University of Ljubljana Faculty for Mathematics and Physics Boˇstjan Golob A Study of the Decays of B0s Mesons produced at the Z0 Resonance at LEP Doctoral Thesis SUPERVISOR: Ass. Prof. Danilo Zavrtanik Ljubljana, 1996 Univerza v Ljubljani Fakulteta za matematiko in fiziko Boˇstjan Golob ˇStudij razpadov mezona B0s na trkalniku LEP Disertacija MENTOR: Doc. dr. Danilo Zavrtanik Ljubljana, 1996 Sˇpeli in Gaji, za dni, ko nismo bili skupaj Abstract In the present work the measurement of the B0 meson lifetime is described. The lifetime was measured in two isolated samples of B meson semileptonic decays recorded by the Delphi spectrometer at LEP. The first data sample consists of a reconstructed ? meson accompanied by a high transverse momentum lepton in the same hadronic jet. In the second sample a D± meson was reconstructed in the same jet as an identified lepton L? of the opposite electric charge. The selection of a high pt lepton enriches the samples in semileptonic decays of hadrons, containing the b quark, and the ? and D± mesons assure a high fraction of B0 mesons in selected events. The ?L final state benefits from a high available statistics which is found to be crucial for the performed lifetime measurement. On the other hand the D± sL? analysis offers a better signal to background ratio and thus lower systematical error of the measurement. The combination of results obtained by the two analyses yields an average lifetime of the B0 meson ?bs = (1.66 ± 0.19) ps. Keywords: LEP, Delphi, B0 meson, lifetime, ? meson, D± meson Povzetek V priˇcujoˇcem deluje predstavljena meritev ˇzivljenjskega ˇcasa mezona B0. Meritev je bila izvedena na dveh vzorcih semilepton-skih razpadov mezonov B zabeleˇzenih s spektrometrom Delphi, ki deluje na trkalniku LEP. Prvi vzorec vsebuje dogodke z rekonstruiranim mezonom ? v hadronskem pljusku skupaj z leptonom visoke transverzalne gibalne koliˇcine. Drugi vzorec je sestavljen iz 7 mezonov D± , ki jih spremlja nasprotno nabiti lepton L?. Izbira leptonov z visoko transverzalno gibalno koliˇcino omogoˇca obogatitev obeh vzorcev s semileptonskimi razpadi hadronov, sestavljenih iz kvarkov b, prisotnost mezonov ? ter D± v konˇcnem stanju pa poveˇca deleˇz mezonov B0 v izbranih dogodkih. Izbrani vzorec s konˇcnim stanjem ?L zagotavlja visoko statistiˇcno moˇc meritve, ki je pri doloˇcanju ˇzivljenjskega ˇcasa mezona B0 bistvenega pomena. Pri analizi dogodkov z mezonom D± je ozadje manjˇse, to pa se zrcali v manjˇsi sistematiˇcni napaki meritve. Kombinacija obeh rezultatov da za povpreˇcni ˇzivljenjski ˇcas mezona B0 vrednost ?bs = (1.66 ± 0.19) ps. Kljuˇcne besede: LEP, Delphi, B0 mezon, ˇzivljenjski ˇcas, ? me z on, D± mezon PACS: 13.20.He Decays of bottom mesons 14.40.Nd Bottom mesons 8 Delo je bilo opravljeno pri eksperimentu Delphi v Evropskem laboratoriju za fiziko delcev CERN v ˇZenevi, v okviru raziskovalnega programa Odseka za eksperimentalno fiziko osnovnih delcev Instituta Joˇzef Stefan. Financiralo ga je Ministrstvo za znanost in tehnologijo Republike Slovenije. Zahvaljujem se svojemu mentorju, doc. dr. Danilu Za-vrtaniku, ki me je navkljub obremenjenosti ves ˇcas zavzeto vodil skozi meniˇse neznano fiziko osnovnih delcev ter mi utiral pot med veˇc kot 500 znanstveniki pogosto razhajajoˇcih se mnenj. Prof. dr. Gabrijelu Kernelu gre zahvala za mnoge razgovore ob poznih popoldnevih med zakljuˇcevanjem tega dela, ki so mi razjasnili marsikatero zmoto. Brez sodelavcev in prijateljev, Andreja Filipˇciˇca, Igorja Mandi´ca in Marka Zavrtanika, bi bili dnevi, preˇziveti v CERN, mnogo daljˇsi. Za kritiˇcno branje rokopisa se zahvaljujem tudi dr. Tomaˇzu Podobniku in prof. dr. Aleˇsu Stanovniku. Vsem sodelavcem odseka sem hvaleˇzen, da delo ni bilo breme, paˇc pa zabava. This work would have never been accomplished without a collective effort of all the members of the Delphi collab-oration. I would like especially to express my gratitude to Prof. Dr. Patrick Roudeau and Dr. Achille Stocchi for their guidance into the physics of B mesons. Nenazadnje naj se zahvalim starˇsem, ki so me razume-vajoˇce spremljali na poti. Barbara, ˇSpela in Gaja, upam da razumete. 10 Contents I Phenomenology of B0s Meson Production and Decay 15 1 Introduction 17 2 Production and DecayofB0s Mesons atLEP 21 2.1 Production of B0s Mesons in Z0 Decays ......... 21 2.2 B0s Meson Decays ...................... 27 2.2.1 Spectator Model ................... 27 2.2.2 QCD Corrections .................. 30 2.2.3 Interference Corrections .............. 33 2.2.4 Annihilation and Exchange Corrections ..... 34 2.2.5 Phenomenological Expectations for the B Me-son Lifetimes .................... 35 II Experimental Environment 39 3 Delphi SpectrometeratLEP 41 3.1 Large Electron Positron Collider ............. 41 3.2 Delphi Spectrometer .................... 43 3.2.1 Tracking Detectors ................. 45 3.2.2 Calorimetry ..................... 52 3.2.3 Charged Particle Identification .......... 54 11 III Data Analysis 63 4 The c/> - L Analysis 67 4.1 Selection of Hadronic Events............... 70 4.2 Selection of Leptons.................... 71 4.3 Selection of c/> Mesons................... 74 4.4 Composition of the Selected Sample .......... 79 4.5 Lifetime Measurement................... 83 4.5.1 Evaluation of the Proper Decay Time...... 83 4.5.2 Likelihood Fit.................... 91 4.5.3 Systematic Errors.................. 94 5 The D* - LT Analysis 97 5.1 Selection of Leptons.................... 98 5.2 Selection of D* Mesons .................. 100 5.3 Composition of the Selected Sample .......... 107 5.4 Lifetime Measurement................... 112 5.4.1 Evaluation of the Proper Decay Time...... 112 5.4.2 Likelihood Fit.................... 121 5.4.3 Systematic errors.................. 123 IV Summary 127 6 Combination of the Results 129 7 Conclusions 133 V Povzetek doktorskega dela 139 8 141 8.1 Uvod.............................. 141 12 8.2 Nastanek in razpad mezonov B0 na trkalniku LEP . . 143 8.3 Analiza podatkov...................... 148 8.3.1 Razpadni kanal c/> - L............... 148 8.3.2 Razpadni kanal D* -LT.............. 151 8.4 Rezultati in zakljuˇcki.................... 155 VI 9 13 Appendices 159 161 9.1 Appendix A ......................... 161 9.2 Appendix B .......................... 165 9.3 Appendix C ......................... 166 14 Part I Phenomenology of B0s Meson Production and Decay 15 Chapter 1 Introduction The Standard Model of strong and electroweak interactions [1] is nowadays one of the most established theories in contemporary particle physics. The first direct experimental verifications of the theory have been obtained by the observation of the neutral weak interaction, propagated by weak bosons, in 1973 [2]. The Standard Model predictions were further confirmed with the discovery of charged and neutral weak bosons by UA1 and UA2 experiments in 1983 [3]. However, high precision measurements of the param-eters of the Standard Model as well as confirmations of many of its predictions have been made possible only six years later, by the construction of the Large Electron Positron Collider (LEP) in the European Laboratory for Particle Physics (CERN) near Geneva in Switzerland. LEP began operation in July 1989. The designed energy of the collider allowed a copious production of the neutral weak boson Z0 , one of the carriers of the electroweak interaction. The study of the Z0 production and decay properties at LEP con-firmed the Standard Model to a yet unsurpassed precision. An important ingredient of the Standard Model is the uni-tary Cabbibo-Kobayashi-Maskawa (CKM) matrix [4]. Nine elements of this matrix describe the strength of the couplings between the 17 18 CHAPTER 1. INTRODUCTION quarks and the charged weak bosons W* and must be determined experimentally. From the three individual elements of the CKM matrix describing the couplings of a bottom (b) quark, Vub, Vcb, and Vtb, the first two can be accessed through the measurements of decay properties of the hadrons containing the b quark. For the extraction of these elements from the measurements of inclusive and exclusive branching ratios of B mesons [5] an accurate knowledge of their lifetimes is required. Since around 22% of the quark-antiquark pairs produced in Z0 decays are bb pairs [6], LEP is also a very convenient apparatus to study hadron systems with the b quark constituents. The physics of b quarks has been explored already before the construction of LEP at the e+e storage rings. Advantages of the LEP environment for the b-physics studies arise from a higher available energy taken by the b-hadrons. This results in measurable decay lengths of these particles and hence in a possibility of lifetime evaluations. Due to the higher available energy also a whole spectrum of b-hadron species is produced in the Z0 decays. At LEP for the first time one can reconstruct B0 mesons and measure the lifetime of Ab baryons. A study of the differences in decay properties of individual species of the b-hadrons can give some insight into the importance of the specific processes involved in the decays. The present work describes a measurement of the B0 meson lifetime. This quantity is known with the lowest precision among the lifetimes of the three different types of B mesons: B0 , B+ and B0d1. Since the B0 meson existence was experimentally verified only a few years ago its decays have not yet been studied to a great ex- 1The symbol B0s is used to describe the meson composed of ab and an s quark. B+ and B0d stand for the bu and bd meson respectively. Unless explicitly indicated otherwise the corresponding statements for the charge conjugate states are always implied. 19 tent. Apart from the reasons given above, the determination of the lifetime is important also for the future extraction of the partial de-cay widths of strange B mesons from the measured branching frac-tions. Probably an even more important motivation for the lifetime measurement is its indispensability for the study of the neutral B meson oscillations, one of the challenges of the b-quark physics. The lifetime measurement was performed on two samples of semileptonic B0s decays, obtained by using different reconstructed final states of the B0s decay products. The two samples differed in statistical power as well as in purity of the B0s meson content. The first part of this work gives a short phenomenological in-troduction of the production and the decay of B0s mesons at LEP. The specifics of the experimental setup with an emphasis on the Delphi spectrometer sub-detectors, used in the analysis, is described in the second part. The third part consists of two chapters, each of them detailing the performed data analysis of the two studied inclusive B0s decay channels. Finally, in part four, the results are combined and conclusions are drawn. 20 CHAPTER1. INTRODUCTION Chapter 2 Production and Decay of B0s Mesons at LEP 2.1 Production of B0s Mesons in Z0 Decays The annihilation of an electron and a positron resulting in a fermion-antifermion pair may proceed through a photon or a neutral weak boson exchange. The corresponding Feynman diagrams are shown in figure 2.1. e + Z e e a) b) Figure 2.1: Feynman diagrams for the annihilation of an electron and a positron into a fermion-antifermion pair through the exchange of a) a neutral weak boson and b) a photon. - e 21 22 CHAPTER 2. PRODUCTION AND DECAY OF B0S MESONS AT LEP The total cross-section for such a process at the centre-of-mass energy ?s is 1 [7] ?(e+e- › ff) = NcffQCD [kvf 4 ?j-Q2 f + 3?2 (s-M2)2+s2MZ 2 + ?f + al)(kvfvf 2 + kaf a 2 f)--------M 4 s-----2l . 96 K (s -MZ2)2+s2lZ (2.1) Qf is the charge of the final fermion in units of e0 and Nc is the number of colours (3 for quarks and 1 for leptons). GF is the Fermi constant that is most accurately obtained from the measurements of the muon lifetime [5] and ? is the fine-structure constant. The fermion axial and vector coupling constants to the Z0 boson are vf = 2T3f - 4Qfsin2?w af = 2T3f . (2.2) T3f represents the third component of the weak isospin for the left-handed part of fermions (+1/2 for neutrinos and uplike quarks and -1/2 for massive leptons and downlike quarks). M Z and TZ are the mass and the width of the Z0 boson and ?w is the Weinberg angle. Factors kvf,a equal unity in the limit of massless fermions: kvf a kf = ß = ß 3 -ß 2 2 3 ß = \ 4m2f 1 s (2.3) 1Throughout this chapter the natural system of units is used with h = c = 1. Hence a decay width is given in units of [T] =GeV and time in units of [t] =GeV-1. 2.1. PRODUCTION OF BS MESONS IN Z0 DECAYS 23 In addition to the tree level Feynman diagrams of figure 2.1 one should consider also higher order processes involving the radiation of photons and gluons from the fermion lines and elec-troweak loop corrections to the vertices and propagators [7] [8]. While the electroweak corrections are absorbed by the use of the Fermi constant in the expression (2.1) and in a small shift of the sin2 0w, the lowest-order corrections due to the gluon radiation are described by 1; leptons ) «s(MZ 2) ; rk (2.4) where as(MZ 2) is the strong interaction coupling constant. The cross-section of equation (2.1) is in the vicinity of the Z0 pole completely dominated by the third term, arising from the exchange of the neutral weak boson. The annihilation into the photon described by the first term of equation (2.1) as well as the interference term represent only a small correction to the overall cross-section. For example, the cross-section for the e+e annihilation into a ja+ja pair via the exchange of the Z0 at s = MZ 2 is —t-----------------¦ (2.25) Tb0 /d m2b td0 For the decay constants fd and fb values of 230 and 180 MeV respectively can be taken. While the value of fd corresponds to the measured decay constant of the D+ meson in the n+v^ decays [18], the fb is the theoretical expectation given in [19]. From the measured lifetimes of D mesons one would obtain the expected lifetime difference between the charged and the neutral B mesons of the order of 10%. A somewhat more accurate prediction is achieved by the calculations referred to in previous sections. The corrections to the spectator model should be applied due to the possible radiation and exchange of the gluons between the quarks. The decay width of charged B mesons is affected by the interference effect which prolongs their lifetime. In the decays of neutral B mesons annihilation and exchange processes take place but their effect on the lifetime is negligible. By introducing the estimated corrections to the spectator model decay width, the expected lifetime ratios of 2.2. B0S MESON DECAYS 37 individual B mesons can be summarized as [17]: ?b+ / fb A2 ---- ? 1 + 0.05 V ?Bd 200Me ?Bd ? ?Bs . (2.26) The notations ?Bs,Bd denote the average lifetime of the two mass eigenstates in neutral B meson systems. 38 CHAPTER 2. PRODUCTION AND DECAY OF B0S MESONS AT LEP Part II Experimental Environment 39 Chapter 3 Delphi Spectrometer at LEP 3.1 Large Electron Positron Collider The Large Electron Positron (LEP) collider is operating at the Eu-ropean Particle Physics Laboratory near Geneva. Four bunches of electrons and the same number of positron bunches circulate in a vacuum beam pipe. The beam pipe has almost 27 kilometres in circumference to reduce energy losses due to a synchrotron radia-tion. Four spectrometers, Aleph, Delphi, Opal and L3, are situated at four out of eight bunch-crossing points on the particles orbit. The map marking the position of CERN and LEP in a vicinity of Geneva is shown in figure 3.1. The acceleration of particles colliding at LEP is performed in several stages. Electrons from an electron gun and positrons from an electron converter are first accelerated to 600 MeV energy in the two linear accelerators followed by an electron-positron accumu-lator which injects the particles into the CERN Proton Synchrotron (PS). At the energy of 3.5 GeV particles are passed over from the PS into the CERN Super Proton Synchrotron (SPS). In the SPS electrons and positrons reach 20 GeV, a starting energy for the injection into LEP. 41 42 CHAPTER 3. DELPHI SPECTROMETER AT LEP Figure 3.1: Surroundings of Geneva with CERN sites and LEP collider with marked experimental points. 3.2. DELPHI SPECTROMETER 43 At LEP, electrons and positrons collide at a beam energy of 45.5 GeV 1. The centre-of-mass energy is thus the same as the mass of the Z0 vector boson (MZ0 = (91.1885 ± 0.0022) GeV/c2 [6]) produced in the collision. By variation of the centre-of-mass energy in an interval of ?E « 6 GeV around the Z0 peak a scanning of the Z0 lineshape and the measurement of cross-sections for Z0 decays into a pair of fermions is performed. At LEP luminosity of 1.1 x 1031 cm~2s_1 and the peak cross section for the process e+e -^Z0 — qq, ?h 0ad = (41.488 ± 0.078) nb [6], one hadronic Z0 decay is produced approximately every two seconds. A typical time of bunch circulating inside LEP is 10 hours, after that the accelerating cycle is repeated. In the running periods from the beginning of 1990 until the end of 1994 the four spectrometers recorded more than 12 x 106 Z0 decays into qq pairs. 3.2 Delphi Spectrometer Delphi (DEtector with Lepton, Photon and Hadron Identification) spectrometer is one of the four spectrometers operating at the LEP collider. A collaboration of physicists gathered around the Delphi spectrometer consists of more than 540 scientists from 53 institutes and 22 countries. Eight physicists from the Experimental Particle Physics Department of the Joˇzef Stefan Institute in Ljubljana are taking part in the collaboration as well. The spectrometer was designed to identify and accurately track particles produced in Z0 decays. It is composed of many detectors structured in a cylindric shape, covering most of the solid angle around the electron-positron interaction point. It consists of 1In October 1995 the collision energy of LEP was raised to 140 GeV and a gradual upgrade of the energy is foreseen until the 1998. The description given here thus corresponds to the LEP collider for data taking periods between 1989 and 1995. 44 CHAPTER 3. DELPHI SPECTROMETER AT LEP the barrel and two end-caps. A schematic view of the cross-section through the spectrometer is shown in figure 3.2. Figure 3.2: Spectrometer Delphi: the cross-section through the barrel and one of the end-caps. The spectrometer as well as the entire collider is installed in a tunnel 100 meters below the ground. Both end-caps of 10 m di-ameter may be independently removed to allow access to specific detector components. The superconducting solenoid produces a uniform magnetic field in the direction of the beam axis. The mag-nitude of the longitudinal component of the field inside the Time Projection Chamber (see description below) is Bz = 1.2334±00..00000110 T [20]. The magnitude of the radial component is less than 0.0005 T. A complete description of the spectrometer may be found in [20]. In the following we will briefly review the most important 3.2. DELPHI SPECTROMETER 45 properties of detectors relevant to this analysis. We shall use the coordinate system with the z-axis parallel to the beam of electrons. The radial coordinate R is measured from the beam axis and the azimuth angle ? in the plane perpendicular to it. ? is the polar angle with respect to the z-axis. 3.2.1 Tracking Detectors • Vertex Detector (VD) Three concentric layers of silicon microstrip detectors [21] are mounted at the average radii of 6.3, 9 and 11 cm surrounding the beam pipe. The interval of polar angles in which the particle originating from the interaction point crosses all three layers is 44° < ? < 136°. Each shell consists of 24 modules of single-sided silicon strip detectors, with a 10% overlap in ? for a careful internal alignment between the shells. For the 1994 data taking the first and the third layer were equipped with double-sided detectors. The sense strips of these detectors are aligned parallel to the beam axis on one side and in the perpendicular direction on the other side. Such a configuration enables the measurement of the (R,?) as well as of the z coordinate. The polar angle coverage of the first layer was also increased to 25° < ? < 155°. A schematic view of one half of the second and the third layer of the VD is shown in figure 3.3 a). The VD enables high precision measurements of a track position in the vicinity of a primary vertex and improves the particle momentum resolution. It is used to reconstruct secondary vertices of long lived particles and is indispensable for lifetime measurements of particles containing the heavy 46 CHAPTER 3. DELPHI SPECTROMETER AT LEP Figure 3.3: a) A schematic view of one half of the second and the third layer of the VD. b) A schematic view of the TPC. 3.2. DELPHI SPECTROMETER 47 quarks. Charged particles, crossing the VD, ionize atoms of the semiconducting material. Coordinates of the tracks are obtained from the division of the charge, released in the semiconductor, among several sense strips of the detector. A single layer of the detector provides a measurement of the track position with a precision of 7.6 µm and approximately 100 µm double track separation in the (R,?) coordinate, averaged over the polar angle [20]. In the z coordinate single hit precision varies from 9 µm at ? = 90? to around 30 µm at ? = 45? [21]. Inner Detector (ID) The detector consists of two concentric layers: the drift chamber giving 24 (R, ?) points per track and 5 cylindrical multi-wire proportional chambers with 192 wires providing the (R, ?) coordinate and 192 cathode strips providing the z coordinate. One quarter of the ID is plotted in figure 3.4 a). The detector is mounted at radii between 12 and 28 cm. The geometrical acceptance of drift chambers in ? is 17? to 163?. The (R,?) coordinate of the track traversing the drift chamber is determined from the measured drift time of electrons from ionization. The achieved single wire resolution in the drift chamber is ?R? = 75 - 125 µm, depending on the drift distance. The drift time does not provide information on the direction of the drift. These inherent left-right ambiguities of drift chambers are resolved by multi-wire proportional chambers. 48 CHAPTER 3. DELPHI SPECTROMETER AT LEP ril Figure 3.4: a) One quadrant of Inner Detector, each ? module composed of a drift chamber and 5 multi-wire proportional chambers (taken from [20]). b) 5 layers of OD drift tubes shown in (R,?) projection. Tubes, traversed by charged particles, are shown in colours. 3.2. DELPHI SPECTROMETER 49 • They also provide the z coordinate measurement. The resolution of a single proportional chamber is ?z = 0.5 - 1.0 mm [20] 2. Time Projection Chamber (TPC) A schematic view of the TPC is shown in figure 3.3 b). From the TPC response a charged track reconstruction usually starts. The detector provides a 3-dimensional measurement of particle trajectories. Electrons, produced in ionization of gas atoms in the TPC by a charged track, drift in the electric field parallel to the beam axis. From the drift time the z coordinate of the trajectory is reconstructed. Precision of the track position measurement in z direction depends crucially on the accurate knowledge of the electron drift velocity. At both end-caps of the TPC, drifting electrons enter the multi-wire proportional chambers. Each chamber is divided into 6 sector plates with 192 sense wires and 16 circular pad rows. The induced electric signal on the cathode pads serves for the measurement of the (R,?) coordinate of the charged track. The granularity of pads determines the spatial resolution of the detector in this coordinate. The pads give up to 16 measurements of the (R,?) coordinate between R ? 35 cm and R ? 111 cm. If one requires at least 3 pad rows to be hit, the angular acceptance of the TPC is between ? = 20? and ? = 160?. The high voltage plane provides an electric field E =187 V/cm [20] resulting in electron drift velocity of vd ? 7 cm/µs at T=29?C. The spatial resolution for a single pad row (measured 2Since the beginning of 1995 the polar angle coverage of the ID has been increased to 15? < ? < 165? and the multi-wire proportional chambers have been replaced by straw tube detectors. 50 CHAPTER 3. DELPHI SPECTROMETER AT LEP for Z0 — µ+µ ) is 250 µm in (R, ?) and 880 µm in z [20]. Signals from two tracks can be separated if the distance between the tracks is at least 1 cm. Beside accurate position measurements the TPC provides also information for particle identification. Each sense wire performs a dE/dx measurement which will be discussed in the section on combined charged particle identification with Delphi. • Outer Detector (OD) The OD completes the tracking in the barrel region. It consists of 24 azimuthal modules, each one containing 145 drift tubes, compounded in 5 layers. Layers of drift tubes are shown in figure 3.4 b). Drift tubes in different layers overlap to give the full azimuthal coverage. The OD considerablly improves the momentum resolution particularly for fast particles. Drift tubes are aligned parallel to the beam axis. While all the layers provide the (R,?) coordinate, three of them measure the z position of a track as well. The z coordinate measurement is obtained by comparing the relative timing of electronic signals at both ends of the drift tube. Drift tubes cover the polar angles from 42° to 138° and are situated at radii between 197 and 206 cm. Single point precisions are ?R?=110 µm and ?z=3.5 cm [20]. The combined tracking performance of the VD, ID, TPC and OD can be illustrated with the average momentum resolution for muons from a dimuon decay of the Z0. The distribution of the inverse momentum can be parametrized with the sum of two Gaussian distributions. The narrower Gaussian, which includes around 3.2. DELPHI SPECTROMETER 51 92% of events, has a width of [20] 1 ?( —) p = 0.57 × 10 3 (GeV/c) 1 . (3.1) The precision of the momentum evaluation for tracks arising from hadronic Z0 decays may be obtained by comparison of the recon-structed and simulated momenta in simulated decays of the Z0. Figure 3.5 [20] shows the relative momentum resolution as a func-tion of polar angle. 10 20 30 40 50 60 70 80 90 en Figure 3.5: Relative precision on the momentum of particles in hadronic Z0 decays as obtained from simulation (adopted from [20]). Beside the momentum another important parameter of a track is the impact parameter with respect to the primary vertex, measured in the (R,?) plane. The performance of the four above-mentioned detectors is expressed with the following expression for the resolution on the impact parameter e [21]: ?e ______________ 65GeV/c 202 + (------------)2 µm . (3.2) pt pt is the transverse momentum of a particle with respect to the z axis. 1 52 CHAPTER 3. DELPHI SPECTROMETER AT LEP 3.2.2 Calorimetry • Electromagnetic Calorimeter (HPC) The barrel electromagnetic calorimeter of the Delphi spectrometer is called High Density Projection Chamber (HPC). HPC, as its name suggests, uses a large number of time-projection chambers for calorimetry measurements. The calorimeter is composed of 144 modules, separated into 6 rings along the beam axis. Each ring includes 24 coaxially arranged modules with an inner radius of 208 cm and an outer radius of 260 cm. Polar angle coverage of the HPC is43° meson in the same jet. In order to perform a lifetime measurement on an isolated sample of such events, contributions of different production mechanisms must be evaluated. The processes that contribute to the --# final state are listed below, with corresponding Feynman diagrams shown in figure 4.1. 1. Direct semileptonic decays of strange and non-strange B mesons (b › L): a. Semileptonic decays of B0 mesons where a c/> meson in the final state originates from an intermediate charmed (D) meson (figure 4.1-1.a). b. Semileptonic decays of non-strange B mesons with a c/> meson in the final state produced in a charmed meson decay (figure 4.1-1.b). 2. Cascade decays of B mesons (b › c › L ) where the lepton and the c/> meson are produced in the semileptonic decay of a D meson. These processes may be subdivided into three categories: 67 68 CHAPTER 4. THE meson arise from the decays of two different D mesons (figure 4.1-2.a). b. The initial B0 meson decays into a Ds meson; the lepton and the c/> are decay products of the latter (figure 4.1-2.b). c. A non-strange B meson produces a Ds meson which is a source of both the lepton and the c/> meson (figure 4.1-2.c). 3. Semileptonic decays of Ds mesons (figure 4.1-3.) produced in Z0 — cc events (c — L ). 4. Events with leptons arising from light hadron decays or misiden-tified leptons. This category will often be addressed as the fake lepton category. 5. Events where the lepton accompanies a c/> meson from the original fragmentation of quarks. 6. Accidental combinations of pairs of charged tracks with the invariant mass in the c/> region, usually called the combinatorial background. At this point, it should be mentioned that the above processes are listed in a decreasing value of the mean transverse momentum of the lepton, with respect to the jet axis. Definition of the transverse momentum, as well as its discriminating power, will be discussed in the section on selection of leptons. Different production mechanisms can be separated into those where the appearance of a 0 meson is induced by b quarks (1.,2.) and those where c/> originates from lighter quarks (3.-5.). Apart from the signal process 1.a all the other categories do not contain a semileptonic decay of B0 meson and are thus classified as 69 v B 1.a D (*) (X) B d,u B , d,u 2.c 4> 4> D n( s*)(X) v 2.a v v d,u 1.b D (*) (X) B ) 3. ^ X v ^ Figure 4.1: Feynman diagrams of different production mechanisms of a 400 MeV/c • 20? < ? < 160? • track length at least 30 cm in the TPC. The total energy of these tracks must exceed 12% of the centre-of-mass energy. For a set of tracks i with momenta pi the thrust axis n is calculated from the definition of the thrust T: T\* i | Y) TJ, I = maxn( J, | in | ). (4.1) The polar angle of the thrust axis of the event is required to satisfy With the described cuts, hadronic events are selected with an efficiency over 95%. The remaining background, arising mainly from Z0 › ?+? events and two photon collisions, is below 0.7%. The number of recorded Z0 hadronic decays with the Delphi spectrometer, selected according to the above criteria, was around 751 × 103 in 1992, 755 × 103 in 1993 and 1484 × 103 in 1994 data taking period. The ? -L analysis includes the 1993 and 1994 data. 4.2 Selection of Leptons According to the simple spectator model, the momentum spectrum of the lepton from semileptonic decay, evaluated in the centre-of-mass system, depends quadratically on the mass of the decaying 72 CHAPTER 4. THE 3 GeV/c. To illustrate the influence of the cut on the pt for the isolation of direct semileptonic B meson decays, one can calculate the increase in the fraction of these decays among the processes of figure 4.3 for a certain minimal required transverse momentum of the lepton. By selecting events with pt > 1.0 GeV/c the fraction of direct B meson decays 74 CHAPTER 4. THE ? - $ ANALYSIS is increased by a factor of approximately 1.7. For the isolation of the ?L sample the identified leptons were selected by requiring the total momentum p greater than 2.0 GeV/c and the transverse momentum pt greater than 1.0 GeV/c. In addition the lepton track had to have at least one associated hit in the Vertex Detector. The last criterion was used in order to assure an accurate determination of the secondary vertex. 1800 1600 1400 1200 1000 800 600 400 200 0 2500 2000 1500 1000 500 0 b) pt [GeV/c] 10 20 30 p [GeV/c] Figure 4.3: a) The reconstructed transverse momentum of leptons from different sources in the Monte Carlo sample of hadronic Z0 decays. pt is calculated with respect to the axis of the jet to which the lepton is attributed. The distribution includes the cut on the total momentum p > 3 GeV/c. b) Same for the total momentum of leptons. 4.3 Selection of ? Mesons ? mesons were reconstructed through the ? -* K+K decay mode which represents (49.1 ± 0.9) % of the total ? meson decay width [5]. Charged particles of the event were separated into two hemispheres defined by a plane perpendicular to the thrust axis of the event. The invariant mass was calculated for all pairs of particles with the opposite charge and in the same hemisphere as the 4.3. SELECTION OF ? MESONS 75 identified lepton by assigning them the mass of the kaon. Figure 4.4 a) shows simulated momentum distributions of the K± candidates with invariant mass in the ? meson region: 1.01 GeV/c2 < M(K+K ) <1.03 GeV/c2. Selection of the high momentum kaons suppresses the number of events with kaon candidates arising primarily from the fragmentation, i.e. the processes 5.-7. listed at the beginning of this chapter. However, the cut on the K± momentum cannot be very tight, not only because of the loss of statistics, but also due to the larger relative contribution of direct charm decays (process 3.) at higher values of pK. Together with cascade decays (process 2.a-c), direct charm events are highly suppressed by the cut on the total and transverse momentum of the lepton as described in the previous section. For the reconstructed ? meson momentum a similar feature as for the pK can be seen in figure 4.4 b). A quantity, which is partially correlated to the momentum of the kaons as well as to the transverse momentum of the lepton, is the invariant mass of the K+K~L± combination, shown in figure 4.4 c). In contrast to the momentum of the kaons and the ? meson, M(?L) enables also a partial suppression of cascade and direct charm events. Because of the Lorentz boost of the ? meson and the phase space limitation for the ? — K+K decay, the momenta of the two kaon candidates are highly correlated as shown in figure 4.5. By imposing a cut on the correlation between the two momenta one suppresses the combinatorial background only above the ? mass region and does not change significantly the signal to background ratio achieved by the other selection requirements. The effect of such a cut on the mass distribution of the combinatorial back- 76 CHAPTER 4. THE 1.5 GeV/c (2.0 GeV/c) • p?> 3.5 GeV/c (4.0 GeV/c) • M(K+K-L±) > 1.7 GeV/c2 (1.9 GeV/c2) • 1.67 pK-(+) +0.31 GeV/c > pK+() > 0.63 pK-(+) -0.07 GeV/c. 6 4 2 0 78 CHAPTER 4. THE ? - $ ANALYSIS In order to further suppress the combinatorial background the kaon candidates had to be also identified by the hadron identification algorithm (see section 3.2.3). A loose kaon tag was required for at least one of the charged tracks used to reconstruct the ? meson (both kaons had to be identified by a very loose tag for the 1993 data). The very loose tag enables the selection of kaons with an efficiency over 90% and with a pion misidentification probability of the order of 30%. The invariant mass distribution for the selected kaon pairs is shown in figure 4.6. It was fitted using a Breit-Wigner function to account for the signal and a polynomial for the parametrization of the combinatorial background. The fitting function describes well the shape of the distribution as obtained from the simulation. The width of the fitted Breit-Wigner function reflects the natural width of ? meson resonance, L? = (4.43 ±0.05) MeV [5], as well as the resolution of spectrometer. A signal of 433±62 events was observed in the mass region 1.008 GeV/c2 < M(K+K ) < 1.030 GeV/c2. The signal is centred at M?=(1.019±0.001) GeV/c2 and has a width of r=(11.0±1.5) MeV. The invariant mass distribution is in agreement with the Monte Carlo prediction shown in the same plot. The value of M? agrees with the nominal mass (1019.413 ± 0.008) MeV/c2 of the ? meson [5]. The relative difference between the simulated and the fitted number of ? mesons in the simulated sample was found to be (Nfit - Nsim)/Nsim ? 6%. This value is below the statistical error of the fit. It can be concluded that no accumulation of combinatorial background in the ? mass region is introduced by the cuts used to select the sample of events. 4.4. COMPOSITION OF THE SELECTED SAMPLE 79 140 120 100 80 60 40 20 - 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 M(K K ) [GeV/c ] Figure 4.6: The invariant mass distribution of the selected kaon pairs. The distribution obtained from the Z0 — qq simulation is shown for comparison. 4.4 Composition of the Selected Sample In order to gain insight into the composition of the selected sample a numerical evaluation of relative contributions described at the be-ginning of this chapter must be performed. Events with the invari-ant mass of the kaon pair between 1.008 GeV/c2 and 1.030 GeV/c2 were considered for the composition study and for the B0s lifetime measurement. Various production rates and branching ratios used in the calculations are summarized in table 4.1 [31]. Selection ef- 80 CHAPTER 4. THE ? - $ ANALYSIS ficiencies for different processes were estimated from the Monte Carlo simulation. Measured quantity Value B1 = Br(b › B0 › DsL+?X) × Br(Ds › ?? ) (3.1 ± 0.5) × 10 4 (3.66 ± 0.22) × 10-3 B2 = Br(b › Bu,d › DsX) × Br(Ds › ??) B3 = Br(b › B0 › DsX) × Br(Ds › ??) Br (b › c › L-) (3.9 ± 0.9) ×10- (8.22 ± 0.42) × 10 2 (10.43 ± 0.24) × 10-2 B bcL Br(b›L+) Pu,d = Br(b › B±) = Br(b › B0d) Ps = Br(b › B0) 0.392 ±0.022 0.100 0.2202 0.022 0.002 Rb › bb)/r(Z0 › qq) r(Z Rc = T(Z0 › cc)/r(Z0 › qq) Br(D± s › ?X)/ Br (Dg — ^tt±) 0.1583 ± 0.0098 4.8 ± 0.5 Br(D0 › ?X ) Br(D+ › ?X) (1.8 ± 0.3) × 10 2 (1.7 ± 0.3) × 10 2 Table 4.1: Measured branching ratios and production rates used for the calculation of the ? meson sample composition [31]. Probabilities for observing a certain category of events in the hadronic decay of the Z0 are given below: 1. a. The probability of observing a semileptonic decay of the B0s meson is P1 =2RbPs[Br(Bs › DsL+?X)Br(Ds › ?X) + + Br(B0 › DnsL+X)Br(Dns › ?X)] ? Br(D- › ?X) ? 2 RbB 1 ------------- ?X) ?? ) . (4.2) In the last line we used an inclusive branching ratio B1 = PsBr(Bs › DsL+?X)Br(Ds › ?? ) defined in table 4.1. The much smaller contribution of non-strange D mesons in (4.2) with regard to B0 › DsL+X decays is a consequence of the ratio Br(Ds › ?X)/Br(D0(+) › ?X) ? 10, and of the fact that non-strange D mesons in the B0 decay 4.4. COMPOSITION OF THE SELECTED SAMPLE 81 are mainly produced via an orbitally excited D** meson. The fraction of D** in B0 decays has not yet been measured but can be inferred from the measured production rate in non-strange B decays where it amounts to approximately 1/3 (see Appendix A). Hence the approximation done in the last line of (4.2) is justified. b. P1B =2RbPu,d[Br(Bu,d ~* D0LX)Br(D0 — ?X) + + Br(Bu,d — D+LX)Br(D + — ?X)] « w4RbPu,dBr(b — L)Br(D0(+) — ?X) (4.3) is the probability for direct semileptonic non-strange B meson decay. Since the inclusive branching ratios for D0 and D+ decays into the ? meson are equal within the errors, instead of [Br(Bu,d — D0LX) + Br(Bu,d — D+LX)] we can use the Br(b — L) in the last line of (4.3). The probability for a DsLT pair occurrence in the Bu,d decay requires an intermediate d**0(+) — DsK decay and was not incorporated into P1 B (see Appendix A). 2. a. The cascade production of ? and L from two different D mesons in non-strange B decay has the probability P B =2RbPu,dBr(Bu,d ~* DsDnsX)[Br(Dns — LX)Br(Ds — ?X) + + Br(Ds — LX)Br(Dns — ?X) « Br(Ds — ?X) ~2RbBbcLB2???r . (4.4) Br(Ds — ) Again the contribution from the D0(+) was neglected in comparison with the Ds contribution. b. The relative occurrence of cascade B0 decays into a strange 82 CHAPTER 4. THE ? - $ ANALYSIS D meson per hadronic Z0 event is PB s =2RbPs[Br(Bs › DsX) -Br(B0 › DsL+X)]Br(Ds › ?L?) = =2Rb(B3 - B1)———^???-. (4.5) Br(Ds › ) From the inclusive branching fraction Br(B0 › DsX) we subtracted the semileptonic branching ratio since the latter decays by definition belong to the category 1.a. c. The last of cascade decays is the process with a non-strange B decay where L and ? are both products of a Ds. The probability for such a production mechanism is P3 B =2RbPu,dBr(Bu,d › DsX)Br(Ds › ?L?) = Br(Ds › ? L?) =2RbPudB2???r . (4.6) Br(Ds › ) 3. The only direct charm decay expected to contribute to the selected sample is a Ds semileptonic decay with the probability P1Ds = 2RcPsBr(Ds › ?L?) . (4.7) The production rate of strange mesons in cc events is assumed to be the same as in bb events. The relative contribution of cascade decays (2.a-c) in the selected sample was determined from the simulation, by comparing the number of cascade decays that pass the selection criteria with the number of all events in the sample. The sum of the three processes yielded N(b › c › L) fbcL =-----------:----------rT" = 0.014 ± 0.004 . (4.8) N(all events in sample) The contribution of direct charm events (process 3.) was found to be negligible (of the order of 5 × 10 3). 4.5. LIFETIME MEASUREMENT 83 Larger fractions of background events were found to originate from the following sources: 4. Fake leptons accompanying the ? meson represented ffake=0.11±0.02 of the simulated sample after the cuts. 5. The relative contribution of ? mesons arising from the primary fragmentation was ffrag=0.030±0.006, evaluated from simulation. 6. The fraction of the combinatorial background was determined directly from the fit to the mass distribution of the K+K invariant mass. It was found to be fcomb=0.629±0.085. The B0 purity of the sample is defined as the fraction of B0 mesons among all the selected B mesons. The kinematics of non-strange B meson decays described by the process 1.b is very similar to the kinematics of B0 decays (process 1.a). No difference in the reconstruction efficiency (Lbs, LBud) for these classes of events was expected and indeed a negligible difference was found from the simulated events. The purity of the sample can thus be calculated from the probabilities derived above: e N(B0) P fbs = ------ 0 ss------- = Bs 1—- = 0.50 ± 0.07 . = ------ 0 ss------- = Bs 1—- = 0.50 ± (4.9) 4.5 Lifetime Measurement 4.5.1 Evaluation of the Proper Decay Time In order to evaluate the proper decay time of a B0 meson two quantities must be determined: the decay length L and the momentum pbs of the decaying particle. The proper decay time is then given 84 CHAPTER 4. THE ? - $ ANALYSIS by t = pppp s (4.10) withMBs = (5375 ± 6) MeV/c2 [5]. The uncertainty of the calculated decay time t is derived as a function of the measurement errors on L and pBs: ?t = MBs 2 + pBs \ ?2 pBs ------ ?L +-------2 pbs Mb t2 . (4.11) The measurement of the decay length L was achieved by the reconstruction of the secondary K+K-L vertex. The fitted vertex position is obtained by minimising the ?2 of the distance between the vertex and three tracks in a 5-dimensional space of track parameters [32]. A loose cut was imposed to the quality of the reconstructed vertex by requiring the ?2 probability of the fit to be larger than 10 4 2. The cut is used for the rejection of badly reconstructed vertices and hence for an improvement of the decay length resolution and to a much smaller extent for the suppression of background events. Since the errors on the measured track parameters are not completely Gaussian the distribution of the ?2 probability is not flat but peaks at lower values. The imposed cut rejects 2% of signal events. The primary vertex of the Z0 decay was fitted from all the charged tracks in the event except the two kaon and the lepton candidates. The interaction region of electrons and positrons, called the beam spot, was used as a constraint in the fit. It is determined for about every 200 recorded hadronic decays. Coordinates of the primary vertex are fitted for these subsamples of events and the 2The number of degrees of freedom for the vertex fit is 2N - 3, where N is the number of outgoing tracks. 4.5. LIFETIME MEASUREMENT 85 mean and the width of coordinate distributions determine the position and the size of the beam spot. The size varies in time with typical values between 100 and 120 µm in one of the coordinates transverse to the direction of the beam and approximately 10 times smaller values in the other direction. If the primary vertex fit failed the position of the beam spot was used for the evaluation of the decay length. For the data taken in 1993 the decay length L followed from the relation Lt L = —? , (4.12) sin Bs where Lt is a measured distance between the primary and the secondary vertex in the (R,?) plane. sin ?bs is the polar angle of the B meson direction approximated by the thrust axis of the jet containing the ? meson and the lepton. For 1994 data the decay length was reconstructed in 3 dimensions due to the measurement of the z coordinate with the upgraded Vertex Detector. The sign of the decay length was determined with regard to the angle between the direction of the momentum of the ?L system, denoted by P in figure 4.7, and the vector joining the reconstructed primary and secondary vertex L. Due to the measurement errors on parameters of tracks and due to the background events the angle can be larger than 90?. A negative decay length was assigned to such events. The average decay length of B0 mesons increases and the angle between the direction of the lepton and the ? meson decreases with increasing momentum of a B0. Since the precision of the vertex reconstruction improves for larger angles between the outgoing tracks the resolution on the decay length ?L was parametrized as 86 CHAPTER 4. THE ? - $ ANALYSIS l- L y7 secondary vertex primary vertex K+ Figure 4.7: Illustration of a ?L event with vector L, joining the primary and secondary vertex, and momentum of the ?L system P. For events, where the angle between the two vectors was larger than 90?, the decay length was signed negatively. a function of the decay length: al + bL +L ; L> 0 aL + bLL; L< 0 . (4.13) The parameters were estimated from simulated B0 decays. For the 1994 data they were found to be aL = (276 ± 12) µm, bL +=(4.43± 0.46)×10 2 and bL - = -0.443 ± 0.023. For the 1993 data the corresponding values were (319 ± 26) µm, (5.09 ± 1.01) × 10 2 and -0.527 ± 0.054 respectively. For each event used for the lifetime measurement the precision on the decay length was calculated according to the above parametrization on the event-by-event basis. For a general illustration, the average difference between reconstructed and generated decay lengths is shown in figure 4.8 a) for the 1994 simulated data. The distribution was fitted with two Gaussian distributions. The width of the narrower Gaussian was < ?L >= (371 ± 28) µm. The wider Gaussian comprised a small 4.5. LIFETIME MEASUREMENT 87 fraction of events (? 6%) with < ?L >? 4 mm. Since kaons and leptons do not originate from the same vertex a small shift in the mean of the distribution was observed. Its value was around 70 µm which is negligible compared to the resolution. In the 1993 the average resolution < ?L > was slightly worse and amounted to (429 ± 97) µm. 70 60 50 40 30 20 10 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Lrec-Lgen [cm] 800 700 600 500 400 300 200 100 0 b) Data MC -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 Lrec [cm] Figure 4.8: a) The average accuracy on the decay length L obtained on the simulated sample of B0s decays. b) The comparison of negative decay length distributions for the simulated and the real data sample enriched in light quark decays. A possible discrepancy between the accuracy of decay length determination for the real data and simulated events was checked by comparing negative decay length distributions for which the res- • 88 CHAPTER 4. THE ? - $ ANALYSIS olution effects are dominating. The sample of events was selected using the same cuts as described above. In addition events were required to have a low probability of originating from Z0 › bb decays [33]. The comparison is shown in figure 4.8 b). The relative difference in the mean of the two distributions is 9.8%. Hence in the final evaluation of systematic errors on the B0 lifetime measurement the resolution ?L was varied for ± 10% to account for the possible difference between the Monte Carlo and the real data. The analysis of the simulated sample showed that an approximate linear relation holds between the fraction of the momentum taken by a ?L system in decays of B0 mesons and the ?L momentum itself. The following B0 momentum estimate was obtained from the fit to the dependence shown in figure 4.9 a): pppppp = ap + bp p(?v) , (4.14) with ap = 0.203 ± 0.065 and bp = (1.83 ± 0.28) × 10 2 c/GeV. The momentum of a B0 meson as well as the accuracy of the momentum evaluation were calculated from the above values of fitted parameters and their errors for each individual event in the data sample. Figure 4.9 b) shows the average relative difference between the reconstructed and the simulated momentum of the B0 meson. The distribution was fitted with the sum of two Gaussians. The parametrization (4.14) resulted in an average error on the B0 momentum of about 16%. In order to check the reliability of the Monte Carlo simulation the comparison of the reconstructed momentum in simulated and real data was performed. The reconstructed momentum of real events in the signal region after the subtraction of the background, taken from the side band in the K+K invariant mass distribution 4.5. LIFETIME MEASUREMENT 89 1 0.8 0.6 0.4 0.2 0 0 60 50 40 30 20 10 0 -1 10 15 20 25 30 35 40 45 50 p(?l) [GeV/c] -0.5 0 0.5 160 r 140 - 120 q 80 r 60 r 40 r 20 r 0 Data MC c) rec gen gen (pBs - pBs )/pBs 0 10 20 30 40 50 pBs [GeV/c] Figure 4.9: a) The simulated fraction of the momentum taken by a ?L system in decays of B0 mesons. b) The difference between the simulated and reconstructed B0 momentum as obtained on the Monte Carlo sample of B0 decays. c) Comparison of the reconstructed momentum for the simulated and real data in the signal region, after the subtraction of the combinatorial background (see text). (1.06 GeV/c2 < M(K+K ) < 1.15 GeV/c2), is shown in figure 4.9 c). The agreement with the momentum reconstructed in the simulated sample is satisfactory. The difference between the mean values of these distributions is 0.3 ± 0.4 GeV/c. The statistical error of this difference was considered in evaluation of the systematic error on the lifetime measurement, arising from the possible difference between the real and simulated data in the B0 momentum estimation. 5 90 CHAPTER 4. THE --# analysis. 5.1 Selection of Leptons By selecting electrons and muons with high total and transverse momentum the obtained sample was enriched in direct semilep-tonic decays of B mesons (see figure 4.3). The transverse momentum calculation was done in the same way as for the previous analysis. However, in order to gain the purity of the sample as high 1As in the -L analysis the process II./8, although induced by a B0 meson, is treated as a background from cascade decays. 5.1. SELECTION OF LEPTONS 99 B II.? D (*) ns V + V I. ß B II. W V (*) - D K (*) l+ v Figure 5.1: Diagrams of the processes contributing to D*-^ correlations in the same hadronic jet of the Z0 decay. The symbol Dn5 is used to represent non-strange charmed mesons. X denotes any hadronic system that can be additionally produced in a process. as possible, the cuts were made tighter. Leptons were required to have at least 1.2 GeV/c transverse momentum and the total mo-mentum above 3 GeV/c. The lepton candidate had to be associated to at least one hit in the VD. The lepton candidates were selected using the tight, standard or loose tag of the muon identification al-gorithm and tight or standard tag for the electron identification. In addition electrons arising from photon conversions were rejected. - 100 CHAPTER 5. THE D± S - L+ ANALYSIS 5.2 Selection of D± s Mesons Strange charmed mesons were reconstructed in the following decay modes: D+ — ??+ ; ? — K+K D+ — K* K+ ; K* — K~?+ (5.1) The corresponding branching ratios of D± decay modes are Br(Ds — ??+) = (3.5 ± 0.4)% and Br(D+ s — K* K+) = (3.3 ± 0.5)% [5]. First the invariant mass of two oppositely charged tracks in the lepton hemisphere, defined by the thrust axis of the event, was calculated. In the ? decay mode both tracks were attributed the kaon mass. In the K*0 decay mode the pion mass was attributed to one of the tracks and mass of the kaon to the other. At this stage of analysis only the specific ionization measured in the TPC was used to resolve the mass ambiguity of the two tracks. The kaon mass was attributed to a track with lower dE/dx measurement in the TPC. The dE/dx distribution of kaons and pions from the K*0 decay is shown in figure 5.2 for particles with momentum larger than 1 GeV/c. If the specific ionization measurement was not available both combinations were taken into account. The calculated invariant mass had to be compatible with a ? or K*0 meson. Since the momenta of decay products exhibit a similar behaviour as could already be seen in figure 4.4 a), the selection of particles with higher momentum suppresses the contribution from the combinatorial background. For the same reason the cut on the reconstructed ? or K*0 momentum can be applied. The third particle with the proper charge was then added to decay products of a vector meson (? or K*0) and the invariant mass 5.2. SELECTION OF D±S MESONS 101 100 80 60 40 20 0 0.8 - ^ I \N\\ L_%> ^ Wi///r \l. YYl'A'A / A //\ , A , A , ,\ , . 0.9 1.1 1.2 1.3 1.4 1.5 1.6 dE/dx [dE/dxm.i.p.] Figure 5.2: The specific ionization measured in the TPC for particles from the K?0 meson decay. The distribution is obtained on the simulated sample by requiring the momentum of a particle to exceed 1 GeV/c. of the three particle KKtt combination was calculated. Figure 5.3 shows the momentum distribution of a bachelor pion from the D± — (pn± and of a bachelor kaon from the D± ^K*0K± decay. It stretches to even higher values than the momenta of particles from the 4> and K*0 decay. Note that distributions of figure 5.3 as well as that of 5.4 were obtained on the dedicated Monte Carlo sample of D± decays and hence the relative fractions of different contributions are not the same as in the real sample of qq events. The energy distribution of pseudo-scalar mesons D± for the signal and for two sources of background is shown in figure 5.4 a). Lds is reconstructed as a square of the quadratic sum of the invariant mass and of the momentum of a K±K+tt± and K±tt+K± combination, for the 4> and K*0 decay mode respectively. The combinatorial background exhibits a lower average energy of the meson while the distributions of the signal and of cascade decays are similar. The separation of the latter source is achieved through cuts on the quantities related to the momentum of the selected lepton. 1 102 CHAPTER 5. THE D± S - L+ ANALYSIS 2 _ 10 - 10 :r:i 2.5 10 10 2 10 — .5 10 p? b [GeV/c] ? Comb. back. \ b › l 10 [GeV/c] Kb Figure 5.3: a) The momentum of a bachelor pion from the D± — 4>tt±. b) The momentum of a bachelor kaon from D± s — K*0K±. Distributions were obtained in the dedicated Monte Carlo sample of Ds decays. Beside the selection requirements onleptons these include the momentum and the invariant mass of a D±-#+ system, shown in figures 5.4 b) and c). In the considered decay modes a pseudo-scalar meson D± decays into a vector (c/>, K*0) and a pseudo-scalar (n±, K± ) meson. Helicity conservation implies that the angle between the direction of vector meson decay products in its rest frame and the flight direction of D± follows a cos2 a dependence (see Appendix C). For the background with accidental track combinations this dependence is more isotropic. The additional suppression of the background is thus obtained by requiring a high | cos a| value. For both decay modes the secondary vertex of D± decay products was fitted. As a result of the fit, the parameters of a D± meson track were calculated for the later estimation of the B0 decay vertex. For more accurate vertex determination, decay products of D± mesons as well as leptons were required to have at least one associated hit in the VD. 1 1 5 5.2. SELECTION OF D±S MESONS 103 3 2 _ 10 T 0 10 p 10 ~ 10 1 j a) Comb. back. b › l 0 10 20 30 40 50 p(D -l) [GeV/c] 10 10 10 1 3 2_ 10 15 20 25 30 35 En [GeV/c] Ds L J M(Ds-l) [GeV/c2 Figure 5.4: a) The energy distribution of a Ds meson for various sources described in the text. b) The reconstructed momentum of a D*-^ system. c) The reconstructed invariant mass of a D*^ system for the signal and the background. All distributions were obtained on the dedicated Monte Carlo sample of Ds decays. In accordance to general kinematical considerations given above a set of cuts was chosen for the selection of DsLT correlations. Since the width of the ? meson resonance is more than ten times smaller than the width of the K*0 [5] the combinatorial background is expected to be larger in the latter decay mode. Some of the cuts were thus tighter for this decay channel. As for the ?--# analysis the cuts were tuned slightly different for different years of data 104 CHAPTER 5. THE D± S - L? ANALYSIS taking. Also the cuts, used to select the D±e? sample, were relaxed with regard to the ones used for the D±µ? sample, due to the lower average identification efficiency and misidentification probability for electrons. The selection criteria are summarized in table 5.1. ’92, ’93 ’94 e µ ? e µ e µ ? e µ pk±(?±) [GeV/c] >1.3 p?b(Kb) [GeV/c] >1.7 >2.0 >1.7 >2.0 >1.5 >2.0 >1.5 >2.0 M(KK?) [GeV /c2] 1.012--1.028 0.855--0.935 1.01--1.03 0.845--0.945 | cos ?| >0.4 Prob(?2) Ds >5 ×10-3 p?(k?0) [GeV/c] >3.3 >3.3 >3.4 >3.2 >3.5 >3.2 >4.0 E Ds [GeV/c] >7.0 >9.0 >7.5 >9.5 >6.5 >8.5 >7.0 >8.5 p(D± sL?) [GeV/c] >10.0 >14.5 >11.5 >14.5 >10.0 >12.0 >10.5 >12.0 M(D±-tf?) [GeV/c2] >3.3 >3.0 Table 5.1: The selection criteria used for the isolation of the D±i,? sample. Beside kinematical cuts different identification requirements, summarized below, were imposed: • For the muon sample both kaon candidates had to be identified with at least a very loose tag by the combined hadron identification algorithm. For the data taken in 1994 this cut was tightened by an additional requirement for one or both tracks to be tagged by at least a loose kaon tag. • For the electron sample only one of the kaons had to be identified by a very loose tag (both kaons for the 1994 data). • Since the above identification criteria are tighter for 1994 data, the compatibility of the kaon candidates specific ionization 5.2. SELECTION OF D± S MESONS 105 with the one expected for kaons of the same momentum was additionally checked for 1992 and 1993 data. Figure 5.5 shows the difference between the reconstructed and the expected dE/dx for the kaon and the pion hypothesis obtained in the Monte Carlo sample. ? is the r.m.s. of the Landau distribution. While the sample of kaons from K*0 decays is compatible with the kaon hypothesis (figure 5.5 a)), the combinatorial background, with the majority of the tracks being pions, is compatible with the pion hypothesis (figure 5.5 b)). The selection required the sum of both plotted quantities (figure 5.5 c)) to be less than 0.3. D± s^ ??± . The events where a bachelor pion was identified as a kaon or a proton with the tight tag were rejected. The signal of D± mesons accompanied with a lepton of the opposite charge (right sign) obtained with the described selection criteria is shown in figure 5.6. The invariant mass distribution was fitted with the sum of an exponential slope for the combinatorial background and two Gaussian distributions. The Gaussians of equal width described the D± and a small contribution from D± — ??±(K*0K±) decays. The D± mass was fixed at a nominal value of (1869.4 ± 0.4) MeV/c2 [5]. 81±13 D± candidates were found within ± 2?ds from the peak centred at Mds = (1.968 ± 0.002) GeV/c2. This value agrees with the nominal mass of a D± meson of (1968.5 ± 0.7) MeV/c2 [5]. The width of the D± meson peak was found to be ?ds = (13.5 ± 2.2) MeV. The same selection was repeated on data for the isolation of the same charge D±^± pairs (wrong sign). The mass distribution 106 CHAPTER 5. THE DS - L+ ANALYSIS 10 10 10 10 -1 -0.5 0 0.5 1 (dE/dx -dE/dx)/a [dE/dx . 1 meas. K< L m.i.p. -1 -0.5 0 0.5 1 (dE/dx -dE/dx )/a [dE/dx . 1 ' meas. TV L m.i.p. 10 10 10 r -2 -1.5 -1 -0.5 0 0.5 1 ((dE/dx -dE/dx J +(dE/dx -dE/dxJ)/a [dE/dx . 1 ' ' meas. K) ' meas. Te) L m.i.p.J Figure 5.5: a) The difference between the measured specific ionization and the one expected for kaons with the same momentum. The dis-tribution for the sample of kaons from K?0 decays is centred at zero while the combinatorial background has a higher mean value. b) Same difference for the pion hypothesis. c) The sum of quantities shown in a) and b). The suppression of the combinatorial background is achieved by selecting tracks with lower values of this sum. All distributions were obtained in the simulated sample of K?0 decays. is shown together with the right sign combinations. No excess of events is found in the D±s mass region, in accordance to the ab-sence of physical processes that might contribute to such charge correlations. Invariant mass distributions for subsamples of the single de-cay mode (?, K?0) and lepton flavour (e±, µ±) are shown separately in figure 5.7. The fitted masses are equal to the one obtained in the full sample and the widths of the signals agree within statistical 5.3. COMPOSITION OF THE SELECTED SAMPLE 107 right sign wrong sign 40 30 - 20 - 10 NM =81±13 Ds 0 1.7 1.75 1.8 1.85 1.9 1.95 2.05 2.1 2.15 M(KK?) [GeV/c ] Figure 5.6: The invariant mass distribution of KK? combinations in the selected D± — ??±, K*0K± decays, accompanied by LT in the same jet. The shadowed region shows the distribution of the same charge D±^± combinations. errors of the fit. 5.3 Composition of the Selected Sample The lifetime measurement was performed on events in the D±s me-son mass region between 1.941 and 1.995 GeV/c2. To correctly construct the likelihood function relative fractions of different pro-cesses contributing to events in this signal region must be calcu- 2 108 CHAPTER 5. THE D± S - L+ ANALYSIS 30 25 20 15 10 5 0 30 25 20 15 10 5 0 1.7 1.7 1.8 1.9 2 2.1 20 15 10 5 0 M(KK?) [GeV/c2 1.7 1.8 1.9 2 2.1 M(KK?) [GeV/c2 M(KK?) [GeV/c Figure 5.7: a) The KK? invariant mass for the D± s ^K*0K± decay mode. b) Same for the D± — ??±. c) The invariant mass distribution for the D± s µT subsample. d) Same for the D± se+ subsample. lated. The derivation of probabilities for an occurrence of a particular process, listed at the beginning of the chapter, per hadronic decay of the Z0 is performed using the measured quantities of table 4.1. For the K*0 decay channel the branching ratio Br(D± s — ??±), used in quantities defined in table 4.1, is replaced by the branching ratio Br(D± s ^K*0K±). 5.3. COMPOSITION OF THE SELECTED SAMPLE 109 I. ?. The probability for the signal process, the semileptonic decay of a B0 meson, may be expressed as P1Bs = 2RbBr(b — B0 — DsL+?X)[Br(Ds — ?? ) + +Br(Ds--K*0?-)] = R B r Br(D<: — ??-) n = 2bBi| 1+-------—s------0-I . (5.2) Br( D _*¦ K ? - ) ß. A non-strange B decay into a DsK pair implies an intermediate D** state. A crude estimation shows that only about 3-4% of non-strange D** mesons produce a Ds meson in the final state (see Appendix A). A more refined analysis [35] results in an upper limit Prob(Z0 — Bu ,d -> DsKL?X) ----------------15---------------- < 10% . (5.3) P 1 Bs Hence this contribution to events in the D*-#+ signal was neglected. II. ?. The relative occurrence of cascade decays of a non-strange B meson per hadronic Z0 decay is 2RbBr(b — Bu,d — DsX)Br(Dns — L?X) P2B = 2RbBr(b — Bu,d — [Br(Ds — ??) + Br(Ds — K*0 ?)] = R Bb n Br(Ds — ??) -. = 2bB2cl| 1 +----------------0 I . (5.4) Br(Ds — K ?) The use of the inclusive semileptonic branching ratio Bbcl, measured in cascade decays, and of the inclusive Bu,d — DsX branching fraction is justified, since Ds mesons are produced mainly through the emission of a W* and only a negligible contribution arises from a D** intermediate state. The production rate (5.4) is similar to the rate of the signal (5.2). However the cut on the transverse momentum 110 CHAPTER 5. THE D± S - L? ANALYSIS largely suppresses the former contribution. The ratio of efficiencies of the selected cuts for direct and cascade decays must be taken into account in order to estimate the fraction of cascade decays in the signal region. The ratio was evaluated on the simulated sample of B meson decays and found to be Rbcl = bcl = 0.142 ± 0.033 . (5.5) ß. Cascade decays of strange B’s into two strange charmed mesons are treated as background. The probability for such process can be estimated from cascade decays of non-strange B mesons as P2 = ttt;—P2 . (5.6) 2Pu,d It is of the order of 10% of the P2B and only of the order of 2% of the signal and thus neglected. III. Fake leptons are expected to contribute the same fraction of events to the signal region in the right and the wrong sign sample. Since no excess of events is observed in the wrong sign mass distribution (figure 5.6), this contribution was neglected. IV. The D+ meson decays into a K ?+?+ final state non-resonantly T?0 and through the K ?+ channel. The resonant decay is around 5 times less frequent than the non-resonant one [5]. Because of the small width of the ? meson resonance the reflections in the Ds mass region, arising from the misidentification of one of the pions, are not seen in the ? decay mode of the simulated sample of D+ decays. In the K?0 decay mode reflections appear as abroad accumulation in the M(KK?) invariant mass distribution. In figure 5.8 one can see that the accumulation 5.3. COMPOSITION OF THE SELECTED SAMPLE 111 is situated in the Ds mass region with the FWHM of around 130 MeV/c2. By applying the selection criteria to simulated bb and qq events, the remaining contribution from the reflections with regard to the signal from the K?0 decay channel was determined to be erefN(ref) LbsN(Bs)k?0 Rre f = = (7.6 ± 3.3) × 10 2 . (5.7) 45 40 35 30 25 20 = FWHM ? 130 MeV/c ^jn - —. a 15 vir^ "i_ru 10 5 =11111111111111111111111, At i i 11111111111111111, 1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 M(KK?) [GeV/c ] Figure 5.8: The invariant mass distribution of a K?? combination by assigning the kaon mass to one of the pions. The distribution was obtained in the simulated sample of Bu,d › D±L?X; D± › K??±?± decays. The above derived probabilities for individual processes, contributing to D±-#? correlations, enable the calculation of the purity in the sample used for the lifetime measurement: LbsN(Bs) LbsN(Bs) + €bclN(bcl) + erefN(ref) B0 fBs + 1 = 0.843 ± 0.034 . + = (5.8) In the second line of equation (5.8) we used the fraction of D± candidates reconstructed in the K?0 decay channel, 112 CHAPTER 5. THE D± S - L? ANALYSIS RK?0 = 0.63 ± 0.20. The fraction of cascade decays in the signal region is fbcl = €bclN(bcl) = LbsN(Bs) + €bclN(bcl) + erefN(ref) R PB = -------------P 2 BP 1-------------- = 1 + Rbcl g^" + Rbcl B2 s" + RrefRK?0 = 0.116 ± 0.030 . (5.9) The fraction of reflections then follows from fref = 1 -fbs - fbcl (5.10) V. The contribution of the combinatorial background to the signal region was obtained from the fit to the invariant mass distribution of the KK? combination. The fraction was found to be fcomb = 0.382 ± 0.044. 5.4 Lifetime Measurement 5.4.1 Evaluation of the Proper Decay Time The proper decay time of the B0 meson was evaluated following the same steps as in the ? - L analysis. The D± track, calculated in the fit of the KK? common vertex, was fitted into a common secondary vertex with the selected lepton. Similarly as in the ? - L analysis, vertices with the ?2 probability less than 5 × 10 3 were rejected in order to improve the decay length resolution. For the 1992 and 1993 data the decay distance L was calculated using the relation (4.12) with the B0 meson direction approximated by the momentum of the D± sL? system. The agreement between the polar angle of this momentum and the polar angle of the simulated B0 flight direction 5.4. LIFETIME MEASUREMENT 113 is shown in figure 5.9. The resolution is the same as for the approximation used for the ?L final state, but the non-Gaussian tails vanish. For the 1994 data the decay length was reconstructed in three dimensions. 300 250 200 150 100 50 0 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 ?cos(?) Figure 5.9: The difference between the simulated polar angle of a B0 flight direction and the reconstructed polar angle of a D*^ system. As already mentioned in section 4.5.1, the precision of the decay length measurement improves for large angles between the final state tracks. In contrast to the ? - L analysis, the resolution on the decay length was now parametrized directly as a function of the angle between the direction of the identified lepton and the reconstructed direction of the Ds meson. This dependence is for positive decay lengths of the 1992 and 1993 simulated data plotted in figure 5.10. The width of the Lgen - Lrec distribution as a function of the cos?(Ds^) was fitted with a linear function. For negative decay lengths a linear function was used to describe the dependence of the resolution on the decay length itself. This gave the following 0 114 CHAPTER 5. THE DS - L+ ANALYSIS 100 90 80 70 60 50 40 30 20 10 0 0.5 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0.86 0.92 $° 0.9 ,o** 0.88 Figure 5.10: The dependence of the difference between the reconstructed and the simulated positive decay length on the angle between the Dsf meson and the LT. The resolution improves for large angles. The distribution was obtained on the 1992 and 1993 simulated sample of B0 decays. overall parametrization of ?L: L > 0: ?L L < 0: r (a+ L + bL + cos?(Ds^) ; cos? cL ; cos?(Ds^) < m? (DsL) > m? ) ?L = aL + b7L (5.11) Parameters obtained in two separated samples of simulated events are given in table 5.2 for ’92-’93 and ’94 data. The average decay length resolution for the ’92 and ’93 sim- 5.4. LIFETIME MEASUREMENT 115 ’92, ’93 ’94 a+ L [103 µm] -2.353 ± 0.247 -3.028 ± 0.314 bL + [103 µm] 2.756 ± 0.257 3.413 ± 0.325 cl [µm] 205 ± 75 162 ± 26 m? 0.928 0.935 a- L [µm] 259 ± 120 180 ± 110 bL - -0.67 ± 0.40 -0.56 ± 0.34 Table 5.2: Parameters for evaluation of ?L given by (5.11). Parameters were obtained by a fit to the simulated sample of B0 decays. ulated data is shown in figure 5.11. It was fitted with two Gaussian distributions, giving the width of approximately 765 µm for around 20% of events and 230 µm for 80% of events. For the ’94 data the width was about 770 µm for 15% and 205 µm for 85% of events. 160 140 120 100 80 60 40 20 r /
    ~ 340 \im r A / X 0 b BW k I i I I I Ml l+H Lw-f^ I I I I I I I I I I fV^fa-M -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 L -L \cm\ rec gen L J Figure 5.11: The average precision of the decay length determination for B0s decays in ’92 and ’93 simulation. A possible discrepancy between the simulation and real data in the decay length resolution has already been shown in figure 4.8 b). In the evaluation of systematic errors of the lifetime measurement ?L was varied for ± 10% to account for this effect. The reconstructed energy of the D± sL? system E(Ds^) was 116 CHAPTER 5. THE DS - L+ ANALYSIS used as a basis for the B0 meson energy estimation. Corrections to this crude approximation were then applied in order to estimate the B0 momentum as accurately as possible. For events with coincident directions of the event thrust axis and of the thrust axis of the lepton jet, one can assume a two jet topology, as sketched in figure 5.12. Figure 5.12: Computer reconstruction of a two- and three-jet event with the Delphi spectrometer. Blue lines represent reconstructed tracks. Parts of detectors are shown in green. Marked thrust axis are not the result of reconstruction and are shown for explanation of the text only. The energy of the beams Eb thus agrees with the energy of the jets. For such events the difference between the beam energy and the reconstructed energy of the jet is expected to be correlated with the energy of a neutrino, produced in a semileptonic B decay. The remainder of the B0 meson energy, not taken by the DsLT system, is shown in figure 5.13 as a function of the difference between the Eb and cpjet. p jet is the sum of momenta of charged tracks in the jet. 5.4. LIFETIME MEASUREMENT 117 The distribution is shown only for events with the angle between the event thrust axis and the lepton jet axis ?tj smaller than 5°. 40 30 20 10 -10 -20 -20 -10 10 20 30 40 50 Eb-cpjet [GeV] Figure 5.13: The distribution of the difference between the B0 meson energy and the energy of the Djf-#+ system versus the difference of the beam energy and the jet momentum multiplied by c. The distribution is obtained from the simulated B0 decays satisfying the requirement ?tj < 5°. The full line represents the energy parametrization described by (5.13). The two dimensional distribution, shown in figure 5.13, was fitted with a double Gaussian of the form Ae - ( ' - ' )2 (y'-y' )2 2? -- (5.12) with coordinates of the rotated system x' = xcosß + ysinß and y' = -xsinß + ycosß. From fitted parameters of (5.12) the estimate of the Ebs was obtained as a function of E(Ds^), Eb and pjet, 0 0 118 CHAPTER 5. THE D± S - L? ANALYSIS for events with ?tj < 5?. For events with the angle between the event thrust axis and the lepton jet axis larger than 5?, the average difference between the Ebs and E(Ds^), found in the simulation, was added to the reconstructed energy of the meson and the lepton. The momentum of the B0 meson, needed for the lifetime calculation from (4.10), is given by cpBs = JeB2s - MBsc4, with the final parametrization of a strange B meson energy E(Ds^) + aE(Eb - cpjet) + bE ; ?tj < 5? j E(Dst) + cE ; ?tj > 5? (5.13) The values of parameters, obtained by the fit to simulated distributions, areaE=0.96±0.02, bE=(-1.83±0.25) GeVandce=(7.1±0.2) GeV. The dependence of the relative energy resolution on the B0 energy is shown in figure 5.14. It was fitted with a linear function E = a?EBs + b? (5.14) Bs with the resulting parameters a?=(-7.0±0.6)×10 3 GeV 1 and b?=0.35±0.02. The average precision of the Ebs estimation was around 9.5%. The resolution on the pBs, which enters the relation (4.11) for the calculation of the precision on the proper decay time evaluation, is ?(pBs) = Ebs ?(Ebs )/pbs . Note that the average value of the Ebs/pbs ratio differs from unity for less than 2%. The resolution ?t was calculated separately for each individual event of the signal region, used for the lifetime measurement, from ?L and ?(pBs). Figure 5.15 shows the comparison of the B0 energy distribution between data and the Monte Carlo simulation. For real data, the distribution obtained in the side bands of the invariant mass 1.7 < M(KK?) < 1.8 GeV/c2 and 2.05 < M(KK?) < 2.15 GeV/c2 5.4. LIFETIME MEASUREMENT 119 0.25 0.2 0.15 0.1 0.05 0 15 20 25 30 35 40 45 EBs [GeV] Figure 5.14: The dependence of the relative B0s energy resolution on the energy itself. The dependence was fitted with a linear function on the selected sample of simulated strange B decays. 40 35 30 25 20 15 10 5 0 10 15 20 25 30 35 40 45 50 EBs [GeV] Figure 5.15: The comparison of the B0 energy distribution for the real and the simulated data sample. The distribution of real events in the signal region is obtained by subtracting the energy distribution of events in the side bands of the M(KK?) distribution. The simulated sample was obtained by applying the described selection criteria to Monte Carlo B0 decays. 120 CHAPTER 5. THE D± S - L+ ANALYSIS was subtracted from the distribution of events in the signal region. The difference in the mean of the two distributions was found to be 0.1 ±0.9 GeV. The smaller number of events obtained in this analysis, compared to the (fiL final state, reflects in a higher statistical error on A{E). The error was again used to estimate the systematic error on the lifetime measurement arising from the possible difference between the simulated and the real sample of events. 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 tgen [ps] Figure 5.16: The dependence of selection criteria efficiency on the proper decay time of simulated B0s decays. Dots represent the efficiency after kinematical cuts and triangles after the inclusion of particle identification requirements described in the text. The influence of the selection criteria on the decay time de-pendence of the efficiency can be seen in figure 5.16. The plot does not include the geometrical acceptance of the VD due to the requirement of at least one associated hit per final state track. The efficiency behaviour is flat apart from the two bins at high proper decay times where the statistics is low. Within the statistical er-ror the efficiency is thus consistent with a constant value over the range of the decay times used for the lifetime measurement. A ?2 fit of a constant to the efficiency of kinematical selection criteria 5.4. LIFETIME MEASUREMENT 121 yields the value (28 ± 9) %. As described in section 4.5.3, possible small deviations were taken into account by studying the fitted lifetime of the simulated sample in the same manner as in the ?--# analysis. 5.4.2 Likelihood Fit The unbinned maximum likelihood fit was performed on D±^? events with 1.941 GeV/c2 < M(KK?) < 1.995 GeV/c2. The likelihood function included all non-negligible contributions evaluated in section 5.3. The form of the function, describing the probability distribution of different sources, was N sig L = n(1 -fcomb)(1 -fbcl-fref)PBs(ti,?ti,?Bs) + i=1 + (1 - fcomb)fbclP bcl(ti,?ti,?bcl) + + (1-fcomb)frefPref(ti,?ti,?ref) + + fcomb P comb (ti,?ti,?+,? -,fcomb,f comb,?comb) (5.15) The probability density distribution for B0 decays P Bs was of the same form as in the ?--# analysis given by (4.16). The convolution of an exponential and a Gaussian distribution was also used for the parametrization of probability functions for cascade decays Pbcl and for the contribution from D+ reflections Pref. The average value of the Bu,d meson lifetime ?b = (1.60 ± 0.04) ps [34] was assumed for the mean lifetime of the reflection contribution ?ref. The mean lifetime of cascade decays ?bcl was obtained from the likelihood fit of the simulated sample. The proper decay time distribution and the result of the fit with P bcl as a fitting function is plotted in figure 5.17 c). The fitted lifetime was ?bcl = (2.2±0.2) ps. This value is higher than the average value ?b due to the underes- 122 CHAPTER 5. THE D± S - L? ANALYSIS timation of a B meson momentum in this category of events. By simply comparing the mean value of the estimated momentum of decaying B mesons for direct and cascade decays one can expect a shift of around 20% to 25% in the lifetime. The value close to 2 ps is thus understandable. The proper decay time distribution of the combinatorial background was taken from the side bands of the right sign M(KK?) distribution, as well as from the entire interval of the wrong sign M(KK?) distribution, shown in figure 5.6. Since no excess of events was found in the D± mass region for the wrong sign combinations, we can conclude that these events have the same origin as the background events in side bands of the right sign invariant mass distribution. A study of decay times for this category of events suggested the following term for the probability density function: Pcomb = fcGombG(ti,?comb) + f+ombE(ti,?+) ?G(ti,?ti) + + (1 - fcomb - fc+omb)E(-ti,?-) ? G(-ti,?ti) (5.16) The form of (5.16) slightly differs from the function (4.17) used to describe the same source of background in the ?L sample. The difference might be attributed to the presence of another charged track used to reconstruct secondary vertices. In the ?L final state the resolution on the decay length for all categories of events, including the combinatorial background, was mainly determined by a high pt track due to the small opening angle of particles forming the ? candidate. Hence the resolution ?t was assumed the same for all types of events. In the present analysis the influence of a high pt track is diluted by the presence of an additional charged track forming a pseudo-scalar meson. For the combinatorial background these tracks do not originate from a common vertex. The consequence is a larger smearing in the secondary vertex recon- 5.4. LIFETIME MEASUREMENT 123 struction. This is described by the first and the last term of (5.16). The first term is a Gaussian, centred at zero. The last term is a convolution of an exponential and a Gaussian, with decay time ? -describing a small tail at negative decay times. The second term of (5.16) describes events with the correctly assigned lepton from long living particles. The decay time distribution of the combinatorial background is shown in figure 5.17 b). The likelihood fit to the distribution yielded fcGomb = 0.401 ±0.063, fc +omb = 0.520 ±0.046, ?+ = (1.28 ± 0.11) ps, ? - = (0.42 ± 0.26) ps and ?comb = (0.283 ± 0.037) ps. Since the sample is enriched in bb decays the value of ?+ is approaching the mean lifetime of B mesons. Due to the miss-assigned tracks, not originating from B decays, the fitted value is slightly lower than ?b. The proper decay time distribution of the selected sample of D±^? events is plotted in figure 5.17 a). The result of the maximum likelihood fit is superimposed, shown together with the contributions from the combinatorial background and cascade decays. The fitted lifetime of B0 mesons was ?bs = (1.41 ± 0.26(stat.)) ps . (5.17) The obtained result was again cross-checked by the fit to the simulated sample of B0 decays, processed through the same selection criteria as applied to the real data. With the generated lifetime of 1.6 ps the result was ?bs = (1.535 ± 0.069) ps. 5.4.3 Systematic errors In order to evaluate systematic errors on the lifetime measurement, first all the parameters entering the likelihood fit were varied by 124 CHAPTER 5. THE DS - L+ ANALYSIS 35 30 25 20 15 10 5 0 160 140 120 100 80 60 40 20 0 • Data Comb. back. b—>c—>l :¦•;....•.* -2 -1 0 1 2 3 4 5 6 7 [ps] -2 0 2 40 35 30 25 20 15 10 5 0 -2 0 Figure 5.17: a) The proper decay time distribution of selected D*-^ events. The result of the maximum likelihood fit is shown superimposed. Shaded regions represent contributions of the combinatorial background and of cascade B decays. b) Same distribution for the combinatorial background obtained in the side bands of the right sign M(KK?) invariant mass and in the wrong sign combinations D*-^. c) The proper decay time distribution for cascade decays obtained on the simulated sample. 5.4. LIFETIME MEASUREMENT 125 one standard deviation from their average value and the fit was repeated. The parameters obtained in a separate fit of the combinatorial background were varied according to the resulting correlation coefficients. As can be seen in table 5.3, the largest contribution from this class of systematic errors arises from parameters describing the combinatorial background, particularly from the fraction fcomb. A similar feature was observed in the ?L sample as well. Due to a smaller relative background contribution the error arising from this source is lower in the present analysis. Source of systematic error ?bs variation [ps] Combinatorial background (fcomb, fcomb, fcomb, ?+, ?-) Cascade decays (fbcl, ?bcl) Reflections (fref, ?ref) +0.062 -0.066 +0.033 -0.034 +0.004 -0.004 Difference data/MC in pBs eval. Difference data/MC in ?l +0.042 -0.040 +0.015 -0.006 Possible analysis bias +0.066 -0.066 Total +0.106 -0.107 Table 5.3: Summary of systematic errors on the B0 lifetime measurement for the D±-? analysis. A possible discrepancy between the Monte Carlo simulation and real data was checked in the same manner as for the ?--# analysis. ?L was varied by ± 10%, as already discussed in section 4.5.1, and the corresponding variation of the fitted lifetime was accounted for in the list of systematic errors. Another source in this class of the systematics is the B0 momentum. It was calculated from the estimated energy Ebs. This quantity was varied in each of the events entering the likelihood fit for ±0.9 GeV, the amount found when comparing real and simulated energy distributions. Due to a larger statistical error of this difference the error from 126 CHAPTER 5. THE D± S - L? ANALYSIS this source is larger than the one in the ?--# analysis. The last class of systematic errors are possible limitations of the modelling used. A hint of a non perfect parametrization of the likelihood function can be seen in the first negative bin of the proper decay time distribution in figure 5.17 a), which deviates from the fitted function by around 2?. Together with the acceptance deviations from the constant and parametrizations of pBs and ?L, these effects were accounted for by the lifetime found in the simulated sample. The measured lifetime was corrected for the ratio of the generated and the fitted lifetime of simulated B0 decays. The ratio rcorr = 1.0423 ± 0.0469 was found to be larger than in the ?L case but still within the statistical error of the fit, performed on the simulated sample. Because of the correction the error ?(rcorr )?bs was added in quadrature to other systematic errors of the measurement and the statistical error was corrected by rcorr. With the inclusion of systematic effects the lifetime of B0 mesons, determined from the D± sL? sample, is ?bs = [1.47 ± 0.27(stat.) ± 0.11(syst.)] ps . (5.18) Part IV Summary 127 Chapter 6 Combination of the Results To combine results of the two analyses, differing in the reconstructed final state, one should first consider the statistical correlation between the two samples of events. From the production mechanism point of view all the events reconstructed in the c/> decay mode of the D± sLT analysis are expected to be included in the (fiL sample. The first effect that lowers this ratio is due to the larger sample of hadronic Z0 decays, used in the reconstruction of D± mesons. Because of the slight differences in cuts, used for the different data taking periods, the 1992 data represent around 1/3 of the final sample. Hence only 2/3 of D± mesons reconstructed through the (pn± decay channel are expected to be in common with the selected events of the --# analysis. Another cause of the dilution of the overlap is the relative fraction of the (pn± decay mode in the overall D± sLT signal. This amounts to approximately 40%. Thus one would expect around 0.4 x 2/3 « 25% of the D± candidates, reconstructed in the c/> decay mode, to be the same as events in the c/> mass region of the (fiL sample. However, differences in the selection criteria, used for isolation of the two samples, can cause different events of the D± — (pn± decay channel to be selected in the two samples. Because of that, the value of around 25% is an 129 130 CHAPTER 6. COMBINATION OF THE RESULTS upper limit for the expected statistical overlap of both signals. By comparing the event numbers in both selected samples and under the reasonable assumption that common events do not originate from non-physical sources of background, e.g. the combinatorial background, 6 ± 3 % of events were found to be the same. The common systematic error of the two measurements might arise due to the used branching ratios in the calculation of specific contributions. This would connect the errors arising from the cascade background in the D± sL? analysis and from the Bs purity in the (fiL final state. These two errors, quoted in tables 5.3 and 4.2, are partially correlated, and the amount of the common part for the two errors was found to be ±0.008 ps. Another source of the common systematics is the tracking performance of the Delphi spectrometer. Since this is the same for the two samples and the same sample of simulated events was used for the study of the possible difference in the decay length resolution, errors due to oz, may be considered completely correlated. The common part of the errors, arising from this source, is +-00056 ps. The difference between the Monte Carlo and real data in the pBs estimation were studied using two different simulated samples. The algorithms of the calculation differ as well and the errors from this source were assumed to be uncorrelated. A small correlation in the systematic errors of the two measurements originates from the use of the average B meson lifetime in both analysis. The error, which is in common to both analyses due to this reason, was found to be ±0.002 ps. Quadratic sum of systematic errors, which are common to both analysis, is +-0..010 ps. This value is small compared to overall systematic errors of the two individual results. It was verified that 131 statistical and systematical correlations are too small to influence the combined result of the measurements, derived below. The most common approach to combining results of different measurements is to calculate the mean of individual results, weighted according to their errors: T = Xi^i Wi = 2 . (6.1) Ti are measurements to be combined, and o~i the corresponding errors. The error on the average value T then follows as dr = (V2)-2 . (6.2) i °i However, since the underlying distribution of lifetime measurements is exponential, with 2:p H/1+W2 T1 + [------------T1 - -----------1 (---) . (6.4) IU2 +lV1^r- W1+W2 T2 In case of correlation, part of the systematic errors, common to both measurements, is quadratically subtracted from the individual errors. Due to the statistical overlap of the two samples, 132 CHAPTER 6. COMBINATION OF THE RESULTS statistical errors are increased by the appropriate factor, consid-ering that the statistical errors scale as 1/ ?T, with N being the number of signal events. The resulting systematic and statistical errors are then used as weights for averaging. Finally, common systematic errors are added in quadrature to the error on the average lifetime. Since the correlation between the two performed measurements is small, the result of such a procedure does not differ from the average, obtained by neglecting the correlation. The combined value of the B0 lifetime from the D±^? and ?L sample is ?bs = (1.66 ± 0.19) ps . (6.5) Chapter 7 Conclusions The present work evolved from the studies of B0 meson properties with the Delphi spectrometer ever since the first evidence for the B0 meson production in Z0 decays in 1992 [37]. Since then, experimental methods have been developed, and the number of accumulated B0 decays has been substantially increased. The measurement of the B0 meson lifetime was performed using two to some extent complementary samples of events. For the first time the sample with the reconstructed ? meson accompanied by a high transverse momentum lepton in the same jet was used for such measurement. The selected inclusive decay channel profits from a relatively high statistics. The statistical power of the sample is of large importance in B0 lifetime measurements since with the sources of B mesons presently available they are still limited by the statistical precision (see the summary of B0 meson lifetime measurements in figure 7.1). The selection of leptons with a high pt enriches the isolated sample in direct semileptonic B decays. The reconstruction of ? mesons in the final state raises the fraction of strange B mesons among all the B mesons from around 11%, found in Z0 › bb decays, to (50 ± 7)%. By the use of the Delphi Vertex Detector the determination of the decay length and hence 133 134 CHAPTER 7. CONCLUSIONS of the proper decay time is possible. The maximum likelihood fit of the decay time distribution for events in the (fiL sample resulted in the B0 lifetime of Tbs = [1.75 ± 0.20(stat.) . (syst.)] ps . (7.1) A more exclusive decay channel, with a reconstructed D± meson and a lepton of the opposite charge in the same jet, offers a higher purity of B0 mesons. A cut on the transverse momentum of the lepton again enriches the sample of events in semileptonic decays of B mesons. The reconstruction of a strange charmed meson through its decays into 4>n and K?0K final states results in the B0 purity of 0.843 ± 0.034. Due to a higher signal to noise ratio the systematic error of the lifetime measurement on the selected sample is somewhat lower than the corresponding error in the (p - L analysis. The result of the maximum likelihood fit of the proper decay time distribution was Tbs = [1.47 ± 0.27(stat.) ± 0.11(syst.)] ps . (7.2) The common statistical and systematical errors of the two measurements were found to be small. The results were combined using the weighted average. The mean B0 lifetime is Tbs = (1.66 ± 0.19) ps . (7.3) Figure 7.1 places the obtained results in the world summary of B0 lifetime measurements. Full error bars show the quadratic sum of statistical and systematical errors while vertical lines indicate the amount of statistical errors alone. Currently the most precise individual measurement is performed by the Aleph collaboration [38] using D± sL? correlations. 135 ALEPH Ds-l ALEPH Ds-h CDF Ds-l CDF J/? ? DELPHI Ds-l DELPHI Ds-h DELPHI ?-l DELPHI Ds-l DELPHI Ds incl. OPAL Ds-l World Average [32] [34] [35] [35] [25] [25] This analysis This analysis [25] [36] In \PS] Figure 7.1: The world summary of published measurements of the B0s lifetime. On the right side references of publications are indicated. Ver-tical lines on the error bars indicate the amount of the statistical error in the quadratic sum of the statistical and systematical error, represented by the full line. The world average was taken from [39]. The small statistical error of the measurement is the result of five D± decay channels, used in addition to the two presented in our analysis. Following in precision is the result of the --# analysis of the current work. The world average was calculated by the LEP B Lifetime Working Group for the 1996 edition of the Review of 136 CHAPTER 7. CONCLUSIONS Particle Properties [39]. For the average calculation the previously published result of the D±-#? analysis of the Delphi collaboration [31] was used. The lifetime obtained on the ?L sample contributes to the world average with a weight of approximately 17%. The ratio of the measured B0 lifetime (7.3) and of the world average for the B0d meson lifetime [39] is calculated to be ?- = 1.06 ± 0.13 , (7.4) neglecting possible systematic correlations. Unfortunately the precision of the B0 lifetime measurement is not sufficient to make any statements on adequateness of the models, described in chapter 2, which are used to predict lifetime differences between individual B meson species. Even if one uses the world average for ?bs, the ratio ? s = 1.03 ± 0.07 (7.5) is completely compatible with unity. The ratio of the measured lifetime and the world average of the charged B meson lifetime is ?b+ ???? = 0.98 ± 0.12 , (7.6) with the error which is larger than the expected digression from unity. With the world average value of ?bs one obtains ?B+ ? = 1.01 ± 0.07 , (7.7) Bs the value which is not so far from the limit, predicted by the non-spectator effects: for fb = 200 MeV one would anticipate the ratio of 1.05. In the future, the ?L inclusive decay channel can be used for the study of Bs meson oscillations. Due to the limited statistics the measurement of the oscillation frequency in the Bs system has 137 not been performed yet and only lower limits exist. They are ad-ditionally lowered by systematic uncertainties, one of which is the uncertainty of the measured B0s lifetime. It is difficult to estimate the amount of these systematic effects produced by the error on the average ?Bs. Hopefully, better knowledge of the lifetime will improve results of such analysis as well as enable the future study of B0s decay properties. 138 CHAPTER7. CONCLUSIONS Part V Povzetek doktorskega dela 139 Poglavje 8 8.1 Uvod Standardni model moˇcne in elektroˇsibke interakcije [1] je dandanes ena najbolj uveljavljenih teorij v fiziki osnovnih delcev. Prve eksperimentalne potrditve modela segajo v leto 1973, z odkritjem nevtralnega ˇsibkega toka, katerega nosilec je nevtralni ˇsibki bo-zoni. V letu 1983 sta skupini znanstvenikov, zbranih ob spektrometrih UA1 ter UA2, uspeli neposredno potrditi obstoj nevtralnih in nabitih ˇsibkih bozonov [3]. Natanˇcne meritve parametrov modela, ter s tem testi njegovih napovedi, pa so bili mogoˇci ˇsele ˇsest let kasneje, z zaˇcetkom obratovanja elektronsko-pozitronskega tr-kalnika LEP v Evropskem laboratoriju za fiziko osnovnih delcev (CERN) v Zˇenevi. Elektroni in pozitroni, pospeˇseni v trkalniku LEP, pri trˇcenju tvorijo nevtralni ˇsibki bozon Z0. Meritve produkcijskih in razpadnih lastnosti bozona Z0 so potrdile napovedi teorije Standardnega modela z izjemno natanˇcnostjo. Bozon Z0 razpada v par leptonov ali par kvarkov. Okoli 22% razpadov Z0 — qq predstavljajo razpadi v pare bb [6]. Zaradi tega je trkalnik LEP primeren za ˇstudij hadronov, sestavljenih iz kvarkov b. Fizika kvarkov b je bila pomembno podroˇcje raziskav na trkalnikih elektronov in pozitronov ˇze pred konstrukcijo trkal-nika LEP. Prednosti slednjega izvirajo iz viˇsje razpoloˇzljive energije 141 142 POGLAVJE 8. trkov. Teziscna energija trka na LEP je namreˇc enaka masi bo-zona Z0 , MZ = (91.1885 ± 0.0022) GeV/c2 [6]. Hadroni, ki nastanejo iz kvarka in antikvarka ob razpadu Z0 , imajo zaradi viˇsje energije daljˇse razpadne razdalje. Rekonstrukcija razpadnih razdalj s pomoˇcjo polprevodniˇskih detektorjev pa omogoˇca meritev ˇzivljenjskih ˇcasov nastalih delcev. Poleg tega razpad nevtralnega ˇsibkega bozona omogoˇca nastanek celotnega spektra hadronov, ki vsebujejo kvark b. S trkalnikom LEP lahko prviˇc rekonstruiramo mezone B0 in merimo ˇzivljenjski ˇcas barionov Ab. Priˇcujoˇce delo opisuje meritev ˇzivljenskega ˇcasa mezonov B0. Ta ˇcas je znan z najmanˇso natanˇcnostjo med ˇzivljenjskimi ˇcasi treh razliˇcnih mezonov B: B0 , B+ in B0d 1. Razpadne lastnosti mezonov B0 niso podrobno izmerjene, saj so bili le-ti eksperimentalno potrjeni ˇsele leta 1992 [37]. Zˇivljenjski ˇcas bo v bodoˇcnosti potreben za izraˇcun razpadnih ˇsirin iz izmerjenih razvejitvenih razmerij. Za razpade hadronov, sestavljenih iz kvarkov b, so odgovorni razliˇcni procesi, njihov prispevek pa je odvisen od sestave posameznega delca. Ob zadostni natanˇcnosti lahko z meritvijo ˇzivljenjskih ˇcasov posameznih hadronov b sklepamo o pomembnosti razliˇcnih procesov, udeleˇzenih pri razpadih. Poznavanje ˇzivljenjskega ˇcasa pa je tudi nujen pogoj za meritev frekvence oscilacij mezonov B0. Meritev te koliˇcine zaenkrat ˇse ni mogoˇca zaradi premajhnega ˇstevila zabeleˇzenih razpadov B0 , predstavlja pa motivacijo za nadaljnje raziskave na podroˇcju fizike mezonov B. Meritev smo izvedli na dveh vzorcih semileptonskih razpadov mezonov B0. Vzorec, sestavljen iz dogodkov z rekonstruiranim mezonom ? in leptonom v istem hadronskem pljusku, omogoˇca 1Mezon B0s je setavljen iz antikvarka b in kvarka s. Mezon B+ sestavlja par bu in mezon B0d par bd. Cˇe ni posebej oznaˇceno, je pri oznakah delcev vedno privzeta ustrezna trditev za nabojno konjugirano stanje. 8.2. NASTANEK IN RAZPAD MEZONOV B0S NA TRKALNIKU LEP 143 eno statistiˇcno najnatanˇcnejˇsih meritev ˇzivljenjskega ˇcasa B0. Dogodki, ki vsebujejo mezon D± in lepton nasprotnega naboja v istem pljusku, pa zagotavljajo veˇcjo ˇcistost vzorca in s tem manjˇso sistematiˇcno napako meritve. 8.2 Nastanek in razpad mezonov B0s na trkalniku LEP Anihilacija elektrona in pozitrona lahko poteka preko izmenjave fotona ali ˇsibkega bozona. Feynmanova diagrama obeh procesov sta prikazana na sliki 8.1. e + Z f e f e a) b) Slika 8.1: Feynmanov diagram anihilacije elektrona in pozitrona preko izmenjave a) ˇsibkega bozona ali b) fotona. Anihilacijski sipalni presek pri teziscni energiji ?s je [7] 4??2 ?(e+e › ff) = NffQCD kf 3?2 + V -Q + v GF? Qfvevf G 2F 3s (s -MZ2)2 +s2 MZ2 (MZ2 - s) m Z a2 -? (v2e+a2e)(kvfv2 + kafa2f 96 f m4 s (s -MZ2)2 + s2 - + Z2 M2 (8.1) Qf oznaˇcuje naboj nastalega fermiona, izrazen v enotah e0, Ncf pa ˇstevilo barv (3 za kvarke in 1 za leptone). ? je konstanta + - e 144 POGLAVJE 8. fine strukture in GF Fermijeva konstanta, najnatanˇcneje doloˇcena iz meritev ˇzivljenjskega ˇcasa mionov [5]. Vektorsko in aksialno sklopitveno konstanto bozona Z0 s fermioni izrazimo kot vf = 2T3f - 4Qfsin2?w af = 2T3f , (8.2) kjer T3f predstavlja tretjo komponento ˇsibkega izospina (+1/2 za nevtrine ter kvarke u, c, t in -1/2 za masivne leptone ter kvarke d, s, b). MZ in IZ sta masa ter ˇsirina bozona Z0 , ?w pa Weinbergov kot. Kinematiˇcni faktorji kvf,a so posledica konˇcne mase nastalih fermionov: 3 - ß2 f 2 ß = 1- 4mf 1--------- . (8.3) s Pri izraˇcunu sipalnega preseka (8.1) je treba poleg diagramov na sliki 8.1 upoˇstevati ˇse procese viˇsjih redov. Moˇznost izsevanja gluonov v primeru hadronskih razpadov Z0 je upoˇstevana s faktorjem 1; leptoni ) ? s(mZ2) ; k rk i . (8.4) Izmerjena razpadna ˇsirina bozona Z0 in deleˇz razpadov v pare bb znaˇsata [6] r(Z0 T(Z0) = (2496.3 ± 3.2) MeV › qq) = (1740.7 ± 5.9) MeV Rb ? -----o›=r = 0.2219 ± 0.0017 . T(Z ›qq) (8.5) 8.2. NASTANEK IN RAZPAD MEZONOV B0S NA TRKALNIKU LEP 145 Primarni kvarki, nastali ob razpaduˇsibkega bozona, se s kvarki, nastalimi v procesu hadronizacije [12] [10] [11], zdruˇzijo v opa-zljive hadrone. Hadroni, ki nastanejo v procesu hadronizacije, so lahko stabilni, ali pa nadalje razpadajo v laˇzje hadrone ter leptone. Razpad mezona B0s je v osnovnem redu na partonskem nivoju prikazan na sliki 8.2. hadronski semileptonski razpadi q 1, V, W+ q2, l + q 3 s ------------------------------ s q1 = u, c q2 = d, s^ l+ = e+, \i+, x+ Slika 8.2: Feynmanov diagram razpada mezona B0. Razpad je lahko hadronski, ˇce se nabiti ˇsibki bozon W+ sklaplja s kvarkoma, ali semileptonski, ˇce W+ razpade v lepton in nevtrino. Osnovni pribliˇzek za razpadno ˇsirino mezona dobimo, ˇce zanemarimo vpliv kvarka, ki spremlja teˇzki kvark v hadronu. V tem primeru je lahki kvark zgolj opazovalec razpada in tak pribliˇzek imenujemo spektatorski model. V tem modelu je razpadna ˇsirina 146 POGLAVJE 8. za hadronski razpad kateregakoli mezona B 192?3 mb (8.6) › cq1q2) = 3 G 3m 5b|Vcb|2|Vq1q2|2I(—-,x2,x 1) . r(b r(b › cq1q2) je sorazmerna peti potenci mase kvarka b. Vqiqj so elementi matrike Cabbibo-Kobayashi-Maskava (CKM) [4], ki opisujejo moˇc sklopitve nabitih ˇsibkih bozonov s kvarki. V izrazu (8.6) smo upoˇstevali le razpade, kjer se kvark b sklaplja s kvarkom c. Kot je razvidno iz slike 8.2 so moˇzni tudi prehodi b›u. Slednji zaradi majhne vrednosti razmerja |Vub/Vcb| = 0.08 ± 0.02 [5] prispevajo zanemarljiv deleˇz k celotni razpadni ˇsirini mezona B. S funkcijo I(x3,x2,x1) [14] upoˇstevamo konˇcne mase nastalih kvarkov. V limiti xi = mi/mb › 0 je vrednost funkcije enaka 1. Celotna razpadna ˇsirina delca je vsota parcialnih razpadnih ˇsirin. Ob pribliˇzku Vcs ? Vud ? 1 dobimo s Spekatorskim modelom naslednji izraz za celotno razpadno ˇsirino mezona B: spec = TBpec ? ——? 3-2.9m 5b|Vcb|2 . (8.7) ?B 192 ˇStevilski faktor 2.9 sledi iz omejitev faznega prostora za posamezne razpade, opisanih s funkcijo I(x3,x2,x1), ˇce za masi kvarkov c in b privzamemo vrednosti mc = 1.6 GeV/c2 in mb = 5 GeV/c2 [5]. Proces na sliki 8.2, ki opisuje spektatorski razpad mezona B0 , ni edini mehanizem odgovoren za razpad. K razpadni ˇsirini prispevajo tudi drugi procesi, katerih prispevek je med drugim odvisen od lahkega kvarka, ki sestavlja mezon. Ti procesi povzroˇcijo razlike med razpadnimi ˇcasi razliˇcnih vrst mezonov B. Skicirani so na sliki 8.3. Razpadi z izmenjavo nabitega ˇsibkega bozona (slika 8.3 a)) so moˇzni le pri nevtralnih, razpadi z anihilacijo kvarkov (slika 8.3 c)) pa le pri nabitih mezonih B. Verjetnost za take razpade je zniˇzana 8.2. NASTANEK IN RAZPAD MEZONOV B0S NA TRKALNIKU LEP 147 b d, s W+ c, u u, c a) b u b u W + b) W+ c) u, c, ?l d, s, l + Slika 8.3: a) Razpad z izmenjavo ˇsibkega bozona W±. b) Razpad nabitih mezonov z dvema identiˇcnima u kvarkoma v konˇcnem stanju. c) Razpad preko anihilacije kvarkov. glede na verjetnost za spektatorski razpad zaradi suˇcnosti nastalih kvarkov in zaradi zahteve po prekrivanju valovnih funkcij zaˇcetnih kvarkov [15]. Popravki k razpadni ˇsirini, ki izvirajo iz omenjenih procesov, so zato majhni. Pri razpadih nabitih mezonov B se v konˇcnem stanju lahko pojavita dva identiˇcna kvarka u (slika 8.3 b)). V tem primeru mora biti valovna funkcija konˇcnega stanja antisimetrizirana glede na njuno zamenjavo. Popravek k spektatorski razpadni ˇsirini nabitih mezonov B, ki izvira iz takega procesa, je negativen [15]. Zato je ˇzivljenjski ˇcas mezonov B± nekoliko daljˇsi kot razpadni ˇcas nevtralnih mezonov B. Podrobnejˇsi raˇcun prispevkov k razpadnim ˇsirinam razliˇcnih mezonov B, ki so posledica zgoraj opisanih procesov, nam da nasled- 148 POGLAVJE 8. nje priˇcakovane vrednosti razmerij ˇzivljenjskih ˇcasov [17]: ?b+ / fb \2 ---- ? 1 + 0.05 V ?Bd 200Me ?Bd ? ?Bs . (8.8) Oznaki ?bs in?Bd oznaˇcujeta povpreˇcni vrednosti ˇzivljenskih ˇcasov dveh masnih lastnih stanj v sistemu nevtralnih mezonov B, fb pa razpadno konstanto mezona B. 8.3 Analiza podatkov 8.3.1 Razpadni kanal ?--# Med razpadi Z0 › qq , zabeleˇzenimi s spektrometrom Delphi v letih 1993 in 1994, smo izbrali dogodke, ki vsebujejo me z on ? in lepton v istem hadronskem pljusku. Hadronski razpadi Z0 se od leptonskih loˇcijo predvsem po ˇstevilu nabitih sledi v konˇcnem stanju. Hadronske dogodke, ki se izbirajo ˇze ob zapisu zabeleˇzenih podatkov na magnetne trakove, je moˇc izbrati z izkoristkom nad 95% [27]. Pri izbiri vzorca ?-# smo upoˇstevali elektrone in mione, identificirane s spektrometrom Delphi [20] [24]. Izbira leptonov temelji na zahtevi po visoki transverzalni gibalni koliˇcini delca glede na os hadronskega pljuska. Transverzalna gibalna koliˇcina leptona pt, ki nastane v semileptonskem razpadu hadrona, je namreˇc odvisna od mase razpadlega delca. Izbira leptonov z visoko pt omogoˇca obogatitev izbranega vzorca s semileptonskimi razpadi mezonov B. Simulirane porazdelitve transverzalnih gibalnih koliˇcin leptonov, nastalih v razliˇcnih procesih, so prikazane na sliki 8.4 a). Lep-toni, ki izvirajo iz neposrednih semileptonskih razpadov mezonov B (b› L), imajo viˇsjo povpreˇcno pt, kot leptoni iz kaskadnih raz- 8.3. ANALIZA PODATKOV 149 padov (b › c › L ), kjer kvark b najprej razpade v kvark c, ta pa je vir leptonov. ˇSe niˇzje transverzalne gibalne koliˇcine imajo leptoni iz neposrednih razpadov kvarkov c (c › L ), nastalih pri procesu Z0 › cc . Ozadje vkljuˇcuje napaˇcno identificirane leptone ter leptone iz razpadov laˇzjih hadronov. Prikazane porazdelitve vkljuˇcujejo zahtevo, daje celotna gibalna koliˇcina leptonov, prikazana na sliki 8.4 b), viˇsja kot 3 GeV/c. 1800 1600 1400 1200 1000 800 600 400 200 0 k a) b —> l i. b^c^l -- L "'i:'^ c —> l r- L| ^D Ozadje ^V ^¦¦^¦¦i-^-i;;;;; fŠ^JS^^k^jr^^fB^ 2500 2000 1500 1000 500 pt [GeV/c] 10 20 p [GeV/c] 30 Slika 8.4: a) Rekonstruirana transverzalna gibalna koliˇcina leptonov glede na smer hadronskega pljuska. Prikazane so porazdelitve za razliˇcne vire leptonov na simuliranem vzorcu hadronskih razpadov Z0. Porazdelitve vkljuˇcujejo zahtevo, da je celotna gibalna koliˇcina delca viˇsja kot 3 GeV/c. b) Enako za celotno gibalno koliˇcino leptonov. Mezone ? smo rekonstruirali v razpadnem kanalu ? › K+K , z razmejitvenim razmerjem (49.1 ± 0.9) % [5]. Izbira kaonov je temeljila na izmerjeni gibalni koliˇcini sledi ter na izraˇcunani gibalni koliˇcini mezona ? in invariantni masi sistema K+K-L. Poleg kine-matiˇcnih rezov smo uporabili detektorje obroˇcev Cˇerenkova, vgrajene v spektrometer Delphi [20]. Meritev kota Cˇerenkova skupaj z meritvijo specifiˇcne ionizacije sledi omogoˇca loˇcevanje pionov in kaonov, s tem pa zniˇzanje kombinatoriˇcnega ozadja v porazdelitvi invariantne mase parov K+K . 150 POGLAVJE 8. Porazdelitev invariantne mase parov K+K za izbrane dogodke je prikazana na sliki 8.5. Vrh porazdelitve je pri M? = (1.019 ± 0.001) GeV/c2, ˇsirina pri poloviˇcni viˇsini pa je T = (11.0±0.5) MeV. V intervalu 1.008 GeV/c2 < M(K+K ) < 1.030 GeV/c2 je 433 ± 62 ? mezonov. M? se ujema z nominalno maso mezona ?, ki znaˇsa (1019.413 ± 0.008) MeV/c2 [5]. 140 120 100 80 - 60 - 40 - 20 0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 M(K +K -) [GeV/c2] Slika 8.5: Porazdelitev invariantne mase izbranih parov K+K-. Na sliki je za primerjavo prikazana tudi enaka porazdelitev za simulirane hadronske razpade Z0. Deleˇze razliˇcnih procesov, ki prispevajo h konˇcnemu stanju ?L, smo izraˇcunali s pomoˇcjo simuliranih razpadov Z0 › qq , ki smo jih izbrali z enakimi rezi kot podatke. Deleˇz mezonov B0 8.3. ANALIZA PODATKOV 151 med vsemi razpadi mezonov B, ki prispevajo k rekonstruiranemu konˇcnemu stanju, znaˇsa 0.50 ± 0.07. Razpadni ˇcas mezona v njegovem lastnem sistemu je zvezan z razpadno razdaljo L in gibalno koliˇcino pBs: t = ppppp- . (8.9) Mbs = (5375 ± 6) MeV/c2 je masa mezona B0 [5]. Razpadno razdaljo mezona smo izmerili s prilagajanjem obeh kaonskih in leptonske sledi v izbranih dogodkih v skupno razpadno vozliˇsˇce. Gibalno koliˇcino smo doloˇcili iz deleˇza pBs, ki jo prevzame sistem ?L, na simuliranih semileptonskih razpadih mezona B0. Za vsak dogodek v intervalu invariantne mase K+K med 1.008 in 1.030 GeV/c2 smo izraˇcunali razpadni ˇcas po enaˇcbi (8.9). Z metodo maksimalne zanesljivosti smo na tako izraˇcunane razpadne ˇcase prilagajali verjetnostno porazdelitev, ki je opisovala semilep-tonske razpade mezonov B0 in prispevke procesov, ki predstavljajo ozadje. Porazdelitev razpadnih ˇcasov je prikazana na sliki 8.6. Rezultat prilagajanja je povpreˇcni ˇzivljenjski ˇcas mezonov B0. Ob upoˇstevanju sistematiˇcnih napak, ki so podrobneje opisane v tabeli 4.2, je izmerjeni ˇzivljenjski ˇcas na izbranem vzorcu dogodkov ?L ?bs = [1.75 ± 0.20(stat.) . (syst.)] ps . 1 )\ (8.10) 8.3.2 Razpadni kanal D± - L? Iz vzorca hadronskih razpadov Z0 , zabeleˇzenih v letih 1992, 1993 in 1994, smo izbrali dogodke ki vsebujejo mezon D± skupaj z lep-tonom nasprotnega naboja v istem hadronskem pljusku. Naˇcin 152 POGLAVJE 8. 100 80 60 40 20 0 350 300 250 200 150 100 50 0 Nap. id. leptoni ¦\'¦:-i Komb. ozadje • Podatki -2 -1 0 t [ps] 450 400 350 300 250 200 150 100 50 0 t [ps] t [ps] Slika 8.6: a) Porazdelitev razpadnih ˇcasov, v lastnem sistemu mezona B0s , za dogodke v izbranem obmoˇcju invariantnih mas K+K-. Osenˇcena obmoˇcja predstavljajo prispevek kombinatoriˇcnega ozadja ter ozadja, ki izvira iz napaˇcno identificiranih leptonov in leptonov iz razpadov lahkih hadronov. Krivulja prikazuje rezultat prilagajanja z metodo maksimalne zanesljivosti. b) Enaka porazdelitev za dogodke v stranskem intervalu invariantnih mas K+K-, ki predstavljajo kombinatoriˇcno ozadje. c) Porazdelitev razpadnih ˇcasov za dogodke z napaˇcno identificiranimi leptoni oziroma z leptoni iz razpadov lahkih hadronov. Porazdelitev je bila rekonstruirana na simuliranem vzorcu razpadov B0s. 8.3. ANALIZA PODATKOV 153 izbire leptonov je bil enak kot v predhodni analizi, izbrani rezi pa nekoliko ostrejˇsi. Zaradi tega je izbrani vzorec vseboval niˇzje deleˇze ozadja kot vzorec dogodkov ?L. Mezone D± smo rekonstruirali preko dveh razpadnih kanalov: D± — ??± in D± ^K*0K±. Razvejitveni razmerji obeh razpadnih naˇcinov sta podobni in znaˇsata (3.5 ± 0.4)% za prvi in (3.3 ± 0.5)% za drugi kanal [5]. Medtem, ko smo mezone ? iskali preko enakega razpadnega kanala kot v prejˇsnji analizi, smo pri mezonih K*0 izkoristili razpad v K ?+. Kinematiˇcni izbirni kriteriji so vsebovali gibalne koliˇcine obeh nabitih kaonov in nabitega piona v konˇcnem stanju, gibalno koliˇcino mezona ? ali K*0, invariantno maso para delcev, ki sta tvorila ? ali K*0, energijo mezona D± in pa gibalno koliˇcino ter invariantno maso sistema D± sLT. Ker gre pri obravnavanih procesih za razpad psevdoskalarnega mezona D± v vektorski mezon ? ali K*0 ter psevdoskalarni mezon ?± ali K± , smo pri izbiri dogodkov upoˇstevali tudi ustrezno kotno porazdelitev nastalih delcev. Poleg kinematiˇcnih zahtev smo tako kot v prejˇsnji analizi uporabili identifikacijo delcev s pomoˇcjo detektorjev Cˇerenkova in meritve specifiˇcne ionizacije. Invariantno maso obeh kaonov in piona za izbrane dogodke kaˇze slika 8.7. V porazdelitvi je izrazit vrh pri masi Mds = (1.968 ± 0.002) GeV/c2. ˇSirina signala je ?ds = (13.5 ± 2.2) Me V. Mds se ujema z nominalno maso mezona D± (1968.5 ± 0.7) MeV/c2. V obmoˇcju ±2?ds je 81 ± 13 mezonov D± . Deleˇz mezonov B0 med vsemi mezoni B, ki prispevajo h konˇcnemu stanju D±^+, je na simuliranih dogodkih, analiziranih na enak naˇcin kot podatki, znaˇsal 0.843 ± 0.034. Za dogodke z invariantno masoM(KK?) med 1.941 in 1.995 GeV/c2 154 POGLAVJE 8. 40 30 - 20 - 10 0 1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 M(KK?) [GeV/c ] Slika 8.7: Porazdelitev invariantne mase delcev KK? za izbrane dogodke D± — ??±, K*0K± z LT v istem hadronskem pljusku. Senˇcena porazdelitev kaˇze invariantno maso dogodkov, kjer je izbrani lepton enakega naboja kot rekonstruirani mezon D± . smo izraˇcunali razpadni ˇcas po enaˇcbi (8.9). Razdaljo med inter-akcijsko toˇcko e+e ter razpadno toˇcko mezona B0 smo dobili s prilagajanjem rekonstruirane sledi D± in leptonske sledi v skupno razpadno vozliˇsˇce. Gibalno koliˇcino mezona B0 smo izraˇcunali iz energije sistema D±^+, kateri smo dodali popravke izvrednotene na simuliranih razpadih. Kot pri analizi s konˇcnim stanjem ?L smo na razpadne case 8.4. REZULTATI IN ZAKLJUCˇKI 155 izbranih dogodkov prilagajali verjetnostno porazdelitev z metodo maksimalne zanesljivosti. Prilagajana funkcija je zopet vsebovala ˇclene za opis signala ter ˇclene, ki so popisovali porazdelitve dogodkov ozadja. Dobljene porazdelitve so skupaj z rezultati prilagajanja prikazane na sliki 8.8. Ob upoˇstevanju sistematiˇcnih napak, opisanih v tabeli 5.3, je konˇcni rezultat prilagajanja ?bs = [1.47 ± 0.27(stat.) ± 0.11(syst.)] ps . 8.4 Rezultati in zakljuˇcki (8.11) Meritvi ˇzivljenjskega ˇcasa mezonov B0 , izvedeni na dveh razliˇcnih vzorcih semileptonskih razpadov mezonov B, lahko zdruˇzimo v enoten rezultat. S primerjavo izbranih dogodkov smo ugotovili obseg statistiˇcnega prekrivanja vzorcev. Analiza virov sistematiˇcnih napak je pokazala, da je majhen del sistematiˇcne napake skupen obema meritvama. Korelacije obeh meritev so majhne in ne vplivajo na povpreˇcen rezultat obeh meritev. Pri izraˇcunu povpreˇcja smo kot uteˇz posamezne meritve upoˇstevali relativne napake, da se izognemo sistematiˇcnemu premiku kombiniranega rezultata [36]. Izmerjeni povpreˇcni ˇzivljenjski ˇcas mezonov B0 je ?bs = (1.66 ± 0.19) ps . (8.12) Slika 8.9 uvrˇsˇca obe izvedeni meritvi v povzetek svetovnih meritev ˇzivljenjskega ˇcasa ?bs. Meritev ˇzivljenjskega ˇcasa na vzorcu dogodkov z mezonom ? in leptonom v istem hadronskem pljusku je ena najnatanˇcnejsih posameznih meritev ?bs. 156 POGLAVJE 8. 35 30 25 20 15 10 5 0 160 140 120 100 80 60 40 20 0 • Podatki i::i: Komb. ozadje I b—>c—>l -2 -1 0 1 2 3 4 5 6 7 40 35 30 25 20 15 10 5 0 [ps] -2 0 2 -2 0 2 Slika 8.8: a) Razpadni ˇcas za dogodke z invariantno maso M(KK?) v obmoˇcju mase mezona D* . Senˇcene porazdelitve kaˇzejo prispevka dveh najpomembnejˇsih virov ozadja: kombinatoriˇcnega ter ozadja z dogodki, ki vsebujejo kaskadne razpade mezonov B. Polna krivulja je rezultat prilagajanja. b) Porazdelitev razpadnih ˇcasov za dogodke v stranskem intervalu mase M(KK?) ter za dogodke z rekonstruiranim mezonom Dsf v pljusku z leptonom enakega naboja. Ti dogodki predstavljajo kombina-toriˇcno ozadje. c) Porazdelitev razpadnih ˇcasov za dogodke s kaskad-nimi razpadi mezonov B. Razpadni ˇcasi so bili izmerjeni na simuliranih razpadih mezonov B. 8.4. REZULTATI IN ZAKLJUCˇKI 157 ALEPH Ds-l ALEPH Ds-h CDF Ds-l CDF J/? ? DELPHI Ds-l DELPHI Ds-h DELPHI ?-l DELPHI Ds-l DELPHI Ds incl. OPAL Ds-l World Average [32] [34] [35] [35] [25] [25] This analysis This analysis [25] [36] In \PS] Slika 8.9: Povzetek meritev ˇzivljenjskega ˇcasa ?bs. Ob desni strani so oznaˇcene reference izvedenih meritev. Prvi del napak, omejen z navpiˇcnimi ˇcrtami, oznaˇcuje deleˇz statistiˇcne napake v kvadratiˇcni vsoti sistematiˇcne in statistiˇcne napake. Svetovno povpreˇcje je povzeto po [39]. Zˇal doseˇzena natanˇcnost meritve ne zadoˇsˇca za preverjanje napovedi razmerja ?Bs/?Bd ali ?b+/?bs, podanima z (8.8). Cˇe uporabimo svetovno povpreˇcje ˇzivljenjskih ˇcasov ?bs in ?Bd [39], dobimo 158 POGLAVJE 8. ob zanemaritvi moˇznih korelacij ??? s = 1.03 ± 0.07 , (8.13) vrednost, kije popolnoma skladna z 1. Podobno dobimo z uporabo svetovnih povpreˇcij ?B+ ???? = 1.01 ± 0.07 . (8.14) Zgornja vrednost tega razmerja ni daleˇc od napovedi (8.8): pri razpadni konstanti fb=200 MeV [19] [18] bi za razmerje priˇcakovali vrednost 1.05. Zˇivljenjski ˇcas ?bs bomo v bodoˇce lahko uporabili pri morebitnih meritvah razpadnih ˇsirin mezonov B0. Posebej zanimiv je vzorec razpadov s konˇcnim stanjem ?L. Zaradi zadovoljive statistiˇcne moˇci ga bo mogoˇce uporabiti pri meritvah frekvence oscilacij v sistemu mezonov B0. Zabeleˇzeno ˇstevilo razpadov B0 zaenkrat omogoˇca zgolj doloˇcanje spodnje meje frekvence oscilacij. Upati je, da se bo ob uporabi omenjenega vzorca dalo dodatno omejiti obmoˇcje moˇznih frekvenc, ter s tem doseˇci globji vpogled v fiziko mezonov B. Part VI Appendices 159 Chapter 9 9.1 Appendix A • D?? production in B semileptonic decays The relative fraction of orbitally excited charm mesons in the B meson semileptonic decays can be calculated from the following measured exclusive branching ratios [5][36]: Bb0 B+ a = Br(B0d › D-L+?L) = (1.9 ± 0.5) × 10-2 = Br(B0d › D?-L+?L) = (4.4 ± 0.4) × 10 2 = Br(B+ › D L+?L) = (1.6 ± 0.7) × 10 2 = Br(B+ › D? L+?L) = (4.7 ± 1.1) × 10 2 . (9.1) The decays into the D?? represent the remainder of the inclusive semileptonic branching fractions [36] sl sl = Br(B0d › L+?LX) = (9.5 ± 1.6) × 10 2 = —— Br(B0d › L+?LX) = (9.9 ± 1.7) × 10 2 . ?bd (9.2) Hence, one can expect the fraction of the D?? in the B0d and 161 162 CHAPTER 9. B± meson decays to be Ba i T)b f?? =1----- sl 0 = 0.34 ±0.13 Ba + B b f?? =1-----+ sl + = 0.36 ±0.17, (9.3) respectively. The average fraction for the non-strange B mesons f ?? = 0.35 ±0.10 (9.4) can be used as an estimate of the D?? production in the B0 decays. • Ds production in the D?? decays Each orbitally excited D meson appears as four states with the spin and parity quantum numbers JP = 0+, 1 + , 1 + , 2+. Due to the isospin conservation the D?? cannot decay into the Ds meson and a single pion. Hence the kinematicaly favoured decay which produces the strange D meson in the final state is D?? › Ds??, with isospin of the pions system equal to 0. In the case of non-strange D meson in the final state the isospin allowed decay is D?? › DnsK. In order to satisfy the Bose-Einstein statistics, the angular momentum of the ?? system must be even, if the isospin equals to 0. For the D?? › Ds?? decay the following conservation laws must be obeyed: PD?? = + 1 = (-1)Ds (-1)L?? (-1)L Jd? s? = (6)ds + (^)?? + ^ . (9.5) The subscripts indicate particles to which the corresponding quantum number belongs to, and L is the relative angular momentum between the Ds meson and the ?? system. It is easy to conclude from the above equations that the only decay expected to 9.1. APPENDIX A 163 contribute to the strange D meson in the final state is the decay of the 1+ state proceeding through an L = 1 angular momentum wave. On a similar ground one can conclude that the decays of the D?? into the non-strange D mesons are possible through the L = 0 wave for the 0+ and 1+ state, while the dominant decay of the 2+ state proceeds through the L = 2 wave. Up to now the only experimentally observed 1+ D?? state is reconstructed through the decays into the non-strange D meson [43][34]. It is therefore reasonable to assume that this decay is the dominant one. The upper limit for the Ds production from the orbitally excited strange D mesons can then be obtained by assuming the other 1+ state to decay dominantly into the Ds?? final state. By assigning the number of different energy levels for the individual JP state simply by spin counting, i.e. as 2J + 1, the fraction of the Ds in the D?? decays is NDs N(J =1) 1 . jD?? N(J = 0) + 2N(J = 1) + N(J = 2) 4 (9 6) • Ds production in the D? n s ? decays The isospin conservation allows the decays of the orbitally excited non-strange D mesons into the D( s ?)K and D(n ? s )? final state. From the parity and angular momentum consideration one can draw the following conclusions: 0+ state can decay into the strange and non-strange D meson ground state through the L = 0 wave. 1+ state exhibit a similar property apart from the fact that the final state charm mesons are now D?. However the mass and 164 CHAPTER 9. the width of the experimentally observed D** state with JP = 1+ is 2423 MeV/c2 and 18 MeV respectively [5]. Since the sum of the masses of the D* and K meson is around 2600 MeV/c2, we conclude that the production of the strange D mesons from this state is negligible. The assumption was thus made that only one of the 1+ states contributes to the Ds in the final state. 2+ state decay into the DsK and Dns? is possible through the L = 2 wave. The measured mass and decay width of the 2+ non-strange D meson state is approximately 2460 MeV/c2 and 20 MeV [5]. The sum of the Ds and kaon masses is close to the value of the quoted D* * mass and hence the decay of this state into the strange D meson was neglected. Beside the above considerations one should note that the decays into the strange D mesons are suppressed also due to the additional ss pair required. The suppression factor can be inferred from the measured probabilities for the b^ Bs and b^ Bu,d fragmentation (see Table 4.1): Prob(ss) Prob(ss) Ps rs = -— _ = ---------— = —— w 0.25 . (9.7) Prob(uu) Prob(dd) Pu,d Finally we obtain for the fraction of the Ds mesons produced in the D** decays NDs N(J = 0) rs N D** N(J = 0) + 2N(J = 1) + N(J = 2) 2 + rs ns + N(J = 1) rs - —-—-----—-----ZT77T-----7T-----T77T-----rr--------- « (3 - 4) X 10 2 , N(J = 0) + 2N(J = 1) + N(J = 2) 2 + rs (9 8) where the first term arises due to the contribution of the 0+ state decays and the second due to the 1+ state decays into the strange charmed mesons. 9.2. APPENDIX B 165 9.2 Appendix B • Convolution of the exponential and the Gaussian distribution 1 ,? (t-u) 2 P(t,?t,?) = ?---- e 2<2 t e - u du = 2??t? J0 1 2 -ir t ____ 2 ? I_____ = ? e2?2 ? 1 + Erf(-?-----------)\ 2 ? 2t 2? The Erf(x) is the error function given by 2 x Erf(x) = ? e z dz . (9.10) ? 0 • Normalisation of the convolution Since the actual interval of the proper decay times used for the lifetime measurement is not infinite, the convolution P(t,?t,?) should be normalised in the finite interval between -|t1 | and t2: 1 ft2 — P(t,?t,?) dt = 1. (9.11) A -|t1| The calculation of the normalisation factor A yields: 1 = 1\Erf(? ?-) +Erf(? ?-)l + A 2 2 | t2t 1 ? 2 r |t |r rr If I n 2 ? 2? ? 2?t - e ? 1 - Erf(—^--?—) . (9.12) ? 2? 2?t In the limit |t1|,t2 › ? the normalisation factor approaches unity. The effect of the normalisation is non-negligible only at large lifetime ?. For example, the normalisation factor of the sum 166 CHAPTER 9. G(t, ?t) + E (t, ?t,?) ? G(t, ?t) for the interval of the decay times used in the described analysis and with the typical achieved resolution ?t differs from unity by less than 2% for the lifetime range up to 2.5 ps. 9.3 Appendix C In figure 9.1 the decay chain D± › ??± ; ? › K+K is sketched in the rest frame of the ? vector meson. Since the D± and the K± are pseudo-scalar mesons, both successive decays proceed through the L = 1 wave. Using the Dirac notation, the state with the total angular momentum J and the z-component of the angular momentum M is represented by |J M >. The decomposition of the D± state into the ? meson spin state |1 M? > and the relative orbital angular momentum between the ? and the ? meson |L M_g > is given by |00 > = ../ —|11 > |1 - 1 >-../—|10 > |10 >+../ —|1 - 1 > |11 > (9.13) The first ket vectors in the above products describe the ? spin state and the second the relative orbital angular momentum between the ? and the ?. Factors 1/3 are the appropriate Clebsch-Gordan coefficients. Since kaons are spinless, the ? meson’s spin wave function in its rest frame equals to the angular part of the K+K system wave function. The angular distribution may be described by the two sets of angles Q,? and Qk, corresponding to the pion and the kaon direction with respect to some arbitrary z axis in the rest frame of the ?. The angular part of the wave function for the final state is 9.3. APPENDIX C 167 K K + a \ k Figure 9.1: The D± decay into the › K+K-, viewed in the rest frame of the 4> meson. then written as LfJ ? Y11(ClK)Y1-1(Cln) - Y10(ClK)Y10(Cln) + Y1-1(ClK)Y11(Cln n)+ (9.14) in accordance with (9.13) and using the spherical harmonics Yf (Q). By squaring the dependence (9.14) one obtains the angular distribution of the decay products in the form dN dQ.KdQ. — ? TT ) 2 (9.15) By choosing the z axis in the direction of the pion 9k becomes just the angle a of figure 9.1 and the distribution is dN dQ ? cos2 a (9.16) 168 CHAPTER 9. Bibliography [1] S.L. Glashow, Partial Symmetries of Weak Interactions, Nuc. Phys. 22 (1961) 579. S. Weinberg, A Model of Leptons, Phys. Rev. Lett. 19 (1967) 1264. A. 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