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Analysis of the decay [pi] --> [rho][nu] Bougerolle, Stephen Edward 1992

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ANALYSIS OF THE DECAY r -* pvbyStephen E. BougerolleB. Sc., The University of Lethbridge, 1987M. Sc., The University of British Columbia, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of PhysicsWe accept this thesis as conformiiigto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJULY 14, 1992© STEPHEN EDwARD BOUGER0LLE, 1992Signature(s) removed to protect privacyIn 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.(Signature)________________________________Department of Prsic5The University of British ColumbiaVancouver, CanadaDate vI i /qDE-6 (2188)Signature(s) removed to protect privacyAbstractAn analysis of the decay r —* pv- has been undertaken in the OPAL experimentat CERN, in Geneva, Switzerland. From the 1990 experimental run of the LEP particleaccelerator, a sample of 3310 e+e r+r events was selected, with an estimatedcontamination of 1.9%. Requirements were applied to select a subsample of r •*decays, resulting in 650 decays being found. From studies with simulated data, the non-pcontamination in this sample was estimated to be approximately 22%, and the p selectionefficiency to be approximately 33%.The Branching fraction for the decay is measured to be B(r —* pvr) = 0.234 +0.009(stat.)+3g(syst.) The mean r polarisation at the peak of the Z° resonance ismeasured to be F,. —0.17+0.10(stat.)+0.08(syst.). From the r polarisation we extracta measurement of the electroweak mixing sin20w = 0.225 + 0.015. The r polarisationasymmetry at the peak of the Z° resonance is also measured, A°, = —0.09 + 0.13 + 0.05.From the polarisation measurement, VT/ar = 0.09 + 0.06, and from the polarisationasymmetry Ve/ac = 0.06 + 0.10. Combined with previous LEP measurements of the rpolarisation, the electroweak mixing is found to be sin2 0w = 0.2308 + 0.0042. Combinedwith measurements from other experiments, one obtains sin20w = 0.2303 + 0.0013.11ContentsAbstractList of Tables vList of Figures viiAcknowledgements viii1 Introduction 12 Theory 72.1 The Standard Model 72.1.1 The Neutral Weak Current 102.2 Electron-Positron Collisions 112.2.1 Forward-Backward Asymmetry 122.3 Polarisation 142.3.1 Measurement 162.3.2 Tau Leptonic Decays 192.3.3 Tau Semi-Hadronic Decays 192.4 Radiative Corrections 243 Experimental Apparatus 253.1 LEP 253.1.1 Injector 253.1.2 Storage Ring 273.2 The OPAL Detector 273.2.1 Inner Detector 293.2.2 Time-of-Flight System 313.2.3 Electromagnetic Calorimetry 313.2.4 Hadronic Calorimetry 323.2.5 Muon Detector 323.2.6 Trigger 333.2.7 Data Readout 341114 Data4.1 Data5 Preselection of e+e — 7+r Events5.1 Data Quality and Status Requirements5.2 Elimination of Cosmic Ray Background5.3 Cone Analysis and Fiducial Acceptance5.4 Elimination of ee—* q Background5.5 Elimination of e+& e+e Background5.6 Elimination of eC —+ e’eX Background5.7 Elimination of ee -4 1cc Background5.8 Summary of Results6 Selection of r± pu decays6.1 Selection Criteria6.2 Summary of Results7 Analysis7.1 Branching Fraction7.1.1 Branching Fraction Uncertainty7.2 Tau Polarisation7.3 Forward-Backward Polarisation Asymmetry .8 Summary of Results8.1 Branching Fraction8.2 Polarisation8.2.1 Polarisation Asymmetry8.3 Lepton Universality8.4 The Weinberg AngleAbstractA Derivation of Polarisation Angle Equations 1004.2 Simulated Data4.2.1 Simulated r-pairs .4.2.2 Other Simulated Data4.3 Reconstruction3636363738394141434347474749495353596161626883898989919194100ivList of Tables1.1 Elementary particles described by the Standard Model 42.1 Standard Model fermions and quantum numbers 82.2 Neutral current couplings 114.1 e+e r+r data samples 374.2 Branching fractions used in simulation 385.1 Contamination in r-pair sample 526.1 r± p+y selection requirements 536.2 Sources of background in selected data 597.1 Alternative branching fraction weighting schemes in Monte Carlo data 667.2 Uncertainties arising from decay-mode weighting 667.3 Uncertainties arising from a1 weighting 677.4 Summary of uncertainties in the branching fraction measurement . . 677.5 Results of simulated analysis with MC data 837.6 Summary of uncertainties in the polarisation measurement 848.1 Past rth pu branching fraction measurements 918.2 Tau polarisation measurements 928.3 OPAL polarisation asymmetry measurements 928.4 Measurements of sin2 0w (for M0 = MH298 = 100 GeV) 95VList of Figures1.1 Diagram for beta decay n —÷ pal6 31.2 Troublesome weak processes (with couplings) 41.3 Higgs boson production 42.1 Feynman diagrams for ee—+ fJ (f $ e) 112.2 Born-level cross-section for r+r production 132.3 Extra Feynman diagram for e+e e+e 142.4 Forward-backward asymmetry, r polarisation and polarisation asymmetry 152.5 X distribution of r leptonic decays 172.6 Possible spin configurations in r+ -÷ 1r±v decay 182.7 Possible spin configurations in r ±Vr decay 182.8 Decay angles in r± P11 202.9 Momentum distribution of r—+ 233.1 Schematic diagram of the LEP injector 263.2 Geographic location of LEP 283.3 The OPAL detector 303.4 1990 ECAL energy resolution 354.1 Feynman diagram for e+e 405.1 1990 distributions of track parameters 425.2 Useful distributions for cosmic ray elimination 445.3 Normalised distributions of r-pair selection quantities (simulated data) 455.4 Bhabha-scattering and two-photon event characteristics (simulated data) 465.5 Two-photon event characteristics (simulated data) 485.6 Di-mnon event characteristics (simulated data) 505.7 Di-mnon event characteristics (simulated data) 516.1 Cone size and neutral cluster requirements 556.2 Values used in neutral cluster requirements 566.3 Cone mass 576.4 Eass/Pctrk 586.5 Agreement between data and Monte Carlo data for significant variables 607.1 Variation of branching fraction with parameter changes 64vi7.27.37.47.57.67.77.87.97.107.117.127.137.147.157.167.177.188.18.28.3cos 0*slices in cos bslices in cosslices in cosslices in cos 0*6570-7-7.737475767778798081858607a’88909396Variation of branching fraction with parameter changesCorrection terms for one-dimensional polarisation measurementData at varying stages of correctionOne- and two-dimensional Ecm corrections (2-D, slices in cosTwo-dimensional Ecm corrections, slices inTwo-dimensional polarisation corrections,Two-dimensional polarisation corrections,Two-dimensional polarisation corrections,Two-dimensional polarisation corrections,Data at varying stages of correction, slices in cosData at varying stages of correction, slices in cosData at varying stages of correction, slices in cos 0*Data at varying stages of correction, slices in cos 0*Variation of r polarisation with parameter changesVariation of r polarisation with parameter changesVariation of r polarisation with parameter changesVariation of r polarisation with parameter changes(i-D fit)(1-D fit)(2-D fit)(2-D fit)—*pth v,- branching fraction measurementsTau polarisation measurementssin2 ow measnrementsviiAcknowledgementsThe list of people whose efforts need acknowledgement is quite long. OPAL is a largecollaboration (around three hundred physicists, plus support staff), all of whom havecontributed in some way to every piece of work we publish. Particular mention must bemade of Keith Riles, Mike Roney, and the other members of the -r polarisation workinggroup. My supervisor Randy Sobie, as well as Richard Keeler, Alan Honma and ChrisOram were of great help in proof- reading my thesis and verifying results. Horst Breukerdevoted a disproportionately large amount of time to finding a nice diagram of LEP, nowincluded in Chapter 3.Back at UBC, I would like acknowledge the past and present members of my examination committee; Drs. D. Axen, F.W. Dalby, M. Halpern, J. Ng, and C. Walthamall contributed many useful comments and suggestions, as did the university examinersN. Weiss and P. Hickson and the external examiner J. Prentice. J.A.R. Coope handled the examination smoothly and was the only person present to comment that thisacknowledgements page was missing.The OPAL experiment has been carried out using facilities of the European Centrefor Particle Physics (CERN), where I myself spent two years working. The two diagramsof LEP in Chapter three are taken from CERN publications. Canadian participationin OPAL has been funded by the Natural Sciences and Engineering Research Councilof Canada (NSERC), and I have been supported in part by two NSERC post-graduatescholarships.vi”Chapter 1IntroductionAs with all work in particle physics, the purpose of this analysis is to further our understanding of fundamental physical interactions. Conventional theory now divides theseinto three categories; gravity, the strong nuclear interaction, and the electroweak interaction. Two older theories describing the weak and electromagnetic interactions havebeen united by Glashow, Salam and Weinberg [1] in the Standard Model of ElectroweakInteractions. Among the results of this work are tests of aspects of the Standard Modeland a measurement of one of its fundamental parameters.The first particle interaction studied was that between electrons and protons, nowdescribed by the theory of electromagnetism. Although electromagnetism provided aqualitatively correct picture of atomic structure, a quantitative explanation required thedevelopment of quantum mechanics. Similarly, electromagnetism correctly explained thewave nature of light but had philosophical problems which were finally resolved with thetheory of special relativity.In 1928, Dirac united the two fields that had developed from electromagnetism,quantum mechanics and special relativity, creating the theory of relativistic quantummechanics[2]. On the basis of this new theory, he predicted the existence of “antipartides”, with the same mass as particles but opposite-sign electric charge. The antiparticleof an electron is a positron (discovered in l932[3]) and the antiparticle of a proton is anantiproton (discovered in 1955[4}). Relativistic quantum mechanics was further expandedthrough the use of field-theory techniques and developed into quantum electrodynamics(QED)[5j, which describes the interactions of electrons and positrons with “field quanta” termed photons. QED reactions can be thought of as occurring by the exchange ofphotons between charged particles, with the strength of these interactions given by thecoupling constant a =2/(47rhc).Concurrent with the development of QED were the discoveries of several new particles. The neutron[6] was the first observed after the electron and proton. Then themuon[7] p, pion[8] it and kaon[9] K were discovered in observations of cosmic-ray particles. Experimenters made these observations while trying to test the Yukawa theoryof strong interactions, which had developed as an attempt to understand the formationand structure of atomic nuclei. Their goal was to discover an exchanged particle similarto the photon, responsible for mediating the strong interaction. Instead they discovered1several new particles. This prompted the development of experiments using particle accelerators; the rate of particle production with these new facilities was much more intensethan the natural cosmic-ray collision rate, so new particle searches could be carried outmore efficiently.The zoo of new particles naturally encouraged particle taxonomy. The muon didnot itself undergo strong interactions, and was classed together with the electron underthe category of “leptons.” The other particles, all subject to the strong interaction,were classed together as “hadrons.” Hadrons were further classed as “mesons,” lighterparticles with integral spin, and “baryons,” heavier particles with half-integral spin[lO].Hadronic structure can be explained by the quark model, in which “quarks” arefermions with fractional charge. Mesons are combinations of a quark and anti-quark(and therefore have integral spin), while baryons are combinations of three quarks orthree anti-quarks (and therefore have half-integral spin). Quark-quark interactions arethought to be well-modelled by a theory called Quantum Chromodynamics or QCD(modelled on Quantum Electrodynamics). Whereas QED describes interactions betweenparticles with electric charge by the exchange of an uncharged photon, QCD describesinteractions between particles with “colour” by the exchange of a particle called a “gluon”which (in contrast to the photon) is itself coloured.The strong and electromagnetic interactions could not account for all phenomena,however. Early on, it was observed that certain nuclei also underwent a process calledBeta decay (for example 3H —* He + c+Energy), the characteristics of which were achange in electric charge of the original particle accompanied by production of an electronor positron plus a certain amount of “missing energy.” The explanation for this was givenby Pauli, who postulated the existence of a new particle carrying the missing energy (theneutrino v), and by Fermi, who postulated a new “weak” force to which neutrinos weresubject.Weak processes were modelled as a four-fermion, “point-like” interaction in whichdoublets of particles were coupled together with a constant strength GF (analogous toa), known as the “Fermi constant.” The possible doublets in question were( ( (He} p) H’where H and H’ were two hadrons. Beta decay was then a reaction between a doublet(p i’i) and a doublet (v e). It was observed that neutrinos are associated with a particularspecies of charged lepton [11]. That is to say, the electron neutrino Ve and muon neutrinoare different. Weak interactions were noted to have another feature; they were notinvariant under parity transformations (inversion of the spatial axes) [12].Fermi’s original point-like coupling presented theoretical difficulties, however. Whileaccurate for low particle energies, the predicted reaction rates became unphysical as theenergy increased. ‘Tb remedy this problem, the existence of an exchanged particle similarto the photon was postulated, a “weak vector boson” with charge +1 and spin 1 (seefigure 1.1). Because of the charge of this W± boson, W-exchange reactions are also called“charged current” reactions.211\ /PiWe Ve A ePointlike Boson exchangeFigure 1.1: Diagram for beta decay n —* peVeAlthough the introduction of the W boson reconciled weak theory with experimentalobservation, it also created new problems; several other processes should be permittedby the new theory, in particular ee -+ W+W (see figure 1.2a). However, the reactionrates calculated for this grow with energy, giving unphysical results. A theoretical solution to this problem was found by postulating the existence of a new class of “neutralcurrent” interactions (see figure 1.2b). If the couplings for the two processes are similar (e gw) the divergences in each process almost cancel. Neutral current reactions(specifically e + V —* C + v) were first observed at CERN in 1973[13].The form of the neutral current is a key feature of the Standard Model. In it, theweak and electromagnetic forces are modelled as combinations of more fundamental interactions, “weak isospin” and “weak hypercharge.” The old coupling constants a andare related to more fundamental constants g and g’ through a new parameter e,called the Weinberg angle. The Weinberg angle can be thought of as giving the relativeproportion of and Z° exchange contributions to electroweak reactions. The electroweakmixing is sin2 0w rather than 19w itself. The structure of the Standard IVlodel is describedmore fully in Chapter 2.If the Standard Model is correct, it should be possible to produce the quanta associated with the charged and neutral currents, Wth and Z° bosons. An observation of theZ° is particularly convincing as its existence is a unique feature of the Standard Model.The Z° was in fact observed at CERN by the UA1[14] experiment in 1983, as were theW+ [15]. This was a major success for the new theory.The introduction of the Z° did not quite cure the problems noted above. In the Standard Model, these are completely eliminated by the introduction of another exchangedquantum called the Higgs boson H° (see figure 1.3). Unlike the \V , the Higgs bosonis uncharged and unlike all bosons we have described so far, it is spinless. It couples inproportion to the mass of the interacting particle. Although the XV and Z° bosons havebeen seen, the existence of the Higgs boson has yet to be verified.At the current time, the Standard Model accurately describes electroweak interactionsamong all the known particles and their antiparticles, listed in table 1.1 (the t quark andv,-, as well as the H°, have yet to be observed). Although the Standard Model has beennotably successful, it has many parameters which must be determined by experiment.These include all the coupling constants (a, GF and sin2 Ow) as well as the individualparticle masses.3+ -UU+ +e ,vv e/e gwe/ \w e/ w(a) Electromagnetic current (b ) Neutral weak currentFigure 1.2: Troublesome weak processes (with couplings)Bosons 7,W± ,Z° ,H°Charged leptons e, r, rNeutrinos Ve, z1, v,Quarks d,u,s,c,b,tTable 1.1: Elementary particles described by the Standard Model+e vv-mweFigure 1.3: Riggs boson production4Accurate measurement of the Weinberg angle is a high priority; whereas the otherfundamental constants are known to better than one part per million, the mixing sin20wis only known to an accuracy of less than one part per thousand. A more accurate measurement is needed for reliable calculations using the Standard Model. When combinedwith other data (such as a measurement of the Z° boson mass) and higher-order calculations, an accurate measurement can be used to place limits on the t quark and Higgsboson masses. The discovery (or non-discovery) of these particles in the correct massrange will provide the best proof to date of the Standard Model’s validity. Measurementsof 0w have been performed by examining neutrino-scattering reactions. Complementarymeasurements are being made using the LEP facility at CERN in Geneva, Switzerland.LEP is built to study electron-positron annihilation into a Z° boson, which decays intoany fermion-antifermion pair, ee —+ Z0 —+ ff with Tflf < rnz/2. The angular distribution and spin polarisation of the outgoing fermions are studied and the Weinberg angleextracted.In the table of particles above, it will be noticed that there is a third charged lepton,the r. The tau, first discovered in 1975[l6], differs from other leptons in its mass anddecay properties. The electron is stable. The muon has a relatively long lifetime, anddecays by u —+ CiYeliM into an electron, the only charged particle lighter than itself. Butthe r, with a mass of 1784 MeV, has a very short lifetime and can decay into manyparticles which are lighter than itself (likely possibilities are—+ P’4 ,—+ e VCVT and r± ièv)The spin polarisation of the r can be studied by measuring the energy distributionof its decay products. From this quantity the electroweak couplings of the Z° to ther and electron can be found. A problem with the measured r lifetime gives reason todoubt that the couplings of the W to the electron and r are identical, and therefore thatthe assumption of “lepton universality” (that charged leptons behave identically in weakinteractions) is valid. A check on the equivalent Z° couplings is a complementary test ofthis hypothesis. If lepton universality is assumed valid, the polarisation measurement canbe used to calculate the Weinberg angle. The theory behind the polarisation measurementand calculation of related quantities is given in Chapter 2.In this work, I have chosen to study a sub-sample of all r lepton decays, those wherethe r± decays into a meson and a neutrino (r± p±v) This in theory provides thebest measurement of the r polarisation possible from any r sub-sample. The experimentalfacilities used are described in Chapter 3. The r± pjj selection process and its resultsare detailed in Chapters 4-6. Other members of the OPAL collaboration have also studiedthe r+ , r+ e± 14 yr , and r+ p±vy sub-samples of r-decays, obtainingindependent polarisation measurements [17].The r—+ 1-1 decay is also important in the field of r physics, distinct from electroweak theory. The fraction of r leptons which decay by this process (its “branchingfraction”) has been measured before, as have the branching fractions of other decays.But when the sums of past measurements are taken together, they do not add up toone within experimental uncertainties. This “branching fraction problem” has becomecontroversial with time and more data are desired to help resolve it. Chapter 7 containsan analysis of the r± sample, with a measurement of its branching fraction as5well as the details of the r polarisation measurement. A summary of the results is givenin Chapter 8.6Chapter 2TheoryThe basic features of the Standard Model were given in Chapter 1. In this chapter,I preseilt the mathematical structure of the theory. The interested reader will find amore thorough discussion in reference [5]. An analysis is given of the reaction e+efJ, detailing the origins of polarisation and its significance. Following this is a briefoverview of the structure of T decays, with emphasis on their angular distributions andthe relevance of these to the T polarisation measurement.2.1 The Standard ModelIt is convenient to describe the Standard Model using the helicity formalism, wherehelicity is defined as the component of spin of a particle along its line of motion. Highenergy spin-i fermions such as those in the Standard Model are helicity eigenstates,with eigenvalues +. Positive helicity states are termed “right-handed,” and negativehelicity states are “left-handed.” It is experimentally observed that the electromagneticinteraction couples equally to right- and left-handed states, the weak charged currentcouples only to left-handed states and the weak neutral current couples differently toright- and left- handed states.Standard Model fermions are classified to reflect experimentally observed symmetries.In much the same way that protons and neutrons are observed to behave similarly instrong interactions, left-handed particles in a weak interactioii doublet are observed tobehave similarly in charged-current weak interactions. For the strong interaction, thisinvariance was reflected in theory by an “isospin” symmetry, modelled on SU(2) spincalculations. The proton and neutron were grouped together in a doublet with isospinI = in which each particle was a distinct eigenstate with component eigenvalues 13 =For charged-current weak interactions, this idea is taken over as “weak isospin.”Analogously, left-handed particles (for example, the eL and lie) are grouped together ina doublet with weak isospin T = in which each particle is a distinct weak eigenstatewith component eigenvalues T3 = +. Where the strong interaction was approximatelyinvariant under SU(2) isospin rotations, the charged-current weak interaction is invariantunder STJ(2)L weak isospin rotations.7Fermion states Q T T3 Y(ve (v (v 0 1/2 1/2 —1e R r —1 1/2 —1/2 —1(u (c (t 2/3 1/2 1/2 1/3d’ Jr s’ U —1/3 1/2 —1/2 1/3CR 12R TR -1 0 0 -2UR CR 1R 2/3 0 0 4/3d bR’ -1/3 0 0 -2/3Table 2.1: Standard Model fermions and qnantum numbersTo incorporate the neutral weak current and electromagnetic interaction, these statesare also labelled according to “weak hypercharge” (another term taken from earlier attempts to model hadronic structure), which is related to the electric charge and weakisospin by(2.1)where Q is the electric charge and Y is the weak hypercharge. These quantum numbersare collected for all the Standard Model fermions in table 2.1. States are labelled with asubscript L or B according to their handedness.While weak interactions between leptons are seen to observe the doublet structureabove, quarks behave differently. Reactions between .s and u quarks are observed, forexample, as well as reactions between d and u quarks (which would otherwise forma natural doublet). In other words, quark mass eigenstates are not weak interactioneigenstates. This difficulty is incorporated in the Standard Model by defining weakinteraction eigenstates ci’, s’ and U which are linear combinations of quark mass states d,.s and b:fd’ fIQa K8 IQ5N fd\K88 Kgb) L) (2.2)6 Kb’3 Kb! b 6The unitary matrix formed by the coefficients Kq is caUed the Cabibbo-KobayashiMaskawa matrix[18]. The primed states are incorporated into the Standard Model, generalising the “generation” structure that was evident for leptons. The Standard Modelplaces no limit on the number of generations, but assuming that neutrinos are massless,recent results from LEP have shown there to be three generations[19].These elements are all tied together in the Standard iViodel Lagrangian. This depictsthe weak and electromagnetic interactions as resulting from more fundamental interactions involving weak isospin and hypercharge:8timt = —igJVt) — iJB (2.3)where g and g’ are new coupling constants relating the strength of the weak hyperchargeand weak isospin interactions, J and J are the weak isospin and weak hyperchargecurrents. W and B represent four “gauge fields” interacting with the currents.The weak isospin and hypercharge fields are connected with the observed electromagnetic and weak boson fields by substituting= (W + iW) (2.4)=B4cosOw+WsinOw (2.5)Z,1 = —B,sinOw+WcosOw (2.6)where W1 are the weak charged boson fields, A is the photon field, and Z representsthe weak neutral boson field. The parameter 0w is the Weinberg angle, mentioned inChapter 1. This demonstrates its role; the Weinberg angle gives the degree of mixingbetween weak hypercharge and weak isospin. The coupling constants g (equivalent toGF) and g’ are related to the electromagnetic coupling a bye=gsin8=g’cosO (2.7)The Standard Model Lagrangian as shown above has a problem; the observed weakbosons have masses, whereas the fields W, and B1 given are massless. This difficulty issolved by adding in another element known as the Riggs mechanism. A full discussion ofthis is outside the realm of this work[5]. We note only its effect; to fix the relation betweenthe gauge boson masses while leaving the weak isospin and hypercharge invariance intact.The Standard Model is a field theory and like all field theories has another potentialproblem: is it possible to make perturbation calculations from it? The answer is yes,but to do so requires renormalisation of the couplings in the theory. As with the Riggsmechanism, a full discussion of this does not belong here. Rowever, one of the majorresults must be mentioned.The weak mixing can be defined in two different ways; in terms of the boson massessin2 ow = 1 — M,/M and in terms of the weak couplings sin20w = g’2/(g + g’2). Atthe lowest level these are equivalent but when higher order corrections are included andthe results renormalised the two definitions are no longer equivalent. The former is takenas the standard definition of sin20w and the latter is taken to define sin20w, the effectiveweak mixing angle at the energy of the experiment. At LEP energies, the two are relatedby sin20w = 1.013 sin2 Ow[32] (assuming the top quark and Riggs boson masses to beeach 100 GeV).92.1.1 The Neutral Weak CurrentAs mentioned in Chapter 1, the form of the neutral current is one of the key features ofthe Standard Model. It is related to the electromagnetic and weak charged currents by1NC y3 2 ,i rem=u11—sin Uj47J11The electromagnetic and weak charged currents, in turn, are= ‘çb7MQ’ (2.9)= X7T3X (2.10)where i/ is a particle spinor, x a left-handed particle doublet, 7 are the Dirac matrices,and is the SU(2)L “Pauli matrix.”The neutral current is conventionally written in a more useful parametrised form:J0 ?k7,(V — a75)b (2.11)The coefficients v and a represent the couplings of the Z° boson to the vector andaxial-vector parts of the current, respectively, and 75 = i70’) 17273. The values of v anda are obtained by comparison with our original interaction Lagrangian:a = (2.12)=T3—2sin2OwQ (2.13)Because sin20w “runs” (varies with energy), these couplings also run. In the sameway we replace sin20w with the effective mixing sin28w, we replace the couplings v anda with and &. These are the couplings actually measured in this analysis.The neutral current can be written in a different form, by separating out the couplingsto left- and right-handed particles:jNC= 7(cR(1 + 75) + cL(1 — 75)) (2.14)The left- and right-handed couplings are related to v and a throughv = (cL + CR) (2.15)a = (cL — CR) (2.16)These coupling constants are listed in table 2.2. The asymmetry between couplings toleft- and right-handed particles is clear. As we will see later, this causes measurable effectsin the reactions studied at LEP. Working backwards, measurement of these asymmetriesallows an estimation of the couplings and from that, a calculation of sin2 Ow.10Constant Ve,iI,L,iIT e,1u,r u,c,t d,s,bv 1 —1 +4sin2Ow 1— sin2Ow —1 + sin2Owa 1 -1 1 -1CL 2 —2 +4sinOw 2— sin2Ow —2+ sin2OwCR 0 4 sin2 Ow — sin20w sin2 °wTable 2.2: Neutral current couplings::><Figure 2.1: Feynman diagrams for ee fJ (f e)2.2 Electron-Positron CollisionsFrom the couplings of leptons to the photon and Z° , it is straightforward to analysethe reaction e+e ff. Here, we discuss only first-order (‘Born level”) calculations.Higher-order corrections are discussed later.From the Feynman diagrams in figure 2.1, one obtains the Born-level differential crosssection for the case where the fermions are not an e+e pair:Bom(scosOp)= [(Fo—pF2)(1+cos2O)+ 2(F1 —pF3)cosO] (2.17)Here = E— + Ee+, cos 0 is the scattering angle of the fermion f with respect tothe e, p is the helicity polarisation of the fermion f, andF0,F123are given by:F0 = qq + 2Re()qeqfvevf + lx2(v + a)(v + a) (2.18)F1 = 2Re()qeqJaeaf + X24veaevfaf (2.19)F2 = 2Re()qeqjveaj + 2(v + a)2vf aj (2.20)F3 = 2Re()qqjavj + + a) (2.21)with11s — M + ifz/Mz(2.22)In the equations, the vector and axial coupling constants ve,vf,ae,af are labelledaccording to the particles to which they couple. The charge of the fermion producedis given by q. The factor of iI’z/Mz in x is inserted to account for the Z° resonancewidth. The effect of this resonance term x is that particle production by Z° exchangeincreases greatly when the centre of mass energy M. This can be seen in figure2.2.The reaction ee —÷ ee (“Bhabha scattering”) is complicated by the fact that theincoming and outgoing particles are identical. Becanse of this it is necessary to includean extra diagram when calculating the cross section (see figure 2.3), and the aboveequation is incorrect. The actual Bhahha-scattering cross section increases strongly atsmall angles.2.2.1 Forward-Backward AsymmetryFor f 0 e, the terms proportional to p cos U above induce an asymmetry between theintegrated cross-sections in the forward (cos U > 0) and backward (cos U < 0) hemispheres.This is given by= 1 [u(cosO >0) — u(cosU < 0)j = (2.23)Thorn 4F0where a represents the integrated cross section, and Born in particular is the integratedtotal cross section. When = Mx, terms proportional to Re(x) drop out, the-exchange term in F0 becomes negligible, and the forward-backward asymmetry is givenby3I2veaeN(2vfaf_N 32 21! 2 2’ AeAf (2.24)4 \V+J \vf+afJ 4In the equation above, we have introduced a term A, defined by2V3ci3 (2.25)= 2(v/a)2 261 + (v/a)2where the subscript x identifies a particular particle and its coupling constants. This isdone for convenience; the functional form of A also occurs later when we discuss the rpolarisation and r polarisation asymmetry.The forward-backward asymmetry is plotted as a function of in figure 2.4. Inthe same diagram are shown the r polarisation and r polarisation asymmetry. It canbe seen that they are inherently more sensitive to changes in sin20w Also, AFB varies12-DC3.532.521 .50.501 Q3v’s (0ev)ir production cross sectionFigure 2.2: Boru-level cross-section for r+r productionPlotted on the vertical scale is the expected Born-level cross section for the interactione+e as a function of centre-of-mass energy. The resonance structure of the Z°is seen in the spike near 90 GeV.1013Figure 2.3: Extra Feynman diagram for ee —* eemore with the energy of the 7--pair and as a result is more sensitive to higher-orderradiative corrections. Furthermore, in the epxected case where Ae = AT, the latter twomeasurements are sensitive to the sign of A, but the forward-backward asymmetry is not.2.3 PolarisationBecause of the difference between the coupling constants CL and CR mentioned above,the outgoing fermions in ec —* ff will be preferentially left-handed, and the outgoingantifermions will be preferentially right-handed. This helicity polarisation is directlydependent on sin20w; measurement of the polarisation provides a measurement of theWeinberg angle.If the produced particles are quarks, the initial helicity state is usually destroyed asthe quarks bind into hadrons. If, however, the particles are p or r pairs, their polarisationcan in theory be measured. For this reason, I limit discussion to the reaction e+e_hereafter, where t is a lepton.By convention, the “polarisation” refers to the mean helicity polarisation of the Lproduced in the collision, taking into account the entire angular range. It is a feature ofelectron-positron collisions that helicity is almost completely conserved. As a result ofthis (and the fact that the Z° has spin 1), the produced t’ has opposite helicity to the Land therefore opposite polarisation. Measurement of one is equivalent to measurementof the other.The cross sections for left- and right-handed charged lepton pair production as afunction of .s are given byJR0LF2 (2.27)Born F0where R and L are the cross sections for the values p = +1, —1, integrated over cos 0.At the peak of the Z° resonance, we have = Mz. Therefore the real part of x vanishes,xI > 1 and the polarisation equation becomes14w=.22 ..... w=.23 w=.24I I I I I I I I I I I I I I0.20—0.2___________________________________________________I I I I I I I I i...___, I I90 90.25 90.5 90.75 91 91.25 9L5 91.75 92o eVvs. CMS EnergyI I I I I I I I I I I I I I I I I I I I I0--0.2’- JHi I I I I I I ii _jI J_._.I I I90 90.25 90.5 90.75 91 91.25 9 .5 91.75 92Ge\JA ° vs. CMS FnercvI I I I I I I I I I I I I I I I0.20--0.2—l II Ii I I I I I I L____i...___...I II90 90.25 90.5 90.75 91 91.25 9.5 91.75 92O eVA8 vs. CMS EnergyFigure 2.4: Forward-backward asymmetry, r polarisation and polarisation asymmetryThe change in analysis variables is plotted as a function of energy. The different linesrepresent different values of w= sin2 Ow (legend at top). The polarisation is seen to beslightly more sensitive than the polarisatiori asymmetry, but both are inherently moresensitive than the forward-backward asymmetry. As well, the latter suffers more fromradiative corrections due to its stronger energy dependence.15=— 2vEat =— (2.28)Vj + ajWe can also calculate the lepton-polarisation asymmetry, defined by= [u(cosO>0,p=+l)—a(cosO>0,p=—1)Born—c(cosO < O,p = +1) + a(cosO <O,p = l)]= (2.29)At the Z° resonance, this equation simplifies in much the same way as that for thepolarisation, giving:= 32veae (2.30)From this, we see that a measurement of the lepton polarisation actually gives theneutral-current couplings to the outgoing lepton, whereas the polarisation asymmetrygives the couplings to the incoming electron. The polarisation, polarisation asymmetryand forward-backward asymmetry are complementary measurements. The first two aremore sensitive and allow a test of lepton universality (in their equality or non-equality),whereas the latter can be measured with more statistical precision.2.3.1 MeasurementFor the polarisation measurement, the r is the most convenient of the leptons tostudy as the polarisation of a r sample can be determined from the energy distributionof its decay products. The angular distribution of decay particles in the r centre-of-massframe is dependent on the r polarisation. When these particles are boosted forward bythe momentum of the r in the lab frame, this distribution in angle becomes a distributionin energy. Thus, knowing the energies of the particles measured in the lab enables usto calculate their decay angle distribution in the r centre-of-mass frame, from which thepolarisation is extracted. While the same is true of a muon, the long lifetime of the p atLEP energies means we do not observe many decays.Also, a measurement of the r polarisation is unique in giving the neutral-current rcouplings. While it is well established that the electron and p have the same couplingconstants, the same can not be said of the r. The r polarisation measurement providesa needed check on the Standard Model.Although the examples given here always refer to the r decay, the distributions forr decay are related by a simple charge-parity conjugation (CP) transformation, in sucha way that the final results differ only by the sign attached to the particle helicities. Butas mentioned above, the helicity H = —H;, with the happy result that these two effectscancel and we can ignore the charge of the r in the polarisation analysis.16\\\\\\\I I I I I 111111110 0.2 0.4 0.6 0.8 1x><z-1.75p=—1 .5- .250.750.50.250Figure 2.5: X distribution of r leptonic decaysThe variable x E/ET is plotted on the horizontal axis. The vertical axis gives thedifferential decay distribution, for different values of the r polarisation.17T helicity T hel.Tho,_______1)1.4 [EEc> ‘T LZZ>—1-i_il u_ ILIT(r Centre-of-mass frame)Figure 2.6: Possible spin configurations iii r+ decay•TH=OTLZE ZZ1‘p(r Centre-of-mass frame)Figure 2.7: Possible spin configurations in r decay182.3.2 Tau Leptonic DecaysThe decays r+ —* e viz, and r± are identical in form to the well-studieddecay j? —* e±vePp. Rather than repeat details, I quote only the final result here.The distribution of number of decays in terms of the final-state lepton energy is givenby= [(5 — 9x2 + 4x3) + P(1 — 9x2 + 8?)] (2.31)where x = Et/Emax, the normalised energy of the decay lepton in the laboratory frame,and P is the r polarisation. In arriving at this expression, the mass of the lepton has beenignored, as have radiative corrections to the decay matrix element. This distribution isplotted in figure 2.5.2.3.3 Thu Semi-Hadronic DecaysSemi-hadronic modes such as q+jj and r± > are valuable for polarisationstudies because of their two-body final state. This means that the angular distribution of semi-hadronic decays is more sensitive to the r polarisation than is the angulardistribution for leptonic decays.The transition amplitude (or “matrix element”) for a two-body decay can be writtenin general form[20]M(O, ) = J2J7 l(DL,)*@, X, _)frn1 (2.32)where in is the component of r spin along its direction of motion (the helicity), m1 andin2 are the helicities of the decay particles, in’ = — in2, and O* are the polarand azimuthal decay angles of the hadron with respect to the r direction of motion.Dmt(a, /3,7) are the rotation matrix elements for a spin-j system through the Eulerangles a,/3,-y, and fmim2 is a kinematic term dependent on the final state spin components.The simplest semi-hadronic r decay is r± . Here there is only one possiblespin configuration for the final state, with in1 = 0, in2 = —1/2 and in’ 1/2 (see figure2.6). There are then two matrix elements, one for each possible value of in:M(in = +1/2,O) = fo_1c 4Mfrn = —1/2, r. ) = f0a(— sin j)et (2.33)The kinematic parts are equal in this case so we can factor them out and write theangular distributions (after integrating out q&):— *= Mfrn = +1/2,O*) = —(1 + coso*)NdcosO 219r rest frame p rest frameFigure 2.8: Decay angles in r± — pvdN_*= IM(m = _1/2,0)j2 = (1 — cosO*) (2.34)NdcosO 2The complete distribution is the mean of the number of decays in each r helicitystate:1 dN — 1 ( N dN AL dALNdcos0— 2 iV +Ndcos0* + N++N_dcosO*1______= 2(1 + N + ALcos0*) (2.35)= (1+Fcos0*)where 0* is the decay angle of the 7r+ in the r± rest frame (as in leptonic decays), andare the numbers of decays with positive and negative r helicities.The case for decays such as pv is complicated by the p having spin 1. In ther rest frame, there are then two allowed final spin states, corresponding to mp = —1 andrn = 0 (see figure 2.7). Proceeding as for r+ irv , we have four matrix elements toconsider:M(m = +1/2, rn = ) = filei* sinM(rn = +1/2, rn = 0,0*, *)= f0,_ cos0*M(rn = —1/2,m = _1,0*,*) = f.1,_cos-- (2.36)M(m = —1/2,m = 0,0*,) = fo,_ei*(_sin)(2.37)VHT20Unlike the r+ R.+v_ decay, here the kinematic parts are not equal. However, fromthe components of the p polarisation vector (given by standard relativistic quantummechanics) we can deduce their relative amplitudes and assign= (2.38)= A(2.39)where A represents some unknown scale.Squaring the matrix elements, we factor out the constant term A. The azimuthalangle 4 integrates out as with r± ?èZ) , and we obtain the decay distributionsWi1 dN — iTL+PoVr. NdcosO— 2M 2—cos1 dN — *TL fiLVi-. Ndcos8* — + cosM2 1NdcosOm =(,)—(1+cos0’) (2.40)1 dN *TR + PL11r NdcosO = 1 — cosO )We can then sum over helicity states, equivalent to the treatment of r± w+v , andobtainNdcos0*= (1+aFcos0*) (2.41)where P is the -r polarisation and a relates the contributions from p helicity states:— 2M— T 242aM22M2As expected, this is of the same form as the distribution for the decay r+ W+l)(see equation 2.36), except for the factor of a, which represents the cancelling effect ofdifferent p helicities. Although the p is a resonance, one can insert a mean mass of 768.3MeV to obtain a = 0.46.The presence of a implies a diminished sensitivity to the 7 polarisation. and forthis reason it has conventionally been thought that these decays were inferior to the-÷ 7r±vT decay as a method of measuring the r polarisation. However, we can makeuse of the fact that the p decays via p± K±71 and determine the decay distribution asa function of cos 9* and cos ‘, where b is the decay angle of the ir± in the centre-of-massframe of the p. with respect to the p helicity axis (see figure 2.8)[21].First, we note that the above matrix elements are calculated in the r centre-of-massframe, where we can not observe the p helicity. To analyse the p± w+KO decay we need21to know the matrix elements in the laboratory frame. This involves a “Wigner rotation”of the original matrix elementsM!?6m (2.43)m’where i is given bym_m+(m+m)cos0*cosq= (2.44)m+m + (m— m)cos0*(for ET >> niT)Re-applying the general decay equation 2.32 for the case of p —÷ irir° gives thespatial components of the p decay matrix element:MP(rnp=O,çb,/,1)= cos,L’M(m = —1,, ‘) = sin (2.45)These distributions are incorporated by re-calculating the matrix element to representthe decay to final-state irs. Integrating out the redundant angle qY as we did earlier forwe obtain the final result:Ndcos0*dcos = 8(m +2m) [(1 +F)W+ +(1 — FT)W_J (2.46)where0 .0* 2W+ = sin b [(mTsinicos -- — mcosisin--)2+ mpsin-i-]+ 2 cos2 (mT cos cos + msin sin 0)2 (2.47)and.0* 0*2 2 20W = sin , [(mT sin sin -- + m cos cos -i-) + m cos+ 2 cos2 (mT cos sin—in sin cos )2 (2.48)This distribution, along with that for cos 0* alone, is plotted in figure 2.9, for P = —1,= +1 and PT = —0.15 (the Standard Model expectation for sin20w = 0.231).22Figure 2.9: Momentum distribution of r±Plotted on the vertical scale is the differential decay distribution, as a function of thedecay angles. The value P = —0.15 is the default in our simulated data set, and isapproximately equal to the polarisation measured later.— I cos(EY) cos(0)2—D, P=—1 2—D, P=—O.15— I CDSLi c,OSJ j2—D, P=-rl 1 —D diszrbution232.4 Radiative CorrectionsTn addition to the Born-level calculations we have been doing so far, there are higher-order corrections which must be taken into account. Some of these are “internal” andresult in renormalisation of the coupling constants, as described above. More serious arethe cases where extra photons are produced along with the final-state r-pair.The terms arising from calculations with these extra photons are called radiativecorrections. There are three types; “initial-state” radiation where the photons are radiated from the colliding e+ and C, “final-state” radiation where the photons are producedalong with the r+r, and “decay-state” radiation where photons appear with the r decayproducts (for example, r± p±7V)For each of these corrections, there is a direct and indirect effect on the r polarisation.Direct effect means a spin flip; the helicity of the r is changed by the addition of aphoton, in effect changing the polarisation. The indirect effect is to distort the amount ofenergy measured for particles in the experiment, causing mis-calculation of the scatteringangles.Estimates have been made of the degree to which the r polarisation is affected bythese corrections[22]. It was found that the direct effects are very small (SF c 0.2%),but the indirect effects are larger (SF 4%). However, it is possible to simulate andcorrect for the indirect effects of radiation very easily, and this is done in my analysis.After these corrections, the remaining uncertainty due to direct radiative effects is sosmall compared to that from the limited sample size in our measurement ( 9%) that Ifeel safe in ignoring the direct effect.24Chapter 3Experimental ApparatusThe parts of the experimental apparatus used fall into two large divisions; one consists ofthe LEP accelerator facilities which annihilate electrons and positrons, the other consistsof the OPAL detector and its subsystems, used to observe the products of this annihilation. A description of LEP is given below, followed by a description of the parts of theOPAL detector relevant to this analysis. Finally, the OPAL trigger system is described;this is the process by which individual particle annihilation ‘events” are selected foranalysis.3.1 LEPThe first proposals for a high-energy electron-positron collider were made at CERN in1976 and after a period of design work, the construction of LEP was authorised in 1981.The original design of LEP was optimised to collide particles at a centre-of-mass (CMS)energy between 80-100 GeV[23]. With the later discovery of the Z° it was decided tooperate LEP at a CMS energy close to the mass of this particle (91.18 GeV), turningLEP into a “Z° Factory.”A schematic view of LEP is shown in Figure 3.1. The facility consists of two mainsections; an Injector which produces electrons and positrons in bunches, and a StorageRing, which circulates and collides the particle bunches. It is the latter which is usuallyreferred to as3.1.1 InjectorThe injector has several components. To produce positrons, a beam of electrons is accelerated in a linear accelerator (linac) to an energy of 200 MeV and then dumped into aconverter target. In the converter target, the impacting electrons radiate bremsstrahlungphotons which “convert” into electrons and positrons by pair-production. The positronsare separated out and, along with electrons from another source, enter a second linacwhere they attain an energy of 600 MeV.251.ThOPAL__/1 ss 1L3 \ I -DELPHI[fR0TONCOflfltt’P—C(e/ectr)Figure 3.1: Schematic diagram of the LEP injectorThe LEP injector encompasses all the accelerators operating at CERN. Four experimentsare situated at different points around the storage ring.26Both electrons and positrons are accumulated in separate bunches in a small racetrack-shaped storage ring called the Electron-Positron Accumnlator (EPA). From the EPAthese bunches are dumped to the CERN Proton Synchrotron (PS), an accelerator ring,in which they are further accelerated to an energy of approximately 3.5 GeV. Then theyare shunted on to the Super Proton Synchrotron (SPS), another accelerator ring, wherethey reach an energy of 20 GeV.From the SPS, the particle bunches are injected finally into the LEP storage ring.3.1.2 Storage RingThe LEP storage ring (actually a storage/accelerator ring in that it both acceleratesand holds particles) was constructed in a tunnel running underneath the Pays de Gexregion of France and the Swiss Canton of Geneva. It is roughly circular, approximately27 km in circumference, situated in such a way as to avoid geological problem areas andpopulation centres (see Figure 3.2).Although approximately circular, LEP is more properly described as a rounded octagon. It consists of eight straight sections, connected by eight rounded corners. Particlescollide in four of the eight straight sections and it is here that the four LEP experiments(OPAL, ALEPH, DELPHI and L3) are located. The other four straight sections areunused by experiments.The particle bunches in LEP rotate approximately every 90 ps, which implies a beam-crossing rate of 45 KHz. Each ideally contains 4.16 x 1011 particles, with expecteddimensions a = 155 — 190pm, c, = 6 — 12pm, u = 1.17cm.The performance of an accelerator is usually measured by giving its luminosity C, aquantity defined in such a way that N = a where a is the cross-section for collision ofthe accelerated particles and N is the frequency of collisions observed. LEP is designedto operate at a luminosity of £ = 1.6 x 1031cm2r’. During the collection of the datahere, the luminosity was typically £ = 0.4 x 10’cm2s3.2 The OPAL DetectorOPAL has been designed with a cylindrical geometry typical of such experiments. It hasa barrel section and two endcaps. The barrel contains a solenoid coil of radius 2.18 m,which generates a magnetic field of 0.435 T parallel to the z axis. Inside the coil isthe Inner Detector, which contains subdetectors used to track particles as they emergefrom the beam interaction point. Outside the coil is the Outer Detector, which containssubdetectors used to identify particles and measure their energies.Each Endcap contains subdetectors which perform the same function as the OuterDetector, and in addition contains a Forward Detector which measures positions andenergies of tracks which emerge at very small angles with respect to the beam line.A complete description of the detector is given elsewhere[24]. For this analysis, theInner Detector and parts of the Outer Detector are relied upon almost exclusively, corresponding to a region of uniformly good performance. My description here focuses on27Plan de SituationOp ‘.GIBARUFigure 3.2: Geographic location of LEPLEP is situated in the Pays de Gex region of France and the Swiss Canton of Geneva.Although the tunnel crosses the international boundary, all access points to LEP are inFrance, as are the four LEP experiments. The city of Geneva is located to the south ofLEP.1:80000hex28these sections of the detector.The axis of the OPAL barrel is nearly that of the LEP beam line. All measurementsare done in a co-ordinate system where the x axis is horizontal and points roughly towardsthe centre of LEP, the y axis is vertical and the z axis points in the direction of motionof the circulating electrons. A spherical polar co-ordinate system is also defined in whichthe polar angle 0 is measured from the z axis, the azimuthal angle is measured fromthe x axis about the z axis, and the radius r = i/x2 + y2 + z2 (see figure 3.3).3.2.1 Inner DetectorThe inner detector is an array of three concentric cylindrical subdetectors sitting insidethe OPAL coil. Electrons and positrons collide at the Interaction Point, inside the LEPbeam pipe. Any charged particles resulting from these collisions fly outwards, tracingout spiral orbits due to the presence of the magnetic field generated by the coil.Innermost of the three subdetectors is the vertex detector, which gives precisionlocations of charged particle tracks close to the interaction point. Next is the jet chamber,which gives momenta and further position data of tracks. Outermost are the Z-chambers,which measure track positions entering the coil. All three of these are wire chambers ofvarying designs, operating in a common pressure vessel at 4 bar gas pressure. Theycomplement each other, providing complete information about a track when their outputis combined.The vertex detector is a cylindrical drift chamber 1 m long and 47 cm in diameter,surrounding the vacuum pipe containing the electron and positron beams as they circulatethrough LEP. It is divided into two layers consisting of 36 cells each, an inner layer with“axial wires” parallel to the z axis and an outer layer with “stereo wires” oriented at anangle of four degrees to the z axis. The axial-wire layout allows a measurement of trackpositions in r and 95 to an accuracy of 50 pm . Resolution in z is approximately 1.5 mm.The jet chamber is 4 m long, with an inner diameter of 0.5 m and an outer diameterof 3.7 m. It is divided into 24 sectors, each of which contains one wire plane with wiresparallel to the z axis. The wire position, drift time, and charge division give coordinatesin r, 95 and z. The average resolution in r and 95 is approximately 135 pm , and in z isapproximately 6 cm. Since the jet chamber provides measurement of track positions atmany points in space, the information it provides can be used to calculate the curvatureof the track. From this, knowing the strength of the OPAL magnetic field, the trackmomentum can be calculated with an average resolution of Sp/p2 = 2.2 x 10 GeV’.There are 24 Z-chambers in the form of a cylinder occupying the space betweenthe coil and jet chamber. Each is 4 m long, 50 cm wide and 59 mm thick, split intoeight cells 50 x 50 cm in size with six wires running in the 95 direction. z positionsare given by the wire hits, and the 95 co-ordinate is read out by charge division. Withthis arrangement, the Z-chambers are able to measure the z co-ordinate of tracks to anaccuracy of approximately 300 pm and the r-95 position to approximately 1.5 cm.29Figure 3.3: The OPAL detectorThe different components of the OPAL detector are visible in this cutaway view. In thebottom left, the co-ordinate axes used for analyses are indicated.C’,LUC)z0DC,20UiC’,UiUi303.2.2 Time-of-Flight SystemThe first of the outer detectors is the Time-of-Flight system (TOF), which is just outsidethe OPAL coil. This consists of 160 scintillating counters, each 6.8 m long, 45 mm thickand approximately 90 mm wide. The counters are placed parallel to each other to forma barrel of mean radius 2.36 m.When a particle passes through the TOP, a pulse of light results and is read out ateach end of the scintillating bar by photomultipliers. This gives a measurement of thetime at which the particle passed, which is useful for elimination of cosmic ray events.As well, the difference in time between the light pulses at each end of a bar can be usedto estimate the z position of the particle. The accuracy of signal timing is approximately220 ps and the accuracy in z measurement is approximately 5.5 cm.3.2.3 Electromagnetic CalorimetryAs charged particles pass through the lead glass they undergo electromagnetic interactions. Photons are radiated from the particles by bremsstrahlung, and electrons areproduced both by ionisation of the atoms in the material and pair-production from thebremsstrahlung photons. This results in a chain of electrons and photons, called anelectromagnetic shower.The electromagnetic calorimeter (ECAL) is optimised for measurement of electromagnetic showers. It is split into a barrel (EB) and two endcaps (EE), each made upof lead glass’ blocks. The depth of glass (approximately 25-27 radiation lengths) andblock size are chosen to correspond to the expected dimensions of an electromagneticshower. Shower electrons emit Cerenkov radiation in the lead glass. This radiated lightis then read out by photomultipliers, giving a measurement of the energy deposited bythe shower particles. The accuracy of the energy measurement is limited by fluctuationsin the number of particles per shower.There are two differences between EE and EB. The first is their geometry; EB issegmented in into 160 blocks and in z into 59 blocks. Each half of EE consists of adome-shaped array of 1132 blocks. In EB the blocks are oriented such that their axespoint roughly to the interaction point. In EE the block axes are parallel to the beamline.The other difference between EB and EE is the way in which the Cerenkov light isread out. EB is outside the OPAL coil and can therefore use photomultiplier tubes,which can operate in the slight residual magnetic field. EE operates in the full magneticfield and therefore must use a new type of photomultiplier called a vacuum photo triode,which is not affected by the field.The lead glass has an intrinsic energy resolution of JE/E = 0.2% + 6.3%/i/p (E in0eV), among the best possible of calorimeter designs. However, this intrinsic resolutionis degraded by the calorimeter’s position behind the coil. The effect of this is to degraderesolution at very low energies (see figure 3.4)175% PbO by weight, refractive index n = 1.8467 at A = 586 nm313.2.4 Hadronic CalorimetryThe hadron calorimeter (HCAL) is designed to provide position and energy measurementsof hadronic showers. These are showers of hadrons resulting from collisions between ahadronic particle and nuclei in the detector material. Typically they take place at muchgreater depths in material than do electromagnetic showers.HCAL is divided into a barrel, two endcaps, and two “pole tips.” The barrel calorimeter is made up of 9 layers of wire chambers alternating with 8 iron slabs, and is positioned from radii 3.39 m to 4.39 m. Each slab is 100 mm thick and each chamber 25 mm.Hadronic showers are initiated in the iron (which also serves as the flux return for thecoil) and their position and energy are read out in the wire chambers. The electromagnetic and hadronic calorimeters combined are approximately 6 nuclear absorption lengthsdeep, corresponding to the expected depth of a hadronic shower.The wire chambers vary in length and width, but each consists of a series of channelsat least 5 m long, running parallel to the z axis, made of PVC plastic with an anode wirerunning down the centre of the channel. The PVC is coated with graphite which servesas a cathode, and the channels are operated in limited streamer mode.The charge deposited by a shower is read out in an arrangement of pads and strips.Pads are large conducting surfaces, roughly the expected size of a hadronic shower. Theysit against and outside the plastic of the chambers. Charge is induced on them from thatdeposited in the chambers and the pads are taken to define towers, the total charge inwhich corresponds to that deposited by a hadronic shower.Strips sit inside the detector plastic and run along one wall of each channel. Thecharge deposited on a strip gives a rough measurement of energy and, by charge-division,gives a more precise measurement of the location of the hadronic shower.The endcap calorimeter is of much the same design, but here there are 8 layers ofchambers and 7 layers of 100 mm thick iron, with 35 mm gaps between the iron. Thechambers are oriented with their wires pointing along the x axis.The pole tip calorimeter is similar to the endcap, but covers the extreme range of 0,0.91 cos 0 < 0.99. It has 10 chambers and 9 layers of iron, each some 70 mm thick.The hadronic calorimeter has an energy resolution varying from CE/F =to CE/F = 140%//E, depending on the energy of the particle initiating the shower.It must be noted, though, that some particles initiate electromagnetic showers beforereaching the hadronic calorimeter, so an accurate energy measurement must sum theenergies in both the electromagnetic and hadronic calorimeters.3.2.5 Muon DetectorThe muon detector consists of large-area drift chambers placed outside all other detectorelements, next to the hadron calorimeter. It has a barrel and two endcaps. Muon identification is accomplished by accurate measurement of the outgoing track direction andcomparison to tracks in the central chamber to measure the angle of multiple scattering.The main purpose of the muon detector is therefore to accurately measure the positionof outgoing tracks.32The barrel part of the detector consists of 110 wire chambers, each 1.2 m wide and90 mm deep. Forty-four are mounted on each side of the barrel, plus 10 in a separatemodule on top and 12 in another on the bottom.The 4’ coordinate is measured by the barrel detector to an accuracy of approximately1.5 mm, and the z coordinate to an accuracy of 2 mm. It has an efficiency for detectingmuons above 3 0eV of approximately 100%, and a probability of mis-identifying anisolated 5 0eV pion as a muon of 1%.Each endcap detector consists of 8 chambers 6 x 6 m in size and 4 patch chambers3 x 2.5 m in size. The chambers contain two layers of streamer tubes, similar in designto those used iii the hadron calorimeter. One layer is oriented horizontally and anothervertically. Instead of a pad-and-strip arrangement, each layer has two sets of strips, oneset parallel to the wires and one perpendicular.The endcap measures positions to an accuracy of approximately 3 mm. Muon misidentification was measured to be 0.2% for 10-50 0eV hadrons passing through the chamberat an angle of 45°.3.2.6 TriggerTo properly identify electron-positron collisions, signals from the different OPAL subdetectors are combined in various ways to produce a logical trigger signal telling whetherthe data in the sub detectors should be recorded or not. The full details of the triggersystem are written elsewhere[25] but I will describe its basic features here.Where possible, signals from the different subdetectors are divided into bins in U and4). In each bin, a trigger signal is generated if certain conditions are met, and the resultsfrom the bins are combined by a trigger processor which decides whether to keep theevent or not. As well, some subdetectors generate independent “direct” trigger signalsnot subject to this binning. Times of all signals are referenced to another signal generatedby LEP at the moment the beams cross.The vertex detector and jet chamber produce a signal called the “track trigger.” Observed wire chamber hits are divided into bins by their values for r/z, which is equivalentto U. A peak in one U bin indicates the presence of a track.The time-of-flight detector produces a trigger by looking for any hit in a block of24 scintillator bars within +50 ns of receiving a beam-crossing signal from LEP. If morethan six such blocks show hits, a trigger is generated.ECAL signals are counted by binning in U and 4) the energy read out from lead-glassblocks. If this energy passes different threshold values, trigger signals are produced.HCAL triggers are produced in much the same way, summing the energy in pads.The muon detector barrel and endcap sections generate separate triggers. In thebarrel, a trigger is generated if hits are observed in at least three of the four layers ofthe detector, in one 4) bin of approximately 15° size. Triggers are generated in the muonendcap by summing the charge collected in 64-128 adjacent strips. If at least two planes ofthe detector show that this charge is above a threshold value, and the measured positionis consistent with a track from the interaction point, a trigger signal is generated.33The Forward Detector trigger compares the energy deposited in the detectors againsta threshold value, and generates a trigger signal if the total energy is above thresholdand is deposited in a configuration consisteut with a physical event. This signal is usedmainly to detect-Bhabha scattering events.In 1990, these signals were combined and an overall trigger was geuerated if any of alist of conditious were met. The most relevant of these for this analysis were:• More than two tracks are observed in the barrel region• More than three tracks are observed anywhere• A track is observed with an associated TOF trigger• Associated TOF and EM triggers are observed• At least one EB bin has 2.6 CeV deposited• 4 GeV energy is deposited in the EM barrel, and there is either an associatedtrack or an associated TOF triggerThese triggers are independent of each other and redundant to a high degree. As aresult, we have a high trigger efficiency. For detection of ee —* r+r events, this wasestimated to be 99.9 + 0.1%.3.2.7 Data ReadoutFor each subdetector, there is a specialised processor which accumulates raw signals fromthe subdetectors and processes them, producing more readily usable output. These arecalled the Local System Crates (LSCs).The processed signals from the LSCs are collected in a parallel-processing systemcalled the Event Builder. This is capable of handling events at a rate of 10 Hz, each ofup to 100 Kbyte in size.From the Event Builder, a special processor called a filter performs a quick analysisof the event and labels it as coming either from an electron-positron collision or fromone of several categories of ‘background” events (ex. cosmic rays, interactions of beamparticles with material in LEP). In 1990, the filter ran in a VME-bus microprocessorsystem, as did the Event Builder and LSCs.Events which make it through the event builder were, in 1990, written out to cartridgetape by a VAX 8700, regardless of whether they passed filter requirements or not. As well,events which were labelled acceptable by the filter were passed on to another computersystem which performed “event reconstruction:” The raw (lata from the detector wasconverted into physically meaningful quantities such as the number of observed particles,their energies and momenta.348760.2Electron energy (GeV) (E112 scale)Figure 3.4: 1990 ECAL energy resolutionThe inherent energy resolution of the electromagnetic calorimeter is seen to decreasebecause of the material situated in front of it.5 10 50100 500cx35Chapter 4DataIn addition to the experimental data, simulated “Monte Carlo” data representing allsignificant processes at LEP are required to estimate efficiency and contamination. Production of these data sets begins with simulation of particle production, including somehigher-order field theoretical corrections. The output of this simulation is then processedby a detector simulation which produces “raw” data iu the same form as actual OPALevents. Finally, both data and Monte Carlo data are passed through the same process ofreconstruction, whereby physical quantities such as particle momenta and directions areobtained from raw data.4.1 DataThe events used for this analysis were collected during the OPAL running period fromMay-August of 1990. They correspond to an integrated luminosity of £ = 5.8pb’, or3310 r-pair events. This sample was taken at different energy values from 88.22 0eV to94.22 0eV, with about two-thirds near the Z° peak at 91.175 0eV (see table 4.1).4.2 Simulated DataSimulation of OPAL data proceeds through two steps. First, the fundamental particledata are created; particle types, their four-momenta, charge, and spin. A number of“generator” computer programs are used for these, according to the specific processbeing simulated.After generation, the interaction of these particles is simulated in the OOPAL[26]computer program. This is built around GEANT{27], a standard modelling system forparticle physics experiments. GEANT tracks the location and momentum of each particle at a certain time, calculates changes after a small increment in time, then repeatsthis procedure, “swimming” the particle through space untit it leaves the region of thedetector. Interactions between particles are taken into account, as are decay lifetimesand particle-detector interactions.36(GeV) Real Events Sim. r-pairs Sim. r— P-’T decays91.16 0 50000 20000088.22 47 0 2000089.22 138 0 2000090.22 151 0 2000091.22 2437 0 2000092.22 234 0 2000093.22 182 0 2000094.21 121 0 20000Table 4.1: e+e —* 7+7 data samplesPhysical objects are modelled by feeding GEANT a series of points indicating theextent of solid objects in space (a “volume”), along with their density. GEANT returnsthe energy and electric charge deposited in each volume, along with positions of theparticles. GOPAL then converts this into simulated track information from the centraldetector and simulated calorimeter energies from the outer detector.4.2.1 Simulated r-pairsr-pairs are generated using the KORALZ[28j 3.7 computer program. This is a standardfor LEP experiments, designed to provide accurate simulation of these effects:• “Initial-state” radiation of photons from the colliding e and e particles• “Final-state” radiation of photons from the produced r+ and c particles• Higher-order weak corrections• Final state spin-angle correlations• Simulation of all significant r decay modes• Effects on decay distributions resulting from r polarisation• Effects on —* P’ decay distributions due to p helicityA main sample of fifty thousand e+e—* r+r events was made and then processedby GOPAL to provide simulated raw data. The parameters generated for each decay(its type, momenta, and decay angles) are stored with each simulated event. By comparing the reconstructed event quantities with those generated, it is possible to calculateefficiencies, final-sample contamination, and correction factors needed for analysis.Three types of weights are applied to each r decay in this sample. Decays are weightedby mode to correct the “missing mode problem” of r physics. World average branching37fractions are used for one-prong modes and OPAL measurements are used for the three-prong and five-prong modes[29]. As the sum of these modes is not equal to one, eachfraction is increased in proportion to its error (roughly 0.8 standard deviations) until thesum of branching fractions for all modes is one. The branching fractions used are listedin table 4.2.Decays proceeding by r+ atzz,. —* w±(2irO)vr are also weighted according to theactual mass of the a1 meson, given by KORALZ, in such a way that the weighted massspectrum has the current best-measured mass and width (the original KORALZ samplecontained less accurate values for these parameters than those that are now known).Finally, each r-pair is weighted according to the helicity of the r. These weightsmay be varied, allowing simulation of any degree of polarisation.Decay mode B.F. (measured[29], %) B.F. (weighted, Ye)T—+ f?L/r 23.1 + 0.6 23.6y— 17.9 + 0.2 18.1T 17.5 + 0.2 17.7r—÷ rr+iri> 0w°)v 15.2 + 0.3 15.4r —* (w, K)v 12.0 + 0.3 12.3r—÷ w(2ir°)ziT 8.3 + 0.7 8.9y— ir(> 3ir°)v 1.6 + 0.6 2.1r —+ (K*)±v 1.4 + 0.2 1.6r—+ irir+irw+wi> Olr°)VT 0.25 + 0.08 0.3Total 97.3 + 1.2 100.0Table 4.2: Branching fractions used in simulationIn addition to the main sample of r-pairs, Monte Carlo subsamples are also generatedconsisting of events where each r+ decays by ± , at a range of r energies (seetable 4.1). These samples are required to calculate correction factors in the polarisationanalysis, where the particle decay angles are needed. As these decays are not usedanywhere else, they are not passed through the entire simulation chain, only producedby KORALZ.4.2.2 Other Simulated DataA sample of 167,000 multi-hadron events was generated using two computer programs,HERWIG[30] and JETSET[31j. While these differ iu their treatment of QCD processes,the difference is of little importance to r studies, where we are interested only in the grossqualities of these events; the number of charged tracks and ECAL clusters produced.Hence, both samples are combined and treated as one.Bhabha scattering is studied using a sample of 24,000 events, limited in range to12.5° 0 167.5° 50 the sample will be useful for background studies in the barrel38region. Di-muon background is estimated from a sample of 20,000 events, generatedusing KORALZ as were the r-pair samples.Last, simulated “two-photon” data sets (e+e_ +e+eX) were generated. The name“two-photon” refers to the photon-fusion process in which extra particles X are created(see figure 4.1). Approximately 19,000 of these events were generated in total, distributedamong several possibilities for the extra particles X (=ee, jc7c, r+r, q4).All these background samples have been processed by GOPAL, following the samechain as the r-pair sample.4.3 ReconstructionRaw information from the Inner Detector subdetectors is combined into a series of quantities corresponding to “charged tracks.” Each of the Outer Detector subdetectors produces a “clnster” by grouping together adjacent hits. A charged track should correspondto the trail left by a particle, and a cluster should correspond to the energy left behindby a shower.Tracks and clusters are associated with each other based on their nearness in 0 and. One cluster may have several associated tracks, but each track is associated with atmost one cluster per subdetector.Unless stated otherwise, the only clusters we make use of here are ECAL clusters.For these, the total energy deposited in a contiguous group of lead-glass blocks (the “rawenergy”) is corrected to account for losses in the coil and to neighbouring blocks. Theresult is the “cluster energy” (or “corrected energy”). Cluster locations in 0 and ó aretaken to be the cluster centroid.After reconstruction, it is observed that the energy and momentum resolution inthe Monte Carlo data are significantly better than in the data. To make the MonteCarlo data more faithfully represent reality, Monte Carlo ECAL energies and chargedtrack transverse momenta are “smeared” random numbers are added to each trackmomentum and cluster energy in such a way that the distributions of each becomebroader. The exact change isSEecai = 0.O3Eecai (4.1)£DL_1 -n in—3 ID-I- \2ctrk — >< IU ><where Eecai is the the corrected energy of an ECAL cluster and F’ is the transversemomentum of a charged track.39exFigure 4.1: Feynmari diagram for e+e —* e+CX40Chapter 5Preselection of e+e rHt EventsA sample of r-pair events is collected using requirements which have been standardised for all analyses in the OPAL ‘r polarisation working group. These ensure minimalcontamination from other processes. r-pair events are selected with approximately 55%efficiency, with little effect on the r+ p±jj branching fraction or polarisation calculations. Although requirements are aimed at eliminating certain types of background,many are effective against more than one type, and an estimate of the background isgiven only when all requirements are combined.5.1 Data Quality and Status RequirementsFor all events, the Jet Chamber and Electromagnetic Calorimetry must have been functional, prodncing both data and triggers. Also, the Forward Detector, Vertex Chamber,Time-of-Flight counter, Hadron Calorimeter Barrel and Muon Detector Barrel all had tobe operational and producing data.In all preselection requirements except those which ellminate cosmic rays, we refer to“good” charged tracks and clusters, which are defined as follows:• Tracks: dol <2 cm, zo 50 cm, N,t 20, P1 0.1 0eV, and R[ 75 cm• ECAL barrel clusters: Eraw 100 MeV• ECAL endcap clusters: Eraw 200 MeV, Nblms 2, Fmax 0.99Here d0 is the distance of closest approach between the track and the z axis, z0 is thez value of the track at the point of closest approach, and P1 is the transverse momentumof the track. Distributions of track parameters in 1990 are shown in figure 5.1.is the radius of the first wire hit in the jet chamber, and Fmax is the fraction ofenergy contributed by the highest-energy block in an electromagnetic calorimeter cluster.Eraw is the total cluster energy before the above-mentioned corrections are applied.These requirements are very loose, intended to be the minimum necessary to ensurethat tracks and clusters are due to physical particles from the interaction point.41(/1 (11- -0.24cQ.24 a0.2o.20.160.160.12 .120,08 0.08004 0.040 0—100—50 100cm cmzo,.‘ r1TTI1I.II1 . IIIu.lO0.140.12E- 0a t.. 001 z0,Q5!0 50 100 150 00PT Number of jet chamber hitsFigure 5.1: 1990 distributions of track parametersThe parameters d0 and z0 are, respectively, the impact parameter and zcoordinate at the point of closest approach to the z axis of the charged track. FT is thetrack’s transverse momentum. Arrows mark positions where requirements are applied.—2 —1 0 1 2 0 50I I I425.2 Elimination of Cosmic Ray BackgroundCosmic rays are eliminated with these requirements:At least one TOF counter must satisfy Itmeag— expI < 10 ns, where texp is the timegiven by the LEP beam crossing signal and tmeas is the time of the TOF signal. This“gate” should be sufficient to include most events arising from e+e annihilation (seefigure 5.2a). If all TOF signals from counter pairs separated by more than 165° havetime differences greater than 10 ns, the event is rejected. For e+e annihilation, thetime difference for these “back-to-back” counters is approximately St = 0 + 6 ns, but forcosmic ray events the time difference is seen to vary between roughly 15 St 25 ns.This can be seen in figure 5.2b, where two peaks are evident, one for cosmic ray particlesand one for annihilation products.At least one track must satisfy dol 0.5 cm and z0 $ 20 cm, and the average z0 forall tracks must satisfy < zo > 20 cm. These two requirements ensure that the eventis consistent with production of particles from the interaction point. Distributions ofI domin vs. Zomin and of < z0> are shown in figure 5.2c,d for events which are rejectedby the TOP requirements (ie cosmic ray events). Cosmic ray events are seen to spreaduniformly, whereas annihilation events are narrowly distributed around the origin.5.3 Cone Analysis and Fiducial AcceptanceAll backgrounds other than that from cosmic rays are eliminated using requirementsbased on a cone analysis. The procedure for determining a cone is as follows:All charged tracks in the event are listed in decreasing order of their momenta. Thehighest-momentum track is taken to define a cone direction, and then the list is searchedfor the next-highest momentum track within 35° of the cone. When it is found, themomenta are added together to define a new cone direction. Then the procedure isrepeated until no new tracks can be added to the cone. New cones are further defined,until all tracks in the event have been accounted for.In a r-pair event, all the r decay products should be collimated due to the highmomentum of the r, producing only two cones. Therefore events are required to haveexactly two cones. Each cone must have at least one charged track and at least 1% of theLEP beam energy. The amount of multi-hadron background deleted by this requirementcan be seen in figure 5.3a.The acolinearity of the two cones must be less than 15° (0.13 rad), to eliminate eventswith highly energetic radiated photojis. The expected distribution of the acolinearity oftwo-cone r-pair events is shown in figure 5.3b. It is estimated that this requirementrejects about 2% of r pairs.____The average cos 01 of the two cones must satisfy cos 0 < 0.68. This requirementeliminates 41% of all r-pair data. This acceptance limit is necessary so that eventsoccur within a region of the detector where response is uniform and well-understood.By increasing the cos 0 limit, one encounters problems in the “overlap region” of thedetector, where the endcap and barrel meet.43120a)>100-DE:3za)0N80604020040200—20—40—20 —10(a)t—t (real daft)CI)a)>I)0a)-o8:3z20nsL;__I___l.__._C___).__J____I.__ I I I I ii—1 —0.5 0 0.5 1d0 (cm)çc)z0 vs d0 (cosmic rays)if)a)>a)0C:-oc1801 601 401 201 00806040200—20 0 20<z > (cm)(d)<z0> (real data5Figure 5.2: Useful distributions for cosmic ray eliminationPlot (a) shows the spread in impact time for particles from annihilation. Plot (b) indicatesthe time difference between hits in opposing TOF sectors. The two peaks indicate dataand cosmic ray times. Plot (c) shows the random locations of cosmic ray tracks, and plot(d) in contrast shows the tight constraints on tracks from annihilation.0 10(b)A (eal data back—to—back hits)PS44Figure 5.3: Normalised distributions of r-pair selection quantities (simulated data)Plot (a) shows the number of charged cones in T-pair and multi-hadron events. Plot(b) shows the acolinearity of two-jet r-pairs. Plots (c) and (d) show, respectively, thenumber of good charged tracks and number of good clusters in T-pairs and multi-hadronevents.D.12CcC0.10.0.c8c.)pzO.060.040.020(I)>80.32NJ0.240.2U20.16U-0.120.080.040.09C0.08-o0.07°0.0680.05U20.040.030.020.010N(c)N,0 0.1 0.2 0.3 0.4rod(b) 40 ci L nec rity0.07°0.06NJ00050-0.040.03ElMH+L10.020.01N(C)Ntrk0 a 10 12id ) N14 16 18 20N4500 20 40 60 80otrk (0eV/c)\C)C>U)<0.12 -U)00.08UCU->U)U-0080604020>U)C)U-1CC80604020020 40 60 80 100Ptrk (0eV/c)(b)CU)>wQQ40.C3C00.02U00.012 3 4E. (0eV)0.040(c) (d)Figure 5.4: Bhabha-scattering and two-photon event characteristics (simulated data)The two upper plots show the dramatic contrast between the characteristic energy deposits of T-pairs (a) and Bhabha-scattering events (b). The two lower plots contrast theenergy seen in the detector for two-photon events (c) and r-pairs (d).80E. (0eV)465.4 Elimination of ee -÷ qq BackgroundMulti-hadron events are characterised by large numbers of particles produced in “jets.”Since one jet corresponds roughly to a cone, some multi-hadron background is rejectedby the two-cone requirement. To further reduce this background, we make two morerequirements:The total number of good charged tracks must satisfy 2 Nctrk 6.The total number of good ECAL clusters must satisy Necai 10.The distribution of the number of good charged tracks for r-pairs and multi-hadronevents is shown in figure 5.3c, and the distribution of the number of good ECAL clustersin figure 5.3d. There is clear separation between the two types of events.5.5 Elimination of e+e e+e BackgroundSince electrons almost always deposit their entire energy in the electromagnetic calorimetry, an e+e e+e event will be characterised by total ECAL energy near that of thecolliding particles.Therefore, events must satisfyEEecai < O.8Ecm, where Ecm is the centre-of-mass energy of the colliding electron andpositronorEEecai +°.3Pctrk <Ecmwhere the sum is over all good charged tracks and clusters in the cone.The distribution of r-pair events in ZEecai and EPctrk is shown in figure 5.4a and thesame distribution for Bhabha-scattering events in figure 5.4b.5.6 Elimination of e+e e+eX BackgroundTwo-photon events e+e are characterised by two cones of secondaryparticles X having very low energy and transverse momenta. To identify them, we definea quantity called the visible energy E3 as the sum over all tracks and clusters of theenergy for each (the energy is calculated assuming the charged track to be a 7è). Whenthe charged track and cluster are associated the maximum of the two energies is taken.The aim of this definition is to accurately estimate the total energy of the particles inthe event.We require E3 > 0.O3EcmIf E8 < O.2Ecm then the track and cluster total transverse momenta must satisfypf’k> 2 GeV or pal > 2 GeV.Figures 5.4c,d show distributions of E3 for two-photon and r-pair events. Figure 5.5shows vs. Ff’c, also for two-photon and r-pair events. In both cases there is clearseparation, as desired.4720C-)>1612 -84O0 4 8 12 16 20+ + - -÷ Pctrk (0eV/c)(a)e e e e ,e e________________________—________U>a) ,. .16....:::.. ...8.. .... •.•.: .4•. ;. :.... .f:.....; .—_______0 12 16 20Perk (0eV/c)(b)TrFigure 5.5: Two-photon event characteristics (simulated data)The momentum profiles for two-photon events (a) and T-pairs (b) show a sharp contrast.485.7 Elimination of ee—+ trFw BackgroundTo be rejected, events must have two charged tracks which can be identified as muons(“muon candidates”). Muon candidates must satisfy these requirements• p-i- > 2 GeV• p>6GeV• Idol < 1 cm• Iz0 < 50 cmTTCV+CJ+CZ•‘‘hits—and at least one of these requirements:• At least two associated hits in MB or ME• At least 4 associated HCAL hits, with 1 in the last 3 layers of RB if cos 01 <0.65• F> 15 0eV and associated ECAL energy less than 3 0eVWe require E0 > O.6Ecm for the event to be rejected, where E0 is the scalar sum oftrack momenta and highest single cluster energy. Also, the azimuthal angles of the twomuon candidates must satisfy 6q5> 0.32 rad for the event to be rejected.Distributions of these quantities for simulated muons and r-pairs are shown in figures5.6 and 5.7.5.8 Summary of ResultsAs estimated from our simulated data, the preselection requirements identify r+r pairswith 54.7+0.7% efficiency (59% from the acceptance requirement alone, as stated above).The final sample contains a contamination of 1.9% non-r pair events, broken down intable 5.1. The fraction of r± pv events in the preselected sample is estimated to beenhanced by a factor of 1.02 + 0.00549Figure 5.6: Dimuon event characteristics (simulated data)The upper two plots compare the number of hits left in the muon barrel detector for(a) muons and (b) r-decay products. The bottom two plots compare the total energyseen for (c) di-muons and (d) r-pairs. The missing energy arising from r decay is clearlyvisible.I0.8ci)>0.6° 0.50.4t 0.30.20.10HHH H(I)CI])I 0 .35- 0.30.25C3 0.2CL0150.10.050[1)Cci)O,16jO.1 2O.O8LI0.040012345678(a)MB nits, /.Ln fl0 1 2 3 45 6(D)MB hits, TT78hil(0eV) (0eV)(c)E101, (d)E101,r509876oo 4•. 3• 2123456789Penetration(a)HCAL hits, tN5045. . 45.2,40a!30 : 3025. 2520 2015••‘510. 10* 50 0Figure 5.7: Di-muon event characteristics (simulated data)The upper two plots compare the hadron calorimeter interaction profiles for (a) a muonand (b) particles from r decay. The penetration (outermost HCAL layer hit) is plottedagainst the number of layers hit in total. These two plots demonstrate the penetratingpower of muons. The lower two plots show the electromagnetic calorimeter interactionprofiles for (c) muons and (d) r decay products. It is clearly seen that the muons depositless energy in ECAL.51z zEEDoDE0876R432UED° ° n D2 ° ° 00I.’0• I • I123456789Penetration(b)-1cA hits, T20 40E (0eV)(d)E00 vs. P,,1,0 20 402 (0eV)vs. Par,, P0(CbCbçt±±±+,CCbCbCbIIII-J-4-1-1-oC±-CI++cECoE9-100r\)C; I CDChapter 6Selection of r± P11T decaysThe complete chain for a T± decay isT± P+VT n7vThe data are subjected to certain requirements, designed to select events where thefinal-state w and O are well separated spatially in the detector. In addition to reducingcontamination these requirements allow accurate calculation of the decay angles cosand cos ‘, necessary for polarisation studies.6.1 Selection CriteriaMy requirements, summarised in table 6.1, are designed to select events where thefinal-state j-+ and r0 are well separated spatially in the detector. Tracks and clusters areexamined within a decay cone of half-angle 0.3 rad around the direction of the cone ascalculated in preselection. The theoretical cone size for r+ p±vT decays is plotted in6.la. This is defined by the angular spread from the direction of the r decay cone (asgiven by preselection) of the final-state + and photons, where the photons are requiredVariable LimitCone size 0.3 radNeutral cluster energy threshold 2.0 GeVNumber of charged tracks in cone 1Minimum # of neutral ECAL clusters 1Maximum # of neutral ECAL clusters 2Minimum cone mass 550 MeVMaximum cone mass 950 MeVMaximum E/p 0.9Table 6.1: T± —* P+VT selection requirements53to have at least 2.0 GeV of energy to be consistent with a “neutral cluster threshold”defined below. As can be seen, the cone size nsed is very conservative.Tracks are defined to be in the decay cone if their initial momentum vector pointswithin the cone. I reqnire there to be exactly one such good charged track. This greatlyreduces background from r decay modes with 3 or 5 charged particles (see figure 6.lb).Events are required to have either one or two neutral ECAL clusters inside the cone,where a neutral cluster is unassociated with the good charged track and must have atleast 2.0 GeV energy before corrections are applied. The particular value of the neutralcluster energy threshold was chosen considering several factors. ECAL energy correctionsare made assuming that the energy deposited is due to an incident electron. For lowenergy photons, the corrections are of little use and consequently the measured energyis in doubt. As well, there is evidence of more general problems with the Monte Carlomodelling of low-energy clusters. This can be seen by comparing the distribution ofneutral-cluster energy for events with one neutral cluster, at varying threshold values.The x2 for this distribution is plotted in figure 6.lc. It is seen to reach a fairly steadylow point at approximately 1.6 GeV. Aside from these measurement difficulties, higherthreshold values also reduce the background from r+ W±L) decays. These very oftenproduce more than one cluster, with secondary clusters typically low in energy (see figures6.2a,b).The distribution of the number of neutral clusters for r± p±jj and the mainbackground sources is shown in figure 6.2c.The four-momenta of the charged track and each neutral cluster are calculated:Fctrk = ((Ptrk +Fecai = (Eecai, Eecaiflecae) (6.1)where m+ is the mass of the irk, J5ctrk is the momentum of the charged track, Eccat theenergy of the neutral cluster, and eca1 the direction vector of the neutral cluster.The cone mass Mcone is calculated as the magnitude of the sum of all four-momentain the eventMcone = (5F0)2 — (SF1)2 (SF2)— (SF3)2 (6.2)I require that this mass be consistent with that of a p meson: 0.55 < Mcone < 0.95GeV/c2. This reduces the background from r± > ir±2wOzi decays and other hadronicmodes. The cone mass distribution is shown after all other cuts for data and simulateddata in figure 6.3a. The distributions for the major background sources are shown infigure 6.3b-d.To further reduce the background from r± —* ié(> 2)ir°v7 modes, I also require thatthe energy in the cluster associated with the charged track satisfy Eass/Fctrk < 0.9c.The aim of this requirement is to remove events where clusters from a ir overlap withthat associated with the charged track. The E/p distribution is plotted after all otherrequirements in figure 6.4a, for data and simulated data. The distributions for the majorbackground sources are shown in figures 6.4b-d.54(1) (1)>N >N80,120.20.10.08 IcO.06 t o.i80.04U-0.020 00 0.125 0.25 0.375 0.5 0123456789rod NCt,k(a)Cane size, r (Mc data) (b)# good charged tracks (MC data)>< 1.3 —1.2 • 80% CL I1.!•.•. 50%C.L.0.9 • •.0.8 . • .• •.• .0.7 ‘0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6Eraw (0ev)(c)Chi—Squared of Neutral Cluster Energy vs. Neutral Cluster ThresholdFigure 6.1: Cone size and neutral cluster requirementsThe cone size (a) is from simulated r± ._+ pu decays, counting the angular separationof the lr± and photons with more than 2.0 0eV of energy from the cone direction. Thedistribution of number of good charged tracks (b) is shown after preselection but before—* p±VT selection. Plot (c) shows the x2 of the neutral cluster energies for eventspassing r 4 pu- selection, as a function of the neutral cluster threshold energy. Thex2 is defined as the sum over 20 energy bins of the difference between the number ofevents given by the data and the number of events predicted from the Monte Carlo data:= Z(N —±EZT—(837T )vj-55(I)>N0C-)0.50.40.30t 02oU0 2 4 6 8 100(0ev)(a)# secondary clusters, r (b)Secondary cluster energy, r0.5>0U1)0.410.3C.2 0.2U00.10Cl)0,36U-0.32hO.280______c024_0t 0.20C--0.1 60.120.080.040N0(c)Number at neutral custers (MC data)Figure 6.2: Values used in neutral cluster requirementsPlot (a) shows the number of ECAL clusters not associated with the one good chargedtrack required, and plot (b) shows the total energy in these clusters. From this it isapparent that a high threshold energy for neutral clusters will eliminate most of thesepion secondary clusters. In plot (c), the distribution of number of neutral clusters isshown after preselection but before r± selection. The distributions shown arenormalised to the branching fractions in table 4.2.fiHIILJ T—>7r(2rr°)vV7/J TflVrI! TP—eu er’r,T H jiv0 1 2 3 4 5 656Figure 6.3: Cone massFigure 2. Cone mass, (a) data and Monte Carlo data, and (b-d) most important sourcesof background. The arrows in plot (a) represent limits required. The Monte Carlo datahave been normalised to the same number of preselected r-pairs as the data. All plots areafter all selection requirements other than those on cone mass. In plot (d), distributionsfrom T± , and r+ 7r±VT are combined. The data are shownas points, Monte Carlo data by the empty histogram, and estimated background by thehatched histogram.57-36322824ICC200175C-0 15012510075502506CU)>0)50C0) 4-oECz200 e\/(a)Cone mass cfte other cuts0eV0CU)>U)S 6CI:(b)T—rr(2v3)LJ (MC)rC eV(d)r (,ere,) (MC)a 0.521.500eV(MC)0(‘.3C31)>31)031)-DE:3(‘3C31)>a)0a)-oE:3z(3)Ca)>a)011).08:3zFigure 6.4: Eass/PtrkFigure 3. Eass/Pctrk, (a) data and Monte Carlo data, and (b-d) most important sourcesof background. The arrow in plot (a) represents the required limit. The Monte Carlodata have been normalised to the same number of preselected rpairs as the data. Allplots are after all requirements other than that on E/p. In plot (d), distributions from—* e VeVT7± /97v1- and T± qrz’T are combined. The data are shown aspoints, Monte Carlo data by the empty histogram, and estimated background by thehatched histogram.0Ca)>C):,zMCReal8kg.1 601 401 201008060402002.421 .61 .20.80.40(c)E/p, ate all equvernents0 0.5 1 . 5 218a1086420a(b)E/p, r2u°) (Monte Carib)7-6H5-4-3E8/p(d)E/p, (Monte Carlo)0 0.5 1 1.5S/p(c)E/p, T (3°) (Monte CaMo)258Background Decay Mode Fraction of candidatesT—+ 7r27r°zJT 16.06 + 1.35r— (K)vT 1.54 ± 0.21T—÷ (K,7r)vT 1.32±0.03T —+ 7r(> 3w°) 1.90 + 0.71:l: ftVZ)T 0.46 + 0.05r —+ el7v,- 0.50 + 0.06Total 21.78 + 1.54Table 6.2: Sources of background in selected data6.2 Summary of ResultsAfter making all these requirements, the selection efficiency for r+ pv decays isestimated to be 32.7 + 0.8%. with contamination 21.8 + 1.5%. The main sources ofbackground are listed in table 6.2.As this analysis depends strongly on the quality of the Monte Carlo data, severalchecks are made on its reliability. There is good agreement on the most basic quantities;the numbers of charged tracks and neutral clusters. This is evident from figures 6.5a andb, where the distributions of the number of charged tracks and the number of neutralclusters are shown respectively, after all other requirements have been applied. To checkthe cone mass, I study the effect of requiring that the good charged track has beenmeasured in the Z-chambers. If there are systematic problems with the measurementof 0 in charged tracks, they may be eliminated by this requirement (at the cost of alower T± > pv_ selection efficiency). No significant difference was seen in the branchingfraction or r polarisation (B = 2.0 x l0, SF,- = 0.014).A variable which provides a useful check on the modelling of ECAL is the anglebetween the ECAL cluster associated to the charged track and the nearest neutral cluster.One sees reasonable agreement between data and simulated data here, as well (see figure6.5c).59DM0 Real D BkqHio2; , N8 N SDZ1010_____ is0123456789 0123456789N ck(c)N+., otter othe recuirement (b)N1, after utner requirements90-80706050z4030201000.2rod(c)Associated—nearest neutral cluster angleFigure 6.5: Agreement between data and Monte Carlo data for significant variablesThe plots show comparisons of data and Monte Carlo data in (a) number of chargedtracks, (b) number of neutral clusters, and (c) the associated-nearest neutral clusterangle. The Monte Carlo data have been normalised to the same number of preselectedr-pairs as the data. Plots (a) and (b) are shown after all requirements other than thoseon the variables shown, plot (c) after all requirements. The data are shown as points,Monte Carlo data by the empty histogram, and estimated background by the hatchedhistogram.{S0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.1860Chapter 7AnalysisFrom the sample of r± —* pv decays selected, several measurements are made. Thebranching fraction is most straightforward and is calculated first. Then the r polarisationanalysis is detailed, with results given both for it and the r polarisation asymmetry A.7.1 Branching FractionThe branching fraction is calculated by:i\TP liPt ± ± N — IVoand — Jbkg —r 11r)= NT 1 — fnofl—TN eGand ‘ J 6kg ! biaswhere Nand is the number of r candidates which pass the 7± 4 PVT selection requirements and Nana is the number of r candidates observed (twice the number of r-pairswhich passed preselection requirements). These two terms refer to the data. The otherterms are all estimated from Monte Carlo data.The two factors fT and f4” are, respectively, the estimated fraction of eventsin the preselected data sample arising from processes other than r-pair production, andthe estimated fraction of decays in the final data sample which arise from processes otherthan r±The term €‘ represents the efficiency for detecting 7*_4 p±7/7 decays after preselection. This is calculated as the ratio of the number of decays which pass all requirementsover the number which pass all preselection requirements.The term F6as represents the artificial enhancement of the r 4 p±VT signal as aresult of applying the preselection requirements. This is the ratio of the relative fractionof r 4 p±VT decays after preselection over the relative fraction before preselection.For this analysis,• Nand = 650• Nand=662O• fnon—P= 0.218 + 0.015(MCstat.) + 0.002(meas.)61• fT = 0.019 + 0.01• = 0.327 + O.007(MG.stat.) + 0.004(conv.)• Fas = 1.021 + 0.005with the resultBQr—+ ,ozzT) = 0.234 + 0.009(stat.) +gg (.sy.st.)7.1.1 Branching Fraction UncertaintyUncertainties in the branching fraction measurement can be divided into three classes.The statistical uncertainty is that arising from the sample size of the data. It is calculatedas11 fflOflP\ 1 1ÔB= I—SBJ = Fe(Br,Ntend)” (7.2)uBr (1fok9 )f HaswhereB —N f/VT 73r GetuP GandLU LfU TiTTUL)r — eij—’r, lvccandandF(R, N) = \/( R) (75)The function Fe gives the “binomial error,” the statistical uncertainty for any fractionR of N events taken from a binomial distribution. Here, the original distribution is thatof all preselected r-pair events, and the final distribution is that of all selected r decays.Inserting the appropriate values (given above), I estimate the statistical uncertainty atc5Bstat = 0.87%.Some uncertainties can be easily estimated from the parameters in the branchingfraction calculation. The uncertainties8f_T and 6F5 are taken from r preselection.The former is based on comparisons between data and Monte Carlo data, the latterpropagated through from Monte Carlo statistics.The non-p background uncertainty Sf’ contains a binomial term (labelled “MCstat.” above) and an additional uncertainty arising from the uncertainty in the measuredbranching fractions for each background decay mode (labelled “meas.” above):(6f73)2= E (fmocze(c5Bmocze)meas)2 (7.6)modep62where the sum is taken over all modes present in the selected data sample (other than4 p14), fmode is the fraction of each in the final sample, and SB is the measuredbranching fraction uncertainty for each mode.Similarly, the efficiency uncertainty Se’ is given by a binomial term “MC stat.” plusan extra relative uncertainty of 2% to allow for inaccuracy due to poor modelling of therate of conversion of photons in the detector (labelled “cony.”). It is estimated thatsome 10% of r± 4 pu decays have such a conversion, with a 20% mismatch in thisrate between estimates from data and Monte Carlo data.These uncertainties are easily propagated through to obtain a total SB. There areother uncertainties which do not lend themselves to this easy estimation from parameters,though. To estimate these I vary analysis parameters and measure the resulting change inthe branching fraction. The spread upwards and downwards from the nominal branchingfraction is taken to give the uncertainty in each direction. The final uncertainty quotedis thus asymmetric.The parameters varied include the neutral cluster threshold, cone size, degree ofpolarisation in the Monte Carlo data, and all the specific values chosen for r+ p±14selection requirements. The cone size is varied by roughly twice its resolution, +0.05 rad(see figure 7.la). The neutral cluster threshold is varied by +200 MeV, also twice itsexpected resolution. As can be seen in figure 7.lb, the variation range chosen for theneutral cluster threshold is high enough to avoid a divergence in the branching fractiondue to the low-energy region of poor neutral cluster modelling mentioned earlier. Theactual range chosen and the variation of branching fraction within this range are plottedin figure 7.lc.The degree of Monte Carlo polarisation is varied by +0.14, an estimate of the finaluncertainty of the polarisation from this analysis. This is shown in figure 7.ld.To vary the values used in r± .4 P14 selection requirements, an estimate is needed ofhow well these quantities are simulated in the Monte Carlo data. The peak position andwidth of the mass distribution are seen to differ in the data and Monte Carlo data by nomore than +10 MeV. From this, I conservatively set the limit on variation of the masscuts at +50 MeV. The results of this variation are shown in figures 7.2a and 7.2b. TheP2/p requirement is varied by +0.3, a factor of roughly three times its resolution, to allowfor systematic effects (see figure 7.2c). Although I see no evidence for problems withthe P2/p distribution here, other OPAL analyses have shown some evidence of modellingproblems at low P2/p values.Another parameter that is varied is a requirement on the angle between the associatedcluster and nearest neutral cluster. This is ordinarily set to zero but is varied up toroughly one ECAL block width, 0.05 rad, to obtain an uncertainty from any problemswith the ECAL modelling of electromagnetic shower shapes. It is expected that suchproblems would be most evident in the merging of the associated and neutral clusters,and that this parameter is a direct way of testing how well this merging is modelled. Theresults of variation are plotted in figure 7.2d.An additional uncertainty is estimated by varying the degree of weighting attachedto different decay modes in the Monte Carlo. Due to the “missing mode problem,” theseweights are somewhat arbitrary. To attach an uncertainty to the weights chosen, I use two63rn 0.246 6 6 6 o o o o o[ p i I I I I0.26 0.28 0.3 0.32 0.34Cone Size (rod)( a)Branching fraction vs. cone size§( § 0 6 6 § § § §11111 III III IF 1 11:111 F 1:1Figure 7.1: Variation of branching fraction with parameter changesPlot (a) shows the change in the branching fraction as the cone size is varied. Plot (b)shows the divergence of the branching fraction as the neutral cluster threshold is lowered,and plot (c) shows the variance of the branching fraction over the range used to calculatethe uncertainty from the neutral cluster threshold. Plot (d) shows the dependence ofthe branching fraction on the degree of polarisation assumed in the simulated data. Theerror bars shown represent the possible variation from point to point accounting forcorrelations.0.23m 0.260.22rn0240.23rn°240.230 QL 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6Eeca (0eV)(b)3ranching frac:on vs. nejtral cluster threshold, long rcnge00 ( §F F F F I F F I I I F F F I I F I F F I F F F F I F F F F I I F I F I F F F F1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2EeCOF (0eV)(c)Branching trocfon vs. neutrci cluster tnreshold, shoE range§§ § § 6069 § IF I IF F F I F F F F F F F F I F F F F I F F F F F ......I..........I... I F F F F F F F—0.28 —0.24 —0.2 —0.16 —0.12 —0.08 —0.04rad(d)Branching fraction vs. degree of MC polar sation64rn°25§0.2.13 I I I I I I I I I I0.5 0.52 0.54 0.56 0.58 0.6Cone mass (0eV/c2)(a)Banching traction vs. low—mass limitrn024022I I I I I I I I I I I I0.9 0.92 0.94 0.96 0.98 1Cone mass (0eV/c2)(b)Branching fraction vs. high—mass limitrn°24f(‘22 I _i_,.,_I0.6 0.7 0,8 0.9 1 1.1 1.2E/ pc(c)Branching fraction vs. E/o requirement0 0 00.23 I I I I I I _.___i._l i0 0.01 0.02 0.03 0.04 0.05ra d(d)Branching fraction vs. ass—nearest neutral cluster requirementFigure 7.2: Variation of branching fraction with parameter changesPlots (a) and (b) show the amount by which the branching fraction varies as the conemass requirements are changed. Plot (c) shows the dependence of the branching fractionon the E/p requirement. Plot (d) shows the branching fraction dependence on the anglebetween the associated and nearest neutral clusters, a variable not used to select eventsbut which gives an indication of the reliability of our simulation of electromagnetic showershapes.65Decay mode BF (%) BF (high signal, %) BF (high noise, %)T —* 23.6 24.1 23.1T —* C7ei1r 18.1 18.0 18.1T—+ 17.7 17.6 17.7r —+ irir+ir(> Oir°)VT 15.4 15.4 15.5T —* (ir, K)v 12.3 12.2 12.3T— r(2w°)vT 8.9 8.8 9.1T — 71-(> 37r°)VT 2.1 2.0 2.27 —* (Kv 1.6 1.5 1.6r±— -ir r+ir_qr+lr_( 0ir°)v 0.3 0.3 0.3Table 7.1: Alternative branching fraction weighting schemes in Monte Carlo dataQuantity Nominal High signal High noise S— S+Branching fraction (%) 23.44 23.62 23.23 0.21 0.18Polarisation (1-D fit) -0.042 -0.031 -0.052 0.010 0.011Polarisation (2-D fit) -0.169 -0.161 -0.176 0.007 0.008Table 7.2: Uncertainties arising from decay-mode weightingdifferent weighting schemes to maximise and then minimise (within reasonable bounds)the signal-to-noise ratio in the final sample. The first set of weights is put together byvarying the p branching fraction up by its best-measured uncertainty, 0.006, then scalingup all other modes in proportion to their uncertainties until the sum of all branchingfractions is one. The second set of weights is put together by varying the p branchingfraction down by 0.006, then scaling all other modes up till the sum of all branchingfractions is one. The weighted branching fractions used are listed in table 7.1. It isseen that the results using the nominal weighting scheme lie between those using the twoextremes (see table 7.2).As mentioned in Chapter 4, weights are also applied to correct the shape of the a1resonance in the Monte Carlo sample. To account for uncertainty in the new parametersof the a1, the mass and width are varied separately and the resulting variations summedquadratically to give a total uncertainty estimate (see table 7.3).All uncertainty estimates for the branching fraction measurement are summarised intable 7.4. The most significant potential sources of error are seen to be those associated with the background, branching fraction weights, and efficiency. The cone massrequirements also contribute significant uncertainty.66Mai (GeV) 1 (GeV) B (%) F1 ] p2 ]1.251 0.599 23.44 -0.042 -0.1691.251 0.500 23.56 -0.054 -0.1701.251 0.700 23.33 -0.034 -0.171SF— 0.11 0.012 0.002Sr+ 0.12 0.008 0.0001.275 0.599 23.54 -0.050 -0.1711.226 0.599 23.32 -0.035 -0.1716M 0.12 0.008 0.002SM+ 0.10 0.007 0.0005— fi 0.16 0.014 0.002s+ fi 0.16 0.011 0.000Table 7.3: Uncertainties arising from a1 weighting[ Source of Uncertainty Variation range SB (%) SB (%) 1Statistics— 0.87 0.87Efficiency uncertainty (conversions)— 0.47 0.47Non-rho bkg. uncertainty (measurement) 0.46 0.46Variation of Mcone lower limit +50 MeV 0.11 0.45Variation of Mcone upper limit +50 MeV 0.34 0.02Efficiency uncertainty (MC stats) 0.29 0.29Non-tau background uncertainty— 0.24 0.24Branching fraction weights see above 0.21 0.18E/p variation +0.30 0.17 0.19Monte Carlo polarisation +0.14 0.16 0.17a1 parameter weighting see above 0.16 0.16Bias uncertainty—— 0.12 0.12Variation of neutral cluster threshold +200 MeV 0.11 0.10Ass.-nearest neutral cluster angle 0-0.05 rad 0.04 0.11Non-rho bkg. uncertainty (MC stats)— 0.05 0.05Cone size variation +0.05 rad 0.04 0.03Total systematic 0.92 0.97Table 7.4: Summary of uncertainties in the branching fraction measurement7.2 Tau PolarisationAs we saw in Chapter 1, the r-polarisation determines the momentum distribution of ther decay products. To measure the polarisation, the decay angles are calculated and theresulting distributions fit to the theoretical functions above. The angles cos 0* and cosare given byo— 2M ( E NM+M (77cos— M2—M \.fJ I 2—MT p beam pandM 2Er+Epcos= (M (M+ + Mo)9’I2 p (7.8)where Ebeam is the energy of the colliding e, with F,- Ebeam.Data are binned two ways; a two-dimensional distribution in cos 0* and cos b (5 x 4bins), and a one-dimensional distribution in cos 0* alone (10 bins). Corrections are thenmade to the data on a bin-by-bin basis to recover the actual distributions from thosemeasured. I discuss only the two-dimensional case here. The one-dimensional case is thesame in method.= C Cf/MCjt±(1—fr)Nm8 (7.9)ft3Most of these terms are analogous to those used for the branching fraction calculation,taken here to apply to the numbers of events per bin instead of the sample as a whole. Allcorrections are calculated from the Monte Carlo sample, using either the actual anglescos 0 and cos as given by KORALZ or the reconstructed angles cos O and cosgiven by the equations above.The term QM represents the change in shape of the distribution resulting fromthe fact that the Monte Carlo data are produced with only one value for the centre-of-mass energy, whereas the data have been collected at a range of values. To calculateit, KORALZ is run, producing cos 0 x cos cba distributions at centre-of-mass energiescorresponding to those scanned in 1990. These are normalised and then their sum istaken, weighting each according to the number of events at each centre-of-mass energy inthe selected data. KORALZ is run again at 91.16 GeV (the Z° mass used in the MonteCarlo sample), and the new distribution is divided by the weighted sum distribution toobtain a correction factor.The preselection bias 0tnj represents the change in shape of the distribution resultingfrom preselection; it is the ratio of the cos 0 x cos ija distribution before preselection overthe distribution after preselection. This calculation is limited to events with cos 0 cone <0.68.The efficiency is obtained analogously to the preselection bias, by taking the ratio ofthe cos 0 x cos ‘çb distribution before r± p±vT selection over the distribution afterselection. A notable bias is seen towards low values of cos 0. This is equivalent to a bias68towards low p+ energies and is an understandable consequence of requiring neutral clusters; the cone size of an event increases as the p energy drops, increasing the probabilityof separate clusters forming.The factor Gjt corrects systematic mis-measurements of cos 0 and cos ba (due toradiative corrections, mass smearing, and possibly other things). The correction is theratio of the cos 0 x cos ‘/,a distribution over the cos 0 x cos br distribution.The background is calculated by taking the ratio of the cos 0 x cos ‘çbr distribution ofnonr± p+jj decays in the final sample over the distribution of all events in the finalsample.The correction factors and data for the one-dimensional fit are shown in figures 7.3-7.5. The correction factors and data for the two-dimensional fit are shown in figures7.5-7.14After all the corrections are made, a x2 fitting procedure is used to extract the polarisation. As above, I detail only the two-dimensional case here; the one-dimensional fitis the same in method. The x2is calculated by— NAe0r9\ 2(7.10)jj Jwhere Ujj represents one of two possible uncertainty estimates (detailed below) and(0*)hi9h ()hiOh= 1Vg7 J J W(0, Ø)d0*db (7.11)(9*)tow (.)lowwhere W(0*, &) is the theoretical decay distribution given above, the high and low superscripts refer to the limits of each bin in 0’ and 5, and N[r is the total number ofselected decays.The x2 is minimised as a function of the polarisation F,-, with the statistical uncertainty in the polarisation given by the change in F,- (as returned from the fit) for a changeof +1 in the x2 The theoretical distribution, all corrections to the real distribution andujj are functions of F,- and change as it is varied.For the two-dimensional fit, events in a bin of range 0.6 cos 0* < 1.0 are combinedwith those in the bin of range 0.2 cos 0* < 0.6 and the same b if the number of eventsin the former (before corrections) is less than 10. This results in the elimination of twobins, so the fit uses 18 points instead of 20.Two fits are done, one to obtain the statistical uncertainty and another to obtainboth the nominal polarisation and the uncertainty from limited Monte Carlo statistics.For the first fit, I use = where is defined by/ r..rCorr \ 1/20— (NMC’VTOt I cBiascEc1McDeth(l — çBkg‘tii i r..rMCJ j 3 jj‘‘Tot / f’sNffc is the number of Monte Carlo events in bin (i,j): N4J and N[r are thenumbers of Monte Carlo and real events passing all requirements, respectively. This isan estimate of the statistical uncertainty in each bin.69jl1.2 0.80.6has 05 Hcos(6) cos(8)Preselection bias correction2‘H E1.75 E10.75 0.40.5on0.25 L_rH...—0.5 0 0.5 1 —i—0.5 0 0.5cos(0) cos(.E)Oetector effects correct}on BackgroundFigure 7.3: Correction terms for one-dimensional polarisation measurementThe horizontal scale shows the cosine of the decay angle of the p+ in the rth rest frame,cos O. The preselection bias correction is the inverse of the r± j) efficiency loss inr-pair preselection. The detector effects correction removes the effects of cos O’ smearingdue to radiative corrections and systematic problems. The efficiency plotted here isthe r± p+vT selection efficiency, and the background is the fraction of nonr±P-’T decays in the final sample. The corrections are applied by first subtracting thebackground, then dividing by the efficiency. The data are then multiplied by the detectoreffects and preselection bias corrections.70Figure 7.4: Data at varying stages of correctionThe vertical scale shows the number of events per bin, as correction factors are appliedin sequence to the data. The horizontal scale shows the cosine of the decay angle of thep± in the r± rest frame, cos 0*. The raw data plot is extended in range to show theslight degree of smearing beyond cos 0* = +1. The hatched histogram represents theestimated background, normalised to the number of selected events. The fit shown inthe final results plot is the integrated number of events per bin.71Z’20 H100 -Z 2402001 601 2080400+á.5Ô 05ccs(0)Background, efficiency corrected806040200Z 24020060120800— +— 0)cas(0,)Raw data (and eet, background)0 oHcae(0)Detectar effects correctedZ 2402001 6012080400coe(ODCorrected datc çand fit)1.6_____________________________1.20.80.40—1 —0.5 0—0.5<cos(’)<0i 1.60 1.2_ _0.5 1—0.5 0cos(8) 0.5<cosQ)<12—Dimensional E correctionFigure 7..5: One- and two-dimensional E corrections (2-D, slices in cosThe horizontal scale shows the cosine of the decay angle of the p± in the T± rest frame,cos 0*. This reflects the difference in polarisation distributions on the Z° peak andscattered over several values of Ecm (as in our data). Error bars are shown but toosmall to be easily visible. These corrections are the result of a high-statistics KORALZanalysis.1.8cJ 1.61.41.20.80.60.40.20 H III ii 11111 1 .1 II 1:11—i —0.75 —0.5 —0.25 0 C.25 0.5 0.75cos(0)1 —Dimensioncl E correctionI I I I I—1 <cos()<—0.5 cos(0’)I I I I0.8 6-OL0.5cos(0)c 1.21 0.80.4.0_——0.5 00<cos()<0.50.5 1cos(0)72Figure 7.6: Two-dimensional E corrections, slices in cosThe horizontal scale shows the cosine of the decay angle of the lr± in the p± rest frame,cos ‘b. This reflects the difference in polarisation distributions on the Z° peak andscattered over several values of Ecm (as in our data). Error bars are shown but toosmall to be easily visible. These corrections are the result of a high-statistics KORALZanalysis.73I IC)C)—1.61.20.80.401.61.20.8O.z:.0—0.5 0 0.5 1—1 <cos(0)<—0.6 cos()1.6 —C)1.2 -0.8 -0.4 -0E1.6C)1.20.80.401.6C-)1.20.80.40—0,5 0 0.5—0.6<oos(O)<—0.2 cosQ/i)—1 —0.5 0 0.5 —i —0.5 0 0.5 1—0.2<cos(8)<0.2 cos() C.2<Dos(8)<0.6 ccs()2—Di--enscnc E,, correctIonI I I—1 —0.5 00.6<cos(S)< 10.5 1cosQ4&)E—1 <cos()<—0.5 cos(0) —0.5<cos(’)<0 cos(0)0.5 0.5 i—-á.56 050<cos(1’)<0.5 cos(0) 0,5<cosQi< 1 cos(0)_____________________________Backgroundo.:—<cosQi)<—0.5 cos(0) —0.5<cos’)<0 cos(0’)i 6.5 o0<cos(1’)<0.5 cos(0) 0.5<cos(1’)< 1 cos(B)EfficiencyFigure 7.7: Two-dimensional polarisation corrections, slices in cosThe horizontal scale shows the cosine of the decay angle of the p+ in the T± rest frame,cos Q* The efficiency plotted here is the r+ selection efficiency, and the background is the fraction of nonr± > decays in the final sample. The corrections areapplied by first subtracting the background, then dividing by the efficiency. Then thedata are multiplied by the detector effects and preselection bias corrections.740 0—1—0.5 0 0.5 —1 —0.5 0 0.5— <cos()<—0.5 cos(0)—0.5<cos(1/i)<0 ccs(0)05 10<cos(&)<0.5 cos(0) 0.5<ccs(fr)<1 ccs(5)Detector efects correcon1- j0 0 . .... ..——- —0.5 0 0.5 —i —0.5 0 0.5—1 <cos()<—C.5 cos(0) —0.5<cos(5)<0 cos(0)1 C) 10.... Q——1—0.5 0 0.5 1 —1—0.5 0.5 10<cos()<0.5 cos(O’) 0,5<cos(’)< 1 cos(0)Preselection bias correctionFigure 7.8: Two-dimensional polarisation corrections, slices in cosThe horizontal scale shows the cosine of the decay angle of the p± in the r± rest frame,cos Q* The preselection bias correction is the inverse of the r+ v efficiency loss inr-pair preselection. The detector effects correction removes the effects of cos O smearingdue to radiative corrections and systematic problems. The corrections are applied by firstsubtracting the background, then dividing by the efficiency. Then the data are multipliedby the detector effects and preselection bias corrections.75o 01—— —0.5 0 0.5 1 —1 —0.5 0 0.5 1—1 <cos(0)<—0.6 cos() —1 <ccs(0)<—O.6 cos()1 10ji 05—0.6<cos(0)<—0.2 cos() —0.6<cos(0)<—0.2 cos(b)— Oil-6—0.2<cos(EY)<0.2 ccsQ)—0.2<cos(0)<0.2 cos(b)OJ165 Q 05’’J0.2<cos(0’)<0.6 cosQ/i) 0.2<cos(0)< 0.6 ccsQ6)L0—0.5 0 0.5 1 —1—0.5 0 0.5 10.6<cos(O)< 1 cosQVi) 0.6<cos(0)< 1 cos()Background EfficiencyFigure 7.9: Two-dimensional polarisation corrections, slices in cosThe horizontal scale shows the cosine of the decay angle of the ii in the p± rest frame,cos ‘. The efficiency plotted here is the r± pu selection efficiency, and the background is the fraction of non.r± pi., decays in the final sample. The corrections areapplied by first subtracting the background, then dividing by the efficiency. Then thedata are multiplied by the detector effects and preselection bias corrections.761.8 1.8J1.2 ft 1.20.6 — 0.60!.. oF—1 —0.5 0 0.5 1 —1 —0.5 0 0.5— I <cos(O)<—0.6 cos(i)— <cos(C)<—0.b cos()1.8 1.8J1.2 1.20.6 0.60 ............ 0——1—0.5 0 0.5 1 —1—0.5 ‘0 0.5—0.6< cost) < —0.2 cos(i) —0.6< cosç <-0.2 cos()1.8 1.81.20.6 0.60 ....‘....‘....‘.... 0——0.5 0 0.5 1 —1—0.5 0 0.5—0.2<cos(8)<C’.2 cos(Ji) —0.2<cos(G$)<0.2 ccs()1.8. 1.810.2 <cos(O) <0.6 cosQi) 0.2<cos(0) <.0.6 cosQI)d6O5,0.6<cos(O)< 1 cosQ’) 0.6<cos(Q)< 1 cosQ,1i)Detector effects correction Preselection bias correctionFigure 7.10: Two-dimensional polarisation corrections, slices in cosThe horizontal scale shows the cosine of the decay angle of the ir in the p± rest frame,cos J.’. The efficiency plotted here is the r± pz selection efficiency, and the background is the fraction of nonr± pv decays in the final sample. The corrections areapplied by first subtracting the background, then dividing by the efficiency. Then thedata are multiplied by the detector effects and preselection bias corrections.77._ §H•-0.50 0.5—1 <cos()<—0.5 cos(0)Ft §,,Figure 7.11: Data at varying stages of correction, slices in cosThe vertical scale shows the number of events per bin, as correction factors are appliedin sequence to the data. The horizontal scale shows the cosine of the decay angle ofthe p± in the T± rest frame, cos 0*. The hatched histograms represent the estimatedbackground, normalised to the number of selected events. The raw data plot is extendedin range to show the slight degree of smearing beyond cos 0* = +1.z 604020C+6040200-1 61 <COS(lJIr)<_0.5 COS(0,)z—1 0—0.5<cosQi1’,)<C ccs(8,)6040200E+z 6040200a 00<cos(c)<0.5 cos(8r) 0.5<coSQii’,)< 1Raw data (and estmcted backqrond)1 2080400cos(8,)—112080400 .: . .t.—1zI zO8040C—0.5 6—05< cos (v’) <0zIscos(Ei’)12080400—0.5 0 0.5 I —1 —0.5 60<cos(b)<0.5 cos(0) 0.5<cos(1i)<Background, efficiency corrected0.5cos(0)78Figure 7.12: Data at varying stages of correction, slices in cosThe vertical scale shows the number of events per bin, as correction factors are appliedin sequence to the data. The horizontal scale shows the cosine of the decay angle of thep in the r rest frame, cos Q*• The fit shown in the final results plot is the integratednumber of events per bin.=60 160120 12080 § 8040 400 0—1—0.5 0 0.5 1—1 <oos()<—0.5 ccs(0)i6C . 160120 2080 804 4:-0.5 0.5—0.5<cos’)<0 cos(0)—1•—0.5 00<cos(1i) <0.5-0.5 —1—0.5 0cos(8) 0.5<cosQ)<1Detector ettects corrected0.5cos(O)180 i605 Q5—1 <cos()<—0.5 cos(0)—0,5<cos()<0 cos(8)160-1208040 o0 ..—1—0.5 00<cosQi)<0.5160-0.5 °i5O 1cos(0) 0.5<cos(0)<1 cos(8)Corrected data (and fit)7918O—___________zl2o60H ‘0[—1 —0,5 0— <coe(0)<—0.620ft—0.5 0 ‘ 0.5—0.6<cos(0)<—C.2 co5().0..._..—0.2<cos,)<0.2 COS(r)+__10,‘.‘..0,___________________—1 0 1D.2<CCS(®•r)<0.6 cos(i,)—1 0 1 10.6<cos(®,)< 1 cos(-Vi,)Raw data (and est. background)Figure 7.13: Data at varying stages of correction, slices in cos 0”The vertical scale shows the number of events per bin, as correction factors are applied insequence to the data. The horizontal scale shows the cosine of the decay angle of the ir+in the p± rest frame, cos cl’. The hatched histograms represent the estimated background,normalised to the number of selected events. The raw data plot is extended in range toshow the slight degree of smearing beyond cos 0*+ 1’F..0.—1 0 1—1 <cos(0’,)<—0.6 cos(’,)00F____0. .0—1 0 1—0.6<cos(8)<—0.2 cos(i/c)0.5co s ()i80z755025075znO250V0 7550250:0 7550250:0 75zDO250+—-0.5 0.5—0.2<cos(8)<O.2 oos()18O120600018012060018012060Oto—0.5 ‘ 0.5O.2<ccs(0)<C.5 casQ+)—0.5 0O.6<cos(0)<10.5Background, efficiency correctedcos()80—1—0.5 0 0.50.6<cos(0’)<l cosQ)Detector effects corrected18OZ120H60501—1 —0.5 0 0.5 1—- <cos.c5)<—0.6 cos(i)•180-50C.5i—0.6<cos(8)<—0,2 cos(J)180z1206050—1 —0.5 0 0.5 1—0.2<cos(0)<0.2 cos(i580120____________6DE I-0.5 0 0.5 10.2<cos(0’ <0.6 cosQ9)80Z2Q0.6<cos(C)<1 cosQ1b)Corrected date (and fit)Figure 7.14: Data at varying stages of correction, slices in cosThe vertical scale shows the number of events per bin, as correction factors are appliedin sequence to the data. The horizontal scale shows the cosine of the decay angle of thein the p rest frame, cos . The fit shown in the final results plot is the integratednumber of events per bin.:1 801 20600 6 • 1—1 <cos(0)<—0.6 cos(V’)18Dz12060—i—0.5 0 0.5—0.6<cos(O)<—0.2 cos(i)180 rz1206050 l—i —0.5 0 0.5—0.2<cos(0’)<0.2 cos(i/)801 2060b—0.5 6 0.5-0.2<cos(0)<0.6 cosQf’)801 2060081For the second fit I define o’jj = u7, where(Tot)2= (?j)2 + (utfCOstat)2 (7.13)The term 8tat is the binomial error on the Monte Carlo correction factors. This isan estimate of the total nncertainty in each bin from statistics and Monte Carlo statistics.The uncertainty of F,- from limited Monte Carlo statistics is taken as the quadraticdifference between the fit uncertainty in the first pass SF,? and that in the second passSF,-:= (5T)2 — (SF,?)2 (7.14)There are further uncertainties not easy to estimate on a bin-by-bin basis. Theuncertainty from non-p background is obtained by varying the amount of each type ofbackground in the final spectrum up and down by the measured error on that background,taking the uncertainty of F’,- itself to he the resulting change in F,-.It is necessary to attach an uncertainty to any systematic problems with calculationof cos 0* and cos b. Since by far the biggest potential source of such a problem is mis-measurement of the p mass, I look for differences between the real and Monte Carlo massmeasurements to estimate this uncertainty. Smearing due to inherently limited massresolution is not at issue here, as this is removed by the detector effects correctionI am only interested in differences between data and Monte Carlo data.It is seen that there could be an additional smearing of up to 1% as well as a possiblepeak shift of up to +10 MeV. To estimate the cos 0 and cos if’ uncertainty I re-run theanalysis, adding in separately these systematic effects in the Monte Carlo data. Theresulting change in F,- taken as an uncertainty estimate.Other parameters can be varied in the same way as was done for the branchingfraction calculation. As can be seen in figures 19a and 19b, the long-range neutral clusterthreshold variations that were troublesome for the branching fraction are not evident inthe polarisation measurement.The uncertainties from all these procedures are summarised in table 7.6. The finalresults from the polarisation measurements are:• Two-dimensional fit: F2 = —0.17 + 0.10 + 0.08 (x2/D = 16.8/17)• One-dimensional fit: F1 = —0.04 + 0.14 + 0.13 (x2/D = 7.9/9)Here D represents the number of degrees of freedom in each procedure. In Chapter8 these results are analysed in the framework of the Standard Model. The difference inthe values returned by each fit is well within the limits of statistical fluctuation. Thishas been checked by processing samples of Monte Carlo data of roughly the same size asthe data set. The results are displayed in table 7.5. The x2 in this table is the sum ofdifferences between (Fft) and (FMc) over the uncertainty (SF1±).82MC events MC events, seL P1 + SP1 P2 + SPfj P1 — P2 [ PMQ12500 667 —0.100 + 0.143 —0.098 + 0.114 0.002 -0.16612500 675 —0.109 + 0.137 —0.091 + 0.110 0.108 -0.18912500 673 —0.011 + 0.142 —0.013 + 0.114 0.002 -0.15512500 674 —0.121 + 0.137 —0.231 + 0.107 0.110 -0.11912500 666 —0.183 + 0.132 —0.300 + 0.105 0.117 -0.16312500 640 +0.052 + 0.133 —0.107 + 0.109 0.059 -0.10012500 650 —0.006 + 0.138 —0.122 + 0.109 0.116 -0.13512500 695 —0.372 + 0.138 —0.391 + 0.100 0.019 -0.194Mean -0.106 -0.169 0.066 -0.153Standard deviation 0.131 0.126 0.052 0.033x2/degree of freedom 5.45/8 9.40/8[ 100000 5340 —0.116 + 0.049 —0.160 + 0.039 0.044 -0.153Table 7.5: Results of simulated analysis with MC data7.3 Forward-Backward Polarisation AsymmetryThe forward-backward polarisation asymmetry is obtained by calculating separately thei- polarisation in the forward (cos°T > 0) and backward (cos 6- <0) hemispheres, thentaking1 /Q12AFOI / fl /7 14creff t•1Owhere c=cos°1max is the acceptance (c=0.68 here), PF and PB are the polarisationresults from using data only in these hemispheres. The factor Fff is a small correctionterm arising from integrating the Born cross-section for T-pair production over 0, takinginto account efficiency variations. I calculate Feff = 0.98 + 0.02.The measured values of these parameters are= —0.24 + 0.15 + 0.08PB = —0.09 + 0.15 + 0.08where the systematic uncertainty of each measurement is taken to be that from the meanpolarisation measurement above. The polarisation asymmetry is then= —0.09 + 0.13 + 0.05The statistical uncertainty of the polarisation asymmetry is obtained by propagatingthrough the statistical uncertainties from the individual polarisation measurements. The83Source of Uncertainty SP SP SP [Statistics 0.136 0.136 0.104 [ 0.104Mcone upper limit 0.112 0.002 0.056 0.011Mcone lower limit 0.014 0.051 0.016 0.048MC statistics 0.036 0.036 0.032 0.032Ass.-neutral cluster angle 0.001 0.046 0.009 0.0298*/b mis-measurement 0.034 0.079 0.024 0.022Neutral cluster threshold 0.013 0.032 0.004 0.020E/p variation 0.022 0.038 0.017 0.015Non-p background measurement 0.042 0.042 0.01.5 0.015B.F. weights 0.010 0.011 0.007 0.008a1 weights 0.014 0.011 0.002 0.000Cone size 0.004 0.000 0.004 0.000Total systematic 0.134 j 0.129 0.075 [ 0.075Table 7.6: Summary of uncertainties in the polarisation measurementsystematic uncertainty is obtained assuming that the uncertainty of the polarisation ineach hemisphere is the uncertainty of their difference, plus a contribution from SFej.84ci0C-0.20—0.20.20—0.20.20—0,2(a) —D Polcriscton vs. cone sze(0eV)Figure 7.15: Variation of T polarisation with parameter changes (1-D fit)Plot (a) shows the change in the polarisation as the cone size is varied. Plot (b) showsthe change in polarisation as the neutral cluster threshold is lowered. The divergenceseen in the branching fraction is absent here. Plot (c) shows the variance over the rangeused to calculate the uncertainty from the neutral cluster threshold.0.26 0.28 03I Ii t_I I I. ,.32 0.34. )i —D Dç otor v. cone s:ze0o Size (rcd)11I I I I I I i_.j._I i ..j_...Li i I I0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6[..111liii II 11111 11111 I 11111 III II I II1 .8 1 .85 1.9 1 .95 2 2.05 2.1 2.15 2.2E (0eV)(c)1 —D Polarization vs. neutral cluster threshold, short range85aE 0.2I I I I I I I I0.5 0.52 0.54 0.56 0.58 0.6Cone moss (0eV/c2)(a)1 —D Polorisation vs. low—mass limito7 0.2I____I I I I I I .._.i_._...____.i... I I0.9 0.92 0.94 0.96 0.98Cone mass (GeV/c2)(b) —D Polariscion vs. nig—rnass IimtaZO.2-- : f f fI I I I I I I I0.6 0.7 0.8 0.9 1.1 1.26/pc(c) —D Poicrisaton vs. 6/p equirernent: } }0 0.01 0.02 0.03 0.04 0.05rod(d)1 —D Polarisation vs. Ass—nearest neutral cluster requrementFigure 7.16: Variation of r polarisation with parameter changes (1-D fit)Plots (a) and (b) show the amount by which the polarisation varies as the cone massrequirements are changed. Plot (c) shows the dependence of the polarisation on the E/prequirement. Plot (d) shows the dependence on the angle between the associated andnearest neutral clusters, a variable not used to select events but which gives an indicationof the reliability of Monte Carlo simulation of electromagnetic shower shapes.860.20.25-0.20Figure 7.17: Variation of T polarisation with parameter changes (2-D fit)Plot (a) shows the change in the polarisation as the cone size is varied. Plot (b) showsthe change in polarisation as the neutral cluster threshold is lowered. The divergenceseen in the branching fraction is absent here. Plot (c) shows the variance over the rangeused to calculate the uncertainty from the neutral cluster threshold.0—0.2—0.4 )I I I iI i I I I I I II0.26 0.28 0.3 0.32 0.34Cone Size (rad)( o)2—D Polarisation vs. cone sizej.Ii Iii.I:I I I I I I I I I I I I i II......._....I.......j.....j.....I I I I I0—0.2—0.40—0.2—0.40 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.66, (0eV)(a)2—D Polarisation vs. cone size--It t t t f t•1 I liii III II III I II liii I__I.....1 .8 1.85 1.9 1 .95 2 2.05 2.1 2.15 2.2EecCI (0eV)(c)2—D Dolarisafion vs. neLtra custer threshold, short range870.5 0.52 0.54 0.56 0.58 0.6Cone mass (6eV/c2)(a)2—D Polarisation vs. ow—mass limit-0.2 L f fI I I I I I I I0.9 0.92 0.94 0.96 0.98 1Cone mass (6eV/c2)(b)2—D Polarisation vs. high—mass limitI I I I I I I I I I I0.6 0.7 0.8 0.9 1 1.1 1.2E/pc(c)2—D Poicisaticn vs. E/p recuiremen:H f—0.4 I I F I I I0 0.0 0.02 0.03 0.04ra d(d)2—D Polanisction vs. Ass.—neares: neutral clusten requirementFigure 7.1.8: Variation of r polarisation with parameter changes (2-D fit)Plots (a) and (b) show the amount by which the polarisation varies as the cone massrequirements are changed. Plot (c) shows the dependence of the polarisation on the E/prequirement. Plot (d) shows the dependence on the angle between the associated andnearest neutral clusters, a variable not used to select events but which gives an indicationof the reliability of Monte Carlo simulation of electromagnetic shower shapes.88Chapter 8Summary of Results8.1 Branching FractionThe measured T± p±i/T branching fraction isB(T —+ PT) = 0.234 + 0.009(stat.) (yst.)Past measurements are listed in table 8.1. These are combined by treating the systematic uncertainties as if they were statistical in origin, and the total uncertainty fora measurement as if it is associated with normally distributed data. The combinedbranching fraction is then given byB______mean — ‘ mean) (3P.2and the combined uncertainty(_____SBmeen = (B) (8.2)From the data in table 8.1 we obtain Bmean = 23.2 + 0.5%. This is to be comparedto 23.1 ± 0.6%, the average without the result presented here. One can also look at thevalue of 22.7 + 0.8% published in 1990 by the Particle Data Group[32]. The individualmeasurements, new average and old average are plotted in figure 8.1. All are seen toagree well within uncertainty limits.8.2 PolarisationFor the nominal polarisation, I take the value returned by the two-dimensional fit, andcombine the systematic and statistical uncertainties:PT = —0.17 + 0.1389PDC 90 average New overage___CELLO__Mark CMark H• Crysta’ EaHALEPHAECUSOPALI I I I0.175 0.2 0.225 0.25 0.275Branching *zctionFigure 8.1: r± p± branching fraction measurementsPublished T± pzI branching fraction measurements, except for that from DASP,which had error bars too large for convenient display. The dotted line indicates thepublished world average in 1990, from the Particle Data Group’s Review of ParticleProperties[32}. The solid line indicates the new average.90Experiment Measurement ( %) ReferenceARGUS 22.3 + 1.0 [33]ALEPH 24.6 + 1.1 [34]Crystal Ball 21.3 ± 2.1 [35]Mark II 25.8 ± 3.0 [36]Mark III 23.0 + 2.1 [37]CELLO 22.2 + 1.7 [38]DASP 24.0 + 9.2 [39]Table 8.1: Past T± —+ P-’T branching fraction measurementsOther measurements of the r polarisation are given in table 8.2. Combining OPALresults, the mean polarisation is found to bepOPAL= —0.06 + 0.07The most accurate measurement can be obtained by combining all results from LEP, listedin table 8.2 and plotted in figure 8.2. As has been remarked earlier, the r polarisation isenergy-dependent so only measurements at the same energy can be averaged. From allLEP measurements, we obtain the polarisation at \/ Mzo:pLEP= —0.12 + 0.048.2.1 Polarisation AsymmetryAlthough the polarisation asymmetry analysis proceeds in parallel with that of the polarisation, other LEP experiments have not presented results for this measurement. OPALmeasurements[17] are summarised in table 8.3. The combined polarisation asymmetry is= —0.17 + 0.08.8.3 Lepton UniversalityFrom Chapter 2, we recall the definition of A and its relation to the effective vector andaxial coupling constants for leptons:A — 2(ve/ae) — 4Ar01 8 3e= 1 + (e/&e)2 — FB1 +(/a)2 = PT (8.4)These equations can be solved to obtain€/&e and &/a,-. From the T polarisation wehave AT = 0.17 + 0.13, yielding91Experiment Decay modes Measurement (%) ReferenceOPAL—> pv —0.17 + 0.13OPAL 4 &7eVr +0.20 + 0.15 [17]OPAL—97v—0.17 + 0.19 [17]OPAL T—f KilT —0.08 ± 0.12 [17]OPAL Combined —0.06 ± 0.07ALEPH—÷ —0.36 + 0.18 [40]ALEPH T—+ /1VIZ/T —0.19 + 0.14 [40]ALEPH-4 ?rll —0.13 + 0.08 [40]ALEPH—* —0.12 + 0.07 [40]ALEPH—+ ir2ir°v —0.15 ± 0.17 [40]ALEPH Combined—0.15 + 0.05 [40]DELPHI—+ ei7CvT —0.10 + 0.22 [41]DELPHI—0.09 + 0.21 [41]DELPHI T —* 1rVT —0.28 ± 0.13 [41]DELPHI T PVT —0.17 + 0.12 [41]DELPHI Combined—0.18 + 0.08 [41]LEP Combined—0.130 + 0.034Table 8.2: Tan polarisation measurementsi ,1 ,4PoliviOue 1FBT -4 eveilT 0.16 ± 0.19T + LV/VT —0.08 + 0.22T —* 7rZlT —0.34 ± 0.16T —* P-’T —0.09 ± 0.14Combined —0.17 + 0.08Table 8.3: OPAL polarisation asymmetry measurements92LEP cvemqeDELPHIDELPHDELPHIDELPHDELPHALEPHALEPHALEPHALEPHALEPHALEPHOPAL C;OPAL TOPAL TOPAL TOPAL Tcorn LinedTTT enellrcorn LinedI > C11I pnIHT1TII > erornbnedT’Figure 8.2: Tau polarisation measurementsThe solid line indicates the average polarisation.-—0.6 —0.4 —0.2 0P.0.2 0.493= 0.09 + 0.06From the polarisation asymmetry we have A8 = 0.12 + 0.19, yielding= 0.06 + 0.10The LEP average polarisation yields A,- = 0.130 + 0.034 and Vt/a,- = 0.065 + 0.017.From the OPAL average polarisation asymmetry, we obtain.\ = 0.23 + 0.11, and= 0.12 + 0.06. Comparing this with the polarisation results, we see reasonableagreement and conclude there is no evidence here for violation of lepton universality.Fnrther data would make it possible to refine this comparison more. However, independent numbers have not been made available by the other LEP experiments.8.4 The Weinberg AngleGiven a value of /d, sin2 Ow is obtained using=1_4sin2Ow (8.5)One then calculates sin2 0w using the relation sin20w = 1.013 sin20w (for assumedHiggs boson and top quark masses of 100 GeV each). Using the value of ‘k3/à derivedfrom the r polarisation, we findsin2 Ow= 0.225+0.015Using /a from the polarisation asymmetry yieldssin2 0w = 0.232 + 0.025These results should be compared to those from other r polarisation measurements,the relative uncertainties of which we have already seen in table 8.2. It is seen thatthis result compares quite favourably with those from other experiments. Combiningall LEP r polarisation measurements yields F,- = —0.130 + 0.034, I/à = 0.065 + 0.017and sin20w = 0.2308 + 0.0042 In comparison, the combined LEP forward-backwardasymmetry measurements give sin20w = 0.2307 + 0.0014[42].Also of interest are measurements of sin20w obtained by other methods, summarisedin table 8.4 and shown in figure 8.3. From the LEP experiments, measurements are alsoobtained from the forward-backward asymmetries for different fermion types and from ameasurement of the mass of the Z°The most accurate measurements of the weak mixing before LEP were made by examining neutral current “deep inelastic” scattering of ji neutrinos off quarks. In this typeof analysis the energy distribution of outgoing neutrinos is fit to a theoretical predictionto obtain sin20w Another, similar, experiment can be done where elastic scattering ofp neutrinos off electrons is studied. In this case, the ratio of reaction rates for incidentneutrinos and antineutrinos gives sin20w94Method Measured sin2 8w [ ReferenceLEP r polarisation 0.2308 + 0.0042—LEP fwd-bwd asymmetry 0.2307 + 0.0014 [42]Mw/Mz 0.219 + 0.009 [32]Deep inelastic scattering 0.233 + 0.006 [32]Elastic ye scattering 0.222 + 0.011 [32]Elastic vp scattering 0.207 + 0.032 [32]Inelastic eN scattering 0.217 + 0.020 [32]Atomic parity violation 0.215 + 0.018 [32][ Combined 0.2303 + 0.0013Table 8.4: Measurements of sin2 Ow (for M0 MHiggs = 100 GeV)Other variations on the scattering method include elastic scattering of neutrinos offprotons rather than electrons, and inelastic scattering of electrons off nuclear targets.In electron scattering, the weak mixing is retrieved by studying the asymmetry in thereaction cross sections for left- and right- handed electrons, a method illherently similarto the analysis of r polarisation.The weak mixing can also be extracted by studying parity violation in atomic systems.Here the Coulomb potential is modified by Z° exchange, causing “forbidden” atomictransitions which can be measured.Combining all results, we obtain the best measurement of the electroweak mixing,sin20w = 0.2303 ± 0.0013. This is to be compared with the electromagnetic couplinga 7.2974 x i0 (known to 0.045 parts per million[32fl and the Fermi constant GF =1.1664 x 10 GeV2 (known to 17 parts per million[32]). As these are the fundamentalparameters of the Standard Model, it is clear that the precision of Standard Modelpredictions is still limited by the accuracy of sin2 Ow.Combining sin2 Ow with the mass of the Z° (91.175 + 0.021 GeV, also from LEP[42])the mass of the W boson is estimated to be 80.01 + 0.32 GeV. This is consistent withdirect measurement. The mass of the top quark is also loosely constrained; includingother results, one finds m0 = 132 + 35 GeV[42], but no tight limit can be placed on themass of the Higgs boson yet. With further work the accuracy of sin2 Ow can be improved,giving better mass limits. The consistency of the Standard Model can then be checkedby direct measurement of these masses.95AverageAtomic paritynefastic eN scott.Dastic cp scott.Efastic e scott.nefostic scott.LEP PH Asymm.LEPi Pal.I I I0.18 0.2 0.22 0.24sin2OFigure 8.3: sin2 0w measurementsThe solid line indicates the average sin2 O.96Bibliography[1] Glashow, S.L., Nuci. Phys. 22(1961)579, Weinberg, S., Phys. Rev. Lett.19(1967)1264, Phys. Rev. D5(1972)1412, Salam, A., Rev. Mod. Phys. 52(1980)525[2] Dirac, P.A.M., Proc. Roy. Soc. A117(1928)610[3] Anderson, C.D., Phys. Rev. 43(1933) 491[4] Chamberlain, et al., Phys. Rev. 100(1955)947[5] See, for example, F. Halzen and A.D. Martin, “Quarks &r Leptons: An IntroductoryCourse in Modern Particle Physics,” John Wiley & Sons, 1984.[6] J. Chadwick, Proc. Roy. Soc. A136(1932)692[7] Street, J.C., and E.C. Stevenson, Phys. Rev. 52(1937)1003, Anderson, C.D., and S.Neddermeyer, Phys. Rev. 51(1937)884; 54(1938)88[8] Lattes, C.M.G., et aL, Nature 159(1947)694[9] Rochester, G.D., and C.C. Butler, Nature 160(1947)855[10] See, for example, Perkins, D., “Introduction to High Energy Physics,” 2nd Ed.,Addison-Wesley, 1982[11] Danby, 0., et aL, Phys. Rev. Lett. 9(1962)36[12] Lee, T.D., and C.N. Yang, Phys. Rev. 104(1956)254, Wu, C.S., et aL, Phys. Rev.105(1957)1413[13] Hasert, F.J. et aL, Phys. Lett. B46 (1973)138, Nuci. Phys. B73(1974)1[14] 0. Arnison et al., Phys. Lett. B126(1983)398[15] G. Arnison et aL, Phys. Lett. B122(1983)103[16] Pen, M.L. et aL, Phys. Rev. Lett. 35(1975)1489[17] M.Z. Akrawy et al., Phys. Lett. B266(1991)201[18] Kobayashi, M., and K. Maskawa, Prog. Theor. Phys. 49(1972)28297[19] “Electroweak Parameters of the Z° Resonance and the Standard Model,” CERNPPE/91-232 (20 December 1991)[20] Martin L. Pen, “High Energy Hadron Physics,” John Wiley & Sons 1974, pp277-81.[21] “Tan decays as polarization analysers,” André Rouge, Invited talk given at theWorkshop on Tau Lepton Physics, Orsay, France, (1990)[22) 5. Jadach et al., “Z Physics at LEP1,” CERN 89-08, ed. 0. Altarelli et al., Vol.1(1989)235-265[23] “LEP Design Report,” CERN-LEP/TH/83-29[24] Ahmet, K., et aL, NueL Iristr. arid Meth. A30.5(1991)275[25] “The Trigger System of the OPAL Experiment at LEP,” CERN-PPE/91-32 (20February 1991)[26] “The Detector Simulation Program for the OPAL Experiment at LEP,” CERNPPE/91-234[27] “Simulation program for particle physics experiments, GEANT: User guide andReference,” CERN-DD/78-2[28] “The Monte Carlo program KORALZ, version 3.8, for the lepton or quark pairproduction at LEP/S LC energies,” CERN-TH/5994-91[29] Branching fractions for the three- and five- prong modes are from the paper “Measurement of the r topological branching fractions at LEP,” which is currently underconsideration by the collaboration. One-prong modes are the world average calculated from the Particle Data Group “Review of Particle Properties” Phys. Lett.B239(1990), with the addition of numbers from OPAL (Fhys.Lett.B266(1991)201),ALEPH (CERN-PPE/91-186), L3 (Phys.Lett.B265(1991)451), ARGUS (DESY 91-084) and Crystal Ball (Fhys.Lett.B259(1991)216).[30] “A Monte Carlo event generator for simulating hadron emission reactions with interfering gluons,” CAVENDISH HEP 90-26a[31] “The Lund Monte Carlo for jet fragmentation and e+e- physics: JETSET version6.2,” LUTP 85-10[32] Particle Data Group, Phys. Lett. B239(1990),[33] “Measurement of exclusive one-prong and inclusive three-prong branching ratios ofthe r lepton,” DESY 91-084[34] “Measurement of Tau Branching Ratios,” CERN-PPE/91-186[35] Crystal Ball Collaboration, Phys. Lett.B259 (1991)21698[36] P.R. Burchat et at. Phys.Rev.D35(1987)27[37] J. Adler et at. Phys.Rev.Lett.59(1987)1527[38] H. J. Behrend et at. Z.Fhys.C46(1990)537[39] R. Brandelik et at. Z.Phys. .C1(1979)233[40] D. Decamp et at., Phys.Lett.B265(1991)201[41] Preprint DELPHI 91-60 PHYS 115 (1991)[42] “Electroweak Parameters of the Z° Resonance and the Standard Model,” the LEPCollaborations: ALEPH, DELPHI, L3 and OPAL. CERN-PPE/91-23299Appendix ADerivation of Polarisation AngleEquationsThe energy of the p in the r decay frame is given byE:=mrnp (A.1)and its momentum byrn2 —2m(A.2)if is the decay angle of the p in the r rest frame, then the component of the pmomentum parallel to the r line of motion is E P, cosBoosting to the lab frame, we haveE =7T(El,3TP,cos8*) (A.3)with= (A.4)Since ET Mz/2, we may safely approximate /3- = 1 and substitute the expressionsfor E and P°, yielding= -(m + rn + (m — m) cos (A.5)Defining X = E/E we re-arrange to give cos2m____cosO*= X— (A.6)rn.—mDerivation of the second decay angle cos 5 proceeds similarly. In the p decay framethe energy of the charged pion is given by100= m + rfl mro (A.7)and its momentum by- (m - (m + mo)(m - (m+-mo)2 ( •)4mSince the masses of the pious are small compared to that of the p we can approximatem+ = rno m. Then these equations reduce to(A.9)F =m-4m (A.1O)We then boost to the lab frame again= -y(E + /3P cos (A.11)Here we can not safely assume j3 1, unlike the case of the r decay.Inserting values for E, P and ‘y,,, = we find= + — 4m cos (A.12)Re-arranging yieldscos=(2E_E (A.13)3prn4m \ E ,‘orcos=2(2E_E) (A.14)—4m Pp101

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