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## Non-perturbative QCD

The non-perturbative part of the fragmentation bridges the gap between the perturbative calculable part and the physical fragmentation function measured by an experiment. It is therefore an important ingredient for comparisons between the predictions of physical observables and their measured values. As has already been mentioned, this part of the fragmentation describes the physics at a low energy scale of LQCD, where the value of aS is large.

During the non-perturbative part of the fragmentation, the production of hadrons from the generated partons takes place. Usually, phenomenological schemes are used to model the carry-over of parton momenta and flavor to the hadrons. It is also possible to extract the nonperturbative component directly, without using any model assumption. This is done by theorists for moments of the fragmentation distribution. In Chapter 6, a method will be proposed to perform this extraction also for the x-dependence of the fragmentation function.

### Hadronization in Monte Carlo Generators

The computation of the hadronization process in Monte Carlo programs is not based on first principals of QCD, but on phenomenological models. The most popular models of hadronization are the string model, implemented in the JETSET [18] and PYTHIA [19] Monte Carlo event generators, and the cluster hadronization of the HERWIG Monte Carlo event generator [20]. The earliest model, the independent hadronization model is not used in this thesis but has great historical importance. The analysis presented in this work relies only on PYTHIA and JETSET. Particular attention will therefore be given to the string hadronization model in the discussion below. The hadronization models differ in how gluons created during the perturbative part are treated; how (if at all) different partons interact during the hadronization; and the number, type and momentum distribution of the created hadrons. The independent and string models use phenomenological parametrizations for the fragmentation variable. These parametrizations, that are sometimes referred to as “fragmentation functions”, will be reviewed in Section 2.3.4.2.

Independent Hadronization This is the simplest scheme for generating hadron distributions from those of partons.In this model, first proposed by Field and Feynman [21], partons are supposed to fragment independently from one another. Each quark in the system after the showering process is combined with an antiquark from a q¯q pair created out of the vacuum to give a “first generation” meson with energy fraction z of the initial quark. The leftover quark, with energy fraction 1−z, is fragmented in the same way, over and over, until the leftover energy falls below a certain threshold. The variable z is distributed according to a given probability density function, which is the “hadronization model” (see Section 2.3.4.2). The most significant drawback of this model is the fact that it is not able to simulate interference effects. This model is still used to simulate events with two well separated jets, where interference factors are less important. The independent fragmentation can also be used by PYTHIA [19] as a non-default option.

Cluster Hadronization Cluster fragmentation is used in the HERWIG Monte Carlo generator [20]. In this model, assuming a local compensation of color, based on the pre-confinement property of perturbative QCD [22], the remaining gluons at the end of the parton shower evolution are split into quark-antiquark pairs. Color singlet clusters of typical mass of a couple of GeV are then formed from quark and antiquark of color-connected splittings. These clusters decay directly into two hadrons unless they are either too heavy (then they decay into two clusters) or too light, in which case a cluster decays into a single hadron, requiring a small rearrangement of energy and momentum with neighboring clusters. The decay of a cluster into two hadrons is assumed to be isotropic in the rest frame of the cluster unless a perturbative-formed quark is involved. A decay channel is chosen based on the phase-space probability, the density of states, and the spin degeneracy of the hadrons. Cluster fragmentation has a compact description with few parameters, due to the phase-space dominance in the hadron formation. A scheme of the cluster fragmentation is shown in Figure 2.7.

String Hadronization A large class of models describes hadronization through the image of color flux tubes, or strings, each spanned between two quarks created during the perturbative part of fragmentation. Usually the motion of a string is described by classical, relativistic dynamics. The treatment of gluons seems more natural in the string picture than in the cluster or independent hadronization schemes: perturbative gluons are simply incorporated into the strings connecting two nearby quarks. They appear as a kink in the string. This complicates the dynamics, but does not lead to fundamentally different effects. Gluons will therefore be neglected in the following considerations. A scheme of the string hadronization is shown in Figure 2.8.

The strength between two color sources, with large mutual distance, is expected to be independent of the distance due to the self-interaction of gluons. The energy density, or string tension k, is constant along the string, assuming the transverse extension of the string is constant over its full length and much smaller than the longitudinal extension. The string is estimated to have a diameter of 1fm and a tension of k 1GeV/fm. As the quarks fly apart and the string is stretched, the energy stored in the string increases according to E(d) µ kd, where d is the distance between the quark and the antiquark situated at both ends of the string, i.e. the length of the string. If a virtual q¯q pair fluctuates out of the vacuum, somewhere along the string, and if this pair has the same color as the endpoint quarks of the string, the color field is locally compensated and the string breaks into two pieces. This process is repeated until the remaining energy of the individual string segments is no longer sufficient to transform another virtual q¯q pair into a real one.

Because the string is assumed to be uniform along its length, the probability of a q¯q pair creation occurring on the string per unit length and per unit time is a constant P0. To obtain the probability that the string breaks due to this pair creation at a given point, the history of this point has to be taken into account: the probability of a string break is proportional to the probability that no previous break occurred in its backward light cone. Had there been any previous break within the light cone, the considered space-time point would not be on the string, since the endpoints of the newly created sub-strings move apart at the speed of light 13 (assuming massless endpoint quarks). The probability for no string break in the backward light cone is given by the Wilson area law [23] dP dA = P0e−P0A.

#### Phenomenological Hadronization Models

There is a number of different phenomenological models that tend to parametrize the energy fraction taken by the b hadron with respect to the energy of the b quark. These models correspond to different functions of the fragmentation variable z and they usually depend on one or two parameters. The models are usually used to describe the non perturbative part of fragmentation, and therefore the quark’s energy is understood as the energy after the perturbative phase. In the framework of Monte Carlo generators, the last statement is synonym to after the parton showering process. The use of the independent or string hadronization models requires the use of one of the phenomenological parametrizations described below. In particular, one must use hem in the JETSET [18] and PYTHIA[19] Monte Carlo generators that have been used in the present analysis. It is a common practice to fit the parameters, those models depend on, when measuring the b fragmentation function in e+e− collisions. This has been done by ALEPH [31] , OPAL [32] and SLD [33] as a part of their b fragmentation analyses, and also by DELPHI [34], previously to the analysis presented in this work.

Phenomenological hadronization models have sometimes been folded with theoretical QCD computations of the perturbative component of the fragmentation function. It will be shown in Some of these functions, derived from phenomenological ideas, have been contributed by Peterson et al. [35], Collins and Spiller [36], and Kartvelishvili et al. [37]. Models that are built on more basic ideas of the string fragmentation process have been suggested by Andersson et al. [38] and Bowler [24]. In the sections below, where these models are presented, the fragmentation variable is noted as z to be consistent with the original publications.

The Peterson Model A simple phenomenological parameterization of the spectra of heavy hadrons was presented in 1983 by Peterson et al [35]. This model relies on simple kinematical arguments. The basic idea behind this model is that when a light quark q combines with a heavy quark Q to form a hadron H = Qq, the heavy quark decelerates only slightly. Thus, Q and Qq should carry almost the same energy; this effect is expected to dominate over more subtle dynamical details. Therefore the gross feature in the amplitude M of the transition Q!Qq+q is supposed to be given by the energetic gain in the process: M µ 1 DE = 1 EH +Eq−E.

**B-hadron Production Rates**

Production rates of the different weakly decaying B-hadron particles have been accurately determined using measurements from LEP and TeVatron experiments. These determinations use also results from B−B mixing obtained at b-factories and at the previous facilities. Several types of data have been combined. They comprise “direct” measurements as the production rates of B-hadrons decaying into a specific final state of known branching fraction (semileptonic decays or exclusive channel). A more inclusive approach was also followed by DELPHI using neural networks to separate charged from neutral b-hadrons. “Indirect” measurements have been used also as the mixing probability c. For b-hadron jets, produced at high energy, this probability corresponds to the average of the contributions from B0d and B0s mesons: c = fdcd + fscs (2.36) in which fq are the fractions of bq mesons in the jet and cd,s are the oscillation probabilities for B0d and B0s meson respectively.

**Table of contents :**

**1 Introduction **

**2 Theory of Bottom Production, Fragmentation and Decay **

2.1 Overview: The Life Story of a Bottom Quark

2.2 Bottom Quark Production in The Hard Process

2.2.1 Bottom Quark Production at LEP

2.2.2 Bottom Quark Production at the TeVatron

2.3 Theoretical Aspects of b Fragmentation

2.3.1 Definitions of Fragmentation Functions

2.3.2 Perturbative and Non-perturbative Parts

2.3.3 Perturbative QCD

2.3.3.1 Theoretical QCD Calculations

2.3.3.2 Parton Showers in Monte Carlo Generators

2.3.4 Non-perturbative QCD

2.3.4.1 Hadronization in Monte Carlo Generators

Independent Hadronization

Cluster Hadronization

String Hadronization

Baryon Production

2.3.4.2 Phenomenological Hadronization Models

The Peterson Model

The Collins-Spiller Model

The Kartvelishvili Model

The Lund Symmetric Fragmentation Function

The Bowler Model

2.4 Excited States

2.5 B-hadron Production Rates

2.6 B Decays

**3 Experimental Framework I- The LEP Collider and the DELPHI Experiment **

3.1 The Large Electron Positron Collider

3.2 The DELPHI Experiment

3.3 Tracking Detectors

3.3.1 The Vertex Detector

3.3.2 The Inner Detector

3.3.3 The Time Projection Chamber

3.3.4 The Outer Detector

3.4 Other Detectors

3.4.1 Ring Imaging Cherenkov Detectors

3.4.2 Electromagnetic and Hadron Calorimeters

3.4.3 Scintillators

3.4.4 Muon Chambers

3.5 Particle Identification and Reconstruction

3.5.1 Track Reconstruction

3.5.1.1 Primary Vertex Reconstruction

3.5.1.2 Impact Parameter Reconstruction

3.5.2 Hadron Identification

3.5.3 Lepton Identification

3.6 DELPHI Monte-Carlo Simulation

3.7 Data Reprocessing

**4 Experimental Framework II- The TeVatron Collider and the CDF Experiment **

4.1 TeVatron – the Source of pp Collisions

4.2 The CDF-II Detector

4.3 Standard Definitions in CDF-II

4.4 Tracking Systems

4.4.1 Silicon Tracking Detectors

4.4.2 Central Outer Tracker

4.4.3 Pattern Recognition Algorithms

4.4.4 Momentum Scale

4.5 Time of Flight

4.6 Calorimeters

4.7 Muon Systems

4.8 Triggering

4.8.1 Level 1 Trigger

4.8.2 Level 2 Trigger

4.8.3 Level 3 Trigger

4.9 Luminosity Measurement

**5 B Fragmentation at DELPHI **

5.1 General Event Selection and Jet Energy Measurement

5.1.1 Data/Monte-Carlo comparison and adjustments

5.1.1.1 Accuracy of track reconstruction

5.1.1.2 Efficiency and track energy distribution

5.1.2 Jet energy reconstruction

5.2 B-Energy Reconstruction

5.3 Selection of B Candidates

5.4 Measurement of the B-Fragmentation Distribution

5.4.1 Fit Results on Real Data Events

5.4.2 Fit Results on Simulated Events

5.5 Systematic Uncertainties

5.5.1 Real Data and Simulation Tuning

5.5.1.1 Energy Calibration

5.5.1.2 Level of the Non-b Background

5.5.1.3 Track Energy and Multiplicity Tuning

5.5.1.4 Jet Multiplicity

5.5.1.5 Summary

5.5.2 Physics Parameters

5.5.2.1 b-Hadron Lifetimes

5.5.2.2 B Production Rate

5.5.2.3 b-Hadron Charged Multiplicity

5.5.2.4 g!bb Rate

5.5.2.5 Summary

5.5.3 Parameters Used in the Analysis

5.5.3.1 Parametrization of the Weight Function

5.5.3.2 b-Tagging Selection

5.5.3.3 Jet Clustering Parameter Value

5.5.3.4 Level of Ambiguous Energy

5.5.3.5 Secondary Vertex Charged Multiplicity

5.5.3.6 Summary

5.6 Comparison with Other Experiments

**6 Extraction of the x-Dependence of the Non-perturbative QCD Component **

6.1 Introduction

6.2 Extracting the x-Dependence of the Non-perturbative QCD Component

6.3 x-Dependence Measurement of the Non-perturbative QCD Component

6.3.1 The Perturbative QCD Component is Provided by a Generator

6.3.2 The Perturbative QCD Component is Obtained by an Analytic Computation Based on QCD

6.4 Results Interpretation

6.4.1 Comparison with Models

6.4.2 Proposal for a New Parametrization

6.5 Checks

6.5.1 The Use of a Fitted Parametrization

6.5.2 The Effect of Parametrization

6.5.3 Number of Degrees of Freedom

6.5.4 Using a Different Tuning of the Monte Carlo

6.6 Combination of Fragmentation Distributions from All Experiments

6.7 Comparison of Results for All Experiments

6.8 Thoughts about Fitting Moments of Fragmentation Functions

6.9 Charm Fragmentation

6.10 Conclusions

**7 B Fragmentation and Related Studies at CDF **

7.1 Introduction

7.2 Data Sample

7.2.1 Reconstruction of B± !J/yK±

7.2.2 Subtracting the Backgrounds in the Data

7.3 Monte Carlo Samples

7.3.1 General Description

7.3.2 PYTHIA Parameters

7.4 Outline of the Analysis Method

7.5 Preliminary Monte Carlo Studies

7.6 Data and Monte Carlo Comparisons

7.6.1 Comparisons with msel=5 Samples

7.6.2 Comparisons with an msel=1 Sample

7.7 A Method of Fitting the Fragmentation Function Parameters

7.8 An Estimate of the b Production Cross Section

7.8.1 Evaluation of Efficiency

7.8.2 The Inclusive b Quark Production Cross Section

7.8.3 Statistical Error Estimation

7.8.4 Systematic Error Estimation

7.8.4.1 Luminosity

7.8.4.2 Branching Ratios and Production Fraction

7.8.4.3 Trigger and Reconstruction Efficiencies

7.8.5 Comparison with Other Measurements and with Theoretical Predictions

**8 Conclusion **

**A The Mellin Transformation **

**B Fitting Histograms of Singular Error Matrices **

**Bibliography **