Measurements of Event Shapes in Deep Inelastic Scattering at HERA with ZEUS docx

Smith At the University of Wisconsin — Madison Mean values and differential distributions of event-shape variables have beenstudied in neutral current deep inelastic scattering using an

Scattering at HERA with ZEUS

by

Adam A Everett

A dissertation submitted in partial fulfillment of the

requirements for the degree of

All Rights Reserved

Inelastic Scattering at HERA with ZEUS

Adam A Everett

Under the supervision of Professor Wesley H Smith

At the University of Wisconsin — Madison

Mean values and differential distributions of event-shape variables have beenstudied in neutral current deep inelastic scattering using an integrated luminosity

of 82.2 inverse pico-barns collected with the ZEUS detector at HERA The kinematicrange was Q-squared from 80 to 20480 GeV-squared and Bjorken-x from 0.0024 to 0.6,where Q-squared is the virtuality of the exchanged boson The Q-dependence is com-pared with a model based a combination of next-to-leading-order QCD calculationswith next-to-lead-logarithm corrections and the Dokshitzer-Webber non-perturbativepower corrections

Mean values and differential distributions of event-shape variables have beenstudied in neutral current deep inelastic scattering using an integrated luminosity of82.2 pb−1 collected with the ZEUS detector at HERA The kinematic range was 80 <

Q2 < 20480 GeV2 and 0.0024 < x < 0.6, where Q2 is the virtuality of the exchangedboson and x is the Bjørken variable The Q-dependence is compared with a modelbased a combination of next-to-leading-order QCD calculations with next-to-lead-logarithm corrections and the Dokshitzer-Webber non-perturbative power corrections

Acknowledgements I would like to thank the High Energy Physics partment at the University of Wisconsin for the opportunity to perform research with

de-an outstde-anding group of physicists at de-an outstde-anding university I would especially like

to acknowledge and thank Wesley Smith and Don Reeder for their guidance, support,and superb dedication

I would also like to thank the members of the ZEUS Collaboration who made datataking possible, and who gave me great advice on all of the details of an analysis

A very special thanks are due to my colleagues from the University of Glasgow, IanSkillicorn and Steven Hanlon, who very patiently taught me the ropes of an eventshape analysis, and shared so much of their knowledge and experience with me And,

of course, I thank Alexandre Savin and Dorian K¸cira for being my ”advisors awayfrom home.”

Thank you to the friends who were always willing to play cards, watch movies, andplay when we were in danger of doing too much work

I give special thanks to my parents and family for their support and teasing And Ioffer a huge thank you to my wife, Jayda, for putting up with me and following mearound the globe I will never be able to thank you enough for all you have done for

me during this adventure

1.1 Standard Model 3

1.1.1 Leptons 3

1.1.2 Quarks 3

1.1.3 Bosons 5

1.1.4 Fields 6

1.2 Quantum Chromodynamics 8

1.2.1 Perturbative Quantum Chromodynamics 12

2 Event Shapes in Deep Inelastic Scattering 15 2.1 Deep Inelastic Scattering 16

2.1.1 Kinematic description 20

2.1.2 DIS Cross Section 21

2.1.3 QCD Radiation 25

2.2 Introduction to Event Shapes in Deep Inelastic Scattering 26

2.2.1 Hadronization 26

2.2.2 Power Corrections 28

2.2.3 Mean Non-Perturbative Calculations 30

2.2.4 Differential Non-Perturbative Calculations 31

2.3 Definition of the Event Shapes 31

2.4 The Breit Frame 35

3 Experimental Setup 39 3.1 Detection of Particle Interactions 39

3.1.1 Definitions 40

3.1.2 Basic Experimental Design 44

3.2 Deutsches Elektronen Synchrotronen 48

3.3 Hadron-Elektron Ring Anlage 49

3.3.1 HERA Injection 50

3.3.2 HERA Experiment Halls 53

3.3.3 HERA Luminosity 53

3.4 The ZEUS Detector 56

3.4.1 The Tracking Detectors 59

3.4.2 The Uranium Calorimeter and Plastic Scintillator 63

3.4.3 Background Rejection 67

3.5 Trigger and Data Acquisition 69

4 Event Simulation 73 4.1 Monte Carlo Event Generation 74

4.1.1 Monte Carlo Input: Parton Distribution Functions and Parton

Evolution 75

4.1.2 Hard Process and Higher Order Effects 76

4.1.3 Soft Process 78

4.2 Monte Carlo Programs in HEP 80

4.3 Detector Simulation 81

4.4 Hadronic Final States 82

5 NLO Calculations 83 5.1 The Running Coupling Constant 83

5.2 NLO Integration Techniques 84

5.3 NLO Parameterization 86

5.3.1 Means 86

5.3.2 Differential Distributions 87

5.4 DISENT, DISASTER, DISPATCH, and DISRESUM Programs 88

5.4.1 Means 88

5.4.2 Differential Distributions 89

6 Event Reconstruction 95 6.1 Particle Track and Vertex Reconstruction 95

6.2 Calorimeter Quantities 97

6.2.1 Calorimeter Cell Removal 98

6.2.2 Calorimeter Energy Corrections 98

6.3 Electron Reconstruction 100

6.4 Hadronic System Reconstruction 102

6.5 Event Kinematics 105

6.5.1 The Electron Method 105

6.5.2 The Jacquet-Blondel Method 107

6.5.3 The Double-Angle Method 107

6.6 Calorimeter Cells and Energy Flow Objects (EFOs) 108

6.7 Boosting to the Breit Frame 110

6.8 Reconstruction Summary 111

7 Event Selection 113 7.1 Online Selection 114

7.1.1 First Level Trigger (FLT) 115

7.1.2 Second Level Trigger (SLT) 116

7.1.3 Third Level Trigger (TLT) 116

7.2 Offline Selection 117

7.2.1 Trigger Bits 117

7.2.2 Electron Selection 118

7.2.3 Background Rejection 119

7.2.4 Kinematic Selection and Phase Space Definition 120

7.2.5 Particle Selection 121

8 Analysis Method 125 8.1 Monte Carlo Description of the Data 126

8.2 Data Corrected to the Hadron Level 133

8.3 Systematic Uncertainties 137

9 Fit Method 191

9.1 Fits to the Means 191

9.1.1 Fit Procedure 191

9.1.2 Systematic Errors 193

9.2 Fits to the Differential Distributions 195

9.2.1 Fit Procedure 195

9.2.2 Fit Range 197

9.3 Results 199

9.3.1 Mean values 199

9.3.2 Differential distributions 205

9.3.3 Y2 and KOUT 207

10 Conclusions 223 10.1 Summary 223

10.2 Outlook 224

A Treatment of Statistical Uncertainties 227 A.1 Differential Distribution 227

A.2 Means 230

List of Tables

3.1 The integrated luminosity delivered by HERA I and HERA II and gated

by ZEUS for each year of running 55

3.2 Active and inactive dimensions of the CTD 60

3.3 Radiation length and interaction length of some common materials 65

8.1 List of systematic checks performed 141

8.2 Efficiencies, purities, and correction factors for mean event shapes 150

9.1 Kinematic bins used in this analysis 209

9.2 Fit ranges used for fits to differential data 209

9.3 Results from the Hessian fit to means for αs(MZ) 210

9.4 Results from Hessian fit to means for α0 210

9.5 Effect of matching methods on extraction of αs(MZ) from differential distributions 211

9.6 Effect of matching methods on extraction of α0 from differential distri-butions 211

9.7 Results from the Hessian fit to distributions for αs(MZ) 212

9.8 Results from the Hessian fit to distributions for α0 221

B.1 Mean data and uncertainties for Q, x, TT, BT, and C 234

B.2 Mean data and uncertainties for Tγ, BT γ, M2, y2 and KOU T /Q 235

B.3 Differential tables for C 236

B.4 Data tables for M2 237

B.5 Data tables for TT 238

B.6 Data tables for BT 239

B.7 Data tables for Tγ 240

B.8 Data tables for Bγ 241

B.9 Data tables for KOU T/Q 242

B.10 Data tables for KOU T/Q 243

B.11 Data tables for y2 244

B.12 Data tables for y2 245

List of Figures

1.1 Different concepts of the atom 2

1.2 The generations of matter 4

1.3 Forces and Bosons 6

1.4 Generations of matter and forces 7

1.5 Diagrams of generic LO, NLO, and NNLO processes in which one initial particle undergoes one split 14

2.1 Generic DIS diagram 15

2.2 Picture of a DIS event 17

2.3 Probe resolution as a function of Q2 19

2.4 Probe resolution of the proton 19

2.5 Illustration of a simple scattering cross section 22

2.6 Diagrams of higher order DIS interactions 24

2.7 Evolution from a hard process to final state hadrons 27

2.8 Diagram of the Breit Frame 35 2.9 Current hemisphere in the Breit frame compared to e+ei annihilation 36

3.1 A cross sectional view of a generic detector and the passage of various

particles through this detector 44

3.2 An aerial view of HERA and PETRA 49

3.3 HERA injection system 50

3.4 HERA tunnel 52

3.5 The integrated luminosity delivered by HERA I and HERA II for each year of running 54

3.6 A 3D cutout of the ZEUS detector 57

3.7 A 2D x-y cross sectional view of the ZEUS detector 58

3.8 Drift chamber 61

3.9 An x-y view of the CTD 62

3.10 An x-y view of the UCAL 63

3.11 Diagram of a BCAL tower 64

3.12 The timing of various events in the ZEUS detector 69

3.13 The ZEUS trigger and DAQ systems 72

4.1 Event generating dice 73

4.2 Event generator stages 75

4.3 Diagrams of simple QED radiation 77

4.4 Diagrams of models used in event simulation 79

5.1 Fit parameters for mean NLO event shapes 90

5.2 Fit parameters for mean NLO event shapes 91

6.1 Helix fit for the CTD 97

6.2 Clustering cells into islands 100

6.3 ZEUS kinematic plane 106

6.4 Matching tracks to islands: EFOs 109

7.1 Event chain 118

7.2 Kinematic bins used in this analysis 124

8.1 Ariadne description of zvtx and E − pz 128

8.2 Ariadne description of electron scattering angle 129

8.3 Ariadne description of Ee0 and Et0 130

8.4 Ariadne description of xDA and Q2 DA 131

8.5 Ariadne description of yEL and yJB 132

8.6 Mean reconstructed MC event shapes compared to data 140

8.7 Measured distributions of TT compared to reconstructed MC 142

8.8 Measured distributions of BT compared to reconstructed MC 143

8.9 Measured distributions of M2 compared to reconstructed MC 144

8.10 Measured distributions of C compared to reconstructed MC 145

8.11 Measured distributions of Tγ compared to reconstructed MC 146

8.12 Measured distributions of Bγ compared to reconstructed MC 147

8.13 Measured distributions of y2 compared to reconstructed MC 148

8.14 Measured distributions of Kout/Q compared to reconstructed MC 149

8.15 Efficiencies according to Ariadne for TT 151

8.16 Efficiencies according to Ariadne for BT 152

8.17 Efficiencies according to Ariadne for M2 153

8.18 Efficiencies according to Ariadne for C 154

8.19 Efficiencies according to Ariadne for Tγ 155

8.20 Efficiencies according to Ariadne for Bγ 156

8.21 Efficiencies according to Ariadne for y2 157

8.22 Efficiencies according to Ariadne for Kout/Q 158

8.23 Purities according to Ariadne for TT 159

8.24 Purities according to Ariadne for BT 160

8.25 Purities according to Ariadne for M2 161

8.26 Purities according to Ariadne for C 162

8.27 Purities according to Ariadne for Tγ 163

8.28 Purities according to Ariadne for Bγ 164

8.29 Purities according to Ariadne for y2,kt 165

8.30 Purities according to Ariadne for Kout/Q 166

8.31 Correction Factors according to Ariadne for TT 167

8.32 Correction Factors according to Ariadne for BT 168

8.33 Correction Factors according to Ariadne for M2 169

8.34 Correction Factors according to Ariadne for C 170

8.35 Correction Factors according to Ariadne for Tγ 171

8.36 Correction Factors according to Ariadne for Bγ 172

8.37 Corrected data compared with hadron level MC for mean event shapes 173 8.38 Corrected data compared to hadron level MC for TT 174

8.39 Corrected data compared to hadron level MC for BT 175

8.40 Corrected data compared to hadron level MC for M2 176

8.41 Corrected data compared to hadron level MC for C 177

8.42 Corrected data compared to hadron level MC for Tγ 178

8.43 Corrected data compared to hadron level MC for Bγ 179

8.44 Corrected data compared to hadron level MC for y2,kt 180

8.45 Corrected data compared to hadron level MC for Kout/Q 181

8.46 Fractional systematic error for mean TT 182

8.47 Fractional systematic error for mean BT 183

8.48 Fractional systematic error for mean M2 184

8.49 Fractional systematic error for mean C 185

8.50 Fractional systematic error for mean Tγ 186

8.51 Fractional systematic error for mean Bγ 187

8.52 Fractional systematic error for mean y2 188

8.53 Fractional systematic error for mean KOU T/Q 189

9.1 Ratio of (NLO+NLL+PC) to (NLO+NLL) 199

9.2 Mean corrected data compared to MC 200

9.3 Mean corrected data fit to NLO + PC 201

9.4 Extraction of (αs, α0) from event shape means 202

9.5 Effects of high Bjørken x kinematic bins and statistical uncertainties only in extracting (αs(MZ), α0) values from event shape means 203

9.6 Differential data compared to Ariadne for M2, C, TT 213

9.7 Differential data compared to Ariadne for Tγ and Bγ 214

9.8 Differential data fit to NLO+NLL+PC for M2, C, and TT 215

9.9 Differential data fit to NLO+NLL+PC for Tγ and Bγ 216

9.10 Extraction of αs and α0, from distributions, using different matching

schemes 217

9.11 Study of differential extraction dependence on Q2 218

9.12 Measurements of y2 compared to NLO and MC 219

9.13 Measurements of KOU T/Q compared to MC 220

10.1 Future of Event Shapes 225

Chapter 1

Introduction

1Science is the observation, identification, description, and explanation of phenomena

In short, science is mankind’s methodical attempt to understand the physical universe.Physics is the branch of science that studies matter, energy, and their interactions;and particle physics is the branch of physics which investigates the basic elements ofmatter as well as the fundamental laws that control them From the early philosopherssuch as Democritus and Aristotle to the more modern scientists such as Galileo andNewton, the study of the universe has progressed from a philosophical basis to a base

of empirical observations merged with mathematical descriptions As understanding

of the macroscopic universe increased, scientists attempted to organize the universeinto fundamental constituents: atomic elements Mendeleev’s organization in 1869

of all known elements according to their atomic mass lead to the realization that theelements could be grouped according to similar chemical properties: the Periodic Table

of Elements The large number of elements and the periodic structure of propertiessuggested a more fundamental substructure within the atom

The first advance in the study of the structure of the atom came with the 1898

1

Reference [1] was useful in compiling this chapter.

discovery by Joseph Thompson of the electron About a decade later, Ernest ford demonstrated that atoms have a small, dense, positively charged nucleus ErnestRutherford found the first evidence of the proton in 1919, and James Chadwick dis-covered the neutron in 1931 The discoveries of the electron and proton helped to lead

Ruther-to Max Born’s declaration that “Physics as we know it will be over in six months.”

Figure 1.1: Conceptual views reflecting our understanding of the atom at different times inhistory Atoms were once thought to be the smallest constituent of matter, but the ability toprobe the atom at higher energies has led to a deeper understanding of its internal structure.Image courtesy of [2]

Physics, however, was not over in six months There were still a few questionsremaining such as: what holds the protons and neutrons together? Similar to scatter-ing experiments used to probe the structure within the atom, accelerator experimentsprobed the nucleus and the interactions between the neutrons and the protons Theresult of these studies the 1950s showed a “particle explosion” in which physicists dis-covered many new particles similar to protons and neutrons: baryons, and a new class

of particles: mesons Then in 1964, Murray Gell-Mann and George Zweig proposed

to explain the structure of the hundred or so baryons and mesons with just a fewmore-fundamental particles: quarks.

The quark theory of Gell-Mann and Zweig has evolved into the Standard Model

of Fundamental Particles and Interactions Despite a few shortcomings, the StandardModel is now the most consistent theory that explains particles and their complexinteractions, or forces In the Standard Model there are three types of elementaryparticles: leptons, quarks, and the force mediators, which are bosons

1.1.1 Leptons

Leptons are a class of spin-12 point particles which do not form composite objects.Three of the leptons have an electrical charge of −1, and their only observable differ-ence is their mass quantum number These leptons are the electron, muon, and tau.These three leptons each have a corresponding neutrino (also a lepton) which have

no charge and very little mass These six leptons are divided into three generations

of leptons: the lepton and its corresponding neutrino Each of these six leptons alsohave a corresponding antimatter antilepton, so that there are a total of 12 leptons

1.1.2 Quarks

Quarks are a class of spin-1

2 particles which form composite objects, but arenever observed independently Whereas, leptons have integer electrical charge, quarkshave fractional electrical charge of 2/3 or −1/3 Each quark has a quantum numberknown as flavor which identifies the quark according to its electroweak transformation

properties So like leptons, quarks are organized into three generations of two quarkflavors with each generation having progressively higher mass Also like leptons, each

of the quarks have a corresponding antiquark

Figure 1.2: The quarks and leptons organized into their families (generations) in order ofincreasing mass Image courtesy of [2]

Quarks carry an additional charge called color The color charges are red, blue,and green The corresponding anti-colors are anti-red, anti-blue, and anti-green Witheach quark and antiquark that can come in three colors, there are a total of 36 quarks.Quarks combine to form composite particles called hadrons which have an integernet electrical charge and have no net color Hadrons are divided into two classes:baryons made of three quarks, and mesons made of one quark and one antiquark, andtheir anitparticles

1.1.3 Bosons

The above mentioned basic particles can interact with each other These teractions include attraction, repulsion, annihilation, and decay; and the force andquantum number exchange are carried by the spin-1 force carrier particles, bosons Aparticular boson can only be produced or absorbed by a particle which is affected bythat particular force For example, only particles that carry electric charge (such asthe electron or proton) can produce or absorb the electromagnetic force carrier par-ticle, whereas particles that carry no electrical charge (neutrinos) cannot produce orabsorb the electromagnetic force carrier

in-Electromagnetic Interaction The electromagnetic force carrier is the photon,which is massless and carries no electric charge

Weak Interaction Another type of interaction is the weak interaction The weakinteraction mediator is the W± or the Z0 The Standard Model unifies the weak andthe electromagnetic interactions into the electroweak interaction

Strong Interaction The interaction which binds quarks together to form hadrons

is the strong force which is mediated by the gluon The gluon mediates forces betweenthe color charged quarks Gluons, however, also have color charge so that gluonscan interact with gluons When two quarks are close to one another, they exchangegluons such that quarks continually change their color charge Because gluon exchangechanges color and since color is conserved, the gluons carry a color and anti-colorcharge The three colors and three anti-colors mean that there are eight combinations

of color and anti-color charges for the gluon.

Gravity The fourth type of interaction is gravity which is believed to be mediated

by the graviton Gravity, however, is not yet included in the Standard Model and willnot be discussed further

Figure 1.3: The bosons and the four fundamental forces which they mediate Each force ismediated by a specific boson(s), and each force is responsible for specific types of interactionswith certain types of particles Image courtesy of [3]

1.1.4 Fields

Basic quantum mechanics deals with the quantized treatment of dynamical tems of particles Similarly, quantum field theory is the application of quantum me-chanics to dynamical systems of fields In this context, fields are mathematical entitiesrepresenting particles Quantized fields are used because they can account for issues of

sys-Figure 1.4: An illustrative description of the fundamental particles and the fundamentalforces The different types of fundamental particles (quarks and leptons) are organized onlevels reflecting the forces which affect them For example, the electromagnetic force affectscharged leptons and quarks but not neutral leptons Image courtesy of [3].

causality, multiparticle states, transition between states with different particle number,antiparticles, and the relationship between spin and statistics

One of the fundamental quantities of classical mechanics is the action, S, which

is the time integral of the Lagrangian, L The Lagrangian in field theory is written asthe spatial integral of the Lagrangian density, L, so that the action can be written as:

La-that is invariant under gauge transformations2 is a gauge theory With the property

of gauge invariance, the field theory can be described mathematically with a groupapproach

The field theory of quantum electrodynamics (QED) describes the netic interaction In terms of group field theory, QED is described by the group U(1).The U signifies that the transformation is unitary: M†M = I The unitary propertyimplies that there is an inherently conservative nature to the transformation The 1signifies that the matrix is of dimension 1

electromag-The success of QED led eventually to the combination of the U(1) description ofQED with an SU(2) description of the weak force to form a SU(2) × U(1) structure for

a combined electroweak theory (S signifies that the matrices be special : determinant

is 1) U(1) requires one generator matrix, and SU(2) requires three generators Inthe combined theory, these generators give rise to two charged and two neutral bosonfields as described for the electromagnetic and weak interactions discussed above.The theory describing strong interactions is known as quantum chromodynamics(QCD), which is a SU(3) group The SU(3) group requires eight generators, whichcorresponds to the number of colorless combinations of two gluons discussed above.Additionally, QCD is non-Abelian which means that the group operations are notcommutative and that the generators can be self interacting as described above

on the vast amount of experimental tests over the last several decades.

Confinement One of the most striking properties of QCD is that single quarks andsingle gluons have never been experimentally observed In fact, the group property ofinvariance under rotation in parameter space leads to this property of confinement:particle states are unchanged by rotations in red, green, blue space In other words,observed particles must be colorless

Asymptotic Freedom The phenomenon of asymptotic freedom refers to the erty that the closer the quarks are to one another, the weaker the color charge Con-versely, as quarks move apart the color force increases Thus, when quarks are veryclose they behave as though they were free particles An analogy with QED will help

prop-in understandprop-ing this seemprop-ingly anti-prop-intuitive result

The electron-electron Coulomb interaction for free electrons is repulsive cause they have the same negative charge This contrasts with the electron-positronCoulomb interaction which is attractive because the electron and positron charges are

be-of opposite signs Recall in quantum field theory that the electron can emit a photonwhich can be reabsorbed or can then split into an electron-positron pair This emis-sion and splitting can surround the electron with a screening cloud of electrons andpositrons Because of the attractive / repulsive Coulomb interactions, the electron-positrons pairs will polarize so that the positrons are closer to the original electronand the electrons are farther from the original electron The total measured charge ofthe electron would then be dependent on the location of the probe within the screen-ing cloud As the probe penetrates the positive screen, the measured charge of the

electron will increase because the effective screening will be reduced In QED, theelectric field strength falls with the well known 1/r2 dependence.

Carrying the analogy to QCD, there is a similar distance dependence for the fieldstrength However, the non-Abelian nature of QCD leads to a dramatic contrast withthe screening effect of QED Quarks can emit gluons which can either annihilate intoquark-antiquark pairs or into gluon-gluon pairs Unlike the Coulomb interactions,the quark-antiquark pairs and the gluon-gluon pairs will not polarize in the samefashion as the electron-positron pairs Gluons carry twice the color charge of quarks sogluon-gluon effects will dominate and effectively spread out the effective color charge.Contrary to QED, a color charge will be preferentially surrounded by like color charges.The total measured charge of the quark would then be dependent on the location ofthe probe within the screening cloud As the probe penetrates the like-color screen,the measured charge of the quark will decrease because the effective antiscreening will

be reduced This antiscreening, known as asymptotic freedom, is essential since it iswhat allows QCD to be a quantitatively calculable theory The antiscreening effectmeans that the strength of the interaction actually increases approximately linearlywith distance

Coupling Constant The strength of the strong coupling is characterized by thecoupling constant αs Mathematically, αs is expressed as:

Λ2 QCD

µR, the renormalization scale, is the scale at which divergences in the theory are tored into the coupling [4] It is customary to use µR= Q in deep inelastic scattering

fac-Clearly at sufficiently low Q2, αs in equation (1.2) becomes large ΛQCD denotes thescale at which this behavior occurs, and this is the energy scale at which quarks can

be considered free particles As discussed in the section about asymptotic freedomabove, αs(Q2) decreases as Q2 increases and thus becomes small for short-distanceinteractions So, for Q  ΛQCD, higher powers of αs will be quite small and pertur-bative calculations in terms of quarks and gluons interacting weakly is appropriate(see Section 1.2.1) For Q of the order of Λ, the quarks and gluons will be in stronglybound hadrons Thus, Λ represents the boundary between “free” partons and ”bound”hadrons This value, Λ must be experimentally determined and is generally accepted

to be between 100 and 500 MeV β(αs) is an equation that describes the running ofthe coupling with Q2:

αs(Q2) = 12π

(11n − 2Nf)(ln(Λ2Q2 )) (1.7)

1.2.1 Perturbative Quantum Chromodynamics

Perturbation theory is a mathematical tool used to describe a complicated tum system in terms of a simpler system Mathematically, the simple system is per-turbed by a weak disturbance to the system In practice, the resultant perturbation isexpressed as a finite power series in a parameter (also known as the coupling constant),α:

quan-T (α) = quan-T0α0+ T1α1+ · · · (1.11)which converges when summed to higher orders (power of the coupling constant) ofthe expansion If the expansion parameter is very small, the relative importance ofhigher order terms in the calculation is small For Q2  ΛQCD, αs  1, and theseries will converge for higher orders of the perturbative expansion This is referred

to as the hard scale or hard interactions In contrast, where Q2 → ΛQCD, αs→ 1 the

perturbative expansion does not converge This is referred to as the soft scale or softinteractions.

A Leading Order (LO) calculation includes only contributions from the lowestorder terms for that process A Next-to-Leading Order (NLO) calculation also in-cludes terms with the second largest order of the coupling constant These usuallyinvolve radiation of gluon from a final state parton which results in an additional finalstate parton and are thus known as real corrections There is also a class of virtualcorrections expressed with loop diagrams which involve the emission and reabsorption

of a gluon, or the temporary fluctuation of a parton into a qq or gg pair

QED and QCD have a special tool, Feynman diagrams, developed by R man to aid in the systematic summation of the power series terms In Feynmandiagrams, the power of the coupling parameter in the series corresponds to the num-ber of corresponding interaction vertices Illustrations of various Feynman diagrams

Feyn-to different orders are shown in Fig 1.5

of particle physics scattering experiments are:

Elastic Scattering: only the momentum of the target and incident particles is changed

The target and incident particles, themselves, are left intact.

Inelastic Scattering: the target particle is excited For example if a nucleus isbombarded by neutrons, it may be excited to some nuclear resonance

Deep Inelastic Scattering: the target (and sometimes the incident particle) is stroyed and completely new particles may be created

de-Mathematic descriptions of scattering involve parameters which (1) describe thedistance of closest approach (how close the incident particle would come to the target

if it moved in a straight line) and (2) the angle of deflection The distribution ofdeflection angles is described by a function known as the differential cross sectionwhich is discussed in Section 2.1.2 In more complicated cases of scattering, such asdeep inelastic scattering of electrons and protons, so-called form factors have to bemultiplied to the scattering formulae, describing the internal structure of the proton(also discussed in Section 2.1.2)

2.1 Deep Inelastic Scattering

Deep Inelastic Scattering is the name given to the scattering process used toprobe the insides of hadrons using leptons The interaction proceeds through the ex-change of a probe Figures 2.1 and 2.2 are representative diagrams of deep inelasticscattering (DIS) events commonly observed at HERA These are divided into two cat-egories The first involves the exchange of a virtual photon or Z0 boson Since neither

of these bosons carry electric charge, this category of scattering events is referred to

as neutral current (NC) The second category is the exchange of a W± boson, whichdoes carry electric charge and is thus referred to as charged current (CC) In the case

Figure 2.2: A conceptual look at neutral current and charged current processes in electronproton collisions At sufficient energy, the electron exchanges a boson with one of the quarks

in the proton while the other quarks continue on their original path The electron and struckquark are deflected from their original paths Image courtesy of [3]

of CC events, the final state neutrino’s momentum cannot be directly measured whichleads to less than accurate reconstruction of the kinematics Therefore, the chargedcurrent processes will not be discussed further.

The exchanged probes are described by their wavelength with the relation

λ = ~

where ~ is Plank’s constant, and Q is related to the momentum of the probe (seeSection 2.1.1) As the momentum of the probe increases, the wavelength of the probedecreases The shorter the probe wavelength, the more detail the probe can resolve

As illustrated in Figs 2.3 and 2.4, increases in Q2 lead to shorter wavelengths of theprobe The long wavelength probe sees the proton as a point particle The shorterwavelength probe is able to resolve the size of the proton As the probe wavelengthdecreases, the probe is able to resolve objects inside the proton: initially three valencequarks, and with smaller wavelength a sea of quarks and gluons

Due to the quantum mechanical uncertainty principle, it is not required thatthe exchange boson obey the laws of conservation of momentum and energy as long

as it lives for a time ∆t < ~/∆E An exchanged boson which doesn’t have thecorrect energy or momentum is termed a virtual boson, and Q2 is referred to as thevirtuality of the interaction Virtual refers to a field excitation mode that appears

in an intermediate step in a calculation The virtuality can range from 0 < Q2 < s(see Section 2.1.1), with greater virtualities corresponding to smaller wavelengths andtherefore a greater resolving power of the exchanged boson

Figure 2.3: As the energy of the exchanged probe (Q2) increases, the wavelength of theprobe decreases Long wavelength are only able to resolve the proton itself Shorter wave-lengths are able to resolve the valence quarks inside the proton Image courtesy of [3].

Figure 2.4: As the wavelength gets smaller, the probe is able to resolve even finer structurewithin the proton Image courtesy of [3]

2.1.1 Kinematic description

In lepton-hadron collisions, the center of mass energy, √

s, can be calculated byexamining the four-momenta as labeled in Fig 2.1

Additionally, the interaction can be characterized by some kinematic quantities Thefirst is the Lorentz invariant momentum transfer, Q, defined as

Q2 = −q2 = −(k − k0)2 = 2k · k0 (2.3)There is another possible Lorentz invariant, W W is the invariant mass of thehadronic final state:

which has the range 0 ≤ W ≤ s W is also the center of mass energy of the proton system If Q2 & 1 GeV and W  mp, the reaction is deep-inelastic scattering(DIS)

boson-In 1969, Bjørken proposed the scaling property: as Q2 → ∞, W2 → ∞, x is fixed

At fixed scaling variable, x, the scattering is independent of q2 This suggests thatthe probing virtual photon in DIS scatters off of something pointlike Since protonsare not point-like particles but do indeed have a complex internal structure, there is adimensionless variable, Bjørken x, which describes to lowest order the fraction of thetotal proton momentum carried by the struck parton: