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A study of the retention of quark quantum numbers in hadron jets Hayward, Scott Kelly 1990

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A S T U D Y O F T H E R E T E N T I O N O F Q U A R K Q U A N T U M NUMBERS IN H A D R O N JETS By Scott Kelly Hayward B.Math., University of Waterloo, 1987 B.Sc, University of Waterloo, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1990 © Scott Kelly Hayward, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of \AV^>\C "S The University of British Columbia Vancouver, Canada Date \Z O C T & ^ & g - j ^ o DE-6 (2/88) Abstract The retention of quantum numbers in hadron jets was studied, and a model was con-structed to infer the quantum numbers of the leading quark and antiquark in the the reaction e+e~ —» qoQo —• hadrons, from the quantum numbers of the final state particles. The method used involved taking a weighted average of the quantum numbers of the hadrons in the jet, with the weight assigned to each particle dependent on its kinemat-ical variables. The model was tested using the Lund Monte Carlo J E T S E T 7.1 at 80 G e V . It was found that in two jet events, it was possible to determine the sign of the charge of the leading partons with SO percent accuracy overall, and with greater than 94 percent accuracy in 41 percent of the events (using cuts on the weighted average to select the events). In light quark events (dd, uu, and ss), it was found that the quark jet and its flavour could be determined with about 48 percent precision and heavy quark tagging of cc and bb could be achieved in 38 and 63 percent of events respectively. In 15 percent of the light quark events, selected by consistency between the two back to back jets, successful determination of the quark jet and its flavour were obtained in 85 percent of events, with heavy quark contamination reduced to 12 and 6 percent of cc and bb events respectively. The model was also run on three jet events at 80 G e V . Although it was possible to identify one jet as a non-gluon jet (in particular the jet which subtends the largest angle with the other two jets) in 90 percent of events, it was not possible to determine which of the two remaining jets was the gluon jet. The sign of the electric charge of the non-gluon jet could be determined in 70 percent of events overall, and in 90 percent of a smaller group of 20 percent of events again using cuts on the weighted charge for the jets. The flavour of the non-gluon jet could be determined in 42 percent of events overall, and in 65 percent of a smaller group of 18 percent of events using consistency between the non-gluon jet and the other two jets, although heavy quark contamination was more significant. Finally, the accuracy with which the sign of the electric charge and flavour could be determined in two jet events at energies in the range 20 to 100 GeV was studied. It was found that as the centre of mass energy increased, the precision with which these properties could be inferred showed a marked increase. Table of Contents Abstract ii List of Figures vi List of Tables vii acknowledgements viii 1 Introduction 1 2 Characteristics of Jets in e+e~ —> hadrons 3 2.1 Distribution of Particles in the Jet 5 2.2 Dynamics of Fragmentation in Space-Time . 7 3 The Lund Monte Carlo Program 10 3.1 Generation of e +e~ Annihi la t ion Events in Lund 11 3.2 Mult i jet Production Schemes 14 3.3 Analysis of Lund Generated Events 16 3.4 Characteristics of the Lund Monte Carlo 17 4 Modelling the Process 22 4.1 Properties of Quarks Produced from the Vacuum 22 4.2 Weighted Average of Jet Quantum Numbers 23 4.3 Extending the Model " 28 4.4 Choosing a Weight Function 32 ,iv 5 Inferring Parton Quantum Numbers in Hadron Jets 36 5.1 Computing the Weighted Charge for the Jet 36 5.2 Electric Charge Retention in Two Jet Events 37 5.3 Quark Flavour Retention in Two Jet Events 43 5.4 Charge Retention in Three Jet Events 49 5.5 Quark Flavour Retention in Three Jet Events . 52 5.6 Effect of Energy on Quantum Number Retention 54 6 Conclusions 58 Bibliography 60 v List of Figures 2.1 Hadron Jet Producing Events 4 2.2 Theoretical Distribution of Charge in Rapidity . 7 2.3 Light Cone Geometry of e+e~ —» hadrons at High Energy 8 3.1 Free Decay and Annihilation of Heavy Quarks ' 15 3.2 Average Number of Particles Per Event as a Function of log(Ejet) . . . . 20 3.3 Distribution of Momentum of the Particle Containing the Leading Parton in the First and Last Generations at 80 GeV 21 4.1 Naive Model of Hadron Production in e+e~ —> qq —> hadrons 24 4.2 Realistic Model of Hadron Production in e+e~ —> qq —> hadrons 29 4.3 Weight Function Calculated Using Particle Distributions 35 5.1 Weighted Charge Distributions For Jets at 80 GeV 40 5.2 Weighted Charge Distributions For Quark, Gluon, and Antiquark Jets at 80 GeV 51 5.3 Energy Effects on Weighted Charge Distribution 57 vi List of Tables 3.1 Number of Jets Generated for Initial Parton Configurations at 80 G e V . 18 3.2 Number of Jets Generated for Initial Parton Configurations in dd Events 19 5.1 Mean and Standard Deviation of the Difference Between Quark and A n -tiquark Jet Weights at SO G e V 38 5.2 Attempts to Guess the Sign of the Electric Charge of Leading Down Quarks in Two Jet Events at 80 G e V 41 5.3 Attempts to Guess the Sign of the Electric Charge of Leading Quarks in Two Jet Events at 80 G e V 42 5.4 Weighted Leading Quark Flavour Statistics at 80 G e V Using w(x) = x 0 - 3 1 45 5.5 Attempts to Guess the Flavour of the Leading Partons in Two Jet Events at 80 G e V 46 5.6 Frequency in which the A Jet is a Quark, Gluon, or Antiquark Jet at 80 G e V 50 5.7 Frequency with which leading partons have the minimum weighted charge at 80 G e V . . 50 5.8 Results of Guessing the Sign of the Electric Charge of the A Jet at 80 G e V 53 5.9 Attempts to Guess A Jet Flavour in Three Jet Events at 80 G e V . . . . 53 5.10 Attempts to Guess the Sign of the Electric Charge of Leading Quarks in Two Jet Events at Various Energies 55 5.11 Attempts to Guess the Flavour of First Generation Leading Fermions in Two Jet Events at Various Energies 56 vi i acknowledgements I would like to thank my thesis supervisor, Dr. Nathan Weiss, for suggesting this project and his guidance throughout the course of my research. In addition I wish to thank Dr. Randy Sobie for helpful discussions on heavy quark tagging, and Dr. Philip Burrows from Stanford Linear Accelerator Centre for his explanation of the jet finding algorithms in Lund. Finally, I would like to thank my parents, step-parents, and my sister and brother-in-law for their continued support over the last two years. V i l l C h a p t e r 1 I n t r o d u c t i o n The idea that additive quantum numbers of a parent quark or gluon in a hadron jet are conserved on average was first suggested by Feynman [1]. Thus, it was postulated, it might be possible to infer some properties of the parent particle from the properties of the hadrons in the jet. It was later shown by several authors [2, 3, 4] that in quark jets, quark quantum number conservation is true only to within an additive constant in cases where sea quarks have some non-zero average for the quantum number in question. In the case of gluon jets however, the conservation of observable quantum numbers should not be affected by sea quark effects. In this paper we wish to reexamine this problem, in light of better detectors and higher energies, in order to determine which properties of the leading quark or gluon may be inferred from the particles seen by a detector in a hadron jet. We wi l l be concentrating on the process e + + e~ —> hadrons at energies ranging from 20 to 100 G e V . Events wi l l be generated by the Lund Monte Carlo Jetset 7.1, and analyzed using several models presented herein of how the sea quarks are arranged in the final state particles. Unless otherwise stated, all parameters in the Monte Carlo program may be assumed to be set to the default values provided by Lund. In Chapter 2, we wi l l describe the generic properties of quark jets. This includes the distribution of particles in jets, the number of particles produced as a function of energy, and the limits which special relativity places on the formation of jets. Chapter 3 discusses the Lund Monte Carlo used to simulate jet production, verifies that it conforms 1 Chapter 1. Introduction 2 to some relevant attributes of jets which are predicted by theory, and points out some areas where additional tests may be warranted. Chapter 4 introduces the idea of taking a weighted average of additive quantum numbers of the jet, and how this might be applied to determination of properties of the leading quark from the hadrons in the jet. The study which follows in Chapter 5 attempts to determine the frequency with which one can correctly guess the sign and possibly the magnitude of the electric charge of the leading parton in a jet, in order to differentiate quark jets from gluon jets, up and charm events from down, strange, and bottom events. In addition, the possibility of determining the flavour of the leading quark wil l be examined. These studies are dependent on both the number of jets and the centre of mass energy of the interaction. . Results for two and three jet events wil l be presented, and a comparison of the accuracy achieved on the range 20 to 100 G e V wi l l be done. Chapter 2 Characteristics of Jets in e + e —> hadrons Jets may be loosely defined in high energy reactions as groups of particles leaving an interaction all within a region swept out by a thin cone whose apex is at the interaction point. The particles typically have high momentum along the axis of the cone, with l im-ited momentum perpendicular to it. They occur in many high energy processes, and are associated with a deep inelastic collision which results in high momentum particles with colour charge (quarks, antiquarks and possibly gluons produced subsequently) moving apart. Such reactions may be grouped into the three generic types listed below, where /, /, and h refer to leptons, antileptons, and hadrons respectively. Specific examples of each of these processes are shown in figures 2.1(a) to 2.1(c). 1. / + h —> / -f hadrons (leptoproduction) 2. h + h —• hadrons (deep inelastic hadron scattering) 3. / + / —• hadrons (annihilation) In each of the first two processes, there is a hard interaction with either a valence quark in the hadron, or a sea quark from the vacuum. In both cases, one quark from the hadron is given high momentum and leaves the interaction region to form a jet. In the third process the lepton and antilepton annihilate into a 7 or ZQ. The 7 or ZQ then bursts into a quark and antiquark which separate with high momenta. This is the simplest of the three interactions, because there is no ambiguity about which quark was hit, and there is no remaining fragment of struck hadrons. In addition there is a symmetry about 3 Chapter 2. Characteristics of Jets in c + t —> hadroi is Figure 2.1: Hadron Jet Producing Events Chapter 2. Characteristics of Jets in e+e —• hadrons 5 the centre of momentum from which the particle and antiparticle emerge, with momenta equal in magnitude and opposite in direction in this frame. 2.1 Distribution of Particles in the Jet In the process e+ + e~ —• hadrons, two to four jets are commonly produced when the C M S energy of the interaction is in the range 20 to 100 G e V . In the case of two jet events, the central axis of the jets have an angular distribution of 1 + cos2(8CM) with respect to the incoming e + and t~. This distribution is consistent with the quarks, being s p i n - | particles. The l imited transverse momentum of hadrons in the jet led Feynman [1] to suggest in 1969 (in reference to jets produced by hadron collisions, as e+e~ collisions were not yet at energies sufficient to produce jets) that the best variables used to describe the particles are the transverse momenta Q, and the ratio of longitudinal momentum to the total longitudinal momentum x. In particular, _ Pz + Po pz . x = — • as Ejet -» oo for pz > pQ (2.1) Pjet + tijei Pjet where (po,p) and (Ejet,0,0,pjet) are the four-momenta for the particle and jet respec-tively. Feynman then showed that if it is assumed that at very high energy, the field energy is distributed among all possible types of particles which may be produced in fixed ratios independent of Ejet, then the probability of finding a particle of type i , wi th mass Hi, in the low momentum region is of the form ft(Q,^-)dpzd*Q dN- = — ^ 2 2) where in the asymptotic l imit , f(Q,x) = Fi(Q) is independent of x, —> ^ , and the x dependence, of dNi becomes Chapter 2. Characteristics of Jets in e+e hadrons 6 dx dNi a — (2.3) x If we now integrate from x = — ^ = — « Tp— up to x = 1 we obtain Ni cx f \ — = / 0 5 ( E j r t ) - / O 0 ( £ ) (2.4) Thus, the number of particles in a jet is expected to grow logarithmically with the energy of the jet. In addition, we may define y = log(x), so that equation 2.3 becomes dN, , . —;— = constant (2.5) dy which shows that the number of particles is constant in y in the low momentum region of phase space. We now recall the definition of x from equation 2.1, and apply a logarithm to it. V = log(x) Pz + Po = l°9l Pjet i rjjei Pjet ~t~ Ejet -i7(l + v)-'jet + Ejet logKr^)*] - log? i — v Pi y = tanh-1(v)-log[Piet + Ejet] (2.6) Pi The last step is a mathematical identity, which can be shown by writing out v in terms of y (using the definition of tanh(y)), and calculating the log(j^-). Rapidity, which is defined to be tanh~l{v), may be shown to be additive under Lorentz boosts, so that y is within a constant equal to the rapidity. Shifting y by a constant factor log[Piet*Eiet] gives the rapidity and is such that as x —+ 0, y —> 0. Thus, the number of particles is constant Chapter 2. Characteristics of Jets in e + t —> hadrons 7 > ' dN < ——==• i> parton cur rent parton fragmentation plateau fragmentation log E * Figure 2.2: Theoretical Distribution of Charge in Rapidity in rapidity in the low momentum region. Figure 2.2 (see [5] and [6]) shows a plot of ^ x as a function of y. showing the fragmentation region where the leading quark is assumed to be, and the current plateau which consists of the low momentum (wee) hadrons. The right half of the graph is for the quark jet, while the left half, for the antiquark jet, is drawn by symmetry. 2.2 Dynamics of Fragmentation in Space-Time Bjorken [5] and Feynman [6] did some early work which described the limits which special relativity places on models of jet fragmentation. Brodsky and Weiss [3] later proposed a model to describe the process of hadronization analyzed in the centre of momentum frame of the interaction. Figure 2.3 (see [5] and [7]) shows the light cone geometry from this frame, in which the quark qQ and antiquark q~o leave the interaction region at nearly the speed of light along the positive and negative z-axes. If the total centre of mass energy of the interaction is 2Ejet, then the leading quark and antiquark will be in frames Figure 2.3: Light Cone Geometry of e+e —+ hadrons at High Energy boosted from the centre of momentum frame by a Lorentz contraction factor -) <x Ejei (in particular = Ejei/mqo). In this boosted frame, the quark and antiquark travel a distance ~)d apart, where d is some hadronic dimension on the order of 1 fm, then begin emitting hadrons. It is this boosted distance over which the leading particles are free that allows them to be considered free fermions in the tree level QED calculation which leads to the 1-\-COS2(6CM) angular distribution of the jets. Lorentz invariance of the process implies that hadron emission occurs near the hyperboloid t2 — z2 — <P (points on this hyperboloid are mapped to other points on it when the coordinates are transformed under a Lorentz boost along the z axis). Since points on the hyperboloid are at spacelike separation from each other, there can be no causal effect on the emission of one hadron by another. Brodsky and Weiss argue that the cause of hadron emission must therefore come from some point t2 — z1 < d2, and proceed on the supposition that at some small time near z = t: = 0, a large number of virtual gluons uniformly distributed in rapidity are emitted. The — spectrum of the gluons with small x in equation 2.3 is consistent with QCD based Chapter 2. Characteristics of Jets in e+e —• hadrons 9 models [3]. These gluons live for some average proper time T — d, then burst to produce a quark-antiquark pair. In the rest frame, the gluon path is z = vgt, and it bursts at time t = -fgd. Squaring both equations and eliminating vg, one finds that t2 = i]S = d2 d2 2 - z 2 = d2 (2.7) (i-S) t Thus the gluon bursts near the hyperboloid. The quark-antiquark pairs formed by the bursting gluons then condense to make hadrons, which are uniformly distributed in rapidity. The path of the leading quark and antiquark meets the hyperboloid at time t = 'yd, which is then about how long the hadronization process takes. In the rest frame, the low momentum hadrons are therefore produced first, and higher and higher momentum ones are produced at later and later times. This process is referred to as an inside-outside cascade [8, 9], because in any frame, those hadrons which have the lowest momentum in the frame will be produced first. It is worth noting that in the analysis done by Cahn and Colglazier [2] in the rest frame of the leading quark, the low momentum hadrons in that frame (corresponding to the high momentum hadrons in the rest frame) are emitted first. It appears in this frame that the leading quark is emitting hadrons sequentially in processes like q —> q + meson and q qq + baryon. It is this model which is used by the Lund Monte Carlo to generate hadron jets from fast moving quarks. Chapter 3 The L u n d Monte Carlo Program The purpose of this study is to examine the retention of quark quantum numbers in hadron jets, and to try to infer some properties of the leading quarks from those jets. In order to gauge the success of such a study, one must know the property of the leading quark being inferred in order to test the method for accuracy. Using real experimental data this is not possible, because there is no other way to accurately determine the characteristics of the leading quark. For this reason, a Monte Carlo program was selected to produce "simulated" events. The properties of the leading quark and antiquark were then conjectured, and the accuracy of these conjectures could be verified because the Monte Carlo program provides all information about the event. Thus we can use the Monte Carlo as an estimate of how successful the quantum number retention results are likely to be on real data. The distributions of weighted averages of electric charge over many events may also provide a new test of the Lund Monte Carlo. There are several Monte Carlo programs on the market which could have been used. The one chosen was Lund Jetset, version 7.1. It is probably the most common, and was therefore easily obtainable. It has proven to be very successful in describing many types of high energy interactions over a large range of energies, and provides many options to change parameters to suit the needs of most applications. In addition, it has extensive analysis routines which conform to analyses done on experimental data. What follows is a discussion of the Lund Monte Carlo [10, 11, 12, 13], and a description of programs used with it. 10 Chapter 3. The Lund Monte Carlo Program 11 3.1 Generation of e+e Annihilation Events in Lund The Lund Monte Carlo program provides a statistical simulation to generate interactions involving Q C D at high energies. It is based partially on perturbative Q C D calculations with the assumption that at high energies, the strong coupling constant decreases, and thus lower order processes wil l give approximate answers to quark gluon production. In addition, Lund has constructed a phenomenological model of the process of fragmentation whereby quarks and gluons hadronize into mesons and baryons. There are three distinct phases in high energy processes in the Lund program. First , there is some hard interaction, with high energy, short distances, and no "confinement" forces. This is often an electrodynamic process, as is the case with e +e~ —* <7o<?o- Second, the quarks and gluons from the first step recede at high energies and the strong force begins to act. A t this step, the Lund fragmentation model generates quark-antiquark pairs, which then hadronize through confinement forces. Finally, the mesons and baryons produced in the second step, many of which are heavy resonances, are allowed to decay to the lighter particles which would be seen by a detector. In the case of e + e " annihilation, the hard interaction is simply e +e~ —* 7 / Z 0 — + Qoqo-If no quark flavour is explicitly specified, then Lund wil l chose one with a probability distribution based on a theoretical model for the process at the energy of the interaction. The manner in which gluons appear in the interaction depends on parameters set in the program. B y default, they are produced by the Lund string fragmentation model, which is a phenomenological model of the strong interactions. However it is possible to have them arise using second order Q C D calculations which have been computed and input into the Lund program, with the direction of the outgoing partons determined by the theoretical angular distribution. In the Lund model, as the quark and antiquark separate, a colour flux tube is formed Chapter 3. The Lund Monte Carlo Program 12 between them with an energy per unit length of K = iGeV/fm — Q.2GeV2 (this is a free parameter in Lund and may be adjusted). As the string stretches, there is some probability that it wi l l break into a quark-antiquark pair. The new quark wi l l then be attracted to the leading antiquark q0, and the new antiquark to the leading quark. The process continues with many quark-antiquark pairs being formed and appearing on average, but with large fluctuations, on the spacetime hyperboloid t2 — z2 = d2. The process is viewed from the frames of the leading quark and antiquark, in which the fastest rest frame hadrons are produced first. This is implemented by first defining the positive (quark) and negative (antiquark) light cone variables W± — E ± pz (with a numerical subscript to denote the iteration). A t iteration zero, they are defined to be W ± ( 0 ) = Ejet ± pz (3.1) This choice of variables is chosen so that the fragmentation scheme is covariant with respect to boosts along the z axis at each step. In addition, at each iteration the product of these is VK+W'- = E2 — p\ = m2 + p\ = m]_, where pL is the transverse momentum. Beginning with W+(0) and W_( 0 ) , the generation of hadrons is done iteratively. Either the quark or antiquark is chosen at random to generate the next particle. A new quark-antiquark pair is generated, with ratio of probabilities of production P(dd) : P{uu) : P(ss) : P(cc) = 1 : 1 : 0.3 : 1 0 - 1 1 for the different flavours (these are the default parameter values for this ratio). Occasionally, a diquark-antidiquark pair is produced in order to generate baryons. In this case, either a lone baryon is produced, or a baryon and meson are generated together in order to split the diquark or antidiquark pair. Given the new particle's quark content, Lund decides whether it is a pseudoscalar or vector meson (or s p i n - | octet or s p i n - | decuplet baryon) if the particle consists only of light quarks. If it contains heavier quarks, a resonance is chosen. The decision as to which particle is Chapter 3. The Lund Monte Carlo Program 13 produced for a given quark content is based on a combination of theoretical prejudices and experimental fitting of particle ratios. Once the identity of the new particle is chosen, its kinematical variables are deter-mined. Suppose that at iteration i , the quark (not the antiquark) produces a new particle, and that its identity has been determined. Then a fraction of W+(,-j is given to the "new" hadron. where this fraction is determined by the probability distribution f(x) defined bv (l-x)a -bm exp[ x x 2 f(x)dx = ^ ^-exp( "" L)dx (3.2) The default values of the dimensionless parameters a and b above are 0.5 and 0.9 respectively. Then the new hadron is given a transverse momentum p±_ wi th a Gaussian distribution fq(p±) where Mpx)dp± = -exp(-^)dP± (3.3) 72 K X Once these have been chosen, the light cone variables for the next iteration are given by W+{i+1) = (1 - x+{i+1))W+(i) (3.4) = W.(t) - m l ( ' - y (3.5) This procedure iterates to produce new particles unti l at some step the remaining energy is small, and W + ( „ ) W _ ( n ) < W^in, where Wmin is a parameter dependent on the flavours of the two remaining quarks. Once this condition has been met, two hadrons are produced, and the second step in the generation of the event is complete. Chapter 3. The Lund Monte Carlo Program 14 The last step in event generation consists of following the particles through their decays, if any. For hadrons consisting of light quarks, most branching ratios have been accurately measured experimentally, and Lund picks the decay and computes the mo-menta of the final states. For some of the less well known resonances of heavy quark states, the quarks within the hadrons are allowed to decay weakly to lighter quarks (with probabilities based on terms in the Kobayashi-Maskawa mixing matrix). Some decays are free decays, while others contain annihilation processes. Examples of each are given below, where q, Q, 1, and v refer to light quarks, heavy quarks, leptons, and neutrinos respectively. Heavy quarks in baryons decay similarly. Figures 3.1(a) to 3.1(d) illustrate examples of the processes. (a) Qqx -> qAq3q2q\ free decay (b) Qq~i — > lvq2q~\ free decay (c) Qq\ —» q3q2 annihilation (d) Qq~i — > Iv annihilation 3.2 Multijet Production Schemes Production of events with three or more jets may be done in either one of two ways with Lund. The default method consists of an iterative parton fragmentation scheme which allows the leading quark and antiquark to undergo a strong fragmentation process pro-ducing gluons and possibly other quark-antiquark pairs. This is done v ia three processes, 9 ~* QQi 9 ~* 99i a n d 9 qq, and may result in the production of many jets. The second possibility uses second order Q C D matrix elements which have been calculated and input into the Monte Carlo program. These have possible final states consisting of qq, qgq, qggq, and qqqq, normally producing two, three, four, and four jets respectively. A t very high energies, the Q C D coupling constant decreases, and so it is Chapter 3. The Lund Monte Carlo Program 15 (c) annihi lat ion (d) annihilation Figure 3.1: Free Decay and Annihilat ion of Heavy Quarks expected that second order perturbation theory will be a reasonable approximation. Neither of the two methods performs perfectly. In particular, the parton fragmen-tation model does not give the correct rate for three jet events, and the Q C D matrix element scheme overestimates the four jet rate. However for the purposes of this study, the Q C D matrix element option was chosen, because it listed the leading partons and their four momenta, and thus provided unambiguously the identity of the parton which started the jets observed in the final state of most events. At 100 G e V , the parton frag-mentation method often produced ten or more partons, and it was not possible to identify which jet was produced by the leading quark or antiquark, and which was produced by a gluon (or to which gluon the additional jet should be attributed). Chapter 3. The Lund Monte Carlo Program 16 3.3 A n a l y s i s of L u n d G e n e r a t e d E v e n t s Among the Lund event analysis routines, there are two which reconstruct jets from the final state hadrons. The first is a cluster algorithm which builds clusters of particles with momenta in the same direction, while the second analyzes energy deposition in calorimeters. A l l jet reconstruction that follows wi l l be done with the former algorithm, called L U C L U S , with options to make it conform to the analysis routine introduced by Bethke [14] at the J A D E collaboration. The algorithm begins by assuming that all particles in the jet are clusters. It then calculates the invariant mass-squared between each pair of clusters, which is defined between the clusters i and j to be If the two clusters with the minimum invariant mass-squared satisfy the condition where ycut is a parameter which may be set, then the two clusters are joined to form a new cluster by adding their four-momenta. This process continues unti l no two clusters satisfy the condition given in equation 3.7. In practice, it is possible that some of the low momentum particles wi l l be assigned to a cluster which is not the one which would be the most appropriate for them. In addition, there are configurations in which a cluster could be built of low momentum hadrons and form a "jet", but which does not contain the high momentum particles along its axis to warrant calling it a jet. Thus, once the final cluster axes have been determined, each individual particle is assigned to the cluster whose axis subtends the smallest angle with its three momentum. Then the longitudinal momentum of each particle is calculated by m2j = E{Ej(l — cosdij) (3.6) mlj < m L = VcutE2t (3.7) Chapter 3. The Lund Monte Carlo Program 17 taking the dot product of the three momenta of the particle and the cluster to which it has been assigned. This process wi l l reduce the number of jets if no particles are assigned to one of the clusters. Note that reassignment of particles to different clusters is rare. Also, the particles reassigned wi l l be nearly perpendicular to the other two jets (otherwise the cluster analysis would put them in the correct cluster), and wil l have small momentum (otherwise, since they are nearly perpendicular to the other jets, they would form a new jet). Thus, changes to the momentum of the clusters from the reassignment of particles wi l l be small. In what follows, reference wil l be made to the first and last generation of particles in an event. The first generation of hadrons is defined to be those produced in the second step of the Lund Monte Carlo program, by the fragmentation of the quarks and gluons and before any decays have taken place. The last generation of hadrons and leptons wi l l refer to those which remain after decays have taken place. However, there are two exceptions. Hadrons which decay into photons only (such as 7r° —> 7 7 ) are left as hadrons, and of course neutrinos are assumed not to be detected. 3.4 Characteristics of the Lund Monte Carlo In this section, we present results from running the Monte Carlo to determine some of its characteristics. First , we examine how well the number of jets found by the jet analysis conforms to the number of partons in the ini t ial state. Next we analyze whether or not the number of final state hadrons produced actually increases as the logarithm of the energy as expected in the result given in equation 2.4. Finally, we look at the momentum distribution of the particle which contains the leading parton. It wi l l be shown later that this distribution is very important in determining the distribution of the weighted average of leading quark quantum numbers. Chapter 3. The Lund Monte Carlo Program 18 Table 3.1: Number of Jets Generated for Initial Parton Configurations at 80 G e V Quark Number of Number of Jets Flavour Partons 2 3 4 > 5 down 2 82241 3415 0 0 down 3 17627 93253 342 0 down 4 132 3428 8518 7 up 2 82300 3458 0 0 up 3 17572 93555 421 0 up 4 128 3394 8365 5 strange 2 82189 3401 0 0 strange 3 17690 93226 390 0 strange 4 121 3466 8344 6 charm 2 83207 3938 0 0 charm 3 16674 95792 455 0 charm 4 119 3159 8706 10 bottom 2 86589 3483 0 0 bottom 3 13285 98556 458 0 bottom 4 126 2326 9434 11 Table 3.1 shows the number of jets produced by init ial configurations of two to four partons when the scaled mass cut-off parameter ycut = 0.02. From the table it is clear that the number of jets produced corresponds well to the number of leading partons. If ycut was lowered, then the conditions required for a cluster to be considered a jet would be weakened, and the number of jets would increase. Conversely, if ycut was raised, then the number of jets would decrease. These changes in ycui would produce shifts of numbers of events in table 3.1 towards the right and left respectively. In two jet events, if the ini t ial state had more than two partons, then the jet which makes the smallest angle with the leading quark is defined as the quark jet, while the other is the antiquark jet. In three jet events, the jet which subtends the smallest angle with the ini t ial quark and antiquark are defined to be the quark and antiquark jets, while the third jet is deemed to be the gluon jet. Chapter 3. The Lund Monte Carlo Program 19 Table 3.2: Number of Jets Generated for Initial Parton Configurations in dd Events C M S Energy Number of Number of Jets (GeV) Partons 2 3 4 > 5 20 2 48577 47785 6011 15 20 3 50633 168690 61140 3257 20 4 790 13537 20853 3803 40 2 68760 15005 39 0 40 3 30871 99910 5600 25 40 4 369 6448 8969 238 60 2 78536 6780 0 0 60 3 21257 89571 1214 0 60 4 207 4191 7743 26 80 2 82241 3415 0 0 80 3 17627 93253 342 0 80 4 132 3428 8518 7 100 2 91122 1966 0 0 100 3 14514 95333 146 0 100 4 109 2701 8755 2 Table 3.2 shows the number of jets produced in dd events with different ini t ial parton configurations and at various energies. From the table it is observed that as the centre of mass energy increases, there is a greater correspondence between the number of ini t ial partons ancl the number of jets produced. It is also observed that at the lower energies, the parameter ycut should be raised because the jet algorithm analysis finds many low parton multiplici ty events with high jet multiplicity. Thus, the jet analysis algorithm using y^t does not provide a measure of the number of jets which is completely independent of the jet energy, although it is quite close. It was shown in equation 2.4 that in the naive model, the number of wee hadrons in a jet is proportional to the logarithm of the jet energy (modulo a constant). In figure 3.2 we plot the mean number of particles as a function of the logarithm of the energy, for 3 x 10 5 light quark events (10 5 of each light flavour). Charm and bottom quark events Chapter 3. The Lund Monte Carlo Program 20 3 3.5 4 4.5 Log(E) Figure 3.2: Average Number of Particles Per Event as a Function of log(Ejet) had slightly larger numbers of particles by factors of about 1.0S and 1.15 respectively, but the numbers are still linear functions of the logarithm of energy. From the diagram, it is apparent that the Lund Monte Carlo conforms to this result. One important factor which determines the viability of determining the properties of the leading parton is the distribution in momentum space of the hadron which contains it. Figure 3.3 shows this distribution for the first and last generation of particles (using 25,000 and 100,000 events respectively). Notice that there is a small probability (about 0.25 percent) that at 80 GeV, the leading parton will eventually end up in a particle which is travelling in a direction away from the initial direction of the leading hadron. Chapter 3. The Lund Monte Carlo Program 21 -0.2 0 0.2 0.4 0.6 0.B Momentum Fraction (First Gen) 600 IT. £ 400 c 3 cf 200 0 -0.2 0 0.2 0.4 0.6 0.6 1 Momentum Fraction (First Gen) -0.2 0 0.2 0.4 0.6 0.8 Momentum Fraction (Last Gen) -0.2 0 0.2 0.4 0.6 0.8 1 Momentum Fraction (Last Gen) -0.2 0 0.2 0.4 0.6 0.8 1 -0.2 0 0.2 0.4 0.6 0.8 1 Momentum Frection (First Gen) Momentum Fraction (Last Gen) Figure 3.3: Distribution of Momentum of the Particle Containing the Leading Parton in the First and Last Generations at 80 G e V Chapter 4 Modelling the Process It was suggested by Feynman [15] that it might be possible to find the quantum numbers of the quarks experimentally by averaging the sum of the quantum numbers of the hadrons in a single jet. This is because the leading quark should be in the fragmentation region, so that the fast hadrons in the jet should retain its quantum numbers. This was later shown to be true to within an additive constant, as will be demonstrated in section 4.2. In addition, it was hoped that by taking a weighted average of the particles in the jet, one could determine the quark parentage of hadron jets on an event by event basis with some accuracy. In what follows we will present two models of hadronization, a very simple model and a more realistic one, and analyze this possibility. 4.1 Properties of Quarks Produced from the Vacuum The leading quark-antiquark pair is produced by the bursting of a high energy photon or ZQ. For energies below the Z0 mass, tree level QED calculations show that the probability that a given quark flavour is produced is proportional to the square of the electric charge of the quark. Thus the ratio of the probabilities of producing the different quark flavours are P{dd) : P(uu) : P(ss) : P(cc) : P{bb) = 1 : 4 : 1 : 4 : 1 . This is not, however the ratio at which quark-antiquark pairs are produced from the vacuum. Here probabilities for production are dependent on the colour charge which is the same for all quarks, but the phase space for production of quarks decreases with mass so that the heavier quarks are produced much less frequently. 22 Chapter 4. Modelling the Process 23 Given some additive quantum number A, we can calculate its expectation value in quarks produced from the sea, by summing over the probabilities of the vacuum producing each possible flavour as follows: < Xv >= ^2 p(flavour)X(flavour) (4-1) flavours where p(flavour) is the probability that a quark-antiquark pair created from the vacuum has a given flavour. In Jetset 7.1, the ratio of the production of light flavours by the vacuum is Pv{dd) : Pv(uu) : Pv(ss) = 1 : 1 : a , where the Lund default value of the parameter a is 0.3. Heavy flavour production from the vacuum is negligible. Thus the average electric charge for a quark created by the vacuum is: < K > ~ (2Tc0 ( _ 3 ) + (2To0 ( 3 ) + (2T^) ( "3 ) - 3 ^ 2 ^ | a = ° - 3 " 69 ( 4 " 2 ) Similarly, one can compute the mean square and variance for the electric charge of sea quarks: , rxehg s2 . 1 / 1 N 2 , 1 A 2 , Q / 1 x,2 _ ( 5 + Q ) i _ <^ ) >-(2 + a)[ 3 j + (2 + a ) l 3 J + (2 + a ) ( 3> ~ 9(2 + a) | o = 0 - 3 " 207 (4.3) { < H G F = < ( K H 9 ? > _ < X F G >*=0S± | Q = Q 3 = i | (4.4) 4.2 Weighted Average of Jet Quantum Numbers In the simplest model of hadronization, as the quark and antiquark separate, they emit a series of gluons between them which subsequently burst into quark-antiquark pairs. Adjacent quarks and antiquarks coming from neighbouring gluons then pair up forming a Chapter 4. Modelling the Process 24 Figure 4.1: Naive Model of Hadron Production in e + e —» qq —> hadrons series of mesons m i to m/v, which for the moment we will assume contain the structured quark content shown in figure 4.1. In addition, it will be assumed that these mesons leave the interaction region in the form of a jet, in which their trajectories define a small angled cone whose axis of symmetry is along the trajectory of the leading quark. The longitudinal momenta (momenta projected onto the axis of the cone) of the mesons m; are assumed to decrease as i increases. We now construct a weighted average of such an additive quantum number for a quark jet. This is done by taking a weighted average of the quantum numbers of the mesons which occur in the final state of the jet. With weight u',- for the ith meson in the jet, m,-, we define A to be the weighted average for the jet, so that: Chapter 4. Modelling the Process 25 A = 5 i^w,-[A(m,-)] (4.5) In the naive model, the meson m, contains quark qi-i and antiquark qt, so that this becomes N A = ][>.-W9.--i)+ *(?;•)] (4.6) i=i Defining a dummy weight u;/v+i = 0, letting A, = A(g,-), and noting that for additive quantum numbers A (eft) — — A(«j,-), this becomes (4.7) t=i When this is averaged over many events, the effect of the sea quarks approaches that of the "average" sea quark (ie. A^  —>'< A^ > ) , and this weighted average becomes < A >= (A0— < Xy >)wi (4.8) Using equation 4.7, we may compute the mean square and variance for A. N N A 2 = wl\20 + 2A 0 i«i k(wi+1 - w{) + tfiwi+i ~ wi)2 i=i i=i N t-1 +2 XiXj(wi+1 - Wi)(wj+i - WJ) t'=l j = l (4.9) N < A 2 >= ( A 0 - < A„ >)2w\ - a2vw\ + 2a2v £ w{{wi - tu<+1) (4,10) a\A) =< A > - < A = a2v t=i TV i=i (4.11) Chapter 4. Modelling the Process 26 In order to find the best weight, we now define the ratio of the standard deviation to the mean, and require that this ratio is a minimum with respect to variations of al l ./V weights. r = 4^  (4.12) < A > Provided that < A > ^ 0, then one can show that: ST Sw, = 0 (An- < A„ >)a2v . S < A 2 > a 2 S < A > < A > — 2 < A J > 8WA SWJ = 0 (4.13) Apply ing this result with the above expressions gives: ST (An- < Ar >)ol Swi (4.14) t = l ST (A 0 - < \ v >)o~l SWJ = 0 wj-i - 2Wj + wJ+1 = 0 for 2 < j < N - 1 (4.15) 1 ST 0 2WN — IWA'-I = 0 (4.16) (A 0 - < A„ >)a2v SwN The condition •—• = 0 implies that IOAT_I = 2 IOJV, while ^ = 0 then implies that U>N-2 = 3 WN, = 0 implies that to/v-3 = 4 tu/v, and so on. Working back this way, sr one finds that j-^ = 0 implies that w\ = Nw^. B y setting iwi = 1, one finds that the weights are given by: Wi = N-i+l N (4.17) This uses the degree of freedom one has in multiplying all weights by a constant, without changing T. In addition, this solution is consistent with the equation for Chapter 4. Modelling the Process 27 given in equation 4.14 above. Apply ing these weight functions to the mean and variance for the jet weighted average gives: < A >= A 0 - < A v > (4.18) r = J ^ > °« * ( 4 . i9 ) < A > (\0-<\v>)y/N Thus, one expects that the weighted average for the quark jet of an additive quantum number A is just Ao, the quantum number of the leading quark in the jet, less < Xv >, the average value of the quantum number for a sea quark. In addition, using the weighting given above, the ratio of standard deviation of the weighted average to its mean decreases as where N is the number of particles in the jet, multiplied by a constant which is dependent on the properties of the vacuum. The average number of particles in a jet increases as the logarithm of energy, so that in this model, it should be possible to determine the properties of the leading quark in the asymptotic l imit (ie high energy, and hence large particle multiplici ty). It may be noted that in the SU(3) symmetric model in which dd, uu, and ss pairs are produced by the vacuum with equal probabilities, then a = 1 and equation 4.2 gives < A ^ 5 >= 0. Then the distribution of weighted charges over many events would be peaked at the charge of the leading quark. In the SU(2) symmetric model, in which only dd and uu pairs are produced, and in equal quantities (so a = 0), equation 4.2 gives < Xlhg >— | . Then the peaks of the weighted charge distribution would be at | — | = | for up quarks and — | — | = — | for down quarks. Thus the u quark peak and the J peaks would be at the same point, and hence u quark jets would be indistinguishable from d antiquark jets. Chapter 4. Modelling the Process 28 This is also evident via the argument [16] that in the absence of strange and heav-ier quarks, all mesons are left invariant under the combined operations of isospin rota-tion and charge conjugation. Under this transformation, the light quarks transform as (d, d, u, u) —• (u, u, d, J) , and leave all pion states are unchanged. If it is claimed that a given jet is a d jet, for example, then applying this transformation it could equally well be claimed that the jet is a u jet. Hence d and d jets are indistinguishable from u and u jets respectively. 4.3 Extending the Model The above model is clearly an oversimplification of what really occurs in the production of a quark jet. It assumes that the leading quark is in the fastest particle in the jet, as well as a very particular structure for the relative positions of quark-antiquark pairs produced from the sea. It does not account for the production of baryons and antibaryons, which depending on the model used for baryon production, may imply that such an ordered structure does not exist. The model neglects the possibility that the mesons and baryons ini t ial ly produced may undergo one or more decays before they reach the detector, and such decays could certainly mix the order of the quarks in the jet. Thus a more realistic model would be that shown in figure 4.2, although it is clearly more complicated. In the models presented in chapter 2, it was noted that gluons burst on average near the hyperboloid t2 — z2 = d2, and the low momentum ones are distributed uniformly in rapidity on average. However in a given event, there may be large deviations from the this behaviour. It is thus possible that quark-antiquark pairs could be formed with very similar momenta, and hadronize without the rigid structure where meson m,- contains <7,_i and q~i. In addition, the leading quark might hadronize with a slower antiquark and not be contained in the fastest meson produced. Chapter 4. Modelling the Process 29 5. First generation of hadrons decays to final state hadrons and leptons. 4. Confinement forces act to condense quarks and gluons to hadrons. 3. Quark and antiquark recede at high velocities, and emit gluons as the strong force begins to act. 2. Photon or 2 bursts to form a quark antiquark pair. Electron and positron annihilate to form a photon or Z boson. Figure 4.2: Realistic Model of Hadron Production in e+e" qq —* hadrons Chapter 4. Modelling the Process 30 The Lund Monte Carlo program produces baryons by assuming that there is some probability that a gluon, instead of bursting into a quark-antiquark pair, wi l l form a diquark-antidiquark pair (similar to the model of Cahn and Colglazier [2]). Then a baryon consists of a quark and a diquark, and an antibaryon consists of an antiquark and an antidiquark. Baryon production may then be accounted for easily by extending the definition of quarks in the vacuum to include antidiquarks, and the definition of antiquarks in the vacuum to include diquarks. Then a baryon can be thought of as a meson in which the antiquark is a diquark, and an antibaryon as a meson in which the quark is an antidiquark. This allows baryon production to be accounted for in the computation of < Xv > and cr2, using equations 4.1 to 4.4 with the extended definition of vacuum quark flavours, without otherwise complicating the model. The only problem that this scheme introduces is that it accounts for the existence of diquark-antidiquark pairs which do not occur in nature, but this is a second order correction in the ratio of baryons to mesons, and hence a small effect. The decay of the first generation of hadrons wi l l have several effects on the jet.-Decays in which the outgoing particles have high momenta in the rest frame of the decaying particle may further change the order of the quarks and antiquarks making up the final state hadrons. In addition, there wil l be an increase in the transverse momentum of particles throughout the jet, and this wi l l particularly affect the low momentum hadrons. Because most decays introduce new7 up and down quarks, but not strange quarks, there wi l l be changes in the numerical values of the vacuum parameters computed in equations 4.2 to 4.4. Finally, some additive quantum numbers are not conserved in hadronic decays. In particular, weak decays do not conserve flavour, and this makes it much more difficult to predict leading quark flavours from final state particles. The detector sees only the final state hadrons so in the following model, we again consider the quarks and antiquarks but make fewer assumptions about the structure of Chapter 4. Modelling the Process 31 the quarks within the jet. It is clearly possible that in the more realistic model, the weight function computed in equation 4.17 wi l l no longer be the best weight function. We now construct another weighted charge for the jet, but allow the weight to be a function of the fraction of the longitudinal momentum of the jet carried by the particle (ie the magnitude, not just the order in momentum). This is done on the assumption that the faster particles are more likely to contain the leaking quark than the slower ones. As before, we construct a weighted average of the additive quantum number A for a jet. However, we now allow the weight assigned to each particle to be dependent on the fraction of the total longitudinal momentum of the jet that it carries, defined for the ith particle in the jih jet by ^ = Pij-Piet ( 4 2 ( ) ) Pjet • Pjet A'; where p~t = Y.P'J (4.21) The weighted average of the jet in the jih event is then given by: Aj = £ w{xt3)[X{q{xtJ)) + \(q{xij))] (4.22) t'=i where X(q(xij)) and X(q(xij)) are the quantum numbers of the quark and antiquark in the ith particle in the jth jet. When averaged over many events, this weighted charge becomes: < A > = ^  jjlEE ">[xij)[K<l(xij)) + KfaiM (4.23) The sum over particles and events can be combined into an integral by introducing a function n(x) which is the number density of particles with a fraction of the jet's longitudinal momentum x. Then 4.23 becomes: Chapter 4. Modelling the Process 32 < A >= Cdx n(x)w(x)[X(q(x)) + X(q(x))] (4.24) Jo We now assume that the probability that a given particle in any event contains the leading quark depends only on the fraction of the longitudinal momentum of the jet that it carries. W7e define the function p(x) to be the probability that a particle with momentum x contains the leading quark. In addition, we allow for the possibility that during the process of hadronization and subsequent decays, the particle containing the leading quark may be moving in a direction opposite to the ini t ial direction of the quark when it was produced, so that p(x) is (an increasing function) defined from -1 to 1. Since the antiquark jet should have an identical particle number density, we expect n(x) to be an even function. Normalization then requires that: ^ dxn(x)p(x) = 1 (4.25) We now write X(q(x)) and A(t7(a;)) in terms of p(x), and apply the results to 4.24. A(«7(x)) = p(x)A 0 + [1 - p(x)} <XV> (4.26) X(q(x)) = p ( - i ) ( - A 0 ) + [1 - p(-x)](- < Xv >) (4.27) < A >= ( A 0 - < Xv >) ( dx n(x)w(x)\p(x) - p(-x)} (4.28) 4.4 Choosing a Weight Function The choice of a weight function is critical in order to provide the possibility of determining any properties of the leading quark. Many different weight functions were tried. The Chapter 4. Modelling the Process 33 results presented are of three types. First , the weight function derived in equation 4.17 was tested. Second, powers of x were used, so that w(x) = xk (4.29) where A; is a constant which could be varied. Values of k between 0.3 and 0.35 showed the greatest success when ycui = 0.02, with higher values of k being more successful for jets defined by a smaller setting of the parameter ycut. Two other investigations into weight functions were also attempted. In the first, the mean < A > and standard deviation a (A) of the weighted average were computed with the more realistic model of hadronization presented in the last section, so that T = z££l could be extremized. However, it was found that <x(A) was dependent on the amount of mixing in the order of vacuum quarks and antiquarks (ie deviations from the order given in the naive model) in the mesons, not just the position of the leading quark. Several models of this mixing were considered, using information that would not be easily obtainable experimentally, but none of the models produced weight functions which were as successful as w(x) = xk. One last possibility is an extension of the naive model which produced the weight function wt = A ^ j " 1 " 1 . When this weight function is applied to equation 4.7, it becomes A = A„4D. (4.30) Thus the contribution to A of each quark-antiquark pair produced by the vacuum is equal. This minimizes the variation in A due to the randomness of the vacuum quarks. We now apply this general principle, but allow the weight function to be a function of x. We therefore require that Chapter 4. Modelling the Process 34 w(x) — w(x*) = constant (4-31) where x" is the longitudinal momentum (averaged in some way) of the next slowest particle to one with longitudinal momentum x. We now define r(x,y) to be the proba-bil i ty that given a particle with longitudinal momentum fraction y, then the next slowest particle wi l l have fraction x. The normalization condition required of r(x,y) is rv / dx r(x,y) — I — paiow(y) (4.32) Jo where psi0w(y) is the probability that a particle with longitudinal momentum fraction y is the slowest particle in the jet. In implementing this method, the momentum region is broken up into a discrete number of bins of equal width labelled 0 to K — 1. Setting the constant in equation 4.31 equal to one, we obtain w(xx) - ] T r(xj, Xi)w(Xj) = 1 for 0 < i < K - 1 (4.33) j=0 The solution of this equation can be easily verified to be W { X i ) = l - r t e . s , - ) ( 4 - 3 4 ) Figure 4.3 shows a plot of a weight function computed with the longitudinal momen-tum divided into K = 256 bins of equals size, based on 250,000 two jet dd events. Results for other flavours are very similar. It should be noted that although the weight function is not monotonically increasing from from zero to one, it is true that for any a;,- and Xj, w(x{) > W(XJ) provided that X{ > Xj and xt- + Xj < 1 (4.35) Chapter 4. Modelling the Process 35 i i i ' ' ' 1 ' I ' ' i i i i i 1 i i i i 1 i i ' i I i 0 50 100 150 200 250 Momentum Fraction x Figure 4.3: Weight Function Calculated Using Particle Distributions Chapter 5 Inferring Parton Quantum Numbers in Hadron Jets In what follows, we wi l l be attempting to infer properties of the leading partons in hadron jets by taking weighted averages of the quantum numbers of the final state particles in those jets. The properties we wil l try to infer are the sign and magnitude of the electric charge of the leading partons, as well as the flavour. We wil l restrict ourselves to two and three jet events, because four jet events consist of either a quark, two gluons, and and antiquark. or two quarks and two antiquarks. From the results to be presented on three jet events, gluons jets are sufficiently difficult to differentiate from quark jets that events with four or more jets are unlikely to produce worthwhile results. 5.1 Computing the Weighted Charge for the Jet Given an event, we compute the weighted down, up, strange, and lepton (combined electron, muon, and tau; neutrinos are not detected) characteristic of both jets using the last generation of particles. The "down characteristic" of an elementary particle is defined to be 1 if it is a d quark, -1-if it is a d antiquark, and 0 otherwise. The down characteristic of composite particles.is defined to be the sum of the down characteristic of its components. For example, the quark contents of the pions are 7 r + = ud 7T° = —j={uu + dd) 7T _ = du (5-1) The down characteristics of these mesons are thus -1, 0, and +1 respectively. The up and strange characteristics are defined in an analogous manner, and the lepton characteristic 36 Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 37 is defined to be equal to the negative electric charge of the lepton (so that it is one for the electron, muon, and tau, and zero for neutrinos). The weighted quark and lepton characteristics of the particles in a jet are then com-puted by taking a weighted average of the corresponding characteristics for the particles in the jet. Thus the down weight of an yV particle jet in which particle m,- has longitudinal momentum xt- would be A' Djet = y£w(xi)d(mi) (5.2) t = i with Ujet, Sjet, and Ljet defined similarly. The weighted electric charge can easily be extracted from the weighted quark and lepton characteristics via / 1 2 Q%t - -^(Djet + Sjet) + gtVjet ~ Ljet (5.3) 5.2 E l e c t r i c C h a r g e R e t e n t i o n i n Two J e t E v e n t s It is clear by symmetry that the magnitude of the electric charge of the leading particle of one jet is opposite to that of the other jet. Thus if the weighted charge of one jet is subtracted from the weighted charge of the other, the result should, on average, have twice the magnitude with only y/2 times the standard deviation. The mean, standard deviation, and their ratio for the weighted electric charge of events of each of the five flavours, using weights Ar~^~1"1 (1), a ; 0 3 1 (2), and the weight function in equation 4.34 (3), is given in table 5.1. From the results in the table, we see that weight function (3) appears to be the best of the three. However as wil l be seen, weight function (2) is the most successful at guessing most properties. In events where one jet has a very fast leading particle (say x > 0.5), if the charge of this leading particle is nonzero, then it should be a good indicator of the Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 38 Table 5.1: Mean and Standard Deviation of the Difference Between Quark and Antiquark Jet Weights at 80 G e V Quark Weighting Mean Standard Ratio Flavour Function Deviation (SD/Mean) down (1) . 0.323 0.427 1.324 down (2) 0.243 0.292 1.203 down (3) 0.349 0.411 1.175 up (1) 0.501 0.426 0.851 up (2) 0.383 0.291 0.759 up (3) 0.525 0.410 0.780 strange (1) 0.365 0.418 1.146 „ strange (2) 0.283 0.285 1.005 " strange (3) 0.380 0.406 1.067 charm (1) 0.523 0.505 0.964 charm (2) 0.382 0.345 0.905 charm (3) 0.611 0.500 0.818 bottom (1) 0.501 0.645 1.287 bottom (2) 0.374 0.439 1.176 bottom (3) 0.505 0.617 1.222 Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 39 charge of the leading parton. However, because weight function (3) assigns a low weight to particles with very high momentum, it relies heavily on the weight of the other jet in the event to guess the jet charge. Thus, it is possible that although it has a smaller ratio of standard deviation to mean, it is not the best weight function to guess the properties of the leading parton. Alternatively, it is possible to imagine a weight function whose weighted average has a very low standard deviation for jets of a fixed number of particles, but jets of different numbers of particles have different mean weighted averages. In this case, the weight function would be very good at predicting the properties of the leading particle in the jet, but would have a relatively high standard deviation. This would occur because when averaged over many events, some events would be clustered around one mean while others would be clustered around another mean. We now consider the distribution of the weighted charge for two jet events started by each of the five quark flavours. These distributions are shown in figure 5.1 with successive distributions shifted vertically. Note that distributions are for positively charged leading partons (ie u and c quarks, and d, s, and b antiquarks) and are based on 25,000 events of each leading quark flavour. From the figure, it is clear that only the sign of the charge of the leading quark could possibly be inferred, and not its magnitude, so that this method could not be used to differentiate u and c jets from d, s, and 6 jets. It is apparent, however, that one should be able to determine which quark in the jet has positive and which has negative charge with some accuracy. In addition, one would expect that in some regions of the distribution, one should be able to identify the positive and negative characteristics of the jets very accurately. In table 5.2, several methods are used to guess the sign of the charge of one of the jets in e +e~ —» dd events. The first four in each flavour are among those suggested by Feynman and Fie ld [17], while the others are from considerations based on the weighted Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 40 - 1 0 1 2 W e i g h t e d E l e c t r i c C h a r g e Figure 5.1: Weighted Charge Distributions For Jets at 80 G e V charge or weighted quark characteristics. Note that the numbers 0.447, 0.323, and 0.475 are approximate averages over the five flavours of the mean weight of positively charged quark or antiquark jets for the given weight functions. From the results in table 5.2, it is clear that the weighted charge methods are more successful than those considering only the leading quark. We also see that weight function (1) is less successful than either of the other two weight functions at guessing the sign of the charge of the leading quark. Final ly, although weight function (3) had a lower standard deviation to mean ratio than weight function (2), it is not significantly more successful at guessing the charge of the leading quark in dd events. W i t h other quark flavours, weight function (2) is more accurate, and thus wil l be used for the rest of the results presented. Table 5.3 presents a similar set of results, showing the results using w(x) — r 0 ' 3 1 but Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 41 Table 5.2: Attempts to Guess the Sign of the Electric Charge of Leading Down Quarks in Two Jet Events at 80 G e V Method of Weight Number Number Fraction Identification Function Correct Guessed Correct Charge of Leading Hadron n / a 35156 58343 0.603 Charge of Leading Hadron, x > 0.5 n /a 11468 16861 0.680 Charge of Leading Hadron n /a 231 308 0.750 Both Jets Agree Charge of Leading Hadron, x > 0.5 n / a 26 28 0.929 Both Jets Agree Sign of Best Quark Guess (1) 81353 100000 0.814 Sign of Best Quark Guess (2) 83401 100000 0.834 Sign of Best Quark Guess (3) 83459 100000 0.835 Sign of Q%t (1) 77710 99846 0.778 Sign of Qfet (2) 79796 99996 0.798 Sign of Qfet (3) 80323 99995 0.803 Sign of Q%„ \QfJ > *f-Sign of Qfet, \Qfet\>*¥* (1) 59013 68726 0.859 (2) 60733 68856 0.882 Sign of Q t , | g& |> A f * (3) 60336 67681 0.891 Sign of Q%t,\Q%t\> 0.447 (1) 38093 41543 0.917 Sign of Qfet, \Qfet\ > 0.323 (2) 38961 41485 0.939 Sign of Qit, \Qfet\ > 0.475 (3) 37716 39807 0.947 Sign of Q ^ , | Q £ t | > 2 x 0.447 (1) 8952 9231 0.970 Sign of Qfet, \Qfet\ > 2 x 0.323 (2) 8416 8564 0.983 Sign of Qfet, > 2 x 0 . 4 7 5 (3) 7256 7367 0.985 Sign of Qfet, \QfJ > 4 x 0.447 (1) 78 79 0.987 Sign of QfeV \QfJ > 4 x 0.323 (2) 31 31 1.000 Sign of Q£t, \Qfet\ > 4 x 0.475 (3) 19 19 1.000 Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 42 Table 5.3: Attempts to Guess the Sign of the Electric Charge of Leading Quarks in Two Jet Events at 80 G e V Method of Quark Number Number Fraction Identification Flavour Correct Guessed Correct Sign of Best Quark Guess down 83401 100000 0.834 Sign of Qfet down 79796 99996 0.798 Sign of Q%v \QfJ > 0.323 down 38961 41485 0.939 Sign of Qfet, \Q%t\ > 2 x 0.323 down 8416 8564 0.983 Sign of Best Quark Guess up 88393 100000 0.884 Sign of Qfet up 90480 100000 0.905 Sign of Qfet, \Qfet\ > 0.323 up 58819 59756 0.984 Sign of Q£t,\Qb\> 2 x 0 . 3 2 3 up 18373 18414 0.998 Sign of Best Quark Guess strange 87275 100000 0.873 Sign of Qfet strange 84066 99998 0.841 Sign of Q%v\Q%t\> 0.323 strange 44081 45597 0.967 Sign of Qfet, | Q £ ( | > 2 x 0.323 strange 10259 10317 0.994 Sign of Best Quark Guess charm 80091 100000 0.801 Sign of Q% charm 87053 100000 0.871 Sign of Qfet, \Q%t\> 0.323 charm 55397 56964 0.972 Sign of Qfet, \Qfet\ > 2 x 0.323 charm 21496 21579 0.996 Sign of Best Quark Guess bottom 80957 100000 0.810 Sign of Q%t bottom 80207 100000 0.802 Sign of Qfet,\QfJ> 0.323 bottom 53063 58028 0.914 Sign of Qfet, \QfJ > 2 x 0.323 bottom 26127 26825 0.974 comparing the accuracy at guessing the different quark flavours. The methods "Sign of Best Quark Guess" refers to guessing the flavour of the leading quark by choosing that flavour (including leptons) whose weight has the largest magnitude, determining the choice of quark/antiquark or lepton/antilepton by the sign of this weight, and using the sign of this quark or lepton as the conjectured sign of the leading quark in the jet. The other methods consist of using the sign of the weighted electric charge for the jets in the event. It is clear that up and charm events are easier to tag than down, strange, or bottom. The reason for this is that (|— < qeJ >) > ( | + < q*1 >), so the peak for charge | Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 43 quarks is farther from the origin than charge — | quark, as can be seen in figure 5.1. It is apparent that we are able to correctly guess the charge of the leading partons in over 80 percent of events. We are also able to correctly identify charge | events with an accuracy of well over 90 percent in more than 40 percent of events, and charge | events wi th an accuracy of over 95 percent in more than 55 percent of events, by guessing only those events in which the magnitude of the weighted average exceeds some cut-off. 5.3 Quark Flavour Retention in Two Jet Events The next question to probe is whether or not it is possible to determine the flavour of the quark and antiquark which started the jet. Again , we subtract the weighted quark and lepton characteristics of one jet from those of the other jet, which by symmetry should on average be equal in magnitude but opposite in sign. The quark flavour characteristic with the maximum magnitude is postulated to be the flavour which began the jet, and the sign of this flavour characteristic determines which jet is postulated to be the quark jet, and which is the antiquark jet. However it quickly becomes apparent that there is a problem with this method. E n -ergies in the range 20 to 100 G e V are certainly high enough that the leading quark-antiquark pair could be either charm or bottom. However, due to the short lifetimes of hadrons containing charm pr bottom, the final state hadrons contain only down, up, and strange quarks. The method thus far presented cannot be used to identify cc or bb events. In addition, contamination from cc or bb events prevents accurate determination of light quark events (although at energies not too far above twice the mass of the heavy quarks, it is possible to tag them by looking for high sphericity events). The source of this problem is that we are attempting to determine a property of the leading quarks (namely flavour) which is not conserved by weak interactions, and such Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 44 interactions occur in many of the decays of the hadrons in the jet. Flavour non-conserving weak interactions result in not just the decays of all heavy quark hadrons before they can be recorded by the detector, but also in the decay of many strange hadrons. The result is not only that the strange characteristic is lost, but along with along with it the down or up characteristics of states are also affected. For example, in the decay K° —* K°, down and strange quark characteristics of +1 and -1 vanish, and in the decay K+ —• p+vfX, up and strange quark characteristics of +1 and -1 vanish. This reduces the mean weighted average of strangeness in 55 events, as is apparent in table 5.4, and increases the standard deviation of all weighted averages. In order to compensate for this, the strange weight of jets is multiplied by a factor 1.25, so that < Sjet > in ss events is similar in magnitude to < Djet > in dd events. If this were not done, the results of guessing the leading quark flavour in dd and uu events would be slightly better, but those for 55 events would be significantly worse. These effects are illustrated in table 5.4, which shows the average and standard de-viations of the weighted flavours of the leading quarks. Although the decays of leading strange quarks result in lower mean weights, the decreased number of strange ss pairs produced in the vacuum also leads to a smaller standard deviation. The reason that the heavy quark weight in bb events is lower than in cc events is that bottom quarks emit both positively and negatively charged fast leptons during their decays, while charm quarks almost exclusively emit positively charged fast leptons. In order to reduce the contamination of light quark events by heavy quark events in which the heavy quarks have decayed, it is noted that some heavy quark events may be tagged by looking for high momentum leptons [18]. Those events with relatively high lepton weight are "tagged" as likely candidates for heavy quark events. In the results shown below, weighted lepton numbers were multiplied by a factor 2.5, then any event in which the weighted lepton value was higher in magnitude than that of any of the Chapter 5. Inferring Parton Quantum Numbers in Hadron Jeis 45 Table 5.4: Weighted Leading Quark Flavour Statistics at 80 G e V Using w(x) = x°" Leading Mean Standard Ratio Quark Weight Deviation (SD/Mean) Down 0.382 0.330 0.864 Up 0.433 0.309 0.713 Strange 0.283 0.292 1.033 Charm -0.226 0.334 -1.473 Bot tom 0.150 0.499 3.330 quark flavours was deemed to be a heavy quark. From the sign of the weighted lepton characteristic, the sign of the charge of the heavy quark was determined. The source of these high momentum leptons is the decay of a W+ or W~ emitted when a heavy quark decays to a lighter quark. Table 5.5 shows how7 events of different flavours were tagged using this method. It should be noted that in events in which no baryons are present, the relation Djet + Ujet + Sjet = 0 should hold because each meson adds some weight to one quark flavour (for the quark), and subtracts the same w7eight from another quark flavour (to account for the antiquark). Many jets do not have any strange quark characteristic, because fewer strange quarks are produced by the vacuum and some of those that are created subsequently decay v ia weak interactions. In such jets, the nonzero quark weightings wi l l be Djet = —Ujet. These events wi l l be tagged as du if Djet > 0, and du if Djet < 0 (provided that they are not tagged as heavy quark events). The results in the table are computed using the weight function w(x) = x 0 3 1 . The two methods used to guess are testing the difference in the lepton and quark weights for the two back to back jets, and considering the two jets separately and guessing only when the results are consistent. The former guesses all events, while the latter guess only a fraction. Several observations may be made about the results in table 5.5. First , strange quark Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 46 Table 5.5: Attempts to Guess the Flavour of the Leading Partons in Two Jet Events at 80 G e V Quark Leading Quark Flavour Guess d d u u s s H(+) H(-) du du d 48398 2013 6056 12051 5682 6889 1661 1639 13190 2421 d 12635 102 678 823 432 911 70 71 9892 1044 u 5297 14039 50564 1107 2933 7999 1577 1521 1841 13122 u 633 1303 12722 46 145 1150 56 53 782 10216 s 20394 3656 3677 16593 46463 1967 1261 1348 3833 808 s 4656 283 388 950 15341 225 47 40 2843 361 c 1787 21866 16266 653 15554 1812 22364 15931 625 3142 c 212 3823 3268 26 4004 211 4548 33 263 2416 b 7422 2023 1628 8592 13105 1237 31235 31999 2214 545 b 1375 148 167 775 3452 115 2755 6363 1455 219 jets are more difficult to identify than down quark jets, which are in turn more difficult to identify than up quark jets. Both of these effects may be traced back to flavour non-conserving weak interactions. The first effect is obvious from the observation that the hadron formed by a leading strange quark has a fairly high probability of decaying before it arrives at the detector. Thus the leading strange quark characteristic is often washed out by some weak interaction. This effect has been partially compensated for by mult iplying the strange weight by 1.25. If the leading quark in a jet is a down quark, then it has probability of about ^ (using the ratio 1:1:0.3 for d:u:s pair production by the vacuum) of being paired with a strange antiquark, with the most probable meson formed being a K°. The quark and antiquark then mix, forming either a K$ or K^, neither of which have net flavour characteristics, so that the flavour of the leading down quark has effectively been lost. If the leading quark is an up quark, it has the same probability of being paired with a strange antiquark, wi th the most probable meson formed being a K+. The mean proper distance travelled by a K+ moving with speed c before it decays is about 371 cm. Thus there is a very Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 47 good chance that a K+ containing a leading up quark wil l not decay. In addition, even if it does decay, the branching ratio to decay into pions with the same up total quark characteristic is just over 28 percent. Since the flavour of a leading up quark is less likely to be washed out by a weak interaction than is a leading down quark, then it is not unexpected that up quark jets are identified correctly somewhat more often than down quark jets. The table also shows that 66 events are much easier to tag as heavy quark events than cc events. There are two main reasons for this. First , the lepton produced by the decay of a bottom quark has, on average, higher momentum than the lepton produced by the decay of a charm quark [18]. Second, if a leading bottom quark decays into a charm quark, which then decays into a light quark, two W bosons are emitted, and either one may decay into a fast lepton which is then used to tag the event as being from a heavy quark. This also explains why in cc events, the charge of the tagging lepton is almost always the same as the leading heavy quark, while in 66 events it is not. From the results in the table, we see that the probability that the flavour of light quark event wi l l be correctly identified (and the quark and antiquark jets correctly identified) is about 48 percent, with another 13 percent of dd and uu events being identified as first generation quark events with the charge correctly tagged. The heavy quark tagging with leptons picks up about 38 percent of cc events and 63 percent of bb events. If no claim is made about those events where the two jets do not agree, then about 80 percent accuracy can be achieved in 16 percent of first generation events, 70 percent accuracy can be achieved in 22 percent of ss events, with about 11.5 percent of cc and 6 percent of 66 events being incorrectly identified as light quark events. One final consideration which should be addressed is the problem of misidentification of particles by the detector. It is not uncommon for a neutral particle (such as a K$ or Ki) to be misidentified as TT 0 . This presents no problem in weighted charge experiments, Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 48 as long as the particle is misidentified as another neutral particle. It also poses l i t t le problem in quark flavour retention, because most neutral particles (mesons in particular) have no net d, u , or s characteristics. For example: TT0 = l/y/2[dd+uu] KGS = 1/V2[ds + sd) In both of the above cases, there is a symmetry between the quarks and antiquarks of each flavour, thus there is no net up, down, or strange characteristic in either of the particles, and hence they do not contribute to the weighted averages for any quark quantum number. Charged kaons, pions or leptons may also be mistaken for one another. This wi l l affect the quark weights, because charged kaons and pions have different quark contents, and leptons do not contain quarks. In guessing leading quark flavours, it is important that the detector is able to correctly identify not just the charge, but also the particle identity. Fortunately, since most errors by the detector are misidentifications of one particle as another particle with the same charge, there wi l l be no effect on weighted electric charge, and hence no effect on charge retention results. Final ly, the detector may also not see some particles. If these are neutral particles, there wi l l be little affect on weighted averages in which the weight assigned is an in-creasing function of momentum (like xk). In this case, the weights assigned to all of the other particles wi l l increase slightly because each wi l l carry a slightly higher fraction of the apparent jet momentum, but it would be rare that this would change the relative weights of the flavours significantly. The weight function in equation 4.17, however, wi l l assign higher weights to all particles with momentum lower than that of the unobserved particle, and wi l l assign lower weights to all particles with momentum higher than that of the unobserved particle (except the fastest particle, which is given the same weight). Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 49 In addition, the shift in weight wi l l be greater for those particles nearest in order (of mo-mentum) to the unobserved particle. This weight function is thus more likely to change the relative weights of the flavours in the jet. If the detector misses charged particles, then this wi l l clearly affect the weighted charge and flavour, especially if the particle not observed has high momentum. However, since charged particles are easier to detect, this should not occur very often. 5.4 C h a r g e R e t e n t i o n i n T h r e e J e t E v e n t s In applying the weighted charge methods to three jet events, we begin by assuming that the gluon is produced by emission from either the quark or antiquark at some time very shortly after they leave the interaction region. It is then reasonable to assume that in most cases, the angle between the gluon jet and the parton from which it was emitted wi l l be the smallest angle between any two of the three jets. We therefore assume that the two jets which subtend the smallest angle between them consist of the gluon jet and either the quark or antiquark jet. These jets wi l l be referred to as the " B " and " C " jets. The other jet, which wi l l now be referred to as the " A " jet, should contain either the quark or antiquark. In the Lund Monte Carlo model, the gluon is emitted from the antiquark jet (although which jet emits the gluon may be ambiguous in practice). As table 5.6 below shows, the assumption that the A jet is not the gluon jet is correct in the vast majority of light quark events, as well as a large majority of heavy quark events. The problem of charge retention in three jet events is very similar to that in two jet events, except that there is no longer a clear symmetry in which the leading particle of one jet is the antiparticle of the leading quark of the other jet. In addition, there is a new type of jet, namely a gluon jet, and there wi l l be interactions between the gluon and quarks which were not considered in the two jet model. We wi l l first attempt to Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 50 Table 5.6: Frequency in which the A Jet is a Quark, Gluon, or Antiquark Jet at 80 G e V Quark Non-Emit t ing Emit ted Emi t t ing Flavour Quark Gluon Antiquark Down 23752 1247 1 U p 23760 1240 0 Strange 23658 1336 6 Charm 21411 3507 82 Bot tom 18922 5771 307 Table 5.7: Frequency with which leading partons have the minimum weighted charge at 80 G e V Quark Non-Emit t ing Emit ted Emi t t ing Flavour Quark Gluon Quark Down 9161 6720 9119 U p 8053 8790 8157 Strange 8868 7168 8964 Charm 8454 8375 8171 Bot tom 7994 8946 8060 determine which of the three jets is the gluon jet. This is done by postulating that since the gluon has zero electric charge, its weighted electric should be smaller than that of the quark and antiquark in the event. Figures 5.2a to 5.2e show the distributions of weighted electric charge for the quark, antiquark, and gluon for 25,000 three jet events of each flavour. In addition, table 5.7 shows the frequency that the smallest weighted charge in magnitude is the quark, antiquark, or gluon jet (assuming that the antiquark emitted the gluon). Clearly from both the figures and the table, this method does not provide accurate determination of which jet is the gluon jet, even in charge | quark jets. The fact that the gluon jet is not necessarily the most likely of the three jets to have the min imum weighted electric charge implies that there must be some interaction between the gluons and quarks in which charge is transferred from one to the other. Note however, that these Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 51 Weighted Electric Charge Figure 5.2: Weighted Charge Distributions For Quark, Gluon, and Antiquark Jets at 80 G e V Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 52 results are from a Monte Carlo based on some model of the hadronization process. In order to determine whether or not these results conform to experimental data, it would be necessary to compare the distribution of weighted charges of the non-gluon jet with those of the other two jets. Table 5.8 shows the results in guessing the sign of the charge of the parton which produced the A jet in events using several methods similar to those applied to two jet events. The results are not as good as in the two jet case, partly due to the events in which the A jet is not the quark jet. but also due to the more complicated interactions in the three parton hadronization process. In some fraction of the events, however, it is possible to accurately determine the sign of the charge of the A jet. 5 . 5 Quark Flavour Retention in Three Jet Events Determination of leading quark flavours is, as would be expected, more difficult than in two jet events. This is partly because we cannot be sure which jet is the gluon jet and which are quark or antiquark jets. As with three jet charge retention, the interactions are more complex, and with three leading partons the events do not possess the same symmetry that two jet events did. Table 5.9 shows the results of attempts to determine the flavour of the quark in the A jet, in which two methods are employed. The first is to guess the flavour based on the sign and magnitude of QA-jet — QB-jet — Qc-jet, while the second is to consider both QA-jet and Qs-jet + Qc-jet separately, and guess only when the two are consistent. Not only are the rates at which guessing is successful lower, but the heavy quark events are also more difficult to identify. Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 53 Table 5.8: Results of Guessing the Sign of the Electric Charge of the A Jet at 80 G e V Method of Identification Quark Flavour Number Correct Number Guessed Percentage Correct Sign of gjf-iet Sign of Qi_jet - Qi_}et - Q'<_jet Sign of Qi_jet - Q*_jet - Q«_jet Magnitude > 0.323 Down Down Down 18259 18132 10775 24827 25000 13017 0.735 0.725 0.828 Sign of Q«_iet Sign of Qi_jei - Qi_jet - Q*_iet Sign of Q*_itt - Qi_iet - Q$_jet Magnitude > 0.323 Up Up Up 20910 20626 14067 24899 25000 15294 0.840 0.825 0.920 S l g n of Q«_jet Sign of Qi_jet - Q%_jei - Qi_3et Sign of Qi_iei - Qi_Jtt - Q$_jut Magnitude > 0.323 Strange Strange Strange 19099 19007 11670 24819 25000 13498 0.770 0.760 0.865 Sign of Q«_jet Sign of Qi_3et - Qi_jet - Q$_jet Sign of Qi_]ei - Qi_jet - Q«_iet Magnitude > 0.323 Charm Charm Charm 20127 20294 13559 24852 25000 15076 0.810 0.812 0.899 Sign of Q«_jet Sign of Qi_jet - Q*'_jet - Q*<_jet Sign of Qi_jet - Qi_3et - Qt_jet Magnitude > 0.323 Bottom Bot tom Bot tom 19418 20034 13676 24832 25000 15679 0.782 0.801 0.872 Table 5.9: Attempts to Guess A Jet Flavour in Three Jet Events at 80 G e V Quark Leading Quark Flavour Guess d d u u s s #(+) H(-) du du d 10362 1655 2913 4984 1134 1427 230 266 1490 539 d 3073 174 504 568 64 142 11 15 1081 284 u 2782 5099 11149 1246 862 1486 224 231 426 1495 u 467 681 3323 125 39 168 10 12 211 1147 s 5146 3071 2157 6447 6584 570 203 191 447 184 s 1359 528 362 1103 1729 55 11 13 316 91 c 1203 6309 4031 1183 2504 425 7575 1254 138 378 c 168 1624 1095 125 491 43 822 11 73 295 b 2418 1624 1092 3362 1859 369 5561 8265 325 125 b 517 226 144 558 321 39 548 1068 198 46 Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 54 5.6 Effect of Energy on Quantum Number Retention It was shown in chapter 4 that in a jet of N particles, the weight function W{ = n~^'t1 had a distribution such that the ratio was proportional to The source of this contribution to was the randomness with which quark-antiquark pairs were created from the vacuum. It was therefore reasoned that, since the number of particles increases with increasing Ejet, in the asymptotic l imit the weighted average should determine the properties of the leading quark and antiquark. Figures 5.3(a) and 5.3(b) show the distribution of weighted electric charge and the ratio a s a function of where A r is the mean number of particles in a jet, for the weight function w(x) = x 0 , 3 1 . The first figure shows that as the energy is increased (and the number of particles as well), the distribution of weighted charge does give rise to a slightly more well defined peak. The second figure shows that does increase as =^= in the range 20 to 100 G e V . Tables 5.10 and 5.11 show results for guessing the leading quark charge and flavour in two jet events at various energies. It is clear that as the energy is increased, the success of using the weighted averages to infer properties of the leading partons increases. As well as more quark-antiquark pairs from the vacuum, at higher energies the outgoing hadrons have a greater separation of energies (since the number of particles only grows more slowly than Ejet), so that the effect of decays mixing the order of the quarks in the jet wi l l be reduced: Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 55 Table 5.10: Attempts to Guess the Sign of the Electric Charge of Leading Quarks in Two Jet Events at Various Energies Method of C M S Quark Number Number Fraction Identification Energy Flavour Correct Guessed Correct Sign of Q%t 20 G e V down 18565 24988 0.743 Sign of Qfet 40 G e V down 19373 24997 0.775 Sign of Qfet 60 G e V down 19664 25000 0.787 Sign of Qfet 80 G e V down 19978 24998 0.799 Sign of Qfei 100. G e V down 20124 25000 0.805 Sign o{Qfet, \Qit\ > 0.323 20 G e V down 10118 11550 0.876 Sign of Ql^, > 0.323 40 G e V down 9972 10881 0.916 Sign of Qfet, \QU > 0.323 60 G e V down 9847 10615 0.928 Sign of Qfeu \QU > 0.323 80 G e V down 9753 10343 0.943 Sign of Qfet, \QU > 0.323 100 G e V down 9653 10209 0.946 Sign of Qfet 20 G e V up 21318 24998 0.853 Sign of Qfet 40 G e V up 22002 24999 0:880 Sign of Q%t 60 G e V up 22375 25000 0.895 Sign of Qfet 80 G e V up 22605 25000 0.904 Sign of Qfet 100 G e V up 22760 25000 0.910 Sign of Qfet, \QU > 0.323 20 G e V up 14245 14906 0.956 Sign of Qfet, > 0.323 40 G e V up 14560 14967 0.973 Sign of Qfet, > 0.323 60 G e V up 14495 14778 0.981 Sign of Qfei, \Qfet\ > 0.323 80 G e V up 14766 15005 0.984 Sign of Qit, \QU > 0.323 100 G e V up 14751 14940 0.987 Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 56 Table 5.11: Attempts to Guess the Flavour of First Generation Leading Fermions in Two Jet Events at Various Energies Energy Leading Quark Flavour Guess Quark d d u u s s #(+) H(-) du du 20d 2352 21 230 114 66 322 24 19 4358 759 40d 2764 29 196 191 88 269 20 25 3283 422 60d 2956 28 199 193 88 234 16 12 2780 320 80d 3169 24 161 213 112 218 24 18 2481 255 lOOd 3278 26 170 215 108 240 15 16 2303 206 20d 9160 596 1762 2671 1026 1997 342 311 5649 1486 40d 10781 570 1618 2891 1239 1857 351 390 4365 938 60d 11509 537 1640 3035 1360 1765 345 421 3684 704 80d 12092 511 1477 3089 1362 1712 412 412 3327 606 lOOd 12341 428 1493 3091 1537 1761 416 386 3062 485 20u 229 138 2450 . 18 48 303 20 19 660 4479 40u 174 256 2815 23 61 283 14 10 359 3343 60u 168 271 3089 10 51 324 14 20 255 2852 80u 154 347 3164 13 38 278 13 11 215 2606 lOOu 158 368 3292 9 37 283 19 8 189 2363 20u 1527 2768 10019 456 772 1859 322 279 1309 5689 40u 1449 3205 11584 340 780 1916 338 354 793 4241 60u 1344 3372 12098 310 743 2043 383 400 578 3729 80u 1307 3545 12634 293 680 1918 371 384 490 3378 lOOu 1296 3533 12959 236 720 1983 380 408 442 3043 Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets 5 7 3000 (a) Weighted Electric Charge 0.2 0.22 0.24 0.26 (b) log(l/sqrt(N)) Figure 5.3: Energy Effects on Weighted Charge Distribution Chapter 6 Conclusions From the study which has been done, several conclusions may be drawn. In two jet events, determination of the sign of the charge of the leading quarks can be achieved with very good accuracy, looking only at the last generation of particles, and there is a substantial fraction of the events in which it becomes very precise. Determination of the leading quark flavour in two jet light quark events is moderately successful, and again there is a fraction of the events in which it is very successful. Difficulties arise from flavour non-conserving weak interactions, which manifest themselves by washing out the flavour of the leading quark, and contaminating the results with heavy quark events. The latter may be partially compensated for by tagging heavy quark events using the weighted lepton characteristic of the jet. Improved results could be expected if other methods of tagging heavy quark events were implemented. In addition, if some of the decays could be traced back in the event, this would decrease the transverse momentum of particles in the jet, and eliminate some of the weak decays. It is worth noting that the mobili ty of the leading quark in momentum space is very important in determining the mean weighted charge. It may therefore be possible that this could be used (computing weighted averages with different weight functions), to determine the mobili ty of the momentum of the leading quark in a process similar to an integral transform. The two jet methods were attempted on three jet events, and found to be less suc-cessful. It was not possible to determine the identity of the gluon jet with reasonable 58 Chapter 6. Conclusions 59 accuracy, which implies that there is some transfer of quantum numbers in the inter-actions between the gluon and the quarks. It was possible, however, to determine the charge and flavour of the quark or antiquark which started the A jet with fair accuracy in a smaller subset of the events. As the C M S energ)' of the interaction was increased, results showed definite improve-ments in both charge and flavour retention determination. A t these higher energies there are more vacuum quark pairs, so that effects due to the statistical nature in which dif-ferent quark-antiquark flavours are created from the vacuum decrease. Also, the decays of the first generation of hadrons are less likely to mix the order of quarks in the mesons at higher energies. Finally, as the energy increases, there is a greater correspondence between the leading partons and the number of jets in the event. Bibliography [1] R . P . Feynman. Very High Energy Collisions of Hadrons. Physical Review Letters, 23(24):1415, 15 December, 1969. [2] R . N . Cahn and E . W . Colglazier. Quantum Numbers and Quark-Parton Fragmenta-tion Models. Physical Review D, 9(9):2658, 1 May, 1974. [3] S.J. Brodsky and N . Weiss. Retention of Quantum Numbers by Quark and M u l t i -quark Jets. Physical Review D, 16(7):2325, 1 October, 1977. [4] G . R . Farrar and J . L . Rosner. Question of Direct Measurement of the Quark Charge. Physical Review D. 7(9):2747, 1 May, 1973. [5] J . D . Bjorken. Hadron Fina l States in Deep Inelastic Processes. In J . G . Korner, G . Kramer, and D . Schildknecht, editors, Current Induced Reactions, Springer-Verlag, 1976. [6] R . P . Feynman. What Neutrinos Can Tell Us About Partons. In A . Frenkel and G . Marx , editors, Neutrino '72, Volume II, Omkdk-Technoinform, 1972. [7] J . Kogut and Leonard Susskind. Vacuum Polarization and the Absence of Free Quarks in Four Dimensions. Physical Review D, 9(12):3501, 15 June, 1974. [8] S . M . Berman, J . D . Bjorken, and J . B . Kogut. Inclusive Processes at High Transverse Momentum. Physical Review D, 4(11):3388, 1 December, 1971. [9] A . Casher, J . Kogut, and Leonard Susskind. Vacuum Polarization and the Absence of Free Quarks. Physical Review D, 10(2):732, 15 July, 1974. 60 Bibliography 61 [10] T. Sjostrand. The Lund Monte Carlo for Jet Fragmentation. Computer Physics Com-munications, 27(3):243, September, 1982. [11] T . Sjostrand. The Lund Monte Carlo for e + e " Jet Physics. Computer Physics Com-munications, 28(3):229, January, 1983. [12] T . Sjostrand. The Lund Monte Carlo for Jet Fragmentation and e+e~ Physics— J E T S E T Version 6.2. Computer Physics Communications, 39(3):347, April, 1986. [13] T. Sjostrand. A Manual to The Lund Monte Carlo for Jet Fragmentation and e +e~ P h y s i c s — J E T S E T Version 7.1. May, 1989. [14] S. Bethke et al. Experimental Investigation of the Energy Dependence of the Strong Coupling Strength. Physics Letters B, 213(2):235, 20 October, 1988. [15] R . P . Feynman. Photon Hadron Interactions. W . A . Benjamin, Inc., 1972. [16] M . J . Teper. Identifying Jet Quantum Numbers Event by Event. Physics Letters B, 90(4):443, 10 March, 1980. [17] R . D . Fie ld and R . P . Feynman. A Parametrization of the Properties of Quark Jets. Nuclear Physics-B, 136:1, 1978. [18] J . H . K u h n and P . M . Zerwas. Heavy Flavours. Z Physics at LEP I, Volume I, C E R N Report 89-08:267, 21 September, 1989. 

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