determination of coupling constant from events deep ... appeared and the contributed paper...

Determination of Strong Coupling Constant from Multi- Jet Events in Deep Inelastic Scattering

at HERA

Kyung Kwang Joo

A thesis submitted in conformity with the requirements for the Degree of Doctor of Philosophy

Graduate Department of Physics University of Toronto

Copyright O 1997 by K.K. Joo

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Abstract

This thesis describes the determination of the strong coupling constant a, from measurements of multi-jet rates in deep inelastic electron-proton scattering at HERA by the ZEUS collaboration for values of the squared four-mornentum transfer e2 between 120 and 3600 G~v'. A total integrated luminosity of 3.2 coliected during 1994, was used for the measurement. Jets are identified with the JADE algorithm. The jet rates were fitted to a next-to-leadhg order O (a:) perturbative QCD cdculation with as left as a free parameter. Using the fitted value for as the perturbative cdculation describes well the measured rates and kinematic properties of multi-jet events. A cut on the angular distribution of parton emission in the y* - parton center of mass is

3 introduced to rernove forward going jets. The value of as is measured in three Q- regions and finally cornbined over the full kinernatic range. The as value decreases for increasing Q', consistent with the ninning of the strong coupling constant, as expected by QCD. The determined value

0.004 is as ( M y , ) = 0.1 17 f 0.005 (stat) + oos (systap) k 0.007 (syst th,v).

Ac knowledgments

My memory goes back to 1993, when I joined and have been a member of ZEUS collaboration. Since then, 1 am happy to able to meet and work with many different people having different interests. Saying a word for their countless helps is rather incornplete, but, aha, it is tirne to express my gratitude to these people from the bottom of my heart.

First of d l , 1 am extremely thankful to my supervisor, R.S. Orr for his endess support and constant encouragement during my stay in Hamburg Gemany. Without his patience, understanding and advice, this work would not cornplete in the way it is. I also thank him for allowing me to join the TLT group within the ZEUS collaboration and making me to feel proud of being TLT member. 1 appreciate his careful reading of the manuscript and wonderful guidance, despite his busy schedule.

A speciai thank goes to Dr. LH. Park, who has been leading me the right direction in my analysis. His enormous passion for physics and sunny or rainy tireless work iike a workaholic make me always awake in spending my time at DESY.

1 also would like to thank J.F. Martin who shows a good leadership and support as a ZEUS Canada leader during my stay in Hamburg.

I'd like to diank al1 of those who have made my stay here a pleasant way to learn experimental high energy physics. Above d l , to TLT group members, 1 offer my thanks and gratitude for their inputs during my TLT period. Without their knowledge and expenence, my work wouid have lacked direction and clarity. Sampa Bhadra and Dinu Bandyopadhyay, thanks for providing good organization in effective way. Richard, thanks for making me TLT expert in a short time and nice companionship al1 the way, including free beer at :he first time when 1 arrived in Hamburg. Cortney, thanks for introducing me to the TLT hardware and making our office cheerful dunng his stay as a captain Canucks. In addition. Gerd Hartner and Yoshihisa Iga, thanks for both providing streamline tracking code in the interface of TLT for saving CPU time and monitoring them in the offline environment. Others - David, Stefan Polenz, thanks for going through al1 odds with me during the hectic time of taking data.

I would like to express my appreciation for al1 of the support 1 have received from the people who have worked in jet analysis group - Jutta, Thomas, Tara, Darin and Tony. 1 thank al1 the comments and good advice from these people.

During rny stay at DESY, 1 am indebted to ail the members of Canadian group for their special feelings acted as one big family. 1 would like to appreciate their endeavour to build this

atmosphere and make office like a home. Mohsen, thanks for teaching me Persian a bit and making good humors. Wai and Pat, thanks for playing soccer. If we were 10 year younger, we would play in the Bundesliga! Mark, thanks for nice system managing and pizza ordering. Rainer, thanks for giving me motivation to leam Gennan. Laurel, th& for the Company with my wife. Dave, thanks for fixing the bugs in my zarah job from tune to time. Al1 others - Michael, Lim, thanks for having nice time together.

Thanks to also Korean group at DESY - Dr. Myoung-Youl Pac, Dr. Shin Woo Nam, Se Byong Lee, Min Kyu Lee, Chin Kyoung Chang, Sang Hoon, Mun Hee, Tak Soo Shin, 1 greatly enjoyed their Company during my year in no sunshine Hamburg.

As for another group at DESY, 1 would like to express my gratitude to my Japanese col- leagues. Dr. Katsuo Tokushuku and Dr. Yuji Yarnazaki, thanks for giving GFLT information. Ichiro Suzuki and Kensuke Hornrna, thanks for chatting.

1 would like to also thank Hyun Soo Kim for shipping and packing al1 my stuffs from Tor- onto to Hamburg during my absence in Toronto. Without his help, I would have had to fly quite often. I appreciate many kind helps from secretaries - Marianne Khurana, Winnie Karn, for managing my stuffs.

1 also thank my family - Kyung Pyo, Kyung Chae, Kyung Rae, Kyung Sirn, In Sim, Young Sim - whose continuous supports have encouraged me to make my way. Thanks, Mom and Dad, for showing me the devoted love and warmth during my whole life. Hye Kyeong's farnily have also been extremely supportive, and 1 thank them, too.

The last, but by no means at least, the most important dedication goes to my wife Hye Kyeong Song. Her companionship, love, and friendship mean more than anything to me. Espe- cially 1 thank her sacrifice and care dunng the stay of last severai months in Hamburg. Without her loving support and comfon, 1 could not be the man 1 am.

Contributions to the ZEUS Experiment

My involvement and experîence as a high energy physicist has begun from FaLi of 1992 after 1 joined the EHEP group of University of Toronto. 1 moved to Hamburg, Germany where EHEP group is perforrning an experiment at the Deutches Elektronen Synchrotron laboratory. Since then, 1 have been a member of ZEUS collaboration and joined the ZEUS Third Level Trig- ger (TLT) group. During this penod, 1 was involved with the TLT system.

After ep collision has been firstly made in 1992, it has been important to anaiyze these data. With more knowledge in the physics filter from each analysis group, it was necessary to develop more user fnendly system in the TLT, so any requests for modification should be instailed in the online data acquisition systern after fuily testing and debugging them in offline environment. It is extremely important to have reliable code management system in both online and offline TLT envirmment for the data reconstruction. During these periods, I was fully responsible for the maintenance and development of the TLT system. 1 made the online code compatible with ornine software during this period. Also data quality monitoring in online reconstruction phase is very important, so 1 have involved in every phase of TLT monitoring system. 1 aiso contributed ro install TLT simulation in Monte Carlo (MC) funnel production.

From the beginning of January of 1995 after two y e m in the TLT group, 1 joined one of subgroups in Deep Inelastic Scattering Group (DIS) - Jet physics group, for my analysis. I rnainly concentrated on the measurement of strong coupling constant as from the multi-jet rates by counting jets. From the measured rate of (2+1) jet events, it is possible to determine the strong coupling constant because jet counting method relies on the fact that the ratio of (2+1) jet cross section to the ( l+l) jet cross section is proportional to ax.

The physics analysis presented in this thesis is based on the data collected in the 1994 running period at HERA. The first results of this andysis were shown at "Workshop on Deep Inelas- tic Scattering and QCD" in Paris (1995). The value of as with systematic checks from ZEUS experiment appeared and the contributed paper "Measurement of the Strong Coupling Constant frorn Multi-jet Deep Inelastic Scattering Events at HERA" was subrnitted to the European Physi- cal Society in Brussel ( 1995).

The final analysis contributed to the publication "Measurement of as from Jet Rates in Deep Inelastic Scattering at HERA" and has been published in Phys. Lett. B363 ( 1995). For more details to the ZEUS users, 1 have made ZEUS note for this analysis - ZEUS-Note 95-134.

S..

Ill

Contents

1 . Introduction ........................................................................................................ 1.1 Motivation .................... ..... .............................................................................. 1.2 HistoricaI Introduction ....................................................................................... 1.3 DIS Experiments ................................................................................................ 1.4 Overview of This Thesis ....................................................................................

2 . The Theory of Perturbative Quantum Chromodynamics (QCD) ... 3.1 The QCD Lagrangian ........................................................................................ 2.2 The Strong Coupling Constant as ..................................................................... 2.3 Renomalization Scheme Dependence ..............................................................

3.3.1 Renormalization Scale Dependence .......................................................... 2.3.2 Optimization of Renormdization Scaie ....................................................

2.4 Asymptotic Freedom and Quark Confinement .................................................. 2.5 The Status of as Measurement .......................................................................... 2.6 How to Measure q at HERA ............................................................................

2.6.1 The as Measurement from let Rate R, + ................................................ 2.6.2 The Differential Jet Rate D I + .................................................................

3 . Deep Inelastic Scattenng (DIS) at HERA ............................................... 3.1 Kinematics and Variables at HERA ................................................................... 3.2 Factorization Theorem in DIS ...........................................................................

3.2.1 Factorization Scale Dependence ............................................................... 3.2.2 Parton Distribution Parameterization ..................................... ..... ..............

4 . Jet Definition and Jet Finding Algorithms .............................................. 4.1 Cluster Algorithm for Jet Finding ......................................................................

4.1.1 The JADE Jet Finding Algorithm in DIS .................................................. 4.1.2 Jet Finding Schemes .................................... ... ...........................................

4.2 Collinear Jet Singularity in ep Scattering ..........................................................

5 . Theoretical Overview of NLO Calcu!ations in DIS ............................

5.1 Two Jet Kinematics ............................................................................................ 35 3

5.2 O (a;) Perturbative QCD Calculations .......................................................... 38

6 . HERA and the ZEUS Experiment ............................................................ 6.1 The HERA Accelerator ...................................................................................... 6.2 The ZEUS Detector ...........................................................................................

6.2.1 Calorimeter ................................................................................................ .................................. 6.2.1.1 Energy Loss of Charged Particles in Matter

6.2.1.2 Electrornagnetic (EM) S howers .............. ... ................................... 6.2.1.3 Hadronic Showers .........................................................................

6.3 The ZEUS Online Trigger System and Data Acquisition ................................ . 6.3.1 Overview of the ZEUS Data Acquisition System ..................................... 6.3.2 Global First Level Trigger (GFLT) ............................................................ 6.3.3 Global Second Level Trigger (GSLT) ....................................................... 6.3.4 Event Builder ( E W ) ................................................................................. 6.3.5 Third Level Trigger (TLT) .........................................................................

6.3 5 1 Online Background Reduction ..................................................... 6.3.5.2 Online Selection of Physics Events ............................................... 6.3.5.3 Online Monitor in the TLT ............................................................

.......................................... 7 . Event Simulation and Hadronization Mode1 7.1 Leading Order (LO) Simulations .......................................................................

7.1.1 First Order Matrix Elements (ME) ............................................................ 7.1.2 The Parton Showers (PS) ........................................................................... 7 . I . 3 Matrix Elements and Parton Showers (MEPS) .........................................

7.2 Color Dipole Mode1 (CDM) .............................................................................. 7.3 Models of Hadronization ...................................................................................

7.3.1 String Fragmentation ................................................................................. 7.3.2 CIuster Fragmentation ...............................................................................

8 . Reconstruction Methods and Event Selection ....................................... 8.1 Electron (EL) Method ........................................................................................ 8.2 Jacquet-Blonbei (IJ'B) Method ............................................................................ 8.3 Double Angle (DA) Method .............................................................................. 8.4 Mixed Method .................................................................................................. 8.5 Electron Identification ..................... ,.. ...........................................................

8.6 Event Selection Cuts .......................................................................................... 70

8.7 General Event Characteristics ............................................................................ 74

9 . Properties of (2+1) Jets and Jet Production Rates .................................................... 9.1 Analysis of (2+1) Jets .................................... .... ................................................

9.1.1 Jet Multiplicities ........................................................................................ 9.1.2 Effect of Parton Shower and Parton Density to Jets .................................. 9.1.3 Separation of Jets from the Beam Remnant Jet ........................................

9.2 Properties of Parton Variables (z, x, ) ................................................................ ..................................................................... 9.3 The Properties of (2+ 1) Jet Events

9.4 Jet Production Rates and Correction Procedure ................................................. ....................................................................................... 9.4.1 Uncorrected Data

9.4.3 Correction for Detector Effec ts ...................... .... .... ............................. ........................................................ 9.4.3 Correction for Hadronization Effects

9.4.4 The Corrected Jet Production Rates ..........................................................

....................... I O . Measurement of the Strong Coupling Constant g ( Q) 103

10.1 The as (Q) Measurement from Jet Rates ................................................. 103

10.2 Running of as ( Q ) at HERA .......................................................................... 104

Systematic Effects on the Determination of as ( Q ) ............................. 1 1.1 Experiment Uncertainties ................................................................................

1 1.1.1 Event Selection ........................................................................................ 1 1.1.2 Energy Scde ............................................................................................ 1 1.1.3 Jet Analysis .............................................................................................. 1 1.1.4 Fitting Method ......................................................................................... 1 1.1 -5 Mode1 Dependence ..................................................................................

1 1.2 Hadronization Correction ................................................................................ 11.3 Dependence of The Result on the Input Parton Density .................................. 1 1.4 Renormalization and Factorization Scherne Dependence ................................

12 . S u m m q of Final Result and Cornparison of ZEUS with Other Measuremen ts .................................................................................................... 121

.................................................................................... 12.1 Presentation of Results 121

12.2 Cornparison with Other Measurements ........................................................... 123

I 3. Conclusion ..........................................................................................................

References ................................ .... .... .. . .. -. -.. ....... ..-... ...............A. - .. . .. .. -. . .. . . . .-. ..-. .. .

vii

List of Figures

Figure 1- 1 :

Figure 1-2:

Figure 1-3:

Figure 1-4:

Figure 1-5:

Figure 1-6:

Figure 2- 1 :

Figure 2-2:

Figure 2-3:

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Figure 3- 1 :

Figure 3-2:

Figure 4- 1 :

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Figure 6-6:

Figure 6-7:

Figure 6-8:

Figure 6-9:

Figure 6- 10:

Figure 7- 1 :

Figure 7-2:

Figure 8- 1 :

Figure 8-2:

An illustration of the vistual photon probe in DIS ........... .. ........................ .................... The lowest order splitting functions P,,, P,,, P,, and P,,

.......................... Diagram for a typical three-jet event in an e'e- collision

Schematic representation of neutrd current DIS leading to event with ( l+ l ) jets ................................................................................................ ...

................................... Feynman diagrams for QCDC and BGF processes

The three kinematic regions for measuring the strong coupling constant as .............................................................................................................. A symbolic representation of the interactions in the QCD Lagrangian .....

.................... The l-loop and 2-loop as expressions at different A values

The variation of as in 2-loop with d5) as a function of Q ..................... ........................ Screening of electromagnetic charge in a dipolar medium

........................ The Iowest order corrections to the quark-gluon coupling

.................................................... A summary of measurement of as (Q)

Inelastic scattering view for one-boson exchange in ep collision at ........................................................................................................ HERA

The short and long range processes of DIS ............................................... An overview of an ep scattenng ................................................................

........................................ Ernission of a collinear gluon in the initial state

............ The HEM accelerator cornplex and the HERA injection system

........................................... Integrated luminosity delivered and collected

................................. The longitudinal cross section of the ZEUS detector

......... Cross section of the ZEUS detector perpendicular to the bearn line

.................................. A tower in the FCAL and RCAL of ZEUS detector

.......................... Layout of the ZEUS trigger and data acquisition system

................ Timing distribution of FCAL-RCAL time vs RCAL time in ns

............................. A beamgas event with track reconstruction in the TLT

Online processing tirne in the TLT ............................................................ Online TLT vertex distribution ..................................................................

.................................... Parton shower evolution in an ep scattering event

................................................ A diagram of cluster fragmentation model

The basic diagram for the DIS process ...................................................... ............. Contours for the scattenng angle and energy of electron and jets

viii

Figure 8-3:

Figure 8-4:

Figure 8-5:

Figure 8-6:

Figure 8-7:

Figure 8-8:

Figure 8-9:

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Figure 9-2:

Figure 9-3:

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Figure 9-8:

Figure 9-9:

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Figure 9- 12:

Figure 9- 13:

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Figure 9- 18:

Figure 9- 19:

Figure 9-20:

Figure 9-2 1 :

Figure 10- 1 :

Figure 10-2:

Figure 11-1:

Figure 1 1-2:

Creation of calorimeter objects via island clustenng ............................ .... The distributions of 2-vertex and FCAL energy ....................................... The distribution of scattered electron ........................................................ E.P. distribution for photoproduction Monte Car10 compared with DIS Monte Carlo .............................................................................................. Typical DIS event observed in ZEUS detector .......................................... Distribution of the double angle DIS event sample in the x . Q~ plane .........................I.I.......................................................*.................................

2 2 The distribution of the kinernatic variables XDA, QDA, YJB and WD, ....... Distribution of multiplicity for ZEUS 1994 data .................................... ...

.............. Effect of parton showers and partoii densities as a function of x

Polar angle of the current jet for ( l+ l ) jet ................................................. Polar angle distribution of rnost energetic jet of (2+1) jet events for y,,, .. The distribution of the corrected (2+ 1) jet rate without z-cut .................... Allowed (2+l) jet region of the parton phase space in y,,, = 0.02 ............ The phase space for the modified JADE jets ............................................. The distributions of parton variables 2 and xp at the detector Ievel ........... Distribution of the corrected parton variables z and xp at the parton level

Plots for mean deviation and root mean square values of parton variable 7 .. ................................................................................................................. The plot of mean deviation and resolution of x,, reconstruction ................

One of (2+l) jet events in ZEUS ............................................................... Effect of increasing z and y cut .................................................................. The distributions of PT and M j j after a z cut at the detector level .............

The distributions of PT and MjJ after a z cut at the parton level ............... Scatter plots for pseudorapidity of (2+1) jets ............................................ Pseudorapidity distribution of the two jets ................................................ Pseudorapidity distribution of the two jets in each kinematic region ........ Uncorrected jet rates in each kinematic region .......................................... The jet production rates of LEPTO 6.1 ..................................................... Jet production rates Rj+! as a function of ycut ................... .. .................. as vs e2 with y... = 0.04 in E665 .............................................................

....................... Measured values of as (Q) for three different e2 regions

The measured differential jet rate DI+! .................................................... The correction factor for the detector effect ..............................................

Figure 1 1-3: The hadronization effect on (2+ 1 ) jer rate for different Monte Carlo ....................................................................................................... rnodels 113

................................ Figure 11-4: Systematic uncertainties in the measured value of A 119

Figure 1 1-5: The cornparison of values <M: , /Q~> . <M,,/Q>. < P T / Q > and 2 3 < P T / Q - > .................................................................................................. 120

2 Figure 12- 1 : Measured values of % for three different Q regions ............................... 123

.......... Figure 12-2: Surnrnary of as measurements from ZEUS and other experiments 124

Figure 12-3 : A surnmary of as from ZEUS and H 1 expenments .................................. 125

... Figure 12-4: Cornparison of ax measurement between ZEUS and LEP experiments 126

List of Tables

Table 1- 1:

Table 1-2:

Table 2- 1 :

Table 2-2:

Table 4- 1 :

Table 5- 1:

Table 6- 1:

Table 6-2:

Table 9- 1 :

Table 9-2:

Table 9-3:

Table 9-4:

Table 9-5:

Table 9-6:

Table 10- 1 :

Table 10-2:

Table 10-3:

Table 11-1:

Table 1 1-2:

Table 1 1-3:

Table 1 1-4:

Table 12- 1:

Table 12-2:

Table 12-3:

.................................................................................... Quarks and leptons

Coupling parameters in each interaction ................................................... The brief comparison of the basic properties and features of QED and QCD ..........................................................................................................

............................................................ A sumrnary of measurement of as

Resolution cnteria and recombinaûon schemes for the jet clustering algorithm ................................................................................................... The differences between DISJET and PROJET NLO QCD theoretical calculation programs ................................................................................. The running parameters dunng luminosity period since 1992 .................. Classification of physics groups and filters implemented in the TLT during 1994 .............................. .. .................................................................. Measured uncorrected jet production rates ................................................ The correction factors in kinematic region 120 c @ c 240 G ~ V ? ............ The correction factors in kinematic region 240 < e2 < 720 G ~ V ~ ............ The correction factors in kinematic region 720 < QI < 3600 G ~ V ' .......... The correction factors in kinematic region 120 c @ < 3600 G ~ V ~ ..........

......................... The jet production rate Ri+/ corrected to the parton level

(2+l) jet production rate corrected to the parton level .............................. The values of <@. R. - + l . . and as (Q) in each three bin

MS .................................................................................................................... The values of ce>. Rz+ , . A!?: . and as ( Q ) in the cornbined bin .................................................................................................................... Surnmary of experimental systematic checks in each kinernatic bin ......... Summary of hadronization systematic checks in each kinematic bin ....... Surnrnary of parton density uncertainty in each kinematic bin ................. Summary of scaie uncertainty in the NLO calculations in each kinematic

.............................................................................................................. bin

The determined values of R2+ I\- and as ( Q ) in each bin MS ....................................................................................................................

The deterrnined values of RZ + , , d? and as (Q) in cornbined bin MS

rneasurernents at LEP with jets in Eo scheme ......................................

1. Introduction

1.1 Motivation

High energy physics deals with the most elementary constituents of rnatter and the forces by which they interact. The present understanding of high energy physics is based on interactions of fundamental spin 112 particles, called fermions, via the exchange of spin 1 particles, the gauge bosons. There are two classes of fermions, quarks and leptons, as shown in Table 1.1. These are classified into three generations and there are six different flavors of both quarks and leptons.

?àbk 1.1: Quark and leptons. u. c and t-quark have charge 2/3 and d, s and b-quark have charge -1/3 of proton. Leptons have electric charge -1. neutrinos have no electric charge.

By the exchange of gauge bosons (y, 2, W' , g ) elernentary particles may interact through the electromagnetic, the weak, and the strong interactions as shown Table 1.2 The strength of the interactions is controlled by coupling constants in each interaction. The values of the coupling constants are not predicted by any theory and rnust be detemiined by experirnent. The theory which descnbes the interactions of photons with electrically charged particles is called Quantum Electrodynarnics (QED). QED is the rnost successful physical theory ever developcd and has been tested to exceptionai precision. The weak force is rnediated by the intermediate vector bosons, and 2. It was proposed in the mid-seventies that the strong interactions are mediated by the exchange of gluons between quarks. nie resulting theory is called Quantum C hromodynarnics (QCD).

1 Interaction 1 Coupling strength 1 Value l

Table 1.2: Coupling parameters in each interaction as quoted by the Particle Data Gmup [ I l . We rise units with c=l and f i =I throughout this thesis.

Elec tromagnetic

1 Weak

ap~o(Mt?)

G~

1/137.0359895k0.0000006 1

1.16639. ~ o - ~ ~ o . O O O ~ . 10-~ G ~ v - ~

Apart from quark masses, QCD has only one free parameter, the strong coupling constant, as. The strong coupling constant is one of fundamental parameters of what is now called the standard model. Its value is much less well known than the electroweak couplings since quarks and gluons cannot be observed directly and their properties and interactions must be studied indi- rectly. Therefore, the measurement of as in different processes at different momentum scales provides a crucial test and consistency check of QCD.

i -2 Wstorical Overview

Why do particle physicists need to accelerate electrons and protons to high energies? His- torically, energetic beams have been used to probe the structure of matter. From various experiments, it tums out that we observe finer structure if we use higher energies to probe smaller distance scaies. The first evidence of point-like constituents of the proton was observed in electron-proton scattering at SLAC. Feynman had proposed that high energy hadron-hadron scattering could be understood in terms of constituents, which he referred to as partons. These constituents were later identified with the quarks which Gell-Mann and independently Zweig introduced in 1964 [SI.

In the simple Quark-Parton Mode1 (QPM) of Deep Inelastic Scattering (DIS), a quark, which carries a fraction x of the proton momentum, is scattered out of proton by the exchange of

2 2 a single virtual boson carrying a four-momentum of q, where Q = -q . The narne "DIS" rneans that large center of mass energy, $. of the virtual photon-proton system makes it possible to probe smail space-time intervals and to have multi-particle production in the final state. Neutra1 Current (NC) DIS is characterized by the exchange of a virtual photon or 20 boson between the incident lepton and proton. whereas Charge Current (CC) reactions involve the exchange of d boson. The proton structure functions embody information about the internai structure of the proton. In 1969 Bjorken [3] predicted that the scaling property of structure functions was expected in deep inelastic electron-nucleon scattering. in the QPM, the structure functions depend only on x ,

and not on Q'. This dependence on a single dimensionless variable is known as scaling. It was confirmed by expenments at SLAC.

In the QPM, the proton is viewed as being composed of quarks having spin 112 and carrying fractionai electric charge. When the momentum carried by the virtuai photon is low in deep inelastic electron-proton scattering, its wavelength is relatively long compared to the dimensions of proton. It will not be able to resolve any structure and will effectively see the proton as a point. With higher rnomentum. the photon will have a wavelength comparable to that of the nucleon. The photon will begin to probe the finite spatial extent of the proton and may resolve the internai structure of the proton as shown Fig. 1.1. In order for the parton model to work, large 3-momentum ApAx -3 is needed to probe small distance. Furthemore, large energy AEAt - f r means

the interaction will occur in a short time, so that we c m ignore the interactions with spectator partons which are not directly involved in the interaction. This is the impulse approximation necessary for the parton mode1 to be valid.

/ Proton Proton

/ Quark

Figure 1.1: An illustration of the virtual photon probe us its wavelength becomes much smaller than the proton diamete,: (a) At very lorv values of momentum transfec the proton behaves as a single object either point-like or rvith a finite size. (b) As mornenturn transfer becornes larger and kirger. a short wavelength photon resolves the quarks within the proton in the DIS process.

The internai structure of the proton is characterized by two measurable structure functions in the case of an exchanged virtual photon. At high energies the differential cross section for DIS mediated by a virtual photon has the form [4]

where y is the ratio of the electron energy transferred to the hadronic system to the total etectron energy in the rest frarne of the proton. F I , depend on x and e2. In the QPM the F depend

2 only on x , and not on Q . Scaling can hold exactly only for non interacting quarks. Clearly, however, the quarks in the proton interact by the exchange of gluons. This gluon radiation will lead to a hreaking of scale invariance which becomes stronger as Q' increases. The breaking of scale invariance can be understood within the context of QCD.

The first QCD correction terms to O (as) are that QCD allows interactions between - - quarks and gluons via gluon emission q -+ qg, q + q g , g + gg and sea quarks produced by g + qq. QCD predicts the probabilities for these branching in terms of the splitting functions. These are given by

P,, represents the probability of a quark emitting a gluon and so becorning a quark with momentum reduced by a fraction z. Pqg is the probability that a gluon annihilates into a qq pair such that the quark has a fraction z of the gluon momentum. The sarne interpretation is applied to Pgq and P,,. The lowest order splitting functions are shown diagrammatically in Fig. 1.2.

Figure 1.2: The lowest order splitting functions, Pqql Pgq, Pqg and Pgg .

What we measure expenmentally are not partons, but jets in the final state. The concept of jets should address the connection between observed hadrons and original partons. What is a jet? Naively speaking, a jet is a group of hadrons, consisting of well-collimated particles. Neither QCD nor experiment provides a clean definition of a jet. Therefore an aigorithm is employed to experimentally define a jet. A jet is defined by a given resolution for a given algorithm so that the nurnber of jets is formed experimentally differcnt according to different jet finding algorithms. A jet should be an experimentally measurable quantity and the definition of jets is applicable in theoretical calculation.

Jet production was first observed at e+e- colliders in 1975. The process e+e- + qq produces hadrons and a pair of clean back-to-back jets was seen at a center of mass energy of 6-8 GeV at SPEAR [5]. With more center of mass energy available, drarnatic examples of three jet events were observed in efe- -t qqg at the PETRA collider [6], where the total center of mass energy was 3 1 GeV. Figure 1.3 shows how a quark may radiate a hard gluon, carrying a fraction of the quark energy at a large angle, and the gluon and quark give rise to separate hadronic jets. The observation of three jet events was direct evidence for the existence of gluons.

Figure 1.3: Diagram for a typical three jet event in an e+e- collision. Al1 rhreejïnal srare partons have appreciable energy and are well separated in angle.

In the QPM of DIS, the process q + q (p = y, f), where is the cxchanged vir- tua1 boson, gives rise to (1+1) jet configuration in the final state where the proton remnant is denoted by "+Y"' The transverse momentum of the scattered electron is balanced by a single jet associated with the struck quark. The proton rernnant carries little transverse momentum. A schernatic view of the (1+ 1) jet configuration is shown in Fig. 1.4.

QCD modifies this picture. Multi-jet production in DIS beyond ( l+ l ) jets provides a good laboratory for testing QCD. In DIS, the zeroth order electroweak process V* q + q is corrected by the first order QCD processes: QCD Compton (QCDC) process of the hard emission of gluon V* q + qg ; and Boson-Gluon Fusion (BGF) to a quark-antiquark pair V* g + qq. The diagrams for the QCDC and BGF processes are shown in Fig. 1 S. The theoretical predictions for the cross section of these processes are available and have been calculated [7] to O (a:). The fraction of (2+1) jets, R2 + I , is defined as the (2+1) jet cross section + , divided by the total cross section otO,. R , + , is a quantity which is calculable in perturbative QCD.

Figtire 1.4: Schernatic representation of neutral current DIS leading to event with (1 + I ) jets. The vil-tuai photon emitted by the electron scatters a quark out of the proton. This is the Born r e m for (1+ 1) jets. The remnant from the proton goes in the original proton direction. A single jet and the proton remnant jet are observed in the fino1 state.

QcD Compton (QCDC) \. Boson-Gluon Fusion (BGF) \

Figure 1.5: Feynman diagrams for QCD Compton scattering and Boson Gluon Fusion process for (2+1) partons in the final state at H E M . Both are leading processes to O (as) .

Jet production in DIS was first observed in the fixed-target expenment E665 at FERMI- LAB at a center of mass energy of 30 GeV [8].

Since 1992 the two detectors HI and ZEUS began collecting data at the ep colliding facility HERA (Hadron Elektron Ring Adage). ZEUS is one of two experiments at HERA. The hadronic final state in deep inelastic neutrai current at HERA provides a new kinematic domain to study jet. Jet production at HERA is complernentary to e+eS experiments. At the much larger center of mass energy of HERA, jet production with large invariant mass and large transverse momentum can be clearly seen.

1.3 DIS Expenments

Figure 1.6: The three kinematic regions, (1, II, III), used in this annlysis for measuring the strong coupling constant, ~ 4 . ïhe kinernatical lirnits for HERA and forfired target experiments are nlso

2 show. The variables are related by e2 = xys in x - Q phase space.

The center-of-mass energy squared at fixed target DIS experiment is given by s = 2MpE for a lepton beam hitting a fixed proton target. For a collider, a center-of-mas energy s = 4EeEp is available. The three different kinematic regions in the x and @ plane used for the study presented in this analysis are shown in Fig. 1.6. The figure includes, for comparison, the regions accessible to the fixed target experiment such as SLAC [9], BCDMS [IO], NMC [Il] , EMC [12] and CCFR [13]. The SLAC data were obtained from scattering electrons from hydrogen with 0.6 I e2 5 30 G ~ v ~ . The Bologna, CERN, Dubna, Munich and Saclay (BCDMS) experiment involved the scattering of muons from a hydmgen target in the kinematic range, 0.06 I x 10.8 and 7 I Q' < 260 G ~ v ~ . The kinematic range covered by the New Muon Collaboration (NMC) is 0.006 5 x 5 0.6 and 0.5 I e2 1 55 Gev2. Here muons of 90 and 280 GeV interact in a liquid hydrogen target. The European Muon Collaboration (EMC), where the kinernatic range is 7 5 e2 5 170 Gev2 and 0.03 < x 5 0.75, scattered muons off a deutenum target. At FNAL, the Chicago, Columbia, F e d l a b and Rockefeller (CCFR) expenment used a neutrino beam on an iron target with 0.015 1 x 5 0.7 and 1.3 1 e2 1200 G~v*, while the E665 experiment used a muon bearn of 490 GeV on various targets.

1.4 Overview of This Thesis

This thesis describes the extraction of as from high e2 deep inelastic neutral current ep collisions. The general outline of this thesis is as follows. Chapter 2 provides a short theoretical description of QCD and the renormalization group equation for the as measurement. In the fol- Iowing chapter, the details of DIS at HERA are reviewed. Various parameterizations of parton distribution inside the proton are also discussed. Chapter 4 presents the JADE jet finding algorithm used in this analysis. The pseudo-particle concept in ep scattering is also reviewed. Next-to-leading order QCD calculations are discussed in Chapter 5. Chapter 6 briefly descnbes the experimen- ta1 setup used in HERA and the ZEUS detector relevant to this analysis. The ZEUS trigger and data acquisition systems are also reviewed. In the following chapter, several Monte Car10 (MC) models that simulate the partonic processes are discussed for detector acceptance and resolution correction. After online data reconstruction, a large fraction of the data collected at ZEUS com- prises background events to the sarnple. Chapter 8 explains the reconstruction of data at HERA and the data selection criteria used to remove backgrounds. The event selection, from the trigger to the final data set, produced a pure sample with little background. Measured distributions of the kinematic variables are shown and compared with Monte Carlo. The jet production rates and the procedure used for acceptance correction for jet rates are given in Chapter 9. The various reconstruction properties of jets are also discussed. The determination of as and the confirmation of its dependence on e2 are reported in Chapter 10. The systematic checks are described in Chapter 11. The ZEUS result is compared with other experiments in Chapter 12. The conclusion is given in Chapter 13.

2. The Theory of Perturbative Quantum Chromodynamics

Quantum Chromodynamics (QCD) [14] is considered to be the fundamental theory that describes the strong interactions of quarks and gluons, where gluons are the exchanged quanta of the strong interaction. Quarks are members of an SU(3), triplet and may carry any of the three coior charges, known as red, green, and blue. The gluons are members of an SU(3), octet, and are bi-colored. Due to the non-abelian nature of QCD, i.e. the generators do not cornmute, gluons exhibit the feature of gluon self coupling, in contrast to the photons of Quantum Electrodynamics (QED) which cannot directly couple to themselves. As a direct consequence, the effective coupling strength as in QCD has a strong dependence on the momentum scale of the interaction. This implies the strong coupling constant as of QCD decreases as the energy scale increases so that perturbative calculations are applicable in high energy processes. In this section we introduce the basics of perturbative QCD.

2.1 The QCD Lagrangian

We begin with the Lagrangian density for QCD for one quark field q :

where a (a = 1, ... 8) are the generators of SU(3), G; are the eight gauge-invariant gluon fields, and g, is the coupling which charactenzes the strength of color interaction. The generators h satisfy

where the factors fobc are the SU(3) structure constants. The kinetic energy term is given by

It is worthwhile to decompose the Lagrangian into its different pieces:

The fint line contains the kinematic ternis for different fields, which give rise to the corresponding propagator. The seconc! Line gives the color interaction between quarks and gluons. Due to the non-abelian character of the color gauge symmetry group, the term generates the cubic and quartic gluon self-interactions that correspond to three and four gluon vertices. The strength of these interactions is given by the coupling g,.

The interactions associated with each of the terms in the Lagrangian of (2.4) are shown schematicaily in Fig. 2.1.

Figure 2.1: A symbolic representation of the interactions contained in the QCD Lagrangian. The diagrams labeled qij and GG are the quark and gluon propagators. The cubic and quartic gluon self-interaction vertices are also shown.

The Lagrangian (2.1) look very simple due to the properties of color syrnrnetry. However, it is rich in physics implications. Al1 interactions are given in terms of a single universal coupling

2 g,. It has become customary to refer to the quantity as = g,/4ic as the strong coupling constant. this is by analogy with the QED electromagnetic coupling constant defined by a = e"4n. The probability of gluon ernission or absorption is proportional to the strong coupling constant as.

Due to the fact that the QCD Lagrangian involves a single coupling g,. al1 strong interaction phenornena should be describable in terms of just one parameter. At O (g,) , or tree-level, it is straightforward to caiculate ail the diagrams involving quarks and gluons. The calculation of perturbative corrections to the tree-level results contains divergent loop integrals. Although 4- rnomentum is conserved at each vertex, there is no restriction on the momentum that appears on the loop and it should be summed over al1 possible values. In order to avoid divergences, it is necessary to find a way to obtain finite results with physicai meaning. In the dimensionai regularization method 1151, introduced by 't Hooft and Veltman, the divergent expressions are made finite temporarily in 4-2e dimensions. Dimensional regularization makes the separation of the divergent and finite parts. The original divergences will manifest themselves as poles in 1 / ~ in the lirnit E + O.

In the next procedure known as renormalization, these regularized divergences of perturbation theory are removed by absorbing them into the definition of physical quantities. Different renormalization treatments with different scaies must lead to the same numencal values for any physicai observable. The equation that expresses the invariance of the physics under changes of the scale pararneter is known as the Renormalization Group Equation (RGE) [16].

2.2 The Strong Coupling Constant a S

The QCD coupling constant, as, of a particular process depends on the square of the mornenturn transfer, This running of q can be conveniently expressed in terms of the P (aS) function contained within the RGE:

with

in the Modified Minimai Subtraction (z) scheme [17].

In this equation, j . ~ is a pararneter with dimensions of m m . It is introduced in order to make the coupling constant dimensionless. N/ is the number of different flavors of virtual qq loops. The coupling constant, as, as a function of p, is obtained by integrating the above equation. A constant of integration, A, is introduced in solving this difFerentiai equation. This constant is the one fundamental constant of QCD that must be determined from experiment. In the 1-loop expression for as, one gets

If one keeps the first two terms of the RGE one gets the well known 2-loop expression for as. The scale dependence of the running coupling to second order is given by

The ninning coupling constant can also be expressed in the scale dependent form,

(2 . IO)

Due to the negative sign of the second term, the 2-loop expression reduces as compared to the 1- loop equation. Figure 2.2 shows the 1-loop and 2-loop expressions for as as a function of p = Q.

(bl

-- A = 1 0 0 MeV

..... A = 200 MeV

-.- A = 300 MeV

Figure 2.2: The 1-loop and 2-loop as expressions ar d%ferent A values. ( a ) as in 1-loop and 2- loop approximations as a function of Q for A- = 300, 200, 100 MeV from top tu bottom. (6) The ratio of 1-hop and 2-loop values for as foyhe three A values as a function of Q .

The coefficients Po and contain Nf, which is the number of active quark flavors. Roughly speaking, if is smailer than the mass of a quark of Ravor f , this quark cannot be produced and it does not count in NT As A depends on Np it must change discontinuously as p increases across a flavor threshold, since as is a continuous function. Experiments at LEP, PEP, PETRA, TRISTAN and SLC are above the b-quark threshold and quote d5) ; those in DIS "su- ally quote d4) because the data are in the energy region where the b-quark is not readily excited. The lattice gauge theory values frorn the system and from T decay are at sufficiently low energy for d3). The boundxy requirement fixes the relation between the A ( N p Fs) values [18], i.e.

This shows the importance of quoting A together with N/. Although the parameter A does not depend on the scale, it is a renormalization scheme dependent quantity.

Even within the 2-loop expression the variation of as with A - is large. Fig. 2.3a shows 5, MS

the ratio of as for A - = 200 and 300 MeV divided by as for A - = 100 MeV as a function 5 , MS 5, MS

of Q. A change in A - frorn 100 to 300 MeV at Q = 10 GeV gives 27% change of as. If we 5, M

take Q = Mf, change wih be 16%, which shows the higher sensitivity to A - at low Q values. 5, MS

Q ( G ~ v ) A.(M eV)

Figure 2.3: (a) irhe variation of as in 2-loop with A(') as a fünction of (6) The relation between h - and A . Note thnt the differences between A(') and Agi are numericolly very signifi$anS. 5 , MS

2.3 Renormalization Scheme Dependence

Denoting by R a generic physical quantity calculable in QCD, this quantity c m be corn-

puted as a perturbative expansion in the form

The coefficients Ro, R I , RZ corne from calculating the appropnate Feynman diagrams. The second and third term represent respectively leading order (LO) and next-to-leading order (NO) contribution to R and so on. The energy scale, where the observable R is considered, is denoted by Q. When calcuiations are performed, various divergences arise and these should be regulated in a consistent way. This requires a particular renormdization scheme (RS). The most commonly used one is the Modified Minimal Subtraction (MS) scheme.

2.3.1 Renormalization Scde Dependence

At second order, the dependence on other RS dependence is completely given by one parameter which can be taken to be the value of the renormalization scale, pK A shift in the scale is equivalent to a change in the coefficient of the NLO term in the perturbation expansion for the process in question. From the LO formula Eq. (2.8). one could derive:

So in the 1-loop approximation as does not change whde changing p2 + $. It c m be s h o w that a change of scale yields a change in as which is of the NLO. Therefore, in the LO perturbation theory one can not specio the scale at which as is evaluated, and it is necessary to go beyond the LO. As a result of this, a theoretical calculation should be done at least up to NLO to be useful in as measurement.

Any physical quantity should be independent of the choice of the RS and in tum the choice of the scale of p. In principle, if a process is calculated to al1 orders in perturbation theory, the result will be independent of the choice of p. The change simply reshuffles the relative size of

each term in the perturbation series. When the calculation is tmncated at a finite order, there will be a renomalization scale dependence which may spoil the convergence of the perturbative expansion if p' is significantly different from p, i.e. an ambiguity in the choice of p is unavoid- able and this makes the prediction unreliabie. It has k e n shown that the dominant uncertainty in 5 measurements arises from renomalization scaie ambiguity [19]. This will be investigated for Our measurement in Chapter I I .

2.3.2 Optimization of Renormalization Scale

Beyond the LO in the as power series of expansion, the rates of events with u+l) jets

(Ri+ i = 0- J + 1 /otO,) are given by

where Q is the typical energy scale of the process and y,,, denotes the jet finding algorithm resolution parameter which will be discussed in Chapter 4. As higher order loop diagrams contain In ( Q ~ / ~ ~ ) te-, NLO coefficients in the equation have a dependence on the renormalizatnon scale. Obviousiy, the observable R would be a renomalization group invariant quantity if we could calculate diagrams to ail orders in g. Equation (2.14) implies that the renormalization scale dependence is a higher order effect, O (a:) , and correspondingly higher order terrns have an explicit In (Q'/$) dependence. When dealing with finite order calculations, then. the question arises as to which is the best scale to use. It has been proposed that the optimal scale c m be chosen by rninirnizing the effect of the neglected higher order terms and is determined by

where R" represents the (2+1) jet rate calculated to order n. This method is cailed as Pnnciple of Minimal Sensitivity (PMS) which looks for a stationary point where the prediction is least sensitive to scale.

2.4 Asymptotic Freedom and Quark Confinement

Consider the lowest order term in the RGE

2 Since Po = i l - - N the coupling as decreases at short distances as p -t w for h j S 16. i.e.

3 /'

This is known as "Asyrnptotic Freedorn", Le. when p » A, as (y2) -t O. At lower energy scales the mnning coupling constant increases. for p + A, g ($) + - and perturbation theory breaks down. The scale A indicates where the suong coupling diverges, so the scale A marks the bound- ary between perturbative and non-perturbative QCD.

Let us compare QCD and QED. The analogous equation to Eq. (2.10) in QED is

for Q' D $, where p is an arbitrary normalization point at which the coupling aQED has been measured. The coefficient we would identiv as pfED is then

The QED running coupling increases with the energy scale, e2 + m. This can be understood by considering a picture of the "charge screening" as shown in Fig. 2.4. In QED. an electron is sur- rounded by a cloud of virtual electron-positron pairs. The pairs are polarized in such a manner that the positrons are closer to the pole electron, and then a probe photon of relatively low energy observes a lower, shielded charge, as it cannot resolve the polarized cloud. As the energy of the photon increases, the distance scale of its interaction shrinks, allowing it to resolve the shielding area, and hence observing a higher charge.

Figure 2.4: Screening of electromagnetic charge in a dipalar medium.

The factor ( 4 3 ~ ) in Eq. (2.18), which arises from e+e- loops, corresponds to the (-

N / / 6 r ) factor in the QCD formula Eq. (2.10). The factor of 2 difference is due to the convention used to define the color matrices. In QCD. as shown in Fig. ZSa, the quark-antiquark vacuum polarization effect shields the color charge just as in the case of QED. If this were the only contribution. the interaction would grow stronger with increasing Q* as in QED.

However in QCD, there is another contribution which makes the behavior of the effective coupling constant different. The factor 33/2 in Eq. (2.10) cornes from a gluon-gluon pair produced in a gluon self-interaction. This diagrarn has no counterpart in the QED, as shown in Fig. 2.5b. It also should be noted that it has the opposite sign to the contribution from quark loops which causes QCD coupling to decrease with increasing e2. Due to the continuous gluon ernission and absorption, the QCD charge of the quarks is diffusely spread out. As one increases the Q', thereby looking at smaller and smaller spatial distances, it becomes less likely to see the charge. This effect is known as "anti-screening" of the color. This effect is bigger than the screening correction.

Figure 2.5: The lowest order corrections to the quark-gluon coupling. (a) quark-antiqrrark vacuum polarization in QCD. (b) anti-screening efftect.

A bnef schematic comparison between QED and QCD is given in TabIe 2.1. This table is rather simplistic, but it will be useful to bear in mind that the formulation of QCD is quite sirnila. to that of QED.

At large distances, or smaU momenturn scales, as becomes infinite, such that perturbative expansion in as is no longer valid. This behavior implies that free, individual, bare quarks are not found in nature. They must be confined within color-neutral hadronic matter, so-called infrared slavery. In this phase space region, non-perturbative methods could be used to describe the strong interaction between quarks and gluons. But the same level of predictive power as perturbation theory has not been achieved in those regions. Understanding non-perturbative effects in QCD is

one of the key tasks in the subject Some phenomenological models have been developed to describe the confinement at s m d scales of p by modelling the process of "hadronization", i-e. the transformation of quarks and gluons into hadrons. Hadronization. thus, is a non-perturbative process which typically happens at srnall energy scales. It was expected that at large enough energies, hadrons would appear as weU collimated jets around the direction of the quark or gluon, having a total Cmomentum approximately equai to that of the original parton. In practice, experimentalists rely on different Monte Car10 (MC) models of the hadronization process in order to relate the observed distributions of hadrons to the underlying parton processes. The Monte Car10 simulation techniques involved will be reviewed in Chapter 7.

1 Gauge Group

Force Couples I to

Exchange

1 Static Potential

Coupling Con- stant

1 Free Particles

Theoretical Calculation

O t hers

- - - -- - - -

Quantum Electrodynamics (QED)

Leptons ( e , p, z)

Electric Charge

Photons (y) (carry no charge)

Leptons ( e , p, z)

Perturbation Theory

Screening Effect Abelian Precision: 1 0 - ~ - IO-'

Quantum Chrornodynamics (QCD)

3 Color Charges (Red, Green, Blue)

Gluons ( g ) (carry 2 color charges) (gluon self-coupling is possible)

. - - - -

aS(e2 = M~ )-0.12 Asyrnptotic Geedorn (Running Phe- nomena)

Hadrons (colorless)

Perturbation Theory up to O (a2) , some to O (a3)

Anti-screening Effect Non-abelian, Quark Confinement

Table 2.1: A brief cornparison of the basic properîies and features of QED and QCD. The fundamental idea of QCD is that the color charge of the quarks act as the source of the strong force between quarks, just as the electric charge acts as the source of the electromagnetic force between electrically charged particles.

2.5 The Status of a Measurement S

Detailed tests of QCD have been made possible by the advent of high energy experiments as well as accurate theoretical caiculations. If QCD is the correct theory of the strong interactions, ail measured observables should lead to the same coupling strength, referred to a comrnon scale.

During the past decade various experimental and theoretical analyses have been performed to determine the strong coupling constant as. Results have been obtained from the ratio of hadronic and electronic branching fractions of T leptons [20], from deep inelastic neutrino scattering [21], from a new analysis of J / Y and Y decays [22], from b-quark production in p p collisions [23], from the hadronic decay width of the 2 boson, rhad [24], and frorn hadronic event shapes and jet production rates in e+e- annihilation at LEP [25] and SLC [26]. These measurements are complementary to older results: from deep inelastic muon and electron scattenng [27]; from W + jet production in p p collisions [28]; and from the total hadronic cross section chad [29] and hadronic event shapes in lower energy e+e- annihilation [30]. Al1 these analyses are based on perturbative QCD calculations which are cornplete to at least NLO in as, defined in the FS scheme. Some observables, like rhad, ohad, and R,, are calculated to complete NNLO, and new QCD calculations which are based on NLO plus resurnmation of the leading and next-to- leading logarithms to d l orders, are available for some hadronic event shape variables in e+e- annihilation. A summary of al1 these results on as is given in Table 2.2.

The breaking of Bjorken scaling in deep inelastic lepton-hadron scattering is one of the most powerful quantitative tests of perturbative QCD. This method does not depend on the measurement of jet cross section. In the leading-logarithm approximation, the measured structure functions Fi ( x , e2) are related to the quark distributions according to QPM. QCD predicts the evolution of structure function. The slope of the structure function evolution with scaie is related to the measurement of as, Le.

The as measurements from DIS[v], DIS[p], NMC expenments are based on this method. The advantage for using this method at HERA compared to NMC result is to provide much larger momentum transfer in the rneasurernent. Furthemore, data have appeared from KERA at rnuch smaller value of x than the previous data [3 11.

Process

DIS [ V ;F2 F3]

DIS [p ; F2 ]

J% , Y decay

e+e- Ehad I e+e- [shapes]

- pp - W jets

O r (Z -) had)

O Z ev. shapes

ALEPH

DELPHI

SLD

O Z ev. shapes

ALEPH

DELPHI

L3 OPAL

Average

0.119+ 0.006

Average

0.123 + 0.005

Resum.

Resum.

Resum.

Resum.

Table 2.2: A summary of measurements of as [32]. The as values are obtained using the solution 3 to the RGE to O (as) . For certain variables. such as thrust, heavy jet man, jet rates in Durham

scheme and energy correlations (for jet rates, only the leading logarithm ternis are resurnmed). it h m been proved possible to sum both the leading and next-to-leading logarithms.

Figure 2.6 surnmarizes the value of the strong coupling as a function of the energy scaie for various measurements. The value of as (m,) , extracted from the hadronic width of the r lepton. provides a very important low energy measurement; although it has a rather large relative error, it implies a very precise prediction ai the M scale, which is in good agreement with the 2 direct measurements of as ( Mp) .

= DIS<E,-SR>

A R 'S iG 'S - SR)

* R,

+ DISIir)

C

A qetates

O j+*v

e*e-(cm)

V e0e'(even t s hapes PETRA)

f e'e-(event shapes TRISTAN)

* pp + bDX

A P R P P + ? X

Figure 2.6: A summary of measurernents of as ( Q ) , cornpared with the QCD predictions of the n'nning as, for 3 difSerent values of A? = 300,200, 100 MeV (Fom top to bottom).

MS

2.6 How to Measure a at FERA S

A wide range of methods is available for measuring the strong coupling constant as as described in section 2.5. In this section, we will briefly review the two methods used to measure the strong coupiing constant as at HERA.

2.6.1 The a Measurement fiom Jet Rate R2 +

S

In deep inelastic scattering a final parton state consisting of the srnick quark, the proton rernnant, and a hard gluon radiated from the struck quark wiii lead to the three jet configuration denoted by (2+1). The probability of radiating the gluon depends on a,. So the measured rate of such events allows one to perfonn an experimental measurement of a,. This jet production rate is the conventional way of determining as due to the direct relationship between jet rate and as. Perturbative QCD predicts the fraction of (2+1) jet events as a function of Q. where Q is the negative 4-momentum transfer of the virtual boson in the ep scattering process. Frorn the rneasured

R, - + jet rate, one cm determine the strong coupling constant because the jet counting method relies on the fact that the ratio of (2+1) jet cross section to the ( 1+1 ) jet cross section is propor- tionai to as in LO. By fitting the R2 + data to the O (a:) QCD calculation in which can be varied, we may measure as. The extracted a, is only meaningful when the (2+1) jet rate calculation is considered up to NLO in QCD, where the renormalization scheme is defined unambigu- ously. This method has been adopted in this analysis.

2.6.2 The Differential Jet Rate D , ,

While as is usually determïned from the jet production rates, another useful method to measure as is using the differential jet rate D I + , , defined by

at HERA. This function is essentially the nurnber of events which flip from a (2+1) jet configuration to a (I+l) jet configuration in a given interval Ay,,, of the jet finding algorithm resolution parameter y,,,. Therefore, each event contributes only once to this distribution. This method has been used at LEP experiments to measure as. It will be discussed in Chapter 11 as a means to check the systematics in this analysis.

3. Deep Inelastic Scattering (DIS) at HERA

3.1 Kinematics and Variables at HERA

The very high energies offered by HERA aliow one to probe deep inside the proton in order to study its constituents, the quarks and gluons. At high e2, the scattering takes place frorn the individual quarks , within proton. This deep inelastic scattering process is illustrated in Fig. 3.1 where k and k represent the four-momenta of the incoming and the scattered electron, respectively, and P that of the initial proton.

, Figure 3.1: Inelastic scattering for one-boson exchange in ep collision at HERA. k and k are four vector rnornenta of electron before and afrer emitting virîual photon.

The square of the total center-of-mass energy is

The Bjorken variable x is *

and y variable which describes the energy transfer to the hadronic final state is

Q' is the negative of the squared four-momentum carried by the virtual photon.

2 e2 = -42 = -(k-k) = ,yxy .

The center of mass energy W of the virtuai photon-proton (y* P) system is

m

where M p , the mass of proton, can usually be neglected.

3.2 Factorization Theorem in DIS

3.2.1 Factorization Scale Dependence

The basic conceptual justification for introducing parton density functions (PDF) is the "factorization theorem" [33]. The cross section for DIS can typically be written as the sum of a convolution integral of a hard scattering cross section (ô) based on basic Feynman diagrams, with parton distribution functions which represent the probability finding the incorning partons in a given particle. This schematically can be expressed as

and this is shown in Fig. 3.2. K.,, ( 6 ) gives the probability of finding a quark type i in the proton which carries a momentum fraction 6 of the proton. Factorkation is valid to al1 orders of as.

Current Jet

P \ J Spectator Jet

Figure 3.2: The short and long range processes of DIS are shown. The short-distance interaction occurs between the photon and quark and it factorizes with the long-distance parton distribution functions. The hadronization process takes place afer the hard interaction.

The factorization scale is the typical scale which divides the hard scattering process from the parton density functions. The PDF are functions of 6 and aiso depend on renomalization scde p, and factorization scale pF. The PDF are universal in the sense that they describe the quark composition of the proton in any hard interaction, so they are independent of the process under consideration. This gives a predictive power to QCD. #en the PDF are known, al1 hadron cross sections are related via the corresponding hard scattering cross section ô. For exarnple, by rneasuring the PDF in ep scattering, the cross section for the Drell-Yan process in pp scattenng can be predicted, or vice versa.

A consequence of factorization is that measunng parton distributions at one momentum scale allows their prediction for any other scale as long as both scales are large enough (a, is small) that perturbative QCD holds. When the proton is probed wiîh an increased resolving power

3 7 Q- » Qo, where Q: is a certain reference scale, say, the photon starts to see evidence for the point-like valence quarks within the proton, one sees more gluons split into a qq pair and more quarks lose part of their momentum by radiating a gluon. The evolution of the PDF is described by the Dokshitzer-Gnbov-Lipatov-AltareIli-Paris (DGLAP) evolution equation [34],

This equation expresses that a quark with momentum fraction 6 could have corne from a parent quark with a larger momentum fraction 5' which has radiated a gluon. The lower bound of the integral is 6 . The evolution kemels Pi, are given by perturbative expansions. In order to evduate the PDF at a scale we need to know the PDF for 5' > 5 at a given scale pi, o. Once the set of PDF at a given starting point is known, everything is predicted by DGLAP equation. The DGLAP equation only describes the scale evolution of PDF, not the 6-dependence. The 6-dependence c m be obtained from measurements by way of pararneterization and fitting to data.

We must choose the factorization scheme where a calculation is performed. Currently WS and DIS-schemes [35] are available. In MS-scheme, constant terms in the dimensional regularization of the hard scattering are absorbed in the long range PDF by a suitable choice of p,. The DIS-scheme is an especiaily convenient scheme for DIS F2 measurements. Since F2 is an observable, the quark density must be finite and well-defined. In this scheme, ail corrections to the stmc- ture function F2 are absorbed into the distributions of the quarks and antiquarks. For a cornparison of data and theory, it is important to use scheme and scale consistently.

LO calculations use the LO tree level cross section 6, the LO DGLAP equation and the LO expression for as. Since ô has no scale dependence, o acquires a net scde dependence through the PDF. To evaiuate a NLO calculation, the NLO cross section, the NLO DGLAP equation and the 2-loop formula for as are used. Schematicdy, the PDF in NLO have the general form:

where the zeroth-order term is the QPM contribution with 6 = x . The momentum fraction 6 carried by the incoming quark is degraded to x by the emission of the gluon so that x < 6 < 1 condition is satisfied.

When convoluting a PDF with cross section in a given scheme in LOMLO, one is forced to use the LONLO PDF in the same scheme. Often it will be convenient to choose the two scales

3 p, and p, to be equal. In this case, a simplification is made by taking pi = pi = e2 which leads to

The effect of this simplification to the andysis will be reviewed in Chapter 1 I .

3.2.2 Parton Distribution Parameterization

Perturbative QCD describes the Q' evolution of the parton distributions through the DGLAP equations. It does not provide a prediction of parton distributions. A method comrnonly used to determine the parton distribution is the global fit method. This method starts with a pararneterization of the parton distributions at a reference scale Q: and evolves the distributions into the region in which the measurement of structure functions is performed. The starting parameters are adjusted until a good fit to the measured structure functions is achieved. In this global fit method, much of the experimental data from al1 deep inelastic expenments and related hard scattering processes are used. To be able to predict the cross section, it is of fundamental importance to have a reliable and precise set of parton density distributions. Many different pararneterizations have been suggested and we review them only as they are related to our andysis.

The factorization theorem (3.6) shows the importance of the parton distribution parameterizations as an input in QCD calculations. The x dependence has to be parameterized and fitted to the various data sets currentiy available. Usually simple functions are used to pararneterize the x dependence of the starting distributions, such as

2 pi x& ( x , Q,) = A,x-~ ' ( 1 - x ) Pi ( x ) (3.10)

at a low scale Q:. Pi ( x ) is typicaliy taken as a logarithmic or tem. This form is physically motivated by the fact that die quark density is dominated by the valence quark density at x = 1, while at small x, sea quarks and gluons play a dominant role. Here the Balitsky-Fadin-Kuraev- Lipatov (BFKL) approximation [36], i.e. the dynamics of a shoa distance strong interaction process by summing (aslog I /X) contributions at small x , may be used as an estimation for the increase of the gluon density. The DGLAP evolution equation is then used to determine fi (x, Q') at al1 the x. Q* values at which data exist, and the parameters (ai, Pi, . . . ) determined by an opti- mum fit to the data.

The Martin-Roberts-Stirling (MRS) sets [37] are obtained from a global NLO QCD fit to data from BCDMS [IO], NMC [Il], EMC [12], CCFR [13], CDHSW[38], WA70 1391, and E605 [10], starting from Q; = 4 G ~ V * with

3 The gluon and sea quark distributions at low x are required to satisfy xf - xa, at e2 = (36. The MRS group produced two parameterizations corresponding to extreme values of a. The MRSD$ uses a constant gluon and sea distribution with cc = O for x << 1. The MRSDI- parameterization

- ln uses a singular distribution for the gluon and the sea, xg ( x ) , x? ( x ) - x as x -t O, leading to a stronger rise in Fz at low x. do and d. are allowed to be different with

The CTEQ group (Coordinated TheoreticaVExperimental Project on QCD Phenomenol- ogy and Tests of the Standard Model) [41] also performs a NLO global QCD fit with starting scale, Q, = 1.6 GeV with

They allow a flavor asymmetric sea Ü ( x ) #d(x) and have a singular gluon behavior with - 1/2

xg (r) - x at low x . There are qualitative differences between MRS and CTEQ parton distributions. These differences primady originate in the different treatment of the sea quark distribu-

tions in the two analyses. Motivated by the NMC resuits [ I l ] for the Gottfried sum rule [42] which predicts the difference between the proton and neutron structure functions, MRS aliows 2 and i to be different according to (3.13). in set do and d.. For the strange sea quark distribution

1. + MRS takes s = - (Ü + d) at Q: = 4 G~v*, motivated by vN + p p.- X data [43]. In contrat, 4

in the CTEQ analysis the Ü, d and s distributions are freely and independently pararneterized.

Glück-Reya-Vogt (GRV) sets [44] start with valence-like parton distribution at a low Q: 2 scale, Qo = 0.2 - 0.3 G ~ v ~ , and demand xf ( x , Q;) + O as x + O. GRV parameterizes

a xf ( x , Q:) = Ax ( 1 - x ) P

with Ü = d sea quarks. The remaining sea quarks (strange, cham, bottom) are assumed to be zero at Q;. Al1 PDF are then evolved to higher value of e2 by DGLAP evolution.

4. Jet Definition and Jet Finding Algorithms

One cannot observe free partons, due to the confining nature of the color force. In the laboratory, partons materialize as groups of correlated hadrons. One refers to such groups of final state hadrons as "jets". There is no unique way of defining a jet, and hence no unique way of con- necting any final state hadron with a particular parton. Jets can o d y be defined in a phenomenological fashion. The same jet definition applied to data and simulated events allows one to make the connection between experirnent and QCD.

Figrire 4.1: An overview of an ep scattering. ntree stages of the event evolution (Partons, Had- rons, Detector objects) are indicated.

. . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . : s :

P

When quarks inside the proton collide with an electron in the process ep -t ex , the bare quarks are not observed as individuai particles. Instead, a process similar to that shown in Fig. 4.1 occtirs. First stage is the perturbative phase, where the e2 of the interactions is sufficiently large that perturbation theory is vaiid (parton level). A quark radiates gluons. This showering process continues until the available energy is degraded towards p2 = h2. At this time, the coupling becomes very large, and the partons are no longer able to exist as freely moving particles. It is

Perturbative Phase : Partons < d

Detector Objects

Hadronization Detcctor Effects

here that the hadronization process occun. After fragmentationhadronization processes, the initial partons are combined and color-singlet hadrons are formed (hadron level). The hadronization of quarks and gluons cannot be calculated in the perturbative regime of QCD. A perturbative treatment of parton production is combined with a non-perturbative mode1 for the fragmentation of a partonic state into hadrons. The difference between parton jets and hadron jets is caused by hadronization and is cailed the ''hadronization effect". We are left with many different hadronic states at the hadron level. The stable hadrons reach ZEUS detector and particles will be detected by experimental observation (detector level). It is this final distribution of hadrons which is observed. Various efTects coming from detector response will cause the difference in jets. This is called the "detector effect". Our airn is to learn how to recombine the hadrons, or the detector objects in order to get the best estimate of the four vectors of original partons.

4.1 Cluster Algorithm for Jet Finding

Jet reconstruction algorithms have been proven to be useful tools when comparing predictions of perturbative QCD with experimental data. The idea is to assign the large arnount of hadronic energy in small angular regions found in the experiment to a jet. The hadronic energy is calculable within the framework of perturbative QCD. But we also should be aware of the non- perturbative hadronization process of which we have Little theoretical understanding. In this respect, a good jet reconstruction algorithm requires that the hadronization effects should be as srnall as possible in jet finding. Equally, the chosen jet algorithm should be suitable for use in perturbative QCD calculations.

The identification of jets in DIS is more complicated than in e+e' experiments where the center of mass system and the laboratory system coincide. In DIS hadrons are strongly boosted dong the proton direction. Most of proton remnants which are the relics of fragmentation will not be observed in the ZEUS detector. In the lab frarne the boost resulting from the unequal energies in ep collision at HERA leads to a loss of particles going down the forward bearn pipe. This loss distorts the continuity in the particle flow between the current jet and any forward going particles. Due to fake jets produced from the particles placed at the end of the broken flow, the result from jet clustering algorithm will change. This problem can be solved by introducing the idea of a pseudo-particle. Furthermore, due to the gluon radiation from the incident proton one of the singularity regions in (2+1) jet production is dong the proton direction.

The cluster type of algorithm was developed in efe' expenments and is based on cornbin- ing particles which are close in phase space into the same jet. The procedure is first to define a distance measure di, (usually denoted by y, = d:/ $) between two clusters. The distance between al1 combinations of two clusters is calculated and the two clusters which have the minimum distance are combined into one cluster. This is then repeated until no two clusters are closer than some distance d,,,. The remaining uncoalesced clusters are considered to be the jets. There exist

a number of such clustering algorithms. They differ from each other in the definition of di, and in the recombination scheme. In the next section the JADE algorithm and various recornbination schemes will be reviewed.

4.1.1 The JADE Jet Finding Algorithm in DIS

In the measurement described in this thesis, the JADE jet finding algorithm [45] was used, since it is currently the only algorithm which allows cornparison of data to QCD NLO calculations. The jet reconstruction procedure of the JADE algorithm stuü by calculating the invariant mass for al1 pairs of particles according to the expression:

where Ei and E, are the energies of particles i and j assuming that these objects are rnassless, and O, is the angle between them. is the squared invariant mass of the hadronic final state. If a minimum value of y, is less than the cut-off parameter y,,,, the two clusters i and j are merged into a new object by adding their four-rnornenta and the process is repeated until ail y, > y,,,. The surviving objects are jets.

In the data, a fictitious cluster, called a pseudo-particle, carrying the rnissing longitudinal momentum in the forward Z - direction is added to each event. It prevents the detected fraction of particles originating from the proton remnant frorn forming spunous jets. In the JADE algorithm. we introduce it as

seudo where p ,, pz, p3, . . ., p, are the four momenta of the particles and d> is the four momentum of the pseudo-particle. The pseudo-particle is treated like any other particle in the jet finding algorithm. In this respect, precisely speaking, our JADE jet finding algorithm is sometimes cailed the modified JADE (mJADE) algorithm.

4.1.2 Jet Finding Schemes

A number of the jet algorithms have been proposed with different mi, definitions and recornbination methods. These schemes, denoted E, Eo, P, Po, are given Table 4.1. The Eo, P, Po

- schemes are variation of the E - scheme. In e+e- experiments where there is a uniquely defined energy scaie, the center of mass energy EL = e2 is the natural choice for the visible (measured) energy scaie.

In the E - scheme, y , is defined as the square of the invariant mass of the pair of particles i and j scaied by the visible energy in the event,

with the recombination

Pt = P i + P j

where pi and pi are four-rnomenta of the particles. Energy and momentum are explicitly conserved in this scheme, and it is entirely Lorentz invariant.

In the Eo - scheme, yij is defined by Eq. (4.4), but the recombination method is defined by

where Ei and Ej are energies, and pi and pi are the three-rnomenta of the particles. The three- momentum F k is rescaled such that particle k has zero invariant mass. Total momentum in this scheme is not conserved. The original JADE scheme is based on the exact four-vector recombination, but neglects explicit mass terms in the definition of mu. It has been shown [46] that in second order perturbation theory the Eo - scheme and the JADE scheme are equivalent.

In the P - scheme, y, is defined by Eq. (4.4) and

by

p, = Pi+$,

the recombination procedure is defined

(4-7)

This scheme conserves the total momentum of an event, but does not conserve the total energy.

The Po - scheme is similar to the P - scheme, but the total energy EVis in Eq. (4.4) is recdculated at each recombination according to

JADE

Recombination Remarks

Neglect masses of objects.

Energy & rnomentum are conserved. Lorentz invariant.

Total momentum is not conserved. Original JADE scheme.

Energy is not conserved.

Sarne as P, but scale is recaiculated after each iteration.

Breit frarne

Table 4.1: Resolution criteria and recombination schemes for the jet clustering algorithm. pi corresponds to 4-vectors of ( E i , pi). Breit frame is the one where no energy transferred to the parton due tu the fact that the photon and proton are collinear and the 3-momentum of the parton is ercactly reversed by the collision. Sometimes it is called the "brick-wall" frame.

The most recently proposed change to the JADE scheme is the "Durham or kr algorithm 2 2 2 [47]. A distance measure rn, = 2min ( Ei , Ej ) ( 1 - cos @ . .) , corresponding to the relative

11 transverse momentum of two objects, is used. Due to the fact that the original JADE algorithm sometimes artificially introduces jets by identiQing very soft gluons with very different direc- tions, the kT - algorithm takes it into account in the jet clustenng procedure. The k , algorithm is an attractive one which largely avoids 'unintuitive' recombinations of particles with large angles between them.

4.2 Collinear Jet Singularity in e p Scattering

Figure 4.2: Emission of a collinear gluon in the initial state. A quark from the proton emits a gluon in the same direction of the proton remnant before it is hit by virtual photon. The gluon and the remnant are collinear.

In ete- annihilation, rnost of outgoing hadrons are within the acceptance of the detector and there is no problem for the outgoing particles close to the direction of the incorning particles. But ep scattering is different. If a parton is scattered out of the proton, it may have emitted another parton with a large momenturn before interacting with the virtual photon. The probability for this process is very large for small emission angle. A pictorial view of collinear gluon ernission is shown in Fig. 4.2. We suppose that the gluon with momentum Pz is emitted from the quark line before the quark interacts with the virtual photon. According to QPM, the incorning quark is collinear with the incoming proton. Therefore the gluon is emitted nearly collinear with respect to the incoming quark. Afier the fragmentation process the jet from the gluon will be collinear with the proton rernnants. For nearly collinear gluons aU these particles will appear in one single jet. Furthemore when the gluon becomes soft, this leads to a problem in perturbative QCD calculations. This is cailed infrared singularity. We must regularize these infrared and collinear singularities in the cross section calculations of e p scattenng. This has been performed within the jet algorithm. Soft and collinear divergences are avoided by requiring a minimum invariant mass

2 of al1 parton pairs: yij = mi,/ > y,,. Here y,, is chosen as srnall as possible to take sorne softer radiation into account. It should be smaller than any experimental jet resolution so that no observable jet ernission is lost. In the jet finding aigorithm, if a low energy particle or collinear particle is added to the final state, the final result should not change abruptly. This is called infrared and collinear safe. The JADE algorithm is known to satisfj these conditions [48]. Finaily the definition m u t be applicable to both expenment and theoretical calculations for cornparison between the two to be meaningful.

5. Theoretical Overview of NLO Calculations in DIS

In order to measure the strong coupling constant as, it is necessary to compare experimental measurements with QCD calculations based on models which fit the experimenral data weil. In this analysis the LEPTO 6.1 mode1 [49] which wïil be reviewed in Chapter 7 describes the data very well and was used for the detector acceptance correction and resolution, hadronization effect. This Monte Car10 does not allow one to extract the fundamental parameter in QCD, A. which is crucial in Our measurement. Furthemore it does not include the exact NLO matrix elernents calculation. Perturbative QCD calculations up to O (a:) are available for this purpose in DIS. In this chapter two QCD calculation programs, DISJET and PROJET, are reviewed. In addition, various kinematir variables related to jets are discussed.

5.1 Two Jet Kinematics

The differential rnatic variables [SOI

cross section for two jet production depends on five independent kine-

where is the azimuth - between the outgoing parton plane and the lepton scattenng plane in the y* - parton center of rnass system. Two parameters, x and e2, are sufficient to describe the QPM ( l + l ) jet process. The three additionai variables (xp, Z, 0) are introduced to describe the (2+1) parton kinematics, where the variables xp and z are Lorenz invariant parton variables. The 0 dependence is often neglected by averaging over @. The x, variable is determined by

where 6 is the fraction of the proton's longitudinal momentum P carried by the incoming parton of momentum p (p = CP) and q is the four-momentum of the exchanged virtual boson and M,, is the invariant mass of the two-jet system. The parton variable z is related to the angle of the jets in the y* - parton center of mass. The advantage of using z is that this variable is not frame dependent, so that this quantity is well defined in the theory and calculated in experiment. The value of z is given by

P ' P l 1 - - (1 -cos@**) 2 1 = - P'4 2

where 0 ,* is the polar angle of the jet in the y* - parton center of mass and O 5 z 5 1. The outgoing four-rnomentum of the parton from the hard scattering is p l . Experimentaily, z is measured from (2+1) jet events in the lab frame

C @ - P z ) jer 1 + jet2

where Z, + z2 = 1 and z,,,,, = O since the remnant jet has no transverse momentum.

The transverse momentum PT of the jets in the y* - parton system is given by [5 11

For neutral current electron-proton scattering, the (2+1) jet differential cross section at the O (as) LO level, expressed in t e m s of the above variables, is given by [50]

where I , and 1, are the gluon-initiated (eg + eqq) and (anti)quark-initiated (eq + eqg,

e t -t eqg ) contributions, respectively.

z , and a, there are singularities in Even though the 3-parton phase space is effectively descnbed by these three variables x,.

the two-jet cross section at the O (aS) tree level given by [50]

and

where zq is Iabelled specifically for the quark jet in the QCDC process. Physically, these singularities correspond to the lirnit where the gluon or the opening angle between the partons vanishes. In the BGF process the singularity is related to the collinear emission of the two quarks. A singularity occurs in the QCDC process if the Cmomentum of the gluon is parallel to that of quark, or if the quark emits a very soft gluon.

In the JADE scheme. the integration in Eq. (5.6) has to be done numerically because of the x-dependent parton density distributions. The integration lirnits for z and xp in the JADE scheme are functions of the scaled invariant mass cutoff y,,, = mi/+. Any pair of partons with a scaled invariant mass below this cutoff are not resolved. Therefore, the theoretical singularities are regulated by a single cutoff parameter y,,,. In the jet definition scheme as defined in Eq. (4.2), the corresponding (2+1) jet region of the upper ( z , , xp, ,,,) and lower (z,,, x,, ,,) limits is given with the following upper and lower integration lirnits [50]:

In LO, the observable (2+1) jet cross section is expressed as,

LU do2 + 1 - - dxdy - '31as

where the coefficient c , contains the hard scattering rnatrix elements and the parton density functions of the incoming proton. The first index, i, stands for the jet multiplicity including the rernnant jet and the second index, j, represents the order of the as.

The NLO (2+ 1) jet cross section is expressed as

do:: - - 2

- '31 as + C32as dxdy

In NLO, do2 + , includes the contribution from unresolved (3+1) jet events as well as a negative correction corning from virtual loops [52]. The (2+1) jet rates are derived from the cross section

- 2 as R2 + - ot + hro1. The resulting O (as ) corrections to RI + I . using the NLO calculation are considerable. They Vary from -20 to +20 % when y,,, is varied between 0.01 and 0.06 in the

kinematic region used for this study [52.53]. M e r integrating over the jet variables (x,, z, a) the coefficients ci, are functions of the event kinematic variables x, y, y,,, and the factonzation scaie pF The parton densities contained in ci, are calculated at the scale Q? In finite order penurbative QCD calculations as depenàs on the renormaiization scale pR. A matrix elements calculation for the full NLO contributions is available in the DISJET program [54] by T. Brodkorb and E. Mirkes, and the PROJET program [55] by D. Graudenz. Furthemore, they include the scale and factorization dependent contributions. Both programs are numerical cross section calculations in deep inelastic electron proton scattering on the bais of Born term and NLO corrections. They agree in their predictions of as for a given jet rate. They also reproduce the shape of the measured jet rate distributions as a function of y,,, very well in Our kinematic region.

5.2 O (a:) Perturbative QCD Calculations

In DIS the O (as) corrections to the O (a:) Born term are well known and extensively discussed [SOI. In addition, the O (a:) term for the production of (3+ 1) jets has been calculated. In this analysis the calculation of DISJET and PROJET is used. Matrix elernents for the complete contributions to (2tl) jet NLO O (a:) corrections were first discussed in [56]. DISJET is based on these matrix elements and also includes the full NLO contribution. NLO calculations allow for the calculation of NLO (1+1), (2+l) and LO (3+l) jet cross section in DIS. In DISJET, al1 relevant helicity contributions to the total cross section are included. Averaging over the azimuthal angle between the parton plane and the lepton plane in the y* - initial parton center of mass system the jet cross section in the photon exchange approximation is given by

where the subscripts U and L refer to the unpolarized and the longitudinal polarization cross sections, respectively. These two cross sections can technically be obtained by the following covari-

* PV ant projections on the hadron tensor H , which is calculated in perturbation theory.

where p denotes the rnomentum of the initial state parton and x, = Q'/ ( 2 p - q) . The O (a:) (2+1) jet matrix elements represent a full NLO calculation including df processes in Eq. (5.17). The following subprocesses contribute to (2+1) jet cross section up to O (a:) ;

and the correspondhg anti-quark process with q tt 4. The second projection (- p g , ) in ou+ in Eq. (5.15) gives a contribution of the order of 15 - 30 % to the (2+l) jet cross sections depending on the kinematic ranges, whereas the contribution from G, in Eq. (5.14) is fairly small. This originates frorn the y dependence coefficients ( 1 + ( 1 - y ) 2, and -y2. since y is peaked at smdl values. The (2+l) jet NLO contributions originating from the projection with -gp, on the hadron tensor in Eq. (5.15) were first presented and discussed in [57]. A complete list of tree level rnatrix elements with up to 4 partons in the final state can also be found in [58]. Based on Eq. (3.6), the general structure of given by

the hadronic jet cross section within the framework of penurbative QCD is

where one surns over a = q, 4, g. Here fa ( 5 , &) is the probability density to find a parton a with fraction 5 in the proton if the proton is probed at a scde &. ôa denotes the partonic cross section which collinear initial state singularities have been factorized out at a scde and implic- itly included in the scde dependent parton densities fa (6 , &) . Note that éf depends in NLO explicitly on the renormalization scale pi and factorization scale j.~:.

In order to calculate the (2+1) jet cross section, we need a jet definition. This could be done by introducing a resolution cntenon. As a jet resolution criterion NLO programs use the invariant mass cut criterion such that

where y,,, is the resolution parameter and si, is the invariant mass of any two final state partons, including the remnant jet with momentum p, = ( 1 - 5) P. Mc is a typical mass scale of the process which defines the jet definition schemes. Mo is a fixed mass cut which we have introduced in

2 order to clearly separate the perturbative and non-penurbative regime in the case, where Mc is small. Table 5.1 shows the main differences between the two NLO QCD caiculations.

I Program I DISJET l PROJET - - - -

Authors 1 E.Mirkes, T. Brodkorb 1 D. Graudenz

NLO

Jet Scheme

NLO

I Jet Kinematics I

E- Scheme E- Scheme

I z Cut

(x,, 5)

Angle Cut

Yes I Yes I

Table 5.1: The differences between DISJET and PROJET NLO QCD theoretical calculation programs. The difference between them is due to the missing NLO corrections ta the longitudinal cross section. PROJET makes an approximation by neglecting the contribution of the longitudinal part of the virtual boson to the (2+I) jet cross section.

No

No

Yes

Yes

6. HERA and the ZEUS Experiment

In this chapter, the HERA accelerator and ZEUS experiment are reviewed. The Hadron Elektron Ring Anlage (HERA - for hadron electron ring installation) is the first electron-proton collider in the world, built at the Deutsches Elektronen Synchrotron (DESY) in Harnburg, Ger- many. HERA was approved in April 1984. Tunneling was finished in 1987 and the first electrons were circulating in August 1988. The first etectron proton collisions were achieved on October 19. 199 1, and in May 1992, collisions began between 26.7 GeV electrons and 820 GeV protons, delivering the first lurninosity to the detectors Hl [59] and ZEUS [60]. At HERA the bearns c m be made to collide at four interaction regions. The two detectors are installed in the north (Hl) and the south (ZEUS) interaction points.

6.1 The H E M Accelerator

The HERA machine itself is shown schematically in Fig. 6.1. The ring has a circumfer- ence of 6.3 km and the HERA ring extends into the surrounding neighborhood by running about 20 m underground.

e s h HERA Injection Scheme

Hill1 Nonh \

HERA

Hall East \

@ * (ZEUS Detecior) @ *

& -

C

Figure 6.1: 7he H E M accelerutor complex and the HERA injection system. HI and ZEUS detectors are located in the north und south hall, respectiveiy. The pre-accelerators are located on the DESY site.

Electrons obtained from a hot cathode are accelerated by a 500 MeV linear accelerator (LINAC II'), and accumulated in the storage ring (PM). They are then injected into the DESY II, where they are accelerated to 7 GeV, and transferred to the PETRA ring. This is repeated until PETRA is filled with 70 bunches, each having 0.4 x 10 ' l electrons. The electrons are then accelerated to 14 GeV in PETRA and transferred to HERA electron storage ting. This procedure is repeated until HERA is filled with 210 bunches. The electmns are then accelerated to the final energy in the HERA ring.

Protons are initially accelerated as negatively charged hydrogen ions in the 50 MeV Iinear accelerator. The electrons are stripped off by passing the protons through a thin alurninum foil. The protons then move into the DESY III storage ring and are accelerated to 7.5 GeV. They are transferred to PETRA II where up to 70 bunches are accelerated to 40 GeV. The protons are then injected into the HERA proton storage ring. This procedure is repeated until 210 bunches are stored in the HERA ring. The protons are then accelerated to the final energy.

Table 6.1 shows the relevant parameters of HERA machine operation. Particles are accelerated in HERA by passing through Radio Frequency (RF) cavities. From the Lorentz force for the motion of a charged particle in a magnetic field one c m show that P = 0.3 B p , where p is the bending radius in meters and B is measured in Tesla. The tunnel diameter and guide field of superconducting magnets (4.68 T) used in the HERA proton ring set the proton energy. For an electron machine in the same tunnel as the proton machine, the maximum energy is set by the practical RF power level available to replace the synchrotron radiated power. The required luminosity deterrnines the nurnber of bunches and the frequency of beam crossings.

Bearn Parameters 1992 1993 1994 Design 1 1 / / (e- period) / (e+ 4od) Goals

& CM energy (GeV) 1 296 1 296 1 300 1 300 1 314 1 # of bunches

Table 6.1: The running parameters during lurninosity period and design paranteters are shown. During the h l y of 1994 running period, HERA switched e- into e+. In this thesis, electron is used in a generic sense to refer to e- or e+.

Luminosity ( ~ r n - ~ s - l ) 1 1.5 x lo2' 9

1.5 x lo30

84

1.5 x lo30 ( 5.2 x 1030

153

1.5 x 103 l

153 200

In addition to the 200 designed colliding electron and proton bunches spaced by 96 ns, 10 unpaired bunches of protons and electrons are also accelented to allow the expenments to study proton-tas and electron-gas background interactions.

A plot of Iuminosity versus time is shown in Fig. 6.2. The incegnted lurninosity delivered by HERA in 1994 was over 6 and ZEUS recorded 3.20 pb-I.

HERA Luminosity 1992- 1994 1 9 9 4 dcli w c r c d

ZEUS 1994 5 Ldt- 3.20 p b - 1

1993 5 Wt - 0.55 ph -'

1 9 9 4 "GooJ" for physics

Figure 6.2: Integrated lwninosity delivered and collected since data taking period of 1 992 as a function of tirne. Since July in 1994 afer e+ was injected, 73e peflonnance in both HERA and ZEUS shows irnprovernent of integrated lurninosiîy.

6.2 The ZEUS Detector

The ZEUS collaboration currently consists of approximately 450 physicists from 49 insti- tutes in 12 countries (Canada, Germany, Israel, Italy, Japan, Korea, the Netherlands, Poland, Rus- sia, Spain, the UK and the USA). A detailed description of the ZEUS apparatus can be found elsewhere [6 11. In this thesis, the main components of the ZEUS detector relevant to this analysis are briefly reviewed.

The ZEUS detector is shown in Figs. 6.3 and 6.4. One of the outstanding features of the ZEUS detector is the calorimeter made of uranium and scintillator. The high resolution uranium

scintiilator calonmeter [62], composed of the Forward Calorirneter (FCAL), Barre1 Calorimeter (BCA4L) and the Rear Calorimeter (RCAL), is the principal component used in this analysis. The

O coverages in pseudo-rapidity q = -ln ( tan ?) are 4.3 2 2 1.1 , 1.1 2 q > -0.75, and -0.75 2 q h -3.8 in each FCAL. BCAL and RCAL region, respectively. Holes of 20 x 20 cm2 in the center of FCAL and RCAL are required to accommodate the HERA beam pipe.

Overview o f Ihe ZEUS De t e c l o r { iongitudinol cul )

Figure 6.3: The longitudinal cross section of the ZEUS detector: The 2-mis points in rhe direction of rhe incoming proton beam. The main component rvhich was used in the analysis is the calorim- ete!:

Charged particles are tracked by the inner tracking detectors which operate in a magnetic field of 1.43 T provided by a thin superconducting coil. Immediately surrounding the bearn pipe is the Vertex Detector (VXD), a cylindrical dnft charnber, which consists of 120 radical cells. each with 12 sense wires [63]. The VXD is designed to measure the secondasr vertices of short-lived particles. Surrounding the VXD is the Central Tracking Detector (CTD) which consists of 72 cylindricai drift chamber layers, organized into 9 'superlayers' [64]. To measure tracks at srnall angles with respect to the bearn direction, ZEUS has the Fonvard Tracking Detector (FTD) and the Rear Tracking Detector (RTD). In this analysis these detectors are primarily used for the deter- rnination of the event vertex.

The energy leakage from the CAL is detected by the Backing Calorimeter (BAC) which is constructed frorn a sandwich of 7.3 cm ihick iron plates and aluminum proportional tube counters.

The iron plates also serve as a retum path for the magnetic flux of the solenoidai coil. The outer- most components are the muon detecting systems consisting of the Forward Muon (MON), Barre1 Muon (BMUON) and Rear Muon (RMUON) detectors. The Veto Wall is an iron wall covered with 2 planes of large scintillation counters located behind the RCAL. It provides background signal from the beam halo accompanying the proton bearn and allows the trigger to veto non-ep events. The C5 counters are two lead scintillator ring counters mounted on the beam pipe behind RCAL. They are used for monitoring the time structure and other properties of the two beams.

There are other detector components which are not shown in Fig. 6.3. Very fonvard scattered protons are detected in the Leading Proton Spectrometer (LPS) Iocated between 25 and 100 m downstream in the proton direction. A Forward Neutron Calorimeter (FNC) is installed at zero degrees in the proton direction approximately 105 m frorn the interaction point. Its purpose is to measure the angle and energy of neutrons produced with small momentum transfer. The furninos- ity is measured from the rate observed in the Luminosity Detestor (LUMI), positioned at down- Stream from ZEUS in the electron direction, of brernsstrahlung photons from the Bethe-Heitler process ep + e'py . The cross section for this process is known very accurately in QED. The final state electron and photon are in coincidence in electromagnetic caiorimeter of the LüMI detector.

Overview o l the ZfUS O e t e c t o r ( cross sec l i o n )

Figure 6.4: Cross section of the ZEUS detector perpendicular fo the beam iine.

6.2.1 Calorimeter

The coverage provided by the ZEUS calonmeter extends from 2.2' to 176.5' in polar angle @ and represents 99.7% of 4 ~ . The ZEUS calorimeter is a sarnpling calorimeter constructed from towers which consist of alternate layers of Depleted Uranium (DU) for absorber and plastic scintillator as an active materiai. Each tower is segmented longitudinally into an Electro- rnagnetic (EMC) section and Hadronic (HAC) sections in FCAL, BCAL and RCAL Both the FCAL and BCAL have two HAC sections while the RCAL has only one as shown in Fig. 6.5. These sections are further subdivided into cells. which are read out on two sides through wavelength shifter bars, light guides and photomultipliers (PMTs). The depth of the EMC cells is 25 radiation lengths corresponding to one interaction length. The total HAC sections in FCAL, BCAL and RCAL are 6.4 and 3 interaction length deep, respectively. The relative thicknesses of uranium and scintillator in the layer structure were chosen to give equal calorimeter response to electrons and hadrons. The resolution has been rneasured in test beams [65] to be o (E) / E = 0.18/b for electrons and o (E) / E = 0 . 3 5 / b for hadrons where E is in GeV. An energy independent term could be added in quadrature to these resolutions at the level of 2%. The calorirneter noise, which is dominated by the uranium radioactivity, is typically 15 MeV in the EMC cells and 25 MeV in the HAC cells. The detail descriptions of calonmeter can be found in reference [66].

FCAL

EMC j

RCAL

Figure 6.5: A rower in the FCAL consists of four 5 x 20 cm2 EMC cells follorved by one 20 x 20 cm2 HACI and one 20 x 20 cm2 HACZ cell. In the RCAL a tower consists of two IO x 20 cm2 EMC cells and one 20 x 20 cm2 HACI.

6.2.1.1 Energy Loss of Charged Particles in Matter

Calorimeters measure the energy of both charged and neuuai particles by generating a rneasurable signal that is proportional to an incident particle energy. When a particle enten a calorimeter, the incident particle interacts with the calonmeter absorber creating secondary particles.

This process is referred to as showering. Electrornagnetic and hadronic showering will be discussed in the following sections. There are two types of calorirneten. A homogeneous calorimeter is made from a materid which both absorbs energy and simultaneously generates a signal. Alter- natively, alternating layers of absorber material and active material for rneasuring signal from the deposited energy is used in sarnpling calorimeter.

6.2.1.2 Electrornagnetic (EM) Showers

Since electrons and photons only interact electromagnetically with matter, the showers initiated by these particles are governed purely by the electromagnetic interaction. Electrons lose their energy through bremsstrahlung at high energies and most of energy loss is due to ionization at lower energies. Since photons are electrically neutral, they do not expenence the Coulomb force. But photons cm interact with matter in a variety of ways that lead to ionization of atorns and to energy deposition in a medium. For high energy photons pair production of electrons and positrons is dominant process which is dnving electromagnetic showers. At lower energies Compton scattering and photoelectric effect dominate.

The average energy loss of a high energy electron due to brernsstrahlung is

where Xo is the radiation length, which is the distance that an electron travels before its energy drop to ë* of its original value. The value of Xo, in terms of the atomic number Z and the atomic rnass A, can be approximated by

which is accurate tu 20% for 13 S 2 2 92. The particle loses energy through radiation until its energy reaches critical energy. The energy at which the loss due to bremsstrahlung is equal to the loss due to ionization is called the critical energy E,. An estimate of the critical energy, with an error of 10% for values of Z between 13 and 92, is given by E, = 550 MeVIZ. Equation (6.1) shows that high energy electrons lose their energy expnnentially with the distance traveled in matter due to bremsstrahlung. Thus energetic electrons will radiate most of energy within several radiation lengths inside medium. This is an important characteristic for design of electromagnetic calorimeter.

The transverse size of the electromagnetic shower determined by multiple Coulomb scattering with the ~ o l i k r e radius is given by

The radius of the cylinder which contains 95% of the shower energy is 2R,.

6.2.1.3 Hadronic Showers

Due to the large number of different reaction processes contributing to the shower development, hadronic showers are more compiicated than electromagnetic showers. When hadrons interact strongly with matter, they break apart the nuclei, producing rnesons and other hadrons. These can in turn interact and deposit energy by ionization in the medium. While charged hadrons Iose energy through ionization, al1 hadrons interact inelastically with the nuclei of the absorber material which in tum Ieads to the shower evolution. When a hadron interaction breaks up a nucleus, energy is Iost to supplying nuclear binding energy. This energy is not visible in the sensitive (sampling) medium of the calorimeter. Neutrino escape also contributes to signal loss or missing energy. Muons are minimum ionizing particles and they leave only part of their energies in the calorimeter. The main effect of the signal loss is that the fluctuations on these energies give an irreducible limitation to the energy resolution of the calonmeter. The loss of visible energy c m be compensated by the use of depleted uranium for the absorber material and plastic scintillator for the active material. Detectable energy is released as photons of a few MeV and neutrons by spallation. Fission and neutron capture. in the case of depleted uranium absorber, are also ways of energy transfer in hadronic showers. These neutrons undergo elastic np scattering in a hydroge- nous active matenal. The recoil protons yield visible signal in the scintillator. This implies that the thickness of the absorber and active materials can be tuned such that an equd response can be obtained for hadron and electromagnetic showers. This self-compensation results in the ZEUS calorimeter having a high resolution for hadrons.

A quantity used to describe the development of hadronic showers is the nuclear interaction length. which is the average distance traversed in the medium by the particle before it is absorbed. The interaction length is given by

where p is the density of the material in g/cm3. A hadronic shower has an electrornagnetic component which is produced by neutral pions which decaying to iro -t W. Fluctuations in the amount of an energy is a major contributor to the resolution. At the energies of 800 GeV a hadronic shower is contained at the 95% level in 8hinl, while at 400 GeV the corresponding containment depth is 7)cint.

6.3 The ZEUS Online Trigger System and Data Acquisition

6.3.1 Overview of the ZEUS Data Acquisition (DAQ) System

At HERA. the design lurninosity is 1.5 x 103' cm-2 s-' with Ip and 1, of 160 rnA and 60 rnA, respectively. At the interaction point (IP) of the ZEUS detector, high rates of background and low physics rates occur. The electron and proton bearn at HERA are powerful sources of backgrounds from synchrotron radiation and bearngas interactions. [n addition. the halo muons that follow the proton bearn can also trigger the expenment. Since the bunch crossing time of 96 ns is much shorter than the time necessary to form a trigger decision, the design of an efficient trigger system at HERA is a challenge. This is effectively handled by storing the data in a pipeline while a decision is formed in each local First Level Tngger (KT). The Global FIT (GFLT) collects the global information from local trigger systems and is designed to reduce a bearngas background rate of 100 lcHz to 1 kHz at the design luminosity. The Global Second Level Trigger (GSLT) is designed to reduce its input rate from the FLT to 100 Hz. The role of the EVent Builder (EVB) 1671 is to combine the data from the detector components into the final format and transport them to the Third Level Trigger (TLT).

The schematic overview of the ZEUS data acquisition system is shown in Fig. 6.6. More than 250.000 readout channels produce 10 GB/s of an initial raw data rate. The interface of EVB and TLT at crate level is shown in the figure. The events flow from EVB into the card labelled 2TP via a transputer (TP) network into a Triple Port Memory (TPM) in each crate. This 2TP rnod- ule has two LNMOS Tg00 transputers connected by way of private ports to the TPM. The TPM also has a port connected to the VME-Bus. The events are read from the TPM of the 2TP module and then analyzed. The hardware configuration of the TLT system is fully described in Ref. [68].

The TLT system is a UNIX based farm of 30 analysis processors and 6 managing processors, having a computing power in excess of 1000 MIPS [69]. For an input of 100 Hz, each pro- cessor has 300 ms for processing a complete event. The task of the TLT is to suppress background events and accept and classiQ ail physics events to match the data writing capability of 5 Hz. A

. two stage triggering filter logics is used to achieve this goal in the TLT. First fast calonmeter tirn- ing and track reconstruction are used to reject many background events. In the second phase, physics candidates are selected frorn the remaining events by various physics filters. In the running period of 1994, the TLT was able to identify and reject background events, and process the remaining to select physics candidate events based on calculation of the global properties of the event.

€th met f

Cornponent Front End

. a .

.c. EVB

4 _ _ _ _ _ _ - - - - - .-.-.

..... .....................

TLT

EVBALT VME cnie

M BIS

Figure 6.6: Layout of the ZEUS trigger and data acquisition system. The ZEUS trigger system is designed os three levels. More processing time is nvailnble in each successive trigger level.

6.3.2 Global First Levei Trigger (GFLT)

The Global First Level Trigger (GFLT) uses information collected from component FLTs such as the calorimeter, the muon chambers, the centrai tracking detector, the luniinosity monitor, and CS bearn monitor counters. The pipeline DAQ system stores the detector information for al1 the bunches, with about 5 ps window. Meanwhile the local K T s caiculate the signals for FLT and GFLT gathers them and evaluates the trigger logic for every crossing. After GFLT decision is made, the corresponding pipeline informations are readout and digitized, then sent to SLT and event buffer. The design output rate of GFLT is 1 kHz and in 1994 actual output rate was a few hundred Hz.

6.3.3 Global Second Level Trigger (GSLT)

The GSLT is based on extensive network of transputers which use the two-transputer VME module (2TP). Since more complete and precise data are available than the FLT processors, trigger selection algorithms such as spark, calorimeter timing, cosmic muon, described in the next section, have been implemented in the GSLT. The main component participating in the SLT is the dorimeter S LT.

6.3.4 Event Builder (EVB)

The ZEUS data structure is ZEBRA with ADAM0 [70] flavor. The EVB merges the data originating from the various detector components into a single event record and transport them to the TLT. Since the EVB is connected with al1 parts of the DAQ system it is an important tool for system andysis and diagnosis at online level. The EVB design requires a total bandwidth of 20 Mbyte/s. The EVB hardware uses a 2TP module for computation and data transfer. Up to 76 events can be constructed at the sarne tirne. The cornplete event is transferred to TLT input buffer after construction.

6.3.5 Third Level Trigger (TLT)

6.3 S. 1 Online Background Reduction

The first step of TLT beamgas identification is done by summing energy in the ZEUS calorimeter. The beamgas interactions which deposit energy above noise in the rear andor forward calorimeter are rejected by taking advantage of the excellent ( < 1 ns ) timing resolution of the calorimeter. This technique of identifjring and rejecting events that are out-of-time with respect to ep collisions is powerful, as c m be seen in Fig. 6.7. Two peaks can be clearly seen. Physics events

are centered at (0.0). Background coming from the interaction between the proton beam and residual beam gas gives rise to the Iarge peak with negative RCAL time. In an ep collision particles are coming from nominal interaction point and arrive at the cells of RCAL, and FCAL with equai times t = O ns, while the time for ap-beamgas interaction coming from upstream of detector is set to TR = - 8 ns. This corresponds to twice the distance between RCAL and interaction point, and an event is rejected as beamgas if I TF - TR - 12 1 < 8 ns and I TR+12 I < 8 ns conditions are satisfied.

ep collision

FCAL RCAL

I C C

! 6CO

' 2CC

500

4CZ

3

p-background

FCAL RCAi

Figure 6.7: Timing distribution of FCAL-RCAL time vs RCAL time in ns.

Dunng the 1994 running period, the event timing was calculated by an error weighted time average using al1 photomultipliers with energy above 200 MeV,

P M T C wi ti

where tyMT is the time of the energy deposit rneasured by photomultiplier (PMT). The weight function, y, is defined as

\Vi =

where Ei is the energy measured by a photomultiplier in GeV. This allowed low energy beamgas events to be rejected. In addition, TLT also used the calorimeter tower around the beam pipe in both the rear and fonvard calorimeter, where the energy of the beamgas interactions is rnostly deposited. Events outside a f 8 ns time window were rejected. A typical beamgas event is shown in Fig. 6.8.

For further reduction of the background events originating from the outside of the nominal interaction point and identification of physics-like events, track and vertex reconstruction are essential tools in TLT online environment. Unlike 1993 run period, which used only 16 2-by-timing layers of tracks using CTD, in 1994 TLT performed pattern recognition and reconsuuction of tracks using CTD Flash Analogue to Digital Converter (FADC) readout. This improved the vertex resolution and alIowed a tighter vertex cut. The 2-by-timing system uses the time difference between signals arriving at the two ends of the wires to determine the Z position of the tracks. The FADC readout system measures the arriva1 time of ionization at sense wires as well as the charge deposited in the CTD. In order to satisb TLT processing time several streamlining options were used for online track reconstruction. VXD data were completely ignored and this removes short lived secondary particles. CTD data corning from superlayer (SL) 6, 7, 8 and 9 were ignored as well. CTD outer tracks that do not reach SLl were rejected because track is not complete. Fur- thennore, SL1 tracks must have at least 5 out of 8 hits and layers with more than 50 hits were not used. Track extrapolated to calorimeter was bypassed. The quality of the vertex distribution was monitored as a potential tool for beamgas rejection at higher lurninosities. Use of the FADC tracking code improved track matching to calorimeter objects such as condensates or clusters for physics classes that use electron-like and muon-like objects.

Besides the bearngas background, the detector is aiso tnggered by sparks in the photomultipliers. Since each calonmeter ce11 is read out by two photomultipliers, a large energy imbalance between the two photomultipliers is the signal for sparks. The original TLT spark rejection dgo- rithm was slightly modified as follows: if an event is not rejected as a spark. then include the energy from sparking cells in al1 calonmeter energy surns. So events are flagged as a date if ECel1 > 1.5 GeV and 1 A ~ y r n m e t r y , , ~ ~ I > 0.9 where asyrnrnetry is defined as

L - R Asymmetry = - IL+Rl

spark candi-

(6.7)

where L and R represent the energy of the left and right photomultiplier. TLT rejects events if number of spark candidates is one and Er,, - EsPark < 2 GeV. This condition has been proved to be quite safe for DIS events.

Cosmic rays and bearn related halo muons were rejected by using calorimeter energy, time, ce11 positions and barre1 muon trigger and track information. The energy deposit in each calonmeter ce11 was required to have the signature of cosmic muons because muons are minimum ionizing particles.

Figure 6.8: A benmgas event with track reconstruction in the TLL

6.3.5.2 Online Selection of Physics Events

The events remaining after background reduction were passed through various physics filters in order to clarify them. Events selected by the physics filters were subjected to prescaies to lirnit the output rate. In addition, there was a prescaled sample of various triggers input to the SLT and TLT that were passed through the trigger systems without rejection for subsequent determination of trigger biases in an offline study of veto cuts and physics filters. For some filters, the TLT algorithm searched for clustered deposits of the calorimeter energy to identiw electron-like, muon-like or jet-like objects. The physics algorithm were then applied to al1 these events surviving the veto cuts, from kinernatic cuts based on calonmeter energy, track counting, or track matching with clusters. Some high rate filters used a cut on the vertex position to reduce background. From offline analysis of the triggen taken in 1993, physics groups on ZEUS have been able to define filters which accept or Save the physics processes of interest. Extensive Monte Car10 studies have also been performed to determine optimal criteria and cuts to ensure a reasonable rate. Most physics filters used simple kinematic cuts in the TLT. Every filter had its own prescale factor, so filters where the rate cannot be safely reduced at the TLT were prescaled. The physics classifications used in 1994 are shown in Table 6.2.

General Physics Groups Fil ter Classification

1 a) Nominal Neutrai Current (NC)

Deep Inelastic Scattering (DIS) / b) High neutrai current

1 C) Charged Current (CC)

Soft Photoproduction (SPP) 1 a) Nominal photoproduction

1 b) Elastic photoproduction

- - -- 1 Search for Exotic (EXO) physics 1 a) EX0 particles 1

Hard Photoproduction (HPP)

Heavy Flavour (HFL) physics

Table 6.2: Classifcation of physics groups and filfers implemented in the TLT during 1994.

a) Hard scattering processes

a) J / W , LI*, semi-leptonic, B E . C E

6.3.5.3 Odine Monitor in the TLT

The online performance was monitored in several ways during the running period of 1994. An imponant part of this monitoring was the collection of histograms from the memory of the individual TLT node machines every 30 seconds. Processing tirne. caiorimeter energy distributions and various histograms were provided.

Figure 6.9 shows the amount of processing time needed per event for analysis dunng the online data reconstruction phase in the TLT. The TLT spent 357 msec per event. The TLT could handle 30/357 = 84 Hz of input rate with 30 processors. In fact. the TZT received 30/95 1 = 32 Hz from SLT output. Frorn these two histograms it is possible to determine if the TLT is causing the dead time in the online data chain. Energy distributions were aiso checked, so strange behavior in the calorimeter could be quickly diagnosed by monitoring these histograms at online reconstmction level.

& O 50000 ZEUS 1994 L Q) Run Number 9804 D

40000 Mean value 357 3 z

30000

20000

10000

O O 500 1000 1500 2000

30000 i Y - C - Q, > i- - (b) w 25000 r - - O F ZEUS 1994 - L Mean value 95 1

1 O 0 0 2000 3000

( m e c ) (msec)

Figure 6.9: Online processing time in the TLT Analyze Event (AE) is the analysis process in the TLT (a) This histogram shows real active time tu process an event in the TLT (b) Total tirne (=active + idle time) of AE in the TLT

The TLT online track reconstruction was used in several physics filters to make a cut on the number of tracks within a certain theta range and perform track matching to energy deposits in the calorimeter. Figure 6.10 shows typical online vertex distributions. Figure 6.10a is online Z

vertex distribution for events after timing, sparks and cosmic cuts in the TLT, but before physics filters. The narrow vertex from ep collisions is superimposed on the proton beamgas background. The long tails are due to e and p gas interactions outside nominal interaction point. The hatched region shows the Z vertex distribution of the events flagged as physics candidates after physics filters in the TLT. Figure 6.10a and 6.10b clearly show how the TLT filter effectively reduces events coming from the beamgas interactions. The advantage of using offline reconstruction program at the TLT cannot be overestimated for these reasons. Figure 6.10~ shows the soft photoproduction events tagged by the LUMI detector. In addition, TLT veto, filter trigger bits were monitored for various triggers. Electron and LüMI energy distribution were also added in the TLT histograms.

Online Track Reconstruction in the TLT

............. JO000 . . . . . =us l S S 4 . .

Run Numbu 9804 ... ................ 25000 ".

Z af Vœrtrr (Full f i t )

i; [ ................................ f.. .........................-.

Figure 6.10: Online TLT vertex distribution. (a) Vertex distribution of sample before/nfter physics filters. (b) Vertex distribution nferjilter with gaussian fitting. (c) Vertex distribution rvith sofi photoproduction tag.

7. Event Simulation and Hadronization Mode1

So far, we have discussed and focussed on QCD processes. In this chapter, we will deal with event simulations to generate DIS events. It is very important to interpret the experimentai data physically and have a mode1 which simulates QCD processes. Monte Carlo (MC) simulations are an essential tool to generate QCD processes at the parton level and final state hadrons, in order to compare the measurements with QCD predictions. There are two general classes of Monte Carlo event generators which have been used: the Matrix Elernents (ME) approach; and the Parton Shower (PS) approach. Only the main features of event ger~erators used are reviewed in this chapter. The details can be found in the corresponding manuals [49].

Once the initial parton distribution is generated it must be converted into the color singlet hadrons which cm be observed by a particle detector. This process is called hadronization, or fragmentation. It cannot be treated by the perturbative QCD because it takes place in the low momentum transfer region where a, becornes large. This process must be phenomenologically modelled. Two classes of models for hadronization will be reviewed.

7.1 Leading Order (LO) Simulation

LEPTO 6.1 [49] is an event generator based on the LO electroweak scattering cross section for y and 9 exchange. This program simulates cornplete lepton-nucleon scattering events and integraies the cross section for the underlying parton level scattering processes. It includes QCD corrections using exact first order matrix elements and higher orders in the leading logarithmic parton shower approach.

7.1.1 First Order Ma& Elements (ME)

LEPTO 6.1 simulates the LO process (y+ q + q ) and the QCDC (yq + q g ) and the BGF (yq -t 94) process to O (aS) . The ME are functions of three variables which relate to energy, polar angle of the two scattered partons and azimuthai angle with respect to the scattered lepton direction. From Eqs (5.7) and (5.8), one sees the two jet cross section for BGF and QCDC processes have singularities.

A Monte Carlo generator avoids these QCD divergences by using a cut-off in some vari- 2 ables. The invariant mass of two massless partons i and j is given by mi, = EiEj ( 1 - cos@,.)

where Oij is the angle between two partons. The cut m i = is applied to al1 pairs of partons in the finai state, including the target remnant system. W is the invariant mass of the hadronic system. The parameter y,,, is then set to control these singularities.

7.1 -2 The Parton S howers (PS)

Even though the ME c m predict the generai feanires of fixed target DIS experiments. the multi-jet events in ete- experiment are not properly described by the second order ME approach. Furthemore high transverse momentum (PT) jets in pp collision could not be explained by ME approach. Motivated by these facts, an alternative to the ME approach, known as the PS approach, has been developed. This PS approach gives a more correct simulation of processes in higher orders of as.

In DIS the incoming parton from the proton is a quark or gluon which can emit partons both before and after the boson vertex giving rise to initial state (1s) and final state (FS) parton showers. In the PS approach one could start with the QPM process. On top of this the initiai state parton shower (ISPS) and the final state parton shower (FSPS) are evolved by possible branchings regulated by the Aitarelli-Parisi formulation, which uses the leading logarithmic approximation. Figure 7.1 shows the time development of PS evolution. This is helpful in understanding the s hower process.

P - Proton Remnant

Figure 7.1: Parton shower evolution in an ep scattering event. ISPS and FSPS are drawn before and aper boson vertex. Proton remnant is also shown.

A parton in the incoming proton can initiate a parton ernission where in each branching 2 2 one parton decreases the virtuality of the parton (O > rnp > . . . > rn ) . it becornes more and more

I Pn

spacelike. The radiated daughters ( r r , . . . r ) are either onshell or have timelike virtualities which result in timelike shower since this process is generating real partons. Wben the parton @,) interacts with the photon at boson vertex, it will be tumed to onshell or to a timeiike virtuality rn' 2 O. It radiates partons (t l 7 t2, . . ., tn) decreasing its virtuality. The virtuality of the shower

P n + 1

is allowed to run down until a branching occurs. Branchings are allowed until mim reaches sorne cut-off value of pi - 1 G ~ v ~ , at which point PS is stopped and the fragmentation process starts. In the final state parton shower branchings, to account properly for the color coherence among the many soft gluons that are emitted, an angular ordenng cm be imposed on the shower such that successive gluon emissions lead to decreased opening angles for ( t l , t2, . . ., t,) . The probability for each branching is mled by the Sudakov form factor, which expresses the probability that a parton does not branch between sorne initial maximum virtuality and some minimum value. This form factor is used to find the mass of the decaying parton, and the momentum fractions in the branching. The shortcoming of the separate evolution between the ISPS and the FSPS is that this approximation violates the gauge invariance because no interference ternis are taken into account between the ISPS and the FSPS.

The amount of parton emission depends on the virtuality scale of the stnick quark just before and after the boson vertex. These virtualities are chosen, using the Sudakov form factor, between the lower cutoff and a maximum vaiue to be given by the momenturn transfer scale in the process. In contrast to e+e- physics where no scale mbiguity exists, there is no unique scale in DIS. It could be any function of e2 and w2. Since the invariant mass of the hadronic final state is

2 1 -x given in terms of x and e2 by # = Q (-) + M:, w2 can be the same order of magnitude as e2 at large x, but is much larger at small x. Ghas been argued that the mean transverse mornentum (P:) is proportional to w2 for not too small x [ïl] and therefore w2 is the preferred scale. For small x it follows » Q* and the proton remnant Pr with Pr = ( 1 - x) P takes a larger energy fraction which contributes to w2, but it does not radiate. ~ a k i & w2 as the scale of maximum virtuality will overestimate the arnount of radiation. In the first order ME (P:) is approxirnately ~ ~ ( 1 - x ) as x + 1, and is approxirnately e21n ( 1 / x ) as x -t O. Based on the limiting behavior of

1 (P:) at low and high x, this leads to the choice given by e2 ( 1 - x ) max ( 1, ln (-) ) . X

7.1.3 Matrix Elements and Parton Showers (MEPS)

LEPTO 6.1 allows the possibility of adding parton showers to the first order matrix elernents, denoted by MEPS. The combination of the two approaches gives the first order parton ernission plus the higher order emissions through parton showers. Lf there is an overlap in phase space between ME and PS this will lead to double counting. The probabilities for al1 partonic subprocesses are matched to avoid this double counting. Our analysis is rnainly based on MEPS mode1 approach.

7.2 Color Dipole Mode1 (CDM)

An alternative approach to QCD showers is given by the color dipole model [72]. The CDM is based on the fact that a gluon emitted fiom a qq pair in an e+e- collision can be treated as radiation from the color dipole between the q and q, and that, to a goad approximation, the emission of a second softer gluon c m be treated as radiation from two independent dipoles, one between the q and g and one between the g and q. In the CDM this is generalized so that the emission of a third, still softer gluon, is given by three independent dipoles, etc. In DIS a color dipole is formed between the stmck quark and the proton rernnant. When this dipole radiates a gluon, it splits into two radiating dipoles. one between the stmck quark and the gluon and the other between the gluon and the remnant. Repeated gluon emission leads to a chain of such dipoles.

Unlike ece- case, one end of the dipole system in DIS is not point-like, since the proton remnant is an extended object. It is known [73] that emissions of srnaII wavelengths from an extended antenna are suppressed and the maximum P; in the hadronic center of mass system for an emitted gluon varies as #13. ARIADNE [74] is a program for the simulation of QCD showers, implementing the color dipole model. It interfaces with LEFTO for the simulation of hard scattering. CDM model was used for corrections for detector acceptance and resolution in this analysis, together with MEPS model.

7.3 Models of Hadronization

7.3.1 String Fragmentation

In the string fragmentation mode1 [75], the axis of the color flux tube which is stretched between two partons moving away from each other is represented by a massless relativistic string with no transverse degree of freedom. As a quark q and an anti-quark ij move apart, the potential energy stored in the string increases, and the string may break by the production of a new q'& so that the system splits into two color singlet systems qq' and q'q. The splitting is done in such a way that energy, momentum, and ail internai quantum numbers are conserved. If the invariant mass of either of these string pieces is large enough, they will break up further until only on-mas- shell hadrons remain, each hadron corresponding to a small piece of string. The string model invokes the idea of quantum mechanical tunneling to produce the quark and anti-quark pairs which leads to string breakups. The tunneling probability, where qij will appear, is proportional to

where K is inversely proportional to the string constant. This formula implies a suppression of

heavy quark production from the sea, u : d : s : c - 1 : 1 : 0.3 : 10-ll. Clearly charm quark production is negligibie in the string breakup. After this fragmentation process, the unstable hadrons are decayed into stable hadrons.

The function f ( 2 ) governs sharing of energy-momentum between the hadrons. The Lund symmetric fragmentation function requires the left-rîght symmetries involved in breaking a string at the quark or the anti-quark side, and takes the form

Here, a and b are parameters which rnust be detennined from experimentai data.

The width of the jets is given by

where o is a free parameter which controls the width of quark transverse momentum. 4

The string model approach becomes more distinct when gluons are present. The gluon is treated as a kink in the string that is stretched between the two quarks. One consequence of this formulation is that the production of hadrons wiil be larger between the gluon jet and the two quarks jets compared with the region between the two quark jets. This is called the string effect.

LEPTO and ARIADNE programs use Lund string fragmentation rnodel. Both programs are interfaced to HERACLES [76] for the electroweak ndiative processes and to JETSET [77] for the hadronization.

7.3.2 Cluster Fragmentation

A diagram of cluster fragmentation is shown in Fig. 7.2. In the cluster fragmentation model, the parton shower evolution is used to produce the underlying structure of the final state hadrons. At the end of shower evolution, gluons are forced to spiit into quarks, the quarks are combined into colorless clusters, which are then decayed isotropically in the rest frame of the cluster. These colorless chsters are called as basic units. The cluster model assumes the hadrons are produced from basic units. A cluster is charactenzed by its total mass and total flavor content. HERWIG [78] is based on the cluster mode1 for the hadronization.

Yc+ X -

X - Tc+

Tt - 7 r +

7 K +

n

K +

7 r 0 Tt"

Tt - T c +

T t 0

Figure 7.2: A diagram of cluster fragmentation model. According to the cluster fragmentation scenario, PS picture is used to produce a partonic confguration (shower evolution). Any gluons are split into 4 4 pairs (QCD branchings). PS evolution should give cluster mass specrrwn strongly damped at masses above a few GeV (cluster formation). Any high mass clusters are allowed to produce more than two lighter clusters (cluster decays). This is typicdly done by allowing branchings of cluster + cluster + hadron or cluster + cluster + cluster.

8. Reconstruction Methods and Event Selection

Approximately 1o6 DIS events were collected by the ZEUS DIS trigger during the 1994 running period of HERA. Al1 these events do not correspond to Neutra1 Current (NC) DIS events of interest for this analysis. Since the tngger criteria at both online and offline reconstruction level are noi designed to be as tight as possible, the data still contain backgrounds such as beam-gas, cosmic ray and photoproduction events. It is the aim of the selection cuts to reduce the levei of background events in the neutral current DIS sarnple by applying filter programs designed to select neutral curren t candidates after the reconstruction phase.

Since the ZEUS detector is nearly hennetic, it is possible to reconstmct the kinematicaiiy interesting quantities x, y and Q' for neutral current DIS using different combinations of the angles and energies of the scattered lepton and of the hadronic system. Several different methods have been used to reconstmct these variables. Only two variables are independent at fixed e p center of mass energy squared.

This chapter describes several reconstruction methods to select neutral current DIS events which are crucial for Our measureinent of strong coupling constant. Especially, in order to tag events as neutral current candidates electron identification is required. The basic idea for electron identification will be reviewed. Finally, several selection criteria and general event characteristics will be discussed.

In order to reconstmct kinematic variables, we take the direction of the initial proton as the positive Z axis in the ZEUS coordinate. The X axis points horizontally towards the center of HERA and Y axis points vertically upwards. We assume that the lepton and proton rest masses are negligible. The Cmornenta of the incoming electron, scattered electron and the incident proton are

respectively. The outgoing electron is scattered at an angle Oc relative to the incorning proton direction. This is shown in Fig. 8.1.

If we could measure the direction and energy of both the hadrons and the scattered electron, the kinematics of the event would be overconstrained. In order to measure the kinematic variables, one could use either the electron energy and angle or the hadronic final state alone.

I

Figure 8.1: The basic diagranz for the DIS process, where k and k denote the 4-rnornenta of- the incoming and outgoing elecrron, P and q thnt of the initial proton and exchanged boson. respectively. The angle of the scattered electron is denoted by Oe, mensured with reference to the posi- t ive Z mis.

8.1 Electron (EL) Method

Using only the measured angle and energy of the scattered electron. the kinematic vari- 3

ables x, y and Q- = sxy, can be calculated as follows.

E: O e 2 1 - - (sin-) Et? 2

According to Eq. (8.4), e2 increases as the angle of the scattered electron becomes srnaller. Due to the geometncal acceptance of the ZEUS detector, the scattered electron enters in the main

2 detector if Q is above a few G ~ v ~ . The experimental resolution in Q:, is good, except for very small scattering angles (large 0) [79].

8.2 Jacquet-Blondel (JB) Method [80]

From the measurement of only the outgoing hadronic system, the four-momentum transfer q can be expressed as

I

where P represents the four-momentum sum of al1 the final state hadrons. y can be calculated by surnming the energies Ei and polar angles Oi of calonmeter cells which are not associated with the scattered electron:

2 - 2 Combining Eq. (8.3) and (8.4) leads to ( 1 - y , / ) = (E&o~) = (P i ) el . With the

assurnption that the total transverse momentum carried by hadrons which escape through beam 7

pipe hole can be neglected, Q' is given by

7 The variable x c m then be determined frorn y and Q-, i.e. x jB = Q:*/ (sy,,) . From the Eqs. (8.6) and (8.7) outgoing hadrons emitted in the fonirard direction only Ieads to a very small change in y or Q'.

8.3 Double Angle (DA) Method

A combination of both the final state electron and hadronic system c m be used in the double angle method. The angle Oc of the electron and the angle yH of the sinick quark are used. The

angle yH is the scattering angle of the struck quark balancing the momentum vector of the electron to satisQ four-momentum conservation in the naive QPM. It can be determined using the equation

where the sums run over al1 calorimeter cells which are not assigned to scattered electron. The kinematic variables can then be determined from the following equations:

siny, + sinOe + sin (Oe + y,) siny, + sinOe - sin (Oe + y,)

sin Oe ( 1 - - - siny, + sinOe - sin (yH + Oe)

The double angle method is iess sensitive to the calorimeter energy scaie since it relies on ratios 7

of energies and only uses the angles 0, and yH to reconstnict (x, Q-), and it gives good accuracy over the whole kinematic region.

8.4 Mixed Method

In this rnethod, the variables of both electron and hadronic methods are combined. The rnixed variables are defined as

To summarize the reconstruction methods, contours for the scattering angle and energy in the x and e2 plane of both the electron and jet are s h o w in Fig. 8.2.

! a) scattered elec:ron energy ..-.

- -

Figure 8.2: The dotted lines show constant values of y for y= I , 0.1 and 0.04. (a) Contours of constant scuttered electron energies. (b) Contours of constant scattering angle are shown. The iine labelled 176.5 is the RCAL beam hole. The scattered electron is lost down the RCAL beam hole for Q' iess thnn about 2 G ~ V ~ and value of x greater than IO"'. (c) Lines of constant jet energy. (d) Lines of constant jet angle.

8.5 Electron Identification

As mentioned earlier, neutral current events in DIS at HERA are charactenzed by the presence of a scattered electron in the final state. Due to conservation of momentum, the scattered electron is bück-to-back in the azimuth with the fragments of the hadronic system. Therefore, the scattered eiectrons are in general well separated in space from the hadronic fragments, except the regions of high x or low x as shown in Fig. 8.2. in order to select neutral current events it is important to identiQ electron in an efficient fashion. In this thesis, the identification of the scattered electron and the measurement of its energy are based entirely on the calorimeter information of the ZEUS detector. An electron impinging on the calorimeter will initiate an electromagnetic shower. Typically al1 of the energy will be absorbed in the EMC section. Several algorithms have been developed within ZEUS and the SINISTRA electron finder [81] used in this analysis is described as shown in Fig. 8.3.

The SINISTRA uses the cluster algorithm based on islands clustering. Single particles or jets of particles from interaction point deposit their energy in cells of the calorimeter. The deposited energy is usually spread over several adjacent cells. Detection of electrons, proton fragments, muons and other particles requires an algorithm which combines the energy deposited in individual units of the calorimeter into physically meaninghl groups, or 'clusters' of energy. The role of the cluster algorithm is to merge cells which belong most likely to the shower of a single particle.

Calorimrirr Towcr

Cluircr 2

Figure 8.3: Creation of calorimeter objects via island clustering. m e boxes schematically represent towers, while the nurnbers represent energy deposits with arbitrary units in each tower: Each tower iypically has 8 neighbors. The arrow points in the direction of the highest energy neighbor for a given towe,: Certain towers have no highest energy neighbor and we assign these towers seed. Clusters can be defned on the basis of these seeds. The darker lines shows the contour of the tower assigned tu the object by island algorithm. A circle represents a seed and the towers assigned ro a cluster are represented by the arrows pointing towards the seed.

Islands are created as follows. First the energy in each tower of the CAL is compared to the energy of its neighbors. A tower becomes a seed for an island if al1 the neighboring towers have lower energy. Otherwise it is assigned a link to the neighboring tower with highest total energy deposition. Each seed will be the center for a different energy cluster. Now we loop over al1 of the towers again such that al1 towers with links leading to the same seed are now assigned to one island.

For clustering, a window consisting of 3 x 3 towers centered on the tower with highest energy deposition is chosen. The choice of this window is motivated by the fact that a single electrornagnetic particle is expected to deposit al1 the energy in the EMC sections of the calonmeter and to be well contained transverseiy within this window. A window of this size also contains cells which were not assigned to the cluster. They are ignored. Two energy sums are made in this

W W window. EEMC represents the energy deposited in the EMC sections of the window and EroT represents the total cluster energy contained in the window. For electrons and photons, the ratio of W Emc to E&, will peak at unity while it is evenly distributed between zero and one for hadrons.

W W The lower values of EEMC/ETOT for the electromagnetic clusters happen if electrons enter the calorirneter between two modules and deposit an unusually high fraction of their energy in the

W W hadronic section of the calorimeter. These events are removed by requiring EEMC/EToT > 0.8. This gives rise to a known inefficiency of the electron finder and leads to a loss of 1 % of electrons while rejecting 84% of the hadronic clusters.

8.6 Event Selection Cuts

In this study, DIS events were selected by using ZEUS calorirneter. Because of the l / ~ " behavior in the propagator in the cross section of neutral current DIS, high rates of photoproduction will occur at srnall Q'. The following selection criteria were applied in filter programs for DIS events.

First, the detector operation was checked to select event candidates. The Central Tracking Detector (CTD), calorimeter, the magnet and the lurninosity monitor should be well functioning during the penod of taking data. Therefore, test uiggers, events with no ep bunch crossing number, and events taken from the run penods when detector was not working properly were removed. Events with cosmic muons and halo muons that follow the proton beam were rejected by the algorithms based on event topology described in Chapter 6. More stringent timing algorithm than used online was applied.

The event vertex was reconstmcted from the tracking information using vertex detector (VXD) and the CTD. In order for an event to corne frorn interaction point, Z position of vertex

t50 cm was required. Energy in the FCAL had to be greater than 1 GeV to remove diffractive events in which no quantum nurnbers are exchanged between the two interacting particles (in this case the photon and the proton) through elastic scattering. Usually diffractive events do not deposit a significant amount of energy in the FCAL. The distribution of Z vertex and FCAL , energy is shown in Fig. 8.4. Elastic QED Compton events, eP + e Py, where the proton is elasti- cally scattered, were rejected by algorithm based on event topology of two calorimeter energy deposits bdanced in transverse momentum.

Figure 8.4: The mensured distributions of the (a) z-vertex and (b) FCAL energy in the range 120 < 9' < 3600 G ~ v ~ . 0.01 < x < 0.1, and 0.1 < y < 0.95 between data and MEPS Monte Carlo are drawn. The peak of 2-vertex is around zero, which means most events came from the nominal interaction point in the ZEUS detector: Once we apply al1 kinematic conditions, no events in tvhich FCAL energy is less than 1 GeVare found in the final samples.

For photoproduction event candidates, an electron scatters through small angle and does not enter into the calorimeter. Therefore finding an electron is important in our sample. Electron identification is based on the transverse and longitudinal shower shape in the caiorimeter. Events were rejected if the scattered electron impacts RCAL inside 16 cm2 in the XY-position, to ensure containment of electron and sufficient reconstruction accuracy of electron in the detector. A scat-

I

tered electron candidate had to be reconstructed with an energy E, greater than 10 GeV to have a high purity. To reject backgrounds from photoproduction events with a fake electron (rnostly no's close to the proton beam) in the FCAL the electron candidate was required to satisQ y, < 0.95. The distributions of the scattered electron is drawn in Fig. 8.5.

a ZEUS 1994

E'. (GeV) 0. (degree)

Figure 8.5: The distribution of scnttered electron. (a ) energy (b) angle in the region 120 < @ < 3600 G ~ v ~ , 0.01 < .Y < 0.1, and 0. I < y < 0.95 berween data and MEPS Monte Cnrlo are drawn.

Furthemore the following cuts were applied for the final data sample. A useful variable that distinguishes DIS events from photoproduction and proton bearn gas is

where the sum runs over al1 measured particles and E is the total energy, P. is the longitudinal - mornentum measured in the calorimeter. If al1 final state particles are measured in the calorimeter. conservation of momentum and energy implies that (E - P.) = ( E p + E,) - ( E p - E,) =

C

2 E, = 55 GeV. Since 6 is conserved quantity. so 35 < 6 < 60 GeV was used for background rejection. For proton beam gas events from inside detector, 6 will be small because most particles are fonvard. For photoproduction events in which the scattered electron escapes down the RCAL , beam hole, 2E will be Iost from 6 distribution. The remaining photoproduction background in

Z this high (x, Q ) data sample was less than 1%. The 6 distribution for a Monte Carlo simulation of photoproduction background and DIS is shown in Fig. 8.6. Events surviving an electron energy cut of 10 GeV are shown in the shaded histogram.

" 1 9 . 1 2 r 1.- . - I ' 0 i 1 . a . -

, ,-1 . r [ 1 I t ) ,O-, 1 ~hot&roduction MC . l &- g a-1 .

I i I

t I

.*., I I I

. .-. 0 . .

- 4 I .... 0.08 F .... . . . : k; DIS MC . - . . . . . . I ' . . . . . . .-. . . : '-1

..... i . . . . . . - i ..... 0.06 I ' . . . . ..... .... 6-Cut rd I I

Figure 8.6: E - P- distribution for photoproduction Monte Carlo compared with DIS Monte Cnrlo. The photoproduction Monte Carlo was generated with MRSA parameterization and dijet events having transverse momenturn PT > 2.5 GeK DIS Monte Carlo uses MEPS option rvith MRSD'- parameterization for parton density. The photoproduction Monte Carlo sample is s h o w as dashed line before any kinematic cuts are applied. Afer a 10 GeV electron energy cut the small fraction of photoproduction events remnin. The input DIS Monte Carlo sample is s h o w in the right penk around 2E, - 55 GeV as dotted 1ine. The result for requiring a 10 GeV electron is shown as light shaded histogram.

Several considerations are taken account of in the selection of the kinematic region used for the ap rneasurement. First. the Jacquet-Blondel method has the best resolution for y and. in order to well measure the hadronic system. the value of ylB was used. The requirement of yDA 2 0.1 was used to suppress events with very forward going jets. Secondly, theoretical uncertainties in the QCD (2+1) jet cross section occur at low x where the parton densities of the proton are less well known. The parton shower uncertainties are also much bigger in this region. Both these considerations led to the restriction .yDA Z 0.01. Additionally, xD, less than O. 1 was required because double angle method resoiution becomes unacceptably large for xDA 2 0.1. Finaily, we constrain the analysis to high Q~ where jets are more energetic, well collirnated and well separated from the bearn remnant. So 120 c e2 c 3600 G ~ V ~ was chosen. In choosing high e2 lirnit, enough statis-

2 tics were considered. Furthemore, in the high Q , clear jet structure is observed because hadronization uncertainties are rninirnized. The kinernatic variables x and (2' were reconstnicted with the double angle method summing over al1 cells in the calonmeter.

One of neutral current DIS evenü in ZEUS detector is shown in Fig. 8.7. The key signature for neutral current scattering is the presence of the scattered electron inside detector.

Figure 8.7: Typical DIS evenr observed in ZEUS derecto,: Calorimeter cells wirh energy deposit are shown. Trncks are reconstructed through hits in the CTD. An elecrron kvas found in the EMC section of BCAL.

8.7 Generd Event Characteristics

Figure 8.8 shows the distribution in the data before selection cuts were applied. The kine- matical bin in 120 < e2 < 3600 G ~ V ~ is drawn. The e2 range is further subdivided into three separate regions to measure strong coupling constant as at increasing momentum transfer scales as a consistency check and as a test for the running of the strong coupling constant o: . These ranges

z' are: 120 < Q* < 240 G ~ v ~ , 240 < Q* < 720 G ~ v ~ , and 720 < e2 < 3600 GeV . Here x and y ranges are 0.0 1 < x < 0.1 and 0.1 < y < 0.95.

Figure 8.8: Distribution of DIS event sample reconstr~icted by Double Angle (DA) method in the 7

-r - Q- plane. Also shown are lines of constant y = I und y = 0.04, the bins (IJIJII) in which strong coupling constant is measured. Ïhe limit of jixed target experirnents is also drawn. The nllmber of evenrs in each region (I, II, III) were 1649. 2048. and 775 respectively.

7 The measured distributions of the variables x,,, yJB, QiA and w:, are plotted in Fig.

8.9. The data are shown together with the distributions of the Monte Carlo with a full detector simulation of neutral current events. The good agreement can be seen between data and Monte Carlo over entire regions in each distribution. The MEPS mode1 is found to reproduce the global event characteristics of the data in a reasonable way [82]. Three different Q' bins (120 < Q\ 240 G~v', 240 c e2 c 720 G ~ v ~ , and 720 < e2 c 3600 G ~ v ~ ) are sub-samples of whole region, so al1 distributions are not drawn in each kinematic bin.

The Monte Car10 events were generated using the MEPS version of LEPTO 6.1 [49] and Lund string fragmentation mode1 [75] for the hadronization together with the HERACLES 4.1 program [76] which calculates the QED radiative corrections. The parton densities of the proton were taken from the M R S ~ set [37]. Al1 parameten of the MEPS Monte Car10 programs were

used with the default setting, since their configuration gives a good global description of the data in the selected kinernatic region. However we used different pararneter value y,, . The pararneter y,*, sets a minimum y, of partons in the first order QCD rnatrix elements [49]. The default value was not used, but instead it was lowered from 0.015 to 0.005 in order to study the measured jet rate as a function of the jet resoluùon parameter ycu, for y,,, > 0.01. The detector simulation program is caiied MOZART, based on a detector simulation package called GEANT [83]. The generated events were processed through GEANT based on the full detector simulation and al1 the cuts, reconstruction methods, and jet analyses applied to the data were also applied to the Monte Car10 simulated events.

- MC MEPS 1 -

2 Figure 8.9: The mensured distribution of the kinernatic variables (a) xDA (b) QDA (c) yJB and Id) Wb, are shown in rhe range 120 < e2 < 3600 G ~ v ~ , 0.01 < x < 0.1, and 0.1 < y < 0.95.

9. Properties of (2+1) Jets and Jet Production Rates

9.1 Analysis of (2+ 1 ) Jets

In this chapter the experimental considerations involved in measuring the relative jet production rates are discussed. In the following sections, for the jet classification in the (2+1) jet events, we will designate each jet as follows. The jet which has the largest angle with respect to the proton direction is called the "Least energetic jet". The other jet, which has a srnaller angle, is called the "most energetic jet". The remaining jet is called the "bearn remnant jet".

9.1.1 Jet Multiplicities

At the detector level, caiorimeter ce11 energy and position information were used for jet clustering. Only calorirneter cells with energies above 150 (200) MeV for EMC (HAC) cells were used. The energy deposited in the calorimeter is grouped into entities known as condensates. which are a contiguous group of energy deposit in the calorimeter cells. Due to the energy threshold cuts used in building condensates, the values of the kinematic variables are different from the ones evaluated at the ce11 level. Due to the broken energy flow around beam pipe in the FCAL at ZEUS the condensate analysis produces an incomplete jet reconstruction [84]. Therefore. the jet clustering algorithm is run on ce11 level information. For the systematic checks. pre-clusters were formed. Pre-clusters mean that very forward region eight cells around FCAL bearn pipe and condensates from other cells in the BCAL and RCAL are rebuilt. Figure 9.1 shows the multiplicity distributions in each level (cell, condensate. pre-cluster, and true hadron). Multiplicity is defined as the number of input particles to the JADE jet finding algorithm. Condensate rnultiplicity is typically 20 in the ZEUS detector. About 40 particles are coming into jet algorithm as an input in preclustering case. Number of particles are about 100 in ce11 level case. In the JADE jet finding algorithm the invariant masses of input particles are compared. Since the number of possible pairs (i. j ) increases rapidly in ce11 level, more computing time per event for jet clustering is needed. The shaded area shows the hadron distribution at the hadron level.

One has to be sure that the energy rneasurement in the ZEUS detector is reliable enough 3

for the purpose of this analysis. The scale pararneter W is taken from calorimeter cells associated with the hadronic system.

where s is the squared total center of mass energy. It reflects the measured hadronic activity. y,,. What is needed of the measurement is a scaled mass squared yu as indicated in Eq. (4.2), not each

jet kinematic variable. This rnakes up for energy rnisrneasurernent on an event by event basis. For input to the jet finding algonthm, the calorimeter ce11 energy and position information are reconstructed. No explicit preclustering is applied in the forward region where the particle density is high. In addition, d l cells corresponding to the electron are rernoved.

il chden.de Distribution

;i - ' 'i a . : . - . f&-Ciuster Distribution

Number of Particles n

Figure 9.1: Distriblrtiort of niriltiplicity for ZEUS 1994 dam. The m t ltiplicity distribirriorz of iripirt ohjects in the JADE jet clirs~ering dgoritlini is shorvn ,nt detector Irvel, cor~dmsczte. prr-clusrer: c z r d cell distribrïtiorr.

9.1.2 Effect of Parton Shower and Parton Density to Jets

The effect of parton shower and parton density distributions on jets was considered in the jet reconstruction. At high x. parton densities of proton are well known; therefore the sensitivity of the theoretical calculations to different parameterizations of the p a o n densities is minimized. Also. the uncertainty stemming from the initial staie parton showers used in the Monte Carlo simulation for correcting the data to the partonic level is minimized. These effects lead to .r > 0.0 1 as shown in Fig. 9.2.

k Parton Showcr Effect

x ml 1

Parton 0ejsity Effsct (b)

1

;+ Use x > 0.01

< t o z !

1 oz 1 0 '

x,

Fiylire 9.2: Effect of pnrton showers and parton clensitirs to jet rate as njrnction of s. (a ) Loirer- ing s doivn to 0.005 increrms the total cross section and (2+ 1 ) jet raie. but pnrtoti shoiver iitzcer- initities due to lziglzer ordrr sofl gluott ernissioii gets large>: (0) The ~incertczirtty due to the difleretit clroices of the pnrion detzsity is not negligiblr at loiver .Y. The systemcitic error on ( 2 + 1) jet nitr is triore than I O % iri the loii*er .r = 0.01 region.

9.1.3 Separation of Jet from the Beam Remnant Jet

In the present kinernatic region ( 120 c Q' < 3600 G~v') , the average value of the polar angle of the reconstructed jet is about 50 degrees for ( l+ 1 ) jet production (QPM) as shown in Fig. 9.3. In this case. there is no experimentd difficulty in isolating the current jet from the proton remnants which do not participate in the collision and are lost down the beam pipe.

In applying the jet finding algorithm to the ZEUS data, there are difficulties in treating the Very Forward Region (VFR) and Very Backward Region (VBR) surrounding the beam pipe. These regions consist of the 8 cells of the FCAL and RCAL immediately adjacent to the beam pipe. None of Monte Carlo models describe data in the VFR for both DIS and photoproduction [Ml. The treatment of these regions is very important as one of the non-remnant jets in (2+1) jet production is typically directed forward because of the forward singularity in the cross section. In the theory. the forward singularity in the (2+l) jet cross section is regulated by the cutoff u,,,, in the jet finding algorithm. There are no theoretical difficulties. The fonvard singularity cannot be regulated only by the ycl,, in the real experiment. Even requiring a large y,,, in the identification

of (2+1) jet events, which requires a large invariant mas between the jets, does not constrain the jets to be farther away from the beam direction in the !ab frame. Instead, a Iarge reduction in ( l + l ) jet statistics is incurred as shown in Fig. 9.4. Unlike e+e-experirnent, the singularity in jets cannot fully be controlled only by the y,,,,. Therefore understanding of this forward region in our kinematic regions is crucial factor in the measurement of the strong coupling constant.

Figlrre 9.3: Polar ongle of the crirrent jet for (i+l) jet. The average of tlris current jet is 52 drgrees in the experiment restilting in no experimentnl problem in separnting cnrre>zt jets froni hearrz reminnt jets.

1 200 - - y'", .z 0.0 1

1000 -.

Figiire 9.4: Polir angle distribrition of most enrrgetic jet of (2+ 1 ) jet eventsfor different r~rrlttes of With increusing y,,,,. ï z large frnctio~z of (2+i) jet events is renioved.

In order to solve the experimentd problems arising in the separation of jets from the bcam rernnant in the very forwardhackward region, several schemes have been proposed. Firstly. a box

cut and an angle cut were introduced. Elhination of the 8 cells in the VFRIVBR is called box cut.

If the box cut is applied to the 8 cells in the VFR of FCAL and VBR of RCAL of the ZEUS detector, then jets are poorly reconstmcted because only a srnaII fraction of true energy is available for jet reconstmction. An angle O cut (O c 10') on the very forward region jets can be applied as was done in the H l experiment [86]. However, an angle is not a Lorentz covariant quantity. Instead, we have imposed a cut on the parton variable z . as defined in Eq. (5.3). for (2+1) jet events. z variable is a Lorentz covariant quantity. Once we have two jets, this z variable c m be easily reconstmcted with a reasonable precision in the experiment. Since forward jets are typi- calIy characterized by smali :, the requirement of large value of : selects (2+ 1) jet events which contain jets relatively far from the beam remnant. The loss of statistics is large, about 50 %. But the fraction of (2+1) jet events containing a fonvard jet with pseudorapidity q . < 2.66

c Je* (0. < 8 ) is reduced from 30% to 10% by requiring 0.1 < : c 0.9. A comprehensive study of

Jet the jet kinematics and jet rates without the application of a : cut has been reported [82] using 0.55

of data collected in 1993.

Figure 9.5: The distribrltiorz of the corrected ( 2 + l ) jet rate withorrt a z crit. (a) 93 dcm irz the kirze- rucitic range 160 < Q' c 1280 G~v'. 0.01 < -Y < 0.1 arid 0.04 < y < 0.95 (b ) 94 data in the kirte- nicitic range 120 < e2 < 3600 G~v-'. 0.OI < .Y < 0. I and 0.1 < y < 0.95. resprctivrl~:

Funhermore. without a c cut. we cannot solve the long standing problem of the discrepancy in dope between data and NLO QCD calculations as shown in Fig. 9.5. We see that two NLO QCD calculations do not describe the data fairly well if a z cut is not applied. We will see that the z-cut results in a significantly improved agreement between the KLO QCD calculations and the data compared to the earlier analysis done without restriction on z [82].

- 50 6. Y - 45

$40;

35

30

25

-

(0)

- t \ O 93 Data \

L '..\\ - - NLO 01SJfT

- . . \ O \

.... . \ NLO PROJET 1 \

\

..:., O 1 GO<@< 1280 G& r . . . \

c. 50 K Y

c 45

;:"O

35

30

25

20

1 (b)

i 0 94 Dato

1 - - NLO DISJET \

.... \

L ..,\, NLO PROJET

. \ . \ ' . . \ \ 120<Q'<3600 ~ e v

. .' '.' . .' 20 ..' . \

...:, . . \ '..\,@

.,\ ..\ . .'

. . . . . -.. . . -. . . . \

... ,.a. ... ... ' . ., *-- --- -----__

E O

15 ... ... .. ._. . 1 15:

0.02 0.04 0.06 0.02 0.04 0.06

10

5

... '. . 1 101 L ..Y.$\ \ - - *----* 5

1- ' , I

5

9.2 Properties of Parton Variables ( z , x ) P

The two dimensional phase space region of 2 versus - y p , as defined in Eq. (5.2) and Eq. (5.3), is shown in Fig. 9.6. Jets are identified using the modified JADE algorithm. The area defined by the curve indicates the kinernatic Iirnit for yod, = 0.02 and four different values of -Y, x

= 0.001. 0.0 1.0.1 and 0.5. The upper and lower lirnits on -yp depend on x. The limit on ,- is given

by -"car ' :min = 1 - - If the invariant mass of two jets is zero. Eq. (5.2) shows thai .yp will be "rnax'

1. which gives a singularity in cross section calculation. This figure shows that the phase space for the (2+ 1 ) jet production becomes larger with higher .r up to -Y - 0.1. but falls above x - 0.2.

Figlue 9.6: Allowed ( 2 + l ) jet region of the pcirton phase spnce in the y,,, = 0.02 for diflerent x valws.

The distribution of z versus xP for the uncorrected 1994 data is shown in Fig. 9.7. The data iypicdly lie close to the 2 limit which is close to the singular regions of the cross section. Boson Gluon Fusion (BGF) and QCD Compton (QCDC) processes have a singularity at ; = O or : = 1. These two singularities are removed by applying O. 1 < z c 0.9 cut which is drawn at z = 0.1 and : = 0.9 as a solid line. The points above .r,, b 0.9 are removed by y,,, as we see in the figure. The points are uncorrected data in the kinernatic range 120 < Q' < 3600 G ~ v ~ , 0.0 1 < x < 0.1. and O. 1 c y < 0.95.

ZEUS 1994

Figlue 9.7: Scntter plot of -Y verslts z iipitlr the »iodijïed JADE (mJADE) jet findirlg algoritli~ri czr [/le delecfor leriel. The solirj>dots are the lnrger angle jets and rlie open squares ( [? ) are r k sinnller angle jets.

The uncorrected r distribution is shown in Fig. 9.8a. Since the two jets satisfy the con- straint :, = 1 - i l , the curve is symetric about : = 0.5. At the detector level. the distribution is reasonably well described by the MEPS model. but a discrepancy exists for z c O. 1 which corresponds to jets close to the forward beam pipe. The measured xp distribution for the two jet events is also shown in Fig. 9.8b at the detector level. The -5 distribution is well described by the MEPS model. Note that the upper limit of xp is sufficiently far from the .Y,, singularity of the QCDC pro-

7 cess in our considered (r. Q-) region. Here R,+ = N, + , / N , , , , where j stands for O, 1. 2. or 3.

N;+ I is the number of (i+I) jet events and Nt,, is the total number of selected DIS events.

1- !=

a 0.35 1 1 X

0.9 (a > u 1 1 > 0.3 F- (b)

0.8 0 ZEUS 1994 1 F t

0.7 - MCMEPS ' I 0.25 1 - r

r 1 0.6 - t

C r-&4+ 8 I

+- , I I 0.2 P

l

Figure 9.8: Th5 distributions of pclrtori wrrinbles : and .yp nt the drtector lrvel in the kiileiircitic mnge 120 < Q- < 3600 ~r v'. 0.0 1 < x < 0.1, nud O. I < y < 0.95. Uiicorrected dtitn is contptr,rd to the MEPS sirndation. (a ) Uncorrectrd 2 distribution of the jets. The dcztn shoiv n large excess ofcvents rvhe~ coinpnred rvitlz the Monte Cnrlo in the Iorver : region corresponding to the singr<- lcrity (h ) Uncorrected .yp distribution of two-jet events rvith O. 1 c r < 0.9 cut. The vczl~ïe of xr, is crboiit 0.5 $ Q ~ - M ~ .

The corrected : and x,, distributions are compared to the NLO QCD calculations in Fig. 9.9. These distributions of parton variables, z and xp, were corrected for detector and hadronization effects with the MEPS Monte Carlo. The MRSD/- parton density parameterization was used in the calculations of NLO programs. The shape of the corrected ; -rp distribution and the NLO calculations is in good agreement. The corrected r distribution is well-described by the calculations for : > 0.1. However, an excrss is observed in data relative to the NLO QCD calculations in the lowest : bin. This is the region with the largest systematic uncenaincies.

?- i-

[+-- (0) i I

l

L I Data ! - - NLO PROJET :

0.8 -

r t

(b) ; - l- Data

Figtlre 9.9: ( a ) Distribution of the corrected pczrtori variable. c, in (2+ 1) jet everlts for IZO < Q' < 3600 G~v ' , 0.01 < x < 0.1. cind 0.1 < y < 0.95, coinpnred tu the NLO QCD ccilcdcrtiorrs (PROJET and DISJET). (b) Distribirtion of the corrected iarinble. -rp, for (2+ 1) jet riwtts tlmt pnss t h ; cctt. 0.1 < r < 0.9. Only stntistical errors are s h o w in al1 the figures. Al1 t h datez poi~trs c m corrected to the pcrrton levrl.

In order to see how well the parton variable r is reconstructed. the resolution plot for : is shown in Fig. 9.10. One clearly sees the correlation between detector and hadron level using MEPS Monte Car10 model. The mean deviations are distributed around zero, which means that there is no significant shift over the entire value of t. Figure 9.1 1 shows the relative error.

- rec [ m e ) / -yp, rrsc and resolution of .yp reconstruction.

o 1

N 0.5 L !- MEPS r C

0.4 i l- i

0.3 C ' . i .

f 0.1 E- Mean Oeviotion N E

(b) -

Figure 9.1 0: Plots for meari clrvintion and Root Mean Sqrtare (RMS) valrtes of partori itnri&le :. MEPS mode1 is ilseci for reconstritction nt hndron leivl. ( a ) Scatter plot of z for the detrctor l e ~ v l vs liadrott leitel. (b ) Relative rrror of : variable. The resolution is drrzoted by the error bar: ( c ) RMS valrte of pcrrtorz variable t. The hori,-ontd bars indicclte the birz size.

L (b) + -

Figure 9.11: ( a ) Plot of the nircrn devinrior1 of the niecrsitred and genernted valrte for .Y,, wu-iczblr. (b) Rrsolittion of xp.

9.3 Properties of (2+ 1) Jet Events

The properties of two jet events are studied in this section for the high II. Q') kinernatic range, 120 < Q' < 3600 G~v' , 0.01 c .r < 0.1. The jet properties were studied at a fixed y,,,, of 0.02. This value was chosen as a compromise between the increase of the higher order corrections at lower yCe, values and the loss of statistics at higher y,,, values for two-jet events. Figure 9-12 shows a typical (2+ I ) jet event at HERA. The well defined energy deposition of electron and two jets c m be clearly seen.

Figure 9.13 shows the effect of increasing value of : and y cuts. As mentioned earlier, the fonvard singularity in the event around FCAL beam pipe can not be regulated by a y,,,, alone in the JADE algorithm. Motivated by this the effect of removing forward jets by means of different 2 and y cut have been studied. Without a O. 1 c : c 0.9 cut. the rate of forward jets peaks very close to the beam pipe (q 2 2.6). This is shown in Fig. 9.13a. If a O. 1 c : < 0.9 cut with y > 0.04 is applied, a large proportion (- 33%) of forward going jets are removed. Furthermore, if the y

variable cut is increased to O. 1. 16% more fonvard going jets are removed when both z and y cuts are applied. The reconstruction of fonvard going jets is maximized in this case as shown in Fig. 9.13b.

€TA PHI UCAL transverse energy ZR

Figure 9.12: One of (2+l) jet everzts in ZEUS. Al1 electrorl ivas found in RCAL cozd trnmivrse ertergy distribution of evertt rvitli jets is shorvn in the (q, $) plcine.

This 0.1 < : < 0.9 cut removes jets with transverse rnomentum PT below about 4 GeV. Here PT is meüsured with respect to the y* direction and is calculated in the y* - parton system. The effect of the cut is shown in Fig. 9.14a and Fig. 9.15a. The average value of <PT> is about 1 1 GeV. and this is sufficientiy large to ensure the validity of a perturbative QCD calculation. The invariant mass distribution of two jets is shown in Fig. 9.14b and Fig. 9.15b. Data are compared with MEPS Monte Car10 and NLO QCD calculations, respectively. The average invariant m a s of two jets system is about 23 GeV, and the invariant mass distribution is well described by .MEPS

- c + V F R L ( 0 )

1 0.01 < x < 0.1

k 120 < Q' < 3600 ~ e v F

1 E t r L

r - - f L _

Figiire 9.13: Eflect of increusing : and y - ciit to (2+1) jet reconstritction. The jets are orderrd in '1 mid only fonvard going jets (= most energetic jets) ore drnrvn in ~ltefgi<rr. ( a ) T$e penk of jets is reconstnicted inside VFR (Very Fonvard Region) ithicli rneritzs q > 2.66 ( O < 8 ). Histogt-ctttts of higher q jets are drarvn as a co~ztinuous line rvitlt/rvitltoiit 0. I < : < 0.9 ciir. Tite penks of (2+1) jet evenrs are rnoviiig oiitside VFR as ive appZy z ciit. (b) y > 0. I condition is appiied rvitii z ciit.

- MC MEPS

O -

10 20 30 40 50 O 20 40 60 80 100

PT (GeV MA &eV)

Figure: 9.11: The distributions of (a ) P crnd (b) MJJ afer a ,- cut nt the detecror 1eveZ in the T kiminatic region, 120 < Q' < 3600 G e r . 0.01 < x < 0. I . and 0. I < y < 0.95. Uncorrectrd data c m cornpared to the MEPS niodel and are rvell described by MEPS titodel.

NLO PROJET

NLO DISJET

- NLO PROJET ; - NLO DISJET 1

Figrire 9.15: The distributions of PT and MJJ after n r cut in the kinetnntic range 120 < Q': < 3600 G ~ v ~ . 0.01 < x < 0. 1. and 0.1 < y < 0.95. The data points are corrected to the partmi lrwl and conzpnred to the NLO QCD progratns. (a ) The jet trnnsverse momentcm PT in the y* - pczrroii certter-of-mm system and (b) Two-jet invariant masses arc sliorvn.

4 k I 5 i ' 3-5 E Lower 7 jet 1 H 4.5 I

l= (a) l c +. Higher 77 jet

I (b)

3 4 E MEPS

Fiqlre 9.16: Smtter plots of pnrton versrts hadron pseuclorcrpidity for (2+ 1 ) jets. ( a ) Correktiori for lobver q jets. (6) Correkltion for higher ri jets. MEPS mode1 rvas rrsed for ee~wucting pïlrtoji w d Iicrdron l e i d variables.

Figure 9.16 shows the correlation between the pseudorapidity, i l j , , , of two jets at the parton and hadron level. The two plots correspond to the higher and lower pseudorapidity aficr the jets have been ordered in this variabIe.

In Fig. 9.17 the distribution in pseudorapidity, qj,,, of the two jets is shown. The predic- rion of the MEPS mode1 describes the data fairly well over the whole region 120 < Q' < 3600 G~v'. The higher q jet is usually found very close to the forward beam pipe. The jets are ordered in pseudorapidity q and the peak rate for the higher rl jet occurs at q = 2.3. This value is away from the very fonvard beam pipe regions (q = 2.66 ) of the ZEUS detector. Most of lower q jets, which are usually well separatrd from the beam pipe region, are found at positive pseudorapidi- ties in the BCAL or in the FCAL.

$ C 0.3 F - ,

\ i 9 ZEUS 1994 +, ' + V F R (O) f 0.25 FI-

c - MC MEPS O i Higher 7 jet

0.2

1. 0.3 c i

t TI t RCAL + t BCAL + t FCAL + > 0.25

L

II" E Lower 7 jet -0 0.2 t - -+-

Figwe 9.17: Pseudo-rapidity distribution of the two jets in the kinernntic range 120 < Q? < 3600 Ge V'. 0.01 < x < 0.1, a~ td 0.1 < y < 0.95. (a ) The higher jet (= most energetic jet) and (6 ) !lie l o w r q jet (= least energetic jet) froni (2+1) jet production. Uncorrected data are coinparrd to MEPS Monte Car20 nlodel. The Jilled circles are ZEUS dcrttr and solid line is Monte Crrrlo. The holindaries of the differenr caloritneter parts are indicrited in the bottuin figrire.

Figure 9.18 shows how the peak rate for the higher q jet moves away from the very ior- ward region as Q' is increased in our kinematic region ( 120 c a2 < 3600 G~v'). The jet structure aIso becomes more pronounced. The figures show that the lower jet is outside the VFR. These rvents are sub-sarnple of the events shown in Fig. 9.17. Al1 distributions are drawn after :-cut applied to the sample. This increase in the separation of the jets from the very forward region with increasing Q' is why high Q' events were chosen in the initial event selection. The high Q' value also ensures that the jets are well defined, as illustrated by the event shown in Fig. 9.12.

-4 -2 O 2 4

Lower 77*

0.5 7 - g = Cd) 5 0.4 / V F R '

& L 120 <0'<240 = 0.3 - - - 0.2

- - - 0.1

= <CI 0.4

r 240<@<720

Figure 9.18: Pseudo-ropiditv distriburion o the trvo jets in tliree kitzernntic birrs (120 < d c 240 I G ~ v ~ . 240 < e2 c 720 G~V', and 720 c Q < 3600 G~v') (a). (b). and (c) is the loiver q jet distribution as incrensing Q-. (& (e). and (B is the distribution of the Iiigher q jet.

9.4 Jet Production Rates and Correction Procedure

The rneasured jet production rates are affected by many effects such as Initial State photon Radiation (ISR). event selection cuts, the detector acceptance. hadronization effects. Due to measurernent errors. events are no& observed at their true values of (.Y, Q". The nurnber of events in each bin in x and Q' must be corrected for the effective migration of events into and out of the bin. Furthemore the theoreticai calculations for the jet rates are available only at the parton level. So. the measured jet ntes must be corrected to the parton level so that a valid cornparison may be made with the NLO QCD calculations. These corrections are extremely important in order to make a valid measurement of us. These corrections were made by Monte Carlo methods. The MEPS Monte Carlo simulation, with hadronization described by the Lund string fragmentation model 1751, was used for these corrections. The jet production rates at the parton level. hadron level and the detector level were made using the JADE algorithm and data selection criteria described in Chapter 8.

9.4.1 Uncorrected Data

Table 9.1 shows measured jet rates R, + , = N, , , i N , , , where j = 0.1,2.3. at each -Y,,,, for each kinematic range. Thrse jet production rates are before any corrections.

Figures 9.19 show the measured jet production rates compared to the MEPS model calculation. The measured rates are shown as function of the JADE algorithm jet resolution parameter y,,,, in each kinernatic region ( 120 c Q? < 240 G~v' , 240 c Q' c 720 G~v ' , 720 < Q' < 3600 G~v', and 120 < Q' < 3600 G~v ' ) at the detector level. The relative jet production rates are nor- malized to the total number of selected events. The statistical uncertainties were calculated using the binomial method; they correspond to the width of the binomial distribution having a mean equal to the observed nurnber of events. The (2+1) jet rate increases with finer jet resolution. smaller y,,,, (from 0.06 to 0.01). The chosen value of y,,,, was restricted to 0.01 - 0.06 because the jets are not experimentally resolvable at very small y,,,. and higher order corrections become important, while at larger values of ycer the terms proportional to y,,, neglected in the calcuiation become significant [52 ] . Moreover, uncertainties in the renormalization scale also becorne larger for y,,,, greater than about 0.06 [53]. Jet rates below y,,, = 0.01 are not shown in the figure. For y,,, = 0.02 about 1 8 of al! events have (3+ 1 ) reconstructed jets. Above this y,,,,, the jet rates of R , + , become statistically too small.

0.010

0.030

II

III

IV

1

III

IV

1

III

IV

I

II

0.050

Table 9.1: Mensurrd iincorrected jet production rares. R, + l . for vnrioits vahes of t,,,, from 0.01 to 0.06. 1. II. III. and IV ore the kinemntic ranges. I2O < Q < 240 G ~ v ~ . 140 < Q- < 720 G~v' . 720 < Q\ 3600 0ev2 . and 120 < Q? < 3600 G ~ v ~ . respecrive. Errors slioivn c m stntisticcd ody. (-) indicntes thnt ihe jet rntes cari not be mensilred at thnt y,,,, dtie to [imited stntistics.

82.9 & 0.8

83.6 t 1.3

83.4 I 0.6

88.8 + 0.8

93.3 I 0.9

94.0 f 0.4

95.9 f 0.5

0.060

89.8 + 1.1

90.8 k 0.4

94.5 I 0.6

93.9 + 0.5

III

IV

1

II

13.9 + 0.8

12.8 + 1.2

12.9 + 0.5

9.8 $ 0.7

6.7 + 0.9

5.9 + 0.4

3.9 -t 0.5

III

IV

1

II

III

IV

3.2 + 0.4

3.6 i 0.7

3.7 + 0.3

1.5 I 0.3

9.2 + 1.0

8.5 + 0.4

5.3 + 0.6

6.1 + 0.5

O. 1 i 0.1

0.1 + 0.1

O. 1 + O. 1

95.4 -t 0.7

95.7 -t 0.3

97.2 + 0.4

96.7 f 0.4

1 .O + 0.4

0.7 k 0.1

0.1 1 0.1

0.1 + 0.1

96.0 f 0.7

96.8 + 0.3

97.9 + 0.4

97.8 I 0.3

97.0 k 0.6

97.7 + 0.2

4.7-+ 0.8

4.3 f 0.3

2.9 + 0.4

3.3 + 0.4

-

0.3 -t O. 1

-

- 4.0 + 0.7

3.2 i 0.3

2.1 k 0.4

2.3 + 0.3

2.4 -t 0.6

2.3 + 0.2

- -

-

-

-

-

o 94 ZEUS (Uncorrected)

- MEPS

120<d<240 GeV

R I O Z ; Y

CE-

&O' P * - w 0 l + l jet

K- u 1,- -

n - - P n v a 1+1 jet

O 94 ZEUS (Uncorrected)

- MEPS

720<4'<3600 G~V'

2+1 jet

1 + 1 jet

$ O 2 2 O 94 f EUS (~ncorrected)

- MEPS

240<d<720 ~ e p

s10 - 2 - u

Y 1 + 1 jet

ce- : O 94 ZEUS (Uncorrected)

- MEPS

120<Q'<3600 G ~ V '

Figure 9.19: The dependence of the uncorrected jet rates on the jet findi~rg czlgoritltin resolirtiotr porcirneter yî l l l for eacli kinernntic region. (a ) I2O < Q' c 240 G ~ V ~ (6) 240 < Q' < 720 G ~ V ~ ( c ) 720 < Q- < 3600 G~V' (dl 120 < Q' < 3600 G d 2 . The resrtlt of MEPS Monte Ccrrlo pro- ,qnrrn is cilso drarvn as n friricrion of y,ll,. Errors slioivn are stcztistical only

9.4.2 Correction for Detector Effects

MEPS Monte Carlo was used to correct for detector effects, and also for ISR effects. This was done by comparing the jet rates with a perfect detector (hadron level) and no ISR effects to the jet rates with ISR obsenled in the ZEUS detector. The correction factor was defined at each

?"-i([ as

reconsrr~~cred where R,+ , is the reconstructed jet rate for Monte Carlo at the detector level and R?:; is jet rate in Monte Carlo events with no detector simulation and no initial photon radiation at hadron level. The following bin-by-bin correction of the jet production rates for each y,,, was applied.

where N denotes the number of ( j + l ) jet events and j is a jet number. First the nurnber olevents was corrected and then jet rates were corrected by

whcre i = 0.1 ,?J. The resulting jet multiplicities satisfy the unitary condition. Le. R, + , = I . i

9.4.3 Correction for Hadronization Effects

The data were further corrected to the parton level in the same way as above. by using the Lund string fragmentation mode1 [75]. The only difference between the hadron and parton events is hadronization effects. The correction factors for the hadronization are defined at each y,,,, as

1 Zrble 9.2: The correction factors in kinenzatic region. 120 < Q- < 240 G ~ v ~ . 6' refrrs to the correction jirctors for derector effécts mid i? refers to the correction factors for Iinùroni:arion. (-)

indicates correction firctors are too Irigh.

Tuble 9.3: The correction firctors 61. t? in the kiriematic region. 240 < Q? < 720 G ~ v ~ . 6 refcrs to the correctio~z firciors for detector effects and ? refcrs to fhe correcrion factors for hadronicci- tion. (-) indicates correction factors are too higiz.

fibk 9.1: The correction /nctors C<'. d' in the kiiiematic regiort. 720 < Q? < 3600 G~v'. r&rs to the correction fircrors for detector eflects cznd t'l refers to the correction fcrcrors for Ircid- ro~zizntion. (-) iridiccitrs correctiorl factors are too high.

Table 9.5: The correction factors 61. C" in the kimwrcztic regiort. 120 < Q' < 3600 G ~ V ? 6 refei-s to the correctio,~ fc~tors for detector effects and d' refers ro the correction fizctors for Irnd- rorz izcztion. (-) itzdicates correctiotz factors are too /zigh.

1+1 jet

(b) 240<Q'<720 GeV

(solid) Detector+lSR

LEmo 6.1 (ME=) (dash) Hadron

LEPTO 6.1 (MEPS) (dosh) Hadron

-102 - - -IoZ:

-10' E l + t jet

w- (d) 1 20<d<3600 CeV

(solid) Detcctor+iSR

Lm0 6.1 (MEPS) (dash) Hadron

!s if-

Figrire 9.20: Tlte jet prodr<ction rates of L E P 7 0 6.1 (MEPS) as ci function of vc,,, nt t h e ùifferent levels (detector:hadron:parton level) in each kinematic re ion, ( a ) 120 < Q ~ - < 240 G~V' (6 ) 240 5 < Q~ < 720 G~V' ( c ) 720 c e2 < 3600 G ~ V ~ Id) I2O < Q < 3600 c e v 2 . respecfiveiy

+ 1 +l jet

(O) 120<42<240 G ~ V '

(solid) ûetector+6R

LEITO 6.1 (MEPS) (dosh) Hadron

(d O t) Parton

L a-

-

10

3+ 1 jet

2+1 jet

1 1

O 0.02 0.04 0.06 O

r

-

L 8

0.02 0.04 0.06

where j is a jet number. The jet clustering algorithm applied at the parton level is exactly same as at the hadron level on an event by event basis. Therefore, the difference is interpreted as non-perturbative hadronization effects. A11 correction factors for <+ , and c;+ are tabulated in Table 9.2 - 9.5 for the each kinernatic region ( 120 < Q? < 140 G~v'. 240 < Q' < 720 G~v' , 720 c Q~ < 3 6 0 0 ~ e ~ ' . 120< ~ ' < 3 6 0 0 ~ e ~ ' ) .

In Fig. 9.20. the jet production rates of LEPTO 6.1 (MEPS). RE,. R::: ,, and R Y : ~ ~ . are shown as a function of y,,,, at the detector, hadron, and parton levels for each Q- region. From these plots. the y,,,, dependent correction factors for the detector effects were estimated.

9.4.4 The Conected Jet Production Rates

The fully corrected data were then obtained in the same way as before at each y,,,, with the corrected number of events

where 6> is correction factor for hadronization effects and 6 is the correction factor for detector effects. For this analysis. we rejected bins with large correction factors. The jet rates corrected by these factors are s h o w in Table 9.6. These rates. which are corrected to the parton level. were used for measuring strong coupling constant. ar.

Figure 9.21 shows the corrected jet rates as a function of y,,,, for data. compared to the DISJET and PROJET NLO QCD calculations for each kinematic region. Only statistical errors are shown. The comected jet rate and the NLO QCD calculations agree well over most of the range in y,,,, and particularly at y,,,, = 0.02, where the strong coupling constant, a.$, was extracted in this analysis. At y,,,, = 0.02 the ( I + l ) and (2+l) jet classes dominate, with R2 + - 7.8 to 9.0 in eüch kinematic region. Both NLO programs agree well in their production of jet rates over al1 y,,,, values.

III

IV

III

IV

86.3 + 1.2

88.1 t 0.5

1 I 1

III 97.5 2 0.6 2.5 I 0.7 I -

90.0k 1.1

90.6 f 0.5

III

I I 1

I III I

Eible 9.6: The jet prodiictim rates corrected to the parton level in each kinernatic regioir. Errors shoivtz are stntistical on [y. (-) indiccites correctioiz factors thnt are ntlncceptnbly higk for a valid measctrement of as.

9.3 f 1.2

10.4 k 0.5

96.1 k 0.7

3.3 4 0.6

1.5 f: 0.2

8.6 I 1.1

8.6 2 0.5

94.5 +: 0.8

3.9 t 0.8

1.7 + 0.5

0.8 ': 0.1

4.6 + 0.8 1 -

(O) . Data - - NLO DISJm . - - - NLO PROJET

t i 20<d<î40 GeV

(b) . oata - - NLO DISJET

NLO PROJET

240<4'<720 GeV' 9 .-:.y*-

-- a... ... - -- .* *<.&

S. - -. . _.. * .: .; , . > . - .. 0

-*.:.< .;. . . - -.. 5

a. .--...a-

' .: - L Ir.

&-, . .. - - - * -.+. _. . . , - :$ ...

* _-.. . L' 3+1 jet

:e î+lijet

3tl jet

1 [ . I

2+1 jet

(c) 0 Data - - NLO OISJET

NLO PROJET

%oz Y

ar'

(d) a Data - - NLO DISJET

NLO PROJET

r a--*+- a - - - - - a - - - - - * - - - - * - - - - 1 +l jet

3+1 jet 3+1 jet

1

2+1 jet 2+1 jet

Figure 9.2 1: Jet prodrcction rates R as n function of the jet resoliitio~l parameter y,,,, for t h J + $ four Q' biits ( a ) I2O < Q' < 240 GeV-. (6) 240 < Q' < 720 G ~ v ~ . ( c ) 720 < Q' < 3600 ~e v2.

cind (d) I 2 O < ~ ? 3 6 0 0 0eV2. Only stnfisticnl errors are sho~vn. T,vo NLO QCD cczlcttlitic~is, DISJET and PROJET, each with the value of A- obrczined froni the fit nt y,,, = 0.02. are d s o slzo\vrz. MS

10. Measurement of the Strong Coupling Constant a S

in this chapter the measurement of the snong coupling constant will be reviewed by comparing the corrected jet rates with the NLO QCD caiculati~ns. Only the statistical error is considered in this chapter. The fuli systematic uncertainty will be discussed in Chapter 1 1.

in determining as from the QCD predictions, it is the rate of muiti-jet events that rnatten, not the number of jets in each event. As shown in Fig. 9.2 1, dl points shown are highly correlated to each other since each of the points at each y,,, is detennined using the whole data set. i.e, at y,,, = 0.02 one point contributes to R, + , , R2 + and Rj + at the same time. The conventional way to remove this correlation is to define the differential jet rate to which an event will contribute only once.

10.1 The cr ( Q> Measurement from Jet Rates S

In Table 10.1 the corrected jet rates R2 + are summarized in three kinernatic regions and full region.

Table 10.1: (2+1) jet production rates, R2 + I . corrected to the parton level in the kinematic range 120 5 e2 < 240 G ~ v ~ , 240 c Q' < 720 G~v', 720 < e2 c 3600 c e v 2 and 120 < Q~ < 3600 Ge vL. Errors shown are statistical only.

The value of as was determined by varying fi) in the QCD calculation until the best fit MS

to the ratio R2 + was obtained at y,,, = 0.02. The slope of the measured R2 + is not steeply fall- ing and agrees with that from the NLO QCD calculations, so the result is not sensitive to the par-

ticular value of y,,, used. We chose y,,, = 0.02 for the fit since the jets are resolvable at this value with a large invariant mass between them and statistics are large. The contribution frorn Rj + , , which is a higher order effect, becomes negligible for value of y,,, h 0.02.

The results of best fit in three kinematic regions and whole kinematic region are shown in Table 10.2 and Table 10.3. respectively.

Table 10.2: The values of <Q>. R A!?) . and a ( Q ) in each three bin (120 < e2 c 240 2 + 4 P ~e v2, 240 < Q? < 720 G ~ v ~ , 720 < Q < !%O Ge V ). n2e errors are only statistical errors.

+ 64 205 -53

g (QI + 0.008 0.148 -a,,

<Q> (GeV)

Table 10.3: The values of c Q > , R2 + l , A!?), and as ( Q ) in the combined region (120 < Q' MS 3600 ~e v2). The errors are only statistical errors.

22.1

10.2 Running of a (Q) at HERA S

The most distinct feature of QCD is its non-abelian nature of SU(3) gauge group. This introduces the decrease of as with Q and the existence of the triple gluon vertex. These characteristics have been extensively tested from the study of multi-jet production in e+e- experiments [31]. Recently E665 experirnent has shown evidence of the running of the strong coupling constant in a single experiment 181. The average squared transverse energy of jets in deep inelastic - muon-nucleon scattenng is measured as a function of the momentum transfer Q', in the range 3 < @ < 25 G ~ v ~ . Perturbative QCD predicts that the squared transverse energy averaged over al1

events is directly proportional to the strong coupling constant as. The strong coupling constant is rneasured by the JADE jet finding algorithm at y,,, = 0.04 as a function of is shown in Fig. 10.1. The data decrease from 0.33 to 0.22 over the range 3 to 25 G~v*. E665 experiment leads to the conclusion that the e2 dependence of the data is consistent with the running of the strong coupling constant expected from the perturbative QCD.

E665 Jet Anolysis

a €665 Data

a qqstotes

! O p**v @ e+e-(0-1

1. B eee'(event shapes)

@ e * e'e'(even t shapes)

Figure 10.1: (a) as vs Q' with y,,, = 0.04 in E665 experiment [a]. nie points are the data und the error bars are statistical only. The dashed curve shows perturbative QCD prediction for the data. (b) The data points are compared with as from other experiments.

In our anaiysis, for the ZEUS experiment, the running of a, cm be observed over a large range of e2. The values are shown in Fig. 10.2 and plotted as a function of Q for the three Q' regions. The values of as were calculated from the fitted value of A!?) . These values are listed in Table 10.2. The dashed curves drawn in Fig. 10.2 correspond to A!? = 100, 200, and 300 MeV.

MS There is a clear decrease in as for increasing scaie Q, consistent with the running of the strong coupling constant.

The value of A presented is A-!?' . The nurnber of quark flavors,_N- enters into the formu- MS

lae as through the formula P as shown in Eq. (2.6) and Eq. (2.7). In MS scheme, it is conventional to take N' to be the number of quarks with mass which is less than the renormalization scale p. If the value of = Q' is used, the nurnber of effective Bavon is 5 in our analysis. Therefore, we have used 5 flavors in the calculation of h because our lower bound of the e2 range is

MS above the b-quark mass threshold.

a ZEUS (Jet Rates)

2 Figure 10.2: Measured values of as (Q) for three diferent Q regions. The statistical errors only are show as thin vertical bar: The solid dash lines represent as with A?.) = 100,200,300 MeV

MS respective ly.

11. Systematic Effects on the Determination of a S

In order to investigate the sensitivity of the results to features of the main analysis, various checks were performed. We discuss a variety of sources of systematic effects resulting in uncertainties in the as determination. which can be grouped in one of the following: experimental errors, hadronization correction. parton density and renornializarion and factorization scale. We

checked the systematic uncertainties separately in the three kinematic regions and then finally in the whole region, Le. 120 < Q' < 240 G~v', 240 < Q' < 720 G~v', 720 < Q\ 3360 G ~ V ~ and 120 < Q' < 3600 G~v' . The central A!? - values in each kinernaticd region are 362 (25 1 )

.ws ( A , 5 ) MeV. 3 17 (2 17) MeV. 137 (86) MeV, and 30< (208) MeV. In the following subsections the nurn-

( 5 ) bers in the parentheses refers to the changes of the A- . M S

1 1.1 Experiment Uncertainties

The experimental systematic errors are subgrouped. by collecting those that are heavily correlated. into: (a) event selection, (b) rnergy scale. (c) jet analysis. (d) fitting method. and (r) model dependence.

The systematic errors from event selection (a) include: different electron finding algorithm; variations in the selection criteria, E - P. - > 45 GeV; y, < 0.7. The errors from energy scalr ( b ) includes: a 5% error assigned to the uncertainty of the calorimeter energy response. The errors from jet analysis (c) include: the choice of the rnass scale, W' = s ( 1 - x,,) yDA and

D!? 9

WjB = .Y ( 1 - x , ~ ) - , ~ , in the JADE scaled rnass definition. ji, = in;/W-: using preclustered cells as input for jet clustering. The errors from the fitting method (d) include: QCD fit at y,,t, = 0.03 instead of 0.02; QCD fit by using differential jet rates; a more restrictive : cut. 0.15 < r < 0.85. Finally the error from model dependence of corrections for detector acceptance and resolution (e) was estimated by using the color dipole mode1 [72] as implemented in the ARIADNE 4.06 Monte Car10 [74]. All rrror sources discussed above are shown in each kinematic region. where a fit value for each type of analysis was compared to the central value. Finally. we took separately the largest positive and negative deviation in each subgroup. and these experimental sources of uncertainties were üdded in quadrature as the experimental systematic error.

1 1.1.1 Event Selection

For neutral current DIS rvents the scattered electron is found by an electron finder in the ZEUS detector. Several electron finders are widely used in ZEUS for this purpose and they have different efficiency and purity for selecting electrons. The SINISTRA electron finder was used in Our main analysis. We chose the EXOTIC electron finder for a consistency check of the final value of A. The difference in the final fit value of d4) (d5)) is found to be +20 (+20) MeV. +23 (+2 1 ) MeV, -32 (-2 1 ) MeV. and +23 (+2 1 ) MeV in each kinematic region.

Although the current data sample is well selected by elirninating background events, e.g., photoproduction, bearn-gas, etc., residual uncertainties in the method of event selection may remain. These uncertainties have been investigated by varying the selection criteria to estimate the sensitivity of the jet production mes and finally to the result of us. The systematic uncertainties on the final jet production rates were found to be srnaller than the statistical errors. These relative rate measurements are rather insensitive to the details of the event selection unlike the case of absolute cross section measurements.

The event selection cuts were varied to see the effect on the final jet rates. E - P- > 45 GeV was applied and this causes -48(-33) MeV, - 17 (- 10) MeV, -36 (-24) MeV, and -25 (- 16) MeV differences of A'") (d5') in each bin. Furthemore, the y, cut was tightened to 0.7. This makes O (+4) MeV, + 13 (+ 13) MeV, +8 (+6) MeV, and + 15 (+15) MeV differences in the A fit.

1 1.1.2 Energy Scale

When we measure the energy in the detector, inactive material inside detector such as support structures may cause a particle to lose energy. This is called dead material. We assumed the uncertainty in the hadronic energy scale due to dead materid in the calorimeter as follows. It

affects energy measurements and influences the final result of A. Decreasing (or increasing) the energy of each ce11 energy by 5% error approximately corresponds to increasing (or decreasing) the dead material by one radiation length. This change is consistent with Our knowledge OF the uncertainty in the hadron energy scale from studying DIS jet balüncing the electron in transverse momentum [87]. The fit value of A"" (d5' ) for increasing energy differs by +79 (+77) MeV. +55 ( d 6 ) MeV, -7 (-4) MeV. and 4 9 ( 4 1 ) MeV. Decreasing energy causes -67 (-47) MeV. -67 (-47) MeV, +25 (+18) ,MeV, and -54 (-38) MeV differences in the fit.

1 1.1.3 Jet Analysis

We have chosen W for the reference m a s scale in the JADE jet definition because it 1 7

reduces detector effects in the main analysis. We used WDA in the formula yi, = mi/ Wb, for jet clustering to investigate the uncertainty in the experimental choice of scale. Furthermore we have used the w ; ~ scale which results in almost the same value as the main analysis. The use of w i A gives the final A'." (A'') ) fit value -47 (-32) MeV, -27 (-17) MeV, -22 (- 14) MeV, and - 15 (-9) MeV in each kinematic region.

The preclustered cells were used as an input for jet clustering for the systematic checks. This results in -7 (-1) MeV, -2 (-2) MeV, -52 (-35) MeV, and -25 (-16) MeV in the final result for

.

1 1.1.4 Fitting Method

The present measurement of A depends on fitting the NLO QCD calculations at y,. = (3) 0.02. If the fit is perfomed nt a different value of ycll, = 0.03. then the vdues of A"" ( A )

change by 4 1 (-28) MeV. +13 (+13) MeV. +34 (+25) MeV, and - 1 1 (-5) MeV in each kinernatic range respective1 y.

The erron shown in Fig. 9.21 are the statistical binomial errors. These are highly correlated, because al1 (2+l) jet events at a given y,,,, are included in the points at smaller y,,,,. These correlations are avoided by redefining the data in terms of the differential jet rate

where every event enters only once. In Fig. 11.1 the differential jet rate D, + , is shown with the statistical errors and is compared to the DISJET and PROJET calculations. The fit range is chosen between y,,,, = 0.02 and 0.06 where the detector acceptance and hadronization effect are not large. If the fit is perforrned on the differential jet rate D , + , . the rneasured values of A"' ( A"' )

Vary by - 12 ( - 5 ) MeV, +42 (+36) MeV. +8 (+6) MeV. and +39 (+33) MeV.

We also tightened the : cut to O. 15 c r < 0.85. This causes -8 (-2) MeV, -23 ( - 14) MeV, -7 (-4) MeV, and -34 (-23 j MeV differences in our andysis.

1 1.1.5 Mode1 Dependence

We have chosen the MEPS model for estimating the detector acceptance effect since it

reasonably produces the global propenies of the data. To study the sensitivity of the resolution and acceptance to models of the final state and fragmentation. we chose an alternative model. the CDM model, as implemented in the ARIADNE Monte Carlo, which also describes the data in a -lobal way. Figure 1 L .Z shows the detector effect. the ratio of the rate at the hadron level to the C

detector level. The correction factor depends on y,,,, and varies below 20%. It is observed that the acceptance correction is not sensitive to the models used as may be expected. The extracted values of A") (d5)) are found to Vary by +18 (+18) MeV. +69 (+57) MeV. -22 (-14) MeV. and +35 (+30) MeV in each region respectively.

Dota - - NLO DISJET

..... NLO PROJfT

9 Oata - - NLO DISJET

..... N t 0 PROJET

8 ' '. 720<@<3600 GeVf

O Data - - " NLO DISJET

4 ..... NLO PROJET

8 (d) I) Data

5 t - - NLO DISJET

4 t ..... NLO PROJET

Figure I L I : The nleasured diqrentinl jet rates D I + , are shoicn Ni the kinernntic rmye (0) 120 < Q- < 240 G~V' (b) 240 < Q- < 720 G~V' ( c ) 720 < Q' < 3600 G~V' (d) 120 c Q- < 3600 G ~ v ~ , respectively. The dntcz are fulh correctecl to the pcirton level. Srutisticcil errors only cire slioicw.

O? \ fi- e-

&

1.8 (a) 1 20<&<240 G&

1.6 MEPS

1.4

1 -2 I

0.8 5 0.6 F 0.4 k 0.2 F

! . ,

O0 I

0.02 0.04 0.06

5 1.8 (b) 240<Q'<720 G e 3 I

(d) 120<Q'<3600 GeVf

MEPS

A 1.2 [r I L - - -

Figlire 11.2: The correction fizctor for the cletrctor eflect estiïnnted /rom the MEPS i?iodel. The ratio of ( 2 + 1 ) jet rate nt rhe delector level ro the Izadron leivl is drnrvn as njtnction of y,,, iri rhe kirirï~ratic rmgr ( (1 ) 120 < e2 < 240 G~V' (b) 240 < Q' < 720 G ~ V ~ (c) 720 < a2 < 3600 G~V' (d) 120 c Q' c 3600 G~v- ' .

I I I

Systematic uncertainties from experimental checks in each kinematic range are tabulated in Table 1 1.1. Most of these changes result in relatively srnall systematic changes in our measurement.

I Central analysis

Electron finder F-

1 Energy scale (-5%)

Method of W cdculation

Preclustering of cells

y,,,, dependence

Differential jet rate met hod

0.15 < z < 0.85

Detector correction (ARIADENE)

7iildr 11.1: Stmi?iczn of experin~eiztd systmntic checks iiz ench ~ i e r ~ i n t i c Diil. Al1 r?al~rrs cire rsprrssed a s A!? (MeV) and the ntonbers in parenthesis refer ro A- (MeV). Energy scczlr crm-

rtf S iiig j k m cdorirneter respoiise is Jmririnnt nnrong experintentnl wfih:nmtic rrrors.

1 1.2 Hadronization Correction

The jet rates have been corrected for the effect of the hadronization since the NLO theoretical calculations are available only at the parton level. Only phenomenological models are available to describe the fragmentation of quark and gluons into final hadrons. In order to evaluate the uncertainty of the hadronization corrections. several aspects of the hadronization scheme were varied.

To evaluate the hadronization correction, the HERWIG model was used in addition to LEPTO 6.1 (MEPS) providing the central value of the analysis. The hadronization scheme used in the HERWIG model is the cluster fragmentation and it is entirely different from the Lund string fragmentation applied in the LEPTO Monte Carlo. We retained the standard detector corrections based on the full simulation of the ZEUS detector using MEPS. These studies were performed at the hadron level.

We checked the effect of hadronization by comparing the ratio of the (2+1) jet fraction measured with a perfect detector (hadron level) to the (2+1) jet fraction at the parton level as a function of y,,,. Ideally, the above ratio should have a good correlation. The hadronization corrections obtained from the two different models Vary from 1 to 20% depending on y,,,, as shown in Fig. 1 1.3.

(O) 120<@<240 Ge\P

MEPS (String)

f - 1.4 HERWIG (Cluster)

(d) 1 20<9'<3600 GeV - - - MEPS (String)

HERWIC (Cluster)

-- - 0.2 c O0 0.02 0.04 0.06

Figitre 11.3: The hndronicotiort effeect on ( 2 + l ) jet rcite for the clnuter frngineritatio~z tnodel m d the string frngmenfafion rnodel. The ratio of (2+ 1 ) jet rate ar the hadron level to rhe pcirtoti Ievel is drnrvn a s n fi~ncfion of y,,, in the kinernnric range ( a ) 120 < & c 240 G ~ V ~ (b) 240 < e2 c 720 G ~ V ~ (c ) 720 < e2 < 3600 G ~ V ~ (d) 120 c @ c 3600 G ~ v ~ . These are the snme valires os i ~ i Table 9.2 - 9.5.

Since iMonte Carlo models used for the hadronization corrections have a large number of free parameters and cutoffs, tuned to fit the experimental data, this can Iead to Vary the theoretical inputs to our models. Therefore the estimates of the uncertainties due to the hadronization are made by varying independently one of the pararneter sets in the Lund string fragmentation model. We considered a few of those which play a dominant role for the shape of the hadronic final state.

* The pararneter "d' in the LUND string model is a syrnrnetric fragmentation function. This regulates the longitudinal quark fragmentation and was varied between 0.1 and 1 . A fragmentation process is described in terms of stnng at the (1 end of the system and fragmenting towards the i . This results in changes of A'.') (d5') of -2 (+3) MeV, -6 ( -1 ) MeV, +1 (+39) MeV and -6 (-2) MeV when "a" is 0.1. If "a" is 1, then this results in +34 (+3 1 ) MeV, +15 (+15) MeV. +53 (+39) MeV. and +26 (+23) MeV respectively.

* The pararneter oq for the width of the Gaussian distribution of p, and p, of the transverse rnomenturn of hadrons, was vmied between 0.25 and 0.45 GeV. This chang& d4) (A'') ) by 4 3 ( 4 2 ) MeV, +5 (+6) MeV, +13 (+IO) MeV, and +18 (+17) MeV in the final result when

oq is 0.25. If oq is 0.45. then the change is + 18 (+ 18) MeV, -5 (- 1) MeV. + 1 1 (+8) MeV. and +3 (+5) MeV.

* The parameter hi, in the LEPTO MEPS model was varied. which regulates the divergence in the Ieading order Matrix Element. It must be kept as small as possible to keep the available phase space of (2+ 1 ) jet as large as possible. The value of y,ni, was varied from 0.005 to 0.015. This results in changes of A") (A'' ' ) by +68 (+58) MeV, +23 (+2 1 ) MeV, +28 (+?O) iMeV, and +37 (+32) MeV.

* The QCD shower rnodels contain a parameter, Q,. which defines the cutoff value on the virtuality of the parton where the QCD shower process is halted and the hadronization process begins. Changing the value of Q, changes the final plinon multiplicity in the shower before hadronization starts; thus the hadronization corrections changes as well. We have repeated Our analysis for different value of Q, in the range of 0.8 to 4 GeV, and take the resulting deviation as the error due to parton virtuality. This results in changes of A'') (A( ' ) ) of -82 (-59) MeV, -47 (-33) MeV, -7 (4) MeV, and -55 (-39) MeV.

* An additional transverse momentum can be given to the hard interacting parton. This momentum is called the intrinsic k,, where k, is defined with respect to the proton direction. The intrinsic transverse momentum k , of the struck parton in the proton has been varied from 0.44 to 0.7 GeV. This results in changes of A'" (A''') by +17 (+22) MeV, +8 (+9) MeV, +43 (+3 1 ) MeV, and + 17 (+ 16) MeV.

The difference between the cluster and string fragmentation rnodels is the dominant systernatic uncertainty and leads to a shift in A@) (d5) ) of +80 (+68) MeV, + 125 (+ 102) MeV, +83 (+60) MeV, and +77 (+63) MeV from our central values, as shown in Figs. 11 -4. Table 1 1.2 shows systematic uncertainty from hadronization corrections in each kinernatic region.

Central analysis 1 362 (251) 1 317 (217) 1 137 (86) 1 305 (208)

op. = 0.25 GeV 1 410 (293) 1 321 (223) 1 150 (96) 1 323 (225)

Q,, = 4 GeV ( 280(192) 1 270(181) 1 130(82) 1 250(169)

op, = 0.45 GeV

- - -. - - - . - -- .

k, = 0.7 GeV 1 385 (273) 1 325 (226) 7 180 ( 1 17) 1 322 (224)

380 (269)

7izhlr II.2: Simrtiar~ oj' litzdmtriratiori correcriori cltecks in rnch kipmcitic biri. AI1 idrres <ire expressrd czs A? (MeV) czrrd tlzr nlotibers iri pcirenrliesis refer ro A!? (MeV). The lnrgest deviez-

1CIS 1WS- rions ji-orri [lie central value of nrise from tlzr clznngr of the h~zdroni,ntiorr niodel.

1 1.3 Dependence of the Result on the Input parton Density

Herwig cluster hadronization

The motivation of the kinematics chosen for the present analysis is two-fold. One is to ensure a pronounced jet structure and to have more jets produced by increasing phase space (high A- and Q'). On the other hand one wants to have a srnall influence of the dependence of the crJ determination on the choice of the parton density in the proton. The parton distribution functions have been provided by several recent parameterizations. Al1 these parameterizations are based on global fits to high-statistics deep inelastic experiments and use the NLO formalism of perturbative QCD. They describe deep inelastic data in the region of x > 0.01 well. In the region of small -r (s

< 0.0 1) predictions of various parameterizations start to differ.

312 (216)

442 (319)

148 (95) 308 (2 13)

442 (3 19) 230 ( 147) 382 (27 1 )

For the O (af) fits, we have used the M R S ~ set for the input structure function of the proton. This set describes the data well in the measurernent of the totd cross section at HERA in the kinematic region -r > However, the lower lirnit in s for the present analysis is not large enough to ignore the effect of the parton density entirely.

We have repeated the analysis with the parton distribution sets MRS, GRV, and CTEQ in the NLO theory, al1 of which describe the data from NMC and HERA well [87]. The differences in A from the centrai value are generally small (c 20 MeV), as shown in Table 11.3 and Figs. 1 1.4. It is noted that the fitted A value does not depend strongly on the one used in the parton distribution parameterizations.

1 Central anaiysis 1 367 (251) 1 317 (217) 1 137 (86) 1 305 (208) 1 1 MRSA 1 390(277) 1 340(238) 1 145(92) ( 327(228) 1

Erhle 11.3: Sirrnrnrrn of pcirton densitr. tincertainiy in ench kirienrcrtic bin. 411 vnlurs arc. rspresseti crs A.!? (MeV) mid the uirr,rbcrs in p~rrentlir.~is t-e/er to A!? (MeV) .

.tlS MS

GRV HO

CTEQ 3M

1 1.4 Renormalization and Factorization Scale Dependence

In order to get a finite answer for physical observables QCD has to be renomalized. This

- - - - ~

352 (247)

350 (246)

introduces a dependence on the renormalization scheme and the renormalization scale in any fixed order perturbation expansion. In the NLO QCD calculations. the renormdization scheme is chosen to be the scheme. This leaves then the dependence on the renormalization scale p R .

3 12 (216)

315 (219)

This is necessary in any perturbative cornputation. In addition one has similar scale ambiguities with factorization which de fines how soft and collinear initial state singularities are absorbed into

136 (86)

140 (89)

the structure function. This results in a dependence on the factorization scale pF

309 ( 2 14)

286 ( 196)

7 We chose pi = Q- for Our main analysis. This scale was chosen for the factorization

scale yF. To investigate the renormalization and factorization scale dependence of the result we performed fits of the data to O (a:) cdculation at different values of

and

ranghg from 0.4 to 2 for the systematic study, whrre the superscript R (F) stands for renormalization (factorîzation) scde. The scale dependence decreases with increasing Q and becomes negligible in the highest Q' interval as seen in the Fig. 1 1.4. It is slightly larger in DISJET than in PROJET. The behavior of these two programs is different, but the two programs give the same value for a in our analysis. We quoted the largest deviation in figure as the scale uncertainty. Table 11.4 shows scale uncertainty of the two NLO QCD calculations in each kinematic region.

Tnhlr 11.4: S~onnicrry of scde rrricerrainty in the NLO QCD cn~cularions. DISJET cind PROJEZ in eciclr kiizentritic bin. AI1 vnl~es are e-rpressed ns A!? (MeV) and the izuiribers in parentlusis rifer lo A?' (MeV). MS

,CI S

Central analysis R

-t-, = x i > 0.4

(DISJET)

sR = X F < 2.0 P P

(DISJET)

= > 0.4

(PROJET)

Y = Y < 2.0

(PROJET)

1 2 0 < ~ ' < 7 4 0

362 (25 1 )

255 ( 173)

466 (338)

485 (354)

355 (249)

2 4 0 < ~ ' < 7 2 0

317 (217)

243 ( 164)

4 1 O (293)

380 (269)

305 (3 11 )

720<&3600

137 (86)

104 (64)

168 (109)

- - -

122 (76)

123 (77)

1 2 0 < ~ ' < 3 6 0 0 '

305 (208)

325 ( 150)

390 (377)

- - - -

390 (277)

300 (207)

(a) 1 20<dG!40 G ~ V ' (b) 240<~'< 720 CeVf (c) 720<~'<3600 ~ e v

Statistical Error

Electron finder

Energy scale (+5%) Energy scale (-5%) Method of W colculation Preclustering of cells

ifferential jet rate method O. l5<z<O.85

(e) Detector correction(AR1ADNE)

Experimental S ystematics

Fragmentation porameter

a,=0.25 GeV a, of fragmentation oR=0.45 GeV function yh=O.O 15 Minimum y, of parton in ME Q,=4 GeV Minimum PS virtuolity k=0.7 GeV lntrinsic k, of struck parton

Herwig cluster hadronization

Had ronization Correction

MRSA GRV HO

CTEQ3M

Parton Density Uncertainty

O . ~ < X ~ = X ~ = ~ ' / Q ' < ~ . O (DISJET) 0 . 4 < x ~ = ~ ~ = ~ ' / ~ ' < 2 . 0 (PROJET)

Scale Uncertainty

Total Error

A(~)Z (GeV)

~ ( ~ l . 5 (GeV)

(d) 1 2 0 < ~ ~ < 3 6 0 0 G ~ V '

Statistical Error

Electron finder E-P, > 45 GeV y, < 0.7 Energy scale (+5%) Energy scole (-5%) Method of W colculotion Preclustering of cells

ifferential jet rate method 0.1 5<z<0.85

( e ) Detector correction(AR1ADNE)

Experimental Systematics

a=0.1 a= 1 .O [ Fragmentation parorneter

a,=0.25 GeV a, of fragmentation 0,=0.45 GeV function y,=0.015 Minimum y, of parton in ME Qo=4GeV MinirnumPSvirtuality b=0.7 GeV Intrinsic kt o f struck parton Herwig cluster hadronization

Hadronization Correction

M RSA GRV HO CTEQ3M

Parton Density Uncertainty

O . ~ < X ~ = X ~ = , L L ~ / Q ~ < ~ . ~ (DISJET) O . ~ < X ~ = X ~ = , U ~ / Q ~ < ~ . O (PROJET)

Scale Uncertainty

Total Error - MS (MeV)

Figirre 22.3: Sysfernnfic irncerfninties in the rnetcsured vniir~ of A for the kiner~intic rcriige ( a ) I Z O < Q' < 240 G ~ V ~ (6) 240 c Q' c 720 c e v 2 (c) 720 < Q- < 3600 G ~ V ~ (d) 120 < QI < 3600 cev2 respectively as exprrssed as n devirrtion from the centrd value in the anaiysis.

It is conventional to choose Q% the "hard scaie" appropriate for rnulti-jet production at 3

larger Q- for DIS events. However jet production in DIS is a multi-scale process. It is not evident that Q' is the brst choice [52] for the renomalization and factorization in the perturbative calcu-

7 lation. Alternative scales have been suggested, e-g. P; of the jets or the square of the invariant

-3

mas. M& of the two jets. DISJET and PROJET do not provide results with these alternative scales. Therefore the ratios <P;/Q'> and <M;,/Q2> were evaluated for the full Q' nnge for (2+1) jet events. These ratios were used to estimate the resultant change of scale and hence the uncertainty in as. For our events with < Q > - 22 GeV these ratios typically lie between 0.4 and 2. i.e. within the range explored in our estimation of the scale uncertainty using DISJET and PROJET. This is shown in Fig. 1 1.5.

Figure 11.5.- The cornparismi of vctlries < M ; ~ / Q ' >. < M J j / Q > . < P T @ > and < P;/Q' > in mir th ree kinenlntic reg ions.

As seen in Fig. L 1.3, al1 values are moving in the same direction. This reflects the same systematic effect in each kinematic region, except the third bin which is statistically Iimited in our analysis. For each group in Fig. 11.4 we quoted the largest deviation from the central value in each direction as the systematic error. The positive and negative deviations were then added in quadrature separately to give the systematic error. The total systematic uncertainty for the value of as is comparable to the statistical errors.

12.

value

QCD

Sumrnary of Final Result and Cornparison of ZEUS with Other Measurements

In the following sections we present contemporary world measurernents of as and our of as is compared with various high precision experiments.

Recent measurements of a, from various expenments are reviewed in the framework of predictions: r lepton decays, deep inelastic muon and neutrino scattering processes, heavy

quarkonia decays, b6 production at p p colliders, total hadronic width of the boson, hadronic event shapes, jet production rates of hadronic final States in ece-. The measurements are al1 based on QCD calculations in the NLO or higher order perturbation theory. Within their quoted experimental and theoreticai uncertainties, the results provide convincing evidence for the energy dependence of as as predicted by QCD.

12.1 Presentation of Results

The determined values of A!? for the three ranges in Q' as well as for the full kinematic S

range are listed in Table 12.1 and ?&le 12.2. Also listed are the values of as for the average Q' of each range and the value extrapolated to Q = M ,. Both statistical and systematic errors are 2 listed. The systematic error was calculated by adding in quadrature the systematic errors from each of the four independent groups: the experimentd uncertainty ( exp) . the hadronization uncertainty (had) , the parton density uncertainty ( p d ) , and the renormaiization and factorization scale uncertainty ( S C ) . The value of as expressed at the rnass of the f boson is consistent between the first two e2 ranges, and third bin statistically limited is also consistent within 2 0 .

Table 12.1: The determined values of R1 + A? (MeV) and as for the three ronges in g2. MS R , + , is calculated at y,,,= 0.02 and onfy statistzcal errors are shown in jet rate calculution. Sta-

tisticul ÿirst) and experimentd systematic (second) errors are listed. The third error corresponds to theoretical systematic uncertain fies (hadronization, parton densiîy und scale uncertain@.

<a R 2 + ~

( 5 ) A-

as ( Q I

as(M2')

13.3 GeV

7.8 f 0.7 + 108+31 + 115

25 1 -97 - 74 - 105 MeV +0.015+0.00~+0.016 -0.017-0.012-0.018

+ 0.007 + 0.002 + 0.007 -0.ûO8-0.006-0.W

20.4 GeV

9.0 + 0.7 +90+76+ 119

217 -74 - 60-67 Mev

0.152 + 0.01 1 + 0.010 + 0.014 -0.01 1 - 0.009 - 0.0 10

0.1 7 + 0.006 + 0.006 + 0.008 -0.007 - 0.005 - 0.006

35.5 GeV

8.6k 1.1

+82+30+61 86 -58-4, - 24 MeV

0.1 18 + 0.013 + 0.006 + 0.010 -0.017 - 0.012 -0.006

+ 0.010 + 0.004 +0.008 0A03 -0.013-0.010-0.004

1 ce> 1 22.1 GeV 1

Tobie 12.2: Al1 values are cnlculnted in the full kinematic range, 120 < Q' < 3600 G ~ v ~ .

1\- MX

as (QI

Our best value for combining the data from the entire kinematic range 120 < Q' < 3600 G ~ v ~ , and when expressed at the rnass of 9 is given by:

+64+57+89 208 -53 - SO - 75 MeV + 0.008 + 0.007 + 0.0 1 1 -o.,-o.,-0.01,

0004 O 005 O001 O 005 as (My) = 0.1 17 k 0.005 (stat) it (exp) t (hnd) f (pd) t ( S C )

Furthemore, if we calculate the systematic error by adding in quadrature the systernatic errors irom each of the independent groups, the as value is given by:

0.w as (Mt,) = 0.117 i 0.005 (stnt) f ws (systeIp) t 0.007 (systlheOF) ( 12.2)

where stat corresponds to the statistical error. The overall systematic error is separated into its experimental (exp) and theoreticai (theory) contribution.

The values of as are plotted in Fig. 12.1 as a function of Q for each three Q' range. They were calculated from the fitted values of A!?) . Both statistical and systernatic errors are shown. In

cMs addition, the measured a, decreases with increasing Q, consistent with the running of the strong coupiing constant. This shows a good agreement between the measured strong coupling constant and the expected value by QCD. One should note that it is possible to make better test of the running of as ( Q ) in a single expenment with more extended kinematic range at HERA.

a ZEUS (Jet Rates)

Figure 12.1: Measured values of as for three dgerent e2 regions. The statistical and systernoric errors are sho wn as thick and thin vertical bars, respectively. The solid dash Zines represent as with id? = 100, 200, 300 MeV The result fmm the ZEUS experirnenr is cornpared with a fit to rhe 2-IL%; solution of Renormalization Group Equation (RGE).

12.2 Cornparison with Other Measurements

Since the running coupling constant is a function of energy, we could compare the results in two ways. One is to see the @ dependence of the coupling at the different scaie where they are measured. The other is to bring aii the measurements to a certain reference scaie where they are cornpared. It is a standard convention to adapt ZS scheme and the 9 mass scale for cornparison.

Thus, the cornparison of al values c m be made at a single value of e2. Figure 12.2 shows n

the summary of as from other experirnents at the mass of the 2" boson. The measurements agree with each other within the overall errors.

r (LEP)

: , :

~6 4 (lattice QCD) : i : - 1 .

i :

J N + Y decays : 1 :

-: l :

: # '

e+e- (ohaa) : 1 : - 1 . -

e'e' (ev.shapes) : # :

1 :

: P .

: ?

Hl. jet rates(NL0) : , ' ' 1 -

ZEUS, jet rates(NL0) A . - .

: I ' 6 '

I '

r (~~-+had.) : :- : , -

ZO ev.shapes(resu mmation) a y -

Figure 12.2: Sumrnary of as measurements from ZEUS and other experiments (881- The vertical dashed line shows the weighted world average value. not including ZEUS value, with the error shown by the dotted lines.

An extraction of % from multi-jet production had been reported in 1993 by H1 expenment [86] using a rnodified JADE jet finding algorithm in which the proton remnant is included in the clustering procedure. An integrated luminosity 0.4 was used. Different treatment of forward region was made. H l experirnent applied angle cut 10' < < 148O which reduces the dependence of forward uncertainties. The comparison of running as between ZEUS and Hl is shown in Fig. 12.3. HI measurement yield as ( M Z J ) = 0.121 + 0.009 ( s tar ) + 0.0 12 ( s y s t ) in the kinernatic range 100 < e2 < 4000 G ~ v ~ . Below Q~ = 100 G ~ v ~ , H l sees that the corrections depend on Monte Carlo rnodels. Therefore only data at Q* > 100 G ~ V ' are used to extract a, in

HI experirnent. Both experiments show that jet rates in DIS can be quantitatively well described by perturbative QCD in NLO and the measured value of a, is in agreement both with detennina- tions from e+e- annihilation at LEP using the same observable and with the world average.

A!'% = 300 MeV

\

\

00 MeV ' , ' . ' . ' . - .

..

8 ZEUS e et Rotes)

O H l (Jet Rotes)

1 O'

Q (GeV)

Figure 12.3: A surnmary of as frmn ZEUS (black dots) and HI (rectangular points) experiments as n function of Q. The thick and thin vertical bars represent the statistical and systematic errors in ZEUS and HI experiment, respectively. The weighted world average. taken from 1881, is quoted ro be as ( M ,) = 0.1 18 I 0.003 which corresponds to A!? =209 +:; MeV as s h o w in the rniddle h e . 6 t h - two lines represent as wiih A- = 10oMfOO MeV

MS

Very precise rneasurernenis have been derived from event shape variables at the 2 pole from e+e- annihilation at LEP. Sirnilar analyses on event shapes have been made at TRISTAN, at PEPPETRA, at CLEO. The errors in the values of as (My) from these shape variables are

totally dorninated by the theoreticai uncertainties associated with the choice of scale, and the effects of hadronization corrections.

The ZEUS measurement has opened the door to a new kinematic domain in DIS. Table 12.3 gives the present status of measurements from jets in the LEP expenments [89]. In Fig. 12.4 the measurement obtained by this analysis is compared to the LEP results. The determined value of as from ZEUS is in agreement with the LEP.

1 / ALEPH 1 DELPHI 1 L3 1 OPAL 1

Table 12.3: as measurements at LEP with jets in Eo scheme using f ied order perturbative QCD ( 3

+ ' 58 Me respectively. predictions. The average value of a, = 0.1 19 + 0.0 10, h =239 - 108 MS

DELPH

Figure 12.4: as measurement from jet rates nt ZEUS. Cornparison was made with LEP experiments. This good agreement between our value and the results obtained using other methods in diflerent kinematic regimes shows a significant test of QCD.

13. Conclusion

This thesis has presented a measurement of the strong coupling constant as from multi-jet events in DIS at HERA during the 1994 mnning penod in which the data were collected. The integrated IuMnosity of 3.2 collected was used in this analysis.

Multi-jet production in ep collisions has been investigated using the JADE jet definition for 120 < Q' < 3600 G ~ v ~ , 0.01 < x < 0.1, 0.1 < y < 0.95. In this kinernatic range, distinct jet structure have been observed. To leading order in as , (2+1) jet production proceeds via QCD- Compton scattering (QCDC) and Boson-Gluon Fusion (BGF). Various measurements of the kinematic properties of the jets have been analyzed. Two jet kinernatics is well described by the Monte Carlo model (MEPS). The large invariant mass and transverse momentum of two jets make perturbative QCD applicable at HERA. The partonic variables z and x, are reconstructed from two jets and well described by NLO QCD calculations.

In ep collisions the application of a single jet resolution parameter y,,, is not sufficient to resvict the phase space of (2+l) jet production expenmentally. We introduce an additional cut on the angular distribution of parton emission in the y* - parton systern, in both theory and experiment, which restricts the problematic region of (2+1) jet phase space, where jets are not well iso- lated in experiment. With this additional cut the jet production rates in DIS are welt reproduced by the perturbative NLO QCD caiculations. In addition, the measured jet rates are affected by the detector effect and hadronization effect, so these corrections to the parton level have been made by using the MEPS Monte Carlo model.

We have detemiined as in the three e2 regions by identical anaiysis in a single experiment and have found their consistency, which is a significant test of QCD. Our final value for as using the data frorn the entire kinernatic range 120 < e2 < 3 6 0 G ~ v ~ , and when expressed at the 9 mass it is given by

0004 0.005 O O01 O 005 as (Mt') = 0.117 I 0.05 (stat) f (exp) + (had) f (pd) + (SC)

0004 = 0.117 + 0.005 (stat) I ioos (systcXp) + 0.007 (syst ,,,,,)

where star refers to the statistical error and the overall systematic error is separated into the experimental and theoretical errors respectively. Moreover, we found a decrease in as for increasing scale Q consistent with the running of the strong coupling constant. The three values of as, expressed at 2 f mass, are consistent within the errors.

Our value of as is consistent with the compilation by the Particle Data Group [ l ] of previous measurements of as (My) using different methods: 0.1 12 t 0.005 (DIS), 0.12 1 k 0.006 (e+e- event shape analysis) and 0.124 f 0.007 (9 width). The good agreement between our value of a, and the results obtained using other methods in different kinematic regimes represents a success for QCD. Finally, Hf and ZEUS experiments show jet rates are consistent with NLO QCD cdculations and allow the study of the ninning of as using one observable at various scales in a single experiment.

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