UNIVERSITY OFf-IAWAI'1 LIBRARY
AN ANALYSIS OF THE K+1l"+1l"- FINAL STATE IN
A DISSERTATION SUBMITTED TO THE GRADUATE DMSION OF THE UNIVERSITY OF HAWAI'I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN
PHYSICS
MAY 2008
By Hulya Guler
Dissertation Committee: Stephen L. Olsen, Chairperson Thomas E. Browder Sandip Pakvasa John M. J. Madey Kathleen C. Ruttenberg We certify that we have read this dissertation and that, in our opinion, it is satisfactory in scope and quality 118 a dissertation for the degree of Doctor of Philosophy in Physics.
DISSERTATION COMMI'ITEE
ii © 2008, HuIya GuIer
iii Anneanneciifime.
Hatice Giir 1919-2007
iv Acknowledgements
I thank Stephen Olsen, Thomas Browder, Sandip PakvaBa, John Madey, and Kathleen Ruttenberg for serving on my committee. I am especially indebted to my advisor Stephen Olsen for his support and for trusting me enough to let me work inde pendently. I also thank ThomaB Browder, Sookyung Choi, ThomaB Dombeck, Fang Fang, Alexei GarmaBh, Li Jin, Michael Jones, Kay Kinoshita, Alexander Kuzmin, Jimmy Macnaughton, Daniel Marlow, Kenkichi MiyabayaBhi, Shohei Nishida, Sandip PakvaBa, Michael Peters, Yoshihide Sakai, Karim Trabelsi, Gary Varner, and Hitoshi Yamamoto for helpful discussions. I thank my mother, Bema Giir, for the sacrifices she haB made to enable me to pursue my interest in physics. I thank Jamal Rorie for being the ideal officemate and for his patience and un derstanding aB I worked on this thesis. I am grateful to my friends who have supported and encouraged me; in particu lar: Gene Altman, Javier Ferrandis, Tamar Friedmann, Jeanne Hogan, Abidin Kaya,
Elena Martin, Miifit Tecimer, and Melek Yal!tmt~. Finally, I thank the KEKB team and the Belle Collaboration for producing the beautiful data that have made this thesis possible.
v Abstract
Inclusive branching ratios have been measured for the decays B+ -+ J /1/JK+7r+7r and B+ -+ '1/1 K+7r+7r- using 492 fb-1 of data collected at the T(4S) resonance by the Belle detector at KEKB. Three-dimensional amplitude analyses have been performed to study the resonant structure of the K+7r+7r- final state in these decay modes.
vi Contents
Acknowledgements v
Abstract . .. vi
List of Tables xi
List of Figures xiii
1 Introduction ...... 1
1.1 The Standard Model 1
1.2 Meson Spectroscopy 2
1.3 Production and Decay Rates . 3
1.4 Motivation.. 5
1.5 Synopsis... 6
2 Experimental Apparatus 7
2.1 KEKB ...... 7
2.2 The Belle Detector . 9
2.2.1 Silicon Vertex Detector . . 9
vii 2.2.2 Central Drift Chamber . . . . 12
2.2.3 Aerogel Cherenkov Counters . 14
2.2.4 Time-of-Flight Counters ... 17
2.2.5 Electromagnetic Calorimeter . 18
2.2.6 Extreme Forward Calorimeter 18
2.2.7 Kdp, Detector ...... 20
2.3 1i.igger, Data Acquisition, and Software . 21
2.3.1 Trigger ...... 21
2.3.2 Data Acquisition 25
2.3.3 Software.. 26
3 Particle Identification 27
3.1 Electron Identification 27
3.2 Muon Identification . . . . 30
3.3 Hadron Identification . . 31
4 Event Selection ...... 35
4.1 Multiply-Reconstructed Tracks 36
4.2 Selection of J N and 1/J' Candidates 38
4.2.1 Leptonic IN and 1/J' Decays .... 38
4.2.2 1/1 Decays to JN7r+7r- . 40
4.3 Selection of K± Candidates 41
4.4 Selection of 7r± Candida.tes . 41
4.5 B-Meson Reconstruction .... 42
viii 4.6 Signal and Sideband Regions. . . . · · 45 4.7 Cut Optimization ...... · · 47
5 Coordinate Transformations 49
5.1 M(K7r7r). 49
5.2 M(K7r) 57 5.3 M(7r7r). . · · 63 5.4 Peak Transformations. . . · · 69
6 Inclusive Branching Fractions 72
6.1 Measurement Technique .. 72
6.2 Systematic Errors ...... 73
6.2.1 Efficiency Errors .. 76
6.2.2 Binning ...... 76
6.2.3 Correction for Over-Subtraction . . 77
6.2.4 Background Shape ...... 77
6.2.5 Determination of Signal and Sideband Regions . . 78
6.2.6 NB ••••••••••••••••• 78
6.2.7 J N and 'I/! Branching Fractions . 78
6.3 Results.... 78
6.4 Cross-Check. 79
7 Amplitude Analyses 82
7.1 Fitting Technique .. 82
7.2 Normalization Procedure .... 83
ix 7.3 Background Functions · · . 85 7.4 Efficiency Functions . · · 91 7.5 Phase-Space Functions · 91 7.6 Signal Functions · 92 7.7 Results . . . . . · 95 7.8 Statistical Errors . . . 101 7.9 Systematic Errors . · ...... 103 7.9.1 Background Parameterization 103
7.9.2 Efficiency ...... 104
7.9.3 Integration Step Size .. 104
7.9.4 Modeling of the Signal . 104
7.10 Discussion ...... 105
7.10.1 Interference Effects . . . 105
7.10.2 The L-Region ..... 109
7.10.3 Deficiencies of the Fits 110
8 Conclusions ...... 112
x List of Tables
1.1 Quarks and Leptons . . . . . 2
2.1 SVDl and SVD2 parameters . 11
2.2 Physical processes of interest and their cross sections 23
4.1 Leptonic J /1/J and 'if! yields . 39
4.2 1/J' -> J/'Ijnr+7r- yields ... 41
4.3 Selection criteria for ~ -> 7r+7r- decays . 42
4.4 Fractions of events with multiple B candidates 44
4.5 Mean of the signal peak, width of the main Gaussian, and background fraction in the signal region, according to I~EI fits for data ...... 47
4.6 Background rejection factors and signal efficiencies for kaon and pion cuts 48
6.1 Measured inclusive branching fractions for B+ -> J/1/JK+7r+7r- and B+-> 1/J' K+7r+7r- ...... • . . . • • ...... 79
6.2 Measured inclusive branching fractions for B+ -> 1/J'K+ 81
7.1 Fitted values of background-function parameters for B+ -> J/1/JK+7r+7r- and B+ -> 1/J' K+7r+7r- ...... • . • . .. 87
7.2 Angular distribution of the K+7r+7r- final state for various combinations of initial and intermediate-state spin parities ...... 94
7.3 Fitted values of signal-function parameters for B+ -> J/1/JK+7r+7r- 96
xi 7.5 Decay fractions for B+ -+ JNK+w+w- submodes . 102
7.6 Decay fractions for B+ -+ 1/J'K+w+w- submodes. . 103
7.7 Masses, widths, and spin-parity values of the resonances included in the fits 105
xii List of Figures
1.1 Masses and widths of observed kaon states. 4
2.1 A conceptual diagram of KEKB ...... 8
2.2 A three-dimensional depiction of the Belle detector 10
2.3 Side-view of the Belle detector ...... 10
2.4 Impact-parameter resolution of SVDl and SVD2 in the r-t/> and z directions 12
2.5 The structure of the CDC ...... 13
2.6 Spatial resolution and transverse-momentum resolution of the CDC . 14
2.7 Configuration of the cathode layers of the CDC 15
2.8 The configuration of the ACC . . 16
2.9 Barrel and endcap ACC modules. 16
2.10 The layout of a TOF module. 17
2.11 The layout of the ECL . . . . 19
2.12 The structure of an ECL counter. 19
2.13 A three-dimensional view of the EFC's crystal structure . 20
2.14 Cross-section of a KLM detector layer; barrel and endcap RPC's 22
2.15 The Belle Level-l trigger system 24
2.16 The Belle DAQ system . . . . . 25
xiii 3.1 dE!d:c as a function of momentum for electrons and pions 29
3.2 Leld distributions for electrons and pions...... • 29
3.3 Muon likelihood distributions, and efficiencies of various muon-likelihood cuts, for simulated muons and pions 31
3.4 Muon efficiency and pion fake rate . 32
3.5 CDC measurements of dE!d:c as a function of momentum for charged tracks in collision data ...... 34
3.6 Charged-hadron meffles, calculated using TOF and CDC information, for data and MC ...... •. 34
4.1 Interpretation of a single track as multiple tracks by the tracking software 37
4.2 Cosine of the opening angle between track pairs in data . 37
4.3 Measured invariant maS!!e8 of electron and muon pairs. . 39
4.4 Measured invariant maffles of '1/1 -+ J Nnr+ 1r- candidates 40
4.5 dE versus MBC for J!'l/JK+1r+1r- and'l/lK+1r+1r- 43
4.6 dE versus M BC after best-candidate selection . 44
4.7 l!;.E distributions for MC . 45
4.8 l!;.E distributions for data 46
5.1 M (K1r1r) distributions for data in the signal and sideband regions . 50
5.2 dE versus M (K1r1r) for data . 52
5.3 dE versus M' (K1r1r) for data 52
5.4 M'(K1r1r) distributions for data in the signal and sideband regions 53
5.5 dE versus M(K1r1r) for generic MC . 54
5.6 dE versus M'(K1r1r) for generic MC . 54
5.7 M(K'lr1r) distributions for generic MC 55
xiv 5.8 M'(K1r1r) distributions for generic Me . 56
5.9 t:.E versus M(K1r) for data 58
5.10 t:.E versus M'(K1r) for data 58
5.11 t:.E versus M(K1r) for generic Me . 59
5.12 t:.E versus M'(K1r) for generic Me 59
5.13 M(K1r) distributions for data in the signa.! and sideband regions 60
5.14 M' (K1r) distributions for data in the signal and sideband regions 60
5.15 M(K1r) distributions for generic Me . 61
5.16 M'(K1r) distributions for generic Me 62
5.17 t:.E versus M (1r1r) for data . 64
5.18 t:.E versus M'(1r1r) for data 64
5.19 t:.E versus M(1r1r) for generic Me 65
5.20 t:.E versus M'(n) for generic Me . 65
5.21 M(1r1r) distributions for data in the signal and sideband regions. 66
5.22 M'(1r1r) distributions for data in the signal and sideband regions 66
5.23 M(1r1r) distributions for generic Me . 67
5.24 M'(1r1r) distributions for generic Me . 68
5.25 Transformation of the K· (892) and DO background peaks. 70
5.26 Transformation of the ~ and l background peaks. •.. 70
6.1 Dependence on .M2(K1r1r), .M2(K1r), and .M2(1r1r) of B+ -> Jj'ljJK+1r+1r- signal efficiency, and corresponding mass spectra for data ...... 74
6.2 Dependence on .M2(K1r1r), .M2(K1r), and .M2(1r1r) of B+ -> 'I/IK+1r+1r- signa.! efficiency, and corresponding mass spectra for data • . . . . . • .. 75
xv 6.3 Dependence on bin size of the efficiency-corrected signal yield for B+ -+ J /'l/JK+7r+7r- and B+ -+ 'I/J' K+1r+1r-. . . . • . . . . . • ...... 77
6.4 Dependence on M2(K1r1r), M2(K1r), and M2(1r1r) of B+ -+ 'I/J'K+, 'if! -+ J N1r+1r- signal efficiency, and corresponding lIlllBS spectra for data. . .. 80
6.5 Dependence on bin size of the efficiency-corrected signal yield for B+ -+ 'if!K+, 'if!-+ J/'I/J7r+1r-. . 81
7.1 Phase-space boundaries 85
7.2 Sideband fits for B+ -+ JNK+1r+1r- and B+ -+ 'I/J' K+1r+1r- data 88
7.3 B+ -+ JNK+1r+1r- sideband fits for slices in M2(K1r1r) 89
7.4 B+ -+ 'if! K+1r+1r- sideband fits for slices in M2(K1r1r} . 90
7.5 Density of states in phase space as a function of M2(K1r1r}, M2(K1r}, and M2(1r1r) for B+ -+ JNK+1r+1r- and B+ -+ 'if!K+1r+1r- . 92
7.6 Signal-region fits for B+ -+ JNK+1r+1r- data...... 97
7.7 B+ -+ JNK+1r+1r- signal-region fits for slices in M2(K1r1r) 98
7.8 Signal-region fits for B+ -+ 'if! K+1r+1r- data...... 99
7.9 B+ -+ 'I/J' K+1r+1r- signal-region fits for slices in M2(K1r1r} 100
7.10 Scatter-plots of M2(K1r), M2(1r7r}, and M2(K1r1r} for data 107
7.11 p-w interference...... 108
xvi Chapter 1
Introduction
1.1 The Standard Model
Nearly all the phenomena that have been observed in high-energy physics experiments can be described within the theoretical framework of the Standard Model of particle interactions.l According to this model, the fundamental constituents of matter are six quarks and six leptons, which can be grouped into three generations, as depicted in Fig. 1.1. Each particle also has an antimatter equivalent with the same mass and opposite charge. Quarks and leptons interact quantum-mechanically by exchanging a set of media tor particles known as gauge bosons. The electromagnetic interaction is mediated by photons, while the weak interaction is mediated by W± and ZO bosons. These two interactions are unified by the electroweak theory. The strong interaction is mediated by gluons and is described by quantum chromo dynamics (QeD). The gravitational force, which has thus far resisted integration into the Standard Model but is negligible at the subatomic level, is believed to be mediated by a hypothetical particle called a graviton.
'For a dlscllBSion of the strengths and wealmesses of the Standard Model, see [1).
1 Flavor Q/lel Mass (MeV/c?) u c t +g 3 1.5-3 1250 ± 90 172500 ± 2700 Quarks d 8 b -31 3-7 95±25 4200±70
e JL T -1 0.511 106 1777 Leptons v. V.,. vI' 0 < 0.000002 <0.19 < 18.2
Table 1.1: Quarks and leptons. Masses are from [2].
While leptons exist 8B free particles, quarks are invariably observed in strongly bound states of integral charge, referred to 8B hadrons. Hadrons can be mesons, which are quark-antiquark pairs, or baryons, which are quark triplets. Recently, there have been experimental suggestions of larger groupings [3], though these are yet to be conclusively established.
1.2 Meson Spectroscopy
Although there are only 21 ways to form quark-antiquark combinations with six quarks, the number of mesonic states observed in experiments is much higher. This cornucopia of mesons can be understood by analogy to the energy levels of hydrogen. Where8B the energy spacings of the hydrogen atom are caused by the electromagnetic force, those of a meson are caused by the strong force and are large compared to the rest m8BB of the ground state. Different excited states of a given quark-antiquark combination are thus different enough in m8BB to be considered distinct particles [4]. Predicting the m8BBe5 of hadrons is a challenging problem in quantum chromody-
2 namics.2 Since the bound-state equations have proven too complicated to be solved analytically, perturbative methods, effective theories, or numerical techniques such as lattice gauge theory are necessary. The situation is especially complicated for mesons composed of 'It, d, and 8 quarks, in which relativistic effects cannot be neglected. The B+ meson,3 which is studied in this thesis, is the ground state of bu. The 3 IN and 7f;', which are more formally known as IN(IS) and 7f;(2S), are the 1 S1 and ~SI states,4 respectively, of cC. The charged pion 1T+ is ud, and the charged kaon K+ is su. Figure 1.1 shows the masses, widths, and spin-parity values of the known kaon states.
1.3 Production and Decay Rates
In addition to predicting the masses and widths of hadrons, the Standard Model must also explain the production and decay mecbaniBTDB of these hadrons. Experimental measurements of hadron decay rates provide a good testing ground for developing the theory. Whether or not a certain particle is produced in a given decay, as well as the interference patterns among different particles that decay to the same final state, can hint at the underlying physics. For example, the breaking of flavor symmetry causes different quark flavors to have different masses and mixes the 3PI and 1 PI states of the kaon system. The physical states K l (1270) and K l (I400) can be expressed in terms of the 13Pl and
2See, for example, [5]. 3Conjugate modes are implied unless stated otherwise. 4The states 13S1 and 3 K(3100) 2.5 ,,".."'. K\ ·~.Th0) K,(2320) 2 ~ K; (2045) ~u ?:- 0.5 K o ? 1 - 1+ 2- 2+ Y Y 4- 4 + Y ? JP Figure 1.1: Masses and widths of observed kaon states, along with their measured spin parity (JP) values. Solid lines denote established states, and broken lines denote resonances requiring confi rmation. Measured widths are indicated by hatching. All masses and widths are the central values from [2J. Note that K\ (1650) and K2(2250) are not resonances but represent various peaks reported in the 1.6-1.9 GeV /2 and 2.15-2.26 GeV /2 regions. 4 II PI states as 3 KI (1270) - K(1 PI) sin9K + K(11 PI) cos9K , (Ll) 3 I KI (1400) - K(1 Pd cos9K - K(I Pdsin9K , (1.2) where 9K is the 3 PI_I PI mixing angle. The value of 9K can be related to the masses of the KI (1270) and KI (I400), to the strong decays of the KI (1270) and KI (I400), and to rates of weak decays to final states involving the K I (1270) and K I (I400) [6,7,8]. The results presented in this thesis are likely to lead to a better determination of 9K • 1.4 Motivation This thesis makes use of the large number of B mesons produced at KEKB to study the structure of the K+7r+7r- final state in the decays B+ -+ J/1/JK+7r+7r- and B+ -+ 1/J'K+7r+7r-. While many of the kaon excitations in Fig. 1.1 can decay to a K 7r7r final state, their large widths and sinIi1ar masses, as well as their common intermediate states, make them difficult to distinguish based on the mass of the K 7r7r system alone.5 To determine the composition of the K+7r+7r- final state, more information is necessary. In this analysis, the data are therefore fitted in the three dinIensions .M2(K7r7r), .M2(K7r), and W(7r7r) , which are the squared invariant masses of the K+7r+7r-, K+7r and 7r+7r- systems, respectively. An unbinned maximum-likelihood fit is performed to extract as much information as possible from the data. The large data size, combined with the clean environment afforded by the presence of a J / 1/J or '1/1 in the final state, allows precise measurements of the structure of the 51n 2001, the Belle Collaboration measured the branching fraction for B+ .... JI?/JK1(1270) with 2% of the data UBed in this thesis. The decay B+ .... JI?/JK+7r+7r- WBB found to be dominated by B+ .... J NKl (1270), and no other structure WBB detected (9). With the current statistics, further structure can be observed around 1.4 GeV / r? 5 K+7r+7r- system in these decays. The spin-dependent angular distributions of the final state are included in the fitting model, and interference among the different decay channels is found to be significant. In both B+ --+ J /'l/JK+7r+7r- and B+ --+ '1/1 K+7r+7r-, the K+7r+7r- final state is found to be dominated by the Kl(1270). The three-dinlensional fit allows a measure ment of the decay fractions for the varions K 1(1270) decays that contribute to the K+7r+7r- final state, with significantly more precision than was previously possible, and with interference effects included. While a large Kw component is measured, only a small Ko(1430)7r component is observed. 1.5 Synopsis Chapter 2 describes the apparatns nsed to produce and detect B-mesons. Chapter 3 outlines the methods nsed for particle identification. Chapter 4 details the proce dures nsed for event selection. Chapter 5 explains the transformations applied to the M2(K7r7r), M2(K7r), and M2(7r7r) coordinates to correct for kinematic differences be tween sideband and signal-region data. Chapter 6 presents measurements of inclusive branching fractions for the decays B+ --+ J/'l/JK+7r+7r- and B+ --+ 1/J'K+7r+7r-, and Chapter 7 presents three-dinlensional amplitude analyses of the K+7r+7r- final state in these decays. Chapter 8 offers concluding remarks. 6 Chapter 2 Experimental Apparatus The B-mesons analyzed in this thesis were produced by the KEKB collider and re constructed by the Belle detector. This chapter summarizes the relevant details of the collider and detector, as well as the framework used to collect and analyze the data.l 2.1 KEKB KEKB is an asymmetric-energy2 e+e- collider, located at the High-Energy Accel erator Research Organization (KEK) in Tsukuba, Japan [10]. Figure 2.1 illustrates the system, which lies 11 m below ground level. A linear accelerator boosts elec trons to 8 GeV and positrons to 3.5 GeV, and injects them into two separate storage rings-the High-Energy Ring (HER) for the electron beam and the Low-Energy Ring (LER) for the positron beam. The rings, which are 3,016 m in circmnference and are kept at a pressure of approximately 10-9 Torr [11], meet at a single interaction point (IP), crossing at a finite angle of ±11 mrad to reduce parasitic collisions. The IThe figures In this chapter have heen taken from the references cited In the text. 2The energy 8BYIIllI1etry is useful for time-dependent CP-vioJation studies and is not Importent for the a.na.\ysis presented In this thesis. 7 center-of-mass (eM) energy of 10.58 GeV maximizes the cross-section for producing the T(4S) resonance, which is a bb bound state that decays predominantly to a pair of B-mesons. Interaction Point (IP) Figure 2.1: A conceptual diagram of KEKB. KEKB was designed to reach a peak luminosity of 1 x 1034 cm-2s-1, with an LER current of 2600 rnA, an HER current of 1100 rnA, and 5000 bunches in each ring, corresponding to a 60-cm or 2-ns spacing between bunches. It has attained, as of March 2008, a world-record peak luminosity of 1.71 x 1034 cm-2s-1, with an LER current of 1662 rnA, an HER current of 1340 rnA, and 1389 bunches. By adopting a continuous-injection scheme in 2004 [12], it has also achieved a daily integrated 1 luminosity record of 1.23 fb- • To date, KEKB has collected an integrated luminosity of 781 fb-1. 8 2.2 The Belle Detector The Belle detector is a large-solid-angle spectrometer that surrounds the IF and records the products of the collisions of electrons and positrons produced by KEKB.3 It comprises several sub-detectors, as shown in Figs. 2.2 and 2.3. A double-walled beryllium beam pipe, with helium gas flowing between the walls for cooling, separates the beam line from the Belle detector. Immediately outside the beam pipe is a Silicon Vertex Detector (SVD), which reconstructs decay vertices. A Central Drift Chamber (CDC) tracks the trajectories of charged particles. A system of Aerogel Cherenkov Counters (ACC) distinguishes high-momentum kaons and pions, and an array of Time-of-Flight counters (TOF) measures the time-of-arrival of charged particles. An electromagnetic calorimeter (ECL) records the energy deposited by photons, leptons, and hadrons. An extreme forward calorimeter (EFC) extends the polar-angle coverage of the ECL. Finally, a KL/p, detector (KLM) identifies muons and KL mesons. All the sub-detectors except the KLM are placed inside a superconducting solenoid 3.4 m in diameter and 4.4 m in length, which produces a 1.5-T magnetic field parallel to the z axis. The curvature of the trajectories of charged particles in this magnetic field allows their transverse momenta to be measured. This section describes each part of the detector, roughly in order of radial distance from the IP. Further details may be found in Ref. [13]. 2.2.1 Silicon Vertex Detector The ability to measure B-decay vertices with a precision of ~ 100 p,m is an important feature of the Belle detector. This is accomplished by the SVD, which provides 3In the following, we make use of a. coordinate system where the origin is the IP, z points toward the outside of the KEKB ring, Y points up, and z is the direction opposite to the e+ beam. The "forward" side of the detector is defined to he its +z region. When convenient, we switch to a cylindrical coordinate system (T, tP, z), where T is the ra.dlal distance from the z axis and tP Is the azimuthal angle; or a spherical coordinate system (p, tP, II), where p Is the ra.dlal distance from the IP and II is the polar angle. 9 SVD CDC -'--i-_ ACC TOF Eel. Figure 2.2: A three-dimensional depiction of the Belle detector. I 1 , --. - Figure 2.3: Side-view of the Belle detector. 10 charged-particle tracking close to the IP. The SVD is made up of a set of rectangular double-sided silicon-strip detectors (DSSD's) arranged in cylindrical layers about the z axis. On one side of each DSSD, p-type Si strips run parallel to the z axis and measure the r-¢J coordinate of a passing charged track. On the opposite side, n-type Si strips run perpendicular to the p-type strips and measure the z coordinate. Signals from the two sides of a given strip can be combined to determine the three-dimensional position of a hit. To achieve good position resolution, it is necessary to place the SVD close to the IP and to reduce multiple scattering by minimizing the amount of material used. Aligmnent is also important, and the positions of all the DSSD's must be known with an accuracy of 10 I'm [14]. Because of the large beam backgrounds near the IP, the SVD must be able to withstand large doses of radiation. The SVD and the beam-pipe were upgraded in October 2003 to achieve a larger acceptance, better vertex resolution, higher radiation tolerance, and faster read out [15]. Table 2.1 lists some of the parameters of the pre-upgrade (SVDl) and post-upgrade (SVD2) versions of the detector. Figure 2.4 shows the improvement in impact-parameter resolution. SVDl SVD2 number of DSSD layers 3 4 number of DSSD's 102 246 polar-angle coverage 23· < e < 139· 17" < e < 150· radiation tolerance up to 1 Mrad up to 20 Mrad readout dead time 128 JtS 25.6 JtS beam-pipe inner radius 20= 15= layer-O radius 30.0= 20.0= layer-l radius 45.5= 43.5= layer-2 radius 60.5= 70.0= layer-3 radius NjA 88.8= Table 2.1: SVDl (16) and SVD2 (15) parameters. 11 314() .. SVD2 Co5mk: .. SVD2 Cosmic •c (17.4±O.3)<1l( 34.3±O.7)lp ~m (26.3±O.4)<1l( 32 .9±O.8Vp ~m ".2 ~I'" ., SVDI Cosmic ., SVOI Cosmic (19.2±O.8)<1l( 54±O.8)tP ~m (42.2±1 )G>(44.3±l )1p ~ 100 100 • .. ' . .. . • , • .. ' . • • • • 4() • • 4() • • • • • • . • • • ...... • • 20 -. • • • • I I I "' . °O~~~~2 ~~J~~4 ~~'~~' ~~'~~' °O~~~~2 ~~J~~4 ~~'~~' ~~7~~' pseudo momentum (GeV/t) pseudo momtnlum (GeV/c) Figure 2.4: Impact-parameter resolution of SVD1 and SVD2, in the r -¢ (left) and z (right) 3 2 directions, as measured with cosmic-ray data [14]. The pseudo-momentum p is p(3 sin / 0 for T-¢ and p(3sin5/ 2 0 for z, where p, (3, and 0 are the momentum, velocity, and polar angle of the track, respectively. The impact-parameter resolution is expressed as a quadratic sum of two terms: one due to the detector's intrinsic resolution, and another due to multiple scattering. 2.2.2 Central Drift Chamber The CDC is used to reconstruct charged-particle trajectories and determine their mo menta based on their curvature in the 1.5-T magnetic fi eld. It also provides trigger information, as well as a precise measurement of dE/ dx for charged-particle identifi cation. The geometry of the CDC is shown in Fig. 2.5. The detector asymmetrically 0 0 covers the polar-angle region 17 ~ B ~ 150 , which corresponds to 92% of the solid angle in t he T (48 ) rest frame. The aluminwn end plates support 8,400 gold-plated tungsten sense wires, each 3D-J.£m thick, and 33 ,344 aluminum field wires, each 126-J.£m thick [17]. The drift volume is filled with a gas composed of equal amounts of helium and methane, the former minimizing multiple scattering and the latter maintaining good dE/ dx resolution. A charged particle traversing the CDC ionizes the gas; the electrons thus freed 12 .... ''''. ,.... .-_ ...... • 17" ------::: .::::-:._------e- -e+ -- 10 y [1~p~~.fl------~I"~w.~.1 y L. L. 1QOmm .... 1QOmm - Figure 2.5: The structure of the CDC. Distances are in rom. drift toward the sense wires, where they initiate avalanches of charge. The difference between the event time and the time at which a signal is received at a sense wire indicates the radial distance of the track from the wire [18]. Measurements from many wires are combined to reconstruct the particle's trajectory using Kalman filtering [19], which corrects for measured non-uniformities in the magnetic field, as well as for the effects of multiple scattering and energy loss to ionization. Figure 2.6 shows the spatial and transverse-momentum resolutions, as measured with cosmic-ray data. If all the wires are aligned with the z axis, they cannot provide information on the z dependence of a track.4 To measure the component of the momentum parallel to the magnetic field, a fraction of the wires in the CDC are inclined at a small stereo angle with respect to the z axis (for an illustration, see Figs. 1 and 2 in [20]). The CDC contains 32 axial layers interspersed with 18 stereo layers. Three cylindrical cathode layers provide further z-coordinate information. These 4lnformation that may be gained by analyzing the size or time-of-arrlval of the signal at each end of the wire Ie Iim1ted by the the length of the wire or by the timing resolution of the electronics, reepectively. 13 _0.4.------. ~0.3S + + .... + • • 0.15 • • • • • •• •• • 0.1 ...... o.os 't10 -8-6-4.Z0 Z 4 6 810 °o~~~~~~~~~·1 2. 3 4 5 SIgDod_rrom ..... _ (mml Pt(GeVlc) Figure 2.6: Left: spatial resolution of the CDC as a function of drift distance. Diffusion limits the resolution at large drift distance. Right: transverse-momentum resolution as a function of transverse momentum. The dotted line is what would be expected for {J = 1 [17]. are glued to the inner cylinder and to a 400-JJm-thick layer of carbon-fiber-reinforced plastic (CFRP) placed between the second and third anode layers, as illustrated in Fig. 2.7. Each cathode layer is divided into eig11t 45° segments in r-rjJ, and 64 (layer 0) or 80 (layers 1 and 2) segments in z. Analyzing the spatial distribution of the mirror charge induced on a cathode layer by an electron avalanche at a nearby sense wire yields the z coordinate of the avalanche. Including cathode-layer information improves the position resolution in the z direction by nearly a factor of five [21]. The inner part of the CDC and the cathode layers were replaced with a small-cell inner chamber in October 2003 to handle high hit rates. 2.2.3 Aerogel Cherenkov Counters In a dielectric medium, light travels at a speed reduced by a factor of n, the index of refraction of the medium. If a charged particle passes through the medium at a velocity greater than that of light (Le., {:J > lin), the material emits what is referred to as as Cherenkov radiaton [18]. The ACC exploits this phenomenon for particle identification. 14 • Sense wire o Field wire 1125" o . 0 0 o . 00 o · layer 1 00 o · Figure 2.7: Configuration of the cathode layers of the CDC. The ACC system [22] is an array of counters configured as shown in Fig. 2.8. The barrel part contains 960 modules arranged in 60 azimuthal segments, and the endcap part contains 220 modules arranged in 5 concentric circles. Typical barrel and endcap modules are depicted in Fig. 2.9. Each module consists of five silica aerogel tiles stacked in a thin aluminum box lined wit h refl ective Gore-Tex material; light from each box is collected by one or two fine- mesh photomultiplier tubes (PMT's), which are capable of operating in strong magnetic fields. Aerogel is a transparent , low-density solid produced by drying off the liquid in a gel and replacing it with a gas. The index of refraction of aerogel can be adjusted by varying its density; it is chosen such that in the momentum range of interest, a charged pion will produce Cherenkov radiation whereas a charged kaon will not. In the barrel, the refractive index is optimized, according to the polar angle of the module, for separating pions and kaons from two-body decays of boosted B mesons; in the endcap, it is optimized for B O fl avor tagging in CP -violation studies. To produce aerogel that has a low index of refraction and is hydrophobic to maintain its transparency over a long period, a special procedure was devised [23]. 15 _ ___~ 1'_'."'_50 IEAC~/Out8ideJ.: ___ 1670 (EACC/inside) 1622 (BACCI Figure 2.8: The configuration of the ACC. Figure 2.9: Barrel (left) and endcap (right) ACC modules. 16 2.2.4 Time-of-Flight Counters The TOF system provides a fast and precise measurement of the time of arrival of charged particles, for purposes of triggering the detector and identifying particles at momenta below 1.2 GeV Ie [24). It is made up of 64 modules arranged cylindricaJly about the z axis, covering the polar-angle range 34° < (J < 120°. Each module com prises two TOF counters and one trigger scintillation counter (TSC), as shown in Fig. 2.10. A TOF counter is a bar of scintillator with a fine-mesh PMT at each end; a TSC counter is a thin strip of scintillator with a fine-mesh PMT at one end. TSC signals are used in coincidence with TOF-counter signals to filter out backgrounds from photons and neutrons: combined with the 1.5-T magnetic field, the 1.5-cm gap that separates the TOF scintillator from the TSC scintillator prevents the charged, low-energy interaction products of these particles from producing hits in both coun ters. The resolution of the TOF system has been measured to be Ut Rl 115 ps for pions and kaons in hadronic events [25). Backward Forward LP.(Z=O) .7Z5 i 1905 ...... ···············io··+······· Ra1210 -PMT- X -PMT-- - --40 --! -TOF.4OT,,60W,,%S5(IL------1- PMT--~ --Ra12ZO ..I.. LS ! Rall"75 •...... ·8LS ·················r TSC.STlI120W,,2630L ~.. .. --ES· RalZ>AJ Ra1175 Figure 2.10: The layout of a TOF module. Distances are given in mm. 17 2.2.5 Electromagnetic Calorimeter Leptons, photons, and hadrons that reach the ECL interact electromagnetically or strongly with the scintillating material of this detector. Analyzing the pattern of sig nals in the ECL can help with particle identification. In particular, the energies and positions of electrons and photons can be determined from the electromagnetic show ers that these particles produce in the ECL through a combination of bremsstrahlung and e+e- pair production. The ECL also provides trigger information and a lumi nosity measurement based on Bhabha events [26]. The ECL [27] is configured as shown in Fig. 2.11. Like the CDC, it covers the polar-angle range 17" < () < 150°. It is made up of 8736 counters: 6624 in the bar rel, 1152 in the forward endcap, and 960 in the backward endcap. Each counter is a block of CsI(TIi) crystal read out by two photodiodes, as illustrated in Fig. 2.12. Each crystal is shaped as a frustum of a square pyramid with a depth of 30-em, which corresponds to 16.1 radiation lengths. The transverse dimensions depend on the loca tion of the counter and are chosen to optimize position resolution while maintaining energy resolution. The counters point approximately toward the IP, with a small tilt to prevent photons originating at the IP from escaping through the gaps between the crystals. The entire structure weighs 43 tons. The energy and position resolutions of the ECL have been measured to be UE/ E = 1 4 1 1 4 (1.3EBO.07/EEBO.8/E / )%, and Upos = (0.27EB3.4/E /2EB1.8/E / ) =, respectively, with E given in GeV [13]. 2.2.6 Extreme Forward Calorimeter The EFC extends the polar-angle coverage of the ECL. The detector, which is depicted in Fig. 2.13, surrounds the beam pipe and is made up of forward and backward parts, each in the form of a cone with an inner radius of 6.5 cm. The forward cone, at z = 60 em, covers the polar-angle range 6.4° < () < 11.5°, while the backward cone, 18 unit (mml I , " I " J I II i I 2.0 III LO III 0.0 ED to ED 2.0 III 3.0 m Figure 2.11: The Ia.yout of the EeL. Side View Top View n n Pruunp Box \j: i ·1 d u u y...AI plate : i DeltlD~ 'I' 'I' ~Tef1OD AI plate .!:=.o... A..,.ute , I'botodlodes ... I'botodlode TeIlon&AI Col(TI) AeryUte HoleCor_toO" preamp box to Al plate Figure 2.12: The structure of an ECL counter. 19 at z = -43.5 em, covers 163.3° < 9 < 171.2°. Each cone hOB five layers of 32 crystals, which point approximately at the IP. The light from each crystal is collected by one photodiode (in the inner two layers) or two photodiodes (in the outer three layers). The depth of the crystals, which is limited by the available space, corresponds to 12 radiation lengths in the forward cone and 11 radiation lengths in the backward cone. Since the EFC is placed in a region of high radiation, it uses Bismuth Germanate (BGO) crystals, which are radiation-hard up to '" 10 Mrad. Due to its position, the EFC shields the CDC from beam backgrounds and provides faBt feedback to KEKB on the background rate and luminosity. • Figure 2.13: A three-dimensional view of the EFC's crystal structure. 2.2.7 KL/JL Detector The KLM [28], which is the only detector system placed outside the superconducting solenoid, is used to identify KL's and muons. Hadrons interact strongly with the material of the KLM, producing showers of charged particles. Muons cannot inter- 20 act strongly and lose energy mainly through ionization, leaving cleaner, penetrating tracks. The longer range of KL's distinguishes them from other hadrons. 0 0 The KLM, which covers the polar-angle range 20 < (J < 155 , has an eight-fold symmetry about the z axis, as can be seen in Fig. 2.2. It is made up of alternating layers of large-area charged-particle detectors and 4.7-cm-thick iron plates. The iron provides material to interact with KL's, as well as a compact return path for the fringe magnetic field of the solenoid. The barrel region has 15 layers of detector and 14 layers of iron, while each endcap has 14 layers each of detector and iron. A layer of detector comprises two resistive-plate counters (RPC's), as shown in Fig. 2.14~ An RPC is made up of two parallel plates of highly resistive glass, across which an 8-kV voltage is applied. The surface of the glass plates, which are held apart by spacers, is painted with India ink (in the barrel) or coated with conducting carbon tape (in the endcaps) to produce a uniform potential across the surface. The volume between the glass plates is filled with a mixture of 30% argon, 8% butane-silver and 62% FC-I34a at atmospheric pressure. When a charged particle traverses the gas, it produces ionization, which stimulates an avalanche that is detected on each side of the detector by 5-cm-wide pickup strips. The strips on the two sides run at right angles to each other to allow the (J and rP coordinates of the discharge to be measured. 2.3 Trigger, Data Acquisition, and Software 2.3.1 Trigger Although beam bunches cross every few nanoseconds, most crossings do not produce events that are interesting from a physics perspective. The purpose of the trigger is to identify events that should be stored for analysis. Table 2.2 lists event types of interest and their cross sections. The challenge is to separate these events from background, which is predominantly caused by spent particles (beam particles that 21 +HV ·HV +HV ~(----- 22IIcm -----~) ·HV Figure 2.14: Left: cross-section of a KLM detector layer. Right: barrel (top) and endcap (bottom) RPC·s. Distances are in mm unless indicated otherwise. 22 Event Type Cross Section (nb) e+e- ...... T(4S) ...... BB 1.2 e+e- ...... qij ...... hadrons 2.8 e+e- ...... ,.,,+,.,,-,7+7- 1.6 e+e- ...... e+e-(11u.b ~ 17") 44 e+ e- ...... 'Y'Y( I1u.b ~ 17") 2.4 21 processes (11u.b ~ 17",Pt ~ 0.1 GeV/c) ~ 15 Table 2.2: Physical processes of interest and their cross sections. scatter against residual gOB molecules in the beam pipe). Belle implements two types of hardware triggers (Levels 0 and 1) and two types of software triggers (Levels 3 and 4). The Level-O pre-trigger is a fOBt trigger bOBed on TO~ and CD(J6 hits. This trigger signal is available approximately 600 ns after the collision and instructs the SVD to hold its analog signals until the Level-l trigger decision. The Level-l trigger, shown OB a block diagram in Fig. 2.15, combines information from the CDC, TOF, ECL, KLM, and EFC in a central trigger system called Global Decision Logic (GDL) [30]. The decision to keep or discard an event is made bOBed on correlations among the sub-detector signals and is available within 2.2 p,s of the collision. The trigger is over 99.5% efficient for hadronic events. In addition to this trigger, the GDL also provides signals to gate the ECL readout and stop the CDC readout. The timing of these signals is determined by the first TSC signal, or by the ECL trigger if there are no TSC hits. The Level-3 trigger [31] is an online software trigger, introduced in January 2001 to further reduce the amount of data to be stored. It employs an ultra-fast track finding algorithm and rejects events that do not contain any tracks originating within "The TOF Level-O trigger was upgraded In 2003 to reduce its dead time from 350 os to 96 os [29]. 6Prlor to the Inner-CDC upgrade In 2003, the TOF alone provided the Leve1-0 trigger. 23 I Cathode Pads ~ :;t Z finder 1tnckc:oont l Stereo Wi.u r I A:d.I Wira TI'ltdI. Segmt.nl r" trat:k count Multiplicity till ... I T.".""" 's;, ~ Timln. .s OllStuc:ount C ~ Timins Trigger Cell J" : ~ 411'4 Sum Hi&h ThJ"C:!ihold ~ I EMfJYSum .- Q Low n.uhold ~ ~ Bhabha Q G'" I Hit f----.j IJ hit I • TriSle ~ Bhabha I Trigger Cell ~e Th~"""~ Two photOfl 2.2 ~ arter event crossing • Figure 2.15: The Belle Level-1 trigger system. Izl < 5 cm of the IP. This trigger rejects 50-60% of the events that pass Levell, while retaining over 99% of hadronic events. Events that survive the Level-3 trigger are written into a raw-data file. The Level-4 trigger [32] is an omine software trigger. Using a fast track-finder, this trigger selects events that have at least one track with transverse momentum Pt > 300 MeV /c and a distance from the IP of Idr l < 1.0 cm and Idz l < 4.0 cm at the point of closest approach. The Level-4 trigger reduces the number of events by a factor of five and is more than 99% efficient for hadronic events. Events that satisfy the Level-4 criteria are fully reconstructed and written to data summary tapes (DSTs). Events that are rejected remain available in the raw-data fil es. Levels 3 and 4 do not remove events that have been identified at Level 1 as Bhabha or " . These are used for measuring luminosity and are pre-scaled in the Level-1 hardware by a factor of 1/ 100. Randomly triggered events, which are used for studying the background, are also saved. 24 2.3.2 Data Acquisition Figure 2.16 shows an overview of the Belle data acquisition (DAQ) system [33J. Sig nals from the seven sub-detectors are processed in parallel. A charg~to-time (Q-to-T) conversion technique is used to read out all the detectors save the KLM and the SVD. In the Q-to-T scheme, charges are stored in capacitors, which are discharged at a con stant rate. Tim~to-d igital converters (TDC's) record the start and stop times of the discharge relative to a common stop time, thereby encoding both the time and the amplitude of the signal. In the case of the KLM, ruts are recorded directly as time signals. The SVD alone is read out by analog-to-digital converters (ADC's). An advantage of the widespread use of time signals for readout is that a single type of module can be used throughout, simplifying system maintenance. library Figure 2.16: The Belle DAQ system. Data from the subsystems are assembled into complete events in an event builder. They are then transferred to the online computer farm for the Level-3 trigger, and 25 subsequently to the oHiine storage facility for the LeveI-4 trigger and further process ing. The Belle DAQ system is capable of operating at an event rate of up to 500 Hz, with a deadtime < 10%. The storage size of a typical event is 30 kB. 2.3.3 Software The Belle Collaboration has developed an extensive body of software, referred to as the Belle Analysis Software Framework (BASF) [34], to integrate the different stages of data processing and analysis. The code, which is written mostly in C++, is made up of modules that are compiled as shared objects and dynamically loaded at runtime. The system is highly flexible: not only can users load specific modules from the BASF library, but they can also download the library itself and modify it to suit their needs. The first step of data processing involves recording digitized detector signals in DST files. In the next step, the signals for each event are analyzed: charged-particle tracks are reconstructed, and correlations are made among signals from different de tectors. Quantities of interest, such as the four-vectors of charged tracks and photons, are recorded in mini-DST (MOST) files. The information is organized into tables us ing a data-management system called PANTHER [35]. Individual users can then read the information in the tables by plugging their own analysis modules into the BASF interface. To minimize duplication of effort, commonly-used analysis utilities are also provided as part of the BASF library. Monte Carlo (MC) simulations are also supported by BASF. Lists of four-vectors for a given decay chain are generated using EvtGen [36]. The detector response is then simulated using GEANT [37], combining randomly-triggered data with the simulated events. The output is saved in the same format as the data in the MDST files, allowing the same analysis code to be applied to MC as to data. The Belle computing system is described in more detail in Ref. [38]. 26 Chapter 3 Particle Identification 3.1 Electron Identification Electrons are identified based on information from the CDC, ECL, and ACC. The following discriminants are used [39):1 • The energy cluster associated with the particle is found by extrapolating its CDC track to the ECL. Since electron showers have significantly better position resolution than hadronie showers, a better fit is more likely to be obtained for an electron than for a hadron. • In the energy range of interest, the electon's mass is a negligible fraction of its total energy. Thus, Elp, where E is the energy measured by the ECL and p is the momentum measured by the CDC, peaks near unity for electrons. It is smaller for muons and hadrons, which are heavier and leave less of their energy in the ECL. Elp is the most powerful discrimi nant at particle momenta above 1.0 GeV Ie. At lower momenta, electrons lose a greater fraction of their energy to interactions with detector mate- 'The plots included in this chapter have been taken from the references cited in the text. 27 rial before reaching the ECL, and their Elp decreases, making them less distinguishable from hadrons. • Electromagnetic and hadronic showers in the ECL have different shapes. Whereas electrons deposit all of their energy in a compact shower in the ECL through the electromagnetic interaction, muons deposit a relatively small amount of energy through ionization, and hadrons leave multiple trails by interacting strongly with the CsI. • At particle momenta greater than 0.3 GeY Ie, dEldx, the rate of energy loss to ionization, can be used to distinguish electrons from muons and hadrons. Figure 3.1 shows dEldx as a function of momentum for electons and pions. • Due to their low mass, electrons are able to produce Cherenkov radiation in the ACC at a lower momentum than hadrons. This can be used to identify electrons in the momentum region below 1.0 GeYIe. For each electron candidate, "electron likelihoods" and "non-electron likelihoods" corresponding to each discriminant are calculated. These are then combined into an overall "electron probability";2 L. Leld= L.+Le' (3.1) where L. is the product of the five electron likelihoods, and Le is the product of the five non-electron likelihoods. Distributions of this quantity for electrons and pions are shown in Fig. 3.2. If charged tracks with momenta between 1.0 GeYIe and 3.0 GeYIe are required to satisfy Leld > 0.5, the efficiency for a single electron track is (92.4 ± 0.4)%, and 2Technica.\ly, L.ld is not a probability, as there may be correlations among the discriminants. 28 e',------, ~ ~ 1.8 . ~ ~ g V 1.6 • electron fI' '_ .. , .------c:I .I,~~... ,.. •• "---• .....,...---,. -..... J. 4 Ifi"...... ! • • pion 1.5 , 1.5 J p.. (GeV/e) F igure 3.1: dE/dx as a function of momentum for electrons and pions. eleClron 10 10 10 10 o 0.2 0.4 0.6 0.8 I eid li kelihood Figure 3.2: Leid distributions for electrons and pions. 29 the probability of misidentifying a pion as an electron is (0.25 ± 0.02)% [39]. In the analysis presented in this thesis, electron candidates are required to have Leid > 0.01.3 3.2 Muon Identification Unlike hadrons, muons do not interact strongly; they therefore penetrate deeper into the material of the detector and leave cleaner tracks. To identify muons, charged tracks reconstructed in the SVD and CDC are extrapolated into the KLM and ass0- ciated with hit positions recorded by the KLM. The quantities used to discriminate muons from hadrons are: • !:J.R, the difference between the expected and measured range of the track in the KLM, and • X~, the reduced x: of the transverse deviations of the recorded KLM hits from the extrapolated track. Probability distributions in !:J.R and ~ for muons, pions, and kaons have been de termined based on 105 simulated events of each type. A two-dimensional probability density is constructed as (3.2) where p can be JI, 7r, or K. For each recorded charged track, a muon likelihood ratio is defined as L - Pp (3.3) muld - Pp + P" + PK ' where Pp, P", and PK are the probabilities, given by Eq. 3.2, that the charged track is a muon, a pion, or a kaon, respectively [40]. "Specifically, the requirement is eld.prob(3, -1, 5) > 0.01. 30 Muon-likelihood distributions for muons and pions are shown in Fig. 3.3, along with efficiencies 88 functions of cut value. Figure 3.4 shows the momentum depen dence of muon efficiency and pion fake rate when a muon-likelihood cut of 0.66 is ap plied. Note that applying a muon-likelihood cut automatically abandons any muons with momenta too sma1l to reach the KLM (Le., < 0.6 GeY Ie). In the analysis presented in this thesis, muon candidates are required to have Lmuld > 0.1. 5 5 !.i 10 10 4 ~ 104 j 10 "0 "0 103 i1:1 i 102 0 0.2 0.4 0.6 0.8 I 0 0.2 0.4 0.6 0.8 I MuoD_likelihood MuoD_likelihood >. I ~ 0.1 r:l .~ 0.8 """'"' .~ 0.08 ~ 0.6 tfl" 0.06 0.4 0.04 0.2 - 0.02 I I , I I 0 00 0.2 0.4 0.6 0.8 I 0 0.2 0.4 0.6 0.8 I MuoD_likelihood MuoD_likelihood Figure 3.3: Muon likelihood distributions (top) and efficiencies as functions of muon likelihood cut value (bottom) for simulated muons (left) and pions (right) in the momentum range 0.7-3.0 GeV Ie and with polar angle between 23° and 150° [4OJ. 3.3 Hadron Identification If the momentum of a particle is known. a measurement of velocity gives its m88S via the equation: p = mc{3'Y, (3.4) 31 0.75 .~----~---~--- '-~ + I i+ r 0.251- ~~I-i- I Ii i °0L-~~~-Lt--~t~~--~Z--~~~~3 00 U t t~ Z ~ 3 P(GeVk) P(GeVl<) Figure 3.4: Muon efficiency as measured with cosmic-rays (left) and pion fake rate as measured with Ks -+ 11"+11"- events (right) [28]. A muon-likelihood cut of 0.66 [28] has been applied. where {3 = vic and 'Y = (1 - (F)-1/2. All three methods of charged-hadron iden tification used at Belle are based on measuring velocity and combining it with the momentum measured in the CDC: • A particle's rate of ionization energy loss, dEldx, depends on its veloc ity (18). The energy loss measured in the CDC can be used to identify charged kaons, pions, and protons at momenta below 1.5 GeV Ie, as shown in Fig. 3.5. • A charged particle's time-of-flight, as measured by the TOF, can be com bined with its momentum to determine its mass, as shown in Fig. 3.6. This method is effective at momenta below 1.2 GeVIe. • As noted in Section 2.2.3, whether Cherenkov radiation is emitted when a particle passes through a material can indicate the particle's velocity. The refractive index of the aerogel used in the ACC is optimized for separating pions and kaons at momenta above 1.2 GeV Ie. 32 Information from the three detectors is combined into a likelihood that the particle is of type i: (3.5) where i can be 71", K, or p. The likelihood that the particle is of type i can then be compared to the likelihood that it is of type j by constructing a likelihood ratio [41 J: ..) P; L pld (~: 3 = D D (3.6) r;+.rj The efficiency and fake rate for a given cut depend on the momentum and polar-angle distributions of the particles in the decay of interest. For the decay D>+ -+ D°7l"+, with DO -+ K7I", the bon efficiency and pion fake rate for a likelihood-ratio cut of Lp1d(K: 71") ~ 0.6 have been measured to be approximately 80% and 10%, respectively. In the analysis presented in this thesis, bon candidates are required to satisfy Lp1d(K: 71") > 0.5, Lp1d(P: K) < 0.95, and Leld < 0.95.4 Pion candidates are required to satisfy Lp1d(K: 71") < 0.95 and Leld < 0.95.5 4Speclfica1ly, the requirement Is: atc_pld(3, 1, 5, 3, 2) > 0.5 and Bt<4Jld(3, 1,5,4,3) < 0.95 and eld.prob(3, -1,5) < 0.95. &Speclfica1ly, the requirement Is: atc_pld(3, 1, 5, 3, 2) < 0.95 and eld.prob(3, -1, 5) < 0.95. 33 x 4 iB -0 3.5 P 3 2.5 2 ·e 1.5 0.5 _1·'"'.5,-'--~~_-o-, ~.L.J.-"":-o . 5 =""-':"0~--'--~'c0"'.5o-'-~--'---!--'-' I09 1O( P (GeV/c) ) Figure 3.5: CDC measurements of dEj dx as a function of momentum for charged tracks in collision data. Solid curves indicate the expected distributions for pions, kaons, protons, and electrons. Tracks independently identified as electrons are shown in blue. 2000 1750 Exp5 data a ([OF) = lOOps 1500 P< 1.25GeV/c 1250 1000 750 K 500 p 250 0 -0.2 0 0.8 1.2 ~olS(Gev) Figure 3.6: Charged-hadron masses, calculated using time-of-Bight measurements from the TOF and momentum measurements from the CDC, for data (points) and MC (histogram). 34 Chapter 4 Event Selection To analyze the processes B+ --> J /7/JK+rr+rr- and B+ --> '1/1 K+rr+rr-, it is necessary to identify events containing B mesons decaying this way (signal) and to distinguish them from all other events (background). The goal is to find a set of criteria that retain the most signal while rejecting the most background. In studying a mode involving a J N or '1/1 in the final state, a key strategy for removing background events is to reconstruct J/7/J's only in their decays to e+e- or 1'+1'-, and 'I/I's only in their decays to e+e-, 1'+1'-, or J /7/Jrr+rr-. Although this means abandoning all but 11% of J N's and 5% of 7/J"s, the background reduction is dramatic. The requirement that each event contain a pair of oppositely-charged tracks that satisfy lepton-identification criteria and have a total invariant mass close to that of a J/7/J or 7/J' reduces continuum backgrounds (i.e., backgrounds from e+e- --> qq, which plague most analyses) to a negligible level. It also greatly suppresses backgrounds from B decays that do not contain a J/7/J or 7/J' in the final state. In the case of a multi-particle final state such as J NK+rr+rr- or 7/J' K+rr+rr-, however, a new challenge arises in the form of multiple B candidates. IT a correctly reconstructed B candidate includes a low-momentum pion, then an additional, false B candidate can be formed by replacing that pion with a low-momentum pion from the 35 other B. As the exchange does not significantly affect the energy or momentum of the B candidate, both candidates can satisfy the kinematic criteria outlined in Section 4.5. Multiple candidates can spoil branching-fraction measurements and distort observed mass spectra. Although a "best candidate" selection is performed for such events, the accuracy of the selection necessarily deteriorates as the number of candidates increases. The event selection procedure described in this chapter is therefore designed to minimize the number of multiple candidates without sacrificing events with low momentum pions. 4.1 Multiply-Reconstructed Tracks Low-momentum charged tracks that curl up in the CDC can be reconstructed multiple times by the track finder, as illustrated in Fig. 4.1. This effect manifests itself in the form of equally-charged tracks with nearly identical three-momenta, or oppositely charged tracks with nearly opposite three-momenta. To ensure that no track is in cluded more than once, criteria similar to those presented in Ref. [42] are used. Any track that has a transverse momentum Pl. < 0.4 GeY Ic is paired with every other track in the event. If the momentum difference Ipl - 1121 is less than 0.1 GeY Ic, the cosine of the opening angle 8 is plotted, as shown in Fig. 4.2. For same-charged pairs with cos(8) > 0.98, and oppositely-charged pairs with cos(8) < -0.98, the track with the larger value of (4.1) is removed. Here, dr and dz describe the track's separation from the event IP at its point of closest approach, and the ratio 1: 30 is the ratio of FWHM for dr and dz distributions in MC. This cut reduces the number of charged tracks in the data sample by 1.4%. 36 true track duplicate track (same charge) x / \ duplicate track Interaction Point (opposite charge) Figure 4.1: The interpretation of a single track as multiple tracks by the tracking software. Ip - P 1< 0.1 GeV/c AND P < 0.4 GeVlc PJI 0.1 AND PI! 0.4 GeV/c , 2 n Ip,· < < ~ ~ .. ~ .. ..'ii, .. - ~ .. "C .. to 40- ,. • _ _ _ _ • u u u u , • _ _ _ _ • U U U U , 00«9) c:Os(9) P 1 ~ " ~tOO i ..:c~------~~ .. ,.- ., .aa ....1 -0.4 -G.2 -0 0.2 0.4 U U t ., 0.2 G.4 0.1 0.1 1 cos(9) COs(e) Figure 4.2: Cosine of the opening angle between equally-charged (left) and oppositely charged (right) track pairs in data. T he top and bottom rows show pairs that are selected and excluded, respectively, by the criteri a Poll < 0.4 GeV Ie and iPI - P2i < 0.1 GeV Ie described in the text. Tracks that are reconstructed more than once are observed as a sharp peak at cos(lJ) '" 1 if they are of equal charge, and at cos(lJ) '" -1 if they are oppositely-cllarged. 37 4.2 Selection of J /'ljJ and 'ljJ' Candidates 4.2.1 Leptonic IN and 1/1' Decays The decays J N --> p,+ p,- and 'IjJ' --> p,+ p,- are reconstructed by combining oppositely charged track pairs that satisfy the muon-identification criteria of Section 3.2. Simi larly, J N --> e+ e- and VI --> e+ e- decays are reconstructed by combining opposite1y charged track pairs that satisfy the electron-identification criteria of Section 3.1. To account for energy losses due to final-state radiation or bremsstrahlung in the de tector, photons detected within 50 mrad of the initial direction of any electron with energy less than 3.5 GeV are also included in the e+e- invariant-mass calculation.l The resulting dilepton mass distributions are shown in Fig. 4.3. The J N peak is modeled as a double Gaussian, and the 'IjJ' peak is modeled as a single Gaussian. In each case, a linear background is assumed. Table 4.1 lists the J / 'IjJ and 'IjJ' yields according to the fits. Muon pairs are discarded unless they have an invariant mass within ±30' of the fitted J N or VI mean.2 Electron pairs are similarly discarded unless they have an invariant mass within the range extending from -40" to +30' of the fitted J N or VI mean. The mass window for electron pairs is asymmetric about the mean so as to include the radiative tails of the J N and VI, which are not completely recovered by the photon addition described above. Lepton track pairs that survive the mass cuts are fitted to a co=on vertex, which is constrained within errors to the measured IP. This vertex is then fixed, and another fit is performed, constraining the dilepton invariant mass to the nominal J /'IjJ or 'IjJ' mass. Since the observed widths of the J N and VI are dominated by measurement error, forcing the masses of the J /'IjJ and VI candidates to be equal to their nominal values improves the mass resolution of the B candidate. lSince an electron with energy greater than 3.5 GeV does not curve a.ppreclably in the l.5-T magnetic field, adding the II&'IOci&ted ECL shower to such an electron would count Its energy twice. 2Here, u Is the fitted width of the peak in question and refers to the width of the narrower Ga.ussIan in the case of the J I'1/J fits. 38 Jill! """ eOeO ",(28) -+ ..... '\, ~~ "• "'" :I ..... :EC! """'" C! ~35000 ~ ...... ; "'" ...... • ~25000 ~• ..... 20000 20000, "'" .... '''''.... "'".... f .. , ." .. us • us .... ' .1 3,71 30. M(.·.), GaVf~ M(e·.,. GeV/~ Jlv ..., 11·).1 0 w(2SH "-". "u "u >45000 • "I ..... :IC! """'" C! ~ ..... ~"'" .;"'" .; ..... wE25000 ~25OCW) ,20000 20000 .... '"'" '''''.... j ''''' + + .... fM ,... .., 3.15 ...... • ...... M{j.t"}t'). GaYle' Figure 4.3: Measured invariant masses of electron (top) and muon (bottom) pairs. Blue arrows indicate the mass windows for J N (left) and .p' (right) candidates. Mode Yield IN -> e+e (4.08 ± 0.08) x 105 J N -> ",+ ",- (4.37 ± 0.07) x 105 'I/J' -> e+e- (8.4 ± 0.8) x 103 'I/J ' -> ",+ ",- (9.7 ± 0.3) x 103 Table 4.1: Inclusive J/.p and .p' yields as calculated by integrating the fitted peaks shown in Fig. 4.3. Errors are statistical only. 39 4 .2.2 'lj;' Decays to J /'lj;7r+7r- To reconstruct ..p' -> J /..p7r+7r - decays, leptonic J /..p candidates are combined wi th a pair of oppositely-charged tracks that satisfy the pion-identification criteria of Sec tion 3.3. Since the 7r+7r- invariant-mass distribution in ..p' -> J /..p7r+7r - decays is known to peak at high values [43], the dipion invari ant mass is required to be greater than 0.4 GeV /c2 . The resulting e+e- 7r+7r- and ",+ ",- 7r+ 7r- mass distributions are shown in Fig. 4.4.3 In each case, the ..p' peak is modeled as a double Gaussian with a single mean on top of a second-order-polynomial background. Table 4.2 lists the ..p' yields according to the fits. Unless they have an invariant mass within ±3a of the fitted ..p' mean, ..p' -> J /..p7r+7r - candidates are discarded 4 0 '+I(2S) -+ J/vIt· I't , JI", -+ • •• . ;- 1\ I:;- ~3500 ~ 1- ~ ~2100 ~ "",... ~\ "" ~ ... I, .1 \ ... ~ ...... , '.n .n . (25) -+ JI. 1(.,(, JI" ... ~'li ' ~ "'" I:- " ~ -r ~ I:- w - ""I:- ''''' E'- ~ j \'i ,... ,...... ,., ~n 2.74 M(}I'li ~1(), GeVle' Figure 4.4: Measured invariant masses of ..p' -> J/1/J7r+7r - cand idates, with J/1/J decaying to electrons (top) and muons (bottom). Blue arrows indicate the mass windows. 3The widths of the peaks are narrow a.. a result of constraining the mass of the J /'I/J as described in Section 4.2.1. 4The fitted width used for the mass cuts is for the narrower Gaussian. 40 Mode Yield 1/1 --+ e+e-1I"+1I"- (2.0 ± 0.3) x 10" 1/1 --+ p,+ p,-1I"+1I"- (2.1 ± 0.2) x 10" Table 4.2: Inclusive 1/1 yields as calculated by integrating the fitted peaks shown in Fig. 4.4. Errors are statistical ouly. 4.3 Selection of K± Candidates Kaon candidates are charged tracks that satisfy the kaon-identification criteria of Section 3.3 and have an impact parameter (i.e., distance of closest approach) with respect to the fitted dilepton vertex of Idrl < 0.4 cm and Idzl < 1.5 cm. 4.4 Selection of 7r± Candidates Pion candidates are charged tracks that satisfy the pion-identification criteria of Sec tion 3.3 and have an impact parameter with respect to the fitted dilepton vertex of Idrl < 0.4 em and Idzl < 1.5 cm.5 Any pion candidate that is identified as the product of a K] --+ 11"+11"- decay is removed from the event. To reconstruct this decay, oppositely-charged pion candi dates are combined, and standard K] selection criteria, summarized in Table 4.3, are applied. Both pions are vetoed if their combined invariant mass lies between 0.482 GeV/il and 0.510 GeV/il, which corresponds roughly to a region extending from -40" to +3u around the nominal K] mass. 5The Impact-parameter requirement is not applied to the pions in '1/1-+ J/'I/nr+1r-. 41 p (GeV/c) dr (cm) l (cm) art> (rad) ZdIst (em) < 0.5 > 0.05 - < 0.3 < 0.8 0.5-1.5 > 0.03 >0.08 < 0.1 < 1.8 > 1.5 > 0.02 > 0.22 < 0.03 < 2.4 Table 4.3: Selection criteria for R1-- 11'+11'- decays [44]. Here, p is the momentum of the R1 candidate, dr is the smaller of the pion ra.dial impact parameters with respect to the IP, I is the flight length of the R1 candidate in the r-t/> plane, dt/> is the angle in the r-t/> plane between the momentum of the R1 candidate and its displa.cement from the IP, and Zdlst is the separation in the z direction of the pion tracks at their point of closest approach. 4.5 B-Meson Reconstruction If an event contains two oppositely-charged pion candidates, a charged bon candi date, and a J Nor 'l/I candidate, all satisfying the criteria discussed above, a B-meson candidate can be reconstructed. Two kinematic variables can be used to identify true B mesons. First, the reconstructed IIlB8S of a true B meson is likely to fall near the nominal B mass. Second, 88 B mesons are produced in the reaction (4.2) the energy of each true B in the T(4S) frame is half that of the T(4S), which, in turn, is the total energy of the electron and positron beams in this frame.6 'Ihm B mesons can therefore be identified 88 a cluster in a plot of the mass M(B) versus the energy E*(B) of B candidates.7 Since the mass, energy, and momentum of the B candidate are interrelated 88 MJ(B) = E*2(B) - P2(B), one can equivalently plot E*(B) versus P*(B). Although the beam energies at KEKB are remarkably constant, slight drifts can 6To boost to the T(48) rest frame, the nominal three-momenta of the electron and positron beams are assumed. As the beam parameters may drift, the frame to which the parametars are boosted is not necessarUy the same as the reat frame of the T(48) that is produced. Since the same boost is applied to all the varlabIea, however, the Inaccuracy Introduced by this assumption is amaIl. 7Starred variables are measured in the T(48) rest frame. 42 I Entries 22878 71 I I Entries 325822 I ' .3 Mac (Gevtc') Mac (Gevtc') Figure 4.5: t.E versus MBe for JN f(+",+", - (left) and .p' f(+",+",- (right) final states. cause the mass of the Y (4S ) that is produced to vary. This smears the signal in the plot of E'(B ) versus P'(B). To reduce the smearing, the variables E'(B) and P'(B ) are recast in forms that are readily corrected for drifts in the beam energy- namely, the energy difference 6.E and beam-constrained mass M Be : 6.E = E'(B) - Ebeam , (4.3) MBe = JE i,';am - P' 2(B ) . (4.4) Here, Eb..m is half the energy of the Y (4S) and is measured independently [45]. For a true B meson, 6. E will peak at 0, and MBe will peak at the nominal B mass. Distributions of 6. E versus M Be for the decay modes under study are shown in Fig. 4.5. As Table 4.4 indicates, many events have more than one reconstructed B can didate; some have ten or more. To ensure that no B decay is counted more than once, a best-candidate selection is performed. First , B candidates are required to 2 have ]6.E] < 0. 2 GeV and MBe > 5.27 GeV /c If, after this requirement, an event 43 B ~ J/V K+ 7t+ If I Entries 45433 1 B .... ",(2S) K' n' n· 1 Entries 0782 1 ~ !:!. w -0.15 5.25 5.26 5.27 5.28 5.lI 5.3 5.21 5.24 5.25 Moe (GeV/c') Moe (GeV/c') Figure 4.6: t:.E versus MBe for J N K+1f+1f - (left) and .p' K +",+",- (right) final states after best-candidate selection. still has multiple B candidates with the same final state, then the charged tracks that make up each B canclidate are fitted to a common vertex. The canclidate whose vertex fit has the smallest r is kept, and the rest are discarded. Accorcling to Me studies, this procedure selects the correct B candidate in approximately 55% of cases where there are multiple canclidates. Figure 4.6 shows 6E versus M Be after the selection. In the case of B + -> J /.pK+1f+7r-, the decay B + -> '1/1' K + -> J /'1/17r+7r- K + is vetoed by rejecting all B candidates that have a J NJ7r+7r invariant mass between 3.675 GeV /e? and 3.695 GeV /2. 16EI < 0.2 GeV It:.EI < 0.2 GeV Decay Mode MBe > 5.2 GeV /2 M Be > 5.27 GeV /e? B + -> J /1/JK+7r+7r 48% 25% B+ -> 1/J' K +7r+7r- 57% 34% Table 4.4: Fractions of events with multiple B candidates for different t:. E and MBe cuts. 44 B+ ~ J/W K'" n+ It =:; 80000 ..:I; 70000 ~ 60000 • • 50000 "~ 40000 30000 20000 10000 ~~o.~~~.~~~~.~~~ ~~.~~O~~O~. OI~O?.~g=~O~.ro~~O.M~~O.M 6 E (GOY) B'" ~ W(2S) K'" It''' 1t > :I.. 25000 ~ 20000 • "E• 1.5000 ill 10000.... -B.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 6 E(GeV) Figure 4.7: Me t:.E distributions for B+ ..... JN K +7r+7r- (top) and B+ ..... I/J'K+7r+7r (bottom). 4.6 Signal and Sideband Regions Distributions of t3.E for B + --+ J /I/J I<+7r+7r- and B + --+ l/J'I<+7r+7r- MC are shown in Fig. 4.7. The signal peak is modeled as a double Gaussian with a single mean, and the background is modeled as a second-order polynomial. Figure 4.8 shows the corresponding distributions for data. The signal is modeled as a double Gaussian with a single mean, fixing the width and relative height of the wider Gaussian to the results of the MC fit. The background is modeled as a first-order polynomial. Using the results of the data fits, the signal region is defined as -3(7 t.E < It3.E - ~t. EI < +3(7 t.E , (4.5) where ~t. E is the mean of the signal peak, and (7 t.E is the width of the narrower 45 -0.05 0 0.0:; 0.1 0.15 d E (GoY) 0.1 0.15 d E (GoY) Figure 4.8: Data 6E distributions for B+ -> J /..p K +7f+7f- (top) and B + -> ..p' K +7f+7f (bottom). Blue and magenta arrows indicate the signal and sideband regions, respectively. Gaussian. The sideband region, which is used to study the background under the signal, is defined as -0.13 < I ~E - J.lt.E1 < - 0.05 and 0.05 < I~E - J.lt.E1 < 0.13 . (4.6) The sideband normalization factor is given by J Pbkgd~E s igna.l IB - (4.7) J Pbkg d6E ' sideband where Pbkg is the polynomial representing the background. T he fraction of signal region events that are background is estim ated as B nB - S IB , (4.8) 46 where S and B are the numbers of events in the signal and sideband regions, respec tively. Table 4.5 lists the values of {J-/j,E, (J /j,E, fB, and nB for both decay modes. Mode {J-/j,E (MeV) (J/j,E (MeV) fB nB B+ -> J NK+7r+7r- -0.902 ± 0.095 5.81 ± 0.11 0.2180 0.266 ± 0.003 B+ -> '1/1 K+7r+7r- -2.29 ± 0.27 4.86 ± 0.33 0.1823 0.345 ± 0.012 Table 4.5: Mean of the signal peak, width of the main Gaussian, sideband normalization factor, and background fraction in the signal region, according to the fits in Fig. 4.8. Errors are statistical only. 4.7 Cut Optimization The cuts presented in this chapter were tuned by studying the fj"E distribution of the JNK+7r+7r- final state after requiring MBc > 5.27 GeV /rJ. Cut efficiencies were optimized for the decay s+ ...... '1/1 K+ ...... J /'ljJ7r+7r- K+, by se lecting events with JN7r+7r- invariant mass between 3.675 GeV /rJ and 3.695 GeV/rJ. The signal efficiencY of each cut was estimated by measuring the height of the fj"E signal peak with and without the cut. The background reduction due to each cut was estimated by vetoing events with JN7r+7r- invariant mass between 3.675 GeV /rJ and 3.695 GeV /rJ and measuring the level of the background under the signal peak with and without the cut. An iterative method was used, starting with an arbitrary set of cuts and varying each cut separately. After each round, cuts were adjusted to increase background rejection while maintaining efficiency. Cuts that did not remove a larger fraction of the background than of the signal were discarded. The procedure was repeated until a stable set of cuts was obtained. Signal efficiencies for the final set of cuts were checked using B+ ...... JNK1(1270)+ MC. Approximate efficiencies and background-rejection factors for the kaon and pion 47 cuts are shown in Table 4.6. Background Data MC Cut Rejection Efficiency Efficiency Idr" I < 0.4 em 42% 84% 84% Idz.rl < 1.5 em IdrKI < 0.4 em 11% 99% 97% IdzKI < 1.5 cm L:;iK: 11") > 0.5 76% 89% 94% L~Id(K: 11") < 0.95 24% 99% 99% L::id < 0.95 L!fd < 0.95 9% 96% 99% L:;d(P: K) < 0.95 Table 4.6: Background rejection factors and signal efficiencies for kaon and pion cuts. AB explained in the text, "data efficiency" is for JJ+ ..... 'Ij/ K+ ..... J/'Ijnr+'/r-K+ data, and "MC efficiency" is for B+ -+ JNKl(1270)+ -+ JNK+'/r+'/r- MC. The L:;d(K: '/r) > 0.5 cut is less efficient for the former than for the latter, because the average momentum of the kaon in B+ -+ 'lj/K+ is higher than that ofthe kaon in B+ ..... JNKl(1270)+, and particle identification is less effective at high momenta. 48 Chapter 5 Coordinate Transformations Signal and sideband regions were defined in Section 4.6. Since the sideband region is used to study the background under the signal, it is essential that the data in the sidebands accurately model the background in the signal region. To this end, a set of transformations is applied to the variables M(K7l'7l'), M(K7l'), and M(7l'7l'). This chapter describes these transformations. 5.1 M(K7r7r) Figure 5.1 shows M(K7l'7l') for events in the signal and sideband regions. There is clearly a problem: signal and sideband data have different maxima in M(K7l'7l'). Plotting AE versus M(K7l'7l') reveals the cause of the discrepancy. As Fig. 5.2 demonstrates, the kinematically allowed range of M(K7l'7l') depends on AE. While the minimum value of M(K7l'7l') is simply M(K7l'7l')mIn = MK + 2M.. , the maximum value varies with AE as M(K7l'7l')max = AE + MB - M.p. Here, MK, M.. , MB and M.p stand for the nominal masses of the subscripted particles.l lThe symbol'" stands for IN In the case of the B+ --> JNK+'ff+'ff- decay and for '1/1 In the case of the B+ --+ '1/1 K+'ff+'ff- decay. 49 ~ - ~ ,~ C------=------, ! ... .~ ,.. ~ ~ N ... ~ ... .. Iw '",.. • U ,~ U tA IA 2 U U u ,.I 2 U U 1 M(KlI:l) (GlYIc ) M(Ku) (GeVIell Figure 5.l: Invariant mass of the K +7r+7r- system in B + -> J/r/J K +7r+7r- (left) and B+ -> ..p' K +7r+7r- (right) data. Blue and red nistograms show events in the signal and normalized sideband regions, respectively. To understand the t:.E dependence of M(K7m)max, note that the maximum value of M(K7m) is attained when both the K +7r+7r- system and the 1/J are at rest in the B-candidate's rest frame. Thus, M(K7r7r)max = M(B) - M"" (5 .1 ) where M(B) is the mass of the B candidate. Recalling that in the T ( 48) rest frame, Eb.am is the energy of a true B meson produced in the reaction T(48) -+ B+B-, Eq. 4.3 may be written as t:.E - E*(B) - Eb."m - J M2(B) + P*2(B) - VM1 + PEl (P*(B))2] (Pa) 2] "" M(B) [1 + 21 M(B) - M a [11 + 2 Ma "" M(B) - M a . (5 .2) Here, P*( B) is the momentum of the B candidate, while M a and Paare the mass and momentum of a true B-meson. As a consequence of the Mac > 5.27 GeV Ie? requirement, Pa < 0.46 GeV Ie. Not only are (P*(B)IM(B))2 and (Pal M aJ2 small , but the difference between them is even smaller. Since the B meson is narrow, Ma "" Ma; thus, combining Eqs. 5.1 and 5.2 gives M(K7r7r)max "" t:.E + Ma - M.p . 50 Transforming M(K1I'1I') as follows removes its dependence on !l.E: Here, M(K1I'1I')mln = MK + 2M" and M(K1I'1I')max = !l.E + MB - Mt/J as before, while M(K1I'1I')::"'" = MB - Mt/J is the value of M(K1I'1I')max at !l.E = O. Figure 5.3 shows !l.E versus the transformed coordinate M'(K1I'1I'). While the minimum value of M(K1I'1I') is unaffected by the transformation, the maximum value is changed in such a way that the maximum of M'(K1I'1I') at any !l.E is equal to the maximum of M(K1I'1I') at!l.E = o. The range of M(K1I'1I') is compressed for positive values of!l.E and stretched for negative values of !l.E.2 An important feature of the transformation is that it does not change M(K1I'1I') at !l.E = O. Thus, although sideband and signal regions are both transformed, the change is mjnimal in the signal region. Figure 5.4 shows M'(K1I'1I') for events in the signal and sideband regions. The problem of Fig. 5.1 has been solved: the endpoints of the transformed signal and sideband distributions match. The transformation has been checked by analyzing generic MC, which simulates T(48) decays to B+ B- and IJO fJO and includes all known decay modes of B mesons. Ai> can be seen in Fig. 5.5, M(K1I'1I') displays the same !l.E-dependence in MC as was observed in data. The transformed scatter-plot is shown in Fig. 5.6. Since the main source of background is misreconstructed B-meson decays, one way to determine whether the transformed sidebands correctly model the background in the signal region is to examine MC events that do not contain B+ decays to '1jJK+1I'+1I'- 2In principle, real B-mesons should have M = 0 on average. Because of experimental errors, however, the observed mean of the M distribution of the signal Is shifted from 0 by 1-2 MeV. For simplicity of presentation, this mean Is assumed to be 0 in the equations of this chapter. In fact, just as the signal and sideband regions are centered around the measured M mean, the transformations are also made about the measured M mean. 51 B' --> ",(25) IC '" 1[" f ' ;;0.15 0.05 - - -<1.1 ~).1 5 - Figure 5.2: /:;.E versus M(K":7r) for B+ ~ J N K +7r+7r- (left) and B+ ~ ,p' K +7r+7r (right) data. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M(K7r7r). M'(K'm) IG."M, M ' (~~ ) IG.,I/o' , Figure 5.3: /:;.E versus M'(K7r7r) for B+ ~ JN K +7r+7r- (left) and B+ ~ ,p'K+7r+7r (right) data. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M'(K7r7r). 52 ~ ~ .------, ~ n. > I ~ i q 0 ". ~ ~ ... N .. ~ .. ~ .. " • 0.' 1~ 1A 1A 2 U U M'(Ku ) (G.V/Cl) Figure 5.4: Transformed invariant mass of the K +7r+7r- system in B + - J /I/JK +7r+7r (left) and B+ - I/J' K +7r+7r- (right) data. Blue and red histograms show events in the signal and normalized sidehand regions, respectively. final states. If the transformed sidebands are suitable, the background events in the sidebands should have t he same M'(KTr7r) dependence as those in the signal region. Figures 5.7 and 5.8 show the M(KTr7r) and M'(Krr7r) distributions, respectively, of the background in the signal and sideband regionsa The transformation clearly im proves the agreement at high M (K 7r7r) . While there is some discrepancy between the background in the signal and sideband regions, thls is independent of the transfor mation4 Note that the generic-MC plots in tllis chapter have significantly greater statistics than data. Compared to the number of events in the data sidebands, t he number of background events in the MC sidebands is larger by a factor of 60 in the case of B + -+ J /'Ij;K +7r+7r- and by a factor of 10 in the case of B + -+ 'Ij;' K +7r+7r-. 3 Although the shape of the Me signal distribution, which is also shown, differs significantly from that of the data, nothing in the analysis depends on the two being the same. 4The small bump near 1.3 GeVIe> in the sideband-subtracted M(K7rrr) and M'(K 7rrr) distri butions of B+ -> JII/J K +rr+rr- is a result of feed-across from KI (1270) decays to non-K+7r+rr final-states. It is less than ~ 0.4% of the [(1(1270) signal. 53 M(K•• ) (Ge'ilc' ) M(K.. ) /G•• VI,,', Figure 5.5: 6.E versus M(K7r7r) for B+ -> J/..p K +7r+7r- (left) and B+ -> ..p' K+7r+7r (right) as reconstructed in generic MC. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M(K7r7r). 2..2 2.4 M'(K.. ) (GeVic' ) Figure 5.6: 6.E versus M'(K7r7r) for B+ -> J/I/JI(+7r+7r- (left) and B+ -> I/J'K+7r+7r (right) as reconstructed in generic MC. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M'(K7r7r). 54 lama..... t:...... '~~~~~~~~Un-~,,~~,~~u~~' _ IOeVlo'l UI •1I!I<=l _ I:: a:: ... t:...... ·'~~~--~~~~u~~u~-+.--~u~~u' ·,~..~~--~~~~~~,.~~.--~u~~· -- II!I<=lIOeVlo'l ) : a :: j ::.. """...... u • u ·'~.. ~~--~~~~~~u~~.--~u~~u' II!I<=lIOeVlo'l ) ... -- a .. I .,I:---"....c~~.=f!L+-r-l .. u u u a u u Figure 5.7: M(K7f7r) distributions for IJ+ -+ J NK+1r+1r- (left) and B+ -+ '1/1-- K+1r+1r (right) as reconstructed in generic Me. The top row shows the signal region for events in which a B+ -+ 1/JK+1r+1r- decay was generated. The second row shows the signal region for events in which a B+ -+ 1/JK+1r+7r- decay was not generated. The third row shows the normalized sideband region for events in which a B+ -+ 1/JK+1r+1r- decay was not generated. Finally, the fourth row shows the result of subtracting the third-row distribution from the second-row distribution. 55 1= 1 iI .... iI:: ~ ..... j:: 1=...... J~ "\ ... • ...... a .. IA • .. •• •• •• •• a IA - 1I'{Ka)(OaV1D') 1I'{Ka)(OaV1D') 1! ...... I .... 1 ..... ii- iI ...to> j:... I ...... '" 0 0 .. •• ...... a .. IA .. •• ...... a___ .. 10') .. •• 1I'{Ka)(OaV1D') 'II'" ... I- .. Ia .., II- II ... ..,... j:.... j ...... " • .. .. •• ...... a .. 0 ...... 1I'{Ka)(OaV1D') 1I'{Ka) (OoWo'l .. I .. I ...... _n • r ~n. iI .., II 0 j ... • -tr~U Iv I 0 ..., :...... •• a__ .. (OoWo'l IA .. • ...... a .. IA Figure 5.8: M'(K7I'7I') distributions for B+ .... JNK+7I'+7I'- (left) and B+ .... 'I/IK+7I'+7I' (right) DB reconstructed in generic Me. The top row shows the signal region for events in which a B+ .... 'l/JK+7I'+7I'- decay W8B generated. The second row shows the signal region for events in which a B+ .... 'l/JK+7I'+7I'- decay W8B not generated. The third row shows the normalized sideband region for events in which a B+ .... 'l/JK+7I'+7I'- decay W8B not generated. Finally. the fourth row shows the result of subtracting the third-row distribution from the second-row distribution. 56 5.2 M(K7r) Just as the range of M(K7r7r) depends on l:J.E, the range of M(K7r) also depends on l:J.E, as Fig. 5.9 demonstrates. While the minimum. value of M(K7r) is simply M(K7r)mIn = MK + M'If' the maximum. depends on l:J.E and is given by M(K7r)max - M(Kn)max - M'If Rj l:J.E + MB - M", - M'/t . (5.4) To correct for this M-dependence, which makes the sidebands different from the signal region, the variable M(K7r) is transformed as follows: with M'(K7r7r) given by Eq. 5.3. It can be shown that this is equivalent to: M'(K7r) = M(K7r)mIn [M(K ) - M(K) 1 M(K7r7r)::"'" - M(K7r7r)mIn (5.6) + 7r 7r min X M(K7r7r)max _ M(K7r7r)mIn . Expressed in this form, the transformation is a function of only M(K7r) and l:J.E. Figure 5.10 shows l:J.E versus the transformed variable M'(K7r). Although the trans formation distorts the K*(892)O and DO backgrounds, this can be accounted for in a straightforward way, as explained in Section 5.4. Figures 5.11 and 5.12 show l:J.E versus M(K7r) and M'(K7r) for generic MO. Histograms of M( K 7r) and M' (K7r) for sideband and signal-region data are plotted in Figs. 5.13 and 5.14. Since there are few events at high M(K7r), the improvement due to the transformation is not apparent in these plots. Figures 5.15 and 5.16 show sideband and signal-region histograms of M(K7r) and 57 ~ 0.2 ;-0.15 O·" H~~s ..· .. H -"H ".1 -0.2 0.6 0.8 1 1..2 1.4 1.6 1.6 2 2.2 M(K. ) (GeVic' ) Figure 5.9: D. E versus M (Krr ) for B+ --> J/.pK +rr+rr- (left) and B+ --> .p' K +rr+rr (right) data. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M(Krr). The lines near 0.9 GeV Ie? and 1.9 GeV le2 in M(K rr) are due to random combinations with K *(892)O and DO, respectively. ~ 0.2 ;;0.15 0.1 0." -0.15 -0.2 0.6 0.8 1.2 1.4 1.6 1.8 2 2.2 M'(K.) (Gel/lc'l M'(K.) (GeV/c' ) Figure 5. 10: D. E versus M'(Krr) for B+ --> JI.pK +rr+rr- (left) and B+ --> .p'K +rr+rr (right) data. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M'(Krr). 58 B' --> ",(25) K' It' . ' M(K.) /G.,,,,', M(K.) (G.',f,o', F igure 5.11: fj,E versus M(K7r) for B+ -> J /1f; K +7r+7r- (left) and B+ -> 1/l'K+7r+7r (right) as reconstructed in generic MC. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M(K7r) . B' --> '1'(25) K' It' It M'(K. ) (Ge'"o" M'(K.) /G.,,,,', Figure 5.12: fj,E versus M'(K7r) for B+ --+ J/1/l K +7r+7r- (left) and B+ --+ 1/.i K +7r+ 7r (right) as reconstructed in generic MC. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M'(K7r ). 59 M'(Krr) , respectively, for generic Me. The sideband-subtracted background distri butions of M(Krr) are slightly but consistently negative at high M(Krr) (around 2 GeV /e?) as a result of the discrepancy in the range of M(Krr) between the side bands and the signal region. The transformation solves this problem. The apparent excess of events near the K '(892)O and DO masses in the bottom row of Fig. 5.16 is caused by the aforementioned distortion of the K'(892)O and DO peaks in the background. 1 'N.----.~------, }! 1200 i ,to :I~ "" ~ n. ~ ... ~ ,,, ! N ~ .. " 1.41.11.12:2.2 l M(KIl) lGeVlc ) Figure 5.13: Invariant mass of the K +rr- system in B+ --> J /1/lK + rr +rr- (left) and B+ --> .p' K +rr+rr- (right) data. Blue and red histograms show events in the signal and normalized sideband regions, respectively. ,.. ~ , :I .. •0 '" ~ '" ! .. w• .. " I ~ lA lA 1~ 2 U I,. fA 1.1 J :u l1li '(1(1;) (~Vlel) M'{K1;) (a-Vlcl ) Figure 5.14: Transformed invariant mass of the K +7f- system in B+ --> J/.p K+rr+rr- (left) and B+ --> .p' K +7f+7f- (right) data. Blue and red histograms show events in the signal and normalized sideband regions, respectively. 60 J=II:: 1=.... I I ...... IR1oV1a'l ....ll11ovto'l ... J ... II ... 1...... ,. I • 1JI I .. IIPCalRloV1a'l ....ll11ovto'l 'lI'" I .... J ... a- I! ... 1= '" 1..... ,. .. • ... .. I .. IIPCalRloV1a'l IIPCal IIIovto'l ... J J '" I! ... a to Rn ~ toO • ~ j j ....u • ... ·teD ...... , 1.lil 'fA UI 1.8 . .. IIPCalRloV1a'l Figure 5.15: M(K1r) distributions for B+ -+ JNK+1r+1r- (left) and B+ -+ 'l/lK+1r+1r (right) as reconstructed in generic Me. The top row shows the signal region for events in which a B+ -+ .pK+1r+1r- decay was generated. The second row shows the signal region for events in which a B+ -+ .pK+1r+1r- decay was not generated. The third row shows the normalized sideband region for events in which a B+ -+ .pK+1r+1r- dec8¥ was not generated. Finally, the fourth row shows the result of subtracting the third-row distribution from the second-row distribution. 61 ~Dm'~--~~------~ l .... :fa:: .... Ia:: I: I: ...... 1.4 t.S t.a 2 U ·~ut-~~~~r-~u~~ur-~u~-i.~~u~ II'(ICa) ,...... , _(Ga,"" I ... a "" j : .. t.a tA 1.8 t.I • ....'.. aa.a _I(CIoV1o') II'(ICa) (GoVio') 1.8 1.. a IJl • • II'(ICa) (Goy..., _I'''''''' 11 .. Ia .. I :~~~~~--~------4 ... 1.2 'lA 1.4 1.8 • u 1.2 'lA 1.4 .... a u _I(GoVio') _I(GoVio') Figure 5.16: M'(K'/I") distributions for B+ -+ Jf'ljJK+'/I"+'/I"- (left) and B+ ...... 1/J'K+'/I"+'/I" (right) as reconstructed in generic Me. The top row shows the signal region for events in which a B+ ...... 1/JK+'/I"+'/I"- dec8¥ was generated. The second row shows the signal region for events in which a B+ ...... 1/JK+'/I"+'/I"- decay was not generated. The third row shows the nonnaJired sideband region for events in which a B+ ...... 1/JK+'/I"+'/I"- dec8¥ was not generated. Finally, the fourth row shows the result of subtracting the third-row distribution from the second-row distribution. 62 5.3 M(1r7r) The situation for M(7l"7l") is similar to that for M(K7l"). The kinematically-allowed range of M(7l"7l") depends on t:.E, as Fig. 5.17 demonstrates. While the mjnimum value of M(n) is simply M(7l"7l")mIn = 2M", the maximum depends on t:.E and is given by M(n)max - M(K7l"7l")max - MK Rj t:.E+MB-M",-MK • (5.7) To correct for this t:.E-dependence, which makes the sidebands different from the signal region, the variable M (7l"7l") is transformed as follows: with M'(K7l"7l") given by Eq. 5.3. It can be shown that this is equivalent to: Expressed in this form, the transformation is a function of only M(7l"7l") and f1E. Figure 5.18 shows t:.E versus the transformed variable M'(7l"7l"). Figures 5.19 and 5.20 show t:.E versus M(7l"7l") and M'(7l"7l") for generic MC. Histograms of M(n) and M'(7l"7l") for sideband and signal-region data are plotted in Figs. 5.21 and 5.22. Since there are few events at high M(7l"7l"), the improvement due to the transformation is not apparent in these plots. Figures 5.23 and 5.24 show sideband and signal-region histograms of M(7l"7l") and M'(7l"7l") , respectively, for generic MC. The sideband-subtracted background distri- 63 ~:,: :>:g~~;~', ~ . 1 -0.15 -().2 0.2 0.4 , ~ 1.4 1~ 1~ M(.. ) (GeVle') M( ••) (GeVle') Figure 5.17: t;. E versus M(7r7r) for B+ --+ J!1/J K+7r+7r - (left) and B+ --+ 1jJ'K+7r+7r (right) data. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M(7r7r) . B+ .... '1'(25) K+ W 1[' -, ,~. :> 0.2 ~ ;;0.15 0." ~ .1 ..(1 .15 1.2 1.4 1.8 1..8 M'(.. ) (Geille') M'(M) (GeV/e' ) Figure 5.18: t;.E versus M'(7r7r) for B + --+ J !1/JK +7r+7r - (left) and B + --+ 1jJ'K+7r+7r (right) data. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M'(7r7r). 64 1.4 1. 1.8 M(•• ) (GoV/e' ) M(.. ) (GeVle' ) Figure 5.19: t:;. E versus M(7m) for B + --> J f.p K +7r+7r- (left) and B + --> ,p' K +7r+7r (right) as reconstructed in generic MC. The signal region is delineated in green, and the sidebands are deli neated in blue. The red lines indicate the minimum and maximum values of M (7r7r) . M'(.. ) (GeVle') ·M'(" ) (GeVle,) Figure 5.20: t:;. E versus M'(7r7r) for B + --> J /,p K +7r+7r- (left) and B + --> ,p'K+7r+7r (right) as reconstructed in generic MC. The signal region is delineated in green, and the sidebands are delineated in blue. The red lines indicate the minimum and maximum values of M '(7r7r). 65 butions of M(trtr) are slightly but consistently negative at high M(trtr) (arotmd 2 1.6 GeV /c ) as a result of the discrepancy in the range of M(trtr) between the side bands and the signal region. The transformation solves this problem. While there is some discrepancy between the background in the signal and sideband regions near the K~ and p masses, this is mostly independent of the transformation and is taken into consideration when calculating systematic errors. ~,-, ~------~------, I",>'" ~ 700 J~'" ... " ~'" .. '" " '~o ~~~ ~~=;;=,.. ,.. ~==0.1 ~~~~1.2 IA~07 1.1 ~U~J '.., 1.21.41.11.1 MCu ) (GeVIc') M( 1lI ) (GeVfc·) Figure 5.21: Invariant mass of the tr+tr- system in B + --+ J /1/; K+tr+tr- (left) and B+ --+ 1/;' K +tr+tr- (right) data. Blue and red histograms show events in the signal and normalized sideband regions, respectively. ~ ,... ;, > ... > 1 1 '" ... ,~ ,., 0 ~• '" ~ '" " .. ~ ... J .. ~ ~ -'" .. '",~ " '., OA OJ U U ... U '.., U ... ,.. ... •• l t M'(III1) (GeV/c ) " '(u) (GeVlc ) Figure 5.22: Transformed invariant mass of the tr+tr- system in B + --+ J/1/;K+tr+tr - (left) and B + --+ 1/;' K +tr+tr- (right) data. Blue and red histograms show events in the signal and normalized sideband regions, respectively. 66 ... -1IIID ... "~~~~~~~~~~~~~~~... , • ...... ,. Mral (08VJo1l _lllla¥lo'l ~~--~------, .. i I ... ii .... i! ... 1- I ...GO ...... ' ... .. ' .. _(GaYJo'l .. _(GaYJo'l I ..... i! ~ ... I ..... OJ .. , 'o. OA ...... _(GaYJo'l -,- I ";. i! ", n ... 11 .-V- j .u l 'E-='---+ ... .. ~ ~ .. , ~ ~ ~ • Figure 5.23: M(1r1r) distributions for B+ -+ Jf'1jJK+1r+1r- (left) and B+ -+ 'I/IK+1r+1r (right) as reconstructed in generic Me. The top row shows the signal region for events in which a B+ -+ 'ljJK+1r+7r- decay was generated. The second row shows the signal region for events in which a B+ -+ 'ljJK+7r+7r- decay was not generated. The third row shows the normalized sideband region for events in which a B+ -+ 'ljJK+7r+1r- decay was not generated. Finally, the fourth row shows the result of subtracting the third-row distribution from the second-row distribution. 67 IlOm I ,a ..... il I::: j ..... ,. ,. ... .,.., (OoVIo') .,.., ((IoVlo') ... I- I ... a .... il OIl r~ j ...... 0 .. ... tA .. .. II'(aJ(OoVIo') .,.., ((IoVlo') I ... i! OIl j ...... ,. 0 .. • .,.., ((IoYIo') .,.."GoYIo'J I .. I .. " r'l il .. ,i! 0 fl-'>- .. ~ l.J j .. I ...·to ~ '"0 ...•1m ... , tA ,...... '" ...... to ... .. II'(aJ(OoVIo') Figure 5.24: M'(ll'll') distributions for B+ -+ JNK+ll'+ll'- (left) and B+ -+ 'l/IK+lI'+lI' (right) DB reconstructed in generic Me. The top row shows the signal region for events in which a B+ -+ 'I/IK+ll'+ll'- decay was generated. The second row shows the signal region for events in which a B+ -+ 'I/IK+ll'+ll'- decay was not generated. The third row shows the normalized sideband region for events in which a B+ -+ 'I/IK+ll'+ll'- decay was not generated. Finally, the fourth row shows the result of subtracting the third-row distribution from the second-row distribution. 68 5.4 Peak Transformations As Figs. 5.9 - 5.12 demonstrate, transforming the M (K'Jr) coordinate distorts the shapes of the K*(892) and DO backgrounds, which are observed both in the signal and sideband regions at constant M(K'Jr). Modeling this distortion is straightforward. In the untransformed coordinate M(K'Jr), the probability density of the K*(892) background is described by a Breit-Wigner as where MK o(892) and rKO(892) are the mass and width, respectively, of the K*(892). Using Eq. 5.6, M(K'Jr) is expressed as a function of M'(K'Jr) and aB. The formula for BWKO(892) is then numerically integrated over the appropriate range of !l.E to obtain the shape of the K*(892) background as a function of M'(K'Jr). The same procedure is performed for the ~ background by first describing it as a Gaussian, G ( (M(K'Jr) - MdJ)2) D" IX exp - 2q2 ' (5.11) where MD" is the mass of the DO and q is its observed width. Figure 5.25 illnstrates the concept. First, the M(K'Jr) distribution of background Me events in the sideband region is fitted to a Breit-Wigner or Gaussian plus a polynomial background term. The peak function is then transformed as described above and superimposed on the M'(K'Jr) distribution. The shape of the transformed peak is modeled accurately in both cases.5 Background ~ and pO peaks, which are apparent in the M('Jr'Jr) distribution of generic-Me events, are similarly distorted by the M('Jr'Jr) transformation. The °In the DO plot, the slight dlsagreement between the M'(K".) distribution of Me events and the superimposed function is a result of not transforming the polynomial background function. While the distortion of the peaks must be taken into account in fitting the sideband data., the distortion of the background does not need to be modeled. 69 ...... MC, Siclebend Region, Before Trtln.formation MC, Sidebllnd Region, Before Trtln.formetlon l~ I~ .... ::l3000 ~ iE_ W,o ... I ".. "" '~ __---'T---~ __---tr--- ~r----t----~~ M(Ka:) (GeV1c" MC, SIct.bend Region, After Tflln.form-.ion MC, Sidebllnd Region, Aft., T ...n . formetion 'l! .... ft.... ~ .... 1~ ... ~ "" ~:KIIXI ~ ,..JOG J."" '''' I ,.. w• , ... w n. '" k---~--~r---Er--~~--r---T..• ~~~r-~r--nr--n~~~~~ , 1, 1 M'(Ka) (Gev/c: ) M'(Ib.) (GeVle~ Figure 5.25: M(K7T) (top) and M'(K7T) (bottom) distributions of K'(892) (left) and DO (right) peaks in generic-MC sidebands. The red curves in the top row show the results of fits. The red curves in the bottom row show the fitted functions of the top row when the peak function is transformed as described in the text. ...... MC Si~nd Region Befcwe T ...n.f ormetion j320l I'- ~- E- !2100 I'- ~,... E- Ie .r'1 rL nn! , -u ~ ".. l'I"'u ., .. , ... ",-: ~-;~-.br~.T-~~-rr-~r-~,t..•.-~r--1 , • M( IUI ) (GeV/e2) MC, Sktebllnd Region, Aft., Tflln.formetion MC, Sideb.nd Region, After nen.formetion ~ 2400 l: !"..~~~>...n ;:3000 :2000 i'C ""I_ w,,,.. ... "" • , " '(n ) (GeVle~ Figure 5.26: M(7T7T) (top) and M'(-7r7T) (bottom) distributions of Kg (left) and pO (right) peaks in generic-MC sidebands. The red curves in the top row show the results of fits . The red curves in the bottom row show the fitted functions of the top row when the peak function is transformed as described in the text. 70 distorted peaks are modeled by describing the ~ 8B a Gaussian and the pO 8B a Breit Wigner in M(n), expressing M('/r'/r) 8B a function of M'('/r'/r) and l:J.E using Eq. 5.9, and integrating over the appropriate region of l:J.E. Figure 5.26 shows the results. Not surprisingly, the effect of the transformation is larger on narrow resonances than on wide resonances. 71 Chapter 6 Inclusive Branching Fractions 6.1 Measurement Technique Branching fractions for B+ decays to J /'l/JK+7r+7r- and '1/1 K+1r+1r- final states are measured using a background-subtraction technique.! For each final state, data events in the signal and sideband regions are distributed into cubic bins in .M2(K1r1r), .M2(K1r), and .M2(n).2 The number of signal events observed in each bin is calculated as Ni = Si - fB Bi , (6.1) where Si and B, are the numbers of signal-region and sideband-region events, respec tively, that fall into the i'th bin, and fB is the sideband normalization factor given by Eq. 4.7. The fraction of charged B mesons that decay to the final state in question can be expressed as B =-1 LN.-, (6.2) NB g. i • 'peaking backgrounds are not expected in these fina.I states and were not seen in generic Me. 2The variables M (K1r1r), M(K ... ), and M (...... ) of this chapter are the transformed variables M'(K1r1r), M'(K... ), and M'( ...1r). respectively, of Chapter 5. 72 where e, is the signal efficiency in bin i, and NB is the total number of charged B-mesons in the data sample. Assuming equal rates for T(4S) ..... B+B- and T(4S) ..... B°{Jo, NB is equal to the number of B-pairs produced, which is moni tored independently [451. To determine the signal efficiency, 10.7 x 106 nonresonant signal Me events are generated in each of the two decay modes of interest. They are then reconstructed, applying the event-selection requirements of Chapter 4. The generated events are binned according to the generated values of M'(Kmr), M'(K7l"), and M'(7l"7l") , and the reconstructed events are binned according to the reconstructed values of M'(K7l"7l"), M'(K 7l"), and M' (7l"7l"). The efficiency in each bin is the ratio of reconstructed to generated events in that bin. Figures 6.1 and 6.2 show the variation in efficiency over the three variables, along with the corresponding mass spectra for data. The overall efficiency is (19.85 ± 0.01)%3 for B+ -> JNK+7l"+7l"- and (6.579 ± 0.008)% for B+ -+ '1/1 K+7l"+7l"-. The number of efficiency-corrected signal events observed is (4.14±0.06)xlW for B+ -+ JNK+7l"+7l"- and (1.12±0.05)x lQ4 for B+ -+ 'I/J'K+7l"+7l"-. This method of measuring inclusive branching fractions automatically corrects for efficiency variations over the phase-space. Fnrthermore, it makes no assumptions as to the shape of the signa1 in tl.E. 6.2 Systematic Errors The following possible sources of systematic error are considered. They are assumed to be uncorrelated and are added in quadrature to estimate the overall systematic error. 3In this chapter, when a single error is given, it is statistical. 73 .!' ---- ~ 1000 ~-- 51 ... +- .------;; 0 .15 ... ., )400 ... '" f 0.25 ~3000 > ~ 0.2 ---- ~-51 ...... o" ., ." + ;e3000 ~~ 51_ ... ;; ., - ,... i 1000 .." ... • • , Figure 6.1: The plots on the left show the dependence on M2(K7r7r), M2(K7r), and M 2(mr) of the B+ -> J /'Ij; K+7r+7r - signal efficiency. The plots on the right show the corresponding data distributions in the signal region (red) and normalized sideband-region (bl ue). 74 ~ D." .i' j > '"... u ... .m • ~ - " ". ... +----- ~ ,,.". ....'" ~ ,.... '.03 ! ...... " ..,. • ... "'{K!CI;) (Gey1Ic·) "'(K~ (G.yl/c') ~ ... j ... .i' > ... u .m • ,.. ~ ------.... "~ .. " + '" .... ~ ... •m ". ... ! ". 0.01 + 50 ~~ •.• , U , U U • ~ , U U ... 4 • W(K!!.) (GeY'Ic')• • W(K Il) (GeY'1c• ) ~ ... ! •.'" .i' ~ 0.0, - - > ,.. ~ • W .... " ... ~ ...... ~ '" ;- .. " ! '" I- ~ ,.. .. " -+- ...... t...;~ • .. , "'(:0:) (GeY·/c') "'(M) (GeY'/c') Figure 6.2: The plots on the left show the dependence on M2(Klr7r) , M2(K7r), and M2(7r7r) of the B + -- ..p' K +7r+7r- signal efficiency. The plots on the right show the corresponding data distributions in the signal region (red) and normalized sideband-region (blue). 75 6.2.1 Efficiency Errors Since the signal efficiency in Eq. 6.2 is determined using Me, any discrepancy in signal-reconstruction efficiency between data and Me will result in a systematic error. In particular, discrepancies can be introduced in track reconstruction and particle identification. Track-Reconstruction Efficiency The agreement between data and Me in track-reconstruction efficiency was studied for pion tracks in two ways [46, 47]. The error was found to be larger at momenta below 300 MeV Ie. Based on the results of these studies, an error of 1% is included for each lepton track, 1.4% for each pion track, and 1.2% for each kaon track. The errors for the different tracks are added linearly. Lepton-Identification Efficiency The agreement between data and Me in lepton-identification efficiency was studied using TY -> l+l- [48]. The Me was found to understimate the lepton-identification efficiency. Based on the results of this study, a correction factor of 0.984 ± 0.019 is applied for each electron track, and 0.962 ± 0.031 for each muon track. Kaon-Identification Efficiency The agreement between data and Me in kaon-identification efficiency was studied using D*+ -> D°7r+ -> K-7r+7r+ [49]. Based on the results of this study, a systematic error of 1% for B+ -> J/1/JK+7r+7r- and 2% for B+ -> 1/IK+7r+7r- is included. 6.2.2 Binning Figure 6.3 shows the dependence of the efficiency-corrected signal yield on the bin size. When the bins are too small, there are few data events in each bin; on the 76 other hand, when the bins get too large, variations in efficiency over the phase space are not properly taken into account. It is therefore desirable to set the bin size as small as possible, while avoiding regions of instability. Based on Fig. 6.3, a bin size of 0.15 Gey 2/ c4 is chosen. The associated error is taken to be the RMS of the signal 4 yield in the region between 0.1 Gey2/c4 and 0.2 Gey2 /c • Figure 6.3: Dependence on bin size of the efficiency-corrected signal yield for B+ -> J N K+7r+7r- (left) and B+ -> ..p' K +7r+7r - (right). 6.2.3 Correction for Over-Subtraction In the f::>.E distributions for nonresonant B+ -> J /1/I K +7r+7r- and B+ -> 1/1' K +7r+7r MC, shown in Fig. 4.7, a polynomial background can be seen under the peak. Since all the events in the MC include a signal decay, this "background" is made up of rnisreconstructed signal events. Although these events are included as signal in the efficiency calculation, they are removed by the background-subtraction procedure. The percentage of signal that is subtracted in tllis way is fowld to be (3.75 ± 0.91)% for B+ -> J /1/I K +7r+7r-, and (5.66 ± 1.23)% for B+ -> 1/1' K +7r+7r-. The observed branching fractions are corrected for tllis effect, and the associated uncertainty, in each case, is included as a systematic error. 6.2.4 Background Shape To determine the sideband normalization factor 1B in Eq. 6.1 , the data f::>.E distribu tion is fi tted as described in Section 4.6. In tills fit, the background under t he signal 77 is parameterized 8B a first-order polynomial. To estimate the error introduced by this 8BSUIIlption, a second fit is performed, parameterizing the background 8B a second order polynomial. The fractional change in the signal yield is taken 8B a systematic error. 6.2.5 Determination of Signal and Sideband Regions Signal and sideband regions are defined b8Bed on the results of fitting the data l:l.E distribution, 8B described in Section 4.6. To determine the error 8BSOciated with the fit, P,I!.E and (TI!.E in Eqs. 4.5 and 4.6 are varied within the fit errors. 6.2.6 NB The total number of B-pairs produced is determined with an accuracy of 1.3% [45]. 6.2.7 J N and 'I/J' Branching Fractions In the Me used for determining the efficiency, J N's from the signal B are forced to decay to e+ e- or p,+ p,-, and 'I/J"s from the signal B are forced to decay to e+ e-, p,+p,-, or J/'l/J1r+1r-. Thus, to obtain branching fractions for B+ --> J/'l/JK+1r+1r- and B+ --> '1/1 K+1r+1r-, the branching fractions meBBUred using Eq. 6.2 must be divided by previously-meBBUred values [2] of these J N and '1/1 decay rates. The uncertainties of these previous meBBUrements are included 8B a systematic error. 6.3 Results The meBBUred inclusive branching fractions are 4 8(B+ --> JNK+1r+1r-) - (7.16 ± 0.10 ± 0.60) x 10- , 4 8(B+ -->'I/J'K+1r+1r-) - (4.42 ±0.21 ±0.51) X 10- , 78 where the first error is statistical and the second error systematic. Table 6.11ists the components of the error. B+ -+ J/?/JK+Tr+1r- B+ -+ 1/IK+1r+1r (x 10-4) (xw-4 ) Branching Fraction 7.164 4.424 Data Statistics 0.098 0.208 MC Statistics 0.013 0.008 Tracking Efficiency 0.430 0.384 Lepton-ID Efficiency 0.366 0.227 Kaon-ID Efficiency 0.072 0.088 Binning 0.010 0.008 Over-subtraction 0.068 0.058 Background Shape 0.115 0.196 Signal/Sideband Regions 0.025 0.054 NB 0.094 0.058 ?/J Branching Fraction 0.051 0.087 Table 6.1: Measured inclusive branching fractions for B+ ...... JNK+1r+1r- and B+ ...... 1/1 K+1r+1r-, along with estimated statistical and systematic errors. The systematic errors are explained in the text. 6.4 Cross-Check AI> a cross-check of the branching-fraction measurements for B+ -+ J /?/JK+1r+1r and B+ ..... 1/1 K+1r+1r-, the branching fraction for B+ ..... 1/1 K+ is also measured, by reversing the 1/1-+ JN1r+1r- veto in the reconstruction of B+ -+ JNK+1r+1r-. To determine the signal efficiency, 10.7 x 104 B+ -+ 1/IK+ signal MC events are generated, where 1/1 is forced to decay to J/'I/J1r+1r-, with IN decaying to e+e- or p,+ p,-. Figure 6.4 shows the variation in efficiency over the three variables, along with the corresponding mass spectra for data. The overall efficiency is (13.6 ± 0.1)%. The 79 J D~'~ 250 5: D.1' + ~ iii 0.14 ------+- ~ ... 0.12., +~--- a ... !- ,"...... ~ ...... ·~--~~~~, ~~··~--~···r-'~~('KU-)7(~r-V·~k·) t' 0 .1• .,,------, j 0." ~* ~~-+- E 0.14 ____------t-+ ~ ... iii 0.12 ~200 •., + - ,...... ! ,...... t .. . .. , , j ::: ~ 110G ::o 1400 i 0.12 .., ~, ...... ~ 1000 .... j: .... ~ ... • m . ~r__,.r_.~r_--cr_,. --~. ~~~~~ ~(ItlI:' (GtlV'/C·' Figure 6.4: The plots on the left show the dependence on M2(J<1r7r), M2(K-rr) , and M2(1r7r) of the B+ -. .p' J<+ -+ J/.p7r+7r- J<+ signal efficiency. The plots on the right show the corresponding data distributions in the signal region (red) and normalized sideband-region (blue). 4 number of effi ciency-corrected signal events observed is (1.24 ± 0.03) x 10 • The kaons and pions in B+ ---> .p' J(+ ---> J /.p1[+1[- J(+ have different momentum distributions than those in B+ ---> J /tj;K+1[+1[- in general. The systematic error associated with track reconstruction is 1% for kaons and 1.2% for pions in this mode. Figure 6.5 shows the dependence of the efficiency-corrected signal yield on the bin size. As the amount of Me used is smaller for tllis mode, a bin size of 0.25 Gey2/c4 is chosen, and the as ociated error is taken to be the RMS of the signal yield in the 4 4 region between 0.2 Gey 2/c and 0.3 Gey2/c . The measured branching fraction is 4 8 (B+ ---> tj;' J(+) = (6.65 ± 0.18 ± 0.55) x 10- , 80 B*.... -+ ",(25) K' .,< ~ ,- """ ''''' III It 'm»or---'orr.,~-'IT-~~~~~OA'---To .~~ Bin WIdth (GliVi /c') Figure 6.5: Dependence on bin size of the efficiency-corrected signal yield for B+ -+ ,p' K+ -+ J /1/nr+7r- K+ where the first error is statistical and the second error systematic. This is consistent with the previously measured value of (6.48 ± 0.35) x 10-4 [2J. Table 6.1 lists the components of the error. B+ -; 1/;'K+ (x 10- 4 ) Branching Fraction 6.650 Data Statistics 0.167 Me Statistics 0.056 Tracking Efficiency 0.399 Lepton-ID Efficiency 0.334 Kaon-ID Efficiency 0.066 Binning 0.017 Over-subtraction 0.063 Background Shape 0.013 Signal/Sideband Regions 0.018 NB 0.088 1/;' Branching Fraction 0. 113 Table 6.2: Measured inclusive branching fractions for B+ -+ ,p' K +, along with estimated statistical and systematic errors. The systematic errors are explained in the text. 81 Chapter 7 Amplitude Analyses To determine the resonant structure of the K+1I"+1I"- final state in the decays B+ --> J /'I/lK+1I"+1I"- and B+ -+ 'I/l' K+1I"+1I"-, amplitude analyses are performed. Using an unbinned maximum-likelihood method, data for each mode are fitted in three dimen sions: W(K1I"1I"), W(K1I"), and W(n).! 7.1 Fitting Technique Signal-region data are fitted by maximizinlf the log-likelihood function, which is given by l(a) = I)np(x;i Ii) , (7.1) i where the sum is over the events in the signal region, Ii is the vector of parameters with respect to which e is maximized, Xi is the vector of coordinates for a given event (i.e., X =[W(K1I"1I"), W(K1I"), Wen)]), and p is the probability-density function (PDF) that is used to model the observed distribution. Maximizing l is equivalent IThe variables M(K1r1r), M(K1I"), and M(1I"1I") of this chapter are the transformed variables M'(K1r1I"), M'(K1I"), and M'(1I"1I"), respectively, of Chapter 5. 2 All mmdmIzatloDB In this chapter are done using standalone MINUIT [50]. 82 to maximizing the probability of drawing the observed distribution of events from a parent distribution given by p( X; it). 3 The distribution of events in the signal region is modeled as __) PB(X) ps(x; it) ( (7.2) P X; a = nB f PB(X) d,3x + ns f ps(x; it) d}lx ' where PB and Ps describe the observed shapes of the background and signal, respec tively. The constants nB and ns are the background and signal fractions in the signal region; the former is given by Eq. 4.8,4 and the latter is 1 - nB. The observed signal distribution Ps can in turn be expressed as Ps(X; it) = e(X)q;(X)s(x; it) , (7.3) where e is the detector efficiency, q; is the phase-space function, and s is the raw signal function. Using nonresonant Me, the detector resolution has been measured to be approx imately 3-4 MeV Ie? in each of the three coordinates M(K7r7r), M(K7r), and M(7r7r). Since this is smaller than the width of any resonance included in the fits, the effect of detector resolution on line shapes is neglected. 7.2 Normalization Procedure The integrations of Eq. 7.2 are performed numerically, using Simpson's rule. A step size of 0.010 GeV2/c4 for B+ -+ Jf'Ij;K+7r+7r- and 0.005 Gey2 Ic4 for B+ -+ 'I/J' K+7r+7r- is used in each dimension.s The three-dimensional region of integration 3For more on the maximum-likelihood method, see [51]. 4The background fraction nB Is corrected for the over-subtraction effect described in Section 6.2.3. SA larger step size Is necessary for B+ -+ JNK+1r+1r- beceuBe of the larger phBBe space. With a processor speed of 1.0 GHz and a step size of 0.005 GeV2!c4, the B+ -+ '1/1 K+1r+1r- fit takes approxlma.tely 3 hours. A Bimila.r fit for B+ -+ J!",K+1r+1r- would require two months. When the 83 can be determined by noting that the minimum and maximum values of W(K7r7r) are given by M2(K7r7r)mIn - (MK + 2M,,)2 , (7.4) M2(K7r7r)max - (MB - M",)2 , (7.5) where MB, MK, M", and M", are the nominal values of the subscripted particles. For a given value of W(K7r7r), the minimum and maximum values of W(7r7r) are M2(7r7r)min _ (2M,,)2, (7.6) M2(n)max - (y'M2(K7r7r) - MK r (7.7) For given W(K7r7r) and W(n), the minimum and maximum values of W(K7r) are W(K7r):::' = ~ [ M2(K7r7r) + Mk + 2~ - M2(7r7r) ± )1- (2M,,/M(7r7r))2 x y'M2(K7r7r) - (MK + M(7r7r))2 x y'M2(Kn) - (MK - M(7r7r))2 ] . (7.8) Figure 7.1 shows the calculated kinematic boundaries for B+ -> J/7/JK+7r+7r-, along with the observed distributions of sideband data, for two slices in W(K7r7r). Events that do not fall within the calculated boundaries are excluded from the fits. Due to the coordinate transformations of Chapter 5, such events are rare: 36 out of the 12913 sideband events and 3 out of the 10594 signal-region events for B+ -+ J /7/JK+7r+7r-, and 3 out of the 2230 sideband events and 0 out of the 1176 signal-region events for B+ -> 7/J'K+7r+7r- fall outside the boundaries. 84 . .~ '.~ . " .~ .' . u .. ... Figure 7.1: Distributions of M2(Krr) versus M2(-rr".) for two M2(K".".) sli ces in sideband data for B+ ~ J /1f; K +".+".- . The blue and red curves show the calculated boundaries corresponding to the low and high edge, respectively, of the plotted M 2(K".".) region. 7.3 Background Functions To determine the three-dimensional shape of the background in the signal region, an unbinned maximum-likelihood fit is performed on the sideband-region data. The log-likelihood function to maximize is given in this case by j e (- ) = " In PB(i ; aB) (7.9) B aB ~. J PB (-x;aB- )d3 x ' J where the sum is over the events in the sideband region. The maximization is per formed by varying the parameters aB, which are then fixed at their optimal values in fitting the signal region. The background is modeled as a combinatorial term plus a set of non-interfering resonances. For B+ -+ J/'l/J f(+n+n-, PB(i;aB) = [tax/T;(X)] x [~ayjTj(y)] x [ ~azkTk( Z) ] + e-2x [aK.(892)BW K·(892)(Y) + aoG o(y) + aKsG Ks(Z) + apBWp(z)] , (7.10) 85 In these equations, Tn represents an nth-order Chebyshev polynomial.6 The variables x, y, and z stand for W(K7l'7l'), W(K7l'), and W(7l'7l'), respectively, and are defined over the interva1s: XmIn = (MK + 2M,,)2 , Xmax = (MB - Mt[J)2 , Ymin = (MK + M,,)2 , Ymax = (MB - Mt[J - M,,)2 , ZmJn = (2M,,)2 , Zmax = (MB - Mt[J - MK)2 . The peak functions BWK .(892), GD , GKs ' and BWp are obtained as described in Section 5.4. Each peak function P(ii!) is normali2ed over the kinematically-allowed phase space to satisfy: (7.12) The factor of e-2z by which the peak functions are modulated is found empirically to produce a good fit to the sideband data. Since there are more low-energy particles than high-energy particles, it is not surprising that background peaks are more pro nounced at low W(K7l'7l'). Combining a K*(892) with a random pion, for example, will tend to produce a low value for W(K7l'7l'). Table 7.1 shows the fitted values of the background-function parameters. The statistical error in each parameter is defined as the change in that parameter required to reduce the log-likelihood by 1/2. The fitted functions, along with the sideband data distributions, are shown pro jected onto the three axes in Fig. 7.2. Figures 7.3 and 7.4 show W(K7l') and Wen) 6For more on Chebyshev polynomials, see [52]. 86 Parameter a.:o 1.0 (fixed) 1.0 (fixed) a,,1 -1.5899 ± 0.0048 -1.238 ± 0.036 a",2 0.9422 ± 0.0085 0.480 ± 0.045 a.,a -0.4736 ± 0.0087 -0.221 ± 0.021 a.,4 0.1778 ± 0.0066 N/A a.,s -0.0488 ± 0.0033 N/A ayo 1.0 (fixed) N/A C!yl 0.088 ± 0.021 N/A aza 1.0 (fixed) N/A a.1 -0.022 ± 0.022 N/A az2 0.129 ± 0.018 N/A aK·(892) 0.0353 ± 0.0065 0.0161 ± 0.0062 aD 0.0007 ± 0.0011 N/A aKs 0.0061 ± 0.0023 N/A ap 0.086 ± 0.012 0.0352 ± 0.0099 fB -21484.6 822.7 x: 1709.5 286.8 Nbins 1707 294 Npar 12 5 Table 7.1: Fitted values of background-function parameters. 87 ...... )! 100 "> 110 t1I , .. >• !! ,.. " .. d '" .. '" ~• --E .. E .. ~ .. ~ " " • , U , ... • , U ... "(I(Q;)• (GoIV·/c·) "IK•a ) (Gel/'ie·) ~IOO ~ 1.0 > tli_ ~ 120 !! !! '" ~)()O d .. E-• .. wo i .. '" '" " .. , • , , • "CIU) lo.V·/e·) "(K. ) (Gel/lie') ...E. 200 700 ~ :: 110 tli", +. " ,.. !! ... ~ ,.. d 0'" - ... -; 100.. !'"0 w ... \ ! .. '" " • U , ... U , OJ .. M"'(IDl) (GeVrlc") (GeVl/c', Wla ) Figure 7.2: Results of sideband fits for B+ - J/ r/J J<+7[+7[- (left) and B + - ..p' K +7[+7[ (right). Data (points) and fits (histograms) are shown projected onto the three axes. The red histograms show the overall background functions. The combinatorial components are shown in yellow, while the J< '(892) , p, J projections for slices in M 2(K-Tm). As a measure of goodness of fit , a X2 variable is calculated by distributing t he data into cubic bins that are 0.1 GeV /c2 wide on each side. T he normalized PDF, with the parameters set to their best-fit values, is integrated over each bin and multiplied by the total number of events to determine the munber of events expected in the bin. Adjacent bins are combined wltil each bin has at least 6 data events. A X2 variable for the multinomial distribution is calcu.l ated as [53J (7.13) 88 0.60 o.v"fe' c M'(Kn ) c 1M o.V'/c' 0.60 o.Y'/Il' c M'(KIIII' c 1M o.V' fe' ~ ". ~ '" c! ... c! '",.. ,.. •~ .. •~ .. • .. • ,'".. -! .. ~ .. .lJ .. " " • • .. • .. U , ... Jr(KJI;) (GeV'le') Jr(u ) (Gev'le') 1M o.V'/c' « "(Ku ) « 2.32 o.V'/Il' 1." o.V'/c' « M'{Kn1 « 2.32 o.V'/c ' :;. ;!' ... > '", > c! ,.. • ,.. " '" ! .. ~ ,.. ,'".. • ~ .. E ,.. w• .. w• .. " ,. .. .. U , ... "(lb.) lOeV'Ie') "(a ) (GeY'Ic') 2.32 o.V'/c' « M'(Kn) 0( :1.11 o.V'Ic' 2.32 o.V'Ic' « M'(Kn ) « 3.11 o.V'/c' .i' ,.. .~ '", > >• ,.. c! '" " ,.. ! '" ~ " .. • ,.. . ~ .. E .. w• .. w• .. " "• • U"(Ib.) (Geyl/c') W(M) lOeY'Ie") :1.11 ,o.V"Ic'.. c " (Ko) c • .04 GoV'Ie' :1.11 o.V"k' c M'{Kn1 c • .04 GeY'/Il' .~ ~ > '" c! .. c! ,.. •~ .. ~ .. • .. I .. E .. w ~ " " ... • ... Jr(KII) lGeY'/c')• Jr(a ) (o.Y'Ie') • .04 o.V'Ie' c " (Ku) c • .10 o.V'Jo ' ..... o.V"Ie' c M'(K ~ c ..., o.Y'Ic' .i' .i' " > .. >• .. c! " ~ .. .. ,; .. ! .. ~ J• .. w• ".. w• ".. • U • "'(K• II) lGeyl/c') "'(a ) (GeV'/c:') Figure 7.3: B + -> J N K +7r+7r- sideband data (points) and fit results (histograms) for sli ces in M2(K7r7r ). The fit components are color-coded as in Fig. 7.2. 89 0.10 o.V'/c' c M'(X n } c 1.00 o.V'/c' CLIO o.V'k' c "(lto l c 1.00 GeV'/c' .. ;!' ~ > .. " 0• ,. ~ .." ! " ! " " • .. j • -! " • W• j " ,. ~ 2 2. • 0. OJ • ";11)(o.V~/e·) W(n )(o.v'/e4) 1.00GeY"Ic ' c IIf(Ku)c UOC.V'Ic · 1.00 0..... 10· c "(Ka) c l AO o.V'/r;' ;. .. ;!' .. > & " .; .. ~ .. 1..-' ~ .. " 0 ,. "• .. II • ~, ~ ~ .. ~ ~ " W " ~ " h. , ,. ~ •• ... • .. OJ ... "(K1I) (o.V'/c·) "(0:) (GliV'/e·, l AOo.v"k' < It{Ko )c 1.IOo.V'/r;' 1.40 0.V"/c' < "(Kn l < 1.10 GeV '/c' -~ .. ;!' .. .;> >• .. ij 0 .. ~ .. ! .. 0 ~, • • .. ~ .. ~ W• W• .. " " /'1, \ ,. • • .. . 4 "(KII) (o.V'Ie'") M'(U)(eo-V'1c ) 1..10 Gev"Io:' c M'(Kn) < 2.20 GeV'Ic' UO 0.'1"10:' c "(Ka) c 2..20 o.V'/r;' ;, .. ;!' ,. ".;> >• .. 0 .. ! .. ~ • .. ~• .. .. E " i W• .. " " OJ • ... 4 • .. t "(ICK) lo.V'/c·) "10:) (GliV Ic ) 2..20 a.V'Ir;' '" Wlltn ) c 2..10 G.V'/t:' 2..20 o.Y"k· '" " (Ka) c 2.10 o.V'/c' ;!' .. ~ > .. .; " 0• .. 3 " .. " .. ~ .. ! " • .. W• ~ " W " • " • U t •• OJ • W(It .. ) leo-V'/c·) "(U ) (GtlVtJC'") Figure 7.4: B+ -+ ,p' K +.".+.".- sideband data (points) and fit results (histograms) for slices in M 2(K ."..".). The fit components are color-coded as in Fig. 7.2. 90 where NbIns is the total number of bins used, '1!.j is the number of observed events in a given bin, and Pi is the number expected in that bin based on the PDF. If the expected distribution Pi were obtained by a binned maximum-likelihood fit of the data distribution '1!.j, the number of degrees of freedom associated with the -K would be reduced by the number of fit parameters Npar and would be given by No.o.F. = Nblns - Npar - 1. If, on the other hand, the two distributions were not correlated by a fit, the number of degrees of freedom would be NO.O.F. = Nblns - 1. Since, in this case, the distributions are related by an unbinned maximum likelihood fit, the true NO.O.F. can be expected to lie between these extremes [54]. 7.4 Efficiency Functions The dependence of the detector efficiency on the kinematic variables is obtained for three-dimensional bins, 0.15 GeV2/c4 on each side. using nonresonant signal Me as described in Section 6.1. The function is implemented as a lookup table: the efficiency for a given point is the efficiency in the corresponding bin. 7.5 Phase-Space Functions Four-body phase space functions for IJ+ ..... J/'1jJK+1r+1r- and B+ ..... '1jJ'K+1r+1r- are obtained by using GENBOD [55] to generate particle four-momenta that are weighted by the density of states in phase space [56]. In each decay mode, 108 events are generated. Event weights are distributed into cubic bins in M'l(K1r1r), M'l(K1r), and M'l(n), with a bin width of 0.02 Gey2/c4• The phase-space function is implemented as a lookup table: the value of the function at a given point is the phase-space weight in the corresponding bin. Figure 7.5 shows projections of the phase-space functions for the two modes onto the three axes. 91 . "'" 4 .... ~,- .... }.., I- !:~ .... I: I: ... "" • 0 II'(Jom) (Qov'lc~ . ,... ~-I .... ~- I:: g ... 1=~ 1- 'ClIO.., ... 0 • • tf """" (Qov'Ic'l ~ .... ~UOl,,- '- I: I- ~ I: 1= ,...... 0 1I'{aa)(...... '1 • • II't=l (Oev'Ic~ Figure 7.5: Density of states in phase space lIB a function of Af2(K7r7r), Af2(K7r), and Af2(7r7r) for B+ -> JNK+7r+7r- (left) and B+ -> 'I/IK+1r+1r- (right). 7.6 Signal Functions The K+1r+1r- final state is modeled as a nonresonant signal plus a superposition of initial-state resonances R I . The latter are presumed to decay through intermediate state resonances R2 as R, -> ~a, R2 -> be, where a, b, and c are the finp.\-state particles. Specifically, 2 s(X; it) = la..rA..r(X)12 + L L aJ,J.AJ,J.(X) (7.14) J, J. Here, J1 and J2 stand for the spin-parity (JP) of RI and ~, respectively. Resonances with different J1 are added incoherently, while those with the same J1 are added coherently. The parameters varied in the fit are the complex coefficients a..r and 92 aJ,J.' While the nonresonant signal is assumed to be constant over the phase space: (7.15) the resonant decay amplitudes AJ,J. are expressed as (7.16) where rR.(moo) is the mass-dependent width: (7.17) and FR is the Blatt-Weisskopf barrier factor: FR = 1 for J2 = 0, _ "';1 + Jl2q3 for J = 1, (7.18) - "';1 + Jl2q2 2 = "';9 + 3Jl2q3 + .R4cfo for J2 = 2. "';9 + 3Jl2q2 + .R4qi The meson radial parameter R is set to 1.5 (GeV/ c) -1. The function QJ,J. describes the spin-dependent angular distribution of the final state and is shown for various combinations of J1 and J2 in Table 7.2. In equations 7.16-7.18 and Table 7.2, the nominal masses of the resonances Rl and R2 are denoted by MR, and MR., and the nominal widths by r R, and r R •• The angle (J is between a and b in the be rest frame and can be expressed as (J _ moo [2 _ 2 (m~ - m!)(m~ - m~)] COB - 4 m"" mab + 2 , (7.19) pq1T!abc moo 93 J1 J2 ClJ,J. 0+ anything 0- 1 0+ 1+ 2 0- 1- (1 + Z2) COS 0 1+ I- 2 1- 1+ 1+ z2cos o 1+ 1+ 2 I- I- l-cos 0 1+ 2+ 2 2 1- 2- (1 + z2) [1 + 3 cos 0 + 9z (cos2 0 - 1/3)2] 2+ 1+ 2- 1- 3+ (1 +4z2)c0s20 2+ 1 l-co820 2- 1+ 2+ 2+ 2- 2- 1 + z2 /9 + (z2/3 - 1) cos2 0 - z2(c0s2 0 - 1/3)2 2+ 2 2 2 2- 2+ 1 + z2 /3 + z2 cos 0 + z4(C08 0 - 1/3)2 Table 7.2: Angular distribution of the K+1r+1r- final state for various combinations of initial and intermediate-state spin parities. See Ref. [57] for derivations and conditions of applicability. 94 and z is given by z =p/mobc' (7.20) The breakup momentum p is the momentum of a or be in the abc rest frame: (7.21) while q is the momentum of b or c in the be rest frame: (7.22) The constant qo is the value of q evaluated at mbc = MR •. 7.7 Results Tables 7.3 and 7.4 list the fitted values of the relative amplitudes and phases that make up the raw signal functions for B+ --> J /'I/1K+7r+7r- and s+ --> '1/1' K+7r+7r-. The fitted PDF is shown projected onto the three axes, along with the data, in Figs. 7.6 and 7.8. The three-dimensional nature of the fits can be seen in Figs. 7.7 and 7.9, which show W(K7r) and W(7r7r) projections for slices in W(K7r7r). Since the components of the signal function that describe the different submodes are not separately normalized over the phase space, it is not meaningful to compare measured relative amplitudes for different submodes. A decay fraction is therefore calculated for each submode by integrating the corresponding component of the signal function over the phase space and dividing by the integral of the full signal function: (7.23) Decay fractions are not branching fractions; indeed, due to interference effects, decay fractions for a given final state may not add up to unity. Tables 7.5 and 7.6 show 95 J1 Submode Amplitude Phase Nonresonant K+7r+7r- 1.0 (fixed) o (fixed) K1(1270) -> K*(892)7r 0.70 ±0.04 ±0.15 o (fixed) K1(1270) -> Kp 1.98 ±0.1O ±0.27 -1.309 ± 0.077 ± 0.068 1+ K1(1270) -> Kw 0.93 ±0.05 ±0.13 1.860 ± 0.084 ± 0.052 K1(1270) -> Ko(I430)7r 0.41 ± 0.15 ± 0.16 2.27 ± 0.45 ± 0.56 K1(1400) -> K*(892)7r 0.34 ±0.05 ±0.11 -0.82±0.17 ±0.48 1- K*(141O) -> K*(892)7r 0.55 ±0.09 ±0.12 o (fixed) K;(1430) -> K*(892)7r 0.71 ±0.07 ±0.11 o (fixed) K;(1430) -> Kp 0.72 ±0.06 ±0.12 -2.BB±0.12 ±0.14 K;(1430) -+ Kw 0.287 ± 0.040 ± 0.054 -0.41 ± 0.20 ± 0.18 2+ K;(1980) -> K*(892)7r 0.77 ±0.06 ±0.11 -1.41±0.13 ±0.19 K;(1980) -> Kp 0.92 ±0.07 ±0.12 1.05±0.16 ±0.19 K;(1980) -> Kw 0.403 ± 0.042 ± 0.053 -2.33±0.17 ±0.17 K(1600) -> K*(892)7r 0.180 ± 0.036 ± 0.033 o (fixed) K(1600) -> Kp 0.164 ± 0.017 ± 0.023 0.80 ± 0.28 ± 0.18 K (1770) -> K*(892)7r 0.186 ± 0.035 ± 0.026 -2.76±0.33 ±0.19 2- 2 K2(1770) -> K;(1430)7r 0.302 ± 0.045 ± 0.045 2.96 ± 0.39 ± 0.24 K2(1770) -> K!2(1270) 0.480 ± 0.068 ± 0.078 -2.96±0.35 ±0.26 K2(1770) -+ Klo(980) 0.115 ± 0.030 ± 0.023 2.95 ± 0.48 ± 0.21 iB -10592.4 ~ 1488.8 NbIm! 1202 Npar 32 Table 7.3: Fitted values of signal-function parameters for B+ -+ J NK+7r+7r-. The first set of errors is statistical. and the second is systematic. 96 Own.. PDF K,(I400) ..... K·(892) . K,('17O) ..... 1('(I92)ll _...... 1<,(1410) ..... K{n2) 1 1(,(1170) ..... ~(1430)1: -~"" Kdl430) -t 1("(192) . K,(,nO) .... K '.(1 270) K,( 1270) .... 1("(892)11 KJl430) .... Kp K,(l nO) .... KI .(980) K.(121O) ..... Kp KJI430H k .. 1(~( I 980I -+ K'(892) K K,(1270) ..... Kf)) 1((1Il00) ..... 1('(192) I: K;(198O) ..... Kp K,(1270) ..... ~(1430). 1((1Il00) ..... Kp K~(1INIO) ..... K . •.!:! ~> 350 CI" 300 0 ;! 250 ci 200 -.,CII ;: 150 c -w 100 50 0 1 1.5 2 2.5 3 3.5 4 4.5 M'(K1tlt) (GeV 2/c' ) • ~ 1000 >., CI 800 ..,0 0 ci 600 -CII ~ 400 E UJ 200 0 0.5 2 2.5 3 3.5 4 M'(Kn) (GeV 2/c') • 1000 ~ >., CI 800 0 ;! ci 600 -In ..,;: 400 E w 200 0 0.5 1 1.5 2 2.5 M'(mt) (GeV 2/c' ) Figure 7.6: Results of signal-region fits for B+ ~ J/1j; K +rr+rr- . Data (points) and fits (histograms) are shown projected onto the three axes. The legend is indicated at the top. 97 o.eo o.V"/c' .. "'(IC g) c 1.... a.V'II; ' O.RGoV"II;' ,, "'{lIt .. ~ .. 1."a.V'/c' ~ to ~ : ~ ~ ~ .. ~ : o .. j : I : .. ".. u u • . ~~~~.~. ----~-----o,.~.---- ~-----c,~.•c--- J ~(K II ) Ia-yl/c") M'( ftlII) (Geyl/C') 1M o.y'/c' c " (lCn l " 2.32 Goy '/c' 1." GoV"/c' c "'(1(0 ) c 2.12 a.v'/c' ~ .oo Loo ~ ... 1'00 1 i : Ia . • .~.,~~~~~,~.---c~--~..~ --~---o.~. ---o.--! • u u'(KJI) (Geyl/c·) 2..32 GeV"/e' .. "'(IC o)" 1.1' o.V'k; ' 11. a.V"/c' .. "'(1(0) c ".04 GoV 'II; ' · ~.~.~~?>--~, .~~~~;;..~~~ ~-u~---T. -J M'(Kft ) (GeY·/e") 4..IW GeV"., /C ' .. " (le n) c 4.to C.V'/c ' i : ~ .. ~ .. j :.. Figure 7.7: B+ -> J fV; I< +7r+7r - signal data (points) and fit results (histograms) for slices in M2(I<7r7r). T he fi t components are color-coded as in Fig. 7.6. 98 OYerIiU PDF K'(1611O) -t K'(I92) . a.ckground K'(1611Oh K P K'(l680) -t K. -K,( 1270) -t K'(192) . K,(1270) -t Kp K,(1270) -t K. •.!:! N 60 > CI" 50 0 ~ 40 0 -CO 30 ;:" E 20 w 10 0 1 1.5 2 2.5 3 3.5 4 4.5 PK(K1t1[) (Gey2/c ') • 140 .!:! N > 120 t CI" 0 100 ~ 0 80 -co 60 ;:" E 40 W 20 0 0.5 2 2.5 3 3.5 4 PK(Klt) (Gey2/c') • .!:! N 120 > CI" 100 0 ...0 80 0 -.. 60 ;:" E 40 W 20 + 0 0.5 1.5 2 2.5 M2(1t1t) (Gey2/c') Figure 7.8: Results of signal-region fits for B+ -> 1// K+7f+7f-. Data (points) and fits (histograms) are shown projected onto the three axes. The legend is indicated at the top. 99 OAO o.V'/e' c "'(KJ . iC-C--...~ ----.-----~, •.• -----.c---~.~.• c--- ~ "(Itt) (GeV'k", 1J1O Gev"./c' < "'(1<'11 ) < 1AO GeV '/c' ~ :: a .. ~ 12 • " j u , ' ~~~~..o- ----.-----~, •---- -.c---~..~ ---J "Co) (Geyl/c:", 1AO a.v",Ic' < M'(Kn) < 1.10 GeV'./c' 1AO GeV',Ic' c M'CK J . .. u ... .. u • OJ "'(0 ) (GeY'lc') 1M c.V'/e' c M"(Ka1 < 2.20 GeV '/e' i : ~ • .. " I 20 " . ~••~~~ ,F§~~" ~--~---"~--~' ~~'~J~~T. ~ I M'(K:K) (GeY~/c:" 2.20 c.V'./c' c WCKu) c 2.60 C.V''''' :uo c.V"/c' < "'(KJ ~ 3Ii ~ ~ ~ : .I " ~ .. •• u , .. . . lL~~~o.~~~"----~, .~----,-----~,,~--~ N'(K:K, (GeV'k') "(0 ) (GeV'/c:') Figure 7.9: B + -t v/ K +7r+7r- signal data (points) and fit results (histograms) for slices in M2(K7r7r). The fit components are color-coded as in Fig. 7.8. 100 J1 Submode Amplitude Phase Nonresonant K+1r+1r- 1.0 (fixed) o (fixed) K1(1270) -+ K*(892)1r 0.213 ± 0.037 ± 0.049 o (fixed) 1+ K1(1270) -+ Kp 0.513 ± 0.070 ± 0.141 -0.66 ±0.26 ±0.1O K1(1270) -+ Kw 0.048 ± 0.041 ± 0.022 -0.37 ± 1.21 ± 0.51 "K*(1680)" -+ K*(892)1r 0.67 ± 0.12 ± 0.13 o (fixed) 1- "K*(1680)" -+ Kp 1.17 ±0.16 ±0.30 1.27±0.24±0.14 "K*(1680)" -+ Kw 0.233 ± 0.097 ± 0.043 -3.06 ± 0.43 ± 0.45 iB 638.3 X2 180.0 Nblns 168 Npar 10 Table 7.4: Fitted values of signal-function parameters for B+ -> 'Ijf K+7r+7r-. The first set of errors is statistical, and the second is systematic. the decay fractions for the best fits. Decay fractions for selected combinations of submodes are also shown. 7.8 Statistical Errors As with the sideband-region fits, the statistical uncertainties in the fit parameters (i.e., amplitudes and phases) are determined by the fitter: the error in a given parameter is the change in that parameter that reduces the log-likelihood by 1/2. The statistical uncertainties in the decay fractions, on the other hand, are more complicated. Since a given decay fraction involves the integral of the full signal function, the error in a single decay fraction incorporates the errors in all of the parameters. To determine the statistical errors in the decay fractions, 1000 sets of correlated signal-function parameters are drawn for each mode from Gaussian distributions using the fitted 101 parameter values and the error matrix.7 Decay fractions are calculated for each set of generated parameters. The resulting distribution for each decay fraction is then fitted to a Gaussian, the width of which is an estimate of the statistical error in the decay fraction. I Submode Decay Fraction Nonresonant K+1r+1r- 0.256 ± 0.019 ± 0.043 K 1(1270) ...... K*(892)1r 0.060 ± 0.006 ± 0.023 K1(1270) ...... Kp 0.268 ± 0.011 ± 0.038 K1(1270) ...... Kw 0.064 ± 0.005 ± 0.008 K1(1270) ...... Ko(1430)1r 0.002 ± 0.002 ± 0.002 K1(1400) ...... K*(892)1r 0.021 ± 0.006 ± 0.014 K*(1410) ...... K*(892)1r 0.037 ± 0.012 ± 0.013 K:i(1430) ...... K*(892)1r 0.065 ± 0.011 ± 0.012 K:i(1430) ...... Kp 0.060 ± 0.009 ± 0.018 K:i(1430) ...... Kw 0.012 ± 0.003 ± 0.004 K(1600) ...... K*(892)1r 0.032 ± 0.012 ± 0.008 K(1600) ...... Kp 0.028 ± 0.005 ± 0.006 K2(1770) ...... K*(892)1r 0.048 ± 0.017 ± 0.009 K2(1770) ...... K:i(1430)1r 0.016 ± 0.004 ± 0.003 K2(1770) ...... Kh(1270) 0.024 ± 0.006 ± 0.003 K2(1770) ...... Kfo(980) 0.005 ± 0.002 ± 0.001 K:i(1980) ...... K*(892)1r 0.133 ± 0.013 ± 0.012 K:i(1980) ...... Kp 0.226 ± 0.020 ± 0.020 K:i(1980) ...... Kw 0.051 ± 0.009 ± 0.005 K1(1270) ...... K*(892)1r, Kp, Kw, Ko(1430)1r 0.394 ± 0.016 ± 0.064 K2 (1430) ...... K*(892)1r, Kp, Kw 0.136 ± 0.014 ± 0.014 Table 7.5: Decay fractions for B+ --> J/1f;K+7r+7r- submodes. The first error is statistical, and the second is systematic. 7Correlated Ga.ussian distributions are genera.ted using CORSET and CORGEN [58J. 102 Submode Decay Fraction Nomesonant K+7r+7r- 0.236 ± 0.039 ± 0.099 K,(1270) -> K·(892)7r 0.052 ± 0.014 ± 0.009 K,(1270) -> Kp 0.146 ± 0.026 ± 0.033 K,(1270) -> Kw 0.001 ± 0.003 ± 0.001 "K·(1680)" -> K·(892)7r 0.125 ± 0.037 ± 0.021 "K·(1680)" -> Kp 0.341 ± 0.059 ± 0.059 "K·(1680)" -> Kw 0.017 ± 0.016 ± 0.009 I K,(1270) -> K·(892)7r, Kp,Kw I 0.199 ± 0.028 ± 0.039 I Table 7.6: Decay fractions for B+ -+ '1/1 K+1r+1r- submodes. The first error is statistical, and the second is systematic. 7.9 Systematic Errors Several sources of systematic error are considered, as described below. They are added in quadrature to obtain the overall systematic errors shown in Tables 7.3-7.6. 7.9.1 Background Parameterization One possible source of systematic error in the fits is the fixed background fraction nB in Eq. 7.2. While the error in nB is small, the correction for the over-subtraction, described in Section 6.2.3, lowers nB by 10.8% for B+ -> JNK+7r+7r- and by 11.4% for B+ -> "p' K+7r+7r-. The systematic error associated with this correction is esti mated conservatively as the change in each parameter when the fits are performed with the uncorrected values of nB. There may be an additional systematic error if the background in the signal region is not correctly parameterized by the shape determined by fitting the sidebands. As noted in Chapter 5, generic-Me studies suggest that not enough of the IG and p background peaks are removed by the sideband subtraction. To estimate this error, a fit is performed in which the coefficients of the background peaks in Eqs. 7.10 and 103 7.11 are doubled. 7.9.2 Efficiency To estimate the error introduced by binning the efficiency information, the fits are repeated using bin sizes of 0.10 GeV2 /e4 and 0.20 GeV2/e4 for the efficiency. The average absolute change in each parameter is the estimate of the error. Another possible source of error is that the Me may not faithfully reproduce the detector efficiency for low-momentum particles. To test for such an effect, a fit is performed in which all charged particles are required to have a momentum greater than 200 MeV/e. In yet another fit, the Idrl and Idzl cuts described in Chapter 4 are loosened from 0.4 em to 0.8 em, and from 1.5 em to 3.0 em, respectively. The changes in each parameter observed in these two fits are added in quadrature to obtain an estimate of the error due to inaccuracies in the efficiency estimation. 7.9.3 Integration Step Size To estimate the error introduced by the finite step size used in the numeral integrals of Sections 7.2 and 5.4, the fits are repeated, using a step size of 0.005 GeV2/e4 for B+ -> JNK+rr+rr- and 0.010 GeV2/e4 for B+ -> 1/J'K+rr+rr-. The change in each parameter is an estimate of the uncertainty associated with the numerical integration. 7.9.4 Modeling of the Signal The mAASes and widths of the resonances included in the fits are based mostly on [2) and are listed in Table 7.7. Many of these have large uncertainties. To estimate the error due to fixing these values at their nominal means, the fits are repeated, varying the mass and width of each resonance in turn within its errors. For each mass or width variation, the absolute average change in each parameter is recorded. The average changes are added in quadrature to estimate the modeling error. 104 Mass Width Resonance JP (MeV/dl) (MeV/dl) pO 775.5 ± 0.4 146.4 ± 1.1 1 w 782.65 ± 0.12 8.49 ± 0.08 1 10(980)° 980 ± 10 50 ± r.l 0+ 12(1270)0 1275.4 ± 1.1 185.2 ± ~:A 0+ K*(892)0 896.00 ± 0.25 50.3 ± 0.6 1 K1(1270)+ 1272 ± 7 90 ± 20 1+ Kl(I400)+ 1402 ± 7 174 ± 13 1+ K*(141O)+ 1414 ± 15 232 ± 21 1 Ko(1430)+ 1414 ± 6 290 ± 21 0+ K2(1430)+ 1425.6 ± 1.5 98.5 ± 2.7 2+ K;(1430)0 1432.4 ± 1.3 109 ± 5 2+ K(1600)+ 1602 100 2- K*(1680)+ 1717 ± 27 322 ± 110 1- K2 (1770)+ 1773 ± 8 186 ± 14 2- K;(1980)+ 1973 ± 26 373 ± 68 2+ Table 7.7: Masses, widths, and spin-parity values of the resonances included in the fits. 7.10 Discussion In choosing the signal components to be included in the fits, the data have been used as a guide. In both B+ -+ J NK+7r+7r- and B+ -+ '1j;' K+7r+7r- data, the Kl (1270) signal is dominant. The initial fits were therefore done with only Kl (1270) ..... K*(892)7r and K1(1270) ..... Kp on top of the nonresonant component. Additional decay channels were added one at a time until a reasonable level of agreement was obtained. 7.10.1 Interference Effects The inclusion of interference among submodes that share the same initial-state spin parity is essential to obtaining good fits to the data. In particular, drantatic interfer- 105 ence effects are observed between K 1(1270) ...... K*(892)7r and K 1(1270) ...... Kp, and between K 1(1270) --+ Kp and K 1(1270) ...... Kw. Figure 7.10 shows scatter-plots of the signal-region data over the three coordinates. Interference between K 1(1270) ...... K'(892)7r and K 1(1270) ...... Kp is responsible for the abrupt weakening of the latter signal at M(K7r) > MKO(892). The data in this region cannot be described accurately if these two modes are added incoherently. Since the previously-measured [2] branching fraction for K 1(1270) ...... Kw is small compared to that for K 1(1270) -+ Kp, and since only 1.7% of w's decay to 7r+7r-,8 one might expect K1(1270) ...... Kw to playa negligible role in this analysis. Nonetheless, the narrow width of the w causes a significant distortion of the observed p line shape through interference [59]. Figure 7.11 shows finely-binned projections of data and fit results onto the W(7r7r) axis. An interference pattern is clearly visible at the w mass and is accurately modeled by the PDF. This dramatic interference makes possible a precise measurement of the K 1(1270) ...... Kw decay fraction in B+ ...... Jf1j;K+7f+7r-. The K1(1270) ...... Kp signal straddles the edge of phase space, as can be seen in the middle panel of Fig. 7.10. It is only in the high-M(K7r7r) tail of the K 1(1270) that interference between K1(1270) ...... Kp and K1(1270) ...... Kw can be measured. The large statistics in B+ ...... Jf1j;K+7r+7r- make the K1(1270) ...... Kw decay fraction measurable. In the case of B+ ...... '1/1 K+7r+7r-, however, there are fewer events, and the measurement of the K 1(1270) ...... Kw decay fraction in this mode is of marginal significance. The p-w interference pattern that can be seen in the W(7r7r) spectrum of B+ ...... ?jJ' K+7r+7r- in Fig. 7.11 is due mainly to a higher K7r7r resonance, whiclt is modeled as the K'(1680), as explained below. 8Although w decays dominantly to to ".+".-,,0, it can a.\so decay to ".+".- through G-parity violation [11, which causes mixing between p and w. An w component is therefore present whenever & particle decays to p. 106 ~ 4,------, ,.~ 4.,------, ~U ..."u~ I· f .~ .',.< • u a a '-••<-, u u :,! ... u a u a U 4 U 1II'(K.m1 aeY'to4 ~ur------, ~ ....,------, ;: , ~ ... 'if a- l ~ " ,,- u- .:. .-~ .,. ~ 4r------, ~ ..., "E- ~ .~ '~E- ~:'{";\:"~ , , , o ... Figure 7.10: Scatter-plots showing M2(K1r) versus M2(K1r1r) (top), M2(1r1r) versus M2(K1r1r) (middle), and M2(K1r) versus M2(1r1r) (bottom) for B+ -> Jf.,pK+1r+1r- (left) and B+ -> 1/J'K+1r+1r- (right) data. Interference between the Kp and K4(892)1r sub modes of the Kl(1270) is responsible for the abrupt fading of the K4(892) signa\ at M(K1r) > MKO(892)' 107 ~ ~ 160 >GI 140 Cl It) 120 c c 100 ci -.. 80 GI .;: 60 c -w 40 20 0 0.2 0.4 0.6 0.8 1 4 M2(Jt1t) (GeV2/c ) ~ 24 .!:! N 22 >GI 20 Cl 18 It) c 16 c ci 14 .. 12 -GI 10 .;: c 8 -w 6 4 2 0 0.6 0.8 4 M2(Jt1t) (GeV2/c ) Figure 7.11: Finely-binned projections of signal-region data onto the M2(11'11') axis, for B+ -t J/'l/J K +11'+11' - (top) and B+ -t 'I/J' K +11'+11'- (bottom), showing p-w interference. The top and bottom plots are color-coded as in Figs. 7.6 and 7.8, respectively. 108 7.10.2 The L-Region The M(K7r7r} region between 1.5 and 2.0 GeVle?, which is known as the L-region, comprises several wide, overlapping resonances [60, 61, 62]. The dearth of knowledge on the resonant structure of this region, as well as the large uncertainties in the masses and widths of the knoWn states, make it difficult to characterize this region in this analysis. The model presented here is not necessarily the only one supported by the data. To describe the structure observed at 2.6 GeVJ Ic4 in the W(K7T7T} distribution of B+ -> Jf'ljJK+7r+7r-, apeak with a mass of 1.6 GeVle? and a width oflOO MeVle? is included in the fit, decaying to K*(892}7T and Kp. This peak, which is referred to as the K(1600} in this thesis, may be the K2(1580}, an as-yet unconfirmed JP = 2- state that has previously been observed decaying to K*(892}7T [61]. AIl can be seen in Fig. 7.7, there are K* and p signals near the high end of the W(K7r7T} spectrum, which cannot be described by the nonresonant component of the signal PDF. To fit the data in this region, a K;(1980} resonance is included. This is another state that requires confirmation. Even alter including K(1600} and K;(1980) components, a slight enhancement 2 4 remains around 3 GeV /c in W(K7T7T). A K2(1770) signal is therefore also included, with its known decay channels to K*(892)7T, K;(1430)7r, Kfo(980), and K!2(1270). Fitting the B+ -+ 1/1 K+7T+7T- mode is more difficult still, since there are fewer events to analyze, and since only a small portion of the L-region is within the kine matic limits of the decay. In addition to the K 1(1270) signal, the W(K7T7r) spec trum has what appears to be the tail of at least one higher-mass resonance. AIl can be seen in Fig. 7.9, there are clear K*(892) and p peaks at high W(K7r7r), which are not reproduced by the PDF if no high-mass resonance is included in the fit. If the enhancement is modeled as a single resonance, the data favor a mass of roughly 1.7 GeV Ie? and a width of 400-500 MeV Ie? In this analysis, the enhancement is 109 modeled DB the K*(1680). It must be emphllBized that the data do not preclude other possibilities, such DB the K 2 (1770). Indeed, the hint of 10(980) in the IDBt U2(K1l'1l') slice in Fig. 7.9, which is not included in the model, cannot come from a 1- state such DB the K*(1680), or from a 2+ state such DB the K;(1430). 7.10.3 Deficiencies of the Fits Due to the difficulties inherent in any study of the wide and overlapping kaon exci tations that decay to K 1l'1l', there are large uncertainties in the m8BSeB and widths of many of the states included in the fits. Although PDG values [2] have been used DB a guide, the data suggest that the true m8BSeB and widths may be diHerent. The top panel in Fig. 7.7, for example, suggests that there should be more K 1(1270) -> Kp at low U2(K1l'1l'). This implies that the mean mass of the K 1(1270) may be lower than assunIed in the fit. The discrepancy may also be due to a distortion effect near the edge of phase space that is not properly taken into account by the fitting model. In B+ -+ 'I/J' K+1l'+1l'-, the low statistics and the kinematic cutoff limit the conclu sions that can be drawn about the signal components. Although the data size is significantly larger in B+ -+ J NK+1l'+1l'-, a further lim itation is imposed by the increase in computation time DB more parameters are added to the fit. Each additional decay channel included in the signal function contributes an amplitude and possibly a phase to be varied in the fit. Since the normalization integral of the signal function in Eq. 7.2 depends on the values of the parameters ii, the integration must be performed for each set of parameters attempted by the fitter. While the step size used in the numerical integration can be increased to speed up the process, it must be small enough to allow the PDF to resolve the structures in the data. In particular, the p-w interference pattern can be fitted with a step size of 4 0.01 Gey2/C\ but not with a step size of 0.02 Gey2/c • Thus, as a consequence of the finite processor speed, not every possible decay channel can be included in the 110 fit. The model is necessarily incomplete. The large nonresonant component observed in both B+ --> J/l/JK+7r+7r- and B+ -> '1/1 K+7r+7r- may be an indication of contributions from additional wide kaon excitations. It may also include some misreconstructed resonant signal. While the nonresonant component is assumed in this analysis to be distributed according to phase space, this may not be accurate. There are currently no accepted models of nonresonant B-meson decays. As more data are collected and processor speeds improve, studies similar to this one can be done to resolve these issues. 111 Chapter 8 Conclusions The decays B+ -> J /7/JK+7r+7r- and B+ -> 7/J' K+7r+7r- have been analyzed using data recorded by the Belle detector. Inclusive branching fractions have been measured (see Table 6.1), and amplitude analyses have been performed in three dimensions (W(K7r7r), W(K7r), and Wen)) to describe the resonant structure ofthe K+7r+7r final state (see Tables 7.3, 7.4, 7.5, and 7.6). Performing an unbinned fit in three dimensions exploits practically all of the information available in the data. While it is challenging to obtain a good fit in three dimensions, the method is powerful and has great discriminating power. A model that produces a good fit in one dimension can easily fail in three dimensions. FUrthermore, accounting for interference effects allows precise measurements of decay fractions. With high statistics, the spin-dependent angular distribution of the final state can also be used to distinguish wide, overlapping resonances. Although more data are required to clarify the structure of the L-region, precise measurements of K1(1270) decay fractions to KO(892)7r, Kp, and Kw have been made, updating previous branching-fraction measurements [63]. It has also been shown that trW interference in the 7r+7r- invariant mass spectrum can be used to measure decay fractions for decays involving an w. 112 The analysis presented in this thesis demonstrates that the decay modes B+ -> Jf'1j;K+7r+7r- and B+ -> 1/J'K+7r+7r- have the potential to provide clean laboratories for spectroscopy of the kaDn excited states. 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