Dalitz Plot Analysis of the Decay B± → K±K±K∓ Citation Dvoretskii, Alexei (2006) Dalitz Plot Analysis of the Decay B± → K±K±K∓. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/7HTG-0W17. https://resolver.caltech.edu/CaltechETD:etd-03312006-135838 Abstract We perform an analysis of the three-body charmless decay B+ -> K+K+K- using a sample of 226.0 +/- 2.5 million BB pairs collected by the BABAR detector and measure the total branching fraction and CP asymmetry to be B = (35.2 +/- 0.9 +/- 1.6) x 10-6 and A = (-1.7 +/- 2.6 +/- 1.5)%. We fit the Dalitz plot distribution using an isobar model and report the measured values of magnitudes and phases of the production coefficients. The decay dynamics is dominated by the K+K- S-wave, for which we perform a partial-wave analysis in the region m(K+K-) < 2 GeV/c2. We find no evidence of CP violation for individual components of the isobar model. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: charmless; dalitz plot; isobar model; partial-wave analysis; three-body Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Hitlin, David G. Thesis Committee: Hitlin, David G. (chair) Kapustin, Anton N. Porter, Frank C. McKeown, Robert D. Defense Date: 20 March 2006 Record Number: CaltechETD:etd-03312006-135838 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-03312006-135838 DOI: 10.7907/7HTG-0W17 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1226 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 03 Apr 2006 Last Modified: 01 Jul 2025 16:35 Thesis Files Preview PDF - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

Dalitz Plot Analysis of the Decay B ± → K ± K ± K ∓ Thesis by Alexei Dvoretskii In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2006 (Defended March 20, 2006) ii c 2006 Alexei Dvoretskii All Rights Reserved iii Acknowledgements I am grateful to my advisor, Professor D. Hitlin, who has guided my graduate research and supported my graduate education through the US Department of Energy grant DE-FG02-92-ER40701. In an experiment as large as BABAR any result is ultimately only possible because of substantial dedicated contributions from many collaborators and the Stanford Linear Accelerator staff in the accelerator, detector, and computing groups, to whom I am deeply grateful. Graduate school is a long experience — in my case, with a duration of almost eight years, particularly so. It is because of the wonderful people whom I have met at Caltech and during my stay in the Bay Area, too many to list here, that I will always remember this time with a lot of fondness. To all of you, thank you for you support and friendship. Finally, I thank my family who have always been supportive and understanding of my desire to pursue a doctoral degree abroad. iv Abstract We perform an analysis of the three-body charmless decay B ± → K ± K ± K ∓ using a sample of 226.0 ± 2.5 million BB pairs collected by the BABAR detector and measure the total branching fraction and CP asymmetry to be B = (35.2 ± 0.9 ± 1.6) × 10−6 and ACP = (−1.7 ± 2.6 ± 1.5)%. We fit the Dalitz plot distribution using an isobar model and report the measured values of magnitudes and phases of the production coefficients. The decay dynamics is dominated by the K + K − S-wave, for which we perform a partial-wave analysis in the region m(K + K − ) < 2 GeV/c2 . We find no evidence of CP violation for individual components of the isobar model. v Contents Acknowledgements iii Abstract iv Table of Contents v List of Figures viii List of Tables xi 1 Introduction 1 2 Theory 4 2.1 The Standard Model of quark weak interactions . . . . . . . . . . . . . . . . . . . . . 4 2.2 Low-energy effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Direct CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Time-dependent CP asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 QCD factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 Perturbative QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 Intrinsic charm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.9 Theoretical predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 PEP-II and the BABAR Detector 3.1 PEP-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 17 vi 3.2 The BABAR detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 The silicon vertex tracker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 The drift chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5 The Cherenkov light detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.6 The electromagnetic calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.7 The instrumented flux return (IFR) . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.8 Trigger and data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Event Selection 32 4.1 Preselection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Continuum background suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Continuum background extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Signal efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Modeling of B Backgrounds 47 5.1 Study of individual B background modes . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Background cocktail study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6 Dalitz Plot Analysis Formalism 55 6.1 Three-body decay kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.2 Partial-wave expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.3 Isobar model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.4 Resonant amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.5 Nonresonant amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.6 Discrete ambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.7 Binned maximum likelihood fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.8 Statistical errors and error propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 74 vii 7 B ± → K ± K ± K ∓ fit 77 7.1 Isobar-model fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.2 Partial-wave analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8 Systematic Uncertainties 90 8.1 Sources of systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.2 Control sample studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.3 Systematic uncertainty in efficiency parameterization . . . . . . . . . . . . . . . . . . 97 8.4 Systematic uncertainty in background estimates . . . . . . . . . . . . . . . . . . . . . 97 8.5 Systematic uncertainty in resonance lineshapes . . . . . . . . . . . . . . . . . . . . . 98 8.6 Tensor resonances f2 (1270) and f20 (1525) . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.7 Systematic uncertainty summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 9 Results 104 Bibliography 107 viii List of Figures 2.1 The Unitarity Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 B ± annihilation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 External and internal W -emission diagrams . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Examples of radiative correction to the tree diagrams . . . . . . . . . . . . . . . . . . 7 2.5 QCD penguin diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 Electroweak penguin diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.7 First-order gluon exchanges contributing to QCDF scattering kernels . . . . . . . . . 14 2.8 Rescattering diagrams with intrinsic charm for the decay B ± → K ± χc0 . . . . . . . . 16 3.1 Schematic drawing of the PEP-II storage ring and the linear accelerator complex . . . 17 3.2 Integrated luminosity for the Run 1–4 data taking periods in 1999–2004 . . . . . . . . 19 3.3 Schematic drawing of the BABAR detector . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Schematic drawing of the SVT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5 Longitudinal section of the DCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.6 Drift chamber dE/dx measurements as a function of track momentum . . . . . . . . . 25 3.7 DIRC schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.8 Kaon identification efficiency as a function of track momentum . . . . . . . . . . . . . 27 3.9 EMC schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.10 IFR schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.11 Cross section of a resistive plate chamber . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.1 Average fractions of misreconstructed simulated B ± → K ± K ± K ∓ signal events . . . 34 ix 4.2 ∆E distributions for different K/π final state particle assignments for simulated signal B ± → K ± K ± K ∓ events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 35 Average fractions of simulated B ± → K ± K ± K ∓ events, for which all final state kaons have been correctly identified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4 Average candidate multiplicity for simulated B ± → K ± K ± K ∓ events . . . . . . . . . 37 4.5 ∆E and mES distributions for simulated signal events, and sideband and off-peak events 38 4.6 Di-kaon invariant mass scatter plots and invariant mass projections for sideband and off-peak events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Distributions of the neural network input variables for simulated signal events, and sideband and off-peak events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 41 Distributions of neural network output for simulated signal events, and sideband and off-peak events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 39 42 Optimum selection boundary in the (, f ) plane and the normalized figure-of-merit of Eq. (4.1) as a function of the selection efficiency . . . . . . . . . . . . . . . . . . . . . 42 4.10 The mES distribution of the 9870 selected events . . . . . . . . . . . . . . . . . . . . . 44 4.11 The qq backgrounds extrapolation factor Rqq as a function of the background function shape parameter ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.12 Efficiency for properly reconstructed signal B ± → K ± K ± K ∓ events . . . . . . . . . . 45 5.1 Di-kaon invariant mass scatter plots of events from a simulated cocktail of generic BB decays selected in the signal region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.1 Countours of kinematic plots for B ± → K ± K ± K ∓ . . . . . . . . . . . . . . . . . . . . 57 6.2 Belle parameterization of the B ± → K ± K ± K ∓ nonresonant component . . . . . . . . 65 6.3 Fajfer et al. parameterization of the B ± → K ± K ± K ∓ nonresonant component . . . . 66 6.4 Cheng and Yang parameterization of the B ± → K ± K ± K ∓ nonresonant component . 68 6.5 Discrete ambiguity example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.1 The Dalitz plot of the candidates selected in the signal region . . . . . . . . . . . . . . 78 x 7.2 Non-uniform binning of the Dalitz plot used in the isobar-model fit . . . . . . . . . . . 78 7.3 Scan of the χ2LLR function in the (ρχc0 , φχc0 ) plane . . . . . . . . . . . . . . . . . . . . 81 7.4 Mass projection of the fit without and with the f0 (980) component near the K + K − threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.5 Mass projections of the best fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.6 The distributions of the residuals from the fit bias study . . . . . . . . . . . . . . . . . 85 7.7 The results of the partial-wave analysis of the K + K − S-wave . . . . . . . . . . . . . . 88 7.8 The results of the partial-wave analysis in the φ(1020) region for the P -wave . . . . . 89 8.1 Charmed tree cascade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.2 The m(K ± π ∓ ), mES , ∆E, and neural network output distributions for the selected B ± → Dπ ± control sample events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.3 The ∆E distribution for B ± → Dπ ± , D → K ± π ∓ control sample events . . . . . . . . 94 8.4 The m(π + π − ), mES , ∆E, and neural network output distributions for the selected B ± → K ± J/ψ control sample events . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.5 Di-kaon invariant mass distribution near the φ resonance . . . . . . . . . . . . . . . . 99 8.6 Di-kaon invariant mass distribution near the χc0 resonance . . . . . . . . . . . . . . . 100 xi List of Tables 1.1 Branching fractions for charmless decays of B mesons to fully-charged two-body and three-body final states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Theoretical predictions of branching fractions and CP asymmetries for charmless quasitwo-body decays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 16 Exclusive B ± decay modes considered, their branching fractions, and the sizes of simulated datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 2 48 Exclusive B 0 decay modes considered, their branching fractions, and the sizes of simulated datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.3 Branching fractions for B ± decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.4 Branching fractions for B 0 decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.5 Expected contributions to the final B ± → K ± K ± K ∓ data sample for exclusive B ± decay modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Expected contributions to the final B ± → K ± K ± K ∓ data sample for exclusive B 0 decay modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 52 52 Expected contributions to the final B ± → K ± K ± K ∓ data sample for exclusive B ± three-body decay modes with an intermediate D0 . . . . . . . . . . . . . . . . . . . . 53 7.1 Fit fractions and phase differences of the best fit . . . . . . . . . . . . . . . . . . . . . 84 7.2 Means and standard deviations of the distributions of the residuals from the fit bias study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 xii 7.3 Fit results for CP violation parameters and symmetric 90%-confidence-level intervals for CP asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 86 Kaon and pion identification efficiencies and their ratios in control samples for data and Monte Carlo–simulated events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.2 Tracking and particle identification efficiency asymmetries in data control samples . . 96 8.3 Estimates of the systematic uncertainties due to the uncertainty in the signal efficiency parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Estimates of the systematic uncertainties due to the uncertainty in the modeling of B backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 99 Estimates of the systematic uncertainties due to neglecting detector resolution effects for the χc0 resonance 8.8 98 Estimates of the systematic uncertainties due to neglecting detector resolution effects and the uncertainty in the Blatt-Weisskopf interaction radius for the φ resonance . . . 8.7 98 Estimates of the systematic uncertainties due to the uncertainty in the sideband extrapolation qq background subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 97 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Estimates of the systematic uncertainties due to to the uncertainty in the f0 (980) lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.9 Estimates of the systematic uncertainties due to to the uncertainty in the f0 (1710) lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.10 Sources of systematic uncertainty for the overall branching fraction and asymmetry . 102 9.1 Production coefficient magnitudes, phase differences, fit fractions, quasi-two-body branching fractions, and CP violation parameters of the individual components of the isobar model for the leading fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 1 Chapter 1 Introduction Rare charmless decays of B mesons provide a rich laboratory for studying different aspects of weak and strong interactions. Many branching fractions and CP asymmetries for such modes have now been measured at the high-luminosity SLAC and KEK B factories. The already considerable precision of such measurements will be further improved with still larger datasets that will become available within the next several years. An understanding of strong interaction effects in B decays is often a prerequisite for precise determination of the weak interaction parameters. The experimental measurements from the B factories have motivated recent theoretical progress in developing phenomenological models of quantum chromodynamics (QCD) effects leading to specific predictions for two-body pseudoscalar-pseudoscalar, B → P P , and pseudoscalar-vector, B → P V , branching fractions and asymmetries [1–6]. Global fits to experimental data in the QCD factorization framework [7–9] show a reasonable agreement between theory and experiment and are already sensitive to the parameters of the Cabibbo-KobayashiMaskawa (CKM) quark mixing matrix. Hadronic uncertainties in these modes can also be controlled by relating measurements in different modes using the SU (2) and SU (3) symmetries in the lightquark sector. In particular, simultaneous measurements of CP asymmetries and branching fractions in isospin or SU (3)-related channels can provide information about the Unitarity Triangle angles α and γ [10–12]. Improved experimental measurements of a comprehensive set of charmless B decays coupled with further theoretical progress hold the potential to provide significant constraints on the CKM matrix 2 Table 1.1: Branching fractions for charmless decays of B mesons to fully-charged two-body and three-body final states [14]. B 0 decay B 0 → π+ π− B 0 → K + π− B0 → K +K − B (4.6 ± 0.4) × 10−6 (18.2 ± 0.8) × 10−6 < 0.6 × 10−6 B ± decay B → π± π± π∓ B ± → K ± π± π∓ B ± → K ± π± K ∓ B± → K ±K ±K ∓ ± B (11 ± 4) × 10−6 (57 ± 4) × 10−6 < 6.3 × 10−6 (30.8 ± 2.1) × 10−6 parameters and to discover hints of physics beyond the Standard Model in penguin-mediated b → s transitions. We study the three-body charmless decay B ± → K ± K ± K ∓ . Experimentally, it is one of the most accessible charmless modes having a relatively large branching fraction (Table 1.1) and a clean final state signature. In addition to measuring the overall branching fraction and CP asymmetry, our goal in this analysis is to describe the dynamics of the decay matrix element and study the K + K − spectrum. Light scalar meson spectroscopy is an area of active experimental and theoretical research. Among the most interesting questions, are the nature of the f0 (980) resonance and searches for a possible light 0++ glueball state [13]. The CLEO collaboration measured the quasi-two-body branching fraction B(B ± → K ± φ) = −6 (5.5+2.1 , using a sample of 9.7 million BB decays [15]. Based on a sample of 88.8 million −1.8 ±0.6)×10 BB decays collected during 1999–2002, BABAR has previously reported the measurements of the total branching fraction and CP asymmetry, B(B ± → K ± K ± K ∓ ) = (29.6 ± 2.1 ± 1.6) × 10−6 , A(B ± → K ± K ± K ∓ ) = (0.02 ± 0.07 ± 0.03) [16]; the quasi-two-body branching fraction and asymmetry −6 B(B ± → K ± φ) = (10.0+0.9 , A(B ± → K ± φ) = 0.04 ± 0.09 ± 0.01 [17]; and the quasi−0.8 ± 0.5) × 10 −6 two-body branching fraction B(B ± → K ± χc0 ) = (1.49+0.36 [18]. In a quasi-two-body −0.34 ± 0.22) × 10 analysis, possible interference effects between different decay channels leading to the same final state are neglected. This introduces a significant systematic uncertainty, thus limiting the applicability of the quasi-two-body approach to low-statistics analyses. A more advanced formalism involves studying the structure of the full matrix element using a Dalitz plot fit. The most recent results published by the Belle collaboration are based on an isobar- 3 model amplitude analysis of the B ± → K ± K ± K ∓ Dalitz plot based on a sample of 152 million BB decays [19]. The total branching fraction, excluding the mode B ± → K ± χc0 , is measured to be B(B ± → K ± K ± K ∓ ) = (30.6 ± 1.2 ± 2.3) × 10−6 . We present herein the first BABAR amplitude analysis of the B ± → K ± K ± K ∓ Dalitz plot based on the dataset collected during Run 1–4 periods in 1999–2004. The dataset consists of 205.4 fb−1 taken on the Υ (4S) resonance corresponding to 226.0 million B +B − decays. 4 Chapter 2 Theory This chapter describes the Standard Model (SM) theory of weak interactions as applied to charmless decays of B ± mesons. While the weak interaction theory is well understood, the hadronic uncertainties due to the non-perturbative nature of QCD are difficult to estimate, complicating the extraction of the weak interaction parameters from the measurements of the branching fractions and CP asymmetries in hadronic decays of B mesons. We describe several phenomenological methods that lead to specific predictions for quasi-two-body branching fractions and asymmetries. 2.1 The Standard Model of quark weak interactions The weak interaction in the quark sector of the SM is governed by the following Lorentz-invariant hermitian Lagrangian: g L = − √ Vij (Ūiα Djα )µV −A Wµ+ + h.c., 2 describing a charged flavor-changing weak current transition between a left-handed up-like, (u, c, t), quark U and a left-handed down-like, (d, s, b), quark D mediated by an intermediate W ± boson. The Greek indices indicate a sum over Nc QCD colors. Left-handed and right-handed quark currents are given by (Ūiα Djα )µV ∓A = Ūiα (1 ∓ γ5 )γ µ Djα . We have written the Lagrangian in terms of the quark mass eigenstates; the unitary CabibboKobayashi-Maskawa (CKM) matrix Vij [20] describes the transformation between the mass eigen- 5 ∗ Vtd Vtb α β ∗ VudVub ∗ Vcd Vcb γ (a) A η ∗ VudVub ∗ |Vcd Vcb | 0 0 Vtd V∗tb ∗| |Vcd Vcb α β γ ρ 1 (b) 7–92 7204A5 Figure 2.1: The Unitarity Triangle. states and the weak interaction flavor eigenstates that define the weak current. Because of the unitarity requirement and the freedom to choose the phases of the mass eigenstates, the three-generation quark mixing CKM matrix has four parameters, three real angles and one complex phase, that must be determined experimentally. The irreducible complex phase is the sole source of CP violation in the SM. A parameterization that makes manifest the relative sizes of the CKM matrix elements is due to Wolfenstein [21]: 2  1 − λ2 λ   Vij =   −λ 1 − λ2 Aλ2 Aλ3 (1 − ρ − iη) −Aλ2 1 Aλ3 (ρ − iη) 2     + O(λ4 ),  where the Cabibbo angle [22], λ = 0.221 ± 0.002, is used as an expansion parameter. For B meson decays the most important unitarity relation is ∗ Vud Vub + Vcd Vcb∗ + Vtd Vtb∗ = 0, which defines the Unitarity Triangle (UT) shown in Fig. 2.1. 6 Figure 2.2: B ± annihilation diagram. Charged B mesons decay via the weak interaction primarily through the b → cW − transition with an average lifetime of 1.5 ps [14]. The relatively long lifetime is due to the large mass of the virtual W meson and the smallness of the Vcb matrix element. 2.2 Low-energy effective Hamiltonian To first order the matrix element of the transition is GF M = √ VU b VU∗0 D0 Q1 + h.c., 2 √ where GF / 2 = g 2 /8m2W is the Fermi constant and Q1 is a transition operator given by Q1 = (b̄α Uα )µV −A (D̄0 β U 0 β )µV −A . The weak decays of the B mesons are dominated by diagrams in which the constituent b quark 7 Figure 2.3: External (left) and internal (right) W -emission diagrams. decays weakly and the light quark is a spectator. The contribution of annihilation topology diagrams (Fig. 2.2), where the charged current is formed by the constituent quarks of the B ± meson, is small because of the small value of |Vub | and the helicity suppression [23]. The two possible spectator topologies corresponding to the b → U D0 Ū 0 transition induced by the Q1 four-quark operator are the external and internal W -emission tree topologies (Fig. 2.3). The gluon exchange lines in Fig. 2.3 indicate the hadronization process into final state mesons. The internal-emission diagram is suppressed by a factor 1/Nc because the colors of the quarks in hadronizing mesons must match. The exchange of hard gluons with virtualities between mW and some relevant hadronic scale µ can introduce new types of operators. Quark color indices can be rearranged via the radiative correction to the W -emission diagram (Fig. 2.4) inducing the operator Q2 = (b̄α Uβ )µV −A (D̄0 β U 0 α )µV −A . This operator corresponds to the same tree diagram topologies as in Fig. 2.3, but here the external- 8 Figure 2.4: Examples of radiative correction to the tree diagrams inducing the Q2 operator. emission diagram is suppressed by 1/Nc with respect to the internal-emission diagram. QCD penguin diagrams (Fig. 2.5), a different type of decay topology, correspond to the b → DQQ̄ transition. As gluons couple to both left-handed and right-handed quarks, they give rise to the following four operators: Q3 = (b̄α Uα )µV −A (Q̄β Qβ )µV −A Q4 = (b̄α Uβ )µV −A (Q̄β Qα )µV −A Q5 = (b̄α Uα )µV −A (Q̄β Qβ )µV +A Q6 = (b̄α Uβ )µV −A (Q̄β Qα )µV +A . The contribution of QCD penguin diagrams is expected to be suppressed relative to the contribution of tree diagrams by a factor of order m2 αs log t2 ∼ 0.03, 12π mb 9 Figure 2.5: QCD penguin diagram. where αs is the strong interaction coupling constant. Electroweak penguin diagram topologies (Fig. 2.6) are even more suppressed by a similar factor due to the smallness of the respective coupling constants. For the B ± → K ± K ± K ∓ decays considered here, the dominant contribution is due to the QCD penguin-mediated b → sss transition described by the Q3−6 operators with the (b̄s)(s̄s) flavor structure. After the QCD effects are taken into account, the low-energy effective Hamiltonian is written as an operator product expansion in terms of four-quark operators Qk , and the Wilson coefficients ck : 6 X GF X ∗ Vqb Vqs ck (µ)Qk (µ) + h.c. Heff = √ 2 q=(u,c,t) k=3 (2.1) The Wilson coefficients account for the short-distance corrections arising from the exchange of gluons with virtualities between µ and mW . They can be computed perturbatively, with the renormalization scale µ chosen to be of order mB for B decays. As the Hamiltonian should not depend on the choice of scale, the scale dependence of the Wilson coefficients must cancel that of the current-current operators Qk . 10 Figure 2.6: Electroweak penguin diagrams. 2.3 Direct CP violation The Hamiltonian of Eq. (2.1) involves sums of products of the CKM matrix elements and the hadronic operators. The matrix element for the decay of a B ± meson into a final state f ± can be written as M(B ± → f ± ) = X |Mj |ei(δj ±φj ) , j where φj are the “weak phases” and δj are the “strong phases”. The weak phases arise because of the irreducible complex phase of the CKM matrix and change sign under CP conjugation. The strong phases are generated by the hadronic operators, e.g., in quark rescattering processes, and do not change sign under CP . The CP asymmetry is defined as A≡ |M(B − → f − )|2 − |M(B + → f + )|2 . |M(B + → f + )|2 + |M(B − → f − )|2 11 The difference between the rates of two CP -conjugate processes is proportional to |M(B − → f − )|2 − |M(B + → f + )|2 = 2 X |Mi ||Mj | sin(δi − δj ) sin(φi − φj ), i6=j hence CP violation can be generated through interference between terms with different weak phases that also acquire different strong phases. This type of CP violation is referred to as CP violation in decay, or direct CP violation, as opposed to CP violation produced in the interference between mixing and decay amplitudes in B 0 and B 0 decays. ∗ In the Hamiltonian of Eq. (2.1), the leading terms are proportional to Vcb Vcs and Vtb Vts∗ , both of order O(λ2 ) and having a zero weak phase. The term with a non-zero weak phase is suppresed ∗ by Vub Vus = O(λ4 ). Direct CP violation in B ± → K ± K ± K ∓ in the SM is therefore expected to be small as has been confirmed experimentally by BABAR in Ref. [16]. 2.4 Time-dependent CP asymmetry The picture of CP violation is somewhat more complicated for decays of neutral B mesons into a CP eigenstate. The two neutral B mesons produced by a decay of the Υ (4S) are in a coherent state. In a time-dependent asymmetry measurement, the first neutral B decays into a tagging mode that identifies its flavor. At that time, taken to be zero, the second neutral B must have the opposite flavor. Because of B 0 B 0 mixing, the flavor eigenstates and the mass eigenstates (measured mass difference ∆m/ΓB = 0.77 ± 0.01) of neutral B mesons are different. Therefore, when the second B decays into a CP eigenstate fCP at time t, the decay amplitude is a superposition of B 0 and B 0 decay amplitudes. The asymmetry is defined in terms of decay rates into a CP eigenstate fCP for time-evolving neutral B mesons that begin at time zero as a B 0 or a B 0 : A(t) = Γ(B 0 (t = 0) → fCP )(t) − Γ(B 0 (t = 0) → fCP )(t) . Γ(B 0 (t = 0) → fCP )(t) + Γ(B 0 (t = 0) → fCP )(t) 12 Defining M(B 0 → fCP ) = X = X |Mj |ei(δj +φj ) , j 0 M(B → fCP ) |Mj |ei(δj −φj ) , j and a direct CP asymmetry: A0 = |M(B 0 → fCP )|2 − |M(B 0 → fCP )|2 , |M(B 0 → fCP )|2 + |M(B 0 → fCP )|2 it can be shown that A(t) = A0 cos(∆mB t) − ηCP =(λ)(1 − A0 ) sin(∆mB t), where ηCP is the CP parity of the final state and λ= Vtd Vtb∗ M(B 0 → fCP ) . Vtd∗ Vtb M(B 0 → fCP ) Even when the direct CP asymmetry is zero, A(t) = −ηCP sin (arg(λ)) sin(∆mB t), showing that there can still be time-dependent CP asymmetry. Further, if the leading terms in the sums of Eq. (2.2) have the same weak phase φ, the hadronic uncertainties cancel giving arg(λ) = −2(β + φ). This is the case for B 0 → J/ψ KS0 , where A(t) ∝ sin2β, and the measurement of the time-dependent asymmetry gives the best determination of the UT angle β to date. Measurements of time-dependent CP asymmetries in charmless final states can be sensitive to the other two angles of the UT. However, for most final states fCP , the contributing diagrams have different weak phases and an understanding of the hadronic uncertainties is needed to relate the asymmetry measurements to the underlying parameters of the weak interaction. This can be 13 done by calculating the hadronic contributions in a model-dependent way using phenomenological approaches, some of which are described in the sections below. Another approach is to use light-quark symmetry to control hadronic uncertainties in a modelindependent way. It was shown in Ref. [10, 11] that by measuring the decay rates in isospin-related B → ρπ channels, hadronic uncertainties due to penguin contributions can be eliminated. It is possible then to determine the UT angle α without ambiguities by a time-dependent Dalitz plot analysis of B 0 → π + π − π 0 . It has also been shown that SU (3) symmetry can be used to bound the hadronic uncertainties in the CP asymmetry measurements in the B 0 → η 0 KS0 and B 0 → φKS0 decays [12]. In all cases, it is necessary to measure decay rates and CP asymmetries in many channels related by isospin or SU (3). A global SU (3) fit in Ref. [24] has shown sensitivity to the UT angle γ and predictions have been made for yet-unobserved channels. 2.5 Factorization At present, the evaluation of hadronic matrix elements is not possible by direct calculation, e.g., by using lattice QCD techniques. This has motivated the development of several phenomenological models. Factorization [25, 26] is a powerful theoretical tool, which has been successfully used to relate the hadronic matrix elements to experimentally measured parameters. The basis of factorization is the color-transparency argument: in two-body decays of B mesons the light quark-antiquark pair that couples to the W boson is created with a high momentum in a color singlet; it leaves the interaction region quickly and then undergoes hadronization. The quark-antiquark pair effectively behaves like a pointlike colorless particle. Soft gluons, therefore, are ineffective in rearranging the quarks. The remaining quark produced by the b quark decay picks up the spectator quark and undergoes hadronization. To first order, there is no interaction between the two components, and the matrix elements are expected to factorize into products of color-singlet quark currents. 14 For a two-body decay, such as B ± → K ± φ, we can then write in the factorization approximation < K ± φ|(b̄s)µ (s̄s)µ |B ± > = < K ± |(b̄s)µ |B ± >< φ|(s̄s)µ |0 > . The factorized matrix elements can be related to experimentally measurable quantities: < φ|(s̄s)µ |0 > = µ fφ mφ , pq < K ± |(b̄s)µ |B ± > = F+ (q 2 )pµ + (F0 (q 2 ) − F+ (q 2 )) 2 q µ , q (2.2) (2.3) where fφ = 237 ± 4 MeV is the φ-meson decay constant describing the strength of the s̄s quark interaction inside the φ, µ is the polarization of the φ vector resonance, pµ ≡ pµB + pµK . Transition form factors, F0BK (q 2 ) and F+BK (q 2 ) [27], are functions of the momentum transfer, q µ ≡ pµB − pµK , and can be meausured in semileptonic decays, such as B → Klν̄. The naive factorization (NF) framework outlined above in many cases provides predictions of the branching fractions that have the correct order of magnitude. However, it has two significant shortcomings. Firstly, the hadronic operator matrix elements evaluated in terms of the form factors of Eq. (2.3) are independent of the renormalization scale µ, and thus cannot cancel the scale dependence of the Wilson coefficients. Secondly, under the assumption of no rescattering, the matrix elements of the hadronic operators are all real. Therefore, under NF, direct CP asymmetries are always zero. Finally, for B ± → K ± χc , where χc are 0++ charmonium states, the prediction for the branching fraction is B(B ± → K ± χc ) ∝ | < χc |(c̄c)V ∓A |0 > |2 = 0 under NF. The form factor vanishes because the di-quark current and the meson have opposite CP parity. The observations of the decay B ± → K ± χc0 by the BABAR and Belle collaborations [18, 19, 28] clearly show that the factorization assumption breaks down in this case. 15 Figure 2.7: First-order gluon exchanges contributing to the Type I (top) and Type II (bottom) QCDF scattering kernels. 2.6 QCD factorization These shortcomings motivated the development of an extended approach, called QCD factorization (QCDF) [29, 30]. In QCDF, hard scattering corrections of order αs and ΛQCD /mb are systematically evaluated. They introduce a scale dependence into the matrix elements of hadronic operators, restoring the overall scale independence, and also give rise to complex scattering phases that can generate direct CP violation. Under NF, the dynamics of the strong interactions is absorbed into products of transition form factors and meson decay constants. In QCDF this product is replaced by a more general expression given by F BK (q 2 )fφ → F BK (q 2 ) Z T I (x)Φφ (x)dx + Z T II (ξ, x, y)ΦB (ξ)ΦK (x)Φφ (y)dxdydξ. The first term describes hard gluon exchanges without the participation of the spectator quark (Fig. 2.7, top). The spectator quark is simply picked up by a final-state kaon through soft gluon exchange, and this process is characterized by the transition form factor. The scattering kernel T I then describes the corrections to the four-quark effective vertex. The integration is performed over light meson distribution functions Φ. The second term, with a scattering kernel T II , describes sixquark vertices where the spectator quark is involved in scattering (Fig. 2.7, bottom), and annihilation type diagrams. The long-distance perturbative effects are absorbed into the transition form factor and provide 16 the overall normalization of the meson distribution functions. The scattering kernels are calculated perturbatively. Formally, in the limit of mb → ∞, QCDF reduces to the NF formula. In the QCDF framework, the strong phases are suppressed by αs , and therefore the predicted CP asymmetries are small. 2.7 Perturbative QCD Perturbative QCD (PQCD) [31] is an alternative theoretical framework that gives somewhat different predictions for charmless decays. The non-perturbative long-distance effects are included in the universal meson wave functions that parameterize the distribution of momentum between the partons inside the meson. Whereas in QCDF the effective theory is defined at the energy scale µ ∼ mb , in PQCD the matching is done at a lower scale. This introduces large double logarithms of order log2 (mb /µ). It is argued that the resummation of these logarithms introduces form factors that suppress the long-distance effects, and that most of the contribution to the transition form factors is from the region where αs /π < 0.3, which justifies a perturbative calculation. In PQCD, therefore, the strong phases arise in the leading order and predicted CP asymmetries can be large. PQCD also predicts larger contributions for penguin and annihilation diagrams due to the enhancement of the corresponding Wilson coefficients evolved to a lower mass scale. 2.8 Intrinsic charm The large observed branching fraction in B ± → K ± χc0 has highlighted the importance of intrinsic charm (IC) contributions to three-body charmless final states [32–35]. Typical rescattering diagrams for the decay B ± → K ± χc0 are shown in Fig. 2.8. These diagrams are formally suppressed, because rescattering is required to produce a charmless final state, but can give competitive contributions because they correspond to CKM-favored tree and penguin b → ccs transitions. 17 B− - Ds∗−¡v - K− ¡ ¡ µ ¡ D0 @ 6 R @ D0 @@v - B− - χc0 Ds− ¡v ¡ µ ¡ D∗0 @ 6 R @ D∗0 @@v - (1) K− ¡ B− - χc0 Ds∗−¡v - K− ¡ ¡ µ ¡ D∗0 @ 6 R @ D∗0 @@v (2) χc0 (3) Figure 2.8: Typical rescattering diagrams with intrinsic charm for the decay B ± → K ± χc0 . The boxes and disks represent effective weak and strong vertices, respectively. Table 2.1: Theoretical predictions of branching fractions and CP asymmetries for charmless quasitwo-body decays. Mode B± → K ±φ B ± → K ± f0 (980) B ± → K ± χc0 2.9 Ref. [36] [1] [2, 37] [3] [4] [24] [5] [6] Method NF QCDF NF QCDF QCDF PQCD SU (3) fit QCDF IC B (2.5 − 15)×10−6 −6 (4.3+3.0 −1.4 )×10 −6 (3.8 − 3.9)×10 (6.2 − 7.2)×10−6 (2.5 − 11.6)×10−6 −6 (10.2+3.9 −2.1 )×10 −6 (8.7 ± 0.4)×10 (5.5 − 10.9)×10−6 (1.3 − 3.5)×10−4 A (1.0 − 1.4)% (0.6 − 1.7)% - Theoretical predictions Significant theoretical progress has been made as outlined above. Global fits to the available data peformed using the QCDF [7, 8] framework show a reasonable agreement between theory and experiment. The experimental and theoretical uncertainties are still large, but global fits performed in Ref. [9] already show some sensitivity to the CKM matrix parameters. Charmless decays therefore have a lot of promise to further constrain the parameters of the weak interaction, once a better understanding of the hadronic phenomenology is achieved. In Table 2.1 we summarize current phenomenological predictions of quasi-two-body branching fractions and asymmetries for decays contributing to the K ± K ± K ∓ final state. 18 Chapter 3 PEP-II and the BABAR Detector In this chapter we briefly discuss the SLAC B factory complex, which consists of the PEP-II collider used to produce a large sample of BB pairs in e+ e− collisions, and the BABAR detector used to detect the products of B meson decays. 3.1 PEP-II The PEP-II [38] is a high-luminosity asymmetric energy e+ e− collider at the Stanford Linear Accelerator Center (SLAC) with two storage rings: the high-energy ring (HER) storing 9.0 GeV electrons and the low-energy ring (LER) storing 3.1 GeV positrons. The two rings are located in the PEP tunnel with a circumference of 2.2 km and are filled from the existing 3 km SLAC linear accelerator complex (Fig. 3.1). The two beams are brought together and collide head-on at Interaction Region 2 where the BABAR detector is located. The nominal center-of-mass energy of the two beams is √ s = 10.580 GeV, just above the BB threshold and corresponding to the peak cross-section for the production of the Υ (4S) bb resonance, σ(e+ e− → Υ (4S)) = 1.1 nb. Other important processes at this energy include the production of lighter quark pairs: σ(e+ e− → dd) = 0.35 nb, σ(e+ e− → uu) = 1.39 nb, σ(e+ e− → ss) = 0.35 nb, σ(e+ e− → cc) = 1.30 nb; leptoproduction σ(e+ e− → µ+ µ− ) = 1.16 nb, σ(e+ e− → τ + τ − ) = 0.94 nb; and Bhabha scattering with σ(e+ e− → e+ e− ) ∼ 0.4 nb within the acceptance of the BABAR detector [39]. A fraction of the data is taken 40 MeV below the Υ (4S) peak, about 20 MeV below the BB threshold, and is used 19 Figure 3.1: Schematic drawing of the PEP-II storage ring and the linear accelerator complex. to study backgrounds due to these processes. The Υ (4S) resonance decays predominantly to a pair of BB mesons each with a momentum of 0.335 GeV in the center of mass. A key feature of the B factory is the boost of βγ = 0.56 between the center-of-mass frame and the lab frame. Because of the boost the typical distance between the decay vertices of the two B mesons ∆z is increased to βγcτ ∼ 250 µm, well within the resolution of the BABAR silicon vertex tracker. Sensitivity to ∆z makes it possible to perform time-dependent analysis of CP asymmetries in decays of B 0 and B 0 . The application of this powerful technique to the B 0 (B 0 ) → J/ψ KS0 decay has allowed the determination of the weak sector mixing matrix parameter sin2β with great precision [40]. This analysis studies decays of charged B mesons, and we do not make use of the ∆z information. However, because the branching fractions for charmless three-body B decays are very small, a critical factor for this analysis is the other unique feature of the B factory — its high luminosity. This analysis is based on a sample of 226.0 ± 2.5 million BB pairs collected during Run 1–4 data-taking periods in 1999–2004. The integrated luminosity delivered by PEP-II and recorded by BABAR 20 Figure 3.2: Integrated luminosity delivered by PEP-II and recorded by BABAR during the Run 1–4 data taking periods in 1999–2004. is shown in Fig. 3.2. The total integrated Run 1–4 luminosity is 210.6 fb−1 taken at the Υ (4S) resonance and 21.6 fb−1 taken 40 MeV below. The dataset used in this analysis corresponds to the integrated luminosity of 205.4 fb−1 and 16.1 fb−1 on-peak and off-peak, respectively. 3.2 The BABAR detector The BABAR detector [41] is a multisystem particle detector optimized for the study of B meson decays at a high-luminosity asymmetric-energy B factory. A schematic drawing of the BABAR detector is shown in Fig. 3.3. The BABAR detector consists of five subdetector systems. They are, in order of increasing distance from the interaction point (IP): the silicon vertex tracker (SVT), the drift chamber (DCH), the detector of internally-reflected Cherenkov light (DIRC), the electromagnetic calorimeter (EMC), and the instrumented flux return (IFR). The first four subsystems are located inside a superconducting solenoid that provides a 1.5 T magnetic field. The data acquisition system (DAQ) provides event triggering, data readout, and detector control and monitoring. 21 Detector CL 0 Scale I.P. 4m BABAR Coordinate System y x Cryogenic z Chimney 1015 1149 1749 4050 Instrumented Flux Return (IFR)) Barrel Superconducting Coil Electromagnetic Calorimeter (EMC) 1149 370 Drift Chamber (DCH) Silicon Vertex Tracker (SVT) Cherenkov Detector (DIRC) IFR Endcap Forward 3045 End Plug 1225 Magnetic Shield for DIRC Bucking Coil 1375 Support Tube 810 e– Q4 e+ Q2 Q1 3500 B1 Floor 3-2001 8583A50 0 IFR Barrel Superconducting Coil Cutaway Section Scale 4m BABAR Coordinate System y z x DIRC EMC DCH SVT IFR Cylindrical RPCs Corner Plates Earthquake Tie-down Gap Filler Plates 3500 Earthquake Isolator Floor 3-2001 8583A51 Figure 3.3: Schematic drawing of the BABAR detector. 22 The BABAR coordinate system is right-handed, with an origin at the nominal interaction point. The z axis is along the axis of the DCH in the direction of the higher-energy electron beam; the y axis points vertically upward; and the x axis points horizontally outward from the center of the storage rings. 3.3 The silicon vertex tracker The SVT and the DCH, the two innermost systems of the BABAR detector, are used jointly for charged particle tracking. For the rare charmless three-body decays considered here, the high efficiency of BABAR particle tracking is critical. Its excellent momentum resolution is essential for reconstructing this and many other exclusive B and D decays and rejecting combinatorial backgrounds; it also enables us to study the dynamic aspects of the matrix element of the decay. Precise tracking is also necessary for extrapolating particle trajectories into the DIRC, EMC, and IFR, and improves the overall performance of those systems as well. Precise vertexing is less important for the decay considered in this analysis. It is critical, however, for time-dependent analyses of CP violation in decays of neutral B. In this analysis most final state particles will have momenta greater than 0.7 GeV/c. In many other analyses, however, particle ID for slow final state particles will be provided by the dE/dx measurements in the tracking detectors. In the 1.5 T magnetic field of the solenoid, the helical trajectory formed by a charged particle with transverse momentum pT > 0.18 GeV/c extends beyond the outermost layer of the DCH. DCH-only tracking is possible for momenta down to pT > 0.12 GeV/c. For transverse momenta pT < 0.12 GeV/c SVT information must be used. The helical trajectory is parameterized by five parameters: the distance of closest approach to the z axis, d0 ; the corresponding azimuthal angle, φ0 ; the corresponding z coordinate, z0 ; the dip angle λ; and the curvature, which is inversely proportional to the transverse momentum ω = 1/pT . The SVT has been designed to provide precise reconstruction of particle trajectories and vertices near the IP. The SVT is also a standalone tracker for charged particles that do not enter the DCH, 23 considerably expanding the transverse momentum coverage down to about 80 MeV/c. The SVT consists of five layers of double-sided silicon microstrip sensors arranged as shown in Fig. 3.4. The segmentation into strips is along the z axis of the detector on the inner side and azimuthally on the outer side. Each SVT module is divided into a forward and a backward half and is read out from both ends. The SVT dominates the resolution of the vertex parameters, (d0 ,φ0 ,z0 ), giving on average σ(d0 ) = 23 µm and σ(z0 ) = 29 µm. 3.4 The drift chamber The DCH is the main charged-particle tracker. The DCH design requirements include: efficient detection of charged particles with high momentum resolution over a wide range of particle momenta; maximum hermiticity; small amount of material in front of the EMC to avoid energy resolution degradation; small outer radius to minimize the cost of the EMC; longitudinal resolution better than 1 mm for KS0 vertex reconstruction; high dE/dx resolution necessary for particle identification for particles with momenta less than 700 MeV/c; fast trigger input to the charged particle trigger; and ability to operate reliably in the presence of large machine backgrounds. The drift chamber (Fig. 3.5) is a cylinder with an inner radius of 23.6 cm and an outer radius of 80.9 cm, offset from the IP by 370 mm in the direction of the electron beam to provide better forward coverage. The chamber consists of 7104 hexagonal drift cells arranged in 40 layers, each 4 layers forming a superlayer. Superlayers 1, 4, 7, and 10 are axial with the wires parallel to the z axis. The other superlayers have wires strung at a small stereo angle enabling z coordinate measurements. The stereo angle of the wires increases from +45 mrad to −76 mrad going outward with alternating positive and negative stereo superlayers. The working gas mixture is a 4:1 mixture of helium and isobutane. Charge particles leave a trace of ionization in the drift cell; the ionization electrons are accelerated towards the sense wires causing an avalanche; the signal on the sense wires is read out by the front-end electronics mounted 24 580 mm Space Frame Bkwd. support cone 520 mrad e- Fwd. support350 mrad cone e+ Front end electronics Beam Pipe Beam Pipe 27.8mm radius Layer 5a Layer 5b Layer 4b Layer 4a Layer 3 Layer 2 Layer 1 Figure 3.4: Schematic drawing of the SVT. 25 630 1015 1749 68 Elec– tronics 485 e– 809 27.4 464 1358 Be 17.2 236 IP e+ 469 1-2001 8583A13 Figure 3.5: Longitudinal section of the DCH. at the back of the DCH. The operating voltage on the sense wires for the majority of the data-taking period was 1930V. The leading edge of the signal is detected to provide a measurement of the drift time, which is converted into a measurement of the closest approach to the sense wire. A pattern-matching algorithm is then used to reconstruct the trajectory of the particle. The timing information is also used for providing the trigger to the other detector subsystems. A truncated mean of the lowest 80% of the total charge deposited by a particle along its trajectory is used to measure dE/dx. The drift chamber provides transverse momentum resolution of σ(pT )/pT = (0.13 ± 0.01)% · pT + (0.45 ± 0.03)%. The rms dE/dx resolution has been measured to be 7.5%, providing reliable K/π separation at a level of 3 standard deviations up to a momentum of about 0.55 − 0.60 GeV/c (Fig 3.6). 26 104 p d dE/dx K π 103 e 10–1 1-2001 8583A20 µ 1 Momentum (GeV/c) 10 Figure 3.6: Drift chamber dE/dx measurements as a function of track momentum with superimposed Bethe-Bloch predictions. 3.5 The Cherenkov light detector The DIRC is the main particle ID detector for charged particles with momenta over 0.7 GeV/c. It provides most of the K/π identification for the final state particles in this analysis. DIRC design was motivated by the need to provide reliable K/π separation for charged particles with momenta larger than 0.7 GeV/c. Reliable identification of kaons with momenta of up to 2 GeV/c produced in b → c → s cascade decays is essential for tagging the flavor of neutral B mesons in time-dependent analyses. The majority of final state mesons in charmless three-body decays have large momenta of up to ∼ 4 GeV; good K/π separation is essential for assigning decay candidates to the correct final state. Similar to the drift chamber, the design requirements also include smallest possible amount of material in front of the EMC, smallest possible outer radius, fast signal response, and ability to operate in the presence of high backgrounds. The DIRC (Fig. 3.7) is made of 144 quartz bars with refractive index n = 1.473 arranged in a 12-sided barrel. A charged particle traveling at a speed of βc > c/n emits Cherenkov photons in a 27 PMT + Base 10,752 PMT's Light Catcher Purified Water 17.25 mm Thickness (35.00 mm Width) Standoff Box Bar Box Track Trajectory PMT Surface Wedge Mirror Bar Window 4.9 m { 4 x 1.225m Bars glued end-to-end { 1.17 m 8-2000 8524A6 Figure 3.7: DIRC schematic. cone with an opening Cherenkov angle given by cos θc = 1/nβ. Some fraction of the photons are trapped inside the bar by total internal reflection and propagate towards the backward end of the bar, in the case of the photons initially propagating forward after being reflected by a mirror at the forward end of the bar. The photons are then allowed to expand in an expansion region filled with 6000 liters of purified water, whose refractive index matches well that of the quartz, toward an array of 10,752 photomultiplier tubes (PMTs) where the light is detected. A quartz wedge is used at the end of each bar to collimate the photons by reflecting the photons traveling inward toward the z axis outward and outward traveling photons at large angles inward, thus reducing the size of the region that needs to be instrumented. The image detected by the PMTs is a conical section from which the Cherenkov angle is inferred. The arrival time of the signal is used to suppress background hits and to associate signal hits with charged tracks. Ambiguities introduced by photon reflections are resolved through the timing information and pattern matching. The DIRC timing resolution and single photon Cherenkov angle resolution have been measured with di-muon events to be 1.7 ns and 10.2 mrad, respectively. K/π separation inferred from the π Mis-ID as K Kaon Efficiency 28 1 0.9 0.8 0.7 0.6 0.2 0.1 0 1 2 3 Track Momentum (GeV/c) Figure 3.8: Kaon identification efficiency and pion-as-kaon misidentification rate as a function of track momentum. di-muon measurements is better than four standard deviations up to 3 GeV/c track momentum. Fig. 3.8 shows the efficiency of kaon identification and the pion-as-kaon misidentification rate as a function of track momentum determined from a control sample of D → K ± π ∓ decays. 3.6 The electromagnetic calorimeter The EMC is a detector of electromagnetic showers produced by photons and electrons. It is not used extensively in this analysis, however, this system is critical for analyses that have π 0 and η mesons in the final state. The EMC also provides electron identification, which is critical for the tagging of neutral B mesons via semileptonic decays; the reconstruction of rare semileptonic decays of B and D mesons, and τ leptons; and the reconstruction of the leptonic decays of charmonium. The EMC was designed to provide excellent resolution in energy and angle over a wide range of energies for incident particles. The lower bound of 20 MeV was set by the need for efficient reconstruction of low-energy π 0 and η produced in B decays; the higher bound by the need to measure the QED processes e+ e− → e+ e− (γ), and e+ e− → γγ needed for calibration and luminosity measurements. 29 2359 1555 1375 920 2295 1127 38.2˚ 1801 558 Interaction Point External Support 26.8˚ 22.7˚ 1979 15.8˚ 1-2001 8572A03 Figure 3.9: EMC schematic. The length unit is 1 mm. The EMC is a hermetic total-absorption calorimeter which is composed of 6580 thallium-doped cesium iodide (CsI(Tl)) crystals, arranged projectively into a 5760 crystal barrel and an 820 crystal forward endcap (Fig. 3.9). The crystal length varies from 16 radiation lengths (29.6 cm) in the backward direction to 17.5 radiation lengths in the forward direction of the higher energy electron beam. The face size of the crystals is about 5 cm × 5 cm, about the size of the Molière radius, providing a fine segmentation. Electromagnetic showers produce scintillation light that propagates toward the back end of the crystal, possibly being reflected by diffuse reflective coating of the crystal wrapping. Each crystal is read out by two PIN diodes with quantum efficiency of 85%. The data from individual crystals are multiplexed and sent to a buffered pipeline. Sums of crystal blocks are formed and sent to the neutral hadware trigger logic. Upon receipt of a trigger, the data samples from the pipeline are read out and feature extraction is performed. The response of individual crystals is calibrated using a light pulser system, the 6.13 MeV N16 gamma-ray line from an activated coolant fluid, and the Bhabha events. Neutral clusters are formed from sets of contiguous crystals. The energy and angular resolution of the calorimeter are parameterized as (2.32 ± 0.30)% σE = p ⊕ (1.85 ± 0.12)%, 4 E E( GeV) 30 Barrel 342 RPC Modules 3200 FW 920 3200 19 Layers 1940 18 Layers BW 1250 432 RPC Modules End Doors 4-2001 8583A3 Figure 3.10: IFR schematic. ! σθ = σφ = 3.87 ± 0.07 p + 0.00 ± 0.04 E( GeV) mrad, respectively. 3.7 The instrumented flux return (IFR) The IFR is the main detector for identifying muons and detecting neutral hadrons (KL0 and neutrons). It also serves as a flux return of the 1.5 T magnetic field and a support structure for the rest of the BABAR detector. The IFR information is not used extensively in this analysis, however, this system is used for the tagging of neutral B mesons via semileptonic decays; the reconstruction of rare semileptonic decays of B and D mesons, and τ leptons; the reconstruction of the leptonic decays of charmonium; and analyses of exclusive final states with KL0 mesons. The IFR is made up of resistive plate chambers (RPCs) interleaved between the steel layers of the flux return. The detector arrangement is a hexagonal barrel with a forward and a backward endcap as shown in Fig. 3.10. H.V. 31 Foam Bakelite Gas Bakelite Foam Aluminum X Strips Insulator Graphite 2 mm 2 mm 2 mm Graphite Insulator Y Strips Spacers Aluminum 8-2000 8564A4 Figure 3.11: Cross section of a resistive plate chamber. Individual RPCs (Fig. 3.11) detect the signal induced by streamers from ionizing particles on capacitive readout strips. An 8 kV voltage difference is applied to graphite-coated Bakelite electrodes. The gap is filled with a mixture of 57% argon, 39% freon-134a, and 4% isobutane. The aluminum readout strips are arranged orthogonally. The signals from the strips are read out and digitized. IFR hit information is also passed to the Level 1 Trigger. The IFR system was comissioned and achieved muon efficiency of 90% and pion misidentification rate of less than 8% over the momentum range between 1.5 to 3.0 GeV/c. The muon detection efficiency degraded substantially during the Run 1–2 period dropping to around 65%. The problems may have been caused by the overheating of the RPCs, but the exact nature of the problem has never been conclusively established. The RPCs in the endcaps were replaced with limited streamer tubes (LSTs) during the post-Run 2 shutdown in summer 2002. 32 3.8 Trigger and data acquisition The main design goal of the BABAR trigger is to select events of physics interest with a high and well-understood efficiency, while rejecting a large fraction of background events and keeping the total event logging rate under 120 Hz. The trigger is implemented as a two-level system: a Level 1 hardware trigger and a Level 3 software trigger. The hardware trigger receives input from the DCH and the EMC for efficient triggering on both charged tracks in the drift chamber and electromagnetic showers in the calorimeter. An input from the IFR is mainly used for diagnostic triggering on µ+ µ− and cosmic events. For each subsystem there is a set of electronics boards that processes the input and emits trigger primitives. A DCH trigger (DCT) algorithm finds track segments based on the pattern of hits in the superlayers of the drift chamber. Track segments that can be linked together to reach superlayers 5 and 10 of the drift chamber form short and long DCT track primitives, respectively. A pattern of segments in the axial superlayers consistent with a track with pT > 0.8 GeV/c forms a high pT track primitive. The EMC trigger (EMT) boards compute energy sums for predefined azimuthal segments of the calorimeter and emit a corresponding primitive if a predefined energy threshold is exceeded. The DCT and EMT triggers provide highly efficient standalone triggering systems and can be used for detailed cross-validation studies. Azimuthal segments of DCT and EMT primitives are further shipped to the global trigger (GLT) where additional primitives are defined corresponding to azimuthally matched objects. Finally, if the primitives match any of a predefined set of configurations a Level 1 Trigger accept is issued. The typical output rate of the Level 1 trigger is 1 kHz. A Level 1 Trigger accept decision causes all the detector subsystems to be read out into the data acquisition system that consists of special-purpose readout modules. The online dataflow system combines the data in individual modules into complete events that are made available to the Online Event Processing (OEP) system running on a farm of UNIX computers. The OEP hosts a Level 3 software trigger that performs quick reconstruction of drift chamber helices and calorimeter clusters and selects events for logging. The output rate of the Level 3 33 trigger is limited to 120 Hz, which includes physics triggers, prescaled Bhabha event triggers for calibration, prescaled pass-through triggers, and diagnostic triggers. Selected events are sent to a logging manager and stored in temporary disk storage. Events from the disk storage are grouped into runs and are archived to tape storage where they become available for data quality checks, offline calibrations and eventual event reconstruction. The OEP also hosts a real-time data quality monitoring system, and an online event display. Other subsystems of the DAQ implement detector control and calibrations. 34 Chapter 4 Event Selection This chapter describes the event selection, whose main goal is to suppress copious backgrounds coming from the light-quark and charm qq continuum production, while retaining a high signal efficiency. The event selection proceeds in stages that follow closely the layout of the BABAR analysis environment. 4.1 Preselection The first step of the analysis chain is event reconstruction. Five-parameter track candidate helices (Sec. 3.3) are fitted to the pattern of hits in the SVT and DCH. Track candidates are classified in the ChargedTracks category if they satisfy the following criteria: • At least 12 hits in the DCH. • Transverse momentum, 0.1 < pT < 10 GeV/c. • The point of closest approach coordinates satisfy |z0 | < 10 cm and |d0 | < 1.5 cm. If the number of DCH hits is greater than 20, the track candidate is upgraded to the GoodTracksLoose category. The overall event shape is characterized using rotationally invariant Fox-Wolfram moments [42], Hl ≡ X pi pj i,j 2 Evis Pl (cos θij ), 35 where the sum is over all charged and neutral decay products, pi and pj are the particle momenta, Pl (cos θij ) are the Legendre polynomials in the polar angle between particles i and j, and Evis is the visible energy of the event. When operating at the Υ (4S) resonance, continuum qq events and Υ (4S) decays have very different event topologies. In the center-of-mass (CM) frame the momenta of the quarks in e+ e− → qq are high, whereas the B mesons in Υ (4S) → B + B − are produced almost at rest. Accordingly, the decay products have a two-jet-like distribution for qq events, but are much more spherically distributed for BB events. The distribution of the ratio H2 /H0 is highly peaked towards 1 for qq events, but it is flat for B decays. Events with at least three GoodTracksLoose tracks and H2 /H0 < 0.98 are classified into the multihadron category. Essentially all of the produced BB pairs are preseved at this stage. The second step is the skim stage, in which a loose selection of multihadron events for three-body charmless analyses is done. Track candidates from the ChargedTracks category are used to form B ± → K ± K ± K ∓ decay candidates. Two nearly independent kinematic variables are used for identifying B decays [43, 44]. The first is ∆E ≡ E − √ s0 /2, the difference between the reconstructed B candidate energy and the beam energy measured in the CM frame. For signal events, the ∆E distribution peaks near zero with a resolution of 21 MeV/c. The second is the energy-substituted mass: mES ≡ q (s0 /2 + p0 · pB )2 /E02 − p2B , where pB is the momentum of the B candidate and (E0 , p0 ) is the four-momentum of the e+ e− initial state, both in the laboratory frame. For signal events, the mES distribution peaks near the B mass with a resolution of 2.6 MeV/c2 . Candidates with |∆E| < 0.45 GeV and |mES − mB | < 0.1 GeV/c2 are selected, where all possible combinations of K/π assignments of the three final state particles are considered. 36 Figure 4.1: Average post-preselection fractions of misreconstructed simulated B ± → K ± K ± K ∓ signal events, shown for different regions of the di-kaon invariant mass scatter plot. The third step is the analysis n-tuple production stage. N-tuples are two-dimensional tables, where each row represents an event and each column a specific analysis variable. The n-tuples are stored in ROOT [45] files and are shared between several analyses of three-body charmless B decays within BABAR. B candidates are formed again using track candidates from the GoodTracksLoose category. Candidates are required to have Evis < 20 GeV, |mES −mB | < 0.1 GeV/c2 , and |∆E| < 0.15 for data, or |∆E| < 0.35 GeV for simulated signal events. For a set of tracks the thrust is defined as P |pi T̂| T ≡ Pi , i |pi | where the thrust axis T̂ is chosen such that T is maximized. In the CM frame, the distribution of the absolute value of the cosine of the angle between the thrust axis of the tracks forming the B candidate and the thrust axis of the remaining tracks, | cos θT |, is strongly peaked towards 1 for qq events. We require | cos θT | < 0.95. We use a sample of 1.314 million Monte Carlo-simulated B ± → K ± K ± K ∓ events, generated with a uniform distribution over the phase space of the decay, to study the signal selection. The average preselection efficiency is evaluated to be 64.2%. Misreconstructed signal candidates are those 37 Figure 4.2: ∆E distributions for different K/π final state particle assignments for simulated signal B ± → K ± K ± K ∓ events. for which one or more daughters used to form the B candidate are decay products of the recoil B. The fraction of misreconstructed events is on average 2% at this stage, but can be significantly larger in the corners of the decay phase space, where one of the B decay daughters is slow (Fig. 4.1). To select a three-kaon final state we require that all the final state tracks are consistent with a kaon hypothesis using a likelihood-based kaon identification algorithm, which uses the information from the SVT, DCH, and the DIRC. Fig. 4.2 shows the ∆E distributions for properly reconstructed signal events satisfying mES ∈ (5.27, 5.29) GeV/c2 for different K/π assignement of the final state particles. The distribution is shifted toward negative values when kaon-as-pion misidentification occurs and to positive values when pion-as-kaon misidentification occurs. Requiring events to be within a narrow ∆E signal window helps eliminate a significant fraction of events with particle misidentification. We study the three-kaon final state selection efficiency using correctly reconstructed simulated signal events with |∆E| < 0.1 GeV and mES ∈ (5.27, 5.29) GeV/c2 . The average particle identification efficiency for kaons is 94%, giving an average signal efficiency of 82% for B ± → K ± K ± K ∓ , with little dependence on the di-kaon invariant mass values (Fig. 4.3). The average pion-as-kaon misidentification probability is 6%. Because of particle misidentification, there can be cross-feed 38 Figure 4.3: Average fractions of simulated B ± → K ± K ± K ∓ events, for which all final state kaons have been correctly identified, shown for different regions of the di-kaon invariant mass scatter plot. between different charmless three-body final states, typically due to one particle being misidentified. The B ± → K ± π ± K ∓ mode is Cabibbo-suppressed and the upper limit of Ref. [16] implies that its cross-feed contribution to the K ± K ± K ∓ final state is no more than 1.3%. We suppress leptonic backgrounds using likelihood-based electron and muon identification algorithms. The average fraction of correctly reconstructed signal events with all final state particles identified as kaons by the kaon-identification algorithm, for which at least one of the daughter particles is identified as an electron or a muon is 0.2% and 1.8%, respectively. We reject B candidates if any daughter track is identified as an electron, or if two daughter tracks are identified as muons. After the preselection is done, a kinematic fit is applied to the selected candidates, with the invariant mass of the three daughter particles constrained to the mass of the decaying B ± meson. The kinematic fit improves the resolution of the di-kaon invariant masses for signal events, and forces the selected candidates to lie within the kinematic boundary of the B ± → K ± K ± K ∓ Dalitz plot. Fig. 4.4 shows the average candidate multiplicity in different regions of the di-kaon invariant mass scatter plot after requiring |∆E| < 0.1 GeV/c and mES ∈ (5.27, 5.29) GeV/c2 . The overall average candidate multiplicity is 1.04, but can be larger in the corners of the plot, where one of the daughter tracks is slow and can be easily substituted by a track from the rest of the event. If there 39 Figure 4.4: Post-preselection average candidate multiplicity for simulated B ± → K ± K ± K ∓ events, shown for different regions of the di-kaon invariant mass scatter plot. are multiple candidates after all selection criteria are applied, a candidate is chosen at random. 4.2 Continuum background suppression Continuum qq events are the dominant source of background. We study these events using two data samples: • The off-peak data taken 40 MeV below the Υ (4S) resonance, for which we require mES ∈ (5.18, 5.27) GeV/c2 and |∆E| < 0.15 GeV. Because of the lower beam energies, the mES distribution for off-peak events is shifted down by 20 MeV with respect to the distribution for on-peak events. Events in the off-peak dataset consist mostly of qq continuum events. The total number of off-peak events is small, since the integrated off-peak luminosity for the BABAR Runs 1–4 data period is only 21.6 fb−1 , much smaller than 210.6 fb−1 collected on-peak. • A sample of on-peak data events selected in the region mES ∈ (5.20, 5.25) GeV/c2 , |∆E| < 0.15 GeV. This mES sideband sample contains mostly qq events and a small fraction of candidates due to misreconstructed B decays. The sideband sample size is large, but an extrapolation into the signal region is required. 40 Figure 4.5: ∆E and mES distributions for simulated signal events (top), and sideband and off-peak events (bottom). In both cases, a random candidate is chosen when multiple candidates are present. Fig. 4.5 shows the mES and ∆E distributions for simulated signal events with |∆E| < 0.1 GeV and mES ∈ (5.27, 5.29) GeV/c2 , sideband events with |∆E| < 0.15 GeV and mES ∈ (5.20, 5.25) GeV/c2 , and off-peak events with |∆E| < 0.15 GeV and mES ∈ (5.18, 5.27) GeV/c2 . Fig. 4.6 shows the dikaon invariant mass scatter plots and their projections for sideband and off-peak events. The qq backgrounds are especially prominent near the boundary of the scatter plot and there is a significant contribution from continuum production of the φ(1020) resonance. Following a previous charmless BABAR analysis [28], we further suppress the continuum background using the output of a multi-layer perceptron neural network with the following four inputs computed in the CM frame: the cosine of the angle between the direction of the B candidate and the beam direction, cos θB,z ; the absolute value of the cosine of the angle between the candidate thrust axis and the beam direction, | cos θT,z |; and momentum-weighted sums over tracks and neutral clusters not belonging to the candidate, L0 = P i pi and L2 = P i pi cos 2 θi , where the angles θi 41 Figure 4.6: Di-kaon invariant mass scatter plots for sideband (top left) and off-peak events (top right). Invariant mass projections for sideband (bottom left) and off-peak events (bottom right). 42 are measured with respect to the candidate thrust axis. Fig. 4.7 shows the distributions of the four neural network input variables for simulated signal events with |∆E| < 0.1 GeV and mES ∈ (5.27, 5.29) GeV/c2 , sideband events with |∆E| < 0.15 GeV and mES ∈ (5.20, 5.25) GeV/c2 , and off-peak events with |∆E| < 0.15 GeV and mES ∈ (5.18, 5.27) GeV/c2 The neural network has a four-neuron input layer, two hidden layers with 3 and 2 neurons, and two output neurons. The response of each neuron in layer L + 1 is modeled as xL+1 = P 1 − exp(−w0 − i wi xL i ) P , 1 + exp(−w0 − i wi xL i ) where the sum runs over all neurons in the preceding layer. The weights w are optimized to maximize the separatation between signal and background in a training sample consisting of 104 correctly reconstructed simulated signal events with mES ∈ (5.25, 5.29) GeV/c2 , |∆E| < 0.1 GeV for the signal category and 104 sideband events with mES ∈ (5.20, 5.25) GeV/c2 , |∆E| < 0.15 GeV for the background category. The response of the neural network, mapped on the interval (−1, +1), is shown in Fig. 4.8. We require ∆E ∈ (∆E min , ∆E max ), and NN > NN0 , where the interval (∆E min , ∆E max ) and NN0 are found via joint optimization. Different (∆E min , ∆E max ) intervals and NN0 values are tried in the range (−0.1 GeV, 0.15 GeV) with a 10 MeV step and in the range (−1, 1) with a 0.05 step, respectively. For each point, we evaluate the signal efficiency, , using properly reconstructed simulated signal events in a region mES ∈ (5.274, 5.284) GeV/c2 , an interval about two standard deviations wide around the mES signal peak. We also use the events from the mES ∈ (5.20, 5.25) GeV/c2 kinematic sideband to evaluate the fraction of eliminated background events, f . The pairs of values (, f ) thus determined form a scatter plot whose boundary (Fig. 4.9) corresponds to the set of possible optimum selection choices. Making a direct connection between a pair of values (, f ()) and the eventual sensitivity of the amplitude fit is difficult. We choose to maximize 43 Figure 4.7: Distributions of the neural network input variables for simulated signal events (left), and sideband and off-peak events (right). 44 Figure 4.8: Distributions of neural network output for simulated signal events (left), and sideband and off-peak events (right). Figure 4.9: Optimum selection boundary in the (, f ) plane (left) and the normalized figure-of-merit of Eq. (4.1) as a function of the selection efficiency for points on the optimum selection boundary (right). 45 the figure-of-merit often used for selection optimization in branching fraction measurements: F () = √ S0 S =p , S+B S0 + (1 − f ())B0 (4.1) where S0 (S) and B0 (B) are the expected numbers of signal and background events in the signal region before (after) applying the final selection. We estimate S0 = 2.5 × 103 , using the preselection efficiency and the BABAR measurement of the total branching fraction [16], and B0 = 7.6 × 103 by extrapolating the number of events from the mES ∈ (5.20, 5.25) GeV/c2 , ∆E ∈ (−0.1, 0.15) GeV sideband, giving F (1) = 24.9. The figure-of-merit is maximized at F ()/F (1) = 1.6 for (, f ) = (0.81, 0.916), corresponding to ∆E ∈ (−40, 40) MeV and NN > −0.2. 4.3 Continuum background extrapolation Figure 4.10 shows the mES distribution of the 9870 events selected in the region mES ∈ (5.20, 5.29), fitted with a sum of a Gaussian distribution and a background function having a probability density: p p(x) ∝ x 1 − x2 exp (−ξ(1 − x2 )), √ where x = 2mES / s0 and ξ is a shape parameter [46]. The fit gives 2394 ± 63 events in the Gaussian peak, having the mean, m0 = 5.280±0.001 GeV/c2 , and a standard deviation, σ = 2.64±0.07 MeV/c2 . We define a signal region (SR) with mES ∈ (5.27, 5.29) GeV/c2 and a sideband (SB) with mES ∈ (5.20, 5.25) GeV/c2 . The ratio of the integrals of the background function over the signal region and the sideband yields an extrapolation coefficient, R p(x)dx Rqq ≡ RSR , p(x)dx SB shown as a function of the shape parameter ξ in Fig. 4.11. The fit gives ξ = 21.1 ± 1.5; the corresponding extrapolation coefficient is Rqq = 0.231 ± 0.007. We use it to estimate the expected SB number of qq background events in the signal region, nSR − nSB ) = 977 ± 34, where qq = Rqq (n BB 46 Figure 4.10: The mES distribution of the 9870 selected events. The solid histogram shows a fit with a sum of a Gaussian distribution (m0 = 5.280 ± 0.001 GeV/c2 , σ = 2.64 ± 0.07 MeV/c2 , N = 2394 ± 63) and an ARGUS background function (dashed histogram, ξ = 21.1 ± 1.5). The shaded regions correspond to the signal region and the mES sideband defined in the text. Figure 4.11: The qq backgrounds extrapolation factor Rqq as a function of the background function shape parameter ξ. 47 Figure 4.12: Efficiency for properly reconstructed signal B ± → K ± K ± K ∓ events in 0.25 × 0.25 GeV2 /c4 bins across the di-kaon invariant mass scatter plot. nSB = 4659 is the number of events in the sideband, from which we subtract the number of non-signal BB background events, nSB = 431 ± 19, estimated using a large number of simulated exclusive B BB decays as described in Chapter 5. 4.4 Signal efficiency We determine the final signal selection efficiency using properly reconstructed simulated signal events generated with a uniform phase-space density. The efficiency is adjusted to account for the observed descrepancies between the tracking and particle identification efficiencies in data and simulation, by applying correction factors derived from data control samples [47, 48]. The efficiency distribution in 0.25 × 0.25 GeV2 /c4 bins across the di-kaon invariant mass scatter plot is shown in Fig. 4.12. It is rather uniform, except near the corners of the plot, where the momentum of one of the particles is small. Misreconstructed events are effectively an undesirable background. Fortunately, their contribution is very small. We determine the overall fraction of misreconstructed events to be 1.38 ± 0.02%, 48 with only 0.34 ± 0.02% for the signal mES bin. 49 Chapter 5 Modeling of B Backgrounds Backgrounds coming from non-signal B decays are studied using Monte Carlo-simulated events. Constructing an adequate model of B backgrounds is a difficult task because many modes can contribute. We first look at simulated events for a number of individual B decay modes to get an understanding of which background modes are important. Our final model of B backgrounds is based on a generic cocktail of exclusive B decays. 5.1 Study of individual B background modes Tables 5.1 and 5.2 show the exclusive B +B − and B 0B 0 background modes considered, their branching fractions taken from Ref. [14], the number of events expected in a sample of 226.0 × 106 BB decays, and the sizes of the simulated datasets. We make reasonable assumptions for the values of the branching fractions where there are no measurements or where only upper limits are available. Tables 5.3 and 5.4 summarize the branching fractions used. In the following, the final state particles likely to be used to form a B candidate that mimics the signal final state are set in boldface, and h stands for either a kaon or a pion. We consider the following sources of backgrounds: • Charmless two-body decays, such as B 0 → h+ h− . Combining the two final state mesons with a charged track from the rest of the event can mimic a three-body final state. In the case of B ± → KS0 h± , KS0 → π + π − there is a direct contribution to the h± π ± π ∓ charmless 50 Table 5.1: Exclusive B ± decay modes considered (charge conjugation implied), their branching fractions, the number of events expected in a sample of 226.0 × 106 BB decays, and the sizes of simulated datasets. The first column (SP-Mode) is the BABAR simulation production unique mode identifier. SP-Mode 991 1049 1215 1216 1217 1453 1454 1508 1509 1542 1765 1940 1970 1972 2355 2356 2390 2392 2393 2394 2395 2421 2422 2423 2432 2435 2436 2441 2626 2627 2628 2629 3018 3019 3135 3172 3173 3174 3175 4731 4732 4733 4734 4735 4736 4737 4738 4750 4751 4752 Decay 0 0 B + → K 0 π + , K 0 → KS , KS → π+ π− 0 0 B + → K + K 0 , K 0 → KS , KS → π+ π− B + → D0 π+ , D0 → K + π− B + → D0 π+ , D0 → K + K − B + → D0 π+ , D0 → π+ π− B + → K1 (1270)+ γ B + → K1 (1400)+ γ B + → η 0 K + , η 0 → ρ0 γ B + → η 0 π + , η 0 → ρ0 γ B + → ηK ∗+ , η → π + π − π 0 , K ∗+ → K + π 0 B + → K2∗ (1430)+ γ B + → D ∗0 e+ ν e , D ∗0 → D 0 γ, D 0 → K + π − π 0 B + → K ∗ (1410)+ γ B + → K ∗ (1680)+ γ B + → K ∗+ ρ0 k, K ∗+ → K + π 0 B + → K ∗+ ρ0 ⊥, K ∗+ → K + π 0 B + → ρ+ ρ0 B + → K ∗+ K ∗0 k, K ∗+ → K + π 0 , K ∗0 → K + π − B + → K ∗+ K ∗0 ⊥, K ∗+ → K + π 0 , K ∗0 → K + π − 0 B + → K ∗+ K ∗0 k, K ∗+ → K 0 π + , K 0 → KS , K ∗0 → K + π − 0 B + → K ∗+ K ∗0 ⊥, K ∗+ → K 0 π + , K 0 → KS , K ∗0 → K + π − B + → D ∗0 π + , D ∗0 → D 0 π 0 , D 0 → K + π − B + → D0 π+ , D0 → K + π− π0 B + → D ∗0 π + , D ∗0 → D 0 π 0 , D 0 → K + π − π 0 B + → D0 K + , D0 → K + π− π0 B + → D0 K + , D0 → K + π− B + → D 0 ρ+ , D 0 → K + π − B + → D 0 ρ+ , D 0 → K + π − π 0 B + → D ∗0 K + , D ∗0 → D 0 γ, D 0 → K + π − B + → D ∗0 K + , D ∗0 → D 0 γ, D 0 → K + π − π 0 B + → D ∗0 π + , D ∗0 → D 0 γ, D 0 → K + π − π 0 B + → D ∗0 π + , D ∗0 → D 0 γ, D 0 → K + π − B + → D0 K + , D0 → K + K − B + → D0 K + , D0 → π+ π− B + → K ∗+ γ B + → D ∗0 K + , D ∗0 → D 0 γ, D 0 → K + K − B + → D ∗0 K + , D ∗0 → D 0 γ, D 0 → π + π − B + → D ∗0 π + , D ∗0 → D 0 γ, D 0 → K + K − B + → D ∗0 π + , D ∗0 → D 0 γ, D 0 → π + π − B + → D 0 e+ ν e , D 0 → K + π − B + → D 0 e+ ν e , D 0 → K + π − π 0 B + → D 0 µ+ ν µ , D 0 → K + π − B + → D 0 µ+ ν µ , D 0 → K + π − π 0 B + → D ∗0 e+ ν e , D ∗0 → D 0 π 0 , D 0 → K + π − B + → D ∗0 e+ ν e , D ∗0 → D 0 π 0 , D 0 → K + π − π 0 B + → D ∗0 µ+ ν µ , D ∗0 → D 0 π 0 , D 0 → K + π − B + → D ∗0 µ+ ν µ , D ∗0 → D 0 π 0 , D 0 → K + π − π 0 B + → D ∗0 e+ ν e , D ∗0 → D 0 γ, D 0 → K + π − B + → D ∗0 µ+ ν µ , D ∗0 → D 0 γ, D 0 → K + π − B + → D ∗0 µ+ ν µ , D ∗0 → D 0 γ, D 0 → K + π − π 0 B (7.52 ± 0.50) × 10−6 (3.45 ± 3.45) × 10−7 (1.87 ± 0.12) × 10−4 (1.91 ± 0.14) × 10−5 (6.78 ± 0.54) × 10−6 (1.00 ± 1.00) × 10−5 (1.00 ± 1.00) × 10−5 (2.30 ± 0.23) × 10−5 (1.03 ± 1.07) × 10−6 (1.96 ± 0.34) × 10−6 (1.40 ± 0.40) × 10−5 (3.27 ± 0.75) × 10−3 (1.00 ± 1.00) × 10−5 (1.00 ± 1.00) × 10−5 (1.83 ± 0.67) × 10−6 (1.83 ± 0.67) × 10−6 (2.60 ± 0.60) × 10−5 (3.89 ± 3.89) × 10−6 (3.89 ± 3.89) × 10−6 (3.89 ± 3.89) × 10−6 (3.89 ± 3.89) × 10−6 (1.08 ± 0.17) × 10−4 (6.48 ± 0.77) × 10−4 (3.76 ± 0.79) × 10−4 (4.88 ± 1.16) × 10−5 (1.41 ± 0.26) × 10−5 (5.11 ± 0.81) × 10−4 (1.77 ± 0.37) × 10−3 (5.23 ± 1.97) × 10−6 (1.81 ± 0.78) × 10−5 (2.31 ± 0.55) × 10−4 (6.68 ± 1.25) × 10−5 (1.44 ± 0.28) × 10−6 (5.11 ± 1.01) × 10−7 (4.03 ± 0.26) × 10−5 (5.35 ± 2.06) × 10−7 (1.89 ± 0.74) × 10−7 (6.84 ± 1.32) × 10−6 (2.42 ± 0.48) × 10−6 (8.19 ± 1.03) × 10−4 (2.84 ± 0.51) × 10−3 (8.19 ± 1.03) × 10−4 (2.84 ± 0.51) × 10−3 (1.53 ± 0.23) × 10−3 (5.31 ± 1.06) × 10−3 (1.53 ± 0.23) × 10−3 (5.31 ± 1.06) × 10−3 (9.44 ± 1.67) × 10−4 (9.44 ± 1.67) × 10−4 (3.27 ± 0.75) × 10−3 Expected 1700 80 42000 4300 1500 2000 2000 5200 200 440 3200 700000 2000 2000 400 400 6000 900 900 900 900 25000 150000 80000 11000 3200 120000 400000 1200 4000 50000 15000 330 120 9100 120 40 1500 500 190000 600000 190000 600000 350000 1200000 350000 1200000 210000 210000 700000 Generated 219000 182000 216000 114000 182000 234000 234000 145000 146000 121000 182000 283000 290000 290000 201000 178000 352000 126000 126000 122000 121000 172000 249000 232000 134000 362000 108494 294000 259500 42000 239000 192500 250000 20000 436000 250000 30000 147000 30000 294000 294000 294000 292000 294000 294000 294000 294000 289000 288000 294000 51 Table 5.2: Exclusive B 0 decay modes considered, their branching fractions, the number of events expected in a sample of 226.0 × 106 BB decays, and the sizes of simulated datasets. SP-Mode 1028 1042 1123 1159 1452 1455 1568 1766 1971 1973 2128 2359 2360 2396 2398 2498 2979 2981 2982 2984 2992 2995 3134 4209 4635 Decay B 0 → K + π− B 0 → π+ π− B 0 → D ∗− π + , D ∗− → D 0 π − B 0 → D − ρ+ B 0 → K1 (1270)0 γ B 0 → K1 (1400)0 γ B 0 → D ∗− ρ+ , D ∗− → D 0 π − B 0 → K2∗ (1430)0 γ B 0 → K ∗ (1410)0 γ B 0 → K ∗ (1680)0 γ B 0 → D ∗0 γ, D ∗0 → D 0 γ, D 0 → K + π − B 0 → K ∗0 ρ0 k, K ∗0 → K + π − B 0 → K ∗0 ρ0 ⊥, K ∗0 → K + π − B 0 → ρ0 ρ0 B 0 → K ∗0 K ∗0 , K ∗0 → K + π − , K ∗0 → K − π + B 0 → ρ+ ρ− B 0 → D0 π0 , D0 → K + π− B 0 → D0 π0 , D0 → K + π− π0 B 0 → D ∗0 π 0 , D ∗0 → D 0 π 0 /γ, D 0 → K + π − B 0 → D ∗0 π 0 , D ∗0 → D 0 π 0 /γ, D 0 → K + π − π 0 B 0 → D 0 ρ0 , D 0 → K + π − B 0 → D ∗0 ρ0 , D ∗0 → D 0 π 0 /γ, D 0 → K + π − B 0 → K ∗0 γ B 0 → K ∗+ ρ− , K ∗+ → K + π 0 B 0 → D− K + , D− → π− π0 B (1.82 ± 0.08) × 10−5 (4.60 ± 0.40) × 10−6 (1.87 ± 0.16) × 10−3 (7.70 ± 1.30) × 10−3 (1.00 ± 1.00) × 10−5 (1.00 ± 1.00) × 10−5 (4.60 ± 0.64) × 10−3 (1.24 ± 0.24) × 10−5 (1.00 ± 1.00) × 10−5 (1.00 ± 1.00) × 10−5 (3.63 ± 3.99) × 10−7 (3.34 ± 3.34) × 10−6 (3.34 ± 3.34) × 10−6 (1.00 ± 1.00) × 10−6 (4.45 ± 4.45) × 10−6 (2.60 ± 0.60) × 10−5 (1.11 ± 0.13) × 10−5 (3.84 ± 0.66) × 10−5 (1.03 ± 0.21) × 10−5 (3.56 ± 0.93) × 10−5 (1.10 ± 0.45) × 10−5 (9.53 ± 9.75) × 10−6 (4.01 ± 0.20) × 10−5 (3.66 ± 1.33) × 10−6 (2.66 ± 1.24) × 10−7 Expected 4100 1040 420000 1700000 2000 2000 1000000 2800 2000 2000 80 800 800 200 1000 6000 2500 9000 2300 8000 2000 2000 9100 800 60 Generated 1873500 2595000 4226000 112000 234000 234000 1052000 184000 288000 286000 183000 202000 182000 203000 122000 4383000 154000 70000 140000 140000 70000 140000 392000 12000 295000 three-body final state. • Charmless decays with a photon in the final state, such as B ± → K ∗± γ. Some of these can produce a charmless three-body final state with an extra photon, e.g., B ± → η 0 K ± , η 0 → ρ0 γ, ρ0 → π + π − . • Charmless four-body decays, such as B ± → K ∗± K ∗0 , K ∗± → K ± π 0 , K ∗0 → K ± π ∓ can mimic a charmless three-body final state when the π 0 is soft. • Semileptonic decays, e.g., B + → D0 `+ ν ` , D0 → K + π − , can mimic a charmless three-body final state due to lepton-as-kaon misidentification. Lepton vetoes described in Section 4.1 suppress these backgrounds. • Numerous multibody decays with charm proceeding through intermediate D0 or D∗0 mesons. These modes can be an important source of backgrounds because of their high branching fractions. The expected numbers of events in the signal region due to individual non-three-body B ± and 52 Table 5.3: B ± decay branching fractions from Ref. [14] (charge conjugation implied), the values used in the generation of the BABAR generic cocktail of B ± decays, and the branching fractions used in the study of individual B background modes. B + decays Γ2 Γ2 Γ3 Γ3 Γ28 Γ31 Γ32 Γ52 Γ54 Γ55 Γ152 Γ154 Γ157 – Γ175 Γ176 Γ179 Γ197 Γ198 Γ204 – Γ205 Γ206 Γ219 Γ225 Mode B + → D0 e+ ν e B + → D 0 µ+ ν µ B + → D∗0 e+ ν e B + → D∗0 µ+ ν µ B + → D0 π+ B + → D 0 ρ+ B + → D0 K + B + → D∗0 π + B + → D∗0 ρ+ B + → D∗0 K + B + → K 0 π+ B + → η0 K + B + → ηK ∗+ B + → K ∗0 ρ+ B + → K ∗+ ρ0 B + → K ∗+ K ∗0 B+ → K +K 0 B + → K ∗+ γ B + → K1 (1270)+ γ B + → K1 (1400)+ γ B + → K ∗ (1410)+ γ B + → K2∗ (1430)+ γ B + → K ∗ (1680)+ γ B + → ρ+ ρ0 B + → η0 π+ B PDG (2.15 ± 0.22)% (2.15 ± 0.22)% (6.50 ± 0.50)% (6.50 ± 0.50)% (4.91 ± 0.21) × 10−3 (1.34 ± 0.18)% (3.70 ± 0.60) × 10−4 (4.60 ± 0.40) × 10−3 (9.80 ± 1.70) × 10−3 (3.60 ± 1.00) × 10−4 (2.18 ± 0.14) × 10−5 (7.80 ± 0.50) × 10−5 (2.60 ± 0.40) × 10−5 – (1.10 ± 0.40) × 10−5 < 7.10 × 10−5 CL = 90% < 2.50 × 10−6 CL = 90% (4.03 ± 0.26) × 10−5 < 9.90 × 10−5 CL = 90% < 5.00 × 10−5 CL = 90% – (1.40 ± 0.40) × 10−5 < 1.90 × 10−3 CL = 90% (2.60 ± 0.60) × 10−5 < 4.50 × 10−6 CL = 90% B cocktail 2.10% 2.10% 5.60% 5.60% 5.30 × 10−3 1.34% 4.40 × 10−4 4.60 × 10−3 1.55% – 1.80 × 10−5 7.50 × 10−5 2.60 × 10−5 1.00 × 10−5 1.00 × 10−5 1.00 × 10−6 1.40 × 10−6 4.50 × 10−5 – – – – – 1.50 × 10−5 2.00 × 10−6 B used (2.15 ± 0.22)% (2.15 ± 0.22)% (6.50 ± 0.50)% (6.50 ± 0.50)% (4.91 ± 0.21) × 10−3 (1.34 ± 0.18)% (3.70 ± 0.60) × 10−4 (4.60 ± 0.40) × 10−3 (9.80 ± 1.70) × 10−3 (3.60 ± 1.00) × 10−4 (2.18 ± 0.14) × 10−5 (7.80 ± 0.50) × 10−5 (2.60 ± 0.40) × 10−5 (1.00 ± 1.00) × 10−5 (1.10 ± 0.40) × 10−5 (3.50 ± 3.50) × 10−5 (1.00 ± 1.00) × 10−6 (4.03 ± 0.26) × 10−5 (1.00 ± 1.00) × 10−5 (1.00 ± 1.00) × 10−5 (1.00 ± 1.00) × 10−5 (1.40 ± 0.40) × 10−5 (1.00 ± 1.00) × 10−5 (2.60 ± 0.60) × 10−5 (3.50 ± 3.50) × 10−6 53 Table 5.4: B 0 decay branching fractions from Ref. [14], the values used in the generation of the BABAR generic cocktail of B 0 decays, and the branching fractions used in the study of individual B background modes. B 0 decays Γ20 Γ23 Γ27 Γ33 Γ84 Γ85 Γ90 Γ91 Γ92 Γ143 Γ158 – Γ174 Γ180 Γ188 Γ193 Γ194 – Γ195 Γ196 Γ202 Γ226 Γ230 Mode B 0 → D − ρ+ B 0 → D− K + B 0 → D∗− π + B 0 → D∗− ρ+ B 0 → D0 π0 B 0 → D 0 ρ0 B 0 → D∗0 γ B 0 → D∗0 π 0 B 0 → D∗0 ρ0 B 0 → K + π− B 0 → K + ρ− B 0 → K ∗+ ρ− B 0 → K ∗0 ρ0 B 0 → K ∗0 K ∗0 B 0 → K ∗0 γ B 0 → K1 (1270)0 γ B 0 → K1 (1400)0 γ B 0 → K ∗ (1410)0 γ B 0 → K2∗ (1430)0 γ B 0 → K ∗ (1680)0 γ B 0 → π+ π− B 0 → ρ0 ρ0 B 0 → ρ+ ρ− B PDG (7.70 ± 1.30) × 10−3 (2.00 ± 0.60) × 10−4 (2.76 ± 0.21) × 10−3 (6.80 ± 0.90) × 10−3 (2.91 ± 0.28) × 10−4 (2.90 ± 1.10) × 10−4 < 5.00 × 10−5 CL = 90% (2.70 ± 0.50) × 10−4 < 5.10 × 10−4 CL = 90% (1.82 ± 0.08) × 10−5 (8.50 ± 2.80) × 10−6 – < 3.40 × 10−5 CL = 90% < 2.20 × 10−5 CL = 90% (4.01 ± 0.20) × 10−5 < 7.00 × 10−3 CL = 90% < 4.30 × 10−3 CL = 90% – (1.24 ± 0.24) × 10−5 < 2.00 × 10−3 CL = 90% (4.60 ± 0.40) × 10−6 < 2.10 × 10−6 CL = 90% (3.00 ± 0.60) × 10−5 B cocktail 6.80 × 10−3 2.00 × 10−4 2.70 × 10−3 6.80 × 10−3 2.40 × 10−4 2.50 × 10−4 – 2.90 × 10−4 2.50 × 10−4 1.80 × 10−5 1.50 × 10−5 2.00 × 10−5 8.00 × 10−7 1.00 × 10−6 4.50 × 10−5 – – – – – 4.70 × 10−6 2.00 × 10−6 3.00 × 10−5 B used (7.70 ± 1.30) × 10−3 (2.00 ± 0.60) × 10−4 (2.76 ± 0.21) × 10−3 (6.80 ± 0.90) × 10−3 (2.91 ± 0.28) × 10−4 (2.90 ± 1.10) × 10−4 (2.50 ± 2.50) × 10−5 (2.70 ± 0.50) × 10−4 (2.50 ± 2.50) × 10−4 (1.82 ± 0.08) × 10−5 (8.50 ± 2.80) × 10−6 (1.10 ± 0.40) × 10−5 (1.00 ± 1.00) × 10−5 (1.00 ± 1.00) × 10−5 (4.01 ± 0.20) × 10−5 (1.00 ± 1.00) × 10−5 (1.00 ± 1.00) × 10−5 (1.00 ± 1.00) × 10−5 (1.24 ± 0.24) × 10−5 (1.00 ± 1.00) × 10−5 (4.60 ± 0.40) × 10−6 (1.00 ± 1.00) × 10−6 (2.60 ± 0.60) × 10−5 54 Table 5.5: Expected contributions to the final B ± → K ± K ± K ∓ data sample for exclusive B ± decay modes. Mode 2436 2422 2432 4738 2629 2421 2626 4732 3174 2423 2441 4733 2392 – Decay B + → D 0 ρ+ , D 0 → K + π − B + → D0 π+ , D0 → K + π− π0 B + → D0 K + , D0 → K + π− π0 B + → D∗0 µ+ ν µ , D∗0 → D0 π 0 , D0 → K + π − π 0 B + → D∗0 π + , D∗0 → D0 γ, D0 → K + π − B + → D∗0 π + , D∗0 → D0 π 0 , D0 → K + π − B + → D∗0 K + , D∗0 → D0 γ, D0 → K + π − B + → D0 e+ ν e , D0 → K + π − π 0 B + → D∗0 π + , D∗0 → D0 γ, D0 → K + K − B + → D∗0 π + , D∗0 → D0 π 0 , D0 → K + π − π 0 B + → D 0 ρ+ , D 0 → K + π − π 0 B + → D0 µ+ ν µ , D0 → K + π − B + → K ∗+ K ∗0 k, K ∗+ → K + π 0 , K ∗0 → K + π − Non-three-body total Expected 17 ± 5 12 ± 3 5±1 4±4 4±1 3±1 2±1 2±2 1±0 1±1 1±1 1±1 1±1 59.3 ± 8.1 Table 5.6: Expected contributions to the final B ± → K ± K ± K ∓ data sample for exclusive B 0 decay modes. Mode 1123 2398 – Decay B 0 → D∗− π + , D∗− → D0 π − B 0 → K ∗0 K ∗0 , K ∗0 → K + π − , K ∗0 → K − π + Non-three-body total Expected 4±1 1±1 6.7 ± 1.9 B 0 background modes after all selection requirements have been applied are shown in Tables 5.5–5.6. 5.2 Background cocktail study The background model used in this analysis is based on a comprehensive cocktail of B +B − and B 0B 0 generic decays composed according to the best available knowledge of the branching fractions and the decay dynamics of individual modes. Samples of 536.3 million B + B − and 541.3 million B 0B 0 decays were used, about 4.8 times the number of BB pairs in the experimental dataset used in this analysis. The original cocktail contains signal events coming from three-body charmless decays, possibly with one or more photons radiated from the charged quark lines in the Feynman diagrams. These events are removed by requiring that the final state of the generated decay be different from 55 Table 5.7: Expected contributions to the final B ± → K ± K ± K ∓ data sample for exclusive B ± three-body decay modes with an intermediate D0 . Mode 3018 1216 2435 1215 – Decay B + → D0 K + , D0 → K + K − B + → D0 π+ , D0 → K + K − B + → D0 K + , D0 → K + π− B + → D0 π+ , D0 → K + π− Charm three-body total Expected 87 ± 17 37 ± 3 13 ± 2 3±1 139.8 ± 17.2 h± h± h∓ + nγ. The di-kaon invariant mass scatter plots of events from the generic cocktail of B ± and B 0 decays selected in the signal region are shown in Fig. 5.1. After adjusting by a luminosity scaling factor, the expected background contributions are 96 ± 6 events for the B +B − decays and 34 ± 4 for B 0B 0 decays. Three-body charmed decays B ± → D0 h± , D0 → h+ h− contribute to the same final state as the signal charmless modes, but because of the long lifetime of the intermediate D0 meson they can be added incoherently in the background model. These decays are initially in the original cocktail, but are removed by the requirement that the final state be different from h± h± h∓ + nγ. They are studied individually as described in Section 5.1 and their contributions (Table 5.7) are added back to the final B background model. The same procedure is used for several modes with an h+ h+ h− + γ final state, such as B ± → D∗0 h± , D∗0 → D0 γ, D0 → h+ h− . We estimate the total expected number of BB background events in the signal region to be nSR = 276 ± 20, with B ± → DK ± decays giving the largest contribution. BB 56 Figure 5.1: Di-kaon invariant mass scatter plots of events from a simulated cocktail of generic B +B − (top) and B 0B 0 (bottom) decays selected in the signal region. 57 Chapter 6 Dalitz Plot Analysis Formalism In this chapter we briefly review the three-body kinematics, the partial-wave analysis method, the isobar model, and amplitude parameterizations. The review is short; more detailed discussions can be found elsewhere [14, 49]. 6.1 Three-body decay kinematics For a decay of a particle having mass M into three particles having masses m1 , m2 , m3 , the three twobody invariant mass combinations (Mandelstam variables): s12 ≡ m212 , s23 ≡ m223 , and s13 ≡ m213 , are related by a kinematic constraint: s12 + s23 + s13 = M 2 + m21 + m22 + m23 . The decay kinematics is therefore completely specified by any pair of the invariant mass combinations; a two-dimensional plot of the event distribution in these variables is called a Dalitz plot. For B ± decays into three charged mesons we use the two opposite-charge combinations, labeled (s23 , s13 ). The kinematically accessible range of values for the Mandelstam variables is s23 ∈ [(m2 + m3 )2 , (M − m1 )2 ], s13 ∈ [(m1 + m3 )2 , (M − m2 )2 ]. 58 For a given value of s23 , s13 = smin 13 (s23 )  1 + cos θ13 2  + smax 13 (s23 )  1 − cos θ13 2  , max where s13 ∈ [smin 13 , s13 ], and θ13 is the helicity angle. In the rest frame of particles (2, 3), θ13 is the angle between particles (1, 3) and 2 2 smin 13 = (E1 + E3 ) − (p2 + p3 ) , 2 2 smax 13 = (E1 + E3 ) − (p2 − p3 ) , where E1 , E3 and p2 , p3 are the energies and momenta of respective particles. The B ± → K ± K ± K ∓ Dalitz plot contour is shown in Fig. 6.1 (a). For modes with identical particles in the final state, we order the daughter particles such that s23 ≤ s13 , giving a folded Dalitz plot shown in Fig. 6.1 (b). Finally to emphasize the regions of smaller invariant mass we use the di-kaon invariant mass scatter plot (m23 , m13 ) shown in Fig. 6.1 (c). For three-body decays the decay rate Γ is given by dΓ = 1 1 |M|2 ds23 ds13 . (2π)3 32M 3 (6.1) The Dalitz plot thus gives a graphical representation of the variation of the square of the matrix element |M|2 over the kinematically accessible phase space of the process (s23 , s13 ). Eq. 6.1 can be rewritten as dΓ = 1 1 |M|2 J dm23 dm13 , (2π)3 32M 3 where J = 8m23 m13 is the Jacobian of the transformation from the Dalitz plot of Fig. 6.1 (a) to the folded di-kaon invariant mass scatter plot of Fig. 6.1 (c). 59 Figure 6.1: Countours of kinematic plots for B ± → K ± K ± K ∓ : (a) Dalitz plot, (b) folded Dalitz plot, (c) di-kaon invariant mass plot. 60 6.2 Partial-wave expansion Decay and scattering amplitudes are closely related [50]. In this analysis, we take the ansatz that the decay matrix element can be parameterized as a sum of two-body amplitudes in the (2,3) and (1,3) channels: M = M23 + M13 . (6.2) Possible interactions between same-sign particles in the (1,2) channel or three-body interactions, beyond a constant term that can be accommodated in Eq. (6.2), are neglected. All particles involved in the decay are spinless and the two-body amplitudes can be written as a partial-wave expansion: M23 = JX max ρJ (s23 )eiφJ (s23 ) PJ (cos θ13 ), (6.3) ρJ (s13 )eiφJ (s13 ) PJ (cos θ23 ), (6.4) J=0 M13 = JX max J=0 where ρJ and φJ are the mass-dependent magnitude and phase of the J-th partial wave and PJ (cos θ) is an order-J Legendre polynomial. The sum is truncated at the partial wave Jmax once a good fit is achieved. For two-body scattering experiments the partial-wave analysis method is used to extract the individual scattering contributions in a model-independent way. A fit to the distribution allows the extraction of the magnitudes of the partial waves, and the phases relative to the phase of the reference wave. However, the variation of the phase of the reference wave as a function of the invariant mass cannot be determined, unless an additional constraint can be imposed, e.g., from unitarity. In the context of a three-body decay the partial-wave analysis is complicated by the interference between the M23 and M13 amplitudes. However, if one of the amplitudes is known, e.g., determined from a fit to a model, the Dalitz plot can be used as an interferometer, and it is possible to perform a partial-wave analysis to extract the magnitudes and phases of the other amplitude. 61 6.3 Isobar model A widely used approximation to the parameterization of the matrix element is the isobar model [51, 52]. Each of the two-body partial-wave amplitudes is decomposed into a sum of components: ρJ (s23 )eiφJ (s23 ) = X ρk eiφk Tk (s23 ), (6.5) ρk eiφk Tk (s13 ), (6.6) k ρJ (s13 )eiφJ (s13 ) = X k where Tk is a two-body amplitude for the k-th component of the partial wave and ρk eiφk is a complex production coefficient. In the isobar model, the individual terms are interpreted as complex production amplitudes for two-body resonances (isobars). The sum can also include terms describing the nonresonant component that cannot be easily identified with a dominant resonance. Combining the expansions of Eqs. (6.3-6.6), M23 = X ρk eiφk Tk (s23 )PJk (cos θ13 ), k M13 = X ρl eiφk Tk (s13 )PJk (cos θ23 ), k and the isobar model matrix element is a sum of components: M= X k Mk = X ρk eiφk Ak k The overall phase of the matrix element is arbitrary, only component phase differences φkl = φl − φk can be measured. For B ± → K ± K ± K ∓ the matrix element must be symmetrized with respect to the {1 ↔ 2} interchange of the identical kaons giving M= X ρk eiφk √ (Tk (s23 )PJk (cos θ13 ) + Tk (s13 )PJk (cos θ23 )) . 2 k (6.7) We adopt a convenient normalization convention such that the expectation of the signal event 62 yield density in the Dalitz plot is given by the square of the matrix element multiplied by the experimental efficiency: dY = (s23 , s13 )|M(s23 , s13 )|2 , ds23 ds13 Z Y = (s23 , s13 )|M(s23 , s13 )|2 ds23 ds13 = ¯ Z |M(s23 , s13 )|2 ds23 ds13 . (6.8) (6.9) Here, because different regions of the Dalitz plot correspond to different event topologies, the efficiency, (s23 , s13 ), can be non-uniform across the Dalitz plot. The isobar-model average efficiency ¯ is defined by Eq. (6.9). As the experimental apparatus has a finite resolution, measurements of kinematic quantities will be different from their true values. In this analysis, we ignore the resolution effects in the fit. This approximation works well when the intrinsic scale of variation for component amplitudes is much larger than the experimental resolution. When this is not the case, we include the effect of neglecting the experimental resolution in the systematic error. Normalization choices for the matrix element often vary between experiments, and normalizationindependent component fractions, R |Mk |2 ds23 ds13 Fk ≡ R , |M|2 ds23 ds13 (6.10) are often reported to facilitate comparisons. Partial branching fractions, Bk ≡ Fk B, (6.11) are defined as the product of the component fractions and the total branching fraction. Interference terms fractions are given by Mk M∗l ds23 ds13 Fkl ≡ 2< R , |M|2 ds23 ds13 R (6.12) with X k Fk + X k<l Fkl = 1. (6.13) 63 P k Bk 6= B because of the interference between the It can be seen from Eq. (6.13) that, in general, components. Direct CP asymmetry is accommodated by using two different isobar models for B + and B − decays. Charge-dependent production coefficients are defined as iφ± k = ρ ρ± k ke r 1 ∓ Ak iφk ∓iδφk /2 e e , 2 (6.14) where Ak is the CP asymmetry in the k-th channel defined by Ak = 2+ ρ2− k − ρk 2− , ρ2+ k + ρk (6.15) + and δφk = φ− k − φk . A fit to the event distribution with respect to Sn + − k=1 (ρk , Ak , φk , φk ) gives a direct measurement of the CP asymmetry for each component. As before, the overall phase for each charge is arbitrary. 6.4 Resonant amplitudes We consider a contribution to the matrix element from an isolated intermediate spin-J resonance with a mass m0 and a total width Γ0 , corresponding to the decay chain B ± → K ± X, X → K ± K ∓ . It is convenient to work in the rest frame of the resonance. Denoting by s the invariant mass of the two kaons that form the resonance, their momentum is given by q q = |q| = s − 4m2K ± 2 ; the corresponding invariant phase space of the two-body decay X → K ± K ∓ is s 2q %= √ = s 1− 4m2K ± ; s 64 and the momentum of the bachelor kaon is q m4B + s2 + m4K ± − 2m2B s − 2m2B m2K ± − 2sm2K ± √ p = |p| = . 2 s The momentum of the bachelor kaon in the rest frame of the decaying B meson is √ p∗ = |p∗ | = s p= mB q m4B + s2 + m4K ± − 2m2B s − 2m2B m2K ± − 2sm2K ± 2mB . The relativistic Breit-Wigner amplitude is given by A(s, cos θ) = FJ (p∗ RB )FJ (qRX ) PJ (cos θ), 2 m0 − s − im0 (Γ0 + ∆Γ(s)) (6.16) where FJ are centrifugal barrier factors for two spinless particles in a state of angular momentum J undergoing scattering with an interaction range R with an asymptotic behavior FJ (x) ∝ xJ for x  1, and FJ (x) ∼ 1 for x  1. We use Blatt-Weisskopf barrier factors [53]: F0 (x) = 1, p √ F1 (x) = x/ 1 + x2 , F2 (x) = x2 / (x2 − 3)2 + 9x2 , etc. The expression in Eq. (6.16) is equivalent to the Zemach angular-momentum tensor formalism [54, 55]. The first three Zemach tensors are given by Z0 (p, q) = 1, Z1 (p, q) = −2p · q, Z2 (p, q) = 4 3
3(p · q)2 − |p|2 |q|2 , and in general ZJ (p, q) ∝ pJ q J PJ (cos θ). The expression in Eq. 6.16 can therefore be rewritten up to a multiplicative constant as A(s, cos θ) = F̃J (p∗ RB )F̃J (qRX ) ZJ (~ p, ~q) m20 − s − im0 (Γ0 + ∆Γ(s))  √ J s , mB (6.17) √ where the modified Blatt-Weisskopf barrier factors are: F̃0 (x) = 1, F̃1 (x) = 1/ 1 + x2 , F̃2 (x) = p √ 1/ (x2 − 3)2 + 9x2 , etc. The last term ( s/mB )J = (p∗ /p)J appears because the Zemach tensors are defined in terms of the bachelor kaon momentum in the rest frame of the resonance, while the 65 Blatt-Weisskopf barrier factor for the B ± → K ± X decay is defined in terms of the bachelor kaon momentum in the rest frame of the decaying B meson. The term ∆Γ(s) appearing in the denominator parameterizes the mass dependence of the total width and can be important near kinematic thresholds. In general, the total width of the resonance is given by Γ0 = P k Γk , ∆Γk (s) = P k ∆Γk (s), where the sums are over all of the decay channels of the resonance. The mass dependence of the partial width of the decay into two pseudoscalars, such as X → K + K − , can be parameterized as  ∆Γk (s) = Γk  % FJ2 (qRX ) − 1 , %0 FJ2 (q0 RX ) (6.18) where %0 and q0 are evaluated for s = m20 . For three-body decays of the X resonance or decays to a final state containing higher-spin particles the expression will be more complicated. Resonance partial decay widths can often be taken to be constant (∆Γk (s) = 0) if the resonance is narrow, and far away from the corresponding decay thresholds. When the resonance is close to, but below the threshold the relativistic Breit-Wigner amplitude can be replaced by the parameterization originally due to Flatté [56]. In this analysis, it is applied to the scalar f0 (980) resonance that can couple to KK and ππ final states. The amplitude is given by A(s) = 1 , m20 − s − im0 (%π gπ + %K gK ) where gπ and gK are the f0 (980) couplings to KK and ππ final states and q q %π = 2/3 1 − 4m2π± /s + 1/3 1 − 4m2π0 /s, q q %K = 1/2 1 − 4m2K ± /s + 1/2 1 − 4m2K 0 /s. (6.19) 66 6.5 Nonresonant amplitude Three-body decays of D mesons have been studied extensively. These decays are typically dominated by light-meson resonances. The nonresonant amplitude contribution is typically small and it can be to a good approximation taken to be constant over the Dalitz plot [57, 58]. This approximation is essentially a zeroth order term in the Taylor expansion of the S-wave scattering: ANR (s23 , s13 ) = ρ23 (s23 )eiφ23 (s23 ) + ρ13 (s13 )eiφ13 (s13 ) . (6.20) The phase space available to the decay of the B meson is larger by a factor of order m4B /m4D and treating the nonresonant component as a constant is not likely to be a good approximation. A better fit can be obtained by continuing the Taylor expansion to include a term linear in s23 and s13 . To ensure that the magnitudes stay positive, we exponentiate and expand the logarithm: 1 ∂ρ δs, ρ ∂s ∂φ φ(s) ≈ φ + δs, ∂s log ρ(s) ≈ log ρ + (6.21) (6.22) giving after appropriate redefinitions: ANR (s23 , s13 ) = ρ23 eiφ23 e(−α23 +iβ23 )s23 + ρ13 eiφ13 e(−α13 +iβ13 )s13 , (6.23) or for B ± → K ± K ± K ∓ , after symmetrization with respect to {1 ↔ 2},  ρeiφ  (α+iβ)s23 ANR (s23 , s13 ) = √ e + e(α+iβ)s13 . 2 (6.24) In general, we expect that the nonresonant amplitude will not have much phase variation and that its magnitude will be a slowly decreasing function of energy as the particles have less time to interact. A similar ansatz has been made in the B ± → K ± K ± K ∓ analysis by the Belle collaboration [59] 67 Figure 6.2: Functional dependence of the square of the nonresonant component used by the Belle collaboration analysis of B ± → K ± K ± K ∓ . where the nonresonant amplitude is parameterized as
ρeiφ ANR (s23 , s13 ) = √ e−αs23 + e−αs13 , 2 (6.25) and the best fit gives α = (0.121 ± 0.014) GeV−2 c4 . The nonresonant component branching fraction (Eq. (6.11)) is evaluated to be BNR = (24.0 ± 1.5 ± 1.8) × 10−6 , or (74.8 ± 3.6)% of the total B ± → K ± K ± K ∓ branching fraction. A contour plot of |ANR /A0 |2 is shown in Fig. 6.2, where A0 is the value of the amplitude evaluated at m23 = 2 GeV/c2 , m13 = 2 GeV/c2 . We have also investigated two theoretical models of the nonresonant component [60, 61], which we will briefly review here. Both models are based on the naive factorization approach of Section 2.5 applied to the operator product expansion effective Hamiltonian of Section 2.2, giving real-valued nonresonant amplitude parameterizations with no phase variation over the Dalitz plot. In the Fajfer, Pham, and Prapotnik model of Ref. [60] the leading contributions to the nonreso- 68 Figure 6.3: Functional dependence of the square of the nonresonant component parameterized according to the model of Fajfer, Pham, and Prapotnik [60]. nant amplitude is ANR (s23 , s13 ) ∝ {w+ (s23 , s13 )(m2B − m2K − s23 )/2 + w− (s23 , s13 )(s23 + 2s13 − m2B − 3m2K )/2} + {1 ↔ 2}, where w± are form factors. The form factors, calculated by applying the Heavy Quark Chiral Perturbation Theory (HQChPT) to both the weak and strong interaction vertices, receive contributions from three-body contact terms and B ∗ pole diagrams. The nonresonant component branching fraction is evaluated to be BNR = 9 × 10−6 , significantly smaller than the value reported by the Belle collaboration. A contour plot of |ANR /A0 |2 is shown in Fig. 6.3, where A0 is the value of the amplitude evaluated at m23 = 2 GeV/c2 , m13 = 2 GeV/c2 . The approach of Ref. [60] has been criticized by Cheng and Yang in Ref. [61]. For three-body B decays the momentum of the final state mesons is comparable to or larger than the chiral symmetry 69 Figure 6.4: Functional dependence of the square of the nonresonant component parameterized according to the model of Cheng and Yang [61]. breaking scale, Λχ ∼ 830 MeV, over much of the Dalitz plot, and in the calculation of form factors w± in Ref [60] the HQChPT theory may have been applied twice beyond the range of its validity. The problem is made manifest by a numeric example; when applied to the B ± → K ± π ± K ∓ decay, the approach of Ref. [60] gives a branching fraction prediction that is much larger than the current experimental upper limit. In the alternative model proposed by Cheng and Yang, the HQChPT is applied to the strong vertex only, and the invariant mass dependence of the weak vertex is parameterized using form factors. The functional dependence of form factors is not known a priori; Fig. 6.4 shows a contour plot of the square of the amplitude parameterized according to the model of Cheng and Yang using an ad hoc parameterization of the form factors suggested in the paper. While the amplitude shape is qualitatively similar to that used by the Belle collaboration [59], this choice of form factors leads to an estimate of BNR /B(B ± → K ± K ± K ∓ ) = 3% for the nonresonant component fraction — much smaller than the value reported by the Belle collaboration. 70 After surveying the theoretical models and comparing them with the experimental results from the Belle collaboration, we conclude that the proposed theoretical models are in disagreement with the data and no successful physics-based parameterization of the nonresonant component in charmless three-body B decays exists to date. In this analysis, we therefore follow the approach used by the Belle collaboration and use an ad hoc parameterization of the nonresonant component, which is motivated by an expansion of the amplitude beyond the usual constant term. 6.6 Discrete ambiguities In certain cases, the production coefficients in an isobar model fit can only be determined up to a discrete ambiguity. As an example, we consider a region of the Dalitz plot where a narrow scalar resonance is interfering with a nonresonant component. The matrix element is modeled as M ∝ ρNR eiφNR + ρeiφ 1 , m20 − s − im0 Γ0 where the nonresonant component is taken to be constant across the narrow resonance and the resonant amplitude is parameterized as a relativistic Breit-Wigner for which the possible mass dependence of the width has been neglected. Introducing the resonant phase shift δ defined by tan δ ≡ m0 Γ0 , m20 − s after appropriate redefinitions M ∝ 1 + ρeiφ sin δeiδ , where ρeiφ is the resonance production coefficient relative to the nonresonant component. The density of observed signal events is proportional to the square of the amplitude: |M|2 ∝ 1 + (ρ2 − 2ρ sin φ) sin2 δ + 2ρ cos φ sin δ cos δ. (6.26) 71 Identical event distributions will be observed for two different production coefficients, ρ1 eiφ1 and ρ2 eiφ2 , as long as the following system of equations is satisfied: ρ21 − 2ρ1 sin φ1 = ρ22 − 2ρ2 sin φ2 , (6.27) ρ1 cos φ1 = ρ2 cos φ2 . Given one solution with the production coefficient ρ1 eiφ1 , the conjugate solution is given by ρ2 = p (ρ1 − 2 sin φ1 )2 + 4 cos2 φ1 , (6.28) cos φ2 = ρ1 /ρ2 cos φ1 . Fig. 6.5 is a graphical illustration of this type of degeneracy for the case of ρ1 eiφ1 = 2e−iπ/6 , √ ρ2 eiφ2 = 2 3eiπ/3 . While the circles traced in the Argand diagram are different for the two generated amplitudes, the square of the amplitude as a function of the phase shift δ is the same in both cases. Another (extreme) example of this degeneracy is the fact that a flat density generated when no resonance is present, ρ1 eiφ1 = 0, can also be a result of the interference between the nonresonant component and a resonance with ρ2 eiφ2 = 2eiπ/2 . The condition for the resonance to be clearly observed is ρ1,2  1, in which case Eq. (6.28) simplifies to ρ2 ≈ ρ1 − 2 sin φ1 , (6.29) φ2 ≈ −φ1 . We finally note that this type of degeneracy is lifted if the resonance has non-zero spin, if it interferes with another non-constant amplitude in the orthogonal channel, or if the nonresonant amplitude has significant variation across the resonance. Even when the exact degeneracy is lifted by one of the mechanisms mentioned above, finite experimental resolution and limited statistics can conspire to produce approximate discrete ambiguities of this type. 72 Figure 6.5: Discrete ambiguity example for the matrix element of Eq. (6.26). The Argand plot diagram (top), and amplitude squared as a function of the phase shift√(bottom) for two conjugate solutions with production coefficients, ρ1 eiφ1 = 2e−iπ/6 and ρ2 eiφ2 = 2 3eiπ/3 . 73 6.7 Binned maximum likelihood fit In this analysis we use the method of binned maximum likelihood for estimating the parameters of the isobar model. The loss of statistical precision due to binning is minimal if the bin sizes are smaller than either the typical experimental resolution or the typical size of the Dalitz plot features. The unique advantages of a binned fit approach are: • Computing the expected number of background events in a binned approach is much easier than constructing functional parameterizations of backgrounds whose rates and composition vary non-trivially over the Dalitz plot. • In a binned fit, there is a close connection between the minimized value and goodness-of-fit, which is lacking for unbinned maximum likelihood fits. • With a large number of events, computing unbinned likelihood becomes computationally expensive. The probability (likelihood) of the observed distribution is given by the product of Poisson likelihoods for individual bins: L= N Y µni e−µi i i=1 ni ! , where µi is the expected number of events in the i-th bin as a function of the estimated parameters and ni is the number of events observed in the i-th bin. Parameter estimates are obtained by maximizing the likelihood; equivalently, we minimize the log of the likelihood-ratio statistic: χ2LLR /2 = N X i=1 µi − ni + ni log ni . µi The variable χ2LLR is asymptotically distributed as a χ2 variate for µi , ni  1 with a number of degrees of freedom equal to the number of Dalitz plot bins minus the number of evaluated model parameters. The properties of the χ2LLR for bins with small numbers of events are discussed in [62]. 74 For CP asymmetry fits the statistic is extended to: χ2LLR /2 = N X + + µ+ i − ni + ni log i=1 N X n− n+ − − − i i + µ − n + n log i i i −. µ+ µ i i i=1 (6.30) We choose a non-uniform binning designed to capture the structure of the observed distribution of events. The expected number of events in each bin µi is the sum of signal and background contributions: µi = si + bi + qi , where the expected number of properly reconstructed signal events si is evaluated by integrating the yield density of Eq. (6.8) over the bin, Z si = (s23 , s13 )|M(s23 , s13 )|2 ds23 ds13 ; i the expected contribution arising from non-signal B decays bi is evaluated using simulated samples of B decays as described in Chapter 5; and the expected contribution of qq backgrounds qi is evaluated using extrapolation from the mES sideband: SB qi = Rqq (nSB i − bi ), where nSB is the number of events observed in the sideband bin, bSB is the expected B background i i contribution in the sideband bin, and Rqq is the extrapolation coefficient described in Sec. 4.3. In order to have a well-defined Poisson likelihood, we require the total expectation µi to be non-negative in each bin. We have neglected the contributions from misreconstructed signal events, which depend nontrivially on the signal model and the misreconstruction mechanism and are difficult to compute. Fortunately, their contributions are small and are largely absorbed in the q-term because of the similar mES shapes for misreconstructed signal events and qq continuum backgrounds. We have also neglected intermode cross-feeds between three-body charmless final states due to particle misidenti- 75 fication because they are small for the Cabibbo-favored B ± → K ± K ± K ∓ mode. As an important technical note, in the isobar model, si = X ρk ρl ei(φk −φl ) Z (s23 , s13 )Ak A∗l ds23 ds13 . (6.31) i k,l When fitting for parameters ρ and φ only, the bin integrals in Eq. (6.31) need only be computed once; at each iteration of the fitting process the expectations si can be computed by simply recalculating the sum. This is not the case when fitting for amplitude shape parameters; the expectations si in that case need to be recalculated at each minimization iteration by numeric integration, which is computationally expensive. Approximations and iterative fitting techniques can be used to mitigate against this problem. The overall branching fraction is given by B= where n = PN i=1 ni , b = PN i=1 bi , q = n−b−q , ¯ PN (6.32) ¯ is the average efficiency of Eq. (6.9). i=1 qi , and  The overall CP asymmetry is given by A= where n± = n − − n+ , n−b−q (6.33) PN ± i=1 ni and an assumption has been made that there is no asymmetry for the back- grounds. 6.8 Statistical errors and error propagation We briefly discuss the evaluation of statistical errors arising from the intrinsic randomness of the observed processes. For efficiency determination, the number of events n selected from a sample of N events and satisfying a certain selection criterion is distributed according to a binomial distribution with probability equal to the selection efficiency . For a binomial distribution, an unbiased estimator of the 76 efficiency is ˆ = n/N and its variance is σˆ2 = (1−)/N . The efficiency and its statistical uncertainty are therefore estimated as n = ± N r n/N (1 − n/N ) . N When the distribution of an estimator is not readily computed, we use error propagation. For a function X(θi ), which depends on a vector of parameters θi , evaluated with an uncertainty given by a covariance matrix Σkl , assuming that the parameter errors are small we can write an estimator: X̂(θi ) = X(θ̂i ), and its variance: 2 σX̂ = X ∂X ∂X kl ∂θk ∂θl Σkl . (6.34) We consider, as an example, a CP asymmetry estimator: Â(n+ , n− ) ≡ n− − n+ , N (6.35) where n± are the numbers of selected B ± events, respectively, and N = n+ + n− . The derivatives are given by ∂  2n∓ =± 2 . ∂n± N (6.36) The covariance matrix assuming independent Poisson processes for n± is   n+ Σ+− =   0  0  ,  n− (6.37) 4n+ n− 1 − Â2 = . 3 N N (6.38) giving 2 σ = 77 The asymmetry and its statistical uncertainty are therefore estimated as A= 1 n− − n+ ±√ N N √ 4n+ n− . N (6.39) In our maximum likelihood fits we use the MINUIT [63] package for numeric minimization of χ2LLR . The covariance matrix of the parameter estimates is asymptotically related to the Hessian of the log-likelihood evaluated at the minimum: −1 Σkl = Hkl = 1 ∂ 2 χ2LLR . 2 ∂θk ∂θl Derived quantities are estimated via error propagation of Eq. (6.34). (6.40) 78 Chapter 7 B ± → K ±K ±K ∓ fit In this chapter we apply the analysis formalism developed in Chapter 6 to the data sample selected in the B ± → K ± K ± K ∓ final state. We perform an isobar-model Dalitz plot fit in Section 7.1 and then a partial-wave analysis in Section 7.2. 7.1 Isobar-model fit The Dalitz plot of the 1769 B + and 1730 B − candidates selected in the signal region is shown in Fig. 7.1. We divide it into 292 non-uniform rectangular bins as shown in Fig. 7.2 and perform a binned maximum likelihood fit using the isobar-model formalism described in Chapter 6. The efficiency is parameterized as a two-dimensional histogram (Fig. 4.12); the expected B backgrounds are determined as described in Chapter 5; and the expected qq backgrounds are extrapolated from the mES sideband as described in Section 4.3. The isobar-model matrix element, parameterized as in Eq. (6.7), includes the following components: • B ± → K ± φ(1020) The contribution from the vector φ(1020) resonance is clearly visible in the Dalitz plot of Fig. 7.1. We model the amplitude using the relativistic Breit-Wigner amplitude parameterization of Eq. (6.16). Because mK ± + mφ  mB , the B decay is far from the kinematic threshold and we approximate the B vertex barrier factor as FJ (p∗ RB ) ≈ 1. The interac- 79 Figure 7.1: The Dalitz plot of the 1769 B + and 1730 B − candidates selected in the signal mES region. Figure 7.2: Non-uniform binning of the Dalitz plot used in the isobar-model fit. Finer binning is used around the φ, D0 , and χc0 resonances, and for smaller values of s23 . 80 tion range Rφ has not been measured, and for the Blatt-Weisskopf barrier factor FJ (qRφ ) we use Rφ = 4.0 GeV−1 , motivated by the recent FOCUS collaboration measurement of −1 RK ∗ = 3.96 ± 0.54+1.31 [64]. The φ(1020) is near the KK threshold and we use −0.90 GeV a mass-dependent width with ∆Γ(s) = ∆Γ1 (s) + ∆Γ2 (s), where Γ1 = Γ0 B(φ → K + K − ), Γ2 = Γ0 B(φ → K 0 K 0 ), and ∆Γ1,2 are given by Eq. 6.18. • B ± → K ± f0 (980). A large B ± → K ± f0 (980) signal measured in B ± → K ± π ± π ∓ [19, 28], and a recent measurement of the ratio of the f0 (980) coupling constants to KK and ππ [65], motivate us to include an f0 (980) contribution using the amplitude parameterization of Eq. (6.19). Define IK /Iπ to be the ratio of the integrals of the square of the f0 (980) amplitude given by Eq. (6.19) over the B → KKK and B → Kππ Dalitz plots. The partial branching fractions for B ± → K ± f0 (980) measured in the K ± K ± K ∓ and K ± π ± π ∓ final states are then related by the ratio R≡ B(B ± → K ± f0 (980), f0 (980) → K + K − )) 3 IK gK = , ± ± + − B(B → K f0 (980), f0 (980) → π π ) 4 Iπ gπ (7.1) where 3/4 is an isospin factor. Experimental measurements of gK /gπ [65–68] have significant uncertainties. The contribution of the f0 (980) resonance was taken to be negligible in the Belle collaboration analysis of B ± → K ± K ± K ∓ [59] motivated by the small gK /gπ value reported by the E791 collaboration, based on an analysis of the Ds± → π ± π ± π ∓ decay with no direct observation of the f0 (980) → K K̄ coupling [68]. We consider the more recent BES collaboration measurement from a joint analysis of J/ψ → φπ + π − and J/ψ → φK + K − to be superior, and in the following adopt their values: m0 = 0.965 ± 0.008 ± 0.006, gπ = 0.165±0.010±0.015 GeV/c2 , and gK /gπ = 4.21±0.25±0.21 [65]. For this choice of parameters, we compute IK /Iπ = 0.29, and R = 0.92 ± 0.07 for the right-hand side of Eq. (7.1), where we have combined the statistical and systematic errors in quadrature. This suggests that the B ± → K ± f0 (980) contribution to B ± → K ± K ± K ∓ may be large. • B ± → K ± X0 (1550). 81 Following Ref. [19], we introduce a broad scalar resonance, whose interference with a slowly varying nonresonant component is used to describe the rapid decrease in event density around m(K + K − ) = 1.6 GeV/c2 . Evidence of a possible resonant S-wave contribution in this region has been reported previously [69, 70], however its attribution is uncertain: the f0 (1370) and f0 (1500) resonances are known to couple to ππ more strongly than to KK [14] and possible interpretations in terms of those states [13] must account for the fact that no strong B ± → K ± f0 (1370) or B ± → K ± f0 (1500) signal is observed in B ± → K ± π ± π ∓ [19, 28]; the contribution of the f0 (1710) resonance is included in the fit as a separate component and is found to be small. In the following, we designate the broad scalar resonance X0 (1550) and determine its mass and width directly from the fit. The amplitude is modeled using the relativistic Breit-Wigner amplitude parameterization of Eq. (6.16). As the X0 (1550) resonance is speculative and its decay modes are unknown, we take its width to be constant. • B ± → K ± f0 (1710). We include a potential contribution from the f0 (1710) resonance, which has been measured to couple more strongly to KK than to ππ [71], and could potentially contribute to B ± → K ± K ± K ∓ without being observed in B ± → K ± π ± π ∓ . The amplitude is modeled using the relativistic Breit-Wigner amplitude parameterization of Eq. (6.16). As the decay modes of the f0 (1710) resonance are not well established, we take its width to be constant. • B ± → K + χc0 The contribution from the χc0 resonance arising from the rescattering diagrams shown in Fig. 2.8 is clearly visible in the Dalitz plot of Fig. 7.1. The amplitude is modeled using the relativistic Breit-Wigner amplitude parameterization of Eq. (6.16). The χc0 resonance has many decay modes. For simplicity we take its width to be constant. • Nonresonant B ± → K ± K ± K ∓ . In Section 6.5 we have surveyed the two published theoretical models [60, 61] for the B ± → K ± K ± K ∓ nonresonant component and concluded that both are in disagreement with the 82 Figure 7.3: Scan of the χ2LLR function in the (ρχc0 , φχc0 ) plane, showing the two solutions from the best-fit doublet. experimental data from the Belle collaboration [19]. We therefore adopt an ad hoc parameterization based on the expansion of Eq. (6.24). A fit to the m23 > 2 GeV/c2 region dominated by the nonresonant component gives α = 0.140 ± 0.019, β = −0.02 ± 0.06, consistent with no phase variation. In the following, we fix β = 0 and thus adopt the same parameterization as used by Belle [19]. We fit for the magnitudes and phases of the decay coefficients, the mass and width of the X0 (1550), and the nonresonant component shape parameter α. As the overall complex phase of the isobar-model amplitude is arbitrary, we fix the phase of the nonresonant contribution to zero, leaving 14 free parameters in the fit. The number of degrees of freedom is 292 − 14 = 278. We perform multiple minimizations with different starting parameter values and cluster the fits using a Euclidean metric in the parameter space, with distance measured in units of expected uncertainty on the fit parameters. We find multiple solutions clustered in doublets, where the solutions within each doublet are very similar, except for the magnitude and phase of the χc0 83 Figure 7.4: Mass projection of the fit without (left) and with (right) the f0 (980) component near the K + K − threshold. The histogram shows the total expected number of events in each bin, with BB and qq background contributions shown in dark and light gray, respectively. production coefficient. The twofold ambiguity arises from the interference between the narrow χc0 and the nonresonant component that is approximately constant across the resonance, as discussed in Section 6.6. The highest-likelihood doublet has χ2LLR = (346.6, 352.0); the second best doublet has χ2LLR = (362.4, 368.7). A two-dimensional scan of χ2LLR in the (ρχc0 , φχc0 ) plane while keeping all the other parameters fixed is shown in Fig. 7.3 for the highest-likelihood doublet. The least significant components are the f0 (980) and the f0 (1710). The omission of the f0 (980) component from the fit degrades the best fit from χ2LLR = 346.6 to 363.9 and the data distribution near the K + K − threshold is not described adequately as shown in Fig. 7.4. The omission of the f0 (1710) degrades the fit to χ2LLR = 360.7. The invariant mass projections of the best fit are shown in Fig. 7.5. The fit gives α = 0.152 ± 0.011 GeV−2 c4 , m0 (X0 ) = 1.539 ± 0.020 GeV/c2 , and Γ0 (X0 ) = 0.257 ± 0.033 GeV/c2 . The fitted values of the shape parameter α and the resonance mass are consistent with the values in Ref. [19], but our preferred value for the width is significantly larger than Γ0 = 0.14 ± 0.02 GeV/c2 , reported by the Belle collaboration [19]. The isobar-model weighted signal efficiency of Eq. (6.9) is ε̄ = 0.282±0.011, where the systematic error is evaluated in Chapter 8. Using Eqs. (6.32, 6.33), we calculate the total branching fraction and asymmetry to be B(B ± → K ± K ± K ∓ ) = (35.2 ± 0.9) × 10−6 and A(B ± → K ± K ± K ∓ ) = 84 Figure 7.5: Mass projections of the best fit. The histogram shows the total expected number of events in each bin, with BB and qq background contributions shown in dark and light gray, respectively. The inset shows a fit projection near the φ resonance. 85 Table 7.1: Fit fractions of Eq. (6.10, 6.12) and phase differences of the best fit. In the top part of the table, the diagonal elements set in boldface correspond to the component fractions (Eq. (6.10)) and the elements above the diagonal correspond to the fractions of interference terms (Eq. (6.12)). The bottom part of the table gives the antisymmetric matrix of phase differences, φij = φj − φi . Fij φ f0 (980) X0 (1550) f0 (1710) χc0 NR φij φ f0 (980) X0 (1550) f0 (1710) χc0 NR φ 11.8 ± 0.9 f0 (980) −0.94 ± 0.18 19 ± 7 X0 (1550) −1.71 ± 0.36 53 ± 12 121 ± 19 f0 (1710) 0.01 ± 0.10 −4.5 ± 2.9 −30 ± 11 4.8 ± 2.7 χc0 0.11 ± 0.02 −0.91 ± 0.19 −1.10 ± 0.31 −0.10 ± 0.07 3.1 ± 0.6 φ 0 2.51 ± 0.17 1.70 ± 0.27 −2.71 ± 0.35 −2.27 ± 0.35 2.99 ± 0.20 f0 (980) −2.51 ± 0.17 0 −0.81 ± 0.25 1.07 ± 0.36 1.50 ± 0.33 0.48 ± 0.16 X0 (1550) −1.70 ± 0.27 0.81 ± 0.25 0 1.88 ± 0.19 2.32 ± 0.22 1.29 ± 0.10 f0 (1710) 2.71 ± 0.35 −1.07 ± 0.36 −1.88 ± 0.19 0 0.44 ± 0.30 −0.59 ± 0.25 χc0 2.27 ± 0.35 −1.50 ± 0.33 −2.32 ± 0.22 −0.44 ± 0.30 0 −1.02 ± 0.23 NR 3.54 ± 0.38 −85 ± 21 −140 ± 26 4±6 3.87 ± 0.37 141 ± 16 NR −2.99 ± 0.20 −0.48 ± 0.16 −1.29 ± 0.10 0.59 ± 0.25 1.02 ± 0.23 0 Table 7.2: Means and standard deviations of the distributions of the residuals obtained in the fit bias study. φ f0 (980) ρ ρ φ ρ (-0.01,1.00) (0.08,1.06) (-0.08,1.04) (0.08,1.06) f0 (1710) χc0 ρ φ ρ φ (0.11,1.03) (-0.08,1.06) (0.12,0.95) (-0.11,1.16) X0 (1550) φ m0 (-0.11,1.10) (-0.04,1.32) NR ρ φ (0.06,1.03) (-0.10,1.06) Γ0 (0.06,1.09) α (0.17,1.02) (−1.7 ± 2.6)%. The fit fractions of Eqs. (6.10, 6.12) and phase differences for the leading solution are shown in Table 7.1. In the top part of the table, the diagonal elements set in boldface correspond to the component fractions (Eq. (6.10)) and the elements above the diagonal correspond to the fractions of interference terms (Eq. (6.12)). Because of the large negative interference, in the scalar sector the sum of the diagonal fractions is substantially larger than 100%. The bottom part of the table gives the antisymmetric matrix of phase differences, φij = φj − φi . For the second solution in the leading doublet, the χc0 fraction is increased to 6.0 ± 0.7%, and the fraction of the term describing the interference with the nonresonant component is reduced to 1.4 ± 1.0%. The χc0 phase changes such that φ(χc0 ) − φ(NR) = 0.29 ± 0.20. 86 Figure 7.6: The distributions of the residuals obtained in the fit bias study. 87 Table 7.3: Fit results for CP violation parameters and symmetric 90%-confidence-level intervals for CP asymmetry. φ f0 (980) X0 (1550) f0 (1710) χIc0 χII c0 NR A 0.00 ± 0.08 −0.31 ± 0.25 −0.04 ± 0.07 0.0 ± 0.5 0.19 ± 0.18 −0.03 ± 0.28 0.02 ± 0.08 (Amin , Amax )90% (−0.14, 0.14) (−0.72, 0.12) (−0.17, 0.09) (−0.66, 0.74) (−0.09, 0.47) (−0.14, 0.18) φ− − φ+ −0.67 ± 0.28 −0.20 ± 0.16 0.02 ± 0.15 −0.07 ± 0.38 0.7 ± 0.5 −0.4 ± 1.3 - To check for possible fit bias, we fit 400 Monte Carlo–simulated data samples generated according to the nominal model of the best fit, and compute the residuals of the fit parameters defined as the difference between the value returned by the fit and the nominal value divided by the fit error. The distributions of the residuals are shown in Fig 7.6. In the limit or large statistics and parabolic Gaussian errors, the residuals are expected to have a normal distribution N (0, 1). Table 7.2 gives the means and the standard deviations, (µ, σ), of the distributions of the residuals, which are all close to their nominal values. To look for possible direct CP violation, we extend the isobar model by defining charge-dependent production coefficients as in Eq. (6.14) and use the log-likelihood of Eq. (6.30). As there is freedom to choose the overall phase for each component, we fixed the phases of the nonresonant contribution to zero for both charges. The results of the fits are summarized in Table 7.3. For the χc0 component we give the results for both solutions from the leading doublet. Table 7.3 also gives symmetric 90%-confidence-level intervals around the nominal CP asymmetry value estimated by fitting Monte Carlo–simulated samples generated according to the parameterized model of the nominal asymmetry fit. All asymmetries A and phase differences φ− − φ+ are consistent with zero. 7.2 Partial-wave analysis We further study the nature of the dominant S-wave component by considering the interference between the low-mass and the high-mass scattering amplitudes in the region m23 ∈ (1.1, 1.8) GeV/c2 , 88 m13 > 2 GeV/c2 . The matrix element is modeled as M= ρS (s23 ) iφS (s23 ) ρNR −αs13 √ e + √ e , 2 2 (7.2) where ρS and φS are taken to be constant within each bin of the s23 variable and the nonresonant amplitude parameterization is taken from the fit to the high-mass region. The partial-wave expansion truncated at the S-wave describes the data adequately; the magnitude of the S-wave in each bin is readily determined. Because of the mass dependence of the nonresonant component, the phase of the S-wave can also be determined, albeit with a sign ambiguity and rather large errors for bins with a small number of entries or small net variation of the nonresonant component. The results are shown in Fig. 7.7, with the S-wave component of the isobar-model fit overlaid for comparison. Continuity requirements allow us to identify two possible solutions for the phase; the solution labeled by black squares is consistent with a rapid counterclockwise motion in the Argand plot around m(K + K − ) = 1.55 GeV/c2 , which is accomodated in the isobar model as the contribution of the X0 (1550). Isospin symmetry relates the measurements in B ± → K ± K ± K ∓ and B 0 → K + K − KS0 [72]. Our results for the K + K − S-wave can therefore be used to estimate a potentially significant source of uncertainty in the measurements of sin2β in B 0 → φ(1020)KS0 [73, 74] due to the contribution of a CP -even S-wave amplitude. We perform a partial-wave analysis in the region m23 (K + K − ) ∈ (1.013, 1.027) GeV/c2 , modeling the matrix element as a sum of an S-wave and a P -wave: ρP (s23 ) iφP (s23 ) ρS e cos θ13 , M= √ + √ 2 2 (7.3) where the S-wave is taken to be constant over the small region of phase space considered. The fit results for the P -wave are shown in Fig. 7.8. Because the phase of the S-wave is taken to be constant, there is a twofold ambiguity in the sign of the P -wave phase. We assign the data points in Fig. 7.8 to two solutions, one labeled by black squares and another labeled by white squares, using continuity requirements. The superimposed curve corresponds to a fit with a Breit-Wigner 89 Figure 7.7: The results of the partial-wave analysis of the K + K − S-wave: (a) magnitude squared, (b) phase. The discrete ambiguities in the determination of the phase give rise to two possible solutions labeled by black and white squares. The curves correspond to the S-wave component from the isobar-model fit. The inset shows the evidence of a threshold enhancement from the fits of the S-wave in the vicinity of the K + K − threshold and in the region around the φ(1020) resonance. 90 Figure 7.8: The results of the partial-wave analysis in the φ(1020) region for the P -wave: (a) magnitude squared, (b) phase. The discrete ambiguities in the determination of the phase give rise to two possible solutions labeled by black and white squares. The curve corresponds to a BreitWigner fit of the φ(1020) resonance. parameterization of the φ resonance. For the S-wave we get ρ2S = (3.4 ± 2.5) × 102 GeV−4 c8 and compute its fraction in this region using Eq. (6.10) to be (9 ± 6)%. We also consider the region 2mK + < m(K + K − ) < 1.006 GeV/c2 , in the immediate vicinity of the K + K − threshold. The contribution of the φ(1020) resonance tail in this region is suppressed by the centrifugal barrier and is estimated to be smaller than 10%. We fit ρ2S = (6.1 ± 1.6) × 102 GeV−4 c8 for the magnitude of the S-wave in this region. The fits in the vicinity of the K + K − threshold and in the region around the φ(1020) resonance indicate a threshold enhancement of the S-wave, which is accommodated in the isobar model by the contribution of the f0 (980) resonance as shown in the inset of Fig. 7.7. 91 Chapter 8 Systematic Uncertainties In this chapter we evaluate the systematic uncertainties for the overall braching fraction and asymmetry, and the fit fractions, phase differences, and direct CP violation parameters of the isobar-model fit. 8.1 Sources of systematic uncertainties We use statistical inference to estimate the parameters of the underlying physics model from the observable quantities meausured by the subsystems of the BABAR detector. Our knowledge of the parameters of the signal model, our estimates of the background processes and our parameterization of the detector response have by necessity a limited precision, giving rise to systematic uncertainties on the values of the measured parameters. We consider the following sources of systematic uncertainty: • Tracking efficiency. We determine the tracking efficiency using a Monte Carlo–simulation of the detector response adjusted by correction factors derived from data control samples. The correction factors have been evaluated by the BABAR Tracking Task Force [47] as functions of the measured track momentum vector and the event track multiplicity with an associated overall uncertainty of 0.8% per track. A conservative assumption of fully correlated tracking uncertainties gives a tracking efficiency uncertainty of 2.4%. 92 • Particle identification efficiency. We model the performance of the particle identification algorithms using a Monte Carlo simulation of the detector response adjusted by correction factors derived from data control samples. The correction factors have been evaluated by the BABAR Particle ID Group as functions of the measured track momentum. We determine the overall particle identifcation efficiency uncertainty independently using data control samples in Section 8.2. • Charge asymmetry in tracking and particle identification. The overall efficiency and its associated uncertainty cancel out in the determination of CP asymmetries. In this analysis, we have assumed equal selection efficiencies for both charges. The tracking and particle identification efficiency can however be charge dependent, e.g., due to the difference in the strong-interaction cross-sections for charged kaons interacting with the detector material. Charge-dependent efficiencies can be defined as ± ≡ (1 ∓ A ), where  is the average efficiency, and A is the efficiency charge asymmetry. A measurement of CP asymmetry is given by: A= N− − − N+ + N+ + + N− − where N± be the number of B ± decays produced, and A0 is the CP asymmetry calculated assuming A = 0. For A0 A  1, A ≈ A0 + A . We evaluate possible efficiency asymmetry using control samples in Section 8.2 and assign an appropriate systematic error. • Selection efficiency. The efficiency of the applied selection is estimated using Monte Carlo–simulated signal events. 93 We use a control sample to check the selection efficiency of the ∆E requirement for data and simulated events. The systematic uncertainties due to the efficiency parameterization are evaluated in Section 8.3. • Background estimates. The systematic uncertainties in evaluation of the backgrounds due to non-signal B decays and qq continuum events are discussed in Section 8.4. • Resonance lineshapes. In this analysis, we have neglected the uncertainties on the Dalitz plot coordinates of the selected events due to detector resolution. This approximation is not expected to affect the components of the isobar model that have an intrinsic scale of variation larger than the detector resolution, but can be important for the narrow φ and χc0 resonances. The fit fractions, phase differences, and CP asymetries of the individual components of the isobar model, while not being sensitive to the overall efficiency, do depend on the component amplitude parameterization used in the fit. In the case of the X0 (1550) and the nonresonant component, the amplitude shape parameters are determined directly from the fit and the associated uncertainties are included in the statistical error. The parameters of the f0 (980), φ(1020), f0 (1710), and χc0 resonances are fixed in the fit and the uncertainty in their parameters is included in the systematic error. The systematic uncertainties due to resolution effects and uncertainties in resonance parameters are evaluated in Section 8.5. 8.2 Control sample studies B ± → Dπ ± , D → K ± π ∓ decays are produced via the charmed tree cascade of Fig. 8.1 and are observed in the K ± π ± π ∓ final state. We perform a dedicated selection by considering triplets of charged tracks with net unit charge. One of the same-charge tracks is assigned a kaon hypothesis; the two other tracks are each assigned a pion hypothesis. The candidates with m(K ± π ∓ ) ∈ 94 Figure 8.1: Charmed tree cascade. Figure 8.2: The m(K ± π ∓ ), mES , ∆E, and neural network output distributions for the selected B ± → Dπ ± control sample events. 95 Figure 8.3: The ∆E distribution for B ± → Dπ ± , D → K ± π ∓ control sample events in data (left); for simulated events (center); and a fit to the data using shifted histogram obtained from Monte Carlo simulation to model the signal peak (right). (1.80, 1.90) GeV/c2 are retained. No particle identification requirements are applied. The distributions of the m(K ± π ∓ ) invariant mass, mES , ∆E, and the output of the neural network in Fig. 8.2 all show a clear signal peak. To further improve the signal purity, we require m(K ± π ∓ ) ∈ (1.85, 1.88) GeV/c2 , mES > 5.27 GeV/c2 , |∆E| < 50 MeV, the neural network output greater than 0.4, and remove the contribution from B ± → K ± J/ψ decays by vetoing the region m(π + π − ) ∈ (3.08, 3.12) GeV/c2 . The final sample purity is estimated to be 95%. The same selection is repeated for Monte Carlo–simulated B ± → Dπ ± events. Backgrounds are estimated and subtracted by extrapolation from the mES sideband. After background subtraction, the number of B ± candidates is n± = 6130/6277; the number of kaon candidates that satisfy the kaon identification algorithm is nK ± = 5316/5461; and the number of pion candidates that fail the kaon identification is nπ± = 11678/11663. We use the B ± → Dπ ± control sample to compare the shape of the ∆E distribution between data and simulated events (Fig. 8.3). We fit both distributions by a sum of a Gaussian, representing the signal, and a first-order polynomial, representing the background. In both cases the Gaussian does not describe the low side of the peak adequately. The width of the peak is similar in both cases, but the peak position for data is shifted by about −5 MeV. The fit is improved significantly when we use a histogram of the ∆E distribution of simulated events shifted by −4 MeV to describe the signal peak. Using simulated events we evaluate that a −5 MeV shift in the ∆E distribution would reduce the selection efficiency for the selection requirement |∆E| < 40 MeV by 1%. 96 Figure 8.4: The m(π + π − ), mES , ∆E, and neural network output distributions for the selected B ± → K ± J/ψ control sample events. 97 Table 8.1: Kaon and pion identification efficiencies and their ratios in control samples for data and Monte Carlo–simulated events. Mode B → DK + B ± → K + J/ψ ± K 0.869 ± 0.003 0.899 ± 0.003 MC K 0.874 0.904 K /MC K 0.994 ± 0.003 0.994 ± 0.004 π 0.941 ± 0.002 - MC π 0.950 - π /MC π 0.991 ± 0.002 - Table 8.2: Tracking and particle identification efficiency asymmetries in data control samples. Mode B ± → DK + B ± → K + J/ψ A 0.01 ± 0.01 0.00 ± 0.01 AK 0.001 ± 0.004 0.001 ± 0.004 Aπ 0.000 ± 0.002 - B ± → K ± J/ψ , J/ψ → `+ `− decays form another high-statistics control sample. We perform a dedicated selection by considering triplets of charged tracks with net unit charge. One of the samecharge tracks is assigned a kaon hypothesis, the two other tracks are each assigned a pion hypothesis. The candidates with m(π + π − ) ∈ (3.05, 3.15) GeV/c2 are retained. No particle identification requirements are applied. The distributions of the m(π + π − ) invariant mass, mES , ∆E and the output of the neural network in Fig. 8.4 all show a clear signal peak. To further improve the signal purity we require m(π + π − ) ∈ (3.08, 3.12) GeV/c2 , mES > 5.27 GeV/c2 , |∆E| < 50 MeV, the neural network output greater than 0.4, and remove the contribution from B ± → DK ± decays by vetoing the region m(K ± π ∓ ) ∈ (1.8, 1.9) GeV/c2 . The final sample purity is estimated to be 94%. The same selection is repeated for Monte Carlo–simulated B ± → K ± J/ψ events. Backgrounds are estimated and subtracted by extrapolating the event counts from the mES sideband. After background subtraction, the number of B ± candidates is n± = 3612/3621 and the number of kaon candidates that satisfy the kaon identification algorithm is nK ± = 3245/3259. Table 8.1 shows the kaon and pion identification efficiencies and their ratios in the control samples for data and simulated MC events. Table 8.2 shows the tracking and particle identification efficiency asymmetries in data control samples. Based on the control sample studies described above, we assign a 0.5% systematic uncertainty per kaon and 1.0% per pion due to the uncertainty in the particle identification efficiency. For final states with an odd number of kaons, we assign a combined 1.5% systematic error due to possible 98 Table 8.3: Estimates of the systematic uncertainties for the fit fractions (upper triangle and diagonal), phase differences (lower triangle), and CP violation parameters due to the uncertainty in the signal efficiency parameterization. φ f0 (980) X0 (1550) f0 (1710) χc0 NR A φ+ − φ− φ 0.1% 0.01 0.01 0.01 0.02 0.01 0.0% 0.01 f0 (980) 0.0% 0.4% 0.02 0.02 0.02 0.01 0.5% 0.01 X0 (1550) 0.0% 0.7% 0.9% 0.01 0.01 0.01 0.1% 0.00 f0 (1710) 0.0% 0.1% 0.4% 0.1% 0.01 0.01 0.5% 0.01 χc0 0.0% 0.0% 0.0% 0.0% 0.0% 0.01 0.3% 0.01 NR 0.0% 1.2% 1.5% 0.2% 0.0% 1.0% 0.2% - kaon charge asymmetries in kaon tracking and particle ID. We assign a 1% systematic uncertainty due to the −5 MeV shift in the ∆E distribution of signal events observed in the data. 8.3 Systematic uncertainty in efficiency parameterization Signal efficiency is parameterized as a binned histogram with efficiency in each bin determined based on the number of simulated signal events that pass the selection criteria. We fluctuate the efficiency according to its statistical error and rerun the fit. This procedure is repeated 100 times; we take the standard deviation of the fit results as an estimate of the systematic uncertainties (Table 8.3). The uncertainty on the overall branching fraction is 0.2%. No significant change in the overall CP asymmetry is observed. 8.4 Systematic uncertainty in background estimates The uncertainty on the contribution of B backgrounds is evaluated by combining in quadrature the uncertainty on the values of branching fractions for individual decay modes with the statistical uncertainty due to limited sizes of simulated B background samples. We vary the bin contributions of the B backgrounds by their uncertainties and rerun the fit. This procedure is repeated 100 times; we take the standard deviation of the fit results as an estimate of the systematic uncertainties (Table 8.4). The uncertainty on the overall branching fraction is 0.6%. No significant change in the 99 Table 8.4: Estimates of the systematic uncertainties for the fit fractions (upper triangle and diagonal), phase differences (lower triangle), and CP violation parameters due to the uncertainty in the modeling of B backgrounds. φ f0 (980) X0 (1550) f0 (1710) χc0 NR A φ+ − φ− φ 0.1% 0.02 0.04 0.07 0.04 0.03 0.1% 0.01 f0 (980) 0.0% 1.1% 0.04 0.07 0.03 0.02 1.8% 0.01 X0 (1550) 0.0% 1.7% 3.0% 0.04 0.03 0.02 0.7% 0.01 f0 (1710) 0.0% 0.5% 2.4% 0.6% 0.06 0.06 4.8% 0.05 χc0 0.0% 0.0% 0.0% 0.0% 0.0% 0.02 0.6% 0.02 NR 0.1% 3.3% 3.4% 1.2% 0.0% 3.1% 0.6% - Table 8.5: Estimates of the systematic uncertainties for the fit fractions (upper triangle and diagonal), phase differences (lower triangle), and CP violation parameters due to the uncertainty in the sideband extrapolation qq background subtraction. φ f0 (980) X0 (1550) f0 (1710) χc0 NR A φ+ − φ− φ 0.0% 0.01 0.01 0.01 0.00 0.00 0.1% 0.00 f0 (980) 0.0% 0.3% 0.01 0.00 0.01 0.01 0.7% 0.00 X0 (1550) 0.0% 0.0% 0.5% 0.01 0.01 0.00 0.1% 0.00 f0 (1710) 0.0% 0.1% 0.0% 0.1% 0.01 0.01 1.4% 0.01 χc0 0.0% 0.0% 0.0% 0.0% 0.0% 0.00 0.7% 0.02 NR 0.0% 1.0% 0.2% 0.2% 0.0% 1.1% 0.3% - overall CP asymmetry is observed. The estimates of qq continuum backgrounds have been obtained by sideband extrapolation as discussed in Section 4.3; we have used the central value of the background extrapolation coefficient, throughout. To evaluate the uncertainty in qq background subtraction, we vary the extrapolation coefficient by its uncertainty and take the changes in the values of the fit fractions, phase differences, and CP violation parameters as estimates of respective systematic uncertainties (Table 8.5). 8.5 Systematic uncertainty in resonance lineshapes The observed width of narrow resonances can be significantly larger than the natural width due to detector resolution. Fig. 8.5 shows the distribution of the di-kaon invariant mass near the mass of the φ resonance for data and simulated B ± → K ± φ events selected in the signal region. The 100 Figure 8.5: Di-kaon invariant mass distribution near the φ resonance for candidates selected in the signal region for data (left) and simulated B ± → K ± φ events (right). Γ is the width of the Gaussian at half maximum. Table 8.6: Estimates of the systematic uncertainties for the fit fractions (upper triangle and diagonal), phase differences (lower triangle), and CP violation parameters due to neglecting detector resolution effects and the uncertainty in the Blatt-Weisskopf interaction radius for the φ resonance. φ f0 (980) X0 (1550) f0 (1710) χc0 NR A φ+ − φ− φ 0.8% 0.05 0.02 0.02 0.02 0.03 0.9% 0.03 f0 (980) 0.1% 2.8% 0.04 0.03 0.04 0.03 6.5% 0.04 X0 (1550) 0.2% 3.7% 2.4% 0.01 0.00 0.01 0.3% 0.04 f0 (1710) 0.0% 0.2% 0.7% 0.1% 0.01 0.01 3.5% 0.06 χc0 0.0% 0.1% 0.0% 0.0% 0.0% 0.01 1.3% 0.04 NR 0.4% 8.1% 1.6% 0.3% 0.1% 5.0% 2.7% - data distribution is fitted with a sum of a Gaussian and a first-order polynomial. The width of the distribution, estimated as the width of the Gaussian at half maximum, is Γ = 5.3 ± 0.3 MeV. The simulated events distribution is fitted by a sum of two Gaussians. The width of the distribution, estimated as the width of the narrow Gaussian at half maximum, is Γ = 4.5±0.2 MeV. In both cases, the distribution widths are somewhat larger than the natural width of the resonance, Γ(φ) = 4.2 MeV. To estimate the effect of neglecting the detector resolution, we change the width of the φ resonance to 5.2 MeV and rerun the fit. In our nominal fit, we have used R = 4 GeV−1 for the Blatt-Weisskopf interaction range parameter. When the interaction range is allowed to float in the fit, we get R = 4.17 ± 0.39 GeV−1 , consistent with the value used in the nominal fit. To estimate the systematic uncertainty due to the choice of the interaction range, we repeat the fit with R = 3.5 GeV−1 . 101 Figure 8.6: Di-kaon invariant mass distribution near the χc0 resonance for candidates selected in the signal region for data (left) and simulated B ± → K ± χc0 events (right). Γ is the width of the Gaussian at half maximum. Table 8.7: Estimates of the systematic uncertainties for the fit fractions (upper triangle and diagonal), phase differences (lower triangle), and CP violation parameters due to neglecting detector resolution effects for the χc0 resonance. φ f0 (980) X0 (1550) f0 (1710) χc0 NR A φ+ − φ− φ 0.0% 0.01 0.03 0.03 0.11 0.02 0.0% 0.02 f0 (980) 0.0% 0.7% 0.04 0.04 0.12 0.03 2.9% 0.01 X0 (1550) 0.1% 1.5% 2.9% 0.00 0.08 0.01 0.2% 0.01 f0 (1710) 0.0% 0.4% 1.1% 0.2% 0.08 0.01 1.9% 0.00 χc0 0.1% 0.5% 0.5% 0.1% 0.2% 0.09 3.8% 0.17 NR 0.0% 1.4% 3.2% 0.5% 1.9% 0.8% 1.1% - We combine the changes in the values of the fit fractions, phase differences, and CP violation parameters in quadrature to get estimates of respective systematic uncertainties (Table 8.6). Fig 8.6 shows the distribution of the di-kaon invariant mass in the region near the mass of the χc0 resonance for data and simulated B ± → K ± χc0 events selected in the signal region. The data distribution is fitted with a sum of a Gaussian and a first-order polynomial. The width of the distribution, estimated as the width of the Gaussian at half maximum, is Γ = 22.3 ± 5.2 MeV. The distribution of simulated events is fitted by a sum of two Gaussians. The width of the distribution, estimated as the width of the narrow Gaussian at half maximum, is Γ = 22.0 ± 0.7 MeV. In both cases due to resolution effects the distribution widths are considerably larger than the natural width of the resonance, Γ(χc0 ) = 10.1 MeV. To estimate the effect of neglecting the detector resolution we change the width of the χc0 resonance to 22. MeV and rerun the fit. We take the changes in 102 Table 8.8: Estimates of the systematic uncertainties for the fit fractions (upper triangle and diagonal), phase differences (lower triangle), and CP violation parameters due to to the uncertainty in the f0 (980) lineshape. φ f0 (980) X0 (1550) f0 (1710) χc0 NR A φ+ − φ− φ 0.1% 0.08 0.03 0.04 0.04 0.03 0.2% 0.01 f0 (980) 0.0% 3.2% 0.09 0.09 0.09 0.07 2.8% 0.01 X0 (1550) 0.0% 5.5% 2.5% 0.01 0.01 0.02 0.6% 0.02 f0 (1710) 0.0% 0.6% 1.0% 0.2% 0.01 0.02 4.8% 0.01 χc0 0.0% 0.1% 0.0% 0.0% 0.0% 0.02 0.2% 0.01 NR 0.1% 10.4% 3.0% 0.5% 0.1% 5.8% 0.4% - Table 8.9: Estimates of the systematic uncertainties for the fit fractions (upper triangle and diagonal), phase differences (lower triangle), and CP violation parameters due to to the uncertainty in the f0 (1710) lineshape. φ f0 (980) X0 (1550) f0 (1710) χc0 NR A φ+ − φ− φ 0.0% 0.01 0.02 0.10 0.01 0.00 0.1% 0.01 f0 (980) 0.0% 0.4% 0.02 0.11 0.01 0.01 0.7% 0.01 X0 (1550) 0.1% 1.2% 2.9% 0.09 0.01 0.01 0.1% 0.01 f0 (1710) 0.1% 0.8% 3.0% 0.4% 0.10 0.10 1.6% 0.03 χc0 0.0% 0.0% 0.0% 0.0% 0.0% 0.00 0.4% 0.01 NR 0.1% 1.6% 3.5% 2.1% 0.0% 1.9% 0.4% - the values of fit fractions, phase differences, and CP violation parameters as estimates of respective systematic uncertainties (Table 8.7). In this analysis we have used the parameters of the f0 (980) resonance measured by the BES collaboration [65], the most reliable measurement to date. We vary the parameters individually by their combined statistical and systematic uncertainties added in quadrature and repeat the fit. We combine the changes in the values of the fit fractions, phase differences, and CP violation parameters in quadrature to get estimates of respective systematic uncertainties (Table 8.8). The width of the f0 (1710) resonance is Γ(f0 (1710)) = 138±9 MeV [14]. To estimate the sensitivity of our results to this parameter we change it to 147 MeV and repeat the fit. We take the changes in the values of fit fractions, phase differences, and CP violation parameters as estimates of respective systematic uncertainties (Table 8.9). 103 Table 8.10: Sources of systematic uncertainty for the overall branching fraction and asymmetry. Tracking efficiency Particle ID efficiency Tracking + PID ∆E shift B backgrounds Sideband extrapolation Signal efficiency N (B + B − ) Total 8.6 B 2.4% 1.5% 3.9% 1.0% 0.7% 1.4% 0.2% 1.1% 4.5% A 1.0% 0.5% 1.5% 1.5% Tensor resonances f2 (1270) and f20 (1525) We do not expect a large contribution from the tensor resonance f2 (1270). It is known to couple preferentially to ππ and the measured branching fraction B(B ± → K ± f2 (1270), f2 (1270) → π + π − ) = (0.75 ± 0.17 ± 0.06) × 10−6 [75] is small. The tensor resonance f20 (1525) is known to couple preferentially to K K̄, and a strong f20 (1525) → K K̄ signal has been observed in different production environments [65, 66, 76]. In this analysis we did not see any evidence of a low-mass D-wave in the partial-wave analysis of Section 7.2. We try including f2 (1270) and f20 (1525) components in our isobar model, and get fit fractions F (B ± → K ± f2 (1270)) = (0.28 ± 0.24)% and F (B ± → K ± f20 (1525)) = (0.08 ± 0.10)%, consistent with zero. We thus do not observe a statistically significant contribution from the tensor resonances in B ± → K ± K ± K ∓ . 8.7 Systematic uncertainty summary Different sources of systematic uncertainty for the overall branching fraction and asymmetry are summarized in Table 8.10. As the uncertainties in the determination of the tracking and particle identification efficiencies may be correlated, we conservatively combine them linearly. All other uncertainties are added in quadrature. The systematic uncertainties for the overall branching fraction and asymmetry are dominated by the combined systematic uncertainty in tracking and particle 104 identification efficiencies. For the fit fractions, phase differences, and CP violation parameters of the individual components of the isobar model, we combine all systematic uncertainties in quadrature, adding 2% in quadrature to all CP asymmetry uncertainties to account for possible charge asymmetry in tracking and particle identification for kaons. 105 Chapter 9 Results Our final results for the total branching fraction is B(B ± → K ± K ± K ∓ ) = (35.2 ± 0.9 ± 1.6) × 10−6 , where the first error is statistical and the second is systematic. The fit fraction of the isobarmodel terms that do not involve the χc0 resonance is (95.0 ± 0.6 ± 1.1)% for the best fit, giving B(B ± → K ± K ± K ∓ ) = (33.5 ± 0.9 ± 1.6) × 10−6 if intrinsic charm contributions are excluded. The − − − + + + + − K K )−B(B →K K K ) = (−1.7 ± 2.6 ± 1.5)%. These results total asymmetry is ACP = B(B →K B(B ± →K ± K ± K ∓ ) are in agreement with the Belle collaboration isobar-model Dalitz plot analysis result B(B ± → K ± K ± K ∓ ) = (30.6 ± 1.2 ± 2.3) × 10−6 [19], and a previous inclusive BABAR measurement, B(B ± → K ± K ± K ∓ ) = (29.6 ± 2.1 ± 1.6) × 10−6 , ACP (B ± → K ± K ± K ∓ ) = 0.02 ± 0.07 ± 0.03 [16]. In both cases, the branching fraction B(B ± → K ± χc0 , χc0 → K + K − ) was not included in the charmless total. Table 9.1 gives the final results for the magnitudes of production coefficients, ρ; the matrix of the differences between the phases of the production coefficients, φij = φj − φi ; the matrix of component fit fractions (Eq. 6.10) and interference term fractions (Eq. 6.12), Fij ; the component CP asymmetries, A, and their symmetric 90%-confidence-level intervals; the charge-dependent production coefficient phase differences for the individual components of the isobar model, δφ = φ− − φ+ ; and the quasi-two-body branching fractions, Fii × B(B ± → K ± K ± K ∓ ). Using B(φ → K + K − ) = 49.2±0.6%, we get B(B ± → K ± φ) = (8.4±0.7±0.7±0.1)×10−6 , where the last error is due to the φ branching fraction uncertainty. The result is in good agreement with the Belle collaboration isobar-model Dalitz plot fit result, B(B ± → K ± φ) = (9.83 ± 0.90 ± 0.90) × 106 10−6 [19], and the result from a previous BABAR quasi-two-body analysis, B(B ± → K ± φ) = (10.0 ± 0.9±0.5)×10−6 [17]. These measurements are in good agreement with the PQCD prediction and are on the higher side of the QCDF predictions in Table 2.1. Our asymmetry result for B(B ± → K ± φ) is in good agreement with the QCDF predictions. The quasi-two-body branching fraction B(B ± → K ± χc0 , χc0 → K + K − ) = (1.10 ± 0.20 ± 0.09) × 10−6 is in good agreement with the Belle collaboration isobar-model Dalitz plot fit result, B(B ± → K ± χc0 , χc0 → K + K − ) = (0.86 ± 0.26 ± 0.06) × 10−6 [19], and the result from a previous BABAR quasi-two-body analysis, B(B ± → K ± χc0 , χc0 → K + K − ) = (1.1 ± 0.2 ± 0.1) × 10−6 [18]. Using B(χc0 → K + K − ) = (6.0 ± 0.9) × 10−3 , we get B(B ± → K ± χc0 ) = (1.84 ± 0.32 ± 0.14 ± 0.28) × 10−4 , where the last error is due to the χc0 branching fraction uncertainty. This is in good agreement with the prediction based on intrinsic charm rescattering [6]. We combine our result, B(B ± → K ± f0 (980), f0 (980) → K + K − ) = (6.5 ± 2.5 ± 1.6), with the BABAR measurement, B(B ± → K ± f0 (980), f0 (980) → π + π − ) = (9.47 ± 0.97 ± 0.46) × 10−6 and estimate B(B ± → K ± f0 (980)) = (27 ± 5 ± 3) × 10−6 , where we have used isospin symmetry and combined statistical and systematic errors in quadrature. This value is significantly larger than the theoretical prediction of Ref. [5], where f0 (980) was considered as a two-quark state. The discrepancy, mostly due to the large branching fraction of f0 (980) → K K̄, suggests that this simple quark model may not describe very well the structure of the f0 (980) resonance. For the ratio of the partial branching fractions we get: B(B ± → K ± f0 (980), f0 (980) → K + K − ) = 0.69 ± 0.32, B(B ± → K ± f0 (980), f0 (980) → π + π − ) where we have combined the statistical and systematic errors in quadrature. This is consistent with 0.92 ± 0.07, the prediction that we obtain by evaluating the right-hand side of Eq. (7.1) using the f0 (980) parameters measured by the BES collaboration [65]. In conclusion, we have measured the total branching fraction and the CP asymmetry in B ± → K ± K ± K ∓ , which is in agreement with previous measurements. An isobar-model Dalitz plot fit φ 1.66 ± 0.06 φ 0 2.51 ± 0.17 ± 0.11 1.70 ± 0.27 ± 0.07 −2.71 ± 0.35 ± 0.13 −2.27 ± 0.35 ± 0.12 2.99 ± 0.20 ± 0.06 φ 11.8 ± 0.9 ± 0.8 φ 4.14 ± 0.32 ± 0.33 φ 0.00 ± 0.08 ± 0.02 (−0.14, 0.14) −0.67 ± 0.28 ± 0.05 ρ φij (rad.) φ f0 (980) X0 (1550) f0 (1710) χc0 NR Fij × 100% φ f0 (980) X0 (1550) f0 (1710) χc0 NR Two-body B F × B × 106 CP A (Amin , Amax )90% φ− − φ+ f0 (980) −0.31 ± 0.25 ± 0.08 (−0.72, 0.12) −0.20 ± 0.16 ± 0.04 f0 (980) 6.5 ± 2.5 ± 1.6 f0 (980) −0.94 ± 0.18 ± 0.11 19 ± 7 ± 4 f0 (980) −2.51 ± 0.17 ± 0.11 0 −0.81 ± 0.25 ± 0.12 1.07 ± 0.36 ± 0.17 1.50 ± 0.33 ± 0.17 0.48 ± 0.16 ± 0.08 f0 (980) 5.2 ± 1.0 X0 (1550) −0.04 ± 0.07 ± 0.02 (−0.17, 0.09) 0.02 ± 0.15 ± 0.05 X0 (1550) 43 ± 6 ± 3 X0 (1550) −1.71 ± 0.36 ± 0.24 53 ± 12 ± 7 121 ± 19 ± 6 X0 (1550) −1.70 ± 0.27 ± 0.07 0.81 ± 0.25 ± 0.12 0 1.88 ± 0.19 ± 0.09 2.32 ± 0.22 ± 0.09 1.29 ± 0.10 ± 0.04 X0 (1550) 8.2 ± 1.1 f0 (1710) 0.0 ± 0.5 ± 0.1 (−0.66, 0.74) −0.07 ± 0.38 ± 0.08 f0 (1710) 1.7 ± 1.0 ± 0.3 f0 (1710) 0.01 ± 0.10 ± 0.03 −4.5 ± 2.9 ± 1.2 −30 ± 11 ± 4 4.8 ± 2.7 ± 0.8 f0 (1710) 2.71 ± 0.35 ± 0.13 −1.07 ± 0.36 ± 0.17 −1.88 ± 0.19 ± 0.09 0 0.44 ± 0.30 ± 0.13 −0.59 ± 0.25 ± 0.11 f0 (1710) 1.22 ± 0.34 χc0 0.19 ± 0.18 ± 0.05 (−0.09, 0.47) 0.7 ± 0.5 ± 0.2 χc0 1.10 ± 0.20 ± 0.09 χc0 0.11 ± 0.02 ± 0.05 −0.9 ± 0.2 ± 0.5 −1.1 ± 0.3 ± 0.5 −0.10 ± 0.07 ± 0.07 3.1 ± 0.6 ± 0.2 χc0 2.27 ± 0.35 ± 0.12 −1.50 ± 0.33 ± 0.17 −2.32 ± 0.22 ± 0.09 −0.44 ± 0.30 ± 0.13 0 −1.02 ± 0.23 ± 0.10 χc0 0.437 ± 0.039 NR 0.02 ± 0.08 ± 0.04 (−0.09, 0.18) - NR 50 ± 6 ± 4 NR 3.54 ± 0.38 ± 0.40 −85 ± 21 ± 14 −140 ± 26 ± 7 4± 6 ±3 3.9 ± 0.4 ± 1.9 141 ± 16 ± 9 NR −2.99 ± 0.20 ± 0.06 −0.48 ± 0.16 ± 0.08 −1.29 ± 0.10 ± 0.04 0.59 ± 0.25 ± 0.11 1.02 ± 0.23 ± 0.10 0 NR 13.2 ± 1.4 Table 9.1: Production coefficient magnitudes, phase differences, fit fractions, quasi-two-body branching fractions, and CP violation parameters of the individual components of the isobar model for the leading fit. The first error is statistical, the second is combined systematic. 107 108 and a partial-wave analysis of the K + K − S-wave show evidence of large contributions from a broad X0 (1550) scalar resonance, a mass-dependent nonresonant component, and the f0 (980) resonance. The quasi-two-body branching fractions for B(B ± → K ± φ, φ → K + K − ) and B(B ± → K ± χc0 , χc0 → K + K − ) are consistent with the previous measurements and with the theoretical expectations. We see evidence of a large B ± → K ± f0 (980) component in the K ± K ± K ∓ final state and combined BABAR results for B ± → K ± f0 (980) are consistent, albeit with large errors, with the measurement of the ratio of the f0 (980) coupling constants, gK /gπ , by the BES collaboration [65]. We see no evidence of direct CP violation for the individual components of the isobar model. One of the main highlights of this analysis is the complicated S-wave model that had to be used to fit the observed event distribution. The identification of the lowest-mass quark model pseudo-scalar meson nonet is still a subject of controversy. In addition to the usual di-quark states, speculative fourquark and glueball [13] interpretations have been suggested. Even the attribution of the well-known resonance f0 (980) is uncertain, even less is known about the speculative σ resonance and other low-mass scalars: f0 (1300), f0 (1500), and f0 (1710). In this analysis, as well as in the preceding analysis by the Belle collaboration [19], a broad scalar resonance, which we label X0 (1550), had to be introduced to describe the observed event distribution. Evidence of a possible resonant Swave contribution around m(KK) = 1.5 − 1.6 GeV has been reported previously [69, 70], but the attribution of the X0 (1550) at this point is uncertain. Finally, we had to use an ad hoc parameterization of the mass-dependent nonresonant component to achieve a good fit. A physics-based model successfully describing the data is clearly desirable, especially in view of the possibility of performing a partial-wave analysis by considering the interference between the low-mass K + K − S-wave and the mass-dependent nonresonant component. The published theoretical models [60, 61], surveyed in Section 6.5, have been found to be in disagreement with the experimental data. 109 Bibliography [1] H.-Y. Cheng and K.-C. Yang, Phys. Rev. D64, 074004 (2001). [2] D. Du, H. Gong, J. Sun, D. Yang, and G. Zhu, Phys. Rev. D65, 094025 (2002). [3] M. Beneke and M. Neubert, Nucl. Phys. B675, 333 (2003). [4] C. Chen, Y. Keum, and H. Li, Phys. Rev. D64, 112002 (2001). [5] H.-Y. Cheng and K.-C. Yang, Phys. Rev. D71, 054020 (2005). [6] P. Colangelo, F. de Fazio, and T. Pham, Phys. Lett. B542, 71 (2002). [7] D. Du, J. Sun, D. Yang, and G. Zhu, Phys. Rev. D67, 014023 (2003). [8] R. Aleksan, P.-F. Giraud, V. Morénas, O. Pène, and A. Safir, Phys. Rev. D67, 094019 (2003). [9] N. de Groot, W. Cottingham, and I. Wittingham, Phys. Rev. D68, 113005 (2003). [10] H. Lipkin, Y. Nir, H. Quinn, and A. Snyder, Phys. Rev. D44, 1454 (1991). [11] A. Snyder and H. Quinn, Phys. Rev. D48, 2139 (1993). [12] Y. Grossman, Z. Ligeti, Y. Nir, and H. Quinn, Phys. Rev. D68, 015004 (2003). [13] P. Minkowski and W. Ochs, Eur. Phys. Journal. C39, 71 (2005). [14] S. Eidelman et al., Phys. Lett. B592 (2004), and 2005 partial update for the 2006 edition available on the PDG WWW pages, URL http://pdg.lbl.gov/. [15] R. Briere et al. (CLEO Collaboration), Phys. Rev. Lett. 86, 3718 (2001). [16] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett. 91, 051801 (2003). 110 [17] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D69, 011102 (2004). [18] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D69, 071103 (2004). [19] A. Garmash et al. (Belle Collaboration), Phys. Rev. D71, 092003 (2005). [20] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973). [21] L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983). [22] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963). [23] M. Bander, D. Silverman, and A. Soni, Phys. Rev. Lett. 43, 242 (1979). [24] C.-W. Chiang, M. Gronau, Z. Luo, J. Rosner, and D. Suprun, Phys. Rev. D69, 034001 (2004). [25] M. Bauer, B. Stech, and M. Wirbel, Z. Phys. C34, 103 (1987). [26] M. Neubert and B. Stech, Adv. Ser. Direct. High Energy Phys. 15, 294 (1998). [27] D. Melikhov and B. Stech, Phys. Rev. D62, 014006 (2000). [28] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D72, 072003 (2005). [29] M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Phys. Rev. Lett. 83, 1914 (1999). [30] M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachrajda, Nucl. Phys. B591, 313 (2000). [31] Y. Keum, H. Li, and A. Sanda, Phys. Rev. D63, 054008 (2001). [32] M. Ciuchini, E. Franco, G. Martinelli, and L. Silvestrini, Nucl. Phys. B501, 271 (1997). [33] C. Isola, M. Ladisa, T. Pham, and P. Santorelli, Phys. Rev. D64, 014029 (2001). [34] S. Brodsky and S. Garnder, Phys. Rev. D65, 054016 (2002). [35] M. Ciuchini, E. Franco, G. Martinelli, M. Pierini, and L. Silvestrini, Phys. Lett. B515, 33 (2001). [36] Y.-H. Chen, H.-Y. Cheng, B. Tseng, and K.-C. Yang, Phys. Rev. D60, 094014 (1999). 111 [37] D. Du, H. Gong, J. Sun, D. Yang, and G. Zhu, Phys. Rev. D66, 079904 (2002). [38] W. Kozanecki, Nucl. Instrum. Meth. A446, 59 (2000). [39] Slac report 504, ”the BABAR physics book. physics at an asymmetric b factory.” (1998). [40] B. Aubert et al. (BABAR Collaboration), Phys. Rev. Lett 89, 201802 (2002). [41] B. Aubert et al. (BABAR Collaboration), Nucl. Instrum. Meth. A479, 1 (2002). [42] G. Fox and S. Wolfram, Phys. Rev. Lett. 41, 1581 (1978). [43] W. Dunwoodie, private communication. http://www.slac.stanford.edu/∼wmd/bbkinematics/kinematics.note. [44] W. Ford, BABAR Analysis Document #53. [45] ROOT. An Object-Oriented Data Analysis Framework. http://root.cern.ch/. [46] H. Albrecht et al. (ARGUS Collaboration), Z. Phys. C48, 543 (1990). [47] T. Allmendinger et al., BABAR Analysis Document #867. [48] S. Mclachlin, BABAR Analysis Document #1056. [49] S. Spanier, BABAR Analysis Document #303. [50] K. M. Watson, Phys. Rev. 95, 228 (1954). [51] G. Fleming, Phys. Rev. 135, B551 (1964). [52] D. Morgan, Phys. Rev. 166, 1731 (1968). [53] J. Blatt and V. Weisskopf, Theoretical Nuclear Physics (New York, J. Wiley, 1952). [54] C. Zemach, Phys. Rev. 133, B1201 (1964). [55] C. Zemach, Phys. Rev. 140, B97 (1965). [56] S. Flatté, Phys. Lett. B63, 224 (1976). 112 [57] E. Aitala et al. (E791 Collaboration), Phys. Rev. Lett. 89, 121801 (2002). [58] S. Kopp et al. (CLEO Collaboration), Phys. Rev. D63, 092001 (2001). [59] A. Garmash et al. (Belle Collaboration), Phys. Rev. D65, 092005 (2002). [60] S. Fajfer, T. Pham, and A. Propotnik, Phys. Rev. D70, 034033 (2004). [61] H.-Y. Cheng and K.-C. Yang, Phys. Rev. D66, 054015 (2002). [62] J. Heinrich, CDF note 5718. [63] F. James, MINUIT Reference Manual. CERN Program Library Long Writeup D506. [64] J. Link et al. (FOCUS Collaboration), Phys. Lett. B621, 72 (2005). [65] M. Ablikim et al. (BES Collaboration), Phys. Lett. B607, 243 (2005). [66] D. Barberis et al. (WA102 Collaboration), Phys. Lett. B462, 462 (1999). [67] M. Achasov et al. (SND Collaboration), Phys. Lett. B485, 349 (2000). [68] E. Aitala et al. (E791), Phys. Rev. Lett. 86, 765 (2001). [69] M. Baubillier et al., Z. Phys. C17, 309 (1983). [70] D. Aston et al., Nucl. Phys. B301, 525 (1988). [71] M. Ablikim et al. (BES Collaboration), Phys. Lett. B603, 138 (2004). [72] M. Gronau and J. Rosner, Phys. Rev. D72, 094031 (2005). [73] B. Aubert et al. (BABAR Collaboration), Phys. Rev. D71, 091102 (2005). [74] K. Abe et al. (Belle Collaboration), Phys. Rev. Lett. 91, 261602 (2003). [75] A. Garmash et al. (Belle Collaboration) (2005). [76] D. Barberis et al. (WA102 Collaboration), Phys. Lett. B453, 305 (1999).