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Measurement of Vub using b semileptonic decay Lu, Jiansen 2002

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MEASUREMENT OF |Vub| USING b SEMILEPTONIC DECAY By Jiansen Lu B.Sc , University of Science and Technology of China,1986 M.Sc , University of Saskatchewan, 1997 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A December, 2001 © Jiansen Lu, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia,, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her' representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Pty$icS W A*W»fr The University of British Columbia Vancouver, Canada Date (ffgQ ^ / ^ ' / DE-6 (2/88) Abstract The magnitude of the C K M matrix element | V u b | is determined by measuring the inclusive charmless semileptonic branching fraction of beauty hadrons at O P A L based on b —> X u ^ event topology and kinematics. This analysis uses O P A L data collected between 1991 and 1995, which correspond to about four mil l ion hadronic Z decays. Bv(b -»• Xu£u) is measured to be (1.63 ± 0.53 ±g;| |) x I O - 3 . The first uncertainty is the statistical error and the second is the systematic error. From this analysis, | V U b | is determined to be: | V u b | = (4.00 ± 0.65 (stat) ±lf6 (sys) ± 0.19 (HQE)) x 10" 3 . The last error represents the theoretical uncertainties related to the extraction of | V U b | from Br(6 —> Xu£u) using the Heavy Quark Expansion. n Table of Contents Abstract ii List of Tables vii List of Figures viii Acknowledgements xi 1 I N T R O D U C T I O N 1 1.1 The C K M matrix in the Standard Model 1 1.2 Measurements of | V u b | 8 1.3 Outline 10 2 b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 12 2.1 Introduction to b physics 12 2.2 b hadron production 13 2.3 b hadron semileptonic decay 19 2.3.1 b to c semileptonic decay 20 2.3.1.1 Exclusive b to c semileptonic decay 20 2.3.1.2 Inclusive b to c semileptonic decay 25 2.3.2 b to u semileptonic decay 28 2.3.2.1 Exclusive b to u semileptonic decay 28 2.3.2.2 Inclusive b to u semileptonic decay 30 2.4 The b ->• XJu hybrid model 31 2.4.1 b to u inclusive model 32 2.4.1.1 A C C M M Model 34 i i i 2.4.1.2 Q C D universal structure function 34 2.4.1.3 Parton Model 35 2.4.1.4 Decay kinematics 35 2.4.1.5 Implementation of inclusive models 36 2.4.2 I S G W 2 exclusive model 37 2.5 Signal and background simulation using J E T S E T 40 2.5.1 J E T S E T 40 2.5.2 Signal and background simulations 42 3 T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 43 3.1 The L E P collider 43 3.2 The O P A L detector 46 3.2.1 Beampipe 48 3.2.2 Subdetectors for particle tracking 49 3.2.2.1 The silicon microvertex detector 49 3.2.2.2 The central vertex chamber 51 3.2.2.3 The jet chamber 52 3.2.2.4 The z chambers 52 3.2.2.5 Magnet 53 3.2.2.6 The muon chambers 53 3.2.2.7 Performance of the tracking system 54 3.2.3 Subdetectors for calorimetry 56 3.2.3.1 The electromagnetic calorimeter 57 3.2.3.2 Hadron calorimeter 60 3.2.4 Luminosi ty monitor • 61 3.2.5 Trigger of the O P A L detector 63 iv 3.2.6 Online R O P E 64 4 E V E N T P R E S E L E C T I O N 66 4.1 A n introduction to Art i f ic ia l Neural Networks 66 4.2 Mult i -hadron selection 71 4.3 b identification 73 4.3.1 Jet finding 73 4.3.2 Pr imary and secondary vertex reconstruction 73 4.3.3 b vertex flavour tagging 77 4.4 Lepton selection 78 4.5 B semileptonic decay selection 86 5 I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 8 8 5.1 b to u neural network 88 5.1.1 Hadronic invariant mass 91 5.1.2 The lepton energy in the b hadron rest frame 97 5.2 Discussion of b to u neural network outputs 99 6 R E S U L T S 109 6.1 The branching ratio of b to u semileptonic decay 109 6.2 Systematic errors and cross check 112 6.2.1 Systematic errors 114 6.2.2 Cross checks of the result 122 6.3 Measurement of | V u b | from the branching ratio of b to u semileptonic decayl23 7 C O N C L U S I O N 1 2 5 APPENDICES 1 2 8 v A M y Contribution to O P A L Collaboration 128 B Glossary 130 References 136 Bibliography 136 vi List of Tables 1.1 The basic particles of the Standard Model 2 2.1 A brief history of b physics 13 2.2 Some existing experiments to study b physics 14 2.3 b hadron production fraction 16 2.4 r th y(ps _ 1) predictions from different form factor models 30 2.5 Experimental fit results of pp and the spectator quark mass m s p in the A C C M M model 34 2.6 Partial widths for b to u semileptonic decays for B mesons in the ISGW2 model 37 2.7 Partial widths for b to u semileptonic decay for B s and B c in the ISGW2 model 38 6.1 Results of different neural network cuts I l l 6.2 Systematic errors for the branching fraction of b to u semileptonic decay. 115 6.3 Systematic errors for the branching fraction of b to u semileptonic decay from b and c hadron decay properties 118 6.4 Systematic errors for the branching fraction of b to u semileptonic decay from D decay multiplicity : 120 7.1 The branching fractions of b —> Xu£v measured from A L E P H , D E L P H I , L3 and this analysis 126 vii List of Figures 1.1 The quark q; weak decay to the quark qj by W boson 3 1.2 The unitarity triangle 6 1.3 Feynman diagrams for B° — B° mixing. . 6 2.1 Kinematics and lowest order Feynman Diagram in multihadron production 15 2.2 The hadron production from annihilation of e + and e~ 16 2.3 The lepton spectrum for b to c semileptonic decay 23 2.4 Lepton spectra from b —> c e ve and b —>• c —>• e . . .- 26 2.5 The uq invariant mass distributions 33 2.6 The electron energy distribution (l/rfree)(dr/dEe) vs electron energy for B ~ —> X u Q e P decay 39 3.1 The layout of the L E P collider and four L E P experiments 44 3.2 The layout of the C E R N injector system for the L E P ring 45 3.3 The O P A L detector 47 3.4 The O P A L beampipe structure 48 3.5 The structure of the O P A L silicon microvertex detector 50 3.6 A schematic diagram for five parameters describing a track 56 4.1 Input, hidden and output layers in a neural network 67 4.2 g(x) as a function of temperature 69 4.3 A 2 jet event from the O P A L event display 74 4.4 A 3 jet event from the O P A L event display 75 4.5 The beam spot, primary vertex and secondary vertex in b decay 76 v i i i 4.6 Distr ibut ion of the vertex tagging variable B for O P A L data and Monte Carlo simulated events 79 4.7 d E / d x from O P A L data for the electron, muon, pion, kaon and proton . 81 4.8 The d E / d x separation power from O P A L data for the electron, muon, pion, kaon and proton 82 4.9 The electron neural network output for O P A L data and Monte Carlo sim-ulated events 83 4.10 The photon conversion neural network output for O P A L data and Monte Carlo simulated events 85 4.11 The b hadron semileptonic decay neural network output distributions . . 87 5.1 Comparison between the signal and the background in the Monte Carlo simulation for the seven b to u neural network input variables 92 5.2 Comparison between O P A L data and Monte Carlo simulated events for the seven b to u neural network input variables 93 5.3 The correlations between different variables 94 5.4 The correlations between different variables (continued) 95 5.5 Fraction of true b decay vs track weight from Monte Carlo simulated events and comparison of the track weight between O P A L data and Monte Carlo simulated events 96 5.6 Fraction of true b decay vs cluster weight from Monte Carlo simulated events and comparison of the cluster weight between O P A L data and Monte Carlo simulated events 98 5.7 Comparison of the b hadron direction between O P A L data and Monte Carlo simulated events 99 ix 5.8 The difference between the reconstructed b hadron direction and the true b hadron direction in Monte Carlo simulated events 100 5.9 The b to u neural network output distributions 101 . 5.10 The figure of merit between the signal and the background vs epochs for the training and test samples and the signal purity and efficiency vs the b to u neural network output for the training sample 102 5.11 Comparison of neural network output distributions for the signal and the background between the training sample and test sample 103 5.12 The neural network output distributions for different signal compositions 104 5.13 The neural network output distributions for I S G W 2 and A C C M M models in the hybrid model 105 5.14 The neural network output distributions for B to D , D* and D** semilep-tonic decays 106 5.15 The hadronic invariant mass and the lepton energy in the b hadron rest frame from O P A L data, with no b to u neural network cut and with b to u neural network cut 0.9 applied 108 6.1 The neural network output distributions for O P A L data and Monte Carlo simulated events 112 6.2 The neural network output distributions for the data after subtracting the background from the Monte Carlo simulated events 113 x Acknowledgements I would like to thank my supervisor Dr . Janis M c K e n n a for her help and guidance throughout the work. Her encouragement and patience have been invaluable in the completion of this project. I have worked at C E R N for more than 14 months for the O P A L online data recon-struction. I had a very pleasant time to work with Gordon Long, Car la Sbarra, Ian Bailey and Brigi t te Vachon for the O P A L online data reconstruction. Thanks for their help and sharing the duty. Thanks to those O P A L physicists: Oliver Cooke, Pauline Gagnon, A l a i n Bellerive, Bob Kowalewski, Rob McPherson and Richard Hawkings for a lot of good discussions for this V u b analysis. Thanks to Christoph Schwick for his help with the b —» Xu£u hybrid model. I would like to thank my P h D committee for al l their help in my academic progress. The financial support from my supervisor's N S E R C grants and U B C Physics and Astronomy Department is gratefully acknowledged. I am grateful to my fellow graduate student Douglas Thiessen for sharing the computer lab teaching assistantship and sharing four years of student life. F ina l ly I must thank my wife Hu i X u . W i t h her encouragement, I continue to stay in my science and research career. I also want to thank my wife again for bringing my son A n d y L u to this world last June. x i Chapter 1 INTRODUCTION This thesis presents a measurement of | V u b | using b semileptonic decay. The data used in this analysis consist of four mil l ion hadronic Z decays collected with the O P A L detector at the L E P accelerator between 1991 and 1995. In this chapter an overview of the Cabibbo, Kobayashi and Maskawa ( C K M ) ma-t r ix [1] in the Standard Model of particle physics is presented. Measurements of | V u b | are discussed. A n outline of this thesis is presented. 1.1 The C K M matrix in the Standard Model The Standard Model is used to describe the fundamental particles of nature and their interactions. According to the Standard Model , matter is built of six quarks and six leptons and their anti-particles, shown in Table 1.1, which interact v ia the exchange of gauge bosons. The Standard Model contains a neutral Higgs boson that is introduced for the SU(2)<g>U(l) symmetry breaking. There is no experimental evidence for the neutral Higgs boson yet. There are four interactions, namely strong, weak, electromagnetic and gravitation, so far as we know. Each interaction is mediated by the exchange of bosons. The strong 1 Chapter 1. I N T R O D U C T I O N 2 interaction is mediated by a gluon. The weak interaction is mediated by and Z bosons. The electromagnetic interaction is mediated by a photon and the gravitational interaction is supposedly mediated by a graviton. The Standard Model unifies the electroweak interaction, electromagnetic interaction and strong interaction based on group theory SU(3)cS>SU(2)<g>U(l). Leptons H V e J ( \ K » t \T) Quarks ( N u ( \ C V s ; Gauge Bosons 7, W± and Z, gluons Table 1.1: The basic particles of the Standard Model . 7 is a photon. The SU(2)<g>U(l) in the Standard Model is the gauge group of the electroweak interac-tion. Quark weak eigenstates are different from their mass eigenstates for their respective interactions. The weak eigenstates and mass eigenstates are connected by the C K M ma-trix. A quark q; decays weakly to a quark qj and a W boson, as shown in Figure 1.1. The decay rate is proportional to the C K M matrix element | V j j | 2 . The interaction Lagrangian for flavour-changing quark transitions coupling to W bosons can be written as: 4nt = ^ §(J"w+ + J^w;)* ' (1.1) where g is a gauge coupling constant. is an operator which annihilates a W + or creates a W ~ and vice verse for W ~ . The charged current is equal to: ^ ^ E ^ a - ^ v u d j , (1.2) i j where the i and j are indices running from 1 to 3. u, is a field operator which creates u, c and t quarks or annihilates their corresponding anti-particles as the index i runs Chapter 1. I N T R O D U C T I O N 3 W 9 i V i j 1j Figure 1.1: The quark q; weak decay to the quark qj by W boson. Vij is a C K M matrix element. from 1 to 3. dj is a field operator which annihilates d, s and b quarks or creates their corresponding anti-particles as the index j runs from 1 to 3. The Vjj is the C K M matrix element. The C K M matrix is defined as following: s vb'y V u d V u s V u b V c d V c s V c b s , (1.3) V vtd vts vtb ) [ b , where d', s' and b' are the weak eigenstates, d, s and b are the mass eigenstates. It is the C K M matr ix which specifies a rotation of basis from mass to weak eigenstates. A common approximate parameterization of the C K M matr ix is the Wolfenstein form [2]: 1 l - A 2 / 2 A A A 3 ( p - i 7 7 ) ^ - A l - A 2 / 2 A A 2 v A A 3 ( 1 -p-iri) - A A 2 1 j where A is the Cabibbo mixing angle and has a value around 0.21. A , p and rj are of order unity. The 90% confidence limits on magnitudes from recent measurements [2] and the Chapter 1. I N T R O D U C T I O N 4 requirement that the C K M matrix be unitary are: ' 0.9742 - 0.9757 0.219 - 0.226 0.002 - 0.005 N 0.219 - 0.225 0 .9734 - 0.9749 0 . 0 3 7 - 0.043 ^ 0.004 - 0.014 0.035 - 0.043 0.9990 - 0.9993 y Measurements of the C K M matr ix elements are described briefly below: | V u d | : | V u d | is precisely determined to be 0.9740 ± 0.0010 from superallowed 0 + —> 0 + nuclear (3 decay compared to the muon decay rate [2]. The uncertainties are mainly from nuclear structure corrections and isospin symmetry breaking corrections. | V u d | can also be obtained from free neutron decay and charged pion decay, but with larger uncertainties. | V U S | : | V U S | is determined to be 0.2196 ± 0.0023 from K° and K + semileptonic decays [3]. | V u b | : | V u b | is measured to be (3.3 ± 0.8) x 10~ 3 from B —>• TT£U and B —> ptv from C L E O measurements [4]. | V u b | can also be measured from inclusive b to u semilep-tonic decays. The average | V u b | from inclusive measurements of A L E P H , D E L P H I and L 3 , the other three L E P experiments, is (4.13 l^? ! ) x 10~ 3 [5]- This current analysis also measures | V u b | using inclusive b to u semileptonic decay. The result wi l l be incorporated with other three L E P experiments and be used to produce a new L E P | V u b | value. | V c d |: | V C d | is measured to be 0.224 ± 0.016 from the differential cross section of z ^ N —>• / i ~ c X , i.e. the charm production rate in neutrino interactions with valence quarks in nucleus [2]. There is a large theoretical uncertainty in predicting this differential cross section due to the sea quarks, especially due to the s quark contributions. | V C S | : | V C S | is determined to be 1.04 ± 0.16 from D -> K£+v [2]. The recent measurement value of | V C S | from W * ->• hadrons from the D E L P H I experiment is 0.94±°0il±0.l3 Chapter 1. I N T R O D U C T I O N 5 [6], where the unitarity of the C K M matrix and three generations of quarks are assumed. | V C S | can also be extracted from f M N —> / / ~ c X , similar to the extrac-tion of | V c d | . Bu t the uncertainty is large due to lack of knowledge regarding the contribution of ss pair to the parton sea. | V c b | : |V C b | can be measured from B —>• Div and B —>• D*£u decays. The average value of | V c b | is 0.0395 ± 0.0017 [2]. | V t d | and |V t s | : | V t d | and | V t s | can be measured from B ° B d ° and B S B S mixing. Using A M B d = (0.472 ± 0.018) p s _ 1 (using convention h = c = 1) from B°B° mixing, where A M B d is the mass difference between the high mass and low mass in B d system, | V t b V t d | is determined to be 0.0084 ± 0.0018 [2]. | V t s | can be obtained by using the unitarity of the C K M matrix: V u s V : b + V c s V c * b + V t s V * b = 0. (1.4) As the first term in Equation 1.4 is tiny, | V t s | is equal to | V c s V * b / V * b | and is approximately equal to | V c b | . | V t b | : | V t b | is measured from t to b semileptonic decay from C D F and DO experiments and the result is [2]: IV I2 1 t b | =0.99 ± 0 . 2 9 . (1.5) | V t d | 2 + | V t s | 2 + | V t b | 2 The requirement of unitarity of the C K M matrix produces: V u d V ^ b + V c d V c * b + V t d V t * b = 0. (1.6) Since V u d and V t b are real numbers and approximately equal to 1 and V c d = —A < 0, the above equation can be written as: V ub + y t d = 1 f l 7) vcdvc*b| vcdvcb. Chapter 1. I N T R O D U C T I O N 6 which can be depicted as the unitarity triangle in a complex plane in Figure 1.2. If a deviation from unitarity could be proven, the physics beyond the Standard Model wi l l be discovered. The angles a , j3 and 7 in the unitarity triangle are related to the phase in the C K M matrix. The observed Charge Pari ty (CP) violation is solely related to a nonzero value of this phase, which can be measured in B° — B° mixing. The Feynman diagrams for B° — B° mixing are shown in Figure 1.3. (P. ii) As C P is not good symmetry of nature, the mass eigenstates in the weak interaction are different from flavour eigenstates. The flavour eigenstate for B° is bd and for B° is bd. Chapter 1. I N T R O D U C T I O N 7 The low mass B i and high mass B 2 eigenstates can be expressed as linear combinations of flavour eigenstates, B° and B°: |B 1 ) = p|B°) + q|B 0 ) , |B2> = p |B°>-q |B°>, where p and q are complex coefficients and obey the normalization condition: | p | 2 + N 2 = l -(1.8) (1.9) (1.10) The coefficients p and q are governed by a time-dependent Schrodinger equation: ,d I P ldT M - f M 1 2 - ^ ^ " ^ V Mh + -2 M - f PV q j ( i . i i ) The time evolution of |B°(t)> and |B°(t)> in terms of basis states |B°) and |B°) can be written as [7]: (1.12) |B°(t)) = g + ( t ) |B 0 ) + ^g_( t ) |B°) , |B°(t)) = Pg_(t)|B 0) + g + ( t ) |B°) , where A m i g +(t) = e - l M t e - c o s ( - ^ t ) , g_(t) = e - i - e ^ s i n ( ^ t ) , (1.13) (1.14) (1.15) where Ame is the mass difference between the high mass |B 2 ) and the low mass |Bi) states. Defining the final states (CP eigenstates) from B° and B°, such as 7 r + 7 r ~ and 3/ipKs as fcp, the decay widths for B° and B° decay to fcp can be written as [7]: » o / + \ , f_ x _ i M2n-rtA + l A l 2 1 - | A | 2 r(B°(t)^fCP) = | A | V l t ( : r ( B ° ( t ) ^ f C P ) = | A | V i t ( 2 . - r t , l + | A |2 1 - I A I 2 cos(M 1 2t) - Im(A) sin(M 1 2 t)), (1.16) cos(Mi 2t) - Im( A ) sin(Mi 2t)), (1.17) Chapter 1. I N T R O D U C T I O N 8 where * = <"8> and A - ( f C p | ^ | B ° ) , (1.19) A = ( f C p | H | B ° ) , (1.20) where % is the Hamil tonian. The time dependent CP asymmetry a(t) can be defined as: For B° —y 7r+7r~ and B° —> J /^Kg decays, Im(A) is equal to sin(2a) and sin(2/3) respec-tively. The angles a and B in the unitarity triangle can be measured from CP asymmetry in the B system and wi l l provide additional information to the unitarity triangle. A n y inconsistency in this unitarity triangle may be a hint or a sign of new physics. 1.2 Measurements of | V u b | V u b , an element of the C K M matrix, describes decays of the b to u quark. Its magnitude, | V u b | , can be calculated by measuring the inclusive b —> u semileptonic decay rate, which refers to the total decay rate with all possible final-state hadrons in b —>• u semileptonic decay, ignoring the detailed breakdown among the individual modes. Given that the branching fraction of inclusive b —> u semileptonic decay is of order 10~ 3 , a large number of b hadrons is required to measure | V u b | . The dominant background to b —> X u £ u comes from b —>• X c £ v decays because the branching ratio of b —> X c£i^ is more than 50 times greater than that of b —> Xud.v. Here the lepton £ refers to either an electron or a muon, and b denotes al l weakly decaying b hadrons^ 1). X u and X c represent hadronic 1 Charged conjugate states are implied if not stated otherwise. Chapter 1. I N T R O D U C T I O N 9 states resulting from a b quark semileptonic decay to a u or c quark respectively. The determination of | V u b | depends on the b to u and b to c semileptonic decay models. The lepton endpoint energy E m a x for b —> Xu£u and b —>• Xc£v can be calculated as: E m « = m ° - < + m ' , (1.22) where m B and are the mass of B meson and the mass of the lepton £ respectively. m x is the mass of X u or X c . The E m a x can be calculated as 2.64 G e V and 2.31 G e V for B —>• ir£v and B —> Div respectively. The inclusive method developed by A R G U S [8] and C L E O [9] is to extract | V u b | / | V c b | from the excess of events in the 2.31 to 2.64 G e V / c region of the lepton momentum spectrum in the B meson rest frame, where the b —>• Xc£v contributions vanish. This technique uses only a small fraction of the lepton phase space and so has considerable model dependence in extrapolating to the entire lepton spectrum in the B rest frame. In addition, since the L E P experiments can not precisely determine the B meson rest frame, this method is not appropriate for the L E P experiments. Instead, at L E P , | V u b | or | V u b | / | V c b | is extracted using a larger portion of the lepton spectrum as well as other kinematic variables. The inclusive measurement of the branching fraction of the b —> Xu£u decay has also been performed at L E P by A L E P H [10], D E L P H I [11] and L3 [12]. The theoretical uncertainty for the value of | V u b | extracted from a measurement of the inclusive b —> Xu£v branching fraction differs from that extracted from measurements of exclusive b —> u semileptonic decay rates. A recent theoretical study concludes that there is a 5% theoretical uncertainty on | V u b | values derived from b —> Xu£v inclusive measure-ments [13], using the Heavy Quark Expansion. There is a 15% theoretical uncertainty associated with | V u b | values extracted from measurements of the exclusive branching fractions B —> -K£U or B —> p£v [14], interpreted within the framework of the Heavy Quark Effective Theory ( H Q E T ) . Chapter 1. I N T R O D U C T I O N 10 The current analysis uses O P A L 1991 to 1995 data, collected near the Z resonance, comprising about four mil l ion hadronic Z decays. Monte Carlo simulated events were gen-erated using the J E T S E T 7.4 [15] generator, with parameters described in [16]. Approxi -mately five mil l ion hadronic Z —> bb decays were generated to study the b —> X c £ u decay and the b —> c —> £ cascade decay. Six mil l ion hadronic Z —>• qq (where q can be u, d, s, c and b) decays were generated to study the leptons from primary charm quarks and light quarks. Two hundred thousand events from a b —> X u £ u hybrid model [17] were produced to simulate the b —> u semileptonic decay. The b —> X u £ u hybrid model wi l l be described in detail in Section 2.4. 1.3 Outline In this thesis, the determination of | V u b | using the inclusive b —> X u £ u decay rate from the O P A L data taken at the center of mass energies near the Z resonance is described. The b semileptonic decay, the O P A L detector, the event preselection, b —» X u £ u decay models and the neural network used to separate b —> X n £ u from the background wi l l be discussed in detail in the following chapters. In Chapter 2 the theory behind b to u semileptonic decay models is discussed. A n overview of the current status of | V u b | measurements is presented. A hybrid model for the b to u semileptonic decay is discussed. In Chapter 3 descriptions of the L E P accelerator, the O P A L detector and the online data reconstruction are presented. O P A L subdetectors are discussed in detail. In Chapter 4 an event preselection is presented. Mul t ihadron selection, b tag, lepton selection and b semileptonic decay selection are discussed in detail. The concept of an artificial neural network is reviewed as neural networks are used several times in this analysis. Chapter 1. I N T R O D U C T I O N 11 In Chapter 5 an identification of the b to u semileptonic decay is presented, b —» Xu£u neural network and its input variables are discussed in detail. The b —» Xu£u neural network output distributions for different signal and background compositions are also discussed. In Chapter 6 the x2 fit method which is used to extract the branching fraction Br(b —> Xu£u ) from the b —> Xu£u neural network is presented. The systematic error analysis and cross checks for Br(6 —» Xu£u ) are presented. The extraction of |V u b | from Br(fr —y Xu£u ) is also discussed. In Chapter 7 the final results for the branching fraction of b —> Xu£u decay and |V u b| are presented. The average of |V u b | values from the four L E P experiments is also presented. In Appendices, my contribution to O P A L collaboration is presented and a glossary of terminology in this thesis is also presented. Chapter 2 b SEMILEPTONIC DECAY AND HYBRID MODEL In this chapter an overview of b quark and b hadron physics is presented. The theory behind b —>• X c £ u and b —>• X u £ v semileptonic decays is presented. A b —>• X u £ v hybrid model is described. 2.1 Introduction to b physics Bound bb states have a rich spectrum and are collectively called T mesons. T(1S) was first observed in 1977 in the n+/J,~ spectrum from 400 G e V protons striking a nuclear target at the Fermi National Laboratory ( F N A L ) under the direction of Dr . Leon Led-erman [18]. One year later, the T(1S) and T(2S) were confirmed in e +e~ annihilation at the D O R I S R ing in Hamburg [19]. The C U S B and C L E O experiments at the Cornell Electron-positron Storage R ing ( C E S R ) at Cornell University also confirmed T(1S) and T(2S) in 1979 and discovered T(3S) and T(4S) in 1980 [20]. Later B mesons were also discovered at C E S R [21]. In 1987, the A R G U S experiment in D E S Y first observed the ev-idence of B°B° mixing [22]. In 1989, evidence for b to u semileptonic decay was observed 12 Chapter 2. b SEMILEPTONIC DECAY AND HYBRID MODEL 13 by the A R G U S and C L E O experiments [8, 9]. Almost at the same time, Isgur and Wise and Voloshin and Shifman developed the Heavy Quark Effective Theory ( H Q E T ) [23, 24] to describe heavy quark transitions. Using H Q E T , the calculation of the transition rate for b semileptonic decay can be simplified. In 1993, B s and A b were discovered by L E P experiments at C E R N [25]. The same year, C L E O found evidence of the penguin decay B —> K*7 [26]. In 1997, a series of B rare decays, including B —>• Kir, were discovered by C L E O at C E S R [27]. A brief summary of the history of b physics is shown in Table 2.1. 1977 Discovery o f T ( l s ) at F N A L 1983 Discovery of B meson at Cornell 1987 Discovery of B°B° mixing at D E S Y 1989 Discovery of B —>• Xu£u at Cornell and D E S Y 1989 Heavy Quark Effective Theory by Isgur, Wise, Voloshin and Shifman 1993 Discovery of B s and A b at C E R N 1993-1994 Discovery of radiative penguin decays B —>• K*7 and B —> Xsj 1997 Discovery of B —> Kir rare decay at Cornell Table 2.1: A brief history of b physics. 2.2 b hadron production The b hadron can be produced by e +e~ or pp collider or fixed target experiments. The three main approaches for b hadron production are: e +e~ —> T(4S) —> B B , e +e~ —> Z —>• bb and pp —> b b X . The competitive detectors and accelerators associated with the b hadron production are shown in Table 2.2. The current analysis uses the O P A L data of Z decay to multihadrons. O P A L is one Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 14 bb production Detectors Accelerator N(bb) 106 e+e- -> T(4S) -> B B C L E O II C E S R 5.1 e+e- T(4S) ->• B B B A B A R P E P II >10 e+e" T(4S) -> B B B E L L E K E K B >10 e+e" ->• Z -> bb A L E P H , DELPHI , OPAL, L3 L E P 0.9 e+e~ -> Z ->• bb SLD SLC 0.08 pp ->• bbX C D F , DO Tevatron 600 Table 2.2: Some existing experiments to study b physics of the four experiments in the Large Electron Positron (LEP) collider at C E R N . From 1990 to 1995, the center of mass energy was around the Z resonance (91.1 GeV) and these data were used in this analysis. After 1995 the center of mass energy was increased to produce W pairs for studying triple gauge boson coupling. The primary quark pair or the lepton pair can be produced from Z decay. The decay of Z into quark pairs results in multihadronic events. The lowest order Feynman diagrams for Z decay to multihadrons are shown in Figure 2.1. The decay rate of Z to fermion pairs, rz_>. ff from the lowest order, using the Born approximation is: with u p — 8 M W sin 2 0 W ' c v = I 3 — 2 sin 2 0wQf > C A = I 3 , Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 15 Figure 2.1: Kinematics and lowest order Feynman Diagrams in the multihadron produc-tion in Z decay, f and f represent fermion pair. where M z is the Z mass, M w is the W mass. 0 W is the Weinberg angle describing the relative strength of the electromagnetic to the weak coupling. Qf is the electric charge of the fermion. I 3 is the third component of the weak isospin of the fermion. G F is the Fermi coupling constant, cy and C A are vector and axial-vector coupling constants. For M z = 91 GeV and sin 2 #w = 0.23, T z - ^ c c and rz_>bb can be calculated as: T(Z cc) « 280 MeV, Br(Z -> cc) « 12%, (2.2) T(Z -> bb) « 360 MeV, Br(Z -> bb) « 15%. (2.3) The multihadronic decay accounts for 69.9% of Z decay. The relative ground state b hadron production fractions from the Z resonance are shown in Table 2.3 [2]. The hadron production in Z decay can be described in the follow-ing four processes, which are shown in Figure 2.2. At first, primary quarks are produced from Z decay by electron and positron annihilation. This process can be precisely calcu-lated by the electroweak theory. The second process is that the primary quarks radiate Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 16 B + 38.9 ± 1.3 % B° 38.9 ± 1.3 % Bs° 10.7 ± 1.4 % Ab 11.6 ± 2.0 % B c + negligible Table 2.3: b hadron production fraction. (1) Electroweak ( 3 ) Non-pertubative QCD (Hadronisation and fragmentation) (4) Decays (2) Perturbative QCD Figure 2.2: The hadron production from annihilation of e + and e [28]. Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 1 7 gluons and gluons can radiate gluons or a quark anti-quark pair. This process is repeated until the parton energy is low relative to the Q C D scale parameter. This process can be calculated using perturbation theory. When the parton energy is low relative to the Q C D scale parameter, the strong coupling constant is large and hadrons are formed by partons. At this point, perturbation theory can not be applied anymore. Thus phenomenological models have to be used to describe this process, called fragmentation. The last step is that unstable hadrons decay to stable particles according to their decay branching fractions. The fragmentation functions can be deduced by analyzing Q —> (Qq) + q. The scaled energy transferred to the hadron Qq can be defined as: z = ( T ^ p ^ ( E + P| |)Q where ( E + P||)Qq is the hadron Qq energy plus its longitudinal momentum relative to initial quark direction. ( E + P||)Q is the primary quark Q energy plus its momentum, z is Lorentz invariant. Several models to describe the z distribution in the fragmentation process are discussed below: • The Peterson fragmentation function [29]: The Peterson function is mainly used in modeling heavy quark fragmentation. The heavy quark Q with momentum p evolves to a hadron Qq with momentum zp and a light quark q with momentum (1 - z)p. The transition amplitude is proportional to the inverse of the energy transfer A E - 1 , where A E = y/voil + p 2 - y/wtn + (zp)2 - ^/m 2 + ((1 - z)p) 2. (2.5) Assuming that the heavy quark mass is small compared to the momentum p, A E can be simplified as: 1 m 2 . . Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 18 Using where the factor \ arises from longitudinal phase space, the Peterson function can be obtained: v 1—z z / m 2 where N is the normalization factor, €Q = mQ is the heavy quark mass and m q Q is the light quark mass. • The fragmentation model of Collins and Spiller [30]: The heavy quark fragmenta-tion function and heavy meson structure function should be the same when z —> 1. For z —> 1, the heavy meson structure function goes as 1 - z, whereas the Peterson function goes as (1 — z) 2 . This difference becomes important only for very high quark mass. Collins and Spiller suggest another fragmentation function, where the fragmentation function becomes 1 - z as z —> 1: f(z) = N ( ± ^ + ^ 6 ) ( 1 + z 2 ) ( l - - L - - - ) - 2 , (2.9) z 1 — z 1 — z z where N is the normalization factor, e is a parameter which is proportional to the inverse mass of the heavy quark. • The Kartvelishvili et al. fragmentation model [31]: Kartvelishvili, Likhoded and Petrov discussed another fragmentation function for charmed mesons: f(z) = N z a ( l - z), (2.10) where N is the normalization factor, a is a free parameter. • The Lund symmetric fragmentation model [32]: The Lund symmetric model is mainly used for light quark fragmentation and can be written as: f ( z ) = N i i _ ^ _ e ^ ^ (2.H) Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 19 where a and b are free parameters. M H is the mass of the hadron. p t is the hadron transverse momentum with respect to the string direction in the Lund symmetric fragmentation model. The z value can not be directly measured from experiments. Instead an accessible variable, the scaled energy of the hadron, x E = E h a d r o n / E b e a m , is used. Here Ehadron is the measured energy of the hadron and E b e a m is the incident beam energy. 2.3 b hadron semileptonic decay Semileptonic decays of B mesons have been extensively studied experimentally and theoretically. The study of b hadron semileptonic decay can help us understand the weak interaction and strong interaction in the Standard Model and is used to measure the fundamental parameters in the Standard Model. The b semileptonic decay width can be calculated from the electroweak theory and QCD. There are large theoretical uncertainties for the non-perturbative Q C D effects. Heavy quark effective theory (HQET) [23, 24] is introduced to analyze the meson containing a heavy quark and a light quark to reduce the theoretical uncertainties. In a heavy quark and a light quark system, the momenta of the light quark and the heavy quark have a scale around A Q C D (0.2 - 0.3 GeV), a typical strong interaction scale. The associated velocity transfer from the light quark to the heavy quark is approximately AQco/mQ. When the heavy quark mass is far greater than the QCD scale, A Q C D (0.2 - 0.3 GeV), the heavy quark behaves as a stationary source of a colour field. Spin decouples from the dynamics as the colour magnetic moment is inversely proportional to the heavy quark mass. In HQET, the decay properties of the hadron with a heavy quark are analyzed in terms of an expansion in E/mQ, Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 20 where E is the kinematic energy of the heavy quark. In contrast to calculations from hadron models, the H Q E T expansion is derived from the fundamental theory of QCD. In b hadron semileptonic decay, the b quark decays to a c quark or u quark via emission of a virtual W - boson, which subsequently decays to a lepton plus an anti-neutrino, b to c and b to u semileptonic decays are discussed in detail below. 2.3.1 b to c semileptonic decay The decay rate of b to c semileptonic decay depends upon the C K M matrix element |VCb|- |V Cb| can be derived from exclusive and inclusive b to c semileptonic decay rates. 2.3.1.1 Exclusive b to c semileptonic decay The b to c semileptonic decay is dominated by a few resonant states, namely the ground state l ^ o (D), the first excited state l 3 S i (D*) and the higher states 13P2, l 3 P i , 1 3 P 0 , l x P i , 2 % and 2 3 Si (collectively referred as D**). The matrix element for B semileptonic decay to a charm hadron can be written as [33]: M = -i^VchC,W, (2.12) where Gp is Fermi coupling constant, is the leptonic current: £ / i = u<7/i(i - 75)v„, (2.13) and W1 is the hadronic current. u> and YV are field operators for the lepton and neutrino. B —> Div decay is a pseudoscalar meson to a pseudoscalar meson decay Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 21 and the hadronic current 'Wx can be written as follows, based on two form factors: W = <D(k)|J"|B(p)> = F ^ q 2 ) ^ + k)" - M | ~ M V ) + F 0 ( q 2 ) M 2 B " 2 M 2 D q ^ , (2.14) where q 2 is the four momentum transfer between B meson and D meson, p and k are the four momenta of the B and D mesons. As cfC^ = 0 when the lepton mass is zero, the hadronic current % ) l can be simplified as M 2 — M 2 W = F 1(q 2)((p + k)" - 2 M V ) . (2.15) q The differential decay rate for B —> D£u can be deduced as: dr G2|vcb|2k3 |F!(q 2 ) | 2 . (2.16) dq 2 24TT3 B —> D*£u decay is a pseudoscalar meson to a vector meson decay and the hadronic current TC1 can be calculated from four form factors [33], A(q 2 ) , Ai (q 2 ) , A 2 (q 2 ) and V(q 2 ) : ft" = ( D * | V - A " | B ) , (2.17) with W £ , k ) r | B , p ) ) = « ! ^ , (2,8) -+- mrj* <D- ( e,k)|A'-|B(p)) = i ( m B + m D . ) A l ( ^ ) e - ^ i A r f ( e ' f P + ^ + ' M ^ D . ( C • p f r me + q (2.19) where e is the D* polarization vector, q 2 is the four momentum transfer between the B meson and D* meson, p B and pp.* are the four momenta for B and D* mesons. Another form factor A 0 is defined, which will be used later, in terms of A + A 3 where A g = (m B + m D . ) A i - (m B - m D . ) A 2 ^ 2 K I D * Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 22 In the heavy quark limit, the form factors in B to D and B to D* semileptonic decay are related and can be expressed by the Isgur-Wise function [34] which contains all the nonperturbative Q C D effects: F i ( q 2 ) = V ( q 2 ) = A o ( q 2 ) = A 2 ( q 2 ) , (2.21) A 2 (q 2 ) = A l ( q 2 ) ( l - f ) = £ ( u , ) ^ ± ^ , (2.22) (m B + m H ) 2 2y/mBmH with m H + m B ° ~ q 2 ,0 0 o N w = v B • v H = , (2.23) 2 m B o m n where H represents the D or D* meson as appropriate. v B and V H are four velocities for B and H mesons. ISGW, ISGW** and ISGW2 models are used to describe b semileptonic decay and are discussed below. I S G W : Isgur, Scora, Grinstein and Wise (ISGW) [34] used a non-relativistic ap-proximation in B decay due to the heavy mass of the b quark. The form factor F(q 2) at the minimum recoil of the final state meson, i.e. maximum q 2 , can be obtained by solving the Coulomb potential plus linear potential for the ground B state in the Schrodinger equation: — An/ V(r) = — - + c + br, (2.24) or where as = 0.5, c = -0.84 GeV and b = 0.18 G e V / c 2 . F(q 2) is modeled to be exponential: F ( q 2 ) o c F ( q 2 n a x ) e x p ( q i ^ q m a x ) (2.25) .2 ^ 2 k n 2 Mmax where k accounts for relativistic effects and is determined to be 0.7 from the measured pion form factor. Chapter 2. b SEMILEPTONIC DECAY AND HYBRID MODEL 23 The ISGW model predicts the relative fractions of the B ->• D£u, B D*£u and B -» D**£v to be 27%, 62% and 11% [34]. The lepton spectra for b to c semileptonic decay predicted by the ISGW model and the free quark model are shown in Figure 2.3. For b to c semileptonic decay, the lepton spectrum T i 1 1 r E e (GeV) Figure 2.3: The lepton spectrum for b to c semileptonic decay from the ISGW model and free quark model [34]. predicted by the ISGW model is close to that predicted by the free quark model. For b to u semileptonic decay, the lepton spectrum from the ISGW model is much softer than that predicted by the free quark model, which will be discussed in the b —» Xu£i/ hybrid model. I S G W * * : Various experiments have measured the sum of exclusive D and D* fractions of the total semileptonic B decay width. The C L E O experiment measured (65 ± 12)% [35] and the A R G U S experiment measured (60 ± 10)% Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 24 [36] significantly smaller than the 89% predicted by the ISGW model. C L E O therefore increased the D** rate from 11% to 32%, which was referred to as the ISGW** model and better fit the lepton spectrum than the ISGW model. ISGW2: The ISGW2 model [37] is an update of the ISGW model for the semilep-tonic meson decay based on the discovery and development of Heavy Quark Symmetry. The Heavy Quark Effective Theory (HQET) treats both the 1 / m Q and the perturbative QCD corrections to the extreme Heavy Quark Symme-try limit. In the low-recoil region, the ISGW model is already consistent with the Heavy Quark limit. In high recoil b —» civ transitions, ISGW2 improved ISGW by including the constraints imposed by Heavy Quark Symmetry, hy-perfine distortions of wavefunctions and form factors with more realistic high recoils behaviors. The ISGW2 model predicts the relative fractions of the B ->• D£v, B -» D*£v and B -+ D**£u to be 29%, 61% and 10%, which is close to the ISGW model. In the exclusive method, the value of | V c b | can be extracted by studying the decay rate of B° —> D*+£~i> as a function of the recoil kinematics of the D * + meson. In b to c semileptonic decay, both b quark and c quark are heavy and the H Q E T can be applied. Using HQET, the differential partial width for B° —>• D*+£~u is given by [7]: — * = / C ( w ) ^ ( w ) | V c b | 2 , (2.26) where )C(u) is a known phase space term and J-(co) is the hadronic form factor for this decay. m B o and mo*+ are the mass of B° and D * + . q 2 is the four momentum transfer between B° and D * + . Although the shape of the form factor T(LO) is not known, its magnitude at zero recoil, LO — 1, can be estimated using HQET. In the heavy quark limit, Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 25 coincides with the Isgur-Wise function which is normalised to unity at the point of zero recoil. Corrections to J-(l) have been calculated to take into account the effects of finite quark masses and QCD corrections, yielding the value and theoretical uncertainty J"( l)=0.913± 0.042 [7]. J-(co) is unknown. There are various parameterizations of T(co). A simple one is T(co) =JF(1)(1 — (co — l)p2) using the Taylor expansion at co = 1, as the range of to is fairly small, between 1 and 1.5, in B° —» D*+£~v decay. 2.3.1.2 Inclusive b to c semileptonic decay In the inclusive b to c semileptonic decay, the final states refer to all possible final-state c hadrons, ignoring the detailed breakdown among the individual modes in b semileptonic decay. Experimentally, only a lepton is identified and this elimi-nates the difficulty to construct each daughter hadron individually. The C L E O experiment fit the electron spectrum to the sum of the shape from b —> ceue and b —> c —> e decay [38] in the region 0.6 - 2.6 GeV/c. The electron momentum is required to be greater than 0.6 GeV/c to eliminate electrons from primary charm quarks. The lepton spectra of b —> cePe and b —> c —>• e decay from the C L E O mea-surement are shown in Figure 2.4. The Br(b —>• Xeu) from C L E O measurements is (10.49 ± 0.17 ± 0.43)%, where X refers to a u quark or a c quark. The A R G U S collaboration used a charge correlation method to separate the contributions from b ->• cev and b -> c -> e and yielded Br(b ->• X£v) as (9.7 ± 0.5 ± 0.4)%. The inclusive b to c semileptonic decay rate can be written as following: F = | V c b | 2 z 0 ( — | ) ? 7 Q C D , (2.27) 1 9 2 7 T 3 •b with z 0(x) = 1 - 8x 2 + 8x 6 - x 8 - 24x4 lnx, (2.28) Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 26 Figure 2.4: Lepton spectra from b —>• c e ve (filled circles) and b —> c —>• e (open circles) from C L E O data [38]. The curves show the best fit to the modified ISGW model with 23% B to D** semileptonic decay. Chapter 2. b SEMILEPTONIC DECAY AND HYBRID MODEL 27 where rrib and m c are the b quark and c quark mass. ? 7 Q C D is a Q C D correction factor. r / | V c b | 2 can be deduced from the combined calculations of Ball et al. and Shifman et al. [39, 40]: = (42.3 ±4 .2 ) ps- 1 . (2.29) lvcb>r Using the average of Br(b —>• Xiv) from C L E O and A R G U S measurements and subtracting the contribution from b —¥ u£v, | V c b | from T(4S*) measurements can be calculated as (3.87 ± 0.09 ± 0.19) x 10~2 [2]. The average of Br(b —>• Xiv) from the four L E P experiments at the Z resonance is (10.56 ± 0.11 ± 0.18)% [5, 41, 42, 43, 44]. The L E P experiments use hemisphere b tagging techniques to select high purity samples of b hadrons. The opposite hemi-sphere is then searched for high momentum lepton candidates and the fraction of these samples which result from semileptonic b decay are determined using a vari-ety of techniques. The precision of these measurements is limited by the modeling uncertainties in the semileptonic decay lepton momentum spectra. The combined L E P result [5] for | V c b | from Br(6 —> Xclv) using Heavy Quark Expansion [45] is: / 1 55 \ \ u2 — 0 5 | V c b | = 0.0411 [Bi{b -+ X c ^ ) ^ j ( l - 0 . 0 2 4 ^ y - ) ( l ± 0 . 0 2 5 Q C D ± 0 . 0 3 5 m b ) , (2.30) where /i2 is the average of the square of the b quark momentum in the b hadron. Using Equation 2.30, | V c b | can be derived from L E P average Bv(b -» Xiv) after subtracting the contribution from b —> ulu and the value is (4.07 ± 0.05 ± 0.24) x 1 0 - 2 [5], where the first uncertainty is statistical and the second uncertainty is systematic. Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 28 2.3.2 b to u semileptonic decay-Exclusive and inclusive b to u semileptonic decays can be used to extract |V u b| , a small and least well known element in the C K M matrix. The branching fraction of b to u semileptonic decay is small. The exclusive and inclusive b to u semileptonic decays are discussed below. 2.3.2.1 Exclusive b to u semileptonic decay Unlike b —> Xc£u, b —> Xu£u decay is distributed over many exclusive modes with no dominant modes. Because of the isospin symmetry, the following relations hold: where T is the decay rate. C L E O has searched for B to TT~ , 7r°, p~, p° and co semileptonic decays [4] from 2 million T(4S) —> B B decays. Neutral pions are reconstructed by pion decay to two photons. Vector mesons are reconstructed by their decay to two pions. The b —> Xc£u background is suppressed by requiring the lepton momentum to be greater than 1.5 GeV/c in B to pion semileptonic decay and 2.0 GeV/c in B to vector meson decay. A simultaneous fit to the B meson mass and the energy difference between the beam energy and B meson energy yields the number of decays to charged pions and vector mesons. For the decay rate of B —> -KQ£V, the isospin symmetry is used. The branching fractions of B° —>• p~~£+v and B° —> 7r~£+v from recent C L E O measurements [2] are: r(B° -» P - £ + v ) = 2r(B~ -> p°rp), (2.31) r(B° -> ir~£+v) = 2T(B- n°ru) (2.32) Br(B° -> p~£+v) = (2.6 t™) x 10 - 4 (2.33) Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 29 Br(B° ->• TI~£+P) = (1.8 ± 0.6) x I O - 4 . (2.34) Similar to B —> Y)£v in Equation 2.14, B° i r ~ £ + u is the transition between a pseudoscalar meson to a pseudoscalar meson semileptonic decay, the hadronic current can be written in terms of two form factors: M 2 — M 2 M 2 — M 2 W = <7r(k)|J"|B(p)> = F 1 ( q 2 ) ( ( p + k ) ^ - l i B q 2 i »q")+F 0 (q 2 ) B q 2 «q", (2.35) where q 2 is the four momentum transfer between B meson and ir meson, p and k are the four momenta for B meson and n meson, respectively. Similar to B —> D * £ u decay, B° —>• p ~ £ + v is a pseudoscalar meson to a vector meson semileptonic decay. The hadronic current can be written in terms of four form factors as in Equation 2.19, just replacing mo* with mp. The differential decay rate of B —> p £ v with p decay to P1P2 can be written as [14]: d F ( V 2 = i r ^ P ^ ~> P i P 2 ) ( | H + ( q 2 ) | 2 + |H_(q 2 ) | 2 + |H 0 (q 2 ) | 2 ) , dq 2 1927f imB (2.36) where p is the p meson momentum. B(p —> P1P2) is the branching fraction for p decay decay to P]P2- H + ( q 2 ) , H_(q 2) and H 0 (q 2 ) are related to helicity +1, -1 and 0 for the p meson and can be expressed in terms of the four form factors [14]: H ± (q 2 ) = (mB + m ^ A ^ q 2 ) T V(q 2 ) 2 l ° B P , (2.37) m B -+- mp Ho(q2) = ^ ^ ( ( m l - m 2 - q 2 ) (m B + m p ) A l ( q 2 ) - ^ ^ ) , (2-38) 2mpvq2 H m B + m,, where p is the p meson momentum in the B rest frame. From lattice QCD (UKQCD [46]), the differential decay rate of B —>• p £ v near maximum q 2 can be simplified as [7]: d r ( y = i o " 1 1 9 2 G J ! i 3 b ' 2 q 2 A i / 2 ( < i ' ) a 2 [ i + b ( q 2 - q L j L < 2 - 3 9 ) Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 30 where A(q2) = ( m | + m 2 - q 2 ) 2 - 4m|m 2 . The constants a = (4.6 ±g;| ± 0.6) GeV, b = (—8 1|) x 10~2 GeV 2 are determined from the lattice calculation. | V u b | can be extracted from T(B —>• plv) based on: '-^f1 = f l h J , (2.40) I V u b | where r t h y is model dependent and is shown in Table 2.4. Form Factor model f t h y C p s - 1 ) ISGW2 [37] 14.2 U K Q C D [46] 16.5 L C S R [47] 16.9 Wise/Ligeti+E791 [48] 19.4 Beyer/Melikhov [49] 16.0 Table 2.4: f t h y (ps *) predictions from different form factor models. 2.3.2.2 Inclusive b to u semileptonic decay The first evidence of b to u semileptonic decay was reported by the A R G U S and C L E O Collaborations in 1990 [8, 9]. The inclusive method developed by A R G U S and C L E O was used to extract the b —> Xulv signal using the difference in the endpoint of the lepton spectrum in b —> Xu£u and b —» Xciu . A R G U S measured the ratio between b —> Xnlu in the lepton spectrum region of 2.3 - 2.6 GeV/c and b —>• Xc£u in the lepton spectrum region of 2.0 - 2.3 GeV/c from T(4S) —> B B and then used the A C C M M model to expand to the whole lepton spectrum to extract | V u b / V c b | [8]. The C L E O collaboration extracted the signal from the region of the lepton spectrum of 2.2 - 2.6 GeV/c [50] and 2.3 - 2.6 GeV/c [51]. Extraction of Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 31 I 'Vub/Vcbl is model dependent. The C L E O Collaboration used the A C C M M model to extract | V u b / V c b | as 0.076 ± 0.008 [50, 51]. In the L E P experiments, the lepton spectrum had to be boosted to the b hadron rest frame. As there is a relatively large uncertainty in the b hadron direction, the endpoint method is not appropriate to L E P experiments. The inclusive methods used in this analysis and other three L E P experiments are to extract | V u b | or | V u b | / | V c b | by using the entire lepton spectrum as well as other kinematic variables. A L E P H and this analysis used a neural network discriminant based on kinematic variables. L3 adopted a sequential cut analysis based on the kinematics of the two leading hadrons produced in the same hemisphere as a tagged lepton. D E L P H I used a classification based on the reconstructed hadronic mass, decay topology and presence of secondary kaons. 2.4 The b -» X J i s hybrid model A b —> Xu£v hybrid model, which combines exclusive and inclusive models, is used to simulate the b —> Xu£u decay in this V u b analysis. Several theoretical models have been proposed for the b —> Xu£u decay. The exclu-sive bound-state models [34, 37, 52, 53] approximate the inclusive b —» Xu£v decay spectrum by summing contributions from all the exclusive final states. The ex-clusive models do not include all the possible exclusive final states nor any non-resonant states and therefore yield an incomplete prediction of the inclusive lepton momentum distribution, especially in the high hadronic invariant mass region. The inclusive free quark models [54, 55, 56, 57, 58] treat the heavy quark as a free quark and the final state as a quark plus gluons. The free quark models are known to give Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 32 poor agreement with experiments at low u quark recoil momentum. Therefore, a hybrid model [17] is proposed to model the b —> Xu£v decay by using the exclusive model in the lower hadronic invariant mass region and using the inclusive model in the higher hadronic invariant mass region. The ISGW2 model [37] is used as the exclusive part of the hybrid model. The A C C M M model [54], combined with the W decay model [59] plus J E T S E T fragmentation, is used as the inclusive part of the hybrid model. Since the ISGW2 exclusive model includes the exclusive resonant final states IS, 2S and IP up to 1.5 G e V / c 2 in the hadronic mass, the bound-ary between the inclusive and exclusive parts of the hybrid model is placed at the hadronic invariant mass of 1.5 G e V / c 2 . The relative normalization of the inclusive and exclusive parts of the hybrid model is determined by the inclusive model. This hybrid model is only applied to the B mesons. There are no theoretical predictions for b to u semileptonic transitions of b baryons. The exclusive transitions of the b baryons in the O P A L tuned J E T S E T 7.4 [15, 16] are used. In order to estimate systematic uncertainties due to modeling of the inclusive spec-trum, alternative models are also studied. Signal events have been generated also with the Q C D universal function [55, 56, 57] and parton [58] models. The invariant mass distribution of the hadronic recoil uq system is shown in Figure 2.5 for the Q C D universal function, A C C M M and parton models. The invariant mass dis-tribution of the hadronic recoil uq system for the hybrid model is also shown in Figure 2.5. 2.4.1 b to u inclusive model In this analysis, the A C C M M model [54] is used as a base model for the inclusive part in the hybrid model. The QCD universal structure function [55, 56, 57] and Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 33 o > U LO o 500 400 .2 300 c ill 200 100 QCD ACCMM Parton CM 1.5 _ 2 2.5 3 23.5 uq mass (GeV/c ) ^ 1000 0 O £ 800 h <2 600 h c HI 400 h 200 h b) exclusive model <-hybrid model inclusive -> model 3.5 uq mass (GeV/c ) Figure 2.5: a,b. The uq invariant mass distributions, a using the Q C D universal function, A C C M M and parton inclusive models, b using the hybrid model. Only the portion of the uq invariant mass above 1.5 G e V / c 2 from the inclusive model in a is used in the hybrid model. The boundary between the exclusive model (left arrow) and the inclusive model (right arrow) in the hybrid model is indicated by the solid line in b. Chapter 2. b SEMILEPTONIC DECAY AND HYBRID MODEL 34 the parton model [58] are used to estimate inclusive model systematic errors in the hybrid model. These inclusive models are described in detail in the following subsections. 2.4.1.1 A C C M M Model In the A C C M M model, the b quark and the spectator quark momenta in the b hadron rest frame follow a Gaussian distribution: (2.41) where the width p F is known as Fermi momentum [54, 60, 61]. The pp values extracted from the fit to the momentum spectrum of leptons from the b —> civX are shown in Table 2.5. A value of p F = 0.5 GeV/c [61] extracted from the b —>• c£uX Channel P F (GeV/c) m s p (GeV/c 2 ) Ref b ->• civX 0.27 ± 0.04 0.30 [60] b ->• civX U - J 1 - 0 . 0 7 0.0 [61] b —> S7 0.45 0.0 [62] B -+ J / ^ X 0.57 0.15 [63] Table 2.5: Experimental fit results of p F and the spectator quark mass m s p in the A C -C M M model in the C L E O data is used in the b —>• Xu£u hybrid model. 2.4.1.2 Q C D universal structure function A universal structure function [55, 56, 57] is used to describe the distribution of the light cone residual momentum of the b quark inside the b hadron. The light cone Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 35 residual momentum k+ can be expressed as the difference between the b quark pole mass and its effective mass m b inside the hadron: k + = m b — m^. The light-cone structure function is suggested as [55]: f(k +) = i ( l - x ) V ^ - x ) 2 > 6 ( l - x ) (2.42) 7 T Z A where x = A = m B m b . m B is the b hadron mass and m b is the b quark pole mass. G is the Heaviside step function. A is treated as a free parameter and is predicted by QCD sum rules [64, 65] to be 0.57 GeV. 2.4.1.3 Parton Model In the parton model [58], the b quark behaves as a free particle carrying a fraction z of the b hadron momentum, i.e. p b = zp B . The z distribution can be described by the Peterson fragmentation function [29]: Nz( l - z) 2 f ( 2 ) = ( ( l - z ) 2 + e b z ) 2 ' ( 2 ' 4 3 ) where eb is a free parameter and its experimental value is 0.0047lo!ooo8 [2]-2.4.1.4 Decay kinematics In the above models, the b quark decay kinematics are required to obtain the decay product u quark's energy and momentum. The u and q quarks' energies and momenta are used to obtain the uq invariant mass. The b quark decays as b —> Wu, and the virtual W boson may be characterized via its effective mass Q 2 . The description of the Q 2 distribution is [59]: Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 36 where x 2 = ^ and m b . F 0 (x) = 2 ( l - x 2 ) 2 ( l + 2x 2), Fx(x) = F O ( X ) ( T T 2 + 2S 1 ) 1 (x 2 ) - 2Si, i( l - x 2)) + 8x 2 ( l - x 2 - 2x4) ln(x), +2(1 - x 2 ) 2 (5 + 4x2) ln(l - x 2) - (1 - x 2)(5 + 9x 2 - 6x 4). Si i(x) is the Nielsen polylogarithm [66, 67] and nib is the b quark mass. as is the Q C D coupling constant at the scale of mb. as has an approximate value 0.24 at the mb scale. To ensure energy conservation, the effective b quark mass is: m 2 = m 2 + m 2 p - 2m B V / p 2 + m s 2 p . (2.45) The m B is the b hadron mass and m s p is the spectator quark mass, pb is the b quark momentum. The effective b quark mass depends on p F and the mass of the spectator quark m s p . Due to the spin of the W boson, the lepton production angular distribution -^^j is (1 + cos#)2 [59]. The angle 6 is the lepton direction in the W rest frame with respect to the W direction in the b quark rest frame. The corresponding azimuthal distribution <j> in the lepton production is isotropic. 2.4.1.5 Implementation of inclusive models Finally, a string is added between the u quark and the spectator quark for the hadronization of the uq system. The string fragmentation in J E T S E T is used, which will be discussed in Section 2.5.1. The lepton's momentum and energy can be obtained from the virtual W boson's momentum and energy which can be calculated from the b quark decay kinematics. This b —> Xu£v decay Monte Carlo model is incorporated into J E T S E T and is referred to as the inclusive model in this analysis. Chapter 2. b SEMILEPTONIC DECAY AND HYBRID MODEL 37 B° -> X u a e ^ e B -> X u f l e ^ e I = 1 I = 0 X mass MeV partial width mass MeV partial width mass MeV partial width 1% 1 3 S ! 1 3 P 2 1 3 Px l 3 P o 2 % 2 3 S i TT(140)+ p(770)+ a2(1320)+ bi(1235)+ ai(1260) + a0(1450)*+ TT(1300)+ p(1450)+ 0.96 1.42 0.33 1.09 0.87 0.05 0.17 0.41 7r(140)° p(770)° a 2(1320)° b 1(1235)° ai(1260)° a0(1450)*° 7r(1300)° p(1450)° 0.48 0.71 0.16 0.54 0.43 0.02 0.08 0.20 77(547)° 7/(958)° cu(782)+ f2(1270)° hi(1170)° fi(1285)° f0(1370)*° ?7(1440)° w(1420)° 0.45 0.28 0.71 0.18 0.57 0.41 0.03 0.08 0.20 total width 5.3 2.6 2.9 Table 2.6: Partial widths for b to u semileptonic decays for B mesons in the ISGW2 model [37]. The notation N 2 S + 1 L j is used here, where N is the energy level quantum number. S, L and J are the spin, the orbital and total angular momentum for the two quark combination. The partial width is in units 10 1 3 |V u b | 2 sec - 1 . 2.4.2 I S G W 2 exclusive model The ISGW2 model can be used to describe exclusive b to c semileptonic decays as mentioned in Section 2.3.1.1. The ISGW2 model can also be used to describe exclusive b to u semileptonic decays and is used as the exclusive part of the b —>• Xu£u hybrid model. The partial widths for b to u semileptonic decays from B, B s and B c mesons in the ISGW2 model are shown in Table 2.6 and Table 2.7. The lepton spectra for B~ -> X u Q e P e predicted by the ISGW2 model are shown in Figure 2.6. Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 38 B° -4 Xu-Seue X u e e i / e X mass partial width mass partial width 1% 1 3 S X 1 3 P 2 1 3 P X l : P i I ' P o 2 % 2 % K(494)+ K(892)*+ K(1430)*+ K 1 A(1270)+ K 1 B(1400)+ K0(1430)*+ K(1460) + K(1580)*+ 0.85 1.14 0.28 1.72 0.08 0.04 0.45 0.54 D(1864)° D(2010)*° D2(2460)*° D 1 A(2420)*° D!(2420)*° D0(2400)*° D(2580)° D(2640)*° 0.30 0.62 0.06 0.62 0.04 0.01 0.46 0.40 partial total 5.1 2.5 Table 2.7: Partial widths for b to u semileptonic decay for B s and B c in the ISGW2 model [37]. The notation N 2 S + 1 L j is used here, where N is the energy level quantum number. S, L and J are the spin, the orbital and total angular momentum for the two quark combination. The partial width is in units 10 1 3 |V u b | 2 sec - 1 . Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 39 0.8 0.7 0.6 0.5 -1 1 free quark decay total IP + i s U + P + r)' + T) + TT r\ + T) + 7T 7? + IT 7T 1 1 1 -0.4 - -0.3 - -0.2 - ^•^ \ -0.1 0 - -0 0.5 1 1.5 2 2.5 3 Ee (GeV) Figure 2.6: The electron energy distribution (l/Tfree)(dr/dEe) vs electron energy for B - —> X u Q e ^ showing contributions of 7r°, n, TJ', p°, u> and the IP and 2S states [37]. l Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 40 2.5 Signal and background simulation using JET-SET Monte Carlo simulation of multihadron production is model dependent because of the difficulties of calculation of the fragmentation phase of the multihadron production shown in Figure 2.2. The Monte Carlo simulated events in this analysis were generated using the J E T S E T 7.4 [15] generator, which is described in detail below. 2.5.1 J E T S E T J E T S E T is a Monte Carlo simulation program for jet fragmentation and e + e -physics based on the Lund Model. O P A L J E T S E T is a Monte Carlo simulation generator to simulate the multi-hadronic event production in electron and positron collisions using the O P A L tune [16]. O P A L J E T S E T describes many aspects of hadronic event production extremely well. In JETSET, e+e~ —> Z —>• qq process is based on the Feynman rules of the elec-troweak theory. The 7 and Z interference, initial state radiation and gluon emission of the final quark are considered. The gluon is radiated by primary quarks from Z decay, and later the gluon splits into gluons or quark antiquark pairs. This process can be described by the perturbative theory until the energy density of the system decreases below a fixed level. Matrix-element and parton shower methods are both used to model the perturbative corrections. In the matrix element method, Feynman diagrams are calculated order by order. The calculation becomes difficult for the higher order Feynman diagrams. In the parton model, one parton decay into two partons or more is used to approach Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 41 the multi jet events. The branching probabilities of q —»• q7, q —> qg, g —> gg and g —> qq, where q represents a quark and g a gluon, are calculated by applying the Altarelli Parisi equations [68]. After the perturbative phase, the coloured partons are transformed into colour-less hadrons. There are three main models to describe this fragmentation process, namely string fragmentation (SF), independent fragmentation (IF) and cluster frag-mentation. In JETSET, the string fragmentation is used. In string fragmentation, the Q C D potential between quarks is described as: 4GI V Q C D = k r - (2.46) where k is a constant and r is the distance between quarks. The string model suggests there is a colour flux tube between quarks. The transverse dimensions of the tube are of typical hadronic sizes, around 1 fm. As the quark q and q move apart, the potential energy stored in the string increases. The string may break up by the production of a new q'q' pair. Then two colour-singlet systems qq' and q'q are formed. The string break-up process is repeated until on-mass-shell hadrons remain. The quark flavour production ratio can be predicted by invoking the quantum mechanical tunneling, in which u : d : s : c = 1 : 1 : 0.3 : 1 0 ~ N . Charm and heavier quarks are not expected to be produced in this soft fragmentation, but only in the perturbative parton-shower process. A large fraction of the particles produced by fragmentation are unstable and sub-sequently decay into stable particles, which are controlled by the decay table in JETSET. The particle mass and decay properties are well defined in the decay table according to recent measurements. Chapter 2. b S E M I L E P T O N I C D E C A Y A N D H Y B R I D M O D E L 42 2.5.2 Signal and background simulations The b hadron was generated from e +e" -» bb according to O P A L J E T S E T [15] generator in the Monte Carlo simulation. For the signal b —> Xu£u Monte Carlo simulation, the b hadron is forced to decay to Xu£u according to the branching fractions predicted by the b —>• Xu£u hybrid model. The energies and momenta of X u , £ and v are calculated from the b —> Xu£u hybrid model. For background Monte Carlo simulation, the A C C M M model [54] is used to describe the lepton spectrum of b —y Xc£v and b —> c —> £ decays. The fragmentation function of Peterson et al. [29] is used to describe the b quark and c quark fragmentation. The branching fractions of B° D~£+u, B° -> D*~£+u, B+ ->• D°£+u, B+ -> D*°£+v, B ->• B**£u and A b —y AcX£u are modified to reproduce those given by the Particle Data Group [2]. The A b lepton momentum spectrum corresponding to -56% polarization [69] is used as a central value. The b to u semileptonic decay and background simulated events are passed through the O P A L detector simulation [70] to produce the corresponding response. The production fractions of B + , B°, Bg and A b in Z decay are chosen to reproduce those given by the Particle Data Group [2]. Chapter 3 T H E L E P COLLIDER AND O P A L DETECTOR The current analysis uses the data taken by the O P A L detector. O P A L is one of the four experiments at the Large Electron and Positron collider (LEP) at C E R N . The L E P collider and O P A L detector are described in this chapter. For the O P A L coordinate system, a right handed coordinate system is used, with positive z along the e~ beam direction and x pointing toward the center of the L E P ring. The polar and azimuthal angles are denoted by 6 and <fi, and the origin is taken to be the center of the detector. 3.1 The LEP collider The L E P collider is located in a 27 km circumference tunnel at C E R N , Geneva, Switzerland. It lies underground at a depth between 50 meters and 130 meters below the surface. The four L E P experiments A L E P H , D E L P H I , L3 and O P A L are equally spaced around the collider as shown in Figure 3.1. The O P A L detector 43 Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 45 lies underground 99 m below the surface. The production of L E P beam is shown in Figure 3.2. The L E P injector linac produces electrons and accelerates them to 200 MeV. Some of these electrons strike a tungsten target and produce positrons. Both electrons and positrons are further accelerated to 600 MeV by a second linac and then stored in the Electron Positron Accumulator ring (EPA). Later these leptons are injected into the Proton Synchrotron (PS) and are accelerated to 3.5 GeV. Electrons and positrons are then passed on to the Super Proton Synchrotron (SPS) and further accelerated to 20 GeV before injection into L E P . The L E P collider Figure 3.2: The layout of the C E R N injector system for the L E P ring [73]. began to operate in 1989 and was shut down in November, 2000. The L E P operation was divided into two phases: LINACS ( L I D LSS 1 Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 46 L E P I (1989-1995): L E P I operated in the center mass of around 91 GeV and was designed for the study of the Z boson. The current analysis uses L E P I data. L E P 11(1996-2000): L E P II was used to search for possible new particles. In 1997, L E P operated at a center of mass energy exceeding 160 GeV and pro-duced the first observation of W + W - pair production. 3.2 The OPAL detector The O P A L detector [71], i.e Omni Purpose Apparatus for LEP , is one of the four multi purpose detectors at LEP . The structure of the O P A L detector is shown in Figure 3.3. The O P A L detector is about 12 m long, 12 m high and 12 m wide and attains almost 47T coverage by closing the barrel with endcap detectors. The main features of the detector are: - the reconstruction of charged particle tracks inside a magnetic field enabling a measurement of momentum, particle identification (by dE/dx) and the re-construction of primary and secondary vertex positions; - the identification of electrons and photons together with a measure of their energy via electromagnetic shower detection; - the measurement of hadronic energy by total absorption; - the detection of Bhabha scattering (e + e _ —>• e + e _ ) events at low angles with respect to the beam line, providing a measurement of the absolute luminosity. The O P A L sub-detectors are discussed in the following sections. Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 47 Figure 3.3: The O P A L detector [74]. Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 48 3.2.1 Beampipe The O P A L beampipe, which is shown Figure 3.4, consists of an inner aluminum tube covered by layers of carbon fiber epoxy of 2.2 mm thickness. The beampipe is the innermost detector component and is used to isolate the electron and positron beams from the 4 bar pressure gas vessel of the vertex chamber and the jet chamber. Before 1991, the inner radius of the pipe was chosen to be 80.2 mm. In 1991 a second beam pipe support pressure vessel carbon-fibre support ring support and cooling ring beryllium shell cable guides/ cooling manifold short outer detector ladder ladder end support long outer detector ladder inner detector ladder ceramic hybrid brake mechanism aluminium support roller balls) radiation monitor/ beryllium beam pipe Figure 3.4: The O P A L beampipe structure [75]. beam pipe at a radius of 53.5 mm consisting of 1.1 mm thick beryllium was added and the silicon microvertex detector was inserted between them. Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 49 3.2.2 Subdetectors for particle tracking The tracking of charged particles in the O P A L detector uses information from the silicon microvertex detector [75], the central vertex chamber [76], the jet cham-ber [77], the z chambers [78] and the muon chamber [79]. The silicon microver-tex detector collects the ionization charges generated by charged particles passing through the silicon wafers and measures the track position. The central vertex chamber, the jet chamber, the z chambers and the barrel muon chambers are all gaseous drift chambers. In drift chambers an electric field causes the ionized elec-trons to drift towards the anode field wires. The ionizing particle position can be detected by measuring the time that electrons need to reach the anode wire, from the moment that the ionizing particle traverses the detector. The endcap muon chambers use streamer tubes. Streamer tubes work in a similar manner to drift chambers but operate in a higher electric field. These subdetectors for the particle tracking are described in detail below. 3.2.2.1 The silicon microvertex detector The B meson decay length is of the order of a few millimeters. With the silicon microvertex detector, the B meson decay point can be precisely measured. The silicon microvertex detector [75] consists of 25 ladders arranged in two barrels of inner radius of 6 cm and outer radius of 7.5 cm, which is shown in Figure 3.5. The inner layer consists of 11 ladders and the outer layer of 14 ladders. Each ladder is 18 cm long and consists of 3 silicon microstrip wafers (3 cm x 6 cm) daisy chained together. Each detector has 629 readout strips. These strips are oriented along the beam axis and have a 50 fim readout pitch in order to measure coordinates in the r — <fi plane with an intrinsic resolution of about 5 pm. The absolute resolution for Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 50 Figure 3.5: The structure of the O P A L silicon microvertex detector [75]. Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 51 the hit position in event reconstruction is 10 /xm. The hit efficiency for the ladder was measured with Z —> events to be 97%. The first silicon microvertex detector, /xvtxl was installed in O P A L in 1991 and had readout in the r — (f> plane only. The single hit coverage in </> is almost 100%. In 1993 an upgraded detector /tvtx2 was installed that had r — <j> and r — z wafers glued back to back, resulting in polar coordinate acceptance to | cos#| < 0.83 for the inner barrel and | cos#| < 0.77 for the outer barrel. In 1995, the detector was further upgraded to /ivtx3. The number of ladders for the inner layer and outer layer was increased to 12 and 15 respectively. The outer layer was also extended from 3 wafers to 5 wafers by an addition of a layer of 2 wafer ladders at the r — z end. In 1996, 2 wafer ladders were added to the inner layer extending polar coordinate acceptance to |cos#| < 0.89 for the inner layer. 3.2.2.2 The central vertex chamber The central vertex chamber [76] is designed to provide precise tracking capabilities for the reconstruction of secondary vertices. The central vertex chamber consists of two 1.0 m cylindrical drift chambers with 36 sectors azimuthally each. The inner chamber contains the axial sectors, where each sector contains a plane of 12 sense wires strung parallel to the beam direction. The wires range radially from 103 mm to 162 mm at a spacing of 5.3 mm. The outer chamber contains the stereo sectors each containing a plane of 6 sense wires inclined at a stereo angle of around 4 degrees. The stereo wires lie between the radii 188 mm and 213 mm at a spacing of 5 mm. These chambers are operated with a gas mixture of 88% argon, 9.4% methane and 2.6% isobutane at a pressure of 4 bars. The precise measurement of the drift time onto axial sector sense wires provides a position resolution of 50 Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 52 microns in the r - (f> plane. Measuring the time difference between signals at two ends of the anode wires allows a fast but relatively coarse z coordinate measurement which is used by the O P A L track trigger in a pattern recognition. A more precise z measurement is made by combining axial and stereo drift time information offline. 3.2.2.3 The jet chamber The central jet chamber [77] is designed to provide precise tracking capabilities for the reconstruction of jet-like events. The central jet chamber is a cylindrical drift chamber of length 4 m with an outer (inner) diameter of 3.7 (0.50) m respectively. The chamber is divided into 24 identical sectors each containing a sense wire plane of 159 wires strung parallel to the beam direction. Each wire is read out at both ends with a 100 MHz flash A D C . The coordinates of wire hits in the r - <t> plane are determined from a measurement of drift time. The central jet chamber measures the three-dimensional coordinates of charged tracks, the particle momentum and the particle energy loss in the gas volume. The resolution in the r - <f> plane is 160 (xm. The z coordinate is measured by collecting the charge at each end of a wire, which produces a resolution of 6.2 cm. The jet chambers are operated with the same gas mixture as the central vertex chamber. 3.2.2.4 The z chambers The z chambers [78] provide a precise measurement of the z coordinate of tracks as they leave the jet chamber. The z chambers consist of a layer of 24 drift chambers, each 400 cm long, 50 cm wide and 5.9 cm thick covering 94% of the azimuthal angle and the polar angle range from 44 to 136 degrees. Each chamber is divided into 8 cells of 50 cm x 50 cm in the z direction, with each cell containing 6 sense wires Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 53 spaced at 0.4 cm and with a stagger of ± 250 /jm in order to resolve the left-right ambiguity. The chambers use the same gas as the jet chamber with a uniform field of 800 V / c m over the full drift distance of 25 cm in z. A F A D C system is employed to determine the drift distance and a charge division technique is used to give a coarse <j> measurement. 3.2.2.5 Magnet The O P A L magnet consists of a solenoid and an iron return yoke. The solenoid was constructed with a water cooled coil of aluminum and glass-epoxy. The magnetic field distribution has to satisfy two main requirements: — high uniformity through the central detector volume. The central detector refers to the silicon microvertex detector, the central vertex detector, the jet chamber and the z chambers. The field in the central detector region is mea-sured to be 0.435 T produced by 7000 A current and to be uniform within 0.5% over the volume of the central detector. The solenoid and pressure vessel together represent about 1.7 radiation lengths of material. — a magnetic field not exceeding a few tens of Gauss in the angular region between the solenoid and the iron yoke to facilitate the operation of photo multiplier tubes in surrounding sub-detectors. For the- O P A L magnet, the stray field outside the solenoid is below 0.01 T because of the soft iron return yoke. 3.2.2.6 The muon chambers The outermost of the O P A L detectors are the muon chambers [79]. The muon detector consists of a barrel and two endcaps and covers the iron yoke almost Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 54 completely. Particles which reach the muon chambers have to traverse 7 interaction lengths (for pions) and therefore the tracks in the central tracking system matched to muon chamber hits can be identified as muons. The probability for an isolated pion of 5 GeV/c being misidentified as a muon is less than 1%. The barrel part of the muon detector consists of 110 large-area drift chambers divided into four modules. Each chamber is 120 cm wide and 9 cm deep. Two modules, each with 44 chambers are mounted on each side of the barrel. The remaining two modules, contain 10 chambers in the top module and 12 chambers in the bottom module, which close the gap and provide full coverage in 4>. The hit in z, with a resolution of 0.2 cm in the muon chamber, can be obtained by measuring the time differences at each end of the sense wires. The resolution in r - (j) is 0.15 cm using drift time information. The barrel part of muon chambers covers | cos#| < 0.68. The endcap muon detectors cover the angular range 0.67 < |cos0| < 0.985. An area of about 150 m 2 at the end of the O P A L detector is covered with four layers of limited streamer tubes which are perpendicular to the beam axis. Each endcap consists of eight quadrant chambers (6 m x 6 m) and four patch chambers (3 m x 2.5 m). Each chamber consists of two layers of streamer tubes, spaced by 1.9 cm, one layer having vertical wires and the other horizontal wires. The position resolution in x and y coordinates is 0.1 cm. 3.2.2.7 Performance of the tracking system The reconstructed tracks are characterized using the following five parameters, where the point of closest approach is with respect to the origin: - k, the track curvature, where \k\ = j - and p is the radius of curvature of the Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 55 track. The sign of k is chosen positive for a particle which is deflected in the direction of increasing </> if traveling along the track direction. Positive values of k correspond to a particle with negative charge; — <f>0, the azimuthal angle of the track tangent at the point of closest approach with respect to the x axis; — CLQ, the impact parameter and d0 = (<f> x d) • z, where <f> is the unit track vector at the point of closest approach and d is the vector from the origin of the O P A L detector to the origin, z is the unit vector along the z axis; — tan A = cot 9, where 9 is the polar angle of the track tangent at the point of closest approach; — z0, the track z coordinate at the point of closest approach in the r - </> plane. A schematic diagram of these five parameters is shown in Figure 3.6. In the O P A L jet chamber, each hit provides a measurement of the specific energy loss of the charged particle. The dE/dx resolution [80] is: a(dE/dx) „ N - O . 4 3 ( 3 1 } dE/dx ' { ' where N is the number of measured dE/dx samples, around 159 for this jet chamber. The dE/dx resolution is measured to be 3.8% for minimum ionizing particles in a jet. The transverse momentum of the track can be calculated from the measured cur-vature of the track in the magnetic field. The resolution [81] of the transverse momentum of the track in the jet chamber can be deduced as: ^ = J(omy + (o.ooi5Pt)2, (3.2) Pt Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 56 z Figure 3.6: A schematic diagram for five parameters describing a track. where p t is given in G e V / c . The momentum dependent term of the resolution can be reduced to 0.00128p t by combining the track information of the other central detectors. 3.2.3 Subdetectors for calorimetry The calorimetry system is designed to measure the particle energies. This is es-pecially important for neutral particles, which are not detected by the tracking detectors. The calorimetry system is composed of the electromagnetic calorimetry and hadron calorimetry, which are described in detail below. Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 57 3.2.3.1 The electromagnetic calorimeter The function of the electromagnetic calorimeter is to detect and identify electrons and photons from tens of MeV to 100 GeV by using lead glass blocks. The elec-tromagnetic calorimeter consists of a barrel, covering angle | cos#| < 0.82, and two end cap arrays, covering angle 0.81 < |cos#| < 0.98. This arrangement, plus two forward lead scintillator calorimeters from the forward detector, makes the O P A L acceptance for electron and photon detection almost 99% of the solid angle. The presence of about 2 radiation lengths of material in front of the calorimeter (due to the solenoid and pressure vessel) results in most electromagnetic showers initiating before reaching the lead glass. Presampling devices are therefore installed in front of the lead glass in the barrel and endcap regions to measure the position and en-ergy of showers to improve overall spatial and energy resolution and give additional 7/71"° and electron/hadron discrimination. The components of the electromagnetic calorimeter are described in detail below: Barrel electromagnetic presampler: The barrel electromagnetic presampler [82] consists of 16 chambers forming a cylinder of radius 239 cm and length 662 cm. Each chamber consists of 2 layers of drift tubes operated in the limited streamer mode with the anode wire running parallel to the beam direction. Each layer of tubes contains 1 cm wide cathode strips on both sides at ± 45 degree to the wire direction. Spatial positions can then be determined by reading out the strips in conjunction with a measurement of the charge collected at each end of the wire to give a z coordinate by charge division. The hit multiplicity is approximately proportional to the energy deposited in the material in front of the presampler allowing the calorimeter shower energy to be corrected with a corresponding improvement in resolution. From test Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 58 beam results the presampler can achieve a spatial resolution for electromag-netic showers in the plane perpendicular to the shower direction of 0.6 to 0.4 cm for incident energies in the range of 6 GeV to 50 GeV. The resolution in z for an isolated charged particle is around 10 cm. Barrel lead glass calorimeter: The electromagnetic barrel lead glass calorime-ter [71] consists of a cylindrical array of 9940 lead glass blocks at a radius of 246 cm and covering |cos#| < 0.82. The lead glass provides an excellent intrinsic energy resolution (CTE/E = 5%/y/E [71], for E in GeV), spatial resolu-tion (around 1 cm) and linear response over a wide dynamic range. To achieve good energy resolution at high energies, shower leakage from the back of the calorimeter must be minimized. This is achieved by using a very dense glass (24.6 radiation lengths) with each block 37 cm in depth and 10 cm x 10 cm in area. In order to maximize detection efficiency the longitudinal axis of each block is angled to point at the interaction region. The focus of this pointing geometry is slightly offset from the e+e~ collision point in order to reduce particle losses in the gaps between blocks. Cerenkov light from the passage of relativistic charged particles through the lead glass is detected by 3 inch diameter photo-tubes at the base of each block. Each photo-tube is shielded from the stray field of the magnet so that operation in magnetic fields up to 100 Gauss is possible with a gain variation of less than 1%. The photomulti-plier signals are digitized by charge integrating 15-bit ADCs. The linearity of the A D C was measured to be better than ± 1 count over the full range of the core channel. In order to ensure the quality of the gain calibration for each phototube over long time periods, a gain monitoring system using a Xenon light source is employed. Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 59 Endcap electromagnetic presampler: The endcap presampler [83] is a multi-wire proportional counter located in the region between the pressure bell and the endcap lead glass detector. The device consists of 32 chambers arranged in 16 sectors covering all <j> and the polar angle range 0.83 < |cos#| < 0.95. Performance results from test beam electrons and pions show that the spatial resolution attained is in agreement with the expected 1 / \ / l 2 of the strip or wire effective pitch, which is 0.2 cm for the anode strips and 0.32 cm for the cathode planes. Endcap electromagnetic calorimeter: The endcap electromagnetic calorime-ter [84] consists of two dome-shaped arrays of 1132 lead glass blocks located in the region between the pressure bell and the pole tip hadron calorimeter. It has an acceptance coverage of the full azimuthal angle and 0.81 < | cos#| < 0.98. Time-of-flight Counters: The time-of-flight system consists of barrel and endcap detectors and provides fast triggering information, effective rejection of cosmic rays and charged particle identification in the low momentum region (0.6 -2.5 GeV/c) . The time-of-flight system consists of 160 scintillation counters forming a barrel layer 6.84 m long with mean radius 2.36 m surrounding the O P A L coil and covering the region | cos 9\ < 0.82. Light is collected from both ends of each counter via Plexiglas light guides glued directly onto phototubes which are shielded from stray magnetic fields. The output signal from each, phototube is then split into two parts. The first, for the timing measurement, goes to a constant fraction discriminator, T D C and mean timer. The second goes to an A D C for a pulse height measurement which can be used to correct the pulse heights of the electromagnetic and hadronic calorimeters. The time-of-flight trigger signals are derived from the mean timers with the requirement that the time of flight was within 50 ns and that discriminators at both ends Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 60 fire within 50 ns of each other. The time-of-flight system achieves a timing resolution in the central region of the counters of 350 ps with a z resolution around 10 cm. The endcap detector was installed in 1997 to provide additional information on minimum ionizing particles in the endcap region. It consists of a 1 cm thick scintillator layer between the endcap presampler and the endcap electromagnetic calorimeter. 3.2.3.2 Hadron calorimeter The hadron calorimeter is built in 3 sections: the barrel, the endcap and the pole-tip. By positioning detectors between the layers of the magnet return yoke a sam-pling calorimeter is formed covering a solid angle of 97% of 4TT and offering at least 4 radiation lengths of iron absorbers to particles emerging from the electromagnetic calorimeter. Essentially all hadrons are absorbed at this stage leaving only muons to pass into the surrounding muon chambers. To correctly measure the hadronic energy, the hadron calorimeter information must be used in combination with that from the preceding electromagnetic calorimeter. This is necessary due to the like-lihood of hadronic interactions occurring in the 2.2 radiation lengths of material that exists in front of the iron yoke. The barrel region contains nine layers of chambers sandwiched with eight layers of 10 cm thick iron. The barrel ends are then closed off by toroidal endcaps consisting of 8 layers of chambers sandwiched with 7 slabs of iron. The chambers themselves are limited streamer tube [85] devices strung with anode wires 10 mm apart in a gas mixture of isobutane (75%) and argon (25%) which is continually flushed through the system. The signals from the wires themselves are used only for monitoring purposes. The chamber signals from induced charge are collected on pads and strips Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 61 which are located on the outer and inner surfaces of the chambers respectively. The layers of pads are grouped together to form towers that divide the detector volume into 48 bins in <j> and 21 bins in 9. The analogue signals from the 8 pads in each chamber are then summed to produce an estimate of the energy in hadron showers which is subsequently digitized by a 12-bit A D C . From pion test beam results, the tower response was found to be linear with energy and has a resolution of <7E :=120%\/E [71] where E is in GeV. The strips are made from aluminum of width 0.4 cm, which runs the full length of the chamber centered above the anode wire positions. Hence they run parallel to the beam line in the barrel region and in a plane perpendicular to this in the endcaps. Strip hits thus provide muon tracking information with positional accuracy limited by the 1 cm wire spacing. Typically, the hadronic shower initiated by a normally incident 10 GeV pion produces 25 strip hits and generates a charge of 600 pC. The barrel detector covers the angular region up to |cos#| < 0.81. The endcap detector extends the angular region up to | cos#| < 0.91. The pole-tip detector [86] extends the angular coverage up to | cos#| < 0.99 using ten thin multi-wire cham-bers with nine 8 cm thick iron plates. 3.2.4 Luminosity monitor The forward detector [71] is used to determine the luminosity delivered to the interaction point. The cross section of Bhabha ( e + e - —> e+e~) events in the forward detector can be calculated as: where a is the electromagnetic coupling constant at the Z resonance, s is the center of mass energy. 9min and 9max define the angular acceptance of the forward 1670*2, 1 1 (3.3) ^Bhabha — Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 62 detector, which are 47 mrad and 120 mrad for this detector. The forward detector consists of a lead glass calorimeter, proportional tube chambers, gamma catcher and far forward monitor, which are described below. Calorimeter: The forward calorimeter consists of 35 sampling layers of lead glass scintillator with a presampler of 4 radiation lengths and the main calorimeter of 20 radiation lengths. Proportional tube chambers: There are three layers of proportional tube cham-bers positioned between the presampler and main sections of the calorimeter. The position resolution is ±0.05 cm. Gamma catcher: The gamma catcher is a ring of lead scintillator of 7 radiation length thickness. It plugs the hole in acceptance between the inner edge of the electromagnetic endcap calorimeter and the start of the forward calorimeter. Far forward monitor: The far forward monitor counters are small lead glass scin-tillator calorimeters, 20 radiation lengths thick, mounted on either side of the beampipe 7.85 m from the intersection region. Electrons which are deflected outwards by the action of L E P quadrupoles are detected in the range 5 to 10 mrad. A silicon tungsten detector was added to improve luminosity measurement precision after 1993. The silicon tungsten detector is a sampling calorimeter designed to detect low angle Bhabha scattering events in order to measure the luminosity. There are 2 calorimeters at ±238.94 cm in z from the interaction point with an angular acceptance of 25 mrad to 59 mrad. Each calorimeter consists of 19 layers of silicon detectors and 18 layers of tungsten. At the front of each calorimeter is a bare layer of silicon to detect preshowering, the next 14 silicon layers are each behind 1 radiation length (0.38 cm) of tungsten and the final 4 layers are behind Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 63 2 radiation lengths (0.76 cm) of tungsten. Luminosity may be determined to a precision of 0.1% using the forward detector and silicon tungsten detector. 3.2.5 Trigger of the OPAL detector Electron and positron bunches collide at the interaction point every 11 ps in the 4 x 4 modes (4 electron bunches and 4 positron bunches) before 1993. After 1993 L E P was changed to 8 on 8 bunches. O P A L decided to modify the pretrigger to keep the old trigger [87]. The trigger is used to decide which events are of physics interest and which should not be considered further during this time interval. The expected physics event rate can be calculated by the luminosity times the interaction cross-section. For L E P I, the luminosity is around 1.6 x 10 3 1 cm~ 2 s _ 1 . The cross section of Z decay to lepton pairs and multi-hadrons is 92 nb, so the expected total physics event rate is around 1.5 Hz. To efficiently select physics events of interest, the trigger should be as loose as possible. However this will increase the background rate and experiment deadtime. The O P A L trigger is designed so that the deadtime of the experiment does not exceed 10% of the trigger rate. As the average dead time for recording an event is 20 ms, the maximum allowed trigger rate can be calculated as 10%/20 ms, i . e. 5 Hz. The general trigger information can be divided into two classes, stand-alone signal and coincidence signal. Stand-alone signals are formed by 1 bit information indi-cating tracking chamber multiplicity counts and total energy summations. For the coincidence signal, each detector is divided into 24 intervals in </) and 6 intervals in 9. A 24 x 6 9 — <f> matrix is formed. A 9 — </3 bin is set if the signal in the cor-responding region of the detector is seen. The standalone trigger and coincidence trigger are analyzed in parallel to provide a faster trigger for physics events. If no Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 64 trigger condition is met, the whole detector is reset for the next bunching crossing. 3.2.6 Online ROPE When the trigger conditions are met, each subdetector is read out separately by its own special front-end readout electronics into its local system crate(s). Then the data are assembled in the event builder and are passed in sequence to the Filter. The Filter is a collection of software that acts as a third level trigger where the events are checked, analyzed, monitored and compressed before being written to disks. Some obvious junk events, such as beam gas backgrounds, typically 15% - 35% of all triggers, are rejected. A completed record of events, including event pointers, event header and calibration information, is written in data files in 20 Mbyte partitions. A l l of these 20 Mbyte files are copied from the Filter disk to the R O P E (Reconstruction of O P A L Events) farm. R O P E is a collection of software designed to reconstruct raw digit events to DST (Data Summary Tape) format which can be analyzed online or offline. The event data files are also spooled off to cartridge as a permanent backup before they are deleted from the Filter disk. The Online R O P E farm provides the following functionality: 1. record online data onto optical disks and tapes; 2. reconstruct the events in real time or almost real time; 3. perform data monitoring; 4. distribute DST files and R O P E log files to computer center, subdetector cali-bration files to the subdetector groups. Finally, the data are stored in DST format for offline analysis. The raw data are also stored for future improvements, such as better calibration and improved Chapter 3. T H E L E P C O L L I D E R A N D O P A L D E T E C T O R 65 reconstruction techniques. In addition, the system is used to perform offline data Re-Ropes. Re-Rope is offline reconstruction performed on raw data with improved calibrations etc. The system runs on 12 dedicated HP workstations. It runs continuously, not knowing about the start or end of runs, and requires no operator intervention. Tapes are automatically mounted by a robot in the C E R N computer center. Chapter 4 E V E N T P R E S E L E C T I O N This chapter describes the techniques to identify candidates of b decay and candi-dates of electron and muon tracks from raw O P A L data in Z decay. A hadronic event selection [88] and detector performance requirements are applied to the data. The thrust axis polar angle |cos#| is required to be less than 0.9 to ensure that the events are well contained within the acceptance of the detector. The selected events must pass the b identification [89], the lepton selection and the b semileptonic decay selection. These preselections use the O P A L standard b tagging (BT) and particle identification (ID) packages and are described in detail in the following sections. After all these preselections, the b —> X u £ u decay purity is 1.3% and the main background is from b —> Xc£i/ decays. An Artificial Neural Network (ANN) algorithm is introduced in this chapter as it is widely used in event selection. 4.1 An introduction to Artificial Neural Networks Artificial Neural Networks are computational models applied to classification, pat-tern recognition and optimization, inspired by biological neural systems. A neural 66 Chapter 4. E V E N T P R E S E L E C T I O N 67 network is constructed based on a series of inputs, one or more outputs and multiple layers of nodes lying between inputs and outputs, an example of which is shown in Figure 4.1. The inputs, outputs and nodes in the neural network are connected by Wii Input layer Hidden layer Output layer Figure 4.1: Input, hidden and output layers in a neural network weights which can be obtained by training of known samples. Feed-forward neural networks, in which the connections are unidirectional, are used in this analysis. Neural network outputs can be calculated from inputs and weights based on the following formula: Yi = g(E wiJ§(E + 0j) + ft). (4-1) j k Chapter 4. E V E N T P R E S E L E C T I O N 68 where y; is the neural network i output. Wjj is the weight between the internal node j and i t h neural network output. W j k is the weight between the internal node j and k t h neural network input. t9j and 9\ are constant threshold values for the nodes, which can be used to adjust neural network performance and are commonly set to zero. The transfer function g controls neural network response and has the form: where T is the temperature of the neural network. The Figure 4.2 shows g(x) as a function of different temperatures. For a higher temperature, g(x) approaches a straight line and the resulting neural network has a linear response. This neural network can be used in function fitting applications. For a lower temperature, g(x) approaches a step function and is used in pattern recognition. The temperature is relatively arbitrary as the neural network can adjust itself accordingly. The temperature is chosen to be 1, which is a common choice for the separation of two samples. The weights and thresholds of the neural network can be obtained by minimizing an error function using a training sample, for which the event classification is known. The error function is defined as: where the sum runs over all training patterns. o p is the output of the neural network from the training sample. t p is the expected output from the training sample. For minimizing the error function, the back propagation technique is used and is introduced as follows. The weights are updated during each iteration of the training pattern p: (4.2) (4.3) p W: r(P+l) _ jk - (4.4) Chapter 4. E V E N T P R E S E L E C T I O N Figure 4.2: g(x) as a function of temperature Chapter 4. E V E N T P R E S E L E C T I O N 70 Aw- 1 / can be obtained from: Awjj? = - t j | ^ + a A w j T ^ (4.5) where rj is the step size parameter controlling the convergence rate, a is the mo-mentum parameter which is between 0 and 1. After the neural network is optimized using the training samples, test samples are used to test the neural network performance. The neural network can learn the specific features of training samples by using a very complicated neural network structure. If neural network output distributions from the test samples are different from those of the training samples, the training sample size, n, a and neural network structure should be adjusted. Here the neural network structure means number of layers, number of input variables and nodes. To optimize the training procedure, the following rules should be considered: - Input variables are chosen based on separation power and minimal correlation. - A l l input variables should be scaled to values of the order of one. If the order of magnitude of the input values varies a lot, the training will be very slow. - The number of training events in the different classes should be equal. The training events of the different classes should be presented to the neural net-work alternately. - For a given number of hidden nodes, twice the number of inputs should be sufficient for any problem, but fewer hidden nodes are often used. For the size of the training sample, the more the better. One thousand training events per hidden node should be sufficient. - Parameters such as the step size n and the momentum a should be optimized. Chapter 4. E V E N T P R E S E L E C T I O N 71 The temperature is not a critical parameter, as long as no extreme values are chosen. 4.2 Multi-hadron selection The current analysis uses O P A L data from Z decays. The multi-hadron Z —> qq selection criteria [90] (where q can be u, d, s, c, b) are applied first to suppress the background events from Z to lepton pair decays. In this multi-hadron event selection, good electromagnetic clusters are defined as having: 1- E R A W > 0.100 GeV in the barrel or E R A W > 0.200 GeV in the endcap, where E r a W is the uncorrected energy of the cluster; 2. Nbiocks > 1 in the barrel or Nbiocks > 2 in the endcap, where N b i o c k s is the number of adjacent blocks in the cluster. Good tracks are defined as having: 1- Nh i t > 20, where N h i t is the total number of hits in the vertex, jet and z tracking chambers; 2. |do| < 2 cm, where |d 0 | is the impact parameter; 3. \z0\ < 40 cm, where \z0\ is the z coordinate of the track at the point of closest approach; 4. p x y > 0.050 GeV/c , where p x y is the track momentum transverse to the z axis; 5. | cos#| < 0.995, where 9 is the azimuthal track angle; 6. Xr-<i> < 999, where Xv-<p i s the %2 calculated from the track fitting in r — (j) plane; Chapter 4. E V E N T P R E S E L E C T I O N 72 7. x\ < 9 9 9 , where x\ is t n e X 2 calculated from the track fitting in r — z plane. Events are selected as multi-hadron events if they satisfy all of the following con-ditions: 1- R-vis > 0 . 1 0 , where R v i s = r a w and the summation runs over all good clusters; 2 . i R b a i l < 0 . 6 5 , where R b a i = c ° s 6 and 9 is the polar angle of the cluster; 3. N t r a c k s > 5, where N t r a C k S is the number of good tracks; 4. N c i u s t e r s > 7 , where N c i u s t e r is the number of good clusters. The main background in the multi-hadron selection is from Z — > • T + T ~ and two-photon events, which account for ( 0 .11 ± 0 . 0 3 ) % and ( 0 .5 ± 0 . 0 2 ) % of selected multi-hadronic events respectively [90]. The multi-hadron selection efficiency is 9 8 . 4 % [90]. The selection of b hadron events uses tighter requirements for tracks, which are: - the number of hits in the jet chamber greater than 2 0 , - the track momentum less than 6 5 GeV/c , - the track transverse momentum to the z axis greater than 0 .15 GeV/c, - | cot t9| less than 1 0 0 , where 9 is the track polar angle from the z axis, - xU < ioo, -xl< ioo. The selection of b hadron events also requires tighter cuts for electromagnetic clus-ters, which are: - the corrected energy of barrel clusters greater than 0 .2 GeV, Chapter 4. E V E N T P R E S E L E C T I O N 73 - the corrected energy of endcap clusters greater than 0.1 GeV. 4.3 b identification A neural network algorithm [89] based on charged particle vertex information is used to separate the b flavour events from the other flavour events in each hemisphere. The O P A L BTag [89, 91] package is used for the thrust and jet finding, primary and secondary vertex reconstruction, b tagging and charged track quality selection. Both hemispheres are searched for lepton candidates. 4.3.1 Jet finding The primary quark and anti-quark pair from electron and positron collision pro-duces a back to back pair of jets due to momentum conservation, as shown in Figure 4.3. The primary quark may radiate gluons and produce additional jet(s), as shown in Figure 4.4. A cone jet finding algorithm [92, 93] is used to define the tracks and clusters into jets. In this analysis, the total energy in a jet greater than 10.0 GeV and a cone of half angle greater than 0.7 radian are used to define b jets. 4.3.2 Primary and secondary vertex reconstruction The primary vertex (b hadron production point) position, shown in Figure 4.5, is calculated using a three-dimensional Y 2 minimization technique. The primary vertex is found by using the tracks from only one hemisphere plus a beam spot constraint. A common vertex is fit to all tracks in the hemisphere and the beam Chapter 4. E V E N T P R E S E L E C T I O N 74 Run:evenf 4243: 25225 Date 930702 Tire 11600Ctrk(N= 22 Sump= 58.9) Ecal(N= 36 SumE= 25.3) Hcal(N=20 Surc£= 14,9) Ebeam 45.605 Evis 85.1 Bniss 6.1 Vtx ( 0.00, 0.09. 1.44) Muon(N= 4) Sec Vlxfffc 0) Fdet(N= 0 SumE= 0.0) Bz=4.350 Thrust=0.8900 Aplan=0.0088 Oblal=0.1583 Spher=0.0491 Figure 4.3: A 2 jet event from the O P A L event display. Tracks within inner detectors are depicted as curved lines. The hits in the calorimeters are depicted as filled rectangles, with energy deposited proportional to the rectangle area. The hits in the muon chambers are shown as arrows and X's at the outermost detector. Chapter 4. E V E N T P R E S E L E C T I O N 75 Run:event 1922 : 53868 Dale 900806 Time 134222Ctrk(N= 75 Sunp= 56.6) Ecal(N= 75 Sunfe 65.8) Heal (N=23 Sur£= 11. Ebeam 45.137 Evis 104.4 Bmiss -14.1 Vtx ( -0,03, 0.08, -0.49) Muon(N= 1) Sec Vtx(N= 8) Fdet(N= 0 Sun£= 0. Bz=4.350 Thrust=0.7381 Apian=0.0617 Oblal=0.2896 Spher=0.4732 , Tmrss: 5 10 20 50 Gev Centre ol screen is ( 0 0000, 0.0000, 0.0000) | | | ] Figure 4.4: A 3 jet event from the O P A L event display. See Figure 4.3 for the explanation of tracks and hits. Chapter 4. E V E N T P R E S E L E C T I O N 70 spot. Tracks contributing a x2 °f greater than four are iteratively removed from the fit. If no tracks remain after this procedure, the beam spot is used instead. The (Primary vertex) Figure 4.5: The beam spot, primary vertex and secondary vertex in b decay. secondary vertex (b hadron decay point) is reconstructed using a similar method to the primary vertex reconstruction. The following track quality constraints are required for secondary vertex track candidates: - the track momentum greater than 0.5 GeV/c , - the impact parameter with respect to the reconstructed hemisphere primary vertex less than 0.3 cm, - the error on the track impact parameter with respect to the reconstructed hemisphere primary vertex less than 0.1 cm. Chapter 4. E V E N T P R E S E L E C T I O N 77 4.3.3 b vertex flavour tagging The relatively long lifetimes of b hadrons, combined with the boost provided in Z decay, give rise to b track lengths of a few millimeters. This lifetime information can be used to select b hadron samples. Typically, either several charged particle tracks with impact parameters with respect to the primary vertex significantly larger than measurement errors are required, or a second vertex is sought with a significant decay length. With the installation of silicon micro-vertex detectors, b vertex tagging becomes possible. The standard O P A L b tag (BT) neural network 5 [89, 91] is used to separate the b flavour events from other flavour events. The B T neural network uses the number of tracks in the secondary vertex, the vertex decay length L, the decay length significance L/cr L , the reduced decay length significance and the critical track discriminant [91] as input variables. The vertex decay length L is calculated as the length of the vector from the primary to the secondary vertex. L is given a positive sign if the secondary vertex is displaced from the primary in the direction of the jet momentum, and a negative sign otherwise. The critical track discriminant represents the probability that a set of tracks with invariant mass greater than the average charm hadron mass is consistent with having originated from the reconstructed secondary vertex. The b decays tend to have a large number of tracks in the secondary vertex due to the high mass of b hadrons, longer decay length and larger decay length significance due to the longer life of b hadrons. A vertex tagging variable B is defined as: |B | = - l n ( l - N N b ) , (4.6) where N N b is the B T neural network output. B is defined as having the same sign as the vertex decay length. The output from the B T neural network is required to be greater than 0.8, corresponding to vertex tagging variable |B | greater than 1.6, Chapter 4. E V E N T P R E S E L E C T I O N 78 which is shown in Figure 4.6, resulting in a b purity of more than 91% and an effi-ciency around 30%. The b purity increases after subsequent b hadron semileptonic decay neural network selection, which is described below. The thrust axis polar angle | cos#| is required to be less than 0.9 so that events will be well contained in the acceptance of the detector. Here the thrust T is defined as: where the summation i runs over all tracks and unassociated clusters in the event. is the momentum of the i t h track or unassociated cluster. The fi which maximizes the thrust value is called the thrust axis. Both hemispheres are searched for electron and muon candidates after the b identification. 4.4 Lepton selection Electrons are identified by the O P A L electron neural network [89, 94] using the track and calorimeter information. Before applying the neural network, the following cuts are required: - Candidate track momentum is required to be greater than 2 GeV/c. - The number of z chamber hits associated with the candidate track is required to be greater than 3. The number of jet chamber hits used to calculate the dE/dx is greater than 40. - The average specific energy loss of ionizing particles can be described by the Bethe-Bloch equation [2]: (4.7) dE/dx = -47rN A r 2 m e c 2 z 2 Z 1 1 2m ec max - £ 2 - ^ ) , (4.8) I 2 Chapter 4. E V E N T P R E S E L E C T I O N 79 Vertex tagging variable B = -ln(l-NNb) Figure 4.6: Distribution of the vertex tagging variable B for O P A L data and Monte Carlo simulated events. The contributions from uds, c and b jets are indicated [89]. A cut 1.6 in B is used, corresponding to B T neural network output N N b cut 0.8. Chapter 4. E V E N T P R E S E L E C T I O N 80 where N A is Avogadro's number, r e is the classical electron radius and m e is the electron mass. z e is the charge of the the incident particle. Z and A are the atomic number and mass of the medium. T m a x is the maximum kinetic energy which can be imparted to a free electron in a single collision. I is the mean ionizing potential and 5 is a density function. The dE/dx distributions for the electron, muon, pion, kaon and proton from O P A L data are shown in Figure 4.7. d E / d x | n o r m is defined as: dE/dx| d E / d x m e a s u r e d dE/dx e xp e C(, e ci ^ ^ a(dE/dxeXpeCted) where d E / d x m e a s u r e d is the raw measured value and dE/dx e x p ected is the ex-pected value according to the Bethe-Bloch equation. ( d E / d x ) n o r m is required to be greater than -2. The separation power S between particle a and b based on dE/dx can be defined as: S = d E / d x a - d E / d x b y a 2 ( d E / d x a ) + a 2 ( d E / d x b ) where dE /dx a and d E / d x b are the dE/dx for particle a and b respectively. cx 2(dE/dx a) and <7 2(dE/dx b) are the measurement errors on the specific ion-ization for particle a and b. The separation power depends on the particle momentum which is shown in Figure 4.8. The six inputs to the electron neural network are: 1. the track momentum; 2. the track polar angle; 3. the electromagnetic energy to momentum ratio; 4. the number of electromagnetic calorimeter blocks contributing to the energy measurement; Chapter 4. E V E N T P R E S E L E C T I O N 81 Figure 4.7: dE/dx from O P A L data for the electron, muon, pion, kaon and proton [80]. The dot points show data. The curves show the Bethe-Bloch predictions. Chapter 4. E V E N T P R E S E L E C T I O N 82 1 10 momentum (GeV/c) Figure 4.8: The dE/dx separation power from O P A L data for the electron, muon, pion kaon and proton [80]. Chapter 4. E V E N T P R E S E L E C T I O N 83 5. the normalized ionization energy loss dE/dx; 6. the error on the normalized ionization energy loss. The electron neural network output is shown in Figure 4.9. The neural network output is required to be greater than 0.9. The resulting electron efficiency is approx-imately 74% with a purity of 94%. Electrons from photon conversions, 7 —> e+e~, NN e output Figure 4.9: The electron neural network output N N e for the true electron signal (hatched area) and for the total signal (solid histogram) in the Monte Carlo simulated events. The arrow shows the selected region. The O P A L data (points) are compared to the Monte Carlo simulated events [28]. Chapter 4. E V E N T P R E S E L E C T I O N 84 contribute a significant background to the prompt electron samples. Another neu-ral network is used to reject this background [89]. The nine input variables to this photon conversion neural network are: — the distance between two tracks at tangency; — for both tracks, the radius of the first measured tracking chamber hit with respect to the center of the O P A L detector; — the radius of the reconstructed vertex of the candidate photon conversion; — the invariant mass of the pair, assuming both tracks to be electrons; — the impact parameter of the reconstructed photon with respect to the primary vertex of the event; — the electron identification neural network output of the partner track; — for both tracks, the product of the momentum and charge. The photon conversion background is reduced by 94% after requiring the photon conversion neural network output less than 0.8, which is shown in Figure 4.10, whilst retaining 98% of the selected prompt electrons. The muon momentum is required to be greater than 3 GeV/c . Muons are identified using reconstructed track segments in the muon chambers [89]. The reconstructed tracks in the central detector are extrapolated to the muon chambers to see if they match the track segments reconstructed in the external muon chambers. The position matching parameter xPos is required to be less than 3, where Xpos is defined as: Xl„ = (^f + (4.11) where A9 and A(p are the differences in 9 and <f> between the extrapolated track position from the central detector and the nearest muon segment. aA9 and oAcf) Chapter 4. E V E N T P R E S E L E C T I O N 85 Figure 4.10: The photon neural network output N N C V for O P A L data (points) and Monte Carlo simulated events (histogram). The darkly hatched region shows the electron from photon conversion. The lightly hatched region shows the prompt electron. The open region shown non-electron candidates. The tracks are selected with N N C V < 0.8 to eliminate the electron background from the photon conversion [28]. Chapter 4. E V E N T P R E S E L E C T I O N 86 are the errors of A# and A</>. The measured energy loss dE/dx is also required to be consistent with the expected value for a muon. The muon selection efficiency is approximately 90% and the muon purity approximately 93%. Electron and muon momenta transverse to the direction of the jet containing the lepton are required to be greater than 0.5 GeV/c in order to reject leptons from light quark decays. The lepton is included in the calculation of the jet direction. 4.5 B semileptonic decay selection A neural network [95] based on lepton information is used to separate the b hadron semileptonic decays, b —> X c £ u and b —>• Xu£z^, from non semileptonic decays. Eight input variables to this B semileptonic decay selection neural network are: — the total momentum of the lepton candidate; — the lepton transverse momentum with respect to the nearest jet axis, where the jet excludes the lepton candidate itself; — the energy of the jet containing the lepton candidate; — sub-jet energy, where the sub-jet contains the lepton candidate. The lepton jet is divided into two sub-jets, as described in [88]; — the scalar sum of transverse momenta of charged tracks in the lepton jet; — the impact parameter of the lepton track with respect to the primary vertex, divided by the error on the distance; — lepton Qjet, i.e. the charge of the lepton candidate multiplied by the jet charge of the jet containing the lepton, including the lepton. The jet charge is the Chapter 4. E V E N T P R E S E L E C T I O N 87 weighted sum of all track charges in the jet, Q * = E i S D X c / 5 ' (4-12) where Q ; is the track charge, p; is the track momentum. The summation runs over all charged tracks in the jet; — opposite Qjet , i.e. the charge of the lepton candidate multiplied by the jet charge of the most energetic jet in the hemisphere opposite to the lepton. The distributions of the neural network output variable are shown in Figure 4.11. A neural network cut greater than 0.8 is applied. After this neural network b semileptonic decay selection, the b hadron semileptonic decay purity is 97% and the efficiency is 65%; the c —> I events, where c is primary quark, and b —> c —> I events are suppressed. Neural Network Output Neural Network Output Figure 4.11: a,b. The b hadron semileptonic decay neural network output distributions, a for the b semileptonic decays and the scaled background from the Monte Carlo simulated events, b comparison between O P A L data and Monte Carlo simulated events. The selected region is shown by the arrow in b. Chapter 5 I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y This chapter describes a neural network to separate the b —> X u £ u decay from the background after the event preselection in Chapter 4. The input variables to the neural network are described in detail. The b —>• X u £ u model uses the b —> X u £ u hybrid model described in Chapter 2. 5.1 b to u neural network It is difficult to extract b —» X u £ u decays from the dominating b —» X c £ u background using only one single kinematic variable. A multi-layered feed-forward artificial neural network based on the J E T N E T 3.0 program [96] is used to extract b —> X u £ u events. There are four layers in this neural network. The neural network structure is 7-10-10-1. In the first layer, seven variables are used as inputs to the neural network. Each of these variables has separation power between b —» X u £ u and b —» X c £ u . The last layer is the neural network output variable which combines the separation power from these seven input variables. A figure of merit [97] is used to determine the discrimination power of 88 Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 89 these seven variables in separating two classes of events, i.e. signal and background. The figure of merit is defined as: ( f i W - f 2 ( x ) ) 2 OL\OL2 J a i i i ( x ) + a i 2 f 2 ( x j where fi(x) and f2(x) are normalized distributions for two classes, class 1 and class 2. a\ and a 2 are the fractions of class 1 and class 2 in the total samples. The higher the figure of merit, the better the separation between the two classes. In selecting these neural network input variables, nineteen variables were considered as input variables to this neural network initially, listed as below: 1. the invariant mass of the most energetic final state particle combined with the lepton, 2. the lepton energy in the b hadron rest frame, 3. the lepton momentum transverse to the jet axis (the jet axis calculation includes the lepton), 4. the transverse momentum of the most energetic particle with respect to the lepton direction, 5. the rapidity of the most energetic final state hadron calculated with respect to the lepton direction (assuming all hadronic particles are pions), the rapidity y is defined as: y = 2 l 0 g E — p ? ( 5 - 2 ) where P L is the longitudinal momentum along the direction of the incident particle and E is the energy, both defined for a given particle, 6. the fraction of the reconstructed b hadron energy carried by the lepton, Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 90 7. the reconstructed hadronic invariant mass, 8. hadronic invariant mass of the charge particles, 9. charged particle multiplicity, 10. invariant mass of the most energetic and the second most energetic particles, 11. invariant mass of the most energetic and the third most energetic particles, 12. sum of the rapidity of the charged particles with respect to the lepton axis, 13. rapidity of the lepton with respect to the hemisphere axis, 14. invariant mass of the most energetic particle and the fourth most energetic particle, 15. invariant mass of the second, the third and the fourth most energetic particles, 16. invariant mass of the first, the third and the fourth most energetic particles, 17. transverse momentum of the second most energetic particle with respect to the lepton direction, 18- Ej p(j)transverse/ Ej p(j), where p(j) is the momentum of the j t h particle and p(j) transverse is the transverse momentum of the j t h particle with respect to the lepton direction, 19- Ej (p(j)transverse) • The last twelve variables are discarded for poor separation power between b —>• Xu£u signal and background or poor agreement between the data and Monte Carlo simulated events. The first seven variables are selected as inputs to the b —> Xulv neural network and the figures of merit for these seven variables are 0.057, 0.034, 0.018, 0.016, 0.012, 0.011 and 0.005 respectively, as ordered in the list above. The figure of merit of the reconstructed Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 91 hadron invariant mass is relatively low because of the poor resolution of the reconstructed hadron invariant mass. The seven input variable distributions for the b —> X u £ u and the background in the Monte Carlo simulation are shown in Figure 5.1. The agreement between Monte Carlo simulated events and O P A L data for these seven variables is shown in Figure 5.2. The correlations of these seven input variables, which are marked as V I to V7 in the same order as list above, are shown in Figure 5.3 and Figure 5.4. The correlations of these seven input variables are not high and these seven variables can be used as inputs to the neural network. The input variables, hadronic invariant mass and the lepton energy in the b hadron rest frame, are discussed in detail in the following sections. 5.1.1 Hadronic invariant mass The reconstructed hadronic invariant mass M x , the last input variable to b —¥ X u £ v neural network, can be obtained from: M x 2 = E ( W i E i ) 2 - £ ( W i P i ) 2 > (5.3) i i where i denotes all hadronic tracks and clusters. W; is the track or neutral cluster weight from O P A L b hadron reconstruction code [98], which is equal to the probability that a hadronic track or a neutral cluster comes from b decay. E; and pi are the energy and momentum of the i t h hadronic track or neutral cluster. The track and cluster weight are used to separate tracks and clusters from the true b production and the fragmentation. The track weight is obtained by combining the track vertex weight and the rapidity-based neural network weight and is shown in Figure 5.5. The track vertex weight and the rapidity-based neural network are described in detail in [98]. Approximately thirty percent of the b hadron energy is in neutral final state particles. A l l clusters in the electromagnetic and hadronic calorimeters are considered in the cluster Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 92 5 6 Mass (GeV/c2) > 0)15000 o m CM ^10000 w •g 5000 c LLI • — j ~ ' b ^ u 2> background 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Lepton Energy (GeV) o > co O d 10000 <g 5000 r c UJ ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 background 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Momentum (GeV/c) u > a> CJ LO O 10000 |g 5000 c UJ 0 1 2 3 4 5 6 7 8 9 10 Transverse Momentum (GeV/c) (N15000 QIOOOO c UJ 5000 M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - b -> u 5) background 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Rapidity LO CM o 6 3000 (A .2 2000 c UJ 1000 '1 1 1 1 11 1 1 1 11 1 1 1 '11 - b ^ u 6> background 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fraction ^10000 co 5000 r co c UJ Mass (GeV/c ) Figure 5.1: Comparison between the signal and the background in the Monte Carlo simulation for the seven b to u neural network input variables. Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 93 O P A L o %1 O in d W c HI 5000 • data ^ Monte Carlo Mass (GeV/O O10000 > O 7500 r CNI ° 5000 r (0 0) ™ 2500 UJ 1 I I I • A 3) data Monte Carlo • ' i •••• i ' i : - 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Momentum (GeV/c) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Lepton Energy (GeV) o ^10000 u in 7500 d 5000 </> 0) "5 2500 -c m 0 I l > l l l l l l l l l l l l l l l l l l l l 4) 4 data Monte Carlo ] r * ^ - * - - - j 2 3 4 5 6 7 8 9 10 Transverse Momentum (GeV/c) m ^10000 r o co 7500 CD C 5000 UJ 2500 0 • i 1 •111 • • • • i • m • i • • • • i • • data 5 ) Monte Carloj 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Rapidity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fraction CM o >10000 2 7500 r CO © 5000 0) 2500 c LU data 7) Monte Carlo 3 4 5 2 Mass (GeV/c) Figure 5.2: Comparison between O P A L data and Monte Carlo simulated events for the seven b to u neural network input variables. Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y Figure 5.3: The correlations between different variables. Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y Figure 5.4: The correlations between different variables (continued). Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 96 co o 1 r o CD 3 0.8 F; 0.6 "5 0.4 .1 0.2 |-u CO I o TJ i i i I i i i i | i i i i | i i i i | i i i i | i i r r - p r T T i | i i i i | i i i i | • Monte Carlo a) y=x r ' 1 • • 1 • • • • 1 • • • • 1 1 • 1 • 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Track Weight S 0.12 E-0.1 b 0.08 0.06 f 0.04 0.02 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 • data b) — Monte Carlo ' ' 1 • 1 • • 1 • • • • 1 • • • • 1 • • • • 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Track Weight Figure 5.5: a,b. a Fraction of true b decay vs track weight from Monte Carlo simulated events; b comparison of the track weight between O P A L data and Monte Carlo simulated events. Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 97 weight procedure. The rapidity of each cluster is calculated and then a weight is assigned according to the parameterization performed in several bins of its momentum. This weight represents the probability that this cluster comes from the b hadron rather than the fragmentation. The cluster weight is shown in Figure 5.6. Most neutral clusters are detected in the electromagnetic calorimeter. Only a small portion of neutral clusters are in the hadronic calorimeter. The parameterization from momentum underestimates the cluster weight in the hadronic calorimeter for the cluster weight of 0.0-0.2 region as shown in the top plot of Figure 5.6. As only a small fraction of neutral clusters is in the first two bins, the reconstructed hadronic invariant mass will not be affected. The hadronic mass of X u system in B —> X u £ u is expected to be smaller than the hadronic mass of X c in B —» X c £ v due to the smaller mass of the u quark. 5.1.2 The lepton energy in the b hadron rest frame The b hadron direction is reconstructed using O P A L b reconstruction code [98]. The re-construction is based on the principle of assigning weights to charged tracks and neutral clusters, which are the same as the weights used in the reconstruction of the hadronic invariant mass Wj in Equation 5.3. The b hadron momentum is then reconstructed by summing the weighted momenta of hadronic tracks and clusters, with the lepton in the hemisphere. The b hadron flight direction is determined by the vector of b hadron mo-mentum. The b hadron flight direction can also be calculated from the primary vertex and the secondary vertex. The b hadron direction resolution can be improved by com-bining the direction obtained from the b hadron momentum and the flight direction. The combination method is simply based on the weight from the angular resolution of the b direction from the momentum sum method and the flight direction. The comparison of the reconstructed b hadron direction for the data and the Monte Carlo simulation is shown in Figure 5.7. The difference between the reconstructed b hadron direction in Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 98 o 1 L 0 ) o 0 8 ^ SI § 0.6 F £ 0.4 .1 0.2 +-> u s o I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Monte Carlo a) ^ • ^ * H T f l --- y=x •'• • • 1 • • • • 1 • • • • 1 • 1 • • 1 • • • 1 1 I i i i i I i i i i I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Cluster Weight CM ° . 0.12 E-o Z 0.1 7 zO.08 7 "~ 0.06 0.04 0.02 0 " ' I ' ' ' • I • " ' I • • " I " 1 1 I " " I 1 " 1 I " "H data b) Monte Carlo 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Cluster Weight Figure 5.6: a,b. a Fraction of true b decay vs cluster weight from Monte Carlo simu-lated events; b comparison of the cluster weight between O P A L data and Monte Carlo simulated events. Chapter 5. IDENTIFICATION OF B TO U SEMILEPTONIC DECAY 99 Monte Carlo and the true b hadron direction in Monte Carlo is shown in Figure 5.8. The lepton energy in the lab frame is then boosted to the b hadron rest frame and is shown in the second plot of Figure 5.1. 1— 0.07 o z 0.06 •o z 0.05 0.04 0.03 0.02 0.01 0 CM O 9 0.035 •a 0.03 z 0.025 0.02 0.015 0.01 0.005 0 p j I I I I I I I I I I I I I I I I I i i i i I i i i i I i i i i I i i i i I i n data Monte Carlo i m i i -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 COS© 11 i i 1 1 i i i i i i T T i I i i i i I i i i i data Monte Carlo E ' • • • i • • • • 1 • • • • 1 • • • • 1 • • • • 1 5 6 7 O (Radian) Figure 5.7: a,b. The b hadron a cos# direction and b <f> direction comparison between O P A L data and Monte Carlo simulated events. 5.2 Discussion of b to u neural network outputs Twelve thousand b —> Xu£v events, which are simulated with the hybrid model described in Chapter 3 and which have passed the event preselection, and as many background Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 100 910000 b o jg 8000 |r | 6000 4000 2000 1 1 1 i i i i i i i i i i i i i i i i i i i i i i i 1 1 1 1 i a) -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 AO (Radian) j-12000 o 510000 = 8000 b LU 6000 E-4000 2000 0 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i_i b> HI 0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 AO (Radian) Figure 5.8: a,b. The difference between the reconstructed b hadron direction and the true b hadron direction in Monte Carlo simulated events for a 8 and b <f>. Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 101 events from the multi-hadron Z —> qq Monte Carlo simulation after the preselection are used to train the b —> X u £ v neural network. Two other samples of signal and background events of the same size are used to test the neural network performance. The neural network output distributions from b —> X u £ v and background are shown in Figure 5.9. Ninety percent of the background in this analysis comes from the b —> X c £ u decay, 6.8% d 4500 -(A 4000 -ffl 3500 -c LU 3000 -2500 -2000 : 1500 -1000 • 500 -a) 1 1 " " i " " i - b ^ u l v background 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neural Network Output ,.10000 d O 8000 1 b) c LU 6000 4000 2000 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 11 • • • • i • 1 1 1 1 • • • • i 1 b—>cl v b—> c —> 1 other c ^ l ^^^^ •1 — 1 — 1 • 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neural Network Output Figure 5.9: a,b. The b to u neural network output distributions, a for b to u signal and background, b for different background components. from the b —> c decay with the c subsequently decaying to a lepton. Another 0.6% comes from the c —> £ decay in which the c quark is the primary quark. Other background processes make up the remaining 2.6%, of which 36% is from the b —> r decay with the T subsequently decaying to an electron or a muon, and most of the rest of the "other" background is from a pion or a kaon misidentified as an electron or a muon. The figure of merit for signal and background vs epochs for the training and test samples are shown in Figure 5.10a. It can be seen that the neural network learns quickly and reaches good performance after several epochs and the figures of merit of the training sample and test sample are similar. The number of training epochs should not be chosen Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 102 too high as the neural network will become over-trained in the high epochs and the figure of merit will drop gradually. The choice of training epochs is not critical and is chosen as 30 in this neural network training. The signal purity and efficiency vs neural network E p o c h Neural Network Output Figure 5.10: a,b. a The figure of merit between the signal and the background vs epochs for the training and test samples; b the signal purity and efficiency vs the b to u neural network output for the training sample. output for the training sample are shown in Figure 5.10b. The agreement between the training sample and the test sample from the b —> X u l v neural network output are shown in Figure 5.11. The b —>• X u £ v signal is divided into B —> irlv, B -> p £ v , B -> r\h> and B —>• utv and the corresponding neural network outputs are shown in Figure 5.12. The neural network outputs for the exclusive model and the inclusive model used in the hybrid model are shown in Figure 5.13. The neural network outputs from B —> D £ u , B —> D*£u and B —> D**£u in the background are shown in Figure 5.14, The signal b —> X u £ v purity in the last bin of the neural network distribution is expected to be higher than in all other bins. This corresponds to a lower hadronic Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 103 0.14 0.12 0.1 0.08 0.06 0.04 0.02 1111111111111111111111111111111111111111111111 T^TT, j ' " . " " ; a) • test sample r^* — training sample..: ' — ' 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Neural Network Output 0.25 0.2 0.15 0.1 0.05 0 i 1 1 1 1 1 11111111111 b) test sample training sample i . . . . i ' — i • 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neural Network Output Figure 5.11: a,b. Comparison of neural network output distributions for a the signal and b the background of the training sample and test sample. Chapter 5. IDENTIFICATION OF B TO U SEMILEPTONIC DECAY 104 15 F 10 k 5 h • PN — ~ £15 r 1^0 l -w 5 ¥ 0.1 0.1 0.1 0.2 0.2 0.2 M • • • • i • • a)B-^ x x 0.3 0.4 i • • • • i • b ) B ^ • * • • • • * • 0.3 0.4 i • • • • i • • c)B-> x X 0.3 0.4 i • • • • i • n d) B-> 1 1 1 1 1 1 1 71 1 V 0.5 0.6 • i 1 • • • i p 1 v 0.5 0.6 • i • • • • i •n i V r _ l — X 0.5 0.6 co 1 v 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neural Network Output Figure 5.12: a,b,c,d. The neural network output distributions for a B to ir, b B to p, c B to rj and d B t o w semileptonic decay. Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 105 ,-350 5 300 8 250 ~200 w 150 100 50 0 i • • • • i • • • • i • • • • i 1 • • • i •1 • • i • •1 11 • • 1 1 1 • • • • i a) ISGW2 model 1 • — • • • • 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neural Network Output 140 t OS120 •2 100 c LU 80 60 40 20 0 i • • • • i • • • • i • • • • i • • • • i • • • • i • • • • i • • • • i b) ACCMM model • • • • • • — • — • — • 1 1 1 1 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neural Network Output Figure 5.13: a,b. The neural network output distributions for a the ISGW2 model b the A C C M M model in the hybrid model. Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 106 ^10000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neural Network Output Figure 5.14: The neural network output distributions for B to D, D* and D** semileptonic decays. Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 107 invariant mass and a higher lepton momentum in the b hadron rest frame in the last bin compared to all other bins from the data, as on average the sample has lower hadronic invariant mass and higher lepton momentum in the b hadron rest frame. This is shown in Figure 5.15. Chapter 5. I D E N T I F I C A T I O N O F B T O U S E M I L E P T O N I C D E C A Y 108 CM O > 0 o ^ -co 0 • mmmm V . +-> C LU 0 o m CM (0 0 0.2 0.175 0.15 0.125 0.1 0.075 0.05 0.025 0 0.25 0.2 0.15 •s 0.05 c LU a) - no cut cut 0.9 0 1 2 3 4 5 6 27 V7, Hadronic Invariant Mass (Gev/c ) • • • i • 1 • • i • • • 11 • 1 • • i • 1 • 11 • • " i • 0.1 \r — no cut cut 0.9 T b) I I I I 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 V2, Lepton Energy in B Rest Frame (GeV) Figure 5.15: a,b. a The hadronic invariant mass from O P A L data, b the lepton energy in the b hadron rest frame from O P A L data, with no b to u neural network cut and with b to u neural network cut 0.9 applied. Chapter 6 R E S U L T S This chapter describes the measurement of Br(b —> Xu£v) using the b -» Xu£u neural network. The systematic error analysis and cross checks of Bv(b —> Xu£u) are presented. The extraction of | V u b | from Br(b —>• Xu£u) is discussed. 6.1 The branching ratio of b to u semileptonic decay The branching fraction of b —> Xn£u decay can be obtained from the best fit of the Monte Carlo simulated events to O P A L data based on the b —> Xu£u neural network output distributions. Bv(b —>• Xu£u) is extracted from the b —>• Xn£v neural network output distributions by minimizing: [Nda t , _ N d a t a ( x f k M C » " + (1 - x)C^)Y k i N k where N £ a t a is the number of events from the data in the k t h bin of the b —> Xu£u neural network output. N d a t a is the total number of events in the data after preselection. The free parameter x is the fraction of signal events in the data after preselection, which can be converted to Br(6 —> Xu£u) based on the number of signal events and the number of background events in the Monte Carlo simulation after preselection. f^ C b u is the 109 Chapter 6. R E S U L T S 110 fraction of simulated signal events in the k bin of the b —>• Xu£u neural network output with respect to the total number of simulated signal events after preselection. f^ C b g is the fraction of simulated background events in the k t h bin of the b —> Xu£v neural network output with respect to the total number of simulated background events after preselection. Here the background includes b —> Xc£u, b—> c ^ £, c—> £ and other contributions. The sum over the index k is performed from the neural network cut to the last bin in the neural network output distribution. The Br(6 —> Xu£u) from the fit result x, as well as its statistical and systematic errors, depends on the b —>• Xu£u neural network cut. The resulting Br(b —> Xu£u) is stable, with variations less than 0.24 x 10~3 as the neural network cut varies in value from 0.3 to 0.8. A neural network cut of 0.7 is chosen to minimize the total relative errors and yields Bv(b -> XJv) = (1.63 ± 0.53) x IO" 3 , where the uncertainty is the statistical error only. The total relative errors are defined as the combination of the statistical error and the systematic error of Br(b —> Xu£u) divided by Br(b —> Xu£v) and are also shown in Table 6.1. In Figure 6.1, the neural network output from the Monte Carlo simulation events with no b —> Xu£v semileptonic decay is shown and the excess of events in the data can be seen in the highest bin. Here the Monte Carlo sample is normalized to the same number of entries as the data. The last bin of neural network output from the data contains 869 events. According to the Monte Carlo simulation, there would be 775 events in the last bin without the b —» Xu£v semileptonic transition. The excess of events in that bin is 94 ± 31. The x 2 / n d f is 14.6/9, which corresponds to a 10% confidence level when one presumes no contributions from b —> Xu£v transition. Here the x 2 is calculated using all bin information. When the b —> Xu£u transition is incorporated in the Monte Carlo simulation with a branching fraction of 1.63 x 10~3, the Monte Carlo simulation agrees Chapter 6. R E S U L T S 111 Neural network Bv(b -+ XJi>) Statistical error Systematic error Total relative cut x l 0 ~ 3 x l O " 3 x l O - 3 error 0 1.45 0.52 +0.82 - 0 . 8 7 + 6 7 % - 7 0 % 0.1 1.51 0.53 +0.62 -0 .73 +54% - 6 0 % 0.2 1.50 0.53 +0.61 -0 .70 +54% - 5 9 % 0.3 1.54 0.53 +0.59 - 0 . 6 7 + 5 1 % - 5 5 % 0.4 1.55 0.53 +0.59 -0 .65 + 5 1 % - 5 4 % 0.5 1.57 0.53 +0.60 -0 .68 + 5 1 % - 5 5 % 0.6 1.60 0.53 +0.57 -0 .66 + 4 9 % - 5 3 % 0.7 1.63 0.53 +0.55 -0 .62 + 4 7 % - 5 0 % 0.8 1.78 0.62 +0.62 -0 .72 + 4 9 % - 5 3 % 0.9 2.03 0.72 +0.72 -0 .80 +50% - 5 3 % Table 6.1: Results of different neural network cuts to get the branching fraction of b to u semileptonic decay. Chapter 6. R E S U L T S 112 much better with the data, as can be seen in the right plot of Figure 6.1. The x 2 /ndf is then 8.3/8, corresponding to a 41% confidence level. W .S>10 k_ c H I 10 a) X 2 /ndf= 14.6/9 • OPAL data * ~ i — M C , no b - > u l v 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neural Network Output w .Sho +* c LU 10 \ b) X7ndf= 8.3/8 OPAL data M C with b -> u 1 v 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Neural Network Output Figure 6.1: a,b. The neural network output distributions for O P A L data and Monte Carlo simulated events, a with no b to u semileptonic transition in the Monte Carlo simulated events, b with a branching fraction of 1.63 x 10~3 b to u semileptonic decay incorporated. The distribution of Monte Carlo simulated events is normalized to the data for both plots. The data after subtracting the background from the Monte Carlo simulated events agree well with the simulated b —> Xu£v signal within statistical errors, which is shown in Figure 6.2. 6.2 Systematic errors and cross check The list of systematic errors on Br(b —> Xu£v) is given in Table 6.2. Unless otherwise specified, the systematic errors are evaluated by varying each parameter described by ± l a and taking the corresponding largest errors. A detailed description of these systematic errors are presented in Section 6.2.1 and a series of cross checks are performed and Chapter 6. R E S U L T S 113 300 © 250 a> 200 100 50 0 -50 i | i i i i | i i i i | i i i i | i i i i | i i i i | i n i | i a) • OPAL data — M C , b - > u l v 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Neural Network Output 200 150 h fcj 100 r a « 50 h 0 r -50 -100 b) OPAL data 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Neural Network Output Figure 6.2: a,b. The neural network output distributions, a Data after subtracting the background from the Monte Carlo simulated event (points) show agreement with the simulated b to u signal (solid histogram), b Data after subtracting the Monte Carlo simulated events with a branching fraction of 1.63 x I O - 3 b to u decay incorporated. Here the error bars include the statistical error from data and Monte Carlo simulated events. Chapter 6. R E S U L T S 114 presented in Section 6.2.2. 6.2.1 Systematic errors The resulting systematic error in Table 6.2 is discussed in detail: b quark fragmentation: Many parameterizations have been suggested to describe the heavy quark fragmentation process. The Peterson function [29] is used here to simulate the b and c quark fragmentation in the Monte Carlo simulation. The systematic error in the b quark fragmentation is estimated by varying the b hadron mean scaled energy (xE)b within the experimental range 0.702 ± 0.008 [99] recom-mended by the L E P Electroweak Working Group [99]. This value is consistent with a recent determination of ( x E ) b = 0.714 ± 0.009 from SLD [100]. The systematic error is also estimated from the Collins and Spiller fragmentation function [30] and Kartvelishvili fragmentation function [31] which are discussed below. The largest deviations are taken as systematic errors. The uncertainties of parameters from different fragmentation models are discussed below, where z is defined in Equation 2.4. 1. The Peterson function [29]: 0.0047 lo]ooo8 [2]- From the O P A L measurement, the b hadron mean scaled energy ( x E ) b is 0.702 ± 0.008, corresponding to a value of eq = 0.0038 Igiooos- F o r c Q u a r k fragmentation, (x E ) c is 0.484 ± 0.008, which corresponds to eq = 0.031 ± 0.006. 2. The function from Collins and Spiller [30]: f(z) = N( 1 - z 2 - z e )( l + z 2 ) ( l z 1 1 - z € (6.3) Chapter 6. R E S U L T S 115 Error Source Variation or ABr(6 ->• XJu) value and variation IO" 3 Fragmentation (xE)b 0.702 ± 0.008 [99] +0.28 -0.32 Lepton spectrum (b —>• c) ISGW** [35], ISGW [34] +0.18 -0.29 M C statistics (see text) ±0.22 b and c hadron semileptonic decay (see text) ±0.19 M C modeling (see text) ±0.19 b —> Xu£u modeling error (hybrid) (see text) ±0.19 b —> Xu£u modeling error (inclusive) Parton [58], Q C D [55] ±0.14 b —> Xn£v modeling error (exclusive) ISGW2 [37], J E T S E T [15] ±0.07 Tracking resolution ±10% [89] ±0.07 c hadron decay multiplicity (see text) ±0.07 Ab production rate (11.6 ± 2.0)% [2] ^0.04 Ab polarization -0.56 +_°0f6 [69] ±0.03 Electron ID efficiency ± 4 % [89] T-0.04 Muon ID efficiency ± 2 % [95] T-0.03 Electron fake rate ± 2 1 % [89] ^0.02 Muon fake rate ± 8 % [89] qFO.Ol Br(fr -> XTVT) (2.6 ± 0.4)% [2] ±0.01 b lifetime (1.564 ± 0.014) ps [2] < 0.01 Rb 0.21644 ± 0.00075 [2] < 0.01 Total +0.55 -0.62 Table 6.2: Systematic errors for the branching fraction of b to u semileptonic decay. Chapter 6. R E S U L T S 116 where N is the normalization factor. For b quark fragmentation, e is equal to (3.42 ± 0.62)xlCT 3 , corresponding to ( x E ) b = 0.698 ± 0.004 [95]. For c quark fragmen-tation, e is equal to 0.059 ± 0.032, corresponding to (x E ) c = 0.473 ± 0.017 [101]. 3. The Kartvelishvili et al. [31] function: f(z) = N z Q ( l - z ) , (6.4) where N is the normalization factor. For b quark fragmentation, a is equal to 10.04 ± 0.57 corresponding to ( x E ) b = 0.720 ± 0.005 [95]. For c quark fragmentation, a is equal to 4.02 ± 0.78 corresponding to (x E ) c = 0.484 ± 0.018 [101]. 4. The Lund symmetric function [32]: 1 b M 2 f(z) = N - ( l - z ) a e x p [ - ^ ] , (6.5) z z where N is the normalization factor, b quark fragmentation for this function is not studied yet. For c quark fragmentation, a is equal to 0.18 and b M ^ is equal to 1.55 ± 0.36 corresponding to (x E ) c = 0.478 ± 0.018 [101]. The systematic error for Br(b —> Xu£u) from c quark fragmentation can be neglected because there is a very small background from c —>• £, where c is a primary quark. 6 —¥ ~Kc£v lepton momentum spectrum modeling: Different decay models are used to predict the lepton spectrum in the b hadron rest frame for the b —> Xc£u decay. Although all models are derived for B° and B + semileptonic decay only, they are extrapolated to the B s and A b semileptonic decays. This will be correct in the simple spectator model and is a reasonable approximation for this analysis. The lepton spectrum from the A C C M M model [54] is used as a base model for the b -» Xc£v decay. In A C C M M model, the spectator quark in the B meson has 2 momentum distribution f(p) = -7^3-e p F , where p E is the Fermi momentum. The Chapter 6. R E S U L T S 117 C L E O Collaboration [103] has fit the A C C M M model to their data to determine p F equal to 298 MeV/c . The systematic errors due to b —> Xc£v lepton momentum spectrum modeling are estimated from the ISGW [34] and ISGW** [35] models. The lepton spectrum from the b —>• c —>• £ decay in the A C C M M model is different from the ISGW model. The systematic error due to the shape of the b —>• c —>• £ lepton spectrum is calculated and is found to be negligible. The lepton spectrum from the c —> £ decay is varied from the A C C M M model to the ISGW model, where the c quark is a primary quark from Z decay. The systematic error is calculated and found to be negligible. Monte Carlo statistics: The systematic uncertainty due to the limited Monte Carlo statistics is ± 0 . 2 2 x l 0 ~ 3 . b and c hadron decay properties: The systematic error is estimated from the uncer-tainties of the branching fractions of B° -> D~£+u, B° ->• D*~£+u: B+ -> D°£+u, B+ B*°£+v, B -»• D**£u and A b ->• AcX£u. There is a 6.8% background contri-bution from the b —> c —>• £ decays and a 0.6% background contribution from the c —>• £ decays. The systematic error is also estimated from the uncertainty of the branching fraction of the b —> c —> £ decays. A summary of the systematic errors from the uncertainties of b hadron and c hadron semileptonic decay branching ratios is shown in Table 6.3. The Br(B —>• D**£v) in Table 6.3 is obtained by averaging the Br(B ->• D**£v) from A R G U S [105], A L E P H [106], D E L P H I [107] and the total B semileptonic decay branching fraction subtracting the contribution from B to D and D* semileptonic decay, described by the LEP, C D F and SLD Heavy Flavour Working Group [5]. For the decay of B —> D**£v, in which D** refers to D i , D2, D 2 and D*, the branching ratio for each specific D** final state is not well measured. Chapter 6. R E S U L T S 118 For this analysis, the narrow final states of D** in B —> D**£v are replaced by the broad states and then vice-versa to check the sensitivity of the Br(b —>• Xu£u) to the relative ratio of the narrow and broad states of D** in B —¥ D**£u. The effect on the Br(b —»• Xu£u) is found to be negligible. Error Source Variation ABr(b X u ^ ) ( 1 0 - 3 ) Br(B° ^-D-£+u) Br(B° -> D*~£+v) B r ( B + -> D°£+u) B r ( B + ->• D*0£+u) Br(B D**&/) Br(b -» c -> £) B r ( A b -» A C X ^ ) (2.10 ± 0.19)% [2] (4.60 ± 0.27)% [2] (2.15 ± 0.22)% [2] (5.3 ± 0.8)% [2] (3.04 ± 0.44)% [5] (8.4 t°Qf9)% [95] (7.9 ±1.9)% [2] T0.02 ±0.03 q=0.06 ±0.04 ±0.16 T0.02 T0.06 Total ±0.19 Table 6.3: Systematic errors for the branching fraction of b to u semileptonic decay from uncertainties of the b hadron and c hadron semileptonic decay branching ratios. Monte Carlo modeling errors: The systematic error for the Monte Carlo modeling errors is estimated by re-weighting each input variable distribution in the Monte Carlo simulation to agree with the corresponding data distributions. A branching fraction of 1.63 x 10~3 for the b —» Xu£u transition is incorporated in the Monte Carlo simulation. This gives a conservative estimate of the systematic uncertainty due to the modeling of the input variables. b —>• ~X.u£v modeling error from the hybrid model: The boundary between the in-clusive and exclusive regions in the hybrid model is varied from 1.5 G e V / c 2 to 0.9 G e V / c 2 . This conservatively estimates the systematic error arising from the place-ment of the boundary between the inclusive and exclusive models. This produces Chapter 6. R E S U L T S 119 an uncertainty of ±0.19x10 3 for Bv(b —> X u £ u ) . b —> Xu£u inclusive model: The QCD universal function model and the parton model are used to evaluate the systematic errors in the inclusive part of the b —> X u £ u hybrid model. This gives a change of -0.14 x 10~3 for the Q C D universal function model and +0.02 x 10~3 for the parton model for the branching ratio of b —> X u £ u . The largest variation is taken as the systematic uncertainty. b -> X u £ i / exclusive model: The model implemented in the O P A L J E T S E T [16] Monte Carlo simulation, which has the u quark and the spectator quark forming one single hadron in the final state, is used to estimate the systematic error in the exclusive part of the b —t X u £ u hybrid model. Tracking resolution: The systematic error due to the uncertainties of the detector resolution is estimated by applying a ±10% variation to the r-<f> track parameters and an independent ±10% variation to the analogous parameters in the r-z plane to the Monte Carlo simulated events. c hadron decay multiplicity: The systematic error of the Br(b —> X u £ v ) associated with the c hadron decay charge multiplicity is estimated using the average charged track multiplicity of D + , D°, D+ decays as measured by M A R K III [102]. The systematic uncertainty of the Br(6 —> X u £ v ) is ± 0 . 0 7 x l 0 - 3 from the uncertainty of c hadron decay multiplicity. The systematic error of Bx{b —>• X u £ u ) associated with D decay charge multiplicity is shown in Table 6.4. Ab production rate: The P D G [2] gives the production fraction of B + , B°, B° and A b in Z decay as (38.9 ± 1.3)%, (38.9 ± 1.3)%, (10.7 ± 1.4)% and (11.6 ± 2.0)%. The neural network output variable distributions among B + , B° and B° are similar and the systematic effects caused by the uncertainties of the production fraction of Chapter 6. R E S U L T S 120 Error Source Variation A B r ( 6 ^ X u l i / ) ( 1 0 - 3 ) D° —> 0 prong 0.054 ± 0.011 [102] ±0.06 D° -> 2 prong 0.634 ± 0.015 [102] <0.01 D° —>• 4 prong 0.293 ± 0.023 [102] <0.01 D° —> 6 prong 0.019 ± 0.009 [102] <0.01 D + —y 1 prong 0.384 ± 0.023 [102] <0.01 D + —>• 3 prong 0.541 ± 0.023 [102] <0.01 D + —>• 5 prong 0.075 ± 0.015 [102] <0.01 —y 1 prong 0.37 ± 0.10 [102] ±0.04 —y 3 prong 0.42 ± 0.15 [102] <0.01 D+ —y 5 prong 0.21 ± 0.11 [102] <0.01 Total ±0.07 Table 6.4: Systematic errors for the branching fraction of b to u semileptonic decay from uncertainties of the D decay multiplicity. Chapter 6. R E S U L T S 121 B + , B ° and B ° are neglected. Due to the difference of the neural network output variable distributions between A b and B mesons, the fraction of A b is varied by a one standard deviation to determine the corresponding systematic error. A b polarization: According to the Standard Model, the b quark is longitudinally po-larized with a large average value of (P£) = -0.94 for a weak mixing angle of sin 2 # w = 0.23 [104] in Z decay. Any b quark polarization information is lost in the B meson due to the fact that the B meson has spin 0. As A b has spin | and the b quark mass is heavy, A b is expected to carry most of the b quark polarization. The lepton momentum spectrum from A b semileptonic decays depends on the degree of A b polarization. The systematic uncertainties are estimated by using the A b polarization range (P£ b ) = - 0 . 5 6 ±°0f6 [69]. Lepton identification efficiency: The number of selected events in the signal and background depends on the electron identification efficiency and the muon iden-tification efficiency. The electron identification efficiency has been studied using control samples of electrons from e+e~ —> e+e~ events and photon conversions, and is modeled to a precision of 4 . 1 % [89]. The muon identification efficiency has been studied using the muon pairs produced in two-photon collisions and Z —>• /J,+fJ,~ events, giving an uncertainty of 1.9% [95]. Lepton fake rate: Fake electrons in the electron sample are primarily from the charged hadrons (mainly charged pions) mis-identified as electrons and from untagged pho-ton conversions. The uncertainty associated with electron mis-identification is ± 2 1 % [89]. The muon fake rate is studied from K g —>• TT+IT~ and r —> 37r de-cay. The uncertainty of the fake muon rate is estimated to be ± 8 % . b —> ~KTVT branching ratio: One important composition in the "other" background in Chapter 6. R E S U L T S 122 Figure 5.9 results from a b quark semileptonic decay to a r lepton with the r lepton subsequently decaying to an electron or a muon. The branching ratio of b —> XTVT is (2.6 ± 0.4)% [2] and the systematic error is estimated using the uncertainties of the b —» XTVT branching ratio. Uncertainty of 6 lifetime: The average b hadron lifetime is measured to be (1.564 ± 0.014) ps [2]. The uncertainty in b lifetime results in a negligible uncertainty in Br{b -» X u l v ) . Uncertainty of R B : The fraction of Z —> bb events in hadronic Z decays, R B , is mea-sured to be 0.21644 ± 0.00075 [2]. The uncertainty in R B results in a negligible uncertainty in the background composition. 6.2.2 Cross checks of the result The fit procedure is applied to the electron sample and the muon sample separately. The Bv(b —>• X u £ v ) fitting result for the electron sample is (1.33 ± 0.86 (stat)) x 10~3 and the result for the muon sample is (2.12 ± 0.83 (stat)) x 10~3. The same procedure is also applied to the data between 1991 and 1993, and the data between 1994 and 1995, respectively. The Bv(b -> X J u ) results are (1.83 ± 0.75 (stat)) x 10" 3 and (1.49 ± 0.87 (stat)) x 1 0 - 3 for the data between 1991 and 1993 and the data between 1994 and 1995 respectively. Chapter 6. R E S U L T S 123 6.3 Measurement of | V U D | from the branching ratio of b to u semileptonic decay The b —> X u £ u decay width can be obtained from the Operator Product Expansion (OPE) [109, 110]: r = M | V u b r ( A 0 ( 1 - 4 = - i ) - 2 A + o ( ; l ) ) , ( a. a ) where G F is the Fermi coupling constant and m b is the b quark mass, /i is the normaliza-tion point for m b . is the average of the square of the heavy quark momentum inside the B hadron. is the expectation value of the chromomagnetic operator. The leading order coefficient A 0 can be calculated from the partonic width free from bound-state nonperturbative corrections. The term starting with 1/m2, is the nonperturbative correc-tions. For the nonperturbative corrections, /iQ is around 0.4 G e V 2 with the conservative estimated accuracy ±25%. is (0.6 ± 0.2) GeV 2 . The perturbative expansion of the b —y X u £ v decay width has the general form [109]: F = %^|V„ b |>(l + ^  + a2 + a3 ( ^ Y + ....), (6.7) 1927Td 7T \ 7T J \ 7T J where jl is the normalization point for as. The value of low scale running mass m b(/i) is translated to the scale \i of 1 GeV and m b ( lGeV) is assumed to be (4.58 ± 0.060) G e V / c 2 [111]. Combining the uncertainties from perturbative and non-perturbative cor-rections, the b —>• X u £ u decay width can be written as: n i 9 , m b ( lGeV) - 4.58GeV N , ' T = 6 6 p s - 1 | V u b | 2 ( l + 0 .065-^ ^~rj ± 0.02 p e r t ± 0.035 n o n p e r t ) . (6.8) The |V u b | can be derived as: | V u b | = 0.00442 x f6^^"^ I " X (! ± °- 0 2 5^ ± °- 0 3 5^ ^ Chapter 6. R E S U L T S 124 where r b is the average b hadron lifetime. The uncertainty 0 . 0 2 5 Q C D combined with the uncertainty from perturbative and non-perturbative corrections. 0.035m b is from the uncertainty of the running of the b quark mass. The L E P working group slightly modified this formula by combining the work from another group [45] and produced [5, 113]: i | V u b j = 0.00445 x (^rt*^ 1-55pS) 2 x ( l ± 0 . 0 1 0 p e r t ± 0 . 0 3 0 1 / m 3 ± 0 . 0 3 5 m b ) . (6.10) y U.002 r b ) b This formula is used in the calculation of | V u b | in this analysis. Chapter 7 C O N C L U S I O N The branching fraction of the inclusive b —> X u £ u decay is measured to be: Br(b - » X J v ) = (1.63 ± 0.53 (stat) 1 ° ; ^ (sys)) x 10~3. (7.1) The first error 0.53 is the statistical error from the data only. The errors associated with the limited statistics of the Monte Carlo sample are included in the systematic error. This result is consistent with similar measurements from A L E P H , D E L P H I and L3, the other three L E P experiments, shown in Table 7.1. In Table 7.1, the first error in Br(b —>• X Q £ v ) combines the statistical error from the data and limited Monte Carlo statistics as well as the uncorrelated systematic uncertainties due to experimental systematic errors, such as detector resolution and lepton identification efficiency. The second error contains the systematic uncertainties from the b —> X c £ u Monte Carlo simulation models. The third error contains the systematic uncertainties from the b —>• X u £ u models. The second and third errors are correlated between the various experiments. The Br(6 —> X u £ u ) from the D E L P H I collaboration is revised by the L E P Heavy Flavour Working Group [5]. |V u b | can be obtained from Br(b —¥ X u £ u ) using Equation 6.10 where the average b hadron 125 Chapter 7. C O N C L U S I O N 126 Experiment Br(6 -+ XJv) (10- 3) Ref A L E P H 1.73 ± 0.56 (stat+det) ± 0.51 (b ->• c) ± 0.2 (b -> u) [10] D E L P H I 1.69 ± 0.53 (stat+det) ± 0.45 (b ->• c) ± 0.2 (b ->• u) [11, 5] L3 3.3 ± 1.3 (stat+det) ± 1.4 (b -> c) ± 0.5 (b -+ u) [12] O P A L (This analysis) 1.63 ± 0.57 (stat+det) (b ->• c) ± 0.25 (b -» u) Table 7.1: The branching fractions of b -> X u & / measured from A L E P H , D E L P H I , L3 and this analysis. lifetime r b is equal to (1.564 ± 0.014) ps[2]. Thus, |V u b | obtained from this analysis is: | V u b | = (4.00 ± 0.65 (stat) ±°0f6 (sys) ± 0.19 (HQE)) x 10~3, (7.2) where the systematic error includes the b to u and b to c semileptonic decay modeling error, and the H Q E error is purely the theoretical error from the Heavy Quark Expansion. This result is consistent with the |V U b| value from the C L E O exclusive measurement of (3.3 ± 0.8 (total)) x 10" 3 [4]. The four measurements of Br(6 —> XJv) from the L E P experiments are averaged using the Best Linear Unbiased Estimate technique [112, 113]. This technique provides an unbiased estimate of L E P Br(6 —> XJv) average result Br LEp: Ef=i EjU B r i (E~ 1 ) i j B r L E p (7.3) E i U E j U E - 1 ) * ' where Br; is the Br(6 —> Xn£v) from the four L E P experiments. E is the error matrix in-cluding the off-diagonal terms giving the correlations between the four L E P experiments. The L E P average Br(b -+ Xu£u) is obtained as [113]: Bv(b -> X J v ) = 1.71 ± 0.31 (stat + det) ± 0.37(b -> c) ± 0.20(b u). (7.4) Using Equation 6.10, the average |V u b| obtained from the four L E P experiments is: | V u b | = (4.09+0.37 (stat + det) ±0.44(b -> c)±0.24(b ->• u)±0.19 (HQE)) x 1 0 - 3 . (7.5) Chapter 7. C O N C L U S I O N 127 The measurements of |V U b| will also be performed by the BaBar experiment at S L A C and the Belle experiment at K E K . Now the L E P average |V u b | is most precise measurement of |V U b| in the world. The more precise |V u b | measurements will be achieved by the BaBar and Belle experiments in the future as there will be more B events accumulated in BaBar and Belle than in L E P . Appendix A My Contribution to OPAL Collaboration I was involved with the O P A L Collaboration from September, 1997 to October, 2001 as a part of University of British Columbia group. I spent a total of 14 months at C E R N during this period. My contributions to O P A L Collaboration are listed below: • I participated in O P A L data taking shifts each year from 1998 to 2000. • As a member of O P A L Online R O P E (Reconstruction of O P A L Events) experts, I was in charge of the O P A L online event processing, data recording and offline event re-processing system. I also developed a graphical interface to execute Online Rope commands to make the operation of R O P E farm smoother and more friendly. • I wrote the Monte Carlo simulation code for the.b —>• X u £ u hybrid model and produced 200,000 Monte Carlo events on the O P A L Monte Carlo farm. • I did a study in b quark fragmentation. The study result is used as a general tool for b fragmentation in O P A L heavy flavour group. 128 Appendix A. My Contribution to OPAL Collaboration 129 • I represented O P A L to present a talk about "Measurement of | V c b | using B° —>• D*£u decays" in year 2000 APS April meeting in Long Beach, USA. I also presented a talk about " Measurement of | V u b | using b semileptonic decay" at APS Northwest Section meeting in May 2001 in Seattle, USA. • I was one of the editorial board members for the O P A L paper " D s —> r v T branching fraction measurement". • Although O P A L is a large Collaboration, this | V u b | analysis is mainly done by me. This result was published in the Eur. Phy. J. C 21 (2001) 399 in August, 2001. • I am a member of the L E P | V u b | working group which combines all L E P | V u b | measurements. Appendix B Glossary A C C M M Altarelli, Cabibbo, Corba, Maiani and Martinelli, a model de-scribing that the b quark and the spectator quark momenta in the b hadron rest frame follow a Gaussian distribution. A L E P H One of the four particle physics experiments at LEP . A N N Artificial Neural Network, a computational model applied to clas-sification, pattern recognition and optimization, inspired by bio-logical neural systems. A R G U S A collaboration that studied b and T physics using the DORIS electron-positron ring at D E S Y in Hamburg, Germany. b and b b represents a b quark, the fifth flavour of quark (in order of increasing mass), with electric charge -1/3. b represents a b hadron in this thesis. BaBar A collaboration that studies millions of B mesons produced by the PEP-II storage ring at S L A C . 130 Appendix B. Glossary 131 B E L L E A collaboration that studies millions of B mesons produced by the K E K b factory in Japan. Bhabha event The event from e+e~ —> e+e~ process. B T B Tag neural network package from O P A L . Calorimeter A composite detector using total absorption of particles to mea-sure the energy and position of incident particles or jets. C D F A detector for proton and anti-proton collider in Fermi National Accelerator Laboratory. C E R N European Organization for Nuclear Research. CESR Cornell Electron Storage Ring. C L E O A collaboration studying the production and decay of beauty and charm quarks and tau lepton produced in the Cornell Electron Storage Ring. C K M matrix Cabibbo, Kobayashi and Maskawa, a matrix describing the relation between quark weak and mass eigenstates. Collider An accelerator in which two beams traveling in opposite directions are steered together to provide high-energy collisions between the particles in one beam and those in the other. Deadtime A span of time during which a detector, or an associated readout system, is unable to record new information. D E L P H I One of the four particle physics experiments at LEP . Appendix B. Glossary 132 D E S Y Deutsches Elektronen-Sychrotron at Hamburg, Germany. The laboratory performs basic research in high-energy and particle physics as well as in the production and application of synchrotron radiation. Drift Chamber A multi-wire chamber in which spatial resolution is achieved by measuring the time electrons need to reach the anode wire, measured from the moment that the ionizing particle traversed the detector. DST Data Summary Tape, a data format in O P A L which can be ana-lyzed online and offline. DO A detector for proton and anti-proton collider in Fermi National Accelerator Laboratory. Feynman diagram Schematic representation of an interaction between par-ticles. Flash A D C Flash Analog to Digital Converter, an A D C whose output code is determined in a single step by a bank of comparators and encoding logic. F N A L Fermi National Accelerator Laboratory in Batavia, Illinois (near Chicago), named for particle physics pioneer Enrico Fermi. Hadron A particle made of strongly-interacting constituents (quarks and/or gluons). These include the mesons and baryons. Such particles participate in strong interactions. Appendix B. Glossary 133 H Q E T Heavy Quark Effective Theory, a theory used to describe heavy quark transitions. ISGW Isgur, Scora, Grinstein and Wise, a model describing B meson decays using form factors based on heavy quark symmetry. Jet Groups of particles moving in roughly the same direction. J E T S E T A Monte Carlo simulation program for jet fragmentation and e+e~ physics. J E T N E T A program used to train the neural network. K E K Koo Energy Ken. The High Energy Accelerator Research Orga-nization, Tsukuba, Japan. K E K B K E K B factory. L E P Large Electron Positron collider at the C E R N laboratory in Geneva, Switzerland. LINAC LINear Accelerator. L3 One of the four particle physics experiments at LEP . M A R K III One of the particle physics experiments at S L A C . O P A L Omni Purpose Apparatus for LEP , one of the four particle physics experiments at LEP . P D G Particle Data Group. P E P Positron Electron Project, a 2.2 km circumference storage ring at S L A C . Appendix B. 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