal. as shown in Figure 4.6. i i :; li :II IIi •B• I B .81 ~ I• t t 1 t~ t I ff I + + I I I I • :- -- -+ -+- t·l., • I .. I f ~ ' J I • I • t . 8 • Figure 4.5: An HCAL noise event. The track requirement for the small fraction of events without TEC informa tion is replaced by d emanding an energy of a t least 20 GeV d eposit ed in the barrel (central ) part of the electromagn0tic calorimet er. Studies of b eam-gate events (events recorded 4.4 Multije t Eve nt Pre s election 45 outside of bunch-nossing gates) showed that this provides an adequate rejection of H CAL noise events. (/) c Q) > • Data 0WWMC .qqMC ~ Other Bckg MC 10 w (/) c 10 • Data 0WWMC .qqMC F22J Other Bckg MC Q) > w 10 10 10 10 10 0 25 50 75 0 100 20 40 60 80 NATR K Figure 4.6: Distribution of thC' muuber of reconstructed energy ftow objects. o cuts arc applied. Various processes not included in th<' MC prediction account for its discrepancy with t he data at this level. 4.4.2 Figure 4.7: Distribution of the number of reconstructed tracks, after the total multiplicity cut has been applied. Visible Energy and Energy Imbalance For the C'vents remaining after the ASRC and track multiplicity cuts. a set of momentum vectors Pi is reconstruct<'<! from the calorimetric dusters and tracks in th<' central tracker and muon chambers, as clescribC'd in Section 3.4. In the following I will refer to these momentum vectors as particles. I define the normalized visible energy E vis and the longitudinal energy imbalance E 11 as E vis ~Pi Vs ~Pi cos(Ji ~Pi ( 4.1) (4.2) 46 Event Selection and Cross Section Measurement where the sums arc over all the reconstructed particles, Js is the nominal LEP centerof-mass energy. The distributions of these two quantities for high multiplicity events selected in 9~ arC' compared with the corresponding Monte Carlo predictions in Figure 4.8. L!) • Data 0WW--?qqqq MC Bilqq MC f8l!)Other Bckg MC C\.1 0 Q 1000 C/) c CD > ::G 1500 0 c) (/) -tii c • Data 0WW--?qqqq MC Bilqq MC f8l!)Other Bckg MC CD w 1000 500 500 0 .6 0.8 1.2 1.4 Figure 4.8: Distributions of the normalized visible energy and longitudinal energy imbalance. The discrepancy at low values of E,,i.• is due to the imperfect modeling of e+e- ~c+c - hadrons. I demand the normalized visible energy Evis to be more than 0. 7 and t he absolute value of the longitudinal en ergy imbalance En to b e smaller than 0.25. These cuts reject most of hadronic radiative returns to Z0 , which have a preferred value for the visible energy of s+~J~ (0.61 - 0.66 for the energies analyzed in this thesis), and 11 events. They also help reduce background coming from semileptonic w+w- ~ qqCv events. as those typically have significant missing energy and energy imbalance due to presence of neutrinos in the final state. 4.4.3 Lepton and Photon Rejection Further suppression of radiative return and w +w - ~ qqfv events is achieved by rejecting events with identified energetic photons and leptons. 4.5 J et and W B oson R econstru ctio n 47 Two r<'quirements arc applied to select electromagnetic ASRCs. Firstly, most electrons and photons arc fully absorbed in the BGO calorimeter. Only very energetic ones with an impact point dose to a crystal edge deposit some energy in the HCAL. I demand an ~~;:.~~ ratio of no more than lOo/c for an ASRC to be considered electromagnetic. Secondly. the shap<' of the ECAL bump must be consistent with the electromagnetic hypothesis. A good discriminating variable sensitive to the bump shape is the ~ ratio, where E 9 ( 25 ) arc estimates of the bump energy in the electromagnetic hy- pothesis. constructed from the crystal energies in the 3x3 (5 x 5) matrix around the bump center of gravity. I require the t;;;- ratio to be more than 0.95 for the sE>lected ASRCs. I reject the event if the most energetic ASRCs satisfying these two criteria has an energy of more than 40 GeV. As shown in Figure 4.9. the high energy part of the elrctromagnetic ASRC energy distribution is dominated by radiative qq{r) <"vents with a smaller contribution from v.,r+vv- ---7 qqev events. The peak around 69 GeV is due to radiative returns to Z0 . where the ISR photon, with an energy of P-r = s;~~ recoiling against a two-fermion system of an invariant mass close to Mz, is seen in the detector. I also select muons which satisfy the quality criteria described in Section 3.4. I reject events which contain a muon with a momentum greater than 20 GeV. Figure 4.9 shows the d istribution of the reconstructed muon momentum in selected high multiplicity events. The high tail of the distribution is dominated b.v w+\v- ---7(1Q/W events. The signal w+w- ---7 qqqq rvcnts arr not expected to have energetic leptons or photons and are practically not affE'ctecl by these cuts. 4.5 ~lost Jet and W Boson Reconstruction vv+vv- ---7 qqqq and high energy qq events survive the cuts described above. A morr detailed analysis of the jet structure and the kinematics of the events is necessary 48 Event Selection and Cross Section Measurement >Q) • Data • Data <.9 1500 0WW~qqqq MC 0WW~qqqq MC ~ c ... !mqq MC fiilqq MC ~Other Bckg MC ~Other Bckg MC Q) Jj 1000 500 10 40 60 80 0 20 40 60 80 max P 11 [GeVJ max Ebump [GeV] Figure 4.9: Distributions of t he maximal EM bump energy and maximal muon momentum. The discontinuity in the P 11 spectrum is due to an artifact of my data analysis software. to bring the background level down to an acceptable level. This section describes the next two steps of my selection of four-jet w +vv- events: the reconstruction of jets and combination of jets into pairs corresponding to the primary W bosons. 4.5.1 Jet Construction Algorithm For each high multiplicity event which satisfied the cuts described so far. I combine particles into jets using the Durham kT algorithm [75]. This iterative combination procedure starts with the list of N reconstructed energy ftow objects and finds the pair (i.j) which minimizes the jet resolution parameter 2min(Ef. Ej)(l- cosfJiJ) YN - l. N L.Ek (4.3) where E i .j are the energies of the particles and f)ij is the 3D angle formed by their momenta vectors. The two particles are combined into a new one with the 4-momentum zl' ( 4.4) 4.5 Jet and W Boson R econ stru ction 49 The succC'ssive combination is repeated unt il t here are exactly four objects left. I call these resulting objects jets in the context of my four-jet event selection. StudiC's of IontC' Carlo w+w- -+ qqqq events have shown that the jet 4-momenta give a good approximation of the energies and clin'ctions of the generated quarks, accurate to approximately 0 (10%) and 0(1 °) respectively, for a typical hadronic jet produced in a vv+vv--qqqq event. I repeated my wholC' selection and measurement of the 'vV mass using othC'r jet algorithms (LUCLUS [7G] and Cambridge [77]). which are significantly different from Durham and each other in t heir approaches to particle combinat ion. I found no statistically significant changes in t h<' expected performance of the analysis. This indicates t hat for my analysis tll<' detector response to hadrons (calorimeter energy resolution and granularity), rather than the effects of particle (mis)assignment to jets, is the fe:u·tor limiting the jet energy and angular resolution. For the reconstructed 4 jets. I compute the parameter y 34 as described above. This separation between the two jets closest in the Durham measure is a valuable discriminator between 4 (or more )-jet events a nd the QCD events with two primary quarks accompanied by hard gluon radiation. In thC' latter case the gluons tend to be emitted with a small transverse momentum with respect to the quark-antiquark axis. This leads to low momentum jets or jets separated by a small angle. Both of these sit uations produce a pair of jets with a sm all Durham distance between them, while in v..r+v..r- -+ qqqq events y 34 is determined by t he initial quark kinematics, which usually leads to well-separated energetic jets and large values of y 31 . The distribution of Yat in selected h adronic events is shown in Figure 4.10. As the jet resolution parameter and, therefore, the combination procedure are not Lorentz-invariant. I do the computation in the event rest frame system. This prevents apparent event boosts. due to jet mismeasurements and undetect0d particles carrying away significant energy, from causing artificial 0ffects in y 3 1 and other event shape variables I use later in my an alysis. I reject the events which have y 34 smaller than 0.0015. This value can be intuitively understood as one rejecting events if the least <'ncrgetic jet is well isolated but has Event Selection and Cross Section Measurement 50 • Data c:::J ww ~qqqq MC ~qqMC [2'&] Other Bckg MC 10 1 -6 -4 -2 In y34 Figure 4.10: Distribution of the y 3"1 parameter for selected h adronic events. an energy of less than rv5 GeV or. alternatively. if all the four jets carry the same energy, but the a ngle between the two closest ones is less than "'10°. 4.5.2 Jet Quality Requireme nts Having reconstructed the jets, I now apply two additional requirements on their quality. The first one is motivated by the fact that my rejection of events with photons performs less than perfectly for events with energetic ISR photons at ECAL edges or in the EGAP calorimeter that fills the gaps between the barrel and endcap parts of the BGO calorimeter. or for events with converted ISR photons. As the radiative return cross section is very high rela tive to the sign aL misidentification of photons as energetic jets even with a small proba bility would lead to a significant additional background. Therefore. I require each jet to have the fraction of its energy associa ted with electromagnE'tic deposits in ECAL less than 98%. The distribution of the m aximal electromagnetic jet energy fraction max 8 "i:tL for selected hadronic events, after applying the y 31 cut. is shown iu Figure 4.11 . A noticeable peak at high values of max EJID:~L is due to misidentified photons discussed above a nd is well described by 4.5 Jet and W Boson Reconstruction 51 the Monte Carlo. C\J 0 0 --c U) U) • Data 150 Q) oww~qqqqM miqq MC ~ Other Bckg MC > w c 600 • Data DWW~qqqqMC Q) > w miqqMC ~Other Bckg MC 400 100 200 50 0 c=====~~~~~~ 0.6 0.7 0.8 0.9 5 10 15 20 25 min N iet Figure 4.11: Distribution of the maximal ele-ctromagnetic jet energy fraction for events which passed the y 31 cut . Figure 4.12: Distribution of the minimal jet multiplicity for events which passed the y 3.1 cut . Another jet quality requirement demands each jet to have at least 4 particles with momenta of at least 300 t-. IeV C'ach. This cut rejects events where one jet is faked by a n ISR photon hitting the HCAL below the ECAL acceptance, or by a possiblymismea.':;ured single isolated energetic particle. The rate of the latter class of events is sensitive to the details of fragmentation and is not supposed to be reproduced reliably by t-.Ionte Carlo codes. The value of the energy cut was chosen after studies of low energy Z0 - qq data. The chosen value optimizes the agreement between data and r-.Ionte Carlo, but the agreenwnt remains imperfect because of the difficulties of properly simulating particle-detector interactions. The distribution of the minimal jet multiplicity min E > JOO 1\IeV eu t. is shown in Figure -! .12. for selected hadronic events. after applying the .Y31 52 Eve nt Selection and Cross Section Measureme nt 4.5.3 Constrained Kinematic Fit of Measured Jet Parameters After applying the rejection described above. I am left mostly with genuine high energy hadronic events without significant initial stat<' radiation. Given an ideal detector, one would expect these events to be reconstructed with t he total energy within "'1 GeV (typical ISR energy loss) of the nominal LEP j$, and very low transverse and longitudinal energy imbalance. In a real dct ector, statistical and systematic uncertainties of particle identification and momentum measurement lead to a rather different picture. with the resolution on t he total energy of "'15 GeV and significant momentum imbalance for high energy hadronic events. I try to partially correct the jet parameter measurement by means of a constrained kinematic fit. I choose a n ew set of values for each jet's energy and direction {E,B,¢} so that they satisfy the energy-momentum conservation constraints =L Ei - .jS 0 (4.5) mo·1 1 - -cos 'f't A.. 2 sin e. t 0 (4.6) 2 mo·1 1- sin B·t sin..+. . 'f'! E Oi2 0 (4.7) Ei 0, (4.8) b..E i= l. I 2 b..Px = L Ei E Oi !=1.4 b..P_v = L Ei i= 1,4 b..P:;- L i= l.cl while minimizing the difference between the new fitted set of parameters and the measured ones {E0 .80 .¢0 } . The measure of the difference I use is a \': 2 defined as L L i=1.4 {E.O ..P} (:r- .ro? a2(E e) . X ( 4.9) 1 where the sum is over the measured parameters of the four reconstructed jets. l\Iy choice of constraints is equivalent to fixing the velocities of the jets to the measured 4.5 Je t and W Boson R econ struction 53 values. This reduces the number of free p arameter~ in thE' fit a nd a ims to approximate th<' correlation intrinsic in the jE't energy and mass m€'asurement . I ignore correlations between measured jet energies and angles and correlations b etween different jets. which is a valid assumption for high multiplicity. collima ted. W<'ll-s€'parated jE'ts. The resolutions CJ7 . dep end on jet energy and polar angle a~ r;::;~E( as+ E 1 + v.c 1 vrr:::; E C£ 7r 2 . (O . 1r - O) +dEIO- ?I) nun - (4.10) -rr 2 nun (O . 1r - O) + dol O- -2 1 ) (-!.11) (.::1o ( + . co + E 1 ao 1 - 1- ~d> c<P 1r 2 a +-(1 + +d 0 - JEsinfJ <P E min(O.-rr- 0) <PI 21 ). (4. 12) Th(' parameters ax. b;c. Cx, d.~. WCr(' obtained from studies of e+('- __, vv+w - l\Iont e Carlo events [78]. As ~hown in SC'ct.ion 4.11 . the L3 detector simulation providE'~ an adequa t e description of tll<' actu al detector response. I do the minimization of the \ 2 by gra dient descent with the help of t he IIKUIT softwar<' packagE' [79]. The constraints are implemented adding a penalty contribution to th€' \ 2 2 !:::,. \: 4C !:::.E 2 + !:::.P} + t:::.P; + !:::.P'j - 2 (4.13) CJo wher<' the parameter CJo is gra dually dE'creased during several iterations of thE' fit. The fit I described abovE' is ref€'rrecl to as a 4C fit, according to th<' number of constraints used. The new jet momentum 4-vcctor~ obtained in the minimization satisfy energymomentum conservation b., . construction a nd greatly improve t h<' measurement of th€' jet-kinem at ics-based quantities which are correla t ed with the total event energymomC'ntum. The large effect of the kinem a tic fit on the averag<' of the reconstructed masses of t he two W bosons in w +w - __, qqqq events is shown in Figure 4.13. Figure 4.14 shows t he distribution. in both data and l\fonte Carlo, of tlw 4C kinematic fit probability P, 2( \ 2 1c;4), defined as the probability of drawing a value Event Selection and Cross Section Measurement 5-1 >Q) ~ WW-tqqqq MC events , before kin.fit ~ 150 (/) ffiZJ WW-tqqqq MC events, after kin. fit c Q) > w 100 50 60 80 100 M reconstructed [GeV] Figure 4. 13: Distribution of the average reconstructed W mass before a nd after 4C kinematic fit for selected w +-vv----+ qqqq Monte Carlo events. The resolution on the average mass is improved from 9 GeV to 1.5 GcV by t he fit. L!) • C\J 0 0 Data ~ WW-tqqqq MC - 1mmJ qq MC • U> cQ) > ~ Other Bckg MC w 0 0.2 0.4 0 .6 0 .8 F igure -1.14: Distribution of the 4C kinematic fit probability. higher than the observed '( 2 from the '( 2 distribution with 4 degrees of freedom. The data arC' WC'll described by the Ionte Carlo in tlw whole range of the fit probability. ThC' flat ness of the distribution for ~~ + w----+ qqqq events indicate's correctness of t hC' 4.5 J et and W Boson R econstruction 55 parametrization of the jet errors I used. The peak at low values of the fit probability consists of events in the non-Gaussian tails of thr measured jrt momentum distribution, mostly clue to particle and showrr losses. and eYents with significant initial state photon or final state gluon radiation, with a small admixture of \V+ w -~ qqTv events with th<> T-lepton decaying hadronically. that passed the rejection described above. I rrjrct such mismeasurements by clrmanding the events I consider in the following analysis to have a P \2 ( \ 2<-~C ; 4) of at least 0.1 o/c. Energy-momentum conservation equat ions ( 4.5-4.8) are not th<> only constraints one ran reasonably apply to a w +y...r- ~ qqqq event. Thr experimental stat istical error on a reconstructed \ \' invariant mass, typically of the order of 10 GeV. is much larg<'r than the Breit-Wigner width of the produced \rY boson. fw ~ 2 GeV. I can exploit this property of genuine w +y...r- events a nd assume. once I have combined the jets into two jet pairs, that the two reconstructed W bosons (jet pairs) have equal masses. This gives me au additional constraint which I use in a 5C fit. The 5C fit is performed for each of the three possible combinations of the jets into pairs. The decision on which pair combination is most likely to represent the two primary W bosons is taken at a later stagr in my selection and is clescribrd in Subsection 4.5.4. I form the ith jrt combinat ion (i=L3) by assuming that jets 1 and i + 1 come from the same W , and the oth er two jets come from the other one. The penalty addition to the ~2 (4. 13) is modified to assure that the two W candidates have equal r<>constructed massrs after thr fit , •) b.\ -iiC.i (4.14) As computed with the jet momenta obtained in the 5C fit. the two clijet pairs have the same mass m 5 c . .1\ ly measurement of theW mass described in Chapter 5 is based on thE' analysis of the distribution of this quantity. The W boson masses I obtain aftrr t lw 4C kinematic fit. two for each jet combination. arc more difficult to use for this purpose because of tlw significant (negative) correlation betweE'n them, introduced by Eve nt Se lection and Cross Se ction M easure m e nt 56 the fit , and non-parabolic errors. introduced by the constraints (non-linear in mfS) . m 5 c efficiently combines information contained in mfS, t aking t he above details into account and providing a single Mw estima tor per jet combina tion. An a dditiona l b enefit of t he 5C fits is the set of the fit probabilities. computed for the :\ 2 distribution with 5 degrees of freedom. The fit prob ability P x2 (x2 5 c; 5) i is a powerful test of t he hy pothesis t hat the i th jet combina tion m aps correctly onto t he original W bosons. The set of the probabilities Px2(x2 5c; 5)i (i=l ,3) can also b e used to solve the combinatorial problem of choosing the combination of jets wit h t he highest likelihood of correct m apping, as discussed in the following Section . 4.5.4 J e t Combination Algorithms A problem unique t o t he w+w - ____, qqqq ch annel is the presence of t hree p ossible ways to form two dijet syst ems in each event. The knowledge of which combination corresp onds to t he t wo qq' pairs coming from the two W b osons is impor tant for my ana lysis and esp ecially for the measurements of t he W m ass and w +v.,r- "''/Z0 couplings. The a mbiguity cannot be resolved for each and every event, but one can devise st atistical met hods t o exploit kinem atic properties of w+w - ____, qqqq events and t o m ake choices with much higher efficiency than the 1 obtained with random picking. In my an alysis several algorithms of find ing t he correct jet combination in w +w -----* qqqq events were used for d ifferent purposes: • The .6.m algorit hm star ts wit h rejecting the combination with t he lowest sum of the reconstructed W m asses E m =mic + m~c, as t his combination is unlikely to represent the two W bosons. Out of t he rem aining two combinations, I choose the one wit h t he sm allest difference of the reconstructed W masses .6.m = \m1c - m~c \. • Alternatively, one can choose the combination with the highest Em as t he most likely to be the correct one. In a w+w - ---t qqqq event , random jet comb inations typically do not have invaria nt masses as high as Mw . 4.5 Jet and W Boson Reconstruction 57 • I can improve the ~m algorithm by using tlw decision variable of a more general form ~'m = et min mfS+max mlS. The parameter o optimizing the performance is found to be close to 2 a t all the energies considered. • The x 2 algorithm orders the combinations according to the probability of the 5C kinematic fit described in Subsection 4.5.3. Incorrect combinations are expected to have low kinematic fit probabilities. as they do not usually satisfy the equal mass constraint. There is a good data-l\IC agreement in the variables these algorithms rely on, as shown in Figure 4.15. JS [GeV] Algorithm .6.m Em E'm \ 2 161.0 55% 81% 83% 52% 172.0 GG% 68% 74% 58o/l 182.7 70% 48% 61 % 61% 189.0 72% 36% 54% 62% Table 4.3: Correct combination efficiency of the jet combination algorithms used in the analysis, at different center-of-mass energies. The performance of the algorithms, defined as the probability of choosing the correct combination in a w+w ---.... qqqq event selected with the cuts described so far in this Chapter. was detNmined with l\Ionte Carlo events generated at different center-of-mass energies. and is shown in Table 4.3. The performance of all considered algorithms varies considerabl~· with energy. becau e of the significant changes in the kinematics of the vV+ \v- ewnts as the center-of-mass energy rises from the threshold to higher values. The .6.m algorithm is t hc most efficient one at higher energies. Closer to the threshold it suffers from the limited phase space which significantly affects the underlying Breit-Wigner distribution. The ~m and ~'m algorithms' performance. while the highest near t hreshold , decreases rapidly as the center-of-mass energy and the rate of random high-invariant-mass combinations increase. I usc the E'm a nd .6.m algorithms in the analysis of the 161. 172 and the 183. Event Selection and Cross Section Measurement 58 >Q) >Q) • Data 0WW~qqqq MC (carr.) ~ww~qqqq MC (incorr.) ~Background MC <.9 750 --L!) ( /) c w 0WW~qqqq MC (carr.) 18a WW~qqqq MC (incorr.) ~Background MC --c "<:t (/) Q) Q) > • Data <.9 600 > w 500 400 250 50 100 150 50 100 150 I~MI [GeVI >Q) • Data <.9 0WW~qqqq MC (carr.) .LM fGeV] L!) :::!:: 300 c Q) > 10 4 ....-------------~ • Data N WW~qqqq MC (incorr.) ~ 0WW~qqqq MC (carr.) U5 ~ WW~qqqq MC (incorr.) ~Background MC c l1il Background MC (/) 200 ~ 10 3 w w 200 100 0 100 200 300 r'M [GeV] 0 0.2 0.4 0.6 0 .8 1 P(x 5~; 5) Figure 4.15: Distributions of (a) ~m , (b) Em, (c) E'm, (d) PA2(x: 2 5c;5) for selected multijet cvents. Three combinations per event arc shown. 189 GeV d ata respectively, to find the preferred jet combination for t he purposes of the -v,r+-yy- product ion cross section measurement and t he analysis of anomalous couplings described in C hapter G. The choice of the combination a lgorit hm is more complicated in t he case of theW mass measurement. and will be discussed in Chapter 5. Th('re is an intrinsic uncertainty in the performance definition due to the ambiguity 4 .6 Preselectio n S um m ary 59 of mapping reconstructed jets onto the generated quark momenta. Using different mapping algorithms, I checked that the uncertainty is of the order of 1Si and can be neglected. 4.6 Preselection Summary The preselect ion I described so far consists of t h<' following requirements: • ASRC :2: 30 :2: 10 or 1\ATRJ,· = 0 and EECA L bane / > 20 GeV • NATRK • E vi.'> > 0.7 • lEu I < 0.25 • max E Ell I < 40 G0V • max~, < 20 G<:>V • Y31 > 0.0015 EJPL • max lt.'£ 1 < 0.98 • min N E>300 l\ lc\' :2: 4 The results of applying t hese cuts to the data and l\Ionte Carlo samples I considered are summarized in Table 4.4. The preselection achieves efficient rejection of the background events clearly incompatible with th<:> w+w ----? qqqq hypothesis. Good agr00ment between the data and }.lontc Carlo is observed at all considered energies. Event Selection and Cross Section Measurement 60 ft ~1c [GeV] c(r+<'- -+ \Y+~I- -tqqqq) [<7<.] c(r+e--+ vV+\iV--+qqev) [o/t.] c(<'+e- -+ w+w- -+q(jjw) [%] E(<'+e- -+ ~r +w - -+qqTv) [o/c] cr(e+e- -+qq)acc [pb] cr(e+r - -+Z 0 Z0 )acc [pb] cr(e+e- -+e+ e- hadrons)acc [pb] N~elected Ns~J exp. 9GA 161.0 94.0 3.1 0.4 2.7 10.5 0.136 0.07 138 135 ± 12 Data period 96B 97 172.0 182.7 93.7 92.2 1.3 1.1 0.1 0.2 2.5 2.3 8.2 6.3 0.132 0.211 0.03 0.03 142 797 142 ± 12 741 ± 27 98 189.0 92.9 1.1 0.1 2.3 5.8 0.393 0.01 2402 2384 ± 49 Table 4.4: Efficirncies for different w+w- event classes. St-.1 background cross sections aud number of data event. accepted by the preselection cuts. Al o given is the standard deviation on the total number of selected events expected in the Sl\1. 4.7 Neural Network Discriminator The cuts described so far reduce the accepted qq cross section by a factor of 17 and the accepted 11 cross section by a factor of 106 . but the backgrounds are still large relative to the w+w- ---+ qqqq signal cross section. :t\Ioreover. after the rejection I am left with a dean sample of high energy hadronic multijet events with no dear-cut characteristics that would allow me to reject them without cutting into the sample of ~~+ ~r- ---+ qqqq events I aim to select. evertheless, one can attempt to combine the information contained in several variables .ri (i=1, ), each not effective enough on its own, into a more efficient multidimensional discriminator F(i). Accepting events with all t he discriminating variables above their thresholds (cuts) is equivalent to using a discriminator with the very special form of a multidimensional 8-function B(x 1 - .r~ut •... , .1· 1 - .r~ 1 ) which ignores potentially important correlations between the variables and is not necessarily optimal. Analysis of the differential cross section d.r~~~d:rN would in principle make full usc of all the available information. but is numrrically complical eel for all but prohibitively small values of :\ by the limited 4.7 N e ural N etwork Discriminator 61 available l'd onte Carlo statistics. A compromise between the two approaches is reached by the artificial neural network method. I chose to use a feed forward neural network with one hidden layer [80] to separate \ i\r+ v:- ~ qqqq from the dominant QCD qq background. ThC' discriminating variable. referred to as the neural network output, is defined as N g( L <rig( L f3ijXj)), t=l.l5 (4.15) j=l.7 whcrC' the activation function g(.r) is 1 g( .r) .ri arC' 7 input variables, (4.1G) 1 + e-.r normalized to be in the [0.1] range, and the weights <r1 , connecting the 15 nodes of the hidden layer with the output, and /Ji1 , connecting thE' hidden layer with the 7 inputs. arC' dC'tC'rmined during the neural network training. The chosen neural net configuration provides for an efficient projection onto the discriminant of thE' information contained in dx 1 ~ad.rN of a very general form, while taking correlations between the input variables into account. I found that increasing further the number of input variables, nodes in the hidden layer or the number of hidden layers does not lead to an increase in the neural network performance. The following 7 variables were idC'ntified as having the most di criminating power and were used as inputs to the neural net: • log y 34 , the logarithm of the Durh am measure introduced in Subsection 4.5.1. • spherocity of the event s . 4 Lparticles Ei sin (}i 2 11 1r mm(- ) , (4.17) Lparticles Ei obtained by thC' minimization of the expression which can be intuitively understood as thE' normalized total transverse momentum relative to the event axis 62 Event Selection and Cross Section Measurement C\J Q ~> • 200 IE:!ffi qq MC t w Data C=:J ww~qqqq MC j Other Bckg MC 100 0 -6 -5 -4 -3 -2 Figure 4.16: Distribution of the y 34 distance after the preselection. which varies during the minimization. Spherocity assumes values from 0, for an event which consists of two exactly back-to-back jets of p articles. to 1, for a completely isotropic event. The spherocity and y 34 were computed in the event r est fram e. C') • 0 .e 200 Data C=:J ww~qqqq MC mmml qq MC CJ) cQ) > ~ Other Bckg MC w 100 0 0 0.2 0.4 0 .6 0 .8 Spherocity Figure 4.17: Distribution of the spherocity aft er t h e preselection. • E 111 ; 11 and Emax. the energies of the least a nd most energetic jet in the <>vent. 4 .7 N e ural N e twork Discriminator 63 after the 4C kinematic fit. In a QCD qq event the least (most) energetic jet is usually due to the gluon (quark) coming from a qq(l) branching. resulting in a typically large energy difference between the two jets. The energy is split more evenly between jets in a vv+vv- - qqqq event . Close to the V\'+V\r- threshold each of t he four jets has ideally an energy dose to !\~~, . • signed m<~c mass difference m 1c 1-m4 c 2 for the jet combination chosen with the .6.m algorithm. This is small in magnitude for signal events and unlimit eel for the background. >(l) • Data (.9 ~ WW-)qqqq MC l{) ~ qq M C ~ Other Bckg MC ~ 200 c (l) > w 100 -50 0 50 100 g M4 6-M 4 [GeV] Figure 4.18: Distribution of the V..' mass difference m IC 1-m4 c 2 for the jet combination found with the .6.m algorithm after the preselection. The distribution is asymmetric because of my ordering of the reconstructed W bosons (the first jet pair contains the most energetic jet by construction). • Om in. the smallest angle found b<>tween any two reconstructed jets. It is expect<>d to be large for ·w +vv- - t qqqq events and small for qq events due to QCD dynamics. • < l\111 > .the average mass of the two jets (hemispheres) reconstructed when the Durham recombination proceeds to combine all t he particles into two jets. For Event Selection and Cross Section Measurement 64 l!) 0 0 150 • -- Data c:::::J ww~qqqq MC en &=·,WI qq MC c Q) > w 100 !Zm Other Bckg MC 50 0 0.5 0 1.5 1 2 2.5 min ejet-jet Figure 4.19: Distribution of thE> minimal jet-jet angle Bmin after th<' preselection. qq <'vents thE> two j0ts will hopefully correspond to the two primary quarks and have relatively low mass each . while it is impossible to split a typical \i\r+ \v- ___.. qqqq event into two low-ma!:is syst0ms. >Q) • Data (.!) c:::::J ww~qqqq Me .£2 en 200 ~qqMC c Q) > ~ Other Bckg MC w 100 0 '----~-~ 0 20 40 60 80 MH [GeVJ Figure 4.20: Distribution of th<' average hemisphere mass after the prE>selection. Distributions of som<' of the neura l network input variables are shown in Figures 4.16-4.20. There is a good agr<'ement b etween the da ta and th<' \1ont e Carlo 4.8 Selection of vV+ \i\7- ----.qqqq events 65 predictions. The nC'ural network is trained. 1.e .. the weights w ={o.i. 8i are optimized. to 1} produce' N values dos<' to 1 for i\Ionte Carlo w+\\"----. qqqq events and 0 for qq events. Training is achi<'ved by using the back propagation algorithm which adjusts the weights in a gradient descent minimiz:ation (r - ()£ w- TJuw (4.1 ) of t lw dassificat ion error (--1.19) /If(' events N'2+ qq /If C' C11C11 I S (4.20) w t vv - /1/C' evcu ts The classification error £ is computed on a large (training) sample of i\Ionte Carlo events with qq a nd \i\'+ \i\'- events mixed in the proportion expected in the selC'cted data samples. The optimization was redone for each energy point used in my analysis, resulting in four different neural networks used for the four data samples I analyzed. The distributions of the neural network output for data events accepted by the preselection at difl'erent energies are compared to the corresponding Monte Carlo predictions in Figure 4.21. The signa l-background s<'paration achieved is superior to that which is possibk with any other single variable. 4.8 Selection of w+w -~ qqqq events The neural network provides an <'ffici<'nt means of selecting the dean four-jet <'veut samples I use for the nl<'asurements of t he \i\1 mass and triple gauge boson couplings described in Chapters 5 and 6. Demanding accepted <'vents to have the n eural network output above some threshold ~11 in reduces th<' amount of background by a further factor of 4-10 and brings the purit.v of the selected \ i\f+ \i\r----+ qqqq events to an acceptable level. The value of Event Selection and Cross Section Measurement 66 l.{) 0 10 l.{) • Data --0 40 c Q) > f.\!lqq MC I!§SIZZMC w 0WW~qqqqMC ( /) •ww~lvXMC c Q) > • Data 0 --- 0WW~qqqq MC ( /) 0 w ~yy~hadrons MC •ww~lvX MC f.\!lqq MC !Jf3l!ZZMC ~yy~hadrons MC 10 20 0 0.2 l.{) 0 0 --- 150 c Q) > --c oww~qqqq Me ( /) 400 Q) > w (l&!lyy~hadrons MC 0.4 0.6 0.8 Neural Network Output • Data 0 0 oww~qqqq Me f.\!lqqMC mzzMc 100 0 .2 l.{) • Data •ww~lvX MC ( /) w 0.4 0.6 0.8 Neural Network Output •ww~ lvXMC f.\!lqqMC mzzMc ~yy~hadrons MC 200 50 0.2 0.4 0.6 0.8 1 Neural Network Output 0.2 0.4 0.6 0.8 Neural Network Output Figure 4.21: Distributions of the neural network output for t he data events selected in (a) 9GA. (b) 9GB. (c) 97. and (d ) 98 and the corresponding l\Ionte Carlo predictions. The Ionte Carlo distributions are norma lized to the? cross sections determined in Section 4.9. theN cut Nmin was chosen at each energy point to optimize the expected statistical <'rror of the w +v,r- ---+ qqqq cross section measurement obtainable with the achieved 4.9 Measurement of the v,r+-yv- ---+qqqq Cross Section 67 selection, ( ~a- ( vv+ v.' - ----+ qqqq) o-(\\' +V,'----tqqqq) ) NW f W_ ( a ccepted N'v+w - accepted + Nqq + Nqq ac('l']Jted - accepted (Nqq ). a ccepted ) (4.21) The optimal value of the cut was found to be dose to 0.6 at all energies considered. The performance of the selections achieved and the results of applying them to the L3 data. are given in Table 4.5. JR W ' [GeV] c(e+e- ---+ -vv+w - ---+qqqq) [%] E( e+ E'- ----+ vV+\v - ----+ qqev) [o/t] c( e+ e-----+ v,r+ w-----+ qq11 v) [o/c.] c (e+e- ---t w+w- ---tqQTV) [%] o-(e+e- ---+qq)acc [pb] o-(e+e----+Z 0 Z0 )ac(' [pb] Xs!'lected ~SI\1 cxp. 96A 161.0 79.4 0.4 0.0 0.5 0.94 0.045 14 26 ± 5 Data period 9GB 97 172.0 182.7 83. 85.2 0.2 0.2 0.0 0.0 0.7 0.7 1.15 1.16 0.067 0.151 57 468 62 ± 8 414 ± 20 98 189.0 84.0 0.3 0.0 0.6 1.13 0.316 1393 1391 ± 37 Table 4.5: Efficiencies for different w+w- event classes, Sl\1 background cross sections and number of data events accepted by the neural network selection. Also given is the standard deviation on the total number of selected events expected in the Sl\ I. 4.9 Measurement of the w +w --+ qqqq Cross Section One could proceed to extract the V.'+\,\ '- ----+ qqqq cross section from the number of evC'nts which passC'd the neural network selection. corrected for accepted background and selection efficiency. That would not be an efficient use of all the available information though , as suc:h an approach effectively treats equally all selected events irrcsp<'Ctivc of their lik<>lihood of being v.r+ v,r- ---t qqqq. This likelihood can indeed G8 Event Selection and Cross Section Measu rement vary from almost 100% for events with aN close to L down to much lower values for events with N just above Nmin· I attempt to use this addit ional knowledge in the determination of the w +v.,r- ---t qqqq cross section by using a m aximum likelihood fit of theN distribution for events which passed my preselection. I divide the [0,1] range into Nbin s = 40 equal bins and find the w +w - ---t qqqq and qq cross sections by maximizing the binned likelihood (4.22) i= l ,40 j = l.NM c i= l ,40 where d; is the number of data ewnts in the ith bin, !lfji is the number of Monte Carlo events of class j in the same bin, and tj i are auxiliary parameters varied during the maximization. In my fit s index j spans the N!l 1c = 5 Monte Carlo event classes which contribute to the neural network output distribution (W+W ----t qqqq. w +w ----t qq(v, qq, Z0 Z0 , and e+e----te+e- hadrons). The total Monte Carlo prediction for bini, Pt is given by (4.23) The chosen form of the likelihood takes into account the statistical fluctuations of t he bin-by-bin :t-.Iontc Carlo predictions due to the finite available l\1onte Carlo statistics. tj i can be interpreted as the (unknown) prediction of l\Ionte Carlo j in bin i, i.e. , the mean of the Poisson distribution a random drawing of which produced !lfji · A very convenient property of the likelihood (4 .22) is tlw possibility of performing the optimization in Nbins x N!ll cs dimensions defined by the tj i variables analytically [81]. The maximization for w +w- ---t qqqq and qq is performed numerically with t he help of the :t-.IINUIT software p ackage. As was noted before, both the w +w - ---t qqqq and qq cross sections were varied in the maximization. The aim of leaving the qq cross section free is to decrease the sensitivity of the measured w +w - ---t qqqq cross section on the QCD prediction for 4 .10 Rarity Analysis 69 the mnltijet qq event rates. In effect. any changes that affect t he rate normalization of preselected qq events without affecting the shape of theN distribution for them. will not introduce any bias into the measurement of the \,V+W --+ qqqq cross section with floating o- (e+e--+ qq). Thr more efficient usc of the information contained in the data N distribution, by thr maximum likelihood fit. leads to a gain of about 5% in the expected statistica l accuracy of the measurement. compared to t he simple event counting measurement. The results of the fit performed on the d a ta are reported in Table 4.6. Data Period 96A 96B 97 98 a-( qqqq) [ph J a-(qqqCI)st.t [pb ] 0.85 _ 0:3 -l 5 . 32+0.90 - 0.83 8 .30+0.47 -0.46 7 . 53+0.26 -0.25 5.67 7.17 7.59 o-(qq) [pb J a-( qCI)st.t [ph J 167.9+ :>. 147 - 1·1.9 121 129.6!~~:~ 108 109.2!~:~ 104.5!U 98 Ta ble 4.6: e+e- -+ \t\'+vv--+ qqqq and c+e- -+ qq cross sections determined in neural network output fits. Statistical and Ionte carlo statistics errors arc combined. Also included arc contributions due to measured cross section correlations. The Standard f\ Iodel cross sections calculated with GE TLE arc also given for comparison. The threshold cross section is strongly dependent on Mw and is not given. There is a good general agreement between th<' measured \V+w - -+ qqqq cross sections and the Standard Model expectations calculated with the GE TLE semianalytical code [65]. 4.10 Rarity Analysis As a cross-check, I use anot her method of measuring the \ V+W - -+ qqqq cross section using the same 7 variables that were used as neural network inputs. I repeat the fit described in the previous section with another multidimensional discriminator, rarity [82]. T he n eural network input variables, already normalized to be in the [0.11 range. an• transformed if necessary as x; -+ 1 - .ri so that they assume high vahws for the w +\"'--+ qqqq signal and lower values for the qq background. An 70 Event Selection and C r oss Section Measu rement auxiliary variable F , computed for an event dis the fraction of V\T+ w - ~ qqqq i'd onte Carlo events with .r, < d, for all i. (4.24) The rarit.v R is defin<.>d as the integral probability of F. (4.25) ThC' rarit.v distribution is fiat. by construction. for t he w +yv- ~ qqqq signal. and is concentrated at low values of R for background. This discriminator is different from the neura l network in its usc of t he information contained in the spectrum d.r,~~d.r,. A difference in t he results of the fits using the two discriminators would indicate incorrect modeling by the l\Iontc Carlos I used of the corr<'lations between the neura l network input variables. Th<' results of thC' fits using the rarity distributions shown in Figure 4.22 are reportC'cl in Table' 4. 7 and are in good agreement wit h the results obtained in the neural network output fit s. Another measurement [83] using the same differential cross section, based on the minimal spanning tree method (a multidimensional generalizat ion of the runs test) gave consistent estimates of the w +w - ~ qqqq cross section and confirmed the compatibility of the data with the l\Ionte Carlo simula tions. 4.11 Analysis Uncertainties The w +vv- ~qqqq cross section m easurement method described in this chapter relics on Monte' Carlo simulation of the neural network output distribution for signal a nd background events, as well as the preselection efficiency for w + V\r- events. Imperfect modeling of these quantities might kad to a bias of the measured cross section. In 4 .11 Analysis U ncert a int ies Data Period 96A 96B 97 98 a(qqqq)Si\1 [pb l 1.76 0.74 _ 0 : ;1 ~ 1 5 .60+0.9" 5.67 - 0.86 8 .32+0.4B 7.17 - 17 ~ +0.26 7.59 ( .60_0.25 a( qqqq) [pb ] 71 a(qq) [pb] 166.0_15:0 125.8~ }~:~ 110.5~~:~ 103.4~~:~ a(qq)Si\1 [pb J 147 121 108 9 Table 4. 7: e+e- ----t w +v,r- ----t qqqq and e+e- ----t qq cross sections determined in rarity fits. Statistical and I\Iontc carlo statistics errors are combined. Also included are contributions due to measured cross section correlations. The Standard Model cross sections calculated with GENTLE are given for comparison. this section I give estimates for the most important efl'ects which might contribute to such a bias. I consider three main categories of systematic errors: the ones due to imperfect modeling of signal event properties. uncertainties of the rate and properties of accepted background events. and uncertainties due to mismodeling of the L3 detector response. common to both signal and background. Most of the uncertainties are expected to be much smaller than tlw statistical errors of the measurements. One cannot evaluate such biases in the usual way of checking the stabilit.v of the result with respect to variations of selection criteria or other technical parameters of the measurement. Such variations would lead to changes of the selected data sample and to statistical variations which would be large compared to the systematic cfl'ects under investigation. Therefore, I estimate all systematic errors by varying parameters of Monte Carlo models and then repeating the measurements with modified I\ Ionte Carlo samples but without changing the data. The errors were reevaluated for each data-taking period. In the text I quote numbers relevant for the measurement at Js = 189 GeV. Estimates for other data periods are very similar and the difl'crences are indicated whenever necessary. Eve nt Se lection and Cross Section Meas ure ment 72 I.{) 0 c) ----cen: 1.() • Data OWW---?qqqq MC .WW---?IvX MC lmqq MC ll&§I ZZ MC ~yy---? hadrons MC 10 Q) > w 0 0 ---cen 60 Q) > w • Data OWW---?qqqq MC .WW---?IvXMC lmqq MC ll&§IZZ MC ~yy---?had rons MC 40 10 20 0 0.2 0 .4 0.6 0 .8 0.2 0.4 0.6 Rarity 0 .2 0.4 0.6 0.8 0.8 Rarity 0.2 Rarity 0.4 0.6 0 .8 Rarity Figure 4.22: Distributions of the event rarity for the d ata events selected in (a) 96A, (b ) 96B, (c) 97, and (d) 98 and the corresponding Monte Carlo predictions. The Monte Carlo distributions are normalized to the cross sections measured in this section. 4 .11.1 Mode ling of the Detector Response I\lismodeling det ector behaviour affects properties of both sign a l and background events. The corresponding systematic errors ar e correlated with t hose of measurem ents using w+w - ----> qqfv events. Fortunately, one can gauge. and put tight limits 4.11 Analysis Uncertainties 73 on such mismodcling using a well-understood control sample of hadronic events. The data collected during calibration LEP runs at the Z0 peak provides such a sample. The Z0 -mediated qq production has a cross section of 30 nb and is the dominant process in e+c,- collisions at ,fS ~ 91 GeV. Employing the event selection described in [84], it is possible to select Z0 ---+ qq events with efficiencies in exe<'ss of 99<7< and with an admixture of background events. mostly from Z0--+ T + T- decays, of approximately 0.3<7<. The selected high statistics. kiiH'matically unbiased. and virtually background-free' samples of qq events arc a gpod probe of the detector response to jC'ts of hadrons. To estimate possible imperfections of the 1\Iontc Carlo description of the detector behavior, I compared the data summarized in Table ..t.8. with the corresponding l\IC predictions. Year 1996 1997 1998 £ selected [pb - I ] 1.17 2.17 3.01 (vs)[GeVJ 91.1100 91.237 91.3123 ~qq " selectro 35519 65324 90512 Table 4.8: Total sclrcted integrated luminosity for the data sets used in thC' analysis. Accepted-luminosity av<'raged center-of-mass energy and number of selected qq events are also given. • In order to judge possible detrctor miscalibration I study the angular dependence of the energy flow using thC' selected sample of qq events. I measure' the total energy of the particles deposited in a unit of solid angle, normalized to the event center-of-mass energy and averaged over the qq sample. I assume cylindrical symmetry of the detector. and sum up the energ~' in rings equidistant in cos 0. The ratio of the distribut ions thus obtained for data and l\IC is shown in Figure 4.23. I estimate the cffC'ct of t he jet energy scalC' miscalibration by redoing the cross sC'ction measuremrnt after correct ing energies of all reconstructed particles in thC' ~IC sample's I use by the found ratio. The procedure leads to a change of 74 Event Selection and Cross Section Measurement the fitted vV+w - - c{qqq CTOSS section of 0.02 pb. 1.1 . - - - - - - - - - - - - - - , ~ 0 ~ ...._ ctl 0 I\.. <ll 1\ :\J- 0 - - ctl ctl (tj 0 t t t t t +t t t t t +t t t t +t t t u.r v • • • • • + + ++++++ + + +• • • • a; w <l v 0.5 0.9 0.8 L-~-.J..._~--'--~---'--~-....J -1 0.5 -0.5 0 0 ~~--~~--~--~~--~-1 -0.5 0 0.5 cosO cosO Figurf' 4.23: Angular distribution of average jet energy in Z0 - <iCI events. R a tio of the da ta a nd l\I C distributions is shown. Figure 4.24: Angular distribution of jet energy variance in Z0---t qq events. Ra tio of the da ta a nd I-.IC distributions is showu. • A similar mism a tch is found in the modeling of jet energy resolutions. Figure 4.24 shows the ra tio of the spreads (va riances) of the energies of jets dep osited in co () rings. I estima te tll<' cfi"ect oft he misma tch by redoing the cross section fit after smearing reconstructed jet energies in .i\IC events with Gaussian ra ndom noise of the m agnitude cho~:>en to reproduce the resolutions found in da ta. This leads to a (negligible) cha nge of the fitt ed \V+vv- - qqqq cross section of 0.003 pb. • Another way to estima te possible system atic cross section biases due to detector miscalibra tion comes from studying the subdet ector composition of jet energies. The corr<>ction factors applied t o raw deposited energies, to compute a particle energy as described in Section 3.4, difi"er in da t a and !\IC by on the order of 3<.X for E CAL and lOCX for HCAL clusters. One can t ake t he difference of the corrections as an estimate of the calibration uncertainty. 4.11 Analysis Uncertainties 75 I redo the analysis after applying the data correction factors to ~Ionte Carlo events, i.e., ignoring this difference, to estimate the possible influence of detector miscalibration/mismodeling. I observe a shift in the fitted w +w - ---tqqqq cross section of 0.04 pb. which I take as an estimate of the systematic error due to uncertainties of jet energy measurement. It is more conservative than but in rough agreement with the two estimates obtained from energy scale and resolution corrections, described above. • An incorrect description of the detector geometry might lead to a systematic mismeasurement of jet directions. This might influence the cross section measurement and the reconstruction of invariant W masses and is especially important for the measurement of theW mass. To quantify possible mismodeling I study the acollinearity between two jets, constructed by the Durham algorithm in the selected Z0 ---t qq events, as a function of the direction of the more energetic jet in the event. Misalignments of jet polar and azimutal angles are studied separately. But for initial state radiation. negligible for events at the Z 0 peak, the two jets are produced back-toback in the laboratory frame, and would have zero acollinearity if reconstructed by a perfect detector. I study both the mean of the distributions in each bin of cos 0, i.e. , the bias of the jet direction determination, and the variance, i.e. , the angular resolution. The agreement between data and Monte Carlo in both quantities is illustrated in Figures 4.25 and 4.26. I estimate the effect of possible jet direction mismeasurement by redoing the cross section fit after applying the found 0-dependent 0 and ¢ corrections to directions of all reconstructed jets in the MC samples. This leads to a change of the fitted w+w- ---tqqqq cross section of 0.01 pb. • A small mismatch is also found in the modeling of the resolution of the jet direction measurements. Figure 4.26 shows the ratio of the spreads (variances) 76 Eve nt S e lection and Cross Section Measure m e nt of the acollineariti0s of jets in two-jet zo___, qq events, as a function of cos (J. I r0clo the cross s<>ction measurement after smearing reconstructed jet directions in IC events with Gaussian random noise of the magnitude equal to the found clata-r..IC differences, resulting in a change of the fitted V..'+w - ____, qqqq cross s<>ction of 0.002 pb. 0 .01 l{) 0.01 l{) 0 0 c) Q 00.005 00.005 -- ~ ~ I I cu cu cu cu 0 0 fl..<ll fl..<ll a;;- t -e-- <l <l 'Z.o.oo5 t 'Z.o.oo5 -0.01 '--~-....&.-~--'--~--'--~--' -1 -0.5 0.5 0 Ht j I I I I I1I I 1I I j 0 -0.01 -1 -0 .5 0 cose 0.5 cose FigurE' 4.25: Angular dependence of reconstruct<>d jet angk biases in Z0 ----tqq events. Differences of the data and r.. IC distributions for (J and <P biases arc shown. 4.11.2 Modeling of the w +w - Signal Properties • The distributions of the kinematic variables of v..r+w- events used in the cross section measurement depend on the properties of t he \V boson. A mismatch b<:>tween the vahws of r..Iw. fw asstm1ed by the r..Ionte Carlo. and the true. unknown values of these parameters could lead to a bias of the cross ection estimation. The size of such effects was evaluated by repeating the measurements using w+w- Mont.e Carlo sets generated a t different values of l\1w. in the 8081 GeV range. and fw. in the 1.51-2.71 GeV range. The obserwd dependence of the fitted cross section on the l\Ionte Carlo V\r mass. of -1.5 ± 1.0 x 10- 3 77 4.11 Analysis Uncertainties 0 ~ 1.5 . . . - - - - - - - - - - - - - - - . - (tj 0 ---ro ~ co co 0 .b Q) + :-:.- + + +++ + ++ + + + + + +++ + + + 0 .b Q) a:> :-:.-e- v v <l • • + ++ + f t f f t t t +++ + + • • <l 0.5 0.5 0 L...-~--'--~-......__~--'-~-~ -1 0.5 -0.5 0 ol...-~--'--~-......__~--'--~-_1 cose cose -1 -0.5 0 0.5 Figure 4.26: Angular dependence of reconstructed jet angle resolutions in Z0 ---t qq events. Ratios of the data and MC distributions for () and ¢ biases arc shown. pb/ GeV, and the W width. of 1.4 ± 1.0 x 10- 3 pb/ GeV , was convoluted with the current uncertainties of these parameters [23] to give the associated cross section uncertainty of 0.01 pb and 0.01 pb respectively. Similarly, i'donte Carlo events generated at different values of ..JS were used to determine the influence of the center-of-mass energy uncertainty of the cross section measurement. For all data sets it was found to be negligible, compared to the i'.Iw , rw uncertainties, due to good knowledge of the LEP beam energy. • KORALW code. used for simulating w +w- events. contains only the CC03 subset of e+e- ---t f 1f 2 f3f..t diagrams. Considering Z0 Z0 four-quark l\Ionte Carlo events as an additional background makes up for most of the deficiency, but still does not take into ac·count the interference between CC03 and the other diagrams. The effect of this simplification was estimated by redoing the cross section fits with w +w - events weighted by ratios of the full and CC03 matrix elements, as described in Section 5.4. The change in the fitted cross section of 0.03 pb is taken as the corresponding systematic error. 78 Event Selection and Cross Section Measurement • It was checked that the selection procedure is not sensitive to the presence of ISR photons in w+w- events. The signal Monte Carlo events were put in one of two samples, ISR-enriched and ISR-depleted, depending on whether ISR photons with an energy of at least 8 GeV were found in the list of generated particles. The ISR-enriched sample selected this way constitutes 10% of selected w+w- events. The difference of t he efficiencies of the neural network selection, evaluated on the two samples, was found to be 0.1 ± 0.3%. This small difference, together with the high theoretical precision of the ISR implementation in the MC, allows me to safely neglect this systematic error. • Also the measured cross section was shown to be insensitive to the fraction of five-or-more jet events in the signal. Simulation of such events, done with parton shower methods, might be imperfect. I compute the Durham parameter y 45 as described in Subsection 4.5.1. This quantity is expected to be larger for events with at least five well-defined jets than for four-or-less jet events. A five-jet-enriched sample was constructed by selecting w+w- Monte Carlo events with the value of y 45 larger than 0.008, about 12% of the total sample of w+w----+ qqqq events. The efficiencies of the selection, evaluated on this sample and the rest of w+w- MC events, agreed to 0.4 ± 0.3%. Therefore, I take 0.004 pb as an estimate of the corresponding systematic error on the measured four-jet cross section. • o precision calculations are available for the non-perturbative process ofhadronization. Various phenomenological models have to be used to simulate fragmentation of the four-quark system and t he following h adron decays. There are subtle differences in the models' predictions of various properties of the resulting hadronic system. Some of those properties, e.g. , charged and neutral multiplicity, or electromagnetic energy fraction. affect the reconstructed jet parameters I use in my selection. This results in an uncertainty of my measurement due to the modeling of fragmentation and hadron decays. I compare the JETSET /LUND and HERWIG [85] models to estimate this uncertainty. Both 4.11 Analysis Uncertainties 79 models give an adequate description of low energy qq event properties [63] and it is not possible to discriminate against one of them on the basis of disagreement with data. I redo the cross section fit using a w+w- MC sample obtained by HERWIG hadronization of four-quark systems generated with KORALW. I compare the result with the base one, obtained with a JETSET and KORALW w +w- Monte Carlo. I assign the found difference, 0.10 pb, to t he systematic error due to fragmentation modeling. • Bose-Einstein correlation is an enhancement in the two-particle correlation function for identical bosons at small distances, experimentally observed in multihadron events [86, 87]. The effect, similar to Hanbury-Twiss interference [88], is due to the symmetry of the multiboson production amplitudes. This purely quantum mechanical phenomenon has not been incorporated in currently available probabilistic Monte Carlo models of parton showering and string fragmentation. Therefore, ad hoc algorithms have to be applied to the generated multihadron events to reproduce this feature of the data. The LUBOEI algorithm [73] is used in the production of standard L3 MC samples for this purpose. It applies corrections to the momenta of particles belonging to low-invariant- mass identical boson pairs, which could be important for low momentum pions or for pions separat ed by a small angle. Such corrections change the structure of jets and introduce cross-talk between the two hadronic systems coming from decaying W bosons. I repeated the cross section fit using a w +w - MC sample generated without applying t he algorithm. and I take the difference of the result with respect to the original fit, found to be 0.03 pb, as the uncertainty due to modeling of Bose-Einstein correlation between identical hadrons. • Another poorly modeled mechanism leading to cross-talk between the two W bosons is due to the interconnection between the two diquark systems during their fragmentation. The typical W boson lifetime, of the order of 1/ rw ~ 0.1 fm, is much smaller than the non-perturbative hadronization scale of 1 fm. Event Selection and Cross Section Measurement 80 Therefore. the hadronization of the two W s cannot be considered independent. Their overlap is difficult to describe in the framework of currently available fragmentation models. In the language of the string model, in the limit of zero \1\T width fw the fragmentation of an w + w - ~q 1 q 2 'q3 q/ event would proceed through stretching and decay of strings (color flux tubes) binding color singlets q 1q 2' and q3q/. In the case of the large W width. an uncertainty arises due to the inability of the model to predict the probability of an alternative configuration. with the two strings connecting pairs q 1 q/ and q 3 q 2 ', referred to as color reconnection. Reconnection probabilities, calculated with various models [89. 90, 91, 92] implemented in L3 event generators. vary over a wide range. The probability is difficult to measure experimentally [93] . Moreover, the dependence of t he measurement bias on the reconnection probability is different for each color reconnection model. Therefore, a wide range of models has to be considered for estimating the systematic error due to color reconnection. I redid t he cross section m easurement with w + w - Monte Carlos implementing Gustafson-Hakkinen and Sjostrand-Khoze (types I, II and II') models of color rcconnection in the LUND , HERWIG. and ARIADNE [94] fragmentation frameworks, as well as Monte Carlos with t he corresponding fragmentation models with no reconnection allowed. The differences range from -0.01 to 0.03 pb. The shifts are correlated with the reconnnection probabilities predicted by the models. I t ake the largest of those. corresponding to the Sjostrand-Khoze I model, as an estimate of the systematic error clue to color reconnection. 4.11.3 Modeling of the Backgrounds • QCD four-j et events constitute the main background for my selection. The events affecting the measurement represent a very small fraction, about 1%, of all qq events and their rate and properties may be modeled imperfectly by the parton shower a lgorithm. While the fit method I usc is not sensitive to the total rate of QCD multijet events, a discrepancy in the neural network 4.11 Analysis Uncertainties 81 output distribution would lead to a bias of the cross section measurement. It is impracticable to study the validity of the parton shower model with the data collected at high energies because of its very limited amount. Therefore, I judge the reliability of the JETSET's description of QCD four-jet events by studying the Z0 peak data collected by L3. As this data was not used to tune the four-jet rate predicted by JETSET, I believe the quality of the description observed with Z0 data can be extrapolated to the high energy data I use in my analysis. To estimate the uncertainty on the cross section, I apply my preselection to a large sample of Z 0 data collected in 1995, and the corresponding Monte Carlo sets, with all energy-scale-dependent parameters reduced by a factor of Mz / 180 GeV. I compute y 3 4-dependent correction factors (weights) which have to be applied to 91 GeV qq Monte Carlo events to reproduce the y 34 distribution observed for events selected in the 1995 Z0 data, dN dN ( - d )data/( - d )MC· Y34 (4.26) Y34 The dependence of the weights on y 34 is shown in Figure 4.27. I redo the cross section fit with the qq Monte Carlo events given weights determined by (4.26), and assign the change in the result. 0.10 pb. to the systematic error due to modeling of QCD background. The reweighting effectively increases the rate of QCD four-jet events which pass the neural network output cut, by about 5%. The result is reproduced when reweighting is done in other variables which reflect the multijet structure of an event, e.g., the neural network output. Also it was checked that the weight function (4 .26) computed with 1995 data is consistent with ones obtained with Z0 data collected during calibration runs in 1996-1998. This indicates that the found data-Monte Carlo disagreement is due to parton shower model deficiencies rather than time-dependent detector miscalibration/mismodeling. • Distributions of some of the variables used for my selection show differences Event Selection and Cross Section Measurement 82 (_) ~ (_) ~ --co cu --co 1.5 cu 0 0 , ........ +!tltll 0.5 0 .5 0 L---~--~--~---L--~--~ -6 -5 -4 -3 In y34 0 L-~~~--L-~-L~~~~~ 0 0.2 0.4 0.6 0.8 Neural Net Output Figure 4.27: The ratio of selected event rates in Z0 data a nd MC as a function of y 34 and neural net output. The combination of data and Monte Carlo statistical errors is shown. in the b ackground-dominated r egions which indicate possible background mismodeling. The most drastic disagreement is found in the distribution of the minimum number of particles in a jet, min NE> 300 t\1eV (see Subsection 4.5.2). The excess in the d ata of events with low-multiplicity jets was also observed with samples of Z0 ----rqq events, and is likely to be due to imperfect modeling of the fragmentation in e+ e- ----r qq events. The importance of this discrepancy for my measurement was evaluated by redoing the analysis while varying the cut on t he minimal jet multiplicity in the [2,6] interval around the default value of 4. This is equivalent to changing the fraction of such mismodeled events in the sample I use for the neural network output fit. The maximal observed cross section shift of 0.08 pb was added in quadrature to the QCD modeling uncertainty evaluated above. to arrive at an estimate of the systematic error due to qq background modeling. • ?vly preselection reduces t h e rate of accepted e+e- ----r e+e- hadrons Monte Carlo events by a huge factor of 3 x 106 . T h e accepted cross section is bound to 4.11 Analysis Uncertainties 83 be unreliable with the properties of two-photon events being poorly modeled by currently available l\Ionte Carlo models. Fortunately, the cross section is ver~' small, and since the accepted events are concentrated at very low values of the neural network output. there is no significant effect on the measured w +w - cross section. Changing the Monte Carlo prediction for the accepted e+e- - t e+e- hadrons cross section by ±100% changes the fit result by 0.002 pb, which I take as an estimate of the corresponding systematic error. The difference is compatible with the estimate I get by comparing fit results obtained with PYTHIA and PHOJET e+e- -e+e- hadrons Monte Carlo samples. • The rate of accepted Z0 Z0 four-jet events has a much greater influence on the fit result , as such events typically have very high values of the neural network output. On the other hand, the cross section and properties of this electroweak process are understood much better. Changing the Monte Carlo prediction for the accepted Z0 Z 0 cross section by ±5% [66]leads to a change of 0.02 pb in the fitted w+w- cross section. 4.11.4 Luminosity • The systematic uncertainty on the integrated luminosity collected by the detector directly translates into an error on the measured cross section. It amounts to 0.2% of the measured cross section at all the energy points I used. 4.11.5 MC Statistics • I compared results of the measurements obtained with MC event sets used for testing and training the neural network, half of all available Monte Carlo events each, to check the effect of limited Monte Carlo statistics on my measurements. The observed difference, of about 0.1% of the fitted cross section, is in good agreement with binomial expectations given the total and accepted number of Monte Carlo events from various sources. This indicates that the the number of MC events used was sufficient to avoid overtraining of the neural network. 84 E ven t Se lection a nd C ross Section M easure m e n t The uncertainty due to limited MC statistics is already implicitly included in the error determined in the maximization of likelihood (4.22) and is not combined with other systematic uncertainties. 4.11.6 Summary of the Systematic Unce rtaintie s The systematic errors from the sources described above were evaluated for each set of data and Monte Carlo used in my analysis. The values for high-energy (above threshold) data sets are in good agreement with each other. The systematic errors are summarized in Table 4.9. Source of uncertainty Jet energy measurement Jet direction measurement Mw, rw , uncertainties CC03 approximation ISR modeling FSR modeling Fragmentation Modeling of BE correlations Modeling of color reconnection qq background (QCD modeling) qq background (other) e+e- --?e+e- hadrons background Z 0 Z0 background cross section Luminosity M C statistics Total vs CJ(W+w - --7 qqqq) error [pb ] Data period 96A 98 0.01 0.04 0.01 0.002 0.002 0.01 0.03 0.01 0 0 0.004 0 0.02 0.10 0.01 0.03 0.03 0.01 0.10 0.06 0.08 0.02 0.002 0 0.02 0 0.002 0.015 0.01 0.003 0.07 0.18 Table 4.9: Systematic errors of the w +w - --7 qqqq cross section measurement itemized by the source of the uncertainty. The quadratic sum of all contributions except the error due to finite MC statistics is also given. The detector miscalibration and w +w- m ismodeling typically lead to similar errors (relative to the measured cross section) at threshold and at higher energies. Dependence of the selection on t he kinematics-affecting parameters M w, r w , ..JS 4.12 Summary of the R esults 85 is naturally stronger at low energy. but is negligible compared to other systematic uncertainties. Background modeling errors arc dominant at threshold. where the signal-background separation is poor. and are significant at a ll other energies. The most important sources of system a tic errors are the modeling of hadronization in ~~+ ~ r- --+qqqq events at the highN C'ncrgies, and thC' modeling of QCD background at threshold. The systematic errors of the measurements are smaller than the statistical ones even for the highest statistics measurement at .jS = 189 Gc.V. As tl1C' same Monte Carlo models and calibration procedures were used throughout my analysis. I assume the systematic uncertaint ies to be fully correlatcd when combining results obtained at different .jS points. 4.12 Summary of the Results The measured w +vv- -. qqqq cross sect ions and their statistical and systematic measurements arc summarized in Table 4.10. The measured cross sections are in overall agreement with the Standard 1\Iodel predictions. The stat istical error due to limited data statistics is dominant at all energy points considered. Data Period 9GA 9GB 97 98 a(w+w---+ qqqq) [pb l 0.85_0 ::3.1 ± 0.07 5.32~8:~~ ± 0.15 0.'17 ± 0 ?Q 8 · 30 +-0.1G ·~ 7.53~g:~g ± 0.18 a(w+w ---+ qqqCi)sr.~ [pb ] 1.76 5.67 7.17 7.59 Table 4.10: The c+c---+ w +W ---+ qqqq cross sE'ctions determined in neural network output fit s together with their statistical and systematic errors. The Standard 1\Iodel cross sections ('alculated with GE JTLE are also given for comparison. 86 Eve nt S e lection and Cross Section M e asure m e nt 4.12.1 Measure m e nt of the Total w+w- Cross Section and the W Branching Ratios The results of my w+w----+ qqqq cross section measurement can be combined with the studies of other final states [95. 96, 97] to obtain the total w+w- production cross sections. Also study of the fractional composition of selected w+w- events allows one to make a measurement of the branching ratios of the W boson. For each L3 data set. dedicated selections were used to select events and measure the \ V+v..'- cross section in the sC'mileptonic and fully leptonic w+v.,r- decay channels (qqev, qq{w. qqrv, and six tvC'v' final states). The results of the selections were combined with the w+w- ---+ qqqq cross section measurement in a max:imization of the total likelihood lnL :2:::= (Ni ln f.Li - J-ti) + ln LN( O"( qqqci)), (4.27) q(jtll,(l>f'v' given by the product of Poissonian probabilities of observing the actual number of <'vents found in data Ni, for the qq[I/ and [I/[v channels, and the likelihood LN(O"(qqqq)) (sec (4.22)) as determined in the neural network output fit of the w +w----+ qqqq cross section. Tlw expectations £( (4.28) qqqq.qqlv,tvf' v' are given by the w +v.,r- channel cross sections O"j to be determined, the matrix E iJ of the efficiencies of selection of final state i to accept events from final state j, and the expected non-\ tV+ v..r- backgrounds a i · As an example, Tables 4.11 and 4.12 show the efficiency matrix. t he expected backgrounds and the number of data events found by the selections used in the analysis of t he L3 ys = 189 GeV data sample [98]. The w +w - ch annel cross sections O"j arc related to the total e+e- ---+ w +w cross section awll' a ud theW decay branching ratios BR(W-+ff') , which are the free 4.12 Summary of the R esults 87 Efficiencies [%] for Selection qqel/ ClQf-W qqTv c+ e- --tqqev 87.2 0.3 2.2 e+ e---.qq;w 0.3 78.5 4.5 0.1 c+e- --.qqTv 4.8 5.7 49.3 0.2 ('+('- --tevev evev eVJW CI/Tl/ 60.9 0.1 10.3 57.4 10.1 1.2 39.0 c+e- --tCI/f-LV c+ c- --tCI/TV 9.4 f-Wf-W (;'+e- --tTl/TV 0.4 4.9 0.5 0.2 2.~ TVTI/ qqqq 0.9 0.2 c+e- --. JWf-LV c+ c- --. f-LVTV f-LVT I/ 9.4 2.3 0.2 10.3 49.1 8.7 1.0 2.3 33.9 8.2 1.0 23.6 Table 4.11: Selection efficiencies for [1/ev. qqev selections used by L3 at .jf; = 189 GeV. parameters of the fit , as CJ(w+w---.qqqq) CJwwBR('N--.qq') 2 (4.29) (4.30) C5(W+w- --tqqCv) - 2fqqeuC51vii ·BR(W--tqq')BR(W--.ev) (J(w+w---. el/ev) - feueuCJII'wBR(W--.ev) CJ(V\T+ w ---. CvC' v') 2 2fcue'u'C511W BR(W--. Cv)BR('vV--t e'v' ). (4.31) (4.32) The correction factors f t , given in Table 4.12. arc necessary to correct for the effects of the Feynman diagrams not present in the CC03 set (CC20 for qqev and CC j l\C56 for {v(v final states), to simplify the comparison with the Standard r..Iodel predictions and the consequent W mass measurement using the w +Vv'- production cross section. The effects of the additional diagrams are negligible a nd no correction is necessary for qQf-W, qqTv. and qqqq final states. The likelihood (4.27) docs not take into accou nt possible dependence of the fitted Event Selection and C ross Section Measurement Selection Ndata Background Conversion [pb] Factor .f e+ c - ---. eve I/ 28 0.014 0.88 c+c - ---.evtw 41 0.020 1.07 e+c - --->evTI/ 50 0.046 1.07 e+ c- ---t f.LV tW 15 0.029 0.96 e+e- ---? f.LVTI/ 36 0.018 1.10 e+e- -. TI/TI/ 16 0.030 0.96 e+e----t qqei/ 358 0.105 1.0 c + e- ---t CJCfJW 3G4 0.083 e+e- ---tqCfTv 313 0.240 Table 4.12: Number of selE'ctecl data events. Ndata, and expected non-w+wbackground cross section for fvCI/, qqCv selections used by L3 at ..jS = 189 GeV. The ratio of tllC' CC03 cross section without cuts and the four-fermion cross section within phase-space cuts, calculated with EXCALIBUR. is listed in the last column. vv+vv- - qqqq cross section on the w+w- - qqCv cross sections, fixed to the Standard l'd odcl values during the neural network output distribution fit. This dependence is expected to be very small because of very low qqqq preselection efficiency for qqt'I/ events and the large difference of the N distribution for t hem and w+v...r- - qqqq €'vents. The largest cross-efficiency. da(;r+ w<- ~qiq<D has been numerically determined · aMc qqrv to be in the 0.2-0.4% rang€' for all energies considered and was neglected. In addition. on€' can demand the equality of the leptonic branching ratios BR(v\·---t (I/) used in the fit . in the hypothesi~ of charged current lepton universality. Fits with and without this assumption were performed using all high energy L3 data collected in 1996-1998. The resulting values of the w+w- cro. s section at each energy ancl vV decay branching ratios arc reported in Table 4.13. The systematic errors were obtained by propagating the system atic errors on individual v...r+w - channel cross ~cction~. taking int.o account their correlations. With the curr€'ntly available statistics, 4.12 Summary of the Results 9 the st a tistical errors arc dominant for all the m easurements. , / ,--, ..0 c.. , / / / 20 / L.........J . .. . ....-... ~ .._... / / / / / I $ $ + I 10 I • Q) +Q) Data Standard Model .._... b --- no ZWW vertex . . . . only v e exchange 0 160 170 180 190 200 ~ [GeV] Figure 4.28: The cross section of the process e+e- ---7 w +vv- ---7 fJ 2 f3L as a function of the centre-of-m ass energy. Thr measurements of aww arc shown as dots with error bars, combining sta tistical a nd system a tic errors in quadra ture. The points at energies higher tha n 189 GeV arc prelimina ry L3 measurements. The solid c urves show the Sta ndard 1\Ioclel exp ect ation a nd its theoretical uncert ainty. The dashed cmve shows the expecta tion if there is no ZWW coupling. The dotted curve shows t he expectation if only t-channel Ve exch ange in V\1 pa ir production is considered. Eve nt Se lection a nd Cross Section M easurem e n t 90 Lepton Lepton Standard Non- U ni versali ty U ni versali ty l\Ioclel 10.43 ± 0.23 ± 0.13 10.83 G7.51 BR(\ \'-'cv) [%] 10.27 ± 0.44 ± 0.17 BR(W-+ l tv) [%] 10.00 ± 0.46 ± 0.17 BR(V\T-->Tv) [~] 11.16 ± 0.63 ± 0.25 BR(W-->Lv) [%] BR(W-->qq') [%] 68.57 ± 0.69 ± 0.38 68.72 ± 0.69 ± 0.38 aww( Ji; = 161.34 GcV) [pb] 2.70~2:~~ ± 0.14 2.6 ~8:~~ ± 0.14 aw11 ·(JS = 172.13 GeV) [pb] 12.10~ ] :~8 ± 0.23 12.04~ : :~~ ± 0.23 12.43 awl\'( Ji; = 182.68 GeV) [pb] 16.52 ± 0.68 ± 0.26 16.45 ± 0.67 ± 0.26 15.72 awll'( Ji; = 188.64 GeV) [pb] 16.36 ± 0.39 ± 0.26 16.28 ± 0.38 ± 0.26 16.65 Table 4.13: Measured \1\T decay branching ratios. a nd total vv+w - cross sections at different energies, derived with and without the assumption of chargcd current lepton universality. The corrE'lations between the leptonic branching fractions arc -0.008, -0.272. -0.286 for (<'It), (eT), (Ia), respectivel.v. Also shown are theW decay branching fractions and the total w +w cross section as expected in the Standard l\ Iodel. The measured values of the \!\. branching ratios for different lepton families are in agreement with each other and with the Standard l\ Iodel expectations [99], and support the lepton universali ty hypothesis. The energy dependencc of the w +\vproduction cross section. shown in Figure 4.28, also agrees well with the Standard l\lodel and conforms to the Sl\1 picturc of gauge cancellations between t and s-channel e+e--+ \ \'+\\'- diagrams. The branching ratio results represent a significant improvement over previous measurE'ments at Fermi lab [23], whil<' the measurement of the e+e- - t YA,'+vv- cross section only became possible with the advent of LEP 2. 4.12.2 Measure m e nt of the I Yes I Matrix Eleme nt The fraction of \ V bosons decaying into hadrons is determined by the sum of squared Cabibbo-Kobayashi-l\ Iaskawa matrix clements over all kinematically allowed quark 4 .12 Summary o f the R esults 91 pairs, BR(Vv'~qq') BR ( W~[ I/) - (1 + as(~Iw)) L IViil2· (4.33) i= u .c:) = d.s.b Consequently. my bra nching ratios measurement puts a constraint on the elements of the CKi\I matrL'C 2.115 ± O.OG8(stat. ) ± 0.038(syst. ). (4.34) i = u .c;j = d.s.b This constraint can be interpret ed as a measurement of a CKl\1 matrix elem ent. The sensitivity is great<'st for the diagonal clements Vud and Vcs· The Vcs matrix element is difficult to measure direct!~' and is the more p oorly known of the two. I find it to be 1.033 ± 0.033 (stat.) ± 0.01 8(s.yst.), (4.35) where the st atitstical error includes the errors on ns and the oth er Vii· but is dominated by th e statist ical error on the W branching r atios. The m easurem ent has significantly larger sensitivity than other m ethods of m easuring Vcs not relying on t h e unitari ty of the CKi\I m a trix. 4 .12.3 Limits on the Invisible D ecays of theW Our m easurem ent of the t ot al w+w- cross sect ion assumes that all W s decay into known fermion d oublet s a nd can be detected. This is not the case in som e extensions of the Standard Model. For example, som e R-parity conserving supersymmetric theories [44. 45] predict a Vv decay into a chargino-neutralino pair. with the subsequent d ecay of the ch ar gino, \ V± ~ A. 0 \ ± ~ \ o \ Op±. In the region of SUSY p aram et er space, esp ecially difficult for direct searches, where the ch argino-n eutra lino m ass difference is small, m\o :::; m\ ± , and the neutralino is the lightest supersymmetric particle (LSP), such d ecays would produc<' no 92 Event Selection and Cross Section Measurement detectable particles besides the low energy leptou and would be virtually invisible. ¥.'+ \v- events in which at least one \i\' decay was of this kind. would be rejected by the standard \V+\·V- channel selections. This would lead to a systematic deviation of the apparent \ V+ \v- c-ross section from the Standarcll\Iodel prediction r S/1/ w 2 ( f S!II + f invisible) · W \V a meosur-Pd \I ' ll' (4.36) when? q~t•istble is tlw ¥.' width corresponding to such invisible decays. Addition- ally, w+w- events with exactly one W decay unobservable would contribute Lo the measured cross section of single-resonant \i\' production e+e----+ Wev as a'neasur-ed ~\ ·e v 2finvisiblefS!Il aS!II \\ .ev w w aS!IJ + (r&' + q~visible)2 ww· (4.37) provided the selection efficiency for them is the same as for genuine 'vVev events. Absence of significant deviations from the Standard Model cross section predictions for either ¥.'+ v,r- or single W production [100] allows me to put a limit on the invisible W width r invtsiblc \V < 17 1\ IeV (4.38) at 95<,/( confidence level, taking into account t he system atic errors of both the w +w and the single W production measurements. Measurement of theW Mass and Width 93 Chapter 5 Measurement of the W Mass and Width The sample of 13 data I obtained. using the selection procedure described above, is enriched in V\' +w- ---> qqq(j events. and this allows me to make precision measurements of the mass and the width of the \\' boson. Three methods have been used at LEP 2 for this purpose [101]. The first one. described in detail in Sections 5.1 and 5.2, C'xploits the strong dependencC' of the w +-vv- cross section in the threshold region vs ~ 2f\Iw. on the V\' mass l\h,·. Due to this dependC'nce. the value of the threshold cross section report.0d in Section 4.12 can b<.' used to give an C'stimate of theW mass. The rest of the chapter is devoted to the second. the most statistically powerful method of measuring I\hv ancl rw, which relies on the direct reconstruction of the invariant masses of the produced " ' bosons. by methods described in the previous chapter. The shape of the reconstructed mass distribution is ensitive to and can be used to measure the V\' boson mass and width with great accuracy. In this chapter I describC' the invariant mass reconstruction aspects of this measurement, as well as the statistical methods of estimating I\ Iw given the reconstructed invariant mass distribution. The results of applying these methods to the selected 13 C'vcnt samples and an analysis of the systematic uncertainties of the measurement are also presented. The third method of mC'asuring I\lv,r employs the fact that in the relatively low \V boost regime at LEP2, the energy of the fermions produced. ~ in the Vv rest frame. lies in a relatively narrow window around '!) in the laboratory frame JS -(1 4 v 1- -4I\l~ -) < 8 Ef <vs(1 - + 4 J 41\.I~") 1---. 8 (5 .1 ) Measurement of the W Mass and Width 94 The distribution is sensitive to T'vl w and can be studied for all flavors ofleptons in both w+w- ---t qq£v and w +w- ---t €v£1/ events [102]. Excellent lepton energy resolution is crucial for this measurement. The method is not practicable for w +w - ---t qqqq events because of the much larger experimental errors of jet energy measurement, and is not considered in this t hesis. The data collected by 1 3 at JS of 189 GeV, together with Monte Carlo sets generated at this energy. is used for data-Monte Carlo comparisons in this chapter as the most statistically significant dataset used in ~y analysis, except where indicated otherwise. 5.1 Threshold Cross Section Fit The energy dependence of t he w +vv- production cross section in the threshold region is mainly determined by t he 2-body Lorentz invariant phase space available to the two W bosons, a( yS, I\Iw) "' -1 s v 4l'vf2 1- ~' s (5.2) in the zero W width limit. This leads to a sharp rise in the cross section at values of ..,fS around ~ 2Mw as t he center-of-mass energy increases. Similarly, the cross section at a fixed JS in this energy region exhibits a strong dependence on Mw. A series of cross section curves aw ~ w - ( ..,JS) for different values of Mw is shown in Figure 5.1. The actual non-negligible W width and initial state radiation decrease the dependence somewhat, but the resulting sensitivity still allows one to transform a measurement of the w+w- cross section into a precise measurement of the W mass, given a Standard Model calculation of aw+w- ( .JS,Mw ). TheW mass is determined by maximizing t he likelihood (4.27) , computed with the 96A data. with the W branching ratios set to their Standard Model expectations, and the total vv+w - cross section replaced by the Mw-dependent Standard Model prediction. The much larger data samples collected at higher energies do not 5.1 Threshold Cross Section Fit -?=: .._.. I s ~ 1' I Q) + Q) .._.. t> 95 10 ~-----------------------------------------Mw = 80.0 GeV 9 Mw = 80.4 GeV 8 Mw = 80.6 GeV 7 ......... ......... .... 6 Mw = 81.0 GeV ....................... . ........... ·· 5 4 ... . ......... .... ·· 3 ....... ... ..... ...... ..... . .. . .. .... ....... .. ... 2 .. .. .. ...-············ ........ ... ··· ..... .. ··· .. .. ... -·· ...... ... ...... 1 0 158 159 160 161 162 163 164 {S [GeV] Figure 5.1: V\"+ \,\ '- cross section in the thrE>shold region. computed with GENTLE. Curves corresponding to differE>nt values of l\Iw are given. contribute to and were not used for the measmement. as the y..r+ w - cross section at higher energies is much less dependent on M,"r. The Sl\ I cross section was calculated by the GENTLE [65] semianalytical code. implementing the CC03 subset of the e+e- ~ f 1f 2 f3 f 4 Feynman diagrams. at the average center-of-mass energy of the 96A data period (y'S) = 161.34 GeV. T he cross section prediction as a function of theW mass. together with the threshold cross section and mass measurements, is shown in Figure 5.2. The fit gives Mw 0 ·50 GeV 80 . 94 + - 0.4~ ' (5.3) when' the errors arc statistical and correspond to a decrease of the likelihood of 0.5 Measurement of the W Mass and Width 96 8 crww = 2.6s~g:~~ ± 0.14 pb ,.........., ...0 0.. '---' 6 -- Mw = 80.94~:!~ ± 0.10 GeV ~ I ~ + 4 ~ I 1' Q) - +Q) 2 b 0 79 80 81 82 83 Mw [GeV] Figure 5.2: V\w w - cro s section at .,fS = 161.34 GeV as a function of l\1.,,·. Th<' horizontal and vertical bands indicate my measurements of a(e+e- --t w+ \v - ) and 1w respectively. relative to the maximal value. The next section discusses the systematic errors of this measurement . 5.2 Threshold Measurement Uncertainties There are two smuces of systematiC' errors for t his measurement. The first one is the systematic uncertainty of the cross section measurement proper. I t contributes an error on the measured \ \' mass of (5.4) 97 5.2 Threshold Measurement Uncertainties where (5.5) jt is the inverse of tlw slop<' of tll<' cross s<'ction curve in Figure 5.2. used to convert vv+\,Y- cross section uncertainties into ~Ivv ones. This error was evaluated to be 88 ~1eV and turns out to be tlw dominant one for this method of measuring :\Iw. Oth<'r systematic errors are due t.o the uncertainties implicit in the Standard ~Iodel calculation. V\' hen interpreting the measured cross section in terms of ~I,v. they translate into the W mass errors with the constant factor 11· • The \·\ '+v;- cross section depends strongly on .Jf;, so any uncertainty on the LEP beam C'nergy directly translates into an error on the measured ~Iw of (5.6) This corresponds to an <'rror of 27 1eV for the 96A data period. • ThC' \\' width is anot her parameter the cross section depends on. Fortunately. its current knowledge fw = 2.06 ± O.OG GC'V [23] is precise enough to make the corresponding systcma tic error (5.7) negligible compared to ones from other sources. • Higher-order radiativ<' corrections neglected in the e+c- --+ w + \1\r- cross section calculations I rely on are assumed to be of the order of (~a)rad.cor. ~ 2o/c at the threshold [29]. The r<'sulting uncertainty in the fitted ~Iw b b.~lw - JW(c' +e - --+ W + W - ) ( -D.cr ) rad.cor. · which corresponds to an C'rror of 34 MeV. (} (5.8) Measurement of the W Mass and Width 9 For the statistiC's of the L3 9GA data sample, all the systematic errors of the measurement are much smaller than the statistical one. Including the systematic errors added in quadrature, the result of the threshold cross section measurement of t-.Iw is 80.94~8:~~(stat.) ± O.lO(syst.) GeV. (5.9) The result is in agreement with the Standard t-. Iodel expectation and other t-.Jw measurements [23]. This determination of the \V mass is based on the assumption that the Standard l\Iodel describes correctly the e+e- --> vV+\i\'- cross section. This qualification is eliminated and the statistical precision of the measurement is greatly improved by the direct reconstruction method a pplied to the L3 data collected at higher energies, as described in t he following sections. 5.3 W Invariant Mass Spectrum Reconstruction The bulk of the V..'+ w - pairs collected by L3 were produced at energies significantly above the threshold. At those energies. the e+e- --> w+ ~r- cross section is not sensit ive to t-.Iw. but the increased <'vent rate a nd decreasing importance of phase space effects make it possible to measure t-.lw studying the resonant shape of the produced ~' bosons· invariant mass distribution. Above the threshold. the invariant mass spectrum of the produced W bosons is mostly determined by theW propagators in t he e+e-- w+w- matrix element, da(e+e-- ~r+ \v -) dm 1dm2 (5.10) Phase space and radiative corrections and selection and reconstruction effects distort the Breit-Wigner distribution. Teverthcless t he prominent invariant mass peak allows one to measure with great accuracy tlw W mass. which defines its position, and the 5.3 W Invariant M ass Spectrum R econ struc tio n 99 W width. which is related to its broadness. Directly at the threshold the W resonance is severely deformed by the phase space factor, which, together with the low w+w- cross section, makes the data collected at 161 GeV of little use in the d irect reconstruction measurement. Only the 96B, 97, and 98 data sets were considered in the subsequent analysis. 5.3.1 Direct W Invariant Mass R e construction Different estimators of the W invariant mass could be chosen to construct the spectrum to be analyzed. I find t h at the average W mass m 5 c obtained with the 5C kinematic fit , as described in Subsection 4.5.3, is the best for this purpose. No improvement in the expected W mass error was found when the whole analysis was repeated with alternative W mass estimators (W masses obtained with 4C kinematic fits, or rescaled 4C-fitted masses [103], or 2D generalizations of the above). 5.3.2 J et Combination Algorithm Another ambiguity in the construction of t he spectrum is the choice of the jet combinations to be used for the fit. I chose the L:'m ( ~m) algorithms for the w +w--+qqqq cross section measurement at 161, 172 (183 , 189) GeV, because they gave the highest rate of correct assignment of jets to primary W bosons (see Subsection 4.5.4). This decision has to be reconsidered for the mass measurement. Even the most efficient pairing algorithms has significant ( rv25%) misassignment probability at all energies. The mass spectrum contribution from the incorrect combinations constitutes a kind of irreducible background. possibly Mw-dependent, which might have a significant effect on the accuracy of the measurement. For that reason, the criterion of the maximal correct combination probability has to be complemented by an attempt to minimize the overlap of the invariant mass distributions for correct and incorrect jet combinations. Repeating the mass measurement using various jet combination schemes, I found the x2 algorithm to provide t he best compromise between these requirements, re- Measurement of the W Mass and Width 100 sulting in the smaller expected statistical error of the mass measurement. As shown in Figure 5.3(a). with the \ 2 algorithm. the incorrect combinations form a fiat distribution covering a wide range of masses. The overlap with the mass peak given by the correct combinations is much larger for the more efficient .6.m algorithm (sec Figure 5.3(c)). In about 409'< of w+V\r - - 4 qqqq events the correct jet combination is not found by the\ 2 algorithm. One could try to recover these events for theW mass fit by considering the combinatious with the second-highest P\2(\ 2 5c;5) as well. approximately 25o/t. of which map correctly onto the underlying W bosons. The m 5 c distribution for the second-best P\2(\ 2 5c; 5) combinations for the selected qqqq events is shown in Figure 5.3(b). The distribution has an acceptable signal-to-background ratio in the peak region and can be used in the mass fit. The combinations with the worst P \2 (-x·2 5c; 5) are seldom the right ones and are not useful for the measurement. 5.3.3 Event Quality Requirements With the exception of a small fraction of badly reconstructed events with low values of the P\2(\" 2 56 5) probability. most of the w+v..;- - 4 qqqq events. in which the right jet combination was found, lie in the fiat part of the kinematic fit probability distribution. This is not the case for the incorrect combinations or for the background events, for which tlle equal mass assumption is not correct and which typically have low kinematic fit probabilities, as shown in Figure 5.4. The expected error of the mass measurement was found to decrease after jet combinations with the fit probabilit.v of less than 5% were removed from the fit. Those are dominated by incorrect combinations which contain no information about the 'N mass. Removing them leads to a reduction of the background and increase of the signal-to-background ratio in the mass peak. The improvement is especially large for the second-best combinations. More than 70% of the best combinations and only 25% of the second-best combinations in genuine w +w- - 4 qqqq events survive the probability cut and are used in theW mass fit. The results of applying the probability 5 .3 W Invariant Mass Spectrum Reconstruction >Q) 101 Data c:=J WW~qqqq MC (corr.) (.9 Ui ~ WW~qqqq MC (incorr.) c Q) JJ 100 03!3 Background MC 50 60 80 m 5 c [GeV] >Q) • Ui c Data c:=J WW~qqqq MC (corr.) (.9 60 Q) ~ WW~qqqq MC (incorr.) E.::!§ Background MC > LU 40 20 60 80 m 5 c [GeVJ >Q) • Data (.9 c:=J WW~qqqq MC (corr.) c ~ WW~qqqq MC (incorr.) Ui 150 ~ Background MC Q) > w 100 50 60 80 m 5 c [GeV] Figure 5.3: Th<' m 5 C' distribution for (a) th<.> best P :. . 2 ( \ 2 5 c; 5) combina tio n. (b ) th<.> second-best P\ 2( \ 2 5c; 5) combina tion . and (c) the combina tion found with the 6.m algorithm. for selected w +w - - t qqqq candida tes. Correct and incorrect jet combina tions are shown separ a tely. M easurement of the W Mass and Width 102 l.() C\J 0 c) ----c: (j) • Data • Data OWW~qqqq MC (corr.) f.l2] WW~qqqq MC (incorr.) 0WW~qqqq MC (corr.) f.l2J WW~qqqq MC (incorr.) filll Background MC filll Background MC Q) > w 10 2 10 10 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 P(x 5 ~; 5) 0.6 0.8 1 P(x 5~; 5) Figure 5.4: Distributions of th<' 5C kinematic fit probability P\2( \' 2 5 c; 5) for the b<'st (a). and secoud-best (b) j<'t combinations in the selected w+w-----t qqqq candidates. cut to the L3 data samples, selected b:v the neural network selection described in Section .J.8. are summarized in Table 5.1. The expected and observed ratios of selected events are in good agreement at all energies considered. The reconstructed invariant mass distributions for the events selected at different energies arc shown in Figure 5.5. These distributions arc used in the v\i mass fit described in the following sections. Ntotal N(P 1 > 0.05 ) 'e:rp(Pl > 0.05) (P 2 > 0.05) 1e.rp(P2 > 0.05) D ata Period 96B 97 9 1393 468 57 333 935 39 4..J. ± 7 289 ± 17 949 ± 31 2 51 142 135 ± 12 40 ± 6 6±2 Table 5.1: ~umber of jet combinations selected in \i\f + ~r- ----t qqqq candidates in data, together \\·ith Sl\I l\Ionte Carlo expectations. Also given is the standard deviation on t.he total number of selected combinations expected in the Sl\1. 5.3 W Invariant Mass Spectrum Reconstruction >Q) --c <.9 ( /) >Q) • Data 7.5 • Data --c 0WW~qqqqMC lil.ilqq MC I§] Other Bckg MC Q) 103 <.9 oww~qqqq Me ( /) millqq MC ~Oth Bckg MC Q) > > w w 5 0.5 2.5 ~;..J 0 40 60 <-. 0 80 40 60 m5c [GeV] >Q) --c <.9 ( /) >Q) • Data 40 Q) --c • Data oww~qqqqMc <.9 oww~qqqqMC ( /) [])qq MC [S;IOther Bckg MC 80 m 5c [GeV] 10 Blqq MC ll190ther Bckg MC Q) > > w LU 20 5 0~~~~~~~~~ 40 60 80 60 m5 c [GeV] >Q) • Data >Q) • Data <.9 100 0WW~qqqqMC <.9 oww~qqqq Me ( /) ffjlqq MC ll190ther Bckg MC --c Q) 80 m 5 c [GeV] (i5 c 20 Blqq MC ll190ther Bckg MC Q) > > w LU 50 10 60 80 m 5 c [GeV] 60 80 m5c [GeV] Figure 5.5: Reconstructed m 5 c invariant mass distributions for the best (a.c,e) and second-best (b,cU ) P\2(\2 5c; 5) combinations selectC'd in 9GB. 97 aud 98 data. 104 5.4 Measurement of the W Mass and Width Monte Carlo Reweighting Method A straightforward approach to estimate l\Iw. given the reconstructed invariant mass spectra. would be to compare tlw distribution observed in data with those obtained by l\rlonte Carlo simulations at different (continuous) values of l\ Iw and choose the V..' mass value ma"Ximizing the likelihood of observing the data with the m 5 c-dependent probability density given by the i'dont e Carlo prediction. This approach is not fca~ible. considering the large amounts of l\Ionte Carlo events necessary for this measurement, but one can obtain l\Ionte Carlo predictions for any value of Mw by using a reweightecl l\lonte Carlo. To each event 1 m a \V+ \i\'- :.route Carlo event sample generated with the \\. mass set to l\I¢t". I assign a weight AI C' 1 2 3 1)j2' IM (l\ I v\' ,JJ, · P, · P; ·Pi (5.11) where J;f is the four-momentum of the jth outgoing fermion in event i , and M(Mw,pf , pf,pt.pJ) is the e+c-~fJ2 hL, matrix element. In the limit of infinite number of l\ Ionte Carlo events. differential cross sect ions corresponding to a new value of the \i\' mass l\hv will be approximated by the reweighted density of the l\Iontc Carlo sample generated at l\I~~c. Many other factors, besides the Born-level rnatrix clement. contribute to the total event weight which dC'termines an event ·s acceptance or rejection in von :-Jeumann 's scheme, such as phase space. initia l and final state radiation, and h adronization. ~one of these additional factors depends on Iw though , which is the basis of the validity of the rewei ghting method . This method of extrapolating l\lonte Carlo predictions as a function of l\hv can be straightforwardly generalized to other W boson parameters which affect the e+C'- ~ w+w- matrix element , such as theW width rw and w+w-,;zo couplings. Extrapolation in more than one parameter simultaneously, e.g., l\ Iv,' and fw. is also feasible Kith this approach. The reweighting method will be the basis of the measurements 5.5 Fit of t h e I nvaria n t Mass Spectra 105 of \\' boson parameters presented here and in Chapter 6. The EXCALIBUR [74] implementation of the four-fermion matrix element I usc for reweighting. allows me to vary the \\' mass and width as w<'ll a a wide range of w+\\'-~tfZ 0 couplings used in the ('akulation of the matrix clements. It also provides <1 choice of the subset of Feymnan diagrams used in the calculation. The fits described i11 the chapter were performed using the CC03 set, the same one used for generating the w +w- ~donte Carlo events I used. The uncertainty in the mass determination due to this simplification is addressed in Section 5.6. Attention should be paid to an important numerical feature of the method. The fluctuations in the fitted value of l\Iw due to finite l\Ionte Carlo. tatistics are controlled by the effective number of events N~{~ = (E wi) 2 / E w?. This number depends on 17 l\Iw and falls quickly as l\hv gets further away from M~~ • This could lead to increased errors due to finite l\IC statistics and. more importantly, to Mw-dependent biases (nonlinearity) of the fits based on the reweighted l\Ionte Carlo. I try to reduce these effects by using very large baseline l\Ionte Carlo samples, and by using mixed samples consisting of events generated with different values of l\1~~" spanning the range of the fit. whenever possible. A shown in the next section. the l\1onte Carlo statistics I used was adequate for m:v measurements. 5.5 Fit of the Invariant Mass Spe ctra An unbinned maximum likelihood fit is used to compare the data m 5 c distribution with reweighted l\Ionte Carlo predictions. Distributions for the two best jet combinations are fitted separately in the [70 GeV, 90 GeV ] range. Events with a reconstructed ma ·s below 70 GeV have a very small chance of coming from a vv+w --+ qqqq event with the correct jet ('ombination found , and do not carry information about the W mass. The likelihood maximized iu the fit is a product of the l\lw-clependent probability 106 Measurement of theW Mass and Width densities in m 5 c L(l\lw) = ( ·( ) dav.:w ( c daog c (5.12) IT c (l\1w )aww1(l\1) · w + a c l\1w d n1 scm ;,1\lw )+ d m sc(m ;)). 5 5 09 evaluated for all data events in the fit rauge. The factor c(l\1w) = a~!(~.sured / avvw(l\1w ) normalizes the Mvv-depenclent accepted ~ '+ ~r- cross section to the measured one. After this correction the results of the fit depend only on the shape of the m 5c distributions and not on the number of events in them. The total and differential cross sections aw\V (l\1w), ~;~t't' (m5 c ;. l\Iv;). and ~:,~t· (m 5 C i) arc evaluated numerically with the box method [104]. A bin (box) with a width 6. 1 is constructed around each data ev<'nt with the reconstructed average invariant mass of m 5 c, . The l\1onte Carlo predict ion for the accepted cross section and the differential cross section at m 5 ci · rcwcighted to lw. are then given by O\.vw (l\1v;) dawv.: dm5C (m 5c ,.1\lw) (5. 13) (5. 14) where the latter sum is over the Monte Carlo events inside the ith box. The weights arc unity for background l\lont<' Carlo events and are given by the reweighting formula (5.11) for \iV+\v- l'..Ionte Carlo events. The reference ~' mass l\ I~/ is chosen to be in the middle of the ~, mass range covered by the w+ ~r- l\Ionte Carlos used in the rewcight ing. The bins [m5ci-~6..m5ci+~6.] were centered around the data events m 5c , and had equal size of 250 i\IeV for th<' w+w - l\1onte Carlo, small enough to provide an adequate description of the Breit-Wigner resonant shape. Bins of 2 GeV were used for background Monte Carlos, large enough to guar antee sufficient l\Ionte Carlo statistics in ec1ch background box. Oth('l' box algorithms. both data-d<'pendcnt and fixed. were tried. and no significant difference was found either in the results of the fits on data, 5.5 Fit of the Invariant Mass Spectra 107 or in expected accuracy of the measurement. I tr<:>at the mass distributions of the best and second best jet combinations as independent. i.e., the likelihood minimized in the fit is given by the product of two likelihoods (5.12). and the l\IL estimate of the statistical error is obtained by solving numerically (5.15) Ou<' can question th<' validity of this approach , as both combinations in an event arc formed with the sanH' jets. and tlH' rc•sttlting two m 5 c masses are strongly correlated on an evC'nt-by-eve11t basis. A priori. the <' correlations should lead to a correlation between the results of the fits of t.h<' two distributions. This turns out not to be the case. At most on<' of the two combinations maps correctly onto the two W bosons and carri<'s information about th<' vV mass. contributing to the result of the fit of the corresponding distribution. Tll<' other one (or both) lie in the flat background of incorrect combinations and do not significantly affect the result of the fit in which it is used. Therefore, I <'xpect the results of the fit s of the first and second combinations to b<' C'ssentially uncorrelated. This property of the fit was confirmed numerically. 50K samples of Js = 1 )9 GeV l\Ionte Carlo events were obtained by randomly combining signal and background l\ Ionte Carlo events in numbers given by the Poissonian distributions with the means corresponding to tll<' Sl\I expectations for the 98 data set. No structure is seen in the scatter plot of the l\ IL fit results of the best and second best jet combinations, performed with the samples, as shown in Figure 5.6. TlH' correlation of the fits of the two jet combinations was evaluated to be l?12 J ((m1 - (m1) )2) (( m2- (m2) )2) - 0.01 ± 0.03, which allows me to safely neglect t h<' correlation. (5.16) M easure m e nt of the W M a ss a nd Width 108 2 ~ ... . ... 0 t- -2 tI I -2 0 I Figure 5.6: Correlation of the best and second best jet combination Mw fit results for ;..route Carlo event samplcs. The ~..I onte Carlo rewcighting f\IL fit described above gives consistent (asymptotically unbiased) estimates of ~Iw and the statistical uncertainty of the measurement. All effccts affecting the resonant shape are taken into account exactly by the fit. provided they are correctly modclled by the f\ Ionte Carlo generators. Thc consistency of the estimators was checked by applying t he fit to a large number of f\Ionte Carlo event 5.5 Fit of the Invariant Mass Spectra 109 samples. constructed in the way described in the previous paragraph, using \h,'+vv1\lC events simulated at different value's of 1\lw. > Q) ~ 0 ~ 100 b !!: ~ 1.02±0.0:2 -+±8 MeV I u ~ !!: ~ ! 0 l I lI 81000 t T t 80250 80500 80750 I I -1 00 80000 MwMC fMeVl Figure 5. 7: Average fitted W mass for small l\lonte Carlo samples with different Mw. Statistical uncertainties of the averages and the results of th<' linC'ar regression are a lso given. The average fitted W mass and its statistical uncertainty for each Mw point used is shown in Figure 5. 7. Linear regression of the data with (1\ l~~/') - 80500 :\leV gives a linearity coefficient a b = 4± a(l\1e~n - 80500 !\leV) + b (5.17) 1.02±0.02 compatible with 1 and a negligible bias leV. At each l\hv point the spread of the fit results was compared to the distribution of the fit errors given by (5.15), as demonstrated in Figure 5.8. The average fit C'lTOr agn'es well with the standard deviation of the fit results at each l'd._,,. point considered, and, therefor<', gives a good estimate of the statistical uncertainty of the mass measurement. The technical checks described above were also performed for the \h.' width fits. Measurement of the W Mass and Width 110 >Q) Mean ~ RMS 80.50 .1240 0 ~ ~ 1000 c.. E cu CJ) 500 0 L---~--~~----~----~----~----~--~-----L~=-~--~ 80 80.2 80.4 80.6 80.8 81 Mmeasured [GeV] >Q) Mean ~ 4000 RMS .1241 .8178E-02 ~ Q) c.. E cu CJ) 2000 0.12 0.14 ~Mmeasured [GeV] Figure 5.8: Distributions of the Mw fit results (a) and fit errors (b) for small !\Ionte Carlo samples corresponding to !\Iw = 80.500 GeV. :\leans and standard deviations of th<:> distributions are given. As shown in Figure 5.9, the results arc again compatible with a negligible bias and a linearity of 1. Also for the \t\' width fits, the fit error gives an unbiased estimate of the statistical uncertainty of the measurement (sec Figure 5.10). The fit procedure described above was applied to the samples of ~r+ vv -- qqqq eandidates selectf'd in 96B. 97 a nd 98 13 data sets, df'scribecl in Table 5.1 , giving the invariant mass spectra shown in Figure 5.5. Th0 results of independent and simultaneous !\Iw and rw fits with these data are summarized in Tables 5.2 and 5.3. The results of independent and simultaneous Mw, rw fits. as well as results 111 5.5 Fit o f t h e Invariant M ass Spectr a >Q) ~ ~ 100 ~ ~ I 0 -- b -- 1.02±0.05 -4 1 ± 26 MeV u ~ r.! 0 -100 1500 2000 2500 r WMC !MeV) Figure 5.9: Average fitted W width for small Monte Carlo samples with different fw. Statistical uncertaintiC's of the average, a nd the results of the linea r regression are also given. Data Period 96B ~ 1. 000 _ 0::1 97 0 98 All data 80 · 587 -O. L2!J 0 603+0.107 . -0. 107 27 U() 563+0.209 . -0.209 +0.130 Table 5.2: Results of independent l'vlw and r w measurements with direct reconstruction of w +w - ___. qqqq events. The errors arc statistical. obtained with different L3 d ata sets. arc in good agreement with each other. The mass-width correlation for simult aneous fit s is smalL which agrees with the intuitive understanding of the fit using different features of the reconstructed mass spectra to determine .l\1\\· and r\.-v· The observed statistical errors a re compa tible with l\IC expect a tions for event samples of the size observed in the data. 112 Measurement of theW Mass and Width ~2000 Mean RMS ~ 0 2.120 .3187 ~ (/) Q) a. E co (f) 1000 3 2 r measured [GeV] >Q) Mean RMS ~ 2000 .3198 .3371E-01 ~ ~ Q) a. E co (f) 1000 0 U-------~--------~----------------~~~------------~ 0.2 0.3 0.4 0.5 ~rmeasured [GeV] Figure 5.10: Distributions of the rw fit results (a) a nd fit errors (b ) for small l'donte Carlo samples corresponding to rw = 2.110 GeV . :t\Iean s and standard deviations of th<' distributions are given. 5.6 Direct Reconstruction Measurement Uncertainties The fit method d escribed c-orrects implicitly for many effect s that affect the invariant m ass distribution provided those arC' modeled correctly by the Monte Carlo simulations. As already discussed in Sect ion 4.11 possible sources of biases of the V\T mass measurement are imperfections of the :t\Ionte Carlo codes used to simula te sign al a nd background events, and incorrect simulation of the detector response. affecting b oth 5. 6 Direct R econstructio n Measu rem e nt U n certaint ies Data Period 9GB 97 98 All data Mw [GcV] 81.032_o::3oo 0 .559+0.203 - 0.202 80 .58o+O .J3 <1 - 0.136 0·107 80 .603+ - 0.108 113 Ql\I\\"r\\ 1.16 ' -1:02·1 0·506 1 .830+ - 0.·137 ? 383+0.381 ~· -0.351 +0.293 2.148_0.272 0.255 0.033 -0.050 -0.030 Table 5.3: Results of simultaneous i\lw and fw measurements with direct reconstruction of vv+w·- - qqqq events. The errors arE' statistical. The correlation between the measurements is also given. kinds of IC events. The corresponding systematic errors were evaluatE'd by performing t lH' mass measurements on large l\lonte Carlo samples generated with modifiE'd model parameters or different l\IC models. with unchanged baseline l\lC samples. The systematic errors were reevaluated for each set of data and l\Ionte Carlo usE'cl in my analysis and were found to be consistent with each other. I assume errors from any given source. with the cxcE'ption of the finite l\IC statistics error. to be fully correlated between data taking periods. I arrive at the combined systematic error by averaging errors for particular data samples, with weights given by the statistical errors of the relevant measurements. Throughout this section I quote estimates of systematic errors for both the \\' mass and W width measurements. with the latter given in parentheses. 5.6.1 Modeling of the D et e ctor R e sponse To evaluate possible systematic errors due to mismodeling the detector response. I follow the method used to calculate similar systematic errors in the measured production cross section. A large sample of hadronic Z0 events was used to evaluate biases and resolution data-MC differences in the determination of jet energies and directions. The measur<."ments were repeated upon applying the resulting corrections to l\lonte Carlo signal and background events. The changes of the fit result that these corrections led to were taken as E'stimates of the corresponding systematic errors. Corrections of the jet energy scale and resolutions led to i\Iw (fw) shifts of 10 (5) M easure m e nt of the W M ass and Width 114 and 0 (10) l\IeV respectively. Alternatively, neglecting data-MC subdetector calibration differences changes the measured V\1 mass (width) by 15 (20) l\ IeV. The latter was chosen a a conservative estimatc of the effect of possible detector miscalibration on t hc measurements. Th<' same procedurc w<'ls applied to angular biases and rcsolutions. e and ¢ corrections to the dirertions of rcconstructed jets change the fitted \\' mass (width) by 7 ( ) l\IeV. Smearing jet directions leads to a change of 13 (26) l\IeV. I usc the quadratic sum of thcse numbers as an estimate of the effect of possible jet dircction mismeasurement. 5.6.2 Mode ling of the w +w - Signal Prope rties T lw unccrtainties of the l\Iontc Carlo models of w+ y.,r- production. which affected the cross section mcasurcment. also wcrc considercd from the point of view of t lw measurement of the \ Y mass. • Thc effect of the CC03 approximation was again evaluated by reweighting Monte Carlo w+ y.,r- events with the ratios of t he full and CC03 matrix elements. and redoing the mass fit with the resulting sample of weighted l\IC events. The result differs by 6 (9) l\ IeV from the one obtained without reweighting. • Initial state radiation has an important influence on the reconstructed \\' mass spectrum. Photons radiated by the incoming lepton beams typically escape detection and carry away on the order of 2 GeV of energy per Vv'+w - event at 189 GeV. This loss is not accounted for by the kinematic fit, which leads to an upward shift of about (5.1 ) in the dijf't mass spectrum. Fortunately, the uncertainty on the fitted \N mass is much smaller than the avcr<'lge energy loss, as reliable QED calculations are available and implemented in the l\Ionte Carlo code that was used to simulate 5.6 Direct R econstruction Measurement Uncertainties 115 w+w- events [29. 101]. These' theoretical calculations have an intrinsic accurac~' on (E1sR) of e:1bout 15 }.leV, which was taken as an estimate of possible' systC'matic error on the \\' mass clue to mismodeling of initial state radiation. DuE' to the effcctiv<' convolution of the ISR sprctrum with the relatively wide Breit-Wigner 'vV resonant peak, I expect the uncertainty of thr measured W width due to thr mismodeling to br negligible. • Similarl:r, emission of hard gluons by the final state quarks aff<>cts the reconstruction of dijet invariant masses. The fo\Jr jet hypothesis. assumed by thr jet combination a lgorithms and kinematic fit I used. doesn't hold for events with significant final state radiation. The succrss rate of thr jC't combination algorithm is significantly lower for the five-or-more jet events. For the subset of events with t he correct jet combination chosen, the reconstructed mass spectrum still differs from the one obtained with four-jet events, due to worse mass resolution. Nevertheless, this doesn't result in large systematic errors as the description of such events by the }.lonte Carlo models I used was found to be adequate for my measurements. To evaluate a possible contribution to the systematic error. I repeat the mass (width) fit rejecting different fractions, up to 12%, of the data events with the highest values of y,15 . This changes the fit result by up to 9±20 (13±54) }.1eV which I take as an estimate of the corresponding systematic error. • As for the cross section measurement. the systematic errors due to modeling of fragmentation and interaction between two W---> qq' hadronic systems wen• evaluated by comparing a large number of }.Iontc Carlo models not rejected by low energy data with the default model used for the measurC'ment. To evaluate the systematic error clue to modeling of hadronization. I perform a W mass fit using <1 HER'vVIG f\Ionte Carlo sample as '·data·· and the default lonte Carlo sample with JETSET / LU ND fragmentation as the baseline f\ IC. I assign the difference of thC' fit result from the nominal \ V mass ns<'d to gencrat<' the HER\ VIG sample. 45±20 (93±59) f\IeV , to the systematic error of the Measurement of the W Mass and Width 116 measured \"/ mass (width) du<> to errors in modeling of fragmentation. In the same manner. the 50±21 (47±61) ~lt>V bias observed when fitting a ~1C sample generated without simulating Bose-Einstein correlations, relative to the default l\lC sample. was taken as an estimate of the corresponding systematic error on theW mass (width). Tht> biases found for various color reconnection models, varied in tlw 30 80 (20 80) MeV range for theW mass (width). each with a statistical uncertainty of approximately 20 (GO) l\IeV. I take the bias observed for the Sjostrand-Khoze I model. 54±20 (73±59) l\1eV, as an estimat<> of the error chw to modeling of color reconnection. 5.6.3 Modeling of the Backgrounds • To evaluate the sensitivity of the measurement to possible mismodeling of QCD background events, I redo the fit after giving qq Monte Carlo <>vents y:j 1-dependent weights determined with Z0 data. as described in Section 4.11. This leads to a change of the fitted \11;' mass (width) of 0 (20) l\leV. Similar shifts are achieved by increasing the accepted qq cross section by 5% in the fit. without changing the reconstructed mass distribution of qq Monte Carlo events, and by replacing the qq I\Ionte Carlo distribution with an appropriately rescaled one obtained with data collected at the Z0 peak. • Also, repeating the fit while changing preselection cuts, which is equivalent to changing the fraction of background events in the fitted sample. led to a 9±21 (23±59) !\leV shift of l\Iw (fw ). I combine this shift with the result of the y 31 reweighting to arrive at an estimate of the systematic error due to possible mismodeling of the qq background. • zozo events constitute a nother non-negligible background component in the event sample used for the W mass fit . For an estimate of the systematic error on ~ Iw (rw) due to modeling of this background, I take the shift of the fitted 5 .6 Direct Reconstruction Measurement Uncertainties 117 W mass (width) of 1 (6) t\IeV which I obtain by changing the Z 0 Z 0 accepted rat<.> by ±5o/c. 5.6.4 Beam Energy Uncertainty A wrong estimate of the e+e- collision energy would force t he kinematic fit to overcorrect and bias tllC' reconstructed dijet masses. Thus an uncertainty of the LEP beam energy directly translates into a systematic error on the W mass 6t\Iw ~ l\Iw - E 6Ebeam· (5.19) beam The beam energy uncertainties reported in Table 3.1 lead to an error of 17 l\IeV on the measured V. 7 mass. This estimate was confirmed by repeating the measurements after changing the value of .jS used by the kinematic fit. by the amount of the LEP bE'am energy uncertaint:v. and observing the resulting change of the measw·ements. This procedure also leads to an estimate of the systematic error on t heW width of 5 l\IcV. 5.6.5 MC Statistics Because of the finite number of t\Ionte Carlo events I used, the box method provides an imperfect approximation of the differential cross section used in the fits. The position and the width of the Vv mass peak is defined by the l\Iontc Carlo with an uncertainty of approximately r,x f.JNMC · where N 11 tc is the MC statistics used. To evaluate the systematic error due to this uncertainty, I split my l\1onte Carlo event samples into several (2 20) parts and repeat th<.> mass fits using t h ese subsC'ts and the full amount of real data. The spread of the result gives an estimate of the systematic error of 7 (25) !\leV. combining all data taking periods. The estimates obtained using different numbers of subsamples are consistent with each other. M ea s ure m en t of t h e W M ass a nd Width 11 5.6.6 Othe r Che cks A number of other checks of the fit were performed. · o statistically significant changes were observed while changing technical parameters of the fit. such as box widths and the algorithms used to construct the boxes. The results were also stable with respect to changes of the sample used for the measurement, namely variations of the neural network output and kinematic fit probability cuts, and the range of the m"c distribution used for the nwasurement. ~o additional systematic errors were assigned as a result of these checks. 5.6. 7 Summary of the S yst e matic Unce rtainties The systematic errors of the W mass and width measurements arc summarized in Table 5.4. Source of uncertainty Jet energy measurement Jet direction measurement CC03 approximation ISR modeling FSR modeling Fragmentation l\Iodeling of BE correlations l\Iodeling of color reconnection qq background ( QCD modeling) qq background (other) Z0 Z0 background Ji; uncertainty l\1C statistics Total l\ Iw error [l\IeV ] 15 15 6 15 9 45 50 54 0 9 1 17 7 93 fw error [l\ IeV ] 20 27 9 0 13 93 47 73 20 23 6 5 25 138 Table 5.4: Systematic errors of the measurements of l\ lw and fw itemized by the source of the uncertainty The quadratic sum of all contributions is also given. Possible detector mismodcling gives rise to small systematic errors owing to tight constraints on detector miscalibration and misalignment that one can obtain from 5. 7 Summary of the R esults 119 low energy qq data. Also small are systematic errors due to uncertainties of the background simulation. due to the high purity of the event sample used for the measurement and thE' low infiuencc of the fiat background m 5 c distribution on thc measuremE'nt of the W mass and, to a smaller degrec. the width. The dominant systematic errors in the determination of both \\' mass and width comc from uncertainties of the simulation of hadronic structure in V\7+w ----+ qqqq events. The fragmentation and color n'cmmection errors are due to poor theoretical understanding of the non-perturbative QCD processes involved, while the BoseEinstein correlations error is caused by the intrinsic difficulty of simulating the quantum mechanical phenomenon with probabilistic ~Ionte Carlo models. The systematic errors are smaller than the statistical ones for m.v measurements, but they will become morE' important for the final analysis of the total LEP 2 data sample. 5.7 Summary of the Results This section summarizes the results of my analysis of the dijet invariant mass spectra reconstructed with four-jet w +V\r- ---+ qqqq events. I measured theW boson mass and width to be l\ I w 80.60 ± O.ll(stat.) ± 0.09(syst.) GeV (5.20) rw 2.15 ± 0.28(stat.) ± 0.14(syst.) GeV. (5.21) Similar measurements were done by L3 wit h W invariant mass distributions r<'constructed with semileptonic V\'+V\'- events [105]. Their results Mw 80.22 ± 0.12(stat.) ± 0.07(syst.) GcV (5.22) r,v 2.22 ± 0.30(stat.) ± 0.17(syst.) GcV (5.23) Measurement of the W Mass and Width 120 arc in agreement with my measurements within their statistical and systematic errors. I combine them to arrive at the best estimate of theW mass and width using the L3 high energy data collected during 1996- 1998. Another independent measurement of these parameters using lepton energy spectra is available [102], but its significantly lower precision and p artial correlation with the direct m 5 c reconstruction qq:fv measurements make its inclusion impractical Some of t he systematic errors of the qqfv measurement, described in [105], are positively correlated with those of the qqqq one, e.g., the LEP beam energy and ISR modeling errors. The contribut ion of such fully correlated errors into the total systematic error is small for both channels. Some larger ones, for example the errors due to the detector response and fragmentation modeling, are related for both measurements, but the relationship is non-trivial and it is impossible to determine either the size or the sign of the possible correlation. The largest errors, due to Bose-Einstein correlations and color reconnection modeling, are present only for the qqqq measurement. Therefore, assuming all systematic errors uncorrelated is a valid approximation when combining the qqqq and qqfv measurements. I checked that this simplification can lead to an underestimate of t he combined systematic error of at most 10%. The result of the combination is Mw 80.43 ± 0.08(stat.) ± 0.06(syst.) GeV (5.24) rw 2.18 ± 0.20 (stat.) ± 0.11(syst.) GeV. (5.25) The direct reconstruction W mass measurement can be combined with the ( uncorrelated) threshold one, (5.9), to obtain the best estimate of theW mass Mw 80.44 ± 0.08(stat.) ± 0.06(syst.) GeV. (5.26) The measured value of the W mass is in good agreement with measurements by other LEP and Tevatron experiments [106, 107, 108, 109, 110. 111], as well as indirect determination using other precision electroweak measurements [112]. 5. 7 Summary of the R esults 121 The measurement of the\\: width agrees well with the Standard l\Iodel prediction (2.13) using the measured value of the\\' mass. and with other direct and indirect measurements [23]. 122 Measurement of the W Mass and Width Search for Anomalous Triple Gauge Boson Couplings 123 Chapter 6 Search for Anomalous Triple Gauge Boson Couplings The vv+w- ---+qqqq-enridwcl event sample I select can also lw used to study triple gauge couplings involving the v\' boson. The total w+\1\r- production cross section already provides some information about the structure of the c+e----+ w+w- matrix clement. Delicate cancellations between the Feynman graphs shown in Figure 2.2 arc neceBsary for good high-energy behavior of the V\7 +\i\7 - cross section. A coupling set different from the one predicted by the Standard 1odel would destroy the gauge cancellations and force some partial-wave amplitudes to violate the unitarity bounds above some energy threshold. Already at the energies available at LEP 2. non-S.l\1 triple gauge boson couplings would result in significant deviations of the measured w+w- cross section from the Sf\I predictions, as illustrated in Figure 4.28. Still more information about the gauge couplings is contained in the differential V\ 7+\l\ 7 - cross section. Change!:> of the couplings affect each e+e----+ w +w - helicity amplitude differently and generally modify the angular dependence of both the production rate and polarization of the \".7 bosons. The polarization of the W bosons determines, in turn. the angular distributions of their decay products relative to th<> W flight direction, through the V-A charged weak current int<:>raction. To access all available information about e+e- ---t w +w - amplitudes, on<:> would study th<> full four-fermion production differential cross sections. In th<> limit of zero W width and no initial state radiation, both \\' momenta in the rest frame and the fermion momenta in theW bosons' rest frames are constants determined by jS and I\1w. Provided the incoming lepton beams have no significant 124 Search for Anomalous Triple Gauge Boson Couplings .0$ • .,·!·. - 8* f e f 8*+ $:0· transverse polarization, the final st a te is invariant with resp ect to rot ations around the bea m axis. With thes<' assumptions the final four fermions' phase space is completely described by the set of t he five deca.v an gles shown in Figure 6.1. The polar angle 8w d etermines the direction of the negatively ch arged V\' boson with resp ect to the direction of the incoming electron. Two more pola r (tJ 1 • tJ2 ) and two azimuthal (¢ 1 • ¢ 2 ) a n gles d escribe the fermion flight direction in the respective Vv boson rest frames. ·w hile one would ideally stucl.v the full fiw-dimensional differentia l cross section da(e 1 e - ~ r, hf3f4) · · · "bl · d cos e we1 cos e1 dt/> 1d cos e2 dt/> 2 , 1t 1s unposs1 e m · to reconstruct a 11 o r· t h e an g 1es m · practice w+w - events of any category. Measurement difficulties are especially large for fourj et events and r estrict the an alysis to the -vv- pola r angle 8w . In semileptonic w +w events it is possible to d etermine the charge of t he lepton with high reliability. and that allows one to reconstruct the decay phase sp ace of the leptonically-decaying Vv boson. Other advantages of this d ass of \J\r+ w - events ar e the ab sence of combinatoria l problems and the straightforward identification of the ncgativel.v charged W boson. \ Vhile both lepton ch arges are known in f v.Cv events, the flight directions of th e \!\' bosons cannot be determined unambiguously, which limits tlw usefulness of su ch events for m easurem ents of anomalous couplings. This ch a pter describes my st udy of t he cos 8w distribution. including the deter- 6.1 Parametrization of Trilinear Couplings of theW Boson 125 mination of the flight direction of the v\'- and the selection of \\'+\\ 7 ----+ qqqq events used for my measurements, as well as the maximum likelihood fit used to obtain estimates of the gauge couplings given the reconstructed distributions in cos 8w. The results of applying these methods to the selected L3 event samples and an analysis of the systematic uncertainties of tllC' measurement are also presented. The data collected by L3 at ..jS of 189 GeV are used for data-Monte Carlo comparisons in this chapter as t he most statistically significant dataset used in my analysis. except where indicated otherwise. 6.1 Parametrization of Trilinear Couplings of the W Boson To check for and quantify possible deviations of the e+ e----+ w +w - matrix element from t h e sum of Sl\I amplitudes shown in F igure 2.2. one has to consider a more general form of the matrix element. The most general form of a Lorentz-invariant Lagrangian, describing triple gauge boson vertices involving the \ V boson [113] . contains seven arbitrary constants for each neutral gauge boson. + ig~' f 1wpu((8PH '- 11 )ll '+" - 'I-F -J.L(8P H!+"))Vu + !i?, · c:'wpu ll ·+ n ·- v - ~ l v- lr+J.Lf"Pa.B y: . 11 v pu ?l\J2 Pll v a,J - - w (6.1) where the coupling strengths for V=1 ,Z0 are 9"1w•w - e = J47raem (6.2) gzow ' \V- ecotBw. (6.3) 126 S earch fo r A n omalo u s Triple Gau ge Boson Couplings The Standard Model Lagra ngian can b e derived from (6.1 ) by assuming (6. 4) and the rest of the couplings vanishing. Linear combinations of the coupling const ants in (6.1 ) det ermine t he static electromagn etic properties of the W boson, n a m ely the electric ch ar ge (6.5) the m agnetic and electric dipole m om ents (6 .6) (6.7) a nd the electric and m agnetic quadrupole moments (6. 8) Afw (6 .9) It is difficult to interpret exp erimental d ata in terms of 14 free complex parameters. Therefore t h e triple gauge b oson couplings par am eter space h as to be r estricted by a pply ing addit ional requirem ents on t h e Lagr angia n (6.1 ). I assume all coup lings to b e real. and disr egard those inducing C or P -violating inter actions. This leaves a set of six p ar am et ers gy , Kv. Av, wh ere V ="(,Z0 . l\lloreover , gJ is fixed to be 1 by t h e electr om agnet ic gauge invaria nce requirem ent. The spa ce of the rem a ining five par am eters can be fur t h er reduced by classifying all possible SU(2)xU(1)-invariant operators involvin g triple ga uge b oson vertices, according to t heir dimension . Operators of higher dimensions ar c exp ected t o be suppressed by p owers of t h e presu mably large "n ew physics" scale. The couplings corresponding to t he rem aining oper ators 6.2 R econstru ction of the w- Direction Distribution 127 become constrained. In the scheme adopted by the LEP experiments one considers models with a relatively light Higgs boson. in which case the ··new physics" and the SU(2)xU(1) breaking happen independently. and operators giving rise to triple gauge boson vertices can be constructed with only Standard Model vector and scalar fields. The relations between lower-dimensional operators of such kind provide two more constraints [114] on the vw+w- couplings: (6.10) (6.11) where b..g,z= 9Jz - 1. (6.12) The additional constraints reduce the number of free parameters to three. I choose the set of b..gf , b..n'Y. and .>."~ to parametrize possible deviations of the triple gauge boson vertices from the Standard Model form. Other parametrizations. inspired by different realizations of "new physics." are possible [115]. 6.2 Reconstruction of the w- Direction Distribution In my analysis I study the angular dependence of w +V\r- production. I reconstruct the distribution of the polar angle of w - bosons in three steps. First I combine the jets into pairs to form candidate primary V\' bosons. Then I try to find the dijet combination which is most likely to form a negatively charged W boson. To that end I estimate the charges of the jet pairs using a jet charge algorithm. At the last step I tighten the event selection requirements to avoid dealing with events where my measurements have especially large uncertainties. The problem of combining jets is simpler for this measurement than for the mea- Search for Anomalous Triple Gauge B oson C o uplings 128 surement of the 'N mass. The distribution in cos 8w given by incorrect jet combinations always has a large overlap with the one given by correct combinations. The best way to reduce the background given by incorrect combinations is to maximize the success rate of the combination algorithm. For the reconstruction of the cos 8v,· distribut ion, I choose the algorithms I used for the w+ -vv -~ qqqq cross section measurement, the I.:'m (.6.m) algorithms a t 172 (183. 189) GeV, as they give the highest rate of correct assignment of jets to primary V..' bo ons (see Subsection 4.5 .4). I don't use the data collected at the W-pair production threshold for my measurements of triple gauge boson couplings. At low energy a W boson is reconstructed as a pair of back-to-back jets with typically low total momentum. Because of that both the magnitude and th<' direction of th<' momentum vector can be measured only with very large errors. 1\loreover. at the threshold th<' s-channel amplitudes containing tripk gauge boson vertices are supprC'ssed by a fact.or of !3w, which makes the total cross section at this energy less scnsitiv<' to possible anomalous vw+w - couplings. On th<' contrary tlw contribution of s-channel diagrams is larg<' at higher energies. The direction of the flight of the V\' can be measured with an uncertainty of only a few degrees. provided the correct jet pairs were constructed by the jet combination algorithm. This uncertainty is small compared to the characteristic width of tll<' cos 8w distribution and doesn't impair the sensitivity of the distribution to anomalous couplings. 6.2.1 J et and W Boson Charge E stimation The most serious uncertainty in the reconstruction of cos 8w. in an event with the correct jet combinat ion found. is du<' to the difficulty of dC'termining which of the two \ V boson candidate's represents t h<' primary v..r- . Incorrect \ i\. charge assignments produce cos 8w ~ cos 8w migration of events. This results in a background whose distribution is fully anticorrelatcd with the one given by the correct combinations, with respect to changet> of gauge boson couplings. Thus a vv- charge confusion probability of x would lead to a decrease in the sensitivity of the measurement by 6.2 Reconstruction of the v..r- Direction Distribution 129 a factor 1-2x, which is very significant for the values of x which can be achieved in practice. The ch arge of a W boson would be trivial to d educe giVen t he char ges of t he quarks int o which it decayed. Assuming quarks i and j were produced in t he d ecay of a Vv boson . the sign of the combina tion of quark char ges qi + qj - I: qk (6.1 3) kf.i,j would coincide with the sign of the charge of t he original W boson. Unfortunately, one can only attempt t o estimate t he quark charges given informa tion about t he d et ected products of their fragmentation. The jet char ge estimat or I used for this purpose is inspired by t he p arton cascad e picture of fr agmentation. In the simplest model of t his kind, t he Field-Feynman independent fragmentation model [116], the informa tion about the ch arge of t he fragment ing quark is carried by a few higher-momentum p articles produced during t he fr agmentation , which have a higher cha nce t o be produced at t he top of the parton cascad e a nd t o a ct ually contain the original quark. Therefore, an estimator of the quark cha rge could b e given by an average of char ges of particles b elonging to a jet, wit h higher weights given t o fast er p articles a nd lower ones t o less ener getic ones. Given the list of p articles combined into a j et by the Durham a lgorit h m (seC' Subsection 4.5 .1). I define t he char ge of the j et as a normalized weighted average of ch ar ges of all tracks in t he list L w(pi) qi L w(pi) ' (6.14) with the weights given by a p ower of the projection of the t rack m oment um on the j et axis (6.15) Only tracks wit h a minima l Pr of 150 MeV are included in t he average. T he optirnal S earch for Anomalous Triple Gauge Boson Couplings 130 value of the exponent was found to be 0.5 in a dedicated study of the jet charge algorithm [117]. The precision of tlw quark charge estimates obtained with this algorithm iH low because of the fluctuations intrinsic in the parton cascade. The performance is further degraded by neutral particles produced in the shower and particle decay products. Nevertheless it still turns out to be possible to correctly identify tlw charges of the W bosons in a large fraction of ~'+ \v----+ qqqq events. 6.2.2 Event Selection and Jet Combination Starting with the sample of four-jet c>vents accepted by the neural network selection described in Section 4.8, I try to increase the fraction of correct ly identified \\·bosons by improving or rejecting some> jet charge measurements which arc likely to be incorrc'ct. First I reject a small fractiou of events without TEC information, recorded in the beginning of each LEP fill. J et charges cannot be reconstructed in such events and they an' not used for the measurement . Also I consider unreliable any jet charge obtained with less than 3 tracks. Such jets are concentrated at low polar angles. which is an area of low efficiency of the tracking system and high particle losses. I try to correct such possible misnH•asurements by assigning to such jets the charge found for t he other jet of t he W boson candidate pair. This has an effect of neglecting the m ismeasured jet charge while estimating thr charge of the jet pair it belongs to. The d istribution of the number of tracks used for jc>t charge measurements, shown in Figure G.2. is in agreement with tlw I\IC prediction. If both jets in a \".' candidate have less than 3 tracks. I consider the jet pair charge undetermined and assign zero to both jet ch arges. This has the effect of using only t he charge of the other W candidatr while constructing the difference (6.13). Finally. I reject eV<'nts in which both ~r candidates have undetermined charges. In the remaining events I compare the two jet pairs found as described in the previous 6.2 Reconstruction of thew- Direction Distribution 131 section. and assume the one with the algebraically smaller value of Q defined by (6.13) to be the w-. Figure G.3 shows the distribution of theW charge estimator Q observed in data. en - 0WW~qqqq MC c: !illlqq MC w 300 ~----------------------~ N • Data -~ 400 ~Other Bckg MC 0 en c ~ • Data 0WW~qqqq MC (corr.charge) 122:JWW~qqqq MC (incorr.charge) ll!iiBckg and WW (incorr.comb.) MC w 200 200 0 0 10 20 30 40 Ntracks Figure 6.2: Distribution of the number of tracks used in jet charge measurement. -2 -1 0 2 Q Figure 6.3: Distribution of the W boson charge sign estimator. Difference of the charges of the forward (cos 0w > 0) and the backward (cos 0w < 0) W candidates is shown. The additional requirements reduce the data statistics, but increase the statistical sensitivity of the cos 0w distribution for the selected sample. The results of applying the jet charge cuts to the L3 data samples I used in my analysis are summarized in Table 6.1. The expected and observed ratios of selected events are in good agreement at all energies considered. I evaluate the performance of the W charge estimator (6.13) by applying it to a large sample of Monte Carlo events. The negatively charged W boson is found by the algorithm in 72% of w+w- ~ qqqq events in which the correct jet pairs were constructed by the jet combination procedure. The estimates of the probability of the correct charge assignment obtained at different energies were found to be consistent with each other. Search for Anomalous Triple Gauge Boson Couplings 132 Data Period Ntotal I\ sPlectul Nexp 9GB 97 57 56 57± 8 468 449 381 ± 20 98 1393 1384 1363 ± 37 Table 6.1: Number of selected w+\\r- - 7 qqqq candidates in data, together with St-.1 t-.1onte Carlo expectations. Also given is the standard deviation on the total number of selected events expected in the St-.l. The charge separation provided by the jet charge algorithm proves to be useful to distinguish two W bosons with a charge difference of 2e . Unfortunately. this is not adequate for the much more challenging task of distinguishing two quarks coming from the same W, with a charge difference of only ~e. Consequently, one cannot determine quark flavors and reconstruct the W decay phase space for W bosons decaying into hadrons. This precludes the usc of information about vV decays, i.e., W polarizations, in my measurement of triple gauge boson couplings using four-jet events. 6.3 Extended Maximum Likelihood Fit To determine the W boson couplings given the reconstructed cos 8w distributions. I use a method similar to the one developed for the W mass fit. A binned maximum likelihood fit is used to compare the data distribution with reweighted t-.Ionte Carlo predictions. The reconstructed cos 8w distributions arc fitted in the [-1. 1] range in bins of equal size of 0.1. .6i = [-1 + 0.1(i- 1). -1 + 0.1i]. The chosen bin size is small in comparison to the characteristic structure size of the cos 8w distribution, and is sufficient to guarantee large enough t-.IC statistics in each bin. Denoting the number of data events found in bin 'i and the vector of the coupling values to be determined [1, the likelihood maximized in the fit is a bin-by-bin product 133 6.3 Extended Maximum Like lihood Fit of Poissonian probabilities (6.16) where the bin content tvionte Carlo expectations are lli = 1 ~1 d daww da&9 e (x, 0) + d e (1·)d:r. cos - w cos - w (6.17) The cross section integrals are <."valuated as sums of weights of the l\tionte Carlo events inside the ith bin. .i ~. d da - e (:r. fl)d.T cos - w (6.18) The background Monte Carlo <'Vents have constant weight independent of the couplings n while the w+-vv- T\IC events are reweight.ed from the Standard Model parameter set 0 0 to 0 according to formula (5.11). EXCALIBUR matrix elements were used to calculate the weights, as described in Section 5.4. Unlike the likelihood (5.12) maximized in the T\Iw fits. the one used here makes use of theW-coupling dependence of the total w +w - production cross section, i.e. , thE' overall normalization of thE' distribution. In this case the information contained in the total event rate is significant compared to the one given by t he shape of the distribution and cannot be ignored. Using the event rate information reduces the statistical uncertainty of the fit results by 20- 40%, depending on the coupling. As in the T\Iw case, the reweighting method corrects implicitly for selection and reconstruction biases as well as their possible dependence on the couplings fitted. As I did not have samples of T\Iontc Carlo w +w ----+ qqqq events generated with various non-zero anomalous triple gauge boson couplings, I used samples of semileptonic w+w- MC events for technical checks of the maximum likelihood estimator. As for the Mw fits, the estimates of both the fitt ed parameters and their uncertainties were found to be unbiased in the whole range of the fit. 134 Search for Anomalous Triple Gauge Boson Couplings c) U; c 10 Q) > w • Data 0WW MC, ~g?=O 1229 Background MC --· All M , ~g?:-0.5 ·· ·· All M , ~g?=+0 .5 c) - U; cQ) > 75 w • Data 0WW MC, ~g?=O 1229 Background MC - - All MC, ~g?:-0.5 ···· All MC, ~g?=+0 . 5 50 5 25 -0.5 0 0.5 co sew --c 0 en Q) dj 200 -0.5 0 0.5 co sew • Data 0WW MC, ~g?=O 1229 Background MC -- All MC, ~g?:-0 .5 ···· All MC, ~g?=+0.5 100 0.5 co sew Figure 6.4: Reconstructed cos 8w distribut ions for events selected in t he 96B. 97 and 98 d ata sets and the corresp onding l'vlonte Carlo predictions. Effect s of a non- zero anomalous coupling 6.g f are a lso shown. The fit procedure d escribed a bove was applied to the samples of w +w - -+ qqqq candidates selected in the 96B, 97 and 98 L3 d ata set s. described in T able 5.1. The reconstructed cos 8w distributions for the events selected at different ener gies are shown in Figure 6.4. The results of t he 6.g f, 6,K,1 . and>.., fits using t hese d istrib u t ions 6.4 Analysis Uncertainties 135 are " z = 0 . 16+0. 13 - 0.20 t...:>.9 l " . t...:>.h.-y = 0 .26+0.24 -0.33 \ A -y = 0. 18+0.13 -0.20 (6.19) As in the case of the Mw fits, the statistical error estimates were obtained by finding parameter ranges in which the logarithm of the likelihood isn't smaller than that at the maximum by more than 4 (see Equation (5.15)). Unlike the Mw fits, the depen- dence of the maximized likelihood on any coupling is non-Gaussian, due to interplay of the cross section and the cos 8w distribution contributions. Therefore, the errors obtained in this way don't necessarily correspond to a coverage of 68% but nevertheless they can be used to indicate the statistical uncertainty of the measurement. The results of the anomalous couplings fits are in good agreement with the Standard Model expectations. The observed statistical errors are compatible with MC expectations for event samples of the size observed in the data. It turns out to be difficult to separate effects due to changes of various gauge cou- plings, and therefore the analysis of results of simultaneous fits of several anomalous couplings is technically complicated because of large correlations between different measured couplings. Dedicated analyses [118. 119] have shown that the agreement with the Standard Model holds also in the full multidimensional space of anomalous couplings. 6.4 Analysis Uncertainties To evaluate the systematic uncertainties of my measurements, I rely on the methods used for the Mw fit, described in Section 5.6. To estimate the errors, I perform the fits on large Monte Carlo samples generated with modified model parameters or different MC models, using the unchanged baseline MC sample. The systematic errors were reevaluated for each set of data and Monte Carlo used in my analysis, and were found to be consistent with each other and independent of 136 Se arch for Anoma lous Triple Gauge Boson Couplings energy. I assign the error estimates obtained for the 189 GeV data samples, to the combination of the fits of all data samples considered. Throughout this section I quote estimates of systematic errors for the D.gf measurement, with the ones for 6."'"' and >."' given in the summary. 6.4.1 Modeling of the Detector Response Corrections of the jet energy scale and resolutions led to a D.gf shift of 0.01. Neglecting data-MC subdetector calibration d ifferences changed the measured D.gf by 0.02. The latter was chosen as a conservative estimate of the effect of possible detector miscalibration on the measurements. Changes of fitted D.gf due to e and ¢ corrections, and smearing of d irections of reconstructed jets, amount to 0.01, added in quadrature. 6.4.2 Modeling of the w+w - Signal Propertie s The uncertainties of the Monte Carlo models of w+w- production which affected the cross section measurement, also were considered from the point of view of the measurement of the W mass. • Reweighting t he Monte Carlo w+w- events with the ratio of the fu ll and CC03 matrix elements was used to evaluate the effect of the CC03 approximation. The reweighting leads to shifts of less than 0.005 for all the fitted couplings. • Initial state radiation changes the w+w - cross section by a few percent and introduces an additional fluctuation of t he w- flight d irection of the order of 1o . The uncertainty of the simulation of t hese effects is much smaller than the (already small) effects themselves. The precision of the QED calculations implemented in the Monte Carlo codes I used [25] allows me to neglect this source of uncertainty. • Emission of hard gluons by the final state quarks affects both the success rate of the jet combination a lgorithm, which is significantly lower for the five-or-more 6.4 Analysis U n certainties 137 jet events, and the W flight direction resolution. To evaluate possible systematic error due to mismodeling of multijet events. I repeat the fits rejecting different fractions. up to 12%. of the data events with highest values of y 45 . This changes the fit result hy up to 0.04±0.04 which I take as an estimate of the error. • To evaluate the systematic error due to modeling of hadronization, I repeated the fits using a HERWIG I\Ionte Carlo sample as data and the default Monte Carlo sample with JETSET / LU ID fragmentation as the baseline MC. I assign the f),_gf bias found. 0.09±0.04. as the systematic error due to modeling of fragmentation. Similarly, the 0.01±0.04 bias observed when fitting a I\IC sample generated without simulating Bose-Einstein correlations with the default one, was taken as an estimate of the corresponding systematic error. The biases found for several color reconnection models varied in the 0.01- 0.04 range. with a statistical uncertainty of approximately 0.02. I take the shift observed for the Sjostrand-Khoze I model, 0.02. as an estimate of the error due to modeling of color rcconncction. • An additional uncertainty in the measurement is due to the use of the total event rate in the fit. The accuracy of t he theoretical prediction of the w+wcross section is approximately 1%. This introduces an uncertainty in relating the measured V\'+w- cross section to the triple gauge boson couplings. To evaluate this error, I repeated the fits changing the vv+v,r- cross section by 1% to obtain a shift of the fitted D.gf of 0.03. 6.4.3 Mode ling of the Backgrounds • Increasing the accepted qq cross section by 5%, which is a realistic estimate of the uncertainty in the background from this source. leads to a change in the fitted D.gf of 0.02. 138 Search for Anomalous Triple Gauge Boson Couplings • Repeating the fit while changing preselection cuts, which is equivalent to changing the fraction of background rvents in the fitted sample. led to small. less than 0.01, shifts in b.gf. I take the quadratic sum of these shifts as an estimate of the systematic error due to possible mismodeling of the qq background. • Z0 Z0 events are the only other non-negligible background in the event sample used for the measurements. I obtain negligible, less than 0.002, shifts when changing their accepted rate by ±5o/c.. 6.4.4 MC Statistics To evaluate the systematic error due to the finite number of Monte Carlo events, I rely on the procedure used for the .t\lw measurement. I split my Monte Carlo event samples into several parts and repeat the fits of real data with these subsets. The spread of the results gives an estimated systematic error of 0.02. 6.4.5 Other Checks The results of the fits werr found to be stable with respect to changes of the bin size and changes of the sample used for the measurement, controlled by the neural network output cut. No additional systematic errors were assigned as a result of these checks. 6.4.6 Summary of the Systematic Uncertainties The systematic errors of the triple gauge boson couplings measurements are summarized in Table 6.2. As was the case for the W mass measurements, uncertainties of detector response and background modeling lead to small systematic errors due to good understanding of the detector behavior achievrd with large a mounts of qq data and high purity of the selected event samples. The uncertainty of the total w +w - cross section also leads to a small error, as the fit results are detennined mostly by the reconstructed vv- 6.5 S ummary of t he R esults 139 Source of uncertainty Jet energy measurement Jet direction measurement vv+-vv cross section CC03 approximation FSR modeling Fragmentation l\Iodeling of BE correlations i\Iodeling of color reconnection qq background Z0 Z0 background MC statistics Total D.gf error b.K, error >-, error 0.02 0.01 0.03 0 0.04 0.09 0.01 0.02 0.02 0.04 0.01 0.05 0 0.07 0.12 0.02 0.01 0.03 0 0.04 0.16 0.02 0.01 0.03 0 0.04 0.08 0.01 0.03 0.02 0 0.02 0.11 () 0.02 0.11 Table 6 .2: Systematic errors of the triple gauge boson couplings measurements itemized by the source of the uncertainty. The quadratic sum of all contributions is also given. angular distribution rather than the accepted event rate. The dominant systematic errors in the determination of the gauge couplings come from uncertainties in the simulation of hadronization of w +w - ~ qqqq events. The statistical errors are dominant for all thC' measurements. 6.5 Summary of the Results This section summarizes the results of my analysis of the distributions of the W flight direction cos 8w reconstructed with four-jet vV+vv- ~qqqq events. I choose the D.gf, b.K-y, >-, parameter set to quantify possible deviations of the triple gauge boson couplings from their Standard Iodcl values. I measured the anomalous couplings to be 0.16~g:~~(stat.) ± 0.11 (8yst.) (6.20) 0.26~g:~~(stat.) ± 0.16(syst.) (6.21 ) 0.18~g:~~(stat. ) ± O.ll (syst .). (G.22) 140 Search for Anomalous Triple Gauge Boson Couplings The results are in agreement with the Standard 1\Iodel expectation of vanishing anomalous couplings. Both the total event rate and the shape of the cos Gw distribution agree with the 1\Ionte Carlo predictions. The results are in agreement with and complementar~' to L3 measurements using semileptonic w+w- and ··single 'N,'' W---+ev events [119, 120, 100]. Summary and Conclusions 141 Chapter 7 Summary and Conclusions This thesis describes the study of the properties of the \1\T boson I performed using e+ e- ---t w+w- ---t qqqq events found in the data collected by the L3 experiment in 1996- 1998. Approximately 250 pb - l of data collected at center-of-mass energies between 161 and 189 GeV was used in my analysis. resulting in a sample of 1,932 selected events. The large sample of high purity well-reconstructed events allowed me to study this novel physics process. and to measure a variety of the properties of the W boson with an accuracy unachievable in the past. The measurements of the W boson parameters have proven to be a sensit ive probe of the Standard Model and to provide information about crucial components of the theory, in particular the electroweak symmetry breaking mechanism and the nonAbelian triple gauge boson interactions. 7.1 Summary of Results I developed a neural network discriminator to distinguish w +w- ---t qqqq events from events from various background sources and used it to measure the e+ e- ---t w +w - ---t 142 Summary and Conclusions qqqq cross section in a wide energy range. 5.32~g:~~(.<>fat.) ± 0.15(syst.) pb, vs 161.34 GeV vs = 172.13 GeV 8.3o~g:~6(stat.) ± 0.20(syst.) pb. ,jS = 182.68 GeV + 0 26 7.53_ 0 :25 (stat.) ± 0.18(syst.) pb. ,fi = 188.64 GcV 0.85~g:X~(stat. ) ± 0.07(syst.) pb. = (7.1) I combined my measurements wit h those using other w+w- decay channels to arrive at m<>asurements of the total vV+ \\'- production cross section at these energies. 2.68~g:~~(stat . ) ± 0.14(syst.) pb, JS = 161.34 GcV 12.04~U~(8tat.) ± 0.23(syst.) pb. JS = 172.13 GeV 16.45 ± 0.67(stat.) ± 0.26(syst.) pb. ,jS = 182.68 GcV 16.28 ± 0.38(stat.) ± 0.26(syst.) pb. ,jS = 188.64 GeV (7.2) and the branching ratios of the \V decays BR (W---t cv) 10.27 ± 0.44(stat.) ± 0 .17(syst.) o/c. (7.3) BR(W---t JW) 10.00 ± 0.46(stat.) ± 0.17(syst.) o/c (7.4) BR(W---trv) 11.16 ± 0.63(stat.) ± 0 .25(syst.) % (7.5) BR(W---t Cv) 10.43 ± 0.23(stat.) ± 0.13(syst.) % (7.6) BR(V\' ---t qq') 68.72 ± 0.69(stat.) ± 0.38(syst.) %. (7.7) The measured values of tlw e+e- ---t v\·+v v- cross section and W branching ratios ar<> in good agreement with t he p redictions of the Standard 1\ Iodel. The V\7+w - results, together wit h measurC'meuts of thP single W boson production <'+<'- ---t Wev. allowed 7.1 Summary of Results 143 me to limit the width of non-S~l invisible decays of the \i\' r~~/'isibte < 17 ~leV (7.8) at 95o/c confidence level, and infer the value of the C.'i element of the CabibboKobayashi-l\laskawa matrix 1.033 ± 0.033(stat.) ± 0.018(8yst.). (7.9) The measurements of the e+ c- ----t \ v +\i\'- cross sections were not feasible before LEP 2 while the determination of the branching ratios is significantly more precise than previous measurements. I interpreted the measured value of the e+e- ----t w +w - cross section at the pro- duction t hreshold JS = 161.3 GeV to obtain an estimate of the mass of theW boson 80 . 94~8:~~(stat.) ± 0.10(syst. ) GeV. (7.10) I performed also a measurement of the mass and width of the 'vV relying on the direct reconstruction of the invariant masses of dijet systems produced in V\T boson decays. Distributions of dijet invariant masses were reconstructed for thC' data collected at energies above the threshold. using kinematic fit and jet combination methods developed for the cross section measurement. A maximum likelihood fit of the distribution, based on ~lonte Carlo reweighting technique, was used to determine the W mass and width l\hv 80.60 ± O.ll(stat.) ± 0.09(8.1JSf.) GeV (7.11) fw 2.15 ± 0 .28(stat.) ± 0.14(syst.) GeV. (7.12) I combined my threshold and direct reconstruction measurements of l\1w and fw with ones performed with scmilcptonic w +w - events recorded by L3 , to arrive at 144 Summary and Conclusions my best estimates of the Vl mass and width fvv 80.44 ± 0.08(stat.) ± 0.06(syst.) GeV (7.13) 2.18 ± 0.20(stat.) ± O.ll(syst.) GeV. (7.14) There is agreement between the direct reconstruction and threshold measurements of I\Iw, and the found values of Mw and fw are consistent with SI\I predictions. The measurements arc competitive with the ones previously obtained by CDF and D0 experiments. I analyzed the distribution of the flight direction of the V\7 bosons, reconstructed using jet charge estimators, as well as their production cross section. to probe the trilinear couplings of theW boson with the photon and Z0 . I investigated the reduced set of anomalous W boson couplings D.g f. b.r;,-y, and A-y, parametrizing possible deviations of the form of the V\r+ w -"1 / Z0 vertex from the one given by the Standard 1\Iodcl. The I\ Ionte Carlo reweighting method was applied to the fit of the reconstructed differential cross section. The anomalous gauge couplings of the W boson were found to be 0.16~8:~6( stat.) ± 0.11 ( syst.) (7.15) 0.26~g:~j(stat.) ± 0.16(syst.) (7. 16) 0.18~8:~6(stat.) ± O.ll(syst.). (7.17) Both the measured V\'+w- cross sections and the reconstructed W boson angular distributions are consistent with the Standard Model couplings of the W boson. The limits on anomalous W couplings are stricter than previously available ones. As my measurements of the couplings arc based on data at a constant momentum transfer scale, they don't require a ny as ·umptions about the energy dependence of possible anomalies. The results and analysis methods developed in this thesis were used for six L3 publications [95, 9G. 97. 121. 122, 123]. 7.2 C ombination wit h O t h e r R esults 145 Errors of all the m easurements described in this thesis were dominat ed by the sta tistical fluctuations due t o finite number of the d a ta samples used. :-Jevertheless th e system a tic uncert ainties of modeling of hadronization of the four-quark syst em are important. a nd more work will b e necessary to reduce them to the level adequat e for the final analysis of the full LEP 2 dat a samples. 7.2 Combination with Other Results A large amount of da t a was collected a t LEP since 1998. a t collision center-of-mass energies from 192 to 208 GeV. Approxima tely 350 pb- 1 of luminosity were delivered to each of the LEP experiments in 1999- 2000. Analysis of the da t a will allow the LEP collaborations to m ake significant improvements in measurem ents similar to the ones described in this th esis. Preliminar y studies using t hese da t a indicate that e+e- ---+ w +v.,r- cross section is in agreement with the Sta ndard I'vlodel predictions up to energies of 208 GeV [124]. Also in agreement with Sl\1 expect ations are the determined values of the VV branching ratios. measured with a 0.1% precision. combining m easurements of all LEP experiments. Triple gauge couplings were found to b e zero within 0.03-0.07, depending on the coupling. The prelim inary measurements of the W boson m ass and width, from the combined LEP 2 data, are J\lw 80.427 ± 0.046 GeV (7.1 8) rvv 2.12 ± 0.11 GeV. (7.19) With the increased precision , no differen ce is observed b etween the values of Mw m easured with hadroniC' and semileptonic w +v,r- events, 5 ±51 J\IeV, (7.20) 146 Summary and Conclusions which indicates that haclronization effects probably lead to small biases of measured l\lvv. Progress is also being madC' on the theoretical side [125]. A new generation of 1\IC models has been developed. that will !C'ad to a more precise theoretical description of w+w- production. Accuracy of the calculated e+e-----> w+w- cross section better than 0.5% seems within their reach. Comparison of the measurement of the W mass with indirect predictions based on other precision electroweak data allows one to check the consistency of the Standard l'vlodel, and estimate the mass of the Higgs boson assuming the validity of the Standard 1\Iodel. The relatively high value of the \N mass I measured indicates a light Higgs boson. with a mass dose to the lower limit allowed by direct searches at LEP (currently 113 GeV [126]). This conclusion holds with the analysis of all the data collected at LEP so far. Figures 7.1 and 7.2 show the results of the fits clone by the LEP Electroweak Working Group of a number of electroweak measurements including the L3 measurements of 1\h,v. The directly measured values of the \A/ boson and top quark masses were found to be in agreement with the ones predicted by the Standarcll\Iodel given other electroweak measurements. Joint analysis of all electroweak data currently available produces an estimate of the Higgs boson mass M 11 < 170 GeV at 95% C.L., (7.21) that is especially promising from the point of view of the Large Hadron Collider program which will undertake extensive searches for the Higgs boson. Such a small mass of the Higgs boson would mean that the next generation of experiments at the LHC is destined either to find it or discredit it as the mechanism of spontaneous SU(2) x U(1) symmetry brC'aking. and possibly to discover an alternative mechanism. 147 7.3 Conclu s ions - LEP1, SLD, vN Data . v:-·; :. ~· 80.5 68% CL 4 > Q) C\1 ~ 80.4 ;>< <l $: E 2 80.3 Preliminary 80.2 -t---,,.....:..,:.~---r=.;:L.,.-~~~~.,....;-.~~ 130 150 170 190 210 10 m1 [GeV] Figure 7.1: The comparison of the direct measurements of Mw , I\lt with the values inferred from other precision electroweak measurements. The band shows the Standard Model prediction of the 1\lw (:t\11 ) dependence for various values of the Higgs boson mass. 7.3 Figure 7.2: The y 2 of the LEP Electroweak Working Group fit as a function of the Higgs boson mass. Also shown are the lower bound on 1\IH given by direct . earches and some systematic uncertainties of the indirect determination of 1\IH . Conclus ions Within the accuracy of my measurements, no deviations from Standard Model expectations were observed and no new physics were necessary to explain a ny features of the data I studied. Yet again t he Standard 1\lodel was able to pass all the tests it was confronted with. It did so in a new setting of e+e- ~ w+w- annihilation, and in ever more precise and direct measurements of its fundamental parameters. ·w hile the Standard 1\lodel is perceived to be incomplete from the theoretical point of view, and is likely to be replaced with a more basic theory eventually. for now it remains a task and a challenge to physicists to perform an experiment whose outcome will not be predicted by the Standard Model. 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