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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 N U M B E R S IN HADRON JETS By Scott Kelly Hayward B.Math., University of Waterloo, 1987 B . S c , University of Waterloo, 1988  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF MASTER OF SCIENCE  in T H E FACULTY OF GRADUATE STUDIES PHYSICS  We accept this thesis as conforming to the required standard  T H E 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  DE-6 (2/88)  \Z  OCT&^&g-j^o  Abstract  T h e retention of quantum numbers in hadron jets was studied, and a model was constructed to infer the quantum numbers of the leading quark and antiquark i n the the reaction e e~ +  —» qoQo —• hadrons, from the quantum numbers of the final state particles.  T h e 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 kinematical variables. The model was tested using the L u n d 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 i n 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 i n 85 percent of events, with heavy quark contamination reduced to 12 and 6 percent of cc and bb events respectively. T h e model was also run on three jet events at 80 G e V . A l t h o u g h 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. T h e sign of the electric charge of the non-gluon jet could be determined in 70 percent of events overall, and i n 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 2  1  Characteristics of Jets in  e e~ +  3  —> hadrons  2.1  Distribution of Particles in the Jet  5  2.2  Dynamics of Fragmentation in Space-Time  .  3 The Lund Monte Carlo Program  7  10  3.1  Generation of e e~ A n n i h i l a t i o n Events in L u n d  11  3.2  Multijet Production Schemes  14  3.3  Analysis of L u n d Generated Events  16  3.4  Characteristics of the L u n d Monte Carlo  17  +  4 Modelling the Process  22  4.1  Properties of Quarks Produced from the V a c u u m  22  4.2  Weighted Average of Jet Q u a n t u m Numbers  23  4.3  Extending the M o d e l  4.4  Choosing a Weight Function  "  28 32  ,iv  5  6  Inferring Parton Quantum Numbers in Hadron Jets  36  5.1  C o m p u t i n g the Weighted Charge for the Jet  36  5.2  Electric Charge Retention in T w o Jet Events  37  5.3  Quark Flavour Retention in T w o Jet Events  43  5.4  Charge Retention in Three Jet Events  49  5.5  Quark Flavour Retention i n Three Jet Events  5.6  Effect of Energy on Q u a n t u m Number Retention  Conclusions  .  52 54 58  Bibliography  60  v  List of Figures  2.1  Hadron Jet Producing Events  2.2  Theoretical Distribution of Charge in Rapidity  2.3  Light Cone Geometry of e e~  3.1  Free Decay and Annihilation of Heavy Quarks  3.2  Average Number of Particles Per Event as a Function of log(Ej )  3.3  Distribution of Momentum of the Particle Containing the Leading Parton  +  4  —» hadrons  .  7  at High Energy  8 '  15 et  . . . .  in the First and Last Generations at 80 GeV  20  21  4.1  Naive Model of Hadron Production in e e~  4.2  Realistic Model of Hadron Production in e e~  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  5.3  24  —> qq —> hadrons  +  +  29  —> qq —> hadrons  80 GeV  51  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  .  3.2  Number of Jets Generated for Initial Parton Configurations in dd Events  5.1  Mean and Standard Deviation of the Difference Between Quark and A n -  19  tiquark Jet Weights at SO G e V 5.2  38  Attempts to Guess the Sign of the Electric Charge of Leading Down Quarks in T w o Jet Events at 80 G e V  5.3  41  Attempts to Guess the Sign of the Electric Charge of Leading Quarks in T w o Jet Events at 80 G e V  42  5.4  Weighted Leading Quark Flavour Statistics at 80 G e V Using w(x) = x -  5.5  Attempts to Guess the Flavour of the Leading Partons in T w o Jet Events  0  3 1  at 80 G e V 5.6  45  46  Frequency in which the A Jet is a Quark, Gluon, or A n t i q u a r k Jet at 80 GeV  5.7  18  50  Frequency with which leading partons have the m i n i m u m 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 i n T w o Jet Events at Various Energies  55  5.11 Attempts to Guess the Flavour of First Generation Leading Fermions in T w o Jet Events at Various Energies  vii  56  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.  Vill  Chapter 1  Introduction  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 i n a hadron jet. We w i l l be concentrating on the process e  +  + e~ —> hadrons at energies ranging  from 20 to 100 G e V . Events will be generated by the L u n d Monte Carlo Jetset 7.1, and analyzed using several models presented herein of how the sea quarks are arranged i n 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 L u n d . In Chapter 2, we will 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 L u n d 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 i n the jet. The study which follows i n Chapter 5 attempts to determine the frequency w i t h 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 will 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 will be presented, and a comparison of the accuracy achieved on the range 20 to 100 G e V will 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. T h e particles typically have high momentum along the axis of the cone, with l i m 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 w i t h 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 i n 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. T h i s 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 +  5  —• hadrons  the centre of momentum from which the particle and antiparticle emerge, with momenta equal i n 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 + cos (8CM) 2  to the incoming e  +  with respect  and t~. This distribution is consistent with the quarks, being s p i n - |  particles. The limited 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,  _  x =  Pz + Po —  •  Pjet + tijei where (po,p) and (Ej ,0,0,pj ) et  p  as E  z  jet  - » oo for  p > p z  Q  . (2.1)  Pj t e  are the four-momenta for the particle and jet respec-  et  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 Ej , et  then the probability of finding a particle of type i , with  mass Hi, in the low momentum region is of the form f (Q,^-)dp d*Q dN- = — ^ t  where i n the asymptotic limit, f(Q,x) x dependence, of dNi becomes  z  = Fi(Q) is independent of x,  2 2) —> ^ , and the  Chapter 2. Characteristics  of Jets in e e  hadrons  +  6  dx  dNi a — x If we now integrate from x = — ^ = — « Tp— up to x = 1 we obtain  Ni cx f \  — = /  (E  0 5  j r t  (2.3)  ) - / 0(£)  (2.4)  O  Thus, the number of particles in a jet is expected to grow logarithmically w i t h the energy of the jet. In addition, we may define y = log(x), so that equation 2.3 becomes  dN, —;— = constant dy  , . (2.5)  which shows that the number of particles is constant i n y i n 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) l  °9l  + Po  Pz  Pjet ~t~ Ej t e  -i7(l  + v)-  'jet +i Ej Pjet rjjei et  logKr^)*] i — v  y = tanh- (v)-log[ 1  - log? Piet  Pi +  Ejet  ]  (2.6)  Pi The  last step is a mathematical identity, which can be shown by writing out v i n terms  of y (using the definition of tanh(y)), defined to be tanh~ {v), l  and calculating the log(j^-).  Rapidity, which is  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  <  ——==• parton fragmentation  current plateau  i>  parton 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 q and antiquark q~o leave the interaction region at nearly Q  the speed of light along the positive and negative z-axes. If the total centre of mass energy of the interaction is 2Ej , then the leading quark and antiquark will be in frames et  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 E  jei  (in particular  =  E /m ). jei  qo  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 Q E D calculation which leads to the 1-\-COS (6CM) 2  angular distribution of the jets. Lorentz invariance of the process implies that hadron emission occurs near the hyperboloid t — z — <P (points on this hyperboloid are mapped 2  2  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 t — z < d , and proceed on the supposition that at some small time near 2  1  2  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  Characteristics  2.  of Jets in  ee +  —•  9  hadrons  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 = v t, and it bursts at g  time t = -f d. Squaring both equations and eliminating v , one finds that g  g  d  2  t = i]S 2  =  d  2  (i-S)  t -z 2  = d  2  2  (2.7)  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  T h e L u n d M o n t e Carlo P r o g r a m  T h e 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. T h e 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 L u n d Monte Carlo. There are several Monte Carlo programs on the market which could have been used. T h e one chosen was L u n d Jetset, version 7.1. It is probably the most common, and was therefore easily obtainable. It has proven to be very successful i n 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. W h a t follows is a discussion of the L u n d Monte Carlo [10, 11, 12, 13], and a description of programs used with it.  10  Chapter 3. The Lund Monte Carlo Program  3.1  Generation of e e +  11  Annihilation Events in Lund  The L u n d 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 will give approximate answers to quark gluon production. In addition, L u n d 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 L u n d 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 L u n d 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 L u n d will chose one with a probability distribution based on a theoretical model for the process at the energy of the interaction. The manner i n which gluons appear in the interaction depends on parameters set i n the program. B y default, they are produced by the L u n d 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 L u n d program, with the direction of the outgoing partons determined by the theoretical angular distribution. In the L u n d 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 free parameter i n L u n d and may be adjusted).  — Q.2GeV (this is a 2  A s the string stretches, there is some  probability that it will break into a quark-antiquark pair.  T h e new quark w i l l then  be attracted to the leading antiquark q , and the new antiquark to the leading quark. 0  The process continues with many quark-antiquark pairs being formed and appearing on average, but with large fluctuations, on the spacetime hyperboloid t — z = d . 2  2  2  The process is viewed from the frames of the leading quark and antiquark, i n 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 ± p (with a z  numerical subscript to denote the iteration). A t iteration zero, they are defined to be  W±(0) = E  jet  ±p  (3.1)  z  This choice of variables is chosen so that the fragmentation scheme is covariant w i t h respect to boosts along the z axis at each step. In addition, at each iteration the product of these is VK+W'- = E — p\ = m + p\ = m]_, where p 2  2  L  is the transverse momentum.  Beginning with W+( ) and W_( ), the generation of hadrons is done iteratively. Either 0  0  the quark or antiquark is chosen at random to generate the next particle. A new q u a r k 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 i n 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, L u n d 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. T h e 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 determined. 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) -bm 2 f(x)dx = ^ ^-exp( exp[ "" )dx x x a  L  (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±_ w i t h a Gaussian distribution f (p±) q  where  7  2  Mpx)dp±  = -exp(-^)d K  X  (3.3)  P±  Once these have been chosen, the light cone variables for the next iteration are given by  W  = (1 - x  +{i+1)  =  W.  (t)  -  +{i+1)  )W  m  (3.4)  +(i)  l  (  ' - y  (3.5)  This procedure iterates to produce new particles until at some step the remaining energy is small, and W ( „ ) W _ ( ) < W^ , where W +  n  in  min  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 i n the generation of the event is complete.  Chapter 3. The Lund Monte Carlo Program  14  T h e 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 L u n d picks the decay and computes the momenta 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 i n the Kobayashi-Maskawa mixing m a t r i x ) . 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)  Qq -> q q3q q\ free decay x  2  A  (b) Qq~i — > lvq q~\ free decay 2  (c)  Qq\ —» q q 3  (d) Qq~i — > Iv  3.2  2  annihilation annihilation  Multijet Production Schemes  Production of events with three or more jets may be done in either one of two ways with L u n d . T h e default method consists of an iterative parton fragmentation scheme which allows the leading quark and antiquark to undergo a strong fragmentation process producing gluons and possibly other quark-antiquark pairs. This is done v i a three processes, 9 ~* QQi 9 ~* 99i  a  n  d 9  qq, and may result in the production of many jets.  T h e second possibility uses second order Q C D matrix elements which have been calculated and input into the M o n t e 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  (c)  annihilation  15  (d)  annihilation  Figure 3.1: Free Decay and Annihilation 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 fragmentation 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 fragmentation 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  3.3  16  A n a l y s i s of L u n d G e n e r a t e d Events  A m o n g the L u n d event analysis routines, there are two which reconstruct jets from the final state hadrons.  T h e first is a cluster algorithm which builds clusters of particles  w i t h momenta in the same direction, while the second analyzes energy deposition i n calorimeters. A l l jet reconstruction that follows will 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  m j = E{Ej(l — cosdij) 2  (3.6)  If the two clusters with the m i n i m u m invariant mass-squared satisfy the condition  m  where y  cut  lj  <  m  L  = VcutE  2 t  (3.7)  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 until no two clusters satisfy the condition given in equation 3.7. In practice, it is possible that some of the low momentum particles will 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  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 will 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 will be nearly perpendicular to the other two jets (otherwise the cluster analysis would put them in the correct cluster), and will 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 will be small. In what follows, reference will be made to the first and last generation of particles in an event. T h e first generation of hadrons is defined to be those produced in the second step of the L u n d Monte Carlo program, by the fragmentation of the quarks and gluons and  before any decays have taken place. T h e last generation of hadrons and leptons  will 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 initial 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 will 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 Flavour down down down up up up strange strange strange charm charm charm bottom bottom bottom  Number of Partons 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4  Number of Jets 2 3 4 82241 3415 0 17627 93253 342 132 3428 8518 82300 3458 0 17572 93555 421 3394 8365 128 82189 3401 0 17690 93226 390 121 3466 8344 83207 3938 0 16674 95792 455 119 3159 8706 86589 3483 0 13285 98556 458 126 2326 9434  > 5 0 0 7 0 0 5 0 0 6 0 0 10 0 0 11  Table 3.1 shows the number of jets produced by initial configurations of two to four partons when the scaled mass cut-off parameter y  cut  = 0.02. From the table it is clear  that the number of jets produced corresponds well to the number of leading partons. If y  cut  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 y  cut  the number of jets would decrease. These changes in y  cui  was raised, then  would produce shifts of numbers  of events in table 3.1 towards the right and left respectively. In two jet events, if the initial 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 initial 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 (GeV) 20 20 20 40 40 40  Number of Partons 2 3 4 2 3 4 2 3 4 2 3 4 2  60 60 60 80 80 80 100 100 100  3 4  2 48577 50633 790 68760 30871 369 78536 21257 207 82241 17627 132 91122 14514 109  Number of Jets 3 4 47785 6011 168690 61140 13537 20853 15005 39 99910 5600 6448 8969 6780 0 1214 89571 4191 7743 3415 0 93253 342 3428 8518 1966 0 95333 146 2701 8755  > 5 15 3257 3803 0 25 238 0 0 26 0 0 7 0 0 2  Table 3.2 shows the number of jets produced in dd events with different initial 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 initial partons ancl the number of jets produced. It is also observed that at the lower energies, the parameter y  cut  should be raised because the jet algorithm analysis finds many low parton  multiplicity 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 i n 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 light quark events (10 of each light flavour). C h a r m and bottom quark events 5  5  20  Chapter 3. The Lund Monte Carlo Program  3  3.5  4  4.5  Log(E) Figure 3.2: Average Number of Particles Per Event as a Function of log(E ) jet  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  -0.2  0 0.2 0.4 0.6 0.8 Momentum Frection (First Gen)  -0.2  0  0.2  0.4  0.6  1  21  -0.2  -0.2  0.B  Momentum Fraction (First Gen)  0 0.2 0.4 0.6 0.8 Momentum Fraction (Last Gen)  0  0.2  0.4  0.6  1  0.8  Momentum Fraction (Last Gen)  600 IT.  £ c  400  3  cf  200 -0.2  0 -0.2  0  0.2  0.4  0.6  0.6  1  0  0.2  0.4  0.6  0.8  Momentum Fraction (Last Gen)  Momentum Fraction (First Gen) Figure 3.3: Distribution of M o m e n t u m of the Particle Containing the Leading Parton i n the First and Last Generations at 80 G e V  1  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 Z mass, tree level QED calculations show that the probability 0  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 i n quarks produced from the sea, by summing over the probabilities of the vacuum producing each possible flavour as follows:  < X >=  p(flavour)X(flavour)  ^2  v  (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 P {dd) : P (uu) : P (ss) = 1 : 1 : a, where the L u n d default value of the v  v  v  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 =  ° - " 69 3  ( 4  "  2 )  Similarly, one can compute the mean square and variance for the electric charge of sea quarks:  , rxehg s2 . <^  ) -(2  <  H  G  1  1  + a)  {  4.2  / N2 ,  1  >  F  =  3  [  <  (  K  j  H  9  A  2  (2 + a ) 3  +  l  ?  > _<  X  F  J  G  , +  Q  / x,2 _ ( + ) i  (2 + a )  5  1  (  >*=0S±  Q  3> ~ 9(2 + a)  |  Q=Q3=  i |  _ | o = 0  - " 207 (4.3) 3  .4)  (4  Weighted Average of Jet Q u a n t u m N u m b e r s  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 i  th  we define A to be the weighted average for the jet, so that:  meson in the jet, m,-,  Chapter  4.  Modelling  the Process  25  (4.5)  A = 5^iw,-[A(m,-)]  In the naive model, the meson m, contains quark qi-i and antiquark q , so that this t  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 >= (A — < 0  Xy  (4.8)  >)wi  Using equation 4.7, we may compute the mean square and variance for A.  N A  2  =  wl\  2  +  0  + 2A i«i 0  N t-1 2  N k(w  - w{) +  i+1  tfi i+i w  i=i  ~  w  i)  2  i=i  XiXj(w -  Wi)(wj+i  i+1  (4.9)  - WJ)  t'=l j = l  N 2 2 < A >= ( A - < A„ >) w\ - a vw\ + 2a £ w {wi - tu< ) 2  2  0  v  {  +1  (4,10)  t=i TV  a\A) =< A > - < A  =a  (4.11)  2 v  i=i  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 m i n i m u m with respect to variations of a l l ./V weights.  r = 4^ < A>  (4.12)  Provided that < A > ^ 0, then one can show that:  ST  Sw,  .  = 0  (An-  < A„ >)a  S< A > 2  < A> —  2 v  8WA  a  2 J  2< A >  S< A> SWJ  = 0  (4.13)  A p p l y i n g this result with the above expressions gives:  ST  0  <\  t=l  r  ST (A -  (4.14)  < A >)ol Swi  (An-  v  wj-i - 2  = 0  >)o~l SWJ  Wj  ST  1  ( A - < A„ >)a Sw 2  0  v  J+1  0  =0  for 2 < j < N - 1  2WN — IWA'-I = 0  (4.15)  (4.16)  N  The condition •—• = 0 implies that U>N-2 = 3 WN,  +w  IOAT_I =  2 IOJV, while  ^ = 0 then implies that  = 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 i n 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 i n equation 4.14 above. A p p l y i n g these weight functions to the mean and variance for the jet weighted average gives:  < A >= A - < A > 0  r  =  J ^ >  <A>  (4.18)  v  °«  *  (  4.i9)  (\ -<\ >)y/N 0  v  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 i n the jet, less < X > , the v  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 i n the jet, multiplied by a constant which  is dependent on the properties of the vacuum. T h e average number of particles i n 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 limit (ie high energy, and hence large particle multiplicity). It may be noted that i n 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 ^ > = 0. Then the distribution of weighted charges over many events would be 5  peaked at the charge of the leading quark. In the SU(2) symmetric model, i n which only dd and uu pairs are produced, and i n equal quantities (so a = 0), equation 4.2 gives  < Xl >— | . T h e n the peaks of the weighted charge distribution would be at | — | = | hg  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 v i a the argument [16] that in the absence of strange and heavier quarks, all mesons are left invariant under the combined operations of isospin rotation 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  T h e above model is clearly an oversimplification of what really occurs in the production of a quark jet. It assumes that the leading quark is i n 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 initially 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 t — z 2  2  = d , and the low momentum ones are distributed uniformly in 2  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 L u n d Monte Carlo program produces baryons by assuming that there is some probability that a gluon, instead of bursting into a quark-antiquark pair, will form a diquark-antidiquark pair (similar to the model of C a h n 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 < X > and cr , using equations 4.1 to 4.4 with the extended definition 2  v  of vacuum quark flavours, without otherwise complicating the model. T h e 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 i n the ratio of baryons to mesons, and hence a small effect. The decay of the first generation of hadrons will 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 will be an increase in the transverse momentum of  particles throughout the jet, and this will particularly affect the low momentum hadrons. Because most decays introduce new up and down quarks, but not strange quarks, there 7  w i 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 i n 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 i n the more realistic model, the weight function computed i n equation 4.17 will 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 i n momentum). T h i s 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 i particle in the j  ih  th  jet by  ^  =  Pij-Piet  (  4  2  (  )  )  Pjet • Pjet A';  where p~ = Y.P'J t  The weighted average of the jet i n the j  Aj = £  ih  event is then given by:  {x )[X{q{x ))  w  t3  (4.21)  tJ  + \(q{xij))]  (4.22)  t'=i  where X(q(xij)) the i  th  and X(q(xij))  particle i n the j  th  are the quantum numbers of the quark and antiquark i n  jet. W h e n averaged over many events, this weighted charge  becomes:  < > = ^ jjlEE ">[xij)[K<l(xij)) + KfaiM A  (4.23)  T h e 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  < A >=  32  Cdx n(x)w(x)[X(q(x)) + X(q(x))] Jo  (4.24)  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.  W e define the function p(x) to be the probability that a particle with 7  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 initial 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 + [1 0  p(x)} <X >  (4.26)  V  X(q(x)) = p ( - i ) ( - A ) + [1 - p(-x)](- < X >) 0  < A >= ( A - < 0  4.4  v  X >) ( dx n(x)w(x)\p(x) - p(-x)} v  (4.27)  (4.28)  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.  M a n y 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  (4.29)  w(x) = x  k  where A; is a constant which could be varied. Values of k between 0.3 and 0.35 showed the greatest success when y  cui  = 0.02, with higher values of k being more successful for  jets defined by a smaller setting of the parameter  y . cut  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 i n the last section, so that T = z££l could be extremized. However, it was found that <x(A) was dependent on the amount of m i x i n g 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) = x . k  One last possibility is an extension of the naive model which produced the weight function w = t  A  ^j" " . W h e n this weight function is applied to equation 4.7, it becomes 1  1  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 i n 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 i n some way) of the next slowest particle to one with longitudinal momentum x. W e now define r(x,y) to be the probability that given a particle with longitudinal momentum fraction y, then the next slowest particle will have fraction x. T h e normalization condition required of r(x,y) is  rv  / dx r(x,y) — I — p i (y) Jo a  (4.32)  ow  where p i w(y) is the probability that a particle with longitudinal momentum fraction y s  0  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 i n equation 4.31 equal to one, we obtain  w(x ) - ] T r(xj, Xi)w( ) = 1 x  Xj  for 0 < i < K - 1  (4.33)  j=0  T h e solution of this equation can be easily verified to be  W  {  X  i  )  =  l-rte.s,-)  ( 4  -  3 4 )  Figure 4.3 shows a plot of a weight function computed with the longitudinal moment u m 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 x - + Xj < 1 t  (4.35)  Chapter 4.  Modelling  the Process  i  i  0  i '  35  ' '  1  50  'I  '  ' i  100  i  i  i  i  1  150  i i  i  i  200  1  i  i  '  i I i  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 will 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 i n those jets. The properties we will try to infer are the sign and magnitude of the electric charge of the leading partons, as well as the flavour. We will 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. T h e 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)  7 T = 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). T h e weighted quark and lepton characteristics of the particles i n a jet are then computed 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 x - would be t  A'  D  = £w(x )d(m )  (5.2)  y  jet  i  i  t=i  with Uj t, Sjet, and L e  jet  defined similarly. T h e weighted electric charge can easily be  extracted from the weighted quark and lepton characteristics v i a /  1  2  Q%t - -^(Djet + Sjet) + gtVjet ~ Ljet 5.2  (5.3)  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. T h e mean, standard deviation, and their ratio for the weighted electric charge of events of each of the five flavours, using weights ~^~ " (1), a ; Ar  1  1  0 3 1  (2), and the weight function i n equation 4.34 (3),  is given i n table 5.1. From the results i n the table, we see that weight function (3) appears to be the best of the three. However as will 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), i f 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 Flavour  Weighting Function  down down down up up up strange strange strange charm charm charm bottom bottom bottom  (1) (2) (3) (1) (2) (3) (1) (2) (3) (1)  (2) (3) (1) (2) (3)  Mean . 0.323 0.243 0.349 0.501 0.383 0.525 0.365 0.283 0.380 0.523 0.382 0.611 0.501 0.374 0.505  Standard Deviation 0.427 0.292 0.411 0.426 0.291 0.410 0.418 0.285 0.406 0.505 0.345 0.500 0.645 0.439 0.617  Ratio (SD/Mean) 1.324 1.203 1.175 0.851 0.759 0.780 1.146 „ 1.005 " 1.067 0.964 0.905 0.818 1.287 1.176 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 i n 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 i n 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 w i t h 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 i n the jet has positive and which has negative charge with some accuracy. In addition, one would expect that i n 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 F i e l d [17], while the others are from considerations based on the weighted  Chapter 5.  Inferring Parton Quantum  -  Numbers  1  40  in Hadron Jets  0  Weighted Electric  1  2  Charge  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.  Finally, 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 will 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 T w o Jet Events at 80 G e V M e t h o d of Weight Identification Function Charge of Leading Hadron n/a Charge of Leading Hadron, x > 0.5 n/a Charge of Leading Hadron n/a B o t h Jets Agree Charge of Leading Hadron, x > 0.5 n/a B o t h Jets Agree Sign of Best Quark Guess (1) Sign of Best Quark Guess (2) Sign of Best Quark Guess (3) Sign of Q% (1) Sign of Qf (2) Sign of Qf (3) Sign of Q%„ \QfJ > *f(1) Sign of Qf , \Qf \>*¥* (2) Sign of Q t , | g & | > A f * (3) Sign of Q% ,\Q% \> 0.447 (1) Sign of Qf , \Qf \ > 0.323 (2) Sign of Qi , \Qf \ > 0.475 (3) t  et  et  et  et  t  Sign Sign Sign Sign Sign  t  et  et  t  et  of Q ^ of Qf of Qf of Qf of Qf  , | Q £ | > 2 x 0.447 , \Qf \ > 2 x 0.323 , >2x0.475 , \QfJ > 4 x 0.447 \QfJ > 4 x 0.323 t  et  et  et  et  eV  Sign of Q£ , \Qf \ > 4 x 0.475 t  et  (1) (2) (3) (1) (2) (3)  Number Correct 35156 11468 231  Number Guessed 58343 16861 308  Fraction Correct  26  28  0.929  81353 83401 83459 77710 79796 80323 59013 60733 60336 38093 38961 37716 8952 8416 7256 78 31 19  100000 100000 100000 99846 99996 99995 68726 68856 67681 41543 41485 39807  0.814 0.834 0.835 0.778 0.798 0.803 0.859 0.882 0.891 0.917 0.939 0.947  9231 8564 7367 79 31 19  0.970 0.983 0.985 0.987 1.000 1.000  0.603 0.680 0.750  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 i n T w o Jet Events at 80 G e V M e t h o d of Identification Sign Sign Sign Sign Sign Sign Sign  of Best of Qf of Q% of Qf , of Best of Qf of Qf ,  Quark Guess  Sign Sign Sign Sign  of Q£ ,\Qb\> 2x0.323 of Best Quark Guess of Qf of Q% \Q% \> 0.323  Sign Sign Sign Sign Sign Sign Sign Sign Sign  of Qf , | Q £ | > 2 x 0.323 of Best Quark Guess of Q% of Qf , \Q% \> 0.323 of Qf , \Qf \ > 2 x 0.323 of Best Quark Guess of Q% of Qf ,\QfJ> 0.323 of Qf , \QfJ > 2 x 0.323  et  v  et  \QfJ > 0.323 \Q% \ > 2 x 0.323 Quark Guess t  et  et  \Qf \ > 0.323 et  t  et  v  et  t  (  et  et  t  et  et  t  et  Quark Flavour  Number Correct  Number Guessed  Fraction Correct  down down down down up up up up strange strange strange strange charm charm charm charm bottom bottom bottom bottom  83401 79796 38961 8416 88393 90480 58819 18373 87275 84066 44081 10259 80091 87053 55397 21496 80957 80207 53063 26127  100000 99996 41485 8564 100000 100000 59756 18414  0.834 0.798 0.939 0.983 0.884 0.905 0.984 0.998  100000 99998 45597 10317 100000 100000 56964 21579 100000 100000 58028 26825  0.873 0.841 0.967 0.994 0.801 0.871 0.972 0.996 0.810 0.802 0.914 0.974  comparing the accuracy at guessing the different quark flavours. T h e 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 i n 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 (|— < q J >) > e  ( | + < q* >), so the peak for charge 1  |  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 w i t h 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  T h e next question to probe is whether or not it is possible to determine the flavour of the quark and antiquark which started the jet. A g a i n , 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 m a x i m u m 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 q u a r k 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. T h e 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). T h e 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. T h e 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 v , up +  fX  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 < Sj > in ss events is similar in magnitude et  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 5 5 events would be significantly worse. These effects are illustrated in table 5.4, which shows the average and standard deviations 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. T h e 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 i n 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 Quark Down Up Strange Charm Bottom  Mean Weight 0.382 0.433 0.283 -0.226 0.150  Standard Deviation 0.330 0.309 0.292 0.334 0.499  Ratio (SD/Mean) 0.864 0.713 1.033 -1.473 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 how events of different flavours were tagged using this method. It 7  should be noted that i n events i n which no baryons are present, the relation Dj  et  +  Ujet + Sj t = 0 should hold because each meson adds some weight to one quark flavour e  (for the quark), and subtracts the same w eight from another quark flavour (to account 7  for the antiquark).  M a n y 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 i a weak interactions. In such jets, the nonzero quark weightings w i l l be Dj  et  = —Ujet. These events w i l l be tagged as du if Dj t > 0, and du i f Dj t < 0 e  e  (provided that they are not tagged as heavy quark events). The results i n the table are computed using the weight function w(x) = x  0 3 1  . The  two methods used to guess are testing the difference i n 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. T h e 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 T w o Jet Events at 80 G e V Quark d d u u s s c c b b  d  d  u  48398 12635 5297 633 20394 4656 1787 212 7422 1375  2013 102  6056 678 50564 12722 3677 388 16266 3268 1628 167  14039 1303 3656 283 21866 3823 2023 148  Leading Quark Flavour Guess u s s H(+) H(-) 1639 5682 6889 1661 12051 432 70 71 823 911 1521 1107 1577 2933 7999 56 46 145 1150 53 16593 46463 1967 1261 1348 950 15341 47 40 225 653 15554 1812 22364 15931 26 4004 211 4548 33 8592 13105 1237 31235 31999 775 3452 2755 6363 115  du  du  13190 9892 1841 782  2421 1044 13122 10216 808 361 3142 2416 545 219  3833 2843 625 263 2214 1455  jets are more difficult to identify than down quark jets, which are i n turn more difficult to identify than up quark jets.  B o t h 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 multiplying 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 m i x , 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, w i t h the most probable meson formed being a K . +  by a K  +  T h e mean proper distance travelled  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  good chance that a K  +  47  containing a leading up quark will 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 w i 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 . This presents no problem in weighted charge experiments, 0  Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets  as long as the particle is misidentified as another neutral particle. It also poses little problem in quark flavour retention, because most neutral particles (mesons in particular) have no net d, u , or s characteristics. For example:  TT = l/y/2[dd+uu] 0  K = 1/V2[ds + sd) G  S  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 w i 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 w i l l be no effect on weighted electric charge, and hence no effect on charge retention results. Finally, the detector may also not see some particles. If these are neutral particles, there will be little affect on weighted averages i n which the weight assigned is an increasing function of momentum (like x ). k  In this case, the weights assigned to all of the  other particles will increase slightly because each will 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. T h e weight function in equation 4.17, however, w i l l assign higher weights to all particles with momentum lower than that of the unobserved particle, and will 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).  48  Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets  49  In addition, the shift i n weight w i l l be greater for those particles nearest in order (of momentum) 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 will 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  Charge Retention in Three Jet Events  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 i n most cases, the angle between the gluon jet and the parton from which it was emitted w i 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 will be referred to as the " B " and " C "  jets. T h e other jet, which will now be referred to as the " A " jet, should contain either the quark or antiquark. In the L u n d Monte Carlo model, the gluon is emitted from the antiquark jet (although which jet emits the gluon may be ambiguous in practice). A s table 5.6 below shows, the assumption that the A jet is not the gluon jet is correct i n the vast majority of light quark events, as well as a large majority of heavy quark events. The problem of charge retention i n three jet events is very similar to that i n 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 w i l l be interactions between the gluon and quarks which were not considered in the two jet model. We w i l l first attempt to  Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets  Table 5.6: Frequency in which the A Jet is a Quark, Gluon, or A n t i q u a r k Jet at 80 G e V Quark Flavour Down Up Strange Charm Bottom  Non-Emitting Quark 23752 23760 23658 21411 18922  Emitted Gluon 1247 1240 1336 3507 5771  Emitting Antiquark 1 0 6 82 307  Table 5.7: Frequency with which leading partons have the m i n i m u m weighted charge at 80 G e V Quark Flavour Down Up Strange Charm Bottom  Non-Emitting Quark 9161 8053 8868 8454 7994  Emitted Gluon  Emitting Quark  6720 8790 7168 8375 8946  9119 8157 8964 8171 8060  determine which of the three jets is the gluon jet. T h i s 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 i n 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 m i n i m u m 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  50  Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets  Weighted Electric Charge  Figure 5.2: Weighted Charge Distributions For Quark, G l u o n , and Antiquark Jets at 80 GeV  51  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 i n 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. T h e results are not as good as i n the two jet case, partly due to the events i n 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 i n two jet events. This is partly because we cannot be sure which jet is the gluon jet and which are quark or antiquark jets. A s 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 i n the A jet, in which two methods are employed. T h e 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 M e t h o d of Identification - Qi_ - Q*_ 0.323  - Q'<_ Q«_  Quark Flavour Down Down Down  - Qi_ - Qi_ 0.323  - Q*_ - Q$_  Up Up Up  Sign of gjf-iet  Sign of Qi_ Sign of Qi_ Magnitude > Sign of Q«_ Sign of Qi_ Sign of Q*_ Magnitude > jet  }et  jet  jet  jet  jet  iet  jei  jet  itt  S n of  iet  iet  jet  Q«_  Strange Sign of Qi_ - Q%_ - Qi_ et Strange Sign of Qi_ - Qi_ - Q$_ Strange Magnitude > 0.323 Sign of Q«_ Charm Sign of Qi_ - Qi_ - Q$_ Charm Sign of Qi_ - Qi_ Q«_ Charm Magnitude > 0.323 l g  jet  jet  jei  iei  Jtt  3  jut  jet  3et  jet  jet  ]ei  jet  iet  Sign of Q«_ Sign of Qi_ - Q*'_ - Q*<_ Sign of Qi_ - Qi_ - Qt_ Magnitude > 0.323 jet  jet  jet  jet  3et  jet  jet  Bottom Bottom Bottom  Number Correct 18259 18132 10775  Number Guessed 24827 25000 13017  Percentage Correct 0.735 0.725 0.828  20910 20626 14067  24899 25000 15294  0.840 0.825 0.920  19099 19007 11670  24819 25000 13498  0.770 0.760 0.865  20127 20294 13559  24852 25000 15076  0.810 0.812 0.899  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 d d u u s s c c  d 10362 3073 2782 467 5146 1359 1203 168  b b  2418 517  d 1655 174 5099 681 3071 528 6309 1624 1624 226  Leading u u 2913 4984 504 568 11149 1246 3323 125 2157 6447 362 1103 4031 1183 1095 125 1092 3362 144 558  Quark Flavour Guess s # ( + ) H(-) s 1134 1427 266 230 142 64 11 15 862 1486 224 231 168 12 39 10 6584 570 203 191 55 11 1729 13 2504 1254 425 7575 43 491 822 11 369 1859 8265 5561 321 39 1068 548  du  du  1490 1081 426 211 447 316 138 73  539 284 1495 1147 184 91 378 295  325 198  125 46  Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets  5.6  54  Effect of Energy on Quantum Number Retention  It was shown in chapter 4 that in a jet of N particles, the weight function had a distribution such that the ratio contribution to  was proportional to  W{ =  ~^'  n  t1  T h e source of this  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 Ej t, in the asymptotic limit the weighted average should determine the e  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  where A is the mean number of particles in a jet, for the  function of  weight function w(x) = x  r  0 , 3 1  . T h e 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. T h e 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. A s 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 Ej ), et  jet will be reduced:  so that the effect of decays mixing the order of the quarks i n the  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 i n T w o Jet Events at Various Energies M e t h o d of Identification Sign of Q% Sign of Qf t  et  Sign Sign Sign Sign  Qf  of et of Qf of Qf o{Qf , et  ei  et  Sign Sign Sign Sign Sign Sign  of of of of of of  Sign Sign Sign Sign Sign Sign Sign  of of of of of of of  \Qit\> > > > >  Ql^,  Qf ,\QU Qf \QU Qf , \QU et  eu  et  0.323 0.323 0.323 0.323 0.323  Qf Qf  et  et  Q% Qf Qf Qf , > Qf , > Qf , > Qf , \Qfet\> > Sign of Qi , t  et  et  et  \QU 0.323  et  et  ei  t  \QU  0.323 0.323 0.323 0.323  CMS Energy 20 G e V 40 G e V 60 G e V 80 G e V 100. G e V 20 G e V 40 G e V 60 G e V 80 G e V 100 G e V 20 G e V 40 G e V 60 G e V 80 G e V 100 G e V 20 G e V 40 G e V 60 G e V 80 G e V 100 G e V  Quark Flavour down down down down down down down down down down up up up up up up up up up up  Number Correct  Number Guessed  Fraction Correct  18565 19373 19664 19978 20124 10118 9972 9847 9753 9653 21318 22002 22375 22605 22760  24988 24997 25000 24998 25000 11550 10881 10615 10343 10209 24998 24999 25000 25000 25000  0.743 0.775 0.787 0.799 0.805 0.876 0.916 0.928 0.943 0.946 0.853 0:880 0.895 0.904 0.910  14245 14560 14495 14766 14751  14906 14967 14778 15005 14940  0.956 0.973 0.981 0.984 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 T w o Jet Events at Various Energies Energy Quark 20d 40d 60d 80d lOOd 20d 40d 60d 80d lOOd 20u 40u 60u 80u lOOu 20u 40u 60u 80u lOOu  Leading Quark Flavour Guess  d  d  u  u  s  s  2352 2764 2956 3169 3278 9160 10781 11509 12092 12341 229 174 168 154 158 1527 1449 1344 1307 1296  21 29 28 24 26 596 570 537 511 428 138 256 271 347 368 2768 3205 3372 3545 3533  230 196 199 161 170 1762 1618 1640 1477 1493 2450 2815 3089 3164 3292 10019 11584 12098 12634 12959  114 191 193 213 215 2671 2891 3035 3089 3091 . 18 23 10 13 9 456 340 310 293 236  66 88 88 112 108 1026 1239 1360 1362 1537 48 61 51 38 37 772 780 743 680 720  322 269 234 218 240 1997 1857 1765 1712 1761 303 283 324 278 283 1859 1916 2043 1918 1983  #(+)  24 20 16 24 15 342 351 345 412 416 20 14 14 13 19 322 338 383 371 380  H(-) 19 25 12 18 16 311 390 421 412 386 19 10 20 11 8 279 354 400 384 408  du  du  4358 3283 2780 2481 2303 5649 4365 3684 3327 3062 660 359 255 215 189 1309 793 578 490 442  759 422 320 255 206 1486 938 704 606 485 4479 3343 2852 2606 2363 5689 4241 3729 3378 3043  Chapter 5. Inferring Parton Quantum Numbers in Hadron Jets  57  3000  0.2 (a) Weighted Electric Charge  0.22 0.24 (b) log(l/sqrt(N))  Figure 5.3: Energy Effects on Weighted Charge Distribution  0.26  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. T h e 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 i n 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 mobility 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 w i t h different weight functions), to determine the mobility of the momentum of the leading quark i n a process similar to an integral transform. T h e two jet methods were attempted on three jet events, and found to be less successful. 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 interactions 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 improvements 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 different quark-antiquark flavours are created from the vacuum decrease. Also, the decays of the first generation of hadrons are less likely to m i x 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 i n 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 . C a h n and E . W . Colglazier. Q u a n t u m Numbers and Quark-Parton Fragmentation Models. Physical Review D, 9(9):2658, 1 M a y , 1974. [3] S.J. Brodsky and N . Weiss. Retention of Q u a n t u m 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 M a y , 1973. [5] J . D . Bjorken. Hadron F i n a l States in Deep Inelastic Processes. In J . G . K o r n e r , G . K r a m e r , and D . Schildknecht, editors, Current Induced Reactions, Springer-Verlag, 1976. [6] R . P . Feynman. W h a t Neutrinos Can Tell Us A b o u t Partons. In A . Frenkel and G . M a r x , editors, Neutrino '72, Volume II, Omkdk-Technoinform, 1972. [7] J . Kogut and Leonard Susskind. V a c u u m Polarization and the Absence of Free Quarks in Four Dimensions. Physical Review D, 9(12):3501, 15 June, 1974. [8] S . M . B e r m a n , J . D . Bjorken, and J . B . Kogut. Inclusive Processes at High Transverse M o m e n t u m . Physical Review D, 4(11):3388, 1 December, 1971. [9] A . Casher, J . K o g u t , and Leonard Susskind. V a c u u m Polarization and the Absence of Free Quarks. Physical Review D, 10(2):732, 15 July, 1974. 60  Bibliography  61  [10] T . Sjostrand. T h e L u n d Monte Carlo for Jet Fragmentation. Computer Physics Communications, 27(3):243, September, 1982. [11] T . Sjostrand. T h e L u n d Monte Carlo for e e " Jet Physics. Computer Physics Com+  munications, 28(3):229, January, 1983. [12] T . Sjostrand. T h e L u n d 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 L u n d Monte Carlo for Jet Fragmentation and e e~ +  P h y s i c s — J E T S E T Version 7.1. M a y , 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 Q u a n t u m Numbers Event by Event. Physics Letters B, 90(4):443, 10 M a r c h , 1980. [17] R . D . F i e l d 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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