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A measurement of the strong coupling constant using two jet event rates at 91 Gev Jones, Matthew 1992

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A MEASUREMENT OF THE STRONG COUPLING CONSTANT USINGTWO JET EVENT RATES AT 91 GEVByMatthew JonesB. Sc. (Physics and Mathematics) University of Victoria, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTERS OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESPHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1992© Matthew Jones, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of	PH Y5T-05(Signature)The University of British ColumbiaVancouver, CanadaDate  g -	 1992 DE-6 (2/88)AbstractAn analysis of hadronic Z ° decays observed in the OPAL detector at LEP ispresented. The analysis involves the selection of hadronic Z ° decays from the set ofOPAL data and the selection of well resolved charged tracks and electromagneticenergy clusters observed in each event. These are then used to determine the rela-tive rates of events with 2-, 3-, 4- and 5-jets using a number of different jet definitionschemes. These rates are then compared with theoretical predictions, computed per-turbatively to second order in the coupling constant a s using the theory of QuantumChromodynamics. By fitting the theoretical predictions to the data, the QCD pa-rameters A and ,u are determined and a value of the strong coupling constant isobtained for each jet definition scheme.In the analysis, the experimental uncertainties in the measured n-jet event ratedistributions are estimated. A particular emphasis is placed on the description ofthe techniques used to correct the measured distributions for the limited acceptanceand response of the detector and for the unknown process of hadronization. Theapplicability of the corrections for detector acceptance and response are justifiedand an examination of the uncertainties in the measured values of a s due to theunknown level of virtuality of the primary partons is presented. It is found thatthe measured values of a s are consisted with other measurements made at LEPenergies and that the values determined in each jet definition scheme are consistentwith each other.iiTable of ContentsAbstract	 iiList of Figures	 vList of Tables	 viiAcknowledgements	viii1 Introduction 12 QCD and n-jet Event Rates 32.1 The Strong Coupling Constant 	 42.2 Hadronic Events and Jets 	 52.3 The JADE-Type Algorithms 	 63 The OPAL Detector and Event Selection 103.1 Central Tracking Chamber 	 103.2 Electromagnetic Calorimeter 	 143.3 Event Selection Criteria 	 143.3.1	 Charged Track Cuts 	 153.3.2	 Electromagnetic Cluster Cuts 	 193.3.3	 Event Selection Cuts 	 204 Experimental n-jet Event Rates and Data Correction 264.1 Bin-by-Bin Corrections 	 264.2 Potential Sources of Bias 	 30iii4.3 Corrected Jet Rate Distributions 	 314.4 Hadronization Corrections 	 335 Experimental Determination of a3 	 95.1 Statistical Correlation in R2 	  395.2 Estimation of Uncertainties in D 2 (y) 	 405.3 Determination of Am---g and ,u 2 from the D2(y) Distributions 	  405.4 Parton Virtuality and Hadronization Effects 	 425.5 Summary of a 3 Measurements	 466 Conclusion	 48Bibliography	 49ivList of Figures2.1 Feynman rules for (a) the quark-gluon vertices in QCD and for comparison withQED, (b) the electron-photon vertex. 	  32.2 A representation of a hadronic event. e+ e — annihilate in the OPAL detectorpossibly with initial state radiation (a) to produce a Z° (b) which subsequentlydecays to a qq, pair. These radiate off gluons to form a parton shower (c). Thefinal state partons form hadrons by way of an incalculable process (d), some of	which decay (e) before being detected    53.1 The OPAL detector 	  113.2 Cross sections of the OPAL detector a) perpendicular and b) parallel to the beamaxis 	 123.3 The two photon process for production of qq states. 	 153.4 Transverse impact parameter distribution for charged tracks. 	  163.5 Transverse impact parameters for real tracks (solid line) and for Monte Carlotracks (dashed line). Arrows indicate the placement of cuts of this parameter. .	 183.6 Longitudinal impact parameters for real tracks (solid line) and for Monte Carlotracks (dashed line). Arrows indicate the placement of cuts of this parameter.	 183.7 Visible energy versus electromagnetic cluster multiplicity. 	 233.8 Momentum balance versus t3 	  24	3.9 Visible energy versus t3    254.1 Experimental n-jet event rate distributions as functions of y,„,t 	  274.2 Event shape distributions for both real events (solid curves) and events simulatedusing the JETSET Monte Carlo followed by a simulation of the OPAL detector(broken curves). (EM-{-CT analysis) 	 324.3 N-jet event rate distributions corrected for detector acceptance and response asfunctions of y,, t . Dashed lines show the uncorrected distributions. (EM clusteranalysis) 	  344.4 Bin coefficients for correcting 2- and 3-jet event rate distributions for detectoracceptance and response. (EM cluster analysis) 	  354.5 N-jet event rate distributions corrected for detector acceptance and responseand for hadronization as functions of yctit . Dashed lines show the uncorrecteddetector level distributions. (EM cluster analysis) 	 374.6 Bin coefficients for correcting 2- and 3-jet event rate distributions for detectoracceptance and response and for hadronization. (EM cluster analysis) 	 385.1 Differential 2-jet event rates compared with fits to 0(a s2 ) calculations. Brokenlines show the one parameter (Am—s-) fits and solid lines shown the two parameter(AKTs-,f) fits. Arrows indicate the regions of y used for the fits. 	 435.2 Variation of a3(Aq0 ) with the parton shower cutoff, Q o , as determined from one-and two-parameter fits to the differential 2-jet distributions	 45viList of Tables2.1 The various 1147j and recombination definitions. In experimental applications,is replaced by Ev2i3 , the visible energy in the event   75.1 Differential 2-jet event rates. 	 415.2 Results of the one parameter fit to D 2 (y) with /22 E Ago . 	 415.3 Results of the two parameter fit to D2(y) with A iw§- and f	 ig/E6m as freeparameters   425.4 Final results of as (M10 ) for different jet definition schemes. 	 46viiAcknowledgementsI would like to express my thanks to the following people:David Axen, for his support in many areas of this research and the graduate programat UBC;Roger Howard, for installing and maintaining the CERN libraries and the OPALsoftware on the IBM 3090 at the university's computing centre;Tom Nicol and Alan Ballard, for maintaining the tape handling software on theIBM 3090;Steve Bougerolle, for copying the 1990-1991 data and sending the tapes from CERNto UBC;Rick Schypit, Randy Sobie and Georges Azuelos, for providing me with informationfrom CERN which proved very useful in understanding this analysis.Without the efforts of any one of them, the work presented here would have notbeen possible.viiiChapter 1IntroductionThe LEP accelerator, which produces Z° particles at rest by colliding high energy electronsand positrons, is perhaps the best facility for studying and testing Quantum Chromodynamics(QCD), the theory of strongly interacting quarks. This is true for a number of reasons. First,due to the narrow width of the Z° resonance and its high cross section, large numbers ofe+ Z° qq events have been produced with well known centre of mass energies. Becauseof the high Z° mass, the velocities of the final state quarks are sufficient to boost them and theirdecay products to form well collimated jets of particles. Also, these quarks have sufficient energyto radiate hard gluons which also form collimated jets of particles. These are the properties ofsuch events which allow one to examine theoretical predictions made about small numbers ofobserved jets rather than very large numbers of observed particles in the detector.One of the fundamental parameters of QCD is the strong coupling constant a 3 which is anal-ogous to the fine structure constant a in QED. In QCD, observables are generally computed as aperturbation series in the coupling constant and hence, a measurement of an observable definedin this way allows one to experimentally determine the value of a,. Furthermore, a 3 is believedto decrease with increasing energy and at ECM = Mzo it should be small enough that observ-ables computed to small orders in a 3 should be able to describe experimental measurementswell. Finally, the presence of jets in hadronic Z ° decays makes it possible to deduce, as directlyas possible, the momentum distributions of the underlying partons, since their momenta areexpected to lie in the directions of the well distinguished jets. This allows QCD predictionsof the momentum distributions of partons to be studied through the observation of jets in thedetector.1Chapter 1. Introduction	 2In this analysis the strong coupling constant was determined by fitting the observed rates oftwo jet events to the QCD predictions. Although a measurement such as this requires a preciseprescription for the way in which two jet events are defined, an unambiguous definition doesnot exist. Instead, a number of algorithms, or schemes, for defining jets were examined anda s was measured using jets defined according to each. These various jet definition schemes,the selection of "good" events for the analysis, the analytical treatement of the measured n-jetevent rate distributions and the determination of a 3 from these distributions are described inthe following chapters.Chapter 2QCD and n-jet Event RatesThe theory of Quantum Chromodynamics (QCD) is believed to describe processes involvingquarks. Within the context of this theory, quarks may be charged with one of three possible"colours" and interact by the exchange of coloured gluons. This is distinctly different fromQuantum Electrodynamics (QED) which describes the interactions of electrons which haveonly one type of electric charge and exchange electrically neutral photons.The Feynman rules for QCD differ from those for QED because of these extra features.The vertices and propagators carry extra factors to keep track of the possible colours of thequarks and gluons and there are additional self-coupling diagrams such as three- and four-gluonvertices. A particular representation of the Feynman rules for QCD may be found in [1] andwill not be reproduced here. However, to introduce the strong coupling constant, it is useful toconsider the vertex factor in the Feynman rules which describe the coupling of quarks to gluonsas shown in figure 2.1.- i .1F-1` ,;5-7 	2*.:)._SULL., a, v	 ,V* 71.,v	 V(a)	 (b)Figure 2.1: Feynman rules for (a) the quark-gluon vertices in QCD and for comparison withQED, (b) the electron-photon vertex.3Chapter 2. QCD and n-jet Event RatesHere, the numerical factor T,ai in the quark-gluon vertex introduces a weight for quarks withcolour indices i and j coupling to a gluon with colour a. The important property of thesediagrams is the presence of the coupling constants a, and a. These are fundamental quantitieswhich characterize the strength of the couplings and are present in all matrix elements andhence, in all expressions for observables computed using the Feynman rules. Their values maybe deduced by actually measuring an observable and solving for a or a,.2.1 The Strong Coupling ConstantWhen Feynman diagrams beyond leading order are considered, processes may be describedby an effective coupling constant which is in fact not constant, but depends on the momentumtransfer, Q 2 involved. In QCD, a 8 (Q 2 ) becomes infinite in the limit as Q 2 — 0 and hence,measurements of a s must be made at some non-zero momentum transfer, u 2 . The expressionfor a 8 (it 2 ) is given to second order in log(a 2 /A 2 ) by[5]:	1 	 log(log(/2/A2)))as(IL 2 ) / A2) ( 1	 bo log(y 2 /A 2) )bo log(it2	 (2.1)33 – 2Nf	where bo127r	(2.2)153 – 19Nfand	 b1	( .3)247 2The QCD scale parameter A is a constant to be determined from experiment and the factor Nfis the number of active light quark flavours which, at V7s = 91 GeV will be treated as Nf = 5.Hence, a measurement of a s at a particular Q 2 = p 2 provides an experimental determinationof the parameter A which may be used to predict the values of a s at different values of Q 2 .As mentioned previously, the coupling constant may be determined by measuring an ob-servable which has been calculated to some order in a, from perturbation theory. However,QCD matrix elements have so far only been computed to second order. and hence can describeprocesses with at most four final state partons. Thus, there is a vast difference in the observ-ables which may be calculated using perturbation theory and those which can be measured,since hadronic Z° decays result in large numbers (>> 4) of hadrons. The useful observables areChapter 2. QCD and n-jet Event Rates 	 5r+ Z°(a)	 (b)K+(d)	 (e)7r —Figure 2.2: A representation of a hadronic event. e+ e — annihilate in the OPAL detector possiblywith initial state radiation (a) to produce a Z ° (b) which subsequently decays to a q4 pair. Theseradiate off gluons to form a parton shower (c). The final state partons form hadrons by way ofan incalculable process (d), some of which decay (e) before being detected.those which are found to be somewhat insensitive to the process by which the initial partonsfragment into the observed hadrons. Examples of such observables are the rates of suitablydefined n-jet events and it is these which are used in this analysis.2.2 Hadronic Events and JetsFigure 2.2 shows a schematic representation of a hadronic Z ° decay and may be describedas follows. An electron and a positron in the LEP ring annihilate in the OPAL detector to forma Z°, after possible emission of initial state radiation. The Z° then decays to a 0 pair whichsubsequently radiate gluons. Since the gluons themselves carry colour charges, they too mayradiate gluons. The resulting process is referred to as a porton shower, in which all partonsChapter 2. QCD and n-jet Event Rates 	 6produced continue to radiate or pair produce more partons. This is believed to continue untilthe invariant masses of the partons drops below some cutoff, Q o at which point the remainingpartons somehow rearrange themselves to form colourless hadronic bound states. The detailsof this hadronization process are not known, but they are modelled in a number of differentways by various Monte Carlo programs.Since 4-momentum is conserved at each parton splitting, the momentum of a primary partonis carried away by the partons into which it splits and hence, by the hadrons into which theyfragment. Since the available Q 2 at each splitting decreases, the transverse momenta of theradiated partons with respect to their parent decreases at each splitting, resulting in collimatedjets of particles whose total momenta reduces to those of the initial partons. Experimentally,this loose description of a jet is unsatisfactory. What is required is a consistent method forpartitioning the observed hadrons into distinct classes which have the desired "jet-like" prop-erties. Since the sum of the momenta of the particles in the jets is expected to represent themomenta of the primary partons, such a definition allows one to compare experimental datawith observables computed from second order QCD matrix elements in cases where fewer thanfive jets are observed.2.3 The JADE-Type AlgorithmsThis section describes the algorithms used to define the jets observed in the detector.The original algorithm was first used by the JADE collaboration[2, 3] in their studies of jetmultiplicities. The algorithms partition an arbitrary list of 4-vectors p2 = (E„ g) into distinctjets by systematically merging pairs of 4-vectors until the scaled invariant mass of any pairbecomes larger than some cutoff parameter, yczit.Chapter 2. QCD and n-jet Event Rates 	 7The details of the algorithms are as follows:i. Compute the "scaled invariant mass" M for each pair of vectors, i,j in the list.Mi2i may be, for example, defined by M i ----- (pi + ps) 2 /E6m .ii. If min is Mi23 is greater than the cutoff, y cut , then terminate the algorithm.iii. Otherwise, "merge" the two vectors pi and pj for which Mid is minimal and replacetheir entries in the list by a new vector pk, thus reducing the number of elementsin the list by one. pk may be defined, for example, simply by pk = pi p3 but otherdefinitions are used.iv. Return to step (i).The scaled invariant mass and the formula for merging two vectors are not uniquely pre-scribed. Instead, there are a number of schemes each with their different prescriptions. Theschemes for defining jets which were examined in this analysis are listed in table 2.1. TheScheme vi Recombination CommentsE (Pi + Pj)2 Is Pk = Pi + pi Lorentz invariantJADE 2EiEs(1 — cos eii)Is Pk = pi + pj Conserves E pP (pi + pj) 2 /s pk = A + 15;Ek = IfklConserves > E but not EP"Durham 2 • min(E,?, ED • (1 — cos eij)IS Pic = P + Pj Conserves E p and avoids ex-perimental problemsGeneva 9E.Ei(1 — cos Oij)/(Ei + Ej) 2 Pk = Pi + pi Conserves E p and avoids ex-perimental problemsTable 2.1: The various /14-223 and recombination definitions. In experimental applications, s isreplaced by E v2, 3 , the visible energy in the event.definitions of MZ which have the centre of mass energy squared, s, in the denominator areslightly modified when the algorithms are applied in experimental situations. In such cases, s isChapter 2. QCD and njet Event Rates 	 8replaced by the visible energy squared Ey2is = (E i Ei ) 2 , computed from the sum of the observedenergies in the list of 4-vectors.Clearly, the number of jets found by the algorithm depends on the quantity ycut . ForYcut << 1, the algorithm will be terminated earlier, leaving a large number of unmerged 4-vectors and hence, a high jet multiplicity. For values of Ycut	1, all vectors will be mergedsince any two tracks from a Z° decay must have a squared invariant mass of less than s =These algorithms allow one to measure the rates of n-jet events as a function of the parameterYcut by counting the number of unmerged vectors which remain at a given value of y cut .To compare these results with the predictions of perturbative QCD, the algorithms may beapplied to the primary partons with momentum distributions computed from the 0(a8 ) QCDmatrix elements. This has been done analytically in the MS renormalization scheme and in thisscheme, the rates of 3 and 4 jet events at a particular scale, maymay be expressed as a functionof y ycut as follows:R3(y) = '73	Ca 27r(12 1 A(y) + (as 27r(P2) ) 2 (A(y) • 2rbo log(,u 2/s) B(y))	 (2.4)atot R4 (y) = aat4at 	( (18(1'121 2C(Y)27r(2.5)where 1)0 was defined in equation 2.2 and the functions A(y), B(y) and C(y) depend on theparticular jet definition used. The two jet event rate is then computed using the unitaritycondition R2 + R3 + R4 = 1 which should be valid only for those values of ycut where thefraction of events with five or more jets is negligible. Thus,R2(y)	(p2 )ot	 2r= 1 — 	 A(y)	 ( a 27r(iu2) ) 2 (A(y) • 2rbolog(,u2/s) B(y) + C(y)) (2.6)at The functions A(y), B(y) and C(y) were tabulated for E, JADE and P schemes in [5] andwere parameterized as fourth order polynomials in log(1/y) with coefficients listed in [6] forthe Geneva and Durham schemes. Thus, by measuring R2 as a function of ycut , the strongcoupling constant, as (,u 2 ) may be obtained as a function of u 2 and AMS by fitting the measureddistributions to those predicted by equation 2.6. The subscript on A indicates the fact that asChapter 2. QCD and n-jet Event Rates 	 9was determined from the rates calculated in the MS scheme and may be different from thosecomputed using different renormalization techniques.Chapter 3The OPAL Detector and Event SelectionThe LEP accelerator produces beams of electrons and positrons and accelerates them toapproximately 45 GeV. These beams circulate in the LEP storage ring and are caused tocollide at four points around the ring where detectors are located. The OPAL detector [4] isone of these and is of a conventional design, intended to serve as a multipurpose apparatus forobserving a wide range of processes. The primary components of the OPAL detector are shownin figures 3.1 and 3.2. The coordinate system used is the polar coordinate system typical ofsuch detectors with e and cb shown in figure 3.1. Positrons enter the detector travelling in thepositive z direction, while electrons travel in the negative z direction.The important components for this analysis are the central tracking chambers and theelectromagnetic calorimeter. These will be described briefly in the following sections. The othercomponents shown in figure 3.1 play no major role in this analysis and will not be describedhere.3.1 Central Tracking ChamberThe central tracking system consists of a high resolution vertex detector, a large volumejet chamber and an array of z-chambers. The vertex detector is located between the beam pipeand the jet chamber while the z-chambers are located between the jet chamber and the coil. Allthree drift chambers are located within a pressure vessel filled with an argon-methane-isobutanemixture at a pressure of 4 bars.The central vertex (CV) chamber is one metre long, 47 cm in diameter and has two layersof drift cells. The inner layer has 36 cells with axial wires and the outer layer has 36 stereo cells10MUON CHAMBERSELECTROMAGNETICCALORIMETERSTIME OF FUGHTAND PRESAMPLERHADRONCALORIMETERSFORWARDDETECTORcrao A - WS! Pei 91 10(16.Pf(rIVIAMP 110y4a)Muon detectorhadron calorimeterReturn yoke*4lead glass calorimeterTime-of-flightSolenoid andPressure vessel_____/Jet-chamberVer tex detectorBeam pipeInteractions regionVertex detectorForward detectorsPre samplerLead glass calorimeterPole tip hadroncalorimeterHadron calorimeterReturn yokeS C quadrupoleMuon detectoraZ-chambersChapter 3. The OPAL Detector and Event Selection	 12Figure 3.2: Cross sections of the OPAL detector a) perpendicular and b) parallel to the beamaxis.Chapter 3. The OPAL Detector and Event Selection 	 13with wires inclined at 4° with respect to the beam axis. Each axial and stereo cell has 12 and6 anode wires, respectively. The anode wires are read at both ends and the z-coordinate of ahit can be determined to an accuracy of 4 cm from the difference in the arrival times of thesesignals. In addition, the stereo wires can determine the z-coordinate of a hit with a resolutionof 700,am and both sets of wires determine the coordinate q with a resolution of 55,am.The central jet (CJ) chamber has a cylindrical active volume about four metres long withinner and outer diameters of 50 and 370 cm, respectively. The chamber consists of 24 sectors,each consisting of a plane of 159 sense wires spaced 10 mm apart between the radii of 255mmand 1835 mm. The position of a hit in the r —0 plane is obtained from the wire position and thedrift time to obtain an average resolution of aro = 135,am. Each sense wire is read out at bothends and the ratio of the integrated charge at each end provides an estimate of the z-coordinateof a hit with an average resolution of a = 6 cm. For tracks with polar angles between 43° and137°, all 159 hits are recorded, provided the track's transverse momentum is sufficient to reachthe outermost layers. In addition, the jet chamber provides dE/dx information, however thisis not used for this analysis.Beyond the jet chamber lie the Z-chambers which provide more precise measurement ofthe z-coordinates of tracks which in turn provides a more accurate measurement of their polarangles. They consist of 24 identical drift chambers, 4 metres long and 50 cm wide. Each chamberis divided into 8 cells in the z-direction with 6 anode wires each, oriented in the 0 directionso that the drift direction is along the z-axis. The z-coordinates of hits are determined withresolutions between 100,am and 200 ,um, depending on the drift distance, while the coordinatein the r — 0 plane is determined by charge division with a resolution of about 1.5 cm. Thepolar acceptance of the z-chambers is 44° < 0 < 136° and tracks with B in this range will haveinformation from all three tracking chambers.Chapter 3. The OPAL Detector and Event Selection 	 143.2 Electromagnetic CalorimeterThe endcap and barrel regions are instrumented with separate, overlapping assemblies oflead-glass blocks fitted with photomultiplier tubes. The barrel has an angular acceptance of35° < 9 < 145° while the endcaps extend this range down to 11° from the beam axis.The barrel assembly is located outside the coil, directly behind the presamplers at a radiusof 245.5 cm. This calorimeter consists of 2455 blocks arranged in an array of 59 blocks in 9by 160 blocks in 0 and give an intrinsic spatial resolution of approximately 11 mm. Theseblocks have an effective depth of 24.6 radiation lengths (X °} and point to positions slightlydisplaced from the interaction point to eliminate cracks through which neutral particles froman event could escape without being detected. The presence of the coil and the pressure vesselcontributes to 2 X 0 of material in front of the calorimeter which degrades energy and spatialresolution of electromagnetic showers.Each endcap assembly consists of 1132 blocks mounted parallel to the beam axis. Theblocks have 52 cm 2 cross sections and effective radiation lengths of typically 22 X o . Theirintrinsic spatial resolution is similar to that of the blocks in the barrel and likewise, there are2 X0 of material in front of these blocks due to the pressure vessel.3.3 Event Selection CriteriaThe multihadronic events recorded by OPAL are often not ideally suited for analysis. Forexample, jets which lie too far forward in 9 may have tracks with far less than the maximum159 hits recorded in the jet chamber which would suffer from poorly measured momentum.Or, a jet might intersect the boundary between the endcap and the barrel — a region wherethe energy resolution is severely degraded. Another possibility is that the multihadronic eventobserved did not have the well defined initial state Z ° q4 but instead took place by way of atwo photon process, e+ e+ + 0 which has the Feynman diagram shown in figure 3.3.Each of these possibilities may be removed from the data sample by applying quality cuts onChapter 3. The OPAL Detector and Event Selection 	 15_e+	 e+Figure 3.3: The two photon process for production of qq states.the charged tracks in the drift chamber and on the energy clusters in the calorimeter and finallyby performing cuts on the overall quality of the events.3.3.1 Charged Track CutsThe cuts used to select good charged tracks are based on the number of hits recorded inthe jet chamber, a minimum momentum transverse to the beam axis and their transverse andlongitudinal impact parameters measured with respect to the event vertex. The intent of thesecuts is to leave a cleaner set of tracks in the event which retain the overall event structure. Forthe analysis performed here, the actual cuts used may not significantly change the shapes of thefinal distributions because of the correction procedures to be described in chapter 4. However,there are obvious cuts which can be made to remove tracks which are clearly not associatedwith a hadronic Z° decay, or which are poorly resolved to the point where their analysis wouldgive little useful information.The impact parameters 4 and 4 are defined as the distances from the event vertex to thepoint of closest approach on the helix fitted to the track's hits in the drift chambers. The signof do is such that 4 < 0 when the track's circle in the r — plane contains the event vertex.*Hence, large, positive values of 4 are associated with secondary decays in the drift chamber*These definitions differ those of d o and zo found in the OPAL DST banks.Chapter 3. The OPAL Detector and Event Selection	 1610-2=10-310-410-5 -10mil -80	 -40	 0 40	 80	 120	 160	 200d0' (cm)Figure 3.4: Transverse impact parameter distribution for charged tracks.and not with the primary Z° decay products. This is clearly shown by the long positive tail ofthe cro distribution shown in figure 3.4.The cuts used in this analysis are similar to those used in other studies of OPAL data. Theyare as follows:i. At least 20 CJ hits.ii. pT > 150 IVIeV/c	 Minimum transverse momentum.iii. 141 < 5 cm	 Maximum transverse impact parameter.iv. 141 < 40cm	 Maximum longitudinal impact parameter.v. 20° < B < 160°	 Polar angular acceptance.Tracks which are poorly reconstructed or which are only partially contained in the jetchamber will have inaccurate momentum measurements. Such tracks are typically associatedwith a low number of hits in the jet chamber and are removed by applying the cut (i) on thenumber of CJ hits. Because the number of drift chamber cells intersected by a track dependson the angle 0, this cut also restricts the polar acceptance of the tracks. The a more definitepolar acceptance cut is provided by (v).Chapter 3. The OPAL Detector and Event Selection 	 17The cut (ii) on transverse momentum effectively rejects tracks which curve in the magneticfield to the point where they do not enter the calorimeter. That is, tracks with transversemomenta less than 150 MeV/c have radii of curvature less than 115 cm and will either spiral inthe drift chamber, or stop in the coil.tCuts (iii) and (iv) are intended to restrict all tracks retained for further analysis to originatefrom either a Z ° decay or from prompt secondary decays. Hence, the cut values must not bechosen too small lest too many tracks from the decays of particles which did originate from theevent vertex be excluded. The distributions are shown in figures 3.5 and 3.6 with the cut valuesindicated. These show the impact parameter distributions for both real and Monte Carlo dataand the approximate agreement between these samples indicates that no significant bias hasbeen introduced by the placement of the cuts.tThe relation pr = (0.2998 MeV/c)Bp is useful here, where p is measured in centimeters and B is in kiloGauss.1010 = -10	 -7.51 2.5 7.5	 10-2.5F-d0' (cm)-110rJ-•-210Chapter 3. The OPAL Detector and Event Selection 	 18Figure 3.5: Transverse impact parameters for real tracks (solid line) and for Monte Carlo tracks(dashed line). Arrows indicate the placement of cuts of this parameter.10 I 	1 h 	 I 	' 	I I  L_J_  i I L 4/' 	1_ 	I	 ; -100	 -75	 -50	 -25	 0	 25	 50	 75	 100zO' (cm)Figure 3.6: Longitudinal impact parameters for real tracks (solid line) and for Monte Carlotracks (dashed line). Arrows indicate the placement of cuts of this parameter.Chapter 3. The OPAL Detector and Event Selection 	 193.3.2 Electromagnetic Cluster CutsThe selection of the significant electromagnetic energy clusters is performed with the sameintent as the selection of charged tracks. That is, an attempt is made to remove clusters whichare not likely to have been caused by the primary Z ° decay products. This is performed by thefollowing cuts which depend on whether the cluster is in the barrel or the endcap calorimeter.i. Erav, > 300 MeV for endcap clusters.Eraw > 100 MeV for barrel clusters.ii. Energy in at least 2 blocks for endcap clusters.iii. When both tracks and clusters are analysed, clusters are required to have no asso-ciated drift chamber track.The cuts (i) on the raw energy measured in the cluster simply remove clusters caused by lowenergy particles or noisy calorimeter channels. The cut (ii) on the number of blocks spanned bythe cluster is not applied to the barrel because of the pointing geometry of the lead-glass blocksin that region. However, there is no pointing geometry in the endcaps and thus, a track fromthe origin would create a cluster spanning more than one block, especially for larger angles O.Cut (iii) suppresses the effects of counting particles twice when they are detected bothin the tracking chambers and in the calorimeter. The track-cluster association is performedoffline after both the charged tracks, presampler clusters and calorimeter clusters have beenreconstructed. Tracks are extrapolated from their last measured point in the drift chamber intothe outer detectors by simulating their propagation using LEANT. At each outer detector, theextrapolated tracks are associated with a cluster if they lie within the angular range spannedby the cluster.Chapter 3. The OPAL Detector and Event Selection 	 203.3.3 Event Selection CutsDue to the long running periods and the nature of the apparatus, not all subdetectors inOPAL remain active all of the time. The redundancy in the OPAL trigger allows runs andthe recording of data to continue even if one of the subdetectors is not active. Before the cutson event quality are made, the status of each of the subdetectors involved in the analysis isdetermined and events are rejected if the essential subdetector elements were not functioningas required. Specifically, events were rejected from the analysis of charged tracks if any ofthe vertex, jet or Z-chambers, were inactive and events from electromagnetic cluster analyseswere rejected if information from the calorimeters or presamplers was not present. With theassurance that the required components were running, events were selected based on the qualityof the data obtained from them.Given that all the required detectors are active the following initial cuts, based on thenumbers of "good" tracks and clusters, are applied:i. At least 5 charged tracks.ii. At least 3 electromagnetic clusters.Cut (i) is applied when the analysis involves charged tracks and (ii) is applied when electro-magnetic clusters are to be analysed. For events which pass these cuts and depending on thetype of analysis to be performed, a list of 4-vectors is constructed from the tracks, clusters ora combination of the two, which have passed their respective quality cuts.Since only momentum is determined from the jet chamber and only energy is measured inthe calorimeter, charged tracks are assumed to be pions and clusters are assumed to be photons.Hence, the 4-vectors (E2, p2) are constructed as follows:E2	\AAP	 for charged tracks, 	 (3.1)=	 E3412'	 for electromagnetic clusters,	 (3.2)where //I is a unit vector from the event vertex to the centre of cluster number j. Once this listChapter 3. The OPAL Detector and Event Selection	 21has been formed, no further distinction is made between electromagnetic clusters and chargedtracks. The remaining cuts are made only on quantities derived from this list of 4-vectors.For electromagnetic clusters, the energy Ei is the corrected energy which is obtained froman empirical function of the polar angle of the cluster, the energy measured in the associatedsubdetectors, such as the presampler, and the raw energy in the lead-glass. The parametersin the empirical formula have been determined from Monte Carlo simulations of the detector'sresponse to electrons in a range of energies and angles. Thus, an estimate of the actual energyof an electromagnetic particle may be obtained even when some energy is not observed sinceshowering may start in the coil or the pressure vessel.While the corrected energy may not be reliable for values below 1 GeV, this should notintroduce any systematic.bias for two reasons. First, the significance of clusters in this analysisis effectively weighted by the cluster energy. Secondly, as will be described in the followingchapter, the measured distributions are corrected for any limited detector response. Thesecorrected distributions should then be free from bias, provided the energy response of thecalorimeter is adequately modelled by the Monte Carlo used in the correction procedure.Next, the following quantities are computed from the contents of the list of 4-vectors:Visible energy: Eiji, = EiMomentum balance: Pbal	 Ei	 >iThrust vector:Invariant hemisphere mass: M+HThe thrust vector is defined as the vector i which maximizes the quantity:T = E	 • il	(3.3)EiFor well collimated two jet events, i points roughly along the jet axis but for events with higherjet multiplicities its direction is less well defined. Even so, it assigns an axis which wouldapproximate the direction of the most well defined jet-like features of the event. Once thethrust vector has been obtained, the tracks in the list may be grouped according to the side ofa plane perpendicular to i on which they lie. The invariant mass in each hemisphere is thenChapter 3. The OPAL Detector and Event Selection	 22calculated as follows: 2	 2	M+(_) = E Ei) - E	 (3.4)+(-)	 +(-)where +(—) denotes the subset of all tracks indices i for which .77, • t > 0(< 0). The followingcuts are then applied to these quantities:iii. Ems > 20 GeV for electromagnetic cluster analysis,Evi, > 20 GeV for charged track analysis,Ev i s > 40 GeV for both CT and EM cluster analysis.iv. Pb a i < 0.4	 Maximum momentum balance.v. 1t31 < cos(43°) = 0.731	 Minimum polar angle of thrust axis.vi. min(M+ , M_) > 2 GeV	 Minimum hemisphere invariant mass.All of these cuts reduce the background from two-photon events which are characterized bylow numbers of tracks or clusters, a thrust axis which lies close to the beam axis and a largemissing energy, since the final state electrons are generally scattered at small angles and escapedown the beam-pipe. Figure 3.7 shows the visible energy versus the number of electromagneticclusters. The large cluster of points in the centre of the plot is due to hadronic Z ° decays whilethe cluster at low energies and low multiplicities corresponds to two-photon events. A similarresult is observed when the visible energy is plotted against the number of charged tracks.These motivate the requirements that events pass cuts (i) and (ii).The correlation between the momentum balance and the direction of the thrust axis areshown in figure 3.8. This shows that events with a large momentum imbalance are generallyassociated events lying close to the beam axis. Such events are generally caused by events whichare not well contained in the detector, or which have a portion of their tracks removed by thetrack quality cuts. This acceptance of partially contained events is prevented by cutting on t 3as shown in the figure. The choice of the t 3 cut value is such that events passing the cut willlie in the barrel region of the calorimeter, where there remains approximately 28° between the60Chapter 3. The OPAL Detector and Event Selection 	 234020EM clusters.	 111 11	 I10	 20	 30	 40	 50	 60Number of clustersFigure 3.7: Visible energy versus electromagnetic cluster multiplicity.Charged tracks + EM clustersFigure 3.8: Momentum balance versus t3.Chapter 3. The OPAL Detector and Event Selection 	 24ca1Erz 0.8E00. -0.75 -0.5 -0.25	 0	 0.25	 0.5120100C.7tem 80ra604020Charged tracks + EM clusters-1Chapter 3. The OPAL Detector and Event Selection 	 25t3Figure 3.9: Visible energy versus t 3 .effective cut on the polar angles of the tracks. This is sufficiently large that entire jets will beretained for further analysis.This effect is also shown clearly in figure 3.9 where the visible energy is plotted against t 3 .This shows a significant drop in visible energy for events with large t 3 which again may beattributed to only partial acceptance of their tracks or clusters. In all, approximately 58,000multihadronic events were analysed of which 40,000 were accepted for the charged track analysis,34,000 for the electromagnetic cluster analysis and 37,000 for the combined analysis.Chapter 4Experimental n-jet Event Rates and Data CorrectionThe lists of good 4-vectors from events which passed the selection cuts were used as inputto the jet finding algorithms based on the E, JADE, P, Geneva and Durham schemes. For theanalysis involving electromagnetic clusters, the measured jet rate distributions as functions ofy,„t in each of these schemes are shown in figure 4.1. These are the raw distributions measuredat the detector level and are biased by the limited detector acceptance and response. Also,they are unsuitable for a direct comparison with the theoretical expressions for R2, R3 and R4found in equations 2.5 and 2.6. This is because the theoretical rates were determined from themomentum distributions of the primary partons, whereas the distributions shown in figure 4.1were measured using the detected particles. These particles are believed to have evolved fromthe initial partons through the incalculable process of hadronization and clearly, the rates atthe detector and parton levels may not necessarily be directly compared.This chapter describes the way in which both the detector and hadronization effects can beremoved by correcting the measured n-jet event rate distributions using Monte Carlo simulationsof the OPAL detector and a model for hadronization. The resulting distributions may becompared directly with the QCD predictions. In principle, the unfolding procedure used tocorrect this data may be applied to any measured distribution and its use and limitations aredescribed in the following sections.4.1 Bin-by-Bin CorrectionsTo illustrate the principle used to correct an arbitrary distribution for limited detectoracceptance and response, it is useful to examine a simple situation. Consider a spectroscopy260.1 0.2y_cutL0	 0.1 0.2y_cutEt	 0.8g	 0.4	 r0E0.80,90.40•-•1 0	 0.1	 0.2y_cut1	. .0 Ls:b-*-4..„,__4_  74-----1-1,----f------4-- -- 4--0.20	 0.1y_cutry40.)E0.8Eea0.40	 0.1	 ---------- 0.2L	y_cut2 jets3 jets• •	 — 4 jets, 	5 jetsChapter 4. Experimental n-jet Event Rates and Data Correction	 27E0.8g 0.4Figure 4.1: Experimental n -jet event rate distributions as functions of yci.t•Chapter 4. Experimental n-jet Event Rates and Data Correction	 28experiment where the energies of gamma rays from a source are measured using a detectorand are histogrammed using some form of multichannel analyser. We assume that the energyspectrum is divided up into N discrete bins and say that the energy E ° of an incident gammaray falls in bin i when Ei_1 < E° < Ez , i 1, N where the Ej's represent the boundariesof the bins in the histogram. With a perfect detector, every incident gamma ray which hasenergy in bin i would contribute one count to bin i in the measured distribution. However, forreal detectors, there are several factors which may cause this gamma ray to contribute to thecontents of some bin other than i. This effect is referred to as migration between bins.First, the intrinsic energy resolution of the detector may cause an incident gamma ray withenergy E° in bin i to be detected in some other bin. The response of such a detector to agamma ray with energy E ° may be modelled, for example, by a Gaussian with a mean atE° and with some width a(E°) representing the detector's intrinsic energy resolution for thatparticular energy. Hence, if this were the only factor limiting the measured energy distribution,monochromatic gamma rays with energy in bin i would sometimes have a measured energy inbin i but could also have energies in bins j i. If, however, a were reasonably narrow, thenone would expect the number of gamma rays with detected energy in bins j < i and j i tobe small.Next, if the detector had some material in front of its active elements, some energy may belost in this material and the detected energy would be less than the incident energy. Thus, formonochromatic gamma rays with energy in bin i, the mean of the detected energy distributiondescribed above would lie in some bin j with j < i.A third factor which could affect the measured distribution is that of limited detectoracceptance. If the detector did not trigger on every gamma ray, then the number of detectedgamma rays would be less than the number which were incident. Furthermore, the acceptancemay be different in different energy regions and a uniform energy distribution of incident gammarays could result in a nonuniform distribution of measured energies.All these detector characteristics may be modelled, in a statistical sense, by a transfer matrixChapter 4. Experimental n-jet Event Rates and Data Correction	 29Gib, the elements of which represent the probability that incident gamma rays with energy inbin j are detected in bin i. Thus, if the source emitted gamma rays with an energy distributiongiven by Ey, then one would detect a distribution given byE fiet = E 	 (4.1)Clearly, if one knew the elements of the matrix G exactly, then one could correct a measuredenergy distribution for the detector's limited acceptance and response to arrive at a distributionEfc)" = E (G-i ) , jErt	 (4.2)Then, one would expect that Ef"r E° 	 the true energy spectrum of the source.One way to determine G -1 would be to simulate the response of the detector to gammarays using a Monte Carlo program. Then, for a large number of simulated incident gamma raywith incident energy in bin i one could count the number detected in bins j for j = 1, . . . , Nand obtain an estimate for the elements Gip With sufficient statistics, one could obtain anestimate for the entire matrix G which could then be inverted to obtain an estimate of G -1 .However, the statistical uncertainties of the matrix elements are only reduced as 1/VIV whenthe number of simulated events, N, becomes large. Due to the large number of arithmeticoperations required to compute the inverse of a matrix, the statistical uncertainties in G -1would be reduced far less rapidly and only for very large Monte Carlo data samples would itbe possible to estimate G -1 reliably in this way.If, however, the matrix G were diagonal, that is, G z3 = b z bi3 , then the inverse G-1 wouldalso be diagonal and could be written (G'), 3 = ci Si3 where c, 1/b,. This matrix couldbe readily determined using the Monte Carlo technique outlined above without the need forenormous sets of simulated data. Furthermore, the bin widths for the energy distribution maybe suitably chosen so that the situation where G is diagonal is realized, at least approximately.The factors c, may then be determined simply as the ratio of the normalized initial distributionand final distributions:(-E;3 / Ezci = d(E2 et Ez E,det)(4.3)Chapter 4. Experimental n-jet Event Rates and Data Correction	 30where EP and Efiet are the simulated incident and detected energy distributions. Once the c i 'shave been determined in this way, the energy spectrum of an arbitrary source may be correctedfor the limited acceptance and response of the detector byEforr = Efiet (4.4)where Elie' now represents the number of real gamma rays detected having energy in bin i.This technique is referred to as bin - by- bin correction.4.2 Potential Sources of BiasThe bin-by-bin correction technique may be reliably applied to distributions for which G isdiagonal. However, in some circumstances, the same technique can be applied to distributionswhere some of the off-diagonal elements of G are non-zero. This situation would arise whenthe bin widths are chosen smaller than the intrinsic resolution of the detector. This doesnot necessarily make the technique outlined above inapplicable, however it does introduce thepossibility of biasing the corrected distributions.This potential for bias arises when the shapes of the simulated and detected distributionsdo not match. To examine this effect more closely let us continue to consider the gamma raysource and detector experiment. When the bin widths are small, there are contributions to thefraction of gamma rays with detected energy in bin i not only from the incident gamma rayswith energy in bin i, but also from those with energies in bins j i which migrate into bini at the detector level. If the fraction which remained in bin i was the same in both the realexperiment and in the simulated data but the fraction migrating into bin i were significantlydifferent, then the correction coefficients ci, computed using equation 4.3 would be determinedincorrectly and would bias the corrected distribution towards the simulated one. This effectgenerally takes place where there are sharp discontinuities occuring near different bins in thedistributions of the simulated and real data. For this reason it is desirable to have the simulateddata model the real data as closely and as smoothly as possible.Chapter 4. Experimental n-jet Event Rates and Data Correction	 31Of course, the bin-by-bin correction procedure may be applied to any measurable distribu-tion and it is used in this analysis of OPAL data. To apply bin-by-bin corrections to arbitrarydistributions measured using the OPAL detector, it is necessary that the simulated propertiesof multihadronic events agree closely with the observed properties. To achieve this agreement,considerable attention has been given to the problem of tuning the parameters of JETSETand other Monte Carlo programs so that they reproduce the overall properties of the observedhadronic events [7]. Figure 4.2 shows several event shape distributions for both measured eventsand events simulated using JETSET 7.2 with suitably tuned parameters. Definitions and a dis-cussion of these parameters may be found, for example, in [8]. In general, the agreement isgood and this suggests that the limited detector acceptance and response may be correctedfor without biasing results when histogram bins are smaller than the intrinsic resolution of thedetector.4.3 Corrected Jet Rate DistributionsThe jet rate distributions shown in figure 4.1 were corrected for the limited acceptance andresponse of the OPAL detector in the manner just described. A first set of hadronic Z° decayswas generated using the JETSET 7.2 Monte Carlo program [9] and the n-jet hadronic eventrates Rnhad(yi) were measured for each jet definition scheme shown in table 2.1. In a secondset, JETSET was used to simulate hadronic Z ° decays including initial state radiation followedby a simulation of the OPAL detector using the COPAL detector simulation program. Then-jet event rates of these simulated events, R nsirn(yi), were then measured in each jet definitionscheme using good 4-vectors from these events which passed the same cuts applied to the realdata described in chapter 3.From these pairs of distributions, the bin-by-bin correction coefficients for the n-jet eventrates were computed using equation 4.5:c(n) = (Riniad(N)INhad) •  (142n(Y•i) Nsim)(4.5)2 	10101 -T-10101 410 -210-3100.25	 0.5	 0.75	 1C parameter0.4Y parameter0	 0.2Chapter 4. Experimental n-jet Event Rates and Data Correction	 320	 0.25	 0.5	 0.75	 1Sphericity0.6	 0.8	 1Thrust0	 0.1	 0.2Aplanarity0.4Oblateness1010.2-110Figure 4.2: Event shape distributions for both real events (solid curves) and events simulatedusing the JETSET Monte Carlo followed by a simulation of the OPAL detector (broken curves).(EM+CT analysis)101-110 —-210101-110-2101	 I 	 1 	Chapter 4. Experimental n-jet Event Rates and Data Correction	 33where Nsim and Nha d were the total number of events in the first and second Monte Carlo datasets, respectively. The corrected rates R„,""(yi) were then computed from these coefficients andthe n-jet rates Rn (yi):Rwrr(m) = ci Rn(yi) (4.6)The n-jet event rates measured using electromagnetic clusters, corrected in this way for detectoracceptance and response, are shown in figure 4.3. Also, the correction coefficients for the twoand three jet events are shown in figure Hadronization CorrectionsThe distributions which have been corrected for the acceptance and response of the detectorare still unsuitable for comparison with equations 2.5 and 2.6 since they have not been correctedfor the hadronization process. To achieve this, the bin-by-bin correction procedure was appliedto the detector level data with the correction coefficients c, computed as follows.The JETSET Monte Carlo was used to simulate the parton shower only, and the 4-vectorsof the partons were used as input to the various jet finding schemes, thus determining the n-jetevent rates at the parton level, Rplart( \y ) In cases where only two partons were generated in theparton shower the event was classified as a two jet event for all values of yi > 0. The same set ofMonte Carlo data which included initial state radiation, hadronization and detector simulationwas used to simulate the detected distributions and the correction coefficients were computedin the usual way:(n) (mart/yz Orpart) Ct — (4.7)(Mim (MiNsim)Then, the measured n-jet event rate distributions, corrected for the limited detector acceptanceand response as well as for hadronization process were computed simply using equation 4.6.This step in the correction procedure requires still more justification. Since the real distribu-tions at the parton level and the process of hadronization are not known there is the possibilityof biasing the corrected distributions using the phenomenological JETSET model or any otherMonte Carlo generator. In general, if it can be shown that the results obtained are insensitive• 1 E-E4);0.8• 0.6AG 0.40.200	 0.1	 0.2y_cut10.80.60.4g 0.2• 0.6g 0.40.20 	 I 0	 0.1	 0.2y_cut_	.-o---'--.----'------•_-	f.---'.--- 70 	 .0	 0.1	 0.2y_cut10.• 1E• 0.8<1,)	 10.81 0.20Chapter 4. Experimental n-jet Event Rates and Data Correction	 340 	0	 0.1	 0.2y_cut0	 0.1	 0.2y_cut2 jets3 jets- 4 jets5 jetsFigure 4.3: N-jet event rate distributions corrected for detector acceptance and response asfunctions of ycut . Dashed lines show the uncorrected distributions. (EM cluster analysis)0 0.1 0.2y_cut0.1 0.2y_cut1.410 .1.2rtx▪ 0.8▪ 0.6I	 I 0.2y_cut0.10- 	1.441 1.2 - 	Ec4I0.80.6Chapter 4. Experimental n-jet Event Rates and Data Correction	 35 .....01.111111111111110 0.1	 0.2y_cut1.4V 1.2 iECI 146 --s-ZA 0.80.6_L I0--0.1	 0.2y_cut2 jets3 jetsFigure 4.4: Bin coefficients for correcting 2- and 3-jet event rate distributions for detectoracceptance and response. (EM cluster analysis)Chapter 4. Experimental njet Event Rates and Data Correction 	 36to the particular Monte Carlo model. or to the parameter values used to simulate the events,then one can assume that modelling the unknown parton distributions in this way will notbias the corrected measurements. Any variation in the parameters deduced from the correcteddistributions when different Monte Carlo models were used, or when a range of Monte Carloparameters were used may be quoted as a systematic uncertainty associated with the correctionprocedure.The n-jet event rates measured using electromagnetic clusters, corrected in this way areshown in figure 4.5 and the correction coefficients c . 2) and c, 3) are shown in figure	 0.1	 0.2	 0	 0.1	 0.2E0. 0.4--....-,0	 \-.....--.____4,___Jt_ -. ----1- ,--..--	0	 __.-__"-1=:t-------4L----I0	 0.1	 0.2	 0	 0.1	 0.2y_cut	 y_cut1At,0.6a>OS0.60.41 0.200 0.1	 0.2y_cutChapter 4. Experimental n-jet Event Rates and Data Correction	 37y_cut	 y_cutFigure 4.5: N-jet event rate distributions corrected for detector acceptance and response andfor hadronization as functions of y,, t . Dashed lines show the uncorrected detector level distri-butions. (EM cluster analysis)2 jets3 jets4 jets------ 5 jetsJ0.2y_cut0	 0.10.1	 0.2y_cutI	 I	  	J0.1	 0.20y_cut1.20.8 L.0.60.2y_cut0	 0.10.2y_cut0.8C.>0.61.4 - 	1.2	1 		A 0.8 	0.60	1.4	 n 		1.2	k 	1	 tv-0.8 Li-0.6I Chapter 4. Experimental n-jet Event Rates and Data Correction	 38•5ACSI2 jets3 jetsFigure 4.6: Bin coefficients for correcting 2- and 3-jet event rate distributions for detectoracceptance and response and for hadronization. (EM cluster analysis)Chapter 5Experimental Determination of a sThe distributions shown in figure 4.5 were corrected for both detector acceptance andresponse and for hadronization. Now the functions given in equations 2.5 and 2.6 may be fit tothese curves to obtain a s and p 2 . In practice, it is more meaningful to fit the differential twojet rate distribution defined byD2(y) R2(y) — R2(y — AY)•Ay (5.1)This, and the details of the procedure used to it the data are described in this chapter. Also,a description of the treatment of the uncertainties in the data and the fitted parameters ispresented here.5.1 Statistical Correlation in R2The n-jet event rate distributions Rn(Y cut) were obtained by counting the number of eventswith n jets at each value of y eut . If y3 _jet and y2—,jet were the smallest values of y eut for whichan event had two and three jets, respectively, then this event would be counted as a three jetevent for all y such that y3—jet < y < Y2—j et and as a two jet event for all y > Y2—jet • Hence,a single event is counted in the R, distributions many times, which introduces correlationsbetween the contents of neighboring bins in y. For this reason, the value of x 2 computed froma fit to these distributions is not well defined in a statistical sense.The correlations betwixt adjacent bins in the 2-jet distribution may be removed by consid-ering the differential 2-jet rates defined in equation 5.1. This new distribution represents thenumber of events which were counted as 3-jet events for y eut < y — Ay and as 2-jet events foryeut > y and hence, each event is counted in this distribution only once. Thus, the statistical39Chapter 5. Experimental Determination of a,	 40errors in D 2 (y) are uncorrelated and a meaningful value of x2 may be obtained from a fit tothis distribution.5.2 Estimation of Uncertainties in D 2 (y)The D2(y) distributions defined above were measured in each of the jet definition schemesusing three sets of 4-vectors obtained from the selection of charged tracks, electromagneticclusters and the combination of these, as described in chapter 3. These distributions werecorrected for detector acceptance and response as well as for hadronization by the techniquesdescribed in chapter 4 using approximately 36,000 Monte Carlo Z° decays with full detectorsimulation and 100,000 events simulated at the parton level.Table 5.1 shows the resulting differential 2-jet event rates obtained for each of these schemes.In each jet definition scheme, the values of D 2 (y) were obtained by the weighted mean of therates D (2°T) (y), DTM) (y) and D (EM+CTE DnY)/(a ( ' ) ) 2 < D2(y)	' E, 1/(a(2))2 (5.2)where the index i runs over the three sets of data, i EM, CT, EM + CT. The statisticaluncertainties of < D2(y) > were obtained by adding those of the Dr ) , Tem) and em+CT)distributions in quadrature and the experimental uncertainty was computed usingaexp(y)< D2(0 2 > — < D2(Y) > 2 2(5.3)The final error quoted for < D 2 (y) > in table 5.1 is the result of these statistical and experi-mental errors, added in quadrature.5.3 Determination of A-f‘-4 and	 from the D2(y) DistributionsWhen /2 2 is constrained by the relation p 2 	114, the strong coupling constant may beparameterized only in terms of .A.Kt as given by equation 2.1. In this form, A iv- may beChapter 5. Experimental Determination of as 	41y Ay< D2(Y) >(E)< D2(Y) >(JADE)< D2(y) >(P)< D2(y) >(Durham)< D2(Y) >(Geneva)0.010 0.005 15.46 + 1.57 23.91 ± 1.08 26.74 + 0.95 26.51 + 0.72 17.59 + 1.130.015 0.005 16.16 + 1.02 18.38 + 0.55 19.94 + 0.56 15.31 + 0.57 17.41 + 0.840.020 0.005 14.03 + 0.62 14.29 + 0.54 15.69 + 0.62 10.13 + 0.43 16.57 + 0.470.030 0.010 11.62 + 0.30 11.01 + 0.30 11.66 ± 0.36 6.52 + 0.23 13.31 + 0.480.040 0.010 9.17 + 0.27 8.69 f 0.28 8.57 + 0.29 4.10 + 0.22 9.41 + 0.450.050 0.010 7.56 + 0.24 6.43 + 0.26 5.79 + 0.24 3.03 + 0.16 6.84 f 0.250.060 0.010 6.05 + 0.22 5.06 f 0.21 4.57 f 0.21 2.17 + 0.16 5.16 f 0.280.080 0.020 4.74 + 0.14 3.62 + 0.12 3.46 + 0.13 1.63 + 0.09 4.08 f 0.150.100 0.020 3.36 + 0.11 2.68 + 0.14 2.34 f 0.11 1.03 ± 0.07 2.72 + 0.120.120 0.020 2.50 + 0.13 1.87 + 0.09 1.51 + 0.08 0.70 + 0.06 2.20 f 0.120.140 0.020 1.84 + 0.10 1.36 + 0.08 1.22 + 0.08 0.55 + 0.05 1.75 + 0.100.170 0.030 1.45 ± 0.07 0.96 f 0.06 0.77 ± 0.05 0.40 ± 0.04 1.35 + 0.070.200 0.030 0.98 + 0.06 0.68 f 0.05 0.49 + 0.04 0.25 + 0.03 0.98 + 0.06Table 5.1: Differential 2-jet event rates.E JADE P Durham GenevaA-0- ( MeV) 793 it +31303 -29 +27245 -25 +40341 -38 +43234 -38a s (Mlo ) +0.0030.146 -0.003 +0.0020.123 -0.002 +0.0020.119 -0.002 +0.0020.126 -0.002 +0.0030.119 -0.003X 2/DOF 4.13/3 3.68/5 5.96/6 2.26/8 0.13/2Table 5.2: Results of the one parameter fit to D 2 (y) with 11 2 -Aq 0determined from a one-parameter fit of the data in table 5.1 to the D2(y) distribution, definedby equations 2.6 and 5.1.In all schemes, the coefficients A(y), B(y) and C(y) were computed using the parameter-ization found in [6]. Because the 0(al) calculations appear to underestimate the productionof 4-jet events when 1.1 2 is constrained to .11q0 [2] this fit was performed in the region whereR4 (y) < 1%. The resulting values of a s , 447 and the corresponding )(2 for each fit are pre-sented in table 5.2. The uncertainties on A-K4- correspond to 68.3% confidence limits and theuncertainties quoted on a s (114) were estimated by recording its maximum and minimum valuesfound within the 68.3% confidence region of A.Chapter 5. Experimental Determination of as 	42One may also write it2 = fE6m in equations 2.6 and 2.1 and treat f as a free parameter.When ,u 2 is given this freedom the 0(ceD calculations can adequately describe the 4-jet eventrates in the region y < 0.06 [12]. However, the 5-jet event rates are not predicted by thesecalculations and for this reason the fits were performed in the region where R 5 (y) < 1%. Theresulting values for A l\-Ts- and f are shown in table 5.3*. Again, the quoted uncertainties of AvigE JADE P Durham GenevaAK-ff ( MeV) +9119 -9 +9121 -9 +13159 -12 +75212 -48 +26145 -16+0.000001 +0.00048 +0.014 +0.0037 +0.064f	 1.1 2/ s 0.000039 -0.000001 0.00241 -0.00041 0.034 -0.012 0.0025 -0.0011 0.031 -0.030+0.028 +0.009 +0.009 +0.056 +0.029,,as (112 ) 0.398 -0.031 0.186 -0.010 0.148 -0.010 0.216 -0.061 0.14l -0.036+0.002 +0.002 +0.002r, +0.008_	 , +0.005„ „a s (M10 ) 0.108 -0.002 0.108 -0.002 0.112 -0.002 U.117 -0.005 0.111 -0.002X 2 /DOF 2.32/6 3.43/8 10.40/9 1.98/9 3.48/5Table 5.3: Results of the two parameter fit to D 2 (y) with Am-g and f	 ,u 2 /E6m as freeparameters.and f correspond to 68.3% confidence intervals. From these fitted parameters, a, is computedin two ways. a 3 (,u 2 ) is simply the value computed using equation 2.1 with both A iv-Ts- andit 2 = f Mlo determined from the fits. a s (M o ) is computed using the value of A TT/ -- determinedfrom the two parameter fits, but with ,u 2 M. The uncertainties on a 3 (,u 2 ) and as(MZ0)were determined from the maximum and minimum values obtained within the 68.3% confidenceintervals of their parameters.The measured differential 2-jet event rate distributions with the both fitted curves super-imposed on the data are shown in figure Parton Virtuality and Hadronization EffectsThe way in which the measured D2(y) distributions are corrected for the limited acceptanceand response of the detector was described in chapter 4. The procedure is justified on thegrounds that the properties of the detector, although complicated, are well understood and*In practice, the parameter log(f) was used to fit the data.F--- E schemeN1	-«.,A	 10 	/'-'------,V-	/	------ ____.1Durham scheme11"	  L 	 I ----I0.1	 0.2A	 -10V1 P schemen 0.1	 0.20Chapter 5. Experimental Determination of as 	43I	 I	 10	0.1	 0.2AJADE schemeevqi 10 —;, v	 .-:. (	 ,,-.--------	------_______1 , I 0	 0.1	 0.2y_cut	 y_cuty_cut y_cutASN ....; ,gz 1	 10 — r----s,:,.= / V	 -_-_Geneva schemef = p 2 ls0.2y_cutA— f = 1ms , —0.1Figure 5.1: Differential 2-jet event rates compared with fits to O(ct s2 ) calculations. Brokenlines show the one parameter (AMA ) fits and solid lines shown the two parameter (A ra-§-.f) fits.Arrows indicate the regions of y used for the fits.Chapter 5. Experimental Determination of as 	44may be modelled in arbitrarily great detail. So, the response of the detector can be, and indeedhas been properly simulated and thus, the bin-by-bin correction coefficients computed usingthe Monte Carlo followed by the detector simulation may be used to correct for the detector'slimited resolution and response.This is not the case when the possible effects of hadronization are considered. Unlike theresponse of the detector, almost all details of the hadronization process are unknown and itis difficult to argue that the bin-by-bin coefficients which are be used to correct the data forhadronization process are correct. In this case, the application of bin-by-bin corrections isjustified only when it is demonstrated that the results are insensitive to the details of thehadronization process. This may be achieved by comparing the results computed using eitherdifferent hadronization models, or a wide range of model parameters. To compare the resultsof different hadronization models, one repeats the entire analysis with correction coefficientsfor detector acceptance and response as well as for the hadronization process computed usinga different Monte Carlo, such as HERWIG [10]. This was not done in this analysis, but othershave estimated the uncertainty in c by such methods and found it to be less than 3% [11].In this study, a s was determined by the methods described in the previous sections exceptwith correction coefficients determined from several sets of JETSET Monte Carlo data generatedwith different values of the parton shower cutoff parameter, Q o . This parameter representsa level of parton virtuality such that partons with invariant masses less than Q o will notsplit. Since the response of the OPAL detector was simulated only for JETSET events withQ o = 1 GeV, the corrected D 2 (y) distributions were computed as shown in equation 5.4DT"(Yi) = cir d (C2o) • et' • D2(Yi) (5.4)where the coefficients c(!et correct only for the detector's acceptance and response for hadronsgenerated with Q o = 1 GeV while C had correct for the hadronization of partons simulated withvarious values of the shower cutoff, Q o . The variation of oe,(1q,) with Q o determined fromboth the one-parameter and the two-parameter fits is shown in figure 5.2.It was found that the Geneva scheme depends only weakly on the parameter f becauseDurham scheme, f from fit= Durham scheme, f 1I	 ,	 IChapter 5. Experimental Determination of a,	 450.16a,,0.120a,a sa,0.16— scheme, f	 1L= E scheme, f from fit-0	 2	 4 6 8 0	 2	 4	 6	 8JADE scheme, f.1 JADE scheme, f from fit2	 4 6 8 0	 2	 4	 6	 8= P scheme, f	 1IP scheme, f from fit,	 #	 _4___.--._- ___+___-4	 4I r	 1	 1	 ,	 12	 4 6 8 2	 4	 6	 80	 2	 4	 6	 8Geneva scheme, f 10.16a s0.120	 2	 4	 6Qo (GeV)0.16a s0.12I	 I	 I 0	 2	 4	 6	 8Qo (GeV)0.160.12Figure 5.2: Variation of a,(.11,_qe ) with the parton shower cutoff, Q o , as determined from one-and two-parameter fits to the differential 2-jet distributions.Chapter 5. Experimental Determination of as 	46change in f may be absorbed in a s by a change in	 since the expression for a, given inequation 2.1 depends only on the ratio ,u 2 /AM. The only explicit dependence on f enters inthe region where the function B(y) in equation 2.6 contributes, which is generally the region ofsmall y. Since the Geneva scheme produces a larger fraction of 4- and 5-jet events, as can beseen from figure 4.5, the region of y used for the fit was typically y > 0.1 and hence the explicitdependence on f is reduced. Thus, it was found that for some sets of Monte Carlo data, theweak dependence of a s on f in the Geneva scheme prevented a minimal x 2 from being foundin a physical region of the parameters. For this reason, the dependence of a 3 on Q o for the twoparameter fit in the Geneva scheme is not presented here and no attempt has been made toestimate the systematic uncertainty due to parton virtuality of a 3 determined in this scheme.5.5 Summary of a 3 MeasurementsIt has become customary to average the values of a s (MMo ) obtained for the one- and two-parameter fits to obtain a quantity /7;040 and quote half the difference of the two values asthe uncertainty in this average due to the ambiguity of the renormalization scale atat which a 3is evaluated. The mean values of a 3 were determined in this way and are presented in table 5.4along with the various estimates of the systematic errors.Scheme cE;(11q0 ) Aa,(exp.) Aa s (scale) Aa3 (Q0) Aa s (tot.)E 0.127 0.004 0.019 0.004 0.020JADE 0.116 0.003 0.008 0.001 0.009P 0.116 0.003 0.004 0.003 0.006Durham 0.122 0.007 0.005 0.002 0.009Geneva 0.115 0.005 0.004 0.006Table 5.4: Final results of (3,-;(11q0 ) for different jet definition schemes.The experimental uncertainties Aa s(exp.) consist of the mean errors on a s (Ago ) fromtables 5.2 and 5.3 added in quadrature while Aa,(scale) was determined from half the differenceChapter 5. Experimental Determination of as	47of these two values. Aa s (Q 0 ) was computed usingo,	2Aas(Q0) = —N E(as(Aq0;Qo 1 ) — ccs(Mz2 *Q (i) ))where oz,(M o ; Qn are the N values of a s (Aqo ) from the two-parameter fit corrected withvarious values of the parton shower cutoff, Qo.(5.5)Chapter 6ConclusionIt has been demonstrated that the data from the OPAL detector is ideal for the determina-tion of the strong coupling constant a s using n-jet event rates. This follows from the fact thatthe high rates and good jet resolution at LEP energies prevent the experimental measurementsfrom being limited by statistics, as may be the case and energies off the Z° resonance. The limi-tations are only those of detector resolution and acceptance and theoretical uncertainties in thehadronization process and the QCD matrix elements. Some of these systematic uncertaintieshave been analysed in detail in this study and have been shown not to be a severe limitationof the precision with which a, can be measured.With the current 0(a 23 ) matrix elements, the rates of 2-, 3- and 4-jet events were predictedusing a number of jet definition schemes and were compared with the rates measured in thisanalysis as functions of the jet resolution parameter, ycut• The agreement among the valuesof a 3 obtained in this way lead one to conclude that this parameter is not sensitive to theparticular scheme used to define jets. Hence it would appear that one has the freedom tochoose any scheme based on its particular merits for other analyses and in particular, one mayquote the value of a 3 which was most precisely measured using these schemes. Thus, one mayconclude that the value of a s measured from n-jet event rates using the techniques describedhere isa3(A410) = 0.116 + 0.006. (6.1)This value is in agreement with 0.120 + 0.006 obtained by a combined analysis of OPAL studiesof event shapes, jet rates and energy correlations correlations [13]. In general, to improve sucha measurement, one needs theoretical calculations to higher orders in a3.48Bibliography[1] Richard D. Field, Applications of Perturbative QCD, Addison-Wesley, 1989.[2] JADE Collaboration, W. Bartel, et. al., Z. Phys. C — Particles and Fields 33 (1986), 23.[3] JADE Collaboration, S. Bethke, et. al., Phys. Lett. B 213 (1988), 235.[4] OPAL Collaboration, K Ahmet, et. al., CERN-PPE/90-114.[5] Kunszt and Nason, Z Physics at LEP 1, CERN Yellow Report 89-08 (1989), Vol. 1.[6] S. Bethke, et al., CERN-TH 6222/91.[7] OPAL Collaboration, M. Z. Akrawy, et. al., Z. Phys. C — Particles and Fields 47 (1990),505.[8] V. Barger and R. Phillips, Collider Physics, Addison-Wesley, 1987.[9] T. SjOstrand, Comp. Phys. Comm. 39(1986) 347, Comp. Phys. Comm. 43(1987) 367,Int. J. of Mod. Phys. A3(1988)751,Z Physics at LEP 1, CERN Yellow Report 89-08 (1989), Vol. 3.[10] B.R. Webber, Nucl Phys B310 (1988) 461, Z Physics at LEP 1, CERN Yellow Report89-08 (1989), Vol. 3.[11] OPAL Collaboration, M. Z. Akrawy, et. al., Z. Phys. C — Particles and Fields 49 (1991)375.[12] S. Bethke, Z. Phys. C — Particles and Fields 43 (1989), 331.[13] S. Bethke and J.E. Pilcher, Tests of Perturbative QCD at LEP, HD-PY 92-06, EFI 92-14.49BIOGRAPHICAL INFORMATION NAME:	 TI MOTHY 4/1AT "CH Ew TON esMAILING ADDRESS:	 '75C)	 titioo 17 LAN D DRive ,CAS T LGG-AR ,	 .C. ,/J ZE9PLACE AND DATE OF BIRTH: VAVQCOUVER , 	 8 . C.seP TeivIBER. 5 2 296'EDUCATION (Colleges and Universities attended, dates, and degrees):UN	 lz.s.r-rY OF ercrogi ,	 1986 - 908 . SC . ( PH Ys Ics Mb NIA-THEMA-TICSPOSITIONS HELD:PUBLICATIONS (if necessary, use a second sheet):AWARDS:Complete one biographical form for each copy of a thesis presentedto the Special Collections Division, University Library.0•6


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