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Determination of the Strong Coupling Constant from Dijet Production in Deep Inelastic Scattering at HERA

l\1athieu Plamondon

April 2005

Department of Physics, Mc Gill University, Montreal, Canada.

A thesis submitted to McGill University in partial fulfilment of the requirements of the degree of Master of Science

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In the present analysis, the constant Œs of Quantum Chromodynamics (QCD) is de­ termined from dijets production in the regime of neutral current deep inelastic scat­ tering (NC DIS) with high momentum transfer Q2. This measurement uses the data collected between the years 1998 to 2000 by the ZEUS detector located on the HERA ring collider in Hamburg, Germany. Corresponding to an integrated luminosity of 82.7 pb-l, the data were produced by the interactions between 27.5 GeV electrons (or positrons) and 920 GeV protons. Defined in a phase space reducing the theoreti­ cal and systematic uncertainties, the dijet production is observed as well as the total inclusive DIS cross-section. The resulting observable, the ratio of dijets over the num­ ber of DIS events, is then compared to the NLO predictions provided by the program DISENT. From a QCD fit performed on four bins in Q2, we obtain a value consistent with the world average: Œs(Mz) = 0.1157 ± 0.0017(stat.)::g:gg~~(syst.)::g:gg~~(th.) Résumé

Dans cette analyse, la constante as de la Chromo dynamique Quantique est déterminée grâce à une analyse de la production d'événements avec deux jets lors de collisions hautement inélastiques et auxquelles est associé un transfert de quan­ tité de mouvement (Q2) élevé. Cette mesure utilise les données recueillies entre les années 1998 et 2000 par le détecteur ZEUS situé sur le collisionneur circulaire HERA à Hamburg (Allemagne). Correspondent à une luminosité intégrée de 82.7 pb-l, ces dernières ont été produites à la suite d'interactions entre des électrons (ou positrons) ayant une énergie de 27.5 GeV et des protons de 920 GeV. Définie dans un es­ pace de phase réduisant les incertitudes théoriques, la production d'événements di­ jets a été mesurée ainsi que la section efficace totalement inclusive dans ce régime.

L'observable choisi, la fraction R 2+1 d'événements avec deux jets ou dijets sur l'ensemble des occurrences avec un haut Q2, a été comparée avec les prédictions au second ordre fournies par le programme DISENT. Grâce à une évaluation exécutée sur quatre intervalles en Q2, une valeur près de la moyenne mondiale a été obtenue:

as(Mz) = 0.1157 ± 0.0017(stat. )~8:gg~~(syst. )~g:gg~~(th.) CONTENTS 1

Contents

Acknowledgements 4

1 Introduction 5

2 Theory 7 2.1 The Strong Coupling Constant. 7 2.1.1 A Parameter in QCD Theory 7 2.1.2 Asymptotic Freedom ..... 8 2.1.3 Solutions to the Renormalization Group Equation 10 2.2 DIS Cross Sections in pQCD ...... 11 2.2.1 Description of the Kinematic Variables 11 2.2.2 General Cross Section in QCD ..... 14 2.2.3 Partonic Cross Sections and Physical Observables 14 2.2.4 Parton Distribution Functions . . 17 2.2.5 NLO Calculations with DISENT 18 2.2.6 Multi-partonic Production 20 2.2.7 Hadronisation ...... 23 2.3 The Experimental Observables. 25 2.3.1 Inclusive DIS Cross Section 25 2.3.2 Dijet Cross Section 30 2.3.3 Breit Frame . 32 2.3.4 Jet Definition 32 2.3.5 Construction of the Observable 35

3 Experimental Setup 39 3.1 HERA Accelerator 39 3.2 ZEUS Detectof .. 41 3.2.1 Central Tracking Detector (CTD) 45 CONTENTS 2

3.2.2 Uranium-Scintillator Calorimeter (CAL) 47 3.2.3 Trigger Chain ...... 52 3.2.4 Luminosity Measurement . 54 3.3 Reconstruction of Physical Objects 55 3.3.1 Generic D IS03 Trigger ... 57 3.3.2 The SINISTRA Electron Finder 60 3.3.3 ZUFOs ...... 61 3.3.4 Reconstruction of the Kinematic Variables 64

4 Description of the Analysis 67 4.1 Monte Carlo Simulations 67 4.2 DIS Selection Cuts 69 4.3 Dijets Selection Cuts 77 4.3.1 Selection at the Detector Level 78 4.3.2 Selection at the Hadron Level 83 4.4 Jet Energy Correction 84 4.5 Correction Procedure. 87 4.5.1 Acceptance Correction 88 4.5.2 QED Correction. . . . 90 4.5.3 Hadronisation Correction . 91 4.6 Experimental Cross Sections and Uncertainties . 93 4.6.1 Statistical Uncertainty ...... 93 4.6.2 Systematic Uncertainties of the DIS selection 95 4.6.3 Systematic Uncertainties of the dijet selection 96

5 Results 98 5.1 Dijet Fraction and NLO QCD Predictions 98

5.2 Determination of Œs...... 100 5.3 Systematic and Theoretical Uncertainties 102 CONTENTS 3

5.4 Discussion of the Results 103

6 Summary and Outlook 107

Glossary 109

References 112 ACKNOWLEDGEMENTS 4

Acknowledgements l would like to thank Prof. François Corriveau for his supervision during my master degree and for the unique experience of participating in ZEUS. l am also grateful to the builders of this experiment and to the members of an international collaboration from whom l could learn a lot. Elisabetta Gallo, of the Jets Subgroup, introduced me to the relevant techniques and allowed me to make great progresses. Thanks to Enrico Tassi for his expertise in my subject, shared during fruitful discussions. l could always rely on the dedicated help provided by Roberval Walsh and the availability of Jeff Standage who let me use sorne of his material. Andrée Robichaud-Véronneau accompanied me throughout this period and her support remains priceless. Aiso contributed Sanjay Padhi whose advice and contagious motivation was essential for the successful complet ion of this work. 1 INTRODUCTION 5

1 Introduction

Since several decades now, the Standard Model remains a privileged candidate to provide a simultaneous explanation of the nature of elementary particles and their interactions. It includes the electroweak theory which is the unification of the elec­ tromagnetic and weak forces based on gauge theories and SU(2) ® U(l) symmetry. On the other hand, the strong interaction is SU(3) symmetric and is embedded in the global picture of Quantum Chromodynamics (QCD). The latter was built in a succession of experimental evidences which required modifications of the very model. During the late 60's, ideas were developed to explain the multiplicity of the recently discovered particles and lead to the first attempts to provide a quantum field de­ scription of the internaI structure of nucleons. Since then, the development of QCD suffered many delays which explain that even nowadays, the model remains under construction. They were primarily caused by technological difficulties to perform the experiments, large-scale projects involving an increasing amount of resources. From the theoretical point of view, many other hurdles had to be overcome in order to perform sensible tests. Forced to observe indirect phenomena, theorists devel­ oped techniques to simulate the transition between the fundamental dynamics at the parton level and the fiow of outgoing hadrons which is detected. Owing to these phenomenological models, remarkable results were obtained despite the lim­ ited predictive power offered by the theory. Divergencies in the QCD calculations of meaningful quantity in QCD arise from integration over unbounded quantities and could be circumvented only after scores of computational achievements. Since the discovery of the sixth flavour as the top quark in 1995, the model remains chal­ lenged by the still undiscovered Higgs boson and waits for clear indications of physics existing beyond its scope.

At the sc ales associated to QCD, any process finds its full description starting from first principles. This last statement cornes of course with restrictions since many 1 INTRODUCTION 6 quantities, in particular the specific masses of the particles, cannot find an explana­ tion within the scope of the present theory. In the close future, the investigation of the Higgs mechanism would signify not only a confirmation, but also the hope for unexpected behaviors, giving new insights about a more general theory. A unique parameter, the strong coupling constant called as, dictates the strength of the strong force in this world. There is more than a mere curiosity behind the knowledge of this constant of nature. It is incorporated in every prediction and will determine at which point the Standard Model will begin to fail. For this reason, we see of course no limit to the precision that has to be reached for its value. Its difficult measurement remains by far the less accurate of aIl the fundamental constants. Among several methods which were attempted, the analysis of jet rates [1, 2] has allowed one of the most precise determinations of the strong coupling constant. This observable is governed by the weight of the different branchings in the Feynman diagrams and is hence directly sensible to the value of as.

The present work will therefore consider the multiplicity of jets produced in deep inelastic scattering and perform a measurement of the strong coupling constant. The electron-proton collider HERA plays a major role in the testing of QCD, its high­ energy collisions revealing even new information on the structure of the proton. AlI the theoretical aspects of the problem will be discussed in the first chapter. Once the choice of our observable is motivated on those bases, the experimental setup, in par­ ticular the ZEUS detector, is described in details. The following section explains step by step the analysis done, the techniques leading to an actual comparison between theory and experiment. The last part is dedicated to the strategy used to extract as, to the consideration of aIl the sources of uncertainty, to the possible cross-checks and to the final results. 2 THE ORY 7

2 Theory

This first chapter establishes the theoretical framework that will ultimately allow a measurement at the small scale involving Quantum Chromodynamics (QCD.) We begin by an in depth understanding of the nature of the strong coupling constant. A description of a general cross-section in perturbative QCD (pQCD) introduces the crucial concepts of parton density functions (PDFs) and hadronisation. The numerical implementation of this model is subject to a detailed explanation. This is a non trivial step before a possible comparison with the experimental data is possible. Motivated by the theory, an observable central to the present analysis is constructed thereafter.

2.1 The Strong Coupling Constant

2.1.1 A Parameter in QCD Theory

In or der to understand the role played by the strong coupling constant, as, a brief overview of Quantum Chromodynamics is required. This gauge field theory was de­ veloped more than thirty years ago to explain the strong interaction according to both experimental and theoretical progresses achieved at that time [3, 4]. Since then, the basic picture of hadrons has not changed: they are colourless bound states of quarks, antiquarks and gluons. The quarks belong to one of the six fiavors {u, d, s, c, b, t} and possess one of the three choices of a quantum number called color (or anti­ color). The strong force between them is mediated by the other group, the gluons. These are spin-one gauge bosons and available in eight colors. However, the SU(3) symmetry on which QCD lS based makes a clear distinction with others field the­ ories like QED: the interaction carriers contain themselves the color charges. In comparison, the mediators of the electromagnetic force, the photons, are electrically neutral. This property of QCD has drastic consequences which are demonstrated by 2 THE ORY 8 the Lagrangian density:

L QCD = -~p~~) p(a)p,1I + i L ?j}~1P,(Dp,)ij1jJ~ - L mq?j}~1jJqi (1) q q where the gauge coupling gs appears in the covariant derivative defined by

(2) and most importantly, in

(3) which is the gluon field strength tensor. In the previous equations, Jabe and Àa correspond respectively to the structure constants and representation matrices of

SU(3), A~ are the 8 Yang-Mills gluon fields and 1jJ~(x) the 4-component Dirac spinors associated with each quark field of color i and flavor q. The effective strong coupling constant is defined in analogy with the fine structure constant a em of QED:

(4)

The nature of as appears then as the unique parameter of the theory that has to be determined by experiment. Referring to the right-hand side of eq. 3, it is this third non-abelian term which allows the well-known multi-gluons vertices. The constant directly tunes the strength of this interaction. The gluon self-interactions provide explanations for two properties of high-energy cross-sections, namely the asymptotic Jreedom and the confinement of quarks (subject developed in section 2.2.7).

2.1.2 Asymptotic Freedom

The asymptotic freedom can be described briefly as the fact that as happens to decrease (vanish) and be sufficiently small at high energies to allow a reliable use of perturbative methods or pQCD. From such a statement, one must point out that 2 THE ORY 9 the coupling constant do es depend on an energy scale, namely the renormalization scale J.1R [5, 6]. This conclusion follows when considering any QCD prediction of physical observables. The Feynman rules,· obtained from the Lagrangian density (eq. 1), allow a calculation of these quantities perturbatively. However, integration in sorne diagrams over unconstraint loop moment a create ultraviolet divergencies that must be circumvented. Renormalization procedures handle successfully these problematic cases at the price of the introduction of an unphysical parameter J.1R' It can be defined in turn as the scale at which the subtractions were performed in order to remove the UV divergencies. However, any physical observable R has to remain independent of such arbitrary value. The renormalization group equation

2 dR [2 a 2 aas a] J.1R-d2 = J.1R-a2 + J.1R-a2 -a R = 0 (5) J.1R J.1R J.1R as provides a mathematical statement for this requirement and clearly shows that as has to vary when considering a different sc ale J.1R. This very dependence is governed by the Callan-Symanzik ,8-function:

(6)

with the coefficients {Ji given by [7,8, 9, 10, 11]

{Jo (7)

(8)

(9)

etc ...

where ni is the number of active fiavors, i.e. fiavors that can be produced at a given scale J.1R. The calculation of the one-Ioop coefficient {Jo in the 1970's together with

ni = 6 has led to the pro perty of asymptotic freedom of QCD:

as J.1R ---t 00 (10) 2 THE ORY 10 which can then be interpreted as follows: when probed at high energies, quarks and gluons behave like quasi-free particles.

2.1.3 Solutions to the Renormalization Group Equation

When solving eq. 6 for as, a constant of integration must be inserted. The value of the coupling constant at a reference sc ale pr;f remains a natural choice for its defi­ nit ion and becomes the fundamental constant of QCD, one that has to be extracted from experiment. The mass of the ZO boson (91.1882 ± 0.0022 GeV [12]) became a standard choice for pr;f because of its advantages of being precisely determined,

lying comfortably in the perturbative regime (as(Mz ) « 1) and far from the quark thresholds (mb « M z « mt).

Furthermore, the constant of integration can be conveniently split into a second dimensional parameter called A. With this procedure, the PR dependence of as can be expressed in an explicit parametrization:

(11)

This solution exhibits clearly the asymptotic freedom property and A sets in a sense the characteristic scale of QCD. In or der to evaluate as at any scale, many strate­ gies exist and the one adopted throughout this analysis makes use of eq. 11. The parameter A is initially evaluated via an iterative procedure and the value found is then used to obtain as(PR).

So far, we have assumed an idealistic situation where the heavy quarks with masses greater than PR are totally neglected. This assumption prevents us from having the whole description of the coupling constant (i.e. at aU scales) as the f3- function coefficients do vary by discrete amounts when a fiavor threshold is crossed.

This problematic procedure is treated by constructing an effective theory with (n f -1) massless fiavours and one heavy fiavour quark. A physical quantity calculated in 2 THEORY 11 this theory is made to match the full nrfiavour theory at a scale comparable to the he avy-quark threshold. This requirement relates the coupling constants by

(12) where M is the mass defined in the MS renormalization scheme [13]. The relation between A[nf-1] and A[nf] follows after considering eq. 11 [14]:

2 963 [4] '" A[5] (~) 25 [21 ( /A[5] )] 14375 A QCD'" QCD + [5] n mb QCD (13) AQCD

2.2 DIS Cross Sections in pQCD

The next step consists of describing the computations of cross sections to the next­ to-leading or der (NLO) i.e. including the second term in the perturbative expansion. After a global introduction to DIS processes, the difficulties encountered in the cal­ culations are exposed and special care is taken to define physical observables which are comput able in pQCD. The related solutions will come along with the description of DISENT [15], the program providing NLO predictions used in this analysis.

2.2.1 Description of the Kinematic Variables

Prior to the consideration of cross sections, the knowledge of electron1-proton col­ lisions involved in HERA and the variables describing them is required. Accord­ ing to Feynman's original model [4], the partons compose the proton as free and point-like particles. In the meanwhile, Gell-Mann had foreseen the nucleon with charged spin-~ constituents, namely the quarks [3]. The experimental verification of the Callan-Gross relation (see section 2.3.1) allowed later to identify Feynman's partons with Gell-Mann's quarks [16] and this lead to the so-called quark-parton model (QPM). Fig. 1 shows the basic features encountered with this approach. An

1 From now on, the term electron will designate any type of initial lepton, electrons or positrons. 2 THE ORY 12 interaction between an incoming high-energy lepton (electron or positron E with 4- momentum k) with a proton is viewed as an exchange of a virtual gauge boson with 4-momentum q. Depending on the charge of the latter, we separate these events in two classes: neutral exchanged particles ("'( or ZO) lead to neutral current deep inelas­ tic scattering (NC DIS) while charged ones (W±) pro duce charged current pro cesses (CC DIS). Only those of the first group are considered in this analysis, characterized by event rates several orders of magnitude larger than the charged current pro cesses and the presence of the same outgoing lepton with 4-momentum k'. Experimentally, this signature facilitates both the identification of these events and the measure­ ment of their kinematic variables. The struck quark carrying a fraction ç of the total momentum P of the hadron system is then scattered from it while the other components remain unaffected, following their way through. In the QPM, two inde­ pendent variables are sufficient to depict the entire kinematics of a process with a given center-of-mass energy Vs. They are normally chosen among the momentum transfer Q2, the Bjorken scaling variable x and the inelasticity y defined by:

Q2 p.q Q2 x=-- y=--=- (14) 2p· q p' k sx where the mass of the leptons is not considered.

The variable y, the inelasticity, is understood in the proton rest frame as a mea­

sure of the energy transferred from the electron during the interaction. A value of y approaching one corresponds to an elastic collision. In the QPM language where the quarks are kept massless, the interpretation of x coincides with that of ç, the frac­ tion of the longitudinal momentum of the hadron system carried out by the struck quark. Also, at fixed value of x, the exchanged boson can probe smaller distances with increasing momentum transfer, its wavelength being inversely proportional to -IQ2. However, in the case of jet studies, a fourth variable is found to be particu­

larly relevant. Diagram 1 gives a spatial interpretation of "'(h, the scattering angle of the hadronic system in the laboratory frame, though it is more generally defined 2 THEORY 13

l(k)

çp

P(p)

Figure 1: Leading-order (O(Q~)) of the neutral current deep inelastic scattering be­ tween a proton and a lepton in the quark-parton model (QPM).

according to the kinematics of the event:

(1 - y)xEp - yEe (15) (1 - y)xEp + yEe where Ep and Ee are associated to the energies of the initial proton and electron respectively. This equation makes sense if the four-momentum transfer q{t originates from the lepton, qO being the energy and qi(i = 1,2,3), the 3-momentum.

With the evidence of neutral components existing in hadrons [17] and the ne­ cessity to explain the binding of quarks, a more complete theory was built. The improved quark-parton model then includes the gluons, intermediates of the strong force between the three valence quarks, and a sea of quark-antiquark pairs that are continually created and annihilated. The process in fig. 1, restricted to the naive

QPM model, appears then as the first-order contribution (O( Q~)) of an expansion 2 THE ORY 14 performed according to the rules of pQCD.

2.2.2 General Cross Section in QCD

Up to this point, our concerns were aimed at the description of cross sections at the level of partons only. Motivated by the picture of nucleons proposed in the last s,ection via the improved quark-parlan model, it is now possible to express the global formula for proc~sses involving a proton:

da = ~ Jdxfa(x,J.L})dâa(xP,Œs(J.LR),J.L~,J.L})· (1 +6had) (16) a=q,q,g

Eq. 16 appears as a convolution performed over the momentum fraction x between the partonic cross sections âa and the parton distribution functions (PDFs) fa. Owing to factorisation theorems of QCD [18J, it reflects our quantitative ability to separate (factorise) the short-range aspect of the interaction from the long-distance structure of the hadron. The first part is characterized by exchanges with high momentum transfer (high Q2) and is therefore calculable in pQCD. On the other hand, the PDFs do not possess this advantage and their treatment will be discussed in sect.2.2.3. Another scale J.LF is introduced there to set a limit between the two regimes.

The factorisation formula 16 includes also an isolated term 6had which accounts for the effects of hadronisation. This subject will be developed later in section 2.2.7, once the perturbative treatment of the final state evolution breaks down.

2.2.3 Partonic Cross Sections and Physical Observables

The main features in the final state of a DIS collision originate' from the hard­ scattering part, the one involving high-Q2 exchanges.' As described previously, the parton level cross sections are obtained by considering the appropriate Feynman di­ agrams in an expansion in Œs. A NLO calculation in pQCD is limited to the first 2 THE ORY 15 two orders in the perturbative series:

(17)

The leading-order term with m partons having momenta Pk (k = 1, ... , m) in the final state corresponds to the Born-Ievel cross section which has to be integrated over the full phase space d(m) of a given observable:

(18)

m In eq. 18, Mm are the QCD matrix elements while F5 ) denote the functional form for the physical quantity of interest. In the current analysis about jet rates, it would stand for the jet algorithm and the selection cuts. Though this expression cannot be expressed in an analytical form, the integration of the Lü part can easily be achieved with the help of numerical techniques.

The NLO term includes various contributions and gives rise to scores of diffi­ culties, complicating the numerical implementation mentioned above. Despite the fact that ultraviolet divergencies have been efficiently cured by a renormalization procedure, the calculation of higher-order corrections still suffers from the presence of soft and collinear singularities. They appear whenever a parton in the initial or final state has another parton emitted softly or collinearly to it. This creates a complex set of single and double poles2 which would prevent one from obtaining any finite result if no cancellation was performed. The choice of the observable itself is of critical importance: it must be chosen such that it permits to factorize the collinear singularities originating from the initial state:

(19)

for Pi ---+ (1- x)Pa

2This explicit expression of the singularities is obtained by regularizing (redefining the space­ time) the phase space integrals in a number of dimensions d = 4 - 2€ other than four [19]. 2 THEORY 16 where Pa corresponds to the initial-state parton collinear to Pi of the final-state. This necessary artefact redefines the density functions fa which becomes dependent on an arbitrary sc ale MF at which the factorization is performed. Its qualitative interpretation links it to a cutoff transverse momentum of the partons, marking a separation between contribution to the PFD or to the final state. Another conse­ quence, a collinear counter-term dô- c must be incorporated in the next-to-Ieading order partonic cross section:

dÔ-~l) = 1 [dô-~ - dô-~] + 1 dô-~ + 1dô-~ + 1dô-~ (MF) (20) m+l m+l m m where dô-:; represents the real contribution with m+ 1 partons and dô- ~, the virtual correction to the Born-Ievel with m partons in the final state. They possess a sim­ ilar structure to the Lü formula (eq. 18) with corresponding matrix elements. We explicitly pointed out that the entire MF - dependence in eq. 20 is contained in the last term on the right-hand side which reads in d - 2E dimensions as [19]:

p~~) being the Altarelli-Parisi splitting functions of the first order (see next section on PD Fs). Regarding the remaining term dô-~, it is the key ingredient used by the program DISENT to get rid of the other class of singularities, those involving the final state partons (the implementation of DISENT will be detailed in section 2.2.5)

We can then state the other general property that Fjm) has to fulfil in order to give a finite value for dÔ-~l): the infmrerF safety. Basically, the observable must be

independent of the removal of a soft parton Pj

(22)

for À -+ 0

3lnfrared refers in this context to long distances, where pQCD breaks down and soft (non­ perturbative) pro cesses take place. 2 THE ORY 17 and of combining collinear partons {Pi, Pj} into the four-momentum p.

A last requirement concerns the Born-Ievel cross section and guarantees its integra- bility:

(24)

for Pi' Pj -+ 0

2.2.4 Parton Distribution Functions

The distribution of the parton's momentum inside the proton cannot be resolved from first principles by the perturbative approach. Only their variation with the sc ale at which they are probed is predicted by a set of integro-differential equations. As a function of Q2, the density functions {fa(x, Q2)} = {qa(x, Q2), g(x, Q2)} of eq .16 are controlled by the Altarelli-Parisi equations:

(25)

og(X,Q2) as(Q) dy [""" ( 2) (/) ( 2) (/)] oln Q2 = ~ 11x Y L qa y, Q Pqq x y + 9 y, Q Pgg x y (26) a where the kernels Pkl(Z) give the probability of the emission of a parton k by the parton l with a fraction z of its momentum. These Altarelli-Parisi splitting functions can be expressed as expansions in as and many approaches exist to de duce the first terms in pQCD. Adding up ladder diagrams of consecutive gluon emission allows the so-called DGLAp4 evolution equations to be derived [20]. The solutions give a global description of the PDFs, provided that they are known at sorne fixed Qo for aIl values of x. This is achieved by parametrizing the density functions at this input sc ale Qo and fitting the evolved ones to the data, generally coming from a set of experiments

4 Dokshitzer-G ri bov-Li patov-Altarelli-Parisi 2 THE ORY 18

( deep-inelastic scattering, jet production, Drell-Yan and prompt-photon). Several distributions are made available and two of them came into play in this analysis. Those presented by the CTEQ (Coordinated Theoretical-Experimental project on QCD) collaboration offer a special interest by coming with different values of as assumed in their evolution. It is this advantage that will be exploited and ultimately allow a determination of the strong coupling constant. The set known as CTEQ4 [21] uses the following parametrizations of the quarks and gluon densities (where the indexed variables are the free parameters of the fit to the data):

xqa(x, Q6) Aax8a(1- x)TJa(l + EaVx + 1'ax) (27)

xg(x, Q6) AoxAl(l - x)A2(1 + A3 XA4 ) (28) while the MRS (Martin-Roberts-Stirling) series [22] consider only the first function (eq. 27) for both cases. Based on similar assumptions and next-to-Ieading or der DGLAP equations, these methods depend strongly on the inputs at Qo. Despite this drawback, the factorisation formula (eq. 16) does not lose its predictive power because of the universality of the parton densities: once they have been determined, their value can be used later in any other calculation of observables.

2.2.5 NLO Calculations with DISENT

The NLO predictions in this analysis were obtained with the use of the program DISENT developed by Seymour and Catani [15]. Before explaining its treatment of the remaining singularities, a description of the machinery in place during the computation is presented. As noted previously, due to the complexity of the phase space, the flexibility offered by the numerical integration techniques is required in practice.

The program starts by generating one m-partons event which accounts for the

2 Born-Ievel cross section. The matrix element is evaluated and this amplitude IM m l 2 THE ORY 19

becomes the weight of the event. It is next submitted to the observable's function F J which takes the form of a user routine in the Implementation. If the event belongs to the phase space considered, then it is histogrammed in the appropriate bin according to its weight convoluted with a PDF.

From this event, two next-to-Ieading order contributions are added, one with m partons and one with m+ 1 in the final-state, and eq. 17 becomes:

dei NLO = dô-(O){m} + dô-(l){m+1} + dô-(l){m} a a a a (29)

Both have their own weight (not necessarily positive definite) calculated and are analysed in a way similar to the one above. Having summarized the procedure, it remains to clarify how these two last weights are obtained, free of all divergencies. The technical trick employed by DISENT was mentioned with the exact identity of eq. 20 and is referred to as the subtraction method. A 'fake' cross section dô-~ is added and subtracted, permitting an analytical handling of the singularities prior to the numerical integration. Its choice must satisfy simultaneously two main conditions. We require from it to first behave as a local counter-term to the real correction with m+ 1 final-state partons:

dô-(l){m+l} = [(dô- R ) - (dô- A ) 1 a 1 a €---+O a €---+O (30) m+l such that the f-poles are removed for this first part, which is numerically computed in four dimensions. The second feature leads to the cancellation of all the singular points from the other contribution dô-à1){m}. The one-dimensional subspace which creates the soft and collinear divergencies has to be Integrable analytically, producing poles balancing exactly those encountered originally in the m-partons kinematics:

(31) dô-i1){m} = L[{ 1dô-~} + dô-;: + dô-~] E---+O

and therefore legitimizes the numerical integration associated with this other next­ to-leading or der term. The strength of the program resides in the dipole formalism 2 THEORY 20 providing a general expression for dâ:. In this scherne, for fixed {Pl . .. Pm} of re­ defined5 rnornenta, the cornbined particle s'ubspace dcjJ(Pk) (which depends only on the dipole {Pi, Pj}) can be factorised. With the universal dipole functions V ij based only on the dynarnics of the tree-Ievel with m partons, it is possible to build the subtraction terrn

2 m dâ: = d(m) ({Pk} )dcjJ( {Pk} ) (Pi + Pj) L IMm( {Pk} )s( {Pk}) 1 ® V ijF5 ) ({Pk}) (32) ij which possesses all the required properties (® denotes colour and helicity correla­ tions). By construction, all the singular regions of IMm+112 are reproduced in V ij to allow the cancellation of soft and collinear divergencies of der;;. This conclusion follows independently of the choice of the observable FJm} , as long as the infrared safety is satisfied. The first terrn of eq. 20 can then be treated nurnerically in a stan­ dard way: each tirne a singular region is approached in a m+ l-state, the subtraction counter-terrn possesses the sarne lirnit and cancels the possibly large contribution in the bin. For the m-partons configuration, the convenient splitting functions accept an analytical integration over the factorized dipole's subspace and a process-independent factor 1 is calculated once and for aIl. Owing to the exact phase space factorisation of eq. 32, it is then convoluted as follows:

(33)

1.e. with a terrn which reproduces the rn-partons Born-Ievel cross section.

2.2.6 Multi-partonic Production

The main features of the finalstate are generally deterrnined by the hard sc ale pro­ cesses that were discussed above. However, the confinement of the quarks do not

5The set {th . .. Pm} are constnlcted by starting with the original m+ 1 configuration and modi­ fying the kinematics in such a way that energy-momentum conservation remains the same after the

removal of one parton: Pt = 0 , Pl + ... + Pm = Pl + ... + Pm+ l 2 THEORY 21

HADRONISA TION

PARTON SHOWER

Figure 2: Schematic representation of the processes ta king place after a DIS collision. allow a direct observation of the outgoing particles. The transformation of these partons into colourless hadrons cannot be predicted by QCD and hence limits our understanding of the theory. Owing to the factorisation theorem [18] for hard pro­ cesses, this generation can be separated artificially in several steps (see fig. 2). Even at the level of the outgoing partons, the topology of the event can be drastically modified by the emission of initial or jinal-state radiation, i.e. from partons pro­ duced respectively before and after the hard scattering. The correct approach would be to calculate the Feynman diagrams thoroughly, for all the branchings up to the N final state partons. Unfortunately, the matrix elements method can only have a limited success in practice since the calculations become rapidly complicated, even for low values of N. Furthermore, there are strong experimental indications that the higher orders mechanisms induce rather large corrections.

Two approximation schemes have been developed to model the multiple-parton emissions. Both schemes successfully describe the e+ e- annihilation into hadrons [23], but the case of deep inelastic scattering presents additional complications. 2 THE ORY 22

In the parton shower approach (PS) [24], the initial- and final-state parton emis­ sions are considered separately, thus neglecting the interference between the two. A parton of the proton close to mass-shell6 can produce a QCD cascade with successive branchings, creating on-shell or time-like virtual (m 2 > 0) partons and also, another space-like (m 2 < 0) parton becoming increasingly off-shell. The latter takes part in the hard scattering with the electroweak boson and sees its virtuality relocated on mass-shell or in the time-like domain. Then, all these generated time-like par­ tons produce showers by branching into daughter partons according to the DGLAP evolution equations which is a procedure motivated by the leading-Iogarithm approx­ imation. A Sudakov form factor [25] gives the probability that no branching occurs between a given virtuality and sorne minimum value. The properties of each daugh­ ter parton can be deduced and this procedure is iterated. The virtualities gradually decrease, together with the opening angles, until they fall below a cut-off sc ale m6 2 around 1 GeV •

The Colour Dipole Model (CDM) [26] considers instead that all the radiations originate from the colour dipole formed by the struck quark and the proton remnant. The emission of a gluon can be treated approximately as being produced by quark­ antiquark pairs. The resulting qqg group is considered after as two independent dipoles, one formed with the quark and the gluon, one with the gluon and the antiquark. This pattern is repeated for point-like actors and the transverse extent of the proton remnant is taken into account by considering that only a part of it actually participates in the emission of a gluon. In order to include pro cesses initiated by a gluon (e.g. Boson Gluon Fusion), the first dipole made out of the struck quark and the proton remnant can also radiate the antipartner of the struck quark according to the BGF matrix element. This matching procedure is still governed by the Sudakov form-factor.

6 A particle on mass-shell satisfies E2 - p2 = m 2 where m is its mass. 2 THE ORY 23

Figure 3: For e+e- ---+ hadrons, schematic representation of the Lund String (left) and cluster (right) hadronisation models following a parton shower.

2.2.7 Hadronisation

Once the low momentum transfer regime is attained at a sc ale m6, pQCD fails to provide quantitative predictions and the experimentalist is then forced to use phenomenological models to simulate the hadronisation. This step takes place as a consequence of the confinement property of the strong force: coloured partons are never observed as free particles in the final state. Two main models exist at the moment to de scribe the formation of color-singlet bound states, the hadrons. Both are built on the same hypothesis that the flow of flavour quantum numbers and energy-momentum at the hadron level follows the one at the parton level. For instance, according to this local parton-hadron duality, a jet initiated by a quark of a given flavour has to contain a hadron close to its axis with the same flavour.

The string fragmentation algorithm is based on the assumption of the linear confinement of the colour field [27]. A quark-antiquark pair moving apart from their common vertex has its interaction flux confined to a uniform tube about 1 fm across. As the relativistic flux tunnel stretches, the energy contained in the field increases 2 THEORY 24 at the expense of the quarks' kinetic energy. In the Lund String model [28], the potential energy grows linearly with distance r:

E(r)="".r (34) with a string constant "" of the or der of 1 Ge V / f m. A quark separation of about 2 - 5 fm typically provokes the fragmentation of the string and the formation of new quark-antiquark pairs. The flux tube splits into two independent colour singlets which evolve afterwards with their own colour vertex line. Several of these break­ ings can occur provided that the invariant masses of the newly formed strings are sufficiently large. The gluons are interpreted as 'kinks' in qi]g systems (see left of fig. 3). With these internaI excitations carrying energy and momentum, the string evolution becomes more complex, fragmenting iteratively into sm aller segments until their individual energy is too low to allow the separation of the partons. At this stage, the outgoing hadrons are formed.

The preconfinement of colour [29] in the branching process is the starting point for the development of cluster hadronisation models. The non-perturbative splitting of

all the gluons into qi] pairs (or di-quark anti-di-quark pairs) [30] occurs at the end of the parton shower. As se en on the right of fig. 3, the quarks combine thereafter with the nearest neighbour and form colour singlets with a resulting mass around two or three times mo. These clusters finally fragment into hadrons. Momentum exchanges can occur between them if one is not massive enough to decay into two hadrons. In such case, its mass is adjusted with this mechanism and it becomes the lightest hadron of the appropriate flavour. Regarding the heavy clusters, they experience iterative fissions down to a threshold mass Mf at which they decay isotropically into pairs of hadrons [31]. The value of Mf is tuned to reproduce the experimental data and two input parameters, CLMAX and CLPOW, determine its value by the formula: (35) 2 THE ORY 25 where ml and m2 are the quark masses forming a given cluster.

2.3 The Experimental Observables

Two cross sections need to be calculated for the purposes of the present study. Their ratio will become the observable to be compared between MC and data when performing the measurement. Many addition al aspects of the latter will be specified in this section: the choice of the Breit frame as the reference frame, the use of the KT-cluster algorithm to define the jets and finally, the experimental cuts which restrict the phase space.

2.3.1 Inclusive DIS Cross Section

The first and most natural way of looking at deep inelastic scattering is by considering its totality, without concerns about the final-state. This point of view gives the fully

7 inclusive DIS cross section which we denote by (hot. The presence of the scattered lepton f is therefore the unique signature necessary to define this type of event. The first contribution cornes from the Lü (O( Œ~)) quark-parton model:

(36) presented in fig. 1. At the next-to-Ieading order (O(Œ!)), there are one-Ioop vir­ tuaI corrections (fig. 4) of the previous QPM pro cess and also the real subprocesses involving two partons in the final-state:

f(k) + qi(PO) ---t f(k') + qi(Pl) + g(P2) (37)

f(k) + g(po) ---t f(k') + qi(Pl) + iii(P2) (38)

7 Inclusive means that only one aspect of the event is considered during the selection, whatever the other characteristics. This is opposed to exclusive which implies that different characteristics of an event must be fulfiUed. 2 THEORY 26

l' l' l'

Po Po Po

Figure 4: Feynman diagrams for the three l-loop O( a;) virtual corrections to the leading-order QPM process, known as the QCD Compton and the boson-gluon fusion (fig. 5). Thanks to its fully inclusive nature, the e± P --t e±X scattering can be expressed in closed analytical form [32] after calculating all the diagrams encountered so far:

2 da(ëp) 41m [ 2 2 2 Y 2 ] dxdQ2 = XQ4 y xF1 (x, Q ) + (1 - y)F2(x, Q ) =t= y(l - 2)xF3 (x, Q ) (39)

where x, y and Q2 and the kinematic variables defined in eq. 14. By defining

FL = F2 - 2xF1 , an equivalent expression is obtained:

( 40) with Y± = 1 ± (1 - y)2 and the {Fi} are called proton structure functions, The second term FL reflects specifically the absorption of longitudinally polarised virtual photons and becomes important only at large y, In DIS pro cesses with large Q2 and small x, this part is expected to be negligible, Furthermore, within the QPM (i.e, at first order) , it must vanish according to the Callan-Gross relation [16]:

( 41)

Also, contributions caused by the existence of the ZO boson are strongly suppressed at low Q2. These consequences are shown by developing the expressions for the 2 THE ORY 27

l' l'

Po Po

(a)

l' l'

q

~--+-P2

Po Po

(b)

Figure 5: Feynman diagrams O(a;) for the QCD Compton (a) and BGF processes (b), structure functions [33]:

L Ai(Q2)xf~LO(X, Q2) ( 42) i=q,q,g L Bi(Q2)xf~LO(x, Q2) ( 43) i=q,q,g

The SUffi runs over the active flavours i, the functions f N LO are related to the parton density functions8 and the coefficients Ai and Bi are given by [35]:

SThe functions fNLO are convolutions of the NLO parton density functions with appropriate perturbative coefficient functions, See [34] for details, 2 THE ORY 28

(45) where v and a represent respectively the vector and axial vector couplings. With e f being the charge of the fermion f, the three terms in eq. 44 correspond to the 1 exchange, the 1 - zO interference and the ZO exchange, respectively. The function P( Q2) depends on the ratio of the two propagators involved:

ZEUS

x=O.OO8 x=O.021

1 --

• ZEUS 96197 __ 0 o H194197 !; Fixed Target NLO QCD Fit • MRST99 / CTEQSD

x=O.OS x=O.18

1

Figure 6: Measurements of F~m from ZEUS (solid lines) for various x values are eompared with other experiments (left). Strong sealing violations are observed for

low and high x. For intermediate x values, F~m is seen to seale at large Q2.

(46) 2 THE ORY 29 making XF3, the parity violation contribution, to vanish for Q2 « .M1. Furthermore, we can reduce our scope to the F~m physics, as only the first electromagnetic compo­ nent of Ai remains. This field of study has become in the last decades the theater of remarkable results. In the low and high x region, a clear pattern of scaling violations was revealed: F2m increases or decreases with Q2 depending on the fixed x value. At low Q2, the virtual photon can only resolve constituents of the size of its wavelength: the valence quarks. As the momentum transfer increases, smaller structures start to be probed: quarks emitting gluons or split gluons. These partons had their mo­ mentum fractions x reduced by the presence of aU these inner radiations. Hence, we observe a steep ri se of the gluon and sea quark densities in this region. On the other hand, at large x, where the valence quark density dominates, the opposite behaviour is expected: a decrease of F~m [36]. The measurements shown in fig. 6 were obtained by the ZEUS collaboration and clearly show this pattern.

0.5

0.4 9 0.3 1 ++ Çl .49+ t + IIW2 0.2 1

0.1 w="4 0 0 3 4 5 6 7 8 Q2 IGeV/c)2

Figure 7: These results from BLAC [37] demonstrate an evident scaling behavior m (here F2 is denoted by I/W2) for w = ~ = 4.

Another important feature of the structure functions shown in fig. 7 was predicted 2 THEORY 30 by Bjorken [38, 39] and has been confirmed at SLAC [37, 40]. The scale invariance states that the structure functions should be governed, at finite x and in the high energy regime, by only one dimensionless variable. This fact,

( 47) can be observed at finite Q2 since point-like constituents keep the same structure, regardless of the resolution reached by the probe.

2.3.2 Dijet Cross Section

P,

~-+---Pl •••••• ~-++-PI ••••••

Po Po Po Po

P,

Pl ...... Pl Pl ...... Pl P, P, P,

Po Po Po Po (a) (h)

Figure 8: Virtual corrections O(a;) to the QCD Compton (a) and BGF processes (b).

The second observable was suitably chosen to depend at the leading order on the strong coupling constant. This sensitivity will be exploited later in order to determine

9 a value of the strong coupling constant. The dijet cross section d(J2+1 , characterized

9This standard notation is interpreted as follows: two jets in the final-state together with the proton remnant "+ 1" which is not detected experimentally. 2 THEORY 31 by the presence of two high transverse energy jets, offers this advantage since its main contributions come from the QCD-Compton and BGF pro cesses described earlier (fig. 5). At the next-to-leading order, the previous mechanisms are corrected for the exchange of a gluon in a loop (fig. 8).

The real correction includes the pro cesses with three partons in the final-state becoming dijets events for one of the following reasons: either a pair remains unre­ solved by the jet algorithm or one of the parton is discarded by the experimental cuts. It should be pointed out that this first option brings a dependence on the way the jets are defined by the algorithm (see section 2.3.4). The relevant diagrams are summarized in fig. 9.

P, P,

P, ...... p, p, ...... p, P, P, P,

P, Po P, (al (bl

't--- P,

d--- P,

Po (cl Po

Figure 9: Real corrections O(Q';) to the QCD-Compton and BGF processes which contribute to the dijet cross section when a pair of jets is unresolved. (Note: among the eight permutations existing in each class, only two diagrams are presented.) 2 THE ORY 32

2.3.3 Breit Frame

Dijet studies suggest an observable at a more exclusive level than with the total DIS cross section, offering several alternatives to its definition. One of these choices concerns the reference frame in which the jet se arch should be performed. A maximal separation between the proton remnant and the high ET particles would facilitate the latter. This situation is exposed in fig. 10, where the Breit frame (or brick-wall frame) appears as a good candidate satisfying this condition. In this frame, a space­ like boson with four-momentum q = (0,0,0, -Q) and emitted along the z-direction collides with the proton face to face and no energy is exchanged between the lepton and the hadron sides. These conditions are enclosed in the mathematical form:

2xp+ q = o. (48)

For the Born pro cess shown, the entire hadronic system lS constrained to the z­

axis. The incoming parton, fixed to p = (Q/2, 0, 0, +Q/2), is back-scattered without

transverse momentum: p' = (Q/2, 0, 0, -Q/2). The proton remnant pursues its way in the opposite direction, bringing on the other side the soft particles not related to the final-state which originates from the hard-scattering. By the perfect balance of the transverse momentum, there is no QCD contribution from the QPM in the

Breit frame. Any PT distribution requires at least one 0(a8 ) hard pro cess to allow partons to populate the x-y plane. This property that allows the QCD-Compton and BGF mechanisms at the leading-order makes the Breit frame a privileged choice when considering dijet studies.

2.3.4 Jet Definition

The observable may be built according to jet distributions, but the very definition of a jet is not universal and changes the intrinsic nature of what is analysed. The jet finding algorithm represents more than a passive tool; its validity and performance 2 THE ORY 33

LABFRAME BREITFRAME

+ -q (' ==:I·:;3~)====:~::::::::::::: .... ~ .. ,.... ;~:~ .. :: - ~::Ë~~~~::Ë'f:::::::::~:::?;:::,:,::::::::::::::::::::~;:;!:::: p .. remnant p remnanf .,." )i.,c ,/ e+ ", + .. «.,t' ,,'

Figure 10: Schematic comparison of the Born process seen in the lab or the Breit frame. being evaluated by many criteria. First of alI, its easy implementation has to handle successfully the divergencies by being collinear and infrared safe. According to the cancellation methods used by DISENT (sect. 2.2.5), two parallel partons must be treated in the same way as a combined one. In arder to deal with soft divergencies, the algorithm must remain insensitive to the emission of low energy particles. The counterpart in terms of the experiment concerns the trigger energy threshold of the calorimeter cells and the background noise which should not alter the result. Furthermore, we require that the jet finding algorithm induces small hadronisation corrections, such that the correspondence between the parton level and the final-state hadrons can be reliably made. Though sorne candidates, like the JADE [41, 42] or co ne types [43], have these aspects sufficiently fulfilIed, the longitudinally invariantlO KT-cluster algorithm developed by Ellis and Soper [44] has bec orne over the years a widely used one. The implementation of the KT-cluster algorithm can be summarized in a limited number of steps:

10 Longitudinally invariant means that the defined distance between the jets includes only vari­ ables invariant under boost along the hadron direction, 2 THE ORY 34

1. For each particle in the final-state (the so-called proto-jets), the quantity

(49)

is evaluated and for each pair of this li st of proto-jets, a distance is defined as follows: (50)

where ETi is the transverse energy and 'r/i, the pseudorapidity (see equation 58 for the definition of this quantity) associated to the i th jet. The resolution parameter R plays the role of a jet radius in the 'r/ - cp plane and is usually set to 1.

2. The smallest entry of the set {dij , di} is found and labeled dmin .

3. If dmin is a dkl belonging to {dij }, then proto-jets k and lare merged into a new one according to the Snowmass convention:

E - E + E _ ETk'r/k + ETI'r/1 ri-. _ ETkCPk + ET/CPI Tkl - Tk Tl , 'r/kl - E + E ' 'Pkl - E + E (51 ) Tk Tl Tk Tl

If instead dmin is a dk belonging to {di}' then the corresponding proto-jet k is removed from the list as being "not mergeable" and is added to the list of jets.

4. Steps 1 to 3 are repeated until an particles (proto-jets) are assigned to jets.

At the end of this iterative procedure, a list of jets with increasing values of di = Efi

is created. However, only those with high ET (the last ones added to the jet list) are of physical importance and kept in the final list. This requirement of a minimum transverse energy,

Ejet > Ejet. > 0 T T,mm (52)

removes the singular regions encountered in the integration of the dijet cross section. Such cut is naturally implemented in the KT-cluster algorithm and allows it to satisfy the safety conditions stated in section 2.2.3. The choice of the Kr-cluster algorithm 2 THE ORY 35 is motivated by both theoretical and phenomenological points of view. This subject has been well documented [45, 19, 46] and the principal advantages can be stated as follows:

• infrared and collinear safety at all orders in as insured by the jet resolution variables in eq. 50 and the minimal energy requirement.

• the minimal relative transverse momentum as resolution variable is naturally suggested in the evolution of partonic systems within pQCD and reduces un­ natural assignment of particles to jets.

• jet cross sections are affected by sm aller hadronisation effects, as demonstrated by Monte Carlo studies.

• The four-moment a of the partons in a NLO program or those reconstructed from energy deposits in the detector are treated on the same footing.

• The algorithm avoids by construction the overlapping of jets.

2.3.5 Construction of the Observable

The choice of the observable is primarily motivated from the theoretical point of view. This approach is based on the common knowledge that the dependence on the renormalization scale clearly overcomes the other uncertainties. Nevertheless, considering ratios like the measured dijet fraction R2+1 (Q2)

(53) offers the experimental advantage of reducing largely various sources of systematic errors. In particular, the luminosity simply cancels at the top and bottom, at once removing the necessity of its measurement. As mentioned previously in section 2.3.2, the presence of the dijet cross section in the numerator insures a high sensibility of 2 THE ORY 36 this observable to the strong coupling constant. A detailed study [47] has been performed in order to understand and optimize any jet study in the Breit frame. Based on its conclusions, the minimal energy associated with any jet should stay above 8 GeV in order to guarantee their good reconstruction. Furthermore, the energy ordering of the two jets becomes a concern since sorne configurations lead to infrared sensitive observables. Referring to the discussion dedicated to this subject in [48], an asymmetric cut scenario

E~jet1 > 12 GeV and E~jet2 > 8 GeV (54) is adopted to allow the cancellation mechanism to take place efficiently. The problem arises in the Breit frame because at the leading-order O(a!) and also for the virtual correction O(a;), the condition E!J,jetl = E!J,jet2 is automatically satisfied. This region of symmetric jet energies has to be avoided, the three-partons phase space (real correction) being insufficient in its vi ci nity for the compensation between the real and virtual contributions. Otherwise, the performances of the NLO program are greatly affected and the predictions would not be reliable.

Two momentum scales mustbe provided in any pQCD calculation of DIS pro­ cesses. Their purely technical nature should not alter the values of the associated observable. However, our incapacity to evaluate quantities to an orders does in­ troduce a dependence to these unphysical choices. The factorisation scale J1F has in practice a limited effect on the calculated cross sections due to the compensa­ tion taking place between the collinear counter-term and the PDFs. Concerning the renormalization scale, a computation reaching at least the second order remains compulsory. This is more clearly seen by writing the NLO cross section of eq. 17 (sect. 2.2.3) for a given parton a with the explicit scale dependencies:

(55) where n equals zero or one for the total inclusive and dijet cross sections, respectively. Without the next-to-Ieading term ê~l) which starts to compensate, the amplitude of 2 THEORY 37

dâa could be varied more or less arbitrarily by a different choice of J1R. Throughout this study, the Physical Bcale Argument [49J has motivated our choice of the renor­ malization scale. Noting that higher order perturbative coefficients are polynomials in In(J1/Q) (eq. 11), predictions are not expected to be strongly J1-dependent in the vicinity of Q unless the series intrinsically misbehaves. Though several other options can find sorne theoretical justification on that matter [50, 51, 52], we will consider the typical energy scale which characterizes the hardness of the process. From this viewpoint, the Q value of the specific event will be selected as a reference (central value) for the renormalization scale (p,r;:' = Q).

In terms of the kinematic variables, the cross sections are evaluated in a region in which their sensitivity on J1R does not override the one due to as:

200 < Q2 < 1000 GeV2 and 0 < y < 1. (56)

A standard procedure to assess the renormalization scale dependence of a given cross section is to vary by a factor of two the value of P,R in the calculations. With this argument, fig. 11 shows that a NLO prediction of R2+1 does not exceed 12% of un­ certainty in the lower Q2 region, level which go es down to 8% in the high-Q2 part. In comparison, the same graph exposes the as variation which is of the same magnitude in this kinematic range. The upper limit of the kinematic range considered is set so that the effects related to the Zo propagator can be safely neglected. Actually, this contribution is expected to become significant only for values of Q2 above a few thousands GeV2 [48J.

In a determination of as (1\1z ), the correlations existing between the proton distri­ bution functions and the partonic cross section must be handled appropriately. The correct procedure consists in obtaining the NLO predictions for various as(Mz ), those assumed in the extraction of the different sets of PDFs. The same values are also used to compute the as-factors at the other scales (refer to eq. 6), prior to the convolution with the different contributions of the event. In a last step, the observed 2 THE ORY 38

~ 0.15~~~-----,~~~---,-~~~--,--~~~----,---~--~---,----~~~--,-----~~~-,---~~----, ...

0.1 ...... : .~_'"..... ~.C"C~~<::::~...:.-..~ ~ ;..~R~=Q~~~2 S::::::~~'~"";"'~"";"';""~"';"~-'~"~"";""~""~"':"'J"~"

i 0.05 .. ~.~~.:~~~C·~~~.~~.~~r·~.·~·~·.~.·.:·~·~·~·~·.~r~.~·.·~·.. ··~I·... u ... uu .. uu ...... u a,=fJ.1212 ' , , , , . § -: O'~'; ...... , .. .

El! ... 05": ..~:~=::L.:.:::::C~:.:.:.: ......

-0.1 ...... -.- .•...... _--.

Figure 11: The JLR - related uncertainty (bold lines) for the dijet fraction R2+1 is compared ta its CYs dependence (hatched lines) as a function of Q2. The relative uncertainties are calculated with respect ta R2+1 (JL R = Q, CYs = 0.1163) ratio R2+1 will be compared to these CYs(Mz)-dependent predictions and a QCD fit will provide the value which reproduces the experimental data. 3 EXPERIMENTALSETUP 39

3 Experimental Setup

Having an ide a of the observable of interest, we present it from the point of view of the experiment. After an overview of the HERA ep collider and the ZEUS de­ tector, the emphasis is given to the components most relevant to this analysis: the central tracking detector (CTD), the calorimeter (CAL) and the trigger system. The information they provide allows the construction of two physical entities: the experi­ mental signatures for the scattered lepton and for the jets. An exhaustive description of the detector can be found in [53].

3.1 HERA Accelerator

HERA InjectionScheme e.penmertt Hall NORTH Hl

Experiment Ha11 EAST HERMES PETRA PETRA HallW HallE HERA 820 GaV protons x 27 5 GaV slsetrons Ecm" 300 GaV L.1.!) 1031 cm-2s- 1 220 bLinches, 29m spaced

Experiment Hall SOUTH ZEUS

Figure 12: Layouts of the HERA collider (right) and its injection facilities (left).

The first lepton-hadron collider in the world, HERA (Hadron Elektron Ring Anlage), is a unique facility located at the DESY (Deutsches Elektronen-SYnchroton) laboratory in Hamburg, Germany. Inside its 6.3 km ring in circumference, situated 15 - 25 meters under the earth's surface, protons are accelerated to 920 GeVll while

llProton energy attained during the 1998-2000 data-taking period. For 1996-1997, they were 3 EXPERIMENTAL SETUP 40 electrons (or positrons) reach 27.5 Ge V in the opposite direction. With this con­ figuration yielding a center-of-mass energy of yS l'V 300 GeV, the high momentum of the hadrons requires superconducting magnets cooled down to 4.2 0 K while, in a different ring, the leptons are steered with conventional magnets.

As shown in the HERA layout of fig. 12, the tunnel consists of four circular arcs (with a radius of 797 m) and four straight segments, each 360 met ers long, where as many interaction points were designed for the experiments. The South and North Halls, siege of head-on collisions between electrons and protons, are occupied by two multi-purpose detectors, ZEUS and Hl respectively. The east-west sections are non-colliding points for the purposes of fixed-target experiments, HERMES and HERA-B.

The le ft diagram of the same figure gives a magnified view of the injection facili­ ties. Before filling the HERA st orage ring, the particles are subject to several phases of pre-acceleration. For the proton, the chain starts in the H- Linac which brings , the ions to 50 MeV. At this point, the electrons are stripped off before entering the proton synchroton DESY III. From the 7.5 GeV reached there, their energy will be further increased to 40 Ge V in PETRA. They are then transferred to HERA where radio frequency cavities complete the acceleration pro cess. The leptons are submitted to a similar series, beginning their course in the linear accelerator LINAC II. Injected into DESY II and after PETRA II, their energy reaches successively 450 MeV, 7.5 GeV and 14 GeV, before being transferred to the HERA storage ring.

10 In HERA, the bunches contain 0(10 ) particles and a distance of 28.8 m sep­ arates them, leading to a bunch crossing time of 96 ns. Among the 210 positions available for the storage, a few are kept empty in or der to study the background. The beam related backgrounds are inspected with bunches in which either the electron or the proton position remains vacant (see fig. 13). AIso, in order to estimate the

accelerated to 820 GeV instead. 3 EXPERIMENTALSETUP 41 effects of cosmic ray muons, a few pilot bunches have neither of the two filled.

Electron Bunch Train r~------~------~ \ up , 00 0 --- 00 0 0 y .... • oC , :: GlUa. ----<~ , : Colllding e bunches 'a "§! 1 § eleelron 1 ----I:>:96ns:<]-- t: 1 I!! .a 1 , :: 1. ~ 'i ~ ZEUS ! Colliding p bunches § ~ § \ , ...... 111 --- 1D n 1 ~~------~------) y HERA Proton Bunch Train center

Figure 13: The bunch distributions in the HERA storage ring (lejt) and the ZEUS coordinate system defined according to the beam directions (right).

3.2 ZEUS Detector

The ZEUS detector was designed to study a wide spectrum of the HERA physics and required the combined efforts of 500 physicists belonging to 50 institutes of 12 nations. Constructed 30 m underground in the South Hall, it occupies a volume of 12 x 11 x 20 m 3 and weights 3600 tons. A cartesian, right-handed coordinate system defines its geometry, the nominal interaction point corresponding to the origin and the z-axis going along the incoming proton direction (see figure 13, right). The x-axis points horizontally towards the center of the HERA ring and the y-axis, upwards. When

polar coordinat es are used, the azimuthal angle

simply under boosts along the z-axis. The rapidity y is such a Lorentz invariant, defined by: = 1 y ~ nE'(E + Pz) (57) 2 - Pz 3 EXPERIMENTAL SETUP 42

Overview of the ZEUS Deteclor ( longitudinal cut )

4

2

-2

1 1 10 m a -5 m

Figure 14: Longitudinal cross section of the ZEUS detector

In the mass m ----+ 0 limit, it becomes the pseudorapidity Tf,

Tf = -ln tanOj2, (58)

which offers boost invariant differences and is still directly obtained from the geom­ etry of the detector (via the angle 0). The boosted center-of-mass also required an asymmetry of the detector in the longitudinal direction. In this view of fig. 14, an imbalance of the components is clearly exhibited, the forward parts (FTD, FCAL or FMUON) possessing a greater thickness. The hermeticity of ZEUS reaches 99.7% of the solid angle and excludes only the very low angles along the beam pipe.

The transverse cross section se en in figure 15 reveals instead a concentric, sym­ metric shape of the central components. Close st to the interaction point are placed the devices specialized for the tracking of charged particles. Since the vertex detector (VXD) has been abandoned in 1996, this task is performed mainly by the central 3 EXPERIMENTALSETUP 43 tracking detector (CTD) and also a new silicon microvertex detector (MVD) which was installed in 2001 close to the beam pipe. They are embedded in a 1.43 Tesla mag­ netic field provided by the superconducting coil around it. The tracking information is complemented in the forward direction by the forward tracking detector (FTD) and in the backward region, by the rear tracking detector (RTD). Both are similar systems of three planar drift chambers, the latter extending the CTD acceptance to

0 0 polar angles between 160 and 170 • Covering a radius of 34 cm around the center of the beam pipe ho le and located between the RTD and the RCAL, the small-angle rear tracking detector (SRTD) measures electrons with even smaller scattering an­ gles. Located around the RCAL beam pipe [54], this detector consists of an array of scintillator strips. Each of its four 24 x 24 cm2 quadrants contains two layers of orthogonally positioned 1 cm wide strips.

Enclosing these chambers, the high resolution uranium Calorimeter (CAL) per­ forms the energy measurements. It is divided in three parts: one for the forward (FCAL), the rear (RCAL) and the central section (BCAL). The latter has a bar­ rel shape and surrounds the 2.46 m long solenoid of 1.91 m in radius. In order to discriminate with more efficiency between low energy « 5 Ge V) electromagnetic and hadronic flows in the rear direction, silicon-diodes coyer the length of the RCAL modules at the shower's maximum position. These devices, located three radiation lengths deep, form the hadron-electron separator (RES). An iron yoke around the whole calorimeter serves as a return path for the solenoid magnetic field, as an ab­ sorber for the backing calorimeter (BAC) which measures the energy leaking from the main calorimetry and as part of the muon spectrometer. In the barrel and rear regions, LSTs (limited strea,mer tubes) are located on the inner si de (BMUI and RMUI) and outer side (BMUO and RMUO) of the yoke for the muon identifica­ tion and their momentum measurement. In order to deal with high muon moment a present in the forward direction, the inner FMUI and outer FMUO incorporate also drift chambers for the tracking. Finally, close to the beam pipe, several detectors are 3 EXPERIMENTALSETUP 44

Overview of the ZEUS Detector ( cross section)

-5 m o Sm

Figure 15: Transverse cross section of the ZEUS detector placed to perform specifie tasks. For instance, an iron-scintillator Vetowall is posi­ tioned 7.5 m upstream of the interaction point to reject beam-related background.

The lead-scintillator C5 beam monitor located at z = -3.15 m analyses the bunch shapes and determines the nominal interaction point from the timing measurements. Several other detectors are located particles scattered at small angles and leaving the main detector: the proton remnant tagger (PRT) at z = -5.1 m, the leading-proton spectrometer (LPS) at intervals between 20 and 90 meters and the forward neutron calorimeter (FNC) about 100 m downstream. Among all these components, only the CTD and the CAL come into play in the jet study performed here and hence merit a more detailed description. 3 EXPERIMENTALSETUP 45

3.2.1 Central Tracking Detector (CTD)

The Central Tracking Detector, a cylindrical drift chamber, determines the momen­ tum, the direction and even (via a ~~ measurement [55, 56]) the nature of charged . . 12 particles for a limited momentum range . Its geometry depends directly on the techniques used to extract these informations. With 205 cm in length, it spans 150 < () < 1640 in angular coverage. In between its inner (18.2 cm) and outer (79.4 cm) radii, it contains 72 layers of sense wires grouped into nine superlayers. The positions are arranged such that high-momentum particles travelling in a straight trajectory me et at least two planes of sense wires in each superlayer traversed and also that one layer has a drift time consistent with the ZEUS data rate (rv 10 MHz). The schematic overview of one octant (fig. 16) gives more insights about the radial distribution. While the superlayers with an odd number (called axial superlayers) have aIl their wires kept parallei with respect to the beam, the even numbered ones, denoted stereo superlayers, have them slightly tilted (±5°) from the chamber axis. This allows a z-position determination with a better resolution (rv 1.4 mm) than the method using the time difference of the pulses at the end of the wires (rv 1 - 3 cm) and from which the polar angle ofthe track is calculated. However, a z-by-timing sys­ tem does equip the first three axial superlayers, this information is obtained quickly enough for first level trigger purposes.

A gas mixture of argon, carbon dioxide and ethane fills the chamber in the ratio 83:5:12. Bubbled through alcohol and kept at 3 mbar above the atmospheric pressure, this configuration was motivated by the detector lifetime and safety requirements13 [57]. While crossing the CTD, a charged particle ionises the gas, creating electron-ion

12Due to the 8-bits read-out, there exists saturation for ~~ digitized values exceeding 200 out of 256 bits for the current resolution. This limits the measurement range for the ionization energy loss. 13 However , a pure 50:50 proportion of argon and methane would offer better resolution and less noise. 3 EXPERIMENTALSETUP 46

Outer electrostatic screen

Figure 16: One octant of the CTD. The angle by which the wires are tilted is indicated below each even superlayer. The line segments represent the way the different types of superlayers interpret a straight (high-momentum) track going radially outwards. pairs along its path. An electric field of 1.82 k V/cm makes the freed electrons drift towards the positive sense wires at a constant velocity ('" 50 JLm/ns) , appropriate for its maximum drifting time (500 ns) and avoiding concerns about the presence of the strong magnetic field. The ions are accelerated in the opposite direction, reaching the negative sense wires. An avalanche process occurs under the action of the electric field in the case of the electrons. Their number is multiplied by a factor

4 of about 10 . This creates sufficiently amplified signaIs, adequate for the readout and digitisation by 8-bit flash analog to digital converters (ADCs).

With the patterns of collected pulses and the knowledge of drift times, the tracks are then fitted according to a 5-parameter helix model [58], typical trajectory of a particle evolving in a magnetic field. The associated curvature is closely related to the transverse momentum PT. In the interval 150 MeV < PT < 5 GeV, the momentum resolution of tracks traversing aU the superlayers was analysed by detailed Monte 3 EXPERIMENTALSETUP 47

Carlo studies [59, 60] and could be parametrized as

a (PT ) = 0.0058pT EB 0.0065 EB 0.0014 in GeV (59) PT PT where the EB sign means summation in quadrature. The first term is attributed to the CTD single-hit position resolution, the constant term to the smearing due to multiple scattering inside the chamber and the final term, more relevant at low PT, to scattering which happens prior the tracking detector.

3.2.2 Uranium-Scintillator Calorimeter (CAL)

Especially in a study concerning jets, the calorimetry plays a crucial role since a sensitivity to both charged and neutral particles is required. In a calorimeter, the identification and the energy measurement of incoming particles is performed by making the incoming radiations interact with sorne heavy material. This general principle requires a thorough understanding of the reactions which occur inside mat­ ter, processes varying according to the type of particle, its energy and the properties of the penetrated volume.

In the case of incident electromagnetic radiations, a cascade of photons, electrons and positrons is created by successive bremsstrahlung emissions and pair productions. The shower pro cess takes place until the critical energy Ec is reached, i.e. when the energy losses through bremsstrahlung and ionisation become equal. At this threshold, the collisions simply overcome the other modes and electron-positron pair production is strongly suppressed.

A hadronic fiow generates a similar showering, governed instead by various inelas­ tic mechanisms leaving the nuclei in an excited state which can decay afterwards. In addition to their more complex nature, the hadronic showers possess different macro­ scopic properties. Propagating much deeper in matter and having a larger transverse spread, the dimension al arguments generally suffices to discriminate their hadronic 3 EXPERIMENTALSETUP 48 origins from electromagnetic sources. Furthermore, the passage of neutrons is mainly disturbed by the elastic or inelastic collisions with the material's nuclei. Because of the short range of the strong force, their efficiency in stopping the neutral particle is limited in comparison with the frequent Coulomb interactions and the possible Cherenkov radiation encountered for charged particles.

0/ p,Y DIMENSIONS: 'Y 6m-6m / l'. " 1 v ' ~~--~~~LI ~~+-~ -I.... l /I-I-+----I

HAC2 HAC1! ~~~..L~1~50~L~E~~~~I~~llll~I~~~~J E /REAR == LE~C~ r- / TRACKING

1----I---E==~3 CENTRAL i TRACKING it~1-1/1 e- i ...--.-p

./ 1 '\. ./ 1 "­ FCAL(7).) BCAL(5}') RCAL(I,}') HAC 1.2: 3.1}' HACI.2: 2.1}' HAC: J.lÀ DEPTH: 1.52m DEPTH: l.08m DEPTH: o.9m

Figure 17: Layout of the CAL and the tracking detectors.

These basic facts are essentially reflected in the design chosen for the CAL. It consists of a sampling calorimeter in which 3.3 mm plates of depleted uranium

238 235 (98.1 % U , 1. 7% Nb and 0.2%U ) and 2.6 mm thick plastic scintillators alter­ nate. This very configuration was finely selected in an effort to render this detector compensating, i.e. to obtain, for the same incident energy, an equal response from the detector for hadrons than for electrons (and hence the electromagnetic parts of the jets). With a ratio of the sampling fractions close to unity (ejh = 1.00 ± 0.02), the CAL guaranties an optimal accuracy in the energy sc ale and a better resolution 3 EXPERIMENTALSETUP 49

in the case of the hadronic showers. The hadronic component of the latter is com­ pensated by enhancing the production of neutrons by the absorber and also their signalleft in the active material through elastic scattering with the free protons. At the same time, the high-Z uranium plates convert most of the low energy photons, reducing the electromagnetic response of the jets in the scintillator plates. With its geometry, the calorimeter offers a good resolution for the electrons

(60)

and worst one for the hadrons

(61) . where the energy is measured in GeV.

As shown in figure 17, the CAL is divided into the forward (FCAL), barrel (BCAL) and rear (RCAL) sections which are associated to the angular ranges given in table 1. Each calorimeter is subdivided into modules (see fig. 18) which are further segmented into 20 x 20 cm2 towers. Longitudinally, the innermost part is occupied by a 25 X o electromagnetic calorimeter (EMC) which fully contains most electron showers. Two hadronic calorimeters (HAC) follow in depth, except for the RCAL which needs only one of those. The longitudinal size of each module is actually de­ termined by requiring a 99% containment of the most energetic showers. Ranging from 30 GeV in the rear to about 800 GeV in the forward direction, they require a depth depending on the polar angle (see table 1). Both the FCAL and RCAL consist of 23 modules, with a height varying between 2.2 m and 4.6 m. The modular struc­ ture of the BCAL is similar though adapted to the central region, with 32 wedge-cut sections tilted by 2.5 0 in

2 sections are projective and vertically segmented into cells of 20 x 5 cm , apart for

the RCAL which has 20 x 10 cm2 rectangle as smallest u~its. With this granularity 3 EXPERIMENTALSETUP 50

C-Ieg

tension 'slrap

PARTICLEV

silicon deteclDr

scintillatQr plaie

DU - plaIe --~~~

EMClower

12t

Figure 18: A calorimeter module of the FeAL.

and the weighted signaIs collected on each side of the cells, the angular precision for particles attains 2 mrad in the e < 30° region and < 5 mrad for e > 30°.

Concerning the readout, the scintillators provide pulses short enough to avoid pile-up effects with a 96 ns bunch crossing time. Furthermore, the timing resolution of the calorimeter being at the nanosecond level, this information can be used to remove efficiently sorne sources of background. A reference time t=O is defined as the time at which particles originating from ep collisions at the interaction point 3 EXPERIMENTALSETUP 51

Section Polar angle Pseudorapidity Depth Position

FCAL 2.2° < () < 39.9° 4.0> 'f} > 1.0 7.1 Àint 234.4 cm

BCAL 36.7° < () < 129.1° 1.1 > 'f} > -0.74 5.3 Àint 134.5 cm

RCAL 128.1° < () < 176.5° -0.72> 'f} > -3.5 4 Àint -160.2 cm

Table 1: Characteristics of the three CAL sections: their respective angular coverage, their maximum depth given in hadronic interaction lengths (À int) and the z-position (minimum distance) relative to the interaction point. arrive at the calorimeter. Co smic rays, for instance, would have energy depositions in the upper and lower parts of the BCAL with a time difference greater than 10 ns. Proton beam-gas collisions taking place behind the RCAL produce particles detected with a negative timing, a signature making these events easy to discriminate against events occurring at the interaction point.

The calibration of the calorimeter is performed in several ways [61]. The electronic system is calibrated alone by known charge injections executed at the level of the front-end cards; this is performed independently of the photomultipliers and the noise contribution. Laser flashes are sent to the PMTs in order to test their response together with the pulse shaping. Adding to these methods the conventional test beams and the use of mobile radioactive sources, the calorimeter can be calibrated on a regular basis. Finally, the natural radioactivity of the uranium pro duces a constant background current in the photomultipliers which provides a channel-by­ channel adjustment of the gain and also a constant monitoring of the detector. The stability of the so-called uranium noise (UNO) allows to identify problematic cells showing deviations from the expected values. 3 EXPERIMENTAL SETUP 52

3.2.3 Trigger Chain

With the high interaction rate given by the 96 ns bunch crossing time of HERA, a full readout of the data cannot be achieved for each event. Furthermore, the high background rate generated reaches 10 -100 kHz and has to be reduced to an output frequency four orders of magnitude smaller (rv 5 - 10 Hz), the one associated with the interesting physics events. This difficult task is performed by the ZEUS trigger system [62, 63] which possesses a complex three-Ievel structure, as represented in figure 19.

The high background rate dominated by proton beam-gas collisions demands a filtering to be performed quickly and at the hardware level. The First Level Trigger (FLT) is designed to lower the raw input rate to 1 kHz. Each detector component has its own FLT implemented in the electronics and stores the data in a 4.4 p,s pipeline.

Within 2 p,s after the bunch crossing, all the local trigger decisions are made and collected by the Global First Level Trigger (GFLT). If the event is accepted, the data (e.g. event vertex, tracks' momenta and calorimeter clusters) is assembled locally in each detector's Second Level Trigger (SLT) and sent to the Global Second Level Trigger (GSLT). Software-based and running on a network of transputers, this step has sufficient time to exploit together the full spatial and energy information of the calorimeter and of the reconstructed tracks. More stringent cuts are imposed against the background and events survive this filter at around 100 Hz. The accepted ones have their information from all the components transferred to an Event Builder which reconstructs the complete events. Then, they are passed to the Third Level Trigger (TLT) which includes parts of the ofRine reconstruction code. Simple jet and electron finders are involved thère, though their attributed efficiency remains high in comparison with other more sophisticated finders (like Sinistra95 in section 3.3.2) which select candidates with greater purity. Running on a computer farm equipped with Intel CPUs, it reduces the rate to 5 - 6 Hz. With a typical size of 150 kB per 3 EXPERIMENTALSETUP 53

ZEUS detector components data rate 10 MHz component front end

4.4 /lSec pipeline readout and 10calFLT

GlobalFLT output rate - 1000 Hz

~~~ ...... ~~

pipelined local SL T

GlobalSLT output rate - 100 Hz

Event BuUder collecting subevents via TP networks - 20 Mbytes/sec

TLT computer farm consisting of 30 Silicon Graphies output rate 5-10 Hz

data transfer to main storage facility 0.5·1 Mbyte/sec

Figure 19: Schematic representation of the ZEUS trigger and the data acquisition system. 3 EXPERIMENTAL SETUP 54 event, this complete information is written on disks via fiber-link connections and available for later omine analysis.

3.2.4 Luminosity Measurement

Though the luminosity is not used during the as determination, it plays an essential role in any cross section calculation and deserves a brief description. Its measurement at HERA [64] is based on the ep-bremsstrahlung process:

e + P -+ e' + 'Y + P (62) where the final-state electron and photon emerge at very smaU angles from the in­ teraction point. With a maximum divergence of 230 J1rad in the horizontal direction

[65], a photon detector is placed along the beam pipe at z = -107 m. It consists

(cm) 50 OS os os BZ BT BU BU BU OR BROR OROR

p , :', , -25 Gamma

BH Detector BH ~:""'~ e -50 OB Electron OL Detector

1 (m)O 1 0 20 30 40 50 60 70 80 90 1 00 11 0

Figure 20: Layout of the detectors performing the luminosity measurement.

of a 18 x 18 cm2 lead-scintillator calorimeter with a total depth around 22 X o and 3 EXPERIMENTALSETUP 55 a resolution of (JE/E = 25%JE(GeV). Referring to fig. 20, this lumi-, detector is protected by a 3 X o lead shield against a large flux of synchrotron radiations (in the keV range). They are produced by the bending dipoles (denoted by BH) which deflect the electrons from the beam pipe towards the center of the HERA ring at z = -22 m. Smalliumi-e calorimeters provide additional information about the par­ ticular bunch crossing (e.g. electron-photon coincidence with E~ + E, = Ee, when a beam's electron with energy Ee emits a photon with energy E, and ends up with

E~), detecting the electron branch with an energy range 0.2Ee ~ E~ ~ 0.9Ee. The luminosity is deduced from R ep , the observed rate of ep-bremsstrahlung events, and

.c = Rep (Jobs (63) bh where (Jb~s is the theoretical cross section corrected for detector effects and cuts acceptance. The latter is semi-classically obtained by the Bethe-Heitler formula [66]:

d(Jbh = 4Ctr2~(Ee + E~ _~) (ln 4EpEeE~ _~) (64) dE, C E,Ee E~ Ee 3 MmE, 2 and within the experimental conditions, it agrees excellently with QED calculations. The radiative corrections to this pro cess amount to -0.3% in the phase space consid­ ered. The uncertainty is hence dominated by the direct measurement of Rep in which the background has to be taken into account. The principal source cornes from beam­ gas bremsstrahlung in which the electron interacts with a nucleus (eP -+ e' Z,). This pro cess possessing a signature identical to ep-bremsstrahlung, its contribution is es­ timated with the help of the pilot electron bunches. The ZEUS integrated luminosity for the 1998-2000 running periods is listed in table 2 and graphically shown in fig. 21.

3.3 Reconstruction of Physical Objects

A parallel can then be established between the entities involved at the level of the outgoing hadrons and what is actually detected. Before considering the two classes of 3 EXPERIMENTALSETUP 56

HERA luminosity 1994 - 2000

";"" 100 100 -..0a. 99-00 e+ ~ ëi) g 80 80 E ::J ....J "0 Q) 60 60 êti.... 0> Q) +-' t: 40 40

20 20

200 400 600 800 Days of running

Figure 21: The luminosity delivered during the 98-00 period.

Period HERA ZEUS ZEUS

1 1 1 6 delivered (pb- ) on tape (pb- ) physics (pb- /10 events) e- 98-99 25.2 17.78 16.67/23.57 e+ 99-00 94.95 73.37 66.04/66.76

Table 2: The luminosity collected by ZEUS in the 1998-2000 data-taking period. The data considered as "useful" for analysis by the offiine quality monitoring is designated by physics. 3 EXPERIMENTALSETUP 57

"physical objects" encountered in this analysis, the most general selection criteria for the events are described. They are generically contained at the TLT level, namely the DIS03 trigger. The energy depositions in the calorimeter are treated with algorithms whose tasks are to discriminate between showers induced by the scattered electron and those forming the proto-jets. These entities are respectively known as Sinistra95 candidates and ZUFOs. Finally, the techniques used to reconstruct the kinematic variables are compared.

3.3.1 Generic DIS03 Trigger

The third level trigger for DIS events consists of 16 subtriggers, but only the generic DIS03 trigger was considered for the current data preselection. Its main features are stated here and details (exact thresholds and box cuts) regarding its complex history can be found on the web page of the High-Q2 Group [67]:

1. One of the relevant FLT slots (30,44, ... ) has fired [68]. The presence of inactive material in front of the calorimeter (see the mapping in fig. 22) makes the scattered electron leave energy deposits in several EMC cells. For instance, three to four cells are typically hit when directed into the RCAL. Patterns are quickly recognized according to the different topologies of the calorimeter parts. The two principal slots are defined by:

• REMC - ISOe x REMCth With slot 30, the REMC-ISOe trigger of the RCAL finds a pattern consistent with an electromagnetic shower, i.e. in up to four trigger towers, a total energy in the EMC section above 2.08 GeV and more than 75% of the energy of the tower. Then, its algorithm considers the surrounding towers which must be considered as "quiet". This requirement is put in coincidence with the REMCth sum condition: the summed energies of the RCAL towers with EMC energy above 625 MeV exceed 3.75 GeV. 3 EXPERIMENTALSETUP 58

• REMC.or.BEMC The slot 44 is fired if the total energy in the RCAL(BCAL) EMC sections is greater than 3.4 GeV(4.78 GeV).

2. At the SLT, global vetoes to the events are applied :

• As mentioned in section 3.2.2, timing cuts help to reject beam-gas events and cosmic rays inconsistent with pro cesses occurring at the interaction

point: ItRCALI < 8 ns, ItFCALI < 8 ns. The latter sources of background

are further suppressed with the ItFCAL - tRCALI < 8 ns and

flIcAL - t13cAL > -10 ns cuts.

• Events induced by CAL photomultiplier sparks are rejected. They are characterized by a large energy deposition in only one of the two PMTs of the cell. By considering also its neighbours and its particular history, the trigger determines whether the signal fakes or reveals an electron. The sparks originate from internaI instabilities, for instance in the high voltage.

• Events accepted by the FLT must not be considered as "empty", i.e.

they satisfy at least one of the following conditions: EREMC > 2.5 GeV,

EBEMC > 2.5 GeV, EFEMC > 2.5 GeV or EFHAC > 2.5 GeV.

• In addition, an E - Pz + 2E')' cut of 29 Ge V is applied where E')' is the measurement in the Lumi-, detector which provides an improved measurement of E - Pz for the SLT. The stilliimited accuracy of E - Pz available at this level justifies the relatively low value for the selection of NC DIS events.

3. At the TLT level, the selection is refined as follows:

• E - Pz + 2E')' > 30 GeV with recalculated values.

• E - Pz < 100 GeV removes overlaid events with very high E - Pz. 3 EXPERIMENTALSETUP 59

• An electron found by ELECT5 or LOCAL [69] with an energy larger than 4 GeV. These electron finders are simpler than Sinistra95 (see next section 3.3.2), but possess higher efficiencies.

• Box eut on the electron position on the RCAL surface around the beampipe hole. For running on the TLT, a special version of the SRTD position reconstruction is used. The actual size and shape of the box cut changes with the running periods, varying between a 12 x 12 cm2 box

and a R = 15 cm circle.

...... ~= 8 ...... 7 ...... 6 ...... 5 ...... 4 ...... 3 2 1 o

3 2 1 0

Figure 22: Mapping of the material in front of the calorimeter in units of radiation lengths (Xo). 3 EXPERIMENTALSETUP 60

3.3.2 The SINISTRA Electron Finder

In our offiine analysis, a more sophisticated algorithm, Sinistra95 [70], is used for the scattered electron recognition. It exploits the fact that the induced electromag­ netic showers possess a distinctive shape. Since this technique requires the ability to distinguish between patterns described by numerous variables, a neural network is adopted as structure. The latter analyses input parameters are extracted from each island [71], an object constructed by energy deposits in adjacent EMC ceIls as depicted in fig. 23. In this pre-processing phase, the center of each island's ceIl is adjusted, along the side with a lower granularity, by weighting the signaIs of the PMTs on both edges. After, the corrected ceIl centers are projected onto a coordi­ nate system which minimises the distorsions caused by the different geometries of the CAL components and the transition regions (super-cracks) between them. The shower axis is determined by joining the event vertex to the center of gravity of these points. Around it, the distribution of the energy deposits is described in terms of 16 moments of Legendre and Zernike polynomials, used respectively for the radial and angular components. Added to the total energy of the island, all these input param­ eters are then analysed by the neural network which determines the probability P that it corresponds to the scattered electron.

Monte Carlo studies (e.g. [72]) have demonstrated that for cluster with a Sinis­ tra95 probability P > 0.9 and an energy above 10 GeV, the selection efficiency with Sinistra95 reaches 95 % in the RCAL and rear BCAL. Towards the forward region and in the super-cracks, it decreases steadily. In particular for high Q2 and high y, sorne contamination originates from neutral pions decaying into two photons and mimicking the showering of an electron. Finally, it is worth mentioning the method

used to determine the position f'clus of the cluster in the CAL since the center of grav­ ity in the projective space is not actually meaningful. A logarithmically weighted 3 EXPERIMENTALSETUP 61 average [73] is performed over the assigned centers of the constituent cells:

L:i W(~ iclus = (65) L:i Wi g max(O, W o + ln L: ~ ) (66) j Ej

th where Ei and Wi are respectively the energy and the weight assigned to the i cell. The parameter Wo is tuned for the different parts of the calorimeter, setting a threshold for the minimum relative energy deposition that a cell must have in order to be included in the cluster. Again for energies above 10 GeV, this position measurement possesses a better resolution than the one provided with the CTD [74].

3.3.3 ZUFOs

When the hadronic system is considered, it has been shown that the use of the tracking information significantly improves its reconstruction [74]. Charged particles with low energy constitute the jets to a large fraction and the superior accuracy of their measurement by the CTD should be exploited. With this principle of combining both CTD and calorimeter informations, an algorithm was developed to create more sophisticated energy fiow objects: the ZUFOs or ZEUS Un-indentified Flow Objects.

Adjacent cells are initially clustered in each separate part of the calorimeter: in EMC, HAC1 and HAC2. The nearest neighbours are connected while those at the corners are not taken in account. The cells islands of the three sections are then combined and form co ne islands which are three dimension al objects and whose positions are determined according to the logarithmic center of gravit y of formula 65. This procedure correctly handles the exponential falloff of the shower energy distribution and avoids biases due to varying cells projectivity as se en by the vertex.

The next step consists in identifying the charged tracks and extrapolating their trajectory to the inner surface of the calorimeter. To be kept in the list of "good" tracks, one has to traverse more than three superlayers of the CTD and possess a 3 EXPERIMENTALSETUP 62

Figure 23: Schematic representation of the information combined into ZUFOs. The clustering of neighbouring calorimeter cells is first performed separately within the EMC and the HAC sections. The cone island, formed with the HACl cell island 1 and with the EMC cell islands 2 and 3, is then matched to tracks. momentum in the range 0.1 < PT < 20 GeV. For those which have passed at least seven superlayers, the upper limit on PT is raised to 25 GeV. Two criteria can signify a matching condition if the distance of closest approach between the extrapolated track and the cone island position is either less than 20 cm or inferior to the maximum radius of the island on a plane perpendicular to the axis drawn from the island center to the vertex. If a co ne island is associated with none of the tracks or more than three, it is simply considered as a neutral energy deposition. On the other hand, a good track which is not matched to any calorimeter object is taken as charged energy coming from a pion.

At this stage, to decide which information is kept becomes a more elaborate issue. In the first case of a one-to-one track-island correspondence, the tracking is privileged provided that both: 3 EXPERIMENTALSETUP 63

• in or der to ensure that the energy deposit in the calorimeter originates from this unique track, we ask for:

Ecal < 1.0 + 1.2 . a( ECal) (67) P P

where the uncertainty is deduced from a(E;al) = E;a1a(p) EB ~a(Ecal). The 20% increase imposed to the ratio resolution is justified by an underestimation of the track's resolution which was determined from test beam data.

• a second requirement is related to the resolution itself, asking for a better performance from the CTD:

a(p) a(Ecal) -- < ---'----'- (68) p Ecal are satisfied. If the shower shape suggests a muon, the object is not treated on the same footing. Instead, the tracking information will be favoured only if Ecal < 5 GeV, Eca/P < 0.25 and PT < 30 GeV. Another exception is applied in the supercracks between the calorimeter regions where the resolution requirement cannot be kept as tight as in eq. 68 and is further loosened by 20%.

The more complex cases with i-to-j island(s)-track(s) matchings are handled in a similar way with the quantities replaced accordingly:

Ecal ---+ L Ecal,i and (69)

and a(p) ---+ L(a(pj))2. (70) j

Regarding the position, the angular information from the trajectory of a single track is kept provided that less than three islands are attached to it. It remains a superior measurement even if the calorimeter energy was preferred. The set of objects which form the output of this procedure are named ZUFOs. 3 EXPERIMENTALSETUP 64

3.3.4 Reconstruction of the Kinematic Variables

Referring to the leading-order process shown in figure 1, only two out of four mea­ sur able quantities are required for a full reconstruction of the kinematic variables.

In the laboratory frame, they are chosen among the set {Be, E~, Ih, Eh}, correspond­ ing respectively to the polar angle and the energy of the scattered electron and the hadronic final state. Actually, the polar angle of the hadronic system coincides with Ih only within the naive quark-parton model. More generally, this quantity is defined by [75] P:],h - 6~ cos Ih = 02' >"2 (71) .LTh+Uh, where the following sums run over the hadronic system only:

# had 6h = L Ei(l - cos Bi) (72) i=l

#had 2 #had 2 P:]"h = ( L EisinBicoscPi) + (L EisinBisin cPi ) , (73) i=l i=l none of them being significantly affected by the loss of particles through the forward beam hole. However, Ih can still be biased by the scattering of particles in the material between the primary vertex and the calorimeter or by backsplash on the calorimeter. To minimise these effects, an algorithm discards CAL clusters below 3 GeV and above an angle Imax. The value of the latter is set to minimise the bias and is determined iteratively with Monte Carlo samples [74].

According to the selected pair of variables, different methods were developed and the possibility to mix them even exists. They aIl possess strengths and drawbacks, making them suitable for sorne specific situations and regimes. For instance, only the Jacquet-Blondel reconstruction method (JB) [76] remains available in the case of charged current DIS events. As the final state neutrino leaves undetected, the hadronic system remains in our hands:

YJB = (74) 3 EXPERIMENTALSETUP 65

(75) where Ee is the incoming electron energy. This first estimation of the inelasticity,

YJB, will become important later in defining the MC validity cut (sect. 4.3). Another approach considers uniquely the electron information E' Yel - 1- 2~ (1 - cos Be) (76) e (77) and its accuracy depends highly on the measurement of the electron's energy, espe­ cially around the kinematic peak region, near the beam energy (27.5 GeV). On the other hand, Yel is found to be a powerful tool in identifying sorne photoproduction background. One option appears particularly efficient for our purposes: the Double Angle method (DA) [75]. It exploits uniquely the angular variables (Be, Ih) which are measured with a better resolution than the energy. Furthermore, it suffers less from detector effects and depends only at the second or der to the energy scale. In this context, the three Lorentz invariants are given by:

sin Be (1 - cos Ih) YDA (78) sin Ih + sin Be - sin( Ih + Be) 4E sin Ih(1 + cos Be) Q1A 2 (79) e sin Ih + sin Be - sin( Ih + Be) Q1A XDA (80) SYDA

One must point out that a sensitivity to the emission of initial (ISR) and final state (FSR) radiations remains. This topie will be treated in details in section 4.2. An important feature offered by the DA method is an alternative expression for the scattered electron four-momentum:

(81) 3 EXPERIMENTALSETUP 66

(E;J"". (GeV)

Figure 24: Monte Carlo comparison of the resolution between the scattered electron energy which is extracted from the DA method (left) and the one measured directly in the calorimeter, but corrected by the presampler and for the presence of dead material

(right). On the x-axis, Etrue corresponds to the energy at the generated level. Only the events without initial-state radiation (ISR) are plotted here. In an event affected by an ISR, the initial electron emits one or many photons before the hard scattering. Its energy lower than the nominal value modifies the whole characteristics (kinematic variables and boost vector) of the DIS event and makes its reconstruction difficult (subject developed in sect. 4.5.2).

(p~,X)DA (E~)DA sin (je cos CPe (82)

(p~,y)DA (83)

(p~,z)DA (E~) DA COS (je (84)

The boost vector which relates the Breit and laboratory frames will be reconstructed by this method. The double angle's performance simply supersedes the direct energy measurement of the scattered electron (see figure 24). 4 DESCRIPTION OF THE ANALYSIS 67

4 Description of the Analysis

This section is dedicated to the various steps which will ultimately lead to: dameas jdQ2 Rmeas(Q2) = __2+'-1-----,._--:- meas (85) 2+1 datot jdQ2 the measured dijet fraction after corrections for known effects. In a sense, the ob­ servable has to be brought to a level at which it can be compared with the NLO predictions. These extrapolations could not be achieved without a necessary tool, the Monte Carlo (MC) simulations. After describing their structure and respective role, the selection procedures leading to the two relevant cross sections are elaborated and MC methods are exploited to apply sever al corrections to these observed quan­ tities. Finally, systematic checks are performed in order to evaluate the amplitude of various sources of experimental uncertainties.

4.1 Monte Carlo Simulations

The limitations of the detector's performance must be taken into account when deal­ ing with recorded data. Unfortunately, neither the complexity of the physics pro­ cesses which are analysed nor their detection allow an analytical treatment of effects like the experimental acceptance and the resolution. For these reasons, Monte Carlo techniques had to come into play by simulating both the events and the response of the detector to them.With this powerful tool, the theories can be tested, their implications being propagated down to the detector level.

The purpose of the MC generators is to create simulated events in which the theoretical models are implemented. Their output consists of a list of four-momenta of aIl the particles belonging to the initial- and final-state. In general, the theory describing the physical interactions can't be included to an orders. Phenomenological models appear then as valid approximations which also reduce the processing time. In this analysis, the Monte Carlo pro gram DJANGO 1.1 served as an interface to 4 DESCRIPTION OF THE ANALYSIS 68 other programs involved in the generation of events. For instance, ARIADNE 4.08 simulates not only the hard scattering, but also the parton cascade according to the colour-dipole model and the hadronisation process with the Lund String model of JETSET 7.41. An alternative consists in the MEPS model of LEPTO 6.5 [77] which uses the exact matrix elements (ME) combined with the parton shower (PS) approach (sect. 2.2.6) ta describe the multi-partonic production. The program HERACLES 4.6.1 includes various electroweak corrections: for single photon emission, for self­ energy and two-bosons exchange diagrams.

An interface called AMADEUS collects aIl these event generators and creates standard MC samples adapted to the ZEUS environment. They are then passed to the program MOZART. Based on GEANT 3.13 [78], it simulates the whole detector response, including the inactive material (fig. 22) and the uranium calorimeter noise. The output of these processed events is given after to a package which reproduces the trigger chain (CZARjZGANA). At this point, the measured and simulated data are treated on the same footing qy ZEPHYR which executes the event reconstruction. In the end, the resulting set of MC events, in the same format as the real data, possesses additionally the true information which initiates them. AlI these steps are accompli shed by the FUNNEL facility which manages the processing operations distributed among several computer farms inside and outside the DESY site. For the omine analysis, a set of libraries called EAZE (for Effortless Analysis of ZEUS Events) was developed in order to facilitate the retrieving of the data and limit the computing effort. Built on this architecture, the ORANGE package was created for the same purpose, incorporating the correction routines and the knowledge acquired by each physic groups. The structure of the ZEUS software for the Monte Carlo production and data analysis is summarized in figure 25. 4 DESCRIPTION OF THE ANALYSIS 69

AMADEUS lliic=J ~ 1 1 physics event generation 1'-c=J 1 ~ 1 MOZART - FLT detector simulation SLT l EVB CZAR: TLT ZGANA + ZGANA_TL T trigger trigger simulation 1 ZEPHYR evcnt reconstruction ~ EAZE/ORANGE

offline event analysis

Figure 25: Schematic representation of the ZEUS software environment.

4.2 DIS Selection Cuts

The selection of NC DIS sample starts with the Data Quality Monitoring (DQM) which discards the real data taken when the ZEUS conditions were not optimal, as judged by the offiine DQM routines (EVTAKE). For instance, 16% of the on-tape events are to be rejected for the 1998-2000 running period. Also, the reconstruction procedure is characterized in ZEUS by e.g. a scaling of the CAL energy [79] and the suppression of the uranium noise. Problematic cells are identified by imbalances of the photomultipliers' signaIs on each side, a high mean energy or an unexpected "firing" frequency.

When considering the collected data (simulated or real) , the events have to fulfil first the DIS03 trigger as described in section 3.3.1. The identification of the scattered 4 DESCRIPTION OF THE ANALYSIS 70 electron remains the main signature of deep inelastic pro cesses which is refined in our omine analysis. In order to insure their good reconstruction and a high efficiency in their selection, the following steps are applied to the candidates found by the Sinistra electron fin der (see section 3.3.2):

1. The candidate electron with the highest probability is taken.

2. If the probability of this candidate satisfies P > 0.9 and its energy is greater than 10 GeV, it remains a potential choice for the DIS electron. Otherwise, the who le event is discarded.

3. In order to remove background from photoproduction, Yele associated to this electron should not exceed a value of 0.95 .

4. The purity and efficiency of the electron identification is further increased by imposing cuts on the energy distribution if its polar angle is in the region

• The size of the shower is controlled by considering the energy depositions

inside a cone of llRe = 0.3 in the 'T] - cp plane14 . If this summed energy is less than 90% of the total energy of the candidate, the event is rejected. In the forward regi9n Be < 30°, this requirement is raised to 98%15.

• The isolation of the candidate is quantified by summing the energy not associated to it within a radius of 0.7. If this amount corresponds to more than 10% of the candidate's energy, the event is discarded.

5. A fiducial cut is applied to the position of the scattered electron. It removes the upper part of the ReAL occluded by the cryogenie supply. This region

14In this plane, the energy distributions are analysed aeeording to the distance to the eandidate's position whieh is defined by b..Re = J ('f/ - 'f/e)2 + (CP - CPe) 2 • 15This eut is included for eonsistency reasons though very low statisties are expeeted in this forward region with Q2 < 1000 GeV2 4 DESCRIPTION OF THE ANALYSIS 71

corresponds to a candidate cluster with Ixl < la cm and y > 80 cm.

Once a valid scattered electron candidate satisfies all these requirements, the general properties of the events are considered in or der to reject various sources of background:

IZvtxl < 50 cm The vertex reconstructed by the CTD has to be located close to the nominal interaction point. Events which are not centered see a different detector and could be biased by this modified geometry. The quality of the position measurement is also checked by requiring at least two associated tracks and a X2 jndf below la from the fitting procedure with ndf as the number of degrees of freedom.

38 < E - Pz < 65 GeY This quantity is defined by the following sum over the calorimeter cells: (86) cells

According to the energy-momentum conservation, it should equal 2Eh e ~ 55 Ge V for an hermetic detector. In photoproduction events where the elec­ tron escapes through the beam hole, this value can be considerably lower. The inferior limit suppresses this source of background and also events with an ISR radiation which cannot be detected either. The upper bound allows to remove cosmic ray background.

~ < 3 Geyl/2 The event net transverse momentum Pt is expected to be a JEt for NC DIS events. Considering that the energy measurement experiences fluctuations of the order of y'1!J;, the ratio should remain below 1 in general. However, for cosmic rays and beam-related backgrounds, large imbalances in the energy detection can be produced.

eP ---Pt eP,,! A second electron candidate allows to identify elastic Compton processes

(eP ---> eP"!). This background is characterized by the presence of only two 4 DESCRIPTION OF THE ANALYSIS 72

• ZEUSDATA 20000 15000 c=J ARIADNE MC 15000 10000 10000

5000 5000

0 0 10 20 30 100 120 140

(E:JDA (GeV) 8 e (deg)

15000 4000

10000

2000 5000

0 0

(E:Jcorr (GeV)

15000

10000

5000

0

E;n (GeV) P trk (GeV)

Figure 26: These graphs compare the data and the MC for our total DIS sample with different variables of the scattered electron. On the left, are shown the distributions of the electron energy, extracted from the Double Angle method or after correction. The lower plot corresponds to the energy associated to the electron within a cone of radius 0.3. The control plots on the right-hand side concern the other properties: the

scattering angles (je and

electromagnetic deposits in the calorimeter. The Sinistra candidate with the second highest probability is taken and must fulfil the same requirements about the energy and isolation than for the first electron. On the other hand, its

probability requirement is now raised 99%, the shower size criterion (with Ein )

is only applied if 20 0 < (Je(2) < 140 0 and Yel is not taken into account. If it satisfies these cuts, the energy deposited in the whole calorimeter but not belonging to the two candidates is considered. If this amount is inferior than 4 GeV, the event is discarded .

• ZEUS DATA c:::==J ARIADNE MC --- - ARIADNE MC (hadron)

-. a,. .... ___ _ _.- ...... - . - _.-.- -

-1 -D.8 -D.6 -0.4 -0.2 o 0.2 0.4 0.6 0.8

25000 20000 15000 10000 5000

00 10 20 30 40 10 20 30 Oh (GeV) Pt,had (GeV)

Figure 27: These variables concerning the hadronic system cumulate all the events of the DIS sample. The first graph also include the distribution of rh at the hadron level which differs notably from the reconstructed level in the region close to -1.

AlI the cuts mentioned in this section finally delimit our so-called DIS sample.

It corresponds actually to (J'f~t, the observed inclusive DIS cross section which still 4 DESCRIPTION OF THE ANALYSIS 74 needs to be corrected for detector and QED effects. The control plots appearing in this section compare the Monte Carlo and the data under different aspects at the detector level. The most relevant distributions are grouped into categories and in an the graphs, the simulated events have been normalised. to data.

• ZEUS DATA CJ ARIADNE MC

200 500 600 700 800 900 1000 cr (Getl)

10

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 YDA (Getl)

10 4

10 3

10 2

10

0.1 0.2 0.3 0.4 0.5

X DA (Getl)

Figure 28: Control plots for the three kinematic variables measured with the double angle method (defined in eq. 78): QbA' XDA and YDA. 4 DESCRIPTION OF THE ANALYSIS 75

• ZEUSDATA ARIADNE MC

-100 100 Zvtx (cm)

5 10

40000 10 4

3 10

20000 10 2

10

0 1

E-pz{GeV) E nin (GeV)

10 4 4 10

3 10

Figure 29: Five variables permitting cleaning cuts are compared between MC and data. In each plot, the distributions left in white correspond to those Just before the specific cut is applied (i. e. all the other cuts are already applied exeept the one described generally by a line.). . They are respectively: the z position of the vertex,

the E - pz distribution, the energy Enin not associated to the scattered lepton within

a radius of 0.7 (sinee this cut actually depends on Enin , the upper and lower sets

compare respectively the distributions before and after this very cut.), Yel estimated from the electron information and ft. 4 DESCRIPTION OF THE ANALYSIS 76

In figure 26, the scattered electron variables are presented. On the left-hand side, three energy measurements are put on the same scale. While either the corrected energy (E~)corr or the one contained in a co ne of radius 0.3 (Ein ) are described adequately except in the peak region, the energy (E~)DA, deduced from the Double Angle method, is reproduced very well over the whole kinematic range. Since a good agreement is also observed for the scattering angles Be and 4>e, the boost vector of eq. 81 is believed to be reliably reconstructed. This fact is further reinforced by considering the angle "'Ih in figure 27 showing the variables related to the hadronic system. Though no change of the reference frame is required for the definition of the inclusive cross section, this topic will be further discussed in the analysis of the dijets where it becomes a central point (see section 4.3). On the other hand, these nice features are also reflected in the excellent reconstruction of the kinematic variables (fig. 28): QbA and XDA. In the case of YDA, a discrepancy at large y is present, revealing the remaining contamination in the data due to photoproduction processes.

The cleaning cuts of fig. 29 discard the regions left in white. The z position of the vertex is accurately modeled by the MC in the center of the detector. The discrep­ ancy se en upstream (~ -70 cm) is caused by residual beam-gas events. Similarly, co smic rays are not included in the simulation and this contamination in the data is clearly distinguished in the high ./Êt region. The energy not assigned to the electron candidate in flRe < 0.7 and Yel are reasonably well described and their respective cuts won't alter the determination of the acceptances. The same conclusion holds for the distribution of E - Pz, though a known problem with the reconstruction in the FCAL affects the data distribution.

In order to assess the importance of the previous cuts, the original sampIe is defined as all the events fulfilling the DIS03 trigger, possessing a reconstructed vertex within 100 cm of the interaction point, satisfying 32 < E - Pz < 75 GeV, possessing at least a SINISTRA candidate with an energy greater than 10 GeV and lying in 4 DESCRIPTION OF THE ANALYSIS 77

Cut Data Data MC MC Individual Cumulative Individual Cumulative

IZvtxl < 50 cm 4.05% 4.05% 4.48% 4.48% 38 < E - Pz < 65 GeV 4.71% 4.50% 4.53% 4.30%

E nin < 10%E~ 1.05% 0.96% 2.05% 1.52% fiducial cut 1.63% 1.51% 1.58% 1.46%

1 2 ...!l. < 3 GeV / v'Et 0.05% 0.04% 0.40% 0.13% Yel < 0.95 0.01% 0.01% 0.53% 0.19%

E in > 90%Ee 0.03% 0.02% 0.54% 0.08%

eP ~ eP,,! 0.01% 0.01% 0.30% 0.27% Total Il 12.44% 11.10% Table 3: The rejection rate of each individual cut is given when applied to the raw data (216751 events) and MC (568628 events) samples. In the cumulative columns, the cuts are performed after all the others ones above it. The NC DIS selection gives a resulting amount of 189797 data events and 505513 simulated events.

2 the selected phase space (200 < QbA < 1000 GeV ). The influence of each cut is summarized in table 3, leading to a global rejection of about 12% of the original number of events for both simulated and real data.

4.3 Dijets Selection Cuts

A measurement involving dijets is performed at a more exclusive level than the previous cross section. A specific subset of the defined DIS sam pIe is selected. This class of events is primarily characterized by the presence of two and only two jets with high transverse energies. The first step consists in applying the KT-cluster algorithm over the hadronic final state. At the detector level, this corresponds to 4 DESCRIPTION OF THE ANALYSIS 78 the so-called ZUFOs, these physical objects combining the calorimeter and tracking informations (section 3.3.3). Prior to their reconstruction, the cells assigned to the DIS electron candidate have to be removed. At the generated level, what actually belongs to the final state may vary, the time evolution being governed by the decay of the different particles. A hadron is considered as stable if its lifetime is larger than a critical time scale Tc. Otherwise, its decay products are rather inserted in the list of the final state particles. Though many values of Tc could be adopted as valid lifetimes, the one chosen here, (87) presents no drawback with regards to the experimental setup [2].

4.3.1 Selection at the Detector Level

The clustering algorithm is performed as described in section 2.3.4 (in the inclusive mode identified as "3212") and in the Breit frame (see section 2.3.3). For this last reason, the ZUFOs must be transformed from the laboratory to this new reference frame. This boost remains a crucial step which can have drastic consequences to the global appearance of an event. Owing to the good reconstruction of aIl the variables involved in the boost vector (eq. 81), this operation can be legitimately done provided that:

-0.7 < cos "th < 0.5 (88)

YJB ~ 0.04 (89)

The first of these cuts in the phase space removes the problematic region where cos "th is close to 1 [47]. The lower limit avoids the necessity of large extrapolation near cos "th rv -1. This fact is clearly se en in the upper plots of fig. 27 or fig. 30, where the distribution of the true cos "th, exactly known at the generated level, is also given. The Jacquet-Blondel variable is used to reject a phase space region which can't be reached by the simulation. 4 DESCRIPTION OF THE ANALYSIS 79

• ZEUS DATA c:::=::J ARIADNE MC ---- ARIADNE MC (hadron)

-0.8 -0.6 -0.4 -0.2 o 0.2 0.4 0.6 0.8

10 20 30 40 10 20 30

Ôh (GeV) Pt,had (GeV)

Figure 30: These variables concerning the hadronic system cumulate all the events of

the dijet sample. In this control plot, the cuts applied to "(h exclu de the problematic regions (left in white) close to 1 and -1. The distribution at the generated level (hadron) is also shawn ta expose this facto

Just before the clustering is performed, the ZUFOs are boosted to the Breit frame and then scaled so that all the particles become massless. This procedure actually defines the transverse energy throughout this analysis, in analogy with the original partons which are not assumed to be massive in the DISENT predictions. The clustering ereates a set of jets on whieh a preselection eut is applied:

E~jet ~ 3 Ge V. (90)

The massless jets are boosted back to the lab frame and the set {E~,jet,T7fet,

1. If a jet is found with E!j!,jet ;::: 5 Ge V and T/~,jet < -2, the who le event is discarded. Photons emitted by the ·electron are likely to be detected in the very backward region as jets [47]. This erroneous identification greatly alter the Breit frame reconstruction and this effect is not worth being corrected for.

2. If a jet is found with E!J,jet ;::: 5 Ge V and ~Re (j et) < 1 (the jet is within an area

of radius 1 around the electron candidate) in the T/ - cp plane of the laboratory frame, the whole event is discarded. Jets close to the electron candidate are very often a sign of wrong identification: the selected electron is not the scattered one, but a product of the jet. Alternatively, a photon radiated by the scattered electron is considered as the jet.

3. The energy sc ale difference between data and simulated events is reduced by correcting the transverse energy of the jets as prescribed by a detailed study of quantities sensitive to differences in the energy scale [2]. These factors depend on the jet's position in the detector and are applied ta the data anly:

• 1.0101 for jets with -2 < T/jet < 0

• 1.0181 for jets with 0 < T/jet < 1

• 1.0056 for jets with 1 < T/jet < 1.5

• 0.9874 for jets with 1.5 < T/jet < 2

16 4. Once these cleaning cuts have been applied , the energy of the jets is corrected for losses in the inactive material in front of the calorimeter. The correction factors, deduced from Monte Carlo simulations, are applied to both simulated and real datâ. A complete description of the procedure is presented in the next section 4.4.

16Each one of two cleaning cuts (1 and 2) have discarded at this stage 2.8% of the events. 4 DESCRIPTION OF THE ANALYSIS 81

600 600

500 500 • ZEUS DATA 400 c::::J ARIADNE MC 400

300 300

200 200

100 100

0 0 10 20 30 10 20 30 8 E (Ge V) E 8 (Ge V) t,jeU t,jet2 500

400

1 2 2 8 8 11 11 jet1 jet2

Figure 31: Properties of the seZected dijets. The ordering of the jets is do ne according to the position in the Breit frame, i. e. the most forward jet gets the index (~et1". 4 DESCRIPTION OF THE ANALYSIS 82

These steps completed, the corrected jets are then ordered by decreasing energy in the Breit frame. The angular selection is considered first for the final observable in order to avoid undesired cuts of events:

B -2 < Tljet < 1.8 (91)

L -1 < Tljet < 2 (92)

E~,jet > 2.5 GeV (93) where the subscripts Land B refer respectively to the laboratory and the Breit frames. For the pseudorapidity distributions, the cut in the Breit frame is justified in [47] for a more reliable reconstruction of the jets. For the same reason, they must be located in a Tlfecrange in which the detector is optimized to provide better mea­ surements. The third requirement concerns the low transverse-energy jets from the sample which are not weIl reconstructed and hence must be removed for consistency. FoIlowing the theoretical motivation of an asymmetric cut in section 2.3.5 and the insights provided by a detailed study in the Breit frame [47], the two remaining jets with highest energies should satisfy simultaneously:

E~jetl > 12 GeV and E~jet2 > 8 GeV (94)

In order to be considered as a dijet event, two and only two jets must fulfil these conditions: Le. a third jet above 8 GeV leads to the rejection of the event. The energy configuration is chosen as a compromise between sufficient statistics and a reliable reconstruction of the jets. The properties of the two jets in the Breit frame are compared in fig. 31 between MC and data. At this point, the jets are not ordered in terms of the highest and second-highest energy, but rather according to their

pseudorapidity such that Tlf!t1 > Tlf!t2. It should be noted that despite a Monte Carlo generator limited to the first order, the distributions are reproduced adequately. Among 189797 DIS events of the 1998-2000 data taking period, an amount of 2503 dijets (1.3%) are selected. Similarly, a small fraction of the 505513 simulated deep 4 DESCRIPTION OF THE ANALYSIS 83 inelastic scatterings possess two jets: 8605 or 1. 7%. The limited statistics would become a serious issue in any attempt to consider higher orders in as. For instance, in our selected phase space, only 401 out of the 2503 dijets possess a third jet with E,f! > 5 GeV and 133 with E,f! > 8 GeV.

4.3.2 Selection at the Hadron Level

The selection of the dijets at the hadron level differs slightly: the cleaning cuts and energy corrections do not apply. The hadrons, chosen according to their lifetime (eq. 87), are first boosted to the Breit frame and kept massive during this process. Since the true information is available, the boost vector is exactly calculated, without bias from the initial- or final-state radiations (see sect. 4.5.2 for more details about ISR and FSR). Furthermore, the assumption of an energy of the incoming electron equal to the nominal energy is abandoned at the hadron level. A decrease in energy due to a ISR is taken into account while computing the transformation matrix. Another consequence of the complete knowledge of the dynamics of the event, the

quantity cos "(h is rather found as in eq. 15:

xEp + q3 COS"(h = E 0 (95) x p+q

where the boson four-momentum qJL is defined as going from positron to the quark.

This definition from the quark-parton model avoids the use of y which is greatly altered in the radiative events. Also, the momentum transfer Q2 is calculated by

contracting qJL given in the table at the generated level:

(96)

The KT-cluster algorithm is performed on the hadrons whose three-momenta have been scaled, as mentioned before, to render the jets massless. The dijets' selection is then simplified to:

-0.7 < COS"(h < 0.5 (97) 4 DESCRIPTION OF THE ANALYSIS 84

-2 < < 1.8 (98)

L -1 < fJjet < 2 (99)

E:'jet1 > 12 GeV and E:'jet2 > 8 GeV (100)

4.4 Jet Energy Correction

The presence of inactive material in front of the calorimeter decreases the measured energy of the jets by an average of ~ 15% 17. The method adopted here to compensate for this effect relies on the detector simulation and is similar to the one depicted in [80]. The correction is performed with the jets pre-selected in the Breit frame with

E!J,jet > 3 Ge V and boosted back to the laboratory frame. This choice is motivated by a better knowledge of the dead material in the reference frame of the detector, where it simply depends on the position and not on the kinematics of the events. A matching condition is determined by calculating the distance between each pair of jets, one at the hadron, one at the detector level:

(101)

in analogy with the clustering algorithm. The minimum value is taken and if this tlRhd(min) is below 1, the associated couple is included in the appropriate correlation plot and removed from the list of jets. This procedure is iterated until no pair has tlRhd < 1. The fJYet range relevant to this analysis is divided into 12 bins with the following edges:

-1.00 , -0.75 , -0.50 , -0.25 , 0.00 , 0.25 , 0.50 , 0.75 , 1.00 , 1.25 , 1.50 , 1. 75 , 2.00

17The previous energy correction we encountered in sect. 4.3.1 was meant to reduce the bias between the simulation and the real detector from the point of view of their response. When we consider the energy losses due to the inactive material, both MC and data must be corrected on the same footing. 4 DESCRIPTION OF THE ANALYSIS 85

S 50 ,------,-----~ S50r------­ III ~ ~45 O.5

/ ••+ 35

30

25 /.f+ 20 / •• t· /~.t 2nd parametrization 15 ~;., 10

5 ,..... J't parametrization

10 20 30 40..J.hstj 50 10 20 30 Et' (GeV)

S 50r------" S 50 III ~ 45 " ~45~ ..J'Lu.... 1 40 ....' "w- 40~ 35 35

30 30

25 25

20 20

15 15

10 10

5

10 20 30 10 20 30 40..J.hstj 50 t:,' (GeV)

Figure 32: Comparison between the jet energies at the generated and detector levels. The fitted curves give the parameters used in the energy correction. Note that the first few points at low energy are discarded because of the bias produced by the E!J > 3 GeV eut. 4 DESCRIPTION OF THE ANALYSIS 86

1 goCI) -(1) 0.95 r- 0 0.9 r- 0 0 • 0

0.85 r- 0 0 0 0 • 0 • 0.8 r- • • • • • • • 0.75 r- • •

0.7 r-

0.65 -1 -0.5 o 0.5 1 1.5 2 L ll

Figure 33: Slopes used in the correction factors for each range of TJJet. Their inverse is an indication of the amount of material in front of the calorimeter, greater in the transition regions of the detector (see mapping of the inactive material in figure 22) at

TJL ~ -0.75 and 7]L ~ + 1.10. The filled dots correspond to the first parametrization at low E~ on each correlation plot.

Each matched pair is inserted in the right bin(plot) according to their transverse energy(pseudorapidity) at the generated level. The energy boundaries,

0.0 , 0.5 , 1.0 , 1.5 , 2.0 , 2.5 , 3.0 , 3.5 , 4.0 , 4.5 , 5.0 , 5.5 , 6.0 , 6.5 , 7.0 , 7.5 , 8.0 , 8.5 , 9.0 , 9.5 , 10.0 , 11.0 , 12.0 , 13.0 , 14.0 , 15.0 , 16.0 , 17.0 , 18.0 , 19.0 , 20.0 , 22.0 , 24.0 , 26.0 , 28.0 , 30.0 , 33.0 , 36.0 . 39.0 , 42.0 , 45.0 , 50.0 , 60.0 , 80.0 , 100.0 in GeV, get increasingly distant as the statistics become lower. The averages (E~:J:/) and (E~:J::) in each bin are computed, the errors used later in the fit procedure. 4 DESCRIPTION OF THE ANALYSIS 87

Among a dozen of "'let ranges, four are presented in figure 32 with these points. The energy los ses are refiected in the fact that aIl the slopes are clearly below one. The strategy for recovering a distribution eentered around the generated values is to exploit parametrizations of the form: EL,det _ EL,had + b T,jet - m T,jet (102) where, on top. of the pseudorapidity, a dependenee on the energy is added whenever a global function is unable to describe the who le '" range. Henee, the parameters m and b vary also according to ET sinee several curves like eq.102 are traeed. This set of correlations is inverted and provides an expression for the corrected transverse energy of the jets: L ( ) _ E~,jet(ree) - b("'7et, E~,jet(rec)) ET,jet earr - (L L ( )) (103) m "'jet' ET,jet ree from which we deduee the appropriate correction factors: L L E~,jet(carr) E, ("'jet' ET,jet) = EL ( )' (104) T,jet ree where "corr" and "rec" refer to the corrected and reconstructed levels respectively.

The latter can be directly applied to the energies in the Breit frame. The con­ venient transformation of the ZUFOs into massless entities avoids the problem of a third boost, the energies simply factorise out of the four-momenta. The factors, applied for both the Monte Carlo and the data, are selected according to the pseudo­ rapidity measured with the detector, this quantity being weIl reconstructed at this level. The set of slopes m( ",L, E~) is plotted in fig. 33. The transition regions (",L around -0.75 and +1.10), which are occluded by more material, are clearly revealed with the lower values, sign of greater corrections.

4.5 Correction Procedure

The observed cross sections must be corrected bin-by-bin for detector, QED radia­ tion, zO contribution and hadronisation effects. The latter three are not included 4 DESCRIPTION OF THE ANALYSIS 88 in the DISENT NLO predictions and phenomenological (Monte Carlo) methods are required to evaluate their importance. In summary, the procedure consists in deter­ mining the individu al factors which account for each effect:

meas CCC C obs Cl data = det· qed· zo· had· Cl data· (105)

4.5.1 Acceptance Correction

1~------~ - Purity (dcrtoldOZ) 0.9t-f-______-""------.. ,-....- .. ,-E-"'-i-ci-en-c..;.y-~-dcr~t:;;;.:o/I...'d_OZ_) ___ ...,

~ ...... ,II.,.,III.,.III"' ... I1 ... ,II"I.,I1 ... ,."II.1I ... , •• 1111 .... "" .. ".''' .. . , .. ,...... ,...... ,...... ,...... "", ...... ,.. , - Purity (dcr +ldOZ) 0.7 ...... ,,, ...... ,...... "',...... Efficiency2 (dcr +ldOZ) 2

0.6 r- 1

1 : ...... 0.5p'······················\ ...... , ...... :

III 1 1 1 1 200 300 400 500 600 700 800 900 1000 OZ (GeYl)

Figure 34: Efficiency and purity of each bin for the total inclusive cross section (top) and the dijet sample (bottom).

The limitations in the detector's design can introduce important biases in any measured observable and require an evaluation of its capacity to measure the specific pro cesses we analyse, i.e. its acceptance. As mentioned previously in section 4.1, these complex effects are corrected by simulating both the geometry and the response of the detector and comparing with the generated level. The correct way to handle the acceptances is performed by a thorough investigation of the sample in each selected bin. Two quantities become handful when justifying the choice of sorne phase space:

# generated and measured in the bin ef ficiency # generated in the bin # generated and measured in the bin purity (106) # measured in the bin 4 DESCRIPTION OF THE ANALYSIS 89

From a different point of view, the purity and efficiency associated to a given bin refiect the migration of events in and out of its range. Deep inelastic scattering and jet production can be analysed under various aspects, but the observable R2+1 is par­ ticularly weIl designed for the selection of a pure sample. The foIlowing configuration of four bins of Q2 with boundaries:

200 , 300 , 425 , 625 ,1000 in Ge V offers the advantages of approximatively constant purities and efficiencies over aIl the bins. This feature is presented in fig. 34 for both the total inclusive and the dijet samples. Once these facts were determined, the correction factors for each bin are found by taking the inverse of the acceptance:

C _ a'M~ _ # generated in the bin det - det - (107) a MC # measured in the bin and their values are plotted in fig. 35. It should be noted that the factors found do not vary much over our kinematic range and are under control, kept under ;S 20 %.

t 1.25,------~ "0 (.) 1.225:: - Inclusive DIS ~--I!-----< 1 1.2- ...... Dijets 1.175::

1.15 :: I ...... ~ ...... •...... •.. ct :::~:·T·········:···········l·············< !

1.05- 1.025-

1 1 1 1 1 1 1 ~oo 300 400 500 600 700 800 900 1000 cr (Ge~)

Figure 35: Correction factors applied to compensate for the detector's limited accep- tance. 4 DESCRIPTION OF THE ANALYSIS 90

4.5.2 QED Correction

80000r------~

70000

60000

50000

40000

30000

20000

10000

00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 'y

Figure 36: Number of events as function of the fraction f'Y of the incoming electron's energy lost through the emission of a photon (ISR). Only the events with a radiation (J'Y > 0) are considered.

E1ectroweak effects and the running of the e1ectromagnetic constant (Œem) can induce large bias in a measured cross section. The emission of a photon by the incoming e1ectron can have dramatic consequences in the reconstruction of an event, in particu1ar for the boost vector which depends linearly on the beam energy. Our cut on E - Pz actually imposes a limit on the fractiona1 energy f'Y that the e1ectron can 100se: f'Y ;S 0.3 for E - Pz > 38 GeV . (108) 4 DESCRIPTION OF THE ANALYSIS 91

The common way to correct these effects requires the generation of two large Monte Carlo samples: one including all the electroweak pro cesses and one based on QCD only. In the former, the exact kinematics are determined by subtracting the four-momentum of any initial-state radiation (ISR) to Ee, the initial electron's energy. According to the funneled sample, around 25% of our DIS events are affected by QED radiations and their distribution in terms of fr is shown in figure 36. On the other hand, the radiation of a photon by the final-state electron (or FSR) could also affect the kinematics in theory, but in general the finite resolution of the detector do es not manage to resolve them and for such reason this small effect is not considered in the present analysis. Furthermore, the phase space restricted to a medium Q2 < 1000 Ge V 2 allows us to neglect safely the contributions from heavy propagators. Rence, no additional Monte Carlo sample with ZO processes has to be produced and Czo is set to 1. By counting the number of DIS events and dijets with and without QED effects, we obtain the correction factors for each cross section:

(J'l:t~ (noqed) C = ....c:...;~:-:-__ qed (109) (J'l:t~ (qed) after an appropriate normalisation of the samples based on their respective luminos­ ity. The influence of these effects reaches 10% in the case of the total inclusive cross section, as presented in fig. 37.

4.5.3 Hadronisation Correction

In order to bring the measured cross section to a level comparable with the NLO predictions, the transformation of the outgoing partons into colorless hadrons has to be considered. As mentioned in section 2.2.7, the hadronisation model chosen will affect the observables which depend on the hadronic flow. In practice, the parton information is extracted from the MC sample' free of QEDeffects. After applying the same clustering and selection steps to both systems (partonic and hadronic), the ratio of the dijet cross sections is used as a measure of the hadronisation effect: 4 DESCRIPTION OF THE ANALYSIS 92 l . . :8 o.98f------+------t------• "2"" 1 1 ------t------_ .. -- ______.... _...... __ ...... ______.... _ g- 0.961- : c: ~ I~ 0.941- ur1'0.92 1-

0.9~1-___·_------Dijet 0.881- - Inclusive DIS

0.861- 1 1 1 1 1 1 1 20~0~~-L3~0=0~~~4~OO~~~~5~00~~~=60~0~~-L~~0=0~-L~8~OO~~~~90~0~~~1~OOO d (Gell)

Figure 37: Correction factors to compensate for the QED effects.

~ 1.06,------, l! :8 ~ 1.05- ~ ~ 1.04- I~ d 1.03 -

1.02-

1.01-

1 1 1 1 1 1 loo 300 400 500 600 700 800 900 1000 d (Gell)

Figure 38: Correction factors for the hadronisation.

_ aMC\noqe.par 1 d) Chad - (110) a'M~ (noqed) .

The uncertainty related to this procedure is often evaluated by applying the same method for several hadronisation models and by taking the spread of the correction factors as the error. However, the impossibility to obtain a sample with the cluster model as implemented in HERWIG or LEPTO-MEPS has forced the use ofthe unique Lund String model as shown in figure 38. Lead by the fact that this correction 4 DESCRIPTION OF THE ANALYSIS 93 remains smaIl, an uncertainty of 2% is assumed, of the order of those found in a similar analysis [1].

4.6 Experimental Cross Sections and Uncertainties

The experimental differential cross sections of figure 39 are obtained by

d abs N ai (b/Q2) i (111) dQ2 P = 12(pb- l ) BWi(Q2)

th where Ni is the number of events measured in the i bin and BWi, its width. In this formula, the amount of background is neglected and the total integrated lumi­

l nosity12 (after EVTAKE routines) for the years 1998 to 2000 amounts to 82.7 pb- .

Though the luminosity measurement cancels in the ratio R2+1' it adds to the overall experimental uncertainty an error which is correlated between aIl the bins. We esti­ mate the latter to be at the level of 2.1% [81], making a weighted average over the 1998-2000 data taking period:

",2000 (812) r 812) = L...-i=1998 L i X I--i ( r ",2000 r (112) 1-- tat L...-i=1998 I--i

The other sources of uncertainty are detailed here in categories and summarized at the end of this chapter in tables 4, 5 and 6.

4.6.1 Statistical Uncertainty

The limitation in the number of events used to measure a given cross section is at the origin of the statistical uncertainties. More than sim ply due to a limited quantity of collected data, it also rises from a limited amount of simulated events to correct the acceptance, QED and hadronisation effects. The fact that the number of events

(instead than the number of jets) is employed as a definition of the R2+1 simplifies the calculations. However, it should be noted that the strong correlations existing

between the numerator (N2+l ) and denominator (Ntat ) of the ratio must be handled 4 DESCRIPTION OF THE ANALYSIS 94

x 10

10000 .~ i:' .::: QI NLODISEN c:: =O.1112 QI ~ --- a S .Q 10 -- a =O.1163 ... S () • ZEUS DATA a ..... a =O.1212 t s .Q - ARIADNEMC 'b • ZEUS DATA ~ § 8000 c:: -8

datot

6000

4000

-1 .... 10 " ". . " .:., .. ".,. 2000 " , . .:., .. ".:..·.:...~a2+1 " , ..,. ::-.

-2 10 '--_----'__ '---"---'-_'---Jc...... L-' 300 400 500 600 700 800900 cr (Getl)

Figure 39: On the left, comparison between MC and data of the number of DIS events and dijets (scaled by a factor of 10) in each Q2 bin. It should be noted that the MC incorporates only terms of the first order. The measured cross section are presented on the right (after all corrections), compared with three NLO predictions of DISENT

assuming different values of Œs. 4 DESCRIPTION OF THE ANALYSIS 95 properly. Assuming a large number of simulated events giving a well-determined acceptance correction C, the dijet fraction

2 1 2 1 R2+1-· - C ---N + - C N + (113) Ntat N2+1 + N R has a statistical uncertainty given by:

(114) which differs from the typical square root of the number of entries. This source of uncertainty is uncorrelated between the data points due to its statistical nature.

4.6.2 Systematic Uncertainties of the DIS selection

Several checks were performed in order to evaluate the amplitude of various system­ atic uncertainties which could affect the inclusive DIS cross section:

• the cut on Yel is lowered to 0.9 .

• the cut on Enin reflecting the isolation of the scattered electron is varied by

±3% W.r.t. its central value (Enin < lO%E~).

• the effect due to the presence of satellite events is assessed by relaxing the cut

on the vertex position to 1Zvtx 1 < 100 cm.

• the E - Pz lower bound is varied between 35 and 40 GeV.

• the entire analysis is repeated after applying a bias of (+3%,+ 1%, + 2%) and ( -3%, -1 %, - 2%) in the data only to the scattered electron 's energy detected in the F /B/ReAL. The amplitude of the shifts are based on the energy resolutions of the different parts of the calorimeter.

• the effects due to the reconstruction of the kinematic variables are taken into account after smearing the two angles involved in the double angle method by their respective resolution. 4 DESCRIPTION OF THE ANALYSIS 96

Q2 (GeV2) 1Q~t (pbjGeV2) Stat. Syst. ( uncorr. ) Syst. (corr.)

200 - 300 12.00 ±0.05 ±0.04 ±0.03 300 - 425 4.83 ±0.03 ±0.02 ±0.01 425 - 625 2.05 ±0.02 ±0.07 ±0.04 625 -1000 0.72 ±0.01 ±0.02·1O-1 ±0.02·1O-1

Table 4: Measured total inclusive cross sections corrected for detector and QED effects. See caption of table 5 for details.

• for the cleaning cut which removes the contamination of photons in the very backward region, the TJL criteria of the reconstructed jets is changed to -3 and -1.5 .

• the E~ cut setting a minimum jet energy is raised to 4 GeV.

4.6.3 Systematic Uncertainties of the dijet selection

The uncertainty of this observable is governed by the absolute energy sc ale of the detector which is strongly correlated between the bins. For this reason, this quantity is often isolated though belonging to the sources of systematic errors:

• the influence of the absolute energy scale on the dijet cross sections was eval­ uated by varying the transverse energy of the jets by ±2% in the MC only.

• the uncertainty related to the boost is estimated by smearing the three angles (cos Ih, Be, CPe) of the boost vector according to their resolution. 4 DESCRIPTION OF THE ANALYSIS 97

Q2 (GeV2) d;~tl (pbjGeV2) Stat. Syst. (uncorr.) Syst. (corr.)

200 - 300 1.14. 10-1 ±0.05·1O-1 ±0.04·1O-1 (!g:6~) . 10-1 1 1 1 (+0.05) . 10-1 300 - 425 0.70.10- ±0.04·1O- ±0.03·1O- -0.04 425 - 625 0040. 10-1 ±0.02·1O-1 ±0.01·1O-1 ±0.03·1O-1 625 -1000 0.17. 10-1 ±0.01 . 10-1 ±0.01·1O-1 ±0.01 . 10-1

Table 5: Measured dijet cross sections after corrections for detector, QED and hadro­ nisation effects. The systematic uncertainties are divided between correlated (energy scale and luminosity) and uncorrelated sources (to a good approximation, all the other listed checks are considered as uncorrelated errors.).

Q2 (GeV2) d;~tl (pbjGeV2) Stat. Syst. (uncorr.) Syst. (corr.) 2 2 (+0.05) . 10-2 (+0.08) . 10-2 200 - 300 0.95.10- ±0.04·1O- -0.06 -0.05 2 2 2 (+0.08) . 10-2 300 - 425 1.45. 10- ±0.07·1O- (!g:g~) . 10- -0.05 2 2 (+0.09) . 10-2 2 425 - 625 1.95. 10- ±0.11 . 10- -0.14 (:~g:gg) . 10- 2 2 (+0.12) . 10-2 (+0.13) . 10-2 625 -1000 2.33.10- ±0.14·1O- -0.06 \-0.12

Table 6: Measured dijet fractions and their uncertainties. Bee caption of table 5 for details. 5 RESULTS 98

5 Results

The observable R2"':ïs has been measured and the remaining step consists in a com­ parison between the experimental results and the prédictions given by the theory.

The actual method used to extract the value of Œs is elaborated in the present section. Various theoretical and systematic uncertainties are investigated at last.

5.1 Dijet Fraction and NLO QCD Predictions

An amount of 2503 dijets were found among 189797 deep inelastic scattering events in our selected phase space:

200 < Q2 < 1000 GeV2 and 0 < y < 1 (115)

The experimental results can be compared to the theory by running the program DISENT [15] which provides QCD partonic cross sections to the next-to-Ieading order in Œs. In or der to guarantee reliable values for the quantities we calculate, a considerable amount of events has been processed for each single prediction. With at least 70 millions events, the results were found to be stable and did not suffer from awkward divergencies which often can't be attenuated when the statistics are insufficient. For practical reasons, the MRST99 package [82] was chosen as the default PDF set which is convoluted with the partonic cross sections. After analysing the events in a way similar to the hadronic system (see section 4.3) but with partons, three predictions for each Q2 bin were obtained according to the value of Œs(Mz ) assumed by the different PDFs: 0.1112 (MRST99li), 0.1163 (MRST central) and 0.1212 (MRST991i). Both data and prediction points are plotted in figure 40. In this graph, the error bars describe the statistical uncertainty while the band is related to the uncertainty in the energy scale of the calorimeter. The experimental results could favour a particular curve and this fact will be exploited for an Œs determination. 5 RESULTS 99

.... 0.028 of. r:z:.C\4 • ZEUS DATA 0.026 NLO DISENT (Jl,,-,.trQ) aiMz}=O.1212 0.024 aiMz}=O.1163 aiMz}=O.1112

0.022 '" .- '" '" ,,- .- ,,- 0.02 ,,- ,,- ,,- ,,- ,,- ,,- 0.018 ,,- ,,- ,,- / / ,,- 0.016 / / / / / 0.014

0.012

0.01

0.00~0':-::0:-'-'---'-~c':::-::-"--'----'---'---:-'::c-:-'-:-'-L...L-=-=-'---'--'----'--.L.-:-'---L:-'-~-:-'--'---'----'---:-':--::-'---'--L---'---'::--:--'---L:-'-L---:'1000

cjl (GeV)

Figure 40: In the four bins of the range 200 < Q2 < 1000 GeV2 , the measured ratio

RH1 is compared with three NLO predictions of DISENT assuming different values of as. The yellow band indicates the uncerlainty due ta the absolute energy scale of the detector and the error bars stand for the statistical uncerlainties. 5 RESULTS 100

5.2 Determination of Œs

The procedure used to extract the value of as which reproduces the data can be summarized as follows:

1. the ratio R~~i is calculated with three different parton distribution functions of the MRST series which assume different values of as(Mz ).

2. for each Q2 bin, the as(Mz ) dependence of the observable, R2+1' is parametrized according to:

where Al and A 2 are fit parameters.

3. the measured values R2':ï s found in the previous section are compared to these parametrizations for each Q2 bin and via a global X2_fit, a value of as(Mz ) is extracted.

Fig. 41 summarizes these steps. The experimental ratio R2':ï s appears as a large horizontal line between two other sm aller ones which delimit the statistical error band. This measurement is compared to the central parametrization (diagonalline) from predictions using the nominal renormalization scale, i.e. J.LR = Q. Step 3 is repeated with the ratios measured after varying the energy sc ale of the detector by ±2%. This gives basically the correlated uncertainty and the result at this stage reads:

as(Mz ) = 0.1157 ± 0.0017(stat.)~g:gg~î(coTT.) where the correlated uncertainty refer to the energy scale dependence. The effect of all the other systematic checks listed in sect. 4.6 will be treated in the next section, together with the theoretical errors. 5 RESULTS 101

-1 x10 .... 0.125 .... i- i- 0.017 ___ 1 ______'- __ _ rz:.C\l 0.12 rz:.C\l j300<(f<5A~S, , GeV '~=0/2 0.115 .., ------.------.--->------, ------_ ,.. _------0.016 , , , , , , , , , 0.11 , , ,

0.105 0.015

0.1 0.014 0.095

0.09 0.013 ------~::'~q--

0.085 0.012 0.08 0.11 0.115 0.12 0.11 0.115 0.12

...... , , , i- i- ..., _------_._-->------'----, , rz:.C\l 0.022 rz:.C\l 0.028 , 2' 2' 62S

0.017 0.022

0.016 0.021

0.11 0.115 0.12 0.11 0.115 0.12

Figure 41: Four plots corresponding ta each Q2 bin are shawn with three parametriza­ tians of the Œs(Mz ) dependence of the ratio R2~î: for J1R equal ta Q, 2Q and Q/2 respectively. The horizontal band represents the ratio Rr-:ls with its statistical uncer­ tainties. Performing a QCD fit between our measurement and the NLO predictions corresponding ta J1R = Q, we find: Œs(Mz ) = 0.1157 ± 0.0017(stat·)~g:gg~î(corr.) 5 RESULTS 102

syst. check 1 Œs(Mz ) 1 uncertainty 1

Yel 0.1157 0.01%

Zvtx 0.1153 0.32%

+0.1177 +0.09% E'e -0.1156 -0.06% E-pz 0.1154 0.22%

+0.1159 +0.16% Enin -0.1156 -0.01% Ein 0.1157 0.04% smear angles 0.1162 0.48%

+0.1159 +0.01% cleaning cut -0.1156 -0.03% Emin 0.1157 0.03% +0.1198 +3.55% simulation model -0.1117 -3.44% +0.1170 +1.17% hadronisation model -0.1143 -1.16%

+3.77% Total 1 0.1157 1 -3.55%

Table 7: The contribution of each systematic check on the uncorrelated uncertainty

of Œs(Mz ).

5.3 Systematic and Theoretical Uncertainties

Each systematic source of uncertainty of the last chapter were considered in or der to assess the uncorrelated error on our measured value of Œs(Mz ). The variation on Rf":fs they represent was included in the extraction of the strong coupling constant and the amplitudes of these uncorrelated errors are tabulated in table 7.

Several sources of theoretical. uncertainties were finally taken into account and are deseribed here:

• as shown in fig. 41, the renormalization se ale was varied by factors of 2 around

its nominal value (i.e. fJ,R = Q). This procedure is motivated by the fact that 5 RESULTS 103

given an observable R in the perturbative approach,

(117)

the renormalization group equation (eq.5),

2 d J-LR-d2 R = 0 (118) J-LR transforms into the following estimate:

N 2 d "'\:"' iR O( N+l) J-L Rd2 ~ Œs i = Œs (119) J-L R i=ü In other words, varying the scale gives an insight about the amplitude of the terms which are neglected in the calculations. It is by far the most important contribution to the theoretical uncertainty with an effect reaching 5.5% of the value of Œs(Mz ).

• the whole Œs(Mz ) determination was repeated with a different set of proton PDFs, CTEQ4. The impact of an alternative proton distribution function amounts to 2.6% and this effect was assumed to be symmetric around the measured value.

• the factorisation scale was varied between J-LF = Q/2 and J-LF = 2Q. Due to the cancellation operating on this variable (sect. 2.3.5), its effect remains limited

• as a cross-check, another functional form was adopted for the parametrizations

RH 1 ( Œs (Mz ) ). Assuming a linear dependence does not make a significant change in the measured value of Œs(Mz ).

5.4 Discussion of the Results

All the previous sources of uncertainty were added in quadrature in their respective categories, giving the final result: 5 RESULTS 104

This result is consistent with the current PDG world average [12J: Œs(Mz ) 0.1181 ± 0.0020. In fig. 42, it is compared with other Œs(Mz ) determinations per­ formed in ZEUS using different methods. The systematic uncertainty in this context corresponds to aIl the related sources of errors (correlated and uncorrelated) added in quadrature.

From the schematic comparison, sorne of the choices made in this analysis be­ come apparent. Using the greater luminosity of the 1998-2000 running period, the statistical uncertainty is of the same order than another recent study on jets with

E~ > 8 GeV [83J. A study based on dijets has also been done in ZEUS in year 2001 and possesses many aspects of the present analysis. Performed at lower ener­ gies (E~jet1 > 8 GeV and E~jet2 > 5 GeV), its systematic uncertainty is stilliower than the one reached here (see the result of [48J in fig. 42). This fact is explained in great part by an accurate estimation of the hadronisation and the simulation models achieved at that time. Restricting ourselves to values of Q2 below 1000 GeV2 insured important statistics in each bin, but limited greatly our ability to decrease the renor­ malisation scale dependence refiected in the large theoretical error. Concerning the systematic uncertainty, the dijet fraction provides small correlated uncertainties at relatively high energies. Unfortunately, this fact cannot be fully exploited because of the impact of the models used in the simulations which has to be reliably estimated.

Further to dijets, the various analysis presented in fig.42 summarize aIl the

Œs(Mz ) determinations which exploit different aspects of jets' nature. In the first alternative [84J, the inclusive cross section in '"YP interactions is considered. Being in the photoproduction regime, extremely small statistical errors were obtained when extracting the strong coupling constant from the measured dd~et. A comparison with ET the results from the inclusive NC deep inelastic scattering [83J remains difficult since different samples and variables U~2) were employed as observables. Nonetheless, we 5 RESULTS 105

Dijet fraction in Ne DIS ZEUS (current analysis) Inclusive jet cross sections in 'YP ZEUS (phys LeU ~ 560 (2003) 7) Subjet multiplicity in CC DIS ZEUS (Eur Phys Jour C 31 (2003) 149) Subjet multiplicity in NC DIS ZEUS (phys LeU B 558 (2003) 41)

Theoretical uncertainty Jet shapes in NC DIS ZEUS (DESY 04-072 - hep-exl0405065) Statistical uncertainty 1-----1 NLO QCDtit Systematic uncertainty 1 1 ZEUS (phys Rev D 67 (2003) 012007) Inclusive jet cross sections in NC DIS ZEUS (phys LeU B 547 (2002) 164) Dijet cross sections in NC DIS ZEUS (phys LeU B 507 (2001) 70) World average (S. Bethke, hep-exlO211012)

0.1 0.12 0.14

Figure 42: Different techniques performing an CYs determination in ZEUS are com­ pared to the PDG world average [12}. 5 RESULTS 106

can point out that a small theoretical uncertainty was attained despite the fact that inclusive cross sections are less sensitive to the value of as than dijet cross sections. The explanation follows from the important statistics available in the high Q2 region (Q2 > 500 Ge V 2) with one-jet events. Working in this phase space rèduced greatly

the residual dependence on ftR, as seen in the figure.

Many observables can be built from the internaI structure of jets rather than

their rates. A common candidate is named the me an subjet multiplicity (nsbj) which reflects the number of structures present in a jet at a given resolution parameter

Ycut of the clustering algorithm. Both studies [85] and [86] make use of this tool, but suffer greatly from the renormalisation scale dependence of the subjet multiplic­ ity. Furthermore, the latter analysis considers charged current (CC) deep inelastic pro cesses with transfer of W± bosons which are far less numerous than NC DIS. Similarly, the mean integrated jet shape ('ljJ(r)) is defined as the averaged fraction of

jet transverse energy inside the cone r:

('ljJ(r)) = _1_ L ETj~) (120) jets t N Je. s ET

where Njets is the total number of jets in the sample [87]. As expressed by their dashed error bars, these two observables cannot provide a reduction of the theoretical uncertainties and the choice of jets rates remains a justified one. Finally, a global

NLO QCD fit was performed in [88]. This combined fit for as(Mz ) and the gluon and quark densities yields a value for as(Mz ) in accordance with the world average. 6 SUMMARY AND OUTLOOK 107

6 Summary and Outlook

In this document, an analysis of the dijet production in ep scattering was performed with data collected by the ZEUS detector at HERA. Measured in the neutral current deep inelastic regime, the total inclusive and dijet cross sections were obtained after correcting for detector, QED and hadronisation effects. These experimental results agreed well with the next-to-Ieading order pQCD calculations in the selected phase space which limits the theoretical uncertainties coming from the renormalization scale dependence of the predictions. An observable was built as the ratio of the two cross sections as a function of Q2, namely the dijet fraction, in order to compare with the NLO predictions. A QCD fit achieved to consistently incorporate the built­ in dependence of the PDFs on the assumed value of the strong coupling constant leading to their determination. From this procedure, the value of cts(Mz ) which gives the best description of the data could be extracted:

in agreement with the world average cts(Mz ) = 0.1181 ± 0.0020.

The observable in this study was especially defined in a phase space with im­ portant statistics and involved high energy jets of striking events. This reliability from an experimental point of view was made at the expense of theoretical accuracy. In an attempt to reduce the dependence on the renormalization scale, the present technique could be improved in two ways. The first option consists in an extension of the phase space to higher value of Q2, where the fLwdependence falls below the 5% level. In this case, the contribution coming from the heavy propagators (ZO) has to be taken into account as well as the limited number of events available in this region. This last problem would be solved by a sample increased in volume, inte­ grating the 96-97 running period and more importantly, the new data taken after the HERA luminosity upgrade. The second improvement would originate from ex- 6 SUMMARY AND OUTLOOK 108 tended predictions giving a next-to-next-to-Ieading order (NNLO) accuracy. In the last few years, remarkable achievements have been made by theorists to provide this complete NNLO frame which would lower significantly the residual scale dependence [89]. These new calculational tools have already been tested in analysis of the higher orders which include three-jets events ([90], to be published). In conclusion, we have reached a point where the high-energy perturbative regime will allow stringent tests challenging our current understanding of QCD and hence also provide valuable information about the internaI structure of the proton. GL OSSA RY 109

Glossary

ADC ...... Analog to Digital Converter ARIADNE ...... Program involved in the event generation which simulates the hard scattering and the parton cascade according to the color-dipole model BCAL ...... Barrel Calorimeter C5 ...... Bearn monitor, installed in a collimator CAL ...... The uranium CALorimeter CTD ...... Central Tracking Detector CTEQ ...... Coordinated Theoretical-Experimental project on QCD DESY ...... Deutsches Elektronen Synchrotron DGLAP ...... Dokshitser-Gribov-Lipatov-Altarelli-Parisi: evolution scheme, named after the authors, used for initial-state radiation im­ plementing ordering in transverse momenta DJANGO ...... Interface to other programs involved in the generation of events DIS ...... Deep inelastic scattering DISENT ...... Program calculating DIS event features to next-to-Ieading order EAZE ...... Effortless Analysis of Zeus Events EFO ...... Energy Flow Object combining track and calorimeter energy information EMC ...... ElectroMagnetic Calorimeter section FDET ...... Forward tracking detector FLT ...... First Level Trigger FSR Final-State Radiation GEANT ...... Detector description and simulation tool GLOSSARY 110

GFLT Global First Level Trigger GSLT Global Second Level Trigger HAC ...... Hadronic Calorimeter section HERA ...... Hadron-Elektron Ring Anlage HERWIG ...... General purpose generator for Hadron Emission Reactions With Interfering Gluons; based on LO matrix elements, par­ ton showers and a cluster model for hadronisation ISR ...... Initial-State Radiation JETSET ...... Program which implementd the Lund String model for the hadronisation LEPTO ...... Generator of events using the MEPS model LO ...... Leading Order LST ...... Limited Streamer Tube L UMI ...... Luminosity Monitor MC ...... Monte Carlo MEPS ...... Matrix Elements plus Parton Shower implemented in the LEPTO program MOZART ...... MOnte carlo for Zeus Analysis, Reconstruction and Trigger MRS ...... Martin-Roberts-Stirling: a proton density parametrisation named after the authors NLO ...... Next to Leading Order QCD calculation PDF ...... Parton Density Function PETRA ...... Positron-Elektron-Tandem-Ring-Anlage PMT ...... Photomultiplier Tube pQCD ...... Perturbative QCD PYTHIA ...... General purpose generator for e+ e-, pp and ep interactions, based on LO matrix elements, parton showers and Lund hadronisation GLOSSARY 111

QCD Quantum Chromodynamics QED ...... Quantum Electrodynamics RCAL ...... Rear Calorimeter SINISTRA ...... Electron Finder program SLT ...... Second Level Trigger SRTD ...... Small angle Rear Tracking Detector TLT ...... Third Level Trigger VXD ...... Vertex Detector ZEPHYR ...... ZEus PHYsics Reconstruction package ZGANA ...... ZEUS trigger simulation program ZUFO ...... ZEUS Un-identified Flow Objects combining track and calorime­ ter energy information REFERENCES 112

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